Orbital structure of self-consistent cuspy triaxial stellar systems

We used a multipolar code to create, through dissipationless collapses of systems of 10⁶ particles, two cuspy self-consistent triaxial stellar systems with γ ≈ 1. One of the systems has an axial ratio similar to that of an E4 galaxy and it is only mildly triaxial (T = 0.914), while the other one is strongly triaxial (T = 0.593) and its axial ratio lies in between those of Hubble types E5 and E6. Both models rotate although their total angular momenta are zero, i.e., they exhibit figure rotation. The angular velocity is very small for the less triaxial model and, while it is larger for the more triaxial one, it is still comparable to that found by Muzzio (Celest Mech Dynam Astron 96(2):85–97, 2006) to affect only slightly the dynamics of a similar model. Except for minor evolution, probably caused by unavoidable relaxation effects of the N-body code, the systems are highly stable. The potential of each system was subsequently approximated with interpolating formulae yielding smooth potentials, stationary in frames that rotate with the models. The Lyapunov exponents could then be computed for randomly selected samples of the bodies that make up the two systems, allowing the recognition of regular and of partially and fully chaotic orbits. Finally, the regular orbits were Fourier analyzed and classified using their locations on the frequency map. Most of the orbits are chaotic, and by a wide margin: less than 30% of the orbits are regular in our most triaxial model. Regular orbits are dominated by tubes, long axis ones in the less triaxial model and short axis tubes in the more triaxial one. Most of the boxes are resonant (i.e., they are boxlets), as could be expected from cuspy systems.     

Regular and chaotic orbits in a self-consistent triaxial stellar system with slow figure rotation

We created a self-consistent triaxial stellar system through the cold disipationless collapse of 100,000 particles whose evolution was followed with a multipolar code. The resulting system rotates slowly even though its total angular momentum is zero, i.e., it offers an example of figure rotation. The potential of the system was subsequently approximated with interpolating formulae yielding a smooth potential stationary in the rotating frame. The Lyapunov exponents could then be computed for a randomly selected sample of 3,472 of the bodies that make up the system, allowing the recognition of regular and partially and fully chaotic orbits. The regular orbits were Fourier analyzed and classified using their locations on the frequency map. A comparison with a similar non-rotating model showed that the fraction of chaotic orbits is slightly but significantly enhanced in the rotating model; alternatively, there are no significant differences between the corresponding fractions neither of partially and fully chaotic orbits nor of long axis tubes, short axis tubes, boxes and boxlets among the regular orbits. This is a reasonable result because the rotation causes a breaking of the symmetry that may increase chaotic effects, but the rotation velocity is probably too small to produce any other significant differences. The increase in the fraction of chaotic orbits in the rotating system seems to be due mainly to the effect of the Coriolis force, rather than the centrifugal force, in good agreement with the results of other investigations.     

Dark halos acting as chaos controllers in asymmetric triaxial galaxy models

We study the regular or chaotic character of orbits in a 3D dynamical model,describing a triaxial galaxy surrounded by a spherical dark halo component.Our numerical experiments suggest that the percentage of chaotic orbits decreases exponentially as the mass of the dark halo increases.A linear increase of the percentage of the chaotic orbits was observed as the scale length of the halo component increases. In order to distinguish between regular and chaotic motion,we chose to use the total angular momentum ...     

Sensitivity of the orbital content of a model stellar system to the potential approximation used to describe it

We have classified orbits in a stationary triaxial stellar system created from a cold dissipationless collapse of 100,000 particles. In order to integrate the orbits, two potential approximations with different fitting functions were used in turn. We found that the relative amount of chaotic versus regular orbits does depend on the chosen approximation of potential, even though both models resulted in very good fits of the underlying exact potential. On the other hand, the content of regular orbits, i.e., its distribution among main families, does not strongly depend of the potential approximation, being therefore a more robust signature of the gravitational system under study.     

Spatial Structure of Regular and Chaotic Orbits in A Self-Consistent Triaxial Stellar System

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.     

A three dimensional investigation of two dimensional orbits

Orbits in the principal planes of triaxial potentials are known to be prone to unstable motion normal to those planes, so that three dimensional investigations of those orbits are needed even though they are two dimensional. We present here an investigation of such orbits in the well known logarithmic potential which shows that the third dimension must be taken into account when studying them and that the instability worsens for lower values of the forces normal to the plane. Partially chaotic orbits are present around resonances, but also in other regions. The action normal to the plane seems to be related to the isolating integral that distinguishes regular from partially chaotic orbits, but not to the integral that distinguishes partially from fully chaotic orbits.     

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.     

An approach to the radial distribution of planetary orbits

The problem of determination of the radial distribution of the planetary orbits is approached under the assumption that the average present radial sizes of the orbits were already determined when the protoplanetary cloud flattened by initial angular momentum aggregated into a set of concentric rings from which the planetary material was ultimately collected. The object of this argument is to derive a consistent stationary distribution of orbits so that the problem of the non-stationary formation of the orbital rings is not here considered. Under the flattening assumption the 3D Poisson equation is replaced by the 2D Helmholtz equation (inhomogeneous) which is solved by use of an averaging theorem generalization of the well-known averaging theorem for the homogeneous Helmholtz equation. Augmenting the ring potentials obtained by specializing the mass distribution in the disk by a solar potential term and a rotational potential, differentiation leads to a generalization of the Kepler 3D law suitable for the many-body problem of a solar system with circular orbits. In this way a system of transcendental equations involving Bessel functions of the first and second kind are obtained which must be satisfied by the orbital radii. Naturally the restriction to circular orbits represents only an approximation to the orbital determination problem, but considering that no arguments have previously been available for the determination even of circular orbits it would seem to represent an advance.     

Dynamical friction on globular clusters in core-triaxial galaxies: is it a cause of massive black hole accretion?

The present work extends and deepens previous examinations of the evolution of globular cluster orbits in elliptical galaxies, by means of numerical integrations of a wide set of orbits in five self-consistent triaxial galactic models characterized by a central core and different axial ratios. These models are valid and complete in the representation of regular orbits in elliptical galaxies. Dynamical friction is definitely shown to be an efficient cause of evolution for the globular cluster systems in elliptical galaxies of any mass or axial ratio. Moreover, our statistically significant sample of computed orbits confirms that the globular cluster orbital decay times are, at least for clusters moving on box orbits, much shorter than the age of the galaxies. Consequently, the mass carried into the innermost galactic region in the form of decayed globular clusters may have contributed significantly to feeding and accreting a compact object therein.     

Dynamical evolution of two associated galactic bars

We study the dynamical interactions of mass systems in equilibrium under their own gravity that mutually exert and ex‐perience gravitational forces. The method we employ is to model the dynamical evolution of two isolated bars, hosted within the same galactic system, under their mutual gravitational interaction. In this study, we present an analytical treatment of the secular evolution of two bars that oscillate with respect to one another. Two cases of interaction, with and without geometrical deformation, are discussed. In the latter case, the bars are described as modified Jacobi ellipsoids. These triaxial systems are formed by a rotating fluid mass in gravitational equilibrium with its own rotational velocity and the gravitational field of the other bar. The governing equation for the variation of their relative angular separation is then numerically integrated, which also provides the time evolution of the geometrical parameters of the bodies. The case of rigid, non‐deformable, bars produces in some cases an oscillatory motion in the bodies similar to that of a harmonic oscillator. For the other case, a deformable rotating body that can be represented by a modified Jacobi ellipsoid under the influence of an exterior massive body will change its rotational velocity to escape from the attracting body, just as if the gravitational torque exerted by the exterior body were of opposite sign. Instead, the exchange of angular momentum will cause the Jacobian body to modify its geometry by enlarging its long axis, located in the plane of rotation, thus decreasing its axial ratios. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

The structure of motion in a 4-component galaxy mass model

We use a composite galaxy model consisting of a disk-halo, bulge, nucleus and dark-halo components in order to investigate the motion of stars in ther-z plane. It is observed that high angular momentum stars move in regular orbits. The majority of orbits are box orbits. There are also banana-like orbits. For a given value of energy, only a fraction of the low angular momentum stars — those going near the nucleus — show chaotic motion while the rest move in regular orbits. Again one observes the above two kinds of orbits. In addition to the above one can also see orbits with the characteristics of the 2/3 and 3/4 resonance. It is also shown that, in the absence of the bulge component, the area of chaotic motion in the surface of section increases, significantly. This suggests that a larger number of low angular momentum stars are in chaotic orbits in galaxies with massive nuclei and no bulge components.     

Determining the type of orbits in the central regions of barred galaxies

We use a simple dynamical model which consists of a harmonic oscillator and a spherical component, in order to investigate the regular or chaotic character of orbits in a barred galaxy with a central spherically symmetric nucleus. Our aim is to explore how the basic parameters of the galactic system influence the nature of orbits, by computing in each case the percentage of chaotic orbits, as well as the percentages of different types of regular orbits. We also give emphasis to the types of regular orbits that support either the formation of nuclear rings or the barred structure of the galaxy. We provide evidence that the traditional x1 orbital family does not always dominate in barred galaxy models since we found several other types of resonant orbits which can also support the barred structure. We also found that sparse enough nuclei, fast rotating bars and high energy models can support the galactic bars. On the other hand, weak bars, dense central nuclei, slowly rotating bars and low energy models favor the formation of nuclear rings.We also compare our results with previous related work.     

Triaxial equilibrium ellipsoids among the asteroids?

We present a physical model to explain the existence of a class of large-lightcurve-amplitude, rapidly rotating asteroids found most commonly among objects in the size range 100–300 km diameter. A significant correlation between rotation period and lightcurve amplitude exists for asteroids in this size range in the sense that those with larger amplitudes spin more rapidly and hence these objects have high rotational angular momenta. Since this is a property of Jacobi ellipsoids, we have investigated whether these asteriods might be examples of triaxial equilibrium ellipsoids. We find that objects rotating with periods of 6 hr must have densities between 1.1 and 1.4 g cm^?3, while those rotating in 4 hr would have densities between 2.4 and 3.2 g cm^?3. If this model is valid then at least some of these asteroids have rather low mean densities. The reality of this result and its interpretation in terms of collisional evolution of the asteroids is discussed.     

