This article was devoted to studying the global dynamics of 3-dimensional galactic models via a powerful numerical method, the Frequency Map Analysis of Laskar (Laskar 1988, 1990, 1993, 1996). This is the continuation of a work which was initiated with the application of the method in the axisymmetric logarithmic potential (Paper 1). Whereas the first attempt was essentially conceived as a test for the applicability of the method in galactic dynamics, this second part aimed in the clarification of many obscure aspects of the dynamics of triaxial galactic models.
Many attempts were made up to now in order to shed light on the 3-dimensional dynamics of galaxies. However, the lack of an appropriate tool for the visualisation of the phase space of these latter systems left many gaps in our knowledge. The frequency map analysis method based on an appropriate Hamiltonian formalism gives a new perspective for the illumination of these issues. The purpose of this work was twofold. First, it was intended to establish the link between modern ideas of the theory of Hamiltonian systems and the dynamics of galaxies. The second objective was to provide a very detailed image of the dynamics of a quite popular galactic potential for a wide range of its shape parameters in order to form a strong basis for future studies, with an eye on the central problem of galactic dynamics, namely the self-consistency of the employed models and the role played by the irregular orbits.
Our study was focused on the interpretation of the dynamics of the triaxial version of the logarithmic potential, a special member of the family of power law galaxies (Evans 1994). These models present common physical characteristics and observable properties (Evans & de Zeeuw 1994). Furthermore, some of their principal dynamical features, as the symmetries of the phase space, the fixed points, the principal periodic orbits and the integrable cases, are identical. Thus, we may conjecture that there exist some topological reasons connected with the form of the isopotential surfaces, for which most of the dynamical aspects presented for a member of this family are generic for any other power law potential, with an appropriate adjustment of the parameters involved (see also de Zeeuw & Pfenniger (1988)). These dynamical characteristics (exposed in Sect. 3) are the main determining factors for the dynamics of the model potential, as illustrated by the four common types of quasi-periodic motion covering the phase space of the models. The fundamental aspect of the frequency map analysis, namely the quasi-periodic approximations of numerically integrated orbits, enabled us to elucidate the connection of each kind of orbit with perturbations of the integrable cases (Sect. 4).
The quasi-periodic approximations are instrumental for the elaboration of the principal aspect of this work, i.e. the study of the global dynamics of the phase space in frequency maps, for a wide number of axial ratios (Sect. 5). The choice of the axial ratios as perturbation parameters is dictated by both dynamical and physical arguments. Besides, there is no loss of information by keeping the energy fixed, as similar dynamical behaviour may occur for any energy level, by a suitable rearrangement of the axial ratios.
The frequency maps for the circulating part of the phase space revealed the large chaotic regions created by the perturbed manifolds of the principal periodic orbits, and especially the small axial periodic orbit. Although the instability of this latter orbit was already known (e.g. Binney 1981), its dynamical influence was never discussed within the framework of modern dynamical systems and its extent was underestimated. In contrast to the restricted 2D system (Paper 1), these chaotic regions are very apparent even for systems close to axisymmetric, as many instabilities are introduced due to the addition of the third dimension. Apart from the fact that the numerical integration assures the accurate distinction between the regular and irregular motion (in contrast to previous studies), the frequency map is the natural space for the visualisation of all these phenomena related with the dynamics of multi-dimensional systems.
The phase space of the system is much more complicated than it was previously imagined. Previous studies accounted only for the periodic orbits (simple resonant points of the frequency plane) and neglected the existence of more complicated resonant conditions between the three fundamental frequencies of motion. These are identified through the lines of resonance on the frequency map and form a complicated network, the system's Arnol'd web. The resonant lines may be conceived as the generalisation of the librating islands of the 2D system, as they are perturbed by the addition of a third dimension. These resonant lines, which correspond to resonant tori of non-maximal dimension, are the principal orbital families affecting the system's dynamical behaviour. The intersections of these lines define all the important periodic orbits of the system. Thus, it is quite easy to define their exact position, as we have a very good initial guess for ensuring and accelerating the convergence of any root finding numerical method.
