 Solution

Only the photon energy above the hydrogen ionization threshold initially becomes photoelectron kinetic energy. The stated mean therefore suppliesLet $\mathcal{R}$ be the local number of ionizations per volume per time. Photoionization equilibrium makes this equal to the recombination rate. If $⟨E_{\mathrm{rec}}(T)⟩$ is the mean kinetic energy removed per recombination, thermal balance reduces toNeither the ionization threshold nor the density remains in this event-by-event balance. The threshold energy is stored as atomic ionization energy and returned in recombination radiation; it is not a net thermal input. This separation between photoelectron heating and kinetic energy lost by recombining electrons is the relevant cooling balance.^[ref]

In the elementary mean-thermal-energy approximation, a captured electron removes $\frac{3}{2}{k}_{B}T$. This gives the concise estimateThat coefficient is a stated approximation, not a universal recombination identity. Real recombination weights the electron distribution by its velocity-dependent capture rate. If the chosen cooling model instead writes $⟨E_{\mathrm{rec}}⟩={\eta }_{\mathrm{rec}}(T)k_{B}T$, the precise answer is ${\eta }_{\mathrm{rec}}(T_{\mathrm{eq}})T_{\mathrm{eq}}=T_{\ast }$. The source supplies no recombination cooling coefficient or capture kernel from which a unique more accurate coefficient can be computed. Its emitted-photon average is also being used as the average of the absorbed ionizing photons; spectral filtering would require a different weighted average.