Impact experiments: 1. Ejecta velocity distributions and related results from regolith targets

Velocity distributions are determined for ejecta from 14 experimental impacts into regolithlike powders in near-vacuum conditions at velocities from 5 to 2321 m/sec. Of the two powders, the finer produces slower ejecta. Ejecta include conical sheets with ray-producing jets and (in the fastest impacts at $\text{V}{}_{\mathrm{<mtext>imp</mtext>}}\text{? 700}\text{m/sec}\text{)}$ high-speed vertical plumes of uncertain nature. Velocities in the conical sheets and jets increase with impact velocity (Sect. 6). Ejecta velocities also increase as impact energy and crater size increase; a suggested method of estimating ejecta velocity distributions in large-scale impacts involves homologous scaling according to R/R_crater, where R is radial distances from the crater (Sect. 7). The data are consistent with Holsapple-Schmidt scaling relationships (Sect. 8). The fraction of initial total impact energy partitioned into ejecta kinetic energy increases from around 0.1% for the slow impacts to around 10% for the fast impacts, with the main increase probably at the onset of the hypervelocity impact regime (Sect. 9). Crater shapes are discussed, including an example of a possible “frozen” transient cavity (Sect. 10). Ejecta blanket thickness distributions (as a function of R) vary with target material and impact speed, but the results measured for hypervelocity impacts agree with published experimental and theoretical values (Sect. 11). The low ejecta velocities for powder targets relative to rock targets, together with the paucity of powder ejecta in low-speed impacts ( < 1 projectile mass for V_imp ≈ 10 m/sec) enhance early planetary accretion effeciency beyond that in some earlier theoretical models; 100% efficient accretion is found for certain primordial conditions (Sect. 12).     

Collisional growth of planetesimals

Safronov's (1972) demonstration that relative velocities of planetesimals would be comparable to the dominant size bodies' escape velocities, combined with a plausible size distribution that has most mass in the largest bodies, yielded his evolution model with limited growth of the largest planetesimal with respect to its next largest neighbors. A numerical simulation of planetesimal accretion (Greenberget al., 1978) suggests that at least over one stage of collisional accretion, velocities were much lower than the escape velocity of the largest bodies, because the bulk of the mass still resided in km-scale bodies. The low velocities at this early stage may conceivably have permitted early runaway growth, which, in turn, would have kept the velocities low and permitted continued runaway growth of the largest bodies.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.     

Orbital evolution of the Gefion and Adeona asteroid families: close encounters with massive asteroids and the Yarkovsky effect

Asteroid families are groupings of minor planets identified by clustering in their proper orbital elements; these objects have spectral signatures consistent with an origin in the break-up of a common parent body. From the current values of proper semimajor axes a of family members one might hope to estimate the ejection velocities with which the fragments left the putative break-up event (assuming that the pieces were ejected isotropically). However, the ejection velocities so inferred are consistently higher than N-body and hydro-code simulations, as well as laboratory experiments, suggest. To explain this discrepancy between today’s orbital distribution of asteroid family members and their supposed launch velocities, we study whether asteroid family members might have been ejected from the collision at low speeds and then slowly drifted to their current positions, via one or more dynamical processes. Studies show that the proper a of asteroid family members can be altered by two mechanisms: (i) close encounters with massive asteroids, and (ii) the Yarkovsky non-gravitational effect. Because the Yarkovsky effect for kilometer-sized bodies decreases with asteroid diameter D, it is unlikely to have appreciably moved large asteroids (say those with D > 15 km) over the typical family age (1-2 Gyr).For this reason, we numerically studied the mobility of family members produced by close encounters with main-belt, non-family asteroids that were thought massive enough to significantly change their orbits over long timescales. Our goal was to learn the degree to which perturbations might modify the proper a values of all family members, including those too large to be influenced by the Yarkovsky effect. Our initial simulations demonstrated immediately that very few asteroids were massive enough to significantly alter relative orbits among family members. Thus, to maximize gravitational perturbations in our 500-Myr integrations, we investigated the effect of close encounters on two families, Gefion and Adeona, that have high encounter probabilities with 1 Ceres, by far the largest asteroid in the main belt. Our results show that members of these families spreads in a of less than 5% since their formation. Thus gravitational interactions cannot account for the large inferred escape velocities.The effect of close encounters with massive asteroids is, however, not entirely negligible. For about 10% of the simulated bodies, close encounters increased the “inferred” ejection velocities from sub-100 m/s to values greater than 100 m/s, beyond what hydro-code and N-body simulations suggest are the maximum possible initial ejection velocity for members of Adeona and Gefion with D > 15 km. Thus this mechanism of mobility may be responsible for the unusually high inferred ejection speeds of a few of the largest members of these two families.To understand the orbital evolution of the entire family, including smaller members, we also performed simulations to account for the drift of smaller asteroids caused by the Yarkovsky effect. Our two sets of simulations suggest that the two families we investigated are relatively young compared to larger families like Koronis and Themis, which have estimated ages of about 2 Byr. The Adeona and Gefion families seems to be no more than 600 and 850 Myr old, respectively.     

