Solar radiation pressure on (99942) Apophis

Near-Earth asteroid (99942) Apophis currently resides among the top positions on the list of objects with small, yet non-zero impact probability with the Earth. For that reason an unusual observational and theoretical effort has been dedicated to precisely characterize its future orbit. Here we discuss orbital perturbation of Apophis due to incident and reflected solar radiation pressure (SRP). We both revisit recent analytical estimate of the SRP effects for this body and also formulate a numerical approach allowing us to compute the SRP orbital perturbation under general assumptions. Contrary to some previous results, we show that SRP has a much smaller effect on the Apophis trajectory than does the thermal re-radiation force which produces the Yarkovsky effect. When the Yarkovsky effect becomes constrained enough in the future, our approach may be used to improve the orbit determination for this asteroid.     

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.     

Utilization of an H-reversal trajectory of a solar sail for asteroid deflection

Near Earth Asteroids have a possibility of impacting the Earth and always represent a threat. This paper proposes a way of changing the orbit of the asteroid to avoid an impact. A solar sail evolving in an H-reversal trajectory is utilized for asteroid deflection. Firstly, the dynamics of the solar sail and the characteristics of the H-reversal trajectory are analyzed. Then, the attitude of the solar sail is optimized to guide the sail to impact the target asteroid along an H-reversal trajectory. The impact...     

On the Possibility of Deflecting an Asteroid from Collision with the Earth Using the Kinetic Method

The possibilities of deflecting an asteroid from its collision course with the Earth by changing its velocity with an impact are considered. Using the asteroid Apophis as an example, the time dependence of the positions and sizes of the keyholes leading to collision is studied. It has been found that the possibility of deflecting this asteroid usually exists, and the impact can be accomplished in principle, given the capabilities of modern space technology. A change in the velocity should be performed before the encounter of 2029 in order to use the gravitational maneuver effect. The possible accuracy of determining Apophis’ orbit and the keyholes that lead to collision and are associated with the resonance returns are considered.     

The influence of the correlations between an asteroid’s orbital parameters on the estimation of the probability of planetary collision by the Monte Carlo method

The probability of an asteroid colliding with a planet can be estimated by the Monte Carlo method, in particular, through the statistical simulation of the possible initial conditions for the motion of an asteroid based on the probability density distribution set by the respective covariance matrix to be further projected with the orbital model onto the supposed time point of the collision. Hence, the collision probability is calculated as the ratio between the number of projected (virtual) asteroids striking the planet and their total number. The main problem is that different elements of the initial conditions (orbit or state vector) are correlated and, therefore, cannot be simulated independently. These correlations are reflected in the nondiagonal covariance matrix of the solution. The matrix is diagonalized by an orthogonal transformation. In the uncertainty domain constructed from the diagonal matrix elements, the initial values for each of the six orbital elements are simulated independently from the other elements, but with the accounting for their normal distribution. The program for calculating the normal distribution is based on the central limit theorem. Each sample of the initial values for the six orbital elements is transferred to the initial reference frame using an inverse transformation. Then, numerical integration is used to track the asteroid’s motion along the respective orbit to predict a possible impact event. Asteroids 99942 Apophis and 2007 WD5 are used as examples to show that disregarding the correlations when diagonalizing the covariance matrix to set the initial conditions may seriously distort the collision probability estimates. The paper gives the probabilities of the collisions of Apophis with the Earth and asteroid 2007 WD5 with Mars calculated by the author from observation sets showing nonzero collision probabilities. The author’s estimates are compared to those calculated by NASA.     

