Foundations of Black Hole Accretion Disk Theory
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Abstract
This review covers the main aspects of black hole accretion disk theory. We begin with the view that one of the main goals of the theory is to better understand the nature of black holes themselves. In this light we discuss how accretion disks might reveal some of the unique signatures of strong gravity: the event horizon, the innermost stable circular orbit, and the ergosphere. We then review, from a firstprinciples perspective, the physical processes at play in accretion disks. This leads us to the four primary accretion disk models that we review: Polish doughnuts (thick disks), ShakuraSunyaev (thin) disks, slim disks, and advectiondominated accretion flows (ADAFs). After presenting the models we discuss issues of stability, oscillations, and jets. Following our review of the analytic work, we take a parallel approach in reviewing numerical studies of black hole accretion disks. We finish with a few select applications that highlight particular astrophysical applications: measurements of black hole mass and spin, black hole vs. neutron star accretion disks, black hole accretion disk spectral states, and quasiperiodic oscillations (QPOs).
Keywords
black hole accretion disks1 Introduction
Because of its firm connection to black holes themselves, black hole accretion disk theory belongs to the realm of fundamental physics. However, the theory itself employs a complicated maze of fluiddynamics results and several phenomenological estimates and guesses known only to specialists. This Living Review aims to give readers a useful guide, a “road map” if you will, through the unavoidable complexity of the subject.
Below we list fourteen “Destinations” on this road map and explain their logical connections. In our opinion, these Destinations are the key issues, or landmarks, of the theory being reviewed. The particular road map that we present is, of course, biased by our own ideas and research histories. However, we are confident that the grand landscape of this field will, nevertheless, shine through in the end.

Destination 1: Quasars and other similar supermassive objects, which are collectively called “active galactic nuclei” (or AGN), having masses^{2} in the range 10^{6} M_{⊙} < M < 10^{9} M_{⊙}. They reside at centers of our and other galaxies. The “active” ones among them are the most powerful steady energy sources known in the universe. Many have radiant powers L in excess of their corresponding Eddington luminosities.^{3} The high efficiency of quasars, η = L/Mc^{2} > 0.1, where Ṁ is the mass supply rate from accretion (“the accretion rate”), is puzzling. Black hole accretion disk theory predicts that L > L_{Edd} would imply small accretion efficiency η ≪ 0.1. However, the famous “Sołtan argument,” based on quasar counts, shows that on a long time average, t ∼ t_{Hubble}, quasars can have both L > L_{Edd} and η ∼ 0.1 [290, 254]. AGN are described in much greater detail in the authoritative monograph by Krolik [163].

Destination 2: Microquasars and similar “stellarmass” black holes, having M ∼ 10 M_{⊙}. The term “microquasars” was invented by Mirabel [206] to convey that these objects, in many regards, behave like scaleddown versions of quasars. A few tens of them have been found in our Galaxy as members of Xray binaries [184]. The natural scaling (time) ∼ (mass) adds importance to observations of microquasar variability, because the same processes that takes hundreds of years (say) in quasars, takes only minutes in microquasars. Particularly interesting are spectral state changes, which occur on timescales ∼ 1 day (see [258] for a review), and quasiperiodic oscillations (QPOs), which have timescales as short as ∼ 1 ms [308, 184, 258]. Quasars and microquasars are among the most intriguing astrophysical objects ever discovered, and the goal of black hole accretion disk theory is, obviously, to explain their observed properties — but one hopes for much more! One hopes that observations of quasars and microquasars, together with their proper theoretical interpretation, would eventually test the very heart of black hole physics itself. When this happens, we may meaningfully constrain our knowledge of the fundamental properties of space and time.^{4}

Destination 3: Event horizon. This is a sphere of radius ∼ GM/c^{2} surrounding the black hole singularity, from within which nothing may emerge — a oneway membrane. Note that this means that black holes have no rigid surfaces. This is a unique signature of black holes; other relativistic features may be observable around nonblack hole objects, specifically sufficiently compact neutron stars, but the event horizon is a defining property of black holes.

Destination 4: Ergosphere. This is a region around a rotating black hole where spacetime itself is dragged along in the direction of rotation at a speed greater than the local speed of light in relation to the rest of the universe. In this region, negative energy states are possible, which means that the rotational energy of the black hole can be tapped through various manifestations of the “Penrose process” [242].

Destination 5: Marginally stable orbit (also called the “innermost stable circular orbit” or ISCO). This is the smallest circle (r = r_{ms}) along which free particles may stably orbit around a black hole. No stable circular motion is possible for r < r_{ms}. This is a unique feature of relativity, as in Newtonian theory, orbits at all radii are possible.^{5}

Destination 6: Angular momentum transport and energy dissipation. The quasisteady accretion of a particle of mass m through a Keplerian disk from a large outer radius, r_{out}, to an inner radius, r_{in}, requires that the particle give up an amount of energy^{8} ∼ 0.1 mc^{2}. To do this, the particle must also give up an amount of angular momentum ∼ (G M r_{out})^{1/2}. We will see in Section 3.2 that viscous stresses within the fluid can facilitate this (mass transfer in, angular momentum transfer out, and energy dissipation). However, the stresses can not come from ordinary molecular viscosity, as this is much too weak in astrophysical accretion disks. Instead, the stresses likely come from turbulence that acts like an effective viscosity.

Destination 7: Radiative processes and radiative transfer. These depend on the thermodynamic state of matter (electron density, ion density, temperature), its motion, and the magnetic field, but most importantly on whether the matter is opaque or transparent to radiation, i.e., whether its optical depth is large, τ ≫ 1, or small, τ ≪ 1. In the general case when the matter is optically thick τ > 1, the accretion disk can be quite luminous and also efficiently cooled by radiation. Accretion disks with τ < 1 are inefficiently cooled and thus less luminous.
 Destination 8: Thick Disk.$$h > 1,\;\dot m \gg 1,\;\tau \gg 1,\;q\sim 1,\;\beta \ll 1,\;{r_{{\rm{in}}}}\sim {r_{{\rm{mb}}}},\;\eta \ll 0.1$$
 Destination 9: Thin Disk.$$h \ll 1,\;\dot m < 1,\;\tau \gg 1,\;q = 0,\;\beta \sim 1,\;{r_{{\rm{in}}}} = {r_{{\rm{ms}}}},\;\eta \sim 0.1$$
 Destination 10: Slim Disk.$$h\sim 1,\dot m \gtrsim 1, \;\tau \gg 1,\;q\sim 1,\;\beta < 1,\;{r_{{\rm{mb}}}} < {r_{{\rm{in}}}} < {r_{{\rm{ms}}}},\;\eta < 0.1$$
 Destination 11: AdvectionDominated Accretion Flow (ADAF).$$h < 1,\;\dot m \ll 1,\;\tau \ll 1,\;q\sim 1,\;\beta = 1,\;{r_{{\rm{mb}}}} < {r_{{\rm{in}}}} < {r_{{\rm{ms}}}},\;\eta \ll 0.1$$
Upper half: The four a priori possible types of dynamical states of accretion structures, corresponding to the division of the dynamical parameter space into fastslow rotation and largesmall pressure. Lower half: A different division of the parameter space, corresponding to highlow accretion rate and largesmall opacity.
Fast rotation (Disk)  Slow rotation (Bondi)  

Large pressure  slim, thick  ADAFs 
Small pressure  thin  freefall 
Accretion rate high  Accretion rate low  
Large opacity  slim, thick  thin 
Small opacity    ADAF 
Interestingly, the parameter space of each of these types of accretion disks overlaps that of other solutions. For instance, ADAF solutions exist that have the same mass accretion rates as thin disk solutions [59]. In such cases, it is not clear how nature might choose one over the other.

Destination 12: Stability. Stability analysis is important because the systematic differential rotation that is one of the defining characteristics of accretion disks is also a potential source of destabilizing energy. On the one hand, this may be essential, as the angular momentum transport and energy dissipation required for accretion may require disks to be mildly unstable. On the other hand, if a model is violently unstable, then the basic assumption of a “steadystate” would be violated.

Destination 13: Oscillations. As with any finite distribution of fluid, accretion disks have natural oscillation modes associated with them. If these modes can be excited at appreciable amplitudes, they may be able to modify the observed light curve of the disk in measurable ways. This makes disk oscillations a leading candidate for explaining the quasiperiodic oscillations (QPOs) that we discuss in Section 12.4. Because of the close observational links between black hole accretion disks and jets [92, 94], we include jets as the final Destination of our review.

Destination 14: Jets. Jets are narrow (opening angle < 5°), long (length > 10^{7} ly in the case of AGN), and fast (v > 0.9c) streams of matter emerging from very compact regions around the black hole, usually in opposite directions, presumably normal to the plane of the accretion disk. Jets can play a significant role in transporting energy and angular momentum away from the accretion disk [48]. They also play an important role in shaping the black hole’s environment far beyond the gravitational reach of the black hole itself, affecting galactic evolution, particle acceleration, and intragalactic ionization.
Going handinglove with analytic models of accretion disks are direct numerical simulations. Although analytic theories have been extremely successful at explaining many general observational properties of quasars and microquasars, numerical simulations can be critically important in at least two respects: 1) as an extension of analytic work, by treating nonlinear perturbations and higher order coupling terms, and 2) in cases that are highly time variable or contain little symmetry, such that the prospects of finding an analytic solution are poor. There is also an important overlap region where various analytic and numerical methods are applicable and can be used to independently validate results. Because of these close connections between analytic and numerical work, we have dedicated Section 11 to the discussion and review of direct numerical simulation of black hole accretion disks.
We finish this Living Review with Section 12, which tries to make some connections between the concepts discussed in earlier sections and actual observational phenomena. We emphasize that we are not aiming to provide a comprehensive review of black hole observations, but rather to highlight a very small subset of these that are of particular relevance to our review.
Throughout this review, we adopt the − + + + metric signature and often use units where c = 1 = G. To make all physical quantities dimensionless, we sometimes also use the mass of the black hole as a unit, M = 1. We use the common Einstein summation convention, where repeated indices in a formula imply summation over the range of that index. We also follow the common convention where Greek (Latin) indices are used for four(three)dimensional tensor quantities.
2 Three Destinations in Kerr’s Strong Gravity

Destination 3: Event Horizon: That radius inside of which escape from the black hole is not possible;

Destination 4: Ergosphere: That radius inside of which negative energy states are possible (giving rise to the potentiality of tapping the energy of the black hole).

Destination 5: Innermost Stable Circular Orbit (ISCO): That radius inside of which free circular orbital motion is not possible;
Our principal question is: Could accretion disk theory unambiguously prove the existence of the event horizon, ergosphere, and ISCO using currently available or future observations?
2.1 The event horizon
The mathematically precise, general, definition of the event horizon involves topological considerations [207]. Here, we give a definition which is less general, but in the specific case of the Kerr geometry is fully equivalent.
2.1.1 Detecting the event horizon
Evidence based on estimating the compactness parameter: A source for which observations indicate ℭ ≈ 1 may be suspected of having an event horizon. Values ℭ ≈ 1 have indeed been found in several astronomical sources. In order to know ℭ, one must know mass and size of the source. The mass measurement is usually a direct one, because it may be based on an application of Kepler’s laws. In a few cases the mass measurement is remarkably accurate. For example, in the case of Sgr A*, the supermassive black hole in the center of our Galaxy, the mass is measured to be M = (4.3 ± 0.5) × 10^{6} M_{⊙} [111].
Until recently, estimates of size were always indirect, and generally not accurate. They are usually based on time variability or spectral considerations. For the former, the measurement rests on the logic that if the shortest observed variability timescale is Δt, then the size of the source cannot be larger than R = cΔt. For the latter, the argument goes like this: If the total radiative power L and the radiative flux F can be independently measured for a blackbody source, then its size can be estimated from L = 4π R^{2} F. Keep in mind that one must know the distance to the source in order to measure L. The flux can be estimated from F = aT^{4}, where T is the temperature corresponding to the peak in the observed intensity versus frequency electromagnetic spectrum.
Evidence based on the “no escape” argument: For accretion onto an object with a physical surface (such as a star), 100% of the gravitational binding energy released by accretion must be radiated away. This does not apply for a black hole since the event horizon allows for the energy to be advected into the hole without being radiated. This may allow for a black hole with an event horizon to be distinguished from another, similarmass object with a surface, such as a neutron star. This argument was first developed by Narayan and collaborators [215, 216, 214]; we describe it in more detail in Section 12.2.
2.2 The ergosphere
The ZAMO frame defines a local standard of rest with respect to the local compass of inertia: a gyroscope stationary in the ZAMO frame does not precess. Considering its kinematic invariants,^{12} one sees that the ZAMO frame is noninertial (a_{μ} ≠ 0), nonrigid (σ_{μν} ≠ 0, Θ = 0), and surfaceforming (ω_{μν} = 0). The ZAMO vectors ῆ^{μ} and N^{μ} are timelike everywhere outside the horizon, i.e., outside the surface 1/g^{tt} = 0. This means that the energy of a particle or photon with a fourmomentum p_{μ} measured by the ZAMO is positive, E_{ZAMO} = N^{μ}p_{μ} > 0.
The ZAVO frame defines a global standard of rest with respect to distant stars: a telescope that points to a fixed star does not rotate in the ZAVO frame. Considering its kinematic invariants one sees that the ZAVO frame is noninertial (a_{μ} ≠ 0), rigid (σ_{μν} = 0, Θ = 0), and not surfaceforming (ω_{μν} ≠ 0). At infinity, i.e., for r → ∞, it is (η^{ν}η_{ν}) = g_{tt} → −1, and therefore n^{μ} → η^{μ}. For this reason, η^{μ} is called the stationary observer at infinity. The ZAVO vectors η^{μ} and n^{μ} are timelike outside the region surrounded by the surface g_{t} = 0, called the ergosphere. Inside the ergosphere η^{μ} and n^{μ} are spacelike. This means that inside the ergosphere, the conserved energy of a particle (i.e., the energy measured “at infinity”), as defined by (7a), may be negative.
2.3 ISCO: the orbit of marginal stability
2.4 The PaczyńskiWiita potential
2.5 Summary: characteristic radii and frequencies
We end this section with a few formulae for the Kerr geometry that we will use elsewhere in this review.
3 Matter Description: General Principles
3.1 The fluid part
3.1.1 Perfect fluid
3.2 The stress part
3.2.1 The alpha viscosity prescription
As we mentioned in Section 1, the viscosity in astrophysical accretion disks can not come from ordinary molecular viscosity, as this is orders of magnitude too weak to explain observed phenomena. Instead, the source of stresses in the disk is likely turbulence driven by the magnetorotational instability (MRI, described in Section 8.2). Even so, one can still parametrize the stresses within the disk as an effective viscosity and use the normal machinery of standard hydrodynamics without the complication of magnetohydrodynamics (MHD). This is sometimes desirable as analytic treatments of MHD can be very difficult to work with and full numerical treatments can be costly.
3.3 The Maxwell part
Magnetic fields may play many interesting roles in black hole accretion disks. Large scale magnetic fields threading a disk may exert a torque, thereby extracting angular momentum [48]. Similarly, large scale poloidal magnetic fields threading the inner disk, ergosphere, or black hole, have been shown to be able to carry energy and angular momentum away from the system, and power jets [49]. Weak magnetic fields can tap the differential rotation of the disk itself to amplify and trigger an instability that leads to turbulence, angular momentum transport, and energy dissipation (exactly the processes that are needed for accretion to happen) [26, 27].
3.3.1 The magnetorotational instability (MRI)
We mentioned in Section 3.2 that a hydrodynamic treatment of accretion requires an internal viscous stress tensor of the form \({{\mathcal T}_{r\phi}} < 0\). However, we also pointed out that ordinary molecular viscosity is too weak to provide the necessary level of stress. Another possible source is turbulence. The mean stress from turbulence always has the property that \({{\mathcal T}_{r\phi}} < 0\), and so it can act as an effective viscosity. As we will explain in Section 8.2, weak magnetic fields inside a disk are able to tap the shear energy of its differential rotation to power turbulent fluctuations. This happens through a mechanism known as the magnetorotational (or “BalbusHawley”) instability [26, 118, 27]. Although the nonlinear behavior of the MRI and the turbulence it generates is quite complicated, its net effect on the accretion disk can, in principle, be characterized as an effective viscosity, possibly making the treatment much simpler. However, no such complete treatment has been developed at this time.
3.4 The radiation part
Radiation is important in accretion disks as a way to carry excess energy away from the system. In geometrically thin, optically thick (ShakuraSunyaev) accretion disks (Section 5.3), radiation is highly efficient and nearly all of the heat generated within the disk is radiated locally. Thus, the disk remains relatively cold. In other cases, such as ADAFs (Section 7), radiation is inefficient; such disks often remain geometrically thick and optically thin.
In the remaining parts of this section we give explicit formulae for the bremsstrahlung and synchrotron emissivities and their Compton enhancements. These sections are taken almost directly from the work of Narayan and Yi [225]. Additional derivations and discussions of these equations in the black hole accretion disk context may be found in [299, 295, 225, 87].
3.4.1 Bremsstrahlung
3.4.2 Synchrotron
3.4.3 Comptonization
4 Thick Disks, Polish Doughnuts, & Magnetized Tori
In this section we discuss the simplest analytic model of a black hole accretion disk — the “Polish doughnut.” It is simplest in the sense that it only considers gravity (Section 2), plus a perfect fluid (Section 3.1.1), i.e., the absolute minimal description of accretion. We include magnetized tori in Section 4.2, which allows for \({(T_\nu ^\mu)_{{\rm{MAX}}}} \neq 0\), but otherwise \({(T_\nu ^\mu)_{{\rm{VIS}}}} = {(T_\nu ^\mu)_{{\rm{MAX}}}}{(T_\nu ^\mu)_{{\rm{RAD}}}} = 0\) throughout this section.
4.1 Polish doughnuts
In real flows, the function ℓ = ℓ(Ω) is determined by dissipative processes that have timescales much longer than the dynamical timescale, and are not yet fully understood. Paczyński realized that instead of deriving ℓ = ℓ(Ω) from unsure assumptions about viscosity that involve a free function fixed ad hoc (e.g., by assuming α(r, θ) = const), one may instead assume the result, i.e., assume ℓ = ℓ(Ω). Assuming ℓ = ℓ(Ω) is not selfconsistent, but neither is assuming α(r, θ) = const.
Before leaving the topic of Polish doughnuts, we should point out that, starting with the work of Hawley, Smarr, and Wilson [125], this simple, analytic solution has been the most commonly used starting condition for numerical studies of black hole accretion.
4.2 Magnetized Tori
5 Thin Disks
Most analytic accretion disk models assume a stationary and axially symmetric state of the matter being accreted into the black hole. In such models, all physical quantities depend only on the two spatial coordinates: the “radial” distance from the center r, and the “vertical” distance from the equatorial symmetry plane z. In addition, the most often studied models assume that the disk is not vertically thick. In “thin” disks z/r ≪ 1 everywhere inside the matter distribution, and in “slim” disks (Section 6) z/r ≤1.
5.1 Equations in the Kerr geometry
 (i)Mass conservation (continuity):where V is the gas radial velocity measured by an observer at fixed r who corotates with the fluid, and Δ has the same meaning as in Section 2.$$\dot M =  2\pi \Sigma {\Delta ^{1/2}}{V \over {\sqrt {1  {V^2}}}},$$(88)
 (ii)Radial momentum conservation:where$${V \over {1  {V^2}}}{{{\rm{d}}V} \over {{\rm{d}}r}} = {\mathcal{A} \over r}  {1 \over \Sigma}{{{\rm{d}}P} \over {{\rm{d}}r}},$$(89)Ã= (r^{2} + a^{2})^{2} − a^{2}Δsin^{2} θ, Ω = u^{φ}/u^{t} is the angular velocity with respect to the stationary observer, \(\tilde \Omega = \Omega  \omega\) is the angular velocity with respect to the inertial observer, \(\Omega _K^ \pm = \pm {M^{1/2}}/({r^{3/2}} \pm a{M^{1/2}})\) are the angular frequencies of the corotating and counterrotating Keplerian orbits, and \(\tilde R = \tilde A/({r^2}{\Delta ^{1/2}})\) is the radius of gyration.$$\mathcal{A} =  {{M\tilde A} \over {{r^3}\Delta \Omega _K^ + \Omega _K^ }}{{(\Omega  \Omega _K^ +)(\Omega  \Omega _K^ )} \over {1  {{\tilde \Omega}^2}{{\tilde R}^2}}},$$(90)
