4. Conclusions

Summarizing the main results of this paper, let us recall especially the following:

• we developed a new, purely numerical procedure for the computation of ``synthetic'' proper elements of asteroids, which consists in: (i) numerical integration of asteroid orbits in the framework of a realistic dynamical model, (ii) online digital filtering of the short-periodic effects, (iii) Fourier analysis to remove main forced terms and find proper modes with given arguments, and (iv) checking the accuracy by means of running box tests;
• by means of this theory, we produced a new set of accurate proper elements and frequencies for 10,256 outer main belt asteroids between 2.5 and 4.0 AU, combining outputs from the 2 Myr integration and from an extended 10 Myr integration, to which we subjected objects with proper elements/frequencies poorly determined in the shorter run;
• for each integrated body we provide an individual estimate of the accuracy of the results in terms of the standard deviations and maximum excursions, as well as an estimate of the maximum Lyapunov Characteristic Exponent; these data enabled us to conduct a comprehensive, yet straightforward statistical analysis of the results, and provided us with a sort of ``dynamical portrait'' of the outer main belt as a whole;
• we achieved an average improvement of the accuracy of new proper elements with respect to previously available ones, computed by means of our analytical theory, by more than a factor of 3. In terms of the relative velocities at breakup this means that the typical accuracy is of the order of $\sim 5$ m/s.
• such a high accuracy of proper elements opens the possibility to study the individual families, their internal structure, boundaries and velocity fields, as well as the phenomena of chaotic diffusion, leakage of the members from the families, etc. In this respect, we offered few examples, including leakage along the resonant lines from the Koronis, Eos and Themis families;
• the availability of individual measures of chaotic behavior for each asteroid provides a basis to study the dynamical structure of the entire region, from the point of view of the long-term stability of motion, e.g., in the Nekoroshev sense.

We would like to emphasize the implications of the availability of an accuracy estimate for each individual asteroid. This means that also the asteroids for which the proper elements are poor can be used in the study of asteroid families, but the users of the proper elements should take into account the uncertainty. The algorithm to do this has not yet been developed, but it is clear that the uncertainty must appear in the distances between sets of proper elements used for the classification procedures. It is obvious that the wealth of data we produced contains plenty of information on the dynamics of the asteroids in the outer main belt, on the stability of their motion, chaos, families, and many other important topics. We plan in the next future to analyze some of these problems. In addition, we intend to extend our analysis to the inner part of the main belt (this will require inclusion of the terrestrial planets in the model and much more complex and time consuming integration/analysis), and to resonant asteroids such as Hildas and Trojans (with appropriately defined proper parameters). On the contrary, we do not intend to consider Near-Earth objects (NEO's) in the framework of this project, as these have to be handled in a very different way (Gronchi and Milani [2000]). Acknowledgements: We are grateful to M. Carpino for his kind assistance with the chapter on digital filtering, and to A. Lemaitre for reviewing the manuscript and suggesting several improvements. This research has been supported through the grants by the Italian Space Agency (ASI), by the Italian Ministry of Foreign Affairs and Yugoslav Ministry of Environment, Science and Development, and by the Ministry of Science and Technology of Serbia through the project 04M04 ``Astrometrical, astrodynamical and astrophysical investigations''.