Supplemental Material: Ray Systems in Granular Cratering

Supplemental Material: Ray systems in granular cratering

1 Planetary ray systems

(a) (b)

(c) (d) Figure S-1: Ray systems in planetary cratering. (a) Kepler crater, Moon; 32 km diameter (Source: www.neophoto.nl). (b) Unnamed crater, Moon; 30 m diameter (Source: NASA/GSFC/Arizona State University). (c) Unnamed crater, Mars; 30 m diameter (Source: NASA/JPL/University of Arizona). (d) Debussy crater, Mercury; 80 km diameter (Source: NASA/JPL/Carnegie Institution of Washington).

1 2 Methods

2.1 Granular experiments We drop a steel ball of diameter D on a granular bed housed in a cylindrical vessel. To drop the ball, we use an electromagnet with a concave holding surface; the ball falls without vibration or rotation. The granular bed is made of spherical glass grains of median diameter d = 45µm, with an upper limit of 63 µm (Product name: J-400 from Potters-Ballotini, Japan). To aid the visualization of the ejecta, we surround the granular bed with a black, flat panel, flush with the surface of the bed. By conducting experiments with and without the panel, we have verified that the panel does not affect the formation of the crater or the ray system. To form a granular bed with a hexagonal undulating surface, we start with a hexagonal grid, which we make using a 3D printer (Zprinter 450). The arms of the hexagon are 3 mm x 3 mm in cross section; the distance between the centers of two adjacent hexagons is λ. We gently press the grid on a granular bed with an evened-out surface. When the grid becomes flush with the surface, we carefully lift and remove the grid. The surface becomes undulating, imprinted with a hexagonal network of valleys of wavelength λ. Our experiments span the following range of parameters: Ball diameter, D 2.54 cm to 10.16 cm Wavelength, λ 0.75 cm to 1.5 cm Drop height of ball 100 cm to 400 cm Vessel diameter 28.2 cm to 37.5 cm Vessel height 15.0 cm to 26.1 cm Initial packing fraction of the granular bed loose to close packed

Further, by repeating the experiments under diﬀerent ambient humidity, we also vary the particle cohesivity.

2.2 Granular simulations We use the discrete-element method as implemented in the open-source pack- age LIGGGHTS [1]. The grains are represented as soft spheres that interact with each other and with the boundaries of the vessel through contact forces. We model the normal repulsive forces using a damped-Hertzian spring, the normal cohesive forces using a modiﬁed Johnson-Kendall-Roberts model, and the tangential forces using Coulomb’s friction law. The geometric and material parameters of the simulations are as follows:

2 Grain diameter d 180 ± 3.6 µm Grain density ρ 2.7 g/cc Young’s modulus 94 MPa Poisson’s ratio 0.17 Coeﬃcient of friction 0.5 Coeﬃcient of restitution 0.4 Cohesive energy density 0.22 J/m3

Exploiting the symmetry along the axis of impact, we reduce the computational costs by simulating a quarter of the spatial domain. In the simulations the granular bed is housed in a sliced, quarter-cylindrical vessel of diameter 10 cm and height 2.8 cm. At the curved outside wall and the flat bottom wall, we set the coefficient of friction to 0.5, the same as that between the grains. Further, to reduce the influence of waves reflected off these walls, we set the coefficient of restitution at the walls to near zero [2]. The radial walls on the sides need special consideration since they are absent in a simulation of the full spatial domain. There we set the coefficient of friction to zero. To form the granular bed, we pour the grains into the vessel and let them settle under gravity. (We conduct all simulations with Earth’s gravity— acceleration due to gravity = 9.81 m/s2.) To form the undulating granular bed, we carve a hexagonal network of valleys of wavelength λ by deleting selected grains from the bed surface and letting the remaining grains settle under gravity. We compute the speed of sound in the granular bed, c, by perturbing a layer of grains at the bottom of the bed and measuring the velocity of the resulting pulse as it travels through the bed. We measure c at a distance D/2 below the surface of the bed. For the material parameters of our simulations, c = 16.9 m/s. In planetary cratering, hypervelocity regime refers to the regime where the impactor velocity exceeds the speed of sound in the planetary crust as well as in the impactor. In the granular model, we call the hypervelocity regime where the impactor velocity exceeds c. For simplicity, we model the impacting ball as a sliced, quarter-spherical mesh that travels through the granular bed at a constant velocity U. The mesh is brought to a halt as soon as it reaches the bottom wall. Despite the simplifications, we have verified that our simulations replicate several experimental trends for ejecta dynamics, such as the spatio-temporal evolution of the ejecta velocity and the ejection angle [3, 4].

