Accretion disc

Unsolved problems in physics

Accretion disc jets: Why do the discs surrounding certain objects, such as the nuclei of active galaxies, emit radiation jets along their polar axes? These jets are invoked by astronomers to do everything from getting rid of angular momentum in a forming star to reionizing the universe (in active galactic nuclei), but their origin is still not understood.

An accretion disc is a structure (often a circumstellar disk) formed by diffuse material in orbital motion around a central body. The central body is typically a star. Gravity causes material in the disc to spiral inward towards the central body. Gravitational forces compress the material causing the emission of electromagnetic radiation. The frequency range of that radiation depends on the central object. Accretion discs of young stars and protostars radiate in the infrared; those around neutron stars and black holes in the x-ray part of the spectrum.

Manifestations

Accretion discs are a ubiquitous phenomenon in astrophysics; active galactic nuclei, protoplanetary discs, and gamma ray bursts all involve accretion discs. These discs very often give rise to jets coming from the vicinity of the central object. Jets are an efficient way for the star-disc system to shed angular momentum without losing too much mass.

The most spectacular accretion discs found in nature are those of active galactic nuclei and of quasars, which are believed to be massive black holes at the center of galaxies. As matter spirals into a black hole, the intense gravitational gradient gives rise to intense frictional heating; the accretion disc of a black hole is hot enough to emit X-rays just outside of the event horizon. The large luminosity of quasars is believed to be a result of gas being accreted by supermassive black holes. This process can convert about 10 percent of the mass of an object into energy as compared to around 0.5 percent for nuclear fusion processes.

In close binary systems the more massive primary component evolves faster and has already become a white dwarf, a neutron star, or a black hole, when the less massive companion reaches the giant state and exceeds its Roche lobe. A gas flow then develops from the companion star to the primary. Angular momentum conservation prevents a straight flow from one star to the other and an accretion disc forms instead.

Accretion discs surrounding T Tauri stars or Herbig stars are called protoplanetary discs because they are thought to be the progenitors of planetary systems. The accreted gas in this case comes from the molecular cloud out of which the star has formed rather than a companion star.

Accretion disc physics

File:Accretion disk.jpg

Artist's conception of a binary star system with one black hole and one main sequence star

In the 1940s, models were first derived from basic physical principles. In order to agree with observations, those models had to invoke a yet unknown mechanism for angular momentum redistribution. If matter is to fall inwards it must lose not only gravitational energy but also lose angular momentum. Since the total angular momentum of the disc is conserved, the angular momentum loss of the mass falling into the center has to be compensated by an angular momentum gain of the mass far from the center. In other words, angular momentum should be transported outwards for matter to accrete. According to the Rayleigh stability criterion, \[\frac{\partial(R^2\Omega)}{\partial R}>0,\] where \(\Omega\) represents the angular velocity of a fluid element and \(R\) its distance to the rotation center, an accretion disc is expected to be a laminar flow. This prevents the existence of a hydrodynamic mechanism for angular momentum transport.

On one hand, it was clear that viscous stresses would eventually cause the matter towards the center to heat up and radiate away some of its gravitational energy. On the other hand, viscosity itself was not enough to explain the transport of angular momentum to the exterior parts of the disc. Turbulence-enhanced viscosity was the mechanism thought to be responsible for such angular-momentum redistribution, although the origin of the turbulence itself was not well understood. The conventional phenomenological approach introduces an adjustable parameter \(\alpha\) describing the effective increase of viscosity due to turbulent eddies within the disc. In 1991, with the rediscovery of the magnetorotational instability (MRI), S. A. Balbus and J. F. Hawley established that a weakly magnetized disc accreting around a heavy, compact central object would be highly unstable, providing a direct mechanism for angular-momentum redistribution.

\(\alpha\)-Disc Model

Shakura and Sunyaev (1973) proposed turbulence in the gas as the source of an increased viscosity. Assuming subsonic turbulence and the disc height as an upper limit for the size of the eddies, the disc viscosity can be estimated as \( \nu=\alpha c_{\rm s}H\) where \(c_{\rm s}\) is the sound speed, \(H\) is the disc height, and \(\alpha\) is a free parameter between zero (no accretion) and approximately one. Note that in turbulent motion \( \nu\approx v_{\rm turb} l_{\rm turb} \), where \( v_{\rm turb} \) is the velocity of turbulent cells relative to the mean gas motion, and \( l_{\rm turb} \) is the size of the largest turbulent cells, which is estimated as \(l_{\rm turb} \approx H = c_{\rm s}/\Omega\) and \( v_{\rm turb} \approx c_{\rm s} \), where \(\Omega = (G M)^{1/2} r^{-3/2}\) is the Keplerian orbital angular velocity, \(r\) is the radial distance from the central object of mass \(M\).

By using the equation of hydrostatic equilibrium, combined with conservation of angular momentum and assuming that the disc is thin, the equations of disk structure may be solved in terms of the \(\alpha\) parameter. Many of the observables depend only weakly on \(\alpha\), so this theory is predictive even though it has a free parameter.

