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.     

Periodic orbits in the Planar General Three-Body Problem

The article contains a numerical study of periodic solutions of the Planar General Three-Body Problem. Several new periodic solutions have been discovered and are described. In particular, there is a continuous family with variable masses, extending all the way from the elliptic restricted problem to the general problem with three equal masses. All our examples have special symmetry properties which are described in detail. Finally we also suggest some important applications to the natural satellites of the solar system.     

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.     

Bifurcations of Plane to 3d Periodic Orbits in the Restricted Three-Body Problem with Oblateness

We consider the bifurcation of 3D periodic orbits from the plane of motion of the primaries in the restricted three-body problem with oblateness. The simplest 3D periodic orbits branch-off at the plane periodic orbits of indifferent vertical stability. We describe briefly suitable numerical techniques and apply them to produce the first few such vertical-critical orbits of the basic families of periodic orbits of the problem, for varying mass parameter and fixed oblateness coefficent A₁ = 0.005, as well as for varying A₁ and fixed = 1/2. The horizontal stability of these orbits is also determined leading to predictions about the stability of the branching 3D orbits.     

Multiple Periodic Orbits in the Ring Problem: Families of Triple Periodic Orbits

The ring problem deals with the motion of a small body which is subjected to the combined gravitational attraction of N massive bodies arranged in an annular configuration. In this paper we study the distribution of the triple periodic orbits in the phase space of the initial conditions and we discuss their evolution and their principal features. 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.     

A Search for Collision Orbits in the Free-Fall Three-Body Problem II

A numerical procedure to systematically find collision orbits in the planar three-body problem has been developed in the preceding paper (Tanikawa et al., 1995). Using this procedure, a search for binary and triple collision orbits has been carried out in the free-fall three-body problem. Some detailed structures of a part of the initial value space are discussed. Various interesting orbits have been found. Examples are oscillatory orbits in which ejected particles change from ejection to ejection, and orbits which are not isosceles initially but nearly isosceles after escape. Some results of isosceles problems (Simó and Martínez, 1988) are extended to non-isosceles problems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Asymptotic Orbits at the Triangular Equilibria in the Photogravitational Restricted Three-Body Problem

We study numerically the asymptotic homoclinic and heteroclinic orbits associated with the triangular equilibrium points L ₄ and L ₅, in the gravitational and the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these critical points, are also presented. Hundreds of asymptotic orbits for equal mass of the primaries and for various values of the radiation pressure are computed and the most interesting of them are illustrated. In the Copenhagen case, which the problem is symmetric with respect to the x- and y-axis, we found and present non-symmetric heteroclinic asymptotic orbits. So pairs of heteroclinic connections (from L ₄ to L ₅ and vice versa) form non-symmetric heteroclinic cycles. The termination orbits (a combination of two asymptotic orbits) of all the simple families of symmetric periodic orbits, in the Copenhagen case, are illustrated.     

Symmetric Periodic Orbits in the Anisotropic Schwarzschild-Type Problem

Studying the two-body problem associated to an anisotropic Schwarzschild-type field, Mioc et al. (2003) did not succeed in proving the existence or non-existence of periodic orbits. Here we answer this question in the affirmative. To do this, we start from two basic facts: (1) the potential generates a strong force in Gordon’s sense; (2) the vector field of the problem exhibits the symmetries S _i , , which form, along with the identity, an Abelian group of order 8 with three generators of order 2. Resorting to S ₂ and S ₃, in connection with variational methods (particularly the classical lower-semicontinuity method), we prove the existence of infinitely many S ₂- or S ₃-symmetric periodic solutions. The symmetries S ₂ and S ₃ constitute an indicator of the robustness of the classical isotropic Schwarzschild-type system to perturbations (as the anisotropy may be considered).     

On the Evolution of Orbits in a Photo-Gravitational Circular Three-Body Problem: The Inner Problem

Astronomy Letters - We consider the spatial restricted circular three-body problem in the nonresonant case. The massless body (satellite) is assumed to have a large sail area and, therefore, the...     

Numerical-Analytical Study of Linked Orbits in the Restricted Elliptic Doubly Averaged Three-Body Problem

Solar System Research - The restricted elliptic doubly averaged three-body problem is considered. The terms up to the second order inclusive with respect to the orbital eccentricity of the...     

Planar Symmetric Periodic Orbits in Four Dipole Problem

We have extend Stormer’s problem considering four magnetic dipoles in motion trying to justify the phenomena of extreme “orderlines” such as the ones observed in the rings of Saturn; the aim is to account the strength of the Lorentz forces estimating that the Lorentz field, co-acting with the gravity field of the planet, will limit the motion of all charged particles and small size grains with surface charges inside a layer of about 200 m thickness as that which is observed in the rings of Saturn. For this purpose our interest feast in the motion of charged particles with neglected mass where only electromagnetic forces accounted in comparison to the weakness of the Newtonian fields. This study is particularly difficult because in the regions we investigate these motions there is enormous three dimensional instability. Following the Poincare’s hypothesis that periodic solutions are ‘dense’ in the set of all solutions in Hamiltonian systems we try to calculate many families of periodic solutions and to study their stability. In this work we prove that in this environment charged particles can trace planar symmetric periodic orbits. We discuss these orbits in details and we give their symplectic relations using the Hamiltonian formulation which is related to the symplectic matrix. We apply numerical procedures to find families of these orbits and to study their stability. Moreover we give the bifurcations of these families with families of planar asymmetric periodic orbits and families of three dimensional symmetric periodic orbits.     

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.     

Multiple Periodic Orbits in the Hill Problem with Oblate Secondary

We study the multiple periodic orbits of Hill’s problem with oblate secondary. In particular, the network of families of double and triple symmetric periodic orbits is determined numerically for an arbitrary value of the oblateness coefficient of the secondary. The stability of the families is computed and critical orbits are determined. Attention is paid to the critical orbits at which families of non-symmetric periodic orbits bifurcate from the families of symmetric periodic orbits. Six such bifurcations are found, one for double-periodic and five for triple-periodic orbits. Critical orbits at which families of sub-multiple symmetric periodic orbits bifurcate are also discussed. Finally, we present the full network of families of multiple periodic orbits (up to multiplicity 12) together with the parts of the space of initial conditions corresponding to escape and collision orbits, obtaining a global view of the orbital behavior of this model problem.     

Properties of New Coordinates for the General Three-Body Problem

This is an analysis of the features of the new coordinate system given by the principal axes of inertia, as determined by Euler angles, and twodistances related to the inertia principal moments and an auxiliar angleas coordinates, for studying the general three-body problem, interactingthrough gravitational forces.The reduction of order is performed in these new coordinates by using the angular velocity vector or the Euler angles.The Eulerian case of collinear motion is revisited from our own perspective.The value of the auxiliar angle is computed for the Sun–Earth–Moon system.     

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.     

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.     

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.     

Some Peculiarities of the Evolution of Orbits in the Satellite Restricted Elliptic Doubly Averaged Three-Body Problem

Solar System Research - The inner (or satellite) version of the restricted elliptic three-body problem is considered. The terms up to the fourth order inclusive in small parameter are retained in...     

On the Evolution of Orbits in the Outer Version of the Restricted Elliptic Doubly Averaged Three-Body Problem

Solar System Research - The outer version of the restricted elliptic three-body problem, where a perturbed body of negligible mass is far from two bodies of finite masses (the main and perturbing...