Material falling towards the compact object will carry some angular momentum from the companion star. Conservation of angular momentum means that this material will collapse into a disc of material around the black hole, known as an accretion disc. The form taken by the disc, and so the accretion flow system observed, depends on whether the material collapsed into an optically thick accretion flow, or whether it remains optically thin (it is possible to observe a combination of these flows, as seen in different accretion states).

Optically Thick Accretion

The standard view of this disc is that published by Shakura & Sunyaev in 1973. The authors general description of accretion portrays a system in which matter is transferred from the companion star, by some form of matter transfer, carrying considerable angular momentum. This would prevent free fall onto the compact object, but at some distance centrifugal forces would be comparable to those of gravitational attraction. Here, the material would begin to form a disc around the compact object, in roughly circular orbits. In order for this matter to accrete toward the black hole or neutron star (or even in some cases a white dwarf), the angular momentum must be transported outwards by some method. The authors make suggestions for possible methods of angular momentum transport that include friction, turbulent motion caused by magnetic fields and viscous stresses, although the actual method was not determined. In fact it is only in the last decade that the origin of the viscous stresses has been identified as the magnetic rotational instability.

The outward movement of angular momentum allows for the release of gravitational energy as material spirals inwards. Some of the energy released is used to increase the kinetic energy of rotation, whilst the rest is radiated away. This radiation is emitted from a disc that is optically thick and geometrically thin, and so emits a blackbody. The emission spectrum from such a source is thermal and can be summarised as follows, by making the following assumptions: energy radiated locally is emitted locally and the accretion rate is constant with radius. The resulting radiation at a given radius (r = R / RG where RG is the gravitational radius, GM / c2) is given by a (quasi) blackbody spectrum of temperature T(r) is proportional to r-3/4. As material moves in through the accretion disc, more luminosity is generated over a smaller region that results in emission at a higher temperature. This can be seen in Figure shown below, where we show this type of a spectrum, generally known as a multicoloured disc blackbody spectrum.

The amount of gravitational energy that can be tapped in this way using Virial theorem is                      . This shows that the emission from the system is also dependent on the mass if the compact object (M) and the rate at which material is accreted through the disc (   ). This tells us that with increased mass accretion rates, come increased luminosities. However, there is a point at which the outward pressure of the accretion luminosity  (radiation pressure) is greater than the inward force of gravity, that is known as the Eddington limit. The resulting limit is shown to be:

where G is the gravitational constant, M is the mass of the central object, mp is the mass of a proton, c is the speed of light and        is the Thomson cross-section for the electron.

In the optically thick regime discussed above, we find that all the energy emitted from the system has been thermalised as a result of numerous collisions within the disc. If the accretion rate is much lower, then the density of the accretion flow will also be lower, and at the lowest rates may become optically thin to electron-proton collisions. This is an important point to consider as it is the protons in this flow that carry the majority of the mass, and as such tap the majority of the available gravitation energy. However, it is the electrons that are more efficient at radiating this energy. In the optically thick regime, collisions between these two particles will result in thermal equilibrium and the emission of a blackbody spectrum. In this case though, we find that incomplete thermalisation leads to a two temperature plasma in which the protons gain most of the gravitational energy and transfer only a little of this to the electrons via Comptonisation, Bremsstrahlung  and synchrotron radiation, whilst the rest is advected in through the accretion disc and lost beyond the event horizon of the black hole. This is known as an advection dominated accretion flow (ADAF).

Optically Thin Accretion

Accretion Instabilities

Variations in the mass accretion rate can change the amount of energy emitted from these systems, but more dramatic changes occur due to instabilities in the accretion disc. This occurs in two major forms; the hydrogen ionisation and radiation pressure instabilities. The first of these occurs at fairly low luminosities, and is linked to the long term outbursting behaviour of X-ray binary systems, whilst the second should occur at higher luminosities.

The hydrogen ionisation instability can occur when the accretion disc surrounding the compact object is quite large, meaning that material towards the outer edge of the disc is cool. At low mass accretion rates the hydrogen in a cool region of the disc can be mostly neutral. Fluctuations in the accretion disc can increase the temperature in a region to a point at which hydrogen ionises. The energy in that particular region can no longer escape, which causes further heating in the disc, and then causes more hydrogen to ionise so that more energy is trapped. This runaway effect only stops when the disc becomes mostly ionised. In this way, the whole disc becomes unstable causing an outburst of energy to occur. This thermal instability causes a viscous instability, such that an increase in temperature will also increase the mass accretion rate through each annulus. As material moves in through the disc, this causes a drop in pressure, which results in a drop in temperature, allowing hydrogen to recombine. This once again triggers an instability causing a run-away cooling of the disc. This causes the mass accretion rate to be reduced and the cycle to start again. This cyclical behaviour, resulting from the hydrogen instability, has been linked with hysteresis.

Instabilities can also occur at higher mass accretion rates/luminosities. At these higher rates we find that the standard Shakura-Sunyaev disc becomes unstable at small radii due to the rapid increase in the heating of the disc as radiation pressure becomes dominant. As a result a small increase in temperature will lead to a large increase in pressure, which will in turn lead to a further increase in temperature creating another runaway effect. It is at this stage that the Shakura-Sunyaev equations become unstable and we must consider other options (possibly super-Eddington accretion?).