Construction of a Nekhoroshev like result for the asteroid belt dynamical system

We investigate the possibility of obtaining a Nekhoroshev like result for the dynamical system describing the motion of an asteroid in the main belt, From the mathematical point of view this is a new result since the problem is degenerate and we want to control also the motion of degenerate actions, We find that there are regions, such as the resonances of low order among the fast angles (mean motion resonances), where a Nekhoroshev like result cannot be proved a priori, Conversely, we are able to confine the motions in the mean motion resonances of logarithmically large order in the perturbation parameters, as well as in the non-resonant region, We discuss also the connection with the existence of invariant tori.     

Probing the Nekhoroshev Stability of Asteroids

We apply the spectral formulation of the Nekhoroshev theorem to investigate the long-term stability of real main belt asteroids. We find numerical indication that some asteroids are in the so-called Nekhoroshev stability regime, that is they are on chaotic orbits but their motion is stable over very long times. We have analyzed the motion of bodies in different regions of the belt, to assess the sensitivity of our method. We found that it allows us to clearly discriminate between different dynamical regimes, such as the one described by the Nekhoroshev stability, the one well described by the KAM theory, and the unstable chaotic regime in which diffusion in phase space can be detected over time spans much shorter than the age of the solar system.     

The dynamical architecture and habitable zones of the quintuplet planetary system 55 Cancri

We perform numerical simulations to study the secular orbital evolution and dynamical structure of the quintuplet planetary system 55 Cancri with the self-consistent orbital solutions by Fischer and coworkers. In the simulations, we show that this sys-tem can be stable for at least 108 yr. In addition, we extensively investigate the planetary configuration of four outer companions with one terrestrial planet in the wide region of 0.790 AU ≤ a ≤ 5.900 AU to examine the existence of potential asteroid structure and Habitable Zones (HZs). We show that there are unstable regions for orbits about 4:1, 3:1 and 5:2 mean motion resonances (MMRs) of the outermost planet in the system, and sev-eral stable orbits can remain at 3:2 and 1:1 MMRs, which resembles the asteroid belt in the solar system. From a dynamical viewpoint, proper HZ candidates for the existence of more potential terrestrial planets reside in the wide area between 1.0 AU and 2.3 AU with relatively low eccentricities.     

On asteroid distribution

The existing explanations for the asteroid distribution in the main belt (between the orbits of Mars and Jupiter) are based on numerical integration of resonance orbits in models with more than two degrees of freedom. We suggest an approach based on the investigation of the families of periodic solutions of the planar circular restricted three-body problem, i.e., a model with two degrees of freedom. This work shows that (a) the distribution of asteroids near the (p + 1)/p resonances and position of the outer boundary of the main asteroid belt can be explained within the planar circular restricted three-body problem and (b) this problem does not explain the asteroid distribution near other resonances.     

Asteroid Mean Elements: Higher Order and Iterative Theories

Mean orbital elements are obtained from osculating ones by removing the short periodic perturbations. Large catalogues of asteroid mean elements need to be computed, as a first step in the computation of proper elements, used to study asteroid families. The algorithms for this purpose available so far are only accurate to first order in the masses of the perturbing planet; the mean elements have satisfactory accuracy for most of the asteroid belt, but degraded accuracy in the neighbourhoods of the main mean motion resonances, especially the 2:1. We investigate a number of algorithms capable of improving this approximation; they belong to the two classes of Breiter-type methods and iterative methods. The former are obtained by applying some higher order numerical integration scheme, such as Runge–Kutta, to the differential equation whose solution is a transformation removing the fast angular variables from the equations; they can be used to compute a full second order theory, however, only if the full second order determining function is explicitly computed, and this is computationally too cumbersome for a complicated problem such as the N-body. The latter are fixed point iterative schemes, with the first order theory as an iteration step, used to compute the inverse map from mean to osculating elements; formally the method is first order, but because they implement a fixed frequency perturbation theory, they are more accurate than conventional single iteration methods; a similar method is already in use in our computation of proper from mean elements. Many of these methods are tested on a sample of asteroid orbits taken from the Themis family, up to the edge of the 2:1 resonance, and the dispersion of the values of the computed mean semimajor axis over 100 000 years is used as quality control. The results of these tests indicate that the iterative methods are superior, in this specific application, to the Breiter methods, in accuracy and reliability. This is understood as the result of the cancellations occurring between second order perturbation terms: the incomplete second order theory, resulting from the use of a Breiter method with the first order determining function only, can be less accurate than complete, fixed frequency theories of the first order. We have therefore computed new catalogues of asteroid mean and proper elements, incorporating an iterative algorithm in both steps (osculating to mean and mean to proper elements). This new data set, significantly more reliable even in the previously degraded regions of Themis and Cybele, is in the public domain. This revised version was published online in July 2006 with corrections to the Cover Date.     

