Physical and Chemical Evolution of Lunar Mare Regolith

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Item Type Article; text

Authors O'Brien, P.; Byrne, S.

Citation O’Brien, P., & Byrne, S. Physical and Chemical Evolution of Lunar Mare Regolith. Journal of Geophysical Research: Planets, e2020JE006634.

DOI 10.1029/2020JE006634

Publisher Blackwell Publishing Ltd

Journal Journal of Geophysical Research: Planets

Download date 26/09/2021 22:26:13

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Link to Item http://hdl.handle.net/10150/660051 RESEARCH ARTICLE Physical and Chemical Evolution of Lunar Mare Regolith 10.1029/2020JE006634 P. O'Brien1 and S. Byrne1 Key Points: 1Lunar and Planetary Laboratory, Tucson, AZ, USA • We present a novel three- dimensional landscape evolution and regolith transport model calibrated to the lunar mare • Gardening by secondary craters is the dominant control on regolith particle surface residence times Abstract The lunar landscape evolves both physically and chemically over time due to impact • Constraints from Apollo samples cratering and energetic processes collectively known as space weathering. Despite returned soil samples and remote sensing indicate space and global remote sensing reflectance measurements, the rate of space weathering in the lunar regolith is weathering maturation timescales of 7–19 Myr of cumulative exposure not well understood. To address this, we developed a novel three-dimensional landscape evolution model to simulate the physical processes that control the burial, excavation, and transport of regolith on airless bodies. Applying this model to the lunar mare, we find that over billions of years of surface evolution, Supporting Information: • Supporting Information S1 material typically spends only a few million years on the surface where it is exposed to the effects of space weathering. The small surface residence times are a result of vigorous mixing by small-scale impacts, predominantly driven by secondary crater formation. We deduce the rate of space weathering Correspondence to P. O'Brien, by comparing our modeled distribution of surface residence times on the lunar mare to measurements [email protected] of space weathering maturity from Apollo soil samples and orbital surface reflectance datasets. These chemical constraints indicate that soil on the lunar mare reaches maturity in 7 Myr of cumulative surface Citation: exposure though due to uncertainties in the rate of small secondary crater production, this timescale could O'Brien, P., & Byrne, S. (2021). Physical be 2–3 times higher. Weathering progresses more rapidly upon initial exposure to space but the surface and chemical evolution of lunar residence time required to achieve maturity is realized over billions of years as regolith is repeatedly mare regolith. Journal of Geophysical Research: Planets, 126, e2020JE006634. buried and exposed by small impacts. https://doi.org/10.1029/2020JE006634 Plain Language Summary On the Moon, large impact craters churn the upper soil layers Received 20 DEC 2019 and micrometeorites, galactic cosmic rays, and the solar wind alter the chemical structure of material Accepted 22 NOV 2020 on the surface in a process called space weathering. The relationship between these processes is crucial to understanding how quickly planetary surfaces are space weathered. To study this problem, we have developed a three-dimensional computer model that simulates lunar-like landscapes that evolve over time from flat surfaces to cratered landscapes. Using this model, we measure how long material spends on the surface where it would be exposed to the space environment. We compared the results of this model to chemical measurements of lunar soil returned by the Apollo astronauts and orbital measurements of the lunar surface. We find that over the last 3.5 billion years most material has been on the surface for only a few million years in total as it is repeatedly buried and excavated by small craters. Space weathering must therefore occur rapidly to produce the observed chemical properties of the lunar soil.

1. Introduction The surfaces of airless bodies like the Moon are directly exposed to the space environment and as a result change both physically and chemically over time due to processes collectively known as space weathering. The rate of space weathering on the lunar landscape is poorly understood because these two facets of surface evolution have yet to be fully linked. Macroscopic physical processes like impact cratering disrupt and overturn the upper surface layer comprised of loose, unconsolidated material known as soil or regolith (Shoemaker et al., 1969). As the landscape evolves and material is transported across the surface, the micro- scopic chemical structure of material on the surface is altered by micrometeorite impacts, cosmic rays, and the solar wind (Hapke, 2001; Pieters & Noble, 2016). These processes are deeply coupled since the amount of time material spends on the surface exposed to the space environment depends on the rate at which material is cycled in and out of the uppermost regolith layer.

© 2020. American Geophysical Union. Analysis of lunar soil properties from Apollo program samples and remote sensing surface reflectance All Rights Reserved. measurements have characterized the degree of space weathering, or maturity, of the lunar regolith (Lucey

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et al., 2000; Morris, 1976). Maturity metrics quantify the amount of weathering products accumulated by a sample, but not the timescale over which those chemical products were accumulated. In the absence of definitive, widespread measurements of cumulative surface exposure ages, the rate at which the surface of the Moon is weathered by the space environment remains unknown. Landscape evolution models, like those applied by Hartmann and Gaskell (1997) and Richardson (2009) to understand cratering equilibrium, can be applied here to investigate the timescales of regolith space weathering exposure on airless bodies.

We have developed such a model and calibrated it to reproduce the histories and topographic statistics of surfaces on the lunar maria by incorporating constraints from high-resolution lunar topography (Henriksen et al., 2017; Robinson et al., 2010; Smith et al., 2010). The model is used to track the trajectories of tracer particles as synthetic lunar surfaces evolve over time and to measure how long particles reside in the upper millimeter of regolith; that is, on the surface (Morris, 1978). By combining the distribution of regolith exposure ages with measurements of soil maturity (Lemelin et al., 2016; Lucey et al., 2000), we link the physical and chemical evolution of the Moon's surface to estimate the rate of space weathering on the lunar maria. While presently applied only to four Apollo landing sites, this work provides a framework for a broader understanding of how the surface of the Moon and other airless bodies, like Mercury, Ceres, and Vesta are modified over time by space weathering.

2. Background In addition to macroscopic crater-forming impacts, the Moon is continually bombarded by micrometeorites, particles from the solar wind, and cosmic rays. These energetic processes, collectively known as space weathering, induce chemical changes in material residing on the surfaces of airless bodies that lack the shielding of an atmosphere or magnetic field (Morris, 1978). Lunar soils that have undergone space weathering show lowered albedo, spectral reddening, and subdued mineral absorption bands (Hapke, 2001; Keller & McKay, 1997; Pieters & Noble, 2016; Pieters et al., 1993, 2000; Taylor et al., 2001). From analysis of returned lunar samples, it was determined that the primary agent of these chemical changes is the formation of nanophase metallic iron and other optically active opaque (OAOpq) particles as a result of the energetic processes in the space environment operating to reduce FeO in the lunar soil (Britt & Pieters, 1994; Cassidy & Hapke, 1975; Hapke, 2001; Pieters & Noble, 2016). These particles form within agglutinitic glasses and on the rims of grains through irradiation and sputtering processes as well as from condensation of vaporized material following micrometeorite impacts (Keller & McKay, 1993, 1997; Noble et al., 2005; Taylor et al., 2001). The degree of space weathering, as characterized by some quantitative measure of the amount of these space weathering products that a surface soil has accumulated, is termed soil maturity (McKay et al., 1974; Morris, 1976). Laboratory techniques can address the rate of space weathering by quantifying the amount of OAOpq particles that have formed in a planetary regolith (e.g., Ramirez & Zega, 2016). Previ- ous studies have placed limits on the space weathering rate using solar flare track densities and amorphized rim widths which accumulate in lunar soil grains and saturate at inferred solar wind exposure ages of a few million years (Keller & Berger, 2017; Keller et al., 2016). Remote sensing reflectance measurements have generated maturity proxies like the optical maturity (OMAT) parameter (Lucey et al., 2000). Though sensitive only to the uppermost few regolith grains, orbital measurements provide information about how maturity varies across the entire lunar surface and relates to age and composition. Space weathering is ubiquitous on airless bodies and much work has been done to characterize and understand the process from Mercury (Domingue et al., 2014; Lucey & Riner, 2011; Noble & Pieters, 2003; Riner & Lucey, 2012; Sasaki & Kurahashi, 2004) to Ceres and Vesta (Blewett et al., 2016; Fulvio et al., 2012; Pieters et al., 2012, 2016; Stephan et al., 2017) and the asteroids (Clark et al., 2002; Gaffey, 2010; Hiroi et al., 2006; Nesvorný et al., 2005; Noguchi et al., 2011). While the surface weathering process operates similarly throughout the inner solar system, there are significant variations in the chemical byproducts and rates of weathering that make the process unique to each space weathered body. These variations arise from differences in surface composition, surface gravity, and the flux of micrometeorites and energetic particles and are beyond the scope of the present study, which focuses on the lunar maria. The mare are composed primarily of basalt and most mare units formed on the nearside of the Moon between 3.3 and 3.8 Ga (Hiesinger et al., 2011).

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Since the emplacement of the maria, the physical evolution of the lunar landscape has been dominated by a relatively small number of processes: topographic relief-creation from impact cratering and relief-reduction from impact gardening and seismic shaking. Over the last few billion years, the initially smooth maria have accumulated craters and the regolith layer in these regions is steadily increasing in thickness as large impacts break up solid basalt and small impacts churn the upper few meters of the surface (McKay et al., 1991; Shoemaker et al., 1969). Impacts expose less-weathered regolith and deposit bright, immature rays, like those surrounding young craters such as Tycho (Dundas & McEwen, 2007; Hawke et al., 2004; Werner & Medvedev, 2010), that darken and fade with time. Smaller impacts repeatedly overturn the upper regolith, continually exposing (and re-exposing) material to the effects of space weathering. Downslope movement of material from impact gardening and seismic shaking both limits and assists weathering by burying surface materials and uncovering fresh material on steep slopes as regolith is transported to topographic lows.

The competing processes of impact cratering and erosion can be reproduced by landscape evolution models and thus can provide a key way to understand the burial, excavation, and transport of regolith and consequently the rate of space weathering on the lunar surface. Landscape evolution models have their origins in the pioneering experiments of the 1970s that formed cratered landscapes by firing projectiles into a sandbox (Gault, 1970; Oberbeck & Quaide, 1967) and the mathematical erosion models of Soderblom (1970) and So- derblom and Lebofsky (1972), which demonstrated that the effect of a steady stream of micrometeorite impacts on the lunar landscape can be modeled as a form of topographic diffusion. Numerical models followed advancements in computing capability, from low-resolution, geometric models of crater rim emplacement and erasure (Woronow, 1977a, 1977b, 1978), to three-dimensional simulations capable of tracking digital elevation models at considerably higher spatial resolutions (Gaskell, 1993; Hartmann & Gaskell, 1997). The Cratered Terrain Evolution Model (CTEM) outlined in Richardson (2009) significantly advanced the state of landscape evolution models by incorporating crater scaling relations and downslope mass movement from seismic shaking associated with each added crater. Additionally, this model includes a human-calibrated crater counting feature, which produces the observed crater size-frequency distribution at a given time step. This last capability highlights the fact that most previous studies using landscape evolution models were focused on addressing questions related to crater saturation/equilibrium or age dating of planetary surfaces (e.g., Chapman & McKinnon, 1986; Gault, 1970; Minton et al., 2015, 2019; Richardson, 2009; Soderb- lom, 1970, etc.). A notable exception is the model of Craddock and Howard, which was the first to explicitly include diffusional effects, as well as other types of erosion, to explore the degradation of lunar craters (Cra- ddock & Howard, 2000) and fluvial erosion on Earth and Mars (Craddock & Howard, 2002). More recently, CTEM has been adapted to study the transport of material across mare-highlands contacts as well as the formation and destruction of impact glass spherules (Huang et al., 2017, 2018). Costello et al. (2018)'s updates to the impact gardening model of Gault et al. (1974) have generated key insights about the 1D vertical mixing of material on airless bodies. Such analytical models provide useful estimates of the time needed for impacts to disturb regolith to a certain depth but are limited in their application to the inherently three dimensional nature of airless landscapes as a whole.

