K. Wünnemann et al. / Proceedings of the 11th Hypervelocity Impact Symposium (2011)

Scaling of impact crater formation on planetary surfaces – insights from numerical modeling

K. Wünnemann1*, D. Nowka1, G.S. Collins2, D. Elbeshausen1, M. Bierhaus1

1Museum für Naturkunde, Leibniz Institut an der Humboldt-Universität Berlin, Invalidenstraße 43, 10115 Berlin, Germany

2IARC, Department of Earth Science and Engineering, Imperial College London, London, SW7 2AZ, United Kingdom

Received 28 December 2009; received in revised form 25 March 2010; accepted 20 September 2010

Abstract

We conducted a suite of more than 150 numerical models of crater formation in targets with different material properties. The study aims to determine important scaling parameters used in scaling laws that predict the crater diameter as a function of impact velocity, gravity, density of the projectile and target, projectile size, and a number of material properties such as the coefficient of friction, cohesion, and porosity. We focus on large-scale craters on planetary surfaces where crater growth is primarily limited by gravity. However, our models show that the resistance against plastic deformation affects the size of the transient crater over a broad range of size-scales of impact events. Generally it can be said the higher the coefficient of friction, the more porous the target, and the larger the cohesive strength the smaller the resulting crater. We provide estimates of scaling parameters applicable for materials of different friction, porosity, and cohesion. By means of the famous Barringer Crater in Arizona we demonstrate the effect of target properties on the size of the projectile required to form a crater of given size.

Keywords: Numerical modelling, Scaling, Impact Cratering, Crater Size

1. Introduction

Craters on planetary surfaces, resulting from collisions with asteroids or comets, exhibit a large variety of sizes and morphologies. In order to assess the role of impact processes in the evolution of planets it is crucial to quantify the cratering process: How much energy is required to produce a crater of a given size and morphology and how does the cratering process depend on the properties of the target? First assumptions can be directly derived from the observation that impact craters fall into two different morphological cases on all planetary surfaces [e.g. 1, 2]. Smaller, so-called (i) simple craters are bowl-shaped, circular depressions with a depth to diameter ratio of roughly 1/5. With increasing crater diameter the morphology changes abruptly at a distinct size into so-called (ii) complex craters

*Corresponding author. Tel.: +49 2093 8857; fax: +49 2093 8565. E-mail address: kai.wuennemann@mfn-berlin.de

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with central peaks or rings, relatively flat crater floors and a much smaller depth to diameter ratio. The sharp transition from simple to complex crater morphology at a well-defined crater size is observed on all planetary bodies and depends on the local gravity g. The transition diameter decreases in proportion to 1/g from approximately 15 km on the Moon to 2-4 km on the Earth [3, 4, 5]. The gravity dependence of the transition diameter indicates that simple and complex crater morphology must be a consequence of the gravity-driven collapse of a transitional cavity, the so-called transient crater, when some strength threshold is exceeded. The size of the final crater and its morphology (simple or complex) reflects the degree of collapse [6]. Complex craters have experienced much more extensive modifications and are larger relative to the size of the transient crater than simple craters. Due to the different degree of crater collapse and involved enlargement of the transient crater between simple and complex craters the size of the final crater structure is an inappropriate measure of the size of an event [7]. This is particularly true when comparing crater sizes on different planets formed in different gravity regimes and in targets with varying material properties.

Consequently, quantifying the size of an impact event, or more precisely the kinetic energy of the projectile required to produce a crater of a given size, has to be carried out in two steps [e.g. 8]. Each step reflects a specific stage of crater formation and may be treated independently from the other stages in the course of the formation of the final crater: (I) First, one has to estimate the size of the transient crater from the morphology and size of the final crater. In other words the degree of collapse during the so-called modification stage needs to be quantified. Based on the reconstruction of the transient craters of complex lunar impact structures scaling laws have been developed relating the size of the final crater with the size of the transient crater [9, 10, 7]. (II) Second, some functional relationship between the kinetic energy and/or momentum of the impactor (in terms of velocity, mass, size, angle of incidence) and the excavation of the transient crater, the so-called excavation stage, has to be found [e.g. 7].

Both stages of crater formation, the excavation stage (formation of the transient crater) and subsequent modification stage (collapse of the transient crater), are strongly affected by the properties of the target in terms of gravity and material properties such as cohesive strength, friction, and porosity [11, 12].

