Treatment of star catalog biases in asteroid astrometric observations

In this paper, we discuss the detection of systematic biases in star positions of the USNO A1.0, A2.0, and B1.0 catalogs, as deduced from the residuals of numbered asteroid observations. We present a technique for the removal of these biases, and validate this technique by illustrating the resulting improvements in numbered asteroid residuals, and by establishing that debiased orbits predict omitted observations more accurately than do orbits derived from non-debiased observations. We also illustrate the benefits of debiasing to high-precision astrometric applications such as asteroid mass determination and collision analysis, including a refined prediction of the impact probability of 99942 Apophis. Specifically, we find the IP of Apophis to be lowered by nearly an order of magnitude to 4.5 × 10⁻⁶ for the 2036 close approach.     

Time evolution of orbital uncertainties for the impactor candidate 2004 AS1

We evaluate the asteroid impact risk from the discovery night onwards using six-dimensional statistical orbit computation techniques to examine the a posteriori probability density of the orbital elements. Close to the discovery moment the observational data of an object are typically exiguous: the number of observations is very small and/or the covered orbital arc is very short. For such data, the covariance matrices computed in the linear approximation (e.g., with the least-squares technique) are known to fail to describe the uncertainties in the orbital parameters. The technique of statistical ranging gives us rigorous means to assess the orbital uncertainties already on the discovery night. To examine the time evolution of orbital uncertainties, we make use of a new nonlinear Monte Carlo technique of phase-space sampling using volumes of variation, which complements the ranging technique for exiguous data and the least-squares technique for extensive observational data. We apply the statistical techniques to the near-Earth Asteroid 2004 AS₁, which grabbed the attention of asteroid scientists because, for one day, it posed the highest and most immediate impact risk so far recorded. We take this extreme case to illustrate the ambiguities in the impact risk assessment for short arcs. We confirm that the weighted fraction of the collision orbits at discovery was large but conclude that this was mostly due to the discordance of the discovery-night observations. This case study highlights the need to introduce a regularization in terms of an a priori probability density to secure the invariance of the probabilistic analysis especially in the nonlinear orbital inversion for short arcs. We remark that a predominant role of the a priori can give indications of the feasibility of the probabilistic interpretation, that is, how reliable the results derived from the a posteriori probability density are. Nevertheless, the strict mathematical definition of, e.g., the collision probability remains valid, and our nonlinear statistical techniques give us the means to always deduce, at the very least, order-of-magnitude-estimates for the collision probability.     

Spectral observations of 19 weathered and 23 fresh NEAs and their correlations with orbital parameters

Results of our visible to near-infrared spectrophotometric observations of 41 near-Earth asteroids (NEAs) are reported. These moderate-resolution spectra, along with 14 previously published spectra from our earlier survey [Hicks, M.D., Fink, U., Grundy, W.M., 1998. Icarus 133, 69-78] show a preponderance of spectra consistent with ordinary chondrites (23 NEAs with this type of spectrum, along with 19 S-types and 13 in other taxonomic groups). There exists statistically significant evidence for orbit-dependent trends in our data. While S-type NEAs from our survey reside primarily in (1) Amor orbits or (2) Aten or Apollo orbits which do not cross the asteroid main-belt, the majority of objects with spectra consistent with ordinary chondrites in our survey are in highly eccentric Apollo orbits which enter the asteroid main-belt. This trend toward fresh, relatively unweathered NEAs with ordinary chondrite type spectra in highly eccentric Apollo orbits is attributed to one or a combination of three possible causes: (1) the chaotic nature of NEA orbits can easily result in high eccentricity orbits/large aphelion distances so that they can enter the collisionally enhanced environment in the main-belt, exposing fresh surfaces, (2) they have recently been injected into such orbits after a collision in the main-belt, or (3) such objects cross the orbits of several terrestrial planets, causing tidal disruption events that expose fresh surfaces.     

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.     

