Stability of the Triangular Libration Points in the Unrestricted Planar Problem of a Symmetric Rigid Body and a Point Mass

In papers (Godziewski and Maciejewski, 1998a, b, 1999), we investigate unrestricted, planar problem of a dynamically symmetric rigid body and a sphere. Following the original statement of the problem by Kokoriev and Kirpichnikov (1988), we assume that the potential of the rigid body is approximated by the gravitational field of a dumb-bell. The model is described in terms of a 2D Hamiltonian depending on three parameters.In this paper, we investigate the stability of triangular equilibria permissible by the dynamics of the model, under the assumption of low-order resonances. We analyze all resonances of order smaller than four, and we examine the stability with application of theorems by Markeev and Sokolsky. These are the possible following cases: the non-diagonal resonance of the first order with two null characteristic frequencies (unstable); resonances of the first order with one nonzero frequency (diagonal and non-diagonal, stable and unstable); the second-order resonance, which is non-diagonal and stable, and the third-order resonance which is generically unstable, except for three points in the parameters' space, corresponding to stable equilibria.We discuss a perturbed version of Kokoriev and Kirpichnikov model, and we find that if the perturbation is small and depends on the coordinates only, the triangular equilibria persist, except if for the unperturbed equilibria the first-order resonance occurs. We show that the resonances of the order higher than two are also preserved if the perturbation acts.     

Existence and Stability of Libration Points in the Restricted Three-Body Problem When the Primaries are Triaxial Rigid Bodies

This paper deals with the stationary solutions of the planar restricted three-body problem when the primaries are triaxial rigid bodies with one of the axes as the axis of symmetry and its equatorial plane coinciding with the plane of motion. It is seen that there are five libration points, two triangular and three collinear. It is further observed that the collinear points are unstable, while the triangular points are stable for the mass parameter 0 < _crit(the critical mass parameter). It is further seen that the triangular points have long or short periodic elliptical orbits in the same range of .This revised version was published online in October 2005 with corrections to the Cover Date.     

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.     

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 Centre Manifold of Collinear Points in the Planar Three-Body Problem

We study the existence of invariant tori in a neighbourhood of the collinear equilibrium points of the planar three-body problem. To this end some properties of the normal form of the Hamiltonian reduced to the 4D central manifold are proved. Using this normal form, we show that the nondegeneracy conditions of KAM theorem are satisfied for all positive masses, including the 2:1 resonance case. The evaluation of the conditions is done numerically.     

Existence and Stability of L4,5 In The Relativistic Restricted Three-Body Problem

The existence and stability of triangular libration points in the relativistic restricted three-body problem has been studied. It is found that L_4,5 are unstable in the whole range 0 ≤ μ ≤ 1/2 in contrast to the classical restricted three-body problem where they are stable for 0 < μ < μ₀, where μ is the mass parameter and μ₀ = 0.03852.... This revised version was published online in July 2006 with corrections to the Cover Date.     

Restricted four-body problem. The case of a central configuration: Libration points and their stability

We consider the problem of the motion of a zero-mass body in the vicinity of a system of three gravitating bodies forming a central configuration.We study the case where two gravitating bodies of equal mass lie on the same straight line and rotate around the central body with the same angular velocity. Equations for calculating the equilibrium positions in this system have been derived. The stability of the equilibrium points for a system of three gravitating bodies is investigated. We show that, as in the case of libration points for two bodies, the collinear points are unstable; for the triangular points, there exists a ratio of the mass of the central body to the masses of the extreme bodies, 11.720349, at which stability is observed.     

Some Equilibrium of the Restricted Three-Body Problem with Axial Symmetric Primaries and a Gyrostat as Infinitesimal Mass

The restricted three-body problem is reconsidered by replacing the point-like primaries of the classical problem by a pair of axisymmetric rigid bodies which have a plane of symmetry perpendicular to their axes, and the infinitesimal mass by a gyrostat. The conditions for the circular motion of the primaries around their center of mass are stated and they yield the classification of all possible orientations of these bodies into four groups according to the value of their angular velocity. Then the equations of motion of the gyrostat are derived and solved for the equilibrium configurations of the system.     

On the Stability of Equilibrium Points in the Relativistic Restricted Three-Body Problem

The equilibrium points of the relativistic restricted three-body problem are considered. The stability of the triangular points is determined and contrary to recent results of other authors a region of linear stability in the parameter space is obtained. The positions of the collinear points are approximated by series by expansions and their stability is similarly determined. It is found that these are always unstable.This revised version was published online in October 2005 with corrections to the Cover Date.     

