The geology and evolution of the Near-Earth binary asteroid system (65803) Didymos

Domain: earth_science
Published: 2024-07-30
Editorial outcome: major_revision

Abstract
Images collected during NASA’s Double Asteroid Redirection Test (DART) mission provide the first resolved views of the Didymos binary asteroid system. These images reveal that the primary asteroid, Didymos, is flattened and has plausible undulations along its equatorial perimeter. At high elevations, its surface is rough and contains large boulders and craters; at low elevations its surface is smooth and possesses fewer large boulders and craters. Didymos’ moon, Dimorphos, possesses an intimate mixture of boulders, several asteroid-wide lineaments, and a handful of craters. The surfaces of both asteroids include boulders that are large relative to their host body, suggesting that both asteroids are rubble piles. Based on these observations, our models indicate that Didymos has a surface cohesion ≤ 1 Pa and an interior cohesion of ∼10 Pa, while Dimorphos has a surface cohesion of < 0.9 Pa. Crater size-frequency analyzes indicate the surface age of Didymos is 40–130 times older than Dimorphos, with likely absolute ages of ∼12.5 Myr and < 0.3 Myr, respectively. Solar radiation could have increased Didymos’ spin rate leading to internal deformation and surface mass shedding, which likely created Dimorphos.

Introduction
The NASA Double Asteroid Redirection Test (DART) spacecraft successfully struck Dimorphos, the moon of the binary asteroid (65803) Didymos, on Sept. 26, 20221. Prior to colliding with the surface, the Didymos Reconnaissance and Asteroid Camera for OpNav (DRACO)2 imaged both Didymos and Dimorphos. These data were followed by images of Didymos and Dimorphos collected by the Italian Space Agency’s (ASI) LICIACube Unit Key Explorer (LUKE) several 10s to 100s after impact3. Although the encounter possessed some viewing limitations, these images provide the first well-resolved look of a small near-Earth binary asteroid system.

Telescopic observations of the near-Earth asteroid (NEA) population have shown that a sizeable fraction exist as binary systems4. These asteroid binaries are hypothesized to form when increases to the spin rate of a single rubble-pile asteroid causes it to mechanical fail, details of which depends on the asteroid’s geophysical properties5. The shape and surface morphology of asteroids are key to interpreting their evolution. However, while previous spacecraft observations provided insights into the geological characteristics of single asteroids such as Bennu and Ryugu e.g., 6,7, the proximity observations of the Didymos system collected by the DART mission provide a unique opportunity for a close-up geological look of an NEA binary system from which we can infer its geophysical properties and expand our understanding on their formation.

Here, we analyze the obtained images and digital terrain model developed therewith to provide constraints on the physical nature of these asteroids and on the geological origin and evolution of the Didymos system. In brief, our observations show that Didymos is flattened relative to other similar-sized asteroids and may have waviness along its equatorial perimeter. Its polar regions are rough and covered with large boulders, craters, and/or a plausible trough, while near its equator, Didymos’ surface is smooth, with few large boulders and craters. Dimorphos possesses an intimate mixture of boulders of all sizes, several asteroid-wide cracks or faults, and a handful of craters. The surfaces of both asteroids include boulders that are large relative to their host body, suggesting rubble-pile interior structures. Spin-up models indicate that to match observations, Didymos’ surface material is weak at ≤ 1 Pa, but its interior is slightly stronger at ∼10 Pa. Likewise, Dimorphos probably possesses weak surface material at < 0.9 Pa. Using these surface strengths and the number of observed craters, we find that Didymos’ surface is 40–130x older than Dimorphos, with likely absolute surface ages of ∼12.5 Myr and < 0.3 Myr, respectively. The evidence is consistent with radiatively-driven spin-up causing internal deformation and limited surface mass shedding of Didymos to form Dimorphos.

