Rushing to equilibrium: a simple model for the collisional evolution of asteroids

A simple mathematical model for the evolution of a system of collisionally interacting bodies—such as the asteroid population—consists of two coupled, nonlinear, first-order differential equations for the abundances of “small” and “big” bodies. The model easily allows us to recover Dohnanyi's value for the exponent of the equilibrium mass distribution. Moreover, the model shows that any initial value for the ratio of “big” to “small” bodies rapidly relaxes to the equilibrium ratio, corresponding to the exponent, and that integrating the evolution equations backward in time—an attractive possibility to investigate the mass distribution of primordial planetesimals—leads to strong numerical instability.     

Multifractal Fits to the Observed Main Belt Asteroid Distribution

Dohnanyi's (J. W. Dohnanyi, 1969, J. Geophys. Res.74, 2531-2554) theory predicts that a collisional system such as the asteroidal population of the main belt should rapidly relax to a power-law stationary size distribution of the kind N(m)∝m^−α, with α very close to 11/6, provided all the collisional response parameters are independent of size. The actual asteroid belt distribution at observable sizes, instead, does not exhibit such a simple fractal size distribution.We investigate in this work the possibility that the corresponding cumulative distribution may be instead fairly fitted by multifractal distributions. This multifractal behavior, in contrast with the Dohnanyi fractal distribution, is related to the release of his hypothesis of self-similarity.     

Fragment ejection velocities and the collisional evolution of asteroids

We have applied the algorithm developed by Petit and Farinella (Celest. Mech. 57, 1–28, 1993) to model the outcomes of impacts between asteroids of different sizes, to show that a crucial feature of these models is the assumed relationship between velocity and mass of fragments ejected after a shattering impact. Not only how the mean velocity depends upon mass is important to determine the extent of fragment reaccumulation, but also the distribution of velocities about the mean values. The available experimental evidence on this issue is still sparse, and does not constrain the collisional models well enough to allow us to make reliable predictions on the outcomes of impacts between bodies of size much larger than the laboratory targets. As a consequence, when the collisional outcome models are used as an input for simulations of the asteroid collisional history since the origin of the solar system, the results show a strong sensitivity to the assumed velocity vs mass relationship. This sensitivity is stronger in the diameter range (a few tens to a few hundreds of km) where the self-gravitational reaccumulation of fragments is most effective, but may also extend to much smaller sizes.     

Collisional history of asteroids: Evidence from Vesta and the Hirayama families

Collisional evolution studies of asteroids indicate that the initial asteroid population at the time mean collisional velocities were pumped up to ～5 km/sec was only modestly larger than it is today; i.e., the asteroid belt was already depleted relative to the mean surface density elsewhere in the planetary region. Numerical simulations of the collisional evolution of hypothetical initial asteroid populations have been run, subject to three constraints: they must (a) evolve to the present observed asteroid size distribution, (b) preserve Vesta's basaltic crust, and (c) produce at least the observed number of major Hirayama families. A “runaway growth” initial asteroid population distribution is found to best satisfy these constraints. A new model is presented for calculating the fragmental size distribution for the disruption of large, gravitationally bound bodies in which the material strength is increased by hydrostatic self-compression. This model predicts that large asteroid behave as intrinsically strong bodies, even if they have had a history of being collisionally fractured. This model, when applied to the breakup of the Themis and Eos family parent bodies, gives size distributions in reasonably good agreement with those observed.     

