Asteroid Impacts: Laboratory Experiments and Scaling Laws

Holsapple et al.: Asteroid Impacts: Experiments and Scaling Laws

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Asteroid Impacts: Laboratory Experiments and Scaling Laws K. Holsapple University of Washington

I. Giblin Planetary Science Institute

K. Housen The Boeing Company

A. Nakamura Kobe University

E. Ryan New Mexico Highlands University

The present states of the small bodies of the solar system are largely an outcome of collisional processes. A rocky main-belt asteroid has endured a multitude of small and large cratering impacts; for example, estimates here show that one starting with a radius of 1 km has been shattered about five times every 106 yr and one with a radius of 10 km has been shattered about every 107 yr, or perhaps even more frequently in the past when collision rates were higher than they are now. All solar system bodies bear the scars and imprints of those impacts. Much has been learned about these topics since publication of the Asteroids II book (Fujiwara et al., 1989). Here we briefly review the previous wisdom, but primarily address new experiments, calculations, and scaling methods.

1.

INTRODUCTION

An understanding of collisional processes and collisional evolution is required in order to interpret observations of the solar system. Although laboratory experiments and computer simulations have provided many insights into these processes, our understanding of energetic impacts is still relatively primitive. Each new view of asteroids provided by spacecraft brings new surprises. For example, the substantial regolith on a small body like Gaspra, the huge closely packed craters on Mathilde, and the block-strewn surface of Eros were entirely unexpected. Nevertheless, experiments and modeling, guided by observations of asteroids, will allow us to deduce much about the collision history of these bodies, including crater sizes and frequency, crater morphology, ejecta block distributions, regolith development, and the formation of asteroid families. Impact processes are very complex and involve extreme ranges of conditions. The initial coupling of energy from a high-speed impactor into another body can occur in microseconds, with the extreme pressures and temperatures sufficient to melt and vaporize the target and projectile material. On large bodies, the latter stages of these processes can continue for hours and involve very low pressures. These conditions range from those encountered in the initial deto-

nation of nuclear bombs down to those involved in the statics of ordinary soil and rock mechanics. The results of a given impact are therefore very difficult to predict, although much progress has been made over the last few decades. The outcome of a collision depends largely on the ratio of the kinetic energy of the impactor to the mass of the impacted body, a specific energy, commonly denoted as Q. Two threshold values of Q are often defined, although the literature is not consistent in terminology or notation. Impacts with small values of Q form craters, but leave the target body largely intact. Larger values of Q can shatter a body into numerous pieces. The specific energy to shatter, Q*S, is defined as the threshold value for which the largest remaining intact piece immediately following a collision has one-half the mass of the original body. We refer to it as the shattering energy. The shattered pieces may reaccumulate or not, depending on their velocity relative to the escape velocity. A higher threshold Q*D is the specific energy such that the largest object following reaccumulation is one-half the mass of the original body. This is called the dispersion energy. The term disruption energy is sometimes used in the literature for either of these thresholds, often in a generic way for any significant breaking and/or dispersion. Numerous unsolved problems remain for collisions in all three regimes. Important questions about cratering in-

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clude the crater size; the shape; the amount of compaction; and the amount, velocity, and fate of ejected material. For shattering impacts, the distributions of velocity, size, shape, and spin of the pieces, and the ultimate fate of those pieces, are still largely unknown. How many fragments remain in place, how many are lofted and reaccumulate, and what is the structure of that modified body? What conditions determine whether a body is shattered but basically intact, as Eros might be, or completely turned to rubble? Finally, for energy above Q*D, when the body is shattered and scattered, significant questions remain about the size distribution of the fragments, the largest piece, and their ultimate fate. There are three interrelated and complementary approaches to the study of these processes. All three approaches suffer from our lack of detailed knowledge about the materials and structure of solar system bodies and asteroids. In addition, each approach also has its own shortcomings. The first approach is to conduct laboratory experiments. Laboratory methods use guns to launch a projectile into a target at speeds up to ~7 km/s, or explosive charges to simulate an impact. While experiments allow the study of actual geological materials, their primary shortcoming is the inability to use targets and projectiles of the size of interest. Laboratory experiments are limited to samples of centimeter size, while the asteroids range to many hundreds of kilometers in size. The response of a small target to impact is dominated by its material strength, while that of a large asteroid is dominated by gravitational forces. For the predominantly unidirectional gravity field that governs surface cratering, the lack of sufficient gravity for the experimental small targets can sometimes be overcome by performing experiments at high artificial gravity using a geotechnical centrifuge, but that tool is of little use for the three-dimensional gravity fields dominating catastrophic disruptions. Additional uncertainties are introduced by an inability to perform experiments at velocities of several tens of kilometers per second. Features that are important at low velocities, such as projectile material and shape, probably are not significant at high velocities. As a consequence of these limitations, laboratory experiments can only probe a small part of the parameter space of interest. The second approach is to use computer calculations based on the underlying physical principles. These methods continue to evolve. Finite-difference, finite-element and, over the last decade, smooth-particle-hydrodynamics (SPH) methods have been particularly useful, giving efficient ways to study impact processes. As computer power grows in leaps and bounds, we can now calculate many more cases in much less time. However, these methods are also limited in important ways, particularly by the relative infancy of material models. While laboratory experiments use actual materials, computer codes must rely on mathematical models of material behavior. The mechanical behavior of geological materials is considerably more difficult to model than the common metals and alloys used in structural applications. This shortcoming is exacerbated by the fact that the processes involve the extreme ranges of conditions men-

tioned above. The variety of physical material response is enormous, and we have at best a very rough understanding of relevant physical models. One must model the material behavior during shock propagation; crushing of voids in the material; and failure, flow, and fracture. Important features such as compaction, strain softening, nonlinearity, and hysteresis are seldom included. We have only a crude understanding of even the nature of these responses; even when complicated mathematical models are hypothesized and constructed, there is seldom enough data to calibrate them, especially for three-dimensional states. Tests of sensitivities to inputs are rarely made. As a result, code calculations must always be questioned, and even more so when few attempts are made to calibrate them against known experimental results. The third major approach uses scaling methods. Scaling theories are developed to predict how collisional processes will depend on the parameters of the problem, including the size, impact velocity, gravity, and material type. They are developed from considerations of similarity analysis applied to experimental results, from code calculations themselves, and from observations of asteroids. However, scaling laws are based on assumptions about the importance of various parameters. They always require some tie to experiments or calculations to determine unknown constants and can lead to erroneous conclusions if important parameters are neglected or results are extrapolated into regimes of new physics. Since the last contributions in Asteroids II, it has become clear that there are at least four major issues about the mechanics of small-body impacts that have not been sufficiently addressed. The first is the effect of substantial porosity. Twelve years ago there was conjecture about the existence of rubble-pile asteroids [the term “rubble piles” was first introduced by Davis et al. (1977)], but no knowledge of the effects of the implied porosity on the shock processes. There were neither analysis nor code calculations for the cratering and disruption of porous asteroids. Now the scientific community seems to have reached consensus that many and maybe even most large asteroids are reaccumulated rubble piles with possibly large porosity. The discovery of low densities in C-type asteroids such as Mathilde and Eugenia strongly indicate high porosity. For example, Mathilde has a bulk density of about 1.3 g/cm3, implying a porosity of 50%. Rocky asteroids are estimated to have been shattered many times over the lifetime of the solar system and may reaccumulate into a low-density state. Comets are also thought to be very low-density conglomerates of ice and dirt. Porosity may be the dominant physical property governing an impact process. The mechanics of impacts into highly porous bodies is substantially different than for lowporosity bodies, due to significant energy losses from the outgoing shock wave as it compacts the target material. Recent observations of the large craters on Mathilde imply substantial differences in basic cratering mechanisms and ejecta existence and fate. While experiments have been conducted in porous materials such as dry soil and sand, the com-


paction processes in those bodies is much less important than in much more porous bodies. This remains an area of extreme uncertainty, although some progress has been made. The second issue deals with rate-dependent strength effects. There were also at the time of Asteroids II simple theories, but no data, that represented the effective strength of rocky asteroids as size and time dependent. The extrapolation from small-scale experiments to large asteroids is determined primarily by the form of those strength dependences, but the exact nature of those forms was not known. This issue has now been addressed both experimentally and using codes, as is summarized below, but many uncertainties remain. Thirdly, the discovery of the Kuiper Belt in 1992, together with a general feeling in the community that the distinction between asteroids and comets may only be one of nomenclature, has prompted a number of studies into the impact behavior of icy materials and the interpretation of these data in the context of a potentially large icy planetesimal population in the greater solar system. We will briefly discuss ice experiments in this chapter and provide references for further study. However, the study of ice under impact is far less complete than that of rocky materials and no coherent models have been developed for the scaling of ice impacts to realistic sizes and conditions. Finally, much remains to be learned about the effects of oblique impacts. To first order the obliquity may be accounted for by a simple reduction of normal component of velocity, but there may be other more subtle effects deserving of further study, particularly for near-grazing impacts. This chapter is to be considered as an addition to the corresponding chapter in Asteroids II (Fujiwara et al., 1989). The previous results are generally not presented again. Here we review some previous wisdom, but primarily address new approaches and results. Mention of all the research in this broad topic over the last 12 years would produce a reference section alone larger than our allocated space, so we apologize to those whose important work we did not mention. 2. 2.1.

LABORATORY EXPERIMENTS

Overview

The 12 years since the publication of the Asteroids II book have seen many new experimental studies relevant to asteroids, and have produced a valuable database of experimental results (Fujiwara et al., 1989; Martelli et al., 1994), helping us to predict such quantities as the energy required for catastrophic breakup and the post-impact fragment shapes, sizes, and velocities. Target materials used in impact experiments have included rock, glass, clay, sand, loose aggregates, ice and ice-silicate mixtures, and artificial materials such as cement mortar, clay, alumina, and plaster. The impacting projectile has consisted of metals (aluminum, steel, iron), Pyrex, mortar, basalt, nylon, polycarbonate, and ice. Projectile velocities have ranged from a few meters per

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second to >6 km/s, and most tests have been conducted in evacuated chambers (Fujiwara et al., 1977; Takagi et al., 1984; Davis and Ryan, 1990; Ryan et al., 1991; Nakamura and Fujiwara, 1991; Nakamura et al., 1992). Some aspects of impact disruption (e.g., fragment velocity) have been studied by multiple researchers, while other aspects (e.g., rotational modes of ejecta) have been measured by only one or two researchers. Extensive use of computerized image processing — essentially impossible in 1989 — has provided significant new data. This section is intended to provide a comprehensive review of significant experimental studies carried out between 1989 and 2001, to summarize the results, and to briefly discuss their relevance. One must carefully note the wide variations in results in different materials due to the large differences in impact velocities. 2.2.

Impact Techniques and Measurements

The methods to launch high-speed projectiles have remained unchanged over several decades, and include powder guns, light-gas guns, and electromagnetic launchers. Velocities range from tens of meters per second to ~7 km/s. Both laboratory and field explosive tests have also been used to study energetic disruptions. Means of measuring the results of such tests have improved in the past few decades, largely due to the availability of fast-framing video cameras and image-processing computer systems. High-speed frame rates between 400 per second (e.g., Giblin et al., 1994a) and 6000 per second (e.g., Nakamura, 1993) have been used. Film or video footage is typically digitized for computer analysis, thus enabling researchers to measure the inflight dynamical properties of fragments. Examples include Nakamura and Fujiwara (1991) and Davis and Ryan (1990). Giblin et al. (1994a) used two cameras to measure fragment velocities in three dimensions. Cintala et al. (1997) used a strobed laser system to measure cratering ejecta velocities. Giblin (1998) discusses methods to recover three-dimensional particle velocity trajectories from filmed records. 2.3.

Disruption of Target Materials

As stated above, the kinetic energy per unit target mass Q, a specific energy, is used to measure collision outcomes. A related measure is the kinetic energy per unit volume of target, an energy density, which has the units of stress. The specific energy Q multiplied by the target mass density gives the energy density of the impact. “Impact strength” is sometimes defined as either the specific energy or the energy density needed to produce a largest intact fragment that contains one-half the target mass (see Fujiwara et al., 1977; Davis and Ryan, 1990; and Ryan et al., 1991). Here the symbol Q is always used for the specific energy and the energy density is written as ρQ. Figure 1 shows many experimental results for the ratio of the mass of the largest postimpact fragment to the initial target mass, as a function of the total impact specific energy

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10 0 Shatter Threshold Metal

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Ice 10 –2 Silicates

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crushed ice target, broken ice projectile solid ice target, aluminum projectile pellet or chipped ice target, aluminum projectile crushed ice target, solid ice projectile solid ice target, solid ice projectile solid ice target, broken ice projectile

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Q = E/M (erg/g) Fig. 1. Summary of disruption (shattering) experiments in various materials. Data from a variety of sources, including Hartmann (1969), Fujiwara et al. (1977), Fujiwara and Tsukamoto (1980, 1981), Lange and Ahrens (1981), Matsui et al. (1982, 1984), Kawakami et al. (1983), Fujiwara and Asada (1983), Takagi et al. (1984), Cintala and Hörz (1984), Cintala et al. (1985), Smrekar et al. (1985), Hartmann (1988), Davis and Ryan (1990), Ryan et al. (1991), and Nakamura and Fujiwara (1991). There is a general grouping by material types, but significant scatter within material types because of differences in impact velocity, temperature, projectile types, target strength, and many other factors.

Q. A value of Q with the ordinate equal to 1/2 as indicated defines the shattering specific energy Q*S. Included are results for ice, silicate, and meteoritic metal targets. The data show that the degree of fragmentation is strongly dependent on the target material. In all materials, increasing collisional energy increases the degree of fragmentation. Experimental studies using rock projectiles to impact solid rock targets reveal that projectile/target density, mass, or strength differences can also have a significant influence on collision outcome, especially for velocities below ~1 km/s (Matsui et al., 1982; Takagi et al., 1984). Kato et al. (1992) noted similar effects in ice targets impacted by ice, aluminum, polycarbonate, and basalt projectiles. Davis and Ryan (1990), Ryan et al. (1991), and Ryan et al. (1999) found that there is systematically less collision damage to the target at the same specific energy as the projectile becomes weaker. A major question about the interpretation of such experiments for asteroids is the possibility of a size- or rate-dependent strength (see Fugiwara et al., 1989), such as from a model based on the growth and coalescence of inherent flaws in natural rock. [Rate dependence gives the same scaling result as size dependence, since for large bodies all impact processes increase in duration with the size of the body, i.e., larger bodies have slower processes. A common trick in

the movie industry is to slow down the depiction of smallscale simulations to make them appear to be large-scale.] A method to scale these experiments assuming such a sizedependent strength based on crack growth was given first by the scaling model of Holsapple and Housen (1986) as discussed below. The computer simulations by Ryan and Melosh (1998) and Benz and Asphaug (1999), using the modeling approach of Melosh et al. (1992), have since replicated aspects of the scaling, a consequence of their use of a strength model that is rate-dependent. Of course, those studies do not prove that rocky materials actually have rate or size-dependent strength. The first experimental data on how target size affects collisional outcome was provided by Housen and Holsapple (1999a,c) using homogeneous granite targets. Target diameters were varied by a factor of 18 (up to a target diameter of 34.4 cm), and specific energy was kept constant as the size scale was increased. The larger bodies were found to be weaker in impacts than the smaller ones. Meteoritic targets have also been used in impact studies. A series of high-velocity impact experiments into cooled Gibeon iron-nickel meteorites were performed by Ryan and Davis (2001). It was found that at asteroid belt temperatures near 167 K, iron meteorites underwent brittle fracture, and


the resultant fragment size distributions had the typical twoslope power law behavior often observed for homogeneous rock targets. The impact strength was determined to be about 500× larger than for basalt. Ejecta velocities were much higher than those observed in rock fragmentation experiments, consistent with predictions of scaling theories in which velocities scale with the square root of the material strength in the strength regime. Ryan and Davis (personal communication, 2001) also conducted impact experiments using iron meteorite targets that were not cooled. For these targets, the crater formed at the impact point had large rimflaps, and a very “plastic” overall morphology. 2.4. Prefragmented and Shattered Targets To model fractured asteroids, Ryan et al. (1991) constructed samples of previously shattered mortar targets by careful reassembly and weak cementation, and reimpacted them. They found no large differences in impact strength between the preshattered targets and the original targets from which they were constructed, even though the preshattered targets had some porosity and a compressive strength that was less than half that of the original strong mortar target. For some unknown reason, the mean ejecta speeds from these bodies were higher than those measured for the original strong homogeneous targets. The resultant fragment size distributions for the preshattered targets were not significantly different from their homogeneous counterparts, i.e., they were not further fractured upon reimpact. Giblin et al. (1994a) tested the effect of void spaces in targets by fabricating a three-section, strong cement mortar body. The sections consisted of a top and bottom spherical cap, with a 1-mm spacer separating these pieces from a middle section. They reported no appreciable difference between the velocity distributions from these targets compared to homogeneous targets of the same composition. They concluded that the 1-mm spacing might not been a large enough to affect shock wave propagation and subsequent target fracture. Nakamura et al. (1994) also examined the effect of reimpacting previously fractured targets. They performed experiments in which a projectile was shot into a “core” fragment produced from a previous impact event. They found that the outcomes, including largest fragment masses, mass distribution of fragments, and size-velocity distributions, were not changed significantly. However, fine dusts were spewed out with a velocity higher than tens of meters per second within a short time after the impact, probably from the interior of surface cracks. 2.5. Disruption of Porous Targets Porous targets including gypsum (Kawakami et al., 1991; Nakamura et al., 1992), porous alumina, cement mortar (Davis and Ryan, 1990), and sandbags (Yanagisawa and Itoi, 1993) have been tested in impact disruption experiments. Exceptionally high impact strengths have been found for such porous target materials. While these porous materials are fragile in terms of tensile or compressive strength,