The model presented in this work implements features from past models as well as advancements in physical realism that make it uniquely suited to the study of surface residence times in the lunar regolith. Improvements over past models include the explicit treatment of secondary craters following empirical observations of secondary production on airless bodies and the calibration of the micrometeorite-driven diffusion rate which leads to model topographies that visually and statistically resemble landscapes on the Moon. Most importantly for the present study, the combination of fully three-dimensional tracer particle tracking with the Maxwell Z-model (Maxwell, 1977) and mixing with a 1D impact gardening model yields the additional capability of measuring the distribution of regolith surface residence times on a planetary body. In this work, particle surface exposure histories are used to investigate the rate of space weathering on the lunar mare. Our model, however, is designed to be both modular and general, meaning that it can be calibrated and run for multiple airless bodies to address a variety of geomorphological processes.

The relative rates of impact cratering and diffusive mass movement control the roughness of the final landscape on an airless body. Analysis of lunar topography has utilized various metrics to distinguish the two types of lunar terrains on the basis of roughness. The maria, the majority of which were volcanically flooded 3–4 Ga (Hiesinger et al., 2011), are relatively smooth and the heavily cratered highlands regions

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are relatively rough. Although the maria are slowly progressing toward the roughness of the highlands, there is not enough time remaining in the lifetime of the solar system to reach this state (assuming the impact rate remains constant). Rosenburg et al. (2011) calculated median slopes and Kreslavsky et al. (2013) examined interquartile ranges of curvature using Lunar Orbiter Laser Altimeter data sets and demonstrated that roughness was correlated with age even within these distinct terrains. Similarly, the roughness of our model surfaces increases over time as they progress from initially flat surfaces to cratered landscapes (Figure 1). As this occurs, individual topographic features, like crater rims, will become smoother as they are degraded by a steady stream of micrometeorites. The rate of diffusion from small impacts, or diffusivity, controls how rapidly topographic features degrade and therefore (in concert with the macroscopic impact rate) sets the overall surface roughness of the final landscape. Fassett and Thomson (2014) studied the profiles of simple lunar craters by modeling their diffusive degradation from idealized initial profiles and estimated the diffusivity on the lunar maria to be 5.5 m2/Myr.

Figure 1. Example model surface evolving over the age of a typical mare Roughness at a given time step, as measured by the median bidirectional unit. Surfaces are illuminated from the upper left. The landscape is seen to slope, is directly related to the diffusivity parameter in our model. We di- become rougher and more heavily cratered with time. rectly calibrated the rate of diffusion to match observed roughness values at the Apollo landing sites at the same spatial scale as our model landscapes. Here we use the calibrated model to track the burial, excavation, and transport of regolith on the lunar mare. With the addition of a 1D vertical mixing model we estimate the distribution of regolith surface residence times and the characteristic cumulative exposure timescale. Finally, we apply the results of our model to the topic of space weathering. By combining our modeled surface residence time distribution with measurements of lunar soil maturity, we derive the rate of space weathering at four Apollo landing sites.

3. Model We model surface residence time in the lunar regolith using a two part Monte Carlo model. In the first part, a 3D landscape evolution model simulates the surface of an airless body evolving over time as impact craters form randomly and topographic features are eroded by impact gardening (Figure 1). The positions of regolith tracer particles are tracked throughout the model run as they are excavated from the subsurface and transported across the model domain. In the second part, tracer particle surface residence times are estimated using a 1D regolith gardening model that implements vertical mixing from small impacts within each time step. A full table of model parameters can be found in Table 1.

3.1. Three-Dimensional Landscape Evolution Model Our model represents a patch of the lunar surface as a grid of square cells that store elevation values at each time step. The grid contains a single layer representing the surface elevation which is modified over time by the formation of craters and diffusive mass movement. Additional layers representing regolith thickness and the depth to unfractured bedrock grow over time and are tracked separately as they influence the size of craters that form on the grid. To study the problem of surface residence in the lunar regolith, we augment this model to track the 3D position of tracer particles that are excavated, transported, and buried by the same processes that are modifying the grid elevations. By assessing the depths of these particles throughout the model run, we can estimate how long lunar regolith particles typically spend near the surface where they are exposed to the effects of space weathering.

The grid parameters are chosen to approximate the conditions of the Apollo landing sites where samples were retrieved. We initialize the model as a 2 × 2 km grid of pixels with a spatial resolution of 4 m/pixel, chosen as a compromise between high-spatial resolution of regolith processes and computational efficiency.

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Table 1 Model Parameters Parameter Value Reference Planetary body Radius 1737.4 km – Surface gravity 1.62 m/s2 – Impactor density 2,700 kg/m3 Marchi et al. (2009) Target density (regolith) 1,500 kg/m3 McKay et al. (1991) Target density (bedrock) 3,150 kg/m3 Kiefer et al. (2012) Target strength (regolith) 1 kPa Mitchell et al. (1972) Target strength (bedrock) 20 MPa Asphaug et al. (1996) Model domain Grid resolution 4 m/pixel – Grid width 2 × 2 km – Grid width (pixels) 500 × 500 – Time step 1 Myr – Total run time 3.48 Gyr Hiesinger et al. (2011) Diffusivity (4 m scale) 0.109 m2/Myr This work Cratering Pi-group scaling constants ν = 0.4 Holsapple and Housen (2007) k = 1.03, μ = 0.41 (regolith) k = 0.93, μ = 0.55 (bedrock) Minimum crater diameter 8 m –

Continuous ejecta blanket radius, Rej = ηR η = 5 Melosh (1989)

Maximum secondary diameter fraction, Ds,max = γD γ = 0.04 McEwen and Bierhaus (2006) Crater depth d = 0.2D Melosh (1989) Crater rim height 33 This work hr  0.2DD 0.05 1532 16   2 2

Transient crater diameter Dtr = D/1.17 Melosh (1989) Excavation depth 1 Melosh (1989) hDexc tr 16 Inheritance parameter 0.9 Howard (2007) Number of primary impact sampling regions 100 – Tracer particles Number of particles per layer 25 – Total number of particles 15,625 – Particle depth (geometrically spaced) 0–43 m – Space weathering penetration depth 1 mm Morris (1978) Particle sampling depth 10 cm – Maxwell Z-parameter Z = 3 Melosh (1989)

The width of the grid is chosen to be the approximate distance traversed by the Apollo astronauts during sample collection. The grid is seeded with tracer particles, spaced evenly across the grid in the horizontal directions. The initial depths of the particles follow a geometric progression starting at the surface and ending at a depth of approximately 40 m, corresponding to the excavation depth of a crater whose diameter is 1/4 of the total grid size. The initial vertical distribution of tracers reflects the fact that particles in

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the upper few meters of the surface will experience most of the effects of cratering and topographic diffusion but particles buried deeper in the subsurface can still be excavated and transported by larger, rarer cratering events. Craters larger than 1/4 of the grid width rarely form on the model domain and thus particles at depths greater than 40 m would rarely be excavated. As the surface evolves, we track the 3D position of particles as they are influenced by the formation of craters on the landscape and the diffusive downslope movement of material. When the grid elevation changes in the vicinity of a tracer particle due to the formation of a crater or downslope movement of material, the particle can be transported ac- Figure 2. (left) Simplified geometry of our crater sampling process for cording to its proximity to the event. one impactor size bin and two impact regions. The grid is shown at the center with filled circles depicting craters (interior plus ejecta blanket). The model is run for a period of 3.48 Gyr, equivalent to the average age Only primary impacts in the inner circular area can directly influence of the mare units sampled by the Apollo program (Hiesinger et al., 2011). the grid elevations. The size of this region is set by the largest possible While our semi-implicit Crank-Nicolson diffusion scheme described in primary crater in the current size bin, Dp,max. The remainder of the lunar Section 3.4 is unconditionally stable for any timestep, the accuracy of the surface is divided into annuli where primaries can only influence the grid by producing secondaries that land on the model domain. (right) Off-grid method still depends on the timestep. To compromise between computa- primaries generate secondaries that form on the grid. The size of the tional efficiency and temporal resolution, the model time step is chosen largest possible secondary, Ds,max, depends on the distance between the grid to be 1 Myr. Shorter timesteps were tested and provided diminishing im- and the secondary-producing primary. provements in accuracy at the expense of longer model run times.

3.2. Impact Cratering At each time step, we determine the effect of all impactors striking the Moon. Unlike previous studies, we do not employ periodic boundary conditions for the grid. Rather, we randomly sample the sizes and locations of impacts across the entire lunar surface and determine the craters (including secondaries) that overlap the small section of the surface represented by our grid.

It is commonly assumed that since the end of the Late Heavy Bombardment, ≈3.5 Ga, the impact flux in the inner solar system has been relatively constant (Neukum et al., 2001; Robbins, 2014), though some re- searchers have argued that the rate of crater production on the Moon has increased (Culler et al., 2000; Fas- sett & Thomson, 2014; Grieve, 1984; McEwen et al., 1997; Shoemaker et al., 1990; Vokrouhlický et al., 2017) or decreased (Craddock & Howard, 2000; Hartmann et al., 2007; Quantin et al., 2007) in the last few Gyr. We adhere to the assumption of a constant flux on the lunar maria and apply a single impactor rate throughout the model.

3.2.1. Primary Craters We randomly sample impacts from lunar impactor size and velocity distributions (Marchi et al., 2009) using a Poisson sampling method. The impactor size-frequency distribution given in Figure 2 of Marchi

et al. (2009) is interpolated into diameter bins with a factor of 2 geometric spacing starting at dmin, the minimum impactor diameter in our model, which is roughly 6 cm for a grid resolution of 4 m/pixel. An impactor of this size at maximum crater conditions (maximum velocity and head-on collision) would produce an 8 m crater, that is, the minimum resolvable crater which is defined to have a diameter of two pixels. The effects of impactors smaller than this diameter are discussed in Section 3.5.4. We do not restrict the size of

the largest possible impactor. Impactors of size, di, that hit the grid are assigned a random impact velocity from the distribution of Marchi et al. (2009) (Figure 1 of that work) and an incidence angle from a sin (2i) distribution (Shoemaker, 1962). We then use the impactor diameter and normal velocity, v , to convert each impact to a final crater diameter using the pi-group scaling method (Holsapple & Housen, ⊥2007).

