Matter Description: General Principles

3 Matter Description: General Principles

Having provided a detailed description of the key signatures associated with a black hole spacetime, we now move into the mirkier realm of the accretion disk itself. We start from the fundamental conservation laws that govern the behavior of all matter, namely the conservation of rest mass and conservation of energy-momentum, stated mathematically as

Here $ρ$ is the rest mass density, $μ u$ is the four velocity of matter, and $μ Tν$ is the stress energy tensor describing properties of the matter. The conservation equations (43

) are supplemented by numerous “material” equations, like the equation of state, prescriptions of viscosity, opacity, conductivity, etc. Several of them are phenomenological or simple approximations. Nevertheless, we can give a GEN-eral form of $Tνμ$ that is relevant to accretion disk theory as a sum of FLU-id, VIS-cous, MAX-well, and RAD-iation parts, which may be written as,

(Tμ) = (T μ) + (T μ) + (Tμ) + (T μ) , (44 ) ν GEN ν FLU ν VIS ν MAX ν RAD (Tνμ)FLU = (ρu μ)(W uν) + δμνP, (45 ) (Tμ) = ν σ μ, (46 ) ν VIS ∗ ν (T μ)MAX = F μαF αν − 1δμF αβF αβ, (47 ) ν 4 ν μ 4- μ μ μ (Tν )RAD = 3Eu uν + u F ν + uνF . (48 )

Here $W$ = enthalpy, $μ δν$ = Kronecker delta tensor, $P$ = pressure, $ν∗$ = kinematic viscosity, $μ σ ν$ = shear, $μν F$ = Faraday electromagnetic field tensor, $E$ = radiation energy density, and $F μ$ = radiation flux. In the remainder of this section we describe these components one by one, including the most relevant details. Most models of accretion disks are given by steady-state solutions of the conservation equations (43

), with particular choices of the form of the stress-energy tensor $Tμ ν$ , and a corresponding choice of the supplementary material equations. For example, thick accretion disk models (Section 4) often assume $(Tμν )V IS = (Tμν )MAX = (T μν )RAD = 0$ , thin disk models (Section 5) assume $(Tμν )MAX = 0$ , and most current numerical models (Section 11) assume $(T μ)RAD = 0 ν$ .

3.1 The fluid part

The one absolutely essential piece of the stress-energy tensor for describing accretion disks is the fluid part, $(T μ)FLU = (ρu μ)(W uν) + δμP ν ν$ . The fluid density, enthalpy, and pressure, as well as other fluid characteristics, are linked by the first law of thermodynamics, $dU = T dS − P dV$ , which we write in the form,

where $U$ is the internal energy, $T$ is the temperature, $S$ is the entropy, and $𝜖 = ρc2 + Π$ is the total energy density, with $Π$ being the internal energy density, and

The equation of state is often assumed to be that of an ideal gas,

with $ℛ$ being the gas constant and $μ$ the mean molecular weight.

Sometimes we may wish to consider a two temperature fluid, where the temperature $Ti$ and molecular weight $μi$ of the ions are different from those of the electrons ( $Te$ and $μe$ ). For such a case

Two-temperature plasmas are critical in advection-dominated flows (discussed in Section 7). Two-temperature fluids are also important when one considers radiation [281 ], as it is the ions that are generally heated by dissipative processes in the disk, while it is generally the electrons that radiate. Ions and electrons normally exchange energy via Coulomb collisions. As this process is generally not very efficient, the electrons in the inner parts of accretion flows are usually much cooler than the ions (Coulomb collisions are not able to heat the electrons as fast as they radiate or advect into the black hole). However, there have been suggestions that more efficient processes may couple the ions and electrons [245 ], such as plasma waves [38 ] or kinetic instabilities [282 ]. At this point, this remains an open issue in plasma physics, so it is difficult to know how much heating electrons experience.

