The present status of periodic orbits

Recent results on periodic orbits are presented and it is shown that the periodic orbits can be used in the study of planetary systems and triple or multiple stellar systems. Triple stellar systems are stable even for close approaches of the three components. Also stable triple systems exist with nearly zero angular momentum. For the planetary systems a global view is obtained from which it is clear which configurations are stable or unstable and also what factors affect the stability. Also, the relation between resonance and instability is studied by making use of periodic orbits.     

Gas-driven evolution of stellar orbits in barred galaxies

We carry out a detailed orbit analysis of gravitational potentials selected at different times from an evolving self-consistent model galaxy consisting of a two-component disc (stars+gas) and a live halo. The results are compared with a pure stellar model, subject to nearly identical initial conditions, which are chosen so as to make the models develop a large-scale stellar bar. The bars are also subject to hose-pipe (buckling) instability which modifies the vertical structure of the disc. The diverging morphological evolution of both models is explained in terms of gas radial inflow, the resulting change in the gravitational potential at smaller radii, and the subsequent modification of the main families of orbits, both in and out of the disc plane.   We find that dynamical instabilities become milder in the presence of the gas component, and that the stability of planar and 3D stellar orbits is strongly affected by the related changes in the potential — both are destabilized, with the gas accumulation at the centre. This is reflected in the overall lower amplitude of the bar mode and in the substantial weakening of the bar, which appears to be a gradual process. The vertical buckling of the bar is much less pronounced and the characteristic peanut shape of the galactic bulge almost disappears when there is a substantial gas inflow towards the centre. Milder instability results in a smaller bulge, the basic parameters of which are in agreement with observations. We also find that the overall evolution in the model with a gas component is accelerated because of the larger central mass concentration and the resulting decrease in the characteristic dynamical time.     

Slow modes in stellar systems with nearly harmonic potentials: II. Gravitational loss-cone instability

Using a consistent perturbation theory for collisionless disk-like and spherical star clusters, we construct a theory of slow modes for systems having an extended central region with a nearly harmonic potential due to the presence of a fairly homogeneous (on the scales of the stellar system) heavy, dynamically passive halo. In such systems, the stellar orbits are slowly precessing, centrally symmetric ellipses (2: 1 orbits). We consider star clusters with monoenergetic distribution functions that monotonically increase with angular momentum in the entire range of angular momenta (from purely radial orbits to circular ones) or have a growing region only at low angular momenta. In these cases, there are orbits with a retrograde precession, i.e., in a direction opposite to the orbital rotation of the star. The presence of a gravitational loss-cone instability, which is also observed in systems of 1: 1 orbits in near-Keplerian potentials, is associated with such orbits. In contrast to 1: 1 systems, the loss-cone instability takes place even for distribution functions monotonically increasing with angular momentum, including those for systems with circular orbits. The regions of phase space with retrograde orbits do not disappear when the distribution function is smeared in energy. We investigate the influence of a weak inhomogeneity of a heavy halo with a density that decreases with distance from the center.     

Slow modes in stellar systems with nearly harmonic potentials: I. Spoke approximation,radial orbit instability

Using a consistent perturbation theory for collisionless disk-like and spherical star clusters, we construct a theory of slow modes for systems having an extended central region with a nearly harmonic potential due to the presence of a fairly homogeneous (on the scales of the stellar system) heavy, dynamically passive halo. In such systems, the stellar orbits are slowly precessing, centrally symmetric ellipses (2: 1 orbits). Depending on the density distribution in the system and the degree of halo inhomogeneity, the orbit precession can be both prograde and retrograde, in contrast to systems with 1: 1 elliptical orbits where the precession is unequivocally retrograde. In the first paper, we show that in the case where at least some of the orbits have a prograde precession and the stellar distribution function is a decreasing function of angular momentum, an instability that turns into the well-known radial orbit instability in the limit of low angular momenta can develop in the system. We also explore the question of whether the so-called spoke approximation, a simplified version of the slow mode approximation, is applicable for investigating the instability of stellar systems with highly elongated orbits. Highly elongated orbits in clusters with nonsingular gravitational potentials are known to be also slowly precessing 2: 1 ellipses. This explains the attempts to use the spoke approximation in finding the spectrum of slow modes with frequencies of the order of the orbit precession rate. We show that, in contrast to the previously accepted view, the dependence of the precession rate on angular momentum can differ significantly from a linear one even in a narrow range of variation of the distribution function in angular momentum. Nevertheless, using a proper precession curve in the spoke approximation allows us to partially “rehabilitate” the spoke approach, i.e., to correctly determine the instability growth rate, at least in the principal (O(α _T^−1/2) order of the perturbation theory in dimensionless small parameter α _T, which characterizes the width of the distribution function in angular momentum near radial orbits.     