Results
The shapes of Didymos and Dimorphos are key to interpreting their evolution. Didymos’s shape is smaller along the polar axis relative to an earlier radar-derived shape model8 and other “spinning top-shaped” asteroids (e.g., Bennu, Ryugu) that have been previously visited by spacecraft (Fig. 1). Using best-fit ellipsoids, Didymos has an intermediate to major axis ratio b/a = 0.96, and a minor to major axis ratio c/a = 0.73. A cluster of generally top-shaped asteroids, which includes Bennu and Ryugu, sits near b/a ≈ 0.95, and c/a ≈ 0.9. We note, however, that radar shapes have larger uncertainties, usually along their z-axes, depending on the viewing geometry. Didymos possesses a ridge at, or near, its equator that is nearly triangular in shape (Fig. 2a). Unlike Ryugu or KW49, Didymos’ perimeter when viewed somewhat obliquely (Fig. 2b) or along its south pole (Fig. 2f) is not particularly circular, a trait that is reminiscent of Bennu, which has longitudinal (N-S) ridges that are apparent when Bennu is viewed from the same direction6. The squashed appearance of Didymos is broken by a few large protruding boulders that are observed on its limb, usually at moderate to high (>15°) latitudes. Some of the boulders are quite large relative to the overall size of Didymos, with the largest measuring ∼160 m across (Fig. 2b) or ∼21% of Didymos’ diameter.

Dimorphos’ uniform oblate (b ≈ a) spheroid1 shape may be unusual relative to other secondaries (Fig. 1), which are more typically elongated with 0.7 < b/a < 0.9, though there may be observational biases toward elongated objects10. Like for Didymos, Dimorphos’ profile is broken up by larger boulders (>10 m in effective diameter) strewn across the asteroid’s surface (Fig. 3).

Didymos can be divided into three geological units (Fig. 2c). In the limited views available of Didymos, regions at higher latitudes (poleward of 20°) are undulating and rough, though these rough regions extend towards the equator east of the prime meridian. These regions possess many large boulders (10–160 m in length)11, degraded craters (up to 270 m in diameter) and/or broad troughs. At lower latitudes between ∼10°S and ∼10°N, and west of ∼20°E, and, the terrain appears smooth at the resolution of the available images and lacks craters. This “smooth” region could be dominated by rocks and boulders that are smaller than the 5-m pixel scale of the best images and resembles what is seen in DRACO images of Dimorphos with comparable pixel scales. Preliminary albedo assessments12 indicate that this smooth terrain is darker than the surrounding terrain. Transition regions separate Didymos’ rough and undulating terrain from the smooth terrain on either side of the equator, showing evidence for regolith mass movements (Fig. 2a), including possible boulder tracks13. The terrains identified are consistent with the spatial distribution of surface roughness measured on Didymos14.

Didymos exhibits an exceptionally bright (∼20% brighter than average15;) spot within the transition region close to the subsolar point (Fig. 2a). This spot is close to the rim of one of the candidate craters and otherwise is not associated with any other prominent topographical or geological anomaly.

To some extent, Didymos’ geology reflects the effects of surface elevation7 (Methods). The rough terrain corresponds to highlands, while smooth terrains are lowlands, very reminiscent of what has been observed at Itokawa16. The importance of elevation was also noted at the fast-rotating Bennu (∼4.3 hr)6, but is less evident at slow-rotating Ryugu (∼7.63 hr)17.

Dimorphos has no distinct geological units and its overall albedo is slightly brighter than the average albedo of Didymos12. Its surface is covered by an intimate mixture of boulders11 (Udden-Wentworth scale; >1 m-in-length) and cobbles (6–100 cm-in-length; Figs. 3 and 4). The largest of the observed boulders is 16 m in length, too big to have been formed by the impact that formed craters observed on Dimorphos, which are either similar or smaller sized than this length. No evidence exists for large collections of fine particles with sizes below the pixel scale of the finest-resolution complete image (5.5 cm). Regions that might be construed as fine-grained in lower-resolution (>100 cm/px) images are revealed in higher-resolution images (< 10 cm/px) to be piles of smaller rocks. The counts of boulders on Dimorphos >2 m-in-length are considered statistically to be complete11, and the total number of boulders >2 m size per 10 degree bin shows no strong dependencies with latitude or longitude (Fig. 5).

Many of Dimorphos’ boulders possess evidence for linear cracks on their surfaces (Fig. 4a). Some of these cracks are likely the product of thermal fragmentation18. Multiple parallel lineaments are visible on a few individual rocks (e.g., Fig. 4c). The presence of these lineaments could be evidence for either layering, shatter cone-like structures, or albedo markings that could be due to shock12. While shatter cones are typically found in fine quartzite and dolomite, they have been identified in basalt19, which is similar to the likely ordinary-chondrite composition of Dimorphos. Some of the boulders possess plausible craters (Fig. 4b), and there may be a camp-fire structure (Fig. 4d) interpreted to be the result of boulder disruption from impact20 or thermal fragmentation.

Some individual rocks and boulders sit atop other boulders (Fig. 4c); however, many rocks lay alongside larger ones (Fig. 4c). This latter observation could be evidence for imbrication21 resulting from surface boulders migrating across the surface of Dimorphos systematically as a result of slope processes. However, these boulders are not preferentially located along one side of the other boulders, nor are they orientated with their long axes in the slope direction (Fig. 6) as would be expected if they had slid downslope. Our preferred interpretation, therefore, is that many of these boulders leaning or laying on other boulders are local debris aprons formed when rocks slid off and accumulated by their sides with no preferred orientation. These rock accumulations do not indicate widespread regional regolith transport.

The debris-apron interpretation is consistent with shallow (<10°) average surface slopes of Dimorphos (Fig. 6). When the roughness generated by the boulders is accounted for in the imaged sections of Dimorphos (Fig. 6b), the slope vectors point in random directions. It is unsurprising that smaller rocks on Dimorphos accumulate in debris aprons around bigger boulders rather than generate imbrication features. The low average global slope and the dominance of local roughness in minimizing local regional surface displacements further suggest that Dimorphos’ current shape and surface are stable, and likely have undergone only minimal changes between the time they achieved this configuration and the DART impact.

Dimorphos shows a few long, noticeable lineaments. A broad trough spans the northern portion of the asteroid. Long and narrow fractures are visible and are usually linear, although one of the most obvious ones is curved. In all cases, these narrow lineaments span significant fractions of the southeastern part of Dimorphos that was observed with DRACO (Fig. 4e, f). These lineaments are not obviously related to any crater candidate seen on the surface. In the best images collected, there also exists a linear fabric (Fig. 4d). Some of the smaller boulders align with it (Fig. 4c) but not in the slope direction11.

We identify sixteen plausible craters on Didymos. They were identified from DRACO and LUKE images, guidance from a global digital terrain model (GDTM) (Methods), a set of reasonable observational criteria (Methods), and consensus from several authors. Most crater candidates are located at higher latitudes. The largest ones (>160 m-in diameter) are degraded and resemble broad circular depressions seen on asteroids Ida (S-type)22, Eros (S-type)23, Itokawa (S-type)24, and Bennu (C-type)25. The general morphology of these larger circular depressions is probably due to the influence of the asteroid’s shape on crater growth25. The smaller craters possess a more classical bowl shape. Evidence for well-preserved, well-defined crater rims and ejecta is difficult to discern in the available images.

The DRACO images and a GDTM of Dimorphos permitted identifying twelve plausible craters on this asteroid using a methodology that differed from what was done at Didymos. Initial identification was performed by inspecting DRACO images projected onto the Dimorphos shape model using the Small Body Mapping Tool (SBMT26), yielding >22 candidate craters. These candidate craters were then vetted by evaluating their depths, diameters, and overall shapes using local DTMs (Methods).

The identified craters are similar to those observed on other asteroids with boulder-dominated surfaces24,27,28. They are depressions filled with a range of smaller rocks—which may represent disrupted material—surrounded by an uneven rim dominated by individual boulders. The mean geometric29 depth-to-diameter ratio of 0.152 ± 0.04 for craters on Dimorphos is similar to that found on the C-type Bennu (0.17 ± 0.04; see Supplementary Fig. 2), but deeper than on C-type Ryugu27 for craters of similar size Fig. 7). We caution against making any conclusive interpretations regarding these similarities or differences, as the quality of the underlying datasets used to produce the crater DTMs here are not as good as those for Bennu and Ryugu.

