Intro to Astronautics

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Welcome To..

Satellite Tool Kit

Astronautics Primerby

Jerry Jon Sellers

Based on

Understanding Space: AnIntroduction to Astronautics

Copyright 1996 McGraw-Hill Inc.

by

Jerry Jon Sellers

Wiley J. Larson (editor)

Your Understanding of Space Starts Here!

COPYRIGHT NOTICE

McGraw-Hill owns the copyright to the book Understanding Space: An Introduction toAstronautics. This primer was adapted from the book by Jerry Jon Sellers and Wiley J.

Larson. Analytical Graphics, Inc. retains copyright for this Primer. It is illegal toreproduce this material without permission.

STK Astronautics Primer

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OBJECTIVE

The objective of this primer is to provide you with a fundamental reference for keyconcepts in astronautics, including:

♦ Space mission architecture

♦ History

♦ Dynamics

♦ Orbital mechanics

♦ Orbital elements

♦ Orbit propagation

♦ Ground tracks

♦ Satellite access

ABOUT THE AUTHOR

Jerry Jon Sellers was born in Harlan, Iowa. He has worked over 13 years at variousastronautics assignments including the NASA Johnson Space Center, where heworked in Space Shuttle Mission Control (guidance and navigation) and the U.S. AirForce Academy where he served on the faculty of the Department of Astronautics. Hewas a distinguished graduate from the U.S. Air Force Academy in 1984 and hasearned a Master’s Degree in Physical Science from the University of Houston, ClearLake, a Master’s Degree in Aeronautics and Astronautics from Stanford Universityand a Ph.D. in Satellite Engineering from the University of Surrey, UK. He currentlyworks as an international research and development liaison officer in London, UK,and continues to write and consult on space mission analysis and design.

CONTENTS

Stepping Into Space

The big picture of why space is important and how the pieces fit together

Exploring Space

Some early “explorers” who’ve shaped our current understanding of orbits

An Introduction to Orbit Motion

Key concepts necessary for understanding orbit motion

Describing Orbits

Understanding orbital elements (two-line element sets), ground tracks and how they relateto space missions


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Predicting Orbits

The “nuts and bolts” of predicting real-world satellite motion using orbit propagators

Satellite Access

The how, when and where of links between ground stations and satellites

Recommended Reading

Other great astronautics references

STEPPING INTO SPACE

Since the dawn of the Space Age only a few decades ago, we have come to rely moreand more on satellites for a variety of needs. Daily weather forecasts, instantaneousworld-wide communication, and a constant ability to keep an eye on not-so-friendlyneighbors are all examples of space technology that we’ve come to take for granted.

The purpose of this brief astronautics primer is to provide the reader with aconceptual overview of important topics in orbital mechanics. Understanding thesekey concepts will enhance your insight into the science behind Satellite Took Kit andbetter equip you to apply these concepts to practical problems in space. We’ll beginwith a brief overview of space, space missions and space history. Then we’ll get intothe details of orbital mechanics to see how you can use STK to plot your path to thestars.

Why is space so useful?

Getting into space is dangerous and expensive. So why bother? Space offers severalcompelling advantages for modern society

♦ A global perspective—the ultimate high ground

♦ A universal perspective—from space we have a clear view of the heavens,unobscured by the atmosphere

♦ A unique environment—free-fall and abundant resources make space thetrue final frontier

Global Perspective

Space offers a global perspective. As you can see in Figure 1, the higher you are, themore you can see. For thousands of years, kings and rulers took advantage of thisfact by putting lookout posts atop the tallest mountains to survey more of their realmand fend off would-be attackers. Throughout history, many battles have been foughtto “take the high ground.” Space takes this quest for greater perspective to itsultimate end. From the vantage point of space, we can view large parts of the Earth’ssurface. Orbiting satellites can thus serve as “eyes in the sky” to provide a variety ofuseful services.


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Figure 1: Global perspective. From space, satellites can observe large-scale featureson the Earth, track weather patterns, monitor the environment and view widely separatedpoints simultaneously, allowing them to communicate.

Universal Perspective

Space offers a clear view of the heavens. When we look at stars in the night sky, wesee their characteristic twinkle. This twinkle, caused by the blurring of “starlight” as itpasses through the atmosphere, is known as scintillation. Not only is the light blurred,but some of it is blocked or attenuated altogether. This attenuation is frustrating forastronomers who need access to all the regions of the spectrum to fully explore theuniverse. By placing observatories in space, we can sit above the atmosphere andgain an unobscured view of the universe, as depicted in Figure 2. The Hubble SpaceTelescope and the Gamma Ray Observatory are armed with sensors operating farbeyond the range of human senses. Already, results from these instruments arerevolutionizing our understanding of the cosmos.

Figure 2: Seeing beyond the clouds. Earth-based astronomy is obscured by theatmosphere. Astronomers don’t like the “twinkle” of star light. Some wavelengths are


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completely blocked. Space-based astronomy opens the door to a whole new perspective onthe universe.

A Unique Environment

Space offers a unique environment with many advantages.

♦ Free-fall environment—enables developing advanced materials

♦ Abundant resources—solar energy and minerals

Gravity makes some manufacturing processes difficult if not impossible. To formcertain new metal alloys, for example, we must blend two or more metals in just theright proportion. Unfortunately, gravity tends to pull the heavier metal to the bottom,making a uniform mixture difficult to obtain. But space offers the solution. Amanufacturing plant in orbit is literally falling toward Earth but never hitting it. Thisis a condition known as free-fall (NOT zero gravity). In free-fall there are no contactforces on an object, so it is said to be weightless. Unencumbered by the weight felt onthe Earth’s surface, factories in orbit can create exotic new metals for computers orother advanced technologies, as well as for promising new pharmaceutical products tobattle disease on Earth.

Figure 3: Early free-fall manufacturing. In the 16th century, Italian weapons makersdeveloped a secret way of making lead shot for muskets. By dropping liquid lead from a“shot tower,” they found near-perfect spheres would form as the molten lead cooled andhardened in free fall.

Space also offers abundant resources. While some on Earth argue about how to carvethe pie of Earth’s resources into smaller and smaller pieces, others have argued that


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example, is known to be rich in oxygen and aluminum. The oxygen could be used asrocket propellant and for humans to breathe. Aluminum is an important metal forvarious industrial uses. These resources, coupled with the human drive to explore,mean the sky is truly the limit!

Space ApplicationsLet’s look at some important application of space that affect all of our lives today.

Communications Satellites

Science/Science Fiction writer Arthur C. Clarke first proposed putting satellites intoorbits with periods of 24 hours, 36,000 km above the equator, exactly matching therotation rate of the Earth. These geostationary orbits could serve as communicationhubs to link together remote parts of the planet. With the launch of the firstexperimental communications satellite, Echo I, into Earth orbit in 1960, Clarke’sfanciful idea showed promise of becoming reality. Although Echo I was little morethan a reflective balloon in low-Earth orbit, radio signals were bounced off it,demonstrating that space could be used to broaden our horizons of communication.An explosion of technology to exploit this idea quickly followed.

Satellites are now used for a large percentage of commercial and governmentcommunications and for most domestic cable television. Through satellite technology,relief workers can now stay in constant contact with their organizations, enablingthem to better distribute aid to refugees hungry for food. In addition, our modernmilitary now relies almost totally on satellites to communicate with forces deployedworld-wide. Without satellites, global communication as we know it today would notbe possible.

Remote Sensing Missions

Satellites operating from the global perspective of space have also made possible thescience of remote sensing. Remote sensing is the act of observing Earth and otherobjects from space. For decades, military “spy satellites” have kept tabs on theactivities of potential adversaries using remote-sensing technology. This sametechnology has been adapted for civilian uses such as

♦ monitoring Earth’s environment

♦ forecasting the weather

♦ managing resources

Satellites can now “spy” on crops, ocean currents, and natural resources to aidfarmers, resource managers, and planners on Earth. In countries where the failure ofa harvest may mean the difference between bounty and starvation, spacecraft havehelped planners manage scarce resources and head off potential disasters beforeinsects or other blights could wipe out an entire crop. Weather forecasting is a furtherapplication of remote-sensing technology—one we’ve all come to rely on. Overall, we’vecome to rely more and more on the ability to monitor and map our entire planet. As


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Satellites can now “spy” on crops, ocean currents, and natural resources to aidfarmers, resource managers, and planners on Earth. In countries where the failure ofa harvest may mean the difference between bounty and starvation, spacecraft havehelped planners manage scarce resources and head off potential disasters beforeinsects or other blights could wipe out an entire crop. Weather forecasting is a furtherapplication of remote-sensing technology—one we’ve all come to rely on. Overall, we’vecome to rely more and more on the ability to monitor and map our entire planet. Asthe pressure builds to better manage scarce resources and assess environmentaldamage, we’ll call upon remote-sensing spacecraft to do even more.

Figure 4: Satellite remote sensing. From the vantage point of space, we can planurban development and plot the course of dangerous storms.

Space-based Navigation

Early seafarers looked to the stars to guide their way. Modern seafarers look only asfar as satellites in Earth orbit. Systems such as the Global Positioning System (GPS),developed by the U.S. military, tell you where you are, in what direction you’reheading and how fast you’re going.


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Figure 5: Global Positioning System (GPS). GPS allows Earth-based users armedwith a simple, hand-held receiver to triangulate from a constellation of 24 satellites. Theycan then determine their location to within a few meters and velocity, and a few m/secanywhere on Earth.

Exploration

Perhaps the most exciting missions are those which explore the unknown. Missionssuch as the Magellan spacecraft that orbited Venus with a powerful radar to peel backthe clouds of this once mysterious planet are a good example. A computer-enhancedimage taken by Magellan is shown in Figure 6. These types of missions push back theboundaries of human knowledge, giving us new insight into planetary formation,weather and other important processes at work back here on Earth.


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Figure 6: Magellan at Venus. The powerful synthetic aperture radar on NASA’sMagellan spacecraft pierced the thick clouds of Venus, giving us the first details of theplanet’s surface. (Photo courtesy of NASA.)

Describing Space Missions

Space missions seem complex, and they are to a certain extent, but if you look atthem logically, you’ll see many similarities. Let’s begin with some key definitions:

♦ Mission Objective - Why we’re going to space and what we’re going to do oncewe get there.

♦ Users - The people or systems that use data or services provided by thesatellite or satellites.

♦ Operators -The people who manage and run the mission from the ground.

♦ Mission Operations Concept -How users, operators, ground and spaceelements all work together to make a mission successful.

All these come together to form the tangible elements of what is collectively called theSpace Mission Architecture. These elements are depicted in Figure 7; each one isdefined in the subsections following.


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The Space Mission

Space Operations

Spacecraft

Payload Bus

Payload

Space Transportation

Communications Network

Trajectories & Orbits

Figure 7: Space mission architecture. The key to understanding how missions arebuilt is to look at the space mission architecture that includes these critical elements.

Space Operations

The term space operations encompasses all activities needed to monitor and controlsatellites and the other elements that make up a space mission. Space operations areperformed by teams of people located at tracking sites and control centers around theworld.

Spacecraft Bus and Payload

A spacecraft has two basic parts, a payload (or payloads) and a bus. The payload includesspace-borne people and instruments that perform the primary mission. The spacecraft busprovides for the care and feeding of the payload—pointing, heating and cooling, structure,transportation and power. A simple analogy of a spacecraft bus and its payload is a goodold-fashioned school bus, as shown in Figure 8. It contains all of the same types ofsystems needed to support a spacecraft.


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Body & frame(Structures)Horn, radio & driver

(communications & data handling)

Steering (space vehicle control)

Battery & alternator (electrical power)

Radiator, air conditioning &heater (environment control &life support )

engine & drive train(space transportation)

Passengers(payload)

Figure 8: The spacecraft “bus.” A spacecraft has all the basic systems found in aregular school bus.

Trajectories and Orbits

A trajectory is any path an object follows through space. An orbit is a special type ofrepeating trajectory. The simplest way to imagine an orbit is to think of a “racetrack”around the Earth which satellites “drive” around, as shown in Figure 9.

Figure 9: Orbit racetrack. An orbit can be thought of as a fixed racetrack around aplanet, where the size of the racetrack depends on the velocity of the object in orbit.

Depending on the altitude of the orbit, a satellite has different perspectives on the Earth.The total fraction of the Earth a satellite can “see” using its onboard sensors is known asthe field of view. The projection of this field of view onto the Earth’s surface creates aswath width for the sensor as it sweeps around the Earth on its orbit. These twoparameters are illustrated in Figure 10.


