Cratering on Asteroids

Simone MarchiSouthwest Research Institute

Clark R. ChapmanSouthwest Research Institute

Olivier S. BarnouinThe Johns Hopkins University Applied Physics Laboratory

James E. RichardsonArecibo Observatory

Jean-Baptiste VincentMax-Planck Institute for Solar System Reasearch

Impact craters are a ubiquitous feature of asteroid surfaces. On a local scale, small craters

puncture the surface in a way similar to that observed on terrestrial planets and the Moon. At the opposite extreme, larger craters often approach the physical size of asteroids, thus globally affecting their shapes and surface properties. Crater measurements are a powerful investigation means. Crater spatial and size distributions inform us of fundamental processes, such as asteroid collisional history. A paucity of craters, sometimes observed, may be diagnostic of mechanisms of erasure that are unique on low-gravity asteroids. By-products of impacts, such as ridges, troughs, and blocks, inform us of the bulk structure.

In this chapter we review the major properties of crater populations on asteroids visited by spacecraft. In doing so we provide key examples to illustrate how craters affect the overall shape and can be used to constrain asteroid surface ages, bulk properties, and impact-driven surface evolution.

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1. INTRODUCTION

Until the space age, craters had been observed on a single astronomical body, the Moon. It was only with the analysis of the first lunar samples, however, that it finally became clear that the vast majority of lunar craters (and a recognizable minority of terrestrial craters) were caused by cosmic impacts and were not generally of volcanic or other endogenic origin (e.g., Wilhelms, 1993). Three decades later, when the Galileo spacecraft flew past Gaspra, then a few years later past Ida, craters were found on asteroids. During the subsequent two decades, spacecraft flybys and dedicated orbital missions have recorded crater populations on many additional asteroids.

While hypervelocity impact by asteroids and comets or by their debris will produce impact craters on any solar system body with a solid surface, there is a fundamental difference between craters on small bodies and those on larger planets and satellites: instead of cratering on semi-infinite surfaces, asteroidal cratering occurs on smaller bodies generally with minimal gravity. So the ejecta from an impact explosion travel far and often escape into independent orbits around the Sun, becoming individual small asteroids.

Collisional fragmentation and cratering are major evolutionary processes for asteroids since the earliest epochs of solar system history and learning about the visible record of surficial cratering can provide vital clues about their evolution and interactions with the space environment. Cratered terrains provide snapshots of collisions that occurred eons ago, and in turn, inform us about the origin of the impactor populations that have shaped the surfaces of all but the most geologically active bodies. Moreover, craters excavate deep to reveal underlying layers, perhaps differing from surface materials, while some of the escaped material can eventually lend on the Earth as meteorites.

The fundamental observable property of a crater is its size, and the fundamental property of a population of craters concerns the ratio of the number of small craters to large craters, that is the size-frequency distribution (SFD). As with crater populations on larger planets and satellites, there are additional factors that interfere with a direct inference of the projectile population from the observed crater SFD. These include saturation of craters (the maximum number of craters that can be accommodated on a given surface), formation of secondary craters (craters made by impact of ejecta rather than from the primary cosmic projectile impacts), as well as often size-dependent processes that erase craters or alter their morphology (downslope mass-wasting, pit-formation by volatile release, etc). Crater SFDs are also affected by the properties of the target material (e.g., hard rock, rubbly megaregolith, icy or volatile-rich material); these can vary not only spatially across the target body's surface but also in the vertical dimension, so that scaling of impactor size to crater size may actually vary across the surface and with impactor size. Furthermore, energetic collisions may drastically alter the bulk properties of asteroids, and scramble their surfaces by producing surface features such as troughs, ridges, and grooves.

All these issues may initially manifest themselves as problems due to our limited knowledge of asteroid properties, but, if one regards them as potentially decipherable challenges, they may eventually enable crater studies to reveal many properties of asteroid interiors, surfaces, and geological processes.

In this chapter, we attempt to summarize the most up-to-date understanding of asteroidal cratering processes, emphasizing presentation and interpretation of the more recent spacecraft data (e.g., from Vesta, Lutetia, and Itokawa), while also updating interpretations of earlier results from Gaspra, Ida, Mathilde, Eros, and some smaller targets of opportunity.

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2. CRATER STATISTICS

The identification of impact craters is a challenging process. One may think that craters should resemble nice, sharp bowls, but observations of craters on terrestrial bodies and asteroids readily show that this is naive. In reality, cratered landscapes evolve over time under various forces, such as cratering itself, mass wasting, and other endogenic geological processes, which especially hamper our ability to identify old, degraded craters. Furthermore, there is no unanimously accepted standard procedure to map craters, and researchers need to rely on their own bag-of-tricks. For instance, when a group of experienced mappers were given the same image from which to count craters, it was found that the results could differ by a factor of two (Robbins et al., 2014).

In this section we introduce the topic of crater statistics, and chiefly the primary diagnostic tool of crater size-frequency distributions (SFDs). Further, we discuss how crater SFDs can reveal important processes that modify or alter the production population of craters − that is, the crater SFD per unit time that results from the mainly asteroidal projectile population − and are a powerful tool to infer relative and absolute ages of various terrains along with aspects of their bulk mechanical properties.

2.1. Crater Size-Frequency Distributions

In this section we review crater SFDs of asteroids visited by spacecraft. As mentioned above, the identification of impact craters can be cumbersome, particularly when they are degraded and heavily modified by post-formation processes. In addition, oddly shaped asteroids often show large facets that sometimes are interpreted to be the result of impact sculpting or to be a consequence of their rubble pile structure. Here we take the approach of showing selected examples from the various asteroids to illustrate key processes, rather then presenting a global compilation of every measured crater SFD. In doing so we opt to show crater SFDs from selected references along with some newly measured crater SFDs, and remind readers that there are additional, and sometimes different, counts in the published literature (e.g., Schmedemann et al., 2014). An important and often neglected factor that makes cratering on asteroids different from cratering of terrestrial body surfaces, is the fact that the physical sizes of visited asteroids vary by more than three orders of magnitude. As a result, craters form and evolve under very different conditions. Here we start our discussion with the largest bodies and continue with smaller asteroids.

2.1.1. Large asteroids. Vesta is the largest asteroid so far visited by a spacecraft (Russell et al., this volume). The NASA/Dawn mission orbited Vesta for more than one year gathering images of 98% of the surface. The large surface and coverage makes Vesta the best example so far to study cratering on a large asteroid. In addition, Vesta formed within a few Myr after the first solar system solids (e.g., McSween and Huss, 2010), implying that its surface has been subject to extensive cratering throughout nearly all of solar system evolution. As anticipated, the surface of Vesta exhibits an extremely diverse set of crater populations. The significant population of craters (~10-15) larger than 50 km including a few old degraded structures witnesses the heavy collisional history, recorded primarily in the northern hemisphere (Marchi et al., 2012a). The southern hemisphere, on the contrary, has been obliterated by the two largest impact structures, the ~400-km Veneneia and ~500-km Rheasilvia basins. As a result the overall spatial distribution of craters is rather heterogeneous and shows a marked north-south asymmetry. This is easily seen in the global crater distribution, and in the resulting global average crater density (Fig. 1). The formation of Veneneia and Rheasilvia had major effects on the whole surface, as manifested by the extensive troughs and voluminous ejecta blanketing (Schenk et al., 2012; Buczkowski et al., 2012; Yingst et al., 2014). Mapping of these and other geological features led to the

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development of a well-defined time-stratigraphic system (Williams et al., 2014). In this system, the youngest epoch − Marcian − begins with the time of formation of the freshest of the large craters, the ~70 km Marcia. The second youngest epoch − Rheasilvian − begins with the formation of the Rheasilvia basin. The relative youthfulness of Marcian and Rheasilvian terrains offers a unique opportunity among asteroids to study the least processed production populations of craters produced by asteroidal bombardment (Marchi et al., 2014) and also reveals the small main belt asteroid SFD. Figure 2a shows the crater SFDs of selected Marcian and Rheasilvian terrains.

The crater SFDs of older vestan terrains are not easily interpretable because they exhibit odd shapes (see Yingst et al., 2014 for details), and also differ from the arguably more pristine Marcian and Rheasilvian crater SFDs. It is probable that the formation of large basins such as Rheasilvia and Veneneia (and possibly also older ones) altered crater populations on older terrains (see Section 2.2). Here we present data from just a relatively small region where the highest crater density has been observed, the so-called heavily cratered terrain (HCT; Fig. 2b). The plot also indicates a curve corresponding to empirical crater saturation, that is, approximately the highest crater density that can be accumulated on a given surface (Hartmann, 1984; see also Section 2.2). Therefore, Vesta HCT seems to be close to or has reached the empirical saturation level.

