Asteroid motion near the 3 : 1 resonance

A mapping model is constructed to describe asteroid motion near the 3 : 1 mean motion resonance with Jupiter, in the plane. The topology of the phase space of this mapping coincides with that of the real system, which is considered to be the elliptic restricted three body problem with the Sun and Jupiter as primaries. This model is valid for all values of the eccentricity. This is achieved by the introduction of a correcting term to the averaged Hamiltonian which is valid for small values of the ecentricity.We start with a two dimensional mapping which represents the circular restricted three body problem. This provides the basic framework for the complete model, but cannot explain the generation of a gap in the distribution of the asteroids at this resonance. The next approximation is a four dimensional mapping, corresponding to the elliptic restricted problem. It is found that chaotic regions exist near the 3 : 1 resonance, due to the interaction between the two degrees of freedom, for initial conditions close to a critical curve of the circular model. As a consequence of the chaotic motion, the eccentricity of the asteroid jumps to high values and close encounters with Mars and even Earth may occur, thus generating a gap. It is found that the generation of chaos depends also on the phase (i.e. the angles andv) and as a consequence, there exist islands of ordered motion inside the sea of chaotic motion near the 3 : 1 resonance. Thus, the model of the elliptic restricted three body problem cannot explain completely the generation of a gap, although the density in the distribution of the asteroids will be much less than far from the resonance. Finally, we take into account the effect of the gravitational attraction of Saturn on Jupiter's orbit, and in particular the variation of the eccentricity and the argument of perihelion. This generates a mixing of the phases and as a consequence the whole phase space near the 3 : 1 resonance becomes chaotic. This chaotic zone is in good agreement with the observations.     

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.     

Chaotic behaviour of trajectories for the asteroidal resonances

A systematic study of the main asteroidal resonances of the third and fourth order is performed using mapping techniques. For each resonance one-parameter family of surfaces of section is presented together with a simple energy graph which helps to understand and predict the changes in the surfaces of section within the family. As the truncated Hamiltonian for the planar, elliptic, restricted three-body problem is used for the mapping, the method is expected to fail for high eccentricities. We compared, therefore, the surfaces of section with trajectories calculated by symplectic integrators of the fourth and six order employing the full Hamiltonian. We found a good agreement for small eccentricities but differences for the higher eccentricities (e 0.3).     

Express delivery of fossil meteorites from the inner asteroid belt to Sweden

Our understanding of planet formation depends in fundamental ways on what we learn by analyzing the composition, mineralogy, and petrology of meteorites. Yet, it is difficult to deduce the compositional and thermal gradients that existed in the solar nebula from the meteoritic record because, in most cases, we do not know where meteorites with different chemical and isotopic signatures originated. Here we developed a model that tracks the orbits of meteoroid-sized objects as they evolve from the ν₆ secular resonance to Earth-crossing orbits. We apply this model to determining the number of meteorites accreted on the Earth immediately after a collisional disruption of a D∼200-km-diameter inner-main-belt asteroid in the Flora family region. We show that this event could produce fossil chondrite meteorites found in an ≈470 Myr old marine limestone quarry in southern Sweden, the L-chondrite meteorites with shock ages ≈470 Myr falling on the Earth today, as well as asteroid-sized fragments in the Flora family. To explain the measured short cosmic-ray exposure ages of fossil meteorites our model requires that the meteoroid-sized fragments were launched at speeds >500 m s⁻¹ and/or the collisional lifetimes of these objects were much shorter immediately after the breakup event than they are today.     

Dynamical erosion of the asteroid belt and implications for large impacts in the inner Solar System

The cumulative effects of weak resonant and secular perturbations by the major planets produce chaotic behavior of asteroids on long timescales. Dynamical chaos is the dominant loss mechanism for asteroids with diameters in the current asteroid belt. In a numerical analysis of the long-term evolution of test particles in the main asteroid belt region, we find that the dynamical loss history of test particles from this region is well described with a logarithmic decay law. In our simulations the loss rate function that is established at persists with little deviation to at least . Our study indicates that the asteroid belt region has experienced a significant amount of depletion due to this dynamical erosion—having lost as much as ∼50% of the large asteroids—since 1 Myr after the establishment of the current dynamical structure of the asteroid belt. Because the dynamical depletion of asteroids from the main belt is approximately logarithmic, an equal amount of depletion occurred in the time interval 10-200 Myr as in 0.2-4 Gyr, roughly ∼30% of the current number of large asteroids in the main belt over each interval. We find that asteroids escaping from the main belt due to dynamical chaos have an Earth-impact probability of ∼0.3%. Our model suggests that the rate of impacts from large asteroids has declined by a factor of 3 over the last 3 Gyr, and that the present-day impact flux of objects on the terrestrial planets is roughly an order of magnitude less than estimates currently in use in crater chronologies and impact hazard risk assessments.     