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field of view

swath width

Figure 10: Field of view and swath width. The height of the orbit and the sensor fieldof view dictates the swath width that can be imaged on the ground.

There are a variety of different types of orbits that can be found in a typical space mission,including parking orbits, transfer orbits and final mission orbits. These are illustrated inFigure 11.

final orbitparking orbit

transfer orbit

Figure 11: Orbit types. Different types of orbits include the parking orbit, the transferorbit and the final or mission orbit. A satellite normally begins its life in a temporaryparking orbit. From there, an upper stage rocket is used to boost the satellite onto atransfer orbit. An additional boost places it into the final mission orbit.

Space Transportation

Space transportation includes all of the systems necessary to deliver our spacecraft to itsfinal mission orbit. Normally, this consists of a booster, such as the Space Shuttle or


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Ariane, an upper stage, such as the Inertial Upper Stage (IUS), and onboard thrusters forfinal maneuvers and station keeping. The Space Shuttle, shown in Figure 12, is one typeof complete space transportation system.

Figure 12: The Space Shuttle. Space transportation includes the systems that put thespacecraft in orbit, keep it there, and rotate and move it if necessary. Spacetransportation systems develop the velocity needed to obtain and stay in orbit. Spaceboosters are divided into stages that provide incremental changes in velocity and arethen discarded.

Communications Network

A space mission is more than just rockets and satellites. An entire system of ground andon-orbit assets are needed to track, command and control all aspects of the mission. Thiscommunications network ties together various links needed to deliver bus telemetry andpayload data to operators and users, as shown in Figure 13.


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primary aircraft

Tracking & DataRelay Satellite(TDRS)

relay satellite

tracking site mission control center tracking site/users

Figure 13: Communications network. The communications network is the “glue”that holds the mission together. The network ties together space assets, groundcontrollers and users in a complex web of links that transfers data among the variousmission element nodes.


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EXPLORING SPACE

Long before rockets and interplanetary probes escaped the Earth’s atmosphere,people explored the heavens with just their eyes and imagination. Later, with the aidof telescopes and other instruments, humans continued their struggle to bring orderto the heavens. With order came some understanding and a concept of our place inthe universe. Thousands of years ago, priests of ancient Egypt and Babylon carefullyobserved the heavens to plan religious festivals, to control the planting and harvestingof various crops, and to understand at least partially the realm in which they believedmany of their gods lived. Later, philosophers such as Aristotle and Ptolemy developedcomplex theories to explain and predict the motions of the Sun, Moon, planets andstars.

The theories of Aristotle and Ptolemy dominated the world of astronomy and ourunderstanding of the heavens well into the 1600s. Combining ancient traditions withnew observations and insights, natural philosophers such as Copernicus and Kepleroffered rival explanations from the 1500s onward. Using their models and IsaacNewton’s new tools of physics, astronomers in the 1700s and 1800s made severalstartling discoveries, including two new planets—Uranus and Neptune. Let’s brieflyexplore some of these major contributors to our early understanding of space andorbits.

Copernicus

With the Renaissance and humanism came a new emphasis on the accessibility of theheavens to human thought. Nicolaus Copernicus (1473–1543), a Renaissancehumanist and Catholic clergyman, reordered the universe and enlarged man’shorizons. He placed the Sun at the center of the solar system, as shown in Figure 14,and had the Earth rotate on its axis once a day while revolving about the Sun once ayear.

Copernicus further observed that, with respect to a viewer located on the Earth, theplanets occasionally appear to back up in their orbits as they move against thebackground of the fixed stars. Ptolemy and others resorted to complex combinationsof circles to explain this backward motion of the planets, but Copernicus cleverlyexplained that this motion was simply the effect of the Earth overtaking, and beingovertaken by, the planets as they all revolved about the Sun.

However, Copernicus’ heliocentric system had its drawbacks. He couldn’t prove theEarth moved, and he couldn’t explain why the Earth rotated on its axis whilerevolving about the Sun. He also adhered to the Greek tradition that orbits followuniform circles, so his geometry was complex and somewhat erroneous. In addition,Copernicus wrestled with the problem of parallax—the apparent shift in the positionof bodies when viewed from different distances. If the Earth truly revolved about theSun, critics observed, a viewer stationed on the Earth should see an apparent shift in


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position of a closer star with respect to its more distant neighbors. Because no onesaw this shift, Copernicus’ Sun-centered system was suspect. In response,Copernicus speculated that the stars must be at vast distances from the Earth, butsuch distances were far too great for most people to contemplate at the time, so thisidea was also widely rejected.

Figure 14: Copernicus redefines the center. Polish astronomer Copernicus reorderedour view of the universe. He promoted a heliocentric (Sun-centered) universe, a simpler,more symmetric approach with all of the planets in circular orbits about the Sun.Unfortunately, these ideas were widely rejected because they disputed religiousteachings of his day.

Kepler

Johannes Kepler (1571 - 1630) revolutionized our understanding of orbits. In CosmicMystery, written before he was age 25, he calculated that the orbit of Mars was notcircular but elliptical. From this work, he developed three important laws of orbitmotion, described in Figures 15 through 17.

Figure 15: Kepler’s 1st law. The orbits of the planets are ellipses with the Sun at afocus.


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Planetarymotion over 30 days

Planetarymotion over30 days

Area 1 = Area 2

Area 1 Area 2

Figure 16: Kepler’s 2nd law. The orbits of the planets sweep out equal areas in equaltime.

Average distance

Figure 17: Kepler’s 3rd law. The square of the orbit period—the time it takes to goaround once—is proportional to the cube of the average distance to the Sun.

Galileo

In 1609, an innovative mathematician, Galileo Galilei (1564–1642), heard of a newoptical device that could magnify objects so they would appear to be closer andbrighter than when seen with the naked eye. Building a telescope that could magnifyan image 20 times, Galileo ushered in a new era of space exploration. He made somestartling telescopic observations of the Moon, the planets, and the stars, therebyattaining stardom in the eyes of his peers and potential patrons. Observing theplanets, Galileo noticed that Jupiter had four moons or satellites (a word coined byKepler in 1611) that moved about it. This disproved Aristotle’s claim that everythingrevolved about the Earth.


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Galileo also took on Aristotle’s physics. He rolled a sphere down a grooved ramp andused a water clock to measure the time it took to reach the bottom. He repeated theexperiment with heavier and lighter spheres, as well as steeper and shallower ramps,and cleverly extended his results to objects in free-fall. Through these experiments,Galileo discovered, contrary to Aristotle, that all objects fall at the same rateregardless of their weight, as shown in Figure 18.

Galileo further contradicted Aristotle as to why objects, once in motion, tend to keepgoing. Aristotle held that objects in “violent” motion, such as arrows shot from bows,keep going only as long as something is physically in touch with them, pushing themonward. Once this push dies out, they resume their natural motion and drop straightto Earth. Galileo showed that objects in uniform motion keep going unless disturbedby some outside influence. He wrongly held that this uniform motion was circular,and he never used the term “inertia.” Nevertheless, we applaud Galileo today forgreatly refining the concept of inertia as we know it today.

Figure 18: Galileo on gravity. Through application of the scientific method, Galileo putAristotle’s ideas to the test and proved Aristotle wrong—all objects fall at the same rate.

Newton

To complete the astronomical revolution, which Copernicus had almost unwittinglystarted and which Kepler and Galileo had advanced, the terrestrial and heavenlyrealms had to be united under one set of natural laws. Isaac Newton (1642–1727)answered this challenge. Newton was a brilliant natural philosopher andmathematician who provided a majestic vision of nature’s unity and simplicity. 1665proved to be Newton’s “miracle year,” in which he significantly advanced the study ofcalculus, gravitation, and optics. Extending the groundbreaking work of Galileo indynamics, Newton published his three laws of motion and the law of universalgravitation in the Principia in 1687. With these laws, you could explain and predictmotion not only on Earth but also in tides, comets, moons, planets—in other words,motion everywhere. Newton’s laws are explained more thoroughly in the next section.


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INTRODUCTION TO ORBITAL MOTION

Orbits are one of the basic elements of any space mission. Understanding a satellitein motion may at first seem rather intimidating. After all, to fully describe orbitalmotion we need some basic physics along with a healthy dose of calculus andgeometry. However, as we’ll see, the complex trajectories of rockets flying into spacearen’t all that different from the paths of baseballs pitched across home plate. In fact,in most cases, both can be described in terms of the single force pinning you to yourchair right now—gravity.

Armed only with an understanding of this single pervasive force, we can predict,explain and understand the motion of nearly all objects in space, from baseballs toentire galaxies. Once we know an object’s position and velocity, as well as the natureof the local gravitational field, we can predict exactly where the object will be minutes,hours or even years from now.

Overview

Math for the Faint-hearted

♦ What is a vector?

♦ What is a derivative?

♦ What is an integral good for?

Key Concepts in Dynamics

♦ What is the difference between mass, inertia and weight?

♦ What is momentum?

♦ What is energy?

♦ What are Newton’s Laws of Motion?

♦ What is gravity?

Orbits Made Simple

♦ OK, forget the math, what is an orbit, really?

♦ How are energy and momentum conserved in an orbit?

Math for the Faint-hearted

Before delving too deeply into a discussion of dynamics, orbital mechanics andpropagators it’s useful to step back and briefly review a bit of math. BUT DON’TPANIC! This section is designed to simplify a few basic concepts and describe the


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notation used. You should walk away from this section with a “Math Survival Kit”that will make the rest of this Primer far more useful and enjoyable.

Vectors and Such

Scalar

A scalar is a quantity that has magnitude only. Speed, energy and temperatureare examples of scalars. None of these quantities has a unique meaning in anycertain direction. A single letter, such as E for Total Mechanical Energy, denotes ascalar quantity.

Vector

A vector is a quantity that has both magnitude and direction. For example, if I askyou where you drove in your car, you might answer: “I went south.” But thiswouldn’t tell me much. If I asked “How far?,” and you said “five miles,” I could puttogether a better picture. By knowing you drove five miles south, I have bothmagnitude and direction. A letter with an arrow over it, such as

vV for the velocity

vector is used to denote a vector quantity.

Unit Vector

A unit vector is a vector having a magnitude of one; it’s used to describe directiononly. For example, if I want to define where north is on a drawing, I could do sowith a unit vector indicating the direction. A letter with a caret or hat over it, such

as $I for the I-direction denotes a unit vector.

Example

We generally describe the velocity of an object in orbit in terms of three unit

vectors, $, $, $I J K . Thus, a typical velocity vector could be written as:

vV I J K= + −30 21 7 4. $ . $ . $ km / sec

This means the velocity is 3.0 km/sec in the I-direction, 2.1 km/sec in the J-direction and 7.4 km/sec in the K-direction.

Calculus—Just the important bits

Calculus was developed to analyze changing parameters. Let’s look at how thosechanges are described.


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Derivative

A derivative represents the rate of change of one parameter with respect toanother. For example, if you’re traveling north in your car, your position ischanging over time. The rate at which your position changes over time is yourvelocity. Thus, if you travel 30 miles north in 30 minutes, your velocity is:

Velocity VChange in

Change in = = = =

vv

position (R)time (t)

30 miles30 minutes

60 mph north

Several methods are commonly used to denote a vector. In this primer, we use twotypes of notation. The first is to represent a derivative as d. So the change inposition over time would be:

vv

VdRdt

=

We also use a “dot” over a symbol to represent the derivative with respect to time. For

example, v&R represents the derivative of the position vector or the rate of change of

position. v&&R is the second derivative of position, or the rate of change of the rate of change

of position (i.e. the acceleration).

Integral

An integral represents the cumulative effect of one parameter changing withrespect to another. If we were to graph both changing parameters, the integral isthe area under the curve. For example, if you drive north at 30 miles per hour for30 minutes, the integral of this velocity is your new position at the end of the time(e.g., 15 miles north of where you started). In other words, you add up all thechanges in position over time to get the total change. An integral is the reverse ofa derivative. Because acceleration is the derivative of velocity over time, theintegral of acceleration over time is velocity.

Key Concepts

Mass

Mass is a measure of how much matter or “stuff” an object possesses. For example, avolleyball and a cannon ball are about the same size, but the cannon ball has farmore mass because it is made of a more dense material.

Inertia

Inertia describes how hard it is to move an object. It is much easier to push a babycarriage than a bulldozer because the bulldozer, being more massive, has moreinertia.


