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 } } \gtrsim 10 ^ { -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 } } \lesssim 10 ^ { -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 can not be neglected in star formation simulations .