Accretion discs/2. Basic physics of accretion discs

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Author: Dr. Marek A. Abramowicz, Physics Department, Göteborg University, Sweden and N. Copernicus Astronomical Center, PAN, Warsaw, Poland
Author: Miss Odele Straub, N. Copernicus Astronomical Center PAN, Warsaw, Poland

Dynamical, thermal and viscous processes

The accretion discs physics is governed by a non-linear combination of many processes, including gravity, hydrodynamics, viscosity, radiation and magnetic fields. According to a semi-analytic understanding of these processes developed over the past thirty years, the high angular momentum of matter is gradually removed by viscous stresses and transported outwards. This allows matter in the accretion disc to gradually spiral down towards the gravity center, with its gravitational energy degraded to heat. A fraction of the heat converts into radiation, which partially escapes and cools down the accretion disc. Accretion disc physics is often described in terms of dynamical, thermal and viscous processes that occur at different timescales $t_{dyn}$ , $t_{the}$ , $t_{vis}$ :

Dynamical processes occur with the timescale $t_{dyn}$ (a time in which pressure force adjusts to combined gravitational and centrifugal forces).

Thermal processes occur with the timescale $t_{the}$ (a time in which the entropy redistribution occurs due to dissipative heating and cooling processes (in particular radiation).

Viscous processes occur with the timescale $t_{vis}$ (a time in which angular momentum distribution changes due to torque caused by dissipative stresses).

In most analytic models it is assumed that $t_{dyn} < t_{the} < t_{vis}$ , and in the thin disc analytic models it is assumed that $t_{dyn} \ll t_{the} \ll t_{vis}$ . Although neither these inequalities, nor even the very existence of the separate timescals $t_{dyn}$ , $t_{the}$ , $t_{vis}$ , could be considered as well established facts (indeed some of the supercomputer simulations of accretion seem to challenge that), the present understanding of the accretion disc physics --- both in general and in details --- is based (explicitly or implicitly) on this separation into dynamical, thermal and viscous processes.

O.M. Blaes, Physics of luminous accretion disks around black holes, Les Houches 2002 (excellent introduction to the accretion disc physics; short, but containing all relevant equations)

Gordon Ogilvie, March 2005, lectures at Cambridge University (very pedagogical, highly recommended for students)

Dynamical processes, with the timescale $t_{dyn}$

Dynamical equilibria of accretion flows are governed by the balance of four forces: gravitational $\bar G$ , centrifugal $\bar C$ , pressure $\bar P$ , and magnetic $\bar M$ . In particular, accretion discs, are characterized by a significant contribution of $\bar C$ . Thus, in accretion discs the angular momentum of matter is high (and therefore dynamically important $\bar C \sim \bar G + \bar P + \bar M$ ) in contrast to another important type of accretion flows --- the quasi-spherical "Bondi" accretion, where the angular momentum is everywhere smaller than the Keplerian (and therefore dynamically unimportant, $\bar C \ll \bar G + \bar P + \bar M$ ). Some authors take this difference as a defining condition: in an "accretion disc" there must be an extended region where the matter's angular momentum is not smaller than the Keplerian angular momentum in the same region. This is illustrated in Figure 6.

"Keplerian" refers to the angular momentum of a fictitious free particle placed on a free circular orbit around the accreting object. According to Newton's theory (applicable to weak gravity), the Keplerian angular momentum at a distance $r$ from a spherical object with the mass $M$ equals $(GMR)^{1/2}$ , i.e. it is monotonically increasing, indicating (Rayleigh's) stability of all orbits. According to Einstein's theory, in the strong gravity near a compact object such as a black hole or a neutron star, the Keplerian angular momentum has a minimum at the radius $r = r_{ISCO}$ (see Figure 1). All orbits with $r > r_{ISCO}$ are stable, all orbits with $r < r_{ISCO}$ are unstable, the orbit at $r = r_{ISCO}$ is called the innermost stable circular orbit (ISCO). Even closer to the black hole, for $r < r_{MB}$ , the unstable orbits are also unbound. For a non-rotating black hole

$r_{ISCO} = 6GM/c^2$ and $r_{MB} = 4GM/c^2$ .

