Numerical determination of families of three-dimensional double-symmetric periodic orbits in the restricted three-body problem,II

New families of three-dimensional double-symmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the vertical-critical orbits ( _v = – 1, b _v – 0) of the basic plane familiesi, g ₁, g₂, c andI. Further, the predictor-corrector procedure employed to reveal these families has been described and interesting numerical results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections.     

Numerical determination of families of three-dimensional double-symmetric periodic orbits in the restricted three-body problem. I.

New families of three-dimensional double-symmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the vertical-critical orbits ( _v = –1,c _v ),c _v=0) of the basic plane familiesi,g ₁,g ₂,h,a,m andl. Further the numerical procedure employed in the determination of these families has been described and interesting results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections.     

Numerical integration of periodic orbits in the main problem of artificial satellite theory

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ ₂ problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ ₂ andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ ₂, as large as |J ₂|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.     

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.     

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.     

General three-body problem: Families of vertical critical periodic orbits

In this paper we present four families of vertical critical periodic orbits found by continuation, with respect to the small massm ₃, of the vertical critical periodic orbitsl1v, ilv, mlv, c3v of the circular restricted problem. The periodic orbits refer to a suitably defined rotating frame of reference.     

The flattening of three-dimensional periodic orbits in the restricted problem

A number of partly known families of symmetric three-dimensional periodic orbits of the restricted three-body (=0.4) problem are numerically continued in both ends until they terminate with orbits in the plane of motion of the primaries. The families of plane symmetric periodic orbits from which they bifurcate are identified and many orbit illustrations are given.     

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.     

Numerical integration of the variational equations of satellite orbits

A recurrent power series (RPS) method is constructed for the numerical integration of the equations of motion together with the variational equations of N point masses orbiting around an oblate spheroid. By the term “variational equations” we mean the equations of the partial derivatives of the bodies’ position and velocity components with respect to the initial conditions, the relative masses and the spheroid's oblateness coefficients J₂ and J₄. The construction of recursive relations for the partial derivatives involved in the variational equations is based on partial differentiation of the corresponding recursive relations for the integration of the equations of motion. Since the number of the auxiliary variables needed for this complex system becomes tremendously large when N>1, special care must be taken during computer implementation, so as to minimize the amount of computer memory needed as well as the cost in CPU time. The RPS method constructed in this way is tested for N=1,…,4 using real initial conditions of the Saturnian satellite system. For various sets of satellites, we monitor the behaviour of all the corresponding partial derivatives. The results show a prominent difference in the behaviour of the partial derivatives between resonant and non-resonant orbital systems.     

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.     

On the continuation of periodic orbits from the planar to the three-dimensional general three-body problem

It is shown that the vertical critical orbits of the general planar problem of three bodies can be used as starting points for finding monoparametric families of three-dimensional periodic orbits. Several numerical examples are given.     

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.     

Hill's problem: Families of three-dimensional periodic orbits (part I)

Starting from the vertical critical periodic orbits of the planar Hill's problem we computed five families of three-dimensional periodic orbits. It is found that their behaviour is qualitatively similar to that of the corresponding families of the restricted problem.     

Bifurcations of planar to three-dimensional periodic orbits in the general three-body problem

We study the generation of three-dimensional periodic orbits of the general three-body problem from special generating plane orbits, the vertical-critical orbits. The bifurcation process is examined analytically and geometrically. A method of obtaining numerically continuous sets of vertical-critical orbits is outlined, and applied for the determination of 16 monoparametric sets including all possible types of such orbits corresponding to all possible types of symmetry of the bifurcating three-dimensional orbits. The stability of all bifurcation orbits is assessed. Examples of three-dimensional periodic orbits generated from the bifurcation orbits are given.     

Numerical determination of asymmetric periodic solutions

A simple predictor-corrector procedure is described for the determination of asymmetric periodic solutions of dynamical systems of two degrees of freedom. An application in the case of the Störmer problem is given. The computed periodic motions of the charged particle are of the open-path type.     

The family of planar periodic orbits generated by the equal-mass four-body Schubart interplay orbit

We locate members of a one-parameter family of equal-mass four-body periodic orbits in the plane. The family begins and ends with the rectilinear four-body equal-mass Schubart interplay orbit and passes through a double choreography orbit. The first-order stability of these orbits is computed. Some members of this symmetric family are stable to symmetric perturbations; however, they are unstable when all perturbations are allowed.     

General three-body problem: Families of three-dimensional periodic orbits (part II)

In this paper we present a two-parametric family of symmetric periodic orbits of the three-dimensional general three-body problem, found numerically by continuation of a vertical critical orbit of the circular restricted three-body problem. The periodic orbits refer to a suitably defined rotating frame of reference.     

Numerical continuation of periodic orbits from the restricted to the general 3-dimensional 3-body problem

In this paper several monoparametric families of periodic orbits of the 3-dimensional general 3-body problem are presented. These families are found by numerical continuation with respect to the small massm ₃, of some periodic orbits which belong to a family of 3-dimensional periodic orbits of the restricted elliptic problem.     

A restricted problem: Families of vertical critical periodic orbits (part II)

In this paper we extend the computation of four families of vertical critical periodic orbitsb _1v ,b _2v ,c _1v ,c _2v , found in part I (Ichtiaroglouet al., 1980), for >0.5. The planar stability of the periodic orbits is also examined.     

Families of three-dimensional axisymmetric periodic orbits in the restricted three-body problem: Sun-Jupiter case

Families of three-dimensional axisymmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the vertical-critical orbits (_v = 1,b _v = 0) of the basic plane familiesi andI. Further the predictor-corrector procedure employed to reveal these families has been described and interesting numerical results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections.