Kirkwood Gaps and stability of conservative periodic systems

Kirkwood (Smithsonian Inst. A. Rep. (1868)) demonstrated that there were gaps in the distribution of asteroids at certain distances from the Sun which occurred at positions where the orbital periods of the asteroids were in simple ratio to the period of Jupiter. It has also been shown that at some ratios closer to Jupiter groups of asteroids occur. Examples of this distribution are the 2/1 resonance called the Hecube Gap and the 3/2 resonance called the Hilda Group. In an attempt to explain this distribution the stability of the orbits has been examined using a theory of stability represented by Hill's equation (Acta. Math.; 8:1 (1886)). In the theoretical work here reported it has been shown that the equation has stable solutions for Hilda-type asteroids and unstable solutions for the Hecuba-type which the author feels accounts for the striking differences in the two types of resonance. (U.K.)