Icarus 329 (2019) 207–221

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Icarus

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Small body shapes and spins reveal a prevailing state of maximum topographic stability T ⁎ James E. Richardsona, , Kevin J. Gravesb, Alan W. Harrisc, Timothy J. Bowlingd a Planetary Science Institute, 1106 Chalfant St, South Bend, IN 46617, USA b Department of Earth, Atmospheric, & Planetary Sciences, Purdue University, 550 Stadium Mall Drive, West Lafayette, IN 47907, USA c More Data!, 4603 Orange Knoll Ave, La Canada, CA 91011, USA d Southwest Research Institute, 1050 Walnut Street, Suite 300, Boulder, CO 80302, USA

ARTICLE INFO ABSTRACT

Keywords: Over the past thirty years, spacecraft missions, Earth-based radar experiments, and telescopic observations have Asteroids revolutionized our knowledge of Main Belt asteroids, near-Earth asteroids, and comet nuclei. As a result of this Rotation effort, we now possess high resolution shape models and well-constrained spin rates, pole orientations, and basic Asteroids surface properties for a few dozen such small bodies. Here we present the results of a geomorphological ex- Surfaces amination of 32 such small body shape models along with their associated spin properties, and show that small Comets body shape, gravity, and spin combine to gradually drive the surface towards a condition of maximum topo- Nucleus Mars graphic stability; that is, a state of low topographic variation, low slopes, and low surface erosion (mass-wasting) Satellites rates. Of the 32 bodies investigated, 15 (47%) reside within this ‘zone of maximum topographic stability’, and when asteroid lightcurve-derived rotation rate and body elongation estimates are included, 1941 (70%) of 2791 well-observed asteroids reside within this zone. This finding indicates that given a mobile surface layer and sufficient time, small body surfaces naturally tend to erode, and their spin states gradually evolve, towards a state of maximum topographic stability, which also corresponds to a state of lowest internal stress. This erosional effect is most prominent on bodies several kilometers and larger in diameter, where YORP induced spin-state changes are small.

1. Introduction looking for common properties and trends with respect to surface properties and processes. Eight shape models derived from spacecraft For many years, the only small bodies for which we had detailed optical data were used, consisting of four asteroids, two comet nuclei, shapes and images were the two Martian moons, Phobos and Deimos, and the two Martian moons (Thomas, 2014). Twenty-four shape models obtained during the Viking missions of the late 1970s (Thomas, 1979). derived from Arecibo and Goldstone planetary radar data were used, Then, in 1991 and 1994, the Galileo spacecraft flew by Main Belt as- consisting of two Main Belt and twenty-two near-Earth asteroids, with teroids 951 Gaspra (Belton et al., 1992) and 243 Ida (Belton et al., two options for asteroid 1950 DA (Benner, 2013). These shape models, 1996), giving us our first close-up views of an asteroid's surface and listed in Table 1, were selected based upon the degree of surface cov- shape. At about the same time, Hudson and Ostro (1994) pioneered erage obtained (> 65% preferred); the shape model resolution techniques for using Earth-based radar systems to obtain detailed achieved (> 1000 polyhedron facets preferred); and the degree to shapes and physical characteristics from near-Earth asteroids, begin- which the body has been characterized in the scientific literature (see ning with 4769 Castalia. These two methods – spacecraft and planetary the reference list in the far-right column of Table 1). Each small body radar observations – have since produced a steady stream (about 1–2 was then analyzed using the surface gravitational properties approach new objects per year) of highly-resolved asteroid surface, shape, and originally presented in Richardson and Bowling (2014), which also spin characterizations. contains a detailed Section 2.1 Background, describing the history and The purpose of this geomorphological study was to investigate the literature leading up to this study. shape, gravity, and spin of 32 publicly available small body datasets,

⁎ Corresponding author. E-mail addresses: [email protected] (J.E. Richardson), [email protected] (K.J. Graves), [email protected] (A.W. Harris), [email protected] (T.J. Bowling). https://doi.org/10.1016/j.icarus.2019.03.027 Received 8 October 2018; Received in revised form 17 January 2019; Accepted 18 March 2019 Available online 07 April 2019 0019-1035/ © 2019 Elsevier Inc. All rights reserved. ..Rcado,e al. et Richardson, J.E.

Table 1 Selected small body shape models.

−3 −3 No. Name a (km) b (km) c (km) Period (h) c/a ωsc Class ρmeasured (kg m) Error ρoptimum (kg m ) Range ρassumed (kg m ) Group Polygons Source

4179 Toutatis 4.26 2.03 1.70 128.8080 0.40 0.0195 Sk –– – – 2300 A 39,996 Ostro et al. (2001) 4486 Mithra 2.35 1.65 1.44 67.5000 0.61 0.0372 S –– – – 2300 A 5996 Brozovic et al. (2010) M2 Deimos 7.8 6.0 5.1 30.3120 0.65 0.1037 C 1471 ± 166 ––1471 A 5040 Thomas (1993) 9P Tempel 1 6.3 5.9 5.2 40.7000 0.83 0.1481 – 400 ± 200 ––400 A 32,040 Thomas et al. (2007) 2100 Ra-Shalom 2.3 2.3 2.0 19.7970 0.87 0.1572 C –– – – 1500 A 2292 Shepard et al. (2000) 4660 Nereus 0.51 0.33 0.24 15.1000 0.47 0.1664 Xe –– – – 2300 A 2292 Brozovic et al. (2009) 2063 Bacchus 1.11 0.53 0.50 15.0000 0.45 0.1676 Sq –– – – 2300 A 508 Benner et al. (1999) 52760 1998 ML14 1.1 1.1 1.0 14.9800 0.91 0.1678 S ––600 200–10,000 2300 B 1092 Ostro et al. (2001) 25143 Itokawa 0.54 0.29 0.21 12.1320 0.39 0.2250 QS 1950 ± 140 ––1950 A 196,608 Abe et al. (2006) – 2002 CE26 Alpha 3.5 3.3 3.1 15.6000 0.89 0.2590 C 890 ± 290 500 300–1500 890 B 2292 Shepard et al. (2006) 8567 1996 HW1 3.8 1.6 1.5 8.7624 0.39 0.2868 Q –– – – 2300 A 2780 Magri et al. (2011) 216 Kleopatra 217 94 81 5.3850 0.37 0.3425 M 4270 ± 860 2900 1900–5500 4270 B 4092 Ostro et al. (2000) 10115 1992 SK 1.39 0.90 0.91 7.3182 0.65 0.3434 S ––960 580–2310 2300 A 1016 Busch et al. (2006)

