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Impact crater formed on sintered snow surface simulating porous icy bodies
Icarus 216 (2011) 1–9
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Impact crater formed on sintered snow surface simulating porous icy bodies Masahiko Arakawa a,⇑, Minami Yasui b a b
Graduate School of Science, Kobe University, 1-1, Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan Organization of Advanced Science and Technology, Kobe University, 1-1, Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
a r t i c l e
i n f o
Article history: Received 2 May 2011 Revised 13 August 2011 Accepted 18 August 2011 Available online 26 August 2011 Keywords: Impact processes Ices, Mechanical properties Cratering
a b s t r a c t To improve the scaling parameter controlling the impact crater formation in the strength regime, we conducted impact experiments on sintered snow targets with the dynamic strength continuously changed from 20 to 200 kPa, and the largest crater size formed on small icy satellites was considered by using the revised scaling parameter. Ice and snow projectiles were impacted on a snow surface with 36% porosity at an impact velocity from 31 m s1 to 150 m s1. The snow target was sintered at the temperature from 5 °C to 18 °C, and the snow dynamic strength was changed with the sintering duration at each temperature. We found that the mass ejected from the crater normalized by the projectile mass, pV, was , where related to the ratio of the dynamic strength to the impact pressure, pY , as follows: pV ¼ 0:01p1:2 Y the impact pressure was indicated by P = qtC0tvi/2 with the target density of qt, when the impact velocity, vi, was much smaller than the bulk sound velocity C0t (typically 1.8 km s1 in our targets). The ratio of the largest crater diameter to the diameter of the target body, dmax/D, was estimated by calculating the crater diameter at the impact condition for catastrophic disruption and then compared to the observed dmax/D of jovian and saturnian small satellites, in order to discuss the formation condition of these large dmax/D in the strength regime. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Cratering experiments on snow by high-velocity impacts are important for studying the geology of small icy bodies such as comets and icy satellites, and for examining their formation processes, because recent planetary explorations for small icy bodies revealed that they had low mean densities. For example, the Cassini spacecraft found that small icy satellites smaller than 100 km in diameter in the saturnian system had a density less than that of water ice (Thomas, 2010). These icy satellites could be composed of mixtures of silicates, ices, and organic materials, with their low mean density indicating that they have a high level of porosity in their interiors and are not compressed by stress causing large plastic deformation that would decrease the porosity (Yasui and Arakawa, 2009, 2010). The Cassini spacecraft also clarified that small icy satellites had large impact craters compared to their sizes, which were about 1/3 of the size for Janus, Epimeteus, and Hyperion (Thomas, 1999; Leliwa-Kopystynski et al., 2008). The origin of these large craters is quite an interesting problem related to impact physics and planetary geology (Burchell and LeliwaKopystynski, 2010).
⇑ Corresponding author. Fax: +81 78 803 6684. E-mail address: [email protected] (M. Arakawa). 0019-1035/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2011.08.018
There have been few reports of cratering experiments on snow. One such study was by Koschny et al. (2001), who conducted cratering experiments on snow with 50% porosity at an impact velocity from 0.9 km s1 to 3.3 km s1. They used a nylon projectile with mechanical strength much larger than that of snow and found that the crater shape was affected by the penetration of the projectile: the depth of snow craters was deeper than that of ice craters (Kato et al., 1995). However, their conclusions suggested that the cratering efficiency on snow was almost the same as that on ice. Although the number of previous works about cratering experiments on snow is quite limited, there are many reports of cratering experiments on ice (Croft et al., 1979; Kawakami et al., 1983; Lange and Ahrens, 1987; Kato et al., 1995; Shrine et al., 2002; Grey and Burchell, 2003). All of these authors concluded that the cratering efficiency on ice was much larger than that on rock such as basalt. This difference could be caused by the large contrast of the material properties between ice and rock, especially with regard to the mechanical strength and the shock impedance. The laboratory data of cratering efficiency described by using a crater diameter and a volume on impact cratering have been used to construct a crater scaling law in the strength regime (e.g., Kawakami et al., 1983; Mizutani et al., 1983; Holsapple, 1993). The formation processes of small craters on a laboratory scale and those of craters formed in a very low gravity condition could be mainly dominated by the material strength: the final crater size would be controlled by the target material strength. The impact condition applicable
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to this formation mechanism is called the strength regime. The craters made on comet nuclei and small icy satellites would be excavated in the strength regime. However, whether the crater is formed in the strength regime or in the gravity regime ultimately depends on the mechanical strength of the constituent materials. The mechanical strength of snow strongly depends on its porosity, and thus the dominant mechanism for crater formation on snow could be controlled by the porosity of icy bodies. In the study of glaciology, the compressive and tensile strengths of snow were found to depend on the snow density and quality. Furthermore, Shimaki et al. (2011) found that the snow strength drastically changed with the degree of sintering, even at the same porosity. Moreover, comet nuclei and small icy satellites are composed of ices and silicates, and it is possible that the silicates affect the mechanical strength of these mixtures, depending on the content (Arakawa et al., 2002; Hiraoka et al., 2007, 2008). Arakawa and Tomizuka (2004) and Yasui and Arakawa (2010) experimentally found that the strength of the mixture with large porosity decreased with the increase of silicate content. Therefore, we suspect there are many physical factors that can change the mechanical strength of snow on icy bodies, so that a scaling law applicable not only to the gravity regime but also to the strength regime is necessary to determine the dominant mechanism to control the crater formation on small icy bodies. A scaling law in the strength regime is especially important for deriving information about the material properties constituting small icy bodies from the craters on their surfaces. One problem is that the effects of porosity and silicate content included in ice–silicate mixtures on the crater formation are too complex to be incorporated into a single crater scaling law, because they affect several elementary processes such as shock wave generation and propagation, and crack nucleation and growth. Therefore, a simple system for conducting cratering experiments is needed to confirm or improve the proposed crater scaling law. We performed impact cratering experiments by using a snow target with the same porosity each time while changing the degree of sintering: the mechanical strength was only changed at the constant porosity to investigate the strength effect on the cratering efficiency systematically. In this study, we focused on the crater formation process at impact velocities lower than the sound velocity of sintered snow, because small icy bodies such as small icy satellites and icy planetesimals would have low escape velocities below 200 m s1 and would collide with each other at the low velocity during their evolution process. The cratering mechanism is well known to depend on impact velocity, and thus we separately discuss the impact phenomena at the impact velocities below and beyond the sound velocity of the target material. At high impact velocity above the sound velocity of the material, there is a superior scaling law for the cratering process based on the ‘‘point-source’’ assumption by Holsapple and colleagues. This law would not be expected to work for low velocity impact, so we attempted to develop a new scaling law for low-velocity impact. We initially examined the proposed scaling law for crater formation by Holsapple (1993) to determine whether it was suitable for porous targets. Then, we focused on the largest crater size in the strength regime formed on a small icy body while taking into consideration the impact condition necessary for the catastrophic disruption of icy bodies.
2. Experimental methods 2.1. Target preparation and impact cratering experiments The snow target was prepared by using ice particles with a diameter of about 500 lm in a large cold room below 0 °C. These
ice particles were made from commercial ice blocks, and the blocks were ground in a blender. The produced ice powder was sieved to sort the particles with a diameter of less than 500 lm. The sorted ice particles were put into a metal cylindrical container with a diameter of 13.5 cm and a height of 10 cm. The target porosity was about 36%. The snow target was kept in a cold room at a temperature from 5 °C to 18 °C for sintering ice particles. The sintering duration was changed from 3 min to 60 h to prepare snow targets with different mechanical strengths at a constant porosity. We used a projectile made of ice and snow with the porosity of about 30%, and the projectile shape was a cylinder with the diameter of 7 mm and a mass of 0.21 (± 0.1) g. The projectile was launched from a He-gas gun at the impact velocity from 31 m s1 to 150 m s1. The gun was set in a horizontal direction, as shown in Fig. 1, so that the impacted surface of the target was set parallel to the Earth’s gravity, which is the normal to the impact direction. All impact experiments were conducted in a large cold room of the Institute of Low Temperature Science, Hokkaido University, and the temperature was from 5 °C to 18 °C. Before and after the shot, we measured the target mass to calculate the excavated mass from the impact crater. The crater volume was then calculated based on this excavated mass and the mean target density. 2.2. Observation of ejecta curtain made on a sintered snow surface The launched projectile before the impact and the ejecta in the cratering process were observed by a high-speed video camera (Memrecam fx-K3, NAC, Inc.) to determine the impact velocity and to observe the ejecta shape for each shot. Fig. 2a shows a snapshot of the crater formation on snow taken by the high-speed video, and for comparison a snapshot of the crater formation on glass beads is shown in Fig. 2b. The shape of the ejecta curtain for the snow target is clearly different from that for the glass beads target. The shape of the ejecta envelop for glass beads is very smooth, with a U shape, and the ejecta envelope makes the form continuously from the target surface distant from the impact point without any distinct gap. On the other hand, the ejecta envelope for snow shows a clear gap between the bottom of the ejecta curtain and the initial target surface. This gap is formed by the discontinuous motion of ice particles constituting the target surface: the ice particles fluidized by shock stress could be ejected to form the envelope, but the ice particles were never fluidized in the region where the shock stress was below the snow strength, so no ejecta was observed at the region on the snow surface. Therefore, this gap is very good evidence in the case of sintered snow that the crater was formed in the strength regime. 2.3. Measurement of dynamic snow strength Sintered snow is so fragile that it is difficult to measure its mechanical strength by using a conventional method such as a tensile test. Thus, to measure the mechanical strength of sintered snow, we performed an impact test by dropping a brass weight with cylindrical shape (a penetrator; diameter 19.0 mm, length 38.2 mm, mass 92.5 g) on the snow surface. Acceleration of the penetrator by the impact on the snow surface was measured by a small accelerator (SV-1111, NEC San-ei Co., Ltd.) embedded in the penetrator, and the dynamic compressive strength of sintered snow was evaluated based on the acceleration of the penetrator according to the following method. The penetrator was dropped from 10 cm above the target surface, and it impacted the target surface in a normal fashion. The accelerator attached to the penetrator measured the acceleration of the penetrator as it impacted the snow surface at a velocity of 1.3 m s1. The acquired data of acceleration was used to calculate the force loaded on the penetrator, and the calculated force was further used to estimate the
M. Arakawa, M. Yasui / Icarus 216 (2011) 1–9
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Fig. 1. A schematic illustration of the experimental setup. All apparatuses were set in a large cold room at temperatures from 5 °C to 18 °C. Ice and snow projectiles were launched horizontally, and a snow target was sintered in a metal container.
