We use a suite of SPH simulations to investigate the susceptibility of protoplanetary discs to the effects of self-gravity as a function of star-disc properties. We also include passive irradiation from the host star using different models for the stellar luminosities. The critical disc-to-star mass ratio for axisymmetry (for which we produce criteria) increases significantly for low-mass stars. This could have important consequences for increasing the potential mass reservoir in a proto Trappist-1 system, since even the efficient Ormel et al. (2017) formation model will be influenced by processes like external photoevaporation, which can rapidly and dramatically deplete the dust reservoir. The aforementioned scaling of the critical $M_d/M_*$ for axisymmetry occurs in part because the Toomre $Q$ parameter has a linear dependence on surface density (which promotes instability) and only an $M_*^{1/2}$ dependence on shear (which reduces instability), but also occurs because, for a given $M_d/M_*$, the thermal evolution depends on the host star mass. The early phase stellar irradiation of the disc (for which the luminosity is much higher than at the zero age main sequence, particularly at low stellar masses) can also play a key role in significantly reducing the role of self-gravity, meaning that even Solar mass stars could support axisymmetric discs a factor two higher in mass than usually considered possible. We apply our criteria to the DSHARP discs with spirals, finding that self-gravity can explain the observed spirals so long as the discs are optically thick to the host star irradiation.
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