Dynamical causes of asymmetry in the arrangement of gaps in the asteroid belt

The centers of the gaps observed in the asteroid belt are displaced toward Jupiter from their positions that correspond to the exact commensurability between the mean motions of an asteroid and Jupiter. Using the current theory of stability and nonlinear oscillations of Hamiltonian systems, we point out the dynamical causes of this asymmetry. Our analysis is performed in terms of the plane circular restricted three-body problem. The orbits that correspond to Poincaré periodic solutions of the first kind are taken as unperturbed asteroid orbits.     

Equilibria and orbits around asteroid using the polyhedral model

In the current study, we use the polyhedral model to compute the potential of the asteroid. There are five equilibrium points in the gravitational field of the asteroid 283 Emma. We concluded that the zero-velocity surfaces and the equilibrium points change with the suppositive variation of the rotational speed of the asteroid. It is found that if the rotational speed equals a half as it is in present, the number of equilibrium points is also five. However, if the rotational speed equals twice as it is in present, there are only three equilibrium points left. Four different periodic orbits are calculated using the hierarchical grid searching method. We calculated characteristic multipliers of periodic orbits to investigate the stability of these periodic orbits. The orbit near the primary's equatorial plane is more likely to be stable when the separation/ primary-radius is a large number.     

Resonant orbits in the vicinity of asteroid 216 Kleopatra

This investigation examines the resonant orbits in the vicinity of asteroid 216 Kleopatra using a precise gravitational model, with emphasis on their crucial role in determining the global orbital behaviors. Three-dimensional Monte Carlo simulations of test particle trajectories are launched to find the condition and probability distribution of resonance. It is revealed the resonant orbits are rich and concentrated in the near-field regime, which provides a short-term mechanism to clear the vicinal ejecta away from the asteroid. The unstable boundary predicted in our calculations is consistent with the observed mutual orbits of satellites S/2008 (216) 1 and S/2008 (216) 2. The probability distribution of resonance is considered as an indicator of the stability of vicinal orbits, and the results are identical to the previous analysis by Scheeres et al. (Icarus 121:67, 1996) for the stability of retrograde orbits around asteroid 4769 Castalia.     

Model orbits of asteroid 3040 Kozai

The orbital evolutions of the asteroid 3040 Kozai and model asteroids with similar orbits have been investigated. Their osculating orbits for an epoch 1991 December 10 were numerically integrated forward within the interval of 20,000 years, using a dynamical model of the solar system consisting of all inner planets, Jupiter, and Saturn.The orbit of the asteroid Kozai is stable. Its motion is affected only by long-period perturbations of planets. With change of the argument of perihelion of the asteroid Kozai, the evolution of the model asteroid orbits changes essentially, too. The model orbits with the argument of perihelion changed by the order of 10% show that asteroids with such orbital parameters may approach the Earth orbit, while asteroids with larger changes may even cross it, at least after 10,000 years. Long-term orbital evolution of asteroids with these orbital parameters is very sensitive on their angular elements.     

Slow Diffusion and Effective Stability of Dust Particles Orbiting Asteroids

This paper presents quantitative estimates of orbital lifetimes for small dust particles in the vicinity of a spherical asteroid orbiting the Sun on a circular orbit. In particular, the modeling accounts for gravitational interactions with the asteroid and the Sun and radiation pressure from the Sun. The analysis focuses on slow drift of particles in a state-space neighborhood of weakly unstable periodic orbits corresponding to the most tightly bound orbits about the asteroid. Through a Hamiltonian normal-form computation, Nekhoroshev-type estimates are derived that yield regions in state space of effective stability, that is, regions in which the grains would remain over significant periods of time in the absence of other depletion and replenishment mechanisms.     

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 observability of radiation forces acting on near-Earth asteroids

We consider the perturbations on near-earth asteroid orbits due to various forces stemming from solar radiation. We find that the existence of precise radar astrometric observations at multiple apparitions, spanning periods on the order of 10 years, allows the detection of such forces on bodies as large as kilometer across. Indeed, the perturbations are so substantial that certain of the forces can be essential to fit an orbit to the observations. In particular, we show that the recoil force of thermal radiation from the asteroid, known as the Yarkovsky effect, is the most important of these unmodeled perturbations. We also show that the effect of reflected light can be important if even moderate albedo variations are present, while moderate changes in oblateness appear to have a far smaller effect. An unexpected result is that the Poynting–Robertson effect, typically only considered for submillimeter dust particles, could be observable on smaller asteroids with high eccentricity, such as 1566 Icarus. Finally, we also study the possibility of improving the orbit uncertainty through well-timed optical observations which might help in better detection of these nongravitational perturbations.     