Interestingly, asteroid Lutetia provides some similarities with Vesta. First, given its large size (~100 km), it may be a primordial object (i.e., not a fragment of some still larger body) according to models of collisional evolution of the main belt (Morbidelli et al., 2009). Furthermore, its surface shows significant features that were used to map and develop a time-stratigraphic system (Thomas et al., 2012; Massironi et al., 2012). However, the relatively small size and partial surface imaged (40%) did not permit detailed investigations of most of the surface. An exception is the relatively flat, coherent unit, called Achaia (Fig. 2b), which has a crater SFD showing a peculiar kink at crater sizes between ~5-8 km (Marchi et al., 2012b; Barucci et al., this volume). Craters larger than ~8 km are close to the Vesta HCT and saturation, while craters smaller than ~5 km are significantly depleted. The latter has been interpreted as due to resurfacing or a variation of the mechanical properties with depth (Barucci et al., this volume).

The smooth appearances of Achaia and Vesta HCT − suggesting the presence of significant regolith − resemble the surface of asteroid Ida, although on a very different size scale. The Ida crater SFD is also close to the saturation curve, over the size range from ~0.2-10 km. A similar conclusion applies to asteroid Mathilde for craters smaller than ~10 km, while for craters larger than ~10 km the crater density is well above the empirical saturation curve. A comparative plot of their crater SFDs is given in Fig. 2b. In conclusion, except for large craters on Mathilde, heavily cratered surfaces of mid-to-large sized asteroids seem to cluster in proximity to the empirical saturation density.

In between the two extremes of low crater density represented by Marcian and Rheasilvian terrains on Vesta and the most heavily cratered terrains, we find a range of other terrains. Examples are the asteroid Gaspra (excluding the large facets of uncertain origin), or Achaia region on Lutetia for craters smaller than 5 km. All these distributions (except for the kinked Achaia crater SFD) share relatively similar slopes, if one considers the statistical uncertainties associated with the measurements.

2.1.2. Small asteroids. Smaller asteroids, however, do not fit the above picture. Consider for instance the well studied case of Itokawa (a similar discussion about Eros can be found in Section 2.2.4). The crater SFD at sizes larger than ~0.02 km has a characteristic slope somewhat shallower than the slopes observed on other asteroids (see Fig. 2c). Unexpectedly however, the crater SFD has a considerably shallower slope at smaller sizes (<0.02 km) than both that of non-saturated terrains (e.g., Marcia and Rheasilvia on Vesta), and likely saturated ones (Ida, Mathilde). Moreover, craters become increasingly rare on Itokawa at smaller sizes and, indeed, the surface becomes dominated by boulders. As a result,

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the Itokawa crater SFD becomes almost flat in a cumulative plot below ~0.01 km (see Fig. 2c). Eros exhibits a similar behavior, with slight shallowing of the crater SFD below ~0.1 km and more pronounced shallowing below ~0.05 km (see Section 2.3.1 for more details on Eros). Several possible interpretations for the lack of small craters were discussed by Chapman et al., (2002) for Eros and by Michel et al., (2009) for Itokawa, including a decrease in the slope of main belt asteroids SFD, erasing of small craters, armoring by boulders, and a cratering hiatus. Similar decreasing frequencies of smaller craters were also seen on Steins and Toutatis, although with poorer imaging resolution (Besse et al., 2012; Barucci et al., this volume). The shallow sloped crater SFDs do not match those of the relatively pristine Marcia and Rheasilvia craters, suggesting that one or more specific processes on small asteroids are responsible. Although several of the ideas listed above may be partly applicable, the dominant cause is likely to be seismic shaking, discussed in detail below (Section 2.2.4). Unfortunately, image resolutions of larger asteroids (e.g., Vesta, Lutetia) do not allow for a comparison with Eros and Itokawa high resolution counts (< 0.1 km). However, there is a hint that the Marcia crater SFD does not start bending to a shallower slope near 0.1 km diameter. The main belt SFD at these impactor sizes (< 0.01 km) is unconstrained by direct observations, though there are some indirect arguments against the shallow slope being an attribute of the asteroidal SFD. Marchi et al., (2014) showed that a current model main belt SFD matches reasonably well the Marcia crater SFD, and such a model does not yield a significant change in slope for smaller asteroids (Bottke et al., 2005). Furthermore, according to recent evolution models (e.g., Bottke et al., this volume), the near-Earth objects (NEO) SFD slope at this size is roughly similar to that of the main belt, and the observed NEO SFD (Harris et al., this volume) does not show such a shallower slope.

2.2. Temporal Evolution of Crater Size-Frequency Distributions

Cratered surfaces provide time-integrated snapshots of the accumulation of craters over a certain period. Therefore, the total number of craters superposed on a given terrain should monotonically increase over time. Thus, the number of observed craters − in ideal conditions − constrains the age of the terrain (see Section 2.3). In practice, however, crater topography is reduced over time by a number of surface processes, and eventually old, degraded craters fade away. As a result, the temporal evolution of a cratered terrain is rather complex and its correct interpretation involves considering various production and obliterating mechanisms.

The production of craters is primarily due to the flux of incoming asteroids and, as such, is susceptible to temporal variations reflecting variation in the impact rates and/or impactor SFD. Crater obliteration is a natural result of the degradation of the topography over time. Contrary to what is observed on geologically active terrestrial bodies, the primary cause for obliteration on asteroids is related to the bombardment itself. Basically a newly formed crater obliterates smaller previously existing craters. Additional processes tend to magnify the erasure and extend it to larger distances. An interesting aspect is that of crater saturation, as anticipated in the previous section.

A newly formed crater obliterates smaller previously existing craters, or parts of larger craters, by geometrical overlap (cookie cutting), by spreading ejecta on top of them, by seismic shaking, and by sand-blasting. As the integrated areas of superposed craters approach the surface area the chances that an earlier crater can still be recognized drop toward zero. A steady-state is eventually reached (saturation in jargon) where the crater spatial density cannot further increase. The manner of saturation depends on the slope of the SFD of the crater production function, reflecting that of the impactor population (cf. Richardson, 2009). A shallow-sloped impactor SFD destroys pre-existing craters primarily by resetting of the surface by creation of very large craters and their ejecta deposits and also by seismic shaking, whereas a steeply sloped impactor SFD erodes the topography of pre-existing

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craters in a kind of sand-blasting effect dominated by tiny projectiles. We amplify on these mechanisms below.

Numerical codes simulating the random formation of craters on a surface and the various obliterating mechanism have been developed (Chapman and McKinnon, 1986; O'Brien et al., 2006; Richardson, 2009; Marchi et al., 2012c). Here we present a series of global and regional scale cratering simulations to illustrate the basic processes. In these simulations old craters are obliterated by the four primary mechanisms mentioned above.

2.2.1. Crater erasure mechanisms: Cookie Cutting. Cookie-cutting describes the most direct way that new craters destroy old craters, by overprinting and geometric overlap (Woronow, 1977). When an impactor strikes a cratered surface, any pre-existing smaller craters within the excavation and rim collapse regions of the new crater are totally obliterated. For cookie-cutting to occur, however, the new crater (or a few of them) must be a sizable fraction of, or larger than, the pre-existing crater. If the overlapping craters are too small, an original large crater may still be recognized as a topographic depression and counted, though its rim and other short-wavelength topography are gone.

If a production function SFD of craters has a shallow cumulative power-law slope that is larger than -2, then large-crater cookie-cutting tends to dominate the crater erasure process and crater density steady state values will continue to reflect, or follow the shape of the production population, as large regions are continuously reset and then repopulated with small craters again (Richardson, 2009). In this case, the terrain reaches what Chapman and McKinnon (1986) described as a quasi-equilibrium state, one in which large portions of the surface are frequently reset by the formation of large craters, and thus the newer cratering continues to reflect the production population, even after a long period of bombardment (Woronow, 1977; Chapman and McKinnon, 1986). In this process, the crater population will gradually approach empirical saturation levels of ~5-10% geometric saturation (Gault, 1970) as small craters accumulate on the surface, only to be suddenly pushed back down again by the cookie cutting effect of the next large crater to form (see Fig. 3).

2.2.2. Crater erasure mechanisms: Sandblasting. The term sand-blasting can be used to describe the process by which many small impacts can gradually erode away topographic features (Chapman, 1976), such that the numerous small craters degrade larger craters to the point that they are not recognizable as a crater by a crater counter (Soderblom, 1970). The multiple small craters rearrange the material that had formed the topography of the larger old crater so that its topography relaxes back toward the mean elevation of the terrain.

If a production function SFD of craters has a steep cumulative power-law slope less than -2, then small-crater sandblasting tends to dominate the crater erasure process (Richardson, 2009). In this case, the terrain will display classic behavior (Gault, 1970), with the smallest craters reaching crater-density equilibrium conditions first at between 5-10% of geometric saturation (with a cumulative power-law slope of about -2), and with successively larger crater sizes reaching equilibrium over time (see Fig. 3).

