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The shapes of fragments from catastrophic disruption events: Effects of target shape and impact speed
Planetary and Space Science 107 (2015) 77–83
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The shapes of fragments from catastrophic disruption events: Effects of 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 States Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Alicante, Spain c Instituto de Física Aplicada a las Ciencias y la Tecnología, Universidad de Alicante, Alicante, Spain d SUNY-Plattsburgh, 101 Broad St, Plattsburgh, NY 12901, United States e Alma College, Alma, MI 48801, United States b
art ic l e i nf o
a b s t r a c t
Article history: Received 30 April 2014 Received in revised form 3 October 2014 Accepted 8 October 2014 Available online 18 October 2014
We conducted impact experiments at the NASA Ames Vertical Gun Range in the context of an ongoing set of experiments to investigate both target shape and impact speed effects on fragment shapes and mass–frequency distributions in collisions on basalt targets. In this work we present the first part of that set, regarding mostly target shape effects. We impacted both irregularly-shaped and spherical basalt targets at speeds ranging from 4–6 km/s. We obtained mass–frequency distributions from fragments recovered from the impact chamber and measured fragments shapes using a combination of image analysis 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 target shape (i.e., flat, ‘shell-like’ fragments are produced in impacts into irregularly-shaped targets as well as spherical ones). We note that many thin, plate-like impact fragments seem to originate from lowerspeed impacts and can originate from the interior of the targets (in addition to the flattened fragments often seen to origin from the near-surface spall zone in cratering impacts). We measure the porosity of aggregates made by artificially (but randomly) reassembling fragments from each impact to be on the order of 50%, significantly larger than that for hexagonal lattice and random packing of equal sized spheres. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Asteroids Fragmentation Collisions Internal structure Gravitational aggregates
1. Introduction Recent experiments showing ‘onion shells’ of tabular-shaped fragments from impacts into spherical targets (Walker et al., 2013; Nakamura, 2014 personal communication) have re-opened the question of the fracture mechanics responsible for determining fragment shape in catastrophic impacts. Fujiwara et al. (1978) presented the first work on the topic of shapes of fragments from impact experiments in the context of asteroid studies. Fujiwara et al. (1989) suggest two modes of catastrophic disruption for targets in laboratory impact experiments: (1) ‘core-type’ fragmentation in the high-speed, highenergy-density regime; and (2) ‘cone-type’ fragmentation in the lowspeed, low-energy-density regime. These effects appear to be scale invariant, at least over the range of cm to m-scale target sizes investigated in laboratory impact experiments conducted to date. Fig. 1a, for example, shows the result of one of the cratering impacts into 1-m-diameter granite spheres reported in Walker et al. (2013). In
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Correspondence author. E-mail address: [email protected] (D.D. Durda).
http://dx.doi.org/10.1016/j.pss.2014.10.006 0032-0633/& 2014 Elsevier Ltd. All rights reserved.
these experiments aluminum spheres of 4.45-cm diameter were impacted into the granite targets at speeds of 2 km/s. Many of the larger fragments recovered from the experiment enclosure exhibited thin, tabular shapes and the fractured walls of the resulting spall craters displayed the same plate-like nature of the many fragments ‘peeled off’ the outer shell of the granite ‘onion’. Very similar fragment morphology is seen at smaller scales as well—Fig. 1b shows the largest remnant of an impact into a 5-cm-diameter soda lime glass sphere at 2 km/s, exhibiting the same flattened, tabular-shaped fragmentation pattern in the near-surface spall zone (Nakamura, 2014 personal communication). Similar behavior is seen in impacts into 6–10cm-diameter basalt spheres, as reported by Fujiwara and Tsukamoto (1980; Fig. 5) who present sketches of cross-sectional views of reconstructed targets showing shell-like fracture surrounding the remnant cores. Because of the propensity of many previous laboratory investigations to focus on idealized spherical targets there exists some ambiguity in decoupling the relative importance/influence of impact speed versus spherical shape in producing the ‘onion shell’ fragment shapes seen in these experiments. If plate-like fragment shapes are due primarily to impact speed/energy density as
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suggested by Fujiwara et al. (1989) this could play an important role in the outcome of impacts onto small monolithic objects in the main asteroid belt due to the non-negligible probability of low-speed (i.e., below about 3–4 km/s—subsonic in rock) impacts there (Bottke et al., 1994). There is growing interest in spall-type
impacts into initially monolithic rock targets due to the focus on smaller near-Earth asteroids (NEAs) – and the blocky fragments in coarse regoliths observed to exist on objects like Itokawa (Nakamura et al., 2008; Noguchi et al., 2010) – as targets for exploration 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 near surface 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 very similar 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. The spherical 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 the spherical shape. The targets shown here were not shot during this experiment run; they remain for future impact experiments.
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Housen, 2013) and the recognition that the cumulative effects of multiple sub-catastrophic impacts can have the same effect as one larger, catastrophic impact (Gault and Wedekind, 1969; Housen, 2009). Also, the low bulk densities of NEAs in the 100 m to tens of km size range suggests that some of these objects may be rubblepiles and that their low densities can be related to the shapes and mass spectrum of their large components and to the way they reassemble as gravitational aggregates following shattering collisions. A secondary goal of the conducted experiments is to investigate the theoretical conjecture, supported by numerical modelling (Tanga et al., 1999; Campo Bagatin and Petit, 2001) involving the role of the target shape in affecting the mass spectrum of fragments created in shattering collisions. We describe here our experiments to investigate the importance of target shape and impact speed in determining the shapes of fragments from catastrophic impacts.