The asteroid ephemerides program at the U.S. Naval Observatory

The U.S. Naval Observatory has begun a program of ephemeris improvement and reference frame determination from the main belt asteroids. The program is, currently, starting out with a limited set of observations of the larger asteroids to determine the equator and equinox corrections for the USNO W1J00 transit circle observations catalog, and, if possible, improve the orbits of these asteroids based on this limited set of observations. For this project, transit circle observations of the Sun and the planets Mercury through Jupiter, are also being used to determine the equator, equinox, and ephemeris corrections, the next goal is to improve the orbits of the larger asteroids in the optical reference frame using observation series that cover a much longer period of time. This will allow the exploration of the differences between the dynamical reference frame based on radar observations of main belt asteroids and its relation with the optical reference frame. Other goals include the exploration of the mass distribution in the main asteroid belt from high precision observations, and the effect of this mass on the ephemerides of the major planets.     

Orbit determination with very short arcs: II. Identifications

When the observational data are not enough to compute a meaningful orbit for an asteroid/comet we can represent the data with an attributable, i.e., two angles and their time derivatives. The undetermined variables range and range rate span an admissible region of Solar System orbits, which can be sampled by a set of Virtual Asteroids (VAs) selected by means of an optimal triangulation [Milani, A., Gronchi, G.F., de' Michieli Vitturi, M., Kne?evi?, Z., 2004. Celest. Mech. Dyn. Astron. 90, 59-87]. The attributable 4 coordinates are the result of a fit and they have an uncertainty, represented by a covariance matrix. Two short arcs of observations, represented by two attributables, can be linked by considering for each VA (in the admissible region of the first arc) the covariance matrix for the prediction at the time of the second arc, and by comparing it with the attributable of the second arc with its own covariance. By defining an identification penalty we can select the VAs allowing to fit together both arcs and compute a preliminary orbit. Two attributables may not be enough to compute an orbit with convergent differential corrections. Thus the preliminary orbit is used in a constrained differential correction, providing solutions along the Line Of Variation which can be used as second generation VAs to further predict the observations at the time of a third arc. In general the identification with a third arc will ensure a well determined orbit, to which additional sets of observations can be attributed. To test these algorithms we use a large scale simulation and measure the completeness, the reliability and the efficiency of the overall procedure to build up orbits by accumulating identifications. Under the conditions expected for the next generation asteroid surveys, the methods developed in this and in the preceding papers are efficient enough to be used as primary identification methods, with very good results. One important property is that the completeness in finding the possible identifications is as good for comparatively rare orbits, such as the ones of Near-Earth Objects, as for main belt orbits.     

Resonant motion in the restricted three body problem

The resonant structure of the restricted three body problem for the Sun- Jupiter asteroid system in the plane is studied, both for a circular and an elliptic orbit of Jupiter. Three typical resonances are studied, the 2 : 1, 3 : 1 and 4 : 1 mean motion resonance of the asteroid with Jupiter. The structure of the phase space is topologically different in these cases. These are typical for all other resonances in the asteroid problem. In each case we start with the unperturbed two-body system Sun-asteroid and we study the continuation of the periodic orbits when the perturbation due to a circular orbit of Jupiter is introduced. Families of periodic orbits of the first and of the second kind are presented. The structure of the phase space on a surface of section is also given. Next, we study the families of periodic orbits of the asteroid in the elliptic restricted problem with the eccentricity of Jupiter as a parameter. These orbits bifurcate from the families of the circular problem. Finally, we compare the above families of periodic orbits with the corresponding families of fixed points of the averaged problem. Different averaged Hamiltonians are considered in each resonance and the range of validity of each model is discussed.     

