"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

List of Figures

View Image Figure 1:
An artist’s rendition of a generic black hole accretion disk and jet. Inset figures include a time sequence of radio images from the jet in microquasar, GRS 1915+105 [204] and an optical image of the jet in quasar, M87 (Credit: J.A. Biretta et al., Hubble Heritage Team (STScI /AURA), NASA).
View Image Figure 2:
Silhouettes of Sgr A* calculated for four optically thin accretion structures, characterized by very different physical conditions. The display is intentionally reversed in black-and-white and saturated in order to better show the less luminous parts. Although “dirty astrophysics” makes the most prominent differences, effects of the “pure strong gravity” are also seen in the form of “the light circle”, a tiny almost circular feature at the center. Its shape and size depends only on the black hole mass and spin. Image reproduced by permission from [297], copyright by ESO.
View Image Figure 3:
Evidence for the existence of the ISCO from data recorded by the Rossi X-ray Timing Explorer satellite from neutron star binary source 4U 1636–536 [33]. The source shows quasi-periodic oscillations (QPOs) with frequencies in the range 650 Hz < ν < 900 Hz. The sharp drop in the quality factor (bottom panel) seen at ∼ 870 Hz may be attributable to the ISCO [34].
View Image Figure 4:
In equilibrium, the equipressure surfaces should coincide with the surfaces shown by the solid lines in the right panel. Note the Roche lobe, self-crossing at the cusp. The cusp and the center, both located at the equatorial plane 𝜃 = π∕2, are circles on which the pressure gradient vanishes. Thus, the (constant) angular momentum of matter equals the Keplerian angular momentum at these two circles, ℓ = ℓ (r ) = ℓ (r ) 0 K cusp K center, as shown in the upper left panel. In this figure W refers to the effective potential. Image reproduced by permission from [98], copyright by RAS.
View Image Figure 5:
Location of the sonic point as a function of the accretion rate for different values of α, for a non-rotating black hole, a = 0, taking M˙Edd = 16LEddc2. The solid curves are for saddle type solutions while the dotted curves present nodal type regimes. Image reproduced by permission from [9], copyright by ESO.
View Image Figure 6:
The number of citations to the Shakura & Sunyaev paper [279] is still growing exponentially, implying that the field of black hole accretion disk theory still has not reached saturation. Image reproduced from the SAO/NASA Astrophysics Data System, URL (accessed 9 Jan 2013): External Link
View Image Figure 7:
The innermost part of the disk. In the Shakura–Sunyaev and Novikov–Thorne models, the locations of the maximum pressure (a.k.a. the center) rcenter and the cusp rcusp, as well as the sonic radius rsound, are assumed to coincide with the ISCO. Furthermore, the angular momentum is assumed to be strictly Keplerian outside the ISCO and constant inside it. In real flows, r ⁄= r ⁄= r ⁄= ISCO center cusp sound, and angular momentum is super-Keplerian between r cusp and rcenter. Image reproduced by permission from [9], copyright by ESO.
View Image Figure 8:
The advection factor (ratio of advective to radiative cooling) profiles for m˙ = 0.01, 1.0 and 10.0 (here, m˙ = M˙ c2∕16L Edd). Profiles for α = 0.01 and 0.1 are presented with solid black and dashed red lines, respectively. The fraction fadv∕(1 + fadv) of heat generated by viscosity is carried along with the flow. In regions with fadv < 0 the advected heat is released. Image reproduced by permission from [269].
View Image Figure 9:
Profiles of the disk angular momentum (uϕ) for α = 0.01 (left) and α = 0.1 (right panel) for different accretion rates (as a reminder, ˙m = M˙c2∕16LEdd), showing the departures from the Keplerian profile. These plots are for a non-rotating black hole. Image reproduced by permission from [270], copyright by ESO.
View Image Figure 10:
Flux profiles for different mass accretion rates in the case of a non-rotating black hole and two values of α: 0.01 (black solid), 0.1 (red dashed lines). For each value of α there are five lines corresponding to the following mass accretion rates: ˙m = 0.01, 0.1, 1.0, 2.0 and 10.0 (as a reminder, 2 m˙ = M ˙c ∕16LEdd). The black hole mass is 10M ⊙. Image reproduced by permission from [270], copyright by ESO.
View Image Figure 11:
Top panel: Luminosity vs accretion rate for three values of black hole spin (a∗ = a∕M = 0.0, 0.9, 0.999) and two values of α = 0.01 (black) and 0.1 (red line). Bottom panel: efficiency of accretion η = (L ∕LEdd )∕ (M ˙ ∕M˙Edd ) (here M˙Edd = 16LEdd∕c2). Image reproduced by permission from [269].
View Image Figure 12:
Profiles of temperature, optical depth, ratio of scale height to radius, and advection factor (the ratio of advective cooling to turbulent heating) of a hot, one-T ADAF (solid lines). The parameters are M = 10M ⊙, M˙ = 10−5LEdd ∕c2, α = 0.3, and β = Pgas∕(Pgas + Pmag) = 0.9. The outer boundary conditions are Rout = 103RS, T = 109 K, and v∕cs = 0.5. Two-T solutions with the same parameters and δ = 0.5 (dashed lines) and 0.01 (dot-dashed lines) are also shown for comparison, where δ is the fraction of the turbulent viscous energy that directly heats the electrons. Image reproduced by permission from [321], copyright by AAS.
