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

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 $c_0^{-2}$.