Normal Form Invariants Around Spin-orbit Periodic Orbits

We consider a model of spin-orbit interaction, describing the motion of an oblate satellite rotating about an internal spin-axis and orbiting about a central planet. The resulting second order differential equation depends upon the parameters provided by the equatorial oblateness of the satellite and its orbital eccentricity. Normal form transformations around the main spin-orbit resonances are carried out explicitly. As an outcome, one can compute some invariants; the fact that these quantities are not identically zero is a necessary condition to prove the existence of nearby periodic orbits (Birkhoff fixed point theorem). Moreover, the nonvanishing of the invariants provides also the stability of the spin-orbit resonances, since it guarantees the existence of invariant curves surrounding the periodic orbit.     

Origin of the Basaltic Asteroid 1459 Magnya: A Dynamical and Mineralogical Study of the Outer Main Belt

The recent discovery of a relatively small basaltic asteroid in the outer main belt with no apparent link to (4) Vesta raised several hypotheses on its origin. We present the results of a dynamical and mineralogical study of the region near (1459) Magnya intended to establish its origin. The dynamical analysis shows that the region is filled with high-order two-body and three-body mean motion resonances and nonlinear secular resonances, which can lead to slow chaotic diffusion. The mineralogical analysis has not identified any other asteroid with a composition similar to Magnya, nor the presence of fragments that could be securely related to the catastrophic disruption of a differentiated parent body. The various scenarios for the origin of Magnya are also discussed in the face of both the results presented here and recently published results.     

The collisional and dynamical evolution of the main-belt and NEA size distributions

The size distribution of main belt of asteroids is determined primarily by collisional processes. Large asteroids break up and form smaller asteroids in a collisional cascade, with the outcome controlled by the strength-size relationship for asteroids. In addition to collisional processes, the non-collisional removal of asteroids from the main belt (and their insertion into the near-Earth asteroid (NEA) population) is critical, and involves several effects: strong resonances increase the orbital eccentricity of asteroids and cause them to enter the inner planet region; chaotic diffusion by numerous weak resonances causes a slow leak of asteroids into the Mars- and Earth-crossing populations; and the Yarkovsky effect, a radiation force on asteroids, is the primary process that drives asteroids into these resonant escape routes. Yarkovsky drift is size-dependent and can modify the main-belt size distribution. The NEA size distribution is primarily determined by its source, the main-belt population, and by the size-dependent processes that deliver bodies from the main belt. All of these effects are simulated in a numerical collisional evolution model that incorporates removal by non-collisional processes. We test our model against a wide range of observational constraints, such as the observed main-belt and NEA size distributions, the number of asteroid families, the preserved basaltic crust of Vesta and its large south-pole impact basin, the cosmic ray exposure ages of meteorites, and the cratering records on asteroids. We find a strength-size relationship for main-belt asteroids and non-collisional removal rates from the main belt such that our model fits these constraints as best as possible within the parameter space we explore. Our results are consistent with other independent estimates of strength and removal rates.     

The primordial excitation and clearing of the asteroid belt—Revisited

We have performed new simulations of two different scenarios for the excitation and depletion of the primordial asteroid belt, assuming Jupiter and Saturn on initially circular orbits as predicted by the Nice Model of the evolution of the outer Solar System [Gomes, R., Levison, H.F., Tsiganis, K., Morbidelli, A., 2005. Nature 435, 466-469; Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461; Morbidelli, A., Levison, H.F., Tsiganis, K., Gomes, R., 2005. Nature 435, 462-465]. First, we study the effects of sweeping secular resonances driven by the depletion of the solar nebula. We find that these sweeping secular resonances are incapable of giving sufficient dynamical excitation to the asteroids for nebula depletion timescales consistent with estimates for solar-type stars, and in addition cannot cause significant mass depletion in the asteroid belt or produce the observed radial mixing of different asteroid taxonomic types. Second, we study the effects of planetary embryos embedded in the primordial asteroid belt. These embedded planetary embryos, combined with the action of jovian and saturnian resonances, can lead to dynamical excitation and radial mixing comparable to the current asteroid belt. The mass depletion driven by embedded planetary embryos alone, even in the case of an eccentric Jupiter and Saturn, is roughly 10-20× less than necessary to explain the current mass of the main belt, and thus a secondary depletion event, such as that which occurs naturally in the Nice Model, is required. We discuss the implications of our new simulations for the dynamical and collisional evolution of the main belt.     

