Numerical investigation of motions of resonant asteroids in the three-dimensional space

Motions of asteroids in mean motion resonances with Jupiter are studied in three-dimensional space. Orbital changes of fictitious asteroids in the Kirkwood gaps are calculated by numerical integrations for 10⁵ – 10⁶ years. The main results are as follows: (1) There are various motions of resonant asteroids, and some of them are very complicated and chaotic and others are regular. (2) The eccentricity of some asteroids becomes very large, and the variation of the inclination is large while the eccentricity is large. (3) In the 3:1 resonance, there is a long periodic change in the variation of the inclination, when (7 : ) is a simple ratio (7: longitude of perihelion, : longitude of node). (4) In the 7:3 resonance, the variation of the inclination of some resonant asteroids is so large that prograde motion becomes retrograde. Some asteroids in the 7:3 resonance can collide with the Sun as well as with the inner planets.     

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 orbital evolution of NEAs and the resonances with Jupiter

two near-earth-asteroids associated with resonances with Jupiter are studied over a time span of 10⁵ yrs. We found that asteroid (887) is temporary trapped in the 3:1 resonance; thus indicating that this resonance could be a source of short-lived NEAs. We also found that asteroid (3552) with a large eccentricity and a high inclination is wandering about the 1:1 resonant region.     

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.     

Meteoritical and dynamical constraints on the growth mechanisms and formation times of asteroids and Jupiter

Thermal models and radiometric ages for meteorites show that the peak temperatures inside their parent bodies were closely linked to their accretion times. Most iron meteorites come from bodies that accreted <0.5 Myr after CAIs formed and were melted by ²⁶Al and ⁶⁰Fe, probably inside 2 AU. Rare carbon-rich differentiated meteorites like ureilites probably also come from bodies that formed <1 Myr after CAIs, but in the outer part of the asteroid belt. Chondrite groups accreted intermittently from diverse batches of chondrules and other materials over a 4 Myr period starting 1 Myr after CAI formation when planetary embryos may already have formed at ∼1 AU. Meteorite evidence precludes accretion of late-forming chondrites on the surface of early-formed bodies; instead chondritic and non-chondritic meteorites probably formed in separate planetesimals. Maximum metamorphic temperatures in chondrite groups are correlated with mean chondrule age, as expected if ²⁶Al and ⁶⁰Fe were the predominant heat sources. Because late-forming bodies could not accrete close to large, early-formed bodies, planetesimal formation may have spread across the nebula from regions where the differentiated bodies formed. Dynamical models suggest that the asteroids could not have accreted in the main belt if Jupiter formed before the asteroids. Therefore Jupiter probably reached its current mass >3-5 Myr after CAIs formed. This precludes formation of Jupiter via a gravitational instability <1 Myr after the solar nebula formed, and strongly favors core accretion. Jupiter probably formed too late to make chondrules by generating shocks directly, or indirectly by scattering Ceres-sized bodies across the belt. Nevertheless, shocks formed by gravitational instabilities or Ceres-sized bodies scattered by planetary embryos may have produced some chondrules. The minimum lifetime for the solar nebula of 3-5 Myr inferred from the total spread of CAI and chondrule ages may exceed the median lifetime of 3 Myr for protoplanetary disks, but is well within the 1-10 Myr observed range. Shorter formation times for extrasolar planets may help to explain their unusual orbits compared to those of solar giant planets.     

Additional effects of resonance in Hilda-asteroid orbits by the combined action of Jupiter and Saturn

Inside the 3/2 mean motion resonance some Hilda-type orbits show effects of a three-body resonance that includes the frequency of the libration due to the 3/2 resonance. A graphical method presents numerical results for such orbits and demonstrates in 6 cases a process of temporary libration, that is ruled by the additional resonance together with the secular period of the eccentricities of Jupiter and Saturn.     

On the origin and evolution of Jupiter and Saturn

Arguments are presented which show that Kumar's (1977) contention that Jupiter and Saturn are unlikely to have formed from an initially extended state by contraction is not valid.     

Librations of central configurations and braided Saturn rings

We give a simple mathematical model for braided rings of a planet based on Maxwell's model for the rings of Saturn.     

