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Simulations of impact ejecta and regolith accumulation on Asteroid Eros
Icarus 171 (2004) 110–119 www.elsevier.com/locate/icarus
Simulations of impact ejecta and regolith accumulation on Asteroid Eros D.G. Korycansky ∗ , Erik Asphaug CODEP, Department Earth Sciences, University of California, Santa Cruz, CA 95064, USA Received 22 December 2003; revised 16 March 2004 Available online 20 July 2004
Abstract We have carried out a set of Monte Carlo simulations of the placement of impact ejecta on Asteroid 433 Eros, with the aim of understanding the distribution and accumulation of regolith. The simulations consisted of two stages: (1) random distribution of primary impact sites derived from a uniform isotropic flux of impactors, and (2) integration of the orbits of test particle ejecta launched from primary impact points until their re-impact or escape. We integrated the orbits of a large number of test particles (typically 106 per individual case). For those particles that did not escape we collected the location of their re-impact points to build up a distribution on the asteroid surface. We find that secondary impact density is mostly controlled by the overall topography of the asteroid. A gray-scale image of the density of secondary ejecta impact points looks, in general, like a reduced-scale negative of the topography of the asteroid’s surface. In other words, regolith migration tends to fill in the topography of Eros over time, whereas topographic highs are denuded of free material. Thus, the irregular shape of Eros is not a steady-state configuration, but the result of larger stochastic events. 2004 Elsevier Inc. All rights reserved. Keywords: Asteroids; Dynamics
1. Introduction The issue of regolith on asteroids has an interesting history. As late as 1990 it was commonly believed that any asteroid smaller than a few 10’s of km would be denuded of regolith, since ejection velocities would exceed the very low escape velocity. Scaling models by Housen et al. (1983) modified in Veverka et al. (1986), concluded that rocky bodies 20 km in diameter and icy bodies 70 km in diameter could have only a thin veneer (millimeters) of regolith. In the subsequent decade there was much controversy around this issue, with many groups (Asphaug and Melosh, 1993; Love and Ahrens, 1996; Melosh and Ryan, 1997; Ryan and Melosh, 1998; Asphaug, 1999; Benz and Asphaug, 1999) finding, through numerical modeling, that asteroids much smaller than a few kilometers diameter were likely to be gravitationally-bound objects, either shattered rocks (zero tensile strength but not disassembled) or “rubble piles” [zero tensile strength and disorganized configuration—see Richardson et al. (2002)]. It was also recognized through * Correspoding author.
E-mail address: [email protected] (D.G. Korycansky). 0019-1035/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2004.03.021
laboratory experiments (Love et al., 1993) that non-coherent aggregates could be much stronger, in terms of surviving a given impact, than a coherent monolithic solid, owing to the dissipation of shock energy. On the scaling front, Housen and Holsapple (1990) found that the incorporation of rate- and size-dependent strength models significantly lowered strength effects, allowing selfgravity to dominate in the collisional evolution of relatively small asteroids. The issue of asteroid interior structure is discussed elsewhere [see, e.g., Asphaug et al. (2002)], but the implication is that regolith should be prevalent even on small asteroids, as their craters would form in the gravity regime, not the strength regime. In the gravity regime most ejecta from a given crater remain bound to the asteroid, and that is the starting assumption for the following investigation. The NEAR mission to 433 Eros returned a wealth of data (Veverka et al., 2000a, 2000b; Robinson et al., 2002; Cheng, 2002) confirming that at least one small asteroid is awash in regolith. Indeed, the absence of craters at 10 meter scales (Cheng, 2002) and the infilling of crater bowls suggests that regolith is not only prevalent on Eros, but also highly mobile. High resolution images of Eros reveal an abundance of evidence for an active and substantial regolith (Zuber et al., 2000; Veverka et al., 2000b; Chapman, 2002; Cheng, 2002;
Impact ejecta on Eros
Sullivan et al., 2002; Robinson et al., 2002). Zuber et al. (2000) suggest a regolith depth of ∼ 100 m with possible hemispheric-scale variations, inferred from the offset between the center of mass and center of figure of the asteroid. Thomas et al. (2002) arrived at similar conclusions and suggest regolith is global, with no distinct individual regolith deposits blanketing specific features. Robinson et al. (2002) conclude that there may be no clear distinction between the asteroid’s regolith and its deep interior. Linear features on Eros are probably surface expressions of fracture planes in a faulted and jointed interior (Prockter et al., 2002). Together with its relatively high density of 2.7 g cm−3 [and thus relatively low porosity; Britt and Consolmagno (2001)], this fracture fabric has led to acceptance of the idea [by, e.g., Asphaug et al. (2002)] that the jumbled and porous regolith of the exterior of Eros transitions at some depth towards an interior that is widely fragmented but not jumbled. The implication is significant, for a well-connected rock mass transmits impact energy globally leading to the possibility of a moderate impact event resetting the cratering record (Cheng et al., 2002b). A disconnected rock mass, on the other hand, attenuates impact energy before it can have much of a global effect. Veverka et al. (2000b) find evidence on Eros for a complex regolith that includes fine material and many meterscale blocks, in high-resolution images (the “landing” sequence, as the orbiter touched down on the surface). The images were taken of a relatively small part of the asteroid lying outside the lowest areas of the asteroid, namely the craters Himeros and Psyche. The images show degraded craters and many blocks, as well as some of the notable “ponds” (Robinson et al., 2001): smooth, almost perfectly flat areas filled with somewhat bluish material, the hue one expects for grain sizes smaller than the optical wavelength. Their view is that the regolith consists of non-uniform blocks, of areal density ranging over an order of magnitude, and a variety of depths of burial, morphologies and apparent characteristics with no strong correlation between appearance and size. On the other hand, the fact that most blocks are unmantled by dust suggests to Chapman et al. (2002), that regolith is either relatively sparse and patchy, or that fine material is a small component in general. Blocks found in Psyche, Himeros, and Shoemaker Regio appear to have been generated and emplaced by the impact that formed Shoemaker Regio (Thomas et al., 2001, 2002). Veverka et al. (2000b) note that obvious bedrock features are hard to find, with no indications of sharp gradients from excavation into a deeper competent layer. There is a lack of small craters (< 100 m diameter) compared to the numbers expected by extrapolation from larger sizes or the Moon (Chapman, 2002; Chapman et al., 2002). Crater densities on Eros are lowest in the low-lying crater features Psyche, Himeros, and Shoemaker Regio (Chapman et al., 2002), which could be the result of those features being younger, or surface processes resulting in enhanced regolith transport. Fresh-looking small craters are rare in
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comparison to the number of craters that appear degraded. Chapman et al. (2002) point out that the expected characteristics of widely dispersed ejecta from larger impacts (or a single large impact) does not match observed small-crater properties and favor an explanation invoking a lack of impactors in the size range (10 m and less) to account for the observations [as proposed by Bell (2001)]. Other explanations for the dearth of small craters include seismic shaking (Asphaug and Melosh, 1993; Nolan et al., 1996; Cheng et al., 2002b) and the “Brazil nut effect” (Asphaug et al., 2001) by which a regolith is made active by granular sorting in response to thermal or seismic effects. Other than the bluish color of the “ponds” (Robinson et al., 2001), which is likely a result of particle size rather than composition, no significant color differences in regions or crater floors have been found. Albedo features on sloped regions are common, suggesting downslope mass wasting (Sullivan et al., 2002) and a relatively rapid degradation of surface color (“weathering”). Robinson et al. (2002) suggest an average regolith depth of ∼ 20–40 m, based on crater volumes and assumptions about ejecta escape. A comparison of the depth of fresh craters to degraded ones yields estimates from a few meters to ∼ 150 m, with other indications, such as mounds of material in Shoemaker and Himeros giving values of ∼ 100 m. Robinson et al. (2002) also suggest a maximum depth of ∼ 200–300 m due to the presence of topographic features such as grooves and ridges, and the excavation of ∼ 150 m boulders by the event that formed the Crater Shoemaker. Cheng et al. (2002a) analyze small-scale surface roughness measurements obtained by laser rangefinder and conclude that a regolith layer of some tens of meters is present. Prockter et al. (2002) estimate a regolith depth of tens of meters near Shoemaker Regio based on observations of grooves into which regolith drainage may have been responsible for their pitted appearance (Horstman and Melosh, 1989). The abundance of regolith and regolith-related features on Eros compels us to address the simple question of where regolith migrates, over time, in response to impact bombardment. In this paper we consider impactors small enough that their craters form entirely in regolith (that is to say, in the gravity regime), and we evaluate whether Eros (given its odd shape and rapid 5.27 hour spin period) tends to accumulate or lose regolith preferentially in certain areas. The calculations we describe are similar in method to those of Thomas (1998), who studied ejecta patterns on the satellites of Mars, Geissler et al. (1996b), who integrated test particles launched into orbit around Asteroid 243 Ida, and Geissler et al. (1996a), who also included a simple model (an ellipsoid) for Eros. We are not attempting to match the characteristics of specific features, but rather conducting a general study of impact ejecta placement and its connection with the regolith on the object. Nonetheless, something should be said about a connection to the physical volume of material that is moved about or escapes the object due to collisions. Geissler et al.
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(1996b) give expressions for the rate of generation of ejecta volume, based on simple assumptions about the population of impactors and impact speed. Adapting those expressions to the present case, we have for the rate of generation of ejecta volume Ve produced by impactors of diameter D: dVe = Da2 P ∗ QAg −α U 2α dt
D max
D 3−α−q dD.
(1)
Dmin
In this expression, Da is the target diameter and P ∗ is the probability per unit time per unit target cross-section of an impact [P ≈ 3 × 10−18 km−2 yr−1 for main-belt objects, which we assume Eros to have been for most of its life (Bottke and Greenberg, 2001; Chapman et al., 2002)]. We have folded in the crater volume V (D) = Ag −α U 2α D 3−α , where A and α are constants, g is the surface gravity, and U is the impact velocity. Also folded into the expression is the assumption of a power-law distribution dN(D)/dD = QN −q for the impactor population. The integration limits range from Dmin , the smallest impactor to make gravityregime craters, to Dmax , the size of an impactor that would catastrophically disrupt the object. Doing the integral then yields an estimate of the amount of regolith that is mobilized per unit time, and hence, a timescale for “gardening” the asteroid, given an estimate for the regolith depth. The timescale is of course sensitive to the parameters chosen for the normalizations, the power-law indices, and the limits of integration. In particular, the result of the integration is an expression that depends on impactor diameter like D (4−α−q) . For popular choices of impactor population index (q = 3.5 or 4), the ejecta generation rate depends on the diameter Dmin of the smallest impactors that are assumed to be significant. Plugging in numbers yields estimated values of dVe /dt of 6 × 102 and 2 × 103 cm3 s−1 for q = 3.5 and 4, respectively, for Dmin = 1.4 × 102 cm (the size of impactor that makes a crater of 0.2 km in diameter and is cleared by ejecta moving at vmin = 100 cm s−1 ). Setting Dmin = 10 cm yields dVe /dt = 1 × 103 and 1 × 104 cm3 s−1 for q = 3.5 and 4. In turn, assuming a 100-m deep regolith gives a regolith volume of 1017 cm3 , and hence timescales from ∼ 3 × 106 to 5 × 107 yr for the range of values quoted. Overall, these are rather short timescales compared to the expected lifetime of Eros in near-Earth space.
2. Monte Carlo simulations Our method in this study is Monte Carlo simulation of the impact ejecta. First, primary impact points are randomly chosen on the surface. To ensure an unbiased selection of such points, we find them using the method employed by Korycansky and Asphaug (2003). We choose an impact point by picking two points from a uniform random distribution on a sphere around the body, and joining them by a straight line and finding the points of intersection with the
Fig. 1. Two views of the 1280-face Eros shape model used in the calculations. Top: view from z-axis. Longitude 0◦ is (in this view) on the right end of the asteroid. Longitude increases counterclockwise toward positive values (+180◦ ) and decreases toward negative values (−180◦ ) clockwise. Bottom: view from y-axis toward Psyche.
asteroid. The reduced amount of sky (< 2π) seen by concave regions is thus automatically taken into account. Tests of the method show that it produces an isotropic distribution on convex bodies and is independent of the radius of the circumscribing sphere. The gravitational field for the test-particle integrations was computed using the “polyhedron gravity” (PG) method developed by Werner (1994) and Werner and Scheeres (1996). This is an algorithm for computing the gravitational field around an arbitrarily-shaped polyhedron. The calculations were done with Eros models of 1280 faces, for the most part, with additional calculations done with a coarser model of 320 faces. The number of faces chosen for the model was a compromise between accuracy and detail in representing Eros’s shape and speed of computation. The bulk of the cpu time was taken up by computation of the gravitational force which scales linearly with the number of faces. The original Eros model (from which the polyhedra models were derived), was supplied by D. Scheeres (private communication). Data upon which this model is based are discussed by Miller et al. (2002). Figure 1 shows two views of the 1280-face Eros model used for most of the calculations. The top view looks toward the north pole of the asteroid. We use a coordinate system in which longitude 0◦ is (in this view) on the right end of the asteroid. Longitude increases counterclockwise toward positive values (“east” longitudes 0◦ –360◦ ). The upper panel of the figure shows the “convex” side of the asteroid with Himeros and Shoemaker; on the bottom can be seen the crater Psyche in profile. For reference, we give the coordinates of these features in the coordinate system we use:
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Fig. 2. Contour map of Eros: contours show radius in km as function of longitude and latitude relative to center of mass. Positions are marked by initials for Himeros (lat. = +21◦ , long. = +78◦ ), Shoemaker (lat. = −16◦ , long. = +30◦ ), and Psyche (lat. = +32◦ , long. = 266◦ ).
Himeros (lat. = +21◦ , long. = +78◦ ), Shoemaker (lat. = −16◦ , long. = +30 ◦ ), and Psyche (lat. = +32◦ , long. = 266◦ ). The lower panel shows an equatorial view along the y-axis, toward the side (near 270◦ ) on which Psyche is visible. For comparison below, we also show (in Fig. 2) a contour plot of elevation (radius) from the center of mass of Eros, generated from the 1280-face shape model shown in Fig. 1. The positions of Himeros, Shoemaker, and Psyche are marked by initials for reference. From each primary impact point we generated ten test particles that were launched with randomly chosen azimuths (0 < φ < 2π) and a launch angle of 45◦ from the local plane defined by the polyhedral face containing the primary impact point. The particle trajectories are integrated in the gravitational field of the polyhedral model. The effect of Eros’s rotation is included in the equations of motion. The angular velocity Ω = 3.31 × 10−4 s−1 corresponds to the rotational period P = 5.27 hours. (We also did non-rotating cases for comparison.) The density of Eros was taken to be ρ = 2.7 gm cm−3 . Test-particle integrations were done by the same algorithm (second-order leapfrog) as used in Korycansky and Asphaug (2003). Further details, including the full set of equations for a particle in a rotating frame and the time-step algorithm, are given in that paper. We used timesteps of 0.01τ and 0.05τ for the integrations, where τ = (Gρ)−1/2 = 2.36 × 103 s is a free-fall timescale for the body. No substantial differences in the overall outcome were found (i.e., the distribution of secondary re-impacts on the surface), but the smaller-timestep calculations were more accurate, as checked by the fractional change of the Jacobi constant CJ = v 2 + 2(Φ + ΦR ), where Φ is the gravitational potential (calculated by the PG algorithm) and ΦR = −1/2Ω(x 2 + y 2 ) is the rotational potential due to the rotating frame. For calculations with
timestep t = 0.01τ , about 90% of the integrations had relative changes in the Jacobi constant CJ /CJ < 0.01, decreasing to ≈ 40% for integrations with t = 0.05τ . Particle velocities were chosen randomly from a power law distribution such that the cumulative number of ejected particles with speed larger than v is proportional to v −1 , compatible with laboratory and theoretical results (Housen et al., 1983). (The exact choice of this distribution exponent does not matter much for the present suite of calculations, although for future calculations that look in detail at mass and angular momentum balance, it will be of greater importance.) The mass of Eros is 6.72 × 1018 gm, and the average radius ≈ 8.4 × 105 cm, yielding an average escape speed of 1.03 × 103 cm s−1 . For various simulations we used different minimum speeds for the velocity distribution vmin = 100, 200, and 500 cm s−1 . Using minimum speeds much less than vmin = 100 cm s−1 would produce calculations with so many slow-moving particles that the results would differ little from the distribution of primary impact points, which is uniform except for concave areas of the surface (Psyche and Himeros). The flight distance r = v 2 /2g for a particle launched at 45◦ in a constant gravity field g is ≈ 1.1 × 104 cm for v = 100 cm s−1 and g = 0.46 cm s−2 , (the area-averaged surface gravity on Eros), which is smaller than the dimension d ≈ 3 × 104 cm of the smallest faces on the 1280-face model. Thus, slow-moving particles (vmin 100 cm s−1 ) land in general on the same face from they were launched. Particles with positive total energy, or ones that reached distances larger than rmax = 108 cm, we considered to be escapers and were not followed any longer. We typically ran cases with ≈ 105 primary impacts, so that ≈ 106 test particles were followed for each case. When a particle was
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determined to have re-impacted the surface, we incremented a counter for the face that it crossed. Since the polyhedron faces have different areas, we normalized the counts for the face by dividing by the face’s area and scaling so that a uniform distribution would correspond to a surface density σ = 1 over the whole asteroid. We assume that re-impacting particles do not bounce or roll any significant distance; reimpacts are treated as being completely inelastic. Likewise, we have neglected other processes that may affect regolith distribution such as downslope motion due to forces described by Scheeres et al. (2002): acceleration due to gravity and rotational forces, friction, and surface shaking from impacts or close flybys of other objects.
3. Results and discussion In principle, the only free parameters in the calculation are those relating to the velocity distribution of the secondary ejecta, in particular vmin . The other parameters (the launch angle of the ejecta and the index of the power-law distribution) are constrained by experimental results (Housen et al., 1983). For general runs we chose vmin so that the launched ejecta will travel at least one crater radius from the impact center, although we also explored the effect of varying vmin . Monte Carlo calculations inevitably come with a certain amount of statistical noise due to the finite size of the sample. In this case, contour plots of the relative number of ejecta (primary or secondary) show this characteristic. Even if enough test particles are integrated so that several hundred land per face, face-to-face fluctuations of 10% or more occur. Thus, robust conclusions can be reliably made about broad features of the contour plots that we show. Plotting results as gray-scale maps helps the eye to see such features. The first question is whether primary impactors have a significant departure from a uniform coverage. We expect a uniform density of primary impacts for a convex body, which Eros largely is. A contour gray-scale contour plot of primary impacts is shown in Fig. 3 for two Monte Carlo simulations using the 1280-face Eros model, and a comparison with a similar calculation involving a model sphere of 1280 faces. The latter calculation, with 107 trials, shows a uniform distribution with small fluctuations of ±3%, equal to what would be expected from a Poisson distribution on the model sphere in which the mean number of impactors per face was N = 107 /1280 = 7.8125 × 103 (also taking into account the areas of individual faces on the model sphere). For the Eros calculations, we see variations (i.e., deficits) due to the concavity of the low, shadowed regions (Shoemaker Regio/Psyche/Himeros), which do not see a complete 2π steradians of sky. The expected range of a Poisson distribution on the 1280-face Eros model with 107 total impactors is ≈ ±8%. By contrast, the lowest density seen in Fig. 3, panel (b) is 0.77. The systematic variation clearest in the 107 -impact map, where the deficiency of the counts in the
Fig. 3. Shaded contour map of primary impact density on 1280-face models of Eros and a sphere, using the Monte Carlo method described in the text. Impact density is plotted relative to the average value that would result from a uniform density of impacts. (a) Primary impactor density on the Eros model resulting from a calculation with 106 impactors. Maximum and minimum gray-scale values are 0.65 and 1.19. (b) Primary impactor density on the Eros model resulting from a calculation with 107 impactors. Maximum and minimum gray-scale values are 0.77 and 1.05. (c) Primary impactor density on a model sphere resulting from a calculation with 107 impactors. Maximum and minimum gray-scale values are 0.97 and 1.03.
concave regions is evident. The same trend can be discerned in the 106 impact calculation, despite the larger fluctuations. (The level of the “noise” also gives an indication of the level of fluctuations and the robustness of the results for secondary impacts to be presented below.) Figure 4 shows the re-impact points for 10,000 test particles launched from random locations with random launch azimuths and 45◦ launch altitudes. The left-hand panels, (a) and (b) show results for a rotating ellipsoid polyhedron model of 1280 faces. The ellipsoid properties (axes, density, and rotation rates) were chosen to match those investigated by Geissler et al. (1996a), who modeled a variety of cases. In this calculation, the particles were launched at the average escape speed from the ellipsoid, 1.175 × 103 cm s−1 . Our results for the ellipsoid are very close to those of Geissler et al. (1996a), (cf. Fig. 3 of that paper), lending confidence to our results. A sizable anisotropy is present on the leading faces of the ellipsoid. A similar anisotropy is seen for the Eros-model calculation, with some differences due to Eros’s particular shape. A slightly larger proportion of such particles (∼ 67%) escape from Eros, as opposed to the ellipsoid (∼ 63%). Simulation results for secondary ejecta for a rotating Eros are shown in Fig. 5 for vmin = 100, 200, and 500 cm s−1 , in
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Fig. 4. Re-impact sites of points launched from random locations and random azimuths from an 1280-face ellipsoid polyhedron model [(a) and (b)] and the 1280-face Eros model used for the calculations. The ellipsoid has radii 18 × 7.5 × 6.5 km and density 2.724 gm cm−3 to match the dimensions and mass of the body studied by Geissler et al. (1996a). Particles were launched at an altitude of 45◦ and a uniform speed corresponding to the average escape velocity for each object: v = 1.175 × 103 and v = 1.024 × 103 cm s−1 . Panels (a) and (b): plots of x and z coordinates of points with y > 0 and y < 0, respectively, for the ellipsoid model. The results can be compared with Fig. 3 of Geissler et al. (1996a). Panels (c) and (d): plots of x and z coordinates of points with y > 0 and y < 0, respectively, for the Eros model.
the form of color-mapped contour plots. This particular set of orbital integrations was done with a time step t = 0.01τ . The plots show contours of ejecta surface density σ normalized to uniform coverage σ = 1. There are different numbers of escapers, as defined above, for each case. For vmin = 100 cm s−1 , about 12% of the particles escaped; for vmin = 200 cm s−1 , about 25%, and for vmin = 500 cm s−1 , about 58% escaped. The overall variation in σ is ≈ 2–3 between the minimum and maximum values. Systematic trends in ejecta density over the surface are apparent. The clearest result for all three of the cases shown [and similar calculations with a different timestep (t = 0.05τ ), or a coarser 320-face shape model] is that ejecta tend to pile up in the low regions of the asteroid (Psyche, Himeros/Shoemaker Regio) and are sparse on the high areas, particularly the “ends” at longitudes near 0◦ and 180◦ . This is consistent with the lower density of smaller craters in Psyche, Himeros, and Shoemaker Regio noted by Chapman et al. (2002), although downslope mass wasting could also exert a similar influence. There are hints of an extension of the Himeroscentered concentration (longitude ∼ +60◦ –90◦ ) into Shoemaker Regio, seen most clearly in the vmin = 100 cm s−1 and vmin = 200 cm s−1 calculations. This would be consistent with observations of regolith on that part of the asteroid (Robinson et al., 2002). However, statistical noise makes that identification uncertain. There are overall differences among the three cases depending on the minimum velocity. Due to the power-law nature of the ejecta velocity spectrum, test particles moving at the lowest speeds dominate the distribution [n(> v) ∝ v −1 as noted above]. The case with the lowest velocity would thus show the weakest “anisotropy” of ejecta coverage, as
more of the particles would move slowly and travel only short distances before re-impacting the asteroid. Thus we would expect larger differences from uniformity for the cases with larger values of vmin , which is largely what we see in panels (b) and (c) of Fig. 5. The vmin = 100 and 200 cm s−1 cases resemble (in reverse) the topographic map presented by Thomas et al. (2002). The resemblance in the 200 cm s−1 case is striking. In this case the gradients outside of Psyche and the east side of Himeros are particularly sharp. Overall in the vmin = 500 cm s−1 case, the concave side of the asteroid appears to pick up a considerable amount more material than the convex side. For this last case, the most obvious result is a strong deficit of material on the ends of the body, compared to the simulations with smaller values of vmin . This is probably due to the fact that a much larger relative proportion of ejecta have speeds comparable to the escape velocity for the this case. Thus, relatively more ejecta escape, and more ejecta are affected by rotational forces. Comparison with calculations for a non-rotating asteroid show essentially the same results as in Fig. 5. Thus, the rotation of Eros, while being about half that of the “critical” rate [e.g., Pravec et al. (2002)] at which a strengthless asteroid would begin to come apart, does not seem to have strong effects on the placement of ejecta. This is at first surprising result, given that the ends of Eros almost intersect its Roche lobe (Scheeres et al., 2002). However, Geissler et al. (1996a) find that ejecta launched at near-escape speed are the only ones that are strongly affected by rotation, at least in the case of Eros. A calculation by them involving particles launched on an ellipsoid at one-half vesc produced essentially isotropic coverage of the surface. Thus, rotational
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Fig. 5. Shaded contour maps of ejecta placement on Eros, from the Monte Carlo calculations described in the text. The integration timestep t = 0.01τ the rotation rate Ω = 3.31 × 10−4 cm s−1 . (a) Minimum secondary test particle velocity vmin = 100 cm s−1 . Total number of secondary impacts = 8.77 × 105 . Minimum and maximum values on the surface are 0.705 and 1.634. (b) Minimum secondary test particle velocity vmin = 200 cm s−1 . Total number of secondary impacts = 7.58 × 105 . Minimum and maximum values on the surface are 0.508 and 1.795. (c) Minimum secondary test particle velocity vmin = 500 cm s−1 . Total number of secondary impacts = 4.23 × 105 . Minimum and maximum values on the surface are 0.319 and 1.622. Note that the color bar scale is different (larger) than in Fig. 3.
