Tidal driving of inertial modes of Maclaurin spheroids

Abstract

We examine the inertial modes of vibration of a Maclaurin spheroid and determine how they are driven by an external tidal field, either Newtonian (gravitoelectric) or post-Newtonian (gravitomagnetic). The context and motivation for this work come from the realization that inertial modes of rotating neutron stars can be resonantly excited in binary inspirals, with a measurable effect on the phasing of the emitted gravitational waves. We aim to provide additional insights into this phenomenon. We calculate how the frequencies of the relevant inertial modes, and their overlap integrals with the tidal forces, vary as the star's rotation rate increases, spanning almost the full range between zero rotation and the mass-shedding limit. We prove that a single inertial mode couples to a Newtonian tidal field; overlap integrals with all other modes vanish. We prove also that four inertial modes couple to a post-Newtonian, gravitomagnetic tidal field; overlap integrals with all other modes vanish. Finally, we determine the rather extreme conditions under which the gravitoelectric driving of inertial modes dominates over the gravitomagnetic driving.