3. Results

The results obtained with the new method to compute asteroid proper elements can be better described by distinguishing the stable and the unstable orbits. A large majority of the 10,265 initial conditions correspond to ostensibly regular orbits (at least over the tested time span), and for these we could determine synthetic proper elements with very good accuracy and stability in time: e.g., in $76\%$ of the cases e_p and $\sin I_p$ had standard deviations (in the running box test) less than 0.001, and a_p less than 0.0003 AU. In $55\%$ of the cases the RMS was even less than 0.0003 in e_p and less than 0.0001 in $\sin I_p$ (see Section 3.3). The typical improvement with respect to the analytically derived proper elements is by a factor of more than 3. In terms of the so-called standard metrics (Zappalà et al. [1990]) used to define asteroid families, this would imply a typical error of the relative velocity of family members with respect to the ``parent body'' on the order of $\approx 5$ m/s (to be compared with $\approx 17$ m/s for the analytical results). On the other hand, in the basic 2 Myr run $9\%$ of the cases had the standard deviations of either e<sub>p</sub> or $\sin I_p$ larger than 0.003, $5\%$ were chaotic with Lyapunov times shorter than 10,000 yr (Section 3.3). At the extreme end of the stability/instability spectrum, we detected 31 ``pathological'' cases for which the changes with time of the proper elements and/or frequencies were excessively large, thus the values as given in the output are unreliable (Section 3.2). Among these pathological orbits there are even 9 cases in which the orbits become hyperbolic, as a result of close approaches to some major planets, and of course we could not derive proper elements at all (Section 3.1). For all the asteroids in our input catalog, with the only exception of the hyperbolic orbits, we have produced proper elements and error estimates as described in Section 2. These data are available online in three separate files: prop.syn, sigma.syn, and delta.syn, containing the proper elements, their standard deviations and maximum excursions, respectively, plus some additional information (LCE's, run code, RMS of residuals for the mean longitude linear fit). They are available² from the AstDyS web site.