Stellar Sizes from Luminosities

If we assume stars to be spherical blackbody radiators we can often infer their sizes from their luminosities. By using the Stefan-Boltzmann Law the total luminosity (energy emitted per second from the surface of the star) is given by multiplying the energy emitted from each square centimeter of the surface per second by the total number of centimeters on the surface:

Luminosity = c x (surface area) x T 4
= c x 4 x pi x R 2 x T 4
where c is a constant and pi = 3.1416, T is the temperature in K, and R is the radius of the star. Thus, the ratio of the luminosities for two stars (labeled by 1 and 2) is

L 1 / L 2 = ( c x 4 x pi x R12 x T14 ) / ( c x 4 x pi x R 22 x T24 )

Cancelling like factors and rearranging, we may thus write

L 1 / L 2 = ( T 1 / T 2 ) 4 x ( R 1 / R 2 ) 2

The utility of this expression is that we can often estimate T for a star from its spectrum and measure its luminosity L; this then allows us to calculate stellar radii (if we assume that they are spherical blackbodies).

  1. Suppose Star 1 and Star 2 have the same surface temperatures, but Star 1 is 100 times more luminous that Star 2. How much larger is Star 1 than Star 2?

  2. Suppose Star 1 and Star 2 have the same luminosities but their surface temperatures are 10,000 K and 5000 K, respectively. What is the ratio of their radii?
In both of these exercises, assume the two stars to be spherical and to be blackbodies.

The Solution