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For homologous gas-pressure-supported stars, hydrostatic equilibrium gives $T_c\propto \mu M/R$ and ${\rho }_c\propto M/R^3$. Radiative diffusion gives $L\propto R{T}_c^4/(\kappa {\rho }_c)$, so the radius cancels and yields the displayed cubic mass scaling. Composition and opacity are held fixed. Strong radiation pressure alters this gas-pressure relation towards the Eddington luminosity scaling.

An isolated compact dust grain of radius $r_d$ and material density ${\rho }_d$ has opacity per unit dust mass ${\kappa }_d=3Q_{\mathrm{pr}}/(4{\rho }_d{r}_d)$, where $Q_{\mathrm{pr}}$ is its radiation-pressure efficiency. Its Eddington luminosity is $4\pi GMc/{\kappa }_d$. Matching the electron-scattering opacity gives $r_{d,\mathrm{crit}}=3Q_{\mathrm{pr}}/(4{\rho }_d{\kappa }_{\mathrm{es}})$, approximately $1.9\mathrm{cm}$ for $Q_{\mathrm{pr}}=1$, ${\rho }_d=1\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, and ${\kappa }_{\mathrm{es}}=0.4\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$. At that electron-scattering luminosity, smaller grains feel a net outward force and larger grains remain attracted. This is an isolated-grain force balance, not the opacity of an arbitrary gas-dust mixture.

An Eddington accretion rate converts the Eddington luminosity into a mass supply rate using an adopted reference efficiency ${\eta }_0$: ${\dot{M}}_{\mathrm{Edd}}=L_{\mathrm{Edd}}/({\eta }_0{c}^2)$. The common choice ${\eta }_0=0.1$ gives ${\dot{M}}_{\mathrm{Edd}}=10L_{\mathrm{Edd}}/c^2$.

At constant radiative efficiency of black-hole accretion $\eta$ and the Eddington luminosity, a black hole grows exponentially with the Salpeter time $t_S=\eta t_E/(1-\eta )$, where $t_E={\sigma }_{T}c/(4\pi G{m}_p)$. Its doubling time is $t_S\log 2$. The factor $1-\eta$ accounts for the supplied rest energy that escapes as radiation instead of increasing the hole's mass.

For fixed efficiency, growth from $M_i$ to $M_f$ within $\Delta t$ at no more than the Eddington luminosity requires $[\eta /(1-\eta )]t_E\log (M_f/M_i)\le \Delta t$. Solving gives an upper bound, because greater radiative efficiency reduces retained mass growth at a fixed luminosity. A duty cycle below unity tightens the requirement; mergers and super-Eddington episodes invalidate its stated premises.

In an optically thick steady spherical stellar wind of constant mean molecular weight ${\mu }_{\mathrm{mol}}$, let $v$ be the outward speed and $a_g^2=k_{B}T/({\mu }_{\mathrm{mol}}{m}_u)$ the isothermal sound speed squared. Mass conservation gives ${\rho }^{\prime }/\rho =-2/r-v^{\prime }/v$, where a prime denotes differentiation with respect to radius $r$. Use $P_{\mathrm{gas}}=\rho a_g^2$ and radiative diffusion to write $P_{\mathrm{rad}}^{\prime }/\rho =-g\Gamma$, where $g=G{M}_r/r^2$ and $\Gamma =L_r/L_{\mathrm{Edd}}$ uses the local Eddington luminosity $L_{\mathrm{Edd}}=4\pi cG{M}_r/\kappa$ and opacity $\kappa$. Substituting into the momentum equation yields the displayed wind equation. Smooth sonic point crossing requires both its denominator and numerator to vanish. The radiation-pressure gradient already supplies radiative acceleration; adding that force separately would count it twice.

