To verify this new criterion, we perform a series of 3d shearing box simulations which covers a large parameter space. Here I shown several examples from the result.
The top two panels show the movies of the midplane dust density for the stable and unstable case; the disk undergoes fragment if the midplane density exceeds the our critical density.
The bottom diagram tracks the maximum dust density in the box as a function of time. We find the disk undergoes gravitational collapse only the density criterion is satisfied.