## 4. Vertical structure of standard diskHereafter, until specially mentioned, we assume that the matter in the disk moves with the Keplerian angular velocity , and its state is governed by the ideal gas equation where and the continuity equation is We assume the radiation transfer equation in the diffusive approximation: where cm s We take the opacities in the form of a power law
where
,
cm We follow Shakura (1972) and Shakura & Sunyaev (1973) in suggesting that the turbulent viscous stress tensor is parameterized by the pressure: where is the kinematic coefficient of turbulent viscosity. The height-integrated viscous stress tensor is given by The energy emitted from the unit surface of one side of the disk is obtained by integrating (12) using (13) and (15): Above equations written for stationary accretion disks hold in a non-stationary case taking into account that the characteristic hydrostatic time of order of is shorter than the time of radial movement in the disk . There are now various works investigating the vertical structure of
the disks. For example, Nakao & Kato (1995) considered
turbulent diffusion in the disk providing the variations of viscous
heating and -parameter along the
height The dimensionless variable is introduced for convenient description of the problem, along with the following variables: , , , , and . The method involves the finding of the eigenvalues of the dimensionless parameters in the differential equations that describe vertical structure of the disk: using the definite boundary conditions in the disk. , , denote the values in the equatorial plane of the disk and . After some algebraic manipulation of the right hand equations in (17) we obtain written in terms of and , which in view of (7) yields: where It is worth noting that
Specific energy dissipation is
defined by the temperature variations over Ketsaris & Shakura (1998) calculated the values of , and . Selected values of and corresponding values of , in the Thomson opacity regime, and some effective optical thickness of the disk , in the free-free regime, are shown in Tables 1, 2. For the full version of list and discussion the reader is referred to the original paper by Ketsaris & Shakura (1998). The parameter was introduced by them for the sake of convenience and denotes the ratio of total scattering optical thickness to that at the thermalization depth: where is the effective optical depth (Zeldovich & Shakura 1969; Mihalas 1978). © European Southern Observatory (ESO) 2000 Online publication: March 28, 2000 |