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 $r\geq r_s$, so it is enough to demand that the black hole be negligible at the sonic point:
$$
\frac{GM}{r_s^2}\ll\frac{8\pi G\Psi_0}{r_s^{3/2}}.
$$
Equivalently,
$$
\boxed{
M\ll8\pi\Psi_0\sqrt{r_s}
=\frac{16\pi^2G\Psi_0^2(9-7\gamma)}{c_0^2}}.
$$
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.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-314/3/d/solution