www.myreaders.info ORBITAL MECHANICS - MODEL & SIMULATION SOFTWARE (OM-MSS) Earth, Sun, Moon & Satellites Motion in Orbit - Model & Simulation Software. RC Chakraborty (Retd), Former Director, DRDO, Delhi & Visiting Professor, JUET, Guna www.myreaders.info, [email protected], http://www.myreaders.info/html/orbital_mechanics.html, Revised Dec. 16, 2015, pp 1 - 402 INTRODUCTION : Orbital Mechanics - Model & Simulation Software (Om-Mss) . A Monograph of Earth, Sun, Moon & Satellites Motion in Orbit with Examples, Problems and Software Driven Solutions. We look into space from Earth, which is 3rd planet from Sun. Earth takes around 365.25 days to moves around Sun in an Elliptical orbit. The average distance from the Earth to the Sun is called one Astronomical Unit (AU); 1 AU = 149,597,870.7 km. Mars, is 4th planet from Sun, that takes 686.971 Earth days to orbit around Sun. The orbital path of Mars is highly eccentric. Mars & Earth move along their orbits, and come near to one another approximately every two years. This approach of coming near facilitate launching of spacecraft every two years, even that takes about eight months to reach Mars. Example : On Apr. 08, 2014, the near or close distance between Mars and Earth was 92.4 million km. Moon moves around Earth in the same kind of orbit. The Moon is the Earth's only natural Satellite. The average distance of the Moon from the Earth is 384,403 km. A Satellite is an artificial object, intentionally placed into orbit. Thousands of Satellites are launched into orbit around Earth. A few Satellites called Space Probes have been placed into orbit around Moon, Mercury, Venus, Mars, Jupiter, Saturn, etc. Understanding the motion of Earth around Sun, and the motion of Moon and Satellites around Earth is of interest to many. Presented here a Monograph of 'ORBITAL MECHANICS - MODEL & SIMULATION SOFTWARE (OM-MSS)', to Simulate Motion of Sun, Earth, Moon & Satellites. The OM-MSS Software is written in 'C' Language, the Compiler used is Dev C++ and the Platform is a Windows 7, 64 bit Laptop. The Source Code, around 30,000 Lines, is Compiled. The 'OM-MSS.EXE' File generated is of Size 1.5 KB. The Executable File, < OM-MSS.EXE >, is RUN Step-by-Step for a Set of Inputs. The Results seen on Computer Screen are put in a File, that in effect becomes 'A Monograph of Orbital Mechanics with Examples, Problems and Software Driven Solutions'. The execution of 'Orbital Mechanics - Model & Simulation Software (OM-MSS)', illustrates its Scope, Capability, Accuracy, and Usage. www.myreaders.info Return to Website
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ORBITAL MECHANICS - MODEL & SIMULATION SOFTWARE (OM-MSS) Earth, Sun, Moon & Satellites Motion in Orbit - Model & Simulation Software. RC Chakraborty (Retd), Former Director, DRDO, Delhi & Visiting Professor, JUET, Guna
INTRODUCTION : Orbital Mechanics - Model & Simulation Software (Om-Mss) . A Monograph of Earth, Sun, Moon & Satellites Motion in Orbit with Examples, Problems and Software Driven Solutions. We look into space from Earth, which is 3rd planet from Sun. Earth takes around 365.25 days to moves around Sun in an Elliptical orbit. The average distance from the Earth to the Sun is called one Astronomical Unit (AU); 1 AU = 149,597,870.7 km. Mars, is 4th planet from Sun, that takes 686.971 Earth days to orbit around Sun. The orbital path of Mars is highly eccentric. Mars & Earth move along their orbits, and come near to one another approximately every two years. This approach of coming near facilitate launching of spacecraft every two years, even that takes about eight months to reach Mars. Example : On Apr. 08, 2014, the near or close distance between Mars and Earth was 92.4 million km. Moon moves around Earth in the same kind of orbit. The Moon is the Earth's only natural Satellite. The average distance of the Moon from the Earth is 384,403 km. A Satellite is an artificial object, intentionally placed into orbit. Thousands of Satellites are launched into orbit around Earth. A few Satellites called Space Probes have been placed into orbit around Moon, Mercury, Venus, Mars, Jupiter, Saturn, etc. Understanding the motion of Earth around Sun, and the motion of Moon and Satellites around Earth is of interest to many. Presented here a Monograph of 'ORBITAL MECHANICS - MODEL & SIMULATION SOFTWARE (OM-MSS)', to Simulate Motion of Sun, Earth, Moon & Satellites.
The OM-MSS Software is written in 'C' Language, the Compiler used is Dev C++ and the Platform is a Windows 7, 64 bit Laptop. The Source Code, around 30,000 Lines, is Compiled. The 'OM-MSS.EXE' File generated is of Size 1.5 KB. The Executable File, < OM-MSS.EXE >, is RUN Step-by-Step for a Set of Inputs. The Results seen on Computer Screen are put in a File, that in effect becomes 'A Monograph of Orbital Mechanics with Examples, Problems and Software Driven Solutions'. The execution of 'Orbital Mechanics - Model & Simulation Software (OM-MSS)', illustrates its Scope, Capability, Accuracy, and Usage.
The OM-MSS Software is quite exhaustive for beginners, experts, researchers & professional in Spherical Astronomy. The source code of OM-MSS Software in full or in parts has a cost if there is buyer. The cost has not been evaluated / decided. The OM-MSS Software includes the following : (a) Astronomical Time Standards and Time Conversions Utilities : GMT - Greenwich Mean Time, LMT - Local Mean Time, LST - Local Sidereal Time, UT - Universal Time, ET - Ephemeris Time, JD - Julian Day, Standard Epoch J2000, Gregorian Calendar date and more. (b) Positional Astronomy of Earth, Sun, Moon, and Satellites Motion in Orbit, includes computations of : * Position of Sun and Position of Earth on Celestial Sphere at Epoch ; * Keplerian elements : Inclination, RA of asc. Node, Eccentricity, Arg. of Perigee, Mean Anomaly, Mean Motion; * Motion Irregularities : Mean, Eccentric and True anomaly in deg; * Precise Time at Earth Orbit Points : Perihelion, Aphelion, Equinoxes, Solstices, Semi-Major & Minor-axis; * Astronomical years : Anomalistic, Tropical, and Sidereal Years; * Four Seasons : Spring, Summer, Autumn and Winter start time and duration; * Position of Satellites around Earth : Keplerian elements and State Vectors at epoch, and computing, Sub-Sat point lat/log, EL & AZ angles, Distances, Velocity, and more; * Satellite Pass, Ground Trace for Earth Stn using NASA/NORAD 2-line bulletins; (c) Customized Utilities and products : On special request either developed or configured and generated. These are Presented in Section - 1 to 8. The Section - 9 Containes References, and Section - 10 Containes few related Diagrams. Next : Content Index Table.
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Index Table : OM-MSS Sections / Sub Sections, Pages & Titles. (the page number is same as seen in the pdf document status bar) SECTIONS PAGES CONTENTS : Sections / Sub Sections Titles 1 6 ASTRONOMICAL TIME STANDARDS AND TIME CONVERSION UTILITIES. 1.1 12 Conversion of Universal Time (year, month, day, hour decimal) To Julian Day. 1.2 13 Conversion of Julian Day To Universal Time (year, month, day, hour decimal). 1.3 14 Conversion of Fundamental Epoch To Julian day and Julian century. 1.4 15 Add or Subtract time (days, hour, minute seconds) to or from input time. 1.5 16 Julian day for start of any Year. 1.6 17 Solar Time : Local Mean Solar Time (LMT) over observer's Longitude, and Greenwich Mean Time (GMT). 1.7 21 Sidereal Time : Greenwich universal time at hour 0.0 (ST0) and Greenwich Mean Sidereal Time (GMST) at input UT. 1.8 24 Sidereal Time : Greenwich Sidereal Time (GST), Greenwich Hour Angle (GHA), and Mean Sidereal Time (MST) at input UT. 1.9 26 Sidereal Time : LMST is Local Mean Sidereal time over observer's Longitude and GMST is Greenwich Mean Sidereal Time. 1.10 28 Time Conversions : LMT to LST, LST to LMT, LMT to LMST, LMST to LMT. 1.11 32 Concluding Time Standards and Time Conversion Utilities (Sections 1.0 to 1.10). 2 33 POSITIONAL ASTRONOMY : EARTH ORBIT AROUND SUN, ANOMALIES & ASTRONOMICAL EVENTS - EQUINOXES, SOLSTICES, YEARS & SEASONS. 2.1 41 Earth Orbit : Constants used in OM-MSS Software. 2.2 43 Earth Mean anomaly and True anomaly at Input UT, Since Standard Epoch J2000, using standard analytical expressions. 2.3 44 Earth Orbit Input Year : Precise Universal Time (UT) at orbit points - Perihelion and Aphelion. 2.4 46 Earth Orbit Input Year : Precise Universal Time (UT) at orbit points - Vernal Equinox and Autumnal Equinox. 2.5 47 Earth Orbit Input Year : Precise Universal Time (UT) at orbit points - Summer Solstice and Winter Solstice. 2.6 48 Earth Orbit Input Year : Precise Universal Time (UT) at orbit points - Semi-Major Axis and Semi-Minor Axis. 2.7 49 Earth Orbit Input Year : Astronomical Years - Anomalistic, Tropical, and Sidereal Years. 2.8 50 Earth Orbit Oblateness : Semi-Major Axis and Semi-Minor Axis. 2.9 52 Earth Orbit Input Year : Mean anomaly, Eccentric anomaly, True anomaly at UT, based on algorithms of iterative method. 2.10 54 Earth Orbit Input Year : Four Seasons - Spring, Summer, Autumn, and Winter. 2.11 56 Concluding Astronomical Events Anomalies, Equinoxes, Solstices, Years & Seasons (Sections 2.0 to 2.10).
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3 57 POSITION OF SUN ON CELESTIAL SPHERE AT INPUT UNIVERSAL TIME (UT). 3.1 59 Sun Positional Parameters on Celestial Sphere : Input Time (UT) Standard Epoch JD2000. 3.2 60 Sun Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Perihelion. 3.3 61 Sun Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Vernal equinox. 3.4 62 Sun Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Summer solstice. 3.5 63 Sun Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Aphelion. 3.6 64 Sun Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Autumnal equinox. 3.7 65 SUN Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Winter solstice. 3.8 66 Concluding Sun Position on Celestial Sphere (Sections 3.0 to 3.7). 4 68 POSITION OF EARTH ON CELESTIAL SPHERE AT INPUT UNIVERSAL TIME (UT). 4.1 71 Earth Positional Parameters on Celestial Sphere : Input Time (UT) Standard Epoch JD2000. 4.2 84 Earth Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Perihelion. 4.3 97 Earth Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Vernal equinox. 4.4 110 Earth Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Summer solstice. 4.5 123 Earth Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Aphelion. 4.6 136 Earth Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Autumnal equinox. 4.7 149 Earth Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Winter solstice. 4.8 162 Concluding Earth Position on Celestial Sphere (Sections 4.0 to 4.7). 5 164 SATELLITES ORBIT ELEMENTS : EPHEMERIS, Keplerian ELEMENTS, STATE VECTORS 5.1 170 NASA/NORAD 'Two-Line Elements'(TLE) Ephemeris data set. 5.2 174 Conversion of Keplerian Element Set to State Vector Set and vice versa. 5.3 186 Satellite Orbit Keplerian element set at Perigee prior to Epoch. 5.4 192 Concluding Satellites Ephemeris Data Set (Sections 5.0 to 5.3).
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6. 193 SATELLITES MOTION AROUND EARTH : ORBITAL & POSITIONAL PARAMETERS AT EPOCH. 6.1 195 LANDSAT 8 : Orbital & Positional parameters corresponding to input 'Two-Line Elements'(TLE) Bulletins. 6.2 207 SPOT 6 : Orbital & Positional parameters corresponding to input 'Two-Line Elements'(TLE) Bulletins. 6.3 219 CARTOSAT 2B : Orbital & Positional parameters corresponding to input 'Two-Line Elements'(TLE) Bulletins. 6.4 231 ISS (ZARYA) : Orbital & Positional parameters corresponding to input 'Two-Line Elements'(TLE) Bulletins. 6.5 243 GSAT-14 : Orbital & Positional parameters corresponding to input 'Two-Line Elements'(TLE) Bulletins. 6.6 255 MOON : Orbital & Positional parameters corresponding to input 'Two-Line Elements'(TLE) Bulletins. 6.7 267 Concluding Satellites Orbital & Positional Parameters At Epoch (Sections 6.0 to 6.6). 7 268 SATELLITE PASS FOR EARTH STN - PREDICTION OF GROUND TRACE COORDINATES, LOOK ANGLES, UNIVERSAL/LOCAL TIME & MORE. 7.1 270 LANDSAT 8 : Sat Pass for Earth Stn - Prediction of Ground Trace, Look Angles & more at Instantaneous Time. 7.2 290 SPOT 6 : Sat Pass for Earth Stn - Prediction of Ground Trace, Look Angles & more at Instantaneous Time. 7.3 310 CARTOSAT 2B : Sat Pass for Earth Stn - Prediction of Ground Trace, Look Angles & more at Instantaneous Time. 7.4 330 ISS (ZARYA) : Sat Pass for Earth Stn - Prediction of Ground Trace, Look Angles & more at Instantaneous Time. 7.5 350 GSAT-14 : Sat Pass for Earth Stn - Prediction of Ground Trace, Look Angles & more at Instantaneous Time. 7.6 370 MOON : Sat Pass for Earth Stn - Prediction of Ground Trace, Look Angles & more at Instantaneous Time. 7.7 390 Concluding Satellites Passes - Prediction of Ground Trace Coordinates, Look Angles & more at Instantaneous Time. 8 391 CONCLUSION : ORBITAL MECHANICS - MODEL & SIMULATION SOFTWARE (OM-MSS). 9 392 REFERENCES : TEXT BOOKS & INTERNET WEB LINKS. 10 399 ANNEXURE : A Collection of few related Diagrams / Help. Move on to Section (1 to 9) While the Executable File, < OM-MSS.EXE >, is RUN for a Set of Inputs. Next Section - 1 Time Conversion Utilities
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OM-MSS Page 6 OM-MSS Section - 1 ------------------------------------------------------------------------------------------------------1 ASTRONOMICAL TIME STANDARDS AND TIME CONVERSION UTILITIES First look into few preliminaries and then move to time conversion utilities. Time Abbreviations : GMT - Greenwich Mean Time, LMT - Local Mean Time, LST - Local Sidereal Time, Sidereal Time, Solar Time, LMST - Local Mean Sidereal Time, GMST - Greenwich Mean Sidereal Time, GAST - Greenwich Apparent Sidereal Time, TAI - International Atomic Time, TT - Terrestrial Time, TDT - Terrestrial Dynamical Time, TCB - Barycentric Coordinate Time, TCG - Geocentric Coordinate Time, UT - Universal Time, UTC - Coordinated Universal Time, JD - Julian Day, ET - Ephemeris Time, BCE - Before the Common/Current/Christian Era, CE - Common Era/Current Era/Christian Era, Gregorian calendar. Time Standards and designations : Time is a dimension in which the events can be ordered from the past through the present into the future. Our clocks are set to run (approximately) on solar time (sun time). For astronomical observations, we need to use sidereal time (star time). Earth rotation is considered relative to the stars. One earth rotation is the time between two successive meridian passages of the same star. One rotation of Earth is one sidereal day, which is a little shorter than a solar day; ie, one mean sidereal day is about 0.99726958 mean solar day. Day is a unit of time. A day is measured from local noon to the following local noon. In common usage, a day consists 24 hours, noon is 12.00 hours. Solar Time is based on the rotation of the Earth with respect to the Sun. There are two types of solar time, - Apparent or True Solar Time is, that measured by direct observation of the Sun. It is not uniform throughout the year; - Mean Solar Time is, that would be measured if the Sun traveled at a uniform apparent speed throughout the year. Our clocks use Mean Solar Time. Solar day is the time for Earth to make a complete rotation on its axis relative to the Sun; - Apparent or True Solar Day varies through the year, that can be 20 seconds shorter in September and 30 seconds longer in December. This variaion is because of inclination (23.4392794383 deg) of Earth's axis of rotation and the elliptical orbit of Earth around the Sun. - Mean Solar Day, is average of true solar day during entire year, contains 86,400 mean solar seconds. The mean solar day is measured from midnight to midnight where midnight is hour 00. The mean solar day is divided into 24 solar hours, while each solar hour is divided into 60 solar minutes, and each solar minute divided into 60 solar seconds.
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OM-MSS Page 7 Equation of time describes the discrepancy between the Apparent Solar time and the Mean Solar time. In a year, compared to Mean Solar time the Apparent Solar time can be ahead (fast) or behind (slow) or near zeros (same). The typical values are : ahead (fast) as much as 16 min 33 sec around 3 Nov., or behind (slow) as much as 14 min 6 sec around 12 Feb., or near zeros (same) around 15 April, 13 June, 1 Sept and 25 Dec. Solar year is average of 400 consecutive civil years having 97 leap years, ie ((400 x 365) + 97) = 146097 / 400 = 365.242500 solar days. Our clocks are set to run on solar time. But for astronomical observations, we need to use sidereal time (star time). Sidereal Time is based on the rotation of the Earth with respect to fixed stars. ST is a measure of the hour angle of the Vernal equinox. - Apparent Sidereal Time is measured, if the hour angle is with respect to the true equinox, - Mean Sidereal Time is measured, if the hour angle is with respect to the mean equinox, Sidereal Day is equal to the interval of time between two successive transits of the vernal equinox. At Vernal equinox transit point hour angle is zero; - Apparent or True sidereal Day is affected by the motion of true equinox due to 'Precession' and 'Nutation'; - Mean Sidereal Day is affected by the motion of mean equinox due to Precession only; Precession is a change in orientation of rotational axis, slowly westward relative to the fixed stars completing one revolution in about 26,000 years. It is caused by the gravity of the Moon and Sun acting on the Earth. Nutation is a small cyclical motion superimposed upon the steady 26,000 year Precession of the Earth's axis of rotation. It is caused by the gravitational effect of the 18.6-year rotation period of the Moon's orbit. Mean sidereal day is 23.9344699 hours or 23 hours, 56 minutes, 4.0916 seconds, (ie, one mean sidereal day is about 0.99726958 mean solar day). A clock regulated to Apparent Sidereal time compared with one regulated to Mean Sidereal time, the diff is 2 to 3 sec in 19 years is no inconvenience. Sidereal year is the time taken by Earth to complete one revolution of its orbit, measured against a fixed frame of reference (as the fixed stars). A Sidereal year is approximately 365.256363 days, is slightly longer than the solar year (365.2425 days). The actual durations differ from year to year because the motion of Earth is influenced by the gravity of Moon and other planets. Sidereal year average duration is precisely 365.256363004 mean solar days (or 365day, 6hr, 9min, 9.76sec in units of mean solar time) at epoch J2000.0, ie YY 2000, MM 1, Hr 12. Thus, the earth mean motion rev per day around sun = 1.0 / 365.256363004. Sidereal time (ST) of a location is defined by its geographical longitude, thus called Local Sidereal time (LST) of the location.
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OM-MSS Page 8 For observers over Greenwich, whose longitude is set at 0 degrees, the local sidereal time (LST) = Greenwich Sidereal Time (GST) . For Observers over other longitude location, the LST is calculated using formula LST = GST + observer's log in hr : min : sec . Earth rotates through 360 deg longitude. The angle between two longitudes is measured in degree or in time, with 24 hr = 360 deg, ie 1 hr = 15 deg. This means Sun moves through 15 deg of longitude in 1 hour, or 15 minutes of arc in 1 minute of time. Hour Angle HA of an object is its Geographic Position (GP), measured around the celestial equator, westward from the observer's meridian. Greenwich hour angle GHA is angular distance of Geographic Position (GP) of a celestial body, measured (in 0-360 deg) westward from Greenwich (0 deg). Sidereal hour angle SHA is angular distance of of Geographic Position (GP) of a body (X), measured (in 0-360 deg) westward from the point of Aries. Relation GHA of a body = SHA of the body + GHA of Aries. Local hour angle LHA is angular distance of between the meridian of the celestial body and the meridian of the observer, Relation LHA of a body) = GHA of the body - Longitude of observer. Note : Longitude is an angular distance in degree, East(+ve) or West(-ve) of the prime meridian. This notation is popular for public use. This means, longitude increases east or west of the prime meridian (from 0 at prime meridian to 180 on other side of Earth). However, representing longitude 0 to +360 deg east only from Greenwich is preferred since the satellites go around and that makes sense for the longitude to keep increasing if the satellite moves forward, else calculations need to track switching of east or west of longitude. Therefore remember, all through the longitude is represented as 0 to +360 deg east only from Greenwich. For more about Sidereal Time, read at http://en.wikipedia.org/wiki/Sidereal_time , http://star-www.st-and.ac.uk/~fv/webnotes . GMT - Greenwich Mean Time, was originally referred to Mean Solar Time at Greenwich, later adopted as a global time standard. GMST - Greenwich Mean Sidereal Time, is mean sidereal time at zero longitude; GMST is in degree. LMST - Local Mean Sidereal Time, is mean sidereal time at your longitude; LMST is in degree. LMT - Local Mean Solar Time; LMT at zero meridian is Universal time (UT1) also called Greenwich mean time (GMT)
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OM-MSS Page 9 UT - Universal Time, is International time standard for astronomy & navigation, introduced in 1928 by the International Astronomical Union (IAU). UT is modern continuation of Greenwich Mean Time (GMT), based on sidereal time at Greenwich, with the day starting at midnight. UT is a time that Earth's rotation determines, no one controls it. The length of a UT second is defined by the period of Earth's revolution around Sun. UT has many versions. UT is also known as UT0. UT becomes UT1 when it is corrected for the irregular movements of the terrestrial poles. UT1 - In conformance with IAU, the Greenwich Mean Sidereal Time GMST is linked directly to Universal Time UT1 through the equation : GMST = 24110.54841 + 8640184.812866 * T + 0.093104 * T^2 - 0.0000062 * T^3 ; where 24110.4581 in seconds is 6h 41m 50.54841 , T is in Julian centuries from epoch JD2000 = 2451545.0, T = d / 36525, d = JD - 2451545.0, JD is input UT in Julian days LMST = GMST + (observer's east longitude); LMST is usually referred as LST in observatory. UTC - Coordinated Universal Time, a variant of UT, replaced Greenwich Mean Time (GMT) on 1 January 1972 as international time reference. UTC is based on atomic measurements rather than the earth's rotation. UTC is a human invention of highly precise clocks that keep time. UTC is primary time standard by which the world regulates clocks and time for : 1. Internet & World Wide Web standards, the network time protocol over internet for the clocks of computers & servers, the online services in aviation, the weather forecasts and others rely on UTC which is universally accepted time. 2. UTC divides time into conventional days, hours, minutes and seconds. Days are identifiable using Gregorian calendar and Julian day numbers. A day contains 24 hours, hour contains 60 minutes. A minute usually contains 60 seconds but rarely adjusted to have a leap second. All smaller time units are of constant duration for sub-microsecond precision. Note (UTC, UT, & leap second) : The length of a UTC second is defined by count of radiation cycles of atomic transition of element cesium, and not related to any astronomical phenomena. Contrary to this, the length of a UT second is defined in terms of period of Earth's revolution around Sun. Thus, the two time scale are indepedent but controlled by international agreement that UTC cannot differ from UT or UT1 by more than 0.9 second. When the difference crosses this limit of 0.9 second, then one-second change called a 'leap second' is added into UTC. This occurs about once a year. TAI - International Atomic Time, is an extreme precise means of time-keeping, deviates only 1 second in about 20 million years. It is based on a continuous counting of the SI second, does not take into account the Earth's slowing rotation which determines the length of a day. TAI is compared to UT1 and before the difference reaches 0.9 seconds, a leap second is added to UTC.
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OM-MSS Page 10 ET - Ephemeris Time, refers to time in any astronomical ephemeris, was introduced in 1950. ET is calculated from the positions of the sun and moon relative to the earth, assuming that Newton's laws are perfectly obeyed. ET, is reckoned from the beginning of the calendar year A.D. 1900, when the geometric mean longitude of the sun was 279 deg, 41 min, 48.04 sec, at this instant Ephemeris Time was 1900 January 0d 12h precisely. Gregorian calendar is an internationally accepted civil calendar, also called Western or Christian calendar, (Year, Months and Days). A year is divided into twelve months. A normal year consists of 365 days and in a leap year a leap day is added as 29 February making the year 366 days. A leap year occurs every 4 years, but the Gregorian calendar omits 3 leap days every 400 years. The start of day is Midnight as 00 hr: 00 mm: 00 sec UT/GMT. Julian Day (JD) : Refers to a continuous count of days since the beginning of the Julian Period. Julian day number is a count of days elapsed since Greenwich mean noon on 1 January 4713 B.C. BCE - Before the Common/Current/Christian Era, CE - Common Era/Current Era/Christian Era. BCE 4713 January 1 00:00:00.0 UT is JD -0.500000 BCE 4713 January 1 12:00:00.0 UT is JD 0.000000 CE 1858 November 16 12:00:00.0 UT is JD 2400000.0 CE 1858 November 17 00:00:00.0 UT is JD 2400000.5 MJD is Modified Julian day begins at midnight civil date, defined as MJD = JD - 2400000.5 An Epoch specifies a precise moment in time. Fundamental epoch is J1900 means Y 1899, M 12, D 31, H 12.0 = 2415020.00 JD is based on Newcomb planatory theory, used till 1984. New standard epoch is J2000 means Y 2000, M 01, D 01, H 12.0 = 2451545.0000000000 JD One Julian century (JC) is 36525 days. The start of New standard epoch is one Julian century (JC) after the Newcomb epoch. Julian epoch for beginning of year 2000 is Y 2000, M 1, D 1, H 12.0 = 2451545.000000000 Calculating Julian Day Number to Gregorian Calendar date or Gregorian Calendar date to Julian Day Number, is easy. The Julian Day Number so calculated is at 00 hours UT/GMT, on that date. For more about Time Scales : read at http://stars.astro.illinois.edu/celsph.html, http://www.ucolick.org/~sla/leapsecs/timescales.html
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OM-MSS Page 11 Concluding : A brief preamble of Time standards and designations were presented. However, unless otherwise specified, in OM-MSS Software : 1. Julian dates are widely used as time variables in the software. A julian day starts at midday 4713 B.C. 2. Standard epoch is Julian Day J2000 which means Y 2000, M 01, D 01, H 12.00 = 2451545.0000000000 3. Year, Month and Days are of Gregorian calendar. The BCE 4713 January 01 12:00:00.0 UT Monday, is JD 0.000000 . 4. A day is Mean Solar Day is average of true solar day during entire year. Our Clock follows Mean Solar Day. 5. A Mean solar day = 24h 00m 00s = 86400 sec is the time for a little more than one earth rotation ie 360.9856473356 deg . 6. Hours, Minutes, Seconds are in universal time UT over Greenwich; Midnight as 00 hr: 00 mm: 00 sec GMT . Year, Month, Day, Hours, Minutes, Seconds are unsiged +ve values. The time addition & subtraction are in Julian days, or in UT days, hours, minute, seconds, where day = 24 x 3600 seconds. 7. A Sidereal day is Mean sidereal day = 23h 56m 4.090538155680s = 86164.090538155680s is time for exact one earth rotation ie 360 deg . All you need to do is, convert universal time to your local time that corresponds to the longitude of your place. 8. Longitude is represented in 0 to +360 deg east only from Greenwich, since satellites go around & longitude keep increasing 0 to +360 deg, End of a brief on Time Designations, Time Standards. Move on to Time Conversion Utilities of OM-MSS Software, Next Precise time conversion utilities are presented in Sections (1.1 to 1.10) respectively : (a) Conversion of Universal Time (year, month, day, hour decimal) To Julian Day; (b) Conversion of Julian Day To Universal Time (year, month, day, hour decimal; (c) Conversion of Fundamental Epoch To Julian day and Julian century; (d) Add or Subtract time (days, hour, minute seconds) to or from input time; (e) Julian day for start of any Year; (f) Solar Time : Local Mean Solar Time (LMT) over observer's Longitude, and Greenwich Mean Time (GMT); (g) Sidereal Time : Greenwich universal time at hour 0.0 (ST0) and Greenwich Mean Sidereal Time (GMST) ; (h) Sidereal Time : Greenwich Sidereal Time (GST), Greenwich Hour Angle (GHA), and Mean Sidereal Time (MST) ; (i) Sidereal Time : Local Mean Sidereal Time (LMST) over observer's Longitude ; (j) Time Conversions : LMT to LST, LST to LMT, LMT to LMST, LMST to LMT ; Next Section - 1.1 Conversion of UT To JD
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OM-MSS Page 12 OM-MSS Section - 1.1 ---------------------------------------------------------------------------------------------------2 Conversion of Universal Time (year, month, day, hour decimal) To Julian Day, while Year 100 to 3500 1. Input UT year = 1899, month = 12, day = 31, hr = 12, min = 0, sec = 0.0000000000 Output julian_day = 2415020.0000000000 2. Input UT year = 2000, month = 1, day = 1, hr = 12, min = 0, sec = 0.0000000000 Output julian_day = 2451545.0000000000 3. Input UT year = 2000, month = 12, day = 31, hr = 11, min = 59, sec = 59.0000000000 Output julian_day = 2451909.9999884260 4. Input UT year = 2001, month = 1, day = 1, hr = 0, min = 0, sec = 0.0000000000 Output julian_day = 2451910.5000000000 5. Input UT year = 2050, month = 1, day = 1, hr = 12, min = 0, sec = 0.0000000000 Output julian_day = 2469808.0000000000 Note : The results verified & validated with those reported (Ref : http://aa.usno.navy.mil/data/docs/JulianDate.php). Next Section - 1.2 Conversion of JD To UT
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OM-MSS Page 13 OM-MSS Section - 1.2 ---------------------------------------------------------------------------------------------------3 Conversion of Julian Day To Universal Time (year, month, day, hour decimal), while Year 100 to 3500 1. Input julian_day = 2415020.0000000000 Output UT year = 1899, month = 12, day = 31, hr = 12, min = 0, sec = 0.0000000000 2. Input julian_day = 2451545.0000000000 Output UT year = 2000, month = 1, day = 1, hr = 12, min = 0, sec = 0.0000000000 3. Input julian_day = 2451909.9999884260 Output UT year = 2000, month = 12, day = 31, hr = 11, min = 59, sec = 59.0000054240 4. Input julian_day = 2451910.5000000000 Output UT year = 2001, month = 1, day = 1, hr = 0, min = 0, sec = 0.0000000000 5. Input Julian day = 2469808.0000000000 Output UT year = 2050, month = 1, day = 1, hr = 12, min = 0, sec = 0.0000000000 6. Input julian_day = 2451909.9999884260 Output UT Year = 2000, Year days decimal = 365.49999, Month = 12, Day = 31 Hours decimal = 11.99972, Hour = 11, Min = 59, Sec = 59.000 Note : The results verified & validated with those reported (Ref : http://aa.usno.navy.mil/data/docs/JulianDate.php). Next Section - 1.3 Conversion of Epoch To JD
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OM-MSS Page 14 OM-MSS Section - 1.3 ---------------------------------------------------------------------------------------------------4 Conversion of Fundamental Epoch To Julian day and Julian century. The Fundamental Newcomb Epoch J1900, was used till 1984, is replaced by New Standard Epoch J2000. The Newcomb Epoch J1900, is beginning of Y 1899, M 12, D 31, H 12.0. The New standard epoch J2000, is beginning of Y 2000, M 1, D 1, H 12.0. A Julian century = 36525 days. The New Epoch J2000, is one Julian century after Epoch J1900. 1. Input Fundamental Epoch J1900, corresponds to year = 1900, month = 1, day of month = 1, hours = 12.0. Output Julian day = 2415020.00000 2. Input New Standard Epoch J2000, corresponds to year = 2000, month = 1, day of month = 1, hours = 12.0. Output Julian day = 2451545.00000 3. A Julian century, 100 years = 36525 days from New Standard Epoch J2000. Prior to year 2000 will have negative Julian century value. Input Julian day = 2415021.0000000000, corresponds to year = 1900, month = 1, day_no_of_month = 1, hour decimal = 12.0000000000 Output Julian day, from Standard Epoch J2000, Converted to Julian century value is = -0.9999726215 Input Julian day = 2451545.0000000000, corresponds to year = 2000, month = 1, day_no_of_month = 1, hour decimal = 12.0000000000 Output Julian day, from Standard Epoch J2000, Converted to Julian century value is = 0.0000000000 Input Julian day = 2488070.0000000000, corresponds to year = 2100, month = 1, day_no_of_month = 1, hour decimal = 12.0000000000 Output Julian day, from Standard Epoch J2000, Converted to Julian century value is = 1.0000000000 Next Section - 1.4 Add or Subtract time To i/p time
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OM-MSS Page 15 OM-MSS Section - 1.4 ------------------------------------------------------------------------------------------------------5 Add or Subtract time (days, hour, minute seconds) to or from input time. 1. Add Days, hour, minute, seconds To any Input UT Year, month, day, hour, minute, seconds Input UT year = 2000, month = 1, day = 1, hr = 0, min = 0, sec = 0.0000000000 Add UT days = 365, hour = 6, minute = 0, seconds = 0.0000000000 Corresponds to days in decimal = 365.2500000000 Output UT year = 2000, month = 12, day = 31, hr = 6, min = 0, sec = 0.0000000000 2. Subtract Days, hour, minute, seconds To any Input UT Year, month, day, hour, minute, seconds Input UT year = 2000, month = 1, day = 1, hr = 6, min = 0, sec = 0.0000000000 Subtract UT days = 1, hour = 6, minute = 0, seconds = 0.0000000000 Corresponds to days in decimal = 1.2500000000 Output UT year = 1999, month = 12, day = 31, hr = 0, min = 0, sec = 0.0000000000 Next Section - 1.5 JD for start of year
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OM-MSS Page 16 OM-MSS Section - 1.5 ------------------------------------------------------------------------------------------------------6 Julian day for start of any Year. 1. Find Julian day for start of Epoch J2000, means the year 2000, M 1, D 1, H 12.0 Input Start of year = 2000, Output the correspnding Julian day is = 2451545.0000000000 2. Find Julian day for start of any other Year, M 1, D 1, H 12.0 Input Start of year = 2001, Output the correspnding Julian day is = 2451911.0000000000 Next Section - 1.6 LMT over Longitude and GMT
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OM-MSS Page 17 OM-MSS Section - 1.6 ------------------------------------------------------------------------------------------------------7 Solar Time : Local Mean Solar Time (LMT) over Observer's Longitude, and Greenwich Mean Time (GMT) over 0 deg Longitude. Our Clocks are set on Solar time (Sun time), based on the rotation of the Earth with respect to the Sun. Solar day is the time for Earth to make a complete rotation on its axis relative to Sun; it varies through +30 to -20 seconds in a year. A Mean Solar Day (midnight as 0 hour), is average of true solar day during entire year, contains 86,400 mean solar seconds; A Day is measured midnight to midnight, is divided into 24 solar hours, each hour is 60 solar minutes, and each minute is 60 solar seconds. Longitude is angular distance in degree, East(+ve) or West(-ve) of the prime meridian. This notation is popular for public use. But for reasons explained before, the longitude is represented as 0 to +360 deg, east only from Greenwich. example, log 48 deg west is (360 - 48) = 320 deg east only from Greenwich; and Log 22.5 deg east is 22.5 deg east only from Greenwich Local Mean solar time (LMT) at 0 deg longitude, is Universal time (UT1), is Greenwich mean time (GMT). Presented below Conversion between UT(GMT) and LMT over Observer's Geographic Longitude. During conversion, a day change may occur. Chosen instances of Time/Longitudes that involve log 0 & 180 deg cross over : 82 deg W, 22.5 deg E, 15 deg W, 15 deg E, 195 deg W, 165 deg E. 1. Observer's Longitude West of Greenwich : 48 deg W meaning 312 deg East of Greenwich. A. Conversion GMT to LMT : Greenwch mean time (UT/GMT) over Greenwich To Local mean time (LMT) over observer's longitude. Input Longitude in deg = 312.00, convert to arc_deg = 312, arc_min = 0, arc_sec = 0.00, corresponds to angle hr = 20, min = 48, sec = 0.00 GMT year = 2011, month = 7, day = 14, hr = 23, min = 31, sec = 25.00000, over 0 deg longitude Output LMT year = 2011, month = 7, day = 14, hr = 20, min = 19, sec = 25.00001, over observer longitude B. Conversion LMT to GMT : Local mean time (LMT) over observer longitude To Greenwch mean time (UT/GMT) over Greenwich. Input Longitude in deg = 312.00, convert to arc_deg = 312, arc_min = 0, arc_sec = 0.00, corresponds to angle hr = 20, min = 48, sec = 0.00 LMT year = 2011, month = 7, day = 14, hr = 20, min = 19, sec = 25.00001, over observer longitude Output GMT year = 2011, month = 7, day = 14, hr = 23, min = 31, sec = 25.00001, over 0 deg longitude
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OM-MSS Page 18 2. Observer's Longitude East of Greenwich : 22.5 deg E meaning 22.5 deg East of Greenwich. A. Conversion GMT to LMT : Greenwch mean time (UT/GMT) over Greenwich To Local mean time (LMT) over observer's longitude. Input Longitude in deg = 22.50, convert to arc_deg = 22, arc_min = 30, arc_sec = 0.00, corresponds to angle hr = 1, min = 30, sec = 0.00 GMT year = 2011, month = 7, day = 14, hr = 23, min = 31, sec = 25.00000, over 0 deg longitude Output LMT year = 2011, month = 7, day = 15, hr = 1, min = 1, sec = 25.00001, over observer longitude B. Conversion LMT to GMT : Local mean time (LMT) over observer longitude To Greenwch mean time (UT/GMT) over Greenwich. Input Longitude in deg = 22.50, convert to arc_deg = 22, arc_min = 30, arc_sec = 0.00, corresponds to angle hr = 1, min = 30, sec = 0.00 LMT year = 2011, month = 7, day = 15, hr = 1, min = 1, sec = 25.00001, over observer longitude Output GMT year = 2011, month = 7, day = 14, hr = 23, min = 31, sec = 25.00001, over 0 deg longitude 3. Observer's Longitude West of Greenwich : 15 deg W meaning 345 deg East of Greenwich. A. Conversion GMT to LMT : Greenwch mean time (UT/GMT) over Greenwich To Local mean time (LMT) over observer's longitude. Input Longitude in deg = 345.00, convert to arc_deg = 345, arc_min = 0, arc_sec = 0.00, corresponds to angle hr = 23, min = 0, sec = 0.00 GMT year = 2011, month = 7, day = 14, hr = 0, min = 0, sec = 1.00000, over 0 deg longitude Output LMT year = 2011, month = 7, day = 13, hr = 23, min = 0, sec = 1.00001, over observer longitude B. Conversion LMT to GMT : Local mean time (LMT) over observer longitude To Greenwch mean time (UT/GMT) over Greenwich. Input Longitude in deg = 345.00, convert to arc_deg = 345, arc_min = 0, arc_sec = 0.00, corresponds to angle hr = 23, min = 0, sec = 0.00 LMT year = 2011, month = 7, day = 13, hr = 23, min = 0, sec = 1.00001, over observer longitude Output GMT year = 2011, month = 7, day = 14, hr = 0, min = 0, sec = 0.99999, over 0 deg longitude
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OM-MSS Page 19 4. Observer's Longitude East of Greenwich : 15 deg E meaning 15 deg East of Greenwich. A. Conversion GMT to LMT : Greenwch mean time (UT/GMT) over Greenwich To Local mean time (LMT) over observer's longitude. Input Longitude in deg = 15.00, convert to arc_deg = 15, arc_min = 0, arc_sec = 0.00, corresponds to angle hr = 1, min = 0, sec = 0.00 GMT year = 2011, month = 7, day = 14, hr = 0, min = 0, sec = 0.99999, over 0 deg longitude Output LMT year = 2011, month = 7, day = 14, hr = 1, min = 0, sec = 0.99998, over observer longitude B. Conversion LMT to GMT : Local mean time (LMT) over observer longitude To Greenwch mean time (UT/GMT) over Greenwich. Input Longitude in deg = 15.00, convert to arc_deg = 15, arc_min = 0, arc_sec = 0.00, corresponds to angle hr = 1, min = 0, sec = 0.00 LMT year = 2011, month = 7, day = 14, hr = 1, min = 0, sec = 0.99998, over observer longitude Output GMT year = 2011, month = 7, day = 14, hr = 0, min = 0, sec = 0.99999, over 0 deg longitude 5. Observer's Longitude West of Greenwich : 165 deg W meaning 195 deg East of Greenwich. A. Conversion GMT to LMT : Greenwch mean time (UT/GMT) over Greenwich To Local mean time (LMT) over observer's longitude. Input Longitude in deg = 195.00, convert to arc_deg = 195, arc_min = 0, arc_sec = 0.00, corresponds to angle hr = 13, min = 0, sec = 0.00 GMT year = 2011, month = 7, day = 14, hr = 3, min = 0, sec = 0.00000, over 0 deg longitude Output LMT year = 2011, month = 7, day = 13, hr = 15, min = 59, sec = 59.99999, over observer longitude B. Conversion LMT to GMT : Local mean time (LMT) over observer longitude To Greenwch mean time (UT/GMT) over Greenwich. Input Longitude in deg = 195.00, convert to arc_deg = 195, arc_min = 0, arc_sec = 0.00, corresponds to angle hr = 13, min = 0, sec = 0.00 LMT year = 2011, month = 7, day = 13, hr = 15, min = 59, sec = 59.99999, over observer longitude Output GMT year = 2011, month = 7, day = 14, hr = 3, min = 0, sec = 0.00000, over 0 deg longitude
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OM-MSS Page 20 6. Observer's Longitude East of Greenwich : 165 deg E meaning 165 deg East of Greenwich. A. Conversion GMT to LMT : Greenwch mean time (UT/GMT) over Greenwich To Local mean time (LMT) over observer's longitude. Input Longitude in deg = 165.00, convert to arc_deg = 165, arc_min = 0, arc_sec = 0.00, corresponds to angle hr = 11, min = 0, sec = 0.00 GMT year = 2011, month = 7, day = 14, hr = 3, min = 0, sec = 0.00000, over 0 deg longitude Output LMT year = 2011, month = 7, day = 14, hr = 14, min = 0, sec = 0.00001, over observer longitude B. Conversion LMT to GMT : Local mean time (LMT) over input longitude To Greenwch mean time (UT/GMT) over Greenwich. Input Longitude in deg = 165.00, convert to arc_deg = 165, arc_min = 0, arc_sec = 0.00, corresponds to angle hr = 11, min = 0, sec = 0.00 LMT year = 2011, month = 7, day = 14, hr = 14, min = 0, sec = 0.00001, over observer longitude Output GMT year = 2011, month = 7, day = 14, hr = 3, min = 0, sec = 0.00000, over 0 deg longitude Note : The results verified & validated with those reported (Ref : http://www.navy.gov.au/reserves/e-docs/DATA/NAVYPUBS/NAVYMAN/BRD45(2)/02.pdf ). Next Section - 1.7 Sidereal Time ST0 and GMST
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OM-MSS Page 21 OM-MSS Section - 1.7 ------------------------------------------------------------------------------------------------------8 Sidereal Time : Greenwich universal time at hour 0.0 (ST0) and Greenwich Mean Sidereal Time (GMST) at input Universal Time (UT). Astronomical observations need Sidereal time (Star time), but Our Clocks are set on Solar time (Sun time). Sidereal Time keeping system is based on Earth's rotation measured relative to fixed stars. To be specific, ST is hour angle of vernal equinox. Mean sidereal day is about 23 hours, 56 minutes, 4.0916 seconds (ie, 23.9344699 hours or 0.99726958 mean solar days). For finding Sidereal Time, there are many analytical equations comprising different constants and reference Julian day. ST0 is sidereal time of Greenwich at universal time (UT) hour 0.0, on 1st January of each year (e.g. ST0 = 100.807 deg for 2013, Jan.1, hr 0.0 ). GMST is Greenwich Mean Sidereal Time defined by Greenwich meridian and vernal equinox. GMST is hour angle of average position of vernal equinox - neglecting short-term effects of nutation. Presented below computed, ST0 and GMST of Greenwich at universal time (UT) hour 0.0, on 1st January of years 1900, 1999, 2000, 2013. ST0 of Greenwich at universal time (UT) hour 0.0, on 1st January of each year 1900, 1999, 2000, 2013 respectively. 1. Find ST0 Sidereal time in deg, of Greenwich Meridian. Input UT/GMT year = 1900, (means start of year on January 1, hr 0.0) Output ST0 in deg = 100.1838236674, converted into time unit hr = 6, min = 40, sec = 44.11768, at Greenwich Meridian 2. Find ST0 Sidereal time in deg, of Greenwich Meridian. Input UT/GMT year = 1999, (means start of year on January 1, hr 0.0) Output ST0 in deg = 100.2061912009, converted into time unit hr = 6, min = 40, sec = 49.48589, at Greenwich Meridian 3. Find ST0 Sidereal time in deg, of Greenwich Meridian. Input UT/GMT year = 2000, (means start of year on January 1, hr 0.0) Output ST0 in deg = 99.9674763217, converted into time unit hr = 6, min = 39, sec = 52.19432, at Greenwich Meridian 4. Find ST0 Sidereal time in deg, of Greenwich Meridian. Input UT/GMT year = 2013, (means start of year on January 1, hr 0.0) Output ST0 in deg = 100.8067795999, converted into time unit hr = 6, min = 43, sec = 13.62710, at Greenwich Meridian
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OM-MSS Page 22 GMST of Greenwich at universal time (UT) hour 0.0, on 1st January of each year 1900, 1999, 2000, 2013 respectively. A day change may occur, therefore GMST with date adjusted to calendar YY MM DD and UT hr mm sec is presented. 1. Find GMST Greenwich mean sidereal time in deg, of Greenwich Meridian, at any input Julian day, since Epoch J2000 Input UT/GMT year = 1900, month = 1, day = 1, hr = 0, min = 0, sec = 0.000, Corresponds to julian_day = 2415020.50000 Output GMST in deg = 100.1837763190, converted into time unit hr = 6, min = 40, sec = 44.10632, of day = 1, month = 1, year = 1900, 2. Find GMST Greenwich mean sidereal time in deg, of Greenwich Meridian, at any input Julian day, since Epoch J2000 Input UT/GMT year = 1999, month = 1, day = 1, hr = 0, min = 0, sec = 0.00000, Corresponds to julian_day = 2451179.50000 Output GMST in deg = 100.2065060288, converted into time unit hr = 6, min = 40, sec = 49.56145, of day = 1, month = 1, year = 1999, 3. Find GMST Greenwich mean sidereal time in deg, of Greenwich Meridian, at any input Julian day, since Epoch J2000 Input UT/GMT year = 2000, month = 1, day = 1, hr = 0, min = 0, sec = 0.00000, Corresponds to julian_day = 2451544.50000 Output GMST in deg = 99.9677946232, converted into time unit hr = 6, min = 39, sec = 52.27071, of day = 1, month = 1, year = 2000, 4. Find GMST Greenwich mean sidereal time in deg, of Greenwich Meridian, at any input Julian day, since Epoch J2000 Input UT/GMT year = 2013, month = 1, day = 1, hr = 0, min = 0, sec = 0.00000, Corresponds to julian_day = 2456293.50000 Output GMST in deg = 100.8071438223, converted into time unit hr = 6, min = 43, sec = 13.71452, of day = 1, month = 1, year = 2013,
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OM-MSS Page 23 GMST of Greenwich at universal time (UT) at hour 1.0, 6.0, 12.0, 18.0, 23.0 respectively of year 2013. A day change may occur, therefore GMST with date adjusted to calendar YY MM DD and UT hr mm sec is presented. 5. Find GMST Greenwich mean sidereal time in deg, of Greenwich Meridian, at any input Julian day, since Epoch J2000 Input UT/GMT year = 2013, month = 1, day = 1, hr = 1, min = 0, sec = 0.00000, Corresponds to julian_day = 2456293.54167 Output GMST in deg = 115.8482124098, converted into time unit hr = 7, min = 43, sec = 23.57098, of day = 1, month = 1, year = 2013, 6. Find GMST Greenwich mean sidereal time in deg, of Greenwich Meridian, at any input Julian day, since Epoch J2000 Input UT/GMT year = 2013, month = 1, day = 1, hr = 6, min = 0, sec = 0.00000, Corresponds to julian_day = 2456293.75000 Output GMST in deg = 191.0535555147, converted into time unit hr = 12, min = 44, sec = 12.85332, of day = 1, month = 1, year = 2013, 7. Find GMST Greenwich mean sidereal time in deg, of Greenwich Meridian, at any input Julian day, since Epoch J2000 Input UT/GMT year = 2013, month = 1, day = 1, hr = 12, min = 0, sec = 0.00000, Corresponds to julian_day = 2456294.00000 Output GMST in deg = 281.2999673747, converted into time unit hr = 18, min = 45, sec = 11.99217, of day = 1, month = 1, year = 2013, 8. Find GMST Greenwich mean sidereal time in deg, of Greenwich Meridian, at any input Julian day, since Epoch J2000 Input UT/GMT year = 2013, month = 1, day = 1, hr = 18, min = 0, sec = 0.00000, Corresponds to julian_day = 2456294.25000 Output GMST in deg = 11.5463792346, converted into time unit hr = 0, min = 46, sec = 11.13102, of day = 2, month = 1, year = 2013, 9. Find GMST Greenwich mean sidereal time in deg, of Greenwich Meridian, at any input Julian day, since Epoch J2000 Input UT/GMT year = 2013, month = 1, day = 1, hr = 23, min = 0, sec = 0.00000, Corresponds to julian_day = 2456294.45833 Output GMST in deg = 86.7517225072, converted into time unit hr = 5, min = 47, sec = 0.41340, of day = 2, month = 1, year = 2013, 10. Find GMST Greenwich mean sidereal time in deg, of Greenwich Meridian, at any input Julian day, since Epoch J2000 Input UT/GMT year = 2013, month = 1, day = 3, hr = 9, min = 11, sec = 56.61639, Corresponds to julian_day = 2456295.88329 Output GMST in deg = 241.1421329901, converted into time unit hr = 16, min = 4, sec = 34.11192, of day = 3, month = 1, year = 2013, Note : The results verified & validated with those reported (Ref : http://www.csgnetwork.com/siderealjuliantimecalc.html ). Next Section - 1.8 Sidereal Time GST, GHA and MST
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OM-MSS Page 24 OM-MSS Section - 1.8 ------------------------------------------------------------------------------------------------------9 Sidereal Time : Greenwich Sidereal Time (GST), Greenwich Hour Angle (GHA), and Mean Sidereal Time (MST) at input Universal Time (UT). Greenwich Sidereal Time (GST) is the Local Sidereal Time (LST) over Greenwich Meridian ie 0 deg longitude. The Local Sidereal Time (LST) over any longitude location = The Greenwich Sidereal Time (GST) + longitude of the location. The Greenwich Hour Angle (GHA) hour is the angle is between Greenwich Meridian and the Meridian of a celestial body, here it is first point of Aries . The Mean Sidereal Time (MST) of a locality is defined by its longitude; the MST over 0 deg longitude is often called Greenwich Mean sidereal Time (GMST). Finding Sidereal Time (ST) over Greenwich Meridian, corresponding to input Local time LMT over Greenwich ie UT/GMT . Note : Here GHA is GHA(Aries), and MST at 0 deg longitude is GMST; the Greenwich apparent sidereal time GAST = GMST + equation of the equinoxes; The angles are measured in deg or time, with 24 hr = 360 deg or 1 hr = 15 deg; thus GST(hour) = GHA(deg)/15. For finding GST and GHA, there are many analytical equations comprising different constants and reference Julian day. Here used two different equations for GST and GHA; the ref. Julian day for GST is JD1900 and for GHA is JD2000. Input UT/LMT year = 1986, month = 10, day = 23, hr = 15, min = 0, sec = 0.0000000000, over Greenwich Output Greenwich sidereal time (GST), Greenwich hour angle (GHA), Mean sidereal time (MST) 1. GST sidereal time in 0 to 360 deg over Greenwich at UT time = 256.7345487561 ie hour = 17, minute = 6, seconds = 56.29170 2. GHA hour angle in 0 to 360 deg over Greenwich at UT time = 256.7435082620 ie deg = 256, arc min = 44, arc sec = 36.62974 3. MST mean sidereal time in deg, over Greenwich at UT time = 256.7348202780 ie hour = 17, minute = 6, seconds = 56.35687 Input UT/LMT year = 2000, month = 1, day = 1, hr = 0, min = 0, sec = 0.0000000000, over Greenwich Output Greenwich sidereal time (GST), Greenwich hour angle (GHA), Mean sidereal time (MST) 1. GST sidereal time in 0 to 360 deg over Greenwich at UT time = 99.9674763217 ie hour = 6, minute = 39, seconds = 52.19432 2. GHA hour angle in 0 to 360 deg over Greenwich at UT time = 99.9674109260 ie deg = 99, arc min = 58, arc sec = 2.67933 3. MST mean sidereal time in deg, over Greenwich at UT time = 99.9677946919 ie hour = 6, minute = 39, seconds = 52.27073
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OM-MSS Page 25 Input UT/LMT year = 2000, month = 1, day = 1, hr = 12, min = 0, sec = 0.0000000000, over Greenwich Output Greenwich sidereal time (GST), Greenwich hour angle (GHA), Mean sidereal time (MST) 1. GST sidereal time in 0 to 360 deg over Greenwich at UT time = 280.4603000000 ie hour = 18, minute = 41, seconds = 50.47200 2. GHA hour angle in 0 to 360 deg over Greenwich at UT time = 280.4674269260 ie deg = 280, arc min = 28, arc sec = 2.73693 3. MST mean sidereal time in deg, over Greenwich at UT time = 280.4606183750 ie hour = 18, minute = 41, seconds = 50.54841 Input UT/LMT year = 2013, month = 1, day = 1, hr = 0, min = 0, sec = 0.0000000000, over Greenwich Output Greenwich sidereal time (GST), Greenwich hour angle (GHA), Mean sidereal time (MST) 1. GST sidereal time in 0 to 360 deg over Greenwich at UT time = 100.8067795999 ie hour = 6, minute = 43, seconds = 13.62710 2. GHA hour angle in 0 to 360 deg over Greenwich at UT time = 100.8066665780 ie deg = 100, arc min = 48, arc sec = 23.99968 3. MST mean sidereal time in deg, over Greenwich at UT time = 100.8071437424 ie hour = 6, minute = 43, seconds = 13.71450 Input UT/LMT year = 2013, month = 1, day = 3, hr = 9, min = 11, sec = 56.6163906455, over Greenwich Output Greenwich sidereal time (GST), Greenwich hour angle (GHA), Mean sidereal time (MST) 1. GST sidereal time in 0 to 360 deg over Greenwich at UT time = 241.1417688342 ie hour = 16, minute = 4, seconds = 34.02452 2. GHA hour angle in 0 to 360 deg over Greenwich at UT time = 241.1471693383 ie deg = 241, arc min = 8, arc sec = 49.80962 3. MST mean sidereal time in deg, over Greenwich at UT time = 241.1421329996 ie hour = 16, minute = 4, seconds = 34.11192 Note : The results verified & validated with those reported (Ref : http://www2.arnes.si/~gljsentvid10/longterm.htm ). Next Section - 1.9 Sidereal Time LMST and GMST
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OM-MSS Page 26 OM-MSS Section - 1.9 -----------------------------------------------------------------------------------------------------10 Sidereal Time : Local Mean Sidereal Time (LMST) and Greenwich Mean Sidereal Time (GMST) at input Universal Time (UT). LMST is Local Mean Sidereal time over observer's Longitude and GMST is Greenwich Mean Sidereal Time, while input is UT/GMT time. Longitude is expressed in 0 to 360 deg only East of Greenwich; not as positive/negative or east/west of Greenwich (0 to 180). Representing longitude 0 to +360 deg east only from Greenwich, is preferred since satellites go around and angle keep increasing 0 to +360. example : log 82.5 deg (corresponds 5hr 30min) is 82.5 deg East, and log 285.0 deg (corresponds 19hr 0min) is 75 deg West of Greenwich. LMST over observer's Log = GMST + Longitude of observer; or simply written as ST over observer's log = GMST + observer's log. Presented below results of LMST Utilities, for observer Longiudes 0 - 90, 90 - 180, 180 - 270, 270 - 360 deg, East of Greenwich Meridian. A day change may occur, therefore LMST with date adjusted to calendar YY MM DD and UT hr mm sec is presented. 1. Finding LMST Local Mean Sidereal Time, over observer Longitude (Greenwich Medirian) at time UT/GMT YY MM DD hr min sec Input Observer Log arc deg = 0, arc min = 0, arc sec = 0.00, in decimal deg = 0.00000, in time unit hr = 0, min = 0, sec = 0.00 UT/GMT year = 2000, month = 1, day = 1, hr = 0, min = 0, sec = 0.00000, in julian_day = 2451544.50000 over Greenwich Output LMST in deg = 99.9677946232, converted into time unit hr = 6, min = 39, sec = 52.27071, of day = 1, month = 1, year = 2000, 2. Finding LMST Local Mean Sidereal Time, over observer Longitude (0-90 deg East of Greenwich Meridian) at time UT/GMT YY MM DD hr min sec Input Observer Log arc deg = 30, arc min = 0, arc sec = 0.00, in decimal deg = 30.00000, in time unit hr = 2, min = 0, sec = 0.00 UT/GMT year = 2000, month = 1, day = 1, hr = 0, min = 0, sec = 0.00000, in julian_day = 2451544.50000 over Greenwich Output LMST in deg = 129.9677946791, converted into time unit hr = 8, min = 39, sec = 52.27072, of day = 1, month = 1, year = 2000, 3. Finding LMST Local Mean Sidereal Time, over observer Longitude (90-180 deg East of Greenwich Meridian) at time UT/GMT YY MM DD hr min sec Input Observer Log arc deg = 150, arc min = 0, arc sec = 0.00, in decimal deg = 150.00000, in time unit hr = 10, min = 0, sec = 0.00 UT/GMT year = 2000, month = 1, day = 1, hr = 0, min = 0, sec = 0.00000, in julian_day = 2451544.50000 over Greenwich Output LMST in deg = 249.9677945673, converted into time unit hr = 16, min = 39, sec = 52.27070, of day = 1, month = 1, year = 2000,
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OM-MSS Page 27 4. Finding LMST Local Mean Sidereal Time, over observer Longitude (180-270 deg East of Greenwich Meridian) at time UT/GMT YY MM DD hr min sec Input Observer Log arc deg = 210, arc min = 0, arc sec = 0.00, in decimal deg = 210.00000, in time unit hr = 14, min = 0, sec = 0.00 UT/GMT year = 2000, month = 1, day = 1, hr = 0, min = 0, sec = 0.00000, in julian_day = 2451544.50000 over Greenwich Output LMST in deg = 309.9677946791, converted into time unit hr = 20, min = 39, sec = 52.27072, of day = 31, month = 12, year = 1999, 5. Finding LMST Local Mean Sidereal Time, over observer Longitude (270-360 deg East of Greenwich Meridian) at time UT/GMT YY MM DD hr min sec Input Observer Log arc deg = 330, arc min = 0, arc sec = 0.00, in decimal deg = 330.00000, in time unit hr = 22, min = 0, sec = 0.00 UT/GMT year = 2000, month = 1, day = 1, hr = 0, min = 0, sec = 0.00000, in julian_day = 2451544.50000 over Greenwich Output LMST in deg = 69.9677945673, converted into time unit hr = 4, min = 39, sec = 52.27070, of day = 1, month = 1, year = 2000, 6. Finding LMST Local Mean Sidereal Time, over observer Longitude (Greenwich Medirian) at time UT/GMT YY MM DD hr min sec Input Observer Log arc deg = 0, arc min = 0, arc sec = 0.00, in decimal deg = 0.00000, in time unit hr = 0, min = 0, sec = 0.00 UT/GMT year = 2013, month = 1, day = 3, hr = 9, min = 11, sec = 56.61639, in julian_day = 2456295.88329 over Greenwich Output LMST in deg = 241.1421329901, converted into time unit hr = 16, min = 4, sec = 34.11192, of day = 3, month = 1, year = 2013, 7. Finding LMST Local Mean Sidereal Time, over observer Longitude (Delhi, India) at time UT/GMT YY MM DD hr min sec Input Observer Log arc deg = 77, arc min = 13, arc sec = 30.11, in decimal deg = 77.22503, in time unit hr = 5, min = 8, sec = 54.01 UT/GMT year = 2013, month = 1, day = 3, hr = 9, min = 11, sec = 56.61639, in julian_day = 2456295.88329 over Greenwich Output LMST in deg = 318.3671630546, converted into time unit hr = 21, min = 13, sec = 28.11913, of day = 3, month = 1, year = 2013, Note : The results verified & validated with those reported (Ref. http://www.csgnetwork.com/siderealjuliantimecalc.html). Next Section - 1.10 Time Conversions More on LMT, LST, LMST
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OM-MSS Page 28 OM-MSS Section - 1.10 ----------------------------------------------------------------------------------------------------11 Time Conversions : 1. LMT to LST, 2. LST to LMT, 3. LMT to LMST, 4. LMST to LMT Conversions, from Local Solar time to Local Sidereal time and back from Local Sidereal time to Local Solar time. Recall that Our Clocks are set on Solar time (sun time) but astronomical observations need Sidereal time (Star time). Local Mean solar time (LMT) at zero meridian is Universal time (UT1) also called Greenwich mean time (GMT). Greenwich Mean Sidereal Time (GMST) is defined by Greenwich meridian and vernal equinox. Greenwich apparent sidereal time (GAST) is obtained by adding a correction to Greenwich mean sidereal time (GMST); the correction term is called the 'Equation of the equinoxes' at mean time interval at UT time; example : YY 2000, MM 1, DD 1 HH 0 , the Equation of the equinoxes in seconds = -0.8758425604 Longitude is expressed in 0 to 360 deg only East of Greenwich; not as positive/negative or east/west of Greenwich (0 to 180). Representing longitude 0 to +360 deg east only from Greenwich, is preferred since satellites go around and angle keep increasing 0 to +360. example : log 82.5 deg (corresponds 5hr 30min) is 82.5 deg East, and log 285.0 deg (corresponds 19hr 0min) is 75 deg West of Greenwich. Presented below Time Conversion Utilities, for observer Longiudes 0 - 90, 90 - 180, 180 - 270, 270 - 360 deg, East of Greenwich Meridian. A day change may occur, therefore the input/output are with date adjusted to calendar YY MM DD and UT hr mm sec are presented. Examples 1 to 10, Time Conversions : LMT to LST, LST to LMT, LMT to LMST, LMST to LMT for observer Longiudes mentioned above. Conversion Forward LMT to LST ie local mean time to local sidereal time; Backward LST to LMT ie local sidereal time to local mean time; Conversion Forward LMT to LMST ie local mean time to local mean sidereal time; Backward LMST to LMT ie local mean sidereal time to local mean time. 1. Conversion of Local mean time (LMT) to Local sidereal time (LST) , Observer Longiude 0-90 deg East of Greenwich Meridian. Input Observer log arc deg = 77, arc min = 13, arc sec = 0.00, in decimal deg = 77.21667, corresponds to angle hr = 5, min = 8, sec = 52.00 LMT year = 2000, month = 1, day = 1, hr = 15, min = 54, sec = 42.00000, over Observer longitude Output GMT year = 2000, month = 1, day = 1, hr = 10, min = 45, sec = 50.00000, over Greenwich GMST year = 2000, month = 1, day = 1, hr = 17, min = 27, sec = 28.36472, over Greenwich LST year = 2000, month = 1, day = 1, hr = 22, min = 36, sec = 20.36470, over Observer longitude
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OM-MSS Page 29 2. Conversion of Local sidereal time (LST) to Local mean time (LMT) , Observer Longiude 0-90 deg East of Greenwich Meridian. Input Observer log arc deg = 77, arc min = 13, arc sec = 0.00, in decimal deg = 77.21667, corresponds to angle hr = 5, min = 8, sec = 52.00 LST year = 2000, month = 1, day = 1, hr = 22, min = 36, sec = 20.36470, over Observer longitude Output GMST year = 2000, month = 1, day = 1, hr = 17, min = 27, sec = 28.36472, over Greenwich GMT year = 2000, month = 1, day = 1, hr = 10, min = 45, sec = 50.00221, over Greenwich LMT year = 2000, month = 1, day = 1, hr = 15, min = 54, sec = 42.00219, over Observer longitude 3. Conversion of Local mean time (LMT) to Local mean sidereal time (LMST) , Observer Longiude 90-180 deg East of Greenwich Meridian. Note - The procedure is same as LMT to LST neglecting the Equation of Equinoxes Input Observer log arc deg = 150, arc min = 0, arc sec = 0.00, in decimal deg = 150.00000, corresponds to angle hr = 10, min = 0, sec = 0.00 LMT year = 2013, month = 1, day = 1, hr = 2, min = 0, sec = 0.00000, over Observer longitude Output GMT year = 2012, month = 12, day = 31, hr = 16, min = 0, sec = 0.00003, over Greenwich GMST year = 2012, month = 12, day = 31, hr = 22, min = 41, sec = 54.86275, over Greenwich LST year = 2013, month = 1, day = 1, hr = 8, min = 41, sec = 54.86274, over Observer longitude 4. Conversion of Local mean sidereal time (LMST) to Local mean time (LMT) , Observer Longiude 90-180 deg East of Greenwich Meridian. Note - The procedure is same as LST to LMT neglecting the Equation of Equinoxes Input Observer log arc deg = 150, arc min = 0, arc sec = 0.00, in decimal deg = 150.00000, corresponds to angle hr = 10, min = 0, sec = 0.00 LMST year = 2013, month = 1, day = 1, hr = 8, min = 41, sec = 54.86274, over Observer longitude Output GMST year = 2012, month = 12, day = 31, hr = 22, min = 41, sec = 54.86275, over Greenwich GMT year = 2012, month = 12, day = 31, hr = 16, min = 0, sec = 0.00123, over Greenwich LMT year = 2013, month = 1, day = 1, hr = 2, min = 0, sec = 0.00122, over Observer longitude Continue Section - 1.10
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OM-MSS Page 30 5. Conversion of Local mean time (LMT) to Local sidereal time (LST) , Observer Longiude 180-270 deg East of Greenwich Meridian. Input Observer log arc deg = 210, arc min = 0, arc sec = 0.00, in decimal deg = 210.00000, corresponds to angle hr = 14, min = 0, sec = 0.00 LMT year = 2013, month = 1, day = 1, hr = 2, min = 0, sec = 0.00000, over Observer longitude Output GMT year = 2013, month = 1, day = 1, hr = 12, min = 0, sec = 0.00000, over Greenwich GMST year = 2013, month = 1, day = 1, hr = 18, min = 45, sec = 11.99217, over Greenwich LST year = 2013, month = 1, day = 1, hr = 8, min = 45, sec = 11.99218, over Observer longitude 6. Conversion of Local sidereal time (LST) to Local mean time (LMT) , Observer Longiude 180-270 deg East of Greenwich Meridian. Input Observer log arc deg = 210, arc min = 0, arc sec = 0.00, in decimal deg = 210.00000, corresponds to angle hr = 14, min = 0, sec = 0.00 LST year = 2013, month = 1, day = 1, hr = 8, min = 45, sec = 11.99218, over Observer longitude Output GMST year = 2013, month = 1, day = 1, hr = 18, min = 45, sec = 11.99217, over Greenwich GMT year = 2013, month = 1, day = 1, hr = 12, min = 0, sec = 0.00052, over Greenwich LMT year = 2013, month = 1, day = 1, hr = 2, min = 0, sec = 0.00054, over Observer longitude 7. Conversion of Local mean time (LMT) to Local mean sidereal time (LMST) , Observer Longiude 270-360 deg East of Greenwich Meridian. Note - The procedure is same as LMT to LST neglecting the Equation of Equinoxes Input Observer log arc deg = 330, arc min = 0, arc sec = 0.00, in decimal deg = 330.00000, corresponds to angle hr = 22, min = 0, sec = 0.00 LMT year = 2013, month = 1, day = 1, hr = 2, min = 0, sec = 0.00000, over Observer longitude Output GMT year = 2013, month = 1, day = 1, hr = 4, min = 0, sec = 0.00003, over Greenwich GMST year = 2013, month = 1, day = 1, hr = 10, min = 43, sec = 53.14040, over Greenwich LST year = 2013, month = 1, day = 1, hr = 8, min = 43, sec = 53.14039, over Observer longitude Continue Section - 1.10
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OM-MSS Page 31 8. Conversion of Local mean sidereal time (LMST) to Local mean time (LMT) , Observer Longiude 270-360 deg East of Greenwich Meridian. Note - The procedure is same as LST to LMT neglecting the Equation of Equinoxes Input Observer log arc deg = 330, arc min = 0, arc sec = 0.00, in decimal deg = 330.00000, corresponds to angle hr = 22, min = 0, sec = 0.00 LMST year = 2013, month = 1, day = 1, hr = 8, min = 43, sec = 53.14039, over Observer longitude Output GMST year = 2013, month = 1, day = 1, hr = 10, min = 43, sec = 53.14040, over Greenwich GMT year = 2013, month = 1, day = 1, hr = 4, min = 0, sec = 0.00023, over Greenwich LMT year = 2013, month = 1, day = 1, hr = 2, min = 0, sec = 0.00021, over Observer longitude 9. Conversion of Local mean time (LMT) to Local mean sidereal time (LMST) , Observer Longiude 270-360 deg East of Greenwich Meridian. Note - The procedure is same as LMT to LST neglecting the Equation of Equinoxes Input Observer log arc deg = 0, arc min = 0, arc sec = 0.00, in decimal deg = 0.00000, corresponds to angle hr = 0, min = 0, sec = 0.00 LMT year = 2013, month = 1, day = 3, hr = 9, min = 11, sec = 56.61639, over Observer longitude Output GMT year = 2013, month = 1, day = 3, hr = 9, min = 11, sec = 56.61639, over Greenwich GMST year = 2013, month = 1, day = 3, hr = 16, min = 4, sec = 34.11192, over Greenwich LST year = 2013, month = 1, day = 3, hr = 16, min = 4, sec = 34.11192, over Observer longitude 10. Conversion of Local mean sidereal time (LMST) to Local mean time (LMT) , Observer Longiude 270-360 deg East of Greenwich Meridian. Note - The procedure is same as LST to LMT neglecting the Equation of Equinoxes Input Observer log arc deg = 0, arc min = 0, arc sec = 0.00, in decimal deg = 0.00000, corresponds to angle hr = 0, min = 0, sec = 0.00 LMST year = 2013, month = 1, day = 3, hr = 16, min = 4, sec = 34.11192, over Observer longitude Output GMST year = 2013, month = 1, day = 3, hr = 16, min = 4, sec = 34.11192, over Greenwich GMT year = 2013, month = 1, day = 3, hr = 9, min = 11, sec = 56.61663, over Greenwich LMT year = 2013, month = 1, day = 3, hr = 9, min = 11, sec = 56.61663, over Observer longitude Note : The results verified & validated with those reported. (Ref. http://koti.mbnet.fi/jukaukor/astronavigation_time.html). Next Section - 1.11 Concluding Time Standards and Conversion Utilities
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OM-MSS Page 32 OM-MSS Section - 1.11 ----------------------------------------------------------------------------------------------------12 Concluding Time Standards and Time Conversion Utilities presented in Sections (1.0 to 1.10). The Time Designations, Time Standards and Time Conversion Utilities were illustrated & accomplished. Our Clocks are set on Solar time (sun time), called local mean time (LMT) at a given location, corrected for its longitude. Mean solar time (LMT) at zero meridian is Universal time (UT1) also called Greenwich mean time (GMT). One Sidereal day is One Rotation of Earth on its axis is the time between two successive Meridian pass over a fixed Star. Longitude is expressed in 0 to 360 deg only East of Greenwich and not as positive/negative or east/west of Greenwich (0 to 180). Representing longitude 0 to +360 deg east only from Greenwich, is preferred since satellites go around and angle keep increasing 0 to +360. In Sections (1.1 to 1.10), all that presented were results of following Time Conversions & Computing Utilities : 1. Conversion of Universal Time, Year, Month, Day, Hour decimal to Julian Day, while Year 100 to 3500, 2. Conversion of Julian Day to Universal Time, Year, Month, Day, Hour decimal, while Year 100 to 3500, 3. Conversion of Fundamental Epoch to Julian day and Julian century. 4. Add or Subtract Time in days, hour, minute seconds. 5. Julian days at the Start of Epoch J2000 or any other year. 6. Find Local Mean Solar Time (LMT) over Observer's Longitude, and Greenwich Mean Time (GMT) over Greenwich Meridian, 0 deg), 7. Find Sidereal time STO & GMST of Greenwich Meridian, 8. Find Greenwich Sidereal Time (GST), Greenwich Hour_Angle (GHA), and Mean Sidereal Time (MST) at any input Universal Time (UT) , 9. Find Local Mean Sidereal Time (LMST) over Observer's Longitude at input UT time over Greenwich Meridian. 10. Conversion of LMT to LST, LST to LMT, LMT to LMST, LMST to LMT. The Results themselves validate, show accuracy and applicability of the OM-MSS software utilities. However, readers may compare the results with those reported elsewhere or at respective references mentioned. End of Time Designations, Time Standards and Time Conversion & Computing Utilities Move on to Positional Astronomy, Motion of Earth Orbit Around Sun, Astronomical Events and more. Next Section - 2 Positional Astronomy and Astronomical Events
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OM-MSS Page 33 OM-MSS Section - 2 -------------------------------------------------------------------------------------------------------13 POSITIONAL ASTRONOMY : EARTH ORBIT AROUND SUN, ASTRONOMICAL EVENTS ANOMALIES, EQUINOXES, SOLSTICES, YEARS & SEASONS. Look at the Preliminaries about 'Positional Astronomy', before moving to the predictions of astronomical events. Definition : Positional Astronomy is measurement of Position and Motion of objects on celestial sphere seen at a particular time and location on Earth. Positional Astronomy, also called Spherical Astronomy, is a System of Coordinates. The Earth is our base from which we look into space. Earth orbits around Sun, counterclockwise, in an elliptical orbit once in every 365.26 days. Earth also spins in a counterclockwise direction on its axis once every day. This accounts for Sun, rise in East and set in West. Term 'Earth Rotation' refers to the spinning of planet earth on its axis. Term 'Earth Revolution' refers to orbital motion of the Earth around the Sun. Earth axis is tilted about 23.45 deg, with respect to the plane of its orbit, gives four seasons as Spring, Summer, Autumn and Winter. Moon and artificial Satellites also orbits around Earth, counterclockwise, in the same way as earth orbits around sun. Earth's Coordinate System : One way to describe a location on earth is Latitude & Longitude, which is fixed on the earth's surface. The Latitudes and Longitudes are presented in several ways. Example, location of Delhi, India, Using degree decimal latitude 28.61 North of Equator, longitude 77.23 East of Greenwich Using degree minutes second latitude 28:36:36 North of Equator, longitude 77:13:48 East of Greenwich Using time zone hour minutes second latitude 28:36:36 North of Equator, longitude 05 hours, 8 min, 55.2 sec East of Greenwich (east of Greenwich, means in Delhi sun will set at 05 hours, 8 min, 55.2 sec before it sets in Greenwich, ie at Delhi UTM is +05 hours, 8 min, 55.2 sec) Continue Section - 2
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OM-MSS Page 34 Laws of Planetary Motion : In the early 1600s, Johannes Kepler proposed three laws of planetary motion. 1st The Law of Ellipses : The orbits of the planets are ellipses, with the Sun at one focus of the ellipse. 2nd The Law of Equal Areas : The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse. 3rd The Law of Harmonies : The ratio of squares of revolutionary periods for two planets is equal to ratio of the cubes of their semi-major axes. Kepler's first law says all planets orbit the sun in a path that resembles an ellipse. Kepler's second law describes the speed at which any given planet will move while orbiting the sun; this speed is constantly changing. A planet moves fastest when it is closest to the sun, and a planet moves slowest when it is furthest from the sun. Kepler's third law compares motion characteristics of different planets; for every planet, the ratio of squares of their periods to the cubes of their average distances from the sun is the same. It implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus, Mercury the innermost planet, takes only 88 days to orbit the Sun but the outermost planet (Pluto) requires 248 years to do the same. Continue Section - 2
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OM-MSS Page 35 Glossary of terms : definitions, meaning and descriptions 1. Celestial Sphere, is an imaginary rotating sphere of infinite radius, concentric with Earth. The poles and equator of the celestial sphere are the projections of earth's poles and equator out into space, called Celestial North Pole, Celestial South Pole and Celestial Equator. 2. Celestial coordinate, is a system for specifying positions of celestial objects, the satellites, planets, stars, galaxies, and more. The common celestial coordinate systems are : Horizontal, Equatorial, and Ecliptic; two others, Galactic and Supergalactic not included here. Each coordinate system is named for its choice of fundamental plane as reference plane. - Horizontal coordinate system uses the observer's horizon as the fundamental reference plane. The coordinates of a point on celestial sphere are Altitude or Elevation and Azimuth. Altitude (Alt) also referred as Elevation (EL) is the angle between the object and the observer's local horizon, expressed as 0 to 90 deg, +ve/-ve. Azimuth (Az) is the angle of the object around the horizon, usually measured from the north increasing towards the east. - Equatorial coordinate system uses the celestial equator as primary reference circle. The coordinates of a point on celestial sphere are Declination and Right ascension, analogous to Latitude-Longitude coordinate system used on Earth. Declination (delta) of an object is angle measured from celestial equator (0 deg declination) along a meridian line through the object. Right ascension (RA) lines on celestial sphere are identical to longitude lines (0-360 deg) on Earth, but the differences are that (i) the RA lines on the celestial sphere divide one rotation into 24 hours (one hour = 15 deg) expressed in terms of hours : minutes : seconds. (ii) for RA the start point for zero hour is vernal equinox or first point of Aries where Sun crosses celestial equator and not 0 deg log. - Ecliptic coordinate system uses the ecliptic, as the fundamental reference plane. The ecliptic is apparent path of Sun on celestial sphere, crosses celestial equator twice in a year, at Autumnal and Vernal equinoxes. Coordinates of a point on celestial sphere are Ecliptic latitude and Ecliptic longitude; distance is also necessary for complete spherical position. Ecliptic latitude (Lat) is the angle between a position and ecliptic, takes values between -90 and +90 deg. Ecliptic longitude (Log) starts from the vernal equinox or first point of Aries as 0 deg and runs to 360 de These three celestial coordinate systems are summarized below.
