On the tidal interaction of massive extrasolar planets on highly eccentric orbits

Abstract

In this paper we develop a theory of disturbances induced by the stellar tidal field in a fully convective slowly rotating planet orbiting on a highly eccentric orbit around a central star. In this case it is appropriate to treat the tidal influence as a succession of impulsive tidal interactions occurring at periastron passage. For a fully convective planet mainly the l = 2 fundamental mode of oscillation is excited. We show that there are two contributions to the mode energy and angular momentum gain due to impulsive tidal interaction: (i) 'the quasi-static' contribution, which requires dissipative processes operating in the planet, and (ii) the dynamical contribution associated with excitation of modes of oscillation. These contributions are obtained self-consistently from a single set of the governing equations. We calculate a critical 'equilibrium' value of angular velocity of the planet Ωcrit determined by the condition that action of the dynamic tides does not alter the angular velocity at this rotation rate. We show that this can be much larger than the corresponding rate associated with quasi-static tides and that at this angular velocity, the rate of energy exchange is minimized. We also investigate the conditions for the stochastic increase in oscillation energy that may occur if many periastron passages are considered and dissipation is not important. We provide a simple criterion for this instability to occur. Finally, we make some simple estimates of the time-scale of evolution of the orbital semimajor axis and circularization of the initially eccentric orbit due to tides, using a realistic model of the planet and its cooling history, for orbits with periods after circularization typical of those observed for extrasolar planets Pobs ≳ 3 d. Quasi-static tides are found to be ineffective for semimajor axes ≳0.1 au. On the other hand, dynamic tides could have produced a very large decrease of the semimajor axis of a planet with mass of the order of the Jupiter mass M J and final period Pobs ∼ 1-4.5 d on a time-scale ≲ a few Gyrs. In this case the original semimajor axis may be as large as ≳ 102 au. For planets with masses ≳5MJ dynamic tides excited in the star appear to be more important than the tides excited in the planet. They may also, in principle, result in orbital evolution in a time less than or comparable to the lifetime of the planetary systems. Finally, we point out that there are several issues in the context of the scenario of the circularization of the orbit solely due to dynamic tides that remain to be resolved. Their possible resolution is discussed.