How Is a Star's Lifetime Calculated Using Mass-Luminosity Relationship?

[ZedCar]

Homework Statement

If the exponent, α, in the mass-luminosity relationship is 3.0, estimate the lifetime
of the above star. (You can assume that the Sun’s lifetime is 1 x 10^10 years).

Homework Equations

L*/Lsun = (M*/Msun)^α
lifetime ∝ M/L ∝ M^-2

The Attempt at a Solution

The solution is given as:

For a 20Msun star

Lifetime = 20^-2 x 1 x 10^10 yrs
= 25 x 10^6 yrs

Could anyone explain to me how this solution has been arrived at?

Thank you.

 

[HallsofIvy]

What you have written doesn't make much sense. Looking at "L*/Lsun = (M*/Msun)^α", since you don't tell us what the letters stand for, my first guess would be that L* is the lifetime sought, that Lsun is the lifetime of the sun (10^10 years), that M* is the mass of the star and Msun is the mass of the sun. But then you say "lifetime ∝ M/L ∝ M^-2" so apparently "L" is NOT the lifetime of the star. Then I don't know what that "L" could stand for. Can you clarify?

 

[ZedCar]

Sorry about that.

L stands for luminosity, as far as I'm aware

L* the luminosityof the star in question. Lsun that of the sun.
'M' I believe stands for the mass.

 

[haruspex]

You don't care about the luminosity here, so the relevant equation is:
lifetime ∝ M^-2
So a star with 20 times that of the sun will have 20^-2 = 1/400 times the lifetime.