Geminid Meteor shower activity 2003–2005 as observed by Gadanki radar

The distribution of meteor signals reflected from a backscatter radar is considered according to their duration. This duration time (T) is used to classify the meteor echoes and to calculate the mass index (S) of different meteoroids of shower plus sporadic background. Observational data on particle size distribution of the Geminid meteor shower are very scarce, particularly at low latitudes. In this paper the observational data from Gadanki radar (13.46°N, 79.18°E) have been used to determine the particle size distribution and the number density of meteoroids inside the stream of the Geminid meteor shower. The mean variation of meteor number density across the stream has been determined for three echo duration classes, T<0.4, T=0.4–1 and T>1 s. We are more interested in the appearance of echoes of various durations and therefore meteors of various masses in order to understand more on the filamentary structure of the stream. It is observed that the faint particle flux peaks earlier than the larger particles. We found a decreasing trend in the mass index values from the day of peak activity to the next observation days. The mass index profile was found to be U-shaped with a minimum value near the time of peak activity. The observed minimum s values are 1.64±0.05 and 1.65±0.04 in the years 2003 and 2005, respectively. The activity of the shower indicates the mass segregation of meteoroids inside the stream. Our results are best comparable with the “scissors” structure model of the meteoroid stream formation of Ryabova [2007. Mathematical modeling of the Geminid meteoroid stream. Mon. Not. R. Astron. Soc. 375, 1371–1380] by considering the asteroid 3200 Phaethon as an extinct comet.     

Colour dependence of zodiacal light models

Colour models of the zodiacal light in the ecliptic have been calculated for both dielectric and metallic particles in the sub-micron and micron size range. Two colour ratios were computed, a blue ratio C_b (0.40 μm, 0.53 μm) and a red ratio, either C_r (0.82 μm, 0.53 μm) or C_r' (0.71 μm, 0.53 μm). The models with a size distribution ∝s^−2.5ds generally show a colour close to the solar colour and almost independent of elongation. Especially in the blue colour ratio there is generally no significant dependence on the lower cutoff size (0.1–1 μm). The main feature of absorbing particles is a reddening at small elongations. The models for size distributions ∝s⁻⁴ds show larger departures from solar colour and more variation with model parameters. Colour measurements, including red and near infra-red, therefore are useful to distinguish between flat and steep size spectra and to verify the presence of slightly absorbing particles.     

Origin of the ten degree Solar System dust bands

The Solar System dust bands discovered by IRAS are toroidal distributions of dust particles with common proper inclinations. It is impossible for particles with high eccentricity (approximately 0.2 or greater) to maintain a near constant proper inclination as they precess, and therefore the dust bands must be composed of material having a low eccentricity, pointing to an asteroidal origin. The mechanism of dust band production could involve either a continual comminution of material associated with the major Hirayama asteroid families, the equilibrium model (Dermott et al. (1984) Nature 312, 505–509) or random disruptions in the asteroid belt of small, single asteroids (Sykes and Greenberg (1986) Icarus 65, 51–69). The IRAS observations of the zodiacal cloud from which the dust band profiles are isolated have excellent resolution, and the manner in which these profiles change around the sky should allow the origin of the bands, their radial extent, the size-frequency distribution of the material and the optical properties of the dust itself to be determined. The equilibrium model of the dust bands suggests Eos as the parent of the 10° band pair. Results from detailed numerical modeling of the 10° band pair are presented. It is demonstrated that a model composed of dust particles having mean semimajor axis, proper eccentricity and proper inclination equal to those of the Eos family member asteroids, but with a dispersion in proper inclination of 2.5°, produces a convincing match with observations. Indeed, it is impossible to reproduce the observed profiles of the 10° band pair without imposing such a dispersion on the dust band material. Since the dust band profiles are matched very well with Eos, Themis and Koronis type material alone, the result is taken as strong evidence in favor of the equilibrium model. The effects of planetary perturbations are included by imposing the appropriate forced elements on the dust particle orbits (these forced elements vary with heliocentric distance). A subsequent model in which material is allowed to populate the inner solar system by a Poynting-Robertson drag distribution is also constructed. A dispersion in proper inclination of 3.5° provides the best match with observations, but close examination of the model profiles reveals that they are slightly broader than the observed profiles. If the variation of the number density of asteroidal material with heliocentric distance r is given by an expression of the form 1/r^τ then these results indicate that γ < 1 compared with γ = 1 expected for a simple Poynting-Robertson drag distribution. This implies that asteroidal material is lost from the system as it spirals in towards the Sun, owing to interparticle collisions.     