 (iii)Angular momentum conservation:where \({\mathcal L} = {u_\phi}\) is the specific angular momentum, γ is the Lorentz factor, Π = 2HP can be considered to be the vertically integrated pressure, is the standard alpha viscosity (Section 3.2.1), and \({{\mathcal L}_{in}}\) is the specific angular momentum at the horizon, which can not be known a priori. As we explain in the next section, it provides an eigenvalue linked to the unique eigensolution of the set of thin disk differential equations, once they are properly constrained by boundary and regularity conditions.$${{\dot M} \over {2\pi}}(\mathcal{L}  {\mathcal{L}_{in}}) = {{{{\tilde A}^{1/2}}{\Delta ^{1/2}}\gamma} \over r}\alpha \Pi,$$(91)
 (iv)Vertical equilibrium:with \({\mathcal E} =  {u_t}\) being the conserved energy associated with the time symmetry.$${\Pi \over {\Sigma {H^2}}} = {{{\mathcal{L}^2}  {a^2}({\mathcal{E}^2}  1)} \over {2{r^4}}},$$(92)
 (v)Energy conservation:where T is the temperature in the equatorial plane, k is the mean (frequencyindependent) opacity,$$ {{\alpha \Pi \tilde A{\gamma ^2}} \over {{r^3}}}{{{\rm{d}}\Omega} \over {{\rm{d}}r}}  {{32} \over 3}{{\sigma {T^4}} \over {\kappa \Sigma}} =  {{\dot M} \over {2\pi r\rho}}{1 \over {{\Gamma _3}  1}}\left({{{{\rm{d}}P} \over {{\rm{d}}r}}  {\Gamma _1}{P \over \rho}{{{\rm{d}}\rho} \over {{\rm{d}}r}}} \right),$$(93)β = P_{gas}/(P_{gas} + P_{rad} + P_{mag}), β_{m} = P_{gas}/(P_{gas} + P_{mag}), β* = β(4 − β_{m})/3β_{m}, and γ_{g} is the ratio of specific heats of the gas.$$\begin{array}{*{20}c} {\quad \;\;{\Gamma _1} = {\beta ^\ast} + (4  3{\beta ^\ast})({\Gamma _3}  1)\;,\quad} \\ {{\Gamma _3} = 1 + {{(4  3{\beta ^\ast})({\gamma _g}  1)} \over {12(1  \beta/{\beta _m})({\gamma _g}  1) + \beta}},} \\ \end{array}$$
5.2 The eigenvalue problem
5.3 Solutions: ShakuraSunyaev & NovikovThorne
Shakura and Sunyaev [279] noticed that a few physically reasonable extra assumptions reduce the system of thin disk equations (88)–(93) to a set of algebraic equations. Indeed, the continuity and vertical equilibrium equations, (88) and (92), are already algebraic. The radial momentum equation (90) becomes a trivial identity 0 = 0 with the extra assumptions that the radial pressure and velocity gradients vanish, and the rotation is Keplerian, \(\Omega = \Omega _k^ +\). The algebraic angular momentum equation (91) only requires that we specify \({{\mathcal L}_{in}}\). The ShakuraSunyaev model makes the assumption that \({{\mathcal L}_{in}} = {{\mathcal L}_k}({\rm{ISCO}})\). This is equivalent to assuming that the torque vanishes at the ISCO. This is a point of great interest that has been challenged repeatedly [164, 104, 25]. Direct testing of this hypothesis by numerical simulations is discussed in Section 11.4.
The righthand side of the energy equation (93) represents advective cooling. This is assumed to vanish in the ShakuraSunyaev model, though we will see that it plays a critical role in slim disks (Section 6) and ADAFs (Section 7). Because the ShakuraSunyaev model assumes the rotation is Keplerian, \(\Omega = \Omega _k^ +\), meaning Ω is a known function of r, the first term on the lefthand side of Eq. (93), which represents viscous heating, is algebraic. The second term, which represents the radiative cooling (in the diffusive approximation) is also algebraic in the ShakuraSunyaev model.
The general relativistic version of the ShakuraSunyaev disk model was worked out by Novikov and Thorne [229], with important extensions and corrections provided in subsequent papers [237, 265, 241]. Here we reproduce the solution, although with a more general scaling: m = M/M_{⊙} and ṁ = Ṁc^{2}/L_{Edd}.
The ShakuraSunyaev and NovikovThorne solutions are only local solutions; this is because they do not take into account the full eigenvalue problem described in Section 5.2. Instead, they make an assumption that the viscous torque goes to zero at the ISCO, which makes the model singular there. For very low accretion rates, this singularity of the model does not influence the electromagnetic spectrum [298], nor several other important astrophysical predictions of the model. However, in those astrophysical applications in which the inner boundary condition is important (e.g., global modes of disk oscillations), the NovikovThorne model is inadequate. Figure 7 illustrates a few ways in which the model fails to capture the true physics near the ISCO.
6 Slim Disks
The problem of accretion with an additional cooling mechanism has to be treated in a different way than radiatively efficient flows. Without the assumptions of radiative efficiency and Keplerian angular momentum, it is no longer possible to find an analytic solution to the system of equations presented in Section 5.1. Instead, one has to solve a twodimensional system of ordinary differential equations (94) with a critical point — the radius at which the gas velocity exceeds the local speed of sound (the sonic radius). This was first done in the pseudoNewtonian limit by Abramowicz [8], who forged the term “slim disks”. It has since been done using a fully relativistic treatment by Beloborodov [40]. Recently, Sadowski [268] constructed slim disk solutions for a wide range of parameters applicable to Xray binaries.
7 AdvectionDominated Accretion Flows (ADAFs)
The ADAF, or advectiondominated accretion flow, solution also involves advective cooling. In fact, it carries it to an extreme — nearly all of the viscously dissipated energy is advected into the black hole rather than radiated. Unlike the slim disk solution, which is usually invoked at high luminosities, the ADAF applies when the luminosity (and generally the mass accretion rate) are low.
The rapid advection in ADAFs generally has two effects: 1) dissipated orbital energy can not be radiated locally before it is carried inward and 2) the rotation profile is generally no longer Keplerian, although Abramowicz [6] found solutions where the dominant cooling mechanism was advection, even when the angular momentum profile was Keplerian. Fully relativistic solutions of ADAFs have also been found numerically [13, 41]. Further discussion of ADAFs is given in the review article by Narayan and McClintock [219].
8 Stability
Having reviewed some of the main analytic models of accretion disks, it is important now to discuss the issue of stability. Since all analytic models presume steadystate solutions, such models are only useful if the resulting solutions are stable. One reason to suspect accretion disks may not be stable is that the systematic differential rotation that is a signature feature of accretion is a potential source of energy, and therefore, of instability. Another is that some level of instability may be essential in accretion disks as it can provide a pathway to the kind of sustained turbulence anticipated by Shakura and Sunyaev (see Section 3.2.1).
8.1 Hydrodynamic stability
As it turns out, the Høiland criterion is a huge disappointment for understanding why turbulence might exist in accretion disks. This is because it indicates that accretion disks with rotation profiles that do not differ too much from Keplerian should be strongly stable!
8.1.1 PapaloizouPringle Instability (PPI)
The Høiland criterion is only a local stability criterion. Flows can be locally stable, yet have global instabilities. An example of this occurs in the Polish doughnut solution (Section 4). Papaloizou and Pringle [238] showed that this solution is marginally stable with respect to local axisymmetric perturbations yet unstable to loworder nonaxisymmetric modes. As with all global instabilities, the existence of the PapaloizouPringle instability (PPI) is sensitive to the assumed boundary conditions [44]. In cases where the disk overflows its potential barrier (Roche lobe) and accretes through pressuregradient forces across the cusp, the PPI is generally suppressed [117].
8.1.2 Runaway instability
Another instability associated with the Polish doughnut is the runaway instability [5]. If matter is overflowing its Roche lobe and accreting onto the black hole, then one of two evolutionary tracks are possible: (i) As the disk loses material it contracts inside its Roche lobe, slowing the mass transfer and resulting in a stable situation, or (ii) as the black hole mass grows, the cusp moves deeper inside the disk, causing the mass transfer to speed up, leading to the runaway instability. Recent numerical simulations show that, while this instability grows very fast, on timescales of a few orbital periods, over a wide range of disktoblack hole mass ratios when ℓ = const., i.e., a constant specific angular momentum profile [98], it is strongly suppressed whenever the specific angular momentum of the disk increases with the radial distance as a power law, ℓ ∝ r^{p} [63]. Even values of p much smaller than the Keplerian limit (p = 1/2) suffice to suppress this particular instability. [This is equivalent to angular velocity profiles, Ω ∝ r^{−q}, with q > 3/2.]
8.2 Magnetorotational instability (MRI)
Although it had long been suspected that some sort of MHD instability might provide the necessary turbulent stresses to make accretion work, the nature of this instability remained a mystery until the rediscovery of the magnetorotational instability by Balbus and Hawley [26, 118, 27]. Originally discovered by Velikhov [309], and generalized by Chandrasekhar [58], in the context of vertically magnetized Couette flow between differentially rotating cylinders, the application of this instability to accretion disks was originally missed.
The instability itself can be understood through a simple mechanical model. Consider two particles of gas connected by a magnetic field line. Arrange the particles such that they are initially located at the same cylindrical distance from the black hole but with some vertical separation. Give one of the particles (say the upper one) a small amount of extra angular momentum, while simultaneously taking away a small amount of angular momentum from the lower one. The upper particle now has too much angular momentum to stay where it is and moves outward to a new radius. The lower particle experiences the opposite behavior and moves to a smaller radius. In the usual case where the angular velocity of the flow drops off with radius, the upper particle will now be orbiting slower than the lower one. Since these two particles are connected by a magnetic field line, the differing orbital speeds mean the field line will get stretched. The additional tension coming from the stretching of the field line provides a torque, which transfers angular momentum from the lower particle to the upper one. This just reinforces the initial perturbation, so the separation grows and angular momentum transfer is enhanced. This is the fundamental nature of the instability.