Global Dynamics in Self-Consistent Models of Elliptical Galaxies

We compare two different N-body models simulating elliptical galaxies. Namely, the first model is a non-rotating triaxial N-body equilibrium model with smooth center, called SC model. The second model, called CM model, is derived from the SC by inserting a central mass in it, so that all possible differences between the two models are due to the effect of the central mass. The central mass is assumed to be mainly due to a massive central black hole of mass about 1% of the total mass of the galaxy. By using the fundamental frequency analysis, the two systems are thoroughly investigated as regards the types of orbits described either by test particles, or by the real particles of the systems at all the energy levels. A comparison between the orbits of test particles and the orbits of real particles at various energy levels is made on the rotation number plane. We find that extensive stable regions of phase space, detected by test particles remain empty, i.e. these regions are not occupied by real particles, while many real particles move in unstable regions of phase space describing chaotic orbits. We run self-consistently the two models for more than a Hubble time. During this run, in spite of the noise due to small variations of the potential, the SC model maintains (within a small uncertainly) the number of particles moving on orbits of each particular type. In contrast, the CM model is unstable, due to the large amount of mass in chaotic motion caused by the central mass. This system undergoes a secular evolution towards an equilibrium state. During this evolution it is gradually self-organized by converting chaotic orbits to ordered orbits mainly of the short axis tube type approaching an oblate spheroidal equilibrium. This is clearly demonstrated in terms of the fundamental frequencies of the orbits on the rotation number plane and the time evolution of the triaxiality index.     

On various approaches to investigating the instability of stellar systems with highly elongated stellar orbits

We study the various approximations used to investigate the eigenmode spectrum for systems with highly elongated stellar orbits. The approximation in which the elongated orbits are represented by thin rotating spokes, with the rotation imitating the precession of real orbits, is the simplest and most natural one. However, we show that using this pictorial approximation does not allow the picture of stability to be properly presented. We show that for stellar systems with a plane disk geometry, this approach does not allow unstable spectral modes to be obtained even in the leading order in small parameter, which characterizes the spread of nearly radial orbits in angular momentum. For spherical systems, where the situation is more favorable, the spectrum can be determined but only in the leading order in this parameter. A rigorous approach based on the solution of more complex integral equations given here should be used to properly investigate the stability of stellar systems.     

Spatial Structure of Regular and Chaotic Orbits in Self-Consistent Models of Galactic Satellites

In several previous papers we had investigated the orbits of the stars that make up galactic satellites, finding that many of them were chaotic. Most of the models studied in those works were not self-consistent, the single exception being the Heggie and Ramamani (1995) models; nevertheless, these ones are built from a distribution function that depends on the energy (actually, the Jacobi integral) only, what makes them rather special. Here we built up two self-consistent models of galactic satellites, freezed theirs potential in order to have smooth and stationary fields, and investigated the spatial structure of orbits whose initial positions and velocities were those of the bodies in the self-consistent models. We distinguished between partially chaotic (only one non-zero Lyapunov exponent) and fully chaotic (two non-zero Lyapunov exponents) orbits and showed that, as could be expected from the fact that the former obey an additional local isolating integral, besides the global Jacobi integral, they have different spatial distributions. Moreover, since Lyapunov exponents are computed over finite time intervals, their values reflect the properties of the part of the chaotic sea they are navigating during those intervals and, as a result, when the chaotic orbits are separated in groups of low- and high-valued exponents, significant differences can also be recognized between their spatial distributions. The structure of the satellites can, therefore, be understood as a superposition of several separate subsystems, with different degrees of concentration and trixiality, that can be recognized from the analysis of the Lyapunov exponents of their orbits.     

Extension of Cassini's Laws

We investigate the Cassini's laws which describe the rotational motion in a 1:1 spin-orbit resonance. When this rotational motion follows the conventional Cassini's laws, the figure axis coincides with the angular momentum axis. In this case we underline the differences between the rotational Hamiltonian for a 'slow rotating' body like the Moon and for a 'fast rotating' body like Phobos. Then, we study a more realistic rotational Hamiltonian where the angle J between the figure axis and the angular momentum axis could be different from zero. This Hamiltonian has not been studied before. We have found a new particular solution for this Hamiltonian which could be seen as an extension of the Cassini's laws. In this new solution the angle J is constant, which is not zero, and the precession of the angular momentum plane is equal to the mean motion of the argument of pericenter of the rotating body. This type of rotational motion is only possible when the orbital eccentricity of the rotating body is not zero. This new law enables describing in particular, the Moon mean rotational motion for which the mean value of the angle J is found to be equal to 103.9±0.7 s of arc.