On gas giant planets, the speed of sound is isotropic and tends to zero at the surface. Geometrically, this corresponds to a Riemannian manifold whose metric tensor has a conformal blow-up near the boundary. The blow-up is tamer than in asymptotically hyperbolic geometry: The boundary is at a finite distance.
We consider the acoustic wave and the acousto-gravitational equations of gas giants. The geometry of acoustic waves is modeled by the above mentioned singular Riemannian, "gas giant" metric. We give an overview of the basic properties of the geometry, including properties of geodesics near the boundary, the Hausdorff dimension of the boundary, and the discreteness of the (acoustic) spectrum of the Laplace–Beltrami operator. We present the spectral analysis of this operator and derive the Weyl law. The involved exponents depend on the Hausdorff dimension, which, in the supercritical case (the relevant case for Jupiter and Saturn), is larger than the topological dimension. The Weyl asymptotics determines the blow-up in the supercritical case.
We then consider various inverse problems for simple gas giant planets, proving that the metric is uniquely determined by its boundary distance data and that the geodesic ray transform is injective. We study the determination of a conformal factor of the metric from boundary distance data using methods involving singular microlocal analysis of the normal operator corresponding to the geodesic ray transform. Moreover, we present the boundary observability of acoustic waves given full and local observations and show how the resulting observability inequality can be turned into a scanning protocol where the observation set moves in time. We conclude with some remarks on inertia-gravity modes on gas planets forming the essential spectrum.
Joint research with Y. Colin de Verdìère, C. Dietze, J. Ilmavirta, A. Kykkänen, R. Mazzeo and E. Trélat.