effected would be reflected by only a small minority of fast moving ejecta (given the power-law distribution we use). In Fig. 6 we look at the relative numbers of escapers from the asteroid. That is, from each location, we kept track of how many particles escaped from that location. As expected, particles tended to escape from the ends (high spots), where the gravitational potential is lower and centrifugal effects are larger. As vmin increases, a higher proportion of particles escape and the salience of the ends (high spots) diminishes, as can be seen in the last panel. The relative number of escapers rises from 0.123 to 0.242 to 0.577 of the total number (106 ) ejecta for the three cases. We can combine the results of Figs. 5 and 7 to calculate a net gain or loss rate of mass. The mass input due to
Fig. 6. Shaded contour maps of the relative number of escapers on Eros, from the Monte Carlo calculations described in the text. The integration timestep t = 0.01τ . (a) Minimum secondary test particle velocity vmin = 100 cm s−1 . Total number of escapers = 1.23 × 105 . Minimum and maximum values on the surface are 0.149 and 1.786. (b) Minimum secondary test particle velocity vmin = 200 cm s−1 . Total number of escapers = 2.42 × 105 . Minimum and maximum values on the surface are 0.092 and 1.920. (c) Minimum secondary test particle velocity vmin = 500 cm s−1 . Total number of escapers = 5.77 × 105 . Minimum and maximum values on the surface are 0.196 and 1.694. Note that the color bar scale is different (larger) than in Fig. 5.
the primary impact is negligible compared to the mass of secondary ejecta in each impact (∼ 1% or less). The mass balance in a small region of the asteroid is then given by the gain (due to incoming ejecta) minus the loss (due to ejecta launched by an impact). The balance is easy to compute from the calculations, and is just equal to the density of secondary impactors minus density of primary impactors times the number of secondary particles launched per impact (10, in this case). The results are shown in Fig. 7 for the three cases of Fig. 5 in a reduced-gray-scale plot. The scale is normalized to the number of secondary ejecta per face of average area, ±106 /1280 = ±781.25. That is, a gain or loss of 781.25 ejecta in the calculation corresponds to ±1 on the gray-scale plot. As noted in the discussion of Fig. 6, there is a net loss of ejecta due to escapers. Most areas of the asteroid
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Fig. 8. Plots of relative density of secondary impacts vs. radius for the three cases shown in Fig. 5. The integration timestep t = 0.01τ and the rotation rate Ω = 3.31 × 10−4 cm s−1 . (a) Minimum secondary test particle velocity vmin = 100 cm s−1 . (b) Minimum secondary test particle velocity vmin = 200 cm s−1 . (c) Minimum secondary test particle velocity vmin = 500 cm s−1 .
Fig. 7. Plots of relative net gain/loss (accretion/erosion) per unit area for the three cases shown in Fig. 5. The integration timestep t = 0.01τ and the rotation rate Ω = 3.31 × 10−4 cm s−1 . The scale is normalized so that the number of secondary ejecta per face of average ±106 /1280 = ±781.25 corresponds to ±1. (a) Minimum secondary test particle velocity vmin = 100 cm s−1 . Minimum and maximum values on the surface are −0.791 and +0.783. Net loss of secondary particles is 1.23 × 105 /106 . (b) Minimum secondary test particle velocity vmin = 200 cm s−1 . Minimum and maximum values on the surface are −0.812 and +0.896. Net loss of secondary particles is 2.42 × 105 /106 . (c) Minimum secondary test particle velocity vmin = 500 cm s−1 . Minimum and maximum values on the surface are −1.05 and 0.313. Net loss of secondary particles is 5.77 × 105 /106 .
lose mass, in general, for the given assumed ejecta massvelocity distribution. Low-lying areas (Psyche and Himeros) would be expected to gain mass from slow-moving ejecta from other parts of the object. Taken at face value, this suggests that the regolith should be thinner or even non-existent on the ends of Eros. In turn, that would affect our assumptions about gravity-regime impacts; conceivably, the impacts on the objects will be in the strength regime and produce strength-dominated craters. Another way of looking at the results is shown in Fig. 8, where we plot the relative density of secondary impacts versus radius, for the same three cases of Fig. 5. In general, the density of ejecta is largest at the low points, i.e., smallest ra-
dius. The correlation is strongest for the vmin = 500 cm s−1 case, where there is a continued drop-off in density for larger radii. Note that we do not include the effect of particle bouncing. Sawai et al. (2001) find that the coefficient of restitution of bouncing material has a significant effect on its final resting place. Since we are looking at stochastic models, the effect would be moot if the bouncing material bounces isotropically. However, bouncing material tends to have a preferential downslope component. Incorporation of bouncing would thus enhance the effect we already observe, namely, that topographic highs lose material over time, and topographic lows accumulate it, despite the unusual shape and rapid rotation state of Eros.
4. Conclusion We have carried out a number of Monte Carlo simulations of impacts on the Asteroid Eros, with the goal of determining where secondary ejecta would tend to land and to estimate the relative depths of regolith on the object. Generally speaking, we find that the relative density of secondary impacts should be highest at the “low spots” (smaller radii) of the asteroid, in particular in the regions in and around the craters Psyche and Himeros. However, relative variations in secondary impact density are not extremely large: less than a factor of 2 relative to the mean, or a factor of ∼ 4 from
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maximum to minimum on the whole. If secondary ejecta dominate total mass in a collision (i.e., are more massive than the primary impactor) then most regions of the asteroid would be expected to experience a net loss of mass. Low-lying regions (Psyche and Himeros) would gain mass due to slow-moving ejecta (∼ 100–200 cm s−1 ) launched from other regions of the object. By contrast, the ends of the object should have a systematically shallower or even nonexistent regolith layer. Whether this is the case is not clear; the impression given by Robinson et al. (2002) is that a regolith is all-pervasive on the asteroid, although the interiors of Psyche, Shoemaker, and Himeros are indeed dominated by regolith features. However, the fact that these low-lying regions have not yet been completely filled with regolith implies that the creation of topography (e.g., by major impact events or by tidal encounters with planets) has occurred faster on Eros than the infilling of topography by stochastic small-scale cratering, which may be consistent with the lack of small recent impacts advocated by Chapman et al. (2002).
Acknowledgments This research was supported for D.G.K. under the NASA Planetary Geology and Geophysics program (NASA grant NAG5-11521-001) and for EIA by a grant from the NASA Discovery Data Analysis Program. We thank the reviewers, P. Thomas and P. Geissler, for helpful comments. Helpful information was provided by the website of D. Fanning (www.dfanning.com) in making Figs. 2, 3, 5, 6, and 7.
References Asphaug, E., 1999. Survival of the weakest. Nature 402, 127–128. Asphaug, E., Melosh, H.J., 1993. The Stickney impact of Phobos—a dynamical model. Icarus 101, 144–164. Asphaug, E., King, P.J., Swift, M.R., Merrifield, M.R., 2001. Brazil nuts on Eros: size-sorting of asteroid regolith. In: Proc. Lunar Planet. Sci. Conf. 32nd. Abstract 1867. Asphaug, E., Ryan, E.V., Zuber, M.T., 2002. Asteroid interiors. In: Bottke, W.F., Cellino, A., Paolicchi, P., Binzel, R.P. (Eds.), Asteroids III. Univ. of Arizona Press, Tucson, AZ, pp. 463–484. Bell III, J.F., 2001. Eros: a comprehensive model. In: Proc. Lunar Planet. Sci. Conf. 32nd. Abstract 1964. Benz, W., Asphaug, E., 1999. Catastrophic disruptions revisited. Icarus 142, 5–20. Bottke, W.F., Greenberg, R., 2001. The impact record on asteroid surfaces. In: Marov, M.Ya., Rickman, H. (Eds.), Collisional Processes in the Solar System. Kluwer Academic, Dordrecht, pp. 51–71. Britt, D.T., Consolmagno, G.J., 2001. Modeling the structure of high porosity asteroids. Icarus 152, 134–159. Chapman, C.R., 2002. Cratering on asteroids from Galileo and NEAR Shoemaker. In: Bottke, W.F., Cellino, A., Paolicchi, P., Binzel, R.P. (Eds.), Asteroids III. Univ. of Arizona Press, Tucson, AZ, pp. 315–330. Chapman, C.R., Merline, W.J., Thomas, P.C., Joseph, P.C., Cheng, A.F., Izenberg, N., 2002. Impact history of Eros: craters and boulders. Icarus 155, 104–118.
