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Observatoire de Paris - Institut de mécanique céleste et de calcul des éphémérides - UMR 8028 du CNRS - 77 Av. Denfert-Rochereau, F-75014 PARIS

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Viscoelastic Tides: Models for Use in Celestial Mechanics

C.G. Ragazzo, Universidade de São Paulo

  • Salle Danjon. Paris.

In this talk I will present equations for the motion of linear viscoelastic bodies interacting under gravity. The equations are fully three dimensional and allow for the integration of the spin, the orbit, and the deformation of each body. The equations are obtained within a finite dimension Lagrangian framework with dissipation function.The main result is an "Association Principle" that establishes a connection between any spring-dashpot rheological model and the rheology of the deformation variables in the time domain. The theory is applied to the Earth (solid part plus oceans) using a Wiechert rheological model. The model will be compared to another one proposedby Boué, Correia, and Laskar in 2016. Finally the effect of deformation inertia on tides will be discussed. Most of this work was done in collaboration with L. Ruiz.

Modélisation stochastique de la dynamique à temps long du système solaire

E. Woillez, ENS Lyon

  • Salle Danjon. Paris.

Le système solaire présente une dynamique chaotique sur des temps supérieurs à 10 millions d’années [2, 3]. Ce résultat induit un changement total de paradigme : la position des planètes ne peut être prédite que de façon probabiliste, sur des temps de l’ordre de l’âge du système solaire. Le système solaire ne présente pas assez de degrés de liberté pour qu’une limite thermodynamique N ->+infini soit pertinente pour la dynamique des planètes. Je montrerai donc que les outils de physique statistique sont utilisables dans une limite asymptotique t ->+infini , c’est-à-dire pour des prédictions à temps long de l’état du système solaire. Mon exposé sera illustré par 2 exemples particuliers. Le premier est l’influence de la ceinture d’astéroïdes sur la dynamique de Mars [4]. On peut montrer que la trajectoire fortement chaotique des astéroïdes induit une dispersion de la longitude moyenne de Mars sur un temps de l’ordre de 20 millions d’années. Je discuterai dans un deuxième temps un travail en cours [1] sur la dynamique de Mercure. On verra comment la technique dite de moyennisation stochastique pourrait permettre de prédire l’entrée en résonance de la planète avec Jupiter, et la déstabilisation de son orbite. L’exposé vise un auditoire familier de la mécanique céleste, mais non-spécialiste de la physique statistique. J’insisterai donc davantage sur les idées principales qui visent à l’application de la physique statistique en mécanique céleste, et j’éviterai au maximum de présenter des formules théoriques. références: [1] K. Batygin, A. Morbidelli, and M. J Holman. Chaotic disintegration of the inner solar system. The Astrophysical Journal, 799(2):120, 2015. [2] J. Laskar. A numerical experiment on the chaotic behaviour of the solar system. Nature, 338(6212):237–238, mar 1989. [3] J. Laskar. The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zones. Icarus, 88(2):266–291, 1990. [4] E. Woillez and F. Bouchet. Long-term influence of asteroids on planet longitudes and chaotic dynamics of the solar system. Astronomy & Astrophysics, 607:A62, 2017.

Why collisions are negligible

A. Knauf, Universität Erlangen-Nürnberg

  • Salle Danjon. Paris.

We give a non-technical overview over the recent papers arXiv:1802.08566 and arXiv:1802.08564 with Stefan Fleischer. The first shows that for a volume preserving dynamical system, the set of wandering points passing through a sequence of Poincaré surfaces whose sizes decrease to zero, has measure zero. In the second this is applied to show that for a large class of n-body systems, the set of phase space points leading to collision has measure zero. Further applications are indicated.

Hill stability in the AMD framework

Antoine Petit, ASD/IMCCE, CEA

  • Salle Danjon. Paris.

In a two-planet system, due to Sundman (1913) inequality, a topological boundary can forbid close encounters between the two planets for infinite time. A system is said Hill stable if it verifies this topological condition. Hill stability is widely used in the study of extra solar planets dynamics. However people often use the coplanar and circular orbits approximation. In this work, we explain how the Hill stability can be understood in the framework of Angular Momentum Deficit (AMD). In the secular approximation, the AMD allows to discriminate between a priori stable systems and systems for which a more in depth dynamical analysis is required. We show that the general Hill stability criterion can be expressed as a function of only the semi major axes, the masses and the total AMD of the system. The proposed criterion is only expanded in the planets-to-star mass ratio $\epsilon$ and not in the semi-major axis ratio, in eccentricities nor in the mutual inclination. Moreover the development in $\epsilon$ remains excellent up to values of about $10^{-3}$ even for two planets with very different mass values. We performed numerical simulations in order to highlight the sharp change of behaviour between Hill stable and Hill unstable systems. We show that Hill stable systems tend to be very regular whereas Hill unstable ones often lead to rapid planet collisions. We also remind that Hill stability does not protect from the ejection of the outer planet. References : Laskar, Petit, AMD-stability and the classification of planetary systems, 2017, A&A Petit, Laskar, Boué, AMD-stability in the presence of first-order MMR, 2017, A&A Petit, Laskar, Boué, Hill stability in the AMD framework, 2018 submitted.

Principe du prolongement en hydrodynamique

T. Tokieda, Stanford University

  • Salle Danjon. Paris.

Comment varie le niveau d'eau d'un écoulement le long d'un canal, quand on met un gendarme couché au fond ? Ce problème sera résolu par un simple dessin inhabituel et, j'espère, intéressant.

Index theory for symplectic paths with applications to linear stability in N-body problem

QingLong Zhou, Université du Zhejiang

  • Salle Danjon. Paris.

I will give an introduction on the Maslov-type index theory for symplectic matrix paths which was defined by C.Conley, E.Zehnder, Y.Long and C.Viterbo. Next, we will study its iteration theory which turned out to be a powerful tool in the study of various problems on periodic solutions of Hamiltonian systems. Using these tools, we will study the linear stability properties of the homographic solutions which are generated from Lagrangian and Euler central configurations.

Effets de marée dans les satellites glacés de Saturne. Le dilemme Encelade versus Mimas.

Sylvio Ferraz-Mello, IAG, São Paulo, Brésil

  • Salle Danjon. Paris.

One of the puzzling phenomena discovered by the Cassini mission is the existence of strong geysers ejecting plumes of water near the South Pole of Enceladus. These plumes evidence the existence of non primordial heat sources in the interior of the satellite, much more efficient than indicated by classical tidal theories. The heat flowing through Enceladus surface is estimated at 5-16 GW and the tidal heat generation estimated by those theories is ~1 GW. However, the creep tide theory (Ferraz-Mello, Cel. Mech. Dyn. Astron. 1913, 1915), shows that the observed heat flow is simply associated with the fact that the outer layers have low viscosity (ice near the melting point). The comparison with Mimas, with no tectonic activity due to internal heating and viscosity at least one order of magnitude larger (ice at temperatures well below the melting point) are a clue for a non-stationary process in which an increase in temperature means a decrease in the viscosity and a larger dissipation. Such a process may have been triggered by some transitory event enhancing the eccentricity of Enceladus and may have been progressing slowly, subsisting even after the eccentricity was damped to its current value.