Combined effect of YORP and collisions on the rotation rate of small Main Belt asteroids

The rotation rate distribution of small Main Belt asteroids is dominated by YORP and collisions. These mechanism act differently depending on the size of the bodies and give rise to non-linear effects when they both operate. Using a Monte Carlo method we model the formation of a steady state population of small asteroids under the influence of both mechanisms and the rotation rate distribution is compared to the observed one as derived from Pravec et al. (Pravec, P. et al. [2008]. Icarus 197, 497-504). A better match to observations is obtained with respect to the case in which only YORP is considered. In particular, an excess of slow rotators is produced in the model with both collisions and YORP because bodies driven to slow rotation by YORP have a random walk-like evolution of the spin induced by repeated collisions with small projectiles. This is a dynamical evolution different from tumbling and it lasts until a large impact takes the body to a faster rotation rate. According to our model, the rotational fission of small asteroids is a very frequent event and might explain objects like P/2010 A2 and its associated tail of millimeter-sized dust particles. The mass loss during fission of small asteroids might significantly influence the overall collisional evolution of the belt. Fission can in fact be considered as an additional erosion mechanism, besides cratering and fragmentation, acting only at small diameters.     

On the planar motion in the full two-body problem with inertial symmetry

Relative motion of binary asteroids, modeled as the full two-body planar problem, is studied, taking into account the shape and mass distribution of the bodies. Using the Lagrangian approach, the equations governing the motion are derived. The resulting system of four equations is nonlinear and coupled. These equations are solved numerically. In the particular case where the bodies have inertial symmetry, these equations can be reduced to a single equation, with small nonlinearity. The method of multiple scales is used to obtain a first-order solution for the reduced nonlinear equation. The solution is shown to be sufficient when compared with the numerical solution. Numerical results are provided for different example cases, including truncated-cone-shaped and peanut-shaped bodies.     

The random walk of Main Belt asteroids: orbital mobility by non-destructive collisions

Non-destructive collisions among Main Belt asteroids have effects on their orbits due to the transmission of linear momentum. The efficiency of this mechanism depends on several parameters which are currently poorly known. The most critical aspects are (i) the inventory and size distribution of small Main Belt asteroids, with sizes well below a few kilometres; (ii) the energy threshold for collisional fragmentation and fragment dispersion and (iii) the efficiency of linear momentum transfer. In spite of these difficulties, a general statistical model of the dynamical effects of non-destructive collisions can be developed, and is presented here. Based on this model, the consequences of different assumptions concerning the asteroid size distribution and collision physics are computed and discussed. Quantitative evaluations of the collisionally induced orbital mobility in different possible scenarios are presented.     

New method of orbit determination from two position vectors based on solving Gauss’s equations

The classic problem of finding the orbit of a celestial body from its two position vectors for two instants of time is considered. A solution to the problem free from uncertainties is obtained which can be applied for all three kinds of Keplerian movement. The main part of the computational procedure is reduced to solving one equation with one unknown. Formulas are derived for the initial value of the equation root, which makes the application of the Newton-Raphson method successful. The efficiency and reliability of the suggested algorithm is illustrated by examples of the orbit determination for the asteroids Adeona and Icarus, as well as for Halley’s Comet and Bowell’s Comet.     

Computation of Partial Derivatives of Perturbed Motion Using the Arcs of Osculating and Superosculating Orbits

The methods for analytical determination of partial derivatives of the current parameters of motion with respect to their initial values are described. The methods take into account principal perturbations and are based on the use of the osculating and superosculating intermediate orbits constructed earlier by the author. These orbits ensure the first-, second-, and third-order contact to the real trajectory at the initial time. The solution for parameters of the intermediate motion and partial derivatives of these parameters is given in a universal closed form. The partial derivatives on long time intervals are computed using a step-by-step procedure combined with the Encke method of special perturbations, in which the intermediate orbits are used as the reference. The numerical results show that the new approach can be efficiently used for solving the problem of differential correction of orbits of asteroids and comets on the basis of observational data.     