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their high porosity readily dissipates the energy delivered in an impact. Thus, unusually large energies are required to damage porous structures. Ryan et al. (1999) impacted porous and solid ice bodies at a temperature of about –15°C with fractured ice, solid ice, and aluminum projectiles at low velocities, and determined that the impact strength of the porous ice was higher by a factor of about 5 than the impact strength of solid ice targets. The degree of fragmentation also increased with the strength of the projectile. The porous ice targets were as strong as silicates when hit with fractured ice projectiles. Therefore, even though the very porous ice targets had a static material strength well below that for solid ice, they were just as resistant to collision as solid ice. Energy is apparently well dissipated by the void spaces within the target. Kawakami et al. (1991) constructed an ellipsoidal sample of gypsum with a porosity of 10–30% to model Phobos, and impacted the body to produce a crater equivalent to the Stickney crater. Fracture patterns appeared similar for both low- and high-velocity impacts, which was attributed to the low shock impedance and porosity of the target (i.e., shock wave attenuation occurred). They concluded that target type significantly affects the resulting mode of fragmentation. The impact strength was found to be comparable to that of basaltic bodies, even though the static strength of gypsum is 1–2 orders of magnitude lower than basalt. To model stone meteorites, Durda and Flynn (1999) conducted a series of ~5-km/s experiments using inhomogeneous, porous targets constructed of two materials having different strengths (porphyritic olivine basalt). The largest fragments generated were representative of the bulk composition, while millimeter-sized fragments were composed of isolated olivine crystals. The latter indicates that the target experienced preferential failure along phenocryst-matrix boundaries. They conclude that collisions involving chondritic asteroids may overproduce olivine-rich material in the millimeter size range, and olivine may be underrepresented at smaller sizes in the primary debris. Nakamura et al. (1992) used gypsum spheres to investigate the fragment velocity distribution of a porous body. The antipodal fragment velocity for the gypsum target was lower than that for a basalt target impacted with a similar energy density. Yanagisawa and Itoi (1993) also found that their sand-bag and porous alumina targets produced fragments with significantly lower velocities than a comparable impacts in nonporous basalt. Love et al. (1993) found that increasing a target’s porosity had the effect of decreasing the speeds of the ejecta for equal collision conditions. None of those studies give information about gravitational effects for large bodies. Scaling theories predict that lithostatic pressure considerably strengthens a body against disruption, which is why disruption specific energy increases markedly in the gravity regime (see section 3 below). Housen et al. (1991) and Housen (1993) simulated the effects of gravity by applying external pressure to small, weakly cemented, porous basalt targets. The largest overpressures corresponded to an average lithostatic stress inside a 460-km-diameter body. They used explosives buried

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at appropriate depths to simulate an impact disruption. They found that as overpressure is increased, more specific energy was required to shatter the body, and that there was a marked increase in the size of the largest fragment, confirming the scaling theory. Also, the specific energies measured for catastrophic disruption of the target body compared well with those estimated from observations of the Themis, Eos, and Koronis asteroid families. 2.6. Cratering and Ejecta in Porous Targets Compaction mechanisms may dominate the cratering processes of highly porous, pristine bodies such as cometesimals (Sirono and Greenberg, 2000) and asteroids such as Mathilde (Housen et al., 1999). Michikami et al. (2001) impacted centimeter-sized glass bead targets of various porosities in an evacuated chamber to view and measure ejecta velocities. The ejecta velocities decreased markedly as the porosity increased. For the most porous targets (80% and 60% porosity), measured velocities were more than 2 orders of magnitude below those measured for rocks, and for a porosity of 60% only 2% of the ejecta had a velocity greater than 10 m/s. For the 33-km crater on Mathilde, a velocity of 10 m/s is required for material to escape the crater, so these experiments suggest that almost no visible ejecta would be present. Love et al. (1993) also performed cratering experiments in porous glass targets, although the craters were dominated by the spall features of small experiments. Housen and Holsapple (1999b) and Housen et al. (1999) report impact experiments at 1.9 km/s into a very porous material with a density of 0.9 g/cm3 and very low crush strength, as a simulant for cratering on Mathilde. The experiments were performed at 500G on a centrifuge in order to reproduce the ejecta ballistics and lithostatic forces involved in large cratering events on Mathilde. Cratering was dominated by compaction, with negligible ejecta. That interpretation was corroborated by post-event computed tomography scans of the targets that clearly showed substantial increases in density below the crater, accounting for the crater volume. Those results implied that the five largest craters on Mathilde would have increased its overall density by about 20%. More recent cratering experiments at 500G in that same porous material were recently reported by Housen and Voss (2001). For porosities of about 50%, only about 10% of the crater mass was ejected; the crater was formed primarily by compaction. Additional experiments at other gravity levels have also been performed to test scaling and gravitystrength transition. The scaling implications of these experiments are under study. Yamamoto and Nakamura (1997) examined the applicability of the Housen et al. (1983) scaling for very-highvelocity ejecta (hundreds of meters per second) from craters generated by oblique impacts into powder glass sphere targets with a porosity of 44%. The laboratory data were estimated to be about an order of magnitude lower than the results extrapolated from the scaling formula. Since the high-velocity ejecta originate from near the impactor the

point-source assumption of the scaling theory is not valid, so the discrepancy is not surprising. Yamamoto (2001) performed further oblique impact cratering experiments into glass particle targets at various impact angles, and determined the impact angle effects. The size distribution of the regolith material on an asteroid surface affects the asteroid observational properties such as albedo, spectra, and thermal emission. Ejection processes due to repetitive impacts of micrometeoroids on the surface can alter the size distribution of the regolith. Yamamoto et al. (personal communication, 2001) investigated the size-velocity relation of ejecta from the surface of glass spheres. They mixed three different size glass spheres as the target, and found that the flux of high-velocity ejecta increases as the particle size decreases. Cintala et al. (1999) analyzed ejecta velocities for craters formed in coarse-grain sand. For impacts between 0.8 and 1.9 km/s, the ejecta velocity distribution was found to be a power law, although the exponent differed significantly from that expected from scaling theory. They suggested this discrepancy might be due to the fact that the sand grains were comparable in size to the impactor. 2.7. Experiments in Ice Experimental studies in ice are important in the light of studies suggesting that collisions are very important evolutionary processes even in the Kuiper Belt (Farinella and Davis, 1996; Davis and Farinella, 1997) and the Oort Cloud (Stern and Weissman, 2001). Croft (1982) describes cratering experiments in porous ice in which polyethylene projectiles from 2.3 to 6.3 km/s impacted sifted granular water ice with a density of 0.5 g/ cm3. He reports formation of “a large hemispherical cup whose walls consist of snow compacted to nearly the density of competent ice.” Clearly compaction played an important role. Mizutani et al. (1982) reported experiments in which aluminum and polycarbonate projectiles were fired into silicate and competent –18°C ice targets at 100–1000 m/s. They concluded that the specific energy required for the catastrophic failure of ice is approximately 2 orders of magnitude smaller than that required for basalts, and that the crater morphology in ice at these impact velocities is strongly dependent upon the projectile shape and material properties. 2.8. Fragment Properties and Distributions An important aspect of impact disruptions and asteroid evolution is the nature of the fragments: sizes, shapes, spins, and numbers. A number of experiments have been performed to measure these quantities. 2.8.1. Fragment sizes. Fragment size distributions following a disruptive impact tend to conform to power laws, although the distribution is often divided into two or three segments with each segment showing a different power-law exponent. The divisions between the segments


generally correspond to different fragmentation regimes in the disrupted target. For example, Di Martino et al. (1990) impacted alumina cement and basalt targets at 9 km/s, and 94–98% of target mass was recovered after the experiments. They found a knee (a bend in the size distribution) at millimeter sizes, which they attributed to the transition from cratering to bulk fragmentation. This process was studied analytically by Bashkirov and Vitzayev (1996). Since 1989 many authors have reported fragment size distribution data that broadly agree with the discussion in Asteroids II (see, e.g., Martelli et al., 1993; Mizutani, 1993; Davis, 1998). 2.8.2. Fragment shapes. Fragment shapes have not generally been reported, although Giblin et al. (1998b) goes into some detail. They compare their data to those listed in Asteroids II and to the semi-empirical model (SEM) of disruption developed by Paolicchi et al. (1989, 1996), and find good agreement. However, in the triaxial ellipsoid model using orthogonal dimensions a, b, c (see, e.g., La Spina and Paolicchi, 1996) the values of (b/a) = 0.60 and (c/a) = 0.45 found by those authors differ from their Asteroids II counterparts of 0.70 and 0.50. One possible explanation for this difference that the experiments were open-air and there was no secondary fragmentation of ejecta, which would tend to increase c/a and b/a. In studying the size and shape distribution in a laboratory experiment, it should be borne in mind that many studies suggest that monolithic asteroids are rare, and that the majority of asteroids are rubble piles that have been disrupted and reaccumulated numerous times in their lifetime [see estimates below and Richardson et al. (2002)]. We must therefore be cautious extrapolating any data on fragment shapes from the laboratory to real asteroids. Models of the initial and evolved asteroid size distribution tend to have trouble reproducing the steep slopes of the observed population, an issue addressed by Tanga et al. (1999); the paper also includes a review of a geometric approach. They conclude that the steep size distributions may be partly explained by their model, which includes a consideration of the geometry of the ejecta, specifically the convexity of largest fragments. 2.8.3. Velocity distributions. Fragment velocity of disruption experiments has received a large amount of attention since the publication of Asteroids II, largely due to the widespread use of image processing and analysis computer systems. The primary conclusions presented in Asteroids II still hold: The fastest fragments originate near the impact point, fragments generally do not collide with one another, and core fragments in core-type disruption tend to be traveling at very low velocity. The fast fragments produced near the impact point usually consist of fine dust traveling an order of magnitude faster than the larger ejecta, typically several kilometers per second in hypervelocity impacts (e.g., Di Martino et al., 1990; Drobyshevski et al., 1994), although the impacts into porous targets discussed above have much smaller velocities. The fine fragments may be the result of jetting. This is studied and discussed by Arakawa and Higa (1995), who found ejection velocities of fine fragments to be 1.7 to 2.9× the impact velocity in low-velocity

