Numerical modelling of the impact crater depth–diameter dependence in an acoustically fluidized target

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Planetary and Space Science 51 (2003) 831 – 845

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Numerical modelling of the impact crater depth–diameter dependence in an acoustically $uidized target K. W(unnemanna;∗ , B.A. Ivanovb b Institute

a Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, UK for Dynamics of Geospheres, Russian Academy of Sciences, Leninsky prospect, 38-6, Moscow 119334, Russia

Received 27 March 2003; received in revised form 29 July 2003; accepted 5 August 2003

Abstract 2D numerical modelling of impact cratering has been utilized to quantify an important depth–diameter relationship for di6erent crater morphologies, simple and complex. It is generally accepted that the 8nal crater shape is the result of a gravity-driven collapse of the transient crater, which is formed immediately after the impact. Numerical models allow a quanti8cation of the formation of simple craters, which are bowl-shaped depressions with a lens of rock debris inside, and complex craters, which are characterized by a structural uplift. The computation of the cratering process starts with the 8rst contact of the impactor and the planetary surface and ends with the morphology of the 8nal crater. Using di6erent rheological models for the sub-crater rocks, we quantify the in$uence on crater mechanics. To explain the formation of complex craters in accordance to the threshold diameter between simple and complex craters, we utilize the Acoustic Fluidization model. We carried out a series of simulations over a broad parameter range with the goal to 8t the observed depth/diameter relationships as well as the observed threshold diameters on the Moon, Earth and Venus. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Impact cratering; Numerical modelling; Crater mechanics; Acoustic $uidization; Crater geometry; Hydrocode modelling; Yield strength

1. Introduction Almost all solid bodies in the solar system exhibit impact craters on their surface. Although it is now generally accepted that the heavily cratered landscapes on planetary bodies, e.g. the Moon, testify that impact cratering is an important geological process in the evolution of all planetary bodies (Melosh and Ivanov,1999), the process of crater formation itself is still not fully understood. Impact craters fall into two di6erent morphological cases: (i) Simple bowl-shaped craters are circular depressions with a depth/diameter ratio of roughly 1/5. (ii) Complex structures are larger than simple craters and show a much smaller depth/diameter ratio. Their shape is not as uniform as in the case of simple craters. Often, a central uplift—the central peak—is surrounded by a $at crater $oor. Terraced rims and outer concentric zones of normal faulting are further typical morphological features of complex craters. Some larger craters are called multi-ring basins, which present ∗

Corresponding author. E-mail addresses: [email protected] (K. W(unnemann), [email protected], [email protected] (B.A. Ivanov). 0032-0633/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2003.08.001

several rings assumed to be formed as surface waves in a $uid (Baldwin, 1949; Van Dorn, 1968; Murray, 1980) or by speci8c faulting (Melosh, 1989). In Melosh (1989) and Grieve (1998), the morphology of the di6erent crater types is described in detail. A number of static and dynamic models have been proposed to explain the formation of impact craters (see for example Melosh and Ivanov, 1999) and the relationship between impact parameters (kinetic energy of the impactor) and the morphology of the resulting crater structure (O’Keefe and Ahrens, 1993, 1999). In principle, all authors agree that the 8nal shape of impact structures results from the gravity-driven collapse of an approximately hemispherical depression—the transient cavity—that is formed immediately after the impact. The size of this initial cavity is essentially determined by the kinetic energy of the impactor (size, density and velocity of the projectile) and the impact angle. In this paper, we analyse only vertical impacts. Simple craters arise if the transient crater is more or less stable in the gravity 8eld. The material strength can resist the gravity forces acting against the crater rim and the shape of the cavity is retained approximately unchanged. Fractured and molten rocks slide down the crater rim and form a

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breccia lens inside it. In contrast to the simple crater modi8cation, the evolution of the transient cavity to complex crater structures is much more extensive. It is known from observed crater morphology, in combination with extensive numerical modelling, that transient cavities are deep in the case of simple as well as in the case of complex craters. If the depth of the transient crater is greater than a particular threshold value, gravity forces exceed the strength properties of the rocks surrounding the crater and the $oor beneath the cavity rises rapidly upwards until strength arrests any further movement. The small depth/diameter ratio for complex impact structures results from crater $oor uplift. Hence, the 8nal depth of complex craters is the di6erence of two large quantities: the maximum transient crater depth and the structural uplift amplitude. For a terrestrial crater of 200 km in diameter (e.g. Chicxulub), the transient crater depth is about 30 km, while the 8nal crater depth, as one can estimate from impact crater on Venus, is less than 1 km. Whereas the formation of simple craters is understandable (Grieve et al., 1989; Melosh, 1989), the mechanical process of the creation of complex craters is controversial (Melosh and Ivanov, 1999). The central problem in understanding complex crater formation is that all attempts to quantify the process of crater formation using typical material properties of rocks, as measured in laboratory experiments, have failed with respect to the observed transition diameters. Melosh (1977) showed that the formation of the observed morphology can be well described by a Bingham material model with a cohesive strength of about 3 MPa, which is far below realistic values for rock. McKinnon (1978) extended this analysis to a material with a dry friction. He showed that the observed crater morphology needs to assume the dry friction as small as ∼ 5◦ (in contrast to ∼ 30◦ for normal fragmented rocks). Thus, it would appear that the large-scale rheology of rock during an impact is complex and cannot be reproduced in the laboratory with measurements on small samples. With respect to complex crater modi8cation, analytical and numerical studies reveal that temporary strength degradation of rocks surrounding the crater must take place to explain the formation of complex craters at the observed threshold diameters. Two di6erent models addressing strength weakening have been proposed: shock weakening is investigated numerically by O’Keefe and Ahrens (1999). The decrease of strength is explained by shock e6ects like heating and fracturing of the material. The present work is based on a di6erent strength-weakening model: acoustic $uidization (AF), which was originally proposed by Melosh (1979). In this model, high-frequency pressure $uctuations generated during the early stages of the impact event in$uence the frictional strength of the target by periodically reducing the overburden pressure. The object of the present paper is to verify the applicability of the AF-model relating to the cratering process by quantifying the formation of complex craters with respect to crater structures on the Moon, Earth and Venus. The