While the first step, scaling the enlargement of the transient crater during the modification stage, is very difficult to reproduce in analog laboratory experiments, the second step, the calculation of the size of the transient crater as a function of impact energy, gravity and target properties, has been studied in much detail by hypervelocity impact experiments at the laboratory scale [13]. To relate small-scale experiments to natural crater dimensions dimensionless ratios and scaling laws have been developed [7, 14, 15]. However, the effect of gravity on crater growth relative to the size of the transient crater is difficult to realize on laboratory-scale and can be investigated experimentally only by conducting impacts in a centrifuge [e.g. 13] where gravity is artificially enhanced. Such studies are limited to non-cohesive, granular materials such as sand. Although the properties of the target material such as porosity or internal friction are somewhat under the experimenter’s control, varying these parameters independently is difficult.

Alternatively, numerical computer models can be used to simulate impact processes and determine crater size as a function of impact energy, momentum, gravity, and other target properties. On the one hand models can investigate the individual effect of any variable in the model and can simulate conditions that are not achievable in the laboratory, on the other hand in order to make confident predictions of crater dimensions the models need to be validated against experiments [16]. The latter is particularly important to verify whether specific material behavior is reproduced by the model.

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Moreover, modeling impacts makes high demands on computer power in terms of computation time and memory depending on the relative size of the impactor and the transient crater [16].

Here we present numerical models of crater formation quantifying the size of the transient crater as a function of impact energy, gravity, and material properties of the target for vertical impacts. We vary the coefficient of internal friction, cohesion, and porosity independently to investigate the effect on the size of the transient crater. The results of our numerical experiments are compared directly with laboratory impact experiments. We provide important scaling parameters used in scaling laws as a function of material properties in order to improve the assessment of the required impact energy for natural impact craters. The impact energy is an important parameter to estimate the general consequences of large impact events in terms of environmental effects.

2. Crater Formation and Size

The impact of a cosmic body on a planetary surface that finally results in the formation of simple or complex craters is best described by a sequence of three different stages [1, 6, 17]. Each stage is dominated by a specific physical process. During the first phase, the contact and compression stage, the projectile penetrates into the target and generates a shock wave at the contact zone. While the shock wave propagates away from the point of impact it transfers energy and momentum to the target. After shock release the material is set into motion leading to the actual opening of the crater – driven by the so-called excavation flow. The shock wave-induced crater excavation is the characteristic process during the second stage of crater formation, the so-called excavation stage. Due to the approximately hemi-spherical expansion of the shock wave the kinetic energy available to displace a mass particle decreases with increasing distance to the point of impact. Eventually, the excavation flow ceases when the kinetic energy falls below a certain threshold. The threshold value corresponds to the energy required to displace target material against its own weight due to gravity and against its resistance to deformation, the cohesive strength. Depending on whether gravity or strength is the dominating process halting crater growth we refer to a gravity- or strength-dominated regime [7, 18]. Note that all craters formed in cohesionless material, such as sand, are gravity-dominated, but this does not imply that such materials are strengthless. In the gravity regime the crater at the end of the excavation stage may undergo further modifications and it is therefore called the transient crater. Gravity-driven collapse of the transient crater is more extensive when the crater is larger, the material is weaker or the gravity is higher. Generally crater collapse leads to an enlargement of the crater diameter while crater depth is reduced. As described in section 1 there exist two distinctly different crater morphologies on planetary surfaces, simple and complex craters, that have experienced a different amount and extent of collapse and, therefore, a more or less pronounced modification of the morphometry and morphology of the transient crater. Note that in both cases the transient crater is limited by gravity (gravity-dominated craters) and simple vs. complex craters should not be mistaken with strength- vs. gravity-dominated craters. Moreover it should be noted that the three stages of crater formation as well as the different regimes and crater morphologies are not as discrete as described here but grade into one another [1]. This is particularly true during the excavation stage in the gravity-dominated regime. Since more kinetic energy is required to displace a mass particle against its own weight at a certain depth, where the lithostatic pressure is higher, crater growth at depth is arrested before crater growth at the surface (the increase of diameter) comes to a halt. Hence, in reality the so-called transient crater at the end of the

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excavation stage may be considered to be a theoretical construct that is never actually reached at one moment in time during crater formation [6]. Due to this circumstance it is difficult to measure the actual extent of the transient crater. In the case of simple craters the degree of modification due to gravity driven collapse is often negligible and the final crater corresponds approximately to the size of the transient crater. This is particularly true for small-scale laboratory cratering experiments in granular material where gravity, if not artificially enhanced by a centrifuge, causes very little collapse. However, computer models of large-scale impacts, where a significant amount of collapse occurs, show that there is a hiatus between the end of crater growth at depth and at the surface [19]. It is therefore not straightforward to determine the size of the transient crater.

