Geometric inverse problems on gas giants

TL;DR

This paper investigates the unique determination of gas giant interior structures through boundary data by analyzing the geometry of manifolds with conformal boundary blow-up and their relation to hydrodynamic PDEs.

Contribution

It introduces a geometric framework for inverse problems on gas giants, relating boundary measurements to interior properties via Riemannian geometry and PDE analysis.

Findings

Interior structure is uniquely recoverable from boundary data.

Characterization of geodesic behavior near the boundary.

Analysis of the Laplace--Beltrami operator on these manifolds.

Abstract

On gas giant planets the speed of sound is isotropic and goes 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 study the differential geometry of such manifolds, especially the asymptotic behavior of geodesics near the boundary. We relate the geometry to the propagation of singularities of a hydrodynamic PDE and we give the basic properties of the Laplace--Beltrami operator. We solve two inverse problems, showing that the interior structure of a gas giant is uniquely determined by different types of boundary data.