Icarus 347 (2020) 113811

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Icarus

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Impact-produced seismic shaking and regolith growth on asteroids 433 Eros, 2867 Šteins, and 25143 Itokawa James E. Richardson a,∗, Jordan K. Steckloff b, David A. Minton c a Planetary Science Institute, 536 River Avenue, South Bend, IN, 46601, USA b Planetary Science Institute, 2234 E. North Territorial Road, Whitmore Lake, MI, 48189, USA c Purdue University, Department of Earth, Atmospheric, and Planetary Sciences, 550 Stadium Mall Drive, West Lafayette, IN, 47907, USA

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Keywords: Of the several near-Earth and Main Belt asteroids visited by spacecraft to date, three display a paucity of Asteroids small craters and an enhanced number of smoothed and degraded craters: 433 Eros, which has a deficit of Surfaces craters ≲ 100 m in diameter; 2867 Šteins, which has a deficit of craters ≲ 500 m in diameter; and 25143 Asteroid eros Itokawa, which has a deficit of craters ≲ 100 m in diameter. The purpose of this work was to investigate Asteroid Itokawa and model topographic modification and crater erasure due to impact-induced seismic shaking, as well as Impact processes Regoliths impact-driven regolith production and loss, on these asteroid surfaces. To perform this study, we utilized the numerical, three-dimensional, Cratered Terrain Evolution Model (CTEM) initially presented in J. E. Richardson, Icarus 204 (2009), which received a small-body (SB) specific update for this work. SBCTEM simulations of the surface of 433 Eros correctly reproduce its observed cratering record (for craters up to ∼ 3 km in diameter) at a minimum Main Belt exposure age of 225 ± 75 Myr using a ‘very weak rock’ target strength of 0.5–5.0 MPa, and producing a mean regolith depth of 80 ± 20 m, which agrees with published estimates of an actual regolith layer ‘‘tens of meters’’ in depth. SBCTEM simulations of the surface of 2867 Šteins correctly reproduce its heavily softened cratering record at a minimum Main Belt exposure age of 175±25 Myr using a ‘weak rock’ target strength of 2.5–5.0 MPa. Extending this simulation produces a mean regolith depth of 155 ± 5 m at a minimum Main Belt exposure age of 475±25 Myr, in good agreement with the 145±35 m actual mean regolith depth estimated from two chains of drainage-pit craters on the surface, with both crater density and regolith depth in an equilibrium state. SBCTEM simulations of the surface of 25143 Itokawa correctly reproduce its extremely sparse cratering record at a minimum Main Belt exposure age of 20 ± 5 Myr, using a ‘soft soil’ target strength of 10-20 kPa, and producing a mean regolith overturn depth of 5.0 ± 1.0 m, given that Itokawa is composed entirely of regolith and boulders. These simulations demonstrate the efficacy of impact-induced seismic shaking to erase small craters on the surface of asteroids ≲ 25 km mean diameter, correctly reproducing their observed cratering records while using a Main Belt impactor population and standard crater scaling relationships. These simulations also show that the observed regolith layer on asteroids in the 5-50 km size range can be generated entirely through impact cratering processes, and indicate that regolith depth can be used as a additional constraint on the surface age of a given asteroid.

1. Introduction et al., 1994; Housen et al., 1979; Veverka et al., 1986). The term ‘re- golith’ in this context is broadly defined as a mantle of unconsolidated, 1.1. Impact-produced regolith on asteroids fragmental debris of relatively low cohesion, overlying a more coher- ent substratum (Shoemaker et al., 1967). However, the Galileo flyby Prior to the 1991 flyby of asteroid 951 Gaspra by the Galileo space- images of 951 Gaspra in 1991 (Belton et al., 1992), followed by a flyby craft, most theoretical and experimental work suggested that asteroids of asteroid 243 Ida in 1993 (Belton et al., 1996), and most notably, the of Gaspra’s size would develop a negligible regolith layer if composed extensive observations of asteroid 433 Eros conducted by the NEAR- of materials as strong as solid rock, or would develop a regolith only Shoemaker spacecraft in 1999–2001 (Veverka et al., 2001), all showed centimeters to meters in depth if composed of weaker materials (Carr that such bodies are capable of generating and retaining a significant

∗ Corresponding author. E-mail addresses: [email protected] (J.E. Richardson), [email protected] (J.K. Steckloff), [email protected] (D.A. Minton). https://doi.org/10.1016/j.icarus.2020.113811 Received 19 July 2019; Received in revised form 3 April 2020; Accepted 6 April 2020 Available online 11 April 2020 0019-1035/© 2020 Elsevier Inc. All rights reserved. J.E. Richardson et al. Icarus 347 (2020) 113811 regolith layer ‘‘tens of meters’’ in depth on average, and even deeper in some regions (Robinson et al., 2002). Additionally, these objects possess complex surface morphologies, including extensive evidence of downslope regolith migration, slumps, slides, and the degradation of craters via regolith coverage and filling (Carr et al., 1994; Sullivan et al., 1996; Robinson et al., 2002). Since the time of these observations, however, relatively little the- oretical or modeling effort has gone into reconciling the above dis- crepancy between theory and observation utilizing impact cratering as the regolith production mechanism (Küppers, 2001). By contrast, the hypothesis that regolith generation on smaller asteroids may be dominated by the thermal cycling of rocks on their surfaces has re- cently been gaining traction (Delbo et al., 2014). Nonetheless, sample regolith grains returned by the JAXA Hayabusa mission to the largely craterless asteroid 25143 Itokawa contain impact-induced shock fea- tures and other indicators that point to impact processing of Itokawa’s outermost regolith layer (Tsuchiyama et al., 2011). Investigating the hypothesis of impact-generated regolith production on asteroid surfaces was previously inhibited by the lack of an adequate model describing impact ejecta production and deposition behavior within a significantly low-gravity surface environment (< 1/1000th Earth 푔), particularly in Fig. 1. (Lower curves) A plot of the size of impactor necessary to produce > 1 푔푎 the low-speed ejecta-deposition region immediately surrounding the accelerations (vertical launching) of an ℎ = 1 m thick, non-cohesive regolith layer resting on a slight 2◦ slope, for a variety of asteroid sizes and internal seismic resulting crater (the ejecta blanket region). As part of our work for properties. In these numerical shake-table experiments (Section 2.3), we used an impact the Deep Impact and Stardust-NExT missions to comet 9P/Tempel seismic efficiency factor of 휂 = 10−6 (Section 2.2.1), a seismic quality factor of 푄 = 1600 1 (A’Hearn et al., 2005; Veverka et al., 2013), we extended the classic (Section 2.2.3), and a variety of seismic diffusivity values (Section 2.2.3) spanning an 2 − 2 − 2 − scaling relationships for both impact crater size (Holsapple, 1993) and order of magnitude: 휉 = 0.5 km s 1 (solid), 휉 = 0.2 km s 1 (dashed), 휉 = 0.1 km s 1 (dot-dashed), and 휉 = 0.05 km2 s−1 (dot-dot-dashed). (Upper solid curves) Minimum impact ejecta ballistics (Housen et al., 1983) into the extremely low- stony impactor diameter necessary to cause disruption of a stony asteroid of given gravity environments of small solar system bodies in a general, broadly diameter, calculated per Melosh and Ryan( 1997) (top), and Benz and Asphaug( 1999) applicable fashion (Richardson et al., 2007; Richardson and Melosh, (bottom). The region where global surface effects can occur from a single impact without 2013). In addition to correctly duplicating the ejecta plume evolution disrupting the asteroid is shown in gray. observed on Tempel 1 as a result of the Deep Impact collision, the efficacy of this model has been verified through direct comparison to the laboratory-impact ejecta plume behavior experiments conducted in its crater counts, instead displaying the typical observation roll-off by Cintala et al.( 1999), Anderson et al.( 2003, 2004), and Barnouin- expected when crater sizes drop below a few pixels in size (Marchi Jha et al.( 2007), as described in Richardson( 2011). As a result, we et al., 2012). now have the ability to revisit this old problem of asteroid regolith To illustrate the effect of asteroid size on its susceptibility to impact- development via impact cratering on asteroid surfaces, which is one induced seismic shaking, Fig.1 shows the results of a series of numeri- of the goals of this study. cal shake-table experiments (Section 2.3) that map out the diameter of −1 impactor, striking the surface at a speed of 푣푖 = 5 km s , necessary to 1.2. Impact-produced seismic shaking on asteroids produce accelerations greater than the asteroid’s surface gravity 푔푎 on the ‘far-side’ of the body, as experienced by an ℎ = 1 m thick regolith One of the key findings of the NEAR-Shoemaker mission to asteroid layer resting on a slight 2◦ slope, thus producing active downslope 433 Eros (16.9 km mean diameter) was the observed dearth of craters motion of the regolith layer. This is shown for a variety of asteroid smaller than ∼ 100 m in diameter relative to the size-frequency dis- diameters and internal seismic properties. The region where global tribution of larger crater sizes (Veverka et al., 2001; Chapman et al., surface effects can occur from a single impact without disrupting the 2002; Cheng, 2004; Thomas et al., 2002; Robinson et al., 2002). One asteroid is shown in gray, for asteroids ≲ 50 km in diameter. Asteroids ≳ proposed explanation for this phenomena is the seismic reverberation 50 km in diameter experience only local seismic effects from individual of the asteroid following impact events, which is potentially capable of impacts, as impacts large enough to induce global seismic shaking destabilizing slopes, causing regolith to migrate downslope, and thus would also disrupt the asteroid. degrade and erase small craters (Veverka et al., 2001). In Richardson Vertical, dashed lines in Fig.1 indicate the mean diameter of the et al.( 2004, 2005), we presented a multistep model that demonstrated the efficacy of impact-induced seismic shaking in the degradation and three asteroids that are thus far known to possess a dearth of small erasure of small craters on the surface of Eros, which included the use craters. These asteroids all lie in the size regime susceptible to globally- of a two-dimensional, geometric Monte-Carlo crater population model effective seismic shaking, suggesting that the observed paucity of small that was able to reproduce the crater counts and paucity of small craters craters is the result of the topographic degradation produced by impact- observed on 433 Eros (Chapman et al., 2002), when the effects of induced seismic shaking. 433 Eros exhibits global seismic shaking from impact-induced seismic shaking were included. impactor diameters ranging from 4–500 m (onset) to 1100–600 m Since the time of this previous work, two more small asteroids have (disruption); 2867 Šteins exhibits global seismic shaking from impactor been observed to possess the same paucity of small craters as that diameters ranging from 0.07–0.5 m (onset) to 200–300 m (disruption); observed on 433 Eros: near-Earth asteroid 25143 Itokawa (0.35 km and 25143 Itokawa exhibits global seismic shaking from impactor di- mean diameter), which was visited by the JAXA Hayabusa mission in ameters ranging from 0.003–0.005 m (onset) to 3.3–7.4 m (disruption). 2005 (Fujiwara et al., 2006), and Main Belt asteroid 2867 Šteins (5.26 The implication of these results is that in order for an asteroid to km mean diameter), which was visited by the ESA Rosetta mission exhibit a noticeable paucity of small craters in its observed crater size- in 2008 (Keller et al., 2010). Interestingly, the Main Belt asteroid 21 frequency distribution (SFD), when the bombarding population has Lutetia (98 km mean diameter), visited by the ESA Rosetta mission a steeply negative, power-law distribution slope, the asteroid should in 2010 (Sierks et al., 2011), did not show a paucity of small craters experience global, impact-induced seismic shaking over a wide range

2 J.E. Richardson et al. Icarus 347 (2020) 113811 of impactor diameters, roughly two or more orders of magnitude: as on their surfaces, several large concavities (presumed to be from im- shown by the gray region of Fig.1 for asteroids ≲ 25 km mean diameter. pacts (Greenberg et al., 1994, 1996)), and highly irregular shapes We therefore suspect that, had the Galileo spacecraft been able to indicative of at least some degree of structural strength (Belton et al., image countable craters on asteroid 243 Ida (31.4 km mean diameter) 1992, 1996; Carr et al., 1994; Sullivan et al., 1996). Rather than being at smaller crater sizes than the achieved ∼ 100 m, Ida would likely single stone monoliths or highly pulverized and rearranged ‘rubble not have displayed a paucity of small craters on its surface, given its piles’, these features suggest that these asteroids are something in apparent crater density equilibrium (empirical saturation) state when between. Further work has characterized an entire spectrum of asteroid visited in 1993 (Belton et al., 1996; Chapman et al., 1996a). On the structural types, called ‘gravitational aggregates’ by Richardson et al. other hand, asteroid 951 Gaspra (24.4 km mean diameter), imaged (2002), which span the extremes from monolith to highly comminuted at similar resolutions that permitted crater counts down to 100 m rubble pile. Britt et al.(2002) further identified a transition group in the diameters (Belton et al., 1992; Chapman et al., 1996b), showed a central region of this spectrum, called ‘fractured monoliths’ and placed heavily softened and degraded ‘ghost’ crater population overlain by a both 951 Gaspra and 243 Ida into this transitional category based upon ‘fresh’ small crater population possessing a steep, negative power-law porosity estimates. size distribution, indicative of a ‘production population’ (Section 4.1) The NEAR-Shoemaker observations of 433 Eros likewise showed an that is not yet in a crater density equilibrium (empirical saturation) asteroid that most likely falls into this transitional category of ‘fractured state (Chapman et al., 1996b). This feature on Gaspra points to a monolith’, based upon several lines of supporting evidence: relatively recent event that has largely ‘reset’ its cratering record, likely Indications of structural control of impact craters and other fea- caused by a very large (but non-disruptive) impact (Greenberg et al., • tures (Prockter et al., 2002), 1994; O’Brien et al., 2006). We therefore would not expect to see a • The presence of a global network of visible pit-crater chains, paucity of small craters (indicative of impact-induced seismic shaking) ridges, and grooves (Zuber et al., 2000; Prockter et al., 2002; until Gaspra once again settles into a state of crater density equilibrium. Buczkowski et al., 2008), • A mean porosity measurement of about 20%–30%, consistent 1.3. Objectives with a fractured rock composition (Wilkison et al., 2002; Britt et al., 2002), The primary objective of this work is to take our previously devel- • A center-of-figure to center-of-mass distance of only 30–60 m; oped Cratered Terrain Evolution Model (CTEM), described in Richard- indicative of a relatively homogeneous structure, without large- son(2009), update it specifically for application to small solar system scale heterogeneities. That is, the measured porosity is likely bodies, and then apply this improved Small Body Cratered Terrain Evo- not due to huge void spaces, but is instead most likely due to lution Model (SBCTEM) to the three asteroids that have been observed a homogeneous fracture and/or pore-space distribution (Thomas to possess a dearth of small craters, presumably due to impact-induced et al., 2002), seismic shaking — looking not just at the evolution of the crater popula- • A highly irregular shape, indicative of some inherent structural tion but also at the evolution of the body’s regolith layer. Fortuitously, strength (Thomas et al., 2002; Robinson et al., 2002). the impact ejecta production and emplacement portions of CTEM were originally developed for application to small system bodies (Richardson While describing the geology of 243 Ida, Sullivan et al.(1996) sug- et al., 2007; Richardson, 2009), such that the only necessary update to gested a likely similarity between the internal structure of a fractured this portion of the model is the inclusion of impact crater breccia lens S-type (stony) asteroid and the uppermost crustal layers of the Earth’s creation as part of the body’s impact-produced regolith (Richardson and Moon (Dainty et al., 1974; Toksoz et al., 1974; Nakamura, 1976). Both Abramov, 2020): this will be presented in Section3. are composed of silicate rock, presumably began as monolithic or near- On the other hand, the expression used to compute the magnitude monolithic structures, and have since been exposed to impactor fluxes of topographic modification that occurs as a result of impact-induced of similar power-law distributions for millions to billions of years, seismic shaking following an individual impact, as implemented in albeit of different overall magnitudes (Ivanov et al., 2002). This shared our original work (Richardson et al., 2004, 2005), was specific to the bombardment history should produce similar stratigraphic fracture surface of 433 Eros and is therefore not applicable to other small structures within each, which we can take advantage of given the bodies. The original CTEM contained a more general solution, but extensive seismic studies performed of the upper lunar crust during the one that we considered to be only a partial and incremental improve- Apollo era (Latham et al., 1970; Dainty et al., 1974; Toksoz et al., 1974; ment (Richardson, 2009). As such, Section2 describes the steps taken Duennebier and Sutton, 1974; Duennebier et al., 1975; Nakamura, to produce a broadly applicable, general topographic modification 1976). expression, operating as a function of a specified impact, on a given The entire cross-section of the upper lunar crust that has been heav- target body, at a particular location on the surface. ily affected by impact cratering is classically termed the ‘lunar megare- In Section3 this update is incorporated into the Small Body Cratered golith’ (Nash et al., 1971; Hartmann, 1973, 2019). In Richardson and Terrain Evolution Model (SBCTEM), along with a few other minor Abramov(2020), we divided this broadly defined ‘lunar megaregolith’ improvements, such as tracking the amount of surface material lost layer into three distinct regions based upon their different formation to space (ejected) as a function of time and cumulative impacts. In processes and characteristics, as initially suggested by Heiken et al. Section4 we present the results of applying this updated SBCTEM to (1991): asteroids 433 Eros, 2867 Šteins, and 25143 Itokawa, simulating both 1.A Surficial Regolith layer, about 5–20 m in depth (Quaide the crater population and regolith layer evolution on each body as a and Oberbeck, 1968; Oberbeck and Quaide, 1968; Bart et al., function of exposure time to a Main Belt impactor flux. The implications 2011; Yue et al., 2019), consisting of loose (low cohesion), of these results are discussed in Section5. unconsolidated fines and breccia, and characterized by frequent overturn and comminution caused by small meteoritic impacts. 2. Topographic modification via seismic shaking With respect to seismic properties, this layer possessed a mean seismic body-wave velocity of 휈 = 0.1–0.5 km s−1, a mean free 2.1. Using the upper lunar crust as an asteroid seismic analog path for scattering of 푙 = 0.1–0.3 km, a seismic diffusivity of 휉 = 0.02–0.03 km2 s−1, and a seismic quality factor of 푄 = The Galileo images of 951 Gaspra and 243 Ida revealed two battered 1600–2300 (Kovach et al., 1973; Toksoz et al., 1974; Nakamura, objects, with visible systems of pit-crater chains, ridges, and grooves 1976).

