Asteroid Impacts: Laboratory Experiments and Scaling Laws

Holsapple et al.: Asteroid Impacts: Experiments and Scaling Laws 443

K. Holsapple University of Washington

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 nation of nuclear bombs down to those involved in the stat- ics of ordinary soil and rock mechanics. The results of a An understanding of collisional processes and collisional given impact are therefore very difficult to predict, although evolution is required in order to interpret observations of much progress has been made over the last few decades. the solar system. Although laboratory experiments and com- The outcome of a collision depends largely on the ratio puter simulations have provided many insights into these of the kinetic energy of the impactor to the mass of the im- processes, our understanding of energetic impacts is still pacted body, a specific energy, commonly denoted as Q. relatively primitive. Each new view of asteroids provided Two threshold values of Q are often defined, although the by spacecraft brings new surprises. For example, the sub- literature is not consistent in terminology or notation. Im- stantial regolith on a small body like Gaspra, the huge pacts with small values of Q form craters, but leave the tar- closely packed craters on Mathilde, and the block-strewn get body largely intact. Larger values of Q can shatter a surface of Eros were entirely unexpected. Nevertheless, body into numerous pieces. The specific energy to shatter, * experiments and modeling, guided by observations of as- QS, is defined as the threshold value for which the largest teroids, will allow us to deduce much about the collision remaining intact piece immediately following a collision has history of these bodies, including crater sizes and frequency, one-half the mass of the original body. We refer to it as the crater morphology, ejecta block distributions, regolith de- shattering energy. The shattered pieces may reaccumulate velopment, and the formation of asteroid families. or not, depending on their velocity relative to the escape * Impact processes are very complex and involve extreme velocity. A higher threshold QD is the specific energy such ranges of conditions. The initial coupling of energy from a that the largest object following reaccumulation is one-half high-speed impactor into another body can occur in micro- the mass of the original body. This is called the dispersion seconds, with the extreme pressures and temperatures suffi- energy. The term disruption energy is sometimes used in cient to melt and vaporize the target and projectile material. the literature for either of these thresholds, often in a ge- On large bodies, the latter stages of these processes can neric way for any significant breaking and/or dispersion. continue for hours and involve very low pressures. These Numerous unsolved problems remain for collisions in conditions range from those encountered in the initial deto- all three regimes. Important questions about cratering in-

443 444 Asteroids III clude the crater size; the shape; the amount of compaction; tioned above. The variety of physical material response is and the amount, velocity, and fate of ejected material. For enormous, and we have at best a very rough understanding shattering impacts, the distributions of velocity, size, shape, of relevant physical models. One must model the material and spin of the pieces, and the ultimate fate of those pieces, behavior during shock propagation; crushing of voids in the are still largely unknown. How many fragments remain in material; and failure, flow, and fracture. Important features place, how many are lofted and reaccumulate, and what is such as compaction, strain softening, nonlinearity, and hys- the structure of that modified body? What conditions de- teresis are seldom included. We have only a crude under- termine whether a body is shattered but basically intact, as standing of even the nature of these responses; even when Eros might be, or completely turned to rubble? Finally, for complicated mathematical models are hypothesized and * energy above QD, when the body is shattered and scattered, constructed, there is seldom enough data to calibrate them, significant questions remain about the size distribution of especially for three-dimensional states. Tests of sensitivi- the fragments, the largest piece, and their ultimate fate. ties to inputs are rarely made. As a result, code calculations There are three interrelated and complementary ap- must always be questioned, and even more so when few proaches to the study of these processes. All three approaches attempts are made to calibrate them against known experi- suffer from our lack of detailed knowledge about the ma- mental results. terials and structure of solar system bodies and asteroids. The third major approach uses scaling methods. Scaling In addition, each approach also has its own shortcomings. theories are developed to predict how collisional processes The first approach is to conduct laboratory experiments. will depend on the parameters of the problem, including the Laboratory methods use guns to launch a projectile into a size, impact velocity, gravity, and material type. They are target at speeds up to ~7 km/s, or explosive charges to simu- developed from considerations of similarity analysis applied late an impact. While experiments allow the study of actual to experimental results, from code calculations themselves, geological materials, their primary shortcoming is the in- and from observations of asteroids. However, scaling laws ability to use targets and projectiles of the size of interest. are based on assumptions about the importance of various Laboratory experiments are limited to samples of centimeter parameters. They always require some tie to experiments or size, while the asteroids range to many hundreds of kilo- calculations to determine unknown constants and can lead to meters in size. The response of a small target to impact is erroneous conclusions if important parameters are neglected dominated by its material strength, while that of a large as- or results are extrapolated into regimes of new physics. teroid is dominated by gravitational forces. For the predomi- Since the last contributions in Asteroids II, it has become nantly unidirectional gravity field that governs surface cra- clear that there are at least four major issues about the tering, the lack of sufficient gravity for the experimental mechanics of small-body impacts that have not been suffi- small targets can sometimes be overcome by performing ciently addressed. The first is the effect of substantial poros- experiments at high artificial gravity using a geotechnical ity. Twelve years ago there was conjecture about the exis- centrifuge, but that tool is of little use for the three-dimen- tence of rubble-pile asteroids [the term “rubble piles” was sional gravity fields dominating catastrophic disruptions. first introduced by Davis et al. (1977)], but no knowledge of Additional uncertainties are introduced by an inability to the effects of the implied porosity on the shock processes. perform experiments at velocities of several tens of kilome- There were neither analysis nor code calculations for the ters per second. Features that are important at low velocities, cratering and disruption of porous asteroids. Now the sci- such as projectile material and shape, probably are not sig- entific community seems to have reached consensus that nificant at high velocities. As a consequence of these limita- many and maybe even most large asteroids are reaccumu- tions, laboratory experiments can only probe a small part of lated rubble piles with possibly large porosity. The dis- the parameter space of interest. covery of low densities in C-type asteroids such as Mathilde The second approach is to use computer calculations and Eugenia strongly indicate high porosity. For example, based on the underlying physical principles. These meth- Mathilde has a bulk density of about 1.3 g/cm3, implying ods continue to evolve. Finite-difference, finite-element and, a porosity of 50%. Rocky asteroids are estimated to have over the last decade, smooth-particle-hydrodynamics (SPH) been shattered many times over the lifetime of the solar sys- methods have been particularly useful, giving efficient ways tem and may reaccumulate into a low-density state. Comets to study impact processes. As computer power grows in are also thought to be very low-density conglomerates of leaps and bounds, we can now calculate many more cases ice and dirt. in much less time. However, these methods are also lim- Porosity may be the dominant physical property gov- ited in important ways, particularly by the relative infancy erning an impact process. The mechanics of impacts into of material models. While laboratory experiments use ac- highly porous bodies is substantially different than for low- tual materials, computer codes must rely on mathematical porosity bodies, due to significant energy losses from the models of material behavior. The mechanical behavior of outgoing shock wave as it compacts the target material. geological materials is considerably more difficult to model Recent observations of the large craters on Mathilde imply than the common metals and alloys used in structural appli- substantial differences in basic cratering mechanisms and cations. This shortcoming is exacerbated by the fact that the ejecta existence and fate. While experiments have been con- processes involve the extreme ranges of conditions men- ducted in porous materials such as dry soil and sand, the com- Holsapple et al.: Asteroid Impacts: Experiments and Scaling Laws 445 paction processes in those bodies is much less important than second to >6 km/s, and most tests have been conducted in in much more porous bodies. This remains an area of ex- evacuated chambers (Fujiwara et al., 1977; Takagi et al., treme uncertainty, although some progress has been made. 1984; Davis and Ryan, 1990; Ryan et al., 1991; Nakamura The second issue deals with rate-dependent strength and Fujiwara, 1991; Nakamura et al., 1992). Some aspects effects. There were also at the time of Asteroids II simple of impact disruption (e.g., fragment velocity) have been theories, but no data, that represented the effective strength studied by multiple researchers, while other aspects (e.g., of rocky asteroids as size and time dependent. The extrapo- rotational modes of ejecta) have been measured by only one lation from small-scale experiments to large asteroids is de- or two researchers. Extensive use of computerized image termined primarily by the form of those strength depend- processing — essentially impossible in 1989 — has pro- ences, but the exact nature of those forms was not known. vided significant new data. This issue has now been addressed both experimentally and This section is intended to provide a comprehensive re- using codes, as is summarized below, but many uncertain- view of significant experimental studies carried out between ties remain. 1989 and 2001, to summarize the results, and to briefly dis- Thirdly, the discovery of the Kuiper Belt in 1992, to- cuss their relevance. One must carefully note the wide varia- gether with a general feeling in the community that the tions in results in different materials due to the large differ- distinction between asteroids and comets may only be one ences in impact velocities. of nomenclature, has prompted a number of studies into the impact behavior of icy materials and the interpretation of 2.2. Impact Techniques and Measurements these data in the context of a potentially large icy planetesimal population in the greater solar system. We will briefly The methods to launch high-speed projectiles have re- discuss ice experiments in this chapter and provide refer- mained unchanged over several decades, and include pow- ences for further study. However, the study of ice under der guns, light-gas guns, and electromagnetic launchers. impact is far less complete than that of rocky materials and Velocities range from tens of meters per second to ~7 km/s. no coherent models have been developed for the scaling of Both laboratory and field explosive tests have also been ice impacts to realistic sizes and conditions. used to study energetic disruptions. Means of measuring the Finally, much remains to be learned about the effects of results of such tests have improved in the past few decades, oblique impacts. To first order the obliquity may be ac- largely due to the availability of fast-framing video cameras counted for by a simple reduction of normal component of and image-processing computer systems. High-speed frame velocity, but there may be other more subtle effects deserv- rates between 400 per second (e.g., Giblin et al., 1994a) and ing of further study, particularly for near-grazing impacts. 6000 per second (e.g., Nakamura, 1993) have been used. This chapter is to be considered as an addition to the corre- Film or video footage is typically digitized for computer sponding chapter in Asteroids II (Fujiwara et al., 1989). The analysis, thus enabling researchers to measure the inflight previous results are generally not presented again. Here we dynamical properties of fragments. Examples include Naka- review some previous wisdom, but primarily address new mura and Fujiwara (1991) and Davis and Ryan (1990). approaches and results. Giblin et al. (1994a) used two cameras to measure fragment Mention of all the research in this broad topic over the velocities in three dimensions. Cintala et al. (1997) used a last 12 years would produce a reference section alone larger strobed laser system to measure cratering ejecta velocities. than our allocated space, so we apologize to those whose Giblin (1998) discusses methods to recover three-dimen- important work we did not mention. sional particle velocity trajectories from filmed records.

