### Scaling by conventional crater scaling law

The pi-scaling law for crater size in the strength regime describes crater size, *R*, using the dimensionless parameters *π*_{R} = *R*(*δ*_{t}/*m*)^{1/3}, *π*_{3} = (*Y*_{t}*/δ*_{t}*V*^{2}), and *π*_{4} = (*δ*_{t}/*δ*_{p}) as

$${pi }_{R}={k}_{1}{{pi }_{3}}^{-mu /2}{{pi }_{4}}^{(1-3nu )/3},$$

(1)

where *δ*_{t}, *m*, *V*, and *δ*_{p} are target density, projectile mass, impact velocity, and projectile density, respectively, and ({k}_{1}), (mu), and (nu) are empirical parameters related to the point source approximation and determined by laboratory experiments (Housen and Holsapple 2011). Since *π*_{R} ~ (*R/D*_{p})π_{4}^{1/3}, where *D*_{p} is the projectile diameter, Eq. (1) becomes

$$left(frac{R}{{D}_{mathrm{p}}}right)sim {left(frac{{delta }_{mathrm{p}}}{{delta }_{mathrm{t}}}right)}^{nu }{left(frac{{Y}_{mathrm{t}}}{{{delta }_{mathrm{t}}V}^{2}}right)}^{-mu /2}.$$

(2)

Figure 4 shows normalized crater size as a function of the right-hand side of Eq. (2). We set the empirical parameters, (mu) and (nu), to typical values previously obtained for hard rocks of 0.55 and 0.4, respectively (Holsapple 1993; Housen and Holsapple 2011), and the projectile density *δ*_{p} to the bulk density of projectiles, *ρ*_{p}. Previous data are also plotted for basalt (Dohi et al. 2012) and porous gypsum (Yasui et al. 2012). For basalt targets, the data of surface diameter *D*_{s}/*D*_{p} in (a) and depth *d*/*D*_{p} in (b) are in good agreement with the previous data and show a linear relationship. For porous gypsum, the data for the surface (spall) diameter *D*_{s}/*D*_{p} are scattered, but the data for inner (pit) diameter *D*_{i}/*D*_{p} and depth *d*/*D*_{p} appear to be linearly correlated and in good agreement with the previous data using different types of projectiles. Thus, even when projectiles have varied shapes, crater sizes in the strength regime can be scaled using a conventional scaling law previously established, using bulk density as *ρ*_{p}. Note that spallation in porous gypsum targets often produces large fragments (e.g., Fig. 3 in Suzuki et al. 2018). Such a process with large crack propagation is highly probabilistic; hence, the surface (spall) diameter *D*_{s}/*D*_{p} for porous gypsum may scatter. This may be the reason why spallation occurred in shot number 0606-2 but not in 0606-3, even though the impact conditions for these shots were almost the same. On the other hand, the reason of no spallation in shot number 0606-4 would be different. The density ratio of projectile to target in shot number 0606-4 was very high ~ 9: such high density ratio generally causes a carrot-shaped crater without spallation (e.g., Love et al. 1993; Kadono et al. 2012). Furthermore, target differences cannot be scaled by only strength and density. Other parameters should be considered, but these are beyond the scope of this paper and are not discussed further.

As an application of our results, we consider the case if the SCI projectile collides with a boulder on Ryugu (e.g., “SB” boulder with a size of 5 m existing in the vicinity of the SCI impact point as shown in Arakawa et al. 2020). The left-hand side of Eq. (2) becomes ~ 6.6, when setting *ρ*_{p} and *V* to 1.74 g/cm^{3} and 2 km/s for the SCI projectile (Arakawa et al. 2020) and *ρ*_{t} and *Y*_{t} to 1 g/cm^{3} and 1.7 MPa for the SB boulder (Kadono et al. 2020). In this case, if porous gypsum can simulate the boulders with high porosity on Ryugu, Fig. 4 indicates that *D*_{i}/*D*_{p} and *d*/*D*_{p} are ~ 1, respectively, suggesting that the crater is much smaller than the actual crater size that formed in the gravity regime, and would have been extremely difficult to find.

