Thin Disks

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) $z∕r ≤ 1$ .

In thin and slim disk models, one often uses a vertically integrated form for many physical quantities. For example, instead of density $ρ(r,z)$ one uses the surface density defined as,

where $z = ±H (r)$ gives the location of the surface of the accretion disk.

5.1 Equations in the Kerr geometry

The general relativistic equations describing the physics of thin disks have been derived independently by several authors [169 , 7 , 13 , 106 , 40 ]. Here we present them in the form used in [268 ]:

Mass conservation (continuity):

where $V$ is the gas radial velocity measured by an observer at fixed $r$ who co-rotates with the fluid, and $Δ$ has the same meaning as in Section 2.
Radial momentum conservation:

where

$&tidle; 2 2 2 2 2 A = (r + a ) − a Δ sin 𝜃$ , $ϕ t Ω = u ∕u$ is the angular velocity with respect to the stationary observer, $&tidle;Ω = Ω − ω$ is the angular velocity with respect to the inertial observer, $Ω± = ±M 1∕2∕(r3∕2 ± aM 1∕2) K$ are the angular frequencies of the co-rotating and counter-rotating Keplerian orbits, and $&tidle; &tidle; 2 1∕2 R = A ∕(r Δ )$ is the radius of gyration.
Angular momentum conservation:

where $ℒ = uϕ$ is the specific angular momentum, $γ$ is the Lorentz factor, $Π = 2HP$ can be considered to be the vertically integrated pressure, $α$ is the standard alpha viscosity (Section 3.2.1), and $ℒin$ 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.
Vertical equilibrium:

with $ℰ = − ut$ being the conserved energy associated with the time symmetry.
Energy conservation:

where $T$ is the temperature in the equatorial plane, $κ$ is the mean (frequency-independent) opacity, $Γ 1 = β∗ + (4 − 3 β∗)(Γ 3 − 1),$

$∗ Γ = 1 + ----(4-−-3β-)(γg-−-1)----, 3 12(1 − β∕ βm)(γg − 1) + β$
$β = Pgas∕(Pgas + Prad + Pmag)$ , $βm = Pgas∕ (Pgas + Pmag )$ , $∗ β = β (4 − βm )∕3βm$ , and $γg$ is the ratio of specific heats of the gas.

5.2 The eigenvalue problem

Through a series of algebraic manipulations one can reduce the thin disk equations to a set of two ordinary differential equations for two dependent variables, e.g., the Mach number $ℳ = − V∕c = − V Σ ∕P S$ and the angular momentum $ℒ = uϕ$ . Their structure reveals an important point here,

d ln ℳ 𝒩 (r,ℳ, ℒ) ------- = --1--------- (94 ) d ln r 𝒟(r,ℳ, ℒ ) dln-ℒ- 𝒩2(r,ℳ,--ℒ)- d ln r = 𝒟 (r,ℳ, ℒ ) . (95 )

In order for this to yield a non-singular physical solution, the numerators $𝒩1$ and $𝒩2$ must vanish at the same radius as the denominator $𝒟$ . The denominator vanishes at the “sonic” radius $rsonic$ where the Mach number is equal to unity, and the equation $𝒟 (r,ℳ, ℒ ) = 0$ determines its location.

The extra regularity conditions at the sonic point $𝒩i(r,ℳ, ℒ) = 0$ are satisfied only for one particular value of the angular momentum at the horizon $ℒin$ , which is the eigenvalue of the problem that should be found. For a given $α$ the location of the sonic point depends on the mass accretion rate. For low mass accretion rates one expects the transonic transition to occur close to the ISCO. Figure 5 shows that this is indeed the case for accretion rates smaller than about $˙ 0.4MEdd$ , independent of $α$ , where we use the authors’ definition of $2 M˙Edd = 16LEdd ∕c$ . At $M˙ = 0.4M˙Edd$ a qualitative change occurs, resembling a “phase transition” from the Shakura–Sunyaev behavior to a very different slim-disk behavior. For higher accretion rates the location of the sonic point significantly departs from the ISCO. For low values of $α$ , the sonic point moves closer to the horizon, down to $∼ 4M$ for $α = 0.001$ . For $α > 0.2$ the sonic point moves outward with increasing accretion rate, reaching values as high as $8M$ for $α = 0.5$ and $M˙ = 100 M˙Edd$ . This effect was first noticed for small accretion rates by Muchotrzeb [212 ] and later investigated for a wide range of accretion rates by Abramowicz [10 ], who explained it in terms of the disk-Bondi dichotomy.

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.