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Weight

Weight actually describes the force produced by gravity acting on a mass. Your weightin various situations is illustrated in Figure 19.

Figure 19: Weight. Weight describes a contact force caused by the effect of gravity onmass. On Earth, your weight is one value. As you move further from the center of theEarth, say in a penthouse at the top of a 250-mile-high skyscraper, your weight would beslightly less. In orbit at 250 miles altitude, the gravitational force is still the same;however, because you are in free-fall and not in contact with the Earth, your weight iszero. In all cases, mass stays the same.

Momentum

Linear momentum describes the resistance a moving object has to changes in eitherdirection or speed. The more massive an object, or the faster it is moving, the harderit is to stop or change its direction of motion. As a result, linear momentum is theproduct of the mass and velocity of an object. Momentum for baby carriages andbulldozers is shown in Figure 20.

m = 25 kgv= 1000 m/smv = 25,000 kg-m/s

m = 25,000 kgv = 1 m/smv= 25,000 kg-m/s

Figure 20: Momentum, bulldozers and baby carriages. Linear momentum is theproduct of mass and velocity. For a baby carriage to have the same linear momentum asa bulldozer, it would have to be traveling at a much higher velocity.


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Angular momentum is a measure of the spinning properties of an object. As Figure 21illustrates, a non-spinning top immediately falls over. However, a spinning top hasangular momentum, which allows it to resist the force of gravity pulling it over (until itfinally slows down due to friction).

non-spinning top spinning top

angular momentum

Figure 21: Angular momentum. A spinning top has angular momentum which keeps itpointing upright even when pulled by outside forces such as gravity.

Energy

Potential energy is a function of an object’s position and mass. The greater the heightan object is raised to, the greater its potential energy. An object at the top of a deepwell, as shown in Figure 22, has more potential energy than an object at the bottom ofthe well.

PE = 0 at R = infinite

PE < 0 at R > 0

PE < 0 at R = 0

R

Figure 22: Potential energy. Because we normally define our coordinate systems aspositive outward from the center of the Earth, we measure potential energy “from thebottom up.” For example, at the top of a deep well we would say the potential energy iszero. As we get closer to the bottom of the well, the potential energy is less, or morenegative.

Kinetic energy is a function of an object’s mass and speed. Like momentum, the moremassive an object is or the faster it travels, the more kinetic energy it has. It is this


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energy that must be transformed in order to stop a speeding bulldozer, as shown inFigure 23. This is accomplished by applying the brakes, which turns kinetic energyinto heat (those brakes get hot!).

m = 25,000 kgv = 1 m/s1/2 mv2 = 12,500 kg-m/s2

Figure 23: Newton’s 1st law. The kinetic energy of a bulldozer moving at only 1 m/s is12,500 kg-m/s2.

Total energy is the sum of kinetic plus potential energy.

Total Energy = Kinetic Energy + Potential Energy

Newton’s Laws

Newton’s First Law

A body remains at rest or in constant motion unless acted upon by external forces.

In other words, if you were to pitch a baseball, it should continue on its path, in astraight line forever, unless disturbed by an outside force such as gravity or airresistance.

Newton’s Second Law

The time rate of change of an object’s momentum is equal to the applied force.

Change momentumChange time

( )

( )= Force Applied

Recall, momentum is the product of mass and velocity. Thus, as long as massstays constant (which it normally does as long as rockets aren’t firing) thisequation can be reduced to:

F ma=

or:

Force applied = mass (m ) times acceleration (a)


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The significance of this relationship can be felt every time you hit the brakes in you car.The more force you apply (the harder you hit the brakes) the faster you stop (the fasteryou decelerate). This principle is illustrated in Figure 24.

stops in 1second

25,000 N

1 m/s

1 m/s6.9 N

stops in 1 hour

Figure 24: Newton’s 2nd law. Force is proportional to acceleration (or deceleration). A25,000 N force is needed to stop a 1 m/s bulldozer in 1 second, while much smaller 6.9N force would take 1 hour to bring it to a stop.

Newton’s Third Law

For every action, there’s an equal and opposite reaction. This basic principle canbe illustrated by two roller-skating astronautics, as shown in Figure 25.

Figure 25: Newton’s 3rd law. If two people on roller skates push against each other,they both move backward. Their acceleration is proportional to their mass.


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Newton’s Law of Universal Gravitation

The force of gravity between two bodies is proportional to the product of themasses and inversely proportional to the square of the distance between them.This is illustrated in Figure 26.

R

FF

m

m

1

2g

g

Figure 26: Gravitational attraction. Two masses in space each exert a force on theother. The magnitude of this force depends on the product of their masses and thesquare of the distance between them.

Newton’s law can be summarized in equation form as follows:

FGM M

Rgravity = 1 2

2

where

Fgravity = Force of gravity (N)

G = Universal gravitational constant = 6.67 x 1011 (Nm/kg)

M1, M2 = Mass 1 and Mass 2 respectively (kg)

R = Distance between the two masses (m)

In other words, the more mass an object has, the more gravitational force itgenerates. Furthermore, the farther apart two objects are, the less the force is, infact, the force decreases with the square of the distance as illustrated in Figure 27.


27

R

FF

m

m

1

2g

g

2R

m

m

1

2

F /4g

gF /4

Figure 27: Gravity and distance. The force of gravitational attraction decreases withthe square of the distance (e.g., if you double the distance, the force decreases by onefourth).

What exactly is gravity? The study of physics is still grappling to reconcile theforce of gravity with the other fundamental forces of nature. Already, extremelyweak “gravity waves” have been detected from distant galaxies. More sensitiveinstruments are being built to understand and quantify this mysterious force.

How strong is gravity? Let’s look at the Earth-Moon system. The force of gravitybetween the Earth and Moon is 1.98 x 1020 Newtons! To put this into perspective,the Space Shuttle generates about 28 million Newtons thrust at lift-off. The Earth-Moon gravity force is more than one trillion times as great as that of the Shuttle!

Regardless of what gravity really is, we know it’s a force that affects anything withmass (and that’s pretty much everything!). While Galileo right in that thegravitational force is greater on heavier objects than lighter objects, he was wrongin predicting the affect this would have on the rate at which they fall. Theacceleration of an object in a gravitational field is independent of its mass.

Figure 28: All things fall at the same rate. It is important to note that all thingsaccelerate at the same rate within a gravitational field. For example, if you drop a


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hammer and a feather, both objects impact the ground at the same time (neglecting airresistance). Of course, Galileo predicted this. Astronaut Dave Scott proved this with anexperiment on the Moon. He dropped a hammer and a feather at the same time. Both hitthe ground at the same instant (there is no air resistance on the Moon)!

Orbits Made Simple

What is an orbit? In the simplest sense, orbits are a type of “racetrack” in space that asatellite “drives” around.

Figure 29: Orbits as racetracks. The simplest way to think of orbits is as giant, fixed“racetracks” on which spacecraft “drive” around the Earth.

Baseballs and Satellites

But what makes these racetracks? Before diving into a complicated explanation, let’sbegin with a simple experiment that illustrates, conceptually, how orbits work. To dothis, we’ll arm ourselves with a bunch of baseballs and travel to the top of a tallmountain. Imagine you were standing on top of this mountain prepared to pitchbaseballs to the east. As the balls sail off the summit, what would you see? Thebaseballs would follow a curved path before hitting the ground. Why is this? The forceof your throw causes them to move outward, but the force of gravity pulls them down.Therefore, the “compromise” shape of the baseball’s path is a curve.


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Figure 30: Baseballs in motion. A baseball-throwing astronaut can be used to illustratethe simple motion of a satellite. Naturally, the harder the baseballs are thrown, thefurther they travel before hitting the ground.

As Figure 30 illustrates, the faster you throw the balls, the farther they travel beforehitting the ground. This could lead you to conclude that the faster you throw them,the longer it takes before they hit the ground. But is this really the case? Let’s tryanother experiment to see. As you watch, two astronauts, standing on flat ground,release baseballs. The first one simply drops a ball from a fixed height. At exactly thesame time, a second astronaut throws an identical ball horizontally as hard aspossible. What do you see? If the second astronaut throws a fast ball, it travels outabout 20 m (60 ft.) or so before it hits the ground. However, the ball dropped by thefirst astronaut hits the ground at exactly the same time as the pitched ball, as Figure31 shows!

Figure 31: Motion and gravity. Two astronauts each have a baseball held at the sameheight above the ground. If the first astronaut drops her baseball while the secondastronaut throws his, both baseballs hit the ground at exactly the same time. Gravity actson both baseballs in the same way, independent of their horizontal motion.


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How can this be? To understand this seeming paradox, we must recognize that, inthis case, the motion in one direction is independent of motion in another. Thus,while the second astronaut’s ball moves horizontally at 30 km/hr (20 m.p.h.) or so,it’s still falling at the same rate as the first ball. This rate is the constant gravitational

acceleration of all objects near the Earth’s surface: 9.798 m/s2. Thus, they hit theground at the same time. The only difference is that the pitched ball, because it hashorizontal velocity, manages to travel some distance before intercepting the ground.

Now let’s return to the top of our mountain and start throwing our baseballs fasterand faster to see what happens. No matter how fast we throw them, the balls still fallat the same rate. However, as we increase their horizontal velocity, they’re able totravel farther and farther before they hit the ground. Because the Earth is basicallyspherical in shape, something interesting happens. The Earth’s spherical shapecauses the surface to drop approximately 5 m for every 8 km we travel horizontallyacross it, as shown in Figure 32.

8 km

5 m

Figure 32: Our spherical Earth. We know the Earth is a nearly perfect sphere. Forevery 8 km of horizontal distance, the Earth curves down about 5 m. In other words, ifyou could lay an 8 km long board tangent to the Earth at one end, at the other end itwould be 5 m off the ground.

So, if we were able to throw a baseball at 8 km/s (assuming no air resistance), itspath would exactly match the rate of curvature of the Earth. That is, gravity wouldpull it down about 5 m for every 8 km it travels, and it would continue around theEarth at a constant height. If we don’t remember to duck, it will hit us in the back ofthe head about 85 minutes later. (Actually, because of the rotation of the Earth, itwould miss your head.) A ball thrown at a speed slower than 8 km/s falls faster thanthe Earth curves away beneath it.

If we analyze our various baseball trajectories, we see a range of different shapes.Only at exactly one particular velocity do we get a circular trajectory. Any slower thanthat and our trajectory impacts the Earth at some point. If we were to project thisshape into the Earth, we’d find the trajectory we see is really just a piece of an ellipse.If we throw the ball a bit harder than the circular velocity, we also obtain an ellipse.An object in orbit is literally falling around the Earth but, because of its horizontalvelocity, it never quite impacts the ground. If we throw the ball too hard, it leaves theEarth altogether on a parabolic or hyperbolic trajectory, never to return.


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circle

hyperbola

parabola

ellipse

Figure 33: Baseballs in orbit . If we throw a baseball fast, but not quite fast enough,eventually the ball will impact the ground (perhaps even on the other side of the Earth)like an ICBM (Intercontinental Ballistic Missile). If we throw it at just the right speed,gravity will cause the ball to fall 5 m for every 8 km of horizontal distance traveled. But,since the Earth also curves down 5 m for each 8 km of horizontal distance, the ball willstay at same the instantaneous height above the ground. We call this a circular orbit . Ifwe throw it faster than the circular orbit speed, the ball will be in an elliptical-shaped orbit.If we throw the ball faster yet, it will escape the Earth’s gravity altogether on a parabolic orhyperbolic trajectory.

Thus, it is important to note that no matter how hard we throw, our trajectoryresembles either a circle, ellipse, parabola or hyperbola. These four shapes are calledconic sections. Why conic sections? Because these are the shapes we get by slicingthrough a cone with a plane at different angles, as illustrated in Figure 34.

circle parabolahyperbolaellipse

Figure 34: Conic sections . The four basic conic sections: circle, ellipse, hyperbola andparabola.


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So how fast how fast do we have to throw our baseball to put it into a circular orbit?Let’s play with some math. The velocity of a satellite in a circular orbit can be foundusing:

VGM

RcircularEarth=

where

Vcircular = Satellite velocity in a circular orbit (km/sec)

G = Universal gravitational constant = 6.67 x 10-11 km2/sec3

MEarth = mass of the Earth = 5.98 x 1015 kg

R = distance from the center of the Earth = 6378 km at the surface

Thus, at the surface of the Earth, the velocity in a circular orbit would be 7.9 km/sec(17,600 mph)! In other words, to move into a circular orbit that stays just above thesurface of the Earth (ignoring air drag) you’d have to throw the baseball at 17,600mph. Notice that the circular orbit speed depends on your distance from the center ofthe Earth. The lower you are, the faster you must travel to achieve a circular orbit.