On the relation between orbital structure and observed bar morphology

We investigate the morphological relation between the orbits of the central family of periodic orbits ( x ₁ family) and the bar itself using models of test particles moving in a barred potential. We show that different bar morphologies may have as a backbone the same set of x ₁ periodic orbits. We point out that by populating initially axisymmetric stellar discs exponentially with test particles in circular, or almost circular motion, we may end up with a response bar which reveals a shape different in crucial details from that of the individual stable x ₁ orbits. For example, a bar model in which the x ₁ orbits are pure ellipses may have a much more complicated response morphology. This depends on the particular invariant curves around x ₁, which are populated in each model.     

Gravitational loss-cone instability in stellar systems with retrograde orbit precession

We study spherical and disc clusters in a near-Keplerian potential of galactic centres or massive black holes. In such a potential orbit precession is commonly retrograde, that is, the direction of the orbit precession is opposite to the orbital motion. It is assumed that stellar systems consist of nearly-radial orbits. We show that if there is a loss-cone at low angular momentum (e.g. due to consumption of stars by a black hole), an instability similar to loss-cone instability in plasma may occur. The gravitational loss-cone instability is expected to enhance black hole feeding rates. For spherical systems, the instability is possible for the number of spherical harmonics   l ≥ 3  . If there is some amount of counter-rotating stars in flattened systems, they generally exhibit the instability independent of azimuthal number m . The results are compared with those obtained recently by Tremaine for distribution functions monotonically increasing with angular momentum.
The analysis is based on simple characteristic equations describing small perturbations in a disc or a sphere of stellar orbits highly elongated in radius. These characteristic equations are derived from the linearized Vlasov equations (combining the collisionless Boltzmann kinetic equation and the Poisson equation), using the action-angle variables. We use two techniques for analysing the characteristic equations: the first one is based on preliminary finding of neutral modes, and the second one employs a counterpart of the plasma Penrose–Nyquist criterion for disc and spherical gravitational systems.     

Periodic orbits

Recent results on periodic orbits are presented. Planetary systems can be studied by the model of the general 3-body problem and also some satellite systems and asteroid orbits can be studied by the model of the restricted 3-body problem. Triple stellar systems and planetary systems with two Suns are close to periodic systems. Finally, the motion of stars in various types of galaxies can be studied by finding families of periodic orbits in several galactic models.     

Radially anisotropic models of collisionless spherical stellar systems without central singularity

A new class of the simplest equilibrium two-parameter distribution functions for spherical stellar systems with a radially anisotropic stellar velocity distribution is investigated. The models under consideration are a less singular counterpart of the so-called generalized polytropes, which in the past were among the most popular models in works on the equilibrium and stability of gravitating systems. In contrast to the well-known generalized polytropes, the proposed models have finite density and potential at the center. The absence of a singularity is necessary for a proper consideration of the radial orbit instability, which is the most important instability of spherical stellar systems. The main observed parameters of the proposed models (potential, density, anisotropy) are compared with those in well-known equilibrium models.     

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.     

Retrograde precession of stellar orbits and gravitational loss-cone instability in galactic centers

We consider disk and spherical subsystems of stars with nearly radial orbits under conditions when the well-known radial orbit instability is not possible. This requires that the precession of stellar orbits be retrograde, i.e., in the direction opposite to the orbital rotation of stars. We show that an instability that is an analogue of the loss-cone instability known in plasma physics can then develop in the presence of a “loss cone” in the angular momentum distribution of stars, which ensures a deficit or even absence of stars with low angular momenta. Examples of systems with a loss cone are the centers of galaxies or star clusters with massive black holes. The instability can produce a flux of stars onto the galactic center, i.e., it can serve as a mechanism of fueling the nuclear activity of galaxies. Mathematically, the problem is reduced to analyzing simple characteristic equations that describe small perturbations in a disk and a sphere of radially highly elongated stellar orbits. In turn, these characteristics equations are derived through a number of successive simplifications of the general linearized Vlasov equations (i.e., the system that includes the collisionless Boltzmann kinetic equation and the Poisson equation) in action—angle variables.     

Investigation of the Galactic bar based on photometry and stellar proper motions

A new method for selecting stars in the Galactic bar based on 2MASS infrared photometry in combination with stellar proper motions from the Kharkiv XPM catalogue has been implemented. In accordance with this method, red clump and red giant branch stars are preselected on the color-magnitude diagram and their photometric distances are calculated. Since the stellar proper motions are indicators of a larger velocity dispersion toward the bar and the spiral arms compared to the stars with circular orbits, applying the constraints on the proper motions of the preselected stars that take into account the Galactic rotation has allowed the background stars to be eliminated. Based on a joint analysis of the velocities of the selected stars and their distribution on the Galactic plane, we have confidently identified the segment of the Galactic bar nearest to the Sun with an orientation of 20°–25° with respect to the Galactic center-Sun direction and a semimajor axis of no more than 3 kpc.     