2. Impact experiments In order to disentangle the potential effects of both target shape and impact speed in affecting the shape of fragments we chose to conduct impact experiments on both spherical and naturally-occurring irregularly-shaped targets of the same basalt material impacted at a range of speeds (Fig. 2). We used basalt samples obtained from a recent lava flow exposed in a road cut near Flagstaff, AZ (lat N351200 59″, long W1111330 55″). The targets ranged in mass, M, from 238 to 534 g (see Table 1). Because we anticipated reconstructing an idealized fragment size distribution from imaged fragment shapes that would then be compared with the measured mass distribution we needed to obtain the mean density of our basalt samples. We measured the bulk density of each basalt sphere and several representative samples of the irregularly-shaped basalt targets using the Archimedean water displacement method. We obtained values of 2.9570.03 g/cm3 (irregular) to 2.9870.05 g/cm3 (spherical) for the samples. We impacted a total of six targets (two spheres and four irregular targets). The mean impact speed for asteroids in the main belt is 5 km/s (Bottke et al., 1994), so we focused on shots with impact speeds nominally in the 4 to 6 km/s range. For each impact the projectile was a 3/16th in. (0.476 cm) diameter, 0.1583 g mass aluminum sphere fired at the specified target using the NASA Ames Vertical Gun Range (AVGR). The targets were each suspended at the center of the AVGR impact chamber from a thin nylon line such that
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the incoming projectile path would pass very roughly through the center of mass of the target. For maximum stability of the target during the hanging process, particularly in the case of the irregularly-shaped targets, the attach point of the support line for each target was placed such the target hung with its longest (a) axis generally in the vertical direction. Because the AVGR has a roughly 1-cm-diameter patch of uncertainty in the targeted impact location of the projectile, we attempted to roughly ‘align’ each irregular target to maximize its collision cross section for the maximum probability of a good central impact during each shot. We set the alignment of each target such that the impact point was lined up more or less with the incoming impactor direction, thus yielding a roughly normal incidence impact relative to the impacted face of the target. Although the impactor incidence angle cannot be controlled very rigorously we note that this may well affect the outcome of any particular single experiment, but this effect should ‘average out’ when considering the outcomes of several experiments. The chamber floor and walls were lined with cloth sheets to provide a soft buffer to help prevent primary eject from suffering subsequent fragmentation after colliding with the chamber. Following each shot, the debris was collected from the floor of the AVGR chamber. This process typically recovered 495% of the target mass. Large fragments that were collected from the chamber were individually weighed (to a sample completion limit at about m 4 0.20 g), representing up to 90% of the original mass. This allowed us to carefully measure the mass–frequency distribution of the largest fragments from each impact experiment. High-speed video of each impact was obtained by five different video cameras (two Phantom V10s, one Phantom V12.1, and two Shimadzu HPV-1s), with frame rates ranging from 1900 to 125,000 frames/s, to aid interpretation of the fragmentation mode of the targets. The details of each shot are summarized in Table 1.
3. Results and discussion Cumulative mass distributions are derived for the six impact experiments and the results are shown in Fig. 3. As a common pattern repeatedly shown by past impact experiments, we observe that for the several largest fragments (depending on each case) no simple analytic description of the shape of the mass distribution can be found. For smaller fragments however, power law mass distributions are found with exponents 0.75oβo1.2 in the relationship N(4m)¼Am β (where m is the fragment mass, β is the power law index, and A is the corresponding constant). This result is found for
Table 1 Shot 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 by the target mass. Brackets (〈 〉) indicate average values for each shot. Shot
130701
130702
130703
130704
130705
130706
Target type Target mass (g) Target diameter (cm) Target b/a, c/a Impact speed (km/s) Projectile mass (g) E/M (J/kg) fL Exponent β 〈b/a〉 〈c/a〉 〈ψ〉 〈ψ10〉 〈F〉 〈F10〉 〈Porosity〉
Irregular 433.00 6.54 0.61, 0.45 4.73 0.1587 4100 0.07326 1.00 0.73 0.41 0.61 0.66 0.46 0.35 0.53
Irregular 534.60 7.02 0.66, 0.53 4.45 0.1582 2930 0.13118 0.82 0.73 0.42 0.62 0.62 0.49 0.50 0.51
Spherical 237.90 5.34 1, 1 3.89 0.1584 5040 0.05944 1.22 0.74 0.41 0.62 0.53 0.44 0.31 0.49
Spherical 342.70 6.03 1, 1 3.59 0.1582 2970 0.18494 0.96 0.72 0.38 0.58 0.53 0.45 0.34 0.54
Irregular 479.10 6.77 0.80, 0.78 3.68 0.1582 2240 0.23446 0.75 0.67 0.34 0.54 0.51 0.49 0.48 0.61
Irregular 451.20 6.64 0.56, 0.46 5.82 0.1582 5940 0.02637 1.03 0.73 0.38 0.60 0.58 0.48 0.38 0.54
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variable intervals of m/M, depending on the circumstances of the shot, but it roughly holds in the range 0.5 10 3 om/Mo0.5 10 2. The mass of the largest fragment, normalized to the original target mass, tends to be smaller as the specific energy of the impact grows. The exponent of each mass distribution is also related to the corresponding specific energy of each impact, as expected (Fujiwara et al., 1989). Also, the mass distributions seem to show slightly larger values of β (i.e., steeper slopes and relatively smaller fragments) in the case of spherical targets with respect to irregular ones, when comparing two sets of close specific energy impacts. However, this behavior needs further sets of impact experiments to properly check the results by Tanga et al. (1999) and Campo Bagatin and Petit (2001). The 36 largest fragments from each shot were photographed and their largest axes (a and b) accurately measured in the image analysis software ImageJ (e.g., Fig. 4). Fragments are digitally imaged on a uniform green background to allow easy chroma keying in Adobe Photoshop. ImageJ fits an ellipse to the silhouette of each fragment, yielding a measurement of the a and b axis length of each fragment. Their shortest axes (c) were measured by means of a digital caliper. We note that the process of arranging the fragments for photography naturally oriented them such that their shortest axes aligned more or less normal to the green background plane. Thus, the photographic analysis via ImageJ measured the a and b axes while the orientation of each fragment was carefully noted during the manual caliper measurement process to ensure proper determination of c.
Fig. 3. Cumulative mass–frequency distributions for each of the impact experiments. M is the mass of the target, m is the mass of the nth fragment. See Table 1 for details of each shot.
