- Boris IvanovBoris IvanovInstitute for Dynamics of Geospheres, Russian Academy of Science
Impacts of small celestial bodies, in terms of energy density, occupy the range between ordinary chemical high explosives and nuclear explosions. The high initial energy density of impact gives them some features of an explosion (shock waves, melting and vaporization, mechanical disruption of target rocks). A near-surface burst creates an explosion crater, and an impact often results in the creation of an impact crater. The chain of processes connected to an impact crater’s formation is named “impact cratering” or simply “cratering.” The initial kinetic energy and momenta of the impacting body (“projectile”) generates shock waves (decaying with propagation to seismic waves), heats the material (at high impact velocities, to melt or to boil target rocks). A part of the kinetic energy is converted to target material motion, creating the crater cavity. The final crater geometry depends on the scale of event—while small craters are simple bowl-shaped cavities, large enough crater transient cavities collapse in the gravity field. If collapse takes place, the final crater has a complex geometry with central peaks and concentric inner rings. The boundary crater diameter, dividing simple and complex craters, varies with target body gravity and rock strength. Comparison of a crater’s morphology on remote planets and asteroids allows us to make some estimates about their mechanical parameters (e.g., strength and friction) even before future sample return missions. On many planets large impact craters can be seen, preserved much better than on the geologically active Earth. These observations help researchers to interpret the geological and geophysical data obtained for the relatively few and heavily modified large impact craters found on continents and (rarely) at the sea bottom.
Impact and Explosion Cratering
Explosion craters have been well known to military and civil engineers for many centuries, as buried explosions have been used in “mine wars” and in mineral mining. When people started to think about lunar craters, the explosion experience was used to understand the origin of lunar craters. The best compendium of explosion data in application to impact crater analysis was collected by Ralf Baldwin in his books The Face of the Moon and The Measure of the Moon (Baldwin, 1949, 1963). These data allowed Baldwin to make well-based estimates of the energy needed to create a crater of a given size (diameter).
The impact/explosion problem was attacked from a different direction by Daniel Barringer, who bought the land around the now-famous Meteor Crater in Arizona and spent the rest of his life trying to understand the physics of this crater’s formation. Barringer was the first to pay small research grants in the scientific area of impact cratering. The dramatic history of the Meteor Crater study is presented in an important book by Hoyt (1987). Today the Barringer company, together with the Meteoritical Society, annually award scientists with the Barringer Medal for impact crater studies (Masaitis, 2006).
The critical leap in the field of impact cratering came in the early 1960s in connection to the Apollo mission preparation. Important ingredients were (1) nuclear explosion experience and related high-pressure rock study, (2) advances in computers, facilitating numerical modeling, and (3) geological studies, revealing (in addition to the Meteor Crater) several dozen terrestrial impact craters (Grieve & Robertson, 1979; Masaitis, 1999; Roddy, Pepin, & Merrill, 1977).
Projectiles and Targets
The formation of terrestrial planets, as seen now, ended ~4.5 Ga ago. A complicated collisional evolution of planetesimals, condensed previously from the gas-dust nebula, resulted in main planet formation. Leftovers of these processes—planetesimals and asteroids not incorporated into main planets—were the possible population of bombardment projectiles, recorded on the early planetary surfaces of Mercury, the moon, and Mars as craters of the late heavy bombardment period (which ended ~3.3 Ga ago). After this period, a slow orbital evolution of small bodies in the solar system continued to support approximately constant bombardment flux. This impact flux, which continues today, resulted in the permanent formation of recent impact craters on planetary surfaces (“targets”), including, of course, the surface of Earth.
Impact craters are footprints of impacts of small bodies, often named “projectiles.” The size spectrum of projectiles is a product of the accumulation/collisional evolution process, whereas the velocity spectrum of impacts is defined by the target planetary body gravity and orbital parameters. On Mercury the average impact velocity of small bodies is about 40 km s−1 (in rare cases—up to 100 km s−1); for asteroids in the Main Belt the average impact velocity is about 5 km s−1 (Werner & Ivanov, 2015). The specific energy in a high explosive (HE; e.g., trinitrotoluene, TNT) corresponds to the specific energy of an impact of 2 km s−1. The same value for a classic fission nuclear device is about 20 km s−1, and a thermonuclear explosion device corresponds to an impactor faster ~1,000 km s−1. So, impacts of small celestial bodies, in terms of their energy density, occupy the range between ordinary HE and nuclear explosions. The high initial energy density of impacts gives the process some features of an explosion (shock waves, melting and vaporization, mechanical disruption of target rocks). As an impact is close to a near-surface explosion, the most visible result is the production of an impact crater. The chain of processes involved in impact crater formation is named “impact cratering” or simply “cratering.”
