Chaotic Routes Between The Asteroid Belt and Earth
By Paolo Farinella

During the recent history of the Earth, on average some 108 kg per year of meteoritic material has been falling onto it, ranging in size from microscopic dust to 10-km diameter asteroids (Ceplecha 1992). Where does this material come from? Until two centuries ago, the fall of stones from the sky was believed by scientists to be just a matter of superstition; later on, on the other hand, meteorites were long believed to be interstellar objects travelling on hyperbolic heliocentric trajectories. Today, the birthplace of the vast majority of meteorites has been recognized to lie in the main asteroid belt, where the growth of a full-sized planet was aborted during the early evolution of the solar system. However, until recently serious problems remained in understanding the asteroid-meteorite connection.

As a consequence of interaster-oidal impact events, fragments are usually ejected from their parent bodies at velocities of the order of 100-200 m/s (Zappalà et al. 1984, 1990), too small for directly causing the drastic orbital changes needed to achieve planet-crossing orbits. On the other hand, impulsive velocity increments of several km/s would normally induce in the meteoritic material strong shock and thermal modifications, which are seldom observed. Thus the interplay between collisions and subtle dynamical mechanisms is required to transport material from the asteroid belt to the Earth. Nowadays it is known that two types of such mechanisms are available, providing two main routes for the meteorites.

The first route is related to the so-called mean-motion resonances. Since more than one century, dynamicists have known that for orbital periods corresponding to simple fractions (4/1, 3/1, 5/2, 7/3, 2/1) times Jupiter's period, very few or no bodies exist in the asteroid belt. Since Kepler's third law relates orbital periods to semimajor axes, these have become known as "Kirkwood gaps". Only recently the modern theory of chaos in dynamical systems has allowed celestial mechanicians to explain the origin of these gaps. They are due to the so-called resonance effects, by which orbits located in the gaps would be subject to strong perturbations by Jupiter, hence would undergo rapid and irregular variations in their orbital elements. In particular, strong increases in the eccentricity would cause the resonant orbits to become planet-crossing, so that these objects would be eliminated by close encounters (with Jupiter, ejecting them into hyperbolic paths) or collisions (with the planets or the Sun). In particular Wisdom (1983) and recently, in a more realistic way, Moons and Morbidelli (1995) have shown that the semimajor axis interval between 2.48 and 2.52 AU, where the Kirkwood gap corresponding to the 3/1 resonance with Jupiter is located, includes a large chaotic region, causing strong eccentricity increases on time scales of the order of 1 Myr.

This result was used by Wetherill (1985, 1987) to argue that the 3/1 resonance provides the most important source of ordinary chondrites. On the basis of his simplified statistical model of orbital evolution,Wetherill showed that asteroid collisional debris produced at the borders of the 3/1 gap with ejection velocities between 50 and 200 m/s has a high probability to be placed into the chaotic region; there the resonance pumps up the orbital eccentricity to 0.6, corresponding to a perihelion distance of about 1 AU, and later on close encounters with the Earth, removing the semimajor axis from the resonant zone, to give rise to a typical Earth-crossing orbit.

Besides the 3/1 Kikwood gap, Wetherill also suggested that another possible source of meteorites lies in the innermost part of the asteroid belt, at semimajor axes between 2.0 and 2.25 AU, thanks to the presence of the so-called n6 secular resonance (Williams 1969; Williams and Faulkner 1981), which provides the inner and high-inclination boundary to the observed distribution of asteroids (see Figure 1). In the case of secular resonances, what matters are not the orbital periods, but rather the periods of time (of the order of tens of thousands of years) over which the orbits change their mutual orientation (i.e., precess) due to planetary perturbations. The n6 resonance affects orbits whose perihelion direction precesses around the Sun at the same rate as that of Saturn. Numerical computations by Scholl and Froeschlé (1991) showed that indeed the n6 resonance, often acting in a synergistic way with the 4/1 mean-motion resonance with Jupiter at semimajor axes between 2 and 2.1 AU, gives raise to a fairly broad region where Earth-crossing chaotic orbits are generated. Other numerical experiments led to the finding that the n6 resonance is probably an efficient fragment collector not only near the inner edge of the asteroid belt, but also in the moderate-inclination (15 to 20 degrees) regions at semimajor axes of about 2.4 and 2.7 AU.

Subsequent work by our group (Farinella et al. 1993a,b; Morbidelli et al. 1994) has shown that the interplay of asteroidal collisions and chaotic dynamics can easily generate significant numbers of Earth-crossing asteroid fragments (see Figure 2). The ejection of fragments from cratering or break-up events involving thousands of numbered main-belt asteroids having semimajor axes inside 2.8 AU was simulated through a numerical model, consistent with the experimental results on the outcomes of high-velocity (some 5 km/s) impacts (Davis et al. 1989; Petit and Farinella 1993). For every asteroid, the flux of fragments dropped into the chaotic regions associated with the 3/1 and n6 resonances has been estimated, providing indications about the most effective potential meteorite-delivering asteroids (Farinella et al. 1993a). Several fictitious fragments whose orbits were tracked by suitable computer simulations over a few million years have displayed orbital evolution patterns closely resembling those of real near-Earth asteroids and of the largest observed fireballs, such as that associated with the Pribram meteorite fall (Jopek et al. 1995).

We have also estimated the near-Earth asteroid/meteorite production potential of the massive fragment ejection events which generated some populous asteroid families. In particular, a significant meteorite yield has been obtained in the case of Vesta, the third biggest asteroid, for which recent spectroscopic observations (Binzel and Xu 1993) have indicated a clear genetic relationship to a number of minor family members having similar orbital elements, and also to a specific group of meteorites (the so-called HED meteorites, i.e. howardites, eucrites and diogenites). It is remarkable that some near-Earth asteroids also share the same spectral features.

This approach has allowed us to identify the best asteroid candidates for being meteorite parent bodies. Actually, several asteroids can inject into the resonances a significant fraction of their impact ejecta, which then have their orbital eccentricities increased up to Mars- and even Earth-crossing values. Some of these bodies _ in particular the largest ones close to the n6 resonance, such as 6 Hebe, 304 Olga, 739 Mandeville and 759 Vinifera _ are almost certainly efficient deliverers of meteorites. The time required for the fragments to be transported into Earth-crossing orbits is typically of the order of 1 Myr for those bodies whose eccentricity is rapidly pumped up by the resonance to values >0.6, but it may be even 100 or 1000 times longer when the peak eccentricity is lower and Mars encounters are needed as an intermediate process. Together with the occurrence of disruptive impacts during their travel to Earth, these diverse dynamical time scales may explain the wide range of cosmic-ray exposure ages observed for meteorites. Other asteroids inject a substantial fraction of their fragments into the Kirkwood gaps corresponding to the 3/1 (623 Chimaera, 1892 Lucienne) and 5/2 (631 Philippina, 907 Rhoda, 1222 Tina) mean-motion resonances with Jupiter.

However, only for a fraction of the candidate asteroids _ typically, the few largest ones _ detailed spectroscopic data are available, allowing one to infer about their mineralogy and chemistry and to identify or reject specific links to different meteorite types. It appears likely that a large fraction of all meteorites and near-Earth asteroids are delivered by a small and possibly nonrepresentative sample of the known main-belt asteroid population, consisting of relatively large bodies located in the vicinity of resonances. If this is the case, the hunt for the meteorite parent bodies is finally open to observers!

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Department of Mathematics

University of Piza, Italy