a b
Fig. 2. A snapshot of the crater formation process taken by a high-speed digital video camera. (a) An ice projectile was impacted at 41 m s1 on the snow target sintered for 15 min at 10 °C. The image was taken at 25 ms after the impact. (b) A nylon projectile with a size of 10 mm was impacted at 28.5 m s1 on 100 lm glass beads target. The image was taken at 4 ms after the impact.
impact stress applied on the front area of the penetrator, 2.8 cm2. The recorded impact stress profiles for snow targets with various sintering duration are shown in Fig. 3. The impact stress profiles show the rapid increase of stress at the impact point, and after that, they decrease gradually or remain constant for a while. The maximum value of each stress profile was used to estimate the dynamic strength of sintered snow, which corresponds to the degree of sintering; we define the maximum stress as the snow dynamic strength in this study. Fig. 4 shows the relationship between snow dynamic strength and sintering duration. We found that the dynamic strength (rmax) increased with the sintering duration according to the following power law relationship:
rmax ¼ 100:760:14 t0:280:03 ; s
ð1Þ
250 Sintering duration, s 180 900 3600 7800 16380 51000 216000
Stress, kPa
200
Fig. 4. Relationship between maximum stress measured by a penetrator and sintering duration of snow targets. The maximum stress was determined from each stress profile in Fig. 3. The target was sintered at 10 °C.
150
where rmax is in the unit of kPa, and ts is the sintering duration in seconds.
100
50
3. Experimental results 3.1. Crater morphology
0 -2
0
2
4
6x10
-3
Time, s Fig. 3. Stress profiles calculated from acquired acceleration data of a penetrator impacting on the target surface sintered for different durations at the same temperature of 10 °C.
The experimental results of the impact cratering on sintered snow are summarized in Table 1. In the temperature range of our experiments, ice particles sintered so quickly that a sintering duration of 3 min was sufficient to gain mechanical strength larger than the stress induced by the gravity force in the target. Fig. 5 shows
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Table 1 Experimental conditions and results. Run No.
T (°C)
Ice projectile 04-8-27-7 04-8-27-8 04-8-27-9 04-8-27-10 04-8-27-11 04-8-27-12 04-8-27-14 04-8-25-1 04-8-25-2 04-8-25-3 04-8-25-4 04-8-25-5 04-8-25-7 04-8-23-1 04-8-23-2 04-8-23-3 04-8-23-5 04-8-24-2 04-8-31-2
5 5 5 5 5 5 5 10 10 10 10 10 10 18 18 18 18 18 10
Snow projectile 04-8-27-1 04-8-27-2 04-8-27-3 04-8-27-4 04-8-27-5 04-8-27-6 04-8-25-8 04-8-25-9 04-8-26-1 04-8-26-2 04-8-26-3 04-8-26-4 04-8-26-5 04-8-30-1 04-8-30-2 04-8-30-3 04-8-30-4 04-8-30-5 04-8-31-1 04-8-31-3
5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10
vi (m s1)
Mcr (g)
Dcr (mm)
tcr (mm)
P0 (MPa)
rmax (kPa)
pV
pY
pY
15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 850
117.6 137.6 145.2 30.8 65.1 77.8 98.5 36.8 67.2 80.1 107.3 118.9 137.0 66.7 55.3 31.0 97.7 27.1 126.9
9.14 8.84 10.35 2.19 5.08 6.99 7.57 4.69 8.03 8.11 13.6 13.93 22.79 9.77 7.74 4.69 12.34 3.35 3.31
52.8 47.1 54.7 26.9 35.2 40.0 40.3 38.0 41.5 43.4 55.8 59.4 74.5 51.4 40.7 34.8 56.4 30.6 33.5
24.5 20.3 31.6 14.1 20.7 23.4 22.3 19.2 20.8 24.7 25.1 24.7 27.0 23.4 24.3 18.8 23.8 16.6 15.0
93.8 110.1 112.6 24.2 51.3 61.2 77.2 28.5 52.0 62.5 82.8 92.5 106.2 50.0 42.4 23.6 75.7 20.9 101.2
51.2 51.2 51.2 51.2 51.2 51.2 51.2 37.6 37.6 37.6 37.6 37.6 37.6 22.9 22.9 22.9 22.9 22.9 114.5
43.5 44.2 51.7 10.0 24.2 35.0 36.0 22.3 36.5 36.9 64.8 69.7 114.0 39.1 38.7 21.3 58.8 16.0 16.6
6.10 103 4.43 103 4.15 103 9.08 102 2.02 102 1.42 102 8.89 103 4.75 102 1.45 102 9.94 103 5.63 103 4.53 103 3.43 103 9.21 103 1.30 102 4.17 102 4.11 103 5.36 102 1.17 102
5.46 104 4.65 104 4.54 104 2.12 103 9.98 104 8.35 104 6.62 104 1.32 103 7.22 104 6.01 104 4.54 104 4.06 104 3.54 104 4.58 104 5.40 104 9.69 104 3.03 104 1.09 103 1.13 103
15 15 15 15 15 15 15 15 15 15 15 15 15 3630 15 60 130 273 850 3
65.2 80.1 102.3 118.6 138.2 31.2 65.8 76.4 35.4 85.1 104.5 121.3 149.6 104.3 102.5 100.0 100.8 97.3 102.3 96.9
4.27 4.96 5.93 6.23 7.3 1.39 5.35 5.07 3.79 7.58 11.71 11.48 12.2 1.68 10.63 7.67 4.33 3.73 3.14 24.16
32.1 32.9 40.6 42.4 44.3 23.8 38.6 35.7 33.2 44.5 57.6 54.3 54.3 26.5 47.9 49.3 32.9 29.3 28.7 73.7
14.3 16.0 12.2 14.4 13.2 12.5 15.0 12.5 16.2 15.3 14.8 20.6 14.6 7.3 19.8 14.8 14.1 11.8 12.5 19.7
36.0 43.0 55.4 64.1 73.9 17.2 35.9 41.4 18.5 45.9 56.8 65.7 81.8 57.6 56.6 54.7 55.5 53.8 56.6 52.0
51.2 51.2 51.2 51.2 51.2 51.2 37.6 37.6 37.6 37.6 37.6 37.6 37.6 171.0 37.6 55.1 68.2 83.7 114.5 24.1
22.5 24.8 28.2 31.1 36.5 6.9 26.7 25.4 18.0 36.1 61.6 57.4 64.2 9.3 53.1 40.4 22.8 18.6 16.5 127.2
1.99 102 1.35 102 8.21 103 6.12 103 4.56 103 8.66 102 1.45 102 1.08 102 5.21 102 8.76 103 5.77 103 4.29 103 2.80 103 2.59 102 5.89 103 9.17 103 1.11 102 1.45 102 1.80 102 4.35 103
1.42 103 1.19 103 9.23 104 7.98 104 6.92 104 2.97 103 1.05 103 9.07 104 2.03 103 8.19 104 6.62 104 5.72 104 4.60 104 2.97 103 6.64 104 1.01 103 1.23 103 1.56 103 2.02 103 4.64 104
ts (min)
T: sintering temperature, ts: sintering duration, vi: impact velocity, Mcr: ejected mass from the crater, Dcr: crater diameter, tcr: crater depth, P0: initial shock pressure, rmax: dynamic strength of the target, pV: non-dimensional crater volume defined by Eq. (6), pY: scaling parameter for crater formation in strength regime defined by Eq. (7), pY : modified scaling parameter for crater formation in strength regime defined by Eq. (12).
the typical crater morphologies obtained for snow targets with various degrees of sintering: the snow projectile was impacted at 100 m s1 and the temperature was 10 °C. It is very clear that the crater size changes with the sintering duration. For a 60-h-sintered target, the crater is small and shallow. The crater size increases with the decrease of the sintering duration, and we see quite a large crater with a spall-like outer boundary on a 3-minsintered target. In this experimental set up in which the projectile is launched horizontally, ejecta and disrupted snow generated during cratering process cannot remain inside the cratered cavity: the ice particles without bounding each other flew out from the cratered cavity along the gravity force. Therefore, the crater volume should correspond to the region disrupted by the impact if we ignore the impact compaction of snow. The morphological features of a recovered crater are often characterized by the depth-to-diameter ratio. Fig. 6 shows the relationship between the depth-to-diameter ratio (tcr/Dcr) and impact velocities (vi). The results for ice and sand targets obtained by Kato et al. (1995) and Mizutani et al. (1983) are shown for reference. The tcr/Dcr = 0.5 means a hemispherical crater shape, and the experimental results at low-impact velocities are near this value. For ice projectiles, the ratio showing the hemispherical shape, 0.5, is maintained until the velocity of about
80 m s1, and then it decreases. However, for snow projectiles the ratio decreases with the increase of the impact velocity monotonically from 30 m s1 to 150 m s1. We can recover almost intact ice projectiles after the impacts on the crater surfaces for all shots. In the shots for snow projectiles impacted at vi > 100 m s1, we can only recover disrupted fragments or observe the relic of the projectile at the center of the crater surface, and we suspect that the snow projectile might be partly ablated during the penetration into the target even at the impact velocity below 100 m s1. Therefore, the disruption and ablation of a snow projectile might have caused the apparent difference in the ratio between ice and snow projectiles. Koschny et al. (2001) studied the impact crater morphology made on snow by using a nylon projectile at the impact velocity from 1 to 4 km s1, and found that tcr/Dcr decreased from 1 to 0.3. Although the impact velocity and the projectile material were very different in between ours and Koschny’s, it was confirmed that the negative trend on the impact velocity and the observed range of tcr/Dcr were almost the same between them. But the critical impact velocity corresponding to the drastic change of tcr/Dout is different in them: they are 2 km s1 for nylon projectile in Koschny et al. (2001) and 80 m s1 for snow projectile. This difference might be caused by the projectile strength.
