Last Week. 1. Kepler’s Laws 2. Artificial Satellites 2. Escape Velocity – planetary atmospheres 3. Tides 4. Synchronous orbits 5. Planetary ring systems.

Last Week.1. Kepler’s Laws2. Artificial Satellites2. Escape Velocity – planetary atmospheres3. Tides4. Synchronous orbits5. Planetary ring systems6.The Roche Limit7.Masses of stars in a binary system

Do the known forces change with time(space)?

Motion of the Earth-Mapping the Sky

• Motion of the Earth is quite complicated:- It rotates around the Sun It rotates around its own axis It precesses It nutates etc. Solar System rotates around c-of-m of Galaxy

• Overall the stars rise in the East and set in the West just like the Sun and Moon.

This is a picture of the Earth in its orbit.The Earth is much exaggerated insize.Note that the spin axis is always tilted at 23.5 degrees to the plane of the orbit.

To Polaris

Present day

13,000 years In future

South Pole

Precession of Earth’s Spin Axis

Earth’s spin axis is not static in direction. It is slowly rotating or “precessing”.Spin axis remains at 23.50 to axis of orbit round Sun but its direction constantly rotates with a period of 26,000 years. [Solar year varies by 0.0142 days]

A

B

C

Co-ordinate system needed

Star North Pole

Object in sky is overhead at A

It will be at an angle to vertical at B

It cannot be seen at C

On surface of Earth we use a standard system of co-ordinates based on latitude and longitude.

Mapping of the Sky• It is important to be able to determine the position of an object in the Sky.This can be done in various ways.

• Here we see the Celestial sphere.The stars are set on the surface of a huge sphere centred on the Earth.The point in the sky directly overhead is called the Zenith and we also refer to the Meridian, a great circle through the Observer’s zenith and intersecting the horizon N. & S.

• The Earth’s equator projected on to thecelestial sphere establishes the celestial equator thus dividing it into N and S hemispheres. Projecting the Earth’s N and S poles into space along the Earth’s axis of rotation gives the N and S celestial poles.The picture gives the basis of the equatorial co-ordinate system.

Mapping of the Sky -The Equatorial Co-ordinate System.

• System based on the idea of latitude & longitude used to fix positions on Earth. System stays fixed whilst Earth rotates.

• Declination() Latitude This is the angular distance N. & S. of the celestial equator measured along a circle passing through both poles. It is measured in degrees.

• Right Ascension() longitude This is measured eastwards along theequator from the VERNAL EQUINOX() to its intersection with the Hour Circle of the object. Right Ascension is measured in Hours,Mins,Secs.

Vernal equinox = Position on the celestial equator that the Sun passes through on March 20th..


March 20th

Mapping the Sky

• The Earth moves about 1 degree round its orbit every day.

• As Earth orbits the Sun our view of distant stars is constantly changing. Accordingly our line of sight to the Sun sweeps through the constellations along a path called the ECLIPTIC.The points where the Ecliptic crosses the celestial equator are the Vernal and Autumnal equinoxes. Spring officially begins at the Vernal Equinox.

Most northerly extension of the Sun along the Ecliptic is at the Summer Solstice. Most southerly is at winter solstice.

The EclipticThe Ecliptic is the annual path of the Sun across the Celestial Sphere and it is sinusoidal about the celestial equator.The shape occurs because the Northern Hemisphere alternately pointstoward and then away from the Sun during the Earth’s annual orbit.

Time

Zero for Right ascension


Mapping and Time

Time is the other co-ordinate of mapping.Some definitions:-

Hour Circle - Great Circle through object and north Celestial Pole.

Sidereal time - Time necessary to make one complete orbit relative to the background of stars.This is based on successive meridian crossings of a star.

Local sidereal Time - The amount of time elapsing since the Vernal equinox last crossed the Meridian.It is equivalent to the Hour Angle of the Vernal Equinox, where it is defined as the angle between a celestial object and the observer’s meridian,measured in the direction of the object’s motion and the Celestial sphere. This is zero when vernal equinox passes observer’s meridian.

Astronomical Timekeeping system-Sidereal time.

Every point on Earth has a meridian = the great circle passing through the N and S poles and the zenith.The sidereal time at any location = length of time since the vernal equinox crossed the local meridian.As a result if there is a star on the meridian of that place its right ascension = sidereal time.

Sidereal time Solar time.

While Earth rotates once w.r.t. stars it also moves1/365 of way round its own orbit. After 1 siderealday arrow A becomes arrow B but 1 solar dayonly passes when B rotates a little further andbecomes C. Difference = 4 mins.

AB C

The Effect of Precession

Precession is the slow wobble of the Earth’s axis due to the planet’s non-spherical shape and its gravitational interaction with Moon & Sun.This is analogous to the precession of a top. Earth’s precessionalperiod is 25,770 years and causes North celestial pole to make a slowcircle through the heavens. Polaris is now 1 degree from the pole. In

13,000 years it will be nearly 47 degrees away.The same effect alsocauses a 50.26 sec per yr.westward motion of the Vernal equinox along theEcliptic.As a result we need tospecify the era or epochwhen you quote astron.Co-ordinates.

