Families of periodic orbits in the three-body problem

We show by a general argument that periodic solutions of the planar problem of three bodies (with given masses) form one-parameter families. This result is confirmed by numerical investigations: two orbits found earlier by Standish and Szebehely are shown to belong to continuous one-parameter families of periodic orbits. In general these orbits have a non-zero angular momentum, and the configuration after one period is rotated with respect to the initial configuration. Similar general arguments whow that in the three-dimensional problem, periodic orbits form also one-parameter families; in the one-dimensional problem, periodic orbits are isolated.     

Asymmetric Periodic Solutions in the Restricted Problem of Three Bodies

A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincaré surface of sections, that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion.     

Existence of periodic orbits of the second kind in the elliptic restricted problem of three bodies

Hale's method is used to show the existence of symmetric periodic orbits of the second kind for the particular case of the elliptic restricted problem of three bodies. In this treatment we also obtain a new proof of the existence of periodic orbits of the first and second kinds in the circular restricted problem.     

Periodic orbits of the elliptic restricted problem for the Sun-Jupiter-Saturn system

A systematic approach to generate periodic orbits in the elliptic restricted problem of three bodies in introduced. The approach is based on (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to periodic orbits in the elliptic restricted problem. Two families of periodic orbits of the elliptic restricted problem are found by this approach. The mass ratio of the primaries of these orbits is equal to that of the Sun-Jupiter system. The sidereal mean motions between the infinitesimal body and the smaller primary are in a 2:5 resonance, so as to approximate the Sun-Jupiter-Saturn system. The linear stability of these periodic orbits are studied as functions of the eccentricities of the primaries and of the infinitesimal body. The results show that both stable and unstable periodic orbits exist in the elliptic restricted problem that are close to the actual Sun-Jupiter-Saturn system. However, the periodic orbit closest to the actual Sun-Jupiter-Saturn system is (linearly) stable.     

Periodic orbits in the gravity field of a fixed homogeneous cube

In the current study, the existence of periodic orbits around a fixed homogeneous cube is investigated, and the results have powerful implications for examining periodic orbits around non-spherical celestial bodies. In the two different types of symmetry planes of the fixed cube, periodic orbits are obtained using the method of the Poincaré surface of section. While in general positions, periodic orbits are found by the homotopy method. The results show that periodic orbits exist extensively in symmetry planes of the fixed cube, and also exist near asymmetry planes that contain the regular Hex cross section. The stability of these periodic orbits is determined on the basis of the eigenvalues of the monodromy matrix. This paper proves that the homotopy method is effective to find periodic orbits in the gravity field of the cube, which provides a new thought of searching for periodic orbits around non-spherical celestial bodies. The investigation of orbits around the cube could be considered as the first step of the complicated cases, and helps to understand the dynamics of orbits around bodies with complicated shapes. The work is an extension of the previous research work about the dynamics of orbits around some simple shaped bodies, including a straight segment, a circular ring, an annulus disk, and simple planar plates.     

Search for and study of nearly periodic orbits in the plane problem of three equal-mass bodies

We analyze nearly periodic solutions in the plane problem of three equal-mass bodies by numerically simulating the dynamics of triple systems. We identify families of orbits in which all three points are on one straight line (syzygy) at the initial time. In this case, at fixed total energy of a triple system, the set of initial conditions is a bounded region in four-dimensional parameter space. We scan this region and identify sets of trajectories in which the coordinates and velocities of all bodies are close to their initial values at certain times (which are approximately multiples of the period). We classify the nearly periodic orbits by the structure of trajectory loops over one period. We have found the families of orbits generated by von Schubart’s stable periodic orbit revealed in the rectilinear three-body problem. We have also found families of hierarchical, nearly periodic trajectories with prograde and retrograde motions. In the orbits with prograde motions, the trajectory loops of two close bodies form looplike structures. The trajectories with retrograde motions are characterized by leafed structures. Orbits with central and axial symmetries are identified among the families found.     

Families of periodic collision orbits in the general three-body problem

In the general three-body problem, in a rotating frame of reference, a symmetric periodic solution with a binary collision is determined by the abscissa of one body and the energy of the system. For different values of the masses of the three bodies, the symmetric periodic collision orbits form a two-parametric family. In the case of equal masses of the two bodies and small mass of the third body, we found several symmetric periodic collision orbits similar to the corresponding orbits in the restricted three-body problem. Starting with one symmetric periodic collision orbit we obtained two families of such orbits. Also starting with one collision orbit in the Sun-Jupiter-Saturn system we obtained, for a constant value of the mass ratio of two bodies, a family of symmetric periodic collision orbits.     

Über periodische Bahnen im eingeschränkten Dreikörperproblem und in der Nähe der Kommensurabilitäten vom Typus (k + 1)/k

The authors, collaborating during the winter 1966/67, performed numerical investigations on form and character of periodic solutions of the restricted problem of three bodies in the Sun–Jupiter-System and in the neighbourhood of the resonances of the (k + 1)/k type. The principal part of these computations which were achieved using a fast IBM-computer refers to the cases 2/1 (Hecuba-gap) and 3/2 (Hilda-group). The results gave new insight into the genealogy of periodic orbits in the regions which do not contain solutions of Poincaré's first genre. Moreover, L. Carpenter tried to develop an analytical theory (here not explicitely described) which leads to convergent trigonometric series for the coordinates of periodic orbits with restricted eccentricities, using the method of general perturbations in rectangular coordinates recently published by P. Musen and himself.     

