A Uniform Spherical Goat (Problem): Explicit Solution for Homologous Collapse's Radial Evolution in Time

Abstract

The homologous collapse from rest of a uniform density sphere under its self gravity is a well-known toy model for the formation dynamics of astronomical objects ranging from stars to galaxies. Equally well-known is that the evolution of the radius with time cannot be explicitly obtained because of the transcendental nature of the differential equation solution. Rather, both radius and time are written parametrically in terms of the development angle $\theta$. We here present an explicit integral solution for radius as a function of time, exploiting methods from complex analysis recently applied to the mathematically-similar 'geometric goat problem'. Our solution can be efficiently evaluated using a Fast Fourier Transform and allows for arbitrary sampling in time, with a simple Python implementation that is $\sim$$100\times$ faster than using numerical root-finding to achieve arbitrary sampling. Our explicit solution is advantageous relative to the usual approach of first generating a uniform grid in $\theta$, since this latter results in a non-uniform radial or time sampling, less useful for applications such as generation of sub-grid physics models.