  In this paper we consider the effects of opacity regimes on the stability of
self-gravitating protoplanetary discs to fragmentation into bound objects.
Using a self-consistent 1-D viscous disc model, we show that the ratio of local
cooling to dynamical timescales Omega*tcool has a strong dependence on the
local temperature. We investigate the effects of temperature-dependent cooling
functions on the disc's gravitational stability through controlled numerical
experiments using an SPH code. We find that such cooling functions raise the
susceptibility of discs to fragmentation through the influence of temperature
perturbations - the average value of Omega*tcool has to increase to prevent
local variability leading to collapse. We find the effects of temperature
dependence to be most significant in the "opacity gap" associated with dust
sublimation, where the average value of Omega*tcool at fragmentation is
increased by over an order of magnitude. We then use this result to predict
where protoplanetary discs will fragment into bound objects, in terms of radius
and accretion rate. We find that without temperature dependence, for radii <
~10AU a very large accretion rate ~10^-3 Msun/yr is required for fragmentation,
but that this is reduced to 10^-4 Msun/yr with temperature-dependent cooling.
We also find that the stability of discs with accretion rates < ~10^-7 Msun/yr
at radii > ~50AU is enhanced by a lower background temperature if the disc
becomes optically thin.