2. LABORATORY EXPERIMENTS 2.3. Disruption of Target Materials

2.1. Overview As stated above, the kinetic energy per unit target mass Q, a specific energy, is used to measure collision outcomes. The 12 years since the publication of the Asteroids II A related measure is the kinetic energy per unit volume of book have seen many new experimental studies relevant to target, an energy density, which has the units of stress. The asteroids, and have produced a valuable database of experi- specific energy Q multiplied by the target mass density mental results (Fujiwara et al., 1989; Martelli et al., 1994), gives the energy density of the impact. “Impact strength” helping us to predict such quantities as the energy required is sometimes defined as either the specific energy or the for catastrophic breakup and the post-impact fragment energy density needed to produce a largest intact fragment shapes, sizes, and velocities. Target materials used in impact that contains one-half the target mass (see Fujiwara et al., experiments have included rock, glass, clay, sand, loose ag- 1977; Davis and Ryan, 1990; and Ryan et al., 1991). Here gregates, ice and ice-silicate mixtures, and artificial mate- the symbol Q is always used for the specific energy and the rials such as cement mortar, clay, alumina, and plaster. The energy density is written as ρQ. impacting projectile has consisted of metals (aluminum, Figure 1 shows many experimental results for the ratio steel, iron), Pyrex, mortar, basalt, nylon, polycarbonate, and of the mass of the largest postimpact fragment to the initial ice. Projectile velocities have ranged from a few meters per target mass, as a function of the total impact specific energy 446 Asteroids III

100 Shatter Threshold

Metal

10–1

target Ice

/M 10–2 Silicates largest crushed ice target, broken ice projectile M solid ice target, aluminum projectile pellet or chipped ice target, aluminum projectile crushed ice target, solid ice projectile –3 10 solid ice target, solid ice projectile solid ice target, broken ice projectile

Q*Ice Q*Silicate Q*Metal

10–4 104 105 106 107 108 109 1010

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 the movie industry is to slow down the depiction of small- * defines the shattering specific energy QS. Included are re- scale simulations to make them appear to be large-scale.] sults for ice, silicate, and meteoritic metal targets. The data A method to scale these experiments assuming such a size- show that the degree of fragmentation is strongly depen- dependent strength based on crack growth was given first by dent on the target material. In all materials, increasing col- the scaling model of Holsapple and Housen (1986) as dis- lisional energy increases the degree of fragmentation. cussed below. The computer simulations by Ryan and Melosh Experimental studies using rock projectiles to impact (1998) and Benz and Asphaug (1999), using the modeling solid rock targets reveal that projectile/target density, mass, approach of Melosh et al. (1992), have since replicated as- or strength differences can also have a significant influence pects of the scaling, a consequence of their use of a strength on collision outcome, especially for velocities below ~1 km/s model that is rate-dependent. (Matsui et al., 1982; Takagi et al., 1984). Kato et al. (1992) Of course, those studies do not prove that rocky materials noted similar effects in ice targets impacted by ice, alumi- actually have rate or size-dependent strength. The first ex- num, polycarbonate, and basalt projectiles. Davis and Ryan perimental data on how target size affects collisional outcome (1990), Ryan et al. (1991), and Ryan et al. (1999) found that was provided by Housen and Holsapple (1999a,c) using there is systematically less collision damage to the target at homogeneous granite targets. Target diameters were varied the same specific energy as the projectile becomes weaker. by a factor of 18 (up to a target diameter of 34.4 cm), and A major question about the interpretation of such experi- specific energy was kept constant as the size scale was in- ments for asteroids is the possibility of a size- or rate-decreased. The larger bodies were found to be weaker in im- pendent strength (see Fugiwara et al., 1989), such as from pacts than the smaller ones. a model based on the growth and coalescence of inherent Meteoritic targets have also been used in impact studies. flaws in natural rock. [Rate dependence gives the same scal- A series of high-velocity impact experiments into cooled ing result as size dependence, since for large bodies all im- Gibeon iron-nickel meteorites were performed by Ryan and pact processes increase in duration with the size of the body, Davis (2001). It was found that at asteroid belt temperatures i.e., larger bodies have slower processes. A common trick in near 167 K, iron meteorites underwent brittle fracture, and Holsapple et al.: Asteroid Impacts: Experiments and Scaling Laws 447

the resultant fragment size distributions had the typical two- their high porosity readily dissipates the energy delivered slope power law behavior often observed for homogeneous in an impact. Thus, unusually large energies are required to rock targets. The impact strength was determined to be about damage porous structures. 500× larger than for basalt. Ejecta velocities were much Ryan et al. (1999) impacted porous and solid ice bodies higher than those observed in rock fragmentation experi- at a temperature of about –15°C with fractured ice, solid ments, consistent with predictions of scaling theories in ice, and aluminum projectiles at low velocities, and deter- which velocities scale with the square root of the material mined that the impact strength of the porous ice was higher strength in the strength regime. Ryan and Davis (personal by a factor of about 5 than the impact strength of solid ice communication, 2001) also conducted impact experiments targets. The degree of fragmentation also increased with the using iron meteorite targets that were not cooled. For these strength of the projectile. The porous ice targets were as targets, the crater formed at the impact point had large rim- strong as silicates when hit with fractured ice projectiles. flaps, and a very “plastic” overall morphology. Therefore, even though the very porous ice targets had a static material strength well below that for solid ice, they 2.4. Prefragmented and Shattered Targets were just as resistant to collision as solid ice. Energy is ap- parently well dissipated by the void spaces within the target. To model fractured asteroids, Ryan et al. (1991) con- Kawakami et al. (1991) constructed an ellipsoidal sample structed samples of previously shattered mortar targets by of gypsum with a porosity of 10–30% to model Phobos, careful reassembly and weak cementation, and reimpacted and impacted the body to produce a crater equivalent to the them. They found no large differences in impact strength Stickney crater. Fracture patterns appeared similar for both between the preshattered targets and the original targets low- and high-velocity impacts, which was attributed to the from which they were constructed, even though the preshat- low shock impedance and porosity of the target (i.e., shock tered targets had some porosity and a compressive strength wave attenuation occurred). They concluded that target type that was less than half that of the original strong mortar significantly affects the resulting mode of fragmentation. target. For some unknown reason, the mean ejecta speeds The impact strength was found to be comparable to that of from these bodies were higher than those measured for the basaltic bodies, even though the static strength of gypsum original strong homogeneous targets. The resultant fragment is 1–2 orders of magnitude lower than basalt. size distributions for the preshattered targets were not sig- To model stone meteorites, Durda and Flynn (1999) con- nificantly different from their homogeneous counterparts, ducted a series of ~5-km/s experiments using inhomoge- i.e., they were not further fractured upon reimpact. Giblin neous, porous targets constructed of two materials having et al. (1994a) tested the effect of void spaces in targets by different strengths (porphyritic olivine basalt). The largest fabricating a three-section, strong cement mortar body. The fragments generated were representative of the bulk compo- sections consisted of a top and bottom spherical cap, with a sition, while millimeter-sized fragments were composed of 1-mm spacer separating these pieces from a middle section. isolated olivine crystals. The latter indicates that the target They reported no appreciable difference between the veloc- experienced preferential failure along phenocryst-matrix ity distributions from these targets compared to homoge- boundaries. They conclude that collisions involving chon- neous targets of the same composition. They concluded that dritic asteroids may overproduce olivine-rich material in the the 1-mm spacing might not been a large enough to affect millimeter size range, and olivine may be underrepresented shock wave propagation and subsequent target fracture. at smaller sizes in the primary debris. Nakamura et al. (1994) also examined the effect of Nakamura et al. (1992) used gypsum spheres to inves- reimpacting previously fractured targets. They performed tigate the fragment velocity distribution of a porous body. experiments in which a projectile was shot into a “core” The antipodal fragment velocity for the gypsum target was fragment produced from a previous impact event. They lower than that for a basalt target impacted with a similar found that the outcomes, including largest fragment masses, energy density. Yanagisawa and Itoi (1993) also found that mass distribution of fragments, and size-velocity distribu- their sand-bag and porous alumina targets produced frag- tions, were not changed significantly. However, fine dusts ments with significantly lower velocities than a comparable were spewed out with a velocity higher than tens of meters impacts in nonporous basalt. Love et al. (1993) found that per second within a short time after the impact, probably increasing a target’s porosity had the effect of decreasing from the interior of surface cracks. the speeds of the ejecta for equal collision conditions. None of those studies give information about gravita- 2.5. Disruption of Porous Targets tional effects for large bodies. Scaling theories predict that lithostatic pressure considerably strengthens a body against Porous targets including gypsum (Kawakami et al., 1991; disruption, which is why disruption specific energy in- Nakamura et al., 1992), porous alumina, cement mortar creases markedly in the gravity regime (see section 3 be- (Davis and Ryan, 1990), and sandbags (Yanagisawa and low). Housen et al. (1991) and Housen (1993) simulated Itoi, 1993) have been tested in impact disruption experi- the effects of gravity by applying external pressure to small, ments. Exceptionally high impact strengths have been found weakly cemented, porous basalt targets. The largest over- for such porous target materials. While these porous mate- pressures corresponded to an average lithostatic stress in- rials are fragile in terms of tensile or compressive strength, side a 460-km-diameter body. They used explosives buried 448 Asteroids III at appropriate depths to simulate an impact disruption. They point-source assumption of the scaling theory is not valid, found that as overpressure is increased, more specific en- so the discrepancy is not surprising. Yamamoto (2001) per- ergy was required to shatter the body, and that there was a formed further oblique impact cratering experiments into marked increase in the size of the largest fragment, con- glass particle targets at various impact angles, and deter- firming the scaling theory. Also, the specific energies mea- mined the impact angle effects. sured for catastrophic disruption of the target body com- The size distribution of the regolith material on an as- pared well with those estimated from observations of the teroid surface affects the asteroid observational properties Themis, Eos, and Koronis asteroid families. such as albedo, spectra, and thermal emission. Ejection processes due to repetitive impacts of micrometeoroids on the 2.6. Cratering and Ejecta in Porous Targets surface can alter the size distribution of the regolith. Yama- moto et al. (personal communication, 2001) investigated the Compaction mechanisms may dominate the cratering size-velocity relation of ejecta from the surface of glass processes of highly porous, pristine bodies such as come- spheres. They mixed three different size glass spheres as tesimals (Sirono and Greenberg, 2000) and asteroids such the target, and found that the flux of high-velocity ejecta as Mathilde (Housen et al., 1999). Michikami et al. (2001) increases as the particle size decreases. impacted centimeter-sized glass bead targets of various po- Cintala et al. (1999) analyzed ejecta velocities for cra- rosities in an evacuated chamber to view and measure ejecta ters formed in coarse-grain sand. For impacts between 0.8 velocities. The ejecta velocities decreased markedly as the and 1.9 km/s, the ejecta velocity distribution was found to porosity increased. For the most porous targets (80% and be a power law, although the exponent differed significantly 60% porosity), measured velocities were more than 2 orders from that expected from scaling theory. They suggested this of magnitude below those measured for rocks, and for a discrepancy might be due to the fact that the sand grains porosity of 60% only 2% of the ejecta had a velocity greater were comparable in size to the impactor. 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 2.7. Experiments in Ice these experiments suggest that almost no visible ejecta would be present. Love et al. (1993) also performed cra- Experimental studies in ice are important in the light of tering experiments in porous glass targets, although the cra- studies suggesting that collisions are very important evo- ters were dominated by the spall features of small experi- lutionary processes even in the Kuiper Belt (Farinella and ments. Housen and Holsapple (1999b) and Housen et al. Davis, 1996; Davis and Farinella, 1997) and the Oort Cloud (1999) report impact experiments at 1.9 km/s into a very (Stern and Weissman, 2001). porous material with a density of 0.9 g/cm3 and very low Croft (1982) describes cratering experiments in porous crush strength, as a simulant for cratering on Mathilde. The ice in which polyethylene projectiles from 2.3 to 6.3 km/s experiments were performed at 500G on a centrifuge in or- impacted sifted granular water ice with a density of 0.5 g/ der to reproduce the ejecta ballistics and lithostatic forces cm3. He reports formation of “a large hemispherical cup involved in large cratering events on Mathilde. Cratering whose walls consist of snow compacted to nearly the den- was dominated by compaction, with negligible ejecta. That sity of competent ice.” Clearly compaction played an im- interpretation was corroborated by post-event computed to- portant role. mography scans of the targets that clearly showed substan- Mizutani et al. (1982) reported experiments in which tial increases in density below the crater, accounting for the aluminum and polycarbonate projectiles were fired into sili- crater volume. Those results implied that the five largest cate and competent –18°C ice targets at 100–1000 m/s. craters on Mathilde would have increased its overall den- They concluded that the specific energy required for the sity by about 20%. catastrophic failure of ice is approximately 2 orders of More recent cratering experiments at 500G in that same magnitude smaller than that required for basalts, and that porous material were recently reported by Housen and Voss the crater morphology in ice at these impact velocities is (2001). For porosities of about 50%, only about 10% of the strongly dependent upon the projectile shape and material crater mass was ejected; the crater was formed primarily properties. by compaction. Additional experiments at other gravity lev- els have also been performed to test scaling and gravity- 2.8. Fragment Properties and Distributions strength transition. The scaling implications of these experiments are under study. An important aspect of impact disruptions and asteroid Yamamoto and Nakamura (1997) examined the appli- evolution is the nature of the fragments: sizes, shapes, cability of the Housen et al. (1983) scaling for very-high- spins, and numbers. A number of experiments have been velocity ejecta (hundreds of meters per second) from craters performed to measure these quantities. generated by oblique impacts into powder glass sphere tar- 2.8.1. Fragment sizes. Fragment size distributions gets with a porosity of 44%. The laboratory data were es- following a disruptive impact tend to conform to power timated to be about an order of magnitude lower than the laws, although the distribution is often divided into two or results extrapolated from the scaling formula. Since the three segments with each segment showing a different high-velocity ejecta originate from near the impactor the power-law exponent. The divisions between the segments Holsapple et al.: Asteroid Impacts: Experiments and Scaling Laws 449

generally correspond to different fragmentation regimes in (150–690 m/s) collisions of ice spheres, independent of tar- the disrupted target. For example, Di Martino et al. (1990) get size and impact velocity. Arakawa (1999) continued the impacted alumina cement and basalt targets at 9 km/s, and program of ice studies by measuring maximum and mini- 94–98% of target mass was recovered after the experiments. mum ejection velocities in oblique impacts, but using rela- They found a knee (a bend in the size distribution) at milli- tively large impactors with 13% of the target mass. He con- meter sizes, which they attributed to the transition from firmed a maximum velocity of ejected fragments as 3× the cratering to bulk fragmentation. This process was studied impact velocity. analytically by Bashkirov and Vitzayev (1996). Since 1989 Data on ejecta velocities have also been reported for a many authors have reported fragment size distribution data wide range of other target types: cement mortar (Davis and that broadly agree with the discussion in Asteroids II (see, Ryan, 1990), pyrophilite (Takagi et al., 1992), alumina and e.g., Martelli et al., 1993; Mizutani, 1993; Davis, 1998). gypsum (Nakamura and Fujiwara, 1991; Nakamura et al., 2.8.2. Fragment shapes. Fragment shapes have not 1992), porous alumina, commercial mortar and sand (Yana- generally been reported, although Giblin et al. (1998b) goes gisawa and Itoi, 1993), limestone, alumina cement (Giblin into some detail. They compare their data to those listed in et al., 1994a,b) and gabbro (Polanskey and Ahrens, 1990). Asteroids II and to the semi-empirical model (SEM) of dis- The three-dimensional velocity data have generally been ruption developed by Paolicchi et al. (1989, 1996), and find collected using a stereoscopic system. Characteristic ejec- good agreement. However, in the triaxial ellipsoid model tion velocities are typically between 0.1 and 0.5% of the using orthogonal dimensions a, b, c (see, e.g., La Spina and impactor velocity for high-speed impacts into rocky targets, Paolicchi, 1996) the values of (b/a) = 0.60 and (c/a) = 0.45 but much slower for porous targets. found by those authors differ from their Asteroids II coun- The antipodal velocity has been measured by a number terparts of 0.70 and 0.50. One possible explanation for this of researchers, since it provides a reasonable and easily difference that the experiments were open-air and there was measured characteristic velocity for an impact. Nakamura no secondary fragmentation of ejecta, which would tend to (1993) tests the scaling of antipodal velocity with both Q increase c/a and b/a. and NDIS (see section 3.1.1 below), finding good agree- In studying the size and shape distribution in a labora- ment with Fujiwara and Tsukamoto (1980) and Davis and tory experiment, it should be borne in mind that many stud- Ryan (1990). Giblin et al. (1998a,b), using a contact charge ies suggest that monolithic asteroids are rare, and that the to simulate the impact of a 6.2-km/s projectile, found anti- majority of asteroids are rubble piles that have been dis- podal velocity to be only one-third of the value predicted by rupted and reaccumulated numerous times in their lifetime Fujiwara et al. (1989), raising questions about the equiva- [see estimates below and Richardson et al. (2002)]. We must lence between a contact charge and an impactor. therefore be cautious extrapolating any data on fragment Martelli et al. (1993) studied the angular distribution of shapes from the laboratory to real asteroids. velocities in several open-air impact experiments and found Models of the initial and evolved asteroid size distribu- evidence for collimated jets and other anisotropies in the tion tend to have trouble reproducing the steep slopes of ejecta field. They discuss these data as a possible mecha- the observed population, an issue addressed by Tanga et al. nism for the formation of asteroid binaries and families. (1999); the paper also includes a review of a geometric 2.8.4. Velocity-mass correlations. The relations be- approach. They conclude that the steep size distributions tween a fragment’s velocity and mass is a key component may be partly explained by their model, which includes a in models of collisional disruption and asteroid evolution consideration of the geometry of the ejecta, specifically the (see, e.g., Petit and Farinella, 1993). Results of detailed convexity of largest fragments. crater ejecta studies (e.g., Gault and Heitowit, 1963; Vickery, 2.8.3. Velocity distributions. Fragment velocity of dis- 1986, 1987) indicated a power-law relationship between frag- ruption experiments has received a large amount of atten- ment velocity and mass in high-velocity cratering of rock- tion since the publication of Asteroids II, largely due to the like materials. Laboratory disruption experiments (Fujiwara widespread use of image processing and analysis computer and Tsukamoto, 1980; Nakamura et al., 1992; Nakamura, systems. The primary conclusions presented in Asteroids II 1993) have suggested that a power law holds to some extent still hold: The fastest fragments originate near the impact (see Fig. 2). point, fragments generally do not collide with one another, However, other studies (e.g., Ryan, 1992; Takagi et al., and core fragments in core-type disruption tend to be trav- 1996; Giblin, 1998) have found little correlation between eling at very low velocity. The fast fragments produced near velocity and mass for similar experimental parameters (see the impact point usually consist of fine dust traveling an Fig. 3). The authors pointed out the lack of complete data order of magnitude faster than the larger ejecta, typically as well as the problem of selection effects in the analysis several kilometers per second in hypervelocity impacts (e.g., of laboratory fragmentation experiments. Di Martino et al., 1990; Drobyshevski et al., 1994), al- 2.8.5. Rotation rate distributions. Very few data on though the impacts into porous targets discussed above have fragment rotation were available at the time of Asteroids II much smaller velocities. The fine fragments may be the publication, but the brief conclusions presented there are result of jetting. This is studied and discussed by Arakawa still valid. Specifically, the fastest rotators originate near the and Higa (1995), who found ejection velocities of fine frag- impact point and, although many fragments have been ob- ments to be 1.7 to 2.9× the impact velocity in low-velocity served rotating quickly [e.g., 100 rotations/s (Fujiwara, 450 Asteroids III

experiments because such fragments have undergone no 102 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 101 the rotation rate of a total of 811 fragments studied across 8 similar experiments. Data on rotation rate distributions are shown in Fig. 4. 100 Basalt Basalt Some asteroids have non-principal-axis rotation (“tum- Basalt Alumina bling”) states (see, e.g., Hudson and Ostro, 1995). Authors Basalt Alumina discussing fragment rotation in impact experiments have not

Velocity (Center of Mass) (m/s) Velocity Basalt Gypsum generally reported any evidence of tumbling fragments. 10–1 10–3 10–2 10–1 This may be due to lack of instrumentation (or time for data Fragment Size (m) reduction). However, Giblin and Farinella (1997) report data on a number of tumbling fragments from the experi- Fig. 2. Collected data on fragment velocities in the center of ments described in Giblin et al. (1994a, 1998b). Their con- mass system vs. fragment size (from Nakamura, 1993). These data clusions support a biased distribution of spin vectors that clearly exhibit the slope of –1/6 reported by Nakamura and Fuji- favors principal axis rotation in ejected fragments. wara (1991). 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 1987)], none of these approach the rotational bursting limit spinning fragments were split into smaller ones and collided for their materials. Generally, large fragments have been with other fragments.” Giblin et al. (1994a) report that they found to rotate more slowly than small ones. observed several ejected fragments splitting just after ejec- The distribution of asteroid rotation rates is often com- tion from the disrupted target. Giblin et al. (1998a) describe pared to a Maxwellian distribution (e.g., Harris and Burns, a very well studied case from the same experiments of ro- 1979; Farinella et al., 1981; Binzel et al., 1989; Fulchignoni tational bursting, where a tumbling fragment travels more et al., 1995). Although considered appropriate for a highly than five target diameters before breaking into two pieces evolved asteroid population (Harris, 1990; Farinella et al., due to rotational stress. Rotational bursting is most likely 1992; Yanagisawa et al., 1991; Yanagisawa and Hasegawa, in the case of a tumbling prefractured fragment since only 1999; Yanagisawa and Hasegawa, 2000; Sirono et al., 1993; in this case will the internal stresses of the body be time- Kadono, 1993), the Maxwellian distribution is not expected varying. With Monte Carlo simulations those authors show to give a good fit to “young” fragments from disruption that a fragment ejected from a body can subsequently place

10.0 (a) (b)

1.0 1.0 Normalized Velocity Normalized Velocity

0.10 0.10 10–5 10–3 10–1 10–4 10–2 1.0 Normalized Mass Normalized Mass

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). Holsapple et al.: Asteroid Impacts: Experiments and Scaling Laws 451

100 200 (a) (b) gradient –0.44 100 80 40

60 20

10 40 6 Rotations/s 4 Shot 92-5 20 Number of Fragments 2 Shot 92-6 Shot 92-7 Shot 92-8 0 1 0 20 40 60 80 10 2030 40 60 80 Rotations/s Fragment Size (mm)

