= Expansion-wave mass budget
{title2=$M_{\rm ew}=2a_s^3t/G$}
For a <singular isothermal sphere> with $\rho=a_s^2/(2\pi Gr^2)$, the initial <mass> inside the outward front $r_{\rm ew}=a_st$ is $M_{\rm ew}=2a_s^3t/G$. This is not all immediately accreted by the central object. The <inside-out collapse of a singular isothermal sphere> instead has $\dot M_*=m_0a_s^3/G$, 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.
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