Abstract– We present results of a numerical model of the dynamics of ejecta emplacement on asteroid 433 Eros. Ejecta blocks represent the coarsest fraction of Eros’ regolith and are important, readily visible, “tracer particles” for crater ejecta-blanket units that may be linked back to specific source craters. Model results show that the combination of irregular shape and rapid rotation of an asteroid can result in markedly asymmetric ejecta blankets (and, it follows, ejecta block spatial distribution), with locally very sharp/distinct boundaries. We mapped boulder number densities in NEAR-Shoemaker MSI images across a portion of a predicted sharp ejecta-blanket boundary associated with the crater Valentine and confirm a distinct and real ejecta-blanket boundary, significant at least at the 3-sigma level. Using our dynamical model, we “back track” the landing trajectories of three ejecta blocks with associated landing tracks in an effort to constrain potential source regions where those blocks were ejected from Eros’ surface in impact events. The observed skip distances of the blocks upon landing on Eros’ surface and the landing speeds and elevation angles derived from our model allow us to estimate the coefficient of restitution, ε, of Eros’ surface for impacts of 10-m-scale blocks at approximately 5 m s−1 impact speeds. We find mean values of ε of approximately 0.09–0.18.
Knowledge of the surface properties of small asteroids and comets is important for relating astronomical observations of these objects to geologic “ground truth.” Regoliths on small bodies represent valuable natural laboratories for evaluating various models of impact cratering processes (e.g., Cintala et al. 1979; Asphaug et al. 1996) because they may present crater structures or ejecta features that either do not form or are hidden on higher gravity bodies like the Moon. Quantifying the extent to which impact processes generate and redistribute regoliths on small body surfaces (such as excavation depths, retained fraction, and turnover timescales) is pivotal to the issue of how to relate meteorite samples to their asteroidal parent bodies when surficial processes (i.e., “space weathering”) may disguise or cover up underlying material and confound the ability of remote sensing techniques to provide reliable mineralogical assays of the parent objects (e.g., McKay et al. 1989; Chapman 2004). Furthermore, a better understanding of the processes at work in these unique environments and the physical characteristics of the surfaces is crucial for designing technologies and techniques for future robotic and human exploration of NEAs, resource utilization, and impact hazard mitigation.
The Near-Earth Asteroid Rendezvous (NEAR-Shoemaker) mission to Eros presents an unprecedented opportunity to gain fundamental new knowledge about the processes governing regolith formation and redistribution on small bodies. NEAR-Shoemaker’s high-resolution images of Eros (Veverka et al. 2000, 2001) make the asteroid a valuable and heretofore unparalleled laboratory for the detailed study of impact ejecta reaccretion and regolith redistribution and evolution on low-gravity (of order 10−3 g) objects.1
Ejecta blocks represent the coarsest fraction of Eros’ regolith and are important, readily visible, “tracer particles” for crater ejecta-blanket units that may be linked back to specific source craters, thus yielding valuable information on physical properties of Eros (e.g., regolith structure and target strength) and constraining various aspects of impact cratering in low-gravity environments (e.g., ejecta mass/speed distributions and amount of retained ejecta).
These blocks, launched from the surface of a small, rapidly rotating, and highly elongated and irregularly shaped body, are subjected to a complex dynamical process. Dynamical models (e.g., Geissler et al. 1996; Durda 2004, 2009; see also Richardson et al. 2007) of reaccretion of impact ejecta thus provide important and necessary tools for a detailed investigation of the distribution and morphology of blocks and finer regolith across Eros’ surface as revealed by the NEAR MSI instrument.