The existence of ISCO makes physics of the inner part of accretion discs in strong gravity fundamentally different from physics of accretion in weak gravity.

The "Bondi-like" and "disc-like" accretion flows

Most of the accretion discs types (except proto-planetary and GRB ones) have a negligible self-gravity: the external gravity of the central accreting object dominates. The external gravity is important in shaping several crucial aspects of the internal physics of accretion discs, including their characteristic frequencies (that are connected to several important timescales) and their size (inner and outer radius). The most fundamental gravity's characteristic frequencies are the Keplerian orbital frequency $\Omega_K$ , the radial epicyclic frequency $\omega_R$ , and the vertical epicyclic frequency $\omega_Z$ . They are directly relevant for motion of free particles and also play a role for determining equilibria and stability of rotating fluids. In both Newton's and Einstein's gravity the three frequencies are derived from the effective potential $U_{eff}(R, j)$ , and given by the same formulae,

$\left[ \left(\frac{\partial U_{eff}}{\partial R}\right)_j = 0\right] \rightarrow \left[ \Omega_K^2 = \Omega_K^2(R) \right], ~~~\omega_R^2(R) = \left(\frac{\partial^2 U_{eff}}{\partial R^2}\right)_j, ~~~\omega_Z^2(R) = \left(\frac{\partial^2 U_{eff}}{\partial Z^2}\right)_j,$

$(3.1)$

where $j$ is the specific angular momentum, and derivatives are taken at the symmetry plane $Z = 0$ . Small (epicyclic) oscillations around the circular orbit $R = R_0 = const$ , $Z = 0$ are governed by $\delta{\ddot R} + \omega^2_R\,\delta R = 0$ , $\delta{\ddot Z} + \omega^2_Z\,\delta Z = 0$ , with solutions $\delta{R} \sim \exp( -i\omega_R t)$ , $\delta{Z} \sim \exp( -i\omega_Z t)$ , which are unstable when $\omega^2_R < 0$ or $\omega^2_Z < 0$ . In Newton's gravity $U_{eff} = \Phi + j^2/2R$ . A spherical Newtonian body has the gravitational potential $\Phi = -GM/R$ . Thus, in this case, $\Omega_K^2 = \omega_R^2 = \omega_Z^2 = GM/R^3 > 0$ , i.e. all slightly non-circular orbits are closed and all circular orbits are stable.

In Einstein's gravity, for a spherical body, it is $\Omega_K^2 = \omega_Z^2 > \omega_R^2$ , i.e. non-circular orbits are not closed. In addition, for circular orbits with radii smaller than $6GM/c^2$ , it is $\omega_R^2 < 0$ , which indicates the dynamical instability of these orbits. We describe this and other aspects of the black hole gravity that are most relevant to the accretion disc physics in sub-section 2.1 The black hole gravity of this Scholarpedia article.

Paczynski and Wiita (1980) realized that by a proper guess of an artificial Newtonian gravitational potential, $\Phi = -GM/(R - R_G)$ (with $R_G = 2GM/c^2$ ), one may accurately describe in Newton's theory the relativistic orbital motion, and in particular the existence of ISCO. Paczynski's model for the black hole gravity became a very popular tool in the accretion disc research. It is used by numerous authors in both analytic and numerical studies. Effects of special relativity have been added to Paczynski's model by Abramowicz et al.(1996), and a generalization to a rotating black hole was done by e.g. Karas and Semerak (1999). Newtonian models for rotating black holes are cumbersome and for this reason not widely used, see Abramowicz (2009).

