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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]

Frank Shu developed a mathematical account of star formation through the gravitational collapse of isothermal gas spheres. His 1977 treatment used self-similarity to describe the flow after a central core had formed, reducing the evolving spatial problem to functions of a scaled coordinate.

In the inside-out collapse solution, an outward-moving expansion wave separates the still-static outer gas from an infalling inner region. Near the center the flow approaches free fall, and accreting material encounters a radiating shock wave at the core. The construction connects the initial cloud structure to the subsequent supply of mass to a growing protostar under explicit symmetry and temperature assumptions.^[ref]

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.

mass and energy signs. The enclosed mass is $M(r)={\int }_0^{r}4\pi s^2\rho (s)ds$, so $dM/dr=+4\pi r^2\rho$. The negative sign printed in that equation cannot describe positive enclosed mass. Use the physical sign and define gravitational potential energy by

Boundary-pressure virial theorem. Multiply hydrostatic equilibrium by $4\pi r^3$ and integrate. The left side is $[4\pi r^{3}p{]}_0^{r_0}-3{\int }_0^{r_0}4\pi r^{2}pdr$, while the right side is $\Omega$. The central boundary contribution vanishes when $r^{3}p\to 0$ and the outer pressure is $p(r_0)=p_0$. ThusA finite external pressure is compatible with a gravitationally self-contained cloud, but is not the zero-surface-force notion of isolation. External ram stress or mass flow would require additional terms.

radiation budget of monatomic cloud collapse. For an ideal gas with fixed $\gamma$, the specific internal energy is $u=p/[\rho (\gamma -1)]$. Therefore the pressure integral is $3(\gamma -1)U$. Cold monatomic hydrogen has $\gamma =5/3$ when only translational degrees of freedom matter. With negligible final surface pressure, $2U+\Omega =0$, so $E_f=U+\Omega =\Omega /2$. If the initial cloud is sufficiently extended and cold that $E_i\simeq 0$, the emitted energy isBeing initially at rest alone does not set its binding energy to zero. If the initial total energy is appreciable, retain $E_{\mathrm{rad}}=E_i-\Omega /2$; a different equation of state also changes the half-energy result.

Inside-out accretion. The combination of $a_T$ and $G$ with dimensions of mass per time iswhere dimensional analysis leaves the dimensionless coefficient $C$ undetermined. An isothermal gas keeps $a_T$ constant, giving constant supply in the self-similar idealization. A front at radius $a_{T}t$ and an isothermal mass scale $M(r)\sim a_T^{2}r/G$ give the same scaling. The front's swept mass is not automatically the accreted central mass; the coefficient requires a dynamical similarity solution.^[ref]

constant-radius accretion energy budget. If all available accretion power is radiated at the fixed radius, $L_{\mathrm{acc}}=GM\dot{M}/r_0$. Neglecting the seed mass,This equals $-\Omega /2$ precisely whenThe positive sign in the final printed energy condition is inconsistent with the earlier use of negative gravitational potential energy. Equivalently its positive right side must denote the binding-energy magnitude $|\Omega |$. A general structural coefficient $\Omega =-qG{M}^2/r_0$ leaves a mismatch unless $q=1$ under these other idealizations. Radiation trapped in the object, outflows or release over a different radius also change the budget.

Which temperature? With $M=M_s+\dot{M}t$ and fixed $r_0$, $L_{\mathrm{acc}}\propto M$. If this power emerges thermally from area $4\pi r_0^2$, the accretion-powered effective-temperature scaling iswhen the seed mass is negligible. This is the radiating surface's effective temperature, inferred from the preceding accretion-luminosity calculation. A homologous ideal-gas interior instead has virial temperature $T_c\sim \mu m_{H}GM/(k_B{r}_0)\propto t$. Thus the surface-effective and interior-virial scalings are $t^{1/4}$ and $t$, respectively, under their stated assumptions. Inferring a detailed interior temperature profile would additionally require a structural and transport model.

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.