Dynamic tides in rotating objects: orbital circularization of extrasolar planets for realistic planet models

We consider the general problem of the tidal capture or circularization from large eccentricity of a uniformly rotating object. We extend the self-adjoint formalism introduced in our recent paper to derive general expressions for the energy and angular momentum transferred when the planet or a star passes through periastron in a parabolic or highly eccentric orbit around a central mass. These can be used without making a low-frequency approximation as was done in Papaloizou & Ivanov. We show how these can be adapted to the low-frequency limit in which only inertial modes contribute for baratropic planet models. In order to make quantitative estimates, we calculate the inertial mode eigenspectrum for planet models of one and five Jupiter masses M-J, without a solid core, with different radii corresponding to different ages. The spectra are found in general to be more complex than of a polytrope with index n = 1.5, considered in Papaloizou & Ivanov, because of the existence global modes associated with the transition from molecular to metallic hydrogen. None the less the main tidal response is still found to be determined by two global modes which have polytropic counterparts. These also determine the uniform angular velocity in a state of pseudo-synchronization, for which the angular momentum transferred during an encounter is zero. This is found to be close to 1.55 times the circular orbit angular velocity at periastron for all models considered. This is in contrast to the situation when only the f mode is considered and the equilibrium angular velocity is found to be much larger. We consider the multipassage problem when there is no dissipation finding that stochastic instability resulting in the stochastic gain of inertial mode energy over many periastron passages occurs under similar conditions to those already found by Ivanov & Papaloizou for the f modes. We find that this requires circularization to start with a semimajor axis exceeding similar to 30 au, for putative final periods of similar to 3 d reducing to similar to 1-2 au for putative final periods similar to 1.2 d. Finally, we apply our calculations of the energy transfer during a periastron passage to the problem of the tidal circularization of the orbits of the extrasolar planets in a state of pseudo-synchronization, expected because of the relatively small inertia of the planet, and find that inertial mode excitation dominates the tidal interaction for 1M(J) planets that start with semimajor axes less than 10 an and end up on circular orbits with final period in the 4-6 d range. It is potentially able to account for initial circularization up to a final 6-d period within a few Gyr. However, in the case of 5M(J) oscillation modes excited in the star are more important.