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.

An outward collapse front and inward central accretion measure different parts of the same mass budget. In the inside-out collapse of a singular isothermal sphere, the unperturbed profile has $\rho =a_s^2/(2\pi G{r}^2)$ and the front is at $r_f=a_{s}t$. Its swept mass rate is therefore $4\pi r_f^2\rho (r_f){\dot{r}}_f=2a_s^3/G$.

The central rate is instead ${\dot{M}}_{\ast }=m_0{a}_s^3/G$, with $m_0$ determined by the dynamical solution. Material already crossed by the front can remain in the infalling envelope, so the two rates need not agree. Equating them would silently assume an empty reservoir between the front and the central object.^[ref]

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 $t^{-1/2}$.