Catastrophic rupture of lunar rocks: Implications for lunar rock size–frequency distributions
Introduction
Blocks (1˗100 m) and smaller boulders (0.1˗1 m) (Bruno and Ruban, 2017) are ubiquitous on planetary surfaces as a result of impact cratering. The common approach to study these features is the measurement of their size–frequency distribution (SFD). Impact ejecta block SFD have been measured extensively on the Moon since the Surveyor era (e.g., Shoemaker and Morris, 1970), using orbital imaging data (e.g., Cintala and McBride, 1995; Bart and Melosh, 2010; De Rosa et al., 2012; Krishna and Kumar, 2016; Pajola et al., 2019; Watkins et al., 2019) and up to recent time by Chinese lander missions (e.g., Di et al., 2016; Li et al., 2017; Li et al., 2018; and Li and Wu, 2018; Wu et al., 2018; Wu et al., 2021). SFD measurement of impact ejecta blocks have been performed on many airless bodies, such as Ida (Lee et al., 1996), Phobos (Thomas et al., 2000), Eros (e.g., Thomas et al., 2001; Dombard et al., 2010; Michikami and Hagermann, 2021), Itokawa (e.g., Saito et al., 2006; Michikami et al., 2008; Mazrouei et al., 2014; DeSouza et al., 2015), Lutetia (Küppers et al., 2011), Toutatis (Jiang et al., 2015a), Vesta (Schröder et al., 2021a), Ceres (Schulzeck et al., 2018; Schröder et al., 2021b), Ryugu (Michikami et al., 2019; Michikami and Hagermann, 2021; Sugimoto et al., 2021), Bennu (DellaGiustina et al., 2019; Burke et al., 2021), comet 67P (e.g., Pajola et al., 2015, Pajola et al., 2016), and Enceladus (Pajola et al., 2021). Power–laws are often fitted to these measurements and cumulative power–index steeper than −2 are measured, consistent with highly fragmented material (e.g., Hartmann, 1969; Dohnanyi, 1971; Michel et al., 2001; Jutzi et al., 2010). Extrapolation of the SFD measurement to small sizes have been attempted using different mathematical description of the SFD shapes, i.e., power–law (e.g., Shoemaker and Morris, 1970; Li et al., 2017; Bandfield et al., 2011; Watkins et al., 2019; Krishna and Kumar, 2016), exponential (Shoemaker and Morris, 1970; Golombek and Rapp, 1997; Di et al., 2016; Li and Wu, 2018), and with a Weibull distribution (e.g., Schröder et al., 2021a, Schröder et al., 2021b; Pajola et al., 2016). These extrapolations have been performed to compare and validate SFD measurement with thermal observations sensitive to cm and m scale blocks (e.g., Bandfield et al., 2011), and for landing site hazard analysis (e.g., Golombek and Rapp, 1997; Golombek et al., 2003; Golombek et al., 2008; Wu et al., 2018; Ruesch et al., 2021). Additionally, the relationship between boulder size and source crater has been investigated (Bart and Melosh, 2007; Jia et al., 2019).
Despite abundant SFD measurements, the understanding of sources, evolution, and relationships to surface age of these distributions is vague. Attempts to relate a block population to its exposure age have considered the population as a whole without distinction of its size distribution (Basilevsky et al., 2013; Ghent et al., 2014; Basilevsky et al., 2015; Li et al., 2018; Watkins et al., 2019; Wei et al., 2020; Ruesch et al., 2020; Bickel et al., 2020). Among these studies, the work of Basilevsky et al. (2013) and Ghent et al. (2014) have revealed how the measured destruction rate, for all size combined, is found to be higher than predicted theoretically in the study of Hörz et al. (1975). This poor understanding is in contrast with the relatively well–known erosive processes at the lunar surface, namely shattering (e.g., Hörz et al., 1975; Hörz, 1977; Cintala and Hörz, 1992; Hörz et al., 2020; Ruesch et al., 2020) and abrasion (e.g., Shoemaker et al., 1970; Gault et al., 1972; Hörz et al., 1974; Cintala and Hörz, 1992; Rüsch and Wöhler, 2022) by impact bombardment. Thus, a natural question arises: How does the SFD of a block population on the lunar surface changes with time, in particular at small sizes? This study addresses this question by demonstrating that the evolution of block SFD can be modeled with sufficient precision (in terms of size distribution and absolute time) to allow meaningful comparison with measured block abundances.
Section snippets
Overview
The model is based on an improvement of the Monte Carlo study of Hörz et al. (1975) and exploits the advances over the last 46 years in the understanding of impact shattering and meteoroid flux. Briefly, the model of Hörz et al. (1975) simulates a surface composed of isolated blocks of the same size, formed all at the same time, and subject to bombardment by meteoroids. It tracks the energy imparted to each block by multiple meteoroids and calculate when a block accumulates sufficient energy
Overall model SFD shapes
The shattering energy functions and the projectile SFDs strongly influence the shape of the block SFD and its changes with time (Fig. 4). As expected, the shattering function of BA99 (Fig. 4a, b, c) requires more energy for shattering than the function of HH99 (Fig. 4d, e, f) and so it relatively hampers the decrease of block abundance with time. The rather small differences in the projectile SFDs (Fig. 3) have great influence in the block SFD shape. The steeper projectile SFD leads to enhanced
Survival times of blocks
The plot presenting the survival times (Fig. 9) is the same as Fig. 11 of Hörz et al. (1975), including now larger diameters. The survival times for targets of about 3 cm in size are the same for both old and updated model. For diameters larger than ~5 cm the destruction rates are ~2 times higher than in the original model, i.e., survival time are shorter. This difference is however within the range of uncertainty on the crater production rates recognized in the original model. The crater
Conclusions
The model of block catastrophic rupture of Hörz et al. (1975) is revisited in light of new understanding of the functions describing the energy necessary for block shattering and improved estimates of the flux and size–frequency distributions (SFDs) of meteoroids hitting lunar blocks. The input functions that best reproduce the number and size–frequency distribution of blocks on the lunar surface are identified. With such functions, the modeled block SFD well reproduces measurements of block
Data and code availability
LROC/NAC image data are available at http://wms.lroc.asu.edu/lroc/search.
The code for the updated model is available upon request at Ottaviano Rüsch.
Declaration of Competing Interest
None.
Acknowledgment
OR, RMM, and MP are supported by a Sofja Kovalevskaja award of the Alexander von Humboldt foundation. The constructive comments by two anonymous referees are acknowledged. The authors are grateful to Friedrich Hörz for a discussion of this study. This article is dedicated to A. Elbakyan.
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