Collisional evolution of the asteroid belt

A new synthesis of asteroid collisional evolution is motivated by the question of whether most asteroids larger than ∼1 km size are strengthless gravitational aggregates (rubble piles). NEAR found Eros not to be a rubble pile, but a shattered collisional fragment, with a through-going fracture system, and an average of about 20 m regolith cover. Of four asteroids visited by spacecraft, none appears likely to be a rubble pile, except perhaps Mathilde. Nevertheless, current understanding of asteroid collisions and size-dependent strength, and the observed distribution of rotation rates versus size, have led to a theoretical consensus that many or most asteroids larger than 1 km should be rubble piles. Is Eros, the best-observed asteroid, highly unusual because it is not a rubble pile? Is Mathilde, if it is a rubble pile, like most asteroids? What would be expected for the small asteroid Itokawa, the MUSES-C sample return target? An asteroid size distribution is synthesized from the Minor Planet Center listing and results of the Sloan Digital Sky Survey, an Infrared Space Observatory survey, the Small Main-belt Asteroid Spectroscopic Survey and the Infrared Astronomical Satellite survey. A new picture emerges of asteroid collisional evolution, in which the well-known Dohnanyi result, that the size distribution tends toward a self-similar form with a 2.5-index power law, is overturned because of scale-dependent collision physics. Survival of a basaltic crust on Vesta can be accommodated, together with formation of many exposed metal cores. The lifetimes against destruction are estimated as 3 Gyr at the size of Eros, 10 Gyr at ten times that size, and 40 Gyr at the size of Vesta. Eros as a shattered collisional fragment is not highly unusual. The new picture reveals the new possibility of a transition size in the collisional state, where asteroids below 5 km size would be primarily collisional breakup fragments whereas much larger asteroids are mostly eroded or shattered survivors of collisions. In this case, well-defined families would be found in asteroids larger than about 5 km size, but for smaller asteroids, families may no longer be readily separated from a background population. Moreover, the measured boulder size distribution on Eros is re-interpreted as a sample of impactor size distributions in the asteroid belt. The regolith on Eros may result largely from the last giant impact, and the same may be true of Itokawa, in which case about a meter of regolith would be expected there. Even a small asteroid like Itokawa may be a shattered object with regolith cover.     

Mass of Mercury from observations of asteroids

The Sun-to-Mercury mass ratio adopted by the International Astronomical Union (6023600 ± 250) was obtained in 1987 by analyzing Mariner 10 observations (Anderson et al. 1987) and since then has not been improved. The large number of asteroids in Mercury-approaching orbits and the ever-increasing accuracy of their observations allow the mass of Mercury to be estimated by a different method. We have improved the orbital parameters of 43 asteroids and obtained 6023440 ± 530 for the Sun-to-Mercury mass ratio through a simultaneous solution based on their optical and radar observations. A further improvement in this estimate is possible in the immediate future owing to the rapid increase in the number of known asteroids whose observations can be used to solve this problem.     

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.     

Are Main-Belt Asteroids a Sufficient Source for the Earth-Approaching Asteroids?

This paper predicts the size distribution of the Earth-approaching asteroids with diameterd= 10 m to 10 km, assuming they originate as the fragments of main-belt asteroids with a cumulative size distribution proportional tod^−2.5and that they have self-similar fragmentation properties. The resulting distribution is dominated by “fast-track” bodies originating from parent asteroids with orbits close to the 3:1 mean-motion resonance with Jupiter. Because the dynamical lifetimes of these Earth approachers are shorter than their collisional lifetimes, their size distribution is nearly proportional tod^−3.0, the production distribution in the main belt. This prediction, however, is at odds with the Spacewatch observations. The observed distribution is relatively flat ford> ∼100 m, and relatively steep ford< ∼100 m, so that the number of Earth approachers withd∼ 10 m to 0.3 km is overestimated. If these populations are predominantly of main-belt origin, then the size distribution in the main belt is not a simple power law. A nonuniform size distribution with wave-like oscillations, possibly caused by a cutoff at small sizes, would lead to Earth approachers with a size distribution in better agreement with the observations. If such wave-like oscillations are realistic, then the main belt is sufficient to supply the observed number of Earth approachers throughout the observed size range.     

