Fragmentation of ring galaxies and transformation to clumpy galaxies.

We study the fragmentation of collisional ring galaxies (CRGs) using our linear perturbation analysis that computes the physical conditions of gravitational instability. We adopt the analysis to our CRG simulations, and show that our analysis accurately characterises the stability and the onset of fragmentation that is determined by the balance of self-gravity against pressure and Coriolis forces. In addition, since the orthodox `density-wave' model is inapplicable to such self-gravitating rings, we devise a simple model that describes the rings propagating as material waves. We find that the toy model can predict the behaviour of the rings in our simulations. We also apply our instability analysis to a CRG discovered at a high redshift $z=2.19$. We find that a quite high velocity dispersion is required for the stability of the ring, and therefore the CRG can be unstable against ring fragmentation. CRGs are rarely observed at high redshifts, and it may be because CRGs are usually too faint. Since the fragmentation can induce active star formation and make the ring bright enough to observe, the instability may explain the rarity. Fragmenting CRGs would evolve into clumpy galaxies with low surface densities in their inter-clump regions.