Impact and intrusion experiments on the deceleration of low-velocity impactors by small-body regolith

Icarus 223 (2013) 222–233

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Impact and intrusion experiments on the deceleration of low-velocity impactors by small-body regolith Akiko M. Nakamura a,⇑, Masato Setoh a, Koji Wada b, Yasuyuki Yamashita c,d, Kazuyoshi Sangen a a

Department of Earth and Planetary Sciences, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan Planetary Exploration Research Center, Chiba Institute of Technology, Japan c Faculty of General Education, Chubu University, Japan d School of Human Development, Nagoya University of Art, Japan b

a r t i c l e

i n f o

Article history: Received 19 March 2012 Revised 23 November 2012 Accepted 27 November 2012 Available online 10 December 2012 Keywords: Regoliths Impact processes Asteroids

a b s t r a c t Previous laboratory impact experiments into sand and glass beads have enriched our understanding of the cratering process on granular media common on asteroids and planetary regolith. However, less attention has been paid to the fate of the projectile, such as its penetration depth in the granular medium, although this may be important for the regolith mixing process. We conducted laboratory experiments on the deceleration of projectiles with low impact velocities to understand the re-accumulation process of ejecta on small asteroids. Glass beads were used as a model of a granular target. Impact experiments using 6-mm plastic projectiles with velocities of 70 m s1 were performed on the Earth’s surface and under microgravity. Measurements of the resistance force of the glass beads against slow intrusion and penetration were also performed. In the impact experiments, the projectiles were decelerated mainly as a result of drag proportional to the square of the velocity. The drag coefficient was 0.9–1.5. Additionally, we found a possible term proportional to the projectile velocity corresponding to the viscous drag with a viscosity up to 2 Pa s. These forces are consistent with numerical simulations that we carried out. The slow intrusion and penetration measurements showed that the velocity-independent resistance force per unit area on a projectile is roughly 20 times larger than the lithostatic pressure. The penetration depth of re-accumulated ejecta was examined based on the drag parameters obtained in this study. A simple configuration was used to visualize the dependence of penetration depth on the drag parameters. The penetration depth was more sensitive to the drag parameters in the case of small particles impacting a relatively small model asteroid. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Regolith particles and boulders are common on the surface of an asteroid (e.g., Veverka et al., 2001; Fujiwara et al., 2006). They are developed and evolved by impact from interplanetary space and the re-accumulation of ejecta (Housen et al., 1979; Hörz and Cintala, 1997). A visual example of re-accumulation of the ejecta on asteroids is seen on Asteroid 433 Eros, where most of the large ejecta blocks are attributed, based on their spatial distribution, to a relatively young large crater (Thomas et al., 2001). Given that the impact velocity of re-accumulation is limited by the escape velocity of the body, it is less than a few 100 m per second even for the largest main-belt asteroids of several 100 km in diameter. The low-velocity re-accumulation process includes the rebound ⇑ Corresponding author at: Department of Earth and Planetary Sciences, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan. Fax: +81 78 803 5791. E-mail address: [email protected] (A.M. Nakamura). 0019-1035/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.icarus.2012.11.038

of the impactors with a non-zero coefficient of restitution (Durda et al., 2011a), compaction (Fujii and Nakamura, 2009), or disruption of the impactors or the surface boulders (Durda et al., 2011b; Güttler et al., 2012), partial burial or penetration of the impactors, and secondary cratering on the surface. Regolith mixing and change in the size distribution proceed by high-speed primary impacts and also by such secondary impacts, i.e., re-accumulation of the ejecta blocks. The re-accumulation process might also affect the reshaping of a rubble-pile asteroid. However, it is not clear what fraction of the ejecta blocks could survive re-accumulation. The penetration depth of the impactors in the bed of regolith and the thickness of the regolith mixing zone are also unknown. To understand the cratering process and to develop scaling relations for crater dimensions and ejecta, laboratory impact experiments onto non-cohesive and cohesive granular materials were performed at impact velocities up to several kilometers per second (Schmidt and Housen, 1987; Housen and Holsapple, 2011). In those studies, the fate of the impactor was largely ignored, because the impactor is severely broken under a high-velocity impact. On the

A.M. Nakamura et al. / Icarus 223 (2013) 222–233

other hand, studies of projectile-impacts in simulated regolith at velocities of less than 1 m s1 were conducted under microgravity conditions to investigate collisions between planetary ring particles and those in proto-planetary disk environments (Colwell and Taylor, 1999; Colwell, 2003). The impactors in these cases were observed to embed themselves in the target material or to rebound with a very low coefficient of restitution. In this study, we focused on the deceleration of the impactor by simulated regolith at velocities of tens of meters per second, velocities that are relevant for the process of re-accumulation onto small bodies. A better understanding of the deceleration process of impactors by a simulated small-body surface will also be useful for interpreting the results of active penetrometry experiments in space missions (Shiraishi et al., 2000; Kömle et al., 2001). The procedure used for impact experiments on the drag force for a projectile in a granular medium is presented in Section 2, as well as the procedure for measurements conducted to determine the velocityindependent resistance force for a projectile in the granular medium. In this study, we used a millimeter-sized plastic projectile as an impactor and tens or hundreds of micron glass beads as a target, representing regolith particles. The materials and the sizes of the projectile and target are very simplified and limited in comparison to real impactors and regolith surfaces. The purpose of the experiment was to determine the basics of how granular targets resist penetration by projectiles. The results of the experiments are described in Section 3. Section 4 presents a discussion of the experimental results and the relevant numerical simulations, and a demonstration of how the drag equation parameters could affect the re-accumulation process on an asteroid surface. A summary of the study is presented in Section 5. 2. Experiments 2.1. Impact experiments Polydisperse soda lime glass beads of diameter d  50 lm and particle density of 2.5 g cm3, used in previous experiments on porous sintered targets (Setoh et al., 2010), were used in this study. Fig. 1 shows a schematic of the target configuration. The glass

Fig. 1. A schematic view of the impact configuration. A cylindrical hollow space of diameter 2h = 5 cm was opened in acrylic plates of thickness l = 0.5, 1.0, and 1.5 cm. The hollow space was filled with glass beads. The muzzle side (right) of the bed of beads was covered by a sheet of paper 0.1 mm thick. The antipodal side (left) was covered by aluminum foil 0.012 mm thick to prevent the beads from spilling.

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beads were packed into a circular hole in the center of an acrylic plate with a diameter 2h = 50 mm. We used three plates of thickness l = 5, 10, and 15 mm, respectively. The muzzle side of the hole was covered by a sheet of paper of thickness 0.1 mm. A projectile would penetrate the paper and impact the center (at the depth z (=h) = 25 mm) of the bead target. The downrange side of the hole was covered by aluminum foil of thickness 0.012 mm to prevent beads from spilling out from the hole. The bulk porosity of the packed bead beds in the plates was 41 ± 5%. This porosity of the glass-bead target was greater than the average macroporosity value estimated for S-class asteroids, but was similar to the average macroporosity value estimated for C-class asteroids based on the bulk densities of asteroids and ordinary and carbonaceous chondrites and on the assumption that respective chondrite classes are the building blocks of the corresponding asteroid classes (e.g., Britt et al., 2002). The macroporosity of the sub-kilometer S-class Asteroid 25143 Itokawa was estimated to be 41% (Fujiwara et al., 2006). The porosity of regolith can be much higher than the macroporosity of bulk asteroids and can be detected by reflectance measurements and modeling (Hapke, 2008; Shepard and Helfenstein, 2011). The near-surface bulk porosity (i.e., the sum of all porosities, including meteorite micro- and macroporosity) for radar-detected asteroid samples was estimated to be moderate, with a mean of 51 ± 14% (Magri et al., 2001). We performed impact experiments using a commercial automatic electric airsoft gun under two different gravitational conditions: normal gravity and microgravity. The projectile was a plastic sphere of diameter Dp = 6.0 mm and mass m = 0.197 g. The microgravity experiment was conducted in an airplane (Gulfstream II, Diamond Air Service, Inc.). The airplane was initially pulled up to approximately 45° in attitude at about 10,000 m in altitude. Next, the pilots stopped the engine of the airplane and let the airplane fly in a parabolic trajectory. Each parabolic flight took approximately 1 min from start to end and enabled us to perform experiments under microgravity (acceleration due to gravity of less than 1 m s2) for 20 s. The parabolic flight was repeated eight times. The ambient pressure was 0.8 atm. The muzzle of the gun and the target of beads were contained in an acrylic box. We measured the velocity of the projectile after penetration using a high-speed camera (EX-F1) placed outside the acrylic box at 1200 fps. The projectile was illuminated by a lamp. In total seven successful shots were performed. The initial velocity vi of the projectile was determined at atmospheric pressure on the ground using high-speed camera images for 19 shots and was 69.4 ± 1.2 m s1. We also conducted calibration shots in a reduced pressure room and found that the projectile velocity at 0.8 atm was on average only 1.0 m s1 less than that under 1 atm conditions, i.e., the velocity difference was within the observed scatter of the initial velocity. Therefore, we considered the value determined in the laboratory under the 1-atm condition to be the initial velocity for shots without an initial velocity measurement. Similar experiments were performed on the ground using the same target system, the same gun, and the same camera for seven shots. We also conducted impact experiments using higher-speed cameras, either the Shimazu HPV-II or the Photron FASTCAM SA1.1, under conditions of backlight illumination. Experiments with target beads of d = 420 lm were also conducted. We conducted calibration shots into the paper and the aluminum foil to estimate the velocity decrease due to each covering. To cover a wider range of impact conditions for the calibration data of the aluminum foil, projectiles of diameter Dp = 3 mm were also used. The projectiles were accelerated up to 320 m s1 using a small gas gun installed at Kobe University (Setoh et al., 2010). The velocities of the projectiles before (vi) and after (vf) perforating the paper or the foil were determined from the high-speed images. We assumed that the deceleration of the projectile was due to the

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sum of fluid dynamic drag and the constant resistance force proportional to the mechanical strength (Tc) of the covering. The deceleration of projectiles in aerogel is well modeled by such a drag equation, in which fluid dynamic drag is effective in the higher-velocity regime, and resistance due to the mechanical strength of the aerogel is effective in the lower-velocity regime (Niimi et al., 2011). The equation of motion for a spherical projectile is given by

m

  dv 1 ¼ S C d c qc v 2 þ T c ; 2 dt

ð1Þ

where S, Cd_c, and qc are the cross-sectional area of the projectile, the drag coefficient, and the density of the covering, respectively. 2.2. Slow intrusion and penetration measurements To measure the velocity-independent resistance force against penetration, we conducted slow intrusion and penetration tests using stainless steel rods as well as the plastic spherical projectiles used in the impact experiments described above. The intrusion tests were performed using stainless steel rods 5, 10, and 15 mm in diameter. Each of the rods was attached to a compression testing machine at Kobe University and intruded ver-

tically into a bed of the 50-lm diameter glass beads used in the impact experiments. Two cylindrical containers with inner diameters of 59 and 94 mm were used. The height of the bead bed was 20 mm for the smaller container and 37 mm for the larger container. The porosity of the bed of 50-lm beads was about 40%. The speed of intrusion varied between 0.01 and 10 mm s1. For comparison, the resistance force was measured for the 420-lm diameter glass beads used in the impact experiment. The porosity of the bed of 420-lm beads was about 37%. The force for the intrusion and the displacement of the rod was recorded until the rod reached a depth z = 10 mm in the beads. The force was then converted into the value per unit area to compare the results of rods of different diameter. Penetration experiments were performed for the 6-mm-diameter plastic projectile in the horizontal direction. Fig. 2 shows the configuration of the penetration test. The sphere was embedded in glass beads with diameters of 50 lm and 420 lm, filling a 95  95  50-mm acrylic container, to half the depth of the container, i.e., z (=h) = 25 mm from the container surface. The container had a small hole for the string in one of the side walls at z = 25 mm. The string was glued to the sphere. The other end of the string was attached to the tensile testing machine. The string was then slowly pulled at speeds of 0.5 and 1 mm s1, and the force and displacement were recorded.