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OM-MSS Page 36 The Summary of Celestial Coordinate Systems : Systems Coordinates Center point Ref. Plane Poles Ref. Direction (Vertical Horizontal) (Origin) (0 deg Vertical) (0 deg horizontal) (a) Horizontal Altitude or Azimuth Observer on earth Horizon Zenith/nadir North or south elevation, point of horizon (b) Equatorial Declination Right ascension or Earth center(geocentric)/ Celestial equator Celestial poles Vernal equinox hour angle Sun center(heliocentric) (C) Ecliptic Ecliptic latitude Ecliptic longitude Earth center(geocentric)/ Ecliptic Ecliptic poles Vernal equinox Sun center(heliocentric) 3. Celestial Orbit in astronomy, is a gravitationally curved path of celestial body around a point in space. The orbits of planets around the sun or the orbits of satellites around planets are typically elliptical, governed by Kepler's laws of motion. The orbit of each planet is influenced by the other planets as well as by the sun, to a small degree, called perturbations. These perturbations are taken into account in calculating planetary orbits. - Heliocentric orbit is an orbit around the Sun. The planets, comets, and asteroids in our Solar System are in such orbits. - Geocentric orbit is an orbit around the Earth. The Moon and all artificial satellites are in such orbits. - Periapsis & Apoapsis, Perihelion & Aphelion, Perigee & Apogee, represent two points on orbit, named differently to identify the body being orbited. The point closest to the orbited body is called the periapsis and the point furthest to the orbited body is called the Apoapsis. Perihelion and Aphelion, refer to orbits around the Sun; here the orbit point closest to Sun is perihelion and point furthest to Sun is aphelion. Perigee and Apogee, refer to orbits around the Earth; here the orbit point closest to Earth is perigee and point furthest to earth is apogee.
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OM-MSS Page 37 4. Orbit Elements or Parameters, uniquely identify a specific orbit. There are different ways to describe mathematically the same orbit. The orbital parameters are usually expressed either by Keplerian elements or by State vectors, each consisting a set of six parameters. The State vectors, also called Cartesian coordinates, are time-independent, represent the 3-D Position and Velocity components of the orbital trajectory. Keplerian elements are valid only for a specific time, describe the size, shape, and orientation of an orbital ellipse. The State Vectors are often not a convenient way to represent an orbit, hence Keplerian elements are commonly used instead. However, State Vectors and Keplerian Elements can be computed from one another. (a) State vectors are, three Positions (x, y, z) and three Velocities (x dot, y dot, z dot) at Epoch time t. - Position vector describe position of the orbiting body in inertial frame of reference, x-axis pointing to vernal equinox and z-axis pointing upwards. - Velocity vector is velocity of the orbiting body derived from orbital position vector by differentiation with respect to time. (b) Kepler elements are, Inclination, Longitude of ascending node, Argument of periapsis, Eccentricity, Semimajor axis, Mean Anomaly at Epoch time t. - Inclination 'i' of the orbit of a planet, is angle between the plane of planet's orbit and the plane containing Earth's orbital path (ecliptic) or with respect to another plane such as the Sun's equator. For Earth-bound observers the ecliptic is more practical; e.g. inclination of earth orbit to elliptic is 0 deg and to Sun's equator is 7.155 deg. - Longitude of the Ascending node, specify orbit of an object in space. For a geocentric orbit, this longitude is called Right Ascension (RA). It is the angle from a reference direction, called the origin of longitude, to the direction of the ascending node, measured in a reference plane. The reference plane for a Geocentric orbit is Earth's equatorial plane, and the First Point of Aries is the origin of longitude. The reference plane for a Heliocentric orbit is Ecliptic plane, and the First Point of Aries is the origin of longitude. The angle is measured counterclockwise from the origin to the object. - Argument of periapsis, specify angle between orbit's periapsis and orbit's ascending node, measured in orbital plane and direction of motion. Angle 0 deg, means orbiting body is at its closest to central body, at that moment it crosses the plane of reference from south to north. Angle 90 deg, means the orbiting body will reach periapsis at its northmost distance from the plane of reference. Adding the argument of periapsis to the longitude of the ascending node gives the longitude of the periapsis. The word periapsis is replaced by perihelion (for Sun-centered orbits), or by perigee (for Earth-centered orbits).
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OM-MSS Page 38 - Eccentricity 'e' of an orbit shows how much the shape of an object's orbit is different from a circle; eccentricity 'e' vary between 0 and 1. For Circular orbit : e = 0, elliptical orbit : 0 < e < 1, parabolic trajectory : e = 1, hyperbolic trajectory : e > 1 . The Earth's orbital eccentricity varies, from a min value = 0.005 (near circular) to maxi value e = 0.057 (quite elliptical), over a period of 92,000 years, due to gravitational force exerted by Jupiter. The eccentricity of the Earth's orbit is currently about 0.016710219. - Semimajor axis is one half of the major axis, is the radius of an orbit at the orbit's two most distant points. The semi-major axis length 'a' of an ellipse is related to the semi-minor axis length 'b' through the eccentricity 'e'. - Mean Anomaly 'M' relates the position and time for a body moving in a Kepler orbit. The mean anomaly of an orbiting body is the angle through which the body would have traveled about the center of the orbit's auxiliary circle. 'M' grows linearly with time. 'M' is a product of orbiting body's mean motion and time past perihelion, where mean motion 'n' = (2. pi / duration of full orbit). - Epoch is a moment in time, a reference point, for time-varying astronomical quantity, like celestial coordinates or elliptical orbital elements. 5. Heliocentric Orbit : An orbit around the Sun; all planets, comets, asteroids in our solar system are in such orbits. Consider the Orbit of Earth around Sun. The one orbit revolution (360 deg), is one sidereal year, occurs every 365.256363 mean solar days, where one solar day = 24h 00m 00s = 24 x 60 x 60 = 86400.00 seconds. The Earth's Orbit Characteristics and Events are : (a) Orbit Characteristics : Epoch at J2000.0, Ref. http://cdn.preterhuman.net/texts/thought_and_writing/reference/wikipedia_2006_CD/wp/e/Earth.htm - Aphelion 152,097,701 km - Perihelion 147,098,074 km - Semi-major axis 149,597,887.5 km - Semi-minor axis 149,576,999.826 km - Axial tilt 23.4392794383 deg - Eccentricity 0.016710219 - Inclination 7.25 deg to Sun's equator - Longitude of ascending node 348.73936 deg - Argument of periapsis 114.20783 deg - Sidereal orbit period 365.256363 days - Orbital circumference 924,375,700 km - average speed 29.78 km/sec Note : Inclination angle, Longitude of ascending node and Argument of perigee describe the orientation of an orbit in space.
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OM-MSS Page 39 (b) Orbit Events : Equinox, Solstice, and Seasons - Equinoxes occur twice a year; Vernal equinox is around 20-21 March and Autumnal equinox is around 22-23 September; When equinox occurs, the plane of Earth's equator passes the centre of Sun, i.e. when subsolar point is on Equator; At equinox time, the tilt of the Earth's axis is inclined neither away from nor towards the Sun, resulting day and night of same length. At equinoxs the Sun is at one of two opposite points on the celestial sphere where the celestial equator (decl 0 deg) and ecliptic intersect; One intersection is called vernal point (RA = 00h 00m 00s and log = 0 deg) and other called autumnal point (RA = 12h 00m 00s and log = 180 deg). - Solstices occur twice a year; Summer solstice is around 21-22 June and Winter solstic is around 21-22 December; On solstices day, Sun appears to have reached its highest or lowest annual altitude in the sky above the horizon at local solar noon; The solstices day is either longest of the year in summer or the shortest of the year in winter for any place outside of the tropics. - Seasons occur because the Earth's axis of rotation is not perpendicular to its orbital plane but makes an angle of about 23.439 deg; The four Seasons, Spring, Summer, Autumn, Winter, are subdivision of a year, connected with the solstices, and equinoxes; The solstices and equinoxs are the four changing points, in the Solar Cycle, that mark the mid-point of the seasons change. 6. Mean anomaly (M), Eccentric anomaly (E), True anomaly (V) In astronomy, the term anomaly means irregularity in the motion of a planet by which it deviates from its predicted position. Therefore, astronomers use term anomaly (instead of angle) when calculating the position of objects in their orbits. Kepler distinguished three kinds of anomaly - mean, eccentric, and true anomaly. (a) True anomaly is observed angle, as seen from the Sun, between the Earth and the perihelion of the Earth orbit. When the True anomaly is equal to 0 degrees, then the Earth is closest to the Sun (ie at Perihelion). When the True anomaly is equal to 180 degrees, then the Earth is furthest from the Sun (ie at Aphelion).
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OM-MSS Page 40 (b) Mean anomaly is calculated angle, what the true anomaly would be if the Earth moved with constant speed along a perfect circular orbit around the Sun in the same time. Like true anomaly, the mean anomaly is equal to 0 in the perihelion and to 180 degrees in the aphelion, but at other points along the Earth's orbit the true and mean anomalies are not equal to one another. The mean anomaly is often used for one of the orbital elements. (c) Eccentric anomaly is an auxiliary angle, used to solve Kepler's Equation to find True anomaly from Mean anomaly. The is related to both the Mean and the True anomaly. Observe equations & relations among Mean anomaly(M), Eccentric anomaly(E), True anomaly(V), Eccentricity(e), Radial disance(r), Semi-major axis(a) - equation M = E-e.sin(E); a relation between Eccentric and Mean anomaly, is Kepler's equation, solved by numerical methods, e.g. Newton-Raphson method; - equation cos(v) = {cos(E)-e}/{1-e.cos(E)} ; a relation between Eccentric anomaly and True anomaly ; - equation r = a {(1-(e sqr)}/{1+e.cos(V)} ; a relation between True anomaly and Radius distance from focus of attraction to the orbiting body. Thus explained few preliminaries about positional astronomy. Move on to Motion Of Earth Around Sun - Prediction of Astronomical Events, Anomalies, Equinoxes, Solstices, Years & Seasons. The precise time of occurrence of following astronomical events are presented in Sections (2.1 to 2.11) respectively : (a) Earth orbit Mean anomaly, Eccentric anomaly, True anomaly; (b) Earth reaching orbit points, Perihelion, Aphelion, Vernal Equinox, Autumnal Equinox, Summer Solstice, Winter Solstice; (c) Earth reaching orbit points, Semi-Major Axis, Semi-Minor Axis; (d) Astronomical years, Anomalistic, Tropical, Sidereal Years; (e) Earth orbit oblateness, Semi-Major Axis, Semi-Minor Axis; (f) Four Seasons, start time of Spring, Summer, Autumn, Winter. Next Section - 2.1 Earth Orbit Constants used in computation
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OM-MSS Page 41 OM-MSS Section - 2.1 -----------------------------------------------------------------------------------------------------14 Earth Orbit : Constants used in OM-MSS Software. Astronomical and other Constants used in OM-MSS Software . Recall the Preliminaries about 'Positional Astronomy', depictions and interpretations, mentioned in section 2 . (a) The International System of Units (SI), the world's most widely used modern metric system, is followed in OM-MSS Software. Greenwich meam time (GMT), is identified as Universal time (UT) based on sidereal time at Greenwich, with day starting at midnight. Standard Epoch J2000, is Julian Day 2451545.00 UT, ie Year 2000, MM 1, hr 12, min 0, sec 0.0 expressed in UT, is Reference point for Time. One solar day is 24h 00m 00s, ie 24x60x60 = 86400 SI is time for slightly more than one earth rotation, ie 360.9856473356 deg. One sidereal day is 23h 56m 4.090538155680s = 86164.090538155680s is time for one earth rotation, by exact 360 deg. One solar year is time period of earth around sun, that vary slightly year to year. Solar year is called astronomical or tropical or civil year. Consecutive 400 civil years have 97 leap years, so one civil year is (400 x 365 + 97 )/400 = 365.2425 mean solar days where Mean solar day is division of time equal to 24 hours representing average length of the period during which earth makes one rotation on its axis. One sidereal year (365.256363004 days) is slightly longer than a mean solar year (365.2425 days). One sidereal year corresponds to 365 day, 6 hr, 9 min, 9.7635456 sec of mean solar time. One Julian year is exactly 365.25 SI days where SI day is 86400 SI seconds, thus Julian year = 31557600 SI seconds. Gregorian year is the mean duration of a year of our calendar is 365.2425 SI days, is 31556952 SI seconds. Earth mean motion rev per day around sun = 1.0 / One sidereal year in days = 1.0 / 365.256363004.
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OM-MSS Page 42 (b) The Other Constants used in OM-MSS software : the values assigned, correspond to Standard Epoch Julian day JD2000, unless otherwise specified. RADIAN 57.29577951308232300 Pi 3.141592653589793100 GM_SUN (Gravitational parameter) 132712440018 km3/sec2 GM_EARTH (Gravitational parameter) 398600.4418 km3/sec2 EARTH_EQ_RAD_KM (Earth equator radious) 6378.144 km EARTH_AVR_RAD_KM (Earth average radious) 6371.0 km SUN_GEOCENTRIC_DIST_km_from_earth_center 149676781.6 km MOON_GEOCENTRIC_DIST_km_from_earth_center 381191.7836 km EARTH_INCLINATION_deg 23.4392794383 deg EARTH_ORBIT_ECCENTRICITY 0.016710219 EARTH_solar_year 365.2425 days EARTH_SIDEREAL_YEAR 365.256363004 days EARTH mean solar day 86400 sec is 24h 00m 00s EARTH mean sidereal day 86164.0905381557 sec is 23h 56m 4.0905381557 EARTH_MEAN_MOTION_rev_per_day 0.0027378030 rev per day EARTH_ROTATIONAL_RATE_rad_per_sec 7.2921151467e-5 rad/sec Move on to Earth Orbit around Sun, : Compute Anomalies, Precise time at Astronomical Events, Years & Seasons. Next Section - 2.2 Earth orbit - mean and true anomaly
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OM-MSS Page 43 OM-MSS Section - 2.2 -----------------------------------------------------------------------------------------------------15 Earth Mean anomaly and True anomaly at Input UT, Since Standard Epoch J2000, using standard analytical expressions. Finding Mean anomaly and True anomaly in deg, since Standard Epoch J2000 = 2451545.0 Julian day. Mean anomaly gives the planet's angular position for a circular orbit with radius equal to its Semi Major Axis. Since Earth orbit is elliptic, the speed of the Earth varies and the mean anomaly is inaccurate with about +/- 2 deg. The mean anomaly M = 357.5291 + (0.98560028 * (ip_julian_day - 2451545.0)), is measured from 0 to 360 deg. During one revolution the mean anomaly values are : at perigee mean anomaly = 0 deg , and at apogee mean anomaly = 180 deg . True anomaly is true positon of the earth relative to its perihelion, is measured from 0 to 360 deg. Note that what is true anomaly, would be Mean anomaly if the Earth moved with constant speed along a perfectly circular orbit around Sun. Equation of Center, is the difference between the true anomaly and the mean anomaly. Standard Epoch J2000, is beginning of Year = 2000, Month = 1, Day of month = 1, Hour decimal = 12.0 . Finding Mean anomaly and True anomaly in deg : 1. Mean anomaly in deg of the earth around sun, for input Julian day, year, month, day, hours, since Standard Epoch J2000 Input UT year = 2000 month = 1 day = 1 hr = 12 min = 0 sec = 0.00000 Corresponds to julian_day = 2451545.0000000000 Output Mean anomaly in deg of the earth around Sun = 357.5291000000, at input Julian day, minimizing into 0-360 deg = 357.5291000000 2. True anomaly in deg of the earth around sun, for input Julian day, year, month, day, hours, since Standard Epoch J2000 Input UT year = 2000 month = 1 day = 1 hr = 12 min = 0 sec = 0.00000 Corresponds to julian_day = 2451545.0000000000 Output True anomaly in deg of the earth around Sun = 357.4447876113, at input Julian day, minimizing into 0-360 deg = 357.4447876113 Note : Reported values are same, Mean anomaly of Earth around Sun g = 357.53, at Epoch J2000; (Ref. Indian Astronomical Ephemeris, Year 2000, IMD, Page 528.) Next Section - 2.3 Earth orbit points - UT at perihelion and aphelion
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OM-MSS Page 44 OM-MSS Section - 2.3 ----------------------------------------------------------------------------------------------------16 Earth Orbit Input Year : Precise Universal Time (UT) at orbit points - Perihelion and Aphelion. Finding Universal Time (UT) at Perihelion and Aphelion point, Sensing parameter is Mean anomaly (ME) in deg. Perihelion and Aphelion describes two specifc points on Earth orbit around Sun. - perihelion is Point on orbit nearest to Sun, is about 147,098,074 km, and sensing parameter ME deg cross over 360 ie 0 deg around January 03. - aphelion is Point on orbit farthest from Sun, is about 152,097,701 km, and sensing parameter ME_deg cross over 180 deg around July 04. This difference in distance to sun, while earth is at perihelion or aphelion. However the difference not enough to affect the earth's climate. Finding Precise time to reach Perihelion and Aphelion (using sensing parameter ME deg), for any Input year . 1. Find Precise time for Earth to reach Perihelion : Input Year = 2013 Output Time at Perihelion : julian_day = 2456295.8832941712, ie year = 2013, month = 1, day = 3, hr = 9, min = 11, sec = 56.61639 2. Find Precise time for Earth to reach Aphelion : Input Year = 2013 Output Time at Aphelion : julian_day = 2456478.5131087117, ie year = 2013, month = 7, day = 5, hr = 0, min = 18, sec = 52.59269 To verify computed Perihelion and Aphelion Time, apply them as input & compute back Mean anomaly, respectively expected values as 0 deg and 180 deg. 3. Find Mean anomaly and True anomaly in deg : Input Julian day = 2456295.8832941712 is precise time for Earth at Perihelion Output Mean anomaly in deg at Perihelion = 5040.0010049824 minimizing into 0-360 deg = 0.0010049824, note error in deg = 0.00100 True anomaly in deg at Perihelion = 5040.0010392859 minimizing into 0-360 deg = 0.0010392859, note error in deg = 0.00104 4. Find Mean anomaly and True anomaly in deg : Input Julian day = 2456478.5131087117 is precise time for Earth at Aphelion Output Mean anomaly in deg at Aphelion = 5220.0010013299 minimizing into 0-360 deg = 180.0010013299, note error in deg = 0.00100 True anomaly in deg at Aphelion = 5220.0009685492 minimizing into 0-360 deg = 180.0009685492, note error in deg = 0.00097 Note 1 : Perihelion & Aphelion time, reported, (Ref. http://www.usno.navy.mil/USNO/astronomical-applications/data-services/earth-seasons). Perihelion, Year 2013, Month 1, Day 02, Hrs 05, UTC ; Aphelion, Year 2013, Month 7, Day 05, Hrs 15, UTC.
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OM-MSS Page 45 Note 2 : For verifying, the anomaly values are calculated using analytical equations which are approximate, giving error at 3rd place of decimal; The algorithmic solutions are offered in later sections that goes through many iterations minimizing the errors. Next Section - 2.4 Earth orbit points - UT at vernal and autumnal equinox
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OM-MSS Page 46 OM-MSS Section - 2.4 -----------------------------------------------------------------------------------------------------17 Earth Orbit Input Year : Precise Universal Time (UT) at orbit points - Vernal Equinox and Autumnal Equinox. Finding Universal Time (UT) at Vernal equinox and Autumnal equinox point, Sensing parameter is Declination (delta). Equinox occurs twice a year, when Earth rotation axis is exactly parallel to the direction of motion of Earth around Sun. Vernal equinox around March 20/21, and Autumnal equinox around September 22/23; at equinox day and night are of same length. Right Ascension (RA or alpha) and Declination (delta) are astronomical terms for coordinates of a point on celestial sphere. Right Ascension (RA), is similar to longitude on Earth is measured in hours (h), minutes (m) and seconds (s). RA around the celestial equator is 24 hours, where 1 h = 15 deg. Unlike longitude (zero deg) on Earth as Prime Meridian, the reference Right Ascension (zero hour) is First Point of Aries in sky where Sun crosses celestial equator called Vernal equinox. Declination (delta), is similar to latitude on Earth is measured in degrees, arc-minutes and arc-seconds. Declination measures how far overhead an object will rise in the sky, measured 0 deg at the equator, +90 deg at North Pole and -90 deg at South Pole. Vernal point : RA = 00h 00m 00s and longitude = 0 deg , the sense parameter Delta_deg sign change -ve to 0 to +ve Autumnal point : RA = 12h 00m 00s and longitude = 180 deg, the sense parameter Delta_deg sign change +ve to 0 to -ve Finding Precise time to reach Vernal_equinox and Autumnal_equinox (using sense parameter Delta_deg), for any Input year are as follows. 1. Find Precise time for Earth to reach Vernal equinox : Input Year = 2013 Output Time at Vernal equinox : Julian day = 2456371.9598282082, ie year = 2013, month = 3, day = 20, hr = 11, min = 2, sec = 9.15719 2. Find Precise time for Earth to reach Autumnal equinox : Input Year = 2013 Output Time at Autumnal equinox : Julian day = 2456558.3650290174, ie year = 2013, month = 9, day = 22, hr = 20, min = 45, sec = 38.50711 Note : Values reported are same, (Ref. http://www.usno.navy.mil/USNO/astronomical-applications/data-services/earth-seasons). Vernal Equinox, Year 2013, Month 3, Day 20, Hrs 11, Min 02 UTC ; Autumnal Equinox, Year 2013, Month 9, Day 22, Hrs 20, Min 44 UTC. Next Section - 2.5 Earth orbit points - UT at summer and winter solstice
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OM-MSS Page 47 OM-MSS Section - 2.5 -----------------------------------------------------------------------------------------------------18 Earth Orbit Input Year : Precise Universal Time (UT) at orbit points - Summer Solstice and Winter Solstice . Finding Universal Time (UT) at Summer and Winter Solstice point, Sensing parameter is Declination (delta). Solstice occurs twice a year, as Sun appears to have reached its highest or lowest annual altitude in the sky at local solar noon. Summer Solstice is around June 21/22, and Winter Solastice is around December 21/22. Solstices, together with the Equinoxes, are connected with the seasons. Summer Solstice : the sense parameter Delta_deg max about +23.44 deg Winter Solstice : the sense parameter Delta_deg min about -23.44 deg Finding Precise time to reach Summer and Winter Solstice (using sense parameter Delta_deg), for any Input year are as follows. 1. Find Precise time for Earth to reach Summer Solstice : Input Year = 2013 Output Time at Summer Solstice : julian_day = 2456464.7092499998, ie year = 2013, month = 6, day = 21, hr = 5, min = 1, sec = 19.19999 2. Find Precise time for Earth to reach Winter Solstice : Input Year = 2013 Output Time at Winter Solstice : julian_day = 2456648.2153210002, ie year = 2013, month = 12, day = 21, hr = 17, min = 10, sec = 3.73442 Note : Values reported is slightly less then 3 min and 1 min, (Ref. http://www.usno.navy.mil/USNO/astronomical-applications/data-services/earth-seasons). Summer Solstice, Year 2013, Month 6, Day 21, Hrs 05, Min 04 UTC ; Winter Solstice, Year 2013, Month 12, Day 21, Hrs 17, Min 11 UTC. Next Section - 2.6 Earth orbit points - UT at semi-major and semi-minor axis
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OM-MSS Page 48 OM-MSS Section - 2.6 -----------------------------------------------------------------------------------------------------19 Earth Orbit Input Year : Precise Universal Time (UT) at orbit points - Semi-Major Axis and Semi-Minor Axis . Finding Universal Time (UT) at Semi-Minor Axis point and Semi-Major Axis point, Sensing parameter is Mean anomaly (ME) in deg . Semi-Minor Axis Point around Apri 01, sensing parameter ME_deg cross over 90 deg, Semi-Major Axis Point around July 05, sensing parameter ME_deg cross over 180 deg. Finding Precise time to reach Semi-Minor Axis and Semi-Major Axis Point (using sensing parameter ME deg), for any Input year are as follows. 1. Find Precise time for Earth to reach Semi-Minor Axis : Input Year = 2013 Output Time at Semi-Minor Axis Point : Julian day = 2456387.1982014412, ie year = 2013, month = 4, day = 4, hr = 16, min = 45, sec = 24.60452 2. Find Precise time for Earth to reach Semi-Major Axis : Input Year = 2013 Output Time at Semi-Major Axis Point : Julian day = 2456478.5131087117, ie year = 2013, month = 7, day = 5, hr = 0, min = 18, sec = 52.59269 Next Section - 2.7 Earth orbit - astronomical years
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OM-MSS Page 49 OM-MSS Section - 2.7 -----------------------------------------------------------------------------------------------------20 Earth Orbit Input Year : Astronomical Years - Anomalistic, Tropical, and Sidereal Years . The Anomalistic, Tropical, and Sidereal Years are Astronomical years. Look at the differeces. Finding Anomalistic, Tropical, and Sidereal Years for the Input Year. 1. For Anomalistic year : find Precise time for Earth to reach Perihelion to Perihelion : Input Year = 2000 Output Perihelion to Perihelion Time in julian_day = 365.2596290808 ie Days 365, hour 6, min 13, sec 51.95258 is Anomalistic year 2. For Anomalistic year : find Precise time for Earth to reach Aphelion to Aphelion : Input Year = 2000 Output Aphelion to Aphelion Time in julian_day = 365.2596290810 ie Days 365, hour 6, min 13, sec 51.95260 is Anomalistic year 3. For Tropical year : find Precise time for Earth to reach Vernal to Vernal equinox : Input Year = 2000 Output Vernal to Vernal equinox Time in julian_day = 365.2423121394 ie Days 365, hour 5, min 48, sec 55.76884 is Tropical year 4. For Tropical year : find Precise time for Earth to reach Autumnal to Autumnal equinox : Input Year = 2000 Output Autumnal to Autumnal equinox Time in julian_day = 365.2423121394 ie Days 365, hour 5, min 48, sec 55.76884 is Tropical year Note : Values reported are almost same, (Ref. http://en.wikipedia.org/wiki/Year , http://www.yourdictionary.com/sidereal-year). Anomalistic year : Days 365, Hrs 06, Min 13, Sec 52.6 UTC ; Tropical year : Days 365, Hrs 05, Min 48, Sec 46 UTC . Earth's one revolution around Sun called Sidereal year = 365.256363004 (Days 365, Hr 06, Min 09, Sec 09.76) in units of mean solar days, at epoch J2000 The difference in days among Anomalistic, Tropical and Sidereal Year are (Anomalistic - Tropical) year = Days 0, hour 0, minute 24, seconds 56.1837373674 (Sidereal - Anomalistic) year = Days 0, hour 0, minute -4, seconds -42.1890329762 (Sidereal - Tropical) year = Days 0, hour 0, minute 20, seconds 13.9947043912 Next Section - 2.8 Earth orbit oblateness - semi-major and semi-minor axis
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OM-MSS Page 50 OM-MSS Section - 2.8 -----------------------------------------------------------------------------------------------------21 Earth Orbit Oblateness : Semi-Major Axis and Semi-Minor Axis. Finding Semi-Major Axis and Semi-Minor Axis in km. The Earth's orbit is an ellipse. The Earth's shape is very close to an oblate spheroid, with a bulge around the equator. GM_SUN is Gravitational parameter of Sun is product of gravitational constant G and mass M of Sun GM_SUN = = 132,712,440,018 km3/sec2 = 132712.440018e6 = 132712.440018 x 10 to pow 6. The Semi-Major Axis value is computed considering using earth mean motion rev per day obtain as (a) EARTH mean motion rev per day = 0.0027377786, as 1.0 / 365.259629080 days is time Perihelion to Perihelion same as Aphelion to Aphelion, year 2000. (b) EARTH mean motion rev per day = 0.0027378030, as 1.0 / 365.256363004 days is sidereal year for Earth making one full revolution around Sun. The Semi-Minor Axis value is calculated considering earth orbit Eccentricity = 0.016710219 1. Earth Semi-Major Axis (SMA) in km, using EARTH mean motion rev per day as 0.0027377786 using diff. of Julian days Perihelion to Perihelion, year 2000 (a) Ignoring Earth oblateness Input Earth mean motion rev per day = 0.0027377786, GM SUN = 132712440018.00000 Output Semi Major Axis in km = 149598616.3114941400, and Semi Minor Axis in km = 149577728.5363029200 (b) Considering Earth oblateness , Inclination , Eccentricity Input Earth mean motion rev per day = 0.0027377786, GM SUN = 132712440018.00, Incl = 23.43928, Ecc = 0.01671, constant_k2 = 65915.34460 Output Semi-Major Axis in km = 149598616.3117182900, and Semi-Miror Axis in km = 149577728.5365270400
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OM-MSS Page 51 2. Earth Semi-Major Axis (SMA) in km, using EARTH mean motion rev per day as 0.0027378030 obtained using sidereal year for Earth one revolution around Sun. (a) Ignoring Earth oblateness Input Earth mean motion rev per day = 0.0027378031, GM SUN = 132712440018.00 Output Semi-Major Axis in km = 149597724.5233797700, and Semi-Minor Axis in km = 149576836.8727048900 (b) Considering Earth oblateness , Inclination , Eccentricity Input Earth mean motion rev per day = 0.0027378031, GM SUN = 132712440018.00, Incl = 23.43928, Ecc = 0.01671, constant_k2 = 65915.34460 Output Semi-Major Axis in km = 149597724.5236039200, and Semi-Miror Axis in km = 149576836.8729290100 Note : Compare with Two different values of Semi-Major Axis reported as Semi-major axis = 149,597,887.5 KM , (Ref. http://simple.wikipedia.org/wiki/Earth) Semi-major axis = 149,598,261 KM , (Ref. http://en.wikipedia.org/wiki/Earth 27s_orbit) Next Section - 2.9 Earth orbit - mean, eccentric and true anomaly at UT
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OM-MSS Page 52 OM-MSS Section - 2.9 -----------------------------------------------------------------------------------------------------22 Earth Orbit Input Year : Mean anomaly, Eccentric anomaly, True anomaly at UT, based on algorithms of iterative method. Finding Mean anomaly, Eccentric anomaly, True anomaly at any UT, year, month, day, hour, minute, seconds. The Mean anomaly and True anomaly values presented before, were calculated using standard analytical expressions in section 2.2. Here the anomalies are computed based on algorithms of iterative method, while Earth moves through the respective orbit points : perihelion, vernal equinox, semi_minoraxis, summer solstice, aphelion, autumnal equinox, winter solstice. The Computed values of Mean, Eccentric, and True anomaly mentioned below show accuracy of the algorithms. The Input Time at each orbit point is same as what were computed before in sections (2.3 to 2.6). Mean anomaly, Eccentric anomaly, True anomaly at Perihelion, Aphelion, Equinoxes, Solstices & Semi-minor axis points : 1. Input time at Perihelion : year = 2013, month = 1, day = 3, hour = 9, minute = 11, seconds = 56.6163906455 Output the Anomalies in deg : Mean anomaly = 0.00000, Eccentric anomaly = 0.00000, True anomaly = 0.00000 2. Input time at Vernal equinox : year = 2013, month = 3, day = 20, hour = 11, minute = 2, seconds = 9.1571885347 Output the Anomalies in deg : Mean anomaly = 74.98105, Eccentric anomaly = 75.90967, True anomaly = 76.84023 3. Input time at Summer solstice : year = 2013, month = 6, day = 21, hour = 5, minute = 1, seconds = 19.1999861598 Output the Anomalies in deg : Mean anomaly = 166.39491, Eccentric anomaly = 166.61653, True anomaly = 166.83637 4. Input time at Aphelion : year = 2013, month = 7, day = 5, hour = 0, minute = 18, seconds = 52.5926899910 Output the Anomalies in deg : Mean anomaly = 180.00000, Eccentric anomaly = 180.00000, True anomaly = 180.00000 5. Input time at Autumnal equinox : year = 2013, month = 9, day = 22, hour = 20, minute = 45, seconds = 38.5071069002 Output the Anomalies in deg : Mean anomaly = 258.70208, Eccentric anomaly = 257.76639, True anomaly = 256.83232 6. Input time at Winter solstice : year = 2013, month = 12, day = 21, hour = 17, minute = 10, seconds = 3.7344172597 Output the Anomalies in deg : Mean anomaly = 347.25855, Eccentric anomaly = 347.04389, True anomaly = 346.82746
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OM-MSS Page 53 7. Input time at Semi-minor axis : year = 2013, month = 4, day = 4, hour = 16, minute = 45, seconds = 24.6045202017 Output the Anomalies in deg : Mean anomaly = 90.00000, Eccentric anomaly = 90.95729, True anomaly = 91.91449 8. The Semi-Major Axis Point is same as Aphelion presented above, repeated here for completeess. Input time at Semi-major axis : year = 2013, month = 7, day = 5, hour = 0, minute = 18, seconds = 52.5926899910 Output the Anomalies in deg : Mean anomaly = 180.00000, Eccentric anomaly = 180.00000, True anomaly = 180.00000 Note : The values computed above are based on iterative algorithm, therefore more accurate and validated against the expected respective values of anomaly as 0 deg, 180 deg, and 90 deg at Perihelion, Aphelion & Semi-minor axis points. The values at Equinoxes and Solstices are also close to those reported mentioned before. Next Section - 2.10 Earth orbit - four seasons
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OM-MSS Page 54 OM-MSS Section - 2.10 ----------------------------------------------------------------------------------------------------23 Earth Orbit Input Year : Four Seasons - Spring, Summer, Autumn, and Winter. Finding Four Seasons : Start Time of Spring, Summer, Autumn, Winter. Seasons are a subdivision of a year. The Earth's rotation axis is tilted by 23.4392794383 degrees with respect to the ecliptic. Because of this tilt in Earth's rotation axis, the Sun appears at different elevations or angle above the horizon, at different times of a year. The variation in the elevation of the Sun over the year is the cause of the seasons. The 0 to 360 deg Sun's longitudes are equally divided among four seasons as : Sun longitudes : 0 - 90 deg (Spring), 90 - 180 deg (Summer), 180 - 270 deg (Autumn), 270 - 360 deg (Winter). The Sun true longitude (Lsun) is derived from Sun mean longitude(Lmean) and Earth mean anomaly(ME). For start time of any season, the sensing parameter is Sun true longitude value which is reached through many iterations. Finding the Start Time of the Seasons - Spring, Summer, Autumn, Winter, and the corresponding Sun true Longitude (Lsun) for the input year 1. Year = 2013, Start Time of Spring : UT year = 2013, month = 3, day = 20, hr = 11, min = 2, sec = 9.15719, & Sun true Log deg = 360.00000 2. Year = 2013, Start Time of Summer : UT year = 2013, month = 6, day = 21, hr = 5, min = 1, sec = 23.88007, & Sun true Log deg = 90.00000 3. Year = 2013, Start Time of Autumn : UT year = 2013, month = 9, day = 22, hr = 20, min = 45, sec = 38.50711, & Sun true Log deg = 180.00000 4. Year = 2013, Start Time of Winter : UT year = 2013, month = 12, day = 21, hr = 17, min = 10, sec = 7.88032, & Sun true Log deg = 270.00000 Finding the Duration of the Seasons - Spring, Summer, Autumn, Winter for the Year = 2013 1. Season Spring Duration Days 92, hour 17, minute 59, seconds 14.72288 2. Season Summer Duration Days 93, hour 15, minute 44, seconds 14.62704 3. Season Autumn Duration Days 89, hour 20, minute 24, seconds 29.37321 4. Season Winter Duration Days 88, hour 23, minute 40, seconds 57.24261
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OM-MSS Page 55 Summary of four seasons, Year 2013 , first day of Spring, Summer, Autumn, and Winter season. - 1st day of Spring season, Mar. 20, Vernal equinox, Sun crosses Equator moving northward, is beginning of a long period of sunlight at Pole. - 1st day of Summer season, Jun. 21, Summer solstice, Sun is farthest north and time between Sunrise and Sunset is longest of the year, - 1st day of Autumn season, Sept. 22, Autumnal equinox, Sun crosses Equator moving southward, is beginning of a long period of darkness at Pole. - 1st day of Winter season, Dec. 21, Winter solstice, Sun is farthest south and time between Sunrise and Sunset is shortest of the year. Note : Values reported are same, (Ref. http://en.wikipedia.org/wiki/Season). Spring season (Vernal equinox) Year 2013, Mar. 20, Hrs 11, Min 02 UTC ; Summer season (Summer solstice) Year 2013, Jun. 21, Hrs 05, Min 04 UTC. Autumn season (Autumnal equinox) Year 2013, Sept. 22, Hrs 20, Min 44 UTC ; Winter season (Winter solstice) Year 2013, Dec. 21, Hrs 17, Min 11 UTC. Next Section - 2.11 Concluding astronomical events
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OM-MSS Page 56 OM-MSS Section - 2.11 ----------------------------------------------------------------------------------------------------24 Concluding Astronomical Events Anomalies, Equinoxes, Solstices, Years & Seasons presented in Sections (2.0 to 2.10). In Sections (2.1 to 2.10), all that presented were prediction / computation of following Astronomical Time Events of Earth Orbit around Sun : 1. Precise value for Mean, Eccentric & True anomalies; 2. Precise time for Earth to reach Perihelion & Aphelion points; 3. Precise time for Earth to reach Vernal & Autumnal equinox points; 4. Precise time for Earth to reach Summer & Winter solstice points; 5. Precise time for Earth to reach Semi-major & Semi-major axis points; 6. Duration of Anomalistic, Tropical & Sidereal years; 7. Start time & durations of seasons - Spring, Summer, Autumn, & Winter. End of Computing Astronomical Events Anomalies, Equinoxes, Solstices, Years & Seasons. Move on to Compute the Position of Sun on Celestial Sphere at input Universal Time (UT). Next Section - 3 Position of Sun on Celestial Sphere at UT
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OM-MSS Page 57 OM-MSS Section - 3 -------------------------------------------------------------------------------------------------------25 POSITION OF SUN ON CELESTIAL SPHERE AT INPUT UNIVERSAL TIME (UT) . Sun is a star at the center of our Solar System. Although stars are fixed relative to each other, but Sun moves relative to stars. Sun follows a circular path on the celestial sphere, once a year. This path is known as the 'Ecliptic', representing the plane of the Earth's orbit. Inclination of the Earth's equator to the Ecliptic (or earth's rotation axis to a perpendicular on ecliptic) is called Obliquity of the ecliptic. The Obliquity of the ecliptic is currently 23.4392794383 deg with respect to the celestial equator, at standard epoch J2000 . The position of any point on the Celestial Sphere is given with reference to the equator or the ecliptic. - reference to Equator, the position is specified by Right ascension and Declination. - reference to Ecliptic, the position is specified by celestial Longitude and Latitude. The Earth moves in an elliptical orbit around the Sun. Therefore the distance from Earth to Sun is not same at all points on the orbit. - distance Earth to Sun (d_sun) calculated as r = a(1-e*e)/(1+e cos(theta)) where a = semi-major axis, e = eccentricity and theta = mean anomaly of sun. - radial distance from Earth to Sun (Rs) calculated as R = 1.00014 - (0.01671 * cos g) - (0.00014 * cos 2g) where g = mean anomaly of sun - mean distance from Earth to Sun = 149,597,870.700 km, called 1 Astronomical Unit, (Ref. http://en.wikipedia.org/wiki/Astronomical_unit) - minimum distance from Earth to Sun = 147,098,074 km or 0.98 AU, and this point is called Perihelion; - maximum distance from Earth to Sun = 152,097,701 km or 1.02 AU, and this point is called Aphelion; - average distance from Earth to Sun (As) = 149,597,887.5 km is the distance (max + min)/2. (Ref http://wiki.answers.com/Q/What_is_the_distance_between_Earth_and_the_Sun). At any input Universal Time, to compute the position of Sun and its related traits, the algorithm goes through following steps : (a) Find Julian day of interest corresponding to the input Universal Time; (b) Find Corresponding Ecliptic coordinates - Mean anomaly of the Sun (actually, Earth orbits around Sun, but here pretends Sun orbits Earth) - Mean longitude of the Sun; - Ecliptic longitude of the Sun; - Ecliptic latitude of the Sun is always nearly zero (the value never exceeds 0.00033 deg)
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OM-MSS Page 58 - Distance of the Sun from the Earth, in astronomical units - Obliquity of the ecliptic (c) Find Corresponding Equatorial coordinates - Right ascension - Declination In addition to these Ecliptic and Equatorial coordinates, computed many other parameters related to Sun's Position on Celestial Sphere. The Position of Sun on Celestial Sphere is represented by computing following parameters : Semi-major axis (SMA), Mean movement per day (n sun), Mean distance (As), Mean anomaly (m sun), True anomaly (T sun), Eccentric anomaly (E sun), Right ascension (Alpha), Declination (Delta), Mean longitude (Lmean), Ecliptic longitude (Lsun), Nodal elongation (U sun), Argument of perigee (W sun), Obliquity of ecliptic (Epcylone), Mean dist (d_sun), Radial distance (Rs). Total 22 parameters computed at Standard Epoch JD2000 (ie YY 2000, MM 1, DD 1, hr 12.00) and at six orbit time event points for YY 2013. The six orbit events points are when Earth reaches Perihelion & Aphelion, Vernal & Autumnal Equinox, Summer & Winter Solstice. For any desired year, first computed the Universal time (UT) for earth to reach the respective orbit events point, then apply the same UT as input time for finding the corresponding orbit parameters at that time instnt. (Note : The orbit events being specific, the values computed can be verified easily with those reported from other sources.) Move on to Find Position of Sun on Celestial Sphere, the Utilities of OM-MSS Software (Sections - 3.1 to 3.8). Computing Sun Position on Celestial Sphere at Seven different Time events, respectively : (a) Time Event - Standard Epoch JD2000 ; (b) Time Event - when Earth at Perihelion ; (c) Time Event - when Earth at Vernal equinox ; (d) Time Event - when Earth at Summer solstice ; (e) Time Event - when Earth at Aphelion ; (f) Time Event - when Earth at Autumnal equinox ; (g) Time Event - when Earth at Winter solstices ; Next Section - 3.1 Position of sun at standard epoch time JD2000
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OM-MSS Page 59 OM-MSS Section - 3.1 -----------------------------------------------------------------------------------------------------26 SUN Positional Parameters on Celestial Sphere : Input Time (UT) Standard Epoch JD2000 1. Finding Position of Sun on Celestial Sphere at Input UT Standard Epoch time JD2000 . Input Universal Time Corresponds to Julian Day JD2000 : year = 2000, month = 1, day = 1, hour = 12, minute = 0, seconds = 0.00000 Output : Sun Position on Celestial Sphere Corresponding to input time, JD2000 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = -2.5081161195 07. Mean anomaly in deg per day from n_sun (m sun) = -2.4719999999 08. Sun Mean longitude in deg (Lmean) = 280.4600000000 09. Earth Mean anomaly in deg (ME) = 357.5280000000 10. Sun Ecliptic longitude in deg (Lsun) = 280.3756801972 11. Obliquity of ecliptic in deg (Epcylone) = 23.4392794444 12. Sun Right ascension in deg (Alpha) = 281.2858630915 13. Sun Declination in deg (Delta) = -23.0337026521 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 147101227.61694 16. Sun Nodal elongation in deg (U sun) = -79.6243198028 17. Sun Mean anomaly in deg (M sun) = 357.5280000002 18. Sun Eccentric anomaly in deg (E sun) = 357.4860040557 19. Sun True anomaly in deg (T sun) = 357.4436516380 20. Sun Argument of perigee in deg (W sun) = 282.9320285593 21. Sun True anomaly in deg from U & W (V sun) = 357.4436516380 22. Sun Distance in km (d sun) = 147101040.52850 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) Next Section - 3.2 Position of sun at time when earth is at perihelion
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OM-MSS Page 60 OM-MSS Section - 3.2 ------------------------------------------------------------------------------------------------------27 SUN Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Perihelion. 2. Finding Position of Sun on Celestial Sphere at Input Universal Time, when Earth is at Perihelion . Input Universal Time Corresponds to Earth at Perihelion : year = 2013, month = 1, day = 3, hour = 9, minute = 11, seconds = 56.61639 Output : Sun Position on Celestial Sphere Corresponding to input time, Earth reaching Perihelion 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = 0.0000000000 07. Mean anomaly in deg per day from n_sun (m sun) = 0.0000000000 08. Sun Mean longitude in deg (Lmean) = 283.1557666033 09. Earth Mean anomaly in deg (ME) = 0.0000000001 10. Sun Ecliptic longitude in deg (Lsun) = 283.1557666033 11. Obliquity of ecliptic in deg (Epcylone) = 23.4375874462 12. Sun Right ascension in deg (Alpha) = 284.2922173002 13. Sun Declination in deg (Delta) = -22.7872783368 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 147098823.43315 16. Sun Nodal elongation in deg (U sun) = -76.8442333967 17. Sun Mean anomaly in deg (M sun) = 0.0000000000 18. Sun Eccentric anomaly in deg (E sun) = 0.0000000000 19. Sun True anomaly in deg (T sun) = 0.0000000000 20. Sun Argument of perigee in deg (W sun) = 283.1557666033 21. Sun True anomaly in deg from U & W (V sun) = 0.0000000000 22. Sun Distance in km (d sun) = 147098790.67105 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) Next Section - 3.3 Position of sun at time when earth is at vernal equinox
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OM-MSS Page 61 OM-MSS Section - 3.3 ------------------------------------------------------------------------------------------------------28 SUN Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Vernal equinox . 3. Finding Position of Sun on Celestial Sphere at Input Universal Time, when Earth is at Vernal equinox . Input Universal Time Corresponds to Earth at Vernal equinox : year = 2013, month = 3, day = 20, hour = 11, minute = 2, seconds = 9.15719 Output : Sun Position on Celestial Sphere Corresponding to input time, Earth reaching Vernal equinox 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = 76.0765340370 07. Mean anomaly in deg per day from n_sun (m sun) = 74.9810547699 08. Sun Mean longitude in deg (Lmean) = 358.1404045779 09. Earth Mean anomaly in deg (ME) = 74.9810547700 10. Sun Ecliptic longitude in deg (Lsun) = 0.0000000000 11. Obliquity of ecliptic in deg (Epcylone) = 23.4375603478 12. Sun Right ascension in deg (Alpha) = 0.0000000000 13. Sun Declination in deg (Delta) = 0.0000000000 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 148989898.67840 16. Sun Nodal elongation in deg (U sun) = 0.0000000000 17. Sun Mean anomaly in deg (M sun) = 74.9810547697 18. Sun Eccentric anomaly in deg (E sun) = 75.9096738744 19. Sun True anomaly in deg (T sun) = 76.8402303407 20. Sun Argument of perigee in deg (W sun) = 283.1597696594 21. Sun True anomaly in deg from U & W (V sun) = 76.8402303407 22. Sun Distance in km (d sun) = 148912015.96700 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) Next Section - 3.4 Position of sun at time when earth is at summer solsticex
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OM-MSS Page 62 OM-MSS Section - 3.4 ------------------------------------------------------------------------------------------------------29 SUN Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Summer solstice . 4. Finding Position of Sun on Celestial Sphere at Input Universal Time, when Earth is at Summer solstice . Input Universal Time Corresponds to Earth at Summer solstice : year = 2013, month = 6, day = 21, hour = 5, minute = 1, seconds = 19.19999 Output : Sun Position on Celestial Sphere Corresponding to input time, Earth reaching Summer solstice 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = 168.8259558287 07. Mean anomaly in deg per day from n_sun (m sun) = 166.3949127126 08. Sun Mean longitude in deg (Lmean) = 89.5586310183 09. Earth Mean anomaly in deg (ME) = 166.3949127126 10. Sun Ecliptic longitude in deg (Lsun) = 89.9999483110 11. Obliquity of ecliptic in deg (Epcylone) = 23.4375273104 12. Sun Right ascension in deg (Alpha) = 89.9999436629 13. Sun Declination in deg (Delta) = 23.4375273104 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 152030583.04072 16. Sun Nodal elongation in deg (U sun) = 90.0000000000 17. Sun Mean anomaly in deg (M sun) = 166.3949127122 18. Sun Eccentric anomaly in deg (E sun) = 166.6165253213 19. Sun True anomaly in deg (T sun) = 166.8363660940 20. Sun Argument of perigee in deg (W sun) = 283.1636339060 21. Sun True anomaly in deg from U & W (V sun) = 166.8363660940 22. Sun Distance in km (d sun) = 152025947.60113 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) Next Section - 3.5 Position of sun at time when earth is at aphelion
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OM-MSS Page 63 OM-MSS Section - 3.5 ------------------------------------------------------------------------------------------------------30 SUN Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Aphelion . 5. Finding Position of Sun on Celestial Sphere at Input Universal Time, when Earth is at Aphelion . Input Universal Time Corresponds to Earth at Aphelion : year = 2013, month = 7, day = 5, hour = 0, minute = 18, seconds = 52.59269 Output : Sun Position on Celestial Sphere Corresponding to input time, Earth reaching Aphelion 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = 182.6298145405 07. Mean anomaly in deg per day from n_sun (m sun) = 180.0000000001 08. Sun Mean longitude in deg (Lmean) = 103.1643684676 09. Earth Mean anomaly in deg (ME) = 180.0000000002 10. Sun Ecliptic longitude in deg (Lsun) = 103.1643684676 11. Obliquity of ecliptic in deg (Epcylone) = 23.4375223935 12. Sun Right ascension in deg (Alpha) = 104.3014954901 13. Sun Declination in deg (Delta) = 22.7863704018 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 152098409.19029 16. Sun Nodal elongation in deg (U sun) = 76.8356315324 17. Sun Mean anomaly in deg (M sun) = 179.9999999997 18. Sun Eccentric anomaly in deg (E sun) = 179.9999999997 19. Sun True anomaly in deg (T sun) = 179.9999999997 20. Sun Argument of perigee in deg (W sun) = 256.8356315327 21. Sun True anomaly in deg from U & W (V sun) = 179.9999999997 22. Sun Distance in km (d sun) = 152098441.95238 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) Next Section - 3.6 Position of sun at time when earth is at autumnal equinox
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OM-MSS Page 64 OM-MSS Section - 3.6 ------------------------------------------------------------------------------------------------------31 SUN Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Autumnal equinox . 6. Finding Position of Sun on Celestial Sphere at Input Universal Time, when Earth is at Autumnal equinox . Input Universal Time Corresponds to Earth at Autumnal equinox : year = 2013, month = 9, day = 22, hour = 20, minute = 45, seconds = 38.50711 Output : Sun Position on Celestial Sphere Corresponding to input time, Earth reaching Autumnal equinox 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = 262.4817348463 07. Mean anomaly in deg per day from n_sun (m sun) = 258.7020766091 08. Sun Mean longitude in deg (Lmean) = 181.8702061020 09. Earth Mean anomaly in deg (ME) = 258.7020766091 10. Sun Ecliptic longitude in deg (Lsun) = 180.0000000001 11. Obliquity of ecliptic in deg (Epcylone) = 23.4374939503 12. Sun Right ascension in deg (Alpha) = 180.0000000001 13. Sun Declination in deg (Delta) = -0.0000000001 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 150128632.16764 16. Sun Nodal elongation in deg (U sun) = -0.0000000001 17. Sun Mean anomaly in deg (M sun) = 258.7020766085 18. Sun Eccentric anomaly in deg (E sun) = 257.7663930098 19. Sun True anomaly in deg (T sun) = 256.8323186392 20. Sun Argument of perigee in deg (W sun) = 103.1676813607 21. Sun True anomaly in deg from U & W (V sun) = 256.8323186392 22. Sun Distance in km (d sun) = 150048057.36583 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) Next Section - 3.7 Position of sun at time when earth is at winter solstice
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OM-MSS Page 65 OM-MSS Section - 3.7 ------------------------------------------------------------------------------------------------------32 SUN Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Winter solstice . 7. Finding Position of Sun on Celestial Sphere at Input Universal Time, when Earth is at Winter solstice. Input Universal Time Corresponds to Earth at Winter solstice : year = 2013, month = 12, day = 21, hour = 17, minute = 10, seconds = 3.73442 Output : Sun Position on Celestial Sphere Corresponding to input time, Earth reaching Winter solstice 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = 352.3320268290 07. Mean anomaly in deg per day from n_sun (m sun) = 347.2585513424 08. Sun Mean longitude in deg (Lmean) = 270.4309127840 09. Earth Mean anomaly in deg (ME) = 347.2585513424 10. Sun Ecliptic longitude in deg (Lsun) = 269.9999511320 11. Obliquity of ecliptic in deg (Epcylone) = 23.4374619456 12. Sun Right ascension in deg (Alpha) = 269.9999467376 13. Sun Declination in deg (Delta) = -23.4374619456 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 147162417.75585 16. Sun Nodal elongation in deg (U sun) = -90.0000000000 17. Sun Mean anomaly in deg (M sun) = 347.2585513416 18. Sun Eccentric anomaly in deg (E sun) = 347.0438922873 19. Sun True anomaly in deg (T sun) = 346.8274560675 20. Sun Argument of perigee in deg (W sun) = 283.1725439325 21. Sun True anomaly in deg from U & W (V sun) = 346.8274560675 22. Sun Distance in km (d sun) = 147158348.89183 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) Thus Computed values for Position of Sun on Celestial Sphere corresponding to Standard Epoch time JD2000, and six astronomical events while earth reaches Perihelion, Vernal equinox, Summer solstice, Aphelion, Autumnal equinox, Winter solstices. Move on to Summary of these Computed values are presented next. Next Section - 3.8 Concluding Position of Sun at six astronomical events.
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OM-MSS Page 66 OM-MSS Section - 3.8 ------------------------------------------------------------------------------------------------------33 Concluding Position of Sun on Celestial Sphere (Sections 3.0 to 3.7) Concluding Sun Position on Celestial Sphere with respect to Earth orbit, in the Year = 2013, at six astronomical events. In previous Sections (3.1 to 3.7), the position of Sun on Celestial Sphere were represented by computing following parameters : Orbit Semi-major axis (SMA), Mean movement per day (n sun), Mean distance (As), Mean anomaly (m sun), True anomaly (T sun), Eccentric anomaly (E sun), Right ascension (Alpha), Declination (Delta), Mean longitude (Lmean), Ecliptic longitude (Lsun), Nodal elongation (U sun), Argument of perigee (W sun), Obliquity of ecliptic (Epcylone), Mean dist (d_sun), Radial distance (Rs). All these parameters were computed while Earth moves around Sun and reaches six astronomical event points : Perihelion, Vernal equinox, Summer solstice, Aphelion, Autumnal equinox, Winter solstice . Summary of Sun Position on Celestial Sphere with respect to Earth while moved around Sun, passed through at six astronomical events. Universal time Mean anom True anom Ecce.anom Right asc. Declina Mean log. Ecli.log. Nodal elon. Arg of peri Obliquity Mean dist. Radial dist. while earth at M_sun T_sun E_sun Alpha Delta L_mean L_sun U_sun w_sun Epcylone d_sun Rs Perihelion 0.00 0.00 0.00 284.29 -22.79 283.16 283.16 -76.84 283.16 23.44 147098790.67 147098823.43 Vernal equinox 74.98 76.84 75.91 0.00 0.00 358.14 0.00 0.00 283.16 23.44 148912015.97 148989898.68 Summer solstice 166.39 166.84 166.62 90.00 23.44 89.56 90.00 90.00 283.16 23.44 152025947.60 152030583.04 Aphelion 180.00 180.00 180.00 104.30 22.79 103.16 103.16 76.84 256.84 23.44 152098441.95 152098409.19 Autumnal equinox 258.70 256.83 257.77 180.00 -0.00 181.87 180.00 -0.00 103.17 23.44 150048057.37 150128632.17 Winter solstice 347.26 346.83 347.04 270.00 -23.44 270.43 270.00 -90.00 283.17 23.44 147158348.89 147162417.76 Orbit Semi-major axis in km (SMA) = 149598616.31172, Eccentricity (e sun) = 0.0167102190 Mean movement deg per day (n sun) = 0.9856003000, Mean distance from earth in km (As) = 149598616.31172 Continue Section - 2.8
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OM-MSS Page 67 In the table above , all angles are in deg and distances in km. The values show consistency. About accuracy, for an input Universal Time, Compared below the computed values against that reported (Ref. http://www.stargazing.net/kepler/sun.html). For Input Universal Time year = 1997, month = 8, day = 7, hour = 11, minute = 0, seconds = 0.0000000000 (under reference) The Output Computed values : Sun Position on Celestial Sphere at Input UT time 07. Sun Mean anomaly in deg per day from n_sun (m sun) = 213.1154702210 08. Sun Mean longitude in deg (Lmean) = 136.0061615585 10. Sun Ecliptic longitude in deg (Lsun) = 134.9782467378 11. Obliquity of the ecliptic plane in deg (Epcylone) = 23.4395921140 12. Sun Right ascension in deg (Alpha) = 137.44256792552 13. Sun Declination in deg (Delta) = 16.3426505298 The Output Reported values : Sun Position on Celestial Sphere at same UT time, (Ref. http://www.stargazing.net/kepler/sun.html). 07. Sun Mean anomaly in deg per day from n_sun (m sun) = 213.11547 08. Sun Mean longitude in deg (Lmean) = 136.00716 10. Sun Ecliptic longitude in deg (Lsun) = 134.97925 11. Obliquity of the ecliptic plane in deg (Epcylone) = 23.439351 12. Sun Right ascension in deg (Alpha) = 137.44352 13. Sun Declination in deg (Delta) = 16.342193 End of Computing Position of Sun on Celestial Sphere at Standard Epoch JD2000 and at Six Astronomical Events. Move on to Compute the Position of Earth on Celestial Sphere at Input Universal Time (UT). Next Section - 4 Position of Earth on Celestial Sphere at UT
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OM-MSS Page 68 OM-MSS Section - 4 --------------------------------------------------------------------------------------------------------34 POSITION OF EARTH ON CELESTIAL SPHERE AT INPUT UNIVERSAL TIME (UT). Earth is a sphere, the third planet from the Sun and the fifth largest of the eight planets in the Solar System. Planets order from the Sun : Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune. (Ref http://nineplanets.org/ ) Earth Rotates on its axis passing through the North and South Poles. The rotation is counterclockwise looking down at North Pole. The time for Earth to make a complete rotation is approximately 24 hours (exactly 23.9344699 hours or 23 hours, 56 minutes, 4.0916 seconds). This rotation results daytime in area facing Sun and nighttime in area facing away from Sun. Since we are on Earth, we do not sense its rotation, but experience by observing the relative motion of the Sun (like from a moving vehicle we see the surroundings move). Earth Revolves around Sun in a counterclockwise direction. The complete orbit (360 deg) is one sidereal year, occurs every 365.256363 mean solar days. The earth's orbit around the sun is not a circle, it is slightly elliptical. Therefore, distance between earth and sun varies throughout the year. The Earth Orbit Characteristics, at Epoch J2000, means Y 2000, M 01, D 01, H 12.00, JD = 2451545.00, (Ref http://en.wikipedia.org/wiki/Planet_of_Water) Eccentricity 0.01671123 Inclination 7.155 deg to Sun's equator Longitude of ascending node 348.73936 deg Mean anomaly 357.51716 Orbital period 365.256363004 days Argument of perihelion 114.20783 deg Aphelion 152,098,232 km Perihelion 147,098,290 km Average orbital speed 107,200 km/h Semimajor axis 149,598,261 km Average distance(AS)149,597,870.700 km from sun Note : Ambiguity exist in values reported for Earth to Sun distances, that are mean, average, maximum, minimum, semi-major axis, aphelion, perihelion. Move on to Compute the Position of Earth on Celestial Sphere at input Universal Time (UT). At any instant, first need to Compute Position of Sun on celestial sphere and then at same instant Compute Position of Earth on celestial sphere. For the Position of Sun on celestial sphere, much has been computed / illustrated in previous section. The Position of Earth on celestial sphere is characterized by computing around 120 orbital parameters. The number is large, because some parameters are computed using more than one model equation, that require different inputs. This helps in validation of results and understanding the different input considerations.
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OM-MSS Page 69 The Orbital Parameters that Characterize the Position of Earth on Celestial Sphere, are put into following groups : 01. GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input UT time YY MM DD HH. 02. Earth, Log in 0 to 360 deg and Lat in +ve or -ve in 0 to 90 deg, pointing to Sun Ecliptic Log (Lsun) at time input UT. 03. LST Local sidereal time using GST over three longitudes, Greenwich log, Sun mean log (Lmean), & Sun epliptic log (Lsun) . 04. ST0 sidereal time over Greenwich longitude = 0.0, at time input Year JAN day 1 hr 00. 05. ST sidereal time, at time input UT, over three log, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun). 06. H hour angle in 0 to 360 deg using ST over five longitudes, Greenwich, Lmean, Lsun, Earth Sub Sun point SS, Earth Observation point EP, at time input UT. 07. Delta E is Equation of Time in seconds, using p_julian_day, n_sun, w_sun at time input UT. 08. GST Greenwich sidereal time, and GHA Greenwich hour angle 0 to 360 deg at time when earth is at perihelion. 09. ST sidereal time & MST mean sidereal time at different instances, using Earth mean motion rev per day and julian century days from YY 2000_JAN_1_hr_1200. 10. Earth orbit radius, sub sun point on Earth surface & related parameters, using SMA, e_sun, T_sun, w_sun etc. 11. Earth center(EC) to Sun center(SC) Range Vector[rp, rq, r] in PQW frame (perifocal coordinate system). 12. Transform_1 Earth position EC to SC Range Vector[rp, rq] in PQW frame To Range Vector[rI, rJ, rK] in IJK frame (inertial system cord). 13. Transform_2 Earth point EP(lat, log, hgt) To EC to SC Range Vector[RI, RJ, RK, R] in IJK frame. 14. Transform_3 Earth position EC to SC Range Vectors [rI rJ rK] & [RI RJ RK] To EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame. 15. Transform_4 Earth point EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame To EP to SC Range Vector[rvS, rvE, rvZ] in SEZ frame. 16. Elevation(EL) and Azimuth(AZ) angle of Sun at Earth Observation point EP 17. Distance in km from Earth observation point(EP) to Sub Sun point(SS) and Earth Velocity meter per sec in orbit at time input UT. 18. Earth State Position Vector [X, Y, Z] in km at time input UT. 19. Earth State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. 20. Earth Orbit Normal Vector [Wx, Wy, Wz] in km and angles Delta, i, RA at time input UT; Normal is line perpendicular to orbit plane. 21. Transform Earth State Vectors To Earth position Keplerian elements. 22. Transform Earth position Keplerian elements To Earth State Vectors . The values of all these parameters are Computed are at Standard Epoch JD2000 and when Earth is at Perihelion, Aphelion, Equinoxes, and Solstices. The time at perihelion, aphelion, equinoxes, and solstices, were computed earlier for the input year .
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OM-MSS Page 70 Move on to Find Position of Earth on Celestial Sphere, at Seven different Time events, the Utilities of OM-MSS Software (Sections - 4.1 to 4.8). (a) Time Event - Standard Epoch JD2000 ; (b) Time Event - when Earth is at Perihelion ; (c) Time Event - when Earth is at Vernal equinox ; (d) Time Event - when Earth is at Summer solstice ; (e) Time Event - when Earth is at Aphelion ; (f) Time Event - when Earth is at Autumnal equinox ; (g) Time Event - when Earth is at Winter solstices ; Move on to Compute Position of Earth on Celestial Sphere at all seven astronomical Time events in orbit, Earth around Sun. Next Section - 4.1 Position of earth at standard epoch time JD2000
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OM-MSS Page 71 OM-MSS Section - 4.1 ---------------------------------------------------------------------------------------------------35 Earth Positional Parameters on Celestial Sphere : Input Time (UT) Standard Epoch JD2000 1. Find Position of Earth on Celestial Sphere at Input UT Standard Epoch time JD2000 . Input UT Time, Standard Epoch time JD2000 : year = 2000, month = 1, day = 1, hour = 12, minute = 0, seconds = 0.00000 Julian Day = 2451545.00000, year_day_decimal = 0.50000, day_hour_decimal = 12.00000 Observation Point on Earth (Bhopal, India) : Lat +ve or -ve 0 to 90 deg = 23.25993 ie deg = 23, min = 15, sec = 35.76 Log 0 to 360 deg = 77.41261 ie deg = 77, min = 24, sec = 45.41 Alt from earth surface in km = 0.49470 First Compute the Sun Position on Celestial Sphere, then Compute the Earth Position on Celestial Sphere. (A) Computed Values for SUN POSITION on Celestial Sphere at Input Ut Time : (Sr. No 1 - 22) 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = -2.5081161195 07. Mean anomaly in deg per day from n_sun (m sun) = -2.4719999999 08. Sun Mean longitude in deg (Lmean) = 280.4600000000 09. Earth Mean anomaly in deg (ME) = 357.5280000000 10. Sun Ecliptic longitude in deg (Lsun) = 280.3756801972 11. Obliquity of ecliptic in deg (Epcylone) = 23.4392794444 12. Sun Right ascension in deg (Alpha) = 281.2858630915 13. Sun Declination in deg (Delta) = -23.0337026521 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 147101227.61694 16. Sun Nodal elongation in deg (U sun) = -79.6243198028 17. Sun Mean anomaly in deg (M sun) = 357.5280000002 18. Sun Eccentric anomaly in deg (E sun) = 357.4860040557 19. Sun True anomaly in deg (T sun) = 357.4436516380 20. Sun Argument of perigee in deg (W sun) = 282.9320285593 21. Sun True anomaly in deg from U & W (V sun) = 357.4436516380 22. Sun Distance in km (d sun) = 147101040.52850 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) These Values are applied as input for Computing Earth Position on Celestial Sphere around Sun at same input UT Time.
