Periodic and Chaotic Trajectories of the Second Species for the n-Centre Problem

For the n-centre problem of one particle moving in the potential of attracting centres of small mass fixed in an arbitrary smooth potential and magnetic field, we prove the existence of periodic and chaotic trajectories shadowing sequences of collision orbits. In particular, we obtain large subshifts of solutions of this type for the circular restricted 3-body problem of celestial mechanics. Poincaré had conjectured existence of the periodic ones and given them the name ‘second species solutions’. This revised version was published online in July 2006 with corrections to the Cover Date.     

Asymptotic and Periodic Orbits Around L 3 in the Photogravitational Restricted Three-Body Problem

Asymptotic motion near the collinear equilibrium points of the photogravitational restricted three-body problem is considered. In particular, non-symmetric homoclinic solutions are numerically explored. These orbits are connected with periodic ones. We have computed numerically the families containing these orbits and have found that they terminate at both ends by asymptotically approaching simple periodic solutions belonging to the Lyapunov family emanating from L₃.     

Periodic Orbits of a Collinear Restricted Three-Body Problem

In this paper we study symmetric periodic orbits of a collinear restricted three-body problem, when the middle mass is the largest one. These symmetric periodic orbits are obtained from analytic continuation of symmetric periodic orbits of two collinear two-body problems.     

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.     

Second Species Periodic Orbits of the Elliptic 3 Body Problem

We consider the plane restricted elliptic 3 body problem with small mass ratio and small eccentricity and prove the existence of many periodic orbits shadowing chains of collision orbits of the Kepler problem. Such periodic orbits were first studied by Poincaré for the non-restricted 3 body problem. Poincaré called them second species solutions.     

Capture in the Circular and Elliptic Restricted Three-Body Problem

In this paper the authors provide a study of the phenomenon of the gravitational capture by using the models of the circular and elliptic restricted three-body problem. In the first part the inadequacy of the circular restricted three-body problem in the study of the phenomenon of the capture in the case of NEAs is shown. In the model of the spatial elliptic restricted three-body problem criteria of the capture are deduced by using the pulsating Hill-regions. This revised version was published online in July 2006 with corrections to the Cover Date.     

Libration Points in Schwarzschild's Circular Restricted Three-Body Problem

The restricted three-body problem in Schwarzschild's gravitational field is analyzed. The existen- ce of the equilibrium points in the orbital plane is discussed and the corresponding positions are established. There are three collinear libration points, and, if they exist, two triangular libration points (situated in the orbital plane of the primaries). If triangular points exist, they may not form equilateral triangles; the triangles are isosceles for equal masses of the primaries, and scalene else.     

Numerical Determination of Homoclinic and Heteroclinic Orbits at Collinear Equilibria in the Restricted Three-Body Problem with Oblateness

Asymptotic motion to collinear equilibrium points of the restricted three-body problem with oblateness is considered. In particular, homoclinic and heteroclinic solutions to these points are computed. These solutions depart asymptotically from an equilibrium point and arrive asymptotically at the same or another equilibrium point and are important reference solutions. To compute an asymptotic orbit, we use a fourth order local analysis, numerical integration and standard differential corrections.     

Periodic Orbits in the Restricted Three Body Problem with Radiation Pressure

This paper deals with the Restricted Three Body Problem (RTBP) in which we assume that the primaries are radiation sources and the influence of the radiation pressure on the gravitational forces is considered; in particular, we are interested in finding families of periodic orbits under theses forces. By means of some modifications to the method of numerical continuation of natural families of periodic orbits, we find several families of periodic orbits, both in two and three dimensions. As starters for our method we use some known periodic orbits in the classical RTBP. This revised version was published online in August 2006 with corrections to the Cover Date.     

Families of Periodic Orbits Emanating From Homoclinic Orbits in the Restricted Problem of Three Bodies

We describe and comment the results of a numerical exploration on the evolution of the families of periodic orbits associated with homoclinic orbits emanating from the equilateral equilibria of the restricted three body problem for values of the mass ratio larger than μ ₁. This exploration is, in some sense, a continuation of the work reported in Henrard [Celes. Mech. Dyn. Astr. 2002, 83, 291]. Indeed it shows how, for values of μ. larger than μ ₁, the Trojan web described there is transformed into families of periodic orbits associated with homoclinic orbits. Also we describe how families of periodic orbits associated with homoclinic orbits can attach (or detach) themselves to (or from) the best known families of symmetric periodic orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

Saari's Conjecture for the Planar Three-Body Problem with Equal Masses

In the N-body problem, it is a simple observation that relative equilibria (planar solutions for which the mutual distances between the particles remain constant) have constant moment of inertia. In 1970, Don Saari conjectured that the converse was true: if a solution to the N-body problem has constant moment of inertia, then it must be a relative equilibrium. In this note, we confirm the conjecture for the planar three-body problem with equal masses. This revised version was published online in July 2006 with corrections to the Cover Date.     

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．     

Symplectic Mapping for Trojan-Type Motion in the Elliptic Restricted Three-Body Problem

A symplectic mapping for Trojan-type motion has been developed in the secularly changing elliptic restricted three-body problem. The mapping describes well the characteristics of Trojan-type dynamics at small eccentricities. By using this mapping the boundary of the stability region has been studied for different values of the initial eccentricities of hypothetical Jupiter's Trojans. It has been found that in the secularly changing elliptic case the chaotic diffusion at the border of the stability region is stronger than simply in the elliptic case. An explanation of this observation might be the destruction of the chain of islands of the 13:1 secondary resonance between the short and long period component of the Trojan-like motion, caused possibly by the indirect perturbations of Saturn.     

The Restricted Three-Body Problem: Comments on the Rotational Effect

The synchronization between the orbital motion and axial rotation of the two component stars of a binary system is reviewed. Some previous published papers are mentioned and the general conclusion is outlined: If we shall use a rotating coordinate system synchronous with one of the two stellar axial rotations, it is not possible to obtain a Jacobi integral and the Roche geometry cannot be further analyzed. In addition, a theoretical approach is summarized in order to use the axial rotations of the two component stars, even if the constants of the stellar structure (k₂)₁, (k₂)₂must be taken into consideration. So it is found that if the stellar angular velocities are higher than the corresponding Keplerian angular velocity (ω_i≫ ω_k, i= ), the problem of the rotational effect could be of practical consideration. Finally, a theoretical relationship between the two constants (k₂)₁and (k₂)₂of the stellar structure is established.     

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.     

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.     

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．     

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.