3 J.E. Richardson et al. Icarus 347 (2020) 113811

2. An Upper Megaregolith layer, about 1–3 km in depth (Short radially outward and upward, ballistic flow of material that excavates and Forman, 1972; Hoerz et al., 1976; Aggarwal and Oberbeck, the crater cavity; and (3) a relatively minor fraction of the impactor 1979; Thompson et al., 1979; Richardson and Abramov, 2020), kinetic energy propagates away from the impact site (within the target consisting of depositional layers of brecciated and/or melted material) in the form of elastic-wave seismic energy (Melosh, 1989). material, and characterized by the transport and deposition of ma- This transition from shock behavior (which leaves behind energy as terial via either transient crater gravitational collapse or impact heated and accelerated material) to elastic energy is not well under- ejecta ballistic sedimentation. With respect to seismic properties, stood, but can be expressed in energy balance form by (Richardson this layer possessed a mean seismic body-wave velocity of 휈 = et al., 2005): 1–4 km s−1, a mean free path for scattering of 푙 = 0.125–2 km, a 1 2 3 2 −1 퐸 = 휂퐸 = 휂휋휌 푣 푑 , (1) seismic diffusivity of 휉 = 0.04–3 km s , and a seismic quality 푖 푘 12 푖 푖 푖 factor of 푄 = 2600–4000 (Pandit and Tozer, 1970; Dainty et al., where 퐸푖 is the seismic energy produced by an impactor, 휂 is called 1974; Toksoz et al., 1974). the impact seismic efficiency factor, 퐸푘 is the kinetic energy of the 3.A Lower Megaregolith layer, about 20–25 km in depth (Dainty impactor, 휌푖 is the impactor bulk density, 푣푖 is the impact speed, and et al., 1974; Toksoz et al., 1974; Wiggins et al., 2019), consisting 푑푖 is the spherical impactor diameter. The impact seismic efficiency of bedrock that has been fractured in place by impacts, but factor 휂, the fraction of the impactor’s kinetic energy that is ultimately not transported, and characterized by a fracture-density and converted to seismic energy within the target body, is an important, fragment-size distribution that decreases rapidly with increasing but poorly constrained parameter. Various values from the literature depth. With respect to seismic properties, this layer possessed a are listed in Table1. mean seismic body-wave velocity of 휈 = 4–6 km s−1, a mean As listed in Table1, the Saturn IVB booster and Lunar Module (LM) free path for scattering of 푙 = 2–5 km, a seismic diffusivity impacts performed as part of the Apollo Passive Seismic Experiment of 휉 = 3–10 km2 s−1, and a seismic quality factor of 푄 = (PSE) gave values of 휂 = 1 × 10−6 to 1 × 10−5 (Latham and McDonald, 4000–5000 (Dainty et al., 1974; Toksoz et al., 1974). 1968; Toksoz et al., 1974). In these experiments, however, only the Due the complete lack of direct, asteroid-derived seismic property long-period (LP) seismometers were used for these determinations, information, in this work, we use the seismic properties of the lunar which had an upper frequency limit of ∼ 1 Hz (Toksoz et al., 1974), Surficial Regolith layer as a rough analog for the seismic properties whereas the peak seismic frequencies produced by an impact generally of a ‘rubble pile’ asteroid, and the seismic properties of the lunar falls within the range of 1–100 Hz (Richardson et al., 2005). Conse- Upper Megaregolith layer as a rough analog for the seismic properties quently, these lunar impact determinations of 휂 = 1 × 10−6 to 1 × of a ‘fractured monolith’ asteroid. These assumed analog values will 10−5 (Latham and McDonald, 1968; Toksoz et al., 1974) sampled only be utilized during this model development phase (Section2) and as the low-frequency ‘wing’ of the impact seismic signal power spectrum. a starting point for our final model application to each individual Nonetheless, these lower 휂 values do have practical application far from asteroid (Section4), after which the best fit seismic properties will the impact site or in highly attenuating target materials, where the be found by fitting and correctly duplicating the observed cratering high-frequency component of an impact seismic signal has significantly record on each body — particularly the paucity of small craters. In decayed, as indicated by such measured values as 휂 = 1 × 10−5 to each case, we expect that the seismic properties of the asteroid will be 5 × 10−5, obtained from White Sands missile impacts Latham and Mc- more attenuating than those of its lunar analog, due to the asteroid’s Donald(1968). Laboratory impact experiments, on the other hand, that much lower gravitational environment and therefore lower confining monitor the impact seismic signal much closer to the source (generally pressures and seismic coupling conditions. < 1 m distance) and therefore obtain a more complete sample of the signal’s power spectrum, have generally produced much higher values 2.2. Synthetic seismogram creation for the impact seismic efficiency factor, most recently 휂 = 4 × 10−4 to 8 × 10−4 (Richardson and Kedar, 2013; Daubar et al., 2018). In order to model the downslope regolith motion that results from With respect to a lunar analog for impacts on an asteroid surface, an impact on an asteroid surface as a function of impactor properties, the lunar surface feature analysis work of Schultz and Gault(1975a,b) target body properties, and distance from the impact site, we must first estimated an impact seismic efficiency factor of 휂 ≈ 1 × 10−4 from be able to synthesize a reasonable approximation of the ground motion large lunar impacts. This is also the value suggested as typical for most experienced at each location on the surface as a result of this impact. impacts in Melosh(1989). As such, we use a value of 휂 = 1×10−4 during This is done using the techniques previously developed and described in this model development phase (Section2) and as a starting point for our Section2 of Richardson et al.(2005), which will be reviewed here. The final model application to each individual asteroid (Section4), after three basic steps for this process are described in the following sections: which the best fit 휂 will be found as a result of correctly duplicating (1) the total elastic seismic energy produced by a particular impact is the observed cratering record on each body — particularly the paucity computed (Section 2.2.1); (2) this energy is distributed as a function of of small craters. frequency using a normalized, impact seismic signal power spectrum (Section 2.2.2); and (3) this power spectrum is then convolved with a 2.2.2. Impact seismic signal frequency spectrum computation of the seismic energy density at a particular location on The seismic energy injected into the asteroid by an impact will be the asteroid surface as a function of time (Section 2.2.3). These steps divided into a wide spectrum of frequencies, each of which must be result in a synthetic seismogram of the ground motion experienced at treated individually with respect to signal attenuation in the target that location. body (Section 2.2.3). As such, we first need to find a ‘typical’ seismic frequency spectrum produced by a small meteoroid impact. However, 2.2.1. Impact generated seismic energy with the single exception of Duennebier and Sutton(1974) for lunar The kinetic energy released in a meteoritic impact is partitioned impacts recorded via the Apollo 14 short-period (SP) PSE, there are in three major ways: (1) A large fraction (∼ 90%) of this energy no published power spectra for impact seismic signals. We therefore is converted by irreversible shock transformation to internal energy obtained our power spectra through a series of numerical simulations, within both the (former) impactor and the target: vaporizing, melting, using the SALES-2 Lagrangian hydrodynamic code package (Amsden and heating material within a few impactor radii of the impact site; (2) et al., 1980; Collins et al., 2002, 2004) in its most basic mode: moni- another significant fraction (∼ 10%) of the impactor kinetic energy is toring the elastic response of rock to an impact source, without viscosity transferred as residual momentum in the target material to produce a or accumulated damage (Hooke’s law response only).

4 J.E. Richardson et al. Icarus 347 (2020) 113811

Table 1 Experimentally measured impact seismic efficiency factors. Source 휂 measurement Experiment type Gault and Heitowit(1963) <1 × 10−2 Small laboratory impacts Titley(1966) 3 × 10−3 to 3 × 10−1 Nuclear and other large explosion McGarr et al.(1969) 1 × 10−6 to 1 × 10−4 Laboratory impact experiments Latham and McDonald(1968) 1 × 10−5 to 5 × 10−5 White Sands missile impacts Latham and McDonald(1968) 1 × 10−6 to 1 × 10−5 Low angle Lunar Module impacts on the Moon Schultz and Gault(1975a) 1 × 10−5 to 1 × 10−3 Laboratory impact experiments Richardson and Kedar(2013) 4 × 10−4 to 8 × 10−4 Laboratory impact experiments

These simulations consisted of a 250 × 500 axially-symmetric mesh the large 60 m impactor, which has an inherently lower injected that is remapped to form a spherical target body. Two target types frequency spectrum, the fractured mesh continues to select out the were utilized: (a) a seismically homogeneous body 1 km in diameter preferred ∼ 10–80 Hz range. The resulting seismograms have a frequency (about the size of 243 Ida’s moon, Dactyl), having a target density of spectrum generally falling between 1 and 100 Hz, with a peak at about −3 휌푡 = 2200 kg m and mean seismic body-wave velocity of 휈 = 2.0 10–20 Hz. This is consistent with other impact seismic studies that km s−1; and (2) a ‘fractured’ 1 km diameter body, consisting of nine have reported peak frequencies of 10–40 Hz (White Sands missile im- −3 axially-symmetric fragments having a density of 휌푡 = 2700 kg m and pacts (Latham and McDonald, 1968)), 20–30 Hz (Lunar Active Seismic mean seismic body-wave velocity of 휈 = 4.0 km s−1, each separated by Experiment (Houston et al., 1973)), and 5–15 Hz (Lunar Passive Seismic three thin, sandwiched ‘fault gouge’ layers of 휌푡 = 2200, 1700, 2200 kg Experiment, short-period instrument (Duennebier and Sutton, 1974)). m−3, respectively, and 휈 = 2.0, 1.0, 2.0 km s−1, respectively, with each layer 2 m in thickness (along with a 4 m thick ‘regolith’ layer 2.2.3. Seismic energy dispersion and attenuation on the surface of the body). Small mesh and cell sizes are required to As described in Section 2.1, the seismic behavior of the upper lunar keep the frequency resolution of the mesh at 125 Hz in the slowest crust in response to impacts was well characterized during the Apollo material. These two mesh setups are shown in Fig.2( upper panels). era, and as such, assuming a lunar analog internal structure provides us The seismic motion of the free surface midway around the spherical with an advantage in modeling the seismicity of both ‘fractured mono- body from the impact site is recorded as a seismogram for 3 s at a lith’ and ‘rubble pile’ asteroids. These lunar seismic studies showed rate of 5000 samples per seconds, as shown in Fig.2( lower panels). that the dispersion of seismic energy in a fractured, highly scattering The frequency power spectrum from these seismograms are taken and medium is a diffusive process (Pandit and Tozer, 1970). Therefore, the normalized using a Fast-Fourier Transform (FFT) procedure, producing seismic energy density 휖 at a given location within a fractured asteroid amplitude 푎(푓) and phase 푝(푓) information as a function of frequency 푠 obeys the diffusion equation (Dainty et al., 1974; Toksoz et al., 1974): from 푓표 = 0.5 Hz to 푓푓 = 2500 Hz, at a frequency resolution of 푑푓 = 0.5 Hz. The seismic energy contained within a particular frequency bin 휕휖 2휋푓휖 푠 = 휉▽2휖 − 푠 , (4) in this power spectrum is thus given by: 휕푡 푠 푄 where 푡 is the time, 휉 is the seismic diffusivity (in either m2 s−1 or km2 퐸(푓) = 퐸푖푎(푓) . (2) s−1), and 푄 is the seismic quality factor, which determines how rapidly We performed hydrocode impact simulations for three different seismic energy is attenuated as a function of time 푡 and frequency 푓. impactor sizes striking both a homogeneous spherical target and a frac- In Richardson et al.(2005), we solved Eq.(4) in Cartesian coordi- tured spherical target (as shown in Fig.2. Fig.3 shows the normalized nates for a rectangular target body of length 퐿, width 푊 , and height 퐻. vertical power spectra for a 0.5 m, 4 m, and 60 m impactor striking We also approximated the initial seismic energy distribution injected by a homogeneous sphere (upper row) and fractured sphere (lower row) an impactor as a delta function: reasonable because the impactor sizes at 0.1 km s−1, where the slow impact speed is necessary to maintain considered here are generally much smaller than the target asteroid. good stability in a Lagrangian mesh for at least 3 s following the This assumption yields the particular solution and Green’s function: impact. This slow impact speed is acceptable because, as was found −2휋푓푡 in the experimental laboratory impacts described in Richardson and 푄 휖푠(푥, 푦, 푧, 푡, 푓) = 푒 Kedar(2013) where impact speeds of 1 to 6 km s −1 were explored, the [ ] ∞ −휉푛2휋2푡 ∑ 푛휋푥표 푛휋푥 frequency spectrum of an impact is independent of the impact speed. ∗ 1 + 2 cos cos 푒 퐿2 퐿 퐿 In the case of a homogeneous target (Fig.2(A)), the seismic signal 푛=1 [ ] produced by an impact closely resembles that of the classic ‘Lamb ∞ −휉푛2휋2푡 (5) ∑ 푛휋푦표 푛휋푦 pulse’ (Kanamori and Given, 1983; Richards, 1979), which consists of ∗ 1 + 2 cos cos 푒 푊 2 푊 푊 a sharp, singular impulse on the surface and which possessed a flat- 푛=1 [ ] ∞ −휉푛2휋2푡 topped, band-pass filter-like frequency spectrum (Fig.3( upper row)). ∑ 푛휋푧표 푛휋푧 ∗ 1 + 2 cos cos 푒 퐻2 , These impact simulations show that the frequency spectrum of an 퐻 퐻 푛=1 impact is dependent upon the diameter 푑푖 of the impactor, with a smaller impactor producing a higher frequency spectrum. This feature where 푛 is the spatial wave number for an elastic body-wave (P- is similar, and may be related to the impact ‘equivalent depth of burial’, wave) in the target medium, and the impact occurs at point 푥표, 푦표, 푧표 which is also roughly independent of the impact speed (Melosh, 1989): (logically, a point on the surface should be chosen). This solution yields a normalized frequency bin energy density 휖 as a function of time, as √ 휌 푠 푖 푥 푦 푧 푑푏푢푟푖푎푙 = 푑푖 , (3) seen at position , , (again, a point on the surface should be chosen). 휌푠 This solution is in good agreement with the two-dimensional solutions where 휌푠 is the mean density of the target’s surface material. described in Carslow and Jaeger(1959) and Toksoz et al.(1974). The With respect to a fractured target (Fig.2(B)), these hydrocode normalization scheme chosen here lets the seismic energy density 휖푠 go simulations indicate that the fracture blocks within the target will act to 1.0 as time goes to infinity when no attenuation is present. As such, as a crude band-pass filter to the injected signal; that is, the fractured the total (non-normalized) seismic energy density contained within a body will preferentially pass those seismic frequencies that are close specified power spectrum frequency bin is given by: to the harmonic frequencies associated with the mean fracture spacing 퐸(푓) within the body (Fig.3( lower row)). Note that even in the case of 휖 (푥, 푦, 푧, 푡, 푓) = 휖 (푥, 푦, 푧, 푡, 푓) . (6) 푖 푠 퐿푊 퐻

5 J.E. Richardson et al. Icarus 347 (2020) 113811

Fig. 2. (upper panels) Pressure contours in a 0.5 km radius, spherically symmetric, SALES-2 hydrocode mesh shown 0.5 s after the impact of a 4 m diameter cylinder at the top of the mesh: (A) shows the resulting hemispherical, rapidly advancing body (P) wave, along with a more slowly advancing surface or Rayleigh (R) wave, moving through a homogeneous target; (B) shows the same impact in a heterogeneous target that has been divided into multiple fracture blocks separated by low-density ‘fault gouge’. (lower panels) The recorded seismic motion of the mesh surface at the position of the bold arrows in the upper panels, shown for 3 s following the impact. The seismogram on the right displays the same gradual buildup and long ‘coda tail’ decay as the seismograms recorded by the Apollo Passive Seismic Experiment (PSE) for impacts on the lunar surface (Dainty et al., 1974; Toksoz et al., 1974).