For an unattenuated isotropic point source, the radiation flux is $L_{\ast }/(4\pi r^2)$. The radiation pressure force per particle divided by its mass $\bar{m}$ gives the outward accelerationThus the effective inward inverse-square coefficient is $GM-\sigma L_{\ast }/(4\pi \bar{m}c)$. Insert it into the preceding wind equation. The radiative shift of an isothermal breeze maximum isEquivalently, with opacity $\kappa =\sigma /\bar{m}$ and an Eddington luminosity $L_{\mathrm{Edd}}=4\pi GMc/\kappa$, this is $r_M^{\prime }=r_M(1-L_{\ast }/L_{\mathrm{Edd}})$. It is an exterior speed maximum only if the radius is positive and lies within the modeled wind region. For $L_{\ast }\ge L_{\mathrm{Edd}}$, the wind equation's right side is positive at every $r>0$, so an always-subsonic outflow decreases in speed and has no such interior maximum.

Because $\sigma$ is a cross section per particle, its conversion to acceleration requires $\bar{m}$; that mass cannot be omitted dimensionally. The requested dependence on the four stated variables therefore assumes a fixed particle species. For the atomic hydrogen of part (e), set $\bar{m}\simeq m_H$ before molecule formation.

An ultraluminous X-ray source is a compact, apparently off-nuclear source in another galaxy with an inferred isotropic X-ray luminosity typically above $10^{32}$ W, exceeding the classical Eddington luminosity of an ordinary stellar-mass accretor. Imaging X-ray surveys revealed unusually bright sources away from galactic nuclei; distance association with the host converted the observed fluxes into surprisingly high luminosities. Better angular resolution helps exclude blends and background active nuclei. The 2014 discovery of coherent pulsations established a neutron star in M82 X-2, so the class cannot be identified exclusively with black holes.^[ref]

An intermediate-mass black hole can radiate approximately isotropically below its own Eddington luminosity: this avoids extreme super-Eddington emission, but the inferred mass requires dynamical or suitable spectral/timing support and is not proved by luminosity alone. A stellar-mass accretor with a supercritical accretion disk and a wind funnel can combine genuinely super-Eddington output with geometrical beaming; this is compatible with a neutron-star accretor but requires a consistent radiation/flow/torque model. Relativistic beaming from a jet is another possibility for some objects, but orientation, spectral and variability evidence must support it. These mechanisms need not apply identically to every member of the observed class.

For spherical radiation with flux $F=L/(4\pi r^2)$, the outward acceleration is $\kappa F/c$. Equating it to $GM/r^2$ givesThe limit assumes the stated opacity, approximate spherical symmetry and coupling between the radiatively accelerated plasma and the accreting matter. It is not a universal ceiling on isotropic-equivalent luminosity for an anisotropic magnetized flow.

Write the available accretion power as $L=\eta \dot{M}{c}^2$. ThenAccretion as the energy source alone does not fix $\eta$. For radiatively efficient Newtonian accretion onto a slowly rotating stellar surface, $\eta \simeq GM/(R{c}^2)$ and the corresponding surface accretion Eddington rate isThis is the appropriate convention for the numerical neutron-star estimates below, rather than imposing a separate reference efficiency of $0.1$.

The main evidence combines a large luminosity with a compact source and a massive central gravitating object. Rapid continuum variability bounds the emitting size by a light-crossing scale, while broad emission lines and reverberation delays constrain the gas orbital speeds and characteristic radii. Their virial masses, together with stellar and gas dynamics in galactic nuclei, imply compact central masses far larger than ordinary stars. Thermal optical/ultraviolet disk emission, a high-energy coronal component, jets and relativistically broadened inner-disk features are naturally associated with an accretion flow. No one signature alone is a unique proof, but the combined compactness, mass and energy requirements strongly favor accretion onto supermassive black holes. Gravitational accretion releases a much larger fraction of rest energy than ordinary stellar burning, without requiring the luminosity to come from the black-hole interior.

