Astron. Astrophys. 326, 113-129 (1997)

1. Introduction and motivation

1.1. On the assumptions of standard galaxy dynamics

The classical assumption often made in galaxy dynamics (Binney & Tremaine 1987 (BT) chapter 4) is that present day galaxies can be treated as collisionless fluids in steady states described by one particle distribution functions obeying the time independent collisionless Boltzmann equation (CBE). In addition, it is often supposed that these distribution functions are completely characterized by the isolating integrals of motion (of single stars).

This implies that almost all orbits are regular and therefore stars conserve as many integrals of motion (in involution) as the number of spatial dimensions they move in. Thus, for a three dimensional system of N stars there are conserved quantities and the N -body problem is solvable by quadratures-or integrable (see e.g., Whittaker 1937 or Goldstein 1980 for discussions using classical analysis: more modern treatments can be found in Arnold 1989 (ARN) or Abraham & Marsden 1978).

Thus in this picture galaxies are modeled as classical fluids with time independent densities which give rise to potentials similar to those for which the Hamilton-Jacobi equation is separable. Obviously these assumptions cannot be strictly satisfied for real galaxies. It is argued however, that due to the large two body relaxation time (e.g., Saslaw 1985)

(where v is the characteristic speed of a star and n is the number density of "field stars" of mass m) that these systems can be treated as effectively collisionless over many Hubble times. However all that Eq. (1) is saying is that the direct impulse from encounters between pairs of stars is small relative to the velocity of a star in the smoothed potential. This is due to the long range nature of gravitational interactions which ensures that the whole system contributes to the mean force on a star at all times. In the approximation leading to (1) however the perturbation to the trajectory of a test star due to a certain field star is only calculated at their closest approach - that is only once during a crossing time. Nevertheless, due to the complicated nature of the solutions of the Newtonian equations for , discreteness can have a major indirect effect; the nonlinearity of the problem prevents the adding up of the motions due to individual binary interactions to each other and to the motion in the mean field since although the forces add up linearly the solutions of the Newtonian equations do not. This suggests that the N -body problem is perhaps best studied in its entirety. In this regard it may be perhaps useful to recall that the collisionless steady-state approximation is not a trivial simplification of the dynamics: It reduces the N -body problem to N one-particle problems in a given potential, thus reducing the number of coupled first order differential equations to be solved from equations to 6 equations.

For general Hamiltonian dynamical systems, regular motions form invariant (under time propagation of the solution) tori in the phase-space. In integrable systems these occupy the whole of that space. Under perturbations, however, some tori are destroyed leaving behind volumes of phase space where irregular (or chaotic) motion can occur. The extent of the chaotic region will depend on the strength of the perturbation. Bounds given by the KAM theorem predict that a positive measure of tori will survive provided the perturbation to the potential , where is a critical amplitude (e.g., Mackay & Meiss 1987). In general, the value of decreases as and has been shown to be irrelevant for many higher dimensional physical systems (Pettini 1993 (P93) and the references therein) while the perturbation due to discreteness noise decreases only as if it is random (e.g., Saslaw 1985). The diffusion time away from the remaining tori is also a decreasing function of N ; (Nekhoroshev 1977; Perry & Wiggins 1994). These results suggest that a system is unlikely to become more regular with increasing N, in direct contradiction with the results of two body relaxation estimates but as expected from considering the complicated nature of solutions of generic N -body systems. In fact, for the particular case of a spherical gravitational system, it was shown by Gurzadyan & Savvidy (1986) (GS) that as N increases this system tends towards an Anosov (1967) C-system with maximal instability in phase-space (if one neglects escapes and direct collisions). Making the same kind of assumptions as those used to obtain Eq. (1) (by considering an infinite and homogeneous medium), GS obtain the following relation for the N -body relaxation time arising from the instability

which is considerably shorter than the binary relaxation time.

C-systems are very irregular - to the point that their evolution can be described as a Markow process (e.g., Pesin 1989). The collisionless approximation however predicts that spherical systems end up in completely integrable steady states - the most regular and predictable systems that can exist. The contradiction can possibly arise because the analysis leading to (1) focuses on the fact that the force function becomes smoother as N increases. It then assumes directly that this implies the solution becoming more regular not taking into account that exponentially smaller perturbations are sufficient with increasing N to make an N -body system unstable. Therefore, while some quantities that do not depend on the exact details of the dynamics may be slowly evolving and not sensitively dependent on the discreteness noise, the solutions themselves are heavily sensitive to noise. For example, the change in the energies of stars that arises from discreteness noise is likely to be much slower than the changes in their trajectories. A rough example is the fact that numerically integrated orbits in fixed (and smooth) potentials can have the energy (and all the Poincaré invariants) conserved along their trajectories to up to ten digits while the trajectories themselves are completely inaccurate perhaps not resembling (even qualitatively) the real ones (El-Zant 1996b). The numerical errors here represent the noise (if they are taken to be random, a proposition still under discussion: McCauley 1993). This is because here energy is a scalar function of the phase space variables and is related to the force by a path independent integral. That path on the other hand is a 6 component vector in phase space. One can thus easilly envisage perturbations that change the trajectory of a particle without changing its energy. A long time-scale of energy relaxation does not therefore imply that the actual detailed dynamics are collisionless in the sense of being stable solutions of the CBE. One would expect that some macroscopic quantities could be affected, for example velocity dispersion or the actual shape of the gravitational system.

