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.     

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.     

Visible spectroscopic and photometric survey of Jupiter Trojans: Final results on dynamical families

We present the results of a visible spectroscopic and photometric survey of Jupiter Trojans belonging to different dynamical families. The survey was carried out at the 3.5 m New Technology Telescope (NTT) of the European Southern Observatory (La Silla, Chile) in April 2003, May 2004 and January 2005. We obtained data on 47 objects, 23 belonging to the L5 swarm and 24 to the L4 one. These data together with those already published by Fornasier et al. [Fornasier, S., Dotto, E., Marzari, F., Barucci, M.A., Boehnhardt, H., Hainaut, O., de Bergh, C., 2004a. Icarus 172, 221-232] and Dotto et al. [Dotto, E., Fornasier, S., Barucci, M.A., Licandro, J., Boehnhardt, H., Hainaut, O., Marzari, F., de Bergh, C., De Luise, F., 2006. Icarus 183, 420-434], acquired since November 2002, constitute a total sample of visible spectra for 80 objects. The survey allows us to investigate six families (Aneas, Anchises, Misenus, Phereclos, Sarpedon, Panthoos) in the L5 cloud and four L4 families (Eurybates, Menelaus, 1986 WD and 1986 TS6). The sample that we measured is dominated by D-type asteroids, with the exception of the Eurybates family in the L4 swarm, where there is a dominance of C- and P-type asteroids. All the spectra that we obtained are featureless with the exception of some Eurybates members, where a drop-off of the reflectance is detected shortward of 5200 Å. Similar features are seen in main belt C-type asteroids and commonly attributed to the intervalence charge transfer transition in oxidized iron. Our sample comprises fainter and smaller Trojans as compared to the literature's data and allows us to investigate the properties of objects with estimated diameter smaller than 40-50 km. The analysis of the spectral slopes and colors versus the estimated diameters shows that the blue and red objects have indistinguishable size distribution, so any relationship between size and spectral slopes has been found. To fully investigate the Trojans population, we include in our analysis 62 spectra of Trojans available in literature, resulting in a total sample of 142 objects. Although the mean spectral behavior of L4 and L5 Trojans is indistinguishable within the uncertainties, we find that the L4 population is more heterogeneous and that it has a higher abundance of bluish objects as compared to the L5 swarm. Finally, we perform a statistical investigation of the Trojans's spectra property distributions as a function of their orbital and physical parameters, and in comparison with other classes of minor bodies in the outer Solar System. Trojans at lower inclination appear significantly bluer than those at higher inclination, but this effect is strongly driven by the Eurybates family. The mean colors of the Trojans are similar to those of short period comets and neutral Centaurs, but their color distributions are different.     

The dynamics of Neptune Trojan – I. The inclined orbits

The stability of Trojan type orbits around Neptune is studied. As the first part of our investigation, we present in this paper a global view of the stability of Trojans on inclined orbits. Using the frequency analysis method based on the fast Fourier transform technique, we construct high-resolution dynamical maps on the plane of initial semimajor axis a ₀ versus inclination i ₀. These maps show three most stable regions, with i ₀ in the range of  (0°, 12°), (22°, 36°)  and  (51°, 59°),  respectively, where the Trojans are most probably expected to be found. The similarity between the maps for the leading and trailing triangular Lagrange points L ₄ and L ₅ confirms the dynamical symmetry between these two points. By computing the power spectrum and the proper frequencies of the Trojan motion, we figure out the mechanisms that trigger chaos in the motion. The Kozai resonance found at high inclination varies the eccentricity and inclination of orbits, while the  ν₈  secular resonance around   i ₀∼ 44°  pumps up the eccentricity. Both mechanisms lead to eccentric orbits and encounters with Uranus that introduce strong perturbation and drive the objects away from the Trojan like orbits. This explains the clearance of Trojan at high inclination  (>60°)  and an unstable gap around  44°  on the dynamical map. An empirical theory is derived from the numerical results, with which the main secular resonances are located on the initial plane of  ( a ₀, i ₀)  . The fine structures in the dynamical maps can be explained by these secular resonances.     

