Non-perturbative computation of normal modes for rotating gas giant planets

The Cassini spacecraft detected waves in Saturn's rings caused by Lindland and vertical resonances between ring particles and normal modes excited in the planet's interior (Mankovich et al., 2019 and references therein). The detection of these waves potentially provides us with a novel way to constrain Saturn's interior structure in the same way helioseismological measurements improved our understanding of the Sun. Perturbation theory along with internal structure modeling has been used to infer the properties of Saturns interior from the measurements of normal-mode frequencies (Mankovich et al., 2019). However, due to the measurements high precision and Saturns rapid rotation, the perturbation theory-based approach does not allow one to fully exploit the normal-mode dataset. In this paper, we circumvent perturbative methods altogether and, instead, implement a non-perturbative approach for computing both the hydrostatic equilibrium structure of a planet and its acoustic normal-mode spectrum. We use the Concentric Maclaurin Spheroids (CMS) method (Hubbard, 2013) to compute the planets interior structure in hydrostatic equilibrium that results from gravitational and centrifugal forces. The resulting interior structure is then employed by a finite element-based code (Shi et al., 2019) to compute the normal-mode frequencies and eigenfunctions. We model gas giant interior using a polytropic equation of state (EOS) with a unity polytropic index. With this simplified EOS, pressure scales as density squared. Polytropic models have previously been used as a first-order approximation of gas giant interior (e.g., Markham et al., 2018, 2020). This simple pressure-density relation allows one to obtain analytic solutions for the density, gravity and sound speed radial profiles for a non-rotating planet. We use the finite-element method to derive normal-mode frequencies and eigenfunctions for a rotating, polytropic planet assuming a range of rotation rates. We hope that our numerical results would serve as benchmarking cases for future normal-mode calculations of rapidly rotating gas giant planets in the Solar System and beyond.