It is useful to divide the cratering process into a chain of “stages,” which differ in the governing process. The most popular division is by stages: (I) the contact and penetration stage, (II) shock wave propagation and transient cavity growth stage, (III) the transient cavity collapse stage, and (IV) the late stage of the final crater readjustment (Melosh, 1989). In some cases, the governing process for each stage may be described analytically (very useful for first-order estimates), while understanding the whole process demands laboratory and natural experimental modeling, including craters formed by artificial projectiles on the Moon (Plescia, Robinson, Wagner, & Baldridge, 2016), as well as those formed by comets and asteroids. In recent decades, powerful numerical models have improved the understanding of high-velocity cratering. However, many details of numerical models need to be calibrated with experiments and natural crater observations. An excellent introduction to the general sequence of impact cratering processes is found in Melosh (1989). Impact cratering as a geologic process is well illustrated by Osinski et al. (2018).
Stages of Cratering
Contact and Penetration Stage (Stage I)
When the projectile touches the target surface, it starts to decelerate, accelerating the target material near the contact point. In solids, the maximum velocity of strong excitation propagation is the shock front velocity. So, from the initial contact point two shock waves begin to propagate: one is the shock wave in the target (compressing and accelerating the target material), and the second one is in the projectile (compressing and decelerating the projectile material. Due to deceleration, a part of the initial projectile kinetic energy converts to internal energy. In solids, internal energy consists of two parts: (1) the “cold” energy of compressed crystalline lattice (roughly, compressed “springs” of inter-atomic forces), and (2) the “hot” (thermal) energy of atoms oscillating around their equilibrium position. Hence, material behind the shock wave is compressed and heated.
For a projectile/target pair of the same material (e.g., basalt asteroid striking basalt surface), the internal energy density (measured in ) is equal to , where is the impact velocity. With this energy density, measured or calculated thermodynamic material properties (equation of state, EOS) allow computing of the pressure, density, and temperature of the initial material compression. In the simplest case of a planar shock wave, the energy is divided in halves between the internal (heating, compression) energy and the kinetic energy of the material motion behind the shock front. If the projectile and the target are made of different materials, the EOS knowledge allows one to construct an “impedance match solution” to compute shock pressure and particle velocity at the contact point (McQueen et al., 1970).
The time scale of the Stage I duration may be estimated as the time needed for a projectile with diameter translates in space for a distance, equal to , or . For a 1 km body at , the “contact time” is about 0.05 to 0.1 s. During this short time the projectile and the adjacent target material are compressed to a pressure of hundreds of GPa (the pressure in the center of Earth is about 375 GPa). We illustrate thermodynamic parameters of the compressed materials with the example of granite EOS (Pierazzo, Vickery, & Melosh, 1997) for an impact of a granitic projectile into a granitic target with velocities .
Maximum temperature in the contact zone reaches 10,000 to 20,000 K. After pressure release, the material temperature decreases, but the compression/decompression cycle results in enhanced temperatures of the released material ( “residual temperature”), , high enough to melt granite at and to partially vaporize it at (Table 1).
Table 1. Thermodynamic Parameters of Compressed Granite
Particle Velocity km s−1
Density, g cm−3
Temperature T, K
Shock Front Velocity, km s−1
Impact 10 km s−1
Impact 19.4 km s−1
Most natural impacts are oblique, implying that the impactor’s trajectory intersects the plane of the target surface at an oblique angle. For celestial bodies impacting planets and asteroids, the most probable impact angle is 45° (Pierazzo & Melosh, 2000). In this case the initial deceleration is mostly due to the impact velocity component that is vertical to the surface. So oblique impacts (Figure 1) result in slightly less pressure and compression near the contact point (in comparison with the head-on collision).
