Distributions of boulders ejected from lunar craters

Icarus 209 (2010) 337–357

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Distributions of boulders ejected from lunar craters Gwendolyn D. Bart a,*, H.J. Melosh b a b

University of Idaho, Department of Physics, Campus Box 440903, Moscow, ID 83844-0903, USA Purdue University, Department of Earth and Atmospheric Sciences, West Lafayette, IN 47907, USA

a r t i c l e

i n f o

Article history: Received 15 June 2009 Revised 24 May 2010 Accepted 25 May 2010 Available online 2 June 2010 Keywords: Moon, Surface Cratering Impact processes

a b s t r a c t We investigate the spatial distributions of boulders ejected from 18 lunar impact craters that are hundreds of meters in diameter. To accomplish this goal, we measured the diameters of 13,955 ejected boulders and the distance of each boulder from the crater center. Using the boulder distances, we calculated ejection velocities for the boulders. We compare these data with previously published data on larger craters and use this information to determine how boulder ejection velocity scales with crater diameter. We also measured regolith depths in the areas surrounding many of the craters, for comparison with the boulder distributions. These results contribute to understanding boulder ejection velocities, to determining whether there is a relationship between the quantity of ejected boulders and lunar regolith depths, and to understanding the distributions of secondary craters in the Solar System. Understanding distributions of blocky ejecta is an important consideration for landing site selection on both the Moon and Mars. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction Boulders are ubiquitous on solid planetary surfaces, as revealed by spacecraft during the last several decades. Ranger photographs first revealed lunar boulders in 1965 (Kuiper, 1965). Both Viking spacecraft photographed martian boulders (Binder et al., 1977). Galileo observed boulders on the Asteroid Ida (Sullivan et al., 1996), and NEAR observed boulders on the Asteroid Eros (Veverka et al., 2000). Most recently, the Huygens lander took images of rounded rocks on Titan; the rocks observed were 10–15 cm in diameter and probably made of water–ice (Tomasko et al., 2005). The Moon’s surface is covered in boulders, and many small, fresh lunar craters are surrounded by boulder fields. The first lunar boulders were observed in 1965 by the Ranger spacecraft. At the time, the lunar surface bearing strength was unknown, and observation of the boulders sitting on the lunar surface indicated that the surface did have some strength. In a Ranger IX photograph, Kuiper (1965) measured the boulders around a 46 m primary crater to have a volume of approximately 0.2 m3. He then used the boulder sizes to calculate a minimum bearing strength of the Moon of 1–2 kg/cm2. These few boulders around this unnamed crater are the first analysis of lunar boulder ejecta. The next opportunity to observe lunar boulders arrived with the Surveyor spacecraft, which landed on the lunar surface. In order to secure a safe landing site, though, the Surveyors landed on plains rather than near bouldery craters. Nevertheless, the pictures revealed surfaces scattered with fragmental debris. The Surveyor I * Corresponding author. E-mail address: [email protected] (G.D. Bart). 0019-1035/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2010.05.023

landing site was scattered with fragmental debris up to one meter in diameter (Surveyor Scientific Evaluation and Analysis Team, 1966). They found that the size–frequency distribution of those fragments could be represented by N = 3  105y1.77, where N is the cumulative number of grains, and y is the diameter of the grains in mm. After Surveyor VII, Shoemaker and Morris (1970) performed similar studies for the fragmental debris at all Surveyor landing sites. They found that the exponent on y varied between 1.8 and 2.6. The fragments themselves and the equation for the size–frequency distribution reflect the distributions of fragments produced by crushing in ball mills and by impact of a rock surface (Surveyor Scientific Evaluation and Analysis Team, 1966). This observation indicates the impact generated nature of the fragments on the lunar surface. Four years later, in 1969, the Apollo 11 lunar module landed near a 180 m diameter bouldery crater (Hess and Calio, 1969), though the distributions of those boulders were not measured. The local surface was comprised of unsorted fragmental debris, ranging in size from 1 m to microscopic. Both rounded and angular rocks were observed, as well as rocks in all degrees of burial. Some rocks had a light colored rind of altered material about 1 mm thick (Hess and Calio, 1969). The rock compositions included various types of basalts and breccias (Aldrin et al., 1969). The small-particle size distribution was measured at the Apollo 12 landing site, and was found to be similar to that at the Surveyor 3 landing site, nearby (Shoemaker et al., 1970). Most of the boulders found on the Moon are ejected in meteorite impacts (Shoemaker, 1965, p. 254; Melosh, 1984). The fate of the ejected boulders depends on their ejection velocity. Large, fast moving ejecta have enough energy to produce a secondary crater

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upon impact, but slower pieces just land on the surface, although they may roll before finding stable positions. Although most lunar boulders are impact ejecta, some are formed in post-excavation crater modification from slumping and gravitational settling of the rock (Melosh, 1989, Chapter 8); these boulders are found inside the craters in which they formed. Boulders can also be destroyed by meteorite impacts, either by the slow erosional effect of micrometeorite impacts or by the catastrophic effect of impact by a projectile of similar size to the boulder. Vickery (1986, 1987) inferred size and ejection velocities of the solid ejecta of large lunar and martian craters from their secondary craters. We extend her work by directly measuring boulder size distributions about small lunar craters with diameters of hundreds of meters. These results contribute to understanding boulder ejection velocities, to determining whether there is a relationship between the quantity of ejected boulders and lunar regolith depths, and to understanding the distributions of secondary craters in the Solar System. Understanding distributions of blocky ejecta is an important consideration for landing site selection on both the Moon and Mars.

Table 2 Numbers of boulders exterior to and inside of each crater. The total number of boulders counted was 13,955.

2. The data

We analyzed boulder fields around 18 lunar craters. These craters were observed in high resolution (1 m) photographs from Lunar Orbiter III and V, and Apollo 17 photographs. We list each crater in Table 1, along with the photograph number, location of the photograph center, crater diameter, terrain type and regolith depth. We were not able to determine the exact location of each crater from the photographic prints, and hence provide the photograph center as an indication of the general area where the crater is located. Table 2 reports the number of boulders observed inside and outside the crater rim. Note that for the crater in photograph III-189H2, due to the great abundance of boulders, we only analyzed boulders from one quarter of the radial area about the crater. Therefore, the total number of boulders about that crater is about 7200. Photographs of the study craters are reproduced in Figs. 1 and 2. The locations of each observed boulder are plotted in Figs. 3 and 4.

