Z. Knezevic1 and A. Milani2 1 University of Belgrad, Yugoslavia 2 University of Pisa, Italy
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. This 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 ≤ 0.0015, and even better in the proper a; Milani
and Knezevic, 1994, ( Icarus 107, 219). Still, there are
some regions of the belt (e.g. near resonances) in which these proper
elements are of degraded accuracy, thus preventing a reliable
definition of asteroid families.
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 large samples of asteroids, as well as the more refined
analysis of their long-term dynamics. Following the approach employed
in the case of major planets by Carpino et al. (1987,
Astron. Astrophys. 181, 182), that is, applying purely numerical
techniques, we produced the so-called ``synthetic'' proper elements
for a sample of 10,256 asteroids. We have taken into account all the
asteroids with osculating semimajor axes between 2.5 AU and 4.0 AU,
with the exception of those for which the initial value of the
quantity (e2 + sin2 I)1/2 > 0.3, and the perihelion distance q
< 1.75 AU.
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 next spectrally
resolved, and the principal harmonics (proper values) extracted from
the time series. For each set of proper elements and associated
fundamental frequencies the corresponding standard and maximum
deviations are supplied too. For 1862 asteroids exhibiting large
standard deviations of proper values due to the chaotic or secular
resonant effects, we have extended the integrations to 10 Myr, and
repeated the analysis. For only 9 asteroids we could not derive the
proper elements in this way because of the hyperbolic divergence of
their orbits.
In 8009 cases 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 that of the proper semimajor
axis better than 0.0003 AU. Out of these, 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, etc.