The resonant structure of Jupiter's Trojan asteroids – I. Long-term stability and diffusion

We study the global dynamics of the jovian Trojan asteroids by means of the frequency map analysis. We find and classify the main resonant structures that serve as skeleton of the phase space near the Lagrangian points. These resonances organize and control the long-term dynamics of the Trojans. Besides the secondary and secular resonances, that have already been found in other asteroid sets in mean motion resonance (e.g. main belt, Kuiper belt), we identify a new type of resonance that involves secular frequencies and the frequency of the great inequality, but not the libration frequency. Moreover, this new family of resonances plays an important role in the slow transport mechanism that drives Trojans from the inner stable region to eventual ejections. Finally, we relate this global view of the dynamics with the observed Trojans, identify the asteroids that are close to these resonances and study their long-term behaviour.     

Reviewing the Yarkovsky effect: New light on the delivery of stone and iron meteorites from the asteroid belt

Abstract— We give a nonmathematical review of recent work regarding the Yarkovsky effect on asteroidal fragments. This effect may play a critical, but underappreciated, role in delivering meteorites to Earth. Two variants of the effect cause drifts in orbital elements, notably semimajor axes. The “classic” or “diurnal” Yarkovsky effect is associated with diurnal rotation at low obliquity. More recently, a “seasonal” effect has also been described, associated with high obliquity. Studies of these Yarkovsky effects are combined with studies of resonance effects to clarify meteorite delivery. If there were no Yarkovsky drift, asteroid fragments could reach a resonance only if produced very near that resonance. However, objects in resonances typically reach Earth-crossing orbits within a few million years, which is inconsistent with stone meteorites' cosmic-ray exposure (CRE) ages (5–50 Ma) and iron meteorites' CRE ages (100–1000 Ma). In the new view, on the other hand, large objects in the asteroid belt are “fixed” in semimajor axis, but bodies up to 100 m in diameter are in a constant state of mixing and flow, especially if the thermal conductivity of their surface layers is low. Thus, small asteroid fragments may reach the resonances after long periods of drift in the main belt. Yarkovsky drift effects, combined with resonance effects, appear to explain many meteorite properties, including: (1) the long CRE ages of iron meteorites (due to extensive drift lifetimes in the belt); (2) iron meteorites' sampling of numerous parent bodies; (3) the shorter CRE ages of most stone meteorites (due to faster drift, coupled with weaker strength and more rapid collisional erosion); and (4) the abundance of falls from discrete impact events near resonances, such as the 8 Ma CRE age of H chondrites. Other consequences include: the delivery of meteorite parent bodies to resonances is enhanced; proportions of stone and iron meteorites delivered to Earth may be different from the proportions at the same sizes left in the belt, which in turn may differ from the ratio produced in asteroidal collisions; Rabinowitz's 10–100 m objects may be preferentially delivered to near-Earth space; and the delivery of C-class fragments from the outer belt may be inhibited, compared to classes in other parts of the belt. Thus, Yarkovsky effects may have important consequences in meteoritics and asteroid science.     

Secular spin dynamics of inner main-belt asteroids

Understanding the evolution of asteroid spin states is challenging work, in part because asteroids have a variety of orbits, shapes, spin states, and collisional histories but also because they are strongly influenced by gravitational and non-gravitational (YORP) torques. Using efficient numerical models designed to investigate asteroid orbit and spin dynamics, we study here how several individual asteroids have had their spin states modified over time in response to these torques (i.e., 951 Gaspra, 60 Echo, 32 Pomona, 230 Athamantis, 105 Artemis). These test cases which sample semimajor axis and inclination space in the inner main belt, were chosen as probes into the large parameter space described above. The ultimate goal is to use these data to statistically characterize how all asteroids in the main belt population have reached their present-day spin states. We found that the spin dynamics of prograde-rotating asteroids in the inner main belt is generally less regular than that of the retrograde-rotating ones because of numerous overlapping secular spin-orbit resonances. These resonances strongly affect the spin histories of all bodies, while those of small asteroids (?40 km) are additionally influenced by YORP torques. In most cases, gravitational and non-gravitational torques cause asteroid spin axis orientations to vary widely over short (?1 My) timescales. Our results show that (951) Gaspra has a highly chaotic rotation state induced by an overlap of the s and s₆ spin-orbit resonances. This hinders our ability to investigate its past evolution and infer whether thermal torques have acted on Gaspra's spin axis since its origin.     