The numerical models in this study cover a large range of impact energies and, hence, the definition of the size of the transient crater has to be chosen with care. We follow the approach by Elbeshausen et al. [19] who suggest that the end of the excavation stage is best approximated by the point in time when the crater volume reaches its (first) maximum. Elbeshausen et al. [19] note that in some very low-strength materials (e.g. water) even larger crater volumes are reached at later times of crater formation, e.g. due to the broad collapse of the crater rim, but the first (local) maximum still appears to be the most meaningful point in time to define the dimensions of the transient crater. Correspondingly, all dimensions stated in the following refer to the size of the crater when the (first) maximum is reached.

3. Pi-Scaling of the Transient Crater

The primary purpose of scaling laws is to meaningfully extrapolate the results of small-scale laboratory impact experiments so that they may be applied to large-scale natural craters [11]. To achieve this, dimensionless ratios are used to estimate the relative importance of different physical processes during crater formation. Dimensionless measures of the properties of impactor and target can be related to scaled crater dimensions implying that the relative crater size is independent of the real size of an impact event.

The most successful approach in dimensional analysis of impact crater scaling is the so-called Pi-group scaling [e.g. 14]. Instead of defining for instance the volume V, diameter D, and depth d of the transient crater as a function of six (or more) target and projectile properties [7],

(1)

where U is impact velocity, ρ is target density, δ is projectile density, Y is strength, g is gravity, and L is projectile diameter. The impact angle is neglected in this study and we refer to Elbeshausen et al. [19] for further details. The use of dimensionless ratios reduces the number of independent variables to three:

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(2)

πV is the so-called crater efficiency that is defined by the ratio of the mass of material originally contained within the crater profile, the crater mass, and the mass of the projectile. Accordingly, πD, πd define the dimensionless ratios for diameter and depth of the transient crater. Note, for δ=ρ, πD≈ D/L, except some constant factor. The importance of gravity is expressed by the inverse Froude number π2, which is the ratio of gravitational and inertial stresses. π2 may be understood as the gravity-scaled size of an impact1. Strength is gauged by π3 the ratio of material strength and inertial stresses (or the initial dynamic pressure ρU2) and π4 is the ratio of target and projectile density. A more detailed description of the dimensionless Pi-group parameters is given in e.g. Holsapple [7] and Melosh [1]. The functional relation in Eqn. 1 or in Eqn. 2 can be approximated by laboratory or numerical experiments of crater formation [13, 20, 21]. Additionally, it is possible to derive scaling laws by theoretical considerations, based on the fundamental assumption that an impact event may be approximated as a stationary point source of energy and momentum buried at a certain depth in the target, analogous to the detonation center of an explosive source [7]. If this assumption holds true for any hypervelocity impact, the kinetic energy (and momentum) of the impactor that is effectively available as an energy (and momentum) point source is defined by the so-called coupling parameter. It is assumed that the coupling parameter combines the properties of the impactor (velocity U, diameter L, density δ) into one scalar parameter: (3) In two theoretical end-member cases, the coupling parameter is exactly proportional to the kinetic energy (where ν=1/3, µ=2/3) or the momentum (ν=1/3, µ=1/3) of the impactor, respectively.

A significant consequence of the point source, defined by the coupling parameter C, is that scaling relationships for crater formation (and many other impact-related processes) have the form of power-laws. For example, the final size of the crater varies as a power of the scaled impact size. As mentioned in the previous section there exist two regimes of crater formation where either strength or gravity is the dominating parameter limiting crater size. According to e.g. Holsapple and Schmidt [15] the scaling law in Eqn. 2 may be simplified by assuming that the effect of gravity is negligible in the strength regime and, conversely, the effect of strength is negligible in the gravity regime. Hence the dimensionless parameters of crater size πV, πD, πd can be expressed by a power law function of π2 in the 1 The factor 1.61 in the definition of π2 corresponds to the cube root of 4π/3. It arises from the original definition of π2 for explosive cratering. See Holsapple [7] for further explanations.

