The Trojan asteroid belt: Proper elements,stability, chaos and families

I have computed proper elements for 174 asteroids in the 1 : 1 resonance with Jupiter, that is for all the reliable orbits available (numbered and multi-opposition). The procedure requires numerical integration, under the perturbations by the four major planets, for 1,000,000 years; the output is digitally filtered and compressed into a synthetic theory (as defined within theLONGSTOP project). The proper modes of oscillation of the variables related to eccentricity, perihelion, inclination and node define proper elements. A third proper element is defined as the amplitude of the oscillation of the semimajor axis associated with the libration period; because of the strong nonlinearity of the problem, this component cannot be determined by a simple Fourier transform to the frequency domain. I therefore give another definition, which results in very good stability with time. For 87% of the computed orbits, the stability of the proper elements-at least over 1M yr-is within the following bounds: 0.001AU in semimajor axis, 0.0025 in eccentricity and sine of inclination. Half of the cases with degraded stability of the proper elements are found to be chaotic, with e-folding times between 16,000 and 660,000yr; in some other cases, chaotic behaviour does not result in a significantly decreased stability of the proper elements (stable chaos). The accuracy and stability of these proper elements is good enough to allow a search for asteroid families; however, the dynamical structure of the Trojan belt is very different from the one of the main belt, and collisional events among Trojans can result in a distribution of fragments difficult to identify. The occurrence of couples of Trojans with very close proper elements is proven not to be statistically significant in almost all cases. As the only exception, the couple 1583 Antilochus — 3801 Thrasimedes is significant; however, it is not easy to account for it by a conventional collisional theory. The Menelaus group is confirmed as a strong candidate collisional family; Teucer and Sarpedon could be considered as significant clusters. A number of other clumps are detected (by the same automated clustering method used for the main belt by Zappalà et al., 1990, 1992), but the total number of Trojans with reliable orbits is not large enough to detect many significant candidate families.     

Chaotic Diffusion And Effective Stability of Jupiter Trojans

It has recently been shown that Jupiter Trojans may exhibit chaotic behavior, a fact that has put in question their presumed long term stability. Previous numerical results suggest a slow dispersion of the Trojan swarms, but the extent of the ‘effective’ stability region in orbital elements space is still an open problem. In this paper, we tackle this problem by means of extensive numerical integrations. First, a set of 3,200 fictitious objects and 667 numbered Trojans is integrated for 4 Myrs and their Lyapunov time, T_L, is estimated. The ones following chaotic orbits are then integrated for 1 Gyr, or until they escape from the Trojan region. The results of these experiments are presented in the form of maps of T_Land the escape time, T_E, in the space of proper elements. An effective stability region for 1 Gyr is defined on these maps, in which chaotic orbits also exist. The distribution of the numbered Trojans follows closely the T_E=1 Gyr level curve, with 86% of the bodies lying inside and 14% outside the stability region. This result is confirmed by a 4.5 Gyr integration of the 246 chaotic numbered Trojans, which showed that 17% of the numbered Trojans are unstable over the age of the solar system. We show that the size distributions of the stable and unstable populations are nearly identical. Thus, the existence of unstable bodies should not be the result of a size-dependent transport mechanism but, rather, the result of chaotic diffusion. Finally, in the large chaotic region that surrounds the stability zone, a statistical correlation between T_LandT_E is found.     

Neptune Trojans: a Revisit and a New Membertwo

Up to now, 17 Neptune Trojan asteroids have been detected with their orbits being well determined by continuous observations. This paper analyzes systematically their orbital dynamics. Our results show that except for two temporary members with relatively short lifespans on Trojan orbits, the vast majority of Neptune Trojans located within their orbital uncertainties may survive in the solar system age. The escaping probability of Neptune Trojans, through slow diffusion in the orbital element space in 4.5 billion years, is estimated to be ～50%. The asteroid 2012 UW177 classified as a Centaur asteroid by the IAU Minor Planet Center currently is in fact a Neptune Trojan. Numerical simulations indicate that it is librating on the tadpole-shaped orbit around the Neptune's L₄ point. It was captured into the current orbit approximately 0.23 million years ago, and will stay there for at least another 1.3 million years in the future. Its high inclination of i ≈ 54^° not only makes it the most inclined Neptune Trojan, but also makes it exhibit the complicated and interesting co-orbital transitions between the leading and trailing Trojans via the quasi-satellite orbit phase.     