Previous investigations carried out in the 1970s and 1980s (e.g., Capaccioni et al., 1986) showed aspect ratios of b/aE0.7 and c/aE0.5. We found the overall average aspect ratios to be b/a¼ 0.7270.13 and c/a¼0.3970.13 (Table 1), with a result for c/a systematically smaller than reported by Capaccioni et al. (1986). This result is quite stable over the different impact experiments and no differences are found in average shapes among spherical and irregular targets, nor for different specific energy up to a factor of 3. Obviously, this does not mean that fragments look like 3-axial ellipsoidal shapes at all; instead they are quite irregular, but their average relative sizes are distributed as described. Fig. 5 shows histograms for each shot for both b/a and c/a. It is interesting to note that the values obtained for b/a are in agreement 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 between boulder shapes on Itokawa compared with fragment shapes from laboratory impact experiments, Michikami et al. (2010) note that the aspect ratios of boulders on the surface of Itokawa do not match well with experimental results. They show b/a generally ranging between 0.62 and 0.68, a difference they found to be statistically significant at 95% confidence level. It is hard to speculate on the reasons for such differences. One possible cause may be thermal fatigue due to temperature changes leading to partial erosion of some of the boulders on Itokawa that may affect the statistical distribution of the aspect ratio. Finally, we note with interest that Tsuchiyama et al. (2014) find that the shape distribution of particles returned from the surface of Itokawa by Hayabusa is consistent with the results of mechanical disaggregation, primarily as a response to impacts, although we caution that these particles/fragments are all so small that there may be other processes at work in determining their shapes. These very small particles/fragments are at the size of individual mineral grains and there the processes that govern grain shapes may be quite different (e.g., involving crystal growth within the rock, etc.) than those that dominate the shapes of macroscopic impact fragments in laboratory experiments or in the fragmentation of bulk rock in asteroids. We have also investigated the shape metrics Ψ¼ (c2/ab)1/3 and F¼(a b)/(a c) derived from the results of our impact experiments. These parameters quantify deviation from spherical shape and relative flatness, respectively (Benn and Ballantyne, 1993; Ehlmann et al., 2008). Values of Ψ (0oΨo1) approaching 1 indicate roundness in the described 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. 4. Example of the process of measuring fragment shapes from photography. Fragments are digitally imaged on a uniform green background to allow easy chroma keying in 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 each fragment, yielding a measurement of the a and b axis length of each fragment.
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Fig. 5. Histograms of b/a and c/a for each of the shots.
Average values for Ψ and F for all measured fragments are quite close to each other in each shot and do not show significant dependence on specific energy, impact speed, or target shape (see Table 1). Their overall average values are Ψ¼0.60 and F¼ 0.47. 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 trend towards more roundish shapes seems to be present as fragment size decreases. More specifically, some of the largest fragments have flat and sometimes plate-like shapes that are less frequently observed among smaller fragments. This is not clearly related to impact velocity or specific impact energy. Fig. 6a shows a comparison of values of Ψ between two shots with similar impact speeds, while Fig. 6b compares F for two shots with similar specific energies. More qualitatively, high speed video from our experiments clearly shows the formation of shell-like fragments in most of the cratering and lower-energy disruptive impacts (e.g., Fig. 7). Examining average values for F and Ψ for the 10 largest fragments (but not including the largest fragment itself, which is often a large and more rounded core), there appears to be a very weak trend toward values closer to 0 (flat shapes) in the lower-speed shots compared to higher-speed ones (Fig. 8). This reinforces our subjective impression from reviewing the high-speed video and from handling fragments from the different shots. Looking at the high-speed videos, shot 130705 shows clearly ‘shelly’ fragments and it appears that shot 130703 does as well. Unfortunately, the lights that illuminate the interior of the impact chamber in support of the high-speed cameras failed to turn on for shot 130704 so we cannot compare that spherical target’s response to the one from shot 130703. In contrast, we do not see any clearly plate-like fragment shapes in the video from shots 130701, 130702, and 130706 (higher impact speeds). Previous impact experiments (Walker et al., 2013; Nakamura et al., 2014 personal communication) similarly show shell-like fragments arising from spallation from the surface of their targets. An interesting finding of this study is that such fragments can be produced in disruptive, shattering events as well and are not limited to spall-type cratering impacts on the target’s surface. Instead, shelllike fragments may form well inside the target structure, specifically right 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 our knowledge, this behavior has not been explicitly reported until now. There is increasing interest in understanding the internal structure 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 space missions and estimated with acceptable accuracy in the case of primaries of binary systems. Accurate shape models can be obtained by space missions and by radar observations. Combined masses and shapes/volumes then allow estimates of asteroid densities in some cases. Compositions inferred from ground based spectroscopic observations allow the densities of monolithic components to be estimated. For a given asteroid, the comparison of the components’ densities inferred from compositions with the measured bulk density of the whole body provides estimations of the large scale porosity (macro-porosity) of the asteroids. Implications for their internal structure can then be inferred (see additional discussion in Flynn et al., 2015). For this reason, we decided to measure the typical bulk density of the ensemble of collected fragments from each impact experiment randomly assembled together into a ‘rubble pile’. To facilitate assembly of each rubble pile we wrapped all the fragments with measured mass in thin plastic film (the kind that is used for wrapping food in domestic refrigerators) by trying to follow carefully the outer surface of the obtained aggregate. Each wrapped aggregate was then suspended from a horizontal support and plunged into water inside a container situated on a weight scale (pan balance method). In this way we measured the macro-porosities of the six sets of randomly aggregated fragments corresponding to the six impact experiments. We found average porosity values for the six sets of fragment aggregates to be 0.53 (σ1 ¼0.10). We repeated this measurement after shuffling and re-wrapping each aggregate and obtained an average value of 0.48 (σ2 ¼0.10). As a reference, a set of spheres in perfect hexagonal packing has 26% porosity; this number rises to about 35% when random packing is allowed. Delaney and Cleary (2010) studied the general problem of packing in superellipsoids by numerical methods and found a range of porosities from 0.20 to 0.36 for biaxial ellipsoidal shapes. Nevertheless, Bezrukov and Stoyan (2006) found synthetic random packing of oblate 2:1 aspect ratio and prolate 3:1 aspect ratio equal ellipsoids having 0.45 and 0.57 porosities,
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respectively. Our experimental values seem consistent with the latter reference, although the fragments in each of our impact experiment have instead irregular shapes (only roughly tri-axial in a mix of prolate and 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 aggregate structures. In the case of asteroid Itokawa (Lowry et al., 2014) and Mars’ satellite Phobos (Pätzold et al., 2014), the large-scale porosity values 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 of circum-core fragments with plate-like, ‘onion shell’ shapes.