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Fig. 5. Photograph of an impact crater formed on a snow target. The snow target was impacted by a snow projectile at the same velocity of 100 m s1. The targets were sintered for different durations at 10 °C before the impact. Sintering durations were (a) 3630 min, (b) 130 min, (c) 15 min, and (d) 3 min.
Fig. 6. Depth (tcr) to diameter (Dcr) ratio of impact crater formed on snow targets. The closed symbols represent the data for ice projectiles impacted on targets sintered at three different temperatures, and the open symbols represent the data for snow projectiles impacted on targets sintered at two different temperatures. The sintering duration of targets was 15 min. The crater morphology was different from others for the data in parentheses: it had a deep pit at the center of the crater, and this pit made a large difference in the depth to diameter ratio.
3.2. Crater size and ejecta mass Figs. 7 and 8 show the relationship between crater diameter (Dcr) or ejecta mass (Mcr) and projectile kinetic energy (Ek). These relationships are fitted by the following equations: n
Dcr ¼ 10ad Ekd ;
ð2Þ
M cr ¼ 10av Enkv ;
ð3Þ
where Dcr is in mm, Mcr is in g, and Ek is in J. The constants ad and av and the power law indexes nd and nv for each target are shown in Table 2. Dcr is proportional to E0:23 in Fig. 7 and Mcr is proportional k to E0:51 in Fig. 8, irrespective of the target sintering duration or the k projectile material. The power law index, nd, for an ice target was
Fig. 7. Relationship between crater diameter and projectile kinetic energy for snow targets sintered for 15 min at different temperatures. The open symbols and the closed symbols show the diameters obtained for a snow projectile and an ice projectile, respectively.
Fig. 8. Ejecta mass of the impact crater formed on snow targets sintered for 15 min at different temperatures. The relationship between the ejecta mass and the projectile kinetic energy were fitted for each data set; the open symbols and the closed symbols show the ejecta mass obtained for a snow projectile and an ice projectile, respectively. The cross symbols show the ejecta mass measured for the targets sintered at 10 °C for different durations from 3 to 3630 min.
Table 2 Fitted parameters, ad, nd, av, and nv, in Eqs. (2) and (3). ad
nd
av
nv
1.64 ± 0.01 1.74 ± 0.02 1.75 ± 0.01
0.23 ± 0.03 0.24 ± 0.06 0.23 ± 0.02
0.87 ± 0.02 1.09 ± 0.04 1.11 ± 0.03
0.49 ± 0.05 0.56 ± 0.09 0.49 ± 0.04
Snow projectile T = 5 °C 1.59 ± 0.01 T = 10 °C 1.68 ± 0.02
0.21 ± 0.02 0.21 ± 0.05
0.75 ± 0.03 0.95 ± 0.03
0.54 ± 0.06 0.48 ± 0.09
Ice projectile T = 5 °C T = 10 °C T = 18 °C
previously derived by Kato et al. (1995) for ice projectile, and it was 0.34 at Ek < 50 J and 0.68 at Ek > 50 J. Our result for sintered snow target is rather lower than that for ice target. This relatively small value of nd for sintered snow is similar to that obtained for non-cohesive targets such as sand and glass beads (Mizutani
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et al., 1983) although the crater size in the case of sintered snow was surely controlled by the strength. Moreover, the difference of nv between ice and snow targets is even larger: nv is 0.91 for ice at Ek < 50 J and it is twice as large as that of sintered snow. These empirical results of Eqs. (2) and (3) are obtained for the fixed projectile mass and the fixed material properties, which could be valid for the velocity dependence but might have projectile mass dependence, which was not examined in this study: These relationships are used only for the projectile mass of 0.21 g for snow and an ice projectile. All shots were conducted for the snow targets sintered for 15 min in Fig. 7. Thus, it is clear that the crater volume at the same Ek depends on the sintering temperature, so that the crater volume was larger as the sintering temperature was lower in the same projectile group. In Fig. 8, the ejecta mass for the snow targets sintered at different sintering durations from 3 to 3630 min at 10 °C and Ek = 1 J are shown for the comparison by a cross symbol, and it is impressive that they scatter more than one order of magnitude. This means that the ejecta mass is quite sensitive to the sintering duration of the target. Furthermore, the crater formed by an ice projectile is systematically larger than that by a snow projectile, as shown in Figs. 7 and 8. Fig. 9 shows the ejecta mass derived from the targets sintered at different durations. All targets were sintered at 10 °C, and the impact velocity was constant, 100 m s1. The ejecta mass decreases with the increase of the sintering duration and these results show a clear difference of ejecta mass between a snow projectile and an ice projectile: more ejecta was generated by the ice projectile. The empirical relationship between Mcr and ts for the snow projectile can be obtained as follows:
M cr ¼ 102:10:1 t 0:360:02 ; s
ð4Þ
where ts is the sintering duration in seconds. Since the target strength was related to the sintering duration as shown in Fig. 4 and obtained by the penetrator, we can expect that the target dynamic strength (rmax) for each target can be substituted for the ts of Eq. (4). The relationship between Mcr and rmax was examined in Fig. 10 for the data sets of snow and ice projectile groups. The empirical equation is derived for the snow projectile as follows: 1:30:1 M cr ¼ 103:10:1 rmax :
ð5Þ
This empirical equation shows the effect of dynamic strength on the crater formation, and we discovered that the ejecta mass was al-
Fig. 10. Relationship between ejecta mass and dynamic strength. The projectiles were impacted at 100 m s1 on the snow targets sintered at 10 °C for different durations. The difference of projectiles is shown by open and closed symbols.
most inversely proportional to the target dynamic strength of sintered snow targets. 4. Discussions 4.1. Scaling law A scaling law for an impact crater is necessary to estimate the size of the impact crater formed by real-scale impact on planetary bodies. Holsapple (1993) proposed a superior scaling law for an impact crater that includes several parameters, impact conditions, and material properties. Theoretically, this scaling law should be applied only for high-velocity impact, because it includes a point-source assumption. Nonetheless, we attempted to use the scaling law to analyze our low-velocity data and examine whether it can be extrapolated to a low-velocity impact. According to the theory, the non-dimensional crater volume, pV, is defined as follows:
pV ¼
qt V cr mp
;
ð6Þ
where qt is the target density, Vcr is the crater volume, and mp is the projectile mass. The crater volume after the impact is usually controlled by the target strength for a cohesive target, in contrast to the gravity for a non-cohesive target such as sand. We call this an impact in the strength regime. Thus, the following non-dimensional scaling parameter, pY, is proposed to control the crater formation mechanism in the strength regime:
pY ¼
Y
qt v 2i
;
ð7Þ
where Y is the target strength, and vi is the impact velocity of the projectile. Many studies have been conducted for various kinds of cohesive targets, and the results have been used to improve the proposed theoretical relationship between pV and pY. The relationship is expressed as follows:
pV ¼ cpdY ; Fig. 9. The ejecta mass of the impact crater formed on targets sintered for various durations. The data for snow and ice projectiles impacted at 100 m s1 are shown by different symbols.
ð8Þ
and there are several pairs of parameters, c and d, for different cohesive targets (Holsapple, 1993). However, the data set is quite limited, because it is difficult to obtain cohesive targets with different strengths suitable for the cratering experiments. In our
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experiments, we systematically changed the Y of snow targets more than one order of magnitude by sintering, and thus our data set is quite useful for checking the proposed crater scaling law in the strength regime. We used rmax instead of Y to calculate pY and obtained qtVcr from Mcr. Fig. 11 shows the relationship between pV and pY for all of our results. There are five data sets in the figure, each of which uses the same projectile and the target sintered at the same duration of 15 min but at different temperatures. These five data sets were obtained by changing the impact velocity, so we call the collection of sets a velocity data set. The data of this velocity data set are almost merged with each other, and they are fitted by one empirical equation, irrespective of sintering temperatures or projectile materials. The parameters c and d of Eq. (8) are determined to be 100.34±0.12 and 0.60 ± 0.06, respectively. However, the data set for the targets sintered for 3–3630 min at the constant temperature of 10 °C and at the constant impact velocity of 100 m s1 shown by in the figure, which we call a strength data set, shows a different trend from that of the velocity data set: it systematically deviates from the fitted equation of the velocity data set. The parameters c and d obtained for the strength data set are 101.21±0.25 and 1.36 ± 0.13, respectively, so we notice that the slope of the strength data set is twice as large as that of the velocity data set. This inconsistency between the velocity data set and the strength data set in Fig. 11 means that the scaling parameters, pY, used in this relationship should be improved to optimize the parameter for sintered porous materials such as snow. The physical meaning of pY is interpreted to be the ratio of the material strength to the shock pressure induced in the target. The approximation of the shock pressure by the form of qt m2i of Eq. (7) could be almost correct for the high-velocity impacts: the impact velocity is enough higher than the bulk sound velocity of the target, and it could be a good approximation for non-cohesive targets such as sand and glass beads. The longitudinal wave velocity (Vp) and the shear wave velocity (Vs) of the sintered snow were measured by using a pulse transmitted method, and the bulk sound velocity was calculated to be 1830 m s1 (Shimaki, 2010). This velocity is much higher than the impact velocities of our experiments (31–150 m s1), and thus the approximation of the impact pressure written by qt m2i is not valid for the sintered snow target. According to a Hugoinot equation of state, the impact pressure induced in the solids is expressed by the following equation:
Fig. 11. Relationship between pV and pY for all data obtained in our experiments. These scaling parameters were calculated by using Eqs. (6) and (7). The number shown next to the cross symbol is the sintering duration of the target used for each experiment.