For a fully ionized pure-hydrogen plasma, each electron receiving Thomson radiation force is coupled electrostatically to a proton carrying almost all the mass. With spherical isotropic luminosity $L$, the outward force per electron is ${\sigma }_{T}L/(4\pi r^{2}c)$, while gravity per electron–proton pair is approximately $GM{m}_p/r^2$. Equality defines the Eddington luminosity:Here $G$ is the gravitational constant, $m_p$ the proton mass, $c$ the speed of light, $M$ the central mass and ${\sigma }_T$ the Thomson cross-section. The estimate assumes electron-scattering opacity, tight electron–ion coupling and no other dominant force. Geometry, beaming, other opacities and photon trapping can change the relevant luminosity limit.

If ${\dot{M}}_0$ is the supplied rest-mass rate and a fraction $ϵ$ of its rest energy is radiated, $L=ϵ{\dot{M}}_0{c}^2$. The associated Eddington accretion rate is consequentlyThis is a supply rate. The black-hole growth rate is ${\dot{M}}_{\mathrm{BH}}=(1-ϵ){\dot{M}}_0$ if the remaining supplied energy increases the hole's mass. Confusing these rates loses a factor relevant to the growth-time calculation.

The order-of-magnitude efficiency $ϵ\sim 0.1$ is motivated by orbital binding energy released near the innermost stable orbit. A Newtonian orbit at $6GM/c^2$ has binding energy per unit mass $c^2/12\simeq 0.083c^2$; the exact nonspinning thin-disk value is $1-\sqrt{8/9}\simeq 0.057$. Black-hole spin changes the innermost stable orbit and can raise or lower the efficiency. Thus $0.1$ is a useful radiatively efficient accretion estimate, not a universal constant; an advective or jet-dominated flow can have a different radiative efficiency.

Using the stipulated luminosity and ${\sigma }_T=6.65\times 10^{-29}{\mathrm{m}}^2$,The source's redshift and age are treated as its supplied data; its record wording is historical, not a present-day claim about the most distant known quasar.

At continuous Eddington luminosity and fixed efficiency, growth is exponential:This is the Salpeter time, including retained-mass efficiency. Even granting all $0.77\mathrm{Gyr}$ since the big bang permits only about 15.4 e-folds, requiring a seed of at least roughly $400M_{\odot }$. A $100M_{\odot }$ seed would take about $0.84\mathrm{Gyr}$ under these assumptions, before allowing any formation delay; a $10^5{M}_{\odot }$ seed takes about $0.49\mathrm{Gyr}$. These numbers motivate early massive seeds, sustained high duty-cycle accretion, lower radiative efficiencies, super-Eddington phases or contributions from mergers. They do not prove a unique seed channel, since the luminosity-to-mass conversion and accretion history are model-dependent.

Supermassive black holes are compact galactic nuclei with masses typically of millions to billions of solar masses. Their gravitational influence is measured from the motion of surrounding matter; accretion luminosity is evidence of feeding, not a necessary property of every black hole. Keplerian stellar orbits near the Galactic Centre and molecular-maser rotation curves in other nuclei provide particularly direct compact-mass constraints. A rough black-hole sphere of influence is $r_{\mathrm{infl}}\sim G{M}_{\bullet }/{\sigma }_{\star }^2$; resolving it helps dynamical mass estimates.

Accretion converts binding energy into radiation,The latter is the Eddington luminosity for isotropic radiation opposed by electron-scattering opacity, not an absolute luminosity ceiling for every geometry. The retained black-hole mass grows at $(1-\eta ){\dot{M}}_{\mathrm{in}}$. Sustained feeding can produce an active galactic nucleus, whereas gas-poor nuclei may be faint. Loss of gas angular momentum is a central feeding problem.

Black-hole mass correlates with classical-bulge stellar velocity dispersion through the M-sigma relation. Accretion-powered radiation, winds and jets can heat or remove gas, linking nuclear growth to galaxy evolution. Such correlations do not establish one universal feedback history, and disks and pseudobulges do not follow the same tight relations as classical bulges. Seed formation, accretion and mergers all contribute; seed masses and feeding histories remain model dependent.^[ref]

The radiative family is a stylized approximation to gas-pressure-supported main sequence stars with mostly radiative interiors and moderately temperature-sensitive hydrogen burning, such as the roughly solar-mass regime. Its opacity exponents are local approximations, not a universal composition-independent opacity law. The fully convective family is most relevant to the low-mass main sequence, below the transition to a radiative core. Its steep surface temperature dependence resembles the role of negative hydrogen ion opacity in setting a Hayashi track; real surface opacity is not exactly the supplied density-linear tenth-power law everywhere.