The origin of chaotic behaviour in the Miranda-Umbriel 3 : 1 resonances

We have investigated the pericentric resonances through which Miranda and Umbriel are believed to have passed when, due to tidal evolution, their orbital mean motions reached a 3 : 1 commensurability. Our investigation is based upon a perturbative treatment. The predictions of this theory are in good agreement with the results of numerical integrations concerning both the extend of the chaotic layers generated by the separatrices of the primary resonances and the location of the secondary resonances. The effect of tidal evolution is discussed on the bases of the adiatatic invariant theory and its extension to separatrix crossing. We recover qualitatively the mean features of the numerical experiments of Tittermore and Wisdom (1988–1989), Dermott et al (1988) and Malhotra and Dermott (1989).     

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.     

Motion near the 3/1 resonance of the planar elliptic restricted three body problem

The global semi-numerical perturbation method proposed by Henrard and Lemaître (1986) for the 2/1 resonance of the planar elliptic restricted three body problem is applied to the 3/1 resonance and is compared with Wisdom's perturbative treatment (1985) of the same problem. It appears that the two methods are comparable in their ability to reproduce the results of numerical integration especially in what concerns the shape and area of chaotic domains. As the global semi-numerical perturbation method is easily adapted to more general types of perturbations, it is hoped that it can serve as the basis for the analysis of more refined models of asteroidal motion. We point out in our analysis that Wisdom's uncertainty zone mechanism for generating chaotic domains (also analysed by Escande 1985 under the name of slow Hamiltonian chaotic layer) is not the only one at work in this problem. The secondary resonance _p = 0 plays also its role which is qualitatively (if not quantitatively) important as it is closely associated with the random jumps between a high eccentricity mode and a low eccentricity mode.     

On the dynamics in the asteroids belt. Part II: Detailed study of the main resonances

The general theory exposed in the first part of this paper is applied to the following resonances with Jupiter's motion : 3/2, 2/1, 5/2, 3/1, 7/2, 4/1; these are the most relevant resonances for the asteroids. The whole analysis is performed in the framework of the spatial problem of three bodies, both in the circular and in the elliptic case. The results are also compared with the observed distribution of the asteroids.     

The Effect of the Great Inequality on the Hecuba Gap

The effects of the "great inequality" (the quasi-resonance between Jupiter and Saturn) on the motion in the 2/1 mean motion resonance with Jupiter (the Hecuba gap) is investigated. We confirm the proposition made by Ferraz-Mello and collaborators that the great inequality generates secondary resonances which are likely to produce the slow diffusion observed in numerical investigations. We identify, in the restricted three body problem, the frequencies responsible for these secondary resonances. This revised version was published online in July 2006 with corrections to the Cover Date.     

Linking the collisional history of the main asteroid belt to its dynamical excitation and depletion