208 951 Gaspra 18.2 10.5 8.9 7.0420 0.49 0.3569 S –– – – 2300 A 32,040 Thomas et al. (1994) M1 Phobos 26.8 22.4 18.4 7.6530 0.69 0.3636 D 1876 ± 10 2200 710–3600 1876 B 32,040 Willner et al. (2010) 16 Psyche 279 232 186 4.1959 0.67 0.4282 M 4500 ± 1400 3200 2200–5500 4500 B 2292 Shepard et al. (2017) 433 Eros 34.4 11.2 11.2 5.2700 0.33 0.4426 S 2670 ± 30 2200 1400–4000 2670 B 129,600 Thomas et al. (2002) 33342 1998 WT24 0.47 0.43 0.40 3.6970 0.85 0.4611 E ––5000 3700–7900 5000 B 7996 Busch et al. (2008) 103P Hartley 2 2.323 0.745 0.723 18.3400 0.31 0.4647 –– – 200 140–350 200 B 20,584 Thomas et al. (2013) 1627 Ivar 15.38 7.28 7.10 4.7952 0.46 0.5027 S ––2450 1700–4100 2500 B 9996 Crowell et al. (2017) 243 Ida 53.6 24.0 15.2 4.6300 0.28 0.5106 S 2600 ± 500 2300 1500–4800 2600 B 32,040 Thomas et al. (1996) 1620 Geographos 5.11 2.10 1.85 5.2233 0.36 0.5160 S ––2000 1300–4000 2000 B 4092 Hudson and Ostro (1999) 1580 Betulia 6.59 5.85 4.19 6.1384 0.64 0.5921 C ––1100 700–2600 1100 B 2292 Magri et al. (2007) 4769 Castalia 1.6 1.0 0.7 4.0700 0.44 0.5923 S ––2500 1600–5100 2500 B 4092 Hudson and Ostro (1994) 6489 Golevka 0.69 0.57 0.49 6.0289 0.71 0.6028 Q ––1100 600–2800 1100 B 4092 Hudson et al. (2000) 101955 Bennu 0.57 0.54 0.51 4.2975 0.90 0.7902 C 1260 70 ––1260 C 2692 Nolan et al. (2013) 341843 2008 EV5 0.42 0.41 0.39 3.7250 0.93 0.8355 C –– – – 1500 C 3996 Busch et al. (2011) 66391 1999 KW4 Alpha 1.53 1.50 1.35 2.7645 0.88 0.9823 S 1970 ± 240 ––1970 C 9168 Ostro et al. (2006) 29075 1950 DA (Pro) 1.28 1.24 1.19 2.1216 0.93 1.0373 EM –– – – 3000 C 2036 Busch et al. (2007) 29075 1950 DA (Retro) 1.60 1.45 1.20 2.1216 0.75 1.0373 EM –– – – 3000 C 1016 Busch et al. (2007) 136617 1994 CC Alpha 0.69 0.67 0.64 2.3886 0.93 1.0833 Sk 2170 ± 610 ––2170 C 3996 Brozović et al. (2011) 54509 YORP (2000 PH5) 0.15 0.13 0.09 0.2029 0.60 10.8461 –– – – – 3000 C 572 Taylor et al. (2007) – 1998 KY26 0.03 0.03 0.03 0.1784 1.00 12.3356 –– – – – 3000 C 4092 Ostro et al. (1999) Icarus 329(2019)207–221 J.E. Richardson, et al. Icarus 329 (2019) 207–221

2. Spin state and topography axis, generally near the equatorial, extreme ends of the body, will mark areas of topographic high (red-pink). As such, loose regolith material on 2.1. Small body spin states the surface of the body will tend to flow downslope from the areas of high topography far from the minor axis towards the areas of low to- While the surface of an asteroid may look rather like the lunar pography close to the minor axis (Guibout and Scheeres, 2003; surface (Veverka et al., 2001), its small size will have a severe effect on Korycansky and Asphaug, 2004). This will tend to shorten the body and the physics of surface processes thereupon. In particular, the magnitude thereby increase its spin rate (conserving angular momentum). and direction of what we tend to think of as ‘gravity’ (actually a com- When the rotation rate ω of the body is ‘fast’, as shown in the right bination of gravitational and rotational forces) can be quite unintuitive. panel of Fig. 1, the situation flips, such that now surface areas near the Because the gravitational force felt on the surface of an asteroid is principal rotation axis, generally near the poles of the body, will mark generally < 1/1000th that felt on the Earth's surface, the rotational (or areas of topographic high (red-pink), while the areas farthest from the centrifugal) force produced by the body's rotation – as experienced principal rotation axis, generally near the equatorial, extreme ends of inside of the non-inertial, rotating body-frame of the asteroid's surface – the body, will mark areas of topographic low (cyan-blue). As such, loose will also strongly affect what direction is ‘up’ at a given location, for the regolith material on the surface of the body will tend to flow downslope determination of slopes; and what direction is ‘uphill’, for the de- from the areas of high topography near the minor axis towards the termination of topography (see Richardson and Bowling, 2014 Section areas of low topography far from to the minor axis (Guibout and 2.2 Theory for a more detailed discussion). Topography on a small body Scheeres, 2003; Korycansky and Asphaug, 2004). This will tend to is typically quantified by computing the difference between the po- lengthen the body and thereby decrease its spin rate (conserving an- tential energy at a given location and the mean potential energy of the gular momentum). entire surface, and then converting this energy difference to a ‘dynamic When the rotation rate ω of the body is ‘optimum’ as shown in the elevation’ hdyn by dividing by the local gravity, a concept initially in- center panel of Fig. 1, gravitational and rotational forces are roughly troduced by Thomas (1993): balanced, such that topographic differences, surface slopes, and erosion (mass-wasting) rates are all minimized across the surface of the body. As UUlocal− mean ff hdyn = , such, di erences in topography are dominated by the geomorphology glocal (1) (structure) of the body's surface, producing a much more intuitive sense of up, down, uphill, and downhill with respect to a given location. The where Umean is the mean [gravitational + rotational] potential over the actual rotation period of 433 Eros is slightly ‘slow’ (at T = 5.27 h), surface of the body, Ulocal is the [gravitational + rotational] potential at producing some higher elevations at the body's elongated ends, as fi a speci c location (generally the center of a shape model polyhedron shown in the upper row of Supplementary Fig. 14, but the body still facet), and glocal is the [gravitational + rotational] acceleration at that meets the criteria for being in an ‘optimum’ rotation state (defined in fi speci c location. The importance of these combined [gravita- the following Section 3). tional + rotational] forces on asteroid topography is illustrated in Fig. 1, where we show the surface topography that results from both increasing and decreasing the rotation rate of asteroid 433 Eros relative 2.2. Topographic correction to its measured bulk density of ρ = 2670 ± 30 kg/m3 and its actual rotation period of T = 5.27 h (Thomas et al., 2002), where the rotation The erosional and spin change behavior described above and illu- rate ω =2π/T. strated in Fig. 1 thus forms a self-correcting (negative-feedback) system, When the rotation rate ω of the body is ‘slow’, as shown in the left whereby a change in either the body's spin rate or its physical structure panel of Fig. 1, surface areas near the principal rotation (minor) axis, will cause a corresponding change in the body's topography, surface generally near the poles of the body, will mark areas of topographic low slopes, and corresponding erosion (mass-wasting) rates, which will then (cyan-blue), while the surface areas farthest from the principal rotation exist until the body has once again settled into an ‘optimum’ rotation

Fig. 1. (left panel) The northern hemisphere of 433 Eros color contoured with respect to dynamic topography (Eq. (1)) and shown for a ‘slow’ rotation period of T = 6.7 h. Topography is colored using a rainbow scale, with highs in red-pink, and lows in cyan-blue. Surface erosion will generally move material from the topographically high elongated ends of the body towards its low mid-section and poles, thus causing its rotation rate to gradually increase. (center panel) The surface of 433 Eros color contoured with respect to dynamic topography, and shown for an ‘optimum’ rotation period of T = 4.7 h. Topography tends to follow variations in surface structure (ridges, crater rims, basins, plains), with slopes and erosion (mass-wasting) rates at a minimum. (right panel) The surface of 433 Eros color contoured with respect to dynamic topography and shown for a ‘fast’ rotation period of T = 3.7 h. Surface erosion will generally move material from the high (red- pink) poles and mid-section of the body towards its low (cyan-blue) elongated ends, thus causing its rotation rate to gradually slow. 433 Eros's actual, current state is shown in Supplementary Fig. 14. (For interpretation of the references to color in this figure legend, the reader is referred to the online version of this chapter.)