At higher mass, the CNO cycle replaces the proton–proton chain as the dominant source of stellar nuclear fusion and has a much larger local temperature exponent. This promotes a convective core. Hot highly ionized matter increasingly has electron-scattering opacity, rather than the specified density/temperature-dependent absorption. Radiation pressure becomes important in the equation of state, changing $T_c\propto M/R$ and eventually making the luminosity relation approach an Eddington luminosity scaling. Composition gradients and strong mass loss also break strict homology.

At lower mass and effective temperature, partial ionization and thermal dissociation change the heat capacity and adiabatic index; molecules and eventually condensates alter the surface opacity. Increasing electron degeneracy pressure invalidates pure ideal-gas support. Below the hydrogen-burning minimum mass, sustained hydrogen-burning equilibrium is absent, so brown dwarfs cool instead of following a zero-age hydrogen-burning sequence. These changes explain why neither derived slope should be extrapolated indefinitely.

An X-ray binary contains a compact accretor supplied by a donor star. Most bright systems contain a neutron star or black hole. Gas arrives by Roche-lobe overflow, a captured stellar wind, or donor outflows, commonly forming an accretion disk. The primary energy source is the release of gravitational binding energy during accretion, with a characteristic accretion luminosityFor a neutron star, $R_{\mathrm{in}}$ can be near its material radius; for a black hole, radiation is produced by the flow outside the horizon. For $M\sim 1.4M_{\odot }$ and $R\sim 10\mathrm{km}$, $GM/(R{c}^2)\sim 0.2$, so accretion can release much more energy per unit mass than hydrogen burning. The hot inner flow, boundary layer or accretion column emits X-rays.

In a high-mass X-ray binary, a massive, relatively short-lived donor often supplies gas through a strong wind or an equatorial outflow; some systems instead undergo Roche-lobe overflow. These objects trace young stellar populations. In a low-mass X-ray binary, a lower-mass donor commonly fills its Roche lobe and feeds a disk. Such systems can persist in older populations. These mass labels refer to the donor, not the compact accretor.

A magnetized neutron star can channel accretion onto its poles, producing an accreting X-ray pulsar as the star rotates. Accretion transfers angular momentum, so changes in spin constrain the flow and magnetic coupling. A Type I X-ray burst has a different origin: accumulated hydrogen or helium undergoes unstable thermonuclear burning on a neutron star surface. It rises rapidly and then cools as the fuel is exhausted. A black hole has no material surface supporting the same surface-fuel mechanism.

Orbital motion and optical observations of the donor constrain masses, with inclination and donor-model uncertainties. A compact mass exceeding plausible stable neutron star masses supports a black hole interpretation. Rapid variability probes the small emitting region, while disk spectra and recurrent outbursts probe accretion and its instabilities. The Eddington luminosity,is obtained by balancing radiative acceleration $\kappa L/(4\pi r^{2}c)$ against $GM/r^2$; it is a useful luminosity scale under the assumed isotropic, opacity-controlled force law. Geometry and time dependence can complicate its application. X-ray binaries reveal compact stars through efficient gravitational accretion, with pulsations, bursts and orbital dynamics distinguishing their physical components.

Outside the thin burning shell, the enclosed mass and luminosity are approximately $M_c$ and $L$. Divide the hydrostatic pressure support equation by radiative diffusion in a star with constant opacity:In the constant-opacity radiative-zero giant envelope approximation, the outer pressure and temperature constants are negligible. Integration givesThe radiation pressure is $a{T}^4/3$, so the gas pressure fraction $\beta$ is constant and satisfies $(1-\beta )\Lambda =a/3$. Thus alsoThe physical branch has positive gas pressure, hence $L<4\pi cG{M}_c/{\kappa }_0$, the constant-opacity Eddington luminosity for the core mass.