Resonant and Secular Families of the Kuiper Belt

We review ongoing efforts to identify occupants of mean-motion resonances(MMRs) and collisional families in the Edgeworth–Kuiper belt. Directintegrations of trajectories of Kuiper belt objects (KBOs) reveal the 1:1(Trojan), 5:4, 4:3, 3:2 (Plutino), 5:3, 7:4, 9:5, 2:1 (Twotino), and 5:2 MMRsto be inhabited. Apart from the Trojan, resonant KBOs typically have largeorbital eccentricities and inclinations. The observed pattern of resonanceoccupation is consistent with resonant capture and adiabatic excitation bya migratory Neptune; however, the dynamically cold initial conditions priorto resonance sweeping that are typically assumed by migration simulationsare probably inadequate. Given the dynamically hot residents of the 5:2 MMRand the substantial inclinations observed in all exterior MMRs, a fraction ofthe primordial belt was likely dynamically pre-heated prior to resonancesweeping. A pre-heated population may have arisen as Neptune gravitationallyscattered objects into trans-Neptunian space. The spatial distribution of Twotinosoffers a unique diagnostic of Neptune's migration history. The Neptunian Trojanpopulation may rival the Jovian Trojan population, and the former's existence isargued to rule out violent orbital histories for Neptune. Finally, lowest-order seculartheory is applied to several hundred non-resonant KBOs with well-measured orbitsto update proposals of collisional families. No convincing family is detected.     

The resonant structure of Jupiter's Trojan asteroids – II. What happens for different configurations of the planetary system

In a previous paper, we have found that the resonance structure of the present Jupiter Trojan swarms could be split up into four different families of resonances. Here, in a first step, we generalize these families in order to describe the resonances occurring in Trojan swarms embedded in a generic planetary system. The location of these families changes under a modification of the fundamental frequencies of the planets and we show how the resonant structure would evolve during a planetary migration. We present a general method, based on the knowledge of the fundamental frequencies of the planets and on those that can be reached by the Trojans, which makes it possible to predict and localize the main events arising in the swarms during migration. In particular, we show how the size and stability of the Trojan swarms are affected by the modification of the frequencies of the planets. Finally, we use this method to study the global dynamics of the Jovian Trojan swarms when Saturn migrates outwards. Besides the two resonances found by Morbidelli et al. which could have led to the capture of the current population just after the crossing of the 2:1 orbital resonance, we also point out several sequences of chaotic events that can influence the Trojan population.     

Stability of Trojan planets in multi-planetary systems

Today there are more than 340 extra-solar planets in about 270 extra-solar systems confirmed. Besides the observed planets there exists also the possibility of a Trojan planet moving in the same orbit as the Jupiter-like planet. In our investigation we take also into account the habitability of a Trojan planet and whether such a terrestrial planet stays in the habitable zone. Its stability was investigated for multi-planetary systems, where one of the detected giant planets moves partly or completely in the habitable zone. By using numerical computations, we studied the orbital behaviour up to 10⁷ years and determined the size of the stable regions around the Lagrangian equilibrium points for different dynamical models for fictitious Trojans. We also examined the interaction of the Trojan planets with a second or third giant planet, by varying its semimajor axis and eccentricity. We have found two systems (HD 155358 and HD 69830) that can host habitable Trojan planets. Another aim of this work was to determine the size of the stable region around the Lagrangian equilibrium points in the restricted three body problem for small mass ratios μ of the primaries μ ≤ 0.001 (e.g. Neptune mass of the secondary and smaller masses). We established a simple relation for the size depending on μ and the eccentricity.     

