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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.

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.

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.

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.

The three principal black-hole seed channels occupy different mass ranges.

For radiative efficiency $\eta =0.1$, growth at Eddington ratio $f_{\mathrm{Edd}}$ obeyswhere $t_E=0.45\mathrm{Gyr}$. Between $z=25$ and GN-z11 at $z=10.6$,so the exponent is $6.04f_{\mathrm{Edd}}$. Equivalently, the Salpeter time is about $0.05\mathrm{Gyr}/f_{\mathrm{Edd}}$.

Reaching $10^6{M}_{\odot }$ requiresRepresentative values areContinuous Eddington-limited growth supplies only $e^{6.04}\simeq 420$, so a $100M_{\odot }$ seed reaches about $4\times 10^4{M}_{\odot }$, whereas a seed above roughly $2.4\times 10^3{M}_{\odot }$ can reach the target at a unit duty cycle. Light seeds require sustained mildly super-Eddington accretion, an earlier start, mergers, or lower effective efficiency. Heavy direct-collapse seeds need only a moderate time-averaged Eddington ratio and are therefore easier to reconcile with the short available time. No channel is ruled out by the mass alone because seed masses, obscuration, duty cycles, super-Eddington episodes, mergers, and the observational mass estimate are uncertain.

photon orbit and the Newtonian limit. Write $r_g=GM/c^2$ and $e=E/(m{c}^2)$. For a timelike circular Schwarzschild geodesic,The denominator tends to zero at $r=3r_g$, so the energy per unit rest mass diverges as a massive circular orbit approaches that radius. The corresponding local orbital speed measured by a static observer satisfies $v_{\mathrm{loc}}^2/c^2=r_g/(r-2r_g)$ and tends to one. This suggests a null circular orbit, but one must not simply insert $m=0$ into a massive-particle formula. For a photon, the radial effective potential is proportional to $(1-2r_g/r)/r^2$; its derivative vanishes at $r=3r_g$ and its second derivative there is negative. Thus the photon sphere is at $3GM/c^2$, and its circular null orbits are unstable.

At $r\gg r_g$, expansion givesThe second term is the total Newtonian circular-orbit energy, kinetic plus potential; the relativistic energy also includes rest energy. Here $E$ is the conserved energy referred to a stationary observer at infinity, not the local kinetic energy.

The stable inner edge. At fixed angular momentum, write $\ell =L/(mc)$. The timelike radial geodesic equation can be writtenThe circular condition $V_{\mathrm{eff}}^{\prime }=0$ gives ${\ell }^2=r_g{r}^2/(r-3r_g)$. Substituting into the second derivative taken at fixed $\ell$ givesTimelike circular orbits exist for $r>3r_g$; they are stable against radial perturbations for $r>6r_g$ and unstable for $3r_g<r<6r_g$. ThereforeThe boundary orbit is marginally stable. Differentiating $e(r)$ also gives a minimum there, consistently with this fixed-angular-momentum stability calculation.

accretion efficiency and black-hole spin. Matter supplied slowly from far away initially brings rest energy $d{m}_0{c}^2$. A radiatively efficient, zero-inner-torque thin disk releases the binding energy down to the innermost stable circular orbit; the remaining energy $e_{\mathrm{ISCO}}d{m}_0{c}^2$ is carried into the hole during the plunge. ThusHere $\eta$ denotes the efficiency called $ϵ$ in this question. The ideal relation assumes negligible additional dissipation during plunge and neglects photon capture and advection. A hot radiatively inefficient flow need not have this efficiency.

For a Kerr black hole, an equatorial prograde disk has a smaller innermost stable circular orbit and greater binding-energy efficiency than the Schwarzschild value; a retrograde disk has the opposite trend. The formal extremal prograde limits are $r_{\mathrm{ISCO}}\to r_g$ and $\eta \to 1-1/\sqrt{3}\simeq 0.423$. These are ideal limiting values, not a universal realized efficiency. photon capture and the accretion geometry affect the achievable black-hole spin and escaping radiation.

radiation force and the limiting luminosity. For isotropic luminosity $L$, the outward radiation momentum flux at radius $r$ is $L/(4\pi r^{2}c)$. In fully ionized hydrogen, Thomson scattering acts mainly on the electron, while electrostatic coupling makes the electron and proton move together. Neglecting the electron mass, one electron–proton pair experiencesEquating them yieldsThis Eddington luminosity uses electron-scattering opacity, spherical or isotropic force balance and negligible relativistic corrections at the force-comparison radius. Composition, other opacity, anisotropy and photon trapping can change the relevant limit.

Exponential growth and doubling. Distinguish rest-mass supply ${\dot{m}}_0$ from black-hole mass growth $\dot{M}$:At the Eddington limit,Using the printed scattering cross section gives $t_E\simeq 4.50\times 10^8\mathrm{yr}$. The Salpeter time $t_S$ is an e-folding time; the Eddington-limited doubling time isfor the Schwarzschild thin-disk efficiency. If one approximates $\dot{M}\simeq {\dot{m}}_0$, the factor $1/(1-\eta )$ is dropped and the estimate is about $1.79\times 10^7\mathrm{yr}$. The retained-energy version is the mass-conserving expression.

The efficiency bound. A growth factor $10^8$ requires $N=\log (10^8)=18.4207$ e-folds. Eddington-limited accretion is the fastest growth allowed by the adopted luminosity bound at fixed efficiency, so growth within a time $\Delta t$ requiresThis efficiency upper bound from Eddington-limited growth is an upper bound: higher efficiency radiates more of each supplied rest mass, and the luminosity ceiling therefore allows less mass to enter per unit time. A duty cycle below unity or sub-Eddington luminosity tightens it. The approximate convention $\dot{M}={\dot{m}}_0$ would instead give $\eta \lesssim 0.121$; both figures refer to the same effect, with different treatment of the radiated mass deficit. Mergers, super-Eddington growth or a different seed history would change the premises.

Coherent prograde accretion spins up the hole, moves the inner edge inward and increases $\eta$. For fixed $L/L_{\mathrm{Edd}}$ the mass-growth e-folding time then grows, so the logarithmic growth curve becomes less steep; one cannot simply retain the initial Schwarzschild efficiency throughout. Episodic misaligned accretion can have a different black-hole spin history. In general,

energy available to affect the galaxy. Let $M_g$ denote galaxy mass, keeping it distinct from black-hole mass. At constant efficiency, negligible seed mass and no merger contribution, the emitted energy in building $M_{\mathrm{BH}}=0.002M_g$ isFor $v=300\mathrm{km}{\mathrm{s}}^{-1}$ and the nonspinning efficiency this is approximately $121$. The common small-efficiency approximation gives $2000\eta \simeq 114$, and a fiducial $\eta =0.1$ would give about $222$ with retained mass included. These differences do not alter the conclusion: the available accretion energy substantially exceeds the stated galactic binding-energy estimate.

Only a coupling fraction of order one percent would then be energetically sufficient to affect the gas strongly. radiation, winds or jets can heat gas, drive outflows and suppress further cooling and star formation, providing active-galactic-nucleus feedback and possible self-regulation of black-hole growth. energy availability alone does not prove that the whole galaxy is expelled: most radiation can escape, jets can be collimated, heated gas can cool, and collisionless stars couple differently from gas.