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Geometric inverse problems on gas giants

Maarten V. de Hoop, Joonas Ilmavirta, Antti Kykkänen, Rafe Mazzeo

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.

Geometric inverse problems on gas giants

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.
Paper Structure (21 sections, 29 theorems, 159 equations)

This paper contains 21 sections, 29 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Gas giant geometry
  3. Geometry from the equation of state
  4. The geometry of gas giant manifolds
  5. Normal forms and asymptotic curvatures
  6. Geodesics
  7. A travel time inverse problem for a gas giant
  8. Geodesic X-ray tomography on a gas giant
  9. Pestov identity with boundary terms on a regular boundary
  10. Proof of theorem \ref{['thm:injectivity-in-simple-geometry']}
  11. Boundary determination
  12. Derivatives of the integral function
  13. Proof of the Pestov identity
  14. The Laplacian of $g$
  15. Original equations for gas giants
  16. ...and 6 more sections

Key Result

Proposition 1

Suppose that $g$ is an $\alpha$-gas giant metric on the interior of some manifold with boundary $M$. Then $g$ is incomplete, and there is a smoothly varying orthonormal basis of sections for $TM$ such that the sectional curvatures for $2$-planes spanned by pairs of these basis vectors are asymptotic

Theorems & Definitions (56)

  • Proposition 1
  • Proposition 2
  • proof
  • proof : Proof of Proposition \ref{['curvasym']}
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • ...and 46 more