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OM-MSS Page 72 (B) Computed Values for EARTH POSITION on Celestial Sphere around Sun at same Input Ut Time : (Sr. No 1 - 22) Input Time year = 2000, month = 1, day = 1, hour = 12, minute = 0, seconds = 0.00000, corresponding Julian Day = 2451545.0000000000 Observation Point on Earth : Lat +ve or -ve 0 to 90 deg = 23.25993, Log 0 to 360 deg = 77.41261, Alt from earth surface in km = 0.49470 Sun position on Celestial sphere at input time, computed above total 22 parameters. Output Earth Position on Celestial sphere around Sun : Computed below around 120 parameters, presented in 1-22 groups. Number is large, because some parameters are computed using more than one model equation, that require different inputs. This helps in validation of results and understanding the different input considerations. 01. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input UT time YY MM DD HH. Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time UT year = 2000, month = 1, day = 1, hour = 12, minute = 0, seconds = 0.00000 Outputs : GST & GHA in 0-360 deg over Greenwich. E01A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 280.46030, hr = 18, min = 41, sec = 50.47200 E01B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 280.46743, deg = 280, min = 28, sec = 2.73693 02. Finding Earth latitude & longitude pointing to Sun Ecliptic longitude(Lsun). Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : Earth lat & log pointing to Lsun. E02A 011. Earth latitude +ve or -ve in 0 to 90 deg at UT time = -23.03 ie deg = -23, min = 2, sec = 1.33 E02B 011. Earth longitude 0 to 360 deg = 0.83 ie deg = 0, min = 49, sec = 32.03 03. Finding LST over three longitudes, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - for LST, used sidereal time at Greenwich GST and desired geogrphic longitude Inputs : At Time input UT - GST, Log of Greenwich, sun mean log Lmean, Sun ecliptic log Lsun. Outputs : LST over Greenwich, Lmean, Lsun . E03A 011. LST Local sidereal time in 0-360 deg, over Greenwich longitude = 280.46030, hr = 18, min = 41, sec = 50.47200
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OM-MSS Page 73 E03B 011. LST Local sidereal time in 0-360 deg, over Sun mean longitude (Lmean) = 200.92030, hr = 13, min = 23, sec = 40.87200 E03C 011. LST Local sidereal time in 0-360 deg, over Sun epliptic longitude (Lsun) = 200.83598, hr = 13, min = 23, sec = 20.63525 04. Finding ST0 sidereal time over Greenwich longitude = 0.0, at time input Year JAN day 1 hr 00. Note - this is sidereal time ST at UT year, month = 1, day = 1, hours decimal = 0.0 and geogrphic longitude = 0.0 Inputs : Time input UT Year, JAN day 1 hr 00, Log 0.0 Outputs : ST0 over Greenwich E04 011. ST0 Sidereal time in 0-360 deg, over Greenwich at input UT year, MM 1, DD 1, HH 00 = 99.96748, hr = 6, min = 39, sec = 52.19432 05. Finding ST sidereal time over three longitudes of, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - this is local sidereal time LST; (LST = GST at UT time + geogrphic longitude). Inputs : At Time input UT - Log 0.0, Log Lmean, Log Lsun Outputs : ST over Greenwich, Lmean, Lsun. E05A 011. ST Sidereal time in 0-360 deg, over Greenwich at input UT time = 280.46030, hr = 18, min = 41, sec = 50.47200 E05B 011. ST Sidereal time in 0-360 deg, over Sun mean longitude (Lmean) at input UT time = 200.92030, hr = 13, min = 23, sec = 40.87200 005C 011. ST Sidereal time in 0-360 deg, over Sun longitude (Lsun) at input UT time = 200.83598, hr = 13, min = 23, sec = 20.63525 06. Finding H hour angle in 0 to 360 deg over longitudes of, Greenwich, Lmean, Lsun, Earth Sub Sun point SS, Earth Obseration point EP. Note - used Sun Right ascension Alpha at input time; (hour angle HA = LST - Alpha). Inputs : At Time input UT - Sun Right ascension Alpha and ST Sidereal time over longitudes 0.0, Lmean, Lsun, SS, EP Outputs : Hour Angles over Greenwich, Lmean, Lsun, SS, EP E06A 011. H hour angle 0-360 deg, over Greenwich, = 359.17444, deg = 359, min = 10, sec = 27.97287 E06B 011. H hour angle 0-360 deg, over Lmean, = 279.63444, deg = 279, min = 38, sec = 3.97287 E06C 011. H hour angle 0-360 deg, over Lsun, = 279.55012, deg = 279, min = 33, sec = 0.42158 E06D 011. H hour angle 0-360 deg, over SS, = 0.00000, deg = 0, min = 0, sec = 0.00000 E06E 011. H hour angle 0-360 deg, over EP, = 76.58705, deg = 76, min = 35, sec = 13.38687
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OM-MSS Page 74 07. Finding Delta E is Equation of Time in seconds, at time input UT. Note - this value in seconds accounts for relative movement of sun in elliptical orbit w.r.t earth and effect of obliquity of the ecliptic; its maximum value is 16 minutes (960 sec.); Delta E is computed using time in days from the perihelion, n_sun_deg and w_sun at input UT. Inputs : Time input UT in JD, time perihelion in JD, Sun mean movement n_sun, Eccentricity of earth orbit E_Sun Outputs : Delta E time_equation in seconds. E07 011. Delta E Time Equation in seconds = 191.54215, hr = 0, min = 3, sec = 11.54215 08. Finding GST Greenwich sidereal time, and GHA Greenwich hour angle 0 to 360 deg at time when earth is at perihelion. Inputs : Time in JD when earth at perihelion YY = 2000, MM = 1, DD = 4, hr = 0, min = 11, sec = 41.23 Outputs : GST & GHA in 0-360 deg over Greenwich when earth is at perihelion E08A 011. GST sidereal time in 0-360 deg over Greenwich at time when earth is at perihelion = 105.85422, hr = 7, min = 3, sec = 25.01305 E08B 011. GHA hour angle in 0-360 deg over Greenwich at time when earth is at perihelion = 105.85427, hr = 7, min = 3, sec = 25.02537 09. Finding ST sidereal time and MST mean sidereal time, over Greenwich, using Earth mean motion rev per day . Inputs : GST when earth at perihelion, earth rotation rate, ref. JD2000, time input UT in JD, time perihelion in JD. Outputs : STP, angle perihelion to input JD, ST over Greenwich, MSTO & MST over Greenwich, solar time E09A 011. STP sidereal time in 0-360 deg over Greenwich when earth at perihelion = 105.85422, hr = 7, min = 3, sec = 25.01305 E09B 011. Angle in 0-360 deg from earth at perihelion to input JD using earth rotational rate = 174.60617, 009C 011. ST in 0-360 deg over Greenwich using STP and angle from perihelion at input JD = 280.46039, hr = 18, min = 41, sec = 50.49311 E09D 011. ST in 0-360 deg over Greenwich using STP and earth rotation at UT time = 280.46039, hr = 18, min = 41, sec = 50.49311 E09E 011. MST0 in deg, over Greenwich using JD century days, ref J2000 to I/P YY, M1, D1 hr 00 = 99.96779, hr = 6, min = 39, sec = 52.27073 E09F 011. MST in deg, over Greenwich using JD century in days from ref J2000 to UT time Y M D H = 280.46062, hr = 18, min = 41, sec = 50.54841 E09G 011. Solar time over Greenwich in JD (GMT or input UT - 12 hr) = YY 2000, MM 1, DD 1, hr 0, min 0, sec 0.000, ie JD 2451544.50000 10. Finding Earth orbit radious using true anomaly, Sub Sun point (SS) on earth surface and related paramters . (a) Finding Earth orbit radious using true anomaly. Inputs : semi-major axis SMA, eccentricity of earth orbit e_sun, sun true anomaly T_Sun Outputs : earth orbital radious EC to SC (earth center to sun center)
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OM-MSS Page 75 E10A 011. earth orbital radious EC to SC km using_true anomaly at UT time = 147101196.6616485400 (b) Finding Sub Sun point (SS) over earth surface (Latitude, Longitude, & Latitude radius) pointing to Sun Ecliptic Log (Lsun), Sun height from earth surface over SS, and LST over SS log at time input UT. Note - for SS Latitude, used earth inclination, sun true anomaly T_sun and sun argument of perigee w_sun. for SS Longitude, used Sun right ascension Alpha and sidereal time at Greenwich GST. Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : SS point Latitude, Longitude, Latitude radious, LST & LMT over SS . E10B 011. SS point Latitude +ve or -ve in 0 to 90 deg at UT time = -23.03 ie deg = -23, min = 2, sec = 1.33 E10C 011. SS point Longitude 0 to 360 deg = 0.83 ie deg = 0, min = 49, sec = 32.03 E10D 011. SS point Latitude radious km at UT time = 6374.8796719602 E10E 011. Sun height km from earth surface over SS at UT time = 147094821.7819765800 E10F 011. LST local sidereal time in 0-360 deg over SS log at UT time, (LST = GST + log east) = 281.286 ie hr = 18, min = 45, sec = 8.68356 LST local sidereal time and LMT local mean time with date adjusted to calendar YY MM DD and UT hr mm sec. E10G 011. LST local sidereal time at Sub Sun point (SS) YY = 2000, MM = 1, DD = 1, hr = 18, min = 45, sec = 8.68 E10H 011. LMT local Mean time at Sub Sun point (SS) YY = 2000, MM = 1, DD = 1, hr = 12, min = 3, sec = 18.14 (c) Finding LST and LMT over Earth point(EP) where Observer is, at time input UT. Inputs : EP point Latitude, Longitude , Outputs : LST & LMT over EP . E10I 011. LST local sidereal time in 0-360 deg at EP log at UT time, (LST = GST + log east) = 357.873 ie hr = 23, min = 51, sec = 29.57601 LST and LMT with date adjusted to calendar YY MM DD and UT hr mm sec. E10J 011. LST local sidereal time at Earth point (EP) YY = 2000, MM = 1, DD = 1, hr = 23, min = 51, sec = 29.58 E10K 011. LMT local Mean time at Earth point (EP) YY = 2000, MM = 1, DD = 1, hr = 17, min = 9, sec = 39.03
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OM-MSS Page 76 Finding Earth to Sun Position Vectors coordinate in PQW, IJK, SEZ frames and the Vector Coordinate Transforms. First defined coordinate systems, PQW, IJK, SEZ, then computed Position & Velocity vectors in these three coordinate systems. (a) Perifocal Coordinate System (PQW), is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. The system is fixed with time (inertial), pointing towards orbit periapsis; the system's origin is Earth center (EC), and its fundamental plane is the orbit plane; the P-vector axis directed from EC toward the periapsis of the elliptical orbit plane, the Q-vector axis swepts 90 deg from P axis in the direction of the orbit, the W-vector axis directed from EC in a direction normal to orbit plane, forms a right-handed coordinate system. (b) Geocentric Coordinate System (IJK), is also an Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). The system is fixed with time (inertial), pointing towards vernal equinox; the system's origin is Earth center (EC), and its fundamental plane is the equator; the I-vector is +X-axis directed towards the vernal equinox direction on J2000, Jan 1, hr 12.00 noon, the J-vector is +Y-axis swepts 90 deg to the east in the equatorial plane, the K-vector is +Z-axis directed towards the North Pole. (c) Topocentric Horizon Coordinate System (SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). The system moves with earth, is not fixed with time (non-inertial), is for use by observers on the surface of earth; the observer's surface forms the fundamental plane, is tangent to earth's surface the S-vector is +ve horizontal-axis directed towards South, the E-vector is +ve horizontal-axis directed towards East, the Z-vector is +ve normal directed upwards on earth surface. Note that axis Z not necessarily pass through earth center, so not used to define as radious vector. 11. Finding Earth center(EC) to Sun center(SC) Range Vector[rp, rq, r] from in PQW frame, perifocal coordinate system. Inputs : Semi-major axis (SMA), Eccentricity of earth orbit (e_sun), Sun eccentric anomaly (E_sun) Outputs : Vector(r, rp rq) in PQW frame E11A 011. r earth pos vector magnitude EC to SC km in PQW frame perifocal cord at UT time = 147101196.66165
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OM-MSS Page 77 E11B 011. rp earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = 146954807.4835177700 E11C 011. rq earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = -6560992.0569400741 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as earth orbital radious computed before using true anomaly. 12. Transform_1 Earth position EC to SC Range Vector[rp, rq] in PQW frame To Range Vector[rI, rJ, rK] in IJK frame, inertial system cord. Inputs : Vector(rp, rq) EC_to_SC km in frame PQW , Alpha rd, w_sun rd, earth_inclination rd , Outputs : Vector(rI, rJ, rK, r) EC_to_SC km in frame IJK E12A 011. rI earth pos vector component EC to SC km frame IJK at UT time = -125003885.4456264400 E12B 011. rJ earth pos vector component EC to SC km frame IJK at UT time = -51961735.9769164550 E12C 011. rK earth pos vector component EC to SC km frame IJK at UT time = -57556656.2358423690 E12D 011. r earth pos vector magnitude EC to SC km frame IJK at UT time = 147101196.6616485400 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as that compued above in PQW frame. 13. Transform_2 Earth point EP(lat, log, hgt) To EC to SC Range Vector[RI, RJ, RK, R] in IJK frame. Inputs : earth equator radious_km, earth point EP(lat deg, log deg, hgt meter), LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time, Outputs : Vector(RI, RJ, RK, R) Range EC to EP in IJK frame E13A 011. RI pos vector component EC to EP km frame IJK at UT time = 5853.1109516962 E13B 011. RJ pos vector component EC to EP km frame IJK at UT time = -217.3619305363 E13C 011. RK pos vector component EC to EP km frame IJK at UT time = 2517.6312937817 E13D 011. R pos vector magnitude EC to EP km frame IJK at UT time = 6375.3134317570 14. Transform_3 Earth position EC to SC Range Vectors [rI rJ rK] & [RI RJ RK] To EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame. Inputs : Vector(rI rJ rK) position EC_to_SC km in frame IJK , Vector(RI RJ RK) range EC to SC km in IJK frame, Outputs : Vector(rvI, rvJ, rvK, rv) range EP to SC in IJK frame E14A 011. rvI range vector component EP to SC km frame IJK at UT time = -125009738.5565781300 E14B 011. rvJ range vector component EP to SC km frame IJK at UT time = -51961518.6149859200 E14C 011. rvK range vector component EP to SC km frame IJK at UT time = -57559173.8671361510 E14D 011. rv range vector magnitude EP to SC km frame IJK at UT time = 147107078.8474394100
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OM-MSS Page 78 15. Transform_4 Earth point EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame To EP to SC Range Vector[rvS, rvE, rvZ] in SEZ frame. Inputs : lat_pos_neg_0_to_90_deg_at_EP_at_time_UT , LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time Vector(rvI, rvJ, rvK, rv) range EP to SC km in IJK frame, Outputs : Vector(rvS, rvE, rvZ, rv) range EP to SC km in SEZ frame E15A 011. rvS range vector component EP to SC km frame SEZ at UT time = 4309690.2262858748 E15B 011. rvE range vector component EP to SC km frame SEZ at UT time = -56564906.7140329630 E15C 011. rvZ range vector component EP to SC km frame SEZ at UT time = -135728886.1871818300 E15D 011. rv range vector magnitude EP to SC km frame SEZ at UT time = 147107078.8474394100 16. Finding Elevation(EL) and Azimuth(AZ) angle of Sun at Earth Observation point EP . Note : Results computed using 4 different formulations, each require different inputs to give EL & AZ angles. For all situations of Object and Observer positions, a combination of latitude N/S & longitude E/W : Method 1 : for both EL & AZ angles, this does not provide correct results ; Method 2 : for only EL angle, this provides consistent, unambiguous correct results. but for AZ angles the results are ambiguous, need corrections by adding or subtracting values as 180 or 360 or sign change. Method 3 : same as method 2, for EL angle, the results are correct, but for AZ angles the results are ambiguous, need corrections. Method 4 : for finding Azimuth and Distance but not for finding Elevation angle; for AZ angles, this provides correct unambiguous results that need no futher corrections. Therefore for Elevation (EL) angle Method 3 results are accepted and for Azimuth (AZ) angle Method 4 results are accepted . Results verified from other sources; Ref URLs http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone . NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , and http://aa.usno.navy.mil/data/docs/AltAz.php Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html Rem: SS point lat deg = -23.03, log deg = 0.83 YY = 2000, MM = 1, DD = 1, hr = 12, min = 3, sec = 18.14 EP point lat deg = 23.26, log deg = 77.41 YY = 2000, MM = 1, DD = 1, hr = 17, min = 9, sec = 39.03 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 1 - computed values may be Ambiguous or Incorrect). Inputs : Vector[rvS, rvE, rvZ] range EP to SC km in SEZ frame Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP
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OM-MSS Page 79 E16A 011. Elevation angle deg of Sun at EP using rv SEZ at UT time = -67.3171664896 E16B 011. Azimuth angle deg of sun at EP using rv SEZ at UT time = 265.6430420660 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 2 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun declination Delta Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16C 011. Elevation angle deg of Sun at EP using Sun declination diff log range EP to SC = 2.38469 E16D 011. Azimuth angle deg of sun at EP using sun declination diff log range EP to SC = -153.63057 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 3 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun hgt from EC, Sun range from EP (Sun hgt from EC = earth orbit radious EC to SC km ; Sun range from EP = rv range vector EP to SC km frame SEZ) Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16E 011. Elevation angle deg of Sun at EP using Sun hgt diff log range EP to SC = 2.38221 ie deg = 2, min = 22, sec = 55.96 E16F 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 243.63057 ie deg = 243, min = 37, sec = 50.05 Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 4 - computed AZ values is unambiguous & correct). Inputs : Time input UT YY MM DD HH, EP lat & log, SS lat & log Outputs : Azimuth(AZ) of Sun at EP E16G 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 243.63057 ie deg = 243, min = 37, sec = 50.05 Due to such incorrect results, finally for Elevation (EL) Method 3 results and for Azimuth (AZ) Method 4 results are accepted. Finally accepted Elevation angle deg of Sun from EP to SC = 2.3822113375 ie deg = 2, min = 22, sec = 55.96 Finally accepted Azimuth angle_deg of Sun from EP to SC = 243.6305706573 ie deg = 243, min = 37, sec = 50.05
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OM-MSS Page 80 Distance in km from Earth observation point(EP) to Sub Sun point(SS) and Earth Velocity meter per sec in orbit at time input UT. 17. Finding Distance in km from Earth observation point(EP) to Sub Sun point(SS) over Earth surface . Inputs : EP lat & log, SS lat & log, Outputs : Distance in km from EP to SS over Earth surface E17A 011. Distance in km Earth observation point(EP) to Sub Sun point(SS) = 9753.27897 Finding Earth Velocity meter per sec in orbit in frame PQW Inputs : semi-major axis SMA, GM_Sun, earth pos r EC to SC frame IJK, eccentricity of earth orbit e_Sun, sun eccentric anomaly E_Sun Outputs : Earth Velocity magnitude and component Xw Yw in frame PQW in meter per sec E17B 011. Velocity magnitude meter per sec using GM, SMA, r earth EC to SC frame IJK at UT time = 30286.0666340612 E17C 011. Velocity component meter per sec in orbit Xw using GM, e_Sun, SMA, E_Sun at UT time = 1328.6358841510 E17D 011. Velocity component meter per sec in orbit Yw using GM, e_Sun, SMA, E_Sun at UT time = 30256.9092745796 Finding Earth Velocity Vector [vX, vY, vZ] in meter per sec in orbit; a Transform of [Xw, Yw] in frame PQW To [vX, vY, vZ] in frame XYZ Inputs : velocity component (Xw, Yw), sun right ascension Alpha, Sun Argument of perigee W_Sun, inclination Epcylone Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ E17E 011. vX earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 10756.7530226526 E17F 011. vY earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = -28227.5170485498 E17G 011. vZ earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 2178.3888188937 E17H 011. vR earth Velocity magnitude meter per sec using Xw Yw frame PQW RA w i at UT time = 30286.0666340612
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OM-MSS Page 81 Earth State Vectors : Position [X, Y, Z] in km and Velocity [Vx, Vy, Vz] in meter per sec, at time input UT. 18. Finding Earth State Position Vector [X, Y, Z] in km at time input UT. Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ E18A 011. State vector position X km at UT time = -125003885.4456264400 E18B 011. State vector position Y km at UT time = -51961735.9769164550 E18C 011. State vector position Z km at UT time = -57556656.2358423690 E18D 011. State vector position R km at UT time = 147101196.6616485400 19. Finding Earth State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ E19A 011. State vector velocity Vx meter per sec at UT time = 10756.7530226526 E19B 011. State vector velocity Vy meter per sec at UT time = -28227.5170485498 E19C 011. State vector velocity Vz meter per sec at UT time = 2178.3888188937 019D 011. State vector velocity V meter per sec at UT time = 30286.0666340612 20. Earth Orbit Normal Vector [Wx, Wy, Wz] in km and angles Delta, i, RA at time input UT; Normal is line perpendicular to orbit plane. Inputs : earth pos r EC to SC frame IJK, inclination Epcylone, sun right ascension Alpha Outputs : earth orbit normal vector (Wx, Wy, Wz, W) in km E20A 011. Earth orbit normal W km using r earth pos frame IJK inclination Alpha = 147101196.6616485400 E20B 011. Earth orbit normal Wx km using r earth pos frame IJK inclination Alpha = -57381991.7843673230 E20C 011. Earth orbit normal Wy km using r earth pos frame IJK inclination Alpha = -11451330.6380786910 E20D 011. Earth orbit normal Wz km using r earth pos frame IJK inclination Alpha = 134962721.1668659400 020E 011. Earth orbit normal Delta W deg using r earth pos frame IJK inclination Alpha = 66.5607205617 E20F 011. Earth orbit normal Inclination i deg using normal_Delta_W = 23.4392794383 E20G 011. Earth orbit normal Alpha W deg using r earth pos frame IJK, inclination, Alpha = 11.2858630915 E20H 011. Earth orbit normal Right ascension of ascending node using normal Alpha, W = 101.2858630915
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OM-MSS Page 82 Transform Earth State Vectors to Earth position Keplerian elements. 21. Finding Earth position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness E21A 011. Keplerian elements year = 2000, days_decimal_of_year = 0.50000, revolution no = 1, node = 2 ie decending E21B 011. inclination_deg = 23.4392794383 E21C 011. right ascension ascending node deg = 281.2858630915 E21D 011. eccentricity = 0.0167102190 E21E 011. argument of perigee_deg = 282.9320285591 E21F 011. mean anomaly deg = 357.5280000003 E21G 011. mean_motion rev per day = 0.0027377786 E21H 011. mean angular velocity rev_per_day = 0.0027377786 E21I 011. mean motion rev per day using SMA considering oblateness = 0.0027377786 Transform Earth position Keplerian elements to Earth State Vectors . 22. Finding Earth position State Vectors, computed using Keplerian elements at time input UT (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] E22A 011. State vectors year = 2000, days_decimal_of_year = 0.50000, revolution no = 1, node = 2 ie decending E22B 011. state vector position X km = -125003885.4456292700, state vector velocity Vx meter per sec = 10756.7530226511 E22C 011. state vector position Y km = -51961735.9769086610, state vector velocity Vy meter per sec = -28227.5170485504 E22D 011. state vector position Z km = -57556656.2358429060, state vector velocity Vz meter per sec = 2178.3888188931 E22E 011. state vector position R km = 147101196.6616483900, state vector velocity V meter per sec = 30286.0666340612
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OM-MSS Page 83 Note : Computation of all above parameters, grouped in 1 to 22, corresponds to time (a) Universal time over Greenwich (UT/GMT) : Year = 2000, Month = 1, Day = 1, Hour = 12, Min = 0, Sec = 0.000 (b) Mean Solar time (MST) over Earth Observation point (EP ) : Year = 2000, Month = 1, Day = 1, Hour = 17, Min = 9, Sec = 39.028 Move on to next Astronomical event in orbit Earth around Sun. Next Section - 4.2 Position of earth at time when earth is at perihelion
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OM-MSS Page 84 OM-MSS Section - 4.2 ---------------------------------------------------------------------------------------------------36 Earth Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Perihelion. 2. Finding Position of Earth on Celestial Sphere at Input Universal Time, when Earth is at Perihelion . Input UT Time, when Earth is at Perihelion : year = 2013, month = 1, day = 3, hour = 9, minute = 11, seconds = 56.61639 Julian Day = 2456295.88329, year_day_decimal = 2.38329, day_hour_decimal = 9.19906 Observation Point on Earth (Bhopal, India) : Lat +ve or -ve 0 to 90 deg = 23.25993 ie deg = 23, min = 15, sec = 35.76 Log 0 to 360 deg = 77.41261 ie deg = 77, min = 24, sec = 45.41 Alt from earth surface in km = 0.49470 First Compute the Sun Position on Celestial Sphere, then Compute the Earth Position on Celestial Sphere. (A) Computed Values for SUN POSITION on Celestial Sphere at Input Ut Time : (Sr. No 1 - 22) 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = 0.0000000000 07. Mean anomaly in deg per day from n_sun (m sun) = 0.0000000000 08. Sun Mean longitude in deg (Lmean) = 283.1557666033 09. Earth Mean anomaly in deg (ME) = 0.0000000001 10. Sun Ecliptic longitude in deg (Lsun) = 283.1557666033 11. Obliquity of ecliptic in deg (Epcylone) = 23.4375874462 12. Sun Right ascension in deg (Alpha) = 284.2922173002 13. Sun Declination in deg (Delta) = -22.7872783368 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 147098823.43315 16. Sun Nodal elongation in deg (U sun) = -76.8442333967 17. Sun Mean anomaly in deg (M sun) = 0.0000000000 18. Sun Eccentric anomaly in deg (E sun) = 0.0000000000 19. Sun True anomaly in deg (T sun) = 0.0000000000 20. Sun Argument of perigee in deg (W sun) = 283.1557666033 21. Sun True anomaly in deg from U & W (V sun) = 0.0000000000 22. Sun Distance in km (d sun) = 147098790.67105 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) These Values are applied as input for Computing Earth Position on Celestial Sphere around Sun at same input UT Time.
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OM-MSS Page 85 (B) Computed Values for EARTH POSITION on Celestial Sphere around Sun at same Input Ut Time : (Sr. No 1 - 22) Input Time year = 2013, month = 1, day = 3, hour = 9, minute = 11, seconds = 56.61639, corresponding Julian Day = 2456295.8832941712 Observation Point on Earth : Lat +ve or -ve 0 to 90 deg = 23.25993, Log 0 to 360 deg = 77.41261, Alt from earth surface in km = 0.49470 Sun position on Celestial sphere at input time, computed above total 22 parameters. Output Earth Position on Celestial sphere around Sun : Computed below around 120 parameters, presented in 1-22 groups. Number is large, because some parameters are computed using more than one model equation, that require different inputs. This helps in validation of results and understanding the different input considerations. 01. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input UT time YY MM DD HH. Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time UT year = 2013, month = 1, day = 3, hour = 9, minute = 11, seconds = 56.61639 Outputs : GST & GHA in 0-360 deg over Greenwich. E01A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 241.14177, hr = 16, min = 4, sec = 34.02452 E01B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 241.14717, deg = 241, min = 8, sec = 49.80962 02. Finding Earth latitude & longitude pointing to Sun Ecliptic longitude(Lsun). Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : Earth lat & log pointing to Lsun. E02A 011. Earth latitude +ve or -ve in 0 to 90 deg at UT time = -22.79 ie deg = -22, min = 47, sec = 20.10 E02B 011. Earth longitude 0 to 360 deg = 43.15 ie deg = 43, min = 9, sec = 1.61 03. Finding LST over three longitudes, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - for LST, used sidereal time at Greenwich GST and desired geogrphic longitude Inputs : At Time input UT - GST, Log of Greenwich, sun mean log Lmean, Sun ecliptic log Lsun. Outputs : LST over Greenwich, Lmean, Lsun . E03A 011. LST Local sidereal time in 0-360 deg, over Greenwich longitude = 241.14177, hr = 16, min = 4, sec = 34.02452
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OM-MSS Page 86 E03B 011. LST Local sidereal time in 0-360 deg, over Sun mean longitude (Lmean) = 164.29754, hr = 10, min = 57, sec = 11.40850 E03C 011. LST Local sidereal time in 0-360 deg, over Sun epliptic longitude (Lsun) = 164.29754, hr = 10, min = 57, sec = 11.40850 04. Finding ST0 sidereal time over Greenwich longitude = 0.0, at time input Year JAN day 1 hr 00. Note - this is sidereal time ST at UT year, month = 1, day = 1, hours decimal = 0.0 and geogrphic longitude = 0.0 Inputs : Time input UT Year, JAN day 1 hr 00, Log 0.0 Outputs : ST0 over Greenwich E04 011. ST0 Sidereal time in 0-360 deg, over Greenwich at input UT year, MM 1, DD 1, HH 00 = 100.80678, hr = 6, min = 43, sec = 13.62710 05. Finding ST sidereal time over three longitudes of, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - this is local sidereal time LST; (LST = GST at UT time + geogrphic longitude). Inputs : At Time input UT - Log 0.0, Log Lmean, Log Lsun Outputs : ST over Greenwich, Lmean, Lsun. E05A 011. ST Sidereal time in 0-360 deg, over Greenwich at input UT time = 241.14177, hr = 16, min = 4, sec = 34.02452 E05B 011. ST Sidereal time in 0-360 deg, over Sun mean longitude (Lmean) at input UT time = 164.29754, hr = 10, min = 57, sec = 11.40850 005C 011. ST Sidereal time in 0-360 deg, over Sun longitude (Lsun) at input UT time = 164.29754, hr = 10, min = 57, sec = 11.40850 06. Finding H hour angle in 0 to 360 deg over longitudes of, Greenwich, Lmean, Lsun, Earth Sub Sun point SS, Earth Obseration point EP. Note - used Sun Right ascension Alpha at input time; (hour angle HA = LST - Alpha). Inputs : At Time input UT - Sun Right ascension Alpha and ST Sidereal time over longitudes 0.0, Lmean, Lsun, SS, EP Outputs : Hour Angles over Greenwich, Lmean, Lsun, SS, EP E06A 011. H hour angle 0-360 deg, over Greenwich, = 316.84955, deg = 316, min = 50, sec = 58.38552 E06B 011. H hour angle 0-360 deg, over Lmean, = 240.00532, deg = 240, min = 0, sec = 19.14529 E06C 011. H hour angle 0-360 deg, over Lsun, = 240.00532, deg = 240, min = 0, sec = 19.14529 E06D 011. H hour angle 0-360 deg, over SS, = 0.00000, deg = 0, min = 0, sec = 0.00000 E06E 011. H hour angle 0-360 deg, over EP, = 34.26217, deg = 34, min = 15, sec = 43.79952
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OM-MSS Page 87 07. Finding Delta E is Equation of Time in seconds, at time input UT. Note - this value in seconds accounts for relative movement of sun in elliptical orbit w.r.t earth and effect of obliquity of the ecliptic; its maximum value is 16 minutes (960 sec.); Delta E is computed using time in days from the perihelion, n_sun_deg and w_sun at input UT. Inputs : Time input UT in JD, time perihelion in JD, Sun mean movement n_sun, Eccentricity of earth orbit E_Sun Outputs : Delta E time_equation in seconds. E07 011. Delta E Time Equation in seconds = 262.40497, hr = 0, min = 4, sec = 22.40497 08. Finding GST Greenwich sidereal time, and GHA Greenwich hour angle 0 to 360 deg at time when earth is at perihelion. Inputs : Time in JD when earth at perihelion YY = 2013, MM = 1, DD = 3, hr = 9, min = 11, sec = 56.62 Outputs : GST & GHA in 0-360 deg over Greenwich when earth is at perihelion E08A 011. GST sidereal time in 0-360 deg over Greenwich at time when earth is at perihelion = 241.14177, hr = 16, min = 4, sec = 34.02452 E08B 011. GHA hour angle in 0-360 deg over Greenwich at time when earth is at perihelion = 241.14717, hr = 16, min = 4, sec = 35.32064 09. Finding ST sidereal time and MST mean sidereal time, over Greenwich, using Earth mean motion rev per day . Inputs : GST when earth at perihelion, earth rotation rate, ref. JD2000, time input UT in JD, time perihelion in JD. Outputs : STP, angle perihelion to input JD, ST over Greenwich, MSTO & MST over Greenwich, solar time E09A 011. STP sidereal time in 0-360 deg over Greenwich when earth at perihelion = 241.14177, hr = 16, min = 4, sec = 34.02452 E09B 011. Angle in 0-360 deg from earth at perihelion to input JD using earth rotational rate = 0.00000, 009C 011. ST in 0-360 deg over Greenwich using STP and angle from perihelion at input JD = 241.14177, hr = 16, min = 4, sec = 34.02452 E09D 011. ST in 0-360 deg over Greenwich using STP and earth rotation at UT time = 241.14177, hr = 16, min = 4, sec = 34.02452 E09E 011. MST0 in deg, over Greenwich using JD century days, ref J2000 to I/P YY, M1, D1 hr 00 = 100.80714, hr = 6, min = 43, sec = 13.71450 E09F 011. MST in deg, over Greenwich using JD century in days from ref J2000 to UT time Y M D H = 241.14213, hr = 16, min = 4, sec = 34.11192 E09G 011. Solar time over Greenwich in JD (GMT or input UT - 12 hr) = YY 2013, MM 1, DD 2, hr 21, min 11, sec 56.616, ie JD 2456295.38329 10. Finding Earth orbit radious using true anomaly, Sub Sun point (SS) on earth surface and related paramters . (a) Finding Earth orbit radious using true anomaly. Inputs : semi-major axis SMA, eccentricity of earth orbit e_sun, sun true anomaly T_Sun Outputs : earth orbital radious EC to SC (earth center to sun center)
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OM-MSS Page 88 E10A 011. earth orbital radious EC to SC km using_true anomaly at UT time = 147098790.6710525200 (b) Finding Sub Sun point (SS) over earth surface (Latitude, Longitude, & Latitude radius) pointing to Sun Ecliptic Log (Lsun), Sun height from earth surface over SS, and LST over SS log at time input UT. Note - for SS Latitude, used earth inclination, sun true anomaly T_sun and sun argument of perigee w_sun. for SS Longitude, used Sun right ascension Alpha and sidereal time at Greenwich GST. Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : SS point Latitude, Longitude, Latitude radious, LST & LMT over SS . E10B 011. SS point Latitude +ve or -ve in 0 to 90 deg at UT time = -22.79 ie deg = -22, min = 47, sec = 20.10 E10C 011. SS point Longitude 0 to 360 deg = 43.15 ie deg = 43, min = 9, sec = 1.61 E10D 011. SS point Latitude radious km at UT time = 6374.9450864748 E10E 011. Sun height km from earth surface over SS at UT time = 147092415.7259660400 E10F 011. LST local sidereal time in 0-360 deg over SS log at UT time, (LST = GST + log east) = 284.293 ie hr = 18, min = 57, sec = 10.21954 LST local sidereal time and LMT local mean time with date adjusted to calendar YY MM DD and UT hr mm sec. E10G 011. LST local sidereal time at Sub Sun point (SS) YY = 2013, MM = 1, DD = 3, hr = 18, min = 57, sec = 10.22 E10H 011. LMT local Mean time at Sub Sun point (SS) YY = 2013, MM = 1, DD = 3, hr = 12, min = 4, sec = 32.72 (c) Finding LST and LMT over Earth point(EP) where Observer is, at time input UT. Inputs : EP point Latitude, Longitude Outputs : LST & LMT over EP . E10I 011. LST local sidereal time in 0-360 deg at EP log at UT time, (LST = GST + log east) = 318.555 ie hr = 21, min = 14, sec = 13.13953 LST and LMT with date adjusted to calendar YY MM DD and UT hr mm sec. E10J 011. LST local sidereal time at Earth point (EP) YY = 2013, MM = 1, DD = 3, hr = 21, min = 14, sec = 13.14 E10K 011. LMT local Mean time at Earth point (EP) YY = 2013, MM = 1, DD = 3, hr = 14, min = 21, sec = 35.64
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OM-MSS Page 89 Finding Earth to Sun Position Vectors coordinate in PQW, IJK, SEZ frames and the Vector Coordinate Transforms. First defined coordinate systems, PQW, IJK, SEZ, then computed Position & Velocity vectors in these three coordinate systems. (a) Perifocal Coordinate System (PQW), is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. The system is fixed with time (inertial), pointing towards orbit periapsis; the system's origin is Earth center (EC), and its fundamental plane is the orbit plane; the P-vector axis directed from EC toward the periapsis of the elliptical orbit plane, the Q-vector axis swepts 90 deg from P axis in the direction of the orbit, the W-vector axis directed from EC in a direction normal to orbit plane, forms a right-handed coordinate system. (b) Geocentric Coordinate System (IJK), is also an Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). The system is fixed with time (inertial), pointing towards vernal equinox; the system's origin is Earth center (EC), and its fundamental plane is the equator; the I-vector is +X-axis directed towards the vernal equinox direction on J2000, Jan 1, hr 12.00 noon, the J-vector is +Y-axis swepts 90 deg to the east in the equatorial plane, the K-vector is +Z-axis directed towards the North Pole. (c) Topocentric Horizon Coordinate System (SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). The system moves with earth, is not fixed with time (non-inertial), is for use by observers on the surface of earth; the observer's surface forms the fundamental plane, is tangent to earth's surface the S-vector is +ve horizontal-axis directed towards South, the E-vector is +ve horizontal-axis directed towards East, the Z-vector is +ve normal directed upwards on earth surface. Note that axis Z not necessarily pass through earth center, so not used to define as radious vector. 11. Finding Earth center(EC) to Sun center(SC) Range Vector[rp, rq, r] from in PQW frame, perifocal coordinate system. Inputs : Semi-major axis (SMA), Eccentricity of earth orbit (e_sun), Sun eccentric anomaly (E_sun) Outputs : Vector(r, rp rq) in PQW frame E11A 011. r earth pos vector magnitude EC to SC km in PQW frame perifocal cord at UT time = 147098790.67105
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OM-MSS Page 90 E11B 011. rp earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = 147098790.6710524900 E11C 011. rq earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = 0.0000000294 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as earth orbital radious computed before using true anomaly. 12. Transform_1 Earth position EC to SC Range Vector[rp, rq] in PQW frame To Range Vector[rI, rJ, rK] in IJK frame, inertial system cord. Inputs : Vector(rp, rq) EC_to_SC km in frame PQW , Alpha rd, w_sun rd, earth_inclination rd , Outputs : Vector(rI, rJ, rK, r) EC_to_SC km in frame IJK E12A 011. rI earth pos vector component EC to SC km frame IJK at UT time = -119085952.6308768100 E12B 011. rJ earth pos vector component EC to SC km frame IJK at UT time = -64886279.6011658240 E12C 011. rK earth pos vector component EC to SC km frame IJK at UT time = -56976844.6160544460 E12D 011. r earth pos vector magnitude EC to SC km frame IJK at UT time = 147098790.6710524900 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as that compued above in PQW frame. 13. Transform_2 Earth point EP(lat, log, hgt) To EC to SC Range Vector[RI, RJ, RK, R] in IJK frame. Inputs : earth equator radious_km, earth point EP(lat deg, log deg, hgt meter), LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time, Outputs : Vector(RI, RJ, RK, R) Range EC to EP in IJK frame E13A 011. RI pos vector component EC to EP km frame IJK at UT time = 4390.4491505062 E13B 011. RJ pos vector component EC to EP km frame IJK at UT time = -3876.8686176516 E13C 011. RK pos vector component EC to EP km frame IJK at UT time = 2517.6312937817 E13D 011. R pos vector magnitude EC to EP km frame IJK at UT time = 6375.3134317570 14. Transform_3 Earth position EC to SC Range Vectors [rI rJ rK] & [RI RJ RK] To EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame. Inputs : Vector(rI rJ rK) position EC_to_SC km in frame IJK , Vector(RI RJ RK) range EC to SC km in IJK frame, Outputs : Vector(rvI, rvJ, rvK, rv) range EP to SC in IJK frame E14A 011. rvI range vector component EP to SC km frame IJK at UT time = -119090343.0800273100 E14B 011. rvJ range vector component EP to SC km frame IJK at UT time = -64882402.7325481700 E14C 011. rvK range vector component EP to SC km frame IJK at UT time = -56979362.2473482270 E14D 011. rv range vector magnitude EP to SC km frame IJK at UT time = 147101610.1930285400
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OM-MSS Page 91 15. Transform_4 Earth point EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame To EP to SC Range Vector[rvS, rvE, rvZ] in SEZ frame. Inputs : lat_pos_neg_0_to_90_deg_at_EP_at_time_UT , LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time Vector(rvI, rvJ, rvK, rv) range EP to SC km in IJK frame, Outputs : Vector(rvS, rvE, rvZ, rv) range EP to SC km in SEZ frame E15A 011. rvS range vector component EP to SC km frame SEZ at UT time = 34055206.8052048240 E15B 011. rvE range vector component EP to SC km frame SEZ at UT time = -127461490.7613908900 E15C 011. rvZ range vector component EP to SC km frame SEZ at UT time = -65059165.2553297430 E15D 011. rv range vector magnitude EP to SC km frame SEZ at UT time = 147101610.1930285100 16. Finding Elevation(EL) and Azimuth(AZ) angle of Sun at Earth Observation point EP . Note : Results computed using 4 different formulations, each require different inputs to give EL & AZ angles. For all situations of Object and Observer positions, a combination of latitude N/S & longitude E/W : Method 1 : for both EL & AZ angles, this does not provide correct results ; Method 2 : for only EL angle, this provides consistent, unambiguous correct results. but for AZ angles the results are ambiguous, need corrections by adding or subtracting values as 180 or 360 or sign change. Method 3 : same as method 2, for EL angle, the results are correct, but for AZ angles the results are ambiguous, need corrections. Method 4 : for finding Azimuth and Distance but not for finding Elevation angle; for AZ angles, this provides correct unambiguous results that need no futher corrections. Therefore for Elevation (EL) angle Method 3 results are accepted and for Azimuth (AZ) angle Method 4 results are accepted . Results verified from other sources; Ref URLs http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone . NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , and http://aa.usno.navy.mil/data/docs/AltAz.php Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html Rem: SS point lat deg = -22.79, log deg = 43.15 YY = 2013, MM = 1, DD = 3, hr = 12, min = 4, sec = 32.72 EP point lat deg = 23.26, log deg = 77.41 YY = 2013, MM = 1, DD = 3, hr = 14, min = 21, sec = 35.64 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 1 - computed values may be Ambiguous or Incorrect). Inputs : Vector[rvS, rvE, rvZ] range EP to SC km in SEZ frame Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP
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OM-MSS Page 92 E16A 011. Elevation angle deg of Sun at EP using rv SEZ at UT time = -26.2490388254 E16B 011. Azimuth angle deg of sun at EP using rv SEZ at UT time = 255.0411079766 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 2 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun declination Delta Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16C 011. Elevation angle deg of Sun at EP using Sun declination diff log range EP to SC = 33.16712 E16D 011. Azimuth angle deg of sun at EP using sun declination diff log range EP to SC = -128.32056 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 3 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun hgt from EC, Sun range from EP (Sun hgt from EC = earth orbit radious EC to SC km ; Sun range from EP = rv range vector EP to SC km frame SEZ) Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16E 011. Elevation angle deg of Sun at EP using Sun hgt diff log range EP to SC = 33.16375 ie deg = 33, min = 9, sec = 49.51 E16F 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 218.31935 ie deg = 218, min = 19, sec = 9.66 Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 4 - computed AZ values is unambiguous & correct). Inputs : Time input UT YY MM DD HH, EP lat & log, SS lat & log Outputs : Azimuth(AZ) of Sun at EP E16G 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 218.31935 ie deg = 218, min = 19, sec = 9.66 Due to such incorrect results, finally for Elevation (EL) Method 3 results and for Azimuth (AZ) Method 4 results are accepted. Finally accepted Elevation angle deg of Sun from EP to SC = 33.1637517343 ie deg = 33, min = 9, sec = 49.51 Finally accepted Azimuth angle_deg of Sun from EP to SC = 218.3193510511 ie deg = 218, min = 19, sec = 9.66
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OM-MSS Page 93 Distance in km from Earth observation point(EP) to Sub Sun point(SS) and Earth Velocity meter per sec in orbit at time input UT. 17. Finding Distance in km from Earth observation point(EP) to Sub Sun point(SS) over Earth surface . Inputs : EP lat & log, SS lat & log, Outputs : Distance in km from EP to SS over Earth surface E17A 011. Distance in km Earth observation point(EP) to Sub Sun point(SS) = 6326.74270 Finding Earth Velocity meter per sec in orbit in frame PQW Inputs : semi-major axis SMA, GM_Sun, earth pos r EC to SC frame IJK, eccentricity of earth orbit e_Sun, sun eccentric anomaly E_Sun Outputs : Earth Velocity magnitude and component Xw Yw in frame PQW in meter per sec E17B 011. Velocity magnitude meter per sec using GM, SMA, r earth EC to SC frame IJK at UT time = 30286.5538639082 E17C 011. Velocity component meter per sec in orbit Xw using GM, e_Sun, SMA, E_Sun at UT time = -0.0000000000 E17D 011. Velocity component meter per sec in orbit Yw using GM, e_Sun, SMA, E_Sun at UT time = 30286.5538639082 Finding Earth Velocity Vector [vX, vY, vZ] in meter per sec in orbit; a Transform of [Xw, Yw] in frame PQW To [vX, vY, vZ] in frame XYZ Inputs : velocity component (Xw, Yw), sun right ascension Alpha, Sun Argument of perigee W_Sun, inclination Epcylone Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ E17E 011. vX earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 13409.1729584945 E17F 011. vY earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = -27017.6074608137 E17G 011. vZ earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 2741.9541597144 E17H 011. vR earth Velocity magnitude meter per sec using Xw Yw frame PQW RA w i at UT time = 30286.5538639082
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OM-MSS Page 94 Earth State Vectors : Position [X, Y, Z] in km and Velocity [Vx, Vy, Vz] in meter per sec, at time input UT. 18. Finding Earth State Position Vector [X, Y, Z] in km at time input UT. Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ E18A 011. State vector position X km at UT time = -119085952.6308768100 E18B 011. State vector position Y km at UT time = -64886279.6011658240 E18C 011. State vector position Z km at UT time = -56976844.6160544460 E18D 011. State vector position R km at UT time = 147098790.6710524900 19. Finding Earth State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ E19A 011. State vector velocity Vx meter per sec at UT time = 13409.1729584945 E19B 011. State vector velocity Vy meter per sec at UT time = -27017.6074608137 E19C 011. State vector velocity Vz meter per sec at UT time = 2741.9541597144 019D 011. State vector velocity V meter per sec at UT time = 30286.5538639082 20. Earth Orbit Normal Vector [Wx, Wy, Wz] in km and angles Delta, i, RA at time input UT; Normal is line perpendicular to orbit plane. Inputs : earth pos r EC to SC frame IJK, inclination Epcylone, sun right ascension Alpha Outputs : earth orbit normal vector (Wx, Wy, Wz, W) in km E20A 011. Earth orbit normal W km using r earth pos frame IJK inclination Alpha = 147098790.6710524900 E20B 011. Earth orbit normal Wx km using r earth pos frame IJK inclination Alpha = -56701506.3572354090 E20C 011. Earth orbit normal Wy km using r earth pos frame IJK inclination Alpha = -14444830.6291954550 E20D 011. Earth orbit normal Wz km using r earth pos frame IJK inclination Alpha = 134960513.7134575500 020E 011. Earth orbit normal Delta W deg using r earth pos frame IJK inclination Alpha = 66.5607205617 E20F 011. Earth orbit normal Inclination i deg using normal_Delta_W = 23.4392794383 E20G 011. Earth orbit normal Alpha W deg using r earth pos frame IJK, inclination, Alpha = 14.2922173002 E20H 011. Earth orbit normal Right ascension of ascending node using normal Alpha, W = 104.2922173002
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OM-MSS Page 95 Transform Earth State Vectors to Earth position Keplerian elements. 21. Finding Earth position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness E21A 011. Keplerian elements year = 2013, days_decimal_of_year = 2.38329, revolution no = 1, node = 2 ie decending E21B 011. inclination_deg = 23.4392794383 E21C 011. right ascension ascending node deg = 284.2922173002 E21D 011. eccentricity = 0.0167102190 E21E 011. argument of perigee_deg = 283.1557666033 E21F 011. mean anomaly deg = 0.0000000000 E21G 011. mean_motion rev per day = 0.0027377786 E21H 011. mean angular velocity rev_per_day = 0.0027377786 E21I 011. mean motion rev per day using SMA considering oblateness = 0.0027377786 Transform Earth position Keplerian elements to Earth State Vectors . 22. Finding Earth position State Vectors, computed using Keplerian elements at time input UT (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] E22A 011. State vectors year = 2013, days_decimal_of_year = 2.38329, revolution no = 1, node = 2 ie decending E22B 011. state vector position X km = -119085952.6308767500, state vector velocity Vx meter per sec = 13409.1729584945 E22C 011. state vector position Y km = -64886279.6011657040, state vector velocity Vy meter per sec = -27017.6074608137 E22D 011. state vector position Z km = -56976844.6160543860, state vector velocity Vz meter per sec = 2741.9541597144 E22E 011. state vector position R km = 147098790.6710523700, state vector velocity V meter per sec = 30286.5538639082
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OM-MSS Page 96 Note : Computation of all above parameters, grouped in 1 to 22, corresponds to time (a) Universal time over Greenwich (UT/GMT) : Year = 2013, Month = 1, Day = 3, Hour = 9, Min = 11, Sec = 56.616 (b) Mean Solar time (MST) over Earth Observation point (EP ) : Year = 2013, Month = 1, Day = 3, Hour = 14, Min = 21, Sec = 35.644 Move on to next Astronomical event in orbit Earth around Sun. Next Section - 4.3 Position of earth at time when earth is at vernal equinox
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OM-MSS Page 97 OM-MSS Section - 4.3 ---------------------------------------------------------------------------------------------------37 Earth Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Vernal equinox. 3. Find Position of Earth on Celestial Sphere at Input Universal Time, when Earth is at Vernal equinox . Input UT Time, when Earth is at Vernal equinox : year = 2013, month = 3, day = 20, hour = 11, minute = 2, seconds = 9.15719 Julian Day = 2456371.95983, year_day_decimal = 78.45983, day_hour_decimal = 11.03588 Observation Point on Earth (Bhopal, India) : Lat +ve or -ve 0 to 90 deg = 23.25993 ie deg = 23, min = 15, sec = 35.76 Log 0 to 360 deg = 77.41261 ie deg = 77, min = 24, sec = 45.41 Alt from earth surface in km = 0.49470 First Compute the Sun Position on Celestial Sphere, then Compute the Earth Position on Celestial Sphere. (A) Computed Values for SUN POSITION on Celestial Sphere at Input Ut Time : (Sr. No 1 - 22) 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = 76.0765340370 07. Mean anomaly in deg per day from n_sun (m sun) = 74.9810547699 08. Sun Mean longitude in deg (Lmean) = 358.1404045779 09. Earth Mean anomaly in deg (ME) = 74.9810547700 10. Sun Ecliptic longitude in deg (Lsun) = 0.0000000000 11. Obliquity of ecliptic in deg (Epcylone) = 23.4375603478 12. Sun Right ascension in deg (Alpha) = 0.0000000000 13. Sun Declination in deg (Delta) = 0.0000000000 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 148989898.67840 16. Sun Nodal elongation in deg (U sun) = 0.0000000000 17. Sun Mean anomaly in deg (M sun) = 74.9810547697 18. Sun Eccentric anomaly in deg (E sun) = 75.9096738744 19. Sun True anomaly in deg (T sun) = 76.8402303407 20. Sun Argument of perigee in deg (W sun) = 283.1597696594 21. Sun True anomaly in deg from U & W (V sun) = 76.8402303407 22. Sun Distance in km (d sun) = 148912015.96700 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) These Values are applied as input for Computing Earth Position on Celestial Sphere around Sun at same input UT Time.
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OM-MSS Page 98 (B) Computed Values for EARTH POSITION on Celestial Sphere around Sun at same Input Ut Time : (Sr. No 1 - 22) Input Time year = 2013, month = 3, day = 20, hour = 11, minute = 2, seconds = 9.15719, corresponding Julian Day = 2456371.9598282082 Observation Point on Earth : Lat +ve or -ve 0 to 90 deg = 23.25993, Log 0 to 360 deg = 77.41261, Alt from earth surface in km = 0.49470 Sun position on Celestial sphere at input time, computed above total 22 parameters. Output Earth Position on Celestial sphere around Sun : Computed below around 120 parameters, presented in 1-22 groups. Number is large, because some parameters are computed using more than one model equation, that require different inputs. This helps in validation of results and understanding the different input considerations. 01. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input UT time YY MM DD HH. Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time UT year = 2013, month = 3, day = 20, hour = 11, minute = 2, seconds = 9.15719 Outputs : GST & GHA in 0-360 deg over Greenwich. E01A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 343.67866, hr = 22, min = 54, sec = 42.87769 E01B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 343.68516, deg = 343, min = 41, sec = 6.56735 02. Finding Earth latitude & longitude pointing to Sun Ecliptic longitude(Lsun). Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : Earth lat & log pointing to Lsun. E02A 011. Earth latitude +ve or -ve in 0 to 90 deg at UT time = 0.00 ie deg = 0, min = 0, sec = 0.00 E02B 011. Earth longitude 0 to 360 deg = 16.32 ie deg = 16, min = 19, sec = 16.83 03. Finding LST over three longitudes, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - for LST, used sidereal time at Greenwich GST and desired geogrphic longitude Inputs : At Time input UT - GST, Log of Greenwich, sun mean log Lmean, Sun ecliptic log Lsun. Outputs : LST over Greenwich, Lmean, Lsun . E03A 011. LST Local sidereal time in 0-360 deg, over Greenwich longitude = 343.67866, hr = 22, min = 54, sec = 42.87769
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OM-MSS Page 99 E03B 011. LST Local sidereal time in 0-360 deg, over Sun mean longitude (Lmean) = 341.81906, hr = 22, min = 47, sec = 16.57479 E03C 011. LST Local sidereal time in 0-360 deg, over Sun epliptic longitude (Lsun) = 343.67866, hr = 22, min = 54, sec = 42.87769 04. Finding ST0 sidereal time over Greenwich longitude = 0.0, at time input Year JAN day 1 hr 00. Note - this is sidereal time ST at UT year, month = 1, day = 1, hours decimal = 0.0 and geogrphic longitude = 0.0 Inputs : Time input UT Year, JAN day 1 hr 00, Log 0.0 Outputs : ST0 over Greenwich E04 011. ST0 Sidereal time in 0-360 deg, over Greenwich at input UT year, MM 1, DD 1, HH 00 = 100.80678, hr = 6, min = 43, sec = 13.62710 05. Finding ST sidereal time over three longitudes of, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - this is local sidereal time LST; (LST = GST at UT time + geogrphic longitude). Inputs : At Time input UT - Log 0.0, Log Lmean, Log Lsun Outputs : ST over Greenwich, Lmean, Lsun. E05A 011. ST Sidereal time in 0-360 deg, over Greenwich at input UT time = 343.67866, hr = 22, min = 54, sec = 42.87769 E05B 011. ST Sidereal time in 0-360 deg, over Sun mean longitude (Lmean) at input UT time = 341.81906, hr = 22, min = 47, sec = 16.57479 005C 011. ST Sidereal time in 0-360 deg, over Sun longitude (Lsun) at input UT time = 343.67866, hr = 22, min = 54, sec = 42.87769 06. Finding H hour angle in 0 to 360 deg over longitudes of, Greenwich, Lmean, Lsun, Earth Sub Sun point SS, Earth Obseration point EP. Note - used Sun Right ascension Alpha at input time; (hour angle HA = LST - Alpha). Inputs : At Time input UT - Sun Right ascension Alpha and ST Sidereal time over longitudes 0.0, Lmean, Lsun, SS, EP Outputs : Hour Angles over Greenwich, Lmean, Lsun, SS, EP E06A 011. H hour angle 0-360 deg, over Greenwich, = 343.67866, deg = 343, min = 40, sec = 43.16538 E06B 011. H hour angle 0-360 deg, over Lmean, = 341.81906, deg = 341, min = 49, sec = 8.62186 E06C 011. H hour angle 0-360 deg, over Lsun, = 343.67866, deg = 343, min = 40, sec = 43.16538 E06D 011. H hour angle 0-360 deg, over SS, = 0.00000, deg = 0, min = 0, sec = 0.00000 E06E 011. H hour angle 0-360 deg, over EP, = 61.09127, deg = 61, min = 5, sec = 28.57938
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OM-MSS Page 100 07. Finding Delta E is Equation of Time in seconds, at time input UT. Note - this value in seconds accounts for relative movement of sun in elliptical orbit w.r.t earth and effect of obliquity of the ecliptic; its maximum value is 16 minutes (960 sec.); Delta E is computed using time in days from the perihelion, n_sun_deg and w_sun at input UT. Inputs : Time input UT in JD, time perihelion in JD, Sun mean movement n_sun, Eccentricity of earth orbit E_Sun Outputs : Delta E time_equation in seconds. E07 011. Delta E Time Equation in seconds = 482.67883, hr = 0, min = 8, sec = 2.67883 08. Finding GST Greenwich sidereal time, and GHA Greenwich hour angle 0 to 360 deg at time when earth is at perihelion. Inputs : Time in JD when earth at perihelion YY = 2013, MM = 1, DD = 3, hr = 9, min = 11, sec = 56.62 Outputs : GST & GHA in 0-360 deg over Greenwich when earth is at perihelion E08A 011. GST sidereal time in 0-360 deg over Greenwich at time when earth is at perihelion = 241.14177, hr = 16, min = 4, sec = 34.02452 E08B 011. GHA hour angle in 0-360 deg over Greenwich at time when earth is at perihelion = 241.14717, hr = 16, min = 4, sec = 35.32064 09. Finding ST sidereal time and MST mean sidereal time, over Greenwich, using Earth mean motion rev per day . Inputs : GST when earth at perihelion, earth rotation rate, ref. JD2000, time input UT in JD, time perihelion in JD. Outputs : STP, angle perihelion to input JD, ST over Greenwich, MSTO & MST over Greenwich, solar time E09A 011. STP sidereal time in 0-360 deg over Greenwich when earth at perihelion = 241.14177, hr = 16, min = 4, sec = 34.02452 E09B 011. Angle in 0-360 deg from earth at perihelion to input JD using earth rotational rate = 102.53422, 009C 011. ST in 0-360 deg over Greenwich using STP and angle from perihelion at input JD = 343.67599, hr = 22, min = 54, sec = 42.23734 E09D 011. ST in 0-360 deg over Greenwich using STP and earth rotation at UT time = 343.67599, hr = 22, min = 54, sec = 42.23734 E09E 011. MST0 in deg, over Greenwich using JD century days, ref J2000 to I/P YY, M1, D1 hr 00 = 100.80714, hr = 6, min = 43, sec = 13.71450 E09F 011. MST in deg, over Greenwich using JD century in days from ref J2000 to UT time Y M D H = 343.67902, hr = 22, min = 54, sec = 42.96527 E09G 011. Solar time over Greenwich in JD (GMT or input UT - 12 hr) = YY 2013, MM 3, DD 19, hr 23, min 2, sec 9.157, ie JD 2456371.45983 10. Finding Earth orbit radious using true anomaly, Sub Sun point (SS) on earth surface and related paramters . (a) Finding Earth orbit radious using true anomaly. Inputs : semi-major axis SMA, eccentricity of earth orbit e_sun, sun true anomaly T_Sun Outputs : earth orbital radious EC to SC (earth center to sun center)
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OM-MSS Page 101 E10A 011. earth orbital radious EC to SC km using_true anomaly at UT time = 148990030.6242361400 (b) Finding Sub Sun point (SS) over earth surface (Latitude, Longitude, & Latitude radius) pointing to Sun Ecliptic Log (Lsun), Sun height from earth surface over SS, and LST over SS log at time input UT. Note - for SS Latitude, used earth inclination, sun true anomaly T_sun and sun argument of perigee w_sun. for SS Longitude, used Sun right ascension Alpha and sidereal time at Greenwich GST. Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : SS point Latitude, Longitude, Latitude radious, LST & LMT over SS . E10B 011. SS point Latitude +ve or -ve in 0 to 90 deg at UT time = 0.00 ie deg = 0, min = 0, sec = 0.00 E10C 011. SS point Longitude 0 to 360 deg = 16.32 ie deg = 16, min = 19, sec = 16.83 E10D 011. SS point Latitude radious km at UT time = 6378.1440000000 E10E 011. Sun height km from earth surface over SS at UT time = 148983652.4802361400 E10F 011. LST local sidereal time in 0-360 deg over SS log at UT time, (LST = GST + log east) = 0.000 ie hr = 0, min = 0, sec = 0.08759 LST local sidereal time and LMT local mean time with date adjusted to calendar YY MM DD and UT hr mm sec. E10G 011. LST local sidereal time at Sub Sun point (SS) YY = 2013, MM = 3, DD = 21, hr = 0, min = 0, sec = 0.09 E10H 011. LMT local Mean time at Sub Sun point (SS) YY = 2013, MM = 3, DD = 20, hr = 12, min = 7, sec = 26.28 (c) Finding LST and LMT over Earth point(EP) where Observer is, at time input UT. Inputs : EP point Latitude, Longitude Outputs : LST & LMT over EP . E10I 011. LST local sidereal time in 0-360 deg at EP log at UT time, (LST = GST + log east) = 61.092 ie hr = 4, min = 4, sec = 21.99289 LST and LMT with date adjusted to calendar YY MM DD and UT hr mm sec. E10J 011. LST local sidereal time at Earth point (EP) YY = 2013, MM = 3, DD = 21, hr = 4, min = 4, sec = 21.99 E10K 011. LMT local Mean time at Earth point (EP) YY = 2013, MM = 3, DD = 20, hr = 16, min = 11, sec = 48.18
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OM-MSS Page 102 Finding Earth to Sun Position Vectors coordinate in PQW, IJK, SEZ frames and the Vector Coordinate Transforms. First defined coordinate systems, PQW, IJK, SEZ, then computed Position & Velocity vectors in these three coordinate systems. (a) Perifocal Coordinate System (PQW), is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. The system is fixed with time (inertial), pointing towards orbit periapsis; the system's origin is Earth center (EC), and its fundamental plane is the orbit plane; the P-vector axis directed from EC toward the periapsis of the elliptical orbit plane, the Q-vector axis swepts 90 deg from P axis in the direction of the orbit, the W-vector axis directed from EC in a direction normal to orbit plane, forms a right-handed coordinate system. (b) Geocentric Coordinate System (IJK), is also an Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). The system is fixed with time (inertial), pointing towards vernal equinox; the system's origin is Earth center (EC), and its fundamental plane is the equator; the I-vector is +X-axis directed towards the vernal equinox direction on J2000, Jan 1, hr 12.00 noon, the J-vector is +Y-axis swepts 90 deg to the east in the equatorial plane, the K-vector is +Z-axis directed towards the North Pole. (c) Topocentric Horizon Coordinate System (SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). The system moves with earth, is not fixed with time (non-inertial), is for use by observers on the surface of earth; the observer's surface forms the fundamental plane, is tangent to earth's surface the S-vector is +ve horizontal-axis directed towards South, the E-vector is +ve horizontal-axis directed towards East, the Z-vector is +ve normal directed upwards on earth surface. Note that axis Z not necessarily pass through earth center, so not used to define as radious vector. 11. Finding Earth center(EC) to Sun center(SC) Range Vector[rp, rq, r] from in PQW frame, perifocal coordinate system. Inputs : Semi-major axis (SMA), Eccentricity of earth orbit (e_sun), Sun eccentric anomaly (E_sun) Outputs : Vector(r, rp rq) in PQW frame E11A 011. r earth pos vector magnitude EC to SC km in PQW frame perifocal cord at UT time = 148990030.62424
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OM-MSS Page 103 E11B 011. rp earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = 33920145.1258548950 E11C 011. rq earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = 145077403.4095308500 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as earth orbital radious computed before using true anomaly. 12. Transform_1 Earth position EC to SC Range Vector[rp, rq] in PQW frame To Range Vector[rI, rJ, rK] in IJK frame, inertial system cord. Inputs : Vector(rp, rq) EC_to_SC km in frame PQW , Alpha rd, w_sun rd, earth_inclination rd , Outputs : Vector(rI, rJ, rK, r) EC_to_SC km in frame IJK E12A 011. rI earth pos vector component EC to SC km frame IJK at UT time = 148990030.6242361100 E12B 011. rJ earth pos vector component EC to SC km frame IJK at UT time = 0.0001317672 E12C 011. rK earth pos vector component EC to SC km frame IJK at UT time = 0.0000285525 E12D 011. r earth pos vector magnitude EC to SC km frame IJK at UT time = 148990030.6242361100 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as that compued above in PQW frame. 13. Transform_2 Earth point EP(lat, log, hgt) To EC to SC Range Vector[RI, RJ, RK, R] in IJK frame. Inputs : earth equator radious_km, earth point EP(lat deg, log deg, hgt meter), LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time, Outputs : Vector(RI, RJ, RK, R) Range EC to EP in IJK frame E13A 011. RI pos vector component EC to EP km frame IJK at UT time = 2831.4036789008 E13B 011. RJ pos vector component EC to EP km frame IJK at UT time = 5127.3099407798 E13C 011. RK pos vector component EC to EP km frame IJK at UT time = 2517.6312937817 E13D 011. R pos vector magnitude EC to EP km frame IJK at UT time = 6375.3134317570 14. Transform_3 Earth position EC to SC Range Vectors [rI rJ rK] & [RI RJ RK] To EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame. Inputs : Vector(rI rJ rK) position EC_to_SC km in frame IJK , Vector(RI RJ RK) range EC to SC km in IJK frame, Outputs : Vector(rvI, rvJ, rvK, rv) range EP to SC in IJK frame E14A 011. rvI range vector component EP to SC km frame IJK at UT time = 148987199.2205572100 E14B 011. rvJ range vector component EP to SC km frame IJK at UT time = -5127.3098090126 E14C 011. rvK range vector component EP to SC km frame IJK at UT time = -2517.6312652292 E14D 011. rv range vector magnitude EP to SC km frame IJK at UT time = 148987199.3300558000
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OM-MSS Page 104 15. Transform_4 Earth point EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame To EP to SC Range Vector[rvS, rvE, rvZ] in SEZ frame. Inputs : lat_pos_neg_0_to_90_deg_at_EP_at_time_UT , LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time Vector(rvI, rvJ, rvK, rv) range EP to SC km in IJK frame, Outputs : Vector(rvS, rvE, rvZ, rv) range EP to SC km in SEZ frame E15A 011. rvS range vector component EP to SC km frame SEZ at UT time = 28442225.2836758790 E15B 011. rvE range vector component EP to SC km frame SEZ at UT time = -130424975.4876815500 E15C 011. rvZ range vector component EP to SC km frame SEZ at UT time = 66163064.8788587080 E15D 011. rv range vector magnitude EP to SC km frame SEZ at UT time = 148987199.3300557700 16. Finding Elevation(EL) and Azimuth(AZ) angle of Sun at Earth Observation point EP . Note : Results computed using 4 different formulations, each require different inputs to give EL & AZ angles. For all situations of Object and Observer positions, a combination of latitude N/S & longitude E/W : Method 1 : for both EL & AZ angles, this does not provide correct results ; Method 2 : for only EL angle, this provides consistent, unambiguous correct results. but for AZ angles the results are ambiguous, need corrections by adding or subtracting values as 180 or 360 or sign change. Method 3 : same as method 2, for EL angle, the results are correct, but for AZ angles the results are ambiguous, need corrections. Method 4 : for finding Azimuth and Distance but not for finding Elevation angle; for AZ angles, this provides correct unambiguous results that need no futher corrections. Therefore for Elevation (EL) angle Method 3 results are accepted and for Azimuth (AZ) angle Method 4 results are accepted . Results verified from other sources; Ref URLs http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone . NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , and http://aa.usno.navy.mil/data/docs/AltAz.php Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html Rem: SS point lat deg = 0.00, log deg = 16.32 YY = 2013, MM = 3, DD = 20, hr = 12, min = 7, sec = 26.28 EP point lat deg = 23.26, log deg = 77.41 YY = 2013, MM = 3, DD = 20, hr = 16, min = 11, sec = 48.18 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 1 - computed values may be Ambiguous or Incorrect). Inputs : Vector[rvS, rvE, rvZ] range EP to SC km in SEZ frame Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP
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OM-MSS Page 105 E16A 011. Elevation angle deg of Sun at EP using rv SEZ at UT time = 26.3648484347 E16B 011. Azimuth angle deg of sun at EP using rv SEZ at UT time = 257.6979108329 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 2 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun declination Delta Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16C 011. Elevation angle deg of Sun at EP using Sun declination diff log range EP to SC = 26.36737 E16D 011. Azimuth angle deg of sun at EP using sun declination diff log range EP to SC = -167.69773 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 3 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun hgt from EC, Sun range from EP (Sun hgt from EC = earth orbit radious EC to SC km ; Sun range from EP = rv range vector EP to SC km frame SEZ) Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16E 011. Elevation angle deg of Sun at EP using Sun hgt diff log range EP to SC = 26.36517 ie deg = 26, min = 21, sec = 54.63 E16F 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 257.69773 ie deg = 257, min = 41, sec = 51.83 Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 4 - computed AZ values is unambiguous & correct). Inputs : Time input UT YY MM DD HH, EP lat & log, SS lat & log Outputs : Azimuth(AZ) of Sun at EP E16G 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 257.69773 ie deg = 257, min = 41, sec = 51.83 Due to such incorrect results, finally for Elevation (EL) Method 3 results and for Azimuth (AZ) Method 4 results are accepted. Finally accepted Elevation angle deg of Sun from EP to SC = 26.3651749316 ie deg = 26, min = 21, sec = 54.63 Finally accepted Azimuth angle_deg of Sun from EP to SC = 257.6977312752 ie deg = 257, min = 41, sec = 51.83
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OM-MSS Page 106 Distance in km from Earth observation point(EP) to Sub Sun point(SS) and Earth Velocity meter per sec in orbit at time input UT. 17. Finding Distance in km from Earth observation point(EP) to Sub Sun point(SS) over Earth surface . Inputs : EP lat & log, SS lat & log, Outputs : Distance in km from EP to SS over Earth surface E17A 011. Distance in km Earth observation point(EP) to Sub Sun point(SS) = 7083.54272 Finding Earth Velocity meter per sec in orbit in frame PQW Inputs : semi-major axis SMA, GM_Sun, earth pos r EC to SC frame IJK, eccentricity of earth orbit e_Sun, sun eccentric anomaly E_Sun Outputs : Earth Velocity magnitude and component Xw Yw in frame PQW in meter per sec E17B 011. Velocity magnitude meter per sec using GM, SMA, r earth EC to SC frame IJK at UT time = 29906.0325858305 E17C 011. Velocity component meter per sec in orbit Xw using GM, e_Sun, SMA, E_Sun at UT time = -29006.4938048237 E17D 011. Velocity component meter per sec in orbit Yw using GM, e_Sun, SMA, E_Sun at UT time = 7279.7048137597 Finding Earth Velocity Vector [vX, vY, vZ] in meter per sec in orbit; a Transform of [Xw, Yw] in frame PQW To [vX, vY, vZ] in frame XYZ Inputs : velocity component (Xw, Yw), sun right ascension Alpha, Sun Argument of perigee W_Sun, inclination Epcylone Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ E17E 011. vX earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 484.7048638742 E17F 011. vY earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 27434.6468079624 E17G 011. vZ earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 11894.3684465402 E17H 011. vR earth Velocity magnitude meter per sec using Xw Yw frame PQW RA w i at UT time = 29906.0325858305
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OM-MSS Page 107 Earth State Vectors : Position [X, Y, Z] in km and Velocity [Vx, Vy, Vz] in meter per sec, at time input UT. 18. Finding Earth State Position Vector [X, Y, Z] in km at time input UT. Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ E18A 011. State vector position X km at UT time = 148990030.6242361100 E18B 011. State vector position Y km at UT time = 0.0001317672 E18C 011. State vector position Z km at UT time = 0.0000285525 E18D 011. State vector position R km at UT time = 148990030.6242361100 19. Finding Earth State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ E19A 011. State vector velocity Vx meter per sec at UT time = 484.7048638742 E19B 011. State vector velocity Vy meter per sec at UT time = 27434.6468079624 E19C 011. State vector velocity Vz meter per sec at UT time = 11894.3684465402 019D 011. State vector velocity V meter per sec at UT time = 29906.0325858305 20. Earth Orbit Normal Vector [Wx, Wy, Wz] in km and angles Delta, i, RA at time input UT; Normal is line perpendicular to orbit plane. Inputs : earth pos r EC to SC frame IJK, inclination Epcylone, sun right ascension Alpha Outputs : earth orbit normal vector (Wx, Wy, Wz, W) in km E20A 011. Earth orbit normal W km using r earth pos frame IJK inclination Alpha = 148990030.6242361100 E20B 011. Earth orbit normal Wx km using r earth pos frame IJK inclination Alpha = 0.0000262177 E20C 011. Earth orbit normal Wy km using r earth pos frame IJK inclination Alpha = -59264802.7950436320 E20D 011. Earth orbit normal Wz km using r earth pos frame IJK inclination Alpha = 136695692.5988357700 020E 011. Earth orbit normal Delta W deg using r earth pos frame IJK inclination Alpha = 66.5607205617 E20F 011. Earth orbit normal Inclination i deg using normal_Delta_W = 23.4392794383 E20G 011. Earth orbit normal Alpha W deg using r earth pos frame IJK, inclination, Alpha = -90.0000000000 E20H 011. Earth orbit normal Right ascension of ascending node using normal Alpha, W = 0.0000000000
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OM-MSS Page 108 Transform Earth State Vectors to Earth position Keplerian elements. 21. Finding Earth position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness E21A 011. Keplerian elements year = 2013, days_decimal_of_year = 78.45983, revolution no = 1, node = 1 ie ascending E21B 011. inclination_deg = 23.4392794383 E21C 011. right ascension ascending node deg = 0.0000000000 E21D 011. eccentricity = 0.0167102190 E21E 011. argument of perigee_deg = 283.1597696594 E21F 011. mean anomaly deg = 74.9810547697 E21G 011. mean_motion rev per day = 0.0027377786 E21H 011. mean angular velocity rev_per_day = 0.0027377786 E21I 011. mean motion rev per day using SMA considering oblateness = 0.0027377786 Transform Earth position Keplerian elements to Earth State Vectors . 22. Finding Earth position State Vectors, computed using Keplerian elements at time input UT (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] E22A 011. State vectors year = 2013, days_decimal_of_year = 78.45983, revolution no = 1, node = 1 ie decending E22B 011. state vector position X km = 148990030.6242359900, state vector velocity Vx meter per sec = 484.7048638741 E22C 011. state vector position Y km = 0.0001320988, state vector velocity Vy meter per sec = 27434.6468079624 E22D 011. state vector position Z km = 0.0000286959, state vector velocity Vz meter per sec = 11894.3684465402 E22E 011. state vector position R km = 148990030.6242359900, state vector velocity V meter per sec = 29906.0325858305
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OM-MSS Page 109 Note : Computation of all above parameters, grouped in 1 to 22, corresponds to time (a) Universal time over Greenwich (UT/GMT) : Year = 2013, Month = 3, Day = 20, Hour = 11, Min = 2, Sec = 9.157 (b) Mean Solar time (MST) over Earth Observation point (EP ) : Year = 2013, Month = 3, Day = 20, Hour = 16, Min = 11, Sec = 48.185 Move on to next Astronomical event in orbit Earth around Sun. Next Section - 4.4 Position of earth at time when earth is at summer solstice
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OM-MSS Page 110 OM-MSS Section - 4.4 ---------------------------------------------------------------------------------------------------38 Earth Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Summer solstice. 4. Find Position of Earth on Celestial Sphere at Input Universal Time, when Earth is at Summer Solstice . Input UT Time, when Earth is at Summer solstice : year = 2013, month = 6, day = 21, hour = 5, minute = 1, seconds = 19.19999 Julian Day = 2456464.70925, year_day_decimal = 171.20925, day_hour_decimal = 5.02200 Observation Point on Earth (Bhopal, India) : Lat +ve or -ve 0 to 90 deg = 23.25993 ie deg = 23, min = 15, sec = 35.76 Log 0 to 360 deg = 77.41261 ie deg = 77, min = 24, sec = 45.41 Alt from earth surface in km = 0.49470 First Compute the Sun Position on Celestial Sphere, then Compute the Earth Position on Celestial Sphere. (A) Computed Values for SUN POSITION on Celestial Sphere at Input Ut Time : (Sr. No 1 - 22) 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = 168.8259558287 07. Mean anomaly in deg per day from n_sun (m sun) = 166.3949127126 08. Sun Mean longitude in deg (Lmean) = 89.5586310183 09. Earth Mean anomaly in deg (ME) = 166.3949127126 10. Sun Ecliptic longitude in deg (Lsun) = 89.9999483110 11. Obliquity of ecliptic in deg (Epcylone) = 23.4375273104 12. Sun Right ascension in deg (Alpha) = 89.9999436629 13. Sun Declination in deg (Delta) = 23.4375273104 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 152030583.04072 16. Sun Nodal elongation in deg (U sun) = 90.0000000000 17. Sun Mean anomaly in deg (M sun) = 166.3949127122 18. Sun Eccentric anomaly in deg (E sun) = 166.6165253213 19. Sun True anomaly in deg (T sun) = 166.8363660940 20. Sun Argument of perigee in deg (W sun) = 283.1636339060 21. Sun True anomaly in deg from U & W (V sun) = 166.8363660940 22. Sun Distance in km (d sun) = 152025947.60113 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) These Values are applied as input for Computing Earth Position on Celestial Sphere around Sun at same input UT Time.
