The Inverse Problem of Dynamics for Systems with Non-Stationary Lagrangian

We construct a non-stationary form of the Lagrangian of a material point with a known integral of motion and given monoparametric family of evolving orbits. An equation for non-stationary space symmetrical ‘potential’ function of such Lagrangian is given and this stands for the analog of Szebehely's (1974) equation. As an application of the problem, an integrable equation from celestial mechanics of variable mass with use of non-perturbed orbits of evolving type is constructed. On its basis adiabatic invariants of non-stationary two-body problem containing a tangential force are found. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Stochastic stellar orbits in a pair of interacting galaxies

A dynamical model composed of a disk galaxy with an elliptic companion, moving in a circular orbit, is used in order to study the stellar orbits in a binary galaxy. Using the Poincare surface of section we study the evolution of the stochastic regions in the primary galaxy considering the mass of the companion or the value of the Jacobi’s integral as a parameter. Our numerical calculations suggest that the regions of stochasticity increase, as the mass of the companion or the value of the Jacobi’s integral increase. An interesting observation is that only direct orbits become stochastic.     

Kinetic effects in spherical gravitating systems

The kinetic processes of collisional evolution in anisotropic spherical gravitating systems are considered. A generalization of the classical formula for the relaxation time in anisotropic gravitating media is derived in the Fokker-Planck-Landau approximation. It is shown for gravitating stellar systems that the larger the degree of their anisotropy, the higher the rate of relaxation processes. In particular, for spherical systems with nearly radial orbits, the characteristic times of these processes can be comparable to the crossing time. The results obtained can be of interest for a great variety of problems in the dynamics of gravitating systems, for example, in the problems of stability for various models of gravitating systems, in investigating the dynamics of young star clusters at early stages of their evolution, the dynamical evolution and stability of highly flattened galactic disks, in the problems of the distribution of stars around a black hole, and in a number of other problems.     

一种质点系统平衡态的非线性稳定性

在具各向异性速度分布的质点系统的总能量小于零时，由大量静态维里定理构造了该系统的一个旋转椭球形平衡态。讨论该平衡态的非线性稳定性。发现在上述平衡态附近旋转随球作拟周期或近拟周期运动，尽管存在混沌运动和轨道扩散行为，但是对实际的时间尺度而言，可以认为该平衡态具有非线性稳定性。有关结果在用N-体数值模拟研究恒星系统振动行为时有一定的应用价值。     

The secular stability of a gravitating stellar ring

The problem of secular stability of a homogeneous and cold differentially rotating gravitating stellar ring in an external gravitational field is considered. For the analysis of stability we use the method of variation of the system energy in the frames of the local hydrodynamical approximation. We show that the considered stellar system is always characterized by secular instability for all non–zero values of the density. This conclusion is valid also in the case of solid–body rotation of the ring.     

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.     

Computing with certainty individual members of families of periodic orbits of a given period

The accurate computation of families of periodic orbits is very important in the analysis of various celestial mechanics systems. The main difficulty for the computation of a family of periodic orbits of a given period is the determination within a given region of an individual member of this family which corresponds to a periodic orbit. To compute with certainty accurate individual members of a specific family we apply an efficient method using the Poincaré map on a surface of section of the considered problem. This method converges rapidly, within relatively large regions of the initial conditions. It is also independent of the local dynamics near periodic orbits which is especially useful in the case of conservative dynamical systems that possess many periodic orbits, often of the same period, close to each other in phase space. The only computable information required by this method is the signs of various function evaluations carried out during the integration of the equations of motion. This method can be applied to any system of celestial mechanics. In this contribution we apply it to the photogravitational problem.     

Effects of stellar wind,dynamical friction,and star mergers on the dynamical evolution of multiple stars

The dynamical evolution of small stellar groups composed of N=6 components was numerically simulated within the framework of a gravitational N-body problem. The effects of stellar mass loss in the form of stellar wind, dynamical friction against the interstellar medium, and star mergers on the dynamical evolution of the groups were investigated. A comparison with a purely gravitational N-body problem was made. The state distributions at the time of 300 initial system crossing times were analyzed. The parameters of the forming binary and stable triple systems as well as the escaping single and binary stars were studied. The star-merger and dynamical-friction effects are more pronounced in close systems, while the stellar wind effects are more pronounced in wide systems. Star-mergers and stellar wind slow down the dynamical evolution. These factors cause the mean and median semimajor axes of the final binaries as well as the semimajor axes of the internal and external binaries in stable triple systems to increase. Star mergers and dynamical friction in close systems decrease the fraction of binary systems with highly eccentric orbits and the mean component mass ratios for the final binaries and the internal and external binaries in stable triple systems. Star mergers and dynamical friction in close systems increase the fraction of stable triple systems with prograde motions. Dynamical friction in close systems can both increase and decrease the mean velocities of the escaping single stars, depending on the density of the interstellar medium and the mean velocity of the stars in the system.     

