Families Of Three-dimensional Periodic Orbits Research Articles

Numerical Investigation for Periodic Orbits in the Hill Three-Body Problem

The current work performs a numerical study on periodic motions of the Hill three-body problem. In particular, by computing the stability of its basic planar families we determine vertical self-resonant (VSR) periodic orbits at which families of three-dimensional periodic orbits bifurcate. It is found that each VSR orbit generates two such families where the multiplicity and symmetry of their member orbits depend on certain property characteristics of the corresponding VSR orbit’s stability. We trace twenty four bifurcated families which are computed and continued up to their natural termination forming thus a manifold of three-dimensional solutions. These solutions are of special importance in the Sun-Earth-Satellite system since they may serve as reference orbits for observations or space mission design.

Families of periodic orbits in a double-barred galaxy model

We reveal the networks of simple symmetric periodic orbits in a double-barred galaxy model. Specifically, we investigate the dependence on the total orbital energy of the positions but also on the stability of the periodic solutions. For every orbital family, we also compute the horizontal and vertical critical parameter values of the system, at which new periodic families bifurcate from. Of particular interest are the vertical critical points which act as starting points for the creation of new families of three-dimensional periodic orbits. The atlas of the simple periodic trajectories is presented in the (x, E) plane and also in the (x, E, z) and (x,E,z˙) spaces, in order to obtain the global parametric evolution of the orbital families. Our analysis suggests that the orbital properties of a double-barred galaxy model are very complicated and at the same time very fascinating.

Periodic solutions around the collinear equilibrium points in the perturbed restricted three-body problem with triaxial and radiating primaries for binary HD 191408, Kruger 60 and HD 155876 systems

The collinear equilibrium points and periodic motion around them are studied in the framework of the restricted three-body problem where the two primaries are triaxial rigid bodies which emit radiation. Firstly, the positions and stability of the collinear equilibria are studied for the HD 191408, Kruger 60 and HD 155876 binary systems. Then, the planar and three-dimensional periodic motion about these points is considered. Our study includes both semi-analytical and numerical determination of these motions. It is found that all families of planar periodic orbits emanating from these points terminate with asymptotic periodic orbits at the triangular equilibrium points while the corresponding families of three-dimensional periodic orbits terminate with planar periodic orbits. Families of Halo orbits bifurcating from the first vertical critical periodic orbit of the three planar Lyapunov families were also considered.

Periodic orbits around the collinear equilibrium points for binary Sirius, Procyon, Luhman 16, α-Centuari and Luyten 726-8 systems: the spatial case

An investigation of three-dimensional periodic orbits and their stability emanating from the collinear equilibrium points of the restricted three-body problem with oblate and radiating primaries is presented. A simulation is done by using five binary systems: Sirius, Procyon, Luhman 16, α-Centuari and Luyten 726-8. Firstly, based on the topological degree theory, the total number of the collinear equilibrium points for the five binary systems were obtained and then, their positions were determined numerically. The linear stability of these equilibrium points was also examined and found to be unstable in the Lyapunov sense. An analytical approximation of three-dimensional periodic solutions around them was established via the Lindstedt–Poincaré local analysis. Finally, using the analytical solution to obtain starting orbits, the families of three-dimensional periodic orbits emanating from these equilibria have been continued numerically.

PERIODIC MOTIONS AROUND THE COLLINEAR EQUILIBRIUM POINTS OF THE R3BP WHERE THE PRIMARY IS A TRIAXIAL RIGID BODY AND THE SECONDARY IS AN OBLATE SPHEROID

ABSTRACT We consider a modification of the restricted three-body problem where the primary (more massive body) is a triaxial rigid body and the secondary (less massive body) is an oblate spheroid and study periodic motions around the collinear equilibrium points. The locations of these points are first determined for 10 combinations of the parameters of the problem. In all 10 cases, the collinear equilibrium points are found to be unstable, as in the classical problem, and the Lyapunov periodic orbits around them have been computed accurately by applying known corrector–predictor algorithms. An extensive study on the families of three-dimensional periodic orbits emanating from these points has also been done. To find suitable starting points, for all the computed families, semianalytical solutions have been obtained, for both two- and three-dimensional cases, around the collinear equilibrium points using the Lindstedt–Poincaré method. Finally, the stability of all computed periodic orbits has been studied.

