= Solution
Let $q_{\rm in}$ be the <photoionization rate> per neutral hydrogen atom just outside the empty cavity. With an isotropic central source and no attenuation inside that cavity,
$$
q_{\rm in}=\int_0^\infty\frac{\sigma_\nu L_\nu^*}{4\pi r_{\rm in}^2h\nu}\,d\nu
=\frac{\langle\sigma\rangle Q}{4\pi r_{\rm in}^2}.
$$
The <cross section> is zero below threshold; this permits the lower integration limit zero. Pure hydrogen with <hydrogen ionization fraction> $\zeta$ has $n_{\rm HI}=n_H(1-\zeta)$ and $n_e=n_p=n_H\zeta$. Local <photoionization equilibrium> is therefore
$$
n_H(1-\zeta_{\rm in})q_{\rm in}=\alpha_rn_H^2\zeta_{\rm in}^2.
$$
Define the dimensionless quantity $\Theta_{\rm phys}=\alpha_rn_H/q_{\rm in}$. Solving the resulting <quadratic equation>, $\Theta_{\rm phys}\zeta_{\rm in}^2+\zeta_{\rm in}-1=0$, gives the physical root
$$
\boxed{\zeta_{\rm in}=\frac{\sqrt{1+4\Theta_{\rm phys}}-1}{2\Theta_{\rm phys}}
=\frac{2}{1+\sqrt{1+4\Theta_{\rm phys}}}.}
$$
The rationalized form avoids numerical cancellation for small $\Theta_{\rm phys}$. Strong illumination gives $\zeta_{\rm in}\simeq1-\Theta_{\rm phys}$; weak illumination gives $\zeta_{\rm in}\simeq\Theta_{\rm phys}^{-1/2}$.
Eliminating $Q$ with the <H-alpha> <luminosity>, $Q=2.2(4\pi D^2F_{\rm H\alpha})\lambda_{\rm H\alpha}/(hc)$, yields
$$
\boxed{\Theta_{\rm phys}=\frac{n_H\alpha_rhc}{2.2\langle\sigma\rangle F_{\rm H\alpha}\lambda_{\rm H\alpha}}
\left(\frac{r_{\rm in}}D\right)^2.}
$$
\b[The printed definition of $\Theta$ is inconsistent with the defined radii.] The source calls $r_*$ a physical stellar radius, but its factor $(2\pi r_*D)^2$ gives $\Theta$ units of length to the fourth power, rather than a dimensionless value. It also omits the physically necessary cavity radius $r_{\rm in}$. For a concrete counterexample, double $r_{\rm in}$ while holding the source, $D$, and $n_H$ fixed: the incident <photoionization rate> falls by four, so $\Theta_{\rm phys}$ rises by four; the printed expression does not change. The root has the requested form with the corrected $\Theta_{\rm phys}$ above. There is no supplied relation allowing $r_{\rm in}$ to be replaced by $r_*$.
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