Stability Bounds for Three-Dimensional Motion Close to Asteroids

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.

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 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.

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.

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.

T HE relative motion of a satellite (follower) with respect to the reference satellite (leader) in a circular orbit is described by autonomous nonlinear differential equations. The linearized equations at the origin are known as Hill–Clohessy–Wiltshire (HCW) equations [1]. The in-plane motion and the out-of plane motion are independent. The latter is a simple sinusoidal motion. The former has also periodic solutions under the so-called Clohessy–Wiltshire condition. Periodic solutions are given by sinusoidal functions, and the relative orbits are ellipses. These orbits are parametrized by two parameters representing size and phase. Because of this simple nature, relative periodic orbits are used for formation flying [2]. Periodic orbits of the nonlinear relative dynamics are also characterized by the parameters of the follower’s elliptic orbit and its inclination angle in an inertial reference frame, and initial conditions for periodic orbits are explicitly given [3]. If the reference orbit is elliptic, the equations of relative motion involve the true anomaly and the radius of the orbit, which are periodic functions. The linearized equations of motion at the origin are known as Tschauner–Hempel (TH) equations. The in-plane motion and the out-of plane motion remain independent. However, the derivation of the state transition matrix in this case is not immediate. In earlier studies [4–6], the true anomaly is used as independent variable, because this simplifies the resulting equations and the transition matrix can be obtained for them. Rendezvous problems have been extensively studied using this representation [7,8]. The out-of-plane motion is a sinusoidal motion, and the inplane motion has periodic solutions [2,9]. The condition, which generalizes the CW condition, for periodic solutions is given in [9]. Periodic orbits of the TH equations are also used for formation flying [2,9,10]. However, they are more complicated compared to ellipses of the HCW equations, and their shape changes with eccentricity. Some of them are very much distorted. Hence, it is not intuitively clear how to design relative orbits. The direct study of the relative dynamics using time as independent variable is given in [11], and a series expansion of the state transition matrix in the eccentricity is obtained in [12]. A closed form of the state transition matrix is recently derived in [13],where a brief history of elliptic rendezvous is also found. The characterization of periodic orbits of the nonlinear relative dynamics as functions of time is extended to the elliptic case, and initial conditions for periodic orbits are explicitly given [14]. The general solution of the relative dynamics based on the orbital elements is obtained in [15], and the relative motion invariant manifold, where the solution lies, is determined. This technical note is concernedwith the generation of all periodic solutions of the in-plane motion of the TH equations. For this purpose, the state transition matrix of [16], which is given as a function of the true anomaly, is used. The solution is parametrized by four constants, and the condition for periodic solution follows from this. As in the case of the HCWequations, size and phase parameters are introduced. The main contributions of this note are summarized as follows. First, the region where all periodic solutions lie is determined by two ellipses, whose size depends on the eccentricity of the leader orbit. Given size and phase parameters, a graphical method to determine the initial point of a periodic orbit at a given true anomaly is proposed. Varying the value of the true anomaly, the whole family of periodic orbits is generated. The time domain versions of the region and periodic orbits are determined by multiplying the radius of the leader orbit. This clarifies how the shape of a periodic orbit changes with eccentricity and why the shapes of some orbits are distorted. As the eccentricity goes to zero, the region of periodic motion shrinks to a single ellipse, which is a periodic orbit of the HCW equations. The size parameter gives the distance between the leader and the follower, and the phase determines the position of the follower. As in [15], the maximum and minimum distances to the leader can be calculated. Hence, our method is useful to design relative orbits with size and position specifications. This note is organized as follows. Section II reviews the equations of relativemotion along an elliptic orbit. Section III gives a geometric method to draw periodic orbits of the TH equations. Finally, Section IV gives conclusions.

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.

The procedure of numerical ‘ascent’ from families of planar to three-dimensional periodic orbits and the subsequent ‘descent’ to the plane is proved efficient in determining new families of planar asymmetric periodic orbits in the restricted three-body problem. Two such families are computed and described for values of the mass parameter for which it has been found that they exist. Two new families of three-dimensional asymmetric periodic orbits are also presented in this paper.

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].

Superscars are quantum eigenstates that are localized in an extended region of a family of parallel periodic orbits. They are prevalent in pseudointegrable systems. Here we consider a pseudo-integrable right-triangular quantum billiard with an acute angle π/5 as an example. In the configuration space, we identify and explain a fringe structure of superscar eigenstates originating from a correlation between the two quantum numbers imposed by the sign change rule of the wave function during unfolding. In the wave-vector space, we unveil unique characteristics of the peaks for superscar states, and in return, exploit these features to determine the family of periodic orbits and the corresponding quantum numbers without ambiguity. We also find that the eigenstates can be superpositions not only of superscar modes corresponding to different families of periodic orbits, but also of superscar modes corresponding to the same family but with different sets of quantum numbers, which poses a challenge for conventional approaches in identifying the superscar modes but can be addressed properly by our analysis. These results are also applicable to superscar states in other pseudointegrable systems.

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.

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.

We study the motions of an infinitesimal mass in the Sitnikov four-body problem in which three equal oblate spheroids (called primaries) symmetrical in all respect, are placed at the vertices of an equilateral triangle. These primaries are moving in circular orbits around their common center of mass. The fourth infinitesimal mass is moving along a line perpendicular to the plane of motion of the primaries and passing through the center of mass of the primaries. A relation between the oblateness-parameter ‘A’ and the increased sides ‘e’ of the equilateral triangle during the motion is established. We confine our attention to one particular value of oblateness-parameter A=0.003. Only one stability region and 12 critical periodic orbits are found from which new three-dimensional families of symmetric periodic orbits bifurcate. 3-D families of symmetric periodic orbits, bifurcating from the 12 corresponding critical periodic orbits are determined. For A=0.005, observation shows that the stability region is wider than for A=0.003.

The three-rotor system concerns equally massive point particles moving on a circle subject to attractive cosine potentials of strength g. The quantum theory models chains of coupled Josephson junctions. Classically, it displays order-chaos-order behavior with increasing energy E along with a seemingly globally chaotic phase for 5.33g≲E≲5.6g. It is also known to admit pendulum and isosceles breather families of periodic orbits at all energies. While pendula display a doubly infinite sequence of stability transitions accumulating at their libration to rotation threshold at E=4g, breathers undergo only one stability transition. Here, we show that these stability transitions are associated with forward and reverse fork-like isochronous and period-doubling bifurcations. The new family of periodic orbits born at each of these bifurcations is found using an efficient search algorithm starting from a transverse perturbation to the parent orbit. The graphs of stability indices of various classes of orbits born at pendulum bifurcations meet at E=4g forming "fans." The transitions in the librational and rotational phases are related by an asymptotic duality between bifurcation energies and shapes of newly born periodic orbits. The latter are captured by solutions to a Lamé equation. We also find and numerically validate values of scaling constants for self-similarity in (a) stability indices of librational and rotational pendula and (b) shapes of newly born orbits as E→4g. Finally, we argue that none of the infinitely many families of periodic orbits we have found is stable for 5.33g≲E≲5.6g, providing further evidence for global chaos in this energy band.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Ernesto A. Lacomba, Jaume Llibre; Periodic orbits and non integrability of autonomous Glydén‐like systems. AIP Conference Proceedings 7 October 2010; 1283 (1): 25–31. https://doi.org/10.1063/1.3506069 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search