Spin rate distribution of small asteroids

The spin rate distribution of main belt/Mars crossing (MB/MC) asteroids with diameters 3-15 km is uniform in the range from f=1 to 9.5 d⁻¹, and there is an excess of slow rotators with f<1 d⁻¹. The observed distribution appears to be controlled by the Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect. The magnitude of the excess of slow rotators is related to the residence time of slowed down asteroids in the excess and the rate of spin rate change outside the excess. We estimated a median YORP spin rate change of ≈0.022 d⁻¹/Myr for asteroids in our sample (i.e., a median time in which the spin rate changes by 1 d⁻¹ is ≈45 Myr), thus the residence time of slowed down asteroids in the excess is ≈110 Myr. The spin rate distribution of near-Earth asteroids (NEAs) with sizes in the range 0.2-3 km (∼5 times smaller in median diameter than the MB/MC asteroids sample) shows a similar excess of slow rotators, but there is also a concentration of NEAs at fast spin rates with f=9-10 d⁻¹. The concentration at fast spin rates is correlated with a narrower distribution of spin rates of primaries of binary systems among NEAs; the difference may be due to the apparently more evolved population of binaries among MB/MC 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.     

Axial rotation of near-earth asteroids: The influence of the YORP effect

The distribution of axial rotation velocities of near-Earth asteroids (NEAs) substantially differs from that of the Main-Belt asteroids by an excess of both quickly and slowly rotating objects. Among the possible causes of this difference is the influence of the solar radiation—the so-called YORP effect—that arises from the absorption of solar energy and its reemission in the thermal range by a rotating body of irregular shape. It is known that the magnitude of this effect depends on the asteroid size and the quantity of received solar energy (the insolation). Analysis of the observational data showed that the mean diameter of NEAs decreases from the middle of the distribution to the edges, i.e., the excess of both slowly (ω ≤ 2 rev/day) and quickly (ω = 8–11 rev/day) rotating objects is formed due to the asteroids with sizes smaller than those in the middle of the distribution, which agrees well with the influence of the YORP effect. Moreover, the dependence of the axial rotation velocity of NEAs on the relative insolation shows that, for the NEAs referred to, both excesses are found in orbits where, on average, they receive 8–10% more solar energy than the NEAs in the middle of the distribution. This result also agrees with the character of the influence of the YORP effect and can be considered as an additional argument in its support. Thus, the study showed that one can infer that the currently available observational data suggest the possible influence of the YORP effect on the axial rotation of the near-Earth asteroids having sizes of D ～ 2 km and less. This is the first attempt to find the influence of the YORP effect on the axial rotation of the NEA family as a whole.     

The Influence of the Orbital Evolution of Main Belt Asteroids on Their Spin Vectors

It was found that certain features in the observed spin vector distribution of main belt asteroids can be explained by the differences in the dynamical spin vector evolution between objects with high and low orbital inclinations. In particular, the deficiency of high-inclination objects whose spin vectors are close to the ecliptic plane can be accounted for.The present spin vector distribution of main belt asteroids is due to several factors connected with their collisional and dynamical evolution. In this paper, the influence of the orbital evolution on the spin axis of asteroids is examined in the case of 25 objects with typical main belt orbital evolution and 125 synthetic objects, during an integration over a time period of 1 Myr. This investigation produced the following general results:• The difference between maximum and minimum obliquity increases in an approximately linear fashion with increasing orbital inclination of the studied objects.• The inclination is the major factor influencing the magnitude of the obliquity variation. This variation is generally larger for asteroids with their initial spin vectors located close to the orbital plane.• In general, the regular obliquity differences are relatively insensitive to differences in the shape, composition, and spin rate of the asteroids.The result is compared with the properties of the observed spin vectors for 73 main belt asteroids and good agreement is found between the above results and the existing spin vector distribution.     

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 main belt as a source of near-Earth asteroids

We investigate the flux of main-belt asteroid fragments into resonant orbits converting them into near-Earth asteroids (NEAs), and the variability of this flux due to chance interasteroidal collisions. A numerical model is used, based on collisional physics consistent with the results of laboratory impact experiments. The assumed main-belt asteroid size distribution is derived from that of known asteroids extrapolated down to sizes of ≈ 40 cm, modified in such a way to yield a quasi-stationary fragment production rate over times ≈ 100 Myr. The results show that the asteroid belt can supply a few hundred km-sized NEAs per year, well enough to sustain the current population of such bodies. On the other hand, if our collisional physics is correct, the number of existing 10-km objects implies that these objects either have very long-lived orbits, or must come from a different source (i.e., comets). Our model predicts that the fragments supplied from the asteroid belt have initially a power-law size distribution somewhat steeper than the observed one, suggesting preferential removal of small objects. The component of the NEA population with dynamical lifetimes shorter than or of the order of 1 Myr can vary by a factor reaching up to a few tens, due to single large-scale collisions in the main belt; these fluctuations are enhanced for smaller bodies and faster evolutionary time scales. As a consequence, the Earth's cratering rate can also change by about an order of magnitude over the 0.1 to 1 Myr time scales. Despite these sporadic spikes, when averaged over times of 10 Myr or longer the fluctuations are unlikely to exceed a factor two.     

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.     

Thermal inertia of near-Earth asteroids and implications for the magnitude of the Yarkovsky effect

Thermal inertia determines the temperature distribution over the surface of an asteroid and therefore governs the magnitude the Yarkovsky effect. The latter causes gradual drifting of the orbits of km-sized asteroids and plays an important role in the delivery of near-Earth asteroids (NEAs) from the main belt and in the dynamical spreading of asteroid families. At present, very little is known about the thermal inertia of asteroids in the km size range. Here we show that the average thermal inertia of a sample of NEAs in the km-size range is . Furthermore, we identify a trend of increasing thermal inertia with decreasing asteroid diameter, D. This indicates that the dependence of the drift rate of the orbital semimajor axis on the size of asteroids due to the Yarkovsky effect is a more complex function than the generally adopted D⁻¹ dependence, and that the size distribution of objects injected by Yarkovsky-driven orbital mobility into the NEA source regions is less skewed to smaller sizes than generally assumed. We discuss how this fact may help to explain the small difference in the slope of the size distribution of km-sized NEAs and main-belt asteroids.     

The main belt as a source of near-Earth asteroids

We investigate the flux of main-belt asteroid fragments into resonant orbits converting them into near-Earth asteroids (NEAs), and the variability of this flux due to chance interasteroidal collisions. A numerical model is used, based on collisional physics consistent with the results of laboratory impact experiments. The assumed main-belt asteroid size distribution is derived from that of known asteroids extrapolated down to sizes of 40 cm, modified in such a way to yield a quasi-stationary fragment production rate over times 100 Myr. The results show that the asteroid belt can supply a few hundred km-sized NEAs per year, well enough to sustain the current population of such bodies. On the other hand, if our collisional physics is correct, the number of existing 10-km objects implies that these objects either have very long-lived orbits, or must come from a different source (i.e., comets). Our model predicts that the fragments supplied from the asteroid belt have initially a power-law size distribution somewhat steeper than the observed one, suggesting preferential removal of small objects. The component of the NEA population with dynamical lifetimes shorter than or of the order of 1 Myr can vary by a factor reaching up to a few tens, due to single large-scale collisions in the main belt; these fluctuations are enhanced for smaller bodies and faster evolutionary time scales. As a consequence, the Earth's cratering rate can also change by about an order of magnitude over the 0.1 to 1 Myr time scales. Despite these sporadic spikes, when averaged over times of 10 Myr or longer the fluctuations are unlikely to exceed a factor two.     

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.     

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.     

Secular spin dynamics of inner main-belt asteroids

Understanding the evolution of asteroid spin states is challenging work, in part because asteroids have a variety of orbits, shapes, spin states, and collisional histories but also because they are strongly influenced by gravitational and non-gravitational (YORP) torques. Using efficient numerical models designed to investigate asteroid orbit and spin dynamics, we study here how several individual asteroids have had their spin states modified over time in response to these torques (i.e., 951 Gaspra, 60 Echo, 32 Pomona, 230 Athamantis, 105 Artemis). These test cases which sample semimajor axis and inclination space in the inner main belt, were chosen as probes into the large parameter space described above. The ultimate goal is to use these data to statistically characterize how all asteroids in the main belt population have reached their present-day spin states. We found that the spin dynamics of prograde-rotating asteroids in the inner main belt is generally less regular than that of the retrograde-rotating ones because of numerous overlapping secular spin-orbit resonances. These resonances strongly affect the spin histories of all bodies, while those of small asteroids (?40 km) are additionally influenced by YORP torques. In most cases, gravitational and non-gravitational torques cause asteroid spin axis orientations to vary widely over short (?1 My) timescales. Our results show that (951) Gaspra has a highly chaotic rotation state induced by an overlap of the s and s₆ spin-orbit resonances. This hinders our ability to investigate its past evolution and infer whether thermal torques have acted on Gaspra's spin axis since its origin.     

A Search for the Collisional Parent Bodies of Large NEAs

A search for the most likely parent bodies of multi-km near-Earth asteroids (NEAs) is attempted, in the framework of a scenario based on a few simple assumptions. (1) Multi-km NEAs are produced by collisional fragmentation of single parent bodies. (2) The fragments are injected into either the 3/1 mean-motion resonance with Jupiter or the ν₆ secular resonance, or they achieve Mars-crossing orbits. (3) The collisional events responsible for the production of multi-km NEAs do not produce observable dynamical families. We show that a limited number of potential parent bodies of multi-km NEAs compatible with the above assumptions do exist in the asteroid Main Belt. It is not clear whether these objects can likely explain the current inventory of known NEAs having sizes around 1-2 km. Our results seem to indicate that the assumed scenario is not completely adequate to justify the number of observed NEAs larger than 2 km. This preliminary analysis must be complemented by a more precise analysis of the rates of occurrence of NEA-feeding events. If present results are confirmed, the conclusion that the origin of multi-km NEAs must be explained by different models, based on long-term dynamical diffusion produced by the interplay of collisional, gravitational, and nongravitational mechanisms in the Main Belt, plus a possible cometary contribution, will be strengthened.     

Binary asteroid population. 2. Anisotropic distribution of orbit poles of small,inner main-belt binaries

Our photometric observations of 18 main-belt binary systems in more than one apparition revealed a strikingly high number of 15 having positively re-observed mutual events in the return apparitions. Our simulations of the survey showed that it cannot be due to an observational selection effect and that the data strongly suggest that poles of mutual orbits between components of binary asteroids in the primary size range 3–8 km are not distributed randomly: The null hypothesis of an isotropic distribution of the orbit poles is rejected at a confidence level greater than 99.99%. Binary orbit poles concentrate at high ecliptic latitudes, within 30° of the poles of the ecliptic. We propose that the binary orbit poles oriented preferentially up/down-right are due to either of the two processes: (i) the YORP tilt of spin axes of their parent bodies toward the asymptotic states near obliquities 0° and 180° (pre-formation mechanism) or (ii) the YORP tilt of spin axes of the primary components of already formed binary systems toward the asymptotic states near obliquities 0° and 180° (post-formation mechanism). The alternative process of elimination of binaries with poles closer to the ecliptic by dynamical instability, such as the Kozai effect due to gravitational perturbations from the Sun, does not explain the observed orbit pole concentration. This is because for close binary asteroid systems, the gravitational effects of primary’s irregular shape dominate the solar-tide effect.     

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.     

Mineralogies and source regions of near-Earth asteroids

Near-Earth Asteroids (NEAs) offer insight into a size range of objects that are not easily observed in the main asteroid belt. Previous studies on the diversity of the NEA population have relied primarily on modeling and statistical analysis to determine asteroid compositions. Olivine and pyroxene, the dominant minerals in most asteroids, have characteristic absorption features in the visible and near-infrared (VISNIR) wavelengths that can be used to determine their compositions and abundances. However, formulas previously used for deriving compositions do not work very well for ordinary chondrite assemblages. Because two-thirds of NEAs have ordinary chondrite-like spectral parameters, it is essential to determine accurate mineralogies. Here we determine the band area ratios and Band I centers of 72 NEAs with visible and near-infrared spectra and use new calibrations to derive the mineralogies 47 of these NEAs with ordinary chondrite-like spectral parameters. Our results indicate that the majority of NEAs have LL-chondrite mineralogies. This is consistent with results from previous studies but continues to be in conflict with the population of recovered ordinary chondrites, of which H chondrites are the most abundant. To look for potential correlations between asteroid size, composition, and source region, we use a dynamical model to determine the most probable source region of each NEA. Model results indicate that NEAs with LL chondrite mineralogies appear to be preferentially derived from the ν₆ secular resonance. This supports the hypothesis that the Flora family, which lies near the ν₆ resonance, is the source of the LL chondrites. With the exception of basaltic achondrites, NEAs with non-chondrite spectral parameters are slightly less likely to be derived from the ν₆ resonance than NEAs with chondrite-like mineralogies. The population of NEAs with H, L, and LL chondrite mineralogies does not appear to be influenced by size, which would suggest that ordinary chondrites are not preferentially sourced from meter-sized objects due to Yarkovsky effect.     

