Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 347 (Astrophysical Black Holes)

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.

The standard Shakura--Sunyaev thin disk assumes: a steady state; axial symmetry; a geometrically thin disk $H/R\ll 1$; Newtonian, nearly Keplerian circular motion outside the inner edge; subsonic radial drift $|u_R|\ll u_{\varphi }$; vertical hydrostatic equilibrium; negligible disk self-gravity; an optically thick, locally thermal spectrum; local radiative balance between viscous heating and cooling; and a local alpha-viscosity stress with a zero-torque inner boundary near the innermost stable circular orbit. These assumptions also exclude dynamically dominant winds and large-scale external torques from the standard solution.

Start from the continuity equationIntegrate vertically, use axial symmetry and steadiness, and define the surface density of a disk $\Sigma =\int \rho dz$. Thenso the inward-positive accretion rate is

In a Keplerian accretion disk, $u_{\varphi }=R\Omega =(GM/R{)}^{1/2}$. Substitution into the steady azimuthal Navier--Stokes equation and cancellation of the common Keplerian factors givesSince $R^2\Omega \propto R^{1/2}$ and $d\Omega /dR=-3\Omega /(2R)$, this reduces to the standard viscous drift formulaThe minus sign describes inward drift when $\nu \Sigma R^{1/2}$ increases outward.

Let $l(R)=R^2\Omega =\sqrt{GMR}$ be the Keplerian specific angular momentum. In a source-free steady interval, the sum of advected and viscously transported angular momentum is constant:Inside the injection radius, $\dot{m}={\dot{m}}_0$. The zero-torque inner boundary condition at $R_{\mathrm{ISCO}}$ sets $C={\dot{m}}_0{l}_{\mathrm{ISCO}}$, and therefore

Outside $R_0$ there is no net mass flow in the stated steady distribution, but it must carry outward the angular momentum deposited by matter moving from $R_0$ to the ISCO. Its constant viscous torque in an accretion disk is thereforeThusThe two expressions agree at $R_0$; the jump in mass flux there is exactly the injected rate.

For Keplerian angular velocity, $R^2(d\Omega /dR{)}^2=9GM/(4R^3)$. The stated $F_{\mathrm{diss}}$ is the dissipation summed over both disk faces, so $F_{\mathrm{diss}}=2{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^4$. Inside $R_0$ this givesDifferentiating the factor $R^{-3}[1-(R_{\mathrm{ISCO}}/R{)}^{1/2}]$ shows that the maximum occurs atwhen this radius lies below $R_0$. In the ordinary inflowing region far from its inner edge, $T_{\mathrm{eff}}\propto R^{-3/4}$.

For the static angular-momentum sink outside $R_0$, part c instead givesso the genuinely large-radius behavior of the complete injected disk is $T_{\mathrm{eff}}\propto R^{-7/8}$.

Integrating $2\pi R{F}_{\mathrm{diss}}dR$ over both regions givesandHencewhich is exactly the loss of Keplerian orbital energy as matter moves from its injection orbit to the inner edge.

For a standard disk extending through a radius $R\gg R_{\mathrm{ISCO}}$,This is three times the binding-energy release $GM\dot{m}/(2R)$ available outside $R$. The excess is energy carried outward by the viscous torque in an accretion disk and dissipated at larger radii. In the injected model the nonaccreting outer disk is an especially direct example: it radiates despite having zero mean radial mass flux because it absorbs the angular momentum and mechanical work exported by the inner disk.

Moving radially outward from the quasar, the standard active-galactic-nucleus wind bubble contains: the freely expanding fast wind; a reverse shock that thermalizes it; a hot shocked-wind bubble; a contact discontinuity; a dense swept-up interstellar shell behind a forward shock; and finally the undisturbed interstellar medium.

Let $M_s(R)$ be the swept-up shell mass. If inverse-Compton and atomic radiative cooling remove the shocked wind's thermal energy faster than the bubble expands, the reverse shock is momentum driven andIf cooling is slow, the shocked wind remains hot and the bubble is energy driven:supplemented by the bubble energy equationThe hot bubble stores wind energy and performs work for much longer than the direct photon momentum-crossing time. Equating wind power with shell kinetic power gives the characteristic momentum boostwhen the shell is much slower than the nuclear wind.

For a dusty shell, direct ultraviolet absorption, infrared trapping, and incomplete ultraviolet absorption give the radiation forceBalancing it against gravity gives the general critical luminosity of a dusty shell

For a singular isothermal sphere,soandIn the infrared-thick limit $F_{\mathrm{rad}}\simeq {\tau }_{\mathrm{IR}}L/c$; in the single-scattering limit it is $L/c$; and in the ultraviolet-thin limit it is ${\tau }_{\mathrm{UV}}L/c$. Therefore

The swept shell has $M_s=Ar$, where $A=2f_{\mathrm{gas}}{\sigma }^2/G$. Its steady radial thin-shell momentum equation isLet $r_0$ be the launch radius and impose $v(r_0)=0$.

In the single-scattering regime, defineThe force difference is constant, and integration givesThe shell accelerates monotonically, but continuous sweeping of $M_s\propto r$ makes the speed approach the finite maximum

For the optically thin ultraviolet regime, define the launch Eddington factorSince $F_{\mathrm{rad}}={\tau }_{\mathrm{UV}}L/c\propto r^{-1}$ while the shell's gravitational force is constant, integration givesIt initially accelerates, reaches its maximum atand then decelerates, formally stalling at $r=r_0(2{\Gamma }_{\mathrm{UV}}-1)$. The contrast is physical: single-scattering transfers the same $L/c$ to the shell at every radius, whereas an ultraviolet-thin shell intercepts a fraction proportional to its declining optical depth, so the isothermal host's gravity eventually wins.