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