If the conditions for the instability are met, the fastestgrowing mode, which dominates the early evolution, has the form of a “channel flow” involving alternating layers of inward and outwardmoving fluid. The amplitude of this solution grows exponentially until it becomes unstable to threedimensional “parasitic modes” that feed off the gradients of velocity and magnetic field provided by the channel flow. The flow rapidly reaches a state of magnetohydrodynamic turbulence [118, 119]. This instability can be selfsustaining through a nonlinear dynamo process [52] — nonlinear because the motion that sustains or amplifies the magnetic field is driven by the field itself through the MRI. A more complete description of the linear and nonlinear evolution of the MRI is provided in the review article by Balbus and Hawley [27]. A general relativistic linear analysis is presented in [20].
8.3 Thermal and viscous instability
It was realized by Shakura and Sunyaev themselves [280], as well as other authors [176, 287], that the ShakuraSunyaev solution (Section 5.3) should be thermally and viscously unstable for disks in which radiation pressure dominates (when the opacity is governed by electron scattering). The most general and elegant arguments are presented in the classic paper by Piran [246]. This discovery started a long debate, which continues unresolved to this day. A recent update is provided in [60].
Note, though, that this argument only applies when the viscous stress is proportional to the total pressure P_{Tot} (α being the proportionality constant). For some time it seemed that a plausible way to avoid this instability was to argue that the stress is proportional instead to the gas pressure P_{gas}. Recent numerical simulations, though, of the magnetorotational instability in radiationpressure dominated disks have shown that the stress is, in fact, proportional to the total pressure [128]. Interestingly, these simulations exhibit no sign of the predicted thermal instability. Most observations also argue against the existence of this instability. In the case of accretion onto black holes, the instability is supposed to set in for luminosities in excess of L > 0.01 L_{Edd}. However, during outbursts, many stellarmass black hole sources cross this limit both during their rise to peak luminosity and on their decline to quiescence, showing no dramatic symptoms (although they do undergo state changes, as described in Section 12.3). On the contrary, observations suggest that disks in black hole Xray binaries are stable up to at least L ≈ 0.5L_{Edd} [80]. Certainly there is no evidence for the sensational behavior anticipated by some models [172, 300].
9 Oscillations
Even when analytic disk solutions are stable against finite perturbations, it is often the case that these perturbations will, nevertheless, excite oscillatory behavior. Oscillations are a common dynamical response in many fluid (and solid) bodies. Here we briefly explore the nature of oscillations in accretion disks. This topic is particularly relevant to understanding the physical mechanisms that may be behind quasiperiodic oscillations (or QPOs, which are discussed in Section 12.4).
There are a number of local restoring forces available in accretion disks to drive oscillations. Local pressure gradients can drive oscillations via sound waves. Buoyancy forces can act through gravity waves. The Coriolis force can operate through inertial waves. Surface waves can also exist, with the restoring force given by the local effective gravity.
Of particular interest are families of low order modes that may exist in various accretion geometries. Such modes will tend to have the largest amplitudes and produce more easily observed changes than their higherorder counterparts. Here we briefly review a couple relatively simple examples for the purpose of illustration. More details can be found in the references given.
9.1 Dynamical oscillations of thick disks
A complete analysis of the spectrum of modes in thick disks has not yet been done. Some progress has been made by considering the limiting case of a slender torus, where slender here means that the thickness of the torus is small compared to its radial separation from the central mass (i.e., the torus has a small crosssectional area). In this limit, the complete set of modes have been determined for the case of constant specific angular momentum in a Newtonian gravitational potential [43]. A more general analysis of slender torus modes is given in [45].
9.2 Diskoseismology: oscillations of thin disks
pmodes are inertial acoustic modes defined by Ψ < ῶ^{2} and are trapped where ω^{2} > ωr^{2}, which occurs in two zones. The inner pmodes are trapped between the inner disk edge and the inner “Lindblad” radius, i.e., r_{i} < r < r_{−}, where gas is accreted rapidly. The outer pmodes occur between the outer Lindblad radius and the outer edge of the disk, i.e., r_{+} < r < r_{o}. The Lindblad radii, r_{−} and r_{+}, occur where ω = ω_{r}. The outer pmodes are thought to be more consequential as they produce stronger luminosity modulations [233]. In the corotating frame these modes appear at frequencies slightly higher than the radial epicyclic frequency. Pressure is the main restoring force of pmodes.
gmodes are inertial gravity modes defined by Ψ > ῶ^{2}. They are trapped where \({\omega ^2} < \omega _r^2\) in the zone r_{−} < r < r_{+} given by the radial dependence of ω_{r}, i.e., gmodes are gravitationally captured in the cavity of the radial epicyclic frequency and are thus the most robust among the thindisk modes. Since this is the region where the temperature of the disk peaks, gmodes are also expected to be most important observationally [244]. In the corotating frame these modes appear at low frequencies. Gravity is their main restoring force.
cmodes are corrugation modes defined by Ψ = ῶ^{2}. They are nonradial (m = 1) and vertically incompressible modes that appear near the inner disk edge and precess slowly around the rotational axis. These modes are controlled by the radial dependence of the vertical epicyclic frequency. In the corotating frame they appear at the highest frequencies.
All modes have frequencies ∝ 1/M. Upon the introduction of a small viscosity (ν ∝ α, α ≪ 1), most of the modes grow on a dynamical timescale t_{dyn}, such that the disk should become unstable. However, evidence for these modes has so far mostly been lacking in MRI turbulent simulations (see Section 11.6). This leaves their relevance in some doubt.
10 Relativistic Jets
Although the main focus of this review is on black hole accretion disk theory, we note that there has long been a strong observational connection between accreting black holes and relativistic jets across all scales of black hole mass. For supermassive black holes this includes quasars and active galactic nuclei; for stellarmass black holes this includes microquasars. However, the theoretical understanding of disks and jets has largely proceeded separately and the physical link between the two still remains uncertain. Therefore, we present only a few brief comments on the subject in this review. More complete discussions of the theory of relativistic jets may be found in [193]. A review of their observational connection to black holes is given in [205].
In Section 2.2, we described the “Penrose process,” whereby rotational energy may be extracted from a black hole and carried to an observer at infinity. To briefly recap, Penrose [242, 243] imagined a freely falling particle with energy E^{∞} disintegrating into two particles with energies \(E_  ^\infty < 0\) and \(E_ + ^\infty > 0\). Then, the particle with negative energy \(E_  ^\infty\) falls into the black hole, and the other one escapes to infinity. Clearly, \(E_ + ^\infty > {E^\infty}\), so that there is a net gain of energy.
It was first suggested by Wheeler at a 1970 Vatican conference and soon after by others [183, 96] that such a Penrose process may explain the energetics of superluminal jets commonly seen emerging from quasars and other black hole sources. However, a number of authors [31, 314, 161] showed that for \(E_ + ^\infty\) to be greater than E^{∞}, the disintegration process must convert most of the rest mass energy of the infalling particle to kinetic energy, in the sense that, in the centerofmass frame, the \(E_  ^\infty\) particle must have velocity v > c/2. The argument of Wald [314] is powerful, short and elegant, so we give it here in extenso.
Replacing particle disintegration with particle collision does not help, even though the centerofmass energy of such a collision happening arbitrarily close to the horizon of the maximally rotating Kerr black hole may be arbitrarily large [247, 28]. This is because the Wald limit of \(1 + \sqrt 2\) still holds [39]. It would seem that even under idealized conditions, the maximal energy of a particle escaping via the Penrose process is only a modest factor above the total initial energy [39].
11 Numerical Simulations
In simulating accretion disks around black holes, there are a number of challenging issues. First, there is quite a lot of physics involved: relativistic gravity, hydrodynamics, magnetic fields, and radiation being the most fundamental. Then there is the issue that accretion disks are inherently multidimensional objects. The computational expense of including extra dimensions in a numerical simulation is not a trivial matter. Simply going from one to two dimensions (still assuming axisymmetry for a disk) increases the computational expense by a very large factor (10^{2} or more). Going to threedimensions and relaxing all symmetry requirements increases the computational expense yet again by a similar factor. Simulations of this size have only become feasible within the last decade and still only with a subset of the physics one is interested in and usually with a very limited time duration.
Another hindrance in simulating accretion disks is the very large range of scales that can be present. In terms of a grid based code, a disk with a scale height of H/R requiring N_{z} zones to resolve in the vertical direction at some radius R_{in}, would require something of the order N_{z}/(H/R) zones to cover each factor of R_{in} that is treated in the radial direction. The azimuthal direction in a full threedimensional simulation would require a comparable number of zones to what is used in the radial direction. Given that a very large calculation by today’s standards is 10^{9}–10^{10} zones, we can see that treating very thin disks (H/R ≲ 0.01) in a full threedimensional simulation, even with a very modest number of zones in the vertical direction (N_{z} ≥ 50) would take all the resources one could muster. However, twodimensional (axisymmetric) simulations of thin disks and threedimensional simulations of thick, slim, or moderately thin disks have now become quite common.