A few good examples of boulders with tracks (Fig. 1) can be found on Eros (Sullivan et al. 2002). None of these appear to be the result of simple downslope rolling, based on locally “flat” topography and/or the morphological similarity of track depressions to secondary craters, suggesting that the boulders were emplaced directly at the termination of their suborbital, impact-induced trajectories. There are also several examples of small, oblong crater-like depressions, which, although not displaying obvious terminal boulders, have morphologies and small sizes suggestive of an origin by oblique, secondary impact. Eros is bombarded by impactors from interplanetary space, at some 5 km s−1, resulting in so-called “primary” impacts. Ejecta from these impacts may escape or may reimpact in a “secondary” impact. On bodies with high gravity, these secondary impacts occur at speeds comparable (in terms of sound speed in the target, and hence crater-formation mechanics and crater morphologies) to primaries. With the low gravity on Eros, these secondary impacts occur at low speeds, resulting in features that more resemble depressions, pits, or gouges than a classical impact crater. If these features seen on Eros’ surface are indeed formed by nonhypervelocity secondary ejecta particle impacts in the low-gravity surface environment on Eros, then those impactors may (1) have skipped far enough downrange so that their association with the reimpact feature is not readily apparent, (2) be below the limit of image resolution, (3) be buried below the terminus of the oblong track, or (4) have been composed of a clod of ejecta too weak to survive reimpact onto Eros’ surface.
These secondary, reimpact boulder tracks provide clues to the terminal trajectories of their parent impactors. In the case of boulders with tracks, the back-azimuth of the boulder’s terminal trajectory relative to Eros surface is unambiguous. For elongated secondaries lacking grossly asymmetric morphologies or not obviously blocked by high terrain or obstacles to one side or another, the incoming trajectory is at least constrained to two azimuths 180° apart. Without detailed knowledge of the state of compaction or cohesive strength of the regolith, track or secondary impact morphology alone is unlikely to be particularly diagnostic of the terminal trajectory elevation angle or impact speed.
However, knowledge of only the terminal trajectory azimuth may allow some constraints to be placed on candidate source-crater regions for terminal boulders or secondary impactors. This in turn can be used to study the process of primary impact-crater excavation and to place additional constraints on regolith and subsurface material properties by providing “ground truth” on boulder size/ejection speed relationships.
Description of Dynamical Model
The dynamical model is that used by Geissler et al. (1996), updated by Durda (2004), and here further modified to “back track” the derived landing trajectories of selected boulders. A similar, more numerically efficient model that uses many of the same scaling relations and experimental constraints is described by Richardson et al. (2007) and has been applied to the Deep Impact ejecta plume on comet Tempel 1. For our use here, particularly where modified and applied to backtracking specific ejecta blocks, our model continues to provide more than sufficient performance.
We now summarize the basic dynamical model. The volume of the shape model for the asteroid (Eros, in this case, defined by a 2° × 2° lat/long grid across the surface, with a radius from the center of the asteroid defined at each lat/long pair; Thomas et al. 2002) is filled with a uniform grid of point masses (2521 points with 1-km spacing) such that the total mass equals the mass of the asteroid for the density of 2.67 g cm−3 (Yeomans et al. 2000). The accelerations of ejecta particles are computed as the vector sum of the accelerations due to each of the point masses, and the positions of the ejecta particles and the (rotating) shape model are updated at each time step. The shape model rotation period is set to 5.27 hr to match Eros’ spin rate. In the original version of the dynamical model, all crater ejecta particles for an impact event were launched at t =0 from a point source (centered at a specified crater location) with a fixed ejection angle of 45° (local rotational vectors are added to launch velocities relative to the asteroid surface). Durda (2004) introduced modifications and improvements to (1) employ a more realistic annular model of the ejecta curtain, because, in reality, excavation flows do not emanate from just a point source, but encompass the entire center-to-rim extent of a forming crater, and (2) allow for a finite crater excavation time (although this feature is not needed or used in the present study).
The initial conditions for ejecta particle velocities and launch locations within the crater “footprint” are determined from the specified crater latitude/longitude and diameter. Ejecta velocity vectors are determined relative to the local Eros surface normal vector at the center of the crater location. That normal vector is computed as the average of the surface normal vectors in the nine 2° × 2° lat/long patches including and surrounding the lat/long patch of the specified crater. The crater “footprint” and ejecta particle launch locations are assumed to lie at the mean radius of the 2° × 2° lat/long bin containing the center of the specified crater; given the relatively coarse resolution of a 2° × 2° shape model, no distinction is thus explicitly made between the precratered surface and the existing crater floors.
The volume of ejecta (i.e., the number of ejecta particles) with speed greater than an arbitrary speed v is determined by the crater ejecta scaling laws of Housen et al. (1983), with the assumption of gravity scaling for Eros craters larger than 1 km in diameter. Specifically, Housen et al. show that for ejecta originating inward of the crater rim, where the ejection speed is large relative to the speed of ejecta near the final crater rim, the volume of ejecta with speeds greater than v scales as v^(-ev) (see their Table 1 and Eq. 28). Ejecta speeds are randomly generated and assigned between 1 and 20 m s−1 (the range of escape speeds on Eros is 3.1–17.2 m s−1 [Yeomans et al. 2000]), according to this power-law relation.
Table 1. Candidate source craters and their properties for boulder/track 1. The mean speed and mean elevation angle are the landing speed and elevation angle above the horizon that ejecta from the specified crater would land with at the boulder/track 1 location. ε is the derived coefficient of restitution of the surface regolith at that location.
Mean speed (m s−1)
Mean elevation angle
Once ejecta speeds are assigned, the ejecta launch locations within the crater footprint are set by the (randomly distributed) launch azimuth and with the radial location distributed from the center to the rim according to the ejecta speed. The scaled launch position within the crater footprint, xo/R (where R is the crater radius, taken here as the final crater radius), is related to the ejecta speed by:
where k1 and ex are constants (Housen et al. 1983, Table 1 and Eq. 17; Cintala et al. 1979, Equation 7 and Figs. 8 and 9). The constants (k1=0.441 and ex = 2.32, implying α = 0.53 and ev = 1.29; see Housen et al. 1983, Eq. 26) are best fits to the experimental data of Cintala et al. (1999) for impacts into coarse sand, extrapolated to an impact speed of 5.3 km s−1 (the mean impact speed between asteroids in the main belt [Bottke et al. 1994], where, because the dynamical lifetime of Eros’ non-main-belt orbit is short compared with its probable collisional age, most of Eros’ large craters were probably formed [Michel et al. 1998]). Richardson et al. (2007) present analytic expressions for these constants that yield k1≈0.514 and ex≈2.44 compared to our 2.32.
Experimental data (e.g., Cintala et al. 1999) show that ejection angles can vary substantially from the often-assumed 45°, with ejection angles decreasing gradually with increasing distance from the crater center, then increasing again near the final crater rim. Fig. 2 shows a fourth-order polynomial fit through the Cintala et al. experimental data that we used to set the ejection angle for each ejected particle after its scaled launch position is determined from its ejection speed.
The model presented here is an idealized representation of the crater excavation process. The model assumes axisymmetry, whereas for actual impacts, oblique impact angles and geological variations/inhomogeneities in the impacted terrain could cause significant asymmetries that would complicate the identification of boulder source regions derived from the model. Our model also makes some simplifying assumptions (e.g., that the ejecta scaling laws are valid as one approaches the crater rim) that probably break down in detail, although these should affect ejecta blanket emplacement in the more proximal portions of the ejecta blankets, not in the more distal regions we are focused on here. We refer the reader to Richardson et al. (2007) who present detailed model refinements addressing these issues.
Results and Discussion
Figure 3 shows an example from our model of the distribution of ejecta from a km-scale crater on a hypothetical asteroid shape model with Eros-like size and shape. Results showing the distribution of crater ejecta for non-rotating and rotating models demonstrate rather clearly that the combination of asymmetric/irregular shape and rapid rotation of an asteroid can result in markedly asymmetric ejecta blankets (and, it follows, ejecta block spatial distribution), with locally very sharp/distinct boundaries. It is thus crucial to include these effects in any analysis seeking to link predictions of ejecta deposit distributions to observations of asteroid surfaces.