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.     

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.     

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.     

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.     

Yarkovsky depletion and asteroid collisional evolution

The orbital parameters of small asteroids change with time, as a consequence of the so-called Yarkovsky effect. This leads to a steady removal of objects from the Main Belt, which takes place when the objects reach one of the major resonant regions in the orbital elements space. The process may influence the evolution of the inventory and size distribution of Main Belt asteroids, but it has not been taken into account by classical models of the collisional evolution of the asteroid population. In this paper we discuss the role of the Yarkovsky effect in producing the current observed size distribution. We show that adding Yarkovsky effect to purely collisional mechanisms may increase the removal of objects at sizes around 1 km by a factor of about 2 with respect to a purely collisional scenario. Moreover, waves in the size distribution may also be produced. However, taking also into account current uncertainties in the efficiency of purely collisional mechanisms, the role of the Yarkovsky effect seems not dominant, and cannot be unambiguously determined.     

The role of collisions with interplanetary particles in the physical evolution of comets

Effects of collisions with interplanetary particles are investigated. To this purpose, collision probabilities for comets with different orbital elements are computed. It is found that collisions may have a non-negligible effect on the physical evolution of comets. In this connection, it is shown that under certain conditions collisional lifetimes may be shorter than dynamical or vaporization lifetimes. In particular, collisional lifetimes are on average shorter for comets in retrograde orbits than those for direct ones. It is further suggested that catastrophic collisions may contribute to prevent long-period comets in retrograde orbits from reaching short-period orbits by orbital diffusion. Collisions may also produce irregularities of the nucleus brightness by leaving exposed regions of fresh volatile material and may in this way lead to a rejuvenation of old dusty short-period comets. Catastrophic collision probabilities are too low to account for the observed comet splittings, so other trigger mechanisms should be at work. However, it is shown that collisional mini-bursts (increases in brightness of one magnitude or so) caused by decimeter-sized bodies may occur rather frequently on short-period comets when they pass through the asteroid belt. The burst observed in comet Tempel-2 at 3 AU in December, 1978 could be an example of such collisional mini-bursts. The systematic observation of periodic comets when they pass through the asteroid belt could give valuable information about the spatial density of decimeter and meter-sized bodies. In particular, collisional effects for comet Halley, for which a continuous surveillance is planned, are evaluated.     

Particles in the nebular midplane: Collective effects and relative velocities

Abstract– In the absence of global turbulence, solid particles in the solar nebula tend to settle toward the midplane, forming a layer with enhanced solids/gas ratio. Shear relative to the surrounding pressure‐supported gas generates turbulence within the layer, inhibiting further settling and preventing gravitational instability. Turbulence and size‐dependent drift velocities cause collisions between particles. Relative velocities between small grains and meter‐sized bodies are typically about 50 m s^?1 for isolated particles; however, in a dense particle layer, collective effects alter the motion of the gas near the midplane. Here, we develop a numerical model for the coupled motions of gas and particles of arbitrary size, based on the assumption that turbulent viscosity transfers momentum on the scale of the Ekman length. The vertical distribution of particles is determined by a balance between settling and turbulent diffusion. Self‐consistent distributions of density, turbulent velocities, and radial fluxes of gas and particles of different sizes are determined. Collective effects generate turbulence that increases relative velocities between small particles, but reduce velocities between small grains and bodies of decimeter size or larger by bringing the layer’s motion closer to Keplerian. This effect may alleviate the “meter‐size barrier” to collisional growth of planetesimals.     

