Astron. Astrophys. 347, 821-840 (1999)

4. Conclusions and discussion

We described some many-particle experiments concerning numerical computations on the dynamics of the stellar layer of a differentially rotating, almost centrifugally-supported galaxy. Our usage of the oversimplified model of the layer (two-dimensional disk) is justified because global N-body simulations have been shown that the inclusion of motions normal to the plane makes little difference to the evolution of the rapidly rotating thin disk (Hohl 1978). We argued that in general computer experiments presented here confirm the predictions of the linearized stability theory of small-amplitude gravity perturbations developed by Bertin, Lau, Lin, Mark, Morozov, Polyachenko, and others: the differentially rotating, marginally Jeans-stable disk of stars (and a planetary disk with rare collisions between mutual-gravitating particles) is dynamically hotter than the original Toomre's local stability criterion predicts. That is, in a nonuniformly rotating disk of stars the critical Toomre's stability parameter is appreciably greater than (although still of the order) unity. In actual galaxies and planetary rings .

A dynamically cold rigidly rotating disk with the initial radial dispersion of random velocities of stars is found to be gravitationally unstable as predicted first by Toomre's (1964) stability analysis. Namely, small-scale almost radial perturbations grow exponentially during the time of the first rotation of the system under consideration. In agreement with the theory, in the numerical model of the warm () rigidly rotating disk the relatively high temperature leads to significant reduction of the growth rate of the Jeans instability; such a disk is near the stability threshold. In the hot numerical model () all Jeans-type gravity perturbations are stabilized.

By way of contrast, even the Jeans-stable (by the original Toomre's criterion) differentially rotating disk is still violently unstable to the relatively large-scale nonaxisymmetric modes when . In such a system the spiral structure develops rapidly during the first rotation of the system only. Finally, differentially rotating, spatially homogeneous models with the initial value of Toomre's stability parameter (or , respectively) show little structure that can be associated with the Jeans instability. This basically agrees with the theory discussed in the Introduction and Appendix A.1.

In both cases, rigidly and differentially rotating systems, some residual instability is observed for Q up to a factor times the critical value . The reason for such a minimally larger value of the critical velocity dispersion might be partly due to the shortcomings of the asymptotic Lin-Shu density wave theory which is used here. Accordingly, we restricted our analysis to the approximation of moderately tightly-wound spirals (Appendix A.1). Indeed, as is known, since all the above results are given for moderately tightly-wound spirals, they are subject to an uncertainty of a factor of , where . Straightforward estimates show that in the case of spirals shown in Figs. 4, 5, 8, and 9 is about ; thus, we can have reasonable confidence in theoretical and experimental results perhaps to within only. In this regard, it is interesting to note that at least for a disk with a constant rotation velocity Polyachenko (1989), who did not use the approximations of the Lin-Shu theory, has found a slightly greater value of the critical velocity dispersion than the criterion (3) gives. ⁹ Interestingly, such a slightly greater value of the critical velocity dispersion is also consistent with the results of Toomre's (1981) numerical experiments with stellar disks, in which the disks with a flat rotation curve became completely stable specifically when .

Also, following Griv (1992), to obtain a more accurate value of critical velocity dispersion one has to consider the next leading order in the asymptotic expansion by including higher-order terms in the epicyclic amplitude.

In addition, the shortcomings of local experiments in Hill's equations context are quite obvious. For instance, almost certainly in contrast to our calculations, one has to include gravitational forces on a given target particle from other particles whose nearest image lies out of the distance min (see Sect. 2). This is because of the long range of gravitational forces. Further theoretical and experimental N-body studies to clarify the problem are desirable. At the present, however, the causes of these relatively small discrepancies between the results of our theory and local N-body simulations are not clear, but may be due to both theoretical and computational factors just mentioned above.

According to Eqs. (27) and (30), the Jeans-unstable perturbations in a spatially homogeneous disk grow aperiodically with the growth rate . This means that as a rule the Jeans instability develops rapidly on a dynamical timescale ; in galaxies , where yr is the Hubble timescale. Inevitably, the velocity dispersion of particles would be expected to increase in the field of unstable waves with an amplitude increasing with time as a result of "hydrodynamic" (nonresonant) collective interactions between Jeans-unstable perturbations and stars: the Jeans instability grows on a dynamical timescale and presumably heats the disk until . In addition, the Jeans instability, which can effectively heat the medium without raising the entropy, leads to the mass redistribution of the system by increasing the central condensation of the disk (and a diffused outer envelope). The diffusion of stars in the velocity space and the coordinate space takes place because stars gain additional oscillatory energy of the gravitational field in the unstable density waves (see Griv et al. 1994 for a discussion).

It is interesting to note that about the same value of brings both the observations of actual rapidly (and nonuniformly) rotating galaxies of stars we are investigating including our own Galaxy (Toomre 1974, 1977; van der Kruit & Freeman 1986; Bottema 1993) and the global N-body simulations (Hohl 1971, 1972, 1978; Sellwood & Carlberg 1984; Griv et al. 1994; Griv & Chiueh 1998). Also observations and local simulations of the Saturnian ring system show about the same value of Q (Lane et al. 1982; Salo 1992, 1995; Griv 1996, 1997; Griv & Yuan 1996). Therefore, we conclude that in general both the theory and our N-body simulations are in agreement with observational data.

In closing, the differences between actual inhomogeneous gravitating systems and computer models used in our simulations may result in ambiguity in the applications of the N-body calculations and the theory to galaxies and planetary rings. In order to resolve the ambiguity, it will be possible in the future to make more realistic simulations of this type and to extend the theory so as to allow for spatial inhomegeneity and a finite thickness of the disk.

© European Southern Observatory (ESO) 1999

Online publication: June 6, 1999
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