3 3 Granular ray systems

3.1 School experiments Creating impact craters on a granular bed is a popular hands-on activity for school students; see, e.g., https://www.nasa.gov/centers/jpl/education/ craters-20090924.html. In a typical experiment, students drop a small ball on the surface of a granular bed (usually a bed of ﬂour). This forms an impact crater and hurls out ejecta that settles around the crater. We show representative images from such experiments in Fig. S-2. Note the hints of ray systems.

(a) (b) (c)

Figure S-2: Top-view of ejecta blanket produced by dropping a ball on a bed of ﬂour topped with a veneer of cocoa powder. These images are from experiments by school students (Source: http://cpsx.uwo.ca/ outreach/student_programs/index.html, http://www.ingridscience. ca/node/519, http://gcostigan.com/communications/). Note hints of ray systems.

3.2 Beds with undulating surfaces In the letter we report experiments with granular beds with hexagonal surface undulations. The results for surface undulations with other shapes are qualitatively the same; see Fig. S-3. In the letter we argued that for granular beds with hexagonal surface undulations, the number of prominent rays, N, depends only on the ratio D/λ. Using the geometric model discussed in the letter, we compute the dependence of N on D/λ for three shapes of surface undulations (see Fig. S- 4). Note that N ∝ D/λ for all shapes, but the dependence is aﬀected by the shape. Of particular note are the cases of hexagonal and square undulations: for a ﬁxed D/λ, whereas the average N is quite close for both cases, the variation in N is considerably larger for the hexagonal undulations.

4 (a) (b)

Figure S-3: Top-view of ejecta blankets for undulating surfaces produced by (a) radial grid, (b) square grid, (c) triangular grid, and (d) Voronoi grid. In the insets we show the pre-impact surfaces.

3.3 A note on ray systems in ejecta curtains A recent study [5] reported that for impacts on evened-out granular beds, the moving ejecta curtain exhibits spatial variations in the grain density, which they construed as being suggestive of ray systems. They argued that inelastic collision amongst ejecta particles causes a uniform ejecta curtain to spontaneously transition to form ray systems. In our experiments of impacts on evened-out granular beds, although we see spatial density variations in the moving ejecta curtain, they are ephemeral and do not persist in the settled ejecta blanket. By contrast, for impacts on granular beds with an undulating surface, the ray system in the moving ejecta curtain persists in the settled ejecta blanket (see, e.g., Supplementary Videos

5 Triangular grid λ λ 50 Square grid

Hexagonal grid 40 λ

30 N

0 0 5 10 15 D/λ Figure S-4: N vs. D/λ for impacts on granular beds with triangular, square, and hexagonal surface undulations. We set the bin-size δ(D/λ) = 1; The vertical bars span two standard deviations of N.

SV2 and SV3).

3.4 Secondary rays In the letter we have focused on the longest and most conspicuous rays—the prominent rays. In addition to these prominent rays, for some experiments and simulations, we also notice shorter, “secondary rays” that are inter- spersed in between the prominent rays (see Fig. S-5a,b). To shed light on the genesis of the secondary rays, we focus on the grains in the prominent rays and secondary rays from one simulation (Fig. S-5c). Tracing the evolution of these gains backward in time, we find that the grains in the prominent rays come from an annulus of valleys that straddle the edge of the impacting ball—as is noted in the letter—while the grains in the secondary rays come from an annulus of valleys that lies outward of the annulus that engenders the prominent rays (inset of Fig. S-5c). The location of these annuli leads to the key difference between the prominent and secondary rays. As discussed in the manuscript, the impact-generated shockwave creates a flow field that gets focused by the valleys to form rays. The strength of the

6 Figure S-5: Top view of secondary rays. (a) Experiment (same as Fig. 1d); (b) simulation (same as Fig. 1h). We mark some of the secondary rays in the images. (c) Simulated quadrant of panel (b), where we mark the grains in the prominent rays in red and the grains in the secondary rays in yellow. Inset: Zoomed-in view of the pre-impact surface showing the initial locations of the red and yellow grains. The blue quarter-circle shows the impacting ball. (d) Radial velocities corresponding to panel (c). Note that the grains in primary rays travel much faster than those in the secondary rays.

7 shockwave decays rapidly with increasing radial distance from the point of impact. Consequently, the ejecta velocities peak close to the edge of the impactor and rapidly decay with increasing radial distance [3]. The prominent rays are therefore more focused and travel at high velocities whereas the secondary rays are less focused and travel at much slower velocities (Fig. S-5d). As a result, the prominent rays are long and conspicuous, while the secondary rays are shorter and less conspicuous. Carrying forward with the above argument yields a cascade of rays 1. An annulus of valleys that lies outward of the annulus that engenders the secondary rays can engender tertiary rays, and so forth with the next annulus of valleys. Issued from a greater radial distance, the tertiary rays would be shorter and less conspicuous than the secondary rays, and so forth for each level of rays in the cascade. While in the present work we have not studied tertiary and higher-order rays, we expect that decreasing λ—which will reduce the distance between the annuli of valleys—and increasing the impact velocity—which will increase the ejecta velocities—would help accentuate the cascade of rays in the ejecta curtain.

