An illustration from Apollo Over the Moon: a View From Orbit, (Harold Masursky, G. W. Colton, and Farouk El-Baz (editors), NASA Scientific and Technical Information Office, 1978) showing a view of the far-side Lunar Highlands taken by the Apollo 15 mapping camara (image AS15-0293(M)). Don E. Wilhelms writes of this image:
The craters in this far-side area come in various shapes, sizes, and degrees of degradation attesting to a variety of formative processes, energies of formation, and ages. Each individual circular crater was probably produced by the impact of a body from interplanetary space- the larger the crater, the higher the energy; that is, the larger the body, or the greater its velocity upon impact. The first and largest such impact erased all earlier features and produced the crater that fills most of the scene, Gagarin, 265 km in diameter (rim crest outlined). A series of smaller craters followed, starting with crater A (46 km), itself heavily cratered, and ending with the sharp funnel-shaped craters and crater C(14 km). The young age of crater C is demonstrated by the sharpness of its rim crest and its halo of extremely fine, fresh ejecta and secondary craters. During the rain of objects from space, clots of lunar material ejected from impact craters outside this area landed here to form irregular secondary craters. Examples (D) are the elongate partly filled crater near the upper right corner and the elongate but deeper craters in the upper left. Conceivably, however, some irregular craters were formed by volcanism, a process that at one time was widely believed to be the cause of most irregular craters on the Moon. At some time late in the history of the region, an even more distant impact hurled a loose cluster of debris to form the group of sharp, circular (high-energy) craters in the left center of the picture.
Properly simulating (via computer) the development and evolution of complex cratered terrains such as the one above remains an ongoing challenge, but one which we have made significant strides toward achieving -- as the below work demonstrates.
Recent advances in computing technology and our understanding of the processes involved in crater production, ejecta production, and crater erasure have permitted us to develop a highly-detailed Cratered Terrain Evolution Model (CTEM), which can be used to investigate a variety of questions in the study of impact dominated landscapes. In this study, we focus on the manner in which crater densities on impacted surfaces attain equilibrium conditions (commonly called crater `saturation') for a variety of impactor population size-frequency distributions: from simple, straight-line power-laws, to complex, multi-sloped distributions. This modeling shows that crater density equilibrium generally occurs near observed relative-density values of 0.1-0.3 (commonly called `empirical saturation'), but that when the impactor population has a variable power-law slope, crater density equilibrium values will also be variable, and will continue to reflect, or follow the shape of the production population long after the surface has been `saturated.' In particular, we demonstrate that the overall level of crater density curves for heavily-cratered regions of the Lunar surface are indicative of crater density equilibrium having been reached, while the shape of these curves strongly point to a Main Asteroid Belt (MAB) source for impactors in the near-Earth environment, as originally stipulated in Strom et al. (2005). This modeling also validates the conclusion by Bottke et al. (2005) that the modern-day MAB continues to reflect its ancient size-frequency distribution, even though severely depleted in mass since that time.
Typical examples of crater density curves, for a variety of inner solar system objects, taken from Chapman & McKinnon (1986). These, and subsequent crater density plots, are shown in standard R-plot format, as described in Arvidson et al. (1979). The upper three curves (for the Lunar Highlands, Mercury, and Mars) represent older, heavily cratered regions, and are sometimes referred to as `Population 1' craters. The lower two curves (for the Lunar Maria and Lunar Post Orientale regions) represent younger, less cratered regions, and are sometimes referred to as `Population 2' craters.
The 11 CTEM matrix layers. (top left) the full 2000 by 2000 DEM. (top middle) a 500 by 500 excerpt of the DEM, the layers of which are depicted in the remainder of this figure. (top right) regolith depth tracking layer. The remaining 9 panes show the three `levels' of the model by row: (2nd row) bottom level, (3rd row) middle level, and (4th row) top level; and the numerical values tracked by column: (left column) crater position, (middle column) original crater profile, and (right column) current profile deviation.
