= Solution
A <singular isothermal sphere> has total density and circular speed
$$
\rho_{\rm tot}(r)=\frac{\sigma^2}{2\pi Gr^2},
\qquad v_c=\sqrt2\,\sigma.
$$
If gas traces the gravitating matter, $\rho_g=f_{\rm gas}\rho_{\rm tot}$. A moving black hole gravitationally focuses that gas, so the relevant estimate is the <Bondi--Hoyle--Lyttleton accretion rate>
$$
\dot M\simeq
\frac{4\pi G^2M_{\rm BH}^2\rho_g}
{(v_c^2+c_s^2)^{3/2}}.
$$
Hydrostatic gas in the isothermal potential has $c_s$ of order $\sigma$, and therefore
$$
\boxed{\dot M\sim
\frac{f_{\rm gas}G M_{\rm BH}^2}{\sigma r^2}}
$$
up to an order-one coefficient. Taking $c_s=\sigma$ and $v_c=\sqrt2\sigma$ gives $2/(3\sqrt3)$ multiplying the displayed expression. The same $G^2M^2\rho/v^3$ 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.
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