Black holes, accretion disks, and photon rings

Chapter opening

Black holes do not present a visible solid surface. What we observe is the structure of matter and light paths around them. Accretion disks produce continuum spectra and stochastic variability. Broad-line regions turn continuum lags into physical radii. Jets project magnetic fields and relativistic motion into strong polarization. Photon rings write strong-gravity multipath propagation into long-baseline visibility and time delay. A photon event table should keep time, frequency, polarization, and interferometric phase as far as possible, because these coordinates decide whether strong gravity, plasma physics, and instrumental effects can be separated in the inference.

Black-hole scales: mass, distance, and the basic coordinates of the event table

The natural units of a black-hole problem are set by mass:

(204)\[r_g=\frac{GM}{c^2},\qquad t_g=\frac{GM}{c^3},\qquad \theta_g=\frac{r_g}{D}.\]

Here \(r_g\) is the gravitational radius, usually quoted in \(\mathrm{km}\) or \(\mathrm{cm}\); \(t_g\) is the light-crossing time of one \(r_g\); and \(D\) is distance. Numerically,

(205)\[r_g=1.477\,{\rm km}\left(\frac{M}{M_\odot}\right),\qquad t_g=4.93\,\mu{\rm s}\left(\frac{M}{M_\odot}\right).\]

Sgr A* has \(M\simeq4\times10^6\,M_\odot\), so \(t_g\simeq20\,\mathrm{s}\), and ISCO-scale structure can change within minutes. M87* has \(M=(6.5\pm0.7)\times10^9\,M_\odot\), so \(t_g\simeq3.2\times10^4\,\mathrm{s}\); the same dimensionless process is stretched to hours or days [Event Horizon Telescope Collaboration et al., 2022, Event Horizon Telescope Collaboration et al., 2019, Event Horizon Telescope Collaboration et al., 2019].

The angular diameter of a black-hole shadow can be estimated as

(206)\[d_{\rm sh}\simeq \kappa\,\theta_g,\qquad \kappa\simeq 10.4 ,\]

where \(\kappa\) changes only at the percent to ten-percent level with Kerr spin and inclination. At \(1.3\,\mathrm{mm}\), the Event Horizon Telescope measured an asymmetric ring around M87* with diameter \(42\pm3\,\mu{\rm as}\) and a central brightness depression deeper than a factor of ten. For Sgr A*, it measured a thick ring with diameter \(51.8\pm2.3\,\mu{\rm as}\), but the source varied noticeably on minute-to-hour timescales during the observing nights [Event Horizon Telescope Collaboration et al., 2022, Event Horizon Telescope Collaboration et al., 2022, Event Horizon Telescope Collaboration et al., 2019, Event Horizon Telescope Collaboration et al., 2019].

Figure 66 Black-hole time scales and angular scales both grow linearly with mass. Sgr A* and M87* differ in mass by about 10³, but their distances also differ by about 10³, leaving both shadows at tens of \(\mu{\rm as}\). Their time scales are not similar: Sgr A* can evolve within a single VLBI scan, whereas M87* is much closer to a static source for EHT imaging.

In the event-table view, the basic coordinates for black-hole observations become

\[(t_i,\nu_i,{\bf b}_i,\eta_i,w_i),\]

where \(t_i\) is arrival time, \(\nu_i\) is frequency or energy channel, \({\bf b}_i\) is projected baseline, \(\eta_i\) is the polarization modulation angle or Stokes channel, and \(w_i\) is a weight. EHT complex visibilities, GRAVITY differential phases, optical monitoring light curves, and polarization surveys are different projections of the same type of table. Keeping time and polarization before projection makes it possible to compare accretion-flow variability, jet disturbances, and strong-gravity multipath propagation.