Oblique impacts often generate jets of material, with a velocity close to the initial horizontal component of the impact velocity. A part of the target material is also included in the jet. Jetting of the target material after an oblique impact is the main candidate mechanism for ejection of lunar and Martian meteorites, found on Earth (more information about early material jetting and impact launch of Martian meteorites can be found in Artemieva & Ivanov, 2004; Johnson, Bowling, & Melosh, 2014; Johnson, Minton, Melosh, & Zuber, 2015). Vaporization of the target material adds more energy to early stage jets (Quintana, Schultz, & Horowitz, 2018; Schultz, 1996).
The general picture described above is valid only for target and projectile materials with comparable density and for a relatively high impact velocity. If a very dense projectile impacts a very low-density substance (like aerogel), the contact pressure is not large enough to deform the projectile, and the projectile slowly decelerates in the loose target, producing the long tunnel. This technique was used to catch cometary particles in aerogel traps and deliver them to Earth (NASA mission Stardust). Also, the contact pressure must be high enough to overcome the target and/or projectile strength. Typically, shock waves are generated if the impact velocity is comparable to, or larger than, the material sound speed. Most rocks and typical metals have a sound speed of 4 to 6 km s−1. These values represent the approximate lower limit of velocities regarded as “high-velocity impacts.” In granular materials like sand and lunar regolith, the sound speed may be as low as 0.1 to 0.5 km s−1. Even 4 to 6 km s−1 impacts are “supersonic” and, in some sense, “high-velocity impacts.” Low-velocity impacts (like rifle bullets or cannonballs with velocities below ~1 km s-1) make a hole in the ground. In such cases it is typical that the strong bullet is deformed but still compactly placed at the end of the penetrated hole.
In the following, mostly high-velocity impacts are discussed, where strength of solids is small in comparison with impact-derived pressures.
Shock Wave Propagation, Transient Cavity Growth, and Excavation Stage (Stage II)
When the shock wave has propagated to a distance roughly equal to the projectile size, the rarefaction waves from the free surface begin to follow the leading shock wave. “Rarefaction waves” originate at the free surface (or at the target/atmosphere boundary) when a compressing shock wave reaches the surface. The signal of no pressure support starts to propagate back to the compressed target, decreasing pressure (Melosh, 1980, 1985).
The leading shock wave engulfs a larger and larger volume of target material, expending some energy to accelerate, compress, and heat the material. In a simple example of a planar steady shock wave, 50% of the energy increase goes to the kinetic energy (acceleration) and another 50% goes to the internal energy (compression and heating). For these two reasons (rarefaction waves and energy loss for compression), as well as for the reason of simple geometric spreading (divergence), the shock wave amplitude decreases (“decays”) with the travel distance (Figure 2a, b).
One should separate the shock front decay with a distance from the pressure decay in time in an individual material particle, once compressed with passing shock front. The pressure release in a particle partially returns the accumulated energy to kinetic energy, but a portion of the total energy remains as thermal energy of the particle even when the pressure in the particle returns to zero (or the ambient pressure). The residual thermal energy (heat) may be large enough to result in vaporization or melting of the material particle. Consequently, impacts with high enough velocities are able to produce impact melt and impact vapors (Abramov, Wong, & Kring, 2012; Bjorkman & Holsapple, 1987; Kraus, Senft, & Stewart, 2011; Wünnemann, Collins, & Osinski, 2008).
For terrestrial impact craters, estimates of the impact melt volume exist, preserved in the crater and observed now by the field work of geologists (Grieve & Cintala, 1992). The reasonably good agreement of these observations and results of the numerical modeling (Bjorkman & Holsapple, 1987; Pierazzo et al., 1997; Werner & Ivanov, 2015) provide more confidence in available models and allow the transfer of the terrestrial experience to other planetary bodies.
Assuming 50% of impact melt could be emplaced inside the final crater with the diameter D (km), the melt volume (km3) is scaled approximately as (Werner & Ivanov, 2015). For the largest recognized terrestrial craters with , the reported preserved melt volume is about 2,000 to 8,000 km3 (Grieve & Cintala, 1992).
The part of the impact energy that is converted to material motion gives rise to the flow of the target material behind the shock wave (the “cratering flow field”). Starting close to a radial direction from the impact point, the material particle velocities begin to turn toward the free surface (where ambient pressure is zero or low), resulting in curved trajectories of motion (Melosh, 1985). This behind-shock-wave motion results in the growth of a cavity in the target. This motion is called “the cratering flow.” The growing hole in the target is called the “transient cavity.” “Transient” here means that the cavity grows until strength and/or gravity stops the cavity growth. During the transient cavity growth a part of material is ejected, forming an “ejecta blanket” after later deposition. The final cavity shape—the final crater—depends on many parameters. For example, the transient cavity in water will collapse, while in metal targets the final crater is close to a hemisphere.
