Dynamic tides in rotating objects: orbital circularisation of extra solar planets for realistic planet models

(abbreviated) We consider the problem of the tidal capture or circularisation from large eccentricity of a uniformly rotating object. We extend the self-adjoint formalism introduced in Papaloizou \& Ivanov 2005 (PI) to derive general expressions for the energy and angular momentum transfered when the planet or a star passes through periastron in a parabolic or highly eccentric orbit around a central mass, without making a low frequency approximation as was done in PI. We show how these can be adapted to the low frequency limit in which only inertial modes contribute to the energy and angular momentum transfer. 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. We consider the multi-passage 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 for the $f$ modes. We apply our calculations to the problem of the tidal circularisation of the orbits of the extra solar planets in a state of pseudo synchronisation, and find that inertial mode excitation dominates the tidal interaction for $1 M_J$ planets that start with semi- major axes less than $10 AU$ and end up on circular orbits with final period in the $4-6$ day range. It is potentially able to account for initial circularisation up to a final $6$ day period within a few $Gyr$ But in the case of $5M_J$ oscillation modes excited in the star are more important.