4 Shockwave dynamics in the granular bed

As discussed in the letter, the impact of the ball on the granular bed gener- ates a shockwave that sweeps through the granular bed (Fig. S-6a,b). The compressive shockwave is reﬂected back as a rarefaction wave from the bed surface. While the impact produces high compressive stresses directly under- neath the ball, rarefaction waves reduce the stresses near the surface, forming a shallow zone of reduced stresses—known as the “interference zone” [6]— just outside the edge of the ball (Fig. S-6c). This is analogous to what is observed in computational simulations of planetary cratering with the “hy- drocode model” [6]. As noted in the letter, our simulations show that the ray-forming grains come from the valleys that straddle the edge of the impacting ball. Here we further note that these grains reside near the surface of these valleys, within a depth of ≈ 5d. The grains from this region, being part of the interference zone, are subjected to reduced stresses. Extrapolating to planetary ray systems, we submit that the ejecta in the rays are composed of weakly-shocked surﬁcial material from the planetary crust.

1We thank an anonymous referee for pointing out the possibility of such a cascade.

8 Figure S-6: Shockwave propagation in a hypervelocity simulation. (a) Hemi- spherical shockwave produced by impact of a ball along the centerline with velocity U = 20m/s. (b) Proﬁles of the grain velocities along the centerline at diﬀerent times t. Note the moving shockwave. (c) Compressive stresses in a cross-sectional slice of the granular bed (marked by red dashed lines in panel a). The stresses are normalized by the stagnation pressure, ρU 2, where ρ is the grain density. Similar to planetary impacts, the maximum stress is of the order of the stagnation pressure [6]. Also note the shallow zone of reduced stresses near the surface, next to the impactor edge.

5 Prominent valleys

As noted in the letter, our experiments on granular bed surfaces with irregular, multiple-wavelength undulations reveal that not all valleys engender prominent rays. Instead, only the prominent valleys engender prominent rays. We count the number of prominent valleys to estimate D for a known value of N, for which we follow the following procedure: For each experiment, we obtain a terrain map of the pre-impact surface using a laser scanner (David-SLS-2). From the terrain map, we extract one-dimensional topographic profiles along concentric circles of diameter Dc, centered at the point of impact. For each profile, we use the Matlab function findpeaks to identify the valleys2. To identify the prominent valleys, we pick a threshold for the peak prominence3 by setting the parameter MinPeakPromi-

2By default findpeaks identifies the peaks of the profile. To identify the valleys, we input the vertically-flipped profile. 3Findpeaks also computes “peak prominence”—a measure of “how much the peak stands out due to its intrinsic height and its location relative to other peaks”—defined as “ the minimum vertical distance that the signal must descend on either side of the peak before either climbing back to a level higher than the peak or reaching an endpoint” (see https://www.mathworks.com/help/signal/ref/findpeaks.html).

9 1.0 MinPeakProminence = 0.0

0.8

0.6

0.4 Normalized height 0.2

0.0 0 50 100 150 200 250 300 Azimuthal angle (degrees) (a) 1.0 MinPeakProminence = 0.05

0.8

0.6

0.4 Normalized height 0.2

0.0 0 50 100 150 200 250 300 Azimuthal angle (degrees) (b)

Figure S-7: Effect of the MinPeakProminence parameter in identifying prominent valleys. We show a one-dimensional topographic profile along a circle from an experiment with irregular surface undulations. The height is normalized to lie between 0 and 1. We mark the valleys identified by findpeaks with red dots; MinPeakProminence = 0 (a) and 0.05 (b). nence. In Fig. S-7 we illustrate the effect of MinPeakProminence. Note that the number of valleys decreases as we increase the value of MinPeakPromi- nence. For a fixed value of MinPeakProminence, we compute the number of valleys Nv in topographic profiles along circles of diameter Dc. From a plot of Nv vs. Dc, we estimate D as the Dc that corresponds to Nc = N. Here, our estimate of D depends on the selected value of MinPeakProminence. To obtain the value of MinPeakProminence that best identifies the ray-forming prominent valleys, we consider all our experiments and pick a single value of MinPeakProminence that minimizes the error between the estimated D and the actual D; see Fig. S-8 for some representative experimental results. This value is 0.05. We use the same value, MinPeakProminence=0.05, to identify prominent valleys and thus estimate D for the lunar craters Tycho and Kepler. To check

10 Observed rays : 15 Observed rays : 14 20 Observed rays : 11 20 Estimated diameter : 50.3 mm 20 Estimated diameter : 47.1 mm Estimated diameter : 50.7 mm 16 16 16

v 12 v N v 12 12 N N

8 8 8

4 4 4 10 30 50 70 10 30 50 70 10 30 50 70 Dc (mm) Dc (mm) Dc (mm) (a) (b) (c)

Figure S-8: Estimating impactor diameter. (First row) Top-view of ejecta blanket produced by dropping a 50.8 mm steel ball on granular beds with irregular surface undulations. The prominent rays are marked by dashed red lines. (Second row) Plots of the number of valleys, Nv vs. the circle diameter Dc. These plots correspond to the experiments shown in the panels above. The red circles correspond to Nv = N; the attendant Dc is the estimate for D.

