Secular Dynamics of Asteroids in the Inner Solar System

In the last three years we have carried out numerical and semi-analytical studies on the secular dynamical mechanisms in the region (semimajor axis a < 2 AU) where the NEA orbits evolve. Our numerical integrations (over a time span of a few Myr) have shown that: (i) the linear secular resonances with both the inner and the outer planets may play an important role in the dynamical evolution of NEAs; (ii) the apsidal secular resonance with Mars could provide an important dynamical transport mechanism by which asteroids in the Mars-crossing region eventually achieve Earth-crossing orbits; (iii) in this region, due to the interaction with the terrestrial planets, the Kozai resonance can occur at small inclinations, with the argument of perihelion ω librating around 0° or 180°, providing a temporary protection mechanism against close approaches to the planets. The location of the linear secular resonances in this zone has also been obtained by an automatic procedure using a semi-numerical method valid for all values of the inclinations and eccentricities of the small bodies, and also in the case of libration of the argument of perihelion. A map of the secular resonances in the (a, i) plane shows — in agreement with the numerical integrations — that all the resonances with the terrestrial and giant planets are present, and also that some of them overlap. Thus the way is now open to fully take into account secular resonances in modelling the dynamical evolution of NEAs. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of Orbital Characteristics and Evolution of Near-Earth Asteroids (10302) 1989 ML and (4660) Nereus

Near-Earth asteroids (10302) 1989 ML and (4660) Nereus have attracted much attention as candidates for the next generation of deep space explorations. In the study, the maximum Lyapunov exponent (MLE) and MEGNO (Mean Exponential Growth factor of Nearby Orbits) index are calculated after considering the effects of major objects in the Solar system, and the stabilities of these two asteroids are discussed. For each asteroid, 1000 clonal particles consistent with the observational uncertainties are generated from a multivariate normal distribution. Statistical results display probably emerging regions of each asteroid within 0.1 million years, and provide distributions of occurrence times in the phase space of semi-major axis versus eccentricity. We estimate the probability of close encounters and collisions between the asteroid and Earth or other planets. Furthermore, secular resonances, Kozai resonance, and mean motion resonances are analyzed for nominal orbits of the two asteroids. We conclude that 1989 ML is in the region dominated by mean motion resonances with terrestrial planets. The probability of close encounters with them is relatively small, therefore its orbit is relatively stable. Nereus is located in a region that can have close-encounters with the Earth, and it has an extremely unstable orbit.     

近地小行星(10302) 1989 ML和(4660) Nereus的轨道特征及演化分析

近地小行星(10302) 1989 ML和(4660) Nereus作为下一代深空探测的候选目标一直备受关注. 在考虑太阳系主要天体的动力学背景下, 通过计算最大Lyapunov指数(MLE)及MEGNO (Mean Exponential Growth factor of Nearby Orbits)指数讨论它们的稳定性. 同时, 对每个小行星, 在其观测误差范围内按多元正态分布各选取1000个克隆粒子, 通过统计分析显示这两个小行星在10万年内可能的运动范围, 给出半长径-偏心率空间中的出现次数分布图, 并统计小行星与地球或其他大行星之间的密近交汇及碰撞的概率. 此外还对这两个小行星的标称轨道进行长期共振、Kozai共振及平运动共振的动力学分析. 综上得出结论, 1989 ML处在平运动共振主导的区域, 发生密近交汇的概率较小, 从而其轨道相对较稳定; 而Nereus处在地球的密近交汇区域, 轨道极不稳定.     

Orbit Determination of Binary Asteroids

In addition to the detection of an asteroid moon or a binary asteroid, the knowledge of the satellite’s true orbit is of high importance to derive fundamental physical parameters of the binary system such as its mass and to shed light on its possible formation history and dynamical evolution (prograde/retrograde orbit, large/small eccentricity or inclination, etc.). A new methodology for preliminary orbit determination of binary asteroids – and visual binaries in general – is proposed. It is based on Thiele–Innes method combined with a ‘trial and error’ Monte-Carlo technique. This method provides the full set of solutions (bundle of orbits, with the 7 orbital elements) even for a reduced number of observations. The mass is a direct by-product of this orbit determination, from which one can next infer the bulk-density and porosity. In addition to the bundle of orbits, the method provides the marginal probability densities of the foreseen parameters. Such error analysis – since it avoids linear approximation – can be of importance for the prediction of the satellite’s position in the plane-of-sky during future stellar occultations or subsequent observations, but also for the analysis of the orbit’s secular evolution. After briefly describing the method, we present the algorithm and its application to some practical cases, with particular emphasis on asteroids binaries and applications on orbital evolution.     

