H II region

An H II region is a volume of ionized hydrogen surrounding one or more sources of ultraviolet photons. Photoionization heating, recombination, gas dynamics, and dust determine its structure and emission spectrum.

An ionization front is the transition between predominantly ionized and predominantly neutral gas. Its motion and thickness depend on photon supply, recombination, gas density, and hydrodynamic response.

In a homogeneous initially neutral medium, an isotropic source with hydrogen-ionizing photon production rate $Q(t)=Q_0{e}^{t/\tau }$ ionizes the displayed comoving inventory when every escaping photon produces one primary ionization and recombinations are absent. The physical front radius is $a(t)X$ in an expanding background. Retarded photon propagation prevents any instantaneous front from travelling faster than light. This photon-budget front is not the equilibrium Strömgren radius, which balances emission against recombinations.

Photoionization equilibrium holds when the local or integrated photoionization rate equals the radiative recombination rate, so the ionized fraction is stationary.

Consider pure hydrogen around an isotropic central source with hydrogen-ionizing photon production rate $Q$. An empty cavity of physical radius $r_{\mathrm{in}}$ delivers an unattenuated direct photon flux $Q/(4\pi r_{\mathrm{in}}^2)$ to its inner boundary. Let $⟨\sigma ⟩$ be the photoionization cross section averaged over the incident ionizing photon-number spectrum. Then the direct photoionization rate per neutral atom is $q=Q⟨\sigma ⟩/(4\pi r_{\mathrm{in}}^2)$. With total hydrogen number density $n_H$ and ionized fraction $\zeta$, the electron and proton number densities are both $n_H\zeta$. In the local Case B recombination approximation, photoionization equilibrium givesThis uses a locally balanced photoionization and radiative recombination event rate; the on-the-spot treatment incorporates immediate reabsorption of ground-state recombination photons into ${\alpha }_B$.^[ref]

Solving the quadratic equation and rationalizing its physical root yieldsFor $\Theta \ll 1$, $\zeta \simeq 1-\Theta$; for $\Theta \gg 1$, $\zeta \simeq {\Theta }^{-1/2}$. Thus stronger incident photoionization drives the boundary closer to complete ionization, while a larger recombination rate leaves more neutral hydrogen.

Inferring $Q$ from a recombination line requires an additional, global photon-budget assumption. Suppose $F_{\ell }$ is the total extinction-corrected observed radiative flux in a recombination line of wavelength ${\lambda }_{\ell }$, at distance $D$. At fixed gas conditions let $y_{\ell }={\alpha }_{\ell }^{\mathrm{eff}}/{\alpha }_B$ be the line-photon yield per Case B recombination. The measured line then gives the absorbed hydrogen-ionizing rateFor a dust-free ionization-bounded nebula in steady photoionization equilibrium, with no competing absorbers, $Q_{\mathrm{abs},\mathrm{H}}=Q$. Under these assumptions,The angular ratio $r_{\mathrm{in}}/D$ makes the expression dimensionless and lets a resolved cavity and its integrated recombination line constrain the local ionized fraction.

If only a fraction $f_H$ of the source's ionizing photons produces hydrogen ionizations, then $Q_{\mathrm{abs},\mathrm{H}}=f_{H}Q$ and the last expression for $\Theta$ must be multiplied by $f_H$. A nonzero ionizing photon escape fraction or absorption by dust can therefore make the line underestimate the central $Q$. Correcting the escaping line for extinction alone does not recover ionizing photons that never produced a recombination line. A Dusty Strömgren sphere exemplifies the distinct absorption correction. The local boundary formula also requires an empty, transparent cavity; any attenuation before the gas changes its incident $q$.

