Photoionization ($h\nu \ge E_{\mathrm{bind}}$)

Photoionization removes a bound electron when an atom or ion absorbs a sufficiently energetic photon. The threshold is its ionization energy. The excess photon energy becomes kinetic energy of the products; subsequent collisions can heat the gas or cause secondary ionizations. A photoionization rate and a photoionization equilibrium are distinct concepts: equilibrium also requires specified recombination and other reaction rates.

The ionization parameter compares the local density of hydrogen-ionizing photons with the hydrogen number density. For an unattenuated isotropic source the displayed form uses its ionizing photon production rate $Q_H$. Ionic fractions depend on $U$ together with the radiation spectrum, temperature, recombination rates and shielding, so $U$ alone does not determine every emission-line ratio.

An electron ejected following photon absorption. For a free residual ion with negligible recoil, the emitted electron's kinetic energy equals incident photon energy minus the ionization binding energy.

For isotropic specific intensity $J_{\nu }$, absorption cross-section ${\sigma }_{\nu }$, and threshold frequency ${\nu }_0$, the photoionization rate per target atom isFor a directed incident flux, replace $4\pi J_{\nu }$ by its spectral flux density. The factor $h\nu$ converts energy flux into photon flux. This rate alone does not imply photoionization equilibrium.

At an unattenuated inner boundary, define photon-weighted cross section moments $⟨{\sigma }^k⟩=Q^{-1}\int {\sigma }_{\nu }^k{L}_{\nu }/(h\nu )d\nu$. Across a layer $dr$, the inverse-square geometrical factor contributes $-2dr/r$, and expanding $e^{-n_H(1-\zeta ){\sigma }_{\nu }dr}$ contributes $-n_H(1-\zeta )dr⟨{\sigma }^2⟩/⟨\sigma ⟩$. Both are first order; a plane-parallel treatment can suppress the former only as a separate geometrical approximation. The second moment occurs because the photoionization rate weights an already attenuated spectrum by another cross section.

Absorption of an ionizing photon leaves energy in a photoelectron above the binding energy. Thermalization supplies heat to the gas. Photoionization heating competes with the astrophysical cooling function, can establish thermal equilibrium, and can raise the pressure enough to inhibit baryon accretion into shallow dark-matter haloes.