Asteroidal catastrophic collisions simulated by hypervelocity impact experiments

We report the results of six impact fragmentation experiments carried out with free-falling macroscopic targets of different compositions and shapes, and with projectile velocities close to 9 km/sec, i.e., significantly higher than the sound velocity in the target materials. The data have been examined by deriving the mass and shape distributions of the fragments, by reconstructing two of the shattered targets in order to study the geometry of the fracture surfaces, and by analyzing the properties of the fine-grained high-velocity ejecta. The fragment mass distributions show clearly that the degree of target fragmentation depends strongly on the impact parameter. Apart from the few largest fragments, these distributions are well represented by two power laws with different exponents, connected at a size of about 1 cm. The fragment shapes are generally in good agreement with those observed in previous experiments, and no significant shape vs size dependence has been found down to sizes of the order of 0.1 mm. The fragments tend to become larger and possibly more irregular in shape when they are generated farther from the impact point. The fracture surfaces are oriented roughly along meridians and parallels (with the pole at the impact point) when the target is spherical, but are clustered around the symmetry planes when the target is ellipsoidal. Fine-grained particles, with typical sizes and velocities of 0.01 cm and 1 km/sec, respectively, are ejected at low-elevation angles and in a rather collimated way, starting both from the neighborhood of the impact point and from regions of incipient cracking. Particular attention has been paid to a comparison between these results and the observed properties of the outcomes of asteroidal catastrophic collisions, like the dynamical families and the small inner planet crossing objects. While the collisional theory for the origin of families is fully consistent with the experimental results (with some indication for a significant role of the parent asteroid's self-gravitation), the elongated shapes of several Apollo-Amor objects are much rarer among the laboratory fragments, and thus appear to require a different explanation.     

Modelling the outcomes of high-velocity impacts between small solar system bodies

We present a self-consistent numerical algorithm aimed at predicting the outcomes of high-velocity impacts between asteroids (or other small bodies of the solar system), based on a set of model input parameters which can be estimated from the available experimental evidence, and including the possible gravitational reaccumulation of ejected fragments whose velocity is less than a suitably defined escape velocity. All the fragment mass distributions are modelled by truncated power laws, and a possible correlation between fragment ejection velocity and mass is taken into account in different ways, including a probabilistic one. We analyze in particular the effectiveness of the gravitational reaccumulation process in terms of different choices of the collisional parameters and the assumed relationship between fragment speed and mass. Both the transition size beyond which solid targets are likely to reaccumulate a large fraction of the fragment mass and the collision energy needed to disperse most of the fragments are sensitive functions of the assumed fragment velocity versus mass relationship. We also give some examples of how our algorithm can be applied to study the origin and collisional history of small solar system bodies, including the asteroid 951 Gaspra (recently imaged by the Galileo probe) and the asteroid families.     