  2 2 (2 ) 2 gd Y   D dk i  (1) ci22  vv  

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Following the procedure of Marchi et al. (2009), we use two sets of values for the coefficients k and μ, one for craters which form entirely within the loose, upper regolith layer and another for impacts which impact into solid bedrock. At the beginning of the model run, the regolith thickness and the depth to unfractured bedrock are both zero. As the run progresses, these layer thicknesses increase with time t1/2 at rates that produce a 5 m thick regolith and 8 km thick fractured bedrock layer after 3.5 Gyr (Marchi ∝et al., 2009; McK- ay et al., 1991). The layer thicknesses grow uniformly across the grid and represent a median depth for each layer as a function of time. At each timestep, the solid rock values of k and μ are used if 10 times the impactor radius is greater than the current thickness of the heavily fractured layer. We use a target rock strength, Y that increases linearly with depth from 1 kPa in regolith at the surface to a constant 20 MPa in unfractured bedrock (Asphaug et al., 1996; Mitchell et al., 1972) and a target density, ρ that similarly increases from a value of 1,500 kg/m3 for regolith at the surface (Carrier et al., 1991) to 3,150 kg/m3, the density of low porosity lunar basalt (Kiefer et al., 2012), below the heavily fractured layer. When calculating crater size, both target strength and density are averaged over a depth of 10 times the impactor radius. Because regolith thickness increases over time, the changing target density and strength profiles are tracked by the model and updated at each timestep to accurately calculate the sizes of craters forming on a layered target (Xie et al., 2019). Finally, the density of the impactor, δ, is fixed at a value of 2,700 kg/m3 (Marchi et al., 2009).

We use primary crater profiles similar to those in Richardson (2009) with a depth-to-diameter ratio of d / D = 0.2 and an elevation profile with a parabolic interior and exterior that falls off as r−3 plus a linear term outside the crater rim. We use a volume-neutral elevation profile that reaches zero elevation at the edge of the continuous ejecta blanket which occurs at a distance of five crater radii (Melosh, 1989). The rim height needed to achieve volume neutrality with this profile is ≈0.05D. Positive elevation beyond the crater rim is formed from a combination of ejecta deposition and uplift of target material. For simplicity, the ejecta blanket thickness in our model is chosen to be 1/3 of the total added elevation at all points along the crater exterior profile (see Supporting Information S1 for more information on simple crater profiles).

Each crater is blended with the preexisting topography according to the method described in Howard (2007). The key parameter in this method is the inheritance, a value between 0 and 1 that controls the profiles of craters that form on slopes. For an inheritance of 0 the crater rim is horizontal whereas for an inheritance of 1 the crater rim slopes parallel to the preexisting local slope. Following Howard (2007), we use an inheritance parameter of 0.9 in our model. While we do not model the crater excavation and modification step, the shape of the transient crater bowl can be inferred from the final crater profile. The transient cavity is a

parabolic bowl with diameter Dtr = 0.85D and a depth-to-diameter ratio of 0.375 (Melosh, 1989). This inter- mediate stage of crater formation is important for understanding how the addition of a crater to the model domain influences target material in the vicinity of the impact. In Section 3.5, we describe how the position of regolith tracer particles within the transient cavity determines the particles' vertical and horizontal transport due to the crater's formation.

For the purposes of impact sampling, we define two cratering regimes: on-grid primaries whose interior bowl and/or ejecta blanket overlap the model domain, and off-grid primaries that form entirely outside the model domain. Craters that occur far outside the grid cannot directly influence the elevation of the grid but those that are sufficiently large and close to the region of interest can produce secondary craters that do form on the model domain. For each impactor diameter bin, we divide the entire lunar surface into a number of concentric regions extending outward from the center of the grid (Figure 2). Knowing the surface area of each region, we use Poisson sampling with the Marchi et al. (2009) impactor size-frequency distribution to determine the number of impacts that occur in each region during one time step. The inner- most region defines the area where impacts of that size could produce craters which overlap the grid (on- grid primaries). The size of this inner region varies for different impactor diameter bins since larger craters can overlap the grid from larger distances. Impacts that occur in this area are assigned impact angles and velocities and added to the grid as outlined above at positions that are uniformly random in azimuth angle and an areally weighted distance from the center of the grid. The remainder of the lunar surface is divided into equal-width annuli. Impacts in these regions can only influence the grid through the generation of secondary craters (off-grid primaries). To efficiently compute the effect of these distant primaries, we make simplifying assumptions. We assume that in a given annulus, each impact occurs at a distance equal to the angular distance from the center of the grid to the center of the annulus. Within these distant annuli, we

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assume that impactors in a given impactor diameter bin form craters of a fixed size. Taking the geometric average of the impactor size bin, this average crater diameter is computed using the most probable velocity from our lunar impact velocity distribution (Marchi et al., 2009) and most probable impact angle (45°, Shoemaker, 1962).

3.2.2. Secondary Craters In this section, we derive the size and number of secondaries produced by both on- and off-grid primaries in our model. Using observational constraints of secondary crater production on airless bodies, we randomly sample the population of secondaries that are generated by primaries of sufficient size and proximity to the grid at each time step. Various studies have examined the number and size of secondary craters on the Moon, Mercury, Mars, and icy satellites and quantified key aspects of the secondary formation process that are implemented in our model (Bart & Melosh, 2010; Hirata & Nakamura, 2006; Singer et al., 2013; Vick- ery, 1986, 1987). Secondaries are permitted to form at any distance from their primary since large impacts are capable of generating many isolated secondaries beyond the proximal ejecta regions and depositing ejecta at the ballistic antipode (Bandfield et al., 2017; Bierhaus et al., 2018; Hood & Artemieva, 2008; McEwen & Bierhaus, 2006). A detailed study of secondaries from Zunil crater on Mars estimated that only 2%–3% of secondaries were concentrated in obvious rays, with an order of magnitude more between the rays and even more beyond the extent of the ray system, though only secondaries within rays can be reliably mapped (McEwen et al., 2005; Preblich et al., 2007). This abundance of distant secondaries has also been suggested for Tycho crater (Dundas & McEwen, 2007) and on Europa (Bierhaus et al., 2001). Despite the potentially global reach of secondary generation from large craters, the size of the largest secondary that forms from a

primary of diameter Dp decreases with distance from the primary. In our model, primaries are only capable of producing secondaries out to the ballistic range where the largest possible secondary becomes unresolv- able (<2 pixels in diameter). The maximum secondary diameter is observed to be ≈4% of the primary crater diameter but because larger secondary-producing fragments are ejected at lower velocities, these largest secondaries only form close to the primary (Bierhaus et al., 2018). The size-velocity relationship of the largest ejected fragments can be expressed as a power law of the form (Vickery, 1987),

 dmax av ej (2)

Using crater scaling relationships and the ballistic range equation on a sphere, we derive an equation for the

maximum secondary crater diameter Dmax as a function of radial distance from the primary (Equation 5). Assuming an ejection angle of 45° for all secondary-producing fragments, the ejection velocity as a function of distance from the point of impact is

r tan Rgp v() r 2 Rg (3) ej p r 1 tan Rgp

Owing to the Moon's lack of an atmosphere, the ejection velocity is equivalent to the impact velocity of the ejected fragment. Using pi-group scaling, the size of the ejected fragment and its ejection (impact) velocity can be related to the diameter of the secondary crater it produces (Equation 4) (Holsapple & Housen, 2007; Vickery, 1986, 1987).

1.277 2 0.277 dfragment  1.003Dsec ( gv / ej ) (4)

By substituting Equation 3 into Equation 2, we get an expression for the diameter of the largest fragment that is ejected to a distance of r. Setting this equal to Equation 4 and again substituting for the relation

between ejection velocity and distance, we can solve for Dmax(r), the maximum secondary crater diameter as a function of distance from the primary (Equation 5). Next, the fragment power law constant, a, from Equation 2 must be normalized to set the upper limit on the largest secondary that can be produced by a

primary of diameter Dp. This normalization specifies that for any primary the largest secondary is some

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fixed fraction, γ, of the primary diameter, where γ is taken to be 0.04 (Bi- erhaus et al., 2018). This largest secondary is assumed to occur at a radial

distance of Rn = 5Dp / 2, though in reality the ballistic range of the largest secondary-producing fragment may be weakly crater size-dependent for large primaries (Allen, 1979; Xie et al., 2020). In our simplified model, the absolute largest secondary, which is 4% of the primary diameter, occurs just outside the continuous ejecta blanket.

As results from Singer et al. (2013) and Bart and Melosh (2010) suggest a scale-dependence for the fragment power law exponent, β (β increasing with primary diameter), we fit a linear-log trend to the exponents obtained for craters of various sizes from multiple studies (Figure 3). The fit is weighted by the uncertainty on each measurement so as to not be disproportionately affected by the relatively large errorbars on measure- Figure 3. Power law exponent of the size-velocity distribution for ments of the largest craters. There is some indication that β can vary with secondary-producing fragments measured for lunar craters over a wide radial distance or azimuth due to asymmetries in the ejecta distribution range of diameters (compiled from Vickery [1987], Bart and Melosh [2010], for oblique impacts (Krishna & Kumar, 2016), but we adopt the stand- and Singer et al. [2013]). The scale-dependence of the exponent is ard approach of studies like Vickery (1987), Bart and Melosh (2010), and determined by a linear-log fit to the data, β(Dp) = 0.754 log 10(Dp) + 0.777. A minimum value for β of 0.555 (corresponding to the fit becoming flat Singer et al. (2013) which fits a single power law to the distribution of below 800 m) is required to ensure that maximum secondary crater size fragments at all ballistic ranges. We use this single value of β as an av- decreases∼ with distance from the primary (see Equation 5). erage fragment size-velocity exponent for each primary crater. See Sup- porting Information S2 for further discussion of this model parameter. To ensure that the size of the largest secondary decreases with distance from the primary, we require that the fragment power law exponent, β, cannot be less than 0.555 (Equation 5).

0.554 Rrn 2.554 tan( ) 1 tan( ) DDpp Drmax ()  Dp  (5) rRn tan( ) 1 tan( ) DDpp

Equation 5 gives the diameter of the largest secondary at the distance of the grid. Because fragment size decreases with ballistic range, this largest grid secondary will be smaller than the absolute largest secondary (fixed at 4% of the primary diameter) at all distances beyond the primary crater's continuous ejecta blan-

ket. We make the assumption that at all distances from the primary, N (≥Dmax(r)) = 1 in an annular area Δr Δr between r  and r  . This means that for a primary that forms at some distance, r, from the grid, 2 2 there is one secondary of diameter Dmax in an annulus that contains the grid. This annulus is distinct from the off-grid primary sampling regions described in Section 3.2.1. Instead of being centered on the grid, the secondary-production annulus is centered on the off-grid primary and the radius of the annulus is the distance between the off-grid primary and the center of the grid. These two geometries (primary production vs. secondary production) are shown in Figure 2. The width of the secondary-production annulus, that is,

Δr, is set by the maximum distance from the grid center at which a crater of diameter Dmax could influence

the grid elevation. The ejecta blanket of a secondary with diameter Dmax that forms centered on either the inner or outer edge of the annulus would just overlap the grid (Figure 2). This single largest crater will rarely affect the grid since, for distant secondary-producing craters, the grid occupies a small fraction (typically 0.005%) of the annulus surface area. However, this constraint on the largest possible secondary allows us to∼ determine the cumulative secondary crater size-frequency distribution at the grid's distance from the primary, which for all on- and off-grid primaries is given as,

N() D cDb (6)

