Finding an expression for the total mass of a star

[Physics Dad]

I have made an attempt at the question below and just wanted to check if my thinking was correct or totally wrong.

Thank you in advance!

1. Homework Statement
In a simple model of a star, the density is described as ρ=ρ₀(1-r/R), where R is the stellar radius, and ρ₀ is the central density.

Find an expression for the total mass of the star M in terms of ρ₀ and R.

Homework Equations

Mass Continuity
dM/dr=4πr²ρ(r)

The Attempt at a Solution

First of all, I know that the density of the star will scale with the radius, as will the mass, so I know I will need to integrate with respect to r from r=0 to r=R (from the centre of the star to the stellar radius).

If I place the equation for ρ₀ into the equation for mass continuity...

dM/dr=4πr²ρ₀(1-r/R)

and then tidy up a little...

dM=4πρ₀(r²-r³/R) dr

then integrate from 0 to R...

M=4πρ₀(R³/3-R⁴/4R)

then tidy up to give...

M=(π/3)ρ₀R³

And as π/3 is constant, I can say that...

M∝ρ₀R³

This seems a little too simplistic, so would really appreciate some feedback.

Thank you all!

 

[Phylosopher]

I don't have a background in astrophysics. But from what is given, your answer is correct.

Nonetheless, your solution have a hidden assumption. Which is that $\rho_{o}$ is independent from $R$.
In other words, your assumption is that the central density $\rho_{o}$ for all stars being modeled is the same, and so the stars central density independent from their radius $R$