Erosion and Accretion by Cratering Impacts on Rocky and Icy Bodies

Cratering impacts lead to an ejection of the target material from the surface. Pioneering studies (Housen et al. 1983; Holsapple & Schmidt 1987; Holsapple 1993; Holsapple & Housen 2007; Housen & Holsapple 2011) predicted m( > v_eje,tar) as a function of the impact velocity v_imp and impact angle θ assuming that the size of the impact crater is much larger than that of the impactor (hereafter M_HH11,esc,tar). ³ The point-source scaling law can potentially predict the escape mass by using the relation, v_eje,tar = v_esc, where v_esc is the escape velocity of the target, as follows (hereafter HH11):

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{\mathrm{HH}11,\mathrm{esc},\mathrm{tar}}(\gt {v}_{\mathrm{esc}})}{{m}_{\mathrm{imp}}}={C}_{\mathrm{HH}11}{\left(\displaystyle \frac{{v}_{\mathrm{esc}}}{{v}_{\mathrm{imp}}\sin \theta }\right)}^{-3{\mu }_{\mathrm{HH}11}},\end{eqnarray} \tag{ 1 }$$

where ${C}_{\mathrm{HH}11}=(3k/4\pi ){C}_{0}^{3{\mu }_{\mathrm{HH}11}}$ and μ_HH11 are constants. From the experiments, C₀ = 1.5 and μ_HH11 = 0.55 for nonporous icy materials and rocky materials, respectively (e.g., Holsapple & Housen 2007). k = 0.2 for ice and k = 0.3 for rock, respectively (Gault et al. 1963). The densities of the impactor and target are assumed to be identical.

The impact shock propagates through the interior of the target after the impactor makes contact with the target surface. The crater forms as the target material is sheared, moving upward and outward along the bowl-shaped crater edge (Figure 1; see also Housen & Holsapple 2011). The speed of the ejecta v_eje depends on its launch position (e.g., Piekutowski 1980), that is, high- and low-speed ejecta are launched closer to and farther from the impact point, respectively (Figure 1).

Zoom In Zoom Out Reset image size

Figure 2 shows the cumulative mass of ejecta originating from the target as a function of ejection velocity v_eje for different impact velocities and impact angles. The ejection velocity is scaled by ${v}_{\mathrm{imp}}\sin \theta$. Note that the data for the ejection velocity with ${v}_{\mathrm{eje}}\lesssim 0.1{v}_{\mathrm{imp}}\sin \theta$ failed to converge during the performed simulation time (i.e., more numerical time is needed for the convergence of smaller v_eje because the lower-speed ejecta is launched farther from the impact point ⁴ ). The point-source scaling law (Equation (1)) is plotted as a black line. The results indicate that the distribution is uniquely scaled by ${v}_{\mathrm{imp}}\sin \theta$, which is in accordance with the basic principles evident in the point-source assumption (Holsapple & Housen 2007).

Zoom In Zoom Out Reset image size

The slope of the ejection velocity distribution is characterized by −3μ_HH11, a unique constant value, in the case of the point-source scaling law (Equation (1)). This agrees with the results obtained from direct impact simulations only for the low-speed ejecta (Figure 2). This indicates that the point-source scaling law is valid only for a limited range of ejection velocity distributions.

The ejecta velocity distribution has a steeper slope for high-speed ejecta than that predicted by the point-source scaling law (see also Figure 1). The ejection velocity distribution for such high-speed ejecta has a unique power-law dependence depending on different impact angles (Figure 2). This result indicates that the point-source scaling law overestimates the mass of the high-speed ejecta with a launch point near the impact point. This was also reported in the case of rocky bodies (Paper I). In the next subsection, updated scaling laws are derived for the high-speed ejecta where the point-source scaling law is not applicable.

Following the methodology of Paper I and from the arguments in the previous subsection, the escape mass of the target material ${M}_{\mathrm{HG}21,\mathrm{esc},\mathrm{tar}}^{* }$ for the high-speed ejecta regime ($\gtrsim 0.1{v}_{\mathrm{imp}}\sin \theta$) is uniquely expressed by a power law as a function of impact velocity and impact angle as follows:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{M}_{\mathrm{HG}21,\mathrm{esc},\mathrm{tar}}^{* }(\gt {v}_{\mathrm{esc}})}{{m}_{\mathrm{imp}}}\\ \quad =\,{C}_{\mathrm{HG}21,\mathrm{tar}}(\theta ){\left(\displaystyle \frac{{v}_{\mathrm{esc}}}{{v}_{\mathrm{imp}}\sin \theta }\right)}^{-3{\mu }_{\mathrm{HG}21,\mathrm{tar}}(\theta )},\end{array}\end{eqnarray} \tag{ 2 }$$

where C_HG21,tar(θ) and μ_HG21,tar(θ) are a new coefficient and a new exponent that depends on the impact angle, respectively.

One may obtain the exact C_HG21,tar(θ) and μ_HG21,tar(θ) at varying impact angles using the numerical results (indicated by blue points in Figure 3). Subsequently, the approximated μ_HG21,tar(θ) and C_HG21,tar(θ) (the black lines in Figure 3) were derived using the quadratic and cubic functions of the impact angle (the black lines in Figure 3) as

$$\begin{eqnarray}&&{\mu }_{\mathrm{HG}21,\mathrm{tar}}(\theta )={a}_{\mathrm{tar}}{\theta }^{2}+{b}_{\mathrm{tar}}\theta +{c}_{\mathrm{tar}}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{C}_{\mathrm{HG}21,\mathrm{tar}}(\theta )=\exp \left({d}_{\mathrm{tar}}{\theta }^{3}+{e}_{\mathrm{tar}}{\theta }^{2}+{f}_{\mathrm{tar}}\theta +{g}_{\mathrm{tar}}\right),\end{eqnarray} \tag{ 4 }$$

where a_tar, b_tar, c_tar, d_tar, e_tar, f_tar, and g_tar are the fitted parameters, respectively (Table 1).

Zoom In Zoom Out Reset image size

Figure 4 shows the escape mass of the target material (target mass with an ejection velocity greater than the escape velocity) as a function of the impact velocity. Points were obtained from the numerical simulations.

Zoom In Zoom Out Reset image size

As Paper I established that for rocky bodies, the original point-source scaling law for icy bodies (HH11; Equation (1)) is valid only at sufficiently large distances from the impact point (i.e., for a sufficiently small ejection velocity). The coefficient and exponent deviate from those of HH11 (compare the solid black line and the other lines in Figure 2) for large ejection velocities (i.e., small distances to the impact point).

As observed in Figure 2 (see also Figure 1), HH11 for icy bodies overestimates the escape mass of the target material, especially for a significant ejection velocity. Therefore, the new scaling law for the escape mass of the target material that combines with HH11 is given by $\min \{{M}_{\mathrm{HG}21,\mathrm{esc},\mathrm{tar}}^{* },{M}_{\mathrm{HH}11,\mathrm{esc},\mathrm{tar}}\}$ and is written as follows (hereafter, HG21):

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{M}_{\mathrm{HG}21,\mathrm{esc},\mathrm{tar}}(\gt {v}_{\mathrm{esc}})}{{m}_{\mathrm{imp}}}\\ \quad =\,\min \left\{{C}_{\mathrm{HG}21,\mathrm{tar}}(\theta ){\left(\displaystyle \frac{{v}_{\mathrm{esc}}}{{v}_{\mathrm{imp}}\sin (\theta )}\right)}^{-3{\mu }_{\mathrm{HG}21,\mathrm{tar}}(\theta )},\right.\\ \qquad \left.{C}_{\mathrm{HH}11}{\left(\displaystyle \frac{{v}_{\mathrm{esc}}}{{v}_{\mathrm{imp}}\sin (\theta )}\right)}^{-3{\mu }_{\mathrm{HH}11}}\right\}.\end{array}\end{eqnarray} \tag{ 5 }$$

The new scaling law (Equation (5)) is plotted by solid and dashed lines in Figure 4. A close match between the updated scaling law and impact simulations (points) for v_imp = 6–62 km s⁻¹ and θ = 15°–90° for v_esc = 1–10 km s⁻¹ was observed.

The point-source scaling law (Equation (1)) has been widely used. As demonstrated in the case of the rocky bodies (Paper I), the point-source scaling law for the escape mass of the target material of icy bodies also agrees with the numerical results when v_imp ≫ v_esc, whereas it overestimates the escape mass when v_imp ≳ v_esc.

Figure 5 shows the overestimation of the point-source scaling law (HH11; Equation (1)) when compared to the results of the direct impact simulations (HG21; Equation (5)) as a function of the impact velocity. Except for impact angles of 15° and 30°, HH11 overestimated the escape mass for v_imp ≲ 9v_esc, which exponentially increased as the impact velocity decreased. As found in the case of the rocky bodies (Paper I), the difference became more significant as the impact angle tended toward a head-on collision. HH11 overestimated the escape mass of the target material by approximately ∼100 times when v_imp/v_esc ∼ 1 at vertical impact (θ = 90°). For the θ-averaged escape mass over the $\sin 2\theta$ distribution (Equations (10)–(11)), HH11 overestimated by a factor of ∼4 larger than HG21 when v_imp/v_esc ∼ 1.

Zoom In Zoom Out Reset image size

Cratering impacts of small bodies on a large planetary body occur much more frequently than collisions between similar-sized bodies. The statistical distribution of the impact angle follows $\sin 2\theta$ with a peak of 45° (Shoemaker 1962).

Using the newly derived scaling laws for the escape mass of icy target material (Equation (5)) and the accretion mass of icy impactor material (Equation (9)), one can derive the θ-averaged scaling laws weighted over the $\sin 2\theta$ distribution (see Paper I for rocky bodies). These are as follows.

The θ-averaged escape mass originating from the icy target as a function of impact velocity was derived as

$$\begin{eqnarray}\begin{array}{rcl}{\left\langle \displaystyle \frac{{M}_{\mathrm{ice},\mathrm{esc},\mathrm{tar}}(\gt {v}_{\mathrm{esc}})}{{m}_{\mathrm{imp}}}\right\rangle }_{\theta } & = & 0.013{\left(\displaystyle \frac{{v}_{\mathrm{imp}}}{{v}_{\mathrm{esc}}}\right)}^{2.29}\\ & & \left[{\rm{Icy}}\,{\rm{bodies;}}\,{v}_{\mathrm{imp}}\lesssim 9{v}_{\mathrm{esc}}\right]\end{array}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\left\langle \displaystyle \frac{\left.{M}_{\mathrm{ice},\mathrm{esc},\mathrm{tar}}(\gt {v}_{\mathrm{esc}}\right)}{{m}_{\mathrm{imp}}}\right\rangle }_{\theta } & = & 0.051{\left(\displaystyle \frac{{v}_{\mathrm{imp}}}{{v}_{\mathrm{esc}}}\right)}^{1.65}.\\ & & \left[{\rm{Icy}}\,{\rm{bodies;}}\,{v}_{\mathrm{imp}}\gtrsim 9{v}_{\mathrm{esc}}\right].\end{array}\end{eqnarray} \tag{ 11 }$$

That for rocky bodies was (derived in Paper I):

$$\begin{eqnarray}\begin{array}{rcl}{\left\langle \displaystyle \frac{{M}_{\mathrm{rock},\mathrm{esc},\mathrm{tar}}(\gt {v}_{\mathrm{esc}})}{{m}_{\mathrm{imp}}}\right\rangle }_{\theta } & = & 0.02{\left(\displaystyle \frac{{v}_{\mathrm{imp}}}{{v}_{\mathrm{esc}}}\right)}^{2.2}\\ & & \left[\mathrm{Rockybodies};{v}_{\mathrm{imp}}\lesssim 12{v}_{\mathrm{esc}}\right]\end{array}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\left\langle \displaystyle \frac{{M}_{\mathrm{rock},\mathrm{esc},\mathrm{tar}}(\gt {v}_{\mathrm{esc}})}{{m}_{\mathrm{imp}}}\right\rangle }_{\theta } & = & 0.076{\left(\displaystyle \frac{{v}_{\mathrm{imp}}}{{v}_{\mathrm{esc}}}\right)}^{1.65}.\\ & & \left[{\rm{Rocky}}\,{\rm{bodies;}}\,{v}_{\mathrm{imp}}\gtrsim 12{v}_{\mathrm{esc}}\right].\end{array}\end{eqnarray} \tag{ 13 }$$

In the above equations, one uses the smaller result derived from the pairs (Equations (10)–(11) or Equations (12)–(13)). Equations (11) and (13) are obtained by averaging the point-source scaling laws. ⁵

The θ-averaged accretion mass originating from the icy impactor as a function of impact velocity is

$$\begin{eqnarray}\begin{array}{rcl}{\left\langle \displaystyle \frac{{M}_{\mathrm{ice},\mathrm{acc},\mathrm{imp}}(\lt {v}_{\mathrm{esc}})}{{m}_{\mathrm{imp}}}\right\rangle }_{\theta } & = & 0.88-0.024{\left(\displaystyle \frac{{v}_{\mathrm{imp}}}{{v}_{\mathrm{esc}}}\right)}^{1.66}.\\ & & \left[{\rm{Icy}}\,{\rm{bodies}}\right]\end{array}\end{eqnarray} \tag{ 14 }$$

