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.     

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.     

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.     

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.     

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.     

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).     

N-body simulations of tidal encounters between stellar systems

N-Body simulations have been performed to study the tidal effects of a primary stellar system on a secondary stellar system of density close to the Roche density. Two hyperbolic, one parabolic and one elliptic encounters have been simulated. The changes in energy, angular momentum, mass distribution, and shape of the secondary system have been determined in each case. The inner region containing about 40 per cent of the mass was found to be practically unchanged and the mass exterior to the tidal radius was found to escape. The intermediate region showed tidal distension. The thickness of this region decreased as we went from hyperbolic encounters to the elliptic encounter keeping the distance of closest approach constant. The numerical results for the fractional change in energy have been compared with the predictions of the available analytic formulae and the usefulness and limitations of the formulae have been discussed.     

Effect of radiation on the stability of a retrograde particle orbit in different stellar systems

We have numerically investigated the stability of retrograde orbits/trajectories around Jupiter and the smaller of the primaries in binary systems RW-Monocerotis (RW-Mon) and Krüger-60 in the presence of radiation. A trajectory is considered as stable if it remains around the smaller mass for at least few hundred binary periods. In case of circular binary orbit, we find that the third order resonance provides the basis for reduction of stability region of retrograde motion of particle in RW-Mon and Sun-Jupiter system both in the presence and absence of radiation. Considering finite ellipticity in Sun-Jupiter system we find that for distant retrograde orbits, radiation from the Sun increases the width of the stable region and covers a significant portion of the region obtained in the absence of solar radiation. Further, due to solar radiation pressure, the stable region in the neighborhood of Jupiter has been found to shift much below the characteristic asymptotic line for the periodic retrograde orbits. In case of Krüger-60 we observe the distant retrograde orbits around the smaller of the primaries get affected considerably with increase in radiation parameter β₁. Further the range of velocities for which stable motion may persist narrows down for distant retrograde orbits in this system.     

The standard systems of stellar rotation measurements (II)

The limb darkening effect on the measurement of stellar rotation is discussed in this paper. It is shown that this effect plays an important role in the measurement of V_e sin i. In the extreme case with the limb darkening coefficient of 1.0, it may cause a difference up to 17%. As a sequel to paper [1], this work presents the following new explanation for the systematic differences between Slettebak's new and old systems. The main causes of the systematic differences are: (1) The old system made an inadequate twice repeated correction for the limb darkening. (2) Owing to the historical reason, the old system used too large limb darkening coefficients.     

转动恒星结构与演化研究

恒星的自转 ,是恒星结构和演化理论的难点。近年来有许多观测事实 ,特别是早型大质量星的观测事实 ,预示恒星的自转效应可能引起恒星内部的物质向外转移 ,造成恒星表面一些元素丰度超丰 ,并且对恒星结构和演化产生重要影响 ,因此 ,恒星的自转问题受到了越来越多的关注。考虑自转效应后 ,恒星结构和演化模型将是二维模型 ,本文综述了诸多作者如何将二维的恒星结构和演化模型简化为一维模型。作者在研究了以上作者的简化方法后 ,提出了一种比较简单的新方法。这种方法基于如下假设 :假设在等势面上的温度 ,密度 ,压强 ,光度 ,化学组成和角速度等物理和化学量近似于均匀分布 ,并且这些量与等价球面上的量相同。 (等价球面是假想的球面 ,它包围的体积与等势面包围的体积相等。)我们在等价球面上推出新的转动恒星结构和演化方程 ,构造出新的演化模型。这个模型与不考虑转动效应的演化模型相比 ,有以下变化 :流体静力学平衡方程变化 ;辐射温度梯度变化 ,并引起对流判据变化 ;星风物质损失和角动量损失增大。作为转动恒星结构和演化模型的应用 ,我们研究了中 ,小质量星中心氦燃烧阶段在赫罗图中的演化轨迹发生来回摆动 (又称为蓝回绕 )的物理机制问题。有诸多作者曾经研究了可以影响蓝回绕的各种因素。但是不知     

Orbital elements of six visual binary stars

Revised orbital elements of the visual binary stars STT 515, BU 4 AB, STF 183 AB, A 207, STT 43 and A 2413 are given. Dynamical parallaxes and total masses of the systems have been calculated.     

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.     

A calculation of structural parameters of spherical stellar systems by means of characteristic functions method. I. Theory

To calculate structural parameters of stellar systems such as an effective radius and central space (or surface) density, the method of characteristic functions is suggested. The characteristic function of the system is a Fourier image of their normalized space density profile f₃(r). In the case of spherical symmetry the probability distribution of r (Q₃(r) = (3/a³)r²f₃(r)) and its orthogonal projections have the same characteristic functions. This fact is used to calculate the effective radii of a few star cluster models (King law, Plummer model and Gausian profile). It is shown, that the characteristic function for King law clusters tends to a finite generalised function if the concentration parameter c is large. The expression for the effective radius (at c ≫ 1) is given. The formula of the effective radius in the Plummer model as well as the relation between the one-dimensional central velocity dispersion and the root mean square velocity are obtained. It is shown, that in the Gaussian model and for King law clusters the effective radius (half-mass visual radius) can differ from the effective (harmonic) radius a few times. This fact should be taken into account in estimating the mass-to-light ratio from the virial mass of such systems using the King radius.     

About the Hill stability of extrasolar planets in stellar binary systems

We use a three dimensional generalization of Szebehely’s invariant relation obtained by us (Makó and Szenkovits, Celest. Mech. Dyn. Astron. 90, 51, 2004) in the elliptic restricted three-body problem, to establish more accurate criterion of the Hill stability. By using this criterion, the Hill stability of four extrasolar planets (γ Cephei Ab, Gliese 86 Ab, HD 41004 Ab and HD 41004 Bb) is investigated.     

Degenerate equilibrium states of collisionless stellar systems

We study spherically symmetrical equilibrium states of collisionless stellar systems confined to a spherical box. These equilibrium states correspond to the statistics introduced by Lynden-Bell in his theory of 'violent relaxation', and are described by a Fermi–Dirac distribution function. We compute the corresponding equilibrium diagram and show that a global entropy maximum exists for any accessible control parameter. This equilibrium state shows a pronounced separation between a degenerate core and a halo. We therefore check that degeneracy is able to stop the gravitational collapse (of a collisionless system), and we propose a simple model for the 'core–halo' structure. We also discuss the relevance of our study for real galaxies or other astrophysical systems such as massive neutrinos.     

Undamped oscillations of collisionless stellar systems: Spheres,spheroids and discs

The collisionless Boltzmann equation governing self-gravitating systems such as galaxies has recently been shown to admit exact oscillating solutions with planar and spherical symmetry. The relation of the spherically symmetric solutions to the Virial theorem, as well as generalizations to non-uniform spheres, uniform spheroids and discs form the subject of this paper. These models generalize known families of static solutions. The case of the spheroid is worked out in some detail. Quasiperiodic as well as chaotic time variation of the two axes is demonstrated by studying the surface of section for the associated Hamiltonian system with two degrees of freedom. The relation to earlier work and possible implications for the general problem of collisionless relaxation in self gravitating systems are also discussed     

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.