Using wavelet approximation to study the influence of Jupiter on the orbital evolution of a distant satellite of Saturn

We suggest a nonstandard methodology for studying the influence of Jupiter on the secular orbital evolution of a distant satellite of Saturn. This influence is tangible only in short time spans near the times of the smallest separation between Jupiter and Saturn, i.e., when the heliocentric longitudes of the two planets coincide. These times are spaced about 20 years apart. To describe the jumplike behavior of perturbations, we suggest approximating the principal part of the perturbing function averaged over the satellite’s motion by a two-parameter exponential wavelet-type (burst) function. The subsequent averaging (smoothing) of the perturbing function allows us to eliminate the 20-year-period terms and obtain an approximate analytical solution in a special case of the problem. The results are illustrated by plots of the variations in the averaged perturbing function and the orbital eccentricity of Saturn’s outer satellite S/2000 S1, which is most strongly perturbed by Jupiter.     

The Mars 1:2 resonant population

An excess of around 400 asteroids in the distribution of the semimajor axes of the asteroids is identified by means of numerical integrations as generated by a population of approximately 1000 asteroids evolving inside the exterior resonance 1:2 with Mars. Approximately 200 asteroids are librating around the asymmetric libration centers and their evolution in a time-scale of 1 million years appears stable but with a strong influence of Mars' eccentricity. The biggest Mars 1:2 resonant asteroid is (142) Polana.     

The composition of M-type asteroids: Synthesis of spectroscopic and radar observations

We have conducted a radar-driven observational campaign of 22 main-belt asteroids (MBAs) focused on Bus–DeMeo Xc- and Xk-type objects (Tholen X and M class asteroids) using the Arecibo radar and NASA Infrared Telescope Facilities (IRTF). Sixteen of our targets were near-simultaneously observed with radar and those observations are described in a companion paper (Shepard, M.K., and 19 colleagues [2010]. Icarus, in press). We find that most of the highest metal-content asteroids, as suggested by radar, tend to exhibit silicate absorption features at both 0.9 and 1.9 μm, and the lowest metal-content asteroids tend to exhibit either no bands or only the 0.9 μm band. Eleven of the asteroids were observed at several rotational longitudes in the near-infrared and significant variations in continuum slope were found for nine in the spectral regions 1.1–1.45 μm and 1.6–2.3 μm. We utilized visible wavelength data (Bus, S.J., Binzel, R.P. [2002b]. Icarus 158, 146–177; Fornasier, S., Clark, B.E., Dotto, E., Migliorini, A., Ockert-Bell, M., Barucci, M.A. [2010]. Icarus 210, 655–673.) for a more complete compositional analysis of our targets. Compositional evidence is derived from our target asteroid spectra using two different methods: (1) a χ² search for spectral matches in the RELAB database, and (2) parametric comparisons with meteorites. This paper synthesizes the results of the RELAB search and the parametric comparisons with compositional suggestions based on radar observations. We find that for six of the seven asteroids with the highest iron abundances, our spectral results are consistent with the radar evidence (16 Psyche, 216 Kleopatra, 347 Pariana, 758 Mancunia, 779 Nina, and 785 Zwetana). Three of the seven asteroids with the lowest metal abundances, our spectral results are consistent with the radar evidence (21 Lutetia, 135 Hertha, 497 Iva). The remaining seven asteroids (22 Kalliope, 97 Klotho, 110 Lydia, 129 Antigone, 224 Oceana, 678 Fredegundis, and 771 Libera) have ambiguous compositional interpretations when comparing the spectral analogs to the radar analogs. The number of objects with ambiguous results from this multi-wavelength survey using visible, near-infrared, and radar wavelengths indicates that perhaps a third diagnostic wavelength region (such as the mid-infrared around 2–4 μm, the mid-infrared around 10–15 μm, and/or the ultraviolet around 0.2–0.4 μm) should be explored to resolve the discrepancies.     

