Long Time Stability for the Main Problem of Artificial Satellites

We investigate the significance of long time stabilty predictions in the light of Nekhoroshev's theory by studying the orbits of artificial satellites. As a simplified model problem we consider the so-called J₂problem for an earth's satellite, neglecting luni-solar perturbations and nonconservative effects. We consider a wide range of orbits, excluding those which are too close to the critical inclination. Most of the orbits turn out to be stable for times larger than the estimated age of the solar system, thus proving that, as far as dissipation can be neglected, stability in Nekhoroshev's sense may be effective for physically realistic systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

On Motions Near the Lagrange Equilibrium Point L 4 in the Case of Routh's Critical Mass Ratio

We deal with the planar restricted circular problem of three bodies. We study trajectories in a small neighborhood of the Lagrange equilibrium point L ₄ when mass ratio is close to Routh's value. In particular, we show that the case of proper degeneracy takes place and for most initial conditions trajectories are conditionally-periodic. We obtain an approximate representation of families of periodic solution emanating from the equilibrium point L ₄. We also show that in the case of instability of L ₄ the trajectories starting in a vicinity of L ₄ remain in a finite domain forever. We give an upper bound of this domain. To carry out our investigation, we analyze the dynamics of a general Hamiltonian system with two degrees of freedom in the case of 1 : 1 resonance in detail.This revised version was published online in October 2005 with corrections to the Cover Date.     

On the Stability of Planar Oscillations and Rotations of a Satellite in a Circular Orbit

We deal with the stability problem of planar periodic motions of a satellite about its center of mass. The satellite is regarded a dynamically symmetric rigid body whose center of mass moves in a circular orbit.By using the method of normal forms and KAM theory we study the orbital stability of planar oscillations and rotations of the satellite in detail. In two special cases we investigate the orbital stability analytically by introducing a small parameter. In the general case, numerical calculations of Hamiltonian normal form are necessary.     

Stability of Co-Orbital Motion in Exoplanetary Systems

The stability of co-orbital motions is investigated in such exoplanetary systems, where the only known giant planet either moves fully in the habitable zone, or leaves it for some part of its orbit. If the regions around the triangular Lagrangian points are stable, they are possible places for smaller Trojan-like planets. We have determined the nonlinear stability regions around the Lagrangian point L₄ of nine exoplanetary systems in the model of the elliptic restricted three-body problem by using the method of the relative Lyapunov indicators. According to our results, all systems could possess small Trojan-like planets. Several features of the stability regions are also discussed. Finally, the size of the stability region around L₄ in the elliptic restricted three-body problem is determined as a function of the mass parameter and eccentricity.     

Stability of the periodic solutions of the restricted three-body problem representing analytic continuations of Keplerian rectilinear periodic motions

The periodic solutions of the restricted three-body problem representing analytic continuations of Keplerian rectilinear periodic motions are well known (Kurcheeva, 1973). Here the stability of these solutions are examined by applying Poncaré's characteristic equation for periodic solutions. It is found that the isoperiodic solutions are stable and all other solutions are unstable.     

Nonlinear Stability of the Lagrangian Libration Points in the Chermnykh Problem

In this paper we consider the problem of motion of an infinitesimal point mass in the gravity field of an uniformly rotating dumb-bell. The aim of our study is to investigate Liapunov stability of Lagrangian libration points of this problem. We analyze the stability of libration points in the whole range of parameters ω, μ of the problem. In particular, we consider all resonance cases when the order of resonance is not greater than five. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Stability of the Terrestrial Planets as Models for Exosolar Planetary Systems

All results, achieved up to now, show the long term stability of our planetary system, although, especially the inner solar system is chaotic, due to some specific secular resonances. We study, by means of numerical integrations, the dynamical evolution of the planetary system where we concentrate on the stability of motion of the terrestrial planets Venus, Earth and Mars. Our model consists of a simplified planetary system with the inner planets Venus, Earth and Mars as well as Jupiter and Saturn. A mass factor was introduced to uniformly change the masses of the terrestrial planets; Jupiter and Saturn were involved in the system with their actual masses. We integrated the equations of motion with a Lie-integration method for a time interval of 10⁷ years. It turned out that when 220 < < 245 and > 250 the system became unstable due to the strong interactions between the planets. We discuss the model planetary systems for small mass-factors 0.5 10 and large ones 160 270 with the aid of several different numerical tools. These results can be applied to recently discovered exoplanetary systems, which configuration is comparable to our own.     

