Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 347 (Astrophysical Black Holes)

Let $u=-v_r>0$ denote the inward speed. Steady spherical mass conservation and the radial momentum equation for an isothermal gas giveEliminating $d\log \rho /dr$ produces the Isothermal Bondi equationA smooth flow can cross Mach number one only where both sides vanish. Its Bondi sonic point is thereforeThe integrated Bernoulli equation which approaches rest and density ${\rho }_{\infty }$ at infinity isAt the sonic point this gives ${\rho }_s=e^{3/2}{\rho }_{\infty }$, and hence the unique regular transonic branch has

This mode is most plausible for a nearly stationary supermassive black hole immersed in the hot, pressure-supported, X-ray-emitting atmosphere of a massive elliptical galaxy. Such gas is comparatively smooth, its random and rotational speeds can be smaller than its sound speed, and radiative cooling can be slow on the inflow scale.

The ideal Bondi accretion assumptions can fail in several independent ways. The gas may carry enough specific angular momentum to circularize into an accretion disk; a galaxy's stars and dark matter can dominate the gravitational potential outside the black hole's sphere of influence; the hole can move relative to the gas; cooling, conduction, or external heating can destroy adiabaticity or isothermality; a multiphase or clumpy medium is not a uniform reservoir; magnetic fields, turbulence, and viscosity add stresses absent from the spherical model; and AGN feedback, winds, or jets can expel or recirculate the inflow. Time dependence and self-gravity provide further failures. Thus a Bondi estimate is best interpreted as an idealized supply rate, not automatically as the rate crossing the event horizon.

If $c_{s,\infty }\gg \sigma$ and $M_{\mathrm{enc}}\to M$, expansion of the denominator givesthe stated model's pressure-dominated Bondi accretion limit. Its order-unity coefficient differs from the $\pi e^{3/2}$ found for the exactly isothermal critical solution because such coefficients depend on the adopted equation of state and interpolation.

If instead $\sigma \gg c_{s,\infty }$,The gas's thermal motion is then negligible beside the galactic velocity dispersion, and the enclosed galactic mass, rather than the black hole alone, focuses the gas. For a singular isothermal sphere, $M_{\mathrm{enc}}(R)=2{\sigma }^{2}R/G$, so this becomes $\dot{M}\simeq 4\pi {\rho }_{\infty }\sigma R^2$. This second limit describes capture controlled by the host potential and is therefore not a genuinely spherical black-hole Bondi solution.

Write $w=1/r$ and $h=r^2\dot{\theta }=b{v}_{\infty }$. Since $\dot{r}=-hdw/d\theta$, the radial equation becomes the Binet equationwhose solution isChoose the incoming asymptote at $\theta =\pi$ and the downstream axis at $\theta =0$. Then $w\to 0$ as $\theta \to \pi$, while $r\sin \theta \to b$. These two conditions giveThe mirror-image streamlines meet on the downstream axis atAt that point each streamline has radial velocity $-v_{\infty }$ and equal and opposite azimuthal velocity. An inelastic collision cancels the latter, so the specific energy afterwards isThe gas is bound when $E_{\mathrm{after}}<0$, orSweeping the corresponding capture cylinder through gas of density ${\rho }_{\infty }$ gives the Bondi--Hoyle--Lyttleton accretion rateUnlike stationary spherical Bondi accretion, this is a directed, supersonic flow with a downstream focusing wake; bulk speed replaces sound speed as the main resistance to capture.

An optically thick annulus radiates as a blackbody from both faces, soFor a steady Keplerian accretion disk with a zero-torque inner boundary condition,ConsequentlyandAway from the inner edge, $T\propto R^{-3/4}$.

Ignoring inclination and distance factors, the multitemperature blackbody disk spectrum isSet $x=h\bar{\nu }/(k_{B}T)$. Since $T\propto R^{-3/4}$, $RdR\propto {\bar{\nu }}^{-8/3}{x}^{5/3}dx$, whereas the Planck function contributes ${\bar{\nu }}^3/(e^x-1)$. In the stated intermediate-frequency range the radial endpoints become $0$ and $\infty$, leaving a frequency-independent convergent integral. Therefore

The standard Shakura--Sunyaev thin disk has three qualitative radial zones. Its hot inner part is dominated by radiation pressure and electron-scattering opacity; farther out, gas pressure overtakes radiation pressure while electron scattering can remain the main opacity; in the cool outer zone, gas pressure remains dominant and free-free opacity becomes important. The transition radii vary with $M$, $\dot{M}$, and $\alpha$.

For the inner zone, hold the surface density $\Sigma$ fixed during a local thermal perturbation. Vertical hydrostatic equilibrium givesso $H\propto T_c^4/\Sigma$. The alpha disk prescription then yieldswhereas optically thick radiative diffusion with nearly constant electron-scattering opacity givesAt equilibrium $Q^{+}=Q^{-}=Q_0$. For net cooling $\dot{Q}=Q^{-}-Q^{+}$,Thus the total-pressure alpha prescription predicts thermal instability of a radiation-pressure-dominated alpha disk: a temperature increase makes heating outrun cooling. The absence of ubiquitous, large-amplitude thermal limit cycles in luminous AGN light curves indicates that this local model omits stabilizing effects, plausibly magnetic pressure and stress, vertical advection, winds, or a stress law that does not simply track total pressure.

