Expansion-front mass flux versus central accretion

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]