Spatial Elliptic Restricted Three-body Problem Research Articles

Relegation-free closed-form perturbation theory and the domain of secular motions in the restricted three-body problem

We propose a closed-form (i.e., without expansion in the orbital eccentricities) scheme for computations in perturbation theory in the restricted three-body problem (R3BP) when the massless particle is in an orbit exterior to the one of the primary perturber. Starting with a multipole expansion of the barycentric (Jacobi-reduced) Hamiltonian, we carry out a sequence of normalizations in Delaunay variables by Lie series, leading to a secular Hamiltonian model without use of relegation. To this end, we introduce a book-keeping analogous to the one proposed in Cavallari and Efthymiopoulos (Celest Mech Dyn Astron 134(2):1–36, 2022) for test particle orbits interior to the one of the primary perturber, but here adapted, instead, to the case of exterior orbits. We give numerical examples of the performance of the method in both the planar circular and the spatial elliptic restricted three-body problem, for parameters pertinent to the Sun-Jupiter system. In particular, we demonstrate the method’s accuracy in terms of reproducibility of the orbital elements’ variations far from mean-motion resonances. As a basic outcome of the method, we show how, using as criterion the size of the series’ remainder, we reach to obtain an accurate semi-analytical estimate of the boundary (in the space of orbital elements) where the secular Hamiltonian model arrived at after eliminating the particle’s fast degree of freedom provides a valid approximation of the true dynamics.

Angular momentum analysis in the elliptic restricted three-body problem

ABSTRACT In this paper, the angular momentum of the test particle during the flyby in the spatial elliptic restricted three-body problem (ERTBP) is investigated. We focus on the flyby around the small primary, and derive approximate expressions to depict the changes of the angular momentum and energy in this process. Numerical computation based on the Earth–Moon ERTBP shows that our expressions can well approximate the real angular momentum and energy processes during the flyby. Hence, the mechanical process of a flyby orbit in the ERTBP is uncovered in detail. Furthermore, we analyse the optimal regions to maximize the changes of the angular momentum and energy. In addition, the inclination of the test particle during the flyby is investigated through the approximate expression of the angular momentum. The optimal flyby condition to change the inclination is obtained and discussed.

On the Lagrange Stability of the Motion in the Spatial Elliptic Restricted Three-body Problem

Abstract We study the Lagrange stability in the spatial elliptic restricted three-body problem. In this study, we essentially use the boundedness of the motion of a small particle with respect to planar coordinates and show that this plays a key role in ensuring the boundedness of the motion in the spatial case. As a result, we obtain the Lagrange stability theorem for an infinitely small particle in the spatial elliptic restricted three-body problem. The proposed approach can then be extended to the case of the general three-body problem.

Connection between Hill stability and weak stability in the elliptic restricted three-body problem

This paper provides a study on the connection between Hill stability and weak stability in the framework of the spatial elliptic restricted three-body problem. We determine a necessary condition for weak stability by giving an upper and a lower bound of qualitative measure of the Hill stability. The sufficient condition for weak stability and the symmetry of weak stable regions around the planets of the Solar System is also investigated.

On the stability of possible Trojan planets in the habitable zone: an application to the systems HD 147513 and HD 210277

In this paper, we study the dynamical stability of fictitious terrestrial planets in the 1:1 mean-motion resonance with a gas giant moving in the habitable zone (HZ). We investigate the stability of Trojan planets both in a general study and for two specific known extrasolar planetary systems (HD 147513 and HD 210277). In the general study, we determine the stability of the Lagrangian point L4. The numerical simulations have been carried out using the spatial elliptic-restricted three-body problem, where we placed the test particles (TPs) exactly at L4, which is located 60° ahead of the gas giant at the same distance to the star. In the stability study, we have concentrated on the dependences between the eccentricity and the inclination of the Trojan planet. Going a step further, we have investigated two specific systems where the known gas giant moves in the HZ. The two specific extrasolar systems are investigated by using the general study to define the region in eccentricity and inclination where, in principle, stable motion is possible. To find out whether the existence of a Trojan planet is possible, in addition we have calculated the stable area in the semimajor axis (aTP) and the argument of perihelion (ωTP) for an actual planetary mass (2 M⊕). Thus, we can conclude that the region around the Lagrangian points L4 and L5 allows stable motion in the system HD 147513, because the gas giant of this system has an eccentricity of 0.26, which lies well within the stable region. The known planet in the system HD 210277 has a much higher eccentricity and thus it cannot harbour any terrestrial planets in the Lagrangian points.

