The equilibrium tide in stars and giant planets: I - the coplanar case

13 pages, 5 figures, Accepted for publication in Astronomy & Astrophysics, DOI 10.1051/0004-6361/201118160International audienceSince 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, and these are still not well understood. 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 by the quality factor Q, as 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. 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 decoupled from the dynamical tide. The tidal kinetic energy is dissipated into heat through 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; it is responsible for the imaginary part of the disturbing function, which is implemented in the dynamical evolution equations, from which one derives characteristic evolution times. The rate at which the system evolves depends on the physical properties of tidal dissipation, and specifically on how the eddy viscosity varies with tidal frequency and on the thickness of the convective envelope for the fluid equilibrium tide. At low frequency, this tide retards by a constant time delay, whereas it lags by a constant angle when the tidal frequency exceeds the convective turnover rate