The target strength properties are very important in controlling transient cavity shape. Metals, for example, may be described with a constant shear strength value. Rock behavior is more complex, including brittle fracturing and conversion to a granular material. In granular materials shear strength depends on the ambient pressure. The proportionality of the shear strength to the ambient pressure is a good simplified model in many cratering applications. This behavior is called “dry friction” (Collins, Melosh, & Ivanov, 2004).
In the most common case in a planetary context—a rocky target with dry friction in fragmented material—the transient cavity starts to grow, having close to a hemisphere shape until the maximum depth is reached. In the following period (five to six times longer), the transient cavity continues to grow in volume and diameter only. At this stage the main “ejecta” volume is launched. Ejecta is the name given to material fragments ejected from the transient cavity along ballistic trajectories, which curve back toward the target in a gravity field. The combined motion of the ejecta forms an inverted cone, known as the ejecta curtain, which expands across the target surface and is deposited as a blanket of ejecta. Figure 3 illustrates with a numerical model of an impact how the transient cavity grows, creating a crater ~120 m in diameter and ~28 m in depth.
The shock wave propagates away from the impact point with a velocity not less than the target material’s sound speed. Velocities in the cratering flow field near the transient cavity are much smaller than the material sound speed, and this flow may be approximately treated as the flow of incompressible material. In the 1970s, this allowed people to find an elegant model to present cratering flow analytically (Maxwell, 1977). Considering the experimental fact that the growing transient cavity is close to a hemisphere with the radius growing in time, and using the assumption of incompressible material flow, we may build a simple flow velocity field. The curved streamlines of this flow are compatible with available experimental data and numerical simulations (Figure 4). The key assumption in this model is the exponential decay of the radial component of the flow field velocity with a distance, as , where is the exponent value, constant in time. By this designation (with the letter “ ”) in the first publication, the model has the name “ -model.” For a couple of decades, the -model was the workhorse for the transient cavity analysis, but in recent years direct computer modeling allows us to make relatively quick direct numerical modeling of cratering without -model. However, the -model continues to be a useful analytical tool (Croft, 1980; Kurosawa & Takada, 2019). Figure 4 compares theoretical Z-trajectories of computed motion of selected tracers in simple crater formation modeling.
With some simplification, the kinetic energy can be said to be transmitted by a shock wave into the cratering flow field and opening the transient cavity decreases in time by way of three channels: (1) working against the strength of the target material, and converting to heat, (2) displacing the target material out of the transient cavity (in a gravity field via conversion to potential energy), and (3) by the kinetic energy of the material ejected out of the transient cavity. In many cases the transient cavity stops growing in depth while its radius continues to increase. Conditionally the “maximum” size of the transient cavity is measured at the moment when the transient cavity reaches the maximum volume. The relations between the transient crater size, projectile size, and target properties are known as “scaling rules” (or “scaling laws”), which help to extrapolate experimental data to higher impact velocities and planetary-sized projectiles (Elbeshausen, Wünnemann, & Collins, 2009; Holsapple, 1993; Housen, Schmidt, & Holsapple, 1983; Prieur et al., 2017; Schmidt & Housen, 1987; Wünnemann, Zhu, & Stöffler, 2016).
Simple estimates may be done following Holsapple and Schmidt (1979). Assume that is the shear strength of the target material, is the target material density, and is the gravity acceleration. Assume that a hemispheric transient cavity due to the crater-forming flow has a radius , increasing in time (first three frames in Figure 3). The volume of this transient cavity . The displaced mass is . The mechanical work against strength forces (“plastic work,” ) is estimated as
The potential energy of the opened transient cavity may be approximated as the mass of the transient cavity , increased to a height, equal to the current transient cavity radius,
For simplicity, excluding temporary kinetic energy of ejecta from the energy balance, it can be assumed that the transient cavity stops to grow at when the initial kinetic energy of the cratering flow field, , would be totally converted to and :
Equation 3 illustrates the important concept of two principal mechanisms of crater scaling in a wide range of a crater size—“strength” and “gravity” regimes. If , the main part of converts into heat via mechanical work against the target strength, while if , the main part of converts into the transient cavity potential energy in the gravity field (as in water). Hence the effective boundary between strength and gravity regimes of impact (and explosion) cratering may be defined by some specific transient size, :
Where is some constant. Experiments and numerical models allow for an estimate of the boundary crater diameter, , for various targets at various gravity accelerations (Holsapple & Schmidt, 1979; Prieur et al., 2017). For example, surface burst explosion craters in the natural terrestrial wet soils have a . If the same soil existed on the Moon (6 times less than gravity), the strength-to-gravity transition would be shifted to . In intact rocks the value of may be as large as 2 to 4 km. However, near-surface rocks on many planetary bodies are heavily fractured, and this value may occur at a smaller .
