## 4 Discussion
### Scaling Laws of Impact Crater Parameters  
The scale analysis is an effective way to analyze the influence of the impact parameters on the impact crater dimensions. In accordance with the point source solutions proposed by [PERSON] [28; 30], the crater parameters can be represented as functions of the impact parameters. The impact parameters embody the projectile diameter \(d\), the projectile density \(\rho_{\rm p}\), the impact velocity \(u\), the target density \(\rho_{\rm t}\), and the target compressive strength \(Y\). In addition, the projectile length \(l\) is added to represent the shape influence of the projectile. Then, the crater depth \(H\) can be expressed as below:  
\[H=F(d,l,u,\rho_{p},\rho_{t},Y,\Pi_{m}) \tag{1}\]  
Figure 11: The influences of \(\rho_{\rm p}/\rho_{\rm t}\) on crater depth, diameter, volume, and crater \(H/D\) ratio: (**a**) dimensionless crater depth versus \(\rho_{\rm p}/\rho_{\rm t}\); (**b**) dimensionless crater diameter versus \(\rho_{\rm p}/\rho_{\rm t}\); (**c**) dimensionless crater volume versus \(\rho_{\rm p}/\rho_{\rm t}\); (**d**) crater \(H/D\) ratio versus \(\rho_{\rm p}/\rho_{\rm t}\). where the general symbol \(F\) represents an arbitrary function, and the symbol \(\Pi_{\text{m}}\) denotes the set of dimensionless material parameters. The \"coupling parameter\" \(C\), in which the projectile length \(l\) is added to represent the effect of the projectile shape, is given as below:  
\[C=du^{\mu}\rho_{p}^{\
u}l^{\xi} \tag{2}\]  
Then, Equation (1) can be rewritten by adopting the coupling parameters as follows:  
\[H=F\Big{(}du^{\mu}\rho_{p}^{\
u}l^{\xi},\rho_{t},Y,\Pi_{\text{m}}\Big{)} \tag{3}\]  
The coupling parameter with length scale is expressed as below:  
\[C=d\Bigg{(}\frac{u\rho_{t}^{\frac{1}{2}}}{Y^{\frac{1}{2}}}\Bigg{)}^{\mu}\Bigg{(} \frac{\rho_{p}}{\rho_{t}}\Bigg{)}^{\
u}\Bigg{(}\frac{l}{d}\Bigg{)}^{\xi} \tag{4}\]  
where the term \(l\)/\(d\) is the length-diameter ratio of the projectile. Then Equation (3) can be scaled using Equation (4) as follows:  
\[ \begin{array}{l}\frac{H}{d\Bigg{(}\frac{u\rho_{t}^{\frac{1}{2}}}{Y^{\frac{1 }{2}}}\Bigg{)}^{\mu}\Bigg{(}\frac{\rho_{p}}{\rho_{t}}\Bigg{)}^{\
u}\Bigg{(} \frac{l}{d}\Bigg{)}^{\xi}}=F\Bigg{(}\frac{\Bigg{(}\frac{u\rho_{t}^{\frac{1}{2} }}{Y^{\frac{1}{2}}}\Bigg{)}^{\mu}\Bigg{(}\frac{\rho_{p}}{\rho_{t}}\Bigg{)}^{ \
u}\Bigg{(}\frac{l}{d}\Bigg{)}^{\xi}}{\Bigg{(}\frac{u\rho_{t}^{\frac{1}{2}}} {Y^{\frac{1}{2}}}\Bigg{)}^{\mu}\Bigg{(}\frac{\rho_{p}}{\rho_{t}}\Bigg{)}^{\
u }\Bigg{(}\frac{l}{d}\Bigg{)}^{\xi}},\frac{\
u}{\rho_{t}},\frac{\
u}{\gamma},\Pi_ {\text{m}}\\ =F(1,1,1,\Pi_{m})\\ =F(\Pi_{m})=k\end{array}  \tag{5}\]  
where \(k\) is defined as a function of \(\Pi_{m}\). Then, Equation (5) can be rewritten in a dimensionless form as follows:  
\[\frac{H}{d}=k\Bigg{(}\frac{u\rho_{t}^{\frac{1}{2}}}{Y^{\frac{1}{2}}}\Bigg{)} ^{\mu}\Bigg{(}\frac{\rho_{p}}{\rho_{t}}\Bigg{)}^{\
u}\Bigg{(}\frac{l}{d} \Bigg{)}^{\xi} \tag{6}\]  
By using the dimensionless parameter \(H\)/\(l\) instead of \(H\)/\(d\), Equation (6) can be rewritten as follows:  
\[\frac{H}{l}=\frac{H}{d}\cdot\frac{d}{l}=k\Bigg{(}\frac{u\rho_{t}^{\frac{1}{2} }}{Y^{\frac{1}{2}}}\Bigg{)}^{\mu}\Bigg{(}\frac{\rho_{p}}{\rho_{t}}\Bigg{)}^{ \
u}\Bigg{(}\frac{l}{d}\Bigg{)}^{\xi-1} \tag{7}\]  
In this study, the length-diameter ratios \(l\)/\(d\) of all the projectiles keep a constant of 2. Then, \((l\)/\(dss)^{\xi-1}\) is also a constant coefficient. Set \(K=k(l\)/\(d)^{\xi-1}\), then Equation (7) can be simplified as follows:  
\[\frac{H}{l}=K\Bigg{(}\frac{u\rho_{t}^{\frac{1}{2}}}{Y^{\frac{1}{2}}}\Bigg{)} ^{\mu}\Bigg{(}\frac{\rho_{p}}{\rho_{t}}\Bigg{)}^{\
u} \tag{8}\]  
The multiple regression analysis was adopted for the experimental data to calculate the coefficient \(K\) and the exponents \(\mu\) and \(\
u\).