When the number of photoionizations equals the number of recombinations, and only those processes exchange thermal energy, equilibrium requires $⟨h\nu -h{\nu }_0{⟩}_{\mathrm{abs}}=⟨E_{\mathrm{rec}}(T)⟩$. Photon absorption and electron capture have their own spectral weights. The density cancels from this event balance, but the actual equilibrium temperature requires the two energy averages.^[ref]

An ionization-bounded nebula contains enough absorbing material to trap the relevant ionizing photons, unlike a density-bounded region that ends before its ionization front. In dust-free hydrogen at steady photoionization equilibrium, the absorbed stellar photon rate equals the integrated Case B recombination rate. Escape, absorption by dust, and competition from other species must be accounted for before identifying the absorbed hydrogen rate with the entire stellar ionizing luminosity.

For fully ionized pure hydrogen of density $n(r)=n_0(r/r_0{)}^{-\beta }$ and constant Case B recombination coefficient $\alpha$, the total recombination rate isThe central integral converges only for $\beta <3/2$; an envelope extending to infinity has finite recombination capacity only for $\beta >3/2$. At $\beta =3/2$, each logarithmic radius interval contributes equally. A cored power-law density profile removes the central singularity.

For the cored power-law density profile that is constant inside $r_c$ and falls as $r^{-\beta }$ outside, with $\beta >3/2$, the maximum integrated Case B recombination rate isA hydrogen-ionizing photon production rate greater than this has no finite radius at photoionization equilibrium. Equality requires an infinite ionized radius. Other core shapes change the coefficient but retain the convergence condition $\beta >3/2$.

The recombination time is the characteristic time for an ionized plasma to recombine after its ionizing source is removed. Uniform hydrogen with electron density $n_e$ and recombination coefficient $\alpha$ has $t_{\mathrm{rec}}=1/(\alpha n_e)$.

Case B recombination excludes direct recombinations to the ground state because their ionizing photons are assumed to be absorbed locally. The coefficient ${\alpha }_B$ therefore counts recombinations to excited states.

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.

Suppose an emitted spectral luminosity is $L_{\nu }=A{\nu }^{-s}$ for every $\nu \ge {\nu }_0$, where $A>0$ and $h{\nu }_0$ is an ionization threshold. The total emitted ionizing luminosity and photon rate use different weights:Thus a finite emitted mean photon energy for this unbounded power law requires the stronger condition $s>1$ and equalsFor $s=2$, the mean is twice the threshold energy. As $s\downarrow 1$, the high-frequency tail makes the mean diverge even though the photon count stays finite. For $0<s\le 1$, substituting into the finite-mean formula would therefore be invalid; for $s\le 0$ 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 $h\nu -h{\nu }_0$, rather than the whole photon energy, is the initial kinetic energy of the photoelectron in the simplest single-ionization event.

A Strömgren sphere is the idealized ionized region around a steady source embedded in uniform hydrogen gas, bounded where integrated recombinations consume the source's ionizing-photon supply.

In a uniform sharp-front approximation, an inner ${\mathrm{He}}^{++}$ zone of radius $R_2$ lies inside a ${\mathrm{He}}^{+}$ shell ending at $R_1$, within the hydrogen ionization front $R_H$. For nuclear densities $n_H,n_{He}$, the electron densities are $n_H+2n_{He}$, $n_H+n_{He}$ and $n_H$ in the three zones. Thus the hydrogen recombination integral is $(4\pi /3){\beta }_H{n}_H[n_H{R}_H^3+n_{He}(R_1^3+R_2^3)]$. The helium integrals are $(4\pi /3){\beta }_{HeI}{n}_{He}(n_H+n_{He})(R_1^3-R_2^3)$ and $(4\pi /3){\beta }_{HeII}{n}_{He}(n_H+2n_{He})R_2^3$. These are balanced against species-specific effective photon budgets, not three independently reused cumulative stellar photon counts.

For ionizing-photon rate $Q_{\mathrm{H}}$, uniform pure-hydrogen density $n$, and Case B recombination coefficient ${\alpha }_B$, the Strömgren radius is

Before hydrodynamic expansion matters, photon counting in a uniform initially neutral medium givesThe ionization front approaches the Strömgren radius on the recombination time.

A dusty Strömgren sphere includes ionizing photons absorbed by dust before they can ionize hydrogen. Dust therefore decreases the ionized radius below the dust-free Strömgren radius at fixed source luminosity and gas density.