Periodic motions in the spatial Chermnykh restricted three-body problem

Three-dimensional motions in the Chermnykh restricted three-body problem are studied. Specifically, families of three-dimensional periodic orbits are determined through bifurcations of the family of straight-line periodic oscillations of the problem which exists for equal masses of the primaries. These rectilinear oscillations are perpendicular to the plane of the primaries and give rise to an infinite number of families consisting entirely of periodic orbits which belong to the three-dimensional space except their respective one-dimensional bifurcations as well as their planar terminations. Many of the computed branch families are continued in all mass range that they exist.

The Sitnikov family and the associated families of 3D periodic orbits in the photogravitational RTBP with oblateness

We consider the photogravitational restricted three-body problem with oblateness and study the Sitnikov motions. The family of straight line oscillations exists only in the case where the primaries are of equal masses as in the classical Sitnikov problem and have the same oblateness coefficients and radiation factors. A perturbation method based on Floquet theory is applied in order to study the stability of the motion and critical orbits are determined numerically at which families of three-dimensional periodic orbits of the same or double period bifurcate. Many of these families are computed.

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.

BIFURCATIONS FROM PLANAR TO THREE-DIMENSIONAL PERIODIC ORBITS IN THE PHOTO-GRAVITATIONAL RESTRICTED FOUR-BODY PROBLEM

We study the bifurcations of three-dimensional periodic motions from two-dimensional orbits of small particles in the neighborhood of a system that consists of three major bodies being always in syzygy. We assume that two of them have equal masses and are located at equal distances from the third body which has a different mass. All or some of the primaries are radiation sources and therefore apart from gravitational forces, we also consider forces that result from radiation. We apply the method of vertical critical stability in order to find the families of three-dimensional periodic orbits that bifurcate from families of planar periodic motions. Consequently, we study the parametric variations of the bifurcation points.

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.

Stability Bounds for Three-Dimensional Motion Close to Asteroids

We use numerically determined periodic orbits to explore the stability of three-dimensional trajectories around asteroids. In a rotating frame, families of three-dimensional periodic orbits are known to appear in the vicinity of planar, resonant, periodic orbits. Thus, by computing several of these resonant families that bifurcate from the plane and studying their stability properties as they evolve, we find that these families of three-dimensional periodic orbits change their stability type at certain critical inclinations. That characteristic enables us to draw a line relating orbital inclination and mean radius of the orbiter. This line separates regions of stable motion from unstable ones and creates a global criterion for the stability of motion.

Bifurcation points and intersections of families of periodic orbits in the three-dimensional restricted three-body problem

Intersections of families of three-dimensional periodic orbits which define bifurcation points are studied. The existence conditions for bifurcation points are discussed and an algorithm for the numerical continuation of such points is developed. Two sequences of bifurcation points are given concerning the family of periodic orbits which starts and terminates at the triangular equilibrium pointsL4,L5. On these sequences two trifurcation points are identified forµ = 0.124214 andµ = 0.399335. The caseµ = 0.5 is studied in particular and it is found that the space families originating at the equilibrium pointsL2,L3,L4,L5 terminate on the same planar orbitm1v of the familym.

On basic families of three-dimensional periodic orbits of three massive bodies and their stability

The basic families of three-dimensional periodic orbits of the general three-body problem are determined numerically in order to obtain a global view of the simpler patterns of periodic three-body motion in three dimensions. The stability of the orbits is also computed. It is found that most of the orbits are unstable but stability intervals do exist for some of the families.

Periodic orbits in three-dimensional planetary systems

Three-dimensional planetary systems are studied, using the model of the restricted three-body problem for Μ =.001. Families of three-dimensional periodic orbits of relatively low multiplicity are numerically computed at the resonances 3/1, 5/3, 3/5 and 1/3 and their stability is determined. The three-dimensional orbits are found by continuation to the third dimension of the vertical critical orbits of the corresponding planar problem

Three-dimensional periodic orbits in barred galaxies

We study the properties of the families of three-dimensional periodic orbits which bifurcate from the vertical critical orbits of the retrograde family of quasi-circular plane periodic orbits which extend from the galactic center up to infinity. We consider the case of a barred galaxy with a strong central bulge. The values of the parameters are chosen in such a way as to cover the cases of a strong or weak bar with a fast or slow rotation.

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, μ=m2/(m1+m2) andm3, 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 ofm3 around the ‘primaries’m1 andm2. The series of these termination orbits, formed when the value ofm3 varies, are also given. The three-dimensional orbits are computed form3=0.1.

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 μ. 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.

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.

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.