OpenOrb: Open‐source asteroid orbit computation software including statistical ranging

Abstract— We are making an open‐source asteroid orbit computation software package called OpenOrb publicly available. OpenOrb is built on a well‐established Bayesian inversion theory, which means that it is to a large part complementary to orbit‐computation packages currently available. In particular, OpenOrb is the first package that contains tools for rigorously estimating the uncertainties resulting from the inverse problem of computing orbital elements using scarce astrometry. In addition to the well‐known least‐squares method, OpenOrb also contains both Monte‐Carlo (MC) and Markov‐Chain MC (MCMC; Oszkiewicz et al. [2009]) versions of the statistical ranging method. Ranging allows the user to obtain sampled, non‐Gaussian orbital‐element probability‐density functions and is therefore optimized for cases where the amount of astrometry is scarce or spans a relatively short time interval. Ranging‐based methods have successfully been applied to a variety of different problems such as rigorous ephemeris prediction, orbital element distribution studies for transneptunian objects, the computation of invariant collision probabilities between near‐Earth objects and the Earth, detection of linkages between astrometric asteroid observations within an apparition as well as between apparitions, and in the rigorous analysis of the impact of orbital arc length and/or astrometric uncertainty on the uncertainty of the resulting orbits. Tools for making ephemeris predictions and for classifying objects based on their orbits are also available in OpenOrb. As an example, we use OpenOrb in the search for candidate retrograde and/or high‐inclination objects similar to 2008 KV₄₂ in the known population of transneptunian objects that have an observational time span shorter than 30 days.     

Possible potentially threatening co‐orbiting material of asteroid 2000EE104 identified through interplanetary magnetic field disturbances

Near‐Earth objects (NEOs) with diameters of <300 m are difficult to detect from the Earth with radar or optical telescopes unless and until they approach closely. If they are on collisional courses with the Earth, there is little that can be done to mitigate the considerable damage. Although destructive collisions in space are rare for 1 km diameter bodies and above, once hit by a sizeable impactor, such a NEO can develop a relatively dense cloud of co‐orbiting material in which destructive collisions are relatively frequent. The gas and nanoscale dust released in the destructive collisions can be detected remotely by downstream spacecraft equipped with magnetometers. In this paper, we use such magnetic disturbances to identify regions of near‐Earth space in which high densities of small objects are present. We find that asteroid (138175) 2000EE104 currently may have a cloud of potentially threatening co‐orbiting material. Due to the scattered co‐orbitals, there can be a finite impact probability whenever the Earth approaches the orbit of asteroid 2000EE104, regardless of the position of the asteroid itself.     

An Asteroid in a Earth-like Orbit

The dynamical evolution of an asteroid with orbital elements strikingly similar to the Earth is analysed. The object, 1991 VG, was discovered by the Spacewatch telescope during a particular encounter with the planet. 1991 VG experienced a temporary satellite capture by the Earth, a phenomena that is recurrent in its dynamical history. The possible origin of this puzzling object is discussed, including the suggestion that 1991 VG could be a piece of lunar ejecta after a great impact. This revised version was published online in July 2006 with corrections to the Cover Date.     

How selection and weighting of astrometric observations influence the impact probability. The case of asteroid (99942) Apophis

The aim of this paper is to show that in the case of a low probability of asteroid collision with the Earth, the appropriate selection and weighting of the data are crucial for the impact investigation and for analysing the impact possibilities using extensive numerical simulations. By means of the Monte Carlo special method, a large number of 'clone' orbits have been generated. A full range of orbital elements in the six-dimensional parameter space, that is, in the entire confidence region allowed by the observational material, has been examined. On the basis of 1000 astrometric observations of (99942) Apophis, the best solutions for the geocentric encounter distance of  6.065 ± 0.081 R_⊕  (without perturbations by asteroids) or  6.064 ± 0.095 R_⊕  (including perturbations by the four largest asteroids) were derived for the close encounter with the Earth on 2029 April 13. The present uncertainties allow for special configurations ('keyholes') during this encounter that may lead to very close encounters in future approaches of Apophis. Two groups of keyholes are connected with the close encounter with the Earth in 2036 (within the minimal distance of  5.7736−5.7763 R_⊕  on 2029 April 13) and 2037 (within the minimal distance of  6.3359–6.3488 R_⊕  ). The nominal orbits for our most accurate models run almost exactly in the middle of these two impact keyhole groups. A very small keyhole for the impact in 2076 has been found between these groups at the minimal distance of 5.97347   R_⊕  . This keyhole is close to the nominal orbit. The present observations are not sufficiently accurate to eliminate definitely the possibility of impact with the Earth in 2036 and for many years after.     