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.     

Survival of Trojan-type companions of Neptune during primordial planet migration

We investigate the survivability of Trojan-type companions of Neptune during primordial radial migration of the giant planets Jupiter, Saturn, Uranus, and Neptune. We adopt the usual planet migration model in which the migration speed decreases exponentially with a characteristic time scale τ (the e-folding time). We perform a series of numerical simulations, each involving the migrating giant planets plus ∼1000 test particle Neptune Trojans with initial distributions of orbital eccentricity, inclination, and libration amplitude similar to those of the known jovian Trojans asteroids. We analyze these simulations to measure the survivability of Neptune's Trojans as a function of migration rate. We find that orbital migration with the characteristic time scale τ=10⁶ years allows about 35% of preexisting Neptune Trojans to survive to 5τ, by which time the giant planets have essentially reached their final orbits. In contrast, slower migration with τ=10⁷ years yields only a ∼5% probability of Neptune Trojans surviving to a time of 5τ. Interestingly, we find that the loss of Neptune Trojans during planetary migration is not a random diffusion process. Rather, losses occur almost exclusively during discrete prolonged episodes when Trojan particles are swept by secondary resonances associated with mean-motion commensurabilities of Uranus with Neptune. These secondary resonances arise when the circulation frequencies, f, of critical arguments for Uranus-Neptune mean-motion near-resonances (e.g., f^UN_1:2, f^UN_4:7) are commensurate with harmonics of the libration frequency of the critical argument for the Neptune-Trojan 1:1 mean-motion resonance (f^NT_1:1). Trojans trapped in the secondary resonances typically have their libration amplitudes amplified until they escape the 1:1 resonance with Neptune. Trojans with large libration amplitudes are susceptible to loss during sweeping by numerous high-order secondary resonances (e.g., f^UN_1:2≈11f^NT_1:1). However, for the slower migration, with τ=10⁷ years, even tightly bound Neptune Trojans with libration amplitudes below 10° can be lost when they become trapped in 1:3 or 1:2 secondary resonances between f^UN_1:2 and f^NT_1:1. With τ=10⁷ years the 1:2 secondary resonance was responsible for the single greatest episode of loss, ejecting nearly 75% of existing Neptune Trojans. This episode occurred during the late stages of planetary migration when the remnant planetesimal disk would have been largely dissipated. We speculate that if the number of bodies liberated during this event was sufficiently high they could have caused a spike in the impact rate throughout the Solar System.     

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.     

(U)BVRI photometry of Trojan L5 asteroids

(U)BVRI photometry of 27 mainly small (≈30 km) Trojans show that the previously reported size-spectral slope trend among Trojans and Hildas must be modified. The small-slope small-bodies gap present in previous works does not exist. This has also been noted by other recent reported observations. While the largest asteroids have slopes less than about 10% kÅ⁻¹ the smallest asteroids have both small and large slopes. The maximum slope is about 15% kÅ⁻¹ for all outer main belt groups (Cybeles, Hildas and Trojans). Combining observations from different authors reveal that these three groups have small asteroids (below 50 km) in the same slope range, while the large Trojans (≈100 km) have in general significantly steeper slopes than Hildas and Cybeles of similar size. The size-slope trend would favor organic materials as mainly responsible for the reflectance properties of the surfaces.     

The dynamics of inclined Neptune Trojans

We investigated the stable area for fictive Trojan asteroids around Neptune’s Lagrangean equilibrium points with respect to their semimajor axis and inclination. To get a first impression of the stability region we derived a symplectic mapping for the circular and the elliptic planar restricted three body problem. The dynamical model for the numerical integrations was the outer Solar system with the Sun and the planets Jupiter, Saturn, Uranus and Neptune. To understand the dynamics of the region around L ₄ and L ₅ for the Neptune Trojans we also used eight different dynamical models (from the elliptic problem to the full outer Solar system model with all giant planets) and compared the results with respect to the largeness and shape of the stable region. Their dependence on the initial inclinations (0° < i < 70°) of the Trojans’ orbits could be established for all the eight models and showed the primary influence of Uranus. In addition we could show that an asymmetry of the regions around L ₄ and L ₅ is just an artifact of the different initial conditions.     

