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*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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