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 = 4R² T⁴,

is the Stefan-Boltzmann constant
(5.67 x 10^-8 Wm^-2K^-4), and T is the star's surface temperature in Kelvin.

R²T⁴)/(4R_s2T_s4) = (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 B-V 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_scan be found from the equation

L/L_s = 2.51^m,

where m = m_s - 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/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!