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