3. Results 3.1. Impact experiment results

Fig. 2. A schematic view of the measurement of drag force for penetration.

Table 1 shows the experimental conditions and the results. In all impact experiments, except for the shots with l = 15 mm and 420-lm beads, the projectiles exited the bed of beads. Fig. 3 shows an example of the images taken for the microgravity experiments. Because the frame rate was not high enough relative to the projectile velocity, the image of the projectile is elongated in the direc-

Table 1 Summary of experiments. Bead diameter, d (lm)

Target thickness, l (mm)

Impact velocity,

vi (m s1)

Condition

Framing interval (ms)

Post-penetration velocity,

vf (m s1) 081217-01s 081217-06s 081217-02m 081217-05m 081217-07m 081217-03l 081217-04l 090510-4s 090510-6s 090510-5m 090510-7m 090510-9m 090510-2l 090510-8l 091030-1s 091030-2m 091030-3l 100709-1s 100709-2s 100709-3s 100709-6m 100709-7m 100709-8m 100709-4l 100709-5l a b c

50

50

50

420

5 5 10 10 10 15 15 5 5 10 10 10 15 15 5 10 15 5 5 5 10 10 10 15 15

69.4a

Microgravity

1/1.2

69.4a

1G

1/1.2

65.9(4.4) 68.0(4.5) 68.8(2.1)

1G

0.032, 0.064b

68.2(2.2) 69.4(1.1) 70.0(1.1) 69.3(0.9) 68.6(0.8) 68.0(0.4) 68.1(1.1) 68.8(1.1)

1G

0.025

Estimated value, as there was no velocity measurement. The framing interval was varied from 32 ls for the earlier frames to 64 ls for later frames in a single image sequence. The projectile did not come out from the beads.

32.8(1.4) 31.2(1.6) 16.3(0.7) 17.0(0.7) 18.4(1.2) 1.26(0.06) 4.88(0.14) 38.8(2.2) 31.1(1.4) 17.4(0.7) 18.0(0.9) 16.3(0.8) 6.3(0.4) 2.6(0.2) 30.6(1.0) 14.6(0.5) 7.9(0.2) 30.7(0.5) 28.6(0.5) 27.8(0.5) 10.33(0.09) 13.4(0.1) 15.1(0.1) c – c –

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( )   vi þ vf 2 dv 1 C d Al qAl m þ T Al ; ¼ S 2 dt 2

ð2Þ

where Cd_Al is the drag coefficient and qAl and TAl are the density and the strength of the aluminum foil, respectively. From Eq. (2) and using the thickness of the foil lfoil, the following relationship was derived:

m

v 2i  v 2f 2

!

( )   vi þ vf 2 1 ¼ Slfoil þ T Al ; C d Al qAl 2 2

mðv i  v f Þ 1 ¼ C d Al qAl 2 Slfoil

Fig. 3. Successive images of microgravity impact experiments (No. 081217-01s). The aluminum foil was at the right edge of the images, i.e., the bead container is not shown in the figure. The top image is the frame before projectile penetration. After penetrating the target, the projectile appears from right to left in the second image and below. The time interval between two successive images is 833 ls (= 1/1200 s). The image of the projectile is elongated along the direction of motion. The illumination was from the camera side and the diffuse scattered light from the wall of the acrylic box is seen even in the pre-impact image. The scale bar in the top image is 6 mm wide.

tion of motion. Fig. 4a–c and d–e shows the images of the impact experiments using 50-lm and 420-lm diameter glass beads, respectively, taken on the ground at a higher frame rate. The projectiles were preceded after penetration by some amount of glass beads. The separation between the leading glass beads and the projectile increased with the layer thickness. 3.1.1. Calibration shots The velocity deceleration due to the thin paper was measured using three calibration shots with an initial velocity of 67.5– 70.1 m s1. The decrease in velocity caused by the paper was found to be 2.0 ± 1.0 m s1. According to this calibration dataset, a projectile with an initial velocity of 69.4 m s1 is expected to have a velocity of 67.4 m s1 after penetrating through the paper. We used plastic (Dp = 6 mm), glass (3 mm), and nylon (3.175 mm) projectiles for aluminum foil calibration shots to determine the mechanical strength and the drag coefficient in Eq. (1). With the plastic projectiles, the velocity change was less than 12% of the initial velocity when the initial velocity was greater than 27 m s1. The minimum impact velocity at which the plastic 6-mm sphere could penetrate the foil was 16.7 m s1, and the maximum impact velocity at which it could not penetrate the foil was 15.4 m s1. That is, the threshold penetration velocity for the foil was about 16 m s1. Given that the velocity change due to the foil perforation is small we approximated Eq. (1) as



vi þ vf 2



 þ T Al

vi þ vf 2

ð3Þ

1 :