209 J.E. Richardson, et al. Icarus 329 (2019) 207–221 state. There are four basic requirements for this ‘topographic correction’ proximity to the Main Belt, where the impact flux in the solar system is surface behavior to occur (Richardson and Bowling, 2014): highest (Bottke et al., 2005). We therefore expect thermally-driven disturbances to dominate regolith motion in the inner solar system, 1. The mean rotational force is a significant fraction of the mean transitioning to impact-induced seismic disturbances in the vicinity of gravitational force on the body's surface, and can therefore have a the Main Belt, and with both mechanisms decreasing gradually in fre- significant effect on global topography, slope directions, and slope quency as one moves into the outer solar system. As such, it is uncertain magnitudes, as to what role (if any) topographic correction plays in the surface 2. A sufficiently thick, low cohesion, mobile regolith layer exists over modification of such distant families as the Trans-Neptunian Objects most of the body's surface, facilitating topographic erosion and slope (TNOs), given that the active surface processes present on these bodies degradation, are only recently being characterized, via the New Horizons spacecraft 3. A downslope flow disturbance source is present, such as volatile mission (Moore et al., 2016). activity on comet nuclei near perihelion (Belton and Melosh, 2009), Finally, an important exception to condition no. 4, that “surface seismic disturbances from small impacts on asteroids in or near the erosion and downslope regolith movement are the primary forces af- Main Belt (Richardson et al., 2005), or severe thermal cycling on fecting the shape and spin of the body”, are those small bodies whose small, inner solar system asteroids (Mantz et al., 2004), spin states are primarily driven by the spin-up or spin-down produced 4. Surface erosion and downslope regolith movement are the primary by the Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect forces affecting the shape and spin of the body. (Rubincam, 2000), generally limited to those bodies with a dia- meter < 6–8 km and possessing orbits that place them in the near-Earth When the above four conditions are met, if a sufficient amount of environment (and thus more susceptible to solar heating). In these time has occurred since the body's last major surface alteration or sig- cases, we expect that while YORP may be the primary driver of the nificant change in spin state, then the body will naturally erode towards body's spin-state, topographic correction should still play an important the optimum shape and spin state for its current angular momentum, secondary role, as the regolith layer may make up a significant fraction thereby reaching a minimum in topography extremes, slope magni- of the body's radius, and frequent surface disturbances will occur. Given tudes, and global erosion (mass-wasting) rates. In our previous study, that 24 of our 32 selected small body shape models fall into this cate- which was limited to a handful of spacecraft-derived shape models, gory, we will discuss this further in Section 6. elongated asteroids 243 Ida, 433 Eros, and comet nucleus 103P Hartley 2 all met the above criteria for residing at or very near an optimum shape-spin state for their assumed density (Richardson and Bowling, 2.4. Evidence for loose regolith 2014). Prior to the 1991 flyby of asteroid 951 Gaspra by the Galileo 2.3. Caveats and exceptions spacecraft, most theoretical and experimental work suggested that as- teroids of Gaspra's size, if composed of materials as strong as solid rock, The four conditions listed above for ‘topographic correction’ to would develop a negligible regolith layer, and if composed of weaker occur come with some notable caveats and exceptions. For example, the materials, would develop a regolith only centimeters to meters in depth time-scale over which topographic correction progresses depends lar- (Carr et al., 1994; Housen et al., 1979; Veverka et al., 1986). However, fl gely upon conditions no. 2 & 3, above. These factors are discussed in- the Galileo yby images of 951 Gaspra in 1991 (Carr et al., 1994), fl dividually below. followed by a yby of asteroid 243 Ida in 1993 (Sullivan et al., 1996), With respect to condition no. 2, that “asufficiently thick, low co- and most notably, the extensive observations of asteroid 433 Eros – hesion, mobile regolith layer exists over most of the body's surface”, the conducted by the NEAR-Shoemaker spacecraft in 1999 2001 (Robinson time-scale of topographic correction will depend upon a combination of et al., 2002), all showed that such bodies are capable of generating and fi the size of the body and the relative thickness of its mobile regolith retaining a signi cant regolith layer up to tens of meters in depth on layer, such that we expect the efficacy of this mechanism to scale in- average, and even deeper in some regions (see Fig. 2). More recently, fl ∼ versely with the size of the body. On the smallest bodies, up to a few the Rosetta spacecraft's 2008 yby of 5 km diameter 2867 Steins fl ∼ kilometers in diameter, wherein the mobile regolith layer may make up (Jorda et al., 2012) and 2010 yby of 100 km diameter 21 Lutetia a significant fraction of the body's radius, shape change should occur on (Sierks et al., 2011) both revealed the presence of an extensive, loose relatively rapid time-scales. On bodies a few tens of kilometers in dia- meter, such as 433 Eros wherein the mobile regolith layer roughly occupies the outermost 1% of the body's radius, overall shape changes will still occur, but on more gradual time-scales. And on bodies a few hundreds of kilometers in diameter, wherein the mobile regolith layer occupies only a thin ‘skin’ with respect to the body's radius, the time- scale of this mechanism should be quite slow and perhaps insignificant. Nonetheless, as shown in Table 1, large Main Belt asteroids 16 Psyche and 216 Kleopatra both still fall into the Group B optimum rotator category, indicative of either internal deformation or external regolith migration (or both). This will be discussed further in Section 6. With respect to condition no. 3, that “a downslope flow disturbance source is present”, we expect the efficacy of topographic correction to scale inversely with the mean heliocentric distance of the body's orbit. Fig. 2. A view of asteroid 433 Eros captured during an October 26, 2000 low- – This is because two of the disturbance sources mentioned volatile altitude flyover by the NEAR-Shoemaker spacecraft. The image covers a region activity on comet nuclei near perihelion (Belton and Melosh, 2009) and about 800 × 400 m in size. Boulders of various sizes and shapes are set on a severe thermal cycling on small, inner solar system asteroids (Mantz gently rolling, cratered surface, while fine debris and regolith buries the et al., 2004) – are both directly proportional to the body's solar proxi- boulders, fills crater bowls, and generally softens the landscape. For scale, the mity, with regolith-moving disturbances becoming more frequent as the prominent boulder near the center of the image is about 25 m in length. Credit: body passes perihelion. The third source, seismic disturbances from NASA Space Science Data Coordinated Archive (NSSDCA), Goddard Space Flight impacts (Richardson et al., 2005), will be proportional to the body's Center.