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.     

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.     

Numerical simulations of collisions and gravitational encounters in systems of non-identical particles

Numerical simulations of 200 mutually colliding non-identical particles indicate that the equipartition of random kinetic energy is possible only in systems having a narrow distribution of particle masses. Otherwise the random energy is concentrated on heavy particles. The form of the velocity distribution versus particle mass depends also on the elastic properties of the particles, and on the relative importance of the particle size. If the coefficient of restitution is a weakly decreasing function of impact velocity, a large difference in the equilibrium velocities of largest and smallest particles is possible. On the other hand, if the elasticity drops to a low level even in the small velocity regime, the dispersion of velocities is maintained by finite size and differential rotation, and the velocities of smallest particles are, at most, slightly larger than those of the largest ones. The results of simulations are consistent with the predictions of the collisional theory of non-identical particles (Hämeen-Anttila, 1984). The application to Saturn's rings indicates that the geometric thickness of cm-sized particles is of the order of 50 m in the rarefied regions of the rings. Without the gravitational encounters a thickness of about 30 m is derived. These estimations are made by using the latest measurements (Bridges et al., 1984) for the restitution coefficient of icy particles.     

Editorial

A simulation of collisional and gravitational interaction in the early solar system generates planets ～500 km in diameter from an initial swarm of kilometer-sized planetesimals, such as might have resulted from gravitational instabilities in the solar nebula. The model treats collisions according to experimental and theoretical impact results (such as rebound, cratering, and catastrophic fragmentation) for a variety of materials whose parameters span plausible values for early solid objects. Ad hoc sticking mechanisms are avoided. The small planets form in ～10⁴ yr, during which time most of the mass of the system continues to reside in particles near the original size. The relative random velocities remain of the order of a kilometer-sized body's escape velocity, with random velocities of the largest objects somewhat depressed because of damping by the bulk of the material. The simulation is terminated when the largest objects' random motion is of smaller dimension than their collision cross sections, so that the “particle-in-a-box” statistical methods of the model break down. The few 500-km planets, in a swarm still dominated by kilometer-scale planetesimals, may act as “seeds” for the subsequent, gradual, accretional growth into full-sized planets.     

Formation of Asteroid Families by Catastrophic Disruption: Simulations with Fragmentation and Gravitational Reaccumulation

This paper builds on preliminary work in which numerical simulations of the collisional disruption of large asteroids (represented by the Eunomia and Koronis family parent bodies) were performed and which accounted not only for the fragmentation of the solid body through crack propagation, but also for the mutual gravitational interaction of the resulting fragments. It was found that the parent body is first completely shattered at the end of the fragmentation phase, and then subsequent gravitational reaccumulations lead to the formation of an entire family of large and small objects with dynamical properties similar to those of the parent body. In this work, we present new and improved numerical simulations in detail. As before, we use the same numerical procedure, i.e., a 3D SPH hydrocode to compute the fragmentation phase and the parallel N-body code pkdgrav to compute the subsequent gravitational reaccumulation phase. However, this reaccumulation phase is now treated more realistically by using a merging criterion based on energy and angular momentum and by allowing dissipation to occur during fragment collisions. We also extend our previous studies to the as yet unexplored intermediate impact energy regime (represented by the Flora family formation) for which the largest fragment's mass is about half that of the parent body. Finally, we examine the robustness of the results by changing various assumptions, the numerical resolution, and different numerical parameters. We find that in the lowest impact energy regime the more realistic physical approach of reaccumulation leads to results that are statistically identical to those obtained with our previous simplistic approach. Some quantitative changes arise only as the impact energy increases such that higher relative velocities are reached during fragment collisions, but they do not modify the global outcome qualitatively. As a consequence, these new simulations confirm previous main results and still lead to the conclusion that: (1) all large family members must be made of gravitationally reaccumulated fragments; (2) the original fragment size distribution and their orbital dispersion are respectively steeper and smaller than currently observed for the real families, supporting recent studies on subsequent evolution and diffusion of family members; and (3) the formation of satellites around family members is a frequent and natural outcome of collisional processes.     

