3. Non-linear problem of evolution of the standard Shakura-Sunyaev disk
The special case when the moment of viscous forces depends linearly on the surface density and has a power law dependence on the radius () was thoroughly investigated by Lynden-Bell & Pringle (1974). In this particular case Eq. (5) is linear and the solution can be presented as the superposition of particular solutions (Green's functions) while the non-linear equations do not allow such solutions. In the paper by LS87 the necessary relation between and F for -disks (Shakura 1972; Shakura & Sunyaev 1973) was derived from the vertical structure equations. Then Eq. (5) acquires the following non-linear form taking into account that :
where D is the dimension constant; , when the Thomson scattering dominates the opacity in the accretion disk, and , when the free-free and free-bound transitions do. The "diffusion constant" D, defined by the specific vertical structure, relates , F, and h:
(see also Filipov 1984). D is a function of , opacity coefficient, and the dimensionless values, which are the combinations of the characteristic physical parameters of the disk. The value of D is to be derived from the consideration of the disk vertical structure. In the following section we obtain its value using the results of the work by Ketsaris & Shakura (1998).
© European Southern Observatory (ESO) 2000
Online publication: March 28, 2000