  We carry out global three-dimensional radiation hydrodynamical simulations of
self-gravitating accretion discs to determine if, and under what conditions, a
disc may fragment to form giant planets. We explore the parameter space (in
terms of the disc opacity, temperature and size) and include the effect of
stellar irradiation. We find that the disc opacity plays a vital role in
determining whether a disc fragments. Specifically, opacities that are smaller
than interstellar Rosseland mean values promote fragmentation (even at small
radii, R < 25AU) since low opacities allow a disc to cool quickly. This may
occur if a disc has a low metallicity or if grain growth has occurred. With
specific reference to the HR 8799 planetary system, given its star is
metal-poor, our results suggest that the formation of its imaged planetary
system could potentially have occurred by gravitational instability. We also
find that the presence of stellar irradiation generally acts to inhibit
fragmentation (since the discs can only cool to the temperature defined by
stellar irradiation). However, fragmentation may occur if the irradiation is
sufficiently weak that it allows the disc to attain a low Toomre stability
parameter.