One of the craters identified presented evidence of a central mound (crater crt10 in Supplementary Table 4). The material comprising the mound appear to be an accumulation of bright and dark boulders. The other craters show no obvious evidence for either a bench or mound.

The crater cumulative size frequency distributions (CSFD) on both Didymos and Dimorphos (Fig. 7) indicate that Didymos has a higher density of craters and thus an older surface. We provide a range of plausible surface ages for both asteroids, using cratering strengths from Y = 10 to 105 Pa based on experimental data that have been used to construct crater-scaling relationships30 and supplemented with recent insight into the material strength of Dimorphos based on an analysis of the outcome of the DART impact31,32. The ages were calculated using the general form of the crater scaling relationship that includes both strength and gravity contributions with crater scaling constants for sand and cohesive soils30. We assume a main asteroid belt (MBA) impactor population for Didymos and a mixed population of NEA and MBA impactors for Dimorphos (see Methods).

We combine geophysical evidence from our surface observations with insights from the outcome of the DART impact experiment31,32 to better constrain the inferred ages of both Didymos and Dimorphos to understand the origin and evolution of these bodies. While there are many definitions of surface strength as applied to rocks, including compressive, tensile strength, and bulk cohesion C, we focus on C and take it to be equivalent to cratering strength, Y, because they are likely similar in magnitude32. Y is a key input needed to define surface ages.

Didymos’ surface cohesion, C can be estimated using a Factor of Safety (FS) analysis that was employed at Bennu33. This FS analysis provided a good match to surface cohesion estimates derived from analyzing interactions between the OSIRIS-REx Touch-And-Go sampling device and Bennu’s surface34,35. Assuming our current estimate of the density of Didymos (2800 ± 280 kg/m3), the rapid 2.26 hr rotation period of Didymos leads to steep slopes between ∼10° and ∼30° latitude (Fig. 8). Assuming that the surface material has friction angles of 35° as inferred on Dimorphos (see discussion), the FS analysis indicates that these slopes should see land sliding of material ≤10 m in thickness with near-surface C ≤ 1 Pa. We choose a 10 m-thick layer because this is approximately the thickness of the more easily mobilized layer on Bennu and Ryugu33,36,37, and when spread evenly across Didymos, provides enough material to generate ∼10x the volume of Dimorphos, far more than needed based on current estimates needed for the moon’s formation38. Thinner layers and a lower bulk asteroid density would require less cohesion33, while thicker layers and larger densities (within ±280 kg/m3) would require more cohesion but always < 2 Pa. For our nominal assumptions (density of 2800 kg/m3 and 10 m thick regolith layer), surface C > 1 Pa would shutdown surface displacements completely. The starts of a few boulder tracks and the source region of one of plausible landslide coincide well with regions where FS < 1 when C ∼ 0.5 Pa (Fig. 8). The maximum allowable C of 1–2 Pa is similar to cratering strength values estimated at the asteroid Ryugu36,39 and Bennu35).

While a near-perfect circular equatorial perimeter would indicate significant interior deformation of a fairly malleable object7, the plausible presence of undulations in Didymos’ equatorial perimeter, which maybe N-S ridges, would indicate that Didymos contains large distinct mass concentrations (large blocks) that influence the asteroid’s shape, perhaps analogous to findings at Bennu40. These mass concentrations likely require some strength to permit the perimeter to protrude in some locations. Other possible explanations for such protrusions and longitudinal (N-S) ridges at Bennu include structural wedges that resulted from an early quasi-disruption of the asteroid41 from spin-up. Such a scenario, however, was found to be unfeasible42: the forces needed to lead to quasi-failure were so large that the asteroid would have disrupted completely. This would be likely true for Didymos as well.

