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.

 

[Nickie]

I would approach this problem by first understanding the mass-luminosity relationship and how it relates to the lifetime of a star. The mass-luminosity relationship states that the luminosity (L) of a star is directly proportional to the mass (M) raised to the power of an exponent (α). This relationship is given by L*/Lsun = (M*/Msun)^α, where L*/Lsun is the luminosity of the star in comparison to the Sun's luminosity.

From this equation, we can see that as the mass of a star increases, its luminosity also increases. This is because larger stars have more mass and therefore more gravitational energy, which is converted into light and heat. However, as a star ages, it begins to run out of fuel and its luminosity decreases. This is known as the main sequence stage of a star's life.

Now, we can use the given exponent, α=3.0, to estimate the lifetime of the star. The exponent relates to the mass of the star, which is given as 20Msun. This means that the star has 20 times the mass of the Sun. Using the mass-luminosity relationship, we can calculate the luminosity of the star as L*/Lsun = (20Msun/Msun)^3 = 8000. This means that the star has 8000 times the luminosity of the Sun.

Next, we can use the fact that the lifetime of a star is inversely proportional to its luminosity. This means that as the luminosity of the star increases, its lifetime decreases. We can use this relationship to estimate the lifetime of the star by comparing it to the Sun's lifetime, which is given as 1 x 10^10 years.

Using the proportionality relationship, we can write:

Lifetime of the star/Lifetime of the Sun = Luminosity of the star/Luminosity of the Sun

Substituting the values we have calculated, we get:

Lifetime of the star/1 x 10^10 years = 8000/1

Rearranging for the lifetime of the star, we get:

Lifetime of the star = 8000 x 1 x 10^10 years = 8 x 10^13 years

However, this is the lifetime of the star in comparison to the Sun's lifetime. To get the actual estimated lifetime of the star, we need to