2.2.3. Crater erasure mechanisms: Ejecta burial. The ejecta from a fresh crater can obliterate old craters beyond its rim by burying them. That is, if the newly produced ejecta blanket is thicker than the depth of a per-existing crater at that location, the crater will be buried, and thus will be invisible to a crater counter. This concept has been employed to estimate the thickness of ejecta deposits from lunar craters (Moore et al., 1974). On a low gravity asteroid this process may be less efficient because ejecta are more widespread, but Vesta and Lutetia clearly show significant ejecta reaccumulation (Schenk et al., 2012; Thomas et al., 2012). On Eros, however, Cheng et al. (2001) consider the presence of ejecta,

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especially for small craters, as minimal whereas Blitz et al. (2009) consider ejecta from larger craters as a major contributor to the several tens of meters of regolith on many parts of Eros. Associated with ejecta burial is the poorly understood scouring process of ballistic sedimentation (Oberbeck, 1975). Ballistic sedimentation involves high-energy deposition of ejecta onto the surface beyond the crater rim which mobilizes regolith where the ejecta land (Melosh, 1989).

2.2.4. Impact-induced seismic shaking on asteroid surfaces. The Ranger images of the Moon revealed downslope flow of material on lunar slopes in the form of slides, slumps, and creep processes, and impact-induced seismic effects were proposed as a potential cause (Titley, 1966). Cintala et al. (1978) published two of the primary reasons why impact-induced seismic shaking of a small body is an important surface modification mechanism. First, the small volume of the target body concentrates seismic energy from an impact within the body even though it has dispersed throughout the body. Second, the low surface gravity ga of asteroids (frequently less than 1/1000th that of the Earth) permits relatively small seismic accelerations to destabilize material resting on slopes, where destabilization begins at 0.2-0.5 ga for loose regolith (Lambe and Whitman, 1979).

Elastic stresses and seismic effects of large impacts on small bodies were first modeled by Fujiwara (1991) and Ivanov (1991), who investigated the formation of the Stickney impact basin on Phobos. Asphaug and Melosh (1993) also performed hydrocode modeling of the impact that produced the Stickney basin, and as a by-product estimated the resulting velocities imparted to a hypothetical regolith layer resting on the surface: an effect called seismic jolt (Nolan et al., 1996). More extensive seismic jolt estimates for large impacts on small asteroids (e.g., Gaspra, Greenberg et al., 1994, and Ida, Greenberg et al., 1996) indicated that they can severely affect their cratering records, erasing most craters below a few hundred meters in diameter when a loose, mobile regolith layer exists.

The high-resolution, global-coverage of Eros by the NEAR-Shoemaker spacecraft greatly advanced studies of seismic shaking. In addition to clear indications of downslope regolith motion, the Eros cratering record also showed a particularly large number of degraded craters, along with a severe paucity of small craters (< 0.1 km in diameter), which were thought to possibly be the result of seismic shakedown (Veverka et al., 2001; Chapman et al., 2002). Richardson et al. (2004, 2005) used a series of linked seismic and geomorphic models to investigate the detailed process of impact-induced seismic shaking on Eros-like bodies. They developed a basic theory for propagating seismic energy in a highly fractured asteroid, drawing upon previous lunar crust seismic propagation theory (Toksöz et al., 1974). Synthetic seismograms were then applied to a model of regolith resting on a slope, and the resulting downslope motion computed for a full range of impactor sizes. This computed downslope regolith flow was then used in a fully three-dimensional model of the body’s surface, with craters formed by impacts and then erased by the effects of superposing craters, ejecta coverage, and seismic shakedown. These simulations agreed with the observed Eros cratering record, including the observed paucity of small craters (see Section 2.3). More recent simulations (Richardson, 2013) showed that for an Itokawa-like body with an extremely small gravity field (< 0.1 mm/s2), impact-induced seismic shaking is especially effective at degrading and erasing craters on very short time-scales, such that the surface for an asteroid like Itokawa should be maintained clean of all but the very most recent impact craters, with all others reduced to little more than a collection of vague, filled circular features, as observed (Abe et al., 2006).

An important feature of impact-induced seismic shakedown is its extreme dependence on the size and surface gravity of the asteroid under bombardment, a direct result of the fact that seismic accelerations of greater than about 0.2-0.5 ga are required in order to effectively produce downslope regolith flow. Lesser seismic accelerations are incapable of breaking the bonds of particle friction and cohesion and causing movement under the force of gravity. Richardson et al., (2005) examined this feature in detail, showing that for a body less than 5 km in diameter, impactors of less than a few

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centimeters in diameter (at typical asteroid impact speeds) are capable of producing a global seismic event producing accelerations larger than ga across the asteroid's surface. For a body in the size range of 10-30 km mean diameter (like Eros), a global seismic event requires an impactor of a few meters in diameter. There is an upper limit on asteroid size for experiencing global, surface-modifying, seismic effects from individual impacts; it is about 70-100 km (depending upon asteroid seismic properties). Larger asteroids will experience only localized (regional) seismic effects from individual impacts.

Beyond effects on small craters, there are catastrophic seismic effects of large impacts, exemplified on Eros and Steins. NEAR-Shoemaker images of Eros (Thomas & Robinson, 2005) revealed differences in crater densities and scattered boulder densities that pointed to Shoemaker, the most recent large impact crater on Eros (~7.6 km diameter), as the source of these disparities. In particular, craters smaller than ~0.6 km in diameter showed a proportional decrease in numbers as the straight-line, chord distance (through the interior of the asteroid) likewise decreased, a decrease that could not be fully attributed to visible ejecta coverage as a result of the impact. This gradient in crater density as a function of linear distance from the center of Shoemaker crater indicated (i) that the interior of Eros was coherent enough to permit the efficient passage of seismic energy to broad regions on the surface of the body, and (ii) that this energy was sufficient to produce noticeable degradation and/or erasure of craters smaller than ~0.6 km in diameter as a function of seismic proximity to the large impact.

As a second example, Steins is particularly interesting from the standpoint of seismic shaking in that it lies midway in size (at ~7 km long) between the ~34 km long Eros studied by NEAR-Shoemaker, and the ~0.5 km long Itokawa studied by Hayabusa. As described earlier, the surface of Steins shows a noticeable paucity of small craters below about ~0.6 km in diameter, increasingly so as one moves to smaller sizes. Marchi et al. (2010) ascribed this general paucity of small craters to one or a combination of two possibilities, either the cumulative effects of (small) impact-induced seismic shaking, or the gradual reshaping of the asteroid due to spin-up and spin-down caused by the Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect (Rubincam, 2000). In addition, observed heterogeneities in the small crater distribution may be due to a singular impact event: the formation of the relatively fresh ~2 km diameter Diamond crater, which had disruptive effects for the rest of the surface (Jutzi et al., 2010a), probably similar to those of the formation of Shoemaker crater apparently had on the surface of Eros.

2.3. Model Production Function and Age Determination

We provide crater retention age estimates for some of the crater SFDs presented in Section 2.1, and discuss the effects of crater obliteration (Section 2.2) for selected cases (see also Barucci et al., this volume, for additional examples).

The assessment of cratering ages requires knowledge of impactor SFD, impact rate, and a crater scaling law (that is, a relationship between impactor sizes and crater sizes). These inputs are used to derive the Model Production Function (MPF), that is, the number of craters per unit time per unit surface (O'Brien et al., 2006; Marchi et al., 2010, 2011, 2012a,b). Among the available crater scaling laws, we implemented the one by Holsapple and Housen (2007). The advantage of this formulation is that it is of general purpose and can be applied both to strength and gravity regimes. Furthermore, it allows full control of target physical properties, which may differ among various asteroids and across a body. On the other hand, crater size may depend strongly on input parameters (mostly the strength of the target, Y), so the cratering retention age may vary depending on the assumptions. Here we restrict our analysis to outline how to obtain crater retention ages and how they depend on the assumed parameters. An important aspect to bear in mind is that assessing crater retention ages is a multiple-step

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problem and, whenever possible, geological analyses need to be implemented to narrow down the possible range of target properties.

We consider two main cases. The first is the so-called hard rock (HR) case. The crater scaling law for hard rock has been extensively applied to terrestrial planets and asteroids (Melosh, 1989), and therefore offers a good term of comparison. Here we adopt a rock strength of 20 MPa (Asphaug et al., 1996). There is little doubt, however, that the HR scaling does not generally apply to asteroids, particularly for small craters formed in regolith or craters formed on a rubble-pile asteroid that are much larger than the components of the rubble. For this reason, we also implement an additional crater scaling for weaker materials; we adopt a scaling (for so-called cohesive soils in Holsapple and Housen, 2007) that has been calibrated using impact experiments on various weaker terrestrial materials, including alluvium, whose strength is below or of the order of 100 kPa at most. Here we assume that this scaling is applicable to craters in the size range 10s-100s m, and we adopt a nominal strength of 2 MPa, justified by the fact the craters under analysis reach a significant depth where an increased strength with respect to lab experiments is expected (we also use different strength values for specific cases). This choice seems reasonable given that lunar regolith has a strength of ~1 kPa (at 2 m depth), and the strength increases with depth. Larger craters, however, reach a much greater depth and it is uncertain whether both the hard rock or weaker scaling – or neither – apply. Our current understanding of asteroid evolution suggests that all except large ones undergo major collisional evolution during their lifetimes (Bottke et al., 2005) to the point that most are probably rubble piles. Larger asteroids (such as Vesta) are certainly not rubble piles; however, their upper layers may well be moderately to highly fractured like megaregolith on the Moon. We generally refer to these situations, as well as to the loose material case, as rubbly material (RM) for which we apply the cohesive soil expression by Holsapple and Housen (2007) for various values of strength, as indicated.