### Numerical simulations

We investigated the crater formation process under the conditions experimented in this study using a general-purpose shock physics code, iSALE-2D (Wünnemann et al. 2006) to confirm the limited dependence of crater size on projectile shape. This code is an extension of the SALE code (Amsden et al. 1980), which is capable of modeling shock processes in geological materials (Melosh et al. 1992; Ivanov et al. 1997; Collins et al. 2004; Wünnemann et al. 2006).

Three types of projectiles were considered: closed copper tubes, open copper tubes, and solid aluminum spheres, corresponding to projectiles described in Figs. 1a, b, and d, respectively. The diameter and height of all the projectiles were set to 3.2 mm. To simulate the impact of projectiles having a similar bulk density, the thicknesses of the closed and open tubes were set to 0.20 and 0.25 mm, respectively, resulting in the bulk densities of the closed and open tubes being 2.53 and 2.58 g/cm^{3}, respectively.

Four types of impacts were simulated: (a) a closed tube impacting on the closed side, (b) a closed tube impacting on the open side, (c) an open tube on one open side, and (d) a solid aluminum sphere. These projectiles collided perpendicularly on basaltic target flat surfaces along the central axis. Impact velocity was set at 2 km/s.

The calculation settings are summarized as follows: we used the two-dimensional version of iSALE, which is referred to as iSALE-Dellen (Collins et al. 2016) to simulate the vertical impacts performed in the experiments. We used the Tillotson equation of state (EOS) for copper and aluminum (Tillotson 1962) and the “Analytical” EOS (ANEOS) (Thompson and Lauson 1972) for basalt (Pierazzo et al. 2005; Sato et al. 2021). We also employed a constitutive model to calculate the elastoplastic behaviors of both shocked projectiles and targets. The Johnson–Cook model (Johnson and Cook 1983) was used for metal projectiles. We used the “ROCK” model implemented in the iSALE for the basalt target, which is a combination of the Drucker–Prager model (Drucker and Prager 1952) for damaged rocks and the Lundborg model for intact rocks (Lundborg 1968). The two models were coupled with a damage parameter ranging from 0 to 1, depending on the total plastic strain (e.g., Ivanov et al. 1997; Collins et al. 2004). It was not feasible for the numerical integrations to continue until the end of crater formation, and hence, we addressed the peak pressure and resultant particle velocity distributions at a given time. It has been shown that iSALE represents the maximum (peak) pressures experienced at each position in the targets caused by the shock wave well (Nagaki et al. 2016). The compression to a sufficient pressure by the shock wave and the release from the pressure by the rarefaction wave cause the fragmentation of the target material. Crater depth would correspond to a position experienced by a critical peak pressure value. On the other hand, crater diameter is directly related to fragmentation at the target surface, caused by a tensile phase due to the rarefaction wave (e.g., Melosh 1989). Since the rarefaction wave leads to upward motion of the target materials, the distribution of particle velocity near the surface should represent the extent of fragmentation near the surface. If the peak pressure and particle velocity distributions do not strongly depend on projectile shape and interior, the dimensions of the final crater are expected to be similar. To accurately reproduce the dependence of peak pressure distribution on projectile shape and interior in the simulation, we divided the wall thickness into 20 cells, resulting in > 160 cells per projectile radius for the three copper tubes. We inserted Lagrangian tracer particles into the computational cells and stored the maximum pressures experienced at a given time and temporal particle velocity in the simulations. The input parameters of the material models and calculation settings are summarized in Tables 2, 3, and 4.

