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

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4. Properties of models with a central point mass

After this digression, we can now proceed with the analysis of the problem as formulated in Sect. 2. In the presence of a central point mass ([FORMULA]) we expect the lengthscale [FORMULA] to mark the transition from a Keplerian disk to a fully self-gravitating disk with flat rotation curve. Surprisingly, in our class of self-regulated accretion disks the role of the disk self-gravity turns out to be significant all the way down to the center.

4.1. Rotation curves

In Fig. 1 we illustrate the behavior of the rotation curves in our class of models for several values of the angular momentum flux parameter [FORMULA]. For comparison, we also show the Keplerian curve [FORMULA] that is obtained by setting [FORMULA]. We note that the difference from the Keplerian decline is significant even at [FORMULA]. For example, for the [FORMULA] case we find [FORMULA] at [FORMULA].

[FIGURE] Fig. 1. Top panel: Rotation curve of the disk in the presence of a central point mass for different values of the angular momentum flux parameter. Bottom panel: Same as top panel, but in logarithmic scale.

4.2. Effective thermal speed

Disks where the angular momentum is transported outwards ([FORMULA]) tend to develop a warmer core, while the opposite trend occurs for disks where the angular momentum is carried inwards ([FORMULA]). This is shown in Fig. 2 (Fig. 2 and the following Fig. 3, Fig. 4, and Fig. 5 are shown in logarithmic scale to better bring out the behavior in the inner parts of the disk). Note that in the case of inward angular momentum flux ([FORMULA]) the effective thermal speed need not vanish at the inner edge of the disk (as it does in our models) where a boundary layer is expected to be generated (see discussion at the end of Sect. 2.3).

[FIGURE] Fig. 2. Equivalent thermal speed of the disk. The two cases ([FORMULA] and [FORMULA]) show opposite behavior in the inner disk.

[FIGURE] Fig. 3. Cumulative mass of the disk relative to that of the central object.

[FIGURE] Fig. 4. Local self-gravity of the disk, as measured by [FORMULA], for different values of the angular momentum flux parameter.

[FIGURE] Fig. 5. Ratio of the vertical scaleheight [FORMULA] to the thickness of the disk for various models.

4.3. Role of the disk self-gravity close to the central point mass

There are several quantities directly related to the disk density distribution in the disk that allow us to characterize the role of the disk self-gravity.

The most natural quantity to consider is the ratio of the mass of the disk to that of the central object. Obviously, for our non-truncated models this quantity is meaningful only when referred to a given radius. Fig. 3 shows how rapidly in radius the system becomes dominated by the mass of the disk. Note that, in any case, [FORMULA] for [FORMULA].

In galactic dynamics the local disk self-gravity is usually measured in terms of the parameter [FORMULA]. The fully self-gravitating self-similar disk (with flat rotation curve) is characterized by [FORMULA]. The profile of this parameter (see Fig. 4) confirms that indeed, close to the center, the influence of the central mass becomes stronger and stronger.

Given the behavior of the profiles [FORMULA] and [FORMULA], one might conclude that the innermost disk should be treated as a standard Keplerian accretion disk. This conclusion is contradicted by the following argument. In setting up the equations of our models, we have taken the vertical equilibrium to be dominated by the disk self-gravity (see Eq. (6)). If the innermost disk were fully Keplerian, at small radii the vertical scaleheight [FORMULA] should become much smaller than the thickness h associated with our models. Instead, a plot of the ratio [FORMULA] only shows that the two scales become comparable to each other (see Fig. 5), thus demonstrating that the influence of the disk self-gravity is significant all the way to the center. This, of course, reflects our choice of imposing the self-regulation prescription (Eq. (8)) at all radii; in the next section we will show that the disk can indeed make a true transition to a Keplerian disk if the self-regulation prescription is suitably relaxed.

In view of the above considerations, in order to check that no major consequences arise from the use of a vertical equilibrium prescription only partly justified for [FORMULA], we have also considered models based on the improved prescription for the disk thickness (see Appendix A):


Note that in the limit [FORMULA], applicable to the self-similar disk, this prescription reduces to Eq. (6). In the innermost region [FORMULA], [FORMULA], so that [FORMULA].

In Fig. 6 we show the rotation curve of the improved model, compared to that of the original one, for the [FORMULA] case. This plot (along with similar results for the other relevant physical quantities of the disk) shows that, qualitatively, the more refined vertical analysis leaves the models basically unchanged.

[FIGURE] Fig. 6. Rotation curves for two different treatments of the vertical structure for the case [FORMULA], [FORMULA].The solid line represents the improved model, based on Eq. (17); the dotted line is the original model, based on Eq. (6).

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999