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b Dmax where c  and Dmax is the size of the largest secondary at the distance of the grid which is a func- Aannulus tion of primary size and ballistic range. Aannulus is the size of the annulus containing the grid which varies

with the distance between the primary and the grid. The power law coefficient, c, therefore depends on the size of the primary and its distance from the center of the model domain since larger, closer primaries will produce larger and more numerous secondaries on the model domain than smaller, more distant ones. The slope of the size-frequency distribution, b, for secondary crater populations across the solar system ranges from 3.3 to 8.0. (McEwen & Bierhaus, 2006; Robbins & Hynek, 2011). We choose a constant value of b = 4, the most typically reported secondary crater size-frequency distribution (CSFD) slope (Bierhaus et al., 2018; McEwen & Bierhaus, 2006; Strom et al., 2015), which is consistent with observations of secondaries around individual lunar craters like Tycho (b = 3.3–4, Dundas & McEwen, 2007; Hirata & Nakamura, 2006) and the 3.8 km crater Censorinus (b ≈ 4, Krishna & Kumar, 2016) as well as the global population of recently formed secondary splotches (b = 4.14, Speyerer et al., 2016). See Supporting Information S3 for further discussion about this choice of model parameter. After sampling the sizes and locations of all global impacts for a time step, we use Equation 5 to compute the size of the largest secondary crater that could potentially form on the grid from each impact. For those impacts where the maximum secondary at the distance of the grid is resolvable, that is, ≥2 grid pixels, we determine the number and size of all secondaries that overlap the model domain using the cumulative

secondary production function defined by Equation 6. We generate crater diameter bins from Dmin to Dmax

with 2 spacing and calculate the expected number of craters in each bin, where again Dmin is the smallest resolvable crater for a given grid resolution. The incremental number of craters produced per km2 is used in a Poisson sampling method that randomly determines the number of craters in each bin that actually form on the grid. Secondary-producing fragments are not explicitly tracked in the model but we assume the standard fragment launch angle of 45° is steep enough so that their flight paths are not obstructed by the preexisting grid topography. Each secondary is added to a random position on the grid with a volume-neutral profile similar to that used for primary craters. Both the rim height and depth of all model secondaries are reduced by a factor that varies between 0.5 and 1, depending on the distance from the primary at which the secondary forms. Secondaries that form at the edge of the continuous ejecta blanket of their primary have a rim height and depth that is half that of our primary crater morphometry (d/D = 0.1). Secondary rim height and depth increase linearly with ballistic range so that secondaries that form at the antipode of their primary have profiles identical to those of primaries. Fits to average depth-to-diameter ratios at three different ranges from the lunar basin Orientale indicate a potential power law relationship between morphology and ejection velocity (Pike & Wilhelms, 1978; Xie et al., 2020). However, this simple linear gradient in crater morphology reflects the general observation that nearby secondaries are significantly shallower than primaries and with increasing distance from the primary, secondaries become increasingly harder to distinguish from primaries on the basis of morphology (McEwen & Bierhaus, 2006; Melosh, 1989; Pike & Wilhelms, 1978; Robbins & Hynek, 2014; Xiao, 2016). On-grid primary craters that are large enough to produce secondaries (4% of their diameter is >2 pixels) are handled in an identical manner to that for off-grid primaries described in this section. Currently, we do not model self-secondaries, craters formed by ejecta blocks launched at near vertical trajectories that land within the primary crater's ejecta blanket (Shoemaker et al., 1969; Zanetti et al., 2017). We also omit secondary crater chains, clusters, rays, or antipodal concentrations which affect localized areas and are concentrated in the proximal ejecta regions. This approach allows us to include all distant secondaries from the global distribution of large impacts. Integrated over long timescales, distributed background secondaries should have a similar effect as a smaller number of spatially concentrated rays overlapping the grid. The crossover diameter where secondaries exceed primaries by number in our model occurs at 28 m (Fig- ure 4). On the lunar surface, secondaries can dominate at any scale in local areas due to the∼ stochastic nature of secondary production in both time and space. This crossover diameter represents the scale of secondary influence when averaged over billions of years and over many Monte Carlo model runs. At spatial scales smaller than this characteristic value, surface evolution is predominantly controlled by secondary crater formation. Subresolution craters (less than two pixels in diameter) are discussed in Section 3.5.4.

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3.3. Comparison of Model Flux to Lunar Production Functions Here we compare our model crater flux with established lunar production functions to test whether the population of generated craters in our model is consistent with observed size-frequency distributions on the maria. Figure 4 shows cumulative crater counts from 50 Monte Carlo cratering histories for both primary and secondary populations as well as the total number of produced craters. These counts are compared to the Neukum Production Function (NPF) (Neukum et al., 2001) normalized to the same time period as our model run, 3.5 Gyr, and computed for crater diameters relevant to our model grid. From Figure 4, we can see that the two production functions are in good agreement. We confirm the result of Marchi et al. (2009) who noted that their lunar impactor population, derived from a theoretical dynamical model and scaled using the Holsapple and Housen (2007) pi-group scaling method, produces a production function that is within a factor of two of the NPF at sizes ranging from 0.01 km, the minimum crater diameter for which the NPF is valid, up to hundreds of kilometers. However, the subtle differences Figure 4. Cumulative crater counts produced by our model by converting the Marchi et al. (2009) lunar impactor flux to final crater dimensions between our crater population and that of Neukum et al. (2001) warrant using the Holsapple and Housen (2007) scaling method with time- further discussion. dependent target strength and density profiles. Symbols and errorbars show the median and interquartile range (25th to 75th percentile) of The NPF is determined using observed craters whereas Figure 4 shows primary, secondary, and total (primary + secondary) craters from 50 Monte all craters produced in our model, some of which will not be visible, that Carlo cratering populations. Dashed line indicates the Neukum lunar is, countable, after 3.5 Gyr of surface evolution. Although the production production function normalized to an equivalent timescale. function may have been constant over the last few billion years, the observed crater size-frequency distribution varies in shape over time due to changes in target density and strength and enlargement of crater diameters due to diffusive degradation by impact gardening (Xie et al., 2019). Our model includes the effects of time-variable target properties with a growing regolith layer that changes the depth profiles of density and strength at each timestep. These time-dependent target properties will influence the final crater diameter for a fixed impactor size and velocity and therefore this effect is included in our produced crater counts shown in Figure 4. Even with the contribution of time-dependent target strength and density, our produced crater densities are lower than those predicted by the NPF by a factor of 2. ∼ There are two additional effects not represented in Figure 4 that would act to bring our production function and the NPF into even better agreement. First, diffusive degradation, or “sandblasting,” by small impacts acts to widen the apparent diameters of craters after their formation. Xie et al. (2017) quantified the effect of this process on crater populations and found that over time the observed crater size-frequency distribution becomes steeper as craters are pushed into larger diameter bins and small craters are enlarged faster than large ones. Sandblasting is included in our model as a spatially uniform diffusion process that widens and infills craters at every timestep. But this process is not reflected in the results shown in Figure 4 because our model does not presently contain a crater counting mechanism and so the observable CSFD, and the effect of crater enlargement by sandblasting on that distribution, cannot be directly measured. The second effect is related to the NPF and the possibility of secondary contamination. It is presumed that the counts used to produce the NPF consist solely of primary craters but due to the difficulty of identifying small, isolated secondary craters (Hirata & Nakamura, 2006; Robbins & Hynek, 2011, 2014; Xiao, 2016), it is likely that there is a nonzero contribution to the NPF from background secondaries. If it were possible to remove this small fraction of background secondaries, the predicted NPF crater densities would decrease. Again, this effect would be more pronounced for smaller craters, acting to make the NPF shallower toward smaller diameters. The combination of these two effects, steepening of our observed crater size-frequency distribution due to sandblasting by small impacts and decreased small crater densities from the secondary-free NPF likely would improve the agreement between the observed CSFD in our model and the NPF to better than a factor of 2. In future work, we will develop a crater counting mechanism to test the effect of these various processes on the observable CSFD in our model (e.g., Hirabayashi et al., 2017; Minton et al., 2015, 2019; Richardson, 2009).

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3.4. Topographic Diffusion Relief-reduction on airless bodies is driven by mechanisms like small impact gardening and seismic shaking that can collectively be modeled as topographic diffusion (Howard, 2007; Richardson, 2009; Soder- blom, 1970), where the quantity that is diffusing over time is the elevation of the grid as loose regolith is transported downslope. Larger impacts also degrade the landscape but we explicitly model all craters greater than 8 m in diameter. Therefore, in our model, diffusive degradation is the cumulative effect of craters smaller than 8 m (not just micrometeorite gardening) and seismic shaking. The diffusion equation (Equation 7) is applied to the grid at each time step to compute the elevation change at each pixel as a result of these processes transporting material down slopes. Following the methods of Pelletier (2008), we implemented a 2D alternating direction semi-implicit scheme for diffusion that is unconditionally stable for any time step or grid resolution.

z 22 zz   22 (7) t xy

The diffusivity, κ, is the only free parameter in Equation 7. This value controls how rapidly topographic features degrade and therefore sets the overall roughness or smoothness of the grid. We can calibrate the diffusivity by selecting a value that produces model landscapes with the same surface roughness as the lunar mare. The roughness metric we choose to calibrate our model diffusivity to is the median absolute slope. This roughness statistic is established in the literature (Shepard et al., 2001) and has successfully been used to characterize the roughness of the lunar surface (e.g., Rosenburg et al., 2011).

High-resolution topography data from the Lunar Reconnaissance Orbiter's Narrow Angle Camera (NAC) permit characterization of surface roughness on the lunar mare at length scales relevant to our model (Rob- inson et al., 2010). NAC digital terrain models (DTMs) for the Apollo 11, 12, 15, and 17 landing sites at 2 m/ pixel (Henriksen et al., 2017) are downsampled by a factor of two to the same resolution as our model surfaces. Median bidirectional slopes at 4 m baselines are computed for each DTM using elevation differentials in two orthogonal directions. We also compute the interquartile range (IQR), or the spread of the middle 50% of elevation values, in a moving 1 km × 1 km window. This measure of the topographic variation in each window highlights contacts between mare and highlands units where the DTM elevation changes substantially over a short distance. Since the soil samples obtained from these sites were taken from the mare units only, we contour the IQR maps to mask out highlands units (and mare units that were not sampled by the astronauts). Landing site DTMs and masked slope maps are shown in Appendix A. Crater counts from Hiesinger et al. (2011) place the formation ages of these units (Mare Tranquilitatis, Oceanus Procellarum, Mare Imbrium, and Mare Serenitatis) within 0.4 Gyr of each other. Since age is the primary control on surface roughness for surfaces outside cratering equilibrium, it is valid to run the model for a fixed time period equal to the average age of these units, 3.48 Gyr, and to calibrate the diffusivity using a single average slope value. The average slope across these four sites at a scale of 4 m/pixel is 3.12°.

The diffusivity parameter in our model is clearly correlated with the surface roughness of the final surface (Figure 5). When diffusivity is high, topographic features are quickly eroded and the final topography is muted with generally low slopes. When diffusivity is low, erosion from micrometeorites and seismic shaking is less efficient, and features such as crater rims remain sharp, leading to a relatively rugged final surface. The dependence of median absolute slope on diffusivity is determined by running our model with a range of diffusivities. At each diffusivity, we run the model 100 times to determine the average roughness of the model surfaces, including uncertainties. Figure 6 shows median absolute slope plotted against diffusivity for a model resolution of 4 m/pixel. We linearly interpolate between the resulting median slopes to determine the diffusivity that corresponds to the maria roughness at that scale. The diffusivity needed to match the observed roughness of the lunar maria, corresponding to a median slope of 3.12°, is 0.109 m2/Myr.

