= Solution
<inside-out collapse of a singular isothermal sphere> starts from an idealized static envelope,
$$
\rho(r)=\frac{a_s^2}{2\pi Gr^2},\qquad
M(r)=\frac{2a_s^2r}{G},
$$
with constant <isothermal sound speed> $a_s$. Collapse begins centrally after formation of a central <mass>. An expansion wave moves outward through the initially static envelope at the <sound speed>:
$$
\boxed{r_{\rm ew}=a_st.}
$$
“Expansion” describes the outward-moving information front; the <gas> behind it moves inward. The exterior remains close to its original $r^{-2}$ equilibrium until the wave reaches it.
In the inner region, the central <mass> dominates and <pressure> becomes less important:
$$
v_r\simeq-\sqrt{\frac{2GM_*}{r}},\qquad
\rho\simeq\frac{\dot M_*}{4\pi r^2|v_r|}\propto r^{-3/2}.
$$
The self-similar <accretion> scale is \b[$\dot M_*=m_0a_s^3/G$], where the <dimensionless> coefficient $m_0$ follows from the dynamical similarity solution, not from counting material swept by the front. Indeed, the initial <mass> enclosed by the front grows at $2a_s^3/G$, and some of that material remains in the infalling envelope. For fixed <mean particle mass>, $\dot M_*\propto T^{3/2}$. The original similarity calculation establishes the static outer and free-fall inner regimes.<Self-Similar Collapse of Isothermal Spheres and Star Formation by Frank H. Shu>[1977, pp. 488–497: expansion-wave collapse after central-core formation and the two radial density asymptotes]{ref}
The model assumes <spherical symmetry>, isothermality and negligible <rotation> or magnetic stresses. An asymmetric <optically thick> <spectral line> alone does not establish infall: an <optically thin> systemic-velocity tracer and spatial maps help distinguish inward motion from <rotation>, foreground <absorption> and outflow.
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