= 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 $\rho=a_s^2/(2\pi Gr^2)$ and the front is at $r_f=a_st$. 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 $\dot M_*=m_0a_s^3/G$, 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.<Self-Similar Collapse of Isothermal Spheres and Star Formation by Frank H. Shu>[Similarity solution and the distinction between expansion wave and central inflow]{ref}
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