The approach of numericists in many ways parallels that of theorists, so we structure this section much the same as the first part of this Living Review. That is, after a brief introduction to numerical techniques, we discuss how the various components of the matter description (à la Section 3) are implemented in numerical simulations. We then review a few special cases illustrating how analytic models (Sections 4–7) and numerical simulations can complement one another. We finish this numerical section with a few topics of special interest.
11.1 Numerical techniques
11.1.1 Computational fluid dynamics codes
There are many numerical codes available today that include relativistic hydrodynamics or MHD that are, or can be, used to simulate accretion disks. A partial list includes: Cosmos++ [18], ECHO [74], HARM [105], and RAISHIN [208] plus the codes developed by Koide et al. [152], De Villiers and Hawley [71], Komissarov [154], Antón et al. [19], and Anderson et al. [16]. This represents tremendous growth in the field as prior to 1998 there were only a couple such codes, largely derived from the early work of Wilson [318] and later developments by Hawley [126, 125]. Those early codes were only available to a handful of researchers. This was simply a reflection of the fact that the computational resources available to most researchers at that time were insufficient to make use of such codes, so there was little incentive to develop or acquire them. This has clearly changed.
Today there are primarily two types of numerical schemes being used to simulate relativistic accretion disks, differentiated by how they treat discontinuities (shocks) that might arise in the flow: the artificial viscosity scheme, still largely based upon formulations developed by Wilson [318]; and Godunovtype approaches, using exact or approximate Riemann solvers. Both are based on finite difference representations of the equations of general relativistic hydrodynamics, although the latter tend to also incorporate finitevolume representations. The artificial viscosity schemes have the advantages that they are more straightforward to implement and easier to extend to include additional physics. More importantly, they are computationally less expensive to run than Godunov schemes. Furthermore, recent work by Anninos and collaborators [17, 18] has shown that variants of this scheme can be made as accurate as Godunov schemes even for ultrarelativistic flows, which historically was one of the principal weaknesses of the artificial viscosity approach. Godunov schemes, on the other hand, appreciate the advantage that they are fully conservative, and therefore, potentially more accurate. They also require less tuning since there are no artificial viscosity parameters that need to be set for each problem. More thorough reviews of these two methods, with clear emphasis on the HighResolution ShockCapturing variant of the Godunov approach, are provided in the Living Reviews by Martí & Müller [182] and Font [97]. Other numerical schemes, such as smoothparticle hydrodynamics (SPH) and (pseudo)spectral methods are less well developed for work on relativistic accretion disks.
11.1.2 Global vs. shearingbox simulations
Along with settling on a numerical scheme, a decision must also be made whether or not to try to treat the disk as a whole or to try to understand it in parts. The latter choice includes “shearingbox” simulations, the name coming from the type of boundary conditions one imposes on the domain to mimic the shear that would be present in a real disk [121]. The obvious advantage of treating the disk in parts is that you circumvent the previously noted problem of the large range in scales in the disk by simply ignoring the large scales. Instead one treats a rectangular volume generally no larger than a few vertical scale heights on a side and in some cases much smaller. In this way, for a moderate number of computational zones, one can get as good, or often much better, resolution over the region being simulated than can be achieved in global simulations. The obvious disadvantage is that all information about what is happening on larger scales is lost. By construction, no structures larger than the box can be captured and the periodic boundaries impose some artificial conditions on the simulations, such as no background gradients in pressure or density and no flux of material into or out of the box, meaning the surface density remains fixed. Perhaps most important in the context of a review on relativity is that the box is treated as a local patch within the disk, with a scale smaller than that of the curvature of the metric. Thus most general relativistic effects are not, and by construction need not be, included in shearing box simulations. Since this is a Living Review in Relativity, we will not dwell much on this class of simulations. Still, there are some significant highlights that should be mentioned.
By far the most extensive work using shearing box simulations has been aimed at better understanding the magnetorotational instability (MRI). Starting from the earliest “proofofconcept” simulations [121, 296], shearing box simulations have been used to demonstrate various properties of the saturated state of MRI turbulence [271, 272, 173, 283] and much about the vertical structure of MRI turbulent disks [46]. Shearing box simulations have also proven valuable in studying radiationdominated disks. For example, Turner [307] studied the vertical structure of radiation dominated accretion disks and Hirose et al. [129] studied the gaspressure dominated case, both including radiation using fluxlimited diffusion. Hirose and collaborators subsequently used shearing box simulations to demonstrate that radiationdominated disks are thermally stable [128], though they had long been held not to be [176, 280, 246]. As we said, though, there are limits imposed by the finite size of the shearing box. For instance, the viscous, LightmanEardley instability [176] can only be studied through global simulations.
11.2 Matter description in simulations
The minimum physics required for a global simulation of an accretion disk are gravity and hydrodynamics (assuming the disk is dense enough for the continuum approximation to hold). Since many disks have masses that are small compared to the mass of the central compact object, the selfgravity of the disk can often be ignored. Therefore, in the next three sections, gravity will simply mean that of the central black hole.
11.2.1 Hydrodynamics + gravity
The first researcher to develop and use numerical algorithms for simulating relativistic accretion flows was Wilson [316], who considered the spherical infall of material with a nonzero specific angular momentum toward a Kerr black hole using the full metric, although restricted to two spatial dimensions. Wilson was able to confirm the additional centrifugal support that the infalling material experienced due to the rotating black hole. For sufficiently high values of angular momentum, the material could not immediately accrete onto the black hole, instead forming a fat disk of the type described in Section 4.
In many ways, Wilson’s pioneering work was at least a decade ahead of its time and there was consequently a lull in activity until Hawley and Smarr collaborated with Wilson to revive his work [126, 125]. As an example of how analytic and numerical work can complement each other, it is worth noting that one of the test cases they used for their new twodimensional relativistic hydrodynamics code was based on the analytic theory for relativistic thick disks, which had been worked out in the time since Wilson’s original simulations. Using the analytic theory, they constructed a series of disks with different (constant) specific angular momenta, from ℓ < ℓ_{ms} to ℓ > ℓ_{mb.} The ℓ > ℓ_{mb} case yielded a static solution as expected and confirmed the ability of their code to accurately evolve such a solution in multidimensions over a dynamical time. The ℓ < ℓ_{mb} cases showed greater time variability and illustrated the power of direct numerical simulations to extend our understanding of black hole accretion.
11.2.2 Magnetohydrodynamics + gravity
Magnetic fields can play many important roles in relativistic accretion disks, from providing local viscous stresses through turbulence that results from the magnetorotational instability (Section 8.2), to providing a mechanism for launching and confining jets (Section 10). Thus, the inclusion of magnetic fields in numerical simulations of relativistic accretion disks is important. Although the techniques for doing this were first described in [317], relatively little work was done in this area until fairly recently. Perhaps the two most important contributions so far from relativistic MHD simulations of accretion disks have been: 1) elucidating the behavior of MRI turbulent disks in the vicinity of black holes, and 2) exploring the many interrelations between magnetic fields and relativistic jets.
In at least one case, the inclusion of MHD + gravity is sufficient to adequately capture the dynamics of a real black hole accretion disk. Sgr A*, the black hole at the center of the Milky Way galaxy has such an anemically low luminosity [194] that numerical simulations that ignore radiation can safely be applied [76], as has been done by many authors [112, 227, 211, 75]. In most other cases, radiative processes must somehow be accounted for.
11.2.3 RadiationMagnetohydrodynamics + Gravity
Probably the most glaring shortcoming of almost all numerical simulations of accretion disks and many other phenomena in astrophysics to date is the unrealistic treatment of radiation, which is most often simply ignored. This is not due to a lack of appreciation of its importance on the part of numericists, but simply a reflection of the fact that there are very few efficient ways to treat radiation computationally in multiple dimensions.
In the opticallythick limit, some progress can be made by employing the fluxlimited diffusion approximation [174]. Global simulations of disks including a fluxlimited diffusion treatment of radiation (but using pseudoNewtonian gravity) are now being done by Ohsuga and collaborators [232, 231]. A few steps toward the goal of relativistic radiation MHD simulations of black hole accretion disks have also been taken in recent years. A method for treating optically thick accretion using a conservative, Godunov scheme was developed by Farris and collaborators [91]. The same basic method has now been used to examine both Bondi [100] and BondiHoyle [326, 266] accretion. Simulations of accretion disks, though, must await the generalization of this method to treat radiation both in the optically thick and thin limits.
11.2.4 Evolving GRMHD
