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 Platy-ridged surfaces in the Elysium Planitia region of Mars exhibit different crater densities on rafted plates and polygonally patterned areas between them. Rather than being indicative of different ages, these differences provide insight into the variable strength of different types of lava surface. The sizes of small craters, and the resulting size-frequency distribution (SFD), depend on the material strength of target surfaces. Brecciated lava surfaces are likely to have higher crater densities than coherent lava.
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 The interpretation of platy-ridged flows near Athabasca Valles and in the greater Elysium-Amazonis region of Mars has been controversial. These surfaces consist of plates that have been rafted apart, often with polygonally textured surfaces between them, and have been proposed to be frozen lava, ice, or mud. Terrestrial analogs, detailed kinematic analysis, compositional constraints, radar sounding, and the broader geologic context of the Elysium volcanic province argue that the surface is lava [e.g., Keszthelyi et al., 2000, 2004; Jaeger et al., 2007, 2010; Boisson et al., 2009]. The proposal that the surface is instead a frozen sea [Murray et al., 2005] and/or contains recent or current ground ice [Burr et al., 2005; Page, 2007; Page et al., 2009] is difficult to reconcile with the fresh meter-scale morphology [e.g., Jaeger et al., 2007] or the instability of ground ice at the equator of Mars in recent history [Chamberlain and Boynton, 2007].
 However, many observations noted in support of non-volcanic interpretations are not immediately consonant with a volcanic surface and require an explanation. In this paper, we focus on previously observed differences in crater populations. Murray et al.  noted that for craters tens of meters in diameter, crater densities on plate surfaces were ∼20% higher than on areas between them. Page et al.  examined smaller craters at a site with similar surface types. They found a factor of 10–20 difference in crater density between the two surfaces for craters less than about 30 m diameter (Figure 1). Both papers argued that these differences demonstrate that the inter-plate (polygonal) surfaces formed much later than the plates, and so the geologic unit cannot be lava. (Early high-resolution crater counts [e.g., Berman and Hartmann, 2002] found low densities but did not divide units so finely.)
 We investigate the alternative hypothesis that the different crater populations reflect variations in target properties that affect crater size scaling. The size of a small crater depends on target properties; consequently, the crater SFD produced by an impactor population also depends on the target [e.g., Chapman et al., 1970]. Terrestrial lava analogs predict different surface types and material properties for the plates and the polygonized lanes between them. We test a model of cratering on several surfaces and find that they can produce substantial crater density differences.
where V is the transient crater volume, mi is the impactor mass, ρi and ρtar are the impactor and target densities, a is the impactor radius, g is the gravitational acceleration, is the effective strength, vi is the vertical component of impactor velocity, D is the transient crater apparent diameter (measured at ground level) and K1 and μ are empirical constants. For dry soil targets K1 = 0.132 and μ = 0.41, while for hard rock K1 = 0.095 and μ = 0.55 [Holsapple, 1993, figures]. Equation (2) assumes a paraboloid transient crater shape with uniform depth/diameter ratio for all targets. We also assume that the final rim-to-rim diameter is 1.3 times the transient-crater diameter calculated from equation (2) [Holsapple, 1993]. The exact value of this multiplier does not affect the general results.
 Large impact crater sizes are controlled by gravity, but small craters are strength-dominated. The relevant strength is that of a ‘rock mass’ of the scale of the crater (10–100 m), much less than the strength of cm-scale samples because of jointing and other natural defects. An additional complication is that the effective strength for cratering is equal to neither the tensile nor the compressive strength, and is incompletely understood [e.g., Richardson et al., 2007]. Holsapple  suggested that an effective strength of 6.9 MPa is appropriate for hard rock. Holsapple and Housen [2007, and references therein] note that desert alluvium has a strength of 65 kPa, and that the lunar regolith cohesion is only a few kPa. (However, cohesion is not identical to effective strength.) Fine-grained soils at the Viking Lander sites on Mars had cohesion < 1 kPa, while a mix of rocks and fine-grained material may have exceeded 10 kPa [Moore et al., 1977].
2.2. Target Materials for Lava Surfaces
Keszthelyi et al.  provided detailed field observations of Icelandic analogs to platy-ridged lava surfaces on Mars. They found that three lava flows in Iceland have large rafted plates of breccia, separated by low-lying polygonal texture in places. The polygonal surfaces resemble lava lakes: coherent basalt with 25–30 vol% vesicles. Cooling joints spaced ∼5–10 meters apart separate the polygons, which are domed up in the center. In contrast, the upper part of many of the rafted plates consists of up to 5 m of breccia. This breccia is unlike classic aa clinker and is instead composed of broken pieces of pahoehoe. The clasts vary from meter-scale intact pahoehoe lobes to fine rock powder, but the bulk of the breccia is composed of weakly welded 10–20 cm clasts of sparsely vesicular microcrystalline lava. Additionally, there is significant inter-clast void space. On Mars, much of the space between the near-surface clasts may be filled by dust settling from the atmosphere.