P ¼ qt ðC 0 þ sup Þup ;
ð9Þ
where s is an empirical parameter determined by laboratory experiments, C0 is the bulk sound velocity of a target, and up is the particle velocity of a target. Here up equals to vi/2 when matching materials collided with each other at the velocity of vi. Therefore, when the impact velocity is rather lower than C0, the impact pressure is approximated as follows:
P ¼ qt C 0 up ;
ð10Þ
and when the materials of the projectile and target are different, the impact pressure at the low-velocity impact is calculated as follows:
P0 ¼
qt C 0t v i ; 1 þ qt C 0t =qp C 0p
ð11Þ
where the suffixes p and t mean the projectile and the target, respectively. We were not sure if that the linear relationship of Eq. (10) for shock pressure estimation was correct even for porous materials, so the shock pressure induced in snow should be measured in the future to test this assumption. We used Eq. (11) instead of qt m2i in the scaling parameter of Eq. (7) and newly defined the pY as follows:
pY ¼
Y : P0
ð12Þ
Fig. 12 shows the relationship between pV and pY for all of our data sets, including both the strength and the velocity data sets. Both data sets are well mixed with each other, and all of the data can be fitted by only one empirical equation, irrespective of the data set difference. The obtained equation is as follows:
pV ¼ 102:00:2 p1:20:1 : Y
ð13Þ
Therefore, we propose that our modified scaling parameter, pY , might be useful to describe the crater volume of sintered snow rather than the traditional pY, and this modified scaling parameter would be effective only at an impact velocity much lower than C0t. 4.2. Implications for large crater formation on small icy satellites The largest crater formed on small bodies has been studied based on the scaling law for the crater formation and the catastrophic disruption; in these calculations, the largest crater size was estimated by calculating the impact condition corresponding
Fig. 12. Revised relationship between pV and pY . pY was an improved version of scaling parameter pY and was calculated by using Eq. (12). The number shown next to the cross symbol is the sintering duration of the target used for each experiment.
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to the beginning of the catastrophic disruption. These results were then compared with the observations of planetary explorations for asteroids and icy satellites. Housen and Holsapple (1990) discussed the ratio of the largest crater size to the target body size (Rct) and estimated it to be Rct = 0.32 in the gravity regime. A more detailed analysis was made in Holsapple (1994) to study the Rct larger than 0.3, which was widely observed for the impact basins of satellites and planets. Leliwa-Kopystynski et al. (2008) also studied the Rct and found that the Rct of rocky bodies was systematically different from that of icy bodies. According to the observations made during planetary explorations, the Rct of rocky satellites and icy satellites are 0.8 and 0.6, respectively. In this study, we estimate the Rct for the craters formed on the surface of porous small icy bodies and then compared it to those on small jovian and saturnian satellites. In previous studies (Housen and Holsapple, 1990; Leliwa-Kopystynski et al., 2008), the Rct was calculated by using the impact strength Q⁄ and the crater scaling law. The Rct was estimated from the crater size at the beginning of the catastrophic disruption for each target. The scaling parameter PI determining the catastrophic disruption was proposed by Mizutani et al. (1990) and recently applied to porous gypsum targets, confirming its usability. According to Yasui and Arakawa (2011), the following scaling parameter is suitable for porous gypsum:
! P0 mp qt ; PI ¼ Y t Mt qp
ð14Þ
where Yt is tensile strength. We use this formula for our sintered snow target. The catastrophic disruption would occur for the condition of PI = 1, so that we can rewrite the impact condition for catam strophic disruption as P0 Mpt ¼ Yk for a like-material collision, where we assume that rmax can be used for Y in this formula and k is the ratio of dynamic strength Y to tensile strength Yt with the value between 1 and 2 (Shimaki et al., 2011). We use Eq. (13) to estimate the crater size formed on the small icy bodies at the impact condition corresponding to PI = 1, and we assume that this crater size is the largest formed on the target bodies because the targets could be shattered into small fragments to erase any crater morphologies beyond the impact condition of PI = 1 in the strength regime. The diameter of the largest crater (dmax) normalized by the target diameter (D) at the impact condition of PI = 1 depends on the impact velocity (vi), and the relationship is written by the following equation when we assume likematerial collisions at low-impact velocity:
" f þ1 #1=3 dmax e 2Y ; ¼ 2 k qt C 0t v i D
Fig. 13. Diameter of the largest impact crater formed on small icy bodies at different impact velocities. The relationship between dmax/D and impact velocity was calculated by using Eq. (15) assuming that the dynamic strengths of the icy bodies were 2 kPa, 20 kPa, and 200 kPa.
Finally, we estimate the largest crater size formed on sintered snow bodies with the diameter from 1 to 100 km in order to compare the results with the observations of the largest craters on jovian and saturnian satellites. The impact strength is well known to depend on the target size and decrease with the increase of the target size, but in the case of porous material like snow the dependence is still unclear, so we assume that the strength Y is constant, irrespective of the size of the icy bodies. Furthermore, we assume that the impact velocity to generate the largest crater is the same as the escape velocity of each target body, which means that we examine the minimum impact velocity to form the largest crater on each target. The impact velocity on icy satellites is usually very high because the central planet gravitationally attracts and accelerates small bodies coming from the outside of the jovian and the saturnian system; the colliding velocities of these small bodies on icy satellites are more than several km s1, and our scaling law for low-velocity impact cannot be applied to such a high velocity. Therefore, we limit our discussion on the low-velocity impact among small bodies around the central planets. According to these assumptions, we can derive the following equation to show the relationship between the normalized largest crater size and the snow body sizes:
ð15Þ
where the parameters e and f are 0.01 and 1.2 from Eq. (13), respectively, and we choose k = 2 for the calculation. Fig. 13 shows the calculated results of Eq. (15) for the selected Y values, 2 kPa, 20 kPa, and 200 kPa. We assume sintered snow bodies composed of water ice with the porosity of 40%, although the actual icy bodies are composed of ice and silicates and have various porosities. We also assume that the icy body has a spherical shape and the crater shape is simplified to be a hemisphere. At the constant Y, the largest crater size hardly depends on the impact velocity in the range from 10 m s1 to 1000 m s1: the value of dmax/D is between 0.3 and 0.4 for Y = 20 kPa. This ratio also hardly depends on the dynamic strength of the target body, that is, the ratio is between 0.3 and 0.5 when the strength is about one order of magnitude higher or lower. We suggest from this result that the effects of impact velocity and target strength of bodies colliding on the small icy bodies are very limited.
" f 1 #1=3 1f 2 dmax e 8 2Y pGqt : ¼ 2 k 3 D qt C 0t D=2
ð16Þ
Fig. 14 shows the calculated results of Eq. (16) for the representative dynamic strength of 2 kPa, 20 kPa, and 200 kPa. The observation results for small icy satellites Janus, Epimeteus, Hyperion, Mimas, Amalthea, and Thebe are plotted in the same figure, and we found that saturnian satellites Janus, Epimeteus and Mimas were almost on the lines calculated for 2 kPa, 20 kPa, and 200 kPa, while Hyperion, Amalthea, and Thebe were above these lines and distant from them. One interpretation of this result is that the largest craters found on small saturnian satellites were formed in the strength regime impact by a projectile body with a velocity near their escape velocities, and that their surfaces were made of materials similar to sintered snow with a dynamic strength between 2 kPa and 200 kPa. However, further studies to
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References
Fig. 14. Diameter of the largest impact craters formed on the icy bodies with various diameters. The relationship between dmax/D and the target diameter was calculated by using Eq. (16), assuming that the dynamic strengths of the icy bodies were 2 kPa, 20 kPa, and 200 kPa. The observed data of dmax/D for jovian and saturnian small icy satellites are shown on the figure for reference.
confirm this conclusion are necessary, because small jovian satellites and middle-sized saturnian satellites cannot be explained in the same scenario, so they might have different target properties such as porosity, dynamic strength, and composition which are not considered in this study, or their largest craters might have been formed in the gravity regime or by high-velocity impacts. Therefore, further studies on the impact cratering for sintered snow targets with different porosities and having the mixture of other materials such as silicate dusts are necessary to clarify the impact condition forming these largest craters on icy satellites. Acknowledgments We thank two anonymous referees on their helpful comments to revise this manuscript and also thank Mr. S. Nakatsubo of the Contribution Division of the Institute of Low Temperature Science, Hokkaido University, for his technical help. This work was supported in part by Grants-in Aid for Scientific Research (17340127 and 20340118) from the Japan Ministry of Education, Culture, Sports, Science and Technology, and a grant for a Joint Research Program from the Institute of Low Temperature Science, Hokkaido University.
