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

Spherical symmetry reduces the Poisson equation for gravity toAfter one integration,The omitted integration constant would represent a point mass and is absent for the dark-matter configuration alone. Choosing the additive constant so that the potential tends to zero at large radius gives

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

Combining the two expressions for the sonic sound speed givesFor a polytropic gas, $c_s^2\propto {\rho }^{\gamma -1}$, and henceThe transonic accretion in a steep dark-matter cusp has mass accretion rateThus $\dot{M}\propto c_0^{-7}$, whereas classical Bondi accretion onto a point mass has ${\dot{M}}_{\mathrm{B}}\propto c_0^{-3}$. The steeper dependence comes from the cusp potential, whose sonic radius scales as $c_0^{-4}$ instead of $c_0^{-2}$.

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.