We define ‘‘boulder” as an apparently intact rock or rock fragment lying on a planetary surface, regardless of emplacement mechanism. We identify boulders in planetary images as positive relief features; in low resolution images they appear as bright, sun-facing pixels adjacent to dark, shadowed pixels. The quantitative definition of a boulder is a rock fragment with a diameter of >25.6 cm (Dutro et al., 1989); the term boulder suffices for all intact rocks seen in the present work because the images do not resolve features smaller than about a meter.

Table 1 For the photograph numbers, III and V indicate Lunar Orbiter III and V photographs; Ap17 indicates an Apollo 17 panoramic camera photograph, with letters designating individual craters in the photograph. For each crater studied, the table gives the photograph number, the location of the photograph center, the crater diameter, terrain type, and regolith depth at that location. Low Sun angle did not permit analysis of regolith depth in the Apollo photograph. Crater

Longitude

Latitude

Diameter (km)

Terrain

Regolith (m)

III-185-H3 III-168-H2 III-186-H3 III-189-H2 V-63-H2 V-82-M V-152-H2

43.6°W 43.6°W 43.6°W 43.6°W 32.8°E 18.1°E 20.2°W

2.0°S 2.0°S 2.0°S 2.0°S 0.4°S 2.7°N 10.1°N

0.290 0.678 0.229 0.537 4.006 27.42 0.506

6.6 4.3 4.3 6.4 10.1 9.1 10.2

V-153-H2

20.2°W

10.1°N

0.881

V-167-H2 V-167-H3 V-199-M V-211-H3 Ap17-Pan2345(a) Ap17-Pan2345(b) Ap17-Pan2345(e) Ap17-Pan2345(f) Ap17-Pan2345(g) Ap17-Pan2345(i)

30.9°W 30.9°W 47.4°W 56.1°W 17.9°E

12.9°N 12.9°N 23.2°N 13.7°N 20.1°N

0.452 0.690 41.2 0.520 1.059

Mare Mare Mare Mare Highlands Highlands Ejecta blanket Ejecta blanket Highlands Highlands Mare Mare Mare

17.9°E

20.1°N

1.299

Mare

n/a

17.9°E

20.1°N

1.415

Mare

n/a

17.9°E

20.1°N

0.730

Mare

n/a

17.9°E

20.1°N

0.674

Mare

n/a

17.9°E

20.1°N

0.591

Mare

n/a

10.2 12.2 12.2 n/a n/a n/a

Crater III-185-H3 III-168-H2 III-186-H3 III-189-H2 V-63-H2 V-82-M V-152-H2 V-153-H2 V-167-H2 V-167-H3 V-199-M V-211-H3 Ap17-Pan-2345(a) Ap17-Pan-2345(b) Ap17-Pan-2345(e) Ap17-Pan-2345(f) Ap17-Pan-2345(g) Ap17-Pan-2345(i) Total

Number external boulders

Number internal boulders

1495 1996 1241 1801 527 315 1218 657 326 346 265 323 272 177 328 370 380 130

52 102 31 11 290 101 244 225 75 115 86 61 113 49 133 48 44 8

12,167

1788

3. Methods To perform our analysis, we measured boulder distances, calculated ejection velocities, and found regolith depths. The following methods are similar to methods used in our other papers (Bart and Melosh, 2007, 2010). 3.1. Distance measurement To analyze the photographic data, we digitally scanned the photographic prints. Distances on the images were determined by the widths of the photographic strips as published by Boeing Company (1969). We analyzed the photographs with the computer program ImageJ (Rasband, 1997–2009). ImageJ has the capability to measure distances; a small modification in the code allows us to record locations in the photograph. Boulders were identified as bright pixel(s) on the sunward side of dark pixel(s) (shadows). Each boulder’s diameter was measured perpendicular to the Sun direction to eliminate uncertainty of the extent of the boulder into its shadow. The center of the diameter measurement was taken to be the location of the boulder. We then found the crater’s center by fitting several 20–30 sided polygons to the rim of the crater, finding the center of those polygons with ImageJ analysis tools, and taking the average of the results. This average polygon technique reduces random measurement errors, but does not eliminate possible systematic error based on identification of rim location. Finally, for each

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Fig. 1. Image (crater diameter) – (A) III-185-H3 (0.290 km). (B) III-168-H2 (0.678 km). (C) III-186-H3 (0.229 km). (D) III-189-H2 (0.537 km). (E) V-63-H2 (4.006 km). (F) V-82M (27.422 km). (G) V-152-H2 (0.506 km). (H) V-153-H2 (0.881 km). (I) V-167-H2 (0.452 km).

boulder we calculated the distance from the center of the crater to the center of the boulder.

3.2. Ejection velocity calculation We calculate the ejection velocity for each boulder based on the transient crater radius, R, assumed to be 0.85 of the measured final crater radius (Melosh, 1989), and the distance of the boulder from the crater center, D. (Fig. 5 illustrates the parameters involved in the ejection velocity calculation, including R and D.) For the small craters in this study, D is not R so we cannot assume that all boulders were ejected from the same radial location within the crater. As a result, we cannot use the simple ballistic equation to

find the ejection velocity. Instead we solve the following equations simultaneously. Since D is much less than the radius of the planet, we use the equation for the ballistic trajectory of an object over a flat plane:

f ¼

v2 g

sinð2hÞ;

ð1Þ

where f is the horizontal distance that the boulder travels, v is the ejection velocity of the boulder, g is the acceleration of gravity, and h is the angle at which the boulder is ejected. We assume that each boulder was ejected at an angle of 45° from the horizontal. This assumption is reasonable because for the primary excavation flow, ejection angles are usually close (±15°) to 45° and over that range the ejection velocity function is only weakly dependent on

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Fig. 2. Image (crater diameter) – (A) V-167-H3 (0.690 km). (B) V-199-M (41.190 km). (C) V-211-H3 (0.520 km). (D) Ap17-Pan-2345(a) (1.059 km). (E) Ap17-Pan-2345(b) (1.299 km). (F) Ap17-Pan-2345(e) (1.415 km). (G) Ap17-Pan-2345(f) (0.730 km). (H) Ap17-Pan-2345(g) (0.674 km). (I) Ap17-Pan-2345(i) (0.591 km).

ejection angle (Cintala et al., 1978). As a result, sin(2  45°) = 1, and the equation simplifies to

f ¼ v 2 =g:

ð2Þ

Next, a gravity-regime scaling relation for crater ejecta gives the ejection velocity (v) of a boulder from position r within a crater of final radius R as

v¼

pffiffiffiffiffiffi pffiffiffiffiffiffi 2 Rg  r  2 Rg  r k  ; ð1 þ Þ R ð1 þ Þ R

ð3Þ

where g is the gravitational acceleration of the planet and  is the material parameter (Housen et al., 1983, Eq. (13); Richardson et al., 2005). In the second term of Eq. (3), k is selected such that the ejection velocity will go to zero as the final crater rim is approached; we set k = 10. For the Moon we use g = 1.62 m/s2 and

 = 1.5 for basalt (Housen et al., 1983).

This equation is valid only for r greater than approximately the projectile radius, because ejecta streamlines originating at smaller r values will never eject material (Melosh, 1989, Sections 5.5.2–5.5.3). Finally, r and f in the preceding equations are simply related to D, the measured distance from the center of the crater to the final boulder location; D equals the distance from the center of the crater to the location from which the boulder was ejected, r, plus the distance from where the boulder was ejected to where the boulder landed, f:

D ¼ r þ f:

ð4Þ

The uncertainty in the calculated ejection velocity stems from the uncertainty in the measurement of the boulder location (±0.4 m) and the location of the center of the crater (±0.8 m), resulting in a

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III-168-H2 2

2

1.5

1.5

1 0.5 0 -0.5 -1

Y location (km)

2

-1.5

1 0.5 0 -0.5 -1 -1.5

-2

0.5 0 -0.5 -1 -2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

X location (km)

X location (km)

X location (km)

V-63-H2

III-189-H2

V-82-M 30

1.5

4

20

1 0.5 0 -0.5 -1

Y location (km)

2

6

Y location (km)

2 0 -2

-2

0 -10

-30 -30 -20 -10

-6

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

10

-20

-4

-1.5

-6

-4

X location (km)

-2 0 2 4 X location (km)

V-152-H2

V-153-H2

6

2

2

1.5

1.5

Y location (km)

2

0.5 0 -0.5 -1 -1.5

1 0.5 0 -0.5 -1 -1.5

-2

X location (km)

20

30

1 0.5 0 -0.5 -1 -1.5

-2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

10

V-167-H2

1.5 1

0

X location (km)

Y location (km)

Y location (km)

1

-1.5

-2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Y location (km)

III-186-H3

1.5

Y location (km)

Y location (km)

III-185-H3

-2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

X location (km)

X location (km)

Fig. 3. Plots of the locations of the boulders with respect to the crater rims, with distances in km. Note that not all the scales are the same.

distance uncertainty of ±1.2 m. At typical crater diameters and distances, this uncertainty results in an uncertainty in the calculated ejection velocity of about 1 m/s. The actual uncertainty may be slightly higher since we cannot directly measure the transient crater radius. The ejection velocity is thus calculated for each visible boulder ejected from a crater and plotted in Figs. 8 and 9. 3.3. Regolith depth To determine regolith depth, we followed the method of Quaide and Oberbeck (1968) (see also Oberbeck and Quaide, 1967, 1968). Through a series of experimental impacts, they showed that although impacts that form in a coherent, uniform substrate produce round, bowl-shaped craters, impacts that form in a substrate covered with several meters of regolith produce a different crater morphology. At least three crater morphologies were produced

by varying experimental impact sizes and ‘‘regolith” depths: flat bottomed, central mound, and concentric. Each of these morphologies is abundantly represented by small (50 m) lunar craters. Furthermore, the morphology of the craters was correlated to the depth of regolith in which they formed. The ratio of the apparent diameter of the crater (DA) to the diameter (DF) of the interior feature (flat floor, mound, or concentric ring) was a function of crater diameter (DA) to regolith thickness. They found the thickness of the regolith was given by

thickness ¼ ðk  DF =DA ÞDA tanðaÞ=2;

ð5Þ

where k is an empirically determined constant (0.86) and a is the angle of repose of the material (31°). Near each bouldery crater studied in this paper, we measured 20–30 such morphologically distinct small craters, and used those measurements to calculate the regolith depth of that terrain. The regolith depths thus determined are listed in Table 1.

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V-199-M 2

1.5

20

1.5

1 0.5 0 -0.5 -1

Y location (km)

30

10 0 -10 -20

-1.5 -2

X location (km)

0.5 0 -0.5 -1 -2

0

10

20

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

30

X location (km)

X location (km)

Ap17-Pan-2345(a)

Ap17-Pan-2345(b)

Ap17-Pan-2345(e)

2

2

1.5

1.5

1.5

1 0.5 0 -0.5 -1

Y location (km)

2

Y location (km)

1 0.5 0 -0.5 -1

-1.5

-1.5

-2

-2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 X location (km)

1 0.5 0 -0.5 -1 -1.5 -2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 X location (km)

Ap17-Pan-2345(f)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 X location (km)

Ap17-Pan-2345(g)

Ap17-Pan-2345(i)

2

2

1.5

1.5

1.5

Y location (km)

2 1 0.5 0 -0.5 -1

Y location (km)

Y location (km)

1

-1.5

-30 -30 -20 -10

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Y location (km)

V-211-H3

2

Y location (km)

Y location (km)

V-167-H3

1 0.5 0 -0.5 -1

-1.5

-1.5

-2

-2

1 0.5 0 -0.5 -1 -1.5 -2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

X location (km)

X location (km)

X location (km)

Fig. 4. Plots of the locations of the boulders with respect to the crater rims, with distances in km. Note that not all the scales are the same.