Conservation of Energy & Momentum

Now that we’ve looked at the simple geometry of an orbit, we can consider howconservation of energy and momentum affects the velocity of satellites. Gravity is aconservation force, which means that an object moving in a gravitational field doesn’tlose any energy through friction or heat, etc. Additionally, total energy is constant, or:

Total Energy = Kinetic (KE) + Potential Energy (PE) = constant

This basic principle is illustrated by the swinging astronaut in Figure 35.


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Maximum PEKE=0

Maximum PEKE=0

Max KEMin PE

PE+KE=constant

Figure 35: Trading kinetic and potential energy. The conservation of energy(Potential Energy + Kinetic Energy = Constant) is illustrated by a simple swing.Neglecting losses from friction, the total energy of the astronaut on the swing is constant.At the low point in the swing, speed (kinetic energy) is greatest and potential energy islowest. As you swing, you trade kinetic energy (speed) for potential energy (height) withthe sum of the two constant.

When applied to an orbit, the same principle applies. Total energy must be conserved,thus the orbit speed varies throughout the orbit as kinetic energy is traded forpotential energy. A satellite travels fastest at perigee—the lowest point in the orbit—and slowest at apogee—the highest point in the orbit. This is shown in Figure 36.

low kineticenergy

apogee

high potentialenergy

low potentialenergy

high kineticenergy

perigee

Figure 36: Energy conservation in orbit. As a satellite moves closer to the Earth in anorbit, it must speed up to conserve total energy. As it gets further away, the satellitetrades kinetic energy for potential energy and slows down.

Angular momentum is also always conserved. Ice skaters use this principle to speedup or slow down as they spin, as shown in Figure 37. Just like a spinning ice skater,an orbit has angular momentum. Because angular momentum is a vector quantity,


34

the direction as well as the magnitude of this momentum stays the same. As a result,even though the Earth rotates under the orbit and the Earth (and the orbit along withit) moves around the Sun, the orbit itself stays fixed in respect to a stationaryreference.

Figure 37: Conservation of angular momentum. Ice skaters use this principle tospeed up or slow down as they spin. When their arms are extended, the moment ofinertia is low, so they spin more slowly. As they draw their arms in, the moment of inertiadecreases and the spin rate increases to keep total angular momentum constant.


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DESCRIBING ORBITS

Overview

Understanding Coordinate Systems

♦ What is a coordinate system?

The Geocentric-Equatorial Coordinate System

♦ What is the most common coordinate system used for satellites?

Classical Orbital Elements

♦ What do all those Greek letters tell me about an orbit?

Ground Tracks

♦ How do those squiggly lines on a map represent the path of a satellite?

Understanding Coordinate Systems

To be valid, Newton’s laws must be expressed in an inertial frame of reference,meaning a frame that is not accelerating. Any reference frame is just a collection ofdefinitions that allow us to describe positions and velocities in a more meaningfulway. For example, if I simply told you a car is traveling south, you wouldn’t have verymuch information. But if I first tell you that our a reference frame is centered onWashington, DC, and then tell you that the car is 30 miles east of the city travelingsouth at 60 mph, you’d know something far more useful.

In defining coordinate systems and describing position and velocities, we makeextensive use of vectors. We use vectors because we want to keep track of theinformation contained in a particular parameter. Specifically, a vector is a parametershaving both magnitude and direction. For example, 60 mph tells you speed(magnitude) without direction. But 60 mph south tells you both magnitude anddirection. In defining coordinate systems, we are sometimes only interested indirection. In that case, we use unit vectors defined to have a specific direction and amagnitude of 1.

Cartesian coordinate systems are laid out with three orthogonal unit vectors—vectorsat right angles to each other. For example, if the origin of a coordinate system werechosen to be in one corner of a room, the floor would be the fundamental plane. Wecould then describe the position of every piece of furniture in the room with respect tothis system. Such a collection of unit vectors allows us to establish the components ofother vectors in 3-D space.


36

To create a coordinate system, we need to specify four pieces of information—anorigin, a fundamental plane, a principle direction, and a third axis, as shown inFigure 38. The origin defines a physically identifiable starting point for the coordinatesystem. The other two parameters fix the orientation of the frame. The fundamentalplane contains two axes of the system. Once we know the plane, we can define adirection perpendicular to that plane. The unit vector in this direction at the origin isone axis. Next, we need a principle direction within the plane. Again, we choosesomething that is physically significant, like a star. Now that we have two directions,the principle direction and an axis perpendicular to the fundamental plane, we canfind the third axis using the right-hand rule. The right-hand rule can be demonstratedby pointing the thumb of your right hand in the direction perpendicular to thefundamental plane. With you fingers pointing in the principle direction, curl yourfingers 90° so that your thumb is pointing in the direction of the third axis.

(1) pick origin

origin

(2) pick fundamental plane& perpendicular to it

origin

fundamentalplane

(3) pick principal direction

fundamentalplane

fundamentalplane

originorigin

principal direction principal direction

3rd axis, foundusing right-handrule

(4) find 3rd axis

Figure 38: Defining a coordinate system.

Remember—coordinate systems are defined to make our lives easier. If we choose thecorrect coordinate system, developing the equations of motion can be simple. If wechoose the wrong system, it can be nearly impossible.

The Geocentric-Equatorial Coordinate System

For Earth-orbiting spacecraft, we’ll choose a tried-and-true system that we knowmakes solving the equations of motion relatively easy. We call this system thegeocentric-equatorial coordinate system, shown in Figure 39. Here’s how it’s defined:


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Origin

The center of the Earth (hence the name geocentric).

Fundamental plane

Earth’s equator (hence geocentric-equatorial), where perpendicular to the fundamentalplane is the direction of the north pole.

Principle direction

Vernal equinox direction, , or the vector pointing to the first point of Aries. Thevernal equinox direction points at the zodiac constellation Aries and is found bydrawing a line from the Earth to the Sun on the first day of spring. While thisdirection may not seem “convenient” to you, it’s significant to the astronomers whooriginally defined the system. Unfortunately, the vernal equinox direction is notperfectly constant. The Earth’s orbit precesses around the Sun and the Sun is movingthrough the galaxy. Therefore, exactly when this direction is defined is extremelyimportant for the definition of the system. We can use two ways of defining thesedirections. The first is to use the mean or average direction at some point in time. Theother is to use the true position at exactly one specific point in time. Variouscombinations of these definitions using different dates gives us several possibilities forcoordinate systems used in STK:

J2000

Defines the mean vernal equinox direction and mean Earth rotation axis onJanuary 1 of the year 2000 at approximately 12:00:00.00 GMT.

Mean of date

Defines the mean vernal equinox direction and mean Earth rotation axis at theorbit epoch time (the time for which the orbital elements being used is true).

Mean of epoch

Defines the mean vernal equinox direction and mean Earth rotation axis at thecoordinate epoch time (time at which the coordinate system being used is defined).

True of date

Defines the true vernal equinox direction and true Earth rotation axis at exactlythe orbit epoch time specified.


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True of epoch

Defines the true vernal equinox direction and true Earth rotation axis at thecoordinate epoch time specified.

B1950

Defines the mean vernal equinox direction and mean Earth rotation axis at thebeginning of the Besselian year 1950. It corresponds to 31 December 1949 at22:09:07.20 Greenwich Mean Time (GMT).

Mean Equinox, True Equator

Defines mean vernal equinox direction and true Earth rotation axis for the orbitepoch time specified.

Third axis

The third axis of the geocentric-equatorial coordinate system is found using the right-hand rule.

I

K

J

Figure 39: The geocentric-equatorial coordinate system. The system is defined by:u Origin - Center of Earth; u Fundamental Plane - Equals equatorial plane;u Perpendicular to Plane - north pole; u Principle Direction - vernal equinox direction.

Classical Orbital Elements

Three pieces of information are needed to fix any point in space; collectively, they’reknown as an object’s position vector,

rR . Three more pieces of information describe its

velocity vector, r

V . One additional item, time, tells us when the information providedis valid. These elements are know as Cartesian elements. While it is often convenientto describe orbit motion using simply position and velocity vectors in a Cartesiancoordinate system, especially for computational work, these vectors provide us littleinsight into the orbit itself. For this reason, astronomers long ago developed orbital


39

elements. Orbital elements give us a short-hand way of expressing orbit size, shapeand orientation, allowing us to tell at a glance the application for a given orbit. Thissection describes the Classical orbital elements, which are sometimes referred to asthe Keplerian elements and are attributed originally to Kepler himself. Variations onthese elements, the commonly used two-line element sets, are described in a latersection.

Orbit Size

How big is an orbit? This depends on how fast we “throw” our satellite into orbit. Thefaster we throw, the more energy an orbit has, and the bigger it is. We express the sizeof an orbit in terms of its semimajor axis, a., as defined in Figure 40.

apogee perigee

2a = major axis

a = semimajor axis

Figure 40: Semimajor axis. The major axis of an elliptical orbit is the distance betweenthe point of closest approach (perigee) and furthest point (apogee). Semimajor axis isone-half this distance.

We can express the semimajor axis in terms of the distance from the center of theEarth to apogee (Rapogee) and perigee (Rperigee). Perigee is the point in an orbit that isclosest to the Earth. Apogee is the point where it is furthest away (apogee is undefinedfor a parabolic or hyperbolic trajectory). The semimajor axis can be found using:

aR Rapogee perigee

=+

2

where

a = semimajor axis (km)

Rapogee = Distance from center of Earth to apogee (km)


40

Rperigee = Distance from center of Earth to perigee (km)

The orbit’s period, P (i.e., how long the satellite takes to travel around the orbit onetime), is proportional to the orbit size:

Pa

GM Earth= 2

3π

For example, a typical Space Shuttle orbit at an altitude of a few hundred kilometershas a period of about 90 minutes. It orbits the Earth about 16 times each day! Forcommunications satellites in geosynchronous orbit at an altitude of 35,780 km, theperiod is exactly 24 hours.

Orbit Shape

The less circular an orbit is, the more eccentric or “imperfect” it is. Eccentricity, e,describes the shape of orbit with respect to that of a circle.

0<e<1 e = 0

ellipse

parabola

hyperbolae = 1

circle

e>1

Figure 41: Eccentricity. A perfectly circular orbit has an eccentricity of 0. Ellipticalorbits have an eccentricity of less than 1. A parabolic orbit has an eccentricity of exactly1. Hyperbolic orbits (or trajectories) have eccentricities of greater than 1. In practice, aperfectly circular or parabolic orbit cannot be achieved.


41

Orbit Orientation

How the orbit is tilted with respect to the equator is called its inclination, i. An orbitthat stays directly over the equator has an inclination of 0° and is called an equatorialorbit. An orbit that goes directly over the north and south poles must have aninclination of exactly 90° and is called a polar orbit.

Different classes of orbits have different inclinations, as shown in the table.

InclinationInclination Orbit TypeOrbit Type DiagramDiagram

0° = i = 180° Equatorial i=0o

i = 90° Polar i=90o

0° < i < 90° Direct or posigrade(moves in direction ofEarth’s rotation)

ascendingnode

90° < i <180° Indirect or retrograde(moves against thedirection of Earth’srotation) ascending

node

We measure how an orbit is twisted by locating its ascending node, the point wherethe satellite crosses the equator moving south to north. This point is referenced to theI-direction, which points in the Vernal Equinox direction. The angle between the I-direction and the ascending node is called the right ascension of the ascending node,RAAN, Ω.


42

ascendingnode

equatorial plane

ΩΩ

i

J^

K

Figure 42: Right Ascension of the ascending node (RAAN), ΩΩ, is the angulardistance from the vernal equinox direction to the ascending node. The ascending nodeof an orbit is the point where it crosses the equatorial plane from south to north.

We describe the orbit’s orientation by locating perigee with respect to the ascendingnode. This angle is called the argument of perigee, ω; it is measured positive in thedirection of satellite motion.


43

ascendingnode

equatorial plane

i

J^

K

ωω

Figure 43: Argument of perigee. The argument of perigee, ω, is the angular distancebetween the ascending node and perigee.

