Ray Systems in Granular Cratering

In classical experiments of granular cratering, a ball dropped on an evened-out bed of grains ends up within a crater surrounded by a uniform blanket of ejecta. In this Letter, we show that the uniform blanket of ejecta changes to a ray system, or set of radial streaks of ejecta, where the surface of the granular bed includes undulations, a factor that has not been addressed to date. By carrying out numerous experiments and computational simulations thereof, we ascertain that the number of rays in a ray system $\propto D/\lambda$, where $\mathit{D}$ is the diameter of the ball and $\lambda$ is the wavelength of the undulations. Further, we show that the ejecta in a ray system originates in a narrow annulus of diameter $\mathit{D}$ with the center at the site of impact. Our findings may help shed light on the enigmatic ray systems that ring many impact craters on the Moon and other planetary bodies.

Figure 1

Top view of ejecta in low-velocity experiments (a)–(d) and hypervelocity simulations (e)–(h). (a) Ejecta blanket in a typical experiment with an evened-out surface. Note the absence of a ray system. (b) Ejecta blanket in an experiment with an undulating surface (see inset). Note the ray system. (c),(d) Ejecta curtains showing ray systems in experiments with hexagonal surface undulations of wavelength $\lambda$ [see inset of panel (c)]; $\mathit{D}/\lambda =2.93$ (c) and 5.86 (d). (e) Ejecta curtain showing the absence of a ray system in a typical simulation with an evened-out surface. (f)–(h) Ejecta curtains showing ray systems in simulations with hexagonal surface undulations of wavelength $\lambda$ [see inset of panel (f)]; $\mathit{D}/\lambda =2.23$ (f), 3.45 (g), and 4.75 (h). Impact $\text{velocity}=20\mathrm{m}/\mathrm{s}$ for the simulations shown here. In all simulations, we simulate one quadrant and mirror the ejecta pattern assuming symmetry. In panels (d) and (h), in addition to the prominent rays, we can also see secondary rays (see Ref. [21]).

Figure 2

$\mathit{N}$ vs $\mathit{D}/\lambda$. Data from 135 low-velocity experiments with hexagonal surface undulations (see inset; blue circle denotes the impacting ball). The size of the data points (circles) is proportional to the number of overlapping points, with the area corresponding to any color proportional to the attendant value of $\mathit{D}$. The dashed line shows the best fit using linear regression ($R^2=0.98$).

Figure 3

Origin of ray systems. (a) A snapshot of grain velocities in the granular bed. At time $t=0$, the ball impacts the bed surface with $\text{velocity}=20\mathrm{m}/\mathrm{s}$. We show the simulated quadrant of the granular bed in the undeformed configuration; $\mathit{D}/\lambda =3.45$ for the hexagonal surface undulations. The gray region marks the current position of the ball inside the granular bed. The sharp interface between moving and stationary grains marks the shock wave. Inset: Azimuthal velocity $V_t$ on the surface patch demarcated by dotted lines. Induced by the surface-normal velocity from the valley sidewalls, $V_t$ focuses the affected grains into the radially outbound ejecta, thereby engendering a ray system. (b) Top view of the ray system produced by the impact shown in panel (a) [cf. Fig. 1]. The grains in the rays are marked in red. Inset: Enlarged view of the preimpact surface. The ray-forming grains (red grains) straddle the edge of the impacting ball (blue quarter circle). (c) $\mathit{N}$ vs $\mathit{D}/\lambda$. We show experimental data of Fig. 2 (red) and predictions from the geometric model (black), both binned with a bin size $\delta (\mathit{D}/\lambda )=1$. The vertical bars span 2 standard deviations of $\mathit{N}$. In the geometric model, for simplicity, we ignore the thickness of the valleys and idealize them as one-dimensional lines (see inset). Counting the number of intersections between the valleys and the edge of the ball (blue circle) yields the prediction for $\mathit{N}$.

Figure 4

Estimating impactor diameter: a granular crater (a)–(c) and the lunar crater Kepler (d)–(f). (a),(d) Ray systems in the ejecta blanket. The prominent rays are marked by dashed red lines. The granular ray system is produced by dropping a 50.8-mm-diameter steel ball on a granular bed with irregular surface undulations. (b),(e) Preimpact surfaces. For the granular crater, we obtain the terrain map of the preimpact surface using a laser scanner (David-SLS-2); the patch size is $10\times 10{\mathrm{cm}}^2$. For Kepler, we obtain the terrain data from the SLDEM 2015 database [33]; the patch size is $10\times 10{\mathrm{km}}^2$. (e),(f) $N_v$ vs $D_c$. For Kepler, this plot is obtained by averaging the $N_v$ vs $D_c$ data from 75 terrain patches. The estimated impactor diameter $\mathit{D}$ is the $D_c$ corresponding to $N_v=\mathit{N}$—these points are marked by red dots.