On equilibrium tides in fully convective planets and stars

Abstract

We consider the tidal interaction of a binary consisting of a fully convective primary star and a relatively compact mass. Using a normal-mode decomposition we calculate the evolution of the primary angular velocity and orbit for arbitrary eccentricity. The dissipation acting on the tidal perturbation is assumed to result from the action of convective turbulence, the effects of which are assumed to act through an effective viscosity. A novel feature of the work presented here is that, in order to take account of the fact that there is a relaxation time tc (the turnover time of convective eddies) associated with the process, this is allowed to act non-locally in time, producing a dependence of the dissipation on tidal forcing frequency. Results are expressed in terms of the Fourier coefficients of the tidal potential, assumed periodic in time. We find useful analytical approximations for these valid for sufficiently large values of eccentricity e > 0.2. We show that in the framework of the equilibrium tide approximation, when the dissipative response is frequency-independent, our results are equivalent to those obtained under the often used assumption of a constant time-lag between tidal response and forcing. We go on to consider the case when the frequency dependence of the dissipative response is ∝ 1/[1 + (ωm,ktc)p], where ωm,k is the apparent frequency (pattern speed ×m) associated with a particular harmonic of the tidal forcing as viewed in the frame corotating with the primary. We concentrate on the case ωm,ktc≫ 1, which is thought to be appropriate to many astrophysical applications. We study numerically and analytically the orbital evolution of the dynamical system corresponding to different values of the parameter p. We present results from which the time to circularize from large eccentricity can be found. We find that when p 1 differs drastically. In that case the system evolves through a sequence of spin–orbit corotation resonances with Ωr/Ωorb=n/2, where Ωr and Ωorb are the rotation and orbital frequencies and n is an integer. When p= 2 we find an analytic expression for the evolution of semimajor axis with time for arbitrary eccentricity assuming that the moment of inertia of the primary is small. We confirm the recent finding of Ivanov & Papaloizou, on the basis of an impulsive treatment of orbits of high eccentricity, that equilibrium tides associated with the fundamental mode of pulsation and dissipative processes estimated using the usual mixing-length theory of convection seem to be too weak to account for the orbital circularization from large eccentricities of extrasolar planets. Generalizations and limitations of our formalism are discussed.