*Astron. Astrophys. 356, 363-372 (2000)*
## 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
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