In Eq.( 5) the key propagation parameters are the seismic diffusivity (as a function of 푄 and time 푡), it also means that the duration of 휉 and the seismic quality factor 푄. The seismic diffusivity is defined seismic shaking experienced at a particular location within that range as (Dainty et al., 1974; Toksoz et al., 1974): will be longer, as the peak seismic energy will take longer to pass 1 through that location. Lunar analog values for the seismic diffusivity 휉 휉 = 휈푙 , (7) 3 from Section 2.1 will be utilized during this model development phase where 휈 is the mean body-wave velocity in the seismic medium and (Section2 ) and as a starting point for our final model application to 푙 is the mean free path for the scattering of seismic waves in the each individual asteroid (Section4 ), after which the best fit 휉 will be medium, such that the seismic diffusivity 휉 is a measure of how rapidly found as a result of correctly duplicating the observed cratering record the seismic energy can travel (disperse) through the medium. Note on each body. that although a low 휉 means that a seismic signal will not be able to With respect to the seismic quality factor 푄, a series of experiments disperse far into the medium before becoming significantly attenuated described in Titley( 1966) and Tittmann( 1977) showed that under

6 J.E. Richardson et al. Icarus 347 (2020) 113811

Fig. 3. The normalized vertical power spectra for a 0.5 m, 4 m, and 60 m impactor striking the homogeneous sphere (upper row) and fractured sphere (lower row) shown in Fig.2 at a speed of 0.1 km s−1. These frequency power spectra show both a dependence on the diameter of the impactor and the mean fracture block spacing of the target. conditions of increasing vacuum and extremely low moisture content, the impact of a 10 m diameter stony impactor, corresponding to the many rocks – including the lunar samples – display an exponentially seismic energy density curve shown by the solid line in Fig.4 (A). The increasing seismic quality factor 푄, up to the values measured in the total displacement 푑푠(푥, 푦, 푧, 푡) from this synthetic seismogram is then lunar seismic experiments. Because S-type asteroids, such as 433 Eros, divided into a radial and vertical component by a user specified angle are likewise composed of silicate rock, reside in a vacuum, and have (generally 45◦ for an equal division), with 0◦ indicating full radial extremely low moisture contents, we adopt lunar-like seismic quality motion. factors in our modeling. These lunar analog values for the seismic quality factor 푄 from Section 2.1 will be utilized during this model 2.3. Numerical shake-table experiments development phase (Section2 ) and as a starting point for our final model application to each individual asteroid (Section4 ), after which To evaluate the effect of impact-induced seismic shaking on asteroid the best fit 푄 will be found as a result of correctly duplicating the surface morphology, a two-dimensional numerical downslope motion observed cratering record on each body. model (numerical ‘shake-table’) was developed in Richardson et al. Applying these assumed values, Fig.4 (A) plots the normalized (2005) which takes the radial and vertical displacements produced seismic energy density as a function of time (Eq.( 5)) for (dotted) a by a synthetic seismogram and applies them to a hypothetical re- receiver located 15.0 km from the impact of a 10 m diameter stony golith layer resting on a slope of variable angle, and placed in a impactor, (dashed) a receiver located 17.5 km from the impact site, user-specified asteroid gravity field. To accomplish this, we utilize a and (solid) a receiver located 20.0 km from the impact site, using the form of Newmark slide-block analysis (Newmark, 1965), which can seismic parameters: 휂 = 10−4, 휉 = 0.5 km2 s−1, and 푄 = 2000. The be applied when the regolith layer thickness under consideration is impact site was located on the top-center of a large 퐿 = 200 km, 푊 = much smaller than the seismic wavelengths involved (Houston et al., 200 km, 퐻 = 100 km ‘half-space’ test medium, such that small volume 1973; Jibson et al., 1998). This assumption works well for all but concentration effects are avoided, and thus making the resulting syn- the highest impact seismic frequencies, which are the most quickly thetic seismograms conservative with respect to small asteroids, where lost due to attenuation as a function of both distance and time. When their small dimensions and volumes may produce higher seismic energy this condition is met, we can approximate the motion of a mobilized density solutions (attenuation dependent). regolith layer by modeling the motion of a rigid block resting on an inclined plane, as described in Newmark( 1965), Houston et al. The final step in producing a synthetic seismogram is to convert (1973), Lambe and Whitman( 1979), Melosh( 1979) and Jibson et al. the seismic energy density 휖 (푥, 푦, 푧, 푡, 푓) from Eq.( 6) to a surface 푖 (1998). In brief, we compute the accelerations imparted to the block by displacement amplitude value and combine the result over all seismic both the asteroid’s surface gravity (static loading) and the seismically frequencies: shaken slope (dynamic loading), to compute the overall block (regolith √ 푓푓 layer) acceleration, velocity, and displacement at each time step. In this ∑ 휖푖(푥, 푦, 푧, 푡, 푓) 푑푠(푥, 푦, 푧, 푡) = cos ((2휋푓푡) + 푝(푓)) (8) simplified model, the regolith layer (rigid block) is assigned a density 휌푠 푓표 휌푠, a mobilized regolith layer depth ℎ, a coefficient of static friction 휇푠, where 휌푠 is the target surface material density. The resulting synthetic a coefficient of dynamic friction 휇푟, and a cohesion value 퐶 between seismogram is shown in Fig.4 (B) for a receiver located 20.0 km from layer and slope which must initially be exceeded before layer relative

7 J.E. Richardson et al. Icarus 347 (2020) 113811

Fig. 4. (A) a computation of the seismic energy envelope experienced at a particular location on an asteroid surface as a function of time, here shown at distances of 15.0 km (dotted), 17.5 km (dashed), and 20.0 km (solid), taking both seismic energy dispersion and attenuation into account. (B) When the solid curve is convolved with an impact seismic signal power spectrum (Fig.3 bottom center), it produces a synthetic seismogram of the ground displacement as a function of time, here shown at a distance of 20 km from the impact for three minutes following the impact. motion can occur, and which will reestablish itself if relative motion possesses a depth-dependent porosity and cohesion gradient, perhaps between layer and slope goes to zero (that is, a ‘sticky’ layer/block). due to compaction from frequent seismic shaking. This characteris- Fig.5 shows a comparison between the classic Goodman and Seed tic will produce lower porosity and higher cohesion with increasing (1966) sand-layer dynamic-loading experiment (left) and our numerical depth (Robinson et al., 2002). When a regolith with a low cohesion shake-table model (right), where both experiment and model display exhibits a marked increase in shear strength with depth, sliding on the same, gradual migration of a sand (regolith) layer moving downs- long slopes tends to occur on relatively shallow planes parallel to the lope under the −0.4 푔 to 0.4 푔, 7 Hz radial oscillations of an applied ground surface (Lambe and Whitman, 1979), regardless of the overall spring (where 푔 is one Earth gravity). This experiment demonstrates regolith depth. This same property was found for the lunar Surficial ‘stick–slip’ downslope regolith motion, which begins when seismic Regolith layer (Section 2.1), wherein Houston et al.( 1973) estimated a mobilized regolith layer depth of ∼ 8 m, within an overall regolith accelerations are generally > 0.2푔푎 (Lambe and Whitman, 1979), where layer depth of about 5–20 m (Quaide and Oberbeck, 1968; Oberbeck 푔푎 is the surface gravitational acceleration of the body under study. and Quaide, 1968; Bart et al., 2011; Yue et al., 2019). Based upon our When vertical accelerations exceed 1푔푎 (which does not occur in this ex- ample), the regolith layer will experience much more efficient ‘hop-slip’ previous findings for 433 Eros (Richardson et al., 2005) we selected a downslope motion. ‘typical’ asteroid surface mobilized regolith layer depth of ℎ = 1 m for The regolith layer used in this numerical shake-table model consists the numerical shake-table modeling described below. of a Mohr–Coulomb, uniform porosity, sand-like layer, to which a A simple method for examining slope stability under conditions of cohesion value 퐶 (in Pa) can be applied. This necessity to add cohesion seismic shaking is to utilize the slope Factor of Safety equation (Hous- to the regolith layer stems from the evidence on 433 Eros for weak ton et al., 1973; Lambe and Whitman, 1979), which looks at the ratio of the stabilizing forces to destabilizing forces exerted on a regolith layer regolith cohesion (non-zero shear strength), most notably in the form resting on a slope: of steep crater walls in small craters which were clearly formed in 퐶 + 휇 휌 푔 ℎ cos 휃 − 휌 푎 ℎ regolith and ponded deposits (Robinson et al., 2001). On the other 퐹 푂푆 = 푠 푠 푎 푠 푣 (9) hand, broader slope studies indicate that on 300-m scales, only 1%–3% 휌푠푔푎ℎ sin 휃 + 휌푠푎ℎℎ of surface slopes are above typical angles-of-repose (about 30◦), with where 퐹 푂푆 is the slope Factor of Safety, 휃 is the slope angle, 푎푣 is the ◦ an observed maximum of 36 (Thomas et al., 2002). Thus, although maximum vertical seismic acceleration, 푎ℎ is the maximum horizontal some features do show indications of cohesion and strength (in the form seismic acceleration, and where the total maximum seismic accelera- √ of steep crater walls, boulders, outcrops, ridges, and groove edges), 2 2 tion 푎푠 = 푎푣 + 푎ℎ: maximum seismic accelerations are assumed to be areas of obvious regolith coverage all tend to lie below typical angles- equal in both the positive and negative directions, and as such, absolute of-repose (Robinson et al., 2002; Thomas et al., 2002). There is also values are used for 푔푎, 푎푣, and 푎ℎ. In this formulation, a ‘stable’ slope ubiquitous evidence of slope destabilization and the downslope migra- will have 퐹 푂푆 ≥ 1. When the 퐹 푂푆 < 1, the regolith layer will begin to tion of regolith over the entire surface of the asteroid. This evidence slide or hop, and begin to move downslope under the force of gravity suggests that although present, existing cohesion forces seem to be 푔푎. Without cohesion 퐶 present, an ℎ = 1 m regolith layer with 휇푠 = relatively easy to overcome, of order 10–100 Pa (Richardson et al., 0.7, and resting on a 10◦ slope will be destabilized and begin to slide 2005; Rozitis et al., 2014). when the seismic acceleration 푎푣 = 푎ℎ = 0.26 푔푎 (consistent with Lambe With respect to the assumed depth of the mobilized regolith layer and Whitman( 1979)). Note, however, that in the 433 Eros surface −2 ℎ, in our previous modeling for asteroid 433 Eros, we found a best fit gravity environment (푔푎 ≈ 5 mm s ), adding just a small amount of ℎ = 0.1–2 m (Richardson et al., 2005). This value is significantly less cohesion, 퐶 = 100 Pa, to the regolith layer increases this threshold for −2 than the estimates of a loose regolith layer thickness of ‘tens of meters’ destabilization and sliding up to 푎푣 = 푎ℎ = 6.0 푔푎 (0.03 m s ). from the NEAR observations (Thomas et al., 2002; Robinson et al., In order to quantify the amount of seismically-triggered downslope 2002). We infer from this difference that much of the regolith layer regolith flow resulting from a specified impact, on a given target

8 J.E. Richardson et al. Icarus 347 (2020) 113811

Fig. 5. (left) The classic Goodman and Seed( 1966) sand-layer dynamic-loading experiment, compared to (right) a nearly-identical numerical shake-table model simulation. This experiment tracks the gradual downslope displacement (lower plots) of a sand layer resting on a 15◦ slope subjected to a 7 Hz, 0.8 푔 peak-to-peak horizontal seismic acceleration (upper plots), indicated by the arrows in diagram (a). In the numerical shake-table experiment shown on the right, ‘horizontal’ refers to motion parallel to the plane of the slope, dashed lines show the acceleration (top) and displacement (bottom) of the rigid shake-table, and solid lines show the acceleration (top) and displacement (bottom) of the mobile sand layer. body, at a particular location on the surface, we applied our synthetic seismograms (Section 2.2) to a numerical shake-table model of an ℎ = 1 m regolith layer resting on a slope, and done over a wide range of slope angles (2◦–30◦). This numerical ‘experiment’ will produce a set of data points giving downslope volumetric regolith flow per impact 3 −2 푞푖 (m m ) as a function of slope ▽푧 = 휕푧∕휕푥, as shown by the blue dots in Fig.6 . Note that our numerical shake-table has a virtually infinite extent, such that the only limitation on the recorded downslope flow 푞푖 is the duration of seismic shaking resulting from the simulated impact. All six of the previously described impact seismic frequency power spectra (Section 2.2.2) shown in Fig.3 were tested using the conditions shown in Fig.6 , with the power spectrum that produced the most conservative results (a 4 m diameter impactor striking a fractured 1 km diameter spherical target) selected as the ‘typical’ power spectrum for all subsequent numerical shake-table experiments. The motion observed in each numerical shake-table experiment is generally representative of non-linear, disturbance-driven downslope regolith transport, and can thus be described by the equation (Roering et al., 1999): 퐾 ▽푧 푞 = 푖 , (10) 푖 [ ]2 1 − ∣▽푧∣ 푆푐 Fig. 6. (blue dots) Numerical shake-table experiment data (blue dots) for a 10 m impactor striking the surface of 433 Eros, with the receiver located at a distance of 4 which is used to fit each set of numerical shake-table experiment data km from the impact site. The bold solid curve shows a fit to this numerical experiment to determine the downslope diffusion constant 퐾 (units of m3 m−2, 푖 series using Eq.( 10), wherein the resulting 퐾푖 = 136.0 m per impact value is used as volumetric flow per impact, or just m, downslope motion per impact) a single data point in the distance 퐷 curve shown in Fig.8 (pointed to by an arrow). The full set of six curves shown here indicate fits to the shake-table data from the and the critical slope 푆푐 . Note that under normal sliding conditions, same experiment using each of the six impact seismic power spectra shown in Fig.3 , the critical slope 푆푐 would be equal to the coefficient of static friction for (dashed curves) a 60 m impactor, (dotted curves) a 0.5 m impactor, (solid curves) assigned to the regolith layer in the model, 휇 = 0.7, but due to the 푠 a 4 m impactor, with the upper curve in each pair using a homogeneous target and ballistic hopping of the layer while in motion (when 푎푣 > 푔푎, this the lower curve using a fractured target. The most conservative downslope flow results parameter is better left as a free parameter in the fit, with values of occur using the impact seismic power spectrum from a 4 m impactor striking the 1 km diameter fractured target. 푆푐 usually falling between 0.55–0.65 (near the assigned coefficient of dynamic friction, 휇푟 = 0.6). Each numerical shake-table experiment, performed using a single synthetic seismogram applied over a series of ◦ ◦ slope values (2 –30 ), yields a single downslope diffusion constant 퐾푖 degradation and erasure of slopes and impact craters. This model is value which will be utilized in the following Section 2.4. based upon an analytical theory of erosion wherein we assume that the downslope flow of regolith is controlled by the downslope trans- 2.4. Topographic diffusion equation development portation rate (transport-limited flow), and not by the regolith supply A key feature of our Small Body Cratered Terrain Evolution Model or production rate (weathering-limited flow), as described in Culling (Section3 ) is the use of a topographic modification model for the (1960) and Nash( 1980). This assumption should hold true for most of