Large craters excavate to greater depths than small craters, regardless of regolith coverage. To account for this effect, we divide the depth of the regolith at each location by the crater’s diameter to isolate the effect of the regolith depth from the effect of the crater size. This allows us to compare the regolith depth for each case in terms of the fraction of the crater diameter that equals the regolith depth.

(boulder ejection point) crater center

f (boulder flight distance)

r

D (distance from crater center to boulder) R (transient crater radius)

4. Boulder distributions 4.1. Qualitative observations Qualitatively, the data (Figs. 6 and 7) show that larger boulders lie preferentially closer to the crater rim, and small boulders occur at all distances, to the limit of resolution. They also show a concen-

Fig. 5. The ejection velocity of a boulder can be calculated from its measured distance from its crater. The circle represents the crater, and the right end of the horizontal line is the location of the boulder whose ejection velocity is being calculated. D, measured and plotted in Figs. 6 and 7, and R, the transient crater radius, assumed to be 0.85 of the measured final crater radius (see Table 1), are used to calculate r, f, and the ejection velocity using crater ejecta scaling laws (Housen et al., 1983; Richardson et al., 2005).

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ther away. Because the larger boulders spread out on the ejection velocity plot, the distribution appears to flatten.

tration of boulders of moderate sizes at moderate distances. The fragment ejection velocity data (Figs. 8 and 9) show that larger boulders are ejected at lower velocities. In these plots we only include boulders outside the crater rim. Although the boulders remaining in the crater likely moved at low velocities, those velocities cannot be calculated by our method. Comparison of the distance plots (Figs. 6 and 7) with the ejection velocity plots (Figs. 8 and 9) reveals that the strong decrease in boulder size with distance from the crater translates to a much weaker decrease of size with ejection velocity. This effect is a result of the non-linear correspondence between distance and ejection velocity, which is illustrated in Fig. 10. Boulders near the crater were ejected at a wider range of velocities than the boulders far-

III-185-H3

4.2.1. Method In an attempt to quantify our data for comparison with previous work, we fit the maximum boulder size as a function of ejection velocity to a power law, dmax = avb, where dmax is the diameter of the largest boulder, v is the ejection velocity, and a and b are constants determined by the fit. First we had to decide which ‘‘maximum boulder size” points to fit. Point selection was non-trivial

III-168-H2

40 30 20 10

60

Boulder Diameter (m)

50

0

50 40 30 20 10 0

0

0.5

1

1.5

2

2.5

3

0.5

1

1.5

2

2.5

3

40 30 20 10 0 0.5 1 1.5 2 2.5 3 Distance from crater center (km)

Distance from crater center (km)

III-189-H2

V-63-H2

60

V-82-M

50 40 30 20 10

600

Boulder Diameter (m)

120

Boulder Diameter (m)

Boulder Diameter (m)

50

0 0

Distance from crater center (km)

(note scale)

100 80 60 40 20

500

300 200 100 0

0

0 0.5 1 1.5 2 2.5 3 Distance from crater center (km)

1

2

3

4

5

6

0 5 10 15 20 25 30 Distance from crater center (km)

Distance from crater center (km)

V-152-H2

V-153-H2

V-167-H2

50 40 30 20 10 0

60

Boulder Diameter (m)

60

Boulder Diameter (m)

60

50 40 30 20 10 0

0

0.5

1

1.5

2

2.5

3

Distance from crater center (km)

(note scale)

400

0

0

Boulder Diameter (m)

III-186-H3

60

Boulder Diameter (m)

60

Boulder Diameter (m)

4.2. Quantitative description

50 40 30 20 10 0

0

0.5

1

1.5

2

2.5

3

Distance from crater center (km)

0 0.5 1 1.5 2 2.5 3 Distance from crater center (km)

Fig. 6. Boulder diameter (m) plotted against the distance that the boulder lies from the center of the crater (km). Nine craters are plotted here, the rest are plotted in Fig. 7. Larger boulders lie closer to the crater rim. The horizontal bands in the data are a result of discrete pixel sizes encountered when measuring the boulder diameters.

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because of the continuous distribution of the velocity data and the small number of data points. In our first attempt we binned the data in uniform velocity increments. We sorted the boulders in each velocity bin by size (small to large) and took the size of the boulder at the 95th percentile in each bin as a point to fit. We selected the boulder at the 95th percentile rather than the largest boulder to fit the bulk of the data and minimize stochasticity. Nevertheless, these plots did not visually appear to fit the data. Because of the small number of data points at larger velocities, the curve rose up to larger boulder sizes again at that point. Next we tried velocity bins that each held the same number of boulders, again taking a fitting point at the size of the boulder at the 95th percentile. In this case, the large bins

V-167-H3

50 40 30 20 10

500

Boulder Diameter (m)

Boulder Diameter (m)

Boulder Diameter (m)

60

600 (note scale)

400 300 200 100 0

0

Ap17-Pan-2345(a)

40 30 20 10 0 2

2.5

50 40 30 20 10

3

0.5

1

1.5

2

2.5

Ap17-Pan-2345(f)

30 20 10 0 3

Distance from crater center (km)

0.5

1

1.5

2

2.5

3

Ap17-Pan-2345(i) 60

50 40 30 20 10 0

2.5

10

Distance from crater center (km)

Boulder Diameter (m)

Boulder Diameter (m)

40

2

20

Ap17-Pan-2345(g)

50

1.5

30

0

60

1

40

3

Distance from crater center (km)

60

0.5

50

0 0

Distance from crater center (km)

0

10

Ap17-Pan-2345(e)

0 1.5

20

60

Boulder Diameter (m)

Boulder Diameter (m)

50

1

30

0 0.5 1 1.5 2 2.5 3 Distance from crater center (km)

60

0.5

40

Ap17-Pan-2345(b)

60

0

50

0

0 5 10 15 20 25 30 Distance from crater center (km)

0 0.5 1 1.5 2 2.5 3 Distance from crater center (km)

Boulder Diameter (m)

V-211-H3

V-199-M

60

Boulder Diameter (m)

at high velocity had the same problem as in the former case. Because the 95th percentile was above the most dense part of the data, we also tried fitting the median boulder size in each bin, but the fit was not improved. We noticed that sometimes to get a good looking power law we would have to include lone boulders at high sizes, so we tried another variation of our first attempt. We binned the boulders in ejection velocity, selected the largest boulder in each bin, and fit a power law to that. In addition, we made the leftmost start of the fit to be the highest point, because the highest point was not always furthest to the left. These plots produced good results in some cases (i.e. III-186-H3) but not for the majority of craters.