0

ð3 Þ

Eq. (30 ) is a linear equation of two unknown parameters, the drag coefficient Cd_Al and the mechanical strength TAl; therefore, we applied linear least squares to the data to determine the two parameters. Fig. 5 shows the data and the curve fit of Eq. (30 ). The velocity change vi  vf rapidly decreases as the average velocity v i þv f increases, indicating that the second term of Eq. (30 ) is negligi2 ble unless the projectile velocity is small. The drag coefficient and the mechanical strength of the aluminum foil were estimated to be 2.7 ± 0.5 and 58.4 ± 3.8 MPa, respectively. The estimated mechanical strength was within a factor of the values of shear and tensile strengths of an aluminum alloy (1100), 60–165 MPa (Pilkey, 1994), though the foil is made of a slightly different aluminum alloy (1N30). The simple model of deceleration of the projectile by the foil seems usable enough for estimation of the deceleration. Based on these values and Eq. (3), we expect that the threshold impact velocity for penetration of the foil is 14.2 ± 0.8 m s1, which is roughly in agreement with our observations. 3.1.2. Deceleration of the projectile in the beads The projectile velocity after penetrating through the bed of beads was determined from the images. Fig. 6a shows the raw results for post-penetration velocity versus target thickness. We found no obvious difference between the data taken under microgravity conditions and the data taken on the ground. In general, the post-impact velocities of shots into 420-lm beads are systematically lower than those into 50-lm beads. Estimating the effect of the aluminum foil at the rear surface of the target beads is not straightforward. As noted previously, as the projectile became embedded it was preceded by some glass beads that it pushed ahead of itself. This means that the foil at the rear of the container was broken not by the projectile but by the projectile-driven beads, although the resistance of the foil might have had some effect on the projectile via the beads until the foil broke. As will be shown in the numerical simulations in Section 4.2, the effect of projectile penetration extends to regions outside the column of the projectile trajectory, i.e., the pressure of the projectile on particles along the trajectory is transferred to particles in a wider region. A larger foil surface area might affected the force chains in the wider region but would act somewhat diffusively and have a minor effect, particularly in the direction of travel of the projectile. Therefore, we assume that the velocity change due to the foil alone gives us the upper limit to the velocity change due to the foil at the rear surface of the beads. Based on the calibration analyses, the upper limits of the velocity change were calculated, i.e., we put the velocity shown in Fig. 6a into vf in Eq. (30 ) and derived vi as the velocity of the projectile before the foil. Fig. 6b shows the derived vi. The threshold velocity for penetration of the aluminum foil, 14.2 m s1 is shown for l = 15 mm case with 420-mm beads, although the data of l = 15 mm overlap with one another in Fig. 6b.

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Fig. 4. Projectile images after penetrating a layer of (a) 50-lm (l = 5 mm) (No. 091030-1s), (b) 50-lm (l = 10 mm) (091030-2m), and (c) 50-lm (l = 15 mm) beads (091030-3l), (d) 420-lm (l = 5 mm) (100709-1s) and (e) 420-lm (l = 10 mm) beads (100709-6 m). The scale bars represent 6 mm. Elapsed times from impact are 1.5, 3.1, 5.5, 0.7, and 1.6 ms, respectively.

for different rod diameters, intrusion speeds, and container sizes. The pressure increases with depth and reaches about 8 kPa at depth of 10 mm, whereas the lithostatic pressure is 0.15 kPa. The linearly extrapolated value of the measured pressure to a depth z = 25 mm is 20 kPa. The data for the 420-lm beads are also presented in Fig. 7a and show resistance that is similar to or slightly larger than the resistance of the 50-lm beads. Fig. 7b shows examples of the resistant force per unit area of the slow penetration measurements. At the beginning, the resistance force per unit area increased rapidly to a maximum and then approached asymptotically to a constant value which may be slightly higher for 420-lm beads. The profile of the force per unit area versus the stroke seemed sensitive to the preparation of the bead bed, such as packing state. The asymptotic value was 6–10 kPa, which is lower than those expected from the intrusion test (20 kPa) but still about 20 times larger than the lithostatic pressure (0.37 kPa) for the 50-lm beads. We will discuss velocity-independent resistance below in the next section. Fig. 5. Calibration data for aluminum foil of thickness lfoil = 0.012 mm. The fitted   v v curve is m Sli f ¼ 12 qAl C d Al ðv i þ v f Þ=2 þ ðv þTvAl Þ=2, where the drag coefficient and the foil

i

f

mechanical strength of the foil are Cd_Al = 2.7 ± 0.5 and TAl = 58.4 ± 3.8 MPa, respectively.

3.2. Results of slow intrusion and penetration Fig. 7a shows the depth–force per unit area data of the rod intrusion test. Within the range of the measurement conditions, the depth–force per unit area data for 50-lm beads are similar within the scatter of the data. That is, no clear differences are discernible between the resistant force per unit area from the beads

4. Discussion 4.1. Experimental data As shown in Fig. 6a and b, our findings were as follows. First, gravity had no obvious effect on the deceleration of the projectile. That is, the effect of gravity on the drag force on the projectile was negligible in this experiment when compared with the scatter of the data. Second, the drag force depends on the particle size. The larger (420 lm) beads imparted a larger drag force on the projectile. Here, we discuss the drag force for the projectile in the beads based both on the impact experiments and on the slow intrusion and penetration measurements.

A.M. Nakamura et al. / Icarus 223 (2013) 222–233

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Fig. 6. Projectile velocity after penetration of glass beads: (a) raw data, (b) upper limit. The curves show the expected deceleration by a drag force due only to a term proportional to the square of projectile velocity.

The deceleration of the projectile by the beads may be expressed as:

m

  dv 1 ¼  C d q t Sv 2 þ k 1 v þ F 0 ; 2 dt

Fig. 7. The resistance force per unit cross-sectional area. (a) Rod intrusion. The beads diameter, rod diameter, intrusion rate, and container size (small or large) are shown. (b) Sphere penetration. Gray and black curves show running means of 10 data points taken with 0.1 mm interval in stroke for six individual measurements for 420 and 50-lm beads, respectively.