Cratering rate on the jovian system: the contribution from Hilda asteroids

We study the dynamical evolution of the Hilda group of asteroids trough numerical methods, performing also a collisional pseudo-evolution of the present population, in order to calculate the rate of evaporation and its contribution to the cratering history of the Galilean satellites. If the present population of small asteroids in the Hilda's region follows the same size distribution observed at larger radii, we find that this family is the main contributor to the production of small craters (i.e., crater with diameters d∼4 km) on the Galilean system, overcoming the production by Jupiter Family Comets and by Trojan asteroids. The results of this investigation encourage further observational campaigns, in order to determine the size distribution function of small Hilda asteroids.     

Asteroidal binary systems: Detection and formation

Recent occultation data and an analysis of some photometric lightcurves have shown the possible existence of asteroidal binary systems.A simple geometrical model taking into account mutual shadowing effects shows some peculiar features of the lightcurve which can be recovered in several previously observed objects; therefore the hypothesis of a relatively high frequency of binary asteroids should be seriously considered.On the other hand, while the rotational period distribution of large asteroids (D>200 km) is sharply peaked at about 5–8 hours, the surprisingly higher dispersion towards longer periods for intermediate size objects (50<D<150 km) could be connected with a larger probability of binary nature within this class.From a theoretical point of view, the collisional fragmentation of asteroids could originate gravitationally bound fragments, with a tidal transfer of rotational into orbital angular momentum, causing a rapid synchronization of the system. This kind of processes could more easily occur for intermediate objects since: (a) for large ones, very massive colliding bodies are needed for fragmentation, that means a very rare event; (b) for smaller asteroids, solid state interactions are stronger than the gravitational ones, so that a breakage probably causes a complete disruption of the gravitational binding. Further collisional events could disintegrate some systems, so that the present frequency of binary asteroids could be lower than that of the objects whose rotational period was increased by such processes.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

Evolutionary principles applied to mission planning problems

The space mission planning process is considered as a hybrid optimal control problem. Hybrid optimal control problems are problems that include categorical variables in the problem formulation. For example, an interplanetary trajectory may consist of a sequence of low thrust arcs, impulses and planetary flybys. However, for each choice of the structure of the mission, for example, for a particular choice of the number of planetary flybys to be used, there is a corresponding optimal trajectory. It is not a priori clear which structure will yield the most efficient mission. In this work we present a mathematical framework for describing such problems and solution methods for the hybrid optimal control problem based on evolutionary principles that have the potential for being a robust solver of such problems. As an example, the methods are used to find the optimal choice of three asteroids to visit in sequence, out of a set of eight candidate asteroids, in order to minimize the fuel required.     

The perturbations of the orbital elements of Trojan asteriods

An asymptotic solution for the cylindrical coordinates of Trojan asteroids is derived by using a three-variable expansion method in the elliptic restricted three-body problem. The perturbations of the orbital elements are obtained from this solution by applying the formulas of the two-body problem. The main perturbations of the mean motion are studied in detail.     

Virial oscillations of celestial bodies

A general approach to the solution of the perturbed oscillation problem for celestial bodies is considered. The solution sought describes unperturbed virial oscillations (zero approximation) affected by external perturbing effects. In the general case, these perturbations can be expressed by an arbitrary given function of time, Jacobi's function and its first derivative. Standard methods and modes of perturbation theory are used for solution of the problem.It is shown that while studying the evolution of a celestial body as a dissipative system in the framework of perturbed virial oscillations, the analytical expression for perturbing function can be derived, assuming the celestial body to be an oscillating electrical dipole emitting electromagnetic energy.The general covariant form of Jacobi's equation is derived and its spur is examined. It is shown that the scalar form of Jacobi's equation appears to be more universal than Newton's laws of motion from which it is derived.     

