Christiane Froeschlé

On the Relationship between Fast Lyapunov Indicator and Periodic Orbits for Continuous Flows

It is already known (Froeschlé et al., 1997a) that the fast Lyapunov indicator (hereafter FLI), i.e. the computation on a relatively short time of a quantity related to the largest Lyapunov indicator, allows us to discriminate between ordered and weak chaotic motion. Using the FLI many results have been obtained on the standard map taken as a model problem. On this model we are not only able to discriminate between a short time weak chaotic motion and an ordered one, but also among regular motion between non resonant and resonant orbits. Moreover, periodic orbits are characterised by constant FLI values which appear to be related to the order of periodic orbits (Lega and Froeschlé, 2001). In the present paper we extend all these results to the case of continuous dynamical systems (the Hénon and Heiles system and the restricted three-body problem). Especially for the periodic orbits we need to introduce a new value: the orthogonal FLI in order to fully recover the results obtained for mappings.

Meteorites from the Asteroid 6 Hebe

We have numerically integrated the orbits of 18 fictitious fragments ejected from the asteroid 6 Hebe, an S-type object about 200 km across which is located very close to the g = g 6 (or v 6) secular resonance at a semimajor axis of 2.425 AU and a (proper) inclination of 15°.0. A realistic ejection velocity distribution, with most fragments escaping at relative speeds of a few hundreds m/s, has been assumed. In four cases we have found that the resonance pumps up the orbital eccentricity of the fragments to values > 0.6, which result into Earth-crossing, within a time span of ≈ 1M yr; subsequent close encounters with the Earth cause strongly chaotic orbital evolution. The closest Earth and Mars encounters recorded in our integration occur at miss distances of a few thousandths of AU, implying collision lifetimes < 109yr. Some other fragments affected by the secular resonance become Mars-crossers but not Earth-crossers over the integration time span. Two bodies are injected into the 3: 1 mean motion resonance with Jupiter, and also display macroscopically chaotic behaviour leading to Earth-crossing. 6 Hebe is the first asteroid for which a realistic collisional/dynamical evolution route to generate meteorites has been fully demonstrated. It may be the parent body of one of the ordinary chondrite classes.

The Secular Resonances in the Solar System

In the last three years new studies on secular resonances have been done. The second-order and fourth-degree secular perturbation theory of Milani and Kneževic allowed to point out the effect of mean motion resonances on the location of the linear and non linear secular resonances. Moreover this theory improved the knowledge of the exact location of the g = g 6 (i.e. v 6) resonance at low inclination. Morbidelli and Henrard revisited the semi-numerical method of Williams, taking into account the quadratic terms in the perturbing masses. They computed not only the location of secular resonances, but also provided a global description of the resonant dynamics in the main secular resonances namely g = g 5 (i.e. v 5), g = g 6 (i.e. v 6) and s = s 6 (i.e. v 16). The resonant proper element algorithm developed by Morbidelli allows to identify the dynamical nature of resonant objects, and is a powerful tool to study the mechanisms of meteorite transport to the inner Solar System. Purely numerical experiments have been done, which show : (i) the complexity of the dynamics when two resonances overlap; (ii) the efficiency of successive crossings of non linear resonances in pumping up the inclination of small bodies; (iii) the efficiency of the secular resonance v 6 as a source of meteorites up to 2.4 AU.

Stellar perturbations on the scattered disk

We present a statistical model for estimating the effects of stellar encounters on orbits in the outer Solar System, focussing on the scattered disk at <10 3 AU from the Sun. We describe a Monte Carlo simulation using those results and apply it to the evolution of the scattered disk over 4 Gyr, finding that a final perihelion distance distribution with an extended tail reaching to very large values is to be expected. This would likely result from a single close stellar encounter, in agreement with the conclusion by Morbidelli & Levison (2004). We estimate that the newly discovered minor planet (90377) Sedna may be a typical representative of such an extended scattered disk and that a few more objects of the same size may reside at similar heliocentric distances. There is a possibility that the bulk of the population, which should have smaller perihelion distances, contains some very large objects that may have contributed to sculpting the Kuiper Belt. We also find that the creation of an extended scattered disk by a stellar encounter should have been accompanied by a huge influx of large objects into the inner Solar System, but the timing of the encounter is constrained by the fact that the scattered disk must still have been quite massive. Thus it likely happened long before the purported late heavy bombardment of the terrestrial planets.

Dynamical Transport to Planet-Crossing Orbits

The study of small bodies on planet-crossing orbits brings out two main issues i.e. the identification of their birth place and how they got their current orbit.

Origin and Dynamical Transport of Near-Earth Asteroids and Meteorites

This chapter describes the most important mechanisms for transporting material from the Asteroid Belt to the Earth’s vicinity and the evolution of bodies on Earth-crossing orbit. Particular care will be paid to evolutionary timescales and statistical aspects of meteoroids dynamics.

Secular Dynamics of Asteroids in the Inner Solar System

In the last three years we have carried out numerical and semi-analytical studies on the secular dynamical mechanisms in the region (semimajor axis a < 2 AU) where the NEA orbits evolve. Our numerical integrations (over a time span of a few Myr) have shown that: (i) the linear secular resonances with both the inner and the outer planets may play an important role in the dynamical evolution of NEAs; (ii) the apsidal secular resonance with Mars could provide an important dynamical transport mechanism by which asteroids in the Mars-crossing region eventually achieve Earth-crossing orbits; (iii) in this region, due to the interaction with the terrestrial planets, the Kozai resonance can occur at small inclinations, with the argument of perihelion ω librating around 0° or 180°, providing a temporary protection mechanism against close approaches to the planets.