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Z. Kne\v zevi\'c (Astron. Obs., Belgrade), A. Milani (Univ. Pisa)
Analytically computed proper elements in the low to moderate inclination and eccentricity region of the asteroid main belt are accurate to a level very close to the fundamental threshold of the accuracy of any analytical theory, which results from the fact that there is an infinite web of resonances and because of the occurence of chaotic motions (typical instability over 5 Myr in the proper e and sin I being \le 0.0015, and even better in the proper a; Milani and Knezevic, 1994, Icarus \bf107, 219). Zappalá et al. (1995, Icarus \bf 116, 291) have shown that this accuracy was enough to reliably identify asteroid families in the sample of 12,487 asteroids and throughout almost the entire belt. However, they have also found that still there are some regions of the belt in which the reliable identification is not possible, either because the proper elements in these regions are of degraded accuracy (e.g. near resonances), or because the background density of the asteroids is too high, smearing the family borders out and masking the clumping of the family.
We have, therefore, tried a different approach to compute the asteroid proper elements, with a goal to further improve their accuracy and thus enable the identification of families in the densely populated zones of even larger samples of asteroids, as well as the more refined analysis of their long-term dynamics. We adopted an approach similar to the one employed in the case of major planets by Carpino et al. (1987, Astron. Astrophys. \bf 181, 182) and, applying purely numerical techniques, we produced the so-called ``synthetic'' proper elements of asteroids. The procedure consisted of simultaneous integration of asteroid orbits for 2 Myr, on-line filtering of the short-periodic perturbations, and computation of Lyapunov Characteristic Exponents to monitor the chaotic behaviors. The output of the integration was spectrally resolved, and the principal harmonics (proper values) extracted from the time series, together with the associated fundamental frequencies and the corresponding standard and maximum deviations.
For 8009, out of 10,256 sampled main belt asteroids, we have determined the proper elements with an accuracy in terms of the standard deviations of proper eccentricity and sine of proper inclination better than 0.001, and of that of the proper semimajor axis better than 0.0003 AU. In 6387 cases the error in proper e was even less than 0.0003 and in proper sin I less than 0.0001. On the other hand, we have identified 913 asteroids with standard deviations of proper eccentricity or proper sine of inclination larger than 0.003, 497 strongly chaotic bodies (Lyapunov times shorter than 10,000 yr), 33 ``pathological'' cases for which the errors of computed elements and/or frequencies were, for different reasons, excessively large and all the proper values highly unreliable, etc. In only 9 cases we could not derive the proper elements by means of this procedure, because of the hyperbolic divergence of their orbits.