In most simulations of accretion disks around black holes, the selfgravity of the disk is ignored. In many cases this is justified as the mass of the disk is often much smaller than the mass of the black hole. This is also much simpler as it allows one to treat gravity as a background condition, either through a Newtonian potential or a relativistic metric (the socalled Cowling approximation in relativity). However, there are plausible astrophysical scenarios in which this approximation is not valid. Two of the more interesting are: 1) a tidally disrupted neutron star accreting onto a stellarmass black hole; and 2) an overlying stellar envelope accreting onto a nascent black hole during the final dying moments of a massive star. Interestingly these scenarios are currently the most popular models of gammaray bursts [82, 319], which are the most powerful explosions in the Universe since the Big Bang and are observed at a rate of about one per day.
Treating a selfgravitating disk in general relativity requires a code that can simultaneously evolve the spacetime metric (by solving the Einstein field equations) and the matter, magnetic, and radiation fields. Codes capable of doing this have very recently become available, divided into the following “flavors”: selfgravity + hydro [24]; selfgravity + MHD [81, 286, 110, 88]; and selfgravity + radiation MHD [91]. More detailed discussion of these methods and their application to problems in astrophysics is provided in the Living Review by Font [97].
The importance of evolving the gravitational field in the case of black hole accretion is well illustrated in the case of the runaway instability, which we described in Section 8.1.2. That instability has now been simulated directly, including evolving the spacetime metric [209, 210, 160]. These studies have confirmed earlier results that the runaway instability does not develop.
Another application of these types of codes that is relevant to our review is the reexamination of the PapaloizouPringle instability (discussed in Section 8.1.1). Through this instability, a massive, selfgravitating Polish doughnut orbiting around a black hole could become a strong emitter of large amplitude, quasiperiodic gravitational waves [149].
11.3 Polish doughnuts (thick) disks in simulations
As we already mentioned, it is computationally less expensive to simulate thick disks than thin, so it comes as no surprise that most of the numerical work for nearly four decades, starting with the work of Wilson [316], has been on thick disks. In this section we touch on a few highlights from this area of research
11.4 NovikovThorne (thin) disks in simulations
Early numerical simulations of black hole accretion disks nearly all focused on thick disks. This was mostly for computational convenience as thicker disks require fewer resources than thin ones. In recent years, though, the resources have become available to start testing thinner disks. This has enabled researchers to begin rigorously testing the NovikovThorne model (Section 5.3), especially the assumption that the internal torque of the disk vanishes at the ISCO. This is important as some researchers have argued that magnetic fields might nullify this hypothesis by maintaining stresses inside the ISCO [164, 104, 25].
We mention that since present day relativistic numerical simulations do not treat the radiation in geometrically thin, optically thick disks, the simulations of NovikovThorne disks have so far employed an ad hoc cooling prescription [278, 226]. This prescription conveniently makes the same assumption that the NovikovThorne model does: that all energy dissipated as heat in the disk is radiated away locally on roughly the orbital timescale. This is probably reasonable for an appropriate range of mass accretion rates, though it will be good to test this assumption with future global radiationMHD simulations.
11.5 ADAFs in simulations
A lot of recent simulation work has been focused on exploring more ADAFlike flows under the action of realistic MRI turbulence [222, 322, 320]. To some extent, this is an extension of the Polishdoughnut simulations of the last decade, yet goes beyond it in at least two important respects: 1) the simulations cover a significantly larger spatial range (a few hundreds of GM/c^{2} versus a few tens); and 2) the simulations explore much longer temporal evolution (hundreds of thousands of GM/c^{3} versus tens of thousands). The result of this is that the simulations are able to explore steadystate accretion out to much larger radii (∼ 100 GM/c^{2} as opposed to ∼ 10 GM/c^{2}). More can be expected from this work in the coming years.
11.6 Oscillations in simulations
Hydrodynamic simulations have also been used extensively to study the natural oscillation modes of relativistic disks, particularly as they might relate to QPOs (Section 12.4). Rezzolla and collaborators [263, 264] identified a mode they referred to as a pmode that occurred in a near 3:2 ratio with the radial epicyclic mode, which was subsequently confirmed through numerical simulations [325, 324] (animations can be viewed at [262]). The 3:2 ratio of these modes is important as the highestfrequency QPOs in black hole lowmass Xray binaries are observed to occur in this ratio. Later, Blaes and collaborators [45] identified a different pair of modes, the vertical epicyclic and axisymmetric breathing mode, that have a near 3:2 ratio for a broader range of parameters. Similar to [325, 324], they also demonstrated that these modes could be identified in numerical simulations.
A number of MHD simulations have also been performed which claim to observe QPOs. Kato [145] performed a global MRI simulation in a pseudoNewtonian potential and claimed to see QPOs in the power spectrum in a measure of the luminosity associated with radial infall. Machida and Matsumoto [179] have reported the formation of a onearmed spiral within the inner torus of a global MRI simulation, and suggested that this could be responsible for the low frequency QPO in the hard state. Schnittman, Krolik, and Hawley [274] have found tentative evidence for high frequency QPO features in light curves generated by coupling a general relativistic ray tracing code to a global GRMHD simulation with an assumed emission model. Henisey and collaborators [127] found similar tentative evidence in a sample of tilted disk simulations, possibly confirming earlier suggestions that disk tilt may be an important mechanism for driving high frequency QPOs in black hole accretion disks [143, 95]. Most recently Dolence and collaborators [78] report evidence for nearinfrared and Xray QPOs in numerical simulations of Sgr A*.
Aside from these few interesting examples, though, no robust QPO signal has generically emerged in MHD simulations [22, 261]. It is unclear what the implications of this are. It may be that some missing physics, such as radiation transport, plays a fundamental role in exciting QPOs. This is an important open problem in numerical simulations of black hole accretion disks.
11.7 Jets in simulations
In recent years several numerical simulations have demonstrated the generation of jets selfconsistently from simulations of disks. A few researchers have claimed to produce jetted outflows from purely hydrodynamic interactions. Nobuta and Hanawa [228], for instance, were able to achieve jetted outflows driven by shock waves when infalling gas with high specific angular momentum collided with the centrifugal barrier. However, such simulations usually require very special starting conditions and the jets tend to be transient features of the flow; therefore, it is unlikely these are related to the wellcollimated expansive jets observed in many black hole systems.^{13}
Among MHD simulations, some distinction should be drawn between those that impose large scale magnetic fields that extend beyond the domain of the simulation and those that, at least initially, impose a selfcontained magnetic field. In the first case, the disk is often treated merely as a boundary condition for the evolution of largescale magnetic field. Shibata and Uchida [284, 285] used this technique to show that jets could be driven both by a pinching of the fields due to radial inflow through the disk and by the j × B force caused when twisted fields unwind.
Tchekhovskoy and collaborators [303] have used GRMHD simulations of black hole accretion disks plus jets to investigate possible explanations for the observed radio loud/quiet dichotomy in AGN.^{14} For a black hole surrounded by a thin disk, the BlandfordZnajek mechanism predicts the luminosity of the jet should scale as \({L_{BZ}} \propto \Omega _H^2\). This limits the range of power expected for realistic AGN spins to a factor of a few tens at most — too small to explain the observed differences. Thicker disks, on the other hand, produce jets whose luminosity can scale as \({L_{BZ}} \propto \Omega _H^4\) or even \(\Omega _H^6\) [303], providing sufficient range to perhaps explain the dichotomy.
There has also been some work recently trying to understand a different jet dichotomy — one that is observed in black hole lowmass Xray binaries (LMXBs). These exhibit radio emission (associated with jets) whose properties change with the observed Xray spectral and, to a less well determined extent, temporal properties of the accretion disk [92, 94]. Briefly, compact, steady jets are observed in the Low/Hard state, whereas jets appear absent in the High/Soft state. Fragile and collaborators [102] used numerical simulations to rule out disk scale height as the controlling factor, suggesting instead that perhaps the jets are intimately connected with the corona or failed MHD wind. Alternatively, it could be that magnetic field topology is the key factor [136, 190].
11.8 Highly magnetized accretion in simulations
Recently, work has begun to focus on highly magnetized disk configurations, for which some modes of the MRI may be suppressed. One motivation is that these might provide an alternate explanation for the Low/Hard state [35] (Section 12.3).
One way a highly magnetized state could come about is as a result of a thermal instability in an initially hot, thick, weaklymagnetized disk [179, 101]. Machida and collaborators [179] simulated this process for an optically thin disk, assuming bremsstrahlung cooling and pseudoNewtonian gravity. They found that, indeed, the densest inner regions of the disk collapse down to a cool, thin, magneticallysupported structure. Fragile and Meier [101] extended these results by including more cooling processes and using a general relativistic MHD code.
12 Selected Astrophysical Applications
12.1 Measurements of blackhole mass and spin