Using the gas part of the equation of state, define $H=\mu \beta \Lambda /\mathcal{R}$, so $\rho =H{T}^3$. Substitution into hydrostatic balance givesConsequently, in the same deep-envelope radiative-zero approximation,Precisely, a finite outer matching point $(R_s,T_s)$ gives $T=T_s+\mu \beta G{M}_c(1/r-1/R_s)/(4\mathcal{R})$. The displayed $1/r$ profile neglects that additive outer-boundary constant; it is appropriate near a compact core when the envelope extends to much smaller temperatures.

To estimate the thin-shell luminosity, write $K_T=\mu \beta G{M}_c/(4\mathcal{R})$, so $T=K_T/r$. The hydrogen-layer heating law givesThis steep radial dependence strongly concentrates the emission near $R_c$. Continuing the outer profile across the thin emitting layer and taking a distant upper limit givesThe same radiative coefficient implies $H=16\pi ac{K}_T/(3{\kappa }_{0}L)$. Hence the high-power shell burning in a constant-opacity giant envelope satisfiesTaking the cube root gives the requested form,Thus $C$ contains only the specified opacity and heating normalizations and physical constants. The explicit coefficient uses the outer radiative profile as a thin-shell estimate; resolving the variation of $L_r$ within the emitting layer can change that coefficient. Also $\beta$ depends on $L$, so the radiation-pressure-inclusive relation is implicit, not a pure explicit eighth-power core-mass law with independently fixed $\beta$.

The high luminosity-to-mass ratio makes radiative acceleration and atmospheric departures from the simple approximations important. For an illustrative fully ionized hydrogen-rich composition with electron-scattering opacity ${\kappa }_e\simeq 0.34\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$, the stellar electron-scattering Eddington factor isThus electron scattering alone removes roughly half of the effective surface gravity. It is below the Eddington luminosity, so electron scattering alone does not prove that a hydrostatic atmosphere is impossible. Line opacity can provide substantially more radiative acceleration and can drive a stellar wind.

For a plane-parallel atmosphere, the geometrical condition is that the line-forming region be thin compared with the radius. In a simple isothermal photospheric estimate, the atmospheric scale height isThis is plane-parallel validity from pressure scale height. The mass and luminosity alone do not specify $R$ or the atmospheric temperature, so they cannot decide the geometry uniquely. For example, if $T_e=3\times 10^4\mathrm{K}$ and $\mu \simeq 0.62$, the luminosity relation gives $R\simeq 37R_{\odot }$ and $H/R\simeq 3\times 10^{-3}$. Plane-parallel geometry could then be adequate for deep weak photospheric lines. It would still fail for lines formed over an extended wind, and a cooler, more extended supergiant needs a separate geometrical assessment. This numerical example is conditional, not an additional datum about the stated star.

LTE requires collisions and local thermal processes to establish the relevant excitation and ionization populations at the local gas temperature. In the dilute outer layers of a luminous hot star, radiative transition rates can dominate collisions, and the radiation field arriving from other depths is not the local Planck function. A non-LTE stellar atmosphere should therefore use statistical equilibrium with radiative and collisional rates coupled to transfer. Deep layers may approach LTE, but surface ionization balances and abundance-sensitive spectral lines are particularly susceptible to departures. An LTE abundance can be systematically wrong even when a plane-parallel approximation is geometrically good.

A static atmosphere requires velocities to be negligible where the diagnostic lines form. Massive luminous stars commonly have line-driven outflows, so static models cannot represent wind-formed profiles or their mass density and velocity gradients. A hydrostatic approximation can still be useful below the sonic region if the chosen lines form there, provided radiative acceleration is included in the effective gravity. Large outflows call for spherical moving-atmosphere models rather than merely a changed abundance in a static model.