11 the sensitivity of the estimated D on the value of MinPeakProminence, we vary the value of MinPeakProminence by ±25%. We ﬁnd that the estimated D remain comparable to those obtained using scaling laws; see Fig. S-9.

6 Impactor diameter and crater scaling laws

To estimate the impactor diameter for lunar craters based on the crater diameter, we use the following crater scaling law from Cintala and Grieve [7]: 0.28 ρi −0.42 1.09 −0.56 0.28 D = 0.828Dsc ( ) Dr vi g , (1) ρt where D is the impactor diameter, Dsc the simple-to-complex transition diameter, ρi and ρt the impactor and target density, Dr the ﬁnal crater diameter, vi the impact velocity and g the gravitational acceleration. This scaling law combines the crater scaling law of Schmidt and Housen [8] for the transient crater diameter with the scaling law proposed by Croft for relating the ﬁnal crater diameter to the transient crater diameter [9]. As in [7], we set Dsc = 18 km and and assume the target and impactor have the same density, ρi = ρt. For lunar craters g = 1.6 m/s2 and we assume an average impact velocity 15 km/s [10]. With these assumptions we estimate D = 7.3 km for Tycho and 2.5 km for Kepler.

References

[1] Christoph Kloss, Christoph Goniva, Alice Hager, Stefan Amberger, and Stefan Pirker. Models, algorithms and validation for opensource DEM and CFD–DEM. Prog. Computat. Fluid Dy., 12(2-3):140–152, 2012.

[2] Koji Wada, Hiroki Senshu, and Takafumi Matsui. Numerical simulation of impact cratering on granular material. Icarus, 180(2):528–545, 2006.

[3] JO Marston, EQ Li, and ST Thoroddsen. Evolution of ﬂuid-like granular ejecta generated by sphere impact. J. Fluid Mech., 704:5–36, 2012.

[4] Brendan Hermalyn and Peter H Schultz. Time-resolved studies of hypervelocity vertical impacts into porous particulate targets: Eﬀects of projectile density on early-time coupling and crater growth. Icarus, 216(1):269–279, 2011.

12 MinPeakProminence = 0.05. ed diameter 3.4 km. MinPeakProminence = 0.05. Estimated diameter 11 km. Estimat MinPeakProminence = 0.0375. ted diameter 2.8 km. 9 MinPeakProminence = 0.0375. Estimated diameter 9.3 km. Estima MinPeakProminence = 0.0625. ted diameter 3.8 km. MinPeakProminence = 0.0625. Estimated diameter 13 km. Estima 14

v 10 v N N

3 2 3.0 6.0 9.0 12.0 15.0 18.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Dc (km) Dc (km) Figure S-9: Sensitivity test for the value of MinPeakProminence on the estimated impactor diameter for lunar craters Tycho (left column) and Kepler (right column). The prominent rays are marked by dashed red lines. The red circles correspond to Nv = N; the attendant Dc is the estimate for D. For comparison, we note that scaling laws yield the estimate D = 7.3 km for Tycho and 2.5 km for Kepler (see section 6).

13 [5] T Kadono, AI Suzuki, K Wada, NK Mitani, S Yamamoto, M Arakawa, S Sugita, J Haruyama, and AM Nakamura. Crater-ray formation by impact-induced ejecta particles. Icarus, 250:215–221, 2015.

[6] H Jay Melosh. Impact cratering: A geologic process. Oxford Monographs on Geology and Geophysics, No. 11. Oxford University Press, 1989.

[7] Mark J Cintala and Richard AF Grieve. Scaling impact melting and crater dimensions: Implications for the lunar cratering record. Meteorit. Planet. Sci, 33(4):889–912, 1998.

[8] Robert M Schmidt and Kevin R Housen. Some recent advances in the scaling of impact and explosion cratering. Int. J. Impact Eng., 5(1):543– 560, 1987.

[9] Steven K Croft. The scaling of complex craters. J. Geophys. Res. Solid Earth, 90(S02), 1985.

[10] Z Yue, BC Johnson, DA Minton, HJ Melosh, K Di, W Hu, and Y Liu. Projectile remnants in central peaks of lunar impact craters. Nat. Geosci., 6(6):435–437, 2013.