A Symplectic Mapping Model as a Tool to Understand the Dynamics of 2/1 Resonant Asteroid Motion

We present a 3-D symplectic mapping model that is valid at the 2:1 mean motion resonance in the asteroid motion, in the Sun-Jupiter-asteroid model. This model is used to study the dynamics inside this resonance and several features of the system have been made clear. The introduction of the third dimension, through the inclination of the asteroid orbit, plays an important role in the evolution of the asteroid and the appearance of chaotic motion. Also, the existence of the secondary resonances is clearly shown and their role in the appearance of chaotic motion and the slow diffusion of the elements of the orbit is demonstrated. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Symplectic Mapping Model for the Study of 2:3 Resonant Trans-Neptunian Motion

A symplectic mapping is constructed for the study of the dynamical evolution of Edgeworth-Kuiper belt objects near the 2:3 mean motion resonance with Neptune. The mapping is six-dimensional and is a good model for the Poincaré map of the real system, that is, the spatial elliptic restricted three-body problem at the 2:3 resonance, with the Sun and Neptune as primaries. The mapping model is based on the averaged Hamiltonian, corrected by a semianalytic method so that it has the basic topological properties of the phase space of the real system both qualitatively and quantitatively. We start with two dimensional motion and then we extend it to three dimensions. Both chaotic and regular motion is observed, depending on the objects' initial inclination and phase. For zero inclination, objects that are phase-protected from close encounters with Neptune show ordered motion even at eccentricities as large as 0.4 and despite being Neptune-crossers. On the other hand, not-phase-protected objects with eccentricities greater than 0.15 follow chaotic motion that leads to sudden jumps in their eccentricity and are removed from the 2:3 resonance, thus becoming short period comets. As inclination increases, chaotic motion becomes more widespread, but phase-protection still exists and, as a result, stable motion appears for eccentricities up to e = 0.3 and inclinations as high as i = 15°, a region where plutinos exist.     

Asteroid motion near the 3 : 1 resonance

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

Stable Chaos versus Kirkwood Gaps in the Asteroid Belt: A Comparative Study of Mean Motion Resonances

We have shown, in previous publications, that stable chaos is associated with medium/high-order mean motion resonances with Jupiter, for which there exist no resonant periodic orbits in the framework of the elliptic restricted three-body problem. This topological “defect” results in the absence of the most efficient mechanism of eccentricity transport (i.e., large-amplitude modulation on a short time scale) in three-body models. Thus, chaotic diffusion of the orbital elements can be quite slow, while there can also exist a nonnegligible set of chaotic orbits which are semiconfined (stable chaos) by “quasi-barriers” in the phase space. In the present paper we extend our study to all mean motion resonances of order q≤9 in the inner main belt (1.9-3.3 AU) and q≤7 in the outer belt (3.3-3.9 AU). We find that, out of the 34 resonances studied, only 8 possess resonant periodic orbits that are continued from the circular to the elliptic three-body problem (regular families), namely, the 2/1, 3/1, 4/1, and 5/2 in the inner belt and the 7/4, 5/3, 11/7, and 3/2 in the outer belt. Numerical results indicate that the 7/3 resonance also carries periodic orbits but, unlike the aforementioned resonances, 7/3-periodic orbits belong to an irregular family. Note that the five inner-belt resonances that carry periodic orbits correspond to the location of the main Kirkwood gaps, while the three outer-belt resonances correspond to gaps in the distribution of outer-belt asteroids noted by Holman and Murray (1996, Astron. J.112, 1278-1293), except for the 3/2 case where the Hildas reside. Fast, intermittent eccentricity increase is found in resonances possessing periodic orbits. In the remaining resonances the time-averaged elements of chaotic orbits are, in general, quite stable, at least for times t∼250 Myr. This slow diffusion picture does not change qualitatively, even if more perturbing planets are included in the model.     

Symplectic Mapping for Trojan-Type Motion in the Elliptic Restricted Three-Body Problem

A symplectic mapping for Trojan-type motion has been developed in the secularly changing elliptic restricted three-body problem. The mapping describes well the characteristics of Trojan-type dynamics at small eccentricities. By using this mapping the boundary of the stability region has been studied for different values of the initial eccentricities of hypothetical Jupiter's Trojans. It has been found that in the secularly changing elliptic case the chaotic diffusion at the border of the stability region is stronger than simply in the elliptic case. An explanation of this observation might be the destruction of the chain of islands of the 13:1 secondary resonance between the short and long period component of the Trojan-like motion, caused possibly by the indirect perturbations of Saturn.     

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.     

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

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

Comparative Study of the 2:3 and 3:4 Resonant Motion with Neptune: An Application of Symplectic Mappings and Low Frequency Analysis

The 2:3 and 3:4 exterior mean motion resonances with Neptune are studied by applying symplectic mapping models. The mappings represent efficiently Poincaré maps for the 3D elliptic restricted three body problem in the neighbourhood of the particular resonances. A large number of trajectories is studied showing the coexistence of regular and chaotic orbits. Generally, chaotic motion depletes the small bodies of the effective resonant region in both the 2:3 and 3:4 resonances. Applying a low frequency spectral analysis of trajectories, we determined the phase space regions that correspond to either regular or chaotic motion. It is found that the phase space of the 3:4 resonant motion is more chaotic than the 2:3 one.     

