Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 54 / 1 / b / i

For the constant-density model, $m(r)=4\pi \rho r^3/3=M{r}^3/R^3$. This is the incompressible Interior Schwarzschild metric, replacing the earlier finite-slope barotropic stellar equation of state. The Tolman–Oppenheimer–Volkoff equation separates asOn the left, partial fractions giveOn the right the antiderivative is $\frac{3}{4\rho }\log (1-8\pi \rho r^2/3)$. Applying the surface condition $p(R)=0$, and writing $s(r)=\sqrt{1-2M{r}^2/R^3}$, $s_R=\sqrt{1-2M/R}$, givesSolving for the pressure recovers the Interior Schwarzschild solution:In particular the central pressure of a constant-density relativistic star isThe denominator must be positive for a regular positive-pressure centre, requiring $M/R<4/9$. As that limiting compactness is approached, the central pressure diverges, in agreement with Buchdahl's theorem.