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

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5. Extensions

5.1. Matching with an inner Keplerian, non-self-regulated accretion disk

If 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 Q profile is argued to be flat in the outer disk, but is thought to increase inwards inside a circle of radius [FORMULA] often identified as the radial scale of influence of the bulge; even in the absence of a bulge, the central parts of the disk are thought to be generally hotter (e.g., see Bertin & Lin, 1996). In our context, we may then imagine an accretion disk where at small radii self-regulation fails, Q becomes large, and, correspondingly, the disk self-gravity ceases to be important.

This discussion suggests that it should be interesting to explore the possibility where Eq. (8) is replaced by a condition of the form [FORMULA], with a Q profile that decreases monotonically with radius and reaches the self-regulation value [FORMULA] only beyond a certain radial scale [FORMULA]. What would be the impact of such a choice on the structure of the accretion disk? What might be the physical arguments leading to the justification of such a profile? In particular, what would set the scale [FORMULA] for given values of the parameters [FORMULA], [FORMULA], [FORMULA], and [FORMULA]? Note again that we are reversing the standard point of view, whereby one may ask what is the Q profile for an accretion disk, based on the structure calculated from a given choice of the energy equations.

Imposing, as we are going to do, a given profile [FORMULA] 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 [FORMULA], [FORMULA], [FORMULA], and [FORMULA] 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" [FORMULA]-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) [FORMULA] 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 [FORMULA], well inside which the local stability parameter behaves approximately as [FORMULA]; as it might have been anticipated, the location where the standard model breaks down coincides with the location where Q becomes approximately equal to unity. In conclusion, the analysis by Bardou et al. (1998) encourages us to consider the following choice of Q profile


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 Q profile will be determined by the detailed energy processes occurring in the inner disk and by the progressively important role of collective instabilities.) If we express the scale [FORMULA] found in that study in terms of our scale [FORMULA], we find


Note that the ratio [FORMULA] decreases while [FORMULA] increases, and that its value is only weakly dependent on [FORMULA] and on [FORMULA]. For parameters typical of an AGN, self-regulation may thus be ensured very far in, while an extrapolation of the above recipe to parameters typical of a disk surrounding a T Tauri star suggests that, for these latter objects, [FORMULA] should become [FORMULA].

Some examples of partially self-regulated models computed on the basis of Eq. (18) (see Fig. 7) show how an outer disk dominated by self-gravity can match in detail with an inner standard Keplerian accretion disk.

[FIGURE] Fig. 7. Rotation curves of partially self-regulated models with the Q-profile given in Eq. (18), for different values of [FORMULA]. The thickness prescription used is that of Eq. (17).

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 halo (even if it may just correspond to the central region of an elliptical galaxy). This will lead to rotation curves otherwise not accessible by our models. On the other hand, it is easily recognized that this natural extension is going to leave the slow density decline of the disk unaltered. Therefore, if we are interested in producing models with finite total mass, we should be ready to impose an outer truncation radius or, which might effectively be equivalent, to relax the self-regulation prescription in the outermost disk.

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: [FORMULA], giving the relative strength of the external field, and [FORMULA], which measures the size of the core radius. In this case it is easy to demonstrate that at large radii the density deviation [FORMULA] approaches [FORMULA] if [FORMULA], and f if [FORMULA].

In Fig. 8 we show examples of the rotation curve of models with a diffuse halo, for the case [FORMULA], [FORMULA]. For the vertical structure, we have referred to the improved vertical prescription of Eq. (A9), with [FORMULA].

[FIGURE] Fig. 8. Rotation curves of models with a diffuse halo, with [FORMULA], and core radius equal to [FORMULA] ([FORMULA]), for different values of [FORMULA].

5.3. Disks with an outer truncation radius

Self-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.

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999