The three assumptions cannot be accepted on the quoted mass and luminosity alone. LTE is especially doubtful in the outer layers; wind-sensitive abundance diagnostics require non-LTE moving models, and geometry must be checked through atmospheric extension and line-formation depth. Reliable stellar chemical abundances should be supported by observed wind signatures, temperature and gravity constraints, and agreement between several suitable ionization stages or weak photospheric lines.

For fully ionized hydrogen, the outward radiation force on a coupled proton-electron pair at radius $r$ is ${\sigma }_{T}L/(4\pi r^{2}c)$, while its inward gravitational force is approximately $GM{m}_p/r^2$. Equating them yields the Eddington luminosity,This uses isotropic radiation, Thomson scattering and efficient momentum coupling between electrons and ions. Composition or other opacity sources alter the corresponding limit.

With radiative efficiency ${ϵ}_r$, a rest-mass supply rate gives $L={ϵ}_r{\dot{M}}_{\mathrm{in}}{c}^2$. In the approximation that supplied mass is added to the hole, continuous accretion at the Eddington luminosity givesHere $t$ is the elapsed time since seed formation. If the radiated rest mass is counted exactly, $\dot{M}=(1-{ϵ}_r){\dot{M}}_{\mathrm{in}}$ and the Salpeter time becomes $t_e={ϵ}_{r}c{\sigma }_T/[(1-{ϵ}_r)4\pi G{m}_p]$. The displayed approximation in the question neglects this correction; retaining it lengthens the growth time.

The required growth factor is $10^7$, corresponding to $\ln (10^7)=16.12$ e-folds. With the supplied efficiency,At $z=8$, the expansion is overwhelmingly matter-dominated. Using total ${\Omega }_{\mathrm{mat}}=0.25$ and $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ givesThe baryon density is part of the total matter density, so it is not added to ${\Omega }_{\mathrm{mat}}$ in this age calculation. Dark-energy and radiation corrections are small for this estimate. Even starting at the earliest possible time, the approximate growth time exceeds the age, and a real stellar seed forms later. A $30M_{\odot }$ seed cannot reach the stated mass through uninterrupted Eddington-limited growth with the stated efficiency. Exactly retaining the radiated mass gives about $806\mathrm{Myr}$ and strengthens this conclusion. A heavier seed, lower radiative efficiency or super-Eddington episodes can relieve the time constraint; interruptions make it harder.

Both gas and radiation pressures are nonnegative, so $0\le \beta \le 1$. Rearranging the constant-fraction result givesThis is the Eddington luminosity for the effective opacity constant of this model. Positive gas support gives a strict inequality, with equality only in the radiation-only limiting case. Exceeding the limit would require a negative gas-pressure fraction, which is incompatible with the assumed equation of state.

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.

Assume a stationary spherical outflow with speed $v>0$, mass-loss rate $\dot{M}>0$, an optically thick thermal radiation field and negligible radiation inertia compared with matter inertia. The stellar wind continuity equation givesUse the printed gas fraction $\beta =P_{\mathrm{gas}}/P$, with $P=P_{\mathrm{gas}}+P_{\mathrm{rad}}$, and assume constant mean molecular weight ${\mu }_{\mathrm{mol}}$. Put $\mathcal{R}=k_B/({\mu }_{\mathrm{mol}}{m}_u)$ and $a_g^2=\mathcal{R}T$ for the isothermal sound speed squared. Then $P_{\mathrm{gas}}=\rho a_g^2$ and $P_{\mathrm{rad}}=a_{\mathrm{rad}}{T}^4/3$. The momentum equation including the radiation-pressure gradient isDo not also add a separate radiative force here: the diffusion radiation-pressure gradient already represents that force.

