Paolo Farinella

The asteroids as outcomes of catastrophic collisions

Abstract The role of catastrophic collisions in the evolution of the asteroids is discussed in detail, employing extrapolations of experimental results on the outcomrs of high-velocity impacts. We determine the range of the probable largest collision for target asteroids of different sizes during the solar systems lifetime, and we conclude that all the asteroids have undergone collisional events capable of overcoming the materials solid-state cohesion. Such events do not lead inescapably to complete disruption of the targets, because (i) for a previously unfractured target, experiments show that fragments of significant size can survive breakup, depending on the energy and geometry of the collision; (ii) self-gravitation can easily cause a reaccumulation of fragments for targets exceeding a critical size, which seems to be of the order of 100 km. In the intermediate diameter range 100⪅D ⪅300 km, where formation of gravitationally bound “rubble piles” is frequent, the transfer of angular momentum can be large enough to produce objects with triaxial equilibrium shapes (Jacobi ellipsoids) or to cause fission into binary systems. In the same size range, low-velocity escape of collisional fragments can also occur, leading to the formation of dynamical families. Asteroids smaller than ∼100 km are mostly multigeneration fragments, while for D⪆300 km the collisional process produces nearly spheroidal objects covered by megaregoliths; whether their rotation is “primordial” or collisionally generated depends critically on the past flux of colliders. The complex and size-dependent phenomenology predicted by the theory compares satisfactorily with the observational evidence, as derived both by a classification of asteroids in terms of their size, spin rate, and lightcurve amplitude, and by a comparison between the rotational properties of family and nonfamily asteroids. The fundamental result of this investigation is that almost all asteroids are outcomes of catastrophic collisions, and that these events cause either complete fragmentation of the target bodies or, at least, drastic readjustments of their internal structure, shape, and spin rate.

Collision rates and impact velocities in the main asteroid belt

Abstract We have computed, using G.W. Wetherills (1967, J. Geophys. Res. 72, 2429–2444) algorithm, mutual collision probabilities and impact velocities for a set of 682 asteroids of diameter > 50 km, intended to represent a bias-free sample of asteroid orbits. For every asteroid, we have obtained the intrinsic collision probability, P i , the average collision velocity, V , and the number of projectile orbits which can intersect the target asteroids orbit, N cross , using the proper orbital elements of A. Milani and Z. Kneževic (1990, Celest. Mech. 49, 247–411). The average values and the corresponding standard deviations for the whole asteroid sample are: 〈 P i 〉 = 2.85 ± 0.66 × 10 −18 km −2 year −1 , 〈 V 〉 = 5.81 ± 1.88 km/sec, 〈 N cross 〉 = 601 (88% of the existing orbit pairs). No significant differences were found in the average values of P i, V , or N cross using osculating elements instead of proper elements, although results for individual asteroids could change by ≈10%. A running box mean of the intrinsic collision probability with semimajor axis shows a peak near 2.7 AU with 〈 P i 〉 = 3.4 × 10 −18 km −2 year −1 followed by a nearly monotonic decrease to 〈 P i 〉 = 1.8 × 10 −18 km −2 year −1 at 3.3 AU. Collision probabilities are nearly independent of eccentricities but show a significant decrease with larger inclinations. As expected, collisional velocities grow rapidly with increasing orbital eccentricities and inclinations, but they show surprisingly little variation across the asteroid belt. Family asteroids undergo collisions with other members of the same family two to three times more frequently than with nonfamily projectiles, but the relative speed of these intrafamily impacts is comparatively low for the Koronis and Themis families.

Short-Period Comets: Primordial Bodies or Collisional Fragments?

Modeling results show that collisions among Edgeworth-Kuiper Belt Objects (EKOs), a vast swarm of small bodies orbiting beyond Neptune, have been a major process affecting this population and its progeny, the short-period comets. Most EKOs larger than about 100 kilometers in diameter survive over the age of the solar system, but at smaller sizes collisional breakup is frequent, producing a cascade of fragments having a power law size-frequency distribution. Collisions are also a plausible mechanism for injecting EKOs 1 to 10 kilometers in diameter into dynamical resonances, where they can be transported into the inner solar system to become short-period comets. The fragmental nature of these comets may explain their physical properties, such as shape, color, and strength.