Figures 4 and 5 show similar model results, but applied to specific craters (Valentine and Tai-yu) for the real Eros. Such maps of ejecta blanket boundaries provide the theoretical foundation and predictive basis for analysis of Eros surface imagery, emphasizing detection of ejecta deposits associated with particular impact craters. Our work here focuses primarily on (1) counts of ejecta blocks across targeted swaths of Eros’ surface where our dynamical simulations indicate that sharp/distinct ejecta deposit boundaries should exist, and (2) on backtracking the landing trajectories of the three boulders shown in Fig. 1.
Valentine Crater Ejecta Blanket Boundary
Here, we evaluate differences in block number density across a well-defined ejecta boundary predicted by our numerical model. This successful case study concerns the crater Valentine (15.7 °N, 207.4 °W, D = 2.34 km). Valentine is a large impact crater greater than 1 km in diameter that our models show should have formed a substantial ejecta blanket with some distinct, sharp boundaries (Figs. 4 and 5).
To locate high-resolution images to study, we searched for sections in our numerical simulation results where sharp ejecta blanket boundaries should have deposited boulders conveniently along one side of a line of longitude (i.e., perpendicular to the equator). We found one such region near 191 °W longitude between roughly 0 and 30 degrees of north latitude, corresponding to a section of ejecta blanket associated with the crater Valentine. Our ejecta models show that in this region, ejecta from Valentine should be tightly confined to longitudes greater than about 191 °W.
We studied the boulder density (i.e., total number of boulders per unit area larger than a specified size) in six images at the proper latitudes within six degrees of 191 °W longitude. We mapped boulder densities across the predicted sharp ejecta boundary at longitude 191°. A context picture of our six frames is shown in Fig. 6. We marked positions and dimensions of boulders within these images using the techniques and software that we developed for craters and have applied to Eros, Mars, Galilean satellites, and Mercury (a locally developed Perl/Tk graphical user interface that leverages the XPA interface to SAOImage DS9 [Joye and Mandel 2003] to measure and visualize manually defined elliptical features). A subsequent tool (the POINTS shape model software [Thomas et al. 2002]) converted the pixel locations of each boulder into latitude and longitude based upon the coordinates of the four image corner points. This procedure should be accurate to within a small fraction of a degree, assuming that the image is flat and has minimal camera skew. All six images meet these criteria.
In Fig. 7, we plot the longitude-binned boulder densities, which clearly rise sharply with increasing longitude within a degree of our target longitude of 191 °W. These results indicate a distinct and real boundary, significant at least at the 3-sigma level. The striking difference in observed boulder density across the ejecta blanket boundary predicted from our numerical model can be seen in Figs. 8 and 9, which respectively show a low-density boulder frame (#4 in Fig. 6) and a high-density boulder frame (#3 in Fig. 6). Boulders are marked in green in the right-hand panels of Figs. 8 and 9, with the unmarked image to the left for comparison.
Ejecta Block Source Craters
In this section, we “back track” the landing trajectories of ejecta blocks in an effort to constrain potential source regions where those blocks were ejected from Eros’ surface in impact events. We have modified the basic dynamical model to (1) reverse the direction of rotation of Eros, (2) limit the launch azimuths of particles to a small range of angles around the nominal terminal trajectory back-azimuth of the boulder track or oblong secondary crater, and (3) adjust the launch elevation above the local surface of the asteroid to a range of somewhat more oblique angles than the usual approximately 45° ejection angle for impact craters (as the morphologies of the boulder tracks and oblong secondary craters suggest a somewhat oblique impact angle; recall that here we are not modeling the process that ejected the boulders from their source craters, but setting up the “reverse” of the landing conditions that emplaced them back onto Eros’ surface).
We performed this “back tracking” process for the three boulder/track examples shown in Fig. 1. We refer to the three boulders and tracks in the left, middle, and right panels of Fig. 1 as boulder/track 1, 2, and 3, respectively.