This best fit mare diffusivity differs considerably from the value of 5.5 m2/Myr found by Fassett and Thom- son (2014) for the same region of the Moon. There are a number of possible reasons for this discrepancy. First, Fassett and Thomson (2014) used the degradation state of many individual craters to derive the diffusivity needed to match each crater's topographic profile given an assumed age. Our model diffusivity is

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Figure 5. Hillshade of three model surfaces with diffusion rates spanning an order of magnitude less than and greater than our best fit value, κmare (illuminated from the upper left). The surface roughness associated with a particular diffusivity value can be quantified by the median absolute slope of many digital elevation maps run with that diffusion rate.

calibrated to match the overall degradation state of regions containing many craters and intercrater areas, essentially constituting an average landscape diffusion rate.

Second, the diameter range of craters contributing to diffusion in the present study is less than that of Fas- sett and Thomson (2014). The median crater diameter analyzed in that study is 800 m, meaning that their diffusivity is reliable for deriving diffusion ages for craters of approximately that∼ size. Xie et al. (2017) note that in such a 1D analytical model, diffusivity is a function of the largest crater contributing to degradation,

Dmax which is some fraction of the diameter of the crater being degraded. This scale-dependence in the degradation rate has been explored in the context of impact gardening as an anomalous diffusion process (Minton & Fassett, 2016). As the size of the crater increases, so does the range of crater sizes contributing to diffusive degradation and therefore so does diffusivity. This is in contrast to our model where diffusivity is fixed to the grid resolution. As mentioned at the beginning of this section, only craters smaller than two

pixels in diameter (8 m) are included in our diffusivity. Because Dmax for 800 m craters is likely to be greater than 8 m, our lower diffusivity is consistent with the results of Fassett and Thomson (2014).

3.5. Regolith Tracer Particles The goal of this study is to investigate the horizontal and vertical transport of regolith and to determine how long material spends within a millimeter of the surface where it is exposed to space weathering. To this end, we track a population of tracer particles which are influenced by the surface processes implemented in our landscape evolution model. These tracers are initially distributed throughout the subsurface where they are subsequently excavated and transported by the formation of impact craters. They can be transported downhill or buried by slope-dependent mass movement calculated using our calibrated lunar mare diffusivity. Particles are also vertically mixed by subresolution impact craters (<8 m) on timescales less than a single timestep. By tracking the positions of these tracer particles as the landscape evolves, we can measure their depths relative to the surface elevation and determine the amount of time each particle spends on the surface. The distribution of tracer particle surface residence times serves to characterize the exposure ages of regolith on the lunar mare.

Figure 6. Median roughness of model surfaces as a function of diffusivity for a resolution of 4 m/pixel. Errorbars show the interquartile range of roughness values over 100 runs at each diffusion rate. The horizontal 3.5.1. Impact Cratering: Maxwell Z-Model dashed line represents the observed roughness of the mare Apollo landing sites at the same baseline. The best fit model diffusivity is shown as a Although each crater is added instantaneously as a geometric eleva- vertical dashed line. Model runs at this value produces surfaces with, on tion profile, we can consider the effect of each crater's formation on the average, the same median slope (triangle and error bars).

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surrounding region and any regolith tracer particles in that area. Our treatment of the crater formation process reproduces the general impact transport behavior of target material using a set of rules and the equations derived in Maxwell (1977). The Maxwell Z-model describes how material is excavated and ejected during the formation of a crater as a flow field forms in the target. The flow field is defined by the parameters α and Z, where α is a measure of the strength of the field and Z determines the shape of the field and the trajectory of material as it is excavated from the target. A time-varying α is most realistic, as the velocity of material flowing away from the impact point should decay with time (Yamamoto et al., 2009). However, most applications of the Z-model, in- Figure 7. Schematic depicting how tracer particles are transported by the formation of a crater following the Maxwell Z-model (shown for a cluding the present study, assume a constant α for simplicity. Similarly, Z value of 3). Particles can be ejected and land downrange on or off the empirical cratering tests and simulations of crater formation are best fit model domain. Subsurface transport follows flowlines that give a particle's using a Z value of 2.7, but the subsequent equations are more tractable position throughout the crater excavation process. Any particles that end using a value of 3 (Melosh, 1989). up between the transient and final bowls participate in the slumping of material toward the crater center during the crater modification stage. Each time a crater is added to the grid, we identify all the particles that are within the crater's zone of influence, a hemispherical region with radius equal to the radius of the continuous ejecta blanket. Particles in this area are transported along flowlines that form during the crater formation process. The excavation cavity is delineated by flowlines that intersect the surface and eject particles from the transient crater cavity. For a value of Maxwell's Z parameter of Z = 3, all ejected particles are launched on ballistic trajectories at an angle of 45° and a vertical velocity that is dependent on distance from the impact point (Equation 8),

 v  z Z (8) R0

where α is a measure of the subsurface velocity flow field generated by the impact and R0 is the particle's distance from the point of impact when ejected. An expression for α can be obtained by setting Equation 9

(Equation 10 in Maxwell [1977]) equal to the transient crater radius, Rtc.

gR7   tc (9) 12

Given the surface gravity and the ejection velocity we can determine where on the postcrater grid each particle lands by computing where the particle's flight path intersects the surface, taking into account preexisting topographic features (rather than simply using the time-of-flight equation). Ejected particles that land within the continuous ejecta blanket are placed at a random vertical position within the ejecta blanket thickness at that point, taken to be 1/3 of the total exterior crater profile elevation. Outside the excavation cavity, particles are transported along flowlines until the transient crater reaches

its final radius at time tflow, at which point all particle movement ceases. Equation 10 (Equation 7 in Max- well [1977]) gives the radius of the expanding transient cavity as a function of time.

Zt1 Rtc ( Z  1)0  ( t ) dt (10)

Using a value of Z = 3 and our assumption of a constant α, we can substitute this expression for Rtc into

Equation 9 and solve for tflow in terms of α.

4/7 1 12 2 t flow   (11) 4 g

We can calculate a particle's 3D position at all points in time as it moves along a flowline (Figure 7). Equa- tions 12 and 13 give the geometry of these flowlines in polar coordinates, where R and θ are the flowlines's

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time-dependent radius and angle from nadir measured from the impact point (Equations 5 and 6 from Maxwell [1977]).

4 1/4 Rt() ( R0  4 t ) (12)

1 Rt() (t ) cos 1  (1 cos (0 )) (13) R0

If a particle is not ejected, its final position is found by applying Equations 12 and 13 at t = tflow. After the flowfield has ceased, particles between the transient and final crater bowls participate in the transient crater collapse. Infilling particles are assigned a random radial position between their postflow position and the crater center and placed at a random vertical position between the transient and final crater depths at that point. An example of the variety of particle trajectories induced by the formation of a crater can be seen in Figure 7.

One unphysical situation arises from our simple treatment of the crater formation process. Following empirical cratering tests, we require that ejected material must originate at a radial distance of less than the transient crater radius. Our use of a constant, rather than decaying, α can lead to particle flow lines inter- secting the surface during crater excavation beyond the transient crater radius. These erroneous particles are placed on the surface at their final position. This unphysical situation arises in a very small number of particles and the population of tracer particle trajectories is not substantially affected by the approximation of a time-independent α.

3.5.2. Diffusive Mass Transport In addition to being transported by the formation of a crater, tracer particles are also affected by the downslope movement of material driven by impact gardening. After elevation changes at each pixel are computed from the diffusion equation (Section 3.4), we check each particle to see if it has been “excavated” by being on a slope where there is a net outflux of material. If the particle is within the volume of material being removed from its current grid cell, the particle follows the movement of material and is transported downhill. The characteristic length scale for diffusion is given by

Lt 4Δ (14)

For a grid resolution of 4 m/pix and our best fit mare diffusivity of 0.109 m2/Myr, the diffusion length, L, is 0.66 m, that is, less than the width of a single grid cell. Therefore, particles moved by diffusion are shifted by only one cell in the steepest downhill direction. Transported particles are placed at a depth corresponding inversely to their initial depth, reflecting the process by which material near the surface will tend to slump downhill first and deeper material will subsequently slump down and bury the initially transported material. Particles in grid cells where a net influx of material flows into the grid cell (increase in elevation) remain at their initial positions and become buried by the infilling material.

3.5.3. Conservation of Tracer Particles Particles' vertical positions are unrestricted and during the model run can reach depths below that of the deepest initial particle depth due to the formation of multiple overlapping craters that are a substantial fraction of the grid width in diameter. Particles can also reach heights greater than the initial grid surface of z = 0 as a result of structural uplift of crater rims and ejecta deposition. Horizontal movement is also unrestricted and it is possible for particles to be transported outside the model domain. The model grid represents a small patch of the entire lunar surface and, as in regards to sampling impact craters (3.2), the grid boundaries are not periodic for tracer particles. Here we discuss how we handle particles that are transported horizontally outside the model domain through impact cratering or diffusive transport of material.

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Lateral transport by macroscopic physical processes moves regolith across large distances on airless bodies and for any region of the lunar surface, material is constantly being transported into and out of the area. Apollo soil samples revealed that substantial amounts of anorthositic highlands material were present at each landing site even at great distances from the nearest mare-highlands contact (Huang et al., 2017; Li & Mustard, 2000). Each of the processes in our landscape evolution model has the potential to transport particles outside the model domain. When particles are lost from the grid, they are replaced to keep the number of particles constant and to ensure that the population of particles sampled at the end of the run is not biased toward particles that did not experience extensive lateral transport. Because the grid boundaries are not periodic, particles are added to random positions on the grid to mimic the stochastic transport of material onto the model domain. The depth of each added particles reflects that of the particle it is replacing. For example, if a particle is ejected off the grid, a particle is added to a random (x,y) position on the surface of the grid. If a particle is transported through the subsurface by an impact and ends up outside the model domain, a particle is added to a random (x,y) position at the depth of the lost particle when the cratering flow field ceased. This allows us to evaluate how the depth of each particle changes over time in a consistent manner. The source and loss of material at any point on the Moon is not perfectly balanced since target material can be vaporized or escape the Moon during large impacts. This effect is unlikely to be important at the scale of our model since both impact vaporization and the fraction of material launched at escape velocity are negligible at the small crater sizes relevant to our model. In the unlikely case that a particle is on the surface at the impact point it is removed and replaced by another particle transported from outside the grid set randomly on the surface.

3.5.4. Subresolution Effects While macroscopic physical processes generate regolith and transport material over relatively large distances, micrometeorite bombardment churns the uppermost regolith and is a strong control on the amount of time material spends within a millimeter of the surface. Impact gardening occurs on spatial scales smaller than our model resolution and on timescales shorter than our model time step. We implement subresolution impact gardening by applying a 1D Monte Carlo mixing model to the output of our landscape evolution model. These processes operating within the pixel scale of the model do not alter the grid elevations but influence the depth history of tracer particles residing within each pixel.

The landscape evolution model stores tracer particles' average depths at each 1 Myr time step. As a result of impact gardening, particles at an average depth of greater than 1 mm can be excavated and spend some amount of time on the surface. Similarly, particles that start and end a 1 Myr period on the surface can be temporarily cycled to deeper depths, reducing their cumulative surface residence time. Therefore, we must consider the effects of subresolution mixing on each particle's depth at every time step.

