Transcript Stellar Parallax

Last Week.
1. Kepler’s Laws
2. Artificial Satellites
2. Escape Velocity – planetary atmospheres
3. Tides
4. Synchronous orbits
5. Planetary ring systems
6.The Roche Limit
7.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 in
size.
Note that the spin axis is always tilted at 23.5 degrees to the plane of the
orbit.
Precession of Earth’s Spin Axis
13,000 years
In future
To Polaris
Present day
South Pole
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]
Co-ordinate system needed
Star
North Pole
A
Object in sky is overhead at A
It will be at an angle to vertical
at B
B
It cannot be seen at C
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 the
celestial 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 the
equator 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
This is a picture of the Earth in its orbit.The Earth is much exaggerated in
size.
Note that the spin axis is always tilted at 23.5 degrees to the plane of the
orbit.
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 Ecliptic
The 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 points
toward and then away from the Sun during the Earth’s annual orbit.
Zero for Right ascension
Time
This is a picture of the Earth in its orbit.The Earth is much exaggerated in
size.
Note that the spin axis is always tilted at 23.5 degrees to the plane of the
orbit.
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.
B
C
A
While Earth rotates once w.r.t. stars it also moves
1/365 of way round its own orbit. After 1 sidereal
day arrow A becomes arrow B but 1 solar day
only passes when B rotates a little further and
becomes C. Difference = 4 mins.
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 precessional
period is 25,770 years and causes North celestial pole to make a slow
circle 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 also
causes a 50.26 sec per yr.
westward motion of the
Vernal equinox along the
Ecliptic.
As a result we need to
specify the era or epoch
when you quote astron.
Co-ordinates.
Precession
Current values of  and  may then be based on the time elapsed since
the reference epoch.
 = ( m + n.sin.tan ).N
 = (n.cos).N
where 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 the
King’s chamber are oriented
towards Orion’s belt.
OSIRIS was associated with
Orion. Thuban was then the
closest 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 a
star, 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.
The angular size of any star is tiny and
r
only a few are resolvable even with the

arc most sophisticated techniques.
Arc = r gives the ratio of stellar diameter to distance.Even if you can
measure 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 Parallax
The most reliable measurement of distance we have is based on
Trignometric 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 top
by observing it from two places separated by a known baseline distance
2B,which can be measured accurately.The angle p can also be measured
with 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
Background
stars
1AU
Star
In essence the same method can be used to measure the distance to a
close-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 be
the diameter of the Earth’s orbit.
Stellar Parallax
Background
stars
1AU
Star
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
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
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.
Parallax
 From Earth we can only use this technique for about 10,000 stars.
 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 pc
is 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 with
8kpc to the centre of the Milky Way.We need other methods!!
Proper Motion of Stars
v
vT
θ
vR
Star
In discussing the parallax method of
measuring stellar distances we have ignored
r
their proper motion. In addition to the annual
back and forth motion many stars very
Earth
slowly change their positions relative to the
background due to their own motion in space.The component of this
motion 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 motion
we 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 Doppler
shift]can be combined to give the star’s velocity in space.
Proper Motion Contd.
v
vT
θ vR
Remember arc = rθ
Star
r
Earth
In a time t the star moves a distance d
at 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 r
Proxima Centauri-Proper Motion 3.85// and parallax 0.77//
Overall this method is limited to about 120,000 stars.
Parallax
 From Earth we can only use this technique for about 10,000 stars.
 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
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
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 4R2 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.I
taking 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 we
can 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 account
of any possible absorption in dust between us and the other star.]
[Note: This not an SI system of units]
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?
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 ) = -26.7
Now 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