Spherical free-fall accretion rate ($\dot{M}=4\pi \rho \sqrt{2GM}{r}^{3/2}$)

A steady full-sphere radial inflow has mass flux $\dot{M}=4\pi r^2\rho |v_r|$ by conservation of mass. With Newtonian free-fall speed from rest at infinity, $|v_r|=\sqrt{2GM/r}$, this becomes the displayed expression. The assumed velocity then requires $\rho \propto r^{-3/2}$ along the steady flow. A local density and speed do not determine the total mass accretion rate of a general nonspherical inflow, and this estimate does not impose the sonic boundary condition of Bondi accretion.