Nonlinear dynamical analysis for displaced orbits above a planet

Nonlinear dynamical analysis and the control problem for a displaced orbit above a planet are discussed. It is indicated that there are two equilibria for the system, one hyperbolic (saddle) and one elliptic (center), except for the degenerate h _z ^max, a saddle-node bifurcation point. Motions near the equilibria for the nonresonance case are investigated by means of the Birkhoff normal form and dynamical system techniques. The Kolmogorov–Arnold–Moser (KAM) torus filled with quasiperiodic trajectories is measured in the τ ₁ and τ ₂ directions, and a rough algorithm for calculating τ ₁ and τ ₂ is proposed. A general iterative algorithm to generate periodic Lyapunov orbits is also presented. Transitions in the neck region are demonstrated, respectively, in the nonresonance, resonance, and degradation cases. One of the important contributions of the paper is to derive necessary and sufficiency conditions for stability of the motion near the equilibria. Another contribution is to demonstrate numerically that the critical KAM torus of nontransition is filled with the (1,1)-homoclinic orbits of the Lyapunov orbit.     

On the determination of potential function

Szebehely's renowned equation given in 1974, allowing for potential determination from a given orbit or family of orbits, is proved to be equivalent with an equation deduced in 1963 by Drǎmbǎ. This basic equation in the inverse problem of dynamics, for which the denomination of Drǎmbǎ –Szebehely equation is proposed, is generalized for the motion in the n-dimensional Euclidean space. A method for the determination of the potential function from motion equations is extended to this space.     

Clustering of non-point mass system of galaxies in the expanding Universe

We derive the cosmic energy equation for the non-point mass system of galaxies (galaxies with halos) by using the adiabatic approximation for the growth of gravitational clustering of galaxies in the expanding Universe. The cosmic energy equation so derived represents the general form of conservation of energy for the expanding volume. Using the derived form of cosmic energy equation we try to study the evolution of correlation potential energy of the system. We also try to explore the condition under which the approximation of extensivity may be applied to the infinite gravitating non-point mass system of galaxies.     

Study of the nearest open clusters and the associated moving clusters by numerical simulations

In our previous papers, we showed that at the final phases of the dynamical evolution of an open cluster, an extended population of stars elongated along its Galactic orbit, the stellar tail of the cluster, is formed. The tail stars that escaped from the cluster at different times move in a common orbit with low relative velocities. Experiencing a weak interaction with Galactic field stars, these objects, the relics of open clusters, can exist for a fairly long time. In this paper, we investigate the structures of such stellar tails in the nearest open clusters: Hyades, Pleiades, Praesepe, Alpha Persei, Coma, IC 2391, and IC 2602. To this end, we performed several numerical simulations of the dynamical evolution of these clusters in the tidal field of the Galaxy. Our computations of the dynamical evolution were based on known cluster age estimates and real Galactic orbits. The initial conditions were chosen in such a way that the parameters of the simulated clusters corresponded to their observed parameters. As a result, we obtained models of the stellar tails for the nearest open clusters and estimated such parameters of the tails as their sizes, densities, locations relative to the solar neighborhood, and others.     

The Expansion of the Distance Between Two Arbitrary Gravitating Points as Applied to Specific Models of Celestial Bodies

For the complete system of biharmonic functions a suitable representation in spheroidal coordinates is found. It is used for expanding the distance between two arbitrary gravitating points and its inverse quantity in appropriate series. Such expansions can be of interest and useful in a number of problems of celestial mechanics and stellar dynamics.     

Secular and tidal evolution of circumbinary systems

We investigate the secular dynamics of three-body circumbinary systems under the effect of tides. We use the octupolar non-restricted approximation for the orbital interactions, general relativity corrections, the quadrupolar approximation for the spins, and the viscous linear model for tides. We derive the averaged equations of motion in a simplified vectorial formalism, which is suitable to model the long-term evolution of a wide variety of circumbinary systems in very eccentric and inclined orbits. In particular, this vectorial approach can be used to derive constraints for tidal migration, capture in Cassini states, and stellar spin–orbit misalignment. We show that circumbinary planets with initial arbitrary orbital inclination can become coplanar through a secular resonance between the precession of the orbit and the precession of the spin of one of the stars. We also show that circumbinary systems for which the pericenter of the inner orbit is initially in libration present chaotic motion for the spins and for the eccentricity of the outer orbit. Because our model is valid for the non-restricted problem, it can also be applied to any three-body hierarchical system such as star–planet–satellite systems and triple stellar systems.     