Physical properties and orbital stability of the Trojan asteroids

All the Trojan asteroids orbit about the Sun at roughly the same heliocentric distance as Jupiter. Differences in the observed visible reflection spectra range from neutral to red, with no ultra-red objects found so far. Given that the Trojan asteroids are collisionally evolved, a certain degree of variability is expected. Additionally, cosmic radiation and sublimation are important factors in modifying icy surfaces even at those large heliocentric distances. We search for correlations between physical and dynamical properties, we explore relationships between the following four quantities; the normalised visible reflectivity indexes (S^′), the absolute magnitudes, the observed albedos and the orbital stability of the Trojans. We present here visible spectroscopic spectra of 25 Trojans. This new data increase by a factor of about 5 the size of the sample of visible spectra of Jupiter Trojans on unstable orbits. The observations were carried out at the ESO-NTT telescope (3.5 m) at La Silla, Chile, the ING-WHT (4.2 m) and NOT (2.5 m) at Roque de los Muchachos observatory, La Palma, Spain. We have found a correlation between the size distribution and the orbital stability. The absolute-magnitude distribution of the Trojans in stable orbits is found to be bimodal, while the one of the unstable orbits is unimodal, with a slope similar to that of the small stable Trojans. This supports the hypothesis that the unstable objects are mainly byproducts of physical collisions. The values of S^′ of both the stable and the unstable Trojans are uniformly distributed over a wide range, from 0%/1000 Å to about 15%/1000 Å. The values for the stable Trojans tend to be slightly redder than the unstable ones, but no significant statistical difference is found.     

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.     

Discovery and rediscovery of Trojan asteroids

During 1987–1994, observational campaigns with different telescopes at several observatories have been initiated by the author in order to discover new Trojans. The importance of Trojan asteroids comes from celestial mechanics, where they represent the physical solution of the famous Lagrange triangular problem. Their importance lies also in the fact, that they may have some relation with comets. Furthermore, the Trojan belt may be as large as the belt of asteroids. Moreover, recently “families” have been discovered between the already well known Trojans. Enough reasons to continue to search for these interesting objects.     

Vesta fragments from v6 and 3:1 resonances: Implications for V-type near-Earth asteroids and howardite,eucrite and diogenite meteorites

Abstract— A large body of evidence, including the presence of a dynamical family associated with 4 Vesta, suggests that this asteroid might be the ultimate source of both the V-type near-Earth asteroids (NEAs) and howardite, eucrite and diogenite (HED) meteorites. Dynamical routes from Vesta to the inner regions of the solar system are provided by both the 3:1 mean-motion resonance with Jupiter and the V₆, secular resonance. For this reason, numerical integrations of the orbits of fictitious Vesta fragments injected in both of these resonances have been performed. At the same time, the orbital evolution of the known V-type NEAs has been investigated. The results indicate that the dynamical half lifetimes of Vesta fragments injected in both the 3:1 and the V₆, resonances are rather short ('2 Ma). The present location of the seven known V-type NEAs is better explained by orbital evolutions starting from the v₆ secular resonance. The most important result of the present investigation, however, is that we now face what we call the “Vesta paradox.” Roughly speaking, the paradox consists of the fact that the present V-type NEAs appear to be too dynamically young to have originated in the event that produced the family, but they are too big to be plausible second-generation fragments from the family members. The cosmic-ray exposure (CRE) age distribution of HED meteorites also raises a puzzle, since we would expect an overabundance of meteorites with short CRE ages. We propose different scenarios to explain these paradoxes.     

Visible spectroscopic and photometric survey of L5 Trojans: investigation of dynamical families

In this paper we present results obtained in the framework of a visible spectroscopic and photometric survey of Trojan asteroids. We concentrated on bodies orbiting at the L5 Lagrangian point of Jupiter that are also members of dynamical families. Spectroscopy is a crucial tool that allows us to characterize the mineralogical composition of families and their parent bodies, gives evidence of ongoing space weathering, and confirms family membership. We have observed 18 objects belonging to the Aneas, Astyanax, Sarpedon, and Phereclos families as defined by Beaugé and Roig (2001, Icarus 53, 391). In addition, we have determined the spectroscopic properties of 8 background Jupiter Trojans. The observed spectra are reddish with a dominance of D-type asteroids. As expected, the spectra of the non-family members are more heterogeneous compared to the spectra of family members, with the exception of the members of the Aneas family. We also confirm the lack of absorption features in the visible region, as already reported by other authors.     

The Uppsala‐DLR Trojan Survey of L4, the preceding Lagrangian cloud of Jupiter

We have used the ESO Schmidt telescope during the apparitions 1996, 1997 and 1998 to study the preceding Lagrangian cloud of Jupiter, L₄. In our observed field of view, about 700 square degrees, during 1996 we found 399 moving objects that we classified as Trojans. From this we conclude that down to absolute magnitude H ≤ 13.0 there are ≈ 1100 Trojans in the L₄ cloud. Proper elements of 696 numbered and multiopposition Trojans were calculated and one of these in the L₅ cloud were found to be a non Trojan. A discussion of the Trojan proper elements is also presented.     

