Vertical-Critical Periodic Orbits in the General Three-Body Problem

We present special generating plane orbits, the vertical-critical orbits, of the coplanar general three-body problem. These are determined numerically for various values of m₃, for the entire range of the mass ratio of the two primaries. The vertical-critical orbits are necessary in order to specify the vertically stable segments of the families of plane periodic orbits, and they are also the starting points of the families of the simplest possible three-dimensional periodic orbits, namely the simple and double periodic. The initial conditions of the vertical-critical periodic orbits of the basic families l, m, i, h, b and c and their stability parameters are determined. This revised version was published online in July 2006 with corrections to the Cover Date.     

Bifurcations of plane with three-dimensional periodic orbits in the elliptic restricted problem

We show that the procedure employed in the circular restricted problem, of tracing families of three-dimensional periodic orbits from vertical self-resonant orbits belonging to plane families, can also be applied in the elliptic problem. A method of determining series of vertical bifurcation orbits in the planar elliptic restricted problem is described, and one such series consisting of vertical-critical orbits (a _v=+1) is given for the entire range (0,1/2) of the mass parameter . The initial segments of the families of three-dimensional orbits which bifurcate from two of the orbits belonging to this series are also given.     

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.     

The stability parameters of three-dimensional periodic three-body orbits

Formulae containing the elements of the variational matrix are obtained which determine the linear isoenergetic stability parameters of three-dimensional periodic orbits of the general three-boy problem. This requires the numerical integration of the variational equations but produces the stability parameters with the effective accuracy of the numerical integration. The conditions for stability, criticality, and bifurcations are briefly examined and the stability determination procedure is tested in the determination of some three-dimensional periodic orbits of low inclination bifurcating from vertical-critical coplanar orbits.     

The stability parameters of three-body periodic orbits

Formulae containing the elements of the variational matrix are obtained which determine the linear iso-energetic stability parameters of periodic orbits of the general three-body problem. This requires the numerical integration of the variational equations but produces the stability parameters with the effective accuracy of the numerical integration. The procedure is applied for the determination of horizontally critical orbits among the members of sets of vertical-critical periodic orbits of the threebody problem. These critical-critical orbits have special importance as they delimit the regions in the space of initial conditions which correspond to possibly stable three-dimensional periodic motion of low inclination.     

Self-resonant bifurcations of the Sitnikov family and the appearance of 3D isolas in the restricted three-body problem

The association of the Sitnikov family with families of multiple three-dimensional periodic orbits is studied. In particular, the families consisting of three-dimensional periodic orbits bifurcating from self-resonant orbits of the Sitnikov family at double, triple and quadruple period of the bifurcation orbit are considered. The branch families close upon themselves and remain 3D up to their terminations having two common members with the Sitnikov family. By varying the mass parameter we also study the evolution of some of the computed families and find that they become isolas and disappear gradually in three-dimensions by shrinking to point size.     

Simple three-dimensional periodic orbits in a galactic-type potential

We study the families of simple periodic orbits in a three-dimensional system that represents the inner parts of a perturbed triaxial galaxy. The perturbations depend on two control parameters. We find the regions where each family is stable, simply unstable, doubly unstable, or complex unstable. the stable and simply unstable families produce other families by bifurcation. Several families reach a maximum (or minimum) perturbation and then are continued by other families. The bifurcations are direct or inverse. The transition from one type of bifurcation to the other is theoretically explained. Another important phenomenon is the splitting of one family into two, or the joining of two families into one. We do not have any complex instability in the limiting cases of two-dimensional motions (when one control parameter is zero).The two main families of periodic orbits are in most cases stable when the energy is smaller than the escape energy. Most high energy orbits are unstable. However, we found stable orbits even for energies about four times larger than the escape energy.     

Three-dimensional periodic motion of three finite masses around collinear equilibrium configurations

Three-dimensional periodic motions of three bodies are shown to exist in the infinitesimal neighbourhood of their collinear equilibrium configurations. These configurations and some characteristic quantities of the emanating three-dimensional periodic orbits are given for many values of the two mass parameters, =m ₂/(m ₁+m ₂) andm ₃, of the general three-body problem, under the assumption that the straight line containing the bodies at equilibrium rotates with unit angular velocity. The analysis of the small periodic orbits near the equilibrium configurations is carried out to second-order terms in the small quantities describing the deviation from plane motion but the analytical solution obtained for the horizontal components of the state vector is valid to third-order terms in those quantities. The families of three-dimensional periodic orbits emanating from two of the collinear equilibrium configurations are continued numerically to large orbits. These families are found to terminate at large vertical-critical orbits of the familym of retrograde periodic orbits ofm ₃ around the primariesm ₁ andm ₂. The series of these termination orbits, formed when the value ofm ₃ varies, are also given. The three-dimensional orbits are computed form ₃=0.1.     

A Note on Parameter Values for Stable Low-Inclination Periodic Motion in the Restricted Three-Body Problem with Radiation

The intervals of the mass parameter (μ) values for possible stability of the basic families of 3D periodic orbits in the restricted three-body problem determined elsewhere are now extended into regions of theμ - q ₁ parameter space of the photogravitational restricted three-body problem, where q ₁ is the radiation factor of m ₁ and it is assumed that m ₂ does not radiate. Several 3D periodic orbits corresponding to these regions are computed and tested for stability and seven regions, corresponding to the vertical-critical orbits l1v, l'1v, l6v, m1v, m2v and i1v, survive this stability test, emerging as the regions allowing the simplest types of stable low inclination 3D motion of the infinitesimal particle. This revised version was published online in July 2006 with corrections to the Cover Date.     

Bifurcations of plane to 3D periodic orbits in the photogravitational restricted three-body problem

We consider bifurcation of 3D periodic orbits from the plane ofmotion of the primaries in the photogravitational restricted three-bodyproblem. The simplest periodic 3D orbits branch from the plane periodicorbits of indifferent vertical stability. We compute the first few suchorbits of the basic families l, m, i, h, a, b, c forvarying mass parameter and for varying radiation coefficient of thelarger primary. The horizontal stability of the orbits is also computedleading to predictions about possible stability of the 3D orbits.     

On the 2/1 resonant planetary dynamics – periodic orbits and dynamical stability

The 2/1 resonant dynamics of a two-planet planar system is studied within the framework of the three-body problem by computing families of periodic orbits and their linear stability. The continuation of resonant periodic orbits from the restricted to the general problem is studied in a systematic way. Starting from the Keplerian unperturbed system, we obtain the resonant families of the circular restricted problem. Then, we find all the families of the resonant elliptic restricted three-body problem, which bifurcate from the circular model. All these families are continued to the general three-body problem, and in this way we can obtain a global picture of all the families of periodic orbits of a two-planet resonant system. The parametric continuation, within the framework of the general problem, takes place by varying the planetary mass ratio ρ. We obtain bifurcations which are caused either due to collisions of the families in the space of initial conditions or due to the vanishing of bifurcation points. Our study refers to the whole range of planetary mass ratio values  [ρ∈ (0, ∞)]  and, therefore we include the passage from external to internal resonances. Thus, we can obtain all possible stable configurations in a systematic way. As an application, we consider the dynamics of four known planetary systems at the 2/1 resonance and we examine if they are associated with a stable periodic orbit.     

The three-dimensional general three-body problem: Determination of periodic orbits

The three-dimensional general three-body problem is formulated suitably for the numerical determination of periodic orbits either directly or by continuation from the three-dimensional periodic orbits of the restricted problem. The symmetry properties of the equations of motion are established and the algorithms for the numerical determination of families of periodic orbits are outlined. A normalization scheme based on the concept of the invariable plane is introduced to simplify the process. All three types of symmetric orbit, as well as the general type of asymmetric orrbit, are considered. Many threedimmensional p periodic orbits are given.     

On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries

The existence of new equilibrium points is established in the restricted three-body problem with equal prolate primaries. These are located on the Z-axis above and below the inner Eulerian equilibrium point L ₁ and give rise to a new type of straight-line periodic oscillations, different from the well known Sitnikov motions. Using the stability properties of these oscillations, bifurcation points are found at which new types of families of 3D periodic orbits branch out of the Z-axis consisting of orbits located entirely above or below the orbital plane of the primaries. Several of the bifurcating families are continued numerically and typical member orbits are illustrated.     