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gravity regime and by a similar functional relationship of π3 in the strength regime. Here we specify only the scaling laws for the dimensionless crater diameter in the gravity and strength regimes and refer to e.g. [7, 15] for further details:

Gravity regime: (4)

Strength regime:

(5)

The scaling parameters K2, K3, µ, and ν have to be determined by laboratory or numerical

experiments of crater formation and depend on material properties such as friction and porosity. However, it is difficult to separate the effect of petrophysical properties in laboratory experiments since they cannot be varied independently. Therefore, the influence of porosity and friction on crater scaling is still not well understood.

Note, µ and ν are the same scaling parameters as in the definition of the coupling parameter C in Eqn. 3. For frictionless and non-cohesive material, e.g. water, µ reaches its largest value (µ =0.56) but is still somewhat smaller than the theoretical limit given by pure energy scaling (µ =2/3). With increasing friction µ decreases but is larger than the value for momentum scaling (µ =1/3).

4. Numerical Model

To quantify the scaling parameters K2, and µ for gravity-dominated crater formation over a broad range of material properties (friction f, cohesion Y0, and porosity φ) we carried out a suite of numerical experiments. We used the iSALE hydrocode [22, 23, 24], a 2D cylindrically symmetric numerical model capable of simulating hypervelocity impact cratering at all speeds in different materials (solid, liquid, granular). The code is well tested against laboratory experiments at low and high strain-rates [23] and other hydrocodes [16]. For porous materials the calculation of the thermodynamic state is based on the assumption that the crushing of pores can be separated from the compression of the solid component [25]. The compression of the solid component is calculated by an ANEOS-derived equation of state (EoS) for quartzite [26, 27]; the crushing of pores is computed by the ε−α compaction model [23, 28]. The resistance of solid material against plastic deformation is complex in particular for heterogeneous geomaterials. For simplicity we model material strength by a Drucker-Prager yield envelope, where the yield strength Y is a linear function of pressure P, Y=Y0+f P. For Y0≈0 this strength model is a reasonable approximation for the behavior of granular materials such as sand and for Y0>0 it is adequate to describe intact rocks or coarse granular material with some cohesion.

In all models the projectile was resolved by 20 cells per projectile radius (CPPR). The total grid consists of a high-resolution zone at least twice as big as the expected crater size and a surrounding zone where the cell size gradually increases. This outer zone is to ensure a sufficiently large area is covered by the computational domain so that any reflections from the boundaries do not affect crater growth. All boundary conditions are chosen as free-slip except the upper boundary where continuous outflow conditions are assumed. Resolution and validation tests in previous studies showed that the

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error in crater diameter is typically within 10% of experiments at a resolution of 10 CPPR [16, 19]; therefore we expect a smaller resolution error for the models presented in this study with 20 CPPR.

To cover a broad range of π2-values we varied the impactor diameter L between 25-40,000 m, gravity was set to g=1.62 m s-2 (the surface gravity on the Moon) and the impact velocity was kept constant at 5 km s-1, a typical impact velocity often used in laboratory experiments of crater formation. The projectile consists of nonporous quartzite in all models in this study. Therefore, for nonporous targets, ρ=δ. With the given parameter range we were able to vary the gravity-scaled size π2 over three orders of magnitude.

In order to investigate the effect of material properties on crater formation we used coefficient of friction values ranging from f=0-0.8, porosities ranging from φ =0-38%, and cohesive strengths ranging from Y0=0-10 MPa. The size of the transient crater was measured when the crater volume reached its (first) maximum as described in section 2. We determined the crater diameter at the pre-impact target surface.