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 (1987) reports that fragment rotational energy is generally is a plausible mechanism for producing parent-satellite sys- less than 1% of kinetic energy in high-velocity disruption; tems such as Ida and Dactyl. Nakamura (1993) reports between 1% and 10%. A similar Fujiwara and Tsukamoto (1981), Nakamura et al. (1992), study by Giblin (1994) reports this ratio to be between 2% and Nakamura (1993) discuss the variation of rotation rate and 4% for four alumina cement targets disrupted using a with size in ejecta from disruption of rocklike targets, find- contact charge. ing a best-fit power-law exponent between –1.5 and –1.0, Sugi et al. (1998) impacted water-ice targets using a but note that their data were limited. Giblin et al. (1998b) copper projectile at speeds from 54 to 329 m/s in order to find a significantly shallower slope in their data. Figure 4b study the degree of impact vaporization. They observed that, shows some of these data; although there is a definite slope, below 100 m/s, less than 0.03% of impact energy went into a power-law fit simply does not provide a good description impact vaporization in their experiments, and that this in- of the data. creased to 18–26% between 100 and 180 m/s. Schultz (1996) found significant amounts of impact kinetic energy went into 2.9. Energy Balances impact vaporization in oblique impacts of an Al sphere into competent ice, in experiments where impact speeds were Energy partitioning is an important consideration in the between 4.7 and 5.9 km/s. effects of any impact. The total energy is divided (generally in decreasing order) between heat, comminution of the 3. SCALING THEORIES, ISSUES, target, target and ejecta kinetic energy and ejecta rotational AND RESULTS energy; and, for sufficiently high-velocity impact, melt and vaporization energies. An important component in the con- Issues of scaling are fundamental to our understanding text of modeling and understanding collisional evolution is and interpretation of impact processes, and are needed to the fraction of impact energy partitioned into ejecta kinetic unravel the meaning of laboratory experiments and code energy, since this determines whether or not a disrupted calculations. Various approaches to scaling have been de- body remains disrupted or reaccumulates. veloped over the years. The result of any impact depends According to the available results in 1989, this fraction on the conditions of the impactor and those of the impacted is less than 3% for high-velocity (core-type) catastrophic body (target), perhaps in complicated ways. A successful impacts (Fujiwara and Tsukamoto, 1980) and less than 1% scaling approach must distill the large number of possibly for high-velocity cratering impacts into semi-infinite basalt relevant physical parameters down to the essential few. The (Gault and Heitowit, 1963). The fraction of energy parti- goal is to have a rule to predict all impact responses (the tioned into ejecta KE is higher for low-velocity impacts; on effects) from the impactor conditions (the cause). the order of 10–20% (Waza et al., 1985). Fujiwara (1987), Nakamura and Fujiwara (1991), and 3.1. Approaches Nakamura et al. (1992) all reported that the largest propor- tion of energy goes into heat, comminution of the target, There are two synergistic approaches to scaling: the theo- and the relatively high velocity of fine fragments. Fujiwara retical approach using similarity and dimensional analyses; 452 Asteroids III and numerical computer calculations of a suite of impacts. dent strength. This is done because, for many geological ma- Each has its strengths and weaknesses. They are reviewed terials, the tensile strength, which is fundamental in disrup- in turn. tion, is observed to be more strongly rate dependent than is the 3.1.1. Theoretical scaling methods. The impactor has shear strength, which is most important in crater formation. dominant measures of size a, velocity U, and mass density Insofar as the scaling forms, any strength measure with δ that determine its energy, mass, and momentum. It also stress units gives the same result. When trying to correlate has an impact angle and many material properties. A com- between results for different materials, some specific meas- mon feature of all scaling theories is that one single measure must be chosen. However, since different strength measure is chosen to represent all properties of the impact initial ures such as compressive, shear, or tensile strength are often conditions. In the historic approaches, the impactor kinetic in about the same ratios for different materials, the choice is energy was used most of the time, although some ques- not so important. The choice becomes more uncertain when tioned whether the measure should be the impactor momen- choosing between a crush strength of a porous material and tum. Beginning with Holsapple (1981, 1983), and in a num- some other strength measure. ber of subsequent papers (Schmidt and Holsapple, 1982; Therefore, there are two measures: one for the impactor Housen et al., 1983; Holsapple, 1987, 1993; Holsapple and and one for the target resistance. Their choice, together with Schmidt, 1987), a different measure (the coupling param- methods of dimensional analysis, lead to definite power- eter) was introduced. Its existence and its form C = aUµδν law algebraic forms for scaling. The reader is referred to was shown to be a consequence of physical and mathemati- examples in the literature such as Housen and Holsapple cal “point-source solutions” for rapid energy deposition in (1990), Mizutani (1993), and Holsapple (1993). These ap- vanishingly small regions for general materials. Therefore proaches are limited to phenomena where the point-source it is a global measure of the impact process for regions away approximation governs. from the immediate details right at the impact site. It gen- 3.1.2. Code calculation approaches. Scaling laws and erally is intermediate to the energy and momentum meas- studies of impacts can also come from the outcomes of code ures, depending on the materials. However, it is only as valid calculations. Numerical simulations have become popular: as the point-source approximation, and cannot be expected At the last two Lunar and Planetary Science Conferences to hold for cratering when, for example, the crater is only there were perhaps 30 abstracts by authors using the codes a little larger than the impactor, or for phenomena (such as CTH, SOVA, an SPH continuum code, an “n-body” code, or melt and vaporization) that occur only on a scale compa- a finite-element code to do either two- or three-dimensional rable to the impactor radius. However, it does seem to work impact calculations. One attractive feature is the ability to surprisingly well over large ranges in velocity and scale. investigate effects of specific shape and structure (Asphaug For example, for transient craters in water it correlates to et al., 1996, 1998), and to look at gravitational assemblages within a few percent experiments with impact velocities (Richardson et al., 1998; Leinhardt and Richardson, 2001). ranging over the extreme range of 1 m/s to 6 km/s (see Initial shock processes and energy coupling are calcu- Fig. 2 in Holsapple, 1993). Code calculations have showed lated in many ways. There are simple equation-of-state that it governs impact-generated flows in regions as near models with no thermodynamic coupling (e.g., Murnaghan), as 1–2 impactor radii. simple analytical models for single solid phases (Mie- A different measure of the impact, the nondimensional Gruneisen and Tillotson), complex analytical models includ- impact stress (NDIS), was introduced by Takagi et al. (1984). ing melt and vapor (ANEOS), or complete tabular databases It was initially based on the maximum pressure generated such as the SESAME library. None of these include any at the impact point. In a later approach, Mizutani et al. kinetic effects, which might be important. Effects of porous (1993) used the stress propagated to the antipodal point of crush-up are almost never modeled, since that greatly in- the target. [Strictly speaking, it is a measure of the effects creases the numerical difficulties. (Calculations with very and not the impactor (the cause). Its utility for scaling may low density that use the standard forms of the equation of rest on an expectation that the transmitted stress is easier state omit the energy dissipation of a crushable material and to predict than the other effects of the impact.] Mizutani do not model porous materials.) Some of these models are (1993) introduced special assumptions about the nature of only appropriate for special cases, although they are some- the stress decay from the source through the body, while times used in other inappropriate cases. recently Mitani (2000) used a numerical calculation to esti- Strength models include none (hydrodynamic), constant mate the transmitted stress. Thus, the NDIS has evolved (shear and/or tensile), Mohr-Coloumb shear strength, rate- from a detailed measure of the impact condition, which is dependent tensile, and complex damage models. Some have probably not important globally, to a global measure of the even included a viscous component to model acoustic flu- response itself. idization. Gravity, either a fixed planar field or a self-grav- In all cases of scaling it is also necessary to choose sim- ity central field, may or may not be included. ple measures of the target resistance. Approaches for scal- Obviously these myriad approaches lead to myriad ing of cratering or disruption typically use some material re-sults. The results vary substantially due to the difficul- strength and/or the gravitational field (determined by body ties of correctly modeling the complex geological materi- size). Cratering scaling generally uses a constant strength, als. A recent statement in the literature that “recent ex- while disruption scaling has used rate- and/or size-depen- ponential increases in computational power have enabled Holsapple et al.: Asteroid Impacts: Experiments and Scaling Laws 453 numerical simulations to become the method of choice to They give algebraic forms and figures for important aspects investigate these issues (planetary impacts) in greater de- of impact cratering events, including volume, diameter, tail” may be premature, although that may become true in depth, rim heights, timescale, and ejecta velocity. Impact the future. experiments can only be conducted at small scales, but The theoretical methods make it clear that any scaling explosive events have been conducted at relatively large outcomes from code calculations are determined primarily scales (megaton nuclear weapons). Therefore much of the by the dominant measures of the impact conditions and the scaling is based on an equivalence between impact and ex- target resistance in the codes. Since the point-source mea- plosive events. sure is firmly rooted in special mathematical solutions to Generally, an impact will impart more of its energy and energy deposition, the codes must reproduce, at least in an more downward momentum to cratering mechanisms than approximate manner, point-source results for any phenom- an explosive on the surface. However, that difference is off- ena away from the immediate impact region. The point- set as an explosive is buried; for very deep burial, the ex- source measure should fail only for phenomena very near plosive has the greater cratering effects. At a depth of burial the impact point (for example, the amount of melt