Dynamical tidal Love numbers of rapidly rotating planets and stars

Tidal interactions play an important role in many astrophysical systems, but uncertainties regarding the tides of rapidly rotating, centrifugally distorted stars and gaseous planets remain. We have developed a precise method for computing the dynamical, non-dissipative tidal response of rotating planets and stars, based on summation over contributions from normal modes driven by the tidal potential. We calculate the normal modes of isentropic polytropes rotating at up to $\simeq90\%$ of their critical breakup rotation rates, and tabulate fits to mode frequencies and tidal overlap coefficients that can be used to compute the frequency-dependent, non-dissipative tidal response (via potential Love numbers $k_{\ell m}$). Although fundamental modes (f-modes) possess dominant tidal overlap coefficients at (nearly) all rotation rates, we find that the strong coupling of retrograde inertial modes (i-modes) to tesseral ($\ell>|m|$) components of the tidal potential produces resonances that may be relevant to gas giants like Jupiter and Saturn. The coupling of f-modes in rapid rotators to multiple components of both the driving tidal potential and the induced gravitational field also affect the tesseral response, leading to significant deviations from treatments of rotation that neglect centrifugal distortion and high-order corrections. For very rapid rotation rates ($\gtrsim 70\%$ of breakup), mixing between prograde f-modes and i-modes significantly enhances the sectoral ($\ell=|m|$) tidal overlap of the latter. The tidal response of very rapidly rotating, centrifugally distorted planets or stars can also be modified by resonant sectoral f-modes that are secularly unstable via the Chandrasekhar-Friedman-Schutz (CFS) mechanism.