Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 347 / 1

At radius $r$, radiation of luminosity $L$ gives matter of flux-mean opacity $\kappa$ the outward accelerationEquating this with the gravitational acceleration $G{M}_{\mathrm{BH}}/r^2$ gives the Eddington luminosity

Let $L=f{L}_{\mathrm{Edd}}$ and let $\mathcal{M}(r)$ be the radiative force multiplier relative to the Compton or electron-scattering acceleration. A steady spherical wind obeys mass conservation ${\dot{M}}_w=4\pi r^2\rho v$ and the radial Euler momentum equationWhen gas pressure is negligible, radiation accelerates the wind locally if $f\mathcal{M}>1$. Electron scattering alone has $\mathcal{M}\simeq 1$, so it launches a wind only near or above the Eddington ratio $f=1$. Bound-free absorption, spectral lines, or dust can give $\mathcal{M}\gg 1$ and allow a sub-Eddington source to launch a wind, provided the gas is not so highly ionized that those extra opacities disappear. A wind that has already acquired sufficient kinetic energy may coast through a region where $f\mathcal{M}<1$, but such a region cannot launch a cold wind from rest.

Write the innermost stable circular orbit asEstimating the emitting area as $4\pi R_{\mathrm{ISCO}}^2$, the Stefan–Boltzmann law givesand henceThus $T_{\mathrm{BB}}\propto M_{\mathrm{BH}}^{-1/4}$ at fixed $f$ and opacity: a larger black hole radiates from an area growing as $M_{\mathrm{BH}}^2$, faster than its Eddington luminosity grows. Prograde black-hole spin moves the ISCO inward and raises the characteristic temperature, whereas retrograde spin moves it outward and lowers the temperature. Relativistic transfer and the radial temperature profile change the numerical coefficient, not these leading scalings.

For radiative efficiency $ϵ$, luminosity and black-hole growth rate are related byIf the absolute accretion rate is constant, the lifetime needed to reach $M_f=5\times 10^8{M}_{\odot }$ isExpressing the constant luminosity as $L=f{L}_{\mathrm{Edd}}(M_f)$ gives, when $M_{\mathrm{seed}}\ll M_f$,Sub-Eddington feeding can therefore last a substantial fraction of the roughly $1\mathrm{Gyr}$ available by $z=6$, while super-Eddington feeding can assemble the hole much faster than $50\mathrm{Myr}$. Because an arbitrary constant rate can be traded against its starting time and black-hole seed mass, the final mass alone provides essentially no separate constraint on either seed mass or seeding redshift.

For continuously Eddington-limited growth the luminosity rises with the mass, so growth is exponential rather than linear:ConsequentlyFor seed masses from roughly $1M_{\odot }$ to $10^6{M}_{\odot }$, the supplied logarithms give about $20$ to $6.2$ e-foldings, or approximately $0.9$ to $0.3\mathrm{Gyr}$ if the common $t_{\mathrm{Sal}}\simeq 0.045\mathrm{Gyr}$ convention neglects the small $(1-ϵ)$ correction. Unlike constant absolute-rate growth, Eddington-limited growth therefore spends many e-folding times at low mass and strongly links viable seed mass to seeding epoch.

A sufficiently deep high-redshift quasar survey measures the numerous faint progenitors that precede the rare luminous objects. Its luminosity function and duty cycle can distinguish long exponential growth from short rapid episodes and constrain the distribution of seed masses, formation redshifts, Eddington ratios, and obscured growth.

Two useful host-mass arguments are the empirical black-hole--bulge mass relation and an independent dynamical or halo estimate. Applying the low-redshift ratio $M_{\mathrm{BH}}/M_{\mathrm{bulge}}\sim 10^{-3}$--$10^{-2}$ suggests $M_{\ast }\sim 5\times 10^{10}$--$5\times 10^{11}{M}_{\odot }$. Spatially resolved line widths and sizes give $M_{\mathrm{dyn}}\sim R{v}^2/G$, while the abundance of such rare quasars and the cosmic baryon fraction constrain their halo's available stellar mass. The existence of these systems before the Universe is one gigayear old implies exceptionally rapid assembly of both stars and black holes; departures from the local mass relation are therefore plausible and scientifically informative.