Complete fragmentation of the parent bodies of Themis,Eos, and Koronis families

The fragmentation of the parent asteroids of the Themis, Eos, and Koronis families is investigated by considering mutual gravitational effects among the fragmented bodies. The masses of the parent asteroids and the kinetic and gravitational energies of the fragmented bodies are estimated. Comparison of these results and data from the laboratory impact experiments leads to the conclusion that the parent asteroids of the three families were completely fragmented at E_p/M of 10⁸ erg/g or more (E_p, impact energy; M, parent mass). However, since most of the fragments had low relative velocities many reaccumulated through mutual gravitation. The larger members in these families should have the rubble pile structures and hydrostatic equilibrium figures.     

Equilibrium velocities of a planetesimal population

The random velocities v(m) of planetesimal populations specified by maximum and minimum masses and a number density n(m) ∝ m^?q are calculated interatively based on two different physical models involving ratios of rates: (1) excitation of kinetic energy by gravitational perturbation and elastic collision equal to damping of kinetic energy by inelastic collisions; and (2) excitation of kinetic energy a ratio b (～3 usually) to doubling of mass. Model (2) follows the theory of Safronov closely. Both physical models are developed approximately, using averaged factors for collision dissipation and velocity ratios, and then more precisely, allowing for reference orbit differences and for plausible variations in collisional energy dissipation with impact velocity and planetesimal mass ratios. The approximate model (2) agrees reasonably well with the results of Safronov. Both precise models are applied to populations approximating those generated by the calculations of Greenberg, Cox, and Wetherill, producing qualitatively similar velocities v(m). These results encourage analytic models of planetesimal population growth, incrementing masses and calculating velocity distributions in alternate steps. The principal improvement needed in the models is more realistic collision energy partitioning.     

Relative velocities of planetesimals and the early accumulation of planets

Safronov's statement that relative velocities of planetesimals are on the order of the escape velocity of the largest body of the population is shown to be correct only when a major part of the total mass resides in several large bodies. In the first stage of accumulation, runaway accretion produces large bodies separated by mass form the remaining population. At this stage, relative velocities of planetesimals are much smaller than those adopted earlier. This requires a modification of Schmidt's scheme of accumulation of the Earth and other terrestrial planets from material in their feeding zones. This also leads to removal of the author's arguments (Levin 1972c) in favor of a protoplanetary nebula with an extended, massive periphery.Paper presented at the Conference on Protostars and Planets, held at the Planetary Science Institute, University of Arizona, Tucson, Arizona, between January 3 and 7, 1978.     

Size-frequency distributions of fragments from SPH/N-body simulations of asteroid impacts: Comparison with observed asteroid families

We investigate the morphology of size-frequency distributions (SFDs) resulting from impacts into 100-km-diameter parent asteroids, represented by a suite of 161 SPH/N-body simulations conducted to study asteroid satellite formation [Durda, D.D., Bottke, W.F., Enke, B.L., Merline, W.J., Asphaug, E., Richardson, D.C., Leinhardt, Z.M., 2004. Icarus 170, 243-257]. The spherical basalt projectiles range in diameter from 10 to 46 km (in equally spaced mass increments in logarithmic space, covering six discrete sizes), impact speeds range from 2.5 to 7 km/s (generally in 1 km/s increments), and impact angles range from 15° to 75° (nearly head-on to very oblique) in 15° increments. These modeled SFD morphologies match very well the observed SFDs of many known asteroid families. We use these modeled SFDs to scale to targets both larger and smaller than 100 km in order to gain insights into the circumstances of the impacts that formed these families. Some discrepancies occur for families with parent bodies smaller than a few tens of kilometers in diameter (e.g., 832 Karin), however, so due caution should be used in applying our results to such small families. We find that ∼20 observed main-belt asteroid families are produced by the catastrophic disruption of D>100 km parent bodies. Using these data as constraints, collisional modeling work [Bottke Jr., W.F., Durda, D.D., Nesvorný, D., Jedicke, R., Morbidelli, A., Vokrouhlický, D., Levison, H.F., 2005b. Icarus 179, 63-94] suggests that the threshold specific energy, , needed to eject 50% of the target body's mass is very close to that predicted by Benz and Asphaug [Benz, W., Asphaug, E., 1999. Icarus 142, 5-20].     

