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.

For a singular isothermal sphere with $\rho =a_s^2/(2\pi G{r}^2)$, the initial mass inside the outward front $r_{\mathrm{ew}}=a_{s}t$ is $M_{\mathrm{ew}}=2a_s^{3}t/G$. This is not all immediately accreted by the central object. The inside-out collapse of a singular isothermal sphere instead has ${\dot{M}}_{\ast }=m_0{a}_s^3/G$, with $m_0$ fixed by its dynamical similarity solution; the remainder can stay in the infalling envelope. Equating front sweeping with central accretion loses that mass budget.

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 $t^{-1/2}$.