Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 314 / 3 / b / Solution

Let $u>0$ denote inward radial speed. Steady spherical mass conservation and the radial momentum equation giveA regular sonic point requires both factors to vanish:so $c_{s,s}^2=4\pi G{\Psi }_0/\sqrt{r_s}$ and ${\Phi }_{\mathrm{dm}}(r_s)=-4c_{s,s}^2$.

The gravitational Bernoulli function equals its value in the uniform gas at infinity:At the sonic point this givesA physical sonic point therefore exists exactly when