The topology of the sonic point is important, because physically acceptable solutions must be of the saddle or nodal type; the spiral type is forbidden. The topology may be classified by the eigenvalues $λ1,λ2, λ3$ of the Jacobi matrix,

⌊ ⌋ ∂∂𝒟r- ∂∂𝒟ℳ- ∂∂𝒟ℒ- 𝒥 = ⌈ ∂𝒩1-∂𝒩1- ∂𝒩1⌉ . (96 ) ∂∂𝒩r2-∂∂𝒩ℳ2- ∂∂ℒ𝒩2- ∂r ∂ℳ ∂ℒ

Because $det(𝒥 ) = 0$ , only two eigenvalues $λ1,λ2$ are non-zero, and the quadratic characteristic equation that determines them takes the form,

The nodal-type solution is given by $λ1λ2 > 0$ and the saddle type by $λ1λ2 < 0$ , as marked in Figure 5

with the dotted and the solid lines, respectively. For the lowest values of $α$ only the saddle-type solutions exist. For moderate values of $α$ ( $0.1 ≤ α ≤ 0.4$ ) the topological type of the sonic point changes at least once with increasing accretion rate. For the highest $α$ solutions, only nodal-type critical points exist.

5.3 Solutions: Shakura–Sunyaev & Novikov–Thorne

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, $Ω = Ω+k$ . The algebraic angular momentum equation (91 ) only requires that we specify $ℒin$ . The Shakura–Sunyaev model makes the assumption that $ℒin = ℒk(ISCO )$ . 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 right-hand side of the energy equation (93 ) represents advective cooling. This is assumed to vanish in the Shakura–Sunyaev model, though we will see that it plays a critical role in slim disks (Section 6) and ADAFs (Section 7). Because the Shakura–Sunyaev model assumes the rotation is Keplerian, $+ Ω = Ωk$ , meaning $Ω$ is a known function of $r$ , the first term on the left-hand 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 Shakura–Sunyaev model.

In addition to being algebraic, these thin-disk equations are also linear in three distinct radial ranges: outer, middle, and inner. Therefore, as Shakura and Sunyaev realized, the model may be given in terms of explicit algebraic (polynomial) formulae. This was an achievement of remarkable consequences – still today the understanding of accretion disk theory is in its major part based on the Shakura–Sunyaev analytic model. The Shakura–Sunyaev paper [279 ] is one of the most cited in astrophysics today (see Figure 6 ), illustrating how fundamentally important accretion disk theory is in the field.

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

http://adsabs.harvard.edu/abs/1973A&A....24..337S.

The general relativistic version of the Shakura–Sunyaev 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 $˙ 2 m˙ = M c ∕LEdd$ .

Outer region: $P = Pgas$ , $κ = κff$ (free-free opacity)

26 −2 −1 −1 − 3 −1 − 1∕2 F = [7 × 10 erg cm s ](m m˙)r∗ ℬ 𝒞 𝒬, Σ = [4 × 105 g cm −2](α−4∕5m2 ∕10m˙70∕∗10)r∗−3∕4𝒜1 ∕10ℬ −4∕5𝒞1∕2𝒟− 17∕20ℰ− 1∕20𝒬7 ∕10, 2 −1∕10 18∕20 3∕20 9∕8 19∕20 −11∕10 1∕2 −23∕40 −19∕40 3∕20 H = [4 × 10 cm ](α m m˙ )r∗ 𝒜 ℬ 𝒞 𝒟 ℰ 𝒬 , ρ0 = [4 × 102 g cm −3](α−7∕10m −7∕10m˙11 ∕20)r−∗ 15∕8𝒜 −17∕20ℬ3 ∕10𝒟 −11∕40ℰ 17∕40𝒬11∕20, T = [2 × 108 K ](α−1∕5m −1∕5 ˙m3∕10)r−3∕4𝒜− 1∕10ℬ −1∕5𝒟 −3∕20ℰ1∕20𝒬3∕10, ∗ β ∕(1 − β) = [3](α −1∕10m −1∕10 ˙m −7∕20)r3∗∕8𝒜 −11∕20ℬ9∕10𝒟7 ∕40ℰ11∕40𝒬− 7∕20, τ ∕τ = [2 × 10 −3](m˙− 1∕2)r3∕4𝒜− 1∕2ℬ2 ∕5𝒟1 ∕4ℰ1∕4𝒬− 1∕2, (98 ) ff es ∗

where $r∗ = rc2∕GM$ .