5. Results of Numerical Experiments

In total we performed over 150 impact simulations with varying impactor diameter, target porosity, friction, and cohesion. The results of all numerical experiments are summarized in Tab. A1 of the appendix. The measured crater dimensions are listed in terms of dimensionless ratios πV, πD, and πd. First we investigated the effect of friction on crater size in nonporous targets. Fig.1a shows πD as a function of π2. According to the scaling law in Eqn. 4 the data plot as straight lines in double logarithmic representation. We fitted power laws (Eqn. 4) to the data and determined the scaling parameters K2 and µ for models with the same friction coefficient. According to our model results, friction has two effects on crater scaling: (i) the scaled crater diameter πD decreases with increasing coefficient of friction, (ii) the slope β (scaling exponent µ, see Eqn. 4) of the scaling lines in Fig. 1a decreases with increasing coefficient of friction. This means that the dimensionless ratio of crater diameter πD depends less on gravity-scaled size π2 for targets with higher coefficient of friction. Similar effects were observed by [19] for 3D simulations of oblique impacts into frictional targets. We did not use friction coefficients f>0.8 and one can speculate whether the scaling exponent µ (slope β) approaches the momentum limit of µ=1/3 (β=1/6) for very large coefficients of friction.

In a second step we investigated how target porosity affects crater size for a constant coefficient of friction f=0.8 (Fig. 1b). An increase of target porosity results in a decrease of the scaled crater diameter πD. The scaling lines shown in Fig. 1b for different porosities are basically shifted towards smaller πD-values (K2 decreases). The slope β (scaling exponent µ) changes only slightly with porosity.

For comparison the scaling lines obtained from analog laboratory cratering experiments [13, 29] in water and sand are also plotted in Fig. 1a and 1b. The impact experiments in water targets agree well with the numerical models of impacts into frictionless and nonporous targets. Slight differences are due to the fact that a thermodynamic state model (EoS) for quartzite was used in the numerical models. However, similar results were achieved in numerical experiments using an EoS model for water. Apparently, thermodynamic properties of the target material, such as compressibility, have little effect on the scaled crater dimensions at the given impact velocity of 5 km/s. Note, the analog experiments for water comprise impact velocities ranging from a few 10s of m/s to 6 km/s [7, 13, 29].

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Since the exact material properties of sand used in laboratory experiments are unknown none of the numerical modeling suites with varying coefficient of friction and porosity match the experimentally determined scaling line for sand precisely. The best agreement between numerical models and analog experiments is given by the scaling line for a coefficient of friction f=0.8 and a porosity of 25%, which are reasonable values for sand.

In order to verify the accuracy of the numerical models we compare a centrifuge impact experiment (personal communication K. Housen) at 464 G with a polyethylene cylinder (12.1 mm high x 12.2 mm diameter) impacting at 1.82 km/s into dry Ottawa Flintshot sand (with an approximate coefficient of f=0.73 and a porosity of 32%) and the equivalent numerical simulation. For further details about the experimental setup see [30]. The final crater profiles for the laboratory and numerical experiments are compared in Fig. 2.

Fig. 1. Gravity scaled size π2 versus scaled crater diameter πD. (a) Scaling lines for nonporous material and different coefficient of friction f. The dashed line corresponds to an experimentally derived scaling line for water [29]. (b) Scaling lines for a friction coefficient of f=0.8 and porosities 0-38%. The dashed line corresponds to an experimentally derived scaling line

for Ottawa sand [13].

Fig. 2. Comparison of final crater profiles between analog centrifuge experiment (red line, personal communication K.

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Housen; Pierazzo et al [30] and numerical model (black line).

The crater radius is almost identical in both experiments; however, crater depth and the height and extent of the crater rim do not agree so well. We presume that the overestimated crater depth and underestimated rim height are due to shear bulking, which is a typical behavior of granular materials but not implemented in the numerical model, yet.

Fig. 3. Gravity scaled size π2 versus scaled crater diameter πD. (a) Scaling lines for nonporous material, coefficient of friction f=0.8, and different cohesive strength Y0. (b) Scaling lines for four selected target types (see legend). Dashed lines indicate the size of the projectile required to form at crater of the size of the Barringer crater, Arizona, USA (D=1183 m in diameter) if the projectile is composed of iron (grey, density δ=7800 kg m-3) or of rock (brown, density δ=2700 kg m-3). The numbers at the

intercept points indicate the projectile diameter in m for the given target and projectile type.

Finally, we investigated the effect of cohesion on the scaled crater dimensions. We kept porosity and friction constant (f=0.8, φ=0) and varied the cohesion Y0=0, 0.1, 1, 10 MPa (Fig. 3). For large π2 values cohesion does not affect the scaled crater diameter πD significantly. Crater growth is limited by gravity (Eqn. 4) and the scaling lines for Y0=0, 0.1, 1 MPa fall almost on top of each other for π2>1×10-4. However, with decreasing π2 the scaling lines deviate from the line for non-cohesive material approaching a constant scaled crater diameter πD. The limiting value of scaled crater diameter depends on the cohesive strength of the material and indicates the gradual transition from gravity into strength-dominated regime. 6. Discussion

The numerical experiments allow for a much more precise determination of the scaling parameters K2 and µ for different coefficients of friction and porosities. Generally it can be said the higher the coefficient of friction and the more porous the target, the smaller the resulting crater. Elbeshausen et al.

