The tidal interaction of an orbiting giant planet with a star near the Kraft break: the excitation of $r$-modes and the retention of orbital and spin angular momenta misalignment

Abstract

In this paper we extend the previous work of Papaloizou \& Savonije on tidal interactions between a solar mass star and a closely orbiting giant planet which is such that the orbital and stellar spin angular momentum directions are misaligned. Here we consider the situation when the central star has a mass of $1.3 M_{\odot}$ and is in the vicinity of the Kraft break. We find and determine the properties of the lowest order $r$ modes and the tidal response arising from the secular non axisymmetric forcing associated with a misaligned orbit. We find that the response of the thin convective envelope, as well as the shift of $r$ mode frequencies from the low rotation frequency, limit can be understood by adopting a vertically averaged model that is similar to the well known one governed by the Laplace tidal equation for an incompressible ocean. From our results we are able to estimate lower bounds on realignment time scales for hot Jupiter systems with orbital periods in the range $2.8-5 d$ and rotation periods in the range $5-31 d$ that indicate the process is indeed markedly less effective than for a solar type star. This is on account of there being less dissipation in a relatively smaller convective envelope as well as the generally faster rotation and hence larger spin angular momentum expected for the more massive star.