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Veverka, J., Thomas, P., Robinson, M., Murchie, S., Chapman, C., Bell, M., Harch, A., Merline, W.J., Bell, J.F., Bussey, B., Carcich, B., Cheng, A., Clark, B., Domingue, D., Dunham, D., Farquahr, R., Gaffey, M.J., Hawkins, E., Izenberg, N., Joseph, J., Kirk, R., Li, H., Lucey, P., Malin, M., Mcfadden, L.A., Miller, J.K., Owen, W.M., Peterson, C., Prockter, L., Warren, J., Wellnitz, D., Williams, B.G., Yeomans, D.K., 2000b. Imaging of small-scale features on 433 Eros from NEAR: evidence for a complex regolith. Science 292, 484–488. Werner, R.A., 1994. The gravitational potential of a homogeneous polyhedron or don’t cut corners. Celest. Mech. Dynam. Astron. 59, 253– 278. Werner, R.A., Scheeres, D.J., 1996. Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of Asteroid 4769 Castalia. Celest. Mech. Dynam. Astron. 65, 313–344. Zuber, M.T., Smith, D.E., Cheng, A.F., Garvin, J.B., Aharonson, O., Cole, T.D., Dunn, P.J., Guo, Y., Lemoine, F.G., Neumann, G.A., Rowlands, D.D., Torrence, M.H., 2000. The shape of 433 Eros from the NEAR-Shoemaker laser rangefinder. Science 289, 2097–2101.
Simulations of impact ejecta and regolith accumulation on Asteroid Eros D.G. Korycansky ∗ , Erik Asphaug CODEP, Department Earth Sciences, University of California, Santa Cruz, CA 95064, USA Received 22 December 2003; revised 16 March 2004 Available online 20 July 2004
Abstract We have carried out a set of Monte Carlo simulations of the placement of impact ejecta on Asteroid 433 Eros, with the aim of understanding the distribution and accumulation of regolith. The simulations consisted of two stages: (1) random distribution of primary impact sites derived from a uniform isotropic flux of impactors, and (2) integration of the orbits of test particle ejecta launched from primary impact points until their re-impact or escape. We integrated the orbits of a large number of test particles (typically 106 per individual case). For those particles that did not escape we collected the location of their re-impact points to build up a distribution on the asteroid surface. We find that secondary impact density is mostly controlled by the overall topography of the asteroid. A gray-scale image of the density of secondary ejecta impact points looks, in general, like a reduced-scale negative of the topography of the asteroid’s surface. In other words, regolith migration tends to fill in the topography of Eros over time, whereas topographic highs are denuded of free material. Thus, the irregular shape of Eros is not a steady-state configuration, but the result of larger stochastic events. 2004 Elsevier Inc. All rights reserved. Keywords: Asteroids; Dynamics
1. Introduction The issue of regolith on asteroids has an interesting history. As late as 1990 it was commonly believed that any asteroid smaller than a few 10’s of km would be denuded of regolith, since ejection velocities would exceed the very low escape velocity. Scaling models by Housen et al. (1983) modified in Veverka et al. (1986), concluded that rocky bodies 20 km in diameter and icy bodies 70 km in diameter could have only a thin veneer (millimeters) of regolith. In the subsequent decade there was much controversy around this issue, with many groups (Asphaug and Melosh, 1993; Love and Ahrens, 1996; Melosh and Ryan, 1997; Ryan and Melosh, 1998; Asphaug, 1999; Benz and Asphaug, 1999) finding, through numerical modeling, that asteroids much smaller than a few kilometers diameter were likely to be gravitationally-bound objects, either shattered rocks (zero tensile strength but not disassembled) or “rubble piles” [zero tensile strength and disorganized configuration—see Richardson et al. (2002)]. It was also recognized through * Correspoding author.
E-mail address: [email protected] (D.G. Korycansky). 0019-1035/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2004.03.021
laboratory experiments (Love et al., 1993) that non-coherent aggregates could be much stronger, in terms of surviving a given impact, than a coherent monolithic solid, owing to the dissipation of shock energy. On the scaling front, Housen and Holsapple (1990) found that the incorporation of rate- and size-dependent strength models significantly lowered strength effects, allowing selfgravity to dominate in the collisional evolution of relatively small asteroids. The issue of asteroid interior structure is discussed elsewhere [see, e.g., Asphaug et al. (2002)], but the implication is that regolith should be prevalent even on small asteroids, as their craters would form in the gravity regime, not the strength regime. In the gravity regime most ejecta from a given crater remain bound to the asteroid, and that is the starting assumption for the following investigation. The NEAR mission to 433 Eros returned a wealth of data (Veverka et al., 2000a, 2000b; Robinson et al., 2002; Cheng, 2002) confirming that at least one small asteroid is awash in regolith. Indeed, the absence of craters at 10 meter scales (Cheng, 2002) and the infilling of crater bowls suggests that regolith is not only prevalent on Eros, but also highly mobile. High resolution images of Eros reveal an abundance of evidence for an active and substantial regolith (Zuber et al., 2000; Veverka et al., 2000b; Chapman, 2002; Cheng, 2002;
Impact ejecta on Eros
Sullivan et al., 2002; Robinson et al., 2002). Zuber et al. (2000) suggest a regolith depth of ∼ 100 m with possible hemispheric-scale variations, inferred from the offset between the center of mass and center of figure of the asteroid. Thomas et al. (2002) arrived at similar conclusions and suggest regolith is global, with no distinct individual regolith deposits blanketing specific features. Robinson et al. (2002) conclude that there may be no clear distinction between the asteroid’s regolith and its deep interior. Linear features on Eros are probably surface expressions of fracture planes in a faulted and jointed interior (Prockter et al., 2002). Together with its relatively high density of 2.7 g cm−3 [and thus relatively low porosity; Britt and Consolmagno (2001)], this fracture fabric has led to acceptance of the idea [by, e.g., Asphaug et al. (2002)] that the jumbled and porous regolith of the exterior of Eros transitions at some depth towards an interior that is widely fragmented but not jumbled. The implication is significant, for a well-connected rock mass transmits impact energy globally leading to the possibility of a moderate impact event resetting the cratering record (Cheng et al., 2002b). A disconnected rock mass, on the other hand, attenuates impact energy before it can have much of a global effect. Veverka et al. (2000b) find evidence on Eros for a complex regolith that includes fine material and many meterscale blocks, in high-resolution images (the “landing” sequence, as the orbiter touched down on the surface). The images were taken of a relatively small part of the asteroid lying outside the lowest areas of the asteroid, namely the craters Himeros and Psyche. The images show degraded craters and many blocks, as well as some of the notable “ponds” (Robinson et al., 2001): smooth, almost perfectly flat areas filled with somewhat bluish material, the hue one expects for grain sizes smaller than the optical wavelength. Their view is that the regolith consists of non-uniform blocks, of areal density ranging over an order of magnitude, and a variety of depths of burial, morphologies and apparent characteristics with no strong correlation between appearance and size. On the other hand, the fact that most blocks are unmantled by dust suggests to Chapman et al. (2002), that regolith is either relatively sparse and patchy, or that fine material is a small component in general. Blocks found in Psyche, Himeros, and Shoemaker Regio appear to have been generated and emplaced by the impact that formed Shoemaker Regio (Thomas et al., 2001, 2002). Veverka et al. (2000b) note that obvious bedrock features are hard to find, with no indications of sharp gradients from excavation into a deeper competent layer. There is a lack of small craters (< 100 m diameter) compared to the numbers expected by extrapolation from larger sizes or the Moon (Chapman, 2002; Chapman et al., 2002). Crater densities on Eros are lowest in the low-lying crater features Psyche, Himeros, and Shoemaker Regio (Chapman et al., 2002), which could be the result of those features being younger, or surface processes resulting in enhanced regolith transport. Fresh-looking small craters are rare in
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comparison to the number of craters that appear degraded. Chapman et al. (2002) point out that the expected characteristics of widely dispersed ejecta from larger impacts (or a single large impact) does not match observed small-crater properties and favor an explanation invoking a lack of impactors in the size range (10 m and less) to account for the observations [as proposed by Bell (2001)]. Other explanations for the dearth of small craters include seismic shaking (Asphaug and Melosh, 1993; Nolan et al., 1996; Cheng et al., 2002b) and the “Brazil nut effect” (Asphaug et al., 2001) by which a regolith is made active by granular sorting in response to thermal or seismic effects. Other than the bluish color of the “ponds” (Robinson et al., 2001), which is likely a result of particle size rather than composition, no significant color differences in regions or crater floors have been found. Albedo features on sloped regions are common, suggesting downslope mass wasting (Sullivan et al., 2002) and a relatively rapid degradation of surface color (“weathering”). Robinson et al. (2002) suggest an average regolith depth of ∼ 20–40 m, based on crater volumes and assumptions about ejecta escape. A comparison of the depth of fresh craters to degraded ones yields estimates from a few meters to ∼ 150 m, with other indications, such as mounds of material in Shoemaker and Himeros giving values of ∼ 100 m. Robinson et al. (2002) also suggest a maximum depth of ∼ 200–300 m due to the presence of topographic features such as grooves and ridges, and the excavation of ∼ 150 m boulders by the event that formed the Crater Shoemaker. Cheng et al. (2002a) analyze small-scale surface roughness measurements obtained by laser rangefinder and conclude that a regolith layer of some tens of meters is present. Prockter et al. (2002) estimate a regolith depth of tens of meters near Shoemaker Regio based on observations of grooves into which regolith drainage may have been responsible for their pitted appearance (Horstman and Melosh, 1989). The abundance of regolith and regolith-related features on Eros compels us to address the simple question of where regolith migrates, over time, in response to impact bombardment. In this paper we consider impactors small enough that their craters form entirely in regolith (that is to say, in the gravity regime), and we evaluate whether Eros (given its odd shape and rapid 5.27 hour spin period) tends to accumulate or lose regolith preferentially in certain areas. The calculations we describe are similar in method to those of Thomas (1998), who studied ejecta patterns on the satellites of Mars, Geissler et al. (1996b), who integrated test particles launched into orbit around Asteroid 243 Ida, and Geissler et al. (1996a), who also included a simple model (an ellipsoid) for Eros. We are not attempting to match the characteristics of specific features, but rather conducting a general study of impact ejecta placement and its connection with the regolith on the object. Nonetheless, something should be said about a connection to the physical volume of material that is moved about or escapes the object due to collisions. Geissler et al.