The presence of 10 Pa strength within Didymos is consistent with the evidence for surface mass-movements and mass shedding. Several studies show that asteroid spin up facilitates mass movement and mass shedding, especially for lower porosity objects like Didymos, if the interior possesses some strength40,43. The interior strength of Didymos, however, must not be so large that it prevents the asteroid from flattening (unless the asteroid were formed in a flattened state to begin with). To permit such internal deformation while also shedding mass, asteroid spin-up calculations first used at Bennu43, but adapted for Didymos, show that an interior average bulk C of ∼10 Pa (shear stress level at zero pressure) is required near the current spin-rate for a bulk density of 2800 kg/m3 and a friction angle of 35° (Fig. 9). The deformation would permit the observed flattening of Didymos and the non-circular equatorial circumference, as any interior mass concentration would shift outwards. Anything stronger than the C of > 20 Pa would prevent any interior deformation until Didymos spins up to < 2.21 hours. Interior strengths similar to the surface C ≤ 1 Pa would see the entire asteroid deform into a near pancake (which is not observed) before mass shedding begins. It should be noted that at the center of Didymos, the overburden stresses would exceed 100 Pa, and friction effects, rather than cohesion, dominate. The spin-up calculations also indicate that 10 m-radius boulders would need to travel more than 100 m along the surface to possess enough momentum to be ejected and enter orbit.

On Dimorphos, rocks and boulders sitting on larger boulders allow estimating strength properties of the unconsolidated surface of this asteroid. The analysis shows (Fig. 6d) that no rock or boulders sits on any slope exceeding 35°. In the case where no surface cohesion exists, this 35° angle must equal the greatest friction angle that can exist between rocks or boulders on Dimorphos. A separate effort that makes a detailed assessment of the shapes of the boulders on Dimorphos arrives at a near identical friction angle44. Nevertheless, since the friction angle has not been directly measured, we also consider the possible presence of cohesion. As an upper estimate of its value, we assume the smallest reasonable static friction angle possible for natural materials to be ∼25°45. Again, using the same FS approach above, for a 5 m-thick debris layer on Dimorphos, which matches the heights of some the largest boulders on the asteroid, we estimate at most a C ∼ 0.3 Pa to maintain the rocks on 35° slopes. Consistent with this estimate of C, calculations which match the observed momentum transfer observed from DART32, show that surface C cannot exceed 0.9 Pa if it is a 15 m-thick layer, while overlaying a stronger interior of Y > 500 Pa.

It should be noted that Y of individual boulders on Dimorphos likely significantly exceeds this 0.9 Pa upper limit for Dimorphos’ surface. We find evidence for at least two small craters on boulders (Fig. 4b). A systematic analysis of craters on boulders can be used to estimate the strength of surface boulders20 and such work is ongoing for Dimorphos. Both candidate boulder-craters possess a region that resembles spall surrounding their crater pit (Fig. 4b), not uncommonly observed for impacts into tough, non-porous rocks46. This likely indicates that the individual boulders on Dimorphos possess reasonable cratering strengths near several 10s of MPa typical of ordinary chondrites (a factor of 10 less than the dynamic compressive strength of an ordinary chondrite47).

We suspect that Y for Dimorphos may be between the strength holding the individual unconsolidated surface boulders together (≤ 0.9 Pa) and the individual boulders comprising the surface (10s of MPa). We interpret the aforementioned lineament fabric, long and narrow lineaments, and trough (Fig. 4d–f) as evidence that Dimorphos can sustain stresses over long (50–80 m), relatively straight distances via interior Y in excess of 0.9 Pa. Studies have shown that damaged rocks, which likely make up Dimorphos, are weak relative to their original, non-damaged constituent rocks, but are strong enough to allow lineaments to form via impact48, and can sustain fracturing from tidal stresses.

The geological evidence for Dimorphos, therefore, suggests that prior to DART’s impact, it possessed a well-connected and/or well-packed set of fragments, below the unconsolidated, low C (≤ 0.9 Pa) surface layer, whose bulk Y likely exceeds the maximum surface C of 0.9 Pa. A strength increase with depth is in line with interpretations obtained at Eros49, Bennu25,33, and Ryugu36. In the case of Eros (33 × 13 × 13 km), it has been proposed49 that the surface strength envelope might change with depth because of a transition from a loose surface regolith to a coarse megaregolith similar to what is seen on the Moon; for Ryugu and Bennu a range of hypotheses have been proposed, including the presence of a layer of cohesive fines near 5–8 m depth37,50. In the case of Ryugu, this layer had a compressive strength of 140–670 Pa34. From telescopic observations of the DART impact51, there is some evidence for fines (<100 μm) produced in telescopic data in volumes that may exceed what could have just been produced from the crushing of Dimorphos surface rocks alone. Perhaps Dimorphos’ apparent strength is the result of the presence of several large but damaged ordinary chondrite-like fragments that become increasingly packed at depth, with some cohesive glue created by fines.