Velocity distributions of high-velocity ejecta from regolith targets

We measured the velocity distributions of impact ejecta with velocities higher than ∼100 m s⁻¹ (high-velocity ejecta) for impacts at variable impact angle α into unconsolidated targets of small soda-lime glass spheres. Polycarbonate projectiles with mass of 0.49 g were accelerated to ∼250 m s⁻¹ by a single-stage light-gas gun. The impact ejecta are detected by thin aluminum foils placed around the targets. We analyzed the holes on the aluminum foils to derive the total number and volume of ejecta that penetrated the aluminum foils. Using the minimum velocity of the ejecta for penetration, determined experimentally, the velocity distributions of the high-velocity ejecta were obtained at α=15°, 30°, 45°, 60°, and 90°. The velocity distribution of the high-velocity ejecta is shown to depend on impact angle. The quantity of the high-velocity ejecta for vertical impact (α=90°) is considerably lower than derived from a power-law relation for the velocity distribution on the low-velocity ejecta (less than 10 m s⁻¹). On the other hand, in oblique impacts, the quantity of the high-velocity ejecta increases with decreasing impact angle, and becomes comparable to those derived from the power-law relation. We attempt to scale the high-velocity ejecta for oblique impacts to a new scaling law, in which the velocity distribution is scaled by the cube of projectile radius (scaled volume) and a horizontal component of impactor velocity (scaled ejection velocity), respectively. The high-velocity ejecta data shows a good correlation between the scaled volume and the scaled ejection velocity.     

Catastrophic disruption of pre-shattered parent bodies

In this paper, we analyze the effect of the internal structure of a parent body on its fragment properties following its disruption in different impact energy regimes. To simulate an asteroid breakup, we use the same numerical procedure as in our previous studies, i.e., a 3D SPH hydrocode to compute the fragmentation phase and the parallel N-body code pkdgrav to compute the subsequent gravitational re-accumulation phase. To explore the importance of the internal structure in determining the collisional outcome, we consider two different parent body models: (1) a purely monolithic one and (2) a pre-shattered one which consists of several fragments separated by damaged zones and small voids. We present here simulations spanning two different impact energy regimes—barely disruptive and highly catastrophic—corresponding to the formation of the Eunomia and Koronis families, respectively. As we already found for the intermediate energy regime represented by the Karin family, pre-shattered parent bodies always lead to outcome properties in better agreement with those of real families. In particular, the fragment size distribution obtained by disrupting a monolithic body always contains a large gap between the largest fragment and the next largest ones, whereas it is much more continuous in the case of a pre-shattered parent body. In the latter case, the ejection speeds of large fragments are also higher and a smaller impact energy is generally required to achieve a similar degree of disruption. Hence, unless the internal structure of bodies involved in a collision is known, predicting accurately the outcome is impossible. Interestingly, disrupting a pre-shattered parent body to reproduce the Koronis family yields a fragment size distribution characterized by four almost identical largest objects, as observed in the real family. This peculiar outcome has been found before in laboratory experiments but is obtained for the first time following gravitational re-accumulation. Finally, we show that material belonging to the largest fragments of a family originates from well-defined regions inside the parent body (the extent and location of which are dependent upon internal structure), despite the many gravitational interactions that occur during the re-accumulation process. Hence fragment formation does not proceed stochastically but results directly from the velocity field imparted during the impact.     

