Free-fall density profile around a growing point mass
= Free-fall density profile around a growing point mass
{title2=$\rho\propto r^{-3/2}$}
Where a central <mass> dominates and radial infall is locally steady, $|v_r|\simeq\sqrt{2GM_\star/r}$. A conserved <mass> flux $\dot M=4\pi r^2\rho|v_r|$ then gives $\rho\simeq\dot M[4\pi\sqrt{2GM_\star}]^{-1}r^{-3/2}$. Constant accretion gives $M_\star\propto t$ and an inner normalization proportional to $t^{-1/2}$.