The surface composition of Jupiter Trojans: Visible and near-infrared survey of dynamical families

Asteroid dynamical families are supposed to be formed from the collisional disruption of parent bodies. As a consequence, the investigation of the surface properties of small and large family members may give some hints on the nature of the dynamical group, the internal composition of the parent body, and the role played by space weathering processes in modifying the spectral behavior of the members' surfaces. In this work we present visible-near-infrared observations of 24 Jupiter Trojans belonging to seven dynamical families of both the L4 and L5 swarms. The most important characteristics we found is the uniformity of the Trojans population. All the investigated Trojans have featureless spectra and a spectral behavior typical of the primitive P and D taxonomic classes. In particular, no signatures of water ice have been found on the spectra of these primordial bodies. From our investigation, the L4 and L5 clouds appear to be compositionally indistinguishable. Tentative models of the surface composition, based on the Hapke theory, are presented and discussed.     

Orbital evolution of the Gefion and Adeona asteroid families: close encounters with massive asteroids and the Yarkovsky effect

Asteroid families are groupings of minor planets identified by clustering in their proper orbital elements; these objects have spectral signatures consistent with an origin in the break-up of a common parent body. From the current values of proper semimajor axes a of family members one might hope to estimate the ejection velocities with which the fragments left the putative break-up event (assuming that the pieces were ejected isotropically). However, the ejection velocities so inferred are consistently higher than N-body and hydro-code simulations, as well as laboratory experiments, suggest. To explain this discrepancy between today’s orbital distribution of asteroid family members and their supposed launch velocities, we study whether asteroid family members might have been ejected from the collision at low speeds and then slowly drifted to their current positions, via one or more dynamical processes. Studies show that the proper a of asteroid family members can be altered by two mechanisms: (i) close encounters with massive asteroids, and (ii) the Yarkovsky non-gravitational effect. Because the Yarkovsky effect for kilometer-sized bodies decreases with asteroid diameter D, it is unlikely to have appreciably moved large asteroids (say those with D > 15 km) over the typical family age (1-2 Gyr).For this reason, we numerically studied the mobility of family members produced by close encounters with main-belt, non-family asteroids that were thought massive enough to significantly change their orbits over long timescales. Our goal was to learn the degree to which perturbations might modify the proper a values of all family members, including those too large to be influenced by the Yarkovsky effect. Our initial simulations demonstrated immediately that very few asteroids were massive enough to significantly alter relative orbits among family members. Thus, to maximize gravitational perturbations in our 500-Myr integrations, we investigated the effect of close encounters on two families, Gefion and Adeona, that have high encounter probabilities with 1 Ceres, by far the largest asteroid in the main belt. Our results show that members of these families spreads in a of less than 5% since their formation. Thus gravitational interactions cannot account for the large inferred escape velocities.The effect of close encounters with massive asteroids is, however, not entirely negligible. For about 10% of the simulated bodies, close encounters increased the “inferred” ejection velocities from sub-100 m/s to values greater than 100 m/s, beyond what hydro-code and N-body simulations suggest are the maximum possible initial ejection velocity for members of Adeona and Gefion with D > 15 km. Thus this mechanism of mobility may be responsible for the unusually high inferred ejection speeds of a few of the largest members of these two families.To understand the orbital evolution of the entire family, including smaller members, we also performed simulations to account for the drift of smaller asteroids caused by the Yarkovsky effect. Our two sets of simulations suggest that the two families we investigated are relatively young compared to larger families like Koronis and Themis, which have estimated ages of about 2 Byr. The Adeona and Gefion families seems to be no more than 600 and 850 Myr old, respectively.     

Transformation of Trojans into quasi-satellites during planetary migration and their subsequent close-encounters with the host planet

