Synthetic proper elements
for outer main belt asteroids

Z. KNEZEVIC¹ and A. MILANI²
$^1\,$Astronomical Observatory, Volgina 7, 11160 Belgrade 74,
Yugoslavia, e-mail: zoran@aob.bg.ac.yu
$^2\,$Department of Mathematics, Via Buonarroti 2, 56127 Pisa
Italy, e-mail: milani@dm.unipi.it

Submitted 31 May 2000; revised version, 3 October 2000

Abstract:

For the orbits with low to moderate inclination and eccentricity, in the asteroid main belt, the analytically computed proper elements are accurate to a level very close to the best result achievable by any analytical theory. This fundamental limitation results from the infinite web of resonances and because of the occurrence of chaotic motions. Still, there are some regions of the belt in which these proper elements are of degraded accuracy, thus preventing a reliable definition of asteroid families and detailed studies of the dynamical structure. We have used a different method to compute asteroid proper elements, following the approach introduced in the LONGSTOP project to describe the secular dynamics of the major outer planets. By applying purely numerical techniques, we produced so-called ``synthetic'' proper elements for a catalog of 10,256 asteroids with osculating semimajor axes between 2.5 and 4.0 AU. The procedure consisted of simultaneous integration of asteroid and planetary orbits for 2 Myr, with online filtering of the short-periodic perturbations. The output of the integration was spectrally resolved, and the principal harmonics (proper values) extracted from the time series. For each asteroid we have also tested the accuracy and stability in time of the proper elements, and estimated the maximum Lyapounov Characteristic Exponent to monitor the chaotic behaviors. This provided information on the reliability of the data for each orbit, in particular allowing to select 1,852 cases for an extended integration (10 Myr) of the orbits showing instability. The results indicate that for more than half of the cases the proper elements have a time stability improved by more than a factor 3 with respect to the elements computed by the previous analytical theory. But of course there are also unstable cases for which the proper elements are less accurate and reliable, the extreme examples being 23 orbits exibiting hyperbolic escape from the solar system. This form of escape from the asteroid belt could be responsible for a significant mass loss over the age of the solar system. Keywords: asteroids, proper elements, chaos, families.