View Image Figure 13:
Poloidal velocity fields (δux, δuy) of the lowest order, non-trivial thick disk modes. Image reproduced by permission from [45].
View Image Figure 14:
Pseudo-color plots of log(T ) with contours of logρ from four different general relativistic MHD simulations. The simulations all begin with the same initial conditions, but have different energy conservation and cooling treatments: The upper-left panel conserves internal energy and ignores cooling; the upper-right panel conserves internal energy and includes cooling; the lower-left panel conserves total energy and ignores cooling; the lower-right panel conserves total energy and includes cooling. The very different end states illustrate the importance of properly capturing thermodynamic processes. Image reproduced by permission from [101], copyright by AAS.
View Image Figure 15:
Time-average mass accretion rate ˙m = M ˙c2∕LEdd as a function of the energy gap Δ Φ for models with a = 0 (circles), a∕M = 0.9 (squares), and a∕M = − 0.9 (triangles). The bars show the variability of ˙m. The lines represent the predicted dependencies γg∕(γg−1) m˙ ∝ (Δ Φ ), where γg = 4∕3 is the adiabatic index. Image reproduced by permission from [137], copyright by RAS.
View Image Figure 16:
Equatorial slice through hydrodynamic tori at saturation of the Papaloizou–Pringle instability showing formation of significant non-axisymmetric (m = 1) overdensity clumps. The density contours are linearly spaced between ρmax and 0.0. This figure represents models A3p (left) and B3r (right) of [69]. Image reproduced by permission, copyright by AAS.
View Image Figure 17:
Specific angular momentum ℓ as a function of radius at t = 0 (thin line) and at t = 10.0 orbits (thick line). The individual plots are labeled by model. In each case the Keplerian distribution for a test particle, ℓKep, is shown as a dashed line. Image reproduced by permission from [72], copyright by AAS.
View Image Figure 18:
Color contours of the ratio of azimuthally averaged magnetic to gas pressure, P ∕P mag gas. The scale is logarithmic and is the same for all panels; the color maps saturate in the axial funnel. The body of the accretion disk is identified with overlaid density contours at 10− 2, 10− 1.5, 10−1, and 10−0.5 of ρmax(t = 0). The individual plots are labeled by model. In all cases, the magnetic pressure is low (Pmag ∕Pgas ≪ 1) in the disk, comparable to gas pressure (Pmag∕Pgas ∼ 1) in the corona above and below the disk, and high (Pmag∕Pgas ≫ 1) in the funnel region. Image reproduced by permission from [72], copyright by AAS.
View Image Figure 19:
On the left, equidensity contours calculated from an analytic Polish doughnut. On the right, equidensity contours from a numerical MHD simulation (model 90h from [99]). Note, though, that the contours on the left are linearly spaced, while those on the right are logarithmically spaced. Thus, the gradients represented on the left are shallower than those on the right. Image reproduced by permission from [253], copyright by ESO.
View Image Figure 20:
Left: Time-averaged rest mass density in the rz plane for four GRMHD simulations with a = 0 and various disk thicknesses. The dashed vertical line marks the ISCO. The disk opening angle, h = H ∕r, and effective Shakura–Sunyaev viscosity, α, are reported in each panel. The top three panels have h ≪ α and the inner edge of the disk is located outside the ISCO. The bottom panel has h ≫ α and the density increases monotonically down to the event horizon. Figure from [241]. Right: Various fluxes as functions of radius for a numerical Novikov–Thorne disk simulation. Top: Mass accretion rate. Second panel: Accreted specific angular momentum. Solid line is simulation data; dashed line gives Novikov–Thorne solution; dotted line is ISCO value. Note that the specific angular momentum does not drop significantly inside the ISCO, unlike for thick disks, such as in Figure 17. Third panel: The “nominal” efficiency, which is the total loss of specific energy from the fluid. Bottom panel: Specific magnetic flux. The near constancy of this quantity inside the ISCO is an indication that magnetic stresses are not significant in this region. Image reproduced by permission from [240].
View Image Figure 21:
On the left is a schematic diagram of the Blandford–Znajek mechanism [49] for an assumed parabolic field distribution. On the right is the result of a numerical simulation from [158] showing a very similar structure. Images reproduced by permission; copyright RAS.
View Image Figure 22:
Top: Distributions of density in the meridional plane at different simulation times, showing a magnetically arrested state (left) and a non-arrested state (right). Bottom: Snapshot of magnetic field lines at the same simulation times. Image reproduced by permission from [135], copyright by AAS.
View Image Figure 23:
Left: Luminosity as a function of accretion rate for neutron star and black hole sources, illustrating that a wider range of luminosities are expected for black holes. Image reproduced by permission from [215], copyright by AAS. Right: Recent data showing that neutron star sources (open symbols) are systematically more luminous than black hole sources (filled symbols) in analogous spectral states. Image reproduced by permission from [171], copyright by Elsevier.