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

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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 ([FORMULA]) 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 [FORMULA] and F for [FORMULA]-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 [FORMULA]:

[EQUATION]

where D is the dimension constant; [FORMULA], [FORMULA] when the Thomson scattering dominates the opacity in the accretion disk, and [FORMULA], [FORMULA] when the free-free and free-bound transitions do. The "diffusion constant" D, defined by the specific vertical structure, relates [FORMULA], F, and h:

[EQUATION]

(see also Filipov 1984). D is a function of [FORMULA], 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).

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© European Southern Observatory (ESO) 2000

Online publication: March 28, 2000
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