          Astron. Astrophys. 345, 307-314 (1999)

## 2. Model and basic equations

We consider an axisymmetric differentially rotating gas disk in the gravitational field of a mass M. Without including self-gravity and relativistic effects, and adopting cylindrical coordinates we have: G is the gravitational constant, is the Keplerian angular velocity.

We use the axisymmetric hydrodynamic equations in view of viscosity. The equations of motion and continuity have the form    where , is the velocity, P is the pressure, is the volume density of matter in the disk, are the components of the symmetric viscous stress tensor ( ).

We add the thermal equation to the system of Eqs. (2)-(5) as where S is the entropy, T is the temperature, and the variable Q defines the sources of heat.

### 2.1. Equilibrium model and viscosity law

We shall assume that the equilibrium velocity in the disk has only r and components: . The components of the viscous stress tensor can be written in the following form: where . As if we ignore the second (volume) viscosity the trace of the viscous tensor is equal to zero ( ), ( ).

The parameters , and determine the level of turbulence in the disk. Moreover and are caused by the shear character of flow in the z-direction, and the value is connected to the differential rotation in the disk plane and coincides with the -parameter of the standard theory of disk accretion (Shakura & Sunyaev 1973). For one can write: where . This is limited to the case of weak dependence of equilibrium velocities ( and ) on the z-coordinate for a correct transition to the thin disk model. This is possible if the following conditions: and are fulfilled. Therefore one can ignore the terms containing and . The conditions and are assumed in most models of accretion disks. With these assumptions the equilibrium balance of the forces is defined by:   The equilibrium functions can be written in the self-similar form: , if it is assumed . The disk z-structure depends on the equation of state and on the energy flux. For the sake of simplicity we limit ourselves to a polytropic model Then the solutions for the pressure and density are: where , , , n is the polytropic index, and h defines the disk boundary - at the points the equilibrium pressure and density are equal to zero. We assume also that .

The relation defines the adiabatic sound speed ( is the adiabatic index). For the equilibrium velocities ( and ) we obtain from Eqs. (9), (10) taking into account Eqs. (13) and (14):  where , , .    © European Southern Observatory (ESO) 1999

Online publication: April 12, 1999 