Impact rates and velocities were specifically computed for each asteroid, using a model main belt population from Bottke et al. (2005). Current impact rates have been extrapolated back in time following a recent model of main belt evolution (O'Brien et al., 2014). In the following discussion we will not consider Mathilde, because the crater scaling law for highly porous bodies is highly uncertain. Also, for sake of simplicity we will not provide error bars for the best fit ages. The formal analytical best fit age uncertainties are typically of the order of a few %. However, the uncertainty resulting from the impactor population and crater scaling law can be as low as ~20-30 % for statistically robust crater SFDs (such as Rheasilvia), and it can be even larger for oddly shaped crater SFDs and/or with low statistics (such as Steins).

2.3.1. Cratering retention ages. For the interesting case of Vesta, we obtain an age of ~1 Gyr ago for Rheasilvia floor's crater SFD. This age is obtained both using the HR and RM (Y=2 MPa) scaling laws. The quality of the fit is almost identical and only one of them is shown in Fig. 4. The MPF slightly overestimates the number of craters on the smooth ejecta unit. This is not surprising because of the different crater size range and nature of the terrains. The difference can be entirely due to variation in the properties of the terrains. It is also possible that the assumed impactor SFD overestimates the real main belt population. The dip at crater sizes ~0.7-2 km is interesting and may represent a real feature in the impactor SFD. Applying the same scaling laws, we derive an age of ~ 50 Myr ago for the Marcia smooth unit (Fig. 4). It should be noted that the target strength has a significant effect on small crater formation (crater size ~Y(0.2−0.3)), and a strength variation of a factor of 10 results in a factor of 2-3 variation in the cratering age.

It is more difficult to derive an age for the heavily cratered terrain. A nominal fit using RM (Y=2 MPa) scaling law gives an age of ~4.2 Gyr ago. This should be regarded as a lower limit, given that crater obliteration is expected to be important for old terrains. An attempt to correct for this process,

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following the numerical approach outlined by O'Brien et al., (2006), gives an age of ~4.4 Gyr ago. These ages are certainly very uncertain; nevertheless, it is possible that much of the collisional history of Vesta may still be recorded in its HCT.

For Gaspra, we obtain an age of ~1.5 Gyr ago for both HR and RM (Y=2 MPa) scaling laws, which may increase to ~1.8 Gyr ago if crater obliteration is considered (Fig. 4). In the case of Ida we obtain an age of 3.3 Gyr ago (HR) and 2.8 Gyr ago (RM, Y=2 MPa), ignoring crater obliteration. Both ages become ~ 3.5 Gyr ago if we correct for crater obliteration, and the crater SFD seems to be close to or being in saturation (Fig. 4).

The case of Eros is interesting for several reasons. First, it is a near-Earth object yet it is generally believed that most observed craters formed while Eros was in the main belt, prior to being deflected into a near-Earth orbit. Here we assume, therefore, average main belt impact rates, and derive an age of about 1.5 Gyr ago for both HR and RM (Y=2 MPa) scaling laws. This age is derived without crater obliteration and it is based on the fit of craters > 0.4 km (using the global catalog shown in Fig. 5a). Figure 5a also provides two additional crater SFDs: (i) a highly cratered terrain (D~0.1-2 km), and (ii) a high resolution count on a smaller area (D~0.01-0.2 km). Interestingly, the global crater SFD lies below the heavily cratered crater SFD for craters in the size range ~0.2-1 km. This depletion has been ascribed to erasure due to the formation of Shoemaker crater, as mentioned in Section 2.2.4 (Thomas & Robinson, 2005). The global crater SFD is used here for age dating because it provides a better statistics at large crater sizes, but it should be noted that the Eros surface is not uniform. The difference between our age estimate with a previous age estimate (Richardson et al., 2005) of ~0.4 Gyr can be understood in light of the different assumptions. In particular, they implemented a different scaling law that results in a much larger crater for the same impactor. Therefore, the resulting age is younger. In fact, we obtain a similar age implementing RM scaling and Y=200 kPa. On the other hand, Richardson et al. (2005) also argued (based on the largest craters) that the cratering age may be as old as 1 to 2 Gyr ago. The second aspect of interest is the availability of high resolution measurements that allow validation of the seismic models. Figure 5b,c shows that the down-slope regolith migration resulting from impact-induced seismic shaking can explain the paucity of small craters. A similar model also provides a good fit of Itokawa crater SFD.

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3. CRATER MORPHOLOGICAL PROPERTIES

Crater morphological properties, e.g., depth-to-diameter ratio, can be used to address degradation processes operating on asteroids. Early observations of lunar terrains (Gilbert, 1893) revealed that craters with diameters below a few tens of kilometers (so-called simple craters) have bowl shaped profiles with the maximum depth at the center of symmetry of the crater. Larger craters (so called complex craters) show progressively higher raised central peaks surrounded by flatter crater floors. The final crater shapes are attained during the so-called modification stage (e.g., Gault et al., 1968) that takes places when the excavation cavity has reached its maximum radial extension, and surrounding highly fractured rocks collapse within the cavity pulled by gravity. The transition from simple to complex craters scales as ~1/ga (Pike, 1980); therefore, on asteroids it takes place at much larger crater sizes (see Section 2) compared with terrestrial bodies. Due to the low gravity of asteroids, the crater modification stage is probably largely affected by target mechanical properties (e.g., strength) and the asteroid's overall shape. Once a crater is formed, additional degradation processes are responsible for the evolution of its morphology, as discussed in Section 2.

In general crater morphologies on asteroids are not fundamentally different from the ones observed on terrestrial bodies, except that craters on all but the largest asteroids generally form in the strength regime and also that they are simple craters (except the largest craters on the largest asteroids, e.g., Rheasilvia basin on Vesta). Also, the morphology of a crater approaching the physical size of the asteroid significantly differs from the final shape described above. We review key morphologic parameters of asteroidal craters, such as depth-to-diameter ratios, which can be measured from individual images or from digital terrain models. Additional useful data are the ellipticity, particularly for craters on slopes, and the overall shape of large depressions that approach an asteroid’s physical size.

3.1. Crater Depth-to-Diameter Ratio

Here we discuss in detail asteroid crater morphology by presenting selected examples from Itokawa, Eros, Lutetia, and Vesta. For all these bodies, crater morphologies are assessed from imaging. Crater diameter is obtained by fitting an ellipse to the crater rim and taking the length of the largest axis (in case of non-georeferenced images) or the average of the two axes (for georeferenced images). It can be measured very precisely, with a typical uncertainty of twice the spatial resolution. Depth can be obtained directly if the spacecraft carried an altimeter, i.e. the NEAR-Shoemaker mission at Eros and the Hayabusa mission at Itokawa. However, in most cases, depth is measured with less accurate indirect techniques such as measuring the length of rim shadows cast inside the crater, or using a digital terrain model (DTM) reconstructed by stereo-photogrammetry (e.g., Vesta) or stereo-photoclinometry (e.g., Lutetia) from the images. Comparisons of the different imaging-based methods show that the shadow technique tends to overestimate the depth while photoclinometry underestimates the depth, especially for the smallest craters. Stereo-photogrammetry is the most accurate technique, but requires multi-imaging of the surface with very specific illumination conditions, not always achievable (especially during flybys).

In addition, for oddly shaped asteroids it is important to try to take into account preexisting topography under the crater when measuring the depth. Due to the non-equilibrium shape of most asteroids, their topographic excursions are generally significant with respect to the radius of the body.

Craters on asteroids display a wide range of shapes, from sharp bowls to very subdued and degraded depressions. This translates into a wide range of crater depths (h). For simple craters this ratio follows a near linear law, h/D~k, where k for asteroids is approximately 0.15 (Grieve, 2007). Figure 6

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shows an example of this linear relationship for Eros, Lutetia, and Vesta. Variations are generally interpreted as reflecting different states of degradation of the surfaces, but different physical properties of the target may also play a role. Consider two examples: target properties (porosity, strength), and impact velocity. Craters in highly porous material have generally a higher depth-to-diameter ratio than those in low porosity material (e.g., Mathilde; Housen and Holsapple, 2003). An interesting characteristic of craters on asteroids, as distinct from the Moon and terrestrial planets, is that they appear deprived of impact melts due to the lower average impact speeds (e.g., Keil et al., 1997; Marchi et al., 2013); therefore, they better show the morphology of the floor and walls.

3.1.1 Small craters. For asteroids visited by space missions, the overall properties of h/D distributions (weighted toward fresher craters, but including somewhat degraded craters, as well) look very similar. Ranging from ~0.12 to ~0.20 (excepting Itokawa), their mean values are centered close to ~0.15 (Carr et al., 1994; Sullivan et al., 1996; Veverka et al., 1999, 2000; Besse et al., 2012; Vincent et al., 2012, 2014), slightly below the canonical value of ~0.2 for planetary surfaces (Melosh, 1989; and references therein). Smaller, rubble-pile asteroids (e.g., Itokawa) tend to have shallower craters than larger bodies (e.g., Lutetia, Vesta).