ð4Þ

where k1 is a constant and F0 corresponds to the velocity-independent resistance force. The second term in the right-hand side, k1v, maybe represent a viscous-like drag, which is 3pgDpv with viscosity (representing the character of the granular material) g. The projectile seemed intact as shown by high-speed images in Fig. 4; therefore, the projectile’s mass m was regarded as constant. On the other hand, some of target beads at the impact point must have been broken at the impact velocity of 70 m s1 from previous experience of impact disruption of glass beads (Machii and Nakamura, 2011). However, we expected that this would have little effect on the motion of the beads and therefore on the projectile. The fraction of the projectile energy partitioned into comminution in the glass bead experiment must be much smaller than would be the case for a hypervelocity disruptive impact into a basalt target, which is 10–24% (Gault and Heitowit, 1963). Therefore, the square of the projectile velocity must be greater than 76–90% and the velocity must be greater than 87–95%, i.e., projectile deceleration resulting from the conversion of kinetic energy into comminution energy must be less than 5–13% of the total energy budget. Additionally, a previous low-velocity cratering experiment showed that fragmentation of the target glass beads had no obvious effect on the motion of the target beads, i.e., the crater grew monotonically with impact velocity and the slope of the crater size did not change with

impact velocity. The velocity ranged from 11 m s–1 (scarce fragmentation) to 329 m s–1 (common fragmentation) (Yamamoto et al., 2006). Therefore, we expected no significant effect due to comminution (of target beads) on projectile deceleration in the right-hand side of Eq. (4). Fig. 6a and b shows deceleration curves with Cd = 1 and 1.5, k0 = k1 = 0, and qt = 1.5 g cm3. The 50-lm and 420-lm bead data for l = 5 and 10 mm in Fig. 6a seem to be on different curves parallel to the dashed and dotted curves, whereas the data for l = 15 mm are lower than these tendencies. By assuming k0 = k1 = 0 and qt = 1.48 g cm3, we derived Cd = 1.34 ± 0.03, 1.31 ± 0.03, and 1.8 ± 0.2 for the 50-lm bead data, including both the microgravity and the ground measurements for l = 5, 10, and 15 mm, respectively. The two former values are, within uncertainty, the same as expected from Fig. 6a. The value for l = 15 is significantly larger and may reflect a term that is proportional to the projectile velocity. Similarly, Cd = 1.49 ± 0.07 and 1.46 ± 0.10 were derived for the 420-lm beads data, assuming qt = 1.58 g cm3 from Fig. 7a for l = 5 and 10, respectively. Given the coincidence of the two values, it seems that the second term on the right-hand side of Eq. (4) is also negligible for deceleration with l = 5, 10 mm for the 420-lm beads. The 50-lm and 420-lm bead data in Fig. 6b do not lie on curves parallel to the two curves in Fig. 6b. We obtained Cd = 1.18 ± 0.08,

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1.05 ± 0.02, and 0.94 ± 0.02 for all 50-lm bead data for l = 5, 10, and 15 mm, respectively, assuming k0 = k1 = 0. The decreasing value of Cd with l indicates that the modified projectile velocity shown in Fig. 7b is overestimated, especially for the l = 15 mm data, if no additional drag was overlooked in Eq. (4). A similar overestimate was detected for the 420-lm bead data in Fig. 7b. The calculated Cd values were 1.29 ± 0.06, 1.10 ± 0.05, and 0.910 ± 0.003 for l = 5, 10, and 15 mm, respectively. Previous low velocity (vi = 0–4 m s1) free-fall experiments of projectiles impacting glass beads have shown that the velocityindependent resistance force is proportional to the depth z in the beads (Katsuragi and Durian, 2007) and is given by

F 0 ¼ kz;

ð5Þ

where k can be expressed as k = k0Sqtg, with g as gravitational acceleration and k0 as a dimensionless parameter that characterizes the granular medium’s porosity, grain size, and particle surface roughness. The resistance force per unit area was larger than the lithostatic pressure, and the value of k0 (20) determined in the present study was on the same order as the value of 36/p derived from the measurement by Katsuragi and Durian (2007). The larger resistance force per unit area was also shown in a previous study for disk-shaped objects in a horizontal flow of granular medium with a constant velocity of 0.2 mm s1 (Albert et al., 2001). Resistance was shown to be proportional to the length of the disk in the direction of the flow. The discrepancy between the intrusion data and the penetration data of our study is probably due to the variation in shape between the cylinder and the sphere. According to soil mechanics, the resistance force per unit area of ground for a shallow foundation is dependent on the embedded length of the foundation Lf and has the form of NqqtgLf, where Nq is a dimensionless factor, called the bearing resistance factor. This factor was found to be dependent on the angle of internal friction / of the soil. A model-derived analytical formula (e.g., Knappett and Craig, 2012) is written as

Nq ð/Þ ¼ expðp tan /Þ tan2



 :

p / 4

þ

2

ð6Þ

Using the values / = 23° and 28° for the 50- and 420-lm beads (Hakura, 2011), Nq(/) becomes 8.7 and 15, respectively, which is within a factor of difference from the value of k0 obtained in the present study and qualitatively in agreement with the possible greater resistance of the 420-lm beads compared with the 50-lm beads. The ratio of the first term and the third term on the right-hand side of Eq. (4) is

Cdv 2 Cd 2 F ; ¼ 2k0 gz 2k0 r

ð7Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi where F r ¼ v 2 =gz is a Froude number. Assuming Cd = 1 and k0 = 20, the value of Eq. (6) is larger than unity when F 2r > 40; that is, v > 3.1 m s1 for g = 9.8 m s2 and z = 25 mm. Therefore, the effect of the third term and thus the gravity dependence cannot appear in the result of the impact experiments of this study. This is consistent with our observations. Because the third term of Eq. (4) is negligible, the relationship between the projectile velocity and the position is derived as follows:

v ¼ k01 þ ðv i þ k01 Þe 0 k1

2k1 ¼ ; C d qt S

C d qt S L 2m

;

