Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 311 / 1

Outside the star, $p=\rho =0$. The first Tolman–Oppenheimer–Volkoff equation gives $m(r)=M$, and the second givesAsymptotic flatness fixes the integration constant, so $e^{2\Phi }=1-2M/r$. The exterior line element is therefore the Schwarzschild metric with mass $M=m(R)$.

Regularity at the center requires $m(0)=0$ and a finite central density ${\rho }_c$. The equation of state fixes $p_c=p({\rho }_c)$, and the regular central series beginsFor each admissible ${\rho }_c$, ordinary-differential-equation uniqueness determines $m,p,\rho$ outward until the first zero of $p$, which defines $R$. The remaining additive constant in $\Phi$ is fixed by matching to the exterior Schwarzschild time coordinate. Thus smooth stars form a one-parameter family labelled uniquely by ${\rho }_c$.

Let $r_0$ be where the outward-decreasing density first reaches ${\rho }_0$ and set $p_0=p({\rho }_0)$. The quoted compactness inequality at $r_0$ bounds $m_0/r_0$ using only $p_0{r}_0^2$. Its right side must be positive. Writing $x=6\pi p_0{r}_0^2$, positivity of $1-x+\sqrt{1+x}$ implies $x<3$, henceBoth the radius and mass of the unknown high-density core are therefore bounded using only the known pressure $p_0$. From $(r_0,m_0,p_0)$ outward, the known low-density equation of state uniquely determines the envelope and adds only a bounded mass before $p$ reaches zero. Maximizing over the resulting bounded set of admissible initial data gives a finite maximum stellar mass independent of the equation of state above ${\rho }_0$.

For constant density,Substitution in the pressure equation makes it separable. Integrating from $r$ to the surface, where $p(R)=0$, gives the Interior Schwarzschild solutionDirect differentiation verifies the TOV equation and the surface boundary condition.

Set $C=M/R$ and $s=\sqrt{1-2C}$. At the center,The barotropic bound $p_c\le \omega {\rho }_0$ impliesThereforeAs $\omega \to \infty$, this approaches $4/9$, the limiting value in Buchdahl's theorem. Constant-density stars with increasingly large allowed central pressure can therefore approach the Buchdahl bound arbitrarily closely.