α-Disk model

Shakura and Sunyaev (1973)^[7] proposed turbulence in the gas as the source of an increased viscosity. Assuming subsonic turbulence and the disk height as an upper limit for the size of the eddies, the disk viscosity can be estimated as ${\displaystyle \nu =\alpha c_{\rm {s}}H}$ where ${\displaystyle c_{\rm {s}}}$ is the sound speed, ${\displaystyle H}$ is the scale height of the disk, and ${\displaystyle \alpha }$ is a free parameter between zero (no accretion) and approximately one. In a turbulent medium ${\displaystyle \nu \approx v_{\rm {turb}}l_{\rm {turb}}}$, where ${\displaystyle v_{\rm {turb}}}$ is the velocity of turbulent cells relative to the mean gas motion, and ${\displaystyle l_{\rm {turb}}}$ is the size of the largest turbulent cells, which is estimated as ${\displaystyle l_{\rm {turb}}\approx H=c_{\rm {s}}/\Omega }$ and ${\displaystyle v_{\rm {turb}}\approx c_{\rm {s}}}$, where ${\displaystyle \Omega =(GM)^{1/2}r^{-3/2}}$ is the Keplerian orbital angular velocity, ${\displaystyle r}$ is the radial distance from the central object of mass ${\displaystyle M}$.^[10] By using the equation of hydrostatic equilibrium, combined with conservation of angular momentum and assuming that the disk is thin, the equations of disk structure may be solved in terms of the ${\displaystyle \alpha }$ parameter. Many of the observables depend only weakly on ${\displaystyle \alpha }$, so this theory is predictive even though it has a free parameter.

Using Kramers' opacity law it is found that

${\displaystyle H=1.7\times 10^{8}\alpha ^{-1/10}{\dot {M}}_{16}^{3/20}m_{1}^{-3/8}R_{10}^{9/8}f^{3/5}{\rm {cm}}}$

${\displaystyle T_{c}=1.4\times 10^{4}\alpha ^{-1/5}{\dot {M}}_{16}^{3/10}m_{1}^{1/4}R_{10}^{-3/4}f^{6/5}{\rm {K}}}$

${\displaystyle \rho =3.1\times 10^{-8}\alpha ^{-7/10}{\dot {M}}_{16}^{11/20}m_{1}^{5/8}R_{10}^{-15/8}f^{11/5}{\rm {g\ cm}}^{-3}}$

where ${\displaystyle T_{c}}$ and ${\displaystyle \rho }$ are the mid-plane temperature and density respectively. ${\displaystyle {\dot {M}}_{16}}$ is the accretion rate, in units of ${\displaystyle 10^{16}{\rm {g\ s}}^{-1}}$, ${\displaystyle m_{1}}$ is the mass of the central accreting object in units of a solar mass, ${\displaystyle M_{\bigodot }}$, ${\displaystyle R_{10}}$ is the radius of a point in the disk, in units of ${\displaystyle 10^{10}{\rm {cm}}}$, and ${\displaystyle f=\left[1-\left({\frac {R_{\star }}{R}}\right)^{1/2}\right]^{1/4}}$, where ${\displaystyle R_{\star }}$ is the radius where angular momentum stops being transported inward.

The Shakura–Sunyaev α-disk model is both thermally and viscously unstable. An alternative model, known as the ${\displaystyle \beta }$-disk, which is stable in both senses assumes that the viscosity is proportional to the gas pressure ${\displaystyle \nu \propto \alpha p_{\mathrm {gas} }}$.^[11][12] In the standard Shakura–Sunyaev model, viscosity is assumed to be proportional to the total pressure ${\displaystyle p_{\mathrm {tot} }=p_{\mathrm {rad} }+p_{\mathrm {gas} }=\rho c_{\rm {s}}^{2}}$ since ${\displaystyle \nu =\alpha c_{\rm {s}}H=\alpha c_{s}^{2}/\Omega =\alpha p_{\mathrm {tot} }/(\rho \Omega )}$.

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.^{[13][14][15][16]}

Magnetorotational instability

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

Balbus and Hawley (1991)^[9] proposed a mechanism which involves magnetic fields to generate the angular momentum transport. A simple system displaying this mechanism is a gas disk 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 disk 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.^[17]

It can be shown that in the presence of such a spring-like tension the Rayleigh stability criterion is replaced by

${\displaystyle {\frac {d\Omega ^{2}}{d\ln R}}>0.}$

Most astrophysical disks 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.^[18]

Magnetic fields and jets

Accretion disks are usually assumed to be threaded by the external magnetic fields present in the interstellar medium. These fields are typically weak (about few micro-Gauss), but they can get anchored to the matter in the disk, because of its high electrical conductivity, and carried inward toward the central star. This process can concentrate the magnetic flux around the centre of the disk giving rise to very strong magnetic fields. Formation of powerful astrophysical jets along the rotation axis of accretion disks requires a large scale poloidal magnetic field in the inner regions of the disk.^[19]

Such magnetic fields may be advected inward from the interstellar medium or generated by a magnetic dynamo within the disk. Magnetic fields strengths at least of order 100 Gauss seem necessary for the magneto-centrifugal mechanism to launch powerful jets. There are problems, however, in carrying external magnetic flux inward toward the central star of the disk.^[20] High electric conductivity dictates that the magnetic field is frozen into the matter which is being accreted onto the central object with a slow velocity. However, the plasma is not a perfect electric conductor, so there is always some degree of dissipation. The magnetic field diffuses away faster than the rate at which it is being carried inward by accretion of matter.^[21] A simple solution is assuming a viscosity much larger than the magnetic diffusivity in the disk. However, numerical simulations and theoretical models show that the viscosity and magnetic diffusivity have almost the same order of magnitude in magneto-rotationally turbulent disks.^[22] Some other factors may possibly affect the advection/diffusion rate: reduced turbulent magnetic diffusion on the surface layers; reduction of the Shakura–Sunyaev viscosity by magnetic fields;^[23] and the generation of large scale fields by small scale MHD turbulence –a large scale dynamo. In fact, a combination of different mechanisms might be responsible for efficiently carrying the external field inward toward the central parts of the disk where the jet is launched. Magnetic buoyancy, turbulent pumping and turbulent diamagnetism exemplify such physical phenomena invoked to explain such efficient concentration of external fields.^[24]