and va- of about 2× the explosive radius, the tamping effects of soil por), for very slow impactors, or for very large (compared above the explosive increases both the momentum and to the target body) impactors. energy in the downward directed cratering flow, and the Also, since the resistance of the target is one of the pri- results of the explosion and impact at the same energy and mary scaling measures, the choice of the gravity level and energy density are similar. That has been validated both by the strength model in the code will determine the form of many experiments and code calculations (e.g., Holsapple, the scaling outcome. Thus, the form of scaling results from 1980; Ryan, 1992). code calculations is simply a reflection of the models cho- Terrestrial explosive events in continental sites have gen- sen. The codes cannot determine what physics are impor- erated craters to ~400 m in diameter, while nuclear events in tant; the creator of the code does that. A good example is the Pacific corals have produced craters in the kilometer range. Benz and Asphaug (1999) paper, in which the slopes ob- Schmidt et al. (1986) present the explosive cratering data- tained from SPH code calculations for disruption match the base and scaling. slopes given in the scaling theories of Housen and Hols- For small impacts (the strength regime) the crater vol- apple (1990) and Holsapple (1994) in both the strength and ume is determined only by target strength. Explosive field gravity regimes. That is because their strength model is data show some increase in crater volume per unit energy based on the same rate-dependent physics that was the basis as the event size increases, a feature attributed to a weak- for the theoretical scaling approach. ening of the target material with increasing size. However, The contributions of codes are then, in principle, to cali- that increase is only apparent for near-surface explosive brate the constants that are unknown from the dimensional energies greater than about 1 t of TNT (4.2 × 109 J), and is analysis approaches, to investigate ranges where the scaling not apparent in buried events. Thus, there is little evidence may not hold, to bridge the gap between scaling regimes, of rate- or size-dependent strength in cratering events, prob- and to investigate ranges of material, shape, and structural ably because they are dominated by shear flows. models. However, there is generally a lack of good material For large events (the gravity regime), the crater volume models and data. The material models usually have many becomes dependent on the surface gravity and is indepen- parameter choices (“knobs to turn”), and often the magni- dent of the target strength. Dry sands, having essentially tude of some primary variable such as some strength is zero cohesion, are always in the gravity regime. In the gravity simply “dialed-in” to make the code match some limited ex- regime the crater volume per unit energy decreases with in- perimental data. Furthermore, there is usually no unique way creasing event energy. The point of transition between the to make that match; other “knobs” might succeed equally strength and gravity regimes has been estimated both by com- well. Other quite different material models might also suc- paring the physical strength of the material to the stresses ceed. Therefore, magnitude calibrations are often specious. of gravity and by the actual data from the large explosive Codes are a poor way to determine material properties. events. For near-surface terrestrial events in hard rocks, this However, having noted the uncertainties, code studies transition occurs at about 1 kt of TNT (4.2 × 1012 J, crater certainly have contributed to the understanding of scaling diameter ~30 m), while for dry soils it occurs at a few tons of issues, and provide ways to study phenomena out of the TNT (crater diameter ~10 m) (see Holsapple, 1993). These reach of experiments. As they mature and better material transition crater sizes apply only for the terrestrial gravity models are developed and implemented, they will become field. If the size or rate effects in cratering are minimal, then even more useful. Important contributions from the codes the transition diameters for other bodies are found by sim- are included in the sections below. ply dividing these sizes by the magnitude of the gravitational acceleration measured in Earth’s gravity (Holsapple 3.2. Crater Size Scaling Results and Schmidt, 1982). It should be emphasized that all these crater-scaling re- 3.2.1. Previous results. The current knowledge of cra- sults are only for common soils and rocks as found on ter scaling in rocks, water, and dry sands was summarized Earth, and are probably not applicable to highly porous in Schmidt and Housen (1987) and in Holsapple (1993). materials. Scaling in materials with high porosity is not yet 454 Asteroids III determined. In the numerous experiments in dry sands, the impact velocity, size, gravity, and material strengths. The target material is at or near its “maximum density” or fully shear strength was modeled by a Mohr-Coloumb plasticity packed state, with about 30% porosity. Substantial compac- model, with no size or rate effect. The calculations were tion from that state can only occur by crushing or shearing carried out to relatively late stages. They tabulate constants individual grains, which requires substantial pressures, on for the scaling of various crater features. The results com- the order of kilobars (Housen and Voss, 2001). Cratering pare very well to the theoretical scaling curves mentioned experiments in dense sands show only a small amount of above that have been calibrated by the terrestrial cratering crushed material remaining in the crater, and no noticeable database. They have been carried to the very late stages of density increases below the crater. Housen and Voss (2001) crater formation to study complex crater morphologies. measured pressure-crush curves for highly porous materi- Housen and Holsapple (2000) also used the CTH code als, and the pressure required for 10% compaction of their to perform cratering calculations, and included a specific 53% porous material was a factor of about 30 less than for porous model for crushing. It was found that the modeling dense Ottawa sand. Porosity may play only a minor role for of the crush behavior was difficult but essential. They found cratering in dense sands, but is thought to play a much that results were very sensitive to the material constants of larger role in a very-low-density material where the mate- the crush model and were able to match some, but not all, rial is easily crushed. aspects of well-documented experiments in dry sand. 3.2.2. Implications of new experiments on cratering.In Rate-dependent models have been available in the codes water ice, new experiments for cratering and ejecta have used in the weapons community since the 1980s. For im- been performed, as are presented in the experiments sec- pact calculations, a rate-dependent strength model was im- tion above. However, important questions remain about plemented in the SALE finite difference code by Melosh their relevance and scaling to the large icy solar system et al. (1992). That model was based on an implicit descrip- bodies. Small craters in ice are typically dominated by sur- tion of the statistics of flaw sizes and growth, an interpre- face spall, as is observed also for centimeter-sized cratering tation of the one-dimensional Grady-Kipp model (Grady in rocks. However, it is known from explosive testing in and Kipp, 1980). It used a scalar damage measure; and de- rocks that spall effects are absent for craters of tens of graded the material stiffness in both tension and shear to meters and larger. Thus, small spall-dominated experiments zero as the damage accumulated. It has no plasticity model. in ice and other brittle materials may involve different phys- It was used to model Stickney Crater on Phobos (Asphaug ics than do larger cases and, if so, are of limited use for and Melosh, 1993), for disruption calculations (Ryan and scaling to large sizes. Melosh, 1998), and to study crater scaling (Nolan et al., In addition, many ice experiments have used water ice 1996). For cratering, Nolan et al. (1996) conclude that, for near its melting point. Crystalline materials are known to large impacts in basalt, the target material is substantially have significant rate effects at near-melt temperatures, which fractured by the outgoing shock wave, and the crater forms are much less important at low temperatures. The materi- within the fractured region just as it would in an entirely als become more brittle at lower temperature. Thus, there strengthless material. may be dominant rate effects in the small-scale experiments Those results should be compared to the terrestrial cra- in warm ice (where the rates are very large), which would tering database and the impact scaling theories mentioned not be present in large craters (where the timescales are above. The equivalent explosive energy for the impacts orders of magnitude larger). The exact form of these tem- studied by Nolan et al. (1996) (5.3 km/s, 1–120-m-diam- perature dependences are not known. For these reasons, the eter impactors) ranges from a few tons of TNT to ~5 Mt. data on cratering and ejecta in ices have not yet been synthe- For tons to a few kilotons of yield, their cratering results sized into an inclusive scaling form, and large gaps in the can be directly compared to large explosive field events, data exist. because in that range the curvature of their target is of little Recently craters have also been studied in highly porous consequence, and the terrestrial events in this range show materials (see section 2.6). A major observation is that, as no effect of gravity. The numerical simulations give a cra- porosity increases, the mechanics of cratering change from ter volume per unit energy of about 5 × 104 ft3/t of TNT one dominated by lateral and upward flow and ejection to one for the smaller impactor and 2.4 × 105 ft3/t for the larger dominated by crushing within the crater. As a consequence, impactors. These values are 2 orders of magnitude or more the amount and speed of ejected material decrease substan- larger than actual large terrestrial explosive events in rocks tially as the porosity increases. However, important ques- (Schmidt et al., 1986). (The report by Schmidt et al., which tions remain about the exact nature of porous materials in presents scaling curves for explosive cratering, has restricted asteroids, and whether the materials of the experiments men- distribution; it is available to U.S. government agencies and tioned above are reasonable simulants for low-density aster- their contractors only. However, the actual explosive crater- oids. In any case, the experimental results have not yet been ing data is mostly in the open literature in the form of gov- synthesized by a reliable scaling theory. Also, there have ernment agency reports. These include significant explosive been few computer calculations for highly porous targets. craters in dry geologies such as Sailor Hat in rock; Stage- 3.2.3. Code calculations of crater scaling. Using the coach and Scooter in alluvium; and the Danny Boy, Sedan, CTH finite-difference code, O’Keefe and Ahrens (1993, 1999, Teapot Ess, JangleU and Johnie Boy nuclear craters, as well 2000) performed many cratering calculations with various as a multitude of smaller field and experimental craters.) Holsapple et al.: Asteroid Impacts: Experiments and Scaling Laws 455