On the Instability of Jupiter's Trojans

We have numerically explored the mechanisms that destabilize Jupiter's Trojan orbits outside the stability region defined by Levison et al. (1997, Nature385, 42-44). Different models have been exploited to test various possible sources of instability on timescales on the order of ∼10⁸ years.In the restricted three-body model, only a few Trojan orbits become unstable within 10⁸ years. This intrinsic instability contributes only marginally to the overall instability found by Levison et al.In a model where the orbital parameters of both Jupiter and Saturn are fixed, we have investigated the role of Saturn and its gravitational influence. We find that a large fraction of Trojan orbits become unstable because of the direct nonresonant perturbations by Saturn. By shifting its semimajor axis at constant intervals around its present value we find that the near 5:2 mean motion resonance between the two giant planets (the Great Inequality) is not responsible for the gross instability of Jupiter's Trojans since short-term perturbations by Saturn destabilize Trojans, even when the two planets are far out of the resonance.Secular resonances are an additional source of instability. In the full six-body model with the four major planets included in the numerical integration, we have analyzed the effects of secular resonances with the node of the planets. Trojan asteroids have relevant inclinations, and nodal secular resonances play an important role. When a Trojan orbit becomes unstable, in most cases the libration amplitude of the critical argument of the 1:1 mean motion resonance grows until the asteroid encounters the planet. Libration amplitude, eccentricity, and nodal rate are linked for Trojan orbits by an algebraic relation so that when one of the three parameters is perturbed, the other two are affected as well. There are numerous secular resonances with the nodal rate of Jupiter that fall inside the region of instability and contribute to destabilize Trojans, in particular the ν₁₆. Indeed, in the full model the escape rate over 50 Myr is higher compared to the fixed model.Some secular resonances even cross the stability region delimited by Levison et al. and cause instability. This is the case of the 3:2 and 1:2 nodal resonances with Jupiter. In particular the 1:2 is responsible for the instability of some clones of the L4 Trojan (3540) Protesilaos.     

Numerical Simulation of the Orbital Evolution of Near-Earth Asteroids Close to Mean Motion Resonances

The dynamics of near-Earth asteroids near mean motion resonances with the Earth or other planets is considered. The probability domains of the motion of some near-Earth asteroids close to low-order resonances are presented. The investigations have been carried out by means of a numerical integration of differential equations, taking into account the perturbations from the major planets and the Moon. For each investigated object an ensemble of 100 test particles with orbital elements nearby those of the nominal orbit has been constructed and its evolution has been retraced over the time interval (–3000, +3000 years). The initial set of orbits has been generated on the basis of probable variations of the initial orbital elements obtained from the least square analysis of observations.This revised version was published online in October 2005 with corrections to the Cover Date.     

Meteor showers associated with 2003EH1

Using the Everhart radau19 numerical integration method, the orbital evolution of the near-Earth asteroid 2003EH1 is investigated. This asteroid belongs to the Amor group and is moving on a comet-like orbit. The integrations are performed over one cycle of variation of the perihelion argument ω. Over such a cycle, the orbit intersect that of the Earth at eight different values of ω. The orbital parameters are different at each of these intersections and so a meteoroid stream surrounding such an orbit can produce eight different meteor showers, one at each crossing. The geocentric radiants and velocities of the eight theoretical meteor showers associated with these crossing points are determined. Using published data, observed meteor showers are identified with each of the theoretically predicted showers. The character of the orbit and the existence of observed meteor showers associated with 2003EH1 confirm the supposition that this object is an extinct comet.     

A Simple Method for Numerical Calculations of the Evolution of Orbits of Near-Earth Asteroids

A simple method for numerical integration of the equations of motion of small bodies of the Solar System is proposed, which is especially efficient in studying the orbits with small perihelion distances. The evolution of orbits of 121 numbered asteroids with perihelion distances q < 1.2 AU is investigated over the time interval of years 2000–2100 with allowance made for the gravitational influence of nine planets and three largest asteroids. The circumstances of close encounters of asteroids with the Earth and other terrestrial planets are presented.     