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OM-MSS Page 111 (B) Computed Values for EARTH POSITION on Celestial Sphere around Sun at same Input Ut Time : (Sr. No 1 - 22) Input Time year = 2013, month = 6, day = 21, hour = 5, minute = 1, seconds = 19.19999, corresponding Julian Day = 2456464.7092499998 Observation Point on Earth : Lat +ve or -ve 0 to 90 deg = 23.25993, Log 0 to 360 deg = 77.41261, Alt from earth surface in km = 0.49470 Sun position on Celestial sphere at input time, computed above total 22 parameters. Output Earth Position on Celestial sphere around Sun : Computed below around 120 parameters, presented in 1-22 groups. Number is large, because some parameters are computed using more than one model equation, that require different inputs. This helps in validation of results and understanding the different input considerations. 01. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input UT time YY MM DD HH. Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time UT year = 2013, month = 6, day = 21, hour = 5, minute = 1, seconds = 19.19999 Outputs : GST & GHA in 0-360 deg over Greenwich. E01A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 344.88872, hr = 22, min = 59, sec = 33.29393 E01B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 344.89162, deg = 344, min = 53, sec = 29.83101 02. Finding Earth latitude & longitude pointing to Sun Ecliptic longitude(Lsun). Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : Earth lat & log pointing to Lsun. E02A 011. Earth latitude +ve or -ve in 0 to 90 deg at UT time = 23.44 ie deg = 23, min = 26, sec = 21.41 E02B 011. Earth longitude 0 to 360 deg = 105.11 ie deg = 105, min = 6, sec = 40.39 03. Finding LST over three longitudes, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - for LST, used sidereal time at Greenwich GST and desired geogrphic longitude Inputs : At Time input UT - GST, Log of Greenwich, sun mean log Lmean, Sun ecliptic log Lsun. Outputs : LST over Greenwich, Lmean, Lsun . E03A 011. LST Local sidereal time in 0-360 deg, over Greenwich longitude = 344.88872, hr = 22, min = 59, sec = 33.29393
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OM-MSS Page 112 E03B 011. LST Local sidereal time in 0-360 deg, over Sun mean longitude (Lmean) = 74.44736, hr = 4, min = 57, sec = 47.36538 E03C 011. LST Local sidereal time in 0-360 deg, over Sun epliptic longitude (Lsun) = 74.88867, hr = 4, min = 59, sec = 33.28153 04. Finding ST0 sidereal time over Greenwich longitude = 0.0, at time input Year JAN day 1 hr 00. Note - this is sidereal time ST at UT year, month = 1, day = 1, hours decimal = 0.0 and geogrphic longitude = 0.0 Inputs : Time input UT Year, JAN day 1 hr 00, Log 0.0 Outputs : ST0 over Greenwich E04 011. ST0 Sidereal time in 0-360 deg, over Greenwich at input UT year, MM 1, DD 1, HH 00 = 100.80678, hr = 6, min = 43, sec = 13.62710 05. Finding ST sidereal time over three longitudes of, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - this is local sidereal time LST; (LST = GST at UT time + geogrphic longitude). Inputs : At Time input UT - Log 0.0, Log Lmean, Log Lsun Outputs : ST over Greenwich, Lmean, Lsun. E05A 011. ST Sidereal time in 0-360 deg, over Greenwich at input UT time = 344.88872, hr = 22, min = 59, sec = 33.29393 E05B 011. ST Sidereal time in 0-360 deg, over Sun mean longitude (Lmean) at input UT time = 74.44736, hr = 4, min = 57, sec = 47.36538 005C 011. ST Sidereal time in 0-360 deg, over Sun longitude (Lsun) at input UT time = 74.88867, hr = 4, min = 59, sec = 33.28153 06. Finding H hour angle in 0 to 360 deg over longitudes of, Greenwich, Lmean, Lsun, Earth Sub Sun point SS, Earth Obseration point EP. Note - used Sun Right ascension Alpha at input time; (hour angle HA = LST - Alpha). Inputs : At Time input UT - Sun Right ascension Alpha and ST Sidereal time over longitudes 0.0, Lmean, Lsun, SS, EP Outputs : Hour Angles over Greenwich, Lmean, Lsun, SS, EP E06A 011. H hour angle 0-360 deg, over Greenwich, = 254.88878, deg = 254, min = 53, sec = 19.61183 E06B 011. H hour angle 0-360 deg, over Lmean, = 344.44741, deg = 344, min = 26, sec = 50.68350 E06C 011. H hour angle 0-360 deg, over Lsun, = 344.88873, deg = 344, min = 53, sec = 19.42575 E06D 011. H hour angle 0-360 deg, over SS, = 360.00000, deg = 360, min = 0, sec = 0.00000 E06E 011. H hour angle 0-360 deg, over EP, = 332.30140, deg = 332, min = 18, sec = 5.02583
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OM-MSS Page 113 07. Finding Delta E is Equation of Time in seconds, at time input UT. Note - this value in seconds accounts for relative movement of sun in elliptical orbit w.r.t earth and effect of obliquity of the ecliptic; its maximum value is 16 minutes (960 sec.); Delta E is computed using time in days from the perihelion, n_sun_deg and w_sun at input UT. Inputs : Time input UT in JD, time perihelion in JD, Sun mean movement n_sun, Eccentricity of earth orbit E_Sun Outputs : Delta E time_equation in seconds. E07 011. Delta E Time Equation in seconds = 99.08293, hr = 0, min = 1, sec = 39.08293 08. Finding GST Greenwich sidereal time, and GHA Greenwich hour angle 0 to 360 deg at time when earth is at perihelion. Inputs : Time in JD when earth at perihelion YY = 2013, MM = 1, DD = 3, hr = 9, min = 11, sec = 56.62 Outputs : GST & GHA in 0-360 deg over Greenwich when earth is at perihelion E08A 011. GST sidereal time in 0-360 deg over Greenwich at time when earth is at perihelion = 241.14177, hr = 16, min = 4, sec = 34.02452 E08B 011. GHA hour angle in 0-360 deg over Greenwich at time when earth is at perihelion = 241.14717, hr = 16, min = 4, sec = 35.32064 09. Finding ST sidereal time and MST mean sidereal time, over Greenwich, using Earth mean motion rev per day . Inputs : GST when earth at perihelion, earth rotation rate, ref. JD2000, time input UT in JD, time perihelion in JD. Outputs : STP, angle perihelion to input JD, ST over Greenwich, MSTO & MST over Greenwich, solar time E09A 011. STP sidereal time in 0-360 deg over Greenwich when earth at perihelion = 241.14177, hr = 16, min = 4, sec = 34.02452 E09B 011. Angle in 0-360 deg from earth at perihelion to input JD using earth rotational rate = 103.74103, 009C 011. ST in 0-360 deg over Greenwich using STP and angle from perihelion at input JD = 344.88280, hr = 22, min = 59, sec = 31.87289 E09D 011. ST in 0-360 deg over Greenwich using STP and earth rotation at UT time = 344.88280, hr = 22, min = 59, sec = 31.87289 E09E 011. MST0 in deg, over Greenwich using JD century days, ref J2000 to I/P YY, M1, D1 hr 00 = 100.80714, hr = 6, min = 43, sec = 13.71450 E09F 011. MST in deg, over Greenwich using JD century in days from ref J2000 to UT time Y M D H = 344.88909, hr = 22, min = 59, sec = 33.38172 E09G 011. Solar time over Greenwich in JD (GMT or input UT - 12 hr) = YY 2013, MM 6, DD 20, hr 17, min 1, sec 19.200, ie JD 2456464.20925 10. Finding Earth orbit radious using true anomaly, Sub Sun point (SS) on earth surface and related paramters . (a) Finding Earth orbit radious using true anomaly. Inputs : semi-major axis SMA, eccentricity of earth orbit e_sun, sun true anomaly T_Sun Outputs : earth orbital radious EC to SC (earth center to sun center)
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OM-MSS Page 114 E10A 011. earth orbital radious EC to SC km using_true anomaly at UT time = 152030553.3844403900 (b) Finding Sub Sun point (SS) over earth surface (Latitude, Longitude, & Latitude radius) pointing to Sun Ecliptic Log (Lsun), Sun height from earth surface over SS, and LST over SS log at time input UT. Note - for SS Latitude, used earth inclination, sun true anomaly T_sun and sun argument of perigee w_sun. for SS Longitude, used Sun right ascension Alpha and sidereal time at Greenwich GST. Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : SS point Latitude, Longitude, Latitude radious, LST & LMT over SS . E10B 011. SS point Latitude +ve or -ve in 0 to 90 deg at UT time = 23.44 ie deg = 23, min = 26, sec = 21.41 E10C 011. SS point Longitude 0 to 360 deg = 105.11 ie deg = 105, min = 6, sec = 40.39 E10D 011. SS point Latitude radious km at UT time = 6374.7700931266 E10E 011. Sun height km from earth surface over SS at UT time = 152024178.6143472800 E10F 011. LST local sidereal time in 0-360 deg over SS log at UT time, (LST = GST + log east) = 90.000 ie hr = 6, min = 0, sec = 0.07427 LST local sidereal time and LMT local mean time with date adjusted to calendar YY MM DD and UT hr mm sec. E10G 011. LST local sidereal time at Sub Sun point (SS) YY = 2013, MM = 6, DD = 22, hr = 6, min = 0, sec = 0.07 E10H 011. LMT local Mean time at Sub Sun point (SS) YY = 2013, MM = 6, DD = 21, hr = 12, min = 1, sec = 45.89 (c) Finding LST and LMT over Earth point(EP) where Observer is, at time input UT. Inputs : EP point Latitude, Longitude Outputs : LST & LMT over EP . E10I 011. LST local sidereal time in 0-360 deg at EP log at UT time, (LST = GST + log east) = 62.302 ie hr = 4, min = 9, sec = 12.40934 LST and LMT with date adjusted to calendar YY MM DD and UT hr mm sec. E10J 011. LST local sidereal time at Earth point (EP) YY = 2013, MM = 6, DD = 22, hr = 4, min = 9, sec = 12.41 E10K 011. LMT local Mean time at Earth point (EP) YY = 2013, MM = 6, DD = 21, hr = 10, min = 10, sec = 58.23
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OM-MSS Page 115 Finding Earth to Sun Position Vectors coordinate in PQW, IJK, SEZ frames and the Vector Coordinate Transforms. First defined coordinate systems, PQW, IJK, SEZ, then computed Position & Velocity vectors in these three coordinate systems. (a) Perifocal Coordinate System (PQW), is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. The system is fixed with time (inertial), pointing towards orbit periapsis; the system's origin is Earth center (EC), and its fundamental plane is the orbit plane; the P-vector axis directed from EC toward the periapsis of the elliptical orbit plane, the Q-vector axis swepts 90 deg from P axis in the direction of the orbit, the W-vector axis directed from EC in a direction normal to orbit plane, forms a right-handed coordinate system. (b) Geocentric Coordinate System (IJK), is also an Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). The system is fixed with time (inertial), pointing towards vernal equinox; the system's origin is Earth center (EC), and its fundamental plane is the equator; the I-vector is +X-axis directed towards the vernal equinox direction on J2000, Jan 1, hr 12.00 noon, the J-vector is +Y-axis swepts 90 deg to the east in the equatorial plane, the K-vector is +Z-axis directed towards the North Pole. (c) Topocentric Horizon Coordinate System (SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). The system moves with earth, is not fixed with time (non-inertial), is for use by observers on the surface of earth; the observer's surface forms the fundamental plane, is tangent to earth's surface the S-vector is +ve horizontal-axis directed towards South, the E-vector is +ve horizontal-axis directed towards East, the Z-vector is +ve normal directed upwards on earth surface. Note that axis Z not necessarily pass through earth center, so not used to define as radious vector. 11. Finding Earth center(EC) to Sun center(SC) Range Vector[rp, rq, r] from in PQW frame, perifocal coordinate system. Inputs : Semi-major axis (SMA), Eccentricity of earth orbit (e_sun), Sun eccentric anomaly (E_sun) Outputs : Vector(r, rp rq) in PQW frame E11A 011. r earth pos vector magnitude EC to SC km in PQW frame perifocal cord at UT time = 152030553.38444
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OM-MSS Page 116 E11B 011. rp earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = -148035744.2735802800 E11C 011. rq earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = 34622356.6463585200 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as earth orbital radious computed before using true anomaly. 12. Transform_1 Earth position EC to SC Range Vector[rp, rq] in PQW frame To Range Vector[rI, rJ, rK] in IJK frame, inertial system cord. Inputs : Vector(rp, rq) EC_to_SC km in frame PQW , Alpha rd, w_sun rd, earth_inclination rd , Outputs : Vector(rI, rJ, rK, r) EC_to_SC km in frame IJK E12A 011. rI earth pos vector component EC to SC km frame IJK at UT time = -139485317.9369729200 E12B 011. rJ earth pos vector component EC to SC km frame IJK at UT time = 137.1515086517 E12C 011. rK earth pos vector component EC to SC km frame IJK at UT time = 60474252.7228181590 E12D 011. r earth pos vector magnitude EC to SC km frame IJK at UT time = 152030553.3844447700 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as that compued above in PQW frame. 13. Transform_2 Earth point EP(lat, log, hgt) To EC to SC Range Vector[RI, RJ, RK, R] in IJK frame. Inputs : earth equator radious_km, earth point EP(lat deg, log deg, hgt meter), LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time, Outputs : Vector(RI, RJ, RK, R) Range EC to EP in IJK frame E13A 011. RI pos vector component EC to EP km frame IJK at UT time = 2722.4931452498 E13B 011. RJ pos vector component EC to EP km frame IJK at UT time = 5185.9603831673 E13C 011. RK pos vector component EC to EP km frame IJK at UT time = 2517.6312937817 E13D 011. R pos vector magnitude EC to EP km frame IJK at UT time = 6375.3134317570 14. Transform_3 Earth position EC to SC Range Vectors [rI rJ rK] & [RI RJ RK] To EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame. Inputs : Vector(rI rJ rK) position EC_to_SC km in frame IJK , Vector(RI RJ RK) range EC to SC km in IJK frame, Outputs : Vector(rvI, rvJ, rvK, rv) range EP to SC in IJK frame E14A 011. rvI range vector component EP to SC km frame IJK at UT time = -139488040.4301181700 E14B 011. rvJ range vector component EP to SC km frame IJK at UT time = -5048.8088745156 E14C 011. rvK range vector component EP to SC km frame IJK at UT time = 60471735.0915243770 E14D 011. rv range vector magnitude EP to SC km frame IJK at UT time = 152032049.8891739000
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OM-MSS Page 117 15. Transform_4 Earth point EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame To EP to SC Range Vector[rvS, rvE, rvZ] in SEZ frame. Inputs : lat_pos_neg_0_to_90_deg_at_EP_at_time_UT , LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time Vector(rvI, rvJ, rvK, rv) range EP to SC km in IJK frame, Outputs : Vector(rvS, rvE, rvZ, rv) range EP to SC km in SEZ frame E15A 011. rvS range vector component EP to SC km frame SEZ at UT time = -81162555.3582670990 E15B 011. rvE range vector component EP to SC km frame SEZ at UT time = 123501405.2118426000 E15C 011. rvZ range vector component EP to SC km frame SEZ at UT time = -35690148.6676749360 E15D 011. rv range vector magnitude EP to SC km frame SEZ at UT time = 152032049.8891738700 16. Finding Elevation(EL) and Azimuth(AZ) angle of Sun at Earth Observation point EP . Note : Results computed using 4 different formulations, each require different inputs to give EL & AZ angles. For all situations of Object and Observer positions, a combination of latitude N/S & longitude E/W : Method 1 : for both EL & AZ angles, this does not provide correct results ; Method 2 : for only EL angle, this provides consistent, unambiguous correct results. but for AZ angles the results are ambiguous, need corrections by adding or subtracting values as 180 or 360 or sign change. Method 3 : same as method 2, for EL angle, the results are correct, but for AZ angles the results are ambiguous, need corrections. Method 4 : for finding Azimuth and Distance but not for finding Elevation angle; for AZ angles, this provides correct unambiguous results that need no futher corrections. Therefore for Elevation (EL) angle Method 3 results are accepted and for Azimuth (AZ) angle Method 4 results are accepted . Results verified from other sources; Ref URLs http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone . NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , and http://aa.usno.navy.mil/data/docs/AltAz.php Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html Rem: SS point lat deg = 23.44, log deg = 105.11 YY = 2013, MM = 6, DD = 21, hr = 12, min = 1, sec = 45.89 EP point lat deg = 23.26, log deg = 77.41 YY = 2013, MM = 6, DD = 21, hr = 10, min = 10, sec = 58.23 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 1 - computed values may be Ambiguous or Incorrect). Inputs : Vector[rvS, rvE, rvZ] range EP to SC km in SEZ frame Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP
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OM-MSS Page 118 E16A 011. Elevation angle deg of Sun at EP using rv SEZ at UT time = -13.5771288889 E16B 011. Azimuth angle deg of sun at EP using rv SEZ at UT time = 56.6879146981 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 2 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun declination Delta Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16C 011. Elevation angle deg of Sun at EP using Sun declination diff log range EP to SC = 64.60870 E16D 011. Azimuth angle deg of sun at EP using sun declination diff log range EP to SC = 5.97289 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 3 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun hgt from EC, Sun range from EP (Sun hgt from EC = earth orbit radious EC to SC km ; Sun range from EP = rv range vector EP to SC km frame SEZ) Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16E 011. Elevation angle deg of Sun at EP using Sun hgt diff log range EP to SC = 64.60783 ie deg = 64, min = 36, sec = 28.18 E16F 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 275.97696 ie deg = 275, min = 58, sec = 37.06 Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 4 - computed AZ values is unambiguous & correct). Inputs : Time input UT YY MM DD HH, EP lat & log, SS lat & log Outputs : Azimuth(AZ) of Sun at EP E16G 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 84.02304 ie deg = 84, min = 1, sec = 22.94 Due to such incorrect results, finally for Elevation (EL) Method 3 results and for Azimuth (AZ) Method 4 results are accepted. Finally accepted Elevation angle deg of Sun from EP to SC = 64.6078265075 ie deg = 64, min = 36, sec = 28.18 Finally accepted Azimuth angle_deg of Sun from EP to SC = 84.0230377796 ie deg = 84, min = 1, sec = 22.94
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OM-MSS Page 119 Distance in km from Earth observation point(EP) to Sub Sun point(SS) and Earth Velocity meter per sec in orbit at time input UT. 17. Finding Distance in km from Earth observation point(EP) to Sub Sun point(SS) over Earth surface . Inputs : EP lat & log, SS lat & log, Outputs : Distance in km from EP to SS over Earth surface E17A 011. Distance in km Earth observation point(EP) to Sub Sun point(SS) = 2826.52550 Finding Earth Velocity meter per sec in orbit in frame PQW Inputs : semi-major axis SMA, GM_Sun, earth pos r EC to SC frame IJK, eccentricity of earth orbit e_Sun, sun eccentric anomaly E_Sun Outputs : Earth Velocity magnitude and component Xw Yw in frame PQW in meter per sec E17B 011. Velocity magnitude meter per sec using GM, SMA, r earth EC to SC frame IJK at UT time = 29304.2989199811 E17C 011. Velocity component meter per sec in orbit Xw using GM, e_Sun, SMA, E_Sun at UT time = -6783.8841216771 E17D 011. Velocity component meter per sec in orbit Yw using GM, e_Sun, SMA, E_Sun at UT time = -28508.2593543566 Finding Earth Velocity Vector [vX, vY, vZ] in meter per sec in orbit; a Transform of [Xw, Yw] in frame PQW To [vX, vY, vZ] in frame XYZ Inputs : velocity component (Xw, Yw), sun right ascension Alpha, Sun Argument of perigee W_Sun, inclination Epcylone Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ E17E 011. vX earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = -104.0347630397 E17F 011. vY earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = -29304.0795566878 E17G 011. vZ earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 45.0920724379 E17H 011. vR earth Velocity magnitude meter per sec using Xw Yw frame PQW RA w i at UT time = 29304.2989199811
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OM-MSS Page 120 Earth State Vectors : Position [X, Y, Z] in km and Velocity [Vx, Vy, Vz] in meter per sec, at time input UT. 18. Finding Earth State Position Vector [X, Y, Z] in km at time input UT. Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ E18A 011. State vector position X km at UT time = -139485317.9369729200 E18B 011. State vector position Y km at UT time = 137.1515086517 E18C 011. State vector position Z km at UT time = 60474252.7228181590 E18D 011. State vector position R km at UT time = 152030553.3844447700 19. Finding Earth State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ E19A 011. State vector velocity Vx meter per sec at UT time = -104.0347630397 E19B 011. State vector velocity Vy meter per sec at UT time = -29304.0795566878 E19C 011. State vector velocity Vz meter per sec at UT time = 45.0920724379 019D 011. State vector velocity V meter per sec at UT time = 29304.2989199811 20. Earth Orbit Normal Vector [Wx, Wy, Wz] in km and angles Delta, i, RA at time input UT; Normal is line perpendicular to orbit plane. Inputs : earth pos r EC to SC frame IJK, inclination Epcylone, sun right ascension Alpha Outputs : earth orbit normal vector (Wx, Wy, Wz, W) in km E20A 011. Earth orbit normal W km using r earth pos frame IJK inclination Alpha = 152030553.3844447700 E20B 011. Earth orbit normal Wx km using r earth pos frame IJK inclination Alpha = 60474252.7227889370 E20C 011. Earth orbit normal Wy km using r earth pos frame IJK inclination Alpha = -59.4624231720 E20D 011. Earth orbit normal Wz km using r earth pos frame IJK inclination Alpha = 139485317.9370403600 020E 011. Earth orbit normal Delta W deg using r earth pos frame IJK inclination Alpha = 66.5607205617 E20F 011. Earth orbit normal Inclination i deg using normal_Delta_W = 23.4392794383 E20G 011. Earth orbit normal Alpha W deg using r earth pos frame IJK, inclination, Alpha = -0.0000563371 E20H 011. Earth orbit normal Right ascension of ascending node using normal Alpha, W = 89.9999436629
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OM-MSS Page 121 Transform Earth State Vectors to Earth position Keplerian elements. 21. Finding Earth position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness E21A 011. Keplerian elements year = 2013, days_decimal_of_year = 171.20925, revolution no = 1, node = 1 ie ascending E21B 011. inclination_deg = 23.4392794383 E21C 011. right ascension ascending node deg = 89.9999436629 E21D 011. eccentricity = 0.0167102190 E21E 011. argument of perigee_deg = 283.1636338813 E21F 011. mean anomaly deg = 166.3949127377 E21G 011. mean_motion rev per day = 0.0027377786 E21H 011. mean angular velocity rev_per_day = 0.0027377786 E21I 011. mean motion rev per day using SMA considering oblateness = 0.0027377786 Transform Earth position Keplerian elements to Earth State Vectors . 22. Finding Earth position State Vectors, computed using Keplerian elements at time input UT (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] E22A 011. State vectors year = 2013, days_decimal_of_year = 171.20925, revolution no = 1, node = 1 ie decending E22B 011. state vector position X km = -139485317.9369768200, state vector velocity Vx meter per sec = -104.0347628376 E22C 011. state vector position Y km = 137.1503847837, state vector velocity Vy meter per sec = -29304.0795566878 E22D 011. state vector position Z km = 60474252.7228198500, state vector velocity Vz meter per sec = 45.0920723503 E22E 011. state vector position R km = 152030553.3844490300, state vector velocity V meter per sec = 29304.2989199802
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OM-MSS Page 122 Note : Computation of all above parameters, grouped in 1 to 22, corresponds to time (a) Universal time over Greenwich (UT/GMT) : Year = 2013, Month = 6, Day = 21, Hour = 5, Min = 1, Sec = 19.200 (b) Mean Solar time (MST) over Earth Observation point (EP ) : Year = 2013, Month = 6, Day = 21, Hour = 10, Min = 10, Sec = 58.228 Move on to next Astronomical event in orbit Earth around Sun. Next Section - 4.5 Position of earth at time when earth is at aphelion
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OM-MSS Page 123 OM-MSS Section - 4.5 ---------------------------------------------------------------------------------------------------39 Earth Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Aphelion. 5. Finding Position of Earth on Celestial Sphere at Input Universal Time, when Earth is at Aphelion . Input UT Time, when Earth is at Aphelion : year = 2013, month = 7, day = 5, hour = 0, minute = 18, seconds = 52.59269 Julian Day = 2456478.51311, year_day_decimal = 185.01311, day_hour_decimal = 0.31461 Observation Point on Earth (Bhopal, India) : Lat +ve or -ve 0 to 90 deg = 23.25993 ie deg = 23, min = 15, sec = 35.76 Log 0 to 360 deg = 77.41261 ie deg = 77, min = 24, sec = 45.41 Alt from earth surface in km = 0.49470 First Compute the Sun Position on Celestial Sphere, then Compute the Earth Position on Celestial Sphere. (A) Computed Values for SUN POSITION on Celestial Sphere at Input Ut Time : (Sr. No 1 - 22) 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = 182.6298145405 07. Mean anomaly in deg per day from n_sun (m sun) = 180.0000000001 08. Sun Mean longitude in deg (Lmean) = 103.1643684676 09. Earth Mean anomaly in deg (ME) = 180.0000000002 10. Sun Ecliptic longitude in deg (Lsun) = 103.1643684676 11. Obliquity of ecliptic in deg (Epcylone) = 23.4375223935 12. Sun Right ascension in deg (Alpha) = 104.3014954901 13. Sun Declination in deg (Delta) = 22.7863704018 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 152098409.19029 16. Sun Nodal elongation in deg (U sun) = 76.8356315324 17. Sun Mean anomaly in deg (M sun) = 179.9999999997 18. Sun Eccentric anomaly in deg (E sun) = 179.9999999997 19. Sun True anomaly in deg (T sun) = 179.9999999997 20. Sun Argument of perigee in deg (W sun) = 256.8356315327 21. Sun True anomaly in deg from U & W (V sun) = 179.9999999997 22. Sun Distance in km (d sun) = 152098441.95238 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) These Values are applied as input for Computing Earth Position on Celestial Sphere around Sun at same input UT Time.
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OM-MSS Page 124 (B) Computed Values for EARTH POSITION on Celestial Sphere around Sun at same Input Ut Time : (Sr. No 1 - 22) Input Time year = 2013, month = 7, day = 5, hour = 0, minute = 18, seconds = 52.59269, corresponding Julian Day = 2456478.5131087117 Observation Point on Earth : Lat +ve or -ve 0 to 90 deg = 23.25993, Log 0 to 360 deg = 77.41261, Alt from earth surface in km = 0.49470 Sun position on Celestial sphere at input time, computed above total 22 parameters. Output Earth Position on Celestial sphere around Sun : Computed below around 120 parameters, presented in 1-22 groups. Number is large, because some parameters are computed using more than one model equation, that require different inputs. This helps in validation of results and understanding the different input considerations. 01. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input UT time YY MM DD HH. Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time UT year = 2013, month = 7, day = 5, hour = 0, minute = 18, seconds = 52.59269 Outputs : GST & GHA in 0-360 deg over Greenwich. E01A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 287.88360, hr = 19, min = 11, sec = 32.06349 E01B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 287.88367, deg = 287, min = 53, sec = 1.21667 02. Finding Earth latitude & longitude pointing to Sun Ecliptic longitude(Lsun). Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : Earth lat & log pointing to Lsun. E02A 011. Earth latitude +ve or -ve in 0 to 90 deg at UT time = 22.79 ie deg = 22, min = 47, sec = 17.06 E02B 011. Earth longitude 0 to 360 deg = 176.42 ie deg = 176, min = 25, sec = 4.43 03. Finding LST over three longitudes, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - for LST, used sidereal time at Greenwich GST and desired geogrphic longitude Inputs : At Time input UT - GST, Log of Greenwich, sun mean log Lmean, Sun ecliptic log Lsun. Outputs : LST over Greenwich, Lmean, Lsun . E03A 011. LST Local sidereal time in 0-360 deg, over Greenwich longitude = 287.88360, hr = 19, min = 11, sec = 32.06349
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OM-MSS Page 125 E03B 011. LST Local sidereal time in 0-360 deg, over Sun mean longitude (Lmean) = 31.04797, hr = 2, min = 4, sec = 11.51192 E03C 011. LST Local sidereal time in 0-360 deg, over Sun epliptic longitude (Lsun) = 31.04797, hr = 2, min = 4, sec = 11.51192 04. Finding ST0 sidereal time over Greenwich longitude = 0.0, at time input Year JAN day 1 hr 00. Note - this is sidereal time ST at UT year, month = 1, day = 1, hours decimal = 0.0 and geogrphic longitude = 0.0 Inputs : Time input UT Year, JAN day 1 hr 00, Log 0.0 Outputs : ST0 over Greenwich E04 011. ST0 Sidereal time in 0-360 deg, over Greenwich at input UT year, MM 1, DD 1, HH 00 = 100.80678, hr = 6, min = 43, sec = 13.62710 05. Finding ST sidereal time over three longitudes of, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - this is local sidereal time LST; (LST = GST at UT time + geogrphic longitude). Inputs : At Time input UT - Log 0.0, Log Lmean, Log Lsun Outputs : ST over Greenwich, Lmean, Lsun. E05A 011. ST Sidereal time in 0-360 deg, over Greenwich at input UT time = 287.88360, hr = 19, min = 11, sec = 32.06349 E05B 011. ST Sidereal time in 0-360 deg, over Sun mean longitude (Lmean) at input UT time = 31.04797, hr = 2, min = 4, sec = 11.51192 005C 011. ST Sidereal time in 0-360 deg, over Sun longitude (Lsun) at input UT time = 31.04797, hr = 2, min = 4, sec = 11.51192 06. Finding H hour angle in 0 to 360 deg over longitudes of, Greenwich, Lmean, Lsun, Earth Sub Sun point SS, Earth Obseration point EP. Note - used Sun Right ascension Alpha at input time; (hour angle HA = LST - Alpha). Inputs : At Time input UT - Sun Right ascension Alpha and ST Sidereal time over longitudes 0.0, Lmean, Lsun, SS, EP Outputs : Hour Angles over Greenwich, Lmean, Lsun, SS, EP E06A 011. H hour angle 0-360 deg, over Greenwich, = 183.58210, deg = 183, min = 34, sec = 55.56862 E06B 011. H hour angle 0-360 deg, over Lmean, = 286.74647, deg = 286, min = 44, sec = 47.29510 E06C 011. H hour angle 0-360 deg, over Lsun, = 286.74647, deg = 286, min = 44, sec = 47.29510 E06D 011. H hour angle 0-360 deg, over SS, = 0.00000, deg = 0, min = 0, sec = 0.00000 E06E 011. H hour angle 0-360 deg, over EP, = 260.99472, deg = 260, min = 59, sec = 40.98262
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OM-MSS Page 126 07. Finding Delta E is Equation of Time in seconds, at time input UT. Note - this value in seconds accounts for relative movement of sun in elliptical orbit w.r.t earth and effect of obliquity of the ecliptic; its maximum value is 16 minutes (960 sec.); Delta E is computed using time in days from the perihelion, n_sun_deg and w_sun at input UT. Inputs : Time input UT in JD, time perihelion in JD, Sun mean movement n_sun, Eccentricity of earth orbit E_Sun Outputs : Delta E time_equation in seconds. E07 011. Delta E Time Equation in seconds = -262.56430, hr = 0, min = 4, sec = 22.56430 08. Finding GST Greenwich sidereal time, and GHA Greenwich hour angle 0 to 360 deg at time when earth is at perihelion. Inputs : Time in JD when earth at perihelion YY = 2013, MM = 1, DD = 3, hr = 9, min = 11, sec = 56.62 Outputs : GST & GHA in 0-360 deg over Greenwich when earth is at perihelion E08A 011. GST sidereal time in 0-360 deg over Greenwich at time when earth is at perihelion = 241.14177, hr = 16, min = 4, sec = 34.02452 E08B 011. GHA hour angle in 0-360 deg over Greenwich at time when earth is at perihelion = 241.14717, hr = 16, min = 4, sec = 35.32064 09. Finding ST sidereal time and MST mean sidereal time, over Greenwich, using Earth mean motion rev per day . Inputs : GST when earth at perihelion, earth rotation rate, ref. JD2000, time input UT in JD, time perihelion in JD. Outputs : STP, angle perihelion to input JD, ST over Greenwich, MSTO & MST over Greenwich, solar time E09A 011. STP sidereal time in 0-360 deg over Greenwich when earth at perihelion = 241.14177, hr = 16, min = 4, sec = 34.02452 E09B 011. Angle in 0-360 deg from earth at perihelion to input JD using earth rotational rate = 46.73542, 009C 011. ST in 0-360 deg over Greenwich using STP and angle from perihelion at input JD = 287.87719, hr = 19, min = 11, sec = 30.52626 E09D 011. ST in 0-360 deg over Greenwich using STP and earth rotation at UT time = 287.87719, hr = 19, min = 11, sec = 30.52626 E09E 011. MST0 in deg, over Greenwich using JD century days, ref J2000 to I/P YY, M1, D1 hr 00 = 100.80714, hr = 6, min = 43, sec = 13.71450 E09F 011. MST in deg, over Greenwich using JD century in days from ref J2000 to UT time Y M D H = 287.88396, hr = 19, min = 11, sec = 32.15131 E09G 011. Solar time over Greenwich in JD (GMT or input UT - 12 hr) = YY 2013, MM 7, DD 4, hr 12, min 18, sec 52.593, ie JD 2456478.01311 10. Finding Earth orbit radious using true anomaly, Sub Sun point (SS) on earth surface and related paramters . (a) Finding Earth orbit radious using true anomaly. Inputs : semi-major axis SMA, eccentricity of earth orbit e_sun, sun true anomaly T_Sun Outputs : earth orbital radious EC to SC (earth center to sun center)
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OM-MSS Page 127 E10A 011. earth orbital radious EC to SC km using_true anomaly at UT time = 152098441.9523840800 (b) Finding Sub Sun point (SS) over earth surface (Latitude, Longitude, & Latitude radius) pointing to Sun Ecliptic Log (Lsun), Sun height from earth surface over SS, and LST over SS log at time input UT. Note - for SS Latitude, used earth inclination, sun true anomaly T_sun and sun argument of perigee w_sun. for SS Longitude, used Sun right ascension Alpha and sidereal time at Greenwich GST. Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : SS point Latitude, Longitude, Latitude radious, LST & LMT over SS . E10B 011. SS point Latitude +ve or -ve in 0 to 90 deg at UT time = 22.79 ie deg = 22, min = 47, sec = 17.06 E10C 011. SS point Longitude 0 to 360 deg = 176.42 ie deg = 176, min = 25, sec = 4.43 E10D 011. SS point Latitude radious km at UT time = 6374.9453113297 E10E 011. Sun height km from earth surface over SS at UT time = 152092067.0070727500 E10F 011. LST local sidereal time in 0-360 deg over SS log at UT time, (LST = GST + log east) = 104.302 ie hr = 6, min = 57, sec = 12.44676 LST local sidereal time and LMT local mean time with date adjusted to calendar YY MM DD and UT hr mm sec. E10G 011. LST local sidereal time at Sub Sun point (SS) YY = 2013, MM = 7, DD = 6, hr = 6, min = 57, sec = 12.45 E10H 011. LMT local Mean time at Sub Sun point (SS) YY = 2013, MM = 7, DD = 5, hr = 12, min = 4, sec = 32.89 (c) Finding LST and LMT over Earth point(EP) where Observer is, at time input UT. Inputs : EP point Latitude, Longitude Outputs : LST & LMT over EP . E10I 011. LST local sidereal time in 0-360 deg at EP log at UT time, (LST = GST + log east) = 5.297 ie hr = 0, min = 21, sec = 11.17894 LST and LMT with date adjusted to calendar YY MM DD and UT hr mm sec. E10J 011. LST local sidereal time at Earth point (EP) YY = 2013, MM = 7, DD = 6, hr = 0, min = 21, sec = 11.18 E10K 011. LMT local Mean time at Earth point (EP) YY = 2013, MM = 7, DD = 5, hr = 5, min = 28, sec = 31.62
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OM-MSS Page 128 Finding Earth to Sun Position Vectors coordinate in PQW, IJK, SEZ frames and the Vector Coordinate Transforms. First defined coordinate systems, PQW, IJK, SEZ, then computed Position & Velocity vectors in these three coordinate systems (a) Perifocal Coordinate System (PQW), is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. The system is fixed with time (inertial), pointing towards orbit periapsis; the system's origin is Earth center (EC), and its fundamental plane is the orbit plane; the P-vector axis directed from EC toward the periapsis of the elliptical orbit plane, the Q-vector axis swepts 90 deg from P axis in the direction of the orbit, the W-vector axis directed from EC in a direction normal to orbit plane, forms a right-handed coordinate system. (b) Geocentric Coordinate System (IJK), is also an Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). The system is fixed with time (inertial), pointing towards vernal equinox; the system's origin is Earth center (EC), and its fundamental plane is the equator; the I-vector is +X-axis directed towards the vernal equinox direction on J2000, Jan 1, hr 12.00 noon, the J-vector is +Y-axis swepts 90 deg to the east in the equatorial plane, the K-vector is +Z-axis directed towards the North Pole. (c) Topocentric Horizon Coordinate System (SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). The system moves with earth, is not fixed with time (non-inertial), is for use by observers on the surface of earth; the observer's surface forms the fundamental plane, is tangent to earth's surface the S-vector is +ve horizontal-axis directed towards South, the E-vector is +ve horizontal-axis directed towards East, the Z-vector is +ve normal directed upwards on earth surface. Note that axis Z not necessarily pass through earth center, so not used to define as radious vector. 11. Finding Earth center(EC) to Sun center(SC) Range Vector[rp, rq, r] from in PQW frame, perifocal coordinate system. Inputs : Semi-major axis (SMA), Eccentricity of earth orbit (e_sun), Sun eccentric anomaly (E_sun) Outputs : Vector(r, rp rq) in PQW frame E11A 011. r earth pos vector magnitude EC to SC km in PQW frame perifocal cord at UT time = 152098441.95238
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OM-MSS Page 129 E11B 011. rp earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = -152098441.9523840500 E11C 011. rq earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = 0.0007434219 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as earth orbital radious computed before using true anomaly. 12. Transform_1 Earth position EC to SC Range Vector[rp, rq] in PQW frame To Range Vector[rI, rJ, rK] in IJK frame, inertial system cord. Inputs : Vector(rp, rq) EC_to_SC km in frame PQW , Alpha rd, w_sun rd, earth_inclination rd , Outputs : Vector(rI, rJ, rK, r) EC_to_SC km in frame IJK E12A 011. rI earth pos vector component EC to SC km frame IJK at UT time = -140226215.3950415300 E12B 011. rJ earth pos vector component EC to SC km frame IJK at UT time = 446.2552520869 E12C 011. rK earth pos vector component EC to SC km frame IJK at UT time = 58911327.9440127610 E12D 011. r earth pos vector magnitude EC to SC km frame IJK at UT time = 152098441.9523840500 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as that compued above in PQW frame. 13. Transform_2 Earth point EP(lat, log, hgt) To EC to SC Range Vector[RI, RJ, RK, R] in IJK frame. Inputs : earth equator radious_km, earth point EP(lat deg, log deg, hgt meter), LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time, Outputs : Vector(RI, RJ, RK, R) Range EC to EP in IJK frame E13A 011. RI pos vector component EC to EP km frame IJK at UT time = 5832.1367811816 E13B 011. RJ pos vector component EC to EP km frame IJK at UT time = 540.6797455991 E13C 011. RK pos vector component EC to EP km frame IJK at UT time = 2517.6312937817 E13D 011. R pos vector magnitude EC to EP km frame IJK at UT time = 6375.3134317570 14. Transform_3 Earth position EC to SC Range Vectors [rI rJ rK] & [RI RJ RK] To EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame. Inputs : Vector(rI rJ rK) position EC_to_SC km in frame IJK , Vector(RI RJ RK) range EC to SC km in IJK frame, Outputs : Vector(rvI, rvJ, rvK, rv) range EP to SC in IJK frame E14A 011. rvI range vector component EP to SC km frame IJK at UT time = -140232047.5318227100 E14B 011. rvJ range vector component EP to SC km frame IJK at UT time = -94.4244935123 E14C 011. rvK range vector component EP to SC km frame IJK at UT time = 58908810.3127189800 E14D 011. rv range vector magnitude EP to SC km frame IJK at UT time = 152102843.7848425200
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OM-MSS Page 130 15. Transform_4 Earth point EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame To EP to SC Range Vector[rvS, rvE, rvZ] in SEZ frame. Inputs : lat_pos_neg_0_to_90_deg_at_EP_at_time_UT , LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time Vector(rvI, rvJ, rvK, rv) range EP to SC km in IJK frame, Outputs : Vector(rvS, rvE, rvZ, rv) range EP to SC km in SEZ frame E15A 011. rvS range vector component EP to SC km frame SEZ at UT time = -109262491.5112520900 E15B 011. rvE range vector component EP to SC km frame SEZ at UT time = 12944885.2533702720 E15C 011. rvZ range vector component EP to SC km frame SEZ at UT time = -105021012.0974219600 E15D 011. rv range vector magnitude EP to SC km frame SEZ at UT time = 152102843.7848425200 16. Finding Elevation(EL) and Azimuth(AZ) angle of Sun at Earth Observation point EP . Note : Results computed using 4 different formulations, each require different inputs to give EL & AZ angles. For all situations of Object and Observer positions, a combination of latitude N/S & longitude E/W : Method 1 : for both EL & AZ angles, this does not provide correct results ; Method 2 : for only EL angle, this provides consistent, unambiguous correct results. but for AZ angles the results are ambiguous, need corrections by adding or subtracting values as 180 or 360 or sign change. Method 3 : same as method 2, for EL angle, the results are correct, but for AZ angles the results are ambiguous, need corrections. Method 4 : for finding Azimuth and Distance but not for finding Elevation angle; for AZ angles, this provides correct unambiguous results that need no futher corrections. Therefore for Elevation (EL) angle Method 3 results are accepted and for Azimuth (AZ) angle Method 4 results are accepted . Results verified from other sources; Ref URLs http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone . NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , and http://aa.usno.navy.mil/data/docs/AltAz.php Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html Rem: SS point lat deg = 22.79, log deg = 176.42 YY = 2013, MM = 7, DD = 5, hr = 12, min = 4, sec = 32.89 EP point lat deg = 23.26, log deg = 77.41 YY = 2013, MM = 7, DD = 5, hr = 5, min = 28, sec = 31.62 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 1 - computed values may be Ambiguous or Incorrect). Inputs : Vector[rvS, rvE, rvZ] range EP to SC km in SEZ frame Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP
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OM-MSS Page 131 E16A 011. Elevation angle deg of Sun at EP using rv SEZ at UT time = -43.6665758927 E16B 011. Azimuth angle deg of sun at EP using rv SEZ at UT time = 6.7566282101 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 2 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun declination Delta Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16C 011. Elevation angle deg of Sun at EP using Sun declination diff log range EP to SC = 1.16686 E16D 011. Azimuth angle deg of sun at EP using sun declination diff log range EP to SC = 24.38662 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 3 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun hgt from EC, Sun range from EP (Sun hgt from EC = earth orbit radious EC to SC km ; Sun range from EP = rv range vector EP to SC km frame SEZ) Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16E 011. Elevation angle deg of Sun at EP using Sun hgt diff log range EP to SC = 1.16517 ie deg = 1, min = 9, sec = 54.60 E16F 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 245.61183 ie deg = 245, min = 36, sec = 42.59 Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 4 - computed AZ values is unambiguous & correct). Inputs : Time input UT YY MM DD HH, EP lat & log, SS lat & log Outputs : Azimuth(AZ) of Sun at EP E16G 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 65.61183 ie deg = 65, min = 36, sec = 42.59 Due to such incorrect results, finally for Elevation (EL) Method 3 results and for Azimuth (AZ) Method 4 results are accepted. Finally accepted Elevation angle deg of Sun from EP to SC = 1.1651677095 ie deg = 1, min = 9, sec = 54.60 Finally accepted Azimuth angle_deg of Sun from EP to SC = 65.6118300528 ie deg = 65, min = 36, sec = 42.59
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OM-MSS Page 132 Distance in km from Earth observation point(EP) to Sub Sun point(SS) and Earth Velocity meter per sec in orbit at time input UT. 17. Finding Distance in km from Earth observation point(EP) to Sub Sun point(SS) over Earth surface . Inputs : EP lat & log, SS lat & log, Outputs : Distance in km from EP to SS over Earth surface E17A 011. Distance in km Earth observation point(EP) to Sub Sun point(SS) = 9888.76838 Finding Earth Velocity meter per sec in orbit in frame PQW Inputs : semi-major axis SMA, GM_Sun, earth pos r EC to SC frame IJK, eccentricity of earth orbit e_Sun, sun eccentric anomaly E_Sun Outputs : Earth Velocity magnitude and component Xw Yw in frame PQW in meter per sec E17B 011. Velocity magnitude meter per sec using GM, SMA, r earth EC to SC frame IJK at UT time = 29290.9998931436 E17C 011. Velocity component meter per sec in orbit Xw using GM, e_Sun, SMA, E_Sun at UT time = -0.0000001456 E17D 011. Velocity component meter per sec in orbit Yw using GM, e_Sun, SMA, E_Sun at UT time = -29290.9998931436 Finding Earth Velocity Vector [vX, vY, vZ] in meter per sec in orbit; a Transform of [Xw, Yw] in frame PQW To [vX, vY, vZ] in frame XYZ Inputs : velocity component (Xw, Yw), sun right ascension Alpha, Sun Argument of perigee W_Sun, inclination Epcylone Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ E17E 011. vX earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 1114.6970357569 E17F 011. vY earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = -29149.2525417419 E17G 011. vZ earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 2653.5262418839 E17H 011. vR earth Velocity magnitude meter per sec using Xw Yw frame PQW RA w i at UT time = 29290.9998931436
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OM-MSS Page 133 Earth State Vectors : Position [X, Y, Z] in km and Velocity [Vx, Vy, Vz] in meter per sec, at time input UT. 18. Finding Earth State Position Vector [X, Y, Z] in km at time input UT. Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ E18A 011. State vector position X km at UT time = -140226215.3950415300 E18B 011. State vector position Y km at UT time = 446.2552520869 E18C 011. State vector position Z km at UT time = 58911327.9440127610 E18D 011. State vector position R km at UT time = 152098441.9523840500 19. Finding Earth State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ E19A 011. State vector velocity Vx meter per sec at UT time = 1114.6970357569 E19B 011. State vector velocity Vy meter per sec at UT time = -29149.2525417419 E19C 011. State vector velocity Vz meter per sec at UT time = 2653.5262418839 019D 011. State vector velocity V meter per sec at UT time = 29290.9998931436 20. Earth Orbit Normal Vector [Wx, Wy, Wz] in km and angles Delta, i, RA at time input UT; Normal is line perpendicular to orbit plane. Inputs : earth pos r EC to SC frame IJK, inclination Epcylone, sun right ascension Alpha Outputs : earth orbit normal vector (Wx, Wy, Wz, W) in km E20A 011. Earth orbit normal W km using r earth pos frame IJK inclination Alpha = 152098441.9523840500 E20B 011. Earth orbit normal Wx km using r earth pos frame IJK inclination Alpha = 58626279.9571206120 E20C 011. Earth orbit normal Wy km using r earth pos frame IJK inclination Alpha = 14945281.0276021960 E20D 011. Earth orbit normal Wz km using r earth pos frame IJK inclination Alpha = 139547604.4858458000 020E 011. Earth orbit normal Delta W deg using r earth pos frame IJK inclination Alpha = 66.5607205617 E20F 011. Earth orbit normal Inclination i deg using normal_Delta_W = 23.4392794383 E20G 011. Earth orbit normal Alpha W deg using r earth pos frame IJK, inclination, Alpha = 14.3014954901 E20H 011. Earth orbit normal Right ascension of ascending node using normal Alpha, W = 104.3014954901
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OM-MSS Page 134 Transform Earth State Vectors to Earth position Keplerian elements. 21. Finding Earth position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness E21A 011. Keplerian elements year = 2013, days_decimal_of_year = 185.01311, revolution no = 1, node = 1 ie ascending E21B 011. inclination_deg = 23.4392794383 E21C 011. right ascension ascending node deg = 104.3014954901 E21D 011. eccentricity = 0.0167102190 E21E 011. argument of perigee_deg = 256.8356315327 E21F 011. mean anomaly deg = 179.9999999997 E21G 011. mean_motion rev per day = 0.0027377786 E21H 011. mean angular velocity rev_per_day = 0.0027377786 E21I 011. mean motion rev per day using SMA considering oblateness = 0.0027377786 Transform Earth position Keplerian elements to Earth State Vectors . 22. Finding Earth position State Vectors, computed using Keplerian elements at time input UT (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] E22A 011. State vectors year = 2013, days_decimal_of_year = 185.01311, revolution no = 1, node = 1 ie decending E22B 011. state vector position X km = -140226215.3950413500, state vector velocity Vx meter per sec = 1114.6970357568 E22C 011. state vector position Y km = 446.2552521827, state vector velocity Vy meter per sec = -29149.2525417419 E22D 011. state vector position Z km = 58911327.9440126870, state vector velocity Vz meter per sec = 2653.5262418839 E22E 011. state vector position R km = 152098441.9523838800, state vector velocity V meter per sec = 29290.9998931436
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OM-MSS Page 135 Note : Computation of all above parameters, grouped in 1 to 22, corresponds to time (a) Universal time over Greenwich (UT/GMT) : Year = 2013, Month = 7, Day = 5, Hour = 0, Min = 18, Sec = 52.593 (b) Mean Solar time (MST) over Earth Observation point (EP ) : Year = 2013, Month = 7, Day = 5, Hour = 5, Min = 28, Sec = 31.620 Move on to next Astronomical event in orbit Earth around Sun. Next Section - 4.6 Position of earth at time when earth is at autumnal equinox
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OM-MSS Page 136 OM-MSS Section - 4.6 ---------------------------------------------------------------------------------------------------40 Earth Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Autumnal equinox . 6. Finding Position of Earth on Celestial Sphere at Input Universal Time, when Earth is at Autumnal equinox . Input UT Time, when Earth is at Autumnal equinox : year = 2013, month = 9, day = 22, hour = 20, minute = 45, seconds = 38.50711 Julian Day = 2456558.36503, year_day_decimal = 264.86503, day_hour_decimal = 20.76070 Observation Point on Earth (Bhopal, India) : Lat +ve or -ve 0 to 90 deg = 23.25993 ie deg = 23, min = 15, sec = 35.76 Log 0 to 360 deg = 77.41261 ie deg = 77, min = 24, sec = 45.41 Alt from earth surface in km = 0.49470 First Compute the Sun Position on Celestial Sphere, then Compute the Earth Position on Celestial Sphere. (A) Computed Values for SUN POSITION on Celestial Sphere at Input Ut Time : (Sr. No 1 - 22) 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = 262.4817348463 07. Mean anomaly in deg per day from n_sun (m sun) = 258.7020766091 08. Sun Mean longitude in deg (Lmean) = 181.8702061020 09. Earth Mean anomaly in deg (ME) = 258.7020766091 10. Sun Ecliptic longitude in deg (Lsun) = 180.0000000001 11. Obliquity of ecliptic in deg (Epcylone) = 23.4374939503 12. Sun Right ascension in deg (Alpha) = 180.0000000001 13. Sun Declination in deg (Delta) = -0.0000000001 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 150128632.16764 16. Sun Nodal elongation in deg (U sun) = -0.0000000001 17. Sun Mean anomaly in deg (M sun) = 258.7020766085 18. Sun Eccentric anomaly in deg (E sun) = 257.7663930098 19. Sun True anomaly in deg (T sun) = 256.8323186392 20. Sun Argument of perigee in deg (W sun) = 103.1676813607 21. Sun True anomaly in deg from U & W (V sun) = 256.8323186392 22. Sun Distance in km (d sun) = 150048057.36583 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) These Values are applied as input for Computing Earth Position on Celestial Sphere around Sun at same input UT Time.
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OM-MSS Page 137 (B) Computed Values for EARTH POSITION on Celestial Sphere around Sun at same Input Ut Time : (Sr. No 1 - 22) Input Time year = 2013, month = 9, day = 22, hour = 20, minute = 45, seconds = 38.50711, corresponding Julian Day = 2456558.3650290174 Observation Point on Earth : Lat +ve or -ve 0 to 90 deg = 23.25993, Log 0 to 360 deg = 77.41261, Alt from earth surface in km = 0.49470 Sun position on Celestial sphere at input time, computed above total 22 parameters. Output Earth Position on Celestial sphere around Sun : Computed below around 120 parameters, presented in 1-22 groups. Number is large, because some parameters are computed using more than one model equation, that require different inputs. This helps in validation of results and understanding the different input considerations. 01. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input UT time YY MM DD HH. Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time UT year = 2013, month = 9, day = 22, hour = 20, minute = 45, seconds = 38.50711 Outputs : GST & GHA in 0-360 deg over Greenwich. E01A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 313.28074, hr = 20, min = 53, sec = 7.37817 E01B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 313.29307, deg = 313, min = 17, sec = 35.04994 02. Finding Earth latitude & longitude pointing to Sun Ecliptic longitude(Lsun). Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : Earth lat & log pointing to Lsun. E02A 011. Earth latitude +ve or -ve in 0 to 90 deg at UT time = -0.00 ie deg = 0, min = 0, sec = 0.00 E02B 011. Earth longitude 0 to 360 deg = 226.72 ie deg = 226, min = 43, sec = 9.33 03. Finding LST over three longitudes, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - for LST, used sidereal time at Greenwich GST and desired geogrphic longitude Inputs : At Time input UT - GST, Log of Greenwich, sun mean log Lmean, Sun ecliptic log Lsun. Outputs : LST over Greenwich, Lmean, Lsun . E03A 011. LST Local sidereal time in 0-360 deg, over Greenwich longitude = 313.28074, hr = 20, min = 53, sec = 7.37817
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OM-MSS Page 138 E03B 011. LST Local sidereal time in 0-360 deg, over Sun mean longitude (Lmean) = 135.15095, hr = 9, min = 0, sec = 36.22763 E03C 011. LST Local sidereal time in 0-360 deg, over Sun epliptic longitude (Lsun) = 133.28074, hr = 8, min = 53, sec = 7.37817 04. Finding ST0 sidereal time over Greenwich longitude = 0.0, at time input Year JAN day 1 hr 00. Note - this is sidereal time ST at UT year, month = 1, day = 1, hours decimal = 0.0 and geogrphic longitude = 0.0 Inputs : Time input UT Year, JAN day 1 hr 00, Log 0.0 Outputs : ST0 over Greenwich E04 011. ST0 Sidereal time in 0-360 deg, over Greenwich at input UT year, MM 1, DD 1, HH 00 = 100.80678, hr = 6, min = 43, sec = 13.62710 05. Finding ST sidereal time over three longitudes of, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - this is local sidereal time LST; (LST = GST at UT time + geogrphic longitude). Inputs : At Time input UT - Log 0.0, Log Lmean, Log Lsun Outputs : ST over Greenwich, Lmean, Lsun. E05A 011. ST Sidereal time in 0-360 deg, over Greenwich at input UT time = 313.28074, hr = 20, min = 53, sec = 7.37817 E05B 011. ST Sidereal time in 0-360 deg, over Sun mean longitude (Lmean) at input UT time = 135.15095, hr = 9, min = 0, sec = 36.22763 005C 011. ST Sidereal time in 0-360 deg, over Sun longitude (Lsun) at input UT time = 133.28074, hr = 8, min = 53, sec = 7.37817 06. Finding H hour angle in 0 to 360 deg over longitudes of, Greenwich, Lmean, Lsun, Earth Sub Sun point SS, Earth Obseration point EP. Note - used Sun Right ascension Alpha at input time; (hour angle HA = LST - Alpha). Inputs : At Time input UT - Sun Right ascension Alpha and ST Sidereal time over longitudes 0.0, Lmean, Lsun, SS, EP Outputs : Hour Angles over Greenwich, Lmean, Lsun, SS, EP E06A 011. H hour angle 0-360 deg, over Greenwich, = 133.28074, deg = 133, min = 16, sec = 50.67250 E06B 011. H hour angle 0-360 deg, over Lmean, = 315.15095, deg = 315, min = 9, sec = 3.41447 E06C 011. H hour angle 0-360 deg, over Lsun, = 313.28074, deg = 313, min = 16, sec = 50.67250 E06D 011. H hour angle 0-360 deg, over SS, = 0.00000, deg = 0, min = 0, sec = 0.00000 E06E 011. H hour angle 0-360 deg, over EP, = 210.69336, deg = 210, min = 41, sec = 36.08650
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OM-MSS Page 139 07. Finding Delta E is Equation of Time in seconds, at time input UT. Note - this value in seconds accounts for relative movement of sun in elliptical orbit w.r.t earth and effect of obliquity of the ecliptic; its maximum value is 16 minutes (960 sec.); Delta E is computed using time in days from the perihelion, n_sun_deg and w_sun at input UT. Inputs : Time input UT in JD, time perihelion in JD, Sun mean movement n_sun, Eccentricity of earth orbit E_Sun Outputs : Delta E time_equation in seconds. E07 011. Delta E Time Equation in seconds = -489.69657, hr = 0, min = 8, sec = 9.69657 08. Finding GST Greenwich sidereal time, and GHA Greenwich hour angle 0 to 360 deg at time when earth is at perihelion. Inputs : Time in JD when earth at perihelion YY = 2013, MM = 1, DD = 3, hr = 9, min = 11, sec = 56.62 Outputs : GST & GHA in 0-360 deg over Greenwich when earth is at perihelion E08A 011. GST sidereal time in 0-360 deg over Greenwich at time when earth is at perihelion = 241.14177, hr = 16, min = 4, sec = 34.02452 E08B 011. GHA hour angle in 0-360 deg over Greenwich at time when earth is at perihelion = 241.14717, hr = 16, min = 4, sec = 35.32064 09. Finding ST sidereal time and MST mean sidereal time, over Greenwich, using Earth mean motion rev per day . Inputs : GST when earth at perihelion, earth rotation rate, ref. JD2000, time input UT in JD, time perihelion in JD. Outputs : STP, angle perihelion to input JD, ST over Greenwich, MSTO & MST over Greenwich, solar time E09A 011. STP sidereal time in 0-360 deg over Greenwich when earth at perihelion = 241.14177, hr = 16, min = 4, sec = 34.02452 E09B 011. Angle in 0-360 deg from earth at perihelion to input JD using earth rotational rate = 72.12977, 009C 011. ST in 0-360 deg over Greenwich using STP and angle from perihelion at input JD = 313.27154, hr = 20, min = 53, sec = 5.16880 E09D 011. ST in 0-360 deg over Greenwich using STP and earth rotation at UT time = 313.27154, hr = 20, min = 53, sec = 5.16880 E09E 011. MST0 in deg, over Greenwich using JD century days, ref J2000 to I/P YY, M1, D1 hr 00 = 100.80714, hr = 6, min = 43, sec = 13.71450 E09F 011. MST in deg, over Greenwich using JD century in days from ref J2000 to UT time Y M D H = 313.28111, hr = 20, min = 53, sec = 7.46617 E09G 011. Solar time over Greenwich in JD (GMT or input UT - 12 hr) = YY 2013, MM 9, DD 22, hr 8, min 45, sec 38.507, ie JD 2456557.86503 10. Finding Earth orbit radious using true anomaly, Sub Sun point (SS) on earth surface and related paramters . (a) Finding Earth orbit radious using true anomaly. Inputs : semi-major axis SMA, eccentricity of earth orbit e_sun, sun true anomaly T_Sun Outputs : earth orbital radious EC to SC (earth center to sun center)
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OM-MSS Page 140 E10A 011. earth orbital radious EC to SC km using_true anomaly at UT time = 150128324.5301490100 (b) Finding Sub Sun point (SS) over earth surface (Latitude, Longitude, & Latitude radius) pointing to Sun Ecliptic Log (Lsun), Sun height from earth surface over SS, and LST over SS log at time input UT. Note - for SS Latitude, used earth inclination, sun true anomaly T_sun and sun argument of perigee w_sun. for SS Longitude, used Sun right ascension Alpha and sidereal time at Greenwich GST. Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : SS point Latitude, Longitude, Latitude radious, LST & LMT over SS . E10B 011. SS point Latitude +ve or -ve in 0 to 90 deg at UT time = -0.00 ie deg = 0, min = 0, sec = 0.00 E10C 011. SS point Longitude 0 to 360 deg = 226.72 ie deg = 226, min = 43, sec = 9.33 E10D 011. SS point Latitude radious km at UT time = 6378.1440000000 E10E 011. Sun height km from earth surface over SS at UT time = 150121946.3861490200 E10F 011. LST local sidereal time in 0-360 deg over SS log at UT time, (LST = GST + log east) = 180.000 ie hr = 12, min = 0, sec = 0.08799 LST local sidereal time and LMT local mean time with date adjusted to calendar YY MM DD and UT hr mm sec. E10G 011. LST local sidereal time at Sub Sun point (SS) YY = 2013, MM = 9, DD = 22, hr = 12, min = 0, sec = 0.09 E10H 011. LMT local Mean time at Sub Sun point (SS) YY = 2013, MM = 9, DD = 22, hr = 11, min = 52, sec = 31.13 (c) Finding LST and LMT over Earth point(EP) where Observer is, at time input UT. Inputs : EP point Latitude, Longitude Outputs : LST & LMT over EP . E10I 011. LST local sidereal time in 0-360 deg at EP log at UT time, (LST = GST + log east) = 30.694 ie hr = 2, min = 2, sec = 46.49378 LST and LMT with date adjusted to calendar YY MM DD and UT hr mm sec. E10J 011. LST local sidereal time at Earth point (EP) YY = 2013, MM = 9, DD = 23, hr = 2, min = 2, sec = 46.49 E10K 011. LMT local Mean time at Earth point (EP) YY = 2013, MM = 9, DD = 23, hr = 1, min = 55, sec = 17.53
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OM-MSS Page 141 Finding Earth to Sun Position Vectors coordinate in PQW, IJK, SEZ frames and the Vector Coordinate Transforms. First defined coordinate systems, PQW, IJK, SEZ, then computed Position & Velocity vectors in these three coordinate systems. (a) Perifocal Coordinate System (PQW), is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. The system is fixed with time (inertial), pointing towards orbit periapsis; the system's origin is Earth center (EC), and its fundamental plane is the orbit plane; the P-vector axis directed from EC toward the periapsis of the elliptical orbit plane, the Q-vector axis swepts 90 deg from P axis in the direction of the orbit, the W-vector axis directed from EC in a direction normal to orbit plane, forms a right-handed coordinate system. (b) Geocentric Coordinate System (IJK), is also an Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). The system is fixed with time (inertial), pointing towards vernal equinox; the system's origin is Earth center (EC), and its fundamental plane is the equator; the I-vector is +X-axis directed towards the vernal equinox direction on J2000, Jan 1, hr 12.00 noon, the J-vector is +Y-axis swepts 90 deg to the east in the equatorial plane, the K-vector is +Z-axis directed towards the North Pole. (c) Topocentric Horizon Coordinate System (SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). The system moves with earth, is not fixed with time (non-inertial), is for use by observers on the surface of earth; the observer's surface forms the fundamental plane, is tangent to earth's surface the S-vector is +ve horizontal-axis directed towards South, the E-vector is +ve horizontal-axis directed towards East, the Z-vector is +ve normal directed upwards on earth surface. Note that axis Z not necessarily pass through earth center, so not used to define as radious vector. 11. Finding Earth center(EC) to Sun center(SC) Range Vector[rp, rq, r] from in PQW frame, perifocal coordinate system. Inputs : Semi-major axis (SMA), Eccentricity of earth orbit (e_sun), Sun eccentric anomaly (E_sun) Outputs : Vector(r, rp rq) in PQW frame E11A 011. r earth pos vector magnitude EC to SC km in PQW frame perifocal cord at UT time = 150128324.53015
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OM-MSS Page 142 E11B 011. rp earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = -34199483.1036060300 E11C 011. rq earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = -146181083.5288748700 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as earth orbital radious computed before using true anomaly. 12. Transform_1 Earth position EC to SC Range Vector[rp, rq] in PQW frame To Range Vector[rI, rJ, rK] in IJK frame, inertial system cord. Inputs : Vector(rp, rq) EC_to_SC km in frame PQW , Alpha rd, w_sun rd, earth_inclination rd , Outputs : Vector(rI, rJ, rK, r) EC_to_SC km in frame IJK E12A 011. rI earth pos vector component EC to SC km frame IJK at UT time = -150128324.5301489500 E12B 011. rJ earth pos vector component EC to SC km frame IJK at UT time = 0.0000000075 E12C 011. rK earth pos vector component EC to SC km frame IJK at UT time = -0.0001395773 E12D 011. r earth pos vector magnitude EC to SC km frame IJK at UT time = 150128324.5301489500 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as that compued above in PQW frame. 13. Transform_2 Earth point EP(lat, log, hgt) To EC to SC Range Vector[RI, RJ, RK, R] in IJK frame. Inputs : earth equator radious_km, earth point EP(lat deg, log deg, hgt meter), LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time, Outputs : Vector(RI, RJ, RK, R) Range EC to EP in IJK frame E13A 011. RI pos vector component EC to EP km frame IJK at UT time = 5036.6074245797 E13B 011. RJ pos vector component EC to EP km frame IJK at UT time = 2989.7725118111 E13C 011. RK pos vector component EC to EP km frame IJK at UT time = 2517.6312937817 E13D 011. R pos vector magnitude EC to EP km frame IJK at UT time = 6375.3134317570 14. Transform_3 Earth position EC to SC Range Vectors [rI rJ rK] & [RI RJ RK] To EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame. Inputs : Vector(rI rJ rK) position EC_to_SC km in frame IJK , Vector(RI RJ RK) range EC to SC km in IJK frame, Outputs : Vector(rvI, rvJ, rvK, rv) range EP to SC in IJK frame E14A 011. rvI range vector component EP to SC km frame IJK at UT time = -150133361.1375735400 E14B 011. rvJ range vector component EP to SC km frame IJK at UT time = -2989.7725118037 E14C 011. rvK range vector component EP to SC km frame IJK at UT time = -2517.6314333590 E14D 011. rv range vector magnitude EP to SC km frame IJK at UT time = 150133361.1884523300
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OM-MSS Page 143 15. Transform_4 Earth point EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame To EP to SC Range Vector[rvS, rvE, rvZ] in SEZ frame. Inputs : lat_pos_neg_0_to_90_deg_at_EP_at_time_UT , LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time Vector(rvI, rvJ, rvK, rv) range EP to SC km in IJK frame, Outputs : Vector(rvS, rvE, rvZ, rv) range EP to SC km in SEZ frame E15A 011. rvS range vector component EP to SC km frame SEZ at UT time = -50980643.1144296600 E15B 011. rvE range vector component EP to SC km frame SEZ at UT time = 76632812.6748455610 E15C 011. rvZ range vector component EP to SC km frame SEZ at UT time = -118610337.6225001200 E15D 011. rv range vector magnitude EP to SC km frame SEZ at UT time = 150133361.1884523300 16. Finding Elevation(EL) and Azimuth(AZ) angle of Sun at Earth Observation point EP . Note : Results computed using 4 different formulations, each require different inputs to give EL & AZ angles. For all situations of Object and Observer positions, a combination of latitude N/S & longitude E/W : Method 1 : for both EL & AZ angles, this does not provide correct results ; Method 2 : for only EL angle, this provides consistent, unambiguous correct results. but for AZ angles the results are ambiguous, need corrections by adding or subtracting values as 180 or 360 or sign change. Method 3 : same as method 2, for EL angle, the results are correct, but for AZ angles the results are ambiguous, need corrections. Method 4 : for finding Azimuth and Distance but not for finding Elevation angle; for AZ angles, this provides correct unambiguous results that need no futher corrections. Therefore for Elevation (EL) angle Method 3 results are accepted and for Azimuth (AZ) angle Method 4 results are accepted . Results verified from other sources; Ref URLs http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone . NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , and http://aa.usno.navy.mil/data/docs/AltAz.php Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html Rem: SS point lat deg = -0.00, log deg = 226.72 YY = 2013, MM = 9, DD = 22, hr = 11, min = 52, sec = 31.13 EP point lat deg = 23.26, log deg = 77.41 YY = 2013, MM = 9, DD = 23, hr = 1, min = 55, sec = 17.53 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 1 - computed values may be Ambiguous or Incorrect). Inputs : Vector[rvS, rvE, rvZ] range EP to SC km in SEZ frame Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP
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OM-MSS Page 144 E16A 011. Elevation angle deg of Sun at EP using rv SEZ at UT time = -52.1886128588 E16B 011. Azimuth angle deg of sun at EP using rv SEZ at UT time = 56.3658187650 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 2 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun declination Delta Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16C 011. Elevation angle deg of Sun at EP using Sun declination diff log range EP to SC = -52.18740 E16D 011. Azimuth angle deg of sun at EP using sun declination diff log range EP to SC = 33.63457 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 3 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun hgt from EC, Sun range from EP (Sun hgt from EC = earth orbit radious EC to SC km ; Sun range from EP = rv range vector EP to SC km frame SEZ) Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16E 011. Elevation angle deg of Sun at EP using Sun hgt diff log range EP to SC = -52.18889 ie deg = -52, min = 11, sec = 20.02 E16F 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 236.36543 ie deg = 236, min = 21, sec = 55.56 Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 4 - computed AZ values is unambiguous & correct). Inputs : Time input UT YY MM DD HH, EP lat & log, SS lat & log Outputs : Azimuth(AZ) of Sun at EP E16G 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 56.36543 ie deg = 56, min = 21, sec = 55.56 Due to such incorrect results, finally for Elevation (EL) Method 3 results and for Azimuth (AZ) Method 4 results are accepted. Finally accepted Elevation angle deg of Sun from EP to SC = -52.1888941383 ie deg = -52, min = 11, sec = 20.02 Finally accepted Azimuth angle_deg of Sun from EP to SC = 56.3654334756 ie deg = 56, min = 21, sec = 55.56
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OM-MSS Page 145 Distance in km from Earth observation point(EP) to Sub Sun point(SS) and Earth Velocity meter per sec in orbit at time input UT. 17. Finding Distance in km from Earth observation point(EP) to Sub Sun point(SS) over Earth surface . Inputs : EP lat & log, SS lat & log, Outputs : Distance in km from EP to SS over Earth surface E17A 011. Distance in km Earth observation point(EP) to Sub Sun point(SS) = 15828.20917 Finding Earth Velocity meter per sec in orbit in frame PQW Inputs : semi-major axis SMA, GM_Sun, earth pos r EC to SC frame IJK, eccentricity of earth orbit e_Sun, sun eccentric anomaly E_Sun Outputs : Earth Velocity magnitude and component Xw Yw in frame PQW in meter per sec E17B 011. Velocity magnitude meter per sec using GM, SMA, r earth EC to SC frame IJK at UT time = 29679.3404099639 E17C 011. Velocity component meter per sec in orbit Xw using GM, e_Sun, SMA, E_Sun at UT time = 29005.5570441560 E17D 011. Velocity component meter per sec in orbit Yw using GM, e_Sun, SMA, E_Sun at UT time = -6288.1561469737 Finding Earth Velocity Vector [vX, vY, vZ] in meter per sec in orbit; a Transform of [Xw, Yw] in frame PQW To [vX, vY, vZ] in frame XYZ Inputs : velocity component (Xw, Yw), sun right ascension Alpha, Sun Argument of perigee W_Sun, inclination Epcylone Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ E17E 011. vX earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 484.6892104138 E17F 011. vY earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = -27226.6334875568 E17G 011. vZ earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 11804.1836852055 E17H 011. vR earth Velocity magnitude meter per sec using Xw Yw frame PQW RA w i at UT time = 29679.3404099639
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OM-MSS Page 146 Earth State Vectors : Position [X, Y, Z] in km and Velocity [Vx, Vy, Vz] in meter per sec, at time input UT. 18. Finding Earth State Position Vector [X, Y, Z] in km at time input UT. Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ E18A 011. State vector position X km at UT time = -150128324.5301489500 E18B 011. State vector position Y km at UT time = 0.0000000075 E18C 011. State vector position Z km at UT time = -0.0001395773 E18D 011. State vector position R km at UT time = 150128324.5301489500 19. Finding Earth State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ E19A 011. State vector velocity Vx meter per sec at UT time = 484.6892104138 E19B 011. State vector velocity Vy meter per sec at UT time = -27226.6334875568 E19C 011. State vector velocity Vz meter per sec at UT time = 11804.1836852055 019D 011. State vector velocity V meter per sec at UT time = 29679.3404099639 20. Earth Orbit Normal Vector [Wx, Wy, Wz] in km and angles Delta, i, RA at time input UT; Normal is line perpendicular to orbit plane. Inputs : earth pos r EC to SC frame IJK, inclination Epcylone, sun right ascension Alpha Outputs : earth orbit normal vector (Wx, Wy, Wz, W) in km E20A 011. Earth orbit normal W km using r earth pos frame IJK inclination Alpha = 150128324.5301489500 E20B 011. Earth orbit normal Wx km using r earth pos frame IJK inclination Alpha = -0.0001280575 E20C 011. Earth orbit normal Wy km using r earth pos frame IJK inclination Alpha = 59717589.8947850020 E20D 011. Earth orbit normal Wz km using r earth pos frame IJK inclination Alpha = 137740056.9311193500 020E 011. Earth orbit normal Delta W deg using r earth pos frame IJK inclination Alpha = 66.5607205617 E20F 011. Earth orbit normal Inclination i deg using normal_Delta_W = 23.4392794383 E20G 011. Earth orbit normal Alpha W deg using r earth pos frame IJK, inclination, Alpha = -89.9999999999 E20H 011. Earth orbit normal Right ascension of ascending node using normal Alpha, W = 0.0000000001
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OM-MSS Page 147 Transform Earth State Vectors to Earth position Keplerian elements. 21. Finding Earth position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness E21A 011. Keplerian elements year = 2013, days_decimal_of_year = 264.86503, revolution no = 1, node = 2 ie decending E21B 011. inclination_deg = 23.4392794383 E21C 011. right ascension ascending node deg = 180.0000000001 E21D 011. eccentricity = 0.0167102190 E21E 011. argument of perigee_deg = 103.1676813607 E21F 011. mean anomaly deg = 258.7020766085 E21G 011. mean_motion rev per day = 0.0027377786 E21H 011. mean angular velocity rev_per_day = 0.0027377786 E21I 011. mean motion rev per day using SMA considering oblateness = 0.0027377786 Transform Earth position Keplerian elements to Earth State Vectors . 22. Finding Earth position State Vectors, computed using Keplerian elements at time input UT (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] E22A 011. State vectors year = 2013, days_decimal_of_year = 264.86503, revolution no = 1, node = 2 ie decending E22B 011. state vector position X km = -150128324.5301487700, state vector velocity Vx meter per sec = 484.6892104141 E22C 011. state vector position Y km = -0.0000012219, state vector velocity Vy meter per sec = -27226.6334875568 E22D 011. state vector position Z km = -0.0001390446, state vector velocity Vz meter per sec = 11804.1836852055 E22E 011. state vector position R km = 150128324.5301487700, state vector velocity V meter per sec = 29679.3404099639
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OM-MSS Page 148 Note : Computation of all above parameters, grouped in 1 to 22, corresponds to time (a) Universal time over Greenwich (UT/GMT) : Year = 2013, Month = 9, Day = 22, Hour = 20, Min = 45, Sec = 38.507 (b) Mean Solar time (MST) over Earth Observation point (EP ) : Year = 2013, Month = 9, Day = 23, Hour = 1, Min = 55, Sec = 17.535 Move on to next Astronomical event in orbit Earth around Sun. Next Section - 4.7 Position of earth at time when earth is at winter solstice
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OM-MSS Page 149 OM-MSS Section - 4.7 ---------------------------------------------------------------------------------------------------41 Earth Positional Parameters on Celestial Sphere : Input Year Time when Earth is at Winter solstice . 7. Finding Position of Earth on Celestial Sphere at Input Universal Time, when Earth is at Winter solstice . Input UT Time, when Earth is at Winter solstice : year = 2013, month = 12, day = 21, hour = 17, minute = 10, seconds = 3.73442 Julian Day = 2456648.21532, year_day_decimal = 354.71532, day_hour_decimal = 17.16770 Observation Point on Earth (Bhopal, India) : Lat +ve or -ve 0 to 90 deg = 23.25993 ie deg = 23, min = 15, sec = 35.76 Log 0 to 360 deg = 77.41261 ie deg = 77, min = 24, sec = 45.41 Alt from earth surface in km = 0.49470 First Compute the Sun Position on Celestial Sphere, then Compute the Earth Position on Celestial Sphere. (A) Computed Values for SUN POSITION on Celestial Sphere at Input Ut Time : (Sr. No 1 - 22) 01. Earth around Sun Mean motion rev per day (mm) = 0.0027377786 02. Semi-major axis in km considering oblateness (SMA) = 149598616.31172 03. Earth mean motion deg per day using SMA (mm) = 0.9856003000 04. Sun mean movement deg per day (n sun) = 0.9856003000 05. Eccentricity of earth orbit (e sun) = 0.0167102190 06. Perihelion to input time diff in Julian days = 352.3320268290 07. Mean anomaly in deg per day from n_sun (m sun) = 347.2585513424 08. Sun Mean longitude in deg (Lmean) = 270.4309127840 09. Earth Mean anomaly in deg (ME) = 347.2585513424 10. Sun Ecliptic longitude in deg (Lsun) = 269.9999511320 11. Obliquity of ecliptic in deg (Epcylone) = 23.4374619456 12. Sun Right ascension in deg (Alpha) = 269.9999467376 13. Sun Declination in deg (Delta) = -23.4374619456 14. Sun Mean distance in km (As) = 149598616.31172 15. Sun Radial distance from earth in km (Rs) = 147162417.75585 16. Sun Nodal elongation in deg (U sun) = -90.0000000000 17. Sun Mean anomaly in deg (M sun) = 347.2585513416 18. Sun Eccentric anomaly in deg (E sun) = 347.0438922873 19. Sun True anomaly in deg (T sun) = 346.8274560675 20. Sun Argument of perigee in deg (W sun) = 283.1725439325 21. Sun True anomaly in deg from U & W (V sun) = 346.8274560675 22. Sun Distance in km (d sun) = 147158348.89183 Sun Ecliptic latitude is always nearly zero (the value never exceeds 0.00033 deg) These Values are applied as input for Computing Earth Position on Celestial Sphere around Sun at same input UT Time.
