On the shapes and spins of “rubble pile” asteroids

We examine the shape of a “rubble pile” asteroid as it slowly gains angular momentum by YORP torque, to the point where “landsliding” occurs. We find that it evolves to a “top” shape with constant angle of repose from the equator up to mid-latitude, closely resembling the shapes of several nearly critically spinning asteroids imaged by radar, most notably (66391) 1999 KW4 [Ostro, S.J., Margot, J.-L., Benner, L.A.M., Giorgini, J.D., Scheeres, D.J., Fahnestock, E.G., Broschart, S.B., Bellerose, J., Nolan, M.C., Magri, C., Pravec, P., Scheirich, P., Rose, R., Jurgens, R.F., De Jong, E.M., Suzuki, S., 2006. Science 314, 1276-1280]. Similar calculations for non-spinning extremely prolate or oblate “rubble piles” show that even loose rubble can sustain shapes far from fluid equilibrium, thus inferences based on fluid equilibrium are generally useless for inferring bulk properties such as density of small bodies. We also investigate the tidal effects of a binary system with a “top shape” primary spinning at near the critical limit for stability. We find that very close to the stability limit, the tide from the secondary can actually levitate loose debris from the surface and re-deposit it, in a process we call “tidal saltation.” In the process, angular momentum is transferred from the primary spin to the satellite orbit, thus maintaining the equilibrium of near-critical spin as YORP continues to add angular momentum to the system. We note that this process is in fact dynamically related to the process of “shepherding” of narrow rings by neighboring satellites.     

The role of regolith redistribution in influencing the evolution of the shapes of asteroids

Abstract— ‐Major surface fissures and relatively large‐scale, angular surface irregularities are expected to have been present on many asteroids at early stages in their histories as a byproduct of at least two processes (impact disruption and reassembly into rubble piles for all classes of asteroid and, for carbonaceous chondrite parent bodies, aqueous alteration) which led to the low bulk densities currently being observed for asteroids. However, in all cases where high‐enough resolution images exist, such abrupt, deep irregularities are not observed. We model the spatial redistribution of impact‐generated regolith on an asteroid with an idealized irregular shape to show how the complex gravitational field of such a body will lead to the systematic infilling of deep valleys in the surface. Our analysis emphasizes the high efficiency with which regolith redistribution can act to disguise the internal structures of asteroids with sizes in the 20–100 km range.     

Dynamical passage to approximate equilibrium shapes for spinning, gravitating rubble asteroids

Many asteroids are thought to be particle aggregates held together principally by self-gravity. Here we study — for static and dynamical situations — the equilibrium shapes of spinning asteroids that are permitted for rubble piles. As in the case of spinning fluid masses, not all shapes are compatible with a granular rheology. We take the asteroid to always be an ellipsoid with an interior modeled as a rigid-plastic, cohesion-less material with a Drucker-Prager yield criterion. Using an approximate volume-averaged procedure, based on the classical method of moments, we investigate the dynamical process by which such objects may achieve equilibrium. We first collapse our dynamical approach to its statical limit to derive regions in spin-shape parameter space that allow equilibrium solutions to exist. At present, only a graphical illustration of these solutions for a prolate ellipsoid following the Drucker-Prager failure law is available [Sharma, I., Jenkins, J.T., Burns, J.A., 2005a. Bull. Am. Astron. Soc. 37, 643; Sharma, I., Jenkins, J.T., Burns, J.A., 2005b. Equilibrium shapes of ellipsoidal soil asteroids. In: García-Rojo, R., Hermann, H.J., McNamara, S. (Eds.), Proceedings of the 5th International Conference on Micromechanics of Granular Media, vol. 1. A.A. Balkema, UK; Holsapple, K.A., 2007. Icarus 187, 500-509]. Here, we obtain the equilibrium landscapes for general triaxial ellipsoids, as well as provide the requisite governing formulae. In addition, we demonstrate that it may be possible to better interpret the results of Richardson et al. [Richardson, D.C., Elankumaran, P., Sanderson, R.E., 2005. Icarus 173, 349-361] within the context of a Drucker-Prager material. The graphical result for prolate ellipsoids in the static limit is the same as those of Holsapple [Holsapple, K.A., 2007. Icarus 187, 500-509] because, when worked out, his final equations will match ours. This is because, though the formalisms to reach these expressions differ, in statics, at the lowest level of approximation, volume-averaging and the approach of Holsapple [Holsapple, K.A., 2007. Icarus 187, 500-509] coincide. We note that the approach applied here was obtained independently [Sharma, I., Jenkins, J.T., Burns, J.A., 2003. Bull. Am. Astron. Soc. 35, 1034; Sharma, I., 2004. Rotational Dynamics of Deformable Ellipsoids with Applications to Asteroids. Ph.D. thesis, Cornell University] and it provides a general, though approximate, framework that is amenable to systematic improvements and is flexible enough to incorporate the dynamical effects of a changing shape, different rheologies and complex rotational histories. To demonstrate our technique, we investigate the non-equilibrium dynamics of rigid-plastic, spinning, prolate asteroids to examine the simultaneous histories of shape and spin rate for rubble piles. We have succeeded in recovering most results of Richardson et al. [Richardson, D.C., Elankumaran, P., Sanderson, R.E., 2005. Icarus 173, 349-361], who obtained equilibrium shapes by studying numerically the passage into equilibrium of aggregates containing discrete, interacting, frictionless, spherical particles. Our mainly analytical approach aids in understanding and quantifying previous numerical simulations.     

The rotation of small asteroids

The addition of the unbiased sample of R. P. Binzel and J. D. Mulholland (Icarus56, 519–533) nearly triples the sample size of photoelectrically determined rotational parameters for main belt asteroids with estimated diameters (D) ≤30 km. Nonparametric stattistical tests which require no assumptions about the distributions or variances of the samples are used to examine rotational parameters for all D ≤ 30 km asteroids. A comparison of photoelectric and photographic results shows that the techniques have a highly significant difference in the range of detected frequencies. This difference does not allow photographic and photoelectric observations to be combined for meaningful statistical tests since a photographic bias toward smaller sample variances can induce statistical results that appear overly significant. Photographic observations also show a highly significant bias toward detecting asteroids with larger lightcurve amplitudes. The fit of a Maxwellian to the observed rotational frequency distribution can be rejected at a highly significant confidence level but the observed distribution can be acceptably fit by two Maxwellian distributions, which is consistent with the hypothesis that there are separate populations of slow and fast rotating asteroids. The frequency distributions of <15 km main belt asteroids and Earth and Mars crossers are not found to differ significantly. However, the larger mean lightcurve amplitude of the Earth and Mars crossing asteroids is found to be statistically significant. This latter result is interesting in view of the lack of any strong inverse amplitude versus diameter relation for small asteroids. No significant diameter dependence on rotational frequency is seen among only D ≤ 30 km asteroids. However, the inverse frequency versus diameter relation for D ≤ 120 km asteroids found by S. F. Dermott, A. W. Harris, and C. D. Murray (Icarus, in press) is found to be statistically significant using a linear least-squares analysis of photoelectric data only. No significant diameter dependence on rotational lightcurve amplitude is seen using linear least-squares analysis of photoelectric data for D≤30 and D≤90 km asteroids. However, a significant inverse amplitude versus diameter relation is found when this analysis is extended to D≤120 km asteroids. This finding may be consistent with the hypothesis of Dermott et al. that near 120 km there is a transition between primordial asteroids and their collisional fragments.     

The R asteroids reconsidered

JHK infrared photometry shows that R asteroids have two distinct infrared color domains. Most R asteroids have JHK and visual colors and albedos that fall amongst those observed for S asteroids, but a small subset is clearly different. These are designated as a new A class of asteroids.     

The internal structures and densities of asteroids

Abstract— Four asteroidal bodies (the Martian satellites Phobos and Deimos and the main-belt asteroids 243 Ida and 253 Mathilde) have now been the subjects of sufficiently close encounters by spacecraft that the masses and sizes and, hence, the densities of these bodies can be estimated to ～10%. All of these asteroids are significantly less dense than most members of the classes of meteorites identified as being compositionally most nearly similar to them on the basis of spectral characteristics. We show that two processes can act, independently or in concert, during the evolutionary histories of asteroids to produce a low bulk density. One of these processes is the result of one or more impact events and can affect any asteroid type, whereas the other can occur only for certain types of small asteroids that have undergone aqueous alteration.     

The axial rotation of asteroids

The observed fact that light changes of the asteroids exhibit no beat periods is interpreted as an indication that they do not wobble in space like spinning tops, but spin about only one axis (possibly — but not necessarily — inclined but little to the plane of their orbits). Since, moreover, the damping of three-dimensional rotation by jovi-solar attraction would require a time which is long in comparison with the age of the solar system, it is concluded that the present uni-axial rotation must represent a property preserved from the time when the asteroids were formed. This would seem to testify against their origin by collisional fragmentation of larger bodies; for in such a case the resulting solid splinters would still today be characterized by a random distribution of their angular momenta in three dimensions.The writer owes this assurance to Dr. Thomas Gehrels (private information).     

