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Introduce the constant $\nu >0$ by $\psi (r,t)=\nu {\rho }_g(r,t)$, and set $q=\gamma E_{\mathrm{SN}}$, the injected energy per unit mass of newly formed stars. Before a local wind switches on, no return or matter exchange impliesIn this independent-volume supernova quenching model, the accumulated injection per volume isFor any specified positive binding energy per unit remaining gas mass $B(r)$, the local threshold isGas conversion must be included in this comparison; replacing the integrated star formation by its initial rate times $t$ is only a short-time approximation.

For a quenching front in a fixed galactic potential, take $B(r)=-{\Phi }_{\mathrm{grav}}(r)=2k[1+\ln (r_0/r)]$. This treats the gravitating profile as fixed: newly formed stars initially replace gas at the same radius, and later changes of the potential caused by expelled gas are neglected. Gas first becomes unbound at the outer edge, atThereafter the inner edge of the quenched exterior region isThere is no quenched radius inside the galaxy before this onset. The remaining star-forming region is $0<r<r_{\mathrm{quench}}$; the front moves inward. The ideal logarithmically divergent central potential delays complete removal to infinite time in this fixed-profile model. It should not be interpreted as a physical unresolved central core.

The local binding-energy allocation in galactic feedback also admits a different convention, because it calls $\varphi$ an energy of a volume element without specifying how the nonlocal self-gravitating energy is assigned to that element. If the intended prescription is instead the shell-assembly contribution from part (a),then $B_{\mathrm{shell}}=2k$ at every radius. The same time $t_{\mathrm{start}}$ applies everywhere: all shells quench simultaneously, and there is no continuously propagating radial front. This is a shell-energy toy model, not the escape potential including exterior shells.

Both results follow from explicit definitions. A convention assigning $\frac{1}{2}{\rho }_g{\Phi }_{\mathrm{grav}}$ locally replaces $B$ by $B/2$ in $Q(t)=B(r)$; it is again an energy allocation rather than test-parcel escape. Updating the potential after gas loss or choosing a fixed initial-gas binding budget instead of the remaining-gas budget would require an additional closure. The stated independent-volume assumptions do not uniquely choose between these. The displayed escape-potential result and shell prescription must therefore not be presented as interchangeable universal laws.