Nolan et al. (1996) 106 104

Wet Soils, U = 5 km/s Hard Rock, U = 5 Wet Soils, U = 10 Hard Rock, U = 10 Wet Soils, U = 20 Hard Rock, U = 20 (1996) Wet Soils, U = 40 Hard Rock, U = 40 Nolan et al. Water 103 104 Rock Curve, U = 5 π Wet Soils, U = 5 V = 0.98 V/m V/m 2 ρ ρ 10 π – 2 0.65 = = V V −α π π 2 + µ πππ= 0.095 + 102 V ( 23µ ( 101 Y = 6.9 mpa, ρ = 2.65 µ = 0.55, α = 0.65 (a) (b) 100 100 10–9 10–7 10–5 10–3 10–9 10–7 10–5 10–3 π 2 π 2 2 = (ga/U ) 2 = (ga/U )

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 Some suggestions about possible shortcomings can be basalt gave a cratering efficiency of 580 ft3/t (Vortman, 1965). given. The primary difference between these recent codes This can also be seen in Fig. 5, where the calculated crater and the earlier ones is the inclusion of the crack growth sizes are compared to the conventional impact scaling (from physics, which are thought to be the primary cause of rate Holsapple, 1993) in a dimensionless form. On the left at effects. The Grady-Kipp rate models apply to one-dimen- small sizes (Fig. 5a), their result is about an order of mag- sional tensile fracture only, and the codes use it to calculate nitude above the value for cratering in wet soils at impacts a single scalar damage measure. The correct measure of rate of 5 km/s (the vertical arrow). At large sizes, in the gravity to use is uncertain: that at the shock front, or one based regime, they are just about on the curve for cratering in on the pulse duration. Then, when a single crack grows to water. Figure 5b shows that the numerical results are about be as large as (roughly) a calculation zone, the damage is a factor of 100 above the curve for the volume of craters assumed to be complete, indicating a fully failed material. in hard rocks in the strength regime and a factor of 10 above The models then assume there is no remaining rigidity in them in the gravity regime. It must be emphasized that these any tension or shear state, i.e., the material behaves totally theoretical scaling curves are based on the terrestrial crater as a fluid. All shear stresses in the zone become zero, and database, and there are actual explosive craters in this size unrestricted shear strain and flow is possible. In reality, there range to compare to. These results are indeed unexpected. should remain substantial shear strength, even along the A different implementation of rate dependence based on fracture plane, whenever there is a compression acting on an explicit (rather than an implicit) flaw distribution has that plane. There should be no loss of strength or stiffness been developed by Benz and Asphaug (1994) and used in for shear stresses perpendicular to a failure plane. Com- a SPH code. They discuss some significant problems of the pletely failed material (even with cracks in all directions) Melosh et al. (1992) numerical implementation of the im- should be modeled using the plasticity models of a dry soil, plicit flaw description. Asphaug et al. (1996) also added the not as a fluid. A limiting behavior of a cohesionless Mohr- newer explicit method to the SALE finite-difference code Coloumb model such as used for dry sands would seem to to study the mechanics of cratering on Ida, and to deter- be more appropriate. Even at full failure, the cratering re- mine curves of crater scaling in the strength regime. How- sults should correspond to the database for dry sands, not ever, their results also give craters about a factor of 100 that for water. Hence, the fact that the numerical results are a larger in volume at the same source energy than the terres- factor of 100 larger than the terrestrial explosion data prob- trial explosive data as presented in Schmidt et al. (1986). ably results from incorrect modeling of the shear strength These comparisons may illustrate some of the difficulties of damaged material. in the application of code calculations for impacts, and the In addition, cratering flows are dominated by shearing caution that must be exercised in their interpretation and flow, not tensile failures. Shear strength does not appear to application. have the strong size dependence that tensile fracture does. 456 Asteroids III

The terrestrial cratering data show only a little evidence of sufficient to exceed the escape velocity on Gaspra-sized as- decreasing strength with event size. The CTH code results teroids. Thus, small, very strong rock asteroids were ex- mentioned above using a rate-independent shear strength pected to retain little regolith. However, if the strength de- compare very well to the terrestrial cratering data. creases considerably with body size, then the laboratory Clearly there is great opportunity for future work here, experiments are misleading. In experiments simulating a including the thorough testing of the codes against the ter- jointed rock, Housen (1992) showed that indeed the ejecta restrial explosive database and other code models. While velocity did decrease as the strength decreased, and that a the codes were compared to disruption experiments, the body the size of Gaspra could retain much of its ejecta and constants of the crack distribution models were adjusted to develop considerable regoliths. make the best fit. Melosh et al. (1992) made comparisons Impact experiments in porous materials were mentioned to the Takagi et al. (1984) disruption experiments in ba- in section 2.6. Ejecta velocities are substantially less than salt. They found that the use in the code of published in competent nonporous materials or even in dry sands. The strength and crack size distribution data for basalt gave a tests used materials of different crush strengths, different poor match to the experimental outcomes, but by adjust- shear strengths, different porosities, different scaled sizes ing the crack size coefficient down by a factor of about 105 (G-level), and used different impact velocities. That large (less cracks of a given size, or a stronger material) they were range of parameters has so far precluded any identification then able to get a good fit to the fragment size distribution of the most relevant variables and the synthesis into a com- of the experiments. However, at a given strain rate, the im- prehensive scaling theory. The one fact that is apparent is plied strength of that code model is then a factor of about that the conventional wisdom on scaling in normal terres- 2 higher than actual data for similar rock materials. Ryan trial materials is not valid. and Melosh (1998) used that same strength model for ba- The images of Mathilde have also raised new questions salt. Also, for comparison to Housen et al. (1991) experi- about the production of ejecta on a porous body. Although ments in a porous grout they again adjusted the Weibull Mathilde has several very large craters as wide as the as- crack distribution parameters of the code until a reasonable teroid mean radius, there are no visible ejecta features. This match to the disruption experiments was obtained. Then lack has been explained in three very different ways. they also made some comparisons to two experiments that Housen et al. (1999) performed experiments in a material were basically cratering events. However, the crater masses with the same porosity as Mathilde, and concluded that they obtained were larger by factors of 3 and 5.5 respec- almost no ejecta are produced because the cratering is domi- tively than the experiments. nated by (downward) crushing, not (outward and upward) Benz and Asphaug (1994) made comparisons of their excavation. The experiments by Michikami et al. (2001) code results to the disruption experiments by Nakamura and mentioned above confirm those interpretations, in a differ- Fujiwara (1991) and were able to obtain good results, but ent but also highly porous material. On the other hand, again made choices of the fracture parameters to give a best Asphaug et al. (1998) performed a code calculation for a fit. Again, the implied strength at a given strain rate is about porous body and deduced that ejecta velocites are greatly 2× published strength data. Later applications of the code enhanced by the porosity, and that all ejecta would escape with the same material fracture parameters were made by the asteroid. Finally, Cheng and Barnouin-Jha (1999) used Asphaug et al. (1996) for cratering on Ida and by Nolan et conventional crater scaling for dense dry sands to apply to al. (1996) for cratering on Gaspra-sized bodies. However, Mathilde and determined that the large crater morphology comparisons to actual cratering data and scaling curves is consistent with oblique impacts. were not made. There are major differences in these approaches. The use The modeling of the crack growth, shear strengths, and of conventional dry sand crater scaling used by Cheng and the inclusion of substantial porosity deserve special atten- Barnouin-Jha (1999) is questionable in view of recent ex- tion for the code methods. The approaches used in the weap- periments in highly porous materials. The fidelity of the ons community, particularly those for impacts into porous sample material constructed by Housen et al. (1999) as a ceramics, should be considered. These issues require much Mathilde analog is uncertain. Finally, the code calculation further study before code methods can be considered to be of Asphaug et al. (1998) included macroporosity but no entirely reliable. microporosity, and therefore did not model the continuum behavior of highly porous materials. What is the most ap- 3.3. Ejecta Scaling propriate model for Mathilde? We do not know. No resolution of these major discrepancies has been made, so at Ejecta scaling laws are important in order to model the present the question of ejecta scaling in porous asteroids is evolution of the regolith on small bodies and the produc- as uncertain as crater size scaling. tion of interplanetary dust particles. Housen et al. (1983) present scaling formulas for amounts and velocities of the 3.4. Scaling of Shattering and Dispersion ejecta from cratering. In the strength regime, the ejecta velocity at a scaled range increases as the square root of the The scaling of disruptions presented in Asteroids II was strength. Laboratory experiments typically show velocities given in Fujiwara et al. (1989, see Fig. 9, p. 259). Since Holsapple et al.: Asteroid Impacts: Experiments and Scaling Laws 457

109

(1999) (disperse)

108 (1990) (disperse) (1985) (disperse) Benz and Asphaug

7 10 Housen and Holsapple (1998) Davis et al. (1994)

(1990) Housen and Holsapple Holsapple Holsapple (shatter) Farinella et al. (1994) Durda et al. (1996) (disperse) (shatter?, below disperse) 106 Ryan (1982) (1992)

Housen and Holsapple (1998) (shatter) Q*, Specific Energy (erg/g)

D Love and Ahrens u rd a 5 e 10 t a l. (1999) (1 99 8) Ryan and Melosh

104 102 103 104 105 106 107

Target Radius (cm)

Fig. 6. Specific energy thresholds, some for shattering and some for dispersion, as presented by various authors. For small asteroids (on the left) the specific energy decreases with increasing target size because of a decreasing asteroid effective strength with size. For large asteroids (on the right), the energy increases with increasing target size because of the increasing role of self-gravitation. Note that these results are for rocky bodies only, and not for porous bodies.

