Final spin states of eccentric ocean planets

Abstract

Eccentricity tides generate a torque that can drive an ocean planet towards asynchronous rotation states of equilibrium when enhanced by resonances associated with the oceanic tidal modes. We investigate the impact of eccentricity tides on the rotation of rocky planets hosting a thin uniform ocean and orbiting cool dwarf stars such as TRAPPIST-1, with orbital periods ~1-10 days. Combining the linear theory of oceanic tides in the shallow water approximation with the Andrade model for the solid part of the planet, we develop a global model including the coupling effects of ocean loading, self-attraction, and deformation of the solid regions. We derive from this model analytic solutions for the tidal Love numbers and torque exerted on the planet. These solutions are used with realistic values of parameters provided by advanced models of the internal structure and tidal oscillations of solid bodies to explore the parameter space both analytically and numerically. Our model allows us to fully characterise the frequency-resonant tidal response of the planet, and particularly the features of resonances associated with the oceanic tidal modes (eigenfrequencies, resulting maxima of the tidal torque and Love numbers) as functions of the planet parameters (mass, radius, Andrade parameters, ocean depth and Rayleigh drag frequency). Resonances associated with the oceanic tide decrease the critical eccentricity beyond which asynchronous rotation states distinct from the usual spin-orbit resonances can exist. We provide an estimation and scaling laws for this critical eccentricity, which is found to be lowered by roughly one order of magnitude, switching from ~0.3 to ~0.06 in typical cases and to ~0.01 in extremal ones.