Tidal dynamics of extended bodies in planetary systems and multiple stars

¹ Laboratoire AIM, CEA/DSM – CNRS – Université Paris Diderot, IRFU/Service d'Astrophysique, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France e-mail: stephane.mathis@cea.fr
² LUTH, Observatoire de Paris – CNRS – Université Paris-Diderot; Place Jules Janssen, 92195 Meudon Cedex, France
³ SYRTE UMR8630, Observatoire de Paris, 61 avenue de l'Observatoire, 75014 Paris, France e-mail: christophe.leponcin-lafitte@obspm.fr

Received: 12 November 2007
Accepted: 8 December 2008

Abstract

Context. With the discovery during the past decade of a large number of extrasolar planets orbiting their parent stars at distances lower than 0.1 astronomical unit (and the launch and the preparation of dedicated space missions such as CoRoT and KEPLER), with the position of inner natural satellites around giant planets in our Solar System and with the existence of very close but separated binary stars, tidal interaction has to be studied carefully.

Aims. This interaction is usually studied with a punctual approximation for the tidal perturber. The purpose of this paper is to examine the step beyond this traditional approach by considering the tidal perturber as an extended body. To achieve this, we studied the gravitational interaction between two extended bodies and, more precisely, the interaction between mass multipole moments of their gravitational fields and the associated tidal phenomena.

Methods. We use cartesian symmetric trace free tensors, their relation with spherical harmonics and Kaula's transform enables us to analytically derive the tidal and mutual interaction potentials, as well as the associated disturbing functions in extended body systems.

Results. The tidal and mutual interaction potentials of two extended bodies are derived. In addition, the external gravitational potential of such a tidally disturbed extended body is obtained, using the Love number theory, as well as the associated disturbing function. Finally, the dynamical evolution equations for such a system are given in their more general form without any linearization. We also compare, under a simplified assumption, this formalism to the punctual case. We show that the non-punctual terms have to be taken into account for strongly deformed perturbers (J₂ ≥ 10^-2) in very close systems (a_B/R_B≤5).

Conclusions. We show how to derive the dynamical equations for the gravitational and tidal interactions between extended bodies and associated dynamics. The conditions for applying this formalism are given.