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

The black-hole acceleration is $GM/r^2$, while the dark-matter acceleration is $8\pi G{\Psi }_0/r^{3/2}$. Their ratio decreases outwards throughout the subsonic region $r\ge r_s$, so it is enough to demand that the black hole be negligible at the sonic point:Equivalently,Under this condition the subsonic solution and its sonic transition are controlled primarily by the dark-matter cusp, justifying the neglect of the central point-mass gravity.