Near-Sun asteroids

As follows from dynamical studies, in the course of evolution, most near-Earth objects reach orbits with small perihelion distances. Changes of the asteroids in the vicinity of the Sun should play a key role in forming the physical properties, size distribution, and dynamical features of the near-Earth objects. Only seven of the discovered asteroids are currently moving along orbits with perihelion distances q < 0.1 AU. However, due to the Kozai–Lidov secular perturbations, the asteroids, having recently passed near the Sun, could by now have moved to orbits farther from the Sun. In this study, we found asteroids that have been recently orbiting with perihelion distances q < 0.1 AU. Asteroids may be on such orbits for hundreds to tens of thousands of years. To carry out astrophysical observations of such objects is a high priority.     

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.     

Are There Many Inactive Jupiter-Family Comets among the Near-Earth Asteroid Population?

We analyze the dynamical evolution of Jupiter-family (JF) comets and near-Earth asteroids (NEAs) with aphelion distances Q>3.5 AU, paying special attention to the problem of mixing of both populations, such that inactive comets may be disguised as NEAs. From numerical integrations for 2×10⁶ years we find that the half lifetime (where the lifetime is defined against hyperbolic ejection or collision with the Sun or the planets) of near-Earth JF comets (perihelion distances q<1.3 AU) is about 1.5×10⁵ years but that they spend only a small fraction of this time (∼ a few 10³ years) with q<1.3 AU. From numerical integrations for 5×10⁶ years we find that the half lifetime of NEAs in “cometary” orbits (defined as those with aphelion distances Q>4.5 AU, i.e., that approach or cross Jupiter's orbit) is 4.2×10⁵ years, i.e., about three times longer than that for near-Earth JF comets. We also analyze the problem of decoupling JF comets from Jupiter to produce Encke-type comets. To this end we simulate the dynamical evolution of the sample of observed JF comets with the inclusion of nongravitational forces. While decoupling occurs very seldom when a purely gravitational motion is considered, the action of nongravitational forces (as strong as or greater than those acting on Encke) can produce a few Enckes. Furthermore, a few JF comets are transferred to low-eccentricity orbits entirely within the main asteroid belt (Q<4 AU and q>2 AU). The population of NEAs in cometary orbits is found to be adequately replenished with NEAs of smaller Q's diffusing outward, from which we can set an upper limit of ∼20% for the putative component of deactivated JF comets needed to maintain such a population in steady state. From this analysis, the upper limit for the average time that a JF comet in near-Earth orbit can spend as a dormant, asteroid-looking body can be estimated to be about 40% of the time spent as an active comet. More likely, JF comets in near-Earth orbits will disintegrate once (or shortly after) they end their active phases.     

Orbits of short period comets captured from the Oort cloud

Oort cloud comets occasionally obtain orbits which take them through the planetary region. The perturbations by the planets are likely to change the orbit of the comet. We model this process by using a Monte Carlo method and cross sections for orbital changes, i.e. changes in energy, inclination and perihelion distance, in a single planet-comet encounter. The influence of all major planets is considered. We study the distributions of orbital parameters of observable comets, i.e. those which have perihelion distance smaller than a given value. We find that enough comets are captured from the Oort cloud in order to explain the present populations of short period comets. The median value of cos i for the Jupiter family is 0.985 while it is 0.27 for the Halley types. The results may explain the orbital features of short period comets, assuming that the active lifetime of a comet is not much greater than 400 orbital revolutions.     

Orbits of short period comets captured from the Oort cloud

Oort cloud comets occasionally obtain orbits which take them through the planetary region. The perturbations by the planets are likely to change the orbit of the comet. We model this process by using a Monte Carlo method and cross sections for orbital changes, i.e. changes in energy, inclination and perihelion distance, in a single planet-comet encounter. The influence of all major planets is considered. We study the distributions of orbital parameters of observable comets, i.e. those which have perihelion distance smaller than a given value. We find that enough comets are captured from the Oort cloud in order to explain the present populations of short period comets. The median value of cos i for the Jupiter family is 0.985 while it is 0.27 for the Halley types. The results may explain the orbital features of short period comets, assuming that the active lifetime of a comet is not much greater than 400 orbital revolutions.     

Numerical studies of chaotic Hilda-type orbits

Earlier work indicates a comparatively rapid chaotic evolution of the orbits of some Hilda asteroids that move at the border of the domain occupied by the characteristic parameters of the objects at the 3/2 mean motion resonance. A simple Jupiter–Saturn model of the forces leads to numerical results on some of these cases and allows a search for additional resonances that can contribute to the chaotic evolution. In this context the importance of the secondary resonances that depend on the period of revolution of the argument of perihelion is pointed out. Among the studied additional resonances there are three-body resonances with arguments that depend on the mean longitudes of Jupiter, Saturn, and asteroid, but on slowly circulating angular elements of the asteroid as well, and the frequency of these arguments is close to a rational ratio with respect to the frequency of the libration due to the basic resonance.     