Karin cluster formation by asteroid impact

Insights into collisional physics may be obtained by studying the asteroid belt, where large-scale collisions produced groups of asteroid fragments with similar orbits and spectra known as the asteroid families. Here we describe our initial study of the Karin cluster, a small asteroid family that formed 5.8±0.2 Myr ago in the outer main belt. The Karin cluster is an ideal ‘natural laboratory’ for testing the codes used to simulate large-scale collisions because the observed fragments produced by the 5.8-Ma collision suffered apparently only limited dynamical and collisional erosion. To date, we have performed more than 100 hydrocode simulations of impacts with non-rotating monolithic parent bodies. We found good fits to the size-frequency distribution of the observed fragments in the Karin cluster and to the ejection speeds inferred from their orbits. These results suggest that the Karin cluster was formed by a disruption of an ≈33-km-diameter asteroid, which represents a much larger parent body mass than previously estimated. The mass ratio between the parent body and the largest surviving fragment, (832) Karin, is ≈0.15-0.2, corresponding to a highly catastrophic event. Most of the parent body material was ejected as fragments ranging in size from yet-to-be-discovered sub-km members of the Karin cluster to dust grains. The impactor was ≈5.8 km across. We found that the ejections speeds of smaller fragments produced by the collision were larger than those of the larger fragments. The mean ejection speeds of >3-km-diameter fragments were . The model and observed ejection velocity fields have different morphologies perhaps pointing to a problem with our modeling and/or assumptions. We estimate that ∼5% of the large asteroid fragments created by the collision should have satellites detectable by direct imaging (separations larger than 0.1 arcsec). We also predict a large number of ejecta binary systems with tight orbits. These binaries, located in the outer main belt, could potentially be detected by lightcurve observations. Hydrocode modeling provides important constraints on the interior structure of asteroids. Our current work suggests that the parent asteroid of the Karin cluster may have been an unfractured (or perhaps only lightly fractured) monolithic object. Simulations of impacts into fractured/rubble pile targets were so far unable to produce the observed large gap between the first and second largest fragment in the Karin cluster, and the steep slope at small sizes (≈6.3 differential index). On the other hand, the parent asteroid of the Karin cluster was produced by an earlier disruptive collision that created the much larger, Koronis family some 2-3 Gyr ago. Standard interpretation of hydrocode modeling then suggests that the parent asteroid of the Karin cluster should have been formed as a rubble pile from Koronis family debris. We discuss several solutions to this apparent paradox.     

The fossilized size distribution of the main asteroid belt

Planet formation models suggest the primordial main belt experienced a short but intense period of collisional evolution shortly after the formation of planetary embryos. This period is believed to have lasted until Jupiter reached its full size, when dynamical processes (e.g., sweeping resonances, excitation via planetary embryos) ejected most planetesimals from the main belt zone. The few planetesimals left behind continued to undergo comminution at a reduced rate until the present day. We investigated how this scenario affects the main belt size distribution over Solar System history using a collisional evolution model (CoEM) that accounts for these events. CoEM does not explicitly include results from dynamical models, but instead treats the unknown size of the primordial main belt and the nature/timing of its dynamical depletion using innovative but approximate methods. Model constraints were provided by the observed size frequency distribution of the asteroid belt, the observed population of asteroid families, the cratered surface of differentiated Asteroid (4) Vesta, and the relatively constant crater production rate of the Earth and Moon over the last 3 Gyr. Using CoEM, we solved for both the shape of the initial main belt size distribution after accretion and the asteroid disruption scaling law . In contrast to previous efforts, we find our derived function is very similar to results produced by numerical hydrocode simulations of asteroid impacts. Our best fit results suggest the asteroid belt experienced as much comminution over its early history as it has since it reached its low-mass state approximately 3.9-4.5 Ga. These results suggest the main belt's wavy-shaped size-frequency distribution is a “fossil” from this violent early epoch. We find that most diameter D?120 km asteroids are primordial, with their physical properties likely determined during the accretion epoch. Conversely, most smaller asteroids are byproducts of fragmentation events. The observed changes in the asteroid spin rate and lightcurve distributions near D∼100-120 km are likely to be a byproduct of this difference. Estimates based on our results imply the primordial main belt population (in the form of D<1000 km bodies) was 150-250 times larger than it is today, in agreement with recent dynamical simulations.     