Rebuttal to the comment by Malhotra and Strom on “Constraints on the source of lunar cataclysm impactors”

?uk et al. (?uk, M. Gladman, B.J., Stewart, S.T. [2010]. Icarus 207 590-594) concluded that the the lunar cataclysm (late heavy bombardment) was recorded in lunar Imbrian era craters, and that their size distribution is different from that of main belt asteroids (which may have been the dominant pre-Imbrian impactors). This result would likely preclude the asteroid belt as the direct source of lunar cataclysm impactors. Malhotra and Strom (Malhotra, R., Strom, R.G. [2011]. Icarus) maintain that the lunar impactor population in the Imbrian era was the same as in Nectarian and pre-Nectarian periods, and this population had a size distribution identical to that of main belt asteroids. In support of this claim, they present an Imbrian size distribution made from two data sets published by Wilhelms et al. (Wilhelms, D.E., Oberbeck, V.R., Aggarwal, H.R. [1978]. Proc. Lunar Sci. Conf. 9, 3735-3762). However, these two data sets cannot be simply combined as they represent areas of different ages and therefore crater densities. Malhotra and Strom (Malhotra, R., Strom, R.G. [2011]. Icarus) differ with the main conclusion of Wilhelms et al. (Wilhelms, D.E., Oberbeck, V.R., Aggarwal, H.R. [1978]. Proc. Lunar Sci. Conf. 9, 3735-3762) that the Nectarian and Imbrian crater size distributions were different. We conclude that the available data indicate that the lunar Imbrian-era impactors had a different size distribution from the older ones, with the Imbrian impactor distribution being significantly richer in small impactors than that of older lunar impactors or current main-belt asteroids.     

Photometry and spin rate distribution of small-sized main belt asteroids

Photometry results of 32 asteroids are reported from only seven observing nights on only seven fields, consisting of 34.11 cumulative hours of observations. The data were obtained with a wide-field CCD (^40.5′×^27.3′) mounted on a small, 46-cm telescope at the Wise Observatory. The fields are located within ±1.5° from the ecliptic plane and include a region within the main asteroid belt. The observed fields show a projected density of ∼23.7 asteroids per square degree to the limit of our observations. 13 of the lightcurves were successfully analyzed to derive the asteroids' spin periods. These range from 2.37 up to 20.2 h with a median value of 3.7 h. 11 of these objects have diameters in order of two kilometers and less, a size range that until recently has not been photometrically studied. The results obtained during this short observing run emphasize the efficiency of wide-field CCD photometry of asteroids, which is necessary to improve spin statistics and understand spin evolution processes. We added our derived spin periods to data from the literature and compared the spin rate distributions of small main belt asteroids (5>D>0.15 km) with that of bigger asteroids and of similar-sized NEAs. We found that the small MBAs do not show the clear Maxwellian-shaped distribution as large asteroids do; rather they have a spin rate distribution similar to that of NEAs. This implies that non-Maxwellian spin rate distribution is controlled by the asteroids' sizes rather than their locations.     

The shape distribution of boulders on Asteroid 25143 Itokawa: Comparison with fragments from impact experiments

Laboratory impact experiments have found that the shape of fragments over a broad size range is distributed around the mean value of the axial ratio 2:√2:1, which is independent of a wide range of experimental conditions. We report the shape statistics of boulders with size of 0.1-30 m on the surface of Asteroid 25143 Itokawa based on high-resolution images obtained by the Hayabusa spacecraft in order to investigate whether their shape distribution is similar to the distribution obtained for fragments (smaller than 0.1 m) in laboratory impact experiments. We also investigated the shapes of boulders with size of 0.1-150 m on Asteroid 433 Eros using a few arbitrary selected images by the NEAR spacecraft, in order to compare those with the shapes on Asteroid Itokawa. In addition, the shapes of small- and fast-rotating asteroids (diameter <200 m and rotation period <1 h), which are natural fragments from past impact events among asteroids, were inferred from archived light curve data taken by ground-based telescopes. The results show that the shape distributions of laboratory fragments are similar to those of the boulders on Eros and of the small- and fast-rotating asteroids, but are different from those on Itokawa. However, we propose that the apparent difference between the boulders of Itokawa and the laboratory fragments is due to the migration of boulders. Therefore, we suggest that the shape distributions of the boulders ranging from 0.1 to 150 m in size and the small- and fast-rotating asteroids are similar to those obtained for the fragments generated in laboratory impact experiments.     