Thereafter, we were able to study the evolution of the dynamics of the system with the variation of the axial ratios: as the perturbation is magnified the instability zones due to the perturbed manifolds of the principal periodic orbits are increased and overlap with the resonant lines, producing large chaotic regions. Moreover, all the dynamical properties of the system viewed through the frequency maps can be directly linked with the physical space, by mapping a certain diffusion coefficient to each initial condition. We were thus able to demonstrate that, at least for moderate values of the perturbation parameters, the chaotic orbits are mostly concentrated near the middle/long axis 2D system. When the perturbation increases, chaotic orbits spread all over the circulating part of the phase space. Especially in the case of tube orbits, it is quite difficult to construct frequency maps due to the inappropriate parametrisation of the phase space by the Cartesian coordinates and their conjugate momenta. Therefore, the initial condition planes were the guide for studying the dynamics of these regions and revealed all the important chaotic zones covering a wide area of this part of the phase space which separates the different types of quasi-periodic motion.
These results confirm the fact that, except for models close to the integrable Stäckel systems, chaos should be intrinsically connected with the addition of a third non-symmetric dimension to a 2-dimensional non-integrable system. Hence, we can deduce that, the existence of chaotic regions should be a generic feature of the dynamics of triaxial galactic systems, not only for the potentials with cusps (Gerhard & Binney 1985; Merritt & Fridman 1996) but also the ones modeling constant density central cores. Another rather unknown aspect of the dynamics of triaxial models, which we did not account here, is the effect of the inclusion of a slow rotating term in the Hamiltonian of the system. In that case, as the principal mirror symmetries are broken, we may surmise that chaotic regions are further enlarged.
Our results, in addition with recent studies (Merritt & Fridman 1996), suggest that the incorporation of chaotic orbits in our models should be a necessity for the construction of a phase space distribution function supporting the self-consistency of the system. This inclusion does not violate the fact that any stationary solution of the Vlassov equation should depend on the integrals of motion (known as Jeans' theorem). Even if the latter was used as an argument for the irrelevance of chaos in stellar systems (Binney 1982), it is evident that any orbit in a Hamiltonian system (regular or not) always respects at least one integral of motion - the energy. As it was already pointed out by Merritt & Fridman (1996) and Merritt & Valluri (1996) the density distribution function can be splited in two independent parts correponding respectively to the regular and chaotic regions of the phase space. The first one depends on the integrals of motion and the second one is uniform (it depends only on the energy).
The inclusion of chaotic orbits in our models is a rather technical problem, as we do not know how to calculate mean densities for these orbits (Petrou 1984; Merritt & Valluri 1996). In fact, depending on their rate of diffusion, irregular orbits can be distinguished in strongly and weakly chaotic.6 A first approximation is to consider the strongly chaotic orbits as ergodic and the weakly chaotic as regular (Merritt & Fridman 1996). However, in order to be fully assured about the characterisation of irregular orbits, we should study in detail their diffusion in phase space. Indeed, despite the fact that some orbits are strongly chaotic, they are stack for a long time around periodic orbits or resonant tori (Sect 5.2.1). This dynamical features which cannot be detected by classical approaches (e.g. calculation of Lyapunov exponents and KS entropy) will be studied with the help of the frequency map analysis in a forthcoming article.
A second problem arises when attempting the numerical construction of self-consistent solutions with the usually employed Schwarzschild's technique (Schwarzschild 1979) or its variants (see Binney & Tremaine (1987) p. 255, for a review). A rather serious inconvenience of the method (Kuijken 1993) is that the solutions provided can be unstable when changing the initial mesh. In fact, the mesh elements, as they are equally spaced in the original "bad" variables, do not correspond to equal tori volumes, thereby causing the latter mentioned instability. Apart from the fact that the frequency map analysis can be a very effective tool for viewing the organisation of different types of orbits in the phase space, it enables us to calculate numerical action variables through its quasi-periodic approximations. Thus, it can provide an alternative solution to the above mentionned problem by computing phase space density distributions on each torus.
Finally, the fundamental question is whether our models which are covered with large chaotic zones together with the triaxiality hypothesis are relevant for modeling real elliptical galaxies. The answer is hard and still remains a matter of debate (see Merritt 1997). It assumes a perfect knowledge of the physical characteristics of the system, a knowledge which is rather poor in the case of galaxies. In near future, the evolution of observational techniques together with additional theoretical efforts are likely to provide new evidence for the clarification of these challenging issues.
© European Southern Observatory (ESO) 1998
Online publication: December 8, 1997