Gravitational Collapse for Polytropic Gaseous Stars: Self-similar Solutions

TL;DR

This paper proves the existence of smooth, radially symmetric self-similar solutions demonstrating gravitational collapse in polytropic gaseous stars within a specific range of polytropic indices, extending understanding of stellar collapse dynamics.

Contribution

It establishes the existence of self-similar solutions with gravitational collapse for the Euler-Poisson system in the supercritical polytropic range, addressing mathematical challenges posed by sonic points.

Findings

Existence of smooth self-similar solutions in the supercritical range.

Solutions exhibit finite-time density blow-up, indicating collapse.

Presence of sonic points introduces mathematical complexities.

Abstract

In the supercritical range of the polytropic indices $\gamma\in(1,\frac43)$ we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler-Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows-up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson-Penston collapsing solutions in the isothermal case $\gamma=1$. They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof.