Calculating the Radius of a Star
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 = 4pR^{2}s T^{4},
Where L is the luminosity in Watts, R is the radius in meters, s is the StefanBoltzmann constant
(5.67 x 10^{8} Wm^{2}K^{4}), and T is the star’s surface temperature in Kelvin.
The temperature of a star is related to its bv magnitude. The table below can help you find the temperature of the star based on its bv magnitude.
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 L_{s} be the luminosity of the Sun, L be the luminosity of another star, T_{s} be the temperature of the Sun, T be the temperature of the other star, R_{s} 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/L_{s} = (4pR^{2}sT^{4})/(4pR_{s}2sT_{s}4) = (R/R_{s})^{2}(T/T_{s})^{4}
Solving for the ratio R/R_{s} yields
R/R_{s} = (T_{s}/T)^{2}(L/L_{s})^{1/2}
The temperatures can be found approximately from the table above by looking at the BV values. To find the ratio L/L_{s}, 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/L_{s} can be found from the equation
L/L_{s} = 2.51^{Dm},
where Dm = m_{s} – m
Let’s look at the star Sirius. It has visual magnitude of 1.44, BV 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/R_{s} = (5800/9500)^{2}(2.512^{3.37})^{1/2} = 1.76
Sirius has a radius approximately 1.76 times that of our Sun!
bv, 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 