Orbital structure in oscillating galactic potentials

This paper focuses on symmetric potentials subjected to periodic driving. Four unperturbed potentials V_0(r) were considered, namely the Plummer potential and Dehnen potentials with \gamma=0.0, 0.5, and 1.0, each subjected to a time-dependence of the form V(r,t)=V_0(r)(1+m_0\sin(\omega t)). In each case, the orbits divide clearly into regular and chaotic, distinctions which appear absolute. In particular, transitions from regularity to chaos are seemingly impossible. Over finite time intervals, chaotic orbits subdivide into what can be termed `sticky' chaotic orbits, which exhibit no large scale secular changes in energy and remain trapped in the phase space region where they started; and `wildly chaotic' orbits, which do exhibit systematic drifts in energy as the orbits diffuse to inhabit different phase space regions. This latter distinction is not absolute, apparently corresponding instead to orbits penetrating a `leaky' phase space barrier. The three different orbit types can be identified simply in terms of the frequencies for which their Fourier spectra have the most power. An examination of the statistical properties of orbit ensembles as a function of driving frequency \omega allows one to identify the specific resonances that determine orbital structure. Attention focuses also on how, for fixed amplitude m_0, such quantities as the mean energy shift, the relative measure of chaotic orbits, and the mean value of the largest Lyapunov exponent vary with driving frequency \omega and how, for fixed \omega, the same quantities depend on m_0.Comment: 16 pages, 9 figures. Accepted for publication to MNRAS. Minor editions and deletions. Updated reference

Energy relaxation in galaxies induced by an external environment and/or incoherent internal pulsations

This paper explores the phenomenon of energy relaxation for stars in a galaxy embedded in a high density environment that is subjected continually to perturbations reflecting the presence of other nearby galaxies and/or incoherent internal pulsations. The analysis is similar to earlier analyses of energy relaxation induced by binary encounters between nearby stars and between stars and giant molecular clouds in that the perturbations are idealised as a sum of near-random events which can be modeled as diffusion and dynamical friction. However, the analysis differs in one important respect: because the time scale associated with these perturbations need not be short compared with the characteristic dynamical time t_D for stars in the original galaxy, the diffusion process cannot be modeled as resulting from a sequence of instantaneous kicks, i.e., white noise. Instead, the diffusion is modeled as resulting from random kicks of finite duration, i.e., coloured noise characterised by a nonzero autocorrelation time t_c. A detailed analysis of coloured noise generated by sampling an Ornstein-Uhlenbeck process leads to a simpling scaling in terms of t_c and an effective diffusion constant D. Interpreting D and t_c following early work by Chandrasekhar (1941) (the `nearest neighbour approximation') implies that, for realistic choices of parameter values, energy relaxation associated with an external environment and/or internal pulsations could be important on times short compared with the age of the Universe.Comment: 7 pages including 4 Figure

Smooth potential chaos and N-body simulations

Integrations in fixed N-body realisations of smooth density distributions corresponding to a chaotic galactic potential can be used to derive reliable estimates of the largest (finite time) Lyapunov exponent X_S associated with an orbit in the smooth potential generated from the same initial condition, even though the N-body orbit is typically characterised by an N-body exponent X_N >> X_S. This can be accomplished either by comparing initially nearby orbits in a single N-body system or by tracking orbits with the same initial condition evolved in two different N-body realisations of the same smooth density.Comment: 9 pages plus 7 figures, expanded version to appear in Astrophysical Journa

Chaos and the continuum limit in nonneutral plasmas and charged particle beams

This paper examines discreteness effects in nearly collisionless N-body systems of charged particles interacting via an unscreened r^-2 force, allowing for bulk potentials admitting both regular and chaotic orbits. Both for ensembles and individual orbits, as N increases there is a smooth convergence towards a continuum limit. Discreteness effects are well modeled by Gaussian white noise with relaxation time t_R = const * (N/log L)t_D, with L the Coulomb logarithm and t_D the dynamical time scale. Discreteness effects accelerate emittance growth for initially localised clumps. However, even allowing for discreteness effects one can distinguish between orbits which, in the continuum limit, feel a regular potential, so that emittance grows as a power law in time, and chaotic orbits, where emittance grows exponentially. For sufficiently large N, one can distinguish two different `kinds' of chaos. Short range microchaos, associated with close encounters between charges, is a generic feature, yielding large positive Lyapunov exponents X_N which do not decrease with increasing N even if the bulk potential is integrable. Alternatively, there is the possibility of larger scale macrochaos, characterised by smaller Lyapunov exponents X_S, which is present only if the bulk potential is chaotic. Conventional computations of Lyapunov exponents probe X_N, leading to the oxymoronic conclusion that N-body orbits which look nearly regular and have sharply peaked Fourier spectra are `very chaotic.' However, the `range' of the microchaos, set by the typical interparticle spacing, decreases as N increases, so that, for large N, this microchaos, albeit very strong, is largely irrelevant macroscopically. A more careful numerical analysis allows one to estimate both X_N and X_S.Comment: 13 pages plus 17 figure