Catastrophic fragmentation and formation of families: Preliminary results from a new numerical model

Preliminary results of an improved version of the semiempirical model for catastrophic break up processes developed by Paolicchi et al., (1989) are presented. Among the several changes with respect to the old version, the most important seem to be related to the new treatment of gravitational effects, including self-compression and reaccumulation of fragments. In particular, the new model is able to analyze processes involving both cm-sized objects, like those studied by means of laboratory experiments, as well as much larger bodies, for which self-gravitational effects are dominant; moreover, in this latter case the model seems in principle adequate to describe with the same physics very different phenomena, like the formation of plausible asteroid families and the creation of single, rapidly spinning, objects. This fact, if confirmed by refined analyses, may be of high importance for our general understanding of asteroid collisional evolution.     

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 shallow magnitude distribution of asteroid families

It is well known that asteroid families have steeper absolute magnitude (H) distributions for H < 12-13 values than the background population. Beyond this threshold, the shapes of the absolute magnitude distributions in the family/background populations are difficult to determine, primarily because both populations are not yet observationally complete. Using a recently generated catalog containing the proper elements of 106,284 main belt asteroids and an innovative approach, we debiased the absolute magnitude distribution of the major asteroid families relative to the local background populations. Our results indicate that the magnitude distributions of asteroid families are generally not steeper than those of the local background populations for H > 13 (i.e., roughly for diameters smaller than 10 km). In particular, most families have shallower magnitude distributions than the background in the range 15-17 mag. Thus, we conclude that, contrary to previous speculations, the population of kilometer-size asteroids in the main belt is dominated by background bodies rather than by members of the most prominent asteroid families. We believe this result explains why the Spacewatch, Sloan Digital Sky Survey, and Subaru asteroid surveys all derived a shallow magnitude distribution for the dimmer members of the main belt population.We speculate on a few dynamical and collisional scenarios that can explain this shallow distribution. One possibility is that the original magnitude distributions of the families (i.e., at the moment of the formation event) were very shallow for H larger than ∼ 13, and that most families have not yet had the time to collisionally evolve to the equilibrium magnitude distribution that presumably characterizes the background population. A second possibility is that family members smaller than about 10 km, eroded over time by collisional and dynamical processes, have not yet been repopulated by the break-up of larger family members. For this same reason, the older (and possibly characterized by a weaker impact strength) background population shows a shallow distribution in the range 15-60 km.     

Catastrophic fragmentation as a stochastic process: sizes and shapes of fragments

It is rather difficult to understand theoretically and to analyse the experimental data concerning the mass and shape distributions of fragments created by catastrophic collisions. The fragmentation process is discussed as being a purely stochastical phenomenon; the size and shape distributions obtained in this way are compared with the results of laboratory experiments. The results are presented of some computer simulations of random volume fragmentation processes; they are a 3-D generalization of the numerical experiments described in Grady and Kipp (J. Appl. Phys. 58(3), 1210–1222, 1985). The features of the size distribution are discussed, comparing it with the expectations of the Mott-Linfoot and Grady-Kipp theories. In the literature the shape of fragments is defined in terms of the ratios B/A and C/A, where A, B, C are defined as the sizes of a fragment along three orthogonal axes. The definition of the shape of a fragment cannot be considered unique, since it is not obvious in which order to define the three axes when the fragments are not ellipsoidal. A few possible methods are introduced explicity, and the resulting differences are discussed. In this light, the shape results (the mean values and the distribution of the axial ratios) obtained in recent laboratory experiments are rediscussed and critically reviewed. For what concerns the stochastical modelling, the results of various simulations, corresponding to different assumptions regarding fragmentation properties are presented. It is shown that the main features of the shape distributions from laboratory experiments cannot be satisfactorily reproduced. Comparison of the results with the outcomes of the semiempirical fragmentation model by Paolicchi et al. (Icarus 121, 126–157, 1996), as well as with some results coming out from hydrodynamical simulations, shows how only a “global” and physical model, not a purely statistical one (neither global nor “local”), can afford to reproduce the observed data.