The effect of extreme ionisation rates during the initial collapse of a
  molecular cloud core
  What cosmic ray ionisation rate is required such that a non-ideal
magnetohydrodynamics (MHD) simulation of a collapsing molecular cloud will
follow the same evolutionary path as an ideal MHD simulation or as a purely
hydrodynamics simulation? To investigate this question, we perform
three-dimensional smoothed particle non-ideal magnetohydrodynamics simulations
of the gravitational collapse of rotating, one solar mass, magnetised molecular
cloud cores, that include Ohmic resistivity, ambipolar diffusion, and the Hall
effect. We assume a uniform grain size of $a_\text{g} = 0.1\mu$m, and our free
parameter is the cosmic ray ionisation rate, $\zeta_\text{cr}$. We evolve our
models, where possible, until they have produced a first hydrostatic core.
Models with $\zeta_\text{cr}\gtrsim10^{-13}$ s$^{-1}$ are indistinguishable
from ideal MHD models and the evolution of the model with
$\zeta_\text{cr}=10^{-14}$ s$^{-1}$ matches the evolution of the ideal MHD
model within one per cent when considering maximum density, magnetic energy,
and maximum magnetic field strength as a function of time; these results are
independent of $a_\text{g}$. Models with very low ionisation rates
($\zeta_\text{cr}\lesssim10^{-24}$ s$^{-1}$) are required to approach
hydrodynamical collapse, and even lower ionisation rates may be required for
larger $a_\text{g}$. Thus, it is possible to reproduce ideal MHD and purely
hydrodynamical collapses using non-ideal MHD given an appropriate cosmic ray
ionisation rate. However, realistic cosmic ray ionisation rates approach
neither limit, thus non-ideal MHD cannot be neglected in star formation
simulations.
