 Solution

A singular isothermal sphere has total density and circular speedIf gas traces the gravitating matter, ${\rho }_g=f_{\mathrm{gas}}{\rho }_{\mathrm{tot}}$. A moving black hole gravitationally focuses that gas, so the relevant estimate is the Bondi--Hoyle--Lyttleton accretion rateHydrostatic gas in the isothermal potential has $c_s$ of order $\sigma$, and thereforeup to an order-one coefficient. Taking $c_s=\sigma$ and $v_c=\sqrt{2}\sigma$ gives $2/(3\sqrt{3})$ multiplying the displayed expression. The same $G^2{M}^2\rho /v^3$ scaling that produces Chandrasekhar dynamical friction therefore sets the likely capture rate as the hole spirals inward. Feedback, angular momentum, a gas core, or a multiphase medium can reduce the actual rate.