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(150–690 m/s) collisions of ice spheres, independent of target size and impact velocity. Arakawa (1999) continued the program of ice studies by measuring maximum and minimum ejection velocities in oblique impacts, but using relatively large impactors with 13% of the target mass. He confirmed a maximum velocity of ejected fragments as 3× the impact velocity. Data on ejecta velocities have also been reported for a wide range of other target types: cement mortar (Davis and Ryan, 1990), pyrophilite (Takagi et al., 1992), alumina and gypsum (Nakamura and Fujiwara, 1991; Nakamura et al., 1992), porous alumina, commercial mortar and sand (Yanagisawa and Itoi, 1993), limestone, alumina cement (Giblin et al., 1994a,b) and gabbro (Polanskey and Ahrens, 1990). The three-dimensional velocity data have generally been collected using a stereoscopic system. Characteristic ejection velocities are typically between 0.1 and 0.5% of the impactor velocity for high-speed impacts into rocky targets, but much slower for porous targets. The antipodal velocity has been measured by a number of researchers, since it provides a reasonable and easily measured characteristic velocity for an impact. Nakamura (1993) tests the scaling of antipodal velocity with both Q and NDIS (see section 3.1.1 below), finding good agreement with Fujiwara and Tsukamoto (1980) and Davis and Ryan (1990). Giblin et al. (1998a,b), using a contact charge to simulate the impact of a 6.2-km/s projectile, found antipodal velocity to be only one-third of the value predicted by Fujiwara et al. (1989), raising questions about the equivalence between a contact charge and an impactor. Martelli et al. (1993) studied the angular distribution of velocities in several open-air impact experiments and found evidence for collimated jets and other anisotropies in the ejecta field. They discuss these data as a possible mechanism for the formation of asteroid binaries and families. 2.8.4. Velocity-mass correlations. The relations between a fragment’s velocity and mass is a key component in models of collisional disruption and asteroid evolution (see, e.g., Petit and Farinella, 1993). Results of detailed crater ejecta studies (e.g., Gault and Heitowit, 1963; Vickery, 1986, 1987) indicated a power-law relationship between fragment velocity and mass in high-velocity cratering of rocklike materials. Laboratory disruption experiments (Fujiwara and Tsukamoto, 1980; Nakamura et al., 1992; Nakamura, 1993) have suggested that a power law holds to some extent (see Fig. 2). However, other studies (e.g., Ryan, 1992; Takagi et al., 1996; Giblin, 1998) have found little correlation between velocity and mass for similar experimental parameters (see Fig. 3). The authors pointed out the lack of complete data as well as the problem of selection effects in the analysis of laboratory fragmentation experiments. 2.8.5. Rotation rate distributions. Very few data on fragment rotation were available at the time of Asteroids II publication, but the brief conclusions presented there are still valid. Specifically, the fastest rotators originate near the impact point and, although many fragments have been observed rotating quickly [e.g., 100 rotations/s (Fujiwara,

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Velocity (Center of Mass) (m/s)

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1987)], none of these approach the rotational bursting limit for their materials. Generally, large fragments have been found to rotate more slowly than small ones. The distribution of asteroid rotation rates is often compared to a Maxwellian distribution (e.g., Harris and Burns, 1979; Farinella et al., 1981; Binzel et al., 1989; Fulchignoni et al., 1995). Although considered appropriate for a highly evolved asteroid population (Harris, 1990; Farinella et al., 1992; Yanagisawa et al., 1991; Yanagisawa and Hasegawa, 1999; Yanagisawa and Hasegawa, 2000; Sirono et al., 1993; Kadono, 1993), the Maxwellian distribution is not expected to give a good fit to “young” fragments from disruption

experiments because such fragments have undergone no further collisional processing. Thus experimental data may be used to test the idea that small asteroids are “young,” in the sense that most of them may have remained close to their original rotational state. Giblin et al. (1998b) report the rotation rate of a total of 811 fragments studied across 8 similar experiments. Data on rotation rate distributions are shown in Fig. 4. Some asteroids have non-principal-axis rotation (“tumbling”) states (see, e.g., Hudson and Ostro, 1995). Authors discussing fragment rotation in impact experiments have not generally reported any evidence of tumbling fragments. This may be due to lack of instrumentation (or time for data reduction). However, Giblin and Farinella (1997) report data on a number of tumbling fragments from the experiments described in Giblin et al. (1994a, 1998b). Their conclusions support a biased distribution of spin vectors that favors principal axis rotation in ejected fragments. Fujiwara and Tsukamoto (1980) study the issue of rotational bursting and conclude that “generally, no collisions among fragments occurred, but in the exceptional case some spinning fragments were split into smaller ones and collided with other fragments.” Giblin et al. (1994a) report that they observed several ejected fragments splitting just after ejection from the disrupted target. Giblin et al. (1998a) describe a very well studied case from the same experiments of rotational bursting, where a tumbling fragment travels more than five target diameters before breaking into two pieces due to rotational stress. Rotational bursting is most likely in the case of a tumbling prefractured fragment since only in this case will the internal stresses of the body be timevarying. With Monte Carlo simulations those authors show that a fragment ejected from a body can subsequently place

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Fig. 3. (a) Measured two-dimensional velocity and estimated mass from four similar experiments where alumina cement targets were catastrophically disrupted using a contact charge to simulate an impact at 6.2 km/s. (b) Measured three-dimensional velocity vs. mass for a subset of the points in (a). Little correlation is apparent. From Giblin (1998).

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Fig. 4. (a) Rotation rate distribution for 386 fragments from catastrophic disruption of alumina cement targets (Giblin et al., 1998b). (b) Size and rotation rate for the same 386 fragments showing considerable dispersion around the best-fit line. The deviation from a Maxwellian distribution is due to an overabundance of slow and fast rotators and a correspondingly depleted peak in the distribution, similar to the data for small asteroids.

some of its mass in stable orbit by rotational bursting. This is a plausible mechanism for producing parent-satellite systems such as Ida and Dactyl. Fujiwara and Tsukamoto (1981), Nakamura et al. (1992), and Nakamura (1993) discuss the variation of rotation rate with size in ejecta from disruption of rocklike targets, finding a best-fit power-law exponent between –1.5 and –1.0, but note that their data were limited. Giblin et al. (1998b) find a significantly shallower slope in their data. Figure 4b shows some of these data; although there is a definite slope, a power-law fit simply does not provide a good description of the data. 2.9. Energy Balances Energy partitioning is an important consideration in the effects of any impact. The total energy is divided (generally in decreasing order) between heat, comminution of the target, target and ejecta kinetic energy and ejecta rotational energy; and, for sufficiently high-velocity impact, melt and vaporization energies. An important component in the context of modeling and understanding collisional evolution is the fraction of impact energy partitioned into ejecta kinetic energy, since this determines whether or not a disrupted body remains disrupted or reaccumulates. According to the available results in 1989, this fraction is less than 3% for high-velocity (core-type) catastrophic impacts (Fujiwara and Tsukamoto, 1980) and less than 1% for high-velocity cratering impacts into semi-infinite basalt (Gault and Heitowit, 1963). The fraction of energy partitioned into ejecta KE is higher for low-velocity impacts; on the order of 10–20% (Waza et al., 1985). Fujiwara (1987), Nakamura and Fujiwara (1991), and Nakamura et al. (1992) all reported that the largest proportion of energy goes into heat, comminution of the target, and the relatively high velocity of fine fragments. Fujiwara

(1987) reports that fragment rotational energy is generally less than 1% of kinetic energy in high-velocity disruption; Nakamura (1993) reports between 1% and 10%. A similar study by Giblin (1994) reports this ratio to be between 2% and 4% for four alumina cement targets disrupted using a contact charge. Sugi et al. (1998) impacted water-ice targets using a copper projectile at speeds from 54 to 329 m/s in order to study the degree of impact vaporization. They observed that, below 100 m/s, less than 0.03% of impact energy went into impact vaporization in their experiments, and that this increased to 18–26% between 100 and 180 m/s. Schultz (1996) found significant amounts of impact kinetic energy went into impact vaporization in oblique impacts of an Al sphere into competent ice, in experiments where impact speeds were between 4.7 and 5.9 km/s. 3. SCALING THEORIES, ISSUES, AND RESULTS Issues of scaling are fundamental to our understanding and interpretation of impact processes, and are needed to unravel the meaning of laboratory experiments and code calculations. Various approaches to scaling have been developed over the years. The result of any impact depends on the conditions of the impactor and those of the impacted body (target), perhaps in complicated ways. A successful scaling approach must distill the large number of possibly relevant physical parameters down to the essential few. The goal is to have a rule to predict all impact responses (the effects) from the impactor conditions (the cause). 3.1.

