Starn's rings and bimodality of Keplerian systems

The correction terms which are introduced by non-zero size of the particles into the mechanics of Keplerian systems can be replaced by relatively simple approximations which agree with computer simulations. The theory of finite particles confirms the bimodality of collisional systems which has previously been discussed in terms of the mass-point approximation. In Saturn's rings the ringlets correspond to the degenerate mode while the matter which fills the gaps is in the non-degenerate state. The predicted volume density of the ringlets (the fraction of space which is occupied by the particles), 0.2, is much higher than the conventional value which follows from the theory of mutual shadowing. Therefore, the opposition effect of Saturn's rings must originate in the particles themselves. The transition from one mode to the other which is needed to create a dense ring in a cloud of small particles follows from the growth of mass in the central body. This may be a recently-formed planet; but, more probably, the transition occurs in a loose pre-planetary disc.