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Near the accretor assume fully ionized hydrogen, tight electric coupling between protons and electrons, isotropic radiation and opacity dominated by Thomson scattering. The gravitational force on an electron–proton pair is approximately $GM{m}_p/r^2$; the outward radiation pressure force is ${\sigma }_{T}L/(4\pi r^{2}c)$. Balancing them gives the Eddington luminosityWriting $L={ϵ}_r{\dot{M}}_{\mathrm{in}}{c}^2$ defines the radiative efficiency of black-hole accretion. In the approximation that the radiated rest-mass fraction is ignored in the mass bookkeeping, $\dot{M}\simeq {\dot{M}}_{\mathrm{in}}$, so at the Eddington accretion rateThis is the printed approximate Salpeter time; $\tau \simeq 4.5\times 10^7\mathrm{yr}$ for the adopted efficiency. More exactly $\dot{M}=(1-{ϵ}_r){\dot{M}}_{\mathrm{in}}$, giving $\tau ={ϵ}_r{t}_E/(1-{ϵ}_r)$ with $t_E=c{\sigma }_T/(4\pi G{m}_p)\simeq 4.5\times 10^8\mathrm{yr}$. The difference is a ten-percent-level convention here, not a change from exponential growth. Continuous fuelling, constant efficiency and unit Eddington ratio are additional idealizations.

Next normalize the ionizing spectrum rather than replacing every photon by a threshold photon. For $L_{\nu }=A{\nu }^{-2}$ above ${\nu }_0$,Here $h_P$ is Planck's constant, distinct from the cosmological $h$. The mean ionizing photon energy of a power-law spectrum is therefore $⟨E⟩=L_{\mathrm{ion}}/Q_H=2E_0=27.2\mathrm{eV}$. With ionizing luminosity fraction $f_{\mathrm{ion}}=0.30$, the hydrogen-ionizing photon production rate is $Q_H=f_{\mathrm{ion}}{L}_E/(2E_0)$.

Assume a sharp spherical ionization front, homogeneous initially neutral hydrogen at the cosmic mean baryon density, escape fraction one, one primary ionization per photon, and no recombinations or secondary ionizations. The ionization-front growth from an exponentially brightening source is then fixed by photon conservation. In the printed approximate mass convention,Take the stated separation to be a proper separation at the seed-formation epoch, and take the distant halo to follow the expansion on this large scale. Its initial separation $R_i$ defines a fixed comoving radius $X=R_i(1+z_i)$. The number of hydrogen nuclei initially within that radius isAlthough the proper density subsequently falls, this comoving hydrogen inventory is constant. Thus the photon count does not require freezing the expansion for the entire growth interval.

The supplied critical density gives ${\Omega }_b{\rho }_{\mathrm{crit},0}=6.86\times 10^9{M}_{\odot }{\mathrm{Mpc}}^{-3}$, so $M_H\simeq 2.87\times 10^{13}{M}_{\odot }$ for the initial proper-distance interpretation. Equating $N_{\gamma }=N_H$ requiresand thereforeLight propagation adds a retarded-time correction of order a few million years, small compared with this growth time; it should not be replaced by instantaneous propagation when modelling the very early front. Expansion increases the proper distance to the comoving halo during the wait, but not the required comoving photon inventory. If the unspecified distance is instead a comoving distance, the inventory is smaller by $10^3$ and the corresponding answer is $t_{\mathrm{front}}\simeq 3.57\times 10^8\mathrm{yr}$. A distance held at a fixed proper radius until arrival would need a different evolving-volume treatment.

Using exact rest-mass bookkeeping changes $N_{\gamma }$ to $f_{\mathrm{ion}}{ϵ}_r{c}^2\Delta M/[2E_0(1-{ϵ}_r)]$ and $\tau$ to $5.0\times 10^7\mathrm{yr}$. The proper-at-formation case then gives about $7.37\times 10^8\mathrm{yr}$, and the comoving-distance case about $3.91\times 10^8\mathrm{yr}$. These modest changes are smaller than the uncertainties of the sharp-front, no-recombination model. In particular real absorption of the high-energy tail, secondary electrons and a nonunit escape fraction would require radiative-transfer corrections.