Fig. 6. (a) Ψ for shots 130703 and 130705: two shots with nearly the same impact speed (3.89 and 3.68 km/s, respectively). (b) F for shots 130702 and 130704: two shots with nearly the same specific energy (2.93 and 2.97 103 J/kg, respectively).
Fig. 8. Average value of Ψ for the 10 largest fragments (excluding the largest fragment itself) as a function of impact speed.
Fig. 9. Typical shapes of the largest fragments resulting from our disruptive impact experiments at AVGR. Tabular, ‘onion shell’ fragments are observed from both irregularlyshaped 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.
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asteroids with inaccurate shapes may have the same kind of bias towards overestimation that we may find in the volume measurements of wrapped aggregates from our own shattering experiments reported here. It is very easy to overestimate volumes. For instance, when we wrap our samples with thin film as tight as we can, we are approximating the volume of the aggregate to the volume inside the film, which is obviously larger than the true volume of the aggregate itself. Our typical gathered sample aggregates have approximately spherical shapes with equivalent radii smaller than about 4 cm. When wrapping them, it is very easy to have the film a few mm above the average surface of the aggregate. A 2- or 3-mm error here translates into 15–20% overestimate of volume and therefore porosity (and this cannot be quantitatively taken into account in the determination of the standard deviation). This may cause a 35–40% ‘true’ porosity showing up as 45–50% porosity in the wrapped measured samples. A similar source for volume overestimation easily arises when approximating poorly determined asteroid shapes with spheres or triaxial ellipsoids, causing estimated porosities to be higher than actual porosities. 4. Conclusions We have started investigations on the effects of target shape and projectile impact speed on the mass–frequency distribution and shape of fragments resulting from impacts into basalt targets. A first run of experiments was carried out at the NASA Ames Vertical Gun Range. Analysis of the results of the 6 shots performed shows negligible dependence of the shape of the mass–frequency distribution or the shapes of fragments on target shape. Our subjective impression from review of the high-speed video and handling of the fragments from the experiments is that the largest several fragments from lower-speed impacts tend to exhibit flatter shapes than those from higher-speed impacts. This trend is only very weakly supported, however, by actual measurements of fragment shapes; the effect of impact speed (sub vs. super-sonic) needs further investigation to be fully assessed and future sets of impact experiments are being planned to better sample the range of possible impact speeds. We see the formation of flattened, plate-like fragment shapes from both irregular and spherical targets and note with interest that these fragments can originate from the interior of the target, near the core (largest fragment), in addition to the flattened fragments often seen in the near-surface spall zone in cratering experiments. The porosity of aggregates made by artificially (but randomly) reassembling fragments has been measured to be on the order of 50%, significantly larger than that for hexagonal lattice and random packing of ellipsoids, as well as for well determined masses and shapes of asteroids. We suggest that the found average high porosity is mainly due to overestimation of the volume in our samples. Volume overestimate may be affecting as well many asteroid volumes with poor shape determinations. We recommend care in the assessment of high asteroid porosities. Our results and conclusions suggest that modeling of both non-spherical shapes and size distributions are needed in order to properly study and understand the internal structures of asteroids (and comets). Acknowledgements This work was supported by the NASA Planetary Geology & Geophysics program, grant NNX11AP22G (to GJF). ACB
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acknowledges grant AYA2011-30106-C02-02 (2012-2014) by the Spanish Ministry of Science and Innovation (now extinct) and grant PRX12/00129 by the Spanish Ministry of Education, Culture and Sports. We thank Chuck Cornelison, Donald B. Bowling, Alfredo J. Perez, and Adam Parrish of the Ames Vertical Gun Range for their assistance in conducting the impact experiments, and J.-P. Wiens for photographic and high-speed video support. We thank A. Nakamura for the images of the glass fragment from an impact experiment conducted with support by the Space Plasma Laboratory, ISAS, JAXA. References Benn, D.I., Ballantyne, C.K., 1993. The description and representation of particle shape. Earth Surf. Processes Landforms 18, 665–672. Bezrukov, A., Stoyan, D., 2006. Simulation and statistical analysis of random packings of ellipsoids. Part. Part. Syst. Charact 23, 388–398. Bottke Jr., W.F., Nolan, M.C., Greenberg, R., Kolvoord, R.A., 1994. Velocity distributions among colliding asteroids. Icarus 107, 255–268. Campo Bagatin, A., Petit, J-M., 2001. Effects of the geometric constraints on the size distributions of debris in asteroidal fragmentation. Icarus 149 (1), 210–221. Capaccioni, F., Cerroni, P., Coradini, M., Di Martino, M., Farinella, P., Flamini, E., Martelli, G., Paolicchi, P., Smith, P.N., Woodward, A., Zappalà, V., 1986. Asteroidal catastrophic collisions simulated by hypervelocity impact experiments. Icarus 66, 487–514. Delaney, G.W., Cleary, P.W., 2010. The packing properties of superellipsoids. EPL 89 (34002-p1/p5). Ehlmann, B.L., Viles, H.A., Bourke, M.C., 2008. Quantitative morphologic analysis of boulder shape and surface texture to infer environmental history: a case study of rock breakdown at the Ephrata Fan, Channeled Scabland, Washington. J. Geophys. Res. 113, F02012. http://dx.doi.org/10.1029/2007JF000872. Flynn, et al., 2015. Hypervelocity cratering and disruption of porous pumice targets 107, 64–76. Fujiwara, A., Kamimoto, G., Tsukamoto, A., 1978. Expected shape distribution of asteroids obtained from laboratory