Phase Mixing in Unperturbed and Perturbed Hamiltonian Systems

This paper summarises a numerical investigation of phase mixing in time-independent Hamiltonian systems that admit a coexistence of regular and chaotic phase space regions, allowing also for low amplitude perturbations idealised as periodic driving, friction, and/or white and coloured noise. The evolution of initially localised ensembles of orbits was probed through lower order moments and coarse-grained distribution functions. In the absence of time-dependent perturbations, regular ensembles disperse initially as a power law in time and only exhibit a coarse-grained approach towards an invariant equilibrium over comparatively long times. Chaotic ensembles generally diverge exponentially fast on a time scale related to a typical finite time Lyapunov exponent, but can exhibit complex behaviour if they are impacted by the effects of cantori or the Arnold web. Viewed over somewhat longer times, chaotic ensembles typical converge exponentially towards an invariant or near-invariant equilibrium. This, however, need not correspond to a true equilibrium, which may only be approached over very long time scales. Time-dependent perturbations can dramatically increase the efficiency of phase mixing, both by accelerating the approach towards a near-equilibrium and by facilitating diffusion through cantori or along the Arnold web so as to accelerate the approach towards a true equilibrium. The efficacy of such perturbations typically scales logarithmically in amplitude, but is comparatively insensitive to most other details, a conclusion which reinforces the interpretation that the perturbations act via a resonant coupling.     

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 idealized as a sum of near-random events which can be modelled 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 modelled as resulting from a sequence of instantaneous kicks, i.e. white noise. Instead, the diffusion is modelled as resulting from random kicks of finite duration, i.e. coloured noise, characterized by a non-zero autocorrelation time t _c. A detailed analysis of coloured noise generated by sampling an Ornstein–Uhlenbeck process leads to a simple scaling in terms of t _c and an effective diffusion constant D . Interpreting D and t _c following early work by Chandrasekhar (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.     

Orbital structure in N-body models of barred-spiral galaxies

We study the orbital structure in a series of self-consistent N -body configurations simulating rotating barred galaxies with spiral and ring structures. We perform frequency analysis in order to measure the angular and the radial frequencies of the orbits at two different time snapshots during the evolution of each N -body system. The analysis is done separately for the regular and the chaotic orbits. We thereby identify the various types of orbits, determine the shape and percentages of the orbits supporting the bar and the ring/spiral structures, and study how the latter quantities change during the secular evolution of each system. Although the frequency maps of the chaotic orbits are scattered, we can still identify concentrations around resonances. We give the distributions of frequencies of the most important populations of orbits. We explore the phase-space structure of each system using projections of the 4D surfaces of section. These are obtained via the numerical integration not only of the orbits of test particles, but also of the real N -body particles. We thus identify which domains of the phase space are preferred and which are avoided by the real particles. The chaotic orbits are found to play a major role in supporting the shape of the outer envelope of the bar as well as the rings and the spiral arms formed outside corotation.     

Tidal tails of star clusters

Based on recent findings of a formation mechanism of substructure in tidal tails by Küpper et al., we investigate a more comprehensive set of N -body models of star clusters on orbits about a Milky Way like potential. We find that the predicted epicyclic overdensities arise in any tidal tail no matter which orbit the cluster follows as long as the cluster lives long enough for the overdensities to build up.
The distance of the overdensities along the tidal tail from the cluster centre depends for circular orbits only on the mass of the cluster and the strength of the tidal field, and therefore decreases monotonically with time, while for eccentric orbits the orbital motion influences the distance, causing a periodic compression and stretching of the tails and making the distance oscillate with time. We provide an approximation for estimating the distance of the overdensities in this case.
We describe an additional type of overdensity which arises in extended tidal tails of clusters on eccentric orbits, when the acceleration of the tidal field on the stellar stream is no longer homogeneous. Moreover, we conclude that a pericentre passage or a disc shock is not the direct origin of an overdensity within a tidal tail. Escape due to such tidal perturbations does not take place immediately after the perturbation but is rather delayed and spread over the orbit of the cluster. All observable overdensities are therefore of the mentioned two types. In particular, we note that substructured tidal tails do not imply the existence of dark matter substructures in the haloes of galaxies.     

Regular motions in double bars – I. Double-frequency orbits and loops

Bars in galaxies are mainly supported by particles trapped around stable periodic orbits. These orbits represent oscillatory motion with only one frequency, which is the bar driving frequency, and miss free oscillations. We show that a similar situation takes place in double bars: particles get trapped around parent orbits, which in this case represent oscillatory motion with two frequencies of driving by the two bars, and which also lack free oscillations. Thus the parent orbits, which constitute the backbone of an oscillating potential of two independently rotating bars, are the double-frequency orbits. These orbits do not close in any reference frame, but they map on to closed curves called loops. Trajectories trapped around the parent double-frequency orbit map on to a set of points confined within a ring surrounding the loop.     

Spin-induced orbital perturbations for different types of planetary bodies

Using numerical simulations, we studied several coupled translational and rotational solutions of the two-finite-body problem with one spherical and one triaxial body. The aim was to investigate which types of orbits and planetary bodies could produce spin-induced orbital perturbations relevant enough to add to models dealing with other perturbations. To fully assess the strengths and consequences of this perturbation, we did not include any other perturbation even when a more realistic scenario would have required it. Interesting results concern planet–star mass ratios like a hot Jupiter or a super-Jupiter around a star like the Sun or the red dwarf Proxima Centauri. The short-period chaotic effect of the gravitational spin–orbit perturbation on highly eccentric orbits in the vicinity of the Roche limit can be a prominent feature. It should be taken into account when studying the tidal evolution of such a planet or its interactions with any companion in the neighborhood of the star.     

Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

In the presence of a strong   m = 2  component in a rotating galaxy, the phase-space structure near corotation is shaped to a large extent by the invariant manifolds of the short-period family of unstable periodic orbits terminating at L ₁ or L ₂. The main effect of these manifolds is to create robust phase correlations among a number of chaotic orbits large enough to support a spiral density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine–Gordon equation. Numerical examples are given in an N -body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar–spiral pattern.     

Thermal instability in a weakly ionized plasma

We revisit the problem of clump formation due to thermal instabilities in a weakly ionized plasma with the help of a linear perturbation analysis, as discussed by Nejad-Asghar & Ghanbari. In the absence of a magnetic field and ambipolar diffusion the characteristic equation reduces to the thermal instability described by Field. We derive the critical wavelengths, which separate the spatial ranges of stability and instability. Contrary to the original analysis of Nejad-Asghar & Ghanbari, perturbations with a wavelength larger than the critical wavelength destabilize the cloud. Moreover, the instability regime of isentropic perturbations is drastically reduced. Isobaric modes with real values of the critical wavelength appear only if the density dependence of the cooling rate is more pronounced than the temperature dependence. Isentropic modes arise only if the power of the density in the cooling rate is smaller than 1/2, which is not fulfilled for CO cooling. We find that ambipolar diffusion is not a dominating heating process in molecular gas.     

Modeling the dynamical profile of a BL Lacertae object

A simple dynamical model for a BL Lacertae active galaxy is presented. The model consists of a logarithmic potential with an additional term representing internal perturbations. The time independent and the evolving model are investigated. In both cases we search for regular and chaotic motion and study the velocity distribution near the centre of the system. Numerical calculations suggest that responsible for the chaotic phenomena is the internal perturbation, the flattening parameter and the dense nucleus. The radius of the nucleus also affects the maximum velocity in the central regions of the galaxy. Our numerical outcomes are supported by theoretical arguments and analytical calculations. A linking of our numerical outcomes to observational data is also presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics of an interacting luminous disc, dark halo and satellite companion