A stability study of asteroid families near the 3:1 and 5:2 resonance with Jupiter

By using theD-criterion Lindblad (1992) has identified 14 asteroid families from a sample of 4100 numbered asteroids with proper elements from Milani and Kneevi (1990). Taxonomic types and other physical properties for a significant number of objects in five of the families show strong homogeneity within each family, further strengthening their internal relationship.To test the hypothesis of a common origin in, e.g., a catastrophic collision event, we have set out to integrate the orbits of the members of the Maria, Dora and Oppavia-Gefion families over some 10⁶ years. The mean distance for the Maria family is close to the 3:1 mean-motion resonance with Jupiter, while the other two families lie close to the 5:2 resonance.We used a simplified solar system model which included the perturbations by Jupiter and Saturn only and implemented Everhart's variable stepsize integrator RA15. All close encounters between the family members (within 0.1 AU) were recorded as well. Preliminary results from integrations over 4×10⁵ years are presented here.The statistics of close encounters show pronounced peaks for several members within each family, while for others no significant levels above the background of random encounters or even very low frequencies were found. This indicates a subclustering within the families. Quite a lot of very close (<0.005 AU) mutual encounters are found, which suggest that, at least for the larger members in a family, the mutual gravitational interactions could be of some importance for the real orbital evolutions.The encounter statistics between the Dora and Oppavia family members suggest a possible interrelationship between this two groups.     

Asteroid Observations at Low Phase Angles: II. 5 Astraea, 75 Eurydike, 77 Frigga, 105 Artemis, 119 Althaea, 124 Alkeste, and 201 Penelope

The results of photometric observations of seven main-belt asteroids with moderate surface albedos are presented. New magnitude-phase dependences were obtained for these asteroids: 5 Astraea (down to a phase angle of 0.3°, S type), 75 Eurydike (0.1°, M), 77 Frigga (0.9°, M), 105 Artemis (0.3°, C), 119 Althaea (0.3°, S), 124 Alkeste (0.1°, S), and 201 Penelope (0.5°, M). The parameters of approximating functions and the amplitudes of the opposition effect of these asteroids were determined. The obtained data allowed us to improve values of the rotation periods for some of them: 5 Astraea (16.815±0.002 h), 105 Artemis (18.56±0.01 h), 119 Althaea (11.466±0.001 h), and 124 Alkeste (9.907±0.001 h).     

Intermittent Trajectories in the 3/1 Jovian Resonance

The statistical behaviour of intermittent trajectories at the 3/1resonance is investigated. The elliptic planar restricted three-body problemis used as a model. Distribution functions for time intervals D betweeneccentricity bursts are obtained and theoretically interpreted. For smallervalues of D the distribution is found to be of Poisson type, while in itstail it is described by a power law D. This change in thedistribution occurs for values of D in the range10⁵10⁶Jupiter periods. The power-lawindexfor the integral distributions lies in the range (-2,-1) and istrajectory dependent. The algebraic decay in the tails of the distributionsis explained by the phenomenon of sticking of orbits to the chaos borderduring long intervals between bursts.     

Inflight Calibration of the NEAR Multispectral Imager: II. Results from Eros Approach and Orbit

During the Near-Earth Asteroid Rendezvous (NEAR) spacecraft's investigation of asteroid 433 Eros, inflight calibration measurements from the multispectral imager (MSI) have provided refined knowledge of the camera's radiometric performance, pointing, and light-scattering characteristics. Measurements while at Eros corroborate most earlier calibration results, although there appears to be a small, gradual change in instrument dark current and flat field due to effects of aging in the space environment. The most pronounced change in instrument behavior, however, is a dramatic increase in scattered light due to contaminants accumulated on the optics during unscheduled fuel usage in December 1998. Procedures to accurately quantify and to remediate the scattered light are described in a companion paper (Li et al. 2002, Icarus155, 00-00). Acquisition of Eros measurements has clarified the relative, filter-to-filter, radiometric performance of the MSI. Absolute radiometric calibration appears very well constrained from flight measurements, with an accuracy of ∼5%. Pointing relative to the spacecraft coordinate system can be determined from the temperature of the spacecraft deck with an accuracy of ∼1 pixel.     

Investigation of Thermal Inertia and Surface Properties for Near-earth Asteroid (162173) 1999 JU3

In order to obtain the substantial information about the surface physics and thermal property of the target asteroid (162173) 1999 JU3, which will be visited by Hayabusa 2 in a sample return mission, with the Advanced Thermal Physical Model (ATPM) we estimate the possible thermal inertia distribution over its surface, and infer the major material composition of its surface materials. In addition, the effective diameter and geometric albedo are derived to be D_eff = 1.13 ± 0.03 km, pv = 0.042 ± 0.003, respectively, and the average thermal inertia is estimated to be about (300 ± 50) J m^-2 s^-0.5 K^-1 According to the derived thermal inertia distribution, we infer that the major area on the surface of the target asteroid may be covered by loose materials, such as rock debris, sands, and so on, but few bare rocks may exist in a very small region. In this sense, the sample return mission of Hayabusa 2 is feasible, when it is performed successfully, it will certainly bring significant scientific information to the research of asteroids.     