Wavy size distributions for collisional systems with a small-size cutoff

Abstract Dohnanyis [ J. geophys. Res . 74 , 2531–2554, 1969; in Physical Studies of Minor Planets (edited by T. Gehrels), pp. 263–295. NASA-SP 267, 1971] theory predicts that a collisional system such as the asteroid population should rapidly relax to a power-law equilibrium size distribution, provided all the collisional response parameters are independent of size. However, we have found that Dohnanyi did not include in a consistent way in the theory the possible occurrence of a small-size cutoff in the distribution. We have carried out a number of numerical simulations of the collisional evolution process, showing that the cutoff results in a wavy pattern superimposed on Dohnanyis equilibrium power law, which affects the distribution up to sizes of tens of km. The pattern arises because particles just above the cutoff are not removed by catastrophic impacts by smaller projectiles, and therefore are created by break-up of larger bodies faster than they are eliminated; larger particles are increasingly depleted up to the size where the smallest shattering projectile exceeds the cutoff, and beyond that the removal rate is reduced and the distribution flattens. Thus, to be effective in producing the waves, the cutoff (or any other persisting “discontinuity” in the particle properties) must be sharp over a size range corresponding to the threshold projectile-to-target ratio for fragmentation. The presence of a small-size cutoff in the real asteroid belt is an open question, since it may be generated by poorly known non-gravitational effects acting on μm-sized dust, and may be affected by influx of cometary debris. However, the observational evidence for a variable characteristic exponent of the size distribution of interplanetary bodies is now strong, and the cutoff effect may provide a simple explanation for this finding.

Secular resonances from 2 to 50 AU

Abstract By means of a new algorithm which generalizes the second-order and fourth-degree secular perturbation theory of Milani and Kneževic (1990, Celest. Mech. 49, 347–411), we find in the a-e-I proper elements space the location of secular resonances between the precession rates of the longitudes of perihelion and node of a minor body and the corresponding eigenfrequencies of the secular perturbations of the four outer planets. Among the most interesting implications of our findings, we can quote: (i) the fact that the g = g6 (or ν6) resonance in the inner asteroid belt lies closer than previously assumed to the densely populated Flora region, providing a plausible dynamical route to inject asteroid fragments into planet-crossing orbits; (ii) the existence of another possible meteorite source near 2.4 AU at moderate inclinations, again through g = g6; (iii) the existence, confirmed by numerical experiments, of a region affected in a chaotic way by the s = s6 (or ν16) resonance at semimajor axis ⋍ 2.2 AU and moderate inclination, where no asteroid is observed; (iv) the possible presence of some low-inclination “rings” between the orbits of the outer planets where no major mean motion or secular resonance lies very near, allowing minor bodies to survive long times without close encounters; (v) the fact that none of the secular resonances considered in this work exists beyond 50 AU, so that these resonances cannot be effective for transporting inward comets belonging to a possible Kuiper flattened disk.

Asteroid collisional evolution: results from current scaling algorithms

Abstract Acritical element for the understanding of asteroid collisional evolution is the scaling law needed to link laboratory impact experiments to the fragmentation of asteroidal bodies, ranging in size from meters to several hundreds of km. Early workers generally assumed a simple energy scaling, augmented by gravitational self-compression. Recent work on scaling theories has produced algorithms for computing the specific energy, Q ∗ , required to fragment bodies of various sizes, based on two approaches: the strain-rate scaling theory of Housen and Holsapple (Icarus, 84, 226, 1990), based on dimensional analysis, and the 2-D hydrocode ealculations of Ryan and Melosh (1994). The strain-rate scaling predicts a deerease of about an Order of magnitude when going from laboratory sized bodies, 10 cm, to bodies a few tens of km in size, whereas for larger sizes Q ∗ , grows due to gravitational self- compression. The hydrocode results show an even stronger dependence on size, with a Q ∗ decrease of 2–3 orders of magnitude between 10 cm and 25 km, depending on the properties of the material. One possible way to discriminate among these different scaling laws is to examine which of them (if any) can predict the observed size distribution of asteroids from arbitrary starting populations and simultaneously satisfy other constraints on asteroid collisional history, such as the preservation of Vestas basaltic crust. We have now explored this problem using the asteroid collisional evolution model of Davis et al. (AsteroidsII, pp. 805–826, University of Arizona Press, Tuscon, 1989), modified to take the different scaling algorithms as an input option. These model calculations show that a comparatively large value of Q ∗ is neede to match the observed size distribution and to preserve Vestas crust. Simple energy scaling with gravitational self-compression in agreement with the laboratory experiments of Housen et al. (Iearus94, 180, 1991) does the best of reproducing the observed asteroid belf. Strain-rate scaling could also match the observations; however, extension of our knowledge of the main-belt population down to sizes of ∼ 1 km would test this agreement. The hydrocode scaling results generally predict weak asteroids and do not reproduce the size distribution, nor do they allow Vestas crust to be preserved except in a highly improbable fashion. The hydrocode scaling of Q ∗ however, provides only a shattering threshold; work to derive the corresponding scaling law for the energy partitioning coefficient, needed to model the dispersal/reaccumulation of fragments, is under way.

