 Mean ionizing photon energy of a power-law spectrum

For $L_{\nu }=A{\nu }^{-s}$ above an ionization threshold ${\nu }_0$, the energy luminosity is $A{\nu }_0^{1-s}/(s-1)$ and the photon rate is $A{\nu }_0^{-s}/(sh)$. Their ratio is the displayed emitted mean energy. For slope two it is twice the threshold energy. Absorption weighting, a high-energy cutoff and secondary ionizations can change the energy deposited per ionization; emitted and locally absorbed spectra must not be conflated.