Here we outline the mixing process for one tracer particle during a single 1 Myr time step. We begin by determining all subresolved craters that form during that period and could influence the particle's depth. Costello et al. (2018) studied impact-driven overturn using an analytical model and concluded that secondaries dominate mixing in the upper meter of the lunar regolith. For completeness, we use a flux that is a sum of primary and secondary craters. The lunar primary impact flux from Marchi et al. (2009) is converted to crater sizes using parameters appropriate for small scales. To obtain a secondary flux, we generate many stochastic 3.5 Gyr cratering histories using the methods described in Section 3.2 and fit a power law size-frequency distribution to the total population of secondary craters produced per km2 per year. These two fluxes are summed and extrapolated to relevant crater sizes. The largest crater considered in the mixing model has a final diameter of two pixels, the smallest crater resolved in the landscape evolution model. The smallest crater has a final diameter of 1 cm. Craters smaller than this induce elevation changes of less than a millimeter and approach the regime where impactors are effectively striking individual regolith grains. Since the crossover diameter at which secondaries, on average, exceed primaries by number in our model occurs at 28 m, secondaries dominate the subresolution effects represented by this Monte Carlo mixing model. At ∼the upper end, the flux of 8 m secondaries is roughly double that of 8 m primaries. At 1 cm, the ratio between primary and secondary craters climbs to 750. Based on these fluxes, it is reasonable to expect that mixing of the upper tens of centimeters of regolith∼ in our model will be controlled by secondary cratering.

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This expectation is supported by analytical overturn models (Costello et al., 2018) and observations of new lunar impact craters (Speyerer et al., 2016) which found rapid mixing timescales for the upper 2 cm of regolith on the order of 100 kyr, driven by secondary impact processes.

Across this size range, the depth-to-diameter ratio of the final crater profile is assigned to be a constant 0.1. Because of the steep slope of the cumulative secondary crater size-frequency distribution, most of the craters at this scale are secondaries, which tend to be shallower than the canonical primary crater profile. Furthermore, it is observed that even small primary craters on the Moon are shallower than their larger counterparts, with fresh crater depth appearing to roughly follow a power law dependence on diameter (Daubar et al., 2014; Stopar et al., 2017). To reflect these observations, crater depth and rim height for both the transient and final profiles are reduced by a factor of 1/2 from the profiles described in Section 3.2. For each crater size bin, we randomly sample the number of craters that form in a circular area where a crater of that size would induce a nonzero elevation change at the position of the particle (the center of the circle). Craters occur in random order and are assumed to occur evenly spaced in time such that the time between 1Myr 4 impacts is . The total number of craters is in all size bins, Ntotal is close to 10 meaning that the average Ntotal time between events in this diameter range is roughly 100 years.

Because of the large number of craters that form within each pixel during any 1 Myr period, it is not compu- tationally feasible to use the full three-dimensional Maxwell Z to model particle transport at this scale. We use a simplified one-dimensional treatment where the effect of each crater on the position of the particle is evaluated based on the particle's depth and radial distance from the impact. The particle is ejected if it is within the excavation cavity, a parabolic bowl with maximum depth

1 hDexc tr (15) 16

where Dtr is the apparent transient crater diameter (Maxwell, 1977). The particle's landing point is related

to its initial position, r1, relative to the radius of the excavation cavity, rexc, since material originating closer to the impact point is ejected with larger velocities (Maxwell, 1977; Richardson et al., 2007; Yamamoto et al., 2009). To simplify the complex relationship between launch position and velocity, we dictate that particles at the impact point land at the edge of the continuous ejecta blanket and particles at the outermost extent of the excavation cavity land on the crater rim. Because more than 90% of ejected material lands within a few crater radii, the volume of material transported beyond the continuous ejecta blanket at these scales is negligible (Melosh, 1989). In our model, the continuous ejecta blanket extends to 5 crater radii so the following expression satisfies our constraints on the transport of material ejected from small craters.

r1 rf54 RR cc  (16) rexc

The particle's final depth is randomly sampled within the ejecta blanket thickness at its final position, taken to be 1/3 of the total positive crater elevation at that radial distance.

Outside of the excavation cavity but within the crater's zone of influence, a hemisphere with radius 5Rc (the extent of the ejecta blanket), the particle is transported through the subsurface using a simplified version of the Maxwell Z-model in which particles move radially away from the impact point (Z = 2). As in the case of ejected particles, our goal is to produce a simple, arbitrary function that reproduces the expected impact transport behavior. In the subsurface, the distance traveled decreases with distance from the impact point such that particles just below the excavation cavity travel to the edge of the transient cavity and particles

along the hemisphere at a distance of 5Rc from the impact do not move at all. The following function (Equa- tion 17) is one way to reproduce this general behavior using a power law that smoothly transitions between two endpoints; material just outside the excavation cavity that must clear the edge of the transient cavity (by definition a parabolic bowl that is void of material when the cratering flow field ceases) and distant material that is not affected by the crater's formation.

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n n rr r rr r Δr  tr exc exc tr exc exc nn  rrrr1 ej (17) exc exc 11  rrej ej

where r1 is the initial radial distance of the particle from the impact point

and rexc, rtr, and rej are the radial distances from the impact point along a line passing through the particle's initial position to the edge of the ex-

cavation cavity, transient cavity, and a hemisphere with radius rej = 5Rc, respectively. The exponent, n, controls how rapidly the crater flow field decays with distance from the impact point. This exponent must be sufficiently large to prevent particles from becoming systematically and un- physically driven deeper into the subsurface by all impacts. By running the mixing model for a range of exponents and identifying the point at which the slope of the average particle depth versus time curve stops changing, we selected a value of n = 5.

Like the excavation cavity, the transient cavity is a parabolic bowl with maximum depth Figure 8. Depth of a particle over 1 Myr as it undergoes mixing from 3 small impacts. This “stock market curve” can be used to measure how long hDtr tr (18) the particle spent at a depth of less than 1 mm where it would be exposed 16 to space weathering (dashed line). If after subsurface flow the particle is between the transient cavity and the final crater bowl, it becomes part of the infilling material and is assigned a random radial position between its postflow position and the center of the final crater. The particle's new depth is selected randomly between the transient and final crater bowls at that point. Particles that do not become part of the infill remain at their final position and have the elevation change from the final crater added to their depth. If under the crater interior, material is removed from atop the particle and if under the crater exterior, material is added (ejecta blanket plus cataclastic dykes). These various regions and the behaviors of particles within them are similar to what is shown in Figure 7 with the exception that particles move on radial rather than curved flow trajectories.

With this framework for how small impacts influence tracer particle depth, we can evaluate the effect of all craters during the 1 Myr time step. Similar to the work of Arnold (1975), we generate, as in Figure 8, the “stock market curve” of particle depth versus time. Since this model is stochastic, each particle has a unique exposure history from which we can measure the cumulative amount of time spent at a depth of less than a millimeter, shown as a dashed line in Figure 8. Our simple 1D model demonstrates an active overturn layer a few centimeters thick. In this region, particles are continuously buried and excavated and the chance of randomly ending a jump at or very near the surface is relatively low.

4. Lunar Soil Maturity The rate of space weathering on airless bodies depends on a number of factors, including, but not limited to, surface composition, heliocentric distance, and the flux of impactors and cosmic rays. Holding these parameters fixed, as for lunar mare basalt over the past few billion years when the impact rate has been relatively constant, the maturity of any given soil grain depends primarily on how long that grain has spent within the upper millimeter of the regolith (Morris, 1978). Various metrics exist to quantify the degree of space weathering in a soil sample, including thickness of amorphous grain rims, agglutinate fraction, and solar flare track density (Berger & Keller, 2015; Keller & McKay, 1993, 1997; Pieters et al., 1993; Zhang & Keller, 2011). To investigate how various space weathering proxies change with increased exposure to the space environment, we compare the surface residence times of our synthetic regolith tracer particles to two generally available soil maturity measurements; laboratory measurements of the ferromagnetic reso-

nance metric, Is/FeO, and remote sensing observations of the optical maturity (OMAT) parameter (Lucey

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et al., 2000; Morris, 1976). Both metrics have been shown to accurately reflect the degree of space weathering exposure but are sensitive to

different aspects of the weathering process. Is/FeO measures only single-domain metallic iron particles less than a few tens of nanometers in diameter (Morris, 1978). OMAT senses optical properties that arise from a combination of fine-grained metal (nanophase iron), dark agglutinate glass, and grain size distribution across a range of spatial scales (Lucey et al., 2000). There is also a difference in the regolith depths probed by these two datasets. Infrared remote sensing measurements are sensitive to approximately the upper micron of the lunar surface whereas laboratory soil measurements yield an estimate of the bulk maturity at whatever depth the sample was taken from which for the Apollo samples is approximately the upper 10 cm (Huang et al., 2018). Figure 9. Histogram of Apollo soil maturities from Morris (1978). Is/ FeO values of mare samples range from 0.3 to 94 units. The immature/ Perhaps the most widely used soil maturity metric is I /FeO, which meas- submature boundary occurs at 30 units and the submature/mature s boundary at 60 units. ures the ratio of ferromagnetic resonance intensity due to fine-grained metal (nanophase iron) in a bulk soil sample to the concentration of FeO in the sample (Morris, 1976). This measurement quantifies how much of the available FeO in the sample has been reduced by energetic space weathering processes to form nanophase iron (Fe2+ Fe0). For the samples measured by Morris (1978), on average only ≈2% of the total available FeO had been→ reduced by exposure to the space environment, mean-

ing that the observed changes in Is/FeO due to space weathering arise from a relatively small volumetric

fraction of nanophase iron. There is no evidence to suggest that Is/FeO saturates with respect to exposure age on the lunar mare in contrast to other metrics like regolith grain rims formed by vapor-deposition and solar wind amorphization which reach a maximum, steady-state thickness after a few million years of sur-

face exposure (Keller et al., 2016). Is/FeO also has been shown to correlate with other metrics of relative exposure age such as concentration of noble gases implanted by the solar wind, indicating that it reliably

serves as a measure of relative exposure to the space environment (higher values of Is/FeO can be assumed to indicate longer surface residence times) (Morris, 1976). The correlation of maturity metrics can also be used to identify measurements that saturate with respect to other metrics. An index such as petrographic

agglutinate fraction, when plotted against measurements of Is/FeO, clearly levels off at a maximum value

as Is/FeO continues to increase (Morris, 1976). Once this saturation value has been reached, the metric no longer provides useful information on soil maturity since any additional exposure to the space environment

will produce no change in the measured agglutinate fraction. Is/FeO is suitable as a generally applicable soil maturity metric because it does not saturate with respect to surface exposure age or with respect to any

other established metrics of soil maturity. Additionally, Is/FeO has been measured in a single laboratory for a large number of lunar soil samples and compiled in Morris (1978). As defined in that work (following the

procedure of McKay et al. [1974]), samples with an Is/FeO value less than 30 units are considered immature,

those with values between 30 and 60 units are considered submature, and those with Is/FeO greater than 60 units are considered mature.

