Constant-radius accretion energy budget ($E_{\mathrm{acc}}=G(M^2-M_s^2)/(2r_0)$)

Integrating $L_{\mathrm{acc}}=GM\dot{M}/r_0$ at fixed radius gives this result. With negligible seed mass it supplies the radiative requirement of monatomic cloud collapse only if ${\Omega }_f=-G{M}^2/r_0$ under the adopted zero-initial-energy approximation. A different density-profile coefficient, retained accretion heat, a wind or a changing radius modifies the budget. Positive binding-energy magnitude and negative potential energy must not be interchanged silently.

For constant supply, negligible seed mass and fixed emitting radius, $M=\dot{M}t$ and $L_{\mathrm{acc}}\propto t$. Thermal emission with $L=4\pi r_0^2{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^4$ therefore gives the quarter-power law. It describes an effective surface temperature. A homologous ideal-gas interior at the same fixed radius instead has virial temperature proportional to mass and hence to time; its absolute temperature requires a structural model.