Fugitive stars in active galaxies

We explore the regular or chaotic nature of orbits moving in the meridional plane of an axially symmetric galactic gravitational model with a disk, a dense spherical nucleus and some additional perturbing terms corresponding to influence from nearby galaxies. In order to obtain this we use the Smaller ALingment Index (SALI) technique integrating extensive samples of orbits. Of particular interest is the study of distant, remote stars moving in large galactocentric orbits. Our extensive numerical experiments indicate that the majority of the distant stars perform chaotic orbits. However, there are also distant stars displaying regular motion as well. Most distant stars are ejected into the galactic halo on approaching the dense and massive nucleus. We study the influence of some important parameters of the dynamical system, such as the mass of the nucleus and the angular momentum, by computing in each case the percentage of regular and chaotic orbits. A second order polynomial relationship connects the mass of the nucleus and the critical angular momentum of the distant star. Some heuristic semi-theoretical arguments to explain and justify the numerically derived outcomes are also given. Our numerical calculations suggest that the majority of distant stars spend their orbital time in the halo where it is easy to be observed. We present evidence that the main cause for driving stars to distant orbits is the presence of the dense nucleus combined with the perturbation caused by nearby galaxies. The origin of young O and B stars observed in the halo is also discussed.

In this work, we try to shed some light to the nature of orbits in a three-dimensional (3D) potential of a perturbed harmonic oscillator with eight possible channels of escape, which was chosen as an interesting example of open 3D Hamiltonian systems. In particular, we conduct a thorough numerical investigation distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering unbounded motion for several values of the energy. In an attempt to discriminate safely and with certainty between ordered and chaotic motion, we use the Smaller ALingment Index (SALI) detector, computed by integrating numerically the basic equations of motion as well as the variational equations. Of particular interest is to locate the basins of escape toward the different escape channels and connect them with the corresponding escape periods of the orbits. We split our study into three different cases depending on the initial value of the \(z\) coordinate which was used for launching the test particles. We found that when the orbits are started very close to the primary \((x,y)\) plane the respective grids exhibit a high degree of fractalization, while on the other hand for orbits with relatively high values of \(z_0\) several well-formed basins of escape emerge thus reducing significantly the fractalization of the grids. It was also observed that for values of energy very close to the escape energy the escape times of orbits are large, while for energy levels much higher than the escape energy the vast majority of orbits escape extremely fast or even immediately to infinity. We hope our outcomes to be useful for a further understanding of the escape process in open 3D Hamiltonian systems.

Chaotic orbits suffer significant changes as a result of small perturbations. One can thus wonder whether the dynamical friction suffered by a satellite on a regular orbit, and interacting with the stars of a galaxy, will be different if the bulk of the stars of the galaxy are in regular or chaotic orbits. In order to check that idea, we investigated the orbital decay (caused by dynamical friction) of a rigid satellite moving within a larger stellar system (a galaxy) whose potential is nonintegrable. We performed numerical experiments using two kinds of triaxial galaxy models: (1) the triaxial generalization of Dehnen's spherical mass model (Dehnen; Merritt & Fridman); (2) a modified Satoh model (Satoh; Carpintero, Muzzio, & Wachlin). The percentages of chaotic orbits present in these models were increased by perturbing them. In the first case, a central compact object (black hole) was introduced; in the second case, the perturbation was produced by allowing the galaxy to move on a circular orbit in a logarithmic potential. The equations of motion were integrated with a non-self-consistent code. Our results show that the presence of chaotic orbits does not affect significantly the orbital decay of the satellite.

Recurrence in the phase space of complex systems is a well-studied phenomenon, which has provided deep insights into the nonlinear dynamics of such systems. For dissipative systems, characteristics based on recurrence plots have recently attracted much interest for discriminating qualitatively different types of dynamics in terms of measures of complexity, dynamical invariants, or even structural characteristics of the underlying attractor's geometry in phase space. Here, we demonstrate that the latter approach also provides a corresponding distinction between different co-existing dynamical regimes of the standard map, a paradigmatic example of a low-dimensional conservative system. Specifically, we show that the recently developed approach of recurrence network analysis provides potentially useful geometric characteristics distinguishing between regular and chaotic orbits. We find that chaotic orbits in an intermittent laminar phase (commonly referred to as sticky orbits) have a distinct geometric structure possibly differing in a subtle way from those of regular orbits, which is highlighted by different recurrence network properties obtained from relatively short time series. Thus, this approach can help discriminating regular orbits from laminar phases of chaotic ones, which presents a persistent challenge to many existing chaos detection techniques.

