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Astron. Astrophys. 329, 451-481 (1998)

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1. Introduction

The existence of irregular orbits in the phase space of Hamiltonian systems modeling the motion of stars in galaxies has been identified since the classical work of Hénon and Heiles (1965). However, the role of chaotic orbits in galactic dynamics is not yet completely clarified (Merritt 1996). From a theoretical point of view, the influence of chaos in the structure of galaxies is associated with one of the fundamental issues of galactic dynamics, the construction of self-consistent galactic models: in brief, the combined mass of the star populated orbits of the system, characterised by a phase space distribution function f, should be equal to the density function [FORMULA] derived by the imposed potential [FORMULA] through Poissons' equation, and vice-versa. As f should represent a stationary solution of the Vlassov (or collisionless Boltzmann) equation, it depends in all the phase space variables through the existing integrals of motion (Jeans 1915). The analytic solution of the problem, in order to recover a (fine-grained) phase space distribution function f, is cumbersome but still possible, when two or three integrals of motion are known explicitly (Dejonghe 1986; Binney & Tremaine 1987; Evans 1993; Hunter & Qian 1993; Gerhard 1994). Nonetheless, in the case of generic Hamiltonian systems corresponding to realistic galactic potentials, we dispose only one explicit integral of motion, i.e. the energy integral defined by the constant value of the Hamiltonian. Thus, we rely mostly on numerical methods of linear programming, first devised by Schwarzschild (1979). However, there are still many unresolved problems regarding the computation of a (coarse grained) phase space distribution function through a reliable orbit based technique for both regular and irregular orbits.

This is indeed a quite difficult task from the point of view of the theory of Hamiltonian dynamical systems. In fact, rigorous answers can be given when we are dealing with systems sufficiently close to the two idealistic situations: an integrable system or a completely "chaotic" one (e.g. the K or the C systems - see Arnol'd & Avez 1967). In both cases, the important dynamical property respected on specific sub-manifolds of the phase space is ergodicity1 (for more details see Cornfeld et al. 1989). In the former case, the existing integrals of motion define invariant manifolds topologically equivalent to tori (provided that the phase space is a compact - see Arnol'd (1976)). Orbits evolving on these surfaces are quasi-periodic and fill them ergodically. On the other hand, in the "completely chaotic" case, the orbits are ergodic on the constant energy manifold. Thus, in both situations, we may be able to calculate a mean phase space density, depending on all the integrals of motion (in the close to integrable case), or only on the energy (in the "chaotic" case). Unfortunately, nature is not as simple as these two extreme cases. Most of the Hamiltonian systems generated by realistic galactic potentials are decomposable (non-ergodic) on the constant energy manifolds, as they contain both regular orbits evolving on KAM invariant surfaces and chaotic orbits which belong to the complementary space.

On the other hand, the exact physical parameters which perturb the Hamiltonian systems in question are not the same for all types of galaxies. For example, in the case of barred-spiral galaxies, the chaotic trajectories may be important due to the rotation of the ensemble, the bar perturbation (Contopoulos & Papayannopoulos 1980; Contopoulos 1983; Athanassoula et al. 1983; Pfenniger 1984) and the effects of the gas joining the bar and the arms (Gontopoulos et al. 1989 - for the influence of the latter on self-consistency see Kaufmann & Contopoulos 1996). In contrast, in the case of elliptical galaxies, where the figures are weakly tumbling and the dynamics is gas-free (de Zeeuw & Franx 1991; Merritt 1993; Gerhard 1994), the existence of chaotic regions should be attributed to their supposedly triaxial shapes. Indeed, nowadays, there is a trend towards the belief that axisymmetry is a rather artificial assumption for elliptical galaxies (e.g. de Zeeuw 1996).