The main belt is believed to have originally contained an Earth mass or more of material, enough to allow the asteroids to accrete on relatively short timescales. The present-day main belt, however, only contains ∼5×¹⁰⁻⁴ Earth masses. Numerical simulations suggest that this mass loss can be explained by the dynamical depletion of main belt material via gravitational perturbations from planetary embryos and a newly-formed Jupiter. To explore this scenario, we combined dynamical results from Petit et al. [Petit, J. Morbidelli, A., Chambers, J., 2001. The primordial excitation and clearing of the asteroid belt. Icarus 153, 338-347] with a collisional evolution code capable of tracking how the main belt undergoes comminution and dynamical depletion over 4.6 Gyr [Bottke, W.F., Durda, D., Nesvorny, D., Jedicke, R., Morbidelli, A., Vokrouhlický, D., Levison, H., 2005. The fossilized size distribution of the main asteroid belt. Icarus 175, 111-140]. Our results were constrained by the main belt's size-frequency distribution, the number of asteroid families produced by disruption events from diameter D>100 km parent bodies over the last 3-4 Gyr, the presence of a single large impact crater on Vesta's intact basaltic crust, and the relatively constant lunar and terrestrial impactor flux over the last 3 Gyr. We used our model to set limits on the initial size of the main belt as well as Jupiter's formation time. We find the most likely formation time for Jupiter was 3.3±2.6 Myr after the onset of fragmentation in the main belt. These results are consistent with the estimated mean disk lifetime of 3 Myr predicted by Haisch et al. [Haisch, K.E., Lada, E.A., Lada, C.J., 2001. Disk frequencies and lifetimes in young clusters. Astrophys. J. 553, L153-L156]. The post-accretion main belt population, in the form of diameter D?1000 km planetesimals, was likely to have been 160±40 times the current main belt's mass. This corresponds to 0.06-0.1 Earth masses, only a small fraction of the total mass thought to have existed in the main belt zone during planet formation. The remaining mass was most likely taken up by planetary embryos formed in the same region. Our results suggest that numerous D>200 km planetesimals disrupted early in Solar System history, but only a small fraction of their fragments survived the dynamical depletion event described above. We believe this may explain the limited presence of iron-rich M-type, olivine-rich A-type, and non-Vesta V-type asteroids in the main belt today. The collisional lifetimes determined for main belt asteroids agree with the cosmic ray exposure ages of stony meteorites and are consistent with the limited collisional evolution detected among large Koronis family members. Using the same model, we investigated the near-Earth object (NEO) population. We show the shape of the NEO size distribution is a reflection of the main belt population, with main belt asteroids driven to resonances by Yarkovsky thermal forces. We used our model of the NEO population over the last 3 Gyr, which is consistent with the current population determined by telescopic and satellite data, to explore whether the majority of small craters (D<0.1-1 km) formed on Mercury, the Moon, and Mars were produced by primary impacts or by secondary impacts generated by ejecta from large craters. Our results suggest that most small craters formed on these worlds were a by-product of secondary rather than primary impacts.     

Review of the dynamics in the Kirkwood gaps

The study of mean motion resonance dynamics was motivated by the search for an explanation for the puzzling problem of the Kirkwood gaps. The most important contributions in this field within the last 32 years are reviewed here. At the beginning of that period, which coincides with the first long-term numerical investigations of resonant motion, different hypotheses (collisional, gravitational, statistical and cosmological) to explain the origin of the gaps were still competing with each other. At present, a general theory, based on gravitational mechanisms only, is capable of explaining in a uniform way all the Kirkwood gaps except the 2/1 one. Indeed, in the 4/1, 3/1, 5/2 and 7/3 mean motion commensurabilities, the overlap of secular resonances leads to almost overall chaos where asteroids undergo large and wild variations in their orbital elements. Such asteroids, if not thrown directly into the Sun, are sooner or later subject to strong close encounters with the largest inner planets, the typical time scale of the whole process being of the order of a few million years. Unfortunately, this mechanism is not capable of explaining the 2/1 gap where the strong chaos produced by the overlapping secular resonances does not attain orbits with moderate eccentricity, of low inclination and with low to moderate amplitude of libration. In the light of the most recent studies, it appears that the 2/1 gap is the global consequence of slow diffusive processes. At present, the origin of these processes remains under study.     

The Maria asteroid family: Genetic relationships and a plausible source of mesosiderites near the 3:1 Kirkwood Gap

We present a mineralogical assessment of 12 Maria family asteroids, using near-infrared spectral data obtained over the years 2000-2009 combined with visible spectral data (when available) to cover the spectral interval of 0.4-2.5 μm. Our analysis indicates the Maria asteroid family, which is located adjacent to the chaotic region of the 3:1 Kirkwood Gap, appears to be a true genetic family composed of assemblages analogous to mesosiderite-type meteorites. Dynamical models by Farinella et al. (Farinella, P., Gunczi, R., Froeschlé, Ch., Froeschlé, C., [1993]. Icarus 101, 174-187) predict this region should supply meteoroids into Earth-crossing orbits. Thus, the Maria family is a plausible source of some or all of the mesosiderites in our meteorite collections. These individual asteroids were most likely once part of a larger parent object that was broken apart and dispersed. One of the Maria dynamical family members investigated, ((695) Bella), was found to be unrelated to the genetic Maria family members. The parameters of (695) Bella indicate an H-chondrite assemblage, and that Bella may be a sister or daughter of Asteroid (6) Hebe.     