impact experiments. Nature 272, 602–603. Fujiwara, A., Cerroni, P., Davis, D., Ryan, E., Di Martino, M., Holsapple, K., Housen, K., 1989. Experiments and scaling laws for catastrophic collisions. In: Binzel, R.P., Gehrels, M.S., Matthews, M.S. (Eds.), Asteroids II. University of Arizona Press, Tucson, pp. 240–265. Fujiwara, A., Tsukamoto, A., 1980. Experimental study on the velocity of fragments in collisional breakup. Icarus 44, 142–153. Gault, D.E., Wedekind, J.A., 1969. The destruction of tektites by meteoroid impact. J. Geophys. Res. 74, 6780–6794. Holsapple, K.A., Housen, K.R., 2013. The third regime of cratering: spall craters. In: Lunar Plan. Sci. XLIV, abstract no. 2733. Housen, K.R., 2009. Cumulative damage in strength-dominated collisions of rocky asteroids: rubble piles and brick piles. Planet. Space Sci. 57, 142–153. Lowry, S.C., Weissman, P.R., Duddy, S.R., Rozitis, B., Fitzsimmons, A., Green, S.F., Hicks, M.D., Snodgrass, C., Wolters, S.D., Chesley, S.R., Pittichová, J., van Oers, P., 2014. The internal structure of asteroid (25143) Itokawa as revealed by detection of YORP spin-up. Astron. Astrophys. 562, 48–57. Michikami, T., Nakamura, A., Hirata, N., 2010. The shape distribution of boulders on Asteroid 25143 Itokawa: comparison with fragments from impact experiments. Icarus 207, 277–284. Nakamura, A.M., Michikami, T., Hirata, N., Fujiwara, A., Nakamura, R., Ishiguro, M., Miyamoto, H., Demura, H., Hiraoka, K., Honda, T., Honda, C., Saito, J., Hashimoto, T., Kubota, T., 2008. Impact process of boulders on the surface of asteroid 25143 Itokawa—fragments from collisional disruption. Earth, Planets Space 60, 7–12. Noguchi, T., Tsuchiyama, A., Hirata, N., Demura, H., Nakamura, R., Miyamoto, H., Yano, H., Nakamura, T., Saito, J., Sasaki, S., Hashimoto, T., Kubota, T., Ishiguro, M., Zolensky, M.E., 2010. Surface morphological features of boulders on Asteroid 25143 Itokawa. Icarus 206, 319–326. Pätzold, M., Andert, T.P., Tyler, G.L., Asmar, S.W., Häusler, B., Tellmann, S., 2014. Phobos mass determination from the very close flyby of Mars Express in 2010. Icarus 229, 92–98. Tanga, P., Cellino, A., Michel, P., Zappalà, V., Paolicchi, P., Dell’Oro, A., 1999. On the size distribution of asteroid families: the role of geometry. Icarus 141, 65–78. Tsuchiyama, A., Uesugi, M., Uesugi, K., Nakano, T., Noguchi, R., Matsumoto, T., Matsuno, J., Nagano, T., Imai, Y., Shimada, A., Takeuchi, A., Suzuki, Y., Nakamura, T., Noguchi, T., Abe, M., Yada, T., Fujimura, A., 2014. Three-dimensional microstructure of samples recovered from asteroid 25143 Itokawa: comparison with LL5 and LL6 chondrite particles. Meteorit. Planet. Sci 49, 172–187. Walker, J.D., Chocron, S., Durda, D.D., Grosch, D.J., Movshovitz, N., Richardson, D.C., Asphaug, E., 2013. Momentum enhancement from aluminum striking granite and the scale effect. Int. J. Impact Eng. 56, 12–18.
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The shapes of fragments from catastrophic disruption events: Effects of 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 States Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Alicante, Spain c Instituto de Física Aplicada a las Ciencias y la Tecnología, Universidad de Alicante, Alicante, Spain d SUNY-Plattsburgh, 101 Broad St, Plattsburgh, NY 12901, United States e Alma College, Alma, MI 48801, United States b
art ic l e i nf o
a b s t r a c t
Article history: Received 30 April 2014 Received in revised form 3 October 2014 Accepted 8 October 2014 Available online 18 October 2014
We conducted impact experiments at the NASA Ames Vertical Gun Range in the context of an ongoing set of experiments to investigate both target shape and impact speed effects on fragment shapes and mass–frequency distributions in collisions on basalt targets. In this work we present the first part of that set, regarding mostly target shape effects. We impacted both irregularly-shaped and spherical basalt targets at speeds ranging from 4–6 km/s. We obtained mass–frequency distributions from fragments recovered from the impact chamber and measured fragments shapes using a combination of image analysis 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 target shape (i.e., flat, ‘shell-like’ fragments are produced in impacts into irregularly-shaped targets as well as spherical ones). We note that many thin, plate-like impact fragments seem to originate from lowerspeed impacts and can originate from the interior of the targets (in addition to the flattened fragments often seen to origin from the near-surface spall zone in cratering impacts). We measure the porosity of aggregates made by artificially (but randomly) reassembling fragments from each impact to be on the order of 50%, significantly larger than that for hexagonal lattice and random packing of equal sized spheres. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Asteroids Fragmentation Collisions Internal structure Gravitational aggregates
1. Introduction Recent experiments showing ‘onion shells’ of tabular-shaped fragments from impacts into spherical targets (Walker et al., 2013; Nakamura, 2014 personal communication) have re-opened the question of the fracture mechanics responsible for determining fragment shape in catastrophic impacts. Fujiwara et al. (1978) presented the first work on the topic of shapes of fragments from impact experiments in the context of asteroid studies. Fujiwara et al. (1989) suggest two modes of catastrophic disruption for targets in laboratory impact experiments: (1) ‘core-type’ fragmentation in the high-speed, highenergy-density regime; and (2) ‘cone-type’ fragmentation in the lowspeed, low-energy-density regime. These effects appear to be scale invariant, at least over the range of cm to m-scale target sizes investigated in laboratory impact experiments conducted to date. Fig. 1a, for example, shows the result of one of the cratering impacts into 1-m-diameter granite spheres reported in Walker et al. (2013). In
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Correspondence author. E-mail address: [email protected] (D.D. Durda).
http://dx.doi.org/10.1016/j.pss.2014.10.006 0032-0633/& 2014 Elsevier Ltd. All rights reserved.
