Impact craters formed by spinning granular projectiles

Craters formed by the impact of agglomerated materials are commonly observed in nature, such as asteroids colliding with planets and moons. In this paper, we investigate how the projectile spin and cohesion lead to different crater shapes. For that, we carried out discrete element method computations of spinning granular projectiles impacting onto cohesionless grains for different bonding stresses, initial spins, and initial heights. We found that, as the bonding stresses decrease and the initial spin increases, the projectile's grains spread farther from the collision point, and in consequence, the crater shape becomes flatter, with peaks around the rim and in the center of the crater. Our results shed light on the dispersion of the projectile's material and the different shapes of craters found on Earth and other planetary environments.

Figure 1

(a) Craters on Earth's moon (in the middle, with a smaller bow-shaped crater inside, is Poinsot crater). (b) Craters on Vesta, with a recent 20-km-diameter crater at the top of the image. (c) Layout of the numerical setup (the $y$ coordinate points downward, and although shown on the bottom, the origin of the coordinate system is on the bed surface centered horizontally in the domain). (d) The 445-km-diameter crater on Saturn's moon Tethys. (e) The 76-mm-diameter crater obtained numerically from the impact of a 25-mm-diameter steel sphere falling from 50 mm onto a bed of particles (glass spheres with a mean diameter of 1 mm). (f) Topography (elevation) of a crater formed by a spinning projectile consisting of bonded grains (we notice at least one internal peak close to the rim). In this panel, the bonding stresses are ${10}^7$ ${\mathrm{N/m}}^2$, the ratio between linear and angular kinetic energies is 1, and the color bar shows the elevation from the undisturbed surface (pointing downward). Images in (a), (b), and (d) are courtesy NASA/JPL-Caltech.

Figure 2

Top view of final positions of grains, showing the final morphology of craters for nonrotating and rotating projectiles with different bonding stresses. For spinning projectiles, $h$ = 0.1 m. The color bar on the right shows the elevation of each grain from the undisturbed surface (coordinate pointing downward). The same figure in gray scale is available in the Supplemental Material [27].

Figure 3

Topography (elevation) of the final craters for nonrotating projectiles with different bonding stresses. The color bar to the right of each panel shows the elevation from the undisturbed surface in meters. The same figure in gray scale is available in the Supplemental Material [27].

Figure 4

Topography (elevation) of the final craters for rotating projectiles with different bonding stresses. The color bar to the right of each panel shows the elevation from the undisturbed surface in meters, and $h$ = 0.1 m for all panels. The same figure in gray scale is available in the Supplemental Material [27].

Figure 5

Profiles of the elevations of the final craters for both nonrotating and rotating projectiles, with different bonding stresses. The heights and rotational energies are shown in the legend, and $h$ = 0.1 m for rotating projectiles. All profiles were plotted in a vertical plane of symmetry (and therefore include the crater center). These profiles include the projectile's grains (see the Supplemental Material [27] for profiles excluding the projectile's grains).

Figure 6

(a) Crater diameter $D_c$, (b) penetration depth $\delta$, and (c) the percentage of broken bonds as a function of the initial height $h$ for a nonrotating projectile; (d)–(f) show $D_c$, $\delta$, and the percentage of broken bonds as a function of $K_{\omega }/K_v$ for spinning projectiles falling from $h$ = 0.1 m, respectively. The graphics are parameterized by the bonding stresses [shown in the legend in (a)], and the results for the solid projectile reported in Carvalho et al. [6] are shown for reference.

Figure 7

Final positions of the projectile's grains after the impact has taken place for ${\sigma }_p$ = 1 $\times {10}^7$ ${\mathrm{N/m}}^2$ and different values of $K_{\omega }/K_v$. (a) Frequencies of occurrence of the projectile's grains in the $r\text{−}\theta$ plane (radius-angle plane, independent of the depth), (b) frequencies of occurrence of final positions in terms of the angle (all depths), (c) frequencies of occurrence of final positions in terms of radius (all depths), and (d) frequencies of occurrence of final positions in the $y$ coordinate (depths for all angles and radii).