On self-gravitating polytropic elastic balls

Abstract

A new four-parameters family of constitutive functions for spherically symmetric elastic bodies is introduced which extends the two-parameters class of polytropic fluid models widely used in several applications of fluid mechanics. The four parameters in the polytropic elastic model are the polytropic exponent $\gamma$, the bulk modulus $\kappa$, the shear exponent $\beta$ and the Poisson ratio $\nu\in (-1,1/2]$. The two-parameters class of polytropic fluid models arises as a special case when $\nu=1/2$ and $\beta=\gamma$. In contrast to the standard Lagrangian approach in elasticity theory, the polytropic elastic model in this paper is formulated directly in physical space, i.e., in terms of Eulerian state variables, which is particularly useful for the applications e.g. to astrophysics where the reference state of the bodies of interest (stars, planets, etc.) is not observable. After discussing some general properties of the polytropic elastic model, the steady states and the homologous motion of Newtonian self-gravitating polytropic elastic balls are investigated. It is shown numerically that static balls exist when the parameters $\gamma,\beta$ are contained in a particular region $\mathcal{O}$ of the plane, depending on $\nu$, and proved analytically for $(\gamma,\beta)\in\mathcal{V}$, where $\mathcal{V}\subset\mathcal{O}$ is a disconnected set which also depends on the Poisson ratio $\nu$. Homologous solutions describing continuously collapsing balls are constructed numerically when $\gamma=4/3$. The radius of these solutions shrinks to zero in finite time, causing the formation of a center singularity with infinite density and pressure. Expanding self-gravitating homologous elastic balls are also constructed analytically for some special values of the shear parameter $\beta$.