Star collapse in general relativity — pressure as a function of star radius

[Lilian Sa]

Homework Statement

How can I find the pressure as a function of the radius of a star that have a constant energy density, spherical symmetric, its initial radius is R and the total mass is M?

Relevant Equations

TOV equations.
$ds^2=-e^{\nu(r)}dt^2+r^{\lambda(r)}dr^2+r^2d\Omega^2$

What I've done is using the TOV equations and I what I found at the end is:
$e^{[\frac{-8}{3}\pi G\rho]r^2+[\frac{16}{9}(G\pi\rho)^{2}]r^4}-\rho=P(r)$
so I am sure that this is not right, if someone can help me knowing it I really apricate it :)

 

[etotheipi]

How can I find the pressure as a function of the radius of a star that have a constant energy density, spherical symmetric, its initial radius is R and the total mass is M?

I haven't worked this through. The TOV equations require gravitational equilibrium, so I presume that's what you're interested and not the actual collapse (which I have no idea how to deal with)? You'd need a barotropic equation of state $p = p(\rho)$; does the problem statement supply one? Then you solve the coupled system\begin{align*}

\frac{dm}{dr} &= 4\pi r^2 \rho \\

\frac{dp}{dr} &= \frac{-(p+\rho)(m + 4\pi r^3 p)}{r(r-2m)}

\end{align*}in $m(r)$ and $p(r)$. You can set $m(0) = 0$