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

2. Dynamical phenomena

Discriminating between regular and chaotic orbits is most usually done by calculating the maximal Lyapunov Characteristic Exponent (LCE), , which is defined as the average asymptotic rate of exponential divergence of initially nearby trajectories. If the orbit is said to be regular, whereas corresponds to a chaotic orbit. The inverse of the LCE is called Lyapunov time, . An attempt to connect the value of the Lyapunov time, , to the `event' time, , at which a main-belt asteroid becomes planet-crosser, was made by Lecar et al. (1992), whose numerical results have shown the existence of a power-law relationship between these two dynamical quantities. However, the discovery of a behavior termed stable chaos (Milani & Nobili, 1992), describing asteroids for which but whose orbital elements are stable for times much longer than the ones predicted by the aforementioned relationship, has put under question the validity of such a `law'. Subsequently, other researchers have shown that the regime of validity of this relationship is limited in the resonance-overlap regime of the dynamical system describing the motion of small bodies in the solar system (Varvoglis & Anastasiadis, 1996; Morbidelli & Froeschlé, 1996; Murray & Holman, 1997). This disagreement lead some researchers to conjecture (Murison et al., 1994; Varvoglis & Anastasiadis, 1996; also Tsiganis et al., 2000) that stable chaos is, in fact, the manifestation of the stickiness effect which is caused in Hamiltonian dynamical systems by the presence of stability islands, and the action of the cantori surrounding them, inside a chaotic domain of the phase space.

Milani (1993) has shown that (1868) Thersites, among other Trojans, is an example of an asteroid in stable chaos (ASC) (see also Milani et al., 1997). For all these Trojans the Lyapunov time is less than years and, yet, they are permanent members of the Trojan swarms. An analysis of these Trojans was made in Pilat-Lohinger et al. (1999a) using results from a 10 Myrs integration of the orbits, where proper elements were also determined. In another paper by Pilat-Lohinger et al. (1999b, hereafter Paper I), it was shown how these proper elements vary slowly, for most of them, for a time interval of 100 Myrs. In their integration, Thersites escaped from the solar system after 26 Myrs. The question, of course, is what could be the dynamical mechanism causing orbits in the vicinity of Thersites, which is librating around the stability point, to become grossly unstable.

In dynamical models consisting of more-than-one perturbing planets the most important dynamical phenomena coming into play are the secular resonances , i.e. the resonances between the precession frequency of an asteroid's longitude of perihelion (or node), (or ), with one (or a linear combination) of the characteristic secular frequencies of the solar system, which describe the precession of the planetary orbits. The important rle of the secular resonances, especially the , and , in the dynamical sculpting of the main belt and the evolution of Near-Earth-Asteroid's (NEA's) was demonstrated recently by several researchers (e.g. Michel and Ch. Froeschlé, 1997, Dvorak and Pilat-Lohinger, 1999). Another important dynamical feature is the so-called three-body mean motion resonance (Nesvorný & Morbidelli, 1998; Murray et al., 1998). This kind of resonance could be one of the main sources of `weak' chaos found in nearly co-planar orbits throughout the asteroid belt (see also Nesvorný & Morbidelli, 1999).

The Trojan asteroids are not exactly in a very `friendly' place. Although most of them seem to be favoured by the (or ) stability region of the 1:1 mean motion resonance with Jupiter, the relatively high inclinations () of many of them - including Thersites - render them as `good candidates' for suffering instabilities induced to their orbits by the proximity of secular resonances. In fact, Milani (1994) has shown that the distribution of the Trojans on the plane is bounded by the secular resonance. In the following, we are going to present numerical results showing that these dynamical phenomena may be the cause of driving Thersites away from the Trojan belt.

© European Southern Observatory (ESO) 2000

Online publication: February 25, 2000
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