Finally, we describe a satellite’s instantaneous position with respect to perigee usinganother angle, true anomaly, ν. It is the angle, measured positive in the direction ofmotion, between perigee and the satellite’s position. Of the six orbital elements, onlytrue anomaly changes continually (ignoring perturbations).

perigee

υυ

R

V

Figure 44: True anomaly. The true anomaly, ν, is the angular distance from perigee to

the orbit position vector, rR .


44

Summary of classical orbital elements

Recall, the reason we wanted to develop the orbital elements in the first place was togive us a short-hand method of describing an orbit. We also wanted to use parametersthat would have some physical meaning we could more easily visualize. The sixclassical orbital elements are summarized below.

NameName SymbolSymbol DescribesDescribes

Semimajor axis a Size (and energy)

Eccentricity e Shape (e = 0 for circle, 0> e >1 forellipse, e = 1 for parabola, e > 1 forhyperbola)

Inclination i Tilt of orbit plane with respect to theequator

Longitude of ascending node Ω Twist of orbit with respect to theascending node location

Argument of perigee ω Location of perigee with respect to theascending node

True anomaly ν Location of satellite with respect toperigee

Satellite Missions

As we already know, varying missions require different orbits, which can be describedusing Classical orbit elements. The table following shows various missions and theirtypical orbits. Technically speaking, a geostationary orbit is a circular orbit with aperiod of exactly 24 hours and an inclination of exactly 0°. A satellite in ageostationary orbit appears to be stationary to an Earth-based observer.Geosynchronous orbits are slightly inclined orbits with a period of 24 hours. Inpractice, it is almost impossible to achieve an orbit with exactly a 24-hour period andan inclination of 0°. Thus, the two terms are frequently used interchangeably. A semi-synchronous orbit has a period of 12 hours. Sun-synchronous orbits are retrogradelow-Earth orbits (LEO) inclined 95° to 105°; they are typically used in remote-sensingmissions to observe Earth. A Molniya orbit is a semi-synchronous, eccentric orbit usedfor some communication missions. Super-synchronous orbits are usually circularorbits with periods longer than 24 hours.

MissionMission Orbit TypeOrbit Type SemimajorSemimajorAxis (Altitude)Axis (Altitude)

PeriodPeriod InclinatioInclinationn

OtherOther

♦ Communications

♦ Early Warning

♦ Nuclear detection

Geostationary 42,158 km

(35,780 km)

24hours

~0° e ≅ 0

♦ Remote sensing Sun-synchronous

~6500-7300 km

(~150-900 km)

~90 min ~95° e ≅ 0


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MissionMission Orbit TypeOrbit Type SemimajorSemimajorAxis (Altitude)Axis (Altitude)

PeriodPeriod InclinatioInclinationn

OtherOther

♦ Navigation (GPS) Semi-synchronous

26,610 km

(20,232 km)

12hours

55° e ≅ 0

♦ Space Shuttle Low-Earthobit

~6700 km(~300 km)

~90 min 28.5° or 57° e ≅ 0

♦ Communication/Intelligence

Molniya 26,571 km(RP = 7971 km);(RA = 45,170 km)

12hours

63.4° ω = 270°

e = 0.7

Ground Tracks

Orbital elements allow us to visualize the shape of an orbit around the Earth. Becausewe use satellites for missions involving specific points on Earth—taking pictures,communications, navigation—we really would like to know what path the satellitetraces over the Earth’s surface. A satellite ground track is the orbit path (usually formultiple orbits) projected onto a flat map of the Earth. These projections becomecomplex because we must account for the satellite circumnavigating the entire Earthduring each orbit while the Earth itself rotates at 1600 km/sec underneath it.

To visualize a satellite’s ground track, let’s begin by assuming the Earth doesn’trotate. Picture an orbit around this nonrotating Earth. Because the orbit plane mustpass through the Earth’s center, the ground track traces a great circle. By definition,a great circle is any circle on a sphere that can be projected through the center. Forexample, all lines of longitude are great circles. The equator is the only line of latitudethat is a great circle—No other line of latitude “slices through” the center of the Earth.

When the Earth is stretched out to a flat-map projection (called an equidistantcylindrical projection), things start to look different. Imagine yourself on the ground,watching the orbit pass by overhead. If the Earth didn’t rotate, the projection of theground track would always look the same—a sine wave over the surface of the Earth,.(If you have trouble picturing why it is a sine wave, roll a piece of paper around asoda can and draw an inclined circle around the can. When you unroll the paper,you’ll see a sine wave just like an orbit ground track!)


46

XX XX

Figure 45: Ground track for a nonrotating Earth. If the Earth didn’t rotate, the groundtrack would always look like a constant sine wave. As the figure shows, the ground trackwould always have the same relative orientation with respect to a stationary observer onthe Earth (shown here as an X in the Pacific ocean).

Now let’s start the Earth rotating again. As you watch the orbit pass overhead,something happens from one orbit to the next—the ground track shifts to the west!What happened? The orbit plane is fixed in inertial space. This means the orbit staysthe same with respect to a stationary observer. However, because the Earth rotates at15° per hour, an observer on the Earth is not stationary. As the Earth (and an Earth-fixed observer) rotates to the east, the satellite ground track shifts to the west fromone orbit to the next, as shown in Figure 46. The amount it shifts depends on itsperiod. The longer the period, the more time the Earth has to rotate betweensuccessive orbits.

XXXX

Figure 46: Satellite ground tracks. Satellite orbits are fixed in space with respect to astationary observer. However, a stationary observer on the Earth is rotating to the east at15° per hour. Thus, each successive orbit ground track shifts to the west.

Let’s look at the ground track of some very different orbits.


47

E

D

C B A

Figure 47: Ground tracks for orbits with different periods. Orbit A: Period=2.67 hr,Orbit B: Period=8 hr, Orbit C: Period=18 hr, Orbit D: Period=24 hr, Orbit E: Period=24hr.


48

PREDICTING ORBITS

One of the most important problems in mission planning and satellite command &control is being able to accurately predict orbital motion. To track satellites throughspace, we need to know where they are now and where they’ll be later so that we canpredict sensor coverage and point our antennas at them to gather data. Although wecan easily predict this motion when the orbit is a circle, the problem becomes morecomplicated when the orbit is an ellipse, and most orbits are at least slightly elliptical.

An orbit propagator is a mathematical algorithm for predicting the future position andvelocity (or orbital elements) of an orbit given some initial conditions andassumptions. There are a wide variety of orbit propagation techniques available withwidely different accuracy and applications. Knowing the assumptions built intodifferent propagation schemes is key to knowing which one to use for a givenapplication.

Overview

Understanding Propagators

♦ How do propagators work?

The Two-Body Propagator

♦ What is meant by a “two-body” propagator?

Orbit Perturbations

♦ What are “J2” and those other things that affect an orbit?

Dealing with Perturbations

♦ How can I model orbit perturbations?

Two-line Element (TLE) Sets

♦ What are TLE set?

STK Propagators

♦ What propagators are available in STK?

Understanding Propagators

To understand the basic problem of orbit propagators, let’s return to the example ofour ball-throwing astronaut shown in Figure 48. What we’re after is a simple


49

mathematical algorithm that will allow us to predict the ball’s position and velocity atany point in time. Whether you’re analyzing the motion of baseballs or galaxies, thefundamental approach is the same. This motion analysis process has three steps:

♦ define a convenient coordinate system

♦ list simplifying assumptions

♦ define initial conditions

x

y

Figure 48: A baseball motion propagator. The simple example of a thrown baseballcan be used to describe the basic problem of orbit propagation.

We can now apply the motion analysis process to describe, and eventually propagate,the motion of the baseball. First, we select a simple, convenient coordinate systemwith its origin at the point of release. The x-direction is defined to be positive to theright in the picture. The y-direction is positive down. Next, we need to make someassumptions to make our lives easier. The major assumption we’ll make is thatEarth’s gravity is the only force acting on the ball, a force which is constant over theflight path we’re concerned with. That is, wind resistance and other forces (thegravitational pull of the Moon and stars, solar pressure, etc.) are negligible. Thisgravitational force can be expressed using Newton’s law, using vector notation, as:

vFgravity = mgy N$

Note $y is a vector notation indicating the force of gravity acts only in the y-direction, i.e.,down. See the section describing vectors for further explanation.


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Finally, we need some initial conditions. Let’s pretend that the ball leaves the pitcher’shand at a velocity of 10 m/s on a horizontal path (i.e., all motion in the x-direction).Symbolically we would say the initial velocity (

rVinitial ) and position (

rRinitial ) are:

V xinitial = 10 $ m / sec

The ball starts out traveling 10 m/sec horizontal to the right

rRinitial 0x 0y m= +$ $

The ball starts out at the origin, in the pitcher’s hand

To derive an expression for the velocity and position of the ball as a function of time,we begin by writing the acceleration as a function of time. Recall we assumed the onlyforce on the ball is due to gravity, which acts to accelerate the ball in the positive y-direction. Thus, we have:

ra x gy= +0 $ $ m / sec2

Acceleration is down at gravitational rate.

where g = gravitational acceleration at the Earth’s surface = 9.798 m/sec2

To obtain the instantaneous velocity at any time (t), we must integrate this equationwith respect to time. (Remember integrals from calculus? Basically, an integral is amathematical means of adding together lots of small changes over time to develop thetotal change.) Thus,

r rV t V gty x gtyinitial( ) $ $ $= + = +10 m / sec

This equation tells us that the ball will keep its initial horizontal velocity constant butwill speed up in the vertical direction due to gravity (which we already knew). Toobtain the instantaneous position of the ball at any time (t), we must once againintegrate this equation with respect to time so that:

r r rR t R V t gt y tx gt yinitial initial( ) $ $ $= + + = +1

210

1

22 2

We can now use these relatively simple equations to propagate the motion of thebaseball. Using a simple spreadsheet, we can determine the position and velocity ofthe baseball for each second for a total 10-second flight. These values and a graphdepicting the trajectory are shown in Figure 49. Notice the trajectory we derived isexactly what we’d expect from experience. Often, people mistakenly refer to the


51

baseball trajectories as “parabolic;” however, as we know, this is actually a smallsection of an ellipse.

TimeTime(sec)(sec)

X (m)X (m) Y (m)Y (m)

0 0 0.01 10 4.92 20 19.63 30 44.14 40 78.45 50 122.56 60 176.47 70 240.18 80 313.59 90 396.8

10 100 489.9

Figure 49: Results of baseball propagator: Using the simple equations of motion wederived for the baseball, we can use a spreadsheet to calculate the x and y positions ateach point in time for a 10-second flight. Plotting these on a graph, we see the shape ofthe trajectory.

Note that the technique we developed here was for analytic propagation. An analyticpropagation technique has a close-form solution. In other words, given the initialconditions, we solve directly for the position and velocity at any future time using astraightforward “plug and chug” of the equations of motion. How accurate is thispropagation technique? As we shall see, this (and any other method) is only as goodas the assumptions we make. For example, we assumed no wind resistance, but weknow from experience that a sudden gust of wind could make this trajectory changeconsiderably. In the next section, we’ll apply this same basic technique to understandthe slightly more complicated motion of a satellite in orbit.

The Two-Body Propagator

Now we’ll develop a simple method we can use to propagate the position and velocityof a satellite known at a given time to predict its position and velocity at some time inthe future. By “simple method” we mean the restrictions placed on the complexity ofthe problem. In this case, one of the primary assumptions we will make is that thereare only two bodies concerned—the Earth and the satellite. Thus, we arrive at theterm used to describe this approach “the restricted two-body problem.”

Trajectory of the Baseball

0.050.0

100.0150.0200.0250.0

300.0350.0400.0450.0500.0

0 10 20 30 40 50 60 70 80 90 100

X (m)

Y (

m)


52

Motion Analysis Process

The approach we’ll take is exactly the same as the one we used to describe the motionof the baseball in the previous section. Using the same motion analysis processdescribed, we can apply the three basic steps of our motion analysis model: define acoordinate system, make assumptions and identify equations of motion. Ourcoordinate system will be the geocentric-equatorial coordinate system described earlier.The assumptions we make will “restrict” our solution to cases in which theseassumptions apply. Fortunately, this includes most of the situations we’ll encounter.We’ll assume that:

♦ satellites travel high enough above the Earth’s atmosphere so that the dragforce is small.

♦ the satellite won’t maneuver or change its path, so we can ignore the thrustforce.

♦ we’re considering the motion of the satellite close to the Earth, so we canignore the gravitational attraction of the Sun, the Moon or any third body.(That’s why we call this the two-body problem.)

♦ compared to Earth’s gravity, other forces such as those due to solarradiation, electromagnetic fields, etc. are negligible.