That for rocky bodies was (derived in Paper I):

$$\begin{eqnarray}\begin{array}{rcl}{\left\langle \displaystyle \frac{{M}_{\mathrm{rock},\mathrm{acc},\mathrm{imp}}(\lt {v}_{\mathrm{esc}})}{{m}_{\mathrm{imp}}}\right\rangle }_{\theta } & = & 0.85-0.071{\left(\displaystyle \frac{{v}_{\mathrm{imp}}}{{v}_{\mathrm{esc}}}\right)}^{0.88}.\\ & & \left[{\rm{Rocky}}\,{\rm{bodies}}\right]\end{array}\end{eqnarray} \tag{ 15 }$$

These θ-averaged scaling laws are shown in Figure 8. In Section 5.3, the typical outcome of a cratering impact at a variety of planetary bodies in the solar system (see Table 2) is discussed.

Zoom In Zoom Out Reset image size

Small impactors are attracted by the gravity of large target bodies (i.e., cratering impacts) and impact velocity becomes greater than the escape velocity of the target (v_imp ≳ v_esc). Our numerical simulations neglected material strength. In this subsection, using a simple but a well-defined analytical argument, we estimate the critical target size below which material strength affects the determination of ejecta having velocity greater than the escape velocity of the target (i.e., ejecta of v_eje ≥ v_esc).

Using the Rankine–Hugoniot relations, the shock pressure P_shock during the cratering impact is given by

$$\begin{eqnarray}&&{P}_{\mathrm{shock}}={\rho }_{0}\left({C}_{0}+{{su}}_{{\rm{p}}}\right){u}_{{\rm{p}}},\end{eqnarray} \tag{ 16 }$$

where ρ₀ is material density before the shock compression, C₀ is bulk sound velocity (C₀ ≃ 1000–3000 m s⁻¹ for rocky and icy materials, respectively; Melosh 1989), u_p is a particle velocity (u_p is related to shock velocity as u_s = C₀ + su_p), and s ≡ du_s/du_p is a constant (s ≃ 1–2; Melosh 1989).

For P_shock > P_HEL, where P_HEL is the Hugoniot elastic limit (HEL), a critical pressure above which a fluid-like response is expected, the ejection process is expected to be unaffected by material strength. In this paper, the minimum ejection velocity of our primary interest is equal to escape velocity (i.e., v_eje = v_esc and a larger ejection velocity accompanies a larger shock pressure). u_eje should be larger than u_p due to acceleration during adiabatic expansion. Approximately u_eje ∼ 2u_p (Melosh 1989), which leads to ${u}_{{\rm{p}}}={v}_{\mathrm{esc}}/2\,=\sqrt{2G\pi {\rho }_{0}/3}{R}_{\mathrm{tar}}$, where R_tar is the target radius.

For C₀ > su_p (where u_p = v_esc/2), we can approximate C₀ + su_p ≃ C₀ to determine the minimum shock pressure experienced by ejecta of v_eje ≥ v_esc during a cratering impact. This condition is rewritten as

$$\begin{eqnarray}&&{R}_{\mathrm{tar}}\lt {C}_{0}/(s\sqrt{2G\pi {\rho }_{0}/3})\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\lt \ 2300\,\,\mathrm{km}\times \left(\displaystyle \frac{{C}_{0}}{3000\,{\rm{m}}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{s}{2}\right)}^{-1}{\left(\displaystyle \frac{{\rho }_{0}}{3000\mathrm{kg}{{\rm{m}}}^{-3}}\right)}^{-1/2}.\end{eqnarray} \tag{ 18 }$$

For a small sized target (Equation (17)), we can determine the shock pressure for the above condition by

$$\begin{eqnarray}&&{P}_{\mathrm{shock}}={\rho }_{0}\left({C}_{0}+{{su}}_{{\rm{p}}}\right){u}_{{\rm{p}}}\simeq {C}_{0}{\rho }_{0}{u}_{{\rm{p}}}\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&=\ \sqrt{2\pi G/3}{C}_{0}{\rho }_{0}^{3/2}{R}_{\mathrm{tar}}\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\simeq \ 600\,\mathrm{MPa}\times \left(\displaystyle \frac{{C}_{0}}{3000\,{\rm{m}}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{{\rho }_{0}}{3000\mathrm{kg}{{\rm{m}}}^{-3}}\right)}^{3/2}\left(\displaystyle \frac{{R}_{\mathrm{tar}}}{100\,\,\mathrm{km}}\right).\end{eqnarray} \tag{ 21 }$$