Arakawa, M., Tomizuka, D., 2004. Ice–silicate fractionation among icy bodies due to the difference of impact strength between ice and ice–silicate mixture. Icarus 170, 193–201. Arakawa, M., Leliwa-Kopystynski, J., Maeno, N., 2002. Impact experiments on porous icy-silicate cylindrical blocks and the implication for disruption and accumulation of small icy bodies. Icarus 158, 516–531. Burchell, M.J., Leliwa-Kopystynski, J., 2010. The large crater on the small Asteroid (2867) Steins. Icarus 210, 707–712. Croft, S.K., Kieffer, S.W., Ahrens, T.J., 1979. Low-velocity impact craters in ice and ice-saturated sand with implications for martian crater count ages. J. Geophys. Res. 84, 8023–8032. Grey, I.D.S., Burchell, M.J., 2003. Hypervelocity impact cratering on water ice targets at temperatures ranging from 100 K to 253 K. J. Geophys. Res. 108, 5019. doi:10.1029/2002JE001899. Hiraoka, K., Arakawa, M., Yoshikawa, K., Nakamura, A.M., 2007. Laboratory experiments of crater formation on ice–silicate mixture targets. Adv. Space Res. 39, 392–399. Hiraoka, K., Arakawa, M., Setoh, M., Nakamura, A.M., 2008. Measurements of target compressive and tensile strength for application to impact cratering on ice– silicate mixtures. J. Geophys. Res. 113, E02013. doi:10.1029/2007JE002926. Holsapple, K.A., 1993. The scaling of impact processes in planetary sciences. Annu. Rev. Earth Planet. Sci. 21, 333–373. Holsapple, K.A., 1994. Catastrophic disruptions and cratering of Solar System bodies: A review and new results. Planet. Space Sci. 42, 1067–1078. Housen, K.R., Holsapple, K.A., 1990. On the fragmentation of asteroids and planetary satellites. Icarus 84, 226–253. Kato, M., Iijima, Y., Arakawa, M., Okimura, Y., Fujimura, A., Maeno, N., Mizutani, H., 1995. Ice-on-ice impact experiments. Icarus 113, 423–441. Kawakami, S., Mizutani, H., Takagi, Y., Kato, M., Kumazawa, M., 1983. Impact experiments on ice. J. Geophys. Res. 88, 5806–5814. Koschny, D., Kargl, G., Rott, M., 2001. Experimental studies of the cratering process in porous ice targets. Adv. Space Res. 28, 1533–1537. Lange, M.A., Ahrens, T.J., 1987. Impact experiments in low-temperature ice. Icarus 69, 506–518. Leliwa-Kopystynski, J., Burchell, M.J., Lowen, D., 2008. Impact cratering and break up of the small bodies of the Solar System. Icarus 195, 817–826. Mizutani, H., Kawakami, S., Takagi, Y., Kato, M., Kumazawa, M., 1983. Cratering experiments in sands and a trial for general scaling law. J. Geophys. Res. 88, A835–A845. Mizutani, H., Takagi, Y., Kawakami, S., 1990. New scaling laws on impact fragmentation. Icarus 87, 307–326. Shimaki, Y., 2010. Experimental Study on the Collisional Accretion Process of Icy Planetesimals in Thermal Evolution. M.S. Thesis, Nagoya University, Japan. Shimaki, Y., Arakawa, M., Yasui, M., 2011. Impact experiments on sintered snowballs. In: Proc. 12th Int. Conf. on Physics and Chemistry of Ice, pp. 379– 386. Shrine, N.R.G., Burchell, M.J., Grey, I.D.S., 2002. Velocity scaling of impact craters in water ice over the range 1–7.3 km s1. Icarus 155, 475–485. Thomas, P.C., 1999. Large craters on small objects: Occurrence, morphology, and effects. Icarus 142, 89–96. Thomas, P.C., 2010. Sizes, shapes, and derived properties of the saturnian satellites after the Cassini nominal mission. Icarus 208, 395–401. Yasui, M., Arakawa, M., 2009. Compaction experiments on ice–silica particle mixtures: Implication for residual porosity of small icy bodies. J. Geophys. Res. 114, E09004. doi:10.1029/2009JE003374. Yasui, M., Arakawa, M., 2010. Rate-dependent strength of porous ice–silica mixtures and its implications for the shape of small to middle-sized icy satellites. Icarus 210, 956–967. Yasui, M., Arakawa, M., 2011. Impact experiments of porous gypsum–glass bead mixtures simulating parent bodies of ordinary chondrites: Implications for reaccumulation processes related to rubble-pile formation. Icarus 214, 754–765.
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Icarus journal homepage: www.elsevier.com/locate/icarus
Impact crater formed on sintered snow surface simulating porous icy bodies Masahiko Arakawa a,⇑, Minami Yasui b a b
Graduate School of Science, Kobe University, 1-1, Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan Organization of Advanced Science and Technology, Kobe University, 1-1, Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
a r t i c l e
i n f o
Article history: Received 2 May 2011 Revised 13 August 2011 Accepted 18 August 2011 Available online 26 August 2011 Keywords: Impact processes Ices, Mechanical properties Cratering
a b s t r a c t To improve the scaling parameter controlling the impact crater formation in the strength regime, we conducted impact experiments on sintered snow targets with the dynamic strength continuously changed from 20 to 200 kPa, and the largest crater size formed on small icy satellites was considered by using the revised scaling parameter. Ice and snow projectiles were impacted on a snow surface with 36% porosity at an impact velocity from 31 m s1 to 150 m s1. The snow target was sintered at the temperature from 5 °C to 18 °C, and the snow dynamic strength was changed with the sintering duration at each temperature. We found that the mass ejected from the crater normalized by the projectile mass, pV, was , where related to the ratio of the dynamic strength to the impact pressure, pY , as follows: pV ¼ 0:01p1:2 Y the impact pressure was indicated by P = qtC0tvi/2 with the target density of qt, when the impact velocity, vi, was much smaller than the bulk sound velocity C0t (typically 1.8 km s1 in our targets). The ratio of the largest crater diameter to the diameter of the target body, dmax/D, was estimated by calculating the crater diameter at the impact condition for catastrophic disruption and then compared to the observed dmax/D of jovian and saturnian small satellites, in order to discuss the formation condition of these large dmax/D in the strength regime. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Cratering experiments on snow by high-velocity impacts are important for studying the geology of small icy bodies such as comets and icy satellites, and for examining their formation processes, because recent planetary explorations for small icy bodies revealed that they had low mean densities. For example, the Cassini spacecraft found that small icy satellites smaller than 100 km in diameter in the saturnian system had a density less than that of water ice (Thomas, 2010). These icy satellites could be composed of mixtures of silicates, ices, and organic materials, with their low mean density indicating that they have a high level of porosity in their interiors and are not compressed by stress causing large plastic deformation that would decrease the porosity (Yasui and Arakawa, 2009, 2010). The Cassini spacecraft also clarified that small icy satellites had large impact craters compared to their sizes, which were about 1/3 of the size for Janus, Epimeteus, and Hyperion (Thomas, 1999; Leliwa-Kopystynski et al., 2008). The origin of these large craters is quite an interesting problem related to impact physics and planetary geology (Burchell and LeliwaKopystynski, 2010).
⇑ Corresponding author. Fax: +81 78 803 6684. E-mail address: [email protected] (M. Arakawa). 0019-1035/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2011.08.018
There have been few reports of cratering experiments on snow. One such study was by Koschny et al. (2001), who conducted cratering experiments on snow with 50% porosity at an impact velocity from 0.9 km s1 to 3.3 km s1. They used a nylon projectile with mechanical strength much larger than that of snow and found that the crater shape was affected by the penetration of the projectile: the depth of snow craters was deeper than that of ice craters (Kato et al., 1995). However, their conclusions suggested that the cratering efficiency on snow was almost the same as that on ice. Although the number of previous works about cratering experiments on snow is quite limited, there are many reports of cratering experiments on ice (Croft et al., 1979; Kawakami et al., 1983; Lange and Ahrens, 1987; Kato et al., 1995; Shrine et al., 2002; Grey and Burchell, 2003). All of these authors concluded that the cratering efficiency on ice was much larger than that on rock such as basalt. This difference could be caused by the large contrast of the material properties between ice and rock, especially with regard to the mechanical strength and the shock impedance. The laboratory data of cratering efficiency described by using a crater diameter and a volume on impact cratering have been used to construct a crater scaling law in the strength regime (e.g., Kawakami et al., 1983; Mizutani et al., 1983; Holsapple, 1993). The formation processes of small craters on a laboratory scale and those of craters formed in a very low gravity condition could be mainly dominated by the material strength: the final crater size would be controlled by the target material strength. The impact condition applicable
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to this formation mechanism is called the strength regime. The craters made on comet nuclei and small icy satellites would be excavated in the strength regime. However, whether the crater is formed in the strength regime or in the gravity regime ultimately depends on the mechanical strength of the constituent materials. The mechanical strength of snow strongly depends on its porosity, and thus the dominant mechanism for crater formation on snow could be controlled by the porosity of icy bodies. In the study of glaciology, the compressive and tensile strengths of snow were found to depend on the snow density and quality. Furthermore, Shimaki et al. (2011) found that the snow strength drastically changed with the degree of sintering, even at the same porosity. Moreover, comet nuclei and small icy satellites are composed of ices and silicates, and it is possible that the silicates affect the mechanical strength of these mixtures, depending on the content (Arakawa et al., 2002; Hiraoka et al., 2007, 2008). Arakawa and Tomizuka (2004) and Yasui and Arakawa (2010) experimentally found that the strength of the mixture with large porosity decreased with the increase of silicate content. Therefore, we suspect there are many physical factors that can change the mechanical strength of snow on icy bodies, so that a scaling law applicable not only to the gravity regime but also to the strength regime is necessary to determine the dominant mechanism to control the crater formation on small icy bodies. A scaling law in the strength regime is especially important for deriving information about the material properties constituting small icy bodies from the craters on their surfaces. One problem is that the effects of porosity and silicate content included in ice–silicate mixtures on the crater formation are too complex to be incorporated into a single crater scaling law, because they affect several elementary processes such as shock wave generation and propagation, and crack nucleation and growth. Therefore, a simple system for conducting cratering experiments is needed to confirm or improve the proposed crater scaling law. We performed impact cratering experiments by using a snow target with the same porosity each time while changing the degree of sintering: the mechanical strength was only changed at the constant porosity to investigate the strength effect on the cratering efficiency systematically. In this study, we focused on the crater formation process at impact velocities lower than the sound velocity of sintered snow, because small icy bodies such as small icy satellites and icy planetesimals would have low escape velocities below 200 m s1 and would collide with each other at the low velocity during their evolution process. The cratering mechanism is well known to depend on impact velocity, and thus we separately discuss the impact phenomena at the impact velocities below and beyond the sound velocity of the target material. At high impact velocity above the sound velocity of the material, there is a superior scaling law for the cratering process based on the ‘‘point-source’’ assumption by Holsapple and colleagues. This law would not be expected to work for low velocity impact, so we attempted to develop a new scaling law for low-velocity impact. We initially examined the proposed scaling law for crater formation by Holsapple (1993) to determine whether it was suitable for porous targets. Then, we focused on the largest crater size in the strength regime formed on a small icy body while taking into consideration the impact condition necessary for the catastrophic disruption of icy bodies.