these experiments aluminum spheres of 4.45-cm diameter were impacted into the granite targets at speeds of 2 km/s. Many of the larger fragments recovered from the experiment enclosure exhibited thin, tabular shapes and the fractured walls of the resulting spall craters displayed the same plate-like nature of the many fragments ‘peeled off’ the outer shell of the granite ‘onion’. Very similar fragment morphology is seen at smaller scales as well—Fig. 1b shows the largest remnant of an impact into a 5-cm-diameter soda lime glass sphere at 2 km/s, exhibiting the same flattened, tabular-shaped fragmentation pattern in the near-surface spall zone (Nakamura, 2014 personal communication). Similar behavior is seen in impacts into 6–10cm-diameter basalt spheres, as reported by Fujiwara and Tsukamoto (1980; Fig. 5) who present sketches of cross-sectional views of reconstructed targets showing shell-like fracture surrounding the remnant cores. Because of the propensity of many previous laboratory investigations to focus on idealized spherical targets there exists some ambiguity in decoupling the relative importance/influence of impact speed versus spherical shape in producing the ‘onion shell’ fragment shapes seen in these experiments. If plate-like fragment shapes are due primarily to impact speed/energy density as
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suggested by Fujiwara et al. (1989) this could play an important role in the outcome of impacts onto small monolithic objects in the main asteroid belt due to the non-negligible probability of low-speed (i.e., below about 3–4 km/s—subsonic in rock) impacts there (Bottke et al., 1994). There is growing interest in spall-type
impacts into initially monolithic rock targets due to the focus on smaller near-Earth asteroids (NEAs) – and the blocky fragments in coarse regoliths observed to exist on objects like Itokawa (Nakamura et al., 2008; Noguchi et al., 2010) – as targets for exploration 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 near surface 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 very similar 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. The spherical 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 the spherical shape. The targets shown here were not shot during this experiment run; they remain for future impact experiments.
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Housen, 2013) and the recognition that the cumulative effects of multiple sub-catastrophic impacts can have the same effect as one larger, catastrophic impact (Gault and Wedekind, 1969; Housen, 2009). Also, the low bulk densities of NEAs in the 100 m to tens of km size range suggests that some of these objects may be rubblepiles and that their low densities can be related to the shapes and mass spectrum of their large components and to the way they reassemble as gravitational aggregates following shattering collisions. A secondary goal of the conducted experiments is to investigate the theoretical conjecture, supported by numerical modelling (Tanga et al., 1999; Campo Bagatin and Petit, 2001) involving the role of the target shape in affecting the mass spectrum of fragments created in shattering collisions. We describe here our experiments to investigate the importance of target shape and impact speed in determining the shapes of fragments from catastrophic impacts.
2. Impact experiments In order to disentangle the potential effects of both target shape and impact speed in affecting the shape of fragments we chose to conduct impact experiments on both spherical and naturally-occurring irregularly-shaped targets of the same basalt material impacted at a range of speeds (Fig. 2). We used basalt samples obtained from a recent lava flow exposed in a road cut near Flagstaff, AZ (lat N351200 59″, long W1111330 55″). The targets ranged in mass, M, from 238 to 534 g (see Table 1). Because we anticipated reconstructing an idealized fragment size distribution from imaged fragment shapes that would then be compared with the measured mass distribution we needed to obtain the mean density of our basalt samples. We measured the bulk density of each basalt sphere and several representative samples of the irregularly-shaped basalt targets using the Archimedean water displacement method. We obtained values of 2.9570.03 g/cm3 (irregular) to 2.9870.05 g/cm3 (spherical) for the samples. We impacted a total of six targets (two spheres and four irregular targets). The mean impact speed for asteroids in the main belt is 5 km/s (Bottke et al., 1994), so we focused on shots with impact speeds nominally in the 4 to 6 km/s range. For each impact the projectile was a 3/16th in. (0.476 cm) diameter, 0.1583 g mass aluminum sphere fired at the specified target using the NASA Ames Vertical Gun Range (AVGR). The targets were each suspended at the center of the AVGR impact chamber from a thin nylon line such that
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the incoming projectile path would pass very roughly through the center of mass of the target. For maximum stability of the target during the hanging process, particularly in the case of the irregularly-shaped targets, the attach point of the support line for each target was placed such the target hung with its longest (a) axis generally in the vertical direction. Because the AVGR has a roughly 1-cm-diameter patch of uncertainty in the targeted impact location of the projectile, we attempted to roughly ‘align’ each irregular target to maximize its collision cross section for the maximum probability of a good central impact during each shot. We set the alignment of each target such that the impact point was lined up more or less with the incoming impactor direction, thus yielding a roughly normal incidence impact relative to the impacted face of the target. Although the impactor incidence angle cannot be controlled very rigorously we note that this may well affect the outcome of any particular single experiment, but this effect should ‘average out’ when considering the outcomes of several experiments. The chamber floor and walls were lined with cloth sheets to provide a soft buffer to help prevent primary eject from suffering subsequent fragmentation after colliding with the chamber. Following each shot, the debris was collected from the floor of the AVGR chamber. This process typically recovered 495% of the target mass. Large fragments that were collected from the chamber were individually weighed (to a sample completion limit at about m 4 0.20 g), representing up to 90% of the original mass. This allowed us to carefully measure the mass–frequency distribution of the largest fragments from each impact experiment. High-speed video of each impact was obtained by five different video cameras (two Phantom V10s, one Phantom V12.1, and two Shimadzu HPV-1s), with frame rates ranging from 1900 to 125,000 frames/s, to aid interpretation of the fragmentation mode of the targets. The details of each shot are summarized in Table 1.
