 Constant-radius accretion energy budget

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.