## 5. Summary - discussionThe 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 years. -
Within our initial distribution, most of the orbits seem also to be chaotic. In particular 20 of them escape within 50 Myrs, of them are clearly chaotic, with inclination `jumps' even larger than occurring within the integration time-span. The remaining 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 ) plane the distinction between effectively stable and grossly unstable orbits is clear. Effectively stable librations take place in the region defined by and . For values outside this region, namely for and 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 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 stability region*. Note, however, that the actual limits of a suitably modified `Rabe's curve' for inclinations of and, even more, for the much more complex OSS model are not known. -
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 resonance. This result explains the large variations in the inclination of the and 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 Myrs, while
the © European Southern Observatory (ESO) 2000 Online publication: February 25, 2000 |