A Local Model for the Spherical Collapse/Expansion Problem

TL;DR

This paper introduces a local model for spherical collapse and expansion in astrophysics, enabling detailed study of small-scale phenomena using a simplified, periodic box approach with derived symmetries and nonlinear solutions.

Contribution

It develops a novel local model for spherical flows, extending existing models and deriving symmetries, conservation laws, and nonlinear solutions specific to collapsing or expanding gas clouds.

Findings

Derived symmetries and conservation laws for the local model.

Identified nonlinear solutions including local analogues of zonal flows.

Showed how energy, density, and vorticity evolve during collapse/expansion.

Abstract

Spherical flows are a classic problem in astrophysics which are typically studied from a global perspective. However, much like with accretion discs, there are likely many instabilities and small scale phenomena which would be easier to study from a local perspective. For this purpose, we develop a local model for a spherically contracting/expanding gas cloud, in the spirit of the shearing box, $\beta$-plane and expanding box models which have had extensive use in studies of accretion discs, planets and stellar winds respectively. The local model consists of a, spatially homogeneous, periodic box with a time varying aspect ratio, along with a scale factor (analogous to that in FRW/Newtonian cosmology) relating the box coordinates to the physical coordinates of the global problem. We derive a number of symmetries and conservation laws exhibited by the local model. Some of these reflect symmetries of the periodic box, modified by the time dependant geometry, while others are local analogues for symmetries of the global problem. The energy, density and vorticity in the box also generically increase(/decrease) as a consequence of the collapse(/expansion). We derive a number of nonlinear solutions, including a local analogue of uniform density zonal flows, which grow as a consequence of angular momentum conservation. Our model is closely related to the accelerated expanding box model of Tenerani \& Velli and is an extension of the isotropic model considered by Robertson \& Goldreich.