that time much has been added. A recent version, Fig. 6, –0.33. Housen and Holsapple (1999a,c) postulate that a shows the specific energy required for shattering and for better value is φ = 6 so the slope is –0.67; furthermore, their dispersion. It is from Benz and Asphaug (1999) but with curve is fitted to their size-dependent granite experimental several curves added. results (those data are off this curve to the left). The curve There are three features that are common among the given by calculations by Ryan and Melosh (1998) for mor- various curves. Each has a negative-slope strength region tar assumes that φ = 6.5 and their slope is –0.61. Thus, in on the left, a positive-slope gravity region on the right, and all cases the slopes obtained are entirely consistent with the a transition size between those two regimes. scaling prediction based on the assumed value of the pa- For the strength regime, the scaling theory of Holsapple rameter φ. and Housen (1986) and Housen and Holsapple (1990) and For large bodies, in the gravity regime, the scaling theo- Holsapple (1994) assumes that tensile failures determine the ries predict that the slope should be 3µ. If µ were 2/3 (the strength, so they use a rate-dependent strength model. The limiting value when the point source measure is the impac- slope of the power-law in the strength regime (Holsapple, tor energy), there would be no decrease of coupled energy 1994) is 9µ/(3 – 2φ) where φ is the exponent of a Weibull with increasing impact velocity, and pure energy scaling flaw-size distribution, and µ is the exponent of impact ve- would hold. Then the slope in the gravity regime on this locity in the point-source coupling parameter measure. (The plot would be 2. The Davis et al. (1985) model assumed exponent µ reflects the decrease in coupling efficiency and such energy scaling, so they have the slope of 2. The value increasing waste heat with increasing impact velocity. It is chosen in Holsapple (1994), µ = 0.55, was determined by determined in the early-time coupling regime, and is pri- a multitude of results for nonporous materials such as rocks marily a consequence of the high pressure-temperature and water (see Holsapple, 1993), which gives a slope of equation of state. A material with no energy dissipation has 1.65. The Benz and Asphaug (1999) curve has essentially µ = 2/3.) that same slope, suggesting that the code calculation repro- Holsapple (1994) assumes that µ = 0.55 and φ = 9 to give duces the expected early-time energy coupling. The slightly the slope as –0.33. The code calculations of Benz and shallower slope of the Love and Ahrens (1996) curve sug- Asphaug (1999) also use φ = 9, and their slope is also about gests more dissipation in the energy coupling. That might 458 Asteroids III be a consequence of low resolution at the shock front, al- of these effects? Most laboratory experiments cannot ad- though Ryan and Melosh (1998) got a much steeper slope dress any of them. with a very-low-resolution calculation. The differences also In the strength regime, the experiments by Housen and might arise from the equation of states used. Holsapple (1999c) show the size dependence, but only over The most striking feature of this plot is the wide discrep- a diameter range of 18:1. We do not know of any other ancy of the values. However, much of this apparent discrep- direct evidence for the size effect. It remains uncertain ancy arises simply because some of the curves define the whether those trends continue to asteroid sizes. Codes re- * condition for shattering, QS, while others apply to the con- produce that dependence, but that is simply a consequence * dition for dispersal, QD (see section 1). In the small-body of their assumed models. strength regime where there is negligible gravity, these two In the gravity range, there is some indirect evidence for values of Q are the same, since any shattered body will dis- the positive slope. First, the estimates of required energy perse. There are substantial differences in the predictions in to form the Themis, Eos, and Koronis families are at least this strength region, although most match the centimeter- consistent with current estimates for dispersion (Housen and sized laboratory data. (The theoretical scaling curves are in Holsapple, 1990), although not definitive by themselves. fact calibrated to those results. The codes are usually “dialed The use of crater scaling applied to the largest observed in” to match those results; see section 3.2.) craters on relatively dense bodies seems to show an increase In the gravity regime there is a substantial difference of energy with size, and was used as a lower limit by Hols- between the shattering and the dispersion energy. The Ryan apple (1994). The experiments by Housen et al. (1991), and Melosh (1998) curves are for the shattering energy only, using external pressure to simulate the gravitational bind- while the Housen and Holsapple (1990), Benz and Asphaug ing, show a definite increase in shattering strength with that (1999), Melosh and Ryan (1997), and Love and Ahrens external pressure, and were used as an upper limit for the (1996) curves are for the dispersion energy. The Holsapple curve by Holsapple (1994). The analysis by Durda et al. (1994) curve was based on the largest observed craters on (1998), based on an evolution calculation, has an even larger various bodies, so it is a lower bound for the dispersion en- upward slope in the gravity regime. Finally, the computer ergy, but in principle could be above the shattering energy. calculations also show the upward slope, and here the re- The Housen and Holsapple (1990) curve was based on sults are in a regime where the uncertainties of strength velocity scaling, assuming one-half the mass had velocity modeling discussed above are not important. Therefore, al- greater than the escape velocity. Melosh and Ryan (1997) though there is general agreement on the upward slope, also used velocity scaling, but in a different way, to get their there remain substantial differences in estimated magni- dispersion curve. tudes. This is certainly an area needing further research. The estimates of the energy to disperse are about a fac- And, finally, all experiments and calculations for impact tor of about 100 above the shattering estimates. Fujiwara disruption consider only nonrotating asteroids. If indeed (1982) and Davis et al. (1983) previously noted the large many are rubble-pile bodies, many are rotating at a signifi- difference “energy gap” between these two thresholds. cant fraction of the maximum allowable rotation for their Durda et al. (1998) obtained a very different curve by shape (Holsapple, 2001). What effect does that have on re- comparing the results of a collisional evolution model with quired disruption energy? One would certainly think that the present asteroid size distribution, and then backing out could have a major effect at lowering the required energy the scaling law that best matched the present distribution. for disruption. A more recent paper by Campo Bagatin and Petit (2001), 3.4.2. Implications of disruption scaling. There are with a different model of evolution of asteroids, did not many important implications of these results, numerous ex- agree with that result. amples of which can be found in the literature. Here only one 3.4.1. Uncertainties in disruption scaling. The very significant one is considered, that of collisional lifetimes. wide diversity of curves for either of shattering or disper- Farinella et al. (1998) give the disruptional lifetime in years –18 2 –1 sion is unsettling. Those estimates have changed a lot in for a target of radius R as tdisr = [2.85 × 10 R N(rdisr)] , form over the last 20 years. It used to be common to adopt where N(rdisr) is the number of asteroids of radius greater * a single value for Q at all sizes, and the distinction between than the radius rdisr (km) of the impactor required to shatter shattering and dispersion was overlooked. Then the increas- the target. They use an asteroid number distribution in the ing slope in the gravity regime was introduced for disper- main belt as N(r) = 3.5 × 105 r–5/2, with asteroid radius r in sion thresholds, a consequence of requiring energy to launch kilometers [see Farinella et al. (1998) for details]. remnants to escape velocities. The strengthening due to litho- Using the relation for the specific energy to either shat- static pressure was then formulated, so the positive slope ter or disperse (a factor of about 100 different) with an applied to shattering also. That was based on the concept assumed velocity of 5 km/s gives, for any asteroid, estimates that the initial compressive stress due to lithostatic pressure of the lifetime between shattering events and between com- must first be overcome before the stress wave could achieve plete dispersion events. The Housen and Holsapple (1990) tensile fracture conditions. Finally, a decrease of required shattering kinetic energy per unit mass in the gravity re- energy in the strength regime was introduced based on rate- gime is given in cgs units as Q* = (5.8 × 10–8)U0.35R1.65. dependent strength. That is currently an accepted feature Assuming an impact velocity of 5 km/s, and equal densities of all present curves. What, if any, evidence is there for any for the impactor and target, gives the required impactor Holsapple et al.: Asteroid Impacts: Experiments and Scaling Laws 459 radius for shattering as that a dispersal event simply pulverizes the parent body. Thus any large remnant of the 15-km asteroid must itself –3 1.55 rs(km) = (2.01 × 10 )R(km) be an aggregate of smaller pieces. How can any 10-km body ever again become a “rock” with density ~3? which gives the time period as One possibility is indicated from Housen et al. (1999), which estimates that the five largest impacts on Mathilde 5 1.875 tshatter(yr) = (1.8 × 10 )R(km) increased its density by 20%. Perhaps a primary effect of impacts into very porous bodies is not to disrupt them at Taking the Housen and Holsapple (1990) dispersion curve all, but simply to pound them back into relatively dense as a factor of 100 larger specific energy gives the required asteroids, to begin the cycle over again. Perhaps they den- impactor radius for dispersion as sify during large impacts by shaking small fragments down into the holes between larger ones (see Britt et al., 2002; –3 1.55 rs(km) = (9.32 × 10 )R(km) Asphaug et al., 2002). Alternatively, other processes could increase the density of asteroids (Consolmagno and Britt, and a lifetime before dispersion as 1999). Such processes are a prime area for future research. The question of the effects of impacts into rubble piles 6 1.875 tdisperse(yr) = (8.4 × 10 )R(km) is the primary outstanding question for which we have no answer. Indeed, the present state of knowledge is very tenu- Three features of these lifetime estimates are very sig- ous: What we know about scaling for rocky bodies seems nificant. First is simply the large uncertainty; the time is to tell us they cannot remain rocky, so the scaling does not proportional to the required Q to the power of 0.833, and apply! As stated in the closing words of a lecture by James Q is uncertain to maybe an order of magnitude. The number Head, “Almost everything is not yet known.” densities are uncertain to perhaps a factor of 2–3, giving a combined uncertainty factor of about 20 for the lifetime. REFERENCES The second feature is the large 1.875 exponent on the asteroid radius R. The paper by Farinella et al. (1998) has Arakawa M. 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