Chaos and chaotic phase mixing in cuspy triaxial potentials

Abstract

This paper continues an investigation of chaos and chaotic phase mixing in triaxial generalizations of the Dehnen potential which have been proposed to describe realistic elliptical galaxies that have a strong density cusp and manifest significant deviations from axisymmetry. Earlier work is extended in three important ways, namely by exploring systematically the effects of (1) variable axis ratios, (2) ‘graininess’ associated, for example, with stars and bound substructures, idealized as friction and white noise, and (3) large-scale organized motions within a galaxy and a dense cluster environment, each presumed to induce near-random forces idealized as coloured noise with a finite autocorrelation time. The effects of varying the axis ratio were studied in detail by considering two sequences of models with cusp exponent Γ= 1 and, respectively, axis ratios a : b : c = 1.00: 1.00 −Δ: 0.50 and a : b : c = 1.00: 1.00 −Δ: 1.00 − 2Δ for variable Δ. Three important conclusions are that (1) not all the chaos can be attributed to the presence of the cusp, (2) significant chaos can persist even for axisymmetric systems, and (3) the introduction of a supermassive black hole can induce both moderate increases in the relative number of chaotic orbits and substantial increases in the size of the largest Lyapunov exponent. In the absence of any perturbations, the coarse-grained distribution function associated with an initially localized ensemble of chaotic orbits evolves exponentially towards a nearly time-independent form at a rate Λ that correlates with the typical values of the finite-time Lyapunov exponents Χ associated with the evolving orbits. Allowing for discreteness effects and/or an external environment accelerates phase-space transport both by increasing the rate at which orbits spread out within a given phase-space region and by facilitating diffusion along the Arnold web that connects different phase-space regions, so as to facilitate an approach towards a true equilibrium. The details of the perturbation appear unimportant. All that really matters are the amplitude and, for the case of coloured noise, the autocorrelation time, i.e. the characteristic time over which the perturbation varies. Overall, the effects of the perturbations scale logarithmically in both amplitude and autocorrelation time. Even comparatively weak perturbations can increase Λ by a factor of three or more, a fact that has potentially significant implications for violent relaxation.