speci8c threshold diameter for the transition between simple and complex crater structures is a function of gravity (Pike, 1977,1988), and varies therefore between di6erent planetary bodies (e.g. on the Moon it is ca. 15 km and on the Earth it is ca. 2 km for sedimentary and ca. 4 km for crystalline targets (Grieve, 1987)). The response on Earth shows that material properties, most probably the resistance of rocks against shear stresses, also a6ect the cratering process. Hence, our modelling work requires a working assumption regarding how the acoustic $uidization parameters (wavelength and duration of the acoustic oscillations) scale with the impact and target properties (projectile size, velocity and target material). We begin with a short description of the basic numerical model and crater scaling (Sections 2 and 3). The general test model without any AF is presented in Section 4. The AF model we use is described in Section 5. Results of the AF-parametric 8tting are presented in Section 6. 2. Hydrocode and constitutive equations To investigate the process of impact cratering, the numerical modelling technique has been utilized in numerous studies (see, for example, Roddy et al., 1987; O’Keefe and Ahrens, 1999). The calculations for the present work have been performed with the well-known hydrocode called SALE (Ivanov et al., 1997; Ivanov and Deutsch, 1999), which was originally developed by Amsden et al. (1980). For the simulation of the cratering process, some signi8cant changes to the code were needed. Because of large deformation taking place during crater formation, all calculations have to be performed in pure Eulerian mode. The thermodynamic state of the material is computed by the Tillotson EOS (Tillotson, 1962), wherein the pressure p is a function of density  and internal energy. The temperature is calculated by an enhancement of the original Tillotson EOS (Ivanov et al., 2002). A description of the model-speci8c options and all changes to the original SALE code are presented in W(unnemann and Lange (2002). Under high shock compression, matter behaves almost purely hydrodynamically, because the driving stresses greatly exceed the material strength. Subsequently, as shock compression decays and the driving stresses become comparable to the material strength, plastic yielding begins to a6ect the rheology of the material. We use the Von Mises yield criterion to compute shear failure; that is, we limit the elastic stress that can be supported by the material using the equation (see, for example, Wilkins, 1964) J2 = (1 − 2 )2 + (2 − 3 )2 + (3 − 1 )2 6 2Y 2 ;

(1)

where i are the principle deviatoric stress components, J2 the second invariant of the stress tensor and Y is the material’s shear strength. In our initial calculations, we assume that the yield strength Y is a constant value ( Section 4.1), but all subsequent models (Sections 4.2 and 6) are carried

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Table 1 Model parameters and material properties of target and projectile

Parameter

Target/asteroid

Tillotson Parameter for crystalline rock see for example Melosh (1989) 2650 Density (kg=m3 ) 15 Impact velocity vi (km/s) 9.81, 1.62 Gravity g(m=s2 ) 2:5 × 104 –2:5 × 107 Cohesion Y0 (Pa) Maximum yield strength Ymax (Pa) 2:0e 9 Dry friction coeOcient 0.0 –1.0 1373 Melt temperature Tm (K) Fraction of melt temperature for strength weakening  0.2

out with Y as a function of pressure p (Mohr-Coulomb material) and temperature T (Cristescu, 1967; O’Keefe and Ahrens, 1999): 2
T − Tm Y = min (Y0 + p; Ymax ) 1 − : (2) Tm (1 − ) A change of pressure causes a linear variation of strength according to internal friction , the coeOcient of dry friction. The cohesion Y0 is the strength at zero pressure. Increasing temperature has the inverse e6ect on plastic yielding of the material through reduction of the shear strength Y . T , Tm and  are temperature, melting temperature and the fraction of melting temperature at which strength starts to degrade (in all models Tm = 293◦ K). Completely molten matter has no strength. These are the usual constitutive equations of rock-like material, utilized in this way or very similar to it in hydrocodes (Melosh and Ivanov, 1999). All our model runs start with the 8rst contact of the projectile and the target. The same material properties are used for both projectile and target, and are summarized in Table 1. A vertical downward gravity 8eld simulates terrestrial gravity g and causes a pressure, and concomitantly, a temperature gradient in the target. All models in the present study are carried out on a nonuniform computational grid that consists of 300 × 400 cells in radial and vertical direction, respectively. The radius of the projectile varies between 10 and 20 cells and the minimum grid spacing is chosen between 2.5 and 500 m depending on the size of the impactor diameter (50 m–20 km). To avoid any interfering re$ections from the grid boundaries, we expand the outer 50 –100 cells in each direction (except the symmetry axis to the left) by a factor of 1.05 to its adjacent neighbour cell. We use a cylinder geometric grid with the left boundary being the axial symmetry axis. The two-dimensional (2D) geometry constrains the model to vertical impacts only. The cratering process is calculated beginning with the penetration of the projectile into the target, the shock wave propagation and the associated $ow 8eld (excavation $ow) following the shock front. The simulation is completed with the creation of the transient crater and, 8nally, the collapse of the cavity caused by gravity-driven motions that end in

Fig. 1. Maximum crater depth normalized by projectile radius versus inverse Froude number. Data are based on numerical models. Yield strength is zero. For coeOcients of the regression line see Table 2. Notice the double logarithmic scaling of the axis.

