  We investigate whether or not the low ionisation fractions in molecular cloud
cores can solve the `magnetic braking catastrophe', where magnetic fields
prevent the formation of circumstellar discs around young stars. We perform
three-dimensional smoothed particle non-ideal magnetohydrodynamics (MHD)
simulations of the gravitational collapse of one solar mass molecular cloud
cores, incorporating the effects of ambipolar diffusion, Ohmic resistivity and
the Hall effect alongside a self-consistent calculation of the ionisation
chemistry assuming 0.1 micron grains. When including only ambipolar diffusion
or Ohmic resistivity, discs do not form in the presence of strong magnetic
fields, similar to the cases using ideal MHD. With the Hall effect included,
disc formation depends on the direction of the magnetic field with respect to
the rotation vector of the gas cloud. When the vectors are aligned, strong
magnetic braking occurs and no disc is formed. When the vectors are
anti-aligned, a disc with radius of 13AU can form even in strong magnetic when
all three non-ideal terms are present, and a disc of 38 AU can form when only
the Hall effect is present; in both cases, a counter-rotating envelope forms
around the first hydrostatic core. For weaker, anti-aligned fields, the Hall
effect produces massive discs comparable to those produced in the absence of
magnetic fields, suggesting that planet formation via gravitational instability
may depend on the sign of the magnetic field in the precursor molecular cloud
core.
