tokens: of a given amount of external irradiation. Fig. 4 illustrates this. It is a disc with an accretion rate of M ⊙ yr −1 , approximately a third lower than in Fig. 3 . In this case (Fig. 4 ), external irradiation has added additional mass to the outer parts of the disc, as the equilibrium disc structure is now more massive for a given accretion rate and radius, although in this case it does not meet the criteria required for it to fragment (which requires that the local α ≳ 0.1). In this disc, the external irradiation is maintaining Q ∼ 2 with little dissipation of the GI. In reality, we should expect Q to increase beyond the marginal limit of self-gravity in the outer parts of such discs. Figure 4. Disc with M ⊙ yr −1 , 33 per cent lower than the maximum it can sustain without truncating. It is irradiated at 10 K, and this small amount of external irradiation has altered the surface density structure of the outer parts of the disc, as the equilibrium disc structure is now more massive for a given accretion rate and radius. The ‘spiky’ features on the boundary between the outer and inner parts of the disc are a numerical artefact due to a small change in surface density from the resolution limit of the grid, and are more pronounced in log space. We should bear in mind that this is a simple model in which no infall is considered. Since, in general, adding mass to the outer regions of a self-gravitating disc encourages fragmentation (Kratter et al. 2010 ; Vorobyov Basu 2010 ; Kratter Murray-Clay 2011 ; Forgan Rice 2012 ), it seems that the Jeans criterion for fragmentation may be satisfied at relatively high accretion rates (of the order of ∼10 −7 M ⊙ yr −1 ) in the presence of sufficiently small (∼10 K) irradiation. Fig. 5 shows a contour plot of the amount of external irradiation required to prevent fragmentation of the disc as a function of accretion rate and radius. Below ∼3.2 × 10 −7 M ⊙ yr −1 , in our local viscous approximation, any disc will be able to regulate itself against collapse. Above this, differing amounts of irradiation can either prevent fragmentation and totally remove spiral features from the disc, or prevent fragmentation whilst allowing spiral features to exist. As accretion rate increases, higher temperatures are required to prevent the disc from fragmenting; additionally, beyond 60 au the determining factor in whether spirals are present or not is temperature, rather than a combination of temperature and accretion rate. Figure 5. Contour plot showing the minimum temperature in (K) of external irradiation required to halt fragmentation as a function of accretion rate and radius. The shaded region with dashed contours shows where both fragmentation is suppressed and spiral features are erased. The unshaded region with dotted contours shows where added irradiation prevents fragmentation but preserves spiral structure. 3.2 Comparison with imposed constant β In our model, unlike in SPH simulations by Dipierro et al. ( 2014 )
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