2. Experimental methods 2.1. Target preparation and impact cratering experiments The snow target was prepared by using ice particles with a diameter of about 500 lm in a large cold room below 0 °C. These
ice particles were made from commercial ice blocks, and the blocks were ground in a blender. The produced ice powder was sieved to sort the particles with a diameter of less than 500 lm. The sorted ice particles were put into a metal cylindrical container with a diameter of 13.5 cm and a height of 10 cm. The target porosity was about 36%. The snow target was kept in a cold room at a temperature from 5 °C to 18 °C for sintering ice particles. The sintering duration was changed from 3 min to 60 h to prepare snow targets with different mechanical strengths at a constant porosity. We used a projectile made of ice and snow with the porosity of about 30%, and the projectile shape was a cylinder with the diameter of 7 mm and a mass of 0.21 (± 0.1) g. The projectile was launched from a He-gas gun at the impact velocity from 31 m s1 to 150 m s1. The gun was set in a horizontal direction, as shown in Fig. 1, so that the impacted surface of the target was set parallel to the Earth’s gravity, which is the normal to the impact direction. All impact experiments were conducted in a large cold room of the Institute of Low Temperature Science, Hokkaido University, and the temperature was from 5 °C to 18 °C. Before and after the shot, we measured the target mass to calculate the excavated mass from the impact crater. The crater volume was then calculated based on this excavated mass and the mean target density. 2.2. Observation of ejecta curtain made on a sintered snow surface The launched projectile before the impact and the ejecta in the cratering process were observed by a high-speed video camera (Memrecam fx-K3, NAC, Inc.) to determine the impact velocity and to observe the ejecta shape for each shot. Fig. 2a shows a snapshot of the crater formation on snow taken by the high-speed video, and for comparison a snapshot of the crater formation on glass beads is shown in Fig. 2b. The shape of the ejecta curtain for the snow target is clearly different from that for the glass beads target. The shape of the ejecta envelop for glass beads is very smooth, with a U shape, and the ejecta envelope makes the form continuously from the target surface distant from the impact point without any distinct gap. On the other hand, the ejecta envelope for snow shows a clear gap between the bottom of the ejecta curtain and the initial target surface. This gap is formed by the discontinuous motion of ice particles constituting the target surface: the ice particles fluidized by shock stress could be ejected to form the envelope, but the ice particles were never fluidized in the region where the shock stress was below the snow strength, so no ejecta was observed at the region on the snow surface. Therefore, this gap is very good evidence in the case of sintered snow that the crater was formed in the strength regime. 2.3. Measurement of dynamic snow strength Sintered snow is so fragile that it is difficult to measure its mechanical strength by using a conventional method such as a tensile test. Thus, to measure the mechanical strength of sintered snow, we performed an impact test by dropping a brass weight with cylindrical shape (a penetrator; diameter 19.0 mm, length 38.2 mm, mass 92.5 g) on the snow surface. Acceleration of the penetrator by the impact on the snow surface was measured by a small accelerator (SV-1111, NEC San-ei Co., Ltd.) embedded in the penetrator, and the dynamic compressive strength of sintered snow was evaluated based on the acceleration of the penetrator according to the following method. The penetrator was dropped from 10 cm above the target surface, and it impacted the target surface in a normal fashion. The accelerator attached to the penetrator measured the acceleration of the penetrator as it impacted the snow surface at a velocity of 1.3 m s1. The acquired data of acceleration was used to calculate the force loaded on the penetrator, and the calculated force was further used to estimate the
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Fig. 1. A schematic illustration of the experimental setup. All apparatuses were set in a large cold room at temperatures from 5 °C to 18 °C. Ice and snow projectiles were launched horizontally, and a snow target was sintered in a metal container.
a b
Fig. 2. A snapshot of the crater formation process taken by a high-speed digital video camera. (a) An ice projectile was impacted at 41 m s1 on the snow target sintered for 15 min at 10 °C. The image was taken at 25 ms after the impact. (b) A nylon projectile with a size of 10 mm was impacted at 28.5 m s1 on 100 lm glass beads target. The image was taken at 4 ms after the impact.
impact stress applied on the front area of the penetrator, 2.8 cm2. The recorded impact stress profiles for snow targets with various sintering duration are shown in Fig. 3. The impact stress profiles show the rapid increase of stress at the impact point, and after that, they decrease gradually or remain constant for a while. The maximum value of each stress profile was used to estimate the dynamic strength of sintered snow, which corresponds to the degree of sintering; we define the maximum stress as the snow dynamic strength in this study. Fig. 4 shows the relationship between snow dynamic strength and sintering duration. We found that the dynamic strength (rmax) increased with the sintering duration according to the following power law relationship:
rmax ¼ 100:760:14 t0:280:03 ; s
ð1Þ
250 Sintering duration, s 180 900 3600 7800 16380 51000 216000
Stress, kPa
200
Fig. 4. Relationship between maximum stress measured by a penetrator and sintering duration of snow targets. The maximum stress was determined from each stress profile in Fig. 3. The target was sintered at 10 °C.
150
where rmax is in the unit of kPa, and ts is the sintering duration in seconds.
100
50
3. Experimental results 3.1. Crater morphology
0 -2
0
2
4
6x10
-3
Time, s Fig. 3. Stress profiles calculated from acquired acceleration data of a penetrator impacting on the target surface sintered for different durations at the same temperature of 10 °C.
The experimental results of the impact cratering on sintered snow are summarized in Table 1. In the temperature range of our experiments, ice particles sintered so quickly that a sintering duration of 3 min was sufficient to gain mechanical strength larger than the stress induced by the gravity force in the target. Fig. 5 shows
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Table 1 Experimental conditions and results. Run No.
T (°C)
Ice projectile 04-8-27-7 04-8-27-8 04-8-27-9 04-8-27-10 04-8-27-11 04-8-27-12 04-8-27-14 04-8-25-1 04-8-25-2 04-8-25-3 04-8-25-4 04-8-25-5 04-8-25-7 04-8-23-1 04-8-23-2 04-8-23-3 04-8-23-5 04-8-24-2 04-8-31-2
5 5 5 5 5 5 5 10 10 10 10 10 10 18 18 18 18 18 10
Snow projectile 04-8-27-1 04-8-27-2 04-8-27-3 04-8-27-4 04-8-27-5 04-8-27-6 04-8-25-8 04-8-25-9 04-8-26-1 04-8-26-2 04-8-26-3 04-8-26-4 04-8-26-5 04-8-30-1 04-8-30-2 04-8-30-3 04-8-30-4 04-8-30-5 04-8-31-1 04-8-31-3
5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10
vi (m s1)
Mcr (g)
Dcr (mm)
tcr (mm)
P0 (MPa)
rmax (kPa)
pV
pY
pY
15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 850
117.6 137.6 145.2 30.8 65.1 77.8 98.5 36.8 67.2 80.1 107.3 118.9 137.0 66.7 55.3 31.0 97.7 27.1 126.9
9.14 8.84 10.35 2.19 5.08 6.99 7.57 4.69 8.03 8.11 13.6 13.93 22.79 9.77 7.74 4.69 12.34 3.35 3.31
52.8 47.1 54.7 26.9 35.2 40.0 40.3 38.0 41.5 43.4 55.8 59.4 74.5 51.4 40.7 34.8 56.4 30.6 33.5
24.5 20.3 31.6 14.1 20.7 23.4 22.3 19.2 20.8 24.7 25.1 24.7 27.0 23.4 24.3 18.8 23.8 16.6 15.0
93.8 110.1 112.6 24.2 51.3 61.2 77.2 28.5 52.0 62.5 82.8 92.5 106.2 50.0 42.4 23.6 75.7 20.9 101.2
51.2 51.2 51.2 51.2 51.2 51.2 51.2 37.6 37.6 37.6 37.6 37.6 37.6 22.9 22.9 22.9 22.9 22.9 114.5
43.5 44.2 51.7 10.0 24.2 35.0 36.0 22.3 36.5 36.9 64.8 69.7 114.0 39.1 38.7 21.3 58.8 16.0 16.6
6.10 103 4.43 103 4.15 103 9.08 102 2.02 102 1.42 102 8.89 103 4.75 102 1.45 102 9.94 103 5.63 103 4.53 103 3.43 103 9.21 103 1.30 102 4.17 102 4.11 103 5.36 102 1.17 102
5.46 104 4.65 104 4.54 104 2.12 103 9.98 104 8.35 104 6.62 104 1.32 103 7.22 104 6.01 104 4.54 104 4.06 104 3.54 104 4.58 104 5.40 104 9.69 104 3.03 104 1.09 103 1.13 103
15 15 15 15 15 15 15 15 15 15 15 15 15 3630 15 60 130 273 850 3
65.2 80.1 102.3 118.6 138.2 31.2 65.8 76.4 35.4 85.1 104.5 121.3 149.6 104.3 102.5 100.0 100.8 97.3 102.3 96.9
4.27 4.96 5.93 6.23 7.3 1.39 5.35 5.07 3.79 7.58 11.71 11.48 12.2 1.68 10.63 7.67 4.33 3.73 3.14 24.16
32.1 32.9 40.6 42.4 44.3 23.8 38.6 35.7 33.2 44.5 57.6 54.3 54.3 26.5 47.9 49.3 32.9 29.3 28.7 73.7
14.3 16.0 12.2 14.4 13.2 12.5 15.0 12.5 16.2 15.3 14.8 20.6 14.6 7.3 19.8 14.8 14.1 11.8 12.5 19.7
36.0 43.0 55.4 64.1 73.9 17.2 35.9 41.4 18.5 45.9 56.8 65.7 81.8 57.6 56.6 54.7 55.5 53.8 56.6 52.0
51.2 51.2 51.2 51.2 51.2 51.2 37.6 37.6 37.6 37.6 37.6 37.6 37.6 171.0 37.6 55.1 68.2 83.7 114.5 24.1
22.5 24.8 28.2 31.1 36.5 6.9 26.7 25.4 18.0 36.1 61.6 57.4 64.2 9.3 53.1 40.4 22.8 18.6 16.5 127.2
1.99 102 1.35 102 8.21 103 6.12 103 4.56 103 8.66 102 1.45 102 1.08 102 5.21 102 8.76 103 5.77 103 4.29 103 2.80 103 2.59 102 5.89 103 9.17 103 1.11 102 1.45 102 1.80 102 4.35 103
1.42 103 1.19 103 9.23 104 7.98 104 6.92 104 2.97 103 1.05 103 9.07 104 2.03 103 8.19 104 6.62 104 5.72 104 4.60 104 2.97 103 6.64 104 1.01 103 1.23 103 1.56 103 2.02 103 4.64 104
ts (min)
T: sintering temperature, ts: sintering duration, vi: impact velocity, Mcr: ejected mass from the crater, Dcr: crater diameter, tcr: crater depth, P0: initial shock pressure, rmax: dynamic strength of the target, pV: non-dimensional crater volume defined by Eq. (6), pY: scaling parameter for crater formation in strength regime defined by Eq. (7), pY : modified scaling parameter for crater formation in strength regime defined by Eq. (12).