 We assume a target density of 2000 kg m−3 for all surfaces, corresponding to a porosity of ∼30%. This is a reasonable value for moderately vesicular basalt or a breccia of sparsely vesicular lava, but a wide range is possible [Jaeger et al., 2010]. Dust mantling and mixing is probably much more significant on the rough breccia surface than on the polygons, where original low relief is still visible. To illustrate the effect of target properties on crater SFD, we consider two models for the polygonal surfaces: hard rock with the nominal effective strength of 6.9 MPa, and dry soil with a rock-like effective strength of 2 MPa. This second model is included because the hard rock parameterization is for targets with minimal porosity [Holsapple and Housen, 2007], inappropriate for vesicular basalt. Impacts into porous targets experience more energy dissipation. Vesicular basalt may easily have porosity comparable to dry soil, but equating the two is somewhat uncertain. A strength of 2 MPa is chosen because vesicularity strongly affects the strength of basalt [Al-Harthi et al., 1999], which is otherwise a typical hard rock. This proportional reduction is comparable to that reported for a vesicularity of 25–30%. We treat the brecciated plate surfaces using the “dry soil” model and consider effective strengths of 65 and 10 kPa. The first is the strength for desert alluvium from terrestrial explosion craters, loosely analogous to a mix of breccia clasts and dust, and the second is a reduced value to illustrate a more extreme case. The breccia is variable, and could range from much stronger than alluvium (many interlocked and/or welded clasts) to quite weak (loose clasts or a large proportion of trapped dust). The true strengths are poorly constrained, since these are effective values derived from experiments on materials that are imperfect analogs for Icelandic or Martian lava flow surfaces. Moreover, real lava flows have vertical variation in vesicularity and brecciation, while we assume uniform properties.
Figure 2a shows final crater diameters for several impactor sizes and velocities, demonstrating the effect of target strength. Since it is the vertical component of velocity that is relevant in equation (1), varying the velocity is equivalent to varying the angle of impact. Figure 2a shows cases with reasonable primary and distant secondary impact velocities (10 and 4 km s−1 respectively). Secondary impacts occur at relatively low velocities and are typically oblique, further reducing the vertical velocity component. A given impactor produces a larger crater in weaker material.
 We then calculated the expected crater SFD for impacts into various targets, assuming the same age and impactor SFD for each surface and impacts at 45° and 10 km s−1. We assumed that impactors striking the surface had a size distribution given by N(d ≥ D) = A*D−3, where N is the number of projectiles with diameter greater than D. The constant A was chosen to give one crater larger than 1 km diameter in the stronger targets, and for plotting we assumed an area of 106 km2. The assumed exponent of −3 for the small-impactor size distribution is also arbitrary, but not unreasonable, and gives crater SFDs with slopes similar to model production functions [Hartmann, 2005] (compare with plotted isochrons) that may include a mix of primary and distant secondary craters.
 The resulting SFDs are shown in Figure 2b, plotted using the method of Hartmann . Isochrons are shown, but we do not attempt to reproduce a particular age or real SFD; the parameter of interest is the ratio of the crater densities between targets. At small diameters, densities on the weaker surfaces (our model plates) are much higher than the strong surfaces. For instance, the alluvium model has 6–7× as many craters as the rock-like dry soil model for craters with diameter less than 30 m. (Note that the rock-like soil model gives crater diameters less than those in hard rock because the high porosity reduces crater size.) The density ratio varies strongly with assumptions about the target properties. At larger diameters, the modeled distributions draw closer as the effect of gravity becomes more important, and the crater density in hard rock can exceed that in weak soils; this is unlikely in reality since larger craters probably excavate a non-uniform range of target materials.
 We assumed that all impactors had the same velocity and impact angle. This is an oversimplification, but 45° is the most probable impact angle [Melosh, 1989] and 10 km s−1 is near the average estimated asteroid impact velocity for Mars [Ivanov, 2001]. Figure 2a shows that variations in impact velocity have a large effect on crater size but a small effect on the ratio of the sizes in comparable targets. Thus, the crater density difference is not an artifact of this assumption. We also assumed that the final rim-to-rim crater diameter is always 1.3 times the transient crater apparent diameter. If this were to vary between targets, weaker targets would probably collapse more than strong, enlarging the ratio of crater diameters produced by a given projectile and thus the crater density difference.
 The crater density difference could be larger than shown in Figure 2b. If the projectile SFD was steeper (i.e., the exponent in the cumulative distribution was less than −3), the density difference for each size range would be larger. Secondary craters from Zunil crater, common over Elysium Planitia, have a SFD with an exponent near −5 for craters 20–200 m diameter [Preblich et al., 2007]. (The exponent of the crater SFD differs somewhat from that of the projectiles.) Such secondaries could give an even larger apparent age difference. A shallow impactor SFD would produce a reduced density difference. However, the populations measured by Page et al.  appear quite steep (Figure 1) at the largest diameters where observational completeness is likely.