♦ the mass of the Earth is much, much larger than the mass of the spacecraft.

♦ the Earth is spherically symmetrical with uniform density and can thus betreated as a point mass.

After all these assumptions, we’re left with gravity as the only force affecting themotion of a satellite for the restricted two-body problem. This can be expressed (usingvector notation) as:

vF

GM M

RRsatellite

Earth satellite

satellite=

−2

$

The force due to gravity on a satellite depends on the mass of both the satellite andEarth and the distance to the Earth’s center. The direction is down, in the minus-Rdirection.

For convenience, we often combine GMEarth to derive an expression µEarth , known as theEarth’s gravitational parameter.

While Newton’s law of gravity describes the force on the satellite, we can use Newton’s2nd law of motion to describe the effect of that force to develop our equations ofmotion. From Newton’s second law, the force on the satellite can be expressed as:


53

F M asatellite satellite satellite=

Setting the two expressions for force on the satellite equal to each other, we developan expression for the satellite’s acceleration:

va

RRsatellite

Earth

satellite=

− µ2

$

A satellite accelerates down [minus-R direction] due to gravity. The further away fromthe Earth’s center, the smaller the gravitational force and, therefore, the smaller thecorresponding acceleration.

This equation says the motion of a satellite depends only on the distance between thecenter of the Earth and the satellite. It is independent of satellite mass. Substitutingthe more common notation for acceleration, we get the two-body Equation of Motion.

..

RmEarth

R2Rv =

− $

where

R = distance from center of Earth (km) to satellite

v&&R = 2nd derivative of position = acceleration (km/sec2)

µEarth

= gravitational parameter (km3/sec2)

$R = unit vector in direction of rR

Two-Body Orbit Propagation

What can the two-body equation of motion tell us about the movement of a satellitearound the Earth? Unfortunately, in its present form—a second-order, non-linear,vector differential equation—it doesn’t help us visualize anything about thismovement. So what good is it? To understand the significance of the two-bodyequation of motion, we must first “solve” it using rather complex mathematical slight-of-hand. When the smoke clears, we’re left with an expression for the position of anobject in space in terms of some variables we already know.

( )R

a e

e=

−

+

1

1

2

cosν


54

where

R = magnitude of rR


e = eccentricity (dimensionless)

ν = true anomaly (deg or radians)

This equation represents the solution to the restricted two-body equation of motionand describes the location, R, of a satellite in terms of a few constants and someinitial conditions. We can now make use of the solution to the two-body equation ofmotion to propagate the position of a satellite to any point in time.

In nice circular orbits, determining how long a satellite takes to travel from an initialposition to a future position is simple, because the satellite is moving at a constantspeed. However, in an elliptical orbit this speed varies (recall a satellite travels fastestat perigee and slowest at apogee, keeping total energy constant). As a result, we don’tknow how the true anomaly, ν, changes with time because it doesn’t changeuniformly. Here’s where Johannes Kepler came to the rescue. He developed thistechnique to describe the orbit of Mars. To describe motion in an elliptical orbit,Kepler began by defining the mean motion, n, which tells us the mean, or average,speed in the orbit. The mean motion is defined as:

nangle

time P a= = =

23

π µ

where

n = mean motion (rad/sec)

P = period (sec)

µ = gravitational parameter (km3/sec2)


Kepler figured out how to move n to a time in the future and, conversely, given afuture n, how to find out how long Mars would take to travel there. Kepler’s approachwas purely geometrical—he related motion on a circle to motion on an ellipse. To dothis, he had to invent a new angle called the mean anomaly, M, defined as:

M = nT (1)

where

M = mean anomaly (rad)


55

n = mean motion (rad/s)

T = the time since last perigee passage (sec)

Mean anomaly is an angle that has no physical meaning and can’t be drawn in apicture. We’ll have to describe it mathematically. Expressing this equation in terms oftwo points in the same orbit:

Mfuture - Minitial = n(tfuture -tinitial) - 2kπ (2)

where

Mfuture = mean anomaly when the satellite is in the future position (rad)

Minitial = mean anomaly when the satellite is in the initial position (rad)

tfuture – tinitial = time of flight (TOF)

tfuture = time when the satellite is in the final position (e.g., 3:47 a.m.)

tinitial = time when the satellite is in the initial position (e.g., 3:30 a.m.)

k = the number of times the satellite passes perigee

To relate elliptical motion to circular motion, Kepler defined another new angle calledthe eccentric anomaly, E, so that he could relate M to E and then E to n. With all ofthese things defined, Kepler was able to develop his now-famous equation, commonlycalled Kepler’s Equation. (For this equation to work, all angles must be in radians.)

M E e E= − sin (3)

where

E = eccentric anomaly (rad)

e = eccentricity

Kepler then related E to n using:

coscos

cosE

ee E

=+

+ν

1(4)

where

ν = true anomaly (rad)

And related ν to E through:


56

coscos

cosν =

−−

E ee E1

(5)

Finally, we have all the equations needed to build a complete two-body propagator.The first problem, and the easiest, is finding the time of flight between two points inan orbit. Given ν

initial and ν

future, we simply go through the following steps:

♦ Use Equation (4) to solve for Einitial and Efuture

♦ Use Equation (3) to solve for Minitial and Mfuture

♦ Use Equation (2) to solve for the time of flight (tfuture – tinitial)

Note that if n is between 0° and 180°, so are E and M.

The second problem we can solve using Kepler’s method is far more practical. Thisinvolves determining a satellite’s position at some future time, tfuture, as shown in

Figure 50.

υυ Future

υυ Initial

Time of Flight

Figure 50: Time of flight on an elliptical orbit. The second problem Kepler tackledwas predicting the future position of a satellite knowing only its initial position.

This second problem is much trickier. We assume that we know where the satellite isat time tinitial, so we know νinitial. We start by finding Einitial, using Equation (4). Then we

find Minitial using Kepler’s Equation (3).

M E e Einitial initial initial= − sin

Now, because we know tfuture, using Equation (2), we can find Mfuture:


57

( )M M n t t kfuture initial future initial− = − − 2 π

Great. We’re now on our way to finding nfuture, which tells us where the satellite will

be. So we go to Kepler’s Equation again to find Efuture. Let’s rearrange this equation

and put E on the left side:

E M e Efuture future future= + sin (6)

OOPS! Efuture is on both sides of the equation. This is called a transcendental equation

and can’t be solved for Efuture directly. In fact, almost every notable mathematician

over the past 300 years has tried to find a direct solution to this form of Kepler’sEquation without success. So we must resort to “math tricks” to solve for Efuture. The

“math trick” we’ll use is called iteration. To see how iteration works, think about thekids’ game Twenty Questions. In this game, your partner thinks of a person, place, orthing and you must guess what he’s thinking of. You’re allowed 20 questions(guesses) to which your partner can answer only “yes” or “no.” In seeking the rightanswer, a good player will systematically eliminate all other possibilities until only thecorrect answer remains.

A mathematical application of this can be seen using another transcendentalequation:

y = cos(y)

Because we can’t solve for y using algebra (we can’t get the y out of the cosinefunction to put all the ys on the left side), we must iterate. Begin by taking a guess atthe value for y, and take the cosine to see how close you were. Then take this as thenew value of y and use it for the next guess, and keep doing this iteration until thenew y equals the old y (or is, say, within 0.000001 radians of the old value).

Let’s try it to see what the answer for y really is. Take out your calculator and use π/4radians as your first guess for y. (Remember to set your calculator to use radians, notdegrees.) Keep pressing the cosine function button and you’ll see the value slowlyconverges to 0.739085 radians (about 43°). Presto—you’ve now solved thetranscendental equation y = cos(y) using iteration!

We can use this same iterative technique to solve Equation (6) for Efuture. It turns out

that the values for M and E are always pretty close together, even for the mosteccentric orbits, so let’s use Mfuture for our first guess at Efuture. Here’s the algorithm:

♦ Use Mfuture for the first E.

♦ Solve Equation (6) for a new Efuture.

♦ Use this new Efuture for the next guess for Equation (6).


58

♦ Keep doing the previous step until Efuture doesn’t change by much (less than

about 0.0001 rad). At this point, the solution is said to have converged.

This brute force iteration method will solve Equation (6), but there are much bettermethods in use in most standard propagators.

Let’s quickly summarize what we’ve learned. If we know where we are in an orbit andwhere we want to be, we can use Kepler’s Equation to solve for the time it takes totravel to the place we want to be. The solution is very straightforward. If, however, weknow where we are and want to know where we’ll be at some future time, we can useKepler’s Equation to find that location only by iterating a transcendental equation foreccentric anomaly.

Orbit Perturbations

In deriving the two-body equation of motion, we had to assume that:

♦ gravity was the only force

♦ the Earth’s mass was much greater than the satellite’s mass

♦ the Earth was spherically symmetric with uniform density, so it could betreated as a point mass

These assumptions led us to the restricted two-body equation of motion:

v&& $RR

R+ =µ

20 (7)

The solution to this equation gives us the six classical orbital elements:

a = semimajor axis

e = eccentricity

i = inclination

Ω = longitude of the ascending node

ω = argument of perigee

ν = true anomaly

Under our assumptions, the first five of these elements remain constant for a givenorbit. Only the true anomaly, ν, varies with time as the satellite travels around itsfixed orbit. What happens if we now change some of our original assumptions? Otherclassical orbital elements besides ν will begin to change as well. Any changes to theseclassical orbital elements due to other forces are called perturbations. To see whichclassical orbital elements will change and by how much, let’s look at our firstassumption—gravity is the only force.


59

Atmospheric Drag

Just as a sudden gust of wind changes the course of a football, atmospheric drag canaffect satellites in low Earth orbit (below about 1000 km). Let’s look at how dragaffects the orbital elements.

Because drag is a nonconservative force, it takes energy away from the orbit in theform of friction on the satellite. Thus, we expect the semimajor axis, a, to decrease.The eccentricity also decreases, since the orbit becomes more circular. Let’s see whythis is so. When a satellite in an elliptical orbit is at perigee, it has a greater speedthan it would if the orbit were circular at that same altitude. The drag decreases thespeed, making it closer to the circular orbit speed. That’s exactly what we see inFigure 51. It’s as if drag were giving the satellite a small negative velocity change, ordelta (∆) V, (slowing it down) each time it passes perigee.

successive orbits

original orbit

Earth’s atmosphere

∆∆V drag

Figure 51: The effect of drag on an eccentric, low-Earth orbit. As a satellite passesthrough the upper atmosphere at perigee, drag acts to gradually slow it down,circularizing the orbit until it eventually decays.

Drag is very difficult to model because of the many factors affecting the Earth’s upperatmosphere and the satellite’s attitude. The Earth’s day-night cycle, seasonal tilt,variable solar distance, the fluctuation of Earth’s magnetic field, the Sun’s 27-dayrotation and the 11-year cycle for Sun spots make precise modeling nearlyimpossible. The force of drag also depends on the satellite’s coefficient of drag andfrontal area, which can also vary widely, further complicating the modeling problem.

The uncertainty in these variables is the main reason Skylab decayed and burned upin the atmosphere several years earlier than first predicted. For a given orbit,however, we can approximate how the semimajor axis and the eccentricity change


60

with time, at least for the short term. Different propagation techniques use differentmethods of estimating drag, with widely varying accuracy.

Earth’s Oblateness—“J2”

Columbus was wrong! The Earth isn’t really round. From space, it looks like a big,blue spherical marble, but if you take a closer look, it’s really kind of squashed. Thus,it can’t most accurately be treated as a point mass, as it is treated in the two-bodyassumption. We call this squashed shape oblateness. What exactly does an oblateEarth look like? Imagine spinning a ball of jello around its axis and you can visualizehow the middle (or equator) of the spinning jello would bulge out—the Earth is fatterat the equator than at the poles. This bulge can be modeled by complex mathematics(which we won’t do here) and is frequently referred to as the J2 effect. J2 is a constantdescribing the size of the bulge in the mathematical formulas used to model the oblateEarth. Why “J2?” This term arises from the mathematical short-hand used todescribe Earth’s gravitational field. (Gravitational acceleration at any point on Earth iscommonly expressed as a geopotential function expressed in terms of Legendrepolynomials and dimensionless coefficients Jn—whew!). J2, J3 and J4 are the zonalcoefficients that depend on latitude. Of these, J2 is by far the most important; it isroughly 1000 times greater than either J3 or J4. However, for more precise modelingof the Earth’s oblateness, all three of these must be taken into account. In addition,other, higher order terms can be included in the model. These terms serve to slice theEarth into wedges that depend on longitude (sectoral terms) and slice it again intoregions of longitude and latitude (tesseral terms).