Accretion-disk continua and stochastic variability

In the standard thin disk, gravitational binding energy is locally converted into radiative flux,

(207)\[F(R)=\frac{3GM\dot M}{8\pi R^3} \left[1-\left(\frac{R_{\rm in}}{R}\right)^{1/2}\right],\]

where \(R\) is disk radius, \(\dot M\) is accretion rate, and \(R_{\rm in}\) is the inner boundary, often of order a few \(r_g\). If the disk is locally close to a blackbody, its temperature is

(208)\[T(R)=\left[\frac{F(R)}{\sigma_{\rm SB}}\right]^{1/4}.\]

Far outside the inner boundary, \(T\propto R^{-3/4}\). The half-light radius that dominates a wavelength therefore scales as

(209)\[R_\lambda \propto (M\dot M)^{1/3}\lambda^{4/3}.\]

For a quasar with \(M\sim10^8\,M_\odot\) and \(L/L_{\rm Edd}\sim0.1\), the optical and near-ultraviolet continuum often comes from radii of order \(0.1\)–10 light-days. Microlensing and multi-band continuum lags often imply optical disks larger than the simplest thin-disk prediction, which is why inhomogeneous-disk and reprocessing models remain central to this problem [Dexter and Agol, 2011, Pereyra et al., 2006, Shakura and Sunyaev, 1973].

Figure 67 Thin-disk scaling gives R_λ ∝ λ^4/3. The orange band in the left panel marks the larger optical emitting areas often suggested by microlensing and some continuum-lag measurements. The right panel compares the orbital and thermal times at the same radius; for α ≃ 0.1, the thermal time is about ten orbital periods.

At a given disk radius, the dynamical and thermal times are approximately

(210)\[t_{\rm orb}=2\pi\left(\frac{R^3}{GM}\right)^{1/2},\qquad t_{\rm th}\simeq \frac{t_{\rm orb}}{\alpha}.\]

\(\alpha\) is the Shakura–Sunyaev viscosity parameter; values from 0.01 to 0.3 are commonly used for order-of-magnitude estimates. In AGN optical disks, \(t_{\rm orb}\) can range from tens of days to years, and \(t_{\rm th}\) is longer. Large optical light-curve samples are often described with a damped random walk,

(211)\[C(\tau)=\sigma^2\exp\left(-\frac{|\tau|}{\tau_d}\right), \qquad P(f)\propto\frac{1}{1+(2\pi f\tau_d)^2}.\]

Here \(C(\tau)\) is the light-curve autocovariance, \(\tau_d\) is the damping time, usually measured in rest-frame days, and \(P(f)\) is the power spectral density. Observationally, \(\tau_d\) correlates with black-hole mass. For supermassive black holes it is often tens to hundreds of days, while white-dwarf accretion disks fall on a related mass-scaled trend [Burke et al., 2021, Mangalam and Wiita, 1993, Muñoz-Darias et al., 2016].

Figure 68 A damped random walk represents stochastic variability with a single correlation time. A short τ_d light curve loses memory quickly and its power spectrum turns down to f⁻² at higher frequency. A long τ_d disk remains correlated for longer, extending the low-frequency plateau.

For photon statistics, the hard part of an accretion disk is multimode dilution. The optical continuum of a \(10^8\,M_\odot\) AGN comes from many independent turbulent regions and many thermal times. Bunching from any single quantum mode is averaged over spatial, frequency, and temporal modes. More accessible observables are excess correlations conditioned on phase or color: for example, comparing the width of \(g^{(2)}(\tau)\) before and after a flare, or splitting polarization channels to estimate the lifetime of turbulent cells. Mean brightness mainly sets photon rate and normalization. Detectability is then controlled by coherence time, sampling window, and how well the classical variability can be modeled.

