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Calculating the Radius of a Star

Calculating a star's radius is a somewhat lengthy process. You have to put together many tools that you have developed in various SkyServer projects. Even the largest star is so far away that it appears as a single point from the surface of the Earth - its radius cannot be measured directly. Fortunately, understanding a star's luminosity provides you with the tools necessary to calculate its radius from easily measured quantities.

A star's luminosity, or total power given off, is related to two of its properties: its temperature and surface area. If two stars have the same surface area, the hotter one will give off more radiation. If two stars have the same temperature, the one with more surface area will give off more radiation. The surface area of a star is directly related to the square of its radius (assuming a spherical star).

The luminosity of a star is given by the equation

L = 4pR2s T4,

Where L is the luminosity in Watts, R is the radius in meters, s is the Stefan-Boltzmann constant
(5.67 x 10-8 Wm-2K-4), and T is the star's surface temperature in Kelvin.

The temperature of a star is related to its b-v magnitude. The table below can help you find the temperature of the star based on its b-v magnitude.

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.