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Astrophysics. E3 Stellar Distances. Stellar Distances Units. Earth in June. Distant stars. Near star. 1 AU. d. θ. Sun. θ. 1 AU. Earth in January. Stellar parallax. d = 1 AU θ. d (parsecs) = 1 p (arc-seconds). Angular Sizes 360 degrees ( 360 o ) in a circle - PowerPoint PPT Presentation
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Stellar Distances
Units
Unit Definition In meters
Astronomical unit
Average distance from the earth to the sun. 1AU ≈ 1.50 x 1011 m
Light Year Distance travelled by light in one year 1 ly = 9.46 x 1015 m
Parsec
Distance from Earth to a star that moves through one second of arc (1/3600 degrees) in half a year
1pc = 3.26 ly
Stellar parallax
1 AU
1 AU
Earth in June
Earth in January
Sun d
Distant starsNear
star
θθ
d = 1 AU
θd (parsecs) = 1
p (arc-seconds)
Angular Sizes
360 degrees (360o) in a circle
60 arcminutes (60’) in a degree
60 arcseconds (60”) in an arcminute
Eventually, the movement of the star is too small to see.
The farther away the star is, the smaller its parallax angle.
Stellar Parallax has its limits
Cephied Variables
The luminosity of a Cepheid varies periodically:
Changes in the surface and atmosphere cause the star to increase in surface area and thus increase in luminosity periodically.
Lum
inos
ity
There is a direct relationship between the peak luminosity of Cepheids and their time periods (‘the luminosity-period relationship’):
P eriod /d ay s
L um ino sity
10000
1000
1001 10 100
Peak luminosity
/ L⊙
The distance to a Cepheid can be found by...
i. Measure the average period of luminosity
ii. Use the relationship graph to find peak luminosity from period
iii. Measure peak apparent brightness on Earth
iv. Find distance using... b = L
4 π d2
Standard Candles
Thus the existence of a Cepheid variable star in a distant galaxy enables the luminosity of all the stars in the galaxy to be determined.
A standard candle is a star of known luminosity. This means that the luminosity of all other stars in its galaxy can be estimated by comparing their apparent brightness with the standard candle.
The Magnitude Scale
Seen from earth, different stars have different brightness...
- Apparent brightness gives us a measurement in standard units.
- Apparent magnitude give us a measurement on a logarithmic scale, relative to other stellar bodies.
Greek Magnitude Scale
The ancient Greeks first came up with a magnitude scale for stars. They classified the apparent magnitudes by rating the brightest star they could see as magnitude 1 and the faintest as magnitude 6, decreasing by a factor of 0.5 each magnitude.
Magnitude Description
1st The 20 brightest stars
2nd Half as bright as the 1st
3rd Half as bright as the 2nd
4th … and so on ...
5th … getting dimmer each time, until
6th … the dimmest stars (32 times dimmer than the brightest… so they thought)
So the 6th magnitude stars were...½ x ½ x ½ x ½ x ½ = (½)5 = 1/32 as bright as the 1st magnitude
More recently it was thought that the brightest stars visible from earth are about 100 times brighter that the dimmest. So, using the same classification of m = 1 to 6:
Fractional change = 5√(1/100)
= 1 / 2.512
Each apparent magnitude on the modern scale is 2.512 times dimmer than the previous
Apparent Magnitud
e (m)
Description
1 Thought to be brightest
2 2.512 times less bright than m=1
3 2.512 times less bright than m=2
4 etc
5 etc
6 Thought to be dimmest
Modern Apparent Magnitude Scale
Over time, adjustments have had to made to account for dimmer and brighter stars.
e.g.
Vega has apparent magnitude 0, and Sirius (the brightest star) has a negative magnitude (-1.4).
So for the apparent magnitude scale...
- the bigger the positive number the dimmer the star
- the brightest stars have negative numbers
Calculations involving apparent brightness and apparent magnitude
For any star, the relationship between its apparent brightness b1 and apparent magnitude m1 is...
b1
b0 = 2.512-m1
b0 = 2.52 x 10-8 Wm-2 i.e. the apparent brightness of a star with apparent magnitude zero.
For a second star with magnitude and brightness m2 and b2...
So the ratio of the brightness of the two stars is given by...
b2
b0 = 2.512-m2
b1
b2 = 2.512m2-m1
Absolute Magnitude
Two stars of the same apparent magnitude (i.e. also of the same apparent brightness) may not actually give out energy at the same rate; it could simply be that one is nearer to Earth than the other, brighter star.
The absolute magnitude scale gives the magnitude of all stars measured at an equal distance of 10 pc from the observer. Thus it is a measure of luminosity of a star.
It can be shown that for a star at distance d from earth of apparent magnitude m (from earth) and absolute magnitude M (from 10 pc)...
... where d is measured in parsecs.
m – M = 5 lg (d/10)
Distance measurement
by parallax
d = 1 / pLuminosity
L = 4πd2 b
apparent brightness
spectrum
Wien’s Law (surface
temperature T)
Chemical composition
of corona
L = 4πR2 σT4
Stefan-Boltzmann
Radius
Distance measured by parallax:
Apparent brightness
Distance (d)
b = L / 4πd2
Luminosity class
spectrum
Surface temperature (T)
Wien’s Law
Chemical composition
Stefan-Boltzmann
L = 4πR2 σT4
Radius
Distance measured by spectroscopic parallax / Cepheid variables:
H-R diagram
Spectral type
Luminosity (L)
Period
Cepheid variable