Spherically-symmetric, cold collapse: the exact solutions and a comparison with self-similar solutions

TL;DR

This paper derives exact solutions for the collapse of spherically symmetric, pressureless clouds with arbitrary initial density profiles, highlighting the transient nature of self-similar solutions and incorporating effects like turbulence, rotation, and magnetic fields.

Contribution

It provides the first general exact solutions for cold collapse beyond self-similarity, including turbulence, rotation, and magnetic effects, offering new constraints for numerical models.

Findings

Exact solutions for arbitrary initial density profiles.

Self-similar solutions are only valid briefly during collapse.

Inclusion of turbulence, rotation, and magnetic fields in the collapse analysis.

Abstract

We present the exact solutions for the collapse of a spherically-symmetric, cold (i.e., pressureless) cloud under its own self-gravity, valid for arbitrary initial density profiles and not restricted to the realm of self-similarity. These solutions exhibit a number of remarkable features, including the self-consistent formation of and subsequent accretion onto a central point mass. A number of specific examples are provided, and we show that Penston's solution of pressureless, self-similar collapse is recovered for polytropic density profiles; importantly, however, we demonstrate that the time over which this solution holds is fleetingly narrow, implying that much of the collapse proceeds non-self-similarly. We show that our solutions can naturally incorporate turbulent pressure support, and we investigate the evolution of overdensities -- potentially generated by such turbulence -- as the collapse proceeds. Finally, we analyze the evolution of the angular velocity and magnetic fields in the limit that their dynamical influence is small, and we recover exact solutions for these quantities. Our results may provide important constraints on numerical models that attempt to elucidate the details of protostellar collapse when the initial conditions are far less idealized.