This paper describes a method, based on linear perturbation theory, to determine the dynamical interaction between extended halo and spheroid components and an environmental disturbance. One finds that resonant interaction between a galaxy and passing interlopers or satellite companions can carry the disturbance inward, deep inside the halo, where it can perturb the disc.    Applied to the Milky Way for example, the LMC and SMC appear to be sufficient to cause the observed Galactic warp and possibly seed other asymmetries. This is a multi-scale interaction in which the halo wake has a feature at roughly half the satellite orbital radius owing to a 2:1 orbital resonance. The rotating disturbance then excites an m  = 1 vertical disc mode which has the classic integral-sign morphology. A polar satellite orbit produces the largest warp and therefore the inferred LMC orbit is nearly optimal for maximum warp production.   Both the magnitude and morphology of the response depend on the details of the disc and halo models. Most critically, a change in the halo profile will shift the resonant frequencies and response location and consequently alter the coupling to the bending disc. Increasing the halo support relative to the disc, a sub-maximal disc model, decreases the warp amplitude.   Finally, the results and prognosis for N -body simulations are discussed. Discreteness noise in the halo, similar to that arising from a population of 10⁶-M⊙ black holes, can produce observable warping.     

Global m= 1 instabilities and lopsidedness in disc galaxies

Lopsidedness is common in spiral galaxies. Often, there is no obvious external cause, such as an interaction with a nearby galaxy, for such features. Alternatively, the lopsidedness may have an internal cause, such as a dynamical instability. In order to explore this idea, we have developed a computer code that searches for self-consistent perturbations in razor-thin disc galaxies and performed a thorough mode-analysis of a suite of dynamical models for disc galaxies embedded in an inert dark matter halo with varying amounts of rotation and radial anisotropy.
Models with two equal-mass counter-rotating discs and fully rotating models both show growing lopsided modes. For the counter-rotating models, this is the well-known counter-rotating instability, becoming weaker as the net rotation increases. The m = 1 mode of the maximally rotating models, on the other hand, becomes stronger with increasing net rotation. This rotating m = 1 mode is reminiscent of the eccentricity instability in near-Keplerian discs.
To unravel the physical origin of these two different m = 1 instabilities, we studied the individual stellar orbits in the perturbed potential and found that the presence of the perturbation gives rise to a very rich orbital behaviour. In the linear regime, both instabilities are supported by aligned loop orbits. In the non-linear regime, other orbit families exist that can help support the modes. In terms of density waves, the counter-rotating m = 1 mode is due to a purely growing Jeans-type instability. The rotating m = 1 mode, on the other hand, grows as a result of the swing amplifier working inside the resonance cavity that extends from the disc centre out to the radius where non-rotating waves are stabilized by the model's outwardly rising Q profile.     

Coupled galactic density-wave modes in a composite system of thin stellar and gaseous discs

A rotating disc galaxy is modelled as a composite system consisting of thin stellar and gaseous discs, which are described by a two-fluid modal formalism. The composite disc system is assumed to retain axisymmetry in the background equilibrium. General density-wave perturbations in the two discs are coupled through the mutual gravitational interaction. We study the basic properties of open and tight spiral density-wave modes in such a composite disc system. Within the Lindblad resonances, perturbation enhancements of surface mass density in stellar and gaseous discs are in phase; this is also true during the initial growth phase of density-wave perturbations. Outside the Lindblad resonances, there exists a possible spiral density-wave branch for which perturbation enhancements of surface mass density in stellar and gaseous discs are out of phase. We discuss implications of these results on the critical parameters for global star formation in barred and normal spiral galaxies and on magnetohydrodynamic density waves within the Lindblad resonances.     