Figure 5 shows the results of the calculations: contours of the peak pressure experienced (right) and particle velocity (left). Note that we only used tracers with upward velocities in this plot (hereafter referred to as upward particle velocity). It appears that in any case, pressure decreases along the central axis in the same way, and that the contour lines of the upward particle velocity are distributed at a similar location. To evaluate the peak pressure and particle velocity more quantitatively, we obtained the profiles of the peak pressure and particle velocity along the central axis and target surface, respectively (the analyses are described in Additional file 1: S1 in detail). Figure 6 shows that the profiles of (a) peak pressure experienced and (b) particle velocity for the results shown in Fig. 5 along the central axis (*Z*-axis) and the target surface (*R*-axis), respectively (strictly a line 5 cells away from the *Z* and *R* axes; see the Additional file 1: S1). These show that the profile for each impact condition decreases with a similar slope at distances greater than projectile diameter *D*_{p} and that the difference between the profiles is within a factor of ~ 2 − 3. This implies that when projectile bulk density and impact velocity are the same, the pressure of shock wave detached from isobaric core is almost independent of the projectile shape and internal structure. Even if projectile has an internal structure (e.g., voids), since shock pressure is much higher than the strength of projectile, its shape and the internal structure are completely crushed, and the compressed density and average shock pressure are independent of initial shape and internal structure. Moreover, attenuation of detached shock waves depends on the geometrical expansion of shock waves and elastic–plastic properties of target materials. Therefore, shock pressure and attenuation are independent of initial projectile structure. Thus, even if projectile shape and internal structure vary, the crater depth and diameter become almost the same when the bulk density and diameter of projectile and impact velocity are the same. Note that the profiles also show that the result for the aluminum solid projectile is similar to those for copper projectiles with internal structure. This suggests that even when projectiles are made of different materials, shock impedance becomes similar if bulk density is the same. More systematic investigation is necessary to understand why similar pressures are generated by porous projectiles with internal structure and solid projectiles of different materials when bulk density is the same.

We also simulated the impacts corresponding to the experiments with shot number 1031-2 (closed copper tube colliding at the closed end), 1031-3 (closed copper tube colliding at the open end), 0925-1 (open copper tube), and 1031-5 (solid aluminum sphere). The results are shown in Fig. 7a–d. The contours of the experienced peak pressure and particle velocity are shown on the right and left halves, respectively. For comparison, the crater profile in the corresponding experiment is overlaid (green curve). The shape of the cavity in the targets is very different from that of the final crater and does not become similar thereafter, although the size is comparable. It seems that it is still difficult to reproduce the exact shape of a crater in targets with strength. Figure 8 shows the profiles of (a) peak pressure experienced and (b) particle velocity for the results shown in Fig. 7 along the central axis and target surface, respectively (strictly a line 5 cells away). The crater depth and radius normalized by *D*_{p} obtained in the experiments are also indicated. The peak pressure and particle velocity corresponding to the experimental results of crater depth and radius are shown in Fig. 9. The depth and radius of the final crater corresponded to ~ 2 GPa and ~ 10 m/s, respectively, in each case. In this figure, the results of the shot number 0925-2 and one of the shots in Dohi et al. (2012) (090528-1) are also plotted. (Note that the calculations for 0925-2 and 090528-1 in iSALE were conducted with low resolution and that nylon projectile was used in 090528-1. These calculations in iSALE with low resolution and nylon are described in Additional file 1: S2 and S3, respectively.) Even though the projectile density is different, the corresponding pressure is similar, and so is the corresponding particle velocity. The averages of the corresponding peak pressure and particle velocity are 1.9 ± 0.6 GPa and 13.0 ± 3.6 m/s, respectively. The compressive strength of basalt has been measured to be several hundred megapascals (e.g., Lockner 1995), which is slightly lower than the pressure corresponding to the crater depth. However, the compressive strength values were obtained using static compression tests, and the effect of strain rate could explain this difference (e.g., Kimberley et al. 2013). Therefore, the maximum depth of the final craters in the strength-dominated regime was determined by the balance between the intensity of the compressive pulse and the compressive strength of the target materials. On the other hand, the upward particle velocity *u*_{p} of ~ 10 m/s corresponds to the tensile stress, *T*, of several tens of megapascals, which is estimated from *δ*_{t}*C*_{0}*u*_{p}, where *δ*_{t} and *C*_{0} are the density and bulk sound velocity of basalt, 2.71 g/cm^{3} (the average value for the targets in our experiments) and 2.60 km/s (e.g., Melosh 1989), respectively. This is similar to the tensile strength of basalt (e.g., Lockner 1995), suggesting that crater diameter is related to the tensile process. The results, namely, that crater depth and diameter are related to the compressive and tensile strengths of the targets, are consistent with our current understanding of the cratering process (e.g., Melosh 1989).

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