Viscous processes, with the timescale $t_{vis}$

Despite the fact that the crucial role of accretion power in quasars and other astrophysical objects was uncovered already forty years ago by Salpeter and Zeldovich, several important aspects of the very nature of accretion discs are still puzzling. One of them is the origin of the viscous stresses. Balbus and Hawley recognized in 1991 that, most probably, viscosity is provided by turbulence, which originates from the magneto-rotational instability. The instability develops when the matter in the accretion disc rotates non-rigidly in a weak magnetic field. There is still no consensus on how strong the resulting viscous stresses are and how exactly they shape the flow patterns in accretion discs. A great part of our detailed theoretical knowledge on the role of this source of turbulence in accretion disc physics comes from numerical supercomputer simulations. The simulations are rather difficult, time consuming, and hardware demanding. Due to mathematical difficulties, in analytic models one does not directly implement a (small scale) magnetohydrodynamical description, but describes the turbulence (or rather the action of a small scale viscosity of an unspecified nature) by a phenomenological "alpha-viscosity prescription" introduced by Shakura and Sunyaev: the kinematic viscosity coefficient is assumed to have the form

$\nu = \alpha H V$ ,

where $\alpha =\,\,$ const is a free parameter, $H$ is a lenght scale (usually the pressure scale), and $V$ is a characteristic speed (usually the sound speed). There are several versions of this prescription, the most often used assumes that the viscous torque $t_{r\phi} = \alpha P$ is proportional to a pressure (either the total, or the gas pressure). The rate of viscous dissipation of energy is

$q = \nu \Sigma (R \frac{d\Omega}{dR})^2$

Developement of the MIR instability in a Polish doughnut, from numerical simulations by J. Hawley.
Credit: J.A. Font, Numerical Hydrodynamics and Magnetohydrodynamics in General Relativity

There is a disagreement between experts on the viscosity prescription issue: some argue that only the hydromagnetic approach is physically legitimate and the alpha prescription is physically meaningless, while others stress that at present the magnetohydrodynamical simulations have not yet sufficiently maturated to be trusted, and that the models that use the alpha prescription capture more relevant physics. All the detailed comparisons between theoretical predictions and observations performed to date were based on the alpha prescription.

THE REVIEW ARTICLE: S.A. Balbus and J.F. Hawley Instability, turbulence, and enhanced transport in accretion disks Rev. Mod. Phys. 70, 1 - 53 (1998)

Discovery article: S.A. Balbus and J.F. Hawley A powerful local shear instability in weakly magnetized disks. I - Linear analysis. II - Nonlinear evolution Ap.J. 376, 214-233, 1991

Thermal processes, with the timescale $t_{the}$

Gravitational and kinetic energy of matter falling onto the central object is converted by dissipation to heat. Heat is partially radiated out, partially converted to work on the disc expansion and (in the case of BH accretion) partially lost inside the hole. The efficiency of accretion disc $\eta$ is defined by $L = \eta {\dot M}c^2$ , where $L$ is the total luminosity (power) of the disc radiation. Sołtan gave a strong observational argument, confirmed and improved later by other authors, that the efficiency of accretion in quasars is $\eta \approx 0.1$ . Note that the efficiency of thermonuclear reactions inside stars is about two orders of magnitude smaller. The theoretically predicted efficiency of geometrically thin and optically thick Shakura-Sunyaev accretion disc around a black hole is $\eta \ge 0.1$ . Thus, Shakura-Sunyaev accretion discs could explain the enegetics of the "central engines" of quasars, which are the most efficient steady engines known in the Universe. Other types of accretion discs models (like adafs and slim discs) are called the "radiatively inefficient flows" (RIFs) because they are radiatively much less efficient.

The energy budget may also include rotational energy that could be tapped from the central object. In the black hole case, this possibility was described in a seminal paper by Blandford and Znajek. The Blandford-Znajek process

is an electromagnetic analogy of the well-known Penrose process. Some of its aspects are not yet rigorously described in all relevant physical and mathematical details, and some remain controversial. It is believed that the Blandford-Znajek process may power the relativistic jets.

> PREVIOUS SECTION 1. Observational evidence...

> FIRST SECTION Scholarpedia Accretion discs

> NEXT SECTION 2.1. The black hole gravity

Invited by:

Eugene M. Izhikevich

Accretion discs/2. Basic physics of accretion discs

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Contents

Dynamical, thermal and viscous processes

Dynamical processes, with the timescale $t_{dyn}$

Viscous processes, with the timescale $t_{vis}$

Thermal processes, with the timescale $t_{the}$

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