To isolate the effects of surface composition and variations in impact flux over time, we select from Morris' measurements the maturities for samples obtained from maria sites. Figure 9 shows a histogram of the Apollo 11, 12, 15, and 17 soil maturities. Samples from these locales are predominantly basalt with varying amounts of contamination from anorthositic highlands material (Huang et al., 2017; Li & Mustard, 2000). As expected, the most mature soils come not from this subset but from the relatively older highlands sam-

ples (Apollo 16 soil 65,701 has Is/FeO = 106 units). Nevertheless, the subset of maria samples spans the full range of maturities, allowing us to probe the entire cycle of space weathering. It has been estimated that

measurements of Is/FeO for any given sample can be in error by as much as 10 units (Lucey et al., 2006), a value we adopt as a characteristic uncertainty for this data set.

Remote sensing metrics for soil maturity are not subject to the limits of sample retrieval and therefore can probe space weathering over the entire lunar surface. Lucey et al. (2000) used Clementine reflectance measurements to derive an OMAT parameter that distinguishes between different terrain types, for example, ejecta from large craters and the surrounding region, based on relative exposure age. This study also

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measured OMAT in the laboratory for Apollo soil samples and showed

that OMAT correlates with Is/FeO with a correlation coefficient of ≈0.6, on par with other maturity metrics like petrographic agglutinate fraction. However, Lucey et al. (2000) note that the trend is nonlinear, with OMAT

saturating at Is/FeO values greater than about 50 units. It is impossible to completely mitigate the difference in space weathering rates between basalt and anorthisitic material as distal ejecta can transport material over long distances crossing mare/highlands contacts (Huang et al., 2017; Li & Mustard, 2000) and Apollo soil samples from mare landing sites were observed in some cases to have significant fractions of highlands material (Fischer & Pieters, 1995; Wood, 1970; Wood et al., 1970). To minimize any compositional dependence between these parameters, we focus on OMAT measurements from the same mare regions where the Apollo soil samples were obtained and assume that, on average, the ratio of mare to highland material in regolith sensed by OMAT is the same as in the returned samples from those locales.

The Kaguya orbiter's Multiband Imager (MI) imaged the lunar surface at five wavelengths allowing for an updated investigation of the Moon's OMAT (Lemelin et al., 2016; Ohtake et al., 2013). OMAT mosaics derived from MI images include improved photometric calibration and are available at substantially higher resolution than the Clementine maps used in the initial derivation of the OMAT parameter (59 m/pixel vs. 7.6 km/ pixel). We extract Kaguya OMAT maps centered on each of the Apollo

landing sites represented in our mare Is/FeO data set (Figure 10). These 34 × 34 pixel grids correspond to the size of our landscape evolution model domain at roughly 2 km on a side. The distribution of OMAT values at these sites is shown in Figure 10. The median OMAT at these four Figure 10. (top) OMAT maps of the Apollo 11, 12, 15, and 17 landing locations is 0.143 and the median absolute deviation of the distribution sites at 60 m/pixel (Lemelin et al., 2016). Note that higher OMAT values is 0.015. correspond to more immature surfaces. (bottom) Distribution of OMAT values. Dashed line indicates the median of the distribution. No set of soil maturity measurements is representative of the entire lunar regolith. Since most of the Apollo samples were obtained from a depth of less than 10 cm (Huang et al., 2018), the tracer particles in our landscape evolution model that are analogous to these samples are those that end the model run at a similar depth, that is, near enough to the surface that they could have been sampled by the Apollo astronauts after 3.5 Gyr of mare surface evolution. Remote sensing reflectance measurements like those used to obtain OMAT are sensitive to approximately the upper micron of the surface and so do not probe the same depth range as the returned samples. Therefore, it is not appropriate to compare OMAT soil maturities to the particle population that is representative of the upper 10 cm of regolith. Instead, when comparing to OMAT, we will only consider particles that, at the end of the model run, are directly on the surface (depth < 1 mm). We will compare the surface residence time of two populations of tracer particles in our model (“sampled” and “surface” particles) to these two soil maturity metrics to investigate how both ferromagnetic resonance intensity and optical reflectance properties vary with exposure time in the lunar regolith.

5. Results 5.1. Surface Residence times Figure 11. Distribution of surface residence times for regolith on the lunar mare, drawn from a sample of 300 Monte Carlo landscape evolution Figure 11 shows the combined histogram of tracer particle surface resi- model runs. These cumulative exposure ages are accumulated over 3.5 dence times from 300 landscape evolution model runs. For a typical run, Gyr of landscape evolution. Dashed line indicates the median of the ≈ distribution. 3%–5% of the initial particle distribution ends the 3.5 Gyr model run at

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a depth of less than 10 cm. In turn, only a few percent of these sampled particles end the model run at the surface, leaving ≈150,000 sampled particles and, of those, ≈6,000 surface particles. All sampled particles are passed through the 1D Monte Carlo mixing model described in Section 3.5.4 to evaluate the cumulative amount of time spent at a depth of less than 1 mm during the run. During a 3.5 Gyr model duration, the median surface residence time is 6.42 Myr and 98% of particles spend less than 20 Myr within a millimeter of the surface. This particle distribution reveals that while large-scale impact processes source material to the upper tens of centimeters of the regolith, small-scale impacts are a dominant control on surface residence time. Over the course of mare surface evolution, most regolith material has likely been exposed to the space environment for only a few million years. This short exposure timescale is consistent with the work of Costello et al. (2018), who found using an analytical mixing model that the upper 2–3 cm of the lunar regolith is thoroughly reworked on timescales of 80,000 years. The timescale and thickness of this active layer is consistent with typical 1 Myr particle mixing trajectories as seen in Figure 8 as well as the resulting surface residence time distribution which is heavily skewed toward low cumulative exposure ages. Recently formed lunar surface features like splotches and cold spots are observed to be reworked into the background over relatively short timescales, necessitating rapid gardening by small impactors or other processes (Bandfield et al., 2014; Speyerer et al., 2016). This result has implications for the rate of space weathering on the Moon. If over the last 3.5 Gyr, regolith has on average spent a short amount of time exposed to the space environment, the majority of soil maturation observed in the lunar regolith must have occurred over cumulative exposure timescales of less than 10 Myr. Because the secondary flux is a critical parameter in modeling the surface residence time of lunar regolith particles, we also considered the possibility that the secondary crater size-frequency distribution (SFD) cannot be extrapolated using a single power law but instead “rolls over” to a shallower slope at small sizes, as has been suggested (e.g., Guo et al., 2018; Werner et al., 2009). We tested a second version of the impact flux in our mixing model where the secondary crater size-frequency distribution is fit by a power law with slope b = 4 down to 8 m and slope b = 3 from 8 m down to 1 cm. In this “shallow” SFD case, there are fewer secondaries forming and gardening of the regolith is less vigorous. The median surface residence increased from 6.42 to 17.11 Myr as particles spend longer on the surface before being cycled back down to greater depths by nearby impacts (see Supporting Information S4). Observations of lunar secondary populations at scales relevant to our landscape evolution model ( 10 m–1 km) are consistent with a single slope power law (Dundas & McE- wen, 2007; Hirata & Nakamura, ∼2006; Krishna & Kumar, 2016). Below this scale, observations are unreliable and the characteristics of the secondary SFD at small sizes remain unknown. In the absence of observational constraints or theoretical predictions for the size-frequency distribution of small background secondaries, we assume that our adopted secondary slope of b = 4 (Bierhaus et al., 2018; McEwen & Bierhaus, 2006; Strom et al., 2015) continues to smaller sizes. Potential deviations from a constant power law slope at small diameters could influence the amount of time a typical regolith particle spends on the surface by a factor of 2–3. This uncertainty is on par with that of other parameters in our model, including the macroscopic impact rate.

5.2. Space Weathering Rates To quantify the rate of space weathering on the lunar maria, we map measurements of soil maturity (the amount of weathering) onto our distribution of modeled surface residence times (the timescale of weathering). The mapping method uses inverse transform sampling to connect two datasets via their cumulative distribution functions (cdf). The cdf of each surface residence time and soil maturity distribution is found by binning the data with a fixed number of bins spanning the minimum and maximum values. Prior to this step, all distributions are truncated at the 0.5 and 99.5 percentiles. The inversion sampling method is robust to outliers which may alter the shape of the cdf but for large sample sizes lie only in the far end of the tails of the distribution. Next, we assume that there is a one-to-one correspondence between soil maturity and surface residence time. This means, for example, that material observed to be in the 90th percentile of soil maturity is also in the 90th percentile of time spent on the surface. For a range of cumulative probabilities between 0 and 1, the associated data value from each cdf is accessed to generate pairs of corresponding surface residence times and soil maturities. Figure 12 demonstrates this method for a single cumulative probability of 0.5. Repeating this mapping for the full range of probabilities produces a representation of the space weathering rate for each soil maturity metric.

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Figure 12. An example of how inverse transform sampling is used to obtain pairs of parameter values from two cumulative distribution functions. Black lines indicate where a cumulative probability of 0.5 intersects the cdf for each particular parameter. This process is repeated for the entire cdf over the cumulative probability range [0,1] to build up a mapping of soil maturity versus surface residence time. Soil maturity datasets come from Morris (1978) (Is/FeO) and Ohtake et al. (2013) (OMAT). Cumulative distribution functions for surface residence time are drawn from 300 Monte Carlo runs of our landscape evolution model.

Figure 13 shows the derived relationship between soil maturity and cumulative surface residence time. These trends indicate that regolith grains weather rapidly when initially exposed on the surface and then mature at a slower rate as they remain on the surface or are repeatedly buried Morri and reexposed by im- 1.25 pacts. We find that the timescale for material to reach submaturity (Is/FeO = 30 units) is 1.810.68 Myr and 3.01 that to reach maturity (Is/FeO = 60 units) is 7.152.30 Myr of cumulative surface residence. In our test of a 3.28 shallow secondary size-frequency distribution in our mixing model, these timescales increase to 3.811.64 10.79 Myr and 19.307.04 Myr, scaling roughly with the increase in median surface residence time. However, the shape of the maturation curve remains quasilinear over most of the weathering process (See Supporting Information S5).

These short maturity timescales are broadly consistent with previous attempts to derive the space weathering rate on the Moon. Keller et al. (2016) found that vapor-deposited and solar wind-amorphized rims accumulate on lunar soil grains and reach a steady state in 1–10 Myr of surface exposure. The comprehensive magnetic resonance maturity data set used here does not saturate with respect to exposure age such that our derived weathering rates provide valuable information over the entire maturation process. Unlike the laboratory measurements of OMAT from Lucey et al. (2000), we find no evidence that OMAT saturates with increasing exposure time. Both maturation curves in Figure 13 turn over at roughly 5 Myr cumulative residence time but neither appears to reach a steady-state value over the

3.5 Gyr period our model explores. If Is/FeO or OMAT saturates with respect to cumulative surface residence time, it does not appear from the present study that such exposure timescales have been widely reached on the mare. The similarity between the median surface residence time in our model and the timescale to reach ferromagnetic resonance maturity is a consequence of the arbitrary maturity class demarcations presented in Morris (1978). These boundaries, shown in Figure 9 were chosen to roughly correspond to the petrographic agglutinate fraction maturity classes of McKay et al. (1974). In deriving the space weathering rate, we

Figure 13. Dependence of soil maturity on time spent within a millimeter map the median surface residence time onto the median Is/FeO value of the surface as measured by the ferromagnetic resonance metric, Is/FeO which, for the subset of mare soil measurements, is 58 units, near the (solid blue line) and the optical maturity parameter, OMAT (dashed orange mature class boundary at 60 units (Figure 12). line).