Asteroids in the Eccentrids meteor system

Among 11 673 of near-Earth objects (NEOs), 52 asteroids are identified, which, together with the Eccentrids meteor system, comprise a single population of small bodies of the Solar System with the smallest orbits of high eccentricity. Some features of this unique system of bodies are discussed in this paper. The distribution of perihelion longitudes is studied for the given group of asteroids and compared to that of the Aten asteroids, which are the most similar to the Eccentrids. The dependence is obtained of the character of perihelion longitude distribution on the eccentricities of the NEO orbits. Eight asteroid stream of the Eccentrids are found. The Eccentrids asteroids approaching the Earth’s orbit along its whole length in their aphelia can pose a certain hazard for the Earth.     

Hidden Mass in the Asteroid Belt

The total mass of the asteroid belt is estimated from an analysis of the motions of the major planets by processing high precision measurements of ranging to the landers Viking-1, Viking-2, and Pathfinder (1976-1997). Modeling of the perturbing accelerations of the major planets accounts for individual contributions of 300 minor planets; the total contribution of all remaining small asteroids is modeled as an acceleration caused by a solid ring in the ecliptic plane. Mass M_ring of the ring and its radius R are considered as solve-for parameters. Masses of the 300 perturbing asteroids have been derived from their published radii based mainly on measured fluxes of radiation, making use of the corresponding densities. This set of asteroids is grouped into three classes in accordance with physical properties and then corrections to the mean density for each class are estimated in the process of treating the observations. In this way an improved system of masses of the perturbing asteroids has been derived.The estimate M_ring≈(5±1)×10⁻¹⁰M_⊙ is obtained (M_⊙ is the solar mass) whose value is about one mass of Ceres. For the mean radius of the ring we have R≈2.80 AU with 3% uncertainty. Then the total mass M_belt of the main asteroid belt (including the 300 asteroids mentioned above) may be derived: M_belt≈(18±2)×10⁻¹⁰M_⊙. The value M_belt includes masses of the asteroids which are already discovered, and the total mass of a large number of small asteroids—most of which cannot be observed from the Earth. The second component M_ring is the hidden mass in the asteroid belt as evaluated from its dynamical impact onto the motion of the major planets.Two parameters of a theoretical distribution of the number of asteroids over their masses are evaluated by fitting to the improved set of masses of the 300 asteroids (assuming that there is no observational selection effect in this set). This distribution is extrapolated to the whole interval of asteroid masses and as a result the independent estimate M_belt≈18×10⁻¹⁰M_⊙ is obtained which is in excellent agreement with the dynamical finding given above.These results make it possible to predict the total number of minor planets in any unit interval of absolute magnitude H. Such predictions are compared with the observed distribution; the comparison shows that at present only about 10% of the asteroids with absolute magnitude H<14 have been discovered (according to the derived distribution, about 130,000 such asteroids are expected to exist).     

Meteor Showers Associated with the Taurid Complex Asteroids

Recent theoretical and observational work has shown that the asteroids belonging to the Taurid meteoroid complex have a cometary nature. If so, then they might possess related meteoroid streams producing meteor showers in the Earth atmosphere. We studied the orbital evolution of ten numbered Taurid complex asteroids by the Halphen-Goryachev method. It turned out that all of these asteroids are quadruple crossers relative to the Earth's orbit. Therefore their proposed meteoroid streams may in theory each produce four meteor showers. The theoretical orbital elements and geocentric radiants of these showers are determined and compared with the available observational data. The existence of the predicted forty meteor showers of the ten Taurid complex asteroids is confirmed by a search of the published catalogues of observed meteor shower radiants and orbits, and of the archives of the IAU Meteor Data Center (Lund). The existence of meteor showers associated with the Taurid Complex Asteroids confirms that, most likely, these asteroids are extinct comets. This revised version was published online in July 2006 with corrections to the Cover Date.     

