SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


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

Previous Section Next Section Title Page Table of Contents

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:

[EQUATION]

G is the gravitational constant, [FORMULA] is the Keplerian angular velocity.

We use the axisymmetric hydrodynamic equations in view of viscosity. The equations of motion and continuity have the form

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA], [FORMULA] is the velocity, P is the pressure, [FORMULA] is the volume density of matter in the disk, [FORMULA] are the components of the symmetric viscous stress tensor ([FORMULA]).

We add the thermal equation to the system of Eqs. (2)-(5) as

[EQUATION]

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 [FORMULA] components: [FORMULA]. The components of the viscous stress tensor can be written in the following form:

[EQUATION]

where [FORMULA]. As if we ignore the second (volume) viscosity the trace of the viscous tensor is equal to zero ([FORMULA]), [FORMULA] ([FORMULA]).

The parameters [FORMULA], [FORMULA] and [FORMULA] determine the level of turbulence in the disk. Moreover [FORMULA] and [FORMULA] are caused by the shear character of flow in the z-direction, and the value [FORMULA] is connected to the differential rotation in the disk plane and coincides with the [FORMULA]-parameter of the standard theory of disk accretion (Shakura & Sunyaev 1973). For [FORMULA] one can write:

[EQUATION]

where [FORMULA]. This is limited to the case of weak dependence of equilibrium velocities ([FORMULA] and [FORMULA]) on the z-coordinate for a correct transition to the thin disk model. This is possible if the following conditions: [FORMULA] and [FORMULA] are fulfilled. Therefore one can ignore the terms containing [FORMULA] and [FORMULA]. The conditions [FORMULA] and [FORMULA] are assumed in most models of accretion disks. With these assumptions the equilibrium balance of the forces is defined by:

[EQUATION]

[EQUATION]

[EQUATION]

The equilibrium functions can be written in the self-similar form: [FORMULA], if it is assumed [FORMULA]. 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

[EQUATION]

Then the solutions for the pressure and density are:

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA], n is the polytropic index, and h defines the disk boundary - at the points [FORMULA] the equilibrium pressure and density are equal to zero. We assume also that [FORMULA].

The relation

[EQUATION]

defines the adiabatic sound speed ([FORMULA] is the adiabatic index). For the equilibrium velocities ([FORMULA] and [FORMULA]) we obtain from Eqs. (9), (10) taking into account Eqs. (13) and (14):

[EQUATION]

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA].

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1999

Online publication: April 12, 1999
helpdesk.link@springer.de