We use numerical integrations to investigate the dynamical evolution of resonant Trojan and quasi-satellite companions during the late stages of migration of the giant planets Jupiter, Saturn, Uranus, and Neptune. Our migration simulations begin with Jupiter and Saturn on orbits already well separated from their mutual 2:1 mean-motion resonance. Neptune and Uranus are decoupled from each other and have orbital eccentricities damped to near their current values. From this point we adopt a planet migration model in which the migration speed decreases exponentially with a characteristic timescale τ (the e-folding time). We perform a series of numerical simulations, each involving the migrating giant planets plus test particle Trojans and quasi-satellites. We find that the libration frequencies of Trojans are similar to those of quasi-satellites. This similarity enables a dynamical exchange of objects back and forth between the Trojan and quasi-satellite resonances during planetary migration. This exchange is facilitated by secondary resonances that arise whenever there is more than one migrating planet. For example, secondary resonances may occur when the circulation frequencies, f, of critical arguments for the Uranus-Neptune 2:1 mean-motion near-resonance are commensurate with harmonics of the libration frequency of the critical argument for the Trojan and quasi-satellite 1:1 mean-motion resonance . Furthermore, under the influence of these secondary resonances quasi-satellites can have their libration amplitudes enlarged until they undergo a close-encounter with their host planet and escape from the resonance. High-resolution simulations of this escape process reveal that ≈80% of jovian quasi-satellites experience one or more close-encounters within Jupiter’s Hill radius (R_H) as they are forced out of the quasi-satellite resonance. As many as ≈20% come within R_H/4 and ≈2.5% come within R_H/10. Close-encounters of escaping quasi-satellites occur near or even below the 2-body escape velocity from the host planet. Finally, the exchange and escape of Trojans and quasi-satellites continues to as late as 6-9τ in some simulations. By this time the dynamical evolution of the planets is strongly dominated by distant gravitational perturbations between the planets rather than the migration force. This suggests that exchange and escape of Trojans and quasi-satellites may be a contemporary process associated with the present-day near-resonant configuration of some of the giant planets in our Solar System.     

Trojans’ Odyssey: Unveiling the early history of the Solar System

In our present understanding of the Solar System, small bodies (asteroids, Jupiter Trojans, comets and TNOs) are the most direct remnants of the original building blocks that formed the planets. Jupiter Trojan and Hilda asteroids are small primitive bodies located beyond the ‘snow line’, around respectively the L₄ and L₅ Lagrange points of Jupiter at ～5.2?AU (Trojans) and in the 2:3 mean-motion resonance with Jupiter near 3.9?AU (Hildas). They are at the crux of several outstanding and still conflicting issues regarding the formation and evolution of the Solar System. They hold the potential to unlock the answers to fundamental questions about planetary migration, the late heavy bombardment, the formation of the Jovian system, the origin and evolution of trans-neptunian objects, and the delivery of water and organics to the inner planets. The proposed Trojans’ Odyssey mission is envisioned as a reconnaissance, multiple flyby mission aimed at visiting several objects, typically five Trojans and one Hilda. It will attempt exploring both large and small objects and sampling those with any known differences in photometric properties. The orbital strategy consists in a direct trajectory to one of the Trojan swarms. By carefully choosing the aphelion of the orbit (typically 5.3?AU), the trajectory will offer a long arc in the swarm thus maximizing the number of flybys. Initial gravity assists from Venus and Earth will help reducing the cruise time as well as the ΔV needed for injection thus offering enough capacity to navigate among Trojans. This solution further opens the unique possibility to flyby a Hilda asteroid when leaving the Trojan swarm. During the cruise phase, a Main Belt Asteroid could be targeted if requiring a modest ΔV. The specific science objectives of the mission will be best achieved with a payload that will perform high-resolution panchromatic and multispectral imaging, thermal-infrared imaging/ radiometry, near- and mid-infrared spectroscopy, and radio science/mass determination. The total mass of the payload amounts to 50?kg (including margins). The spacecraft is in the class of Mars-Express or a down-scaled version of Jupiter Ganymede Orbiter. It will have a dry mass of 1200?kg, a total mass at launch of 3070?kg and a ΔV capability of 700?m/s (after having reached the first Trojan) and can be launched by a Soyuz rocket. The mission operations concept (ground segment) and science operations are typical of a planetary mission as successfully implemented by ESA during, for instance, the recent flybys of Main Belt asteroids Steins and Lutetia.     

Where are the Uranus Trojans?