We created a triaxial stellar system through the cold dissipationless collapse of 100,000 particles whose evolution was followed with a multipolar code. Once an equilibrium system had been obtained, the multipolar expansion was freezed and smoothed in order to get a stationary smooth potential. The resulting model was self-consistent and the orbits and Lyapunov exponents could then be computed for a randomly selected sample of 3472 of the bodies that make up the system. More than half of the orbits (52.7 % ) turned out to be chaotic. Regular orbits were then classified using the frequency analysis automatic code of Carpintero and Aguilar (1998, MNRAS 298(1), 1–21). We present plots of the distributions of the different kinds of orbits projected on the symmetry planes of the system. We distinguish chaotic orbits with only one non-zero Lyapunov exponent from those with two non-zero exponents and show that their spatial distributions differ, that of the former being more similar to the one of the regular orbits. Most of the regular orbits are boxes and boxlets, but the minor axis tubes play an important role filling in the wasp waists of the boxes and helping to give a lentil shape to the system. We see no problem in building stable triaxial models with substantial amounts of chaotic orbits; the difficulties found by other authors may be due not to a physical cause but to a limitation of Schwarzschild’s method.

We use a composite gravitational galactic model consisting of a disk, a halo, a massive nucleus and a strong nuclear bar, in order to study the connections between global and local parameters in a realistic dynamical system. The local model is constructed from a two-dimensional perturbed harmonic oscillator and can be derived by expanding the global model in the vicinity of the central stable Lagrange equilibrium point. The frequencies of oscillations are not arbitrary, but they are connected with all the parameters involved with the global model. Moreover, the value of the local energy is also connected with the value of the global energy. Low and high energy stars in the global model display chaotic motion. Comparison with previous research reveals that the presence of the massive nucleus is responsible for the chaotic motion of the low energy stars. In the local motion, the low energy stars show interesting resonance phenomena, but the chaotic motion, if any, is negligible. On the contrary, the high energy stars do not show bounded motion in the local model. This is an indication of particular activity near the center of galaxies possessing massive nuclei.

A 3D dynamical model with a quasi-homogeneous core and a disk component is used for the chaos control in the central parts of elliptical galaxy. Numerical experiments in the 2D system show a very complicated phase plane with a large chaotic sea, considerable sticky layers and a large number of islands, produced by secondary resonances. When the mass of the disk increases, the chaotic regions decrease gradually, and, finally, a new phase plane with only regular orbits appears. This evolution indicates that disks in elliptical galaxies can act as the chaos controllers. Starting from the results obtained in the 2D system, we locate the regions in the phase space of the 3D system, producing regular and chaotic orbits. For this we introduce and use a new dynamical parameter, the S(w) spectrum, which proves to be useful as a fast indicator and allows us to distinguish the regular motion from chaos in the 3D potentials. Other methods for detecting chaos are also discussed.

We explore the classical dynamics of atoms in an optical dipole trap formed by two identical Gaussian beams propagating in perpendicular directions. The phase space is a mixture of regular and chaotic orbits, the latter becoming dominant as the energy of the atoms increases. The trapping capabilities of these perpendicular Gaussian beams are investigated by considering an atomic ensemble in free motion. After a sudden turn on of the dipole trap, a certain fraction of atoms in the ensemble remains trapped. The majority of these trapped atoms has energies larger than the escape channels, which can be explained by the existence of regular and chaotic orbits with very long escape times.

We construct models of triaxial galactic nuclei containing central black holes using the method of orbital superposition, then verify their stability by advancing N-body realizations of the models forward in time. We assume a power-law form for the stellar density, rho ~ 1/r and 1/r^2; these correspond approximately to the nuclear density profiles of bright and faint galaxies respectively. Equidensity surfaces are ellipsoids with fixed axis ratios. The central black hole is represented by a Newtonian point mass. We consider three triaxial shapes: almost prolate, almost oblate and maximally triaxial. Two kinds of orbital solution are attempted for each mass model: the first including only regular orbits, the second including chaotic orbits as well. We find that stable configurations exist in the maximally triaxial and nearly-oblate cases; however steady-state solutions in the nearly-prolate geometry could not be found. A large fraction of the mass, of order 50% or more, could be assigned to the chaotic orbits without inducing evolution. Our results demonstrate that triaxiality may persist even within the sphere of influence of the central black hole, and that chaotic orbits may constitute an important building block of galactic nuclei.