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.     

Yarkovsky/YORP chronology of asteroid families

Asteroid families are the byproducts of catastrophic collisions whose fragments form clusters in proper semimajor axis, eccentricity, and inclination space. Although many families have been observed in the main asteroid belt, only two very young families, Karin and Veritas, have well-determined ages. The ages of other families are needed, however, if we hope to infer information about their ejection velocity fields, space weathering processes, etc. In this paper, we developed a method that allows us to estimate the ages of moderately young asteroid families (approximately in between 0.1 and 1 Gyr). We apply it to four suitable cases—Erigone, Massalia, Merxia, and Astrid—and derive their likely ages and approximate ejection velocity fields. We find that Erigone and Merxia were produced by large catastrophic disruption events (i.e., parent body ?100 km) that occurred approximately 280 and 330 Myr ago, respectively. The Massalia family was likely produced by a cratering event on Asteroid (20) Massalia less than 200 Myr ago. Finally, the Astrid family, which was produced by the disruption of a 60-70 km asteroid, is 100-200 Myr old, though there is considerable uncertainty in this result. We estimate that the initial ejection velocities for these families were only a few tens of meters per second, consistent with numerical hydrocode models of asteroid impacts. Our results help to verify that asteroid families are constantly undergoing dynamical orbital evolution from thermal (Yarkovsky) forces and spin vector evolution from thermal (YORP) torques.     

Interacting ellipsoids: a minimal model for the dynamics of rubble-pile bodies

A simple mechanical model is formulated to study the dynamics of rubble-pile asteroids, formed by the gravitational re-accumulation of fragments after the collisional breakup of a parent body. In this model, a rubble-pile consists of N interacting fragments represented by rigid ellipsoids, and the equations of motion explicitly incorporate the minimal degrees of freedom necessary to describe the attitude and rotational state of each fragment. In spite of its simplicity, our numerical examples indicate that the overall behavior of our model is in line with several known properties of collisional events, like the energy and angular momentum partition during high velocity impacts. Therefore, it may be considered as a well defined minimal model.     

Impact and explosion crater ejecta,fragment size,and velocity

A model was developed for the mass distribution of fragments that are ejected at a given velocity for impact and explosion craters. The model is semiempirical in nature and is derived from (1) numerical calculations of cratering and the resultant mass versus ejection velocity, (2) observed ejecta blanket particle size distributions, (3) an empirical relationships between maximum ejecta fragment size and crater diameter, (4) measurements of maximum ejecta size versus ejecta velocity, and (5) an assumption on the functional form for the distribution of fragments ejected at a given velocity. This model implies that for planetary impacts into competent rock, the distribution of fragments ejected at a given velocity is broad; e.g., 68% of the mass of the ejecta at a given velocity contains fragments having a mass less than 0.1 times a mass of the largest fragment moving at that velocity. Using this model, we have calculated the largest fragment that can be ejected from asteroids, the Moon, Mars, and Earth as a function of crater diameter. The model is unfortunately dependent on the size-dependent ejection velocity limit for which only limited data are presently available from photography of high explosive-induced rock ejecta. Upon formation of a 50-km-diameter crater on an atmosphereless planet having the planetary gravity and radius of the Moon, Mars, and Earth, fragments having a maximum mean diameter of ≈30, 22, and 17 m could be launched to escape velocity in the ejecta cloud. In addition, we have calculated the internal energy of ejecta versus ejecta velocity. The internal energy of fragments having velocities exceeding the escape velocity of the moon (～2.4 km/sec) will exceed the energy required for incipient melting for solid silicates and thus, the fragments ejected from Mars and the Earth would be melted.