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[19] suggest a linear relationship between the scaling exponent µ and the coefficient of friction. They determined µ by fitting the scaling law for crater efficiency πV to three-dimensional numerical modeling data of impacts into targets with varying coefficient of friction (f=0-0.7). Generally, our results support the assumption of a linear relationship (Fig. 4a), but our data are shifted towards smaller µ-values in comparison to Elbeshausen et al. [19]. This may be due to the much higher resolution in the models in this study (CPPR=20) in comparison to the three-dimensional models by Elbeshausen et al. (CPPR=8). Moreover, in the three-dimensional models a simple Tillotson EoS [31] for granite was used. However, as noted previously in section 5 the thermodynamic behavior of the material has little effect on crater formation at the low impact velocity of 5 km/s.

We also find slightly different µ-values when fitting crater efficiency πV (see Tab. A1 in the appendix) and scaled crater diameter data πD (Fig.4a, Tab. A1). We presume this is because our choice of when to measure the transient crater is not the most appropriate. As discussed above, we define the transient crater diameter as the diameter of the crater when it reaches its maximum volume. More detailed inspection of the data, however, show that a somewhat earlier point in time during the course of crater formation appears to be a more appropriate time to measure the diameter of the transient crater. Using this revised definition would reduce the transient crater diameter in particular for larger π2-values. However, note that it is very difficult to define a more accurate procedure to determine the exact dimensions of the transient crater. In principle, the point in time when the excavation flow is halted either by gravity or strength, varies for every single point of the crater wall. This is most apparent when comparing the end of crater growth at depth and at the surface (diameter).

In all our models the scaling exponent µ varies, according to the theory, between values close to the energy scaling limit of µ =2/3 and the momentum scaling limit µ=1/3 (Fig. 4a). Despite some inaccuracies in the determination of µ Fig. 4a may be used to establish the appropriate scaling exponent for crater scaling in a material of known coefficient of friction.

In contrast to friction porosity has only a minor effect on the scaling exponent. The slope of the scaling lines β varies between 0.16 and 0.18 (µ =0.38-0.44) with no recognizable trend and agrees well with experimentally determined values for sand: β =0.16-0.17. However, the scaling lines are shifted downwards with increasing porosity. Accordingly, the scaling factor K2 decreases with increasing porosity as shown in Fig. 4b. There may be a linear relationship between scaling factor K2 and porosity as implied by the line in Fig. 4b; however, further numerical experiments of impacts into targets with different coefficient of friction and porosity are required to confirm the functional relationship between porosity and the scaling factor K2.

The scaling laws given in Eqn. 4 and 5 are not suitable to fit the data from our numerical experiments with cohesive material. If the target material has some cohesive strength the scaling lines gradually approach a constant scaled crater diameter that is proportional to the cohesion of the material. If the density of the projectile and the target are equal Eqn. 4 and 5 can be merged into a single scaling law for both gravity and strength dominated regimes [7]:

(6)

For a single scaling line in Fig. 3 π2 is constant and depends only on cohesion Y0. Using the scaling parameters µ and K2 determined in the numerical experiments with non-cohesive target material Eqn. 6

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agrees well with the modeled scaled crater diameters for impacts into targets with cohesive strength (solid lines in Fig. 3). If Y0=0 or π3 is very small relative to π2 Eqn. 6 simplifies to Eqn. 4 (scaling law for gravity dominated craters). Conversely, if the gravity scaled size of an impact event π2 is small in comparison to scaled strength π3 Eqn. 6 simplifies to Eqn. 5. A similar formulation of the scaling laws for strength and gravity dominated regime has been suggested previously [7, 12]. The results from the numerical experiments in this study confirm the applicability of Eqn. 6 over a range of typical material parameters and size scales of natural impact events. Note, for a cohesion of Y0=10 MPa crater size is affected by material strength over a broad range of π2-values and cannot be neglected when estimating the size of the transient crater.