Using Kramers' law for the opacity it is found that \[H=1.7\times 10^8\alpha^{-1/10}\dot{M}^{3/20}_{16} m_1^{-3/8} R^{9/8}_{10}f^{3/5} {\rm cm}\]
\[T_c=1.4\times 10^4 \alpha^{-1/5}\dot{M}^{3/10}_{16} m_1^{1/4} R^{-3/4}_{10}f^{6/5}{\rm K}\]
\[\rho=3.1\times 10^{-8}\alpha^{-7/10}\dot{M}^{11/20}_{16} m_1^{5/8} R^{-15/8}_{10}f^{11/5}{\rm g\ cm}^{-3}\] where \(T_c\) and \(\rho\) are the mid-plane temperature and density respectively. \(\dot{M}_{16}\) is the accretion rate, in units of \(10^{16}{\rm g\ s}^{-1}\), \(m_1\) is the mass of the central accreting object in units of a solar mass, \( M_\bigodot\), \(R_{10}\) is the radius of a point in the disc, in units of \(10^{10}{\rm cm}\), and \(f=\left[1-\left(\frac{R_\star}{R}\right)^{1/2} \right]^{1/4}\), where \(R_\star\) is the radius where angular momentum stops being transported inwards.

The Shakura-Sunyaev \(\alpha\)-Disc model is both thermally and viscously unstable. An alternative model, known as the \(\beta\)-disk, which is stable in both sense assumes that the viscosity is proportional to the gas pressure \(\nu \propto \alpha p_{\mathrm{gas}}\). Note that in the standard Shakura-Sunyaev model, viscosity is proportional to the total pressure \( p_{\mathrm{tot}} = p_{\mathrm{rad}} + p_{\mathrm{gas}} = \rho c_{\rm s}^2\) since \(\nu = \alpha c_{\rm s} H = \alpha c_s^2/\Omega = \alpha p_{\mathrm{tot}}/(\rho \Omega)\) .

The Shakura-Sunyaev model assumes that the disk is in local thermal equilibrium, and can radiate its heat efficiently. In this case, the disk radiates away the viscous heat, cools, and becomes geometrically thin. However, this assumption may break down. In the radiatively inefficient case, the disk may "puff up" into a torus or some other three dimensional solution like an Advection Dominated Accretion Flow (ADAF). The ADAF solutions usually require that the accretion rate is smaller than a few percent of the Eddington limit. Another extreme is the case of Saturn's rings, where the disk is so gas poor that its angular momentum transport is dominated by solid body collisions and disk-moon gravitational interactions. The model is in agreement with recent astrophysical measurements using gravitational lensing.

Magnetorotational instability

Main article: Magnetorotational instability

File:Protoplanetary disk HH-30.jpg

HH-30, a Herbig–Haro object surrounded by an accretion disc

Balbus and Hawley (1991) proposed a mechanism which involves magnetic fields to generate the angular momentum transport. A simple system displaying this mechanism is a gas disc in the presence of a weak axial magnetic field. Two radially neighboring fluid elements will behave as two mass points connected by a massless spring, the spring tension playing the role of the magnetic tension. In a Keplerian disc the inner fluid element would be orbiting more rapidly than the outer, causing the spring to stretch. The inner fluid element is then forced by the spring to slow down, reduce correspondingly its angular momentum causing it to move to a lower orbit. The outer fluid element being pulled forward will speed up, increasing its angular momentum and move to a larger radius orbit. The spring tension will increase as the two fluid elements move further apart and the process runs away.

It can be shown that in the presence of such a spring-like tension the Rayleigh stability criterion is replaced by \[ \frac{d\Omega^2}{d \ln R}>0.\] Most astrophysical discs do not meet this criterion and are therefore prone to this magnetorotational instability. The magnetic fields present in astrophysical objects (required for the instability to occur) are believed to be generated via dynamo action.

Analytic models of sub-Eddington accretion discs (thin discs, ADAFs)

Unsolved problems in physics

Accretion disc QPO's: Quasi-Periodic Oscillations happen in many accretion discs, with their periods appearing to scale as the inverse of the mass of the central object. Why do these oscillations exist? Why are there sometimes overtones, and why do these appear at different frequency ratios in different objects?

When the accretion rate is sub-Eddington and the opacity very high, the standard 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 in the past thirty years many 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.

When the accretion rate is sub-Eddington and the opacity very low, an ADAF is formed. This type of accretion disc was predicted in 1977 by Ichimaru. Although Ichimaru's paper was largely ignored, some elements of the ADAF model were present in the influential 1982 ion-tori paper by Rees, Phinney, Begelman and Blandford.

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 in 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.

Analytic models of super-Eddington accretion discs (slim discs, Polish doughnuts)

The theory of highly super-Eddington black hole accretion, M>>M_Edd, was developed in the 1980s by Abramowicz, Jaroszynski, Paczyński, 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, M≥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.

See also

• Accretion
• Circumstellar disk
• Solar Nebula
• Dynamo Theory
• Planetary ring
• Singularity
• Spin-flip

References

• Frank, Juhan; Andrew King; Derek Raine (2002), Accretion power in astrophysics (Third ed.), Cambridge University Press, ISBN 0-521-62957-8
• Krolik, Julian H. (1999), Active Galactic Nuclei, Princeton University Press, ISBN 0-691-01151-6

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External links

16x16px Media related to Accretion disks at Wikimedia Commons

• Professor John F. Hawley homepage
• Nonradiative Black Hole Accretion
• Accretion Discs on Scholarpedia
• Magnetic fields snare black holes' food – New Scientistar:قرص مزود