Collisional evolution of the asteroid belt

A new synthesis of asteroid collisional evolution is motivated by the question of whether most asteroids larger than ∼1 km size are strengthless gravitational aggregates (rubble piles). NEAR found Eros not to be a rubble pile, but a shattered collisional fragment, with a through-going fracture system, and an average of about 20 m regolith cover. Of four asteroids visited by spacecraft, none appears likely to be a rubble pile, except perhaps Mathilde. Nevertheless, current understanding of asteroid collisions and size-dependent strength, and the observed distribution of rotation rates versus size, have led to a theoretical consensus that many or most asteroids larger than 1 km should be rubble piles. Is Eros, the best-observed asteroid, highly unusual because it is not a rubble pile? Is Mathilde, if it is a rubble pile, like most asteroids? What would be expected for the small asteroid Itokawa, the MUSES-C sample return target? An asteroid size distribution is synthesized from the Minor Planet Center listing and results of the Sloan Digital Sky Survey, an Infrared Space Observatory survey, the Small Main-belt Asteroid Spectroscopic Survey and the Infrared Astronomical Satellite survey. A new picture emerges of asteroid collisional evolution, in which the well-known Dohnanyi result, that the size distribution tends toward a self-similar form with a 2.5-index power law, is overturned because of scale-dependent collision physics. Survival of a basaltic crust on Vesta can be accommodated, together with formation of many exposed metal cores. The lifetimes against destruction are estimated as 3 Gyr at the size of Eros, 10 Gyr at ten times that size, and 40 Gyr at the size of Vesta. Eros as a shattered collisional fragment is not highly unusual. The new picture reveals the new possibility of a transition size in the collisional state, where asteroids below 5 km size would be primarily collisional breakup fragments whereas much larger asteroids are mostly eroded or shattered survivors of collisions. In this case, well-defined families would be found in asteroids larger than about 5 km size, but for smaller asteroids, families may no longer be readily separated from a background population. Moreover, the measured boulder size distribution on Eros is re-interpreted as a sample of impactor size distributions in the asteroid belt. The regolith on Eros may result largely from the last giant impact, and the same may be true of Itokawa, in which case about a meter of regolith would be expected there. Even a small asteroid like Itokawa may be a shattered object with regolith cover.     

The asteroid belt and its evolution

We have reinvestegated the suggestion that collisional fragmentation in the asteroid belt can account for its present luminosity function. We suggest, based on the usual Boltzmann-type equation for this process, that for the brightest asteroids the time scale for a catastropic collision is 1.2 × 10⁹yr. However, the assumption of molecular chaos is not valid in the asteroid belt and we demonstrate a new method to determine the necessary corrections. We then obtain, using the new procedure, a lower limit for a collision time. For the above sample it is 2 × 10¹¹yr. This, we believe, rules out collisional evolution of the asteroid belt since its formation. Finally, we also show histograms of eccentricity, inclination, absolute magnitude, height above the ecliptic plane, and argument of perihelion for the 2829 asteroids with well-determined orbits. This represents a synthesis of the numbered asteroid and PLS data.     

Planets in the asteroid belt

Abstract— The main asteroid belt has lost >99.9% of its solid mass since the time at which the planets were forming, according to models for the protoplanetary nebula. Here we show that the primordial asteroid belt could have been cleared efficiently if much of the original mass accreted to form planetsized bodies, which were capable of perturbing one another into unstable orbits. We provide results from 25 N‐body integrations of up to 200 planets in the asteroid belt, with individual masses in the range 0.017–0.33 Earth masses. In the simulations, these bodies undergo repeated close encounters which scatter one another into unstable resonances with the giant planets, leading to collision with the Sun or ejection from the solar system. In response, the giant planets' orbits migrate radially and become more circular. This reduces the size of the main‐belt resonances and the clearing rate, although clearing continues. If ～3 Earth masses of material was removed from the belt this way, Jupiter and Saturn would initially have had orbital eccentricities almost twice their current values. Such orbits would have made Jupiter and Saturn 10–100x more effective at clearing material from the belt than they are on their current orbits. The time required to remove 90% of the initial mass from the belt depends sensitively on the giant planets' orbits, and weakly on the masses of the asteroidal planets. 18 of the 25 simulations end with no planets left in the belt, and the clearing takes up to several hundred million years. Typically, the last one or two asteroidal planets are removed by interactions with planets in the terrestrial region     

Mass distribution in the asteroid belt

The dependence of the cumulative number of numbered asteroids (up to 3720) on their absolute magnitude is investigated. The differential mass index k is derived from these relations for fainter asteroids. A steeper slope (2.2 < k < 2.4) is found in the four most populous asteroid familes (Flora, Koronis, Eos and Themis) and a flatter slope (1.3 < k < 1.6) for non-family asteroids. This indicates that there are two different asteroid populations in the asteorid belt. Total masses of the asteroid families may be greater than it is commonly accepted.     

