The shapes of fragments from catastrophic disruption events: Effectsof target shape and impact speed

Daniel D. Durda a,n, Adriano Campo Bagatin a,b,c, Rafael A. Alemañ b,c, George J. Flynn d,Melissa M. Strait e, Angela N. Clayton e, Emma B. Patmore e

a Southwest Research Institute, 1050 Walnut Street, Suite 300, Boulder, CO 80302, United Statesb Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Alicante, Spainc Instituto de Física Aplicada a las Ciencias y la Tecnología, Universidad de Alicante, Alicante, Spaind SUNY-Plattsburgh, 101 Broad St, Plattsburgh, NY 12901, United Statese Alma College, Alma, MI 48801, United States

a r t i c l e i n f o

Article history:Received 30 April 2014Received in revised form3 October 2014Accepted 8 October 2014Available online 18 October 2014

Keywords:AsteroidsFragmentationCollisionsInternal structureGravitational aggregates

a b s t r a c t

We conducted impact experiments at the NASA Ames Vertical Gun Range in the context of an ongoingset of experiments to investigate both target shape and impact speed effects on fragment shapes andmass–frequency distributions in collisions on basalt targets. In this work we present the first part of thatset, regarding mostly target shape effects. We impacted both irregularly-shaped and spherical basalttargets at speeds ranging from �4–6 km/s. We obtained mass–frequency distributions from fragmentsrecovered from the impact chamber and measured fragments shapes using a combination of imageanalysis and manual measurements with a caliper. We find that the characteristics of the mass–frequency distributions and the range of fragment shapes show no significant dependence on targetshape (i.e., flat, ‘shell-like’ fragments are produced in impacts into irregularly-shaped targets as well asspherical ones). We note that many thin, plate-like impact fragments seem to originate from lower-speed impacts and can originate from the interior of the targets (in addition to the flattened fragmentsoften seen to origin from the near-surface spall zone in cratering impacts). We measure the porosity ofaggregates made by artificially (but randomly) reassembling fragments from each impact to be on theorder of 50%, significantly larger than that for hexagonal lattice and random packing of equal sized spheres.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Recent experiments showing ‘onion shells’ of tabular-shapedfragments from impacts into spherical targets (Walker et al., 2013;Nakamura, 2014 personal communication) have re-opened the ques-tion of the fracture mechanics responsible for determining fragmentshape in catastrophic impacts. Fujiwara et al. (1978) presented the firstwork on the topic of shapes of fragments from impact experiments inthe context of asteroid studies. Fujiwara et al. (1989) suggest twomodes of catastrophic disruption for targets in laboratory impactexperiments: (1) ‘core-type’ fragmentation in the high-speed, high-energy-density regime; and (2) ‘cone-type’ fragmentation in the low-speed, low-energy-density regime. These effects appear to be scaleinvariant, at least over the range of cm to m-scale target sizesinvestigated in laboratory impact experiments conducted to date.Fig. 1a, for example, shows the result of one of the cratering impactsinto 1-m-diameter granite spheres reported in Walker et al. (2013). In

these experiments aluminum spheres of 4.45-cm diameter wereimpacted into the granite targets at speeds of 2 km/s. Many of thelarger fragments recovered from the experiment enclosure exhibitedthin, tabular shapes and the fractured walls of the resulting spallcraters displayed the same plate-like nature of the many fragments‘peeled off’ the outer shell of the granite ‘onion’. Very similar fragmentmorphology is seen at smaller scales as well—Fig. 1b shows the largestremnant of an impact into a 5-cm-diameter soda lime glass sphere at2 km/s, exhibiting the same flattened, tabular-shaped fragmentationpattern in the near-surface spall zone (Nakamura, 2014 personalcommunication). Similar behavior is seen in impacts into �6–10-cm-diameter basalt spheres, as reported by Fujiwara and Tsukamoto(1980; Fig. 5) who present sketches of cross-sectional views ofreconstructed targets showing shell-like fracture surrounding theremnant cores.

Because of the propensity of many previous laboratory inves-tigations to focus on idealized spherical targets there exists someambiguity in decoupling the relative importance/influence ofimpact speed versus spherical shape in producing the ‘onion shell’fragment shapes seen in these experiments. If plate-like fragmentshapes are due primarily to impact speed/energy density as

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Planetary and Space Science

http://dx.doi.org/10.1016/j.pss.2014.10.0060032-0633/& 2014 Elsevier Ltd. All rights reserved.

n Correspondence author.E-mail address: durda@boulder.swri.edu (D.D. Durda).