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OM-MSS Page 150 (B) Computed Values for EARTH POSITION on Celestial Sphere around Sun at same Input Ut Time : (Sr. No 1 - 22) Input Time year = 2013, month = 12, day = 21, hour = 17, minute = 10, seconds = 3.73442, corresponding Julian Day = 2456648.2153210002 Observation Point on Earth : Lat +ve or -ve 0 to 90 deg = 23.25993, Log 0 to 360 deg = 77.41261, Alt from earth surface in km = 0.49470 Sun position on Celestial sphere at input time, computed above total 22 parameters. Output Earth Position on Celestial sphere around Sun : Computed below around 120 parameters, presented in 1-22 groups. Number is large, because some parameters are computed using more than one model equation, that require different inputs. This helps in validation of results and understanding the different input considerations. 01. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input UT time YY MM DD HH. Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time UT year = 2013, month = 12, day = 21, hour = 17, minute = 10, seconds = 3.73442 Outputs : GST & GHA in 0-360 deg over Greenwich. E01A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 347.94656, hr = 23, min = 11, sec = 47.17421 E01B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 347.95673, deg = 347, min = 57, sec = 24.23424 02. Finding Earth latitude & longitude pointing to Sun Ecliptic longitude(Lsun). Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : Earth lat & log pointing to Lsun. E02A 011. Earth latitude +ve or -ve in 0 to 90 deg at UT time = -23.44 ie deg = -23, min = 26, sec = 21.41 E02B 011. Earth longitude 0 to 360 deg = 282.05 ie deg = 282, min = 3, sec = 12.20 03. Finding LST over three longitudes, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - for LST, used sidereal time at Greenwich GST and desired geogrphic longitude Inputs : At Time input UT - GST, Log of Greenwich, sun mean log Lmean, Sun ecliptic log Lsun. Outputs : LST over Greenwich, Lmean, Lsun . E03A 011. LST Local sidereal time in 0-360 deg, over Greenwich longitude = 347.94656, hr = 23, min = 11, sec = 47.17421
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OM-MSS Page 151 E03B 011. LST Local sidereal time in 0-360 deg, over Sun mean longitude (Lmean) = 258.37747, hr = 17, min = 13, sec = 30.59328 E03C 011. LST Local sidereal time in 0-360 deg, over Sun epliptic longitude (Lsun) = 257.94651, hr = 17, min = 11, sec = 47.16248 04. Finding ST0 sidereal time over Greenwich longitude = 0.0, at time input Year JAN day 1 hr 00. Note - this is sidereal time ST at UT year, month = 1, day = 1, hours decimal = 0.0 and geogrphic longitude = 0.0 Inputs : Time input UT Year, JAN day 1 hr 00, Log 0.0 Outputs : ST0 over Greenwich E04 011. ST0 Sidereal time in 0-360 deg, over Greenwich at input UT year, MM 1, DD 1, HH 00 = 100.80678, hr = 6, min = 43, sec = 13.62710 05. Finding ST sidereal time over three longitudes of, Greenwich log, Sun mean log (Lmean), and Sun epliptic log (Lsun) . Note - this is local sidereal time LST; (LST = GST at UT time + geogrphic longitude). Inputs : At Time input UT - Log 0.0, Log Lmean, Log Lsun Outputs : ST over Greenwich, Lmean, Lsun. E05A 011. ST Sidereal time in 0-360 deg, over Greenwich at input UT time = 347.94656, hr = 23, min = 11, sec = 47.17421 E05B 011. ST Sidereal time in 0-360 deg, over Sun mean longitude (Lmean) at input UT time = 258.37747, hr = 17, min = 13, sec = 30.59328 005C 011. ST Sidereal time in 0-360 deg, over Sun longitude (Lsun) at input UT time = 257.94651, hr = 17, min = 11, sec = 47.16248 06. Finding H hour angle in 0 to 360 deg over longitudes of, Greenwich, Lmean, Lsun, Earth Sub Sun point SS, Earth Obseration point EP. Note - used Sun Right ascension Alpha at input time; (hour angle HA = LST - Alpha). Inputs : At Time input UT - Sun Right ascension Alpha and ST Sidereal time over longitudes 0.0, Lmean, Lsun, SS, EP Outputs : Hour Angles over Greenwich, Lmean, Lsun, SS, EP E06A 011. H hour angle 0-360 deg, over Greenwich, = 77.94661, deg = 77, min = 56, sec = 47.80490 E06B 011. H hour angle 0-360 deg, over Lmean, = 348.37753, deg = 348, min = 22, sec = 39.09093 E06C 011. H hour angle 0-360 deg, over Lsun, = 347.94656, deg = 347, min = 56, sec = 47.62898 E06D 011. H hour angle 0-360 deg, over SS, = 0.00000, deg = 0, min = 0, sec = 0.00000 E06E 011. H hour angle 0-360 deg, over EP, = 155.35923, deg = 155, min = 21, sec = 33.21890
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OM-MSS Page 152 07. Finding Delta E is Equation of Time in seconds, at time input UT. Note - this value in seconds accounts for relative movement of sun in elliptical orbit w.r.t earth and effect of obliquity of the ecliptic; its maximum value is 16 minutes (960 sec.); Delta E is computed using time in days from the perihelion, n_sun_deg and w_sun at input UT. Inputs : Time input UT in JD, time perihelion in JD, Sun mean movement n_sun, Eccentricity of earth orbit E_Sun Outputs : Delta E time_equation in seconds. E07 011. Delta E Time Equation in seconds = -92.54574, hr = 0, min = 1, sec = 32.54574 08. Finding GST Greenwich sidereal time, and GHA Greenwich hour angle 0 to 360 deg at time when earth is at perihelion. Inputs : Time in JD when earth at perihelion YY = 2013, MM = 1, DD = 3, hr = 9, min = 11, sec = 56.62 Outputs : GST & GHA in 0-360 deg over Greenwich when earth is at perihelion E08A 011. GST sidereal time in 0-360 deg over Greenwich at time when earth is at perihelion = 241.14177, hr = 16, min = 4, sec = 34.02452 E08B 011. GHA hour angle in 0-360 deg over Greenwich at time when earth is at perihelion = 241.14717, hr = 16, min = 4, sec = 35.32064 09. Finding ST sidereal time and MST mean sidereal time, over Greenwich, using Earth mean motion rev per day . Inputs : GST when earth at perihelion, earth rotation rate, ref. JD2000, time input UT in JD, time perihelion in JD. Outputs : STP, angle perihelion to input JD, ST over Greenwich, MSTO & MST over Greenwich, solar time E09A 011. STP sidereal time in 0-360 deg over Greenwich when earth at perihelion = 241.14177, hr = 16, min = 4, sec = 34.02452 E09B 011. Angle in 0-360 deg from earth at perihelion to input JD using earth rotational rate = 106.79243, 009C 011. ST in 0-360 deg over Greenwich using STP and angle from perihelion at input JD = 347.93420, hr = 23, min = 11, sec = 44.20855 E09D 011. ST in 0-360 deg over Greenwich using STP and earth rotation at UT time = 347.93420, hr = 23, min = 11, sec = 44.20855 E09E 011. MST0 in deg, over Greenwich using JD century days, ref J2000 to I/P YY, M1, D1 hr 00 = 100.80714, hr = 6, min = 43, sec = 13.71450 E09F 011. MST in deg, over Greenwich using JD century in days from ref J2000 to UT time Y M D H = 347.94693, hr = 23, min = 11, sec = 47.26242 E09G 011. Solar time over Greenwich in JD (GMT or input UT - 12 hr) = YY 2013, MM 12, DD 21, hr 5, min 10, sec 3.734, ie JD 2456647.71532 10. Finding Earth orbit radious using true anomaly, Sub Sun point (SS) on earth surface and related paramters . (a) Finding Earth orbit radious using true anomaly. Inputs : semi-major axis SMA, eccentricity of earth orbit e_sun, sun true anomaly T_Sun Outputs : earth orbital radious EC to SC (earth center to sun center)
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OM-MSS Page 153 E10A 011. earth orbital radious EC to SC km using_true anomaly at UT time = 147162430.9672899200 (b) Finding Sub Sun point (SS) over earth surface (Latitude, Longitude, & Latitude radius) pointing to Sun Ecliptic Log (Lsun), Sun height from earth surface over SS, and LST over SS log at time input UT. Note - for SS Latitude, used earth inclination, sun true anomaly T_sun and sun argument of perigee w_sun. for SS Longitude, used Sun right ascension Alpha and sidereal time at Greenwich GST. Inputs : earth inclination, sun true anomaly T_Sun, sun argument of perigee W_Sun, sun right ascension Alpha, earth equator radious, GST at input UT, log SS & EP, earth orbit radious EC to SC Outputs : SS point Latitude, Longitude, Latitude radious, LST & LMT over SS . E10B 011. SS point Latitude +ve or -ve in 0 to 90 deg at UT time = -23.44 ie deg = -23, min = 26, sec = 21.41 E10C 011. SS point Longitude 0 to 360 deg = 282.05 ie deg = 282, min = 3, sec = 12.20 E10D 011. SS point Latitude radious km at UT time = 6374.7700931266 E10E 011. Sun height km from earth surface over SS at UT time = 147156056.1971968100 E10F 011. LST local sidereal time in 0-360 deg over SS log at UT time, (LST = GST + log east) = 270.000 ie hr = 18, min = 0, sec = 0.07540 LST local sidereal time and LMT local mean time with date adjusted to calendar YY MM DD and UT hr mm sec. E10G 011. LST local sidereal time at Sub Sun point (SS) YY = 2013, MM = 12, DD = 21, hr = 18, min = 0, sec = 0.08 E10H 011. LMT local Mean time at Sub Sun point (SS) YY = 2013, MM = 12, DD = 21, hr = 11, min = 58, sec = 16.55 (c) Finding LST and LMT over Earth point(EP) where Observer is, at time input UT. Inputs : EP point Latitude, Longitude Outputs : LST & LMT over EP . E10I 011. LST local sidereal time in 0-360 deg at EP log at UT time, (LST = GST + log east) = 65.360 ie hr = 4, min = 21, sec = 26.29001 LST and LMT with date adjusted to calendar YY MM DD and UT hr mm sec. E10J 011. LST local sidereal time at Earth point (EP) YY = 2013, MM = 12, DD = 22, hr = 4, min = 21, sec = 26.29 E10K 011. LMT local Mean time at Earth point (EP) YY = 2013, MM = 12, DD = 21, hr = 22, min = 19, sec = 42.76
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OM-MSS Page 154 Finding Earth to Sun Position Vectors coordinate in PQW, IJK, SEZ frames and the Vector Coordinate Transforms. First defined coordinate systems, PQW, IJK, SEZ, then computed Position & Velocity vectors in these three coordinate systems. (a) Perifocal Coordinate System (PQW), is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. The system is fixed with time (inertial), pointing towards orbit periapsis; the system's origin is Earth center (EC), and its fundamental plane is the orbit plane; the P-vector axis directed from EC toward the periapsis of the elliptical orbit plane, the Q-vector axis swepts 90 deg from P axis in the direction of the orbit, the W-vector axis directed from EC in a direction normal to orbit plane, forms a right-handed coordinate system. (b) Geocentric Coordinate System (IJK), is also an Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). The system is fixed with time (inertial), pointing towards vernal equinox; the system's origin is Earth center (EC), and its fundamental plane is the equator; the I-vector is +X-axis directed towards the vernal equinox direction on J2000, Jan 1, hr 12.00 noon, the J-vector is +Y-axis swepts 90 deg to the east in the equatorial plane, the K-vector is +Z-axis directed towards the North Pole. (c) Topocentric Horizon Coordinate System (SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). The system moves with earth, is not fixed with time (non-inertial), is for use by observers on the surface of earth; the observer's surface forms the fundamental plane, is tangent to earth's surface the S-vector is +ve horizontal-axis directed towards South, the E-vector is +ve horizontal-axis directed towards East, the Z-vector is +ve normal directed upwards on earth surface. Note that axis Z not necessarily pass through earth center, so not used to define as radious vector. 11. Finding Earth center(EC) to Sun center(SC) Range Vector[rp, rq, r] from in PQW frame, perifocal coordinate system. Inputs : Semi-major axis (SMA), Eccentricity of earth orbit (e_sun), Sun eccentric anomaly (E_sun) Outputs : Vector(r, rp rq) in PQW frame E11A 011. r earth pos vector magnitude EC to SC km in PQW frame perifocal cord at UT time = 147162430.96729
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OM-MSS Page 155 E11B 011. rp earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = 143290324.9509236200 E11C 011. rq earth pos vector component EC to SC km in PQW frame perifocal cord at UT time = -33536008.4634748730 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as earth orbital radious computed before using true anomaly. 12. Transform_1 Earth position EC to SC Range Vector[rp, rq] in PQW frame To Range Vector[rI, rJ, rK] in IJK frame, inertial system cord. Inputs : Vector(rp, rq) EC_to_SC km in frame PQW , Alpha rd, w_sun rd, earth_inclination rd , Outputs : Vector(rI, rJ, rK, r) EC_to_SC km in frame IJK E12A 011. rI earth pos vector component EC to SC km frame IJK at UT time = -135018902.5487828300 E12B 011. rJ earth pos vector component EC to SC km frame IJK at UT time = 125.5141403824 E12C 011. rK earth pos vector component EC to SC km frame IJK at UT time = -58537825.7429337430 E12D 011. r earth pos vector magnitude EC to SC km frame IJK at UT time = 147162430.9672938900 Note - r earth pos vector magnitude EC to SC km in PQW frame is same as that compued above in PQW frame. 13. Transform_2 Earth point EP(lat, log, hgt) To EC to SC Range Vector[RI, RJ, RK, R] in IJK frame. Inputs : earth equator radious_km, earth point EP(lat deg, log deg, hgt meter), LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time, Outputs : Vector(RI, RJ, RK, R) Range EC to EP in IJK frame E13A 011. RI pos vector component EC to EP km frame IJK at UT time = 2441.9770992295 E13B 011. RJ pos vector component EC to EP km frame IJK at UT time = 5323.8052057294 E13C 011. RK pos vector component EC to EP km frame IJK at UT time = 2517.6312937817 E13D 011. R pos vector magnitude EC to EP km frame IJK at UT time = 6375.3134317570 14. Transform_3 Earth position EC to SC Range Vectors [rI rJ rK] & [RI RJ RK] To EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame. Inputs : Vector(rI rJ rK) position EC_to_SC km in frame IJK , Vector(RI RJ RK) range EC to SC km in IJK frame, Outputs : Vector(rvI, rvJ, rvK, rv) range EP to SC in IJK frame E14A 011. rvI range vector component EP to SC km frame IJK at UT time = -135021344.5258820700 E14B 011. rvJ range vector component EP to SC km frame IJK at UT time = -5198.2910653470 E14C 011. rvK range vector component EP to SC km frame IJK at UT time = -58540343.3742275240 E14D 011. rv range vector magnitude EP to SC km frame IJK at UT time = 147165672.9912639600
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OM-MSS Page 156 15. Transform_4 Earth point EP to SC Range Vector[rvI, rvJ, rvK] in IJK frame To EP to SC Range Vector[rvS, rvE, rvZ] in SEZ frame. Inputs : lat_pos_neg_0_to_90_deg_at_EP_at_time_UT , LST_local_sidereal_time_in_0_to_360_deg_at_EP_log_at_UT_time Vector(rvI, rvJ, rvK, rv) range EP to SC km in IJK frame, Outputs : Vector(rvS, rvE, rvZ, rv) range EP to SC km in SEZ frame E15A 011. rvS range vector component EP to SC km frame SEZ at UT time = 31550012.9630649570 E15B 011. rvE range vector component EP to SC km frame SEZ at UT time = 122724394.7305014100 E15C 011. rvZ range vector component EP to SC km frame SEZ at UT time = -74840195.9312917890 E15D 011. rv range vector magnitude EP to SC km frame SEZ at UT time = 147165672.9912639300 16. Finding Elevation(EL) and Azimuth(AZ) angle of Sun at Earth Observation point EP . Note : Results computed using 4 different formulations, each require different inputs to give EL & AZ angles. For all situations of Object and Observer positions, a combination of latitude N/S & longitude E/W : Method 1 : for both EL & AZ angles, this does not provide correct results ; Method 2 : for only EL angle, this provides consistent, unambiguous correct results. but for AZ angles the results are ambiguous, need corrections by adding or subtracting values as 180 or 360 or sign change. Method 3 : same as method 2, for EL angle, the results are correct, but for AZ angles the results are ambiguous, need corrections. Method 4 : for finding Azimuth and Distance but not for finding Elevation angle; for AZ angles, this provides correct unambiguous results that need no futher corrections. Therefore for Elevation (EL) angle Method 3 results are accepted and for Azimuth (AZ) angle Method 4 results are accepted . Results verified from other sources; Ref URLs http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone . NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , and http://aa.usno.navy.mil/data/docs/AltAz.php Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html Rem: SS point lat deg = -23.44, log deg = 282.05 YY = 2013, MM = 12, DD = 21, hr = 11, min = 58, sec = 16.55 EP point lat deg = 23.26, log deg = 77.41 YY = 2013, MM = 12, DD = 21, hr = 22, min = 19, sec = 42.76 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 1 - computed values may be Ambiguous or Incorrect). Inputs : Vector[rvS, rvE, rvZ] range EP to SC km in SEZ frame Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP
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OM-MSS Page 157 E16A 011. Elevation angle deg of Sun at EP using rv SEZ at UT time = -30.5668840082 E16B 011. Azimuth angle deg of sun at EP using rv SEZ at UT time = 255.5825938098 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 2 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun declination Delta Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16C 011. Elevation angle deg of Sun at EP using Sun declination diff log range EP to SC = -67.40419 E16D 011. Azimuth angle deg of sun at EP using sun declination diff log range EP to SC = -174.61010 Elevation(EL) & Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 3 - computed AZ values may be ambiguous & incorrect). Inputs : Time input UT YY MM DD HH, Equator radious, EP lat & log, SS lat & log, Sun hgt from EC, Sun range from EP (Sun hgt from EC = earth orbit radious EC to SC km ; Sun range from EP = rv range vector EP to SC km frame SEZ) Outputs : Elevation(EL) & Azimuth(AZ) of Sun at EP E16E 011. Elevation angle deg of Sun at EP using Sun hgt diff log range EP to SC = -67.40528 ie deg = -67, min = 24, sec = 19.03 E16F 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 95.39461 ie deg = 95, min = 23, sec = 40.61 Azimuth(AZ) angle of Sun at Earth Observation point EP, (method 4 - computed AZ values is unambiguous & correct). Inputs : Time input UT YY MM DD HH, EP lat & log, SS lat & log Outputs : Azimuth(AZ) of Sun at EP E16G 011. Azimuth angle deg of sun at EP using sun hgt diff log range EP to SC = 264.60539 ie deg = 264, min = 36, sec = 19.39 Due to such incorrect results, finally for Elevation (EL) Method 3 results and for Azimuth (AZ) Method 4 results are accepted. Finally accepted Elevation angle deg of Sun from EP to SC = -67.4052849205 ie deg = -67, min = 24, sec = 19.03 Finally accepted Azimuth angle_deg of Sun from EP to SC = 264.6053872100 ie deg = 264, min = 36, sec = 19.39
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OM-MSS Page 158 Distance in km from Earth observation point(EP) to Sub Sun point(SS) and Earth Velocity meter per sec in orbit at time input UT. 17. Finding Distance in km from Earth observation point(EP) to Sub Sun point(SS) over Earth surface . Inputs : EP lat & log, SS lat & log, Outputs : Distance in km from EP to SS over Earth surface E17A 011. Distance in km Earth observation point(EP) to Sub Sun point(SS) = 17522.14782 Finding Earth Velocity meter per sec in orbit in frame PQW Inputs : semi-major axis SMA, GM_Sun, earth pos r EC to SC frame IJK, eccentricity of earth orbit e_Sun, sun eccentric anomaly E_Sun Outputs : Earth Velocity magnitude and component Xw Yw in frame PQW in meter per sec E17B 011. Velocity magnitude meter per sec using GM, SMA, r earth EC to SC frame IJK at UT time = 30273.6689861666 E17C 011. Velocity component meter per sec in orbit Xw using GM, e_Sun, SMA, E_Sun at UT time = 6788.3947484369 E17D 011. Velocity component meter per sec in orbit Yw using GM, e_Sun, SMA, E_Sun at UT time = 29502.7580172325 Finding Earth Velocity Vector [vX, vY, vZ] in meter per sec in orbit; a Transform of [Xw, Yw] in frame PQW To [vX, vY, vZ] in frame XYZ Inputs : velocity component (Xw, Yw), sun right ascension Alpha, Sun Argument of perigee W_Sun, inclination Epcylone Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ E17E 011. vX earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 104.0469608256 E17F 011. vY earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = -30273.4565603953 E17G 011. vZ earth Velocity vector meter per sec using Xw Yw frame PQW RA w i at UT time = 45.1220543039 E17H 011. vR earth Velocity magnitude meter per sec using Xw Yw frame PQW RA w i at UT time = 30273.6689861666
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OM-MSS Page 159 Earth State Vectors : Position [X, Y, Z] in km and Velocity [Vx, Vy, Vz] in meter per sec, at time input UT. 18. Finding Earth State Position Vector [X, Y, Z] in km at time input UT. Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ E18A 011. State vector position X km at UT time = -135018902.5487828300 E18B 011. State vector position Y km at UT time = 125.5141403824 E18C 011. State vector position Z km at UT time = -58537825.7429337430 E18D 011. State vector position R km at UT time = 147162430.9672938900 19. Finding Earth State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ E19A 011. State vector velocity Vx meter per sec at UT time = 104.0469608256 E19B 011. State vector velocity Vy meter per sec at UT time = -30273.4565603953 E19C 011. State vector velocity Vz meter per sec at UT time = 45.1220543039 019D 011. State vector velocity V meter per sec at UT time = 30273.6689861666 20. Earth Orbit Normal Vector [Wx, Wy, Wz] in km and angles Delta, i, RA at time input UT; Normal is line perpendicular to orbit plane. Inputs : earth pos r EC to SC frame IJK, inclination Epcylone, sun right ascension Alpha Outputs : earth orbit normal vector (Wx, Wy, Wz, W) in km E20A 011. Earth orbit normal W km using r earth pos frame IJK inclination Alpha = 147162430.9672938900 E20B 011. Earth orbit normal Wx km using r earth pos frame IJK inclination Alpha = -58537825.7429084480 E20C 011. Earth orbit normal Wy km using r earth pos frame IJK inclination Alpha = 54.4170093468 E20D 011. Earth orbit normal Wz km using r earth pos frame IJK inclination Alpha = 135018902.5488411500 020E 011. Earth orbit normal Delta W deg using r earth pos frame IJK inclination Alpha = 66.5607205617 E20F 011. Earth orbit normal Inclination i deg using normal_Delta_W = 23.4392794383 E20G 011. Earth orbit normal Alpha W deg using r earth pos frame IJK, inclination, Alpha = -0.0000532624 E20H 011. Earth orbit normal Right ascension of ascending node using normal Alpha, W = 89.9999467376
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OM-MSS Page 160 Transform Earth State Vectors to Earth position Keplerian elements. 21. Finding Earth position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness E21A 011. Keplerian elements year = 2013, days_decimal_of_year = 354.71532, revolution no = 1, node = 2 ie decending E21B 011. inclination_deg = 23.4392794383 E21C 011. right ascension ascending node deg = 269.9999467376 E21D 011. eccentricity = 0.0167102190 E21E 011. argument of perigee_deg = 283.1725439084 E21F 011. mean anomaly deg = 347.2585513649 E21G 011. mean_motion rev per day = 0.0027377786 E21H 011. mean angular velocity rev_per_day = 0.0027377786 E21I 011. mean motion rev per day using SMA considering oblateness = 0.0027377786 Transform Earth position Keplerian elements to Earth State Vectors . 22. Finding Earth position State Vectors, computed using Keplerian elements at time input UT (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] E22A 011. State vectors year = 2013, days_decimal_of_year = 354.71532, revolution no = 1, node = 2 ie decending E22B 011. state vector position X km = -135018902.5487863400, state vector velocity Vx meter per sec = 104.0469606286 E22C 011. state vector position Y km = 125.5152014755, state vector velocity Vy meter per sec = -30273.4565603953 E22D 011. state vector position Z km = -58537825.7429352700, state vector velocity Vz meter per sec = 45.1220542184 E22E 011. state vector position R km = 147162430.9672977600, state vector velocity V meter per sec = 30273.6689861658
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OM-MSS Page 161 Note : Computation of all above parameters, grouped in 1 to 22, corresponds to time (a) Universal time over Greenwich (UT/GMT) : Year = 2013, Month = 12, Day = 21, Hour = 17, Min = 10, Sec = 3.734 (b) Mean Solar time (MST) over Earth Observation point (EP ) : Year = 2013, Month = 12, Day = 21, Hour = 22, Min = 19, Sec = 42.762 Thus Computed values for Position of Earth on Celestial Sphere corresponding to Standard Epoch time JD2000, and at six Astronomical Events while Earth reaches Perihelion, Vernal equinox, Summer solstice, Aphelion, Autumnal equinox, Winter solstices. A Summary of these Computed values are presented next. Next Section - 4.8 Concluding position of earth at six astronomical events
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OM-MSS Page 162 OM-MSS Section - 4.8 -----------------------------------------------------------------------------------------------------42 Concluding Position of Earth on Celestial Sphere (Sections 4.0 to 4.7). Summary of Earth Position on Celestial Sphere with respect to Sun, at Std Epoch J2000 and at six astronomical events in the Year = 2013. In previous Sections (4.1 to 4.7), presented Earth Position on Celestial Sphere with respect to Sun, at Std Epoch J2000 and in year 2013 when Earth was at Perihelion, Vernal equinox, Summer solstice, Aphelion, Autumnal equinox, Winter solstices. The Earth Observation point (EP) considered as : latitude deg = 23.25993, longitude deg = 77.41261, height km = 0.49470 . The Summary of Computed values presented below, are put in three parts as : - UT/GMT, LMT at Lat/Log of SS, Sun EL & AZ angles at EP, Surface distance EP to SS, Earth Vel; - Earth position Keplerian elements : Inclination, RA of asc. node, Eccentricity, Arg. of Perigee, Mean Anomaly, Mean Motion ; - Earth State Vectors : Position (X, Y, Z) & Velocity (Vx, Vy, Vz) . (a) UT/GMT, LMT at Lat/Log of SS, Sun EL & AZ angles at EP, Surface distance EP to SS and Earth Vel, values at different orbit points Earth Orbit Universal time SubSun Point (SS) Sun hgt at SS Earth Point (EP) Sun angle at EP Dist.EP to SS Earth Vel. Events (GMT Dt/time) (LMT Dt/time lat/log) (km) (LMT Dt/time) (EL/AZ deg) (km) (m/s) Epoch J2000 2000.01.01/ 2000.01.01/ -23.034 147094821.78 2000.01.01/ 2.382 9753.28 30286.06663 12:00:00 12:03:18 0.826 17:09:39 243.631 Perihelion 2013.01.03/ 2013.01.03/ -22.789 147092415.73 2013.01.03/ 33.164 6326.74 30286.55386 09:11:56 12:04:32 43.150 14:21:35 218.319 Vernal equinox 2013.03.20/ 2013.03.20/ 0.000 148983652.48 2013.03.20/ 26.365 7083.54 29906.03259 11:02:09 12:07:26 16.321 16:11:48 257.698 Summer Solstice 2013.06.21/ 2013.06.21/ 23.439 152024178.61 2013.06.21/ 64.608 2826.53 29304.29892 05:01:19 12:01:45 105.111 10:10:58 275.977 Aphelion 2013.07.05/ 2013.07.05/ 22.788 152092067.01 2013.07.05/ 1.165 9888.77 29290.99989 00:18:52 12:04:32 176.418 05:28:31 245.612 Autumnal equinox 2013.09.22/ 2013.09.22/ -0.000 150121946.39 2013.09.23/ -52.189 15828.21 29679.34041 20:45:38 11:52:31 226.719 01:55:17 236.365 Winter solstice 2013.12.21/ 2013.12.21/ -23.439 147156056.20 2013.12.21/ -67.405 17522.15 30273.66899 17:10:03 11:58:16 282.053 22:19:42 95.395
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OM-MSS Page 163 (b) Earth Position Keplerian Elements , values at different orbit points Earth Orbit Epoch Inclination RA of asc. node Eccentricity Arg. of Perigee Mean Anomaly Mean Motion Events (year, day of year) (deg) (deg) (deg) (deg) (deg) (deg) Epoch J2000 2000 0.50000 23.43928 281.28586 0.0167102190 282.93203 357.52800 0.0027377786 Perihelion 2013 2.38329 23.43928 284.29222 0.0167102190 283.15577 0.00000 0.0027377786 Vernal equinox 2013 78.45983 23.43928 0.00000 0.0167102190 283.15977 74.98105 0.0027377786 Summer Solstice 2013 171.20925 23.43928 89.99994 0.0167102190 283.16363 166.39491 0.0027377786 Aphelion 2013 185.01311 23.43928 104.30150 0.0167102190 256.83563 180.00000 0.0027377786 Autumnal equinox 2013 264.86503 23.43928 180.00000 0.0167102190 103.16768 258.70208 0.0027377786 Winter solstice 2013 354.71532 23.43928 269.99995 0.0167102190 283.17254 347.25855 0.0027377786 (c) Earth State Vectors - position & velocity , values at different orbit points Earth Orbit Position X Position Y Position Z Position R Velocity Vx Velocity Vy Velocity Vz Velocity V Events vector(km) vector(km) vector(km) mag.(km) vector(m/s) vector(m/s) vector(m/s) mag.(m/s) Epoch J2000 -125003885.45 -51961735.98 -57556656.24 147101196.66 10756.75 -28227.52 2178.39 30286.07 Perihelion -119085952.63 -64886279.60 -56976844.62 147098790.67 13409.17 -27017.61 2741.95 30286.55 Vernal equinox 148990030.62 0.00 0.00 148990030.62 484.70 27434.65 11894.37 29906.03 Summer Solstice -139485317.94 137.15 60474252.72 152030553.38 -104.03 -29304.08 45.09 29304.30 Aphelion -140226215.40 446.26 58911327.94 152098441.95 1114.70 -29149.25 2653.53 29291.00 Autumnal equinox -150128324.53 -0.00 -0.00 150128324.53 484.69 -27226.63 11804.18 29679.34 Winter solstice -135018902.55 125.52 -58537825.74 147162430.97 104.05 -30273.46 45.12 30273.67 The computed values presented in table (a) (b) (c) show consistency. The angles are expresed in deg and distances in km. End of computing position of Earth on Celestial Sphere at input UT Standard Epoch time JD2000 and at six astronomical events. Move on to Satellites in Orbit around Earth, Ephemeris Data Sets. Next Section - 5 Satellites 0rbit around Earth : Ephemeris data set.
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OM-MSS Page 164 OM-MSS Section - 5 -------------------------------------------------------------------------------------------------------43 SATELLITES ORBIT ELEMENTS : EPHEMERIS, Keplerian ELEMENTS, STATE VECTORS Satellite Ephemeris is Expressed either by 'Keplerian elements' or by 'State Vectors', that uniquely identify a specific orbit. A satellite is an object that moves around a larger object. Thousands of Satellites launched into orbit around Earth. First, look into the Preliminaries about 'Satellite Orbit', before moving to Satellite Ephemeris data and convesion utilities of the OM-MSS software. (a) Satellite : An artificial object, intentionally placed into orbit. Thousands of Satellites have been launched into orbit around Earth. A few Satellites called Space Probes have been placed into orbit around Moon, Mercury, Venus, Mars, Jupiter, Saturn, etc. The Motion of a Satellite is a direct consequence of the Gravity of a body (earth), around which the satellite travels without any propulsion. The Moon is the Earth's only natural Satellite, moves around Earth in the same kind of orbit. (b) Earth Gravity and Satellite Motion : As satellite move around Earth, it is pulled in by the gravitational force (centripetal) of the Earth. Contrary to this pull, the rotating motion of satellite around Earth has an associated force (centrifugal) which pushes it away from the Earth. The centrifugal force equals the gravitational force and perfectly balance to maintain the satellite in its orbit. For a satellite travelling in a circular orbit at altitude 'h' with velocity 'V', these two forces are expressed as : - the centrifugal as F1 = (m * V^2) / (Re + h) and - the gravitational force as F2 = (G * m * Me) /(Re + h )^2 . Where G * Me = 3.99 x 10^14 m^3 / s^2, m is mass of satellite, G is gravitational constant, Me is mass of earth, Re is earth radius. (c) Velocity equations : The two forces F1 and F2 are equal, therefore (m * V^2) / (Re + h) = (G * m * Me) / (Re + h )^2 . Thus satellite velocity 'V' is related to its altitude 'h'; 'V' is constant at all circular orbit points, but vary at elliptical orbit points. Assuming the orbit is Circular, the Satellite Velocty is expressed as V = ((G * Me)/(Re + h))^0.5 , which is simply written as as V = ((G * Me) / r)^0.5, where 'r' = (Re + h) is the distance from satellite to earth centre. Assuming that orbit is elliptical, the satellite Velocty is expressed as V = ((G * Me) * ((2/(Re + h)) - 1/a))^0.5 , which is simply written as as V = ((G * Me) * (2/r - 1/a))^0.5, where 'r' = (Re + h) is distance from satellite to earth centre, and a = (rp + ra)/2 is semi-major axis, interpreted as orbit mean distance from earth center; variables rp & ra are perigee & apogee distances from earth center.
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OM-MSS Page 165 Note that the velocity of a satellite in circular or elliptical orbit depends on its altitude 'h' at that point; secondly, the mass of satellite does not appear in its velocity equations; thus satellite velocity in its orbit is independent of its mass; further, a satellite in elliptical orbit moves faster when closer to earth (near perigee) and moves slower when farther from earth (near apogee). Examples of orbit altitude (km) Vs. velocity (meters/sec), for circular orbits. The typical values are : (1) altitude 200 (km) corresponding velocity 7790 (meters/sec). (2) altitude 500 (km) corresponding velocity 7610 (meters/sec), (3) altitude 800 (km) corresponding velocity 7450 (meters/sec). (4) altitude 35786 (km) corresponding velocity 3070 (meters/sec). (5) altitude Moon 384000 (km) corresponding velocity 1010 (meters/sec). (d) Attitude control : Satellite attitude is defined in terms of three axis - Roll, Pitch, & Yaw. These three axis form the local orbital reference system defined at each point of the orbit by three unit vectors. These vectors are derived from the satellite position and velocity vectors. is fully controlled relative to three axes, at each point of the orbit while the satellite moves in its orbit. - the yaw axis is colinear with the earth's center and the satellite, called the 'geocentric direction' - the roll axis is in the direction of the movement (velocity) of satellite, is perpendicular to geocentric axis, - the pitch axis is perpendicular to the orbital plane. Note, the roll axis does not coincide exactly with the velocity vector due to the eccentricity of the orbit. All three axis pass through the center of gravity of the satellite. The satellite attitude control is important. It helps the communication satellite antennas point towards the region of interest where ground stations are located. Similarly, by maintaining orientation, the solar panel are steered such that the panel surface is normal to Sun to generate full power. Futher, the remote sensing satellite is able to acquire images free from distortion and blurring effects. (e) Time period : One orbit time is called time period of a satellite, calculated as the distance travelled by the satellite divided by its velocity. For Circular Satellite Orbit : time period Pc = circumference of a circle of radius(Re + h) divided by velocity of satellite. = 2 * pi * (((Re + h)^3) / (G * Me))^0.5 . = 2 * pi * ((r^3) / (G * Me))^0.5 , where 'r' is radius of circular orbit from earth ceter. For Elliptical Satellite Orbit : time period Pe = 2 * pi * ((a^3) / (G * Me))^0.5 , where 'a' is semi-major axis of the elliptical orbit. Note : The equations show that the satellite orbit time period increases with increase in altitude. A satellite orbit altitude about 36000 km has its orbit time period roughly 1440 minutes or one sidereal day, ie (23hr,56 min,4sec) for earth to complete one orbit rotation.
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OM-MSS Page 166 Examples of orbit Time period (minutes) Vs. altitude (km), for circular orbits. The typical values are : (1) altitude LEO 200 (km) corresponding Time period 88 (minutes). (2) altitude LEO 500 (km) corresponding Time period 94 (minutes) , (3) altitude LEO 800 (km) corresponding Time period 100 (minutes). (4) altitude Geo 35786 (km) corresponding Time period 1436 (minutes) . (5) altitude Moon 384000 (km) corresponding Time period 40461 (minutes) ie roughly 28 days. (f) Orbits : Two types of orbits, either Circular or Elliptical orbit, are assumed while formulating satellite velocity and its time period. Any satellite can achieve orbit of any distance from the earth if its velocity is sufficient to keep it in orbit or prevent from falling to earth. Thus, the altitudes at which satellites orbit the earth are split into three categories : Low earth orbit, Medium earth orbit, & High earth orbit. Each category serve different purpose. Some orbits provide a constant view of one face of Earth, while others circle over different places in a day. Some orbits are specifically named as Geosynchronous orbit (GSO), Geostationary (GEO), Equatorial Orbit, Polar Orbit, and Sunsynchronous. The Orbit altitudes (km), Time period (minutes) and the usage/mission are briefly mentioned below. (1) Low earth orbit (LEO) : Altitude 160-2,000 (km), Time period 87-127 (min), Used for most Earth Observation Remote Sensing satellites (EORSS), the International Space Station (ISS), the Space Shuttle, and the Hubble Space Telescope (HST). A satellite at 300 (km) altitude has orbital period about 90 (min). In 90 (min), the earth at equator rotates about 2500 (km) . Thus, the satellite after one time period, passes over equator a point/place 2500 (km) west of the point/place it passed over in its previous orbit. To a person on the earth directly under the orbit, a satellite appears above horizon on one side of sky, crosses the sky, and disappears beyond the opposite horizon in about 10 (min). It reappears after 80 (min), but not over same spot, since the earth has rotated during that time. (2) Medium earth orbit (MEO) : Altitude 2000-35,780 (km), Time period 87-1435 (min), Used for navigation satellites of Global positioning called, GPS (20,200 kilometers), Glonass (19,100 kilometers) and Galileo (23,222 kilometers) constellations. The orbit time periods are about 12 hours. The Communications satellites that cover the North and South Pole are also put in MEO. (3) High earth orbit (HEO) : Altitude > 35,780, Time period > a sidereal day (23hr,56 min,4sec), Used to provide coverage over any point on globe, for astronomical work and for the communication in areas usually not possible from other type of orbits. The orbits are highly inclined and highly elliptical, characterized by low-altitude perigee and extremely high-altitude apogee. Such highly elliptical orbits, provide coverage over polar and near polar areas needed in countries, like USA & Russia. Two satellites in any orbit can provide continuous coverage but disadvantage is, satellite position from a point on Earth does not remain same.
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OM-MSS Page 167 (4) Geosynchronous orbit (GSO) : Altitude about 36000 (km), Time period is same as the earth's rotational period (23hr,56 min,4sec). The word 'synchronous' means, for an observer at a fixed location on earth, the satellite returns to exactly same place and same time each day. GSO can be circular or elliptical and of any inclination. The inclination, 0 deg and 90 deg are the special cases of GEO. (5) Geostationary (GEO) : A special case of a Circular GSO orbit in the equatorial plane (inclination zero deg), called Equatorial Orbit. A satellite in this orbit stays fixed relative to earth surface directly over a point on equator. To an observer on ground, the satellite appears notion less. Only in an equatorial orbit, a satellite remain stationary over a point. (6) Equatorial Orbit : Same as Geostationary (GEO); or a special case of a geosynchronous orbit (GSO) that is circular (or nearly circular) and inclination zero (or nearly zero) deg, ie directly above the equator. An orbit directly above equator will have an inclination of 0 deg or 180 deg . (7) Polar Orbit : A strictly defined polar orbit means inclination is 90 deg. If inclination is within a few degrees of 90, then called a near polar orbit. Usually, Polar Orbit refers to near-polar inclination and an altitude of 700 to 900 km. Satellites in polar orbit pass over the equator and every latitude on the Earth's surface at the same local time each day. This means the satellite passes overhead at any location is essentially at the same time throughout the year. This orbit enables regular data collection at consistent times and is useful for long-term comparisons. The weather, environmental and national security related monitoring satellites are placed in polar orbits. (8) Sun-synchronous orbit : This orbit is a special case of the polar orbit, where the orbit inclination and the altitude combines in such a way that the satellite passes over any given point of on the earth surface at the same local solar time, meaning same sunlight. In other words, the surface illumination angle is nearly the same every time. This consistent lighting is a useful characteristic for satellites that image the Earth's surface in visible or infrared wavelengths and for those remote sensing satellites carrying ocean and atmospheric sensing instruments that require sunlight. The Sun-synchronous orbits are useful for the imaging, spy, and weather satellites. Summary of Orbits - Satellites arround Earth (values indicated are approximate) Orbits type Mission Altitude (km) Period Tilt(deg) Shape LEO . Polar sun-sync Remote sensing, Weather 150 - 900 98 - 104 minuts 98 circular . Inclined non-polar International Space Stn 340 91 minuts 51.8 circular . Polar non-sun-sync Earth observing, Scientific 450 - 800 90 - 101 minuts 80 - 94 circular
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OM-MSS Page 168 MEO . Semi synchronous Navigation, Communication, 20,000 12 hours 55 circular Space environment GEO . Geo-synchronous Communication, Early warning 35,786 24 hrs (23hr,56 min,4sec) 0 circular . Geo-stationary Nuclear detection, weather HEO . Geo-synchronous Communication Varies from 12hrs (11hr,58 min) 63.4 long elipse 400 to 35,587 (g) Satellite Ephemeris data : Expressed either by 'Keplerian elements' or by 'State Vectors', that uniquely identify a specific orbit. The Keplerian elements are descriptive of the size, shape, and orientation of an orbital ellipse. The State Vectors represent the 3-D Position and Velocity components of the orbital trajectory at a certain time. These two have their unique advantages. The State Vectors are excellent tool for pre-launch orbit predictions. Keplerian elements, called NASA/NORAD 'Two-Line Elements'(TLE) Ephemeris data set, acquired on a particular day are applied as input to tracking software for accurate predictions of satellite position in next 5 to 7 days. Thereafter a new set of Keplerian elements are acquired. (h) Satellite Orbit Keplerian Element Set : The traditional orbital elements are the six Keplerian elements, that uniquely identify a specific orbit. The Keplerian elements are distributed as NASA/NORAD 'Two-Line Elements'(TLE) Ephemeris data set. The NASA/NORAD 'Two-Line Elements'(TLE) Ephemeris data set are explaned in next section. (i) Satellite Orbit State Vectors Set : It is another common form of Satellite Orbital Element Set. State Vectors represents, Position (X, Y, Z) and Velocity (Vx, Vy, Vz) of a Satellite orbital trajectory in time. Vectors are excellent tool for a pre-launch or any past or future time, prediction of satellite position in orbit. The conversion of Keplerian elements (NASA/NORAD 'Two-Line Elements') to state vectors are presented in next section . (j) Ground Trace : A ground track or ground trace is the path on the surface of the Earth directly below an satellite. It is the projection of the satellite's orbit onto the surface of the Earth. Thus completed, the few preliminaries about 'Satellite Orbit' around earth.
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OM-MSS Page 169 Move on to Satellites Orbit Elements : NASA/NORAD 'Two-Line Elements, Keplerian Element and State Vectors. (Sections - 5.1 to 5.4). First presented NASA/NORAD 'Two-Line Elements'(TLE) Ephemeris data set. This is followed by Conversion of the Keplerian Element Set to State Vector Set and vice versa. Later, Compute Keplerian Element at Perigee prior to Epoch, that can be used as new Ephemeris if perigee is start point for satellite pass. Next Section - 5.1 NASA/NORAD 'Two-Line Elements'(TLE) Ephemeris data set.
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OM-MSS Page 170 OM-MSS Section - 5.1 -----------------------------------------------------------------------------------------------------44 NASA/NORAD 'Two-Line Elements'(TLE) Ephemeris Data Set The Keplerian elements are encoded as text in different formats. The most common format is NASA/NORAD 'Two-Line Elements'(TLE). TLE format consists 3 lines : First line call 'Line 0' contains satellite name, followed by standard two lines call 'Line 1' & 'Line 2' of Orbital Element. TLE is distributed as NASA/NORAD 'Two-Line Elements'(TLE) Ephemeris data set. Example of a TLE for the International Space Station 'ISS (ZARYA)'. There are three lines; line 0 format AAAAAAAAAAA is title 'satellite name', where a disagreement exists about how wide name may be, 11 or 12, or 24 Columns. ISS (ZARYA) 1 NNNNNU NNNNNAAA NNNNN.NNNNNNNN +.NNNNNNNN +NNNNN-N +NNNNN-N N NNNNN line 1 format Total Columns 67, blanks 2 9 18 33 44 53 62 64 . 1 25544U 98067A 08264.51782528 -.00002182 00000-0 -11606-4 0 2927 line 1 with Orbital Element put/filled in. 2 NNNNN NNN.NNNN NNN.NNNN NNNNNNN NNN.NNNN NNN.NNNN NN.NNNNNNNNNNNNNN line 2 format Total Columns 67, blanks 2 8 17 26 34 43 52 . 2 25544 51.6416 247.4627 0006703 130.5360 325.0288 15.72125391563537 line 2 with Orbital Element put/filled in . The format of line 1 & line 2 are explained below. LINE 1 Field Columns Blank Content Values from the above example 1 01-01 02 Line number 1 2 03-07 Satellite number 25544 3 08-08 09 Classification (U = Unclassified) U 4 10-11 International Designator (Last two digits of launch year) 98 5 12-14 International Designator (Launch number of the year) 067 6 15-17 18 International Designator (Piece of the launch) A 7 19-20 Epoch Year (Last two digits of year) 08 8 21-32 33 Epoch (Day of the year and fractional portion of the day) 264.51782528 9 34-43 44 First Time Derivative of the Mean Motion divided by two -.00002182 10 45-52 53 Second Time Derivative of Mean Motion divided by six 00000-0 11 54-61 62 BSTAR drag term -11606-4
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OM-MSS Page 171 12 63-63 64 number 0 (Originally Ephemeris type) 0 13 65-68 Element set number. incremented when a new TLE is generated 292 14 69-69 Checksum (Modulo 10) 7 LINE 2 Field Columns Blank Content Values from the above example 1 01-01 02 Line number 2 2 03-07 08 Satellite number 25544 3 09-16 17 Inclination deg. 51.6416 4 18-25 26 Right Ascension of the Ascending Node (deg) 247.4627 5 27-33 34 Eccentricity (decimal point assumed) 0006703 6 35-42 43 Argument of Perigee (deg) 130.5360 7 44-51 52 Mean Anomaly (deg) 325.0288 8 53-63 Mean Motion (Revs per day) 15.72125391 9 64-68 Revolution number at epoch (Revs) 56353 10 69-69 Checksum (Modulo 10) 7 Note : 1. Where decimal points are assumed, it is leading decimal points, for example Line 1, Field 11 (-11606-4) translates to -0.11606E-4. 2. The orbit number at the epoch time is counted at ascending equator crossings or at perigee. 3. Checksum is computed as follows : Start with zero. For each digit in the line, add the value of the digit. For each minus sign, add 1. For each plus sign, add 2. For each letter, blank, or period, don't add anything. Take the last decimal digit of the result (ie, take the result modulo 10) as the check digit. 4. The International Designator, Line 1 fields (4, 5, 6) are usually blank in the NASA Prediction Bulletins issued. A credible, regular, updated free service source for NASA/NORAD 'Two-Line Elements'(TLE) Bulletins : The Center for Space Standards & Innovation (CSSI) provide worldwide standard data & educational materials to space community. The Data services are offered from CSSI's freely, which includes Two-Line element sets, precision orbit ephemerides, solar weather data, etc. However, for all satellites, you can download the NASA/NORAD 'two-line elements'(TLE) from CelesTrak Web site, URL http://celestrak.com/NORAD/elements/ . See below, for five satellites, the NASA/NORAD 'two-line elements'(TLE) download on May 28, 2014, from site URL http://celestrak.com/NORAD/elements/
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OM-MSS Page 172 The NASA/NORAD 'two-line elements'(TLE) for five satellites download on May 28, 2014, from site URL http://celestrak.com/NORAD/elements/ (a). American Remote Sensing satellite launched on February 11, 2013, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST LANDSAT 8 1 39084U 13008A 14148.14086282 .00000288 00000-0 73976-4 0 4961 2 39084 98.2215 218.5692 0001087 96.5686 263.5699 14.57098925 68534 (b). French Remote Sensing satellite launched on September 9, 2012, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST SPOT 6 1 38755U 12047A 14148.14295346 .00000295 00000-0 73402-4 0 9574 2 38755 98.1987 215.8134 0001368 80.3963 279.7434 14.58528066 91251 (c). Indian Remote Sensing satellite launched on July 12, 2010, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST CARTOSAT 2B 1 36795U 10035A 14148.12955979 .00000641 00000-0 94319-4 0 3461 2 36795 97.9448 207.1202 0016257 44.4835 315.7690 14.78679483209252 (d). International Space Stn launched on Nov. 20, 1998, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST ISS (ZARYA) 1 25544U 98067A 14148.25353351 .00006506 00000-0 11951-3 0 3738 2 25544 51.6471 198.4055 0003968 47.6724 33.3515 15.50569135888233 (e). Indian Geo Comm. Sat launched on Jan. 05, 2014, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST GSAT-14 1 39498U 14001A 14146.03167358 -.00000092 00000-0 00000+0 0 1238 2 39498 0.0049 223.9821 0002051 110.2671 354.6468 1.00272265 1407 (f) Natural satellite Moon moves around Earth in the same kind of orbit as the artificial satellites. For Moon, the NASA/NORAD 'Two-Line Elements'(TLE) Bulletins are not easily available or offered regularly in public domain. Giving Keplerian elements for the Moon is much more difficult. The Moon's orbital plane wobbles around that changes inclination about 18 to 28 degrees. The Moon's orbit is also severely perturbed. The perturbations come from Sun, Earth (not being exact sphere) and from major planets. (Ref. http://www.amsat.org/amsat/archive/amsat-bb/200107/msg00247.html ).
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OM-MSS Page 173 However, for the natural satellite Moon, the Keplerian elements set down load on Jun 14, 2014, 16:14 hrs IST, from site URL http://www.logsat.com/lsp-keplerian-elements.asp & http://www.logsat.com/pub/Jun14.txt is : Satellite: Moon , Catalog number: 000000 , Epoch time: 14143.16621081682 , Element set: 357 , Inclination: 18.7965 deg , RA of node: 352.4777 deg , Eccentricity: 0.0512 Arg of perigee: 316.1136 deg , Mean anomaly: 40.2074 deg , Mean motion: 0.036600996000 rev/day , Decay rate: 0 , Epoch rev: 0 , Checksum: 000 This is rewritten in NASA/NORAD 'Two-Line Elements'(TLE) Bulletins form : MOON (natural satellite of earth) 1 00000U 00000A 14143.16621081 .00000000 00000-0 00000-0 0 3574 2 00000 18.7965 352.4777 0512000 316.1136 40.2074 00.036600996 0006 Thus, explained with examples, the format of NASA/NORAD 'Two-Line Elements'(TLE) as the satellite orbital parameters called Keplerian elements and the source for obtaining them for the satellites launched by any country. Move on to Conversion of Keplerian Element Set to State Vector Set and vice versa, applied to six satellites. These six satellites are - LANDSAT 8, SPOT 6, CARTOSAT-2B, ISS (ZARYA), GSAT-14, and Moon . Next Section - 5.2 Conversion of Keplerian Element Set to State Vector Set and vice versa.
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OM-MSS Page 174 OM-MSS Section - 5.2 -----------------------------------------------------------------------------------------------------45 Conversion of Keplerian Element Set to State Vector Set and Vice versa. This utility is applied to six satellites, LANDSAT 8, SPOT 6, CARTOSAT-2B, ISS (ZARYA), GSAT-14, and Moon . The input is respective Satellite's NASA/NORAD 'Two-Line Elements'(TLE). The output is corrsponding State Vector Set for the respective Satellites. Satellite LANDSAT 8 : Conversion of Keplerian Element Set to State Vector Set and vice versa. (a) LANDSAT 8 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on February 11, 2013 1 39084U 13008A 14148.14086282 .00000288 00000-0 73976-4 0 4961 2 39084 98.2215 218.5692 0001087 96.5686 263.5699 14.57098925 68534 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 39084, LANDSAT 8 , (i) Conversion Utility : Keplerian Elements to State Vectors ( Forward Conversion ) Input : The Keplerian Elements Set at Epoch, extracted from 'Two-Line Elements'(TLE) . EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1408628200 EPOCH_inclination_deg = 98.2215000000 EPOCH_right_asc_acnd_node_deg = 218.5692000000 EPOCH_eccentricity = 0.0001087000 EPOCH_argument_of_perigee_deg = 96.5686000000 EPOCH_mean_anomaly_deg = 263.5699000000 EPOCH_mean_motion_rev_per_day = 14.5709892500 EPOCH_revolution = 6853 EPOCH_node_condition = 1 Output : The Computed corresponding State Vectors Set, Position vector(X, Y, Z) and Velocity vector(Vx, Vy, Vz) at Epoch. EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1408628200 X = -5535.2447229896 Y = -4411.0085700927 Z = 15.4200278230 R = 7077.8646869006 Vx = -655.5016695670 Vy = 849.8345806371 Vz = 7427.2400585557 V = 7504.3851274216 EPOCH_revolution = 6853 EPOCH_node_condition = 1
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OM-MSS Page 175 (ii) Conversion Utility : State Vectors to Keplerian Elements ( Backward Conversion ) Input : The State Vectors Set at Epoch, computed just above from 'Two-Line Elements'(TLE). EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1408628200 X = -5535.2447229896 Y = -4411.0085700927 Z = 15.4200278230 R = 7077.8646869006 Vx = -655.5016695670 Vy = 849.8345806371 Vz = 7427.2400585557 V = 7504.3851274216 EPOCH_revolution = 6853 EPOCH_node_condition = 1 Output : The Computed corresponding Keplerian Elements Set at Epoch. EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1408628200 EPOCH_inclination_deg = 98.2215000000 EPOCH_right_asc_acnd_node_deg = 218.5692000000 EPOCH_eccentricity = 0.0001087000 EPOCH_argument_of_perigee_deg = 96.5686000000 EPOCH_mean_anomaly_deg = 263.5698999999 EPOCH_mean_motion_rev_per_day = 14.5709892500 EPOCH_revolution = 6853 EPOCH_node_condition = 1 Continue To next Satellite
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OM-MSS Page 176 Satellite SPOT 6 : Conversion of Keplerian Element Set to State Vector Set and vice versa. (b) SPOT 6 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on September 9, 2012 1 38755U 12047A 14148.14295346 .00000295 00000-0 73402-4 0 9574 2 38755 98.1987 215.8134 0001368 80.3963 279.7434 14.58528066 91251 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 38755, SPOT 6 , (i) Conversion Utility : Keplerian Elements to State Vectors ( Forward Conversion ) Input : The Keplerian Elements Set at Epoch, extracted from 'Two-Line Elements'(TLE) . EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1429534600 EPOCH_inclination_deg = 98.1987000000 EPOCH_right_asc_acnd_node_deg = 215.8134000000 EPOCH_eccentricity = 0.0001368000 EPOCH_argument_of_perigee_deg = 80.3963000000 EPOCH_mean_anomaly_deg = 279.7434000000 EPOCH_mean_motion_rev_per_day = 14.5852806600 EPOCH_revolution = 9125 EPOCH_node_condition = 1 Output : The Computed corresponding State Vectors Set, Position vector(X, Y, Z) and Velocity vector(Vx, Vy, Vz) at Epoch. EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1429534600 X = -5736.9414700815 Y = -4136.9553443077 Z = 15.1814434008 R = 7072.9857505978 Vx = -612.4123815408 Vy = 878.2634599352 Vz = 7430.3591738511 V = 7507.1055062891 EPOCH_revolution = 9125 EPOCH_node_condition = 1 (ii) Conversion Utility : State Vectors to Keplerian Elements ( Backward Conversion ) Input : The State Vectors Set at Epoch, computed just above from 'Two-Line Elements'(TLE). EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1429534600 X = -5736.9414700815 Y = -4136.9553443077 Z = 15.1814434008 R = 7072.9857505978 Vx = -612.4123815408 Vy = 878.2634599352 Vz = 7430.3591738511 V = 7507.1055062891 EPOCH_revolution = 9125 EPOCH_node_condition = 1 Output : The Computed corresponding Keplerian Elements Set at Epoch.
OM-MSS Page 178 Satellite CARTOSAT 2B : Conversion of Keplerian Element Set to State Vector Set and vice versa. (c) CARTOSAT 2B 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on July 12, 2010 1 36795U 10035A 14148.12955979 .00000641 00000-0 94319-4 0 3461 2 36795 97.9448 207.1202 0016257 44.4835 315.7690 14.78679483209252 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 36795, CARTOSAT 2B , (i) Conversion Utility : Keplerian Elements to State Vectors ( Forward Conversion ) Input : The Keplerian Elements Set at Epoch, extracted from 'Two-Line Elements'(TLE) . EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1295597900 EPOCH_inclination_deg = 97.9448000000 EPOCH_right_asc_acnd_node_deg = 207.1202000000 EPOCH_eccentricity = 0.0016257000 EPOCH_argument_of_perigee_deg = 44.4835000000 EPOCH_mean_anomaly_deg = 315.7690000000 EPOCH_mean_motion_rev_per_day = 14.7867948300 EPOCH_revolution = 20925 EPOCH_node_condition = 1 Output : The Computed corresponding State Vectors Set, Position vector(X, Y, Z) and Velocity vector(Vx, Vy, Vz) at Epoch. EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1295597900 X = -6231.7560551250 Y = -3189.4018492384 Z = 14.8069953230 R = 7000.5204759091 Vx = -453.7396013124 Vy = 940.0898291212 Vz = 7477.6527638575 V = 7550.1615459169 EPOCH_revolution = 20925 EPOCH_node_condition = 1 (ii) Conversion Utility : State Vectors to Keplerian Elements ( Backward Conversion ) Input : The State Vectors Set at Epoch, computed just above from 'Two-Line Elements'(TLE). EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1295597900 X = -6231.7560551250 Y = -3189.4018492384 Z = 14.8069953230 R = 7000.5204759091 Vx = -453.7396013124 Vy = 940.0898291212 Vz = 7477.6527638575 V = 7550.1615459169 EPOCH_revolution = 20925 EPOCH_node_condition = 1 Output : The Computed corresponding Keplerian Elements Set at Epoch.
OM-MSS Page 180 Satellite ISS (ZARYA) : Conversion of Keplerian Element Set to State Vector Set and vice versa. (d) ISS (ZARYA) 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on November 20, 1998 1 25544U 98067A 14148.25353351 .00006506 00000-0 11951-3 0 3738 2 25544 51.6471 198.4055 0003968 47.6724 33.3515 15.50569135888233 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 25544, ISS (ZARYA) , (i) Conversion Utility : Keplerian Elements to State Vectors ( Forward Conversion ) Input : The Keplerian Elements Set at Epoch, extracted from 'Two-Line Elements'(TLE) . EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.2535335100 EPOCH_inclination_deg = 51.6471000000 EPOCH_right_asc_acnd_node_deg = 198.4055000000 EPOCH_eccentricity = 0.0003968000 EPOCH_argument_of_perigee_deg = 47.6724000000 EPOCH_mean_anomaly_deg = 33.3515000000 EPOCH_mean_motion_rev_per_day = 15.5056913500 EPOCH_revolution = 88823 EPOCH_node_condition = 1 Output : The Computed corresponding State Vectors Set, Position vector(X, Y, Z) and Velocity vector(Vx, Vy, Vz) at Epoch. EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.2535335100 X = 311.7253734371 Y = -4283.4907194611 Z = 5261.0200081909 R = 6791.4502853764 Vx = 7415.4532574405 Vy = 1686.8647169809 Vz = 936.2139516379 V = 7662.3074951298 EPOCH_revolution = 88823 EPOCH_node_condition = 1 (ii) Conversion Utility : State Vectors to Keplerian Elements ( Backward Conversion ) Input : The State Vectors Set at Epoch, computed just above from 'Two-Line Elements'(TLE). EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.2535335100 X = 311.7253734371 Y = -4283.4907194611 Z = 5261.0200081909 R = 6791.4502853764 Vx = 7415.4532574405 Vy = 1686.8647169809 Vz = 936.2139516379 V = 7662.3074951298 EPOCH_revolution = 88823 EPOCH_node_condition = 1 Output : The Computed corresponding Keplerian Elements Set at Epoch.
