Stability of Asteroid Motions

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 0 because of 2:3 mean motion resonance and the argument of perihelion of Pluto librates around 90 0. 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. According to theories of secular perturbations of asteroids the semi-major axes, a, are constant after averaging the disturbing function with respect to fast moving angular variables. And when it is assumed that both the eccentricity, e, and the inclination, i, are very small, by adopting the variables = e cos $ and = e sin $ as well as p = sin i cos and q = sin i sin with the longitudes of the perihelion and the ascending node, $ and , the diierential equations of motion are reduced to two sets of linear systems with constant coeecients, one for and and one for p and q. The solutions are expressed as sums of free oscillations and forced ones due to the secular variations of orbital elements of disturbing planets. Therefore, it is concluded that both eccentricity and inclination cannot take any very large values. However, when the eccentricity and/or inclination are not small, the equations are not linear and there are interaction terms between (,)-and (p, q)-sets. However, when it is assumed that all the disturbing planets are moving along circular orbits on the same plane, the equations are expressed as a canonical set of one degree of freedom with an energy integral. In fact when Delaunay variables,