For a steady-state thin accretion disk, it is easy to place a lower limit on the surface temperature of the disk at each radius. This lower limit is set solely by the rate at which mass flows through the disk. Because the disk is assumed to be in a steady state, the rate at which mass flows towards the central source is constant across the disk. The lower limit on temperature arises because there is an upper limit on how much light a body of a given temperature can radiate. This limit, set by the black body spectrum, is proportional to T4. A body can release less energy than this if it is sufficiently thin, but not more.
In a steady state accretion disk, the gravitational potential energy released as mass flows inward must be counterbalanced by the energy carried away as radiation. For a Keplerian disk, the combined gravitational potential energy and orbital kinetic energy of a particle (electron, hydrogen atom, etc.) orbiting the central object is given by
where G is the gravitational constant, M is the mass of the central object, m is the mass of the particle, and R is the distance from the accretion disk center. As a particle drifts inward, it loses energy. The amount of potential and orbital kinetic energy lost per unit distance traveled inward is given by dE/dR = GMm/2R2. Multiplying this by the thickness of the ring dR and replacing the particle mass with the rate at which mass flows through the disk, dM/dt, and we have the rate at which potential and orbital kinetic energy is convert into thermal energy in a ring of thickness dR. Expressed in terms of the free-fall luminosity, the energy loss rate per unit radius is
In this equation, L is the free-fall luminosity, which desribes the maximum amount of power that can be released from the gas falling onto the central object, R is the accretion disk radius at the point of interest, and Rc is the radius of the central source.
This equation shows that most of the energy of an accretion disk is released at its inner edge. For an accretion disk with inner radius R0 and an outer radius so large as to be considered infinite, the fraction of energy released by the portion of the disk outside of radius R is
From this equation, we see that half of the energy liberated in an accretion disk is release before the matter is at twice the disk's inner radius.
This rate of energy release can give us a lower limit on the temperature of the accretion disk's photosphere if we relate it to the flux of black-body radiation. This temperature limit is
|T||≥||1.4 [ (L/1038ergs s-1) (Rc/106cm) (106 cm/R3)]1/4 keV.|
In this equation, Rc is the radius of the central object. The various parameters in this equation are normalized to characteristic values encountered with an accreting neutron star. The equation shows that a large change in the accretion luminosity has a small effect on the minimum possible surface temperature. The surface temperature, however, drops significantly as one moves farther out in the accretion disk. The minimum surface temperature at the inner edge of the disk is 68% higher than the minimum surface temperature at twice the inner-edge radius.
For a neutron star, we see that accretion at close to the Eddington limit produces a disk with a surface temperature at its inner edge of over 1 keV, which places its radiation in the soft x-ray band. Dropping the luminosity by a factor of 100, which is a very common accretion luminosity in neutron star close binary systems, and we see that the surface temperature drops to 0.5 keV. Since the free-fall temperature is in the 100 keV range, a disk supported by gas pressure would have a thickness that is 10-3 times the radius, assuming that the surface temperature is equal to the temperature at the center of the accretion disk. But at 10-2 of the Eddington luminosity, radiative pressure balances the tidal force at z ∼ 10-2 Rc. This gives a thin disk, but not as thin as gas pressure alone would give. radiative pressure becomes less important, so the gas pressure becomes dominant in the outer regions of the accretion disk.
Degenerate dwarfs, which have about the mass of the Sun confined to a volume of Earth, do not have accretion disks that are bright in the x-ray. Bright cataclysmic variables, which are compact binary systems that contain a degenerate dwarf star, have accretion luminosities of 1034 ergs s-1, or about 10-4 times the Eddington limit. Combined with a radius of about 109cm, and we see that the characteristic temperature is above 5 eV, which places this disk in the ultraviolet. Gas pressure gives a disk thickness of around 10-2 times the inner disk radius, which is larger than the ratio of the accretion luminosity to the Eddinton luminosity. This means that radiation pressure is never important in the accretion disks of these objects.
One complication ignored in all of this is the effects of outside heating of the disk. There are instances when the radiation from the central source or from the companion star heat portions of the accretion disk. This additional energy source will increase the temperature of the disk, which will made the disk thicker and more luminous than it would otherwise be.