Broad-line regions: combining time delays and angular offsets

Traditional reverberation mapping of a broad-line region writes

(212)\[L(v,t)=\int \Psi(v,\tau)\,C(t-\tau)\,d\tau ,\]

where \(C(t)\) is continuum variability, \(L(v,t)\) is line variability at velocity \(v\), and \(\Psi(v,\tau)\) is the transfer function. A lag \(\tau\) corresponds to \(R\simeq c\tau\), while the line width gives an orbital speed in the gravitational potential. Empirically, the H\(\beta\) radius and the \(5100\text{\AA}\) continuum luminosity follow

(213)\[R_{\rm BLR}\simeq 33.7\,{\rm light\,days} \left[\frac{\lambda L_\lambda(5100\text{\AA})}{10^{44}\,\mathrm{erg\,s^{-1}}}\right]^{0.533}.\]

The scatter in this relation comes from host-galaxy subtraction, sampling window, anisotropic illumination, line species, and differences in BLR geometry. The black-hole mass is often written as

(214)\[M_{\rm BH}=f_{\rm vir}\frac{R_{\rm BLR}\Delta V^2}{G},\]

where \(\Delta V\) is line width and \(f_{\rm vir}\) absorbs inclination and velocity-field geometry. Seyfert lags can be days to tens of days. In luminous quasars, lags can reach hundreds of days, so monitoring baseline and seasonal gaps become major systematics [Bentz et al., 2013, Blandford and McKee, 1982, Peterson, 1993].

Interferometric phase gives a second projection of the BLR. If a velocity channel of the line has photocenter offset \(\boldsymbol{\delta\theta}(v)\) relative to the continuum, then the differential phase on baseline \({\bf B}\) is approximately

(215)\[\Delta\varphi(v)\simeq 2\pi\frac{{\bf B}\cdot\boldsymbol{\delta\theta}(v)}{\lambda}.\]

Here \(\Delta\varphi\) is in radians, \({\bf B}\) is in meters, \(\lambda\) is in meters, and \(\boldsymbol{\delta\theta}\) is in radians. GRAVITY measured the Pa\(\alpha\) line of 3C 273 and found a red-blue photocenter gradient roughly perpendicular to the jet. The model gave \(R_{\rm BLR}=46\pm10\,\mu{\rm as}\), about \(145\) light-days, and a thick, nearly face-on rotating geometry [Gravity Collaboration et al., 2018].

Figure 69 Time and angular information probe complementary parts of the BLR. The left panel shows the Hβ radius-luminosity relation, where time lag gives a physical radius. The right panel sketches how the red and blue wings of a rotating BLR produce differential phases with opposite signs on a baseline; if the baseline is poorly matched to the rotation geometry, the phase amplitude is suppressed.

If intensity interferometry is applied to a line-emitting region, it measures the narrow-band angular structure through \(|V|^2\), not the differential phase measured by GRAVITY. It is best suited to bright, low-redshift targets with strong broad lines and large angular scales. For \(R_{\rm BLR}=100\) light-days, the angular diameter is

(216)\[\theta_{\rm BLR}\simeq 2\frac{R_{\rm BLR}}{D_A} \simeq 11\,\mu{\rm as} \left(\frac{R_{\rm BLR}}{100\,{\rm light\,days}}\right) \left(\frac{100\,{\rm Mpc}}{D_A}\right),\]

which already demands hundred-meter to kilometer baselines and very high photon flux. A reverberation radius \(R_{\rm BLR}\) plus an interferometric angular diameter \(\theta_{\rm BLR}\) would form a geometric distance \(D_A=2R_{\rm BLR}/\theta_{\rm BLR}\). In practice, inclination, BLR anisotropy, line-region thickness, and host contamination all enlarge the error budget.

Jets and polarization: magnetic-field geometry in photon statistics

Rotating black holes threaded by magnetic flux can power jets. A common scaling for the Blandford–Znajek mechanism is

(217)\[P_{\rm BZ}\sim \frac{\kappa}{4\pi c}\Phi_{\rm BH}^2\Omega_H^2,\]

where \(\Phi_{\rm BH}\) is the magnetic flux through the black hole, \(\Omega_H\) is the horizon angular velocity, and \(\kappa\) is a dimensionless coefficient that depends on field geometry. For low-accretion-rate radio galaxies such as M87*, EHT polarization indicates ordered magnetic fields near the horizon scale, consistent with magnetically arrested accretion flows and jet-power models [Blandford and Königl, 1979, Blandford and Znajek, 1977, Event Horizon Telescope Collaboration et al., 2021].