210 J.E. Richardson, et al. Icarus 329 (2019) 207–221 regolith layer on the surface of each body. Even tiny (∼330 m mean rotational potential to the mean gravitational potential for the body. diameter) near-Earth asteroid 25143 Itokawa, visited by the JAXA For a perfectly spherical shape, this ratio is given by: Hayabusa spacecraft in 2005, displayed a cobble-rich, mobile regolith U 1 rω22 layer (Abe et al., 2006; Fujiwara et al., 2006). rot = , U 4 rGπρ2 Additionally, the NEAR-Shoemaker images of asteroid 433 Eros grav (2) revealed direct evidence of downslope regolith movement in several where Urot is the mean rotational potential, Ugrav is the mean gravita- forms: slumps and debris aprons at the base of steep slopes, bright tional potential, r is the radius of the body, ω is its rotation rate, G is the streaks of freshly exposed material on crater walls, the pooling of re- gravitational constant, and ρ is the bulk density of the body. Due to the golith in topographic lows, a large number of degraded craters, and a r2 term in both numerator and denominator, this ratio is invariant with deficit of craters less than ∼100 m in diameter as extrapolated from respect to body size, a feature that holds true for even more exotic body larger crater sizes (Chapman et al., 2002; Cheng et al., 2002; Robinson shapes. A simplified form of the above ratio was presented by Holsapple et al., 2002; Thomas et al., 2002; Veverka et al., 2001). The most (2004) and called the scaled spin ωsc: plausible explanation for these phenomena is seismic reverberation of ω the asteroid following impact events, which is capable of destabilizing ωsc = . Gπρ (3) slopes, causing regolith to migrate downslope, and degrading or erasing small craters (Richardson et al., 2004; Richardson et al., 2005). Scaled spin is the first key parameter in this study. It is important to It is important to note that these observations of 433 Eros also show recognize that because it is essentially a ratio between the mean rota- evidence for weak regolith cohesion (non-zero shear strength), most tional potential and the mean gravitational potential on the surface of notably in the form of steep crater walls in small craters which were the body, both the body's rotation rate and its density will greatly affect clearly formed in regolith and ponded deposits (Robinson et al., 2001). its value, where each can be treated as an independent variable in

On the other hand, broader slope studies indicate that on 300-meter producing a given scaled spin ωsc value. scales, only 1–3% of slopes are above typical angles-of-repose The second key parameter in this study is the body aspect ratio, (30°–35°), with an observed maximum of 36° (Thomas et al., 2002). given as the body's minor axis length c divided by its major axis length Thus, although some features do show indications of cohesion and a. In most cases, the body's minor c axis is also its principal rotation strength (in the form of crater walls, boulders, outcrops, ridges, and axis, although this is not always the case (as with asteroid 4179 groove edges), areas of obvious regolith coverage all tend to lie below Toutatis). typical angles-of-repose (Robinson et al., 2002; Thomas et al., 2002). The third, and most important key parameter in this study is the

There is also ubiquitous evidence of slope destabilization and the body's topographic variation hvar which is a measure of the slope and downslope migration of regolith over the entire surface of the asteroid, erosional stability of the surface. This is quantified by computing the as mentioned above. This evidence suggests that although present, ex- squared, normalized standard deviation (variance) in [gravita- isting cohesion forces seem to be relatively easy to overcome. tional + rotational] potential values across the surface. For a shape Analytical calculations of slope angle-of-stability and angle-of-re- model composed of N polyhedron facets, this is defined in Richardson pose indicate that directly applying a lunar regolith properties model and Bowling (2014) as: (taken from Houston et al., 1973) in the extremely weak Eros surface 2 −2 Ui gravity field, glocal = 0.0024–0.0056 m s (Thomas et al., 2002), re- ∑ Ai − 1 2 1 ()Umean sults in a regolith layer that can maintain nearly vertical (> 85°) slopes hσvar ==norm , N − 1 ∑ Ai (4) up to depths of 20 m (Richardson et al., 2005). Thus, using the lunar regolith cohesion values directly results in a regolith layer that con- where Ui is the potential at the center of the ith facet of the shape tradicts the visible evidence from Eros, where most regolith slopes lie at model, and Ai is the surface area of that facet. below typical values (< 35°) for the angle-of-repose (Robinson et al., Fig. 3 uses the above study parameters to examine the three spin- 2002; Thomas et al., 2002). In order to bring their model more in-line related dynamic topography states shown in Fig. 1. The topographic with the actual evidence, Richardson et al. (2005) reduced the cohesion variation hvar plot (on the right) can be produced in one of two ways, by formula derived for the lunar regolith (Houston et al., 1973)byan either holding the rotation period T constant and varying the bulk order of magnitude, which gave estimated values of regolith cohesion density ρ of the body, or by holding the bulk density ρ constant and for 433 Eros of about 10–150 Pa (increasing with depth) for the first varying the rotation period T of the body (see Eq. (3)). Both methods 20 m of regolith depth, which is sufficient to permit angles-of-stability generate the same curve, with each unique body shape producing a of > 70° for shallow depths (up to a few meters), but also gave angles- likewise unique topographic variation hvar curve as a function of scaled of-repose of ∼35°–45° for regolith depths up to 20 m. This large dif- spin ωsc. ference between the regolith angle-of-stability and angle-of-repose is at Because each point on a topographic variation hvar curve is non- least somewhat consistent with the observations described by unique with respect to rotation period T and bulk density ρ, associating (Robinson et al., 2001), without violating the larger scale regolith a point on this curve with a particular value of one variable requires properties described by Thomas et al. (2002). This modeling indicated knowing the value of the other to some degree of accuracy. In general, that in an extremely low asteroid surface gravity environment, regolith the rotation period T of the body is much better known than its bulk compression, and thus regolith cohesion values will be significantly density ρ (see Table 1). As such, holding the rotation period constant at lower than those found for the lunar surface environment, by at least an the current value permits exploring the current position of the body on order of magnitude. its topographic variation hvar curve by varying the bulk density over a reasonable range of values consistent with the object's identified spec- 3. Study parameters tral class, if the bulk density has not been determined through other means. For example, the three topography maps shown in Fig. 1 and the For the determination of small body surface gravitational proper- colored curves & points shown in Fig. 3, that correspond to rotation ties, we make use of the polyhedron surface-integral technique of periods T of 6.7 h (blue), 4.7 h (green), and 3.7 h (red) when holding ρ Werner (1994) to compute the gravitational potential and acceleration constant at the measured 433 Eros bulk density of ρ = 2670 ± 30 kg/ at the center of each triangular shape model facet. Additionally, the m3 (Thomas et al., 2002), likewise correspond to bulk density ρ values rotational potential and acceleration at the center of each shape model of 4740 kg/m3 (blue), 2150 kg/m3 (green), and 1490 kg/m3 (red), when facet is computed analytically as a function of distance from the body's holding T constant at the measured 433 Eros rotation period of 5.27 h principal rotation axis. Of particular interest is the ratio of the mean (Thomas et al., 2002). Of these three bulk densities, only the optimum

211 J.E. Richardson, et al. Icarus 329 (2019) 207–221

Fig. 3. (left panel) This plot shows the normalized slope distribution for the three spin states for asteroid 433 Eros presented in Fig. 1, color-coded with respect to whether the body is in either a slow (blue), optimum (green), or fast (red) rotation state. The lowest slope distribution occurs when the body is in an optimum rotation state, such that either slowing down or speeding up the rotation rate will cause an overall increase in slopes across the surface. (right panel) The topographic variation curve (Eq. (4)) for the shape model of asteroid 433 Eros, plotted as a function of scaled spin (Eq. (3)). The three large dots show the position on this curve of the three rotation states presented in Fig. 1, for an assumed bulk density of ρ = 2670 ± 30 kg/m3 (Thomas et al., 2002). A body is considered to be in an ‘optimum’ rotation state when its topographic variation hvar ≤ [curve minimum + 0.01]. 433 Eros's actual, current state is shown in Supplementary Fig. 14. (For interpretation of the references to color in this figure legend, the reader is referred to the online version of this chapter.)