Examples of the impact crater scaling-relationship (from impactor to crater diameter) used by the CTEM, for the Lunar surface (left) and the surface of asteroid 433 Eros (right). The Lunar impacts assume an impact speed of 17.5 km/s (45 deg. incidence) and a surface gravity of 1.63 m/s^2, while the Eros impacts assume an impact speed of 5.0 km/s (45 deg. incidence) and a surface gravity of 0.00638 m/s^2. Target material properties are the same in each case, with the primary variable being effective target strength Y-bar, where each line indicates: (bold solid) 0 Pa Y-bar or pure gravity-dominated cratering, (dashed) 5 kPa Y-bar, (one-dot) 50 kPa Y-bar, (two-dot) 500 kPa Y-bar, (three-dot) 5 MPa Y-bar, and (four-dot) 50 MPa Y-bar or pure strength-dominated cratering. The importance of target strength in the low-gravity Eros environment is well demonstrated by this general crater scaling-law. Note also the transition from strength- to gravity-dominated cratering as one moves toward larger impactor sizes, for all but the highest target strengths.
Examples of the ejecta blanket thickness algorithm used by the CTEM, for a 100 meter diameter impactor striking the Lunar surface (left) and the surface of asteroid 433 Eros (right). The Lunar impact assumes an impact speed of 17.5 km/s (45 deg. incidence) and a surface gravity of 1.63 m/s^2, while the Eros impact assumes an impact speed of 5.0 km/s (45 deg. incidence) and a surface gravity of 0.00638 m/s^2. Target material properties are the same in each case, with the primary variable being effective target strength Y-bar, where each line indicates: (bold solid) 0 Pa Y-bar or pure gravity-dominated cratering, (dashed) 5 kPa Y-bar, (one-dot) 50 kPa Y-bar, (two-dot) 500 kPa Y-bar, (three-dot) 5 MPa Y-bar, and (four-dot) 50 MPa Y-bar or pure strength-dominated cratering. The importance of target strength in the low-gravity Eros environment is again well demonstrated. Note that far-field ejecta is less affected by target strength, such that ejecta blanket thicknesses as a function of distance from the impact eventually merges with that expected from a purely gravity-dominated crater. For Eros, the sharp cut-off in ejecta at about 17.5 km from the impact site indicates the point where ejecta velocities are greater than the escape velocity of the target body. Note also that the model uses flat-plane ballistics equations, such that target body curvature is not included, since the vast majority of impact ejecta lands within 3 crater radii of the impact site (Melosh, 1989), where target body curvature is negligible for all but the largest impacts.
The solid line shows a typical crater cross-sectional profile produced by the model, for a simple crater having a depth to diameter d/D ratio of 0.2 and a rim height to diameter h_r/D ratio of 0.04. Following initial crater production, the CTEM then tracks the deviation in elevation of each pixel making up the crater, from its original profile position. When a pixel-element deviates by more than 75% from its reference value, that pixel is considered to be no longer recognizable as part of that crater. These limits, above and below the reference profile, are shown by the dotted and dashed lines. The dashed line, in particular, shows the typical crater profile produced by downslope regolith motion, such that when the crater has been degraded to a d/D ratio of 0.05, it can no longer be counted.
Overhead view of a 250-m-diameter crater, depicted at four different times and showing its gradual degradation and erasure by impact-induced seismic shakedown, utilizing the downslope-diffusion theory developed in Section 2.6. Note the rapid initial degradation while slopes are still relatively high, followed by a more gradual degradation as slopes flatten. Compare this Eulerian, finite-difference modeling method to the analytical model developed and shown in Fig. 15 of Richardson et al. (2005).
Four stages of a Lunar-surface CTEM run for an impactor population having a steep, cumulative, power-law slope -2.5. In this case, `sandblasting' by small craters dominate, and the run displays classic Gault (1970) behavior: with small craters reaching equilibrium first (at about 5-10% of geometric saturation, and possessing a cumulative power-law slope of roughly -2), followed by successively larger craters. In this instance, the position of the `knee' in the curve can be used as an indication of relative surface age. The run has a pixel-scale of 50 m, and depicts an area 100 km by 100 km in size. R-plot Legend: (bold solid) `observed' crater counts, (thin solid) actual crater counts, (dot-dash) production population, (horizontal dash) geometric saturation, (horizontal dotted) lines are shown at 1%, 5%, and 10% of geometric saturation.
Four stages of a Lunar-surface CTEM run for an impactor population having a shallow, cumulative, power-law slope -1.5. In this case, `cookie-cutting' by large craters dominate, and the resulting crater population continues to reflect its parent impactor population even after equilibrium conditions have been established, as described in Chapman & McKinnon (1986). This model run illustrates well the downslope regolith motion produced by the collapse of unstable slopes and seismic shaking, on the walls of large craters. The run has a pixel-scale of 50 m, and depicts an area 100 km by 100 km in size. R-plot Legend: (bold solid) `observed' crater counts, (thin solid) actual crater counts, (dot-dash) production population, (horizontal dash) geometric saturation, (horizontal dotted) lines are shown at 1%, 5%, and 10% of geometric saturation.