isostatic equilibrium, when the main dynamic movements have ceased. In order to quantify the e6ect of crater formation on the stratigraphy of the target, we utilize massless (Lagrangian) tracer particles, placed inside the target and in the projectile. Each tracer records the thermodynamic state of the computational grid cells it passes during the simulation. 3. Crater-scaling relationships The cratering process is essentially controlled by the kinetic energy of the projectile, the gravity and the strength properties of the target. In purely hydrodynamic (strengthless) targets, the maximum transient crater depth, dmax , for a given impact energy and gravity can be scaled by the following equation (Holsapple and Schmidt, 1987; O’Keefe and Ahrens, 1993):
 dmax gr ; (3) = F − =  r vi2 where F is the Froude number (a dimensionless measure of inertial versus gravitational stresses), r is a size scale that is 8rst set to the radius of the projectile, g is gravity and vi the impact velocity,  and  are empirically determined constants that can be constrained by experimental or numerical studies. To test our numerical method and to quantify the parameters  and , we performed a number of numerical experiments for di6erent values of F, varying gravity and size of the impactor (Fig. 1). The values for  and  are speci8ed in Table 2 and compared to those of previous experimental and numerical studies. Our modelling method compares very favourably with previous work, in terms of this test. An impact velocity of 15 km=s is used in all models, which corresponds to the mean impact velocity on the Moon (O’Keefe and Ahrens, 1994). At 15 km=s, the volume of vaporized material is relatively small. Higher impact velocities

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Table 2 Scale dependence of maximum “strengthless” transient crater depth of the transient cavity in comparison to di6erent authors and di6erent methods of investigation (experimental data are taken from Melosh (1989))





Numerical data This work O’Keefe and Ahrens (1993) O’Keefe and Ahrens (1999)

1.3 1.2 0.96

−0:2 −0:22 −0:22

Experimental data Water Quarz Sand Ottawa Sand Rock

1.88 1.4 1.68 1.6

−0:22 −0:16 −0:17 −0:22

result in signi8cant amounts of vaporized matter. Computations of the expansion of the vapour plume usually consume a lot of CPU time and have little e6ect on the late stages of crater formation. In the present study, only the late stage mechanics of the cratering process are investigated in detail so as to establish the scaling laws. During the late stages of crater formation, the dominant forces are due to gravity and strength. Hence, it is insightful to de8ne a dimensionless measure of the relative importance of these two factors. First strength can be expressed in a similar dimensionless form as the Froude number. The Cauchy number is the ratio between internal stresses and material strength (Melosh, 1989): C=

vi2 : Y

(4)

To balance both forces, we de8ne a parameter S as the ratio of the two dimensionless parameters, the planetary material strength normalized by the overburden pressure at maximum “strengthless” transient crater depth: S=

Y F = : C gdmax

(5)

Thus, one can view the parameter S as a kind of “scaled strength”. The size parameter r (see de8nition of F in Eq. (3) is now set to the maximum depth of the transient cavity dmax . For a given impact energy, S expresses whether gravity or strength is the dominating force or, in other words, whether simple or complex crater modi8cation takes place. The abrupt change from simple to complex morphology at a characteristic threshold diameter suggests that, at a corresponding depth of the transient cavity, a de8nite strength is exceeded. The size of the characteristic threshold strength Ycrit can be roughly estimated by dividing the negative buoyancy force FB = 1=8gD2 dmax associated with the crater cavity, by the area of the hemisphere enclosing the crater A = 1=2D2 (Melosh, 1977): Ycrit =

1 FB 1 = pgdmax ⇒ Scrit = : A 4 4

(6)

For S ¡ 0:25, gravity controls the modi8cation process and a complex crater arises and, conversely, for S ¿ 0:25, strength can resist gravity forces and a simple crater is formed. The transition diameter from simple to complex craters on the Moon is ca. 15 km (Pike, 1977). If it is presumed that the transient crater has more or less the shape of a hemisphere, the maximum transient crater depth dmax is in accordance with 1/2 of the diameter, that is 7:5 km. Inserting this value in Eq. (6), Ycrit takes on the value of 8 MPa for g=1:62 m=s2 and =2700 kg=m3 . For the Earth with a transition diameter of 2–4 km ( Grieve, 1987), Ycrit results in a similar value of 10 MPa. In both cases, the target strength is approximately one order of magnitude below the yield strength of most rocks, which is typically in the range of 100 MPa. In our numerical study, besides a constant yield strength Y (Section 4.1), the Mohr–Coulomb constitutive equation (Eq. (2) is used in most calculations (Sections 4.2 and 6) to describe the strength conditions in the target, where Y is a function of pressure and depth, respectively. If the 8rst term in Eq. (2) is inserted in Eq. (5), it leads to the following expression: S=

Y (p) Y0 + p Y0 = = + ; gdmax gdmax gdmax

(7)

where the pressure p is replaced by the hydrostatic pressure gdmax . Hence, using Eqs. (7) and (3), the value of S can be calculated for each model setting. It should be pointed out that S is limited by the value of , if the cohesion Y0 becomes zero. Since is measured to be larger than 0.5, S would always be larger than the threshold value Scrit = 0:25. The in$uence of thermal softening as quanti8ed in the second term of Eq. (2) is not taken into account in this expression, although thermal softening is used in all computations (Sections 4.2 and 6) utilizing the Mohr–Coulomb constitutive equation (Eq. (2)). 4. Modelling crater formation: classical strength conditions Before introducing the AF model, we carried out a series of simulations for di6erent values of S, by varying only the size of the impactor. 4.1. Constant yield strength First the internal friction coeOcient is set to zero and the in$uence of shock-induced thermal softening is neglected (the second term of Eq. (2) is equal to 1). Thus, Y takes on a constant value of 10 MPa (in Eq. (2), Y0 = 10 MPa, = 0), in accordance with the assumptions of the previous section, where a constant strength of 8–10 MPa is estimated to explain the transition from simple to complex crater morphology at the observed diameters on the Earth and Moon. Fig. 2 illustrates the crater depth as a function of time; the parameter d is a dimensionless measure of the crater

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Fig. 2. Crater depth versus time for di6erent scaled strength (S = 0:05–1.0). The length parameter is normalized by the maximum “strengthless” crater depth dmax , time is scaled by the ratio of impact velocity and dmax . The yield strength is set to 10 MPa for all models. Examples of the resulting crater morphology are shown above and below the diagram (no AF).

