 Solution

If $c_{s,\infty }\gg \sigma$ and $M_{\mathrm{enc}}\to M$, expansion of the denominator givesthe 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 $\sigma \gg c_{s,\infty }$,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, $M_{\mathrm{enc}}(R)=2{\sigma }^{2}R/G$, so this becomes $\dot{M}\simeq 4\pi {\rho }_{\infty }\sigma R^2$. This second limit describes capture controlled by the host potential and is therefore not a genuinely spherical black-hole Bondi solution.