 Variations such as those in Figure 2b could easily be misinterpreted as age variations. The nature of the surface can have a strong effect on the small-crater SFD and the apparent age of the surface. This idea is not novel [e.g., Chapman et al., 1970; Croft et al., 1979] and has been discussed in some derivations of crater SFDs [e.g., Ivanov, 2001]. However, it has frequently been ignored in practice; Hartmann  noted that this might increase the “noise level” without improving the model. This effect is indeed often incidental, since the properties of regoliths probably do not vary too radically and large crater dimensions are dominated by gravity. Nevertheless, it must be considered for small craters where there is reason to consider a target particularly strong or weak, and especially when comparing crater densities on such surfaces.
 As many lines of evidence indicate that Martian platy-ridged terrain is lava, the crater density variations observed there indicate target property variations rather than different ages. Desired crater density ratios could be produced by tuning the target properties. The modest difference reported by Murray et al.  could be explained by a breccia that is more coherent than alluvium, consistent with a large proportion of angular, interlocked and even welded clasts. Alternatively (or additionally), the polygonal areas could be weaker than the cases considered, perhaps due to high vesicularity. Greater variations suggest a locally weak breccia or some of the other factors discussed below. However, such tuning is of little practical value. The material parameters, particularly effective strength, are poorly constrained in detail and probably vary vertically and laterally. The impactor SFD may vary depending on the proportions of primary and secondary craters, and any history of burial or exhumation is not known. The significant result is that crater density variations such as those observed are a likely consequence of the variable properties of rubbly pahoehoe surfaces.
 Bombing various lava targets in Hawaii with identical bombs (although varied fuse timing) produced craters ranging from <5 to >25 m apparent diameter [Lockwood and Torgerson, 1980]. Targets included weak, low-density spatter and hence give a large range in crater size, but a factor of five variation in crater size would give two orders of magnitude difference in apparent crater age using the model above. While our lava models are subject to uncertainty, these experiments demonstrate that large crater size variations are plausible.
 The crater density contrast reported by Page et al.  is an order of magnitude or more, while the contrast reported by Murray et al.  from slightly larger craters is ∼20%. We do not attempt a complete reconciliation of this difference here, but we note some possible contributing factors. Larger projectiles will experience a smaller strength difference; breccias examined by Keszthelyi et al.  were usually a few meters thick, so craters larger than a few tens of meters would excavate through them and be affected by the dense interior of the lava flow, reducing the crater density difference for larger craters. The counting area of Murray et al.  was >100× larger than that of Page et al.  comparing plates and inter-plate areas; extreme differences in the small area could be a local effect, such as a particularly weak breccia. The breccia is probably variable, as described by Keszthelyi et al. . Secondary craters could cause highly localized extreme variations, particularly if the secondary SFD is steep. Page et al.  argued that the craters they observed could not be secondaries because of the partitioning between surfaces, but strength scaling could produce such partitioning. Corinto crater rays show a crater density change as they cross from older terrain to young lava, attributed to this strength effect [Jaeger et al., 2010].
 It is unlikely that either plate or inter-plate areas have been much eroded, since the current morphology closely resembles the uneroded surface of terrestrial lava analogs. However, the surface may have been buried in the past. A possible example of this is seen in Figure 3, where a minimally cratered area with well-defined polygons transitions into a region of muted topography and more craters. Burial by dust or volcanic ash could preferentially screen craters from the polygonal surface; this surface is smoother and will retain less interstitial dust following mantle erosion, but if the region was once blanketed thickly enough to bury both surfaces, the polygons would be more deeply and completely buried since they are lower. Rough surfaces could more effectively trap dust and prevent its later removal.
 Since such small craters are in the strength-dominated regime, lateral variations in strength will influence the final crater shape. Such variations could readily arise, particularly in breccia; Keszthelyi et al.  note a “blade” of lava and other intrusions of coherent lava from the flow interior into the overlying breccia.
 Variations in crater SFD due to target properties could affect a number of other questions relating to small craters; we note a few relevant to Mars. This effect could contribute to low crater densities on fine-layered deposits, although higher eolian erosion rates are probably more significant [McEwen et al., 2005]. It should also influence the cratering record of frozen ground, glacial surfaces and the polar layered deposits, as suggested by Croft et al. . Modest changes in crater density based on small changes in strength could frustrate efforts to correlate surface ages with obliquity variations, independent of the uncertainties about the small-crater production function noted by McEwen et al. . New craters used to constrain the current impact rate on Mars are generally observed in dusty regions [Malin et al., 2006], where the surface layer may be weak. This could yield craters larger than would be produced by impact into a generic regolith, leading to an overestimate of the crater production rate on other surfaces. However, dust deposits could have a very high porosity; scaling for such materials is not well-understood [Holsapple and Housen, 2007].
 Crater density variations between different surface types in platy-ridged terrain can be explained by differences in the material properties of those surfaces, particularly strength. Terrestrial rubbly pahoehoe lava is consistent with an enhancement in crater density on plate surfaces in comparison with polygonal areas, supporting this analog. Large differences in the apparent crater retention age can occur on coeval surfaces if the target properties are sufficiently distinct.
 We thank Kevin Housen and Boris Ivanov for helpful reviews, and the Mars Reconnaissance Orbiter Project and the Mars Data Analysis Program for funding.