A Time-Transformed Leapfrog Scheme

We present a time-transformed leapfrog scheme combined with the extrapolation method to construct an integrator for orbits in N-body systems with large mass ratios. The basic idea can be used to transform any second-order differential equation into a form which may allow more efficient numerical integration. When applied to gravitating few-body systems this formulation permits extremely close two-body encounters to be considered without significant loss of accuracy. The new scheme has been implemented in a direct N-body code for simulations of super-massive binaries in galactic nuclei. In this context relativistic effects may also be included.     

天体运动的轨道偏心率研究概述---从恒星系统到行星系统

偏心率是描述天体运动轨道的重要参数之一, 能够为揭示天体的动力学演化提供重要线索, 进而帮助理解天体形成与演化的过程及背后的物理机制. 随着天文观测技术的不断发展, 人们对于天体运动轨道的研究已经走出太阳系, 包含的系统也从大质量端的恒星系统延伸到了低质量端的行星系统. 聚焦天体轨道偏心率研究, 回顾了目前在恒星系统(包括主序恒星、褐矮星以及致密星)和行星系统(包括太阳系外巨行星以及``超级地球''、``亚海王星''等小质量系外行星)方面取得的进展, 总结了不同尺度结构下偏心率研究的一些共同之处和待解决的问题. 并结合当下和未来的相关天文观测设备和项目, 对未来天体轨道偏心率方面的研究工作进行了展望.     

Analysis of the structure and dynamics of the stellar tails of open star clusters

We have analyzed the formation, structure, and dynamical evolution of the population of stars that escaped from open clusters by numerical simulations using S. Aarseth’s modified NBODY6 code. In the Galactic tidal field, the population of stars that escaped from a cluster is shown to be elongated along the orbit of the cluster symmetrically about its core in the form of stellar tails of increasing sizes. We analyze the parameters of stellar tails as a function of such initial simulation conditions as the number of stars, the cluster density, the eccentricity of the Galactic cluster orbit in the plane of the Galactic disk, and the z velocity component. As a result, we constructed a grid of model stellar tails of open clusters. The grid includes such time-dependent parameters of the stellar tails as the length, the cross section, the number of stars, the velocity distribution, etc. Our simulations allow us to clarify the origin of moving clusters and stellar streams and to assess the role of star clusters in forming the stellar velocity field in the solar neighborhood.     

Regular and irregular periodic orbits

We distinguish between regular orbits, that bifurcate from the main families of periodic orbits (those that exist also in the unperturbed case) and irregular periodic orbits, that are independent of the above. The genuine irregular families cannot be made to join the regular families by changing some parameters. We present evidence that all irregular families appear inside lobes formed by the asymptotic curves of the unstable periodic orbits. We study in particular a dynamical system of two degrees of freedom, that is symmetric with respect to the x-axis, and has also a triple resonance in its unperturbed form. The distribution of the periodic orbits (points on a Poincaré surface of section) shows some conspicuous lines composed of points of different multiplicities. The regular periodic orbits along these lines belong to Farey trees. But there are also lines composed mainly of irregular orbits. These are images of the x-axis in the map defined on the Poincaré surface of section. Higher order iterations of this map , close to the unstable triple periodic orbit, produce lines that are close to the asymptotic curves of this unstable orbit. The homoclinic tangle, formed by these asymptotic curves, contains many regular orbits, that were generated by bifurcation from the central orbit, but were trapped inside the tangle as the perturbation increased. We found some stable periodic orbits inside the homoclinic tangle, both regular and irregular. This proves that the homoclinic tangle is not completely chaotic, but contains gaps (islands of stability) filled with KAM curves.     

The violent past of Cygnus X-2

Cygnus X-2 appears to be the descendant of an intermediate-mass X-ray binary (IMXB). Using Mazzitelli's stellar code we compute detailed evolutionary sequences for the system and find that its prehistory is sensitive to stellar input parameters, in particular the amount of core overshooting during the main-sequence phase. With standard assumptions for convective overshooting a case B mass transfer starting with a 3.5-M_⊙ donor star is the most likely evolutionary solution for Cygnus X-2. This makes the currently observed state rather short-lived, of order 3 Myr, and requires a formation rate > 10⁻⁷–10⁻⁶ yr⁻¹ of such systems in the Galaxy. Our calculations show that neutron star IMXBs with initially more massive donors (≳4 M_⊙) encounter a delayed dynamical instability; they are unlikely to survive this rapid mass transfer phase. We determine limits for the age and initial parameters of Cygnus X-2 and calculate possible dynamical orbits of the system in a realistic Galactic potential, given its observed radial velocity. We find trajectories which are consistent with a progenitor binary on a circular orbit in the Galactic plane inside the solar circle that received a kick velocity ≤200 km s⁻¹ at the birth of the neutron star. The simulations suggest that about 7 per cent of IMXBs receiving an arbitrary kick velocity from a standard kick velocity spectrum would end up in an orbit similar to Cygnus X-2, while about 10 per cent of them reach yet larger Galactocentric distances.