Multi-body spherically symmetric steady states of Newtonian self-gravitating elastic matter

TL;DR

This paper introduces a new framework for analyzing static, spherically symmetric self-gravitating elastic bodies in Newtonian gravity, establishing existence results for solutions including multi-body configurations.

Contribution

It presents a novel Eulerian definition of homogeneous elastic bodies and proves existence of solutions for the Seth model, including multi-body configurations.

Findings

Existence of single-body solutions

Existence of multi-body solutions

Equivalence of Eulerian and Lagrangian formulations

Abstract

We study the problem of static, spherically symmetric, self-gravitating elastic matter distributions in Newtonian gravity. To this purpose we first introduce a new definition of homogeneous, spherically symmetric (hyper)elastic body in Euler coordinates, i.e., in terms of matter fields defined on the current physical state of the body. We show that our definition is equivalent to the classical one existing in the literature and which is given in Lagrangian coordinates, i.e., in terms of the deformation of the body from a given reference state. After a number of well-known examples of constitutive functions of elastic bodies are re-defined in our new formulation, a detailed study of the Seth model is presented. For this type of material the existence of single and multi-body solutions is established.