ð8Þ ð9Þ

where L represents the penetration distance of the projectile. With Cd = 1.32 ± 0.03, determined from all the data for l = 5 and 10 mm for the 50-lm beads, we obtain k1/qtS = 3.0 ± 0.9 m s1, which is

much less than the projectile velocity after penetrating the l = 10 mm and therefore consistent based on the 50-lm bead data for l = 15 mm. This corresponds to a viscosity g = 2.2 ± 0.7 Pa s. As we have no post-penetration velocity data for l = 15 mm, we cannot provide further constraints on the value of k1 for the 420-lm beads. The ratio of the second and the third terms in Eq. (4) is

k1 v 12gv ¼ ; k0 Sqt gz k0 Dp qt gz

ð10Þ

which is larger for smaller projectiles and a lower gz condition for the fixed material constants, g and k0. The ratio of the second and the first terms in Eq. (4) is

k1 v C d qt Sv 2 =2

¼

24g ; C d qt Dp v

ð11Þ

which is also larger for smaller projectiles. Consequently, the dominant drag for a projectile impacting into the beads in this study is the term proportional to the square of the projectile velocity. The drag coefficient is 0.9–1.5, similar to the value of 1.1 ± 0.1 for the projectile penetrating aerogel (Niimi et al., 2011) but smaller than 2.1 ± 0.2 in glass beads (Katsuragi and Durian, 2007). The values of the velocity-independent resistance force observed by slow intrusion and penetration measurements are roughly in agreement with each other and also with previous results. The gravity independence observed in the impact experiments can be explained by the results of the slow intrusion and penetration. The larger drag force exerted by the 420-lm beads may be due to a larger drag coefficient or to a larger contribution of the linear velocity-dependent term. 4.2. Numerical model Our experimental setup was limited, and it was difficult to directly measure the drag force acting on the projectile during penetration. To confirm the drag force measured in the present experiments, we carried out three-dimensional numerical simulations of impacts into a granular target with a kind of N-body code, Distinct Element Method (DEM). We were able to directly observe the drag force in the simulation. This simulation was based on the previous work of Wada et al. (2006), in which impact cratering processes were well reproduced with DEM code. In the basic DEM, elastic spherical particles are assumed, and each of their motions is calculated, taking into account mechanical interactions between particles in contact (e.g., Cundall and Strack, 1979; Wada et al., 2006). In our DEM code, the mechanical interaction forces between contact particles come from the elastic force and friction expressed by the Voigt-model, which consists of a spring and dashpot pair. The spring provides elastic force based on the Hertzian elastic contact theory. The dash-pot expresses energy dissipation during contact to realize inelastic collision with a given coefficient of restitution e. For the tangential direction between contact particles, a friction slider is introduced to express Coulomb’s friction law with a given coefficient of friction l. See Wada et al. (2006) for a detailed description of the code. We prepared a granular target consisting of 384,000 spherical particles of 420 lm in diameter (we assumed quartz-like material properties: density of 2.5 g cm3, Young’s modulus of 94 GPa, Poisson’s ratio of 0.17). These particles were randomly dropped into a rectangular container with a base of 42 by 42 mm. The resultant height of the granular layer was 15 mm and its porosity was 43%, as shown in Fig. 8a. A projectile particle with a diameter Dp = 6 mm (we assumed polystyrene-like material properties: density of 1.7 g cm3, Young’s modulus of 5 GPa, Poisson’s ratio of 0.34) was impacted horizontally into the target at a velocity of 70 m s1. To observe the influence of the target walls, we set two

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Fig. 8. Cross-sectional views of snapshots of a numerical simulation of the wall-free nominal case at (a) 0 s and (b) 20 ls after impact. Each panel has a side view and a top view. To clearly show the cross sections, we show only the particles whose centers are located within a thin cross sectional region (810 lm thick). The particles are colored, depending on the total amount of elastic energy stored in each particle, as indicated in the scale bar. A 6-mm-diameter projectile is represented by a large circle with an arrow indicating the impact direction.

types of wall configurations: wall free and wall confined. For the wall-free target, all walls of the target, including the base, were removed during the impact simulation. For the wall-confined target, only the two lateral walls perpendicular to the impact direction were removed; the top and bottom of the target were covered by walls. The values of e and l between particles (and a projectile) were nominally set to 0.4 and 0.5, respectively. We also conducted several simulations with various values of e and l but found no significant dependence on e and l. Therefore, we show here the results only for the case of e = 0.4 and l = 0.5, as the nominal case. Walls had e = 0 to suppress the reflection wave generated at the walls as much as possible. The whole process from target preparation to the end of the impact simulation was carried out under normal gravity (1G). Figs. 8–10 show snapshots of the wall-free nominal case in cross-sectional views (side and top views). Particles in these figures are colored, depending on the total amount of elastic energy E stored in each particle, as indicated in the scale bar, where the elastic energy was counted for only that stored in springs connecting particles in the normal direction and was equally partitioned into each of the contact particles. The normalizing factor E0 was set to half of the initial relative kinetic energy of a pair of target particles in collision at a collision velocity of 10 m s1. Since the elastic energy reflects the force acting on each particle, these colored views enable us to see how the grain-bridges or the forcechains work in the granular target. In the initial stage of penetration, a hemispherical region in which a large amount of force was exerted on particles propagated like a shock wave (Fig. 8b). After a cratering hole began to open, the projectile was pushed only by particles touching the small head region of the projectile (Fig. 9). We also observed a precursor wave detached from the near-field region of the projectile. This detached wave is particular to granular materials that are modeled by a chain of elastic beads without tensional forces (Hascoët et al., 1999). We observed its propagation speed to be approximately 400 m s1, close to sound speed in the case of such chains. The colored particles in Fig. 9