In fact, simulations of single particle motions in fixed potentials show that the time-averaged phase-space density distribution of individual chaotic trajectories, as well as their statistical properties, change significantly over a time-scale when the potential is given some graininess (Pfenniger 1986; Udry & Pfenniger 1988), or when the trajectories are given small random kicks without changing their total energies significantly (Kandrup 1994; Merritt & Valluri 1996; El-Zant 1996b). The full N -body problem would be expected to be much more irregular and prone to evolution on small (compared to ) time-scales. Such effects are actually observed in N -body simulations of up to heavily softened particles but are assumed to be due to the relatively small number of particles present, even though the two body relaxation time in these simulations is still usually much larger than the time-scales considered. (a review of such occurrences is given by Hernquist & Ostriker 1992: see also van Albada 1986; Sellwood 1987: Zhang 1996 presents a detailed study of an example of a case where discreteness noise interacts with global irregularities in the density thus triggering evolution). In addition, processes involving non-stellar objects such as interactions with giant molecular clouds in galactic disks or black holes in halos, small dissipative perturbations (e.g., Pfenniger & Norman 1990) etc. can cause even more serious trouble for the collisionless approximation.

Even if the collisionless approximation does hold, this does not guarantee that a given density distribution would not evolve over a Hubble time. This is because individual trajectories, even in a smooth steady-state potential, can have time dependent density distributions over such a time-scale. This would be the case in general non-spherical potentials as noted by Binney (1982). Hasan et al. (1993) describe such behaviour for barred spirals while Merritt & Fridman (1996) show that it may also be important for ellipticals. El-Zant (1996b) examines the case of trajectories started near the symmetry plane of disks embedded in triaxial halos. Finally, it seems that perhaps there is also some observational evidence for evolutionary phenomena occuring in galaxies (Wielen 1977; Pfenniger et al. 1994; Courteau et al. 1996; Pucacco 1992).

The above considerations suggest (although in no way prove) that even though gravitational systems obey the CBE in the infinite N limit (e.g., Braun & Hepp 1977) the existence and stability (against discreteness noise) of steady state solutions of that equation are in question when a significant amount of chaos is present. This means that while in some cases the classical theory may still hold over a few Gyr (considering the primitive evolutionary state of cold disks in some spiral galaxies one has to admit that this could often be the case) it does not always obviously do so.

1.2. How chaos drives evolution

A distribution of stars giving rise to an integrable potential can either oscillate coherently or reach a macroscopic steady state through phase mixing. This is a trivial form of relaxation which is simply due to stars moving at different angular frequencies on their respective KAM tori in the 6-dimensional phase-space - the motion in the -dimensional phase-space being a torus characterized by the integrals. Phase mixing conserves the action variables characterising this torus and which are crucial in determining the physical state of a system. Most galaxies are not believed to be undergoing significant large scale oscillations, we therefore conclude that if they are integrable dynamical systems they must be in a steady state. Moreover, if they are sufficiently far from any chaotic system they must be stable to small perturbations (KAM theory). This is what is assumed in the classical theory.

A necessary condition for evolution therefore is the non-integrability of the system. This condition is satisfied for N -body gravitational systems since there are no global integrals of motion other than the classical ones (Poincaré 1889). However, not all non-integrable systems exhibit interesting behaviour (different from classical theory) in the time-scales of interest. We need the property of phase-space mixing - that is the spread of localized volume elements (corresponding to sets of initial conditions) to cover large areas of the phase-space (while still conserving their original Lebesgue measure: e.g., Sagdeev et al. 1988). Phase-space mixing leads to diffusion in the action variables and may therefore lead to evolution in a system's physical parameters. For this process to be efficient it is necessary that the system be sufficiently chaotic so that the diffusion occurs over short enough time-scales and covers a large range of initial states, only then can we say that the macroscopic state corresponding to a given set of micro-states can evolve over a Hubble time.

It is therefore important to examine different methods for detecting these processes and the time-scales associated with them in practical situations where the predictions can be tested. The rest of this study is a step in this direction. In the next section we describe why certain subtleties related to N -body gravitational systems require local methods to be used in the quantification of chaotic behaviour. One such approach based on the geometry of the configuration manifold is then described. In Sect. 3 we describe some of the possible applications of this approach while in Sect. 4 results of some numerical experiments testing the method are reported.

© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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