= Solution
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 $\sigma_TL/(4\pi r^2c)$, while gravity per <electron>–proton pair is approximately $GMm_p/r^2$. Equality defines the <Eddington luminosity>:
$$
\boxed{L_{\rm Edd}=\frac{4\pi Gm_pcM}{\sigma_T}.}
$$
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 $\epsilon$ of its rest <energy> is radiated, $L=\epsilon\dot M_0c^2$. The associated <Eddington accretion rate> is consequently
$$
\boxed{\dot M_{0,\rm Edd}=\frac{4\pi Gm_pM}{\epsilon c\sigma_T}.}
$$
This is a supply rate. The black-hole growth rate is $\dot M_{\rm BH}=(1-\epsilon)\dot M_0$ 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 $\epsilon\sim0.1$ 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> $c^2/12\simeq0.083c^2$; the exact nonspinning thin-disk value is $1-\sqrt{8/9}\simeq0.057$. 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 $\sigma_T=6.65\times10^{-29}\,\mathrm{m^2}$,
$$
\frac{L_{\rm Edd}}{M/M_\odot}\simeq1.26\times10^{31}\,\mathrm W\simeq3.29\times10^4L_\odot,
\qquad
\boxed{M\simeq\frac{6.3\times10^{13}}{3.29\times10^4}M_\odot\simeq1.92\times10^9M_\odot.}
$$
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:
$$
M(t)=M_{\rm seed}e^{\Delta t/t_S},\qquad
\boxed{t_S=\frac{\epsilon}{1-\epsilon}\frac{\sigma_Tc}{4\pi Gm_p}\simeq5.00\times10^7\,\mathrm{yr}\quad(\epsilon=0.1).}
$$
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 $10^5M_\odot$ 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.
\Image[/past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2016-eddington-growth-window.png]
{title=Time needed to build the inferred quasar mass at the Eddington limit}
{description=Original constant-efficiency growth calculation with efficiency 0.1 and the retained-mass factor included. The age line grants the full quoted cosmic age; actual seed formation leaves less time.}
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