Resonant structure of the outer asteroid belt

An analysis of ordered and chaotic regions of motion in the outer asteroid belt has shown that once the eccentricity of Jupiter is introduced the chaotic regions of the circular model are quite easily depleted. This suggests that also objects in neighbouring regions must be strongly perturbed. Therefore it is not surprising that many outer belt asteroids have been reported in the literature as resonant or anyway dynamically protected. By using the planar elliptic restricted 3-body model we have investigated the motion of outer belt asteroids which had not been suspected to librate. We find 3 cases of libration and 11 cases of e, coupling that can be explained within the theory of secular resonances. It is thus established that in the outer belt only resonant and dynamically protected asteroids can have lifetimes of the same order as the age of the Solar System.     

Dynamical behaviour of the primitive asteroid belt

In this paper we consider the dynamical evolution and orbital stability of objects in the asteroid belt. A simple physical model, including full gravitational perturbations from both giant planets, is used to compute the dynamical evolution of 1000 test particles simulating the primitive asteroids. The criterion of planet crossing (or close approach in the case of resonant objects) is used to reject particles from the simulation. 44 per cent of the particles survived for the whole time-span covered by the numerical integration (∼10⁹ yr).
The 4:1, 3:1 and to a lesser extent the 2:1 Kirkwood gaps are formed in ∼10⁷ yr of evolution, representing direct numerical evidence about their gravitational origin.
We found that the rms eccentricity and inclination of the sample experience a fast increase during the first 10⁶ yr. The final rms eccentricity is 0.11, ∼60 per cent smaller than the present rms eccentricity (0.17). Nevertheless, the gravitational action of the giant planets suffices to prevent the formation of large objects, allowing catastrophic collisions and the subsequent depletion of material from this zone of the Solar system. The excited eccentricity by Jupiter and Saturn may favour mutual encounters and the further increase of the relative velocities up to their present values.     

The large-scale structure of the asteroid belt

We examine the distributions of 2888 numbered minor planets over orbital inclination, eccentricity, and semimajor axis, and define 19 zones which we believe adequately to isolate the selection biases in survey programs of the physical properties of minor planets. Six numbered asteroids have exceptional orbits and fall into no zone. We also call attention to rather sharp upper limits, which become increasingly stringent at larger heliocentric distances, on orbital inclinations and eccentricity.     

Collisional origin of the asteroid families: Mass and velocity distributions

The origin of asteroid families in terms of collisional breakup is analyzed using the data by Williams (1979, in Asteroids (T. Gehrels, Ed.), pp. 1040–1063, Univ. of Arizona Press, Tucson). The distributions of mass and relative velocity of the minor family members with respect to the largest body are derived. These ditributions are then compared with the outcomes of catastrophic impacts, predicted from theoretical arguments and observed from laboratory experiments. The general features of the mass distributions can be interpreted in terms of collisional disruption of a parent body followed by self-gravitational reaccumulation on the largest remnant; no decisive evidence for multiple reaccumulations is found. The typical ejection velocities of the family members are of the same order as those of laboratory fragments; since the definition of families is based on purely dynamical arguments, this results gives direct observational support to the collisional formation hypothesis. The transition between collisional outcomes dominated by solid-state forces and by self-gravitatation, expected to occur at diameters D ～ 100 km on the basis of rotational studies and of theoretical estimates, is clearly confirmed by the present analysis. A “morphological” classification into three broad classes (asymmetric, dispersed, and intermediate) is introduced; it is based on the distribution of mass vs relative velocity, taking also into account the parent body's (and the largest remnant's) escape velocity. Finally some results are outlined which apparently do not fit the theoretical predictions: (1) the degree of fragmentation in real families is generally lower than that observed for experimental targets and (2) when the relative velocities are computed, including also proper eccentricity and inclination differences, values higher by about a factor 4 than those derived from semiaxes differences only are found. Further studies are proposed, including more observations, better proper elements computation and classification methods, and new investigations on the physics of hypervelocity impacts.     

Formation of the Kirkwood gaps in the asteroid belt

A possible mechanism to explain the depletion of the Kirkwood gaps in the asteroid belt would be the slow dissipation of the solar nebula at the origin of the Solar System. The effects of this dissipation on a uniform distribution of asteroids are explored by means of the adiabatic invariant theory for the 2/1, 3/1 and 5/2 resonance cases. The framework is the restricted, circular and planar three body problem.     

