A unified normal mode approach to dynamic tides and its application to rotating Sun-like stars

Abstract

We determine the response of a uniformly rotating star to tidal perturbations due to a companion. General periodic orbits and parabolic flybys are considered. We evaluate energy and angular momentum exchange rates as a sum of contributions from normal modes allowing for dissipative processes. We consider the case when the response is dominated by the contribution of an identifiable regular spectrum of low-frequency modes, such as rotationally modified gravity modes. We evaluate this response in the limit of very weak dissipation, where individual resonances can be significant and also when dissipative effects are strong enough to prevent wave reflection from the neighbourhood of either the stellar surface or stellar centre, making radiation conditions more appropriate. The former situation may apply to Sun-like stars with radiative cores and convective envelopes and the latter to more massive stars with convective cores and radiative envelopes. We provide general expressions for transfer of energy and angular momentum that can be applied to an orbit with any eccentricity. Detailed calculations require knowledge of the mode spectrum and evaluation of the mode overlap integrals that measure the strength of the tidal interaction. These are evaluated for Sun-like stars in the slow rotation regime where centrifugal distortion is neglected in the equilibrium and the traditional approximation is made for the normal modes. We use both a Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) procedure and a direct numerical evaluation which are found to be in good agreement for regimes of interest. The former is used to provide expressions for the mode spectrum and overlap integrals as a function of mode frequency and stellar rotation rate. These can be used to find the tidal energy and angular momentum exchange rates and hence the orbital evolution. Finally we use our formalism to determine the evolution time scales for an object, in an orbit of small eccentricity, around a Sun-like star in which the tidal response is assumed to occur. Systems with either no rotation or synchronous rotation are considered. Only rotationally modified gravity modes are taken into account under the assumption that wave dissipation proceeds close to the stellar centre. It is noted that inertial waves excited in the convective envelope may produce a comparable amount of tidal dissipation in the latter case for sufficiently large orbital periods.