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.     

On the Stability of High Inclined L4 and L5Trojans

The orbits of real asteroids around the Lagrangian points L₄ and L ₅of Jupiter with large inclinations (i > 20°) were integrated for 50 Myrs. We investigated the stability with the aid of the Lyapunov characteristic exponents (LCE) but tested also two other methods: on one hand we integrated four neighbouring orbits for each asteroid and computed the maximum distance in every group, on the other hand we checked the variation of the Delaunay element H of the asteroid. In a second simulation – for a grid of initial eccentricity versus initial inclination – we examined the stability of the orbits around both Lagrangian points for 20° < i < 55° and 0.0 < e < 0.20. For the initial semimajor axes we have chosen the one ofJupiter(a = 5.202 AU). We determined the stability with the aid of the LCEs and also the maximum eccentricity of the orbits during the whole integration time. The region around L₄ turned out to be unstable for large inclinations and eccentricities (i > 55° and e > 0.12). The stable region shrinks for orbits around L₅: we found that they become unstable already for i > 45° and e > 0.10. We interpret it as a first hint why we observe more Trojans around the leading Lagrangian point. The results confirm the stability behaviour of the real Trojans which we computed in the first part of the paper.     

Dynamics of Mars Trojans

In this paper we explore the dynamical stability of the Mars Trojan region applying mainly Laskar's Frequency Map Analysis. This method yields the chaotic diffusion rate of orbits and allows to determine the most stable regions. It also gives the frequencies which are responsible for the instability of orbits. The most stable regions are found for inclinations between about 15° and 30°. For inclinations smaller than 15°, we confirm, by applying a synthetic secular theory, that the secular resonances ν₃, ν₄, ν₁₃, ν₁₄ rapidly excite asteroid orbits within a few Myrs, or even faster. The asteroids are removed from the Trojan region after a close encounter with Mars. For large inclinations, the secular resonance ν₅ clears a small region around 30° while the Kozai resonance rapidly removes bodies for inclinations larger than 35°. The dynamical lifetimes of the three L5 Trojans, (5261) Eureka, 1998 VF31, 2001 DH47, and the only L4 Trojan 1999 UJ7 are determined by numerically integrating clouds of corresponding clones over the age of the Solar System. All four Trojans reside in the most stable region with smallest diffusion coefficients. Their dynamical half-lifetime is of the order of the age of the Solar System. The Yarkovsky force has little effect on the known Trojans but for bodies smaller than about 1-5 m the drag is strong enough to destabilize Trojans on a timescale shorter than 4.5 Gyr.     

Considerations on the magnitude distributions of the Kuiper belt and of the Jupiter Trojans

By examining the absolute magnitude (H) distributions (hereafter HD) of the cold and hot populations in the Kuiper belt and of the Trojans of Jupiter, we find evidence that the Trojans have been captured from the outer part of the primordial trans-neptunian planetesimal disk. We develop a sketch model of the HDs in the inner and outer parts of the disk that is consistent with the observed distributions and with the dynamical evolution scenario known as the ‘Nice model’. This leads us to predict that the HD of the hot population should have the same slope of the HD of the cold population for 6.5<H<9, both as steep as the slope of the Trojans' HD. Current data partially support this prediction, but future observations are needed to clarify this issue. Because the HD of the Trojans rolls over at H∼9 to a collisional equilibrium slope that should have been acquired when the Trojans were still embedded in the primordial trans-neptunian disk, our model implies that the same roll-over should characterize the HDs of the Kuiper belt populations, in agreement with the results of Bernstein et al. [Bernstein, G.M., and 5 colleagues, 2004. Astron. J. 128, 1364-1390] and Fuentes and Holman [Fuentes, C.I., Holman, M.J., 2008. Astron. J. 136, 83-97]. Finally, we show that the constraint on the total mass of the primordial trans-neptunian disk imposed by the Nice model implies that it is unlikely that the cold population formed beyond 35 AU.     

