Regular and Chaotic Dynamics of Triaxial Stellar Systems

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Abstract
We use Laskar's frequency mapping technique to study the dynamics of triaxial galaxies with central density cusps and nuclear black holes. For ensembles of ~104 orbits, we numerically compute the three fundamental frequencies of the motion, allowing us to map out the Arnold web. We also compute diffusion rates of stochastic orbits in frequency space. The objects of greatest importance in structuring phase space are found to be the three-dimensional resonant tori, regions where the fundamental frequencies satisfy a relation of the form 0 = lω1 + mω2 + nω3 with integer (l, m, n). When stable, resonant tori generate phase-space regions in which the motion is regular; these regions are not necessarily associated with a stable periodic orbit as in systems with only 2 degrees of freedom. Boxlike orbits are generically stochastic, but some tube orbits are stochastic as well. The spectrum of diffusion rates for boxlike orbits at a given energy is well approximated as a power law over at least 6 decades. Models with high central concentrations—steep central cusps or massive black holes—exhibit the most stochasticity. Even a modest black hole, with a mass of ~0.5% the mass of the galaxy, is as effective as the steepest central density cusp at inducing stochastic diffusion. There is a transition to global stochasticity in boxlike phase space when the mass of the central black hole exceeds ~2% of the galaxy mass. We estimate the dependence of orbital evolution rates on galaxy structural parameters. We predict a greater average degree of dynamical evolution in faint elliptical galaxies because of their high central densities and short crossing times. The evolution time is estimated to be shorter than a galaxy lifetime for absolute magnitudes fainter than about -19 or -20, consistent with the observed change in many elliptical galaxy properties at this luminosity.

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