Mass-Luminosity-Time relation (Astronomy)

[Unto]

Homework Statement

The lifetime of the sun on the main sequenceof the hertzprung-russel diagram is estimated to be 10^10 years. Assuming that the luminosities of stars on the upper diagram main sequence vary as mass^4, estimate the solar mass of a star for which it's main squence lifetime is 10^6 years.

Homework Equations

L = k M^b
T = kM / L = k M^1-b
T = k M^-2

'=k' is a subsitute for the proportional symbol
b = beta

The Attempt at a Solution

I know the star is going to be heavier, but I'm not given any values for luminosity. And when I try to equate time to mass I get funny answers. A push in the right direction here?

 

[Unto]

K the sun's luminosity is 3.9E26 W, I figured I will use this to help me solve the answer.

 

[Jenab6]

The mass luminosity relation isn't a physical model; it's an empirical curvefit. The best fit isn't necessarily a pure power function.

It can be

L/L๏ = K (M/M๏)^a

But K isn't required to be 1.

It can also be more complicated, such as

L/L๏ = K₁ (M/M๏)^a₁ + K₂ (M/M๏)^a₂

For example, the mass luminosity relation for the mass interval from 2 suns to 20 suns is approximately

L/L๏ = 1.505964 (M/M๏)^3.5 − 0.0252982 (M/M๏)^4.5