depth, d = d=dmax and t  is dimensionless time, t  = vi t=dmax where t is the time after impact. Using dimensionless ratios instead of real parameters makes the model runs comparable for di6erent-sized projectiles. In addition, a large parameter range can be covered with less computations. The residual shape of the crater at the end of the calculations, when the main dynamic motions have ceased, is indicated at the top and at the bottom of Fig. 2. The maximum depth of the transient cavity is reached rapidly after the 8rst contact of the projectile and the target surface. For S ¡ 0:25, the maximum depth d is approximately 1, which is in accordance with d ≈ dmax . The collapse of the crater $oor is indicated by decreasing the crater depth. The uplift of the crater $oor leads to the formation of a complex structure with a typical central peak (bottom right, Fig. 2). For S ¿ 0:25, the excavation depth d of the transient crater is less than 1 (d ¡ dmax ). The 8nal depth of the cavity remains almost unchanged, because strength resists gravity forces. The resulting crater morphol-

ogy is a simple one with the exception of the existence of a breccia lens (upper left, Fig. 2). For S close to 0.25, transitional types of crater morphology are formed, with a less bowl-shaped (S = 0:25) or $attened (S = 0:20) crater $oor, indicating the beginning of the gravity collapse. The cratering models con8rm the assumptions about the conditions for the formation of complex craters presented in the previous section. 4.2. Mohr–Coulomb yield strength Next, the in$uence of an increase of strength with depth, and shock-induced thermal softening on the process of crater modi8cation is examined (Eq. (2)). Fig. 3 presents the time–depth evolution of the crater for di6erent values of S (Eq. (7)), varying the coeOcient of internal friction and cohesive strength. As in the case of

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Fig. 3. Crater depth versus time for di6erent scaled strength (S = 0:15–1.03). The length parameter is normalized by the maximum “strengthless” crater depth dmax , time is scaled by the ratio of impact velocity and dmax . Yield strength is a function of pressure and temperature (Eq. (2)). The coeOcient of dry friction is varied between 0.1 and 1.0 (no AF).

constant yield strength (Fig. 2), the transition from simple to complex morphology takes place at S =0:25. Although all curves in Fig. 3 show a reduction of depth with time, crater formation models with S 6 0:25 show that collapse occurs immediately after they have reached their maximum depth, whereas models with S ¿ 0:25 exhibit a delay between excavation and collapse. The crucial di6erence between the collapse process in these two cases is the presence of uplift of the crater $oor during complex crater collapse. Hence, it is important to distinguish the reduction of the crater depth by rim slumping from the collapse of the crater $oor in the case of complex crater modi8cation that appears immediately after the crater depth has reached dmax . The formation of simple craters where S = 1:03 is illustrated in a series of snapshots in Fig. 4. The slumping of the crater rim is visualized by vertical columns of tracer particles on the left side of each frame. After the crater has reached its maximum depth (t  = 96), the material starts to slide down along a de8nite shear fault (dashed line) directed towards the crater centre, where it is pushed upwards (t  = 192). The involved matter is partially molten as indicated in the temperature plots on the right side of each frame. Hence, it can easily $ow because thermal softening reduces strength according to Eq. (2). Finally, the material is deposited inside the crater (t  = 417), whereby the depth/diameter ratio of the crater is reduced to 1/4, which is in good agreement with the observed depth/diameter ratios of simple craters on Moon and Earth with 1/4 –1/5. The e6ect of thermal softening and increasing strength with depth is only apparent in the nature of rim slumping, illustrated by the formation models with S ¿ 0:25 (S = 0:4 and 1.03, Fig. 3). Rim slumping is more pronounced if the

cohesive strength is small and the friction coeOcient large, which would be the case assuming that most of the material surrounding the crater is heavily fractured by unloading from high shock pressure (S = 1:03; = 1:0; Y0 = 1 MPa). Our numerical modelling provides an extremely good explanation for the formation of simple craters. Furthermore, the simulations with a constant cohesion and with pressure-dependent strength support the idea that the transition takes place at S = 0:25. However, the existence of complex craters at a transition diameter of 15 km on the Moon and 2–4 km on the Earth is only understandable if a reduction of the strength properties of the target material takes place during the cratering process. Choosing strength properties in a more realistic order of magnitude for intact rocks (e.g. 100 MPa), neglecting dry friction ( = 0) the threshold diameter would be approximately 100 km on the Moon and 15 km on Earth (Eqs. (5) and (6)). This means that one needs to modify the rock rheology to simulate the observed depth–diameter relationships. Thermal softening as the main weakening mechanism is not e6ective enough in this regard as the modelling results show. 5. The AF model To explain the formation of complex crater structures at the observed rim diameters, the speci8c mechanical properties of the target material in the vicinity of the crater must be assumed to be analogous to a liquid (Melosh and Ga6ney, 1983). Hence, the solid matter of the target must adopt a $uid-like rheology for a short time period during which crater collapse occurs. Melosh (1979) 8rst suggested the

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Fig. 4. Series of snapshots illustrates the formation of a simple crater (S = 1:02; = 1:0). Length parameters are normalized by the maximum “strengthless” crater depth dmax , time is scaled by the ratio of impact velocity and dmax . The left frame of each snapshot shows tracer particles arranged in vertical layers to point out the slumping of the crater rim. On the right frame, a temperature plot (K) is shown. The de8nition of crater depth, d, and diameter, D, is illustrated at the lower panel (no AF).

idea that fractured rock debris surrounding the crater structure will behave like a viscous liquid if they are excited by a strong, short-wavelength acoustic 8eld (Melosh, 1996). In this case, the $uidization of the target material is temporary due to the rapid decay of the impact-generated pressure vibrations. The basic idea of acoustic $uidization is that the vibration of fragments within the target results in

$uctuations of the overburden pressure, which leads to a reduction of the yield strength below the Coulomb threshold for periods when pressure decreases (lowering vibration phase). Local slip between fragments may then result in situations where the driving stresses exceed the temporary reduced strength conditions (see yield criterion in Eq. (1)). From a macroscopic point of view, the time- and