The effect of dense cores on the structure and evolution of Jupiter and Saturn

We have calculated evolutionary and static models of Jupiter and Saturn with homogeneous solar composition mantles and dense cores of material consisting of solar abundances of SiO₂, MgO, Fe, and Ni. Evolutionary sequences for Jupiter were calculated with cores of mass 2, 4, 6, and 8% of the Jovian mass. Evolutionary sequences for Saturn were calculated with cores of mass 16, 18, 20, and 22% of total mass. Two envelope mixtures, representative of the solar abundances were used: X (mass fraction of hydrogen) = 0.74, Y (mass fraction of helium) = 0.24 and X = 0.77 and Y = 0.21. For Jupiter, the observations of the temperature at 1 bar pressure (T_1bar), radius and internal luminosity were best fit by evolutionary models with a core mass of ～6.5% and chemical composition of X = 0.77, Y = 0.21. The calculated cooling time for Jupiter is approximately 4.9 × 10⁹ years, which is consistent, within our error bars, with the known age of the solar system. For Saturn, the observations of the radius, internal luminosity and T_1BAR can be best fit by evolutionary models with a core mass of ～21% and chemical composition of X = 0.77, Y = 0.21. The cooling time calculated for Saturn is approximately 2.6 × 10⁹ years, almost a factor 2 less than the present age of the solar system. Static models of Jupiter and Saturn were calculated for the above chemical compositions in order to investigate the sensitivity of the calculated gravitational moments, J₂ and J₄, to the mass of the dense core, T_1BAR and hydrogen/helium ratio. We find for Jupiter that a model having a core mass of approximately 7% gives values of J₂, J₄, and T_1BAR that are within observational limits, for the mixture X = 0.77, Y = 0.21. The static Jupiter models are completely consistent with the evolutionary results. For Saturn, the quantities J₂, J₄, and J₆ determined from the static models with the most probable T_1BAR of 140°K, using modeling procedures which result in consistent models for Jupiter, are considerably below the observed values.     

SOS: the visible spectroscopic survey of 820 asteroids

We present the results of a visible spectroscopic survey of 820 asteroids carried on between November 1996 and September 2001 at the 1.52 m telescope at ESO (La Silla). The instrumental set-up allowed an useful spectral range of about 4900 Å<λ<9200 Å. The global spatial distribution of the observed asteroids covers quite well all the region between 2.2 and 3.3 AU though some concentrations are apparent. These are due to the fact that several sub-sets of asteroids, such as families and groups, have been selected and studied during the development of the survey. The observed asteroids have been classified using the Tholen and the Bus taxonomies which, in general, agree quite well.     

Stability of jets on Jupiter and Saturn

The linear stability of a zonal jet that decays with depth is investigated under the assumption that the thermal stratification is very small. A westerly cosine jet is found to be more stable than it is in a thin fluid shell with two-dimensional flow. This is in agreement with observations of Jupiter and Saturn, where jet curvature exceeds the barotropic stability criterion. This result constitutes an alternative hypothesis to that of Ingersoll and Pollard [Icarus 52 (1982) 62], who showed that deep jets extending through the interior are also more stable than thin shell jets. The flow regime assumed in the present work requires that a small stratification can exist and persist even in the presence of horizontal temperature gradients. Further work will be needed to test whether this is realistic.     

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.     

Origin and sustainability of the population of asteroids captured in the exterior resonance 1:2 with Mars

At present, approximately 1500 asteroids are known to evolve inside or sticked to the exterior 1:2 resonance with Mars at a ? 2.418 AU, being (142) Polana the largest member of this group. The effect of the forced secular modes superposed to the resonance gives rise to a complex dynamical evolution. Chaotic diffusion, collisions, close encounters with massive asteroids and mainly orbital migration due to the Yarkovsky effect generate continuous captures to and losses from the resonance, with a fraction of asteroids remaining captured over long time scales and generating a concentration in the semimajor axis distribution that exceeds by 20% the population of background asteroids. The Yarkovsky effect induces different dynamics according to the asteroid size, producing an excess of small asteroids inside the resonance. The evolution in the resonance generates a signature on the orbits, mainly in eccentricity, that depends on the time the asteroid remains captured inside the resonance and on the magnitude of the Yarkovsky effect. The greater the asteroids, the larger the time they remain captured in the resonance, allowing greater diffusion in eccentricity and inclination. The resonance generates a discontinuity and mixing in the space of proper elements producing misidentification of dynamical family members, mainly for Vesta and Nysa-Polana families. The half-life of resonant asteroids large enough for not being affected by the Yarkovsky effect is about 1 Gyr. From the point of view of taxonomic classes, the resonant population does not differ from the background population and the excess of small asteroids is confirmed.     

Rotational distortion on the surfaces of Jupiter and Saturn

Having formulated the Clairaut second-order differential equations up to the fourth order in superficial distortion due to Hensen's coefficients in the previous article (El-Sharawyet al., 1989 III, hereafter denotes by SM3), we are now in a position to solve them. In this paper we shall discuss the methods of solving the Clairaut theory, to give an explicit form about the distortion of the surfaces of Jupiter and Saturn, numerically up to the fourth-order.