Dynamics of triaxial galaxies with a central density cusp

Cuspy triaxial potentials admit a large number of chaotic orbits, which moreover exhibit extreme "stickiness" that makes the process of chaotic mixing surprisingly inefficient. Environmental effects, modeled as noise and/or periodic driving, help accelerate phase space transport but probably not as much as in simpler potentials. This could mean that cuspy triaxial ellipticals cannot exist as time-independent systems

Chaos and the continuum limit in the gravitational N-body problem II. Nonintegrable potentials

This paper continues a numerical investigation of orbits evolved in `frozen,' time-independent N-body realisations of smooth time-independent density distributions corresponding to both integrable and nonintegrable potentials, allowing for N as large as 300,000. The principal focus is on distinguishing between, and quantifying, the effects of graininess on initial conditions corresponding, in the continuum limit, to regular and chaotic orbits. Ordinary Lyapunov exponents X do not provide a useful diagnostic for distinguishing between regular and chaotic behaviour. Frozen-N orbits corresponding in the continuum limit to both regular and chaotic characteristics have large positive X even though, for large N, the `regular' frozen-N orbits closely resemble regular characteristics in the smooth potential. Viewed macroscopically both `regular' and `chaotic' frozen-N orbits diverge as a power law in time from smooth orbits with the same initial condition. There is, however, an important difference between `regular' and `chaotic' frozen-N orbits: For regular orbits, the time scale associated with this divergence t_G ~ N^{1/2}t_D, with t_D a characteristic dynamical time; for chaotic orbits t_G ~ (ln N) t_D. At least for N>1000 or so, clear distinctions exist between phase mixing of initially localised orbit ensembles which, in the continuum limit, exhibit regular versus chaotic behaviour. For both regular and chaotic ensembles, finite-N effects are well mimicked, both qualitatively and quantitatively, by energy-conserving white noise with amplitude ~ 1/N. This suggests strongly that earlier investigations of the effects of low amplitude noise on phase space transport in smooth potentials are directly relevant to real physical systems.Comment: 20 pages, including 21 FIGURES, uses RevTeX macro

Chaotic mixing in noisy Hamiltonian systems

This paper summarises an investigation of the effects of low amplitude noise and periodic driving on phase space transport in 3-D Hamiltonian systems, a problem directly applicable to systems like galaxies, where such perturbations reflect internal irregularities and.or a surrounding environment. A new diagnsotic tool is exploited to quantify how, over long times, different segments of the same chaotic orbit can exhibit very different amounts of chaos. First passage time experiments are used to study how small perturbations of an individual orbit can dramatically accelerate phase space transport, allowing `sticky' chaotic orbits trapped near regular islands to become unstuck on suprisingly short time scales. Small perturbations are also studied in the context of orbit ensembles with the aim of understanding how such irregularities can increase the efficacy of chaotic mixing. For both noise and periodic driving, the effect of the perturbation scales roughly in amplitude. For white noise, the details are unimportant: additive and multiplicative noise tend to have similar effects and the presence or absence of a friction related to the noise by a Fluctuation- Dissipation Theorem is largely irrelevant. Allowing for coloured noise can significantly decrease the efficacy of the perturbation, but only when the autocorrelation time, which vanishes for white noise, becomes so large that t here is little power at frequencies comparable to the natural frequencies of the unperturbed orbit. This suggests strongly that noise-induced extrinsic diffusion, like modulational diffusion associated with periodic driving, is a resonance phenomenon. Potential implications for galaxies are discussed.Comment: 15 pages including 18 figures, uses MNRAS LaTeX macro

Chaos and Noise in a Truncated Toda Potential

Results are reported from a numerical investigation of orbits in a truncated Toda potential which is perturbed by weak friction and noise. Two significant conclusions are shown to emerge: (1) Despite other nontrivial behaviour, configuration, velocity, and energy space moments associated with these perturbations exhibit a simple scaling in the amplitude of the friction and noise. (2) Even very weak friction and noise can induce an extrinsic diffusion through cantori on a time scale much shorter than that associated with intrinsic diffusion in the unperturbed system.Comment: 10 pages uuencoded PostScript (figures included), (A trivial mathematical error leading to an erroneous conclusion is corrected