Computer simulation of the formation of asteroid belt structure

The system of two gravitational centers with variable separation between components one of which (the primary) loses its mass onto another (the secondary) is investigated under condition of total mass and angular momentum conservation. When the primary/secondary mass ratio becomes about that of Jupiter/Sun the small bodies ejected with the gaseous matter through the inner Lagrange point from the Roche lobe of the primary form a ring similar to the asteroid belt of the solar system. The formation of ring structure is calculated by numerical integration of Newtonian equations of N-body problem in orbital plane of the gravitational centers. The results are compared with the planar subsystem of the asteroid belt. The presence of the main gaps in the distribution of their mean motions at 2/1, 3/1, 5/2 and some other commensurabilities with the primary mean motion is found. More fine details of the belt structure are obtained, e.g. the gap asymmetry and a qualitative agreement with the eccentricity distribution. Within the scope of the same model the external part of the ring is investigated all the pairwise interactions being included. The clustering of bodies near 3/2 commensurability isolated from the main belt by the wide gap centered at 5/3 commensurability is obtained. It is supposed that the ring structure and the interplanetary spacing law for the terrestrial planets are due to the same mechanism.     

The fossilized size distribution of the main asteroid belt

Planet formation models suggest the primordial main belt experienced a short but intense period of collisional evolution shortly after the formation of planetary embryos. This period is believed to have lasted until Jupiter reached its full size, when dynamical processes (e.g., sweeping resonances, excitation via planetary embryos) ejected most planetesimals from the main belt zone. The few planetesimals left behind continued to undergo comminution at a reduced rate until the present day. We investigated how this scenario affects the main belt size distribution over Solar System history using a collisional evolution model (CoEM) that accounts for these events. CoEM does not explicitly include results from dynamical models, but instead treats the unknown size of the primordial main belt and the nature/timing of its dynamical depletion using innovative but approximate methods. Model constraints were provided by the observed size frequency distribution of the asteroid belt, the observed population of asteroid families, the cratered surface of differentiated Asteroid (4) Vesta, and the relatively constant crater production rate of the Earth and Moon over the last 3 Gyr. Using CoEM, we solved for both the shape of the initial main belt size distribution after accretion and the asteroid disruption scaling law . In contrast to previous efforts, we find our derived function is very similar to results produced by numerical hydrocode simulations of asteroid impacts. Our best fit results suggest the asteroid belt experienced as much comminution over its early history as it has since it reached its low-mass state approximately 3.9-4.5 Ga. These results suggest the main belt's wavy-shaped size-frequency distribution is a “fossil” from this violent early epoch. We find that most diameter D?120 km asteroids are primordial, with their physical properties likely determined during the accretion epoch. Conversely, most smaller asteroids are byproducts of fragmentation events. The observed changes in the asteroid spin rate and lightcurve distributions near D∼100-120 km are likely to be a byproduct of this difference. Estimates based on our results imply the primordial main belt population (in the form of D<1000 km bodies) was 150-250 times larger than it is today, in agreement with recent dynamical simulations.     

The nekhoroshev theorem and the asteroid belt dynamical system

The present paper reviews the Nekhoroshev theorem from the point of view of physicists and astronomers. We point out that Nekhoroshev result is strictly connected with the existence of a specific structure of the phase space, the existence of which can be checked with several numerical tools. This is true also for a degenerate system such as the one describing the motion of an asteroid in the so called main belt. The main difference is that in some parts of the belt, the Nekhoroshev result cannot apply a priori. Mean motion resonances of order smaller than the logarithm of the mass of Jupiter and first order secular resonances must be excluded. In the remaining parts, conversely, the Nekhoroshev theorem can be proved, provided someparameters, such as the masses, the eccentricities and the inclinations of the planets are small enough. At the light of this result, a massive campaign of numerical integrations of real and fictitious asteroids should allow to understand which is the real dynamical structure of the asteroid belt.     

On the transport of bodies within and from the asteroid belt

Abstract— This paper explores two processes, sweeping secular resonance (Ward, 1981) and gas drag (Lecar and Franklin, 1997), at work during the dispersal of the solar nebula. we have two aims not previously considered for the two mechanisms: (1) to explain the likely depletion, by a factor of 1000 or so, of the rocky material in the inner belt (2.0 < a < 3.2 AU); (2) to introduce a means for providing—or contributing to—the dispersion in semimajor axis of the various asteroidal taxonomic classes. We suggest that large asteroids with birthplaces separated by an astronomical unit or more can be finally deposited, owing to drag, at the same semimajor axis. For example, we find that bodies with radii up to 100 km can be transferred by gas drag from the outer belt (a > 3.3 AU) well into the inner one, and that an object already in the inner belt as large or even larger than Vesta (r = 250 km)—thought to be the parent body of many meteorites—can be inwardly displaced by as much as an astronomical unit if the nebula dispersal times lie close to 10⁵ years. For such times, a large fraction of the inner belt's primordial mass can be ejected, with most of it passing into the inner solar system.     