Planetary and Space Science 107 (2015) 77–83

suggested by Fujiwara et al. (1989) this could play an importantrole in the outcome of impacts onto small monolithic objects inthe main asteroid belt due to the non-negligible probability oflow-speed (i.e., below about 3–4 km/s—subsonic in rock) impactsthere (Bottke et al., 1994). There is growing interest in spall-type

impacts into initially monolithic rock targets due to the focus onsmaller near-Earth asteroids (NEAs) – and the blocky fragments incoarse regoliths observed to exist on objects like Itokawa(Nakamura et al., 2008; Noguchi et al., 2010) – as targets forexploration missions and mitigation activities (Holsapple and

Fig. 1. (a) Cratering experiments on 1-m-diameter granite spheres (Walker et al., 2013) showing a trend toward flat, plate-like shapes for fragments spalled from the nearsurface regions surrounding the crater. The impact speed was 2 km/s. (b) Disruptive impacts into 5-cm-diameter soda-lime glass spheres at the same speed display a verysimilar fragmentation pattern (Nakamura, 2014 personal communication).

Fig. 2. Our typical irregularly-shaped and spherical basalt targets. The natural, irregularly-shaped targets are hand specimens collected in the field from the sample site. Thespherical targets we prepared from larger fragments of the same basalt; the darker color is due to infusion of lubricating oil used during the milling process to obtain thespherical shape. The targets shown here were not shot during this experiment run; they remain for future impact experiments.

D.D. Durda et al. / Planetary and Space Science 107 (2015) 77–8378

Housen, 2013) and the recognition that the cumulative effects ofmultiple sub-catastrophic impacts can have the same effect as onelarger, catastrophic impact (Gault and Wedekind, 1969; Housen,2009). Also, the low bulk densities of NEAs in the 100 m to tens ofkm size range suggests that some of these objects may be rubble-piles and that their low densities can be related to the shapesand mass spectrum of their large components and to the way theyreassemble as gravitational aggregates following shatteringcollisions.

A secondary goal of the conducted experiments is to investigatethe theoretical conjecture, supported by numerical modelling (Tangaet al., 1999; Campo Bagatin and Petit, 2001) involving the role of thetarget shape in affecting the mass spectrum of fragments created inshattering collisions.

We describe here our experiments to investigate the importance oftarget shape and impact speed in determining the shapes of frag-ments from catastrophic impacts.

2. Impact experiments

In order to disentangle the potential effects of both target shapeand impact speed in affecting the shape of fragments we chose toconduct impact experiments on both spherical and naturally-occ-urring irregularly-shaped targets of the same basalt material impactedat a range of speeds (Fig. 2). We used basalt samples obtained from arecent lava flow exposed in a road cut near Flagstaff, AZ (latN35120059″, long W111133055″). The targets ranged in mass, M, from238 to 534 g (see Table 1). Because we anticipated reconstructing anidealized fragment size distribution from imaged fragment shapesthat would then be compared with the measured mass distributionwe needed to obtain the mean density of our basalt samples. Wemeasured the bulk density of each basalt sphere and several repre-sentative samples of the irregularly-shaped basalt targets using theArchimedean water displacement method. We obtained values of2.9570.03 g/cm3 (irregular) to 2.9870.05 g/cm3 (spherical) for thesamples.

We impacted a total of six targets (two spheres and four irregulartargets). The mean impact speed for asteroids in the main belt is�5 km/s (Bottke et al., 1994), so we focused on shots with impactspeeds nominally in the �4 to 6 km/s range. For each impact theprojectile was a 3/16th in. (0.476 cm) diameter, �0.1583 g massaluminum sphere fired at the specified target using the NASA AmesVertical Gun Range (AVGR). The targets were each suspended at thecenter of the AVGR impact chamber from a thin nylon line such that

the incoming projectile path would pass very roughly through thecenter of mass of the target. For maximum stability of the targetduring the hanging process, particularly in the case of the irre-gularly-shaped targets, the attach point of the support line for eachtarget was placed such the target hung with its longest (a) axisgenerally in the vertical direction. Because the AVGR has a roughly1-cm-diameter patch of uncertainty in the targeted impact locationof the projectile, we attempted to roughly ‘align’ each irregular targetto maximize its collision cross section for the maximum probabilityof a good central impact during each shot. We set the alignment ofeach target such that the impact point was lined up more or lesswith the incoming impactor direction, thus yielding a roughlynormal incidence impact relative to the impacted face of the target.Although the impactor incidence angle cannot be controlled veryrigorously we note that this may well affect the outcome of anyparticular single experiment, but this effect should ‘average out’when considering the outcomes of several experiments. The cham-ber floor and walls were lined with cloth sheets to provide a softbuffer to help prevent primary eject from suffering subsequentfragmentation after colliding with the chamber.

Following each shot, the debris was collected from the floor ofthe AVGR chamber. This process typically recovered 495% of thetarget mass. Large fragments that were collected from the cham-ber were individually weighed (to a sample completion limit atabout m40.20 g), representing up to 90% of the original mass.This allowed us to carefully measure the mass–frequency distribu-tion of the largest fragments from each impact experiment.

High-speed video of each impact was obtained by five differentvideo cameras (two Phantom V10s, one Phantom V12.1, and twoShimadzu HPV-1s), with frame rates ranging from 1900 to 125,000frames/s, to aid interpretation of the fragmentation mode of thetargets. The details of each shot are summarized in Table 1.

3. Results and discussion

Cumulative mass distributions are derived for the six impactexperiments and the results are shown in Fig. 3. As a commonpattern repeatedly shown by past impact experiments, we observethat for the several largest fragments (depending on each case) nosimple analytic description of the shape of the mass distributioncan be found. For smaller fragments however, power law mass dis-tributions are found with exponents 0.75oβo1.2 in the relationshipN(4m)¼Am�β (where m is the fragment mass, β is the power lawindex, and A is the corresponding constant). This result is found for

Table 1Shot details for the impact experiments, including impact conditions and outcomes. Target diameter for the irregular targets is the diameter of a sphere with the same mass.Target b/a and c/a for the irregular targets determined from pre-impact side and top views in the high-speed video frames. fL is the mass of the largest fragment divided bythe target mass. Brackets (⟨ ⟩) indicate average values for each shot.

Shot 130701 130702 130703 130704 130705 130706

Target type Irregular Irregular Spherical Spherical Irregular IrregularTarget mass (g) 433.00 534.60 237.90 342.70 479.10 451.20Target diameter (cm) 6.54 7.02 5.34 6.03 6.77 6.64Target b/a, c/a 0.61, 0.45 0.66, 0.53 1, 1 1, 1 0.80, 0.78 0.56, 0.46Impact speed (km/s) 4.73 4.45 3.89 3.59 3.68 5.82Projectile mass (g) 0.1587 0.1582 0.1584 0.1582 0.1582 0.1582E/M (J/kg) 4100 2930 5040 2970 2240 5940fL 0.07326 0.13118 0.05944 0.18494 0.23446 0.02637Exponent β 1.00 0.82 1.22 0.96 0.75 1.03⟨b/a⟩ 0.73 0.73 0.74 0.72 0.67 0.73⟨c/a⟩ 0.41 0.42 0.41 0.38 0.34 0.38⟨ψ⟩ 0.61 0.62 0.62 0.58 0.54 0.60⟨ψ10⟩ 0.66 0.62 0.53 0.53 0.51 0.58⟨F⟩ 0.46 0.49 0.44 0.45 0.49 0.48⟨F10⟩ 0.35 0.50 0.31 0.34 0.48 0.38⟨Porosity⟩ 0.53 0.51 0.49 0.54 0.61 0.54

D.D. Durda et al. / Planetary and Space Science 107 (2015) 77–83 79

variable intervals ofm/M, depending on the circumstances of the shot,but it roughly holds in the range 0.5�10�3om/Mo0.5�10�2. Themass of the largest fragment, normalized to the original target mass,tends to be smaller as the specific energy of the impact grows. Theexponent of each mass distribution is also related to the correspond-ing specific energy of each impact, as expected (Fujiwara et al., 1989).Also, the mass distributions seem to show slightly larger values ofβ (i.e., steeper slopes and relatively smaller fragments) in the case ofspherical targets with respect to irregular ones, when comparing twosets of close specific energy impacts. However, this behavior needsfurther sets of impact experiments to properly check the results byTanga et al. (1999) and Campo Bagatin and Petit (2001).

The 36 largest fragments from each shot were photographedand their largest axes (a and b) accurately measured in the imageanalysis software ImageJ (e.g., Fig. 4). Fragments are digitallyimaged on a uniform green background to allow easy chromakeying in Adobe Photoshop. ImageJ fits an ellipse to the silhouetteof each fragment, yielding a measurement of the a and b axislength of each fragment.

Their shortest axes (c) were measured by means of a digitalcaliper. We note that the process of arranging the fragments forphotography naturally oriented them such that their shortest axesaligned more or less normal to the green background plane. Thus,the photographic analysis via ImageJ measured the a and b axeswhile the orientation of each fragment was carefully noted duringthe manual caliper measurement process to ensure proper dete-rmination of c.

Previous investigations carried out in the 1970s and 1980s(e.g., Capaccioni et al., 1986) showed aspect ratios of b/aE0.7and c/aE0.5. We found the overall average aspect ratios to beb/a¼0.7270.13 and c/a¼0.3970.13 (Table 1), with a result for c/asystematically smaller than reported by Capaccioni et al. (1986).This result is quite stable over the different impact experiments andno differences are found in average shapes among spherical andirregular targets, nor for different specific energy up to a factor of 3.Obviously, this does not mean that fragments look like 3-axialellipsoidal shapes at all; instead they are quite irregular, but theiraverage relative sizes are distributed as described. Fig. 5 showshistograms for each shot for both b/a and c/a.

It is interesting to note that the values obtained for b/a are inagreement with the aspect ratios of boulders on the surface of Eros(Michikami et al., 2010). On the contrary, although Nakamura et al.(2008) point out the remarkably good qualitative match betweenboulder shapes on Itokawa compared with fragment shapes fromlaboratory impact experiments, Michikami et al. (2010) note thatthe aspect ratios of boulders on the surface of Itokawa do not matchwell with experimental results. They show b/a generally rangingbetween 0.62 and 0.68, a difference they found to be statisticallysignificant at 95% confidence level. It is hard to speculate on thereasons for such differences. One possible cause may be thermalfatigue due to temperature changes leading to partial erosion ofsome of the boulders on Itokawa that may affect the statisticaldistribution of the aspect ratio. Finally, we note with interest thatTsuchiyama et al. (2014) find that the shape distribution of particlesreturned from the surface of Itokawa by Hayabusa is consistent withthe results of mechanical disaggregation, primarily as a response toimpacts, although we caution that these particles/fragments are allso small that there may be other processes at work in determiningtheir shapes. These very small particles/fragments are at the size ofindividual mineral grains and there the processes that govern grainshapes may be quite different (e.g., involving crystal growth withinthe rock, etc.) than those that dominate the shapes of macroscopicimpact fragments in laboratory experiments or in the fragmentationof bulk rock in asteroids.

We have also investigated the shape metrics Ψ¼(c2/ab)1/3 andF¼(a�b)/(a�c) derived from the results of our impact experiments.These parameters quantify deviation from spherical shape and relativeflatness, respectively (Benn and Ballantyne, 1993; Ehlmann et al.,2008). Values of Ψ (0oΨo1) approaching 1 indicate roundness in thedescribed fragments whereas low values of Ψ closer to 0 indicate flat(oblate) objects. F (0oFo1) is defined when the object is not round.Values of F close to 1 appear for elongated, cigar-like (prolate), bodies,while values of F close to 0 account again for oblate objects.

Fig. 3. Cumulative mass–frequency distributions for each of the impact experi-ments. M is the mass of the target, m is the mass of the nth fragment. See Table 1for details of each shot.

Fig. 4. Example of the process of measuring fragment shapes from photography. Fragments are digitally imaged on a uniform green background to allow easy chroma keyingin Adobe Photoshop (left). The dark bar at the top is an image scale fiducial marker (15 cm long in this case). Analysis in ImageJ (right) fits an ellipse to the silhouette of eachfragment, yielding a measurement of the a and b axis length of each fragment.

D.D. Durda et al. / Planetary and Space Science 107 (2015) 77–8380

Average values for Ψ and F for all measured fragments are quiteclose to each other in each shot and do not show significantdependence on specific energy, impact speed, or target shape (seeTable 1). Their overall average values are Ψ¼0.60 and F¼0.47.Dispersion is larger for F values (⟨σF⟩¼0.22) than for Ψ (⟨σΨ⟩¼0.12),indicating that a wide range of shapes is possible. A loose trendtowards more roundish shapes seems to be present as fragment sizedecreases. More specifically, some of the largest fragments have flatand sometimes plate-like shapes that are less frequently observedamong smaller fragments. This is not clearly related to impact velocityor specific impact energy. Fig. 6a shows a comparison of values of Ψbetween two shots with similar impact speeds, while Fig. 6b com-pares F for two shots with similar specific energies.