Approaches

There are two synergistic approaches to scaling: the theoretical approach using similarity and dimensional analyses;

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and numerical computer calculations of a suite of impacts. Each has its strengths and weaknesses. They are reviewed in turn. 3.1.1. Theoretical scaling methods. The impactor has dominant measures of size a, velocity U, and mass density δ that determine its energy, mass, and momentum. It also has an impact angle and many material properties. A common feature of all scaling theories is that one single measure is chosen to represent all properties of the impact initial conditions. In the historic approaches, the impactor kinetic energy was used most of the time, although some questioned whether the measure should be the impactor momentum. Beginning with Holsapple (1981, 1983), and in a number of subsequent papers (Schmidt and Holsapple, 1982; Housen et al., 1983; Holsapple, 1987, 1993; Holsapple and Schmidt, 1987), a different measure (the coupling parameter) was introduced. Its existence and its form C = aUµδν was shown to be a consequence of physical and mathematical “point-source solutions” for rapid energy deposition in vanishingly small regions for general materials. Therefore it is a global measure of the impact process for regions away from the immediate details right at the impact site. It generally is intermediate to the energy and momentum measures, depending on the materials. However, it is only as valid as the point-source approximation, and cannot be expected to hold for cratering when, for example, the crater is only a little larger than the impactor, or for phenomena (such as melt and vaporization) that occur only on a scale comparable to the impactor radius. However, it does seem to work surprisingly well over large ranges in velocity and scale. For example, for transient craters in water it correlates to within a few percent experiments with impact velocities ranging over the extreme range of 1 m/s to 6 km/s (see Fig. 2 in Holsapple, 1993). Code calculations have showed that it governs impact-generated flows in regions as near as 1–2 impactor radii. A different measure of the impact, the nondimensional impact stress (NDIS), was introduced by Takagi et al. (1984). It was initially based on the maximum pressure generated at the impact point. In a later approach, Mizutani et al. (1993) used the stress propagated to the antipodal point of the target. [Strictly speaking, it is a measure of the effects and not the impactor (the cause). Its utility for scaling may rest on an expectation that the transmitted stress is easier to predict than the other effects of the impact.] Mizutani (1993) introduced special assumptions about the nature of the stress decay from the source through the body, while recently Mitani (2000) used a numerical calculation to estimate the transmitted stress. Thus, the NDIS has evolved from a detailed measure of the impact condition, which is probably not important globally, to a global measure of the response itself. In all cases of scaling it is also necessary to choose simple measures of the target resistance. Approaches for scaling of cratering or disruption typically use some material strength and/or the gravitational field (determined by body size). Cratering scaling generally uses a constant strength, while disruption scaling has used rate- and/or size-depen-

dent strength. This is done because, for many geological materials, the tensile strength, which is fundamental in disruption, is observed to be more strongly rate dependent than is the shear strength, which is most important in crater formation. Insofar as the scaling forms, any strength measure with stress units gives the same result. When trying to correlate between results for different materials, some specific measure must be chosen. However, since different strength measures such as compressive, shear, or tensile strength are often in about the same ratios for different materials, the choice is not so important. The choice becomes more uncertain when choosing between a crush strength of a porous material and some other strength measure. Therefore, there are two measures: one for the impactor and one for the target resistance. Their choice, together with methods of dimensional analysis, lead to definite powerlaw algebraic forms for scaling. The reader is referred to examples in the literature such as Housen and Holsapple (1990), Mizutani (1993), and Holsapple (1993). These approaches are limited to phenomena where the point-source approximation governs. 3.1.2. Code calculation approaches. Scaling laws and studies of impacts can also come from the outcomes of code calculations. Numerical simulations have become popular: At the last two Lunar and Planetary Science Conferences there were perhaps 30 abstracts by authors using the codes CTH, SOVA, an SPH continuum code, an “n-body” code, or a finite-element code to do either two- or three-dimensional impact calculations. One attractive feature is the ability to investigate effects of specific shape and structure (Asphaug et al., 1996, 1998), and to look at gravitational assemblages (Richardson et al., 1998; Leinhardt and Richardson, 2001). Initial shock processes and energy coupling are calculated in many ways. There are simple equation-of-state models with no thermodynamic coupling (e.g., Murnaghan), simple analytical models for single solid phases (MieGruneisen and Tillotson), complex analytical models including melt and vapor (ANEOS), or complete tabular databases such as the SESAME library. None of these include any kinetic effects, which might be important. Effects of porous crush-up are almost never modeled, since that greatly increases the numerical difficulties. (Calculations with very low density that use the standard forms of the equation of state omit the energy dissipation of a crushable material and do not model porous materials.) Some of these models are only appropriate for special cases, although they are sometimes used in other inappropriate cases. Strength models include none (hydrodynamic), constant (shear and/or tensile), Mohr-Coloumb shear strength, ratedependent tensile, and complex damage models. Some have even included a viscous component to model acoustic fluidization. Gravity, either a fixed planar field or a self-gravity central field, may or may not be included. Obviously these myriad approaches lead to myriad re-sults. The results vary substantially due to the difficulties of correctly modeling the complex geological materials. A recent statement in the literature that “recent exponential increases in computational power have enabled


numerical simulations to become the method of choice to investigate these issues (planetary impacts) in greater detail” may be premature, although that may become true in the future. The theoretical methods make it clear that any scaling outcomes from code calculations are determined primarily by the dominant measures of the impact conditions and the target resistance in the codes. Since the point-source measure is firmly rooted in special mathematical solutions to energy deposition, the codes must reproduce, at least in an approximate manner, point-source results for any phenomena away from the immediate impact region. The pointsource measure should fail only for phenomena very near the impact point (for example, the amount of melt and vapor), for very slow impactors, or for very large (compared to the target body) impactors. Also, since the resistance of the target is one of the primary scaling measures, the choice of the gravity level and the strength model in the code will determine the form of the scaling outcome. Thus, the form of scaling results from code calculations is simply a reflection of the models chosen. The codes cannot determine what physics are important; the creator of the code does that. A good example is the Benz and Asphaug (1999) paper, in which the slopes obtained from SPH code calculations for disruption match the slopes given in the scaling theories of Housen and Holsapple (1990) and Holsapple (1994) in both the strength and gravity regimes. That is because their strength model is based on the same rate-dependent physics that was the basis for the theoretical scaling approach. The contributions of codes are then, in principle, to calibrate the constants that are unknown from the dimensional analysis approaches, to investigate ranges where the scaling may not hold, to bridge the gap between scaling regimes, and to investigate ranges of material, shape, and structural models. However, there is generally a lack of good material models and data. The material models usually have many parameter choices (“knobs to turn”), and often the magnitude of some primary variable such as some strength is simply “dialed-in” to make the code match some limited experimental data. Furthermore, there is usually no unique way to make that match; other “knobs” might succeed equally well. Other quite different material models might also succeed. Therefore, magnitude calibrations are often specious. Codes are a poor way to determine material properties. However, having noted the uncertainties, code studies certainly have contributed to the understanding of scaling issues, and provide ways to study phenomena out of the reach of experiments. As they mature and better material models are developed and implemented, they will become even more useful. Important contributions from the codes are included in the sections below. 3.2. Crater Size Scaling Results 3.2.1. Previous results. The current knowledge of crater scaling in rocks, water, and dry sands was summarized in Schmidt and Housen (1987) and in Holsapple (1993).

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They give algebraic forms and figures for important aspects of impact cratering events, including volume, diameter, depth, rim heights, timescale, and ejecta velocity. Impact experiments can only be conducted at small scales, but explosive events have been conducted at relatively large scales (megaton nuclear weapons). Therefore much of the scaling is based on an equivalence between impact and explosive events. Generally, an impact will impart more of its energy and more downward momentum to cratering mechanisms than an explosive on the surface. However, that difference is offset as an explosive is buried; for very deep burial, the explosive has the greater cratering effects. At a depth of burial of about 2× the explosive radius, the tamping effects of soil above the explosive increases both the momentum and energy in the downward directed cratering flow, and the results of the explosion and impact at the same energy and energy density are similar. That has been validated both by many experiments and code calculations (e.g., Holsapple, 1980; Ryan, 1992). Terrestrial explosive events in continental sites have generated craters to ~400 m in diameter, while nuclear events in Pacific corals have produced craters in the kilometer range. Schmidt et al. (1986) present the explosive cratering database and scaling. For small impacts (the strength regime) the crater volume is determined only by target strength. Explosive field data show some increase in crater volume per unit energy as the event size increases, a feature attributed to a weakening of the target material with increasing size. However, that increase is only apparent for near-surface explosive energies greater than about 1 t of TNT (4.2 × 109 J), and is not apparent in buried events. Thus, there is little evidence of rate- or size-dependent strength in cratering events, probably because they are dominated by shear flows. For large events (the gravity regime), the crater volume becomes dependent on the surface gravity and is independent of the target strength. Dry sands, having essentially zero cohesion, are always in the gravity regime. In the gravity regime the crater volume per unit energy decreases with increasing event energy. The point of transition between the strength and gravity regimes has been estimated both by comparing the physical strength of the material to the stresses of gravity and by the actual data from the large explosive events. For near-surface terrestrial events in hard rocks, this transition occurs at about 1 kt of TNT (4.2 × 1012 J, crater diameter ~30 m), while for dry soils it occurs at a few tons of TNT (crater diameter ~10 m) (see Holsapple, 1993). These transition crater sizes apply only for the terrestrial gravity field. If the size or rate effects in cratering are minimal, then the transition diameters for other bodies are found by simply dividing these sizes by the magnitude of the gravitational acceleration measured in Earth’s gravity (Holsapple and Schmidt, 1982). It should be emphasized that all these crater-scaling results are only for common soils and rocks as found on Earth, and are probably not applicable to highly porous materials. Scaling in materials with high porosity is not yet