OM-MSS Page 182 Satellite GSAT-14) : Conversion of Keplerian Element Set to State Vector Set and vice versa. (e) GSAT-14 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:14 hrs IST, Satellite launched on January 05, 2014 1 39498U 14001A 14146.03167358 -.00000092 00000-0 00000+0 0 1238 2 39498 0.0049 223.9821 0002051 110.2671 354.6468 1.00272265 1407 From this TLE, the data relevant for the purpose are manually interpreted & extracted: Satellite number 39498, GSAT-14 , (i) Conversion Utility : Keplerian Elements to State Vectors ( Forward Conversion ) Input : The Keplerian Elements Set at Epoch, extracted from 'Two-Line Elements'(TLE) . EPOCH_year = 2014 EPOCH_days_decimal_of_year = 145.0316735800 EPOCH_inclination_deg = 0.0049000000 EPOCH_right_asc_acnd_node_deg = 223.9821000000 EPOCH_eccentricity = 0.0002051000 EPOCH_argument_of_perigee_deg = 110.2671000000 EPOCH_mean_anomaly_deg = 354.6468000000 EPOCH_mean_motion_rev_per_day = 1.0027226500 EPOCH_revolution = 140 EPOCH_node_condition = 1 Output : The Computed corresponding State Vectors Set, Position vector(X, Y, Z) and Velocity vector(Vx, Vy, Vz) at Epoch. EPOCH_year = 2014 EPOCH_days_decimal_of_year = 145.0316735800 X = 36095.3223130873 Y = -21779.4122304999 Z = 3.4839025250 R = 42157.0290951495 Vx = 1588.6953038064 Vy = 2633.0807004007 Vz = -0.0676820819 V = 3075.2344215913 EPOCH_revolution = 140 EPOCH_node_condition = 1 (ii) Conversion Utility : State Vectors to Keplerian Elements ( Backward Conversion ) Input : The State Vectors Set at Epoch, computed just above from 'Two-Line Elements'(TLE). EPOCH_year = 2014 EPOCH_days_decimal_of_year = 145.0316735800 X = 36095.3223130873 Y = -21779.4122304999 Z = 3.4839025250 R = 42157.0290951495 Vx = 1588.6953038064 Vy = 2633.0807004007 Vz = -0.0676820819 V = 3075.2344215913 EPOCH_revolution = 140 EPOCH_node_condition = 1 Output : The Computed corresponding Keplerian Elements Set at Epoch.
OM-MSS Page 184 Satellite MOON : Conversion of Keplerian Element Set to State Vector Set and vice versa. (f) MOON 'Two-Line Elements'(TLE) downloaded on Jun 14, 2014, 16:14 hrs IST, Natural Satellite 1 00000U 00000A 14143.16621081 .00000000 00000-0 00000-0 0 3574 2 00000 18.7965 352.4777 0512000 316.1136 40.2074 00.036600996 0006 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 00000, MOON , (i) Conversion Utility : Keplerian Elements to State Vectors ( Forward Conversion ) Input : The Keplerian Elements Set at Epoch, extracted from 'Two-Line Elements'(TLE) . EPOCH_year = 2014 EPOCH_days_decimal_of_year = 142.1662108168 EPOCH_inclination_deg = 18.7965000000 EPOCH_right_asc_acnd_node_deg = 352.4777000000 EPOCH_eccentricity = 0.0512000000 EPOCH_argument_of_perigee_deg = 316.1136000000 EPOCH_mean_anomaly_deg = 40.2074000000 EPOCH_mean_motion_rev_per_day = 0.0366009960 EPOCH_revolution = 0 EPOCH_node_condition = 1 Output : The Computed corresponding State Vectors Set, Position vector(X, Y, Z) and Velocity vector(Vx, Vy, Vz) at Epoch. EPOCH_year = 2014 EPOCH_days_decimal_of_year = 142.1662108168 X = 365705.5648844948 Y = -46450.6213911481 Z = 620.9529484744 R = 368644.2811134813 Vx = 161.8765603889 Vy = 989.7819390712 Vz = 341.1953415596 V = 1059.3802758296 EPOCH_revolution = 0 EPOCH_node_condition = 1 (ii) Conversion Utility : State Vectors to Keplerian Elements ( Backward Conversion ) Input : The State Vectors Set at Epoch, computed just above from 'Two-Line Elements'(TLE). EPOCH_year = 2014 EPOCH_days_decimal_of_year = 142.1662108168 X = 365705.5648844948 Y = -46450.6213911481 Z = 620.9529484744 R = 368644.2811134813 Vx = 161.8765603889 Vy = 989.7819390712 Vz = 341.1953415596 V = 1059.3802758296 EPOCH_revolution = 0 EPOCH_node_condition = 1 Output : The Computed corresponding Keplerian Elements Set at Epoch.
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OM-MSS Page 185 EPOCH_year = 2014 EPOCH_days_decimal_of_year = 142.1662108168 EPOCH_inclination_deg = 18.7965000000 EPOCH_right_asc_acnd_node_deg = 352.4777000000 EPOCH_eccentricity = 0.0512000000 EPOCH_argument_of_perigee_deg = 316.1136000000 EPOCH_mean_anomaly_deg = 40.2074000000 EPOCH_mean_motion_rev_per_day = 0.0366009960 EPOCH_revolution = 0 EPOCH_node_condition = 1 End of the Conversion of the Keplerian Element Set to State Vector Set and vice versa for six Satellites. If any need arise, the above utility can accurately convert the Keplerian Element set into State Vector Set, or the reverse. The source for Keplerian Element set, is NASA/NORAD 'Two-Line Elements'(TLE), mentioned before. Next Section - 5.3 Satellite Orbit Keplerian element at Perigee, prior to Epoch.
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OM-MSS Page 186 OM-MSS Section - 5.3 -----------------------------------------------------------------------------------------------------46 Satellite Orbit Keplerian Element set at Perigee, prior to Epoch. Compute Satellite Orbit Keplerian element set at Perigee prior to Epoch. This utility is applied to six satellites, LANDSAT 8, SPOT 6, CARTOSAT-2B, ISS (ZARYA), GSAT-14, and Moon . The input is of the respective Satellite's NASA/NORAD 'Two-Line Elements'(TLE). The outputs is corrsponding Keplerian element set at Perigee prior to Epoch, for the respective Satellites. (a) LANDSAT 8 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on February 11, 2013 1 39084U 13008A 14148.14086282 .00000288 00000-0 73976-4 0 4961 2 39084 98.2215 218.5692 0001087 96.5686 263.5699 14.57098925 68534 From this TLE, the Keplerian Element manually interpreted & extracted : Satellite number 39084, LANDSAT 8 , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1408628200 EPOCH_inclination_deg = 98.2215000000 EPOCH_right_asc_acnd_node_deg = 218.5692000000 EPOCH_eccentricity = 0.0001087000 EPOCH_argument_of_perigee_deg = 96.5686000000 EPOCH_mean_anomaly_deg = 263.5699000000 EPOCH_mean_motion_rev_per_day = 14.5709892500 EPOCH_revolution = 6853 EPOCH_node_condition = 1 Computed Satellite Orbit Keplerian Element Set at PERIGEE prior to Epoch from NORAD TLE as Epoch. PERIGEE_year = 2014 PERIGEE_days_decimal_of_year = 147.0905854502 PERIGEE_inclination_deg = 98.2215000000 PERIGEE_right_asc_acnd_node_deg = 218.5196064188 PERIGEE_eccentricity = 0.0001087000 PERIGEE_argument_of_perigee_deg = 96.7242739798 PERIGEE_mean_anomaly_deg = 359.9999999029 PERIGEE_mean_motion_rev_per_day = 14.5619910304 PERIGEE_revolution = 6853 PERIGEE_node_condition = 1 Move on to next Satellite computing values of 'Keplerian element set at Perigee prior to Epoch' Continue To next Satellite
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OM-MSS Page 187 (b) SPOT 6 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on September 9, 2012 1 38755U 12047A 14148.14295346 .00000295 00000-0 73402-4 0 9574 2 38755 98.1987 215.8134 0001368 80.3963 279.7434 14.58528066 91251 From this TLE, the Keplerian Element manually interpreted & extracted : Satellite number 38755, SPOT 6 , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1429534600 EPOCH_inclination_deg = 98.1987000000 EPOCH_right_asc_acnd_node_deg = 215.8134000000 EPOCH_eccentricity = 0.0001368000 EPOCH_argument_of_perigee_deg = 80.3963000000 EPOCH_mean_anomaly_deg = 279.7434000000 EPOCH_mean_motion_rev_per_day = 14.5852806600 EPOCH_revolution = 9125 EPOCH_node_condition = 1 Computed Satellite Orbit Keplerian Element Set at PERIGEE prior to Epoch from NORAD TLE as Epoch. PERIGEE_year = 2014 PERIGEE_days_decimal_of_year = 147.0896431408 PERIGEE_inclination_deg = 98.1987000000 PERIGEE_right_asc_acnd_node_deg = 215.7608394403 PERIGEE_eccentricity = 0.0001368000 PERIGEE_argument_of_perigee_deg = 80.5618466274 PERIGEE_mean_anomaly_deg = 0.0000002622 PERIGEE_mean_motion_rev_per_day = 14.5762585790 PERIGEE_revolution = 9125 PERIGEE_node_condition = 1 Move on to next Satellite computing values of 'Keplerian element set at Perigee prior to Epoch' Continue To next Satellite
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OM-MSS Page 188 (c) CARTOSAT 2B 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on July 12, 2010 1 36795U 10035A 14148.12955979 .00000641 00000-0 94319-4 0 3461 2 36795 97.9448 207.1202 0016257 44.4835 315.7690 14.78679483209252 From this TLE, the Keplerian Element manually interpreted & extracted : Satellite number 36795, CARTOSAT 2B , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1295597900 EPOCH_inclination_deg = 97.9448000000 EPOCH_right_asc_acnd_node_deg = 207.1202000000 EPOCH_eccentricity = 0.0016257000 EPOCH_argument_of_perigee_deg = 44.4835000000 EPOCH_mean_anomaly_deg = 315.7690000000 EPOCH_mean_motion_rev_per_day = 14.7867948300 EPOCH_revolution = 20925 EPOCH_node_condition = 1 Computed Satellite Orbit Keplerian Element Set at PERIGEE prior to Epoch from NORAD TLE as Epoch. PERIGEE_year = 2014 PERIGEE_days_decimal_of_year = 147.0702033672 PERIGEE_inclination_deg = 97.9448000000 PERIGEE_right_asc_acnd_node_deg = 207.0616328473 PERIGEE_eccentricity = 0.0016257000 PERIGEE_argument_of_perigee_deg = 44.6751258926 PERIGEE_mean_anomaly_deg = 359.9999995488 PERIGEE_mean_motion_rev_per_day = 14.7774423308 PERIGEE_revolution = 20925 PERIGEE_node_condition = 1 Move on to next Satellite computing values of 'Keplerian element set at Perigee prior to Epoch' Continue To next Satellite
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OM-MSS Page 189 (d) ISS (ZARYA) 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on November 20, 1998 1 25544U 98067A 14148.25353351 .00006506 00000-0 11951-3 0 3738 2 25544 51.6471 198.4055 0003968 47.6724 33.3515 15.50569135888233 From this TLE, the Keplerian Element manually interpreted & extracted : Satellite number 25544, ISS (ZARYA) , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.2535335100 EPOCH_inclination_deg = 51.6471000000 EPOCH_right_asc_acnd_node_deg = 198.4055000000 EPOCH_eccentricity = 0.0003968000 EPOCH_argument_of_perigee_deg = 47.6724000000 EPOCH_mean_anomaly_deg = 33.3515000000 EPOCH_mean_motion_rev_per_day = 15.5056913500 EPOCH_revolution = 88823 EPOCH_node_condition = 1 Computed Satellite Orbit Keplerian Element Set at PERIGEE prior to Epoch from NORAD TLE as Epoch. PERIGEE_year = 2014 PERIGEE_days_decimal_of_year = 147.2475593942 PERIGEE_inclination_deg = 51.6471000000 PERIGEE_right_asc_acnd_node_deg = 198.4350551515 PERIGEE_eccentricity = 0.0003968000 PERIGEE_argument_of_perigee_deg = 47.6503677902 PERIGEE_mean_anomaly_deg = 0.0000008019 PERIGEE_mean_motion_rev_per_day = 15.5074083546 PERIGEE_revolution = 88823 PERIGEE_node_condition = 1 Move on to next Satellite computing values of 'Keplerian element set at Perigee prior to Epoch' Continue To next Satellite
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OM-MSS Page 190 (e) GSAT-14 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:14 hrs IST, Satellite launched on January 05, 2014 1 39498U 14001A 14146.03167358 -.00000092 00000-0 00000+0 0 1238 2 39498 0.0049 223.9821 0002051 110.2671 354.6468 1.00272265 1407 From this TLE, the Keplerian Element manually interpreted & extracted : Satellite number 39498, GSAT-14 , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 145.0316735800 EPOCH_inclination_deg = 0.0049000000 EPOCH_right_asc_acnd_node_deg = 223.9821000000 EPOCH_eccentricity = 0.0002051000 EPOCH_argument_of_perigee_deg = 110.2671000000 EPOCH_mean_anomaly_deg = 354.6468000000 EPOCH_mean_motion_rev_per_day = 1.0027226500 EPOCH_revolution = 140 EPOCH_node_condition = 1 Computed Satellite Orbit Keplerian Element Set at PERIGEE prior to Epoch from NORAD TLE as Epoch. PERIGEE_year = 2014 PERIGEE_days_decimal_of_year = 144.0492548835 PERIGEE_inclination_deg = 0.0049000000 PERIGEE_right_asc_acnd_node_deg = 223.9952481674 PERIGEE_eccentricity = 0.0002051000 PERIGEE_argument_of_perigee_deg = 110.2408036654 PERIGEE_mean_anomaly_deg = 0.0000000029 PERIGEE_mean_motion_rev_per_day = 1.0027598249 PERIGEE_revolution = 140 PERIGEE_node_condition = 1 Move on to next Satellite computing values of 'Keplerian element set at Perigee prior to Epoch' Continue To next Satellite
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OM-MSS Page 191 (f) MOON 'Two-Line Elements'(TLE) downloaded on Jun 14, 2014, 16:14 hrs IST, Natural Satellite 1 00000U 00000A 14143.16621081 .00000000 00000-0 00000-0 0 3574 2 00000 18.7965 352.4777 0512000 316.1136 40.2074 00.036600996 0006 From this TLE, the Keplerian Element manually interpreted & extracted : Satellite number 00000, MOON , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 142.1662108168 EPOCH_inclination_deg = 18.7965000000 EPOCH_right_asc_acnd_node_deg = 352.4777000000 EPOCH_eccentricity = 0.0512000000 EPOCH_argument_of_perigee_deg = 316.1136000000 EPOCH_mean_anomaly_deg = 40.2074000000 EPOCH_mean_motion_rev_per_day = 0.0366009960 EPOCH_revolution = 0 EPOCH_node_condition = 1 Computed Satellite Orbit Keplerian Element Set at PERIGEE prior to Epoch from NORAD TLE as Epoch. PERIGEE_year = 2014 PERIGEE_days_decimal_of_year = 139.1147315693 PERIGEE_inclination_deg = 18.7965000000 PERIGEE_right_asc_acnd_node_deg = 352.4777171774 PERIGEE_eccentricity = 0.0512000000 PERIGEE_argument_of_perigee_deg = 316.1135684193 PERIGEE_mean_anomaly_deg = 359.9999999986 PERIGEE_mean_motion_rev_per_day = 0.0366010099 PERIGEE_revolution = 0 PERIGEE_node_condition = 1 End of the computing values of 'Keplerian element set at Perigee prior to Epoch' for the six Satellites. The Epoch corresponds to NORAD TLE (epoch year & epoch day fraction of the year) of the respective satellites. The Keplerian element at Perigee are often adopted as start point for 'Satellite Pass - Prediction of Ground Trace'. Next Section - 5.4 Concluding NASA/NORAD 'Two-Line Elements, Keplerian Element & State Vectors.
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OM-MSS Page 192 OM-MSS Section - 5.4 -----------------------------------------------------------------------------------------------------47 Concluding Satellite Ephemeris Data, NASA/NORAD 'Two-Line Elements, Keplerian Element & State Vectors. (Sections 5.0 to 5.3) In previous Sections (5.0 to 5.3), the following were presented : 1. Few preliminaries about Satellite Orbit around earth, that included Earth Gravity and Satellite Motion, Velocity equations, Attitude control, Orbit Time period, Orbits types and categories, Ephemeris data and Ground Trace. 2. Explained with examples, the format of NASA/NORAD 'Two-Line Elements'(TLE) as the satellite orbital parameters for six satellites - LANDSAT 8, SPOT 6, CARTOSAT 2B, ISS (ZARYA), GSAT-14 and Moon. 3. Conversion of Keplerian Element Set to State Vector Set and Vice versa, applied to all six satellites mentioned above. 4. Computed Satellite Orbit Keplerian element set at Perigee prior to Epoch for all six satellites, using respective satellite's TLE. End of preliminaries about 'Satellite Orbit' around earth and Satellite Ephemeris data and convesion utilities of the OM-MSS software. Move on to satellites motion around earth, computing Orbital & Positional Parameters at Epoch. Next Section - 6 Satellites Motion in Orbit around Earth.
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OM-MSS Page 193 OM-MSS Section - 6 -------------------------------------------------------------------------------------------------------48 SATELLITES MOTION AROUND EARTH : ORBITAL & POSITIONAL PARAMETERS AT EPOCH The Satellites Orbit around Earth, Counterclockwise, in the same way as Earth orbits around Sun. In the previous section, the Preliminaries about 'Satellite Orbit' followed by NASA/NORAD 'Two-Line Elements'(TLE) were presented. Here Presented Satellites Motion around Earth : Computing Orbital & Positional parameters, the OM-MSS Software Utility. This utility is applied one-by-one to six Satellites, LANDSAT 8, SPOT 6, CARTOSAT-2B, ISS (ZARYA), GSAT-14, and Moon . The Input is NASA/NORAD 'Two-Line Elements'(TLE) Bulletin of the respective Satellite. The Output is the corresponding Satellite's Motion around Earth the Orbital & Positional parameters. Satellite Motion around Earth is represented by computing around 120 orbital parameters, put into 28 groups. The number is large, because some parameters are computed using more than one model equation, that require different inputs. This confirms accuracy & validation of results and understanding the different input considerations. Satellite Orbital & Positional parameters for computation purpose are put into following groups : 01. UT Year and Days decimal of year : Convert into UT YY MM DD hh min sec & Julian day. 02. Satellite orbit Semi major axis in km, Ignoring and also Considering earth oblatenes. 03. Satellite Mean motion in rev per day, Ignoring and also Considering earth oblatenes. 04. Satellite Orbit Time Period in minute at time_t Considering earth oblatenes. 05. Satellite Rate of change of Right Ascension and Argument of Perigee in deg per_day at time_t. 06. Satellite Mean anomaly, Eccentric anomaly, True anomaly in deg at time_t considering earth oblateness. 07. Satellite position vector[rp, rq] from Earth center(EC) to Sat in PQW frame, perifocal coordinate system. 08. Satellite Position Range Vector from Earth Ceter(EC) to Satellite(SAT) - finding Range Vector[rI rJ rK r] Components in km in frame IJK 09. GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input at time_t. 10. Satellite(SAT) Orbit point direction : Finding Right Ascension(Alpha) deg and Declination(Delta) deg using angles 11. Satellite Longitude & Latitude in deg at time_t; (ie Sub-Sat point log & lat on earth surface ). 12. Satellite height in km from EC to Sat and from Earth surface to Sat at time_t. 13. Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface at time_t.
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OM-MSS Page 194 14. Local sidereal time(LST) and Local mean time(LMT) over Sub-Sat point Longitude on earth. 15. Local sidereal time(LST) and Local mean time(LMT) Over Earth stn (ES) or Earth point(EP) Longitude. 16. Earth Stn Position Vector from Earth Center(EC) to Earth Stn(ES) : Finding Range Vector[RI, RJ, RK, R] Components in IJK frame. 17. Satellite Position Range Vector from Earth Stn(ES) to SAT : finding Range Vector[rvI, rvJ, rvK, rv] Components in km in IJK frame 18. Satellite Position Range Vector from Earth Stn(ES) to SAT : finding Range Vector[rvS, rvE, rvZ, rv] Components in km in SEZ frame 19. Elevation(EL) and Azimuth(AZ) angle of Satellite at Earth Observation point ES or EP. 20. Satellite Velocity meter per sec in orbit. 21. Satellite Velocity Vector [vX, vY, vZ] in meter per sec in orbit in frame XYZ. 22. Satellite Pitch and Roll angles. 23. Satellite State Vectors - Position [ X, Y, Z ] in km and velocity [ Vx, Vy, Vz ] in meter per sec at time_t. 24. Satellite Direction ie right ascension alpha deg and declination delta deg from sat position vector. 25. Satellite Angular momentum km sqr per sec : finding Hx Hy Hz H from state vector pos and vel. 26. Satellite Orbit normal Vector : finding Wx Wy Wz W Delta Alpha from r_sat_pos frame IJK, i, RA. 27. Satellite Position Keplerian elements computed using State Vector, at time input UT. 28. Satellite position State Vectors, computed using Keplerian elements at time input UT Computing Orbital & Positional parameters, for following Satellites respectively : (a) LANDSAT 8 : American satellite launched on February 11, 2013, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST (a) SPOT 6 : French satellite launched on September 9, 2012, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST (a) CARTOSAT 2B : Indian satellite launched on July 12, 2010, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST (a) ISS (ZARYA) : International Space Stn launched on Nov. 20, 1998, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST (a) GSAT-14 : Indian Geo Comm. Sat launched on Jan. 05, 2014, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST (a) Moon : Natural satellite, moves around Earth , the Keplerian elements set down load on Jun 14, 2014, 16:14 hrs IST Input NASA/NORAD 'TWO-LINE ELEMENTS' of respective Satellite, and Earth stn Latitude & Longitude in deg and Height in meter. Move on to all six respective Satellites, one-by-one for computing Orbital & Positional parameters in Section (6.1 to 6.7). Next Section - 6.1 Computing Orbital & Positional parameters for Satellite LANDSAT 8 .
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OM-MSS Page 195 OM-MSS Section - 6.1 ---------------------------------------------------------------------------------------------------49 Satellite LANDSAT 8 : Computing Orbital & Positional parameters corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins. (a) LANDSAT 8 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on February 11, 2013 1 39084U 13008A 14148.14086282 .00000288 00000-0 73976-4 0 4961 2 39084 98.2215 218.5692 0001087 96.5686 263.5699 14.57098925 68534 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 39084, LANDSAT 8 , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1408628200 EPOCH_inclination_deg = 98.2215000000 EPOCH_right_asc_acnd_node_deg = 218.5692000000 EPOCH_eccentricity = 0.0001087000 EPOCH_argument_of_perigee_deg = 96.5686000000 EPOCH_mean_anomaly_deg = 263.5699000000 EPOCH_mean_motion_rev_per_day = 14.5709892500 EPOCH_revolution = 6853 EPOCH_node_condition = 1 Earth_stn_latitude_deg = 23.25993 Earth stn longitude_deg = 77.41261 Earth surface height_meter = 494.70000 Earth stn tower_height_meter = 15.00000 Earth stn_height_meter = 509.70000 Earth stn min EL look_angle_deg = -5.00000 Note : EPOCH Corresponds to UT year = 2014, month = 5, day = 28, hr = 3, min = 22, sec = 50.54765 which is Greenwich meam time ie GMT Converted to Local Meam Time at Earth Stn Longitude, as regular Calender date : year = 2014, month = 5, day = 28, hr = 8, min = 32, sec = 29.58 At this instant Sun Position as seen from Earth Stn Longitude : Sun angles EL deg = 42.87, AZ deg = 81.74, Sun Surface distance km = 5246.15, Radial km = 151596372.18, Sun Rise D:28, H:05, M:17, S:03 Sun Set D:28, H:18, M:40, S:20 Move to Compute Satellie Orbital parameters corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins, and Earth Stn location. 01. Input EPOCH_year and EPOCH_days_decimal_of_year, Converted into UT YY MM DD hh min sec & Julian day. S01 011. Input UT year = 2014, month = 5, day = 28, hr = 3, min = 22, sec = 50.54765, and julian_day = 2456805.6408628202 02. Finding Satellite orbit Semi major axis in km, Ignoring and also Considering earth oblatenes . (a) Semi major axis (SMA) km at time t Ignoring earth oblatenes . Inputs : SAT mean_motion rev per day at time t, GM_EARTH . Page 196
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OM-MSS Page 196 Outputs : SAT Orbit semi major axis km in km and constant_A, constant_k1 . S02A 011. Satellite orbit Semi major axis in km at time t, Ignoring earth oblatenes = 7080.69383, (b) Semi major axis (SMA) in km at time_t Considering earth oblatenes Inputs : SAT mean_motion rev per day at time_t, GM_EARTH , inclination deg at time_t, eccentricity at time_t, constant_k2 Outputs : SAT Orbit semi major axis km in km at time_t and constant_A, constant_k1, constant_k3 . S02B 011. Satellite orbit Semi major axis km at time t, Considering earth oblatenes = 7077.77844, 03. Finding Satellite Mean motion in rev per day, Ignoring and also Considering earth oblatenes . (a) Nominal mean motion rev per day at time_t Ignoring earth oblateness . Inputs : SAT Orbit semi major axis in km ignoring oblateness at time_t, constant_k1 . Outputs : SAT nominal mean motion in rev_per_day at time_t, ignoring earth oblateness S03A 011. Satellite Nominal Mean motion in rev_per_day at time_t using SMA Ignoring earth oblateness = 14.57099, (b) Mean motion rev per day at time_t Considering earth oblatenes Inputs : SAT nominal mean motion rad per day at time_t Ignoring_oblateness, constant_k2, constant_k3, SAT orbit semi major axis in km considering oblateness at time_t. Outputs : SAT mean motion rev per day at time_t, considering earth oblatenes . S03B 011. Satellite Mean motion in rev_per_day at time t using SMA Considering earth oblatenes = 14.56199, Note - This calculted value is slightly less than the mean motion rev_per_day as EPH input from NORAD TLE 04. Finding Satellite Orbit Time Period in minute at time_t Considering earth oblatenes . Inputs : SAT orbit semi major axis in km considering oblateness at time_t, GM_EARTH . Outputs : SAT orbit time period in minute at time_t considering earth oblatenes . S04 011. Satellite orbit Time Period in minute at time_t using SMA Considering earth oblatenes = 98.76548, 05. Finding Satellite Rate of change of Right Ascension and Argument of Perigee in deg per_day at time_t. (a) Rate of change of Right Ascension in deg per day at time_t . Inputs : SAT mean motion rev per day at time_t considering earth oblatenes, constant_k2, SAT orbit eccentricity at time_t, SAT semi major axis km considering oblateness at time_t, SAT orbit inclination deg at time_t .
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OM-MSS Page 197 Outputs : SAT rate of change of right ascension in deg per day at time_t and constant_k_deg_per_day S05A 011. Satellite Rate of change of Right Ascension in deg per day at time_t = 0.98640, (b) Rate of change of Argument of Perigee in deg per_day at time_t . Inputs : SAT orbit constant_k_deg_per_day, SAT orbit inclination deg at time_t SAT orbit semi major axis km considering oblateness at time_t Outputs : SAT rate of change of argument of perigee in deg per day at time_t S05B 011. Satellite Rate of change of Argument of Perigee in deg per day at time_t = -3.09630, 06. Finding Satellite Mean anomaly, Eccentric anomaly, True anomaly in deg at time_t considering earth oblateness. Inputs : SAT mean anomaly rad at time_t, mean_motion_rad_per_day_at_time_t considering_oblateness, SAT orbit eccentricity_at_time_t Outputs : SAT mean anomaly, eccentric anomaly, and true anomaly in deg at time_t onsidering earth oblateness. S06A 011. Satellite Mean anomaly in deg at time_t = 263.56990, same as EPH mean anomaly S06A 011. Satellite Eccentric anomaly in deg at time_t = 263.56371, S06A 011. Satellite True anomaly in deg at time_t = 263.55752, Note - 1. Here after, the Earth Oblateness is always considered for the computation of satellite orbit parameters, and not repeatedly mentioned. Satellite to Earth, the Position Vectors coordinate and the Vector Coordinate Transforms are in PQW, IJK, SEZ frames . - Perifocal Coordinate System (PQW) is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. - Geocentric Coordinate System(IJK) is Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). - Topocentric Horizon Coordinate System(SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). Each of these coordinate system were explained in detail before and therefore not repeated any more. 07. Finding Satellite position vector[rp, rq] from Earth center(EC) to Sat in PQW frame, perifocal coordinate system. Inputs : SAT orbit semi-major axis (SMA), SAT orbit eccentricity, SAT eccentric anomaly, SATtrue anomaly at time_t Outputs : Vector(r, rp rq) in PQW frame S07A 011. r Satellite pos vector magnitude EC to Sat km in PQW frame perifocal cord at time_t = 7077.86469 S07B 011. rp Satellite pos vector component EC to Sat km in PQW frame perifocal cord at time_t = -794.1764390079 S07C 011. rq Satellite pos vector component EC to Sat km in PQW frame perifocal cord at time_t = -7033.1680137615
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OM-MSS Page 198 08. Satellite Position Vector Earth Ceter(EC) to Satellite(SAT) - finding Range Vector(rI rJ rK r) Components in km in frame IJK Note - Transform_1 : EC to SAT Vector(rp, rq) in frame PQW To EC to SAT Vector(rI rJ rK r) in frame IJK. Inputs : Vector(rp, rq) EC to Sat in km in frame PQW, SAT right ascension of ascending node at time_t, SAT argument of perigee rad at time_t, SAT orbit inclination rad at time_t . Outputs : Vector(rI, rJ, rK, r) EC to SAT in km in frame IJK S08A 011. rI Satellite pos vector component EC to Sat km frame IJK at at time_t = -5535.2447229896 S08B 011. rJ Satellite pos vector component EC to Sat km frame IJK at at time_t = -4411.0085700927 S08C 011. rK Satellite pos vector component EC to Sat km frame IJK at at time_t = 15.4200278230 S08D 011. r Satellite pos vector magnitude EC to Sat km frame IJK at at time_t = 7077.8646869006 Note - r Satellite pos vector magnitude EC to Sat km in PQW frame is same as that computed above in PQW frame. 09. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input at time_t . Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time t UT year = 2014, month = 5, day = 28, hour = 3, minute = 22, seconds = 50.54765 Outputs : GST & GHA in 0-360 deg over Greenwich. S09A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 296.30768, hr = 19, min = 45, sec = 13.84416 S09B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 296.30959, deg = 296, min = 18, sec = 34.52861 10. Satellite(SAT) Orbit point direction : Finding Right Ascension(Alpha) deg and Declination(Delta) deg using angles Inputs : SAT orbit inclination deg at_time_t, EPH right ascension ascending node deg, SAT argument of perigee deg at time_t, SAT true anomaly deg calculated at time_t Outputs : SAT Right Ascension(Alpha) and Declination(Delta) in deg at time_t S10A 011. SAT Right Ascension(Alpha) in deg = 218.5872355801 S10B 011. SAT Declination(Delta) in deg = 0.1248262368 11. Finding Satellite Longitude & Latitude in deg at time_t; (ie Sub-Sat point log & lat on earth surface ). Inputs : SAT right ascension ascending node deg at time_t, GST in 0-360 deg over Greenwich at time_t Outputs : Satellite (Sub-Sat point) longitude 0 to 360 deg at time_t.
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OM-MSS Page 199 S11A 011. Satellite longitude 0 to 360 deg at time_t = 218.59 ie deg = 218, min = 35, sec = 14.05 Inputs : argument of_perigee rad at_time_t, inclination rad at time_t, true anomaly rad calculated at time_t. Outputs : Satellite (Sub-Sat point) latitude +ve or -ve in 0 to 90 deg at time_t. S11B 011. Satellite latitude +ve or -ve in 0 to 90 deg at time_t = 0.12 ie deg = 0, min = 7, sec = 29.37 12. Finding Satellite height in km from EC to Sat and from Earth surface to Sat at time_t. (a) Satellite height in km from EC to Sat; (ie Sat orbit radius EC to Sat in km at time_t). Note - This is SAME as r sat pos vector magnitude EC to Sat in frame IJK calculated above in TRANSFORN_1 . Inputs : SAT true anomaly at time_t, semi_major_axis_km, inclination at time_t . Outputs : Sub-Sat point longitude 0 to 360 deg at time_t. S12A 011. Satellite orbital radious EC to SAT in km using SAT true anomaly at time_t = 7077.86 (b) Satellite height in km from earth surface. Inputs : Sub-Sat point latitude +ve or -ve in 0 to 90 deg at time_t, earth_equator_radious_km . Outputs : Sat height in km from earth surface. S12B 011. Satellite height in km from earth surface at time_t = 699.72 13. Finding Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface at time_t . Inputs : Sub-Sat point lat & log, ES lat & log . Outputs : Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface S13 011. Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface = 15098.60365 14. Finding Local sidereal time(LST) and Local mean time(LMT) over Sub-Sat point Longitude on earth . (a) Local sidereal time(LST) in 0 to 360 deg over Sub-Sat point Longitude on earth . Inputs : GST sidereal_time in 0 to 360 deg at Greenwich at time_t, satellite log in 0 to 360 deg at time_t . Outputs : LST local sidereal time in 0 to_360 deg at SAT longitude at time_t. S14A 011. Local sidereal time(LST) over Sub-Sat point Longitude at time_t = 154.89 (b) Local sidereal time(LST) and local Mean time(LMT) adjusted to calendar date(CD) over Sub-Sat point Longitude on earth . Note - The LST and LMT in hr min sec with YY MM DD adjusted to calendar date (CD) of longitude at time_t .
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OM-MSS Page 200 Here LST in deg is re-calculated alterately in terms of Julian Day (JD) that account for calendar date (CD) of longitude. Inputs : GMT_JD, Sat longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at SAT longitude at time_t. S14B 011. Local sidereal time(LST) in 0 to 360_deg over Sub-Sat point Longitude at time_t = 154.90 S14C 011. LST over Sub-Sat Log, with date adj to CD expressed in JD = 2456805.93, ie YY = 2014, MM = 5, DD = 28, hr = 10, min = 19, sec = 34.87 S14D 011. LMT over Sub-Sat Log, with date adj to CD expressed in JD = 2456805.25, ie YY = 2014, MM = 5, DD = 27, hr = 17, min = 57, sec = 11.48 15. Finding Local sidereal time(LST) and Local mean time(LMT) Over Earth stn (ES) or Earth point(EP) Longitude . (a) Local sidereal time(LST) in 0 to 360 deg over Earth stn(ES) Longitude . Inputs : GST sidereal_time in 0 to 360 deg at Greenwich at time_t, Earth stn(ES) longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at Earth stn(ES) longitude at time_t. S15A 011. Local sidereal time(LST) over Earth stn(ES) Longitude at time_t = 13.72 (b) Local sidereal time(LST) and local Mean time(LMT) adjusted to calendar date(CD) over Earth stn(ES) Longitude . Note - The LST and LMT in hr min sec with YY MM DD adjusted to calendar date (CD) of longitude at time_t . Here LST in deg is re-calculated alterately in terms of Julian Day (JD) that account for calendar date (CD) of longitude. Inputs : GMT_JD, Earth stn(ES) longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at Earth stn (ES) longitude at time_t. S15B 011. Local sidereal time(LST) in 0 to 360_deg over Earth stn (ES) Longitude at time_t = 13.72 S14C 011. LST over ES Log, with date adj to CD expressed in JD = 2456806.54, ie YY = 2014, MM = 5, DD = 29, hr = 0, min = 54, sec = 52.96 S15D 011. LMT over ES Log, with date adj to CD expressed in JD = 2456805.86, ie YY = 2014, MM = 5, DD = 28, hr = 8, min = 32, sec = 29.58 16. Earth Stn (ES) Position Vector from Earth Center(EC) to Earth Stn(ES) : Finding Range Vector(RI, RJ, RK, R) Components in IJK frame Note - Transform_2 : ES position cord(lat, log, hgt) To EC to ES position Vector(RI, RJ, RK, R) in frame IJK . Inputs : ES latitude positive_negative 0 to 90 deg, ES longitude in 0 to 360_deg, ES height in meter (is earth surface + tower hgt), LST in 0 to 360 deg at ES log at time_t . Outputs : ES Position Vector(RI, RJ, RK, R) Components EC to ES in km in IJK frame . S16A 011. RI_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 5690.0261762829 S16B 011. RJ_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 1389.2147306682
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OM-MSS Page 201 S16C 011. RK_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 2517.6372173288 S16D 011. R_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 6375.3284317570 17. Satellite(SAT) Position Range Vector from Earth Stn(ES) to SAT : Finding Range Vector(rvI, rvJ, rvK, rv) Components in km in IJK frame Note - Transform_3 SAT Pos Vct(rI rJ rK) and ES Pos Vct(RI RJ RK) To SAT Pos Vct(rvI, rvJ, rvK, rv) Inputs : SAT Position Vector(rI rJ rK) EC to Sat in km in frame IJK , ES Position Vector(RI RJ RK) EC to ES in km in IJK frame. Outputs : SAT Position Range Vector(rvI, rvJ, rvK, rv) Components ES to Sat in km in IJK frame S17A 011. rv_I range vector component ES to SAT km frame IJK = -11225.2708992725 S17B 011. rv_J range vector component ES to SAT km frame IJK = -5800.2233007609 S17C 011. rv_K range vector component ES to SAT km frame IJK = -2502.2171895058 S17D 011. rv range vector component ES to SAT km frame IJK = 12880.6206358313 18. Satellite(SAT) Position Range Vector from Earth Stn(ES) to SAT : Finding Range Vector(rvS, rvE, rvZ, rv) Components in km in SEZ frame Note - Transform_4 SAT Pos Vct(rvI, rvJ, rvK, rv) in IJK frame To SAT Pos Vct(rvS, rvE, rvZ, rv) in SEZ frame Inputs : ES latitude positive_negative 0 to 90_deg, LST in 0 to 360 deg at ES longitude at time_t SAT PositionRange Vector(rvI, rvJ, rvK, rv) ES to Sat in km in IJK frame Outputs : SAT Position Range(rvS, rvE, rvZ, rv) Components ES to Sat in km in SEZ frame S18A 011. rvS range_vector component ES to SAT km_frame SEZ = -2550.8312634026 S18B 011. rvE range_vector component ES to SAT km_frame SEZ = -2972.2788352154 S18C 011. rvZ range_vector component ES to SAT km_frame SEZ = -12270.6644626761 S18D 011. rv range_vector component ES to SAT km_frame SEZ = 12880.6206358313 19. Finding Elevation(EL) and Azimuth(AZ) angles of Satellite and Sun : Steps 1, 2, 3 AT UT TIME t Rem: Sub-SAT point lat deg = 0.12, log deg = 218.59 YY = 2014, MM = 5, DD = 27, hr = 17, min = 57, sec = 11.48 ES or EP point lat deg = 23.26, log deg = 77.41 YY = 2014, MM = 5, DD = 28, hr = 8, min = 32, sec = 29.58 Note : Step 1 is for Satellite EL & AZ angles. Steps 2 & 3 are for Sun EL & AZ angles Results verified from other sources; Ref URLs Geoscience Australia http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone ,
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OM-MSS Page 202 NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html Elevation(EL) & Azimuth(AZ) angle of SAT at Earth Observation point EP : Step 1. Inputs : Range vector component ES to SAT rv_S, rv_E, rv_Z, in frame_SEZ, EP and Sub_sat point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of SAT at EP S19A 011. Elevation angle deg of Satellte at Earth point EP at time_t = -72.29698 ie deg = -72, min = -17, sec = -49.12 S19B 011. Azimuth angle deg of Satellte at Earth point EP at time_t = 310.63643 ie deg = 310, min = 38, sec = 11.16 Elevation(EL) & Azimuth(AZ) angle of SUN at Sub_Satellite point on earth surface : Step 2. Inputs : Range vector component Sub_sat to Sun rv_S, rv_E, rv_Z, in frame_SEZ, Sub_sat point and Sun_Sun point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of Sun at Sub_Sat S19C 011. Elevation angle deg of SUN at Sub_Sat point at time_t = 0.0508847018 S19D 011. Azimuth angle deg of SUN at Sub_Sat point at time_t = 291.4359560384 Elevation(EL) & Azimuth(AZ) angle of SUN at Satellite height : Step 3. Inputs : Range vector component Sub_sat to Sun rv_S, rv_E, rv_Z, in frame_SEZ, Sub_sat point and Sun_Sun point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of SUN at SAT S19E 011. Elevation angle deg of Sun at Satellite height at time_t = 0.05062 S19F 011. Azimuth angle deg of Sun at Satellite height at time_t = 291.43596 20. Finding Satellite Velocity meter per sec in orbit . Note : Results computed using 2 different formulations, each require different inputs. (a) Inputs : SAT orbit semi-major axis SMA, GM_EARTH, sat_orbit pos r_vector_EC_to_SAT_km_frame_IJK. Outputs : SAT Velocity magnitude and component Xw Yw in frame PQW in meter per sec S20A 011. Satellite Velocity magnitude meter per sec at UT time = 7504.3851274216
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OM-MSS Page 203 (b) Inputs : SAT orbit semi-major axis SMA, GM_EARTH, SAT orbit eccentricity_at_time_t, SAT eccentric anomaly deg calculated at time_t. Outputs : Satellite Velocity components Xw Yw in frame PQW in meter per sec S20B 011. Satellite Velocity components Xw in frame PQW in meter per sec = 7457.0858889990 S20C 011. Satellite Velocity components Yw in frame PQW in meter per sec = -841.2289728478 21. Finding Satellite(SAT) Velocity Vector (vX, vY, vZ) in meter per sec in orbit in frame XYZ. Note - Transform_5 SAT Vel Vct(Xw, Yw) in frame PQW To SAT Vel Vct(vX, vY, vZ) in frame XYZ Inputs : SAT velocity vectors(Xw, Yw), SAT Right Ascension Alpha, SAT Argument of perigee, SAT orbit inclination at_time_t, SAT eccentric_anomaly_deg_calculated_at_time_t. Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ S21A 011. vX Sat Velocity vector component in meter per sec = -655.5016695670 S21B 011. vY Sat Velocity vector component in meter per sec = 849.8345806371 S21C 011. vZ Sat Velocity vector component in meter per sec = 7427.2400585557 S21D 011. vR Sat Velocity magnitude meter in meter per sec = 7504.3851274216 22. Finding Satellite(SAT) Pitch and Roll angles Inputs : Earth equator radious km, ES lat, ES log, Sub_Sat point lat, Sub_Sat point log, Sat_orbit pos r_vector_EC_to_SAT_km_frame_IJK, Outputs : SAT Pitch and Roll angles S22A 011. pitch_angle_deg_for_earth_stn_at_sat_lat_log_at_time_t = 18.0730937568 S22B 011. roll_angle_deg_for_earth_stn_at_sat_lat_log_at_time_t = 11.5734570323 23. Finding Satellite State Vectors - Position [ X, Y, Z ] in km and velocity [ Vx, Vy, Vz ] in meter per sec at time_t . Note - same as values of rI rJ rK r for pos and vX vY vZ vR for vel (a) Satellite State Position Vector [X, Y, Z] in km at time_t Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ S23A 011. State vector position X km at time_t = -5535.2447229896
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OM-MSS Page 204 S23B 011. State vector position Y km at time_t = -4411.0085700927 S23C 011. State vector position Z km at time_t = 15.4200278230 S23D 011. State vector position R km at time_t = 7077.8646869006 (b) Satellite State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ S23E 011. State vector velocity Vx meter per sec at time_t = -655.5016695670 S23F 011. State vector velocity Vy meter per sec at time_t = 849.8345806371 S23G 011. State vector velocity Vz meter per sec at time_t = 7427.2400585557 S23H 011. State vector velocity V meter per sec at time_t = 7504.3851274216 24. Satellite Direction ie right ascension alpha deg and declination delta deg using sat position vector . Note - results same as that using angles incl, RA, Aug of peri, true anomaly Inputs : SAT State Vectors Position(X, Y, Z, R) in km , in frame XYZ Outputs : SAT right ascension alpha deg and declination delta deg S24A 011. Satellite direction right_asc alpha deg at time_t = 218.5511644199 S24B 011. Satellite direction_declination_delta_deg_at_time_t = 0.1248262368 25. Satellite Angular momentum km sqr per sec : finding Hx Hy Hz H from state vector pos and vel . Inputs : SAT State Vectors Position(X, Y, Z, R) in km and Velocity (vX, vY, vZ, vR) meter per sec, in frame XYZ Outputs : SAT angular momentum (Hx Hy Hz H) componts in km sqr per sec S25A 011. Satellite angular momentum Hx in km sqr per sec at time_t = -32774.7240233036 S25B 011. Satellite angular momentum Hy in km sqr per sec at time_t = 41101.4834865149 S25C 011. Satellite angular momentum Hz in km sqr per sec at time_t = -7595.4658600556 S25D 011. Satellite angular momentum H in km sqr per sec at time_t = 53115.0221804269 26. Satellite Orbit normal Vector : finding Wx Wy Wz W Delta Alpha from r_sat_pos frame IJK, i, RA Inputs : SAT Position Vector(rI rJ rK) EC to Sat in km in frame IJK , inclination_deg_at_time_t, right_ascension_ascending_node_deg_at_time_t
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OM-MSS Page 205 Outputs : SAT orbit normal vector (Wx, Wy, Wz, W) in km , RA , i S26A 011. Satellite Orbit normal_W in km = 7077.8646869006 S26B 011. Satellite Orbit normal_Wx in km = -4367.4096755425 S26C 011. Satellite Orbit normal_Wy in km = 5476.9955204054 S26D 011. Satellite Orbit normal_Wz in km = -1012.1370072826 S26E 011. Satellite Orbit normal_Delta_W in deg = -8.2215000000 S26F 011. Satellite Orbit interpreted inclination i = 98.2215000000 S26G 011. Satellite Orbit normal_Alpha_W in deg = -51.4308000000 S26H 011. Satellite Orbit interpreted RA of asc node = 218.5692000000 Transform Satellite State Vectors to Keplerian elements. 27. Finding Satellite position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness S27A 011. Keplerian elements year = 2014, days_decimal_of_year = 147.14086, revolution no = 6853, node = 1 ie ascending S27B 011. inclination_deg = 98.2215000000 S27C 011. right ascension ascending node deg = 218.5692000000 S27D 011. eccentricity = 0.0001087000 S27E 011. argument of perigee_deg = 96.5686000000 S27F 011. mean anomaly deg = 263.5699000001 S27G 011. mean_motion rev per day = 14.5709892500 S27H 011. mean angular velocity rev_per_day = 14.5799930299 S27I 011. mean motion rev per day using SMA considering oblateness = 14.5619910304 Transform Satellite Keplerian elements to State Vectors . 28. Finding Satellite position State Vectors, computed using Keplerian elements at time input UT
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OM-MSS Page 206 (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] S28A 011. State vectors year = 2014, days_decimal_of_year = 147.14086, revolution no = 6853, node = 1 ie decending S28B 011. state vector position X km = -5535.2447229901, state vector velocity Vx meter per sec = -655.5016695622 S28C 011. state vector position Y km = -4411.0085700921, state vector velocity Vy meter per sec = 849.8345806409 S28D 011. state vector position Z km = 15.4200278287, state vector velocity Vz meter per sec = 7427.2400585557 S28E 011. state vector position R km = 7077.8646869005, state vector velocity V meter per sec = 7504.3851274216 Move on to next Satellite. Next Section - 6.2 Computing Orbital & Positional parameters for Satellite SPOT 6
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OM-MSS Page 207 OM-MSS Section - 6.2 ---------------------------------------------------------------------------------------------------50 Satellite SPOT 6 : Computing Orbital & Positional parameters corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins. (b) SPOT 6 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on September 9, 2012 1 38755U 12047A 14148.14295346 .00000295 00000-0 73402-4 0 9574 2 38755 98.1987 215.8134 0001368 80.3963 279.7434 14.58528066 91251 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 38755, SPOT 6 , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1429534600 EPOCH_inclination_deg = 98.1987000000 EPOCH_right_asc_acnd_node_deg = 215.8134000000 EPOCH_eccentricity = 0.0001368000 EPOCH_argument_of_perigee_deg = 80.3963000000 EPOCH_mean_anomaly_deg = 279.7434000000 EPOCH_mean_motion_rev_per_day = 14.5852806600 EPOCH_revolution = 9125 EPOCH_node_condition = 1 Earth_stn_latitude_deg = 23.25993 Earth stn longitude_deg = 77.41261 Earth surface height_meter = 494.70000 Earth stn tower_height_meter = 15.00000 Earth stn_height_meter = 509.70000 Earth stn min EL look_angle_deg = -5.00000 Note : EPOCH Corresponds to UT year = 2014, month = 5, day = 28, hr = 3, min = 25, sec = 51.17894 which is Greenwich meam time ie GMT Converted to Local Meam Time at Earth Stn Longitude, as regular Calender date : year = 2014, month = 5, day = 28, hr = 8, min = 35, sec = 30.21 At this instant Sun Position as seen from Earth Stn Longitude : Sun angles EL deg = 43.56, AZ deg = 81.94, Sun Surface distance km = 5169.96, Radial km = 151596361.23, Sun Rise D:28, H:05, M:17, S:03 Sun Set D:28, H:18, M:40, S:20 Move to Compute Satellie Orbital parameters corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins, and Earth Stn location. 01. Input EPOCH_year and EPOCH_days_decimal_of_year, Converted into UT YY MM DD hh min sec & Julian day. S01 011. Input UT year = 2014, month = 5, day = 28, hr = 3, min = 25, sec = 51.17894, and julian_day = 2456805.6429534601 02. Finding Satellite orbit Semi major axis in km, Ignoring and also Considering earth oblatenes . (a) Semi major axis (SMA) km at time t Ignoring earth oblatenes . Inputs : SAT mean_motion rev per day at time t, GM_EARTH . Outputs : SAT Orbit semi major axis km in km and constant_A, constant_k1 .
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OM-MSS Page 208 S02A 011. Satellite orbit Semi major axis in km at time t, Ignoring earth oblatenes = 7076.06773, (b) Semi major axis (SMA) in km at time_t Considering earth oblatenes Inputs : SAT mean_motion rev per day at time_t, GM_EARTH , inclination deg at time_t, eccentricity at time_t, constant_k2 Outputs : SAT Orbit semi major axis km in km at time_t and constant_A, constant_k1, constant_k3 . S02B 011. Satellite orbit Semi major axis km at time t, Considering earth oblatenes = 7073.14938, 03. Finding Satellite Mean motion in rev per day, Ignoring and also Considering earth oblatenes . (a) Nominal mean motion rev per day at time_t Ignoring earth oblateness . Inputs : SAT Orbit semi major axis in km ignoring oblateness at time_t, constant_k1 . Outputs : SAT nominal mean motion in rev_per_day at time_t, ignoring earth oblateness S03A 011. Satellite Nominal Mean motion in rev_per_day at time_t using SMA Ignoring earth oblateness = 14.58528, (b) Mean motion rev per day at time_t Considering earth oblatenes Inputs : SAT nominal mean motion rad per day at time_t Ignoring_oblateness, constant_k2, constant_k3, SAT orbit semi major axis in km considering oblateness at time_t. Outputs : SAT mean motion rev per day at time_t, considering earth oblatenes . S03B 011. Satellite Mean motion in rev_per_day at time t using SMA Considering earth oblatenes = 14.57626, Note - This calculted value is slightly less than the mean motion rev_per_day as EPH input from NORAD TLE 04. Finding Satellite Orbit Time Period in minute at time_t Considering earth oblatenes . Inputs : SAT orbit semi major axis in km considering oblateness at time_t, GM_EARTH . Outputs : SAT orbit time period in minute at time_t considering earth oblatenes . S04 011. Satellite orbit Time Period in minute at time_t using SMA Considering earth oblatenes = 98.66860, 05. Finding Satellite Rate of change of Right Ascension and Argument of Perigee in deg per_day at time_t. (a) Rate of change of Right Ascension in deg per day at time_t . Inputs : SAT mean motion rev per day at time_t considering earth oblatenes, constant_k2, SAT orbit eccentricity at time_t, SAT semi major axis km considering oblateness at time_t, SAT orbit inclination deg at time_t . Outputs : SAT rate of change of right ascension in deg per day at time_t and constant_k_deg_per_day
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OM-MSS Page 209 S05A 011. Satellite Rate of change of Right Ascension in deg per day at time_t = 0.98594, (b) Rate of change of Argument of Perigee in deg per_day at time_t . Inputs : SAT orbit constant_k_deg_per_day, SAT orbit inclination deg at time_t SAT orbit semi major axis km considering oblateness at time_t Outputs : SAT rate of change of argument of perigee in deg per day at time_t S05B 011. Satellite Rate of change of Argument of Perigee in deg per day at time_t = -3.10534, 06. Finding Satellite Mean anomaly, Eccentric anomaly, True anomaly in deg at time_t considering earth oblateness. Inputs : SAT mean anomaly rad at time_t, mean_motion_rad_per_day_at_time_t considering_oblateness, SAT orbit eccentricity_at_time_t Outputs : SAT mean anomaly, eccentric anomaly, and true anomaly in deg at time_t onsidering earth oblateness. S06A 011. Satellite Mean anomaly in deg at time_t = 279.74340, same as EPH mean anomaly S06A 011. Satellite Eccentric anomaly in deg at time_t = 279.73567, S06A 011. Satellite True anomaly in deg at time_t = 279.72795, Note - 1. Here after, the Earth Oblateness is always considered for the computation of satellite orbit parameters, and not repeatedly mentioned. Satellite to Earth, the Position Vectors coordinate and the Vector Coordinate Transforms are in PQW, IJK, SEZ frames . - Perifocal Coordinate System (PQW) is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. - Geocentric Coordinate System(IJK) is Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). - Topocentric Horizon Coordinate System(SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). Each of these coordinate system were explained in detail before and therefore not repeated any more. 07. Finding Satellite position vector[rp, rq] from Earth center(EC) to Sat in PQW frame, perifocal coordinate system. Inputs : SAT orbit semi-major axis (SMA), SAT orbit eccentricity, SAT eccentric anomaly, SATtrue anomaly at time_t Outputs : Vector(r, rp rq) in PQW frame S07A 011. r Satellite pos vector magnitude EC to Sat km in PQW frame perifocal cord at time_t = 7072.98575 S07B 011. rp Satellite pos vector component EC to Sat km in PQW frame perifocal cord at time_t = 1195.1237964113 S07C 011. rq Satellite pos vector component EC to Sat km in PQW frame perifocal cord at time_t = -6971.2844253704 08. Satellite Position Vector Earth Ceter(EC) to Satellite(SAT) - finding Range Vector(rI rJ rK r) Components in km in frame IJK
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OM-MSS Page 210 Note - Transform_1 : EC to SAT Vector(rp, rq) in frame PQW To EC to SAT Vector(rI rJ rK r) in frame IJK. Inputs : Vector(rp, rq) EC to Sat in km in frame PQW, SAT right ascension of ascending node at time_t, SAT argument of perigee rad at time_t, SAT orbit inclination rad at time_t . Outputs : Vector(rI, rJ, rK, r) EC to SAT in km in frame IJK S08A 011. rI Satellite pos vector component EC to Sat km frame IJK at at time_t = -5736.9414700815 S08B 011. rJ Satellite pos vector component EC to Sat km frame IJK at at time_t = -4136.9553443077 S08C 011. rK Satellite pos vector component EC to Sat km frame IJK at at time_t = 15.1814434008 S08D 011. r Satellite pos vector magnitude EC to Sat km frame IJK at at time_t = 7072.9857505978 Note - r Satellite pos vector magnitude EC to Sat km in PQW frame is same as that computed above in PQW frame. 09. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input at time_t . Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time t UT year = 2014, month = 5, day = 29, hour = 3, minute = 25, seconds = 51.17894 Outputs : GST & GHA in 0-360 deg over Greenwich. S09A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 297.06238, hr = 19, min = 48, sec = 14.97000 S09B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 297.06431, deg = 297, min = 3, sec = 51.52460 10. Satellite(SAT) Orbit point direction : Finding Right Ascension(Alpha) deg and Declination(Delta) deg using angles Inputs : SAT orbit inclination deg at_time_t, EPH right ascension ascending node deg, SAT argument of perigee deg at time_t, SAT true anomaly deg calculated at time_t Outputs : SAT Right Ascension(Alpha) and Declination(Delta) in deg at time_t S10A 011. SAT Right Ascension(Alpha) in deg = 215.8311188173 S10B 011. SAT Declination(Delta) in deg = 0.1229796485 11. Finding Satellite Longitude & Latitude in deg at time_t; (ie Sub-Sat point log & lat on earth surface ). Inputs : SAT right ascension ascending node deg at time_t, GST in 0-360 deg over Greenwich at time_t Outputs : Satellite (Sub-Sat point) longitude 0 to 360 deg at time_t. S11A 011. Satellite longitude 0 to 360 deg at time_t = 215.83 ie deg = 215, min = 49, sec = 52.03
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OM-MSS Page 211 Inputs : argument of_perigee rad at_time_t, inclination rad at time_t, true anomaly rad calculated at time_t. Outputs : Satellite (Sub-Sat point) latitude +ve or -ve in 0 to 90 deg at time_t. S11B 011. Satellite latitude +ve or -ve in 0 to 90 deg at time_t = 0.12 ie deg = 0, min = 7, sec = 22.73 12. Finding Satellite height in km from EC to Sat and from Earth surface to Sat at time_t. (a) Satellite height in km from EC to Sat; (ie Sat orbit radius EC to Sat in km at time_t). Note - This is SAME as r sat pos vector magnitude EC to Sat in frame IJK calculated above in TRANSFORN_1 . Inputs : SAT true anomaly at time_t, semi_major_axis_km, inclination at time_t . Outputs : Sub-Sat point longitude 0 to 360 deg at time_t. S12A 011. Satellite orbital radious EC to SAT in km using SAT true anomaly at time_t = 7072.99 (b) Satellite height in km from earth surface. Inputs : Sub-Sat point latitude +ve or -ve in 0 to 90 deg at time_t, earth_equator_radious_km . Outputs : Sat height in km from earth surface. S12B 011. Satellite height in km from earth surface at time_t = 694.84 13. Finding Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface at time_t . Inputs : Sub-Sat point lat & log, ES lat & log . Outputs : Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface S13 011. Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface = 14843.67830 14. Finding Local sidereal time(LST) and Local mean time(LMT) over Sub-Sat point Longitude on earth . (a) Local sidereal time(LST) in 0 to 360 deg over Sub-Sat point Longitude on earth . Inputs : GST sidereal_time in 0 to 360 deg at Greenwich at time_t, satellite log in 0 to 360 deg at time_t . Outputs : LST local sidereal time in 0 to_360 deg at SAT longitude at time_t. S14A 011. Local sidereal time(LST) over Sub-Sat point Longitude at time_t = 152.89 (b) Local sidereal time(LST) and local Mean time(LMT) adjusted to calendar date(CD) over Sub-Sat point Longitude on earth . Note - The LST and LMT in hr min sec with YY MM DD adjusted to calendar date (CD) of longitude at time_t . Here LST in deg is re-calculated alterately in terms of Julian Day (JD) that account for calendar date (CD) of longitude. Inputs : GMT_JD, Sat longitude_in_0_to_360_deg .
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OM-MSS Page 212 Outputs : LST local sidereal time in 0 to_360 deg at SAT longitude at time_t. S14B 011. Local sidereal time(LST) in 0 to 360_deg over Sub-Sat point Longitude at time_t = 152.89 S14C 011. LST over Sub-Sat Log, with date adj to CD expressed in JD = 2456805.92, ie YY = 2014, MM = 5, DD = 28, hr = 10, min = 11, sec = 34.53 S14D 011. LMT over Sub-Sat Log, with date adj to CD expressed in JD = 2456805.24, ie YY = 2014, MM = 5, DD = 27, hr = 17, min = 49, sec = 10.658 15. Finding Local sidereal time(LST) and Local mean time(LMT) Over Earth stn (ES) or Earth point(EP) Longitude . (a) Local sidereal time(LST) in 0 to 360 deg over Earth stn(ES) Longitude . Inputs : GST sidereal_time in 0 to 360 deg at Greenwich at time_t, Earth stn(ES) longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at Earth stn(ES) longitude at time_t. S15A 011. Local sidereal time(LST) over Earth stn(ES) Longitude at time_t = 14.47 (b) Local sidereal time(LST) and local Mean time(LMT) adjusted to calendar date(CD) over Earth stn(ES) Longitude . Note - The LST and LMT in hr min sec with YY MM DD adjusted to calendar date (CD) of longitude at time_t . Here LST in deg is re-calculated alterately in terms of Julian Day (JD) that account for calendar date (CD) of longitude. Inputs : GMT_JD, Earth stn(ES) longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at Earth stn (ES) longitude at time_t. S15B 011. Local sidereal time(LST) in 0 to 360_deg over Earth stn (ES) Longitude at time_t = 14.48 S14C 011. LST over ES Log, with date adj to CD expressed in JD = 2456806.54, ie YY = 2014, MM = 5, DD = 29, hr = 0, min = 57, sec = 54.09 S15D 011. LMT over ES Log, with date adj to CD expressed in JD = 2456805.86, ie YY = 2014, MM = 5, DD = 28, hr = 8, min = 35, sec = 30.21 16. Earth Stn (ES) Position Vector from Earth Center(EC) to Earth Stn(ES) : Finding Range Vector(RI, RJ, RK, R) Components in IJK frame Note - Transform_2 : ES position cord(lat, log, hgt) To EC to ES position Vector(RI, RJ, RK, R) in frame IJK . Inputs : ES latitude positive_negative 0 to 90 deg, ES longitude in 0 to 360_deg, ES height in meter (is earth surface + tower hgt), LST in 0 to 360 deg at ES log at time_t . Outputs : ES Position Vector(RI, RJ, RK, R) Components EC to ES in km in IJK frame . S16A 011. RI_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 5671.2345911565 S16B 011. RJ_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 1464.0401861835 S16C 011. RK_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 2517.6372173288 S16D 011. R_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 6375.3284317570
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OM-MSS Page 213 17. Satellite(SAT) Position Range Vector from Earth Stn(ES) to SAT : Finding Range Vector(rvI, rvJ, rvK, rv) Components in km in IJK frame Note - Transform_3 SAT Pos Vct(rI rJ rK) and ES Pos Vct(RI RJ RK) To SAT Pos Vct(rvI, rvJ, rvK, rv) Inputs : SAT Position Vector(rI rJ rK) EC to Sat in km in frame IJK , ES Position Vector(RI RJ RK) EC to ES in km in IJK frame. Outputs : SAT Position Range Vector(rvI, rvJ, rvK, rv) Components ES to Sat in km in IJK frame S17A 011. rv_I range vector component ES to SAT km frame IJK = -11408.176061238 S17B 011. rv_J range vector component ES to SAT km frame IJK = -5600.9955304911 S17C 011. rv_K range vector component ES to SAT km frame IJK = -2502.4557739280 S17D 011. rv range vector component ES to SAT km frame IJK = 12952.9887237367 18. Satellite(SAT) Position Range Vector from Earth Stn(ES) to SAT : Finding Range Vector(rvS, rvE, rvZ, rv) Components in km in SEZ frame Note - Transform_4 SAT Pos Vct(rvI, rvJ, rvK, rv) in IJK frame To SAT Pos Vct(rvS, rvE, rvZ, rv) in SEZ frame Inputs : ES latitude positive_negative 0 to 90_deg, LST in 0 to 360 deg at ES longitude at time_t SAT PositionRange Vector(rvI, rvJ, rvK, rv) ES to Sat in km in IJK frame Outputs : SAT Position Range(rvS, rvE, rvZ, rv) Components ES to Sat in km in SEZ frame S18A 011. rvS range_vector component ES to SAT km_frame SEZ = -2615.9229941625 S18B 011. rvE range_vector component ES to SAT km_frame SEZ = -2571.6444671914 S18C 011. rvZ range_vector component ES to SAT km_frame SEZ = -12422.701336675 S18D 011. rv range_vector component ES to SAT km_frame SEZ = 12952.9887237367 19. Finding Elevation(EL) and Azimuth(AZ) angles of Satellite and Sun : Steps 1, 2, 3 AT UT TIME t Rem: Sub-SAT point lat deg = 0.12, log deg = 215.83 YY = 2014, MM = 5, DD = 27, hr = 17, min = 49, sec = 10.65 ES or EP point lat deg = 23.26, log deg = 77.41 YY = 2014, MM = 5, DD = 28, hr = 8, min = 35, sec = 30.21 Note : Step 1 is for Satellite EL & AZ angles. Steps 2 & 3 are for Sun EL & AZ angles Results verified from other sources; Ref URLs Geoscience Australia http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone , NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html
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OM-MSS Page 214 Elevation(EL) & Azimuth(AZ) angle of SAT at Earth Observation point EP : Step 1. Inputs : Range vector component ES to SAT rv_S, rv_E, rv_Z, in frame_SEZ, EP and Sub_sat point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of SAT at EP S19A 011. Elevation angle deg of Satellte at Earth point EP at time_t = -73.54866 ie deg = -73, min = -32, sec = -55.19 S19B 011. Azimuth angle deg of Satellte at Earth point EP at time_t = 315.48904 ie deg = 315, min = 29, sec = 20.53 Elevation(EL) & Azimuth(AZ) angle of SUN at Sub_Satellite point on earth surface : Step 2. Inputs : Range vector component Sub_sat to Sun rv_S, rv_E, rv_Z, in frame_SEZ, Sub_sat point and Sun_Sun point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of Sun at Sub_Sat S19C 011. Elevation angle deg of SUN at Sub_Sat point at time_t = 1.9151336743 S19D 011. Azimuth angle deg of SUN at Sub_Sat point at time_t = 291.4445892810 Elevation(EL) & Azimuth(AZ) angle of SUN at Satellite height : Step 3. Inputs : Range vector component Sub_sat to Sun rv_S, rv_E, rv_Z, in frame_SEZ, Sub_sat point and Sun_Sun point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of SUN at SAT S19E 011. Elevation angle deg of Sun at Satellite height at time_t = 1.91487 S19F 011. Azimuth angle deg of Sun at Satellite height at time_t = 291.44459 20. Finding Satellite Velocity meter per sec in orbit . Note : Results computed using 2 different formulations, each require different inputs. (a) Inputs : SAT orbit semi-major axis SMA, GM_EARTH, sat_orbit pos r_vector_EC_to_SAT_km_frame_IJK. Outputs : SAT Velocity magnitude and component Xw Yw in frame PQW in meter per sec S20A 011. Satellite Velocity magnitude meter per sec at UT time = 7507.1055062891 (b) Inputs : SAT orbit semi-major axis SMA, GM_EARTH, SAT orbit eccentricity_at_time_t, SAT eccentric anomaly deg calculated at time_t. Outputs : Satellite Velocity components Xw Yw in frame PQW in meter per sec
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OM-MSS Page 215 S20B 011. Satellite Velocity components Xw in frame PQW in meter per sec = 7398.9909470324 S20C 011. Satellite Velocity components Yw in frame PQW in meter per sec = 1269.4747135282 21. Finding Satellite(SAT) Velocity Vector (vX, vY, vZ) in meter per sec in orbit in frame XYZ. Note - Transform_5 SAT Vel Vct(Xw, Yw) in frame PQW To SAT Vel Vct(vX, vY, vZ) in frame XYZ Inputs : SAT velocity vectors(Xw, Yw), SAT Right Ascension Alpha, SAT Argument of perigee, SAT orbit inclination at_time_t, SAT eccentric_anomaly_deg_calculated_at_time_t. Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ S21A 011. vX Sat Velocity vector component in meter per sec = -612.4123815408 S21B 011. vY Sat Velocity vector component in meter per sec = 878.2634599352 S21C 011. vZ Sat Velocity vector component in meter per sec = 7430.3591738511 S21D 011. vR Sat Velocity magnitude meter in meter per sec = 7507.1055062891 22. Finding Satellite(SAT) Pitch and Roll angles Inputs : Earth equator radious km, ES lat, ES log, Sub_Sat point lat, Sub_Sat point log, Sat_orbit pos r_vector_EC_to_SAT_km_frame_IJK, Outputs : SAT Pitch and Roll angles S22A 011. pitch_angle_deg_for_earth_stn_at_sat_lat_log_at_time_t = 19.4439603698 S22B 011. roll_angle_deg_for_earth_stn_at_sat_lat_log_at_time_t = 11.6748374344 23. Finding Satellite State Vectors - Position [ X, Y, Z ] in km and velocity [ Vx, Vy, Vz ] in meter per sec at time_t . Note - same as values of rI rJ rK r for pos and vX vY vZ vR for vel (a) Satellite State Position Vector [X, Y, Z] in km at time_t Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ S23A 011. State vector position X km at time_t = -5736.9414700815 S23B 011. State vector position Y km at time_t = -4136.9553443077 S23C 011. State vector position Z km at time_t = 15.1814434008 S23D 011. State vector position R km at time_t = 7072.9857505978
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OM-MSS Page 216 (b) Satellite State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ S23E 011. State vector velocity Vx meter per sec at time_t = -612.4123815408 S23F 011. State vector velocity Vy meter per sec at time_t = 878.2634599352 S23G 011. State vector velocity Vz meter per sec at time_t = 7430.3591738511 S23H 011. State vector velocity V meter per sec at time_t = 7507.1055062891 24. Satellite Direction ie right ascension alpha deg and declination delta deg using sat position vector . Note - results same as that using angles incl, RA, Aug of peri, true anomaly Inputs : SAT State Vectors Position(X, Y, Z, R) in km , in frame XYZ Outputs : SAT right ascension alpha deg and declination delta deg S24A 011. Satellite direction right_asc alpha deg at time_t = 215.7956811827 S24B 011. Satellite direction_declination_delta_deg_at_time_t = 0.1229796485 25. Satellite Angular momentum km sqr per sec : finding Hx Hy Hz H from state vector pos and vel . Inputs : SAT State Vectors Position(X, Y, Z, R) in km and Velocity (vX, vY, vZ, vR) meter per sec, in frame XYZ Outputs : SAT angular momentum (Hx Hy Hz H) componts in km sqr per sec S25A 011. Satellite angular momentum Hx in km sqr per sec at time_t = -30752.3974013968 S25B 011. Satellite angular momentum Hy in km sqr per sec at time_t = 42618.2383781588 S25C 011. Satellite angular momentum Hz in km sqr per sec at time_t = -7572.0687396947 S25D 011. Satellite angular momentum H in km sqr per sec at time_t = 53097.6497915837 26. Satellite Orbit normal Vector : finding Wx Wy Wz W Delta Alpha from r_sat_pos frame IJK, i, RA Inputs : SAT Position Vector(rI rJ rK) EC to Sat in km in frame IJK , inclination_deg_at_time_t, right_ascension_ascending_node_deg_at_time_t Outputs : SAT orbit normal vector (Wx, Wy, Wz, W) in km , RA , i S26A 011. Satellite Orbit normal_W in km = 7072.9857505978 S26B 011. Satellite Orbit normal_Wx in km = -4096.4387213100 Page 217
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OM-MSS Page 217 S26C 011. Satellite Orbit normal_Wy in km = 5677.0533902628 S26D 011. Satellite Orbit normal_Wz in km = -1008.6535752265 S26E 011. Satellite Orbit normal_Delta_W in deg = -8.1987000000 S26F 011. Satellite Orbit interpreted inclination i = 98.1987000000 S26G 011. Satellite Orbit normal_Alpha_W in deg = -54.1866000000 S26H 011. Satellite Orbit interpreted RA of asc node = 215.8134000000 Transform Satellite State Vectors to Keplerian elements. 27. Finding Satellite position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness S27A 011. Keplerian elements year = 2014, days_decimal_of_year = 147.14295, revolution no = 9125, node = 1 ie ascending S27B 011. inclination_deg = 98.1987000000 S27C 011. right ascension ascending node deg = 215.8134000000 S27D 011. eccentricity = 0.0001368000 S27E 011. argument of perigee_deg = 80.3963000000 S27F 011. mean anomaly deg = 279.7434000000 S27G 011. mean_motion rev per day = 14.5852806600 S27H 011. mean angular velocity rev_per_day = 14.5943083253 S27I 011. mean motion rev per day using SMA considering oblateness = 14.5762585790 Transform Satellite Keplerian elements to State Vectors . 28. Finding Satellite position State Vectors, computed using Keplerian elements at time input UT (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness
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OM-MSS Page 218 Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] S28A 011. State vectors year = 2014, days_decimal_of_year = 147.14295, revolution no = 9125, node = 1 ie decending S28B 011. state vector position X km = -5736.9414700815, state vector velocity Vx meter per sec = -612.4123815411 S28C 011. state vector position Y km = -4136.9553443077, state vector velocity Vy meter per sec = 878.2634599350 S28D 011. state vector position Z km = 15.1814434004, state vector velocity Vz meter per sec = 7430.3591738511 S28E 011. state vector position R km = 7072.9857505978, state vector velocity V meter per sec = 7507.1055062891 Move on to next Satellite. Next Section - 6.3 Computing Orbital & Positional parameters for Satellite CARTOSAT 2B
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OM-MSS Page 219 OM-MSS Section - 6.3 ---------------------------------------------------------------------------------------------------51 Satellite CARTOSAT 2B : Computing Orbital & Positional parameters corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins. (c) CARTOSAT 2B 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on July 12, 2010 1 36795U 10035A 14148.12955979 .00000641 00000-0 94319-4 0 3461 2 36795 97.9448 207.1202 0016257 44.4835 315.7690 14.78679483209252 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 36795, CARTOSAT 2B , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1295597900 EPOCH_inclination_deg = 97.9448000000 EPOCH_right_asc_acnd_node_deg = 207.1202000000 EPOCH_eccentricity = 0.0016257000 EPOCH_argument_of_perigee_deg = 44.4835000000 EPOCH_mean_anomaly_deg = 315.7690000000 EPOCH_mean_motion_rev_per_day = 14.7867948300 EPOCH_revolution = 20925 EPOCH_node_condition = 1 Earth_stn_latitude_deg = 23.25993 Earth stn longitude_deg = 77.41261 Earth surface height_meter = 494.70000 Earth stn tower_height_meter = 15.00000 Earth stn_height_meter = 509.70000 Earth stn min EL look_angle_deg = -5.00000 Note : EPOCH Corresponds to UT year = 2014, month = 5, day = 28, hr = 3, min = 6, sec = 33.96586 which is Greenwich meam time ie GMT Converted to Local Meam Time at Earth Stn Longitude, as regular Calender date : year = 2014, month = 5, day = 28, hr = 8, min = 16, sec = 12.99 At this instant Sun Position as seen from Earth Stn Longitude : Sun angles EL deg = 39.18, AZ deg = 80.63, Sun Surface distance km = 5657.39, Radial km = 151596423.96, Sun Rise D:28, H:05, M:17, S:03 Sun Set D:28, H:18, M:40, S:20 Move to Compute Satellie Orbital parameters corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins, and Earth Stn location. 01. Input EPOCH_year and EPOCH_days_decimal_of_year, Converted into UT YY MM DD hh min sec & Julian day. S01 011. Input UT year = 2014, month = 5, day = 28, hr = 3, min = 6, sec = 33.96586, and julian_day = 2456805.6295597898 02. Finding Satellite orbit Semi major axis in km, Ignoring and also Considering earth oblatenes . (a) Semi major axis (SMA) km at time t Ignoring earth oblatenes . Inputs : SAT mean_motion rev per day at time t, GM_EARTH . Outputs : SAT Orbit semi major axis km in km and constant_A, constant_k1 .