PrecessionCurrent values of and may then be based on the time elapsed sincethe reference epoch. = ( m + n.sin.tan ).N = (n.cos).Nwhere N = no.of years between the desired time and the reference epoch(+ve or -ve). If 1950 is the ref. Epoch then m = 3.07327 per year and n = 20.0426// per year.

Great Pyramid of Giza:-was erected about 2600 B.C.It was carefully oriented and base is a nearly perfect square.The “air shafts” to theKing’s chamber are orientedtowards Orion’s belt.OSIRIS was associated withOrion. Thuban was then theclosest to the North Pole.

Units of distance

Astronomical Unit = Mean Earth-Sun distance = 149.6 million km.

Light year = distance travelled in vacuum by light in 1 year. = 9.46 x 1012 km

= 63,240 AU

Up to now we know how to determine a) Surface temperature of astar, b) Masses of stars. Now we consider how to measure distance.As we will see soon this will mean that we also have to worry about the brightness of stars.

The Measurement of Distance in Astronomy• The measurement of distances to astronomical objects is at the heart of our understanding of astronomical objects together with how bright they are and how massive they are etc. Unless one can quantify these things Astronomy hardly exists.

• Various methods - they depend on the distances involved. They have a common feature-a yardstick.

r

arc

The angular size of any star is tiny andonly a few are resolvable even with themost sophisticated techniques.

Arc = r gives the ratio of stellar diameter to distance.Even if you canmeasure the angular size one only gets this ratio but one would have to assume that the Sun was a standard size. This is wholly unjustified.

• Remember one unit of distance is the Astronomical Unit(AU) which is the Mean Earth- Sun distance = 1.496 x 1011 m. This is useful for objects close to us and in the method of Stellar Parallax.

Trignometric ParallaxThe most reliable measurement of distance we have is based onTrignometric parallax. It is only suitable for objects in our locality and is related to the methods used by surveyors to measure the distance to a mountain top.

In the surveyor’s method they find the distance d to the mountain topby observing it from two places separated by a known baseline distance2B,which can be measured accurately.The angle p can also be measuredwith precision.

From simple trignometry tan p = B/d Hence we obtain d.

Note: If p is very small then tan p = p = B/d

Stellar Parallax

In essence the same method can be used to measure the distance to aclose-lying star if we have a long enough baseline.The method relies on the fact that most of the stars are so far away that they do not appear to us to move at all. As the Earth moves in its orbit a local star will appear to move back and forwards against the backdrop of more distant stars.As we go round it will go back and forth.The Earth’s radius is too small for the purpose.The baseline has to bethe diameter of the Earth’s orbit.

Background stars

Star

1AU

Stellar Parallax

Star

Background stars

A measurement of the parallax angle p = one half of the max. change in angular position allows us to determine d d = 1AU/tan p AU/p since we can write p = tan p.Now 1 radian = 57.30 = 2.063 x 106// [ 2 radians = 3600] so d = 2.063 x 106 AU

P//

If we now define a new unit of distance, the parsec(1pc) =2.063x106 AU d = 1/p// pc

1AU

Units of distance



= 63,240 AU

Parsec = is the distance from which the Earth’s orbit subtends an angle of 1//. (1 arcsecond)

1 pc = 3.26 ly = 3.086 x 1016 m = 206,280 AU.

Note:- kiloparsecs, Megaparsecs and gigaparsecs are all in use.

From Earth we can only use this technique for about 10,000 stars.

Parallax

HIPPARCOS (High Precision Parallax Collecting Satellite) - was launched by ESA in 1989. - measured parallax for 118,000 stars to 1-2 milliarcseconds to an error of 10%. - This is for all stars within 300 ly.

We need other methods to measure the distance to objects further away.

It turns out that we need a whole series of different methods.

This simple, direct method gives us a reasonably solid base for further measurements and where we can measure the distance by other methods it gives us a consistency check.

Stellar Parallax

Star

By definition when p is one arcsecond the distance d is 1pc.Thus the pcis the distance from which the Earth’s orbit subtends an angle of 1//.In terms of the light year(ly) we have 1pc = 3.262 ly = 3.1 x 1016m.

Only a small no.of stellar distances can be measured this way.Example - Proxima Centauri (nearest star) has p = 0.77//.

The best that we do is p = 0.001// = 1kpc.This is small compared with8kpc to the centre of the Milky Way.We need other methods!!

Proper Motion of Stars

In discussing the parallax method of measuring stellar distances we have ignored their proper motion. In addition to the annualback and forth motion many stars very slowly change their positions relative to the

background due to their own motion in space.The component of thismotion at rt. angles to our line of sight is called their Proper Motion.It is measured in arcsecs per year. In order to measure the proper motionwe have to measure the position over a long period and then subtract out the periodic variation due to the Earth’s orbit.The proper motion and the radial velocity [measured from the Dopplershift]can be combined to give the star’s velocity in space.

θ

r

Earth

vR

v

vT

Star

Proper Motion Contd.