Track 1 is located near 2.06 °N latitude, 213.48 °W longitude and appears as a very elongated, ellipsoidal crater with major axis oriented approximately 15 °E of north. With a large (approximately 40-m diameter), well-defined terminal boulder resting at the edge of a second, more circular secondary crater just north of the elongated track, it is clear that boulder 1 landed from the south. Potential source regions for this block, as determined by the “back tracking” process described above, are shown by the distribution of tracer-particle landing locations in Fig. 10a. The “footprint” of these locations represents the locus of potential source regions in which the launching of impact ejecta could have resulted in arrival at the boulder/track 1 site, along the landing azimuth inferred from the morphology of the track and the position of its associated boulder.
On the basis of scaling laws for the largest block ejected from a given size crater (Lee et al. 1996), we limited the search for candidate source craters within the potential source regions to craters larger than about 1.4 km across. Table 1 lists the 15 D >1.4 km candidate source craters and their properties, along with the mean ejection speed and elevation angle needed to reach those crater locations along the “up track” azimuth direction from the track 1 location. The launch speeds and elevation angles are, inverting to the landing conditions for boulder 1, the impact speeds and incoming elevation angle that boulder 1 would have impacted Eros’ surface if the boulder was ejected from each candidate source crater.
The oblong secondary crater and associated boulder track shown in the middle panel of Fig. 1 (boulder/track 2) are located near −2.41 °S latitude 179.17 °W longitude, within one of the “ponds” of fine regolith material within the topographic low of a small crater on Eros. The major axis of the large elongated secondary crater in this track is oriented approximately 1.5° north of east; the apparent small (approximately 2-m diameter) terminal boulder indicates that the boulder landed from the west. The potential source regions are shown in Fig. 10b. Due to ejecta dynamics near the elongated end of Eros, the source region is very tightly constrained. This small source footprint, the very small size of the terminal boulder, and results of impact experiments involving blocks in regolith simulants led Durda et al. (2011) to suggest that this fragment and its associated landing scars may have originated as a “chip” off a nearby impacted block. We note here that the source region footprint for boulder/track 2 does still include a subset of candidate source craters for boulder/track 1, however. Table 2 lists the derived mean landing speeds and elevation angles for those candidate source craters.
Table 2. Candidate source craters and their properties for boulder/track 2.
Mean speed (m s−1)
Mean elevation angle
Similarly, the source regions for boulder/track 3 also overlap those for boulders/tracks 1 and 2 to some degree (Fig. 10c; Table 3). Boulder 3, located near 18.48 °N latitude 188.29 °W longitude, is about 25 m in diameter and its associated track suggests a landing from the southwest on a landing trajectory oriented with an azimuth of about 45°. The size of boulder 3 suggests an origin from a crater larger than about 1 km in diameter. Although all three boulders may have originated in separate impact events, the relative paucity of other such clear examples of boulders with landing tracks across Eros’ surface suggests the possibility that all three are related to one another and originated in a single relatively recent impact event (recent because few other such regolith scars are seen, suggesting removal by ongoing erasure of small-scale regolith features) from one of the candidate source craters common to all three boulders/tracks. There appears to be only one crater common to all three potential source footprints: a 1.9-km-diameter crater near 4.32 °N latitude 29.24 °W longitude. This crater does not appear particularly fresh, however. Boulders/tracks 1 and 3 share a significantly larger subset of candidate source craters, suggesting that the hypothesis of origin in a common event may still apply to them and perhaps strengthening the idea that boulder 2 was formed as a fragment from a smaller scale impact into some other block on Eros’ surface close to its present location.
Table 3. Candidate source craters and their properties for boulder/track 3.
Mean speed (m s−1)
Mean elevation angle
The large crater Shoemaker (D =7.6 km) has not been considered in the discussion so far, but also is a potential source crater for boulders/tracks 1 and 3, as it is for many of the other boulders observed on Eros (Thomas et al. 2001). We suggest, however, that if blocks 1 and 3 share a recent common origin with most of the other boulders on Eros, and that all these originated from the Shoemaker impact as suggested by Thomas et al. (2001), then one might expect a greater frequency of observed tracks, given the vast number of boulders scattered across Eros’ surface. For this reason, we suggest a more recent, non-Shoemaker origin for boulders/tracks 1 and 3.
