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

Accretion discs are flattened astronomical objects made of rapidly rotating gas which slowly spirals onto a central gravitating body.

The accretion discs physics is governed by a non-linear combination of many processes, including gravity, hydrodynamics, viscosity, radiation and magnetic fields. The gravitational energy of infalling matter extracted in accretion discs powers stellar binaries, active galactic nuclei, proto-planetary discs and some gamma-ray bursts. The black hole accretion in quasars is the most powerful and the most efficient engine known in the whole Universe.

Contents

1 Accretion discs in the Universe
2 Basic physics of accretion discs
3 Analytic models of accretion discs
4 Numerical simulations
5 The present status of the accretion disc research
6 References
7 See Also

Accretion discs in the Universe

Young stellar objects (YSOs)

During star formation, the central part of a dense molecular cloud collapses to a proto-star with a gaseous envelope that finally settles to a rotating proto-planetary accretion disc. Sedimentation and self-gravity in such discs trigger the formation of planets and planetary systems.

Based on the spectral energy distribution in the infrared and visible light, YSOs are divided into five classes (0-IV), associated with their evolutionary stages. Class 0 refers to collapsing molecular clouds, proto-planetary discs exist in classes I-III, and class IV contains the zero-age main-sequence star.

Cataclysmic variables (CVs)

Interacting close binary systems comprising an accreting white dwarf (WD) and a red giant companion star are known as CVs. Unless the WD is strongly magnetised (e.g., in polars with dipole-like magnetic fields and accretion columns) an accretion disc is formed by mass transfer through the first Lagrang point $L_1$ of the rotating binary system when gas from the outer layers of the secondary overflows to the primary star (Roche lobe overflow). Du the orbital motion the gas possesses angular momentum and forms thus a ring around the compact object and as matter spirals inwards the disc is formed. In case of weakly magnetised WDs (e.g., in intermediate polars) accretion discs may form but they are truncated in the inner regions where the magnetic field pressure dominates the gas pressure.

Some CVs appear as dwarf novae (DNe) i.e., with frequently occurring small outbursts (smaller than classical novae, hence the name). Typically, the outbursts last for about a week are separated by weeks to months of quiescence. The mechanism behind a DN is believed to be an accretion disc instability, that is a sudden increase in the effective disc temperature, primarily caused by viscous heating, tidal torques and mass-transfer fluctuatio DN-type instabilities are called "local limit-cycle instabilities" and are also observed in binary systems containing neutron stars or black holes (soft X-ray transients).

X-ray binaries

Main-sequence secondary stars in orbit with accreting neutron stars or black holes (neutron star binaries or black hole binaries, respectively) are common objects with accretion discs in the Galaxy. Neutron stars are often magnetised, especially young ones, such that their accretion discs are disrupted by magnetic field lines or do not exist at all. Matter in these cases is lead by partial or total column accretion to the compact object. In comparison to CVs their spectral energy distribution is observed up to the X-ray regime, since neutron stars and black holes own a much stronger gravitational potential. X-ray binaries are, depending on the mass of the companion star, roughly divided into two categories, the low-mass X-ray binaries (LMXBs) and the high-mass X-ray binaries (HMXB), where soft X-ray transients and X-ray pulsars are respective sub-classes. Soft X-ray transients with both NSs and BHs show quasi-periodic outbursts. Many, if not all, black hole X-ray binaries exhibit in addition relativistic twin jets that propagate along the rotational axis of the compact object and are called microquasars.

Active galactic nuclei (AGNs)

Active galaxies have relatively small but enormously luminous cores (nuclei) which are in fact so luminous that they usually outshine their hosts. The activity going on is most probably due to the presence of a supermassive black hole and an accretion disc.

Figure 1: AGN unification scheme. Green arrows indicate the AGN type that is seen from a certain viewing angle

Figure 1: AGN unification scheme. Green arrows indicate the AGN type that is seen from a certain viewing angle

Presumably, the disc is surrounded by a hot corona made of fast moving gas (broad line region, BLR) and a shell of slower moving gas clouds (narrow line region, NLR), both pushed towards the rotational axis into a conical shape. Well outside the accretion disc in the equatorial plane a geometrically and optically thick dust torus borders the core (see Figure 1). There is no single way to define the classical qualities of AGNs, but generally they can be divided into two families, the radio loud and the radio quiet AGNs, depending on whether they exhibit jets or not. The viewing angle then gives rise to several AGN types that are distinguished by their emission properties. Among them, the spiral galaxies with broad and narrow emission lines (Seyfert 2 galaxies) or just narrow emission lines (Seyfert 1 galaxies, where the dust torus obscures the BLR); the counterparts in the radio loud family are broad (BLRG viewing angle above ~60°) and narrow line radio galaxies (NLRG viewing angle below ~60°) with jets and radio lobes perpendicular to the accretion disc; and radio loud and radio quiet quasars. The latter are the most luminous beacons in the universe and observed up to highest redshifts, implying their enormous distance and gigantic energy output. Despite their name, quasi-stellar objects are thus anything but stars. However, because quasars shine at such large distances, it is not possible to resolve the bright core. Recent observations detect jets and nebulosity around some of them.