Frequency analysis of the stability of asteroids in the framework of the restricted three-body problem

The stability of some asteroids, in the framework of the restricted three-body problem, has been recently proved in (Celletti and Chierchia, 2003), by developing an isoenergetic KAM theorem. More precisely, having fixed a level of energy related to the motion of the asteroid, the stability can be obtained by showing the existence of nearby trapping invariant tori existing at the same energy level. The analytical results are compatible with the astronomical observations, since the theorem is valid for the realistic mass-ratio of the primaries. The model adopted in (Celletti and Chierchia, 2003), is the planar, circular, restricted three-body model, in which only the most significant contributions of the Fourier development of the perturbation are retained. In this paper we investigate numerically the stability of the same asteroids considered in (Celletti and Chierchia, 2003), (namely, Iris, Victoria and Renzia). In particular, we implement the nowadays standard method of frequency-map analysis and we compare our investigation with the analytical results on the planar, circular model with the truncated perturbing function. By means of frequency analysis, we study the behaviour of the bounding tori and henceforth we infer stability properties on the dynamics of the asteroids. In order to test the validity of the truncated Hamiltonian, we consider also the complete expression of the perturbing function on which we perform again frequency analysis. We investigate also more realistic models, taking into account the eccentricity of the trajectory of Jupiter (planar-elliptic problem) or the relative inclination of the orbits (circular-inclined model). We did not find a relevant discrepancy among the different models.     

Rotational properties of asteroids: Correlations and selection effects

Rotational data on 321 asteroids observed as of late 1978 are analyzed. Selection effects within the sample are discussed and used to define a data set consisting of 134 main-belt, nonfamily asteroids having reliably determined periods and amplitudes based on photoelectric observations. In contrast to A. W. Harris and J. A. Burns (1979, Icarus40, 115–144) we found no significant correlation between rotational properties and compositional type. Smaller asteroids have a greater range of rotational amplitudes than the largest asteroids but are not, on the average, appreciably more elongated. While no definite relationship between asteroid size and rotation rate was found the distribution is not random. The largest asteroids have rotation periods near 7 hr compared with 10 hr for the smaller. A group of large, rapidly rotating, high-amplitude asteroids is recognized. A pronounced change in rotational properties occurs near this size range (diam = 200 ± 50 km) which also corresponds to the size at which a change of slope occurs in the size frequency distribution. We believe this size range represents a transition region between very large, rapidly rotating, low-amplitude (primordial?) objects and smaller ones having a considerably greater range of periods and amplitudes. Asteroids in this transition size range display an increase in rotational amplitude with increasing spin rate; other than this, however, there is no correlation between period and amplitude. The region of low spatial density in the asteroid belt centered near 2.9 AU and isolated from the inner and outer belt by the 2:5 and 3:7 commensurabilities is shown to be a region in which non-C or -S asteroids are overrepresented and which have marginally higher rotational amplitudes than those located in more dense regions. We attribute disagreements between our results and other studies of this type to the inclusion of non-main-belt asteroids and photographic data in the earlier analyses.     

Thermal inertia of main belt asteroids smaller than 100 km from IRAS data

Recent works have shown that the thermal inertia of km-sized near-Earth asteroids (NEAs) is more than 2 orders of magnitude higher than that of main belt asteroids (MBAs) with sizes (diameters) between 200 and 1000 km. This confirms the idea that large MBAs, over hundreds millions of years, have developed a fine and thick thermally insulating regolith layer, responsible for the low values of their thermal inertia, whereas km-sized asteroids, having collisional lifetimes of only some millions years, have less regolith, and consequently a larger surface thermal inertia.Because it is believed that regolith on asteroids forms as a result of impact processes, a better knowledge of asteroid thermal inertia and its correlation with size, taxonomic type, and density can be used as an important constraint for modeling of impact processes on asteroids. However, our knowledge of asteroids’ thermal inertia values is still based on few data points with NEAs covering the size range 0.1-20 km and MBAs that .Here, we use IRAS infrared measurements to estimate the thermal inertia values of MBAs with diameters and known shapes and spin vector, filling an important size gap between the largest MBAs and the km-sized NEAs. An update to the inverse correlation between thermal inertia and diameter is presented. For some asteroids thermophysical modeling allowed us to discriminate between the two still possible spin vector solutions derived from optical lightcurve inversion. This is important for (720) Bohlinia: our preferred solution was predicted to be the correct one by Vokrouhlický et al. [2003. The vector alignments of asteroid spins by thermal torques. Nature 425, 147-151] just on theoretical grounds.