Let’s concentrate on the simplest and most profound case, J2. What effect does J2have on the orbit? Let’s look at Figure 52. Here it’s shown exaggerated; actually thebulge is only about 22 km thick. That is, the Earth’s radius is about 22 km longeralong the equator than through the poles.


61

FJ2

FJ2

Figure 52: Diagram of Earth oblateness. The Earth’s oblateness, shown here as abulge at the equator (highly exaggerated to demonstrate the concept) causes a twistingforce on satellite orbits that change various orbital elements over time.

Let’s see if we can reason out how this bulge will affect the orbital elements. The forcecaused by the equatorial bulge is still gravity. Recall that gravity is a conservativeforce; therefore, the total mechanical energy in an orbit must be conserved. Totalmechanical energy depends on the orbit’s semimajor axis. Thus, as long as energyremains constant (i.e., no drag or other forces adding or stealing energy), thesemimajor axis also remains constant. It turns out that the eccentricity, e, alsodoesn’t change, although the explanation for this is beyond the scope of ourdiscussion here. Although you might expect the inclination to change because thebulge pulls on our orbit, it doesn’t! However, it does affect the orbit by changing theright ascension of the ascending node, Ω, and moving the argument of perigee, ω,within the plane. That’s not very intuitive, but it’s like a force acting on a spinningtop. If you stand a nonspinning top on its point, gravity causes it to fall over. If youspin the top first, gravity still tries to make it fall but, because of its angularmomentum, it begins to swivel—this motion is called precession. Let’s examine theeffect of precession on the ascending node and the argument of perigee more closely.

How J2 Affects the Right Ascension of the Ascending Node, Ω

The gravitational effect of this equatorial bulge slightly perturbs the satellitebecause the force no longer originates from the center of the Earth. This causesthe plane of the orbit to precess (like the spinning top), resulting in a movement ofthe ascending node, ∆Ω. This motion is westward for posigrade orbits (inclination<90°) and eastward for retrograde orbits (inclination > 90°).


62

Figure 54 shows this nodal regression rate, &Ω , as a function of inclination andorbital altitude. Let’s look more closely at this figure. It shows that the higher thesatellite is, the less effect the bulge has on the orbit. This makes sense becausegravity decreases with the inverse square of the distance (see Newton’s Law ofGravitation). It also says that if the satellite is in a polar orbit (center of the graph),the bulge has no effect. The greatest effect occurs at low altitudes with lowinclinations. This makes sense, too, because the satellite travels much closer tothe bulge during its orbit, and thus is pulled more by the bulge. For low-altitudeand low-inclination orbits, the ascending node can move as much as 9° per day(lower left corner and upper right corner of Figure 53).

Nodal Regression Rate as a Function of Inclination and Eccentricity

-10.0

-5.0

0.0

5.0

10.0

0 20 40 60 80 100 120 140 160 180

Inclination (deg)

No

dal

Reg

ress

ion

Rat

e (d

eg/d

ay)

100km altitude circular orbit

4000km x 100km altitude elliptical orbit

Figure 53 Nodal Regression Rate, &Ω . The nodal regression rate caused by the Earth’sequatorial bulge. Positive numbers represent eastward movement; negative numbersrepresent westward movement. The less inclined an orbit is to the equator, the greaterthe effect of the bulge. The higher the orbit, the smaller the effect.

How J2 Affects the Argument of Perigee, ω

Figure 54 shows how perigee location rotates for an orbit with a perigee altitude of100 km depending on the inclination for various apogee altitudes. This perigeerotation rate, &ω , is difficult to explain physically, but it could be derivedmathematically from the equation for J2 effects on perigee location. With thisperturbation, the major axis, or line of nodes, rotates in the direction of satellitemotion if the inclination is less than 63.4° or greater than 116.6°. It rotatesopposite to satellite motion for inclinations between 63.4° and 116.6°.


63

Perigee Rotation Rate as a Function of Inclination and Eccentricity

-5.0

0.0

5.0

10.0

15.0

20.0

0 20 40 60 80 100 120 140 160 180

Inclination (deg)

Per

igee

Ro

tati

on

Rat

e (d

eg/d

ay)

100km altitude circular orbit

4000km x 100km altitude elliptical orbit

Figure 54: Affects of J2 on argument of perigee. The perigee rotation rate caused bythe Earth’s equatorial bulge depends on inclination and altitude at apogee.

Sun-Synchronous and Molniya Orbits

The effects of the Earth’s oblateness on the node and perigee positions give rise totwo unique orbits that have very practical applications. The first of these, the Sun-synchronous orbit, takes advantage of eastward nodal regression at inclinationsgreater than 90°. Looking at Figure 55, we see that the ascending node moveseastward about 1° per day at an inclination of about 98° (depending on thesatellite’s altitude).

Coincidentally, the Earth also moves around the Sun about 1° per day (360° in365 days), so at this Sun-synchronous inclination, the satellite’s orbital plane willalways maintain the same orientation to the Sun. This means the satellite can seethe same Sun angle when it passes over a particular point on the Earth’s surface.As a result, the Sun shadows cast by features on the Earth’s surface won’t changewhen pictures are taken days or even weeks apart. This is important for remote-sensing missions such as reconnaissance, weather and monitoring of the Earth’sresources, because they use shadows to measure an object’s height. Bymaintaining the same Sun angle day after day, observers can better track changesin weather, terrain or man-made features.


64

Sun lineSun line

Sun line

Sun line

Sun angle

Sun angle

Sun angle

Sun angle

orbit plane

Earth moves around theSun at ~1 deg/day

orbit planerotates at ~1 deg/day due to Earth’soblateness

Figure 55: Sun-Synchronous Orbit. Sun-synchronous orbits take advantage of the rateof change in right ascension of the ascending node caused by the Earth’s oblateness. Bycarefully selecting the proper inclination and altitude, we can match the rotation of Ω withthe movement of the Earth around the Sun. In this way, the same angle between theorbit plane and the Sun can be maintained without using rocket engines to change orbit.Such orbits are very useful for remote sensing missions that want to maintain the sameSun angle on targets on the Earth’s surface.

The second unique orbit is the Molniya orbit, named after the Russian word forlightning (as in “quick-as-lightning”). This is a 12-hour orbit with high eccentricity(about e = 0.7) and a perigee location in the Southern Hemisphere. The inclinationis 63.4°—why? Because at this inclination, the perigee doesn’t rotate so thesatellite “hangs” over the Northern Hemisphere for nearly 11 hours of its 12-hourperiod before it whips “quick as lightning” through perigee in the SouthernHemisphere. Figure 56 shows the orbit and ground tracks for a Molniya orbit. TheRussians used this orbit for their communications satellites because they didn’thave launch vehicles large enough to put them into geosynchronous orbits fromtheir far northern launch sites. Molniya orbits also offer better coverage oflatitudes above 80° north.


65

Figure 56: Molniya orbit and ground tracks. Molniya orbits take advantage of the factthat ω, due to Earth’s oblateness, is zero at an inclination of 63.4°. Thus, apogee staysover the Northern Hemisphere, covering high latitudes for 11 hours of the 12-hour orbitperiod.

Other Perturbations

Other perturbing forces can affect a satellite’s orbit and its orientation within thatorbit. These forces are usually much smaller than the J2 (oblate Earth) and dragforces but, depending on the required accuracy, satellite planners may need toanticipate their effects. These forces include:

♦ Solar radiation pressure, which can cause long-term orbit perturbations andunwanted satellite rotation.

♦ Third-body gravitational effects (Moon, Sun, planets, etc.), which can perturborbits at high altitudes and on interplanetary trajectories.

♦ Unexpected thrusting caused by either out-gassing or malfunctioningthrusters, which can perturb the orbit and cause satellite rotation.

Dealing with Perturbations

Understanding and modeling orbit perturbations is one of the primary activities ofastrodynamics. Even very early space pioneers such as Kepler and Newton spentconsiderable effort grappling with the various forces that disturb a satellite from puretwo-body motion. Let’s begin by classifying perturbations with respect to their relativeeffects on orbital elements. Perturbations can cause both secular and periodicchanges to orbital elements. Secular perturbations are those that cause elements tosteadily diverge over time. Periodic perturbations are those that impart a sinusoidalvariation in elements over time. Short-term periodic perturbations are those with aperiod less than the orbit period. Long-term periodic perturbations are those with a


66

period greater than one orbit period. We can now look at two techniques for modelingboth secular and periodic perturbations. The relative effects of these different typesare illustrated in Figure 57.

Long term effects on orbital elements for various types of perturbations

0

1

2

3

4

5

6

7

8

90 1 2 3 4 5 6 7 8

Orbit Periods

Orb

ital

ele

men

t va

riat

ion

(a

rbit

rary

un

its)

Secular

Short- term Periodic

Long-termPeriodic

Figure 57: Types of orbit perturbations. Orbit perturbations are categorized based ontheir long-term effects on orbital elements.

General Perturbations Techniques

General perturbations techniques are those that generalize the effects on orbitalelements in order to develop analytic expressions allowing for direct computation. Inthe grossest sense, general perturbation techniques apply “fudge factors” to thesimple two-body solution to account for the effect of different perturbation sources.For example, returning to our baseball-throwing astronaut scenario, we could modelthe drag on the baseball using:

D V A=1

22ρ

where

D = Drag (N)

ρ = air density (kg/m3)

V = baseball velocity (m/sec)


67

A = baseball cross-sectional area (m2)

We could substitute this expression into our baseball equations to derive newequations of motion that would account for the general effects of drag. Even with thisadditional complexity, the equations could still be solved analytically.

One of the most widely used propagators was developed by the North AmericanAerospace Defense Command (NORAD) to track the 8000-plus satellites and spacejunk in orbit around the Earth. Called Merged Simplified General Perturbations-4,MSGP-4, this technique uses the generalized approach to model orbit perturbations.

Special Perturbations Techniques

In contrast, special perturbation techniques are based on special case assumptionsabout the orbit scenario that allow for more detailed modeling of individualperturbation sources.

Two-line Element Sets

One of the most commonly used methods of communicating orbital parameters is the2-line element sets generated by NORAD in Cheyenne Mountain, Colorado (literally, inCheyenne Mountain!). It is important to note that TLEs were developed specifically foruse with the MSGP-4 propagator! Using TLEs with any other propagator mayinvalidate some of the built-in assumptions.

These elements contain many of the same elements as the classical orbital elements,along with some additional parameters for identification purposes and for use inmodeling perturbations in the MSGP-4 propagator.

STK Propagators

In selecting the “best” propagator to use for a given application, it is important toconsider the assumptions on which they are based. The temptation is to use the most“accurate” propagation model available. However, this can lead to false accuracy,especially for very long term propagation over which time even the best models canbreak down. The relative accuracy among various propagators can vary widelydepending on the scenario. For example, a geostationary spacecraft is well above mostatmospheric drag and J2 perturbations. Therefore, a short-term difference betweenthe two-body propagator and a more complex technique could be relatively small.However, for a spacecraft in low-Earth orbit, the short-term differences between thetwo solutions could be significant. The objective is to choose the most appropriatepropagation scheme for a given application. Unfortunately, any propagation techniqueis simply an attempt to model events in the real world. Regardless of the techniquechosen, only frequent tracking of an orbit can guarantee that the predicted orbitalparameters will match the real world.


68

Two-Body

The two-body propagator or Keplerian motion propagator uses the same basictechnique outlined in the two-body equation of motion development. This techniqueassumes the Earth is a perfect sphere and the only force acting on a satellite isgravity. This propagator doesn’t account for any perturbations.

J2

The J2 propagator accounts for the 1st order effects of J2 Earth oblateness. This effectcauses secular changes to the orbital elements over time.

J4

The J4 propagator accounts for 1st and 2nd order J2 effects as well as 1st order J4effects. J3, which causes long-term periodic effects, is not modeled. Because the 2nd

order J2 and 1st order J4 effects are very small, you’ll see very little differencesbetween the J2 and J4 propagators for most orbits considered.

MSGP-4

MSGP-4 stands for Merged Simplified General Perturbations-4. It is one of the mostwidely used propagators in the industry. This technique uses the generalizedapproach to model orbit perturbations, including both secular and periodic variationssuch as Earth oblateness, solar and lunar gravitational effects and drag.

