Critical phenomena in Newtonian gravity

TL;DR

This paper analyzes the stability of self-similar solutions in Newtonian gravitational collapse, identifying unstable modes and proposing the Larson-Penston solution as the physically realized outcome, with implications for critical phenomena.

Contribution

It demonstrates the stability properties of key solutions and identifies the Hunter (A) solution as a critical solution with a calculable critical exponent in Newtonian gravity.

Findings

Hunter solutions are highly unstable

Larson-Penston solution is stable and likely realized

Hunter (A) solution has a single unstable mode and is a critical solution

Abstract

We investigate the stability of self-similar solutions for a gravitationally collapsing isothermal sphere in Newtonian gravity by means of a normal mode analysis. It is found that the Hunter series of solutions are highly unstable, while neither the Larson-Penston solution nor the homogeneous collapse one have an analytic unstable mode. Since the homogeneous collapse solution is known to suffer the kink instability, the present result and recent numerical simulations strongly support a proposition that the Larson-Penston solution will be realized in astrophysical situations. It is also found that the Hunter (A) solution has a single unstable mode, which implies that it is a critical solution associated with some critical phenomena which are analogous to those in general relativity. The critical exponent $\gamma$ is calculated as $\gamma\simeq 0.10567$. In contrast to the general relativistic case, the order parameter will be the collapsed mass. In order to obtain a complete picture of the Newtonian critical phenomena, full numerical simulations will be needed.