There must be a source of material close to the compact object in order for accretion to take place. The amount of gravitational potential energy that can be converted into radiation clearly depends on the rate at which matter can be accreted down onto the compact object, and on the depth of the gravitational potential well (M/R). Greater mass accretion rates should surely give rise to greater luminosity output. This is only true up to a point, known as the Eddington limit. This is reached when the accreting material produces so much outward flowing radiation that it pushes any infalling material away, preventing further accretion and so limiting the luminosity. The Eddington Luminosity is LEdd~1.3 x 1038 M/MSun ergs s-1.
The rich environment of a galactic centre with its clouds of gas and dust can provide enough material to power the extreme luminosities of the Active galaxies and Quasars, but otherwise the interstellar medium is generally too tenous to produce much luminosity. But most stars are born in binary systems rather than in isolation so when the more massive star has evolved to form a compact object, then the companion star can form the source of accreting material. There are two ways in which the compact object can accrete from the companion star. If the companion star is of a type (generally high mass) which loses some fraction of its mass through a stellar wind then some of this will approach the compact object closely enough so that it can be captured. This gives rise to a chaotic accretion flow which is rather complex. Alternatively, if the orbit is close enough then the outer layers of any companion star can be more strongly attracted to the compact object than to their own stellar core. The outer layers then literally fall off, a process called Roche lobe overflow. This results in a rather narrow stream of material, with a large amount of angular momentum from the orbital velocity. The angular momentum means that the material cannot fall far into the the gravitational potential, in the same way that the Earths orbital velocity means that it cannot fall into the Sun. Instead it forms a ring orbiting around the compact object. But gravitatational orbits have differential velocities - material closer to the center moves more rapidly. Thus there will be viscous frictional forces (based on a nastily complex magnetohydrodynamic dynamo) between the inner and outer parts of the ring which will act to remove angular momentum from the inner edge, slowing it down. This lower angular momentum material can then spreads inwards, while the outer parts of the ring gain the angular momentum and spread outwards. This eventually forms an accretion disk, and such disks are also the way to get galactic material which has high angular momentum down to a supermassive black hole. White dwarfs and neutron stars which accrete via Roche lobe overflow have the further possibility that the angular momentum of the accreting material can also be removed by interactions with the magnetic field. In the gravitational collapse then the stellar core magnetic field is boosted dramatically (by the square of the decrease in radius), so neutron stars can be born with with very high magnetic fields, of order 1013$ Gauss. These fields are much larger than anything that can be made in current laboratory experiments. By contrast, white dwarfs are 2000x larger than neutron stars, so their fields are a million times weaker. However, because the neutron star is much smaller, its magnetic field has to operate over much larger distances compared to the radius of the compact star. The pressure the field exerts on the infalling material is a very strong functions of this distance, so the neutron star is less able to disrupt the outer parts of the disk. For the most highly magnetised white dwarfs the magnetic pressure can be strong enough to completely prevent disk formation, leading to a very simple accretion stream geometry, while lower field white dwarfs and the highly magnetic neutron stars accrete from a disrupted disk. Not all neutron stars in binaries are highly magnetic. Old accreting neutron stars have (relatively) low magnetic fields, for reasons which are not well understood. Probably the best idea for this at present is that the magnetic field trapped in the superconducting neutron star core is dragged to the surface and dissipated as the neutron star spin changes in certain stages of the binary system evolution. However it happens, old neutron stars in binaries have weak fields, which at most can only disrupt the very inner disk. These evolved systems tend to have the neutron star accreting from a low mass companion, contrasting with young systems where the companion is a Roche lobe filling massive star and where the neutron star can still have a high magnetic field. Some of these objects are clearly more complicated than others: accretion disks and stream flows are much simpler than chaotic accretion, extreme magnetic fields and magnetically truncated disks. Here we will concentrate on the simple accretion geometries so as to try to understand the fundamental properties of gravitational energy release.