20.3. Antti Kykkänen: Geometric inverse problems on gas giant planets
Abstract:
In this talk, a gas giant planet is a manifold equipped with a Riemannian metric that is singular at the boundary in a specific way. I will explain how our novel geometric model is derived and I will discuss our recent results on geometric inverse problems on gas giants.
There is one key difference between terrestrial planets, like the Earth, and gas giants, like Jupiter. In gas, the density of matter goes to zero at the surface, whereas in rock, the sound speed is bounded from below by a positive constant even at the surface. I will explain how the standard Riemannian models for rocky planets have to be modified to account for the vanishing of density. Surprisingly little is known about the geometry of the arising Riemannian metrics, which we call gas giant metrics.
We will see an overview of the geometry of gas giants. In particular, we will compute the Hausdorff dimension of a gas giant. We will consider two inverse problems with origins in seismology and prove the following two results. Up to natural obstructions, a gas giant metric is uniquely determined by certain travel time measurements. A smooth function on a gas giant is uniquely determined by the knowledge of its integrals over the maximal geodesics of a gas giant.
The talk is based on joint work with Maarten de Hoop, Joonas Ilmavirta and Rafe Mazzeo. A preprint is available at https://arxiv.org/abs/2403.05475.
In this talk, a gas giant planet is a manifold equipped with a Riemannian metric that is singular at the boundary in a specific way. I will explain how our novel geometric model is derived and I will discuss our recent results on geometric inverse problems on gas giants.
There is one key difference between terrestrial planets, like the Earth, and gas giants, like Jupiter. In gas, the density of matter goes to zero at the surface, whereas in rock, the sound speed is bounded from below by a positive constant even at the surface. I will explain how the standard Riemannian models for rocky planets have to be modified to account for the vanishing of density. Surprisingly little is known about the geometry of the arising Riemannian metrics, which we call gas giant metrics.
We will see an overview of the geometry of gas giants. In particular, we will compute the Hausdorff dimension of a gas giant. We will consider two inverse problems with origins in seismology and prove the following two results. Up to natural obstructions, a gas giant metric is uniquely determined by certain travel time measurements. A smooth function on a gas giant is uniquely determined by the knowledge of its integrals over the maximal geodesics of a gas giant.
The talk is based on joint work with Maarten de Hoop, Joonas Ilmavirta and Rafe Mazzeo. A preprint is available at https://arxiv.org/abs/2403.05475.
