"Foundations of Black Hole Accretion Disk Theory"
Marek A. Abramowicz and P. Chris Fragile 
1 Introduction
2 Three Destinations in Kerr’s Strong Gravity
2.1 The event horizon
2.2 The ergosphere
2.3 ISCO: the orbit of marginal stability
2.4 The Paczyński–Wiita potential
2.5 Summary: characteristic radii and frequencies
3 Matter Description: General Principles
3.1 The fluid part
3.2 The stress part
3.3 The Maxwell part
3.4 The radiation part
4 Thick Disks, Polish Doughnuts, & Magnetized Tori
4.1 Polish doughnuts
4.2 Magnetized Tori
5 Thin Disks
5.1 Equations in the Kerr geometry
5.2 The eigenvalue problem
5.3 Solutions: Shakura–Sunyaev & Novikov–Thorne
6 Slim Disks
7 Advection-Dominated Accretion Flows (ADAFs)
8 Stability
8.1 Hydrodynamic stability
8.2 Magneto-rotational instability (MRI)
8.3 Thermal and viscous instability
9 Oscillations
9.1 Dynamical oscillations of thick disks
9.2 Diskoseismology: oscillations of thin disks
10 Relativistic Jets
11 Numerical Simulations
11.1 Numerical techniques
11.2 Matter description in simulations
11.3 Polish doughnuts (thick) disks in simulations
11.4 Novikov–Thorne (thin) disks in simulations
11.5 ADAFs in simulations
11.6 Oscillations in simulations
11.7 Jets in simulations
11.8 Highly magnetized accretion in simulations
12 Selected Astrophysical Applications
12.1 Measurements of black-hole mass and spin
12.2 Black hole vs. neutron star accretion disks
12.3 Black-hole accretion disk spectral states
12.4 Quasi-Periodic Oscillations (QPOs)
12.5 The case of Sgr A*
13 Concluding Remarks

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 test-bed 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, quasi-periodic 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 inflow-outflow solution) [47Jump To The Next Citation Point] 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 [47Jump To The Next Citation Point] 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 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 semi-analytic models of accretion disks. Some are closely related to the models we have already discussed. For example, the CDAF (convection-dominated accretion flow) [217, 221] is another variant on the ADAF, in which long-wavelength 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 Lense–Thirring precession of the central (rotating) black hole. Then there is the exact solution for stationary, axisymmetric non-circular, 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 quasi-spherical (low angular momentum) accretion flows [250, 249] and convection-dominated 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: Black-Hole 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 non-technical 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].

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