Anelastic tidal dissipation in multi-layer planets

¹ LUTH, Observatoire de Paris – CNRS – Université Paris Diderot, 5 place Jules Janssen, 92195 Meudon Cedex, France
e-mail: francoise.remus@obspm.fr; jean-paul.zahn@obspm.fr
² IMCCE, Observatoire de Paris – UMR 8028 du CNRS – Université Pierre et Marie Curie, 77 avenue Denfert-Rochereau, 75014 Paris, France
e-mail: lainey@imcce.fr
³ Laboratoire AIM Paris-Saclay, CEA/DSM – CNRS – Université Paris Diderot, IRFU/SAp Centre de Saclay, 91191 Gif-sur-Yvette, France
e-mail: stephane.mathis@cea.fr
⁴ LESIA, Observatoire de Paris – CNRS – Université Paris Diderot – Université Pierre et Marie Curie, 5 place Jules Janssen, 92195 Meudon, France

Received: 6 December 2011
Accepted: 24 February 2012

Abstract

Context. Earth-like planets have viscoelastic mantles, whereas giant planets may have viscoelastic cores. The tidal dissipation of these solid regions, which are gravitationally perturbed by a companion body, strongly depends on their rheology and the tidal frequency. Therefore, modeling tidal interactions provides constraints on planets’ properties and helps us to understand their history and evolution, in either our solar system or exoplanetary systems.

Aims. We examine the equilibrium tide in the anelastic parts of a planet for every rheology, and by taking into account the presence of a fluid envelope of constant density. We show how to obtain the different Love numbers describing its tidal deformation, and discuss how the tidal dissipation in the solid parts depends on the planet’s internal structure and rheology. Finally, we show how our results may be implemented to describe the dynamical evolution of planetary systems.

Methods. We expand in Fourier series the tidal potential exerted by a point mass companion, and express the dynamical equations in the orbital reference frame. The results are cast in the form of a complex disturbing function, which may be implemented directly in the equations governing the dynamical evolution of the system.

Results. The first manifestation of the tide is to distort the shape of the planet adiabatically along the line of centers. The response potential of the body to the tidal potential then defines the complex Love numbers, whose real part corresponds to the purely adiabatic elastic deformation and the imaginary part accounts for dissipation. The tidal kinetic energy is dissipated into heat by means of anelastic friction, which is modeled here by the imaginary part of the complex shear modulus. This dissipation is responsible for the imaginary part of the disturbing function, which is implemented in the dynamical evolution equations, from which we derive the characteristic evolution times.

Conclusions. The rate at which the system evolves depends on the physical properties of the tidal dissipation, and specifically on (1) how the shear modulus varies with tidal frequency, (2) the radius, and (3) the rheological properties of the solid core. The quantification of the tidal dissipation in the solid core of giant planets reveals a possible high dissipation that may compete with dissipation in fluid layers.