Secular resonances in the asteroid belt: Theoretical perturbation approach and the problem of their location

In this paper a theoretical perturbation approach to the problem of the dynamics in secular resonance is exposed. This approach avoids any expansion of the main term of the Hamiltonian (linear term in the masses) with respect to the eccentricity or the inclination of the asteroid, in order to achieve results valid for any value of these variables. Moreover suitable action-angle variables are introduced to take properly into account the dynamics related to the motion of the argument of perihelion of the asteroid, which is relevant at high inclination. A class of secular resonances wider than that usually considered is found. An explicit computation of the location of the main secular resonances, estimating also the contribution of the quadratic term in the masses by means of classical series expansion, is reported in the last sections. The accuracy of computations obtained by series expansion is discussed in the paper.     

Radar observations and the shape of near-Earth asteroid 2008 EV5

We observed the near-Earth ASTEROID 2008 EV5 with the Arecibo and Goldstone planetary radars and the Very Long Baseline Array during December 2008. EV5 rotates retrograde and its overall shape is a 400 ± 50 m oblate spheroid. The most prominent surface feature is a ridge parallel to the asteroid’s equator that is broken by a concavity about 150 m in diameter. Otherwise the asteroid’s surface is notably smooth on decameter scales. EV5’s radar and optical albedos are consistent with either rocky or stony-iron composition. The equatorial ridge is similar to structure seen on the rubble-pile near-Earth asteroid (66391) 1999 KW4 and is consistent with YORP spin-up reconfiguring the asteroid in the past. We interpret the concavity as an impact crater. Shaking during the impact and later regolith redistribution may have erased smaller features, explaining the general lack of decameter-scale surface structure.     

A dynamical solution of the triple asteroid system (45) Eugenia

We present the first dynamical solution of the triple asteroid system (45) Eugenia and its two moons Petit–Prince (diameter ∼ 7 km) and S/2004 (45) 1 (diameter ∼ 5 km). The two moons orbit at 1165 and 610 km from the primary, describing an almost-circular orbit (e ∼ 6 × 10⁻³ and e ∼ 7 × 10⁻² respectively). The system is quite different from the other known triple systems in the main belt since the inclinations of the moon orbits are sizeable (9° and 18° with respect to the equator of the primary respectively). No resonances, neither secular nor due to Lidov–Kozai mechanism, were detected in our dynamical solution, suggesting that these inclinations are not due to excitation modes between the primary and the moons. A 10-year evolution study shows that the orbits are slightly affected by perturbations from the Sun, and to a lesser extent by mutual interactions between the moons. The estimated J₂ of the primary is three times lower than the theoretical one, calculated assuming the shape of the primary and an homogeneous interior, possibly suggesting the importance of other gravitational harmonics.     

The size distributions of asteroid families in the SDSS Moving Object Catalog 4

Asteroid families, traditionally defined as clusters of objects in orbital parameter space, often have distinctive optical colors. We show that the separation of family members from background interlopers can be improved with the aid of SDSS colors as a qualifier for family membership. Based on an ∼88,000 object subset of the Sloan Digital Sky Survey Moving Object Catalog 4 with available proper orbital elements, we define 37 statistically robust asteroid families with at least 100 members (12 families have over 1000 members) using a simple Gaussian distribution model in both orbital and color space. The interloper rejection rate based on colors is typically ∼10% for a given orbital family definition, with four families that can be reliably isolated only with the aid of colors. About 50% of all objects in this data set belong to families, and this fraction varies from about 35% for objects brighter than an H magnitude of 13 and rises to 60% for objects fainter than this. The fraction of C-type objects in families decreases with increasing H magnitude for H>13, while the fraction of S-type objects above this limit remains effectively constant. This suggests that S-type objects require a shorter timescale for equilibrating the background and family size distributions via collisional processing. The size distribution varies significantly among families, and is typically different from size distributions for background populations. The size distributions for 15 families display a well-defined change of slope and can be modeled as a “broken” double power-law. Such “broken” size distributions are twice as likely for S-type familes than for C-type families (73% vs. 36%), and are dominated by dynamically old families. The remaining families with size distributions that can be modeled as a single power law are dominated by young families (<1 Gyr). When size distribution requires a double power-law model, the two slopes are correlated and are steeper for S-type families. No such slope-color correlation is discernible for families whose size distribution follows a single power law. For several very populous families, we find that the size distribution varies with the distance from the core in orbital-color space, such that small objects are more prevalent in the family outskirts. This “size sorting” is consistent with predictions based on the Yarkovsky effect.     

