High order symplectic integrators for perturbed Hamiltonian systems

We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form $H=A+\epsilon B$. We give a constructive proof that for all integer $p$, there exists an integrator with positive steps with a remainder of order $O(\tau^p\epsilon +\tau^2\epsilon^2)$, where $\tau$ is the stepsize of the integrator. The analytical expressions of the leading terms of the remainders are given at all orders. In many cases, a corrector step can be performed such that the remainder becomes $O(\tau^p\epsilon +\tau^4\epsilon^2)$. The performances of these integrators are compared for the simple pendulum and the planetary 3-Body problem of Sun-Jupiter-Saturn.Comment: 24 pages, 6 figurre

On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited

In the framework of the planar and circular restricted three-body problem, we consider an asteroid that orbits the Sun in quasi-satellite motion with a planet. A quasi-satellite trajectory is a heliocentric orbit in co-orbital resonance with the planet, characterized by a non zero eccentricity and a resonant angle that librates around zero. Likewise, in the rotating frame with the planet it describes the same trajectory as the one of a retrograde satellite even though the planet acts as a perturbator. In the last few years, the discoveries of asteroids in this type of motion made the term "quasi-satellite" more and more present in the literature. However, some authors rather use the term "retrograde satellite" when referring to this kind of motion in the studies of the restricted problem in the rotating frame. In this paper we intend to clarify the terminology to use, in order to bridge the gap between the perturbative co-orbital point of view and the more general approach in the rotating frame. Through a numerical exploration of the co-orbital phase space, we describe the quasi-satellite domain and highlight that it is not reachable by low eccentricities by averaging process. We will show that the quasi-satellite domain is effectively included in the domain of the retrograde satellites and neatly defined in terms of frequencies. Eventually, we highlight a remarkable high eccentric quasi-satellite orbit corresponding to a frozen ellipse in the heliocentric frame. We extend this result to the eccentric case (planet on an eccentric motion) and show that two families of frozen ellipses originate from this remarkable orbit.Comment: 30 pages, 13 figures, 1 tabl

Spin-orbit coupling and chaotic rotation for coorbital bodies in quasi-circular orbits

Coorbital bodies are observed around the Sun sharing their orbits with the planets, but also in some pairs of satellites around Saturn. The existence of coorbital planets around other stars has also been proposed. For close-in planets and satellites, the rotation slowly evolves due to dissipative tidal effects until some kind of equilibrium is reached. When the orbits are nearly circular, the rotation period is believed to always end synchronous with the orbital period. Here we demonstrate that for coorbital bodies in quasi-circular orbits, stable non-synchronous rotation is possible for a wide range of mass ratios and body shapes. We show the existence of an entirely new family of spin-orbit resonances at the frequencies $n\pm k\nu/2$, where $n$ is the orbital mean motion, $\nu$ the orbital libration frequency, and $k$ an integer. In addition, when the natural rotational libration frequency due to the axial asymmetry, $\sigma$, has the same magnitude as $\nu$, the rotation becomes chaotic. Saturn coorbital satellites are synchronous since $\nu\ll\sigma$, but coorbital exoplanets may present non-synchronous or chaotic rotation. Our results prove that the spin dynamics of a body cannot be dissociated from its orbital environment. We further anticipate that a similar mechanism may affect the rotation of bodies in any mean-motion resonance.Comment: 6 pages. Astrophysical Journal (2013) 6p

Rigorous treatment of the averaging process for co-orbital motions in the planetary problem

We develop a rigorous analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. By constructing a complex domain of holomorphy for the planetary Hamilto-nian, we estimate the size of the transformation that maps this Hamil-tonian to its first order averaged over one of the fast angles. After having derived an integrable approximation of the averaged problem, we bound the distance between this integrable approximation and the averaged Hamiltonian. This finally allows to prove rigorous theorems on the behavior of co-orbital motions over a finite but large timescale

The family of Quasi-satellite periodic orbits in the circular co-planar RTBP

In the circular case of the coplanar Restricted Three-body Problem, we studied how the family of quasi-satellite (QS) periodic orbits allows to define an associated libration center. Using the averaged problem, we highlighted a validity limit of this one: for QS orbits with low eccentricities, the averaged problem does not correspond to the real problem. We do the same procedure to L 3 , L 4 and L 5 emerging periodic orbits families and remarked that for very high eccentricities F L4 and F L5 merge with F L3 which bifurcates to a stable family

