  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.