The area of stable motion for fictitious Trojan asteroids around Uranus’ equilateral equilibrium points is investigated with respect to the inclination of the asteroid’s orbit to determine the size of the regions and their shape. For this task we used the results of extensive numerical integrations of orbits for a grid of initial conditions around the points L ₄ and L ₅, and analyzed the stability of the individual orbits. Our basic dynamical model was the Outer Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations of motion of fictitious Trojans in the vicinity of the stable equilibrium points for selected orbits up to the age of the Solar system of 5 × 10⁹ years. One experiment has been undertaken for cuts through the Lagrange points for fixed values of the inclinations, while the semimajor axes were varied. The extension of the stable region with respect to the initial semimajor axis lies between 19.05 ≤ a ≤ 19.3 AU but depends on the initial inclination. In another run the inclination of the asteroids’ orbit was varied in the range 0° < i < 60° and the semimajor axes were fixed. It turned out that only four ‘windows’ of stable orbits survive: these are the orbits for the initial inclinations 0° < i < 7°, 9° < i < 13°, 31° < i < 36° and 38° < i < 50°. We postulate the existence of at least some Trojans around the Uranus Lagrange points for the stability window at small and also high inclinations.     

Implications of the Galilean satellites ice envelope explosions III. The origin of the Trojans and of some comets

Secondary explosions of the primary ice fragments ejected in the explosion of the electrolyzed massive ice envelopes of the Galilean satellites are capable of imparting velocities of up to ~5km s^–1 to the secondary fragments. As a result, the secondary fragments can enter the orbits of the irregular satellites (Agafonova and Drobyshevski, 1984b) and the Trojan libration orbits. In the latter case a perturbation velocity of V 0.3–2 km s^–1 is sufficient.The primary fragments ejected by the gravitational perturbations due to the Galilean satellites sunward from Jupiter's sphere of action move faster relative to the Sun than Jupiter does and therefore reach their first aphelion ahead of Jupiter in the neighborhood of L ₄. At the same time the fragments propelled from Jupiter's sphere of action beyond the planet's orbit approach it again in their perihelia behind Jupiter in the region of L ₅. The concentration of the fragments and, hence, the frequency of their collisions and explosions at L ₄ turn out to be much greater than those at L ₅. As a result, the number of the secondary fragments of diameter 15 km captured into libration orbits ahead of Jupiter can be as high as many hundreds and should exceed by more than a factor 3.5 that captured behind Jupiter.Since the icy mix of the fragments contains hydrocarbons and particulate material (silicates and the like), after ice sublimation from the surface layers the Trojans should reveal type C and RD spectra typical for Jupiter's irregular satellites, comet nuclei and other distant ice bodies of similar origin. Among the Trojans there cannot be rocky or metallic objects which are known to exist in the main asteroid belt.It is shown that a velocity perturbation of 150–200 m s^–1 resulting from a purely mechanical impact of two bodies may be sufficient to move collision fragments from the orbits of the Trojans to horseshoe-shaped trajectories with a subsequent transfer to the cometary orbits of Jupiter's family.     

The effect of orbital evolution on the Haumea (2003 EL61) collisional family

The Haumea family is currently the only identified collisional family in the Kuiper belt. We numerically simulate the long-term dynamical evolution of the family to estimate a lower limit of the family’s age and to assess how the population of the family and its dynamical clustering are preserved over Gyr timescales. We find that the family is not younger than 100 Myr, and its age is at least 1 Gyr with 95% confidence. We find that for initial velocity dispersions of 50–400 m s^?1, approximately 20–45% of the family members are lost to close encounters with Neptune after 3.5 Gyr of orbital evolution. We apply these loss rates to two proposed models for the formation of the Haumea family, a graze-and-merge type collision between two similarly sized, differentiated KBOs or the collisional disruption of a satellite orbiting Haumea. For the graze-and-merge collision model, we calculate that >85% of the expected mass in surviving family members within 150 m s^?1 of the collision has been identified, but that one to two times the mass of the known family members remains to be identified at larger velocities. For the satellite-break-up model, we estimate that the currently identified family members account for ～50% of the expected mass of the family. Taking observational incompleteness into account, the observed number of Haumea family members is consistent with either formation scenario at the 1σ level, however both models predict more objects at larger relative velocities (>150 m s^?1) than have been identified.     

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 200km across which is located very close to theg=g ₆ (orv ₆) secular resonance at a semimajor axis of 2.425AU and a (proper) inclination of 15° .0. A realistic ejection velocity distribution, with most fragments escaping at relative speeds of a few hundredsm/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 1Myr; 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 ofAU, implying collision lifetimes <10⁹ yr. 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 evolutionroute to generate meteorites has been fully demonstrated. It may be the parent body of one of the ordinary chondrite classes.