The Recovery as an Important Part of NEA Astrometric Follow-up

The number of known near-Earth asteroids (NEAs) has increased rapidly in recent years due to large surveys. This discovery process has to be followed by follow-up observations to obtain a sufficient number of precise astrometric data needed for an accurate orbit determination of newly discovered bodies.Accurate orbit determination requires observations from at least two oppositions. If asteroids are not found in the next apparition, different from the discovery apparition, then they can be considered lost. This is particularly embarrassing for NEAs. If data for different apparitions are not found in the course of precovery surveys or in other archive data, then it is necessary to prepare targeted observations of a particular NEA in the second convenient apparition. Therefore NEA recovery is a very important part of NEA follow-up.We discuss here methods, techniques, and results of planned recoveries at the Klet' Observatory using a 0.57-m telescope equipped with a CCD detector. The Klet' NEA recovery subprogram has brought 21 planned NEA recoveries since 1997, including seven NEAs belonging to the potentially hazardous asteroid category.We briefly mention the overall work on NEA recoveries provided by several NEO follow-up programs as well as the need for communication resources supporting astrometric observers. Finally we present here a planned extension of the Klet' NEA recovery subprogram to fainter objects by means of a new 1.06-m reflector.     

Mining the CFHT Legacy Survey for known Near Earth Asteroids

The Canada‐France‐Hawaii Legacy Survey (CFHTLS) comprising about 25 000 MegaCam images was data mined to search for serendipitous encounters of known Near Earth Asteroids (NEAs) and Potentially Hazardous Asteroids (PHAs). A total of 143 asteroids (109 NEAs and 34 PHAs) were found on 508 candidate images which were field corrected and measured carefully, and their astrometry was reported to Minor Planet Centre. Both recoveries and precoveries (apparitions before discovery) were reported, including data for 27 precovered asteroids (20 NEAs and 7 PHAs) and 116 recovered asteroids (89 NEAs and 27 PHAs). Our data prolonged arcs for 41 orbits at first or last opposition, refined 35 orbits by fitting data taken at one new opposition, recovered 6 NEAs at their second opposition and allowed us to ameliorate most orbits and their Minimal Orbital Intersection Distance (MOID), an important parameter to monitor for potential Earth impact hazard in the future (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of the YORP effect on rotation rates of near‐Earth asteroids

The distribution of near‐Earth asteroid (NEA) rotation rates differs considerably from the similar distribution of Main Belt asteroids (MBAs) by the presence of excesses of fast and slow rotators, which are not observed or not so prominent in the distribution for MBAs. Among possible reasons for the difference, there can be influence of solar radiation on spin rate of small NEAs, the so‐called “YORP effect,” which appears due to reflection, absorption, and IR re‐emission of the sunlight by an irregularly shaped rotating asteroid. It is known that the YORP‐effect action strongly depends on the amount of solar energy obtained by the body (insolation), its size, and albedo. The analysis of observation data has shown that: (1) the mean diameter of NEAs decreases from the middle of the distribution to its ends, that is, the excesses of slow rotators (ω ≤ 2 rev day^?1) and fast rotators (ω ≥ 8 rev day^?1) are composed of smaller NEAs than in the middle of the distribution; (2) NEAs of both excesses are in the orbits where their insolation is about 8–10% larger than that of NEAs in the middle of the distribution; and (3) the objects in both excesses have a little lower albedo on average than that of objects in the middle of the distribution. All these results qualitatively agree well with the YORP‐effect action and may be considered as independent arguments in favor of it.     

Steady-state size distributions for collisional populations:: analytical solution with size-dependent strength

The steady-state population of bodies resulting from a collisional cascade depends on how material strength varies with size. We find a simple expression for the power-law index of the population, given a power law that describes how material strength varies with size. This result is extended to the case relevant for the asteroid belt and Kuiper belt, in which the material strength is described by 2 separate power laws—one for small bodies and one for larger bodies. We find that the power-law index of the small body population is unaffected by the strength law for the large bodies, and vice versa. Simple analytical expressions describe a wave that is superimposed on the large body population because of the transition between the two power laws describing the strength. These analytical results yield excellent agreement with a numerical simulation of collisional evolution. These results will help to interpret observations of the asteroids and KBOs, and constrain the strength properties of those objects.