More qualitatively, high speed video from our experimentsclearly shows the formation of shell-like fragments in most of thecratering and lower-energy disruptive impacts (e.g., Fig. 7). Exam-ining average values for F and Ψ for the 10 largest fragments (butnot including the largest fragment itself, which is often a large andmore rounded core), there appears to be a very weak trend towardvalues closer to 0 (flat shapes) in the lower-speed shots comparedto higher-speed ones (Fig. 8). This reinforces our subjective impres-sion from reviewing the high-speed video and from handlingfragments from the different shots. Looking at the high-speedvideos, shot 130705 shows clearly ‘shelly’ fragments and it appearsthat shot 130703 does as well. Unfortunately, the lights thatilluminate the interior of the impact chamber in support of thehigh-speed cameras failed to turn on for shot 130704 so we cannotcompare that spherical target’s response to the one from shot130703. In contrast, we do not see any clearly plate-like fragmentshapes in the video from shots 130701, 130702, and 130706 (higherimpact speeds).

Previous impact experiments (Walker et al., 2013; Nakamuraet al., 2014 personal communication) similarly show shell-likefragments arising from spallation from the surface of their targets.An interesting finding of this study is that such fragments can beproduced in disruptive, shattering events as well and are not limitedto spall-type cratering impacts on the target’s surface. Instead, shell-like fragments may form well inside the target structure, specificallyright around the core, where the largest fragment is often created.This result seems to be independent of the target shape itself as it

shows up both in irregular and spherical targets (Fig. 9). To ourknowledge, this behavior has not been explicitly reported until now.

There is increasing interest in understanding the internal struc-ture of asteroids (and comets), especially in the case of NEAs,motivated by space exploration and hazard mitigation strategies.Asteroid mass measurements can be obtained by sporadic spacemissions and estimated with acceptable accuracy in the case ofprimaries of binary systems. Accurate shape models can be obtainedby space missions and by radar observations. Combined masses andshapes/volumes then allow estimates of asteroid densities in somecases. Compositions inferred from ground based spectroscopicobservations allow the densities of monolithic components to beestimated. For a given asteroid, the comparison of the components’densities inferred from compositions with the measured bulkdensity of the whole body provides estimations of the large scaleporosity (macro-porosity) of the asteroids. Implications for theirinternal structure can then be inferred (see additional discussion inFlynn et al., 2015).

For this reason, we decided to measure the typical bulk density ofthe ensemble of collected fragments from each impact experimentrandomly assembled together into a ‘rubble pile’. To facilitate assem-bly of each rubble pile we wrapped all the fragments with measuredmass in thin plastic film (the kind that is used for wrapping food indomestic refrigerators) by trying to follow carefully the outer surfaceof the obtained aggregate. Each wrapped aggregate was then sus-pended from a horizontal support and plunged into water inside acontainer situated on aweight scale (pan balancemethod). In this waywe measured the macro-porosities of the six sets of randomlyaggregated fragments corresponding to the six impact experiments.We found average porosity values for the six sets of fragmentaggregates to be 0.53 (σ1¼0.10). We repeated this measurement aftershuffling and re-wrapping each aggregate and obtained an averagevalue of 0.48 (σ2¼0.10). As a reference, a set of spheres in perfecthexagonal packing has 26% porosity; this number rises to about 35%when random packing is allowed. Delaney and Cleary (2010) studiedthe general problem of packing in superellipsoids by numericalmethods and found a range of porosities from 0.20 to 0.36 for biaxialellipsoidal shapes. Nevertheless, Bezrukov and Stoyan (2006) foundsynthetic random packing of oblate 2:1 aspect ratio and prolate3:1 aspect ratio equal ellipsoids having 0.45 and 0.57 porosities,

Fig. 5. Histograms of b/a and c/a for each of the shots.

D.D. Durda et al. / Planetary and Space Science 107 (2015) 77–83 81

respectively. Our experimental values seem consistent with the latterreference, although the fragments in each of our impact experimenthave instead irregular shapes (only roughly tri-axial in a mix of prolateand oblate shapes) and a size distribution.

Estimates of asteroid porosity vary widely in the 0.2 to 0.6 range,mostly due to poor estimation of mass and, in particular, volume.There are very few accurate determinations of both mass and shape(volume) of small rocky bodies believed to have gravitational aggre-gate structures. In the case of asteroid Itokawa (Lowry et al., 2014) andMars’ satellite Phobos (Pätzold et al., 2014), the large-scale porosityvalues range 0.30 to 0.40. Previous estimates of large porosities in

Fig. 7. Still frame from a high-speed video sequence showing the occurrence ofcircum-core fragments with plate-like, ‘onion shell’ shapes.