Small craters exhibit a range of measured h/D, while larger craters tend to have similar h/D ratios. In general, the dispersion in h/D values decreases with increasing crater diameter. This may be the result of several factors: i) Measurement uncertainty is larger for smaller craters, especially when their diameters are only a few times the image resolution, but the dispersion in h/D is still visible when removing craters smaller than ten times the resolution; ii) Fresh craters are more likely to be found among small craters than larger ones, thus small craters can be found in all states of degradation while very young large craters are generally absent; iii) Finally, small craters are most efficiently altered, even to the point of erasure, by post formation processes, such as ejecta blanketing from nearby craters or seismic shaking (see Section 2), whereas the largest craters remain visible, despite degradation, through the history of the asteroid’s surface (unless large craters are saturated).

Relationships between strength, gravity, and formation regime were tested on Vesta by Vincent et al. (2013). The transition between strength and gravity regime is given by Dtrans = 0.8 Y/(ρga) (Asphaug, et al., 1996), or Dtrans ~25 km for Vesta (Y=20 MPa, ρ=2600 kg/m3, g=0.25 m/s2), typically where we start to see a narrowing of the h/D distribution toward the typical ratio of 0.15. The transition between simple and complex craters scales with the inverse of the gravity, so simple craters should transition to complex craters for D~160 km for Vesta. The transition size is fuzzy and, indeed, a crater as small as the ~70 km diameter crater Marcia shows evidence of incipient central peak formation.

If the average initial h/D is similar for all asteroids, recent work on high-resolution images and DTMs has shown that this picture might be too simple, either because of the low resolution of previous studies or the limited number of craters observed. While the overall distribution is generally peaked near a particular value, more complete statistics show that the average h/D can differ from region to region. This is very prominent on Vesta where a double peak is visible in the global distribution of h/D (Vincent et al., 2013) with about 25% difference between the peaks at 0.15 and 0.19. The same effect can be seen on Lutetia with up to 40% difference in average h/D between different regions (Vincent et al., 2012).

It is also valuable to compare the cumulative probability distribution of h/D for different regions and bodies. Figure 7 shows an example of such comparison for Lutetia and Vesta. On both bodies measurements were obtained on the oldest, heavily cratered regions found on the northern hemisphere and on younger regions of Vesta mostly from the southern hemisphere, as well. By comparing the cumulative probability distributions of these broadly different terrains we find that the oldest regions of Vesta and Lutetia display very similar distributions, while younger vestan areas significantly lack

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shallow craters. Overall, Vincent et al. (2013) found a very good correlation between h/D variations and vestan geologic units (Yingst et al., 2014). This is interpreted as evidence of the resurfacing event triggered by the giant impacts in the southern hemisphere of Vesta, which reset the polar surface and partially erased craters even approaching the equatorial regions.

Though most small craters on asteroids resemble those observed on rocky planets, there are a few peculiar morphologies that seem more common on asteroids. For instance many examples of what are termed bi-modal craters were discovered and analyzed on asteroids Eros (Mantz et al., 2004), Lutetia (Vincent et al., 2012), and Vesta (Krohn et al., 2014). They are characterized by a sharp rim on one side and a less defined rim on the other, often associated with important mass wasting. Those craters are typically found on slopes with respect to the gravity field with the sharper rim always on the more elevated side. This peculiar morphology is explained by slumping of material from the upper rim toward the lower rim combined with failure of the lower rim. Material accumulates on the lower edge of the impact feature, smoothing the topography. As discussed earlier, such features are more common on asteroids due to the stronger changes in topography and gravity with respect to the general curvature of the body. Figure 8a,b presents a few examples of these craters.

On areas of relatively homogeneous physical properties variations in crater morphology cannot be due to variations in such properties. Additionally for surfaces unaffected by a massive, local resurfacing event, we can assume that differences in h/D for craters of similar sizes can measure the degradation rate, as the variation of degradation state is only a function of age. For instance, the dispersion of h/D for small craters on regions of similar physical properties on Vesta implies a degradation rate of ~10-7 m/yr (Vincent et al., 2013), comparable with an estimate of boulder erosion on Lutetia (Kueppers et al. 2012), but considerably higher than for the Moon (Fassett and Thomson, 2014).

The morphologies of craters on Eros have been investigated by several authors (Veverka et al., 2000; Chapman et al., 2002; Robinson et al., 2002). Measurements that make use of the NEAR Laser Rangefinder data and a careful assessment of crater freshness indicate that many of the freshest craters on Eros typically possess h/D near 0.2 for D > 100 m (Figure 6). These ratios match results reported on the Moon (Pike, 1977) and Ida (Sullivan et al., 1996). For D <100 m, fresh craters on Eros are shallower than those on the Moon. The average h/D of all craters on Eros indicate that most are degraded (Fig. 8c), and follow a pattern akin to what is observed on Vesta, Steins, and Lutetia.

Craters on Itokawa are very different from most other asteroidal craters. The best defined craters (Hirata et al., 2009) are generally very shallow, with h/D ratios below 0.1 (Fig. 8d). Seismic shaking on this small body, as discussed in the previous section, would surely rapidly degrade any craters that formed on its surface. But, in addition, two other factors might contribute to causing these low h/D ratios: (i) the presence of numerous large blocks can depress crater formation as the impact energy fractures a boulder rather than excavating a depression (Chapman et al., 2002) or otherwise leading to formation of flat shallow craters (e.g., Güttler et al., 2012), and (ii) the large curvature of Itokawa relative to some of the diameters of the large candidate craters proposed by Hirata et al. (2009) can yield very shallow craters, with the crater floor exceeding the height of the crater rims in some rare instances (Fujiwara et al., 1993; Asphaug et al., 1996). The shallow nature of many of Itokawa's craters can therefore be attributed in part to factors influencing their initial formation as well as to subsequent degradation by seismic shaking.

Finally, it is worth considering the spatial variation of crater morphologies across small body surfaces. Most of the differences can be linked to aging processes (erosion, infilling, rim collapse, seismic shaking). However, when statistics are significant enough to build a map of h/D or measure it on various regions of an asteroid, it appears that regions of similar age can show different average h/D. This was determined on Lutetia and Vesta (Vincent et al., 2012, 2013) where a link between variations

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of crater morphologies and geological units might indicate differences in their physical properties.

3.1.2 Large Craters. The largest craters on asteroids can have diameters approaching or even exceeding the asteroid’s radius (see Fig. 9). Most of these craters can be grossly approximated as simple bowl shaped craters, with a few exceptions (e.g., the Rheasilvia basin on Vesta). For these simple giant craters where their h/D can be measured, this ratio is typically less than 0.2 (Thomas et al., 1999). An important reason why many of these giant craters are shallow is illustrated by measurements of h/D of three giant Eros craters both along and perpendicular to the greatest radius of curvature of the asteroid. For example, Himeros crater has a h/D of 0.17 along the long axis of Eros, but a much shallower h/D of 0.1 when measured perpendicular to the long axis of the asteroid. These craters are shallower when the local radius of curvature is more important.

The asteroid Mathilde is unique in that it possesses five giant craters on its imaged side, none of which have led to its disruption. Equally remarkable, given what is now apparent on Eros and Vesta, none of these craters seem to disturb their neighbor. These observations have been attributed to Mathilde’s high porosity of 50% (Cheng and Barnouin, 1999; Housen et al., 1999), which can better absorb the energy of these impacts, making the asteroid harder to disrupt and each crater less likely to influence its neighbor.

The 500±20 km Rheasilvia basin is unique among asteroidal craters (Thomas et al., 1997; Schenk et al., 2012). It possesses a broad central uplift whose height is similar to its rims. The central uplift is remarkably broad compared with other craters on the Moon or Mercury and exhibits extensive slumping and sliding in its interior. In addition, radial curved ridges near the central mound may be the result of Corolis forces acting on inward collapsing material during the modification stage (Otto et al., 2013). However, unlike similar sized craters on other planets, little impact melt is observed, probably due to the comparatively low impact velocities in the asteroid belt which do not efficiently generate melt (e.g., Keil et al., 1997; Marchi et al., 2013).

To date only the relatively recent, giant craters Rheasilvia on Vesta and Shoemaker on Eros show quantitative evidence for resurfacing a significant fraction of the surface of either asteroid. In the case of Rheasiliva, its ejecta have hidden most craters within tens of km of its rim and influenced the crater SFD and h/D of craters further beyond (see Fig.1; Marchi et al., 2012a; Schenk et al., 2012). Unlike Rheaslivia, Shoemaker crater probably removed craters mainly by seismic shaking (Thomas and Robinson, 2005). Indeed the density of craters ranging from ~0.2 to ~1 km diameter increases with radial distance from the center of Shoemaker crater rather than with distance from the crater rim as would be expected for ejecta. Further, the density of the largest blocks seen on Eros that have been attributed to ejecta from Shoemaker crater (Thomas et al., 2001) do not correspond with the locations where the dearth of craters on Eros has been observed.

