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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.

inside-out collapse of a singular isothermal sphere starts from an idealized static envelope,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:“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:The self-similar accretion scale is ${\dot{M}}_{\ast }=m_0{a}_s^3/G$, 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, ${\dot{M}}_{\ast }\propto T^{3/2}$. The original similarity calculation establishes the static outer and free-fall inner regimes.^[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.

A dense cloud core first collapses when self-gravity overwhelms its effective support. While radiation escapes readily, compression can remain approximately isothermal and the Jeans mass decreases as density rises, encouraging gravitational fragmentation. Angular momentum usually leads to a disk rather than unrestricted spherical infall.

As the central optical depth increases, cooling becomes slower than compression and pressure produces a first hydrostatic core. molecular hydrogen dissociation near $2000\mathrm{K}$ absorbs compressional energy and lowers the effective adiabatic response, permitting a second collapse of a protostar. After dissociation the gas stiffens and a much more compact stellar core forms. An accretion shock and surrounding disk then convert infall energy into radiation. A small central core forms before most of the envelope arrives, as demonstrated in radiation-hydrodynamic collapse calculations.^[ref]

The isothermal inside-out collapse of a singular isothermal sphere model provides one idealized supply law, $\dot{M}\sim a_T^3/G$, but rotation, magnetic fields and an unsteady reservoir modify real accretion. Its luminosity scale is $L_{\mathrm{acc}}\sim GM\dot{M}/r_{\ast }$. Dust can absorb the emerging short-wavelength radiation and re-emit it in the infrared, so an embedded source need not have a directly visible stellar photosphere.

As envelope accretion declines, gravitational contraction becomes important on the Kelvin-Helmholtz cooling time $t_{\mathrm{KH}}\sim G{M}^2/(r_{\ast }L)$. deuterium-burning energy-generation rate can temporarily influence the radius; sustained core hydrogen burning eventually establishes the main sequence if the final mass is sufficient. A low-mass convective object approaches along a Hayashi-like track, while more massive objects need not share that track. accretion, contraction and nuclear burning are distinct energy sources, and a protostar is not generally a fixed-mass object.

For inside-out collapse of a singular isothermal sphere, the expansion front is at $r_f=a_{T}t$. The original mass swept by this front isThus the only time-independent accretion scale is $a_T^3/G$. The front-sweeping estimate has coefficient two, but gas takes time to fall from the front to the object; it is not an exact central accretion coefficient. Writefor fixed mean particle mass $\mu m_H$. Determining the dimensionless coefficient $m_0$ requires the collapse solution; it is not fixed by front propagation alone and is not needed for the requested temperature scaling.^[ref]

Well inside the front, where the point mass dominates and infall is approximately pressureless and locally steady, $|v_r|\simeq \sqrt{2G{M}_{\star }/r}$. Conserving the radial mass flux gives the free-fall density profile around a growing point mass:The inner slope is $-3/2$, versus $-2$ in the undisturbed outer cloud. For constant ${\dot{M}}_{\star }$ and negligible initial object mass, $M_{\star }\propto t$ and the inner normalization varies as $t^{-1/2}$. These are inner asymptotes, not an assertion of an abrupt exact power-law join at the front.