The equilibrium tide in stars and giant planets

I. The coplanar case

¹ LUTH, Observatoire de Paris, CNRS, Université Paris Diderot, 5 place Jules Janssen, 92190 Meudon, France
e-mail: [email protected]; [email protected]
² IMCCE, Observatoire de Paris, UMR 8028 CNRS, UPMC, USTL, 77 avenue Denfert-Rochereau, 75014 Paris, France
³ Laboratoire AIM, CEA/DSM, CNRS, Université Paris Diderot, IRFU/SAp Centre de Saclay, 91191 Gif-sur-Yvette, France
e-mail: [email protected]
⁴ LESIA, Observatoire de Paris, CNRS, UPMC, Université Paris Diderot, 5 place Jules Janssen, 92190 Meudon, France

Received: 27 September 2011
Accepted: 9 May 2012

Abstract

Context. Since 1995, more than 500 extrasolar planets have been discovered orbiting very close to their parent star, where they experience strong tidal interactions. Their orbital evolution depends on the physical mechanisms that cause tidal dissipation, which remain poorly understood.

Aims. We refine the theory of the equilibrium tide in fluid bodies that are partly or entirely convective, to predict the dynamical evolution of the systems. In particular, we examine the validity of modeling the tidal dissipation using the quality factor Q, which is commonly done. We consider here the simplest case where the considered star or planet rotates uniformly, all spins are aligned, and the companion is reduced to a point mass.

Methods. We expand the tidal potential as a Fourier series, and express the hydrodynamical equations in the reference frame, which rotates with the corresponding Fourier component. 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 star or planet adiabatically along the line of centers. This generates the divergence-free velocity field of the adiabatic equilibrium tide, which is stationary in the frame rotating with the considered Fourier component of the tidal potential; this large-scale velocity field is decoupled from the dynamical tide. The tidal kinetic energy is dissipated into heat by means of turbulent friction, which is modeled here as an eddy-viscosity acting on the adiabatic tidal flow. This dissipation induces a second velocity field, the dissipative equilibrium tide, which is in quadrature with the exciting potential; this field is responsible for the imaginary part of the disturbing function, which is implemented in the dynamical evolution equations, from which one derives the characteristic evolutionary times.

Conclusions. The rate at which the system evolves depends on the physical properties of the tidal dissipation, and specifically on both how the eddy viscosity varies with tidal frequency and the thickness of the convective envelope for the fluid equilibrium tide. At low frequency, this tide is retarded by a constant time delay, whereas it lags behind by a constant angle when the tidal frequency exceeds the convective turnover rate.