Computer simulation of the formation of asteroid belt structure

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

Local And Global Diffusion Along Resonant Lines in Discrete Quasi-integrable Dynamical Systems

We detect and measure diffusion along resonances in a quasi-integrable symplectic map for different values of the perturbation parameter. As in a previously studied Hamiltonian case (Lega et al., 2003) results agree with the prediction of the Nekhoroshev theorem. Moreover, for values of the perturbation parameter slightly below the critical value of the transition between Nekhoroshev and Chirikov regime we have also found a diffusion of some orbits along macroscopic portions of the phase space. Such a diffusion follows in a spectacular way the peculiar structure of resonant lines.     

Measure of the exponential splitting of the homoclinic tangle in four-dimensional symplectic mappings

Using four-dimensional symplectic maps as a model problem, we numerically compute the unstable manifolds of the hyperbolic manifolds of the phase space related to the single resonances. We measure an exponential dependence of the size of the lobes of these manifolds through many orders of magnitude of the perturbing parameter. This is an indirect numerical verification of the exponential decay of the normal form, as predicted by the Nekhoroshev theorem. The variation of the size of the lobes turns out to be correlated to the diffusion coefficient.     

小行星主带上单个行星的质量上限

介绍了N体模拟的Hermite算法,并利用该算法研究了不同质量行星在小行星主带上轨道的演化情况．采用的演化模型是太阳系N体模型(N=7),即把水星、金星、地球的质量加到太阳上,忽略冥王星,同时在小行星主带附近增加一个假想行星,系统演化时间为1亿年．数值模拟显示能够稳定存在于小行星主带上的单个天体的质量上限其量级为10~(25)kg．模拟同时还显示在某些情况下,假想行星与木星之间的低阶共振可以增强系统的稳定性．     

The use of radar observations of Near-Earth Asteroids in the determination of the dynamical equinox

Near-Earth Asteroids (NEAs) are Solar system special class objects attracting the attention of astronomical community especially during several of the last decades. To some extent the NEAs have an advantage over the minor planets of the main belt: due to close and regular approaches to the Earth the radar observations of NEAs can be obtained for the greater number of objects than for those of the main belt of the minor planets. In this paper the use of all available radar observations together with optical ones resulting in the combined data analysis solution is discussed for different problems such as the asteroid orbits and catalog orientation parameters determination. In particular the problem of the motion of the dynamical equinox in the Hipparcos reference system is considered. About 13000 radar and optical observations of 24 NEAs and main belt minor planets have been used to obtain the precise asteroid orbits, FK5 catalogue orientation parameters and the motion of the dynamical equinox from 1750 till 2000 in the Hipparcos system.     

Stable Manifolds and Homoclinic Points Near Resonances in the Restricted Three-Body Problem

The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses 1− μ and μ that circle each other with period equal to 2π. For small μ, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers p and q, if its period around the heavier primary is approximately 2π p/q, and by its approximate eccentricity e. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with q/p equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any μ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for μ small enough in the unaveraged restricted three-body problem.     

Massive identification of asteroids in three-body resonances

An essential role in the asteroidal dynamics is played by the mean motion resonances. Two-body planet–asteroid resonances are widely known, due to the Kirkwood gaps. Besides, so-called three-body mean motion resonances exist, in which an asteroid and two planets participate. Identification of asteroids in three-body (namely, Jupiter–Saturn–asteroid) resonances was initially accomplished by Nesvorný and Morbidelli (Nesvorný D., Morbidelli, A. [1998]. Astron. J. 116, 3029–3037), who, by means of visual analysis of the time behaviour of resonant arguments, found 255 asteroids to reside in such resonances. We develop specialized algorithms and software for massive automatic identification of asteroids in the three-body, as well as two-body, resonances of arbitrary order, by means of automatic analysis of the time behaviour of resonant arguments. In the computation of orbits, all essential perturbations are taken into account. We integrate the asteroidal orbits on the time interval of 100,000 yr and identify main-belt asteroids in the three-body Jupiter–Saturn–asteroid resonances up to the 6th order inclusive, and in the two-body Jupiter–asteroid resonances up to the 9th order inclusive, in the set of ～250,000 objects from the “Asteroids – Dynamic Site” (AstDyS) database. The percentages of resonant objects, including extrapolations for higher-order resonances, are determined. In particular, the observed fraction of pure-resonant asteroids (those exhibiting resonant libration on the whole interval of integration) in the three-body resonances up to the 6th order inclusive is ≈0.9% of the whole set; and, using a higher-order extrapolation, the actual total fraction of pure-resonant asteroids in the three-body resonances of all orders is estimated as ≈1.1% of the whole set.     

Resonant structure of the outer asteroid belt

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