## 5. Extensions## 5.1. Matching with an inner Keplerian, non-self-regulated accretion diskIf we take an astrophysical situation (such as an AGN or a
protostar) with specific physical conditions, it is likely that the
arguments that support the adoption of the self-regulation constraint,
followed in this paper, fail outside a well-defined radial range,
either at small or at large radii. For example, we may recall that in
the context of the dynamics of spiral galaxies the relevant This discussion suggests that it should be interesting to explore
the possibility where Eq. (8) is replaced by a condition of the form
, with a Imposing, as we are going to do, a given profile may, at first sight, appear to be arbitrary. In reality, the freedom in the choice of the profile allows us to test quantitatively how the dynamical characteristics , , , and of the disk change when the self-regulation constraint is partially relaxed, in the inner disk, in a variety of ways. It is up to us to test the different possibilities (which may correspond to completely different sets of energy balance equations in the inner disk) that might be considered. For our purposes, since we wish to study the deviations from the standard "Keplerian" case, we only need to take into account that the relevant physical processes match so that in the outer disk self-regulation is enforced. Note that in the transition region where matching between the inner and the outer disk occurs, for reasons expressed in Sect. 3, it may be practically impossible to define, from first principles, a satisfactory set of energy equations able to include all the desired radiation processes and the non-linear effects associated with Jeans instability. This procedure draws considerable support from a recent analysis of
"standard" disks (Bardou et al. 1998) aimed at detecting evidence for
the importance of disk self-gravity in the outer disk. Based on an
extension of the "standard" -disks
(characterized by Kramers' opacity and by neglect of radiation
pressure), the effects of the disk self-gravity have been here
incorporated by means of an improved thickness prescription (somewhat
in the spirit of our Appendix A) and of a modified viscosity
prescription, but the (Keplerian)
profile is left unaltered. Therefore, this study is ideally suited to
describe the conditions of our inner disk, as we intend to partially
relax the self-regulation requirement. A very important result of the
analysis by Bardou et al. (1998) is that their "standard" description
breaks down beyond a radius , well
inside which the local stability parameter behaves approximately as
; as it might have been anticipated,
the location where the standard model breaks down coincides with the
location where to be used instead of Eq. (8). (The formula is meant to be used as
a semi-empirical tool; one should keep in mind that the exact form of
the Note that the ratio
Some examples of
## 5.2. The effect of a diffuse "halo"So far we have considered the case where the mass is all
distributed in a disk (either at the center, as a point mass, or in
diffuse form). In view of possible applications to AGN configurations
or to the general galactic context, it is important to consider an
extension of the models to the case where part of the gravitational
field is determined by a diffuse spherical component, which we will
call We have thus considered a set of models where the field external to the disk is produced by the joint contribution of a central point mass and of a halo (which, for simplicity, we take to be spherical). In view of the case of a disk embedded in an elliptical galaxy, we have modeled the halo as approximately isothermal, with a finite core radius. In this case the dimensionless equation giving the rotation curve (Eq. (13)) is modified as follows: We see that now the equations depend on two additional parameters:
, giving the relative strength of
the external field, and , which
measures the size of the core radius. In this case it is easy to
demonstrate that at large radii the density deviation
approaches
if
, and In Fig. 8 we show examples of the rotation curve of models with a diffuse halo, for the case , . For the vertical structure, we have referred to the improved vertical prescription of Eq. (A9), with .
## 5.3. Disks with an outer truncation radiusSelf-regulated accretion disks with finite mass can be easily constructed by imposing the existence of an outer truncation radius. Either the study of the collapse of a gas cloud with finite mass or the consideration of the physical conditions in the outer parts of some astrophysical objects will naturally bring us to address such models. © European Southern Observatory (ESO) 1999 Online publication: October 4, 1999 |