50 40 30 20 10 0

0

0.5

1

1.5

2

2.5

3

Distance from crater center (km)

0 0.5 1 1.5 2 2.5 3 Distance from crater center (km)

Fig. 7. Boulder diameter (m) plotted against the distance that the boulder lies from the center of the crater (km). Nine craters are plotted here, the rest are plotted in Fig. 6. Larger boulders lie closer to the crater rim. The horizontal bands in the data are a result of discrete pixel sizes encountered when measuring the boulder diameters.

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Upon further consideration, we determined that the visually observed trend follows the density of the data points rather than their overall height. We created a grid of 15 diameter bins and 20 velocity bins. We tried presenting the number of boulders in each bin in contour (with the boulder numbers as ‘‘elevations”) and 3D plots (with the number of boulders in each bin represented by the length of a line extending into the third dimension). These plots provided a good qualitative understanding of the bulk data shape, but did not aid in quantifying the data. Our final solution for quantifying the data is the following. The data are divided into velocity bins, and each of those bins is divided into diameter bins. We select a fitting point in each

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velocity bin at its uppermost diameter bin that contains at least 1% of the total number of boulders observed at that crater. Although these points provide better fits to the bulk of the data, they do not take into consideration that there is a cutoff in ejection velocity. That is, the diameter data do not decrease to the limit of resolution, rather, no more boulders are observed beyond a certain distance. Thus, the fits are only valid to the cutoff velocity. If the fits were valid beyond that point, it would indicate that there should be more boulders at higher ejection velocities, which we do not observe. The fits obtained using this method are illustrated in Figs. 11 and 12, and the values for those fits are given in Table 3.

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4.2.2. Analysis We compare the boulder distributions from the craters using the exponent (b) of the power law fit to the size–velocity data. (Hereafter, we call b the ‘‘velocity exponent”.) The velocity exponent describes the change of the maximum boulder diameter with ejection velocity. A smaller (more negative) exponent indicates that the largest boulders are preferentially ejected at low velocities (closer to the crater), whereas a larger exponent indicates that the larger boulders are ejected over a wider range of ejection velocities. Note that our data are limited by resolution and other factors (Bart and Melosh, 2010). Therefore our b values do not accurately represent a theoretical boulder distribution from a uniform strength and density impactor into a flat and uniform strength and density target. In particular, the high velocity ends of the fit frequently appear to be higher than they would otherwise be in

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Fig. 11. A power law fit, dmax = avb, is shown here in red, superimposed on the data presented in Fig. 8. dmax is the diameter of the largest boulder, v is the ejection velocity, and a and b are constants determined by the fit. The red squares are the points selected by the method explained in this section, and the red line is the fit to that data. The fit parameters are given in Table 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

such an idealized case. Nevertheless, comparison of these b values provides us with a mechanism for comparison of the boulder distributions in these various cases. Plots of the b and a values of the fits vs. crater diameter are shown in Figs. 13 and 14. In each plot the craters are distinguished by terrain type. Fig. 13 shows a widely scattered but generally decreasing b value with increase of crater diameter. Fig. 14 does not show any trends in a value with crater diameter, but the terrain type does seem to have an effect. The craters with the largest a values are the craters observed in the craters observed in the Apollo images, followed by some highlands craters. All the Lunar Orbiter mare craters have the lowest a values. This result is consistent with

the findings of Bart and Melosh (2007), who found generally larger boulder diameters for these particular Apollo image craters (presumed to be secondary craters) over the Lunar Orbiter mare craters (presumed to be primary craters), since larger a values correlate with larger boulder sizes. 4.2.3. Comparison with published data Vickery (1987) performed a similar analysis for large craters. She calculated fragment sizes and ejection velocities for ejecta of large craters by measuring secondary craters of 3 mercurian craters, 5 lunar craters, and 4 martian craters. The primary craters had diameters from 26 km to 227 km. She measured the diameters

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of the secondary craters and used scaling laws (Holsapple and Schmidt, 1982; Schmidt and Holsapple, 1982) to estimate the size of the fragment that created each one. Next she measured the distance of each secondary crater from its primary and used the physics of ballistic travel over a spherical planet to calculate each fragment’s ejection velocity. Her data showed that the size of an ejected boulder decreases as ejection velocity increases. The largest fragment sizes as a function of ejection velocity fit to a power law, similar to our fits above. Although the size of the largest fragment depended on crater size, the slope of the size–velocity relation appeared independent of primary crater size. However, when com-

bined with our data for smaller craters, a definite downward trend appears (Fig. 15). We roughly calibrate our methods by finding a velocity exponent for the Vickery data. For the data presented in Fig. 2a of Vickery (1986) we derive an exponent (b value) of 2.3. This value lies well in the range of the exponent values tabulated in Vickery (1987): 2.75 to 1.46. Therefore the difference in value is not a result of a systematic error between our methods. Thus there is a large scale trend in which the small craters have much larger (less negative) velocity exponents than the large craters. The shape of this large scale trend is an envelope rather than a

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G.D. Bart, H.J. Melosh / Icarus 209 (2010) 337–357 Table 3 Values for the fits shown in Figs. 11 and 12. The fits are a power law given by dmax = avb, where dmax is the diameter of the largest boulder, v is the ejection velocity, and a and b are constants determined by the fit. a value

a error

b value

b error

III-185-H3 III-168-H2 III-186-H3 III-189-H2 V-63-H2 V-82-M V-152-H2 V-153-H2 V-167-H2 V-167-H3 V-199-M V-211-H3 Ap17-Pan-2345(a) Ap17-Pan-2345(b) Ap17-Pan-2345(e) Ap17-Pan-2345(f) Ap17-Pan-2345(g) Ap17-Pan-2345(i)

17 31 20 9.7 710 1.0  105 18 136 80 190 4.9  109 52 290 520 260 140 190 1200

10 18 15 6.2 810 2.0  105 12 90 51 170 2.3  1010 36 270 500 250 120 110 630

0.51 0.58 0.53 0.32 0.88 1.4 0.44 0.70 0.70 0.93 3.7 0.62 0.73 0.92 0.82 0.64 0.74 1.3

0.19 0.19 0.25 0.20 0.31 0.45 0.23 0.21 0.21 0.31 1.0 0.23 0.29 0.31 0.30 0.29 0.18 0.20

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Fig. 16. Maximum and median boulder diameters are plotted vs. crater diameter. We exclude the Apollo image mare craters from the median fit because those data have lower resolution. There is a power law relationship between boulder ejection distance and crater diameter; the equation is given on the plot.