The small halo initially holds cool gas in hydrostatic equilibrium. Photoionization deposits energy far above its virial binding scale. With the emitted mean energy, the excess per primary ionization is $13.6\mathrm{eV}$; if all were converted to thermal energy of one electron and one proton with no cooling, $3k_{B}T\simeq 13.6\mathrm{eV}$ would give about $5\times 10^4\mathrm{K}$. Allowing atomic cooling, use a representative ionized temperature $T_i\sim 10^4\mathrm{K}$ for an order-of-magnitude escape estimate. Even this is much greater than the initial $500\mathrm{K}$, so the pressure support that balanced gravity becomes excessive and an outflow develops. The baryons are photoheated and undergo photoevaporation of a dark-matter minihalo; the shallow halo loses much of its gas and its future star formation is suppressed.

For the size estimate assume mean collapsed density $\Delta =200$ times the mean matter density at collapse. Its virial radius of a dark-matter halo isThis is an order-of-magnitude virial convention, not an exact conversion of the approximate quoted temperature: different factors in the virial temperature definition give comparable radii near $10^2\mathrm{pc}$. Keep the radius of the bound halo fixed after its stated collapse, rather than rescaling it with the later cosmic mean density. Fully ionized pure hydrogen has mean molecular weight ${\mu }_i=1/2$. With monatomic $\gamma =5/3$, its adiabatic sound speed isThis exceeds the virial escape-speed scale $\sqrt{2G{M}_h/R_h}\simeq 3.0\mathrm{km}{\mathrm{s}}^{-1}$. A sound-crossing outflow estimate givesmeasured after the ionization front arrives. Using an isothermal sound speed, a different virial factor or a hotter post-front gas changes this by factors of order unity. It is not a second hundreds-of-millions-of-years black-hole growth interval.

The crossing-time estimate assumes the relevant gas can be ionized without a long trapped-front delay. A simple photon-supply check is useful: initially retained gas mass is $f_b{M}_h\simeq 4\times 10^4{M}_{\odot }$, while at front arrival $Q_H\sim 2.4\times 10^{55}{\mathrm{s}}^{-1}$. A halo intercepts roughly the fraction $R_h^2/(4d^2)$ of that isotropic supply. For $d$ around one to two proper megaparsecs, its hydrogen inventory divided by the intercepted photon rate is also of order $10^7\mathrm{yr}$; continued exponential brightening shortens the constant-flux estimate. Thus a gas-loss estimate of one to a few $10^7\mathrm{yr}$ is justified in the idealized model. A detailed evaporation time is not uniquely determined without the gas profile, post-ionization temperature and shielding; retained recombinations would further delay a front in dense gas.

Write $Q={\dot{N}}_{\mathrm{ion}}$ for the constant hydrogen-ionizing photon production rate. Assume an initially neutral gas of hydrogen atoms, a sharp spherical ionization front, complete ionization inside it, and a fixed gas density. The Electron and proton number densities inside are then both $n$. Take the recombination coefficient $\alpha$ to be constant, with the usual Case B recombination interpretation: photons from recombination directly to the ground state are absorbed locally. Neglect absorption by dust, pressure-driven gas motion, and light-travel delays.

The total rate of Case B recombination within radius $R$ isAt photoionization equilibrium, $Q=\mathcal{R}(R_S)$, giving the Strömgren radiusThe square of the density appears because recombination requires encounters between an Electron and a proton.

During Ionization-front growth of a Strömgren sphere, each photon either replaces an ion lost through Case B recombination or ionizes an additional atom. There are $N_i=(4\pi /3)n{R}^3$ ions inside the ionization front, so photon counting givesThe recombination time is $t_{\mathrm{rec}}=(\alpha n{)}^{-1}$. Solving this first-order differential equation with $R(0)=0$ givesFor an initial radius $R_0$, the corresponding result is $R^3(t)=R_S^3+(R_0^3-R_S^3)e^{-t/t_{\mathrm{rec}}}$. Initially, the idealized photon-counting solution has $R\simeq [3Qt/(4\pi n){]}^{1/3}$; at late times, $R/R_S\simeq 1-\frac{1}{3}{e}^{-t/t_{\mathrm{rec}}}$. The initially divergent front speed in this approximation is a consequence of neglecting light-travel delays; the formula describes the growth once that approximation is applicable.

For a power law density, specify a reference radius $r_0$ so that $n(r)=n_0(r/r_0{)}^{-\beta }$. With an inner cutoff $r_{\mathrm{in}}>0$, the recombination integral for a power-law nebula isEquivalently, a logarithmic interval in radius contributes $d\mathcal{R}/d\ln r=4\pi \alpha n_0^2{r}_0^{2\beta }{r}^{3-2\beta }$. For $\beta <3/2$, the central integral converges as $r_{\mathrm{in}}\to 0$, and the recombination rate grows as $R^{3-2\beta }$. At $\beta =3/2$, each logarithmic interval contributes equally and the centre gives a logarithmic divergence. For $\beta >3/2$, the unmodified profile gives a power divergence at the centre, while its outer recombination integral converges. The change at $\beta =3/2$ is set by the competition between the spherical volume element and the squared density.