Stability of L4 for radiated rigid primaries in resonant cases

The non-linear stability of the triangular libration point L₄ of the restricted three-body problem is studied under the presence of third- and fourth-order resonances, when the more massive primary is a triaxial rigid body and source of radiation. In this study, Markeev's theorems are applied with the help of Moser's theorem. It is found that the stability of the triangular libration point is unstable in the third-order resonance case and in the fourth-order resonance case, this is stable or unstable depending on A₁ and A₂, and a source of radiation parameter α, where A₁, A₂ depend upon the lengths of the semi-axes of the triaxial rigid body.     

Stability Regions and Quasi-Periodic Motion in the Vicinity of Triangular Equilibrium Points

The regions of quasi-periodic motion around non-symmetric periodic orbits in the vicinity of the triangular equilibrium points are studied numerically. First, for a value of the mass parameter less than Routh's critical value, the stability regions determined by quasi-periodic motion are examined around the existing families of short (L^s ₄) and long (L^l ₄) period solutions. Then, for two values of μ greater than the Routh value, the unified family L^sl ₄, to which, in these cases, L^s ₄ and L^l ₄ merge, is considered. It is found that such regions surround in general the linearly stable segments of the corresponding families and become smaller as the mass ratio increases. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Systematic Study of the Stability of Symmetric Periodic Orbits in the Planar, Circular, Restricted Three-Body Problem

We investigate symmetric periodic orbits in the framework of the planar, circular, restricted, three-body problem. Having fixed the mass of the primary equal to that of Jupiter, we determine the linear stability of a number of periodic orbits for different values of the eccentricity. A systematic study of internal resonances, with frequency p/q with 2p 9, 1 q 5 and 4/3 p/q 5, offers an overall picture of the stability character of inner orbits. For each resonance we compute the stability of the two possible periodic orbits. A similar analysis is performed for some external periodic orbits.Furthermore, we let the mass of the primary vary and we study the linear stability of the main resonances as a function of the eccentricity and of the mass of the primary. These results lead to interesting conclusions about the stability of exosolar planetary systems. In particular, we study the stability of Earth-like planets in the planetary systems HD168746, GI86, 47UMa,b and HD10697.     

有摄圆型限制性三体问题平动点稳定的充要条件及应用

摘要给出了一个判断有摄圆型限制性三体问题平动点稳定性的充要条件．该条件只依赖于平动点变分方程的特征方程系数的一个简单关系，使用很方便．用所得到的条件，讨论了任意外力摄动对经典圆型限制性三体问题三角平动点稳定性的影响和惯性阻力摄动对Robe圆型限制性三体问题主要平动点的稳定性的影响．     

A Simplified Kinetic Element Formulation for the Rotation of a Perturbed Mass-Asymmetric Rigid Body

Euler's equations, describing the rotation of an arbitrarily torqued mass asymmetric rigid body, are scaled using linear transformations that lead to a simplified set of first order ordinary differential equations without the explicit appearance of the principal moments of inertia. These scaled differential equations provide trivial access to an analytical solution and two constants of integration for the case of torque-free motion. Two additional representations for the third constant of integration are chosen to complete two new kinetic element sets that describe an osculating solution using the variation of parameters. The elements' physical representations are amplitudes and either angular displacement or initial time constant in the torque-free solution. These new kinetic elements lead to a considerably simplified variation of parameters solution to Euler's equations. The resulting variational equations are quite compact. To investigate error propagation behaviour of these new variational formulations in computer simulations, they are compared to the unmodified equations without kinematic coupling but under the influence of simulated gravity-gradient torques.     

The Non-Linear Stability of L4 in the Restricted Three-Body Problem when the Bigger Primary is a Triaxial Rigid Body

The non-linear stability of L ₄ in the restricted three-body problem has been studied when the bigger primary is a triaxial rigid body with its equatorial plane coincident with the plane of motion. It is found that L ₄ is stable in the range of linear stability except for three mass ratios:

受摄限制性三体问题平动点线性稳定性的判断条件及在Robe问题的应用

给出了受摄限制性三体问题平动点线性稳定性的一个判断条件．条件只与平动点切映像的特征方程系数有关，使用方便．用判断条件，讨论了Robe问题平动点在阻力摄动下的线性稳定性，得到了Hallan等给出的Robe问题平动点在阻力摄动下的线性稳定范围．并改进了Giordanoc等的结果．     

Effect of Radiation on the Nonlinear Stability Zones of the Lagrangian Equilibrium Points

The nonlinear stability zones of the triangular Lagrangian points are determined numerically and the effect of radiation of primaries is considered, in addition to the known effect of mass distribution, using the photogravitational restricted threebody problem model. It is found that radiation also has a considerable effect reducing the stability zones. In cases of resonances, these zones are reduced to negligible size for some parameter values within the linear stability regions.