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(1996b) give expressions for the rate of generation of ejecta volume, based on simple assumptions about the population of impactors and impact speed. Adapting those expressions to the present case, we have for the rate of generation of ejecta volume Ve produced by impactors of diameter D: dVe = Da2 P ∗ QAg −α U 2α dt
D max
D 3−α−q dD.
(1)
Dmin
In this expression, Da is the target diameter and P ∗ is the probability per unit time per unit target cross-section of an impact [P ≈ 3 × 10−18 km−2 yr−1 for main-belt objects, which we assume Eros to have been for most of its life (Bottke and Greenberg, 2001; Chapman et al., 2002)]. We have folded in the crater volume V (D) = Ag −α U 2α D 3−α , where A and α are constants, g is the surface gravity, and U is the impact velocity. Also folded into the expression is the assumption of a power-law distribution dN(D)/dD = QN −q for the impactor population. The integration limits range from Dmin , the smallest impactor to make gravityregime craters, to Dmax , the size of an impactor that would catastrophically disrupt the object. Doing the integral then yields an estimate of the amount of regolith that is mobilized per unit time, and hence, a timescale for “gardening” the asteroid, given an estimate for the regolith depth. The timescale is of course sensitive to the parameters chosen for the normalizations, the power-law indices, and the limits of integration. In particular, the result of the integration is an expression that depends on impactor diameter like D (4−α−q) . For popular choices of impactor population index (q = 3.5 or 4), the ejecta generation rate depends on the diameter Dmin of the smallest impactors that are assumed to be significant. Plugging in numbers yields estimated values of dVe /dt of 6 × 102 and 2 × 103 cm3 s−1 for q = 3.5 and 4, respectively, for Dmin = 1.4 × 102 cm (the size of impactor that makes a crater of 0.2 km in diameter and is cleared by ejecta moving at vmin = 100 cm s−1 ). Setting Dmin = 10 cm yields dVe /dt = 1 × 103 and 1 × 104 cm3 s−1 for q = 3.5 and 4. In turn, assuming a 100-m deep regolith gives a regolith volume of 1017 cm3 , and hence timescales from ∼ 3 × 106 to 5 × 107 yr for the range of values quoted. Overall, these are rather short timescales compared to the expected lifetime of Eros in near-Earth space.
2. Monte Carlo simulations Our method in this study is Monte Carlo simulation of the impact ejecta. First, primary impact points are randomly chosen on the surface. To ensure an unbiased selection of such points, we find them using the method employed by Korycansky and Asphaug (2003). We choose an impact point by picking two points from a uniform random distribution on a sphere around the body, and joining them by a straight line and finding the points of intersection with the
Fig. 1. Two views of the 1280-face Eros shape model used in the calculations. Top: view from z-axis. Longitude 0◦ is (in this view) on the right end of the asteroid. Longitude increases counterclockwise toward positive values (+180◦ ) and decreases toward negative values (−180◦ ) clockwise. Bottom: view from y-axis toward Psyche.
asteroid. The reduced amount of sky (< 2π) seen by concave regions is thus automatically taken into account. Tests of the method show that it produces an isotropic distribution on convex bodies and is independent of the radius of the circumscribing sphere. The gravitational field for the test-particle integrations was computed using the “polyhedron gravity” (PG) method developed by Werner (1994) and Werner and Scheeres (1996). This is an algorithm for computing the gravitational field around an arbitrarily-shaped polyhedron. The calculations were done with Eros models of 1280 faces, for the most part, with additional calculations done with a coarser model of 320 faces. The number of faces chosen for the model was a compromise between accuracy and detail in representing Eros’s shape and speed of computation. The bulk of the cpu time was taken up by computation of the gravitational force which scales linearly with the number of faces. The original Eros model (from which the polyhedra models were derived), was supplied by D. Scheeres (private communication). Data upon which this model is based are discussed by Miller et al. (2002). Figure 1 shows two views of the 1280-face Eros model used for most of the calculations. The top view looks toward the north pole of the asteroid. We use a coordinate system in which longitude 0◦ is (in this view) on the right end of the asteroid. Longitude increases counterclockwise toward positive values (“east” longitudes 0◦ –360◦ ). The upper panel of the figure shows the “convex” side of the asteroid with Himeros and Shoemaker; on the bottom can be seen the crater Psyche in profile. For reference, we give the coordinates of these features in the coordinate system we use:
Impact ejecta on Eros
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Fig. 2. Contour map of Eros: contours show radius in km as function of longitude and latitude relative to center of mass. Positions are marked by initials for Himeros (lat. = +21◦ , long. = +78◦ ), Shoemaker (lat. = −16◦ , long. = +30◦ ), and Psyche (lat. = +32◦ , long. = 266◦ ).
