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Astron. Astrophys. 350, 694-704 (1999)

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2. Self-regulated accretion disks

2.1. Basic equations and parameter space

We consider a steady, axisymmetric, geometrically thin accretion disk, with constant mass and angular momentum accretion rates, which we call [FORMULA] and [FORMULA], respectively. The conservation laws for mass and angular momentum, in cylindrical coordinates, are:

[EQUATION]

[EQUATION]

where [FORMULA] is the surface density of the disk, u is the radial velocity, [FORMULA] is the angular velocity, and [FORMULA] is the viscosity coefficient. Note that we take [FORMULA] in the case of inflow. Usually on the right hand side of Eq. (2) the two terms have opposite signs; the first term describes convection of angular momentum, while the second (negative in the common situation where [FORMULA]) accounts for the angular momentum transport due to viscous torques. Thus [FORMULA] corresponds to a net angular momentum influx.

For a cool, slowly accreting disk, the radial balance of forces requires:

[EQUATION]

where [FORMULA] is the mass of the central object and [FORMULA] is the disk contribution to the gravitational potential. The radial gravitational field generated by the disk can be written as:

[EQUATION]

where [FORMULA] and [FORMULA] are complete elliptic integrals of the first kind, and [FORMULA] (see Gradshteyn & Ryzhik 1980). The field [FORMULA] in the equatorial plane is obtained by taking the limit [FORMULA]. Here we have preferred to refer directly to the field (and not to the potential, as was done by Bertin 1997) and to the formulae applicable to the case [FORMULA]; this representation is more convenient in view of the numerical investigations that we have in mind (see Sects. 4 and 5).

For the viscosity we follow the standard prescription (Shakura & Sunyaev 1973):

[EQUATION]

where c is the effective thermal velocity and h the half-thickness of the disk. Here [FORMULA] is a dimensionless parameter, with [FORMULA]; in the following we take it to be a constant.

In the case of dominant disk self-gravity, one may adopt the requirement of hydrostatic equilibrium in the z direction for a self-gravitating slab model, which gives

[EQUATION]

A more refined analysis of the vertical equilibrium is provided in Appendix A. Substituting Eq. (6) into Eq. (5) we obtain [FORMULA], which inserted in Eq. (2) yields:

[EQUATION]

In order to close the set of equations (Eq. (3), Eq. (4), and Eq. (7)), we need an additional relation. In standard studies this is provided by an energy transport equation (see, for example, Pringle 1981; Narayan & Popham 1993; Narayan & Yi 1994). Here we consider an alternative scenario (Bertin 1997) where the energy equation is replaced by the requirement of marginal Jeans stability:

[EQUATION]

where [FORMULA] is the epicyclic frequency. This assumes the presence of a suitable self-regulation mechanism (see Sect. 3 below). For a given value of [FORMULA] and [FORMULA], we now have a complete set of equations, depending on the three parameters [FORMULA], [FORMULA], and [FORMULA]. We recall that the parameter [FORMULA] is the fluid analogue of the axisymmetric stability parameter introduced and described by Toomre (1964) in his analysis of the stability of a stellar disk.

2.2. The self-similar pure disk solution

In the special case of [FORMULA] and [FORMULA], the problem is solved by the self-similar solution with flat rotation curve (Bertin 1997), characterized by:

[EQUATION]

The other properties of the disk are specified by: [FORMULA], [FORMULA], [FORMULA], [FORMULA]. Note that Eq. (9) describes the self-similar disks found by Mestel (1963) in the non-accreting case.

The same solution is also valid asymptotically at large radii even when [FORMULA] and [FORMULA] do not vanish. In fact, the effect of a non-zero [FORMULA] should be unimportant for [FORMULA] while the effects of a non-zero [FORMULA] should be unimportant for [FORMULA]. Here the definitions of [FORMULA] and [FORMULA] are slightly different from those adopted earlier (Bertin 1997).

2.3. The iteration scheme for the general case

For a given value of [FORMULA] and [FORMULA] and for a specified accretion rate [FORMULA], the general case is a problem with a well-defined physical lengthscale [FORMULA] and one dimensionless parameter [FORMULA]. In the following we wish to explore the properties of the mathematical solutions in the entire parameter space available (in particular, for both positive and negative values of [FORMULA]). The knowledge of these solutions is a prerequisite for a discussion of any specific astrophysical application. At a later stage, the physical problem under investigation is expected to restrict the relevant parameter space.

In order to calculate such one-parameter family of solutions, we start by writing the relevant equations in dimensionless form, in such a way that the self-similar solution described above is easily recognized. We thus introduce three deviation functions [FORMULA], [FORMULA], and [FORMULA], which are related to the dimensional disk density [FORMULA], rotation curve V, and effective thermal speed c by the following definitions:

[EQUATION]

[EQUATION]

[EQUATION]

with [FORMULA]. Clearly the functions [FORMULA], [FORMULA], and [FORMULA] are all positive definite; the self-similar solution corresponds to [FORMULA], [FORMULA], [FORMULA].

In terms of the dimensionless radial coordinate [FORMULA], the basic set of equations (corresponding to Eqs. (3), (4), (7), and (8)) thus become:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

with [FORMULA], in the limit [FORMULA].

In the case of outward angular momentum flux ([FORMULA]), the above equations can be readily solved by iteration in the following way. An initial seed solution [FORMULA] is inserted in Eqs. (15) and (16), thus leading to a first approximation to the density deviation [FORMULA]. This is inserted in Eq. (14), then producing (via Eq. (13)) a new expression for the rotation curve deviation [FORMULA]. Typically three or four iterations are sufficient to reach a satisfactory convergence.

For the case when the net angular momentum flux is inwards ([FORMULA]), Eq. (15) shows that we may run into a difficulty at small radii, where the quantity [FORMULA] changes sign. Here we may proceed by analogy with standard studies (e.g., see Pringle 1981), by restricting our analysis to the outer disk defined by [FORMULA], with the condition [FORMULA], i.e. [FORMULA]. This is taken to occur at a point where the angular velocity [FORMULA] reaches a maximum (so that [FORMULA] and [FORMULA] may remain finite); such maximum is identified as a location, in the vicinity of the surface of the central object, to which the accretion disk is imagined to be connected by a relatively narrow boundary layer. Of course, in the boundary layer the physical processes will be different from those described in our model (for example, there will be effects due to pressure gradients and the disk will be thick). Thus we will not be able to follow the associated (inwards) decline of the angular velocity away from the Keplerian profile. As a result, in our calculation we let the effective thermal speed vanish at [FORMULA]; this unphysical behavior would be removed when one describes the boundary layer with more realistic physical conditions (see Popham & Narayan 1991).

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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