− (green) value is consistent with that of an S-type asteroid, indicating types, ρ = 3000 ± 500 kg m 3; and M (metallic) related types, − that 433 Eros is likely near an optimum rotation state, if we did not ρ = 4000 ± 500 kg m 3. Within our sample of 32 small-bodies and know its actual bulk density (Richardson and Bowling, 2014). Slope using these assigned densities, we identify 10 Group A slow rotators distribution and topographic variation plots for both the actually (31%), 15 Group B optimum rotators (47%), and 7 Group C fast rotators measured rotation period and bulk density of asteroid 433 Eros are (22%). These classifications are summarized in Table 2, where each shown in the top row of Supplementary Fig. 14. small body is also identified by body type, either bifurcated (bi-lobate),

As shown in Fig. 3, the topographic variation hvar curve for a given prolate, irregularly shaped, or oblate. small body shape can be divided into three distinct segments, corre- Additionally, Table 1 lists the minimum hvar ‘optimum’ bulk density sponding to the three rotation conditions shown in Fig. 1. At low values ρ associated with each identified Group B object's measured rotation of scaled spin ωsc, the body will occupy the left-hand side of its U- period T, where the ‘range’ column lists the bulk densities ρ associated shaped topographic variation curve and is classified as a Group A ‘slow’ with the two hvar = [minimum + 0.01] points that mark the boundary rotator, with its elongated, equatorial ends marking topographic highs between each group classification (A–B and B–C). As presented in and its middle, polar regions marking topographic lows. At high values Richardson and Bowling (2014), this optimum density may be used as a of scaled spin ωsc, the body will occupy the right-hand side of its U- rough inference of the body's bulk density, when ρ has not been de- shaped topographic variation curve and is classified as a Group C ‘fast’ termined through other means and the optimum bulk density is con- rotator, with its middle, polar regions marking topographic highs and sistent with the body's observed spectral class. However, great care its elongated, equatorial ends marking topographic lows. At optimum should be taken in using this method, as this is simply an inference and values of scaled spin ωsc, the body will occupy the middle, minimum not an actual measurement. region in its U-shaped topographic variation curve and is classified as a Group B 'optimum' rotator, with topographic highs and lows tending to follow variations in physical structure. For the purpose of this study we 4. Erosion index define ‘optimum’ values of topographic variation to be within 0.01 of the shape's minimum value of topographic variation hvar. In addition to investigating the shape, topography, and topographic Supplementary Figs. 10–17 show the topographic variation curve variation curve for each body, we also examined its normalized slope (right panel) for the thirty-two objects included in this study. Note that distribution at the resolution of the shape model, where the slope angle highly elongated bodies (low c/a) produce a steeply sloped, narrow U- ϕ at a specific triangular shape model facet is defined as the angle shaped topographic variation curve, while more spherical bodies (high between the triangular facet's interior-facing normal vector and the c/a) produce a gently sloped, broad U-shaped curve, particularly Table 2 shallow at low values of scaled spin ωsc. The large dots placed on these fi curves show an estimate of the currently known position of each body Selected shape model asteroid classi cations. on its topographic variation curve, using its current rotation period and Group Bifurcated Prolate Irregular Oblate Total Fraction either a measured bulk density (12 objects), or a range of assumed bulk densities (20 objects) based upon the body's spectral classification. In A (slow) 3 3 3 2 10 0.31 B (optimum) 3 5 2 5 15 0.47 the later instance, C (carbonaceous chondrite) related types were given C (fast) 0 0 1 6 7 0.22 −3 an assumed bulk density value of ρ = 1500 ± 300 kg m ; S (stony) Total 6 8 6 12 32 – − related types, ρ = 2300 ± 300 kg m 3; E (enstatite chondrite) related Fraction 0.19 0.25 0.19 0.38 ––

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Fig. 4. (a) A plot of the globally experienced, downslope regolith movement resulting from the seismic shaking produced by a single 10 m impactor striking the surface of asteroid 433 Eros at 5 km/s, plotted as a function of slope on a surface of infinite extent, and computed using the methods described in Richardson et al.

(2005). The solid curve shows a fit to the numerical model using Eq. (5), assuming a regolith critical slope of ϕc = 35° (Carson and Kirkby, 1972). The dashed line shows the more traditional assumption of downslope flow that is linearly proportional to slope, with the overlap region between the two downslope flow regimes labeled “linear region”. (b) Erosion index plotted as a function of scaled spin ωsc for 30 of our 32 examined shape models, color-coded according to rotation group classification: Group A slow rotators (blue), Group B optimum rotators (green), and Group C fast rotators (red). The erosion index tends to decrease as one moves from the slow rotators to the optimum rotators, reaching a minimum near a scaled spin ωsc value of about 0.35, following which the erosion index increases rapidly as one moves towards faster rotation rates. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