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OM-MSS Page 220 S02A 011. Satellite orbit Semi major axis in km at time t, Ignoring earth oblatenes = 7011.63247, (b) Semi major axis (SMA) in km at time_t Considering earth oblatenes Inputs : SAT mean_motion rev per day at time_t, GM_EARTH , inclination deg at time_t, eccentricity at time_t, constant_k2 Outputs : SAT Orbit semi major axis km in km at time_t and constant_A, constant_k1, constant_k3 . S02B 011. Satellite orbit Semi major axis km at time t, Considering earth oblatenes = 7008.67563, 03. Finding Satellite Mean motion in rev per day, Ignoring and also Considering earth oblatenes . (a) Nominal mean motion rev per day at time_t Ignoring earth oblateness . Inputs : SAT Orbit semi major axis in km ignoring oblateness at time_t, constant_k1 . Outputs : SAT nominal mean motion in rev_per_day at time_t, ignoring earth oblateness S03A 011. Satellite Nominal Mean motion in rev_per_day at time_t using SMA Ignoring earth oblateness = 14.78679, (b) Mean motion rev per day at time_t Considering earth oblatenes Inputs : SAT nominal mean motion rad per day at time_t Ignoring_oblateness, constant_k2, constant_k3, SAT orbit semi major axis in km considering oblateness at time_t. Outputs : SAT mean motion rev per day at time_t, considering earth oblatenes . S03B 011. Satellite Mean motion in rev_per_day at time t using SMA Considering earth oblatenes = 14.77744, Note - This calculted value is slightly less than the mean motion rev_per_day as EPH input from NORAD TLE 04. Finding Satellite Orbit Time Period in minute at time_t Considering earth oblatenes . Inputs : SAT orbit semi major axis in km considering oblateness at time_t, GM_EARTH . Outputs : SAT orbit time period in minute at time_t considering earth oblatenes . S04 011. Satellite orbit Time Period in minute at time_t using SMA Considering earth oblatenes = 97.32259, 05. Finding Satellite Rate of change of Right Ascension and Argument of Perigee in deg per_day at time_t. (a) Rate of change of Right Ascension in deg per day at time_t . Inputs : SAT mean motion rev per day at time_t considering earth oblatenes, constant_k2, SAT orbit eccentricity at time_t, SAT semi major axis km considering oblateness at time_t, SAT orbit inclination deg at time_t . Outputs : SAT rate of change of right ascension in deg per day at time_t and constant_k_deg_per_day
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OM-MSS Page 221 S05A 011. Satellite Rate of change of Right Ascension in deg per day at time_t = 0.98670, (b) Rate of change of Argument of Perigee in deg per_day at time_t . Inputs : SAT orbit constant_k_deg_per_day, SAT orbit inclination deg at time_t SAT orbit semi major axis km considering oblateness at time_t Outputs : SAT rate of change of argument of perigee in deg per day at time_t S05B 011. Satellite Rate of change of Argument of Perigee in deg per day at time_t = -3.22839, 06. Finding Satellite Mean anomaly, Eccentric anomaly, True anomaly in deg at time_t considering earth oblateness. Inputs : SAT mean anomaly rad at time_t, mean_motion_rad_per_day_at_time_t considering_oblateness, SAT orbit eccentricity_at_time_t Outputs : SAT mean anomaly, eccentric anomaly, and true anomaly in deg at time_t onsidering earth oblateness. S06A 011. Satellite Mean anomaly in deg at time_t = 315.76900, same as EPH mean anomaly S06A 011. Satellite Eccentric anomaly in deg at time_t = 315.70395, S06A 011. Satellite True anomaly in deg at time_t = 315.63886, Note - 1. Here after, the Earth Oblateness is always considered for the computation of satellite orbit parameters, and not repeatedly mentioned. Satellite to Earth, the Position Vectors coordinate and the Vector Coordinate Transforms are in PQW, IJK, SEZ frames . - Perifocal Coordinate System (PQW) is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. - Geocentric Coordinate System(IJK) is Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). - Topocentric Horizon Coordinate System(SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). Each of these coordinate system were explained in detail before and therefore not repeated any more. 07. Finding Satellite position vector[rp, rq] from Earth center(EC) to Sat in PQW frame, perifocal coordinate system. Inputs : SAT orbit semi-major axis (SMA), SAT orbit eccentricity, SAT eccentric anomaly, SATtrue anomaly at time_t Outputs : Vector(r, rp rq) in PQW frame S07A 011. r Satellite pos vector magnitude EC to Sat km in PQW frame perifocal cord at time_t = 7000.52048 S07B 011. rp Satellite pos vector component EC to Sat km in PQW frame perifocal cord at time_t = 5005.0016826023 S07C 011. rq Satellite pos vector component EC to Sat km in PQW frame perifocal cord at time_t = -4894.6138857698 08. Satellite Position Vector Earth Ceter(EC) to Satellite(SAT) - finding Range Vector(rI rJ rK r) Components in km in frame IJK
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OM-MSS Page 222 Note - Transform_1 : EC to SAT Vector(rp, rq) in frame PQW To EC to SAT Vector(rI rJ rK r) in frame IJK. Inputs : Vector(rp, rq) EC to Sat in km in frame PQW, SAT right ascension of ascending node at time_t, SAT argument of perigee rad at time_t, SAT orbit inclination rad at time_t . Outputs : Vector(rI, rJ, rK, r) EC to SAT in km in frame IJK S08A 011. rI Satellite pos vector component EC to Sat km frame IJK at at time_t = -6231.7560551250 S08B 011. rJ Satellite pos vector component EC to Sat km frame IJK at at time_t = -3189.4018492384 S08C 011. rK Satellite pos vector component EC to Sat km frame IJK at at time_t = 14.8069953230 S08D 011. r Satellite pos vector magnitude EC to Sat km frame IJK at at time_t = 7000.5204759091 Note - r Satellite pos vector magnitude EC to Sat km in PQW frame is same as that computed above in PQW frame. 09. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input at time_t . Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time t UT year = 2014, month = 5, day = 28, hour = 3, minute = 6, seconds = 33.96586 Outputs : GST & GHA in 0-360 deg over Greenwich. S09A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 292.22745, hr = 19, min = 28, sec = 54.58857 S09B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 292.22920, deg = 292, min = 13, sec = 45.10952 10. Satellite(SAT) Orbit point direction : Finding Right Ascension(Alpha) deg and Declination(Delta) deg using angles Inputs : SAT orbit inclination deg at_time_t, EPH right ascension ascending node deg, SAT argument of perigee deg at time_t, SAT true anomaly deg calculated at time_t Outputs : SAT Right Ascension(Alpha) and Declination(Delta) in deg at time_t S10A 011. SAT Right Ascension(Alpha) in deg = 207.1371128409 S10B 011. SAT Declination(Delta) in deg = 0.1211879852 11. Finding Satellite Longitude & Latitude in deg at time_t; (ie Sub-Sat point log & lat on earth surface ). Inputs : SAT right ascension ascending node deg at time_t, GST in 0-360 deg over Greenwich at time_t Outputs : Satellite (Sub-Sat point) longitude 0 to 360 deg at time_t. S11A 011. Satellite longitude 0 to 360 deg at time_t = 207.14 ie deg = 207, min = 8, sec = 13.61
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OM-MSS Page 223 Inputs : argument of_perigee rad at_time_t, inclination rad at time_t, true anomaly rad calculated at time_t. Outputs : Satellite (Sub-Sat point) latitude +ve or -ve in 0 to 90 deg at time_t. S11B 011. Satellite latitude +ve or -ve in 0 to 90 deg at time_t = 0.12 ie deg = 0, min = 7, sec = 16.28 12. Finding Satellite height in km from EC to Sat and from Earth surface to Sat at time_t. (a) Satellite height in km from EC to Sat; (ie Sat orbit radius EC to Sat in km at time_t). Note - This is SAME as r sat pos vector magnitude EC to Sat in frame IJK calculated above in TRANSFORN_1 . Inputs : SAT true anomaly at time_t, semi_major_axis_km, inclination at time_t . Outputs : Sub-Sat point longitude 0 to 360 deg at time_t. S12A 011. Satellite orbital radious EC to SAT in km using SAT true anomaly at time_t = 7000.52 (b) Satellite height in km from earth surface. Inputs : Sub-Sat point latitude +ve or -ve in 0 to 90 deg at time_t, earth_equator_radious_km . Outputs : Sat height in km from earth surface. S12B 011. Satellite height in km from earth surface at time_t = 622.38 13. Finding Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface at time_t . Inputs : Sub-Sat point lat & log, ES lat & log . Outputs : Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface S13 011. Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface = 14014.66346 14. Finding Local sidereal time(LST) and Local mean time(LMT) over Sub-Sat point Longitude on earth . (a) Local sidereal time(LST) in 0 to 360 deg over Sub-Sat point Longitude on earth . Inputs : GST sidereal_time in 0 to 360 deg at Greenwich at time_t, satellite log in 0 to 360 deg at time_t . Outputs : LST local sidereal time in 0 to_360 deg at SAT longitude at time_t. S14A 011. Local sidereal time(LST) over Sub-Sat point Longitude at time_t = 139.36 (b) Local sidereal time(LST) and local Mean time(LMT) adjusted to calendar date(CD) over Sub-Sat point Longitude on earth . Note - The LST and LMT in hr min sec with YY MM DD adjusted to calendar date (CD) of longitude at time_t . Here LST in deg is re-calculated alterately in terms of Julian Day (JD) that account for calendar date (CD) of longitude. Inputs : GMT_JD, Sat longitude_in_0_to_360_deg .
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OM-MSS Page 224 Outputs : LST local sidereal time in 0 to_360 deg at SAT longitude at time_t. S14B 011. Local sidereal time(LST) in 0 to 360_deg over Sub-Sat point Longitude at time_t = 139.36 S14C 011. LST over Sub-Sat Log, with date adj to CD expressed in JD = 2456805.89, ie YY = 2014, MM = 5, DD = 28, hr = 9, min = 17, sec = 27.58 S14D 011. LMT over Sub-Sat Log, with date adj to CD expressed in JD = 2456805.20, ie YY = 2014, MM = 5, DD = 27, hr = 16, min = 55, sec = 6.87 15. Finding Local sidereal time(LST) and Local mean time(LMT) Over Earth stn (ES) or Earth point(EP) Longitude . (a) Local sidereal time(LST) in 0 to 360 deg over Earth stn(ES) Longitude . Inputs : GST sidereal_time in 0 to 360 deg at Greenwich at time_t, Earth stn(ES) longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at Earth stn(ES) longitude at time_t. S15A 011. Local sidereal time(LST) over Earth stn(ES) Longitude at time_t = 9.64 (b) Local sidereal time(LST) and local Mean time(LMT) adjusted to calendar date(CD) over Earth stn(ES) Longitude . Note - The LST and LMT in hr min sec with YY MM DD adjusted to calendar date (CD) of longitude at time_t . Here LST in deg is re-calculated alterately in terms of Julian Day (JD) that account for calendar date (CD) of longitude. Inputs : GMT_JD, Earth stn(ES) longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at Earth stn (ES) longitude at time_t. S15B 011. Local sidereal time(LST) in 0 to 360_deg over Earth stn (ES) Longitude at time_t = 9.64 S14C 011. LST over ES Log, with date adj to CD expressed in JD = 2456806.53, ie YY = 2014, MM = 5, DD = 29, hr = 0, min = 38, sec = 33.70 S15D 011. LMT over ES Log, with date adj to CD expressed in JD = 2456805.84, ie YY = 2014, MM = 5, DD = 28, hr = 8, min = 16, sec = 12.99 16. Earth Stn (ES) Position Vector from Earth Center(EC) to Earth Stn(ES) : Finding Range Vector(RI, RJ, RK, R) Components in IJK frame Note - Transform_2 : ES position cord(lat, log, hgt) To EC to ES position Vector(RI, RJ, RK, R) in frame IJK . Inputs : ES latitude positive_negative 0 to 90 deg, ES longitude in 0 to 360_deg, ES height in meter (is earth surface + tower hgt), LST in 0 to 360 deg at ES log at time_t . Outputs : ES Position Vector(RI, RJ, RK, R) Components EC to ES in km in IJK frame . S16A 011. RI_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 5774.4514013983 S16B 011. RJ_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 980.8294793588 S16C 011. RK_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 2517.6372173288 S16D 011. R_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 6375.3284317570
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OM-MSS Page 225 17. Satellite(SAT) Position Range Vector from Earth Stn(ES) to SAT : Finding Range Vector(rvI, rvJ, rvK, rv) Components in km in IJK frame Note - Transform_3 SAT Pos Vct(rI rJ rK) and ES Pos Vct(RI RJ RK) To SAT Pos Vct(rvI, rvJ, rvK, rv) Inputs : SAT Position Vector(rI rJ rK) EC to Sat in km in frame IJK , ES Position Vector(RI RJ RK) EC to ES in km in IJK frame. Outputs : SAT Position Range Vector(rvI, rvJ, rvK, rv) Components ES to Sat in km in IJK frame S17A 011. rv_I range vector component ES to SAT km frame IJK = -12006.207456523 S17B 011. rv_J range vector component ES to SAT km frame IJK = -4170.2313285972 S17C 011. rv_K range vector component ES to SAT km frame IJK = -2502.8302220058 S17D 011. rv range vector component ES to SAT km frame IJK = 12953.9185555289 18. Satellite(SAT) Position Range Vector from Earth Stn(ES) to SAT : Finding Range Vector(rvS, rvE, rvZ, rv) Components in km in SEZ frame Note - Transform_4 SAT Pos Vct(rvI, rvJ, rvK, rv) in IJK frame To SAT Pos Vct(rvS, rvE, rvZ, rv) in SEZ frame Inputs : ES latitude positive_negative 0 to 90_deg, LST in 0 to 360 deg at ES longitude at time_t SAT PositionRange Vector(rvI, rvJ, rvK, rv) ES to Sat in km in IJK frame Outputs : SAT Position Range(rvS, rvE, rvZ, rv) Components ES to Sat in km in SEZ frame S18A 011. rvS range_vector component ES to SAT km_frame SEZ = -2650.7074843436 S18B 011. rvE range_vector component ES to SAT km_frame SEZ = -2100.8060786626 S18C 011. rvZ range_vector component ES to SAT km_frame SEZ = -12504.573946983 S18D 011. rv range_vector component ES to SAT km_frame SEZ = 12953.9185555289 19. Finding Elevation(EL) and Azimuth(AZ) angles of Satellite and Sun : Steps 1, 2, 3 AT UT TIME t Rem: Sub-SAT point lat deg = 0.12, log deg = 207.14 YY = 2014, MM = 5, DD = 27, hr = 16, min = 55, sec = 6.87 ES or EP point lat deg = 23.26, log deg = 77.41 YY = 2014, MM = 5, DD = 28, hr = 8, min = 16, sec = 12.99 Note : Step 1 is for Satellite EL & AZ angles. Steps 2 & 3 are for Sun EL & AZ angles Results verified from other sources; Ref URLs Geoscience Australia http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone , NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html
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OM-MSS Page 226 Elevation(EL) & Azimuth(AZ) angle of SAT at Earth Observation point EP : Step 1. Inputs : Range vector component ES to SAT rv_S, rv_E, rv_Z, in frame_SEZ, EP and Sub_sat point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of SAT at EP S19A 011. Elevation angle deg of Satellte at Earth point EP at time_t = -74.86473 ie deg = -74, min = -51, sec = -53.02 S19B 011. Azimuth angle deg of Satellte at Earth point EP at time_t = 321.60158 ie deg = 321, min = 36, sec = 5.68 Elevation(EL) & Azimuth(AZ) angle of SUN at Sub_Satellite point on earth surface : Step 2. Inputs : Range vector component Sub_sat to Sun rv_S, rv_E, rv_Z, in frame_SEZ, Sub_sat point and Sun_Sun point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of Sun at Sub_Sat S19C 011. Elevation angle deg of SUN at Sub_Sat point at time_t = 14.4723644816 S19D 011. Azimuth angle deg of SUN at Sub_Sat point at time_t = 292.1397432144 Elevation(EL) & Azimuth(AZ) angle of SUN at Satellite height : Step 3. Inputs : Range vector component Sub_sat to Sun rv_S, rv_E, rv_Z, in frame_SEZ, Sub_sat point and Sun_Sun point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of SUN at SAT S19E 011. Elevation angle deg of Sun at Satellite height at time_t = 14.47214 S19F 011. Azimuth angle deg of Sun at Satellite height at time_t = 292.13974 20. Finding Satellite Velocity meter per sec in orbit . Note : Results computed using 2 different formulations, each require different inputs. (a) Inputs : SAT orbit semi-major axis SMA, GM_EARTH, sat_orbit pos r_vector_EC_to_SAT_km_frame_IJK. Outputs : SAT Velocity magnitude and component Xw Yw in frame PQW in meter per sec S20A 011. Satellite Velocity magnitude meter per sec at UT time = 7550.1615459169 (b) Inputs : SAT orbit semi-major axis SMA, GM_EARTH, SAT orbit eccentricity_at_time_t, SAT eccentric anomaly deg calculated at time_t. Outputs : Satellite Velocity components Xw Yw in frame PQW in meter per sec
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OM-MSS Page 227 S20B 011. Satellite Velocity components Xw in frame PQW in meter per sec = 5272.7792443276 S20C 011. Satellite Velocity components Yw in frame PQW in meter per sec = 5403.9558112581 21. Finding Satellite(SAT) Velocity Vector (vX, vY, vZ) in meter per sec in orbit in frame XYZ. Note - Transform_5 SAT Vel Vct(Xw, Yw) in frame PQW To SAT Vel Vct(vX, vY, vZ) in frame XYZ Inputs : SAT velocity vectors(Xw, Yw), SAT Right Ascension Alpha, SAT Argument of perigee, SAT orbit inclination at_time_t, SAT eccentric_anomaly_deg_calculated_at_time_t. Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ S21A 011. vX Sat Velocity vector component in meter per sec = -453.7396013124 S21B 011. vY Sat Velocity vector component in meter per sec = 940.0898291212 S21C 011. vZ Sat Velocity vector component in meter per sec = 7477.6527638575 S21D 011. vR Sat Velocity magnitude meter in meter per sec = 7550.1615459169 22. Finding Satellite(SAT) Pitch and Roll angles Inputs : Earth equator radious km, ES lat, ES log, Sub_Sat point lat, Sub_Sat point log, Sat_orbit pos r_vector_EC_to_SAT_km_frame_IJK, Outputs : SAT Pitch and Roll angles S22A 011. pitch_angle_deg_for_earth_stn_at_sat_lat_log_at_time_t = 24.0229508063 S22B 011. roll_angle_deg_for_earth_stn_at_sat_lat_log_at_time_t = 12.1020326731 23. Finding Satellite State Vectors - Position [ X, Y, Z ] in km and velocity [ Vx, Vy, Vz ] in meter per sec at time_t . Note - same as values of rI rJ rK r for pos and vX vY vZ vR for vel (a) Satellite State Position Vector [X, Y, Z] in km at time_t Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ S23A 011. State vector position X km at time_t = -6231.7560551250 S23B 011. State vector position Y km at time_t = -3189.4018492384 S23C 011. State vector position Z km at time_t = 14.8069953230 S23D 011. State vector position R km at time_t = 7000.5204759091
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OM-MSS Page 228 (b) Satellite State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ S23E 011. State vector velocity Vx meter per sec at time_t = -453.7396013124 S23F 011. State vector velocity Vy meter per sec at time_t = 940.0898291212 S23G 011. State vector velocity Vz meter per sec at time_t = 7477.6527638575 S23H 011. State vector velocity V meter per sec at time_t = 7550.1615459169 24. Satellite Direction ie right ascension alpha deg and declination delta deg using sat position vector . Note - results same as that using angles incl, RA, Aug of peri, true anomaly Inputs : SAT State Vectors Position(X, Y, Z, R) in km , in frame XYZ Outputs : SAT right ascension alpha deg and declination delta deg S24A 011. Satellite direction right_asc alpha deg at time_t = 207.1032871591 S24B 011. Satellite direction_declination_delta_deg_at_time_t = 0.1211879852 25. Satellite Angular momentum km sqr per sec : finding Hx Hy Hz H from state vector pos and vel . Inputs : SAT State Vectors Position(X, Y, Z, R) in km and Velocity (vX, vY, vZ, vR) meter per sec, in frame XYZ Outputs : SAT angular momentum (Hx Hy Hz H) componts in km sqr per sec S25A 011. Satellite angular momentum Hx in km sqr per sec at time_t = -23863.1594587127 S25B 011. Satellite angular momentum Hy in km sqr per sec at time_t = 46592.1893691367 S25C 011. Satellite angular momentum Hz in km sqr per sec at time_t = -7305.5684084861 S25D 011. Satellite angular momentum H in km sqr per sec at time_t = 52855.0264339402 26. Satellite Orbit normal Vector : finding Wx Wy Wz W Delta Alpha from r_sat_pos frame IJK, i, RA Inputs : SAT Position Vector(rI rJ rK) EC to Sat in km in frame IJK , inclination_deg_at_time_t, right_ascension_ascending_node_deg_at_time_t Outputs : SAT orbit normal vector (Wx, Wy, Wz, W) in km , RA , i S26A 011. Satellite Orbit normal_W in km = 7000.5204759091 S26B 011. Satellite Orbit normal_Wx in km = -3160.6177819132 S26C 011. Satellite Orbit normal_Wy in km = 6171.0228468759
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OM-MSS Page 229 S26D 011. Satellite Orbit normal_Wz in km = -967.6048747356 S26E 011. Satellite Orbit normal_Delta_W in deg = -7.9448000000 S26F 011. Satellite Orbit interpreted inclination i = 97.9448000000 S26G 011. Satellite Orbit normal_Alpha_W in deg = -62.8798000000 S26H 011. Satellite Orbit interpreted RA of asc node = 207.1202000000 Transform Satellite State Vectors to Keplerian elements. 27. Finding Satellite position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness S27A 011. Keplerian elements year = 2014, days_decimal_of_year = 147.12956, revolution no = 20925, node = 1 ie ascending S27B 011. inclination_deg = 97.9448000000 S27C 011. right ascension ascending node deg = 207.1202000000 S27D 011. eccentricity = 0.0016257000 S27E 011. argument of perigee_deg = 44.4835000000 S27F 011. mean anomaly deg = 315.7690000000 S27G 011. mean_motion rev per day = 14.7867948300 S27H 011. mean angular velocity rev_per_day = 14.7961532483 S27I 011. mean motion rev per day using SMA considering oblateness = 14.7774423308 Transform Satellite Keplerian elements to State Vectors . 28. Finding Satellite position State Vectors, computed using Keplerian elements at time input UT (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ]
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OM-MSS Page 230 S28A 011. State vectors year = 2014, days_decimal_of_year = 147.12956, revolution no = 20925, node = 1 ie decending S28B 011. state vector position X km = -6231.7560551250, state vector velocity Vx meter per sec = -453.7396013125 S28C 011. state vector position Y km = -3189.4018492384, state vector velocity Vy meter per sec = 940.0898291212 S28D 011. state vector position Z km = 14.8069953229, state vector velocity Vz meter per sec = 7477.6527638575 S28E 011. state vector position R km = 7000.5204759091, state vector velocity V meter per sec = 7550.1615459169 Move on to next Satellite. Next Section - 6.4 Computing Orbital & Positional parameters for Satellite ISS (ZARYA)
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OM-MSS Page 231 OM-MSS Section - 6.4 ---------------------------------------------------------------------------------------------------52 Satellite ISS (ZARYA) : Computing Orbital & Positional parameters corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins. (d) ISS (ZARYA) 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on November 20, 1998 1 25544U 98067A 14148.25353351 .00006506 00000-0 11951-3 0 3738 2 25544 51.6471 198.4055 0003968 47.6724 33.3515 15.50569135888233 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 25544, ISS (ZARYA) , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.2535335100 EPOCH_inclination_deg = 51.6471000000 EPOCH_right_asc_acnd_node_deg = 198.4055000000 EPOCH_eccentricity = 0.0003968000 EPOCH_argument_of_perigee_deg = 47.6724000000 EPOCH_mean_anomaly_deg = 33.3515000000 EPOCH_mean_motion_rev_per_day = 15.5056913500 EPOCH_revolution = 88823 EPOCH_node_condition = 1 Earth_stn_latitude_deg = 23.25993 Earth stn longitude_deg = 77.41261 Earth surface height_meter = 494.70000 Earth stn tower_height_meter = 15.00000 Earth stn_height_meter = 509.70000 Earth stn min EL look_angle_deg = -5.00000 Note : EPOCH Corresponds to UT year = 2014, month = 5, day = 28, hr = 6, min = 5, sec = 5.29526 which is Greenwich meam time ie GMT Converted to Local Meam Time at Earth Stn Longitude, as regular Calender date : year = 2014, month = 5, day = 28, hr = 11, min = 14, sec = 44.32 At this instant Sun Position as seen from Earth Stn Longitude : Sun angles EL deg = 80.01, AZ deg = 98.36, Sun Surface distance km = 1111.87, Radial km = 151595496.79, Sun Rise D:28, H:05, M:17, S:03 Sun Set D:28, H:18, M:40, S:20 Move to Compute Satellie Orbital parameters corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins, and Earth Stn location. 01. Input EPOCH_year and EPOCH_days_decimal_of_year, Converted into UT YY MM DD hh min sec & Julian day. S01 011. Input UT year = 2014, month = 5, day = 28, hr = 6, min = 5, sec = 5.29526, and julian_day = 2456805.7535335100 02. Finding Satellite orbit Semi major axis in km, Ignoring and also Considering earth oblatenes . (a) Semi major axis (SMA) km at time t Ignoring earth oblatenes . Inputs : SAT mean_motion rev per day at time t, GM_EARTH . Outputs : SAT Orbit semi major axis km in km and constant_A, constant_k1 .
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OM-MSS Page 232 S02A 011. Satellite orbit Semi major axis in km at time t, Ignoring earth oblatenes = 6793.20027, (b) Semi major axis (SMA) in km at time_t Considering earth oblatenes Inputs : SAT mean_motion rev per day at time_t, GM_EARTH , inclination deg at time_t, eccentricity at time_t, constant_k2 Outputs : SAT Orbit semi major axis km in km at time_t and constant_A, constant_k1, constant_k3 . S02B 011. Satellite orbit Semi major axis km at time t, Considering earth oblatenes = 6793.70175, 03. Finding Satellite Mean motion in rev per day, Ignoring and also Considering earth oblatenes . (a) Nominal mean motion rev per day at time_t Ignoring earth oblateness . Inputs : SAT Orbit semi major axis in km ignoring oblateness at time_t, constant_k1 . Outputs : SAT nominal mean motion in rev_per_day at time_t, ignoring earth oblateness S03A 011. Satellite Nominal Mean motion in rev_per_day at time_t using SMA Ignoring earth oblateness = 15.50569, (b) Mean motion rev per day at time_t Considering earth oblatenes Inputs : SAT nominal mean motion rad per day at time_t Ignoring_oblateness, constant_k2, constant_k3, SAT orbit semi major axis in km considering oblateness at time_t. Outputs : SAT mean motion rev per day at time_t, considering earth oblatenes . S03B 011. Satellite Mean motion in rev_per_day at time t using SMA Considering earth oblatenes = 15.50741, Note - This calculted value is slightly less than the mean motion rev_per_day as EPH input from NORAD TLE 04. Finding Satellite Orbit Time Period in minute at time_t Considering earth oblatenes . Inputs : SAT orbit semi major axis in km considering oblateness at time_t, GM_EARTH . Outputs : SAT orbit time period in minute at time_t considering earth oblatenes . S04 011. Satellite orbit Time Period in minute at time_t using SMA Considering earth oblatenes = 92.87941, 05. Finding Satellite Rate of change of Right Ascension and Argument of Perigee in deg per_day at time_t. (a) Rate of change of Right Ascension in deg per day at time_t . Inputs : SAT mean motion rev per day at time_t considering earth oblatenes, constant_k2, SAT orbit eccentricity at time_t, SAT semi major axis km considering oblateness at time_t, SAT orbit inclination deg at time_t . Outputs : SAT rate of change of right ascension in deg per day at time_t and constant_k_deg_per_day
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OM-MSS Page 233 S05A 011. Satellite Rate of change of Right Ascension in deg per day at time_t = -4.94720, (b) Rate of change of Argument of Perigee in deg per_day at time_t . Inputs : SAT orbit constant_k_deg_per_day, SAT orbit inclination deg at time_t SAT orbit semi major axis km considering oblateness at time_t Outputs : SAT rate of change of argument of perigee in deg per day at time_t S05B 011. Satellite Rate of change of Argument of Perigee in deg per day at time_t = 3.68794, 06. Finding Satellite Mean anomaly, Eccentric anomaly, True anomaly in deg at time_t considering earth oblateness. Inputs : SAT mean anomaly rad at time_t, mean_motion_rad_per_day_at_time_t considering_oblateness, SAT orbit eccentricity_at_time_t Outputs : SAT mean anomaly, eccentric anomaly, and true anomaly in deg at time_t onsidering earth oblateness. S06A 011. Satellite Mean anomaly in deg at time_t = 33.35150, same as EPH mean anomaly S06A 011. Satellite Eccentric anomaly in deg at time_t = 33.36400, S06A 011. Satellite True anomaly in deg at time_t = 33.37651, Note - 1. Here after, the Earth Oblateness is always considered for the computation of satellite orbit parameters, and not repeatedly mentioned. Satellite to Earth, the Position Vectors coordinate and the Vector Coordinate Transforms are in PQW, IJK, SEZ frames . - Perifocal Coordinate System (PQW) is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. - Geocentric Coordinate System(IJK) is Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). - Topocentric Horizon Coordinate System(SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). Each of these coordinate system were explained in detail before and therefore not repeated any more. 07. Finding Satellite position vector[rp, rq] from Earth center(EC) to Sat in PQW frame, perifocal coordinate system. Inputs : SAT orbit semi-major axis (SMA), SAT orbit eccentricity, SAT eccentric anomaly, SATtrue anomaly at time_t Outputs : Vector(r, rp rq) in PQW frame S07A 011. r Satellite pos vector magnitude EC to Sat km in PQW frame perifocal cord at time_t = 6791.45029 S07B 011. rp Satellite pos vector component EC to Sat km in PQW frame perifocal cord at time_t = 5671.3601057139 S07C 011. rq Satellite pos vector component EC to Sat km in PQW frame perifocal cord at time_t = 3736.2376169157 08. Satellite Position Vector Earth Ceter(EC) to Satellite(SAT) - finding Range Vector(rI rJ rK r) Components in km in frame IJK
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OM-MSS Page 234 Note - Transform_1 : EC to SAT Vector(rp, rq) in frame PQW To EC to SAT Vector(rI rJ rK r) in frame IJK. Inputs : Vector(rp, rq) EC to Sat in km in frame PQW, SAT right ascension of ascending node at time_t, SAT argument of perigee rad at time_t, SAT orbit inclination rad at time_t . Outputs : Vector(rI, rJ, rK, r) EC to SAT in km in frame IJK S08A 011. rI Satellite pos vector component EC to Sat km frame IJK at at time_t = 311.7253734371 S08B 011. rJ Satellite pos vector component EC to Sat km frame IJK at at time_t = -4283.4907194611 S08C 011. rK Satellite pos vector component EC to Sat km frame IJK at at time_t = 5261.0200081909 S08D 011. r Satellite pos vector magnitude EC to Sat km frame IJK at at time_t = 6791.4502853764 Note - r Satellite pos vector magnitude EC to Sat km in PQW frame is same as that computed above in PQW frame. 09. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input at time_t . Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time t UT year = 2014, month = 5, day = 28, hour = 6, minute = 5, seconds = 5.29526 Outputs : GST & GHA in 0-360 deg over Greenwich. S09A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 336.98019, hr = 22, min = 27, sec = 55.24463 S09B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 336.98371, deg = 336, min = 59, sec = 1.37032 10. Satellite(SAT) Orbit point direction : Finding Right Ascension(Alpha) deg and Declination(Delta) deg using angles Inputs : SAT orbit inclination deg at_time_t, EPH right ascension ascending node deg, SAT argument of perigee deg at time_t, SAT true anomaly deg calculated at time_t Outputs : SAT Right Ascension(Alpha) and Declination(Delta) in deg at time_t S10A 011. SAT Right Ascension(Alpha) in deg = 274.1622871100 S10B 011. SAT Declination(Delta) in deg = 50.7736185669 11. Finding Satellite Longitude & Latitude in deg at time_t; (ie Sub-Sat point log & lat on earth surface ). Inputs : SAT right ascension ascending node deg at time_t, GST in 0-360 deg over Greenwich at time_t Outputs : Satellite (Sub-Sat point) longitude 0 to 360 deg at time_t. S11A 011. Satellite longitude 0 to 360 deg at time_t = 274.16 ie deg = 274, min = 9, sec = 44.23
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OM-MSS Page 235 Inputs : argument of_perigee rad at_time_t, inclination rad at time_t, true anomaly rad calculated at time_t. Outputs : Satellite (Sub-Sat point) latitude +ve or -ve in 0 to 90 deg at time_t. S11B 011. Satellite latitude +ve or -ve in 0 to 90 deg at time_t = 50.77 ie deg = 50, min = 46, sec = 25.03 12. Finding Satellite height in km from EC to Sat and from Earth surface to Sat at time_t. (a) Satellite height in km from EC to Sat; (ie Sat orbit radius EC to Sat in km at time_t). Note - This is SAME as r sat pos vector magnitude EC to Sat in frame IJK calculated above in TRANSFORN_1 . Inputs : SAT true anomaly at time_t, semi_major_axis_km, inclination at time_t . Outputs : Sub-Sat point longitude 0 to 360 deg at time_t. S12A 011. Satellite orbital radious EC to SAT in km using SAT true anomaly at time_t = 6791.45 (b) Satellite height in km from earth surface. Inputs : Sub-Sat point latitude +ve or -ve in 0 to 90 deg at time_t, earth_equator_radious_km . Outputs : Sat height in km from earth surface. S12B 011. Satellite height in km from earth surface at time_t = 426.15 13. Finding Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface at time_t . Inputs : Sub-Sat point lat & log, ES lat & log . Outputs : Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface S13 011. Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface = 11633.16757 14. Finding Local sidereal time(LST) and Local mean time(LMT) over Sub-Sat point Longitude on earth . (a) Local sidereal time(LST) in 0 to 360 deg over Sub-Sat point Longitude on earth . Inputs : GST sidereal_time in 0 to 360 deg at Greenwich at time_t, satellite log in 0 to 360 deg at time_t . Outputs : LST local sidereal time in 0 to_360 deg at SAT longitude at time_t. S14A 011. Local sidereal time(LST) over Sub-Sat point Longitude at time_t = 251.14 (b) Local sidereal time(LST) and local Mean time(LMT) adjusted to calendar date(CD) over Sub-Sat point Longitude on earth . Note - The LST and LMT in hr min sec with YY MM DD adjusted to calendar date (CD) of longitude at time_t . Here LST in deg is re-calculated alterately in terms of Julian Day (JD) that account for calendar date (CD) of longitude. Inputs : GMT_JD, Sat longitude_in_0_to_360_deg .
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OM-MSS Page 236 Outputs : LST local sidereal time in 0 to_360 deg at SAT longitude at time_t. S14B 011. Local sidereal time(LST) in 0 to 360_deg over Sub-Sat point Longitude at time_t = 251.14 S14C 011. LST over Sub-Sat Log, with date adj to CD expressed in JD = 2456806.20, ie YY = 2014, MM = 5, DD = 28, hr = 16, min = 44, sec = 34.28 S14D 011. LMT over Sub-Sat Log, with date adj to CD expressed in JD = 2456805.52, ie YY = 2014, MM = 5, DD = 28, hr = 0, min = 21, sec = 44.24 15. Finding Local sidereal time(LST) and Local mean time(LMT) Over Earth stn (ES) or Earth point(EP) Longitude . (a) Local sidereal time(LST) in 0 to 360 deg over Earth stn(ES) Longitude . Inputs : GST sidereal_time in 0 to 360 deg at Greenwich at time_t, Earth stn(ES) longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at Earth stn(ES) longitude at time_t. S15A 011. Local sidereal time(LST) over Earth stn(ES) Longitude at time_t = 54.39 (b) Local sidereal time(LST) and local Mean time(LMT) adjusted to calendar date(CD) over Earth stn(ES) Longitude . Note - The LST and LMT in hr min sec with YY MM DD adjusted to calendar date (CD) of longitude at time_t . Here LST in deg is re-calculated alterately in terms of Julian Day (JD) that account for calendar date (CD) of longitude. Inputs : GMT_JD, Earth stn(ES) longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at Earth stn (ES) longitude at time_t. S15B 011. Local sidereal time(LST) in 0 to 360_deg over Earth stn (ES) Longitude at time_t = 54.39 S14C 011. LST over ES Log, with date adj to CD expressed in JD = 2456806.65, ie YY = 2014, MM = 5, DD = 29, hr = 3, min = 37, sec = 34.36 S15D 011. LMT over ES Log, with date adj to CD expressed in JD = 2456805.97, ie YY = 2014, MM = 5, DD = 28, hr = 11, min = 14, sec = 44.32 16. Earth Stn (ES) Position Vector from Earth Center(EC) to Earth Stn(ES) : Finding Range Vector(RI, RJ, RK, R) Components in IJK frame Note - Transform_2 : ES position cord(lat, log, hgt) To EC to ES position Vector(RI, RJ, RK, R) in frame IJK . Inputs : ES latitude positive_negative 0 to 90 deg, ES longitude in 0 to 360_deg, ES height in meter (is earth surface + tower hgt), LST in 0 to 360 deg at ES log at time_t . Outputs : ES Position Vector(RI, RJ, RK, R) Components EC to ES in km in IJK frame . S16A 011. RI_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 3410.1853501576 S16B 011. RJ_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 4762.0322691327 S16C 011. RK_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 2517.6372173288 S16D 011. R_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 6375.3284317570
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OM-MSS Page 237 17. Satellite(SAT) Position Range Vector from Earth Stn(ES) to SAT : Finding Range Vector(rvI, rvJ, rvK, rv) Components in km in IJK frame Note - Transform_3 SAT Pos Vct(rI rJ rK) and ES Pos Vct(RI RJ RK) To SAT Pos Vct(rvI, rvJ, rvK, rv) Inputs : SAT Position Vector(rI rJ rK) EC to Sat in km in frame IJK , ES Position Vector(RI RJ RK) EC to ES in km in IJK frame. Outputs : SAT Position Range Vector(rvI, rvJ, rvK, rv) Components ES to Sat in km in IJK frame S17A 011. rv_I range vector component ES to SAT km frame IJK = -3098.4599767204 S17B 011. rv_J range vector component ES to SAT km frame IJK = -9045.5229885937 S17C 011. rv_K range vector component ES to SAT km frame IJK = 2743.3827908621 S17D 011. rv range vector component ES to SAT km frame IJK = 9947.2654283333 18. Satellite(SAT) Position Range Vector from Earth Stn(ES) to SAT : Finding Range Vector(rvS, rvE, rvZ, rv) Components in km in SEZ frame Note - Transform_4 SAT Pos Vct(rvI, rvJ, rvK, rv) in IJK frame To SAT Pos Vct(rvS, rvE, rvZ, rv) in SEZ frame Inputs : ES latitude positive_negative 0 to 90_deg, LST in 0 to 360 deg at ES longitude at time_t SAT PositionRange Vector(rvI, rvJ, rvK, rv) ES to Sat in km in IJK frame Outputs : SAT Position Range(rvS, rvE, rvZ, rv) Components ES to Sat in km in SEZ frame S18A 011. rvS range_vector component ES to SAT km_frame SEZ = -6137.0343240468 S18B 011. rvE range_vector component ES to SAT km_frame SEZ = -2747.3972741696 S18C 011. rvZ range_vector component ES to SAT km_frame SEZ = -7330.5325471668 S18D 011. rv range_vector component ES to SAT km_frame SEZ = 9947.2654283333 19. Finding Elevation(EL) and Azimuth(AZ) angles of Satellite and Sun : Steps 1, 2, 3 AT UT TIME t Rem: Sub-SAT point lat deg = 50.77, log deg = 274.16 YY = 2014, MM = 5, DD = 28, hr = 0, min = 21, sec = 44.24 ES or EP point lat deg = 23.26, log deg = 77.41 YY = 2014, MM = 5, DD = 28, hr = 11, min = 14, sec = 44.32 Note : Step 1 is for Satellite EL & AZ angles. Steps 2 & 3 are for Sun EL & AZ angles Results verified from other sources; Ref URLs Geoscience Australia http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone , NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html
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OM-MSS Page 238 Elevation(EL) & Azimuth(AZ) angle of SAT at Earth Observation point EP : Step 1. Inputs : Range vector component ES to SAT rv_S, rv_E, rv_Z, in frame_SEZ, EP and Sub_sat point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of SAT at EP S19A 011. Elevation angle deg of Satellte at Earth point EP at time_t = -47.47135 ie deg = -47, min = -28, sec = -16.87 S19B 011. Azimuth angle deg of Satellte at Earth point EP at time_t = 335.88313 ie deg = 335, min = 52, sec = 59.26 Elevation(EL) & Azimuth(AZ) angle of SUN at Sub_Satellite point on earth surface : Step 2. Inputs : Range vector component Sub_sat to Sun rv_S, rv_E, rv_Z, in frame_SEZ, Sub_sat point and Sun_Sun point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of Sun at Sub_Sat S19C 011. Elevation angle deg of SUN at Sub_Sat point at time_t = -17.5724321856 S19D 011. Azimuth angle deg of SUN at Sub_Sat point at time_t = 5.9788181882 Elevation(EL) & Azimuth(AZ) angle of SUN at Satellite height : Step 3. Inputs : Range vector component Sub_sat to Sun rv_S, rv_E, rv_Z, in frame_SEZ, Sub_sat point and Sun_Sun point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of SUN at SAT S19E 011. Elevation angle deg of Sun at Satellite height at time_t = -17.57259 S19F 011. Azimuth angle deg of Sun at Satellite height at time_t = 5.97882 20. Finding Satellite Velocity meter per sec in orbit . Note : Results computed using 2 different formulations, each require different inputs. (a) Inputs : SAT orbit semi-major axis SMA, GM_EARTH, sat_orbit pos r_vector_EC_to_SAT_km_frame_IJK. Outputs : SAT Velocity magnitude and component Xw Yw in frame PQW in meter per sec S20A 011. Satellite Velocity magnitude meter per sec at UT time = 7662.3074951298 (b) Inputs : SAT orbit semi-major axis SMA, GM_EARTH, SAT orbit eccentricity_at_time_t, SAT eccentric anomaly deg calculated at time_t. Outputs : Satellite Velocity components Xw Yw in frame PQW in meter per sec
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OM-MSS Page 239 S20B 011. Satellite Velocity components Xw in frame PQW in meter per sec = -4213.9331945858 S20C 011. Satellite Velocity components Yw in frame PQW in meter per sec = 6399.5096047658 21. Finding Satellite(SAT) Velocity Vector (vX, vY, vZ) in meter per sec in orbit in frame XYZ. Note - Transform_5 SAT Vel Vct(Xw, Yw) in frame PQW To SAT Vel Vct(vX, vY, vZ) in frame XYZ Inputs : SAT velocity vectors(Xw, Yw), SAT Right Ascension Alpha, SAT Argument of perigee, SAT orbit inclination at_time_t, SAT eccentric_anomaly_deg_calculated_at_time_t. Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ S21A 011. vX Sat Velocity vector component in meter per sec = 7415.4532574405 S21B 011. vY Sat Velocity vector component in meter per sec = 1686.8647169809 S21C 011. vZ Sat Velocity vector component in meter per sec = 936.2139516379 S21D 011. vR Sat Velocity magnitude meter in meter per sec = 7662.3074951298 22. Finding Satellite(SAT) Pitch and Roll angles Inputs : Earth equator radious km, ES lat, ES log, Sub_Sat point lat, Sub_Sat point log, Sat_orbit pos r_vector_EC_to_SAT_km_frame_IJK, Outputs : SAT Pitch and Roll angles S22A 011. pitch_angle_deg_for_earth_stn_at_sat_lat_log_at_time_t = -7.7991142227 S22B 011. roll_angle_deg_for_earth_stn_at_sat_lat_log_at_time_t = 11.2997817577 23. Finding Satellite State Vectors - Position [ X, Y, Z ] in km and velocity [ Vx, Vy, Vz ] in meter per sec at time_t . Note - same as values of rI rJ rK r for pos and vX vY vZ vR for vel (a) Satellite State Position Vector [X, Y, Z] in km at time_t Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ S23A 011. State vector position X km at time_t = 311.7253734371 S23B 011. State vector position Y km at time_t = -4283.4907194611 S23C 011. State vector position Z km at time_t = 5261.0200081909 S23D 011. State vector position R km at time_t = 6791.4502853764
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OM-MSS Page 240 (b) Satellite State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ S23E 011. State vector velocity Vx meter per sec at time_t = 7415.4532574405 S23F 011. State vector velocity Vy meter per sec at time_t = 1686.8647169809 S23G 011. State vector velocity Vz meter per sec at time_t = 936.2139516379 S23H 011. State vector velocity V meter per sec at time_t = 7662.3074951298 24. Satellite Direction ie right ascension alpha deg and declination delta deg using sat position vector . Note - results same as that using angles incl, RA, Aug of peri, true anomaly Inputs : SAT State Vectors Position(X, Y, Z, R) in km , in frame XYZ Outputs : SAT right ascension alpha deg and declination delta deg S24A 011. Satellite direction right_asc alpha deg at time_t = 274.1622871100 S24B 011. Satellite direction_declination_delta_deg_at_time_t = 50.7736185669 25. Satellite Angular momentum km sqr per sec : finding Hx Hy Hz H from state vector pos and vel . Inputs : SAT State Vectors Position(X, Y, Z, R) in km and Velocity (vX, vY, vZ, vR) meter per sec, in frame XYZ Outputs : SAT angular momentum (Hx Hy Hz H) componts in km sqr per sec S25A 011. Satellite angular momentum Hx in km sqr per sec at time_t = -12884.8928004185 S25B 011. Satellite angular momentum Hy in km sqr per sec at time_t = 38721.0063135077 S25C 011. Satellite angular momentum Hz in km sqr per sec at time_t = 32289.8637426827 S25D 011. Satellite angular momentum H in km sqr per sec at time_t = 52038.1791853826 26. Satellite Orbit normal Vector : finding Wx Wy Wz W Delta Alpha from r_sat_pos frame IJK, i, RA Inputs : SAT Position Vector(rI rJ rK) EC to Sat in km in frame IJK , inclination_deg_at_time_t, right_ascension_ascending_node_deg_at_time_t Outputs : SAT orbit normal vector (Wx, Wy, Wz, W) in km , RA , i S26A 011. Satellite Orbit normal_W in km = 6791.4502853764 S26B 011. Satellite Orbit normal_Wx in km = -1681.5943650662 S26C 011. Satellite Orbit normal_Wy in km = 5053.4394841355
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OM-MSS Page 241 S26D 011. Satellite Orbit normal_Wz in km = 4214.1175529755 S26E 011. Satellite Orbit normal_Delta_W in deg = 38.3529000000 S26F 011. Satellite Orbit interpreted inclination i = 51.6471000000 S26G 011. Satellite Orbit normal_Alpha_W in deg = -71.5945000000 S26H 011. Satellite Orbit interpreted RA of asc node = 198.4055000000 Transform Satellite State Vectors to Keplerian elements. 27. Finding Satellite position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness S27A 011. Keplerian elements year = 2014, days_decimal_of_year = 147.25353, revolution no = 88823, node = 1 ie ascending S27B 011. inclination_deg = 51.6471000000 S27C 011. right ascension ascending node deg = 198.4055000000 S27D 011. eccentricity = 0.0003968000 S27E 011. argument of perigee_deg = 47.6723999999 S27F 011. mean anomaly deg = 33.3515000001 S27G 011. mean_motion rev per day = 15.5056913500 S27H 011. mean angular velocity rev_per_day = 15.5039745355 S27I 011. mean motion rev per day using SMA considering oblateness = 15.5074083546 Transform Satellite Keplerian elements to State Vectors . 28. Finding Satellite position State Vectors, computed using Keplerian elements at time input UT (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ]
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OM-MSS Page 242 S28A 011. State vectors year = 2014, days_decimal_of_year = 147.25353, revolution no = 88823, node = 1 ie decending S28B 011. state vector position X km = 311.7253734381, state vector velocity Vx meter per sec = 7415.4532574405 S28C 011. state vector position Y km = -4283.4907194609, state vector velocity Vy meter per sec = 1686.8647169816 S28D 011. state vector position Z km = 5261.0200081910, state vector velocity Vz meter per sec = 936.2139516370 S28E 011. state vector position R km = 6791.4502853764, state vector velocity V meter per sec = 7662.3074951298 Move on to next Satellite. Next Section - 6.5 Computing Computing Orbital & Positional parameters for Satellite GSAT-14
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OM-MSS Page 243 OM-MSS Section - 6.5 ---------------------------------------------------------------------------------------------------53 Satellite GSAT-14 : Computing Orbital & Positional parameters corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins. (e) GSAT-14 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:14 hrs IST, Satellite launched on January 05, 2014 1 39498U 14001A 14146.03167358 -.00000092 00000-0 00000+0 0 1238 2 39498 0.0049 223.9821 0002051 110.2671 354.6468 1.00272265 1407 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 39498, GSAT-14 , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 145.0316735800 EPOCH_inclination_deg = 0.0049000000 EPOCH_right_asc_acnd_node_deg = 223.9821000000 EPOCH_eccentricity = 0.0002051000 EPOCH_argument_of_perigee_deg = 110.2671000000 EPOCH_mean_anomaly_deg = 354.6468000000 EPOCH_mean_motion_rev_per_day = 1.0027226500 EPOCH_revolution = 140 EPOCH_node_condition = 1 Earth_stn_latitude_deg = 23.25993 Earth stn longitude_deg = 77.41261 Earth surface height_meter = 494.70000 Earth stn tower_height_meter = 15.00000 Earth stn_height_meter = 509.70000 Earth stn min EL look_angle_deg = -5.00000 Note : EPOCH Corresponds to UT year = 2014, month = 5, day = 26, hr = 0, min = 45, sec = 36.59731 which is Greenwich meam time ie GMT Converted to Local Meam Time at Earth Stn Longitude, as regular Calender date : year = 2014, month = 5, day = 26, hr = 5, min = 55, sec = 15.62 At this instant Sun Position as seen from Earth Stn Longitude : Sun angles EL deg = 7.79, AZ deg = 70.35, Sun Surface distance km = 9151.22, Radial km = 151543715.47, Sun Rise D:26, H:05, M:17, S:33 Sun Set D:26, H:18, M:39, S:23 Move to Compute Satellie Orbital parameters corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins, and Earth Stn location. 01. Input EPOCH_year and EPOCH_days_decimal_of_year, Converted into UT YY MM DD hh min sec & Julian day. S01 011. Input UT year = 2014, month = 5, day = 26, hr = 0, min = 45, sec = 36.59731, and julian_day = 2456803.5316735799 02. Finding Satellite orbit Semi major axis in km, Ignoring and also Considering earth oblatenes . (a) Semi major axis (SMA) km at time t Ignoring earth oblatenes . Inputs : SAT mean_motion rev per day at time t, GM_EARTH . Outputs : SAT Orbit semi major axis km in km and constant_A, constant_k1 .
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OM-MSS Page 244 S02A 011. Satellite orbit Semi major axis in km at time t, Ignoring earth oblatenes = 42164.59740, (b) Semi major axis (SMA) in km at time_t Considering earth oblatenes Inputs : SAT mean_motion rev per day at time_t, GM_EARTH , inclination deg at time_t, eccentricity at time_t, constant_k2 Outputs : SAT Orbit semi major axis km in km at time_t and constant_A, constant_k1, constant_k3 . S02B 011. Satellite orbit Semi major axis km at time t, Considering earth oblatenes = 42165.63953, 03. Finding Satellite Mean motion in rev per day, Ignoring and also Considering earth oblatenes . (a) Nominal mean motion rev per day at time_t Ignoring earth oblateness . Inputs : SAT Orbit semi major axis in km ignoring oblateness at time_t, constant_k1 . Outputs : SAT nominal mean motion in rev_per_day at time_t, ignoring earth oblateness S03A 011. Satellite Nominal Mean motion in rev_per_day at time_t using SMA Ignoring earth oblateness = 1.00272, (b) Mean motion rev per day at time_t Considering earth oblatenes Inputs : SAT nominal mean motion rad per day at time_t Ignoring_oblateness, constant_k2, constant_k3, SAT orbit semi major axis in km considering oblateness at time_t. Outputs : SAT mean motion rev per day at time_t, considering earth oblatenes . S03B 011. Satellite Mean motion in rev_per_day at time t using SMA Considering earth oblatenes = 1.00276, Note - This calculted value is slightly less than the mean motion rev_per_day as EPH input from NORAD TLE 04. Finding Satellite Orbit Time Period in minute at time_t Considering earth oblatenes . Inputs : SAT orbit semi major axis in km considering oblateness at time_t, GM_EARTH . Outputs : SAT orbit time period in minute at time_t considering earth oblatenes . S04 011. Satellite orbit Time Period in minute at time_t using SMA Considering earth oblatenes = 1436.14327, 05. Finding Satellite Rate of change of Right Ascension and Argument of Perigee in deg per_day at time_t. (a) Rate of change of Right Ascension in deg per day at time_t . Inputs : SAT mean motion rev per day at time_t considering earth oblatenes, constant_k2, SAT orbit eccentricity at time_t, SAT semi major axis km considering oblateness at time_t, SAT orbit inclination deg at time_t . Outputs : SAT rate of change of right ascension in deg per day at time_t and constant_k_deg_per_day
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OM-MSS Page 245 S05A 011. Satellite Rate of change of Right Ascension in deg per day at time_t = -0.01338, (b) Rate of change of Argument of Perigee in deg per_day at time_t . Inputs : SAT orbit constant_k_deg_per_day, SAT orbit inclination deg at time_t SAT orbit semi major axis km considering oblateness at time_t Outputs : SAT rate of change of argument of perigee in deg per day at time_t S05B 011. Satellite Rate of change of Argument of Perigee in deg per day at time_t = 0.02677, 06. Finding Satellite Mean anomaly, Eccentric anomaly, True anomaly in deg at time_t considering earth oblateness. Inputs : SAT mean anomaly rad at time_t, mean_motion_rad_per_day_at_time_t considering_oblateness, SAT orbit eccentricity_at_time_t Outputs : SAT mean anomaly, eccentric anomaly, and true anomaly in deg at time_t onsidering earth oblateness. S06A 011. Satellite Mean anomaly in deg at time_t = 354.64680, same as EPH mean anomaly S06A 011. Satellite Eccentric anomaly in deg at time_t = 354.64570, S06A 011. Satellite True anomaly in deg at time_t = 354.64461, Note - 1. Here after, the Earth Oblateness is always considered for the computation of satellite orbit parameters, and not repeatedly mentioned. Satellite to Earth, the Position Vectors coordinate and the Vector Coordinate Transforms are in PQW, IJK, SEZ frames . - Perifocal Coordinate System (PQW) is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. - Geocentric Coordinate System(IJK) is Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). - Topocentric Horizon Coordinate System(SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). Each of these coordinate system were explained in detail before and therefore not repeated any more. 07. Finding Satellite position vector[rp, rq] from Earth center(EC) to Sat in PQW frame, perifocal coordinate system. Inputs : SAT orbit semi-major axis (SMA), SAT orbit eccentricity, SAT eccentric anomaly, SATtrue anomaly at time_t Outputs : Vector(r, rp rq) in PQW frame S07A 011. r Satellite pos vector magnitude EC to Sat km in PQW frame perifocal cord at time_t = 42157.02910 S07B 011. rp Satellite pos vector component EC to Sat km in PQW frame perifocal cord at time_t = 41973.0106897037 S07C 011. rq Satellite pos vector component EC to Sat km in PQW frame perifocal cord at time_t = -3934.6506542903 08. Satellite Position Vector Earth Ceter(EC) to Satellite(SAT) - finding Range Vector(rI rJ rK r) Components in km in frame IJK
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OM-MSS Page 246 Note - Transform_1 : EC to SAT Vector(rp, rq) in frame PQW To EC to SAT Vector(rI rJ rK r) in frame IJK. Inputs : Vector(rp, rq) EC to Sat in km in frame PQW, SAT right ascension of ascending node at time_t, SAT argument of perigee rad at time_t, SAT orbit inclination rad at time_t . Outputs : Vector(rI, rJ, rK, r) EC to SAT in km in frame IJK S08A 011. rI Satellite pos vector component EC to Sat km frame IJK at at time_t = 36095.3223130873 S08B 011. rJ Satellite pos vector component EC to Sat km frame IJK at at time_t = -21779.4122305000 S08C 011. rK Satellite pos vector component EC to Sat km frame IJK at at time_t = 3.4839025250 S08D 011. r Satellite pos vector magnitude EC to Sat km frame IJK at at time_t = 42157.0290951495 Note - r Satellite pos vector magnitude EC to Sat km in PQW frame is same as that computed above in PQW frame. 09. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input at time_t . Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time t UT year = 2014, month = 5, day = 26, hour = 0, minute = 45, seconds = 36.59731 Outputs : GST & GHA in 0-360 deg over Greenwich. S09A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 254.92064, hr = 16, min = 59, sec = 40.95379 S09B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 254.92098, deg = 254, min = 55, sec = 15.51883 10. Satellite(SAT) Orbit point direction : Finding Right Ascension(Alpha) deg and Declination(Delta) deg using angles Inputs : SAT orbit inclination deg at_time_t, EPH right ascension ascending node deg, SAT argument of perigee deg at time_t, SAT true anomaly deg calculated at time_t Outputs : SAT Right Ascension(Alpha) and Declination(Delta) in deg at time_t S10A 011. SAT Right Ascension(Alpha) in deg = 328.8938068030 S10B 011. SAT Declination(Delta) in deg = 0.0047349853 11. Finding Satellite Longitude & Latitude in deg at time_t; (ie Sub-Sat point log & lat on earth surface ). Inputs : SAT right ascension ascending node deg at time_t, GST in 0-360 deg over Greenwich at time_t Outputs : Satellite (Sub-Sat point) longitude 0 to 360 deg at time_t. S11A 011. Satellite longitude 0 to 360 deg at time_t = 328.89 ie deg = 328, min = 53, sec = 37.70
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OM-MSS Page 247 Inputs : argument of_perigee rad at_time_t, inclination rad at time_t, true anomaly rad calculated at time_t. Outputs : Satellite (Sub-Sat point) latitude +ve or -ve in 0 to 90 deg at time_t. S11B 011. Satellite latitude +ve or -ve in 0 to 90 deg at time_t = 0.00 ie deg = 0, min = 0, sec = 17.05 12. Finding Satellite height in km from EC to Sat and from Earth surface to Sat at time_t. (a) Satellite height in km from EC to Sat; (ie Sat orbit radius EC to Sat in km at time_t). Note - This is SAME as r sat pos vector magnitude EC to Sat in frame IJK calculated above in TRANSFORN_1 . Inputs : SAT true anomaly at time_t, semi_major_axis_km, inclination at time_t . Outputs : Sub-Sat point longitude 0 to 360 deg at time_t. S12A 011. Satellite orbital radious EC to SAT in km using SAT true anomaly at time_t = 42157.03 (b) Satellite height in km from earth surface. Inputs : Sub-Sat point latitude +ve or -ve in 0 to 90 deg at time_t, earth_equator_radious_km . Outputs : Sat height in km from earth surface. S12B 011. Satellite height in km from earth surface at time_t = 35778.89 13. Finding Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface at time_t . Inputs : Sub-Sat point lat & log, ES lat & log . Outputs : Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface S13 011. Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface = 11907.14680 14. Finding Local sidereal time(LST) and Local mean time(LMT) over Sub-Sat point Longitude on earth . (a) Local sidereal time(LST) in 0 to 360 deg over Sub-Sat point Longitude on earth . Inputs : GST sidereal_time in 0 to 360 deg at Greenwich at time_t, satellite log in 0 to 360 deg at time_t . Outputs : LST local sidereal time in 0 to_360 deg at SAT longitude at time_t. S14A 011. Local sidereal time(LST) over Sub-Sat point Longitude at time_t = 223.81 (b) Local sidereal time(LST) and local Mean time(LMT) adjusted to calendar date(CD) over Sub-Sat point Longitude on earth . Note - The LST and LMT in hr min sec with YY MM DD adjusted to calendar date (CD) of longitude at time_t . Here LST in deg is re-calculated alterately in terms of Julian Day (JD) that account for calendar date (CD) of longitude. Inputs : GMT_JD, Sat longitude_in_0_to_360_deg .
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OM-MSS Page 248 Outputs : LST local sidereal time in 0 to_360 deg at SAT longitude at time_t. S14B 011. Local sidereal time(LST) in 0 to 360_deg over Sub-Sat point Longitude at time_t = 223.81 S14C 011. LST over Sub-Sat Log, with date adj to CD expressed in JD = 2456804.12, ie YY = 2014, MM = 5, DD = 26, hr = 14, min = 55, sec = 15.56 S14D 011. LMT over Sub-Sat Log, with date adj to CD expressed in JD = 2456803.45, ie YY = 2014, MM = 5, DD = 25, hr = 22, min = 41, sec = 11.11 15. Finding Local sidereal time(LST) and Local mean time(LMT) Over Earth stn (ES) or Earth point(EP) Longitude . (a) Local sidereal time(LST) in 0 to 360 deg over Earth stn(ES) Longitude . Inputs : GST sidereal_time in 0 to 360 deg at Greenwich at time_t, Earth stn(ES) longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at Earth stn(ES) longitude at time_t. S15A 011. Local sidereal time(LST) over Earth stn(ES) Longitude at time_t = 332.33 (b) Local sidereal time(LST) and local Mean time(LMT) adjusted to calendar date(CD) over Earth stn(ES) Longitude . Note - The LST and LMT in hr min sec with YY MM DD adjusted to calendar date (CD) of longitude at time_t . Here LST in deg is re-calculated alterately in terms of Julian Day (JD) that account for calendar date (CD) of longitude. Inputs : GMT_JD, Earth stn(ES) longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at Earth stn (ES) longitude at time_t. S15B 011. Local sidereal time(LST) in 0 to 360_deg over Earth stn (ES) Longitude at time_t = 332.33 S14C 011. LST over ES Log, with date adj to CD expressed in JD = 2456804.42, ie YY = 2014, MM = 5, DD = 26, hr = 22, min = 9, sec = 20.07 S15D 011. LMT over ES Log, with date adj to CD expressed in JD = 2456803.75, ie YY = 2014, MM = 5, DD = 26, hr = 5, min = 55, sec = 15.62 16. Earth Stn (ES) Position Vector from Earth Center(EC) to Earth Stn(ES) : Finding Range Vector(RI, RJ, RK, R) Components in IJK frame Note - Transform_2 : ES position cord(lat, log, hgt) To EC to ES position Vector(RI, RJ, RK, R) in frame IJK . Inputs : ES latitude positive_negative 0 to 90 deg, ES longitude in 0 to 360_deg, ES height in meter (is earth surface + tower hgt), LST in 0 to 360 deg at ES log at time_t . Outputs : ES Position Vector(RI, RJ, RK, R) Components EC to ES in km in IJK frame . S16A 011. RI_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 5187.4709555096 S16B 011. RJ_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = -2719.6434583296 S16C 011. RK_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 2517.6372173288 S16D 011. R_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 6375.3284317570
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OM-MSS Page 249 17. Satellite(SAT) Position Range Vector from Earth Stn(ES) to SAT : Finding Range Vector(rvI, rvJ, rvK, rv) Components in km in IJK frame Note - Transform_3 SAT Pos Vct(rI rJ rK) and ES Pos Vct(RI RJ RK) To SAT Pos Vct(rvI, rvJ, rvK, rv) Inputs : SAT Position Vector(rI rJ rK) EC to Sat in km in frame IJK , ES Position Vector(RI RJ RK) EC to ES in km in IJK frame. Outputs : SAT Position Range Vector(rvI, rvJ, rvK, rv) Components ES to Sat in km in IJK frame S17A 011. rv_I range vector component ES to SAT km frame IJK = 30907.8513575778 S17B 011. rv_J range vector component ES to SAT km frame IJK = -19059.7687721703 S17C 011. rv_K range vector component ES to SAT km frame IJK = -2514.1533148038 S17D 011. rv range vector component ES to SAT km frame IJK = 36399.0525712011 18. Satellite(SAT) Position Range Vector from Earth Stn(ES) to SAT : Finding Range Vector(rvS, rvE, rvZ, rv) Components in km in SEZ frame Note - Transform_4 SAT Pos Vct(rvI, rvJ, rvK, rv) in IJK frame To SAT Pos Vct(rvS, rvE, rvZ, rv) in SEZ frame Inputs : ES latitude positive_negative 0 to 90_deg, LST in 0 to 360 deg at ES longitude at time_t SAT PositionRange Vector(rvI, rvJ, rvK, rv) ES to Sat in km in IJK frame Outputs : SAT Position Range(rvS, rvE, rvZ, rv) Components ES to Sat in km in SEZ frame S18A 011. rvS range_vector component ES to SAT km_frame SEZ = 16614.7553985861 S18B 011. rvE range_vector component ES to SAT km_frame SEZ = -2529.1545484668 S18C 011. rvZ range_vector component ES to SAT km_frame SEZ = 32286.9061446925 S18D 011. rv range_vector component ES to SAT km_frame SEZ = 36399.0525712011 19. Finding Elevation(EL) and Azimuth(AZ) angles of Satellite and Sun : Steps 1, 2, 3 AT UT TIME t Rem: Sub-SAT point lat deg = 0.00, log deg = 328.89 YY = 2014, MM = 5, DD = 25, hr = 22, min = 41, sec = 11.11 ES or EP point lat deg = 23.26, log deg = 77.41 YY = 2014, MM = 5, DD = 26, hr = 5, min = 55, sec = 15.62 Note : Step 1 is for Satellite EL & AZ angles. Steps 2 & 3 are for Sun EL & AZ angles Results verified from other sources; Ref URLs Geoscience Australia http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone , NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html
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OM-MSS Page 250 Elevation(EL) & Azimuth(AZ) angle of SAT at Earth Observation point EP : Step 1. Inputs : Range vector component ES to SAT rv_S, rv_E, rv_Z, in frame_SEZ, EP and Sub_sat point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of SAT at EP S19A 011. Elevation angle deg of Satellte at Earth point EP at time_t = 62.50188 ie deg = 62, min = 30, sec = 6.78 S19B 011. Azimuth angle deg of Satellte at Earth point EP at time_t = 188.65531 ie deg = 188, min = 39, sec = 19.13 Elevation(EL) & Azimuth(AZ) angle of SUN at Sub_Satellite point on earth surface : Step 2. Inputs : Range vector component Sub_sat to Sun rv_S, rv_E, rv_Z, in frame_SEZ, Sub_sat point and Sun_Sun point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of Sun at Sub_Sat S19C 011. Elevation angle deg of SUN at Sub_Sat point at time_t = -61.9436264693 S19D 011. Azimuth angle deg of SUN at Sub_Sat point at time_t = 319.8944977041 Elevation(EL) & Azimuth(AZ) angle of SUN at Satellite height : Step 3. Inputs : Range vector component Sub_sat to Sun rv_S, rv_E, rv_Z, in frame_SEZ, Sub_sat point and Sun_Sun point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of SUN at SAT S19E 011. Elevation angle deg of Sun at Satellite height at time_t = -61.94999 S19F 011. Azimuth angle deg of Sun at Satellite height at time_t = 319.89450 20. Finding Satellite Velocity meter per sec in orbit . Note : Results computed using 2 different formulations, each require different inputs. (a) Inputs : SAT orbit semi-major axis SMA, GM_EARTH, sat_orbit pos r_vector_EC_to_SAT_km_frame_IJK. Outputs : SAT Velocity magnitude and component Xw Yw in frame PQW in meter per sec S20A 011. Satellite Velocity magnitude meter per sec at UT time = 3075.2344215913 (b) Inputs : SAT orbit semi-major axis SMA, GM_EARTH, SAT orbit eccentricity_at_time_t, SAT eccentric anomaly deg calculated at time_t. Outputs : Satellite Velocity components Xw Yw in frame PQW in meter per sec
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OM-MSS Page 251 S20B 011. Satellite Velocity components Xw in frame PQW in meter per sec = 286.9628865990 S20C 011. Satellite Velocity components Yw in frame PQW in meter per sec = 3061.8162991033 21. Finding Satellite(SAT) Velocity Vector (vX, vY, vZ) in meter per sec in orbit in frame XYZ. Note - Transform_5 SAT Vel Vct(Xw, Yw) in frame PQW To SAT Vel Vct(vX, vY, vZ) in frame XYZ Inputs : SAT velocity vectors(Xw, Yw), SAT Right Ascension Alpha, SAT Argument of perigee, SAT orbit inclination at_time_t, SAT eccentric_anomaly_deg_calculated_at_time_t. Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ S21A 011. vX Sat Velocity vector component in meter per sec = 1588.6953038064 S21B 011. vY Sat Velocity vector component in meter per sec = 2633.0807004007 S21C 011. vZ Sat Velocity vector component in meter per sec = -0.0676820819 S21D 011. vR Sat Velocity magnitude meter in meter per sec = 3075.2344215913 22. Finding Satellite(SAT) Pitch and Roll angles Inputs : Earth equator radious km, ES lat, ES log, Sub_Sat point lat, Sub_Sat point log, Sat_orbit pos r_vector_EC_to_SAT_km_frame_IJK, Outputs : SAT Pitch and Roll angles S22A 011. pitch_angle_deg_for_earth_stn_at_sat_lat_log_at_time_t = -7.2287185764 S22B 011. roll_angle_deg_for_earth_stn_at_sat_lat_log_at_time_t = 3.2458714002 23. Finding Satellite State Vectors - Position [ X, Y, Z ] in km and velocity [ Vx, Vy, Vz ] in meter per sec at time_t . Note - same as values of rI rJ rK r for pos and vX vY vZ vR for vel (a) Satellite State Position Vector [X, Y, Z] in km at time_t Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ S23A 011. State vector position X km at time_t = 36095.3223130873 S23B 011. State vector position Y km at time_t = -21779.4122305000 S23C 011. State vector position Z km at time_t = 3.4839025250 S23D 011. State vector position R km at time_t = 42157.0290951495
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OM-MSS Page 252 (b) Satellite State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ S23E 011. State vector velocity Vx meter per sec at time_t = 1588.6953038064 S23F 011. State vector velocity Vy meter per sec at time_t = 2633.0807004007 S23G 011. State vector velocity Vz meter per sec at time_t = -0.0676820819 S23H 011. State vector velocity V meter per sec at time_t = 3075.2344215913 24. Satellite Direction ie right ascension alpha deg and declination delta deg using sat position vector . Note - results same as that using angles incl, RA, Aug of peri, true anomaly Inputs : SAT State Vectors Position(X, Y, Z, R) in km , in frame XYZ Outputs : SAT right ascension alpha deg and declination delta deg S24A 011. Satellite direction right_asc alpha deg at time_t = 328.8938068030 S24B 011. Satellite direction_declination_delta_deg_at_time_t = 0.0047349853 25. Satellite Angular momentum km sqr per sec : finding Hx Hy Hz H from state vector pos and vel . Inputs : SAT State Vectors Position(X, Y, Z, R) in km and Velocity (vX, vY, vZ, vR) meter per sec, in frame XYZ Outputs : SAT angular momentum (Hx Hy Hz H) componts in km sqr per sec S25A 011. Satellite angular momentum Hx in km sqr per sec at time_t = -7.6993205387 S25B 011. Satellite angular momentum Hy in km sqr per sec at time_t = 7.9778661408 S25C 011. Satellite angular momentum Hz in km sqr per sec at time_t = 129642.7464875925 S25D 011. Satellite angular momentum H in km sqr per sec at time_t = 129642.7469616872 26. Satellite Orbit normal Vector : finding Wx Wy Wz W Delta Alpha from r_sat_pos frame IJK, i, RA Inputs : SAT Position Vector(rI rJ rK) EC to Sat in km in frame IJK , inclination_deg_at_time_t, right_ascension_ascending_node_deg_at_time_t Outputs : SAT orbit normal vector (Wx, Wy, Wz, W) in km , RA , i S26A 011. Satellite Orbit normal_W in km = 42157.0290951495 S26B 011. Satellite Orbit normal_Wx in km = -2.5036532129 S26C 011. Satellite Orbit normal_Wy in km = 2.5942302435
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OM-MSS Page 253 S26D 011. Satellite Orbit normal_Wz in km = 42157.0289409842 S26E 011. Satellite Orbit normal_Delta_W in deg = 89.9951000000 S26F 011. Satellite Orbit interpreted inclination i = 0.0049000000 S26G 011. Satellite Orbit normal_Alpha_W in deg = -46.0179000000 S26H 011. Satellite Orbit interpreted RA of asc node = 223.9821000000 Transform Satellite State Vectors to Keplerian elements. 27. Finding Satellite position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness S27A 011. Keplerian elements year = 2014, days_decimal_of_year = 145.03167, revolution no = 140, node = 1 ie ascending S27B 011. inclination_deg = 0.0049000000 S27C 011. right ascension ascending node deg = 223.9821000000 S27D 011. eccentricity = 0.0002051000 S27E 011. argument of perigee_deg = 110.2671000002 S27F 011. mean anomaly deg = 354.6467999998 S27G 011. mean_motion rev per day = 1.0027226500 S27H 011. mean angular velocity rev_per_day = 1.0026854765 S27I 011. mean motion rev per day using SMA considering oblateness = 1.0027598249 Transform Satellite Keplerian elements to State Vectors . 28. Finding Satellite position State Vectors, computed using Keplerian elements at time input UT (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ]
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OM-MSS Page 254 S28A 011. State vectors year = 2014, days_decimal_of_year = 145.03167, revolution no = 140, node = 1 ie decending S28B 011. state vector position X km = 36095.3223130876, state vector velocity Vx meter per sec = 1588.6953038064 S28C 011. state vector position Y km = -21779.4122304995, state vector velocity Vy meter per sec = 2633.0807004007 S28D 011. state vector position Z km = 3.4839025250, state vector velocity Vz meter per sec = -0.0676820819 S28E 011. state vector position R km = 42157.0290951495, state vector velocity V meter per sec = 3075.2344215913 Move on to next Satellite. Next Section - 6.6 Computing Orbital & Positional parameters for Satellite MOON
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OM-MSS Page 255 OM-MSS Section - 6.6 ---------------------------------------------------------------------------------------------------54 Satellite MOON : Computing Orbital & Positional parameters corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins. (f) MOON 'Two-Line Elements'(TLE) downloaded on Jun 14, 2014, 16:14 hrs IST, Natural Satellite 1 00000U 00000A 14143.16621081 .00000000 00000-0 00000-0 0 3574 2 00000 18.7965 352.4777 0512000 316.1136 40.2074 00.036600996 0006 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 00000, MOON , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 142.1662108168 EPOCH_inclination_deg = 18.7965000000 EPOCH_right_asc_acnd_node_deg = 352.4777000000 EPOCH_eccentricity = 0.0512000000 EPOCH_argument_of_perigee_deg = 316.1136000000 EPOCH_mean_anomaly_deg = 40.2074000000 EPOCH_mean_motion_rev_per_day = 0.0366009960 EPOCH_revolution = 0 EPOCH_node_condition = 1 Earth_stn_latitude_deg = 23.25993 Earth stn longitude_deg = 77.41261 Earth surface height_meter = 494.70000 Earth stn tower_height_meter = 15.00000 Earth stn_height_meter = 509.70000 Earth stn min EL look_angle_deg = -5.00000 Note : EPOCH Corresponds to UT year = 2014, month = 5, day = 23, hr = 3, min = 59, sec = 20.61457 which is Greenwich meam time ie GMT Converted to Local Meam Time at Earth Stn Longitude, as regular Calender date : year = 2014, month = 5, day = 23, hr = 9, min = 8, sec = 59.64 At this instant Sun Position as seen from Earth Stn Longitude : Sun angles EL deg = 51.13, AZ deg = 85.66, Sun Surface distance km = 4326.36, Radial km = 151461044.87, Sun Rise D:23, H:05, M:18, S:26 Sun Set D:23, H:18, M:37, S:54 Move to Compute Satellie Orbital parameters corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins, and Earth Stn location. 01. Input EPOCH_year and EPOCH_days_decimal_of_year, Converted into UT YY MM DD hh min sec & Julian day. S01 011. Input UT year = 2014, month = 5, day = 24, hr = 3, min = 59, sec = 20.61457, and julian_day = 2456800.6662108167 02. Finding Satellite orbit Semi major axis in km, Ignoring and also Considering earth oblatenes . (a) Semi major axis (SMA) km at time t Ignoring earth oblatenes . Inputs : SAT mean_motion rev per day at time t, GM_EARTH . Outputs : SAT Orbit semi major axis km in km and constant_A, constant_k1 .