Optical and near-infrared jet emission is usually synchrotron radiation. An electron with Lorentz factor \(\gamma_e\) in magnetic field \(B\) has characteristic frequency

(218)\[\nu_{\rm syn}\simeq 4.2\times10^6\,{\rm Hz}\, \frac{\delta}{1+z} \left(\frac{B}{1\,{\rm G}}\right)\gamma_e^2 ,\]

where \(\delta\) is the Doppler factor and \(z\) is redshift. For \(B\simeq0.1\)–1 \(\mathrm{G}\) and \(\delta\simeq10\), optical emission near \(\nu\sim5\times10^{14}\,\mathrm{Hz}\) requires \(\gamma_e\sim10^4\). A power-law electron distribution \(N(E)\propto E^{-p}\) has maximum linear polarization

(219)\[\Pi_{\rm max}=\frac{p+1}{p+7/3},\]

which gives \(\Pi_{\rm max}\simeq70\%\)–75% for \(p=2\)–3. Real blazars usually show only a few percent to a few tens of percent, because many magnetic cells, shocks, shear layers, and turbulent zones add Stokes vectors with different directions. When multi-frequency campaigns find an optical polarization-angle rotation, a \(\gamma\)-ray flare, and a radio knot ejection at compatible times, they usually point to the same jet disturbance crossing different emission zones [Marscher et al., 2008, Urry and Padovani, 1995].

A polarization-resolved second-order correlation can be written as

(220)\[g^{(2)}_{QU}(\tau)= \frac{\langle \delta Q(t)\delta U(t+\tau)\rangle} {\sigma_Q\sigma_U},\]

which measures how the random process of magnetic-field direction is delayed between two Stokes components. If a shock-in-jet model is correct, the correlation time of \(Q\) and \(U\) should be close to the time it takes a disturbance to cross the optical emitting region. If turbulent cells dominate, the correlation function should decay more rapidly and should not be perfectly synchronized across colors.

Photon rings: images, visibilities, and time structure

An interferometer measures Fourier components of the sky brightness:

(221)\[V({\bf u})=\int I({\boldsymbol\theta}) e^{-2\pi i{\bf u}\cdot{\boldsymbol\theta}}\,d^2\theta, \qquad {\bf u}=\frac{{\bf B}_\perp}{\lambda}.\]

For a uniform thin ring, the visibility is approximately

(222)\[V(u)\simeq J_0(\pi d u),\]

where \(d\) is the ring diameter in radians. M87* has \(d\simeq42\,\mu{\rm as}\), so Earth-size \(1.3\,\mathrm{mm}\) VLBI can already see the first visibility null and the ring-like morphology. If the ring has finite width \(w\), the long-baseline oscillations are damped by an envelope. A broad accretion flow is resolved out faster than a narrow photon ring [Event Horizon Telescope Collaboration et al., 2019, Event Horizon Telescope Collaboration et al., 2019, Johnson et al., 2020].

Figure 70 Ring-like structure produces oscillations in long-baseline visibility. The left panel compares rings of different width at the same \(42\,\mu{\rm as}\) diameter: the narrower ring retains oscillations to longer baselines. The right panel sketches how photon-subring width and flux decrease with orbit number. Higher-order subrings are faint, but on sufficiently long baselines they can leave nearly universal structure.