The asteroid belt and its evolution

We have reinvestegated the suggestion that collisional fragmentation in the asteroid belt can account for its present luminosity function. We suggest, based on the usual Boltzmann-type equation for this process, that for the brightest asteroids the time scale for a catastropic collision is 1.2 × 10⁹yr. However, the assumption of molecular chaos is not valid in the asteroid belt and we demonstrate a new method to determine the necessary corrections. We then obtain, using the new procedure, a lower limit for a collision time. For the above sample it is 2 × 10¹¹yr. This, we believe, rules out collisional evolution of the asteroid belt since its formation. Finally, we also show histograms of eccentricity, inclination, absolute magnitude, height above the ecliptic plane, and argument of perihelion for the 2829 asteroids with well-determined orbits. This represents a synthesis of the numbered asteroid and PLS data.     

A former asteroidal planet as the origin of comets

The idea of a missing planet between Mars and Jupiter has been with us since the formulation of the Titius-Bode law. The discovery of the asteroid belt in that location led to speculation about a planetary breakup event. Both ideas remained conjectures until Ovenden's finding in 1972, from which it could be derived that the mass of the missing planet was about 90 Earth masses and that its breakup was astronomically recent. Apparently much of that mass was blown out of the solar system during the disruption of the planet. Because of the action of planetary perturbations, only two types of orbits of surviving fragments could remain at present-asteroid orbits and once-around very-long-period elliptical orbits. Objects in the latter type of orbit are known to exist-the very-long-period comets. A large number of these are on elliptical trajectories with periods of revolution of 5 million years; yet they are known to have made no more than one revolution in an orbit passing close to the Sun. By direct calculation it is possible to predict the distribution of the orbital elements of objects moving on long-period ellipses which might have originated in a breakup event in the asteroid belt 5 million years ago. The comet orbits have the predicted distribution in every case where a measure is possible. Some of the distribution anomalies, such as a bias in the directions of perihelion passage, are statistically strong and would be difficult to explain in any other uncontrived way. In addition, a relative deficiency of orbits with perihelia less than 1 AU indicates that the comets must have had small perihelion distances since their origin, rather than that they have been perturbed into small perihelion orbits from a distant “cloud” of comets by means of stellar encounters. The comet orbital data lead to the conclusion that all comets originated in a breakup event in the asteroid belt (5.5±0.6) × 10⁶ years ago. Asteroid and meteoritic evidence can now be interpreted in a way which not only is supportive but also provides fresh insights into understanding their physical, chemical, and dynamical properties. Particularily noteworthy are the young cosmic-ray exposure ages of meteorites, evidence of a previous high-temperature/pressure environment and of chemical differentiation of the parent body, and compositional similarities among comets, asteroids, and meteorites. Certain “explosion signatures” in asteroid orbital element distributions are likewise indicative. Tektites may also have originated in the same event; but if so, there are important implications regarding the absolute accuracy of certain geological dating methods. Little is known about possible planetary breakup mechanisms of the requisite type, though some speculations are offered. In any case, the asteroid belt is an existing fact; and the arguments presented here that a large planet did disintegrate 5 million years ago must be judged on their merits, even in the absence of a suitable theory of planetary explosions.     

The Flora Family: A Case of the Dynamically Dispersed Collisional Swarm?

Asteroid families are believed to originate by catastrophic disruptions of large asteroids. They are nowadays identified as clusters in the proper orbital elements space. The proper elements are analytically defined as constants of motion of a suitably simplified dynamical system. Indeed, they are generally nearly constant on a 10⁷-10⁸-year time scale. Over longer time intervals, however, they may significantly change, reflecting the accumulation of the tiny nonperiodic evolutions provided by chaos and nonconservative forces. The most important effects leading to a change of the proper orbital elements are (i) the chaotic diffusion in narrow mean motion resonances, (ii) the Yarkovsky nongravitational force, and (iii) the gravitational impulses received at close approaches with large asteroids. A natural question then arises: How are the size and shape of an asteroid family modified due to evolution of the proper orbital elements of its members over the family age? In this paper, we concentrate on the dynamical dispersion of the proper eccentricity and inclination, which occurs due to (i), but with the help of (ii) and (iii). We choose the Flora family as a model case because it is unusually dispersed in eccentricity and inclination and, being located in the inner main belt, is intersected by a large number of effective mean motion resonances with Mars and Jupiter. Our results suggest that the Flora family dynamically disperses on a few 10⁸-year time scale and that its age may be significantly less than 10⁹ years. We discuss the possibility that the parent bodies of the Flora family and of the ordinary L chondrite meteorites are the same object. In a broader sense, this work suggests that the common belief that the present asteroid families are simple images of their primordial dynamical structure should be revised.