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Fig. 18. Maximum and median boulder ejection velocity are plotted vs. crater diameter. We exclude the Apollo image mare craters from the median fit because those data have lower resolution. There is a power law relationship between boulder ejection velocity and crater diameter; the equation is given on the plot.

Table 4 The maximum and median boulder diameter, distance, and ejection velocity exhibit power law relationships with crater diameter: y = axb, where y is either boulder diameter (m), boulder distance (m), or ejection velocity (m/s), x is the crater diameter (m), and a and b are constants determined by the fit to the data. The data are shown graphically in Figs. 16 and 18. Because the median value depends on resolution of the image, the Apollo data are not included in the median calculation.

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(Fig. 17), and boulder ejection velocity (Fig. 18) vs. crater diameter. Because the largest distance or ejection velocity represents only a single data point for each crater, and because its size could be fairly stochastic, these figures also plot the median distance and ejection velocity for each crater. The median value is resolution dependent; if the smallest observable boulder is four times greater in one image than another, the median value of those boulders will be higher. Therefore, the Apollo images were not included in the calculation of median value. These plots show that the maximum/median boulder distances and ejection velocities are functions (power laws) of crater diameter. The fit values are given in Table 4.

line. There is significant scatter over small changes in crater diameter, which is probably due to the varied geologic settings of each impact site, although the difficulty of obtaining a good fit may also contribute. 5. Effect of crater diameter: boulder distance and ejection velocity Next we consider the effect of crater diameter on the boulder distance and ejection velocity. (The effect of crater diameter on boulder size was discussed in Bart and Melosh (2007).) To this end, we plot boulder diameter (Fig. 16), boulder distance

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Fig. 19. Plot of boulder diameter, corrected for crater diameter by dividing by crater diameter to the 0.62 power, vs. distance of the boulder from the center of the crater, corrected for crater diameter by dividing by crater diameter to the 0.86 power. These corrections allow comparisons of boulder distributions independent of effects of crater diameter. The lower frame of Fig. 20 shows all the lunar mare and highlands data plotted together.

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H3, III-168-H2). This determination is not straightforward, though. Several craters (e.g. V-82-M, V-167-H2) have one or two boulders with high normalized ejection velocity values, while the bulk of the data lie much lower. These variations do not correspond with regolith depth or terrain type. It is unknown at this time whether these are stochastic differences or whether some other factor may be causing them.

These relationships show that the changing crater diameters lead to systematic variations in the boulder distributions. We can thus use these results to remove the crater diameter bias in the distributions. To do this, we divide the distances and ejection velocities by the crater diameter raised to the median b value. Figs. 19 and 20 show the corrected diameter/distance distributions, and Figs. 21 and 22 show the corrected diameter/velocity distributions. These normalized plots show some variations among the craters. Although, overall, the larger boulders have lower ejection velocities, some of the ejection velocity plots (e.g. Fig. 21, craters III-185-H3 and V-152-H2) are nearly flat, whereas others (e.g. III168-H2, V-63-H2) show a strong decrease of boulder size with ejection velocity. The maximum normalized ejection velocity of some (e.g. V-167-H3, V-152-H3) is lower than others (e.g. III-186-

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Fig. 21. Plot of boulder diameter, corrected for crater diameter by dividing by crater diameter to the 0.62 power, vs. boulder ejection velocity, corrected for crater diameter by dividing by crater diameter to the 0.24 power. These corrections allow comparisons of boulder distributions independent of effects of crater diameter. The lower frame of Fig. 22 shows all the lunar mare and highlands data plotted together.

shown in Fig. 23 and Table 5. There are at least two possible explanations for deviations from this increase. The craters could be excavating more regolith and less bedrock, or, for an old crater, micrometeorites may have eroded some of the boulders. Hartmann and Barlow (2006) and Wilcox et al. (2005) suggested that regolith depth can be determined by the quantity of boulders about the crater. My data do not support that hypothesis. A plot of number of boulders vs. regolith depth (Fig. 24) does not show more boulders correlated to less regolith. In fact, the craters with the most boulders occur on the highlands, which has more regolith than the mare. Furthermore, comparison of narrow columns in Fig. 23 shows that mare craters (less regolith) in a given size range do not consistently have more boulders. Therefore, a

simple comparison of the number of boulders a crater has is not a good indicator for regolith depth.

6.2. Interior boulders Fig. 25 and Table 6 show that the number of boulders located inside each crater is greater for the highlands craters than for the mare craters. This observation is consistent with some interior boulders being made by crater wall slumping in the modification stage of crater formation. The highlands, having an older, more broken surface, would more readily have boulders falling into the crater than craters formed in the stronger mare basalt.

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Ejection Velocity/(Cr. Diam)^0.32 Fig. 22. Plot of boulder diameter, corrected for crater diameter by dividing by crater diameter to the 0.62 power, vs. boulder ejection velocity, corrected for crater diameter by dividing by crater diameter to the 0.24 power. These corrections allow comparisons of boulder distributions independent of effects of crater diameter. The lower frame shows all the lunar mare and highlands data plotted together.

