 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:For uniform, almost completely ionized pure hydrogen, $n_e=n_p=n_H$. HenceThe final approximation requires $r_{\mathrm{in}}\ll r_S$. 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 $n_e{n}_p$.