Dependence of fragmentation in self-gravitating accretion discs on small-scale structure

We propose a framework for understanding the fragmentation criterion for self-gravitating discs which, in contrast to studies that emphasize the 'gravoturbulent' nature of such discs, instead focuses on the properties of their quasi-regular spiral structures. Within this framework there are two evolutionary paths to fragmentation: (i) collapse on the free-fall time, which requires that the ratio of cooling time to dynamical time (beta) < 3 and (ii) quasi-static collapse on the cooling time at a rate that is sufficiently fast that fragments are compact enough to withstand disruption when they encounter spiral features in the disc. We perform 2D grid simulations which demonstrate numerically converged fragmentation at beta < 3 (in good agreement with Paardekooper, Baruteau & Meru, and others) and argue that this is a consequence of the fact that such simulations smooth the gravitational force on the scale H, the scaleheight of the disc. Such simulations thus only allow fragmentation via route (i) above since they suppress the quasi-static contraction of fragments on scales H; the inability of fragments to contract to significantly smaller scales then renders them susceptible to disruption at the next spiral arm encounter. On the other hand, 3D simulations indeed show fragmentation at higher beta via route (ii). We derive an analytic prediction of fragmentation by route (ii) when beta a parts per thousand(2) 12, based on the requirement that fragments must contract sufficiently to withstand disruption by spiral arms. We also discuss the necessary numerical requirements on both grid-based and smoothed particle hydrodynamics codes if they are to model fragmentation via route (ii).