Using radiative diffusion in a star,Write $g=G{M}_r/r^2$ and $\Gamma =L_r/L_{\mathrm{crit}}=\kappa L_r/(4\pi cG{M}_r)$, the local ratio to the Eddington luminosity. Substituting continuity into the gas-pressure gradient gives the gas and radiation pressure stellar wind equationThe denominator changes sign at the sonic point $v=a_g$. A smooth transonic branch must have its right-hand numerator vanish there too. In a subsonic flow the denominator is negative, so acceleration requires a negative numerator; for a supersonic flow it requires a positive numerator. A regular subsonic-to-supersonic branch must therefore pass through a compatible common zero, with a finite slope obtained by differentiating the equations. Boundary conditions and that regularity requirement normally select the mass flux. Other branches can remain subsonic, remain supersonic or encounter a singular gradient. In the isothermal nonradiating limit the numerator is $2a_g^2/r-GM/r^2$, recovering the Parker wind equation and critical radius $GM/(2a_g^2)$.

To expose the role of $\beta$, rewrite the diffusion temperature gradient asHenceAt the stipulated local value $\Gamma =1$, with $0<\beta <1$, its numerator isThe gas-pressure contribution is not removed by setting the luminosity equal to the Eddington value. Instead the negative temperature gradient makes the thermal term in the numerator positive as well. A subsonic outflow then has $dv/dr<0$, and no common numerator/denominator zero is available for smooth sonic passage. This is the sonic-point obstruction at the Eddington luminosity.

Thus a subsonically launched wind cannot accelerate smoothly to supersonic speed under these diffusion assumptions at $L_r/L_{\mathrm{crit}}=1$. It is not a claim that every initially supersonic solution decelerates; those have positive denominator and accelerate in this equation. Even in the radiation-dominated limit $\beta \to 0$, the spherical thermal term remains positive for $a_g^2>0$. The singular pressureless limit is not a finite-temperature sonic crossing. Finally $a_g$ is the thermal sound scale of this diffusion-controlled stationary equation, not the full adiabatic sound speed of a tightly trapped gas-plus-radiation perturbation. Optically thin winds or different heating/driving mechanisms require another transport model.

Treat each grain as an isolated compact sphere and take its wavelength-independent radiation-pressure efficiency to be $Q_{\mathrm{pr}}=1$. At distance $R$, its mass and radiation-force cross-section areThe inward gravitational force and outward radiation pressure force areThus the opacity per unit dust mass and the dust-grain Eddington limit areTo equal the Eddington luminosity based on electron-scattering opacity, ${\kappa }_d$ must equal ${\kappa }_{\mathrm{es}}$. ThereforeFor fully ionized hydrogen, ${\kappa }_{\mathrm{es}}={\sigma }_T/m_p\simeq 0.40\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$, and the stated ${\rho }_d=1\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$ givesUsing $0.34\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$ for a roughly solar-composition gas instead gives $2.2\mathrm{cm}$. This composition-dependent normalization should be stated because the problem does not fix it. For a different constant radiation-pressure efficiency, the critical radius is multiplied by $Q_{\mathrm{pr}}$.

At $L=L_{\mathrm{Edd},\mathrm{es}}$, the grain's radiation-pressure coefficient isConsequently, smaller grains experience a net outward force, larger grains remain gravitationally attracted, and critical-size grains have zero net radial force. Grains released at rest are therefore size-sorted into expelled and infalling populations. With an initial inward velocity, an outward force does not guarantee reversal before the grain reaches the hole; trajectories also depend on initial conditions. Gas drag and dust destruction have been omitted from this isolated-grain calculation. Ordinary interstellar grains are much smaller than a centimetre, so their direct radiative acceleration is very strong in this idealized model.

Use a geometrically thin spherical shell with constant mass $M_{\mathrm{sh}}$ and assume that its dust transfers momentum efficiently to its gas. Define the ultraviolet and infrared opacities per unit total shell mass by ${\kappa }_{\mathrm{UV}}$ and ${\kappa }_{\mathrm{IR}}$. Its dust optical depths areFor a central ultraviolet luminosity $L$, the usual dusty radiation-pressure-driven shell model absorbs a fraction $1-e^{-{\tau }_{\mathrm{UV}}}$ of the incident momentum and adds an infrared trapping contribution ${\tau }_{\mathrm{IR}}L/c$. Neglecting swept-up mass, shell self-gravity, and other gravitating matter, its thin-shell momentum equation isBalancing these terms gives the critical luminosity of a dusty shell:This is a local force-balance luminosity, so it can depend on radius even when $L$ is constant.

