and the Stokes number $\varOmega\tau_{\rm f}$ is proportional to particle radius [*squared*]{}, so $\varOmega\tau_{\rm f}=0.25,0.50,0.75,1.00$ correspond to physical particle sizes ranging from $40\,{\rm cm}$ to $80\,{\rm cm}$ (see online supplement of J07). Scaling [Eq. (\[eq:Sigma\])]{} to more distant orbital locations gives smaller physical particles and a gas column density closer to the Minimum Mass Solar Nebula value, since self-gravity is more efficient in regions where the rotational support is lower.

There are several points to be raised about our protoplanetary disc model. The high self-gravity parameter that we use implies not only a very high column density, but also that the gas component is close to gravitational instability. The self-gravity parameter $\tilde{G}$ is connected to the more commonly used $Q$ [where $Q>1$ is the axisymmetric stability criterion for a flat disc in Keplerian rotation, see @Safronov1960; @Toomre1964] through $\tilde{G}\approx1.6 Q^{-1}$. Thus we have $Q\approx3.2$, which means that gas self-gravity should start to affect the dynamics (the disc is not formally gravitationally unstable, but the disc is massive enough to slow down the propagation of sound waves). Another issue with such a massive disc is our assumption of ideal MHD. The high gas column density decreases the ionisation by cosmic rays 