Interestingly, the width of emission lines generally refers not only to the gas velocity but relates also to the age of the source. And it is speculated whether quasars could be the progenitors of radio galaxies and whether they in turn evolve to Seyferts.

Gamma ray bursts (GRBs)

The most energetic explosions in the universe are gamma-ray bursts. Models that describe GRBs as a result of merging compact objects or failed supernova (collapsars) generally predict a similar configuration of the end products: the formation of a solar-mass black hole surrounded by a massive debris disc with a huge accretion rate. According to Narayan et al. (2001), such discs are cooled by neutrino cooling rather than radiation or advective cooling and are hence called neutrino dominated accretion flows (NDAFs).

The dominant term $Q_{\nu}^-$ in the cooling rate $Q^-$ is composed of the electron-positron capture rate by a nucleon, the electron-positron pair annihilation rate the nucleon-nucleon bremsstrahlung rate and the rate of plasmon decay (a plasmon is a quasiparticle resulting from the quantisation of plasma oscillations).

After a NDAF has cooled down it may become a geometrically thin standard disc.

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 1.

Figure 2: The "Bondi-like" and "disc-like" accretion flows

Figure 2: 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 the ISCO makes physics of the inner part of strong and weak gravity accretion discs fundamentally different.

The accretion rate

The accretion rate ${\dot M}(r, t)$ is defined as the instantaneous mass flux through a spherical surface $r =\,\,$ const inside the disc. In non-stationary accretion discs ${\dot M}(r, t)$ depends on both time $t$ and location $r$ , 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 ${\dot M}$ is smaller or larger than the characteristic Eddington accretion rate, ${\dot M}_{Edd}$ 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.

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.

MRI instability, turbulence, and viscous stresses

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 Zełdovich, 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.

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.

Analytic models of accretion discs

Assumptions adopted in analytic models of accretion discs

The present understanding of accretion discs rests in its major part on studies of analytic models that assume stationary and axially symmetric accretion. In particular, this assumption was explicitly made in all serious comparisons between theory and observations by a detailed spectral fitting to observed continuous spectra and line profiles.

Analytic models usually assume in addition that $t_{dyn} \ll Min(t_{the}, t_{vis})$ . Here $t_{dyn}$ is the dynamical timescale in which pressure force adjusts to the balance of gravitational and centrifugal forces, $t_{the}$ is the thermal timescale in which the entropy redistribution occurs due to dissipative heating and cooling processes, and $t_{vis}$ is the viscous timescale in which angular momentum distribution changes due to torque caused by dissipative stresses.

However, it is still unknown whether it is physically legitimate to make all these assumptions and suppose that in some "averaged" sense accretion flows may be (approximately) described in terms of stationary and axially symmetric dynamical equilibria. While observations seem to suggest that many real astrophysical sources experience periods in which this may be quite reasonable, several authors point out that results of recent numerical simulations indicate that the MRI and other instabilities could make the accretion flows genuinely non-steady and non-symmetric, and that the very concept of separate timescales may be questionable in the sense that locally it could be $t_{dyn} \approx t_{the} \approx t_{vis})$ .

A Newtonian black hole model: the Paczynski-Wiita potential

For free particles, both Newton's and Einstein's orbital dynamics are described by the same principle: the orbital Keplerian frequency follows from the first derivative, and the epicyclic frequencies follow from the second derivatives of the effective potential. Thus (as Paczynski realized), by a proper guess of an artificial Newtonian potential, one should be able to accurately describe in Newton's theory the relativistic orbital motion. Paczynski's own guess proved to be the simplest and most practical, $\Phi = -GM/(r - 2r_G)$ with $r_G = GM/c^2$ . It is used by numerous authors.

The inner edge, and the inner boundary condition

Analytic models describe accretion discs down to a certain "inner edge" $r_{in}$ which locates close to the central accreting object. The inner edge is a theoretical concept introduced for convenience, because at $r \approx r_{in}$ the accretion flow changes its character. In the case of the black hole accretion, the change goes from almost circular orbits to almost radial free fall. It is therefore convenient to separately model the two regions: $r > r_{in}$ where matter moves on circular orbits, and $r < r_{in}$ where matter free falls. Of course, in reality the situation is more complicated, as the change of the flow character occurs smoothly in an extended region on both sides of $r_{in}$ .

For black hole accretion, $r_{MB} \le r_{in} \le r_{ISCO}$ . For very efficient Shakura-Sunyaev discs, $r_{in} \approx r_{ISCO}$ , while for RIFs $r_{in} \approx r_{MB}$. For stellar accretion, $r_{in}$ is located near the surface of the star and the flow there is described by a boundary layer model.

