The response of self-gravitating protostellar discs to slow reduction in cooling time-scale: the fragmentation boundary revisited

A number of previous studies of the fragmentation of self-gravitating protostellar discs have involved suites of simulations in which radiative cooling is modelled in terms of a cooling time-scale (t(cool)) which is parametrized as a simple multiple (beta(cool)) of the local dynamical time-scale. Such studies have delineated the 'fragmentation boundary' in terms of a critical value of beta(cool)(beta(crit)) such that 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 in a state with beta(cool) < beta(crit). Here we adopt the more realistic approach of effecting a gradual reduction in beta(cool), as might correspond to changes in thermal regime due to secular changes in the disc density profile. We find that the effect of gradually reducing beta(cool) (on a time-scale longer than t(cool)) is to stabilize the disc against fragmentation, compared with models in which beta(cool) is reduced rapidly (over less than t(cool)). We therefore conclude that the ability of a disc to remain in a self-regulated, self-gravitating state (without fragmentation) is partly dependent on the disc's thermal history, as well as its current cooling rate. Nevertheless, the effect of a slow reduction in t(cool) appears only to lower the fragmentation boundary by about a factor of 2 in t(cool) and thus only permits maximum 'alpha' values (which parametrize the efficiency of angular momentum transfer in the disc) that are about a factor of 2 higher than determined hitherto. Our results therefore do not undermine the notion that there is 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.