The effects of opacity on gravitational stability in protoplanetary discs

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.