The orbit of 2010 TK7: possible regions of stability for other Earth Trojan asteroids

Recently the first Earth Trojan has been observed (Mainzer et al., ApJ 731) and found to be on an interesting orbit close to the Lagrange point L4 (Connors et al., Nature 475). In the present study we therefore perform a detailed investigation on the stability of its orbit and moreover extend the study to give an idea of the probability to find additional Earth-Trojans. Our results are derived using different approaches: a) we derive an analytical mapping in the spatial elliptic restricted three-body problem to find the phase space structure of the dynamical problem. We explore the stability of the asteroid in the context of the phase space geometry, including the indirect influence of the additional planets of our Solar system. b) We use precise numerical methods to integrate the orbit forward and backward in time in different dynamical models. Based on a set of 400 clone orbits we derive the probability of capture and escape of the Earth Trojan asteroids 2010 TK7. c) To this end we perform an extensive numerical investigation of the stability region of the Earth's Lagrangian points. We present a detailed parameter study in the regime of possible stable tadpole and horseshoe orbits of additional Earth-Trojans, i.e. with respect to the semi-major axes and inclinations of thousands of fictitious Trojans. All three approaches underline that the Earth Trojan asteroid 2010 TK7 finds himself in an unstable region on the edge of a stable zone; additional Earth-Trojan asteroids may be found in this regime of stability.

Stable and unstable orbits around Mercury

This paper provides a study of the stable and unstable regions around the smaller primary in the framework of the spatial elliptic restricted three-body problem. The definitions and methods used to determine stable and unstable regions are extended to three dimensions. New results concerning the stable and unstable regions around Mercury are obtained in the Sun–Mercury system.

How 'New Horizons' will see the Pluto-Charon system

We present a detailed survey of the dynamical structure of the phase space around the new moons of the Pluto-Charon system. We investigated the system in the framework of the spatial elliptic restricted three-body problem. Stability maps were created using chaos indicators both on the semimajor axis – eccentricity and semimajor axis – inclination plane. The structures related to the 4:1 and 6:1 mean motion resonances are clearly visible on the maps, but the detailed investigation of the resonant arguments shows no evidence of the resonances. We showed the possibility that Nix might be in the 4:1 resonance if its argument of pericenter or longitude of node falls in a certain range. Recently we improved our model to the general four-body problem, which can provide more accurate information about the stability, motion and possible resonances between the moons and Charon, which is relevant, since the 'New Horizons' mission will reach and study the system in 2015.

Families of first kind periodic solutions of the spatial and planar elliptic restricted three-body problem with collision

We consider a restricted three-body problem where the primaries are moving in an elliptic collision orbit and the infinitesimal mass moves in a three dimensional space. This paper is devoted to prove analytically the existence of several families of symmetric periodic solutions as continuation of Keplerian circular orbits. In our approach the perturbing parameter is related with the energy of the primaries.

Detailed survey of the phase space around Nix and Hydra

We present a detailed survey of the dynamical structure of the phase space around the new moons of the Pluto - Charon system. The spatial elliptic restricted three-body problem was used as model and stability maps were created by chaos indicators. The orbital elements of the moons are in the stable domain both on the semimajor axis - eccentricity and - inclination spaces. The structures related to the 4:1 and 6:1 mean motion resonances are clearly visible on the maps. They do not contain the positions of the moons, confirming previous studies. We showed the possibility that Nix might be in the 4:1 resonance if its argument of pericenter or longitude of node falls in a certain range. The results strongly suggest that Hydra is not in the 6:1 resonance for arbitrary values of the argument of pericenter or longitude of node.

A new class of symmetric periodic solutions of the spatial elliptic restricted three-body problem

We show that there exists a new class of symmetric periodic solutions of the spatial elliptic restricted three-body problem. In such a solution, the infinitesimal body is confined to the vicinity of a primary and moves on a nearly circular orbit. This orbit is almost perpendicular to the orbital plane of the primaries, where the line of symmetry of the orbit lies. The existence is shown by applying a corollary of Arenstorf’s fixed point theorem to a periodicity equation system of the problem. And this existence doesn’t require any restriction on the mass ratio of the primaries, nor on the eccentricity of their relative elliptic orbit. Potential relevance of this new class of periodic solutions to real celestial body systems and the follow-up studies in this respect are also discussed.

A stability study of Kuiper Belt binaries

AbstractThe dynamical structure of the orbital element space of seven Kuiper Belt binary systems is studied by numerical methods in the model of the spatial elliptic restricted three‐body problem. It is shown that three systems have an extended region of stability where additional satellites could exist. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

From the circular to the spatial elliptic restricted three-body problem

We deal with the study of the spatial restricted three-body problem in the case where the small particle is far from the primaries, that is, the so-called comet case. We consider the circular problem, apply double averaging and compute the relative equilibria of the reduced system. It appears that, in the circular problem, we find not only part of the equilibria existing in the elliptic case, but also new ones. These critical points are in correspondence with periodic and quasiperiodic orbits and invariant tori of the non-averaged Hamiltonian. We explain carefully the transition between the circular and the elliptic problems. Moreover, from the relative equilibria of elliptic type, we obtain invariant 3-tori of the original system.