Yet, in the previous decade, besides the few triaxial models which were perturbations of the integrable class of Stäckel systems (Kuzmin 1956; de Zeeuw & Lynden-Bell 1985), most of the studied potentials simulating the motion of stars in elliptical galaxies respected rotational symmetries or were close to axisymmetric. Thus, it was argued that chaos is unlikely to be relevant for real galaxies (e.g. Richstone 1982; Binney & Spergel 1982, Binney 1982), with the notable exception of Goodman & Schwarzschild (1981), who claimed that, although chaotic orbits may exist, they seem regular for astronomically interesting time scales. As a result, many "self-consistent" models have been constructed, using the Schwarzschild's linear programming approach or variants of this method, either by using only the quasi-periodic orbits (Schwarzschild 1979; Statler 1987) or by employing both regular and irregular orbits without making a clear distinction between them (Kuijken 1993; Schwarzschild 1993).

Nevertheless, for the central parts of ellipticals, the approximation of irregular orbits by regular ones may be dubious, since the existence of cusps or massive black holes may shorten the dynamical times of stars which pass close to these regions. Consequently, the consideration of chaotic orbits should be inevitable (Gerhard and Binney 1985; Udry & Pfenniger 1988; Lees & Schwarzschild 1992). The work of Merritt and Fridman (1996) contributed to the clarification of the latter issues, as they constructed two self-consistent models of elliptical galaxies with steep and shallow cusps by including irregular orbits in their self-consistent solutions. A particular remark, springing from their analysis was that the extent of the chaotic regions was underestimated in most of the previous studies of triaxial systems. Indeed, the principal planes of motion, which are actually axisymmetric restrictions of the 3-dimensional problem, are expected to be poor guides for the understanding of the behaviour of the general system. Moreover, all the classical methods used so far in order to distinguish regular from irregular motion, as the calculation of the Lyapunov exponents, fail to provide a global view of the dynamics.

In the present study, we will attempt to generalise the previous results in the case of a classic galactic core potential, by displaying a clear image of its 3-dimensional dynamics with the help of the frequency map analysis method. The studied model is the softened version of the widely-known logarithmic potential (Richstone 1980, 1982, 1984; Binney 1981; Binney & Spergel 1982, 1984; Magnenat 1982; Levison & Richstone 1987; Binney & Tremaine 1987; de Zeeuw & Pfenniger 1988; Miralda-Escudé & Schwarzschild 1989; Patsis & Zachilas 1990; Contopoulos & Seimenis 1990; Lees & Schwarzschild 1992; Kuijken 1993; Schwarzschild 1993; Evans 1993, 1994; de Zeeuw et al. 1996). This potential has found an ample number of applications in galactic dynamics due to the fact that although simple, it is capable to reproduce the dynamics of elliptical galaxies or galactic haloes (in its scale free form). The frequency analysis method was already exploited for studying the dynamics of the 2-dimensional restriction of the system (Papaphilippou & Laskar 1996, Paper I). In that work, we verified the suggestions that chaos exists in a rather restricted scale, at least for the physically acceptable values of the axial ratio. Nevertheless, it seems natural that the addition of one degree of freedom changes dramatically the dynamics of the system. The frequency maps provide a clear representation of the triaxial system's regular and chaotic regions, and their evolution with respect to the related shape parameters. It enables us to depict graphically the geography of resonances, leading thus to a profound understanding of the system's global dynamics. Based on the overwhelming efficiency of the method, in comparison with the classical approaches used in galactic dynamics, we will be able to recover many new features of the 3-dimensional dynamics of galactic models and their effects to the system's physical space.

The article is organised in quite the same way as Paper I: in Sect. 2 we present briefly the frequency map analysis, as well as some new rigorous results regarding the method's precision. Next, (Sect. 3) we give a brief description of the Hamiltonian structure of the system, reviewing and illuminating certain aspects of its orbital dynamics. The quasi-periodic approximations given by the frequency map analysis help us to sharpen our understanding regarding the principal families of regular orbits of the logarithmic potential (Sect. 4). The following section (Sect. 5) is devoted to studying the global dynamics of the system through the frequency maps, for a number of axial ratios. All these dynamical aspects are also viewed in the system's physical space. The last section sums up the principal results and points out some perspectives for future studies.

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© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997