Creation,detection, and evolution of Jupiter Trojan families

An investigation is carried out looking at correlations between the orbital elements of collisional targets and projectiles, estimating the number of interlopers in Trojan collisional families, and at the possibility of determining the ages of the Jupiter Trojan families by orbital integration. Real Trojans are integrated and close encounters are recorded in order to evaluate collisional circumstances between Trojans. Fictitious collisional families are created and integrated for 10 MJyr (million Julian years) forward in time and back again to the time of the collision in order to check the performance of the integrator, and the behaviour of the fictitious collisional fragments. Proper elements are calculated for the detection of family clustering using the hierarchically clustering method. This method presents little difficulty finding fictitious families in the Trojan swarms even in areas with densely populated backgrounds. However, even when the background is relatively sparse in objects, several interlopers can be connected to the family at velocity differences below 100 m s^–1. On the other hand, in densely populated backgrounds the contamination of interlopers should be less than 30%. Providing gravity is the only significant force acting on the Trojans and resonance effects are weak, the shape the collision fragments create in the proper element space are preserved on the GJyr scale, and collisions can be tracked with orbital integrations for ages of at least 100 MJyr. However, the shape of artificial families does not correspond to suggested real families. This points to the need of including non‐gravitational forces such as the Yarkovsky effect in order to simulate the family evolution. As a consequence age determination by orbital integration might be severely restricted and previous investigations involving long term orbital integrations might have tobe recalculated (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

UBV photometry of small and distant asteroids

Photoelectric magnitudes and colors on the UBV system are presented for 65 minor planets, including four Mars crossers, six Trojans, and main-belt objects down to 6 km in diameter. The Trojans all have very similar colors not characteristics of the main-belt population. A paucity of S-type asteroids at the smallest diameters, predicted from trends seen at larger sizes, is not observed. The newly available color data for small objects ranging from 1.0 to 5.2 AU in heliocentric distance show the main belt to be a transition zone between predominantly silicate and carbonaceous compositions.     

Phase curves of nine Trojan asteroids over a wide range of phase angles

We have observed well-sampled phase curves for nine Trojan asteroids in B-, V-, and I-bands. These were constructed from 778 magnitudes taken with the 1.3-m telescope on Cerro Tololo as operated by a service observer for the SMARTS consortium. Over our typical phase range of 0.2-10°, we find our phase curves to be adequately described by a linear model, for slopes of 0.04-0.09 mag/° with average uncertainty less than 0.02 mag/°. (The one exception, 51378 (2001 AT33), has a formally negative slope of −0.02 ± 0.01 mag/°.) These slopes are too steep for the opposition surge mechanism to be shadow-hiding (SH), so we conclude that the dominant surge mechanism must be coherent backscattering (CB). In a detailed comparison of surface properties (including surge slope, B-R color, and albedo), we find that the Trojans have surface properties similar to the P and C class asteroids prominent in the outer main belt, yet they have significantly different surge properties (at a confidence level of 99.90%). This provides an imperfect argument against the traditional idea that the Trojans were formed around Jupiter’s orbit. We also find no overlap in Trojan properties with either the main belt asteroids or with the small icy bodies in the outer Solar System. Importantly, we find that the Trojans are indistinguishable from other small bodies in the outer Solar System that have lost their surface ices (such as the gray Centaurs, gray Scattered Disk Objects, and dead comets). Thus, we find strong support for the idea that the Trojans originally formed as icy bodies in the outer Solar System, were captured into their current orbits during the migration of the gas giant planets, and subsequently lost all their surface ices.     

