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.     

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 two-body problem with drag and radiation pressure

The Kepler problem including radiation pressure and drag is treated. The equation of the orbit is derived and the scalar and vector integrals of motion are obtained by direct operation on the vector form of the equation of motion.     

Simulations of impact ejecta and regolith accumulation on Asteroid Eros

We have carried out a set of Monte Carlo simulations of the placement of impact ejecta on Asteroid 433 Eros, with the aim of understanding the distribution and accumulation of regolith. The simulations consisted of two stages: (1) random distribution of primary impact sites derived from a uniform isotropic flux of impactors, and (2) integration of the orbits of test particle ejecta launched from primary impact points until their re-impact or escape. We integrated the orbits of a large number of test particles (typically 10⁶ per individual case). For those particles that did not escape we collected the location of their re-impact points to build up a distribution on the asteroid surface. We find that secondary impact density is mostly controlled by the overall topography of the asteroid. A gray-scale image of the density of secondary ejecta impact points looks, in general, like a reduced-scale negative of the topography of the asteroid's surface. In other words, regolith migration tends to fill in the topography of Eros over time, whereas topographic highs are denuded of free material. Thus, the irregular shape of Eros is not a steady-state configuration, but the result of larger stochastic events.     

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.     

Some results from the National Observatory of Turkey, Kazan State University, and Nikolaev Astronomical Observatory on small bodies of the solar system

A scientific collaboration between TÜB?TAK National Observatory (Turkey), Kazan State University (Russia) and Nikolaev Astronomical Observatory (Ukraine) involves observations of minor planets and near-Earth asteroids (NEAs) with the 1.5 m Russian-Turkish telescope (RTT150). Regular observations of selected asteroids in the range of 11-18 magnitudes began in 2004 with the view of determining masses of selected asteroids, improving the orbits of the NEAs, and studying physical characteristics of selected asteroids from photometric observations. More than 3000 positions of 53 selected asteroids and 11 NEAs have been obtained with an internal error in the range of 30-300 mas for a single determination. Photometric reductions of more than 4000 CCD frames are in progress. Masses of 21 asteroids were estimated through dynamical method using the ground-based optical observations, mainly from the RTT150 and Minor Planet Center. A comparison of the observational results from the RTT150 in 2004-2005 with observations of the same objects at other observatories allows us to conclude that RTT150 can be used for ground-based support in astrometry for the space mission GAIA.     

Impact evolution of asteroid shapes: 1. random mass redistribution

We explore whether the cumulative effect of small-scale meteoroid bombardment can drive asteroids into nonaxisymmetric shapes comparable to those of known objects (elongated prolate forms, twin-lobed binaries, etc). We simulate impact cratering as an excavation followed by the launch, orbit, and reimpact of ejecta. Orbits are determined by the gravity and rotation of the evolving asteroid, whose shape and spin change as cratering occurs repeatedly. For simplicity we consider an end-member evolution where impactors are all much smaller than the asteroid and where all ejecta remain bound. Given those assumptions, we find that cumulative small impacts on rotating asteroids lead to oblate shapes, irrespective of the chosen value for angle of repose or for initial angular momentum. The more rapidly a body is spinning, the more flattened the outcome, but oblateness prevails. Most actual asteroids, by contrast, appear spherical to prolate. We also evaluate the timescale for reshaping by small impacts and compare it to the timescale for catastrophic disruption. For all but the steepest size distributions of impactors, reshaping from small impacts takes more than an order of magnitude longer than catastrophic disruption. We conclude that small-scale cratering is probably not dominant in shaping asteroids, unless our assumptions are naive. We believe we have ruled out the end-member scenario; future modeling shall include angular momentum evolution from impacts, mass loss in the strength regime, and craters with diameters up to the disruption threshold. The ultimate goal is to find out how asteroids get their shapes and spins and whether tidal encounters in fact play a dominant role.     

Dynamical erosion of the asteroid belt and implications for large impacts in the inner Solar System

The cumulative effects of weak resonant and secular perturbations by the major planets produce chaotic behavior of asteroids on long timescales. Dynamical chaos is the dominant loss mechanism for asteroids with diameters in the current asteroid belt. In a numerical analysis of the long-term evolution of test particles in the main asteroid belt region, we find that the dynamical loss history of test particles from this region is well described with a logarithmic decay law. In our simulations the loss rate function that is established at persists with little deviation to at least . Our study indicates that the asteroid belt region has experienced a significant amount of depletion due to this dynamical erosion—having lost as much as ∼50% of the large asteroids—since 1 Myr after the establishment of the current dynamical structure of the asteroid belt. Because the dynamical depletion of asteroids from the main belt is approximately logarithmic, an equal amount of depletion occurred in the time interval 10-200 Myr as in 0.2-4 Gyr, roughly ∼30% of the current number of large asteroids in the main belt over each interval. We find that asteroids escaping from the main belt due to dynamical chaos have an Earth-impact probability of ∼0.3%. Our model suggests that the rate of impacts from large asteroids has declined by a factor of 3 over the last 3 Gyr, and that the present-day impact flux of objects on the terrestrial planets is roughly an order of magnitude less than estimates currently in use in crater chronologies and impact hazard risk assessments.     