The Poincaré recurrence statistic (PRS) measures the probability that a trajectory initiated in a phase space region will first return to that region, as a function of time. For deterministic, area-preserving maps with a mixture of regular and chaotic orbits, the stickiness of invariant tori and islands is responsible for a power-law decay in the PRS. We show that noise perturbations allow trajectories to access the interior of islands, enhancing their trapping effect and causing many orbits to take longer to return to a neighborhood of their initial conditions. The noisy PRS can then exhibit an extended tail on an intermediate time scale, which is eventually followed by an asymptotic exponential decay. We study a typical example, the Harper map, and compare distributions of trapping and visit times to islands with recurrence times to show the importance of noise in creating tails in the PRS. A simple finite-state Markov model of the dynamics confirms how this slower decay can be caused by noise permitting entry to previously inaccessible regions.

Two self-consistent (N-body) non-rotating equilibrium models of elliptical galaxies with smooth central density profiles (called ‘Q’ and ‘C’ models) are constructed, starting from quiet and clumpy cosmological initial conditions, respectively. Both models are triaxial. The Q model has an E7 maximum ellipticity in the inner parts and tends to E6 or E5 maximum ellipticity in the outer parts. The C model has a maximum ellipticity E4 in the inner parts and tends to an E2 or E1 in the outer parts. For each model, we identify the particles moving in chaotic orbits with the Lyapunov number exceeding a particular threshold (namely, 10 −2.8 , in units of the inverse radial periods of the particular orbits). At energy levels in the deepest 30 per cent of the potential well, no chaotic orbits were detected in the above limit of chaoticity. In the Q model, the detected chaotic part is 32 per cent of the total mass. This part has a nearly spherical distribution. It imposes limitations on the maximum ellipticity of the system, in spite of the fact that only a part of less than about 8 per cent of the total mass moves in chaotic orbits and is able to develop chaotic diffusion within a Hubble time. In the C model, the detected chaotic part is about 26 per cent of the total mass, but only less than 2 per cent can develop chaotic diffusion within a Hubble time. These chaotic components produce surface density profiles flatter than the profiles of the rest of the mass, particularly in the Q model. The two profiles intersect at a given distance, where the overall profile forms an observable hump, especially if the surface density profiles are taken along the shortest axis of the projection.

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.

A direct numerical method for quantifying regular and chaotic orbits

We review the dynamical mechanisms we have found to support the morphological features in barred-spiral galaxies based on chaotic motions of stars in their gravitational fields. These morphological features are the spiral arms, that emerge out of the ends of the bar, but also shape the bar itself. The potentials used have been estimated directly from near-infrared images of barred-spiral galaxies. In this paper, we present the results from the study of the dynamics of the potentials of the galaxies NGC 4314, NGC 1300 and NGC 3359. The main unknown parameter in our models is the pattern speed of the system Ωp. By varying Ωp, we have investigated several cases trying to match the results of our modeling with available photometrical and kinematical data. We found realistic models with stars on spirals in chaotic motion, while their bars are built by stars usually on regular orbits. However, we also encountered cases, where a major part of trajectories of the stars even in the bar is chaotic as well. Finally, we examined the gas dynamics of barred-spiral systems, and found that the presence of gas reinforces the intensity of the "chaotic" spiral arms.

The 2:3 and 3:4 exterior mean motion resonances with Neptune are studied by applying symplectic mapping models. The mappings represent efficiently Poincare maps for the 3D elliptic restricted three body problem in the neighbourhood of the particular resonances. A large number of trajectories is studied showing the coexistence of regular and chaotic orbits. Generally, chaotic motion depletes the small bodies of the effective resonant region in both the 2:3 and 3:4 resonances. Applying a low frequency spectral analysis of trajectories, we determined the phase space regions that correspond to either regular or chaotic motion. It is found that the phase space of the 3:4 resonant motion is more chaotic than the 2:3 one.