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determined. In the numerous experiments in dry sands, the target material is at or near its “maximum density” or fully packed state, with about 30% porosity. Substantial compaction from that state can only occur by crushing or shearing individual grains, which requires substantial pressures, on the order of kilobars (Housen and Voss, 2001). Cratering experiments in dense sands show only a small amount of crushed material remaining in the crater, and no noticeable density increases below the crater. Housen and Voss (2001) measured pressure-crush curves for highly porous materials, and the pressure required for 10% compaction of their 53% porous material was a factor of about 30 less than for dense Ottawa sand. Porosity may play only a minor role for cratering in dense sands, but is thought to play a much larger role in a very-low-density material where the material is easily crushed. 3.2.2. Implications of new experiments on cratering. In water ice, new experiments for cratering and ejecta have been performed, as are presented in the experiments section above. However, important questions remain about their relevance and scaling to the large icy solar system bodies. Small craters in ice are typically dominated by surface spall, as is observed also for centimeter-sized cratering in rocks. However, it is known from explosive testing in rocks that spall effects are absent for craters of tens of meters and larger. Thus, small spall-dominated experiments in ice and other brittle materials may involve different physics than do larger cases and, if so, are of limited use for scaling to large sizes. In addition, many ice experiments have used water ice near its melting point. Crystalline materials are known to have significant rate effects at near-melt temperatures, which are much less important at low temperatures. The materials become more brittle at lower temperature. Thus, there may be dominant rate effects in the small-scale experiments in warm ice (where the rates are very large), which would not be present in large craters (where the timescales are orders of magnitude larger). The exact form of these temperature dependences are not known. For these reasons, the data on cratering and ejecta in ices have not yet been synthesized into an inclusive scaling form, and large gaps in the data exist. Recently craters have also been studied in highly porous materials (see section 2.6). A major observation is that, as porosity increases, the mechanics of cratering change from one dominated by lateral and upward flow and ejection to one dominated by crushing within the crater. As a consequence, the amount and speed of ejected material decrease substantially as the porosity increases. However, important questions remain about the exact nature of porous materials in asteroids, and whether the materials of the experiments mentioned above are reasonable simulants for low-density asteroids. In any case, the experimental results have not yet been synthesized by a reliable scaling theory. Also, there have been few computer calculations for highly porous targets. 3.2.3. Code calculations of crater scaling. Using the CTH finite-difference code, O’Keefe and Ahrens (1993, 1999, 2000) performed many cratering calculations with various

impact velocity, size, gravity, and material strengths. The shear strength was modeled by a Mohr-Coloumb plasticity model, with no size or rate effect. The calculations were carried out to relatively late stages. They tabulate constants for the scaling of various crater features. The results compare very well to the theoretical scaling curves mentioned above that have been calibrated by the terrestrial cratering database. They have been carried to the very late stages of crater formation to study complex crater morphologies. Housen and Holsapple (2000) also used the CTH code to perform cratering calculations, and included a specific porous model for crushing. It was found that the modeling of the crush behavior was difficult but essential. They found that results were very sensitive to the material constants of the crush model and were able to match some, but not all, aspects of well-documented experiments in dry sand. Rate-dependent models have been available in the codes used in the weapons community since the 1980s. For impact calculations, a rate-dependent strength model was implemented in the SALE finite difference code by Melosh et al. (1992). That model was based on an implicit description of the statistics of flaw sizes and growth, an interpretation of the one-dimensional Grady-Kipp model (Grady and Kipp, 1980). It used a scalar damage measure; and degraded the material stiffness in both tension and shear to zero as the damage accumulated. It has no plasticity model. It was used to model Stickney Crater on Phobos (Asphaug and Melosh, 1993), for disruption calculations (Ryan and Melosh, 1998), and to study crater scaling (Nolan et al., 1996). For cratering, Nolan et al. (1996) conclude that, for large impacts in basalt, the target material is substantially fractured by the outgoing shock wave, and the crater forms within the fractured region just as it would in an entirely strengthless material. Those results should be compared to the terrestrial cratering database and the impact scaling theories mentioned above. The equivalent explosive energy for the impacts studied by Nolan et al. (1996) (5.3 km/s, 1–120-m-diameter impactors) ranges from a few tons of TNT to ~5 Mt. For tons to a few kilotons of yield, their cratering results can be directly compared to large explosive field events, because in that range the curvature of their target is of little consequence, and the terrestrial events in this range show no effect of gravity. The numerical simulations give a crater volume per unit energy of about 5 × 104 ft3/t of TNT for the smaller impactor and 2.4 × 105 ft3/t for the larger impactors. These values are 2 orders of magnitude or more larger than actual large terrestrial explosive events in rocks (Schmidt et al., 1986). (The report by Schmidt et al., which presents scaling curves for explosive cratering, has restricted distribution; it is available to U.S. government agencies and their contractors only. However, the actual explosive cratering data is mostly in the open literature in the form of government agency reports. These include significant explosive craters in dry geologies such as Sailor Hat in rock; Stagecoach and Scooter in alluvium; and the Danny Boy, Sedan, Teapot Ess, JangleU and Johnie Boy nuclear craters, as well as a multitude of smaller field and experimental craters.)


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Nolan et al. (1996)

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Fig. 5. The cratering results of Nolan et al. (1996) compared to existing scaling curves based on terrestrial explosive craters and laboratory impact craters in rocks, soils and water (from Holsapple, 1993). The vertical arrows indicate the difference between the code results and the scaling curve in the strength regime. (a) The code gravity-regime results fall just about on the cratering curves for water, a decade above the gravity curves for dry sands and soils. (b) The calculations are two decades above rock scaling curves.

For example, the 500-t Sailor Hat explosive event in basalt gave a cratering efficiency of 580 ft3/t (Vortman, 1965). This can also be seen in Fig. 5, where the calculated crater sizes are compared to the conventional impact scaling (from Holsapple, 1993) in a dimensionless form. On the left at small sizes (Fig. 5a), their result is about an order of magnitude above the value for cratering in wet soils at impacts of 5 km/s (the vertical arrow). At large sizes, in the gravity regime, they are just about on the curve for cratering in water. Figure 5b shows that the numerical results are about a factor of 100 above the curve for the volume of craters in hard rocks in the strength regime and a factor of 10 above them in the gravity regime. It must be emphasized that these theoretical scaling curves are based on the terrestrial crater database, and there are actual explosive craters in this size range to compare to. These results are indeed unexpected. A different implementation of rate dependence based on an explicit (rather than an implicit) flaw distribution has been developed by Benz and Asphaug (1994) and used in a SPH code. They discuss some significant problems of the Melosh et al. (1992) numerical implementation of the implicit flaw description. Asphaug et al. (1996) also added the newer explicit method to the SALE finite-difference code to study the mechanics of cratering on Ida, and to determine curves of crater scaling in the strength regime. However, their results also give craters about a factor of 100 larger in volume at the same source energy than the terrestrial explosive data as presented in Schmidt et al. (1986). These comparisons may illustrate some of the difficulties in the application of code calculations for impacts, and the caution that must be exercised in their interpretation and application.

Some suggestions about possible shortcomings can be given. The primary difference between these recent codes and the earlier ones is the inclusion of the crack growth physics, which are thought to be the primary cause of rate effects. The Grady-Kipp rate models apply to one-dimensional tensile fracture only, and the codes use it to calculate a single scalar damage measure. The correct measure of rate to use is uncertain: that at the shock front, or one based on the pulse duration. Then, when a single crack grows to be as large as (roughly) a calculation zone, the damage is assumed to be complete, indicating a fully failed material. The models then assume there is no remaining rigidity in any tension or shear state, i.e., the material behaves totally as a fluid. All shear stresses in the zone become zero, and unrestricted shear strain and flow is possible. In reality, there should remain substantial shear strength, even along the fracture plane, whenever there is a compression acting on that plane. There should be no loss of strength or stiffness for shear stresses perpendicular to a failure plane. Completely failed material (even with cracks in all directions) should be modeled using the plasticity models of a dry soil, not as a fluid. A limiting behavior of a cohesionless MohrColoumb model such as used for dry sands would seem to be more appropriate. Even at full failure, the cratering results should correspond to the database for dry sands, not that for water. Hence, the fact that the numerical results are a factor of 100 larger than the terrestrial explosion data probably results from incorrect modeling of the shear strength of damaged material. In addition, cratering flows are dominated by shearing flow, not tensile failures. Shear strength does not appear to have the strong size dependence that tensile fracture does.