The numerical modeling by Prieur et al. (2017) illustrates how various mechanical target properties (porosity, friction, cohesion) influence the value at a constant impact velocity of 12.7 km s−1 at lunar gravity . This numerical model is used to illustrate the scaling approach discussed above, plotting a small subset of data from Prieur et al. (2017) for a basalt target with 20% porosity and friction coefficient of 0.6. For three values of the model cohesion (strength at zero ambient pressure), (effectively zero), 0.05 MPa, and 0.5 MPa, covering the wide range of rock strength from a dry sand to unfractured rocks, Figure 5 presents a plot of the ratio of the transient cavity diameter to the basaltic projectile diameter.
It can be seen that small 10-m lunar craters are 40 times (and more for larger impact velocities) larger than the projectile. In contrast, transient cavity for the largest craters (Figure 5) are only six to eight times larger than the crater-forming projectile. Please note that lunar craters with diameters larger than 12 to 15 km experience a serious modification in gravity field, forming “complex” craters.
Modification Stage of the Transient Crater (Stage III)
The subsequent evolution of the transient cavity depends on the crater size and strength properties of the target material. In liquids, the transient cavity collapses in a gravity field, producing a splash (like a pebble dropped in a pond). In metals, the transient cavity is “frozen” by the high material strength—in this case the final crater shape is almost identical to the transient cavity shape at the end of the cratering flow. In materials with internal friction (such as sand or fragmented rocks, the most typical case on planetary surfaces), the maximal transient cavity evolves differently depending on its size.
Small-size cavities (on Earth, cavities with diameters below 2 to 4 km) mostly experience wall slumping (Figure 3), forming simple bowl-shaped “simple” craters where the “true” transient cavity bottom is buried under fragmented material from the partially collapsed crater wall (the breccia lens). Large-scale cavities evolve in an enigmatic way—the cavity starts to collapse as in water, and the true transient cavity bottom starts to move up before the wall landslides cover it. Later in time, the liquid-like collapse stops, and the final (“complex”) crater exhibits a central uplift of rocks, initially buried at a depth of D/10 (D is the visible final crater diameter). For example, rocks initially located ~4 km below the pre-impact level are observed in the center of terrestrial craters of 40 km in diameter.
In terrestrial craters the true bottom of a transient cavity is covered with breccia and impact melt. In simple craters, we find melted rocks under material slid from the crater walls (Dence, 2017; Grieve, Dence, & Robertson, 1977). In complex craters with a central uplift, melted rocks flow down from the central uplift, forming ring-shaped deposits halfway from the crater center to the crater rim (Onorato, Uhlmann, & Simonds, 1978).
The uplift process is referred to as “enigmatic” for the following reason: the main strength parameter of fractured and pulverized rocks under a crater is their dry friction coefficient. The modeling of the transient cavity collapse shows that with a “normal” dry friction coefficient, the transition from simple to complex crater should occur at a much larger crater diameter than is observed on Earth and other planetary bodies (Melosh, 1977). To explain the observed transition crater size, where central uplift is first observed, one needs to assume a dramatic decrease of rock friction (McKinnon, 1978)—a dry friction coefficient below 0.05 in contrast to the “normal” value of 0.5 to 0.6. However, inspection and drilling in terrestrial craters show the presence of “normal” rocks with normal frictional properties. To put these facts into one process, the model of temporary dynamic friction decrease has been proposed. In general, this model assumes that some dynamic processes around the growing transient cavity result in a temporary decrease in friction. Later in time, when the central uplift moves up (as in water), the normal dry friction properties are gradually restored and finally “freeze” the final shape of a complex crater (Melosh & Ivanov, 1999).