The two impactor populations used for the model runs discussed in Section 3.2, shown in terms of cumulative impacts per year per square kilometer of target surface area. The solid `concave-up' population has a steep, cumulative, power-laws slope of -3 for impactors < 100 m in diameter; and a shallow, cumulative, power-law slope of -1 for impactors > 100 m in diameter. The dashed `concave-down' population is the exact opposite, and has a shallow, cumulative, power-laws slope of -1 for impactors < 100 m in diameter; and a steep, cumulative, power-law slope of -3 for impactors > 100 m in diameter.
Four stages of a Lunar-surface CTEM run for an impactor population having a shallow, cumulative, power-law slope of -3.0 for impactors < 100 m, and -1.0 for impactors > 100 m in size. In this case, `cookie-cutting' by large craters dominate, and the resulting crater population continues to reflect its parent impactor population even after equilibrium conditions have been established. This run has a pixel-scale of 100 m, and depicts an area 200 km by 200 km in size. R-plot Legend: (bold solid) `observed' crater counts, (thin solid) actual crater counts, (dot-dash) production population, (horizontal dash) geometric saturation, (horizontal dotted) lines are shown at 5%, and 10% of geometric saturation. The (slanted dotted) line indicates where a single crater of the given size would plot on the graph.
Four stages of a Lunar-surface CTEM run for an impactor population having a shallow, cumulative, power-law slope of -1.0 for impactors < 100 m, and -3.0 for impactors > 100 m in size. In this case, `sandblasting' by small craters dominate, and the resulting crater population continues to reflect its parent impactor population even after equilibrium conditions have been established. This run has a pixel-scale of 100 m, and depicts an area 200 km by 200 km in size. R-plot Legend: (bold solid) `observed' crater counts, (thin solid) actual crater counts, (dot-dash) production population, (horizontal dash) geometric saturation, (horizontal dotted) lines are shown at 5%, and 10% of geometric saturation. The (slanted dotted) line indicates where a single crater of the given size would plot on the graph.
The Main Asteroid Belt (MAB) population, as determined through the collisional evolution modeling work of O'Brien et al. (2005) (solid) and Bottke et al. (2005) (dashed), and placed in terms of cumulative impacts per year per square kilometer of target surface area, for a target exposed to the mean MAB impactor flux (Bottke et al., 1993)).
Four stages of a large-scale, Lunar surface CTEM run using the MAB impactor populations derived by O'Brien et al. (2005) (left) and Bottke et al. (2005) (right), with both compared to the crater-count data presented in Fig. 1 of Strom et al. (2005). This run has a pixel-scale of 3079.37 m, and depicts the entire Lunar surface area. A good match (in both runs) first occurs at an MAB exposure age of 10^{9.7} yrs (second row), and remains until the end of the runs at 10^{10.8} yrs.
Four stages of a medium-scale, Lunar surface CTEM run using the MAB impactor populations derived by O'Brien et al. (2005) (left) and Bottke et al. (2005) (right), as compared to the crater-count data presented in Fig. 2 of Hartmann (1995). This run has a pixel-scale of 307.9 m, and depicts 1/100 of the Lunar surface area. An acceptable match (in both runs) first occurs at an MAB exposure age of 10^{9.4} yrs (second row), and remains until the end of the runs at 10^{10.0} yrs.
Four stages of a small-scale, Lunar surface CTEM run using the MAB impactor populations derived by O'Brien et al. (2005), as compared to the crater-count data presented in Figs. 1 & 2 of Hartmann & Gaskell (1997). This run has a pixel-scale of 30.8 m, and depicts 1/10000 of the Lunar surface area. The `elbow' of the production population merges with the Hartmann & Gaskell (1997) counts at an MAB exposure age of 10^{9.1} yrs (upper right), with Hartmann's crater counts displaying a slightly steeper power-law distribution than that produced by our MAB impactor population (to the left of the `elbow'), likely due to the contribution of secondary craters in Hartmann's counts. Also compare this plot to the Lunar Maria and Lunar Post Orientale curves depicted in Fig. 1.
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