Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 14B / a / Solution

At the centre, spherical symmetry and regularity require a finite central density $\rho (0)={\rho }_c$ andAt the stellar surface $r=R$, the pressure vanishes:For a stellar polytrope these become the dimensionless conditions $\theta (0)=1$, ${\theta }^{\prime }(0)=0$, and the surface is the first zero $\theta ({\xi }_1)=0$; specifying ${\rho }_c$ fixes the overall scale.