Steady mass conservation gives $\dot{M}=-2\pi R\Sigma u_R$. Substituting this into the angular-momentum equation and integrating from the innermost stable circular orbit with zero torque givesThereforeUsing $\nu =\alpha c_{s}H$, $c_s\simeq H\Omega$, $l=R{u}_{\varphi }$, and $Rd\Omega /dR$ of order $-\Omega$ givesup to the order-unity Keplerian factor $3/2$.

At the sonic transition, $|u_R|\simeq c_s\simeq (H/R)u_{\varphi }$. Hencefor a geometrically thin disk with $\alpha \ll 1$. The specific angular momentum therefore differs only fractionally from $l_{\mathrm{ISCO}}$ before the gas enters the plunging region of a black-hole accretion disk. Its much shorter inflow time then prevents appreciable viscous transport, justifying angular-momentum conservation across the ISCO and the zero-torque boundary condition.

The maximal radiative efficiency of black-hole accretion is the fraction of rest-mass energy available if all binding energy released before capture escapes as radiation. In a Newtonian disk ending at $R_{\mathrm{in}}$,because a circular orbit has specific binding energy $GM/(2R)$. In relativity, ${\eta }_{\max }=1-E_{\mathrm{ISCO}}$. Black-hole spin moves the innermost stable circular orbit inward for prograde flow and outward for retrograde flow, increasing or decreasing this maximum respectively.

Since $L=\eta \dot{M}{c}^2$, a source of fixed luminosity requires $\dot{M}=L/(\eta c^2)$, while black-hole mass grows at approximately $(1-\eta )\dot{M}$. The actual radiative efficiency of black-hole accretion can lie below the maximum when energy is advected through the horizon or carried away mechanically. A low-density, optically thin advection-dominated accretion flow stores dissipated energy in ions, while a high-rate slim accretion disk traps photons and advects their energy inward; both are radiatively inefficient flows.

For an adiabatic Bondi accretion flow, $\dot{M}\propto M^2{\rho }_{\infty }{c}_{s,\infty }^{-3}$. Inside the Bondi radius the speed and ion temperature are approximately virial, so $v\propto (M/r{)}^{1/2}$ and $T\propto M/r$, while mass conservation gives $n\propto \dot{M}{M}^{-1/2}{r}^{-3/2}$. The frequency-integrated thermal bremsstrahlung emissivity is proportional to $n^2{T}^{1/2}$. Its volume integral is dominated by the inner flow and consequently scales asSince the Eddington luminosity is proportional to $M$, this may be written $L/L_{\mathrm{Edd}}=C(\dot{M}{c}^2/L_{\mathrm{Edd}}{)}^2$. With the standard fully ionized-plasma constants and the Bondi profiles, $\sqrt{C}=9\times 10^{-3}$. Eliminating $\dot{M}$ from $\eta =L/(\dot{M}{c}^2)$ then givesThe cancellation of $M$, ${\rho }_{\infty }$, and $c_{s,\infty }$ expresses the scale-free character of the ideal flow. More physically, two-body emission scales as density squared, so an increasingly dilute flow radiates a progressively smaller fraction of its available accretion power.

The Soltan argument compares the time-integrated luminosity density of the cosmological AGN population with the present comoving mass density in dormant supermassive black holes. If $U_{\mathrm{AGN}}$ is the emitted energy density corrected for obscuration and bolometric output, accretion with population-averaged efficiency $\bar{\eta }$ predictsIn practice $U_{\mathrm{AGN}}$ comes from integrating AGN luminosity functions over luminosity and cosmological redshift, with corrections for obscured sources and missed wavebands, while ${\rho }_{\mathrm{BH}}$ is inferred from local galaxy--black-hole scaling relations.

The inferred efficiency is of order the canonical thin-disk value, about ten per cent, so most cosmic black-hole mass was accumulated in radiatively efficient, optically thick accretion episodes. Radiatively inefficient flows can dominate low-luminosity activity or brief extreme phases, and mergers redistribute existing mass, but neither naturally accounts for the observed integrated AGN radiation while supplying most of the final mass.

For a steady axisymmetric disk,where factors of order unity depend on the vertical density profile. In a geometrically thick advection-dominated accretion flow, $H\sim R$, $c_s\sim v_K$, and the alpha disk estimate gives $|u_R|\sim \alpha v_K$. Thus the inflow time is $t_{\mathrm{in}}\sim R/(\alpha v_K)$ andSubstitution into the given electron--proton thermal equilibration time yieldsUsing ${\dot{M}}_{\mathrm{Edd}}=10L_{\mathrm{Edd}}/c^2=40\pi GM{m}_p/({\sigma }_{T}c)$ and a mildly relativistic electron temperature $k_B{T}_e/(m_e{c}^2)\simeq 0.1$ givesBelow this rate, Coulomb collisions cannot transfer the ions' viscously generated heat to radiating electrons before inflow. The resulting two-temperature accretion flow advects most of that energy through the horizon, so its efficiency is well below the canonical $\eta \simeq 0.1$ of a thin alpha disk and decreases with accretion rate.