Hydrogen-ionizing photon production rate ($Q_{\mathrm{H}}$)

The hydrogen-ionizing photon production rate is the number of photons emitted per unit time above the hydrogen ionization threshold:

A spherical blackbody star emits $Q(E_0)=8{\pi }^2{R}_s^2(k{T}_s{)}^{3}F(E_0/k{T}_s)/(h^3{c}^2)$ photons per time above threshold $E_0$. Integrating specific intensity over the outward hemisphere gives flux $\pi B_{\nu }$; dividing the resulting spectral luminosity by photon energy and substituting $x=h\nu /k{T}_s$ proves the formula. The convergent series $F(y)={\sum }_{m\ge 1}{e}^{-my}(y^2/m+2y/m^2+2/m^3)$ follows by integrating the geometric-series expansion termwise.

The ionizing photon escape fraction is the fraction of photons capable of photoionization that leave their source environment. Only these escaping photons contribute to ionizing distant intergalactic medium. Its value can depend on frequency, source geometry, and time.

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.