It is important to understand the purpose for which MSGP-4 was developed. NORADwanted a simple propagator that would provide acceptable results for a wide variety oftracking tasks, from tracking high-priority military satellites to keeping tabs on spacejunk. Given the over 8000 objects NORAD must track, a technique was needed thatwould not be computationally intensive (is was first developed back when computerswere much slower than today). Furthermore, there are far fewer ground tracking sitesthan there are objects to track. Thus, it is important that the propagated solution begood enough to ensure the tracking radar can find a specific object the next time itgets around to tracking it (which, for some very low priority objects like pieces ofrocket boosters, may be days or even weeks).

Because MSGP-4 is a generalized approach, it is specifically tailored to a given set ofinputs: the TLE sets that contain parameters that make the analytic calculationsvalid. For best results, MSGP-4 should always be used with TLEs. Likewise, NORAD-generated TLEs should only be used in the MSGP-4 propagator.


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HPOP

HPOP is the High Precision Orbit Propagator. As its name implies, it uses a powerfulpropagation technique to incorporate sophisticated orbit perturbation models. HPOPuses a variety of high-fidelity models including:

♦ Joint Gravity Model (JGM) 2—a highly precise model of the Earth’soblateness.

♦ Lunar/solar gravitational effects—based on U.S. Naval Observatory data.Accurate to within 0.03 arc seconds.

♦ Atmospheric drag effects— using either the 1971 Jacchia or the HarrisPriester model, which takes into account daily variations in the height of theatmosphere due to solar heating among other parameters.

♦ Solar radiation pressure—yes, sunlight produces a small force on any exposedsurface. This force varies depending on how reflective the surface is—amirrored surface is more reflective than a black surface.

Depending on the application, HPOP can deliver accuracy on the order of 10 metersper orbit. But beware of false accuracy, always remember—“garbage in, garbage out.”To get this level of accuracy, your initial orbital elements must be at least thisaccurate to start with. Putting NORAD-generated TLEs into HPOP will not necessarilygive you a better solution. No propagator can create accuracy, at best it can onlyminimize the long term dispersions due to inherent limitations in our ability to modelthe effects of perturbations.

Great Arc

The Great Arc propagator allows the user the model the flight path of a vehicle flyingclose to Earth. By providing way points and speed, STK uses Great Arc to predictwhere and when it will be next. The propagation scheme is essentially the same as theTwo-Body propagator, no perturbations are assumed.

Ballistic

The Ballistic propagator is a variation of the Two-Body propagator for use withballistic trajectories. These are the trajectories used by artillery shells, suborbitalsounding rockets and ballistic missiles, allowing the user to predict impact points ordetermined required velocity to reach a certain point. The propagation “engine” is thesame as the Two-Body propagator, no perturbations are modeled.

LOP

The Long-term Orbit Predictor (LOP) allows accurate prediction of a satellite’s orbitover many months or years. This is often used for long duration mission design, fuelbudget definition, and end-of-life studies. For performance reasons, it is impractical


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to compute the long-term variation in a satellite’s orbit using high accuracy, smalltime step, propagators that compute a satellite’s position as it moves through itsorbit. LOP exploits a “variation of parameters” approach which integrates analyticallyderived equations of motion computing the average effects of perturbations over anorbit. This approach allows large multi-orbit time steps and typically improvescomputational speed by several hundred times while still offering high fidelitycomputation of orbit parameters.

Lifetime

Lifetime estimates the amount of time a low Earth orbiting satellite can be expected toremain in orbit before the drag of the atmosphere causes reentry. While thecomputational algorithms are similar to those implemented in the Long-term OrbitPredictor, there are some important differences. First, a much more accurateatmospheric model is implemented to compute the drag effects. The gravitationalmodel for the Earth, however, is significantly simplified since the inclusion of thehigher order terms doesn’t impact orbit decay estimates.


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SATELLITE ACCESS

Overview

Line of sight

♦ Why is line of sight important for satellite viewing?

Communication Architecture

♦ What are the elements that make up a space mission communicationarchitecture?

Communication Links

♦ What are the communication paths used by satellites and groundstations?

Understanding Access

♦ What is meant by “satellite access?”

Describing Access

♦ How do I explain and quantify satellite access?

Line of Sight

Standing on a beach, looking out over the ocean on a clear day, you can seeright to the edge of the horizon, which is about 8 miles away. If you were towatch a ship sailing away from you, you would notice that the hull woulddisappear first, followed by the top of the mast The taller the mast, the furtherthe ship could be from the shore before disappearing completely from sight. Anobject is in your line of sight if you can draw a straight line between yourselfand the object without any interference, such as a mountain or a bend in aroad. An object beyond the horizon is below our line of sight and, therefore, canbe difficult to communicate with. Early methods of long-distancecommunications increased the effective line of sight by employing methodssuch as smoke signals or other means. Because the line of sight was raised sothat others could see, or receive the message being sent, communicationamong objects that didn’t really have a direct line of sight was achieved.

At the beginning of the 20th Century, radio engineers discovered that certainfrequencies could be bounced off the ionosphere, greatly extending the effectiveline of sight for communications and creating a new “radio horizon” far beyondthe more limited visual horizon. Today’s communications satellites take thisbasic principle to the extreme. Ground-based operators can “bounce” radio


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communications off satellites stationed in geosynchronous orbit, creating avirtual line of sight extending half a world away. Furthermore, by bouncingsignals between satellites, this virtual line of sight can be extended to cover theentire global community.

Satellite access is the problem of determining when, where and for how long asatellite (or any number of objects you may be interested in) is within line ofsight of other objects.

Communications Architecture

To understand the satellite access problem more clearly, let’s begin byreviewing the players in the access problem. Figure 58 illustrates the elementsthat make up a communications architecture.

crosslink

crosslink

return link

forward link

forward link

return linkdownlink

uplin

k

Figure 58 Communications Architecture. The communications architectureconsists of space and ground-based elements tied together by communicationspaths or links.

The communications architecture has four elements:

Spacecraft

The spaceborne elements of the system.


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Ground stations

The Earth-based antennas and receivers that talk to the spacecraft. These aretypically remote tracking sites or users of mission data.

Control Center

The command center that controls the spacecraft and all other elements of thesystem.

Relay satellites

Additional satellites that link the primary spacecraft with the ground stationsand control center.

Communications Links

Information moves among the elements of the communications architectureusing various communication paths or links.

Uplink

Data sent from a ground-based station to the primary satellite.

Downlink

Data sent from the primary satellite to a ground station.

Forward link

Data sent from a ground station to the primary satellite via a relay satellite.

Return link

Data sent from the primary satellite to a ground station via a relay satellite.

Crosslink

Data sent through either the forward or return link between the primary satelliteand a relay satellite.

Understanding Access

The simplest example of a satellite access problem is that between a satelliteTV dish and a direct-broadcast geosynchronous satellite. As a user, you justwant to point your dish and start watching the big game. Thus, you’re onlyinterested in downlink.


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Figure 59 Downlinking. As a direct TV subscriber, you would only be interestedin downlinking.

For satellites in geostationary orbit, the geometry and dynamic nature of bothuplink and downlink is very stable. It is this stability that make geostationarysatellites so useful for point-to-point message relaying. You can set up youdish, point it at a pre-determined point in the sky and pretty much forget aboutit. Because the ground track of a geosynchronous orbit is at most a tiny figure-8 centered on the equator, a line of sight between the dish and the satellite isalmost constant (at least constant enough to ensure uninterrupted broadcast ofthat big game!).

Describing Access

So how do you know where to point the satellite dish? This depends on twoimportant pieces of information:

♦ Your location (latitude and longitude)

♦ The satellite’s location (orbital elements)

From this information, STK can determine the azimuth and elevation settingsfor your dish. These two important parameters are defined below.

♦ Azimuth - The compass direction between the ground site and thesatellite direction, e.g., due south would be 180°.

♦ Elevation - Angle measured from the local horizontal to the satellitedirection, e.g., directly overhead would be 90°.

In addition, STK also computes range—the distance between the dish and thesatellite. While range is not so important to the average satellite TV user, itbecomes very important for communications engineers who must ensure thereis sufficient transmission power to effectively carry the signal across thisdistance.

These three parameters are fairly constant for a geosynchronous satellite;however, for any other satellite, we must include an additional parameter—


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time. Imagine it is exactly 13:20 (1:20 p.m.) local time and you want to point aradar antenna at an airplane flying over your position. The plane is initiallydue south of you, flying north but out of sight below the horizon. As the planefirst comes into view, azimuth will be 180° and the elevation 0°. As itcontinues to fly north, azimuth will stay constant (disregarding Earth rotation)and the elevation angle will increase. As it flies overhead, the azimuth anglewill switch around to 0° (it is now north of your position) and the elevationangle will gradually decrease until the plane once again drops from view belowthe horizon. If we kept track of the azimuth and elevation viewing angles to theplane at 10-minute intervals we could build a simple table, or access report, asshown below.

Time Azimuth(deg)

Elevation(deg)

Comment

13:20 180 -45 Below horizon (south of you)

13:30 180 0 Just coming into view

13:40 180 45 Well above horizon

13:50 ---- 90 Directly overhead (azimuth isundefined)

14:00 0 45 Azimuth has switched around, plane isnow north of your position.

14:10 0 0 Plane drops below horizon, out of view

The geometry with respect to a plane flying directly overhead is relatively easyto visualize. However, if you’re faced with the problem of a satellite in a highlyeccentric orbit flying over a position northwest of you on a descending node,things become much more complicated. Fortunately, STK works out thegeometry for you. Figure 60 shows a relatively simple satellite access geometry.


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Figure 60 Satellite Access. Satellite access refers to the problem ofdetermining the geometry and timing of line-of-sight between various ground andspace-based objects.

Access reports can be easily generated using STK; the report provides azimuth,elevation and range (AER) data for specified time intervals between whateverobjects you choose to define. Cumulative access time or duration can also bereported.

Access information becomes even more complicated when multiple objectsmust be taken into account. For example, if you are relaying informationbetween various ground stations and relay satellites or using an entireconstellation of satellites, the geometry can become very complex. To handlethese tasks, STK allows you to link together various objects to create a “chain”for which access information can be determined. Figure 61 shows a morecomplex series of “chained” objects. If one of these “objects” is the Sun, orbitlighting—a critical parameter for power and thermal management—can becomputed.

Figure 61 Chained objects. A series of ground and space-based objects canbe “chained” together, allowing STK to compute access information between allof them. In this picture, access among a facility, relay satellite and a secondfacility is shown.

RECOMMENDED READING

For a more detailed explanation of the topics in this primer as well as anintroduction to the space environment, spacecraft design, rockets and systems,we recommend:


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Understanding Space: An Introduction to Astronautics, Sellers, 1994,McGraw-Hill.

To purchase a copy of Understanding Space, please contact McGraw-Hill atwww.mcgraw-hill.com, www.mhhe.com, or 800-338-3987 : The McGraw-HillCompanies, >Order Services, PO Bos 545, Blacklick, Ohio. ISBN: 0-07-057027-5.

Understanding Space is part of the Space Technology Series, a cooperativeactivity of the United States Department of Defense and National Aeronauticsand Space Administration. Series editor is Dr. Wiley J. Larson. Other books inthe series include:

Fundamentals of Astrodynamics and Applications, Vallado, 1997, McGraw-Hill.

Space Mission Analysis and Design, 2nd edition, Larson & Wertz (ed.), 1996,Kluwer and Microcosm.

Space Propulsion Analysis and Design, Humble & Larson, 1995, McGraw-Hill.

Reducing Space Mission Cost, Larson & Wertz (ed), 1996. Kluwer andMicrocosm.

Cost-Effective Space Mission Operations, Boden and Larson, 1996, McGraw-Hill.

Spacecraft Structures and Mechanisms: From Concept to Launch, Sarafin andLarson, 1995, Kluwer and Microcosm.

For more information about these books, please contact Kluwer atwww.wkap.nl, McGraw-Hill at www.mcgraw-hill.com or 800-338-3987, orMicrocosm at www.microcosm.com.

Future books in the series:

Modeling and Simulation: In Integrated Approach to Development anOperations, Cloud and Rainey.

Human Space Mission Analysis and Design, Connally, Giffen and Larson.

Other recommended reading:

Fundamentals of Astronautics, Bate, Mueller & White, 1971, Dover.

1997 Microcosm Directory of Space Technology Data Sources, 1997, Wertzand Dawson, Kluwer and Microcosm.

Intro to Astronautics

Documents