Problems of asteroid–comet hazard

The most relevant tasks associated with the problem of asteroid–comet hazard are discussed. A review is given of the respective research at the Institute of Applied Astronomy. The institute is currently implementing a project to create an information center whose tasks include the collection and processing of optical and radar observations of small solar system bodies, determination of their orbits, and assessment of the impact hazard of the newly discovered small bodies with respect to the earth, moon, and the other major planets and their satellites.     

On the residuals in asteroid catalogs and a possible nongravitational effect

Based on numerical calculations of the evolution of asteroid orbits, we concluded in one of our previous papers that a nongravitational effect causing the spatial separation of asteroids with different albedos and with sizes of tens of kilometers could act in the Solar system. To find new arguments, we have analyzed the residuals in the catalogs of asteroid orbits for 1996 and 2006. The dependences pointing to the possibility of a nongravitational effect for orbits with larger residuals have been found to be more distinct than those for orbits with smaller residuals. This result may be considered as an indirect confirmation of the above nongravitational effect.     

Asteroid identification at discovery

We present a novel method for the search of linkages among astrometric observations of asteroids, that is, tentative identifications among asteroids observed. Having two different master sets of asteroid observations each containing a number of separate subsets, we define a linkage as a pair of subsets residing in separate master sets that can be tied together with an orbit for given observational errors. To find linkages among a wealth of observations we use an efficient stepwise filtering approach. First, we start with what we call phase-space address comparison. The first step substantially reduces the initially huge amount of pairs by requiring that pairs to be subjected to further analysis have similar geocentric spherical coordinates at common epochs (for example, at three epochs). Second, we search for orbits for each of the selected pairs of subsets. Succeeding in the effort proves that a linkage exists. If there are contradictions among linkages found—for example, a single subset being linked to several mutually exclusive subsets—additional new or archive observations are usually needed to discard erroneous linkages. The new method is built on six-dimensional statistical orbital inversion (Ranging), and is therefore particularly suitable for analyzing objects with the shortest observational arcs, that is, newly discovered asteroids (and comets). Results from extensive and successful tests on simulated survey observations are presented and discussed. Theoretical and empirical scaling results show that the method is applicable to future large-scale surveys that will increase the rate of asteroid discovery by at least two orders of magnitude. The successful linking of faint single-night observation sets obtained with the Very Large Telescope are briefly reviewed.     

Analytical method for the effects of the asteroid belt on planetary orbits

Analytic expressions are derived for the perturbation of planetary orbits due to a thick constant density asteroid belt. The derivations include extensions and adaptations of Plakhov's analytic expressions for the perturbations in five of the orbital elements for closed orbits around Saturn's rings. The equations of Plakhov are modified to include the effect of ring thickness and additional equations are derived for the perturbations in the sixth orbital element, the mean anomaly. The gravitational potential and orbital perturbations are derived for the asteroid belt with and without thickness, and for a hoop approximation to the belt. The procedures are also applicable to Saturn's rings and the newly discovered rings of Uranus.The effects of the asteroid belt thickness on the gravitational potential coefficients and the orbital motions are demonstrated. Comparisons between the Mars orbital perturbations obtained using the analytic expressions and those obtained using numerical integration are discussed. The effects of the asteroid belt on the Earth based ranging to Mars are also demonstrated.     

Significance analysis of asteroid pairs

We have studied statistical significance of asteroid pairs residing on similar heliocentric orbits with distances (approximately the current relative encounter velocity between orbits) up to in the five-dimensional space of osculating elements. We found candidate pairs from the Hungaria zone through the entire main belt as well as outside the main belt, one among Hildas and one in the Cybele zone. We first determined probability that the candidate pairs are just coincidental couples from the background asteroid population. Those with estimated probability <0.3 were further investigated. In particular we computed synthetic proper elements for the relevant asteroids and used them to determine the three-dimensional distance of the members in candidate pairs. We consider small separation in the proper-element space as a signature of a real asteroid pair; conversely, cases with large separation in the proper-element space were rejected as spurious. Finally, we provide a list of candidate pairs that appear real, genetically related, to facilitate targeted studies, such as photometric and spectroscopic observations. As a by-product, we discovered six new compact clusters of three or more asteroids. Initial backward orbit integrations suggest that they are young families with ages <2 Myr.