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Fig. 5. Rheology of an acoustically $uidized matter. Stress and strain rate are scaled by dimensionless ratios (see text). Solid line represents the original AF model (Eq. (8)), dashed line the block model (Eq. (9)).

space-averaged result of these small-scale failure events is bulk viscous $ow of the total matter. The theory of the AF rheology is derived at full length in Melosh (1979). Therein, the strain rate ˙ is a function of the applied shear stress , the intensity of the vibration (pressure $uctuation) Vp and the overburden pressure p. In terms of dimensionless parameters, the rheology equation can be expressed as follows: −1
 2 ˙ = ; (8) cs erfc[(1 − ")=#] with # = Vp =p and " = =( p). cs is the velocity and  the dominating wavelength of the acoustic 8eld. Eq. (8) is plotted for di6erent vibration amplitudes # in Fig. 5, whereas ˙ is normalized by ˙n = ( p)=(cs ). In addition to the original model, there exists a simpli8ed version of the AF model, the so-called “block model” (Ivanov and Kostuchenko, 1997; Melosh and Ivanov, 1999). The rheology of the block model is presented in Fig. 5. This variant is derived in a more straightforward way with a single vibrating block of a characteristic size h. The overburden pressure p causes a friction force YB = (p − Vp ) that impedes the movement of the block if a stress  is acting. The block can only move if  ¿ YB . For a system of blocks and sinusoidal vibration, this simple model leads to a similar rheology equation as given in Eq. (8):   ( − YB )T 1+X X −1 ˙ = − cos X cos−1 X; 22 h2 1−X 1−X (9)

with the vibration period T and X = (1 − ")=#). In Fig. 5, ˙ is normalized by ˙n = ( Tp)=(h2 ), respectively. For a detailed discussion of both models, we refer to Melosh and Ivanov (1999). To utilize the idea of AF model in our numerical model, we assume some further simpli8cations. Eqs. (8) and (9) express that the rheology of rock fragments under vibration is similar to the one of a viscous $uid. A material that behaves elastically up to a certain yield point and then $ows plastically with the viscosity & can be described by a Bingham $uid. The di6erence between the original AF model (or block model) and a Bingham $uid consists in the non-linearity of the AF rheology in comparison to the Bingham rheology. For the numerical models in the present work, we assume, that rock debris under vibration behaves like a Bingham substance with a cohesion YB = (p − Vp ) and a viscosity &. The vibration state of the fragmented rocks is a temporal feature. A reasonable approach to implement the AF process into numerical models is to assume, that Vp decreases exponentially with time (like a free damped oscillation, Melosh and Ivanov(1999); Collins et al. (2002)): Vp = Vp; t=0 exp(−t=Tdec ):

(10)

Vp; t=0 is a function of maximum particle velocity, density and sound speed. Tdec is a damping factor (decay time) which has to be determined. The viscosity & is assumed to be proportional to density , sound speed cs , and a length parameter, comparable to the block size h (in the AF model & = cs for " = 1, in the block model & = 2h2 =T for  ¿ YB , Melosh and Ivanov, 1999). Deep drilling of the 40 km Puchezh Katunki impact structure reveals a system of blocks with an average size of 100 m (Ivanov et al., 1996). Changes in the dominating fragment size at di6erent-sized craters (Kocharyan et al., 1996) suggest that the block length parameter for the determination of & can be scaled by some linear scale. Here we have some uncertainty: if the block system under a growing crater is activated during the crater growth process, the block characteristic size seems to be proportional to the transient cavity depth (Ivanov and Artemieva, 2002) or diameter (Collins, 2001). However, the transient cavity depth itself depends on the value of &. Here we assume that for a given impact velocity (vi = const) the projectile size may serve as a good 8rst guess for the fragment-size scale of a cratering event. This means that the impact of a large projectile leads to a fragmentation into larger blocks and higher viscosity than impacts of smaller projectiles does. In the model, we assume a linear proportionality: & = '& (cs r):

(11)

Thus, the length parameter (block size or wave length) is scaled by the projectile radius r and a dimensionless scaling parameter '& . To estimate the duration and the damping of the vibration, we assume that there is also a linear correlation

K. W/unnemann, B.A. Ivanov / Planetary and Space Science 51 (2003) 831 – 845

Fig. 6. Simpli8ed AF rheology (Bingham rheology) for di6erent vibration amplitudes: 0=large amplitude, 3=small amplitude. Stress is bounded by a “static” (no AF) strength Y (p; T ) (Eq. (2)).

between the damping factor T and the radius of the projectile with the factor 'T : Tdec = 'T (r=cs ):

(12)

For this reason, larger impactors lead to higher viscosities and also to longer lasting vibrations than smaller projectiles do, due to the larger size of vibrating blocks. Melosh (1989) speci8es a Bingham viscosity of approximately 1:1 Gpa s for the formation of central peak craters on the Earth with diameters D ¿ 2:5 km and 3:4 GPa s on the Moon for crater diameters of D ¿ 10 km. The outcome of this viscosity range is a value of 1–1.5 for '& using Eqs. (11), (3) and dmax ≈ (1=2)D. The resulting rheologic model implemented into the SALE code is summarized in Fig. 6, where stress is plotted as a function of strain rate for di6erent amplitudes of vibrations (0=large amplitude, shortly after generation of vibration−3 = small amplitude, near 8nal decay of vibration). Moreover, the stress is bounded by the Coulomb yield strength Y , which depends on pressure and temperature as described in Section 2. Whereas the application of a temporally varying Bingham rheology is a simpli8cation of the original AF model, the assumption that the rheology parameters '& and 'T can be scaled linearly with the projectile radius is more or less speculative (W(unnemann et al., 2002). In the following, we vary these parameters inside reasonable boundaries to compute crater shapes in accordance to the observed impact structure morphometries on the Moon. 6. Crater formation in an acoustically &uidized target The AF model as described in the last section is controlled by two parameters: '& , the viscosity-scaling measurement and 'T , the decay time (see Eqs. (11) and (12)). The size of both parameters, '& and 'T , is unknown and can only be estimated to lie inside reasonable boundaries; probably, '& can be delimited to a range of 1–1.5 (see Section 5). All other