Models of the DART impact indicate that the momentum change observed50 would require an Y anywhere between a few Pa to 10 kPa. A Y below ∼5 Pa implies cratering on Dimorphos is effectively in the gravity crater regime51.

The DART experiment31,32 and asteroid near-surface strength assessments on other asteroids33–36, provide an opportunity to connect surface strengths similar to those derived from observations of slope and numerical models to the Y used in scaling relationships from which absolute surface ages of asteroids are determined. By matching simulated values of the momentum enhancement factor, β, with the outcome of the DART experiment52, we31,32 were able to make predictions for the strength of the Dimorphos surface and the size of the DART crater. Here, we input the known DART impact energy1 and the predicted diameter range for the DART crater, D_DART = 30, 60, and 80 m, into the crater-scaling relationships to determine corresponding cratering strengths for Dimorphos, Y = 1300, 40, and 5 Pa, respectively. We note these values for Y reflect input parameter values for the strength scaling relationships, rather than a physically measured value of the strength of Dimorphos at the DART impact site, and do not account for effects of surface curvature.

The surface strengths inferred for both Didymos and Dimorphos indicate that smaller Ys are probably more reasonable. In the case of Didymos, we find a surface layer with C of ≤ 2 Pa covered by a slightly stronger interior near 10 Pa. The Didymos surface age estimate calculated with similar Y (1 to 10 Pa) yield a surface age near 12.5 Myr.

In the case of Dimorphos, the craters are small (≤10 m) in diameter suggesting that the impact energy required to create them is not very high. The impactors for these craters probably never penetrated below a few meters in depth, and so are less likely to have encountered a stronger substrate inferred by the lineaments. The lower-end of strength (0–0.1 kPa) estimates determined by numerical simulations of the DART impact31,32 and inferences from Ryugu are, therefore, preferred. Consequently, the surface age of Dimorphos must also be well-modeled by impacts in a weak (near the gravity regime) target, with a surface age of 0.09–0.3 Myr (Fig. 7c).

Due to the high boulder number density on the surface of Dimorphos, we cannot discount the possibility that impact armoring53 played a role in reducing the number of observable craters on the hemisphere observed by DRACO. A reduction in the population of craters <2 m was reported on Bennu28. As the number density of boulders on Dimorphos is larger than on Bennu11, we might expect that the transition into the armoring regime occurs at a larger diameter, maybe even at ∼5 m where we see the break in slope of the Dimorphos CFSD (Fig. 7). But this transition could be offset by the greater yield strength of individual S-type boulders relative to more crushable C-type ones that may reduce the efficiency of armoring54. The exact consequence on the true surface age of Dimorphos is, therefore, not exactly known and is likely a second-order effect compared to the poorly understood cratering strength.

Methods
Encounter conditions
The geometry of both the DRACO and LUKE data acquisitions limit the geological interpretations possible for both Didymos and Dimorphos. DRACO images of Didymos and Dimorphos were all collected on a fixed intercept geometry near 60° solar phase, which is suitable for morphological assessments of their surfaces, but less suitable for interpreting their surface albedos. Both asteroids were viewed at a single solar illumination. The limited phase and illumination changes that are observed in the DRACO images are only due to the surface curvature of both asteroids.

In the case of Didymos, the DART spacecraft flew past the object, collecting images with all range of emission angles that provided some stereo parallax. LUKE collected images with phase angles ranging from 60–120°, as the LICIACube spacecraft flew past and observed the southern (-Z) hemispheres of the two asteroids ≈130 s after DART’s impact3,71. About 40% of the total surface area of Didymos was observed with both cameras at a similar pixel scale of 5 m/pixel, with the best-resolved Didymos images collected by DRACO at ≈3 m/pixel.