3.1.1 Perfect fluid

In the case of a perfect fluid, the whole stress-energy tensor (44 ) is given by its fluid part (45 ), and all other parts vanish, i.e., $(T μ)GEN = (Tμ )F LU ν ν$ . In this particular case, one can use $μ ν ν μ μν ∇ μ(Tν η ) = η ∇ μ(Tν ) + T (∇ μην) = 0 + 0 = 0$ , and similarly derived $μ ν ∇μ (T ν ξ ) = 0$ , to prove that

are constants of motion. We can identify $ℬ$ as the Bernoulli function and $𝒥$ as the angular momentum. Their ratio is obviously also a constant of motion,

identical in form with the specific angular momentum (7c), which is a constant of geodesic motion.

3.2 The stress part

In the stress part $(Tμ) = ν σ μ ν VIS ∗ ν$ , the shear tensor $σμ ν$ is a kinematic invariant (cf. Footnote 12). It is defined as

where the symbol $⊥$ denotes projection into the instantaneous 3-space perpendicular to $uμ$ in the sense that $(X μ) = X α(δμ + uμu ) ⊥ α α$ . The other kinematic invariants are vorticity,

and expansion,

In the standard hydrodynamical description (e.g. [168 ]), the viscous stress tensor, $S μν$ , is proportional to the shear tensor,

The rate of heat generation by viscous stress in a volume $V$ is then given by

In addition, the rates of viscous angular momentum and energy transport across a surface $S$ , with a unit normal vector $μ N$ , are

For the case of purely circular motion, where $i i i u = A (η + Ωξ )$ , the kinematic invariants are

where $Ψ2 = g2 − g g tϕ ttϕϕ$ . It is a general property that $(X μ) u = 0 ⊥ μ$ , and so for purely circular motion, one has,

From Eqs. (60

) and (62

) one deduces that for purely circular motion, the rates of energy and angular momentum transport are related as

where $ΩS$ is the angular velocity averaged on the surface $S$ . From this, one sees that as angular momentum is transported outward, additional energy is carried inward by the fluid.

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 magneto-rotational 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.

For these reasons, the Shakura–Sunyaev “alpha viscosity” prescription [279 ] still finds application today. It is an ad hoc assumption based on dimensional arguments. Shakura and Sunyaev realized that if the source of viscosity in accretion disks is turbulence, then the kinematic viscosity coefficient $ν∗$ has the form,

where $l0$ is the correlation length of turbulence and $v0$ is the mean turbulent speed. Assuming that the velocity of turbulent elements cannot exceed the sound speed, $v0 < cS$ , and that their typical size cannot be greater than the disk thickness, $l0 < H$ , one gets

where $0 < α < 1$ is a dimensionless coefficient, assumed by Shakura and Sunyaev to be a constant.

For thin accretion disks (see Section 5) the viscous stress tensor reduces to an internal torque with the following approximate form [see Eqs. (55 ) and (58 )]

However, for thin disks, $r(∂Ω ∕∂r) ≈ − Ω$ and $c ≈ (P ∕ρ)1∕2 ≈ ΩH S$ , so Shakura and Sunyaev argued that the torque must have the form $𝒯rϕ = − αP$ . A critical question that was left unanswered was what pressure $P$ one should consider: $Pgas$ , $Prad$ , or $PTot = Pgas + Prad$ ? This question has now been answered using numerical simulations [128

], so that we now know the appropriate pressure to be $PTot$ . Typical values of $α$ estimated from magnetohydrodynamic simulations are close to 0.02 [122 ], while observations suggest a value closer to 0.1 (see [148 ] and references therein).

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 ].

In most black hole accretion disks, it is reasonable to assume ideal MHD, whereby the conductivity is infinite, and consequently the magnetic diffusivity is zero. Whenever this is true, magnetic field lines are effectively frozen into the fluid. A corollary to this is that parcels of fluid are restricted to moving along field lines, like “beads” on a wire. In ideal MHD, the Faraday tensor obeys the homogeneous Maxwell’s equation

where $∗Fμ ν$ is the dual. If we define a magnetic field 4-vector $bμ ≡ uν∗F μ ν$ , then using $bμu = 0 μ$ one can show that

Using this, it is easy to show that the spatial components of (67

) give the induction equation

while the time component gives the divergence-free constraint

where $Bi = utbi − uibt$ , and $g$ is the 4-metric determinant.