The three useful limiting Eddington ratios areThe infrared limit assumes ${\tau }_{\mathrm{IR}}\gg 1$ and ultraviolet absorption; the single-scattering limit assumes ${\tau }_{\mathrm{UV}}\gg 1\gg {\tau }_{\mathrm{IR}}$; the ultraviolet-thin limit assumes ${\tau }_{\mathrm{UV}}\ll 1$ with infrared opacity negligible compared with ultraviolet opacity. In the ultraviolet-opaque region the two radiative contributions add, and the effective ratio is ${\Gamma }_{\mathrm{IR}}+{\Gamma }_{\mathrm{ss}}(R)$.

The ultraviolet dust transparency radius for this constant-mass shell isFor the terminal speed of a constant-mass dusty shell, approximate the ultraviolet absorption factor by $1$ inside $R_{\mathrm{UV}}$ and by ${\tau }_{\mathrm{UV}}$ outside. The infrared contribution outside that radius is neglected; it is small there if ${\kappa }_{\mathrm{IR}}\ll {\kappa }_{\mathrm{UV}}$. Define ${\Gamma }_{\mathrm{ss}}={\Gamma }_{\mathrm{ss}}(R_0)$, so this otherwise radius-dependent ratio in the paper's final formula is an initial value.

Inside $R_{\mathrm{UV}}$ the acceleration isIntegrating from $R_0$ to $R_{\mathrm{UV}}$ givesOutside $R_{\mathrm{UV}}$, $vdv/dR=G{M}_{\mathrm{BH}}({\Gamma }_{\mathrm{UV}}-1)/R^2$. Its integral to infinity supplies $2G{M}_{\mathrm{BH}}({\Gamma }_{\mathrm{UV}}-1)/R_{\mathrm{UV}}$. Combining the two integrals yieldsThis assumes that the outward trajectory actually reaches the transparency radius; a positive formal endpoint energy alone does not rule out an intermediate turning point.

Put $X=R_{\mathrm{UV}}/R_0=\sqrt{{\tau }_{\mathrm{UV}}(R_0)}$ and $v_{\mathrm{esc},0}^2=2G{M}_{\mathrm{BH}}/R_0$. The ratios obey ${\Gamma }_{\mathrm{UV}}={\Gamma }_{\mathrm{ss}}{X}^2$. The preceding expression becomesIf $X\gg 1$ and infrared work is subdominant,For a luminosity of order the initial single-scattering Eddington luminosity, ${\Gamma }_{\mathrm{ss}}\sim 1$, this implies $v_{\infty }/v_{\mathrm{esc},0}\sim \sqrt{2}{\tau }_{\mathrm{UV}}(R_0{)}^{1/4}\gg 1$. The shell continues intercepting approximately $L/c$ of momentum per unit time over a large distance while gravity weakens as $R^{-2}$. The resulting radiative work is not restricted to its initial gravitational binding energy: an Eddington luminosity is a local force threshold, not a universal cap on outflow speed.

If the unqualified $L_{\mathrm{Edd}}$ instead means the electron-scattering value, the statement needs an additional condition. For $L=L_{\mathrm{Edd},\mathrm{es}}$, ${\Gamma }_{\mathrm{UV}}={\kappa }_{\mathrm{UV}}/{\kappa }_{\mathrm{es}}$, andThus speeds greatly above escape velocity require ${\kappa }_{\mathrm{UV}}/{\kappa }_{\mathrm{es}}\gg \sqrt{{\tau }_{\mathrm{UV}}(R_0)}$, together with a viable launch. Large ultraviolet optical depth by itself does not prove the conclusion at fixed electron-scattering luminosity and opacity. The two interpretations must not be conflated.