 Free-fall density profile around a growing point mass

Where a central mass dominates and radial infall is locally steady, $|v_r|\simeq \sqrt{2G{M}_{\star }/r}$. A conserved mass flux $\dot{M}=4\pi r^2\rho |v_r|$ then gives $\rho \simeq \dot{M}[4\pi \sqrt{2G{M}_{\star }}{]}^{-1}{r}^{-3/2}$. Constant accretion gives $M_{\star }\propto t$ and an inner normalization proportional to $t^{-1/2}$.