Discovery of an asteroid and quasi‐satellite in an Earth‐like horseshoe orbit

Abstract— The newly discovered asteroid 2002 AA₂₉ moves in a very Earth‐like orbit that relative to Earth has a unique horseshoe shape and allows transitions to a quasi‐satellite state. This is the first body known to be in a simple heliocentric horseshoe orbit, moving along its parent planet's orbit. It is similarly also the first true co‐orbital object of Earth, since other asteroids in 1:1 resonance with Earth have orbits very dissimilar from that of our planet. When a quasi‐satellite, it remains within 0.2 AU of the Earth for several decades. 2002 AA₂₉ is the first asteroid known to exhibit this behavior. 2002 AA₂₉ introduces an important new class of objects offering potential targets for space missions and clues to asteroid orbit transfer evolution.     

Impact orbits of the asteroid 2009 FJ with the Earth

We show how to calculate the impact orbits of dangerous asteroids using the freely available the OrbFit software, and compare our results with impact orbits calculated using Sitarski??s independent software (Sitarski, 1999; 2000; 2006). The new method is tested on asteroid 2009 FJ. Using the OrbFit package to integrate alternate orbits along the line of variation (Milani et al., 2002; 2005a; 2005b), we identify impact orbits and can plot paths of risk for the Earth or any other body in the Solar System. We present the orbital elements of asteroid 2009 FJ and its ephemerides, along with uncertainties, for the next 100 years. This paper continues a long-term research program on impact solutions for asteroids (Wlodarczyk, 2007; 2008; 2009).     

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.     

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.     

How precise is the orbit of asteroid (99942) Apophis and how probable is its collision with the Earth in 2036–2037?

The nearest in time close approach of potentially hazardous asteroid (99942) Apophis with the Earth will take place on April 13, 2029, when the minimum distance of the asteroid from the Earth’s center will be as small as 38 000 km. Such a close approach will result in substantial transformation of the asteroid’s orbit. The value of the perturbations depends on the minimum distance between the bodies during the approach. Among possible transformations of the orbit are those which result in new dangerous approaches and even in probable Apophis collisions with the Earth starting from 2036. At present, at least four solutions are known for the Apophis orbit which were obtained using all radar and most of available optical observations. The procedures of assigning weights to conditional equations and the models of the asteroid’s motion have differed to some extent when finding these solutions. Of considerable interest is the comparison of the found orbital parameters with the estimates of their accuracy, since small distinctions in their values result in considerable distinctions in the forecast of Apophis’ motion after 2029 and beyond. It is shown in the paper that the estimates of the probability of an Apophis collision with the Earth in 2036 differ by some orders of magnitude, according to various solutions. The influence of factors which were disregarded in the models of motion even more increases the uncertainty in forecasting the motion after 2029. More accurate forecasting can be achieved as a result of additional optical and, to a greater extent, a series of radar observations in 2013 and then in 2020–2021, and/or as a result of processing radio signals of the transmitter delivered to the Apophis surface or to the orbit of its artificial satellite, as it was proposed in a number of papers.     

Influence of the largest asteroids on the orbital motions of terrestrial planets: application to the Earth and Mars

The level of precision of modern numerical ephemeris of the Solar System necessitates taking into account the gravitational influence of the largest asteroids on the terrestrial planets. This can be done in a straightforward manner when assuming that the mass of the asteroid is well known. Nevertheless, this is rarely the case, even for the largest asteroids. In this paper, we use recent determinations of the masses of Ceres, Pallas, and Vesta to both qualitatively and quantitatively determine the action of these asteroids on the orbital parameters of the Earth and Mars. This is done by the numerical integration by comparing the orbital motions of the perturbed planet when adding or not the perturbing asteroid to the classical 9 bodies problem (the Sun + the eight planets). Some preliminary results are discussed. Published in Russian in Astronomicheskii Vestnik, 2009, Vol. 43, No. 1, pp. 83–86. The text was submitted by the autors in English.     