Evolutionary Search for Deep-Space Science Mission Orbits

We present an application of an evolutionary programming method, niching genetic algorithms, to the search for orbits in the spatial elliptic restricted three-body problem (ER3BP) that is suitable for deep-space science missions. The niching method used is deterministic crowding, which renders a global optimization while permitting for several optimal and suboptimal solutions to coexist. This novel approach yields diverse probing of the state space of the ER3BP. From the practical standpoint, the orbits found remain within a bounded distance from Earth, thus allowing high data-rate communication while ensuring safe operational environment, far from thermal perturbations and visual occultation as well as Earth’s magnetic and radiation fields.

Searching for periodic orbits of the spatial elliptic restricted three-body problem by double averaging

We calculate new families of periodic orbits of the spatial elliptic restricted three-body problem. The original system has three degrees of freedom and depends on time explicitly, but surprisingly after doubly averaging the Hamiltonian over the time and the mean anomaly the system gets reduced to one degree of freedom. This fact allows us to determine families of periodic orbits of the original system. We compute the relative equilibria, study their linear and Lyapunov stability and reconstruct the corresponding invariant manifolds in the original Hamiltonian. Additionally, we determine six families of periodic orbits with large semimajor axes having any eccentricity. Making use of adequate variables, we easily see that four families lie on a plane perpendicular to the plane of the primaries, two of them being stable and the other two unstable. The other two families are stable families of closed orbits coplanar with the primaries.

Capture in the Circular and Elliptic Restricted Three-Body Problem

In this paper the authors provide a study of the phenomenon of the gravitational capture by using the models of the circular and elliptic restricted three-body problem. In the first part the inadequacy of the circular restricted three-body problem in the study of the phenomenon of the capture in the case of NEAs is shown. In the model of the spatial elliptic restricted three-body problem criteria of the capture are deduced by using the pulsating Hill-regions.

Niching genetic algorithms-based characterization of geocentric orbits in the 3D elliptic restricted three-body problem

This paper investigates the application of niching genetic algorithms to the characterization of families of geocentric orbits in the Sun–Earth spatial elliptic restricted three-body problem (ER3BP). The niching method used is deterministic crowding, which renders global optimization while permitting for several optimal and sub-optimal solutions to co-exist. This novel approach yields diverse probing of the state-space of the ER3BP. From the practical standpoint, the orbits found remain within a bounded distance from Earth, thus allowing high data-rate communication while ensuring safe operational environment far from thermal perturbations and visual occultations as well as Earth’s magnetic and radiation fields.

A Symplectic Mapping Model for the Study of 2:3 Resonant Trans-Neptunian Motion

A symplectic mapping is constructed for the study of the dynamical evolution of Edgeworth-Kuiper belt objects near the 2:3 mean motion resonance with Neptune. The mapping is six-dimensional and is a good model for the Poincare map of the ‘real’ system, that is, the spatial elliptic restricted three-body problem at the 2:3 resonance, with the Sun and Neptune as primaries. The mapping model is based on the averaged Hamiltonian, corrected by a semianalytic method so that it has the basic topological properties of the phase space of the real system both qualitatively and quantitatively. We start with two dimensional motion and then we extend it to three dimensions. Both chaotic and regular motion is observed, depending on the objects' initial inclination and phase. For zero inclination, objects that are phase-protected from close encounters with Neptune show ordered motion even at eccentricities as large as 0.4 and despite being Neptune-crossers. On the other hand, not-phase-protected objects with eccentricities greater than 0.15 follow chaotic motion that leads to sudden jumps in their eccentricity and are removed from the 2:3 resonance, thus becoming short period comets. As inclination increases, chaotic motion becomes more widespread, but phase-protection still exists and, as a result, stable motion appears for eccentricities up to e = 0.3 and inclinations as high as i = 15°, a region where plutinos exist.

Periodic solutions in the spatial elliptic restricted three-body problem

We show the existence of a new class of periodic orbits in the three-dimensional elliptic restricted three-body problem in the case of equal masses of the primaries. The doubly symmetric periodic solutions found are perturbations of very large circular Keplerian orbits lying in a plane perpendicular to that of the primaries. They exist for a discrete sequence of values of the mean motion, irrespective of the values of the eccentricity of the primaries orbit.