In this paper it is explained how most of asteroids can avoid very close approach to Jupiter, to the earth for earth orbit crossing asteroids, and to Neptune for Kuiper-belt asteroids by mechanisms which work also for Neptune-Pluto system. In fact the mutual distance of the planets cannot become very small as the critical argument librates around 180° because of 2:3 mean motion resonance and the argument of perihelion of Pluto librates around 90°. And it is found that among nearly 40 Kuiper-belt asteroids discovered in recent years 40% have orbits similar to Pluto. For main-belt asteroids the distribution with respect to the semi-major axes has peculiar characteristics and the author tries to explain how their peaks and gaps are created. It is also found that 30% of 80 earth orbit crossing asteroids which have minimum perihelion distances less than 1.04AU have no chance to collide with the earth. Still 30% of them have a few probability to collide with the earth as they have dynamical characteristics of short-periodic comets.

• KOZAI YOSHIHIDE(National Astronomical Observatory)