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The terrestrial cratering data show only a little evidence of decreasing strength with event size. The CTH code results mentioned above using a rate-independent shear strength compare very well to the terrestrial cratering data. Clearly there is great opportunity for future work here, including the thorough testing of the codes against the terrestrial explosive database and other code models. While the codes were compared to disruption experiments, the constants of the crack distribution models were adjusted to make the best fit. Melosh et al. (1992) made comparisons to the Takagi et al. (1984) disruption experiments in basalt. They found that the use in the code of published strength and crack size distribution data for basalt gave a poor match to the experimental outcomes, but by adjusting the crack size coefficient down by a factor of about 105 (less cracks of a given size, or a stronger material) they were then able to get a good fit to the fragment size distribution of the experiments. However, at a given strain rate, the implied strength of that code model is then a factor of about 2 higher than actual data for similar rock materials. Ryan and Melosh (1998) used that same strength model for basalt. Also, for comparison to Housen et al. (1991) experiments in a porous grout they again adjusted the Weibull crack distribution parameters of the code until a reasonable match to the disruption experiments was obtained. Then they also made some comparisons to two experiments that were basically cratering events. However, the crater masses they obtained were larger by factors of 3 and 5.5 respectively than the experiments. Benz and Asphaug (1994) made comparisons of their code results to the disruption experiments by Nakamura and Fujiwara (1991) and were able to obtain good results, but again made choices of the fracture parameters to give a best fit. Again, the implied strength at a given strain rate is about 2× published strength data. Later applications of the code with the same material fracture parameters were made by Asphaug et al. (1996) for cratering on Ida and by Nolan et al. (1996) for cratering on Gaspra-sized bodies. However, comparisons to actual cratering data and scaling curves were not made. The modeling of the crack growth, shear strengths, and the inclusion of substantial porosity deserve special attention for the code methods. The approaches used in the weapons community, particularly those for impacts into porous ceramics, should be considered. These issues require much further study before code methods can be considered to be entirely reliable. 3.3. Ejecta Scaling Ejecta scaling laws are important in order to model the evolution of the regolith on small bodies and the production of interplanetary dust particles. Housen et al. (1983) present scaling formulas for amounts and velocities of the ejecta from cratering. In the strength regime, the ejecta velocity at a scaled range increases as the square root of the strength. Laboratory experiments typically show velocities

sufficient to exceed the escape velocity on Gaspra-sized asteroids. Thus, small, very strong rock asteroids were expected to retain little regolith. However, if the strength decreases considerably with body size, then the laboratory experiments are misleading. In experiments simulating a jointed rock, Housen (1992) showed that indeed the ejecta velocity did decrease as the strength decreased, and that a body the size of Gaspra could retain much of its ejecta and develop considerable regoliths. Impact experiments in porous materials were mentioned in section 2.6. Ejecta velocities are substantially less than in competent nonporous materials or even in dry sands. The tests used materials of different crush strengths, different shear strengths, different porosities, different scaled sizes (G-level), and used different impact velocities. That large range of parameters has so far precluded any identification of the most relevant variables and the synthesis into a comprehensive scaling theory. The one fact that is apparent is that the conventional wisdom on scaling in normal terrestrial materials is not valid. The images of Mathilde have also raised new questions about the production of ejecta on a porous body. Although Mathilde has several very large craters as wide as the asteroid mean radius, there are no visible ejecta features. This lack has been explained in three very different ways. Housen et al. (1999) performed experiments in a material with the same porosity as Mathilde, and concluded that almost no ejecta are produced because the cratering is dominated by (downward) crushing, not (outward and upward) excavation. The experiments by Michikami et al. (2001) mentioned above confirm those interpretations, in a different but also highly porous material. On the other hand, Asphaug et al. (1998) performed a code calculation for a porous body and deduced that ejecta velocites are greatly enhanced by the porosity, and that all ejecta would escape the asteroid. Finally, Cheng and Barnouin-Jha (1999) used conventional crater scaling for dense dry sands to apply to Mathilde and determined that the large crater morphology is consistent with oblique impacts. There are major differences in these approaches. The use of conventional dry sand crater scaling used by Cheng and Barnouin-Jha (1999) is questionable in view of recent experiments in highly porous materials. The fidelity of the sample material constructed by Housen et al. (1999) as a Mathilde analog is uncertain. Finally, the code calculation of Asphaug et al. (1998) included macroporosity but no microporosity, and therefore did not model the continuum behavior of highly porous materials. What is the most appropriate model for Mathilde? We do not know. No resolution of these major discrepancies has been made, so at present the question of ejecta scaling in porous asteroids is as uncertain as crater size scaling. 3.4. Scaling of Shattering and Dispersion The scaling of disruptions presented in Asteroids II was given in Fujiwara et al. (1989, see Fig. 9, p. 259). Since

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that time much has been added. A recent version, Fig. 6, shows the specific energy required for shattering and for dispersion. It is from Benz and Asphaug (1999) but with several curves added. There are three features that are common among the various curves. Each has a negative-slope strength region on the left, a positive-slope gravity region on the right, and a transition size between those two regimes. For the strength regime, the scaling theory of Holsapple and Housen (1986) and Housen and Holsapple (1990) and Holsapple (1994) assumes that tensile failures determine the strength, so they use a rate-dependent strength model. The slope of the power-law in the strength regime (Holsapple, 1994) is 9µ/(3 – 2φ) where φ is the exponent of a Weibull flaw-size distribution, and µ is the exponent of impact velocity in the point-source coupling parameter measure. (The exponent µ reflects the decrease in coupling efficiency and increasing waste heat with increasing impact velocity. It is determined in the early-time coupling regime, and is primarily a consequence of the high pressure-temperature equation of state. A material with no energy dissipation has µ = 2/3.) Holsapple (1994) assumes that µ = 0.55 and φ = 9 to give the slope as –0.33. The code calculations of Benz and Asphaug (1999) also use φ = 9, and their slope is also about

–0.33. Housen and Holsapple (1999a,c) postulate that a better value is φ = 6 so the slope is –0.67; furthermore, their curve is fitted to their size-dependent granite experimental results (those data are off this curve to the left). The curve given by calculations by Ryan and Melosh (1998) for mortar assumes that φ = 6.5 and their slope is –0.61. Thus, in all cases the slopes obtained are entirely consistent with the scaling prediction based on the assumed value of the parameter φ. For large bodies, in the gravity regime, the scaling theories predict that the slope should be 3µ. If µ were 2/3 (the limiting value when the point source measure is the impactor energy), there would be no decrease of coupled energy with increasing impact velocity, and pure energy scaling would hold. Then the slope in the gravity regime on this plot would be 2. The Davis et al. (1985) model assumed such energy scaling, so they have the slope of 2. The value chosen in Holsapple (1994), µ = 0.55, was determined by a multitude of results for nonporous materials such as rocks and water (see Holsapple, 1993), which gives a slope of 1.65. The Benz and Asphaug (1999) curve has essentially that same slope, suggesting that the code calculation reproduces the expected early-time energy coupling. The slightly shallower slope of the Love and Ahrens (1996) curve suggests more dissipation in the energy coupling. That might

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be a consequence of low resolution at the shock front, although Ryan and Melosh (1998) got a much steeper slope with a very-low-resolution calculation. The differences also might arise from the equation of states used. The most striking feature of this plot is the wide discrepancy of the values. However, much of this apparent discrepancy arises simply because some of the curves define the condition for shattering, Q*S, while others apply to the condition for dispersal, Q*D (see section 1). In the small-body strength regime where there is negligible gravity, these two values of Q are the same, since any shattered body will disperse. There are substantial differences in the predictions in this strength region, although most match the centimetersized laboratory data. (The theoretical scaling curves are in fact calibrated to those results. The codes are usually “dialed in” to match those results; see section 3.2.) In the gravity regime there is a substantial difference between the shattering and the dispersion energy. The Ryan and Melosh (1998) curves are for the shattering energy only, while the Housen and Holsapple (1990), Benz and Asphaug (1999), Melosh and Ryan (1997), and Love and Ahrens (1996) curves are for the dispersion energy. The Holsapple (1994) curve was based on the largest observed craters on various bodies, so it is a lower bound for the dispersion energy, but in principle could be above the shattering energy. The Housen and Holsapple (1990) curve was based on velocity scaling, assuming one-half the mass had velocity greater than the escape velocity. Melosh and Ryan (1997) also used velocity scaling, but in a different way, to get their dispersion curve. The estimates of the energy to disperse are about a factor of about 100 above the shattering estimates. Fujiwara (1982) and Davis et al. (1983) previously noted the large difference “energy gap” between these two thresholds. Durda et al. (1998) obtained a very different curve by comparing the results of a collisional evolution model with the present asteroid size distribution, and then backing out the scaling law that best matched the present distribution. A more recent paper by Campo Bagatin and Petit (2001), with a different model of evolution of asteroids, did not agree with that result. 3.4.1. Uncertainties in disruption scaling. The very wide diversity of curves for either of shattering or dispersion is unsettling. Those estimates have changed a lot in form over the last 20 years. It used to be common to adopt a single value for Q* at all sizes, and the distinction between shattering and dispersion was overlooked. Then the increasing slope in the gravity regime was introduced for dispersion thresholds, a consequence of requiring energy to launch remnants to escape velocities. The strengthening due to lithostatic pressure was then formulated, so the positive slope applied to shattering also. That was based on the concept that the initial compressive stress due to lithostatic pressure must first be overcome before the stress wave could achieve tensile fracture conditions. Finally, a decrease of required energy in the strength regime was introduced based on ratedependent strength. That is currently an accepted feature of all present curves. What, if any, evidence is there for any