Numerical periodic orbits of charged grains around magnetic planets

We apply a numerical searching method to investigate three-dimensional periodic orbits of charged dust particles in planetary magnetospheres. A classic generalized Stormer model of magnetic planets along with the parameters of Saturn is employed. More periodic orbits are found, besides the already known circular periodic orbits in or parallel to the equatorial plane. We divide all these orbits into six categories based on their appearances. By calculating the characteristic multipliers of the orbits, we investigate the stabilities of these periodic orbits.     

Regular and irregular periodic orbits

We distinguish between regular orbits, that bifurcate from the main families of periodic orbits (those that exist also in the unperturbed case) and irregular periodic orbits, that are independent of the above. The genuine irregular families cannot be made to join the regular families by changing some parameters. We present evidence that all irregular families appear inside lobes formed by the asymptotic curves of the unstable periodic orbits. We study in particular a dynamical system of two degrees of freedom, that is symmetric with respect to the x-axis, and has also a triple resonance in its unperturbed form. The distribution of the periodic orbits (points on a Poincaré surface of section) shows some conspicuous lines composed of points of different multiplicities. The regular periodic orbits along these lines belong to Farey trees. But there are also lines composed mainly of irregular orbits. These are images of the x-axis in the map defined on the Poincaré surface of section. Higher order iterations of this map , close to the unstable triple periodic orbit, produce lines that are close to the asymptotic curves of this unstable orbit. The homoclinic tangle, formed by these asymptotic curves, contains many regular orbits, that were generated by bifurcation from the central orbit, but were trapped inside the tangle as the perturbation increased. We found some stable periodic orbits inside the homoclinic tangle, both regular and irregular. This proves that the homoclinic tangle is not completely chaotic, but contains gaps (islands of stability) filled with KAM curves.     

The manifolds of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem

This paper deals with the Sitnikov family of straight-line motions of the circular restricted three-body problem, viewed as generator of families of three-dimensional periodic orbits. We study the linear stability of the family, determine several new critical orbits at which families of three dimensional periodic orbits of the same or double period bifurcate and present an extensive numerical exploration of the bifurcating families. In the case of the same period bifurcations, 44 families are determined. All these families are computed for equal as well as for nearly equal primaries (μ = 0.5, μ = 0.4995). Some of the bifurcating families are determined for all values of the mass parameter μ for which they exist. Examples of families of three dimensional periodic orbits bifurcating from the Sitnikov family at double period bifurcations are also given. These are the only families of three-dimensional periodic orbits presented in the paper which do not terminate with coplanar orbits and some of them contain stable parts. By contrast, all families bifurcating at single-period bifurcations consist entirely of unstable orbits and terminate with coplanar orbits.     

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.     

Straight-line oscillations generating three-dimensional motions in the photogravitational restricted three-body problem

The Sitnikov configuration is a special case of the restricted three-body problem where the two primaries are of equal masses and the third body of a negligible mass moves along a straight line perpendicular to the orbital plane of the primaries and passes through their center of mass. It may serve as a toy model in dynamical astronomy, and can be used to study the three-dimensional orbits in more applicable cases of the classical three-body problem. The present paper concerns the straight-line oscillations of the Sitnikov family of the photogravitational circular restricted three-body problem as well as the associated families of three-dimensional periodic orbits. From the stability analysis of the Sitnikov family and by using appropriate correctors we have computed accurately 49 critical orbits at which families of 3D periodic orbits of the same period bifurcate. All these families have been computed in both cases of equal and non-equal primaries, and consist entirely of unstable orbits. They all terminate with coplanar periodic orbits. We have also found 35 critical orbits at which period doubling bifurcations occur. Several families of 3D periodic orbits bifurcating at these critical Sitnikov orbits have also been given. These families contain stable parts and close upon themselves containing no coplanar orbits.     

Numerical investigation of the manifold I qp in the Sun-Jupiter case of the restricted three-body problem

Message and Taylor (1978) have given values of the mean eccentricities and commen-surabilities which correspond to bifurcation orbits of families of symmetric periodic orbits with families of asymmetric periodic orbits in the limit as the mass ratio tends to zero. These bifurcations have been given in a way that they seem to be isolated and unrelated from the whole structure of the periodic orbits of the system.In this paper a numerical investigation of the horizontal stability of the family I and its branches reveals the above bifurcations orbits in the Sun-Jupiter case of the restricted three-body problem and associates these orbits with the whole structure of the system, giving extensive information on them.     

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.