Various physical and mechanical models have been proposed to explain this temporary dynamic friction decrease. Examples include models of local friction melting and acoustic fluidization of fragmented rocks around a crater. However, the clear mechanics of the process is still unclear. In the most advanced (and widely used in computer modeling) acoustic fluidization model, key model parameters must be estimated by a set of model runs targeted to fit the final crater geometry or subsurface structure (Ivanov, 2005; Riller et al., 2018; Wunnemann & Ivanov, 2003).
Figure 6 illustrates the formation of a complex terrestrial crater, with the Popigai Crater (North Siberia, Russia) as an example, modeled by Ivanov (2005). The transient cavity collapse makes the final crater larger than the transient cavity. The simple relation has been proposed to estimate the transient crater diameter, , for a given final rim crater diameter, (Croft, 1985):
where is the maximum diameter of a simple crater stable in the gravity field, often called the “simple/complex” transitional diameter. For terrestrial crystallin targets, is about 4 km. On other planets, varies approximately in inverse proportion to the surface gravity acceleration (i.e., larger on bodies with smaller gravity). Many papers analyze details of a simple/complex transition. For a good compendium of data, including asteroids and icy satellites of giant planets, see Hiesinger et al. (2016).
The time scale of transient cavity growth and collapse can be estimated as a double free fall time from a height, equal to the transient crater depth, (Melosh, 1989). Taking for estimates , we can estimate the time scale of a gravity crater formation as
For the terrestrial gravity acceleration , craters with diameters D from 10 to 100 km are formed from 60 to 200 s (1 to 3 minutes). The giant impact basins on the Moon and Mars 1,000 to 3,000 km took a few hours to be formed (Ivanov, Melosh, & Pierazzo, 2010; Potter, Kring, Collins, Kiefer, & McGovern, 2013).
The late stage of the final crater readjustment includes many geologic and geophysical processes (new thermal equilibrium, slow creep of rocks etc.).
Impact cratering is an important process that forms impact structures visible on solid surfaces of planetary bodies in the solar system. The study of these structures in comparison with terrestrial impact craters and their erosion remnants is based on the universality of cratering processes. Cratering research allows for the combining of numerical models, laboratory experiments, explosion experience, and geological and geophysical investigations on Earth to decipher remote sensing data from other planetary bodies.
The progress in this field may be awaited in classical scientific routes—better observations, better experiments, better models. Observations are coming from two main directions—space missions and terrestrial geological studies. Space missions are delivering new (and better) images where impact craters created in various extraterrestrial conditions are seen: low gravity on asteroids (Barucci, Fulchignoni, Ji, Marchi, & Thomas, 2015; Krohn et al., 2014; Stephan et al., 2019; Vincent et al., 2014) and frozen target materials on icy satellites and Kuiper belt bodies (Robbins et al., 2017, 2018; Vincent, Oklay, Marchi, Höfner, & Sierks, 2015). New observed crater shapes and morphometry are enhancing researchers’ understanding of impact cratering processes.
The most exciting geological and geophysical information in recent decades involves new terrestrial crater recognition and deep drilling in known terrestrial impact craters. The primary analysis of new drilling data is accompanied by numerical models to verify and improve the understanding of these craters. New data promote rethinking of the acoustic fluidization model (Rae, Collins, Grieve, Osinski, & Morgan, 2017; Riller et al., 2018). This model is widely used but is still questionable (Collins & Melosh, 2002).
Experimental works are very important to expanding knowledge about impact cratering. After many decades of field and laboratory experiments with impacts and explosions, there is still a wide field for new experimental study; for example, recognition of artificial impact craters on the Moon (Plescia et al., 2016), oblique impacts (Burchell et al., 2015; Quintana et al., 2018), and impacts in porous targets (Chocron et al., 2017; Housen, Sweet, & Holsapple, 2018; Nakamura, 2017).
Numerical models evolve quickly due to modern computers and hydrocodes. The most important progress is anticipated in 3D modeling with high spatial resolution. The initial results are very impressive (Davison, Collins, Elbeshausen, Wünnemann, & Kearsley, 2011; Elbeshausen et al., 2009; Gisler, Heberling, Plesko, & Weaver, 2018; Shuvalov, 2011). However, fine 3D details, such as radial fracturing (Ivanov, Deniem, & Neukum, 1997) or radial folding during transient cavity collapse (Kenkmann & von Dalwigk, 2000), still await direct numerical modeling.
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