Sur la dynamique coorbitale : du mouvement des troyens de Jupiter à la rotation des plan etes coorbitales

Le premier chapitre présentera les principales techniques sur lesquelles sont basés les travaux exposés dans ce mémoire : l'étude de l'application fréquence, la mesure de la diffusion par analyse en fréquence et l'application de ces dernières à l'étude de la dynamique globale d'un système planétaire. Dans le deuxième chapitre, je proposerai une introduction à la résonance coorbitale. Après avoir exposé quelques aspects historiques de l'existence d'orbites remarquables dans le problème des trois corps (non restreint), je rappellerai certains points concernant la stabilité de ces solutions, puis présenterai des résultats originaux ayant trait au problème moyen ainsi qu'aux bifurcations des équilibres de Lagrange et d'Euler. Après avoir abordé la résonance coorbitale dans le cadre du problème des 3-corps, j'exposerai, dans le troisième chapitre, les résultats obtenus sur la dynamique des troyens de Jupiter, ainsi que certaines de leurs extensions. Les trajectoires des troyens seront étudiés à l'aide du problème restreint de (n+2)-corps, n correspondant au nombre de planètes du système auxquelles on ajoutera le Soleil et une particule test représentant le troyen. L'essentiel du travail consistera à comprendre les structures dynamiques des essaims dans l'espace des fréquences. Les résultats découleront naturellement de cette étude. Je donnerai d'abord une classification des principales résonances conditionnant la dynamique des essaims. Je montrerai ensuite comment cette structure résonante influence la dynamique à long terme des essaims de Jupiter et en particulier leur érosion engendrée par des phénomènes de diffusion. Je terminerai ce chapitre en abordant la question de la modification de la structure résonante, et par conséquent de la stabilité de l'essaim, sous l'effet de la modification de la géométrie du système planétaire.Le quatrième chapitre regroupera mes travaux effectués sur la rotation des corps en résonance coorbitale.Je débuterai ce chapitre en présentant l'approche générale de la rotation en résonance coorbitale qui nous a conduit à distinguer trois grandes classes dynamiques de systèmes en fonction de leurs masses et des valeurs des moments d'inertie des corps qui les composent. Pour deux de ces classes apparaît un nouveau type de résonance spin-orbit qui peut conduire à des rotations chaotiques. Pour l'autre classe, à laquelle appartiennent les satellites de Saturne Janus, Epiméthée, Hélène, Polydeuces, Téthys et Calypso, on retrouvera les résonances spin-orbit habituelles perturbées par le mouvement coorbital. C'est à la rotation de ces six satellites que seront consacrés les derniers paragraphes du quatrième chapitre. Je terminerai cet exposé en proposant quelques extensions et applications possibles des travaux et méthodes présentés dans ce mémoire

Secular Dynamics of S-type Planetary Orbits in Binary Star Systems: Applicability Domains of First- and Second-Order Theories

We analyse the secular dynamics of planets on S-type coplanar orbits in tight binary systems, based on first- and second-order analytical models, and compare their predictions with full N-body simulations. The perturbation parameter adopted for the development of these models depends on the masses of the stars and on the semimajor axis ratio between the planet and the binary. We show that each model has both advantages and limitations. While the first-order analytical model is algebraically simple and easy to implement, it is only applicable in regions of the parameter space where the perturbations are sufficiently small. The second-order model, although more complex, has a larger range of validity and must be taken into account for dynamical studies of some real exoplanetary systems such as $\gamma$-Cephei and HD 41004A. However, in some extreme cases, neither of these analytical models yields quantitatively correct results, requiring either higher-order theories or direct numerical simulations. Finally, we determine the limits of applicability of each analytical model in the parameter space of the system, giving an important visual aid to decode which secular theory should be adopted for any given planetary system in a close binary.Comment: 32 pages, 8 figures, accepted for publication in Celestial Mechanics and Dynamical Astrophysic