of these effects? Most laboratory experiments cannot address any of them. In the strength regime, the experiments by Housen and Holsapple (1999c) show the size dependence, but only over a diameter range of 18:1. We do not know of any other direct evidence for the size effect. It remains uncertain whether those trends continue to asteroid sizes. Codes reproduce that dependence, but that is simply a consequence of their assumed models. In the gravity range, there is some indirect evidence for the positive slope. First, the estimates of required energy to form the Themis, Eos, and Koronis families are at least consistent with current estimates for dispersion (Housen and Holsapple, 1990), although not definitive by themselves. The use of crater scaling applied to the largest observed craters on relatively dense bodies seems to show an increase of energy with size, and was used as a lower limit by Holsapple (1994). The experiments by Housen et al. (1991), using external pressure to simulate the gravitational binding, show a definite increase in shattering strength with that external pressure, and were used as an upper limit for the curve by Holsapple (1994). The analysis by Durda et al. (1998), based on an evolution calculation, has an even larger upward slope in the gravity regime. Finally, the computer calculations also show the upward slope, and here the results are in a regime where the uncertainties of strength modeling discussed above are not important. Therefore, although there is general agreement on the upward slope, there remain substantial differences in estimated magnitudes. This is certainly an area needing further research. And, finally, all experiments and calculations for impact disruption consider only nonrotating asteroids. If indeed many are rubble-pile bodies, many are rotating at a significant fraction of the maximum allowable rotation for their shape (Holsapple, 2001). What effect does that have on required disruption energy? One would certainly think that could have a major effect at lowering the required energy for disruption. 3.4.2. Implications of disruption scaling. There are many important implications of these results, numerous examples of which can be found in the literature. Here only one significant one is considered, that of collisional lifetimes. Farinella et al. (1998) give the disruptional lifetime in years for a target of radius R as tdisr = [2.85 × 10 –18 R2N(rdisr)]–1, where N(rdisr) is the number of asteroids of radius greater than the radius rdisr (km) of the impactor required to shatter the target. They use an asteroid number distribution in the main belt as N(r) = 3.5 × 105 r –5/2, with asteroid radius r in kilometers [see Farinella et al. (1998) for details]. Using the relation for the specific energy to either shatter or disperse (a factor of about 100 different) with an assumed velocity of 5 km/s gives, for any asteroid, estimates of the lifetime between shattering events and between complete dispersion events. The Housen and Holsapple (1990) shattering kinetic energy per unit mass in the gravity regime is given in cgs units as Q* = (5.8 × 10 –8)U0.35R1.65. Assuming an impact velocity of 5 km/s, and equal densities for the impactor and target, gives the required impactor


radius for shattering as rs(km) = (2.01 × 10 –3)R(km)1.55 which gives the time period as tshatter(yr) = (1.8 × 105)R(km)1.875 Taking the Housen and Holsapple (1990) dispersion curve as a factor of 100 larger specific energy gives the required impactor radius for dispersion as rs(km) = (9.32 × 10–3)R(km)1.55 and a lifetime before dispersion as tdisperse(yr) = (8.4 × 106)R(km)1.875 Three features of these lifetime estimates are very significant. First is simply the large uncertainty; the time is proportional to the required Q to the power of 0.833, and Q is uncertain to maybe an order of magnitude. The number densities are uncertain to perhaps a factor of 2–3, giving a combined uncertainty factor of about 20 for the lifetime. The second feature is the large 1.875 exponent on the asteroid radius R. The paper by Farinella et al. (1998) has only a power of 0.5, since they assumed a constant dispersion specific energy. The effect of the increasing energy with size in the gravity region is significant. The difference between a 10-km object and a 100-km asteroid is, by the results here, a factor of 80 in both the shattering or dispersion lifetimes: The larger one lasts 80× longer. From these estimates, a 100-km-radius asteroid has a shattering lifetime of ~1 b.y. Its dispersion lifetime is 2 orders of magnitude greater, so it will likely never be impacted by the 20-km-diameter body required to disperse it. Few large asteroids will ever be shattered. Then consider a 10-km-radius main-belt asteroid, assuming it somehow is an intact rock body. The impactor that will shatter it at 5 km/s has a diameter of ~150 m. Dispersal requires a 700-m-diameter impactor. The lifetime against shattering is about 107 yr, while that for dispersal is 50× longer. Therefore asteroids are predicted to average perhaps 50 shattering events with various amounts of disruption before a single dispersion event. Thus a 10-km object cannot remain intact for very long. Various amounts of disruption will occur about every 10 m.y., ranging from just breaking it throughout (like Eros?), to almost complete dispersion where lots of small fragments are ejected but later reaccumulate. A 1-km asteroid is shattered maybe five times every 106 yr. How could any present 1–10-km rocky asteroid not be a rubble-pile? A small coherent asteroid cannot escape shattering for very long. Could a 10-km body be a piece from a larger body, e.g., a 15-km body? Dispersion of that 15-km body requires a factor of 100 in energy over its shattering threshold, so all its pieces before reaccumulation would initially be very small. Benz and Asphaug (1999) determine

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that a dispersal event simply pulverizes the parent body. Thus any large remnant of the 15-km asteroid must itself be an aggregate of smaller pieces. How can any 10-km body ever again become a “rock” with density ~3? One possibility is indicated from Housen et al. (1999), which estimates that the five largest impacts on Mathilde increased its density by 20%. Perhaps a primary effect of impacts into very porous bodies is not to disrupt them at all, but simply to pound them back into relatively dense asteroids, to begin the cycle over again. Perhaps they densify during large impacts by shaking small fragments down into the holes between larger ones (see Britt et al., 2002; Asphaug et al., 2002). Alternatively, other processes could increase the density of asteroids (Consolmagno and Britt, 1999). Such processes are a prime area for future research. The question of the effects of impacts into rubble piles is the primary outstanding question for which we have no answer. Indeed, the present state of knowledge is very tenuous: What we know about scaling for rocky bodies seems to tell us they cannot remain rocky, so the scaling does not apply! As stated in the closing words of a lecture by James Head, “Almost everything is not yet known.” REFERENCES Arakawa M. (1999) Collisional disruption of ice by high-velocity impact. Icarus, 142, 34–45. Arakawa M. and Higa M. (1995) Measurement of ejection velocities in collisional disruption of ice spheres. Planet. Space Sci., 44, 901–908. Asphaug E. and Melosh H. J. (1993) The Stickney impact of Phobos: A dynamical model. Icarus, 101, 144–164. Asphaug E., Moore J. M., Morrison D., Benz W., Nolan M. C., and Sullivan R. J. (1996) Mechanical and geological effects of impact cratering on Ida. Icarus, 120, 158–184. Asphaug E., Ostro S. J., Hudson D. J., Scheeres D. J., and Benz W. (1998) Disruption of kilometer-sized asteroids by energetic collisions. Nature, 393, 437–440. Asphaug E., Ryan E. V., and Zuber M. T. (2002) Asteroid interiors. In Asteroids III (W. F. Bottke Jr. et al., eds.), this volume. Univ. of Arizona, Tucson. Bashkirov A. G. and Vitzayev A. V. (1996) Statistical mechanics of fragmentation processes in ice and rock bodies. Planet. Space Sci., 44, 909–915. Benz W. and Asphaug E. (1994) Impact simulations with fracture: I. Method and tests. Icarus, 107, 98–116. Benz W. and Asphaug E. (1999) Catastrophic disruptions revisited. Icarus, 142, 5–20. Binzel R. P., Farinella P., Zappalà V., and Cellino A. (1989) Asteroid rotation rates: Distributions and statistics. In Asteroids II (R. P. Binzel et al., eds.), pp. 416–441. Univ. of Arizona, Tucson. Britt D. T., Yeomans D., Housen K., and Consolmagno G. (2002) Asteroid density, porosity, and structure. In Asteroids III (W. F. Bottke Jr. et al., eds.), this volume. Univ. of Arizona, Tucson. Campo Bagatin A. and Petit J.-M. (2001) How many rubble piles are in the asteroid belt? Icarus, 149, 198–209. Cheng A. F. and Barnouin-Jha O. S. (1999) Giant craters on Mathilde. Icarus, 140, 34–48. Cintala M. J. and Hörz F. (1984) Catastrophic rupture experiments: Fragment-size analysis and energy considerations (abstract). In Lunar and Planetary Science XV, pp. 158–159. Lunar and

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