9 J.E. Richardson et al. Icarus 347 (2020) 113811 the surfaces studied here, due to the ability of ‘impact gardening’ to In all, over 300 numerical shake-table experiment series (like that produce a continuous supply of loose regolith around each new crater. shown in Fig.6) were performed, with the results shown in Fig.7, In this process, classically modeled by Ross(1968), Soderblom(1970), which groups together the seismic energy production terms (휂, 푣푖, and Shoemaker et al.(1970) and Gault et al.(1974) for the lunar surface: 푑푖) from Eq.(1); Fig.8, which groups together the seismic energy (1) small impacts will tend to only overturn and ‘recycle’ the existing attenuation terms (휉, 푄, and 퐷) from Eq.(5); and Fig.9, which regolith layer, with some loss of material to space after each impact groups together the slope stability terms (푔푎 and 퐶) from Eq.(9). due to ejection at greater than the escape velocity 푣푒푠푐 of the body; For experiment consistency, two sets of diverse ‘standard’ conditions while (2) ‘fresh’ regolith is only generated by large impacts, which can were utilized in these numerical shake-table experiments: (1) a 10 m penetrate below the existing regolith layer to excavate a portion of diameter impactor striking the surface of 433 Eros as experienced at 5 the underlying, fractured bedrock. It is this constant regolith recycling, km distance from the impact, shown in the left column of Figs.7–9; with occasional replenishment, that allows us to assume a transport- and (2) a 100 m diameter impactor striking the lunar (Moon) surface limited regolith supply. To develop this erosion theory, we begin with as experienced at 10 km distance from the impact, shown in the right an expression for the conservation of mass on an infinitely small portion column of Figs.7–9. That is, while each one of these eight terms were of a hillslope, in Cartesian coordinates (Culling, 1960): being tested over a wide range of values, the other seven terms were [ ] held constant at their ‘standard’ value for each of the two selected 휕푧 휕푓푥 휕푓푦 = − + , (11) surface environments (Eros and Moon). 휕푡 휕푥 휕푦 We then performed a linear least-squares fit to the 퐾푖 values deter- where 휕푧∕휕푡 is the change in elevation per unit time, and 푓푥 and 푓푦 are mined from each of these 16 numerical shake-table experiment series, the flow rates of regolith in the 푥 and 푦 directions, respectively. If the done over the linear (in log–log space) portion of each curve shown in regolith layer is isotropic with regard to material flow, and the flow Figs.7–9, in order to find the power-law exponent ( log–log slope) of rate is linearly proportional to the slope’s gradient then: each curve. These exponent fits are listed in Table2, along with the 휕푧 휕푧 mean value of each Eros-Moon environment pair. 푓 = −퐾 , and 푓 = −퐾 , (12) 푥 푑 휕푥 푦 푑 휕푦 These power-law fits were combined to produce an empirically- where 휕푧∕휕푥 and 휕푧∕휕푦 are the slope in the 푥 and 푦 directions, respec- derived analytical equation used to compute the amount of downslope diffusion per impact 퐾푖 at each pixel location in an SBCTEM simulation, tively, and 퐾푑 is a downslope diffusion constant, which has units of m3 m−2 s−1 (volume flux per unit time) or simply m s−1 (downslope under a variety of impactor, target, and seismic conditions: motion per unit time). Substituting Eq.(12) into Eq.(11) yields the 휂0.25푑0.75푣0.5푄1.8푔0.5 퐾 = 퐾 푖 푖 푎 , (14) classic topographic diffusion equation (Culling, 1960): 푖 푙푖푛푒푎푟 휉0.3퐷0.5 [ 2 2 ] 휕푧 휕 푧 휕 푧 2 where 퐾 = 0.15 is a linear proportionality constant averaged over = 퐾푑 + = 퐾푑 ▽ 푧 . (13) 푙푖푛푒푎푟 휕푡 휕푥2 휕푦2 all 16 sets of numerical shake-table experiments. The above exponential This expression can be applied directly in our Small Body Cratered constant values (labeled as ‘Adopted’ in Table2) were selected to keep Terrain Evolution Model (SBCTEM) using finite-difference techniques, the resulting 퐾푖 value as consistent as possible across all applicable provided that we can determine the proper diffusion constant 퐾푑 to use circumstances, and to also keep them consistent with their counterparts at each mesh element (pixel location) on the modeled surface. Note that in the 퐾푖푐 ‘‘roll-off’’ correction function (Eq.(15)). Note that although 퐾 the topographic diffusion constant 퐾푑 used in Eq.(13) is not a measure tested, the downslope diffusion constant 푖 did not show a dependence of horizontal regolith migration, but is instead used to compute the upon regolith cohesion 퐶, as shown in Fig.9, wherein an exponent amount of vertical change (휕푧∕휕푡) in the topography as a function of value of zero was found (no dependence observed). What this shows is the elevation gradient (▽2푧) at that location. that because our synthetic seismograms (Fig.4) reach their maximum Also note that the above equations are traditionally a function of displacement, velocity, and acceleration values relatively early on in time, applicable when the diffusion constant 퐾푑 is some constant rate the process, the cohesion of the regolith is usually overcome relatively with respect to time (units of either m3 m−2 s−1 or simply m s−1). In our quickly and the regolith layer experiences about as much downslope case, however, we must define a specific amount of downslope diffusion flow as it would have had it been non-cohesive 퐶 = 0 Pa. However, at 3 −2 some point, the selected cohesion 퐶 value is so high that the seismic per impact 퐾푖, with units of m m per impact or simply m per impact, due to the seismic vibrations produced by a specified impact, on a accelerations experienced at some distance 퐷 are not able to overcome given target body, at a particular location on the surface, as estimated the cohesion at all, and no downslope motion occurs, resulting in the using the numerical shake-table experiments described in Section 2.3. steep roll-off shown by the black equation fit lines shown in Fig.9. Because each impact on the surface of an SBCTEM simulation will As shown in Figs.7–9, the 퐾푖 data points derived from our nu- merical shake-table experiments frequently display a roll-off in values produce a unique downslope diffusion per impact 퐾푖 value at each mesh element (pixel location), the goal of this section is to produce an (toward zero) at some point in the series, where the values no longer fit a linear slope (in log space). This occurs when the vertical seismic analytical expression for computing 퐾푖 at each pixel location following every simulated impact. accelerations 푎푣 on the tested slope drop below 1 푔푎 and the downslope In Richardson et al.(2004, 2005) we developed a simple expression motion of the regolith layer changes from ‘hop-slip’ motion to the less efficient ‘stick–slip’ motion, which itself dies off as the total seismic for computing 퐾푖 as a function of a single parameter, the impactor acceleration 푎 drops below about 0.2 푔 . Therefore, in addition to diameter 푑푖, which was only applicable to ‘global’ or ‘far-side’ seismic ℎ 푎 effects on the surface of 433 Eros. In Richardson(2009), we expanded Eq.(14), we also needed to develop an estimate of 퐾푖 under the conditions of 푎 < 푔 . To do this, we utilize a simplified form of the this expression to compute 퐾푖 as a function of impactor velocity 푣푖, 푠 푎 slope Factor of Safety equation (Eq.(9)) to produce a semi-empirical impactor diameter 푑푖, target surface gravity 푔푎, and distance from the impact site 퐷. In Richardson(2013), Marchi et al.(2015), and in roll-off correction function: this work, we have successively added terms for the impact seismic √ ⎡ ⎡ √ 푎 − 퐶 ⎤ ⎤ √ 푠 휌 efficiency factor 휂, the target seismic diffusivity 휉, the target seis- 퐾 퐾 ⎢ ⎢ . √ 푠 , ⎥, ⎥ 푖푐 = 푖 ∗ min ⎢max ⎢0 1 0⎥ 1⎥ (15) mic quality factor 푄, and have explored the effect of adding regolith 푔푎 ⎢ ⎢ ⎥ ⎥ cohesion 퐶. The goal has been is to create an empirically derived, multi- ⎣ ⎣ ⎦ ⎦ term, power-law expression for the purpose of analytically estimating where 퐾푖푐 is the corrected downslope diffusion constant. In other the value of 퐾푖 at a given location under a wide variety of conditions words, when the acceleration ratio 푎푠∕푔푎 is between 0 and 1, the and target bodies. amount of downslope diffusion becomes roughly proportional to the

10 J.E. Richardson et al. Icarus 347 (2020) 113811

Fig. 7. Numerical shake-table experiments testing the downslope diffusion equation’s seismic energy production terms: 휂, 푑푖, and 푣푖 (top to bottom). Each blue point marks the 퐾푖 value estimated by a single numerical shake-table experiment series (like that shown in Fig.6 ), while the black line indicates the analytical fit computed from Eq.( 15). The left column shows experiments performed in the Eros standard environment, while the right column shows experiments performed in the lunar standard environment.

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Fig. 8. Numerical shake-table experiments testing the downslope diffusion equation seismic energy attenuation terms: 휉, 푄, and 퐷 (top to bottom). Each blue point marks the

퐾푖 value estimated by a single numerical shake-table experiment series (like that shown in Fig.6 ), while the black line indicates the analytical fit computed from Eq.( 15). The left column shows experiments performed in the Eros standard environment, while the right column shows experiments performed in the lunar standard environment. The arrow points to the 퐾푖 data point produced from the numerical shake-table experiment shown in Fig.6 .

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Fig. 9. Numerical shake-table experiments testing the downslope diffusion equation slope stability terms: 퐶 and 푔푎 (top to bottom). Each blue point marks the 퐾푖 value estimated by a single numerical shake-table experiment series (like that shown in Fig.6 ), while the black line indicates the analytical fit computed from Eq.( 15). The left column shows experiments performed in the Eros standard environment, while the right column shows experiments performed in the lunar standard environment.

Table 2

퐾푖 expression (Eq.( 14)) power-law exponent determinations. Variable Range tested Eros standard value Moon standard value Eros fit Moon fit Mean Adopted Impact seismic efficiency 휂 1 × 10−7–5 × 10−4 1 × 10−4 1 × 10−4 0.253 0.334 0.294 0.25

Impactor diameter 푑푖 0.1–1000 m 10 m 100 m 0.805 0.822 0.814 0.75 −1 −1 −1 Impact velocity 푣푖 1–50 km s 5 km s 5 km s 0.470 0.499 0.485 0.5 Target seismic diffusivity 휉 0.005–50 km2 s−1 0.5 km2 s−1 0.5 km2 s−1 −0.280 −0.321 −0.301 −0.3 Target seismic quality factor 푄 10–10 000 2000 2000 1.754 1.856 1.805 1.8 −2 −2 −2 Surface gravity 푔푎 0.001–10 m s 0.005 m s 1.62 m s 0.482 0.492 0.487 0.5 Distance from impact 퐷 1–50 km 5 km 10 km −0.591 −0.521 −0.556 −0.5 Regolith cohesion 퐶 10 Pa–1 MPa 0 Pa 0 Pa 0.000 −0.014 −0.007 0.0 square root of the acceleration ratio. The proportionality constant of 0.1 1995): was found through trial-and-error in working with the numerical shake- −2푓퐷2 휉휋푄 table results shown in Figs.7 –9, with the goal of matching the Eros 퐸ℎ = 퐸푖푒 , (16) surface environment results (shown on the left) as closely as possible. with the seismic energy density 휖ℎ given by: The fits are not as good with respect to the lunar surface environment 퐸 휖 = ℎ . (17) results (shown on the right), indicating room for future improvement. ℎ 2휋퐷2

We can roughly estimate the seismic acceleration amplitude ex- Finally, to convert this seismic energy density 휖ℎ to an approximate sur- perienced at some distance 퐷 from an impact by assuming that the face acceleration amplitude, we use the expression (Lay and Wallace, seismic energy propagates outward from the impact site as a thin, 1995): hemispherical shell. As such, the total seismic energy contained in that 16휋2푓 2 푎 = 휖 . (18) shell 퐸 at some distance 퐷 can be approximated by (Lay and Wallace, 푠 ℎ √ ℎ 2휌푠

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Fig. 10. (left panel) A plot of the cumulative amount of impact-induced topographic diffusion 퐾푐 necessary to degrade a fresh impact crater of diameter 퐶푓 (and 푑∕퐶푓 ≈ 0.2) to the point of erasure, defined as 푑∕퐶푓 ≤ 0.01. This is shown for a mobilized regolith layer depth ranging from ℎ = 0.2 to 5 m, with ℎ = 1 m indicated by the dashed line. (right panel) A finite-differencing application of Eq.( 13) within SBCTEM showing the gradual degradation of a simple crater of diameter 퐶푓 = 250 m, following the accumulation of a topographic diffusion 퐾푐 value of 0 m, 100 m, 1000 m, and 5000 m, which soften the topography and result in a crater 푑∕퐶푓 ratio of 0.18, 0.15, 0.09, and 0.02, respectively. where the dominant seismic frequency 푓 is assumed to be 10–20 Hz imagine making a crater-shaped dimple in a bowl of dry flour, and then (Section 2.2.1). While only a rough estimate, this corrected downslope repeatedly tapping the side of the bowl with your finger. While each diffusion constant 퐾푖푐 does a reasonable job of approximating the 퐾푖 individual tap does relatively little to alter the shape of the dimple, the roll-off observed in our numerical shake-table experiments, as indicated cumulative effect of many taps will be to gradually soften and flatten by the black fit lines in Figs.7 –9, particularly as a function of the the dimple’s form to the point of non-recognizability. most important variable in our SBCTEM simulations, distance from the impact 퐷 (bottom row of Fig.8 ). It is also reassuring to note that 3. Small body cratered terrain evolution model our correction function contains several variable exponents that are identical to the ones derived experimentally (Table2 and Eq.( 14)), Monte-Carlo (or stochastic) cratering models have long been a 1∕4 1∕2 3∕4 −1∕2 namely 휂 , 푣푖 , 푑푖 , and 퐷 ; that is, the seismic energy production mainstay in the study of cratered terrains. The majority of these mod- variables and the distance 퐷. The final result is an analytical expression els have been geometric in nature, with crater rims represented by for computing the downslope diffusion constant 퐾푖푐 resulting from a circles in a two-dimensional matrix, and which employ mathematical specified impact, on a given target body, at a particular location on the functions to determine the effects of ejecta blanket coverage and the surface of our Small Body Cratered Terrain Evolution Model (Section3 ). erosion of large craters by smaller ones (Woronow, 1978; Chapman In Sec. 3.3 of Richardson et al.( 2005), we developed a simple ap- and McKinnon, 1986; Hartmann and Gaskell, 1997; Richardson et al., proximation for the amount of cumulative, impact-induced topographic 2005). In Richardson( 2009), we presented a new Cratered Terrain 퐾 퐶 diffusion 푐 necessary to degrade a fresh impact crater of diameter 푓 , Evolution Model (CTEM) which took advantage of recent advances in 푑∕퐶 ≈ 0.2 and initial depth/diameter ratio of 푓 , to the point of ‘erasure’, (1) the impact cratering scaling relationships as applied to small solar which we define here by a final crater 푑∕퐶 0.01 (Robinson et al., 푓 ≤ system bodies (Richardson et al., 2007; Richardson, 2011), and (2) our 2002). Revising Eq. 31 from Richardson et al.( 2005) to correspond to understanding of seismically-induced crater degradation (Richardson a value of 푑∕퐶 = 0.01 (from its original 푑∕퐶 = 0.004) yields: 푓 푓 et al., 2004, 2005) to produce a fully three-dimensional numerical 2 model of crater production and erasure on a given target surface, 3.1퐶푓 퐾푐 , (19) including automatic crater counting. Substantial improvements in code ≥ 16ℎ modularization, parallelization, and running efficiency were introduced where ℎ is the depth of the mobilized regolith layer (Section 2.3). in Minton et al.( 2015), with improved crater forms (especially for The left panel of Fig. 10 illustrates this approximation applied using complex craters) and impact crater breccia lens creation introduced the range of mobilized regolith layer depths explored in Richardson in Richardson and Abramov( 2020). et al.( 2005), ℎ = 0.2 to 5 m. The right panel of Fig. 10 demonstrates Each SBCTEM simulation uses Monte-Carlo sampling of the finite-differencing application of Eq.( 13) utilized within SBCTEM (Section3 ), showing the gradual degradation of a fresh, simple crater 1. A user-supplied impact velocity distribution, taken from Minton to the point of near-erasure (푑∕퐶푓 = 0.02). Using Eq.( 19), a 퐶푓 = 250 m and Malhotra( 2010) for these simulations, and shown in Fig. 11 crater will reach 푑∕퐶푓 = 0.01 following about 퐾푐 = 12 000 m of gradual, (left), uniformly applied, cumulative topographic diffusion across its surface 2. The spherical target body impact angle distribution, described area. in Gilbert( 1893), Shoemaker( 1962), Pierazzo and Melosh Note that the amount of topographic diffusion that occurs at a (2000), and shown in Fig. 11( right), and 퐾 particular location is cumulative, either as a function of time (using 푑 ) 3. A user-supplied impactor population, described in Section 4.1, or repeated nearby impacts (using 퐾푖푐 ), until the next disruptive event occurs there, such as the formation of a new crater at that location. A to bombard an initially clean bedrock target surface as a function of simple analogy for the process depicted in the right panel of Fig. 10 is to time, and then generate surface terrain maps, regolith depth maps, and

