Isothermal Bondi accretion rate ($\dot{M}=\pi e^{3/2}{G}^2{M}^2{\rho }_{\infty }/c_s^3$)

The regular spherical isothermal inflow from mass density ${\rho }_{\infty }$ at rest at infinity has sonic mass density $e^{3/2}{\rho }_{\infty }$. Combining it with the Bondi sonic point and mass conservation gives $\dot{M}=\pi e^{3/2}{G}^2{M}^2{\rho }_{\infty }/c_s^3$.