*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
helpdesk.link@springer.de |