= Solution
Introduce the constant $\nu>0$ by $\psi(r,t)=\nu\rho_g(r,t)$, and set $q=\gamma E_{\rm SN}$, the injected energy per unit mass of newly formed <stars>. Before a local wind switches on, no return or matter exchange implies
$$
\rho_g(r,t)=\rho_0(r)e^{-\nu t},\qquad
\rho_s(r,t)=\rho_0(r)(1-e^{-\nu t}).
$$
In this <independent-volume supernova quenching> model, the accumulated injection per volume is
$$
u_{\rm SN}(r,t)=\int_0^t\gamma E_{\rm SN}\psi(r,t')\,\mathrm dt'
=q\rho_0(r)(1-e^{-\nu t}).
$$
For any specified positive <binding energy> per unit remaining gas mass $B(r)$, the local threshold is
$$
u_{\rm SN}=\rho_g B(r)
\quad\Longleftrightarrow\quad
\boxed{Q(t)\equiv q(e^{\nu t}-1)=B(r).}
$$
Gas 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 $B(r)=-\Phi_{\rm grav}(r)=2k[1+\ln(r_0/r)]$. 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, at
$$
t_{\rm start}=\nu^{-1}\ln\left(1+\frac{2k}{q}\right).
$$
Thereafter the inner edge of the quenched exterior region is
$$
\boxed{r_{\rm quench}(t)
=r_0\exp\left[1-\frac{q}{2k}(e^{\nu t}-1)\right],
\qquad t\geq t_{\rm start}.}
$$
There is no quenched radius inside the <galaxy> before this onset. The remaining star-forming region is $0<r<r_{\rm quench}$; 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 $\phi$ 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),
$$
\phi_{\rm shell}=-\frac{GM_0(r)}r\rho_g=-2k\rho_g,
$$
then $B_{\rm shell}=2k$ at every radius. The same time $t_{\rm start}$ applies everywhere: \b[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 $\tfrac12\rho_g\Phi_{\rm grav}$ 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.
\Image[/past-exam-of-the-formation-of-structure-in-the-universe-course-of-the-university-of-cambridge/2016/paper-1-quenching-fronts.png]
{title=Original comparison of quenching thresholds for two initial density slopes, distinguishing escape potential from shell-energy allocation}
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