Spectroscopy and photometry of Mars Trojans

Mars is the only terrestrial planet known to have co-orbiting “Trojan” asteroids. We have obtained visible and near-IR reflectance spectra of three of these objects: 5261 Eureka and 1998 VF31 in the L5 region and 1999 UJ7 in the L4 region. We also obtained JHK spectrophotometry and a visible lightcurve for 5261 Eureka. The asteroid 5261 Eureka has a visible spectrum that is classified as Sr in the Bus taxonomy, and has infrared colors consistent with the A-class asteroids. The data for 1998 VF31 have a restricted wavelength range, but are most consistent with the Sr or Sa class, though we note a marginal consistency with the D class. We can rule out a C-class classification. 1999 UJ7 has an X-class or T-class spectrum, which is unlike that of the other two Mars Trojans. The photometric data for Eureka are limited, but we can constrain the period to longer than 5 hours (likely 5.5-6 hours) and lightcurve amplitude of at least 0.15 magnitude at this viewing geometry. The spectral differences among the Mars Trojans suggests that either they did not all form at their present solar distances or that they have not always been at their present sizes.     

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.     

Origin of the structure of the Kuiper belt during a dynamical instability in the orbits of Uranus and Neptune

We explore the origin and orbital evolution of the Kuiper belt in the framework of a recent model of the dynamical evolution of the giant planets, sometimes known as the Nice model. This model is characterized by a short, but violent, instability phase, during which the planets were on large eccentricity orbits. It successfully explains, for the first time, the current orbital architecture of the giant planets [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461], the existence of the Trojans populations of Jupiter and Neptune [Morbidelli, A., Levison, H.F., Tsiganis, K., Gomes, R., 2005. Nature 435, 462-465], and the origin of the late heavy bombardment of the terrestrial planets [Gomes, R., Levison, H.F., Tsiganis, K., Morbidelli, A., 2005. Nature 435, 466-469]. One characteristic of this model is that the proto-planetary disk must have been truncated at roughly 30 to 35 AU so that Neptune would stop migrating at its currently observed location. As a result, the Kuiper belt would have initially been empty. In this paper we present a new dynamical mechanism which can deliver objects from the region interior to ∼35 AU to the Kuiper belt without excessive inclination excitation. In particular, we show that during the phase when Neptune's eccentricity is large, the region interior to its 1:2 mean motion resonance becomes unstable and disk particles can diffuse into this area. In addition, we perform numerical simulations where the planets are forced to evolve using fictitious analytic forces, in a way consistent with the direct N-body simulations of the Nice model. Assuming that the last encounter with Uranus delivered Neptune onto a low-inclination orbit with a semi-major axis of ∼27 AU and an eccentricity of ∼0.3, and that subsequently Neptune's eccentricity damped in ∼1 My, our simulations reproduce the main observed properties of the Kuiper belt at an unprecedented level. In particular, our results explain, at least qualitatively: (1) the co-existence of resonant and non-resonant populations, (2) the eccentricity-inclination distribution of the Plutinos, (3) the peculiar semi-major axis—eccentricity distribution in the classical belt, (4) the outer edge at the 1:2 mean motion resonance with Neptune, (5) the bi-modal inclination distribution of the classical population, (6) the correlations between inclination and physical properties in the classical Kuiper belt, and (7) the existence of the so-called extended scattered disk. Nevertheless, we observe in the simulations a deficit of nearly-circular objects in the classical Kuiper belt.     

Long term stability of Earth Trojans

We explore the long-term stability of Earth Trojans by using a chaos indicator, the Frequency Map Analysis. We find that there is an extended stability region at low eccentricity and for inclinations lower than about $50^{\circ }$ even if the most stable orbits are found at $i \le 40^{\circ }$ . This region is not limited in libration amplitude, contrary to what found for Trojan orbits around outer planets. We also investigate how the stability properties are affected by the tidal force of the Earth–Moon system and by the Yarkovsky force. The tidal field of the Earth–Moon system reduces the stability of the Earth Trojans at high inclinations while the Yarkovsky force, at least for bodies larger than 10 m in diameter, does not seem to strongly influence the long-term stability. Earth Trojan orbits with the lowest diffusion rate survive on timescales of the order of $10^9$  years but their evolution is chaotic. Their behaviour is similar to that of Mars Trojans even if Earth Trojans appear to have shorter lifetimes.