= Solution
{c}
Write $Q=\dot N_{\rm ion}$ for the constant <hydrogen-ionizing photon production rate>. Assume an initially neutral gas of <hydrogen atoms>, a sharp spherical <ionization front>, complete ionization inside it, and a fixed gas density. The <electron> and <proton> number densities inside are then both $n$. Take the recombination coefficient $\alpha$ to be constant, with the usual <Case B recombination> interpretation: photons from recombination directly to the ground state are absorbed locally. Neglect absorption by dust, pressure-driven gas motion, and light-travel delays.
The total rate of <Case B recombination> within radius $R$ is
$$
\mathcal R(R)=\int_0^R\alpha n_e n_p\,4\pi r^2\,dr
=\frac{4\pi}{3}\alpha n^2R^3.
$$
At <photoionization equilibrium>, $Q=\mathcal R(R_S)$, giving the <Strömgren radius>
$$
\boxed{R_S=\left(\frac{3Q}{4\pi\alpha n^2}\right)^{1/3}.}
$$
The square of the density appears because recombination requires encounters between an <electron> and a <proton>.
During <Ionization-front growth of a Strömgren sphere>, each photon either replaces an ion lost through <Case B recombination> or ionizes an additional atom. There are $N_i=(4\pi/3)nR^3$ ions inside the <ionization front>, so photon counting gives
$$
\frac{dN_i}{dt}=Q-\alpha nN_i,
\qquad
4\pi nR^2\frac{dR}{dt}=Q-\frac{4\pi}{3}\alpha n^2R^3.
$$
The <recombination time> is $t_{\rm rec}=(\alpha n)^{-1}$. Solving this first-order <differential equation> with $R(0)=0$ gives
$$
N_i(t)=Qt_{\rm rec}\left(1-e^{-t/t_{\rm rec}}\right),\qquad
\boxed{\frac{R(t)}{R_S}=\left(1-e^{-t/t_{\rm rec}}\right)^{1/3}.}
$$
For an initial radius $R_0$, the corresponding result is $R^3(t)=R_S^3+(R_0^3-R_S^3)e^{-t/t_{\rm rec}}$. Initially, the idealized photon-counting solution has $R\simeq[3Qt/(4\pi n)]^{1/3}$; at late times, $R/R_S\simeq1-\tfrac13e^{-t/t_{\rm rec}}$. The initially divergent front speed in this approximation is a consequence of neglecting light-travel delays; the formula describes the growth once that approximation is applicable.
For a <power law> density, specify a reference radius $r_0$ so that $n(r)=n_0(r/r_0)^{-\beta}$. With an inner cutoff $r_{\rm in}>0$, the <recombination integral for a power-law nebula> is
$$
\mathcal R(R;r_{\rm in})=4\pi\alpha n_0^2r_0^{2\beta}
\begin{cases}
\displaystyle\frac{R^{3-2\beta}-r_{\rm in}^{3-2\beta}}{3-2\beta},&\beta\ne\frac32,\\[6pt]
\displaystyle\ln\frac{R}{r_{\rm in}},&\beta=\frac32.
\end{cases}
$$
Equivalently, a logarithmic interval in radius contributes $d\mathcal R/d\ln r=4\pi\alpha n_0^2r_0^{2\beta}r^{3-2\beta}$. For $\beta<3/2$, the central integral converges as $r_{\rm in}\to0$, and the recombination rate grows as $R^{3-2\beta}$. At $\beta=3/2$, each logarithmic interval contributes equally and the centre gives a logarithmic divergence. For $\beta>3/2$, the unmodified profile gives a power divergence at the centre, while its outer recombination integral converges. The change at $\beta=3/2$ is set by the competition between the spherical volume element and the squared density.
To regularize the centre, choose the following explicit <cored power-law density profile>, taking $r_0=r_c$ and $n_0$ as the core density:
$$
n(r)=\begin{cases}
n_0,&0\leq r\leq r_c,\\
n_0(r/r_c)^{-\beta},&r>r_c.
\end{cases}
$$
For $R\leq r_c$, $\mathcal R(R)=(4\pi/3)\alpha n_0^2R^3$. For $R>r_c$, put $u=R/r_c$ and integrate the core and envelope separately:
$$
\mathcal R(R)=4\pi\alpha n_0^2r_c^3
\begin{cases}
\displaystyle\frac13+\frac{u^{3-2\beta}-1}{3-2\beta},&\beta\ne\frac32,\\[6pt]
\displaystyle\frac13+\ln u,&\beta=\frac32.
\end{cases}
$$
This rate is continuous and strictly increasing with $R$. If $\beta>3/2$, it has a finite supremum, the <critical ionizing photon rate of a cored power-law nebula>:
$$
\boxed{Q_{\rm crit}=4\pi\alpha n_0^2r_c^3
\left(\frac13+\frac1{2\beta-3}\right)
=\frac{8\pi\alpha\beta n_0^2r_c^3}{3(2\beta-3)}.}
$$
For $0<Q<Q_{\rm crit}$ there is a unique finite radius at <photoionization equilibrium>. At $Q=Q_{\rm crit}$ the equality is approached only as $R\to\infty$; for $Q>Q_{\rm crit}$ even ionizing the whole profile cannot consume all photons through <Case B recombination>, so no equilibrium radius exists. For $\beta\leq3/2$, the envelope integral grows without bound and there is no finite critical rate in this infinite-extent model.
For a concrete choice, take any $n_0>0$, $r_c>0$, and $\beta=2$. Then $Q_{\rm crit}=16\pi\alpha n_0^2r_c^3/3$. In the envelope, defining $q=Q/(4\pi\alpha n_0^2r_c^3)$ gives $R=r_c/(4/3-q)$ for $1/3<q<4/3$, explicitly showing the divergence as the critical rate is approached. A different core shape changes the numerical coefficient of $Q_{\rm crit}$, while the outer-slope condition $\beta>3/2$ remains the same.
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