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4. LINEAR SURFACE STRUCTURES AND BLOCKS

Many asteroids possess various types of linear surface structures (Jaumann et al., 2012; Prockter et al., 2002; Sullivan et al., 1996; Thomas et al., 2012; Thomas et al., 1979; Veverka et al., 1994). They have been identified on most asteroids visited by spacecraft with the possible exception of Steins (Besse et al., 2012; Barucci et al., this volume), and Mathilde, which possesses some limited evidence for them (Thomas et al., 1999), although those two asteroids were imaged with relatively poor resolution. In most cases, when sufficiently good image resolution is available, the genesis of these lineaments can be attributed in part, and sometimes entirely, to a source crater. Lineaments can concur to obliterate preexisting topography, including craters (Prockter et al., 2002; Jaumann et al., 2012), indicating that the formation of lineaments by large impacts can drastically reset the surface of an asteroid (see Section 2). The large extent of many lineaments, and their link to the largest craters on asteroids, even when these lineaments are located far from these craters, indicate that internal properties of most of the asteroids visited must be capable of transmitting sufficient impact energy through their interior to cause near-surface failure that can be linked to a parent crater.

4.1. Types of Lineaments

Lineaments on asteroids have various styles (Fig. 10). The most common are shallow troughs, grooves, and pit chains (Thomas and Prockter, 2011). Shallow troughs are linear depressions, with a flat floor and no raised rim. Grooves are usually V-shaped while pit chains comprise a series of small depressions that are somewhat circular, with uniform sizing and spacing. Less commonly observed lineaments include fractures and broad flat troughs (Fig. 10e-f). Fractures have been reported only on Eros, perhaps because the imaging resolution for other asteroids did not easily permit their identification. The morphology of the observed fractures looks like a fan of cracks emanating from a ridge (Buczkowski et al., 2008; Prockter et al., 2002). On Vesta, broad troughs appear to be graben, with their floor tilting toward the dominant fault (Buczkowski et al., 2012). Such tilting is fairly common for graben that form in sedimentary environments on Earth, and may reflect the abundance of regolith on Vesta.

Ridges are another linear structure seen on asteroids that are also less common than negative relief linear features (Fig. 10d) indicative of extension. Ridges, however, appear to have formed through some displacement along a pre-existing zone of weakness, either because of translation and/or compression. Rahe Dorsum on Eros is one of the best examples of such a structure.

Shallow troughs have been observed on Ida, Gaspra, Eros, Lutetia, and Vesta. Grooves have been reported on Gaspra, Eros, Lutetia, and Vesta, while pit chains have been found on Steins, Eros, Lutetia, and – with extraordinary richness – on Vesta (Carsenty et al., 2013). Broad flat troughs are on Eros, Lutetia, and Vesta, while ridges are on Gaspra, Eros, Lutetia, Vesta and even rubble-pile Itokawa. In the latter case, these appear to be very limited in length (Barnouin et al., 2014).

4.2. Cratering Origin of Lineaments

Studies of the asteroids Eros, Vesta, and Lutetia show that there is a close correlation between the observed lineaments and craters, as had been suspected but not easily confirmed by earlier investigations of Ida and Gaspra (Thomas and Prockter, 2011). A detailed mapping of over 1500 lineaments on Eros indicates that the longest, dominant set of lineaments is likely associated with the formation of large craters. Most of these long lineaments, sometimes in excess of tens of km, are circumferential about impacts in the regions separating Himeros from Psyche crater. Fitting planes to

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the surface expressions of these lineaments provides the normal vectors to these planes, which preferentially point to either Himeros or Psyche crater (Fig. 11). It is difficult to assign which crater is directly responsible for most of the lineaments because they are essentially 180 degree away from each other on the opposite sides of the asteroid. Interestingly, many of the circumferential lineaments appear subdued, and are probably not very young. Although it is impossible to distinguish whether or not Himeros crater formed before Psyche, the large size of Himeros, and its degraded appearance relative to Psyche, suggest that it might have formed first. In such a scenario, it is plausible that Himeros could have initially generated most of the long circumferential lineaments, probably due to impact-generated tensile hoop stresses (Asphaug et al., 1996). Later craters such as Psyche, and perhaps Shoemaker, may have only re-activated them, causing the subdued appearance of many of the lineaments. Such a possibility is consistent with a recent numerical study, which models self-consistent dynamically interacting crack distributions and pressure dependent granular flow of a highly damaged material (Tonge and Ramesh, 2014). These calculations investigate the evolution of porosity and damage in an asteroid. Damage is usually expressed on the surface of an asteroid as cracks. These calculations show for the simplistic case where Eros is assumed to be a basaltic monolith that Himeros could easily create a 20-25% porous Eros, and generate lineament distributions comparable to those we see today. Further, the model indicates that subsequent formation of Psyche and Shoemaker craters would have had minor global effects, with little change to the overall asteroid porosity and global surface cracking. These later craters limit their damage to local regions near the craters. However, the calculations also indicate that for a rubble pile Eros struck by a projectile with the same size and speed as employed for the strong case, any global link between surface cracks to Himeros' formation is lost. Cracks are formed at all scales and in all directions, with little directional connection to the parent crater.

The internal structure of Eros is, however, not known. The largest lineament on Eros, Rahe Dor-sum, along with two other sets of smaller lineaments that are either parallel or in conjugate orientation to Rahe Dorsum, seem to have no parent crater and could be a remnant from Eros’ original parent body (Buczkowski et al., 2009). Their presence suggests that Eros must have been somewhat damaged to be-gin with, perhaps as the result of the collision on a larger parent body. Such a view is substantiated by analyses, which indicate that Eros is probably best represented by a lunar-like megaregolith covered by a regolith, but with some large interlocking aggregates or fragments, to account for the low measured porosity (Robinson et al., 2002). Such large scale fragments may also explain the presence of the long linements described above. Support to this scenario may come from a smaller subset of lineaments that are radial to many of the smaller (≤5 km) craters (Buzkowski et al., 2008), indicating that the character-istic size for the putative internal rubbles is likely to be comparable or larger than these craters. There-fore, although a monolithic Eros seems unlikely, it still may have significant internal coherence to ex-plain some of the surface features.

The analysis of the 3D orientation used to understand the origin of structural lineaments on Eros was also employed on Vesta and Lutetia. In the case of Vesta, the two main sets of impressive circumferential lineaments are related to the formation of the two south polar basins Rheasilvia and Veneneia (Jaumann et al., 2012; Fig. 12a). The older set is clearly related to the older Veneneia, while the younger features are associated with the younger Rheasilvia. These faults are much more impressive than on Eros, with absolute vertical and horizontal displacements comparable with the size of Eros itself. Preliminary numerical investigations (Buczkowski et al., 2012) suggest that these likely reflect the differentiated internal structure of Vesta that results in shock impedance differences that can accentuate the formation of large surface faults. Further, a stronger metallic interior core may also contribute to localize seismic energy (Bowling et al., 2014). The formation of circumferential lineaments on Vesta was also investigated with the help of laboratory experiments and three-dimensional CTH simulations (Stickle et al., 2015). The results show that large impacts induce wide-

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spread subsurface damage, and that surface lineaments may be the manifestation of faulting due to shear localization in subsurface failure planes. Furthermore, the offset of the pole of the lineaments to the crater center can be used to constrain the geometry of the impact, such as the impact angle, which for the case of Rheasilvia was inferred to be at least ~50 deg from the normal to the surface (Stickle et al. 2015). This process may also explain the set of pit chains seen prevailing in E-W direction that is parallel to the major set of faults due to the formation of Rheasilvia basin.

On Lutetia, the lineaments analysis (Besse et al., 2014) shows that both radial and circumferential lineaments observed can be attributed to craters, as proposed in recent modeling (Jutzi et al., 2013). Most of the circumferential features can be attributed to the north crater complex (formed by 2 to 4 overlapping large craters; see Barucci et al., this volume), but two other sets of lineaments have also been identified (Fig. 12b,c). One set can be linked to the 60-km Massilia crater, and the other to the southern region of the asteroid not imaged by the Rosetta mission (roughly corresponding to Massilia's antipodes). Given the apparent link between many of these lineaments and craters on Eros, Vesta, and Lutetia, this last set of lineaments on Lutetia could indicate the presence of another sizable crater in the southern hemisphere, although we cannot rule out its association with Massilia crater.

Asteroidal ridges do not seem to be directly related to any particular craters. While some studies have invoked an internal process for their formation (Greenberg, 2008), others indicate that these might be the result of collisional processes that are associated with the formation of the asteroids. As mentioned previously, Buczkowski et al. (2009) show that on Eros, the Rahe Dorsum has a set of conjugate fractures that are likely formed coevally with the Dorsum. Neither one of these features is obviously associated with any surface crater, which may suggest that they resulted from cracking during the disruption of Eros’ parent body. Itokawa has a set of ridges that circumscribe the large lobe, but not the small lobe, which also do not have any obvious association with any of the known candidate craters located on the surface of the asteroid (Barnouin et al., 2008; Barnouin et al., 2014). These ridges may reflect some displacement along the edges of interior chunks in the body of Itokawa that were mobilized at some point in Itokawa’s formation and evolution.