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OM-MSS Page 256 S02A 011. Satellite orbit Semi major axis in km at time t, Ignoring earth oblatenes = 383183.29902, (b) Semi major axis (SMA) in km at time_t Considering earth oblatenes Inputs : SAT mean_motion rev per day at time_t, GM_EARTH , inclination deg at time_t, eccentricity at time_t, constant_k2 Outputs : SAT Orbit semi major axis km in km at time_t and constant_A, constant_k1, constant_k3 . S02B 011. Satellite orbit Semi major axis km at time t, Considering earth oblatenes = 383183.39622, 03. Finding Satellite Mean motion in rev per day, Ignoring and also Considering earth oblatenes . (a) Nominal mean motion rev per day at time_t Ignoring earth oblateness . Inputs : SAT Orbit semi major axis in km ignoring oblateness at time_t, constant_k1 . Outputs : SAT nominal mean motion in rev_per_day at time_t, ignoring earth oblateness S03A 011. Satellite Nominal Mean motion in rev_per_day at time_t using SMA Ignoring earth oblateness = 0.03660, (b) Mean motion rev per day at time_t Considering earth oblatenes Inputs : SAT nominal mean motion rad per day at time_t Ignoring_oblateness, constant_k2, constant_k3, SAT orbit semi major axis in km considering oblateness at time_t. Outputs : SAT mean motion rev per day at time_t, considering earth oblatenes . S03B 011. Satellite Mean motion in rev_per_day at time t using SMA Considering earth oblatenes = 0.03660, Note - This calculted value is slightly less than the mean motion rev_per_day as EPH input from NORAD TLE 04. Finding Satellite Orbit Time Period in minute at time_t Considering earth oblatenes . Inputs : SAT orbit semi major axis in km considering oblateness at time_t, GM_EARTH . Outputs : SAT orbit time period in minute at time_t considering earth oblatenes . S04 011. Satellite orbit Time Period in minute at time_t using SMA Considering earth oblatenes = 39343.20661, 05. Finding Satellite Rate of change of Right Ascension and Argument of Perigee in deg per_day at time_t. (a) Rate of change of Right Ascension in deg per day at time_t . Inputs : SAT mean motion rev per day at time_t considering earth oblatenes, constant_k2, SAT orbit eccentricity at time_t, SAT semi major axis km considering oblateness at time_t, SAT orbit inclination deg at time_t . Outputs : SAT rate of change of right ascension in deg per day at time_t and constant_k_deg_per_day
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OM-MSS Page 257 S05A 011. Satellite Rate of change of Right Ascension in deg per day at time_t = -0.00001, (b) Rate of change of Argument of Perigee in deg per_day at time_t . Inputs : SAT orbit constant_k_deg_per_day, SAT orbit inclination deg at time_t SAT orbit semi major axis km considering oblateness at time_t Outputs : SAT rate of change of argument of perigee in deg per day at time_t S05B 011. Satellite Rate of change of Argument of Perigee in deg per day at time_t = 0.00001, 06. Finding Satellite Mean anomaly, Eccentric anomaly, True anomaly in deg at time_t considering earth oblateness. Inputs : SAT mean anomaly rad at time_t, mean_motion_rad_per_day_at_time_t considering_oblateness, SAT orbit eccentricity_at_time_t Outputs : SAT mean anomaly, eccentric anomaly, and true anomaly in deg at time_t onsidering earth oblateness. S06A 011. Satellite Mean anomaly in deg at time_t = 40.20740, same as EPH mean anomaly S06A 011. Satellite Eccentric anomaly in deg at time_t = 42.17705, S06A 011. Satellite True anomaly in deg at time_t = 44.18593, Note - 1. Here after, the Earth Oblateness is always considered for the computation of satellite orbit parameters, and not repeatedly mentioned. Satellite to Earth, the Position Vectors coordinate and the Vector Coordinate Transforms are in PQW, IJK, SEZ frames . - Perifocal Coordinate System (PQW) is Earth Centered Inertial coordinate frame defined in terms of Kepler Orbital Elements. - Geocentric Coordinate System(IJK) is Earth Centered Inertial (ECI) frame, a Conventional Inertial System (CIS). - Topocentric Horizon Coordinate System(SEZ), is Non-Inertial coordinate frame, known as Earth-Centered Earth-Fixed Coordinates (ECEF). Each of these coordinate system were explained in detail before and therefore not repeated any more. 07. Finding Satellite position vector[rp, rq] from Earth center(EC) to Sat in PQW frame, perifocal coordinate system. Inputs : SAT orbit semi-major axis (SMA), SAT orbit eccentricity, SAT eccentric anomaly, SATtrue anomaly at time_t Outputs : Vector(r, rp rq) in PQW frame S07A 011. r Satellite pos vector magnitude EC to Sat km in PQW frame perifocal cord at time_t = 368644.28111 S07B 011. rp Satellite pos vector component EC to Sat km in PQW frame perifocal cord at time_t = 264348.1020492621 S07C 011. rq Satellite pos vector component EC to Sat km in PQW frame perifocal cord at time_t = 256941.0184081715 08. Satellite Position Vector Earth Ceter(EC) to Satellite(SAT) - finding Range Vector(rI rJ rK r) Components in km in frame IJK
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OM-MSS Page 258 Note - Transform_1 : EC to SAT Vector(rp, rq) in frame PQW To EC to SAT Vector(rI rJ rK r) in frame IJK. Inputs : Vector(rp, rq) EC to Sat in km in frame PQW, SAT right ascension of ascending node at time_t, SAT argument of perigee rad at time_t, SAT orbit inclination rad at time_t . Outputs : Vector(rI, rJ, rK, r) EC to SAT in km in frame IJK S08A 011. rI Satellite pos vector component EC to Sat km frame IJK at at time_t = 365705.5648844948 S08B 011. rJ Satellite pos vector component EC to Sat km frame IJK at at time_t = -46450.6213911481 S08C 011. rK Satellite pos vector component EC to Sat km frame IJK at at time_t = 620.9529484744 S08D 011. r Satellite pos vector magnitude EC to Sat km frame IJK at at time_t = 368644.2811134813 Note - r Satellite pos vector magnitude EC to Sat km in PQW frame is same as that computed above in PQW frame. 09. Finding GST Greenwich sidereal time and GHA Greenwich hour angle in 0 to 360 deg, at input at time_t . Note - for GST, the year 1900 JAN day_1 hr 1200 is ref for time difference in terms of julian_century, for GHA, the year 2000_JAN_day_1 hr_1200 is ref for time difference in terms of julian days. Inputs : Time t UT year = 2014, month = 5, day = 23, hour = 3, minute = 59, seconds = 20.61457 Outputs : GST & GHA in 0-360 deg over Greenwich. S09A 011. GST Greenwich sidereal time in 0-360 deg, over Greenwich = 300.52971, hr = 20, min = 2, sec = 7.13045 S09B 011. GHA Greenwich hour angle in 0 to 360 deg, over Greenwich = 300.53198, deg = 300, min = 31, sec = 55.13594 10. Satellite(SAT) Orbit point direction : Finding Right Ascension(Alpha) deg and Declination(Delta) deg using angles Inputs : SAT orbit inclination deg at_time_t, EPH right ascension ascending node deg, SAT argument of perigee deg at time_t, SAT true anomaly deg calculated at time_t Outputs : SAT Right Ascension(Alpha) and Declination(Delta) in deg at time_t S10A 011. SAT Right Ascension(Alpha) in deg = 352.7612555840 S10B 011. SAT Declination(Delta) in deg = 0.0965103811 11. Finding Satellite Longitude & Latitude in deg at time_t; (ie Sub-Sat point log & lat on earth surface ). Inputs : SAT right ascension ascending node deg at time_t, GST in 0-360 deg over Greenwich at time_t Outputs : Satellite (Sub-Sat point) longitude 0 to 360 deg at time_t. S11A 011. Satellite longitude 0 to 360 deg at time_t = 352.76 ie deg = 352, min = 45, sec = 40.52
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OM-MSS Page 259 Inputs : argument of_perigee rad at_time_t, inclination rad at time_t, true anomaly rad calculated at time_t. Outputs : Satellite (Sub-Sat point) latitude +ve or -ve in 0 to 90 deg at time_t. S11B 011. Satellite latitude +ve or -ve in 0 to 90 deg at time_t = 0.10 ie deg = 0, min = 5, sec = 47.44 12. Finding Satellite height in km from EC to Sat and from Earth surface to Sat at time_t. (a) Satellite height in km from EC to Sat; (ie Sat orbit radius EC to Sat in km at time_t). Note - This is SAME as r sat pos vector magnitude EC to Sat in frame IJK calculated above in TRANSFORN_1 . Inputs : SAT true anomaly at time_t, semi_major_axis_km, inclination at time_t . Outputs : Sub-Sat point longitude 0 to 360 deg at time_t. S12A 011. Satellite orbital radious EC to SAT in km using SAT true anomaly at time_t = 368644.28 (b) Satellite height in km from earth surface. Inputs : Sub-Sat point latitude +ve or -ve in 0 to 90 deg at time_t, earth_equator_radious_km . Outputs : Sat height in km from earth surface. S12B 011. Satellite height in km from earth surface at time_t = 362266.14 13. Finding Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface at time_t . Inputs : Sub-Sat point lat & log, ES lat & log . Outputs : Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface S13 011. Distance of Sub-Sat point To Earth Stn(ES) in km over Earth surface = 9467.59402 14. Finding Local sidereal time(LST) and Local mean time(LMT) over Sub-Sat point Longitude on earth . (a) Local sidereal time(LST) in 0 to 360 deg over Sub-Sat point Longitude on earth . Inputs : GST sidereal_time in 0 to 360 deg at Greenwich at time_t, satellite log in 0 to 360 deg at time_t . Outputs : LST local sidereal time in 0 to_360 deg at SAT longitude at time_t. S14A 011. Local sidereal time(LST) over Sub-Sat point Longitude at time_t = 293.29 (b) Local sidereal time(LST) and local Mean time(LMT) adjusted to calendar date(CD) over Sub-Sat point Longitude on earth . Note - The LST and LMT in hr min sec with YY MM DD adjusted to calendar date (CD) of longitude at time_t . Here LST in deg is re-calculated alterately in terms of Julian Day (JD) that account for calendar date (CD) of longitude. Inputs : GMT_JD, Sat longitude_in_0_to_360_deg .
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OM-MSS Page 260 Outputs : LST local sidereal time in 0 to_360 deg at SAT longitude at time_t. S14B 011. Local sidereal time(LST) in 0 to 360_deg over Sub-Sat point Longitude at time_t = 293.29 S14C 011. LST over Sub-Sat Log, with date adj to CD expressed in JD = 2456801.31, ie YY = 2014, MM = 5, DD = 23, hr = 19, min = 33, sec = 9.92 S14D 011. LMT over Sub-Sat Log, with date adj to CD expressed in JD = 2456800.65, ie YY = 2014, MM = 5, DD = 23, hr = 3, min = 30, sec = 23.32 15. Finding Local sidereal time(LST) and Local mean time(LMT) Over Earth stn (ES) or Earth point(EP) Longitude . (a) Local sidereal time(LST) in 0 to 360 deg over Earth stn(ES) Longitude . Inputs : GST sidereal_time in 0 to 360 deg at Greenwich at time_t, Earth stn(ES) longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at Earth stn(ES) longitude at time_t. S15A 011. Local sidereal time(LST) over Earth stn(ES) Longitude at time_t = 17.94 (b) Local sidereal time(LST) and local Mean time(LMT) adjusted to calendar date(CD) over Earth stn(ES) Longitude . Note - The LST and LMT in hr min sec with YY MM DD adjusted to calendar date (CD) of longitude at time_t . Here LST in deg is re-calculated alterately in terms of Julian Day (JD) that account for calendar date (CD) of longitude. Inputs : GMT_JD, Earth stn(ES) longitude_in_0_to_360_deg . Outputs : LST local sidereal time in 0 to_360 deg at Earth stn (ES) longitude at time_t. S15B 011. Local sidereal time(LST) in 0 to 360_deg over Earth stn (ES) Longitude at time_t = 17.94 S14C 011. LST over ES Log, with date adj to CD expressed in JD = 2456801.55, ie YY = 2014, MM = 5, DD = 24, hr = 1, min = 11, sec = 46.25 S15D 011. LMT over ES Log, with date adj to CD expressed in JD = 2456800.88, ie YY = 2014, MM = 5, DD = 23, hr = 9, min = 8, sec = 59.64 16. Earth Stn (ES) Position Vector from Earth Center(EC) to Earth Stn(ES) : Finding Range Vector(RI, RJ, RK, R) Components in IJK frame Note - Transform_2 : ES position cord(lat, log, hgt) To EC to ES position Vector(RI, RJ, RK, R) in frame IJK . Inputs : ES latitude positive_negative 0 to 90 deg, ES longitude in 0 to 360_deg, ES height in meter (is earth surface + tower hgt), LST in 0 to 360 deg at ES log at time_t . Outputs : ES Position Vector(RI, RJ, RK, R) Components EC to ES in km in IJK frame . S16A 011. RI_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 5572.3086624356 S16B 011. RJ_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 1804.3535200274 S16C 011. RK_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 2517.6372173288 S16D 011. R_earth_stn_pos_vector_component_EC_to_ES_km_frame_IJK = 6375.3284317570
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OM-MSS Page 261 17. Satellite(SAT) Position Range Vector from Earth Stn(ES) to SAT : Finding Range Vector(rvI, rvJ, rvK, rv) Components in km in IJK frame Note - Transform_3 SAT Pos Vct(rI rJ rK) and ES Pos Vct(RI RJ RK) To SAT Pos Vct(rvI, rvJ, rvK, rv) Inputs : SAT Position Vector(rI rJ rK) EC to Sat in km in frame IJK , ES Position Vector(RI RJ RK) EC to ES in km in IJK frame. Outputs : SAT Position Range Vector(rvI, rvJ, rvK, rv) Components ES to Sat in km in IJK frame S17A 011. rv_I range vector component ES to SAT km frame IJK = 360133.2562220591 S17B 011. rv_J range vector component ES to SAT km frame IJK = -48254.9749111755 S17C 011. rv_K range vector component ES to SAT km frame IJK = -1896.6842688544 S17D 011. rv range vector component ES to SAT km frame IJK = 363356.7148849698 18. Satellite(SAT) Position Range Vector from Earth Stn(ES) to SAT : Finding Range Vector(rvS, rvE, rvZ, rv) Components in km in SEZ frame Note - Transform_4 SAT Pos Vct(rvI, rvJ, rvK, rv) in IJK frame To SAT Pos Vct(rvS, rvE, rvZ, rv) in SEZ frame Inputs : ES latitude positive_negative 0 to 90_deg, LST in 0 to 360 deg at ES longitude at time_t SAT PositionRange Vector(rvI, rvJ, rvK, rv) ES to Sat in km in IJK frame Outputs : SAT Position Range(rvS, rvE, rvZ, rv) Components ES to Sat in km in SEZ frame S18A 011. rvS range_vector component ES to SAT km_frame SEZ = 131173.4240060395 S18B 011. rvE range_vector component ES to SAT km_frame SEZ = -156850.6627498179 S18C 011. rvZ range_vector component ES to SAT km_frame SEZ = 300365.6183411674 S18D 011. rv range_vector component ES to SAT km_frame SEZ = 363356.7148849697 19. Finding Elevation(EL) and Azimuth(AZ) angles of Satellite and Sun : Steps 1, 2, 3 AT UT TIME t Rem: Sub-SAT point lat deg = 0.10, log deg = 352.76 YY = 2014, MM = 5, DD = 23, hr = 3, min = 30, sec = 23.32 ES or EP point lat deg = 23.26, log deg = 77.41 YY = 2014, MM = 5, DD = 23, hr = 9, min = 8, sec = 59.64 Note : Step 1 is for Satellite EL & AZ angles. Steps 2 & 3 are for Sun EL & AZ angles Results verified from other sources; Ref URLs Geoscience Australia http://www.ga.gov.au/geodesy/astro/smpos.jsp#intzone , NOAA Research http://www.esrl.noaa.gov/gmd/grad/solcalc/ , Xavier Jubier, Member IAU http://xjubier.free.fr/en/site_pages/astronomy/ephemerides.html
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OM-MSS Page 262 Elevation(EL) & Azimuth(AZ) angle of SAT at Earth Observation point EP : Step 1. Inputs : Range vector component ES to SAT rv_S, rv_E, rv_Z, in frame_SEZ, EP and Sub_sat point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of SAT at EP S19A 011. Elevation angle deg of Satellte at Earth point EP at time_t = 55.75523 ie deg = 55, min = 45, sec = 18.85 S19B 011. Azimuth angle deg of Satellte at Earth point EP at time_t = 230.09443 ie deg = 230, min = 5, sec = 39.95 Elevation(EL) & Azimuth(AZ) angle of SUN at Sub_Satellite point on earth surface : Step 2. Inputs : Range vector component Sub_sat to Sun rv_S, rv_E, rv_Z, in frame_SEZ, Sub_sat point and Sun_Sun point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of Sun at Sub_Sat S19C 011. Elevation angle deg of SUN at Sub_Sat point at time_t = -33.8802777238 S19D 011. Azimuth angle deg of SUN at Sub_Sat point at time_t = 64.9100330114 Elevation(EL) & Azimuth(AZ) angle of SUN at Satellite height : Step 3. Inputs : Range vector component Sub_sat to Sun rv_S, rv_E, rv_Z, in frame_SEZ, Sub_sat point and Sun_Sun point latitude & longitude Outputs : Elevation(EL) & Azimuth(AZ) of SUN at SAT S19E 011. Elevation angle deg of Sun at Satellite height at time_t = -33.99390 S19F 011. Azimuth angle deg of Sun at Satellite height at time_t = 64.91003 20. Finding Satellite Velocity meter per sec in orbit . Note : Results computed using 2 different formulations, each require different inputs. (a) Inputs : SAT orbit semi-major axis SMA, GM_EARTH, sat_orbit pos r_vector_EC_to_SAT_km_frame_IJK. Outputs : SAT Velocity magnitude and component Xw Yw in frame PQW in meter per sec S20A 011. Satellite Velocity magnitude meter per sec at UT time = 1059.3802758296 (b) Inputs : SAT orbit semi-major axis SMA, GM_EARTH, SAT orbit eccentricity_at_time_t, SAT eccentric anomaly deg calculated at time_t. Outputs : Satellite Velocity components Xw Yw in frame PQW in meter per sec
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OM-MSS Page 263 S20B 011. Satellite Velocity components Xw in frame PQW in meter per sec = -711.8057186065 S20C 011. Satellite Velocity components Yw in frame PQW in meter per sec = 784.6140374578 21. Finding Satellite(SAT) Velocity Vector (vX, vY, vZ) in meter per sec in orbit in frame XYZ. Note - Transform_5 SAT Vel Vct(Xw, Yw) in frame PQW To SAT Vel Vct(vX, vY, vZ) in frame XYZ Inputs : SAT velocity vectors(Xw, Yw), SAT Right Ascension Alpha, SAT Argument of perigee, SAT orbit inclination at_time_t, SAT eccentric_anomaly_deg_calculated_at_time_t. Outputs : earth velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ S21A 011. vX Sat Velocity vector component in meter per sec = 161.8765603889 S21B 011. vY Sat Velocity vector component in meter per sec = 989.7819390712 S21C 011. vZ Sat Velocity vector component in meter per sec = 341.1953415596 S21D 011. vR Sat Velocity magnitude meter in meter per sec = 1059.3802758296 22. Finding Satellite(SAT) Pitch and Roll angles Inputs : Earth equator radious km, ES lat, ES log, Sub_Sat point lat, Sub_Sat point log, Sat_orbit pos r_vector_EC_to_SAT_km_frame_IJK, Outputs : SAT Pitch and Roll angles S22A 011. pitch_angle_deg_for_earth_stn_at_sat_lat_log_at_time_t = -0.9076449658 S22B 011. roll_angle_deg_for_earth_stn_at_sat_lat_log_at_time_t = 0.3917892859 23. Finding Satellite State Vectors - Position [ X, Y, Z ] in km and velocity [ Vx, Vy, Vz ] in meter per sec at time_t . Note - same as values of rI rJ rK r for pos and vX vY vZ vR for vel (a) Satellite State Position Vector [X, Y, Z] in km at time_t Inputs : position vector(rI, rJ, rK, r) in frame IJK values assiged to state position vector Outputs : State Position Vector(X, Y, Z, R) in km, frame XYZ S23A 011. State vector position X km at time_t = 365705.5648844948 S23B 011. State vector position Y km at time_t = -46450.6213911481 S23C 011. State vector position Z km at time_t = 620.9529484744 S23D 011. State vector position R km at time_t = 368644.2811134813
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OM-MSS Page 264 (b) Satellite State Velocity Vector [Vx, Vy, Vz] in meter per sec at time input UT. Inputs : velocity vector(vX, vY, vZ, vR) meter per sec in frame XYZ values assiged to state velocity vector Outputs : state velocity vector(Vx, Vy, Vz, V) meter per sec, frame XYZ S23E 011. State vector velocity Vx meter per sec at time_t = 161.8765603889 S23F 011. State vector velocity Vy meter per sec at time_t = 989.7819390712 S23G 011. State vector velocity Vz meter per sec at time_t = 341.1953415596 S23H 011. State vector velocity V meter per sec at time_t = 1059.3802758296 24. Satellite Direction ie right ascension alpha deg and declination delta deg using sat position vector . Note - results same as that using angles incl, RA, Aug of peri, true anomaly Inputs : SAT State Vectors Position(X, Y, Z, R) in km , in frame XYZ Outputs : SAT right ascension alpha deg and declination delta deg S24A 011. Satellite direction right_asc alpha deg at time_t = 352.7612555840 S24B 011. Satellite direction_declination_delta_deg_at_time_t = 0.0965103811 25. Satellite Angular momentum km sqr per sec : finding Hx Hy Hz H from state vector pos and vel . Inputs : SAT State Vectors Position(X, Y, Z, R) in km and Velocity (vX, vY, vZ, vR) meter per sec, in frame XYZ Outputs : SAT angular momentum (Hx Hy Hz H) componts in km sqr per sec S25A 011. Satellite angular momentum Hx in km sqr per sec at time_t = -16463.3436446211 S25B 011. Satellite angular momentum Hy in km sqr per sec at time_t = -124676.5173935477 S25C 011. Satellite angular momentum Hz in km sqr per sec at time_t = 369488.0299592165 S25D 011. Satellite angular momentum H in km sqr per sec at time_t = 390303.3178906982 26. Satellite Orbit normal Vector : finding Wx Wy Wz W Delta Alpha from r_sat_pos frame IJK, i, RA Inputs : SAT Position Vector(rI rJ rK) EC to Sat in km in frame IJK , inclination_deg_at_time_t, right_ascension_ascending_node_deg_at_time_t Outputs : SAT orbit normal vector (Wx, Wy, Wz, W) in km , RA , i S26A 011. Satellite Orbit normal_W in km = 368644.2811134813 S26B 011. Satellite Orbit normal_Wx in km = -15549.7460677369 S26C 011. Satellite Orbit normal_Wy in km = -117757.8642545590
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OM-MSS Page 265 S26D 011. Satellite Orbit normal_Wz in km = 348984.0924757304 S26E 011. Satellite Orbit normal_Delta_W in deg = 71.2035000000 S26F 011. Satellite Orbit interpreted inclination i = 18.7965000000 S26G 011. Satellite Orbit normal_Alpha_W in deg = 82.4777000000 S26H 011. Satellite Orbit interpreted RA of asc node = 352.4777000000 Transform Satellite State Vectors to Keplerian elements. 27. Finding Satellite position Keplerian elements computed using State Vector, at time input UT. Inputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ] Outputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness S27A 011. Keplerian elements year = 2014, days_decimal_of_year = 142.16621, revolution no = 0, node = 1 ie ascending S27B 011. inclination_deg = 18.7965000000 S27C 011. right ascension ascending node deg = 352.4777000000 S27D 011. eccentricity = 0.0512000000 S27E 011. argument of perigee_deg = 316.1136000000 S27F 011. mean anomaly deg = 40.2074000000 S27G 011. mean_motion rev per day = 0.0366009960 S27H 011. mean angular velocity rev_per_day = 0.0366009821 S27I 011. mean motion rev per day using SMA considering oblateness = 0.0366010099 Transform Satellite Keplerian elements to State Vectors . 28. Finding Satellite position State Vectors, computed using Keplerian elements at time input UT (computed again to validate model equations, Keplerian elements to State Vectors & back) Inputs : Keplerian elements : year, days decimal of year, revolution, node, inclination, right ascension, eccentricity, argument of perigee, mean anomaly, mean motion rev per day, mean angular velocity rev per day, mean motion rev per_day from SMA considering oblateness Outputs : State vector year, days decimal of year, revolution, node, State Position Vector [ X, Y, Z ], State Velocity Vector [ Vx, Vy, Vz ]
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OM-MSS Page 266 S28A 011. State vectors year = 2014, days_decimal_of_year = 142.16621, revolution no = 0, node = 1 ie decending S28B 011. state vector position X km = 365705.5648844945, state vector velocity Vx meter per sec = 161.8765603889 S28C 011. state vector position Y km = -46450.6213911481, state vector velocity Vy meter per sec = 989.7819390712 S28D 011. state vector position Z km = 620.9529484743, state vector velocity Vz meter per sec = 341.1953415596 S28E 011. state vector position R km = 368644.2811134810, state vector velocity V meter per sec = 1059.3802758296 End of Computing Orbital & Positional Parameters of Six Satellites. A Summary of Computing Orbital & Positional Parameters are presented next. Next Section - 6.7 Concluding Orbital & Positional parameters of the six Satellites
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OM-MSS Page 267 OM-MSS Section - 6.7 -----------------------------------------------------------------------------------------------------55 Summary of Orbital & Positional parameters of the six Satellites computed above. Summary of Computed Orbital & Positional parameters of six Earth Orbiting Satellites in previous Sections (6.1 to 6.6) : (a) Computed the Orbital & Positional parameters for four LEO, one GEO and one natural satellite around earth. The Input for each satellite, is NASA/NORAD 'Two-Line Elements'(TLE) Bulletin and the Earth Stn Location. The Output is Satellite around earth the Orbital & Positional parameters at Epoch, considering earth oblateness . (b) The Orbital & Positional parameters computed were : Orbit Semi major axis in km, Sat Mean motion in rev per day, Sat Time Period in minute, Sat Rate of change of Right Ascension & Argument of Perigee in deg per_day, Sat Mean, Eccentric & True anomalies in deg, Sat Position, Range & Velocity Vectors from earth center & earth station in PQW, IJK & SEZ frams, Sat Altitude (Alt) & Distance in km over Earth surface, SAt Elevation(EL) & Azimuth(AZ) angles in deg at Earth station, Sun Elevation(EL) & Azimuth(AZ) at Sub-Sat point on Earth surface, Local sidereal time(LST) & Local mean time(LMT) over Sub-Sat point & Earth station Longitude, Keplerian elements & State Vectors, and more. (c) The parameters are large in numbers. Few parameters are recalculated from different input considerations. Readers may examine the parameter values computed for different satellites and compare the variations, that would answer to their many specific questions. End of Computing Six Satellites Orbital & Positional parameters at Epoch corresponding to input NASA/NORAD 'Two-Line Elements'(TLE) Bulletins. Move on to Satellite Pass for Earth Stn - Prediction of Ground Trace Coordinates & Look Angles. Next Section - 7 Satellite Pass for Earth Stn, Prediction of Ground Trace & Look Angles at UT and Local Time.
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OM-MSS Page 268 OM-MSS Section - 7 -------------------------------------------------------------------------------------------------------56 SATELLITE PASS FOR EARTH STN - PREDICTION OF GROUND TRACE COORDINATES, LOOK ANGLES, UNIVERSAL & LOCAL TIME AND MORE. The Satellites, look like slow-moving Stars, are most visible when they are in Sunlight while the viewer is in darkness. A typical Satellite in low Earth orbit (LEO) circles the Earth about sixteen times each day. The Orbital Velocity of a LEO satellite is about 7500 meters/sec. The Orbital Velocity of a Geo-stationary satellite is about 3007 meters/sec. The Moon, the only natural satellite of earth has orbital velocity about 1003 meters/sec. Satellite Pass for Earth Stn is OM-MSS utility is applied to six satellites, LANDSAT 8, SPOT 6, CARTOSAT-2B, ISS (ZARYA), GSAT-14, and Moon. The input is of respective Satellite's NASA/NORAD 'Two-Line Elements'(TLE). The Satellite Pass goes through a Time_Step of 2 minutes (120 sec). For Moon the Time_Step is of 1 hr (3600 sec). The Output is the predictions of instantaneous ground trace coordinates, look angles at each Time_Step on computer screen, in a Table form. The Table form filled with an example of one 'time line' of computed values is described below. ORBIT, INPUT TIME, POSIION, SAT VELOCITY, GROUND TRACE, RANGE, DIST, PITCH/ROLL, EL/AZ, ACCESS EL/AZ, DATA SOLAR TIME EL/AZ, DIST Line 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 @ES 18 19 20 @SS 21 22 23 24 25 26 27 @ES 28 29 30 No. Node GMT Anomaly Height At Vel Lat(SAT)Log Slant S-Sat Pitch Roll EL(SAT)AZ Control EL(SUN)AZ Acq. LMT(ES) EL(SUN)AZ S-Sun Nos D H M S deg km meter deg deg km km deg deg deg D H M S deg km 06853 A 29 03 32 50 299.96 706.66 7504.88 36.08 224.62 11592.1 12585.5 15.1 11.4 -57.70 28.41 OBC 6.1 292.4 VBC 29 08 42 29 45.1 82.2 4992.5 1 Format describtion : 1st line Columns - group name, 2nd line Columns - nos 1-29, 3rd & 4th line Columns - name & Unit, all subsequent lines are 'time lines' of Sat Pass Prediction Values. For each Satellite the 'time lines' are limited to around 800, which means a Satellite pass of around 6 hrs and for Moon around 33 days. In short, the 'time lines' dislays Artificial Satellite's Pass predictions of around 6hrs and that for Moon about one month since Epoch. Acronyms : A is Ascending, D is Decending, Pri is Perigee, Eqa is Equator, Apo is Apogee, LOS is Line Of Sight, OBC is On Board Computer, VBC is Visible Band Camera, NVD / NVD is Night Vision Device / Camera. Values : Col 1 - Orbit no, Col 2 - Node Ascending or Decending, Col 3 to 6 - Input time GMT D H M S, Col 7 - True Anomaly, Col 8 - Sat Height from earth surface, Col 9 - Sat at Perigee, Equator, or Apogee, Col 10 - Sat Velocity, Col 11 , 12 - Latitude & Longitude at sub-satellite point on earth surface,
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OM-MSS Page 269 Col 13 - Sat Slant Range from earth stn, Col 14 - Distance of sub-satellite point from earth stn, Col 15 , 16 - Sat Pitch & Roll angles, Col 17 , 18 - Sat Elevation & Azimuh angles at earth stn, Col 19 - Access to Sat through On Board Computer or Direct Line Of Sight based on elevaion angle at ES, Col 20 , 21 - Sun Elevation & Azimuh angles at sub-Satellite point on earth surface, Col 22 - Data Acquisition using Visible Band Camera or Night Vision Devices as per illumination over observed surface , Col 23 , 24 , 25 , 26 - Local Mean Time at earth stn, Col 27 & 28 - Sun Elevation & Azimuh angles at earth stn, Col 29 - Distance of sub-Sun point on earth surface from earth stn, Col 30 - Line number . Move to Compute Satellites Pass for Earth Stn. The input is NASA/NORAD 'Two-Line Elements'(TLE) Bulletins of respective Satellites. Computing Six Satellite's Pass round Earth - Prediction of Ground Trace, Look Angles & Time for Earth Stn, in Sections (7.1 to 7.6). (a) LANDSAT 8 : American satellite launched on February 11, 2013, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST (a) SPOT 6 : French satellite launched on September 9, 2012, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST (a) CARTOSAT 2B : Indian satellite launched on July 12, 2010, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST (a) ISS (ZARYA) : International Space Stn launched on Nov. 20, 1998, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST (a) GSAT-14 : Indian Geo Comm. Sat launched on Jan. 05, 2014, the NASA/NORAD TLE down load on May 28, 2014, 18:13 hrs IST (a) Moon : Natural satellite, moves around Earth , the Keplerian elements set down load on Jun 14, 2014, 16:14 hrs IST Input NASA/NORAD 'TWO-LINE ELEMENTS' of respective Satellite, and Earth stn Latitude & Longitude in deg and Height in meter. Next Section - 7.1 Satellite Pass for Earth Stn - Prediction of Ground Trace for Satellite LANDSAT 8
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OM-MSS Page 270 OM-MSS Section - 7.1 -----------------------------------------------------------------------------------------------------57 Satellite LANDSAT 8 : SAT PASS FOR EARTH STN - Prediction of Ground Trace Coordinates, Look Angles & more at Instantaneous Time. (a) LANDSAT 8 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on February 11, 2013 1 39084U 13008A 14148.14086282 .00000288 00000-0 73976-4 0 4961 2 39084 98.2215 218.5692 0001087 96.5686 263.5699 14.57098925 68534 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 39084, LANDSAT 8 , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.1408628200 EPOCH_inclination_deg = 98.2215000000 EPOCH_right_asc_acnd_node_deg = 218.5692000000 EPOCH_eccentricity = 0.0001087000 EPOCH_argument_of_perigee_deg = 96.5686000000 EPOCH_mean_anomaly_deg = 263.5699000000 EPOCH_mean_motion_rev_per_day = 14.5709892500 EPOCH_revolution = 6853 EPOCH_node_condition = 1 Earth_stn_latitude_deg = 23.25993 Earth stn longitude_deg = 77.41261 Earth surface height_meter = 494.70000 Earth stn tower_height_meter = 15.00000 Earth stn_height_meter = 509.70000 Earth stn min EL look_angle_deg = -5.00000 Input UT Year and Days decimal of year : Convert into UT YY MM DD hh min sec & Julian day Start Time UT year = 2014, month = 5, day = 28, hr = 3, min = 22, sec = 50.54765, and julian_day = 2456805.6408628202 Sat_motion_direction = Forward Sat_motion_Time_Step_in_sec_pos_or_neg = 120.00000 seconds
OM-MSS Page 330 OM-MSS Section - 7.4 -----------------------------------------------------------------------------------------------------60 Satellite ISS (ZARYA) : SAT PASS FOR EARTH STN - Prediction of Ground Trace Coordinates, Look Angles & more at Instantaneous Time. (d) ISS (ZARYA) 'Two-Line Elements'(TLE) downloaded on May 28, 2014, 18:13 hrs IST, Satellite launched on November 20, 1998 1 25544U 98067A 14148.25353351 .00006506 00000-0 11951-3 0 3738 2 25544 51.6471 198.4055 0003968 47.6724 33.3515 15.50569135888233 From this TLE, the data relevant for the purpose are manually interpreted & extracted : Satellite number 25544, ISS (ZARYA) , EPOCH_year = 2014 EPOCH_days_decimal_of_year = 147.2535335100 EPOCH_inclination_deg = 51.6457000000 EPOCH_right_asc_acnd_node_deg = 56.4734000000 EPOCH_eccentricity = 0.0003968000 EPOCH_argument_of_perigee_deg = 86.4957000000 EPOCH_mean_anomaly_deg = 16.4380000000 EPOCH_mean_motion_rev_per_day = 15.5027620300 EPOCH_revolution = 90394 EPOCH_node_condition = 1 Earth_stn_latitude_deg = 23.25993 Earth stn longitude_deg = 77.41261 Earth surface height_meter = 494.70000 Earth stn tower_height_meter = 15.00000 Earth stn_height_meter = 509.70000 Earth stn min EL look_angle_deg = -5.00000 Input UT Year and Days decimal of year : Convert into UT YY MM DD hh min sec & Julian day Start Time UT year = 2014, month = 5, day = 28, hr = 6, min = 5, sec = 5.29526, and julian_day = 2456805.7535335100 Sat_motion_direction = Forward Sat_motion_Time_Step_in_sec_pos_or_neg = 120.00000 seconds
OM-MSS Page 336 SATELLITE PASS - GROUND TRACE [acronym - Line Of Sight (LOS), On Board Computerr (OBC), Visible Band Camera (VBC), Night Vision Devices (NVD) ]
OM-MSS Page 389 SATELLITE PASS - GROUND TRACE [acronym - Line Of Sight (LOS), On Board Computerr (OBC), Visible Band Camera (VBC), Night Vision Devices (NVD) ] ORBIT, INPUT TIME, POSIION, SAT VELOCITY, GROUND TRACE, RANGE, DIST, PITCH/ROLL, EL/AZ, ACCESS EL/AZ, DATA SOLAR TIME EL/AZ, DIST Line 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 @ES 18 19 20 @SS 21 22 23 24 25 26 27 @ES 28 29 30 No. Node GMT Anomaly Height @ Vel Lat(SAT)Log Slant S-Sat Pitch Roll EL(SAT)AZ Control EL(SUN)AZ Acq. LMT(ES) EL(SUN)AZ S-Sun Nos D H M S deg km meter deg deg km km deg deg deg D H M S deg km 00001 A 22 20 59 20 90.70 376040.59 1021.96 13.59 37.72 382327.0 4313.1 0.6 0.4 0.34 75.75 LOS -52.2 348.0 NVD 23 02 08 59 -34.1 35.6 13809.3 742 00001 A 22 21 59 20 91.25 376228.83 1021.46 13.71 38.27 381073.1 4252.3 0.6 0.4 13.45 81.01 LOS -52.1 11.4 NVD 23 03 08 59 -24.9 47.4 12788.9 743 00001 A 22 22 59 20 91.80 376417.02 1020.96 13.83 38.82 379878.4 4191.6 0.6 0.4 26.79 86.06 LOS -46.4 31.5 NVD 23 04 08 59 -14.0 56.4 11579.6 744 00001 A 22 23 59 20 92.35 376605.15 1020.45 13.96 39.38 378831.5 4131.0 0.5 0.4 40.25 91.47 LOS -36.8 45.8 NVD 23 05 08 59 -2.1 63.3 10252.0 745 00001 A 23 00 59 20 92.90 376793.22 1019.95 14.08 39.93 378012.3 4070.5 0.5 0.4 53.72 98.30 LOS -25.1 55.3 NVD 23 06 08 59 10.5 68.9 8849.2 746 00001 A 23 01 59 20 93.44 376981.19 1019.45 14.19 40.48 377486.2 4010.0 0.5 0.4 66.89 109.58 LOS -12.1 61.7 NVD 23 07 08 59 23.5 73.5 7397.4 747 00001 A 23 02 59 20 93.99 377169.05 1018.95 14.31 41.03 377299.8 3949.6 0.5 0.4 78.35 139.99 LOS 1.5 66.2 NVD 23 08 08 59 36.9 77.4 5913.0 748 00001 A 23 03 59 20 94.54 377356.79 1018.46 14.43 41.58 377477.1 3889.3 0.5 0.4 78.83 217.72 LOS 15.5 69.2 VBC 23 09 08 59 50.4 80.8 4406.9 749 00001 A 23 04 59 20 95.09 377544.38 1017.96 14.54 42.13 378018.1 3829.1 0.5 0.4 67.63 250.49 LOS 29.7 71.0 VBC 23 10 08 59 64.1 83.9 2886.6 750 00001 A 23 05 59 20 95.63 377731.82 1017.46 14.66 42.69 378898.6 3768.9 0.5 0.4 54.53 262.26 LOS 44.0 71.4 VBC 23 11 08 59 77.8 86.6 1358.1 751 00001 A 23 06 59 20 96.18 377919.08 1016.96 14.77 43.24 380072.2 3708.8 0.5 0.4 41.13 269.28 LOS 58.2 69.2 VBC 23 12 08 59 88.4 276.4 175.5 752 00001 A 23 07 59 20 96.72 378106.14 1016.47 14.88 43.79 381473.6 3648.8 0.5 0.4 27.74 274.82 LOS 71.9 59.3 VBC 23 13 08 59 74.7 273.9 1706.5 753 00001 A 23 08 59 20 97.27 378293.00 1015.97 14.99 44.34 383023.3 3588.9 0.5 0.4 14.51 279.99 LOS 81.5 8.5 VBC 23 14 08 59 61.0 276.7 3233.6 754 Next End of Computing Six Satellites Passes for Earth Stn - Prediction of Ground Trace Coordinates, Look Angles & more at Instantaneous Time. A Summary of Computing Satellites Passes for Six Satellites are presented next. Next Section - 7.7 Concluding Satellites Passes - Prediction of Ground Trace of Six Satellites
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OM-MSS Page 390 OM-MSS Section - 7.7 -----------------------------------------------------------------------------------------------------63 Concluding Satellites Passes - Prediction of Ground Trace Coordinates, Look Angles & more at Instantaneous Time. In previous Sections (7.1 to 7.6), Six Satellites, LANDSAT 8, SPOT 6, CARTOSAT-2B, ISS (ZARYA), GSAT-14, and Moon, Passes - Prediction of Ground Trace, Look Angles & more for the Earth Station. A Summary of those Six Satellites Passes - Prediction of Ground Trace, Look Angles & more for the Earth Stn, are presented below : (a) The Satellites considered were four LEO, one GEO and one natural satellite repectively around earth considering earth oblateness. (b) Input NASA/NORAD 'TWO-LINE ELEMENTS' of respective Satellite, and Earth stn Latitude & Longitude in deg and Height in meter. The Satellites Passed passed through a Time_Step of 2 minutes (120 sec). For Moon the Time_Step was of 1 hr (3600 sec). The Output is the predictions of Ground Trace, Look Angles & more for Earth Stn at each Time_Step put on computer screen. (C) The Satellite Pass, the Prediction of Ground Trace parameters included : Sat Orbit no, Sat pass Input time TEL GMT, Sat Position True Anomaly in deg, Sat Height in km from earth surface, Sat at Perigee or Equator or Apogee cross over, Sat Velocity in meters, Sat Ground trace Latitude & Longitude in deg at sub-satellite point on earth surface, Sat Slant Range in km from earth stn, Sat Distance in km of sub-satellite point from earth stn, Sat Pitch & Roll angles in deg, Sat Elevation & Azimuh angles in deg at earth stn, Access to Sat through On Board Computer or Direct Line Of Sight based on elevaion angle at ES, Sun Elevation & Azimuh angles in deg at sub-Satellite point on earth surface, Sat Data Acquisition using Visible Band Camera or Night Vision Devices as per illumination over observed surface , Local Mean Time at earth stn, Sun Elevation & Azimuh angles in deg at earth stn, Distance of sub-Sun point on earth surface from earth stn, and Sat pass Time Line number. End of Computing Six Satellites Passes for Earth Stn - Prediction of Ground Trace Coordinates, Look Angles & more at the Earth station. Next Section - 8 Concluding Orbital Mechanics - Model & Simulation Software (OM-MSS).
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OM-MSS Page 391 OM-MSS Section - 8 -------------------------------------------------------------------------------------------------------64 CONCLUSION : ORBITAL MECHANICS - MODEL & SIMULATION SOFTWARE (OM-MSS). The 'ORBITAL MECHANICS - MODEL & SIMULATION SOFTWARE (OM-MSS)', to the Simulate Motion of Sun, Earth, Moon and Satellites were presented. This Software is written in 'C' Language, the Compiler used is Dev C++ and the Platform is a Windows 7, 64 bit Laptop. The Source Code around 30,000 Lines is Compiled. The 'OM-MSS.EXE' File generated is of Size 1.5 KB. The Executable File, < OM-MSS.EXE >, is RUN Step-by-Step for a Set of Inputs. The Outputs on Computer Screen were put in a File, Which in effect became 'A Monograph of Orbital Mechanics with Examples, Problems and Software Driven Solutions'. The same was presented in Seven Sections. Section 1. [ 1.1 to 1.7 ] Astronomical Time Standards And Time Conversion Utilities. Section 2. [ 2.1 to 2.11] Positional Astronomy : Earth Orbit around Sun, Anomalies & Astronomical Events - Equinoxes, Solstices, Years & Seasons. Section 3. [ 3.1 to 3.8 ] Position Of Sun On Celestial Sphere At Input Universal Time (Ut). Section 4. [ 4.1 to 4.8 ] Position Of Earth On Celestial Sphere At Input Universal Time (Ut). Section 5. [ 5.1 to 5.4 ] Satellites In Orbit Around Earth : Ephemeris Data Set. Section 6. [ 6.1 to 6.7 ] Satellites Motion Around Earth : Orbital & Positional Parameters At Epoch. Section 7. [ 7.1 to 7.7 ] Satellite Pass For Earth Stn - Prediction Of Ground Trace Coordinates, Look Angles, Ut & Local Time. The readers, who are beginners, who have little to no knowledge about Positional Astronomy of Earth, Sun, Moon, and Satellites Motion in Orbit, will have no difficulty in understanding. Each section starts first with a tutorial to understand the repective topics under consideration and then presented the computed values of the parameters.
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OM-MSS Page 392 OM-MSS Section - 9 -------------------------------------------------------------------------------------------------------65 REFERENCES : TEXT BOOKS & INTERNET WEB LINKS. Books 1. Dennis Roddy, 'Satellite Communication', Third Edition, McGraw Hill, chap. 2 - 3, pp 21-86, Jan 2001. 2. Gerard Maral, Michel Bousquet, 'Satellite Communications Systems', Fifth Edition, John Wiley & Sons, chap. 2, pp 19-97, 2002. 3. Hannu Karttunen, Pekka Kroger, et al, 'Fundamental Astronomy', Springer, 5th Edition, pp 1 - 491, 2007. 4. Vladimir A. Chobotov,'Orbital mechanics', American Institute of Aeronautics and Astronautics, pp 1 - 447, 1996. 5. Howard Curtis, 'Orbital Mechanics: For Engineering Students', Aerospace Engineering, Butterworth-Heinemann, , pp 1 - 704, 2004. 6. Howard D. Curtis, 'Orbital Mechanics For Engineering Students, Solutions Manual', Embry-Riddle Aeronautical University, Florida. Internet Weblinks Ref. Sec 1 Astronomical Time 1. Ephemeris.com, 'solar system', URL http://www.ephemeris.com/solar-system.html 2. Ephemeris.com, 'Space and Time', URL http://www.ephemeris.com/space-time.html 3. Wikipedia, 'Time standard', URL http://en.wikipedia.org/wiki/Time_standard 4. Digplanet, 'Time standard', URL http://www.digplanet.com/wiki/Time_standard 5. Yost, Daunt, 'Timekeeping', URL http://csep10.phys.utk.edu/astr161/lect/time/timekeeping.html 6. Eric Weisstein's World of Astronomy, 'Time standard', URL http://scienceworld.wolfram.com/astronomy/topics/TimeStandards.html 7. J. Richard Fisher, 'Astronomical Times', National Radio Astronomy Observatory, URL http://www.cv.nrao.edu/~rfisher/Ephemerides/times.html 8. Kaye and Laby, 'Astronomical and atomic time systems', National Physical Laboratory, URL http://www.kayelaby.npl.co.uk/general_physics/2_7/2_7_1.html 9. Kaye and Laby, 'Astronomical units and constants', National Physical Laboratory, URL http://www.kayelaby.npl.co.uk/general_physics/2_7/2_7_2.html 10. Kaye and Laby, 'The Solar System', National Physical Laboratory, URL http://www.kayelaby.npl.co.uk/general_physics/2_7/2_7_3.html
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OM-MSS Page 393 Ref. Sec 2 Positional Astronomy 1. Fiona Vincent, 'Positional Astronomy', University of St.Andrews, Revised and updated November 2003, URL http://star-www.st-and.ac.uk/~fv/webnotes/index.html 2. Robert A. Braeunig, 'ORBITAL MECHANICS', Rocket and Space Technology, Basics of Space Flight Part III, URL http://www.braeunig.us/space/orbmech.htm 3. James B. Calvert, 'Celestial Mechanics', Physics, Mechanics and Thermodynamics, University of Denver, URL http://mysite.du.edu/~jcalvert/phys/orbits.htm 4. Vladislav Pustonski, 'Orbital Elements & Types of orbits', Introduction to Astronautics, Tallinn University of Technology, 2009-2010, URL http://www.aai.ee/~vladislav/Astronautics_Lecture4.pps 5. PhysicalGeography.net, 'Earth-Sun Geometry', Fundamentals Of Physical Geography, 2nd Edition, Energy and Matter, chap. 6, URL http://www.physicalgeography.net/fundamentals/6h.html 6. Wikipedia, the free encyclopedia, 'Orbital elements', Last Modified on January 24, 2015, URL http://en.wikipedia.org/wiki/Orbital_element 7. Keith Burnett, 'Approximate Astronomical Positions', Last Modified on January 12, 2000, URL http://www.stargazing.net/kepler/ 8. The Physics Classroom, 'Kepler's Three Laws', Circular Motion and Satellite Motion - Lesson 4 - Planetary and Satellite Motion, URL http://www.physicsclassroom.com/Class/circles/U6L4a.cfm 9. Wikipedia, the free encyclopedia, 'celestial coordinate system', Last Edited on MAY 19, 2015, URL http://en.m.wikipedia.org/wiki/Celestial_coordinate_system 10. Denny Sissom, 'Understanding Orbital Mechanics', Elmco INC, pp 1-27, on May 2003, URL http://www.agi.com/downloads/support/productsupport/literature/pdfs/casestudies/at_051303_0830_orbital_mechanics_viasbirs.pdf 11. Athropolis, 'Guide to the Equinoxes and Solstices', Athropolis Productions Limited (Canada), accessed on May, 2015, URL http://www.athropolis.com/sunrise/def-sol2.htm 12. Wikipedia, the free encyclopedia, 'Season', last modified on May 20, 2015, URL http://en.wikipedia.org/wiki/Season 13. Paul Schlyter, 'How to compute planetary positions', Introductory Astronomy : The Celestial Sphere URL http://astro.wsu.edu/worthey/astro/html/lec-celestial-sph.html
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OM-MSS Page 394 Ref. Sec 3 Position Of Sun On Celestial Sphere 1. Wikipedia, the free encyclopedia, 'Position of the Sun', last modified on April 28, 2015, URL http://en.wikipedia.org/wiki/Position_of_the_Sun 2. William B. Stine and Michael Geyer, 'The Sun’s Position', Power From The Sun, Chap 3, URL http://www.powerfromthesun.net/Book/chapter03/chapter03.html 3. Paul Schlyter, 'How The Sun's position', URL http://www.stjarnhimlen.se/comp/tutorial.html#5 4. Paul Schlyter, 'How to compute rise/set times and altitude above horizon', URL http://www.stjarnhimlen.se/comp/riset.html 5. Ed Williams, 'Sunrise/Sunset Algorithm', Aviation Formulary V1.46, accessed on March 2002, URL http://williams.best.vwh.net/sunrise_sunset_algorithm.htm 6. Wikipedia, the free encyclopedia, 'Sunrise equation', Last Modified on May 8, 2015, URL http://en.m.wikipedia.org/wiki/Sunrise_equation Ref. Sec 4 Position Of Earth On Celestial Sphere 1. David R. Williams, 'Earth Fact Sheet', NASA Goddard Space Flight Center, last updated on March 2, 2015, URL http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html 2. Wikipedia, the free encyclopedia, 'Earth's orbit', last modified on 20 May 2015, URL http://en.wikipedia.org/wiki/Earth%27s_orbit 3. Keith Burnett, 'Converting RA and DEC to ALT and AZ', Approximate Astronomical Positions, Last Modified May 27, 1998, URL http://www.stargazing.net/kepler/altaz.html 4. Tom Chester, 'How To Calculate Distances, Azimuths and Elevation Angles Of Peaks', last updated on August 22, 2006, URL http://tchester.org/sgm/analysis/peaks/how_to_get_view_params.html 5. W. D. Komhyr, 'Introduction To Principles Of Astronomy', Operations Handbook, Appendix H, pp 116-121, on June, 1980, URL http://www.esrl.noaa.gov/gmd/ozwv/dobson/papers/report6/apph.html 6. Hartmut Frommert, 'General Coordinate Systems', Index of /spider/ScholarX, spider.seds.org/spider, URL http://spider.seds.org/spider/ScholarX/coord_bas.html
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OM-MSS Page 395 Ref. Sec 5 Satellites In Orbit Around Earth : Ephemeris Data Set. 1. Roman Starkov, 'Orbital elements', OrbiterWiki, Last modified on October 22, 2008 URL http://www.orbiterwiki.org/wiki/Orbital_elements 2. Amiko Kauderer, NASA, 'Definition of Two-line Element Set Coordinate System', Last updated on September 23, 2011, URL http://spaceflight.nasa.gov/realdata/sightings/SSapplications/Post/JavaSSOP/SSOP_Help/tle_def.html 3 T.S. Kelso, 'NORAD Two-Line Element Sets Current Data', Center for Space Standards & Innovation (CSSI), Last updated on May 21, 2015, URL http://celestrak.com/NORAD/elements/ 4. [email protected], 'Brief Introduction To TLEs And Satellite IDs', Visual Satellite Observation, Description of an orbital element set, URL http://www.satobs.org/element.html 5. Keith Burnett, 'Moon Ephemeris', Last modified on February 22, 2000, URL http://www.lunar-occultations.com/rlo/ephemeris.htm 6. Webmaster, 'Date/Epoch Converter', E.S.Q. Software, URL http://www.esqsoft.com/javascript_examples/date-to-epoch.htm Ref. Sec 6 Satellites Motion Around Earth 1. Kimberly Strong, 'Section 2 - Satellite Orbits', Phy 499S - Course Notes, Atmospheric Physics Group, Last updated on April 20, 2005, URL http://www.atmosp.physics.utoronto.ca/people/strong/phy499/section2_05.pdf 2. Ian Poole, 'Satellite Orbit Types & Definitions', Satellite Orbits Tutorial, accessed on May 22, 2015, URL http://www.radio-electronics.com/info/satellite/satellite-orbits/satellites-orbit-definitions.php 3. Michael E Brink. HND, 'An overview of celestial mechanics as applied to the process of satellite tracking', Space Week 2007, URL http://www.parc.org.za/attachments/satnews/Celestial.pdf 4. Vardan Semerjyan, 'Kepler’s equation Solver', Small Satellites, accessed on January 17, 2013, URL http://smallsats.org/2013/01/17/ 5. Christopher D. Hall, 'Orbits', Satellite Attitude Dynamics, Appendix A , accessed on January 12, 2003, URL http://www.aoe.vt.edu/~cdhall/courses/aoe4140/a_orbits.pdf 6. RPC Telecommunications, 'Introduction', Satcom Online, Lecture 2, Space Segment, Submitted on June 06, 2001, URL http://www.satcom.co.uk/print.asp?article=29 7. RPC Telecommunications, 'Kepler And Satellite Ephemeris Formulae', Lecture 2, Space Segment, Submitted on June 06, 2002, URL http://www.satcom.co.uk/article.asp?article=29§ion=2 8. RPC Telecommunications, 'Satellite Orbits', Satellite School / Satellite Orbits, Submitted on November 11, 2002, URL http://www.satcom.co.uk/article.asp?article=11 9. 'Section 4: The Basics of Satellite Orbits', accessed on May 22, 2015, URL http://www.amacad.org/publications/Section_4.pdf
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OM-MSS Page 396 Ref. Sec 7 Satellite Pass For Earth Stn - Prediction Of Ground Trace 1. Patrick Minnis, NASA Official, 'NASA LaRC Satellite Overpass Predictor', Last Updated on Jan 16, 2007, URL http://cloudsgate2.larc.nasa.gov/cgi-bin/predict/predict.cgi 2. Sebastian Stoff, 'Orbitron - Satellite Tracking System orbitron 3.71', Orbitron website, 2007, URL http://www.stoff.pl/ 3. 'Satellite Pass Predictions', Hudson Valley Satellite Net, accessed on May 22, 2015, URL http://hvsatcom.org/ 4. Earth Orbit Objects, 'EOO_VER700_SETUP.exe', Last Updated on Aug 11, 2014, URL http://www3.telus.net/public/boucherd/eoo.html 5. T.S. Kelso, 'Orbital Coordinate Systems, Part I', Last updated on May 17, 2014, URL http://www.celestrak.com/columns/v02n01/ 6. T.S. Kelso, 'Orbital Coordinate Systems, Part II', Last updated May 17, 2014, URL http://www.celestrak.com/columns/v02n02/ 7. T.S. Kelso, 'Orbital Coordinate Systems, Part III', Last updated May 17, 2014, URL http://celestrak.com/columns/v02n03/ Ref. Calculators for Astronomical Time Conversions and Computing Position of Earth, Sun, Moon, and Satellites Motion in Orbit. 1. 'Coordinates-DMS-to-Decimal-To-DMS', www.coolconversion.com, URL http://coolconversion.com/Coordinates-Degree-minute-second-DMS-to-Decimal-To-DMS 2. 'Degrees, Minutes, Seconds to/from Decimal Degrees', www.transition.fcc.gov, URL http://transition.fcc.gov/mb/audio/bickel/DDDMMSS-decimal.html 3. 'Convert epoch to human readable date and vice versa', www.epochconverter.com, Epoch & Unix Timestamp Conversion Tools, URL http://www.epochconverter.com/ 4. 'Converting Addresses to/from Latitude/Longitude/Altitude', www.stevemorse.org, Stephen P. Morse, URL http://stevemorse.org/jcal/latlon.php 5. 'Distance and Azimuths Between Two Sets of Coordinates', www.fcc.gov, URL http://www.fcc.gov/encyclopedia/distance-and-azimuths-between-two-sets-coordinates 6. 'Distance Calculator', www.geodatasource.com, GeoDataSource, URL http://www.geodatasource.com/distance-calculator 7. 'Compute the Sun & Moon position and eclipse irradiance reduction from time and location', www.nrel.gov, MIDC SAMPA Calculator, URL http://www.nrel.gov/midc/solpos/sampa.html
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OM-MSS Page 397 8. 'Sun's Position to High Accuracy' Sun elevation Azimuth angle, www.pveducation.org, URL http://www.pveducation.org/pvcdrom/properties-of-sunlight/sun-position-high-accuracy 9. 'Find Sunrise, Sunset, Solar Noon and Solar Position for Any Place on Earth', www.esrl.noaa.gov, NOAA Solar Calculator, http://www.esrl.noaa.gov/gmd/grad/solcalc/ 10. 'Sun and Moon Positions, Rise and Set Times', www.largeformatphotography.info, The Sun/Moon Calculator, Jeff Conrad, URL http://www.largeformatphotography.info/sunmooncalc/ 11. 'Compute Sun and Moon Azimuth & Elevation', www.ga.gov.au, Geoscience Australia, Astronomical Information, URL http://www.ga.gov.au/geodesy/astro/smpos.jsp 12. 'Sun & Moon Position Calculator', www.satellite-calculations.com, Jens T. Satre, URL http://www.satellite-calculations.com/Satellite/suncalc.htm 13. 'Calculating the Moon's Position', www.internationalastrologers.com, LukeS, URL http://www.internationalastrologers.com/calculating_moon_position.htm 14. 'Sun & Moon Positions', sky.ccny.cuny.edu, Calculation Tools, http://sky.ccny.cuny.edu/mn/pub/sunmooncalculator2.php 15. 'Sunrise Sunset Times by Longitude-Latitude and UTC', www.calculatorsoup.com, URL http://www.calculatorsoup.com/calculators/time/sunrise_sunset.php 16. 'Calculate eccentric anomaly & true anomaly', www.jgiesen.de, Solving Kepler's Equation of Elliptical Motion, URL http://www.jgiesen.de/kepler/kepler.html 17. 'Orbit calculations', www.calsky.com, Cal Sky, Sun, Moon, Satellites, URL http://www.calsky.com/cs.cgi/Intro?obs=19155525889164 18. 'NASA LaRC Satellite Overpass Predictor', www-angler.larc.nasa.gov, NASA.gov, URL http://cloudsgate2.larc.nasa.gov/cgi-bin/predict/predict.cgi 19. 'Satellite tracker based on two line elements', www.satellite-calculations.com, Jens T. Satre, URL http://www.satellite-calculations.com/Satellite/SatTracker/sattracker.php?25544?http://www.celestrak.com/NORAD/elements/stations.txt 20. 'Satellite Look Angle Calculator', www.satellite-calculations.com, Jens T. Satre, URL http://www.satellite-calculations.com/Satellite/lookangles.htm 21. 'Satcom Calculators', www.satcom.co.uk, Satcom Online, Calculate Azimuth And Elevation Angles, URL http://www.satcom.co.uk/article.asp?article=1 22. 'AMSAT Online Satellite Pass Predictions', www.amsat.org, The Radio Amateur Satellite Corporation, URL http://www.amsat.org/amsat-new/tools/predict/ 23. 'Satellite Azimuth And Elevation Position Calculator', www.csgnetwork.com, URL http://www.csgnetwork.com/antennasatelazposcalc.html
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OM-MSS Page 398 24. 'Celestial to Horizon Co-ordinates Calculator', www.convertalot.com, URL http://convertalot.com/celestial_horizon_co-ordinates_calculator.html 25. 'Calculates the orbital radius and period, and flight velocity from the orbital altitude', keisan.casio.com, Keisan Online Calculator, URL http://keisan.casio.com/exec/system/1224665242 26. 'Latitude, Longitude, Height to/from ECEF (X,Y,Z)', www.oc.nps.edu, U.S. Navy website, URL http://www.oc.nps.edu/oc2902w/coord/llhxyz.htm 27. 'Converting Velocity and Position Vectors to Longitude and Latitude', stackoverflow.com , URL http://stackoverflow.com/questions/11677565/converting-velocity-and-position-vectors-to-longitude-and-latitude End of execution of 'Orbital Mechanics - Model & Simulation Software (OM-MSS)', illustrated its Scope, Capability, Accuracy, and Usage. Next Section - 10 ANNEXURE : A Collection of few related Diagrams / Help , appended offline.
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OM-MSS Page 399 OM-MSS Section - 10 -------------------------------------------------------------------------------------------------------66 ANNEXURE : A Collection of few OM-MSS related Diagrams / Help.
Fig-1. Solar System
A collection of celestial bodies composed of a star called the Sun and
eight known planets in elliptical orbits bound by the force of gravity.
Earth rotates around sun with a period of approximately 365.25 days following an Ellipse of Eccentricity 0.01673 and Semi-major axis
149597870 km, which defines the Astronomical unit of distance (AU). Around 2 January, Earth is nearest from sun called Perihelion
while around 5 July it is farthest from Sun called Aphelion (around 152100000 km). The other events point are Vernal equinox around 21 March,
Autumnal equinox around 23 September, Summer solstice around 21 June and Winter solstic around 21 December. The plane of the orbit is called the plane
of the Ecliptic that makes an angle 23.44 deg (the Obliquity of the Ecliptic) with the mean Equatorial plane.
Source Book by Gerard Maral, Michel Bousquet, 'Satellite Communications Systems', Fifth Edition, John Wiley & Sons, chap. 2, Pg 29, 2002.
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Fig- 4 & 5 Positioning of Orbit in Space
Orbit Position in Space at Epoch is defined by the Values of Kepler Orbit elements : (definations apply to both planets & Satellits) 1. Inclination 'i' of the orbit of a planet, is angle between the plane of planet's orbit and the plane containing Earth's orbital path (ecliptic).
2. Right ascension ' ' of the ascending node is the angle taken positively from 0 to 360 deg in the forward direction, between the reference direction and the
ascending node of the orbit (the intersection of the orbit with the plane of the equator crossing this plane from south to north).
3. Argument of Perigee ' ', specify angle between orbit's perigee and orbit's ascending node, measured in orbital plane and direction of motion.
4. Eccentricity 'e' of an orbit shows how much the shape of an object's orbit is different from a circle;
5. Mean Anomaly 'v' relates the position and time for a body moving in a Kepler orbit. The mean anomaly of an orbiting body is the angle through which the body
would have traveled about the center of the orbit's auxiliary circle. 'M' grows linearly with time.
A knowledge of above five parameters completely defines the trajectory of an object or satellite in space. However, the Nodal angular elongation 'u' can also be used
to define the position of the satellite in its orbit. This is the angle taken positively in the direction of motion from 0 to 360 deg between the direction of the
ascending node and the direction of the satellite (u = + v ).
Source Book by Gerard Maral, Michel Bousquet, 'Satellite Communications Systems', Fifth Edition, John Wiley & Sons, chap. 2, Pg 29, 2002. & http://www.britannica.com/EBchecked/topic/101285/celestial-mechanics/images-videos/2285/orbital-element-keplers-laws-of-planetary-motion
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Fig-5. Three Successive Satellite Passes/Orbits Around Earth
The satellites move around the Earth, taking about 95 minutes to complete an orbit.
A satellite at 300 (km) altitude has orbital period about 90 (min). In 90 (min), the earth at equator rotates about 2500 (km).
Thus, the satellite after one time period, passes over equator a point/place 2500 (km) west of the point/place it passed over in
its previous orbit. To a person on the earth directly under the orbit, a satellite appears above horizon on one side of sky,
crosses the sky, and disappears beyond the opposite horizon in about 10 (min). It reappears after 80 (min), but not over same
spot, since the earth has rotated during that time.