In the case of Dimorphos (Supplementary Fig. 1), the impact of the DART spacecraft allowed for very limited stereo parallax1. In total, DRACO observed about 23% of the asteroid in solar illumination at a pixel scale of ≤0.29 m. The best-resolved, complete DRACO image was ≤0.055 m/pixel of a location surrounding the impact point. LUKE images of Dimorphos had fairly coarse pixel scales of 5.5 to 7 m/pixel and were heavily influenced by bright ejecta generated by the DART impact. As a consequence, they did not drive our geological interpretations of Dimorphos.

The images obtained by DRACO and LUKE were used to build global digital terrain models1,10 of both asteroids using stereophotoclinometry (SPC)72. Because of the lack of global coverage, the SPC-derived models were further informed by observed limbs, and the location of the observed terminator1. Reflected light from Didymos and the ejecta produced by the DART impact itself revealed portions of the shapes of both asteroids not directly illuminated by the Sun (Fig. 3).

Didymos digital terrain model
The Didymos GDTM was generated using stereophotoclinometry72 combining images from both DRACO and LUKE. Although DRACO data were key to establish the initial scale of Didymos because the DART trajectory and pointing were better known than LICIACube’s, the best and most consistent solutions for Didymos’ shape were obtained by weighing equally DRACO and LUKE data within the SPC framework. The DRACO data provided the initial scale of Didymos and some stereo that constrains the visible equatorial perimeter of Didymos. These data allowed registering the LUKE data obtained in a similar geometry. Additional LUKE images whose geometry differed from DRACO (imaging mostly the south pole of the asteroid) were then added to the system to complete the process of generating a GDTM and producing a well registered set of LUKE images. Note that DRACO limb information was not used in building the Didymos GDTM presented, which explains why the triangular shape of the ridge and boulder profiles visible in the limb of Fig. 2b were not captured in the shape (Fig. 2d and e). The original GDTM produced did have a center-of-mass (COM) to center-of-figure (COF) offset of (16, 42, −33) m. Given the uncertainties in spacecraft state (position and attitude) associated with the data that went into the model, we did not put much stake in this offset, and re-centered this COM-COF to zero. This re-centering of the Didymos GDTM was further justified given evidence that the unlit portions of the asteroid underestimated the extent of the asteroid limbs seen in back-lit images (Fig. 3b). The consequence of this underestimate is that the Didymos GDTM is slightly too small (in the a-b plane). The GDTM of Didymos produced here is 11% smaller in volume relative to the prior radar model, just within the 12% uncertainty (1-σ) associated with the radar model4. Supplemental Table M1 provides additional characteristics of the model presented. The uncertainties presented are the result of analyzes1,10,73–75, using limb and terminator assessments and surface feature location differences seen in the original images and renderings of the shape model.

Definition of surface elevation
Surface elevation (Fig. 2d) is computed as the difference between the gravitational surface potential and a reference potential divided by the magnitude of the local gravitational acceleration, g, considering reasonable estimates of the asteroid’s bulk density (2800 ± 280 kg/m3;31,76) and spin period (2.26 h77) at the time of writing. The uncertainty in bulk density will be resolved by the Hera mission and arises because of remaining uncertainty in the size of Didymos76.

Didymos crater identification
The identification of Didymos candidate craters was challenging due to two factors: 1) the fairly low imaging resolution available from both DART and LICIACube (Supplementary Fig. 1), and 2) the apparent ongoing surface mass motion that is probably a consequence of Didymos’ rapid rotation (2.26 hrs). In order to develop a plausible list of candidate craters, several of us independently assessed all the available images for Didymos with the aid of a Didymos GDTM. Although the GDTM employed was not good for measuring depths of plausible crater candidates, the model did provide key qualitative data on whether or not the candidate craters had a circular appearance, were deeper relative to their surroundings, or had other attributes associated with craters at other asteroids. The craters were identified using the images and GDTM and a set of observational criteria listed in Supplemental Table 2. These were intended to provide an objective means for establishing confidence in the craters identified, which totaled 16 (Supplemental Table 3).

