b-v |
Surface Temperature
(Kelvin) |
-0.31 |
34,000 |
-0.24 |
23,000 |
-0.20 |
18,500 |
-0.12 |
13,000 |
0.0 |
9500 |
0.15 |
8500 |
0.29 |
7300 |
0.42 |
6600 |
0.58 |
5900 |
0.69 |
5600 |
0.85 |
5100 |
1.16 |
4200 |
1.42 |
3700 |
1.61 |
3000 |
The calculation is actually somewhat easier
if we try to find the ratio of another star's radius to that of our Sun.
Let Ls be the luminosity of the Sun, L be the luminosity of
another star, Ts be the temperature of the Sun, T be the
temperature of the other star, Rs be the radius of the Sun, and
R be the radius of the other star.
We can then write the ratio of their luminosities as
L/Ls = (4pR2sT4)/(4pRs2sTs4)
= (R/Rs)2(T/Ts)4
Solving for the ratio R/Rs yields
R/Rs = (Ts/T)2(L/Ls)1/2
The temperatures can be found approximately from the
table above by looking at the B-V values. To find the ratio L/Ls,
we can use the absolute magnitudes of the stars. The magnitude scale
is a logarithmic scale. For every decrease in brightness of 1
magnitude, the star is 2.51 times as bright. Therefore, L/Ls
can be found from the equation
L/Ls = 2.51Dm,
where Dm = ms - m
Let's look at the star Sirius. It has visual
magnitude of -1.44, B-V of .009, and a parallax of 379.21 milli arc
seconds. Finding its distance from its parallax yields
d = 1/p = 1/.37921 = 2.63 parsecs.
Its absolute magnitude is
M = m - 5 log d + 5 = -1.44 - 5 log (2.63) + 5 = 1.46
We know the temperature of the Sun is 5800K. From the chart, the
temperature of Sirius is about 9500K. Our Sun has an absolute
magnitude of 4.83. The difference in magnitude is
3.37. Putting everything together yields
R/Rs = (5800/9500)2(2.5123.37)1/2
= 1.76
Sirius has a radius approximately 1.76 times that of our Sun!
Supplemental
Exercise 1. Find the
radii for the stars given below. Look up their v, b-v, and
parallax in the
Hipparcos database.
Star Name |
RA |
Dec |
Radius |
Betelgeuse |
88.79 |
7.41 |
|
Barnard's Star |
269.5 |
4.6 |
|
Vega |
279.23 |
38.78 |
|
Polaris |
37.95 |
89.26 |
|
|
Supplemental Exercise 2.
Look up several stars in the
Hipparcos database. Try to find a variety of stars
ranging from white dwarfs to supergiants. Find the radii of
these stars. |