by Maarten V. de Hoop, Joonas Ilmavirta, Antti Kykkänen and 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.
Reference:
Geometric inverse problems on gas giants (Maarten V. de Hoop, Joonas Ilmavirta, Antti Kykkänen and Rafe Mazzeo), 2024.
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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.
[arXiv]
Bibtex Entry:
@unpublished{gas-giant-1,
author = {Maarten V. de Hoop and Joonas Ilmavirta and Antti Kykk\"{a}nen and Rafe Mazzeo},
title = {{Geometric inverse problems on gas giants}},
month = mar,
year = 2024,
arxiv = {2403.05475},
url = {http://users.jyu.fi/~jojapeil/pub/gas-giant-1.pdf},
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.}
}