The Limit Problems for the Equation of Oscillations of a Satellite

We consider the ordinary differential equation of the second order, which describes oscillations of a satellite with respect to its mass center moving along an elliptic orbit with eccentricity e. The equation has two parameters: e and μ. It is regular for 0 ≤ e < 1 and singular when e = 1. For 1 we obtain three limit problems. Their bounded solution to the first limit problem form a two-dimensional (2D) continuous invariant set with a periodic structure. Solutions to the second limit problem form 2D and 3D manifolds. The μ-depending families of odd bounded solutions are singled out. One of the families is twisted into a self-similar spiral. To obtain the limit families of the periodic solutions to the original problem match together the odd bounded solutions to the first and the second limit problem. The point of conjunction is described by the third (the basic) limit problem. The limit families are very close to prelimit ones computed in earlier studies. This revised version was published online in July 2006 with corrections to the Cover Date.     

Long-Term Stability Analysis of Quasi Integrable Degenerate Systems Through the Spectral Formulation of the Nekhoroshev Theorem

We describe a numerical application of the Nekhoroshev theorem to investigate the long-term stability of quasi-integrable systems. We extend the results of a previous paper to a class of degenerate systems, which are typical in celestial mechanics.     

Stability and Geometry of Third-Order Resonances in four-Dimensional Symplectic Mappings

We analyze four-dimensional symplectic mappings in the neighbourhood of an elliptic fixed point whose eigenvalues are close to satisfy a third-order resonance. Using the perturbative tools of resonant normal forms, the geometry of the orbits and the existence of elliptic or hyperbolic one-dimensional tori (fixed lines) is worked out. This allows one to give an analytical estimate of the stability domain when the resonance is unstable. A comparison with numerical results for the four-dimensional Hénon mapping is given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Normal form,Lie-Poisson structure and reduction for the Henon-Heiles system

The reduced Henon-Heiles system is investigated as a Hamiltonian dynamical system obtained by applying the normalization of the HamiltonianH=1/2(p ₁ ² +p ₂ ² +q ₁ ² +q ₂ ² )+1/3q ₁ ³ –q ₁ q ₂ ² to fourth-degree terms. The related equations of motion are bi-Hamiltonian and possess the Lie-Poisson structure. Each Lie-Poisson structure possesses an associated Casimir function. When reduced to level sets of these functions, the equations of motion take various symplectic forms. The various reductions give different coordinate representations of the solutions. These coordinate representations are used to seek the simplest representation of the solutions.     

Normalization of Hamiltonian in the Generalized Photogravitational Restricted Three Body Problem with Poynting–Robertson Drag

We have performed normalization of Hamiltonian in the generalized photogravitational restricted three body problem with Poynting–Robertson drag. In this problem we have taken bigger primary as source of radiation and smaller primary as an oblate spheroid. Whittaker’s method is used to transform the second order part of the Hamiltonian into the normal form.      

Central configurations of the five-body problem with equal masses

In this paper we present a complete classification of the isolated central configurations of the five-body problem with equal masses. This is accomplished by using the polyhedral homotopy method to approximate all the isolated solutions of the Albouy-Chenciner equations. The existence of exact solutions, in a neighborhood of the approximated ones, is then verified using the Krawczyk method. Although the Albouy-Chenciner equations for the five-body problem are huge, it is possible to solve them in a reasonable amount of time.     

Probing the Nekhoroshev Stability of Asteroids

We apply the spectral formulation of the Nekhoroshev theorem to investigate the long-term stability of real main belt asteroids. We find numerical indication that some asteroids are in the so-called Nekhoroshev stability regime, that is they are on chaotic orbits but their motion is stable over very long times. We have analyzed the motion of bodies in different regions of the belt, to assess the sensitivity of our method. We found that it allows us to clearly discriminate between different dynamical regimes, such as the one described by the Nekhoroshev stability, the one well described by the KAM theory, and the unstable chaotic regime in which diffusion in phase space can be detected over time spans much shorter than the age of the solar system.     

