= Solution
If $c_{s,\infty}\gg\sigma$ and $M_{\rm enc}\to M$, expansion of the denominator gives
$$
\boxed{\dot M\simeq
\pi\frac{G^2M^2\rho_\infty}{c_{s,\infty}^3},}
$$
the stated model's pressure-dominated <Bondi accretion> limit. Its order-unity coefficient differs from the $\pi e^{3/2}$ found for the exactly isothermal critical solution because such coefficients depend on the adopted equation of state and interpolation.
If instead $\sigma\gg c_{s,\infty}$,
$$
\boxed{\dot M\simeq
\pi\frac{G^2M_{\rm enc}^2\rho_\infty}{\sigma^3}.}
$$
The gas's thermal motion is then negligible beside the galactic <velocity dispersion>, and the enclosed galactic mass, rather than the black hole alone, focuses the gas. For a <singular isothermal sphere>, $M_{\rm enc}(R)=2\sigma^2R/G$, so this becomes $\dot M\simeq4\pi\rho_\infty\sigma R^2$. This second limit describes capture controlled by the host potential and is therefore not a genuinely spherical black-hole Bondi solution.
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