4.3. Impact Derived Surface Blocks

Besides forming craters and contributing to lineament formation, impacts also generate blocks on the surfaces of most small bodies. While large cratering events are the sources of many blocks on asteroids, some surface blocks may result from the disruptive origins of the asteroids, creating rubble piles, and may provide additional constraints on their make-up (Lee et al., 1986, 1996; Thomas et al., 2012; Kueppers et al., 2012; Thomas et al., 2001; Nakamura et al., 2008; Michikami et al., 2008; Mazaroei et al., 2014).

High resolution imaging data (<100 m/pixel) from asteroids Vesta, Lutetia, and Eros confirm the assumptions made in earlier efforts at Ida (Lee et al., 1996) that blocks on asteroids are the result of impact cratering. In many instances, the size frequency distributions of blocks found to be associated with craters (e.g., Chapman et al., 2002; Kueppers et al., 2012), possess similar cumulative power-law exponents (slopes) with an average value of about -4. Although a preliminary analysis of the block distributions on Itokawa's surface (Michikami et al., 2008) found a shallow slope of -3, a recent more complete evaluation of all of the largest blocks (>5 m) on Itokawa (Mazrouei et al., 2014) show little difference from distributions derived from crater ejecta. The range of observed slopes on asteroids is comparable to that found for the block distributions of lunar impact craters, which cluster between -3 and -4 (e.g., Bart and Melosh, 2010). However, the blocks on Itokawa may more likely be the result of impact disruption and re-accretion (e.g., Fujiwara et al., 2006) rather than crater ejecta, despite the similar sloped SFDs. Despite the overall agreement, it is important to realize that lunar block SFDs

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often exhibit slopes as shallow as -2 and, in a few cases, also as steep as -5 (Bart and Melosh, 2010). Such a significant spread in the slope can be due to differences in terrain properties and/or preservation. In fact, it should be noted that once blocks are exposed on the surface, they undergo erosion and break-up as a result of a number of processes, such as small impacts and thermal fatigue (e.g., Delbo et al., 2014), which may alter their SFD.

The overall similarity of the asteroidal block size-frequency distributions likely reflects a common formation process, a similar range of material properties, and similar preservation. Most of these blocks appear to be derived by impact cratering (Lutetia, Eros) or disruption (Itokawa) of pre-existing regolith and rubble, that was created by brittle fracture. That some individual fragments fail brittlely is suggested by several studies (Nakamura et al., 2008; Michikami et al., 2010) that show that the blocks on Itokawa possess morphologies, shapes, and aspect-ratios very similar to fragments generated in small scale laboratory experiments produced by brittle failure of individual rocks (Tsuchiyama et al., 2011). Once blocks are exposed on the surface they follow similar evolutionary pathways probably as a consequence of thermal fatigue and impacts so they maintain similarities in their size distributions.

4.4. Implications for the Internal Structures

The analyses of lineaments on Vesta, Lutetia, and perhaps Eros indicate that these objects must possess some internal ability to transfer impact energy over long global distances. Even the rubble pile Itokawa has ridges, which indicate the presence of localized aggregates that have slid past each other, although not at the global scales seen elsewhere. Blocks observed on Vesta, Lutetia, Eros, and Itokawa provide further evidence for the presence of subsurface rocks that have been broken brittlely by impact. Gaspra and Ida also possess lineaments, and the latter also surface blocks. Ida's measured density of 2600±500 kg/m3 is similar to that for Eros given its proposed composition, and suggests that its interior is capable of transferring impact energy over long distances (Asphaug et al., 1996). These observations are consistent with their interiors – at least to the depth reached by cratering – being structured and made of materials that are connected and can fail brittley. Of course, Vesta, and possibly Lutetia, are large enough to be primordial differentiated objects that have not been fully damaged and are surely competent (e.g., Pätzold et al., 2011; Russell et al., 2012; Park et al., 2014).

However, collisional evolution models strongly suggest that smaller asteroids, like Eros and Ida, should be rubble piles (Michel, 2001; Durda et al., 2007). The low observed porosity and global lineament patterns observed for some asteroids might be indicators of rubble piles with good internal connectivity. Numerical simulations of impacts into loosely bound aggregates show that well connected aggregates are probably necessary to generate the stresses that produce lineament patterns like those observed on Eros (Crawford and Barnouin-Jha, 2003). Alternatively, Asphaug (2009) proposed that the surface morphology and density of Eros could be explained by a cohesive sand pile although the apparent relationship of many of the lineaments to large craters (in the same way lineaments are related to large craters on Vesta and Lutetia) casts doubt on a sand-pile model. Furthermore, surface roughness assessments indicate some bedrock could be protruding through regolith on slopes (Cheng et al., 2001), which suggest that these asteroids have competent components that may even retain some memory of their relationships on their parent bodies from which they were liberated by catastrophic collisions (Michel et al., this volume).

Mathilde is an exception to what we observe for many of the other asteroids. Unlike Eros and Ida, as well as many of the larger bodies, this C-type asteroid has a limited number of lineaments identified (perhaps due to poor imaging resolution) and high porosity (~50%; Yeomans et al., 1997). Mathilde also possesses an inordinate number of large craters relative to other asteroids (e.g., Cheng

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and Barnouin, 1999; Chapman et al., 1999). It is well known that high porosity limits the extent impact-derived shocks can travel through an asteroid (Love et al., 1993; Cheng and Barnouin, 1999; Housen and Holsapple, 2003; Schultz et al., 2007; Jutzi et al., 2010b). This can limit the extent of lineament formation (Jutzi et al., 2010b), and permit the formation of many large craters without disrupting Mathilde. This difference might be a reflection of the physical differences between ordinary and carbonaceous chondrites (Britt et al., 2002).

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5. CONCLUSIONS

Cratering of asteroids was a topic that didn’t exist until 1991, when the Galileo spacecraft imaged the asteroid Gaspra during a close flyby. Continuing opportunistic flybys as well as dedicated missions to asteroids have now revealed numerous asteroids, all of whose surfaces are cratered to varying degrees. Future missions already launched or under development (such as Hayabusa 2 and OSIRIS-Rex), and higher resolution radar delay-doppler mapping, promise a continuing source of cratering data.

Given the lack of endogenic geological activity on most asteroids, it is no surprise that the dominant evolutionary process influencing the surface of asteroids is impacts by other asteroids and comets. These result in cratered surfaces, although larger impacts penetrate the interiors of asteroids giving rise to a variety of linear features, ridges, and trenches, also expressed on the observed surfaces. Large impacts can also trigger mass-wasting and, in other ways, degrade and eventually erase pre-existing craters and landforms. Asteroids imaged at sub-meter resolutions (Itokawa, Eros) present boulder-strewn surfaces with fewer obvious craters as the inherent rubble-pile constituents and more effective crater destruction processes tend to dominate over formation and retention of craters.

Size-frequency distributions (SFDs) of craters tend to vary not only from asteroid to asteroid, but also on different geological units on the same asteroid. Nevertheless, crater SFDs tend to initially reflect a single Model Production Function (that of the collisionally relaxed SFD for the main-belt asteroid population, see Bottke et al., this volume), subsequently modified by formation of secondary craters in some cases and by crater degradation processes. The degree to which these crater SFDs are undersaturated or instead approach or reach saturation enables us to date relative age differences on an individual asteroid, like Lutetia and Vesta, but also to approximately provide absolute ages for surfaces of different asteroids. Such absolute ages depend heavily on model parameters, including models for the rate of decline of the impact flux from the earliest times, assumed crater scaling laws, and assumed target properties (strength, density, porosity). Given the uncertainties involved in cratering ages, it is important to compare with other methods, whenever possible.

On Vesta, crater retention ages range from ~1 Gyr ago for terrain associated with the Rheasilvia basin to perhaps as much as ~4.4 Gyr ago for the most heavily cratered terrains, although the older estimate is highly uncertain. There have been arguments from studies of the dynamics of the Vesta asteroid family (see Nesvorny et al., this volume) that Rheasilvia, presumably the scar from the impact that created the family, is ~1 Gyr in age (Milani et al., 2014), though other analyses pose a lower limit of ~0.5 Gyr (Nesvorny et al., this volume). The case of Vesta is particularly important because we have samples of it in our meteorite collections: the howardite, eucrite, and diogenite (HEDs) meteorites. Meteorites can retain a record of their parent body collisional history in their isotopic signatures (e.g., Bogard, 2011). Particularly useful for this purpose is the K-Ar system that can be reset during collisions. HED meterorites show a characteristic K-Ar age spectrum indicating numerous resetting events older than ~3.5 Gyr ago (Bogard, 2011; Marchi et al., 2013). Most HEDs are thought to originate from the formation of the Rheasilvia basin on Vesta, although older basins may also contribute. Interestingly, a recent high spatial resolution analysis of feldspar grains in Kapoeta, a howardite, yielded ages ranging from 0.8 to 1.2 Gyr ago (Lindsay et al., 2015), compatible with Rheasilvia's cratering age.