Mass loss from the region of Mars and the asteroid belt

Models of the solar nebula suggest that the mass of solid matter which condensed in the region of Mars and the asteroids was much greater than the amount now present. Bombardment by a primordial population of asteroidal bodies originating near Jupiter's orbit could preferentially remove matter from this region, without significant effects in the Earth's zone. A “critical velocity” exists, for which they can be ejected from the solar system by Jupiter. The minimum perihelion attainable at this velocity lies between the orbits of Mars and the Earth. The lifetimes of Mars-crossing bodies are limited by collisions with Jupiter; Earth-crossers are ejected on a much shorter time scale. The total bombardment flux was at least two orders of magnitude greater in the zone of Mars than in that of the Earth. The flux at Venus and Mercury from this source was negligible. The cratering rate for Mars may have differed greatly from those of the other terrestrial planets for a significant fraction of the age of the solar system.     

Orbital and temporal distributions of meteorites originating in the asteroid belt

Abstract— The recent discovery of the importance of Sun-grazing phenomena dramatically changed our understanding of the dynamics of objects emerging from the asteroid belt via resonant phenomena. The typical lifetimes of such objects are now expected to be <10 Ma, thus demanding a reassessment of our general picture of the meteorite delivery process. By analysing direct numerical integrations of ～2000 test particles beginning in the v₆, 3:1, and 5:2 resonances in the main belt, we have reexamined the orbital and temporal distribution of meteoroids that journey to Earth. Comparing the results with fireball data, we find that the orbital distribution of Earth-impacting chondrites is consistent with a steady-state injection of meteoroids into the 3:1 and v₆, resonances. Because this is the most complete and unbiased data set concerning Earth-impacting meteoroids, the agreement leads us to believe that our model is accurate. The simulations predict a P.M. fall ratio for chondrites ～14% lower than the observed value of ～68%, which argues for a moderate bias being present in this statistic. Most interestingly, the typical meteorite transfer times predicted by our models are several factors lower than the typical chondrite exposure ages, which implies that these meteorites acquired most of their exposure in the main belt before entering the resonances. We discuss some processes that would allow such preexposure. The case of achondrites and iron meteorites is also briefly discussed.     

On three types of fragmentation processes observed in the asteroid belt

From comparison of the mass ratio and velocity dispersion between the asteroid family members, we find that the fragmentation processes in the asteroid belt could be generally classified into three types: (a) surface cratering; (b) spallation; and (c) complete breakup as observed in the laboratory hypervelocity impact experiments. Whether there is ongoing accretion for large bodies like Ceres, as an additional type of collision process, is still an open question to be answered by further study.     

Analytical method for the effects of the asteroid belt on planetary orbits

Analytic expressions are derived for the perturbation of planetary orbits due to a thick constant density asteroid belt. The derivations include extensions and adaptations of Plakhov's analytic expressions for the perturbations in five of the orbital elements for closed orbits around Saturn's rings. The equations of Plakhov are modified to include the effect of ring thickness and additional equations are derived for the perturbations in the sixth orbital element, the mean anomaly. The gravitational potential and orbital perturbations are derived for the asteroid belt with and without thickness, and for a hoop approximation to the belt. The procedures are also applicable to Saturn's rings and the newly discovered rings of Uranus.The effects of the asteroid belt thickness on the gravitational potential coefficients and the orbital motions are demonstrated. Comparisons between the Mars orbital perturbations obtained using the analytic expressions and those obtained using numerical integration are discussed. The effects of the asteroid belt on the Earth based ranging to Mars are also demonstrated.     

Collisional evolution of rotating,non-identical particles

Hämeen-Anttila's (1984) analytical treatment of self-gravitating collisional particle disks is extended to include the particle spin. The equations derived for the coupled evolution of random velocities and spins indicate that friction and surface irregularity typically reduce the local velocity dispersion. Friction, and especially irregularity, also transfer significant amounts of random kinetic energy, E _kin, to rotational energy, E _rot. The equilibrium ratio E _rot/E_kin = 2/(14 – 5) if the particles are spherical, and 2(1 + )/7 if they are irregular but frictionless, and being the coefficients of restitution and friction. These results are not only exact for identical, non-gravitating mass points, but are rather accurate even if finite size, self-gravitational forces, or size distribution are included. Applications to the dynamics of Saturn's rings suggest that the inclusion of rotation is able to reduce the geometrical thickness of the layer of centimeter-sized particles to about one half, at most. Large particles are less affected.On leave from Dept. of Astronomy, University of Oulu, 90570 Oulu, Finland.