Himeros (lat. = +21◦ , long. = +78◦ ), Shoemaker (lat. = −16◦ , long. = +30 ◦ ), and Psyche (lat. = +32◦ , long. = 266◦ ). The lower panel shows an equatorial view along the y-axis, toward the side (near 270◦ ) on which Psyche is visible. For comparison below, we also show (in Fig. 2) a contour plot of elevation (radius) from the center of mass of Eros, generated from the 1280-face shape model shown in Fig. 1. The positions of Himeros, Shoemaker, and Psyche are marked by initials for reference. From each primary impact point we generated ten test particles that were launched with randomly chosen azimuths (0 < φ < 2π) and a launch angle of 45◦ from the local plane defined by the polyhedral face containing the primary impact point. The particle trajectories are integrated in the gravitational field of the polyhedral model. The effect of Eros’s rotation is included in the equations of motion. The angular velocity Ω = 3.31 × 10−4 s−1 corresponds to the rotational period P = 5.27 hours. (We also did non-rotating cases for comparison.) The density of Eros was taken to be ρ = 2.7 gm cm−3 . Test-particle integrations were done by the same algorithm (second-order leapfrog) as used in Korycansky and Asphaug (2003). Further details, including the full set of equations for a particle in a rotating frame and the time-step algorithm, are given in that paper. We used timesteps of 0.01τ and 0.05τ for the integrations, where τ = (Gρ)−1/2 = 2.36 × 103 s is a free-fall timescale for the body. No substantial differences in the overall outcome were found (i.e., the distribution of secondary re-impacts on the surface), but the smaller-timestep calculations were more accurate, as checked by the fractional change of the Jacobi constant CJ = v 2 + 2(Φ + ΦR ), where Φ is the gravitational potential (calculated by the PG algorithm) and ΦR = −1/2Ω(x 2 + y 2 ) is the rotational potential due to the rotating frame. For calculations with
timestep t = 0.01τ , about 90% of the integrations had relative changes in the Jacobi constant CJ /CJ < 0.01, decreasing to ≈ 40% for integrations with t = 0.05τ . Particle velocities were chosen randomly from a power law distribution such that the cumulative number of ejected particles with speed larger than v is proportional to v −1 , compatible with laboratory and theoretical results (Housen et al., 1983). (The exact choice of this distribution exponent does not matter much for the present suite of calculations, although for future calculations that look in detail at mass and angular momentum balance, it will be of greater importance.) The mass of Eros is 6.72 × 1018 gm, and the average radius ≈ 8.4 × 105 cm, yielding an average escape speed of 1.03 × 103 cm s−1 . For various simulations we used different minimum speeds for the velocity distribution vmin = 100, 200, and 500 cm s−1 . Using minimum speeds much less than vmin = 100 cm s−1 would produce calculations with so many slow-moving particles that the results would differ little from the distribution of primary impact points, which is uniform except for concave areas of the surface (Psyche and Himeros). The flight distance r = v 2 /2g for a particle launched at 45◦ in a constant gravity field g is ≈ 1.1 × 104 cm for v = 100 cm s−1 and g = 0.46 cm s−2 , (the area-averaged surface gravity on Eros), which is smaller than the dimension d ≈ 3 × 104 cm of the smallest faces on the 1280-face model. Thus, slow-moving particles (vmin 100 cm s−1 ) land in general on the same face from they were launched. Particles with positive total energy, or ones that reached distances larger than rmax = 108 cm, we considered to be escapers and were not followed any longer. We typically ran cases with ≈ 105 primary impacts, so that ≈ 106 test particles were followed for each case. When a particle was
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determined to have re-impacted the surface, we incremented a counter for the face that it crossed. Since the polyhedron faces have different areas, we normalized the counts for the face by dividing by the face’s area and scaling so that a uniform distribution would correspond to a surface density σ = 1 over the whole asteroid. We assume that re-impacting particles do not bounce or roll any significant distance; reimpacts are treated as being completely inelastic. Likewise, we have neglected other processes that may affect regolith distribution such as downslope motion due to forces described by Scheeres et al. (2002): acceleration due to gravity and rotational forces, friction, and surface shaking from impacts or close flybys of other objects.
3. Results and discussion In principle, the only free parameters in the calculation are those relating to the velocity distribution of the secondary ejecta, in particular vmin . The other parameters (the launch angle of the ejecta and the index of the power-law distribution) are constrained by experimental results (Housen et al., 1983). For general runs we chose vmin so that the launched ejecta will travel at least one crater radius from the impact center, although we also explored the effect of varying vmin . Monte Carlo calculations inevitably come with a certain amount of statistical noise due to the finite size of the sample. In this case, contour plots of the relative number of ejecta (primary or secondary) show this characteristic. Even if enough test particles are integrated so that several hundred land per face, face-to-face fluctuations of 10% or more occur. Thus, robust conclusions can be reliably made about broad features of the contour plots that we show. Plotting results as gray-scale maps helps the eye to see such features. The first question is whether primary impactors have a significant departure from a uniform coverage. We expect a uniform density of primary impacts for a convex body, which Eros largely is. A contour gray-scale contour plot of primary impacts is shown in Fig. 3 for two Monte Carlo simulations using the 1280-face Eros model, and a comparison with a similar calculation involving a model sphere of 1280 faces. The latter calculation, with 107 trials, shows a uniform distribution with small fluctuations of ±3%, equal to what would be expected from a Poisson distribution on the model sphere in which the mean number of impactors per face was N = 107 /1280 = 7.8125 × 103 (also taking into account the areas of individual faces on the model sphere). For the Eros calculations, we see variations (i.e., deficits) due to the concavity of the low, shadowed regions (Shoemaker Regio/Psyche/Himeros), which do not see a complete 2π steradians of sky. The expected range of a Poisson distribution on the 1280-face Eros model with 107 total impactors is ≈ ±8%. By contrast, the lowest density seen in Fig. 3, panel (b) is 0.77. The systematic variation clearest in the 107 -impact map, where the deficiency of the counts in the
Fig. 3. Shaded contour map of primary impact density on 1280-face models of Eros and a sphere, using the Monte Carlo method described in the text. Impact density is plotted relative to the average value that would result from a uniform density of impacts. (a) Primary impactor density on the Eros model resulting from a calculation with 106 impactors. Maximum and minimum gray-scale values are 0.65 and 1.19. (b) Primary impactor density on the Eros model resulting from a calculation with 107 impactors. Maximum and minimum gray-scale values are 0.77 and 1.05. (c) Primary impactor density on a model sphere resulting from a calculation with 107 impactors. Maximum and minimum gray-scale values are 0.97 and 1.03.
concave regions is evident. The same trend can be discerned in the 106 impact calculation, despite the larger fluctuations. (The level of the “noise” also gives an indication of the level of fluctuations and the robustness of the results for secondary impacts to be presented below.) Figure 4 shows the re-impact points for 10,000 test particles launched from random locations with random launch azimuths and 45◦ launch altitudes. The left-hand panels, (a) and (b) show results for a rotating ellipsoid polyhedron model of 1280 faces. The ellipsoid properties (axes, density, and rotation rates) were chosen to match those investigated by Geissler et al. (1996a), who modeled a variety of cases. In this calculation, the particles were launched at the average escape speed from the ellipsoid, 1.175 × 103 cm s−1 . Our results for the ellipsoid are very close to those of Geissler et al. (1996a), (cf. Fig. 3 of that paper), lending confidence to our results. A sizable anisotropy is present on the leading faces of the ellipsoid. A similar anisotropy is seen for the Eros-model calculation, with some differences due to Eros’s particular shape. A slightly larger proportion of such particles (∼ 67%) escape from Eros, as opposed to the ellipsoid (∼ 63%). Simulation results for secondary ejecta for a rotating Eros are shown in Fig. 5 for vmin = 100, 200, and 500 cm s−1 , in
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Fig. 4. Re-impact sites of points launched from random locations and random azimuths from an 1280-face ellipsoid polyhedron model [(a) and (b)] and the 1280-face Eros model used for the calculations. The ellipsoid has radii 18 × 7.5 × 6.5 km and density 2.724 gm cm−3 to match the dimensions and mass of the body studied by Geissler et al. (1996a). Particles were launched at an altitude of 45◦ and a uniform speed corresponding to the average escape velocity for each object: v = 1.175 × 103 and v = 1.024 × 103 cm s−1 . Panels (a) and (b): plots of x and z coordinates of points with y > 0 and y < 0, respectively, for the ellipsoid model. The results can be compared with Fig. 3 of Geissler et al. (1996a). Panels (c) and (d): plots of x and z coordinates of points with y > 0 and y < 0, respectively, for the Eros model.