Evolution and instabilities of disks harboring super massive black holes

The bar formation is still an open problem in modern astrophysics. In this paper we present numerical simulations performed with the aim of analyzing the growth of the bar instability inside stellar-gaseous disks, where the star formation is triggered, and a central black hole is present. The aim of this paper is to point out the impact of such a central massive black hole on the growth of the bar. We use N-body-SPH simulations of the same isolated disk-to-halo mass systems harboring black holes with different initial masses and a different energy feedback on the surrounding gas. We compare the results of these simulations with the one of the same disk without a black hole in its center. We make the same comparison (disk with and without black hole) for a stellar disk in a fully cosmological scenario. A stellar bar, lasting 10 Gyrs, is present in all our simulations.     

Imaging planets around nearby white dwarfs

We suggest that Jovian planets will survive the late stages of stellar evolution, and that white dwarfs will retain planetary systems in wide orbits (≳5 au). Utilizing evolutionary models for Jovian planets, we show that infrared imaging with 8-m class telescopes of suitable nearby white dwarfs should allow us to resolve and detect companions ≳3 M _JUP. Detection of massive planetary companions to nearby white dwarfs would prove that such objects can survive the final stages of stellar evolution, place constraints on the frequency of main-sequence stars with planetary systems dynamically similar to our own and allow direct spectroscopic investigation of their composition and structure.     

Dark stars at the Galactic Centre – the main sequence

In regions of very high dark matter density such as the Galactic Centre, the capture and annihilation of WIMP dark matter by stars has the potential to significantly alter their evolution. We describe the dark stellar evolution code D ark S tars , and present a series of detailed grids of WIMP-influenced stellar models for main-sequence stars. We describe the changes in stellar structure and main-sequence evolution which occur as a function of the rate of energy injection by WIMPs, for masses of  0.3–2.0 M_⊙  and metallicities   Z = 0.0003–0.02  . We show what rates of energy injection can be obtained using realistic orbital parameters for stars at the Galactic Centre, including detailed consideration of the velocity and density profiles of dark matter. Capture and annihilation rates are strongly boosted when stars follow elliptical rather than circular orbits. If there is a spike of dark matter induced by the supermassive black hole at the Galactic Centre, single solar mass stars following orbits with periods as long as 50 yr and eccentricities as low as 0.9 could be significantly affected. Binary systems with similar periods about the Galactic Centre could be affected on even less eccentric orbits. The most striking observational effect of this scenario would be the existence of a binary consisting of a low-mass protostar and a higher mass evolved star. The observation of low-mass stars and/or binaries on such orbits would either provide a detection of WIMP dark matter, or place stringent limits on the combination of the WIMP mass, spin-dependent nuclear-scattering cross-section, halo density and velocity distribution near the Galactic Centre. In some cases, the derived limits on the WIMP mass and spin-dependent nuclear-scattering cross-section would be of comparable sensitivity to current direct-detection experiments.     

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.     

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.     

GG Tau: the ringworld and beyond

In binary stellar systems, exoplanet searches have revealed planetary mass companions orbiting both in circumstellar and in circumbinary orbits. Modelling studies suggest increased dynamical complexity around the young stars that form such systems. Circumstellar and circumbinary disks likely exhibit different physical conditions for planet formation, which also depends on the stellar separation. Although binaries and higher order multiple stars are relatively common in nearby star-forming regions, surprisingly few systems with circumbinary distributions of proto-planetary material have been found. With its spectacular ring of dust and gas encircling the central triple star, one such system, GG Tau A, has become a unique laboratory for investigating the physics of circumsystem gas and dust evolution. We review here its physical properties.     

The β Cephei instability strip

We carried out a series of linear stability analyses of the radial and low-degree non-radial p modes for stellar models with initial masses of     . The stellar models were computed by using convective overshoot distance     , 0.25 and 0.40  H _P. Our numerical results show that the β Cephei instability strip forms a horn-shaped region pointing upwards near the main sequence on the Hertzsprung–Russell diagram (HRD). The lower part of the instability strip for the radial modes join the zero-age main-sequence (ZAMS) at     , while the top of the instability strip extends up to     . The instability strip for the non-radial modes is even wider. The overall instability strip is dominated by the radial and non-radial fundamental modes. The first overtone (the radial-order index     is also pulsationally unstable. We have shown that the β Cephei stability is almost independent of the overshoot parameter d _over used for the stellar models, while it depends critically on the metal abundance. With decreasing metal abundance, the instability region shrinks and eventually disappears for     .