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

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.