the typical crater morphologies obtained for snow targets with various degrees of sintering: the snow projectile was impacted at 100 m s1 and the temperature was 10 °C. It is very clear that the crater size changes with the sintering duration. For a 60-h-sintered target, the crater is small and shallow. The crater size increases with the decrease of the sintering duration, and we see quite a large crater with a spall-like outer boundary on a 3-minsintered target. In this experimental set up in which the projectile is launched horizontally, ejecta and disrupted snow generated during cratering process cannot remain inside the cratered cavity: the ice particles without bounding each other flew out from the cratered cavity along the gravity force. Therefore, the crater volume should correspond to the region disrupted by the impact if we ignore the impact compaction of snow. The morphological features of a recovered crater are often characterized by the depth-to-diameter ratio. Fig. 6 shows the relationship between the depth-to-diameter ratio (tcr/Dcr) and impact velocities (vi). The results for ice and sand targets obtained by Kato et al. (1995) and Mizutani et al. (1983) are shown for reference. The tcr/Dcr = 0.5 means a hemispherical crater shape, and the experimental results at low-impact velocities are near this value. For ice projectiles, the ratio showing the hemispherical shape, 0.5, is maintained until the velocity of about
80 m s1, and then it decreases. However, for snow projectiles the ratio decreases with the increase of the impact velocity monotonically from 30 m s1 to 150 m s1. We can recover almost intact ice projectiles after the impacts on the crater surfaces for all shots. In the shots for snow projectiles impacted at vi > 100 m s1, we can only recover disrupted fragments or observe the relic of the projectile at the center of the crater surface, and we suspect that the snow projectile might be partly ablated during the penetration into the target even at the impact velocity below 100 m s1. Therefore, the disruption and ablation of a snow projectile might have caused the apparent difference in the ratio between ice and snow projectiles. Koschny et al. (2001) studied the impact crater morphology made on snow by using a nylon projectile at the impact velocity from 1 to 4 km s1, and found that tcr/Dcr decreased from 1 to 0.3. Although the impact velocity and the projectile material were very different in between ours and Koschny’s, it was confirmed that the negative trend on the impact velocity and the observed range of tcr/Dcr were almost the same between them. But the critical impact velocity corresponding to the drastic change of tcr/Dout is different in them: they are 2 km s1 for nylon projectile in Koschny et al. (2001) and 80 m s1 for snow projectile. This difference might be caused by the projectile strength.
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Fig. 5. Photograph of an impact crater formed on a snow target. The snow target was impacted by a snow projectile at the same velocity of 100 m s1. The targets were sintered for different durations at 10 °C before the impact. Sintering durations were (a) 3630 min, (b) 130 min, (c) 15 min, and (d) 3 min.
Fig. 6. Depth (tcr) to diameter (Dcr) ratio of impact crater formed on snow targets. The closed symbols represent the data for ice projectiles impacted on targets sintered at three different temperatures, and the open symbols represent the data for snow projectiles impacted on targets sintered at two different temperatures. The sintering duration of targets was 15 min. The crater morphology was different from others for the data in parentheses: it had a deep pit at the center of the crater, and this pit made a large difference in the depth to diameter ratio.
3.2. Crater size and ejecta mass Figs. 7 and 8 show the relationship between crater diameter (Dcr) or ejecta mass (Mcr) and projectile kinetic energy (Ek). These relationships are fitted by the following equations: n
Dcr ¼ 10ad Ekd ;
ð2Þ
M cr ¼ 10av Enkv ;
ð3Þ
where Dcr is in mm, Mcr is in g, and Ek is in J. The constants ad and av and the power law indexes nd and nv for each target are shown in Table 2. Dcr is proportional to E0:23 in Fig. 7 and Mcr is proportional k to E0:51 in Fig. 8, irrespective of the target sintering duration or the k projectile material. The power law index, nd, for an ice target was
Fig. 7. Relationship between crater diameter and projectile kinetic energy for snow targets sintered for 15 min at different temperatures. The open symbols and the closed symbols show the diameters obtained for a snow projectile and an ice projectile, respectively.
Fig. 8. Ejecta mass of the impact crater formed on snow targets sintered for 15 min at different temperatures. The relationship between the ejecta mass and the projectile kinetic energy were fitted for each data set; the open symbols and the closed symbols show the ejecta mass obtained for a snow projectile and an ice projectile, respectively. The cross symbols show the ejecta mass measured for the targets sintered at 10 °C for different durations from 3 to 3630 min.
Table 2 Fitted parameters, ad, nd, av, and nv, in Eqs. (2) and (3). ad
nd
av
nv
1.64 ± 0.01 1.74 ± 0.02 1.75 ± 0.01
0.23 ± 0.03 0.24 ± 0.06 0.23 ± 0.02
0.87 ± 0.02 1.09 ± 0.04 1.11 ± 0.03
0.49 ± 0.05 0.56 ± 0.09 0.49 ± 0.04
Snow projectile T = 5 °C 1.59 ± 0.01 T = 10 °C 1.68 ± 0.02
0.21 ± 0.02 0.21 ± 0.05
0.75 ± 0.03 0.95 ± 0.03
0.54 ± 0.06 0.48 ± 0.09
Ice projectile T = 5 °C T = 10 °C T = 18 °C
previously derived by Kato et al. (1995) for ice projectile, and it was 0.34 at Ek < 50 J and 0.68 at Ek > 50 J. Our result for sintered snow target is rather lower than that for ice target. This relatively small value of nd for sintered snow is similar to that obtained for non-cohesive targets such as sand and glass beads (Mizutani
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et al., 1983) although the crater size in the case of sintered snow was surely controlled by the strength. Moreover, the difference of nv between ice and snow targets is even larger: nv is 0.91 for ice at Ek < 50 J and it is twice as large as that of sintered snow. These empirical results of Eqs. (2) and (3) are obtained for the fixed projectile mass and the fixed material properties, which could be valid for the velocity dependence but might have projectile mass dependence, which was not examined in this study: These relationships are used only for the projectile mass of 0.21 g for snow and an ice projectile. All shots were conducted for the snow targets sintered for 15 min in Fig. 7. Thus, it is clear that the crater volume at the same Ek depends on the sintering temperature, so that the crater volume was larger as the sintering temperature was lower in the same projectile group. In Fig. 8, the ejecta mass for the snow targets sintered at different sintering durations from 3 to 3630 min at 10 °C and Ek = 1 J are shown for the comparison by a cross symbol, and it is impressive that they scatter more than one order of magnitude. This means that the ejecta mass is quite sensitive to the sintering duration of the target. Furthermore, the crater formed by an ice projectile is systematically larger than that by a snow projectile, as shown in Figs. 7 and 8. Fig. 9 shows the ejecta mass derived from the targets sintered at different durations. All targets were sintered at 10 °C, and the impact velocity was constant, 100 m s1. The ejecta mass decreases with the increase of the sintering duration and these results show a clear difference of ejecta mass between a snow projectile and an ice projectile: more ejecta was generated by the ice projectile. The empirical relationship between Mcr and ts for the snow projectile can be obtained as follows:
M cr ¼ 102:10:1 t 0:360:02 ; s
ð4Þ
where ts is the sintering duration in seconds. Since the target strength was related to the sintering duration as shown in Fig. 4 and obtained by the penetrator, we can expect that the target dynamic strength (rmax) for each target can be substituted for the ts of Eq. (4). The relationship between Mcr and rmax was examined in Fig. 10 for the data sets of snow and ice projectile groups. The empirical equation is derived for the snow projectile as follows: 1:30:1 M cr ¼ 103:10:1 rmax :
ð5Þ
This empirical equation shows the effect of dynamic strength on the crater formation, and we discovered that the ejecta mass was al-
Fig. 10. Relationship between ejecta mass and dynamic strength. The projectiles were impacted at 100 m s1 on the snow targets sintered at 10 °C for different durations. The difference of projectiles is shown by open and closed symbols.
most inversely proportional to the target dynamic strength of sintered snow targets. 4. Discussions 4.1. Scaling law A scaling law for an impact crater is necessary to estimate the size of the impact crater formed by real-scale impact on planetary bodies. Holsapple (1993) proposed a superior scaling law for an impact crater that includes several parameters, impact conditions, and material properties. Theoretically, this scaling law should be applied only for high-velocity impact, because it includes a point-source assumption. Nonetheless, we attempted to use the scaling law to analyze our low-velocity data and examine whether it can be extrapolated to a low-velocity impact. According to the theory, the non-dimensional crater volume, pV, is defined as follows:
pV ¼
qt V cr mp
;
ð6Þ
where qt is the target density, Vcr is the crater volume, and mp is the projectile mass. The crater volume after the impact is usually controlled by the target strength for a cohesive target, in contrast to the gravity for a non-cohesive target such as sand. We call this an impact in the strength regime. Thus, the following non-dimensional scaling parameter, pY, is proposed to control the crater formation mechanism in the strength regime:
pY ¼
Y
qt v 2i
;
ð7Þ
where Y is the target strength, and vi is the impact velocity of the projectile. Many studies have been conducted for various kinds of cohesive targets, and the results have been used to improve the proposed theoretical relationship between pV and pY. The relationship is expressed as follows:
pV ¼ cpdY ; Fig. 9. The ejecta mass of the impact crater formed on targets sintered for various durations. The data for snow and ice projectiles impacted at 100 m s1 are shown by different symbols.