Planar periodic orbits in three dipole problem

A systematic and detailed discussion of planar periodic orbits, of a charged particle moving under the influence of an electromagnetic field of three celestial bodies, is given for the first time. In this problem the periodic orbits are all asymmetric. Numerical procedures are applied to find the families of these orbits and to study their stability. Moreover, the bifurcations of these families with families of three dimensional asymmetric periodic orbits are given.     

On the existence of periodic solutions of Poincare's third sort in the general problem of three bodies in three dimensions

A proof is offered of the existence of periodic solutions of the general problem of three bodies, of the third sort envisaged by Poincaré, that is, arising by analytic continuation from unperturbed keplerian motion of each of two bodies about a primary, in which the two orbits are of commensurable periods, of zero eccentricity, but lying in different planes, provided that the inclination of the two planes is sufficiently small (but not zero).Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.     

Keplerian periodic orbits in the isosceles problem

The planar isosceles three-body problem where the two symmetric bodies have small masses is considered as a perturbation of the Kepler problem. We prove that the circular orbits can be continued to saddle orbits of the Isosceles problem. This continuation is not possible in the elliptic case. Their perturbed orbits tend to a continued circular one or approach a triple collision. The basic tool used is the study of the Poincaré maps associated with the periodic solutions. This revised version was published online in July 2006 with corrections to the Cover Date.     

Surface fitting of three-dimensional closed solutions of the restricted three-body problem

The orbits of a family of three-dimensional periodic orbits in the restricted problem of three bodies form a surface. In this paper we determine the equation of this surface in the case of the orbits of double symmetry of the family which emanates from the equilibrium pointL ₁. This equation is obtained numerically by a least squares approximation method.     

Families of Asymmetric Periodic Orbits in the Restricted Three-body Problem

This paper studies the asymmetric solutions of the restricted planar problem of three bodies, two of which are finite, moving in circular orbits around their center of masses, while the third is infinitesimal. We explore, numerically, the families of asymmetric simple-periodic orbits which bifurcate from the basic families of symmetric periodic solutions f, g, h, i, l and m, as well as the asymmetric ones associated with the families c, a and b which emanate from the collinear equilibrium points L ₁, L ₂ and L ₃ correspondingly. The evolution of these asymmetric families covering the entire range of the mass parameter of the problem is presented. We found that some symmetric families have only one bifurcating asymmetric family, others have infinity number of asymmetric families associated with them and others have not branching asymmetric families at all, as the mass parameter varies. The network of the symmetric families and the branching asymmetric families from them when the primaries are equal, when the left primary body is three times bigger than the right one and for the Earth–Moon case, is presented. Minimum and maximum values of the mass parameter of the series of critical symmetric periodic orbits are given. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, we regularize the equations of motion of the problem using the Levi-Civita transformations.     

Periodic orbits of the general three-body problem for the Sun-Jupiter-Saturn system

Two families of symmetric periodic orbits of the planar, general, three-body problem are presented. The masses of the three bodies include ratios equal to the Sun-Jupiter-Saturn system and the periods of the orbits of Jupiter and Saturn are in a 25 resonance. The (linear) stability of the orbits are studied in relation to eccentricity and mass variations. The generation of the two families of periodic orbits follows a systematic approach and employs (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to the elliptic restricted problem and from the circular and elliptic problems to the general problem through bifurcation phenomena relating the three dynamical systems. The approach also provides insight into the evolutionary process of periodic orbits continued from the restricted problems to the general problem.     

On the dynamics in the asteroids belt. Part I : General theory

The classical problem of the dynamics in the asteroids belt is revisited in the light of recently developed perturbation methods. We consider the spatial problem of three bodies both in the circular and in the elliptic case, looking for families of periodic or quasi periodic orbits. Some criteria for deciding the stability of these families are also indicated.     

A note on the elliptic restricted three-body problem

This work considers periodic solutions, arc-solutions (solutions with consecutive collisions) and double collision orbits of the plane elliptic restricted problem of three bodies for =0 when the eccentricity of the primaries,e _p , varies from 0 to 1. Characteristic curves of these three kinds of solutions are given.     

Examples of predominantly two-body orbits in the Sun-Jupiter asteroid system

Accurate numerical continuation of families of plane symmetric direct periodic orbits around the large primary in the Sun-Jupiter case of the restricted problem of three bodies allows the determination of the vertical branching points where families of three-dimensional symmetric periodic orbits bifurcate from the planar ones. Three families of plane periodic orbits, and the initial segments of ten bifurcating families of three-dimensional ones are determined. The stability of these families is examined and examples of their orbits are illustrated.     

Families of periodic orbits in the restricted four-body problem

In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented. Three bodies of masses m ₁, m ₂ and m ₃ (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal and finally when we have three unequal masses. Series, with respect to the mass m ₃, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of the mass parameters are also calculated.     

Numerical investigation of the planar restricted three-body problem

In this article a method is described for the determination of families of periodic orbits, of the restricted problem of three bodies, as branchings of a given family of stable periodic orbits. Poincaré's method of successive crossings of a surface of section is applied for a value of the mass parameter corresponding to the Sun-Jupiter case of the restricted problem. New families are found, of the type of direct asteroids, having long periods and closing in space after many revolutions of the third body about the Sun. Their stability parameters are also given. The generating family, from which they branch, seems to have special significance for stability considerations.