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Fig. 11. (left) The asteroid-to-asteroid impact velocity distribution used in the SBCTEM simulations described in this work, derived from an N-body simulation of the dynamical diffusion of Main Belt asteroids into the Near Earth region (Minton and Malhotra, 2010). The Root Mean Squared (RMS) velocity of this distribution is 5.2 km s−1.(right) The impact angle distribution used in our SBCTEM simulations, derived in Pierazzo and Melosh( 2000) for an isotropic population of impactors striking the surface of a spherical target body. crater counts at specified time intervals throughout the model run. At In its current configuration, the SBCTEM has three methods for each time step, the model crater size-frequency distribution (SFD) is covering a pixel area with new regolith, and two methods for removing compared to the observed crater SFD for the target body, using a 휒2 that regolith. With respect to covering a pixel area with new regolith, goodness of fit test. This surface is usually 1000 × 1000 or 2000 × 2000 regolith can be (1) generated by crater breccia lens creation, which (user-specified) pixel elements in size, with a user-specified pixel-scale occurs when the transient crater created by an impact gravitationally (meters per pixel). The simulated surface is continuous in nature, collapses into its final form; (2) deposited by impact ejecta emplace- possessing periodic boundary conditions from side-to-side and top-to- ment (ballistic sedimentation), or (3) created by the downslope motion bottom. The user also specifies a mean surface gravity 푔푎 (as corrected of regolith from upslope of the pixel in question. With respect to by the mean rotational force on the surface) and mean target body ra- removing regolith from a pixel area, regolith can be removed by (1) the dius 푟푎 (which becomes the model depth), along with standard material process of crater excavation, or (2) the downslope motion of regolith to properties for the terrain, both the ‘bedrock’ substrate and overlying ‘re- regions downslope of the pixel in question. Note that upturned crater golith’ layer (when produced), as well as their impact crater scaling and rims are not considered to be regolith in the model, but instead act as seismic propagation properties. These values are selected or estimated a source region for loose regolith, since this rock has been uplifted, from the scientific literature relative to the object or type of object broken, and is severely weakened. To be considered as ‘regolith’ in under study, in order to achieve the most accurate simulation possible. this model, material must be transported, either ballistically or via mass Note that in its current configuration, the model does not include wasting. the differential effects of either body shape or rotation as a function In the present work, we have made several modifications to CTEM of model surface location, and thus represents the body under study in order to make it more applicable and accurate to the simulation in a uniform, mean surface force fashion. For a complete listing and of crater populations on small asteroid bodies, thus creating what we description of the equations underlying SBCTEM, with the exception call the Small Body Cratered Terrain Evolution Model (SBCTEM), in of impact-induced seismic shaking (which is described in this work), order to distinguish it from the more lunar-surface specific CTEM ver- see Richardson and Abramov( 2020). sion currently maintained at Purdue University (Minton et al., 2015). As mentioned in Section 2.4, a key component of SBCTEM is the This small-body specific SBCTEM includes the following improved or inclusion of downslope regolith migration, triggered either by slope additional features: instability or by the seismic motion generated by nearby impacts, • improved regolith motion computation due to impact induced thus incorporating crater degradation as a function of age and local seismic shaking (Section2 ), impact history within the model. Following each impact, the simulation • escaped ejecta tracking, computing the target volume lost per enters an Eulerian phase wherein seismic shaking induced regolith flow individual impact, between matrix pixels occurs, computed in finite-differencing fashion • improved tracking of bedrock layer and regolith layer changes (Eq.( 13)) using a corrected topographic diffusion value 퐾 at each 푖푐 over time, pixel location. That is, to compute 퐾푖푐 (using Eqs.( 14)–(18)): (1) the • improved mesh stability checks and corrections during the Eule- target-specific parameters, surface gravity 푔 , impact seismic efficiency 푎 rian phase, and 휂, seismic diffusivity 휉, and seismic quality factor 푄 are held constant a simplified crater tracking and counting algorithm (described during the simulation; (2) the impact-specific parameters, impact speed • below). 푣푖 and impactor diameter 푑푖 are Monte-Carlo sampled as described above; and (3) the resulting 퐾푖푐 value is then computed as a function To achieve automatic crater tracking and counting, the SBCTEM of distance 퐷 from the impact site for each matrix pixel location. Ad- uses three auxiliary matrix layers to track the creation and degradation ditionally, the affected area around each new impact crater is checked of each pixel in a particular crater, where layer (a) contains a unique for slopes above the angle-of-repose, which are then collapsed to below crater identifier, layer (b) contains the original crater profile (in meters the angle-of-stability, again using Eq.( 13) and a fixed, small diffusion of height) relative to the mean topographic surface present at the time constant. Each Eulerian phase is broken up into a series of small, incre- of the crater’s formation, and layer (c) contains the deviation of the mental steps to ensure good mesh stability throughout the topographic current crater surface (in meters of height) from its original profile diffusion process that follows each simulated impact. Fig. 12 shows the position. As specified by a user-supplied parameter, when a particular evolving profile (as a function of time and subsequent impacts) of a crater pixel deviates by more than 90%–100% from its original profile typical simple crater resulting from this gradual degradation process. elevation, that crater pixel is permanently ‘erased’ from the three

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Fig. 12. (upper row) Cross-section of a simple crater automatically generated within the Small Body Cratered Terrain Evolution Model (SBCTEM), with the ‘bedrock’ layer shown in gray and the ‘regolith’ layer (both breccia lens and ejecta blanket) shown in black. This ∼ 400 m diameter crater resulted from the impact of a 푑푖 = 20 m stony impactor striking −2 −1 ̄ an Eros-like surface (푔푎 = 5 mm s ) at 푣푖 = 5 km s , having an assumed bedrock target strength of 푌 = 0.5 MPa (near that of weak sandstone). (middle and bottom rows) These show the degradation of this crater as a function of time and subsequent impacts, shown at 10 Myr and 30 Myr later, demonstrating how impact-induced seismic shaking causes the gradual filling and smoothing of the crater rim and bowl, until the feature can no longer be recognized as an impact crater. Note that early in a simulation, smaller craters tend to cluster on the ejecta blanket of a larger crater, due to its much weaker target strength as compared to the surrounding, intact bedrock. tracking layers. Due to the ability of smaller craters to be nested within 4. Specific asteroid surface simulations larger craters, a total of seven sets of three auxiliary tracking layers (21 total) are employed by the model, in order to ensure that all crater 4.1. Main Belt impactor population pixels are tracked properly. In order for a crater to be considered Using our updated Small Body Cratered Terrain Evolution Model ‘observable’ or ‘countable’, at least 40%–50% of a crater’s original pixel (SBCTEM), we performed a systematic series of asteroid surface sim- area must remain, as determined by a second user-supplied parameter. ulations, looking in particular at the evolution of the asteroid’s crater Although SBCTEM does create and track craters down to a single pixel population, and the evolution of an impact-generated regolith layer on in diameter, such that ‘sandblasting’ – the erasure of large craters by the surface. With respect to the selection of an impactor population for this modeling work, Strom et al.( 2005) showed that when mapped small craters – is included in the model (see Section 1.1 of Richardson through the impact cratering scaling relationships (Holsapple, 1993), (2009)), only craters of at least three pixels in diameter are used in the the modern-day asteroid population yields a very good match to the crater counts presented in this study. cratering record on the lunar surface — thus pointing to the asteroid belt as the primary source of impactors in the inner solar system. As currently configured, the Small Body Cratered Terrain Evolution In the same year, O’Brien and Greenberg( 2005) and Bottke et al. Model (SBCTEM) utilizes the Interactive Data Language (IDL) package (2005b) both published the results of collisional evolution models for (Harris Geospatial Solutions) for overall simulation control and dis- the Main Belt asteroids, showing that while greatly depleted over play purposes, while the program engine is written and compiled in the time since its formation, the Main Belt continues to reflect its Late-Heavy Bombardment (LHB) size-frequency distribution; that is, it Fortran 90/95. This program engine utilizes the OpenMP (Open Multi- displays a ‘fossilized’ size-frequency distribution. This is because the Processing) application programming interface (API) to parallelize the shape of the size-frequency distribution curve for a collisionally evolved engine modules wherever possible Minton et al.( 2015). Running on a population (such as the asteroids) is more determined by material 16-processor LINUX modeling computer, a particular simulation may properties and impact parameters than by the starting, or original take from a few hours up to a few weeks to complete, depending size-frequency distribution of the population (Bottke et al., 2005b). Given the above determinations, we utilize the Main Belt asteroid upon the mesh resolution specified by the user, and the degree to population as the source of impactors in our SBCTEM simulations, as which impact induced seismic shaking affects the small body under modeled by O’Brien and Greenberg( 2005) and Bottke et al.( 2005b) study, wherein each simulation generally involves tens of millions of down to sizes smaller than can be actually observed, since we re- individual impacts. quire an impactor range of 0.01 m to 1000 m in diameter for these

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mean flux present in the modern-day main asteroid belt. As such, a direct application of the Main Belt impactors to our asteroid surfaces does not take into account the dynamical filtering that will occur in moving objects from the Main Belt to the Near-Earth Object (NEO) environment (Bottke et al., 2005b). Nonetheless, the cumulative power- law slope of −2.6 over the four orders-of-magnitude from 0.01 m to 100 m does remain approximately the same for impactors in the near-Earth environment (Bottke et al., 2005a), thus permitting us to continue to use a Main Belt SFD for our impactor population for NEO asteroids, with the understanding that the modeled impactor flux level will be higher than actual and thus the resulting surface exposure ages will be lower than actual. The direct application of the Main Belt impactors to our asteroid surfaces does permit an ‘apples to apples’ comparison between the three asteroids studied here, both with respect to the size- frequency distribution of the bombarding population and the exposure rate (impacts per year per km2) from that population.

4.2. 433 Eros modeling Fig. 13. Three crater ‘production populations’ for asteroid 433 Eros using a main Belt impactor population, shown for target effective strength 푌̄ values of 500 kPa (small dashed), 1 MPa (dashed), and 5 MPa (solid) after 100 Myr of exposure. Note that Asteroid 433 Eros was orbited by the NEAR-Shoemaker spacecraft increasing the effective target strength 푌̄ tends to shift the crater production population from 2000–2001, which achieved 100% coverage of the surface with down and to the left, as well as greatly reducing the amount of regolith produced surface pixel scales at better than 10 m resolution (Veverka et al., and retained per impact. These plots display the crater counts in a relative ‘R-plot’ 2001; Thomas et al., 2002). This former Main Belt (now near-Earth) format (Crater Analysis Techniques Working Group et al., 1979). asteroid has dimensions 34.4×11.2×11.2 km (16.9 km mean diameter), possesses a slightly curved, elongated shape, and has a rotation period of 5.27 h (Yeomans et al., 2000; Thomas et al., 2002). Eros was also simulations. Over this size range, the Bottke et al.( 2005b) impactor observed to possess a significant regolith layer, with extensive surface distribution follows a straight-line, cumulative power-law slope of −2.6 coverage, ‘‘tens of meters’’ in depth on average, and even deeper in over the four orders-of-magnitude from 0.01 m to 100 m, then becomes some regions (Robinson et al., 2002). Thanks to radio-science data from shallower over the one order-of-magnitude from 100 m to 1000 m, orbits of the NEAR-Shoemaker spacecraft around the asteroid, Eros’s reaching a cumulative power-law slope of −1.6 at 1000 m. −3 bulk density has been constrained to a value of 휌푎 = 2670 ± 30 kg m , Fig. 13 shows the resulting craters produced as a function of time −2 and its mean surface gravity to a value of 푔푎 = 4.6 mm s (Yeomans for asteroid 433 Eros (its crater ‘production population’ or ‘production et al., 2000; Thomas et al., 2002). The NEAR-Shoemaker observations function’), using this Main Belt impactor population and shown after of Eros likewise showed an asteroid that most likely falls into this 100 Myr of exposure. The plot shows all craters produced without transitional category of ‘fractured monolith’, based upon several lines regard to crater overlapping or other crater erasure mechanisms, and of supporting evidence: (a) indications of structural control of impact is displayed here using a Relative density, or ‘R-plot’ format, which is craters and other features (Prockter et al., 2002), (b) the presence of given by Crater Analysis Techniques Working Group et al.( 1979) as: a global network of visible joints, ridges, and grooves (Zuber et al., 퐶̄ 3푁 2000; Prockter et al., 2002), (c) a mean porosity measurement of about ̄ 푖 푖 푅(퐶푖) = ( ) , (20) 20%–30%, consistent with a fractured rock composition (Wilkison 퐴푐푛푡 퐶푖 − 퐶푖 −1 et al., 2002; Britt et al., 2002), (d) a center-of-figure to center-of- where 푁푖 is the number of craters in bin 푖, 퐴푐푛푡 is the surface area of the mass distance of only 30–60 m; indicative of a relatively homogeneous portion of the surface where crater counts were conducted, 퐶푖 and 퐶푖−1 structure (Thomas et al., 2002), and (e) a highly irregular shape, ̄ are the crater diameters at the bin boundaries, and 퐶푖 is the geometric indicative of some inherent structural strength (Robinson et al., 2002; mean crater diameter of the bin, given by: Thomas et al., 2002). In particular, Wilkison et al.( 2002) concluded ̄ √ that although Eros is a heavily fractured body, they found no evidence 퐶푖 = 퐶푖퐶푖 . (21) −1 that it was ever catastrophically disrupted and reaccumulated into a where in the crater counts presented in this study, we use the standard rubble pile. In a separate analysis, Cheng( 2004) found that crater √ bin size 퐶푖 = 2퐶푖−1 specified in Crater Analysis Techniques Work- morphologies suggested a target strength near the surface of a few ‘‘tens ing Group et al.( 1979). The primary advantage in using a Relative of kilo-Pascals’’. density plot is that most crater populations have cumulative SFD slope Multiple partial and complete SBCTEM simulations of the surface of indices within the range of −2 ± 1, such that they will plot as non- asteroid 433 Eros were performed using a 2000 × 2000 element matrix, sloping (horizontal) or moderately sloping lines on an R-plot. The having a 10.0 m pixel resolution on a 400 km2 surface area. These shallow average slopes of the lines on an R-plot thus make any changes simulations covered only about one-third of the actual Eros surface in the SFD more obvious and facilitate identifying differences in the area of 1125 km2 (Robinson et al., 2002), in order to properly resolve distribution function and densities among crater populations (Crater most of the smallest crater size bins counted by Chapman et al.( 2002). Analysis Techniques Working Group et al., 1979). The other advantage As such, the largest craters on the surface of Eros (≳ 3 km) were not of the Relative density plot is that when a crater population reaches included in these simulations. These simulations were run searching a state of crater density equilibrium (empirical saturation), the R- for (1) the best 휒2 fit to the Chapman et al.( 2002) crater counts, values will generally level off at 0.1 and 0.3, and have an overall ∼ -2 particularly the paucity of small craters ≲ 100 m in diameter, and (2) cumulative power-law slope, which plots as a roughly horizontal line a regolith ‘‘tens of meters’’ in depth on average, and even deeper in on an R-plot (Gault, 1970; Richardson, 2009). some regions, as reported in Robinson et al.( 2002). Using a directed It is important to note that the surface ages estimated in the fol- parameter search process, the primary model parameters explored were lowing SBCTEM model runs are not actual surface ages, but should the effective target strength, varied from 푌̄ = 100 kPa to 푌̄ = 10 MPa; be thought of as ‘Main Belt exposure ages,’ that is, the surface age the impact seismic efficiency factor, varied from 휂 = 1 × 10−7 to 휂 = 1 × obtained when the flux of impactors is constant and equal to the 10−4; the seismic quality factor, varied from 푄 = 1000 to 푄 = 3000; and

