On equilibrium tides in fully convective planets and stars

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, being the turn-over 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)), where ωm,k is the apparent frequency ((patternspeed)×m) associated with a particular harmonic of the tidal forcing as viewed in the frame corotating with the primary. We concentrate on the