Analytic models assume the zero torque inner boundary condition: the viscous torque at the inner edge is vanishingly small. This assumption has been recently challenged by Krolik and others (see Agol, E., and Krolik, J.H. 2000, Ap.J., 528, 161) but defended by Paczynski and collaborators (see Afshordi, N., and Paczynski, B. 2003, Ap.J., 592, 354).

The unresolved issue of the zero torque inner boundary condition is technically rather difficult, mostly because the coupled non-linear differential equations that describe the accretion flow have multiple critical points near $r_{in}$$ . The issue is also fundamentally important because all detailed comparisons between theory and observations performed to date were based on the correctness of the zero torque inner boundary condition. A related fundamental problem, whether a non-zero torque at the event horizon may couple black hole with infalling matter, is also unresolved. While some authors claim that an electrodynamic torque is possible, others argue that "black holes have no hair" and thus, in particular, do not anchor magnetic fields. Therefore, any change in the three properties of a black hole (mass, angular momentum, charge) may occur only by a direct capture of matter by the black hole. Matter may gain some rotational energy of the black hole only if a particle or photon with negative energy would cross the event horizon.

Sub-Eddington models (thin discs, adafs)

When the accretion rate is sub-Eddington and the opacity very high, the standard Shakura-Sunyaev thin accretion disc is formed. It is geometrically thin in the vertical direction (has a disc-like shape), and is made of a relatively cold gas, with a negligible radiation pressure. The gas goes down on very tight spirals, resembling almost circular, almost free (Keplerian) orbits. Thin discs are relatively luminous and they have thermal electromagnetic spectra, i.e. not much different from that of a sum of black bodies. Radiative cooling is very efficient in thin discs. The classic 1974 work by Shakura and Sunyaev on thin accretion discs is one of the most often quoted papers in modern astrophysics. Thin discs have been independently worked out by Lynden-Bell, Pringle and Rees. Pringle contributed many other key results to accretion disc theory, and wrote the classic 1981 review that for many years was the main source of information about accretion discs, and is still very useful today.

Figure 3: Electromagnetic spectrum of SgrA*: a fit to an "adaf" model

Figure 3: Electromagnetic spectrum of SgrA*: a fit to an "adaf" model

When the accretion rate is sub-Eddington and the opacity very low, an adaf is formed. This type of accretion disc was prophesied in 1977 by Ichimaru in a paper that was ignored almost by everybody for twenty years. Some elements of the adaf model were present in the influential 1982 ``ion-tori paper by Rees, Phinney, Begelman and Blandford, however. Adafs started to be intensely studied by many authors only after their rediscovery in the mid 1990 by Narayan and Yi, and independently by Abramowicz, Chen, Kato, Lasota (who coined the name adaf), and Regev. Most important contributions to astrophysical applications of adafs have been made by Narayan and his collaborators. Adafs are cooled by advection (heat captured by moving matter) rather than by radiation. They are very radiatively inefficient, geometrically extended, similar in shape to a sphere (or a "corona") rather than a disc, and very hot (close to the virial temperature). Because of their low efficiency, adafs are much less luminous than the Shakura-Sunyaev thin discs. Adafs emit a power-law, non-thermal radiation, often with a strong Compton component.

Super-Eddington models (slim discs, Polish doughnuts)

The theory of highly super-Eddington black hole accretion, ${\dot M} >> {\dot M}_{Edd}$ , was developed in the 1980s by Abramowicz, Jaroszynski, Paczynski, Sikora and others in terms of "Polish doughnuts" (the name was coined by Rees). Polish doughnuts are low viscosity, optically thick, radiation pressure supported accretion discs cooled by advection. They are radiatively very inefficient. Polish doughnuts resemble in shape a fat torus (a doughnut) with two narrow funnels along the rotation axis. The funnels collimate the radiation into beams with highly super-Eddington luminosities.

Slim discs (name coined by Kolakowska) have only moderately super-Eddington accretion rates, ${\dot M} \approx {\dot M}_{Edd}$ , rather disc-like shapes, and almost thermal spectra. They are cooled by advection, and are radiatively ineffective. They were introduced by Abramowicz, Lasota, Czerny and Szuszkiewicz in 1988.

Analytic models on the accretion rate vs. surface density plane

Other analytic models (cadaf, discs with coronae, magnetized discs)

Predictions of the analytic models

Numerical simulations

The present status of the accretion disc research

References

Original papers

Monographs and textbooks

Abramowicz M.A., Björnsson G., Pringle J.E., 1999, Theory of Black Hole Accretion Discs, Cambridge University Press, Cambridge
Frank J., King A., Raine D.J., 2002, Accretion Power in Astrophysics: Third Edition, Cambridge University Press, Cambridge,
Kato S., Fukue J., Mineshige S., 1998, Black-hole accretion disks, Kyoto University Press, Kyoto

External links

Accretion disk (Wikipedia)

See Also

Invited by:

Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the free peer reviewed encyclopedia

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Categories: Computational Astrophysics | Dynamical Systems | Stellar Astrophysics

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