The Impact Effect Callculator of Celestial Body Impacts to the Earth: The Constructor of Hazardous Orbits

Quick assessment of hazardous effects from impacts of large celestial bodies is achieved through the development of a new consequence calculator. A distinctive feature of this calculator is a new block, the Hazardous-Orbit Constructor, which simulates the conditions of entry of a celestial body into the Earth’s atmosphere and determines the orbital parameters of the body based on given atmospheric entry conditions. This block is used to simulate the atmospheric entry conditions of known asteroids and meteoroids and to determine the orbital parameters of known bolides leading to meteorite fall events. For the case of asteroid 2008 TC3 and the P?ibram meteorite, it is shown that within the potential impact area of the celestial body, the atmospheric entry angle may vary considerably.     

Geometric characterization of the Arjuna orbital domain

Arjuna‐type orbits are characterized by being Earth‐like, having both low‐eccentricity and low‐inclination. Objects following these trajectories experience repeated trappings in the 1:1 commensurability with the Earth and can become temporary Trojans, horseshoe librators, quasi‐satellites, and even transient natural satellites. Here, we review what we know about this peculiar dynamical group and use a Monte Carlo simulation to characterize geometrically the Arjuna orbital domain, studying its visibility both from the ground and with the European Space Agency Gaia spacecraft. The visibility analysis from the ground together with the discovery circumstances of known objects are used as proxies to estimate the current size of this population. The impact cross‐section of the Earth for minor bodies in this resonant group is also investigated. We find that, for ground‐based observations, the solar elongation at perigee of nearly half of these objects is less than 90°. They are best observed by space‐borne telescopes, but Gaia is not going to improve significantly the current discovery rate for members of this class. Our results suggest that the size of this population may have been underestimated by current models. On the other hand, their intrinsically low encounter velocities with the Earth induce a 10–1000‐fold increase in the impact cross‐section with respect to what is typical for objects in the Apollo or Aten asteroid populations. We estimate that their probability of capture as transient natural satellites of our planet is about 8 %. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Orbital YORP and asteroid orbit evolution, with application to Apophis

Photon thrust from shape alone can produce quasi-secular changes in an asteroid's orbital elements. An asteroid in an elliptical orbit with a north–south shape asymmetry can steadily alter its elements over timescales longer than one orbital trip about the Sun. This thrust, called here orbital YORP (YORP = Yarkovsky–O'Keefe–Radzievskii–Paddack), operates even in the absence of thermal inertia, which the Yarkovsky effects require. However, unlike the Yarkovsky effects, which produce secular orbital changes over millions or billions of years, the change in an asteroid's orbital elements from orbital YORP operates only over the precession timescale of the orbit or of the asteroid's spin axis; this is generally only thousands or tens of thousands of years. Thus while the orbital YORP timescale is too short for an asteroid to secularly journey very far, it is long enough to warrant investigation with respect to 99942 Apophis, which might conceivably impact the Earth in 2036. A near-maximal orbital YORP effect is found by assuming Apophis is without thermal inertia and is shaped like a hemisphere, with its spin axis lying in the orbital plane. With these assumptions orbital YORP can change its along-track position by up to ±245 km, which is comparable to Yarkovsky effects. Though Apophis' shape, thermal properties, and spin axis orientation are currently unknown, the practical upper and lower limits are liable to be much less than the ±245 km extremes. Even so, the uncertainty in position is still likely to be much larger than the 0.5 km “keyhole” Apophis must pass through during its close approach in 2029 in order to strike the Earth in 2036.     

Earth impact probability of the Asteroid (25143) Itokawa to be sampled by the spacecraft Hayabusa

The Japanese spacecraft Hayabusa is planed to reach the Asteroid Itokawa in September 2005, and to bring back some samples of its surface to Earth in 2007. We have studied the future possible evolution of this asteroid by integrating numerically over 100 Myr a set of 39 initially indistinguishable orbits (clones), obtained either by small variations of the nominal initial conditions, or by using different computers (introducing different round-off errors). The results indicate that an Earth impact of this 500-m-size asteroid is likely within a million years, which is only a factor of four larger than the average impact frequency of asteroids of this size. The mission Hayabusa may thus sample a good candidate for being among the next 500-m-size Earth impactors.