17 J.E. Richardson et al. Icarus 347 (2020) 113811 the seismic diffusivity, varied from 휉 = 0.125 km2 s−1 to 휉 = 0.750 km2 by Chapman et al.(2002). As such, the largest craters on the surface of s−1. Complete simulations proceeded until the point of crater density Eros were not included in these simulations, and the obtained minimum equilibrium (empirical saturation) was reached for crater sizes ≲ 1 km Main Belt exposure age of 225 ± 75 Myr applies only to properly in diameter, and a reasonable match had been achieved for craters reproducing the crater population ≲ 3 km in diameter. Using SBCTEM ∼ 1–3 km in diameter, given the reduced simulation area. Stochastic to perform a full surface area simulation of 433 Eros (pixel scale = variation in the production of craters ∼ 1–3 km diameter, between 16.77 m) requires ∼ 400–600 Myr of mean Main Belt exposure time, individual simulations, was the primary contributor to uncertainty in for craters up to ∼ 5 km in diameter (that of Psyche crater), which is the Main Belt exposure age that best matches the observed crater consistent with the 400±200 Myr age found using the simpler, geometric counts. Due to the long run time for each complete simulation, up to surface model presented in Richardson et al.(2004, 2005). several days for Eros, only a small number of completed runs could be In order to also include the ∼ 8 km diameter Shoemaker Regio and conducted (10 total). ∼ 10 km diameter Himeros depressions in the simulation requires a Our best fit simulations occur at an assumed bedrock target strength much longer, highly variable, ∼ 1–2 Gyr of mean Main Belt exposure ̄ of 푌 = 0.5–5.0 MPa (that of ‘very weak rock’ (NZGS et al., 2005)), time. Doing so, however, causes a problem in that these 1–2 Gyr long which is consistent with a highly fractured underlying bedrock, where simulations also bring craters in the ∼ 1–6 km diameter size range into an example of one of these runs is displayed in Fig. 14, shown with crater density equilibrium (Richardson et al., 2005), which does not impact-induced seismic shaking turned both off and on to demonstrate match the observations (Chapman et al., 2002). The crater production its efficacy in small crater erasure. The seismic properties that best function for 433 Eros (Section 4.1) indicates that on average, an impact reproduce the Chapman et al.(2002) crater counts at small crater sizes large enough to produce Psyche crater will occur once every 0.4– −6 include an impact seismic efficiency factor of 휂 = 1 × 10 , a seismic 0.7 Gyr, an impact large enough to produce the Shoemaker Regio 2 −1 quality factor of 푄 = 1500, and a seismic diffusivity of 휉 = 0.5 km s , depression will occur once every 0.8–1.4 Gyr, and an impact large indicating that seismic energy transmission through the interior of 433 enough to produce the Himeros depression will occur once every 1.4– Eros is roughly 1/10th–1/100th as efficient as transmission through the 2.0 Gyr, assuming an effective target strength range of 푌̄ = 1–5 MPa. lunar Upper Megaregolith layer (Section 2.1), our selected ‘fractured These differences in cratering record reproduction times remain as an monolith’ asteroid seismic analog. outstanding problem with respect to understanding the full collisional Fig. 15 shows a plot of the mean regolith depth (solid line) and and surface history of asteroid 433 Eros. ejected mass depth (dashed line) as a function of time for the simulation performed at an effective target strength of 푌̄ = 1.0 MPa, where regolith 4.3. 2867 Šteins modeling in the simulation is generated by impact crater breccia lens creation, impact ejecta ballistic emplacement, and the gradual erosion of high topography (such as crater rims). In addition to generating regolith, On September 5, 2008, the European Space Agency’s (ESA) Rosetta each impact also causes a small amount of the asteroid’s mass to be spacecraft flew by small Main Belt asteroid 2867 Šteins, coming within −1 803 km at the point of closest approach and imaging the body with lost to space; that is, ejected from the surface at > 푣푒푠푐 = 10.3 m s . The three blue dots on the regolith growth curve shown in Fig. 15 its OSIRIS narrow-angle and wide-angle cameras (Keller et al., 2010). indicate the range of Main Belt impact exposure ages at which (a) These images, covering about 65% of the asteroid’s surface, revealed crater density equilibrium is initially achieved (Richardson, 2009) and Šteins to be an ‘oblate top’ shaped asteroid with dimensions 6.83 (b) the observed crater counts (Chapman et al., 2002) are well fit by km × 5.70 km × 4.42 km (5.26 km mean diameter), a rotation pe- the simulation, particularly for craters > 1 km in diameter. riod of 6.049 h, and dominated by a large 2.1 km diameter impact crater located at its south pole, with prominent, linear faults radi- ̄ • In the case of a 푌 = 0.5 MPa bedrock, the best fit minimum Main ating from it (Keller et al., 2010; Jorda et al., 2012). The highest Belt exposure age is 200 ± 50 Myr, during which time a regolith resolution images covered about 25% of the surface, and permitted layer 77 ± 15 m in mean depth is created, with a mean of 65 ± 35 cataloguing 42 crater-like features (Besse et al., 2012), with 21 of these m of additional surface material lost to space. accepted as true impact craters and ranging in size from 0.24 km to ̄ • In the case of a 푌 = 1.0 MPa bedrock, the best fit minimum Main 2.1 km (Marchi et al., 2010). This crater population displays a severe Belt exposure age is 250 ± 50 Myr, during which time a regolith paucity of small craters for diameters ≲ 500 m, which was hypothesized layer 66 ± 12 m in mean depth is created, with a mean of 54 ± 25 to be due to either impact-induced seismic shaking, active resurfacing m of additional surface material lost to space. due to the Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effect (Ru- ̄ • In the case of a 푌 = 5.0 MPa bedrock, the best fit minimum Main bincam, 2000), or the recent formation of the large, 2.1 km diameter Belt exposure age is 250 ± 50 Myr, during which time a regolith Diamond crater on the body’s south pole (Keller et al., 2010; Marchi layer 87 ± 16 m in mean depth is created, with a mean of 120 ± 14 et al., 2010; Jorda et al., 2012) – or some combination thereof. m of additional surface material lost to space. Although 2867 Stein’s ‘oblate-top’ shape and equatorial bulge are Taking into account the full range of these simulations, we estimate indicative of a high body rotation rate at some point in Stein’s past, its that on 433 Eros, a mean regolith layer depth of 80 ± 20 m, having a current 6.049 h rotation period puts this asteroid into the ‘optimum’ large standard deviation across the entire surface of 68 ± 9 m, will be rotation state category, given its body aspect ratio of 푐∕푎 = 0.65 and produced over a Main Belt exposure age of 225 ± 75 Myr. scaled spin value of 휔푠푐 = 0.36–0.45 (assumed bulk density 휌푎 = 2000– −3 These best fit simulations are all consistent with the spacecraft 3000 kg m ), utilizing the rotation-state analysis technique described observations that 433 Eros possesses a regolith layer ‘‘tens of meters’’ in Richardson et al.(2019). This optimum rotation state means that in depth on average, and even deeper in some regions (Robinson et al., gravitational and rotational forces are roughly balanced, such that 2002), wherein the regolith depth at best fit is just beginning to reach topographic differences, surface slopes, and erosion (mass-wasting) a plateau level, at which point the amount of regolith lost to space due rates are all minimized across the surface of this body. So while this to small impacts (that only affect the regolith layer) becomes roughly small body may have experienced periods of high YORP spin-up in its equal to the amount of new regolith created by the occasional large past, putting it into a fast rotation state with rapid regolith migration impact that can reach below the regolith layer into the underlying, towards its equatorial bulge (Harris et al., 2009; Richardson et al., fractured bedrock. 2019), for the purposes of our modeling here, we assume that Šteins has As mentioned above, these 433 Eros simulations covered only 400 been in a more quiescent, near-optimum rotation state long enough for km2 of Eros’s actual surface area of 1125 km2 (Robinson et al., 2002), its current cratering record to have reached a crater density equilibrium in order to properly resolve most of the smallest crater size bins counted (empirical saturation) state.

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Fig. 14. The results of two SBCTEM simulations showing crater population creation and erasure on the surface of asteroid 433 Eros for an effective target strength of 푌̄ = 5 MPa and Main Belt exposure age of 298 Myr. Crosses indicate the observed crater counts (Chapman et al., 2002), while a solid line indicates the simulation’s crater counts, shown using a relative ‘R-plot’ format (Crater Analysis Techniques Working Group et al., 1979). (upper panel) This simulation was performed without any seismic effects on the topography, with crater counts leveling off at between 0.1–0.3 in typical Gault( 1970) fashion (horizontal dotted lines), and having a slope index of −2.0 with 88 727 countable craters ≥ 3 pixels in diameter. (lower panel) This simulation was performed with seismic shaking effects on the topography and at crater density equilibrium, displays the observed, suppressed counts for craters ≲ 100 m in diameter, having a shallow slope index of −1.7 with 17 887 countable craters ≥ 3 pixels in diameter.

Using the model of Scheeres( 2007), the YORP-driven rate of change we use orbital elements of 푎 = 2.364 AU, 푒 = 0.146, 푖 = 9.93◦, ◦ ◦ of the spin, 휔, of an asteroid of radius 푟푎 and bulk density 휌푎, averaged 훺 = 55.37 , 휛 = 306.3 , and a pole direction given by right ascension ◦ ◦ ̄ over an orbit with semimajor axis 푎, eccentricity 푒, and inclination 푖 of 91.6 and declination of −68.2 . This gives a value of 0 ≈ 0.00095, can be expressed as: though this value may be highly sensitive to small-scale details of the shape (Statler, 2009), which are poorly resolved in the available shape 퐺1 3 ̄휔̇ = ̄ , (22) −3 √ 2 0 model. Assuming a bulk density of 휌푎 = 2000–3000 kg m we use 2 2 4휋휌푎푟 푎 1 − 푒 푎 Eq.( 22) to estimate its current spin rate change of ̄휔̇ = 0.8–1.3 rad 17 2 ̄ −2 where 퐺1 ≃ 10 kg m/s and 0 is a dimensionless coefficient which s . If we define a spin timescale as 휏푠푝푖푛 = 휔∕ ̇휔 then the YORP spin-up is derived from the asteroid’s shape and spin pole orientation relative timescale for present-day Šteins is 휏푠푝푖푛 ≃ 700–1000 Myr. to its solar orbit. ̄ To estimate the YORP coefficient, 0, for 2867 Šteins based on its 4.3.1. Pit-crater chain analysis current shape and spin axis orientation, we used the Small Body Geo- Many of the craters on the surface of Šteins possess a degraded metric Analysis Tool (SBGAT) to compute the Fourier coefficients of the appearance, indicative of a sufficiently thick, mobile regolith layer ca- YORP model as a function of solar latitude (McMahon and Bercovici, pable of softening and filling in these features due to an active process 2019). We used the Rosetta-derived shape model given by Farnham of downslope regolith migration. In addition, a linear chain of eight and Jorda( 2013). To compute the orbit average of the solar latitude, pit-craters extend northward away from the 2.1 km diameter Diamond

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Table 3 2867 Šteins identified pit-craters from Besse et al.( 2012). No. Name Long. Lat. Diameter (m) Estimated ℎ (m) 17 Agate 1.8◦ +57.6◦ 500 175 18 Amethyst 343.2◦ +28.9◦ 450 160 19 Citrine 340.6◦ +13.7◦ 610 215 20 Opal 337.8◦ +1.1◦ 400 140 21 Garnet 338.3◦ −8.6◦ 390 135 22 Jade 335.3◦ −18.0◦ 260 90 23 Peridot 336.0◦ −25.1◦ 270 95 24 Lapis 329.7◦ −31.5◦ 440 155 A Emerald 139.3◦ −3.9◦ 520 180 B Aquamarine 120.3◦ +7.2◦ 370 130 C – 104.9◦ +11.7◦ 350 125 D – 97.8◦ +16.1◦ 360 125 E – 88.4◦ +19.7◦ 380 135

Fig. 15. A 433 Eros SBCTEM simulation plot of the mean regolith depth (solid) and mean ejected mass depth (dashed) as a function of time for the simulations performed depth estimated from Chain 17–24. As such, we combine the estimated ̄ at effective target strengths of 푌 = 1.0 MPa. The three blue dots on each regolith regolith depths from all 13 pit-craters of Chain 17–24 and Chain A- growth curve indicate the range (low, middle, high) of Main Belt impact exposure ages E to produce a final mean regolith depth estimate for 2867 Šteins of at which (a) crater density equilibrium is initially achieved (Richardson, 2009) and (b) the observed crater counts (Chapman et al., 2002) are well fit, with this error range ℎ = 145 ± 35 m. based upon stochastic differences between individual simulations. 4.3.2. SBCTEM simulations Multiple partial and complete SBCTEM simulations of the surface crater (Fig. 16(a)), nearly reaching the body’s north pole (Keller et al., of asteroid 2867 Šteins were performed using a 1000 × 1000 element 2 2010; Marchi et al., 2010). These pit-craters are presumed to be non- matrix, having a 9.32 m pixel resolution on a 86.92 km surface area, impact in nature, formed by the opening of a fracture beneath the corresponding to the surface area of a sphere having Stein’s mean 2 regolith layer (Fig. 16(b)) due the formation of Diamond crater (Barucci radius of 2.63 km, which is near the actual surface area of 97.6 km et al., 2015). The presence of this prominent pit-crater chain on the computed from the shape-model (Marchi et al., 2010; Jorda et al., 2 surface of Šteins gives us a method for estimating the regolith depth on 2012). These simulations were run searching for (1) the best 휒 fit to the body, based upon the experimental work of Horstman and Melosh the (Marchi et al., 2010) crater counts, particularly the paucity of small (1989) and applied to the Stickney crater on Mar’s moon Phobos. craters ≲ 500 m in diameter, and (2) a regolith depth of ℎ = 145 ± 35 The eight pit-craters in this chain are listed as numbers 17–24 in m as determined in Section 4.3.1. Using a directed parameter search Table3 , which takes its information directly from Table2 of Besse et al. process, the primary model parameters explored were the effective 푌̄ = 100 kPa 푌̄ = 10 MPa (2012). If these circular features are indeed the result of subsidence into target strength, varied from to ; the impact seismic efficiency factor, varied from 휂 = 1 × 10−7 to 휂 = 1 × 10−4; a radial extensional fracture (or fractures) resulting from the formation the seismic quality factor, varied from 푄 = 1000 to 푄 = 3000; and the of 2.1 km diameter Diamond crater, this fissure activation only permit- seismic diffusivity, varied from 휉 = 0.125 km2 s−1 to 휉 = 0.750 km2 ted a partial draining of the overlying regolith, such that the process s−1. Stochastic variation in the production of large craters, between stopped with the formation of a string of unconnected ‘dimples’ on the individual simulations, is the primary contributor to uncertainty in the surface (Melosh, 2011), wherein material drained into the widening Main Belt exposure age that best matches the observed crater counts. fracture along shear planes having a dip angle roughly equal to the Due to the long run time for each complete simulation, up to a few regolith material’s angle of internal friction 휙: thus establishing the weeks for Šteins, only a small number of completed runs could be diameter of the pit-crater early on in the drainage process (Smart et al., conducted (8 total). 2011). As such, we can estimate the regolith depth ℎ at each pit-crater Our best fit simulations occur at an assumed bedrock target strength location by measuring each pit-crater’s radius 푟, using an assumption of 푌̄ = 2.5–5.0 MPa (that of ‘weak rock’ (NZGS et al., 2005)), which for the regolith’s angle of internal friction of 휙 = 35◦ ± 10◦ (Carson and is consistent with a highly fractured underlying bedrock, where an Kirkby, 1972; Richardson et al., 2019), and applying the trigonometric example of one of these runs is displayed in Fig. 17, shown with impact- relationship shown in Fig. 16(b): induced seismic shaking turned both off and on to demonstrate its ℎ = 푟 tan 휙 , (23) efficacy in crater erasure. The seismic properties that best reproduce the (Marchi et al., 2010) crater counts include an impact seismic a relationship that is supported by both the experimental work of efficiency factor of 휂 = 1 × 10−6, a seismic quality factor of 푄 = Horstman and Melosh( 1989) and numerical modeling work of Smart 2000, and a seismic diffusivity of 휉 = 0.5 km2 s−1, indicating that et al.( 2011). Assuming that these pit-craters, which we name Chain similar to 433 Eros (Section 4.2), seismic energy transmission through 17–24, provide a representative sample of regolith depth across the the interior of 2867 Šteins is roughly 1/10th–1/100th as efficient as body, this chain yields an estimated mean regolith depth for Šteins of transmission through the lunar Upper Megaregolith layer (Section 2.1), ℎ = 145 ± 70 m. our selected ‘fractured monolith’ asteroid seismic analog. With similar In addition to Chain 17–24, Besse et al.( 2012) identify a second seismic propagation properties but about 1/3 the diameter, the signif- pit-crater chain on the surface of Šteins, which they designated craters icantly enhanced effectiveness of impact-induced seismic shaking on A-E in their Table2 , as listed here in Table3 . This pit-crater chain 2867 Šteins as compared to 433 Eros is evident in both the R-plot and is less obviously associated with the formation of Diamond crater surface map shown in the lower panel of Fig. 17, where the roll-off of (neither noticeably radial nor concentric to it), and may mark the small craters begins at roughly 500 m and less, whereas on Eros, this location of an underlying fracture that pre-existed Diamond, but was only occurs at craters sizes of 100 m and less. nonetheless extensionally opened (reactivated) during the Diamond- In these simulations, we assume a very simple E-type asteroid bulk −3 forming impact. This pit-crater chain, which we name Chain A-E, is density of 휌푎 = 3000 kg m , consistent with that of the enstatite more regional in extent than Chain 17–24, and thus yields an estimated achondrite (Aubrite) meteorites (Macke et al., 2011), for which the E- regional regolith depth of ℎ = 140 ± 25 m, quite similar to the regolith type asteroids are believed to be the parent bodies. This assumed value

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Fig. 16. (a) The linear chain of eight pit-craters that extend northward (downward in this image) away from the 2.1 km diameter Diamond crater at the body’s south pole, nearly reaching its north pole (Keller et al., 2010; Marchi et al., 2010). (b) These pit-craters are presumed to be non-impact in nature, formed as a radial, extensional feature caused to the opening of a fissure beneath the regolith layer due the formation of Diamond crater (Barucci et al., 2015). This fissure activation only permitted a partial draining of the overlying regolith, such that the process stopped with the formation of a string of unconnected ‘dimples’ on the surface (Melosh, 2011). We can estimate the regolith depth ℎ by measuring each pit-crater’s radius 푟 and using an assumption for the regolith’s angle of internal friction 휙. Drawing based upon Fig. 8.8(a) of Melosh( 2011).

