Accretion disc
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An accretion disc (or accretion disk) is a structure formed by material falling into a gravitational source. Conservation of angular momentum requires that, as a large cloud of material collapses inward, any small rotation it may have will increase. Centripetal force causes the rotating cloud to collapse into a disc, and tidal effects will tend to align this disc's rotation with the rotation of the gravitational source in the middle. Viscosity within the disc generates heat and saps orbital momentum, causing material in the disc to spiral inward until it impacts in an accretion shock on the central body if the body is a star, or slips toward the event horizon if the central body is a black hole.
Accretion discs are a ubiquitous phenomenon in astrophysics; active galactic nuclei, protoplanetary discs, and gamma ray bursts are only a few phenomena in which they are thought to occur. These discs very often give rise to jets coming out of the axis of rotation of the disc. The mechanism that produces these jets is not understood.
The most spectacular accretion discs found in nature are those of active galactic nuclei and 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 huge luminosity of quasars is believed to be a result of friction caused by gas and dust falling into the accretion discs of supermassive black holes, which can convert about half of the mass of an object into energy as compared to a few percent for nuclear fusion processes.
Often, in binary systems with one black hole, observations show matter being pulled from the visible star when it exceeds its roche lobe and falling into the black hole's accretion disc. The largest and most voracious black holes known are those which form the cores of quasars, whose accretion discs emit more radiation than entire galaxies of stars.
Protoplanetary discs are referred to as accretion disks when viewed as material falling into the central protostar.
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<math>\alpha</math>-Disc Model
The viscosity in an accretion disc cannot be ordinary gas viscosity, as this is too small by many orders of magnitude. Shakura and Sunyaev (1973) constructed a simple prescription which parametrised the ignorance of exactly what was causing the viscosity <math>\nu</math> into a parameter, <math>\alpha</math> so
- <math> \nu=\alpha c_{\rm s}H</math>
where <math>c_{\rm s}</math> is the sound speed, and <math>H</math> is the disc thickness. This assumption can be derived by assuming that the accretion disc is highly turbulent, noting that the size of the largest turbulent cells is of the order of the disk height, and observing that the turbulent velocities must be less than the sound speed.
By using the equation of hydrostatic equilibrium, combined with conservation of angular momentum and noting that the disc is thin, the equations of disk structure may be solved in terms of the <math>\alpha</math> parameter. Many of the observables depend only weakly on <math>\alpha</math>, so this theory is predictive even though it has a free parameter.
Using Kramers' law for the opacity it is found that
- <math>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}</math>
- <math>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}</math>
- <math>\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}</math>
where <math>T_c</math> and <math>\rho</math> are the mid-plane temperature and density respectively. <math>\dot{M}_{16}</math> is the accretion rate, in units of <math>10^{16}{\rm g\ s}^{-1}</math>, <math>m_1</math> is the mass of the central accreting object in units of a solar mass, <math> M_\bigodot</math>, <math>R_{10}</math> is the radius of a point in the disc, in units of <math>10^{10}{\rm cm}</math>, and <math>f=\left[1-\left(\frac{R_\star}{R}\right) \right]^{1/4}</math>, where <math>R_\star</math> is the radius where angular momentum stops being transported inwards.
Magneto-Rotational Instability
The Rayleigh stability criterion,
- <math> \frac{\partial(R^2\omega)}{\partial R}>0</math>
holds everywhere in an accretion disc with a Keplerian angular velocity profile. This means that the disk is stable to hydrodynamic perturbations, and the fluid flow is expected to be laminar. For there to be turbulence, as required for the <math>\alpha</math>-disc model this implies that there is some form of nonlinear hydrodynamic instability, or angular momentum transport is due to some other mechanism.
Balbus and Hawley (1991) proposed a mechanism which involves magnetic fields to generate the turbulence. The key point is that magnetohydrodynamics is subtly different from that of hydrodynamics. The tension forces of a magnetic field have no correspondence in the hydrodynamic regime.
A weak magnetic field acts like a spring. If there is a weak radial magnetic field in an accretion disc, then two gas volume elements will experience a force acting on them. The inner element will have a force acting to slow it down. This causes it to lose energy and angular momentum and move inwards, where paradoxically, due to orbital mechanics it speeds up. The reverse happens to the outer gas element, which moves outwards and slows down. As a consequence, the magnetic field 'spring' is stretched, transferring angular momentum in the process. The radial magnetic field is eventually wound into a toroidal field as the disk rotates differentially.
The Parker instability causes regions with higher than average magnetic flux to be buoyant. Thus the toriodal field will tend to rise out of the disc plane, forming a poloidal component. The radial instability then causes small radial kinks in the poloidal field to grow exponentially, completing the dynamo. This process is called the magneto-rotational instability (MRI). This instability has timescale approximately the same as the disc orbital time scale.
Unfortunately, since the MRI is global in character it makes analytic models of accretion discs difficult to obtain. Instead, people now concentrate on numerical magnetohydrodynamic simulations to discover the workings of these astrophysical objects.
See also
References
- {{cite book
| author=Juhan Frank, Andrew King, Derek Raine | title=Accretion power in astrophysics | publisher=Cambridge University Press | edition=Third Edition | year=2002 | id =ISBN 0521629578}}
- {{cite book
| author=Julian H. Krolik | title=Active Galactic Nuclei | publisher=Princeton University Press | year=1999 | id =ISBN 0691011516}}de:Akkretionsscheibe
es:Disco de acrecimiento fr:Disque d'accrétion it:Disco di accrescimento ja:降着円盤 pl:Dysk akrecyjny ru:Аккреция sk:Akréčny disk sv:Ackretionsskiva