| Abstract
| - Using the synthetic theory, we computed proper elements for asteroids in the inner part of the main belt and in the trans-Neptunian region. We present the new results and we discuss some important implications of both their rapidly increasing number and the improved accuracy. We give complete information on the availability of asteroid proper elements through the AstDys web service, with particular emphasis on the content and update strategy of proper element catalogs. Next, we discuss the most interesting features of some important asteroid families and regions (Vesta, Phocaea, TNO). For the Vesta family we provide further evidence that the observed spread of the family members is probably due to diffusion in the proper elements caused by non-gravitational effects. We shed additional light on the long-standing question of whether Phocaea is a genuine family or an island isolated by resonances, and we show the impact of nearby secular resonances on the local dynamics. Finally, we analyzed trans-Neptunian objects by using a “dual” of the main asteroid belt, to show that the currently known population of TNOs corresponds to the unstable outer asteroid belt beyond the 2/1 mean motion resonance with Jupiter. To discuss some of the main features of the dynamics of trans-Neptunians we use two examples, one stable over at least 100 Myr, but with proper elements slightly affected by a nonlinear secular resonance $g+s-g_8-s_8$, and the other chaotic, with a Lyapounov time of the order of 4000 yr, already exhibiting macroscopic instability after 6 Myr.
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