839

parameters (sound speed cs , projectile radius r, vibrational pressure Vp , density ) are derived directly from the model. In the following discussion, we present several model runs varying '& and 'T to 8nd at least a parameter pair with the best 8t between the modelled depth/diameter ratios and those observed on the Moon. Furthermore, we attach great importance in maintaining the correct transition diameters between simple and complex craters, for both the Moon and Earth. Besides varying '& and 'T , we have also experimented with di6erent values of Y0 , the cohesive strength at zero pressure. The complete process of complex crater modi8cation of the transient cavity by gravity in an acoustically $uidized target is visualized in Fig. 7. Each snapshot shows the shape of the crater, shading indicates the strength (Y and YB , respectively) of the target. The approximate beginning of collapse can be seen on the right side at t  = 145, where Y increases with depth according to Eq. (2) with a cohesive strength of 25 MPa (Y = Y0 , for p = 0) near the surface and approximately 200 MPa at a depth of d=dmax = 2:8 (which corresponds to a depth of approximately 3 km for Earth gravity and a projectile radius of 300 m). The 8rst snapshot (t  =58) presents the state of maximum transient crater depth. The black-shaded area surrounding the cavity indicates that the material is acoustically $uidized with YB close to zero (YB Y ). In the following snapshots, the spatial dimension of the $uidized region decreases, and simultaneously YB increases inside this area, because vibration amplitudes decay (see Eq. (10)). Ejected fragments impacting again on the surface (secondary impacts) generate new vibrations that lead again to a $uidization of the target (t  = 318). Finally, all vibrations have ceased and the crater shape is retained by strength (t  = 370; YB ¿ Y ). Horizontal layers of tracers on the left side of the last snapshot indicate a stratigraphic uplift at the centre of the structure and an inverse stratigraphy at the rim zone. The resulting morphology is similar to the one of complex impact structures with a $at crater $oor and a central peak. Fig. 8 shows the crater depth as a function of time after the impact (compare to Figs. 2 and 3) for various-sized craters and therefore di6erent values of S (Y0 =25 MPa; = 1). '& is assumed to be 0.1 and 'T is 150. The residual crater morphology for some selected values of S according to di6erent crater types (simple, transitional, complex and complex with peak ring) is indicated above and below the diagram. The transition between simple and complex craters takes place approximately at S = 1:55, which corresponds to a crater diameter of approximately D = 15– 20 km (see Eq. (7) with D ≈ 2dmax and g = 1:62 m=s2 . This is very close to the observed threshold diameter of 15 km on the Moon and much smaller than a transition diameter of 100 km calculated with the classical strength model presented in the previous section. The exact reproduction of morphology of observed craters needs more tuning of the model. Here we concentrate mostly on the depth–diameter relation.

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K. W/unnemann, B.A. Ivanov / Planetary and Space Science 51 (2003) 831 – 845

Fig. 7. Series of snapshots illustrate the formation of a complex crater (S = 1:23; = 1:0; Y0 = 25 MPa). Shading represents the strength of the target. Di6erent stages indicate the spatial decrease of the zone of AF. In the last sketch, the 8nal state of the crater is shown. The horizontal layering of tracer particles points out the structural uplift of stratigraphy. Length parameters and time are dimensionless (see Fig. 4).

To 8nd out the exact threshold diameter for a given set of model parameters and to verify whether the modelled crater geometries 8t with the observed morphometries of crater structures on the Moon, depth/diameter ratios are plotted versus crater diameters in Fig. 9. The accuracy in measuring the crater geometry of the model results is assumed to be in the range of the grid resolution, which varies depending on the size of the impactor and lies between 2.5 and 500 m. For comparison, the best-8t curves (solid black line) for observed depth/diameter ratios of lunar craters

taken from Pike (1977) are shown. If AF model is not used ('& = 0, 'T = 0; dotted line) no good accordance to the best-8t curves is achieved. Choosing AF parameters '& , 'T di6erent from zero leads to a much better adjustment to the observed crater geometries, whereas more or less similar results arise from either low viscosity ('& = 0:1) lasting only for a short time ('T = 150) or high viscosity lasting much longer ('& = 0:8; 'T = 400). The threshold diameter can be picked between 8 and 17 km, which is close to the observed size of transition craters between simple and complex

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841

Fig. 8. Crater depth versus time for di6erent impact conditions (S = 1:02– 4.46). The length parameter is normalized by the maximum “strengthless” crater depth dmax , time is scaled by the ratio of impact velocity and dmax . The cohesive yield strength Y0 = 25 MPa, = 1:0 and the AF parameters are '& = 0:1, 'T = 150. Examples of the resulting crater morphology are shown above and below the diagram.

morphology on the Moon. Only the depth/diameter ratios of simple craters are too large, which can be explained by the fact that the models of small simple craters do not exhibit a breccia lens (see Fig. 8, simple crater, upper left frame). Assuming a cohesive strength Y0 = 25 MPa for rock debris surrounding the crater does not allow any slumping along the transient crater rim. Reducing the strength at zero pressure (Y0 = 25 kPa) enables sliding of rim walls into the crater interior and a breccia lens formation. Further reduction of the cohesion down to Y0 = 0, which is most often assumed for strength properties of fractured rocks, leads to even more pronounced rim slumping but also results in less good adjustment of the depth/diameter ratios for complex craters. The scale dependence of the modelled crater geometry with reduced cohesive strength for two di6erent sets of AF parameters ('T = 150, 400 and '& = 0:1, 0.8) is also shown in Fig. 9. They 8t much better to the observed crater morphometry especially for small craters than the models with larger cohesive strength do. Reducing cohesive strength implies a much smaller strength all over the target and therefore smaller values of S for the sim-