Coefficient of Restitution of Eros’ Surface
The observed skip distances of the blocks and the landing speeds and elevation angles derived from our models allow us to estimate the coefficient of restitution, ε, of Eros’ surface for impacts of 10 m scale blocks at about 5 m s−1 impact speeds. The ratio of the “skip” speed, Vs, to the initial landing speed, Vi, calculated from our models is a measure of the coefficient of restitution of the asteroid’s surface:
We make the assumption that “the angle of incidence equals the angle of reflection” for the process of an obliquely landing boulder skipping off Eros’ surface, and utilize the ballistic range equation,
where R is the range (in this case, the skip distance from the landing track to the boulder), V is the launch speed (i.e., in this case the “skip” speed, Vs), Θ is the launch angle relative to horizontal, and g is the local surface gravity, to calculate the track/boulder “skip” speed. For very long skip distances, we would need to employ the same dynamical model we used earlier to account for potentially complex boulder trajectories after skipping off the surface, but for the very short skips here relative to Eros’ global environment, the approximation of the simple ballistic range equation should suffice.
The skip distances for boulders 1, 2, and 3 are measured from the images to be approximately 225, 14, and 41 m, respectively. The number of secondary crater marks in track 3 is somewhat unclear given the image resolution, but whether the total skip distance is measured from the first certain large skip mark to the terminal boulder or between the two possible smaller initial skip marks at the limit of resolution, the skip distance relevant for the coefficient of restitution calculation is the same, about 41 m distance. We calculated g at each boulder/track location including the centrifugal effects of Eros’ rotation, finding g =4.4017 × 10−3, 2.7008 × 10−3, and 4.2367 × 10−3 m s−2 at boulder/track location 1, 2, and 3, respectively.
The resulting derived coefficient of restitution for each potential source crater and boulder/track location combination is listed in the last column of Tables 1–3.
The mean coefficient of restitution of Eros’ surface at boulder/track locations 1, 2, and 3 are derived to be approximately 0.173, 0.107, and 0.094, respectively. If we make the assumption that boulders/tracks 1 and 3 originated from a common impact event and limit the calculation of mean coefficient of restitution to the candidate source craters common to those boulders/tracks from Tables 1 and 3, we find mean values of ε at those locations of 0.181 and 0.089, respectively.
For comparison, the Hayabusa spacecraft unexpectedly measured the coefficient of restitution when it bounced on the blocky surface of Itokawa as approximately 0.84 for an impact speed of 6.9 cm s−1 (Yano et al. 2006). The relatively high coefficient of restitution of Hayabusa’s low-speed “impact” into Itokawa’s surface might be interpreted as regolith components acting as nearly solid rock at the size scale of the spacecraft contact area, not inconsistent with the appearance of the asteroid’s surface at those size scales. The ejecta blocks we examined are significantly larger (and more massive) than the Hayabusa spacecraft and impacted at speeds nearly two orders of magnitude greater than the spacecraft bounce speed. The observed “gouge” morphology of the tracks we examined suggests that the impacting blocks plowed at least several meters deep into Eros’ regolith at these locations, and it appears that considerably more impact energy was dissipated in these impacts than in the Hayabusa bounce. Conversely, Colwell and Taylor (1999) and Colwell (2003) investigated normal-incidence impacts of cm-scale Teflon and quartz spheres into powdery regolith simulants at speeds of 1–100 cm s−1, and found much smaller coefficients of restitution in the range approximately 0.01–0.03, nearer to the values we have derived.
Hayabusa images of the much smaller (approximately 530 m long) NEA 25143 Itokawa represent another incredibly rich source of data on the evolution of coarse regoliths.
Acknowledgments–– This work was supported by NASA Discovery Data Analysis program grants NNG06GF92G and NNG05GB96G. We thank Robert Gaskell for help in image interpretation and for providing his GLATLON code for calculating Eros’ surface gravity. This research has made use of SAOImage DS9, developed by Smithsonian Astrophysical Observatory. We thank Jim Richardson and Derek Richardson for their detailed and helpful reviews.