Non-schubart periodic orbits in the rectilinear three-body problem

In the present paper, in the rectilinear three-body problem, we qualitatively follow the positions of non-Schubart periodic orbits as the mass parameter changes. This is done by constructing their characteristic curves. In order to construct characteristic curves, we assume a set of properties on the shape of areas corresponding to symbol sequences. These properties are assured by our preceding numerical calculations. The main result is that characteristic curves always start at triple collision and end at triple collision. This may give us some insight into the nature of periodic orbits in the N-body problem.     

Optimal periodic relative orbit and rectilinear relative orbits with eccentric reference orbits

The problem of two-body linearized periodic relative orbits with eccentric reference orbits is studied in this paper. The periodic relative orbit in the target-orbital coordinate system can be used in fly-around and formation-flying orbit design. Based on the closed-form solutions to the Tschauner–Hempel equations, the initial condition for periodic relative orbits is obtained. Then the minimum-fuel periodic-orbit condition with a single impulse is analytically derived for given initial position and velocity vectors. When considering the initial coasting time, the impulse position of the global minimum-fuel periodic orbit is proved to be near to the perigee of the target and can be obtained by numerical optimization algorithms. Moreover, the condition for a special periodic orbit, i.e., the rectilinear relative orbit in the target-orbital frame, is obtained. Numerical simulations are used to demonstrate the efficacy of the method, and show the geometry of the periodic relative orbit and the rectilinear relative orbit.     

Symmetric doubly-asymptotic periodic orbits at collinear equilibria

A systematic search for periodic orbits doubly-asymptotic to the collinear equilibrium points of the restricted three-body problem is carried out and many such orbits are found, each of them existing for a specific value of the mass parameter. These may be useful as reference orbits and seem to be special limit orbits representing period discontinuities in the evolution of the families of periodic orbits.     

The rectilinear restricted problem of three bodies

In this paper we discuss some aspects of the isosceles case of the rectilinear restricted problem of three bodies, where two primaries of equal mass move on rectilinear ellipses, and the particle is confined to the symmetry axis of the system. In particular, the behaviour near a collision of the primaries and also near a collision of all three bodies is investigated. It is shown that this latter singularity is a triple collision in the sense of Siegel's theory. Furthermore, asymptotic expansions for the particle's motion during a parabolic and a hyperbolic escape are derived.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

A two-parameter survey of periodic orbits in the restricted problem of three bodies

Within the context of the restricted problem of three bodies, we wish to show the effects, caused by varying the mass ratio of the primaries and the eccentricity of their orbits, upon periodic orbits of the infinitesimal mass that are numerical continuations of circular orbits in the ordinary problem of two bodies. A recursive-power-series technique is used to integrate numerically the equations of motion as well as the first variational equations to generate a two-parameter family of periodic orbits and to identify the linear stability characteristics thereof. Seven such families (comprised of a total of more than 2000 orbits) with equally spaced mass ratios from 0.0 to 1.0 and eccentricities of the orbits of the primaries in a range 0.0 to 0.6 are investigated. Stable orbits are associated with large distances of the infinitesimal mass from the perturbing primary, with nearly circular motion of the primaries, and, to a slightly lesser extent, with small mass ratios of the primaries.Conversely, unstable orbits for the infinitesimal mass are associated with small distances from the perturbing primary, with highly elliptic orbits of the primaries, and with large mass ratios.     

Almost rectilinear halo orbits

Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional halo periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L₃-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L₁-family and L₂-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.     

A grid search for families of periodic orbits in the restricted problem of three bodies

A method is described for the numerical determination of families of periodic orbits in the planar restricted problem of three bodies. The families are sought in their representation as curves in a two-dimensional space of parameters. A grid search is applied to the study of the evolution of satellite motion when the mass parameter is varied. Only that part of the space of parameters is investigated for which one of them, the relative energy constant, takes values larger than that corresponding to the inner Lagrangian pointL ₂. Critical values of the mass parameter are determined for which new families of simple or double periodic orbits appear inside the closed ovals of zero velocity.     

Three-dimensional periodic orbits about the triangular equilibrium points of the restricted problem of three bodies

The third-order parametric expansions given by Buck in 1920 for the three-dimensional periodic solutions about the triangular equilibrium points of the restricted Problem are improved by fourthorder terms. The corresponding family of periodic orbits, which are symmetrical w.r.t. the (x, y) plane, is computed numerically for =0.00095. It is found that the family emanating from L₄ terminates at the other triangular point L₅ while it bifurcates with the family of three-dimensional periodic orbits originating at the collinear equilibrium point L₃. This family consists of stable and unstable members. A second family of nonsymmetric three-dimensional periodic orbits is found to bifurcate from the previous one. It is also determined numerically until a collision orbit is encountered with the computations.     