The angular momentum of the asteroids

Consideration of the basic physics involved in the structure of the object are used to obtain relationships for the radius, period, angular momentum, etc. of a typical asteroid. The mass-angular momentum relation for asteroids would tend to favour the fragmentation hypothesis.     

The number density of main-belt asteroids

In this paper we have studied how the number density of asteroids varies as a function of the mean distance from the sun.     

The magmatic activity on asteroids

The tidal effects on a fractured asteroid are considered. The asteroid is assumed to consist of two parts. In gravitational field of another body the motion of one part of the asteroid in relation to second part may be initiated. The necessary conditions for this motion are determined and amount of heat that can be generated is calculated for some cases. It is suggested that metamorphic episodes found in some meteorites are the results of such heating.     

The absence of satellites of asteroids

A CCD-imaging survey was made for satellites of minor planets at distances of about 0.1 to 7 arcmin from 1 Ceres, 2 Pallas, 4 Vesta, 6 Hebe, 7 Iris, 8 Flora, 15 Eunomia, 29 Amphitrite, 41 Daphne, and 44 Nysa, with cursory inspection of 192 Nausikaa. Satellites larger than 3 km were not found in this work, nor in previous photographic surveys. Not finding them appears to be consistent with theoretical studies of collisions in the asteroid belt by several authors. The satellites would have to be larger than at least 30 km to be collisionally stable. Tidal effects would lead to synchronous rotation and therefore long periods of rotation (several days), which are not generally observed. Taking tidal stability into account, we conclude that the only possible satellites for main-belt asteroids, with stability over eons, are near-contact binaries. The only other rare possibility for a satellite might be a piece of debris from a recent collision, and it would now be chaotic and collisionally unstable.     

The Yarkovsky-driven origin of near-Earth asteroids

We investigate the relevance of the Yarkovsky effect for the origin of kilometer and multikilometer near-Earth asteroids (NEAs). The Yarkovsky effect causes a slow migration in semimajor axis of main belt asteroids, some of which are therefore captured into powerful resonances and transported to the NEA space. With an innovative simulation scheme, we determine that in the current steady-state situation 100-160 bodies with H < 18 (roughly larger than 1 km) enter the 3/1 resonance per million years and 40-60 enter the ν₆ resonance. The ranges are due to uncertainties on relevant simulation parameters such as the time scales for collisional disruption and reorientation, their size dependence, and the strength of the Yarkovsky and YORP effects. These flux rates to the resonances are consistent with those independently derived by Bottke et al. (2002, Icarus 156, 399-433) with considerations based only on the NEA orbital distribution and dynamical lifetime. Our results have been obtained assuming that the main belt contains 1,300,000 asteroids with H < 18 and linearly scale with this number. Assuming that the cumulative magnitude distribution of main belt asteroids is N(< H) ∝ 10^γ′H with γ′ = 0.25 in the 15.5 < H < 18 range (consistent with the results of the SDSS survey), we obtain that the bodies captured into the resonances should have a similar magnitude distribution, but with exponent coefficient γ = 0.33-0.40. The lowest value is obtained taking into account the YORP effect, while higher values correspond to a weakened YORP or to YORP-less cases. These values of γ are all compatible with the debiased magnitude distributions of the NEAs according to Rabinowitz et al. (2000, Nature 403, 165-166), Bottke et al. (2000b, Science 288, 2190-2194), and Stuart (2001, Science 294, 1691-1693). Hence the Yarkovsky and YORP effects allow us to understand why the magnitude distribution of NEAs is only moderately steeper than that of the main belt population. The steepest main belt distribution that would still be compatible with the NEA distribution has exponent coefficient γ′ ∼ 0.3.     

