Topology of the Relative Motion: Circular and Eccentric Reference Orbit Cases

This paper deals with the topology of the relative trajectories in flight formations. The purpose is to study the different types of relative trajectories, their degrees of freedom, and to give an adapted parameterization. The paper also deals with the research of local circular motions. Even if they exist only when the reference orbit is circular, we extrapolate initial conditions to the eccentric reference orbit case.This alternative approach is complementary with traditional approaches in terms of cartesian coordinates or differences of orbital elements

The three-body problem and the Hannay angle

The Hannay angle has been previously studied for a celestial circular restricted three-body system by means of an adiabatic approach. In the present work, three main results are obtained. Firstly, a formal connection between perturbation theory and the Hamiltonian adiabatic approach shows that both lead to the Hannay angle; it is thus emphasised that this effect is already contained in classical celestial mechanics, although not yet defined nor evaluated separately. Secondly, a more general expression of the Hannay angle, valid for an action-dependent potential is given; such a generalised expression takes into account that the restricted three-body problem is a time-dependent, two degrees of freedom problem even when restricted to the circular motion of the test body. Consequently, (some of) the eccentricity terms cannot be neglected {\it a priori}. Thirdly, we present a new numerical estimate for the Earth adiabatically driven by Jupiter. We also point out errors in a previous derivation of the Hannay angle for the circular restricted three-body problem, with an action-independent potential.Comment: 11 pages. Accepted by Nonlinearit

Théorie du Mouvement du Satellite Artificiel :<br />Développement des Equations du Mouvement Moyen -<br />Application à l'Etude des Longues Périodes -

The aim of this work is to elaborate a method permitting to obtain geodynamical information from long period variations in the motion of an artificial satellite. Short period phenomena are sometimes difficult to be modelised with precision and are at the origin of numerical difficulties while integrating the differential system of the motion over long periods of time. This is the reason why a direct study of the part of motion which does not contain the short period variations (periods shorter than 1 day) isproposed. In the evolution of variablesdescribing the satellite's motion it is possible to artificially separate the short period variations from the long period ones (including secular variations). The latter describe an imaginary motion designated by mean centered motion. A given algorythm permits the construction of the differential system describing the mean centered motion by analytical transformation of the initial system. For most of the perturbations it was possible to use the hamiltonian formalism and the theory of canonical transformations with the Lie method . Thus were treatedperturbations due to the terrestrial potential, to a third body, to ocean and solid earth tides. This required quite important algebric manipulations which were carried out with a series processor.The mean differential system obtained can be numerically integrated with avery large step (about 12 hours, instead of a few minutes for the initial system). The computation time is thus divided by a factor superior to 10 (the gain differs according to the perturbations) and the numerical errors are insignificant. Precise calculations on continuous arcs of several tens of years then become more reachable. The results yielded by integration can be compared with observations previously treated in an adequate way. Simulated observations permitted the validation of the method.L'objet de ce travail est l'élaboration d'une méthode permettant d'obtenirdes informations géodynamiques à partir des variations à longues périodesdu mouvement du satellite artificiel. Les phénomènes à courtes périodessont parfois difficiles à modéliser précisément, et sont toujoursà l'origine de difficultés numériques lors de l'intégration du sytème différentiel du mouvement sur de longues périodes de temps. C'est pourquoi nous proposons d'étudier directement la partie du mouvement ne contenant pas de variations à courtes périodes (périodes inférieures à 1 jour). Dans l'évolution des variables qui décrivent le mouvement du satellite, il est possible de séparer artificiellement les variations à courtes périodes et les variations à longues périodes (y compris les variations séculaires). Ces dernières décrivent un mouvement fictif que nous désignons par mouvement moyen centré. Nous donnons un algorithme permettant de construire le système différentiel décrivant le mouvement moyen centré par transformation analytique du système initial. Pour la plupart des perturbations, nous avons pu utiliser le formalisme hamiltonien et la théorie des transformations canoniques par méthode de Lie. Nous avons ainsi traité les perturbations dues au potentiel terrestre, à un troisième corps (Lune, Soleil, planétes), aux marées terrestres et océaniques. Cela a nécessité des calculs algébriques très volumineux que nous avons réalisés à l'aide d'un manipulateur de séries.Le système différentiel moyen obtenu peut être intégré numériquement avec un très grand pas de calcul (environ 12 heures au lieu de quelques minutes pour le sytème initial). Le temps de calcul est ainsi divisé par un facteur supérieur à 10 (le gain diffère selon les perturbations) et les erreurs numériques sont insignifiantes. Des calculs précis sur des arcs continus de plusieurs dizaines d'années deviennent alors beaucoup plus accessibles. Les résultats issus de l'intégration peuvent être comparés à des observationsconvenablement prétraitées. Des observations simulées nous ont permisde valider la méthode

A few lessons from MICROSCOPE

International audienc

RESONANCES DUE TO THIRD BODY PERTURBATIONS IN THE DYNAMICS OF MEOS

National audienceThe dynamics of Medium Earth Orbits (MEO) sees nowadays a renewed interest because of the development of satellites constellations (GNSS), that raises the problem of parking orbits for satellites at end of life. Numerical evidence shows that the resonances related to the presence of a third body can affect the stability of MEO orbits. The goal of our work is to study the effects of resonances on the stability of MEO orbits, over long or very long term (hundreds of years). For orbits above 20,000 km altitude, the perturbation due to a third body (the Moon or the Sun) is not fully negligible, so we take into account the third body perturbation on the secular evolution of the angular variable of the satellite orbits, and thus of the resonant angle. We estimate numerically the evolution of the resonant inclination. We study the stability of some resonances analytically and numerically, and in particular the resonance associated to the operational inclination of the Galileo satellites

DEVELOPMENT OF TECHNIQUES TO STUDY THE DYNAMIC OF HIGHLY ELLIPTICAL ORBITS

National audienceMany spacecrafts are or will be placed in highly eccentric orbits around telluric planets of the Solar system. Such eccentricities allow to cover a wide range of altitudes, mainly for planetology purposes. There are also orbits with very high eccentricity around the Earth, especially the GTO (Geostationary Transfer Orbit) and orbits of some space debris. In this case, the motion is strongly perturbed by the lunisolar attraction. For various reasons which will be recalled, the traditional tools of celestial mechanics are not well adapted to the particular dynamic of highly eccentric orbits. Therefore, it is necessary to develop specific techniques for this configuration. This concerns numerical as well as analytical tools. We will show how to construct the expression of the disturbing function due to the presence of an external body, wellsuited for highly eccentric orbits. Expansion of the elliptic motion in closed-form by using Fourier series in multiple of the eccentric anomaly will be presented. On the other hand, classical methods of numerical integration have often a poor efficiency. We will show the interest of geometric integrators and in particular the variational integrators

ASTRONOMY, SPACE GEODESY AND FUNDAMENTAL PHYSICS EXPERIMENTS IN SPACE : PRESENT STATUS AND PROJECTS

International audienceIn the past, several geodetic space missions revealed to be excellent laboratories for some experiments of fundamental physics (e.g. detection of the Lense-Thirring effect using precise satellite orbits). Today, we note the emergence of an ambitious fundamental physics program in space, thanks to the scientific prospective of CNES or ESA. The proposals of measurements linked to the theory of gravitation were the first to appear during the nineties and the French projects MicroSCOPE and PHARAO the first to be selected. At present, the implication of specialists of fundamental astronomy and space geodesy in these projects is effective due to their know-how and their resources in key techniques such us : reference systems, orbital mechanics, and associated techniques (laser, high angular resolution, accelerometry, clocks...). We will review here the contribution of fundamental astronomy and space geodesy to some projects and missions dedicated to fundamental physics

MICROSCOPE: past, present and future

International audienceThe MICROSCOPE mission tested the Weak Equivalence Principle (WEP) with an unprecedented precision of order 10−15, two orders of magnitude better than the previous best lab experiments. While the WEP, the cornerstone of General Relativity (GR), does not sway, the decade-long problems faced by fundamental physics stay still: how can we unify GR with the Standard Model, and how can we explain the acceleration of the cosmological expansion? As most beyond-GR models predict a violation of the WEP, albeit at an unknown level, it remains critical to even better test the WEP. In this paper, we review the MICROSCOPE mission, give its final constraint on the WEP, and build on its experimental limitations to show how we could improve them by a further two-order of magnitude in the precision of the test of the WEP

THE MICROSCOPE SPACE MISSION TO TEST THE EQUIVALENCE PRINCIPLE

The MICROSCOPE mission tested the Weak Equivalence Principle (WEP) with an unprecedented precision of order 10-15 , two orders of magnitude better than the previous best lab experiments. While the WEP, the cornerstone of General Relativity (GR), does not sway, the decade-long problems faced by fundamental physics stay still: how can we unify GR with the Standard Model, and how can we explain the acceleration of the cosmological expansion? As most beyond-GR models predict a violation of the WEP, albeit at an unknown level, it remains critical to even better test the WEP. In this paper, we review the MICROSCOPE mission, give its final constraint on the WEP, and build on its experimental limitations to show how we could improve them by a further two-order of magnitude in the precision of the test of the WEP

Dynamique des orbites fortement elliptiques

Les méthodes de développement des théories utilisent toujours un autre type d'approximation : la dépendance temporelle explicite de l'hamiltonien est négligée dans la résolution de l'équation aux dérivés partielles qui engendre le générateur du changements de variables. Cette thèse est consacrée au développement d'outils permettant de surmonter ces limitations. Dans un premier temps, nous développons la fonction perturbatrice de troisième corps en utilisant des séries de Fourier en multiples de l'anomalie excentrique du satellite (à la place de l'anomalie moyenne). Nous procédons à une normalisation du Hamiltonien ainsi développé, dans le but d'éliminer tous les termes périodiques. Pour y parvenir, nous appliquons un changement de variables canoniques construit à l'aide des transformées de Lie dépendantes de temps. La construction de la fonction génératrice du changement de variables nécessite la résolution d'une équation aux dérivées partielles (EDP) par rapport aux variables angulaires du troisième corps et du satellite. A notre connaissance cette EDP n'a pas de solution exacte. Habituellement, à cette étape, on fait une approximation qui consiste à négliger les termes de l'EDP liés au troisième corps afin de la résoudre. Nous montrons comment cette approximation peut être évitée, en proposant une méthode de résolution de l'EDP itérative, qui revient à effectuer un développement en série de puissances d'un rapport de fréquences petit devant 1. De plus, pour une cohérence globale de la théorie nous avons dû reconsidérer la solution classique de potentiel central et en particulier J . Finalement nous obtenons une théorie qui permet d'extrapoler le mouvement osculateur (et pas seulement le mouvement moyen) sur de longues durées (des dizaines d'années) de façon efficace et avec une excellente précision y compris pour des orbites très excentriques (e>0.8).The methods for developing theories always use another type of approximation. Indeed, the explicit time dependence of the Hamiltonian is neglected for solving the partial differential equations that give the generator of the change of variables. This PhD thesis is devoted to propose several tools to overcome these limitations. Firstly, we expand the third-body disturbing function using Fourier series in multiples of the satellite's eccentric anomaly (instead of the mean anomaly). We then perform a normalization to expanded Hamiltonoan, which aims to eliminate all periodic terms. To this end, we apply a change of canonical variables based on time-dependent Lie transforms. The construction of the generating function of the change of variables requires solving a partial differential equations (PDE) with respect to the angular variables of the third body and the satellite. To our knowledge the PDE has no exact solution. An approximation is usually done at this step to solve it by neglected the terms related to the third body. We show how this approximation can be avoided by providing an interative method for solving the PDE. This amounts to carrying out a power series expansion of a small ratio of frequencies 0.8). Moreover, since the traditional numerical integration methods are not very effective for highly elliptical orbits, even with adaptive variable-step size, we show the benefits of the so-called geometric integrators, especially the variational integrators. To this end, we present a high-order numerical scheme and we test its performance.PARIS-Observatoire (751142302) / SudocSudocFranceF