The accretion disks we encounter around degenerate dwarfs, neutron stars, and the smaller (several solar masses) black hole candidates that are in compact binary systems are thin steady-state Keplerian disks, as are the accretion disks around newly-formed stars. These disks do not have sufficient mass to be self-gravitating; a Keplerian disk spins in a gravitational field that is set by the object at its center. The discussion that follows concentrates on compact objects in close binary systems.
Many of the characteristics of a Keplerian disk are set by the simple Newtonian mechanics of a low-mass object orbiting in a circle around a star. These characteristics are independent of how the gravitational potential energy is converted into thermal energy. The rotation of a Keplerian accretion disk is given from Kepler's law of orbital motion. The gas in the disk orbits in a circle. The time to complete one orbit is proportional to the radius to the 3/2 power. The structure of the disk is determined by the disk temperature, which can be related to the accretion luminosity, which is the maximum power that can be generated from material falling onto the central object.
The free-fall velocities at the surfaces of degenerate dwarfs, neutron stars, and black holes are extremely high. The lowest velocities are associated with degenerate dwarfs, which are stellar remnants held up by electron degeneracy pressure that are about the mass of the Sun the radius of the Earth; the free-fall velocity for these stars is about 2% of the speed of light, which releases 1.5×10-4 of the rest mass energy of material falling onto the star's surface. For a neutron star, which has a mass somewhat larger than the Sun and a radius of order 106 cm, the free-fall velocity is around half the speed of light, and the amount of energy released is over 10% of the rest mass energy. The energy released around a black hole can be greater than this, although the amount will remain below the rest-mass energy of the matter.
The maximum amount of energy liberated through free-fall onto a compact object can be expressed as a temperature. This can be expressed as
where M is the mass of the central source, m is the average mass of the particles in the gas, Rc is the radius of the central source (assumed not to be a black-hole), and G is the gravitational constant. For a degenerate dwarf, the temperature for hydrogen in free-fall is of order 100 keV (109°K), which is in the hard x-ray/soft gamma-ray frequency range, and for a neutron star, it is of order 100 MeV (109°K), which is in the high-energy gamma-ray range. These temperatures are the maximum temperatures possible for the inner edges of accretion disks—radiative cooling prevents an accretion disk from ever realizing them.
While this exercise of deriving free-fall temperatures may appear academic, it actually plays a role in defining the structure of an accretion disk. The thickness of an accretion disk is set by the balancing of pressure within the disk against the gravitational tidal force. As in the case of a star, this structure depends on how the temperature of the gas varies with altitude above the plane of the disk. This in turn depends on the viscosity mechanisms that heat the disk and the radiative processes and the convection that transport the energy to the surface of the disk.
We can easily derive the structure of a disk if we make the temperature independent of altitude above the disk plane and if we only let the gas provide the pressure. This gives us a minimum thickness for the disk, since it ignores an important source of pressure, radiative pressure, that would assist in counteracting the gravitational tidal force. The gas density varies with altitude as
ρ = ρ0 e-G M m z2/2 k T R3
where ρ0 is the mass density at the center of the disk, z is the altitude above the disk plane, R is the disk radius at this point, T is the gas temperature, and k is the Boltzmann constant. The remaining variables are as defined earlier. For a constant-temperature stellar atmosphere, the exponent would have the factor z/R rather than the factor z2/2 R2, so the atmosphere of the accretion disk falls much faster than the atmosphere of a star.
The disk is thin if z/R ≪ 1 when the exponent in the density equation is of order unity. This means that the disk is thin when the characteristic temperature of the disk is much less than the free-fall temperature of the material in the disk. The altitude confining most of the matter is given by
z/r = (kT R/G M m)1/2
The disk is therefore only thin if the temperature in the disk is much smaller that the magnitude gravitational potential energy. In other words, the disk is thin when the temperature of the gas is much less that the free-fall temperature.
One thing that changes this picture is the effects of radiation pressure. Light carries momentum, so when it strikes an object, it exerts a pressure on that object. Normally this pressure is insignificant, but in bright astronomical sources, this pressure can blow away the outer layers of a star. The Eddington luminosity is a measure of when this effect is important. It is derived by calculating when the radiative pressure exerted on a fully ionized hydrogen gas equals the force of gravity. The Eddington limit, which is independent of the radius of the star, is given by
LE = 1.25×1038(M/Msun) ergs/s,
which is 32,000 (M/Msun) times the luminosity of the Sun. This is the maximum luminosity that can occur before gas is driven off; generally a wind is radiatively driven off of a star at lower luminosities than this because radiation exerts a greater pressure on an unionized gas than on ionized gas.
The binary systems with accretion disks that we see in astronomy are very bright, with luminosities that can hover around the Eddington limit. This means that the structure of an accretion disk at times is set not by the gas pressure, but by the radiative pressure. In some cases, the disk is thick, with a thickness of order the radius, and with a wind driven off of its surface by the radiation.
For an accretion disk, the comparison is between the radiative force and the tidal force on a fully-ionized gas. Radiative pressure becomes important when the accretion luminosity satisfies the limit
L/LE > z/Rc,
where L is the accretion luminosity and Rc is the radius of the central object. This limit shows that an accretion disk can radiate at well below the Eddington limit and still have an impact on the structure of the accretion disk. If this inequality is satisfied, the radiation will puff the disk up until the ratio of the thickness to the radius equals the ratio of the luminosity to the Eddington luminosity.
The radiative pressure can play an important role in the structure of an accretion disk around a neutron star or black hole candidate, particularly if the temperature in the disk is very low, so that z/R is very small. The role of radiative pressure is less important in an accretion disk surrounding a degenerate dwarf, which generally have accretion luminosities of 1034 ergs s-1 or less.