"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

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 μ)MAX ⁄= 0 ν, but otherwise (Tμ )V IS = (Tμ)MAX = (T μ)RAD = 0 ν ν ν throughout this section.

4.1 Polish doughnuts

Paczyński and his collaborators developed, in the late 1970s and early 1980s, a very general method of constructing perfect fluid equilibria of matter orbiting around a Kerr black hole [139, 236, 235, 234]. They assumed for the stress energy tensor and four velocity,

Tμ = (Tμ ) = ρW uμu + P δμ , ν ν FLUμ μ ν μ ν u = A (η + Ωξ ), (80 )
and derived from ∇ Tμ = 0 μ ν that,
ℓ∇-νΩ-- 1- ∇ ν lnA − 1 − ℓΩ = ρ∇ νP . (81 )
In the case of a barytropic fluid P = P(𝜖), the right-hand side of Eq. (81View Equation) is the gradient of a scalar function, and thus the left-hand side must also be the gradient of a scalar, which is possible if and only if
ℓ = ℓ(Ω ). (82 )
This statement is one of several useful integrability conditions, collectively called von Zeipel theorems, found by a number of authors [51, 29, 1, 156Jump To The Next Citation Point].

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 self-consistent, but neither is assuming α (r,𝜃 ) = const.

In Boyer–Lindquist coordinates, the equation for the equipressure surfaces, P = P (r,𝜃) = const, may be written as rP = rP (𝜃 ), with the function rP(𝜃) given by

− drP-= ∂𝜃P- = (1-−-ℓΩ-)∂-𝜃 lnA-+-ℓ∂𝜃Ω. (83 ) d𝜃 ∂rP (1 − ℓΩ )∂r lnA + ℓ∂rΩ
Using the expressions for A = A (r,𝜃), Ω = Ω(r,𝜃), and ℓ = ℓ(r,𝜃) from Section 2 (Eqs. 21View Equation and 22View Equation), one can integrate Eq. (83View Equation) to get the equipressure surfaces. A description of how to do this for both Schwarzschild and Kerr black holes is given in [57]. Figure 4View Image illustrates the simplest (and important) case of ℓ = ℓ(Ω) = ℓ0 = const.
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 [98Jump To The Next Citation Point], copyright by RAS.

Another useful way to think about thick disks is from the relativistic analog of the Newtonian effective potential Φ,

∫ P dP Φ − Φin = − ----, (84 ) 0 ρW
where Φin is the potential at the boundary of the thick disk. For constant angular momentum ℓ, the form of the potential reduces to Φ = ln(− u ) t. Provided ℓ > ℓ ms, the potential Φ(r,𝜃) will have a saddle point Φcusp at r = rcusp, 𝜃 = π∕2. We can define the parameter ΔΦ = Φin − Φcusp as the potential barrier (energy gap) at the inner edge of the disk. If Δ Φ < 0, the disk lies entirely within its Roche lobe, whereas if Δ Φ > 0, matter will spill into the black hole even without any loss of angular momentum.

Before leaving the topic of Polish doughnuts, we should point out that, starting with the work of Hawley, Smarr, and Wilson [125Jump To The Next Citation Point], this simple, analytic solution has been the most commonly used starting condition for numerical studies of black hole accretion.

4.2 Magnetized Tori

Komissarov [156Jump To The Next Citation Point] was able to extend the Polish doughnut solution by adding a purely azimuthal magnetic field to create a magnetized torus. This is possible because a magnetic field of this form only enters the equilibrium solution as an additional pressure-like term. For example, the extended form of Eq. (81View Equation) is

ℓ∇ νΩ ∇ νP ∇ν(Ψ2Pmag ) ∇ ν ln A − -------= -----+ -----2------, (85 ) 1 − ℓΩ ρ Ψ ρ
where Ψ2 = gtϕgtϕ − gttgϕϕ and Eq. (84View Equation) becomes
∫ ∫ &tidle; P -dP- Pmag d &tidle;Pmag Φ − Φin = − ρW − Ψ ρW , (86 ) 0 0
where 2 &tidle;Pmag = Ψ Pmag. Komissarov [156] gives a procedure for solving the case of a barotropic magnetized torus with constant angular momentum (ℓ = const.).

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