Fig. 4. Scaling parameters µ and K2 as a function of coefficient of friction f and porosity φ. (a) The scaling exponent µ was determined by fitting Eq. 4 to the scaled crater diameter πD (red line) and crater efficiency data πV (black line) from numerical impact experiments into material with different coefficient of friction, zero cohesion, and zero porosity. The dashed line was

taken from [19]. (b) The scaling factor K2 was determined by fitting Eq. 4 to the scaled crater diameter from numerical impact experiments into material with different porosities, a coefficient of friction f=0.8, and zero cohesion.

7. Example for the Determination of the Projectile Size from Crater Diameter

By means of the famous Barringer (or Meteor) Crater in Arizona [32] we want to demonstrate the effect of target properties (friction, porosity, and cohesion) on the size of the projectile required to form the given crater of 1186 m in diameter [32]. The Barringer crater has a simple crater morphology. For simplicity we assume that the transient crater is similar to the final crater size.

Where the dashed lines in Fig. 3b intersects the scaling lines the projectile size can be read from the diagram for four selected specific target properties. For example: If the crater was formed in a frictionless target with zero porosity the scaled crater diameter πD would have been 36.8 and an iron projectile 28 m in diameter had been sufficient to form the crater. We assume an impact velocity of 18

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km/s, Earth gravity, and a density of the iron projectile of δ=7800 kg/m3. Note, that in the literature a very similar scaling line as the one for frictionless, non-cohesive, nonporous material in Fig. 3b was proposed to estimate the crater size in competent rock [1]. It appears more realistic to assume that the target material does not behave as though it were completely strengthless. According to our numerically-derived scaling lines in Fig 3b we find different projectile sizes if we assume a coefficient of friction of f=0.8 (L=65 m, iron), a porosity of 25% (L=91 m, iron), and a cohesive strength of Y0=1 MPa (L=108 m). If we assume instead of an iron projectile a stony meteorite with a density of δ=2700 kg/m3 the projectile has to be somewhat larger (see numbers along brown dashed line in Fig. 3b) to form a crater of the same size.

The example shows that the projectile size varies by almost a factor four depending on the target properties and the density ratio between projectile and target.

8. Conclusion

Numerical models allow for a detailed determination of scaling parameters that depend on target properties. We assume a simplified elasto-plastic behavior of typical materials for planetary surfaces described by a Drucker-Prager failure model and a porosity compaction model. The model is a reasonable approximation of the strength properties of granular material with some cohesive resistance against plastic deformation. Rocks in the vicinity of impact craters are thought to behave in a similar manner as they have undergone substantial fracturing during the passage of a shock wave. Moreover, on the scale of natural craters on planetary surfaces rocks are usually not pristine containing some pre-existing fractures that make them effectively much weaker. The important model parameters are the coefficient of friction, the cohesive strength and the initial porosity. Other material properties affecting the thermodynamic response are neglected in our study; however previous studies have shown that at relatively low velocities (we used a constant impact velocity of U=5000 m/s) similar results are obtained for different material types such as water and rock. We determined the scaling exponent µ and the scaling factor K2 as a function of friction, cohesion and porosity to predict crater diameter in the gravity regime and in the transition from gravity to strength regime. The scaling parameters may also apply for the strength regime; however this was not tested against laboratory data. In an example study we demonstrate that projectile size to produce a crater of a given size may vary by a factor of four depending on target properties.

Although we conducted more than 150 numerical experiments we do not cover the parameter space sufficiently. In particular the effect of porosity on crater size is not fully investigated yet. Furthermore, it is not known whether this effect stays the same for oblique impacts. In our study we focus on crater diameter; however, scaling of crater efficiency and crater depth may be different in porous targets. We observe that the depth to diameter ratio changes significantly depending on porosity. Further modeling and laboratory experiments are required for a better understanding of crater formation and scaling of cratering processes in porous targets.

Acknowledgements

We thank K. Housen for providing data of a centrifuge impact experiment. KW, DN, DE were

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funded by DFG grant: WU 355/5-2, WU 355/6-1, and the Helmholtz-Alliance HA-203 “Planetary Evolution and Life” WP3200. GSC was funded by NERC Fellowship grant: NE/E013589/1.

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Appendix

Table 1. Results of numerical models in dimensionless ratios.

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Table. 1 (continued)

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