Self-consistent response of a galactic disc to vertical perturbations

We study the self-consistent, linear response of a galactic disc to vertical perturbations, as induced, say, by a tidal interaction. We calculate the self-gravitational potential corresponding to a non-axisymmetric, self-consistent density response of the disc using the Green's function method. The response potential is shown to oppose the perturbation potential because the self-gravity of the disc resists the imposed potential, and this resistance is stronger in the inner parts of a galactic disc. For the   m = 1  azimuthal wavenumber, the disc response opposes the imposed perturbation up to a radius that spans a range of 4–6 disc scalelengths, so that the disc shows a net warp only beyond this region. This physically explains the well known but so far unexplained observation that warps typically set in beyond this range of radii. We show that the inclusion of a dark matter halo in the calculation only marginally changes (by ∼10 per cent) the radius for the onset of warps. For perturbations with higher azimuthal wavenumbers, the net signature of the vertical perturbations can only be seen at larger radii – for example, beyond 7 exponential disc scalelengths for   m = 10  . Also, for the high- m cases, the magnitude of the negative disc response due to the disc self-gravity is much smaller. This is shown to result in corrugations of the mid-plane density, which explains the puzzling scalloping with   m = 10  detected in H  i in the outermost regions ∼30 kpc in the Galaxy.     

The origin and formation of cuspy density profiles through violent relaxation of stellar systems

It is shown that the cuspy density distributions observed in the cores of elliptical galaxies can be realized by dissipationless gravitational collapse. The initial models consist of power-law density spheres such as ρ ∝ r ⁻¹ with anisotropic velocity dispersions. Collapse simulations are carried out by integrating the collisionless Boltzmann equation directly, on the assumption of spherical symmetry. From the results obtained, the extent of constant density cores, formed through violent relaxation, decreases as the velocity anisotropy increases radially, and practically disappears for extremely radially anisotropic models. As a result, the relaxed density distributions become more cuspy with increasing radial velocity anisotropy. It is thus concluded that the velocity anisotropy could be a key ingredient for the formation of density cusps in a dissipationless collapse picture. The velocity dispersions increase with radius in the cores according to the nearly power-law density distributions. The power-law index, n , of the density profiles, defined as ρ ∝ r ^{− n} , changes from n ≈2.1 at intermediate radii to a shallower power than n ≈2.1 toward the centre. This density bend can be explained from our postulated local phase-space constraint that the phase-space density accessible to the relaxed state is determined at each radius by the maximum phase-space density of the initial state.     

The structure of spiral galaxies — II. Near-infrared properties of spiral arms

We have imaged a sample of 45 face-on spiral galaxies in the K band, to determine the morphology of the old stellar population, which dominates the mass in the disc. The K -band images of the spiral galaxies have been used to calculate different characteristics of the underlying density perturbation such as arm strengths, profiles and cross-sections, and spiral pitch angles. Contrary to expectations, no correlation was found between arm pitch angle and Hubble type, and combined with previous results this leads us to conclude that the morphology of the old stellar population bears little resemblance to the optical morphology used to classify galaxies. The arm properties of our galaxies seem inconsistent with predictions from the simplest density wave theories, and some observations, such as variations in pitch angle within galaxies, seem hard to reconcile even with more complex modal theories. Bars have no detectable effect on arm strengths for the present sample. We have also obtained B -band images of three of the galaxies. For these galaxies we have measured arm cross-sections and strengths, to investigate the effects of disc density perturbations on star formation in spiral discs. We find that B -band arms lead K -band arms and are narrower than K -band arms, apparently supporting predictions made by the large-scale shock scenario, although the effects of dust on B -band images may contribute towards these results.     

Determining the nature of orbits in a three‐dimensional galaxy model hosting a BL Lacertae object

A three‐dimensional dynamical model for a galaxy hosting a BL Lacertae object is constructed. The model consists of a logarithmic potential representing an elliptical host galaxy with a bulge of radius cb and a dense massive nucleus. Using numerical experiments, we try to distinguish between regular and chaotic motion in both 2D and 3D system. In particular, we investigate how the basic parameters of our model, such as the mass of the nucleus, the internal perturbation and the flattening parameters influence the amount and the degree of chaos. Interesting correlations are presented for both 2D and 3D dynamical models. Our numerical results are explained and supported using elementary theoretical arguments and analytical calculations. Of particular interest is the local integral of motion which have been found to exist in the vicinity of stable periodic points. The obtained numerical outcomes of the present research are linked and also compared with several data derived from observations. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gravitational collapse in an expanding background and the role of substructure – II. Excess power at small scales and its effect on collapse of structures at larger scales