−3 is slightly higher than the measured value of 휌푎 = 2670 ± 30 kg m the point at which crater density equilibrium is initially achieved with obtained for S-type asteroid 433 Eros (Yeomans et al., 2000), and is a good match to the observed crater counts. This match to the equi- consistent with our best fit bedrock target strength of 푌̄ = 2.5–5.0 MPa, librium crater counts will continue indefinitely (with some jumpiness −2 and results in a mean surface gravity of 푔푎 = 2.2 mm s . This bulk as large impacts occur), while the regolith layer continues to grow in density selection will be discussed further in Section5 . depth until it too reaches an equilibrium level where production (due to Fig. 18 show a plot of the mean regolith depth (solid line) and occasional large impacts) and loss (due to escaped ejecta) are roughly ejected mass depth (dashed line) as a function of time for the lower in balance. Continuing the simulation using an assumed effective target panel simulation shown in Fig. 17, where regolith in these simulations strength of 푌̄ = 5 MPa, this regolith depth equilibrium level is first is generated by impact crater breccia lens creation, impact ejecta bal- reached at a minimum Main Belt exposure age of 475±25 Myr, indicated listic emplacement, and the gradual erosion of high topography (such by the green dots in Fig. 18. The simulated regolith depth equilibrium as crater rims). In this case, where the best fit values for the effective occurs at a mean value of 155±5 m, in good agreement with the actual bedrock target strength of 푌̄ = 2.5–5.0 MPa are high relative to the 145 ± 35 m mean regolith depth estimated from the pit-crater chains, −2 body’s low surface gravity of 푔푎 = 2.2 mm s , craters will be highly and possesses a large standard deviation across the entire surface of strength-dominated and impact ejecta production per impact will be 115 ± 10 m. quite low (Richardson et al., 2007; Richardson, 2011), making breccia Note that this Main Belt exposure age represents a lower limit lens creation the dominant means of regolith production on this body, for the surface, given that both crater density and regolith depth will in agreement with Hirabayashi et al.( 2017). In addition to generating continue indefinitely in equilibrium until some large, disruptive event regolith, each impact also causes a small amount of the asteroid’s mass occurs. This indefinite equilibrium state was verified by extending the to be lost to space; that is, ejected from the surface at > 푣푒푠푐 = 3.4 m Šteins SBCTEM simulation out to a Main Belt exposure age of 600 s−1. Myr, with both the crater counts and regolith depth remaining roughly The three blue dots on the regolith growth curve shown in Fig. 18 constant. indicate the range of Main Belt impact exposure ages at which (a) crater density equilibrium is initially achieved (Richardson, 2009) and 4.4. 25143 Itokawa modeling (b) the observed crater counts (Marchi et al., 2010) are well fit by the simulation. Near-Earth asteroid 25143 Itokawa was joined in a solar co-orbit by the Hayabusa spacecraft on September 12, 2005, with landing and • In the case of a 푌̄ = 2.5 MPa bedrock, the best fit minimum Main surface sample attempts made on November 20 and 25 of the same Belt exposure age is 175 ± 25 Myr, during which time a regolith year. While approaching the asteroid over two months, the surface was layer 88 ± 11 m in mean depth is created, with a mean of 27 ± 7 extensively mapped and observed, with 100% of the surface imaged at m of additional surface material lost to space. ∼ 70 cm/pixel (Fujiwara et al., 2006). This tiny asteroid has dimensions • In the case of a 푌̄ = 5 MPa bedrock, the best fit minimum Main 0.535 × 0.294 × 0.209 km (0.350 km mean diameter), is highly irregular Belt exposure age again is 175 ± 25 Myr, during which time a in shape, and has a rotation period of 12.132 h (Fujiwara et al., 2006). regolith layer 82 ± 12 m in mean depth is created, with a mean of With respect to an observed regolith layer, Itokawa appears to be 28 ± 6 m of additional surface material lost to space. constructed almost entirely of regolith mixed with larger boulders: Note that both of the estimated Main Belt exposure ages of 175 ± 25 a true rubble pile, although its surface does appear to be composed Myr are at the high end of, but still overlap with the 154 ± 35 Myr of somewhat coarser grained material than that observed on asteroid Nolan Scaling Law (NSL) surface exposure age determined in Keller 433 Eros (Miyamoto et al., 2007). Thanks to radio-science data from et al.( 2010), Marchi et al.( 2010). the approach and landing of the Hayabusa spacecraft, Itokawa’s bulk −3 These crater-count best fit simulations produce a regolith depth that density has been constrained to a value of 휌푎 = 1950 ± 140 kg m , −2 is ∼ 60% of the 145 ± 35 m regolith depth estimated in Section 4.3.1. and its mean surface gravity to a value of 푔푎 = 0.1 mm s (Abe et al., This deeper actual regolith depth implies an older actual surface age 2006). Unlike Eros, Itokawa possesses a very rough slope distribution, for Šteins, since the estimated exposure age of 175 ± 25 Myr represents with a notable amount of its surface above typical angles-of-repose

21 J.E. Richardson et al. Icarus 347 (2020) 113811

Fig. 17. The results of two SBCTEM simulations showing crater population creation and erasure on the surface of asteroid 2867 Šteins for an effective target strength of 푌̄ = 5 MPa and Main Belt exposure age of 168 Myr. Crosses indicate the observed crater counts (Marchi et al., 2010), while a solid line indicates the simulation’s crater counts, shown using a relative ‘R-plot’ format (Crater Analysis Techniques Working Group et al., 1979). (upper panel) This simulation was performed without any seismic effects on the topography, with crater counts leveling off at between 0.1–0.3 in typical Gault( 1970) fashion (horizontal dotted lines), and having a slope index of −1.9 with 13 463 countable craters ≥ 3 pixels in diameter. (lower panel) This simulation was performed with seismic shaking effects on the topography and at crater density equilibrium, displays the observed, heavily suppressed counts for craters ≲ 500 m in diameter, having a very shallow slope index of −1.1 with only 590 countable craters ≥ 3 pixels in diameter. within the resolution limits of the shape-model. Much of the surface corresponding to the surface area of a sphere having Itokawa’s mean appears to be covered with unconsolidated millimeter-sized and larger radius of 0.175 km, which is near the actual surface area of 0.393 gravels, wherein morphologic characteristics of this gravel indicate km2 computed from the shape-model (Demura et al., 2006). These that Itokawa has experienced a considerable degree of disturbance- simulations were run searching for the best 휒2 fit to the Michel et al. triggered downslope flow (Miyamoto et al., 2007). A survey of potential (2009) crater counts, particularly the severe paucity of small craters impact features conducted by Hirata et al.( 2009) that included circular ≲ 100 m in diameter. Using a directed parameter search process, the depressions, circular features with flat floors or convex floors, and primary model parameters explored were the effective target strength, circular features with smooth surfaces, identifying 38 crater candidates varied from 푌̄ = 10 kPa to 푌̄ = 100 kPa; the impact seismic efficiency −8 −6 with an average 푑∕퐶푓 ratio of 0.08 ± 0.03: extremely shallow relative factor, varied from 휂 = 1×10 to 휂 = 1×10 ; the seismic quality factor, to craters observed on other asteroids. These candidate craters range varied from 푄 = 1000 to 푄 = 2500; and the seismic diffusivity, varied in diameter from 13 m to 200 m, with an extreme paucity of craters from 휉 = 0.001 km2 s−1 to 휉 = 0.250 km2 s−1. Stochastic variation in evident in all but the largest 100–200 m diameter range, dropping the production of large craters, between individual simulations, is the rapidly as one moves to smaller sizes. primary contributor to uncertainty in the Main Belt exposure age that Multiple partial and complete SBCTEM simulations of the surface of best matches the observed crater counts. Due to the long run time for asteroid 25143 Itokawa were performed using a 1000 × 1000 element each complete simulation, up to a several days for Itokawa, only a small matrix, having a 0.62 m pixel resolution on a 0.385 km2 surface area, number of completed runs could be conducted (9 total).

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Fig. 18. A 2867 Šteins SBCTEM simulation plot of the mean regolith depth (solid) and mean ejected mass depth (dashed) as a function of time for the simulations performed at effective target strengths of 푌̄ = 5.0 MPa. The three blue dots on the regolith growth curve indicate the range (low, middle, high) of Main Belt exposure ages at which crater density equilibrium is initially achieved (175 ± 25 Myr). The three green dots on the regolith growth curve indicate the range of Main Belt exposure ages at which regolith depth equilibrium is initially achieved (475 ± 25 Myr), with this error range based upon stochastic differences between individual simulations. A large, ∼ 3 km diameter, Diamond-like impact occurs at 376 Myr, producing a sudden 16 m increase in the mean regolith depth. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Our best fit simulations occur at an assumed bedrock layer target With respect to Itokawa, this regolith depth is more properly termed the strength of 푌̄ = 10–20 kPa (that of ‘soft soil’ (NZGS et al., 2005)), which regolith overturn depth, since Itokawa is composed entirely of regolith, is consistent with Itokawa’s obvious ‘rubble pile’ structure, where an breccia, and fracture blocks. As such, the plot in Fig. 20 more correctly example of one of these runs is displayed in Fig. 19, shown with impact- shows the current depth of overturned regolith in the simulation, rather induced seismic shaking turned both off and on to demonstrate its than the creation of fresh regolith. efficacy in crater erasure. The seismic properties that best reproduce The three blue dots on the regolith growth curve shown in Fig. 20 the Michel et al.( 2009) crater counts include an impact seismic effi- indicate the range of Main Belt impact exposure ages at which (a) ciency factor of 휂 = 1 × 10−7, a seismic quality factor of 푄 = 1500, crater density equilibrium is initially achieved (Richardson, 2009) and and a tiny seismic diffusivity of 휉 = 0.002 km2 s−1, indicating that (b) the observed crater counts (Michel et al., 2009) are well fit by the seismic energy transmission through the interior of 25143 Itokawa simulation. is roughly 1/100th–1/1000th as efficient as transmission through the In the case of a 푌̄ = 10 kPa bedrock, the best fit minimum Main lunar Surficial Regolith layer (Section 2.1), our selected ‘rubble pile’ • Belt exposure age is 20±5 Myr, during which time a regolith layer asteroid seismic analog. Despite the presence of a highly scattering and 4.6 ± 1.2 m in mean depth is overturned, with a mean of 0.7 ± 0.2 attenuating seismic medium, impact-induced seismic shaking is still m of additional surface material lost to space. extremely effective in softening, filling, and erasing all impact craters In the case of a 푌̄ = 20 kPa bedrock, the best fit minimum Main that occur on the surface of Itokawa, on rapid time scales, wherein • Belt exposure age again is 20±5 Myr, during which time a regolith impactors of only a few centimeters in diameter can still create a layer 4.5 ± 1.3 m in mean depth is overturned, with a mean of ‘global’ seismic event (see Fig.1 ). 0.7 ± 0.2 m of additional surface material lost to space. Note that in the simulated crater population R-plot shown in the lower panel of Fig. 19, the left-hand, negatively sloped portion, for Taking into account the full range of these simulations, we estimate that craters of 0.001–0.01 km (1–10 m) diameter, is an artifact, due to on 25143 Itokawa, a mean regolith overturn depth of 5 ± 1 m, having using a model with a hard-limit on the smallest impactor size sampled a large standard deviation across the entire surface of 5 ± 0.01 m, will by the Monte-Carlo algorithm; that is, the size of impactor needed to be overturned over a Main Belt exposure age of 20 ± 5 Myr. produce a crater of diameter equal to the pixel-scale of the surface. Our mean regolith overturn depth, a feature of ‘impact gardening’ This hard-limit on small impactors means that the smallest craters (see Section 2.4), is a few times larger than the impact-induced seismic on the simulated surface lack the population of even-smaller craters shaking ‘convective cell’ roll size of ∼ 1 m depth estimated by Yamada necessary to effectively erase them, either due to ‘sandblasting’ or et al.( 2016). Note that both of the estimated Main Belt exposure ages impact-induced seismic shaking. As such, the smallest craters in the of 20 ± 5 Myr are about one-fourth to one-third that of the minimum simulation are artificially preserved and more directly reflect their exposure age of 75 Myr estimated in Michel et al.( 2009), due to our ‘production population’ (Fig. 13). On the actual 25143 Itokawa surface, use of a ‘standard’ Main Belt exposure age in all simulations, as opposed the crater population would lack the left-hand side of this ‘check-mark’ to an estimate of the much lower impactor flux at the current orbital shaped curve, and would instead continue down and to the left. This location of 25143 Itokawa. Our Main Belt exposure age for Itokawa’s model feature is more severe on Itokawa (although still visible in both surface of 20 ± 5 Myr is also about half that of the ∼ 40 Myr regolith the Eros and Šteins simulations shown in Figs. 14 and 17) because convection ‘resurfacing’ age estimated by Yamada et al.( 2016), which impact-induced seismic shaking is extremely effective and important is again likely due to our use of a Main Belt impactor flux as opposed in producing Itokawa’s observed, extremely suppressed and softened to the more sparse near-Earth environment impactor flux. crater population. Fig. 20 shows a plot of the mean regolith depth (bold line) and 5. Discussion ejected mass depth (dashed line) as a function of time for the simulation performed at effective target strength of 푌̄ = 20 kPa, where regolith Table4 summarizes the results of the Small Body Cratered Terrain in this simulation is generated by impact crater breccia lens creation, Evolution Model (SBCTEM) simulations described in Section4 . Fol- impact ejecta ballistic emplacement, and the gradual erosion of high lowing a partial-simulation, directed parameter search process, about topography (such as crater rims). In addition to generating regolith, 8–10 completed simulations were performed for each body studied, each impact also causes a small amount of the asteroid’s mass to be with an example of the final results shown in Figs. 14–20. Other −1 lost to space; that is, ejected from the surface at > 푣푒푠푐 = 0.183 m s . user-specified parameters that were applied in all of these simulations

23 J.E. Richardson et al. Icarus 347 (2020) 113811

Fig. 19. The results of two SBCTEM simulations showing crater population creation and erasure on the surface of asteroid 25143 Itokawa for an effective target strength of 푌̄ = 10 KPa and Main Belt exposure age of 20.6 Myr. Crosses indicate the observed crater counts Michel et al.( 2009), while a solid line indicates the simulation’s crater counts, shown using a relative ‘R-plot’ format (Crater Analysis Techniques Working Group et al., 1979). (upper panel) This simulation was performed without any seismic effects on the topography, with crater counts leveling off at between 0.1–0.3 in typical Gault( 1970) fashion (horizontal dotted lines), and having a slope index of −2.1 with 9189 countable craters ≥ 3 pixels in diameter. (lower panel) This simulation was performed with seismic shaking effects on the topography and at crater density equilibrium, displays the observed, extremely suppressed counts for craters ≲ 100 m in diameter, having a very shallow slope index of −1.2 with only 424 countable craters ≥ 3 pixels in diameter.