ple/complex transition. For these models, Scrit is about 1.001 (see Eq. (7)). In Fig. 10, the crater morphology of di6erent-sized structures and two di6erent sets of AF-model parameters are shown. The cohesive strength Y0 is 25 kPa. Whereas the depth/diameter ratios for the two sets are nearly the same (see Fig. 9), the stratigraphic structure beneath the 8nal crater is di6erent. On the right side ('T = 150; '& = 0:1), the modi8cation process of the transient crater seems to be much more distinct than on the left ('T = 400; '& = 0:8). This results in a de8nitive typical crater morphology which becomes particularly clear for the central peak crater. The central high is preserved better (right side), if the vibration state does not last as long, as in the case shown on the left side of Fig. 10, where the decay time is much longer. Moreover, the depth of the transient cavity varies between the di6erent sets of AF parameters for all shown crater sizes. This is because of the lower viscosity during the vibration state of the models shown on the right side of each frame in Fig. 10 ('& = 0:1). Hence, it is veri8ed that the transient cavity depths depends

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Fig. 9. Depth/diameter ratio versus crater diameter for lunar craters with di6erent sets AF parameters. For comparison, the best-8t curve of observed crater geometries taken from Pike (1977) is shown.

on the value of & as already mentioned in the previous section. In Table 3, all parameter choices are summarized and an assessment for each parameter set is stated. Di6erent sets of strength and AF parameters reproduce satisfyingly the observed crater morphometries on Moon. Considering the assumption that the target material during the cratering process is temporarily $uidized by a strong acoustic wave and that the $uidization parameters can be scaled by the size of the projectile like shown for the example of the Moon (Fig. 9), the present model should lead to the right threshold diameter on any other terrestrial planet, e.g. the Earth or Venus changing only the gravity according to the chosen planetary body. Fig. 11 shows the modelled crater depth/diameter relations for the Earth (g = 9:81 m=s2 ) and the Moon (g = 1:62 m=s2 ). The AF parameters are set to '& = 0:1, 'T = 150 (compare to Fig. 9). The cohesive strength is 25 kPa. The threshold diameter can be picked at ca. 3.2 and 14 km, respectively, which is in good agreement with the observed transition diameter of 2– 4 and 10 –15 km on Earth and Moon. Because Venus has approximately the same gravity as Earth (g = 8:87 m=s2 ), some observed depth/diameter ratios are also recorded in the diagram to be compared to the calculated geometries. 7. Summary and conclusion We have numerically modelled the formation of impact craters for a broad range of parameters (impact energy, gravity and strength) a6ecting the cratering process. All parameters are balanced by the ratio S = Y=gdmax , with dmax as the maximum transient crater depth if strength is non-existent.

dmax can be scaled by a power low of the kinetic energy of the impactor and gravity g (O’Keefe and Ahrens, 1993). The strength Y is derived by a rheology law, wherein Y is a linear function of pressure p with the proportionality coef8cient , the coeOcient of dry friction, and temperature T . For zero cohesion, S is limited by the coeOcient of dry friction. With the assumptions that rocks surrounding the crater are heavily fractured, and that thermal weakening starts to degrade strength at 293 K up to melt temperature, where rocks lose any resistance against shear stresses, we 8nd a threshold value of approximately S = 0:25 for the transition between simple and complex crater formation. This is in accordance with estimations by Melosh (1977), who analysed the balance between gravity and strength forces for the collapse of a cavity from the static point of view. If realistic strength properties of approximately 100 MPa of target rocks were assumed, the transition value of S = 0:25 would correspond to a threshold diameter of 100 km on the Moon and 15 km on Earth. A possible weakening mechanism to explain the observed transition from simple to complex morphology at 15 and 2–4 km on Moon and Earth is the AF of fragmented target rocks during the cratering process. In the present study, we try to 8nd a set of parameters describing the $uidization model that allows simulating the observed crater morphologies and morphometries. Fluidization parameters must di6er depending on the size of the generated crater. We assume for a constant impact velocity that the $uidization parameters can be scaled by the size of the impactor. This means that the impact of a large projectile is accompanied by a more viscid $uidization state of the target than it is caused by a smaller impacting body. The problem here is that scaling the $uidization parameters by the

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843

Fig. 10. Final crater morphology of di6erent-sized impactors (50, 100, 200, 2000 m) for two di6erent sets of AF parameters (left: 'T = 400, '& = 0:8 and right: 'T = 150, '& = 0:1). The gravity is 9:81 m=s2 according to conditions on Earth. Dashed line shows the size of the transient cavity. All distances are normalized by the maximum depth of the transient cavity if strength is nonexistent (see Eq. (3)).

projectile size is the same as scaling the properties of the $uidized matter by the maximum transient crater depth (if impact velocity is constant), but the transient crater depth itself depends on the viscosity of the target as shown in Fig. 10. The present study represents the 8rst attempt of a quantitative description of di6erent crater morphologies at di6erent sizes on di6erent planetary bodies with varying gravity conditions in the frame of one model by using appropriate scaling laws. For the example of the Moon, a good agreement between calculated and observed crater geometries can be found for

di6erent sets of parameter choices. If the same model is assigned to Earth-like conditions (g=9:81 kg=m3 ), a transition diameter between simple and complex crater morphology of 3:2 km is derived, which is in good agreement with the observed 2–4 km on Earth. Thus, despite the simple nature of our AF model, we achieve excellent quantitative agreement with the simple to complex transition. However, while our AF-parameter 8t allows us to reproduce the measured depth/diameter ratio as well as the simple/complex transition on the Moon and Earth, the crater structure in each model run may not be reproduced in the best way.

