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 ϩ q)/p resonance, the integral of motion N ϭ ͙Ȑa resonances. The diffusion in the most stable 2/1-resonant region [( p ϩ q)/p Ϫ ͙1 Ϫ e 2 ] (Ȑ 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 ϭ surability there are two possible locations for long-surviving 0) divides the phase space into the manifolds N ϭ const asteroids: the one centered at an eccentricity of 0.3 near the on which the motion takes place. The trajectories on N ϭ libration stable centers with small libration amplitude and the const can be described by the pair ( ϭ (( p ϩ q)/q) 1 Ϫ other at a slightly lower eccentricity with a moderate libration p/q Ϫ , e), where 1 and are the mean longitudes of amplitude (ȁ90؇). Surprisingly, all asteroids observed in the 2/1 resonance (8 numbered and multi-opposition objects in Jupiter and an asteroid and 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. Con-N ϭ 0.457 and N ϭ 0.51 for the 3/2 resonance. For N ϭ cerning the 4/3 resonance, the only asteroid in the correspond-0.457 and the asteroid inside the resonance, 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 (, e) ϭ (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.