A dynamical solution of the triple asteroid system (45) Eugenia

We present the first dynamical solution of the triple asteroid system (45) Eugenia and its two moons Petit–Prince (diameter ∼ 7 km) and S/2004 (45) 1 (diameter ∼ 5 km). The two moons orbit at 1165 and 610 km from the primary, describing an almost-circular orbit (e ∼ 6 × 10⁻³ and e ∼ 7 × 10⁻² respectively). The system is quite different from the other known triple systems in the main belt since the inclinations of the moon orbits are sizeable (9° and 18° with respect to the equator of the primary respectively). No resonances, neither secular nor due to Lidov–Kozai mechanism, were detected in our dynamical solution, suggesting that these inclinations are not due to excitation modes between the primary and the moons. A 10-year evolution study shows that the orbits are slightly affected by perturbations from the Sun, and to a lesser extent by mutual interactions between the moons. The estimated J₂ of the primary is three times lower than the theoretical one, calculated assuming the shape of the primary and an homogeneous interior, possibly suggesting the importance of other gravitational harmonics.     

Asteroid mass determination with the Gaia mission: A simulation of the expected precisions

The ESA astrometric mission Gaia, due for a launch in late 2011, will observe a huge number of asteroids (∼350,000 brighter than V<20) with an unprecedented positional precision (at the sub-milliarcsecond level). This precision will play an important role for the mass determination of about hundred minor planets with a relative precision better than 50%. Presently, due primarily to their perturbations on Mars, the uncertainty in the masses of the largest asteroids is the limiting factor in the accuracy of the solar system ephemerides. Besides, such high precision astrometry will enable to derive direct measurements of the masses of the largest asteroids which are of utmost significance for the knowledge of their physical properties. The method for computing the masses is based on the analysis of orbital perturbations during close encounters between massive asteroids (perturbers) and several smaller minor planets (targets). From given criteria of close approaches selection, we give the list of asteroids for which the mass can be determined, and the expected precision of these masses at mission completion. We next study the possible contribution of the ground-based observations for the mass determination in some special observation cases of close approaches.     

Dynamics of the Hungaria asteroids

To try to understand the dynamical and collisional evolution of the Hungaria asteroids we have built a large catalog of accurate synthetic proper elements. Using the distribution of the Hungaria, in the spaces of proper elements and of proper frequencies, we can study the dynamical boundaries and the internal structure of the Hungaria region, both within a purely gravitational model and also showing the signature of the non-gravitational effects. We find a complex interaction between secular resonances, mean motion resonances, chaotic behavior and Yarkovsky-driven drift in semimajor axis. We also find a rare occurrence of large scale instabilities, leading to escape from the region. This allows to explain the complex shape of a grouping which we suggest is a collisional family, including most Hungaria but by no means all; we provide an explicit list of non-members of the family. There are finer structures, of which the most significant is a set of very close asteroid couples, with extremely similar proper elements. Some of these could have had, in a comparatively recent past, very close approaches with low relative velocity. We argue that the Hungaria, because of the favorable observing conditions, may soon become the best known sub-group of the asteroid population.     

The effect of solar radiation pressure on the Lagrangian points in the elliptic restricted three-body problem

In this paper the effect of solar radiation pressure on the location and stability of the five Lagrangian points is studied, within the frame of elliptic restricted three-body problem, where the primaries are the Sun and Jupiter acting on a particle of negligible mass. We found that the radiation pressure plays the rule of slightly reducing the effective mass of the Sun and changes the location of the Lagrangian points. New formulas for the location of the collinear libration points were derived. For large values of the force ratio β, we found that at β=0.12, the collinear point L₃ is stable and some families of periodic orbits can be drawn around it.     