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Given how Is/FeO and OMAT vary with time spent on the surface, we can also consider how these two maturity metrics relate to each other (Fig- ure 14). As lunar regolith is exposed to the space environment, different complex chemical effects of space weathering like accumulation of nanophase iron, breakdown of crystal lattices, and changes in the grain size distribution proceed at different rates with time and depth. Our results

show that as these changes are occurring, the relationship between Is/ FeO and OMAT is nearly linear. Figure 14 shows the covariance of these two soil maturity metrics with shaded regions representing the maturity classes of Morris (1978) based on ferromagnetic resonance intensity values. The immature class corresponds to OMAT values greater than 0.16 and the submature class to OMAT between 0.16 and 0.14. Finally, mature mare soils are predicted to have OMAT values below 0.14. This is consistent with the OMAT of mature mare surfaces which have typical OMAT values of 0.13–0.15 (Hawke et al., 2004). We find that the exposure age 5.70 needed for material to reach OMAT maturity (OMAT = 0.14) is 8.115.20 Myr, slightly longer than for ferromagnetic resonance maturity. The larg- Figure 14. Covariance of OMAT and ferromagnetic resonance intensity. The curve is obtained by plotting the nth percentile OMAT value versus er uncertainty in this value is due to our formulation of the OMAT error, the nth percentile Is/FeO for n in the range [0,1] from the data shown in described in Section 4. There appears to be a jump in OMAT at early sur- Figure 13. Shaded regions denote the soil maturity stages of immature, face residence times, suggesting that when regolith is first exposed to the submature, and mature, respectively, as shown in Figure 9. space environment, there is a sharp decrease in OMAT without a com-

parable increase in Is/FeO. Beyond Is/FeO = 20, realized in a few Myr of surface exposure, the two metrics scale in a linear-like way. Aside from the potential difference in magnetic resonance versus optical maturation for material immediately following its initial exposure to the space environment, these two metrics are nearly linearly related over most of the mare soil maturation process.

It is important to note that regolith surface residence times are achieved over billions of years of surface evolution. Within the upper tens of centimeters, material that finds its way to the relatively thin surface layer will be rapidly cycled back into the subsurface by vigorous impact mixing, mainly driven by secondaries. In this environment, space weathering exposure is accumulated over many brief near-surface excursions separated by long periods of burial in the subsurface. We examined the connection between model age and cumulative surface residence time by investigating the tracer particle exposure time distribution at every time step. In Figure 15, we plot the average and spread of the distribution as a function of model time for particles that end up on the surface after 3.5 Gyr of model evolution. The exposure histories of these particles are most analogous to remote sensing observations for interpreting the optical properties of the present day lunar surface.

The curve in Figure 15 shows the absolute time at which a typical regolith particle achieves a given cumulative exposure age. This average surface residence time increases steadily over time in the mare regolith, reaching 7.5 Myr of cumulative exposure after 3.5 Gyr. The dashed line represents our∼ derived surface residence time needed to reach a mature OMAT value of 0.14. Although the average regolith particle on the lunar mare has not reached OMAT maturity at 3.5 Gyr of surface evolution, the spread in particle exposure ages demonstrates that even before the average particle has reached maturity, a fraction of particles at the surface will have undergone substantial space weathering and exhibit different optical properties than the background soil. Understanding both the rate of space weathering maturation and the fraction of regolith that has reached OMAT at a given time is important for interpreting young lunar surface features such as crater rays.

Crater rays on the Moon form from the excavation and deposition of ejec- Cumulative surface exposure age as a function of model time Figure 15. ta that can appear bright due to the presence of unweathered material for surface particles (final depth <1 mm). Solid line is the average particle surface residence time throughout the 3.5 Gyr model duration. Boxes show (“immaturity” rays) or from the emplacement of anorthositic highlands the interquartile range of the residence time distribution at intervals of 0.5 material onto a relatively darker basaltic mare surface (“compositional” Gyr and whiskers denote the 0.5 and 99.5 percentiles. Dashed lines indicate rays) (Hawke et al., 2004). Surrounding large craters, these conspicuous our derived timescales for material to reach submaturity and maturity.

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features can serve as a chronometer for the optical changes induced by space weathering as immature rays darken and fade into the surrounding mature landscape with time. The Copernican-Eratosthenian boundary at 1.1 Gyr (Wilhelms, 1987) is commonly considered to be the fading timescale for crater rays since younger craters can still have immature ray systems while older craters have only compositional rays that are unaffected by space weathering (Hawke et al., 2004). Hawke et al. (2004) found that immature rays of Messier crater in Mare Fecunditatis had OMAT values of 0.18–0.20 compared to 0.13–0.14 on nearby, mature mare background surfaces. Meanwhile, the bright rays of Lichtenberg crater, mapped as a Coperni- can-age impact structure (Wilhelms, 1987), were suggested to be fully mature on the basis that the OMAT value observed on the rays was 0.14–0.15, in line with mature mare surfaces in the region. In this case, the optically mature ejecta deposits appear bright solely due to the compositional difference between highlands-rich ejecta and the darker surrounding mare terrain.

Using our results shown in Figure 15, we can investigate the absolute time required for regolith tracer particles to weather from an immature OMAT value of 0.2 to a mature value of 0.14. Our results suggest that over the ray fading timescale of 1.1 Gyr, only a small fraction of surface soil material has been exposed to the space environment long enough to experience significant chemical changes such as darkening and spectral reddening. This dark material is then thoroughly reworked into the brighter ray material, changing the optical properties of the entire feature. After 1.1 Gyr of mare surface evolution, 0.2% of regolith particles have reached OMAT. Werner and Medvedev (2010) argued that the crater ray retention∼ age for medi- um-sized (5–50 km) craters on the mare could be as low as 650 Myr. For this lower estimate, the percentage of mature regolith decreases to <0.1%. In either case, the optical properties of the small fraction of mature regolith are likely to be substantially different than the bulk immature soil. This effect can be seen in the 750 nm reflectance associated with a fresh, immature mare crater which is roughly a factor of 2 greater than the mature, background soil surrounding it (≈0.12 vs. ≈0.06, from Figure 2 in Pieters and Noble [2016]). For crater rays, the presence of minor amounts of mature material intimately mixed into immature regolith by small impacts may be sufficient to cause the entire surface of the ray to appear dark so that it becomes indistinguishable from the background landscape after 1 Gyr of surface evolution. Alternatively, if such small amounts of mature regolith cannot darken the entire∼ ray then these results may indicate another mechanism is needed, for example, small impacts may mix in mature regolith from adjacent to or below the ray.

6. Conclusions We present a general purpose Monte Carlo landscape evolution model for airless bodies calibrated to match the topographic statistics of the lunar mare. Using this model, we estimate the distribution of regolith surface residence times in this region by incorporating horizontal transport and vertical mixing of tracer particles by impact processes. Over the 3.5 Gyr history of a typical mare unit, material typically spends little time within a millimeter of the surface where it is exposed to the bulk of energetic space weathering processes. For particles that end up in the top 10 cm of regolith, the median surface residence time is determined to be just 6.42 Myr in 3.5 Gyr of surface evolution. This tendency toward small surface residence times in the lunar regolith is a result of thorough gardening by small impacts dominated by secondary crater formation which has been inferred from both analytical overturn models (Costello et al., 2018) and observations of the rate at which young lunar surface features disappear into the background landscape (Bandfield et al., 2014; Speyerer et al., 2016). Our modeled regolith surface residence times are compared to empirical metrics of lunar soil maturity to provide a key constraint on the timescales over which space weathering products accumulate.

The rates of maturation for mare regolith inferred from both laboratory measurements of ferromagnetic

resonance intensity, Is/FeO (Morris, 1978) and the OMAT parameter, OMAT, derived from Kaguya Multib- and Imager reflectance data (Lucey et al., 2000; Ohtake et al., 2013) indicate that material weathers rapidly when first exposed to the space environment and accumulates space weathering products more slowly as it remains on the surface or is continually buried and re-exposed by small impacts. We find that the timescale over which mare soils reach maturity is approximately 7–8 Myr of cumulative surface residence time as characterized by both ferromagnetic resonance and OMAT metrics. Uncertainty in the overall lunar cratering rate, particularly for small secondaries, can impart a factor of 2–3 variation in the median surface residence and consequently the timescale to reach space weathering maturity but does not affect the qual-

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itative conclusion that soil maturity increases quasilinearly over most of the maturation process. We also demonstrate how surface exposure accumulates over billions of years of surface evolution. At previously estimated crater ray retention ages, small amounts of regolith have reached advanced stages of maturity and, if intimately mixed with immature ray material, could be sufficient to alter the optical properties of the entire feature, making it indistinguishable from the background mare surface. Maturity timescales on the order of a few Myr are in agreement with detailed studies of lunar soil grains that found certain maturity metrics to reach a steady state over similar timescales (Keller & Berger, 2017; Keller et al., 2016). By using soil maturity metrics that do not saturate with respect to exposure age, the space weathering rates derived here are applicable across the entire lifespan of the lunar maria. Unlike previous studies (e.g., Lucey et al.,2000) we do not find evidence that the OMAT parameter saturates with respect to the commonly used

Is/FeO metric. While saturation may occur on the ancient lunar highlands terrain, both measurements can be used to characterize the space weathering rate on the relatively young mare.

The method described in this work can be extended to other regions of the Moon and to other airless bodies. Next steps for our general landscape evolution model include a treatment of the Late Heavy Bombardment impact flux and updated material parameters to derive the rate of space weathering in the older, compo- sitionally distinct lunar highlands. Topography and OMAT maps exist for Mercury's northern hemisphere following the MESSENGER mission (Becker et al., 2016; Blewett et al., 2014). The European Space Agency's ongoing Bepicolombo mission will significantly improve our understanding of the roughness, composition, and maturity of Mercury's surface (Benkhoff et al., 2010). Lastly, the recently concluded Dawn mission af- fords the opportunity to estimate surface residence times for regolith on the asteroids Vesta and Ceres (Piet- ers et al., 2012). This work provides a basis for unraveling the complex interplay between the processes and products of chemical space weathering and the physical impact processes that control the exposure of planetary regolith to the space environment. In future work, we will adapt our landscape evolution to explore regolith surface exposure on these other airless bodies. More work is needed to unravel the chemical com- plexities of space weathering for objects of different composition and in different energetic space environ- ments, without which comparison of surface weathering rates across the solar system remains challenging.

Appendix A

Figure A1. (left) Colored hillshade of the Apollo 11 landing site. (right) Bidirectional slope map. The median slope at this site is 2.51°.

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Figure A2. (left) Colored hillshade of the Apollo 12 landing site. (right) Bidirectional slope map. The median slope at this site is 2.33°.

Figure A3. (left) Colored hillshade of the Apollo 15 landing site. (right) Masked bidirectional slope map of mare unit. The median slope at this site is 3.85°.

Figure A4. (left) Colored hillshade of the Apollo 17 landing site. (right) Masked bidirectional slope map of mare unit. The median slope at this site is 3.82°.

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Data Availability Statement The model presented in this work can be found in O'Brien and Byrne (2020a). Data sets for this research are available in O'Brien and Byrne (2020b).

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