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

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

The resonant structure of Jupiter's trojan asteroids-II. What happens for different configurations of the planetary system.

In a previous paper, we have found that the resonance structure of the present Jupiter Trojan swarms could be split up into four different families of resonances. Here, in a first step, we generalize these families in order to describe the resonances occurring in Trojan swarms embedded in a generic planetary system. The location of these families changes under a modification of the fundamental frequencies of the planets and we show how the resonant structure would evolve during a planetary migration. We present a general method, based on the knowledge of the fundamental frequencies of the planets and on those that can be reached by the Trojans, which makes it possible to predict and localize the main events arising in the swarms during migration. In particular, we show how the size and stability of the Trojan swarms are affected by the modification of the frequencies of the planets. Finally, we use this method to study the global dynamics of the Jovian Trojan swarms when Saturn migrates outwards. Besides the two resonances found by Morbidelli et al (2005) which could have led to the capture of the current population just after the crossing of the 2:1 orbital resonance, we also point out several sequences of chaotic events that can influence the Trojan population

Spin-orbit resonances and rotation of coorbital bodies in quasi-circular orbits

The rotation of asymmetric bodies in eccentric Keplerian orbits can be chaotic when there is some overlap of spin-orbit resonances. Here we show that the rotation of two coorbital bodies (two planets orbiting a star or two satellites of a planet) can also be chaotic even for quasi-circular orbits around the central body. When dissipation is present, the rotation period of a body on a nearly circular orbit 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 further show that the rotation becomes chaotic when the natural rotational libration frequency, due to the axial asymmetry, is of the same order of magnitude as the orbital libration frequency

On the co-orbital motion of two planets in quasi-circular orbits

We develop an analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. The Hamiltonian, averaged over one of the planetary mean longitude, is expanded in power series of eccentricities and inclinations. The model, which is valid in the entire co-orbital region, possesses an integrable approximation modeling the planar and quasi-circular motions. First, focusing on the fixed points of this approximation, we highlight relations linking the eigenvectors of the associated linearized differential system and the existence of certain remarkable orbits like the elliptic Eulerian Lagrangian configurations, the Anti-Lagrange (Giuppone et al., 2010) orbits and some second sort orbits discovered by Poincar\'e. Then, the variational equation is studied in the vicinity of any quasi-circular periodic solution. The fundamental frequencies of the trajectory are deduced and possible occurrence of low order resonances are discussed. Finally, with the help of the construction of a Birkhoff normal form, we prove that the elliptic Lagrangian equilateral configurations and the Anti-Lagrange orbits bifurcate from the same fixed point L4.Comment: 25 pages. Accepted for publication in Celestial Mechanics and Dynamical Astronom