7. Boulders as a source of secondary craters Because ejected boulders are the source of secondary craters, we compare our boulder populations with secondary crater populations observed on the Moon. Crater distributions are often presented as cumulative plots (Arvidson et al., 1979). A cumulative plot shows, for each crater size, the number of craters of that size and larger, plotted on a log–log plot. The plot, therefore, has a negative slope; the small craters will have the largest values because all the other objects are larger. Shoemaker (1965) showed that cumulative plots of all lunar craters P1 km diameter (primary

and secondary) on Mare Cognitum showed a slope a bit shallower than 2. However, plots of secondary craters had a slope of about 4. Secondary craters are produced by high-speed ejected boulders, so we compare our boulder distributions with the secondary crater distributions. To do so we use scaling laws (Holsapple and Schmidt, 1982) to correlate boulder size with secondary crater size. An analysis similar to that in Vickery (1986, p. 227–228) shows that, for a given target strength, impact velocity, and surface gravity, projectile size is proportional to the resulting crater diameter. The slope of a log–log plot is unchanged when two x values are directly pro-

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portional to each other; therefore we simply plot the boulders on the cumulative log–log plot and compare the slopes with those found by Shoemaker (1965). The cumulative plots of the boulders are shown in Figs. 26 and 27. The boulders consistently plot along slopes steeper (more negative than) 2. Many plot close to 4, and several are steeper yet. This result is consistent with the idea that crater ejecta, and hence secondary craters, plots more steeply than the primary impactor distribution. At the smallest boulder diameters the plot flattens out, which we attribute to resolution effects; small boulders are more difficult to identify and measure. Interestingly, we did identify eight secondary craters around the 502 m diameter crater, photograph III-189-H2, on the Moon. They lie about 2.5 crater radii from the rim. These few secondary craters lie amid the several thousand boulders surrounding the crater. The blocks which formed them may have been ejected from the crater early in the crater formation process at high speeds and steep angles, providing them with enough energy to form secondary craters among other blocks that just came to rest on the surface. This observation is not unique; Morris and Shoemaker

Table 5 Data for each crater listing the crater diameter, the number of boulders greater than 9.0 m, and the regolith depth. Regolith depth could not be measured for all craters. Crater diameter

Number of boulders greater than 9.0 m

Regolith depth (m)

0.290 0.678 0.229 0.537 4.006 27.422 0.506 0.881 0.452 0.690 41.190 0.520 1.059

8 59 17 75 316 315 43 284 23 39 265 7 183

6.6 4.3 4.3 6.4 10.1 9.1 10.2 10.2 12.2 12.2 – – –

1.299

141

–

1.415

220

–

0.730

197

–

0.674

169

–

0.591

44

–

Fig. 24. The number of boulders ejected from a crater does not decrease with regolith depth. However, the craters with the most boulders are also the largest craters, which we have just shown have more boulders. Hence, the size of the crater has a much greater effect on the number of boulders than the regolith depth.

1000

# interior boulders > 2.4 m

Fig. 23. The number of boulders ejected from a crater increases with crater diameter. Here we include only the boulders greater than 9 m, because not all photographs could resolve boulders smaller than that. Craters are separated by type.

III-185-H3 III-168-H2 III-186-H3 III-189-H2 V-63-H2 V-82-M V-152-H2 V-153-H2 V-167-H2 V-167-H3 V-199-M V-211-H3 Ap17-Pan2345(a) Ap17-Pan2345(b) Ap17-Pan2345(e) Ap17-Pan2345(f) Ap17-Pan2345(g) Ap17-Pan2345(i)

10

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Fig. 25. The highlands craters show generally more interior boulders than the mare craters do, which indicates that the highlands surface may have more preexisting fractures than the mare surfaces.

Table 6 The highlands craters show generally more interior boulders than the mare craters do, which indicates that the highlands surface may have more preexisting fractures than the mare surfaces.

Highlands V-167-H2(b) V-152-H2 V-167-H3 V-153-H2 V-63-H2 V-82-M

Crater diameter (km)

Number of interior boulders >2.4 m diameter

0.451 0.505 0.690 0.881 4.006 27.42

66 186 104 211 288 101

Apollo image mare Ap17-Pan0.591 2345(i) Ap17-Pan0.674 2345(g) Ap17-Pan0.730 2345(f) Ap17-Pan1.059 2345(a) Ap17-Pan1.299 2345(b) Ap17-Pan1.415 2345(e) Lunar Orbiter mare III-186-H3 0.228 III-185-H3 0.28 V-211-H3 0.519 III-189-H2 0.536 III-168-H2 0.678 V-199-M 41.190

8 44 48 113 49 133

16 50 60 9 99 86

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lith depths, and to understanding the distributions of secondary craters in the Solar System. The boulder distributions show that larger boulders lie preferentially closer to the crater rim, and small boulders occur at all distances, to the limit of resolution. They also show a concentration of boulders of moderate sizes at moderate distances. Understanding these distributions of blocky ejecta is an important consideration for landing site selection on both the Moon and Mars (Schroeder and Golombek, 2003; Marlow et al., 2006). Vickery (1986), studying only large craters, was not able to detect any change in ejection velocity distribution with crater diameter. But by combining that data with ours, a distinct, though scattered, trend appears. This trend indicates that the large craters eject their large boulders only at the very lowest ejection veloci-

(1970) also document several small, low velocity secondary craters. 8. Conclusions Computers facilitated this analysis of these 18 craters – an analysis that would have been prohibitively difficult 30 years ago when the lunar photographs were taken. By digitizing the photographs, we could record the location and size of each of 13,955 boulders. These data, in conjunction with data from the literature, provide new insights into the impact crater ejection process and its implications. These results provide data important to the understanding boulder ejection velocities, to determining whether there is a relationship between the quantity of ejected boulders and lunar rego-

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Fig. 26. Cumulative plots of boulders ejected from the indicated craters.