° [gravitational + rotational] acceleration vector at the center of that ϕc =35. We term this ratio the ‘erosion index’ for the small body under same triangular facet. These slope distributions are displayed in the examination. In Fig. 4(b) we show the erosion indices for 30 of our 32 center panels of Supplementary Figs. 10–17. Assuming a gradual, dis- small-bodies (two Group C fast rotators are off-scale high), showing the turbance-driven, downslope regolith flow mechanism over the body's general trend in surface erosion index as one moves up in scaled spin surface (condition no. 3 in Section 2.2), we can relate these slope dis- ωsc. The dashed line in Fig. 4(b) represents a fifth-order polynomial fit tributions to a mean downslope erosion rate by utilizing the non-linear to these 30 data points, showing the general trend. The Group A slow downslope flow equation developed by Roering et al. (1999): rotators generally have slope distributions greater than our standard ° ϕm =10 body, with moderate erosion indices. The Group B optimum Kϕi tan rotators generally have the lowest slope distributions, either near or qi = 2 , tan ϕ ϕ ° 1 − ⎡ ⎤ lower than that of our standard m =10 body, and thus possess the ϕ ⎣ tan c ⎦ (5) lowest surface erosion indices. The Group C fast rotators generally have slope distributions much greater than our standard ϕ =10° body, with where q is the downslope flow distance per disturbance, K is a m i i very high erosion indices. The asymmetrical nature of this curve in- downslope material flow diffusion constant, ϕ is the slope angle, and ϕ c dicates that while the effect of slowing a body's rotation rate and is the regolith's critical slope angle (angle-of-repose). An example of an moving it towards gravity-dominated topography does cause a moderate application of this equation is shown in Fig. 4(a), for the global, seismic increase in its erosion index as compared to an optimum rotator, the response of a regolith layer resting on the surface of asteroid 433 Eros opposite effect of raising the body's rotation rate and moving it towards due to 10 m impactor striking the asteroid at a speed of 5 km/s spin-dominated topography is much more severe: causing much higher (Richardson et al., 2004; Richardson et al., 2005). Note that at low erosion indices. slope angles (< ∼ 10°) the amount of resulting downslope flow is A qualitative measure of the erosion index for a particular body can roughly linear with respect to slope, but that at angles > ∼ 10° the be gained from inspecting a map of the slopes over its surface (limited amount of downslope regolith flow becomes increasingly non-linear, by the resolution of the shape model). This is illustrated in Fig. 5, where especially as the critical slope angle ϕ is approached. c polyhedron facet slopes are mapped over the surface of six of our se- The use of a typical geologic material critical slope value of lected small body shape models (this is different from the topography ϕ =35° ±10° (Carson and Kirkby, 1972) is supported by examining c that is mapped in the other shape model figures shown in this work). On the slope distributions of our 15 Group B optimum rotators (center the two Group B optimum rotators shown in the middle of Fig. 5, most column of Supplementary Figs. 10–17), which have a mean slope of of the body is at low slopes (cyan-blue), with structural topography ϕ = 12.0° ± 3.7° and show a strong concentration of slopes below m marking areas of moderate (green) slope. The low slope distributions on angles of about 30°–35°; that is, among these 15 Group B bodies, these bodies is indicative of ongoing slope degradation and a mobile 97.0% ± 5.9% of the shape model slopes are < 30°, and regolith layer, a condition known as ‘transport-limited’ downslope flow 98.5% ± 4.1% of the shape model slopes are < 35°. (Carson and Kirkby, 1972). On the left side of Fig. 5 are two Group A In order to compare the average regolith movement rate on each of slow rotators, with a higher proportion of moderate (green) slopes and the small bodies included in this study, we apply Eq. (5) to each body ϕ some high (red-pink) slopes as compared to two the Group B objects, using its mean slope m, and then eliminate the unknown Ki constant by ff ‘ ’ although the di erence between a Group A and Group B object can be dividing this expression by Eq. (5) applied to a standard body with a ffi ° di cult to detect via slope mapping alone. More dramatically, the two mean slope of ϕm =10, with both expressions using a critical slope of

213 J.E. Richardson, et al. Icarus 329 (2019) 207–221

Fig. 5. Surface slope maps for 6 representative examples from our sample of 32 small body shape models, plotted using a rainbow color scale with blue indicating0° slope, yellow indicating slopes near an assumed angle-of-repose of 30°–35°, and pink indicating slope ≥60°. The two Group A slow rotators and two Group B optimum rotators show signs of a mobile regolith and active slope degradation, resulting in low slope, smooth regions of light green to cyan-blue, interrupted by occasional ridges, rims, and other projections of higher topography and steeper slopes. The two Group C fast rotators displayed on the right show signs of either “scree” slopes of relatively loose, rapidly replaced material, currently resting at the angle-of-repose (1999 KW4 Alpha), or ubiquitous very high slopes and a complete lack of a mobile regolith layer (54509 YORP). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Group C fast rotators shown on the right side of Fig. 5 depict bodies about 1/2 of these bodies are Group B optimum rotators (47%); about where most of the slopes on the body are at or above typical angles-of- 1/3 of the bodies are Group A slow rotators (31%); and about 1/5 of the repose for geologic materials (shown in yellow for slopes 30°–35°). On bodies are Group C fast rotators (22%). some of these bodies, such as 1999 KW4 Alpha, much of the surface is at One interesting way of presenting these results is shown in Fig. 6, or very near the angle-of-repose, indicative of an active “scree” slope of which shows all 32 small bodies studied by plotting their determined loose or lightly cohesive material, wherein each time downslope flow is scaled spin (Eq. (3)) as a function of body aspect ratio c/a: a pre- triggered (by a disturbance) that would otherwise lower this slope, new sentation method originally presented in Holsapple (2004) and what we material moves in from the poles to replace it until the slope once again will call a SSAR (Scaled Spin vs Aspect Ratio) plot. On this plot, each sits at the angle-of-repose (Harris et al., 2009). On other bodies, such as body is color-coded according to its rotation group classification. The 1998 KY26, the rotation rate is so high that all slopes are well above the (green) Group B optimum rotators form an arc on this plot, with the angle-of-repose, such that loose, moveable material will quickly flow Group A (blue) slow rotators located below this arc and the Group C downslope (and escape the body entirely when the slope is > 90°), (red) fast rotators located high and to the right. leaving behind only cohesive, more solid material: a condition known In Fig. 6, the uncertainty in each body's c/a aspect ratio is depen- as ‘weathering-limited’ downslope flow (Carson and Kirkby, 1972). dent upon the accuracy of the body's shape model. The eight spacecraft observation derived shape models, while variable in resolution and object geographic coverage, generally produce c/a values with an un- 5. Scaled spin vs aspect ratio certainty of roughly ± 0.01–0.02 or less (Thomas, 2014). The twenty- four radar derived shape models, also quite variable in resolution and 5.1. Actual small body shape models object geographic coverage, generally produce less rigorous c/a values with an uncertainty of roughly ± 0.05–(0.10 Benner, 2013). Table 2 summarizes the number of body types present in our se- The uncertainty in each body's scaled spin ωsc value depends, in lection of 32 small body shape models, listed as a function of shape-spin large part, upon whether the body is a slow, optimum, or fast rotator Group (A-slow, B-optimum, or C-fast), within which two trends are (see Eq. (3)). For example, the Group A slow rotator 2603 Bacchus, with readily apparent. First, some shape-sorting is visible in our sample, such a rotation period of T = 15.0 h and an estimated scaled spin ωsc value of that the Group A slow rotators contain a relatively even distribution of 0.1676, has an ωsc uncertainty of +0.022/−0.016 for a bulk density ρ each body type, the Group B optimum rotators display a small, but 3 uncertainty of ± 500 kg/m ; the Group B optimum rotator 1627 Ivar, noticeable prevalence of prolate and oblate bodies, while the Group C with a rotation period of T = 4.80 h and an estimated scaled spin ωsc fast rotators are almost exclusively oblate in shape. Second, there is a value of 0.5027, has an ωsc uncertainly of +0.059/−0.044 for a bulk notable prevalence of bodies in an optimum rotation state, such that

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Fig. 6. A plot of scaled spin ωsc vs body aspect ratio c/a for the 32 small body shape models used in this study, color coded according to the determined shape-spin state for the object: Blue = Group A slow rotators; Green = Group B optimum rotators; Red = Group C fast rotators. The slow rotators tend to occupy the central and lower portions of this plot, while the optimum rotators form an arc above them, and the fast rotators tend to occupy the region above those and to the right. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) density ρ uncertainty of ± 500 kg/m3; and the Group C fast rotator A topographic variation curve (Eq. (4)) was generated for each 29075 1950 DA, with a rotation period of T = 2.12 h and an estimated synthetic shape model, producing a family of 22 curves for the prolate scaled spin ωsc value of 1.0373, has an ωsc uncertainly of +0.099/ bodies and 23 curves for the oblate bodies. Fig. 7 shows a contour fitto −0.077 for a bulk density ρ uncertainty of ± 500 kg/m3. the family of topographic variation curves resulting from the collection of 22 prolate shape models, plotted in three dimensions as a function of