The rotation of LAGEOS and its long-term semimajor axis decay: A self-consistent solution

We develop a self-consistent model for the evolution of the spin axis of LAGEOS and the related long-term semimajor axis perturbations, due to asymmetric emission/reflection of radiation from the satellites surface. We show that the theory developed by Bertotti and Iess [1991] for the evolution of LAGEOSs rotation under magnetic and gravitational torques, which we have somewhat generalized here, can lead to a successful fit of the observed semimajor axis residuals, provided the correct initial conditions for the direction of the spin axis are chosen. The remaining residuals have an rms dispersion of 0.50 × 10−12 m/s2, comparable to that of previous solutions, based on purely empirical fits of the spin axis direction as a function of time. The spectrum of the residuals indicates that they are probably due to unmodeled radiation forces (e.g., from Earth albedo and/or penumbra passages). Our solution allows us to predict the future evolution of LAGEOSs rotation for about another decade in the future, until the spin rate will become so slow that some basic assumptions of the theory will fail. A similar model can also be used to model/predict the coupled spin-orbit evolution of the LAGEOS II satellite, launched in late 1992, although the available data still cover a span of time too short for reaching significant quantitative conclusions concerning this satellite.

Orbital evolution around irregular bodies

The new profiles of the space missions aimed at asteroids and comets, moving from fly-bys to rendezvous and orbiting, call for new spaceflight dynamics tools capable of propagating orbits in an accurate way around these small irregular objects. Moreover, interesting celestial mechanics and planetary science problems, requiring the same sophisticated tools, have been raised by the first images of asteroids (Ida/Dactyl, Gaspra and Mathilde) taken by the Galileo and NEAR probes, and by the discovery that several near-Earth asteroids are probably binary. We have now developed two independent codes which can integrate numerically the orbits of test particles around irregularly shaped primary bodies. One is based on a representation of the central body in terms of “mascons” (discrete spherical masses), while the other one models the central body as a polyhedron with a variable number of triangular faces. To check the reliability and performances of these two codes we have performed a series of tests and compared their results. First we have used the two algorithms to calculate the gravitational potential around non-spherical bodies, and have checked that the results are similar to each other and to those of other, more common, approaches; the polyhedron model appears to be somewhat more accurate in representing the potential very close to the body’s surface. Then we have run a series of orbit propagation tests, integrating several different trajectories of a test particle around a sample ellipsoid. Again the two codes give results in fair agreement with each other. By comparing these numerical results to those predicted by classical perturbation formulae, we have noted that when the orbit of the test particle gets close to the surface of the primary, the analytical approximations break down and the corresponding predictions do not match the results of the numerical integrations. This is confirmed by the fact that the agreement gets better and better for orbits farther away from the primary. Finally, we have found that in terms of CPU time requirements, the performances of the two codes are quite similar, and that the optimal choice probably depends on the specific problem under study.

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 main belt as a source of near-Earth asteroids

We investigate the flux of main-belt asteroid fragments into resonant orbits converting them into near-Earth asteroids (NEAs), and the variability of this flux due to chance interasteroidal collisions. A numerical model is used, based on collisional physics consistent with the results of laboratory impact experiments. The assumed main-belt asteroid size distribution is derived from that of known asteroids extrapolated down to sizes of ≈ 40 cm, modified in such a way to yield a quasi-stationary fragment production rate over times ≈ 100 Myr. The results show that the asteroid belt can supply a few hundred km-sized NEAs per year, well enough to sustain the current population of such bodies. On the other hand, if our collisional physics is correct, the number of existing 10-km objects implies that these objects either have very long-lived orbits, or must come from a different source (i.e., comets). Our model predicts that the fragments supplied from the asteroid belt have initially a power-law size distribution somewhat steeper than the observed one, suggesting preferential removal of small objects. The component of the NEA population with dynamical lifetimes shorter than or of the order of 1 Myr can vary by a factor reaching up to a few tens, due to single large-scale collisions in the main belt; these fluctuations are enhanced for smaller bodies and faster evolutionary time scales. As a consequence, the Earths cratering rate can also change by about an order of magnitude over the 0.1 to 1 Myr time scales. Despite these sporadic spikes, when averaged over times of 10 Myr or longer the fluctuations are unlikely to exceed a factor two.