= Spherical free-fall accretion rate
{title2=$\dot M=4\pi\rho\sqrt{2GM}\,r^{3/2}$}
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>.
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