both scaled spin ωsc and body aspect ratio c/a. The foremost curve, at an 5.2. Synthetic small body shape models aspect ratio of c/a = 0.999, is that of a spherical body, with increasing body elongation shown as one moves back and to the upper right. What In order to investigate the clustering shown in Fig. 6 further, we we define as the ‘zone of maximum topographic stability’ resides at the synthesized a series of 22 increasingly prolate (a ≥ b = c) spheroidal bottom of this 3D surface, color-coded in ‘cool’ blues, at topographic shape models, and a series of 23 increasingly oblate (a = b ≥ c) variation h values of < 0.04. Increasing surface instability, in the spheroidal shape models, both ranging in aspect ratio c/a from 0.001 to var form of larger topographic extremes, higher slopes, and higher erosion 0.999 in increments of 0.05, with 2–3 finer increments added below and regolith migration rates, occurs as one moves vertically up this 0.05 and above 0.95 for increased detail in these regions. These syn- surface, towards the ‘hot’ orange-red regions, corresponding to topo- thetic shape models were produced with 2° latitude-longitude resolu- graphic variation h values of > 0.10. As seen in Supplementary tion, each containing 32,040 polygon facets. As with the actual small var Figs. 10–17, the more elongated the body (lower c/a values), the body shape models examined in this study, each synthetic body was steeper are the sides of the body's respective topographic variation then analyzed using the surface gravitational properties approach ori- curve. ginally presented in Richardson and Bowling (2014), using an assumed Looking at this 3D plot from directly overhead, Fig. 8 shows the stony asteroid density of ρ = 2500 kg/m3. The baseline spherical body resulting contours of topographic variation hvar for our 45 synthetic (c/a = 1.0) for both the prolate and oblate synthetic series has a dia- prolate and oblate bodies, plotted by temperature-scale color contour as meter of 4.5 km, although as pointed on in Section 3, the scaled spin ωsc a function of both aspect ratio c/a and scaled spin ωsc. The blue ‘cold’ and resulting topographic variation hvar curve are invariant with respect region of this plot (topographic variation hvar values < 0.04) denotes to the overall size of the body: the curve is shape dependent, not size the region of lowest topographic variation, lowest slopes, and lowest dependent.

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Fig. 7. A three-dimensional surface plot of the family of topographic variation curves (Eq. (4)), plotted as a function of both scaled spin ωsc and body aspect ratio c/a for the 22 synthetic prolate shape model generated for this study. This surface is color-coded using a ‘temperature’ scale, with low values represented in blues and high values represented in reds. (For interpretation of the references to color in this figure legend, the reader is referred to the online version of this chapter.) surface erosion rates; that is, the ‘zone of maximum topographic sta- broadly applied, conservative density estimates thus act as the largest bility’ for each family of shapes. In general, the disk-shaped oblate source of uncertainty in the estimated scaled spin ωsc of these lightcurve bodies are stable at slightly higher values of scaled spin ωsc than are observed asteroids, where this uncertainty will follow the same pattern their blimp-shaped prolate body counterparts. In both cases, however, as that described in Section 5.1, with increasing uncertainty as one the ‘zone of maximum topographic stability’ narrows significantly as moves to faster body rotation rates. one moves towards lower values of body aspect ratio. The body aspect ratio of each object was estimated using an in- verted form of Eq. (1) from Masiero et al. (2009): 5.3. Asteroid lightcurve data 2 ⎛ 1 ⎞ 2 ⎜⎟Δm − cos θ In order to investigate the larger asteroid population beyond those c e 2.5 = ⎝ ⎠ , 2 included in our shape model study, we also estimated the scaled spin ωsc a sin θ and body aspect ratio c/a of 2791 asteroids for which observational (6) lightcurve data was available. This data was obtained from the Warner et al. (2009) Light Curve Database (LCDB) located at: http://www. where Δm is the measured lightcurve amplitude (in units of magnitude) MinorPlanet.info/lightcurvedatabase.html. from the lightcurve database, and θ is the angle of the body's spin vector The 2018 June 23 release of this database includes a frequency- with respect to the line-of-sight of the observations. While θ actually diameter (F-D) “Basic” table that lists lightcurve information for 6755 varies from 0 to π, lightcurve observations will see this only as a var- objects, ranging in size from tiny Near-Earth Asteroids (NEA) to large iation from 0 to π/2 and back to 0 (as the squared values of cos θ and Trans-Neptunian Objects (TNO). This selection of objects includes only sin θ in Eq. (6) indicate). Here we make the simplifying assumption that those lightcurve-observed bodies that have a data reliability factor of the asteroids in this data set are distributed roughly isotropically with U > 1+, which the authors consider to be acceptable for statistical respect to spin vector orientation θ, and as such, we use an assumed analysis (Warner et al., 2009). From this dataset, we selected 2791 non- mean value of θ = π/4 in Eq. (6). This assumption produces the largest TNO asteroids having estimated diameters (computed from their ab- source of uncertainty in the estimated body aspect ratio c/a of these solute magnitude and albedo) of > 10 km, in order to avoid the smaller lightcurve observed asteroids. objects (generally less than a few kilometers in diameter) whose rota- This uncertainty in c/a values estimated using Eq. (6) comes pri- tion rates can be heavily affected by the spin-up or spin-down produced marily from the fact that a near-spherical, high c/a body will produce a by the Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect lightcurve with a small amplitude Δm, regardless of the viewing angle (Rubincam, 2000), thus competing with the erosional effects being θ, whereas an elongated, low c/a body will produce a lightcurve with explored in this work. an amplitude Δm that varies widely over a range of values, depending The scaled spin (Eq. (3)) of each object was estimated using the upon the viewing angle θ. As such, by using a mean observation angle rotation period T determined from the asteroid lightcurve's periodicity of θ = π/4 to compute c/a, we unavoidably underestimate the number and an assumed bulk density for the body. To obtain this assumed bulk of low c/a objects and overestimate the number of high c/a objects. density, we divided this lightcurve dataset into three groups by asteroid That is, the estimated c/a values will be pushed to the right on a SSAR spectral classification: (1) the M-type asteroids and related families (E, plot, depending upon the actual c/a distribution of this observed as- − M, P, and X) were assigned a ρ = 3500 m 3, (2) the C-type asteroids teroid population (for whom we have only lightcurve observations). and related families (B, C, F, and G) were assigned a bulk density of Fig. 9 shows the result of this exercise, plotting all 2791 selected − ρ = 1500 m 3, and (3) the S-type asteroids and related families (Q, S, lightcurve-observed asteroids in SSAR plot format. We include on this − and all others) were assigned a bulk density of ρ = 2500 m 3. These plot our 32 selected small body shape models from Fig. 6 (color-coded

216 J.E. Richardson, et al. Icarus 329 (2019) 207–221

Fig. 8. Contour plots of topographic variation (Eq. (4)) for our 22 prolate (above) and 23 oblate (below) synthetic spheroid bodies, shown as a function of scaled spin