The CM carbonaceous chondrite regolith Diepenveen

A carbonaceous chondrite was recovered immediately after the fall near the village of Diepenveen in the Netherlands on October 27, 1873, but came to light only in 2012. Analysis of sodium and poly‐aromatic hydrocarbon content suggests little contamination from handling. Diepenveen is a regolith breccia with an overall petrology consistent with a CM classification. Unlike most other CM chondrites, the bulk oxygen isotopes are extremely ¹⁶O rich, apparently dominated by the signature of anhydrous minerals, distributed on a steep slope pointing to the domain of intrinsic CM water. A small subset plots closer to the normal CM regime, on a parallel line 2 ‰ lower in δ¹⁷O. Different lithologies in Diepenveen experienced varying levels of aqueous alteration processing, being less aqueously altered at places rather than more heated. The presence of an agglutinate grain and the properties of methanol‐soluble organic compounds point to active impact processing of some of the clasts. Diepenveen belongs to a CM clan with ~5 Ma CRE age, longer than most other CM chondrites, and has a relatively young K‐Ar resetting age of ~1.5 Ga. As a CM chondrite, Diepenveen may be representative of samples soon to be returned from the surface of asteroid (162173) Ryugu by the Hayabusa2 spacecraft.     

The space distribution and orbital resonances of asteroids

We have made a detailed numerical analysis of the various orbital resonances of the asteroids and examined the variation in the distribution of trajectories near the equilibrium points in the phase plane, and hence arrived at a preliminary explanation of the features (gaps and clusters) in their distribution in space.     

The three-dimensional motion of Trojan asteroids

The problem is considered within the framework of the elliptic restricted three-body problem. The asymptotic solution is derived by a three-variable expansion procedure. The variables of the expansion represent three time-scales of the asteroids: the revolution around the Sun, the libration around the triangular Lagrangian pointsL ₄,L ₅, and the motion of the perihelion. The solution is obtained completely in the first order and partly in the second order. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter. As an example for the perturbations of the orbital elements the main perturbations of the eccentricity, the perihelion longitude and the longitude of the ascending node are given. Conditions for the libration of the perihelion are also discussed.     

Nitrogen and argon release profiles in Luna 16 and Luna 24 regolith samples: The effects of regolith reworking

Abstract— Fines, microbreccias and agglutinates from the Luna 16 mature regolith 1635 and fines from the immature/submature Luna 24 regolith have been analysed for N and argon isotopes in order to understand the origin of isotopically distinct N released at different temperatures. All high‐resolution runs reveal a similarity in the release of ³⁶Ar, ⁴⁰Ar and N over a wide temperature interval. The similarity in the ⁴⁰Ar and ³⁶Ar releases and the near coincidence in the 1635 agglutinates implies that the implanted species were redistributed and homogenised during regolith processing such that, regardless of the huge difference in ion implantation energy between solar ³⁶Ar and non‐solar ⁴⁰Ar, their present distribution and their release temperatures are now essentially equal. A small amount of ⁴⁰Ar released in the lower temperature steps with elevated ⁴⁰Ar/³⁶Ar is considered to be trapped after reworking. While such mixing and homogenisation may also be expected for N components of different origins, to date all known stepped runs regularly demonstrate a reproducible variation in δ¹⁵N, suggesting no homogenisation. We consider regolith N to be a mixture of several components trapped at different times, and some nitrogen that was not involved in the reworking. Relatively heavy N released around 500 °C appears to be the most pure form of the component trapped after reworking, probably from accreted meteoritic matter. Middle‐temperature isotopically lighter N appears to be a mixture of solar and non‐solar N largely homogenised, and therefore solar N can not be seen in its pure form. Bulk δ¹⁵N as well as formally deconvoluted δ¹⁵N thermal profiles imply that the non‐solar N has a variable δ¹⁵N value. Several non‐solar N sources are considered with their input resulting in increasing regolith δ¹⁵N with time. Because N from meteorites and interplanetary dust particles appear to be dominant, a mechanism is required to reduce the C/N ratio typical of meteoritic matter to that approaching the low value observed in the lunar regolith. Preferential loss of methane appears to be a viable explanation, following generation either by proton sputtering or in reducing vapour plumes.     

The cool surfaces of binary near-Earth asteroids

Here we show results from thermal-infrared observations of km-sized binary near-Earth asteroids (NEAs). We combine previously published thermal properties for NEAs with newly derived values for three binary NEAs. The η value derived from the near-Earth asteroid thermal model (NEATM) for each object is then used to estimate an average thermal inertia for the population of binary NEAs and compared against similar estimates for the population of non-binaries. We find that these objects have, in general, surface temperatures cooler than the average values for non-binary NEAs as suggested by elevated η values. We discuss how this may be evidence of higher-than-average surface thermal inertia. This latter physical parameter is a sensitive indicator of the presence or absence of regolith: bodies covered with fine regolith, such as the Earth’s moon, have low thermal inertia, whereas a surface with little or no regolith displays high thermal inertia. Our results are suggestive of a binary formation mechanism capable of altering surface properties, possibly removing regolith: an obvious candidate is the YORP effect.We present also newly determined sizes and geometric visible albedos derived from thermal-infrared observations of three binary NEAs: (5381) Sekhmet, (153591) 2001 SN₂₆₃, and (164121) 2003 YT₁. The diameters of these asteroids are 1.41 ± 0.21 km, 1.56 ± 0.31 km, and 2.63 ± 0.40 km, respectively. Their albedos are 0.23 ± 0.13, 0.24 ± 0.16, and 0.048 ± 0.015, respectively.     