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K. W/unnemann, B.A. Ivanov / Planetary and Space Science 51 (2003) 831 – 845

Table 3 Threshold diameters for di6erent acoustic $uidization parameter sets for Earth and Moon

Model parameters

Calc. threshold diameter for Earth

Scrit

Assessment

'T

'&

Y0

Moon

150

0.1

25 Mpa

8

2.456

Crater too deep, no formation of breccia lens, distinct typical morphology, threshold diameter too small

400

0.8

25 Mpa

17

1.685

Crater too deep, no formation of breccia lens, least depth of transient cavity, threshold diameter in observed range

150

0:1

25 kpa

3:2

14

1:001

Best 8tting, distinct typical morphology, deepest transient cavity, threshold diameter very close to observed size, smaller value of Scrit

400

0.8

25 kpa

3.7

18

1.001

Good 8tting, less distinct morphology, less deep transient cavity, threshold diameter close to observed size, small value of Scrit

Fig. 11. Depth/diameter ratio versus crater diameter on Moon, Earth and Venus. The AF parameters are 'T = 150 and '& = 0:1. For comparison, the best-8t curve of observed crater geometries on Moon taken from Pike (1977) is shown. Because of similar gravity conditions on Venus (8:87 m=s2 ) in comparison to Earth, some observed data from di6erent authors (Ivanov and Ford, 1993; Cochrane, 2003) are presented to add depth measurements to well-known simple/complex transition diameter on Earth.

It is perhaps not surprising that we do not achieve exact quantitative agreement between our model results and observation, given the simple nature of our model assumptions. Closer agreement may simply require a more sophisticated construction of the model target. However, increasing the complexity of the model itself may also allow us to reproduce the exact morphology of a visible natural crater as well as the depth/diameter ratio. For example, so far uniform $uidization parameters are used for the whole target. For further improvements, the full AF model should be implemented to include spatial di6erences in AF-model parameters, because the fragmentation mechanism and the size of

fragments will vary across the crater. Rocks lining the transient crater are fractured in much smaller pieces featuring very low cohesive strength than rocks farther away from the point of impact. As shown in our simulations, a very low cohesive strength for material just outside the crater cavity is needed to explain the slumping of the crater rim in the case of simple craters exhibiting a breccia lens. Further exploration data of large crater structures on Earth, e.g. Chicxulub, will provide more information about material properties beneath crater structures as a function of distance to the point of impact and will help to improve the understanding of crater mechanics.

K. W/unnemann, B.A. Ivanov / Planetary and Space Science 51 (2003) 831 – 845

Acknowledgements K. W(unnemann thanks the “Deutsche Forschungsgemeinschaft” for providing a scholarship to stay at the Imperial College, London. B.A. Ivanov thanks the Alexander-von-Humboldt-Stiftung for a Research Award, which makes this collaboration possiblfe. The cooperation of the authors started at the University of M(unster, Germany and was mainly organized by A. Deutsch. We want to thank him for his support. We also express our sincere gratitude to G. Collins for his very careful and patient scienti8c and linguistic reviews of the manuscript. The authors appreciate the valuable editing by J. Morgan, V. Shuvalov and J. Melosh. References Amsden, A.A., Ruppel, H.M., Hirt, C.W., 1980. SALE: a simpli8ed ALE computer program for $uid $ows at all speeds, LA-8095 Report, Los Alamos, NM, Los Alamos National Laboratories, 101pp. Baldwin, R.B., 1949. The Face of the Moon, (Vol. III). Chicago Press, Chicago, 239pp. Cochrane, C.G., 2003. Crater morphometry on Venus (abs). Lunar and Planetary Science Conference XXXIV, Houston, TX, abstract #1173. Collins, G.S., 2001. Hydrocode simulations of complex crater collapse (abs). Lunar and Planetary Science Conference XXXII, Houston, TX, abstract #1752. Collins, G.S., Melosh, H.J., Morgan, J.V., Warner, M.R., 2002. Hydrocode simulations of Chicxulub Crater collapse and peak-ring formation. Icarus 157, 24–33. Cristescu, N., 1967. Dynamic Plasticity. North-Holland Publishing Company, Amsterdam, pp. 509 –558. Grieve, R.A.F., 1987. Terrestrial impact structures. Annu. Rev. Earth Planet. Sci. 15, 245–270. Grieve, R.A.F., 1998. Extraterrestrial impacts on earth: the evidence and consequences. In: Grady, M.M., Hutchison, R., McCall, G.J.H., Rothery, D.A. (Eds), Meteorites: Flux with Time and Impact E6ect, (Vol. 140), Geological Society, London, pp. 105 –131. Grieve, R.A.F., Garvin, J.B., Coderre, J.M., Rupert, J., 1989. Test of geometric model for the modi8cation stages of simple impact crater development. Meteoritics 24, 83–88. Holsapple, K.A., Schmidt, R.M., 1987. Point-source solutions and coupling parameters in cratering mechanics. J. Geophys. Res. 92, 6350–6376. Ivanov, B.A., Artemieva, N.A., 2002. Numerical modeling of the formation of large impact craters. In: Koeberl, C., McLeod, K.G. (Eds.), Catastrophic Events and Mass Extinctions: Impact and Beyond (Special Paper 356). Geological Society of America, Boulder, CO, pp. 619 – 630. Ivanov, B.A., Deutsch, A., 1999. Sudbury impact event: cratering mechanics and thermal history. In: Dessler, B., Grieve, R.A.F. (Eds.), Large Meteorite Impacts and Planetary Evolution II (Special Paper 339). Geological Society of America, Boulder, CO, pp. 389 –397. Ivanov, B.A., Ford, P.G., 1993. The depths of the largest impact craters on Venus (abs). Lunar and Planetary Science Conference XXIV, Houston, TX, pp. 689 – 690. Ivanov, B.A., Kostuchenko, V.N., 1997. Block oscillation model for impact crater collapse (abs). Lunar and Planetary Science Conference XXIX, Houston, TX, abstract #1654. Ivanov, B.A., Deniem, D., Neukum, G., 1997. Implementation of dynamic strength models into 2D hydrocodes, Applications for atmospheric break-up and impact cratering. Int. J. Impact Eng. 17, 375–386.

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