Low-speed impacts between rubble piles modeled as collections of polyhedra, 2

We present the results of additional calculations involving the collisions of km-scale rubble piles. In new work, we used the Open Dynamics Engine (ODE), an open-source library for the simulation of rigid-body dynamics that incorporates a sophisticated collision-detection and resolution routine. We found that using ODE resulted in a speed-up of approximately a factor of 30 compared with previous code. In this paper we report on the results of almost 1200 separate runs, the bulk of which were carried out with 1000-2000 elements. We carried out calculations with three different combinations of the coefficients of friction η and (normal) restitution ?: low (η=0,?=0.8), medium (η=0,?=0.5), and high (η=0.5,?=0.5) dissipation.For target objects of ∼1 km in radius, we found reduced critical disruption energy values in head-on collisions from 2 to 100 J kg⁻¹ depending on dissipation and impactor/target mass ratio. Monodisperse objects disrupted somewhat more easily than power-law objects in general. For oblique collisions of equal-mass objects, mildly off-center collisions (b/b₀=0.5) seemed to be as efficient or possibly more efficient at collisional disruption as head-on collisions. More oblique collisions were less efficient and the most oblique collisions we tried (b/b₀=0.866) required up to ∼200 J kg⁻¹ for high-dissipation power-law objects. For calculations with smaller numbers of elements (total impactor or 200 elements) we found that collisions were more efficient for smaller numbers of more massive elements, with values as low as for low-dissipation cases. We also analyzed our results in terms of the relations proposed by Stewart and Leinhardt [Stewart, S.T., Leinhardt, Z.M., 2009. Astrophys. J. 691, L133-L137] where where Q_R is the impact kinetic energy per unit total mass m_i+m_T. Although there is a significant amount of scatter, our results generally bear out the suggested relation.     

Radar observations and the shape of near-Earth asteroid 2008 EV5

We observed the near-Earth ASTEROID 2008 EV5 with the Arecibo and Goldstone planetary radars and the Very Long Baseline Array during December 2008. EV5 rotates retrograde and its overall shape is a 400 ± 50 m oblate spheroid. The most prominent surface feature is a ridge parallel to the asteroid’s equator that is broken by a concavity about 150 m in diameter. Otherwise the asteroid’s surface is notably smooth on decameter scales. EV5’s radar and optical albedos are consistent with either rocky or stony-iron composition. The equatorial ridge is similar to structure seen on the rubble-pile near-Earth asteroid (66391) 1999 KW4 and is consistent with YORP spin-up reconfiguring the asteroid in the past. We interpret the concavity as an impact crater. Shaking during the impact and later regolith redistribution may have erased smaller features, explaining the general lack of decameter-scale surface structure.     

Effects of thermal radiation on the dynamics of binary NEAs

The Yarkovsky force, produced when thermal radiation is re-emitted asymmetrically, causes significant orbital evolution of asteroids in the 10 m-10 km size range. When acting on a non-spherical body, the momentum carried by this radiation generally produces a torque, called the YORP effect, which may be important in re-orienting asteroidal spins. Here we explore a related effect, the “binary YORP” (BYORP), that can modify the orbit of a synchronously rotating secondary in a binary system. It arises because a locked secondary is effectively an asymmetric appendage of the primary. It should be particularly important for Near-Earth Asteroids (NEAs) owing to their small sizes, proximity to the Sun, and benign collisional environment. To estimate BYORP's strength, we subjected 100 random Gaussian spheroids to the thermal radiation model of Rubincam (2000, Radiative spin-up and spin-down of small asteroids, Icarus, 148, 2-11). For most shapes, a significant torque arose on the secondary's orbit, typically modifying the orbit's size, eccentricity and inclination in less than 10⁵ years, for components of 1 and 0.3 km radii, separated by 2 km, at 1 AU, each of density 1750 kg m⁻³. Together YORP and BYORP are capable of synchronizing secondaries and circularizing orbits, making tidal dissipation unnecessary to explain the evolved state of observed NEA pairs. However, BYORP's rapid timescale poses a problem for the abundance of observed NEA binaries, since their formation rate is thought to be much slower. We consider and reject the following resolutions of this quandary: (i) the approximation using Gaussian spheroids inadequately models YORP; (ii) most secondaries are not synchronous, but inhabit other spin-orbit resonances (very unlikely); (iii) tidal dissipation is much more efficient than previously estimated, and thus capable of stabilizing observed systems; and (iv) moderately close encounters with planets can re-orient secondaries, turning BYORP into a slower, random-walk process. Finally, we speculate that most observed binary NEAs are in a stable state in which the obliquity-changing torques of YORP (acting on the primary) and BYORP cancel out, and that those systems must be close to 55° inclination, where the momentum-changing torques of both YORP and BYORP tend to be very small. Some retrograde systems might develop such that the nodes precess at a Sun-synchronous rate, while some prograde ones might move into the “evection” resonance. All three of these hypotheses can be tested directly by comparison with the i, Ω and ? observed for NEA binaries.     

Zero secular torque on asteroids from impinging solar photons in the YORP effect: A simple proof

YORP torques, where “YORP” stands for “Yarokovsky-O’Keefe-Radzievskii-Paddack,” arise mainly from sunlight reflected off a Solar System object and the infrared radiation emitted by it. We show here, through the most elementary demonstration that we can devise, that secular torques from impinging solar photons are generally negligible and thus cause little secular evolution of an asteroid’s obliquity or spin rate.     