the form of color-mapped contour plots. This particular set of orbital integrations was done with a time step t = 0.01τ . The plots show contours of ejecta surface density σ normalized to uniform coverage σ = 1. There are different numbers of escapers, as defined above, for each case. For vmin = 100 cm s−1 , about 12% of the particles escaped; for vmin = 200 cm s−1 , about 25%, and for vmin = 500 cm s−1 , about 58% escaped. The overall variation in σ is ≈ 2–3 between the minimum and maximum values. Systematic trends in ejecta density over the surface are apparent. The clearest result for all three of the cases shown [and similar calculations with a different timestep (t = 0.05τ ), or a coarser 320-face shape model] is that ejecta tend to pile up in the low regions of the asteroid (Psyche, Himeros/Shoemaker Regio) and are sparse on the high areas, particularly the “ends” at longitudes near 0◦ and 180◦ . This is consistent with the lower density of smaller craters in Psyche, Himeros, and Shoemaker Regio noted by Chapman et al. (2002), although downslope mass wasting could also exert a similar influence. There are hints of an extension of the Himeroscentered concentration (longitude ∼ +60◦ –90◦ ) into Shoemaker Regio, seen most clearly in the vmin = 100 cm s−1 and vmin = 200 cm s−1 calculations. This would be consistent with observations of regolith on that part of the asteroid (Robinson et al., 2002). However, statistical noise makes that identification uncertain. There are overall differences among the three cases depending on the minimum velocity. Due to the power-law nature of the ejecta velocity spectrum, test particles moving at the lowest speeds dominate the distribution [n(> v) ∝ v −1 as noted above]. The case with the lowest velocity would thus show the weakest “anisotropy” of ejecta coverage, as
more of the particles would move slowly and travel only short distances before re-impacting the asteroid. Thus we would expect larger differences from uniformity for the cases with larger values of vmin , which is largely what we see in panels (b) and (c) of Fig. 5. The vmin = 100 and 200 cm s−1 cases resemble (in reverse) the topographic map presented by Thomas et al. (2002). The resemblance in the 200 cm s−1 case is striking. In this case the gradients outside of Psyche and the east side of Himeros are particularly sharp. Overall in the vmin = 500 cm s−1 case, the concave side of the asteroid appears to pick up a considerable amount more material than the convex side. For this last case, the most obvious result is a strong deficit of material on the ends of the body, compared to the simulations with smaller values of vmin . This is probably due to the fact that a much larger relative proportion of ejecta have speeds comparable to the escape velocity for the this case. Thus, relatively more ejecta escape, and more ejecta are affected by rotational forces. Comparison with calculations for a non-rotating asteroid show essentially the same results as in Fig. 5. Thus, the rotation of Eros, while being about half that of the “critical” rate [e.g., Pravec et al. (2002)] at which a strengthless asteroid would begin to come apart, does not seem to have strong effects on the placement of ejecta. This is at first surprising result, given that the ends of Eros almost intersect its Roche lobe (Scheeres et al., 2002). However, Geissler et al. (1996a) find that ejecta launched at near-escape speed are the only ones that are strongly affected by rotation, at least in the case of Eros. A calculation by them involving particles launched on an ellipsoid at one-half vesc produced essentially isotropic coverage of the surface. Thus, rotational
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Fig. 5. Shaded contour maps of ejecta placement on Eros, from the Monte Carlo calculations described in the text. The integration timestep t = 0.01τ the rotation rate Ω = 3.31 × 10−4 cm s−1 . (a) Minimum secondary test particle velocity vmin = 100 cm s−1 . Total number of secondary impacts = 8.77 × 105 . Minimum and maximum values on the surface are 0.705 and 1.634. (b) Minimum secondary test particle velocity vmin = 200 cm s−1 . Total number of secondary impacts = 7.58 × 105 . Minimum and maximum values on the surface are 0.508 and 1.795. (c) Minimum secondary test particle velocity vmin = 500 cm s−1 . Total number of secondary impacts = 4.23 × 105 . Minimum and maximum values on the surface are 0.319 and 1.622. Note that the color bar scale is different (larger) than in Fig. 3.
effected would be reflected by only a small minority of fast moving ejecta (given the power-law distribution we use). In Fig. 6 we look at the relative numbers of escapers from the asteroid. That is, from each location, we kept track of how many particles escaped from that location. As expected, particles tended to escape from the ends (high spots), where the gravitational potential is lower and centrifugal effects are larger. As vmin increases, a higher proportion of particles escape and the salience of the ends (high spots) diminishes, as can be seen in the last panel. The relative number of escapers rises from 0.123 to 0.242 to 0.577 of the total number (106 ) ejecta for the three cases. We can combine the results of Figs. 5 and 7 to calculate a net gain or loss rate of mass. The mass input due to
Fig. 6. Shaded contour maps of the relative number of escapers on Eros, from the Monte Carlo calculations described in the text. The integration timestep t = 0.01τ . (a) Minimum secondary test particle velocity vmin = 100 cm s−1 . Total number of escapers = 1.23 × 105 . Minimum and maximum values on the surface are 0.149 and 1.786. (b) Minimum secondary test particle velocity vmin = 200 cm s−1 . Total number of escapers = 2.42 × 105 . Minimum and maximum values on the surface are 0.092 and 1.920. (c) Minimum secondary test particle velocity vmin = 500 cm s−1 . Total number of escapers = 5.77 × 105 . Minimum and maximum values on the surface are 0.196 and 1.694. Note that the color bar scale is different (larger) than in Fig. 5.
the primary impact is negligible compared to the mass of secondary ejecta in each impact (∼ 1% or less). The mass balance in a small region of the asteroid is then given by the gain (due to incoming ejecta) minus the loss (due to ejecta launched by an impact). The balance is easy to compute from the calculations, and is just equal to the density of secondary impactors minus density of primary impactors times the number of secondary particles launched per impact (10, in this case). The results are shown in Fig. 7 for the three cases of Fig. 5 in a reduced-gray-scale plot. The scale is normalized to the number of secondary ejecta per face of average area, ±106 /1280 = ±781.25. That is, a gain or loss of 781.25 ejecta in the calculation corresponds to ±1 on the gray-scale plot. As noted in the discussion of Fig. 6, there is a net loss of ejecta due to escapers. Most areas of the asteroid
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Fig. 8. Plots of relative density of secondary impacts vs. radius for the three cases shown in Fig. 5. The integration timestep t = 0.01τ and the rotation rate Ω = 3.31 × 10−4 cm s−1 . (a) Minimum secondary test particle velocity vmin = 100 cm s−1 . (b) Minimum secondary test particle velocity vmin = 200 cm s−1 . (c) Minimum secondary test particle velocity vmin = 500 cm s−1 .
Fig. 7. Plots of relative net gain/loss (accretion/erosion) per unit area for the three cases shown in Fig. 5. The integration timestep t = 0.01τ and the rotation rate Ω = 3.31 × 10−4 cm s−1 . The scale is normalized so that the number of secondary ejecta per face of average ±106 /1280 = ±781.25 corresponds to ±1. (a) Minimum secondary test particle velocity vmin = 100 cm s−1 . Minimum and maximum values on the surface are −0.791 and +0.783. Net loss of secondary particles is 1.23 × 105 /106 . (b) Minimum secondary test particle velocity vmin = 200 cm s−1 . Minimum and maximum values on the surface are −0.812 and +0.896. Net loss of secondary particles is 2.42 × 105 /106 . (c) Minimum secondary test particle velocity vmin = 500 cm s−1 . Minimum and maximum values on the surface are −1.05 and 0.313. Net loss of secondary particles is 5.77 × 105 /106 .
lose mass, in general, for the given assumed ejecta massvelocity distribution. Low-lying areas (Psyche and Himeros) would be expected to gain mass from slow-moving ejecta from other parts of the object. Taken at face value, this suggests that the regolith should be thinner or even non-existent on the ends of Eros. In turn, that would affect our assumptions about gravity-regime impacts; conceivably, the impacts on the objects will be in the strength regime and produce strength-dominated craters. Another way of looking at the results is shown in Fig. 8, where we plot the relative density of secondary impacts versus radius, for the same three cases of Fig. 5. In general, the density of ejecta is largest at the low points, i.e., smallest ra-
dius. The correlation is strongest for the vmin = 500 cm s−1 case, where there is a continued drop-off in density for larger radii. Note that we do not include the effect of particle bouncing. Sawai et al. (2001) find that the coefficient of restitution of bouncing material has a significant effect on its final resting place. Since we are looking at stochastic models, the effect would be moot if the bouncing material bounces isotropically. However, bouncing material tends to have a preferential downslope component. Incorporation of bouncing would thus enhance the effect we already observe, namely, that topographic highs lose material over time, and topographic lows accumulate it, despite the unusual shape and rapid rotation state of Eros.
4. Conclusion We have carried out a number of Monte Carlo simulations of impacts on the Asteroid Eros, with the goal of determining where secondary ejecta would tend to land and to estimate the relative depths of regolith on the object. Generally speaking, we find that the relative density of secondary impacts should be highest at the “low spots” (smaller radii) of the asteroid, in particular in the regions in and around the craters Psyche and Himeros. However, relative variations in secondary impact density are not extremely large: less than a factor of 2 relative to the mean, or a factor of ∼ 4 from
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maximum to minimum on the whole. If secondary ejecta dominate total mass in a collision (i.e., are more massive than the primary impactor) then most regions of the asteroid would be expected to experience a net loss of mass. Low-lying regions (Psyche and Himeros) would gain mass due to slow-moving ejecta (∼ 100–200 cm s−1 ) launched from other regions of the object. By contrast, the ends of the object should have a systematically shallower or even nonexistent regolith layer. Whether this is the case is not clear; the impression given by Robinson et al. (2002) is that a regolith is all-pervasive on the asteroid, although the interiors of Psyche, Shoemaker, and Himeros are indeed dominated by regolith features. However, the fact that these low-lying regions have not yet been completely filled with regolith implies that the creation of topography (e.g., by major impact events or by tidal encounters with planets) has occurred faster on Eros than the infilling of topography by stochastic small-scale cratering, which may be consistent with the lack of small recent impacts advocated by Chapman et al. (2002).
Acknowledgments This research was supported for D.G.K. under the NASA Planetary Geology and Geophysics program (NASA grant NAG5-11521-001) and for EIA by a grant from the NASA Discovery Data Analysis Program. We thank the reviewers, P. Thomas and P. Geissler, for helpful comments. Helpful information was provided by the website of D. Fanning (www.dfanning.com) in making Figs. 2, 3, 5, 6, and 7.
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