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Register a new userChaos and chaotic phase mixing in cuspy triaxial potentials
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This paper investigates chaos and chaotic phase mixing in triaxial Dehnen potentials which have been proposed to describe realistic ellipticals. Earlier work is extended by exploring the effects of (1) variable axis ratios, (2) `graininess associated with stars and bound substructures, idealised as friction and white noise, and (3) large-scale organised motions presumed to induce near-random forces idealised as coloured noise with finite autocorrelation time. Three important conclusions are: (1) not all the chaos can be attributed to the cusp; (2) significant chaos can persist even for axisymmetric systems; and (3) introducing a supermassive black hole can increase both the relative number of chaotic orbits and the size of the largest Lyapunov exponent. Sans perturbations, distribution functions associated with initially localised chaotic ensembles evolve exponentially towards a nearly time-independent form at a rate L that correlates with the finite time Lyapunov exponents associated with the evolving orbits. Perturbations accelerate phase space transport by increasing the rate of phase mixing in a given phase space region and by facilitating diffusion along the Arnold web that connects different phase space regions, thus facilitating an approach towards a true equilibrium. The details of the perturbation appear unimportant. All that matters are the amplitude and the autocorrelation time, upon which there is a weak logarithmic dependence. Even comparatively weak perturbations can increase L by a factor of three or more, a fact that has potentially significant implications for violent relaxation.
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(Abridged) This paper studies chaotic orbit ensembles evolved in triaxial generalisations of the Dehnen potential which have been proposed to model ellipticals with a strong density cusp that manifest significant deviations from axisymmetry. Allowance is made for a possible supermassive black hole, as well as low amplitude friction, noise, and periodic driving which can mimic irregularities associated with discreteness effects and/or an external environment. The degree of chaos is quantified by determining how (1) the relative number of chaotic orbits and (2) the size of the largest Lyapunov exponent depend on the steepness of the cusp and the black hole mass, and (3) the extent to which Arnold webs significantly impede phase space transport, both with and without perturbations. In the absence of irregularities, chaotic orbits tend to be extremely `sticky, so that different pieces of the same chaotic orbit can behave very differently for 10000 dynamical times or longer, but even very low amplitude perturbations can prove efficient in erasing many -- albeit not all -- these differences. The implications thereof are discussed both for the structure and evolution of real galaxies and for the possibility of constructing approximate near-equilibrium models using Schwarzschilds method. Much of the observed qualitative behaviour can be reproduced with a toy potential given as the sum of an anisotropic harmonic oscillator and a spherical Plummer potential, which suggests that the results may be generic.
This paper discusses three new issues that necessarily arise in realistic attempts to apply nonlinear dynamics to galaxy evolution, namely: (i) the meaning of chaos in many-body systems, (ii) the time-dependence of the bulk potential, which can trigger intervals of {em transient chaos}, and (iii) the self-consistent nature of any bulk chaos, which is generated by the bodies themselves, rather than imposed externally. Simulations and theory both suggest strongly that the physical processes associated with galactic evolution should also act in nonneutral plasmas and charged particle beams. This in turn suggests the possibility of testing this physics in real laboratory experiments, an undertaking currently underway.
This paper summarises an investigation of chaos in a toy potential which mimics much of the behaviour observed for the more realistic triaxial generalisations of the Dehnen potentials, which have been used to model cuspy triaxial galaxies both with and without a supermassive black hole. The potential is the sum of an anisotropic harmonic oscillator potential, V_o=(1/2)*(a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2}), and a spherical Plummer potential, V_o=-M_{BH}/(r^{2}+e^{2})^{1/2} with r^{2}=x^{2}+y^{2}+z^{2}. Attention focuses on three issues related to the properties of ensembles of chaotic orbits which impact on chaotic mixing and the possibility of constructing self-consistent equilibria: (1) What fraction of the orbits are chaotic? (2) How sensitive are the chaotic orbits, i.e., how large are their largest (short time) Lyapunov exponents? (3) To what extent is the motion of chaotic orbits impeded by Arnold webs, i.e.,, how `sticky are the chaotic orbits? These questions are explored as functions of the axis ratio a:b:c, black hole mass M_BH, softening length e, and energy E with the aims of understanding how the manifestations of chaos depend on the shape of the system and why the black hole generates chaos. The simplicity of the model makes it amenable to a perturbative analysis. That it mimics the behaviour of more complicated potentials suggests that much of this behaviour should be generic.
This paper explores how orbits in a galactic potential can be impacted by large amplitude time-dependences of the form that one might associate with galaxy or halo formation or strong encounters between pairs of galaxies. A period of time-dependence with a strong, possibly damped, oscillatory component can give rise to large amounts of transient chaos, and it is argued that chaotic phase mixing associated with this transient chaos could play a major role in accounting for the speed and efficiency of violent relaxation. Analysis of simple toy models involving time-dependent perturbations of an integrable Plummer potential indicates that this chaos results from a broad, possibly generic, resonance between the frequencies of the orbits and harmonics thereof and the frequencies of the time-dependent perturbation. Numerical computations of orbits in potentials exhibiting damped oscillations suggest that, within a period of 10 dynamical times t_D or so, one could achieve simultaneously both `near-complete chaotic phase mixing and a nearly time-independent, integrable end state.
This talk provides a progress report on an extended collaboration which has aimed to address two basic questions, namely: Should one expect to see cuspy, triaxial galaxies in nature? And can one construct realistic cuspy, triaxial equilibrium models that are robust? Three technical results are described: (1) Unperturbed chaotic orbits in cuspy triaxial potentials can be extraordinarily sticky, much more so than orbits in many other three-dimensional potentials. (2) Even very weak perturbations can be important by drastically reducing, albeit not completely eliminating, this stickiness. (3) A simple toy model facilitates a simple understanding of why black holes and cusps can serve as an effective source of chaos. These results suggest that, when constructing models of galaxies using Schwarzschilds method or any analogue thereof, astronomers would be well advised to use orbital building blocks that have been perturbed by `noise or other weak irregularities, since such building blocks are likely to be more nearly time-independent than orbits evolved in the absence of all perturbations.
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