Astrophysical black holes are not charged, and thus are characterized only by their mass and spin. Measurements of black hole mass are generally straightforward, requiring only the observation of an orbital companion and an application of Kepler’s laws. Current mass estimates for stellarmass black holes (in particular microquasars) are reviewed by McClintock & Remillard [184] and for supermassive black holes by Kormendy & Richstone [159] (see also [196] for somewhat more recent data).
Nevertheless, there remain several fundamental questions connected with the mass of black holes, and some of them are directly connected to accretion disk theory. One of them is the question of ultraluminous Xray sources (ULXs) [61, 181]. ULXs are powerful Xray sources located outside of galactic nuclei, which have luminosities in excess of the Eddington limit for M = 10^{2} M_{⊙}, assuming isotropic emission. The huge luminosities of ULXs lead some to conclude that they are accreting intermediatemass black holes, with masses Mulx > 10^{2} M_{⊙} (e.g., [165, 198]). Others think that they are stellar mass black holes with M_{ULX} ∼ 10 M_{⊙}, either exhibiting beamed emission [108] or surrounded by disks that are somehow able to produce highly superEddington luminosities (e.g., [147, 36]). At this time, at least one ULX (ESO 24349/HLX1) has been convincingly demonstrated to have an intermediate mass (∼ 500 M_{⊙}) [90, 67].
As difficult as it is to nail down the mass on some of these systems, it is even more difficult to measure black hole spin, even though it plays a direct, and important, role in accretion disk physics. One obvious example of the role of spin is the dependence of r_{ms}, the coordinate radius of the ISCO, on spin. In the symmetry plane of the black hole r_{ms} = 6r_{G} for a_{*} = a/M = 0 (nonrotating), 1r_{G} for a_{*} = 1 (maximal prograde rotation), and 9r_{G} for a_{*} = −1 (maximal retrograde rotation) (Section 2.5). It is believed that the inner edge of the accretion disk will be similarly affected. In addition, for rapidly rotating black holes, the BlandfordZnajek mechanism, and similar processes which depend on spin, may account for a fair share of the global energetics, comparable to that of accretion itself (see, e.g., [155, 189] and references therein). Therefore, measuring the spin of accreting black holes is integrally tied up in understanding black hole accretion generally.
Four methods for determining black hole spin have been proposed in the literature. With references to some of the earliest results, they are: 1) fitting the continuum spectra of microquasars observed in the thermally dominant state using disk emission models [65, 185, 197, 277]; 2) fitting observed relativistically broadened iron line profiles with theoretical models [142, 315, 201, 202, 260]; 3) matching observed QPO frequencies to those predicted by theoretical models [62, 11, 259, 306]; and 4) analyzing the “shadow” a black hole makes on the surface of an accretion disk [301]. The first three methods have been the most commonly applied to date. Although there have been some glaring discrepancies in the spin estimates published to date (e.g., one group claiming that Cyg X1 has a nearzero spin, a_{*} = 0.05±0.01 [200], and then later claiming it has a nearmaximal spin, a_{*} > 0.95 [89]), there appears to be a settling of values in recent years and a growing confidence in the methods, particularly the continuum fitting. There are still some concerns, however. For the continuum fitting method, the main issue is that the inclination of the Xray emitting region must be measured by some independent means. This is because the effect of the inclination on the spectrum is degenerate with the effect of spin [175], so both can not be accounted for within the continuum fitting method. In cases where such an independent measure is available (e.g. [292]), the continuum fitting method appears robust. For the relativisticallybroadened iron line method, there are difficulties in properly estimating the extent of the “red” wing, which is most directly related to the spin of the black hole, and in modeling the hard Xray source photons and the disk ionization, both of which strongly affect the reflection spectrum. The reviews by Remillard, McClintock, and collaborators [258, 186] give more complete introductions to the topic of measuring black hole spin, with emphasis mainly on the continuum fitting method.
12.2 Black hole vs. neutron star accretion disks
Little of what we have said so far has depended on whether the central compact object is a black hole or neutron star, provided only that the neutron star is compact enough to lie inside the inner radius of the disk r_{in}. In this case, its presence will not be noticed by the disk except through its gravity, which will be practically the same as for a black hole (an exception would be if the neutron star is strongly magnetized [113]). However, this does not mean that accreting black hole and neutron star sources will be indistinguishable, as we have not yet fully addressed the question of what happens to energy advected past r_{in}. For optically thick, geometrically thin ShakuraSunyaev disks (Section 5.3), a significant fraction of the gravitational energy liberated by advection is radiated by the gas prior to it passing through r_{in}. Thus, the total luminosity of thin disks will not depend sensitively on the nature of the central object. However, this is not the case for the ADAF solution (Section 7), for which much of the thermal energy gained by the gas from accretion is carried all the way in to the central object. Narayan and his collaborators [225, 215, 195, 107] have convincingly argued that this may allow observers to distinguish between black hole and neutron star sources.
12.3 Blackhole accretion disk spectral states
Black hole accretion disks, particularly in Xray binaries, exhibit complex spectra composed of both thermal and nonthermal components. During outbursts, the relative strengths of these components change frequently in concert with changes in luminosity and the characteristics of the radio features (i.e., jets). Astronomers have developed a set of empirical spectral classification states to broadly characterize these observations. These spectral states likely reveal important information about the underlying physical state of the system; therefore it is worth summarizing the states and their observed properties here.
Probably the easiest state to connect with a theoretical model is the “High/Soft” (HS) or “Thermally Dominant” state. As the name implies, the spectrum in this state is dominated by the thermal component. This state is best explained as ∼ 1 keV thermal emission from a multitemperature accretion disk, as predicted by the ShakuraSunyaev (thin) disk model (Section 5.3). However, most sources spend the majority of their lifetimes in the “Low/Hard” (LH) or even “quiescent” state. The quiescent state is characterized by exceptionally low luminosity and a hard, nonthermal spectrum (photon index Γ = 1.5–2.1). As the luminosity increases the sources usually enters the Hard state. Here the 2–10 keV intensity is still comparatively low and the spectrum is still nonthermal. This spectrum is best fit with a power law of photon index Γ ∼ 1.7 (220 keV). In this state, the thermal component is either not detected or appears much cooler, indicating the thin disk may truncate further out than in the Thermally Dominant state, although see [199] for claims that the thin disk extends all the way to the ISCO even in the Hard state. Observations suggest the region interior to the thin disk may be filled with a hot (presumably thick), opticallythin plasma, which accounts for the nonthermal part of the spectrum. This is the picture suggested by the “truncated disk model” [85, 86, 84], which pictures the Hard state as a truncated thin (ShakuraSunyaev) disk adjoined with an inner thick (ADAFlike) flow. This model has shown tremendous phenomenological success [79], although other models for the Hard state still abound [323, 101, 136]. The Hard state is also linked with observations of a persistent radio jet that is not seen in other states (see Section 10). The final spectral state, which is referred to as the “Very High” (VH) or “Steep Power Law” state, is characterized by the appearance of highfrequency QPOs (Section 12.4) and the presence of both disk and powerlaw components, each of which contributes substantial luminosity. In this state, the powerlaw component is observed to be steep (Γ ∼ 2.5), giving the state its name. This state is sometimes associated with intermittent jets.
These states, their distinguishing observational properties, and a sampling of observations from various black hole Xray binaries is presented in the review by McClintock and Remillard [184].
12.4 QuasiPeriodic Oscillations (QPOs)
Einstein’s general theory of gravity has never been tested in its strong field limit, characteristic of the region very near black holes (or neutron stars), i.e., a few gravitational radii away from these sources. Soon VLBI measurements may be able to resolve these scales for the supermassive black hole at the center of our galaxy. However, for most sources, resolution in time seems to be a more practical approach. To zeroth order, the light curves from accreting black holes vary in a chaotic manner, resembling a loud noise. However, Fourier analysis of the light curve reveals stinkingly regular patterns buried in the noise. For Galactic black hole and neutron star sources, these quasiperiodic oscillations (QPOs) have frequencies of a few hundred Hz.
Frequencies in the range 100–1000 Hz formally correspond to orbital frequencies a few gravitational radii away from a stellarmass object. The focus on orbital frequencies is further motivated by the stability of observed QPO frequencies over very long periods of time. For example, QPOs of 300 Hz and 450 Hz were observed from the microquasar GRO J1655–40 during its 1996 outburst and again almost nine years later during its 2005 outburst. This strongly suggests that the frequencies cannot depend on quantities such as magnetic field, density, temperature, or accretion rate, as these all vary greatly in time. The only parameters of a black hole accretion system that do not vary over a nine year period are the mass and the spin of the central black hole. Thus, the oscillation frequencies must only depend on these two parameters, and only frequencies connected to orbital motion have the property that they depend only on mass and spin. Thus, the possible frequencies are: the Keplerian frequency, the two epicyclic frequencies (as originally suggested by Kluźniak and Abramowicz [11]), the LenseThirring frequency (as originally suggested by Stella and Vietri [294]), and their combinations (e.g., [145, 143]).
Frequency ratio of the “twin peak” QPOs in all four microquasars where they have been detected.
Microquasar  Frequency ratio 

