Orbital structure of self-consistent triaxial stellar systems

We used a multipolar code to create, through the dissipationless collapses of systems of 1,000,000 particles, three self-consistent triaxial stellar systems with axial ratios corresponding to those of E4, E5 and E6 galaxies. The E5 and E6 models have small, but significant, rotational velocities although their total angular momenta are zero, that is, they exhibit figure rotation; the rotational velocity decreases with decreasing flattening of the models and for the E4 model it is essentially zero. Except for minor changes, probably caused by unavoidable relaxation effects, the systems are highly stable. The potential of each system was subsequently approximated with interpolating formulae yielding smooth potentials, stationary for the non-rotating model and stationary in the rotating frame for the rotating ones. The Lyapunov exponents could then be computed for randomly selected samples of the bodies that make up the different systems, allowing the recognition of regular and partially and fully chaotic orbits. Finally, the regular orbits were Fourier analyzed and classified using their locations on the frequency map. As it could be expected, the percentages of chaotic orbits increase with the flattening of the system. As one goes from E6 through E4, the fraction of partially chaotic orbits relative to that of fully chaotic ones increases, with the former surpassing the latter in model E4; the likely cause of this behavior is that triaxiality diminishes from E6 through E4, the latter system being almost axially symmetric. We especulate that some of the partially chaotic orbits may obey a global integral akin to the long axis component of angular momentum. Our results show that is perfectly possible to have highly stable triaxial models with large fractions of chaotic orbits, but such systems cannot have constant axial ratios from center to border: a slightly flattened reservoir of highly chaotic orbits seems to be mandatory for those systems.     

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.     

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.     

Realistic triaxial density–potential–force profiles for stellar systems and dark matter haloes

Popular models for describing the luminosity–density profiles of dynamically hot stellar systems (e.g. Jaffe, Hernquist, Dehnen) were constructed to match the deprojected form of de Vaucouleurs' R ^1/4 light-profile. However, we now know that elliptical galaxies and bulges display a mass-dependent range of structural profiles. To compensate this, the model in Terzić & Graham was designed to closely match the deprojected form of Sérsic R ^{1/ n} light-profiles, including deprojected exponential light-profiles and galaxies with partially depleted cores. It is thus applicable for describing bulges in spiral galaxies, dwarf elliptical galaxies, both 'power-law' and 'core' elliptical galaxies, also dark matter haloes formed from Λ cold dark matter cosmological simulations. In this paper, we present a new family of triaxial density–potential–force triplets, which generalizes the spherical model reported in Terzić & Graham to three dimensions. If the (optional) power-law core is present, it is a five-parameter family, while in the absence of the core it reduces to three parameters. The isodensity contours in the new family are stratified on confocal ellipsoids and the potential and forces are expressed in terms of integrals which are easy to evaluate numerically. We provide the community with a suite of numerical routines for orbit integration, which feature: optimized computations of potential and forces for this family; the ability to run simulations on parallel platforms; and modular and easily editable design.     

Chaos in cuspy triaxial galaxies with a supermassive black hole: a simple toy model

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, ${\text{V}}_{\text{0}} = \frac{1}{2}\left( {a^2 x^2 + b^2 y^2 + c^2 z^2 } \right)$ , and aspherical Plummer potential, ${\text{V}}_{\text{P}} = M_{BH} /\sqrt {r^2 + \varepsilon ^2 } $ , with $r^2 = x^2 + y^2 + z^2$ . Attention focuses on three issues related tothe 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, that is, how large are their largest (short time) Lyapunov exponents? (3) To what extent is the motion of chaotic orbits impeded by Arnold webs, that is, 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 ε, and energy E with the aims of understanding how the manifestations of chaos depend onthe 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.     

Analytic stellar structure

This paper continues to elaborate on analytic methods to construct models for the internal structure of solar-type stars. Since a detailed stellar model is desired, a nonlinear analytic density distribution in terms of a two-parameter family of models has been assumed. Hydrostatic equilibrium and energy conservation determine the conditions in the gravitationally stabilized stellar fusion reactor. The results show once more that methods of differential and integral calculus provide a laboratory for the application of special functions of mathematical physics in stellar structure.Paper presented at the Second International Conference on Industrial and Applied Mathematics, July 8–12 1991, Washington D.C., U.S.A.     

Color functions of stellar systems

Model calculations of the photometric evolution of rather dense stellar systems, such as globular clusters, are presented. On “luminosity-effective temperature” diagrams of these systems, low-mass stars are concentrated near the minimum and maximum temperatures for a given luminosity and are deficient in the intermediate region. This sort of double-peaked distribution of the stars can be avoided in open models with ejection of excess metals into the surrounding medium. The distributions of the stars with respect to effective temperature on a “ luminosity-effective temperature” diagram are sensitive to the history of star formation in the system and to possible time variations in the initial mass function. In open systems with a single-peak distribution function, the asymmetry in the distribution varies over wide limits with the lower bound for the initial mass function and this can be used to establish whether the first generations of stars might have been more massive than in the present epoch. __________ Translated from Astrofizika, Vol. 49, No. 1, pp. 139–150 (February 2006).     

Dynamics of plance stellar systems

The evolution of initially balanced rotating disks of stars is investigated with a computer model for isolated disks of stars. An isolated, initially cold balanced disk is found to be violently unstable. Adding a sufficient amount of velocity dispersion will stabilize all small-scale disturbances. However, the disks are still unstable against slowly growing long wave-length modes and after about two rotations most disks tend to assume a bar-shaped structure. It is found that the final mass distribution over most of the disk can be closely approximated by an exponential variation, irrespective of the initial mass distribution. The gravitational two-stream instability is investigated by means of a modified computer model for infinite doubly periodic stellar systems.     

Tomography of collisionless stellar systems

In this paper the concept of tomography of a collisionless stellar system of general shape is introduced, and a generalization of the Projected Virial Theorem is obtained. Applying the tomographic procedure we then derive a new family of virial equations which coincides with the already known ones for spherically symmetric systems. This result is obtained without any use of explicit expressions for the line-of-sight velocity dispersion, or spherical coordinate system.     

Orbital motion of a massive object in a dense stellar system

In this paper we show the positional oscillation of a massive object in a dense stellar system by numerical N -body simulations. We found that the central massive object, which at first is placed at rest at the centre of the surrounding spherical stellar system, promptly departs from the centre and rotates in accordance with the rotation of the stellar system, if the stellar system has an appreciable rotation. This oscillatory motion continues for a long time because of the absence of dynamical friction. Such a long-lasting oscillation may explain the asymmetric structure observed in the centres of M31 and NGC 4486B, may cause the secular flow of gaseous elements distributed in the central regions of galaxies on to the massive object, and may ignite activity in the centres of galaxies.     

Collisional processes in stellar systems

The theory of collisional relaxation in stellar systems is discussed in terms of an expansion in powers of 1/N, the inverse of the total number of stars. The results are expressed in terms of the concept of gravitational polarization.     

Relative chaos in stellar systems

Statistical properties of many-dimensional dynamical systems-stellar systems of different types, are investigated by means of new definition of relative chaos based on the estimation of the Ricci curvature in the direction of the velocity of geodesics. Numerical experiment is performed to calculate the Ricci and scalar curvatures for systems with equal total energy. The results of calculations enable one to obtain schematic classification of stellar systems by increasing degree of chaos.