We find ages for Gaspra and Ida of ~1.8 Gyr ago and ~3.5 Gyr ago, respectively (the age estimate for Ida would be a lower limit if Ida’s surface is saturated with craters). Since Gaspra is probably a member of the Flora family (here we assume it is of rocky composition, not metallic as has been suggested by Chapman 1996c) and Ida is quite certainly a member of the Koronis family, it is interesting to compare our results with dynamical ages for those families, ~1 Gyr and ~2.5 Gyr,

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respectively (Dykhuis et al., 2014; Nesvorny et al., this volume); these are reasonably consistent, given the approximate nature of both cratering and dynamical ages. We have no independent check on our estimate of ~1.5 Gyr ago for the crater retention age on Eros, since Eros cannot be linked to a specific family. Age estimates for other non-family asteroids (Lutetia, Steins, Toutatis), based on the same methodology, are presented in a separate chapter (Barucci et al., this volume). Because of great uncertainty about the applicable crater scaling law for Mathilde and the strength of that C-type asteroid, we do not propose a cratering age for it.

We have described depth-diameter measurements for craters on several asteroids. The great diversity of crater morphologies exhibited on asteroids imaged up close (e.g. the bimodal craters formed on steep slopes on Lutetia and Vesta) suggest that more complete analyses than have been done to date of the statistics of craters of different morphologies (e.g., across the range of degradational states) may illuminate issues of crater formation, degradation, and obliteration, which are the dominant processes shaping asteroid surfaces. In a similar way, studies of impact-induced lineaments, such as ridges and trenches, can help to reveal aspects of the bulk structure of asteroids by providing constraints on how shock waves travel through asteroids.

As Dawn approached Ceres in early 2015, there were ideas that this largest asteroid or dwarf planet might exhibit large craters, like those on Vesta and Lutetia, or might not, because of its different composition, which may affect the formation, preservation, and/or survivability of craters. For example Tyre on Europa and palimpsests on Callisto show little relief. Approach images suggest that the answer is ____ [to be filled in at page-proof stage in late February or afterwards]. Impact cratering on asteroids appears to be a very complex process for which many facets are not yet fully understood despite the significant progress made over the last decade. As we have indicated, craters and impact-induced surface features potentially hold a great wealth of information about the past evolution and internal structures of asteroids. Future missions will certainly improve our understanding of these processes and, undoubtedly, provide new challenges.

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Figure 1. Vesta's global crater catalog. (a): The map shows a cylindrical projection of all mapped craters larger than 4 km in diameter (more than 2500 are shown on the map) overlaid on a digital terrain model. The two lines encompassing the south pole are the projections of best fit circles to Rheasilvia and Veneneia basins. (b): Crater areal density (in units of number of craters per 104 km2). The map is produced averaging craters over a radius of 80 km. Due to limited imaging coverage at high northern latitudes, the map is not reliable for latitudes above ~75 N deg.

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Figure 2. Selected crater SFDs of various asteroids. The data are shown in the form of cumulative numbers of craters per unit area as a function of crater diameter. The thick black lines indicate the level of empirical saturation corresponding to 10% of geometric saturation (Gault, 1970; Melosh, 1989 pg. 192; see Section 2.2). (a): Crater SFDs of representative terrains within the Marian and Rheasilvian units on Vesta. Marcia crater SFD is obtained from high resolution counts within the rim of Marcia crater (from Marchi et al., 2014). The SFDs for craters on the floor of Rheasilvia basin and on its proximal ejecta blanket update those presented by Marchi et al., 2014. (b): A comparative view of crater SFDs of Lutetia (Achaia region, from Marchi et al., 2012b), Ida (Chapman et al., 1996a), Gaspra (Chapman et al., 1996b), Mathilde (Chapman et al., 1999), and the Vesta heavily cratered terrain (HCT; count updated from Marchi et al., 2012a). For a better comparison with the left panel, Marcia crater SFD is plotted here again. (c): Crater SFDs of Itokawa (Hirata et al., 2009), Steins (Marchi et al., 2010), and Eros (updated from Chapman et al., 2002). For a comparison with crater counts shown in panels (a) and (b), the crater SFDs of Marcia and Rheasilvia are also shown.

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Figure 3. A demonstration of crater sandblasting (panels a and b) and cookie-cutting (panels c and d) in a small-scale lunar surface simulation. Beginning with a pristine, large crater in (a), the surface is bombarded with a steeply sloped impactor population rich in small impactors. Individual impacts have very little effect on the large crater's topography and visibility. However, over time and cumulative impacts (b), the large crater is eroded and filled in, as the numerous small impacts redistribute the material making up the walls and rim of the large crater. In the panel (d), a pair of large, simple craters have cookie-cut out (erased) all of the much smaller craters that previously occupied their location in panel (c). In this manner, large craters can severely alter small crater populations and counts over the affected region.

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Figure 4. Examples of Model Production Function (MPF) best fits (solid and dashed thick black curves) for some of the crater SFDs presented in Fig. 2. (a): Vestan terrains, (b) Ida and Gaspra. Note that the change in slope of the MPF is primarily due to similar changes in slope in the assumed impactor SFD. The crater scaling law may also introduce minor modifications in the MPF from one asteroid to another depending on the target strength and gravity, particularly for large craters on large asteroids that may approach the gravity regime. For details on the parameters used for the fits see the text. The thick solid line that bounds the gray area defines the empirical saturation level as in Fig. 2.

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Figure 5. Various Eros cumulative crater SFDs are reported in panel (a): They are from Chapman et al. (2002) (high resolution); Veverka et al. (2001) (heavily cratered), and Thomas and Robinson (2005) (global). The latter is an average over the whole surface, including regions where small craters were erased by formation of Shoemaker, and therefore it contains fewer craters than the heavily cratered counts in the size range 0.1-1 km. The thick solid line that bounds the gray area defines the empirical saturation level as in Fig. 2. (b): Same distribution as in (a) but now depicted in relative-density plot fashion (CATWG, 1979), which is better suited to show fine scale variations in the distributions, showing the increased paucity of small craters as one goes below about 0.1 km crater diameter. The solid curve shows the crater counts for the model simulation shown in panel (c).

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Figure 6. Distributions of crater depth vs crater diameter for asteroids Eros, Lutetia, and Vesta. Although the three asteroids are very different, one can notice similar behavior in their crater morphologies. The average relation between crater depth and diameter is close to a linear law with a slope of ~0.15, although degradation state is an important contributor.

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Figure 7. Cumulative distribution function of h/D on Eros, Lutetia, and Vesta (N: northern hemisphere, S: southern hemisphere), emphasizing the relative differences in the distribution of depth to diameter ratios. One can see that old surfaces like Lutetia or the northern hemisphere of Vesta tend to have significantly more shallow craters than recently reset areas such as the southern hemisphere of Vesta. This degradation with age is even more pronounced in the case of Eros where additional effects such as seismic shaking smooth the topography. About 80% of Eros’ craters have a h/D less than the typical value of 0.15, 70% on Lutetia and Vesta N, and only 20% in Vesta S. Asteroidal h/D average values are 0.12 for Gaspra (Carr et al., 1994), 0.15 for Ida (Sullivan et al., 1996), 0.2 for Mathilde (Veverka et al., 1999), 0.14 for Eros (Veverka et al., 2000), 0.08 for Itokawa (Hirata et al., 2009), 0.12 for Steins and Lutetia (Besse et al., 2012; Vincent et al., 2012), and 0.15 and 0.19 for Vesta N and Vesta S, respectively (Vincent et al., 2013).

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Figure 8. Peculiar crater morphologies on asteroids. (a) Bimodal crater on Lutetia and (b) Vesta. (c) Eroded impact crater on Eros, and (d) a shallow impact feature on Itokawa.

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Figure 9. Examples of larger features on asteroids. (a) Vesta, (b) Itokawa, (c) Mathilde, (d) Gaspra, (e) Eros.

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Figure 10. Common lineaments observed on asteroids: (a) grooves on Ida; (b) pit chains on Eros; (c) shallow troughs on Lutetia. Less common linear structures: (d) ridges seen on Eros; (e) broad troughs on Vesta; and, (f) heavy modification of topography (a crater rim) on Eros by grooves and fractures (images from Prockter and Barnouin, 2003).

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Figure 11. Global distribution of lineaments on Eros. The dashed black line indicates the average planes derived from a subset of about 1500 lineaments. Perpendicular to the dashed lines are seen the large craters Himeros (above line) and Psyche (below line), which are possible sources for some of the lineaments. The two views correspond to a rotation of 180 deg in longitude.

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Figure 12. Location of poles (black pluses) defined by best fit planes to circumferential lineaments seen on (a) Vesta (modified from Jaumann et al., 2012) and (b-c) Lutetia (modified from Besse et al., 2014). The black lines mark major craters. The center of the the large Rheasilvia and Veneneia basins on Vesta are also indicated (black dots) to highlight the slight misalignment with the poles of the equatorial troughs system associated with both basins.

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