show that the projectile was not supported by the entire of the target particles but only by the particles in the vicinity of the projectile. Additionally, particles that seem to be in contact with the projectile are not necessary colored, showing that only several force chains supported the projectile. This inhomogeneity cannot be seen in continuum bodies and is particular to granular materials. The projectile penetrated deeply into the granular target (Fig. 10). Since there were no walls and the vertical thickness of the target was thin compared with the projectile size, the target particles were blown away in the vertical direction (Figs. 9 and 10). On the other hand, as shown in the top views, the target particles were almost not blown away in the horizontal direction because the target was sufficiently wide compared with the projectile size. The wall-free configuration was different from the experimental setup. However, in the wall-confined simulations, we observed the same phenomena, such as an initial shock-like wave, detached wave propagation, and deep penetration of the projectile, suggesting that the wall effect was not essential in the experiments or in the numerical simulation. The horizontal velocity v of the projectile decreased monotonically as it penetrated in the target, as shown in Fig. 11a, in which the penetration distance L of the projectile is plotted with respect to v for the wall-free nominal case. The penetration distance is defined by the horizontal distance from the original impact point to the projectile head. The projectile velocities at L = 5 and 10 mm were about 27.9 and 18 m s1, respectively, which are in rather better agreement with the experimental data of 420-lm beads shown in Fig. 6a than those in Fig. 6b. The force F acting on the penetrating projectile in the horizontal direction was obtained from the simulation data and plotted as a function of v in Fig. 11b. Judging from Fig. 11a and b, it is clear that significantly large F acts until half of the projectile penetrates into the target. This occurs because the projectile is pushed by a large number of target particles in the initial stage, but the projectile then makes a cratering hole and is pushed by only a small number of particles in the later stage. These are indicated by the snapshots of this initial stage (see Figs. 8b and

230

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Fig. 9. Cross-sectional views of snapshots of the same simulation shown in Fig. 8 at (a) 110 ls and (b) 350 ls after impact.

Fig. 10. Cross-sectional views of snapshots of the same simulation shown in Fig. 8 at (a) 750 ls and (b) 1.35 ms after impact.

9). After this initial stage of penetration, in the range v < 30 m s1, F decreases and seems to be proportional to v2, apart from some small fluctuation in F. We plotted three lines corresponding to the hydrodynamic drag force, F ¼ 12 C d qt Sv 2 , with C d ¼ 0:5; 1; and 2 in Fig. 11b. These lines likely fit the numerical data in the range v < 30 m s1, suggesting that the penetration resistance force on average is first-approximately given by the hydrodynamic drag force with Cd = 0.5–1, at least for the velocity range v < 30 m s1. This is consistent with the experimental results obtained in the present study, although the value obtained for Cd

by the simulation is lower than the experimental values. The small fluctuation seen in Fig. 11b would be due to the inhomogeneous distribution of particles pushing the projectile, as shown in Figs. 9 and 10. It is notable that the drag force in average is represented by such a hydrodynamic force in spite of the inhomogeneous force distribution in the granular target. The residual force obtained by subtracting the hydrodynamic drag force with C d ¼ 0:7 from F is plotted against v in Fig. 11c. Although the data are widely scattered, the residual force seems to be dependent on v. In Fig. 11c, we also show three lines

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to the viscous drag force with g  2 Pa s, although the slope for the data seems slightly higher than those of the viscous drag lines. Numerical results for a simulation with the wall-confined target are shown in Fig. 12 to illustrate the wall effect on the drag force. Compared with the wall-free target case (Fig. 11), the drag force results are qualitatively the same. Only the value of Cd is a bit large (1–2) for the wall-confined case. As shown in Figs. 9 and 10,

Fig. 11. Results obtained from the numerical simulation of the wall-free nominal case. (a) Penetration distance L of the projectile normalized by the projectile radius Dp/2 as a function of the horizontal velocity v. (b) The horizontal force F acting on the projectile versus v. The three lines represent the hydrodynamic drag force F = (1/2)CdqtSv2 with Cd = 0.5, 1, and 2, respectively. (c) The residual force obtained by subtracting the hydrodynamic drag force with Cd = 0.7 with respect to v. The three lines represent the viscous drag force F = 3pgDpv with g = 0.5, 1, and 3 Pa s, respectively.

corresponding to the viscous drag F = 3pgDpv with g = 0.5, 1, and 3 Pa s. The data are distributed around these lines, supporting the experimental result that the resistance force has a term similar

Fig. 12. Same as Fig. 11, but for the case of the wall-confined target. In panel (c), Cd = 1.5 is used.

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the projectile was supported by target particles in the vicinity of the projectile. This may be the reason why the wall effect was small in our simulation. Therefore, we conclude that the target walls do not play a major role in our simulations and that the drag forces obtained from our laboratory experiments are consistent with our numerical simulations. 4.3. Re-accumulation process on the surface of an asteroid Based on the results of this study, we consider the re-accumulation of particles ejected by high-velocity impact from the surface of asteroids. Although our understanding of the penetration process of an impactor into a granular surface is in the preliminary stage, a demonstration of how the re-accumulation process can be affected by the drag equation and its parameters will be useful for understanding the re-accumulation and regolith-mixing pro-

cess. Fig. 13a and b show the results of model calculations for a particle (impactor) of density 1.75 g cm3 with different re-accumulation velocities onto model 30-km- and 300-m-diameter asteroids. We assumed that the asteroids were composed of the glass beads used in this study, although the mechanical response of regolith particles on the surface of an asteroid is expected to be different from that of glass beads because of differences in the angle of internal friction, cohesion, and packing density. For the drag equation, we use Eq. (4) but include gravitational acceleration of the asteroids:

m

  dv 1 C d qt Sv 2 þ k1 v þ F 0 ; ¼ mg  2 dt

0

ð4 Þ

where k1 = 3pgDp. The bulk density of the asteroid surface and the drag coefficient were fixed to 1.5 g cm3 and unity, respectively, and we varied k0 and g. The impact angle at re-accumulation was assumed to be normal to the surface. In reality, oblique incidence would be expected, and the penetration depth would be shallower than for normal impact. Moreover, even ricochet of the impactor would occur at shallow incidence (Gault and Wedekind, 1978). For both model asteroids, the penetration depth of a 6-m diameter particle is up to only three times the particle diameter, regardless of the values assumed for k0 and g. The result of shallow burial depth compared with the diameter of the projectile particle is robust as long as the drag coefficient is unity or larger and the density of the surface is not extremely low. This is because deeper penetration is not possible when the drag force is proportional to the square of the particle velocity. Oblique incidence and larger k0, expected for angular regolith on an asteroid surface, would make the penetration depth even shallower. Isolated large blocks seen on the smooth terrains of Asteroid 25143 Itokawa (e.g., the blocks on the bottom left of Fig. 1 of Nakamura et al. (2008)) could be such ejecta blocks that landed on or did not deeply penetrate the surface at re-accumulation. On the other hand, the penetration depth of small particles largely depends on the parameters of the drag equation and on how the coefficient k1 depends on particle diameter Dp. The projectile particles are barely embedded in the subsurface if the drag consists of a term proportional to the particle velocity. The penetration depth, and hence the mixing zone, is much shallower for the smaller model asteroid. Therefore, the regolith mixing process on smaller asteroids is different from the process on larger asteroids. We assumed that the coefficient k1 depends on particle diameter Dp, despite a lack of investigation in this study into the particle-diameter dependence. Further study of the drag on small impactors is anticipated. 5. Summary

Fig. 13. Penetration depth calculated with different drag parameters for reaccumulation of 6-mm- and 6-m-diameter particles onto the surface of a (a) 30km- and (b) 300-m-diameter model-asteroid. The impact is assumed to be normal to the surface. Note that the drag parameters assumed here were based on the values determined from the present study with a plastic projectile and a glass bead bed of a fixed porosity. These parameters could be significantly different from the drag parameters for real asteroids.

We conducted laboratory experiments on the deceleration of projectiles with low impact velocities to understand the re-accumulation process of ejecta on small asteroids consisting of regolith particles. Glass beads of 50 and 420 lm diameter were used to simulate asteroid regolith, although actual granular asteroid material would be much more complex, with an irregular shape and various particle sizes. The porosity of the bead beds was fixed at 40%. Impact experiments using 6-mm plastic projectiles with an initial velocity of 70 m s–1 showed that the dominant drag term was proportional to the square of the projectile velocity, with coefficients of 0.9–1.3 and 0.9–1.5 for 50-lm and 420-lm beads, respectively. A term may appear that is proportional to the projectile velocity corresponding to a viscous drag with a viscosity up to 2 Pa s. A similar term was also detected in a numerical simulation. It should be noted that ‘‘viscosity’’ as used in our paper does not necessarily correspond to the viscosity that is normally used in

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hydrodynamics. We use ‘‘viscosity’’ only as a coefficient of the drag proportional to projectile velocity. We should further explore the physical meaning of this term in the future. The slow intrusion and penetration experiments of this study showed that the velocity-independent resistance per unit area acting on a projectile is roughly 20 times greater than the lithostatic pressure. The deceleration caused by the granular media on an asteroid surface determines the fate of the re-accumulated ejecta. We estimated the penetration depth of an impactor into a granular surface based on the drag parameters derived in this study. The penetration depth was found to be sensitive to the parameters, especially for small particles impacting a small model asteroid. Our present work suggests that the regolith mixing process varies with asteroid size. Our future work will include a wider range of impactor material properties and target granular materials, and include experiments conducted at different size scales. Acknowledgments We are grateful to the enlightening reviews of two anonymous reviewers. A.M.N. and K.W. acknowledge Institute of Space Science (ISSI) for the discussion at the team meetings. M.S. acknowledges the support from the Japan Society for the Promotion of Science (JSPS). The experiments in the parabolic flight were conducted as Student Zero-Gravity Flight Experiment Contest by JAXA and JSF. We thank K. Kogure of JSF for her support in operation of experiments, and also thank Diamond Air Service staffs for their support in operation of experiments and making experimental condition. We also thank T. Katsura, A. Takabe and S. Takasawa for their contribution in the preparation of the microgravity experiments and M. Kiuchi and H. Nagaoka for reduced pressure experiments. The authors are thankful to Center for Planetary Science (CPS) for the usage of the Shimazu HPV-II camera. This research was supported by grants-in-aid for science research from the Japanese Society for the Promotion of Science (Nos. 40260012 and 24540459) of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. References Albert, I., Sample, J.G., Morss, A.J., Rajagopalan, S., Barabási, A.-L., Schiffer, P., 2001. Granular drag on a discrete object: Shape effects on jamming. Phys. Rev. E 64, 061303 (4 pages). Britt, D.T., Yeomans, D., Housen, K., Consolmagno, G., 2002. Asteroid density, porosity, and structure. In: Asteroids III. University of Arizona Press, Tucson, pp. 485–500. Colwell, J.E., 2003. Low velocity impacts into dust: Results from the COLLIDE-2 microgravity experiment. Icarus 164, 188–196. Colwell, J.E., Taylor, M., 1999. Low-velocity microgravity impact experiments into simulated regolith. Icarus 138, 241–248. Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for granular assemblies. Géotechnique 29 (1), 47–65. Durda, D.D., Movshovitz, N., Richardson, D.C., Asphaug, E., Morgan, A., Rawlings, A.R., Vest, C., 2011a. Experimental determination of the coefficient of restitution for meter-scale granite spheres. Icarus 211, 849–855.

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