In a time t the star moves a distance dat right angles to the line of sight.So d = vt. t

If r is the distance to the star then the angular change in position on the Celestial sphere is given by = d = vt. t

r r

Then the star’s proper motion is related to its transverse velocity by Proper Motion = = vt

t rProxima Centauri-Proper Motion 3.85// and parallax 0.77//

Overall this method is limited to about 120,000 stars.

r

Earth

vR

v

vTθ

Star

Remember arc = rθ

From Earth we can only use this technique for about 10,000 stars.

Parallax

HIPPARCOS (High Precision Parallax Collecting Satellite) - was launched by ESA in 1989. - measured parallax for 118,000 stars to 1-2 milliarcseconds to an error of 10%. - This is for all stars within 300 ly.

This sounds a large number but remember the numbers of stars in typical galaxies( 1011 stars)

Units of distance



= 63,240 AU

Parsec = is the distance from which the Earth’s orbit subtends an angle of 1//. (1 arcsecond)

1 pc = 3.26 ly = 3.086 x 1016 m = 206,280 AU.

Note:- kiloparsecs, Megaparsecs and gigaparsecs are all in use.

Distance and Brightness

Stellar parallax is a reasonably secure method of measuring distance but it is limited to a relatively few stars close to us.

We need other methods of measuring distances to stars. Many of them rely on measurements of brightness. We must now consider brightness before we can look at other measures of distance

Luminosity = power radiated by a star.

Stellar Brightness

• Among the most basic observations are distance, brightness(luminosity), surface temperature etc.• Luminosity- once we know the distance of a star we can determine the luminosity, a measure of the total power emitted by the star.

2R

Star radiates isotropically (reasonable assumption). The energy is spread evenly over the surface of a concentric spherical shell centred on the star. Now the surface of a sphere is 4R2 so the flux at any point on the concentric shell is inversely R2, the distance from the star.

• If a star’s apparent intensity, how bright it appears from the Earth, can be measured and its distance is known then its absolute intensity can be calculated.

Stellar Brightness• Stellar separations and intensities vary over many orders-of magnitude. As a result it is convenient to use logarithmic scales.• Astronomers use relative measures of Intensity. The system is based on the assumption that iVEGA = 1.0 and the apparent intensities of all other stars (i) are measured relative to the intensity of Vega. We define the apparent magnitude (m) of a star as m = -2.5log10 i ----- definition. Here m is related to how bright the star appears in the night sky.

• Since iVEGA = 1.0 it has m = 0.0 There are a few stars brighter than Vega in the sky and they all have -ve apparent magnitudes. The brightest of all is Sirius at m = -1.5. An object ten times brighter than Vega has m = -2.5

• One peculiarity of this system is that dimmer stars have larger apparent magnitudes. Thus i = 0.1 has m = + 2.5 10 times dimmer than Vega i = 0.01 has m = +5.0 100 times dimmer than Vega Note:-Eye can detect stars to m = 6 or 7. [System due to Hipparchus.]

Stellar Brightness

• Conversion to Absolute Magnitude( M ):- If we have an absolute intensity (I ) then i I i = C.I

d2

where C is a constant of proportionality.

So i d2 = C.Itaking logarithms to base 10 and multiplying by -2.5 gives

-2.5 log10 id2 = -2.5 log10CI -2.5 log10i -2.5log10d2 = -2.5log10C -2.5log10I

If we then write M = -2.5log10I as for apparent magnitude then wecan write m -5log10d = M - 2.5log10C To connect m and M we must decide on the constant term.

Stellar Brightness

We do this with the following arbitarary definition:-M = m when the star is viewed from a distance d = 10 pc.

Then M = m -5 log10d + 5

We now have a link between M,m and d where d is in parsecs.

[Note: we have assumed that the inverse square law is the only reason for the dimming of the light from the star.It takes no accountof any possible absorption in dust between us and the other star.][Note: This not an SI system of units]

Example:- From the Earth the Sun is 4.8 x 1010 times brighter than Vega.What is the Sun’s absolute magnitude given that Vega is 8.1 pc from Earth?

How can we use this equation?

Example:- From the Earth the Sun is 4.8 x 1010 times brighter than Vega.What is the Sun’s absolute magnitude given that Vega is 8.1 pc from Earth?

i = 1.0 for Vega and 4.8 x 1010 for the Sun - Apparent magnitudes mSUN = -2.5 log (4.8 x 1010 ) = -26.7Now 1pc = 2.06 x 105 AU so for Sun d = 1/ 2.06 x 105 pc Therefore MSUN = mSUN -5log10(1/ 2.06 x 105 ) + 5 = +4.9

What is the Absolute magnitude for Vega? MVEGA = mVEGA - 5log10dVEGA + 5 = 0 - 5log10( 8.1 ) + 5 = 0.5

Hence since M = -2.5 log10I and 0.5 = -2.5 log10IVEGA , 4.9 = -2.5 log10IS

4.9 - 0.5 = -2.5( log10ISUN - log10IVEGA )

So IVEGA = 10(4.9 - 0.5)/2.5.ISUN = 57 ISUN

Last Week. 1. Kepler’s Laws 2. Artificial Satellites 2. Escape Velocity – planetary atmospheres 3. Tides 4. Synchronous orbits 5. Planetary ring systems.

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