GRO J165540  300/450=0.66 
XTE 1550564  184/276=0.66 
H 1743322  166/240=0.69 
GRS 1915+105  113/168=0.67 
It was realized [150] that the behavior of the observed QPO frequencies and amplitudes in both neutron star binaries and microquasars is typical for a certain type of nonlinear resonance. Indeed, it may be observed when a properly tuned spring is attached to a pendulum with a properly chosen length. Such a system oscillates in two “modes”: the pendulum mode and the spring mode.^{15} Mostly due to efforts of Rebusco and Horák [256, 130, 131, 132], a mathematical resonance model was developed to describe an arbitrary system oscillating in two modes near a 2/3 nonlinear resonance. The model’s predictions for the frequency and amplitude behaviors are strikingly similar to the ones observed in Xray binary QPOs, suggesting that they may indeed be explained as a nonlinear resonance of two modes of oscillation. The model does not, however, explain what these modes are, how they are excited, nor what energy reservoir they tap. Only when these questions are answered satisfactorily could one say that the QPO puzzle is solved.^{16}
As a final note, a crucial discovery by Barret and collaborators [32, 33, 34] concerning the behavior of the quality factor of twin peak QPOs proves that they are disk oscillations and cannot be explained by kinematic (Doppler) effects due to the presence “hot spots” on the accretion disk surface. These effects are, however, important in modulating the QPO’s signal [55]. It is not clear, though, whether the oscillations are explained by the discoseismic modes discussed in Section 9.2 (see, e.g., [313] for references).
12.5 The case of Sgr A*
As already mentioned in Section 2.1.1, there is a very good chance that the first direct evidence for a black hole event horizon will come from Sgr A*, the compact, supermassive object at the center of the Milky Way. There have already been a number of strong, indirect arguments in favor of the black hole nature of Sgr A* [54, 53], but no direct evidence yet. Sgr A* is also of interest because it represents a unique case of black hole accretion, having by far the lowest (scaled) mass accretion rate and radiative efficiency of any known source. We anticipate greatly expanding this section in the near future as new results become available.
13 Concluding Remarks
Since the early 1970s, the study of black hole accretion disks has yielded to remarkable successes. And yet, as in most fields of research, each step forward has been met by new questions. In this article, we have tried to give a tour of some of the successes, such as the many disk models (thick, thin, slim, ADAF, …) that have given a firm foundation on which to work, the many studies of disk instabilities and oscillations that help us to understand the ways in which real disks can deviate from the simplistic models, and the numerical simulations that come as close as possible to an experimental testbed for black hole accretion. We have also tried to indicate what we believe are some of the most pressing challenges of the day, including matching our theoretical knowledge to actual observed phenomena such as black hole spectral states, quasiperiodic oscillations, and relativistic jets. There are also observational challenges to find direct evidence of black hole event horizons and definitively constrain black hole spins. We hope, as we continue to update this Living Review, to be able to report on future discoveries in these areas, just as we expect to report new puzzles we have yet to encounter.
Clearly our tour has been incomplete. For instance, despite their prominent role in nature, e.g., as an AGN feedback mechanism, outflows are not accounted for in any of the four main accretion models we presented, as the models all assume that the accretion rate is constant with radius. In reality, outflows may be triggered by any of three mechanisms: thermal, radiative, or centrifugal. Thermal winds are expected to result from heating of the outer regions of an accretion disk by its hot inner region. Radiative winds are driven by radiative flux acting on line opacities. Centrifugal acceleration of particles can take place along magnetic field lines which are sufficiently inclined to the disk plane.
The ADIOS (adiabatic inflowoutflow solution) [47] is a generalization of the ADAF solution that actually has much of the mass, energy, and angular momentum of the accretion “disk” carried away in the form of winds, rather than being advected into the black hole as in a normal ADAF. However, the argument behind the ADIOS model is flawed [14]. Blandford and Begelman [47] claim that black hole accretion flows with small radiative efficiency must necessarily experience strong outflows because the matter all has a positive Bernoulli constant. Yet a positive Bernoulli constant is only a necessary, but not sufficient, condition for outflows. For example, the classical Bondi accretion solution has \({\mathcal B} > 0\) everywhere and yet experiences no outflows. Furthermore, low efficiency accretion flows do not really have a positive Bernoulli constant everywhere, as boundary conditions (ignored in [47]) will impose some regions of negative Bernoulli constant. A more recent look at the ADIOS solution is presented in [37].
There are several other analytic and semianalytic models of accretion disks. Some are closely related to the models we have already discussed. For example, the CDAF (convectiondominated accretion flow) [217, 221] is another variant on the ADAF, in which longwavelength convective instabilities transport angular momentum inward and energy outward. Other models relax some of the standard assumptions about accretion disks. For example, Bardeen and Petterson [30] and others [167, 273, 177] relaxed the assumption that disks are axisymmetric by considering tilted accretion disks, acted on by the LenseThirring precession of the central (rotating) black hole. Then there is the exact solution for stationary, axisymmetric noncircular, accretion flows found by Kluźniak and Kita [151].
On the numerical side, too, there are many interesting accretion configurations that have been identified, but are not included in this review. Some examples include quasispherical (low angular momentum) accretion flows [250, 249] and convectiondominated disks [138].
For those who wish for more details or a different perspective, we can recommend several excellent text books and review articles devoted, partially or fully, to black hole accretion disks. We recall here some of the most often quoted. The oldest, but still very useful and informative, is the classic review by Pringle [248]. The most authoritative text book on accretion is Accretion Power in Astrophysics by Frank, King and Raine [103]. Two monographs devoted to black hole accretion disks are: BlackHole Accretion Disks by Kato, Fukue and Mineshige [144], and Theory of Black Hole Accretion Disks by Abramowicz, Björnsson, and Pringle [3]. Lasota [170] also wrote an excellent nontechnical Scientific American article on black hole accretion in microquasars. Finally, there is a nice series of lecture notes by Ogilvie available on the web [230].
Footnotes
 1.
 2.
M_{⊙} = 1.99 × 10^{33} [g] denotes the mass of the Sun, used in astrophysics as a mass unit.
 3.
Radiant power in astrophysics is traditionally called “luminosity.” At the Eddington luminosity, L_{Edd} ≡ 1.2 × 10^{38} M/M_{⊙} [erg/s], radiation force balances the gravity of the central object (with mass M). In the case of stars, “superEddington” luminosities, L > L_{Edd}, are not possible, as this would mean radiation pressure would blow the star apart.
 4.
 5.
Strictly speaking, this statement is only true for nearly spherical gravity sources. Higher order (octopole) moments allow for the formation of an ISCO, even in Newtonian theory [15].
 6.
Astrophysical black holes do not themselves radiate. The temperature associated with Hawking radiation is T_{H} = (ħc^{3})/(8πGMk_{B}). For a stellarmass black hole T_{H} ∼ 10^{−8} [° K]. Thus, Hawking radiation is completely suppressed by the thermal bath of the 3 [° K] cosmic background radiation. For supermassive black holes, the Hawking temperature is at least five orders of magnitude smaller still.
 7.
The rate Ṁ_{*} at which matter accretes is therefore regulated by internal torques and radiative processes. Only if Ṁ_{*} = Ṁ_{0} = const everywhere, with Ṁ_{0} being the outside mass supply, can the accretion process be stationary. Since the internally determined Ṁ_{*} may change due to instabilities, limit cycles, etc., an occurrence of a really longterm steady accretion flow should be considered a finetuned eigenstate. Note, too, that in many astrophysical situations Ṁ_{0} is also genuinely variable.
 8.
This makes black hole accretion the most efficient energy generation process in the universe, short of matterantimatter annihilation.
 9.
Note that there is an analogy with stars, where there are three timescales — dynamical, thermal, and nuclear — which for most of the life of a star obey t_{dyn} ≪ t_{th} ≪ t_{nuc}.
 10.
The slow rotation cases are sometimes referred to as “Bondi flows” in honor of Bondi’s pioneering works on sphericallysymmetric (nonrotating) accretion [50].
 11.
“Well, in our country,” said Alice, still panting a little, “you’d generally get to somewhere else if you run very fast for a long time, as we’ve been doing.” “A slow sort of country!” said the Queen. “Now, here, you see, it takes all the running you can do, to keep in the same place.”
 12.
For a congruence of observers (or particles or photons) with four velocity U^{μ}, the kinematic invariants fully describe their relative motion. Consider those that, in a particular moment s_{0}, occupy the surface of an infinitesimally small sphere. Now, consider the deformation of that surface at a later moment s_{0} + ds. The volume change dV/ds = Θ is called expansion. The shear tensor σ_{μν} measures the ellipsoidal distortion of this sphere, and the vorticity tensor ω_{μν} describes its rotation (i.e., three independent rotations around three perpendicular axes). Expansion, shear, and vorticity are determined by the tensor X_{μν} = Δ_{μ}U_{ν} in the following way: \(\Theta = (1/3)X_\mu ^\mu\), σ_{μν} = h^{αμ} h^{βν} (1/2)(X_{αβ} + X_{βα}) − Θ h_{μν}, and ω_{μν} = h^{αμ} h^{βν} ((1/2)(X_{αβ} − X_{βα}). Here h^{αμ} = δ^{αμ} + U^{α} U_{μ} is the projection tensor. The acceleration a_{μ} is also considered a kinematic invariant.
 13.
For the benefit of readers who are unfamiliar with the phenomenology of relativistic jets we mention that many jets demonstrate incredibly consistent collimation, even along hundreds or thousands of kiloparsecs (e.g. [114]). From this we can infer that such jets have maintained their orientation and outflow for millions of years.
 14.
Radio loud AGN are ∼ 10^{3}–10^{4} times brighter in radio than radio quiet AGN of comparable optical luminosities [289].
 15.
It is quite remarkable that the mathematical theory describing resonances makes some very general predictions about the behavior of frequencies and amplitudes, even if the specific physical properties of the oscillating objects are not known. To some extent it is possible to accurately describe how things oscillate without even knowing what is oscillating.
 16.
The mathematical foundations of the resonance model are described and discussed in a special issue of the Astronomische Nachrichten [2].
Notes
Acknowledgements
MAA gratefully acknowledges supporting grants from Sweden (VR Dnr 62120063288) and Poland (UMO2011/01/B/ST9/05439). MAA also thanks the College of Charleston for hosting him during a portion of his work on this review. PCF gratefully acknowledges supporting grants from the National Science Foundation under Grant No. NSF PHY1125915, the College of Charleston, and the South Carolina Space Grant Consortium. PCF also enjoyed the hospitality of NORDITA and Göteborg University while working on this review. Computational support was provided under the following NSF programs: Partnerships for Advanced Computational Infrastructure, Distributed Terascale Facility (DTF) and Terascale Extensions: Enhancements to the Extensible Terascale Facility.
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