Seminars ASD

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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Relative equilibrium configurations of gravitationally interacting rigid bodies

Rick Moeckel, University of Minnesota

  • salle RDC bâtiment B . Paris.

Consider a collection of n rigid, massive bodies interacting according to their mutual gravitational attraction. A relative equilibrium motion is one where the entire configuration rotates rigidly and uniformly about a fixed axis — all of the bodies are phase locked. Such a motion is possible only for special positions and orientations of the bodies. A minimal energy motion is one which has the minimum possible energy in its fixed angular momentum level. While every minimal energy motion is a relative equilibrium motion, the main result here is that a relative equilibrium motion of n >= 3 disjoint rigid bodies is never an energy minimizer. Since energy minimizers are the expected final states produced by tidal interactions, phase locking of 3 or more bodies will not occur.

Hyperbolic scattering in the N-body problem

Rick Moeckel, University of Minnesota

  • salle 1525-502 . Jussieu.

It is a classical result that in the N-body problem with positive energy, all solutions are unbounded in both forward and backward time. If all of the mutual distances between the particles tend to infinity with nonzero speed, the solution in called purely hyperbolic. In this case there is a well-defined asymptotic shape of the configuration of N points. We consider the scattering problem for solutions which are purely hyperbolic in both forward and backward time: given an initial shape at time minus infinity, which final shapes at time plus infinity can be reached via purely hyperbolic motions ? I will describe some promising, preliminary work on this problem using a variation on McGehee's blow-up technique. After a change of coordinates and timescale we obtain a well-defined limiting flow at infinity and use it to get Chazy-type asymptotic estimates on the positions of the bodies and to study scattering solutions near infinity. This is joint work in progress, with G. Yu, R. Montgomery and N. Duignan.

Asynchronously rotating ocean planets of low eccentricities

Pierre Auclair-Desrotour, Université de Berne

  • Salle Danjon. Paris.

Eccentricity tides generate a torque that can drive an ocean planet towards asynchronous rotation states of equilibrium when enhanced by resonances associated with the oceanic tidal modes. We investigate the impact of eccentricity tides on the rotation of rocky planets hosting a thin uniform ocean and orbiting cool dwarf stars such as TRAPPIST-1, with orbital periods ~1-10 days. Combining the linear theory of oceanic tides in the shallow water approximation with the Andrade model for the solid part of the planet, we develop a global model including the coupling effects of ocean loading, self-attraction, and deformation of the solid regions. We derive from this model analytic solutions for the tidal torque exerted on the planet. These solutions are used with realistic values of parameters provided by advanced models of the internal structure to explore the parameter space both analytically and numerically. Our model allows us to fully characterize the frequency-resonant tidal response of the planet, and particularly the features of resonances associated with the oceanic tidal modes (eigenfrequencies, resulting maxima of the tidal torque and Love numbers) as functions of the planet parameters (mass, radius, Andrade parameters, ocean depth and Rayleigh drag frequency). Resonances associated with the oceanic tide decrease the critical eccentricity beyond which asynchronous rotation states distinct from the usual spin-orbit resonances can exist. We provide an estimation and scaling laws for this critical eccentricity, which is found to be lowered by one order of magnitude, switching from ~0.3 to ~0.05 in typical cases and to 0.01 in extremal ones.

An application of Groebner bases theory to the problem of orbit determination with very short arcs of observations

Giovanni Federico Gronchi, Université de Pise

  • Salle Danjon. Paris.

We consider the orbit computation problem for a Solar system body observed from a point on the surface of the Earth. In our case the available data are two ‘attributable’ vectors, each containing two angular positions $\alpha, \delta$ and their angular rates $\dot\alpha, \dot\delta$ at two different epochs. The unknowns are the radial distances and velocities $\rho_1$, $\rho_2$, $\dot{\rho}_1$, $\dot{\rho}_2$ of the observed body at these epochs. Using the first integrals of Kepler's motion we can write algebraic equations for this problem, which can be put in polynomial form. From these we obtain a univariate polynomial equation of degree 9 in one of the radial distances. Using Groebner bases theory we show that this equation has the minimum degree among the univariate polynomial equations in $\rho_1$ or $\rho_2$ that are consequence of the conservation laws of Kepler’s problem, provided that we drop the dependence between the inverse of the heliocentric distance $1/|r|$ of the observed body and the unknown radial distance. Some applications of this method using real observations of asteroids will also be shown.

Les invariants de type J+ d'Arnold pour les systèmes de Stark-Zeeman à deux centres

Lei ZHAO, Université d'Augsbourg

  • Salle Danjon. Paris.

Pour une courbe plongée dans le plan, Arnold a défini trois invariants parmi lesquels le $J^+$, qui a été adapté par Cieliebak-Frauenfelder-van Koert 2017 en deux invariants pour les orbites périodiques des systèmes de Stark-Zeeman, qui généralisent par exemple le problème de Kepler plan dans un repère tournant ou le problème restreint plan des trois corps. Ces invariants sont invariants par homotopie de la famille générique des orbites périodiques des systèmes de Stark-Zeeman. Dans cet exposé je vais expliquer une construction d'invariants de type $J^+$ pour les systèmes de Stark-Zeeman à deux centres. Il s'agit d'une collaboration avec Kai Cieliebak et Urs Frauenfelder.

