1/1 resonant periodic orbits in three dimensional planetary systems

We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted TBP, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances and thus, new families, continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consist of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far.

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 $50^\circ$ - $60^\circ$, which may be related with the existence of real planetary systems.

In this work, our target is to analyze the dynamics around the $1:-1$ resonance which appears when a family of periodic orbits of a real analytic three-degree of freedom Hamiltonian system changes its stability from elliptic to a complex hyperbolic saddle passing through degenerate elliptic. Our analytical approach consists of computing, up to some given arbitrary order, the normal form around that resonant (or <em> critical</em>) periodic orbit. <br> Hence, dealing with the normal form itself and the differential equations related to it, we derive the generic existence of a two-parameter family of invariant 2D tori which bifurcate from the critical periodic orbit. Moreover, the coefficient of the normal form that determines the stability of the bifurcated tori is identified. This allows us to show the Hopf-like character of the unfolding: elliptic tori unfold "around'' hyperbolic periodic orbits (case of <em> direct</em> bifurcation) while normal hyperbolic tori appear "around'' elliptic periodic orbits (case of <em>inverse</em> bifurcation). Further, the parametrization of the main invariant objects as well as a global description of the dynamics of the normal form are also given.

This paper investigates the global structures of periodic orbits that appear in Rayleigh–Bénard convection, which is modelled by a two-dimensional perturbed Hamiltonian model, by focusing upon resonance, symmetry and bifurcation of the periodic orbits. First, we show the global structures of periodic orbits in the extended phase space by numerically detecting the associated periodic points on the Poincaré section. Then, we illustrate how resonant periodic orbits appear and specifically clarify that there exist some symmetric properties of such resonant periodic orbits which are projected on the phase space; namely, the period m and the winding number n become odd when an m-periodic orbit is symmetric with respect to the horizontal and vertical centre lines of a cell. Furthermore, the global structures of bifurcations of periodic orbits are depicted when the amplitude ɛ of the perturbation is varied, since in experiments the amplitude of the oscillation of the convection gradually increases when the Rayleigh number is raised.

The rectilinear elliptic restricted Three Body Problem (TBP) is the limiting case of the elliptic restricted TBP when the motion of the primaries is described by a Keplerian ellipse with eccentricity $e'=1$, but the collision of the primaries is assumed to be a non-singular point. The rectilinear model has been proposed as a starting model for studying the dynamics of motion around highly eccentric binary systems. Broucke (1969) explored the rectilinear problem and obtained isolated periodic orbits for mass parameter $\mu=0.5$ (equal masses of the primaries). We found that all orbits obtained by Broucke are linearly unstable. We extend Broucke's computations by using a finer search for symmetric periodic orbits and computing their linear stability. We found a large number of periodic orbits, but only eight of them were found to be linearly stable and are associated with particular mean motion resonances. These stable orbits are used as generating orbits for continuation with respect to $\mu$ and $e'<1$. Also, continuation of periodic solutions with respect to the mass of the small body can be applied by using the general TBP. FLI maps of dynamical stability show that stable periodic orbits are surrounded in phase space with regions of regular orbits indicating that systems of very highly eccentric orbits can be found in stable resonant configurations. As an application we present a stability study for the planetary system HD7449.

We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit.

We study the mechanism of bifurcation of three-dimensional periodic orbits from the plane of motion of the primaries. The branchings of simple and double periodic three-dimensional orbits occur at the plane periodic orbits of indifferent vertical stability. These vertical-critical plane orbits are doubly important. Not only are they necessary in order to specify the vertically stable segments of the families of plane periodic orbits, but they are also the starting points of the families of the simplest possible three-dimensional periodic orbits, namely, the simple and double periodic. We describe here suitable numerical techniques for their accurate determination and for their continuation through the range of the mass parameter μ of the problem. We use these techniques to determine the first few vertical-critical orbits of the basic families of periodic orbits of the problem, namely, families l, m, i, h, a, b and c, for the entire range of μ. The initial conditions of over three hundred vertical-critical orbits are given, classified in twenty-one series. Their horizontal stability is also examined and this leads to predictions of where in the range of μ stable three-dimensional periodic orbits may not branch from these basic families.

The purpose of this series of lecture notes is to give an outline of the basic tools required to show the occurrence of chaotic motions in the simplest non--integrable problems in Celestial Mechanics, such as the circular restricted planar 3--body problem. No formal proofs will be given here; they can be found in the references given for each section. Section 1 describes the linear and local theory of ordinary differential equations in the neighbourhood of a fixed point; the problems arising in the embedding of invariant stable and unstable manifolds are also discussed. Section 2 is about periodic orbits; the subjects discussed include variational equations, surfaces of section, the continuation of periodic orbits in the restricted 3--body problem, and bifurcation of hyperbolic periodic orbits from resonant periodic orbits. Section 3 covers fundamental models of resonance, the global behaviour of separatrices, and their intersections; all this allows to give at least an outline of the proof of the fundamental result --presented by Poincaré in his book Les méthodes nouvelles de la mécanique céleste-- by which homoclinic points must necessarily occur in the restricted problem. A short conclusion underscores the fact --shown in a rigorous vay much later-- that this in turn implies that chaos in the strongest possible sense occurs in the restricted problem, and is an essential feature of every non--integrable system, even very simple ones with only two degrees of freedom. I apologize for reporting here my lectures in a very short format, almost without comments in between the formulas and statements of the main results; my understanding of the purpose of this notes is that they should serve as a reminder of the existence of many subjects to be studied, rather than a complete presentation which could not be contained in this format.

Numerical analysis of orbital motion around a contact binary asteroid system

The stability of the asteroid orbits has been studied by the method of surface of section. Families of simple symmetric periodic orbits of the asteroid and their stability have been computed and this served as a guide for the selection of the energy levels for the surface of section. In this way all possible cases for the structure of phase space have been obtained. It was found that the region in phase space around the resonant orbits at the resonances 1/3, 3/5, 5/7,.... is unstable, but small stability regions of doubly symmetric periodic orbits near the above resonances are also present. At the resonances 1/2, 2/3, 3/4, .... it was found that there exist two separate regions in phase space at about the same resonance 1/2, 2/3, 3/4,...., respectively, one being stable and the other unstable. At certain energy levels only the stable region appears. The above results are consistent with the observed distribution of the asteroids.

O NE of the earliest investigations of the Poincare surfaces of section method [1] to discover periodic orbits in the restricted three-body problemwas byHenon [2–5] in a number of publications. Jefferys [6] also used this method to evolve a large number of periodic orbits in the restricted three-body problem. After that, this technique has been used by many researchers to find periodic orbits in the restricted three-body problem. Some of the important contributions are by Smith [7], Smith and Szebehely [8], Tuckness [9,10], Scott and Spencer [11], Howell and Kakoi [12], Demeyer and Gurfil [13], Guilera [14], and Kolemen et al. [15]. The motivation for the present study comes from the work of Winter [16], which deals with a family of simply symmetric retrograde periodic orbits around the moon in the rotating Earth–moon–particle system in the framework of the planar circular restricted three-body problem for the Earth–moon mass ratio (0.01215). This family of periodic orbits is one particular case of the family of periodic orbits classified by Broucke [17].

In this work, the perturbed equations of motion of the infinitesimal body are constructed in the framework of the circular restricted three-body problem when the main two bodies are oblate and radiating. Under the perturbations effects of the oblateness and the radiation pressure the positions of collinear Lagrange points are evaluated, the interior and exterior first-order resonant periodic orbits are also studied. In addition, the initial positions of the periodic orbits and the size of loops have been estimated under these effects. Thus, the characteristics of periodic orbits have been studied under the combine effects of two, three and four perturbations for all the possible combinations of the perturbed parameters. The different order of resonant periodic orbits have been also analysed under the effects of Jacobi constant, mass factor, order of resonance and number of loops.

In recent years, three-dimensional periodic and quasi-periodic orbits near the collinear libration points in the Sun-Earth/Moon three-body problem have been the focus of great interest for space mission design. Thus, more effort in the astrodynamics community has been directed toward analysis and computation of families of such orbits. But families of periodic orbits in the three-body problem have been the subject of much study for many decades. The first such orbit to be employed for a spacecraft trajectory (ISEE-3) was a member of a particular type of simply symmetric three-dimensional family, i.e., a halo orbit. Thus, attention shifted to further analysis of these periodic halo families. Some of the significant work in the development of periodic solutions in the three-body problem has been reviewed and a number of the highlights from the analysis and eventual numerical computation of halo families is presented here. The halo families of periodic orbits extend from each of the libration points to the nearest primary; they appear to exist for all values of the mass ratio. Thus, this further understanding may serve to support future spacecraft mission planning as well.

A new approach to the study of the Solar System and planetary systems in general is proposed, through the use of periodic planetary-type orbits of the general N-body problem. In such an orbit, one body (called Sun) has a large mass and the rest N-l bodies (called planets) have small but not negligible masses and it can be proved that monoparametric families of periodic orbits of the N-body problem exist in a rotating frame of reference, all being of the planetary type.Two cases are studied in detail, N=3 and N=4. In N=3, apart from a general discussion, we present a detailed analysis of the Sun-Jupiter-Saturn system and a study is made on which configurations with the masses of these two planets, or a multiple of them, are stable or unstable. Also, part of a family is shown to represent the Jupiter family of comets. It was found that commensurabilities are not in general associated with instabilities. For N=4 we present three families of periodic orbits. The motion corresponding to a branch of one of the above families has many similarities with the actual motion of the three inner satellites of Jupiter.It is shown that there exist many commensurable cases in the obtained periodic orbits and that the resonant orbits increase as the number of bodies increases. Based on these results, an attempt is made to explain the existence of commensurabilities in the Solar System.Finally, it is mentioned that a periodic motion of the planetary type can be used as a reference orbit for accurate computations for the actual motions of the planets or satellites of the Solar System. In this way the small divisor difficulties existing in the classical approach will not appear.

High-order resonant orbit manifold expansions for mission design in the planar circular restricted 3-body problem