Resonant Periodic Motion and the Stability of Extrasolar Planetary Systems

Families of nearly circular periodic orbits of the planetary type are studied, close to the 3/1 mean motion resonance of the two planets, considered both with finite masses. Large regions of instability appear, depending on the total mass of the planets and on the ratio of their masses.Also, families of resonant periodic orbits at the 2/1 resonance have been studied, for a planetary system where the total mass of the planets is the 4% of the mass of the sun. In particular, the effect of the ratio of the masses on the stability is studied. It is found that a planetary system at this resonance is unstable if the mass of the outer planet is smaller than the mass of the inner planet.Finally, an application has been made for the stability of the observed extrasolar planetary systems HD82943 and Gliese 876, trapped at the 2/1 resonance.     

Periodic rotations of a rigid body located at the libration point

Periodic rotations of a rigid body close to the flat motions were found. Their orbital stability was investigated. Analysis was done up to second order of the small parameter. It was proved that solutions found are orbitally stable except of the third order resonance case. This resonance do not appear if terms up to the first order of small parameter are considered only.     

Stability of equilibrium solutions in the critical case of even-order resonance in periodic Hamiltonian systems with one degree of freedom

The stability in the Lyapunov sense of an equilibrium position in a periodic Hamiltonian system with one degree of freedom is studied. It is assumed that the equilibrium is stable in the first approximation and that there exists an even resonance of order $k$ k , arbitrary. The critical case is considered, i.e., when the system of parameters is such that, in order to draw rigorous conclusions about the stability of the equilibrium position in the Lypaunov sense, terms or order higher than three in the series expansion of the Hamiltonian function must be taken into account. Sufficient conditions are derived for stability and instability.     

平动点线性稳定条件及阻力对限制性三体问题三角平动点线性稳定性的影响

给出了受摄限制性三体问题平动点线性稳定性的一些判断条件，条件只与相应的平动点切映像的特征方程系数有关，使用方便，用这些判断条件，讨论了一些阻力对经典限制性三体问题三角平动点线性稳定性的影响，改进了Murray等的一些结果。     

On the Stability of High Inclined L4 and L5Trojans

The orbits of real asteroids around the Lagrangian points L₄ and L ₅of Jupiter with large inclinations (i > 20°) were integrated for 50 Myrs. We investigated the stability with the aid of the Lyapunov characteristic exponents (LCE) but tested also two other methods: on one hand we integrated four neighbouring orbits for each asteroid and computed the maximum distance in every group, on the other hand we checked the variation of the Delaunay element H of the asteroid. In a second simulation – for a grid of initial eccentricity versus initial inclination – we examined the stability of the orbits around both Lagrangian points for 20° < i < 55° and 0.0 < e < 0.20. For the initial semimajor axes we have chosen the one ofJupiter(a = 5.202 AU). We determined the stability with the aid of the LCEs and also the maximum eccentricity of the orbits during the whole integration time. The region around L₄ turned out to be unstable for large inclinations and eccentricities (i > 55° and e > 0.12). The stable region shrinks for orbits around L₅: we found that they become unstable already for i > 45° and e > 0.10. We interpret it as a first hint why we observe more Trojans around the leading Lagrangian point. The results confirm the stability behaviour of the real Trojans which we computed in the first part of the paper.     

时间序列数字滤波后自由度的确定--应用于日长变化与南方涛动指数的相关分析

原始观测时间序列进行数字滤波后，其自由度将大大降低，对各种数字滤波器都可以通过Monte Carlo方法模拟获得对应置信水平的相关系数临界值，从而得到时间序列滤波后的自由度，如果滤波后的频段宽度与Nyquist频率之比为Z，那么对于理想的单边和双边滤波器，时间序列滤波后的自由度为原始值的Z／2和Z倍，非理想滤波器一般不符合此规律。