Sur la divergence des formes normales de Birkhoff

Raphael Krikorian, UPMC

  • rez-de-chaussée, bât. B. Paris.

Un hamiltonien réel analytique (ou un difféomorphisme symplectique) admettant un point fixe elliptique non-résonnant est toujours formellement conjugué à un hamiltonien intégrable formel, sa forme normale de Birkhoff. On sait depuis Siegel (1954) que la conjugaison formelle réduisant le hamiltonien est en général divergente et Hakan Eliasson a demandé si la forme normale de Birkhoff elle-même pouvait être divergente. Perez-Marco a démontré en 2001 que si l'on fixe la partie quadratique du hamiltonien (ou de façon équivalente le vecteur de fréquences à l'origine) on avait la dichotomie suivante : soit la forme normale de Birkhoff est toujours convergente, soit elle diverge génériquement. Il est par ailleurs possible de donner des exemples de parties quadratiques pour lesquels la seconde partie de la dichotomie précédente (la divergence générique) est vraie. Le but de cet exposé est de présenter et de donner la preuve du théorème suivant : pour tout vecteur de fréquences à l'origine diophantien, la forme normale de Birkhoff d'un hamiltonien réel analytique générique (admettant deux degrés de liberté ou plus) est divergente. La démonstration repose sur le fait que la convergence de l'objet formel qu'est la forme normale de Birkhoff a des conséquences dynamiques : par exemple, pour des hamiltoniens réels analytiques à deux degrés de liberté dont le vecteur des fréquences à l'origine est diophantien, une abondance anormale de tores invariants.

Generic Finiteness for a class of symmetric central configurations of the six-body problem

Thiago Dias , UFRPE, Recife, Brésil

  • Salle Danjon. Paris.

Central configurations are important objects in celestial mechanics. In particular, they are initial conditions for the only explicit solutions of the n-body problem. One of the leading open questions in the central configurations theory is the finiteness problem: “Consider n bodies with positive masses m1, ...,mn, is the number of central configurations finite?” Chazy and Wintner proposed this question which appears in the Smale’s list for the Mathematicians of the 21st century. In 2006 Moeckel and Hampton used the BKK method to prove finiteness for planar central con?gurations of the four-body problem. In 2012 Albouy and Kaloshin proved the generic finiteness of planar central configurations of the five-body problem. They studied the behavior of the unbounded singular sequences of complex central configurations. In the planar case, there are no general results if the number of bodies is 6 or more. Another possible tool to study the finiteness problemis the complex and computational algebraic geometry. The goal is to compute the dimension of an algebraic variety derived from the central configuration system. The Jacobian criterion reduces the determination of the dimension to the computation of the rank of the Jacobian matrix. Applying Laura-Andoyer equations to study classes of symmetrical central configurations we get polynomial systems with “many” variables and “few” terms. In these cases the Jacobian method is useful. We will use it and a relaxed concept of Groebner basis to prove the generic ?niteness for symmetrical central configurations of the 6-body problem in the case where four bodies are in its symmetry line. This is a joint work with Bo-Yu Pan - National Tsing Hua University, Taiwan.

Unraveling the interior structure of planets and exoplanets using ab initio equations of states

Stéphane Mazevet, LUTH, Observatoire de Paris

  • Salle Danjon. Paris.

Using ab initio molecular dynamics simulations, we recently calculated the equations of statefor the main constituents of planetary interiors: H, He, H2O, MgSiO3(MgO,SiO2) and Fe. These equations of states are multi-phases, include liquid and solid phases, and aim at building benchmark planetary and exoplanetary interior models solely based on ab initio predictions. This talk will concentrate on Jupiter. We will review how our current understanding of the behavior of these basic constituents at extreme density temperature conditions modifies our current understanding of Jupiter interior, not only for the envelop where metallization of hydrogen and hydrogen-helium demixing is the issue, but also for the core where the high-pressure melting properties of iron, water, and silicates bring a new understanding on the nature of giant planet cores. This work is supported by the University Paris Sciences et Lettres (PSL) and the IRIS project Origines and conditions for the emergence of life, and the the French Agence National de la Recherche under contract PLANETLAB ANR-12-BS04-0015. L. Caillabet, et al., Phys. Rev. B 83, 094101 (2011). F. Soubiran et al., Phys. Rev. B 87,165114 (2013). J. Bouchet et al., Phys. Rev. B 86, 115102 (2013). A. Denoeud et al., Phys. Rev. Lett. 113, 116404 (2014). S. Mazevet et al., Phys. Rev. B 92, 014105 (2015). M. Harmand et al., Phys. Rev. B 92, 024108 (2015). A. Denoeud, S. Mazevet et al., Phys. Rev. E 94, 031201 (2016). G. Chabrier, S. Mazevet, F. Soubiran, accepted A&A (2018). S. Mazevet, A. Licari, G. Chabrier, A. Potekin, accepted A&A (2018). R. Musella, S. Mazevet, F. Guyot, accepted PRB (2018).