3.4 Global view of the outer main belt

A global view of the outer main asteroid belt is given on Figures 7 and 8 in the (a,e) and $(a,\sin I)$ planes, respectively. Only asteroids with good proper elements are shown on these plots (the actual criteria were exactly opposite from those for the inclusion in the 10 Myr run, that is $\sigma a < 0.0003$, $\sigma e < 0.003$, $\sigma \sin I < 0.001$ and $LCE < 5 \times 10^5$). The plots are made with the data from the joint catalog.

  

Figure: An overview of the asteroid outer main belt in proper semimajor axis vs. proper eccentricity plane. Only asteroids with accurately determined proper elements are shown.

A prominent feature in Figure 8 is a ``line'' of objects at proper inclination $\simeq 0.37$ and with 2.6<a_p<2.7 AU. These objects all have a low proper eccentricity, but only for a subset the values of e_p are in a narrow range, indicating a possible Hansa family (from 480 Hansa, the lowest numbered asteroid in this group); such a Hansa family had already been proposed by Hergenrother et al. ([1996]) and studied by A. Lemaitre by means of semianalytic proper elements (A. Lemaitre, personal communication). It is likely that such a family exists, that is, that these asteroids are collisionally related; however, without a detailed study we cannot exclude that such grouping is only apparent, due to an island of stability surrounded by secular resonances.

  

Figure: An overview of the asteroid outer main belt in proper semimajor axis vs. sine of proper inclination plane. Only asteroids with accurately determined proper elements are shown.

In Figure 9 the troublesome cases are shown in the same planes (a_p,e_p) and $(a_p,\sin I_p)$. The chaotic asteroids (either $\sigma a > 0.0003$ or $LCE > 5 \times 10^5$) are shown on the left: these two plots illustrate again the fact that the mean motion resonances are the main source of chaotic behavior for main belt asteroids. The chaotic objects are distributed along numerous easily recognizable resonant strips; chaotic diffusion takes place along the resonance lines (see Section 3.5). The most affected region is between 3.0 and 3.2 AU, even if in that region there are also many fairly stable asteroids (see Figures 7 and 8) and the region does not appear to be a continuous chaotic sea. On the right of Figure 9 we show the orbits affected by the secular resonances (either $\sigma e > 0.003$ or $\sigma \sin I > 0.001$). These two plots illustrate the effects of secular resonances, the most prominent feature being the outline of a group of overlapping nonlinear secular resonances between 2.5 and 2.8 AU, at low to moderate inclination and eccentricity. The location of these resonances as revealed by the numerical integration coincides with the analytical prediction (see Figures 7-11 in Milani and Knezevic [1994]). The numerical output not only indicates the exact location of these resonances, but can be used to estimate their width, providing in the same time a clear indication of the level of degradation of the proper elements due to the resonances themselves.

  

Figure: A composite figure showing chaotic asteroids (left) and asteroids with large standard deviations of eccentricity/inclination, affected by the secular resonances in the zone between 2.5 and 2.8 AU (right) in the proper semimajor axis vs. eccentricity (up), and semimajor axis vs. sine of inclination (bottom) plane.

The comparison of the plots presented on Figure 9, which has been prepared with the composite catalog, with the analoguous plots produced with the data from the 2 yr run, shows an additional feature. Quite a number of objects found to be chaotic in the short run and thus appearing on the plots on the left, appear also on the other pair of plots on the right in the long run, even if the objects are not close to any significant secular resonance. As discussed above, chaotic perturbations affect immediately the semimajor axis, but on the long run chaotic diffusion also affects the other proper variables.