Physical study of asteroids: Ligthcurves and rotational periods of six asteroids

V band photoelectric lightcurves and rotational periods are presented for six asteroids: 150 Nuwa, 203 Pompeja, 336 Lacaderia, 545 Messalina, 984 Gretia, and 1240 Fantasia. Except for 984 Gretia, none of these asteroids has been previously observed. The observations were obtained during September 1983 at the Astrophysical Observatory of Catania and are part of a program devoted to increase the present data set of asteroids' rotational properties. For 336 Lacaderia and 984 Gretia the magnitude-phase relations, in terms ofQ_v and β_v, were also obtained.     

The shape of asteroids: Theoretical considerations

Theoretical consideration and observations by other authors indicate that small asteroids are capable of maintaining irregular shapes, notably the shape of a cigar and even of a dumb-bell. This paper presents a model which describes the changes in the shape of an asteroid due to collisions of smaller objects (meteoroids) with the asteroid. The following assumptions must be approximately valid: (1) collisions are not uncommon; (2) collisions between a (relatively) large asteroid and small objects (meteroids) are more common than collisions between asteroids; (3) the cumulative probability of the collision of a meteoroid on a point on the surface of an asteroid is proportional to the zenith angle of the horizon as seen by that point; (4) obliquities of all but the major asteroids are random, so that there is not a preferred side on which collisions occur; (5) a considerable percentage of collision ejecta achieves escape velocity; and (6) the rate of erosion of each point on the surface of an asteroid is proportional to the cumulative probability of collision.Generalized conclusions that are obtained from the computer running of the model indicate that both cigars and dumb-bells are possible outcomes. Sharp corners are smoothed away, the radius of curvature of rounded surfaces increases to the point of going from convexity to concavity, and flat surfaces develop into gentle concavities.Collisions of an asteroid with an object of sufficient size such that the impact causes the breakage of the asteroid or the formation of a large crater, are not discussed in this paper. Previous work, however, suggests that the crater will undergo geomorphological changes of different geometry than a similar crater on the Moon.     

The asteroids as outcomes of catastrophic collisions

The role of catastrophic collisions in the evolution of the asteroids is discussed in detail, employing extrapolations of experimental results on the outcomrs of high-velocity impacts. We determine the range of the probable largest collision for target asteroids of different sizes during the solar system's lifetime, and we conclude that all the asteroids have undergone collisional events capable of overcoming the material's solid-state cohesion. Such events do not lead inescapably to complete disruption of the targets, because (i) for a previously unfractured target, experiments show that fragments of significant size can survive breakup, depending on the energy and geometry of the collision; (ii) self-gravitation can easily cause a reaccumulation of fragments for targets exceeding a critical size, which seems to be of the order of 100 km. In the intermediate diameter range 100?D ?300 km, where formation of gravitationally bound “rubble piles” is frequent, the transfer of angular momentum can be large enough to produce objects with triaxial equilibrium shapes (Jacobi ellipsoids) or to cause fission into binary systems. In the same size range, low-velocity escape of collisional fragments can also occur, leading to the formation of dynamical families. Asteroids smaller than ～100 km are mostly multigeneration fragments, while for $\text{D?300}\text{km}$ the collisional process produces nearly spheroidal objects covered by megaregoliths; whether their rotation is “primordial” or collisionally generated depends critically on the past flux of colliders. The complex and size-dependent phenomenology predicted by the theory compares satisfactorily with the observational evidence, as derived both by a classification of asteroids in terms of their size, spin rate, and lightcurve amplitude, and by a comparison between the rotational properties of family and nonfamily asteroids. The fundamental result of this investigation is that almost all asteroids are outcomes of catastrophic collisions, and that these events cause either complete fragmentation of the target bodies or, at least, drastic readjustments of their internal structure, shape, and spin rate.