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

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

Q_{H} = \int_{ν_{0}}^{\infty} \frac{L _{ν}}{h ν} d ν .

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Blackbody ionizing photon function ( $F (y) = \int_{y}^{\infty} x^{2} / (e^{x} - 1) d x$ )

A spherical blackbody star emits

Q (E_{0}) = 8 π^{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

π B_{ν}

; dividing the resulting spectral luminosity by photon energy and substituting

x = h ν / k T_{s}

proves the formula. The convergent series

F (y) = \sum_{m \geq 1} e^{- m y} (y^{2} / m + 2 y / m^{2} + 2/ m^{3})

follows by integrating the geometric-series expansion termwise.

Ionizing photon escape fraction ( $f_{esc}$ )

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.

Mean ionizing photon energy of a power-law spectrum ( $⟨ E ⟩ = \frac{s}{s - 1} h ν_{0} (s > 1)$ )

Suppose an emitted spectral luminosity is

L_{ν} = A ν^{- s}

for every

ν \geq ν_{0}

, where

A > 0

and

h ν_{0}

is an ionization threshold. The total emitted ionizing luminosity and photon rate use different weights:

L_{ion} = \int_{ν_{0}}^{\infty} L_{ν} d ν = \frac{A ν _{0}^{1 - s}}{s - 1}, s > 1,

\dot{N}_{ion} = \int_{ν_{0}}^{\infty} \frac{L _{ν}}{h ν} d ν = \frac{A ν _{0}^{- s}}{h s}, s > 0.

Thus a finite emitted mean photon energy for this unbounded power law requires the stronger condition

s > 1

and equals

⟨ E ⟩_{em} = \frac{L _{ion}}{N ˙ _{ion}} = \frac{s}{s - 1} h ν_{0} .

For

s = 2

, the mean is twice the threshold energy. As

s ↓ 1

, the high-frequency tail makes the mean diverge even though the photon count stays finite. For

0 < s \leq 1

, substituting into the finite-mean formula would therefore be invalid; for

s \leq 0

even the unbounded photon-rate integral diverges.

A finite high-frequency cutoff changes both integrals and removes these upper-tail divergences. The mean energy deposited in a nebula also need not equal the emitted mean: absorption weights the spectrum with its photoionization cross section, and energy above threshold can be distributed among heat, excitation and secondary ionizations. The excess

h ν - h ν_{0}

, rather than the whole photon energy, is the initial kinetic energy of the photoelectron in the simplest single-ionization event.

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