3. Results and discussion Cumulative mass distributions are derived for the six impact experiments and the results are shown in Fig. 3. As a common pattern repeatedly shown by past impact experiments, we observe that for the several largest fragments (depending on each case) no simple analytic description of the shape of the mass distribution can be found. For smaller fragments however, power law mass distributions are found with exponents 0.75oβo1.2 in the relationship N(4m)¼Am β (where m is the fragment mass, β is the power law index, and A is the corresponding constant). This result is found for
Table 1 Shot 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 by the target mass. Brackets (〈 〉) indicate average values for each shot. Shot
130701
130702
130703
130704
130705
130706
Target type Target mass (g) Target diameter (cm) Target b/a, c/a Impact speed (km/s) Projectile mass (g) E/M (J/kg) fL Exponent β 〈b/a〉 〈c/a〉 〈ψ〉 〈ψ10〉 〈F〉 〈F10〉 〈Porosity〉
Irregular 433.00 6.54 0.61, 0.45 4.73 0.1587 4100 0.07326 1.00 0.73 0.41 0.61 0.66 0.46 0.35 0.53
Irregular 534.60 7.02 0.66, 0.53 4.45 0.1582 2930 0.13118 0.82 0.73 0.42 0.62 0.62 0.49 0.50 0.51
Spherical 237.90 5.34 1, 1 3.89 0.1584 5040 0.05944 1.22 0.74 0.41 0.62 0.53 0.44 0.31 0.49
Spherical 342.70 6.03 1, 1 3.59 0.1582 2970 0.18494 0.96 0.72 0.38 0.58 0.53 0.45 0.34 0.54
Irregular 479.10 6.77 0.80, 0.78 3.68 0.1582 2240 0.23446 0.75 0.67 0.34 0.54 0.51 0.49 0.48 0.61
Irregular 451.20 6.64 0.56, 0.46 5.82 0.1582 5940 0.02637 1.03 0.73 0.38 0.60 0.58 0.48 0.38 0.54
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variable intervals of m/M, depending on the circumstances of the shot, but it roughly holds in the range 0.5 10 3 om/Mo0.5 10 2. The mass of the largest fragment, normalized to the original target mass, tends to be smaller as the specific energy of the impact grows. The exponent of each mass distribution is also related to the corresponding specific energy of each impact, as expected (Fujiwara et al., 1989). Also, the mass distributions seem to show slightly larger values of β (i.e., steeper slopes and relatively smaller fragments) in the case of spherical targets with respect to irregular ones, when comparing two sets of close specific energy impacts. However, this behavior needs further sets of impact experiments to properly check the results by Tanga et al. (1999) and Campo Bagatin and Petit (2001). The 36 largest fragments from each shot were photographed and their largest axes (a and b) accurately measured in the image analysis software ImageJ (e.g., Fig. 4). Fragments are digitally imaged on a uniform green background to allow easy chroma keying in Adobe Photoshop. ImageJ fits an ellipse to the silhouette of each fragment, yielding a measurement of the a and b axis length of each fragment. Their shortest axes (c) were measured by means of a digital caliper. We note that the process of arranging the fragments for photography naturally oriented them such that their shortest axes aligned more or less normal to the green background plane. Thus, the photographic analysis via ImageJ measured the a and b axes while the orientation of each fragment was carefully noted during the manual caliper measurement process to ensure proper determination of c.
Fig. 3. Cumulative mass–frequency distributions for each of the impact experiments. M is the mass of the target, m is the mass of the nth fragment. See Table 1 for details of each shot.
Previous investigations carried out in the 1970s and 1980s (e.g., Capaccioni et al., 1986) showed aspect ratios of b/aE0.7 and c/aE0.5. We found the overall average aspect ratios to be b/a¼ 0.7270.13 and c/a¼0.3970.13 (Table 1), with a result for c/a systematically smaller than reported by Capaccioni et al. (1986). This result is quite stable over the different impact experiments and no differences are found in average shapes among spherical and irregular targets, nor for different specific energy up to a factor of 3. Obviously, this does not mean that fragments look like 3-axial ellipsoidal shapes at all; instead they are quite irregular, but their average relative sizes are distributed as described. Fig. 5 shows histograms for each shot for both b/a and c/a. It is interesting to note that the values obtained for b/a are in agreement 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 between boulder shapes on Itokawa compared with fragment shapes from laboratory impact experiments, Michikami et al. (2010) note that the aspect ratios of boulders on the surface of Itokawa do not match well with experimental results. They show b/a generally ranging between 0.62 and 0.68, a difference they found to be statistically significant at 95% confidence level. It is hard to speculate on the reasons for such differences. One possible cause may be thermal fatigue due to temperature changes leading to partial erosion of some of the boulders on Itokawa that may affect the statistical distribution of the aspect ratio. Finally, we note with interest that Tsuchiyama et al. (2014) find that the shape distribution of particles returned from the surface of Itokawa by Hayabusa is consistent with the results of mechanical disaggregation, primarily as a response to impacts, although we caution that these particles/fragments are all so small that there may be other processes at work in determining their shapes. These very small particles/fragments are at the size of individual mineral grains and there the processes that govern grain shapes may be quite different (e.g., involving crystal growth within the rock, etc.) than those that dominate the shapes of macroscopic impact fragments in laboratory experiments or in the fragmentation of bulk rock in asteroids. We have also investigated the shape metrics Ψ¼ (c2/ab)1/3 and F¼(a b)/(a c) derived from the results of our impact experiments. These parameters quantify deviation from spherical shape and relative flatness, respectively (Benn and Ballantyne, 1993; Ehlmann et al., 2008). Values of Ψ (0oΨo1) approaching 1 indicate roundness in the described 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. 4. Example of the process of measuring fragment shapes from photography. Fragments are digitally imaged on a uniform green background to allow easy chroma keying in 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 each fragment, yielding a measurement of the a and b axis length of each fragment.
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Fig. 5. Histograms of b/a and c/a for each of the shots.