Numerical explorations of the rectilinear problem of three bodies

We present some results of a numerical exploration of the rectilinear problem of three bodies, with the two outer masses equal. The equations of motion are first given in relative coordinates and in regularized variables, removing both binary collision singularities in a single coordinate transformation. Among our most important results are seven periodic solutions and three symmetric triple collision solutions. Two of these periodic solutions have been continued into families, the outer massm ₃ being the family parameter. One of these families exists for all masses while the second family is a branch of the first at a second-kind critical orbit. This last family ends in a triple collision orbit.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

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.     

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.     

Symmetric horseshoe periodic orbits in the general planar three-body problem

We present some families of horseshoe periodic orbits in the general planar three-body problem for the case of two equal masses. The considered system is a symmetric version of the one formed by Saturn, Janus and Epimetheus. We use a mass ratio equal to 35×10⁻⁵, corresponding to 10⁵ times the Saturn-Janus mass parameter of the restricted case; for this mass ratio the satellites have a significantly bigger influence on the planet than in the classical Saturn, Janus and Epimetheus system. To obtain periodic orbits, we search those horseshoe orbits passing through two reversible configurations. A particular kind of periodic orbits where the minor bodies follow the same path is discussed.     

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.     

The rectilinear three-body problem using symbol sequence II: role of the periodic orbits

We study the change of phase space structure of the rectilinear three-body problem when the mass combination is changed. Generally, periodic orbits bifurcate from the stable Schubart periodic orbit and move radially outward. Among these periodic orbits there are dominant periodic orbits having rotation number (n − 2)/n with n ≥ 3. We find that the number of dominant periodic orbits is two when n is odd and four when n is even. Dominant periodic orbits have large stable regions in and out of the stability region of the Schubart orbit (Schubart region), and so they determine the size of the Schubart region and influence the structure of the Poincaré section out of the Schubart region. Indeed, with the movement of the dominant periodic orbits, part of complicated structure of the Poincaré section follows these orbits. We find stable periodic orbits which do not bifurcate from the Schubart orbit.     

Three dimensional periodic orbits in three dipole problem

In the three dipole problem where enormous electromagnetic forces obstruct the three dimensional movement of the charged particle we determined for the first time families of three dimensional asymmetric periodic orbits. We study how these families appear, branching from the planar motion and we develop the procedures we have followed to determine them numerically. Also we give their characteristics and the conical projections and plottings of some orbits.     

Almost-square orbits in the restricted problem of three bodies

In the restricted problem of three bodies, it has been discovered that, in a nonrotating frame, almost perfectly square orbits can result. Numerical investigations of these orbits have been made, and a brief explanation of their behaviour is given.     

Evolution of a ‘Class Two’ family of periodic orbits in the general planar problem of three bodies

The elliptic restricted problem of three bodies with unit eccentricity of the primaries is used to generate a family of periodic orbits in the general problem of three bodies. The parameter of the family is the mass of one of the participating bodies. This varies from zero to a termination value. The mass ratio of the primaries of the unperturbed problem (three to five) is maintained throughout the generation of the family. In this way an asymmetry is introduced generalizing the Copenhagen elliptic problem as the generating model. All members of the family experience a close approach and a collision between the primaries during half of the period of the orbit, therefore, the family is classified as Class Two.     

Irregular periodic orbits

We study the peculiarities of irregular periodic orbits, i.e. orbits belonging to families not connected with the main families or their bifurcation, of Hamiltonian systems of two degrees of freedom. Families of irregular periodic orbits appear in triplets which are either closed or extend to infinity. If these triplets form an infinite sequence they surround an escape region. It seems probable that in general regions covered by irregular families are of high degree of stochasticity.     

The symplectic character of periodic orbits in the three dipole problem

In this work we reveal for the first time that in the three dipole problem only asymmetric periodic orbits exist.For these periodic orbits — planar and three dimensional — of a charged particle moving under the influence of the electromagnetic field of the three dipoles we give their symplectic relations using the Hamiltonian formulation which is related to the symplectic matrix. Also we study the properties of the symplectic matrix and we give the relations there are among the variations of a periodic solution. These relations have been used to check the accuracy of numerical integration of equations of first order variations.     

2/1 resonant periodic orbits in three dimensional planetary systems

We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to 50^?–60^?, which may be related with the existence of real planetary systems.