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．     

On The Concept Of Periapsis In Hill’s Problem

The concept of closest approach is analyzed in Hill’s problem, resulting in a partitioning of the position space. The different behavior between the direct and retrograde motion is explained analytically, resulting in a simple estimate of the variation of Hill’s periodic and quasi-circular orbits as a function of the Jacobi constant. The local behavior of the orbits on the zero velocity surfaces and an analytical definition of local escape and capture in Hill’s problem are also given.     

Global Bifurcations of Periodic Solutions of the Restricted Three Body Problem

We describe global bifurcations from the libration points of non-stationary periodic solutions of the restricted three body problem. We show that the only admissible continua of non-stationary periodic solutions of the planar restricted three body problem, bifurcating from the libration points, can be the short-period families bifurcating from the Lagrange equilibria L ₄, L ₅. We classify admissible continua and show that there are possible exactly six admissible continua of non-stationary periodic solutions of the planar restricted three body problem. We also characterize admissible continua of non-stationary periodic solutions of the spatial restricted three body problem. Moreover, we combine our results with the Déprit and Henrard conjectures (see [8]), concerning families of periodic solutions of the planar restricted three body problem, and show that they can be formulated in a stronger way. As the main tool we use degree theory for SO(2)-equivariant gradient maps defined by the second author in [25].This revised version was published online in October 2005 with corrections to the Cover Date.     

The Parabolic Three-Body Problem

In this communication we present an analytical model for the restricted three-body problem, in the case where the perturber is in a parabolic orbit with respect to the central mass. The equations of motion are derived explicitly using the so-called Global Expansion of the disturbing function, and are valid for any eccentricity of the massless body, as well as in the case where both secondary masses have crossing orbits. Integrating the equations of motion over the complete passage of the perturber through the system, we are then able to construct a first-order algebraic mapping for the change in semimajor axis, eccentricity and inclination of the perturbed body.Comparisons with numerical solutions of the exact equations show that the map yields precise results, as long as the minimum distance between both bodies is not too small. Finally, we discuss several possible applications of this model, including the evolution of asteroidal satellites due to background bodies, and simulations of passing stars on extra-solar planets.     

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Numerical Exploration of Chermnykh’s Problem

We study the equilibrium points and the zero-velocity curves of Chermnykh’s problem when the angular velocity ω varies continuously and the value of the mass parameter is fixed. The planar symmetric simple-periodic orbits are determined numerically and they are presented for three values of the parameter ω. The stability of the periodic orbits of all the families is computed. Particularly, we explore the network of the families when the angular velocity has the critical value ω = 2√2 at which the triangular equilibria disappear by coalescing with the collinear equilibrium point L₁. The analytic determination of the initial conditions of the family which emanate from the Lagrangian libration point L₁ in this case, is given. Non-periodic orbits, as points on a surface of section, providing an outlook of the stability regions, chaotic and escape motions as well as multiple-periodic orbits, are also computed. Non-linear stability zones of the triangular Lagrangian points are computed numerically for the Earth–Moon and Sun–Jupiter mass distribution when the angular velocity varies.     

On final evolutions in the restricted planar parabolic three-body problem

In this paper, we prove the existence of special type of motions in the restricted planar parabolic three-body problem, of the type exchange, emission–capture, and emission–escape with close passages to collinear and equilateral triangle configuration, among others. The proof is based on a gradient-like property of the Jacobian function when equations of motion are written in a rotating–pulsating reference frame, and the extended phase space is compactified in the time direction. Thus a phase space diffeomorphic to -coordinates (θ, ζ, ζ′) is obtained with the boundary manifolds θ = ± π/2 corresponding to escapes of the binaries when time tends to ± ∞. It is shown there exists exactly five critical points on each boundary, corresponding to classic homographic solutions. The connections of the invariant manifolds associated to the collinear configurations, and stable/unstable sets associated to binary collision on the boundary manifolds, are obtained for arbitrary masses of the primaries. For equal masses extra connections are obtained, which include equilateral configurations. Based on the gradient-like property, a geometric criterion for capture is proposed and is compared with a criterion introduced by Merman (1953b) in the fifties, and an example studied numerically by Kocina (1954).     

Homoclinic and heteroclinic orbits in the photogravitational restricted three-body problem

We study numerically the asymptotic homoclinic and heteroclinic orbits around the hyperbolic Lyapunov periodic orbits which emanate from Euler's critical points L ₁ and L ₂, in the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these Lyapunov orbits, are also presented. Poincaré surface of sections of these manifolds on appropriate planes and several homoclinic and heteroclinic orbits for the gravitational case as well as for varying radiation factor q ₁, are displayed. Homoclinic-homoclinic and homoclinic-heteroclinic-homoclinic chains which link the interior with the exterior Hill's regions, are illustrated. We adopt the Sun-Jupiter system and assume that only the larger primary radiates. It is found that for small deviations of its value from the gravitational case (q ₁ = 1), the radiation pressure exerts a significant impact on the Hill's regions and on these asymptotic orbits.     

The restricted two-body problem in constant curvature spaces

We perform the bifurcation analysis of the Kepler problem on and . An analog of the Delaunay variables is introduced. We investigate the motion of a point mass in the field of a Newtonian center moving along a geodesic on and (the restricted two-body problem). For the case of a small curvature, the pericenter shift is computed using the perturbation theory. We also present the results of numerical analysis based on an analogy with the motion of a rigid body.     

Stable Manifolds and Homoclinic Points Near Resonances in the Restricted Three-Body Problem

The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses 1− μ and μ that circle each other with period equal to 2π. For small μ, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers p and q, if its period around the heavier primary is approximately 2π p/q, and by its approximate eccentricity e. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with q/p equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any μ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for μ small enough in the unaveraged restricted three-body problem.