Average values for Ψ and F for all measured fragments are quite close to each other in each shot and do not show significant dependence on specific energy, impact speed, or target shape (see Table 1). Their overall average values are Ψ¼0.60 and F¼ 0.47. 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 trend towards more roundish shapes seems to be present as fragment size decreases. More specifically, some of the largest fragments have flat and sometimes plate-like shapes that are less frequently observed among smaller fragments. This is not clearly related to impact velocity or specific impact energy. Fig. 6a shows a comparison of values of Ψ between two shots with similar impact speeds, while Fig. 6b compares F for two shots with similar specific energies. More qualitatively, high speed video from our experiments clearly shows the formation of shell-like fragments in most of the cratering and lower-energy disruptive impacts (e.g., Fig. 7). Examining average values for F and Ψ for the 10 largest fragments (but not including the largest fragment itself, which is often a large and more rounded core), there appears to be a very weak trend toward values closer to 0 (flat shapes) in the lower-speed shots compared to higher-speed ones (Fig. 8). This reinforces our subjective impression from reviewing the high-speed video and from handling fragments from the different shots. Looking at the high-speed videos, shot 130705 shows clearly ‘shelly’ fragments and it appears that shot 130703 does as well. Unfortunately, the lights that illuminate the interior of the impact chamber in support of the high-speed cameras failed to turn on for shot 130704 so we cannot compare that spherical target’s response to the one from shot 130703. In contrast, we do not see any clearly plate-like fragment shapes in the video from shots 130701, 130702, and 130706 (higher impact speeds). Previous impact experiments (Walker et al., 2013; Nakamura et al., 2014 personal communication) similarly show shell-like fragments arising from spallation from the surface of their targets. An interesting finding of this study is that such fragments can be produced in disruptive, shattering events as well and are not limited to spall-type cratering impacts on the target’s surface. Instead, shelllike fragments may form well inside the target structure, specifically right 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 our knowledge, this behavior has not been explicitly reported until now. There is increasing interest in understanding the internal structure 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 space missions and estimated with acceptable accuracy in the case of primaries of binary systems. Accurate shape models can be obtained by space missions and by radar observations. Combined masses and shapes/volumes then allow estimates of asteroid densities in some cases. Compositions inferred from ground based spectroscopic observations allow the densities of monolithic components to be estimated. For a given asteroid, the comparison of the components’ densities inferred from compositions with the measured bulk density of the whole body provides estimations of the large scale porosity (macro-porosity) of the asteroids. Implications for their internal structure can then be inferred (see additional discussion in Flynn et al., 2015). For this reason, we decided to measure the typical bulk density of the ensemble of collected fragments from each impact experiment randomly assembled together into a ‘rubble pile’. To facilitate assembly of each rubble pile we wrapped all the fragments with measured mass in thin plastic film (the kind that is used for wrapping food in domestic refrigerators) by trying to follow carefully the outer surface of the obtained aggregate. Each wrapped aggregate was then suspended from a horizontal support and plunged into water inside a container situated on a weight scale (pan balance method). In this way we measured the macro-porosities of the six sets of randomly aggregated fragments corresponding to the six impact experiments. We found average porosity values for the six sets of fragment aggregates to be 0.53 (σ1 ¼0.10). We repeated this measurement after shuffling and re-wrapping each aggregate and obtained an average value of 0.48 (σ2 ¼0.10). As a reference, a set of spheres in perfect hexagonal packing has 26% porosity; this number rises to about 35% when random packing is allowed. Delaney and Cleary (2010) studied the general problem of packing in superellipsoids by numerical methods and found a range of porosities from 0.20 to 0.36 for biaxial ellipsoidal shapes. Nevertheless, Bezrukov and Stoyan (2006) found synthetic random packing of oblate 2:1 aspect ratio and prolate 3:1 aspect ratio equal ellipsoids having 0.45 and 0.57 porosities,
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respectively. Our experimental values seem consistent with the latter reference, although the fragments in each of our impact experiment have instead irregular shapes (only roughly tri-axial in a mix of prolate and 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 aggregate structures. In the case of asteroid Itokawa (Lowry et al., 2014) and Mars’ satellite Phobos (Pätzold et al., 2014), the large-scale porosity values 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 of circum-core fragments with plate-like, ‘onion shell’ shapes.
Fig. 6. (a) Ψ for shots 130703 and 130705: two shots with nearly the same impact speed (3.89 and 3.68 km/s, respectively). (b) F for shots 130702 and 130704: two shots with nearly the same specific energy (2.93 and 2.97 103 J/kg, respectively).
Fig. 8. Average value of Ψ for the 10 largest fragments (excluding the largest fragment itself) as a function of impact speed.
Fig. 9. Typical shapes of the largest fragments resulting from our disruptive impact experiments at AVGR. Tabular, ‘onion shell’ fragments are observed from both irregularlyshaped 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.
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asteroids with inaccurate shapes may have the same kind of bias towards overestimation that we may find in the volume measurements of wrapped aggregates from our own shattering experiments reported here. It is very easy to overestimate volumes. For instance, when we wrap our samples with thin film as tight as we can, we are approximating the volume of the aggregate to the volume inside the film, which is obviously larger than the true volume of the aggregate itself. Our typical gathered sample aggregates have approximately spherical shapes with equivalent radii smaller than about 4 cm. When wrapping them, it is very easy to have the film a few mm above the average surface of the aggregate. A 2- or 3-mm error here translates into 15–20% overestimate of volume and therefore porosity (and this cannot be quantitatively taken into account in the determination of the standard deviation). This may cause a 35–40% ‘true’ porosity showing up as 45–50% porosity in the wrapped measured samples. A similar source for volume overestimation easily arises when approximating poorly determined asteroid shapes with spheres or triaxial ellipsoids, causing estimated porosities to be higher than actual porosities. 4. Conclusions We have started investigations on the effects of target shape and projectile impact speed on the mass–frequency distribution and shape of fragments resulting from impacts into basalt targets. A first run of experiments was carried out at the NASA Ames Vertical Gun Range. Analysis of the results of the 6 shots performed shows negligible dependence of the shape of the mass–frequency distribution or the shapes of fragments on target shape. Our subjective impression from review of the high-speed video and handling of the fragments from the experiments is that the largest several fragments from lower-speed impacts tend to exhibit flatter shapes than those from higher-speed impacts. This trend is only very weakly supported, however, by actual measurements of fragment shapes; the effect of impact speed (sub vs. super-sonic) needs further investigation to be fully assessed and future sets of impact experiments are being planned to better sample the range of possible impact speeds. We see the formation of flattened, plate-like fragment shapes from both irregular and spherical targets and note with interest that these fragments can originate from the interior of the target, near the core (largest fragment), in addition to the flattened fragments often seen in the near-surface spall zone in cratering experiments. The porosity of aggregates made by artificially (but randomly) reassembling fragments has been measured to be on the order of 50%, significantly larger than that for hexagonal lattice and random packing of ellipsoids, as well as for well determined masses and shapes of asteroids. We suggest that the found average high porosity is mainly due to overestimation of the volume in our samples. Volume overestimate may be affecting as well many asteroid volumes with poor shape determinations. We recommend care in the assessment of high asteroid porosities. Our results and conclusions suggest that modeling of both non-spherical shapes and size distributions are needed in order to properly study and understand the internal structures of asteroids (and comets). Acknowledgements This work was supported by the NASA Planetary Geology & Geophysics program, grant NNX11AP22G (to GJF). ACB
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