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Fig. 27. Cumulative plots of boulders ejected from the indicated craters.

ties, whereas the smaller craters eject their larger boulders over a wider range of ejection velocities, though still lower than the velocity of the bulk of the boulders ejected from that crater. The boulder distributions also indicate that the number of exterior boulders cannot be used to determine regolith depth, as craters with more boulders are not consistently found in areas with thinner regolith. Furthermore the number of interior boulders reflects the terrain type, with the more fractured highlands terrain producing more internal boulders, presumably as slump blocks in the modification stage of crater formation. Finally, cumulative plots of the boulders plot steeper than a 2 slope, just as secondary craters (products of crater ejecta) plot steeper than the 2 slope of primary craters. This result confirms that secondary craters are produced from the ejected boulders of large primary craters. These data support the idea that the steeper slope

of the size–frequency-diagram of craters less than 1 km is due to the addition of numerous secondary craters at that size. References Aldrin Jr., E.E., Armstrong, N.A., Collins, M., 1969. Crew observations. In: Apollo 11 Preliminary Science Report. NASA SP-214, Washington, DC, pp. 35–39. Arvidson, R.E., and 11 colleagues, 1979. Standard techniques for presentation and analysis of crater size–frequency data. Icarus 37, 467–474. Bart, G.D., Melosh, H.J., 2007. Using lunar boulders to distinguish primary from distant secondary impact craters. Geophys. Res. Lett. 34, L07203. Bart, G.D., Melosh, H.J., 2010. Impact into lunar regolith inhibits high velocity ejection of large blocks. J. Geophys. Res., in press. Binder, A.B., Arvidson, R.E., Guinness, E.A., Jones, K.L., Mutch, T.A., Morris, E.C., Pieri, D.C., Sagan, C., 1977. The geology of the Viking Lander 1 site. J. Geophys. Res. 82, 4439–4451. Boeing Company, 1969. Lunar Orbiter III Photographic Mission Summary. NASA CR – 1069.

G.D. Bart, H.J. Melosh / Icarus 209 (2010) 337–357 Cintala, M.J., Head, J.W., Veverka, J., 1978. Characteristics of the cratering process on small satellites and asteroids. Lunar Planet. Sci. 9, 3803–3830. Dutro, J.T.J., Dietrich, R.V., Foose, R.M., 1989. AGI Data Sheets: For Geology in the Field, Laboratory, and Office, third ed. American Geological Institute, Alexandria, VA. Hartmann, W.K., Barlow, N.G., 2006. Nature of the martian uplands: Effect on martian meteorite age distribution and secondary cratering. Meteor. Planet. Sci. 41 (10), 1453–1467. Hess, W.N., Calio, A.J., 1969. Summary of scientific results. In: Apollo 11 Preliminary Science Report. NASA SP-214, Washington, DC, pp. 1–7. Holsapple, K.A., Schmidt, R.M., 1982. On the scaling of crater dimensions. 2. Impact processes. J. Geophys. Res. 72, 1849–1870. Housen, K.R., Schmidt, R.M., Holsapple, K.A., 1983. Crater ejecta scaling laws – Fundamental forms based on dimensional analysis. J. Geophys. Res. 88, 2485– 2499. Kuiper, G.P., 1965. The surface structure of the Moon. In: The Nature of the Lunar Surface. The Johns Hopkins Press, Baltimore, pp. 99–105. Marlow, J.J., Klein, C.R., Martinez, M.M., McGrane, B.S., Golombek, M.P., 2006. Boulder hazard assessment of potential phoenix landing sites. In: Mackwell, S., Stansbery, E. (Eds.), 37th Lunar and Planetary Science Conference. Abstract 1094. Melosh, H.J., 1984. Impact ejection, spallation, and the origin of meteorites. Icarus 59, 234–260. Melosh, H.J., 1989. Impact Cratering: A Geologic Process. Oxford University Press, New York. Morris, E.C., Shoemaker, E.M., 1970. Geology: Craters. Icarus 12, 167–172. Oberbeck, V.R., Quaide, W.L., 1967. Estimated thickness of a fragmental surface layer of oceanus procellarum. J. Geophys. Res. 72 (11), 4697–4704. Oberbeck, V.R., Quaide, W.L., 1968. Genetic implications of lunar regolith thickness variations. Icarus 9, 446–465. Quaide, W.L., Oberbeck, V.R., 1968. Thickness determinations of the lunar surface layer from lunar impact craters. J. Geophys. Res. 73 (16), 5247–5270.

357

Rasband, W.S., 1997–2009. ImageJ, US National Institutes of Health, Bethesda, MD, USA. . Richardson, J.E., Melosh, H.J., Artemieva, N.A., Pierazzo, E., 2005. Impact cratering theory and modeling for the deep impact mission: From mission planning to data analysis. Space Sci. Rev. 117, 241–267. Schmidt, R.M., Holsapple, K.A., 1982. Estimates of crater size for large body impact: Gravity scaling results. Geol. Soc. Am. Spec. Pap. 190, 93–102. Schroeder, R.D., Golombek, M.P., 2003. Mars Exploration Rover landing site boulder fields. Lunar Planet. Sci. 34, 1975. Shoemaker, E.M., 1965. Preliminary analysis of the fine structure of the lunar surface in Mare Cognitum. In: Hess, W.N., Menzel, D.H., O’Keefe, J.A. (Eds.), The Nature of the Lunar Surface. The Johns Hopkins Press, Baltimore, pp. 23–77. Shoemaker, E.M., Morris, E.C., 1970. Geology: Physics of fragmental debris. Icarus 12, 188–212. Shoemaker, E.M., and 12 colleagues, 1970. Preliminary geologic investigation of the Apollo 12 landing site. Part A: Geology of the Apollo 12 landing site. In: Apollo 12: Preliminary Science Report. NASA SP-235, pp. 113–156. Sullivan, R., and 17 colleagues, 1996. Geology of 243 Ida. Icarus 120, 119–139. Surveyor Scientific Evaluation and Analysis Team, 1966. Size distribution of debris on the lunar surface and the characteristics of the fine matrix. In: Lunar and Planetary Programs Division, Office of Space Science and Applications (Ed.), Surveyor 1: A Preliminary Report. NASA SP-126, pp. 31–32. Tomasko, M.G., and 39 colleagues, 2005. Rain, winds and haze during the Huygens probe’s descent to Titan’s surface. Nature 438, 765–778. Veverka, J., and 32 colleagues, 2000. NEAR at Eros: Imaging and spectral results. Science 289, 2088–2097. Vickery, A.M., 1986. Size–velocity distribution of large ejecta fragments. Icarus 67, 224–236. Vickery, A.M., 1987. Variation in ejecta size with ejection velocity. Geophys. Res. Lett. 14, 726–729. Wilcox, B.B., Robinson, M.S., Thomas, P.C., Hawkes, B.R., 2005. Constraints on the depth and variability of the lunar regolith. Meteor. Planet. Sci. 40 (5), 695–710.