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

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4. Vertical structure of standard disk

Hereafter, until specially mentioned, we assume that the matter in the disk moves with the Keplerian angular velocity [FORMULA], and its state is governed by the ideal gas equation

[EQUATION]

where µ and [FORMULA] erg mol-1K-1 are the molecular weight of the gas and the molar gas constant, respectively. Along the Z coordinate the hydrostatic equilibrium takes place:

[EQUATION]

and the continuity equation is

[EQUATION]

We assume the radiation transfer equation in the diffusive approximation:

[EQUATION]

where [FORMULA] cm s-1 is the light velocity, [FORMULA] erg cm-3K-4. The vertical gradient of the radiation flux Q is proportional to the energy release per unit volume in the disk; that is,

[EQUATION]

We take the opacities in the form of a power law [FORMULA] where [FORMULA], [FORMULA] cm2g-1 if [FORMULA] and [FORMULA], [FORMULA], [FORMULA] cm5K[FORMULA]g-2 if [FORMULA]. Generally, in the optically thick disks the energy release can be described as a power law of temperature and density (Tayler 1980). In a sense the calculation of the disk structure resembles the calculation of stellar internal structure. In the present study two cases are considered: the energy release [FORMULA] is proportional to (a) the pressure [FORMULA], (b) the density [FORMULA] alone. The thermal energy release is due to the differential rotation of a viscous disk:

[EQUATION]

We follow Shakura (1972) and Shakura & Sunyaev (1973) in suggesting that the turbulent viscous stress tensor is parameterized by the pressure:

[EQUATION]

where [FORMULA] is the kinematic coefficient of turbulent viscosity. The height-integrated viscous stress tensor is given by

[EQUATION]

The energy emitted from the unit surface of one side of the disk is obtained by integrating (12) using (13) and (15):

[EQUATION]

Above equations written for stationary accretion disks hold in a non-stationary case taking into account that the characteristic hydrostatic time of order of [FORMULA] is shorter than the time of radial movement in the disk [FORMULA].

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 [FORMULA]-parameter along the height Z. The vertical structure of the disks including radiative and convective energy transfer was investigated by Meyer & Meyer-Hofmeister (1982). They investigated two types of viscosity, proportional to the gas pressure or to the total pressure. We adopt the result of Ketsaris & Shakura (1998) who proposed a new method of calculating the vertical structure of optically thick [FORMULA]-disks assuming power [FORMULA]- and T- dependences for the opacity and the energy release.

The dimensionless variable

[EQUATION]

is introduced for convenient description of the problem, along with the following variables: [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. The method involves the finding of the eigenvalues of the dimensionless parameters in the differential equations that describe vertical structure of the disk:

[EQUATION]

using the definite boundary conditions in the disk. [FORMULA], [FORMULA], [FORMULA] denote the values in the equatorial plane of the disk and [FORMULA].

After some algebraic manipulation of the right hand equations in (17) we obtain [FORMULA] written in terms of [FORMULA] and [FORMULA], which in view of (7) yields:

[EQUATION]

where

[EQUATION]

It is worth noting that D depends on [FORMULA] very weakly: to a power of [FORMULA] or [FORMULA]. This fact is believed to reduce the effect of uncertainties in our knowledge of the real law of opacity. The combination of [FORMULA] in (18) varies slightly with the optical depth [FORMULA], i.e. along the radius of the disk (see Tables 12). Thus, factor D in the basic equation of time-dependent accretion (6) is considered to be constant.


[TABLE]

Table 1. Vertical structure parameters in the Thomson opacity regime



[TABLE]

Table 2. Vertical structure parameters in the free-free opacity regime


Specific energy dissipation [FORMULA] is defined by the temperature variations over Z. In principle, the intensive stirring in the disk can account for the situation when the energy release per unit mass does not depend on the height Z. This refers to the case (b) mentioned above where [FORMULA] is the function of density. In this situation the temperature dependence disappears in the energy production equation (third line of (17)) and [FORMULA].

Ketsaris & Shakura (1998) calculated the values of [FORMULA], and [FORMULA]. Selected values of [FORMULA] and corresponding values of [FORMULA], in the Thomson opacity regime, and some effective optical thickness of the disk [FORMULA], in the free-free regime, are shown in Tables 12. For the full version of [FORMULA] list and discussion the reader is referred to the original paper by Ketsaris & Shakura (1998). The parameter [FORMULA] was introduced by them for the sake of convenience and denotes the ratio of total scattering optical thickness [FORMULA] to that at the thermalization depth:

[EQUATION]

where [FORMULA] is the effective optical depth (Zeldovich & Shakura 1969; Mihalas 1978).

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© European Southern Observatory (ESO) 2000

Online publication: March 28, 2000
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