We study the interplay of clumping at small scales with the collapse and relaxation of perturbations at larger scales using N -body simulations. We quantify the effect of collapsed haloes on perturbations at larger scales using the two-point correlation function, moments of counts in cells and the mass function. The purpose of the study is twofold and the primary aim is to quantify the role played by collapsed low-mass haloes in the evolution of perturbations at large scales; this is in view of the strong effect seen when the large scale perturbation is highly symmetric. Another reason for this study is to ask whether features or a cut-off in the initial power spectrum can be detected using measures of clustering at scales that are already non-linear. The final aim is to understand the effect of ignoring perturbations at scales smaller than the resolution of N -body simulations. We find that these effects are ignorable if the scale of non-linearity is larger than the average interparticle separation in simulations. Features in the initial power spectrum can be detected easily if the scale of these features is in the linear regime; detecting such features becomes difficult as the relevant scales become non-linear. We find no effect of features in initial power spectra at small scales on the evolved power spectra at large scales. We may conclude that, in general, the effect on the evolution of perturbations at large scales of clumping on small scales is very small and may be ignored in most situations.     

The fine-grained phase-space structure of cold dark matter haloes

We present a new and completely general technique for calculating the fine-grained phase-space structure of dark matter (DM) throughout the Galactic halo. Our goal is to understand this structure on the scales relevant for direct and indirect detection experiments. Our method is based on evaluating the geodesic deviation equation along the trajectories of individual DM particles. It requires no assumptions about the symmetry or stationarity of the halo formation process. In this paper we study general static potentials which exhibit more complex behaviour than the separable potentials studied previously. For ellipsoidal logarithmic potentials with a core, phase mixing is sensitive to the resonance structure, as indicated by the number of independent orbital frequencies. Regions of chaotic mixing can be identified by the very rapid decrease in the real-space density of the associated DM streams. We also study the evolution of stream-density in ellipsoidal NFW haloes with radially varying isopotential shape, showing that if such a model is applied to the Galactic halo, at least 10⁵ streams are expected near the Sun. The most novel aspect of our approach is that general non-static systems can be studied through implementation in a cosmological N -body code. Such an implementation allows a robust and accurate evaluation of the enhancements in annihilation radiation due to fine-scale structure such as caustics. We embed the scheme in the current state-of-the-art code gadget -3 and present tests which demonstrate that N -body discreteness effects can be kept under control in realistic configurations.     

On the treatment of entropy mixing in numerical cosmology

For simulations of fluid dynamics in astrophysics, physical viscosity and diffusion are typically neglected. However, in this high Reynolds number regime, real fluids become highly turbulent and turbulent processes mediate substantial transport of momentum and heat that is diffusive in nature. In the absence of models for these processes, code-dependent numerical effects dominate how diffusion operates and may lead to physically incorrect simulation results. We highlight the qualitative difference in these numerical effects for smooth particle hydrodynamics (SPH) and grid-based Eulerian codes using two test problems: a buoyant gas bubble and gas in a galaxy cluster. Grid codes suffer from numerical diffusion in the absence of explicit terms, and small-scale diffusion of heat is completely absent in the Lagrangian SPH method. We find that SPH with heat diffusion added at a level similar to that expected from turbulence diffusion generates more physically appealing results. These results suggest, but do not confirm, that a flat entropy core is to be expected for gas in an idealized galaxy cluster (i.e. one without physics beyond that of a non-radiating gas). A goal of this work is thus to draw attention to the as yet unfulfilled need for models of turbulent diffusive processes in compressible gases in astrophysics.