A closer look at main belt asteroids 1: WF/PC images

We present new reconstructions of images of main belt Asteroids 9 Metis, 18 Melpomene, 19 Fortuna, 216 Kleopatra, and 624 Hektor, made with the uncorrected Wide-Field/Planetary Camera (WF/PC) on the Hubble Space Telescope (HST). Deconvolution with the MISTRAL algorithm demonstrates that these asteroids are clearly resolved. We determine diameters, albedos, and lower limits to axial ratios for these bodies. We also review the process used to restore the aberrated images. No surface features or companions are found, but the rotation of 216 Kleopatra is clearly seen. The asteroidal albedos are similar to those determined by other procedures.     

Near-Earth Asteroid 2005 CR37: Radar images and photometry of a candidate contact binary

Arecibo (2380 MHz, 13 cm) radar observations of 2005 CR37 provide detailed images of a candidate contact binary: a 1.8-km-long, extremely bifurcated object. Although the asteroid's two lobes are round, there are regions of modest topographic relief, such as an elevated, 200-m-wide facet, that suggest that the lobes are geologically more complex than either coherent fragments or homogeneous rubble piles. Since January 1999, about 9% of NEAs larger than ∼200 m imaged by radar can be described as candidate contact binaries.     

A Perturbative Treatment of The Co-Orbital Motion

We develop a formalism of the non-singular evaluation of the disturbing function and its derivatives with respect to the canonical variables. We apply this formalism to the case of the perturbed motion of a massless body orbiting the central body (Sun) with a period equal to that of the perturbing (planetary) body. This situation is known as the co-orbital motion, or equivalently, as the 1/1 mean motion commensurability. Jupiter's Trojan asteroids, Earth's co-orbital asteroids (e.g., (3753) Cruithne, (3362) Khufu), Mars' co-orbital asteroids (e.g., (5261) Eureka), and some Jupiter-family comets are examples of the co-orbital bodies in our solar system. Other examples are known in the satellite systems of the giant planets. Unlike the classical expansions of the disturbing function, our formalism is valid for any values of eccentricities and inclinations of the perturbed and perturbing body. The perturbation theory is used to compute the main features of the co-orbital dynamics in three approximations of the general three-body model: the planar-circular, planar-elliptic, and spatial-circular models. We develop a new perturbation scheme, which allows us to treat cases where the classical perturbation treatment fails. We show how the families of the tadpole, horseshoe, retrograde satellite and compound orbits vary with the eccentricity and inclination of the small body, and compute them also for the eccentricity of the perturbing body corresponding to a largely eccentric exoplanet's orbit.This revised version was published online in October 2005 with corrections to the Cover Date.     

Low-eccentricity motion of asteroids near the 2/1 Jovian resonance

(903) Nealley moves on an orbit of low eccentricity with a mean motion that is slightly larger than the 2/1 value of resonance. This orbit and some related fictious orbits are studied by numerical integrations of the four-body problem Sun-Jupiter-Saturn-asteroid over an interval of 110000 yr. The author's experience on related cases of resonance allows a study of the variation of suitably defined orbital parameters. The long-term evolution of the orbits is compared with earlier predictions. Some of the librating orbits are temporarily captured in a secondary resonance that refers to three-dimensional motion and is demonstrated by a special example.     

Periodic orbit families near the 4:1 jovian resonance

An enlarged averaged Hamiltonian is introduced to compute several families of periodic orbits of the planar elliptic 3-body problem, in the Sun–Jupiter–Asteroid system, near the 4:1 resonance. Four resonant critical point families are found and their stability is studied. The families of symmetric periodic orbits of the elliptic problem appear near the corresponding fixed points computed in this model. There is a good agreement for moderate eccentricity of the asteroid for three of these families, whereas the remaining family cannot be considered as a family of periodic orbits of the real model. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Dynamical Behavior of Asteroids Near Resonance: the 4:1 gap and the 7:2 Group

A comparative study of the evolution of the Sun–Jupiter–Asteroid system near the 4:1 and 7:2 resonances is performed by means of two techniques that proceed differently from the Hamiltonian corresponding to the planar restricted elliptic three-body problem. One technique is based on the classical Schubart averaging while the other is based on a mapping method in which the perturbing part of the Hamiltonian is expanded and the resulting terms are ordered according to a weight function that depends on the powers of eccentricities and the coefficients of the terms. For the mapping method the effect of Saturn on the asteroidal evolution is introduced and the degree of chaos is estimated by means of the Lyapunov time. Both methods are shown to lead to similar results and can be considered a suitable tool for describing the evolution of asteroids in the Kirkwood gap and the group corresponding to the 4:1 and 7:2 Jovian resonances, respectively. This revised version was published online in July 2006 with corrections to the Cover Date.