The bright ring in an EHT image contains the direct image, once-lensed image, and higher-order photon subrings; it is not the mathematically infinitesimal photon ring. Accretion-flow thickness, optical depth, turbulence, and time averaging broaden the narrow strong-gravity structure. Gralla et al. emphasized that the higher-order photon ring can contain only a small fraction of the total flux. Johnson et al. showed that sufficiently long baselines resolve out the broad accretion flow, leaving the universal oscillations of narrow subrings. A useful scaling for the width and flux of the \(n\)th subring is

(223)\[w_n\simeq w_0 e^{-\gamma n},\qquad F_n\simeq F_0 e^{-\beta n},\]

where \(\gamma\) is related to the Lyapunov exponent of the photon orbit, while \(\beta\) also depends on emission and absorption. This scaling describes the exponentially nested angular structure of multipath strong-gravity images. Real observations still have to attach that structure to emission, absorption, and time-averaging models [Gralla et al., 2019, Gralla et al., 2020, Johannsen, 2013].

Multipath propagation also creates time structure. If the mean delay difference between the \(n\)th and \((n+1)\)th loops is approximately the photon-orbit period \(T_\gamma\), the correlation function may show weak peaks near

(224)\[C(\tau)=\langle \delta I(t)\delta I(t+\tau)\rangle \quad\hbox{at}\quad \tau\simeq mT_\gamma .\]

For Sgr A*, \(T_\gamma\) is minutes, overlapping the intrinsic flare time of the accretion flow. For M87*, it is days, making the source more stable but the observing schedule harder. Interpreting such peaks as strong-gravity multipath signals requires excluding jet shocks, disk turbulence, scattering screens, sampling-window artifacts, and calibration errors. The polarization version can compare the delay structure of \(Q\), \(U\), and \(V\). EHT polarization images of M87* and Sgr A* already show the value of this extra information [Chen et al., 2022, Gan et al., 2021, Himwich et al., 2020].

Hawking temperature and boundaries on new physics

The previous sections concerned ordinary radiation organized by strong gravity, plasma, and magnetic fields near black holes. Quantum radiation from the black hole itself starts at a completely different scale. The Hawking temperature is

(225)\[T_H=\frac{\hbar c^3}{8\pi G M k_B} \simeq 6.17\times10^{-8}\,{\rm K} \left(\frac{M_\odot}{M}\right).\]

For stellar-mass black holes, \(T_H\) is already far below the \(2.725\,\mathrm{K}\) cosmic microwave background. For Sgr A* and M87*, it falls to \(10^{-14}\)–\(10^{-17}\,\mathrm{K}\). The evaporation time is roughly

(226)\[t_{\rm evap}\simeq 2.1\times10^{67}\,{\rm yr} \left(\frac{M}{M_\odot}\right)^3 .\]

For ordinary astrophysical black holes, Hawking radiation is far below any foreseeable telescope sensitivity. Only very light primordial black holes would be hot enough to radiate in high-energy bands; those constraints belong to cosmology and high-energy transient searches. For stellar-mass and supermassive black holes, the observational frontier remains accretion flow, polarization, visibility, and time structure [Hawking, 1974, Hawking, 1975].

Figure 71 Hawking temperature decreases inversely with mass, while evaporation time grows as mass cubed. Stellar-mass and supermassive black holes are far colder than the cosmic microwave background and have evaporation times vastly longer than the age of the Universe. Near-term black-hole observations therefore come mainly from accretion flow, polarization, visibility, and time structure.

New physics near black holes can enter the data only through measurable deviations. EHT polarization can constrain some polarization rotations caused by axion-like particles, but plasma Faraday rotation, electron temperature, magnetic-field structure, and imaging priors must be modeled together [Chen et al., 2022, Chen et al., 2020]. Photon-ring shape can test strong-field metrics, but broad accretion flow and sparse baselines limit the precision. Time correlations can search for multipath delays, but accretion-flow turbulence is usually stronger. The same logic applies to transients: unless the event table, background, selection function, and systematic errors are specified, a deviation cannot be reliably assigned to strong-field gravity or a new particle.