Middle region: $P = P gas$ , $κ = κ es$ (electron-scattering opacity)

F = [7 × 1026 erg cm −2 s− 1](m −1m˙)r−3ℬ −1𝒞− 1∕2𝒬, ∗ Σ = [9 × 104 g cm −2](α−4∕5m1 ∕5m˙3 ∕5)r−∗3∕5ℬ−4∕5𝒞1∕2𝒟 −4∕5𝒬3 ∕5, H = [1 × 103 cm ](α−1∕10m9∕10m˙1 ∕5)r21∕20𝒜 ℬ −6∕5𝒞1∕2𝒟 − 3∕5ℰ −1∕2𝒬1 ∕5, 1 −3 −7∕10 −7∕10 ∗2∕5 −33∕20 −1 3∕5 −1∕5 1∕2 2∕5 ρ0 = [4 × 10 g cm ](α m m˙ )r∗ 𝒜 ℬ 𝒟 ℰ 𝒬 , T = [7 × 108 K ](α −1∕5m −1∕5m ˙2 ∕5)r−9∕10ℬ −2∕5𝒟 −1∕5𝒬2 ∕5, −3 −1∕10 −1∕10 −4∕5∗21∕20 − 1 9∕5 2∕5 1∕2 −4∕5 β ∕(1 − β) = [7 × 10 ](α m ˙m )r∗ 𝒜 ℬ 𝒟 ℰ 𝒬 , τff∕τes = [2 × 10 −6](m˙−1)r3∗∕2𝒜 −1ℬ2𝒟1 ∕2ℰ1∕2𝒬− 1, (99 )

Inner region: $P = Prad$ , $κ = κes$

26 −2 −1 − 1 − 3 −1 −1∕2 F = [7 × 10 erg cm s ](m ˙m )r∗ ℬ 𝒞 𝒬, Σ = [5 g cm− 2](α− 1 ˙m −1)r3∗∕2𝒜 −2ℬ3 𝒞1∕2ℰ𝒬 −1, 5 2 −3 1∕2 −1 − 1 H = [1 × 10 cm](m˙)𝒜 ℬ 𝒞 𝒟 ℰ 𝒬, ρ0 = [2 × 10−5 g cm −3](α −1m −1m˙−2)r3∗∕2𝒜− 4ℬ6𝒟 ℰ2𝒬 −2, 7 − 1∕4 −1∕4 −3∕8 − 1∕2 1∕2 1∕4 T = [5 × 10 K](α m )r∗ 𝒜 ℬ ℰ , β∕(1 − β) = [4 × 10−6](α−1∕4m −1∕4 ˙m −2)r21∗∕8𝒜 −5∕2ℬ9∕2𝒟 ℰ5∕4𝒬− 2, (τ τ )1∕2 = [1 × 10−4](α−17∕16m − 1∕16m˙−2)r93∕32𝒜 −25∕8ℬ41∕8𝒞1∕2𝒟1 ∕2ℰ25∕16𝒬 −2. (100 ) ff es ∗

The radial functions $𝒜, ...,𝒬$ that appear in Eqs. (98

), (99

), (100

), are defined in terms of $y = (r∕M )1∕2$ and $a∗ = a∕M$ as [237 ]:

𝒜 = 1 + a2y− 4 + 2a2y −6, ℬ = 1 + a y−3, ∗ ∗ ∗ 𝒞 = 1 − 3y− 2 + 2a∗y −3, 𝒟 = 1 − 2y−2 + a2∗y− 4, 1 + a∗y−3 ℰ = 1 + 4a2∗y −4 − 4a2∗y−6 + 3a4∗y−8 𝒬0 = --------−-2-------−3-1∕2, y(1 − 3y + 2a ∗y )

[ 3 ( y ) 3(y1 − a∗)2 ( y − y1) ] 𝒬 = 𝒬0 y − y0 − --a∗ln -- − --------------------ln ------- [ 2 y0 (y1(y1 − y)2)(y1 − y3) y0 − y1 ( ) ] ----3(y2 −-a∗)2----- -y-−-y2 ----3-(y3-−-a∗)2----- y-−-y3- − 𝒬0 y (y − y )(y − y ) ln y − y − y (y − y )(y − y ) ln y − y 2 2 1 2 3 0 2 3 3 1 3 2 0 3

Here $y0 = (rms∕M )1∕2$ , and $y1$ , $y2$ , and $y3$ are the three roots of $y3 − 3y + 2a∗ = 0$ ; that is

−1 y1 = 2cos[(cos a ∗ − π )∕3], y2 = 2cos[(cos−1a ∗ + π )∕3 ], −1 y3 = − 2cos[(cos a∗)∕3].

For the numerical solutions reproduced in Figure 7

, the opacities were assumed to be $κes = 0.34 cm2 g−1$ and $κff = 6.4 × 1022ρcgsT −7∕2 cm2 g−1 K$ , where $ρcgs$ is the density in $g cm −3$ and $TK$ is the temperature in Kelvin.

The Shakura–Sunyaev and Novikov–Thorne 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 Novikov–Thorne model is inadequate. Figure 7 illustrates a few ways in which the model fails to capture the true physics near the ISCO.

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.

	Abstract
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
	Acknowledgements
	References
	Footnotes
	Figures
	Tables