** Next:** 1. Introduction

# Synthetic proper elements

for outer main belt asteroids

**Z. KNEZEVIC**^{1} and A. MILANI^{2}

Astronomical Observatory, Volgina 7, 11160 Belgrade 74,

Yugoslavia, e-mail: zoran@aob.bg.ac.yu

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.

** Next:** 1. Introduction
*Andrea Milani*

*2000-10-03*