Typically P_HEL ≃ 2–5 GPa for rocky bodies (Sekine et al. 2008) and P_HEL ≃ 150–300 MPa for icy bodies (Stewart & Ahrens 2003). The critical size for P_shock = P_HEL is readily obtained from Equation (19). For icy bodies, P_shock > P_HEL (i.e., strength is negligible) when R_tar ≳ 600 km (with P_HEL ∼ 300 MPa, C₀ ∼ 1300 m s⁻¹, and ρ₀ ∼ 1000 kg m⁻³). For rocky bodies, P_shock > P_HEL when R_tar ≳ 1000 km (with P_HEL ∼ 5 GPa, C₀ ∼ 2600 m s⁻¹, and ρ₀ ∼ 2900 kg m⁻³).

We have ignored the effects of material strength in our study. However, further research needs to be carried out on the effects of material strength on the erosion mass and accretion mass.

In our solar system, a variety of orbits exist for planetary bodies including small bodies (asteroids and comets). For example, the orbital semimajor axis ranges from a ∼ 0.4 to 40 au for Mercury and Pluto, respectively. Cratering impacts inevitably occur in different contexts and epochs for a typical impact velocity range of v_imp ∼ 1–100v_esc (Table 2). High-speed cratering impacts would occur during the instability phase (Mojzsis et al. 2019; Brasser et al. 2020); these include the "Nice model" (e.g., Gomes et al. 2005; Tsiganis et al. 2005), the "Grand-tack" hypothesis (e.g., Walsh et al. 2011), and the "early instability" scenario (e.g., Clement et al. 2018). Giant impacts, such as those that formed the Moon (Bottke et al. 2015) and/or the Martian moons (Hyodo & Genda 2018), would distribute impact debris throughout the inner solar system, and high-speed collisions between the debris and asteroids (and/or planets) may correspondingly occur (v_imp ≫ 5 km s⁻¹).

Secular collisional evolution over billions of years and/or a rise of impact flux during the instability phase are expected to characterize the surface morphology and composition of solar system bodies. Figure 8 illustrates examples of typical outcomes of cratering impacts that would potentially occur on different planetary bodies.

Comparing rocky and icy bodies, rocky target bodies are, in general, more eroded than icy target bodies (left panel in Figure 8), while less icy impactor material is accreted on the icy target surface than in the case of rocky bodies for v_imp ≳ 4v_esc (middle panel in Figure 8).

Cratering impacts on rocky terrestrial planets, except Mercury, generally lead to a net mass increase as their v_imp/v_esc ratios are typically v_imp/v_esc ≲ 5 even in an instability phase. Mercury is the innermost planet in the solar system, leading to the highest orbital Keplerian velocity. Together with its small size, Mercury, cratering impacts on average lead to a net erosion (see also Hyodo et al. 2021). The typical impact velocity on the Moon in an instability phase is similar to that on Earth, whereas its mass is ∼0.01 Earth mass. This leads to v_imp/v_esc ∼ 8, and cratering impacts are generally erosive for the Moon.

The largest asteroids are Ceres (∼500 km in radius) and Vesta (∼260 km in radius). The current average impact velocity among asteroids is ∼5 km s⁻¹ (e.g., Bottke et al. 1994). Therefore, v_imp/v_esc ∼ 10 and v_imp/v_esc ∼ 14 for Ceres and Vesta, respectively. Such a high-velocity impact is expected to be erosive. However, we note that our scaling laws might not be applicable to these asteroids because we have ignored material strength in our study (Section 5.2).

Moons around giant planets and trans-Neptunian objects, such as Rhea, Europa, and Pluto, are icy bodies. The strong gravity of giant planets attracts small bodies as high-speed impactors (e.g., v_imp/v_esc ∼ 23.6 for the largest Saturn's regular moon, Rhea; Wong et al. 2020) and erosive cratering impacts on the moons occur. The typical impact velocity on Pluto is v_imp/v_esc ∼ 1.7 (Dell'Oro et al. 2013), which leads to a net accretion of cratering impacts.