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 .
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