Spherically symmetric self-gravitating elastic bodies: A numerical investigation

Abstract

We investigate numerically the existence and properties of static self-gravitating elastic balls in the Euler formulation of continuum mechanics. In this formulation sufficient conditions for the existence of finite radius balls were recently derived for the Saint Venant-Kirchhoff, quasi-linear John, and Hadamard material models. Some problems were left open regarding whether or not the sufficient conditions for existence are also necessary. We find numerical evidence suggesting that the hyperbolicity condition at the center is indeed necessary, but that several other conditions on the central densities are stronger than necessary and can be replaced by weaker ones. We also find evidence of finite radius balls existing in the quasilinear Signorini model. Furthermore, the properties of static balls in the recently introduced polytropic elastic material model are investigated and we find numerical evidence of the existence of static spherically symmetric balls, shells and multibodies. Mass-radius diagrams are constructed, some of which admit spiral type curves. Finally, we investigate numerically the existence and properties of timedependent homologous solutions to the Cauchy-Poisson system for polytropic elastic balls and find that solutions exist but do not conserve mass unless static.