It is often useful to treat planets as geometric objects where the geometry is defined so that the distance is measured as the travel time of seismic waves. This definition turns a planet into a Riemann or Finsler manifold with boundary. The propagation of seismic waves on rocky planets is well modelled by the geodesic flow on such a manifold, noting that the speeds of seismic waves remain bounded below up to the boundary. On gas giants, however, the (acoustic) wave speed goes to zero at the boundary, at a specific range of rates, which has a significant impact on the geometry. In fact, this property leads to a subset of conformally compact geometry. The curvature is unable to stay bounded and the boundary is a fractal in the sense of Hausdorff dimension. We present the acoustic-gravitational system of equations describing seismology, elements of the relevant geometry and show the presence of a discrete spectrum consistent with observations through Saturn's ring seismology. Moreover, we discuss some results pertaining to tomography on gas giants.

 

Joint research with J. Ilmavirta, A. Kykk\"{a}nen and R. Mazzeo