−3 1.8 −0.3 0.25 include: impactor density 휌푖 = 2500 kg m , impact crater scaling combined use in Eq.( 14), wherein 퐾푖 ∝ 푄 , 퐾푖 ∝ 휉 , and 퐾푖 ∝ 휂 parameters 퐾1 = 0.2, 휇 = 0.5 (Richardson et al., 2007; Richardson, (in order of sensitivity). This codependency is reflected in the 휂, 휉, and 2009); transient crater 푑∕퐶푓 ratio = 0.285; final crater 푑∕퐶푓 ratio 푄 error bars listed in Table4 , which were computed using an estimated = 0.19, final crater rim height/depth ratio = 0.025, where these last 푄 error of ±100. Given this lack of constraint, we began each simulation two are both pre-ejecta blanket placement, such that ejecta blanket series using the applicable lunar analog values selected in Section 2.1), emplacement in a loose target material brings these values to 0.2 and and then adjusted gradually from there, keeping each parameter as −3 0.04, respectively (Melosh, 1989); regolith density 휌푠 = 1500 kg m , reasonable as possible. Due to the cumulative effect of millions of regolith cohesion 퐶 = 100 Pa (see Section 2.3), and a regolith effective simulated impacts on a particular asteroid body, (1) the roll-over point target strength of 푌̄ = 1 kPa (that of ‘very soft soil’ (NZGS et al., 2005)). at which the equilibrium crater counts begin to display an increasing deficit with decreasing size for small craters, and (2) the slope of that 5.1. Asteroid seismic properties equilibrium small crater size-frequency distribution, are both extremely sensitive to the selection of these target seismic properties. This extreme It is important to recognize that with respect to the asteroid seismic sensitivity permits us to fine-tune these parameters to each body’s properties listed in Table4 , the system is underdetermined, having only crater population and thus gain some insight as to the nature of the one constraint (the observed crater counts) and three unknowns. As underlying seismic medium. In future work, we plan to explore this such, the determined values for 휂, 휉, and 푄 are non-unique due to their parameter space more fully, particularly with respect to asteroids 433

24 J.E. Richardson et al. Icarus 347 (2020) 113811

Table 4 Summary of Small Body CTEM results. Parameter 433 Eros 2867 Šteins 25143 Itokawa Model surface area 400.0 km2 86.92 km2 0.385 km2 Model side length 20 km 9.32 km 0.62 km Model pixel length 10 m 9.32 m 0.62 m Model depth 8.46 km 2.63 km 0.175 km −2 −2 −2 Model surface gravity 푔푎 4.6 mm s 2.2 mm s 0.1 mm s −3 −3 −3 Model bedrock density 휌푎 2500 kg m 3000 kg m 2000 kg m Effective bedrock strength 푌̄ 0.5–5.0 MPa 2.5–5.0 MPa 10–20 kPa

Impactor diameter range 푑푖 0.1–700 m 0.1–500 m 0.01–10 m Crater diameter range 푑푓 10–10 000 m 9.3–4600 m 0.62–310 m Impact seismic efficiency 휂 1.0 × 10−6 ± 0.5 × 10−6 4.0 × 10−6 ± 2.0 × 10−6 1.0 × 10−7 ± 0.5 × 10−7 Seismic diffusivity 휉 0.5 ± 0.2 km2 s−1 0.5 ± 0.2 km2 s−1 0.002 ± 0.001 km2 s−1 Seismic quality factor 푄 1500 ± 100 2000 ± 100 1500 ± 100 ̄ −1 −1 −1 Topographic diffusion rate 퐾푑 149 ± 54 m Myr 1295 ± 540 m Myr 168 ± 9.5 m Myr Main Belt exposure age 225 ± 75 Myr 475 ± 25 Myr 20 ± 5 Myr Mean regolith depth 80 ± 20 m 155 ± 5 m 5.0 ± 1.0 m Mean escaped depth 60 ± 30 m 100 ± 5 m 1.0 ± 0.5 m

Fig. 20. A 25143 Itokawa SBCTEM simulation plot of the mean regolith depth (bold) and mean ejected mass depth (dashed) as a function of time for the simulation performed at effective target strengths of 푌̄ = 20 kPa. The three blue dots on the regolith growth curve indicate the range (low, middle, high) of Main Belt impact exposure ages at which (a) crater density equilibrium is initially achieved (Richardson, 2009) and (b) the observed crater counts (Michel et al., 2009) are well fit by the simulation, with this error range based upon stochastic differences between individual simulations. Fig. 21. A plot of estimated crater ‘seismic erasure times’, given in Myr, as a function of ̄ crater diameter 퐶푓 for the three asteroids investigated in this study, shown at 퐾푑 ± 1휎 (listed in Table4 ) for 433 Eros (solid lines), 2867 Šteins (dashed lines), and 25143 Itokawa (dotted lines). For comparison, an estimate of crater seismic erasure times Eros and 25143 Itokawa, for which extensive spacecraft observation for asteroid 243 Ida is also shown (dot-dashed lines). The light gray band indicates datasets now exist (Veverka et al., 2001; Abe et al., 2006). the approximate threshold below which craters will display a noticeably degraded population and crater count paucity, due to impact-induced seismic shaking. Somewhat surprisingly, we find that S-type asteroid 433 Eros and E-type asteroid 2867 Šteins possess very similar seismic energy produc- tion and attenuation properties (Table4 ). Seismic energy transmission 5 through the interior of both 433 Eros and 2867 Šteins is roughly replenished quickly on time-scales of 1 × 10 yr. This extreme seismic 1/10th–1/100th as efficient as transmission through the lunar Upper sensitivity is consistent with the crater counting results of Hirata et al. Megaregolith layer (Section 2.1), our selected ‘fractured monolith’ (2009), in which extremely subdued and subtle circular features had asteroid seismic analog. Likewise, the very similar best fit effective to be included in order to produce the observed ‘crater’ population target strengths for Eros, at 푌̄ = 0.5–5.0 MPa, and Šteins, at 푌̄ = (Section 4.4). 2.5–5.0 MPa, indicate that like Eros, Šteins is most likely a ‘fractured In addition to the matrix element maps within SBCTEM that track monolith’ asteroid (see Section 2.1), rather than a true ‘rubble pile’ digital elevation, regolith coverage, and crater form (Section3 ), for this asteroid, as proposed by Keller et al.( 2010), Marchi et al.( 2010) and study we included a new map for tracking the cumulative amount of Jorda et al.( 2012). topographic diffusion 퐾푐 that occurs at each pixel location over the Conversely, obvious ‘rubble pile’ asteroid 25143 Itokawa requires course of the model run. Utilizing these maps, Table4 lists the mean, ̄ highly unfavorable seismic energy production and attenuation proper- cumulative topographic diffusion rate 퐾푑 (in meters per Myr) for each ties, indicative of a highly-fractured, highly-attenuating interior struc- simulated asteroid surface, as generated by impacts that produce craters ture (as compared to asteroids 433 Eros and 2867 Šteins). Seismic at least one pixel width in diameter. Utilizing Eq.( 19), we can divide energy transmission through the interior of 25143 Itokawa is roughly the cumulative topographic diffusion 퐾푐 required to erase a crater of 1/100th–1/1000th as efficient as transmission through the lunar Sur- diameter 퐶푓 (shown in Fig. 10) by the mean topographic diffusion rate ̄ ficial Regolith layer (Section 2.1), our selected ‘rubble pile’ asteroid 퐾푑 determined for a particular body, and thus estimate the ‘seismic seismic analog. Increasing the seismic energy transmission parameters erasure time’ (or’seismic lifetime’) for a crater of diameter 퐶푓 on the of Itokawa by even a small amount causes the model to retain no surface of that body, using an assumed mobilized regolith depth of ̄ countable craters at all, except at the very smallest sizes, which are ℎ = 1 m. These 퐾푐 ∕퐾푑 estimates are shown in Fig. 21.

25 J.E. Richardson et al. Icarus 347 (2020) 113811

̄ Note that the cumulative topographic diffusion rates 퐾푑 and crater bedrock. As illustrated in Fig. 18, the regolith depth on an asteroid will seismic erasure times shown here are dependent upon the asteroid’s eventually reach an equilibrium state, wherein these production and surface being exposed to a mean Main Belt impactor flux. Exposure loss terms are roughly balanced. As such, if the regolith depth on a to a lower impactor flux, such as that in the near-Earth environment, particular asteroid can be reasonably estimated, this gives us a second will result in both a lower impact cratering rate and a correspondingly constraint (in addition to its cratering record) on the surface age of lower impact-produced topographic diffusion rate. As shown in Fig. 21 the body, up to the level of regolith depth equilibrium: after which using our ‘standard’ mean Main Belt conditions, a 100 m diameter the estimated surface age represents only a lower-limit, similar to the crater on 433 Eros has an expected seismic lifetime of ∼ 13 Myr, a 100 surface ages estimated from the crater density equilibrium. m diameter crater on 25143 Itokawa has a similar expected seismic lifetime of ∼ 11 Myr, while on 2867 Šteins, a 100 m diameter crater 6. Conclusion has a short expected seismic lifetime of only ∼ 1.5 Myr. Since the crater seismic lifetime is proportional to 퐶2 (Eq.(19)), a 1 km diameter crater 푓 Of the several near-Earth and Main Belt asteroids visited by space- . on 433 Eros has a significantly longer seismic lifetime of ∼ 1 3 Gyr, craft to date, three display a paucity of small craters and an enhanced while on 2867 Šteins, a 1 km diameter crater has an expected seismic number of smoothed and degraded craters: 433 Eros, which has a lifetime of ∼ 150 Myr. Notably, the 2.1 km Diamond crater on 2867 deficit of craters ≲ 100 m in diameter (Veverka et al., 2001; Chapman Šteins has an expected seismic lifetime of ∼ 660 Myr, and given its et al., 2002; Cheng, 2004; Thomas et al., 2002; Robinson et al., 2002); generally fresh appearance, this implies that its formation is a relatively 2867 Šteins, which has a deficit of craters ≲ 500 m in diameter (Keller recent event on this asteroid. et al., 2010; Marchi et al., 2010; Jorda et al., 2012); and 25143 Itokawa, Given our conjecture regarding asteroid 243 Ida in Section 1.2, we which has a deficit of craters ≲ 100 m in diameter (Fujiwara et al., performed a single SBCTEM run for the surface of Ida at a pixel scale 2006; Hirata et al., 2009). The purpose of this work was to investigate of 20 m (1600 km2 surface area), using the bedrock effective target and model topographic modification and crater erasure due to impact- strength 푌̄ = 5 MPa and determined seismic properties from our 433 induced seismic shaking, as well as impact-driven regolith production Eros simulations. When exposed to a mean Main Belt impactor flux, Ida and loss, on these three asteroid surfaces. To perform this study, we experiences a low mean topographic diffusion rate of 퐾̄ ≈ 2 m Myr−1, 푑 utilized the numerical, three-dimensional, Small Body Cratered Terrain with correspondingly longer crater seismic erasure times, as shown in Evolution Model (Richardson, 2009; Minton et al., 2015; Richardson Fig. 21. The implication is that if there is a paucity of small craters on and Abramov, 2020), which received a small-body specific update for 243 ida, it is likely only for craters ≲ 20 m diameter, which have seismic this work. In Sections2 and3, we described the development and im- lifetimes of ≲ 42 Myr, on the same order-of-magnitude as the seismic plementation of a broadly applicable, general topographic modification lifetimes for 100 m craters on Itokawa (∼ 11 Myr), 100 m craters on Eros (∼ 14 Myr), and 500 m craters on Šteins (∼ 33 Myr), the largest expression, used to compute downslope regolith diffusion as a function diameters at which a paucity of small craters first becomes evident on of a specified impact, on a given target body, at a particular location those bodies (indicated by the light gray band in Fig. 21). on the surface. In Section 4.2, we used SBCTEM to reproduce the cratering record 5.2. Asteroid impact-generated regolith properties of asteroid 433 Eros for craters up to ∼ 3 km in diameter, including the paucity of small craters ≲ 100 m in diameter, at a minimum Main Belt With respect to regolith production and retention on these small exposure age of 225 ± 75 Myr with a best fit effective target strength ̄ bodies, Figs. 15, 18, and 20 show that 2867 Šteins manages to retain of 푌 = 0.5–5.0 MPa (that of ‘very weak rock’ (NZGS et al., 2005)). a significantly deeper regolith layer than that currently on 433 Eros, These Eros simulations produced a mean regolith depth of 80 ± 20 m, despite Stein’s smaller size. The amount of impact ejecta retained in which agrees with published estimates that Eros possesses a regolith each impact is roughly proportional to the escape velocity of the body, layer ‘‘tens of meters’’ in depth on average, and even deeper in some which can be estimated by: regions (Robinson et al., 2002). √ √ In Section 4.3, we used SBCTEM to reproduce the cratering record 2퐺푚푎 2퐺휌푎 of asteroid 2867 Šteins, including its paucity of small craters ≲ 500 m in 푣푒푠푐 = = 2푟푎 , (24) 푟푎 3 diameter, at a minimum Main Belt exposure age of 175±25 Myr using a target strength of 2.5–5.0 MPa (that of ‘weak rock’ (NZGS et al., 2005)). where 푚푎 is the asteroid’s mass and 퐺 is the gravitational constant. Eq.(24) shows that for two asteroids of about the same mean radius Extending this simulation produces a mean regolith depth of 155±5 m at a minimum Main Belt exposure age of 475±25 Myr, in good agreement 푟푎, the asteroid’s bulk density 휌푎 becomes the primary factor affecting with the 145 ± 35 m actual mean regolith depth estimated from two the escape velocity 푣푒푠푐 of the body. As such, the higher assumed bulk −3 chains of drainage-pit craters on the surface, with both crater density density for E-type asteroid 2867 Šteins of 휌푎 = 3000 kg m contributes notably to its ability to retain a higher fraction of ejecta volume per and regolith depth in an equilibrium state. Our analysis also shows impact and thus develop a thicker regolith layer when compared to a Šteins to currently be in an ‘optimum’ rotation state with stable surface similarly-sized S-type asteroid. Nonetheless, in all three cases examined slopes and minimized regolith migration rates (Richardson et al., 2019), in this study, S-type asteroid 433 Eros, E-type asteroid 2867 Šteins, such that its ‘oblate-top’ shape and equatorial bulge are indicative of and S-type asteroid 25143 Itokawa, the amount of regolith produced past YORP spin-up episodes (Harris et al., 2009), but not its current and retained via impact processes alone is sufficient and reasonably condition. Additionally, the surface ages estimated from our SBCTEM consistent with the spacecraft observations of each body, without the crater population and regolith depth modeling are consistent with our need to invoke other forms of regolith production. estimate for the YORP spin-up timescale for present-day Šteins of 휏푠푝푖푛 ≃ In Section 2.4, we briefly described the process of ‘impact garden- 700–1000 My. ing’, classically modeled by Ross(1968), Soderblom(1970), Shoemaker In Section 4.4, we used SBCTEM to reproduce the cratering record et al.(1970) and Gault et al.(1974) for the lunar surface. This process of asteroid 25143 Itokawa, including the paucity of small craters ≲ 100 contains both a loss term, wherein small impacts tend to only overturn m in diameter, at a minimum Main Belt exposure age of 20 ± 5 Myr and recycle the existing regolith layer, with some loss of material to with a best fit effective target strength of 푌̄ = 10–20 kPa (that of ‘soft space after each impact due to ejection at greater than the escape soil’ (NZGS et al., 2005)). These Itokawa simulations produced a mean velocity 푣푒푠푐 of the body; and a production term, wherein ‘fresh’ re- regolith overturn depth of 4.5 ± 1.3 m, given that Itokawa is composed golith is only generated by large impacts that can penetrate below the entirely of regolith and boulders. This mean regolith overturn depth existing regolith layer to excavate a portion of the underlying, fractured due to ‘impact gardening’ (see Section 2.4), is a few times larger than

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