Astrophysics and astronomy equations

Question

Star A and star B are both on the main sequence. Star A is 56 times more luminous than star B. What is the ratio of their main-sequence lifetimes? Refer to the stellar life expectancies equation

Answer 1

I search the web for "stellar life expectancies equation" and find t = t_solar / M^2.5, where M is mass in units of solar masses. So we need to use mass-luminosity relationship, which Wikipedia gives as L / L_solar = (M/M_solar)^a, where exponent a = 3.5 for most of the main sequence.

With a luminosity ratio of 56 between stars A and B, we can set up a ratio in which M_solar cancels out: L_A/L_B = (M_A/M_B)^3.5 yielding M_A/M_B = (56)^(1/3.5) = 3.16.

Now do the ratio of lifetimes t_A / t_B = (M_B^2.5) / (M_A^2.5) = (M_A/M_B)^-2.5 = 0.056. The more luminous and more massive star of course has a shorter lifetime.