Fig. 8. Average value of Ψ for the 10 largest fragments (excluding the largestfragment itself) as a function of impact speed.

Fig. 6. (a) Ψ for shots 130703 and 130705: two shots with nearly the same impactspeed (3.89 and 3.68 km/s, respectively). (b) F for shots 130702 and 130704: twoshots with nearly the same specific energy (2.93 and 2.97�103 J/kg, respectively).

Fig. 9. Typical shapes of the largest fragments resulting from our disruptive impact experiments at AVGR. Tabular, ‘onion shell’ fragments are observed from both irregularly-shaped and spherical targets (see also Fig. 7). In both of the panels above, the top left-most fragment is the largest remnant, typically a more spherically-shaped central core.

D.D. Durda et al. / Planetary and Space Science 107 (2015) 77–8382

asteroids with inaccurate shapes may have the same kind of biastowards overestimation that we may find in the volume measure-ments of wrapped aggregates from our own shattering experimentsreported here. It is very easy to overestimate volumes. For instance,when we wrap our samples with thin film as tight as we can, we areapproximating the volume of the aggregate to the volume inside thefilm, which is obviously larger than the true volume of the aggregateitself. Our typical gathered sample aggregates have approximatelyspherical shapes with equivalent radii smaller than about 4 cm. Whenwrapping them, it is very easy to have the film a few mm above theaverage surface of the aggregate. A 2- or 3-mm error here translatesinto 15–20% overestimate of volume and therefore porosity (and thiscannot be quantitatively taken into account in the determination ofthe standard deviation). This may cause a 35–40% ‘true’ porosityshowing up as 45–50% porosity in the wrapped measured samples.A similar source for volume overestimation easily arises whenapproximating poorly determined asteroid shapes with spheres ortriaxial ellipsoids, causing estimated porosities to be higher thanactual porosities.

4. Conclusions

We have started investigations on the effects of target shape andprojectile impact speed on the mass–frequency distribution andshape of fragments resulting from impacts into basalt targets. A firstrun of experiments was carried out at the NASA Ames Vertical GunRange. Analysis of the results of the 6 shots performed showsnegligible dependence of the shape of the mass–frequency distribu-tion or the shapes of fragments on target shape. Our subjectiveimpression from review of the high-speed video and handling of thefragments from the experiments is that the largest several fragmentsfrom lower-speed impacts tend to exhibit flatter shapes than thosefrom higher-speed impacts. This trend is only very weakly supported,however, by actual measurements of fragment shapes; the effect ofimpact speed (sub vs. super-sonic) needs further investigation to befully assessed and future sets of impact experiments are beingplanned to better sample the range of possible impact speeds.

We see the formation of flattened, plate-like fragment shapesfrom both irregular and spherical targets and note with interestthat these fragments can originate from the interior of the target,near the core (largest fragment), in addition to the flattenedfragments often seen in the near-surface spall zone in crateringexperiments.

The porosity of aggregates made by artificially (but randomly) rea-ssembling fragments has been measured to be on the order of 50%,significantly larger than that for hexagonal lattice and random packingof ellipsoids, as well as for well determined masses and shapes ofasteroids. We suggest that the found average high porosity is mainlydue to overestimation of the volume in our samples. Volume over-estimate may be affecting as well many asteroid volumes with poorshape determinations. We recommend care in the assessment of highasteroid porosities. Our results and conclusions suggest that modelingof both non-spherical shapes and size distributions are needed inorder to properly study and understand the internal structures ofasteroids (and comets).

Acknowledgements

This work was supported by the NASA Planetary Geology &Geophysics program, grant NNX11AP22G (to GJF). ACB

acknowledges grant AYA2011-30106-C02-02 (2012-2014) by theSpanish Ministry of Science and Innovation (now extinct) andgrant PRX12/00129 by the Spanish Ministry of Education, Cultureand Sports. We thank Chuck Cornelison, Donald B. Bowling,Alfredo J. Perez, and Adam Parrish of the Ames Vertical Gun Rangefor their assistance in conducting the impact experiments, and J.-P.Wiens for photographic and high-speed video support. We thankA. Nakamura for the images of the glass fragment from an impactexperiment conducted with support by the Space Plasma Labora-tory, ISAS, JAXA.

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