Astrophysics

E3 Stellar Distances

Stellar DistancesUnits

Unit Definition In metersAstronomical 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

Spectroscopic ParallaxText

Cephied VariablesThe 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’):

Period /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 ScaleSeen 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.

Asteroid ‘65 Cybele’ and 2 stars with their apparent magnitudes labelled

Greek Magnitude ScaleThe 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 magnitudeFor 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 MagnitudeTwo 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)

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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

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