3.3.1 The magneto-rotational instability (MRI)

We mentioned in Section 3.2 that a hydrodynamic treatment of accretion requires an internal viscous stress tensor of the form $𝒯 < 0 rϕ$ . 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 $𝒯rϕ < 0$ , 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 magneto-rotational (or “Balbus–Hawley”) instability [26 , 118 , 27 ]. Although the non-linear 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 (Shakura–Sunyaev) 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 optically thin limit, the radiation emissivity $f$ has the following components: bremsstrahlung $fbr$ , synchrotron $fsynch$ , and their Comptonized parts $fbr,C$ and $fsynch,C$ . In the optically thick limit, one often uses the diffusion approximation with the total optical depth $τ = τ + τ abs es$ coming from the absorption and electron scattering optical depths. In the two limits, the emissivity is then

where $σ$ is the Stefan–Boltzmann constant. In the intermediate case one should solve the transfer equation to get reliable results, as has been done in [288 , 66

]. Often, though, the solution of the grey problem obtained by Hubeny [133 ] can serve reliably:

In sophisticated software packages like BHSPEC, color temperature corrections in the optically thick case (the “hardening factor”) are often applied [66 ].

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

Thermal bremsstrahlung (or free-free emission) is caused by the inelastic scattering of relativistic thermal electrons off (nonrelativistic) ions and other electrons. The emissivity (emission rate per unit volume) is $fbr = fei + fee$ . The ion-electron part is given by [225 ]

where $n e$ is the electron number density, $¯n$ is the ion number density averaged over all species, $−25 2 σT = 6.62 × 10 cm$ is the Thomson cross section, $αf = 1∕137$ is the fine structure constant, $2 𝜃e = kBTe ∕mec$ is the dimensionless electron temperature, and $kB$ is the Boltzmann constant. The electron-electron part is given by [225

]

where $r = e2∕m c2 e e$ is the classical radius of the electron.

3.4.2 Synchrotron

Assuming the accretion environment is threaded by magnetic fields, the hot (relativistic) electrons can also radiate via synchrotron emission. For a relativistic Maxwellian distribution of electrons, the formula is [225 ]

where $e$ is the electric charge, $B$ is the equipartition magnetic field strength, and $xM$ is the solution of the transcendental equation

where the radius $r$ must be in physical units and $K2$ is the modified Bessel function of the second kind. This expression is valid only for $𝜃e > 1$ , but that is sufficient in most applications.

3.4.3 Comptonization

The hot, relativistic electrons can also Compton up-scatter the photons emitted via bremsstrahlung and synchrotron radiation. The formulae for this are [225 ]

{ [ ] } η1xc 3η1 3−(η3+1) − (3𝜃e)−(η3+1) fbr,C = fbr η1 − -3𝜃--− ----------η--+-1----------- (77 ) e 3 fsynch,C = fsynch[η1 − η2 (xc∕𝜃e)η3]. (78 )

Here $η3 η = 1 + η1 + η2(x ∕𝜃e)$ is the Compton energy enhancement factor, and

hν hν x (x − 1 ) x = ---2, xc = ---c2, η1 = --2--1-----, mec mec 1 − x1x2 x = 1 + 4𝜃 + 16𝜃2, η = −-η1, 1 e e 2 3η3 x2 = 1 − exp (− τes), η3 = − 1 − lnx2∕ ln x1, (79 )

where $h$ is Planck’s constant and $νc$ is the critical frequency, below which it is assumed that the emission is completely self-absorbed and above which the emission is assumed to be optically thin.

	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