To regularize the centre, choose the following explicit cored power-law density profile, taking $r_0=r_c$ and $n_0$ as the core density:For $R\le r_c$, $\mathcal{R}(R)=(4\pi /3)\alpha n_0^2{R}^3$. For $R>r_c$, put $u=R/r_c$ and integrate the core and envelope separately:This rate is continuous and strictly increasing with $R$. If $\beta >3/2$, it has a finite supremum, the critical ionizing photon rate of a cored power-law nebula:For $0<Q<Q_{\mathrm{crit}}$ there is a unique finite radius at photoionization equilibrium. At $Q=Q_{\mathrm{crit}}$ the equality is approached only as $R\to \infty$; for $Q>Q_{\mathrm{crit}}$ even ionizing the whole profile cannot consume all photons through Case B recombination, so no equilibrium radius exists. For $\beta \le 3/2$, the envelope integral grows without bound and there is no finite critical rate in this infinite-extent model.

For a concrete choice, take any $n_0>0$, $r_c>0$, and $\beta =2$. Then $Q_{\mathrm{crit}}=16\pi \alpha n_0^2{r}_c^3/3$. In the envelope, defining $q=Q/(4\pi \alpha n_0^2{r}_c^3)$ gives $R=r_c/(4/3-q)$ for $1/3<q<4/3$, explicitly showing the divergence as the critical rate is approached. A different core shape changes the numerical coefficient of $Q_{\mathrm{crit}}$, while the outer-slope condition $\beta >3/2$ remains the same.

A star-forming galaxy contains short-lived, massive O and B stars whose hot photospheres dominate the ultraviolet and blue continuum and ionize surrounding gas. Once star formation ceases, these stars disappear quickly and an older, cooler stellar population produces a redder spectrum with stronger stellar absorption features and a prominent 4000-angstrom break.

In star-forming regions, direct stellar continuum is accompanied by nebular free-bound and free-free continuum, hydrogen and helium recombination lines, collisionally excited metal lines, and infrared emission from dust that absorbed shorter-wavelength photons. Supernova remnants and cosmic rays add synchrotron radio emission, while hot shocked gas can emit X-rays.

The Strömgren sphere model assumes a steady ionizing source in uniform, static, pure hydrogen of number density $n_H$, with a sharp ionization front enclosing fully ionized gas. If $N_i=(4\pi /3)R^3{n}_H$ is the number of ions, photon conservation givesThe equilibrium Strömgren radius and recombination time areConsequently the radius obeysWriting $x=(R/R_S{)}^3$ turns this into $dx/d(t/t_{\mathrm{rec}})=1-x$. For an initially neutral medium, $x(0)=0$, and the Ionization-front growth of a Strömgren sphere isThe front initially expands rapidly because few ions are recombining and asymptotically approaches $R_S$ as recombinations balance ionizations. This photon-counting solution precedes any pressure-driven hydrodynamic expansion of the H II region.

Let the source's hydrogen-ionizing photon production rate bewhere $h{\nu }_0$ is the hydrogen ionization energy. The idealized Strömgren sphere has an almost fully ionized interior and a thin ionization front. In photoionization equilibrium, every ionization is balanced by a Case B recombination, so spherical symmetry givesFor pure hydrogen of constant number density $n$, the interior has $n_e=n_p=n$, and the Strömgren radius is therefore

Now let the effective number of dust grains per hydrogen nucleus be $f_d$, so the dust absorption coefficient is $a(r)=f_d{\sigma }_{d}n(r)$. If $Q(r)$ is the ionizing-photon rate crossing the sphere of radius $r$, recombinations and dust absorption give the linear ordinary differential equationIf $f_d$ denotes a dust mass fraction instead, the grain mass and gas mean particle mass are simply absorbed into the effective product $f_d{\sigma }_d$. Define the dust optical depthMultiplication by the integrating factor $e^{{\tau }_d(r)}$ and integration to the dusty front gives its governing equationThe exponential function weights recombinations at large optical depth by the extra source photons that dust must remove before those photons reach that radius.

For constant $n$, put $a=f_d{\sigma }_{d}n$ and $\tau =a{R}_d$. The elementary integralreduces the equation toEquivalently, with $y=R_d/R_S$ and ${\tau }_S=a{R}_S$,Its dust-free limit is $y\to 1$, while absorption makes $R_d<R_S$ for nonzero dust abundance.