Impact evolution of asteroid shapes: 1. random mass redistribution

We explore whether the cumulative effect of small-scale meteoroid bombardment can drive asteroids into nonaxisymmetric shapes comparable to those of known objects (elongated prolate forms, twin-lobed binaries, etc). We simulate impact cratering as an excavation followed by the launch, orbit, and reimpact of ejecta. Orbits are determined by the gravity and rotation of the evolving asteroid, whose shape and spin change as cratering occurs repeatedly. For simplicity we consider an end-member evolution where impactors are all much smaller than the asteroid and where all ejecta remain bound. Given those assumptions, we find that cumulative small impacts on rotating asteroids lead to oblate shapes, irrespective of the chosen value for angle of repose or for initial angular momentum. The more rapidly a body is spinning, the more flattened the outcome, but oblateness prevails. Most actual asteroids, by contrast, appear spherical to prolate. We also evaluate the timescale for reshaping by small impacts and compare it to the timescale for catastrophic disruption. For all but the steepest size distributions of impactors, reshaping from small impacts takes more than an order of magnitude longer than catastrophic disruption. We conclude that small-scale cratering is probably not dominant in shaping asteroids, unless our assumptions are naive. We believe we have ruled out the end-member scenario; future modeling shall include angular momentum evolution from impacts, mass loss in the strength regime, and craters with diameters up to the disruption threshold. The ultimate goal is to find out how asteroids get their shapes and spins and whether tidal encounters in fact play a dominant role.     

V-type asteroids in the middle main belt

V-type asteroids are bodies whose surfaces are constituted of basalt. In the Main Asteroid Belt, most of these asteroids are assumed to come from the basaltic crust of Asteroid (4) Vesta. This idea is mainly supported by (i) the fact that almost all the known V-type asteroids are in the same region of the belt as (4) Vesta, i.e., the inner belt (semi-major axis 2.1<a<2.5 AU), (ii) the existence of a dynamical asteroid family associated to (4) Vesta, and (iii) the observational evidence of at least one large craterization event on Vesta's surface. One V-type asteroid that is difficult to fit in this scenario is (1459) Magnya, located in the outer asteroid belt, i.e., too far away from (4) Vesta as to have a real possibility of coming from it. The recent discovery of the first V-type asteroid in the middle belt (2.5<a<2.8 AU), (21238) 1995WV7 [Binzel, R.P., Masi, G., Foglia, S., 2006. Bull. Am. Astron. Soc. 38, 627; Hammergren, M., Gyuk, G., Puckett, A., 2006. ArXiv e-print, astro-ph/0609420], located at ∼2.54 AU, raises the question of whether it came from (4) Vesta or not. In this paper, we present spectroscopic observations indicating the existence of another V-type asteroid at ∼2.53 AU, (40521) 1999RL95, and we investigate the possibility that these two asteroids evolved from the Vesta family to their present orbits by a semi-major axis drift due to the Yarkovsky effect. The main problem with this scenario is that the asteroids need to cross the 3/1 mean motion resonance with Jupiter, which is highly unstable. Combining N-body numerical simulations of the orbital evolution, that include the Yarkovsky effect, with Monte Carlo models, we compute the probability that an asteroid of a given diameter D evolves from the Vesta family and crosses over the 3/1 resonance, reaching a stable orbit in the middle belt. Our results indicate that an asteroid like (21238) 1995WV7 has a low probability (∼1%) of having evolved through this mechanism due to its large size (D∼5 km), because the Yarkovsky effect is not sufficiently efficient for such large asteroids. However, the mechanism might explain the orbits of smaller bodies like (40521) 1999RL95 (D∼3 km) with ∼70-100% probability, provided that we assume that the Vesta family formed ?3.5 Gy ago. We estimate the debiased population of V-type asteroids that might exist in the same region as (21238) and (40521) (2.5<a?2.62 AU) and conclude that about 10 to 30% of the V-type bodies with D>1 km may come from the Vesta family by crossing over the 3/1 resonance. The remaining 70-90% must have a different origin.