ð8Þ
and there are several pairs of parameters, c and d, for different cohesive targets (Holsapple, 1993). However, the data set is quite limited, because it is difficult to obtain cohesive targets with different strengths suitable for the cratering experiments. In our
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experiments, we systematically changed the Y of snow targets more than one order of magnitude by sintering, and thus our data set is quite useful for checking the proposed crater scaling law in the strength regime. We used rmax instead of Y to calculate pY and obtained qtVcr from Mcr. Fig. 11 shows the relationship between pV and pY for all of our results. There are five data sets in the figure, each of which uses the same projectile and the target sintered at the same duration of 15 min but at different temperatures. These five data sets were obtained by changing the impact velocity, so we call the collection of sets a velocity data set. The data of this velocity data set are almost merged with each other, and they are fitted by one empirical equation, irrespective of sintering temperatures or projectile materials. The parameters c and d of Eq. (8) are determined to be 100.34±0.12 and 0.60 ± 0.06, respectively. However, the data set for the targets sintered for 3–3630 min at the constant temperature of 10 °C and at the constant impact velocity of 100 m s1 shown by in the figure, which we call a strength data set, shows a different trend from that of the velocity data set: it systematically deviates from the fitted equation of the velocity data set. The parameters c and d obtained for the strength data set are 101.21±0.25 and 1.36 ± 0.13, respectively, so we notice that the slope of the strength data set is twice as large as that of the velocity data set. This inconsistency between the velocity data set and the strength data set in Fig. 11 means that the scaling parameters, pY, used in this relationship should be improved to optimize the parameter for sintered porous materials such as snow. The physical meaning of pY is interpreted to be the ratio of the material strength to the shock pressure induced in the target. The approximation of the shock pressure by the form of qt m2i of Eq. (7) could be almost correct for the high-velocity impacts: the impact velocity is enough higher than the bulk sound velocity of the target, and it could be a good approximation for non-cohesive targets such as sand and glass beads. The longitudinal wave velocity (Vp) and the shear wave velocity (Vs) of the sintered snow were measured by using a pulse transmitted method, and the bulk sound velocity was calculated to be 1830 m s1 (Shimaki, 2010). This velocity is much higher than the impact velocities of our experiments (31–150 m s1), and thus the approximation of the impact pressure written by qt m2i is not valid for the sintered snow target. According to a Hugoinot equation of state, the impact pressure induced in the solids is expressed by the following equation:
Fig. 11. Relationship between pV and pY for all data obtained in our experiments. These scaling parameters were calculated by using Eqs. (6) and (7). The number shown next to the cross symbol is the sintering duration of the target used for each experiment.
P ¼ qt ðC 0 þ sup Þup ;
ð9Þ
where s is an empirical parameter determined by laboratory experiments, C0 is the bulk sound velocity of a target, and up is the particle velocity of a target. Here up equals to vi/2 when matching materials collided with each other at the velocity of vi. Therefore, when the impact velocity is rather lower than C0, the impact pressure is approximated as follows:
P ¼ qt C 0 up ;
ð10Þ
and when the materials of the projectile and target are different, the impact pressure at the low-velocity impact is calculated as follows:
P0 ¼
qt C 0t v i ; 1 þ qt C 0t =qp C 0p
ð11Þ
where the suffixes p and t mean the projectile and the target, respectively. We were not sure if that the linear relationship of Eq. (10) for shock pressure estimation was correct even for porous materials, so the shock pressure induced in snow should be measured in the future to test this assumption. We used Eq. (11) instead of qt m2i in the scaling parameter of Eq. (7) and newly defined the pY as follows:
pY ¼
Y : P0
ð12Þ
Fig. 12 shows the relationship between pV and pY for all of our data sets, including both the strength and the velocity data sets. Both data sets are well mixed with each other, and all of the data can be fitted by only one empirical equation, irrespective of the data set difference. The obtained equation is as follows:
pV ¼ 102:00:2 p1:20:1 : Y
ð13Þ
Therefore, we propose that our modified scaling parameter, pY , might be useful to describe the crater volume of sintered snow rather than the traditional pY, and this modified scaling parameter would be effective only at an impact velocity much lower than C0t. 4.2. Implications for large crater formation on small icy satellites The largest crater formed on small bodies has been studied based on the scaling law for the crater formation and the catastrophic disruption; in these calculations, the largest crater size was estimated by calculating the impact condition corresponding
Fig. 12. Revised relationship between pV and pY . pY was an improved version of scaling parameter pY and was calculated by using Eq. (12). The number shown next to the cross symbol is the sintering duration of the target used for each experiment.
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to the beginning of the catastrophic disruption. These results were then compared with the observations of planetary explorations for asteroids and icy satellites. Housen and Holsapple (1990) discussed the ratio of the largest crater size to the target body size (Rct) and estimated it to be Rct = 0.32 in the gravity regime. A more detailed analysis was made in Holsapple (1994) to study the Rct larger than 0.3, which was widely observed for the impact basins of satellites and planets. Leliwa-Kopystynski et al. (2008) also studied the Rct and found that the Rct of rocky bodies was systematically different from that of icy bodies. According to the observations made during planetary explorations, the Rct of rocky satellites and icy satellites are 0.8 and 0.6, respectively. In this study, we estimate the Rct for the craters formed on the surface of porous small icy bodies and then compared it to those on small jovian and saturnian satellites. In previous studies (Housen and Holsapple, 1990; Leliwa-Kopystynski et al., 2008), the Rct was calculated by using the impact strength Q⁄ and the crater scaling law. The Rct was estimated from the crater size at the beginning of the catastrophic disruption for each target. The scaling parameter PI determining the catastrophic disruption was proposed by Mizutani et al. (1990) and recently applied to porous gypsum targets, confirming its usability. According to Yasui and Arakawa (2011), the following scaling parameter is suitable for porous gypsum:
! P0 mp qt ; PI ¼ Y t Mt qp
ð14Þ
where Yt is tensile strength. We use this formula for our sintered snow target. The catastrophic disruption would occur for the condition of PI = 1, so that we can rewrite the impact condition for catam strophic disruption as P0 Mpt ¼ Yk for a like-material collision, where we assume that rmax can be used for Y in this formula and k is the ratio of dynamic strength Y to tensile strength Yt with the value between 1 and 2 (Shimaki et al., 2011). We use Eq. (13) to estimate the crater size formed on the small icy bodies at the impact condition corresponding to PI = 1, and we assume that this crater size is the largest formed on the target bodies because the targets could be shattered into small fragments to erase any crater morphologies beyond the impact condition of PI = 1 in the strength regime. The diameter of the largest crater (dmax) normalized by the target diameter (D) at the impact condition of PI = 1 depends on the impact velocity (vi), and the relationship is written by the following equation when we assume likematerial collisions at low-impact velocity:
" f þ1 #1=3 dmax e 2Y ; ¼ 2 k qt C 0t v i D
Fig. 13. Diameter of the largest impact crater formed on small icy bodies at different impact velocities. The relationship between dmax/D and impact velocity was calculated by using Eq. (15) assuming that the dynamic strengths of the icy bodies were 2 kPa, 20 kPa, and 200 kPa.
Finally, we estimate the largest crater size formed on sintered snow bodies with the diameter from 1 to 100 km in order to compare the results with the observations of the largest craters on jovian and saturnian satellites. The impact strength is well known to depend on the target size and decrease with the increase of the target size, but in the case of porous material like snow the dependence is still unclear, so we assume that the strength Y is constant, irrespective of the size of the icy bodies. Furthermore, we assume that the impact velocity to generate the largest crater is the same as the escape velocity of each target body, which means that we examine the minimum impact velocity to form the largest crater on each target. The impact velocity on icy satellites is usually very high because the central planet gravitationally attracts and accelerates small bodies coming from the outside of the jovian and the saturnian system; the colliding velocities of these small bodies on icy satellites are more than several km s1, and our scaling law for low-velocity impact cannot be applied to such a high velocity. Therefore, we limit our discussion on the low-velocity impact among small bodies around the central planets. According to these assumptions, we can derive the following equation to show the relationship between the normalized largest crater size and the snow body sizes:
ð15Þ
where the parameters e and f are 0.01 and 1.2 from Eq. (13), respectively, and we choose k = 2 for the calculation. Fig. 13 shows the calculated results of Eq. (15) for the selected Y values, 2 kPa, 20 kPa, and 200 kPa. We assume sintered snow bodies composed of water ice with the porosity of 40%, although the actual icy bodies are composed of ice and silicates and have various porosities. We also assume that the icy body has a spherical shape and the crater shape is simplified to be a hemisphere. At the constant Y, the largest crater size hardly depends on the impact velocity in the range from 10 m s1 to 1000 m s1: the value of dmax/D is between 0.3 and 0.4 for Y = 20 kPa. This ratio also hardly depends on the dynamic strength of the target body, that is, the ratio is between 0.3 and 0.5 when the strength is about one order of magnitude higher or lower. We suggest from this result that the effects of impact velocity and target strength of bodies colliding on the small icy bodies are very limited.
" f 1 #1=3 1f 2 dmax e 8 2Y pGqt : ¼ 2 k 3 D qt C 0t D=2
ð16Þ
Fig. 14 shows the calculated results of Eq. (16) for the representative dynamic strength of 2 kPa, 20 kPa, and 200 kPa. The observation results for small icy satellites Janus, Epimeteus, Hyperion, Mimas, Amalthea, and Thebe are plotted in the same figure, and we found that saturnian satellites Janus, Epimeteus and Mimas were almost on the lines calculated for 2 kPa, 20 kPa, and 200 kPa, while Hyperion, Amalthea, and Thebe were above these lines and distant from them. One interpretation of this result is that the largest craters found on small saturnian satellites were formed in the strength regime impact by a projectile body with a velocity near their escape velocities, and that their surfaces were made of materials similar to sintered snow with a dynamic strength between 2 kPa and 200 kPa. However, further studies to
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References
Fig. 14. Diameter of the largest impact craters formed on the icy bodies with various diameters. The relationship between dmax/D and the target diameter was calculated by using Eq. (16), assuming that the dynamic strengths of the icy bodies were 2 kPa, 20 kPa, and 200 kPa. The observed data of dmax/D for jovian and saturnian small icy satellites are shown on the figure for reference.
confirm this conclusion are necessary, because small jovian satellites and middle-sized saturnian satellites cannot be explained in the same scenario, so they might have different target properties such as porosity, dynamic strength, and composition which are not considered in this study, or their largest craters might have been formed in the gravity regime or by high-velocity impacts. Therefore, further studies on the impact cratering for sintered snow targets with different porosities and having the mixture of other materials such as silicate dusts are necessary to clarify the impact condition forming these largest craters on icy satellites. Acknowledgments We thank two anonymous referees on their helpful comments to revise this manuscript and also thank Mr. S. Nakatsubo of the Contribution Division of the Institute of Low Temperature Science, Hokkaido University, for his technical help. This work was supported in part by Grants-in Aid for Scientific Research (17340127 and 20340118) from the Japan Ministry of Education, Culture, Sports, Science and Technology, and a grant for a Joint Research Program from the Institute of Low Temperature Science, Hokkaido University.
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