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Astron. Astrophys. 354, 1091-1100 (2000)

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5. Summary - discussion

The conclusions of this paper can be summarized as follows:

  • The result of Milani et al. (1997), based on which Thersites was classified as an ASC, is also confirmed in this paper. Thersites lies on a chaotic orbit with [FORMULA] years.

  • Within our initial distribution, most of the orbits seem also to be chaotic. In particular 20[FORMULA] of them escape within 50 Myrs, [FORMULA] of them are clearly chaotic, with inclination `jumps' even larger than [FORMULA] occurring within the integration time-span. The remaining [FORMULA] of our orbits are `effectively stable' for this time interval, but with erratic variations in the eccentricity also present for most of them.

  • On the [FORMULA]) plane the distinction between effectively stable and grossly unstable orbits is clear. Effectively stable librations take place in the region defined by [FORMULA] and [FORMULA]. For values outside this region, namely for [FORMULA] and [FORMULA] the orbits are unstable. The escaping orbits lie well above these limits, which constitute a set of escaping parameters for orbits initially placed in the vicinity of (1868) Thersites.

  • For the STB-orbits D and [FORMULA] are almost constant. If we consider these values to be a good approximation of proper elements and compares them with previously known results on the stability of the Trojans, we see that, even for these `stable' orbits the values are right at the limit of Rabe's stability curve (Rabe, 1967; see also Fig. 1 in Levison et al., 1997). Therefore, one can conclude that Thersites is somehow `trapped' on the edge of the [FORMULA] stability region . Note, however, that the actual limits of a suitably modified `Rabe's curve' for inclinations of [FORMULA] and, even more, for the much more complex OSS model are not known.

  • [FORMULA] TFA can also be used to distinguish between stable and unstable orbits. The most important features seen in these plots are (i) the disappearance and reappearance of modes for the unstable (and escaping) orbits (ii) a characteristic `drift' of the both the h- and p-spectrum for the unstable orbits which escape, (iii) a broader p-spectrum of the unstable orbits and (iv) large amplitude variations in the h-spectrum of the unstable orbits.

  • The TFA results indicate possible action of secular resonances. An analysis of the corresponding critical arguments has shown that, indeed, secular resonances involving the nodes of the outer planets are responsible for the chaotic behavior of Thersites, the most prominent features being associated to one of the multiplets of the [FORMULA] resonance. This result explains the large variations in the inclination of the [FORMULA] and [FORMULA]orbits (the classification is again justified) which preceed the eccentricity increase. It is interesting to study whether stable chaos in the Trojan swarms is in fact related to high-order secular resonances, much like main-belt ASC's are associated to high-order mean motion resonances (see Milani et al. 1997).

A very interesting result in our study is also the unusual escape path that one of our escaping orbits follows. Such a stickiness effect has already been discovered in a model Hamiltonian system (the so-called Sitnikov Problem, a special case of the spatial elliptic restricted problem, where two equally massive bodies are involved) by Dvorak et al. (1998), but it is much easier to observe in area-preserving maps. These results demonstrate the sticky properties of the island boundaries, which may be the mechanism delaying the transport of chaotic orbits and producing what is called stable chaos. The fact that this phase-space region is highly complicated but also sticky is supported by the fact that the escape time for chaotic orbits may vary by several tens of Myrs. In our results, the smaller value of escape time is [FORMULA] Myrs, while the UNS-orbits do not escape even after 50 Myrs. Moreover, most of the STB-orbits have `stable' orbital elements (compared to the other groups), while the TFA results indicate that there are large frequency[FORMULA]amplitude variations with time. We did not manage, however, to associate any secular resonance (up to the order tested) with these variations. Extending our integration would most probably result to the ejection of most of our fictitious asteroids from the solar system.

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© European Southern Observatory (ESO) 2000

Online publication: February 25, 2000