ωsc and body aspect ratio c/a, and colored scaled using a ‘temperature’ gradient (see legends). according to shape-spin group, and the topographic variation contour maximum topographic stability’ and are presumed to be Group A slow lines derived from our 22 synthetic prolate body shape models and rotators; and 276 (10%) lie above the ‘zone of maximum topographic shown in Fig. 8. The prolate body contours were selected for display stability’ and are presumed to be Group C fast rotators. These results are here because, as listed in Table 1 and shown in Supplementary summarized in Table 3. Figs. 10–17, oblate bodies tend to be smaller in diameter and faster rotating as compared to their prolate body counterparts (see Table 2). 6. Discussion As such, our selected 2791 lightcurve-observed asteroids, deliberately limited to estimated diameters of > 10 km to eliminate those most af- As previously described in Section 2, our so-called ‘zone of max- fected by YORP, are also more likely be prolate in shape. This as- imum topographic stability’ exists in a self-correcting form, such that sumption is supported by the observation that only 40 of the 2791 changes in either the body's rotation rate or bulk density (due to im-

(1.4%) lie above the upper hvar = 0.1 prolate body contour line in pacts, for example) will produce deviations in the body's scaled spin ωsc Fig. 8, and those roughly follow that line. that will potentially move the body further away from the minimum Bearing in mind the uncertainties discussed above, 1941 (70%) of region of its respective topographic variation curve. This move up the the lightcurve-observed asteroids lie within the boundaries of the two curve will push the surface of the body towards higher topographic 0.04 prolate body topographic variation contours, and are presumed to differences, higher slopes, and thus higher erosion rates (see Figs. 1 & be Group B optimum rotators; 574 (21%) lie below this ‘zone of 3). As a result, if surface erosion is the primary mechanism altering the

217 J.E. Richardson, et al. Icarus 329 (2019) 207–221

Fig. 9. Contours of topographic variation (Eq. (4)) for prolate spheroid bodies plotted as a function of scaled spin ωsc and body aspect ratio c/a, displayed in 0.02 increments from 0.02 to 0.10. The central bold contour line indicates the point of minimum topographic variation hvar (generally between 0.01 and 0.02), while the upper and lower bold lines indicates a topographic variation of hvar = 0.04, which bound the ‘zone of maximum topographic stability’ as a function of scaled spin ωsc and body aspect ratio c/a. Against this backdrop are plotted our 32 examined small body shape models, color-coded according to rotation group classification: Group A slow rotators (blue), Group B optimum rotators (green), and Group C fast rotators (red). Additionally, we include the estimated position of 2791 asteroids for which high quality lightcurve data is available (small purple points). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 3 likely Group B body lies within the ‘zone of maximum topographic Selected lightcurve asteroid classifications. stability’ between these contour lines. For roughly prolate bodies, such fi Group S # S % C # C % M # M % Total Fraction as asteroid 433 Eros (Supplementary Fig. 14), these two de nitions are self-consistent, while for other bodies, such as asteroid 2100 Ra-Shalom A (slow) 271 0.097 257 0.092 46 0.020 574 0.206 (Supplementary Fig. 11), the correspondence is not so good. As such, B (optimum) 714 0.256 1003 0.359 224 0.080 1941 0.695 the somewhat arbitrary nature of these classifications should always be C (fast) 134 0.048 142 0.051 0 0.000 276 0.099 kept in mind, and each small body treated as an individual case, Total 1119 0.401 1402 0.502 270 0.097 2791 – whenever possible. Those readers familiar with the asteroid shape-spin internal stability shape of the body, slow rotating asteroids below the ‘zone of maximum work of Holsapple (2001), Holsapple (2004), Holsapple and Housen topographic stability’ on a SSAR plot, will tend to become less prolate/ (2007), Sharma (2013), and Sharma et al. (2009) will recognize the oblate over time, thus moving them up and to the right on a SSAR plot similarity of Figs. 8 & 9 in this work to Figs. 8 & 11 of Holsapple (2004), (Fig. 9) until the ‘zone of maximum topographic stability’ is once again Fig. 7 of Holsapple and Housen (2007), Figs. 3–7ofSharma et al. reached. Fast rotating asteroids, above the ‘zone of maximum topo- (2009), and Figs. 2–7&9–12 of Sharma (2013). This was a pleasantly graphic stability’ on a SSAR plot, will tend to become more prolate/ surprising result, given that this work looks only at geomorphology and oblate over time, thus moving them down and to the left on this plot the surface processes associated with topography, slope, and the mi- until the ‘zone of maximum topographic stability’ is once again gration of loose regolith over the surface of a small body – where the reached. Note that the contour spacing in Fig. 9 is much closer for ‘surface stability’ referred to here is the stability of local topography elongated objects with low aspect ratios, indicating that these bodies and slopes with respect to downslope movement (mass-wasting). have a steeper gradient in topographic variation as one moves away The reason for this similarity is because this asteroid geomor- from the minimum, such that these bodies have a correspondingly more phology work, on the one hand, and Holsapple and Sharma's asteroid severe surface response to changes in scaled spin ωsc. As a result, the shape-spin internal stability work (listed above), on the other hand, negative-feedback resulting from a drift away from the ‘zone of max- essentially present opposite sides of the same coin. For example, the imum topographic stability’ should likewise be more severe, increasing ‘zone of maximum topographic stability’ for small bodies shown in the inherent topographic stability of more elongated bodies. Figs. 8 & 9 here corresponds with the limits of internal stability (failure It should be noted that our identified Group A ‘slow’ rotators, Group curves) for asteroids composed of the weakest possible materials (in- B ‘optimum’ rotators, and Group C ‘fast’ rotators exist on a spectrum, ternal friction angles of 0°–8°) shown in Holsapple (2004), Holsapple without clear boundaries between them. As such, these classification and Housen (2007). Another way to think about this is that the ‘zone of boundaries were selected to be as simple and consistent in use as pos- maximum surface stability’ described in this work corresponds with the sible, while acknowledging their somewhat arbitrary nature. On the one ‘zone of lowest internal stress’ for asteroids that are composed of more hand, we define the classification of a small body possessing a high- typical geologic materials, those that have internal friction angles of ° ° quality shape model based upon its current position on its topographic 25 –45 (Carson and Kirkby, 1972). The similarities between the figures variation curve with respect to that curve's minimum, wherein a likely presented in this work and those presented in Holsapple and Sharma's

Group B body has an hvar ≤ [minimum + 0.01]. On the other hand, we work occur because internal stress in an asteroid's interior is a function define the classification of a small body possessing only high quality of the potential gradient (due to gravity and rotation) in the interior of lightcurve data based upon its position on a SSAR plot with respect to the body, while topographic variation hvar is a function of the potential the two hvar = 0.04 contour lines for ideal prolate bodies, wherein a gradient on the surface of the body (see Eqs. (1) & (4)). As such, there

218 J.E. Richardson, et al. Icarus 329 (2019) 207–221 are potentially two ways in which a given body can be pushed towards Nakamura, R., Scheeres, D.J., Yoshikawa, M., 2006. Mass and local topography the ‘zone of maximum topographic stability’ on a SSAR plot, either by measurements of Itokawa by Hayabusa. Science 312, 1344–1349. https://doi.org/10. 1126/science.1126272. surface erosion and mass-movement, as described in this work, or by Belton, M.J.S., Melosh, J., 2009. Fluidization and multiphase transport of particulate internal failure and rearrangement, as described in Holsapple (2001), cometary material as an explanation of the smooth terrains and repetitive outbursts Holsapple (2004), Holsapple and Housen (2007), Sharma (2013), and on 9P/Tempel 1. Icarus 200, 280–291. https://doi.org/10.1016/j.icarus.2008.11. 012. Sharma et al. (2009). 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