Ecole Doctorale

Thèse Dirigée Par

François Mignard

Daniel Hestroffer

Anne Lemaître

Sergei Klioner

Président Du

Rapporteur Rapporteur

INVESTIGATIONS ON THE DYNAMICS OF MINOR PLANETS WITH GAIA Orbits, masses and fundamental physics

Gaia is an astrometric cornerstone mission of the European Space Agency. With a launch due in late 2011, Gaia will hâve a much more ambitious program than its precursor Hipparcos: to create a "3D map" of our galaxy from astrometric, photometric and spectroscopic observations. It will pinpoint one billion targets with an unprecedented positional pré cision (at the sub-milliarcsecond level for single observation) and will also observe about 350,000 asteroids (mainly main belt asteroids) brighter than V = 20.

The aim of this thesis is the investigation of the science that Gaia will allow with such précisé observations of asteroids and the élaboration of réduction methods for treating this future huge pool of data, in particular regarding mass détermination. In the first part, we relate the characteristics of the Gaia mission, science in general which will ensue, and the st.ate-of-the-art of the current asteroid mass détermination using direct techniques, which are detailed. sont très précises, elles ne couvrent qu'une durée très restreinte par rapport aux archives des données au sol, ce qui nécessite la mise en oeuvre de nouvelles méthodes d'analyse.

Une des contributions importantes de Gaia à partir de l'étude de la dynamique des astéroïdes sera la détermination d'un grand nombre de leur masse. Les répercussions dans la recherche astronomique seront significatives au vu des nombreux intérêts portés pour les masses. D'un point de vue dynamique, leur connaissance pour les astéroïdes les plus massifs permettront d'obtenir des orbites à long terme très précises des petits astéroïdes qu'ils perturberont.

Elles rendront possible l'amélioration des éphémérides planétaires, sachant qu'actuellement leur précision est limitée par l'incertitude qui règne sur l'exactitude des masses des plus gros. Mars, planète la plus sensible à leurs perturbations, sera la principale bénéficiaire de ces déterminations de masses. D'autre part, elles aideront d'une manière indirecte, à mesurer avec plus d'exactitude certains paramètres du système solaire, comme l'aplatissement dynamique du Soleil et certains paramètres relativistes. D'un point de vue physique, l'estimation des masses associée aux dimensions des astéroïdes nous livrera leur densité volumique. Cette dernière comprend une partie du vide qui compose l'astéroïde et que désigne la notion de porosité. Actuellement, son calcul impose la recherche des météorites de composition similaire à l'astéroïde étudié. La porosité est évaluée directement à partir de la densité de grains (sans aucun vide) du météorite et de la densité volumique de l'astéroïde. Elle est un élément clé pour connaître la structure interne des petits corps du Système Solaire, comprendre leur origine ainsi que leur processus de formation. De plus, les différentes densités que nous déterminerons mettront peut-être en évidence, une relation entre densité et classe taxonomique, et nous informeront sur l'existence d'une relation génétique entre certaines classes de météorites et groupes d'astéroïdes. Les masses sont estimées par comparaison des positions observées aux positions calculées aux mêmes dates, d'un astéroïde cible perturbé par un plus massif. On fait l'hypothèse que ces deux trajectoires dépendent de mêmes paramètres (position-vitesse au temps initial x0 = (xo, yo, zq), x0 = (rr'o, ÿo, z'o) et la masse du perturbateur mp mais avec des valeurs différentes. On note u'=(x/0, x'o, m'p) ces valeurs aux positions observées, et u=(xo, xo, mp) aux positions calculées. Ensuite on fait l'hypothèse qu'il

existe une relation linéaire entre les O-C, différence entre les positions observées et les positions calculées, et AU = u' -u, différence entre les valeurs des paramètres à ajuster: O -C = A AU où A est la matrice aux dérivées partielles. Cette matrice donne les variations des po sitions r=(x, y, z) en fonction des paramètres ajustables u. Elle est évaluée à partir de la résolution des équations variationnelles, qui consistent à intégrer numériquement les équations régissant les variations de r en fonction de u, d2 / <9r\ dr dF dt2 \<9u/ <9u <9u où F est la fonction de force de l'équation du mouvement de l'astéroïde: r = F. En règle générale, la détermination de masse a toujours considérée seulement un perturbateur pour un ou plusieurs astéroïdes cibles, due en grande partie au faible nombre de rencontres proches intéressantes entre les astéroïdes observées depuis le sol. L'immense quantité d'astéroïdes que pourra observer Gaia nous a amenés à repenser la question du traitement des données en vue des déterminations de masses. Le nombre très important de rencontres proches qu'il sera possible d'analyser impose un traitement global de l'ensemble des perturbateurs avec les astéroïdes cibles pour un ajustement unique de l'ensemble des masses des perturbateurs. Ainsi notre matrice des équations aux observations A qui servira à l'obtention des masses mais aussi des positions-vitesses à un temps de référence sera unique. Le modèle dynamique utilisé (fonction de force F) intègre les perturbations des 8 planètes, Pluton et pour chaque astéroïde cible, d'une liste particulière de perturbateurs définie au cours de l'étape de sélection des passages proches précédant l'évaluation des masses. Les effets relativistes et l'aplatissement dynamique du soleil ont aussi été modélisés ainsi que les corrections dues au déplacement du photocentre. Pour certains astéroïdes, la force due à l'effet yarkovsky peut aussi être prise en compte. L'étape préliminaire à l'évaluation de la matrice aux dérivées partielles est la sélection des passages proches, et par conséquent des astéroïdes cibles que nous con sidérons pour évaluer les masses des astéroïdes perturbateurs. Les critères classiques pour les sélectionner procèdent d'une représentation par l'approche impulsionnnelle qui assimile la trajectoire perturbée de l'astéroïde cible à une hyperbole. Ainsi l'angle de déflection 9 (Eq. 1) de sa trajectoire par rapport au perturbateur peut ainsi être estimé à partir de la masse des deux astéroïdes m et M, de la vitesse de rencontre v et du paramètre d'impact 6, et appliqué comme critère de sélection.

9 tan -2 G{m + M) v2b (1) Le paramètre d'impact 6, déterminé en pratique par la distance minimale entre les trajectoires non perturbées des deux astéroïdes a aussi été employée comme un second critère. Même si cette approximation n'est pas adaptée à certains types de rencontres proches (durée de perturbation longue ...), en prenant des valeurs très larges pour ces critères, il est possible de sélectionner la majorité des passages proches les plus importants. Ainsi on a procédé à la sélection des rencontres proches entre les 20 000 premiers astéroïdes répertoriés avec une masse supérieur à 10~13MQ, considérés comme des perturbateurs potentiels, et les 350 000 premiers pour les astéroïdes cibles. Les passages proches retenues se produiront entre 2010.5 et 2016 avec une distance minimale inférieure à 0.5 UA et un angle de déflection plus grand que 1 mas. Les résultats de cette sélection sont donnés dans le tableau 1.

En intégrant tous ces passages proches pour l'évaluation de la matrice aux dérivées Table 1: Statistiques des passages proches avec une distance minimale < 0.5 UA, un angle de déflection > 1 mas et la limite inférieure pour la masse des perturbateurs > 10"1:JM©. Nombre d'astéroïdes perturbateurs 602 Nombre d'astéroïdes cibles 43513 Nombre d'astéroïdes perturbateurs et cibles1 435 Nombre de passages proches 78768 1 Ces astéroïdes appartiennent aux deux catégories des astéroïdes perturbateurs et cibles. partielles, laquelle permet d'obtenir la matrice de variance-covariance, il a été possible d'estimer les déviations standard formelles que l'on pourrait obtenir sur les masses des perturbateurs à partir de dates d'observations simulées. Ces dernières prennent en compte l'orbite du satellite autour du point de Lagrange L2 et de sa loi complexe d'observation. Le résultat, donné dans le tableau 2, est très encourageant: 36 masses pourraient être estimées avec une précision relative inférieure à 10% et 149 à moins de 50%, sachant qu'avec Gaia, les erreurs systématiques ne figurant pas dans les déviations standard, seront beaucoup moins importantes que pour les déterminations à partir d'observations au sol. Comme Gaia déterminera la masse d' astéroïdes binaires pour lesquels une masse est déjà connue très précisément par optique adaptative, il sera possible d'apprécier la réduction des erreurs systématiques.

Une étude a aussi été menée sur la possible contribution des observations au sol, pour Table 2: Nombre de masses en fonction des précisions relatives de leur détermination. Nombre de perturbateurs Total 602 er(m)/m < 0.1% 2 <j(m)/m < 1% 3 cr(m)/m < 10% 36 cr(m)/m < 15% 59 er(m)/m < 20% 75 cr(m)/m < 30% 106 a(m)/m < 40% 135 <r(m)/m < 50% 149 compléter celle d'astéroïdes cibles impliqués dans des rencontres proches se produisant aux bords de leur intervalle d'observation par Gaia. En étudiant le cas d'observations supplémentaires avant et après la mission, nous avons trouvé un peu plus d'une vingtaine de masses qui pourraient être estimées avec une précision inférieure à 50% à partir d'une centaine d'observations supplémentaires au sol.

Ce réservoir immense d'astéroïdes qui seront observés par Gaia, nous a amenés à faire une analyse de variance-covariance, pour estimer les précisions que l'on pourrait obtenir sur un certain nombre de paramètres, intervenant dans l'expression de forces perturbant la trajectoire des astéroïdes. Ainsi, la détermination de l'aplatissement dynamique du soleil J2 et de certains paramètres relativistes (PPN) 0 et 7 a été étudiée. De plus, la dynamique des astéroïdes nous a permis d'envisager d'autres tests: relativiste avec le test de violation du principe d'équivalence "faible" à travers l'estimation du paramètre de Nordtvedt 77, et cosmologique avec la possible variation en temps de la constante de gravitation G/G. Dans le tableau 3, nous avons réuni la valeur de référence de ces paramètres utilisés dans nos simulations, et la précision formelle attendue avec Gaia à partir de dates réalistes d'observations simulées pour un échantillon de 350 000 astéroïdes entre 2010.5 et 2016. De plus, la précision du satellite Gaia sera si élevée que nous avons envisagé la possible détection de la force non-gravitationnelle due à l'effet, Yarkovsky. En menant une analyse sur 1366 NEOs avec des dates d'observations réalistes entre 2010.5 et 2016, pour ajuster certaines de leur propriétés physiques (dont dépend l'effet), nous avons trouvé 14 astéroïdes

Table 3: Résultats pour rajustement du J2, /3/y, 77 et G/G Paramètre Valeur initiale Précision Gaia J2 2. x 10"7 1.86 x 10"7 P 1 1.89xl0"3 7 1 1.1 x 1CT3 n 0 6.2 x 1(T4 G/G Ü 3.92 x KT12 [yr~l] pour lesquels on pourrait ajuster leur diamètre avec une précision inférieure à 50%. Tous ces résultats nous encouragent à poursuivre les recherches sur ces possibles déterminations. Comme nous l'avons précisé au début, les masses d'astéroïdes sont très impor tantes pour différents domaines de l'astronomie tels que la construction des éphémérides planétaires et la physique des astéroïdes. Deux études nous ont permis d'entrevoir la future contribution des masses qui seront déterminées avec Gaia: la première est consacrée aux astéroïdes perturbant le plus l'orbite de Mars et la seconde aux densités qui pourront être estimées à partir des masses. La première étude montre que sur 52 astéroïdes, que nous avons répertoriés comme perturbant le plus l'orbite de Mars, parmi plus de 2000 astéroïdes faisant partie des plus massifs, 43 pourraient avoir une masse déterminée avec une précision relative inférieure à 50%. Ces masses amélioreront nettement la solution des éphémérides de Mars. Ceci permettra aussi de manière indirecte d'apprécier les résultats sur les masses. Néanmoins, il est très probable que certains astéroïdes passent dans des zones très résonantes (avec Mars, Jupiter), et que la connaissance précise de leur masse ne sera pas suffisante pour améliorer la solution de Mars si leur orbite n'est pas bien connue. La seconde étude indique que l'ensemble des densités qui pourront être obtenues, réuniront des astéroïdes appartenant à 17 classes taxonomiques différentes. Elles permettront d'affiner la loi hypothétique entre la densité et la classe taxonomique d'un astéroïde. Cette dernière est très importante pour les éphémérides planétaires, qui ont besoin d'intégrer un nombre élevé d'astéroïdes dans leur système pour lesquels la masse n'est souvent pas connu par des méthodes directes, et que l'on doit estimer à partir de leur densité déduite de leur classe taxonomique. 1.1. The Gaia mission 14

of 25 fias at V=15 and 300 fias at V=20 at the terni of its operational life. It will observe 1 billion objects, among them stars, planets, galaxies, quasars, and Solar System objects.

The stars will make up the majority of the observed sources and the astrometric part of Gaia will provide 5 out of 6 parameters defining the stellar kinematic, among which the right ascension, the déclination (a, AE) and the associated proper motion components (na,n,5) according to these two angles and the parallax. The sixth parameter will be brought by the spectroscopy, which will dérivé the radial velocities at 2-10 km.s-1 for V < 17. The spectrophotometry will be performed in about 25 colours for V < 20 and will bring information about the stellar physical properties like the température, gravity, colour etc...

A complété catalogue will resuit from this mission in 2020. It. will be at that time a new breakthrough in the astrometric catalogue history. The story began in 129 B.C with the Greek astronomer Hipparchus and the thousand stars of his catalogue that he compiled in specifying their relative brightness and position with an accuracy of about one degree.

The measurement accuracy then progressed slightly until the next leap in accuracy which occurred in 16th century with the Danish astronomer, Tycho Brahe (1546-1601), attaining a précision on the stars position of about a minute of arc. A long period of steady progress

The second part describes the preliminary step for mass détermination, namely the sé lection software and the criteria used for retaining the interesting close approaches between the few massive asteroids (perturbers) and numerous smaller asteroids (targets) in order to analyse the resulting orbital perturbations and thus, to dérivé the masses of the per turbers. With Gaia, the very high number of significant close encounters involving many perturbers imposes global processing of the whole set of perturbers and target asteroids in a single fît that takes the masses as global parameters. The third chapter is devoted to the description of the method used for the global treatment, as well as the complementary éléments, such as the dynamical model used, the systematic errors, the expected précisions on the orbital éléments of the targets. The expected number of asteroid masses and their associated précisions are given from realistic simulated data over a very likely period of the mission lifetime, between 2011 and 2016. The increase in number of the masses to be estimated is also envisaged from ground-based observations. The fourth and fîfth chapters open tracks on the fundamental science from the asteroid positions at the level of accuracy of Gaia. Thus, we study the possible dérivation of the solar quadrupole moment J2, the PPN parameters (3 and 7, the Nordtvedt parameter 77 (test of the weak Equivalence Principle) as well as a possible variation in time of the gravitational constant G/G. In addition, the study of non-gravitational forces is tackled with the so-called Yarkovsky effcct and the possibilities for deriving sonie physical properties of a sample of NEOs, which will be observed by Gaia. In the same way as for asteroid masses, the expected précisions on ail these parameters are given from realistic simulated data.

The two last chapters illustrate the contribution of asteroid mass détermination by Gaia to several fields of the astronomy. The first study is an illustration of the impact on the improvement of the ephemerides of the inner planets through the example of Mars.

We hâve searched the most influent asteroids for its orbit, and then compared them to the list of those for which Gaia will allow to dérivé a mass. The second study describes the large spectrum of the taxonomie classes swept by the asteroids of the list previously mentioned, and which will be used for the validation and the refinement of the hypothetical relationship between density and taxonomie class. This latter is very largely used for estimating the unknown masses of asteroids modelled in planetary ephemerides.

RÉSUMÉ

Depuis la découverte du premier astéroïde Ceres la première nuit de l'année 1801, plus de 380 000 nouveaux ont été observés. Ces petits corps sont les débris les plus primitifs laissés à la suite de la formation de notre système solaire il y a 4,6 milliards d'années. Ils portent en eux de précieux indices sur son origine et les processus à partir desquels il s'est formé. L'étude de la structure interne des astéroïdes et de leurs caractéristiques orbitales fournissent des informations sur leur évolution dynamique et leur histoire marquée souvent par de très nombreuses collisions avec d'autres astéroïdes. Du fait de leur diversité en composition et en caractéristiques orbitales, ils représentent une source intarissable de contraintes pour les modèles tentant de retracer les processus de formation du système solaire et en particuliers des planètes et de leurs satellites.

Leur étude dynamique permet de déterminer des paramètres caractéristiques du système solaire intervenant dans l'expression de forces perturbatrices pour l'orbite des astéroïdes. De plus, la détection de certains astéroïdes croisant l'orbite de la Terre et la détermination de leur trajectoire sont devenues une nécessité pour préserver la vie sur Terre. L'exploitation de leur dynamique passe par leur observation et c'est sur ce point que la mission Gaia permettra un nouveau bond dans notre connaissance du système solaire.

Figure 1: Précision sur la position des astéroïdes en longitude Gaia (along scan) aal en fonction de leur magnitude apparente.

INTRODUCTION

The science of asteroids is intimately linked to the question of life. The nebular theory, first proposed by Emanuel Swedenborg (1734), States that the Solar System formed from the gravitational collapse of a giant, molecular cloud. The remnants of the pristine blocks from which the planets formed 4.6 billion ago, called asteroids or cornets, hâve experienced numerous collisions, dynamic and thermal events. Guardians of the initial conditions of the birth of planets, their shape and orbital characteristics carry the traces of these events which shaped the Solar System as we know it at the présent time. The very high number of these objects and their diversity in composition and orbital distribution make them inescapable in the investigation of the formation process of the Solar System and of its components, and consequently, on the origin of life. In addition, their st.udy is motivated by the fact they could put an end to life on Earth. Some asteroids cross the Earth's orbit, and we know that human life could pass away after a collision with a body of several kilo métrés of diameter. As a conséquence, our survival will require the détection of the most threatening objects and the study of their physical properties to elaborate some means to avoid they reach to the Earth.

Since the discovery of the first asteroid Ceres in 1801, more than 380,000 asteroids hâve been referenced thanks to the technological progress. The full exploitation of this tremendous pool of information is mainly limited by the measurements accuracy and the availability to observe them ail. It is about this point and also by the very large number of observed objects that the Gaia mission will represent a real breakthrough in the sci ence based on the dynamics of asteroids. This astrometric mission of the European Space

Agency is due for a launch in 2011, and will hâve a much more ambitious program than its precursor Hipparcos: to create a "3D map" of our galaxy from astrometric, photometric and spectroscopic observations. It will pinpoint one billion targets with an unprecedented positional précision (at the sub-milliarcsecond level for single observation), and will also observe about 350,000 asteroids (mainly main-belt asteroids) until the magnitude 20. It will necessarily bring us to rethink astrometry and its associated methods. The main part of this thesis, the asteroid mass détermination, is a perfect illustration. The analysis of the gravitational perturbations during a close encounter between two asteroids has for long been the most productive method in masses détermination. The accuracy of the groundbased observations has allowed only to consider the very high gravitational signature, and consequently, only one target asteroid is very often used for deriving the mass of a per turber. Therefore, the number of masses derived by direct measurements is low, and the measures are very often corrupted by systematic errors, which do not appear in the given standard déviations. Thus, our knowledge of asteroid masses is still very poor, while their interest grows ever greater for planetary ephemerides and the physics of asteroids.

At présent, the précision of the Mars'ephemerides is limited by the uncertainty on the asteroid masses, and their influence on the other inner planets is non negligible, and will become inescapable when more précisé observations will be obtained from space missions.

In addition, the hypothetical relationship between bulk density (depending directly on the mass) and taxonomie class is often used for evaluating the masses which we do not know otherwise, in order to compute the solution of the inner planets ephemerides. The problem is that this relation is estimated only from a weak sample of asteroids. Furthermore, masses are key éléments for estimating the part of void, or any other buried constituent making up the asteroids, and which prove to be a precious due for knowing their internai structure, and consequently the history of the multiple events they hâve experienced.

The level of accuracy of Gaia, as well as the nurnber of observed objects will give access to tens of thousands of close approaches involving several hundreds masses to be estimated. Thus, the target asteroids meeting several different perturbers, which can be in turn perturbed by other asteroids encourage us to solve the problem in a global way.

The treatment of the perturbers and target asteroids will hâve to be led simultaneously.

The model used for solving the numerous équations of conditions will be of an unprecedented completeness, and will reduce in an important way the systematic errors knowing that most of the masses will be obtained from a lot of encounters with different target asteroids. Moreover, the very good distribution of the observations for a great part of the targets will provide précisé orbits, which are essential for good mass détermination. The results will be more accurate, and the associated précisions more realistic, what is non negligible to better assess some results depending on their values.

In addition, many parameters of the Solar System act, upon the orbit of the aster oids and the future observations from Gaia open new perspectives for their détermination.

Thus, the solar quadrupole moment J% as well as some relativistic parameters appearing in the post-Newtonian formulation could be the object of a fît. Moreover, other tests could be turned on hypothetical phenomena as the violation of the weak Equivalence Principle through the Nordtvedt parameter, or the variation in time of the constant of gravitation.

Moreover, the précision of Gaia is so high that the détection of non-gravitational forces, such as the so-called Yarkovsky effect at the origin of NEAs, could be possible and integrated to the dynamical model, and used for fitting some physical parameters of some of the asteroids.

Through this thesis work, we intend to study the dynamics of the asteroids in the Gaia context, and the science which will ensue from it. The main objective is to construct the method which will be used for the détermination of asteroid masses by analysing a very large number of close approaches between asteroids, and also to study the possibility to fit other parameters to which the asteroids are sensitive.

In the fîrst chapter, we présent the Gaia mission and its observational characteristics. A general and brief overall of the scientific objectives of this mission is given. The state-of-the-art of the asteroid mass détermination is presented in detail with the different computation methods, the importance of masses in different fîelds of astronomy, and a current list of the main-belt asteroids for which an estimation of their mass was obtained by direct measurement.

The second chapter is devoted to the sélection of the close approaches, which is the preliminary step to mass détermination. The impulse approach for characterising the im portance of a close encounter is presented through different représentations. This approach is tested with a large sample of close approaches that Gaia will be able to study, and a perspective for developing an analytical characterisation based on the three body problem is presented at the end.

The third chapter présents the global method which will be used for estimating the masses, as well as other global parameters. The expected number of asteroid masses with their standard déviations are given from realistic simulated data of the Gaia mission.

Moreover, the détection of the systematic errors in the réduction of the Gaia data is tackled. The chapter ends with the contribution of ground-based observations to the increase in number of the masses by completing the Gaia observations of certain target asteroids.

The expected précisions on some others parameters such as the solar quadrupole mo ment J2, the relativistic PPN parameters, the Nordtvedt parameter and the variation of the constant of gravitation are then studied. Furthermore, a feasibility study is given on the possibility to detect the Yarkovsky effect on some particular asteroids.

The two last chapters are about the future impact of the asteroid masses from Gaia. A study about the most influent asteroids upon Mars'orbit is led, and an overview is given about the densities and their précisions, which could be used for testing and refining the very important but hypothetical relation between density and taxonomie class.

Part I CONTEXT Chapter 1

The Gaia mission and the asteroid The observation from the Lagrangian point L2 20 Based on the same observation principle as its precursor Hipparcos, Gaia will scan the whole sky down to magnitude V=20 over a period of 5 years. The importance of this mission résides in its unequalled astrometric per formance, which completed by an excellent spectroscopic and photometric measures will reveal to us our galaxy at an unprecedented level of knowledge. Virtually every fields of stellar and galact.ic astrophysics will be impacted by the results of Gaia and this is indeed the raison d'être overview). Gaia will hâve an astrometric accuracy in the instrument and the réduction of the observations took place again until the next breakthrough in 1989 with the launch of the first astrometric satellite, Hipparcos, named in honour of the Greek astronomer. The resulting catalogues allowed us to access to accuracies of the order of 10 mas for a very large number of stars (~3 millions) and 1 mas for the 100,000 stars observed with the main instrument. The Gaia mission will deliver the stellar positions, parallaxes and proper motions for one billion stars with a précision at least 100 times bett.er than Hipparcos, and so will mark a new breakthrough in the catalogue history. bit from 315 to 22,300 miles altitude, ail of the scient ific goals were accomplished. The mission was dedicated to the measurement of stellar parallax and their proper motions. More precisely, the program was divided into two parts: the first experiment part had to measure the five astrometric parameters of some 120,000 stars to a précision of some 2 to 4 milli-arcsec. The second one, named Tycho, was designed to obtain astrometric and two colour photometric measurements in B and V bands of some 400,000 additional stars to a somewhat lower précision, which proved to be of 20-30 milliarcsec astrometry. Two final catalogues were compiled in August 1996: the Hipparcos catalogue brought together the astrometric data of 120,000 stars with one milliarcsec level astrometry, and the Tycho catalogue, the astrometric and bi-colour photometric data. In 2000, another catalogue was compiled from the same observations used to iriake up the Tycho catalogue but by using a more advanced réduction technique. The resuit was a catalogue of two-and-a-h al f million of the brightest stars with a bit more than Tycho, including 99% of ail stars down to magnitude 11.

The astrometric measurement principle was based on the simultaneous observation in two widely separated directions by a scanning satellite which led to a global réduction of the observations. Spinning slowly on its axis, Hipparcos carried out a systematic survey of the sky, through two télescopes separated by an angle of 58 degrees. The care exercised to keep this angle very stable associated to the exploitation of the observations with the aim of precisely determining the position of the space probe assured the accuracy of the measures. A self-consistent 3-step approach using great-circles was processed for extracting the information of the spacecraft together with the 1-D position of the stars at a the observation time.

Gaia will inherit the observation concept which proved so successful with Hipparcos, but will benefit, from recent technologies that will play an important part in the increase of performance over its predecessor. Table 1.1 lists the essential différences of employed technologies and performance between the two satellites. The use of CCD detector is a major factor of the leap in accuracy that Gaia will perform by reaching the expected end-of mission astrometric précision of 25 /ras at V=15 and a number of observed objects of 1 bil lion. In addition to astrometry, spectroscopic and photometric measures will complété the mapping of the Milky Way. Thus, a description of the kinematics in three dimensions of the Galaxy as well as its physical characteristics will becorne possible. Consequently, Gaia will hâve an important impact on many different parts of astronomical and astrophysical science which are overviewed in section 1.1.5.

Table 1.1: OverView of the main différences between Hipparcos and Gaia satellites. The Gaia space probe, based on a modular concept, is made up of two sections: the payload module and the service module. The latter is surrounded by a thermal shield partially coated with solar panels. Their first function will be to shade the optical System with the two télescopes in the payload module. The aim is to assure their thermal stability to -100 °C in order to provide the expected astrometric accuracy. The second function will consist in generating power. The solar panels are made up of 12 separate pans which will be unfolded into a circular dise of 10 meters in diameter once the observation point of the spacecraft reached.

Situated between the sunshield and payload module, the service module will contain the attitude and control, propulsion System, on-board data-handling and power Systems.

It will also house an essential component: the communications unit, that will allow the spacecraft to function and return data to Earth on an average of eight-hours every day.

The data will be transmitted by an extremely high data rate (~5 Mbit per second) from the Earth-Sun Lagrangian L2 point, over a distance of 1.5 million kilométrés. ES A will intercept. the faint signal with its 35-metre radio dishes in Cebreros (Spain) and New Norcia (Australia).

The heart of Gaia, named the payload module, will be made up of two optical télescopes that will be able to precisely pinpoint the location of stars and focus light on one common focal plane, where the information from three different instruments will be treated. The astrometric part will measure the celestial objects positions and brightness.

The light will be split into a spectrum for photometric and spectrométrie analysis. The photometric instrument will bring us information on the colour and brightness of objects and especially of ail stars. The spectroscopic instrument will be designed for providing the radial and rotational velocities, atmospheric parameters like the effective température, surface gravity, metallicity and abundances of Chemicals relevant for tracing the Galaxy évolution.

During its life, the spacecraft. will spin slowly, and the two fields of its télescopes will sweep the celestial sphere. The details of the scanning law are given in the following Section 1.1.4.

As we can see on Fig. 1.5, each telescope will be composed in part of two rectangular main mirrors (Ml, Ml') collecting the incoming signal from the two lines of sight (LOS1 and LOS2) on an area of 1.45x0.50m2. Then, the two light beams will be brought by several relay mirrors (M2, M2', M3, M3' to the combiner (M4, M4'), and sent on two rectangular fiat folding mirrors (M5, M6) to focus the starlight on the same focal plane instruments consisting of the CCD array (Fig. 1.7). We will briefly explain how the two sources from the two lines of sight are discriminated.

The Gaia focal plane will be the largest focal plane ever conceived for a space mission The Gaia mission 19 SM2). Composed of two rows of 7 CCDs, each one is assigned to one of two fields of view.

Its function will be to indicate by which telescope the object is viewed. In conséquence, the two signais will be treated at different times by the same rows of CCDs of the three instruments as it is sliown at the bottom of Figure 1.7 with the time required to reach the different positions of the focal plane according to the origin of the light beam.

Then, the object image will first cross the main Astrometric Field (AM), being responsible for astrometric measurements. It will be sampled by 62 CCDs distributed among 9 columns and each one read out in TDI (time-delayed intégration) at a constant rate of 1.018 pixels per second, following in this way the scanning rotation of the satellite.

Next, the light received from the telescope will go through prisms in order to create a spectral distribution before reaching the Blue Photometer (BP) over the wavelength range 330-680 nm and Red Photometer (RP) over the wavelength range 650-1050 nm.

This part will be sampled by 14 CCDs which will provide low resolution spectrophot.ometric measurements for each object.

Finally, the light will be dispersed by a grating with prisms before being analysed by the Radial Velocity Spectrometer (RVS), a near IR (847-874 nm) spectrograph of medium resolution (R=11500), which will collect spectra with 12 CCDs. spécifie window will be chosen for each observed source by the on-board System in order to transmit only small patches of the CCD surface.

1.1.

The Gaia mission 20 1.1.4 The observation from the Lagrangian point L2

Gaia will observe the sky from its platform located in the vicinity of the Sun-Earth La grange point L2, approximately 1.5 million km away from the Earth in the Sun-Earth direction. This région provides a stable radiation and thermal environment to the spacecraft with relatively easy link to the ground stations. Besides, the observation from this région will be very efficient knowing that the Sun, Earth and Moon will be behind the telescope's field of view. Around the L2 point, Gaia will follow a large Lissajous or bit (~300,000 km) in order to avoid the éclipsé of the Sun by the Earth so as to maintain a constant power supply and thermal flux. A Soyuz Fregat (Fig. 1.8) will be used to transfer the spacecraft into its final destination around L2. The spacecraft will be fitted in the large Soyuz fairing in the restartable upper stage (Fig. 1.10) which will be fired after one révolution in a low altitude parking orbit. Then, the satellite will carry out a cruise of about 3 months to L2 by its own Chemical propulsion. During the operations, the two astrometric fields of Gaia (separated by an angle of 106.5 degrees) will scan the sky at a constant rate of 60 arcsec.s-1. As a resuit, over a periocl of 6 hours (the spin period), the two fields of view will scan across ail objects located along a great. circle perpendicular to the instantaneous spin axis. This latter will precess with a period of 63 days around the solar direction, keeping a fixed angle of 45 degrees with the solar direction and so its orientation will change over one year. This combination of these three rotational movements is illustrated on Fig. 1.11. The resuit will be that the sky will be entirely covered in 6 months and each object will be observed about 86 times on average during the mission lifetime of five years. Figure 1.12 shows the number of scans on the sky and their variation as a function of ecliptic latitude. We can notice that the majority of the Solar System objects, among them the asteroids, will be observed about 50 to 60 times, lying close to the ecliptic. The position of objects observed by Gaia will be expressed in longitude À and latitude /3 on a reference great circle RGC (see Fig. 1.13), computed from the mean positions of instant.aneous scanning circles (their rotation pôle z being never fixed) over one period.

These observations will be used to describe the RGC attitude during the mission. Alongscan observation (and the RGC longitude) are much more accurate than in the transverse direction (i.e. RGC latitude) and for that reason in the following we hâve only considered the astrometric information coming out from the along-scan direction.

Figure 1.13: The Gaia longitude A and latitude (3 of an object position projected on the reference great circle RGC. The frame is centred on the Gaia satellite G. The vector z is the satellite spin axis and u is the direction vector toward the observed asteroid A from the Gaia spacecraft.

Science with Gaia

The main objective of the Gaia mission is to establish a "three dimensional" mapping of the structure of our Galaxy through the 6 phase-space of more than one billion stars. It.s structure can be described through its stellar populations, families of stars possessing similar intrinsic physical propert.ies such as âge, mass, kinematics, space distribution, abundance and Chemical composition. The measurements accuracy will allow a more refined séparation of these different populations.

Knowing that the Galaxy is split int.o different areas which are the bulge, the thin dise, the thick dise and the halo, and that each one is defîned by the orbits and Chemical composition of stars, Gaia will enable astronomers to better ident.ify these régions and to answer questions about t.heir size, âge, radial profiles and shapes and also on the number of arms that the Milky way possesses (Robin [START_REF] Robin | A Complété Census down to Magnitude 20: Stellar Population Properties[END_REF]). The spherical Halo arouses much interest; the astronomers wonder if the distribution in the spherical halo is triaxial, which would hâve an impact on the préservation of the spiral arms.

Besides, in combination with the multi-colour photometry of stars that will be observed and the data processing, Gaia will be in a position to provide their astrophysical characterisation, an element of cardinal importance to unravel the details of the formation history and the évolution of the Galaxy, and beyond, of other galaxies (Brown [2]).

Precious facts will be brought concerning the origin of the different régions of our galaxy

The Gaia mission 23

in order to reply to questions such as: was the central node formed before or after the disk?

The discrimination of the different générations of stars through their composition in heavy éléments will give a picture of the évolution of the Galaxy: at the end of their lives, the stars expel the gas from clouds that they processed in their hearts, and so, enrich with heavier Chemical éléments these same clouds which will be at the origin of future stars.

In addition, the st.udy of the distribution of stars, which is linked through gravitational forces, the star formation history and the Chemical abundance distributions will be very signifîcant to estimate the part played by the dynamic processes in the modifica tion of the initial star distribution. So, it may be possible to know if the Milky Way was formed by the collapsing of a cloud of matter or by merging much smaller components.

Stellar astrophysics

The study of stellar structure and évolution is important because it gives precious informa tion about the properties of matter in extreme physical conditions and, on a larger scale, on the évolution of galaxies and cosmology. In con séquence, the astrometric and photometric data on a very huge pool of stars that Gaia will return will be idéal to treat these different topics. More precisely, Gaia will give us information on their effective température Tejy, metallicity [Fe/H],

and the surface gravity log g obtainable thanks to the spectral energy distribution (SED). With the parallel détermination of the line-of-sight interstellar absorption Av (estimated by photometry) for each star and parallax, luminosity will be estimated, and then radius, mass and âge from theoretical models.

The impact of these data will be very great for stellar science. Ail data will be used for of stellar convective cores, defining the amount of nuclear material available to sustain the luminosity, could be investigated with accurate estimâtes of stellar global parameters from Gaia combined with asteroseismic data (obtained from the Ground or the COROT mission) [START_REF] Esa | GAIA: Composition, Formation and Evolution of the Galaxy[END_REF].

Stellar binarity

As said earlier, Gaia will observe multiple Systems. Even if most of them will remain undetectable by ast.rometry, the techniques based on the composite spectral energy distributions will detect the unresolved ones. Consequently, Gaia will discover about 60 million binary Systems of which about ~35% of them at 1 pc (Arenou et al. [START_REF] Arenou | Duplicity and Masses[END_REF]).

These objects are of fundamental importance in stellar physics because their masses and eventually radii can be derived by direct measurements contrary to single stars. These physical parameters are crucial for checking and calibrating the theoretical models for stellar interiors. Moreover, some of these binaries will be interacting binary Systems and in some favourable cases where their mass will be constrained, we will be allowed to address questions about the mass transfer and mass loss, accretion rates, kinematic behaviour, which are key éléments necessary to understand the origin of the evolutionary history of these objects but also of Type I supernova, millisecond pulsar [START_REF] Esa | GAIA: Composition, Formation and Evolution of the Galaxy[END_REF] ... In addition, the binary rate and the distribution of mass ratio between components will give information about the link between the stellar formation and the place where stars formed (Arenou et al. [START_REF] Arenou | Duplicity and Masses[END_REF]).

brown dwarfs and exo-planets

A Brown dwarf is a failed star in the sense that it, is not massive enough to ignite and to maintain the nuclear fusion reactions of hydrogen in its core. The value of their mass is located between that of large gas-giant planets and the lowest mass stars (between 75 and 80 Jupiter masses). Their intrinsic faintness and their very red colours make difhcult the détermination of good parallaxes not to mention that they are observable only at âges < 1 billion years because of their quick cooling and diminishing luminosity. So, the limit distance for observing them with Gaia can be expressed as a function of their mass and âge. Nevertheless, Gaia should observe between 10,000 and 100,000 brown dwarfs over a wide range of masses and âges1. The high précision of parallaxes from Gaia will be useful for estimating âge and mass by theoretical sequence fitting in the HR diagram (Haywood et al. [6]). Gaia will contribute to the substellar astrophysics by making a detailed spatial and kinematic map of brown dwarfs among clusters with well determined âge and metallicity. The study on their mass ségrégation and éjection will become possible as well as the formation mechanism of substellar mass objects. The multi-band photometry of Gaia will provide physical parameters of these objects like the effective température and associated with the absolute magnitudes (obtained from the précisé Gaia parallaxes) will help to detect the astrometric unresolved binaries.

The substellar science:

Kholo: I SA

The Gaia mission 25

The brown dwarfs in a binary System will be a timely opportunity to learn about their intrinsic physical properties of which very little is known because of the earlier mentioned reasons about their faintness... The mass is the searched physical parameter since in these Systems, it is conserved and can be derived by direct measurements. The method consists of studying the gravitational effect on the primary component, which is observable over a much longer period of time. We hope to observe them in these configurations in spite of the paucity of observations of such binaries. In conséquence, through analysis of the astrometric orbits, Gaia will be able to obtain the mass-distribution of brown dwarf binaries with 1-30 year periods [START_REF] Esa | GAIA: Composition, Formation and Evolution of the Galaxy[END_REF].

As of May 2007, the number of known exoplanets rises to 2332.

Gaia will be able to detect a high number of them. It will employ mainly two methods. The microarcsecond-precision of Gaia on stellar positions will allow it to detect the tiny wobbles of the star's position due to the perturbation by a planet in orbit around it. The second method will be based on the photometry of planetary transits. Gaia will observe transits of extra-solar planets across the dise of their parent star in the spécifie case of an alignment of their orbit to Earth. Furthermore, the photometry will reveal supplementary information on known planets about the orbital period and will allow to lift the lower limit on the mass and to estimate the planet radius.

Estimâtes suggest, that Gaia will detect between 10 000 and 50 000 planets beyond our Solar System. These computations, made by McEvoy [START_REF] Mcevoy | SAG-AC-001[END_REF], took into account. different assumptions on the détection probability for different masses, distances and periods with the expected population of main sequence stars within 500 pc... Moreover, 5 % of the stars were considered to hâve planets of 1 Jupiter mass. Gaia is expected to find every Jupiter-sized planets with an orbital period of 1.5-9 years.

The size of the sample of planetary Systems will allow to elaborate a classification of these Systems and will bring new constraints for the theoretical models trying to model the formation, migration, and dynamic évolution of planetary Systems. In addition, the orbital parameters that we will obtain for a large number of planetary Systems will reveal information about their distribution with respect to the type of parent stars and their âge (Mignard [8]). Indirectly, perhaps Gaia will reveal the existence of habitable Earth-size planets which would be protected from cometary impacts by other planet with the size of Jupiter orbiting at more than 3 AU (Wetherill [START_REF] Wetherill | Possible conséquences of absence of jupiters in planetary Systems[END_REF]).

http: / / exoplanet .eu / Beyond the Milky Way Knowing that the structure of the Local Group has been affected by mutual interactions between galaxies, the mean motions of these last will provide éléments to understand the influence of these interactions on their évolution and the structure of the Local Group.

We will be able to decipher the history of this group of galaxies as well as its future, and to know how the dark matter is distributed on small scales (Ibata [START_REF] Ibata | Local Group Dynamics with GAIA[END_REF]). In addition, the very accurate parallaxes of Cepheids and RR Lyrae stars from Gaia will put an end to the controversy about the zéro point of the period-luminosity-colour relationship and improve significantly this relation for galaxies with different metallicities.

Gaia will hâve also a very important impact on two crucial parameters used to describe the Universe and its évolution, which are the Hubble constant and its âge. The trigonométrie distance measure to the Cepheids and stars in the Large Magellanic Cloud (LMC) will reduce the uncertainty on the most important cosmological parameter, the Hubble constant, which is dominated by the error on the distance to the LMC. With regards to the lower limit of the âge of the Universe, it will be provided by the measured distances of the subdwarf stars and globular clusters, the oldest objects in the Galaxy, through the fit to theoretical models of stellar évolution 3.

Lastly, estimâtes suggest that some 100,000 supernovae will be detected in distant galaxies before they reach their maximum brightness. This will allow us to know when they attain their peak brightness, an estimation of their distance and in conséquence those of the host galaxies [START_REF] Esa | GAIA: Composition, Formation and Evolution of the Galaxy[END_REF]. The astrometric programme will also provide a census of ~500,000 quasars and will detect and observe, by multi-colour, about 3 million galaxies brighter than I~17. The quasars will be used to to establish the relativistic reference frame [START_REF] Esa | GAIA: Composition, Formation and Evolution of the Galaxy[END_REF]. Gaia will be able to detect this deflection of light rays due to the Sun and very probably will détermine the PPN parameter 7 with a précision between 10~6 and 10-7, much better than any other détermination expected by 2015 (Mignard [11]). In addition, the asteroids will be useful for determining others parameters which influence their dynamics.

For instance, a certain set of these bodies with a favourable combination of distances and eccentricities will allow to dérivé the relativistic precession, due to several factors like the PPN parameters /?, represent.ing the non-linearity of gravitation, 7 for the curvature of the space as well as the solar quadrupole J2. Section 4.2 présents these parameters acting on the asteroid orbits and which could be fitted from the asteroids positions. Preliminary results of their formai précisions will be given in order to know if there is an interest to investigate their détermination through the asteroid orbits from Gaia.

Another category of very interesting objects, the white dwarfs, could be very valuable objects for investigating a possible variability of the strength of gravity G, considered for the first time by Dirac [START_REF] Dirac | A New Basis for Cosmology[END_REF], through the variation of the luminosity. More precisely, the conséquence of G decreasing is the expansion of a star and the accélération of its cooling.

As the évolution of white dwarfs is just a cooling process, as soon as the energy sources are exhausted, the influence of a change in G is noticeable and so, Gaia's census of white dwarfs will give their luminosity function per bin and will reveal any change in gravity (Isern et al. [START_REF] Isern | White Dwarfs as Tools of Fundamental Physics: the Gravitational Constant Case[END_REF])

where M is the mass of the perturbing object, r the distance between this latter and the ob server, £ the angular séparation at which the pho-

1.1.6

The solar System objects Hundreds of thousands of solar System objects will also be observed by Gaia. Among them will be cornets and natural planetary satellites, but the majority of them will be made up of minor planets lying in the main belt between the orbits of Mars and Jupiter.

Endowed with capacity for detecting faint object moving quickly, Gaia will likely detect several thousand Near-Earth Objects (NEOs). At a much larger distance, beyond the orbit of Neptune, Gaia is expected to observe the largest objects in the Kuiper belt.

The estimat.ed level of completeness of current surveys (Fig. 1. [START_REF] Bus | Phase II of the Small Main-Belt Asteroid Spectroscopic SurveyA Feature-Based Taxonomy[END_REF]) and the limiting magni tude of Gaia at V ae 20 lead us to predict that about 350,000 asteroids will be observed by Gaia, mainly from the main belt. This prédiction is based on the estimation of the appar ent magnitude V by Eq. 1.1 for various solar élongations of objects of known orbits and absolute magnitude from ASTORB Database available online (Bowell4). Knowing that the spacecraft will be situated at only 1.5 million km from the Earth, the mean distance of asteroids to the Sun and the Earth hâve been considered.

V = H + 5 log(r. A) + f(a) (1.1)
where r and A are respectively the heliocentric and geocentric distances and f(ot) is the phase photometric factor.

Figure 1.19: Cumulative distribution in apparent V magnitude for ail known objects with q > 1.

The inset gives the distribution in absolute magnitude. The surface density £ on the left label is a mean based on an ecliptic band of ±30 degrees. Average apparent magnitudes are computed for 3 different solar élongations.

An analysis of the yearly évolution of the content of the database as a function of the absolute magnitude has shown that one can consider that completeness is achieved at absolute magnitude H = 14 and close to 80% at H = 15. Extrapolation of the curves considering a power low of index 2.5 gives the predicted number of objects for each solar élongation observable by Gaia. Were ail objects observed regularly at 135 degrees of élongation, Gaia would be able to observe about 600,000 planets. The actual geometry is not so favourable and the current guess is that Gaia will hâve a good time sampling about 350,000 asteroids. Most of them will already be known at the moment Gaia will operate.

The discovery of new asteroids will probably concerri the minor planets located inside the orbit. of the Earth and which are in conséquence hard to observe from the Earth.

The précision of position

Conséquence of Geometry of observation

The spécial geometry of the scanning law presented in Sec. 1.1.4, makes impossible observations around conjunction and opposition but will allow to benefit from an observable région between 45 to 135 degrees of élongation in better conditions than from the Earth (Fig. 1.21). This large area of observation at the L2 point will be significant in the discovery of small bodies orbiting inside the Earth's orbit,. Furthermore, Gaia will observe the sky far from the ecliptic, the privileged ground-based observation area, and may find asteroids moving on unusual orbits.

This geometry of observation will hâve an important impact on the observation distribution.

As for the observations of minor planets, Fig. 1.22 illustrâtes with the asteroid 1000 Piazzia the time distribution of observations and the orbit coverage that Gaia will permit for most of the bodies of the main-belt. The mission lifetime of five years matching fairly well the typical orbital period of the main-belt asteroids and the regular return of the pointing to a target during this period are both very favourable circumstances for solar System science with Gaia.

Concerning the near-Earth asteroids (NEAs), we expect to observe about 1300, but, for most of them their large apparent magnitude excursion will be the cause of a fairly irregular coverage. However, several will be well observed over their orbit as shown with the example of 3200 Phaeton in Fig. 1.23. A few additional Trojans and TNOs will also be observed t.ogether.

• * •• * N .• <• 2 * 2 # TT £ o fl " 0 • >- >• -2 * *2 •* * -4 . 1 1 . . . . 1 « , . . 1 . . ! i L_ -4 -1 1 . . I i i . 1 . i 1

Diversity and science

The main interest of asteroids cornes from their status as the remnant débris resulting from the formation process of the Solar System. Most, of them are located in the main asteroid belt between Mars and Jupiter. They are very probably représentative of the population of planetesimals which formed in that part of the Solar System and experienced lirnited thermal évolution. Ail information about their masses, densities, sizes, shapes and taxonomie classes related to their location in the Solar System and more particularly, in the nrain belt and the trojans clouds, will be very precious to investigate the origin and the évolution of the Solar System.

The accuracy of Gaia on the positions and velocities of these objects will lead to a very précisé knowledge of their orbits, a key element for identifying asteroid fanrilies according to dynamic criteria. In addition, the orbits of asteroids are very useful for predicting the stellar occultations by small objects, yielding information on the sizes and shapes, and so on the densities when the mass is known. The next section, devoted to the mass déter mination of these objects will develop the interest represented by the knowledge of their The Gaia mission densifies. With regards to the photometry, it will reveal the properties of their surface by analysing the light réflectance in different colours. In addition, the spectrophotometry will bring important information about their mineralogical composition and allow to dérivé a gradient of their composition in function with heliocentric distance.

r-T î T '-"t-- T (yrl 2 1-1-T-T-1-I-T-'-T-1-1-1-T-»-1-I-1-«-'-'-1- . •* » ** \ • * 1 •* t • • * O as 0 • >- -1 « . 1 . * . « 1 1 . . « 1 A .1 2 -I-.--.-»-1-. . . * 1 . i i .
The NEOs are objects possessing an orbit close to that of the Earth. It is believed that they are comets and asteroids that the gravitational perturbations during close encounters with planets hâve dragged int.o the Earth neighbourhood. The Yarkovsky effect, introduced in Chapter 5, is believed to hâve a rôle in the change from main-belt asteroids to near-Earth asteroids. Consequently, thanks to these data collected about the physical properties of asteroids, it will be possible to discover the root of meteorites among the Near-Earth objects.

The Trojans, which will be observed by Gaia, belong to a category presenting an interest as remnant débris from the formation of the inner planets. They are a large group of small bodies sharing the orbit of the planet Jupiter around the Sun with a semi-major axis between 5.05 AU and 5.40 AU. They appear in libration between one of the two stable Lagrangian points, L4 and L5. A few Mars Trojans are also known to exist, whereas, for the other planets, none hâve been found so far.

This group of asteroids is the source of many questions. First, about of their origin, astronomers wonder if they resuit from the accretion of planetary grains in the same région where they are today or if they were trapped there in the early epoch of the Solar System formation. In addition, concerning their classification, we do not know if they are a well distinct group of intermediate objects between the main-belt and trans-Neptunian objects, or a simple sub-class of asteroids. The classification of asteroids according to their physical properties from Gaia observations will be fundamental in answering to these questions.

In addition, systematic différences between the two clouds of Trojans, each one distributed around one of the two Lagrangian points, are often observed, whereas, in principle, there should be no différences in the dynamics of the t.wo groups. The L4 objects are 1.5 times as numerous as those at L5 and the distribution of orbital inclinations is not identical between the two clouds. The very précisé orbits of these objects from Gaia data will lead to the refinement of the statistics of the distribution of orbital éléments and will shed light on the existence or not of these différences, différences which could also be spectral and that the spectrophotometry data from Gaia will make possible to explore. Furthermore, the photometric data will provide estimations of rotation periods, spin axis orientations, and overall shapes of a statistically significant sample of Trojans.

Among the objects beyond Neptune, Gaia will find trans-Neptunian objects, moving on orbits with a semi-major axis between 30 and 50 AU. The belt that these objects form was suggested independently by Edgeworth and Kuiper by observing the distribution of short-period cometary orbits. In addition, Centaur objects (or scattereddisk objects) having a semi-major axis > 50 AU and perihelion between the orbits of Jupiter and Neptune will be detected by Gaia.

Today, more than 800 small bodies hâve been detected in this région of the solar System but, being very faint objects due to their great distance from the Sun, few of them (about 50) will be seen by Gaia. In spite of this low number, the contribution of Gaia to the study of the outer solar System will be important. The survey will be made down to magnitude 20 in the whole sky also attaining the high inclination contrary to groundbased observations. The provided information will be fundamental in understanding the formation mechanism of these distant objects and the outer solar System. The size and albedos of the largest Centaur objects will be determined by direct measurements.

Moreover, 10 to 20 percent of the objects that Gaia will observe will be binaries and their masses, a key element for determining the bulk densities, will be determined thanks to its very précisé astrometry.

1.2

The asteroid masses

1.2.1
The importance of the masses From a dynamic point of view, the knowledge of the asteroid masses will allow to obtain précisé long-term orbit of several smaller objects that are perturbed by larger asteroids.

More important, it will enable to increase the accuracy of the solar System ephemeris, knowing that today the limiting factor is not the dynamic modelling or the observation fitting, but the lack of reliable masses of the largest asteroids and more globally that of the main belt perturbing the motion of the planet like a ring of matter. This uncertainty is particularly relevant for Mars (Standish et al. [START_REF] Standish | Accuracy limit of modem ephemerides imposed by the uncertainties in asteroid masses[END_REF]). This important scientific issue is considered with soine details in Sec. 6.1

From a physical point of view, the mass and size (the latter being given by IRAS and many other techniques such as occultations, high angular resolution, etc., but also from Gaia) of an asteroid yield its bulk density, which accounts for the amount of matter making up the body and the space occupied by its pores and fractures. The estimation of the porosity, giving an idea of the part of void which makes up the asteroid, is of great interest to understand the formation of these small bodies. It directly dépends on the ratio between the bulk density of the asteroid and the sample density of its material, where the latter can be estimated from the grain density of a meteorite possessing a similar composition. The macroporosity is a récurrent phenomenon among the asteroids:

except for the three cohérent ones with a diameter larger than 500 km (Ceres, Pallas and Vesta), it is rare to find an asteroid with a low macroporosity, as we can see on Fig. 1.26.

At, présent, we can distinguish three groups: The first is composed of the three cohérent asteroids cited just before. The second one gathers the moderate macroporosity among thern, the 433 Eros asteroid is a perfect représentation with its block-strewn (Fig. 1.25).

This porosity seems to be due to subséquent, impact not sufficiently strong to disrupt the asteroid. The third group can be represented by 253 Mathilde, a highly porous object whose surface is covered by huge closely packed craters (Fig. 1.25). The macroporosity of these asteroids is superior to 30 % and in some cases it can reach 50 %. They hâve probably been disrupted and rearranged by gravitational forces.

Knowing that during a collision, the internai structure of an asteroid acts upon the stress-wave propagation, which in turn, modifies its constitution, the porosity becomes a precious indicator of the internai structure of asteroids and of their collisional history (Britt et al. [15]). We guess the important link which can exist between the latter and macroporosity. More precisely, the impact of collision can be very different according to the porosity of the target asteroid. The high porosity of asteroids tends to quickly attenuate the shock by dispersing the shockwave through pore spaces. It. also damps the ejecta velocities and forms craters by compaction process. Consequently, this characteristic makes the asteroid more capable of outliving disruption. In turn, collisions modify the internai structure of the asteroid in a different way according to the présent pile structures possessing the high macroporosity of the third asteroids group (represented in Fig. 1.26). Then, the future shocks in these asteroids will tend to compact the material and to reduce the pore spaces size and so, their high porosity (Britt et al. [START_REF] Britt | Asteroid density, porosity, and structure[END_REF]). However, we must not neglect the asteroid composition which can hâve an important influence on the response to a collision. As we can see on Fig. 1.26, C-type asteroids tend to hâve macroporosities greater than S-type ones, which lead us to think that S-type asteroids are often less affected by catastrophic collisions.

In spite of laboratory experiments and computer simulation, the study of collisions is still in its infancy, suffering from the lack of thorough knowledge about the composition and the internai structure of minor planets and also the difficulty to reproduce the different forces with the right scale acting on the asteroid (Holsapple et al. [START_REF] Holsapple | Asteroid impacts: Laboratory experiments and scaling laws[END_REF]).

In addition, a measure of the bulk density will shed light on the relationship between the spectroscopic taxonomie class (also available from Gaia), directly related to surface mineralogical composition and the density, or possibly the porosity. Albedo, which is the ratio of the light reflected by a body to the incoming light, is another important parameter related to the mineralogical composition and texture of the surface layers.

It will be measured by Gaia and allow to refine the study on this possible relationship between taxonomie class and density, knowing that albedo is useful for separating different taxonomie classes that the spectrophotometric data do not permit. The classes E, M (introduced by Bowell et al. [START_REF] Bowell | Taxonomy of asteroids[END_REF]) and P, which belong to the 14 classes of the most widely used of the asteroid taxonomies, that of Tholen [START_REF] Tholen | Asteroid taxonomy from cluster analysis of Photometry[END_REF], are a perfect example.

Even if the recent new taxonomie classification from SMASSII (Bus and Binzel [START_REF] Bus | Phase II of the Small Main-Belt Asteroid Spectroscopic SurveyA Feature-Based Taxonomy[END_REF])

does not use it to define taxonomie categories because of the lack of albedo information for the majority of the sampled asteroids, the authors insist on its importance as an additional parameter useful for shedding light on the the relationships between the subtle spectral characteristics and albedos observed among the X-type asteroids. Furthermore, albedo and more precisely, the géométrie albedo expressed by the ratio of a body's brightness at zéro phase angle to the brightness of a perfectly diffusing disk with the same cross-section, is also used to dérivé the asteroid size by the radiometric method.

The fîrst radius computed by this method was published by Allen [START_REF] Allen | Infrared Diameter of Vesta[END_REF] and concerned the asteroid 4 Vesta. The principle of this technique, outlined by Morrison [21] consists of the estimation of the diameter and the géométrie albedo of the asteroid by determining the intersection point of two different functions defined by these two same parameters (Delbo [START_REF] Delbô | The nature of near-earth asteroids from the study of their thermal infrared émission[END_REF]). One of these is the expression of the diameter as a function of the géométrie albedo and the absolute magnitude of the asteroid. The second one is a relationship between the diameter, the measured infrared flux, which is the reemission of the absorbed energy as a thermal infrared radiation, and the référencé flux depending on the géométrie albedo.

Such progress on the establishment of the relationship density -taxonomie classalbedo could provide information on their origin and on the formation process of the Solar System. If a well defined relationship can be evidenced, we could make reliable density estimâtes for objects of known taxonomie classes. However, the porosity will be another signifîcant factor to be taken into account in this study knowing that it has a direct influence on the density and that, as we hâve seen previously, asteroids from the same taxonomie class can belong to different macroporosity groups.

Similarly, the comparison between these densities for objects belonging to different taxonomie classes and the density of meteorites may strengthen our ideas about the genetic relationship existing between different classes of meteorites and their supposed parent asteroids (Zappalà et al. [START_REF] Zappalà | The impact of gaia in our knowledge of asteroids[END_REF]). sectionibus conicis solem ambientium [START_REF] Gauss | Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium[END_REF]. The Frenchman Adrien-Marie Legendre independently developed the same method in 1805, as may hâve also the American Robert Adrain in 1808. Consequently, with the least-squares method employed to solve the conditional équations treating their mutual perturbation due to their masses, Gauss had brought together almost ail components to make the first mass détermination, save one of a cardinal importance, sufficiently accurate observations. Therefore, the first détermination of an asteroid mass took place only about 160 years later with the asteroid 4 Vesta. In 1966, Hertz analysed its perturbation on 197 Arete [START_REF] Hertz | The Mass of Vesta[END_REF], which was particular since the ratio of their mean motions is almost 5/4, meaning an encounter between the two asteroids every 18 years and a perturbation which accumulated at each approach [START_REF] Hertz | Mass of Vesta[END_REF]. Between the first détermination and 1988, three other asteroid masses (1 Ceres, 2 Pallas and 10 Hygiea) were derived by observing their perturbations on other asteroids. The year 1989 marks for Hilton [START_REF] Hilton | Asteroid Masses and Densities[END_REF] The kind of close approach First, a différence is noticeable when we consider the kind of close approach. The first mass détermination is a perfect example of the use of résonances in the asteroid motions for which Scholl [START_REF] Scholl | Résonances in the Motions of Minor Planets and Their Use for the Dé termination of Masses[END_REF] showed a great interest in comparison with single encounters. He explains that the accumulation of repeated perturbations with similar geometry during a very long time interval provides more information on the perturbing masses than one encounter having the same encounter distance. The interest of this method is that it permits to consider close approaches with an effect much less intense than in the case of one single encounter. Nevertheless, it requires a pair of asteroids in résonant motion and secondly, the effect of repeated perturbations on the target asteroid is always very difficult.

to evaluate because of numerous perturbations for which it is subjected between two encounters with its perturber. Moreover, with the increase of observational précision, the threshold of sensitivity of gravitational effects continues to increase and the interesting single close approach becomes more and more numerous. However, we can also search the asteroids which will be involved in several encounters without being in résonant orbits.

In this case, that implies a systematic search of these multiple encounters for each pair of asteroids among a set. Sec. 2.2.1 describes briefly the fast and efficient software devoted to the sélection of valuable close approaches developed by F. Mignard.

The duration of close approach Among these encounters between asteroids, we can distinguish two cases according to the duration of the encounter. The case of long-lasting encounters is a fundamental element for Gaia. Its very high astrometric précision will allow to use this kind of close approach whose entailed positional shifts would be hardly noticeable with ground-based astrometric observations. At the présent, there is not really a satisfactory method to characterise the true effect of such close approaches on the target orbit. This subject will be tackled in Sec. 2.1.4. In addition, the mass détermination requires that one takes into considération the other perturbing effects with great care. Secs. 1.2.4 and 3.5 deal with the different causes limiting the mass détermination and especially, the completeness of the dynamic model employed.

As we hâve seen in the asteroid-asteroid perturbation analysis, the encounter type can be different and consequently, can impose sonie particular constraints on the interval of target observation time to be considered and the dynamic model used.

The number of target asteroids

Another différence between ground-based and space observations with Gaia cornes from the number of usable close encounters between one perturber and the fainter targets. As a rule, the mass détermination from the ground relies primarily on one particularly strong encounter between the massive asteroid and the small target. This is not a choice, but just a conséquence of the astrometric précision which limited the exploitable encounters to those producing a strong astrometric signature on the target position. Then, the idea to consider in the conditional équation several target asteroids, appeared in the 1990's. We can cite for example the mass of Ceres obtained by Bowell et al. [START_REF] Bowell | Asteroid mass détermination from multiple asteroid-asteroid encounters[END_REF] proceeded from an analysis of the amplitude of their perturbation in comparison with the measures précision of the Mars orbit. This question had been discussed by Bretagnon [START_REF] Bretagnon | Iinprovement of analytic planetary théories[END_REF] and Williams [START_REF] Williams | Determining asteroid masses from perturbations on Mars[END_REF] in 1984, who had studied for that the near commensurability between the orbital period of Mars and that of asteroids, knowing that this particular configu ration results in an increase of the perturbations amplitude, not only depending on the mass. Furthermore, Williams had led this study in order to détermine asteroid masses from perturbative effects on Mars. In 2001, Standish [35] updated his computation by adding new data including the ranging and Doppler data from the Mars Pathfinder and Mars Global Surveyor spacecraft. This method has proved to be précisé but at présent, it, is limited to the three largest asteroids, that are able to lead to perturbations sufficiently great upon Mars'orbit to be detected.

The spacecraft also played the rôle of target in mass détermination. Even if earlier in 1971, we were able to find a paper by Anderson [START_REF] Anderson | Feasibility of Determining the Mass of an Asteroid from a Spacecraft Flyby[END_REF] about the feasibility of such projects, we had to wait until 1997 to wit.ness the first mass détermination by observing the per turbation of the NEAR spacecraft by the asteroid 253 Mathilde (Yeomans et al. [START_REF] Yeomans | Estimating the Mass of Asteroid 253 Mathilde from Tracking Data During the NEAR Flyby[END_REF]).

Then, the mass of 433 Eros (Yeomans et al. [START_REF] Yeomans | Radio Science Results During the NEAR-Shoemaker Spacecraft Rendezvous with Eros[END_REF]) was determined in 2000 and computed again in 2002 (Miller et al. [39]) with an impressive précision of 0.05%. The mass of Vesta (Konopliv, personal communication 2002) has already been derived indirectly from its perturbation on Eros which was observed by the NEAR Shoemacker spacecraft. The advantage of this method is its excellent précision; it is the most. précisé of them, since the mass of 253 Ida has been obtained with a déviation of 0.4% and 433 Eros at 0.05% (see Tab. 1.4). Moreover, it allows to collect other interesting information about the solar radiation pressure, the surface landmark locations and spin characteristics of the aster oid. In addition, the spacecraft can also be a privileged observer of dynamic phenomenon susceptible to reveal the mass of an asteroid as was the case in 1995 with the first mass détermination in situ (Chapman et al. [START_REF] Chapman | Discovery and Physical Properties of Dactyl a Satellite of Asteroid 243 Ida[END_REF]) when the GALILEO spacecraft confirmed for the first time the presence of a satellite, 243 Dactyl orbiting around an asteroid, 243 Ida.

From analysis of appropriate images for binary Systems that we will develop briefly later, the mass of Ida was derived. Nevert.heless, such a technique requires a long préparation for establishing the mission and financial important means, which explain that the number of masses to be derived is a bit limited.

Very recent ly, the NE A 25143 Itokawa was observed by the Hayabusa spacecraft which permitted to obtain information about its mass (Abe et al. [START_REF] Abe | Mass and Local Topography Measurements of Itokawa by Hayabusa[END_REF]), topography and compo sition. The new interest for Earth-approaching asteroids is increasing and, inspired by the successful mission NEAR, future missions is in the making: The NE AP mission (Benson [START_REF] Benson | Near-Earth Asteroid Prospector and the Commercial Development of Space Resources[END_REF]) (Near-Earth Asteroid Prospector), due for a launch between 2009-2011, should reach a NEO as target. The HERA mission (Sears et al. [START_REF] Sears | The Hera mission: multiple nearearth asteroid sample return[END_REF]), will be aiming to visit three near-Earth asteroids and to return samples to the Earth. So, their bulk properties (size, mass ...) will be derived. The ESA 's Don Quijote mission (Harris et al. [START_REF] Harris | Mitigation-relevant science with Don Quijote -a European-led mission to a near-Earth asteroid[END_REF]) is designed to assess and validate the technology to deflect an asteroid and will be in position to derive the mass of its target asteroid, which will play the rôle of a threatening near-Earth asteroid. However, not only NEAs will be investigated by spacecraft in the future; Dawn spaceprobe (Russell et al. [START_REF] Russell | Dawn Discovery mission to Vesta and Ceres: Présent status[END_REF]), launched on September 27 of this year is scheduled to be put in orbit around 4 Vesta and 1 Ceres for orbiting them. This mission will allow to precisely measure the mass of these asteroids and to characterise their internai structure.

The following paragraph is devoted to the asteroid mass détermination when we are in the presence of a binary asteroid or a multiple System.

The observation of asteroids moons

The presence of asteroid satellites is an opportunity to obtain precious information about the intrinsic properties of asteroids such as the mass and bulk density. The first indisputable détection of an aster oid companion goes back in 1993, where the Galileo spacecraft imaged the satellite of the asteroid 243

Ida called Dact.yl (Belton and Carlson [47]). At that time, the ground-based adaptive optics were the main tool to detect and to observe the moons of aster oids. The first binary observed from Earth was 45

Eugenia around which orbited its small moon Petit-Prince (Merline et al. [START_REF] Merline | Discovery of a moon orbiting the asteroid 45 Eugenia[END_REF]). The radar instruments at Goldst.one and Arecibo were also used and allowed to identify and observe the first binaries among the near-Earth asteroids such as 1999 KW4 (Benner et al. [49]), 2000 DP107 (Ostro et al. [50]) and 2000 UG11 (Nolan et al. [START_REF] Nolan | UG_11[END_REF]). This technique, which is linked to the Doppler shift of the écho of the satellite relative to this of primary, is applicable to the closest objects.

The interest in binary Systems is the possibility to know their total mass (M + m) only from the séparation or semi major axis a and the orbital period P of about their common centre of mass. This information is obtained by adaptive Optics or radar images, which by fîtting to a preliminary trajectory, allow to dérivé the orbit of the secondary, as was done for the moon of 45 Eugenia, Petit Prince (Fig. 1.30). The mass of two components is then given by the third Kepler law (Eq. 1.2),

G(M + m) -^-^«3 (1.2)
where G is the constant of gravitation. This method is very often much more précisé than the dynamic method based on the gravitational perturbation analysis during close approaches but its field of investigation is limited to asteroids for which one or several moons can be observed.

The photometric lightcurves of these asteroids hâve recently become a very important tool of study when their duplicity is known. Even if it, allows only to suggest the multiplicity of a System without giving the undeniable proof as the adaptive optics imagery does, their study in the spécifie case of mutual occultations between the components has revealed to be a fundamental source of information about their three-dimensional shape, orbital period, rotational State and surface photometric characteristics. In Fig. (1.31), we can see the évolution of the photometric lightcurves of 90 Antiope with respect to the symbolic représentation of the relative motion of its two components.

In addition, a reconstruction of the global shape of components can be made by inversion of these lightcurves (Kaasalainen et al. [START_REF] Kaasalainen | Optimization Methods for Asteroid Lightcurve In version. I. Shape Détermination[END_REF], [START_REF] Kaasalainen | Optimization Methods for Asteroid Lightcurve Inversion. IL The Complété Inverse Problem[END_REF]) taking into account the phase effect, the projected of the asteroid surface and the scattering law by this latter. Figs. (1.32-1.33) show the example of the modelling of the global shape of 22 Kalliope. However.

this method requires observations at various geometries. Then, being in possession of the size of the components and the total mass (from methods mentioned earlier), it becomes [START_REF] Descamps | Figure of the double Asteroid 90 Antiope from adapt.ive optics and lightcurve observations[END_REF]). We can see the évolution of its lightcurves with respect to the position of both components drawn in red and blue, in mutual occultations.

possible to evaluate the mass of each component assuming a same bulk density. In consé quence, a value of their density can be computed.

The astrophysical methods

A second type of method, not based on the dynamic perturbation between two bodies, but on physical considération of an asteroid is used to estimate asteroid masses. The astrophysical method allows to compute them from the asteroid's size (e.g its radius for a considered spherical shape) and its density. As for the radius, which can be estimated by different methods, among which one was discussed briefly at the end of Section 1.2.1, the methods of density estimation are numerous as we are going to see in the following paragraphs.

A classical estimation consists in determining the taxonomie class of the asteroids and associating them with the mean density derived from asteroids belonging to the same taxonomie class and having a well determined mass as well as the volume (e.g radius). In the case where this class does not hâve a defined density, the Ceres density of 2.5 g.cmT3 is often chosen. However, this relationship between the density and the taxonomie class has never been proven. This part will be developed in Section 7 where we will talk about the contribution of Gaia to asteroid physics.

Other methods were developed, being based on the hydrodynarnic equilibrium figures which dépend on the internai structure of the asteroid (represented in large part by the density) and its rotation velocity. In 1970, Gehrels [START_REF] Gehrels | Minor planets and related objects. IV. Asteroid (1566[END_REF] made a density estimation of 1566

Icarus from its fast rotation rate and theoretical considérations seen in papers by Alfven [START_REF] Alfven | On the Origin of the Asteroids[END_REF]. Alfven gave the minimum period T from the expression of the angular velocity co (Eq. 1.

3) to conserve the stability of the rotating body. and which is expressed by the Eq.

(1-4),

J1 = 27rGpcj) (1-3) T = 2.7
TpW/5

[hr] (1.4) where p is the density of the asteroid in g.crn~3, G the gravitational constant and (j) a constant depending on the model and for which Jeans [START_REF] Jeans | Astronomy and Cosmogony[END_REF] found two range bounds values.

A density of 3 g.cm~3 was found for Icarus.

Always through the rotational properties, the density of asteroids were estimated by investigating their shape. The airn was to search ellipsoidal equilibrium figures for aster oids with a high rotational angular momenta and by the assumption of a homogeneous mass subjected to self-gravitational interactions. Farinella et al. [START_REF] Farinella | Triaxial equilibrium ellipsoids among the asteroids[END_REF] were interested in biaxial Maclaurin ellipsoids and triaxial Jacobi ellipsoids for fînding a shape permitting to keep an equilibrium between gravitational and centrifugal forces. Knowing that the asteroid shape is determined by its spin angular momentum, which can be expressed as a function of the dimensionless parameter u2/irGp, the density estimation becomes possible when the angular velocity uj is known. In this way, the authors found density ranges for objects having diameters between 100 and 300 km according to their rotation periods.

Cellino [START_REF] Cellino | Flattening, pôle, and albedo features of 4 Vesta from photometric data[END_REF], by fitting the numerical simulations of lightcurves, determined some physical characteristics of 4 Vesta, among them its flattening (initial assumption of the asteroid géométrie model). Then by considering the rotational equilibrium figure of a homogeneous body with a biaxial shape, its density was estimated to 2.4 g.cm~3.

The density of binary Systems was also investigated with the same methods. Equilib rium binary models were proposed for 624 Hektor and 216 Kleopatra by Weidenschilling [START_REF] Weidenschilling | Hektor -Nature and origin of a binary asteroid[END_REF]: the principle is that the tidal forces that each component exerts on each ot.her, distort them into ellipsoid shapes. So, the density can be computed knowing that the ellipsoid shapes and séparation of these two components dépend on the parameter uj2/7rGp and the ratio of their masses. The author computed the density of these asteroids in the case of equal-sized components, developed by Chandrasekhar [START_REF] Chandrasekhar | Ellipsoidal Figures of Equilibriurn[END_REF]. These equilibrium models then were discussed by Leone et al. [START_REF] Leone | Equilibriurn models of binary asteroids[END_REF] for the more general case of unequal masses, by using the classical theory of Roche, which considers only the tidal forces from a gravity gradient of a spherical body and neglects the part of this from the ellipsoid shape. This Roche ellip soid theory was used very recently in the case of a synchronous binary System, 90 Antiope, the first discovered of this type. More interesting is that the modelled lightcurves from the Roche formalism shapes of two components are similar to these observed. Consequently, the fitting of the shapes from Roche ellipsoid theory (depending on the density) to this one obtained by the inversion of the observed lightcurves gave a density of 1.25 g.cm~3

(Descamps et al. [START_REF] Descamps | Figure of the double Asteroid 90 Antiope from adapt.ive optics and lightcurve observations[END_REF]).

We are well aware that this type of method proceeds from many simplifying hypothesis

1.2.

The asteroid masses 44 like the shape of bodies (spliere, ellipsoid) and their structure (considered as homogeneous)

for methods based on the bodies rotation or more from no proven density-taxonomy relationship. Nevertheless, they remain very interesting for several reasons. They allow to estimate much more masses t.han by dynamic method, knowing that, at présent, 2141 radii are available in the ASTORB data base5 allowing to hâve an approximation of their volume. The théories of Solar System ephemerides need to represent the perturbations from asteroids under the forrn of punctual masses and a ring of matter. Furthermore, the astrophysical and dynamic methods are supplementary and should be a good way to control each other as for the method of observing companion and the dynamic method based on the close approaches analysis when they consider the same binary System.

1.2.3

The method we used

The method we hâve developed is based on the same principle as the dynamic analysis of close approaches between asteroids discussed in Section 1.2.2. As a rule, mass détermina tions hâve always considered only a perturber with one or several target asteroids (mainly due to the weak number of valuable close approaches observable from Earth). Thanks to its very high accuracy, the very huge pool of asteroids that Gaia will observe has led us to rethink the question of the data treatment. for mass détermination. First, the number of close approaches, which will hâve a gravitational signature détectable by Gaia, will be very great. Each perturbation between two asteroids will modify more or less their trajectory, and their accumulation will hâve, from a certain moment, a non negligible influence on their future encounters. Moreover, among these perturbers that Gaia will follow, many of them will be, during the 5 year mission, alternat.ely perturbers, target asteroids, or both at the same time. For ail these reasons, the very high number of close approaches détectable by Gaia between asteroids will more or less link ail asteroids involved in these encounters.

As a resuit, we hâve been led to consider ail future perturbers and target asteroids as an indissociable whole and, for the first time, to treat the problem in a global forrn. Consequently, our matrix of équation of conditions used to compute not only the masses, but also the positions and velocities at a reference time, will be unique. The method will be explained in more details in Section 3.1.

1.2.4

The state-of-the-art

As said earlier, the first mass détermination of an asteroid goes back only to 1966, although the principles of assessing masses of major planets or satellites from gravitational perturbations had been applied for more than a century. Hertz [START_REF] Hertz | The Mass of Vesta[END_REF] estimated the mass of 4 Vesta from the analysis of its gravitational pull on a smaller target asteroid, 197 Arete.

This method was applied successfully for the first time to an asteroid and has remained the only one widely used. In 2002, Hilton [START_REF] Hilton | Asteroid Masses and Densities[END_REF] reported 24 asteroids whose mass had been det.ermined by considering the perturbation of a test asteroid, a planet (Mars) or a spacecraft, as well as the observation of the companion for binary asteroids. Twenty six recently determined new masses by the classical dynamic method (see Table 1.4, in bold), must be added today to the Hilton list: ten were estimated by Kochet.ova [START_REF] Kochetova | Détermination of Large Asteroid Masses by the Dynamical Method[END_REF] and four by Baer and Chesley [START_REF] Baer | Astrometric Masses of 21 Asteroids, and an Integrated Asteroid Ephemeris[END_REF], two by Kuznetsov [START_REF] Kuznetsov | Détermination of masses of 108 asteroids[END_REF] and Aslan et al. [START_REF] Aslan | Preliminary Results in Asteroid Mass Détermination[END_REF]. In addition six more précisé masses hâve been derived for the following asteroids: 10 Hygiea by Chesley et al. [START_REF] Chesley | The Mass of Asteroid 10 Hygiea[END_REF], 15 Eunomia by Vitagliano [START_REF] Vitagliano | New mass détermination of (15) Eunomia based on a very close encounter with (50278) 2000CZ12[END_REF], 52 Europa, 444 Gyptis and 511 Davida by Kochetova [START_REF] Kochetova | Détermination of Large Asteroid Masses by the Dynamical Method[END_REF], 16 Psyché by Kuzmanoski et al. [START_REF] Kuzmanoski | Motion of the asteroid (13206) 1997GC22 and the mass of (16) Psyché[END_REF] and 433 Eros by Miller et al. [START_REF] Miller | Détermination of Shape, Gravity, and Rotational State of Asteroid 433 Eros[END_REF].

The observation of a satellite for binary asteroids in the main belt also brought 10 new URL:ftp://ftp. lowell.edu/pub/elgb/astorb. html masses with 4 estimated by Behrend et al. [START_REF] Behrend | Four new binary minor planets[END_REF], five by Marchis et al. [START_REF] Marchis | A three-dimensional solution for the orbit of the asteroidal satellite of 22 Kalliope[END_REF], [START_REF] Marchis | A low density of 0.8gcra"3 for the Trojan binary asteroid 617Patroclus[END_REF], and one by Descamps et al. (personal communication). We hâve filled Table 1.4 with the mass déterminations having in general the best formai précisions by considering separately the three dynamic method (close approaches, fly by and observation of satellite). The aim is to hâve a more correct view of the standard déviation values by comparing the value of computed masses from different methods. Furthermore, we hâve restricted our list of asteroid masses to t.hose lying in the main belt.

The distribution of the relative précisions on the masses in Table 1.4 shows that our knowledge of the asteroid masses is still very poor. Even if the masses of the largest asteroids are derived with a relatively good précision, sometimes reaching the 0.2 percent level (Ceres), the uncertainty in the value of masses from estimâtes given by Hilton [START_REF] Hilton | Asteroid Masses and Densities[END_REF] are generally between 10 and 20 percent. The scatter in the estimâtes of the mass of a given body derived during several independent encounters is a first hint to the external accuracy of the détermination, and not surprisingly this scatter is larger than the typical dispersion expected from the formai accuracies. As an illustration, we hâve considered [START_REF] Bus | Phase II of the Small Main-Belt Asteroid Spectroscopic SurveyA Feature-Based Taxonomy[END_REF] independent results for the mass of Ceres (see Table 1.3) among them, the 13 mass déterminations and their associated précision referenced in Hilton's paper [START_REF] Hilton | Asteroid Masses and Densities[END_REF], as well as 6 more recent ones from Kovacevic and Kuzmanoski [START_REF] Kovacevic | The Mass of (1) Ceres Obtained from Four Close Encounters[END_REF], [START_REF] Kovacevic | A New Détermination of the Mass of (1) Ceres[END_REF].

£i=l wi °i

where n is the number measurements (19 in our case).

Then, the value that we get is fiw = 4.767 x 10~10Mq which happens to be quite close to the médian mi2 -4.76 x 1O~1OM0. In with the weighted mean are easily spotted. Three values (~16 % of total) appear to be outliers above the 3a level. We can wonder if the 19 computed a^do not take into account.

an intrinsic error that we can consider with the same déviation Ointr for ail measures and distributed according to a Gaussian law. So we consider that each variable m* is distributed according to a normal law for the same unknown mean fi but with a variance a2n(r + a2. To evaluate this intrinsic déviation a^r, we assume that the standard déviation associated to variables ra* are ail identical to a . This latter value is estimated from the mean of the variances a2. Consequently, we can evaluate crintr from the expression,

1 n atot = 2 = aintr + °'2 (1-6)
where fi is the empirical mean n i= 1

We find ciintr < 0 and in conséquence, we can consider that the déviation a* associated to masses m, are well computed.

The évaluation of the standard déviation of the weighted mean fiw, by using the ex pression, In some cases, the corrélation which can exist between the a priori mass of certain target asteroids and the perturber can add to a further error on the computed mass. Hilton [START_REF] Hilton | Asteroid Masses and Densities[END_REF] cites the example of Pallas and tells that this corrélation is due to the similarity in the mean motions and mean longitude in the case of Ceres perturbing Pallas. Goffin's analysis [START_REF] Goffin | New détermination of the mass of Pallas[END_REF] puts forward other causes: the high inclination and eccentricity of Pallas would be the origin of the small number of significant, close approaches for the mass détermination.

In addition to these sources of errors which do not allow to estimate the true déviation associated to each measure m,, the standard déviation âflw of the weighted mean is wellknown to underestimate the variance Zhang [START_REF] Zhang | The uncertainty associated with the weighted mean of measurement data[END_REF].

In conséquence, it would be préférable to estimate the uncertainty on the mass déter mination by using the following formula (1.8), also employed by Michalak [START_REF] Michalak | Détermination of asteroid masses I. (1) Ceres, (2) Pallas and (4) Vesta[END_REF],

a^w (1.8) where,

1 n s2 = --'Y^wi(mi -fiw)2 2=1
The use of this formula requires a sufficiently large number of individual détermination rrii and a weighting inversely proportional to a2 (Van der Waerden [START_REF] Van Der Waerden | Mathematical Statistics[END_REF]). We find for the estimation of the déviation of the weighted mean afiw = 0.021 x 10~1oMq.

Nevertheless, even if we can estimate the standard déviation by a more realistic formula like Eq. (1.8) employed by Michalak [START_REF] Michalak | Détermination of asteroid masses I. (1) Ceres, (2) Pallas and (4) Vesta[END_REF], it does not to take into account the bias of each method. So, Viateau and Rapaport [START_REF] Viateau | The orbit of (2) Pallas[END_REF] for the mass of Ceres, hâve increased the standard déviation a -0.1 x 10~10 by a factor two to compensate for the underestimation of the evaluated uncertainty. However the problem remains and it is in practice very difficult, to estimate the value of the uncertainty in the dérivation of the masses of the asteroids by gravitational means.

Another way to obtain a more consistent measure of the mass and its uncertainty, is to repeat the calculation of the mean (Eq. 1. This time, the formai standard déviation is slightly larger than the scatter a^w. The resuit points to a better homogeneity of the individual mass détermination but that does not rule out the possibility that the true value of the Ceres mass is not among the three computed masses excluded from these calculations.

Part II

THE CLOSE APPROACHES

Chapter 2

The efficiency characterisation and As seen in Section 1.2.2, a close approach between two asteroids suitable for mass détermination can be defined as an encounter which produces a measurable gravitational signature. Thus, we guess that this encounter does not dépend only on the géométrie configuration (minimal distance, relative velocity, résonant orbits ...) but also on the measurement errors. Therefore, the accuracy level of Gaia will push away the limit of the géométrie parameters currently selected for ground-based observations. These last impose very large perturbing effects, and in this condition, the impulse approximation (Section 2.1) for characterising the encounter efficiency with mass détermination in mind, seems to be suited. But for the Gaia accuracy, does it allow still to define correctly the importance of a close approach or more exactly, what is the validity area to use this kind of charac terisation? This question will be treated in Section 2.1.4. Nevertheless, it is true that the analytical characterisation are not fundamental since nowadays, the numerical intégrations permit to consider more realistic orbits (with perturbations of planets, relativistic effects ...) and also to integrate the geometry of observations in order to obtain the deflection of the orbit through the angular coordinates. However, the numerical intégrations can be much longer if the nurnber of potential t.arget asteroids is important and do not allow to the conceptualise the encounters as the analytical treatment. In the last section, we detail briefiy a track that we wish to explore for defining another kind of characterisation which would integrate the Sun perturbation during the whole period of the encounter. The impulse approach is based on the approximation that the perturbed body follows a hyperbolic trajectory. From this représentation, the well-known deflection angle can be estimated in the reference frame centred on the perturber. A clear démonstration is written in Binney and Tremaine book "Galactic Dynamics'1'1 [START_REF] Binney | Galactic Dynamics[END_REF]. This formula is derived frorri the équation of a Keplerian orbit in polar coordinates (r, (fi) (Fig. 2.1),

d2u + u U G(M + ni) ZÂ 1 (2.1)
where (fi is the azimuthal angle in the orbital plane, M is the mass of perturber and m that of the target asteroid. L is the conserved angular moment.urn per unit mass associated with the motion of the target,

L = bV
where, b is the impact parameter and V the initial velocity. Theoretically, b is the initial length of the séparation between the two asteroids and perpendicular to V. In practice, b is reduced to the minimal distance between the two asteroid trajectories in the case where no scattering occur, and V is the relative velocity at this instant.

The solution of the équation (2.1) is given by,

1 " . , , . G(M + m) .. -= Ccos(^-0o) + pp- <2-2)
where the constants C and (fio are determined by the initial conditions. Differentiating Equation (2.2) with respect to time we obtain, --= Cr24> sin ((fi -(fio) -CbV sin (<fi -<fio)

knowing that in polar coordinates, the angular momentum may be written as, L -r2cfi

(2.3) 2.1.

The impulse approach 55

By considering that for 0 = 0, r is direct.ed toward the target asteroid as t-» -oo, we obtain for the équations (2.2) and (2.3),

= 0 = Ccos(« + ^-^(2.4) (^) = -V = CbV (2.5)
As the orbit is symmetrical about the point of the closest approach, we hâve:

7T 9 4,0 = 2 + 2
and thus,

tan </>o = tan 6 = cot - 2 bV2 G{M + m) (2.6) 6 tan - 2 G(M + m) bV2 (2.7)
The formula of the deflection angle is the basis of analytical approaches from the two-body problem.

The impulse approach can be expressed by other parameters. For instance, the varia tion of the velocity vector AV of an asteroid of mass m due to its encounter with another asteroid of mass M can be estimated by,

AV = 2 GM ~W (i*m ~rm) (2.8)
where tm and rm are respectively the heliocentric position vectors of the perturber (of mass M) and the target asteroid (of mass m). The associated deflection angle 6\ in the heliocentric reference frame may be approximated by,

|$i| = tan 1 (2.9)
where Vm is the heliocentric velocity vector of the target asteroid at the encounter time.

We can also estimate the change in the mean motion |An| of the perturbed asteroid (Carpino and Knezevic [START_REF] Carpino | Asteroid mass détermination: (1) Ceres[END_REF]) (the orbital eccentricity being neglected) using the formula

(2-7), |An| 6 Gme abV (2.10)
where a is the orbital semi-major axis and e the energy efficiency defined by,

IVm • AV| IVm| -|AV| (2.11)

The Ôpik formalism

Opik developed a theory of encounters in his book "Interplanetary encounters" [START_REF] Opik | Inteiplanetary encounters : close-range gravitational interactions[END_REF]. The author uses a different encounter geometry from that previously detailed by using a hodograph (x,y,z) referred to the perturber's centre. In fact, The direction of the perturber motion dérivés the y -axis, while 2 -axis is parallel to the planet's angular momentum vector and x -axis points away from the Sun. The encounter (relative) velocity V is 2.1.

The impulse approach 56 defined by two angles ip and d (see Fig. 2.2). Thus, the variation of the velocity vector V' -V is computed from the change of angles, Ai/j = ifr' --0, Ad = d' -d

The différence with the représentation detailed in the previous section (see Eq. 2.8) is that here the norm of the relative velocity vector remains the same. This formalism was used for modelling the close encounters of comets with Jupiter (Valsecchi et al. paper [START_REF] Valsecchi | Modelling close encounters with Opik's theory[END_REF]).

The hodograph frame is defined by, In this frame, the relative velocity of the target asteroid may be written as,

The impulse approach 57

where ip is the angle between sides d and the deflection angle 9 (see Fig. Besides, we know the direction of the vector V' -V which is along the position vector rM -rm (see Eq. 2.8). Thus, by projecting the vector tm -rm on the axis £ and £, we

(2.17)

For the computation of the second angle ip', we use the intermediary angle x -V' ~V;/ (see Fig. (2.18) the angle x is now defined and consequently ip' -il> -Thus, the new relative velocity vector of the target asteroid V' may be written as,

V'x = |V| sin d'sin i/j' V'y = |V| cos $ V'z -|V| sin cos ^(2.19) 2.

The Born-approximation

A new perturbation theory based on the Born approximation for the reference orbit was used by Anderson and Giampieri in order to simulate the gravitational field of an asteroid flown by a spacecraft. in analysing the perturbation of its orbit [START_REF] Anderson | Theoretical Description of Spacecraft. Flybys by Variation of Parameters[END_REF]. In this part, we summarise the main formula used for the détermination of the perturbed orbit, which are detailed in the Anderson and Giampieri paper [START_REF] Anderson | Theoretical Description of Spacecraft. Flybys by Variation of Parameters[END_REF]. The new orbit is derived by the principle of variation of parameters. More exactly, the unperturbed reference orbit of the target asteroid is described by a straight line referred to the perturber,

r = b(t)P(t)+ri(t)Q(t) (2.20) r = v(t) Q(t) (2.21)
where P(0) is the unit vector directed from the perturber to the target at the time of the closest approach (t=0), Q(0) is along the motion of target and 6(0) the impact parameter that we defined in the previous section.

Then, by variation of parameters and considering that the perturbed orbit is hyperbolic, the new trajectory at the time t is defined by the new variables 6(t), rj(t), v(t), P (t),

Q (t) which are defined as, b{t) v(t) v{t) bo Vo v0 1 + 2 \-i 2 e + 2èo log ( rp + r/o \ \ b0 J
where, 9 is the deflection angle (see Eq. 2.7), bo -6(0), rjo = r/(0), ro = r(0) and vp -u(0). With regards to the new vectors P (t) and Q (t), they are derived by introducing the Euler's angles (p,q,w) describing the rotations around the three axes, P, Q, W such as W = P x Q. As the perturbed motion of the target is assumed to remain in the plane (P, Q) the two angles p and q are constant. Thus, we hâve

P (t) = P(0) -Q(OMt) Q (t) = Q(0)+P(0 )w{t) w(t)

The impulse approach 59

This method was tested with various cases of close approaches but it proved to be unsuitable for the kind of close approaches that Gaia will observe. It is suited for the case where the target enters in the action sphere of the perturbers, its trajectory being thus mainly derived by the perturber perturbations. This situation rarely occurs between asteroids . The problem is that the perturbation of the Sun is not taken into account between the initial and the end time. Moreover, the différence of the orbital inclination between the two bodies is not considered. The addition of one third dimension poses a problem to hâve a sufficient number of équations for obtaining ail parameters.

2.1.4

The limits of such approach

The objective of this section is to catch a glimpse of the validity area of these characterisations by the impulse approach. Thus, we implemented a test for comparing the analytical solution with a numerical one. As the two methods tackled in Secs. 2.1.1 and 2.1.2 are similar enough (from various tested cases), both based on the deflection angle formula, we considered only one (the first one). First, we describe the test and more particularly, the parameter that we compare. We next show the sample of close approaches for this study and the density distribution with respect to two characteristic parameters, the im pact parameter b scaled in radius of the perturber action sphere Da, namely h, and the encounter velocity V. We then présent the proportion of good approximations by the impulse approach (Sec. 2.1.1) according to these parameters.

Description of the test

The parameter for comparing the two solutions, which is obtained analytically and numerically, is the length of the arc 7 projected on the unit sphere between the two positions of the target asteroid from the unperturbed and perturbed trajectories at a time tc after the encounter. It is obtained by, These positions are taken 3 years after the encounter time te between the two asteroids in order to consider the whole perturbation. For the numerical solution, the intégration starts 1000 days before the time of the encounter for the same reason mentioned just.

before. The positions from the impulse approach are derived from the formula giving the variation of the velocity vector of the target asteroid AV (Eq. 2.8). At the moment of the closest approach, the new osculating éléments of the asteroid are derived from its new velocity V + AV and thus, its position can be analytically evaluated at the time of the comparison tc. The results will be shown with respect to the encounter velocity V (~ 2.1.

The impulse approach 60 duration of the perturbation) and the impact parameter b scaled in sphere action radius Da (~magnitude of the perturbation).

The expression of the sphere action radius Da cornes from the Opik définition [START_REF] Opik | Inteiplanetary encounters : close-range gravitational interactions[END_REF],

where r is the heliocentric distance of the perturber and mp its mass. This radius was defined as the distance "at which solar radial perturbation in ail directions is less or equal to the planetary accélération of gravity" [START_REF] Opik | Inteiplanetary encounters : close-range gravitational interactions[END_REF]. Tisserand was the first to introduce this notion [START_REF] Tisserand | Traité de Mécanique Céleste[END_REF] by computing uthe distance to which the relative perturbation of the heliocentric orbit by the Sun equals the relative perturbation of the heliocentric orbit by the planet" [START_REF] Opik | Inteiplanetary encounters : close-range gravitational interactions[END_REF]. In our case, its expression, Dt = r rrip''5

is close to Da, Dt/Da = 1.24 mj/15

For the most massive asteroid (m ~4.75 x 1O"1OM0), we hâve Dt/Da ~0.239

The sample used for the test This test was applied for a very large sample of close approaches, one million in to tal. They are derived from a software, which is described in Section 2.2.1. The sélec tion is made over a period of 15 years between the 20,000 first numbered asteroids and the 379,000 ones referenced in Astorb data base. The sélection filters used were an im pact parameter smaller than 0.5 AU and a deflection angle (Eq. 2.7) superior to 0.1 mas. The minimal mass of perturbers for the close approach is retained is 1O~13M0.

Thus, about one million possible encounters were obtained. Fig. 2.4 represents the density distribution of these close approaches with respect to the scaled impact pa rameter h = b/Da and the relative velocity V for three levels: 1 (weak density), 2

(medium) and 3 (high). The interest of this sample is that it covers a large spectra in h (0 < h < 5000) and V (0 < V < 24 km.s~l). Fig. 2.4 shows that the most frequently encounters met are located between 400 < h < 1300 and 2.5 < V < 5 km.s~l.

Results

The main results are expressed as the normalised error e between the arc length found with the numerical intégration yn and the impulse approach y* in part of yn, e = \ln ~7i| 7n Table 2.1 gives according to different ranges of the encounter velocity, the proportion of close approaches well-defined by the impulse approach for 5 levels of e errors (5-10-20-30 and 50%). The number in parenthesis indicates the proportion of the cases where the impulse approach underestimates the impact of the perturbation. In the same way, Table 2.2 gives the same type of distribution but with respect to different ranges of the scaled impact parameter h. Table 2.1 illustrâtes very well the characteristic of the impulse 2.1.

The impulse approach 61 approach: the higher encounter velocities are the better the characterisation is. With velocities superior to 3 fcm.s-1, the errors smaller than 50% concern about half of the encounters, with 34% being underestimated, and they reach 80% for V > 10 km.s~1. In addition, the number of these close approaches (V > 3 km.s~1) represents about 64% of the total and thus, 53% of the whole sample is characterised by the impulse approach with an error below the 50% mark. With regards to Table 2.2, in the category bringing together 77% of the total (h > 500), 49,6% of the close approaches are estimated with an error better than 50% (30% with e < 30%) with a majority underestimated. We can note that the impulse approach is really suited for the case where the scaled impact parameter is small: 19,9% of the encounters defined by 0 < h < 100 are estimated with an error e < 5%.

Because of the great diversity of the close approaches that Gaia will observe, the impulse approach will be very probably used as a mean for selecting a large set of interesting encounters but the refinement will be made numerically. As seen earlier, this approach has its limits: it estimâtes about the half of the encounters with an error e smaller than 50%. However, the most interesting criteria for mass détermination over a short period (11.7) 30.7 (20.2) 49.6 (32.9) 77.2 (some years, such as the Gaia mission) is a small impact parameter, which provides the most of information on the masses of perturbers. The justification appears in the équation which gives the variation of the position of the target asteroid with respect to the mass of the perturber (see Sec. 3.4.1): it is in 1/d5 with d the distance between the two objects.

In the case where h < 50, we find 65% of encounters evaluated with a relative error smaller than 50%. As we hâve told, in the Gaia context, the impulse approach seems to be suited for making a pre-selection of interesting close approaches with tolérant enough parameters, but the method for refining it will corne from the numerical intégration. In fact, the tools (Chapter 3), which were developed for responding to the question of mass détermination with Gaia are a first solution, since it allows to evaluate the information on the perturber mass contained in a close approach (variance-covariance matrix, Eq. 3.15, Sec. 3.1). This sélection method will hâve to be applied when the data from Gaia are available but for the moment, the objective of the thesis is to construct and to check the method for mass détermination. In our case, we used the impulse approach because it proves to be so far much quicker than ail methods. Even if we do not take into account ail very interesting close approaches, the majority of the information is concentrated in the multiple close approaches selected by the analytical methods. Thus, it is possible to check the validity of the algorithm by choosing parameters allowing the sélection of a large sample, and to dérivé a valuable number of masses that Gaia could estimate. In addition, the encounters with a small impact parameter are estimated with a dynamic model which does not reproduce ail perturbing forces, and their orbits used for the sélection are not always well-defined. Thus, it is possible that certain encounters marked by a very small impact parameter reveal to be less interesting than expected. Moreover, other unsure parameters, such as the start of the mission and its duration do not allow to State on the exact number of masses which will be derived. Nevertheless, in Section 3.6, we présent, the main resuit with the number of masses according to the estimated précisions from a sélection by impulse approach as well as a possible number of supplementary masses from very small encounters which would escape to the first sélection.

Statistics of the close approaches observable by Gaia 2.2.1

The sélection Software

The target asteroids are selected by using a software initially developed for the Gaia mission by F. Mignard [START_REF] Mignard | Program Close Approaches[END_REF] which searches systematically ail the close approaches within a set of body over a prescribed duration. The program starts from a list of orbital éléments at a given epoch and détermines the positions of the asteroids with a two body problern between two epochs. Then it computes for each pair of selected asteroids the minimum distance by an itérative vector parabolic ht. In practice, this software proves to be very fast and précisé enough to locate the close approach for a period not exceeding 10 years in the case where the same osculating éléments are used for ail simulations. Once a close approach has met the criterion on the minimal distance or impact parameter 6, the angular dehection is evaluated by (2.7) and a second filter is applied by imposing a threshold value below which the close approach is not taken into account for the mass détermination.

The two hrst, parameters used for selecting the valuable encounters are (i) the minimum distance between the target asteroid and the perturber and (ii) the dehection angle 6 of the path of the perturbed asteroid.

Statistics with Gaia

The parameters of the close approach sélection used for predicting mass déterminations with Gaia are realistic values but yet, arbitrary. The target asteroids are selected among the hrst 346,481 numbered asteroids while the perturbers belong to the hrst 20,000 numbered ones with a mass greater than 1O~13M0. This given mass threshold corresponds to an asteroid with a diameter about 50 km and a bulk density of 2.5 g.cm~3.

They are big asteroids with diameters relatively well-dehned and consequently, the order scale on the mass is enough well estimated. Besides, the results presented in Sec. 3.6

giving the expected précisions on the masses with Gaia conhrm that this threshold on the mass is greatly sufficient. In fact, a mass is affected to each asteroids: we hâve considered those obtained from the ground-based observations with a relative error better than 10% (see Tab. 1.4). The others were evaluated by indirect methods from physical properties such as size, magnitude. For ail assigned masses a density of 2.5 g/cm3 has been adopted.

The period of sélection is from 2010.5 to 2016, which is a legitimate choice knowing that the launch of tlie spacecraft is planned for 2011 at latest for a duration of 5 years. The two criteria for selecting a close approach are an impact parameter smaller than 0.5 AU and a deflection angle superior to 1 mas. Even if these two fîlters are arbitrary, that does not mean they are not realistic: from ground-based observations (~50 to 100 mas of astrometric précision) the est.irnated deflection angle limit is always superior to 100 mas (for a simple encounter); Gaia will hâve an accuracy inferior to 2 mas, so 1 mas seems to be a valuable threshold.

The global statistics of these encounters over the whole period of Gaia are gathered in Tables 2.3, 2.4 and 2.5. The first table shows the number of perturbers and target asteroids that were found, as well as the number of bodies which are both perturber and targets. It gives a scale order of the number of asteroids that Gaia will hâve to treat for mass détermination, about more than 40,000 ones involving slightly more than 600 perturbera. The two following Tables 2.4 and 2.5 concern the quality of these encounters. They detail them with respect to the value of our two sélection criteria, that are the deflection angle 9 and the impact parameter b between both asteroids. Even if as we hâve seen previously, the impulse approach is not as précisé as the numerical intégrations for characterising the intensity of encounters, it gives a good glimpse of their importance for mass dérivation. The most interesting encounters appears in Table 2.5

of their distribution according to their impact parameter. We find 646 encounters with a minimal distance smaller than 0.001 AU, which is very interesting for Gaia because of its high astrometric précision. We can count more than 13,000 encounters which will occur between 0.1 and 0.01 AU. Thus, the large number will allow to accumulate of the information for the most perturbing asteroids. With regards to the deflection angle, the major part of the encounters are marked by an angle between 1 and 50 mas, deflection which is non negligible for Gaia. 1 These asteroids belong to both categories of perturbing asteroids and target asteroids Nevertheless, it is well known that the number of observations and their distribution during these encounters hâve an important impact on the précision of the resuit. Therefore, we hâve drawn two histograms representing the number of asteroids that Gaia will observe from 2010.5 to 2016 among sets of 10,000 asteroids distributed according to their IAU number (Fig. 2.5). In Fig. 2.6, we hâve studied the average of observations. We notice that from the IAU number 140,000, the number of observed asteroids falls under the level of 80% and from the 280,000 IAU one under the 50% mark. Finally, the number of observed potential target asteroids is high enough but we can wonder about the number of observations per asteroid and their précision by way of the magnitude at the observa- Besicles, Table 2.6, which gives the distribution of asteroids in percent according their magnitude, shows that the sample of 346,481 asteroids brings only a very weak proportion of asteroids with a magnitude between 15 and 18. Consequently, the information on the masses is unevenly distributed among the first 346,481 numbered asteroids and is found among the about 20,000 first ones, which are the best and the more observed asteroids.

Table 2.6: Distribution of the asteroids according to their magnitude and their IAU number.

The magnitude V<15 corresponds to a précision on the position of a ~310 /.las and at V =18, a ~lmas.

Proportion of asteroids with respect to their magnitude V n°IAU V < 15 15 < V < 18 18 < V < 20

2.3

Perspective: the three-body approach

As mentioned previously, the astrometric accuracy of Gaia will allow to also consider of more distant close approaches, during which the perturbation often lasts for a long time.

This kind of encounters can be diffîculty described by the impulse approach which is suited for the case where the perturbation is transmitted very quickly. Thus, it seems logical to be heading toward the three-body problem to find an analytical formula integrat.ing the Sun perturbation, which has a bigger influence when the encounter lasts. The development of a such solution imposes to State simplifying assumptions représentative of the characteristics of the studied Systems. As a rule, during an encounter, the minimal distance is often small in comparison with the heliocentric distance. Furthermore, the asteroids masses are negligible compared to that of the Sun. Therefore, the Hill's problem seems to be the obvious choice. We présent in this part, the future work that we hâve set about in collaboration with Tamara Ivanova (Institut.e of applied astronomy, St Petersburg).

The elliptic Hill problem

Définition and notations The Hill's problem is a particular case of the three problem where we consider one main mass and two infinitésimal masses separated by a distance rnuch smaller than that which separate them of the main mass. Let M, rai, m2 three niasses under the influence of their mutual gravitational forces such as, rni -C M, ni2 -C M and rai ~7712

Besides, the distance r between the two masses m\ and is negligible compared to the distance R of the main mass M to their centre of mass Cmr/R <C 1

Consequently, the ternis of an order superior or equal to 2 in r/R will be neglected.

Equation in the inertial frame

As a conséquence of the hierarchy on the niasses, we consider that the motion of M is not. perturbed by the masses m\ and m2 and that M is the centre of an inertial frame in which we can write the following équation of motion,

d2Ri dtt2 d2R2 dt2 -Gvÿ -V -GlvAE + Gmi- /?2 H (2.23)
As regards to the hierarchy of distances, the Taylor sériés of Rj/Rf (i -1,2) by considering only the terms of order one in ri/R is given by (Eq. 2.24), Ri R?

from the Taylor sériés of Ri ,

R73 R rj R • rj R R3 1 R3 ~** R2 ' #3 (2.24)
where Ri = (R2+r2i-2 Ri r,)1/2

In conséquence, when the terms of order superior or equal to (r/R)2 are disregarded, the motion of the two infinitésimal masses rai and m2 is described by two équations, the équation of motion of their centre of mass Cm (Eq.2. In conséquence, r is written by,

-(Rx) = Rx + 2Rx + Rx -2 (Ryf + Rÿf'j -Ryf -Rxf* ^2 (Ry) = Rÿ + 2Rÿ + Ry +2 (Rxf + Rxf) -Rxf -Ryf2 df dt2 (Rz) = Rz + 2Rz + Rz (2.

29)

We can introduce the true anomaly / as the independent variable from the expression, d_ _ df_d_ dt dt df

The dérivative with respect to the true anomaly being noted by prime, the équation (2.29

) becomes, dt2 ^R -(/w = « d2 r •• • 1 ^{Rz) = R z'f + z"f2 + 2Rz'f + Rz x'f + x"f2] + 2Rx'f + Rx -2 [Rfy + Rfy'] -Ryf -Rxf y'f + y" f + 2Ry f + Ry + 2 + AE/V + Rxf -Ryf (2.30) so, we hâve -(R dt2 -(r dt2 d1 dt2 = Rf2 [x" -2 + Rf + 2Rf 4-x R -Rf2 -y Rf + 2Rf = Rf2[y" + 2x'] +Rf + 2Rf + y R -Rf2 + x Rf + (Rz) = Rfz" + z! [Rf + 2Rf R-Rr + zRf (2.31)
However, the équation of the centre of mass Cm of m\ and m2 (Eq. 2.25) is the solution of the two-body problem. In conséquence, we hâve the following relations, the intégral of angular momentum (Eq. 2.32) and its dérivative (Eq. 2.33):

(R2f) = PGM = a(l -e2)GM Rf + 2Rf = 0
The accélération of Cm along the radius vector is

R ~RJ = GM (2.
-(Rx) = Rf [x" -2y'] -x- R2 -(Ry) = Rf [y" + 2x'] -y°M d2 Rfz" -z GM R2 + zRf 62 (2.35)
Under the choice of unit of measurements G = M = 1 and by using Eq. (2.35), the équation (2.26) can be written according to pulsating coordinates x,y,z,

Rp[x"-2y'} = Rp[y"+ 2x'] = Rp[z" + z\ = -Gm x R2 p3 -Gm -Gm y R2 p3 z R2 p3
(2.36)

Besides, the mathematical form of Kepler's second law gives

R2/ = Vp In conséquence, Rf = ±-.l = A J R? R R2
At last, the elliptic Hill problem takes the form by using Eq. (2.37),

x" -2y' - (2.41)

The analytical solution

The starting point is the paper of Brumberg and Ivanova [START_REF] Brumberg | Encounter in the Keplerian field -Analytical treatment[END_REF], which is an extension of the results of Henon and Petit [START_REF] Petit | Simulation numérique des anneaux de Saturne[END_REF], [START_REF] Henon | Sériés expansion for encounter-type solutions of Hill's problem[END_REF], [START_REF] Petit | Satellite encounters[END_REF] of the Hill's problem by analytical The first work will be to evaluate the interest of such représentation for close approaches between asteroids by using the already computed solutions for simple cases (circular planar orbits). The main difficulty lies in the fact that the solution dépends on the three constants (for E, e and k) to be computed which hâve to be suited for this kincl of problem. If the response is positive, it will be interesting to reconstruct and to implement. the whole method for deriving the general solution and to adapt to the encounters between asteroids.

We can cite another very interesting paper of Moons et al. [START_REF] Moons | Elliptical Hill's problem (large and small impact parameters)[END_REF], which also has not been applied to a real case yet. This paper gives an analytical solution to the changes in orbital éléments during a close approach between two particles of a ring. The Hill's problem is used once more to formulate the problem, and the solution is treated according to the two cases of a small and large impact parameter from the Lie's transform perturbation method. This method confronts us with a problem of constant to be derived for being able to apply to the case of close approaches but it merits careful thought.

3.7

The expected précision on the target orbital éléments and may be written as, AC0 = (A<A)-1At(0 -C).

(3.5)

The matrix (A^A)-1 is a symmetric square matrix, which is named the normal équations matrix. When A is a full rank matrix, the inverse matrix of (A1 A) can be evaluated.

The applied inversion method will be developed in the next Section 3.1.2.

More precisely, the vector ACo is made up of Au°=(Au5,.., Au°)f which are the corrections to the initial conditions of the n asteroids with Au£ = (Ar°,Ar°) standing for the corrections on the position and velocity of asteroid k at the reference time to with r = r(t) = (rc(t), y(t), z(t)) , r = r(t) = (x(t), ÿ{t), z(t)), r°= r(t0). Then 5mp=(AE?ni, ..,<5mPY are the corrections of the masses of the p perturbers.

The matrix A contains the partial dérivatives of the longitudes À of ail minor planets involved in a close approaches with a massive asteroid (necessary for mass where Ai is the longitude of the asteroid k for the ith observation.

Bk represents the variations of the longitude A of the asteroid k with respect to the p perturbing masses. Bk is a nkbs x p matrix.

[Bk]îj=i,..,p -dXj \ ' dnip)

Weighting of the équations Previously, the problem is treat.ed as if each observation has the same measurement. error, whereas they vary with respect to the apparent magnitude of the objects and its phase angle. Thus, the weighting of the observations allows to take into account, these disparities in astrometric précision. Each line of the matrix A corresponding to the ith observation of the kth asteroid has to be weighted according to the error on the position aBy making the (very reasonable) assumption that the measurement errors on the positions can be considérée! as random centred quantities (E(orfc)j)=0 Vfc, i) and are independent (E(<Jkj,<Jij)=0 for i ^j and l ^k) and adding just a scale factor cro to express the variance per unit weight <7g, the matrix W with the weights is diagonal and can be written, The final solution then becomes AC0 = (AtWA)"1A1W(0 -C).

(3.10)

The explicit general solution

The computation of the solution of the least-squares estimation ACo requires that of the inverse of the normal matrix (A(WA) (see Eq. 3.10). However, as we hâve mentioned in Section 1.2.3, our objective is to treat the problem in an unique way, by considering simultaneously ail asteroids involved in the encounters (perturbers and targets) and of whicli the resulting gravitational perturbations will bring information about the mass of the most. massive ones (those which are labelled perturbers hereafter). Consequently, the size of the partial dérivative matrix A will be very large within the scope of the Gaia mission, predicting the observations of hundreds of thousands asteroids. A formulation of the inversion of this matrix simplifying the calculations was described in Eichborn and Russel's paper [START_REF] Eichhorn | An algorithm for the statistieally rigorous and noniterative computation of trigonométrie parallaxes[END_REF]:

From notations (3.6, 3.7), the matrix A^WA can be written as, The inverse of AtA is obtained by [START_REF] Eichhorn | An algorithm for the statistieally rigorous and noniterative computation of trigonométrie parallaxes[END_REF],

(AtWA)"1 (T -SG"1St)~1 T-1S(StT-1S-G)-1 \ (StT"1S -G)"1StT"1 (G -StT~1S)"1 J (3.12)
The corrections to the initial conditions ACo = (Auo,Amp)( expressed in Eq. Thus, the inversion of matrices is necessary for obtaining the solution of our problem.

Numerous methods exist but the Singular Value Method (SVD), described in the next section, is probably the most useful because it is not a simple inversion method. It provides precious indications on the quality of the solution in the perspective that the problem is linearised.

The inversion by the Singular Value Décomposition

As said earlier, the inversion by the Singular Value Décomposition (SVD) is not only a simple matrix inversion method, it is also a mean to diagnose and give a numerical answer about the quality with which the solution is estimated. More precisely, it indicates if the partial dérivative matrix A defining the linearised équations of the least squares estimation (Eq. 3.10) is is well-or ill-conditioned or even singular. The phenomenon of ill-conditioning shows itself as a small variation of the (O -C) vector yielding a large variation of the correction to the initial conditions ACo-The cause is the existence of collinearities between some columns of the partial dérivatives matrix which can be due to a very weak number of observations or a bad distribution of these last.

The inversion by SVD is based on the theorem of linear algebra which says that any p\ x matrix (p\ is the rows number) A, p\ > P2 can be written as the product of,

A = UDV*
where U is an p\ xp2 column-orthogonal matrix (tVU = Ip2, IP2 being the identity matrix of rank P2), D an P2 XP2 diagonal matrix with positive or zéro singular values u>j=i,..(P2 and V an P2 x P2 orthogonal matrix (VV4 = V'V = IP2). Consequently, the inverse matrix

A-1 is equal to, a-1 = vrr1Ŵ
hen pi < P2-, the décomposition is also possible but the singular values cjj are zéro for j -pi + 1,..., P2 as well as the columns Uij=Pl+iv..iP2.

The diagnosis of the inversion is provided by the values of the singular values ojj. When one of them is so small that it is dominated by roundoff error, the partial dérivative matrix A is clearly ill-conditioned. So, the condition number, which is known as the ratio between the largest L0max and the smallest ujmin singular value, indicates that the partial dérivative matrix A is ill-conditioned when it is superior to I/o where a is the machine accuracy (~ 10"12 for double précision variables). An useful solution consists in replacing the singular values ujj for which ujj/ijrnax < a' by zéro (Lawson [112]), a' being chosen greater than a of some order as a précaution (a' = 10"10 in our simulation). The matrix thus becomes singular since its rank is strictly inferior to p2, the dimension of the nullspace being thus different to zéro. In fact, the SVD method constructs orthonormal bases for the nullspace (columns of V whose same-numbered éléments ujj are zéro) and range (columns of U whose same-numbered éléments ujj are not zéro) of the partial dérivative matrix A [START_REF] Press | The Art of Scientific Computing[END_REF]. When zeroing a singular value, one linear combination of the set of the équations of conditions is lost but being strongly influenced by roundoff error, it is better to discard it. The details of simplifications can be fourid in Hestroffer pH.D [START_REF] Hestroffer | Astrométrie et photométrie des astéroïdes observés par le satellite Hipparcos. Apport à l'élaboration d'un système de référence dynamique[END_REF].

The computation of the corrections to the initial conditions (Eqs. 3.13, 3.14, 3.

the associated standard déviations (Eqs. 3.23, 3.24) and their corrélations (Eq. 3.25)

dépend on the évaluation of the partial dérivatives matrix A (Eq.

3.2).

It can be computed by different methods, which will be detailed in the next Section 3.2

The partial dérivatives matrix

The main part in the linearised least-squares estimation is the évaluation of the partial dérivative matrix A which in our case is written as (Eq.

Analytical method

Brouwer and Clemence [START_REF] Brouwer | Methods of Celestial Mechanics[END_REF] gave analytical expressions for finding the variation of the rectangular coordinates of the asteroids with respect to their osculating éléments (ao, eo,

Numerical methods

The components dY{t)/dC3Q (Eq. 3.26) can be obtained by numerical intégrations, there are two methods: the first presented uses only multiple intégrations of the équations of motion, the second solves an auxiliary System of differential équations giving directly the values of <9r(£)/<9Cq at the time t.

Différence of direct intégrations

The classical method consists in performing two numerical intégrations of the équation of motion (Eq. 3.31) differing from a slight quantity in the initial condition Cq. From the équation of motion of an asteroid, where F is the force function which acts on its orbit and r = (x, y, z) the position vector, the approximation of dr(t)/ôCq is estimated by,

ï(t) = F(t) (3.31) (3.32)
where r* can be expressed as, ri(...,CJ0+ t) -r<(...,C{J, ...,*)

(3.33)
The value of e is chosen by successive tests until to find a stationary value for the quantity dri{t)/dC3Q. This value should not, be too small to limit the roundoff errors and not too large which would resuit in a large truncation error.

The variational équations 3.3.

The realistic simulation 84

The variational équations are second order équations describing the behaviour of dri(t)/dCJQ (Kulikov [START_REF] Kulikov | Hucjiennue Memodu neôecnou MexanuKU e npuMeneuuu k usyueHio deuoteenusj VIII cnymuuKa wnumepa[END_REF]), which are integrated together with the équation of motion (Eq. 3.31). Under the assumption that Cq does not dépend on the time, the variational équations can be expressed by (Herget [117]), d2 ( dri \ 9 (" \ dt2 \dCÎ ) dCl {Tl) dFj / dF^dr^0Fl drn \ dCi h\drndCl drndcl)

(3.34)
The quantifies drn/dCl and dfn/dCJ0 (n=l,..,3) are the resuit of the numerical intégration The intégration is carried out with a fifteenth order Everhart predictor-corrector integrator.

Each step, defined by the integrator with respect to its prédictions, is divided in sub-steps based on Gauss-Radau spacings. This numerical algorithm is a good compromise between accuracy and speed (Everhart [START_REF] Everhart | An efficient integrator that uses Gauss-Radau spacings[END_REF]).

Comparison and our choice

After comparing these methods on a test case with a fit of few parameters from a simulated orbit over a period of 5 years, the method of integrating numerically the variational équations has proved to be the most efficient, combining speed with précision. Even if the analytical method is so far the quickest, it lacks of précision because it considers only the secular effects on a Keplerian orbit, is a large inconvénient to obtain accurate fits of orbits and especially with observations over a short time (~5 years for Gaia mission). However, it is interesting for preliminary study because it allows to détermine very quickly the expected standard déviation of a parameter playing a rôle in a force with secular effects.

Besides, it brings us to look into the force under a more physical représentation. As re gards to the other numerical method (Eq. 3.32) consisting in integrating two trajectories with values of the initial conditions slightly different, it is also précisé that the variational équations but présents two major disadvantages: it is much less quick than both others and requires an empirical détermination of the small variation e for each fitted parameter.

However, this method is a very good tool to check easily the validity of the computed variational équations.

The realistic simulation

The simulations we hâve run use realistic simulation of Gaia observations. For each known asteroid referenced in Astorb database, programs were written by Tanga and Mignard polynomials (Mignard). Thus, at each observation date, the position of the asteroid from the spacecraft is known and consequently, an approximation of its apparent magnitude V. This latter gives us access to an estimation of the measurements précision in Gaia longitude a\ through the formula defined in section 1.1.6, cja = 310 yas for V < 15

(j\ = i0(°147Vr~2-71) mas for V > 15 (3.35) The weighting matrix (Eq. 3.7) can thus be defined, as well as the partial dérivative matrix (Eq. 3.6).

As we said in Section 3.2, the éléments of this latter, dX/dCq, are decomposed according to the rectangular coordinates (x, y, z) and the part

d\(t) _ fdX(t) ÔX{t) dX{t)\ <9r(t) \dx(t) ' dy(t) ' dz(t) )
is analytically estimated. Let u be the position vector of the asteroid A referred to the spacecraft G (see Fig. 3.2) and z the unit vector of the pôle of the RGC mentioned in section (1.1.4). We can define the variation of the vector dk, which is the projection of an infinitésimal variation of the asteroid heliocentric position dr on an intermediary circle RGC' parallel to the RGC plane and Crossing the asteroid,

dk = Z X U • dr Z X U
The angle a between the asteroid satellitocentric position u and the plane of the RGC is defined by,

cos a = sin ~= Z X u Z X u |Z||U| |U|
Thus, the variation in Gaia longitude dX with respect to the variation of the position dr where r=(x, y, z). In the following sections, we will describe the different perturbing forces we hâve modelled, namely the Newtonian gravitational forces F^, the force due to J2 the oblateness of the Sun F , the relativistic forces in the parametrised post-Newtonian formalism F as well as their different dérivatives useful for evaluating the variational équations. The équations will be written in the ecliptic heliocentric coordinates System.

In addition, the photocentre offset will also be integrated to the dynamical model and for some asteroids, the NEOs in particular, the force due to the Yarkovsky. This peculiar case will be developed in Chapter 5 in which are presented preliminary results on the possibility to dérivé some asteroid physical parameters from this non-gravitational force.

The gravitational perturbations

From the Newton's laws of motion and the law of gravitation, we hâve

-G(M® + m) y~+ G ^rrij f- lrl ,=1 VlTi (3. 36
)
where M© is the mass of the Sun, m that of the asteroid for which the équation of motion is written, rrij and rj=(rij, r2j, ) are respectively the mass and the position vector of the perturbing body j. Among the perturbers included, there are the eight planets, Pluto and a list of asteroids perturbing the target asteroid. This list is previously established in the process of the close approaches sélection. Thus, the orbit of perturbers, which mutually perturbed each other, can also be computed and interpolated in Chebyshev polynomials with the aim of having only the trajectory of the studied asteroid to be computed. The solar quadrupole J2

The solar gravitational quadrupole moment J2 is a component of the gravitational field of the Sun due to its oblateness e (the déviation from a sphere), itself caused by its rotation.

J2

The force F ' is derived from the potential function Vj2 = GMq ( ae \ 2 J2 P2(sin0) (3.37) where ae is the Sun radius, P2 the 2 order Legendre polynomial, </> is the latitude of the The perturbations due to General Relativity were modelled in the post-Newtonian ap proximation, which consists of the formalism of the Newtonian theory to which the post-Newtonian corrections are added. These last corne from simplifications based on simultaneously expansion in the following parameters, which are considered as small in the solar System, and that are, the Newtonian potential, ||||, the velocity to which the matter is generated by solar-system gravity, v2, the stress divided by baryon density po, \Tjk\/po, and the internai energy density per po, IL Each post-Newtonian approximation proceeds from its own metric theory. In spite of their great, différences, these approximations remain very similar and are United under the name of "Parametrized Post -Newtonian (P P N) Formalism". Each one is characterised by its set of PPN parameters (Will and Nordtvedt [START_REF] Will | Conservation Laws and Preferred Frames in Relativistic Gravity. I. Preferred-Frame Théories and an Extended PPN Formalism[END_REF]) and their values are taken to be one exact solution of the Einstein field équations in the Post-Newtonian limit [START_REF] Misner | Gravitation[END_REF].

As only relativistic effects from the gravitational field of the Sun are significant to represent the motion of the Solar System objects, we hâve considered the Schwarzschild metric, which describes the gravitational field produced by a fixed body of spherical structure.

This metric defines one exact solution of the Einstein field équations under the assump- 7, and (3 are two PPN parameters [START_REF] Will | Conservation Laws and Preferred Frames in Relativistic Gravity. I. Preferred-Frame Théories and an Extended PPN Formalism[END_REF] to which Section 4.2 will be devoted. The tensor metric g^can be expressed as, The computation of the contravariant tensor metric gPf> for obtaining the expression of the Christoffel symbols (Eq. 3.43) is performed in the linearised theory. This simplification can be applied because the corrections hpi/ (Eq. 3.41) are sufficiently small to neglect. their squares and products. From the expansion in powers of m/r of the tensor metric g^w (Eq. The photocentre offset

The photocentre offset was also implemented in the dynamic model. This phenomenon produces a différence bet.ween the centre of mass of the asteroid Po and the barycentre of light of the observed part P (Fig. 3.5). This displacement d = P -Po dépends on the shape of the asteroid, its réflectance properties described by the albedo and the scattering law of light at the surface, as well as its position with respect to these of the observer and the source of light. This latter is responsible for the partial illumination of the surface facing the observer and is expressed as a function of the phase angle (between the directions asteroid-Sun and asteroid-observer).

The reason for having introduced this effect, whereas, it does not play a rôle in the variance-covariance analysis for mass détermination, is presented in the following section 3.5. The aim is to estimate a minimal sizc of asteroids from which it is not possible to neglect the correction due to photocentre offset at the risk of hindering by mistakes the mass détermination at the astromet.ric précision level of Gaia.

This effect is estiinated under the simplification that the asteroid is a sphere of radius R with a completely homogeneous surface. Even if the photocentre offset on an ellipsoidal model was already st.udied by other authors such as Lupishko et al. [START_REF] Lupishko | Asteroid photocentre displace ment : influence of the scattering law[END_REF] and which will be used in the Gaia data réduction Processing, the approximation by a sphere seems to be sufficient for the future tests on the validity of the dynamic model in Section 3.5. With regards to the scattering law of the asteroid surface, we hâve taken a standard linear combination (Tanga and Kaasalainen [START_REF] Kaasalainen | Photocentre offset in ultraprecise astrometry: Im plications for barycentre détermination and asteroid modelling[END_REF]) of the Lambert and Lommel-Seeliger's law, respectively affected by the coefficient 0.1 and 0.9 and producing a displacement di, and dis. Nevertheless, certain authors (Lupishko et al. [START_REF] Lupishko | Asteroid photocentre displace ment : influence of the scattering law[END_REF]) recommend to use other laws such as the Akimov [START_REF] Akimov | Light reflection by the moon[END_REF] or Hapke [START_REF] Bowell | Application of photometric models to asteroids[END_REF] laws. Nevertheless, this latter is defined by many parameters, not well known and often correlated.

Firstly, the estimation of the photocentre offset d = P -P0 is made in the tangent plane (x,y) to the vector observer-asteroid -k (see Fig. 3.6). In fact, the direction of the photocentre offset is born by the vector Sun-observer x, defined as the intensity equator (Lindegren [START_REF] Lindegren | Meridian observations of planets with a photoelectric multislit micrometer[END_REF]). For their évaluation, we follow Hapke in " Theory of réflectance and emittance spectrocopy" [START_REF] Hapke | Theory of réflectance and emittance spectroscopy[END_REF] and the illustrative figure 3.6. We suppose an element of area da of the sphere surface by a light beam of irradiance J. The irradiance is the average energy Crossing unit area perpendicular to the direction of propagation per unit time. The incident light beam inakes an angle j with the normal vector n perpendicular to da, which forms an angle e with the observer. The phase angle is mentioned by i. Thus, the réflectance function of each element of the sphere is defined by, dl = JQsY(j,e,i)da (3.54) where Y (j, e, i) described the way of which each element scatters the incident light beam and Qs the absorption coefficient of the incident light, related to the albedo. As this latter does not hâve influence on the estimation of the photocentre offset, we take Qs = 1. For the Lambert's law, the scattering function may be written as,

YlU, e, i) = -cos j cose (3.55) 7r

From the previous expression (Eq. 3.55) and without considering the direction of the observer, thus Yi/ cose, we notice that the law represent.s the surface for which the brightness is only proportional to cos F This law was derived from empirical observations of objects with the particularity that the brightness of their surface did not dépend on the incident angle e and the phase angle i. The Lambert's law is especially used for reproducing the scattering behaviour of surfaces covered by bright, flat paints. Thus, it where p& is parameter describing the asteroid physical characteristics (porosity etc...).

This model is more correct for low albedo surface than other very used law, the Lumme and Bowell's one [START_REF] Bowell | Application of photometric models to asteroids[END_REF] applying to high albedo surfaces.

The Minnaert law, considered as a généralisation of the Lambert's law, can be defined by the scattering law, Ym(j, e, i) = Bq cosk i c,osk 1 e

where Bq and k are the Minnaert empirical coefficients, dépend ing on the phase angle i and the wave length. Bq is named as the Minnaert albedo and k the limb-darkening. This model was empirically shown that it can describe the scattering of light by most spécial material and planets at small phase angles (Veverka et al. [START_REF] Veverka | Near-opposition lirnb darkening of solids of planetary interest[END_REF]) and its use is by some difficulties detailed in paper (Esposito et al. [START_REF] Esposito | Albedo and photometric study of Mars with the Planetary Fourier Spectrometer onboard the Mars Express mission[END_REF]).

3.4.

The dynamical model 96

The coordinates (x, y, z) in the planetesimal frame (x, y, k), moving on the sphere of radius R, can be expressed in the following spherical coordinates, from the longitude A and the latitude P, The vector representing the photocentre offset in the ecliptic heliocentric coordinates Sys tem can be easily obtained by expressing the planetesimal reference frame (x,y,k) in our reference frame. From the observer-asteroid vector u and Sun-asteroid s, we hâve k = " where dx is defined by Eq. 3.53.

3.5.

The systematic errors in mass détermination 99

3.4.5

The non-gravitational forces Among the non-gravitational forces, we hâve begun to implement the force due to the Yarkovsky effect in collaboration with Marco Delbo (Nice observatory). The modelling is described in the chapter 5, which is entirely devoted to this effect and more precisely, to the physical properties of certain asteroids which could be deduced from this latter.

Nevertheless, this effect will be taken into account in the computation of the (O -C) only for the most sensitive asteroids, which are among the NEOs.

The systematic errors in mass détermination

The objective of this section is double: the fîrst one is to test the validity of the method for a large sample of asteroids. Therefore, we rebuilt the conditions of mass détermination from Gaia simulated observations with the aim of comparing the results given by our global method with the exact solution. The simulation, as well as the checked values are detailed in the next Section (3.5.1). The second objective is to spot the perturbing forces which could hâve a non negligible impact on the mass détermination at the Gaia précision level.

Thus, we présent in Section 3.5.3 the studies to be led when we will be in possession of the whole Gaia observations, for identifying possible weakness of the dynamic model for different groups of asteroids. These results will allow to complété the information on the uncertainties of the computed masses by providing refined confidence intervals. The aim is to give more exact results and thus, to reduce the possible mistakes in the interprétation of results, which would dépend on the asteroid masses.

The Monte-Carlo simulation test

The solution of mass détermination resulting from the linearised least-squares estimation is described by a normal law since the error measurements are assumed also to follow the same type of distribution. Thus, the value of the masses and their associated standard déviations would be the resuit of an experiment, which would consist in repeating a large number of times the same mass détermination with randomly generated measurement errors. Therefore, we constructed a Monte-Carlo simulations by generating 100,000 possible set of observations, so 100,000 different measures nii of each mass, from which the mean fim and the standard déviation ay is then derived. Thus, the possibility is offered to study the systematic error esys = |m -nm\ of the mass détermination, where m is the exact solution of the problem. In the case where esys is only due to the roundoff error, we can check for each asteroid the theoretical standard déviation from the variance-covariance matrix and the one derived from the 100,000 estimated masses mi (thus, test the method). As a conséquence, the fîrst. objective mentioned previously (test of the robustness) can be carried out. and will be treated in Section 3.5.2. Furthermore, an addit.ional information can be given with the maximal error among the 100,000 ones, which provides an indication on the confidence area of the solution.

With regards to the second objective, we use the same Monte-Carlo simulations but this time, the dynainical model is the varying parameter: each perturbing force described in Section 3.4 is alternatively taken into account for the simulation of the observations, but rernoved in the dynamical model used for fitting the masses. Thus, by analysing the new values of systematic errors esys, the forces and other effects, e.g photocenter offset, necessary to carry out a good mass détermination with the Gaia précision level can be detected, and will be a source of information about the computed uncertainties.

In addition, the semi-major axis of asteroids at the initial time is modified by a term of PARU 3.5.

The systematic errors in mass détermination 100 about 10 6 AU: the aim is to fît in realistic conditions by deriving, simultaneously to the masses, the positions and velocities at the initial time of asteroids.

The observations of asteroids in Gaia longitude À on the RGC are constructed from the modelling of the or bit and the complex scanning law of the satellite (by Mignard). Thus, as the two rotation angles allowing the transformation from the ecliptic to the satellitocentric reference frame are known at each instant, we can obtain the rectangular position (x, y, z) in this latter and consequently, the longitude A in the RGC from the formula,

The error on the positions is derived from the satellitocentric position of the asteroid, its apparent magnitude V at this moment (the formula is detailed later in Eq. 3.69, Sec.

3.8.2) which is linked by an empirical law (including essentially photon and read-out noise)

to the Gaia longitude précision (Eq. 3.35, Sec. 3.3). Thus, it permits to introduce the random noise for each observation date. Hm is the mean of the n estimated masses m, from Monte-Carlo simulations in the case of a perfect dynamic model. a^is the associated empirical standard déviation. mmax is the largest interval between the exact solution of the mass and the computed ones rnt

Density

(i = 1 ,..n).
m', a') are the sarrie variables previously defîned in the case of an incomplète dynamic model. Thus, a discrepancy can appear between the exact solution for the mass and the mean of the computed values, which is well-known as the systematic error esys. This quantity will be the key point to estimate the importance of certain forces or effect corrections in the dynamic model used for mass détermination in the Gaia context.

For the tests, we retain the asteroids involving in close approaches which will occur between 2010.5 and 2016. The sélection criteria are a minimal distance between perturber and the target asteroid smaller than 0.5 AU and a deflection angle superior to 3.5.

The Systematic errors in mass détermination 101 5 mas. Moreover, the perturbers posses a mass estimated greater than 10~13M© among the 1,000 first numbered asteroids, while the target asteroids belong to the 5,000 first ones. Thus, the used sample of asteroids is made up of:

Number of perturbing asteroids 248 Number of target asteroids 1384

Number of perturber-targets asteroids1 164

1 These asteroids belong to both categories of perturbing asteroids and target asteroids

A life size test of the mass détermination

An important number of target asteroids, 1384 in total, is considered in our simulation for the potential détermination of 248 masses. As written earlier, the test of the global method requires that the dynamic model used for the least-squares estimation and for the construction of the observations is the same. The main resuit is described by Table 3.1, which is limited to the asteroids with a formai standard déviations on mass inferior to 50%

(limit error for which the estimation of the mass keeps in importance). For each asteroid, the relative formai (from variance-covariance matrix) and computed (from Monte-Carlo) standard déviations, o(m)/m and ay/m are indicated. In addition, the systematic errors esys are expressed in part of the formai standard déviation ay and the maximal error on the mass is derived from the 100,000 Monte-Carlo runs. Table 3.1 shows that, as expected, the formai standard déviations are in good agreement with the computed ones. Moreover, the systematic errors are relatively small in comparison to the standard déviations (ail located under the 0.8% mark) due to the roundoff errors. Thus, the global method is validated with a large number of asteroids and its interest will appear in the following Section 3.5.3. The systematic errors in mass détermination 103

Testing the dynamic model

The completeness

The importance of the completeness of the dynamical model can be illustrated with the same previous Monte-Carlo simulations but removing a perturbing force in the leastsquares estimation, force which is considered in the process of the simulation of the obser vations. Thus, we tested the solar quadrupole moment J2, the relativity expressed in the post-Newtonian approximation, the correction due to the photocentre offset for asteroids with a diameter inferior to 10 km and 1 km as well as the number of perturbers for each asteroid. For the last case, the observations are simulated by considering further close approaches with a deflection angle limited to 1 mas and to 0.1 mas.

The précision on the perturber position

This case was proceeded from the hypothesis where Gaia will not observe the asteroids larger than one hundred mas, so about 400 km of diameter at 2.5 AU. Thus, only for the construction of Chebyshev polynomials from the perturbers positions, which are used for the évaluation of the variational équations with respect to the perturbing masses, we introduce an additional variation of the semi-major axis of the concerned perturbers. This variation is such that the added error on their position is about a few mas, précision to which the orbit of the big asteroids is derived from the ground-based observations.

The reference mass

The case of the influence of the reference mass was envisaged. The main effect is that from a certain error on this mass, the solution obtained from the first itération gain in error but is corrected during the following itérations. In our case where the initial error on the mass is 15 %, a very good estimation is obtained from the second itération where the errors are minimal. However, the influence of the reference masses is much less important that a non realistic modelling of the dynamic model. In fact, a tool exists to consider this kind of error. It is an estimation with a priori information, which can be treated by conditional équations in addition to the observational ones. We guess that the target asteroids with very few observations are the most open to provide a more erroneous information on the perturber masses. Nevertheless, these last are very often derived from a large number of target asteroids well-observed by Gaia.

Results

The results are gathered in Table 3.2, where the distribution of the relative systematic errors esys/ay is represented for the case where the theoretical précision is smaller than 50%, resulting in 46 objects. We can notice that in neglecting the photocentre offset for the asteroids with a diameter superior to 1 km, or the encounters between asteroids characterised by a small deflection angle of 0.1 mas or yet the relativity, that can produces large systematic errors on the derived mass . Thus, in the évaluation of the (O-C), ail these effects and perturbing forces need to be considered and, st.udied in order to evaluate the influence of the errors on the laws used (photocentre offset), on the parameters (PPN parameters for relativity, parameters of the scattering laws for photocentre shift and on the masses of asteroids involved in the close approaches of weak intensity). Furthermore, the importance to consider many encounters, even small, is well-illustrated by Table 3.2.

As a conséquence, the global method is essential for treating the asteroid masses with the whole Gaia data. As the level of précision will allow to detect small perturbations, many asteroids will be indirectly linked together since certain will perturb asteroids, which in turn will perturb others. Besides, the importance of the relativistic model for reducing the Gaia data is well illustrated by final results (tab. 3.2). 2 The précision on the perturbers positions with a diameter superior to 400 km is known with an error of a few mas.

3 The tested effect is the photocentre offset when the correction is not considered in the least-squares solution for the asteroids with a diameter inferior to the mentioned limit contrary to the simulation of observations where it is systematically applied.

4 The sélection of the perturbations taken into account in the construction of the observations is defîned by the limit on the deflection angle while for the least-squares estimation, it is always to 5 mas.

3.6.

The expected précision on the masses with Gaia 105

3.6

The expected précision on the masses with Gaia

Input data

The prédiction on the mass précision was made from realistic simulated data and a sé lection of close approaches which will occur between 2010.5 and 2016, corresponding approximately to the expected observing period of Gaia. We recall that the target asteroids belong to the first 346,481 numbered asteroids while the perturbers are among the first 20,000 ones and possess a mass greater than 10"13M©. The statistics of these encounters as well as the way by which was estimated the reference mass of ail asteroids are detailed in Section 2.2.2. The target asteroids thus found will expérience at least one close approach with a perturber with a minimal distance < 0.5 AU and a deflection angle > 1 mas (Eq.

2.7, Sec. 2.1.1). Table 3.3 provides the number of close approaches for a set of criteria, the number of perturbers and target asteroids that were found, as well as the number of bodies which are both perturbers and targets. Here only asteroids observed by Gaia are considered.

Table 3.3: Close approaches overall statistics with a minimal distance < 0.5 AU, a deflection angle > 1 mas and a perturber mass > 10"UMQ.

Number of perturbing asteroids 602

Number of target asteroids 43513

Number of perturber-targets asteroids1 435

Number of close approaches 78768

1 These asteroids belong to both categories of perturbing asteroids and target asteroids

Results

Using these set of close approaches with their strength, this can be introduced in the covariance matrix U^1 (Eq. 3.20, Sec. 3.1.3) to evaluate the final précision achievable on the masses of the perturbers. More precisely, these last are given by the diagonal components of U^1, of which the detailed computation is described in Sec 3.1.3 by the formula 3.23.

The main resuit is shown in Table 3.4 with the statistical distribution of the précision on the masses. There are about 36 asteroids for which the mass can be ascertained to better than 10% and this number rises to 149 with a 50% précision. Table 3.5 gives the resuit with a sélection of close approaches extended to the case of minimal distance inferior to 0.01 AU and which was not selected by the sélection software (see Sec. 2.1.4). Finally, we gain only 3 masses what is very small in comparison of 149 found. The intensity of a close approach is not the only condition for a good mass détermination, the observations of the target asteroid has to be numerous enough and well distributed in the time. Moreover, the asteroids with a good expected précision on the mass are very often massive objects meeting numerous target asteroids, which will be well observed by Gaia. Table 3.6 lists the individual objects having at the end a mass recovered with a relative précision smaller than 10%. We can see that the first two, 1 Ceres and 4 Vesta are determined with a formai précision of 0.07% and 0.09%. In addition, the number of targets for each mass détermination is given as well as the number of close approaches which will be out of the Gaia observations range, the formai précision and, the reference masses used in this work and finally, the relative précision in percent. The non observable close approaches correspond to the encounters between a perturber and a target occurring during the mission (here 2010.5-2016) but before the first or after the last observation of that particular target. So they hâve no influence of the fit of the orbital éléments and does not contribute to the équations of condition for the mass of the perturber. The column of close approaches not observed by Gaia reveals a high number of these events which in the case of exceptional close approaches (impact parameter very small and relative velocity not too high) is a large lost of information on the pert.urbing mass. Therefore, in Section 3.8 we hâve investigated the possible contribution of ground-based observations to the mass détermination by Gaia. The expected précision on the target orbital éléments 108

3.7

The expected précision on the target orbital éléments

In the previous section, we hâve seen the important number of mass déterminations that Gaia will allow to accomplish with 5 years of observations. Moreover, many of them will be more précisé and less corrupted by the systematic errors than the estimations from ground-based observations, since the astrometric accuracy of Gaia will allow to consider a dynamic model much more complété. In addition, the unprecedented précision reached for the asteroid orbits and more particularly, for the target asteroids will be a key point for the results of mass déterminations. Therefore, we présent two figures, which represent the standard déviation for the semi-major axis and the eccentricity (Tab. 3.9), the inclination and the longitude of node of targets (Tab. 3.10). These formai standard déviations are obtained from the formula 3.24 of Section 3.1.3, for which the partial dérivatives representing the variation of the rectangular coordinates with respect to the position and velocity at the reference time, are re-written in order to hâve the variation according to where r = (x,ÿ, i). The procedure is the saine for the other osculating éléments.

As we can see on the 4 figures, the majority of the target asteroids will hâve a very welldetermined orbit from Gaia observations with a précision on the semi-major axis smaller than 1CT8 AU, on the eccentricity better than 10~8, on the inclination about 1CP7 degree and on the longitude of node inferior to 10~6 degree. Thus, Gaia will provide accurate masses of perturbers, acting on numerous smaller asteroids for the most of which a very précisé orbit will be derived. With regards to the other asteroids, for which a very poor précision is expected on their orbits, the cause is the very weak number of observations and their distribution. It would hâve been possible to increase artihcially the précision by diminishing the value from which the singular values are replaced by zéro in the process of inversion of the partial dérivatives matrix (see Sec. 3.1.2) but, the problem is that the found orbit will be only one among many others but necessarily the most probable. These cases will be treated separately by algorithms in charge of finding the most plausible orbit from only some observations. That is is one of the tasks of the GAIA working group CU4, which is managed by Karri Muinonen. Nevertheless, these asteroids, very poorly observed, will bring almost any valid information on the mass of perturbers, and will be probably removed of the input data for mass détermination.

Figure 3.9: Density distribution of the standard déviations on the semi-major axis a and the eccentricity e of the target asteroids with respect to the value of these two osculating éléments.

There is 3 levels of density based on the maximal number nmax of standard déviations belonging to one bin. The level "1" means that the bin contained a number of standard déviations n inferior to nmax/3, the level "2" corresponds to nmax/3 < n < 2/3 nmax etc...

3.7.

The expected précision on the target orbital éléments 110 îo4 a(i) 3.9.

3.8.

The contribution of ground-based observations 111

The contribution of ground-based observations

As noticed in the previous Section 3.6, among the close approaches selected for mass détermination dnring the Gaia mission, some will occur at the edge of the observational coverage of the perturbed asteroid, i.e. the t.imespan between the first and last observation of this object. Thus, an important part of the information contained in these encounters will be lost. Therefore, the possibility to increase the précision of certain déterminations from further ground-based observations of particular targets before and after the mission was investigated. The main results is presented in the form of a list of target asteroids to be observed and the resulting benefit in précision on the perturber masses. The number of targets was restricted to the few ones for which adding ground-based observations maximises the scientific return.

Description of simulation

We hâve run the same simulations described in section 3.6 by adding new observation epochs which will be a priori possible from Earth. We hâve separated the case where the observations could be made before the Gaia mission and after. The reason is that chance to observe target asteroids after the mission is higher than before, the time to préparé the observation campaigns being much longer.

With regards to the observations before the mission, we hâve investigated the period For the précision of measurements, we hâve chosen 50 mas as standard value for ail observations as a resuit of discussions with Ukrainian observers and may be future collaborators in this project. We were told that the mean précision of their observations with the RTT-150 telescope in Antalya is about 70 mas (nearly constant with respect to magnitude) in both coordinates. Thus, 50 mas appears to us to be an acceptable value. The perturbers which appear in these tables hâve at least an expected relative précision smaller than 50% for the best one. Besides, we focused on the asteroids for which the précision on the mass, actual or future with Gaia, is smaller than 50%. A more complété list of target asteroids will also be available without any constrain on the perturbers.

However, in the présent case, the interest is turned towards the new masses, a parameter considered of greater interest for the improveinent of the planetary ephemeris or for the bulk composition of the asteroids.

The complementary results are listed in Tables 3.9 and 3.10, where one gives for each perturber, the target asteroids to be observed from the ground to attain the formai précisions listed in Tables 3.7, 3.8. Some additional information about these last are listed such as the period of observable time and the associated mean apparent magnitude.

The apparent magnitude is computed from the formula [START_REF] Bowell | Application of photometric models to asteroids[END_REF] where H is the absolute magnitude (from Astorb database), r and A the heliocentric and geocent.ric distance of the asteroid and i the phase angle (Fig. 3.11).

V = H + 5 log{r • A) + f(i) (3.69) f(i) = -2.5 log [(1 -G) * fa (i) + G * 02(*)]
where G is the slope parameter of the asteroid (from Astorb database). List of the target asteroids (before the mission) Except for the asteroid 92 Undina for which the mass is estimated witli 61.5% of error (current mass, Table 1.4), the mass of the 25 additional asteroids (Tables 3.7, 3.8) could be estimated with a précision better than 50% from a contribution of ground-based ob servations. Moreover, these masses are important because they are not known. Thus, the impact will be very significant since about 20% of additional masses will add to the 149 ones expected with the Gaia observations alone. We can notice that the most efficient encounters hâve a very small impact paraineter because in this type of configuration, the perturbation is transmitted with a minimal time; thus, the conséquence of the perturba tion on the target position is more important than in the case where the perturbation operated over a much longer time.

3.8.

The contribution of ground-based observations 115

List of the target asteroids (after the mission)

4.1

The solar quadrupole J2

The state-of-the-art Currently, scientists agréé that the solar quadrupole moment J2 is the order of 10~7.

However there is still a large scatter in the published values of J2 with clearly systematic effects dépending on the methods (optical observations, helioseismology, models) applied to this dérivation and no firm consensus lias been reached.

The anomaly of the perihelion motion of Mercury has been known since 1859 thanks to Leverrier [131], but the idea of a distorted Sun for explaining the cause appeared only in 1890s, with Paul Harzer paper in 1891 [START_REF] Harzer | Uber die Bewegung des Mercurperihels[END_REF] and the independent investigation of Newcomb in 1895 [START_REF] Newcomb | The Eléments of The Four Inner Planets[END_REF]. However, this assumption (at least with the magnitude required to account for Mercury anomaly) was ruled out from direct measurements of the oblateness carried during the Venus transits of 1874 and 1882. It can be told that the détermination of the solar quadrupole J2 has really begun since 1966, when Dicke and Goldenberg carried out observations with the Princeton Solar Distortion Telescope, which allowed the measure of the solar ellipticity, the first at the time. The revised value of their estimation [START_REF] Dicke | Differential Rotation and the Solar Oblateness[END_REF] gave J2=(2.47±0.23)xl0~5 [START_REF] Dicke | The Oblateness of the Sun[END_REF], value much larger than those currently derived. Besides, the astrophysical conséquences that it generated were difficulty acceptable, but nevertheless it rekindled the interest for this subject and had been at the root of new investigations of solar interior models.

These last can be divided as for asteroid mass détermination, that are the astrophysical methods based on the physical properties of the studied body, and the dynamic methods analysing the gravitational perturbations from J2 on the motion of solar System objects 4.1.

The solar quadrupole J2 120 (planets, asteroids ...) to fit its value.

Astrophysical methods

Among the astrophysical methods, there are those which consist, in finding the solar quadrupole moment J2 from the geometry of the Sun. Thus the measure of the solar oblateness e or the excess équatorial radius Ar, leads to the value of J2 from a Clairaut-like relat.ionship as described in paper of Dicke [START_REF] Dicke | The Solar Oblateness and the Gravitational Quadrupole Moment[END_REF]. Upon the same principle, Sofia et al.

[137] estimated J2 to: J2 = (3 ± 6) x 1CT7 (1994) from the Solar Disk Sextant (S.D.S), which provided solar angular diameter according to different orientations. The solar quadrupole J2 was inferred from the estimated oblateness, by considering solar surface angular rotation data. The SDS data allowed the détermination of new values of J2: in 1996, J2 was estimated to J2 = 1.8 x 10"7 (Lydon and Sofia [START_REF] Lydon | A Measurement of the Shape of the Solar Disk: The Solar Quadrupole Moment, the Solar Octopole Moment, and the Advance of Perihelion of the Planet Mercury[END_REF]) and using a constrain on the rotational State of the central région from helioseismic data IRIS, Paterno et al.

[139] computed an average J2 equal to: J2 = 2.22 x 10"7.

The helioseismology, is a powerful tool to dérivé (but with a model) the solar quadrupole moment. The principle is the décomposition of the oscillations of the Sun into its pulsation eigenmodes, as résonant modes for which a frequency is measured. The solar rotation produces oscillation frequencies which are split into multiplets which spread in frequency dépends strongly on the internai structure of the Sun in its outer layers. Thus, the internai rotation rate of the Sun can be inferred by inverse techniques and then, the solar quadrupole J2 from an intégral relation with the internai rotation of the Sun (Goldreich and Schubert [START_REF] Goldreich | A Theoretical Upper Bound to the Solar Oblateness[END_REF]). So, Gough in 1982, found a solar quadrupole moment J2 ~3.6 x 10~6 [START_REF] Gough | Internai rotation and gravitational quadrupole moment of the sun[END_REF]. We can also cite [START_REF] Brown | Inferring the sun's internai angular velocity from observed p-mode frequency split.tings[END_REF] estimating J2 = 1.7 x 10~7 [START_REF] Brown | Inferring the sun's internai angular velocity from observed p-mode frequency split.tings[END_REF] with a standard déviation which can be reduced by perhaps 10% according to the authors'words, and Pijpers (1998) with a mean J2 equal to: J2 = (2.18 ± 0.06) x 10~7 [START_REF] Pijpers | Helioseismic détermination of the solar gravitational quadrupole mo ment[END_REF] from GONG (Global Oscillation Network Group) data and the space observations SoHO (Solar Heliospheric Observatory).

Another important method is based on the figure of the Sun, which is defined in our case by the level surface where the gravitat.ional potent.ial is constant. As the Sun is a rotating fluid body, the principle of this method is to take into account the differential rotation in the expression of the potential. The order of the theory means the order of the development in /, / being the fiattening of the équation of the ellipsoid. A relation is then found between the solar quadrupole moment J2 and the fiattening / from the modified Clairaut's équations, establishing the pressure equilibrium of the Sun's surface in the case of hydrostatic equilibrium [START_REF] Rozelot | The Figure of the Sun, Astrophysical Conséquences. A Tutorial[END_REF], From this method, we can give the results of Ulrich and Hawkins [START_REF] Ulrich | The Solar Gravitational Figure -j2 and J4[END_REF] who estimated an interval for J2 of 1.0 x 10~7 < J2 < 1.5x10"' in 1981, with a differential rotational law. This latter denied by helioseismic data allowed [START_REF] Godier | Quadrupole moment of the Sun. Gravitational and rotational potentials[END_REF] to state J2 = 1.6 x 10~7 [START_REF] Godier | Quadrupole moment of the Sun. Gravitational and rotational potentials[END_REF].

Dynamical methods

The dynamical methods lay in the analysis of the perturbing effects due to the solar quadrupole moment J2 upon the motion of celestial bodies such as the asteroids, the Moon and the planets. The dynamical method has the great advantage to be directly sensitive to J2, regardless of any assumption 011 the internai structure. More precisely, the ellipsoidal gravitational field acts upon the longitude of node O, the argument of perihelion lo and the mean anomaly M of their orbit. Certain asteroids are priviledged targets because of the resulting secular effect, dépends on the semi-major axis and the 4.1.

The solar quadrupole J2 121 eccentricity of the perturbed object, as it will be shown at t.he end of the section.

Thus, in 1969, Lieske and Null [START_REF] Lieske | Icarus and the Détermination of Astronomical Con stants[END_REF] fitted the value of J2 to: J2 = (1.8 ± 2.0) x 10"5 from astrometric observations of the minor planet 1566 Icarus, which posses a small semi-major axis and a high eccentricity, making it very sensitive to the solar J2. Using the same type of observations from 1949 to 1987, Landgraf found in 1992 a négative solar quadrupole moment J2 = (-6 ± 58) x 10"' [START_REF] Landgraf | An estimation of the oblateness of sun from the motion of Icarus[END_REF], probably caused by the weak précision of observations and their unequal distribution over a long period. This case of négative values is not rare when it is inferred from the perihelion shift of Mercury. In paper of Pireaux and Rozelot,[START_REF] Pireaux | Solar quadrupole moment and purely relativistic grav itation contributions to Mercury's perihelion advance[END_REF], a very complété review of solar quadrupole détermination lists the values of J2 inferred from the Mercury perihelion shift. In this list, 10 négative values out of 23 can be read, the lack of observational précision being probably the main cause.

From the Lunar Laser Ranging data allowing study with précision the Moon's physical librations, Bois and Girard [START_REF] Bois | Impact of the Quadrupole Moment of the Sun on the Dy namics of the Earth-Moon System[END_REF] defined an interval for the possible value of J2 in 1999: 2.0 x 10"7 < J2 < 3.0 x 10"6. With regards to the planetary ephemerides, the solar quadrupole moment J2 is fitted from its effects upon the orbital éléments of ail planets. In this way, Pitjeva estimated J2 = (1.9 ± 0.3) x 10"' [START_REF] Pitjeva | High-Precision Ephemerides of Planets EPM and Détermination of Some Astronomical Constants[END_REF] more précisé value of J2 that are (J2 = (2.0 ± 0.4) x 10-7) with a particular method detailed in Rozelot and Pireaux paper [START_REF] Pireaux | Solar quadrupole moment and purely relativistic grav itation contributions to Mercury's perihelion advance[END_REF] and (J2 = (1.9 ± 0.16) x 10"7).

Conclusion and perspectives

As mentioned earlier, the order of the solar quadrupole moment J2 equal to 10"' seems to be well-determined but its value differs from different methods of détermination (Fig. 4.1).

Each method suffers from the lack of précision of observations, astrometric for dynamical methods and angular for the géométrie shape of the Sun which is not symmetric (North-South sunspots asymmetry). The poor knowledge of the non constant rotation rate varying with latitude is an important hint for methods based on the figure of the Sun. Furthermore, the completeness of ail these dynamic models is an other obstacle: the différence between the planetary ephemerides DE414, and EMP04, INPOP06, both using a slightly different dynamic model than the first, is revealing, as well as the complexity of the differential rotation of the matter hard to be modelled. However, the investigation of the J2 is very important.

A reliable value of J2 would be a serious and valuable constrain for solar interior model, a fact hard to take into account today because of the lack of trust in the published values.

In addition, the physics of the Sun is very strongly connected to its geometry and vice versa, and thus a précisé dérivation of the solar J2 would give precious information about the physical processes occurring in its interior [START_REF] Rozelot | The Figure of the Sun, Astrophysical Conséquences. A Tutorial[END_REF]. Concerning the dynamics of solar System objects, it becomes essential to know precisely the solar quadrupole moment because of the précision level of observations attained current.ly on the planets ephemerides and satellites, such as Mercury, Venus and Mars and the Moon. Moreover, as the relativistic parameters are strongly correlated to the solar J2, their fît from astrometric observations requires a good knowledge of this latter.

In view of numerous interests to know precisely the solar quadrupole moment, future missions were or are being studied. The PICARD mission [START_REF] Picard Team | Simultaneous measurement of the total solar irradiance and solar diameter by the PICARD mission[END_REF] due to launch at the end of 2008, will carry out the diameter and the asphericity of the Sun, as well as helioseismologic observations and the total irradiance. Thus, as seen earlier, these data could be used for The solar quadrupole J2 123 planet to the Sun. Mercury is the most sensitive planet to the solar J2, expressing itself by the precession of its perihelion. Thus, from data of this mission, the détermination of solar J2 is expected to better than 10-8(1). In the following section 4.1, we envisage the détermination of the solar quadrupole moment in the Gaia context from astrometric observations of asteroids which will be observed among the 350,000 numbered ones. The results of the variance-covariance analysis are given in Section 4.3. The cases of fit with and without that of the relativistic parameters are viewed.

The solar quadrupole moment détermination with Gaia

The extraction of the solar quadrupole J2 from Gaia data on astetroids follows exactly the same overall principles as for the masses. The variational équations required are, by considering the same notations in the section 3.4. We hâve also made the variance-covariance analysis by using the analytical formula of the secular variations upon the asteroid orbits. The aim is to compare the resuit with numerical intégration of the variational équations, which also take into account the periodic effects.

Writing of the analytical partial dérivatives for the secular effects

For evaluating the solar quadrupole J2 from the analytical method (Sec. 3.2.1), we express the variation of the positions of the asteroid at the time t with respect to this parameter by using the expression of the secular variations on its osculating éléments, that are the longitude of node fi, the argument of the perihelion w and the mean anomaly M. From the expression of the potential function of the solar quadrupole J2 (Eq. 3.37)

with respect to the mean values of the osculating éléments (â, ê, î, fi,ü>, M) in the solar

équatorial coordinates System, Vj2 = GMq J2 2| T --sin2ï\ (1 -ê2)"3/2
we obtain the variation of these éléments due to the solar quadrupole .12 by writing the average Lagrangian équations [START_REF] Duriez | Cours de Mcaniqe Cleste[END_REF],

where, n is the mean value of the mean motion. As the secular effects from the solar quadrupole J2 on the semi-major axis, the eccentricity and the inclination are null we hâve, a = ao, e = eo, i = io and n = no = / G(Mq + ni)

V ÛQ 1/2
From Equations ( Thus, the variation of the rectangular coordinates Ar' in the solar équatorial system with respect that of the solar J2, A J2, by using the formula (Eqs. 3.28, 3.30, Sec. 3.2.1), may be written as,

Ar' ^CLil{t tref) dr' dito T nw(i tre/) dr1 duo T H'A/ (t tref ) dr' \ dM0) A J2
We make it clear that the longitude of node ito and the argument of perihelion uq are not our usual osculating éléments (in the ecliptic) but those which describe the orbit of the asteroid in the solar équatorial coordinates System. To go back in the ecliptic coordinates System, the variation due to J2 in the rectangular heliocentric coordinates r requires the évaluation of the rotation matrix M2, allowing the change from the solar equator coordinates system to the ecliptic one, M2 = R1(-e)R3(^)Ri(/)

where, Ri(-e), R3($), Ri(/) are the rotation matrix defined in Sec.

The relativistic parameters

In the Eddington-Robertson form of the PPN formalism, that we use for describing the relativistic effects, only the parameters 7 and (3 are considered. Thus, in this section, we investigate the possibility to test with Gaia data the good agreement of their value with General relativity, consequently 7 = (3 = 1. Furthermore, other relativistic effects for which any convincing proof of their existence has not been found, is also studied. Thus the violation of the Principle of Equivalence, pillar of the theory of the General Relativity, as well as the variation of the gravitational constant could be checked by Gaia.

The PPN parameters

The parameter 7

The parameter 7 can be seen heuristically as the measure of the curvature of the space by unit rest mass [START_REF] Will | Conservation Laws and Preferred Frames in Relativistic Gravity. I. Preferred-Frame Théories and an Extended PPN Formalism[END_REF]. The main tests of this parameter are based on the bending of light, which will be also derived by Gaia (Sect. 1.1.5), and the tirne delay of light.

Until 1968, every experiments for computing the defiection of light were made during total solar éclipsé. The results were affected by many errors from weather, the change of température leading to optical distortions etc... In 1964, Dicke [START_REF] Dicke | The Theoretical Significance of Experimental Relativity[END_REF] assessed the State of these measurements and estimated their uncertainty between 20 and 40%. Then, in the early 1970's the development of radio-interferometry allowed estimations to précision of about 0.001 arcseconds: the angular séparation of two quasars was monitored during the occultation of one of the pair by the Sun, in order to measure the defiection of its emitted radio waves. Thus, Hill [START_REF] Hill | A measurement of the gravitational deflection of radio waves by the Sun[END_REF] measured in 1971 the value of (1 -f 7)/2 to: (1 +7)/2 = 1.07 ± 0.17. The very-long-baseline radio interferometry (VLBI) then became the most accurate tool and reached accuracies better than 100 /j,as in the détermination of angular séparations and and their variations. The most précisé measure of 7 -1 was made from almost 2 million of VLBI observations (from 541 radio sources spread out among 87 VLBI sites): 7 -1 = (-1.7 ± 4.5) x 10~4 [START_REF] Shapiro | Measurement of the Solar Gravitational Deflection of Radio Waves using Geodetic Very-Long-Baseline Interferometry Data, 1979[END_REF].

The other main test of 7 is the measure of the time delay of light, which is an additional delay to that of the Newtonian framework, in the round-trip travel time of a radar signal going past close to massive planets. This effect was discovered by Shapiro [157] in 1964.

Two types of experiments hâve been used, which differ from the refiector, taking the form either of the surface of a planet such as Venus or Mercury, either of an electronic equipment on board of a spac.ecraft in charge of the retransmission of the signal to Earth.

Among the most précisé measure, we can cite that made with Cassini experiment, from Doppler tracking of the spacecraft, which was heading for Saturn [START_REF] Bertotti | A test of general relativity using radio links with the Cassini spacecraft[END_REF]. The final resuit gave: 7 -1 = (2.1 ± 2.3) x 10"5. With regards to the fit performed within the planetary ephemerides, the estimated uncertainty on the value of 7 is about 10~4 -10~5 but it dépends on the dynamic model used.

Future results should be soon provided by Gravity Probe B (NASA-launch in 2004), for the measure of 7 with a précision of ~6.10~5. Furthermore, many future missions devoted to the 7 measure are expected: Thus, BepiColombo, from the dérivation of the 4.2.

The relativistic parameters 126 perihelion shift of Mercury orbit as well as measurements of time delay and Doppler shift should allow a précision of 10"6. Among other concepts of missions, there are SORT (Solar Orbit Relativity Test) [START_REF] Turyshev | The laser astrometric test of relativity mission[END_REF] for which the measure of 7 is expected to 10"7, by combining a time delay experiment (via laser signal by two satellites orbiting the Sun) with a light deflection experiment (interferometric measurements of two signais from both satellites).

The LATOR mission seems to be the most promising and predicts an accuracy of 10~8 [START_REF] Turyshev | New concept for testing General Rel ativity: the Laser Astrometric Test Of Relativity (LATOR) mission[END_REF]: the deflection of light will be measured by laser interferometry between two microspacecrafts, whose lines of sight, pass close by the Sun. However, before the ends of these missions, Gaia could provide a very compétitive value of 7 with a précision of 10"6-10~7 from bending of light experiment. Let, remind that at the time, the astrometric data from Hipparcos allowed the best détermination of 7 in the visible from the same experiment designed for Gaia, with 7 = 0.997 ± 0.003 [START_REF] Froeschle | Détermination of the PPN Parameter gamma with the HIPPARCOS Data[END_REF].

The parameter ( 3The parameter (3 is described as the measure of the nonlinearity in the superposition law for gravity [START_REF] Will | Conservation Laws and Preferred Frames in Relativistic Gravity. I. Preferred-Frame Théories and an Extended PPN Formalism[END_REF]. The primary source for its assessement follows from the secular effect on the argument of perihelion uj of the objects orbiting the Sun. Its secular rate in the PPN approximation can be expressed as (Brumberg [162]), . _

where, a is the semi-major axis of the perturbed body, e its eccentricity, n its mean motion, and c the velocity of light. Its secular effect. shows itself in an important way in the perihelion shift of Mercury through the coefficient T -(2 + 27 -(3)/3. In addition, a secular effect also exists on the mean anomaly M, which can be approximated by (Iorio [START_REF] Iorio | On the possibility of measuring the solar oblateness and some relativistic effects from planetary ranging[END_REF] ), M = -A with A = (2 + 47 + 3/3)/9 (4.5) a(l -e^)1/^cz Thus, the fit of the parameter T, and therefore (3 (the value of 7 being derived from another methods), can be made from perturbations analysis on the planet and asteroid orbits.

The Mercury observations set a limit to the uncertainty in (3 to about 3 x 10~3 (Will [START_REF] Will | The Confrontation between General Relativity and Experiment[END_REF] ). With regards to the planetary ephemerides, the fitting simultaneously of (3 and the solar quadruple moment J2 provided (3-1 ~10~4 (Fienga, personal communication).

Nevertheless, the corrélation between the two parameters is important and generate a systematic error, which is hard to estimate.

An indirect test of (3 consists of measuring the Nordtvedt parameter r/ (the topic of the next section), under the assumption of a linear relation between these last, which may be written as [START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF], r] = 4/3 -7 -3

Thus, from different estimations of 7, the limit for the f3 value is 2.3 x 10"4.

Their détermination from asteroid observations by Gaia

The same principle used for the the solar quadrupole J2 détermination (Sect. -2 7 ^r, + 2(7 + 1) (r ' *}y r" *

The PPN parameters 7 or P can be derived from the secular variations of asteroids orbit.s.

Thus, the partial dérivatives to be estimated are expressed from the formulae (

. The choice to estimate particular linear combinations of P and 7, that are T or A, or to consider only the two PPN parameters 7 or P is an option in the implémentation.

Knowing that these parameters are strongly correlated, they hâve to be separately fitted.

The secular variations they produce upon the asteroid orbits can be estmited from the variations of the argument of the perihelion or the mean anomaly. Thus, the partial

where, Uq and Mq are the initial argument of perihelion and mean anomaly in the ecliptic.

The Nordtvedt effect 77

The Principle of Equivalence, which States the identity between the inertial mass rn, and the gravitational mass mg, is the basis of General Relativity. The inertial mass mi measures the résistance of a body to a change of its motion, whereas the gravitational mass mg scales the magnitude of the interaction due to a gravitational field, as we can see in the équation of motion, m, * a = mg * g where a is the accélération of body. More precisely, The fundamental principle of the Universality of Free Fall (UFF) says that : "the world line of a freely falling test body is independent of its composition or structure" (this is the weak équivalence principle). In Newtonian language, any test body falls with the sanie accélération in a given external gravitational field. Thus, the question made scientists wonder if the gravitational energy which binds the constituent éléments of the body (neutrons, protons and électrons) together, through the strong and electromagnetic forces, affect the gravitational and inertial masses in the same way. A négative response would be a violation of the Strong Equiva lence Principle and would imply that. a test or massive body in a gravitational field would not move on a géodésie. This predicted departure from a géodésie was first envisaged by Dicke [START_REF] Dicke | Remarks on the observational basis of general relativity[END_REF] but it is Nordtvedt, who independently discovered and studied it in great details [START_REF] Nordtvedt | Equivalence principle for massive bodies. I. Phenomenology[END_REF], [START_REF] Nordtvedt | Equivalence principle for massive bodies. II. Theory[END_REF], [START_REF] Nordtvedt | The fourth test of General Relativity[END_REF] . Thus, the Nordtvedt effect hypothesis is expressed through the équation,

mg_ mi 1+7] e9 mrc2 1 + T]Aq (4.10)
where, rj is a dimensionless constant, called the Nordtvedt parameter, and c the velocity of light.

mr is the rest mass (inertial mass when it is in rest), mr = J p(x) d3x

with p(x) the mass density of the body.

Eg is the body's self gravitational energy, In Nordtvedt's paper [START_REF] Nordtvedt | Equivalence principle for massive bodies. I. Phenomenology[END_REF], the correction terni A0 is estimated from a solar density distribution fitted to standard star model (Schwarzschild [START_REF] Schwarzscliild | Structure and évolution of the stars[END_REF]), p(x) = e~4x/ae, and would be equal to, 15 GMq

4 ae c2 -0.8 x kt5 A© -
where ae is the solar radius. Nevertheless, the value taken for A© is obtained from 2.E" and, A© = 1.59 x 10"5

This is the value that we considered for our simulations. We can easily refind the standard déviation for the Nordtvedt parameter for an other value of A© that we note A'., by multiplying it with the ratio of A©/A© knowing the exact value is not really known.

The first test devised by Baron Lorand von Eôtvôs and redesigned by the Princeton group of Dicke, consisted in researching the accélération différence between two bodies by balancing the accélération of one against the other. A précisé description can be found in the book entitled "Gravitation" (p. 15) [START_REF] Misner | Gravitation[END_REF]. They found that the gravitational accélérations on gold or aluminum are equal to 10~1] or better. Nevertheless, the objects tested in laboratory hâve a ratio gravitational self-energy Eg -rest mass mr too small, \Eg/mr\ < KG27, for having a chance to observe a violation of the Principe Equivalence, or more exactly the "weak" Principle of Equivalence. This last is applied only to the "laws of motion of freely falling particles" contrary to the strong principle, which is a généralisation 4.2.

The relativistic parameters 129 at ail physical Systems. On the other hand, the ratio \Eg/mr\ is much larger for massive astronomical bodies, such as the Sun with \Eg/mr\ ~3.6 x 10~6, 10~8 for Jupiter and 4.6 x 10"10 for Earth [START_REF] Will | The Confrontation between General Relativity and Experiment[END_REF], Therefore, the experiments for test.ing Nordtvedt effect were applied to the most known both astronomical bodies, Earth and Moon, knowing that in the case where the Nordtvedt parameter is non zéro, Earth should fall toward the Sun with an accélération different from that of Moon. The effect would be a polarisation of the Earth-Moon orbit, finding expression in range oscillation of the distance Earth-Moon [START_REF] Will | Theory and experiment in gravitational physics[END_REF],

ôr(t) = 13.1 T] cos(cuo -ojs)t [métré]
where cuo and ujs are the angular frequencies of the orbits of the Moon and Sun around the Earth. Thus, since the first measure (1969) of the round-t.rip travel times of laser puises between Earth and Moon, this method has remained the most précisé for the Nordtvedt parameter estimation. The précision currently reached with the Lunar Laser Ranging experiment is: |r;| = (4.4 ± 4.5) x 10-4.

Other experiments were elaborated but they proved to be less précisé than the previous one. In 1992, Plastino and Vucetich suggested the détermination of the Nordtvedt param eter 77 from radar observations of 2:1 résonant asteroids with Jupiter, which were predicted to be very sensitive to a non-zero of 77 [START_REF] Plastino | Résonant, asteroids and the équivalence principle[END_REF]. Nordtvedt suggested in his first paper [START_REF] Nordtvedt | Equivalence principle for massive bodies. I. Phenomenology[END_REF],

the observations of trojan asteroids could be a possible experiment: the amount of the angular movement toward Jupiter of the Lagrangian points L4-L5 due to this effect was estimated, 86 ~-A0 ~l" for 77 ~1 [START_REF] Nordtvedt | Equivalence principle for massive bodies. I. Phenomenology[END_REF] O Thus, Orellana and Vucetich fitted this parameter from observations of 6 trojans [START_REF] Orellana | The Nordtvedt Effect in the Trojan Asteroids[END_REF].

Satura having a great influence on the geometry of these Lagrangian points, its mass was also fitted. The value found by the authors was: 77 = 0. ± 0.5. However, the number of observations of asteroids was from 72 to 196 distributed over a minimal period of 64 years, reducing greatly the validity of the dynamical model. With regards to the Gaia mission, about 350,000 asteroids will be observed over a period of 5 years with a mean précision of the order of 1 mas, on each of the ~70 observations to be carried out over this period. One can then hâve good expectations on the capability of Gaia in testing again this effect.

For the variance-covariance analysis, the perturbing force Fr/ due to this effect is considered only through Jupiter, because it is too small for the other planets. Thus, it can be expressed as, F!' = -GMj t, AQ ^(4.12) kjl3 where, Md is the mass of Jupiter and r , its heliocentric position. Therefore, the variational équations is written from the following expressions, Several methods exist for constraining the magnitude of G/G. Among which, the LLR measurements provides the best estimate if we consider the précision of measures and the validity of the model. The best resuit gives: G/G = (4 ± 9) x 10"13 yr~l [START_REF] Williams | Progress in Lunar Laser Ranging Tests of Relativistic Gravity[END_REF]. In Will's paper [START_REF] Will | The Confrontation between General Relativity and Experiment[END_REF], numerous methods are mentioned, such as the study of the Big-Bang Nucleosynthesis, the Helioseismology, the pulsar timing data and the planetary radar-ranging measurements. This latter allowed the estimation of G/G to (2 ± 4) x 10"12 yr~l [START_REF] Hellings | Experimental test of the variability of G using Viking lander ranging data[END_REF] from the observations of Mars (Viking landers, Marineer 9), Venus and Mercury (radar measurements) and LLR measurements. The authors mentioned that the major uncertainty in their détermination was from the uncertainty in the masses of the asteroids. Even if the results from LLR measurements are much better, Will [START_REF] Will | The Confrontation between General Relativity and Experiment[END_REF] thinks that the improvements in the knowledge of the asteroid belt combined with radar observations of planets and spacecraft would give the possibility to attain the précision of 10"13 yr~l. Thus, the asteroid mass détermination from Gaia data will be significant in improving this précision. In addition, G/G will be also evaluated by least-squares fit of ail asteroid that Gaia will observe. The next section provides the preliminary resuit on the expected précision.

The modelling of the variability of G is made through the replacement of G by an effective constant of gravitation Geff -G T G/G(t -to) in the équation of motion (to is the reference time). However, not having the proof of the existence of G/G, this latter Writing of the analytical partial dérivatives

In the literature, we can find some formula about the secular variations of the osculating éléments due to the gravitational "constant" decreasing in time, such as the semi-major axis a, the mean motion n and the mean longitude À. They are based on a property demonstrated by Mestschersky in 1893 [START_REF] Mestschersky | Ein Specialfall des Gyldén'schen Problems (A. N[END_REF], [START_REF] Mestschersky | Uber die Intégration der Bewegungsgleichungen im Problème zweier Kôrper von verânderlicher Masse[END_REF], which says that the classical two-body It would be interesting to use an approximation of the secular effect due to G for estimating its value from the future Gaia observations. Therefore, we can consider the variation in time of G (Eq. 4.13) through a force which would proceed from the potential, uô = G(t -to) (4.14) To find the formula for the first-order secular perturbations in the osculating éléments, we hâve to average the potential Uq over the fast changing angular variables for removing the periodic effects due to 1/r. Thus, the average potential can be expressed as, The resuit is direct if one considers in a first time that G(t -to) is constant. In fact, our period of fit. is about 5 years and a recent estimation of G/G gives: (4. ± 9) x 10"13 by a year [START_REF] Williams | Progress in Lunar Laser Ranging Tests of Relativistic Gravity[END_REF], which is very small in comparison of r. Thus, by applying the effective general averaging principle with slowly varying parameters [START_REF] Arnold | Mathematical methods of classical mechanics[END_REF], which is very effective but of which the field of use is not clearly defined yet, we obtain, Pù\ ~-(4-16)

We next inject, this average potential in the Lagrange's planetary équations. When seeing the form of these équations, we can remark that the variation in time of G modifies the semi-major axis, eccentricity, argument of perihelion and mean anomaly. For the three first osculating éléments, the force is followed by a periodic function, which tends to minimise the secular effect of the force. On the other hand, the expression for the mean anomaly is written as, dM _ 2 d[Uô\ dt na da (4.17) We guess that the major secular effect on the osculating éléments will occur on the mean anomaly. Thus, the variation of the rectangular coordinates of ast.eroids at the time t with respect to G is, the other listed in this section. In this case, the corrélations are also indicated.

The solar quadrupole moment J2 The variation of the gravitational constant G/G Planetary ephemerides

(1.9 ±0.16) x 10-7

~8.4%

[']

1 Standish, pers.comm.

Prospectives

The interesting parameters to be fitted with the future Gaia data from solar System obser vations seem to be the PPN paramater 0, the variation of G and the Nordtvedt, parameter r/. Eveil if the expected précisions from Gaia are inferior to the best measures as shown in Table 4.8 2 *, the method used from asteroids positions is new in view of the size of the sample. Furthermore, the Gaia précisions remain close enough to the best ones. However, regarding 0, we can see in Table 4.3 that, its corrélation with the solar quadrupole moment J2 is high (p=0.65) and conséquently, the derived value will dépend on strongly of this latter. Moreover, the solar J2 détermination seems to be difficult to estimate with Gaia since the expected uncertainty is superior to 95%. It will be préférable to fix its value from the results of the future missions such as LATOR, predicting an accuracy of 10~8.

Furthermore, these missions will end before the final Gaia catalogue. As to the Nordtvedt parameter, the predicted précision appear the most promising since it is almost equal to the best measures from LLR experiments. Nevert.heless, the estimation of this parameter 2However, the Gaia stellar observations will give the best précision never reached, in the 7 détermination from the bending of light experiment.

Chapter 5

The Yarkovsky effect Moscow-Brest (Beekman [START_REF] Beekman | Yarkovsky and the Discovery of 'his' Effect[END_REF]). In his spare time, lie worked on physical problems and in a pamphlet dated from 1901 [START_REF] Yarkovsky | The density of luminiferous ether and the résistance it offers to motion[END_REF], he wrote its remarks about the possible effect of the diurnal heating of a rotating small object in space. However, it was Ernst J. Opik (1893[START_REF] Opik | Collision probabilities with the planets and the distribution of interplanetary matter[END_REF], who around 1909, read the Yarkovsky's work and more than forty years later brought it out of oblivion by discussing its possible importance for orbital motion of meteoroids [START_REF] Opik | Collision probabilities with the planets and the distribution of interplanetary matter[END_REF]. Why study the Yarkovsky effect? The Yarkovsky effect is thought to play a rôle in shaping some characteristics of the asteroid dynamics (Bottke et al. [START_REF] Bottke | The Effect of Yarkovsky Thermal Forces on the Dynamical Evolution of Asteroids and Meteoroids[END_REF]). In the case of the origin of NEOs, the Yarkovsky effect is responsible for delivering asteroids with diameters smaller than 20 km from their original orbits into the powerful résonances which eventually modify their orbits into Earth-crossing ones (Morbidelli and Vokrouhlickÿ Description of the Yarkovsky effect 138 [START_REF] Morbidelli | The Yarkovsky-driven origin of near-Earth as teroids[END_REF]). This scénario has previously hypothesised by Peterson [186] in 1976 and Afonso et al. [START_REF] Afonso | Asteroid fragments in Earth-crossing orbits[END_REF] in 1995. Thus, the Yarkovsky effect is claimed to be an important cause of the dispersion of asteroid dynamical families. In addition, a complementary effect, namely the so-called YORP, is also capable to change the spin rates and spin axis of irregular asteroids with a diameter not larger than few kilométrés. Asteroid rotation rates are spin up our down by YORP, this effect could explain the large number of asteroids with a very fast and very slow rotational speed (Kaasalainen et al. [START_REF] Kaasalainen | Accélération of the rotation of asteroid 1862 Apollo by radiation torques[END_REF], Lowry et al. [START_REF] Lowry | Direct Détection of the Asteroidal YORP Effect[END_REF]).

What will Gaia be able to do? A measurement of the Yarkovsky accélération allows one to dérivé some physical parameters of asteroids. In this chapter, we présent the results of a variance-covariance analysis performed from realistic simulations of observation dates by Gaia for the 4599 known NEAs as of April 2007. Among them, only 1368 NEAs will be observed at least one time by Gaia. However, we hâve retained the asteroids observed more than 10 times since the orbit is not constrained with few observations. On the other hand, we will evaluate when the Yarkovsky effect needs to be included in the dynamic model (though not in the inversion procedure). If it is détectable by Gaia, we will try to dérivé the physical parameters. The details of the on-going work are given in Sec. 5.4.

Description of the Yarkovsky effect

The Yarkovsky effect is the change of the orbital éléments of asteroids and meteoroids, mainly their orbital semi-major axis, due to anisotropic reemission of heat (thermal infrared photons) received from the Sun in the visible. The infrared photons, when they leave the body, carry away momentum. The momentum carried away by thermal infrared photons is stronger in the directions of the warmest hemisphere. The asteroid expériences for each re-radiated photon a force directed in the inverse sense of the re-emission. In the case where there is no thermal inertia, the température distribution is symmetrical about the sub-solar point and the force due to heat émission is thus radial and directed perpendicular to the orbital velocity vector outward the asteroid orbit. However, thermal inertia of asteroids is never null, causing a lag in the re-emission of photons with respect to their absorption. Therefore, the force is not solely radial, and secular variations in the orbital semi-major axis and to lesser degree in the eccentricity appear. The latter variations, over a long period, can hâve non negligible effects in the orbit of some asteroids.

We can separate the Yarkovsky effect, for convenience, into a diurnal and a seasonal component. The diurnal effect is due to a force in the plane perpendicular to the spin axis. It differs from the seasonal component by its dependence on the body's spin rate and consequently on the longitudinal température distribution. Because of the re-emission delay of the photons r due to the thermal inertia and the rotation of the body, the angle between the incident photons and the re-emitted ones becomes non-zero. This effect is dominant in comparison with the seasonal effect for bodies larger than ~100 métrés and with a rotation of the order of the thermal relaxation time r. By définition, the diurnal component of the Yarkovsky effect reaches its maximal intensity when the spin axis of the body is perpendicular to the orbital plane. On the other hand, this effect becomes small when the rotational speed is high so that irradiated part by the Sun does not hâve the time to cool down during the night side and the longitudinal température distribution tends to be constant along the longitude, leading to a longitudinal uniform diffusion of photons. The major effect of the diurnal component of the Yarkovsky effect is a variation of the orbital semi-major axis which increases when the rotation is prograde and decreases for rétrogradé rotators (Bottke et al. [START_REF] Bottke | The Effect of Yarkovsky Thermal Forces on the Dynamical Evolution of Asteroids and Meteoroids[END_REF]). The net force due to the seasonal component. of the Yarkovsky effect, along the plane containing the spin axis, dépends on the body's mean motion and thermal inertia (its latitudinal température distribution). Because of the thermal relaxation time r, the pho ton are re-radiated in the sense of the orbital motion, resulting in a recoil accélération which has a component along the orbital velocity vector. Thus, the semi-major axis of the body decreases, and the object spirals inward to the Sun. For asteroids with size >100 métrés, the diurnal component dominâtes the seasonal one which, by définition, vanishes when the spin axis is perpendicular to the orbital plane. However, the seasonal effect can become important when the conditions miniinising the diurnal effect appears: the thermal relaxation time r much longer than the rotational period of the body or the spin axis in the orbital plane.

Model of the net force due to asteroid thermal émission

The modelling of the force for the numerical intégration was a part of a Master training period carried out by M. Havel [START_REF] Havel | Possibilits de dtection de l'effet Yarkovsky avec la mission Gaia[END_REF] under the supervision of A. Morbidelli and with the active collaboration of M. Delbo. The model also required the création of a database of orbital and physical parameters for each NEO: physical characteristics play a rôle in the Yarkovsky effect. The force was written on the basis of Vokrouhlicky's papers [START_REF] Vokrouhlickÿ | The Yarkovsky Seasonal Effect on Asteroidal Fragments: A Nonlinearized Theory for the Plane-Parallel Case[END_REF], [START_REF] Vokrouhlickÿ | A complété linear model for the Yarkovsky thermal force on spherical asteroid fragments[END_REF] who uses a linear theory of the Yarkovsky effect. The force is given in a frame centred on the asteroid (X, Y, Z) suitable for expressing the two components of the effect, the diurnal and seasonal ones. The axis Z coincides with asteroid's spin axis, direction according to which the seasonal effect, independent of the spin rate, acts through the projected force fz The diurnal effect. is given by a force in the équatorial plane (X, Y). The axis X is defined such that the position vector Sun-asteroid is contained in the plane (X, Z). The axis (X, Y, Z) can be written in the heliocentric référencé System, and consequently the force from the Yarkovsky effect. $ is the radiation pressure parameter on a spherical body (Burns et al. [START_REF] Burns | Radiation forces on small particles in the solar System[END_REF]), insolation with re-radiated thermal radiation (Lebofsky and Spencer [START_REF] Lebofsky | Radiometry and thermal modeling of asteroids[END_REF]), 7r R2aS = ijecrR2 T4 (A, 5) cos <5 dS d\ (5.4) where, o: is the absorption coefficient, that we write (1 -A) with A the bolometric Bond albedo of the body (specifying the fraction of the total incident radiation energy reflected by the asteroid).

e is the emissivity of material, defined as the ratio of energy radiated by this latter to energy radiated by a black body at the same température.

a is the Stefan-Boltzmann constant.

T(A, 5) is the température model at longitude A and latitude <5. The parameter r/ is a scaling factor to match observed fluxes. Its necessity is detailed in Spencer et al. [START_REF] Spencer | Systematic biases in radiometric diameter déterminations[END_REF].

The subsolar equilibrium température Tss is obtained from Eq. 5.4 assuming a smooth non rotating body p -1 in the Standard Thermal Model [START_REF] Lebofsky | Radiometry and thermal modeling of asteroids[END_REF] and the following température model assuming that the surface is in instantaneous equilibrium with insolation [START_REF] Lebofsky | Radiometry and thermal modeling of asteroids[END_REF],

T(\,6) Tss cos1/4 À cos1/4 ô for À < 7r/2 0 for À > 7t/2

From the Eq. ( 5.4) and the température model (5.5), we obtain

T.* = 1/4
(5.5)

(5.6) 6 is the solar colatitude (Fig. 5.4) at a given time, the angle between the spin axis and the direction to Sun from the asteroid.

0 is the thermal parameter, interpreted by Farinella et al. [START_REF] Farinella | Meteorite Delivery via Yarkovsky Orbital Drift[END_REF] as the ratio of the température relaxation time to the rotation period, 0 = r y/qj where T is the thermal inertia defined as, r = (p, K)1'2

(5.7)

with ps the bulk density of the body of the surface, C its spécifie heat (the ratio of the amount of heat required to raise the température of a unit mass of a substance by one unit of température) and K the thermal conductivity. It is expressed in SI = J.ra~2.s-1/2.K~l.

These quantities enter in the expression of the équation describing the heat conductivity (Landau and Lifchitz [197]),

» c f =

This latter is solved by Vokrouhlickÿ to compute the force which causes the Yarkovsky effect, under the assumption that the température distribution on the surface of the spherical asteroid can be written in ternis of a perturbing function plus a constant average value [START_REF] Vokrouhlickÿ | The Yarkovsky Seasonal Effect on Asteroidal Fragments: A Nonlinearized Theory for the Plane-Parallel Case[END_REF].

R' is the radius of the body scaled tration depth of the thermal wave, \psCuJ K\, K2, k3 are functions of R'. In our model we assumed «r = K2 -«3 =0.5. This is suited for bodies larger than some hundreds métrés (Vokrouhlicky [START_REF] Vokrouhlickÿ | The Yarkovsky Seasonal Effect on Asteroidal Fragments: A Nonlinearized Theory for the Plane-Parallel Case[END_REF]).

The Yarkovsky effect Fy in the heliocentric ecliptic coordinates System is obtained by, The component fz is not the true component of the seasonal effect. This is because the component of the force along the spin axis is computed using a colatitude of the Sun 6 estimated punctually, whereas in the case where the seasonal effect dominâtes, Vokrouhlicky suggests to consider a variable colatitude over an average value. In our case, the above représentation of the seasonal component of the Yarkovsky effect works fine, because the diurnal effect is always dominant above the seasonal effect..

Variance-covariance analysis

From the previous sections describing the force proceeding from the Yarkovsky effect, the most influential parameters which can be retained are the thermal inertia F of the asteroid (Eq. 5.7), its diameter d = 2R, and the three components s = of its spin axis in our reference frame. Therefore, the variance-covariance analysis was independently led for each previously mentioned parameters because of their strong corrélation as resuit of their belonging to the same force and so of their numerous common parameters.

As for the mass détermination and the others global parameters of the Chapter 4, the fit method based on the variational équations was used. This time, we spare the reader detailed équations which are long enough partly due to the fact the force is hrst expressed in a reference frame proper to the asteroid and evolving with time.

The simulation is concerned with 1386 near-Earth asteroids for which observation dates were made from the realistic simulations described in Section 3.3. At the start, the checked set of NEAs was composed of 4599 known objects. The Table 5 First, we présent in Table 5.2 the number of NEAs in each class of the relative précision for the üve tested parameters. In Tables 5.3 and 5.4 are listed the objects with at least one parameter derived with a relative précision smaller than 10%. For each object is specified the number of Gaia observations, the relative précision of each tested parameters as well as the semi-major axis, eccentricity, diameter and thermal inertia of the asteroid. Some éléments of comparison were aclded with the Figure 5.5 giving the distribution of the 666

NEAs (number of observations > 10) according to their semi-major axis, and Tables 5.5

and (5.6) providing the distribution according to the diameter and the thermal inertia.

Comments

Table 5.2 illustrâtes well the difhculty to detect the Yarkovsky effect which in general is relatively small in comparison with the Gaia précision: out of 1386 NEOs, only 14 would be sufhciently sensitive to this effect for allowing a fit of their diameter with a relative précision below 50% in the case where no systematic errors occur. In Table 5.3, we can notice that these 14 selected NEOs présent similar characteristics: the semi-major axis is rarely superior to 1 AU, the solar radiation flux being thus strong during some periods of time. Moreover, the diameter is small enough and does not exceed 1.5 km; for the NEAs with the lesser observations (2000 WN 10, 2005 WK4), the diameter is about 0.25 km and belong to the 35 out of 459 possessing a diameter smaller than 0.5 km (see Table 5.5). In the saine way, the thermal inertia F (at the origin of the delay of re-emission of photons) is never weak (see Table 5.6), it. is very often superior to 300 SI. Among the asteroids selected in Table 5.3, some seem to be more interesting: The asteroid 2000 WN10, combining a small semi-major axis and diameter with an important thermal inertia, would

give with only 17 observations the opportunity to fit separately five physical parameters with précision lower than 25 %. 2062 Aten is the second asteroid in view of the précision on the diameter and the thermal inertia too.

These results are encouraging to keep on the st.udy of this effect in the Gaia context.

Keeping in mind that Gaia will observe ot.her unknown NEAs and maybe among them, an asteroid will combine exceptional characteristics, orbital and also physical, rnuch better than those shown in Table 5.4. Furthermore, certain physical parameters seem to be linked by empirical law such as the diameter d and the thermal inertia F. Thus, in Delbo et al.

paper [START_REF] Delbô | Thermal inertia of near-Earth asteroids and implications for the magnitude of the Yarkovsky effect[END_REF], this law applying to NEAs, can be expressed to the first order as, T = g d~t (5.9)

where g -300 ± 47, £ = 0.48 ± 0.04, d is written in km and T in SI. Nevertheless, the work to be accomplished in collaboration with M. Delbo remains important and is detailed in the following section. Semi-major axis a

Perspectives

As said in Section 5.2, the modelling of the seasonal effect applied here is not a good approximation and it would be necessary to refine its expression in the perturbing force knowing that Gaia will discover other NEAs, which would be dominated by this effect.

Furthermore, the Yarkovsky effect is not the only non-gravitational phenoinenon which could hâve an significant influence on the orbit of some asteroids. It would be interesting to complété the list of these non-gravitational forces, to study the ability of Gaia to detect them and the possibility to fit some physical parameters. As mentioned in the Chapter 3, the correctness of the dynamical model is fundamental for the dérivation of these parameters, however the uncertainties of some data, such as the positions of planets, the masses of some main-belt perturbers in view of the high eccentricities of NEAs (see Table 5.4), the physical data of asteroids (shape, density, thermal inertia, spin rate and axis ...) could hâve an important impact on the accuracy of results and especially when the studied effect is small. Therefore, Monte-Carlo simulations for estimating the systematic errors due to these uncertainties seems to be inescapable. In addition, the most of these asteroids combine small orbital semi-major axes with high eccentricities, and thus are very sensitive to relativistic effects and to the solar quadrupole moment. Therefore, the fit of some parameters appearing in the expression of previously mentioned forces will probably require to include the Yarkovsky effect for some asteroids in order to obtain good (O-C).

a précision of 7 métrés. In 1989, the mass of the three bigs were determined by Standish and Hellings [32].

At présent, the list of Williams [START_REF] Williams | Determining asteroid masses from perturbations on Mars[END_REF] has remained a reference. The planetary ephemerides makers always take into account these asteroids and lead studies about their influence on the planetary solutions, and by extrapolation on the planets orbits. In 2003, Standish and Fienga [14] extended this kind of study to the four inner planets. Thus, The following studies to detect the most influent asteroids upon the Mars orbit over a short period (~50 years) were led from the orbital osculating éléments of 2010.5 derived from Astorb data base. 2218 asteroids were selected for this study among them we hâve the most massive asteroid completed with those of which the IRAS diameter is superior to 10 km.

Monte

A revised list of Williams

Our analytical investigation was led in the spirit, of Williams paper [START_REF] Williams | Determining asteroid masses from perturbations on Mars[END_REF]: from a software developed by Simon [START_REF] Simon | Computation of the first and second dérivatives of the Lagrange équa tions by harmonie analysis[END_REF], [START_REF] Simon | Perturbations du premier ordre et dérivées premières des équations du mouvement pour un couple de plantes: formulaires et programmes de calcul[END_REF], which permits to express the theoretical perturbations Aak of an asteroid (to the first order in mass) on the six Mars equinoctial éléments (Jk=i,..6 -(a, A, k, h, q,p) during 1000 years. The method is based on a harmonie décom position with respect to the mean mean longitude of Mars X and the mean mean longitude of the asteroid A', a being the orbital semi-major axis, A = w + u with w the longitude of perihelion and v the true anomaly, k = ecosro, h -esinw, q = sin(î/2) cos 17, p -sin(i/2) sin 17, and 17 the longitude of the ascending node, e the eccentricity and i the inclination.

Acrfc can be written by the formula, A<r*(Â, X') = crskect -I-^a^j cos (iX + jX') + sin (iX -I-jX') i,j

We hâve applied this décomposition for each perturbation of a large number of massive asteroids, and hâve focused on those which act upon the semi-major axis and longitude of r -i 1/2

Mars. We hâve the asteroids by selecting first the two amplitudes c*) = K)2+($)2 The two thresholds on the amplitudes, ca and c\, are derived with the same filter on the period.

In Tables 6.1 and 6.2, we give for each selected asteroid, the number of coefficients obtained in the expression of their perturbation on the semi-major axis Aclm and the longitude AAm of Mars for two filters on the period P < 10 years and P < 50 almost the same asteroids among the most perturbing ones. The more interesting is that a great majority will hâve a mass estimated by Gaia with often a good précision and better than presently known. Because of the planetary perturbations, the orbital semi-major axis of asteroids can vary in an important way, and consequently change their mean motion. Therefore, certain amplitudes associated to a particular combination of their longitude with that of Mars can increase brutally: showing they are likely to lie in an area where the résonances are close each other and thus, the transition of one to another very easy. We led several harmonie décompositions similar to the previous ones by sweeping the range of possible semi-major axes for each asteroid due to the planetary perturbations, which was obt.ained by numerical intégrations over a period of 100 years. 30 new semi-major axis were tested for these last. The Table 6.4 lists the asteroids for which we obtained great variations in the amplitude, already strong in the reference case, with the number of the amplitudes concerned for the semi-major axis and the longitude of Mars. We found 13 asteroids, three of which (Pallas, Irene and Bamberga) are listed among the rnost important perturbers of Mars (Tabs. 6.1, 6.2). This table gives a glimpse of some asteroids which could pose a problem for the fit of the Mars orbit if their orbit is itself not well-defined. Of course, the study does not take into account ail other possibilities of résonances, in particular with Jupiter, which could hâve similar conséquences and maybe to shift or accentuate possible résonances with Mars. Thus, the contribution of Gaia for asteroid masses will not solve the problem entirely; the good fit of the orbit of certain asteroids will be probably at least as important as the précision on their masses. provide precisely the impact of a perturbation in many cases. Nevertheless, the encounter velocities of our selected asteroids with Mars are relatively high in comparison with those between asteroids, and the impulse approach proved to give good approximations for this kind of encounters. Thus, from the formula (2.9) (Sec. 2.1.1),

9\ = tan 1 /JAVm|\ V |VM| )
where, AVm is the variation vector of the velocity of Mars due to a gravitational pertur bation and Vm its heliocentric velocity. The sum of the multiple deflections of the Mars orbit ^2,91 caused by each asteroid is computed and used to select the most perturbing asteroids. Thus, in Table 6.5, the asteroids are listed according to three levels of pertur bation intensity. Among the first two groups bringing together the asteroids producing the largest deflections, we meet a lot of asteroids listed in Tables 6.1 and 6.2 (t.heoretical perturbations by harmonie décomposition).

In addition, we give in Table 6.6 a list of minimal distances between these asteroids and Mars over the saine period, in order to detect those for which a large uncertainty on the mass has important conséquence on the Mars orbit over a short period. It, is interesting to note that the most minimal distance found is 0.25 AU over 50 years. It is small but not small enough to hâve to consider the mutual perturbations of asteroids in the computation of their positions required by the sélection software. Some asteroids appear several times in Table 6.6, such as 654 Zelinda, 7 Iris, 324 Bamberga, which increases their importance during this 50 years to corne. We find again the asteroids already revealed by the harmonie décomposition (Sec. 6.1.2) as very perturbing, which become inescapable asteroids for the improvement, of the Mars orbit solution. Furthermore, it is important to notice that for many of these asteroids, the mass will not be or not precisely derived: as a conséquence, we will hâve to search future important close approaches for deriving their mass from ground-based observations.

The secular effect

In this part, we intend to study the secular effects of asteroids on the orbital semi-major axis, eccentricity and inclination of Mars. Thus, we dérivé 100 values of these three Mars osculating éléments uniformly distributed on a period of 50 years by numerical intégration by considering different dynamic models of Mars. The reference orbit considers only the perturbations of the others planets while for each test orbit is added the gravitational perturbation of one of the tested asteroids. From a linear régression, the three slopes obtained from 100 values of three osculating éléments are compared to that of the reference orbit in order to spot the biggest différences. The results are shown in Table 6.7 with the ranking of the asteroids with respect to the biggest différence with the reference solution for each one of the three osculating éléments of Mars: the semi-major axis a, the eccentricity e and the inclination i. For each one, we select the 40 first asteroids (arbitrary number) but in verifying that the ratio with the biggest différences is inferior to 100. We find the three bigs are always at the top of the table. For the others, the ranking can be different enough for each osculating éléments but however, a majority of asteroids belong to the three columns and many already appeared in the results of the previous sections. détermination with a précision better than 50%. In addition, the perturbations of asteroids do not only concern the planet Mars, but ail inner planets: the Earth, Venus and Mercury will benefit from the masses that Gaia will dérivé knowing that in the future, we will hâve more précisé observations of these planets and the modelling of their orbits will greatly dépend on our knowledge on the dynamics of the asteroids.

Chapter 7

The physical point of view

The bulk density

As mentioned in Section 1.2.1, the Gaia mission will also play an important rôle in the knowledge of the physical properties of asteroids. The masses will be a determining factor for the estimation of densities, which is the key of numerous scientific investigations on the history and the évolution of asteroids, as well as their internai structure. Furthermore, the rnean bulk densities, usually estimated for three taxonomie classes -in planetary ephemeris studies-, is at présent, the only mean to estimate the unknown masses from an estimation of the volume. These as seen in the previous section, are essential to dérivé very précisé ephemerides of some planets such Mars, Earth etc... at the current, précision level of observations. When looking at IRAS database in details, 617 asteroids arc listed with a diameter d > 50 km (~10"13Mq assuming a bulk density of 2.5 g.cm-3) and 217 with d >100 km (~6.6 x 10 13M0), one guesses the importance to evaluate the level of confidence we can attach to this hypothetical relationship density-taxonomy and maybe to expand it. to other classes. Thus, in an indirect way, Gaia could allow to bring many masses outside those from direct measurements, and thus to fulfil a breakthrough in planetary ephemerides.

Accurate density

Gaia, through précisé mass déterminations, will increase signifîcantly the précision on the bulk density of the asteroids for which the diameter and possibly shape are well-derived.

Table 7.1 was drawn to illustrate the previous words. An analysis is performed between every currently known masses (see Sec. 1.2.4, Table 1.4) of asteroids with a well-derived diameter, and the masses which will be estimated from the Gaia data (see Sec. 3.6.2, Table 3.6). The SMASSII taxonomie class (Bus and Binzel [START_REF] Bus | Phase II of the Small Main-Belt Asteroid Spectroscopic SurveyThe Observations[END_REF], [START_REF] Bus | Phase II of the Small Main-Belt Asteroid Spectroscopic SurveyA Feature-Based Taxonomy[END_REF]) of each asteroid is listed as well as the mass m, the IR AS diameter d 1, and the associated standard déviation

For 253 Mathilde, we used the NEAR Shoemaker data (Thomas et al. [START_REF] Thomas | Mathilde: Size, Shape, and Geology[END_REF])

7.1.

The bulk density 166 a(d). The density p is given for a spherical asteroid with the associated errors Ap and A'p. These last are respectively computed with the standard déviation on the masses referenced in Table 1.4 (known masses) and in Table 7.1 (expected précisions with Gaia).

More Precisely, the density error Ap is usually estimated from the formula (Birlan [START_REF] Birlan | Dynamic and Physical Considérations on the Asteroids Density[END_REF]), Am rriAV m

Ap= -+ -ÿï-Wlth P=v (7.1) where Arn is the 1-sigma on the mass, m the mass referenced in Table 3.6, V the volume of the asteroid and its error AV obtained from the 1-sigma on the diameter cr(d),

AV -7T v(d) 2 d2
Except for the binary Systems (marked by **) for which the mass and shape are very well-derived from the adaptive optic data, the asteroid 253 Mathilde visited by the space probe NEAR Shoemaker, 11 Parthenope and 2 Pallas, ail the others densities (22 out of 32) will be improved by the mass détermination with Gaia, among which, 8 déterminations will be more précisé by a factor greater than 2.

Gaia will also bring a numerous new masses for which bulk densities will be estimated with a relatively good précision. Therefore, in Table 7.2, we hâve established the statistical distribution of the précision on the densities from the available IRAS diameter and the Gaia expected précision on the masses listed in Table 3.6 (Sec. 3.6.2). We recall that the used reference masses for which any direct measure was obtained with a précision better than 10% -from ground-based observations-, are estimated for spherical objects with a density of 2.5 g/cm?. When seeing the density error formula (7.1), we guess the prépondérance of the uncertainty on the diameter for most of the density estimations, which dépend on the inverse of the cube of diameter. Therefore, we added a supplementary information in parenthesis in Table 7.2 which spécifiés the number of asteroids for which the uncertainty on the diameter dominâtes the error on the mass according the formula (7.1). An important number of précisé densities will be provided by Gaia: in Table 7.2, about 37 densities are expected with a relative précision lower than 20% and 96 until the 50% mark. For the more précisé ones, we notice the importance of the uncertainty on the diameter in the density error: it is dominating for over half of the déterminations under the 20% mark. Therefore, the knowledge of the asteroid masses from Gaia will be increased in scale when it will be associated with future missions or ground-based observations devoted to the imagery of the asteroids shape. Thus, the mass associated with the volume,-which will be much better determined-, will allow a good approximation of porosities, a precious due for the investigation of their collisional history. Table 7.3 lists the ranked expected densities of 10 to 20% (knowing the 7 smaller than 10% are in Table 7.1) with the SMASSII taxonomie class of the asteroid, its mass m, its IR AS diameter d and the associated standard déviation cr(d), the expected Gaia error on the mass er(ra) and the density error A'p.

Remark on the density error formula

The previous formula (7.1) used for estimating the density error is not based on a statistical law. However, the error on the masses cr(ra) is given in term of 1-sigma of a Gaussian distribution: this approximation is realistic enough if the problem is almost linear, and observational noise Gaussian. As to the error on the diameter a(d) obtained by the IRAS measurements, it is also given as 1-sigma uncertainty due solely to the précision of the IRAS flux data. Thus, the error in volume can be also described by a normal law with a mean pv = n * d^/ft and a standard déviation cr(V) = 7r/6 * (3d2a(d) + 3da2(d) + a3(d)). 1 The SMASSII taxonomie classification (Bus and Binzel [19]).

As a conséquence, the density being defined by the ratio of two Gaussian variables, the density follows a statistical law, which can be complex in the case of non zéro means of two distributions. In the case of zéro means, the density probability function would hâve followed the general Cauchy distribution, f{%)

1 Q 7T X2 + a2
with a = a(m) for which however, any moment (in particular, mean and standard déviation) is defined.

In our case, the complex formula was given by David Hinkley in 1969 [START_REF] Hinkley | On the ratio of two correlated normal random variables[END_REF], This density probability function does not allow to easily compute the density probability. Therefore, we hâve simulated by Monte-Carlo the densities for a subset of asteroids from Table 7.1. The asteroids with a standard déviation on the mass greater than 30% were excluded in order to limit the number of négative densities in the simulations. For each asteroid, 100,000 densities pc were created.

The first Table 7.4 gives an intuitive représentation of the density estimation and its error when using the formula (7.1). Thus, we hâve listed for each asteroid, the adopted density value p, the mean pp of 100,000 simulated densities as well as the médian po.5-

The two following columns give the density error Ap and the standard déviation a(p).

The two groups of the simulated densities, (A~) and (A+), such as we hâve respectively p -Ap < pc < p and p < pc < p + Ap were compared. The proportion of each group n+ /rit and n~/rit out of the total number of simulated densities is listed, as well as the absolute value of the distance mean between the densities pc and p, We note that for each asteroid, the reference value of the density p is almost equal to the médian po.5 for each asteroid and remains close enough to the mean pp. In addition, the mean intervals pA-, pA+ are very close. Therefore, the error formula (7.1) is a legitimate estimation even if nevertheless, the proportion of both groups A~, A+ out of the total number simulated densities can be very different between each asteroid. As a conséquence, the strict comparison of the different density précisions from the error Ap is not possible. The Table 7.5 is the illustration of the previous conclusion on the ranking from standard density error formula. For each asteroid its adopted density p is written, the relative errors Ap/p in % and (ai, cr2,0-3) which were derived to hâve respectively 68.2%, 95.4% and 99.7% of the computed densities pc contained in the intervals p ± cri, p ± cr2, p ± cr3 by analogy with a Gaussian distribution. We can notice that the ranking of asteroids according to the error Ap and the pseudo-standard déviations cri, <72,<t3 can be different as for the asteroids 3 Juno and 4 Vesta ... From a Kendall's test rank, it is possible to measure the degree of correspondence r between the two rankings according Ap and a\ [START_REF] Stuart | Kendall's Advanced Theory of Statistics: Classical Inference and the Linear Model[END_REF]. Thus, we consider two rankings of two variables x and y. The rank corrélation coefficient r is defined by, where n is the number of samples for x and y and, _ f 1 when (xi -Xj)(yi -yj) < 0 13 1 0 when (x* -xj)(yl -yj) > 0

The Kendall coefficient r is: r = 0.85 with 26 discordant pairs < 0). This resuit indicates a good agreement (1 being the perfect one) between the two rankings but not enough good for obtaining a strict ranking that from the errors Ap. In spite of this latter, the density error formula (7.1) remains a valid error estimation. The previous section illustrâtes the impact of the Gaia mission on the asteroid density déterminations, which will be more précisé but also more numerous. Thus, the spectra of taxonomie classes swept across by these asteroids will be very large. Therefore, Gaia will be an opportunity to truly check the possible existence of a relationship between the asteroid bulk density and certain taxonomie classes. At the présent time, this relation is roughly used for estimating the unknown masses of the asteroids belonging to the taxo nomie classes S, C, and X and for which at least one diameter has been estimated. It is also used in the fitting of planetary ephemerides (see Sec. 6.1). The mean density affected to each class is often derived with a weak number of asteroids, as a conséquence of the actual poorness which prevails in the knowledge of their masses.

Table 7.7 was conceived for catching a glimpse of the importance of Gaia on the vali dation or réfutation of this hypothetical relation, or more exactly on the confidence level to be attached to it for a correct use. This table represents the distribution of the two previous categories of asteroids, those with masses currently known and the future ones obtained by Gaia, with respect to the SMASSII taxonomie classes for three classes of rel ative précision on the mass. We chose the comparison in function of the mass précision and not the density because some asteroids referenced in Table 1.4 (future mass détermi nations from Gaia, Sec. 1.2.4) hâve not a derived diameter yet. However, the diameter déterminations are expected during the Gaia mission and probably, ail the asteroids for which a mass will be evaluated, will also hâve an estimated diameter.

The Bus and Binzel taxonomie classification [START_REF] Bus | Phase II of the Small Main-Belt Asteroid Spectroscopic SurveyA Feature-Based Taxonomy[END_REF], as obtained from SMASSII data, This work has allowed to foresee the scientific richness contained in the dynamics of the asteroids, and the potential of the Gaia mission to reveal it. First, the évolution of astrometry, through more and more précisé measures and the huge quantity of asteroids observed, gave the opportunity to rethink the methods. Thus, the détermination of asteroid masses was envisaged in a new way through a global treatment. The whole set of protagonists involved in a close approach are integrated in the unique matrix of the linearised équations of observations giving a global solution for the mass of asteroids. Therefore, the systematic errors will be greatly reduced thanks to a very complété dynamical model. From realistic simulated data taking into account the observational characteristics of the Gaia spacecraft over a likely period of the mission, it was possible to predict a scale order of the number of masses and their associated standard déviation that Gaia could provide. The results are very encouraging since 149 masses could be estimated with a relative précision better than 50%. Furthermore, the contribution of ground-based observations before and after the mission was investigated for a very small sample of target asteroids for which the Gaia-alone coverage of the encounter with a perturber is insufficient. The above number of masses could be increased by 20%, what would be a non negligible gain knowing that many of these niasses hâve never been measured.

Even if the systematic errors will be much less important with the Gaia data than for ground-based observations, they will not be completely absent. Thus, their estimation will be as important as those of the masses. The définition of confidence intervals of the estimated values will allow to obtain more realistic standard déviations and consequently, better interprétations of results depending on these values. Therefore, we hâve begun to implement the method that we will hâve to use for defining these confidence intervals.

Moreover, Gaia will dérivé the mass of some binary Systems for which a mass is already very well known by adaptive optics, and will allow to assess the importance of these sys tematic errors.

As we saw in the last two chapters, the quantity and the précision of such measures on the masses will be fundamental to improve the ephemerides of the inner planets: 43 perturbers out of the 52 identified as the most influent upon Mars orbit will hâve a mass derived with a précision smaller than 50% by Gaia. Besides, the hypothetical relationship between density and taxonomie class of asteroids could be better tested, and refined know ing that the whole set of asteroids for which a mass détermination is expected, will cover a spectrum of 17 taxonomie classes (frorn Bus classification). Furthermore, these density measurements will allow to estimate the porosity of a large sample of asteroids and will give some information on their collisional history, and on the origin and the formation process in the Solar System.

Another point that Gaia has brought us to rethink is the way to characterise the efficiency of the close approaches for mass détermination. The accuracy of Gaia will allow to study very weak perturbations between two asteroids, which cannot be modelled with the impulse approach représentation. Even if the numerical treatment is the most précisé mean to estimate the interest of an encounter, which in addition allows to consider the distribution and the geometry of observations, the research of an analytical characterisation remains interesting. The possibility to apply analytical solutions of the Hill problem to real cases of close approaches between asteroids seems to be promising .

The number of asteroids that Gaia will observe is so important and the observations so précisé that it seemed interesting to study the détermination of many others parameters of the Solar System in spite of the short period of the mission (5 years). Thus, from simulated data for the Gaia observation dates, the détermination of the solar quadrupole moment In addition, the level of accuracy of Gaia also allowed to think about the détection of the weak non-gravitational force due to the Yarkovsky effect. Thus, by performing an anal ysis on a sample of more than thousand of NEAs, the most sensitive objects to this effect could hâve been revealed, and we hâve found 14 asteroids for which the diameter (which scales linearly with the effect) could be fitted with a précision smaller than 50%. The data from Gaia could allow to dérivé some physical characteristics of certain asteroids solely from the analysis of their perturbations by the Yarkovsky effect. This resuit is sufficiently promising to continue the study by improving the modelling of this force and integrating it in the dynamical model. The future work will hâve to study the different sources of systematic errors in order to dérivé some physical properties of particular NEAs. In ad dition, Gaia will observe new NEAs and probably, some will présent exceptional physical characteristics for this study.

Keeping in mind that the précision on the position of asteroids will probably be better than the law we hâve used and consequently, the results found on the expected précisions will be likely better. Besides, the Dawn mission will give before the launch of Gaia a very précisé measurement of the masses of Ceres and Vesta and thus, will reduce the number of unknown parameters having a great importance on the System of équations, because they are involved in numerous close approaches, and will improve the solution on the other masses, and indirectly on the other fitted parameters.

Figure 2 :

 2 Figure 2: Distribution de la densité des déviations standard sur le demi-grand axe d'un large échantillon d'astéroïdes en fonction de leur demi-grand axe. Il y a 3 niveaux de densité déterminés à partir du tiertile de la densité maximale trouvée parmi tous les éléments de la grille.

Figure 1 . 1 :

 11 Figure 1.1: The Gaia spacecraft.

Figure 1 . 2 :Figure 1 . 3 :

 1213 Figure 1.2: Précision of astrometric catalogues in the course of History. Image based on an original figure by Erik Hpg.

  Photometry

Figure 1 . 4 :

 14 Figure 1.4: The modular concept of the Gaia spacecraft.

Figure 1 . 5 :

 15 Figure 1.5: The payload module: the optical bench and the focal plane.

Figure 1 . 6 :

 16 photometer detectors RVS detectors

  . 1.6 and 1.7) is composed of five distinct areas. The first part is made up of detectors for auxiliary instruments: the basic angle monitoring (BAM) System to monitor fluctuations in the basic angle, and the wavefront sensors (WFS) to measure the optical quality of each telescope. The four other parts will be devoted to the signal treatment. The light beam of a single viewing direction (FOV1 or FOV2) will first pass across the Sky Mappers (SMI or 1.1.

Figure 1 . 7 :

 17 Figure 1.7: The CCD receiver.

Figure 1 . 9 :

 19 Figure 1.9:The sunshield is folded during launch and is deployed later en route to L2.

Figure 1 . 8 :Figure 110 ;

 18110 Figure 1.8: The Soyuz launch vehicle is a four-stage rocket consisting of four boosters (stage I). a central cote engine (stage II), a Figure 110; Gaia spacecraft inside tlnrd stage (another engine), and the Fregat the goyuz fairing restartable upper stage.

Figure 1 . 11 :Figure 1 . 12 :

 111112 Figure 1.11: Gaia's two fields-of-view scan the sky according to a carefully prescribed scanning law. The constant spin rate of 60 arcsec s-1 corresponds to 6-hour great-circle scans. The angle between the slowly precessing spin axis and the Sun is maintained at 45°and the basic angle is 106.5°.

Figure 1 . 14 :

 114 Figure 1.14: Image of Gaia observing our Galaxy.

  Figure 1.15: Hertzsprung-Russell Dia gram adapted from R. Powells diagram. This diagram gathers 22000 stars from the Hipparcos Catalogue together with 1000 low-luminosity stars (red and white dwarfs) from the Gliese Catalogue of Nearby Stars.

Figure 1 . 16 :

 116 Figure 1.16: Artistic view of an exoplanet going past its stars.

Figure 1 . 17 :

 117 Figure 1.17: The spiral Andromeda Galaxy, located at approximately 2.5 mil lion light-years is one of both largest galaxy of the Local Group.

2 Figure 1 . 18 :

 2118 Figure 1.18: The light deflection under the influence of a massive object. ton passes the object and 7 one of two main Parametrised Post-Newtonian parameters.

1 Figure 1 . 20 :

 1120 Figure 1.20: Précision of the asteroid position (along scan) gal with respect to its apparent magnitude.

Figure 1 . 22 :

 122 Figure 1.22: Time and space distribution of Gaia observations for one arbitrary typical main-belt asteroid (1000 Piazzia).

Figure 1 . 23 :

 123 Figure 1.23: Time and space distribution of Gaia observations for one arbitrary near-Earth asteroid (3200 Phaeton).

Figure 1 . 24 :

 124 Figure 1.24: The main belt asteroids (image from Skybot 3D)

Figure 1 . 28 :

 128 Figure1.28: The close approach that Gauss predicted in 1802, a short time after the discovery of the asteroid Pallas. The characteristics of the encounter (the impact parameter, the relative velocity) were little favourable in producing a strong perturbation of orbits (~5.7 mas) but their mean motions being very close (résonant orbits), Ceres and Pallas were brought to meet about every 4.6 years at almost the same place. Thus, their mutual perturbations were directed in the same direction, enhancing the perturbations upon their orbits.

Figure 1 . 29 :

 129 Figure 1.29: The first triple aster oid, 87 Sylvia and its moonlets. The image was taken on 9 August 2004 in the I< band using the adaptive Op tics (AO) System of the Very Large Telescope (VLT) at Cerro Paranal of the European Southern Observatory (Marchis et al. [46]).

Figure

  Figure 1.30: A sample of three observations of 45 Eugenia with its companion Petit-Prince from

Figure 1 .

 1 Figure 1.32: AO image of 22 Kalliope from the T8m ESO-VLT on 28 June 2004 by the IMCCE-Berkeley team.

Figure 1 . 33 :

 133 Figure 1.33: Reconstruction of the global shape of 22 Kalliope from AO images made by Kaasalainen (not yet published).

Figure 1 . 34 :

 134 Figure 1.34: Different simulations of Roche figures for 2 binary Systems which hâve already been observed.

  Merline et al.[START_REF] Merline | Asteroids Do Hâve Satellites. Asteroids III[END_REF] Fr ostia**(1.06 ±0.95) x 10"15 89.6 % observation of satellite3 Behrend et al. [71] Tama** (8.9 ±3.2) x 10"16 35.9 % observation of satellite3 Behrend et al. [71] Berna** (2.25 ±2.01) x 10"15 89.3 % observation of satellite3 Behrend et al. [71] Balam** (7.50 ±0.3) x 10"17 10.5 % observation of satellite Marchis (2005)6 Debussy** (2.97 ± 2.6) x 10"16 87.5 % observation of satellite3 Behrend et al. [71] Personal communication 5 http://quasar.ipa.nw.ru/PAGE/DEPFUND/LSBSS/engmasses.htm 6 http://astro.berkeley.edu/ fmarchis/Science/Asteroids/Balam.htmlThe mass was estimated with an eccentricity e=0.From these different mass déterminations, we can obtain an estimation of Ceres mass by computing the usual weighted mean /iw defined by,

Figure 1 .

 1 35, we hâve plotted these different mass estimations m* with their error bars <7*. The weighted mean is shown with the horizontal dashed line at 4.767 x 1O~1OM0. The histogram gives for each mass its reduced value |(raj -p,)|/cq so that the déterminations in strong disagreement

1 Figure 1 .

 11 Figure 1.35: top: Different mass déterminations of 1 Ceres. The masses are in 10"10M© unit.Bottom: Distribution of the different measures ntii of the mass of 1 Ceres according to the normalise variable |1m% -gw\\/ai However, this standard déviation is underestimated because the déviations ci are formai errors and cannot account for systematic errors which become relevant when comparing a set of measurements. The systematic errors may hâve many origins like the incompleteness of the dynamic model, whose effect is shown by Michalak[START_REF] Michalak | Détermination of asteroid masses I. (1) Ceres, (2) Pallas and (4) Vesta[END_REF]. He illustrâtes this point by deriving the masses of three massive asteroids with or without including perturbing asteroids in its dynamic model. The results show that the formai errors of each détermination remain quite similar in both cases while the masses themselves change with the model.

5 )

 5 and variance (Eqs. 1.7 and 1.8) by filtering out outliers at 3G{ at each step. By considering again the mass of Ceres we obtain l±w

  and Vm are the heliocentric position and velocity vectors of the perturber.

Figure 2 . 2 :

 22 Figure 2.2: Direction change of the relative velocity vector of the target asteroid after the encounter represented in a hodograph.

2 . 2 )

 22 . To compute the value of </?, we hâve to define a new frame from the plane containing the relative velocity V (Valsecchi et al.[START_REF] Valsecchi | The Distribution of Energy Perturbations at Planetary Close Encounters[END_REF]). Let (<Ç, C) be the three axis of this frame. They can be expressed as,

  Fig. 2.2). With help of the two following spherical trigonométrie relations in the triangle ABC,

  where (x, y, z) and (xy', z') are respectively the position of the target asteroid without and with the gravitational perturbation of the perturber. (A, 5) and (A', ô') are the associated angular coordinates. For reasons of simplification, we chose only one parameter to be compared and the angle 7 which takes into account two angular coordinates close to those used for observations, seemed to be to us a good choice.

0. 1

 1 AU < b < 0.5 AU 64965 0.01 AU < b < 0.1 Figure 2.6 shows that the average of observations falls quickly under the 30 observations from the 70,000 IAU number and keeps on decreasing until 10 observations.

Figure 2 . 5 :Figure 2 . 6 :

 2526 Figure 2.5: Distribution of the number of asteroids that Gaia will observe from 2010.5 to 2016 with respect to their IAU number. The IAU number in abscissa correspond to bins in bold.

Figure 2 . 7 :

 27 Figure 2.7: Relation between the inertial frame (Xi,yl,Zi) and the synodic frame (£, rç, 0-

&

  development. By sériés expansions, they developed solutions for representing the asymptotic motion of two particles before and after their encounter. The context of this study was the gravitational interaction of particles in Saturn's rings. The solution was treated different.ly according to the value of the impact parameter (small or large). However, only the planar case (Eqs. 2.40 and 2.41) was considered but the reduced eccentricity of the relative motion, which connects the eccentricities of two bodies and the orientations of their semi-major axes was integrated. Thus, Brumberg and Ivanova extended the results by generalising the solution to the elliptic Hill's problem: it is expressed as sériés in powers of the eccentricity of the perturber E and t.hat of the relative motion e as well as the spatial inclination k from the (Henon and Petit) intermediary solution (£o, Vo) in the case E = e = k = 0, of the équations also expressed as a power sériés of a parameter depending on the impact parameter, the true anomaly as well as a constant. More exactly, the Brumberg Sz Ivanova general solution consists in finding the expression of the perturbations 5Ç, ôrj, ÔÇ caused by the orbital parameters E, e and k by sériés in power of these last, such as, £ = £o + 5Ç, T] = î]o + ÔTI, ( = are the solution of the general Hill's problem,

Figure 3 . 1 :

 31 Figure 3.1: Fit of the initial conditions Cq by computing the corrections ACo obtained from least-squares method.

  détermination) with respect to their initial parameters. A is block diagonal and one block has the following form, n^bs x 6 matrix, nkobs being the number of observations for the asteroid k. It.represents the variations of the longitude À of the asteroid k with respect to its position and velocity in rectangular coordinates at the reference time to for each observation. The éléments of this matrix can be written as, r_ , /dXi d\i dXi dXi dXi dXi\ |Dk],j=i".6 = (^ô. &p' âï» ' W âi® J

  conséquence, from the general solution of the least squares problem (Eq. 3.5) we obtain, AC0 = (A!,A«r1Ai,(O-C)w. (3.8) where A and (O-C) were replaced by, Aw = W1/2A and (O -C)w = W1/2(0 -C) (3.9)

1 (

 1 DkWkDk)-1Dkwy2 [wi/2(0 -C)k -Wk/2Bk • Am, n Y [BfWiB, -B[WiD1(DfWiDi)"1Di WjBi]

3. 1 . 3 1 ( 3 22 )

 131322 The variance-covariance matrix Another important resuit is obtained by the estimation of the covariance of the leastsquares solution Cov( ACo, ACo). It provides the probability distribution of the leastsquares estimation only corisidering a normal distribution of measurement errors with a mean of zéro. Thus, the dynamical model, which represents the forces responsible of the motion of the asteroid, is regarded as the exact représentation of the reality. In addition, the variance-covariance matrix Cov( AC0, AC0) is also used for evaluating the corrélation between the different fitted parameters, that is to quantify the intensity of their connection through a linear law.If the normal matrixA'A is invertible, we obtain a relation between the variancecovariance matrix of the fitted parameters Cov(ACo, ACo) and the variance-covariance matrix Cov(0-C,0-C). More precisely one lias, Cov(AC0,AC0) = E[(ACo-E(ACo))(ACo-E(AC0))(] (3.16) As the solution of the least-squares estimation (Eq. 3.10) may be written as, AC0 = M(0 -C) and (O -C) is a vector of random variables we hâve the relation, Cov( AC0, AC0) = MCov ((O -C), (O -C))Ml (3.17) By inserting the solution of the least-squares (Eq. 3.10) in the expression (Eq. 3.16), we find, Cov(AC0,AC0) = (AlWA)"'A;W [Cov (O -C, O -C)] WA(A'WA)-, the variance-covariance matrix Cov(ACo, ACo) is given by, Cov(AC0, AC0) = <7'o(AtWA)"1 (3.20) The diagonal éléments of the variance-covariance matrix give the standard déviation of the jth fitted parameter Cq, a(AC30) = CW(ACq, ACj() the corrélation between the ith and jth fitted parameters Cov(ACq, ACq), To be more explicit, from submatrices (Eq. 3.6) which make up the covariance matrix Oo(AtWA)~1 and the matrix Un (Eq. 3.15), we obtain the expressions for the variancecovariance of the different adjustable parameters (Auj,..,Au®,Amp) CT2(Amp) -Cov(Amp, Amp) = crgUn1 (3.23) a2(Au2) Cov(Au®, Au°) (3.24) ^(D^Dk)"1 + (DtDk)-1DtkBk a2(Amp) ((D' Bk)' Cov(Amp, Auk) -a02U-1B'Dk(DtDk)-1 (3.25)

28 )

 28 «o, flo, u)0i Mo) at the reference time (éléments of the matrix Dj<) in the case of a two-body problem where, a is the semi-major axis e is the eccentricity z is the inclination fl is the longitude of node u> is the argument of perihelionM is the mean anomalyThe differential correction of the heliocentric coordinates (Ax, Ay, Az) may be written with respect to these oncs of the initial éléments (Aao, Aeo, Azo, A00, A0®, A0g) where (A0°, A0°, A0g) are the three components of a rotation in the coordinates System (x, y, 3 Ax = (x --tx)--+ (Hx + Kx)Aeo + -AMo + 0 A0°+ 2 A0°-y A0o 2 ao n Ay = {y -7;tÿ)--+ (Hy + Kÿ)Aeo + -AMo -2 A0? + 0 A<f>% + X A03 2 ao n Az = (z -Mi)-^+ {Hz + Kz)Aeo + -AMo + y A0? -x A0°+ 0 The corrections (A0j, A0°, A0g) are then expressed with (Azo, Aflo , Auzo) as A0i = cos n0Ai0 + sin zo sin ffo Au;o A 02 = sinOoAzo -sinzocosfloAcJo (3.29) A03 = cos zo Au;q + Aflo The differential corrections of the heliocentric coordinates (Ax, Ay, Az) are thus given by, Ax = (x -Mx)--4-(Hx + Kx)Aeo + -AMo 2 ao n + sin f2o 2 Azo -y Af^o + (-sin zo cos flo z -cos zo y) Au;o Ay = -+ {Hy+ Kÿ) Aeo+ -AMo Ai0 + x AQ0 + (-sin ïq sin Üq z + cos îq y) Acu0 (3.30) Az -(z --tz)--4-(Hz + Kz)Aen 4--AMq +(cosf20 y -sinQo x) Azq + (sinzosin^o y -sinzocosQo x) Acuq This analytical formulation, based on a two-body problem, is useful for deriving global parameters taking a part in the expression of forces having secular variations on certain osculating éléments of asteroids such as these due to the solar quadrupole moment J2 or the post-Newtonian approximation of the General Relativity's laws. Some applications are shown in Chapter 4.

  starting from at the reference time. On the other hand, for ail modelled forces F, one has to write the expression of the quantifies,

3. 4 .dynamical model 85 (

 485 The OCA/Nice observatory) for computing the different observation dates by the Gaia spacecraft between 2010 and 2016. These simulations take into account the future space probe motion around the Lagrangian point L2 and its complex scanning law described in Sec tion 1.1.4. The position of the spacecraft were computed and interpolated in Chebyshev

Figure 3 . 2 :

 32 Figure 3.2: The angular variation of the Gaia longitude dX projected on the reference great circle RGC (1 AU of diameter) of pôle z with respect to an infinitésimal variation of the asteroid position dr. The frame is centred on the Gaia satellite G and u is the direction vector toward the observed asteroid A from the Gaia spacecraft.

  of the intégration, the initial variables dri/dCJQ, dfijdCJQ are,

-Figure 3 . 3 Figure 3 . 3 :

 3333 Figure3.3 is given by its right ascension ao and its déclination Sq (Davies et al.[START_REF] Davies | Report of the IAU/IAG/COSPAR Working Group on Cartographie Coordinates and Rotational Eléments of the Planets and Satellites[END_REF]):

Figure 3 . 4 :

 34 Figure 3.4: Définition of the North pôle of the Sun in the ecliptic coordinates System.

  tions of time independence and spatial isotropy. Even if these conditions do not mirror the reality in the Solar System, the Schwarzschild metric gives a very good approximation of the relativistic effects upon the motion of solar System objects. It, may be expressed in the post-Newtonian approximation as, g e {0,1,2,3} and the Latin ones (i,j) 6 {1,2,3}. xM = (et, xl, x2, x3) are curvilinear coordinates in space-time and c the speed of light. r is the triplet of the space coordinates (ad, x2, x3). m = GMq/c2 is the gravitational radius of the Sun. a is the coordinate parameter defining spécifie coordinate conditions; a -0 is the case of the isotropie coordinates used for the modelling of the Solar System objects motion. a = 1 is the case of the standard coordinates. a -2 is the case of the Painlevé's coordinates.

7 = P = 1

 1 in the case of General Relativity Theory.

  motion of a particle are defined by the équation of the géodésie, A are the Christoffel symbols of the second kind defined by, contravariant, metric tensor, which is given by the équations, 9PP9up -àpv (3.44) and VpvX are the Christoffel symbols of the first kind, With regards to the équation of the géodésie (3.42) for the variable x°= et, from the approximations ds2 = -c2dt2 + gn dx1 .dxl + 522 dx2.dx2 + <733 dx3 .dx3 --(c2 -v2)dt2 ~~c2dt2 (v2 « c2) where v is the velocity in the three-dimensional space (xl ,x2,x3), we obtain ds = icdt and as a conséquence, we hâve the . (3.42), we obtain the final forrn,

  previous expressions(3.49) in the Equation(3.47) giving the accélération of a particle in the gravitational field of the Sun in the post-Newtonian approximation, is the post-Newtonian approximation of the force due to the gravitational field of the Sun. variational équations The expression of the relativistic effects from the post-Newtonian approximation in the variational équations for the détermination of the fit.ted parameter G® is, , -v (r r)^n + rn n -2 7^3 r, +2(7+i) 3.4.4

Figure 3 . 5 :

 35 Figure 3.5: Two images of Vesta with two different illuminations. On the left, the phase angle is at zéro (unreal view of Vesta). On the right, the phase angle is of 30°and the Sun slightly above the equator of the asteroid (10°in geocentric latitude).

3. 4 .dynamical model 95 Figure 3 . 6 : 56 )

 4953656 Figure 3.6: Scattering of light beam from Sun by a spherical schematic asteroid.

  area element, da is, do = R2 cos /3d\d/3(3.58) In addition, the law of cosines to the spherical triangle SEder (Fig. 3.6) gives cos j = cos(A + i) cos P (3.59) and to the triangle OEdo, cos e -cos A cos P (3.60) The évaluation of the quantities dli and dRs from Eqs. (3.54, 3.55, 3.56, 3.57, 3.59, 3.60)is led through the calculation of intégrais over the visible and illuminated area of the surface. This latter is defined by, the displacement d£ along with x due to the Lambert's law, we hâve dIL(x,y) = /TT/Z-l

Figure 3 . 7 :

 37 Figure 3.7: Définition of the planetesimal reference frame in our reference System.

Figure 3 . 8 :

 38 Figure 3.8: Symbolic représentation of the least-squares solution. The density of the measures m, is represented with respect to the value of the mass.

3. 6 . 106 Table 3 . 4 : 149 Table 3 . 5 :

 61063414935 The expected précision on the masses witi-i Gaia Number of masses determined in each class of the relative précision Number of masses determined in each class of the relative précision by completing the selected close approaches with encounters with a minimal distance smaller than 0.01 AU and for which the deflection angle is estimated under the 1 mas.

Figure 3 . 10 :

 310 Figure 3.10: Density distribution of the standard déviations on the inclination i and the longitude of node O of the target asteroids with respect to its inclination. The colour code for density is the same that in Table 3.9.

  spreading from 2008 to the first Gaia observations of the target asteroid while in the post mission case, the period is bounded by its last observation and extends until 2021. An observation epoch is considered valid if the solar élongation £ with the Earth (Fig.3.11) lies in the interval 180°±45. Thus, for each target asteroid, we derived the periods of time meeting the élongation condition and we then distributed uniformly n0&. observations. The ground-based observations are considered in geocentric longitude, latitude (a,[START_REF] Haywood | Field Brown Dwarfs GAIA[END_REF] coordinates, a = arctan ^-j ô = arcsin where, (x, y, z)~(r cos a cos Ô, r sin a cos <5, rsinAE) are the geocentric coordinates of the asteroid, r the geocentric distance. Thus, the analytical part of the variational équations (Sec. 3.2, Eq. 3.26) for the ith observation reads,

Figure 3 . 11 :

 311 Figure 3.11: Coordinates of the ground-based observations;

Ai = 3

 3

  in her planetary ephemerides EMP04, Fienga et al. (personal communication) found a similar value with J2 -(1.95 ± 0.55) x 10"' for INPOP06 whereas JPL's ephemerides DE414, its value is J2 = (2.3 ± 2.5) x 10"7 probably due to a différence in the perturbing asteroids model on Mercury and Venus. Nevertheless, Standish gave in a private communications (2000)

Figure 4 . 1 :

 41 Figure 4.1: Recent déterminations of the solar quadrupole moment J2 obtained from various methods.

  ://www.rssd.esa.int/index.php?project=BEPICOLOMBO&page=Presentations 4

  , the secular variations on the longitude of node AQo> the argument of the perihelion Auq and the mean anomaly AMq at the time t in the équatorial System from the osculating éléments at the reference time trej is given by the linear functions in time t,

  4.1) is applied by solving the variational équations,

  that the non nullity of the Nordtvedt parameter 77 is not proven, 77 = 0. The results of the variance-covariance analysis are given in section 4.3.

  constant of gravitation G Dirac was among the first to point out a possible variation of the constant of gravitation G. The origin of this hypothesis cornes from a numerical relation between the mass of a typical elementary particle and the following quantities, G, h (=h/27r) (the Dirac's constant) h being the Planck's constant, the velocity of light c and the Hubble constant Hq (Weinberg [174]), ~G^J ~m« This relation is problematic for many cosmological models, which consider Ho as no con stant but a function of the âge of the Universe. Instead of reformulating the atomic and nuclear physics, Dirac chose one of the other constants, G, for defining it as a function decreasing with the time according to the amount, | = -3 H0 = -tZl where tu is the âge of the Universe. Sonie théories of gravitation inspired by the idea of the gravitational constant varying with the time were attempted. The most accomplished one was proposed by Brans and Dicke [175], who replaced 1/G by a scalar field which can vary from place to place and with time.

  is zéro in the dynamical model. As regards to the variational équations to be solved for fînding the value of G/G, T = + y' (0Fi8r" + j dt2 \dG/GJ dG/G + G1'» dG/G + 9r" dG/G ) the éléments to be computed are,

  problem can be réductible to quadratures if the "constant" of gravitation is expressed as, G = (&i + b'2t) 1where b\ and 62 are constants. Thus, Vinti found the following expressions by assuming G(t) ~(k + t)-1, our case, we express G varying linearly in time as, Geff(t) -G + G{t -t0)(4.13)

2 From 18 )

 218 the following formula (3.30) (Sec. 3.2.1) Ax = -AM; A y = -AM, A z = -The variance-covariance analyses are based upon the realistic simulations of the asteroid observations (see Sec.

3 . 3 )

 33 by Gaia during the period from 2010.5 to 2016. For each parameter, we indicate the standard déviation in two cases: the case A, where a single paramet.er is fitted and the case B with the resuit given for this parameter fitted with ail

Figure

  Figure of the Sun

  net force due to asteroid thermal émission ...astrometric accuracy of the Gaia mission allows to think of studying the possibility of detecting non-gravitational forces acting on the motion of small asteroids. These non gravitational forces are in general very weak in comparison with the gravita tional ones. However, certain asteroids présent physical characteristics making them very sensitive to these non grav itational effects: e.g very small size, or very close to the Sun. The so called Yarkovsky effect is a graduai change of the orbital éléments of asteroids and meteoroids due to the momentum carried off by the photons emitted anisotropically in the thermal infrared from the warm surfaces of these bodies. It was discovered by the Polish civil engineer Ivan Osipovich Yarkovsky (1844-1902) who worked for the Alexandrovsk railway company

Figure 1902 ). 5 . 1 :

 190251 Figure 1902).

5. 1 .

 1

Figure 5 . 2 :

 52 photon

Figure 5 . 3 :

 53 photon

  radius of the body, m its mass and c the speed of light. E is the solar radiation flux at a given heliocentric distance, E = Sq r2 with Sq the bolometric solar flux at 1 AU (~1373 W.m 2) and r the heliocentric distance of the body. The solar radiation flux E allows the computation of the subsolar tempéra ture, Tss, from the energy balance équation which assumes an equality between the solar

FyFigure 5 . 4 :

 54 Figure 5.4: The intermediary reference frarne (X. Y, Z)

Figure 5 . 5 :

 55 Figure 5.5: Distribution of the predicted observable NEAs by Gaia with respect to their semimajor axis.

-

  Carlo simulations were developed with the aim of comparing the different planetary solutions with respect to different sets of asteroid masses. They arrived at the conclusion that the uncertainties into the Earth and Mars ephemeris can be several kilométrés over a period of about 10 years.The study from planetary ephemerides is very interesting because it allows to dérivé the most influent asteroids in the scope of their dynamic model and thus brings a supplementary information on the limit of these représentation. As an illustration, during a ESOC workshop 2007 in Darmstadt, Folkner presented a study of perturbers of Mars from JPL planetary ephemerides. He computed two solutions for Mars orbit over a determined period differing from the set of the considered asteroids: one of the samples brought together 300 massive asteroids while the other was made up of a particular list (not given), a few different of theWilliams list (1984). Then, the comparison of the two solutions to the Mars observations dating from the fitting period showed discrepancies: the rms obtained for the first set (300 asteroids) was very bad in comparison with the second one (Folkner's list).

(k - 1 , 2 )

 12 greater than two threshold values. These last are defined as fractions of two coefficients found for Ceres -a major perturber of Mars -ca concerning the semi-major axis of Mars (k = 1), and c\ for its longitude (k -2), such as, ca = max(cjij C\ = max(cy) 6.1.The Mars ephemeris and the asteroid masses[START_REF] Duriez | Cours de Mcaniqe Cleste[END_REF] The second filter consists in excluding alternately the associated period Pjj superior to P\im = 10 years (Tab. 6.1), 50 years (Tab. 6.2), which is defined by 2ir i x riM + j x n' with i € {0,1,..., 2p -1} and j G {0,1,..., 2p1 -1} where tim and n' are respectively the mean motion of Mars and the considered asteroid, and p and p' two fixed integers.

/

  are respectively the means of the mass and volume distributions and crm,

  ! I Pc -P\ With Pc € AP i= 1

  . It is completed with the spectroscopic measures made by Tholen[START_REF] Tholen | Asteroid taxonomy[END_REF] for their two common taxonomie classes, S and C. In fact, certain asteroids were not observed during the SMASSII campaign whereas their taxonomie class has already been derived over the last years such as the asteroid 8 Flora. The choice of the classes S and C was justified by a statistical comparison between the Tholen database and that from SMASSII data. We considered only the asteroids simultaneously belonging to the two classifications and to their common 10 classes (Fig.7.6). Thus, we estimated the number of asteroids affected to the sanie taxonomie class (exactly) for the both classifications or to a sub-class of the parent (sub-class), and the number of discordant assignations (discordant). The classes C and S are the only one to hâve a much larger number of similar inferences than

  J2 and the PPN parameter /3 hâve proven to be interesting since we can expect précisions close to the best current ones. Moreover, the completeness of the dynamic model used for Gaia will allow to hâve more exact measures by a dynamic method different of those giving the best results. The estimation of the other parameters representing hypothetical phenomena such as the violation of the weak Equivalence Principle (Nordtvet parameter) or the variation in time of the gravitational constant G was envisaged. The prédiction on the précision of their détermination with Gaia encourage us to continue the research to dérivé them.

Table 1 .

 1 2 gives its main characteristics. The focal plane

Table 1 .

 1 2: Characteristic of the focal plane and its components.

Table 1 .

 1

	Index	Mass rrii	error ai	Reference	Year
	i	[10~loM©]	[1O-1OM0]
	1	4.7	0.3	Goffin [76]	1991
	2	4.796	0.085	Sitarski et al. [77]	1992
	3	4.8	0.22	Williams [78]	1992
	4	4.622	0.071	Sitarski et al. [79]	1995
	5	5.0	0.2	Viateau et al. [80]	1995
	6	4.71	0.09	Carpino et al. [81]	1996
	7	4.26	0.09	Kuzmanoski [82]	1996
	8	4.785	0.039	Viateau et al. [83]	1997
	9	4.759	0.023	Viateau et al. [84]	1998
	10	4.39	0.04	Hilton [85]	1999
	11	4.7	0.04	Michalak [86]	2000
	12	4.76	0.02	E.M.Standish1	2001
	13	4.81	0.01	Pitjeva [87]	2001
	14	4.63	0.26	Kovacevic et al. [74]2 2005
	15	4.6	0.29	Kovacevic et al. [74]2	2005
	16	5.02	0.44	Kovacevic et al. [74]2 2005
	17	5.05	0.76	Kovacevic et al. [74]2 2005
	18	4.73	0.029	Kovacevic et al. [75]2	2007
	19	4.79	0.077	Kovacevic et al. [75]2 2007
	1 personal communication 2001

3: Different mass déterminations of Ceres computed between 1991 and 2007.

2 The computed masses differ from the studied target asteroids.

Table

1

.4: The best précisions of asteroid mass déterminations. The new asteroids for which the mass has been determined by dynamic method since Hilton's paper

[START_REF] Hilton | Asteroid Masses and Densities[END_REF]

in 2002, appear in boldface. n Asteroid

Table 2 . 1 :

 21 Distribution of the relative précision of the impulse approach according to the encounter velocity. The part of the close approaches for which the impulse approach underestimates the impact of the encounter is indicated in parenthesis. The last column gives the percentage of each category out of the total number of encounters (=1 million).

	Relative Vel.					Error e <					Part
	V	5			10		20		30		50
	[km. s-1]	[%]		[%]		[%]		[%]		[%]	[%]
	V < 1	1.3	(0.6)	2.7	(1.3)	5.6	(2.7)	8.5	(4.1)	13.8	(6.4)	2.7
	1 < V < 2	4.1	(2.0)	8.2	(4.2)	16.5	(8.6)	24.5	(13.0) 39.2 (21.1) 11.6
	2 < V < 3	4.4	(2.3)	8.9	(4.7)	18.6	(10.6) 28.9 (17.2) 46.5 (28.2) 21.6
	3 < V < 4	4.3	(2.3)	8.9	(4.8)	20.2	(12.3) 32.8 (21.2) 52.9 (34.6) 22.8
	4 < V < 5	4.4	(2.3)	9.3	(5.1)	23.3	(15.1) 38.7 (26.8) 60.5 (41.8) 17.0
	5 < V < 6	4.6	(2.4)	9.9	(5.6)	27.7	(18.9) 44.9 (31.9) 66.5 (47.0) 10.8
	6 < V < 7	5.0	(2.7)	11.7	(7.1)	32.8	(23.3) 50.5 (36.7) 71.0 (51.1)	6.2
	7 < V < 8	6.1	(3.4)	14.7	(9.3)	38.2	(27.3) 55.6 (40.1) 74.2 (53.0)	3.3
	8 < V < 9	7.9	(4.7)	19.8	(13.2) 44.4 (31.7) 60.2 (43.2) 77.2 (55.0)	1.7
	9 < V < 10	10.9	(6.5)	24.0	(15.8) 49.5 (34.2) 64.3 (44.5) 79.5 (54.4)	0.9
	V > : 10	12.9	(7.4)	27.2	(16.8) 52.1 (35.0) 66.6 (44.9) 80.2 (52.8)	1.4

Table 2 .

 2

							Error e <		Part
				5		10		20	30	50
				[%]		[%]		[%]	[%]	[ %]	[%]
		h < 50	8.5	(4.2)	19.6	(11.0) 39.0 (22.6) 51.4 (30.1) 65.3 (38.0)	0.3
	50 <	h < 100	11.-4 (6.1)	24.9	(13.9) 51.4 (31.1) 66.2 (40.1) 81.5 (48.3)	0.9
	100 <	h < 200	9.0	(4.7)	20.8	(11.6) 47.6 (29.4) 63.3 (39.5) 78.6 (48.5)	3.4
	200 <	h < 300	6.3	(3.4)	14.8	(8.5)	39.2	(25.5) 55.6 (36.1) 73.0 (46.8)	5.3
	300 <	h < 500	5.1	(2.7)	11.2	(6.5)	31.0	(21.1) 48.3 (33.2) 67.9 (45.1) 12.9
		h > 500	4.2	(2.2)	8.5	(4.7)	18.9

2: Distribution of the relative précision of the impulse approach according to the scaled impact parameter h. The proportion of the underestimated close approaches is given in parenthesis.

The last column gives the percentage of each category out of the total number of encounters (=1 million). h b/Da

Table 2 .

 2

	Number of perturbing asteroids	602
	Number of target asteroids	43513
	Number of perturber-targets asteroids1	435
	Number of close approaches	78768

3: Close approaches main statistics with a minimal distance < 0.5 AU, a deflection angle > 1 mas and a perturber mass > 10~13M©.

Table 2 .

 2 4: Distribution of the close approaches according to the value of the deflection angle 6.

	Number of close approaches
	1 mas < 9 <	50 mas	75867
	50 mas < 9 < 100 mas	1633
	100 mas < 9 < 500 mas	1107
	500 mas < 9		161
	Total		78768

Table 2 .

 2 5: Distribution of the close approaches according to the value of the impact parameter b.

	Number of close approaches

  can be made by several methods which will be set out in the next Sections 3.2.1, 3.2.2.

	can be expressed analytically from the rectangular coordinates of the observer and the
	asteroid at. the time t. For the computation of the part
	dr(t)	( dx(t) dy(t) c>2(t)\
	Hc[~\âc['~dc['~âc[)	(3.27)
						3.6),
		D,	0		0	B, \
	_	0	E>2	•	:	B2
		:			0	:
		0	0		Dn b" y
	where,
		/ dXi d\i d\i dXi dXi dXA
	k i,j-1,-6		QyO' Qz0' Qj.0' QÿQ '	J
						dXj\
					' dmp)
	Firstly, the variation of the longitude A at a time t of an asteroid with respect to the value
	of the parameter Cq at a référencé time to is expressed by combining the variation of A
	with respect to the rectangular heliocentric coordinates r(t) = (x(t),y(t),z(t)) with that
	of the unknowns Cq as,
							(3.26)

dX(t) _ A ÔX(t) drjjt) dCl ~^dri(t) dCJQ In the équation (3.26), the part made up of dX (t) / dX (t) dX (t) dX (t)\ dr(t) \5x(t) ' dy(t) ' dz(t) J it

Table 3 .

 3

		1: Comparison between the theoretical relative précision and the standard déviation from
	the 100,000 Monte-Carlo measures ra*. Furthermore, the systematic errors appear in part of the
	standard déviation while the maximal error is written in percent according to the exact solution
	on the mass m.
		steroid	a{m)/m	(Tfi/m	^sys/^n	|mmax -m\/m
	n°A name	Theoretical	Monte-Carlo	Monte-Carlo	Monte-Carlo
			[%]	[%]	[%]	[%]
	1	Ceres	0.173	0.173	0.3	0.8
	4	Vesta	0.197	0.197	0.2	0.8
		Irene	1.204	1.206	0.3	5.2
	2	Pall&s	1.373	1.371	0.0	6.3
		Euterpe	1.439	1.439	0.5	6.5
		Hygiea	1.673	1.674	0.1	8.2
		Métis	3.875	3.876	0.2	20.0
		Amphitrite	5.265	5.263	0.1	22.7
		Germania	5.669	5.672	0.2	27.8
		Eunomia	6.753	6.760	0.1	34.8
		Aurora	7.290	7.271	0.8	31.7
	203	Pompeja	7.819	7.831	0.5	33.6
		Psyché	8.653	8.628	0.5	38.6
		Europa	9.477	9.516	0.4	42.8
	704	Interamnia	11.755	11.730	0.3	55.6
		Thisbe	11.871	11.874	0.3	52.6
		Fortuna	11.986	12.002	0.2	56.1
	120	Lachesis	12.063	12.053	0.0	59.8
		Parthenope	13.847	13.791	0.4	63.0
	3	J u no	13.992	14.000	0.7	62.3
	209	Dido	14.608	14.594	0.1	67.6
		Doris	15.350	15.339	0.2	66.2
		Victoria	16.798	16.744	0.1	75.7
		Panopaea	17.190	17.190	0.3	77.1
	511	Davida	17.512	17.605	0.3	77.2
	423	Diotima	18.231	18.245	0.5	86.9
	444	Gyptis	20.797	20.840	0.0	100.4
	128	Nemesis	21.192	21.159	0.8	89.8
	7	Iris	22.647	22.635	0.0	95.9
	165	Loreley	23.199	23.218	0.1	99.3
	200	Dynamene	23.547	23.539	0.2	105.4
		Thémis	23.649	23.701	0.3	111.6
	451	Patientia	24.637	24.573	0.2	112.5
		Elpis	24.786	24.858	0.4	113.1
	107	Camilla	27.183	27.181	0.6	120.3
		Galatea	28.218	28.098	0.1	119.7
		Angelina	28.988	29.062	0.4	123.1
		Eugenia	31.029	30.895	0.2	138.5
	566	Stereoskopia	32.539	32.628	0.4	150.8
		Melpomene	37.373	37.431	0.0	180.3
	121	Hermione	38.246	38.178	0.0	185.9
	532	Herculina	39.202	39.304	0.2	181.3
	349	Dembowska	41.797	41.809	0.0	215.3
		Beatrix	43.054	43.136	0.6	202.6
		Aegle	44.496	44.416	0.0	192.7
	147	Protogeneia	45.447	45.528	0.2	204.5

Table 3 .

 3 2: Distribution of the relative systematic errors according to the tested perturbing forces and effect corrections that we do not take into account into the dynamic model.

	Tested effect

1 Référencé distribution when the dynamic model is the exact représentation of the reality.

Table 3 .

 3 6: The formai précisions on the perturbers masses. The binary asteroid are marked by **. The mass given in the next to last column is that used in our simulation.

	Asteroid	nuinber of	number of	sigma	mass	sigma
	n°namc	target	close approaches	[M0]	[Mo]	[%]
		asteroids	not observed
	Ceres	6007	1688	3.19X10"13	4.75xlO-10	0.067
	Vesta	6685	2065	1.23x 10"13	1.36xlO"10	0.091
	Hygiea	3180	642	3.42xl0~13	4.54x 10"11	0.753
	Irene	1973	413	1.53x 10~13	1.41xl0-11	1.090
	Psyché	4699	1627	3.91xl0"13	3.38xl0-11	1.160
	Euterpe	1053	204	6.30x 10~14	5.38xl0"12	1.170
	Iris	1821	470	l.SOxlO-13	1.41xl0-11	1.270
	Pallas	1194	83	1.35x 10"12	l.OOxlO"10	1.350
	Métis	3046	918	2.10xl0"13	1.45xl0~n	1.450
	Eunomia	1859	580	2.82x 10~13	1.64xl0-11	1.720
	Europa	2155	811	3.73x 10~13	1.82x 10"11	2.050
	Angelina	417	150	5.08x 10~14	2.13X10"12	2.380
	Amphitrite	1633	416	1.70xl0~13	6.29xl0~12	2.700
	Davida	1163	303	7.39x 10~13	2.40X10"11	3.080
	Eugenia	547	184	8.80x 10"14	2.82xl0-12	3.120
	Thisbe	929	227	1.79x 10~13	5.31 x 10~12	3.360
	Hestia	249	51	5.10x 10~14	1.26xl0~12	4.050
	Diotima	650	123	2.47X10"13	5.99x 10~12	4.120
	Interamnia	1035	100	8.69x 10~13	2.09xl0-u	4.160
	Juno	458	107	3.81 x 10~13	8.42x 10~12	4.520
	Fortuna	1163	424	2.09xl0"13	4.49xl0~12	4.650
	Aurora	636	183	2.89x 10~13	5.66xl0~12	5.110
	Nemesis	993	198	2.58x 10~13	4.39xl0~12	5.890
	Germania	535	92	1.93x 10"13	3.17xl0"12	6.100
	Doris	1122	292	4.53x 10"13	7.18xl0"12	6.310
	Pompeja	281	72	6.78xl0~14	1.04xl0"12	6.510
	Parthenope	685	207	1.73xl0~13	2.56x 10"12	6.750
	Patientia	578	144	5.11xl0~13	7.50x 10"12	6.820
	Euphrosyne	78	3	7.80x 10~13	l.IOx 10-11	7.090
	Bamberga	268	87	5.71 xl0~13	7.94xl0"12	7.190
	Alexandra	153	36	2.33x 10~13	3.00x 10~12	7.760
	Massalia	407	122	1.60x 10~13	2.03xl0"12	7.900
	Herculina	347	101	5.95xl0~13	7.22xl0~12	8.240
	Julia	75	24	1.92x 10~13	2.29x 10-12	8.370
	Elpis	424	124	2.68x 10~13	2.95x 10~12	9.100
	Daphné	43	10	3.32xl0~13	3.47xl0~12	9.570

Table 3 .

 3 7: The formai précisions on the perturber masses from the Gaia observations and different number of additional ground-based observations over a period of about 2.5 years before the Gaia

	observations.
	Perturbers			cr(m)/m	[%]
	n°name	Gaia	+25 obs.	+50 obs.	+75 obs.	+ 100 obs.
	389	Industria	12.9	2.9	2.1	1.8	1.6 (x 8.32)
	516	Amherstia	176.0	61.4	45.3	37.7	33.0 (x 5.33)
	23	Thalia	250.0	91.8	67.4	55.8	48.8 (x 5.12)
	66	Maja	98.0	68.2	55.7	48.4	43.5 (x 2.25)
	814	Tauris	94.7	68.4	57.5	51.5	47.7 (x 1.99)
	129	Antigone	79.0	63.5	54.7	49.2	45.3 (x 1.74)
	410	Chloris	69.2	58.4	51.7	47.1	43.5 (x 1.59)
				After the mission

Table 3 .

 3 8: The formai précisions on the perturbers masses from the Gaia observations and different number of ground-based observations over a period of about 5 years after the Gaia observations.

	Perturbers			a(m)/m [%}
	n°name	Gaia	+25 obs.	+50 obs.	+75 obs.	+100 obs.
		Iclea	1190.	20.5	14.7	12.3	10.8 (x 110.19)
		Bredichina	415.	61.6	45.2	38.3	34.3 (x 12.10)
		Hilda	439.	65.8	47.2	39.7	34.9 (x 12.58)
		Una	116.	28.3	20.4	17.	15. (x 7.73)
		Eleutheria	138.	34.8	25.3	21.1	18.6 (x 7.42)
	23	Thalia	250.	91.1	66.2	55.	48.1 (x 5.20)
		Bertha	78.7	30.3	23.4	20.	17.9 (x 4.40)
		Faina	65.3	40.6	31.9	27.3	24.3 (x 2.69)
		Eunike	112.	76.8	62.9	54.6	49.2 (x 2.28)
	38	Leda	56.9	39.5	32.4	28.3	25.5 (x 2.23)
		Siwa	56.3	39.1	32.	27.8	24.9 (x 2.26)
		Atala	102.	71.2	58.3	51.	45.4 (x 2.25)
		Bertholda	57.1	40.5	33.5	29.6	27. (x 2.11)
		Lumen	68.4	50.6	44.7	40.8	37.9 (x 1.80)
	92	Undina	59.3	44.8	38.3	34.7	32.2 (x 1.84)
	63	Ausonia	61.9	52.2	46.2	41.8	38.7 (x 1.60)
		Elektra	73.8	62.5	54.7	49.5	45.6 (x 1.62)
		Klymene	58.5	51.5	47.	43.5	40.9 (x 1.43)
	28	Bellona	51.	45.4	41.2	38.1	35.7 (x 1.43)

Table 3 .

 3 9: The list of target asteroids which could allow to reach the formai précisions mentioned in Table3.7. The date of the close approach is given in the column 'date' and is close to the start of the Gaia observing program (2011.5 in this work).

	Perturbers	Target s	Date	Dmin	Range obs.	Mag. V
	n°name	n°name	JJ-MM-YY	[AU]	[Days]	moy
	389	Industria	33537	1999 HJ10	3-8-2011	0.0001	150.	18.4
	516	Amherstia	57101	2001 OC70	3-2-2011	0.0002	145.	19.0
	23	Thalia	4214	Veralynn	16-10-2010	0.0062	145.	16.5
	66	Maja	182662	2001 TT62	25-8-2010	0.0010	145.	19.0
	66	Maja	71327	2000 AA85	14-8-2011	0.0049	150.	19.4
	814	Ta u ris	85886	1999 CN35	8-3-2011	0.0025	140.	19.2
	129	Antigone	50824	2000 FU35	25-2-2011	0.0104	95.	18.9
	410	Chloris	116055	2003 WQ107	29-3-2011	0.0077	90.	18.6
	410	Chloris	32052	2000 JJ37	8-11-2010	0.0119	70.	17.6

Table 3 .

 3 10: The list of target asteroids which could allow to reach the formai précisions mentioned in Table3.8.

	Perturbers	argets	Date	Dmin	Range obs.	Mag. V
	n°name	n°T	name	JJ-MM-YY	[AU\	[Days\	moy
	286	Iclea	12531	1998 KQ51	19-12-2014	0.0018		17.4
	786	Bredichina	37278	2000 YE7	27-12-2014	0.0036		18.6
	786	Bredichina	127740	2003 FS9	10-2-2015	0.0061		18.8
	153	Hilda	105757	2000 SF100	30-4-2015	0.0114		18.8
	153	Hilda	163335	2000 TV53	5-2-2015	0.0208		17.7
	160	Una	118743	2000 QV146	3-3-2015	0.0023		18.8
	160	Una	29824	Kalmancok	20-11-2015	0.0093		19.2
	567	Eleutheria	28003	1997 WT52	23-2-2014	0.0055		17.3
	567	Eleutheria	34783	2001 RB75	21-9-2013	0.0067		18.8
	23	Thalia	10524	Maniewski	25-2-2015	0.0092		16.6
	23	Thalia	481	Emita	15-10-2014	0.0203		12.6
	154	Bertha	31735	1999 JJ72	15-5-2014	0.0140		16.9
	154	Bertha	83684	2001 TX51	12-12-2014	0.0214		18.8
	751	Faina	23311	2001 AM29	17-3-2014	0.0124		16.8
	751	Faina	39977	1998 HR9	15-1-2015	0.0150		18.0
	185	Eunike	20020	1991 VT	5-8-2014	0.0127		17.4
	185	Eunike	7318	Dyukov	30-5-2014	0.0234		16.1
	38	Leda	43896	1995 UL4	10-6-2014	0.0045		17.4
	38	Leda	9688	Goudsmit	27-7-2014	0.0054		17.6
	140	Siwa	9146	Tulikov	25-10-2014	0.0024		17.0
	140	Siwa	4161	Amasis	12-2-2015	0.0127		17.8
	152	Atala	97265	1999 XM125	21-3-2015	0.0246		18.6
	152	Atala	57236	2001 QG80	21-12-2015	0.0314		17.2
	420	Bertholda	203129	2002 EB91	28-6-2015	0.0199		19.6
	141	Lumen	84403	2002 TA181	13-10-2013	0.0128		19.3
	141	Lumen	113753	2002 TP167	29-5-2015	0.0159		18.1
	92	Undina	82731	2001 PQ59	5-1-2015	0.0175		18.2
	92	Undina	18160	Nihon Uchu.	5-6-2014	0.0178		17.8
	63	Ausonia	6101	1993 EG	12-11-2014	0.0071		16.7
	63	Ausonia	1717	Arlon	10-10-2014	0.0095		15.6
	130	Elektra	23400	A913 CF	16-10-2015	0.0122		18.3
	130	Elektra	169656	2001 FT115	27-9-2013	0.0279		19.4
	104	Klymene	6957	1988 HA	23-7-2014	0.0092		17.2
	104	Klymene	55066	2001 QF81	28-6-2015	0.0111		18.1
	28	Bellona	48155	2001 GU4	5-12-2014	0.0109		17.8
	28	Bellona	5116	Korsor	18-2-2014	0.0131		16.2

  3.4.2 (Eq. 3.38).

	4.2.	The relativistic parameters		125
	and consequently,
		Ar	,, nn{t -tref) -+ nu(t ~tref) ^+ nM{t -tref) -AJ2 , N dr dr dr
	Thus, we obtain
			<9r	A/r ^r'	dr	, _ dr'	dr	dr'
			âr^= M2âfV	=		dM~0=M2dMo

Table 4 .

 4 1: Variance-covariance analysis on the solar quadrupole moment J2 parameter.

	Expected précision a : (.72=2x10 7)
						Corrélation: G/G
	Case A	Analytical	1.41 x 10"7	(~70.5%)	(case B, Numerical)
		Numerical	1.42 x 10"7	(~70.5%)
	Case B	Analytical	1.91 x 10-7	(~95.5%)	(3 : 0.65	7 : 0.03	G/G : -0.04
		Numerical	1.86 x 10-7	(~93.0%)
			The PPN parameters (3 and 7

Table 4 .

 4 2: Variance-covariance analysis on the PPN parameter (3.

		Expected précision a: {(3=1)
				Corrélation: G/G
	Case A Analytical 1.45 x 10	(~0.15%)	(case B, Numerical)
		Numerical 1.44 x 10~3	(~0.14%)
	Case B	Analytical 1.96 x 10~3	(~0.20%)	J2 0.65 7 : 0.05 G/G : -0.08
		Numerical 1.89 x 10~3	(~0.19%)

Table 4 .

 4 3: Variance-covariance analysis on the PPN parameter 7.

		Expected précision cr: (7==1)
					Corrélation: G/G
	Case A	Analytical	7.3 x 10-4		(case B, Numerical)
		Numerical	7.5 x 10"4
	Case B	Analytical	9.8 x 10"4	(~0.10%)	J2 : -0.66	7 : 0.01	G/G : -0.04
		Numerical	1.1 x 10~3	(~0.11%)
			The Nordtvedt parameter 7
		Table 4.4: Variance-covariance analysis on the Nordtvedt parameter 7.
	Expected précision a: (7=0)		Corrélation: G/G
			'		(case B, Numerical)
	Case A Numerical 6.2 x 10 4
	Case B Numerical 6.2 x 10~4		(3 : 0.05 J2 : 0.03 G/G : 0.06

Table 4 .

 4

			5: Variance-covariance analysis on G/G one year.
	Expected précision a: (G/G=0)
	Case A	Analytical Numerical	3.36 x KT12 3.89 x 1CT12	Corrélation: G/G (case B, Numerical)
	Case B	Analytical	3.42 x 1(T12	0 : -0.08	J2 : -0.04	7] : 0.06
		Numerical	3.92 x 1(T12
		Table 4.6: Sumrnary for the solar quadrupole J2 détermination
		Future missions	date of launch	Précision
		Gaia mission		2011	~1.86 x 10-7
		Bepicolombo mission	2013	~io~8
		Method		J2 measure	Précision	Ref.
		Geometry of the Sun	(3 ±6) x 10~7	~200%	[137]
		Helioseismology	(2.18 ±0.06) x 10~7	~2.75%	[143]

  .1 illustrâtes well the difficulty to observe the NEAs: among the 1386 NEAs expected to be observed by Gaia, only 666 hâve a number of dates superior to 10 with a great majority not. exceeding 20

observations.

Table 5 .

 5 1: Distribution of 1386 NEAs with respect to the number of predicted observations by

	Gaia.
	Number of
	observations n	asteroids
	0 < n < 10	720
	10 < n < 20	326
	20 < n < 30	131
	30 < n < 40	71
	40 < n < 50	48
	50 < n < 60	44
	60 < n < 80	32
	80 < n < 100	7
	100 < n < 132	7

Table 5 .

 5 2: Number of NE As with at least one physical parameter Ç determined in each class of the relative précision.

	5.3.	Variance-covariance analysis		144
			Number of NEAs
			Diameter	Thermal inertia
		c	d	r
		Total	1386	1386
		a(C)/C < 10%	1	0
		cr(C)/C < 20%	4	1
		<t(0/C < 30%	8	2
		<t(C)/C < 40%	11	2
		<t(0/C < 50%	14	2
			Number of NEAs
			Spin components
		C	^i	U>2	W3
		Total	1386	1386	1386
		<y(QK< io%	0	0	1
		a(C)/< < 20%	1	0	1
		<t(0/C < 30%	2	1	1
		<7(C)/C < 40%	2	1	2
		a{Ç)/C < 50%	2	2	4

Table 5 .

 5 3: The formai précisions on the six physical parameters ranked according to the best précisions on the diameter. When the précision exceeds the level of 100 %, it is replaced by

	n°IAU	Asteroid name	Gaia obs.	a(d) [%]	*(P) [%]	a(coi) [%]	a(tü2) [%]	b	.
	138852	2000 WN10	17 obs	9.6	98.7	10.2	22.4	7.6
	2062	Aten	132 obs	10.7	21.9	-	-	51.0
	99907	1989 VA	37 obs	19.4	64.6	-	-	-
	33342	1998 WT24	26 obs	19.8	-	-	44.1	40.6
	169239	2000 BM19	64 obs	23.3	-	99.0	-	68.0
	320072	2005 WK4	16 obs	23.9	58.8	-	-	-
	293810	2005 G021	34 obs	26.7	61.1	26.6	-	33.9
	215135	2002 JC	47 obs	27.7	-	-	-	-
	233385	2002 XH4	41 obs	33.9	66.0	-	-	-
	4034	1986 PA	62 obs	35.3	-	-	-	-
	66391	1999 KW4	38 obs	36.2	70.5	-	-	-
	250343	2003 UC20	25 obs	43.0	-	82.0	-	57.0
	346971	2006 R036	40 obs	46.9	-	-	-	-
	188422	2001 RR17	96 obs	48.8	-	-	-	-
	7889	1994 LX	58 obs	52.3	-	-		_
	212606	2002 FB3	51 obs	52.3	-	-	-	-
	342403	2006 KM103	18 obs	57.4	11.3	-	-	-

Table 5 .

 5 4: The characteristics éléments of asteroids listed in Table5.3: number of Gaia observa tions, semi-major axis a, eccentricity e, diameter d and thermal inertia T.

		Asteroid		Gaia	a	e	d	r
	n°IAU	name		obs.	[AU]		[km]	[SI]
	138852	2000 WN10		17 obs	1.00	0.30	0.272	560.2
	2062	Aten	132 obs	0.97	0.18	0.910	313.9
	99907	1989 VA		37 obs	0.73	0.59	0.683	360.1
	33342	1998 WT24		26 obs	0.72	0.42	0.412	459.2
	169239	2000 BM19		64 obs	0.74	0.36	0.706	354.5
	320072	2005 WK4		16 obs	1.01	0.24	0.258	574.9
	293810	2005 G021		34 obs	0.75	0.34	1.310	263.6
	215135	2002 JC		47 obs	0.82	0.39	0.877	319.5
	233385	2002 XH4		41 obs	1.61	0.27	0.266	566.4
	4034	1986 PA		62 obs	1.06	0.44	0.420	454.9
	66391	1999 KW4		38 obs	0.64	0.69	1.500	247.0
	250343	2003 UC20		25 obs	0.78	0.34	0.646	370.1
	346971	2006 R036		40 obs	0.91	0.23	0.609	380.7
	188422	2001 RR 17		96 obs	1.55	0.49	0.894	316.6
	7889	1994 LX		58 obs	1.26	0.35	2.161	207.3
	212606	2002 FB3		51 obs	0.76	0.60	2.193	205.8
	342403	2006 KM103	18 obs	1.57	0.38	0.153	739.5
	Table 5.5: Distribution of 459 NEAs (number of observation n > 15) with respect to their
	diameter
		Diameter d		Number of
			[km]			asteroids
			< d <	0.5	35
		0.5	< d <	1	114
		1	< d <	1.5	121
		1.5	< d <	2	65
		2	< d <	5	101
		5	< d <	10	17
		10	< d <	40	6

Table 5 .

 5

	6: Distribution of 459 NEAs (number of observation n > 15) with respect to thermal
	inertia.
	Thermal inertia T	Number of
		[SI\		asteroids
		< r <	100	6
	100	< r <	200	81
	200	< r <	300	221
	300	< r <	400	107
	400	< r <	500	27
	500	< r <	600	13
	600	< r		4

  years. The amplitudes are counted according to two minimal thresholds: one tenth and one hundredth of ca and c\. The last column gives the expected formai précision on the masses with Gaia (see Sec. 3.6). The major rôle played by the three bigs, Ceres, Pallas and Vesta on the Mars orbit appears clearly in these Tables, they possess by far the greater number of amplitudes for the semi-major axis and longitude of Mars. The amplitudes superior to the second limit ca/100 and c\/100 allow to dérivé a second group of non negligible asteroids: among them, we can cite 7 Iris, 14 Irene, 9 Métis, 27 Euterpe, 15 Eunomia and 324 Bamberga. Except for the number of asteroids, the two tables list

Table 6 .

 6 1: Ranking of the asteroids according to the theoretical perturbations on the semi-major axis and longitude of Mars with periods smaller than 10 years.

	Asteroid		number of coeff.		Comm.	mass
	n°name	Aaw	AAm	A a	AA m	P\i ' Pa	m
		Ca/10	cx/10	ca/100	ca/100		[M©]	[%]
	Ceres	12	12	44	41	-	4.75X10-10	0.07
	Vesta	10	10	41	33	-	1.36X10"10	0.09
	Pallas	6	7	56	48	-	l.OOxlO-10	1.02
	Iris	0	3	35	33	-	1.41X10-11	4.04
	Irene	0	2	17	16	-	1.41 x 10-"	1.09
	Métis	0	1	14	15	-	1.45x10-"	1.45
	Bamberga	0	1	13	16	7: 3	7.94X10-12	7.19
	Eunomia	0	1	11	11	-	1.64x10""	2.66
	Euterpe	0	0	17	18	-	5.38xl0"12	1.17
	Hygiea	0	0	9	9	-	4.54x10-"	0.77
	Psyché	0	0	8	8	-	3.38x10-"	3.80
	Juno	0	0	7	8	-	8.42x 10~12	4.52
	Davida	0	0	7	8	3: 1	2.40x10-"	3.25
	Zelinda	0	0	4	7	-	1.36x 10"12	-
	Eui'opa	0	0	4	6	-	1.82x10-"	2.05
	Daphné	0	0	2	7	-	3.47xl0"12	9.57
	Interamnia	0	0	4	5	-	2.09x10-"	4.16
	Herculina	0	0	3	5	-	7.22x 10"12	8.24
	Hebe	0	0	2	4	2: 1	4.18X10-12	11.50
	Flora	0	0	2	4	-	1.65x 10-12	12.10
	Melpomene	0	0	2	4	-	1.83X10-12	21.20
	Fortuna	0	0	2	4	-	4.49x 10-12	4.65
	Euphrosyne	0	0	2	4	3: 1	1.10x10-"	7.09
	Amphitrite	0	0	2	3	13: 6	6.29X10-12	2.70

Table 6 .

 6 2: Ranking of the asteroids according to the theoretical perturbations on the semi-major axis and longitude of Mars with periods smaller than 50 years.

	Asteroid		number of coeff.		Cornm.	mass	a(m)/rn
	n°name		AAm	A a	AAm	PM : Pa	m
			Ca/10	ca/10	ca/100	ca/100		[Mol	[%]
	1	Ceres	15	2	44	14	-	4.75xlO"10	0.07
	2	Pallas	9	2	51	13	-	l.OOxlO"10	1.02
	4	Vesta	10	1	41	9	-	1.36xlO"10	0.09
	7	Iris	4	2	37	7	-	1.41xl0-11	4.04
	9	Métis	2	1	17	3	-	1.45xl0-11	1.45
	27	Euterpe	1	0	19	4	-	5.38xl0~12	1.17
	14	Irene	0	0	18	3	-	1.41 x 10~n	1.09
	324	Bamberga	0	0	15	2	7: 3	7.94xl0~12	7.19
	15	Eunomia	0	0	13	3	-	1.64xl0-11	2.66
	654	Zelinda	0	0	10	2	-	1.36xl0~12	-
	16	Psyché	0	0	9	0	-	3.38xl0"u	3.80
	3	Juno	0	0	8	0	-	8.42x 10"12	4.52
	10	Hygiea	0	0	7	0	-	4.54xl0-11	0.77
	41	Daphné	0	0	5	2	-	3.47x 10"12	9.57
	532	Herculina	0	0	6	1	-	7.22xl0~12	8.24
	13	Egeria	0	0	4	1	11: 5	5.89xl0~12	10.30
	19	Fortuna	0	0	4	1	-	4.49xl0~12	4.65
	344	Desiderata	0	0	3	2	-	1.52x 10~12	-
	511	Davida	0	0	5	0	3: 1	2.40xl0"n	3.25
	704	Interamnia	0	0	4	1	-	2.09xl0-u	4.16
	18	Melpomene	0	0	3	0	-	1.83xl0~12	21.20
	29	Amphitrite	0	0	2	1	13: 6	6.29x 10~12	2.70
	52	Europa	0	0	3	0	-	1.82xl0"n	2.05
	6	Hebe	0	0	2	0	2: 1	4.18xl0"12	11.50
	8	Flora	0	0	2	0	-	1.65xl0"12	12.10
	51	Nemausa	0	0	1	1	-	2.13xl0"12	11.50
	105	Artémis	0	0	1	1	-	l.llxlO-12	-
	187	Lamberta	0	0	2	0	12: 5	1.49x 10~12	-
	192	Nausikaa	0	0	1	1	2: 1	7.25x 10~13	40.90
	275	Sapientia	0	0	1	1	-	1.17xl0-12	26.80
	313	Chaldaea	0	0	1	1	-	5.88xl0~13	-
	356	Liguria	0	0	1	1	-	1.49xl0~12	18.60
	405	Thia	0	0	2	0	-	1.28x 10-12	-
	419	Aurélia	0	0	1	1	-	1.41xl0"12	31.80
	675	Ludmilla	0	0	1	1	-	1.53xl0~12	42.80
	747	Winchester	0	0	2	0	-	3.33xl0"12	30.40
	409	Aspasia	0	0	1	0	11: 5	2.78x 10"12	23.00
	451	Patientia	0	0	1	0	-	7.50xl0"12	6.82
	521	Brixia	0	0	1	0	-	1.02xl0"12	-
	1

Table 6 .

 6 .he surn of the deflections of the Mars orbit 9\ (in the heliocentric reference frame) over a period of 50 years. The orbit of asteroids is computed from the perturbations of ail planets and that of Mars is directly obtained with the solution of JPL planetary ephemerides De405. As already seen in Sec. 2.1.4, the impulse approach does not.

	Asteroid	number of coeff.	cr(m)/m
	n°name	A a	AAm	[%]
	2	Pallas	1	9	1.02
	14	Irene	0	13	1.09
	324	Bamberga	0	1	7.19
	41	Daphné	4	0	9.57
	13	Egeria	1	1	10.30
	344	Desiderata	0	1	-
	51	Nemausa	0	1	11.50
	28	Bellona	0	1	-
	275	Sapientia	1	3	26.80
	356	Liguria	3	0	18.60
	393	Lampetia	0	3	-
	405	Thia	0	2	-
	675	Ludmilla	0	3	42.80
					V

3: Asteroids for which some coefficients with a period smaller than 100 years, varied at least 100% with the variation of the semi-major axis. The standard déviations marked by mean they are superior to 50%.

Table 6 .

 6

	6.2.	TWO NEW LISTS

Table 6 .

 6 5: Ranking of the asteroids according to the cumulative deflections 0\ (in mas) of the Mars orbit over a period of 50 years from 2010.

	Asteroid	Y	Asteroid	Y
	n°name	[mas]	n°name	[mas]
	1	Ceres	15.853	18	Melpomene	0.092
	4	Vesta	7.374	194	Prokne	0.084
	10	Hygiea	1.339	120	Lachesis	0.083
	2	Pal las	1.267	144	Vibilia	0.076
				747	Winchester	0.075
	16	Psyché	0.840	444	Gyptis	0.073
	7	Iris	0.792	259	Aletheia	0.069
	27	Euterpe	0.761	804	Hispania	0.061
	15	Eunomia	0.504	5	Astraea	0.060
	52	Europa	0.445	89	Julia	0.057
	511	Davida	0.383	165	Loreley	0.050
	9	Métis	0.332	268	Adorea	0.049
	29	Amphitrite	0.246	211	Isolda	0.047
	13	Egeria	0.231	91	Aegina	0.044
	324	Bamberga	0.231	488	Kreusa	0.041
	3	Juno	0.199	12	Victoria	0.040
	65	Cybele	0.164	20	Massalia	0.037
	48	Doris	0.154	404	Arsinoe	0.036
	64	Angelina	0.142	216	Kleopatra	0.036
	6	Hebe	0.138	114	Kassandra	0.030
	31	Euphrosyne	0.133	152	Atala	0.016
	128	Nemesis	0.128	58	Concordia	0.014

Table 6 .

 6 6: List of close approaches between Mars and the asteroids ordered by increasing minimal distance. The standard déviations marked by mean they are superior to 100%.

	Asteroid	Date	djnin	mass	cr(m)/m
	n°name	JJ-MM-YY	[AU]	M©	[%]
		Zelinda	12-10-2043	0.25	1.36xl0~12	65.30
		Erigone	1-8-2017	0.34	2.52xl0-13	-
		Euterpe	30-11-2026	0.34	5.38xl0~12	1.17
		Zelinda	5-5-2019	0.36	1.36xl0-12	65.30
		Iris	3-11-2054	0.36	1.41xl0-n	1.27
		Chaldaea	30-10-2017	0.37	5.88xl0~13	65.40
		Ariadne	15-6-2014	0.40	1.88xl0~13	-
		Iris	24-9-2058	0.40	1.41xl0-11	1.27
		Desiderata	24-7-2029	0.40	1.52xl0~12	98.80
		Flora	4-1-2021	0.41	1.65xl0"12	12.10
		Melpomene	25-3-2013	0.42	1.83xl0"12	21.20
		Diana	21-8-2049	0.44	1.15xl0~12	24.10
		Artémis	9-7-2057	0.44	l.llxlO-12	-
		Bamberga	2-10-2035	0.44	7.94xlO"~12	7.19
		Bamberga	22-8-2022	0.45	7.94xl0-12	7.19
		Flora	23-5-2034	0.45	1.65xl0-12	12.10
		Hamburga	5-5-2023	0.45	4.13xl0"13	30.60
		Ambrosia.	29-3-2038	0.46	1.33x 10"13	-
		Tamara	23-9-2040	0.46	5.29xl0~13	-
		Hamburga	12-10-2047	0.46	4.13xl0"13	30.60
		Klio	9-5-2054	0.47	3.27xl0~13	-
		Prymno	21-6-2055	0.47	8.68xl0"14	-
		Chaldaea	29-12-2013	0.48	5.88xl0~13	65.40
		Virginia	15-5-2043	0.48	6.54xl0"13	-
	41	Daphné	23-2-2031	0.48	3.47xl0"12	9.57
		Peraga	22-4-2051	0.48	5.80xl0~13	10.80
		Thyra	10-10-2026	0.48	3.34xl0~13	-
		Tamara	11-7-2012	0.49	5.29xl0~13	-
		Palatia	10-9-2058	0.49	2.92xl0"13	-
		Klio	29-4-2058	0.49	3.27xl0"13

-

Table 6 .

 6 7: Ranking of asteroids according to the différence between the reference and the tested orbit for the semi-major axis, eccentricity and inclination of Mars.This Table cannot, be used as a strict ranking because it counts the apparition in the tables of resnlts, which partially reflects the importance of an asteroid upon Mars orbit. The rank is not taken into account, because it would be difficult to weight it knowing that the methods used are very different. Nevertheless, even if the three bigs are not the first in this table-we hâve not found very close approaches with Mars (Tab. 6.6)-, they are the most perturbing because they are almost always at the three first ranks of the other tables far above the others. It is interesting to see the impressive proportion of the asteroids that Williams had already selected, 30 out of 52 with the data of which he had at his disposai at this time. Two asteroids, not listed by Williams, clearly appear as very important for the fît of Mars orbit, 27 Euterpe and 654 Zelinda, which will expérience very close approaches with Mars during the future 50 years. A very interesting prédiction is that Gaia should estimate the mass of 27 Euterpe with about 1% of précision. Furthermore, among the additional asteroids to Williams'list, we can mention 105 Artémis and 313 Chaldea, both belong to the list of very close approaches. The importance of Gaia appears in the last column, since we find 43 ones out of 52, for which we expect a mass

	Semi-major axis	Eccentricity	Inclination
	a		e		i
	n°name	n°name	n°name
	7	Iris	4	Vesta	1	Ceres
	2	Pallas	1	Ceres	2	Pallas
	1	Ceres	2	Pallas	4	Vesta
	324	Bamberga	10	Hygiea	324	Bamberga
	4	Vesta	7	Iris	15	Eunomia
	14	Irene	9	Métis	511	David a
	3	Juno	19	Fortuna	10	Hygiea
	9	Métis	14	Irene	14	Irene
	511	David a	52	Europa	704	Interamnia
	52	Europa	6	Hebe	7	Iris
	192	Nausikaa	3	Juno	532	Herculina
	31	Euphrosyne	24	Thémis	16	Psyché
	78	Diana	31	Euphrosyne	52	Europa
	532	Herculina	15	Eunomia	3	Juno
	27	Euterpe	511	David a	41	Daphné
	24	Thémis	51	Nemausa	194	Prokne
	372	Palma	94	Aurora	372	Palma
	8	Flora	11	Parthenope	747	Winchester
	216	Kleopatra	230	Athamantis	6	Hebe
	747	Winchester	532	Herculina	18	Melpomene
	405	Thia	488	Kreusa	29	Amphitrite
	654	Zelinda	27	Euterpe	89	Julia
	145	Adeona	192	Nausikaa	185	Eunike
	5	Astraea	63	Ausonia	451	Patientia
	505	Cava	154	Bertha	702	Alauda
	51	Nemausa	130	Elektra	51	Nemausa
	98	Ianthe	372	Palma	54	Alexandra
	94	Aurora	324	Bamberga	173	Ino
	105	Artémis	16	Psyché	31	Euphrosyne
	313	Chaldaea	41	Daphné	386	Siegena
	11	Parthenope	106	Dione	354	Eleonora
	10	Hygiea	105	Artémis	96	Aegle
	488	Kreusa	259	Aletheia	8	Flora
	194	Prokne	554	Peraga	130	Elektra
	109	Félicitas	313	Chaldaea	654	Zelinda
	20	Massalia	8	Flora	804	Hispania
	415	Palat.ia	704	Interamnia	247	Eukrate
	63	Ausonia	120	Lachesis	27	Euterpe
	230	Athamantis	29	Amphitrite	375	Ursula

Table 7 .

 7 1: Comparison between the précision of the densities from the actual errors on the niasses (ri and the future ones from Gaia data cr2. The binary asteroids are marked by **.

	Asteroid	Taxon.	Mass m	d	a{d)	P	Ap	A'p	Ap/ A'p
	n°name	class1	[Me]	[km]	[km]	[g/cm3]	[%]	[%]
	Ceres	C	4.81xlO~10	848.4	19	3.01	7.2	7.0	1.0
	Pallas	B	l.OOxlO-10	498.1	18	3.09	12.3	12.3	1.0
	Juno	Sk	2.09xl0-11	233.9	11	6.24	31.1	18.9	1.6
	Vesta	V	1.36xlO-10	468.3	26	5.06	17.8	17.2	1.0
	Hebe	s	6.90xl0~12	185.2	2	4.15	17.7	16.2	1.1
	Iris	s	1.41xl0~n	199.8	10	6.75	24.9	19.1	1.3
	Hygiea	c	4.54xl0~n	407.1	6	2.57	7.9	5.8	1.4
	Parthenope	Sk	2.56xl0~12	153.3	3	2.71	8.8	13.4	0.7
	Eunomia	s	1.64xl0~n	255.3	15	3.76	21.3	20.3	1.1
	Psyché	X	3.38X10-11	253.2	4	7.96	13.0	8.5	1.5
	Massalia	s	2.44xl0"12	145.5	9	3.03	35.6	27.1	1.3
	Kalliope**	X	4.00xl0~12	181.0	4	2.58	10.7	20.7	0.5
	Amphitrite	s	7.70xl0~12	212.2	6	3.08	26.5	12.3	2.2
	Euphrosyne	Cb	9.40xl0~12	255.9	11	2.14	68.8	20.6	3.3
	Eugenia**	c	2.82xl0~12	214.6	4	1.09	7.0	7.2	1.0
	Doris	Ch	6.10xl0~12	221.8	7	2.14	59.2	16.5	3.6
	Europa	c	1.27X10-11	302.5	5	1.75	25.0	7.4	3.4
	Cybele	Xc	5.80xl0"12	237.3	4	1.66	31.2	18.2	1.7
	Sylvia**	X	7.39xl0~12	260.9	13	1.59	15.7	25.7	0.6
	Thisbe	B	7.00xl0"12	200.6	5	3.31	21.7	10.8	2.0
	Antiope**	C	4.14xl0-13	120.1	4	0.91	11.2	39.2	0.3
	Hermione**	Ch	2.70xl0~12	209.0	4	1.13	7.3	31.7	0.2
	Loreley	Cb	1.57xl0~n	154.8	4	16.17	73.0	23.4	3.1
	Mathilde	Cb	5.19xl0~14	53.0	3	1.30	15.4	1163.4	0.0
	Bamberga	-	2.29X10"11	229.4	7	7.24	26.3	16.9	1.6
	Huenna**	C	2.38x 10~13	92.3	1	1.15	16.0	189.5	0.1
	Gyptis	C	6.30xl0~12	163.1	10	5.55	37.4	31.0	1.2
	Patientia	-	3.21xl0~n	225.0	4	10.77	28.3	12.7	2.2
	David a	c	2.40x 10-11	326.1	5	2.64	14.9	8.1	1.8
	Herculina	s	1.68xl0-11	222.4	4	5.83	22.3	13.9	1.6
	Interanmia	B	3.50X10-11	316.6	5	4.21	30.6	9.1	3.4
	Pulcova**	-	1.28xl0~12	137.1	3	1.90	8.6	31.1	0.3

Table 7 .

 7 2: Number of asteroids in each class of the relative précision on their densities p.

	Relative	Number of
	Précision	asteroids
	Ap/p < 10%	7 (7)
	Ap/p < 20%	37 (21)
	A p/p < 30%	72 (29)
	A p/p < 40%	95 (34)
	A p/p < 50%	96 (34)

Table 7 .

 7 3: Prédiction on the Gaia précision of densities between 10 and 20%.

	Asteroid	Taxonomie	Mass m	diameter d	a{d)	a(m)	A\P)
	n°name	class1	[Mo]	[km]	[km]	[%]	[%]
	Aurora	C	5.66xl0"12	204.9	3	5.1	10.4
	Thisbe	B	7.00xl0"12	200.6	5	3.4	10.8
	Diotima	C	5.99x 10"12	208.8	4	4.1	11.2
	Germania	B	3.17x 10"12	168.9	3	6.1	11.6
	Nemesis	C	4.39x 10"12	188.2	4	5.9	12.3
	Amphitrite	S	7.70x 10~12	212.2	6	2.7	12.3
	Pallas	B	l.OOxlO-10	498.1	18	1.0	12.3
	Patientia	-	3.21xl0-11	225.0	4	6.8	12.7
	Hestia	Xc	1.26xl0"12	124.1	3	4.0	12.7
	Pompeja		1.04xl0"12	116.2	2	6.5	13.0
	Parthenope	Sk	2.56xl0~12	153.3	3	7.3	13.4
	Herculina	S	1.68xl0-11	222.4	4	8.2	13.9
	Alexandra	c	3.00xl0~12	165.8	3	7.8	13.9
	Julia	K	2.29x 10"12	151.5	3	8.4	14.5
	Hebe	S	6.90xl0"12	185.2	2	11.5	16.2
	Nemausa	Ch	2.13xl0~12	147.9	2	11.5	16.4
	Doris	Ch	6.10xl0"12	221.8	7	6.3	16.5
	Bamberga	-	2.29xl0-11	229.4	7	7.2	16.9
	Flora	-	1.65xl0"12	135.9	2	12.1	17.2
	Vesta	V	1.36xlO"10	468.3	26	0.09	17.2
	Lachesis	c	3.47x 10"12	174.1	2	12.5	17.5
	Scheila	T	9.57xl0"13	113.3	2	11.6	17.7
	Cybele	Xc	5.80xl0"12	237.3	4	12.9	18.2
	Dido	Xc	2.69x 10~12	159.9	3	12.9	18.7
	Juno	Sk	2.09x 10-11	233.9	11	4.5	18.9
	Iris	S	1.41xl0"n	199.8	10	4.0	19.1
	Nuwa	Cb	2.27x 10"12	151.1	4	10.7	19.6
	Medea	-	1.66xl0"12	136.1	2	14.2	19.7
	Boliviana	X	1.37xl0"12	127.6	2	14.8	20.0

Table 7 .

 7 4: Statistical analysis of density Monte-Carlo simulation. Unrealistic densities arise frorn bad mass estimâtes frorri ground-based observations.

	Asteroid	P	Pp	P0.5	A p	°{p)		Ap vs
	n°name						n~/nt	n+/nt
			[g/cm3]		[g/< cm3]	[%]		[g/<cm3]
	Ceres	3.01	3.02	3.01	0.21	0.21	35.7	33.1	0.098	0.096
	Pallas	3.09	3.12	3.09	0.39	0.36	38.9	34.6	0.179	0.174
	Juno	6.24	6.32	6.23	1.18	0.97	42.3	36.8	0.522	0.508
	Vesta	5.06	5.15	5.05	0.87	0.91	37.5	31.6	0.411	0.388
	Hebe	4.15	4.16	4.15	0.67	0.52	40.8	39.9	0.292	0.292
	Iris	6.75	6.85	6.75	1.10	1.06	38.9	33.4	0.508	0.490
	Hygiea	2.57	2.57	2.57	0.15	0.13	38.4	36.0	0.067	0.066
	Parthenope	2.71	2.72	2.71	0.35	0.25	43.2	40.9	0.148	0.147
	Eunomia	3.76	3.84	3.76	0.73	0.70	39.5	33.4	0.337	0.322
	Psyché	7.96	7.97	7.96	0.47	0.39	39.4	38.0	0.209	0.207
	Massalia	3.03	3.10	3.02	0.82	0.67	44.4	36.7	0.362	0.349
	Kalliope	2.58	2.59	2.58	0.52	0.38	42.5	40.3	0.224	0.222
	Amphitrite	3.08	3.10	3.08	0.38	0.31	41.2	37.3	0.169	0.165
	Euphrosyne	2.14	2.17	2.14	0.44	0.33	43.9	38.6	0.191	0.186
	Eugenia	1.09	1.09	1.09	0.10	0.07	42.5	40.1	0.042	0.042
	Doris	2.14	2.15	2.13	0.35	0.26	43.8	39.3	0.152	0.149
	Europa	1.75	1.76	1.75	0.13	0.10	41.3	39.2	0.057	0.056
	Cybele	1.66	1.66	1.66	0.30	0.23	41.2	39.8	0.131	0.130
	Sylvia	1.59	1.61	1.58	0.51	0.38	45.2	39.0	0.219	0.214
	Thisbe	3.31	3.33	3.31	0.36	0.27	42.2	39.4	0.157	0.152
	Loreley1	16.17	16.27	16.15	3.79	2.77	43.2	40.1	1.624	1.594
	Bamberga	7.24	7.29	7.23	1.22	0.89	44.1	39.8	0.522	0.514
	Gyptis	5.55	5.68	5.54	1.72	1.30	45.4	38.3	0.743	0.714
	Patientia1	10.77	10.80	10.77	1.37	0.97	43.0	41.2	0.584	0.574
	David a	2.64	2.65	2.64	0.21	0.15	42.7	40.6	0.090	0.090
	Herculina	5.83	5.85	5.83	0.81	0.58	42.8	41.0	0.348	0.344
	Interamnia	4.21	4.22	4.21	0.38	0.27	43.2	40.9	0.162	0.163
	1

Table 7 .

 7 5: Détermination of new sigma by analogy with Gaussian distribution.

	Asteroid	P	Ap/p	<?i{p)/p	(p)/p	°z(p)/p
					(~68.2%)	(~95.4%)	(~99.7%)
	n°name	[.g/cm3]	[%]	[%]	[%]	[%]
		Hygiea	2.57	5.76	5.05	10.16	15.55
		Psyché	7.96	5.90	4.83	9.78	14.75
		Ceres	3.01	7.03	6.93	14.05	21.98
		Europa	1.75	7.41	5.68	11.44	17.58
		Davida	2.64	7.96	5.73	11.48	17.53
		Eugenia	1.09	8.99	6.57	13.26	20.67
		Interarnnia	4.21	9.09	6.41	12.93	19.67
		Thisbe	3.31	10.84	8.09	16.49	26.26
		Amphitrite	3.08	12.31	9.91	20.15	33.12
		Pallas	3.09	12.67	11.32	23.16	39.21
		Patientia	10.77	12.69	8.94	18.02	27.53
		Parthenope	2.71	12.82	9.05	18.20	27.73
		Herculina	5.83	13.91	9.93	20.04	30.47
		Hebe	4.15	16.20	12.40	24.89	37.37
		Iris	6.75	16.28	14.94	31.30	55.94
	48	Doris	2.14	16.45	11.87	24.22	39.95
		Bamberga	7.24	16.87	11.99	24.38	39.39
	4	Vesta	5.06	17.20	16.77	36.04	66.89
	65	Cybele	1.66	18.21	13.83	27.92	42.17
	3	J uno	6.24	18.88	14.90	31.16	54.57
		Eunomia	3.76	19.34	17.52	37.63	69.44
	22	Kalliope	2.58	20.32	14.73	29.77	46.17
	31	Euphrosyne	2.14	20.57	15.02	31.13	54.55
		Loreley	16.17	23.40	16.86	34.24	53.67
	20	Massalia	3.03	27.08	20.46	44.47	84.21
	444	Gyptis	5.55	31.00	22.01	47.10	88.19
	87	Sylvia	1.59	32.39	22.68	47.28	81.79

Table 7 .

 7 6: Comparison of the common taxonomie classes between the Bus-Binzel and Tholen

La mission astrométrique Gaia de l'agence spatiale européene (ESA) arpentera la galaxie pour en donner une carte en trois dimensions à partir de mesures astrométriques de très hautes précisions complétées par des observations photométriques et spectroscopiques. Son lancement est prévu en 2011 pour une durée de 5 ans. Il observera plus d'un milliard d'objets jusqu'à la magnitude 20, qui seront en grande majorité des étoiles. Les retombées scientifiques seront multiples et toucheront de nombreux domaines astronomiques et astrophysiques comme la physique stellaire, l'étude des naines brunes et des planètes extra-solaires, l'astronomie extragalactique, la physique fondamentale etc ... Le satellite Gaia observera aussi un grand nombre d'objets du système solaire avec plus de 350 000 astéroïdes. Les précisions sur les positions seront excellentes (entre 310 fias et 1.6 mas) comme on peut le remarquer sur la figure1, représentant la précision sur la coordonnée longitudinale pour un transit en fonction de la magnitude apparente de l'objet. Cette relation a été obtenue à partir de simulations intégrant les propriétés des CCDs qui seront utilisés par Gaia. A ces précisions, il faut ajouter la très bonne couverture de l'orbite par les futures observations du satellite, dans l'espace et le temps, pour une grande partie des astéroïdes, et plus particulièrement ceux de la ceinture principale. Ainsi, Gaia permettra de déterminer les orbites de beaucoup d'astéroïdes avec une très grande précision comme l'illustre le graphique 2 représentant la densité de distribution des précisions obtenues sur le demi-grand axe en fonction de ce dernier pour un très large échantillon d'astéroïdes.On peut remarquer que la majorité des déviations standard est située sous les 10"8 UA, ce qui laisse présager de très intéressants ajustements de masses d'astéroïdes et de paramètres globaux à partir de la dynamique de ces petits corps.Les capacités de détection du satellite Gaia ainsi que la précision astrométrique obligent à repenser des questions aussi classiques que la détermination des orbites et la détermination des masses des petits corps du système solaire. En effet, la précision des mesures de GAIA impose de baser les analyses orbitales uniquement sur les observations spatiales, les observations au sol n'amenant finalement qu'une information résiduelle, ce en dépit de leur avantage en terme de base de temps. Mais si les mesures depuis l'espace

The mass of

Ida was estimated by numerical simulations considering that its orbit is stable and not from observations. 2 URL:http://ast.ro.berkeley.edu/ fmarchis/Science/Asteroids/Huenna.html 3 The authors gave only an interval for the mass value. We considered a Gaussian distribution to compute the a of the mass détermination

24.50

Time delay of light (2.1 ± 2.3) x 10-5 (Cassini exp.) [START_REF] Bertotti | A test of general relativity using radio links with the Cassini spacecraft[END_REF] Bending of light (-1.7 ±4.5) x 10"4 ('VLDI) [START_REF] Shapiro | Measurement of the Solar Gravitational Deflection of Radio Waves using Geodetic Very-Long-Baseline Interferometry Data, 1979[END_REF] 1 Bending of light Table 4.8: Summary for the estimation of (3 (PPN), 77 (Nordtvedt) and G/G. Asteroids perturbations -0.±0.5 [START_REF] Orellana | The Nordtvedt Effect in the Trojan Asteroids[END_REF] may be the most difficult, because of numerous systematic errors which could corrupt the resuit due to the simplistic writing of this effect in the dynamic model. A more advanced study should be led for the rewriting the équation of motion in the post-Newtonian ap proximation, necessary to obtain correct (O-C). In addition, the study should also focus on the different group of asteroids, which would be more sensitive to this effect, and of which the dynamic model would be the most complété.

Part V

PROSPECTIVES ON THE

MASSES CONTRIBUTION

Chapter 6

The dynamical point of view Contents 6.1

The Mars ephemeris and the asteroid masses 151

Synthesis and prospectives 162

The importance of the asteroid masses was already tackled in Section 1.2.1, through the possible contributions of Gaia in the physics of asteroids and in the planetary ephemerides.

It is well-known that the uncertainty of the Mars orbit mainly proceeds from the lack of accuracy on the masses of more massive asteroids. Thus, in this chapter, we intend to illustrate the future impact of Gaia for Mars ephemerides by searching for the most perturbing asteroids and to list those with an expected mass estimation by Gaia. This work was led a few in the same spirit of the first exhaustive study on the subject, published by Williams in 1984 [START_REF] Williams | Determining asteroid masses from perturbations on Mars[END_REF].

6.1

The Mars ephemeris and the asteroid masses The first discussions on the perturbations of the Mars orbit by the asteroids appeared in 1984 with the papers of Bretagnon [START_REF] Bretagnon | Iinprovement of analytic planetary théories[END_REF] and Williams [START_REF] Williams | Determining asteroid masses from perturbations on Mars[END_REF]. Bretagnon, through his analytical planetary ephemerides provided the perturbations of the mean longitude of Mars by the three "bigs" (Ceres, Pallas and Vesta). The perturbations were expressed to the first and second order in asteroid mass, under the form of Fourier sériés from the longitude of Mars and the minor planet as argument, the associated amplitude and period. As for Williams, he gave a list of massive asteroids near a commensurability with the Mars period, but more interesting, he selected 36 ones from a study on the theoretical perturbations of Mars mean longitude perturbations by arbitrarly separating the short period (P < 4 years) from the long one. As Williams said, this list can be viewed as "an opportunity for measuring masses" or "a list of source noise for orbit détermination". The importance of asteroid masses on the inner planet orbits (especially Mars) was so seriously considered Synthesis and prospectives 162

Synthesis and prospectives

In this section, we summarise the whole of the previous results by ranking the asteroids with respect to the number of their apparition in the tables of results. Thus, in Table 6.8, the IAU number and the name of the asteroid are indicated and the methods from which it was selected are marked by a cross. The column H.D corresponds to the theoretical détermination by the harmonie décomposition for the two used filters 10 and 100 (Period P < 50 years). The column ^9i, corresponds to Tab. 6.5 and its three hiérarchie groups, dmin to Table 6.6, and Linear Reg. to Table 6.7. The last one shows the importance of Gaia for improving the Mars orbit through the expected standard déviations on the mass. cr(m) 125 [START_REF] Mignard | Fundamental Physics with GAIA[END_REF] In this latter table, we can notice that for the class of relative précision a(m)/m smaller than 10%, 32 asteroids are distributed over 11 taxonomie classes whereas those from ground-based observations 15 masses cover 8 classes. At the 30% mark, Gaia will bring 92 asteroid masses over 15 taxonomie classes with a net proportion for the classes C, B, S, Ch and X while at présent, only 30 asteroids are distributed over 10 classes. Thus, Gaia will expand the spectra of the studied taxonomie classes and maybe give substance to a possible relationship between certain taxonomie classes and the bulk density of asteroids.

In addition, Gaia will give access to their porosity, wliich will provide information on their internai structure and consequently on their history.