The response of self-graviting protostellar discs to slow reduction in cooling timescale: the fragmentation boundary revisited

Abstract

A number of previous studies of the fragmentation of self-gravitating protostellar discs have modeled radiative cooling with a cooling timescale (t_{cool}) parameterised as a simple multiple (beta_{cool}) of the local dynamical timescale. Such studies have delineated the `fragmentation boundary' in terms of a critical value of beta_{cool} (beta_{crit}), where the disc fragments if beta_{cool} < beta_{crit}. Such an approach however begs the question of how in reality a disc could ever be assembled with beta_{cool} < beta_{crit}. Here we adopt the more realistic approach of gradually reducing beta_{cool}, as might correspond to changes in thermal regime due to secular changes in the disc density profile. We find that when beta_{cool} is gradually reduced (on a timescale longer than t_{cool}), the disc is stabilised against fragmentation, compared with models in which beta_{cool} is reduced rapidly. We therefore conclude that a disc's ability to remain in a self-regulated, self-gravitating state (without fragmentation) is partly dependent on its thermal history, as well as its current cooling rate. Nevertheless, a slow reduction in t_{cool} appears only to lower the fragmentation boundary by about a factor two in t_{cool} and thus only permits maximum alpha values (parameterising the efficiency of angular momentum transfer in the disc) that are about a factor two higher than determined hitherto. Our results therefore do not undermine the notion of a fundamental upper limit to the heating rate that can be delivered by gravitational instabilities before the disc is subject to fragmentation. An important implication of this work, therefore, is that self-gravitating discs can enter into the regime of fragmentation via secular evolution and it is not necessary to invoke rapid (impulsive) events to trigger fragmentation.