Combining the two expressions for the sonic sound speed gives
$$
\boxed{
r_s=\left[\frac{2\pi G\Psi_0(9-7\gamma)}{c_0^2}\right]^2}.
$$
For a polytropic gas, $c_s^2\propto\rho^{\gamma-1}$, and hence
$$
\rho_s=\rho_0\left(\frac{2}{9-7\gamma}\right)^{1/(\gamma-1)},
\qquad
u_s=c_0\left(\frac{2}{9-7\gamma}\right)^{1/2}.
$$
The <transonic accretion in a steep dark-matter cusp> has <mass accretion rate>
$$
\boxed{
\dot M=4\pi r_s^2\rho_su_s
=64\pi^5G^4\Psi_0^4\rho_0c_0^{-7}
(9-7\gamma)^4
\left(\frac{2}{9-7\gamma}\right)^{\frac1{\gamma-1}+\frac12}}.
$$
Thus $\dot M\propto c_0^{-7}$, whereas classical <Bondi accretion> onto a point mass has $\dot M_{\rm B}\propto c_0^{-3}$. The steeper dependence comes from the cusp potential, whose sonic radius scales as $c_0^{-4}$ instead of $c_0^{-2}$.

Solution

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

{title2=Transonic accretion in an $r^{-5/2}$ density cusp}

For a background gravitating density $\rho_{\rm dm}=\Psi_0r^{-5/2}$, the potential is $\Phi=-16\pi G\Psi_0r^{-1/2}$. A steady polytropic inflow has a regular sonic point only for $1<\gamma<9/7$, and its accretion rate scales as $\dot M\propto\rho_0G^4\Psi_0^4c_0^{-7}$.

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 Incoming links: Transonic accretion in a steep dark-matter cusp