Accretion discs/Basic physics of accretion discs

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    Basic physics of accretion discs

    According to a semi-analytic understanding 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. The only information that we have about accretion disc physics comes from this radiation, when it reaches radio, optical and X-ray telescopes, allowing astronomers to analyse its electromagnetic spectrum and its time variability. Neither the observed spectra, nor the observed variability, agree satisfactory with those predicted by the present-day accretion disc theory. There is an impressive qualitative and a good quantitative agreement, but several important details fit poorly.

    Keplerian angular momentum, Bondi accretion and accretion discs

    In accretion discs the angular momentum of matter is high and dynamically important in contrast to the quasi-spherical (Bondi) accretion, where the angular momentum is everywhere smaller than the Keplerian one and dynamically unimportant. Some authors take this difference as a defining condition: {\it 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.

    Figure 1: The "Bondi-like" and "disc-like" accretion flows

    "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 accretion rate: sub- and super-Eddington accretion

    Figure 2: Radio maps of SS433, a source containing a super-Eddington accretion disc. SS433 may be a Galactic prototype of the ultraluminous X-ray sources (ULXs) found in other galaxies.

    The accretion rate is defined as the instantaneous mass flux through a spherical surface \(r =\,\,\)const inside the disc. In non-stationary accretion discs accretion rate depends on both time and location, but in stationary disc models with no substantial outflows (no strong winds) it is

    \[{\dot M}(r, t) = const.\]

    Accretion discs may be divided into two classes, depending on whether accretion rate is much smaller than, or comparable to the characteristic Eddington accretion rate, that depends only on the mass of the central accreting object \(M\ ,\)

    \[ {\dot M}_{Edd} = {L_{Edd}/c^2} =1.5 \times 10^{17}\,({M /M_0})\,[{\rm g}/{\rm sec}]. \]

    Here \(M_0 = 2 \times 10^{33}\,\)[g] denotes the mass of the Sun, and \(L_{Edd}\) is the Eddington luminosity (radiation power), familiar from the theory of stellar equilibria: at the surface of a star shining at the Eddington rate, the radiation pressure force balances the gravity force.

    Figure on the left shows radio maps of SS433, a well-known Galactic object with a super-Eddington accretion disc.

    A rather common belief that a black hole cannot accrete at a rate higher than the Eddington one is wrong. In particular, the Eddington rate is not a limit for the mass growth rate of a black hole due to accretion, \(dM/dt\ .\) It could be that \(dM/dt \gg {\dot M}_{Edd}\ .\) This is relevant for modeling the cosmological evolution of black holes.

    Viscous stresses, turbulence and MRI instability

    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).

    There is an acute 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.

    Figure 3: 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

    Energy sources and efficiency of accretion

    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.

    Radiative cooling, radiative transfer

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