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

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.