Radiation pressure correction to meteor orbits

We present a method to calculate the radiation pressure force to gravity ratio on meteoroids from their atmospheric flight. Radiation pressure corrections to meteor orbits are negligible for fireballs; of the order of or less than the measurement errors ( 1%) for photographic meteors; of the order of and in some cases substantially larger than the measurement errors ( 10%) for radar meteors.     

On the detection of the Yarkovsky effect on near-Earth asteroids by means of Gaia

The Yarkovsky effect, which causes a slow drifting of the orbital elements (mainly the semimajor axis) of km-sized asteroids and meteors, is the weak non-gravitational force experienced by these bodies due to the emission of thermal photons. This effect is believed to play a role in the delivery of near-Earth asteroids (NEAs) from the main belt, in the spreading of the orbital elements of asteroid families, and in the orbital evolution of potentially hazardous asteroids.Here we present preliminary results of simulationing indicating that the perturbations induced by the Yarkovsky effect on the positions of some tens of NEAs can be observed by means of the high-precision astrometric observations that will be provided by the ESA mission Gaia.     

The collisional and dynamical evolution of the main-belt and NEA size distributions

The size distribution of main belt of asteroids is determined primarily by collisional processes. Large asteroids break up and form smaller asteroids in a collisional cascade, with the outcome controlled by the strength-size relationship for asteroids. In addition to collisional processes, the non-collisional removal of asteroids from the main belt (and their insertion into the near-Earth asteroid (NEA) population) is critical, and involves several effects: strong resonances increase the orbital eccentricity of asteroids and cause them to enter the inner planet region; chaotic diffusion by numerous weak resonances causes a slow leak of asteroids into the Mars- and Earth-crossing populations; and the Yarkovsky effect, a radiation force on asteroids, is the primary process that drives asteroids into these resonant escape routes. Yarkovsky drift is size-dependent and can modify the main-belt size distribution. The NEA size distribution is primarily determined by its source, the main-belt population, and by the size-dependent processes that deliver bodies from the main belt. All of these effects are simulated in a numerical collisional evolution model that incorporates removal by non-collisional processes. We test our model against a wide range of observational constraints, such as the observed main-belt and NEA size distributions, the number of asteroid families, the preserved basaltic crust of Vesta and its large south-pole impact basin, the cosmic ray exposure ages of meteorites, and the cratering records on asteroids. We find a strength-size relationship for main-belt asteroids and non-collisional removal rates from the main belt such that our model fits these constraints as best as possible within the parameter space we explore. Our results are consistent with other independent estimates of strength and removal rates.     

Equilibrium shapes of rubble-pile binaries: The Darwin ellipsoids for gravitationally held granular aggregates

Binaries are in vogue; many minor-planets like asteroids are being found to be binary or contact-binary systems. Even ternaries like 87 Sylvia have been discovered. The densities of these binaries are often estimated to be very low, and this, along with suspected accretionary origins, hints at a rubble interior. As in the case of fluid objects, a rubble-pile is unable to sustain all manners of spin, self-gravitation, and tidal interactions. This motivates the present study of the possible ellipsoidal shapes and mutual separations that members of a rubble-pile binary system may achieve. Conversely, knowledge of a granular binary’s shape and separation will constrain its internal structure - the ability of the binary’s members to sustain elongated shapes and/or maintain contact will hint at appreciable internal frictional strength. Because the binary’s members are allowed to be of comparable mass, the present investigation constitutes an extension of the second classical Darwin problem to granular aggregates.General equations defining the ellipsoidal rubble-pile binary system’s equilibrium are developed. These are then specialized to a pair of spin-locked, possibly unequal, prolate ellipsoidal granular aggregates aligned along their long axes. We observe that contact rubble-pile binaries can indeed exist. Further, depending on the binary’s geometry, an equilibrium contact binary’s members may, in fact, disrupt if separated. These results are applied to four suspected or known binaries: 216 Kleopatra, 25143 Itokawa, 624 Hektor and 90 Antiope. This exercise helps to bound the shapes and/or provide information about the interiors of these binaries.The binary’s interior will be modeled as a rigid-plastic, cohesionless material with a Drucker-Prager yield criterion. This rheology is a reasonable first model for rubble piles. We employ an approximate volume-averaging procedure that is based on the classical method of moments, and is an extension of the virial method (Chandrasekhar, S., 1969. Ellipsoidal Figures of Equilibrium. Yale University Press, New Haven, CT) to granular solid bodies. The present approach also helps us present an incrementally consistent approach to investigate the equilibrium shapes of fluid binaries, while highlighting the inconsistencies and errors inherent in the popular “Roche binary approximation”.