 Inside-out collapse of a singular isothermal sphere

A collapse front propagates out through an initially singular isothermal envelope at $r_f=a_{T}t$. The only steady accretion scale is $\dot{M}\propto a_T^3/G\propto T^{3/2}$ for fixed mean particle mass. The front sweeps original mass at $2a_T^3/G$, but the exact central accretion coefficient requires the dynamical similarity solution and cannot be identified with front sweeping alone.