On the Asteroidal Population of the First-Order Jovian Resonances

extensive set of trajectories in both resonances. Moreover, we have included the 4/3 resonance in our study. The frequency map analysis was applied to the fairly realistic models of the 2/1, 3/2, and 4/3 jovian resonances and the Figure 1 shows the basic characteristics of the first-order results were compared with the asteroidal distribution at these mean–motion resonances in the averaged, planar circular commensurabilities. The presence of the Hecuba gap at the 2/1 three-body model. The Hamiltonian of this model is of and of the Hilda group in the 3/2 is explained on the basis two degrees of freedom and is separable. Considering the of different rates of the chaotic transport (diffusion) in these (p 1 q)/p resonance, the integral of motion N 5 Ïea resonances. The diffusion in the most stable 2/1-resonant region [(p 1 q)/p 2 Ï1 2 e2] (e is the product of the gravitation is almost two orders in magnitude faster than the diffusion in constant and the mass of the Sun, a and e are the semithe region which accommodates the Hildas. In the 2/1 commenmajor axis and eccentricity of an asteroid, inclination i 5 surability there are two possible locations for long-surviving 0) divides the phase space into the manifolds N 5 const asteroids: the one centered at an eccentricity of 0.3 near the on which the motion takes place. The trajectories on N 5 libration stable centers with small libration amplitude and the const can be described by the pair (s 5 ((p 1 q)/q)l1 2 other at a slightly lower eccentricity with a moderate libration p/ql 2 Ã, e), where l1 and l are the mean longitudes of amplitude (p908). Surprisingly, all asteroids observed in the 2/1 resonance (8 numbered and multi-opposition objects in Jupiter and an asteroid and à is the asteroid longitude Bowell’s catalog from 1994) occupy the moderate-libration area of perihelion. and avoid the area in a close vicinity of the libration stable As an example, Fig. 2 shows two sets of trajectories with centers. Possible explanations of this fact were discussed. ConN 5 0.457 and N 5 0.51 for the 3/2 resonance. For N 5 cerning the 4/3 resonance, the only asteroid in the correspond0.457 and the asteroid inside the resonance, s librates ing stable region is 279 Thule, in spite of the fact that this around zero and e oscillates approximately around 0.22. region is almost as regular (although not as extensive) as the The point (s, e) 5 (0, 0.22) is the stable stationary solution one where the Hilda group in the 3/2, with 79 members, is (libration center) and the set of these points for different found.  1997 Academic Press N (for instance, (s, e) 5 (0, 0.4) for N 5 0.51) forms