= Solution
Write $Q=\dot N_\gamma$ for the <hydrogen-ionizing photon production rate>, counting only <photons> above the hydrogen <ionization energy>, not all stellar <photons>. In steady <photoionization equilibrium>, the external supply balances the volume integral of net <Case B recombination>:
$$
Q=\int_{r_{\rm in}}^{r_S}\alpha_rn_en_p\,4\pi r^2dr.
$$
For uniform, almost completely ionized pure hydrogen, $n_e=n_p=n_H$. Hence
$$
Q=\frac{4\pi}{3}\alpha_rn_H^2(r_S^3-r_{\rm in}^3),\qquad
\boxed{r_S=\left(r_{\rm in}^3+\frac{3Q}{4\pi\alpha_rn_H^2}\right)^{1/3}
\simeq\left(\frac{3Q}{4\pi\alpha_rn_H^2}\right)^{1/3}.}
$$
The final approximation requires $r_{\rm in}\ll r_S$. The <Strömgren sphere> also assumes spherical symmetry, a sharp <ionization front>, a steady source older than the <recombination time>, uniform <number density> and <temperature> so that $\alpha_r$ is constant, negligible dust absorption of ionizing <photons>, negligible leakage, and no important helium correction or <collisional ionization>. Clumping would replace $n_H^2$ by a volume average of $n_en_p$.
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