An image of each crater is shown in Supplemental Material Table 3, along with its location and confidence rating.

Dimorphos crater identification and morphological assessment
We identified, mapped, and assessed crater candidates on Dimorphos through a series of three steps: 1) initial mapping, 2) a triage step using the global DTM, 3) morphological assessments using local DTMs of candidate craters that passed step 2.

Step 1. Craters were identified in DRACO images of Dimorphos by projecting images onto the Dimorphos v003 shape model1 using the SBMT26. The pixel scales of the images used to map craters are between 0.05 m/px and 0.23 m/px. Our initial search yielded 22 mapped candidate craters. Mapped craters had diameters that ranged from 2.9 to 11.1 m and were identified as circular features composed of smaller interior particles encircled by larger cobbles or boulders. These crater characteristics are similar to those of small (<10 m) craters on the rough surfaces of Bennu and Ryugu25,28,78.

Step 2. Using the highest resolution Dimorphos v003 shape model (average facet area of 0.024 m2), three linear cross-cutting profiles were mapped onto each of the 22 candidate craters. The mapped profiles provided the variation in height as a function of distance across each crater. These data were inspected to determine if any of the three profiles had a classical bowl shape or the presence of a central depression. Of the 22 candidate craters, 15 had at least one bowl-shaped profile.

Step 3. Local DTMs were constructed for the 15 candidate craters which passed the triage step. Using the DTM, the crater diameter, D, and depth, d, were measured using 8 profiles that cross-cut the center of the crater. The geometric height was used to measure the depth-to-diameter ratio of each crater. Of the 15 candidate craters, 12 were verified to be craters through this morphological assessment. These are shown in Supplemental Table 4 along with additional information on their diameter and depth.

Assumed impactor populations for calculating surface age
The surface ages presented here take into account the impact history of each body due to collisions with other smaller asteroids. Didymos likely originated in the main asteroid belt (MBA), where the population of impactors is 2–3 orders of magnitude greater than that in near-Earth space. As recorded by the CSFD of its largest craters, the surface age of Didymos most likely reflect that MBA impactor population79 before Didymos became a near-Earth asteroid. We use, therefore, an MBA impactor population distribution with intrinsic collisional probability, <P_i> = 2.86 × 10^−18 km^−2 yr^−1, and a fixed average representative impact velocity, U = 5.6 km s^−1 80. We also note that the CSFD of craters on Didymos with a diameter > 100 m follow reasonably well the MBA distribution with a −2.4 ± 0.2 slope in a log-log plot. Following the above assumptions, and considering a bulk density of 2800 kg m^−3, we calculate that Didymos’ surface age ranges between 12.5 and 675 Myr (Fig. 7).

In Dimorphos’ case, we consider a mixed population of near-Earth asteroids (NEA) and MBA impactors49, given its residence in the inner main belt (semimajor axis > 2.1 AU) for approximately a third of its current orbit. Our assumption is that the small CSFD observed on Dimorphos must have occurred recently and may reflect the time since Didymos rotationally disrupted through radiative torques (i.e., the YORP effect, see the Discussion section). Despite spending the majority of its orbit outside the main belt, the impact flux onto Dimorphos is still dominated by MBAs. Furthermore, NEAs with orbits that cross into the inner main belt can experience larger average impact speeds and impact probabilities81, and an asteroid with Dimorphos’ orbital parameters would thus have intrinsic collisional probability, <P_i> = 11.2 × 10^−18 km^−2 yr^−1, and a fixed average representative impact velocity, U = 7.1 km s^−1. We find that Dimorphos’ crater CSFD has a slope of −2.4 ± 0.5 for craters > 5 m-diameter, which falls within that of the MBA distribution, though the uncertainty is quite large due to small number statistics. Following the above assumptions, and considering a bulk density of 2800 kg m^−3 1, we calculate that Dimorphos’ surface age ranges between 0.09 and 11 Myr for the same Y between 10 and 10^5 Pa, respectively (Fig. 7). The surface age values we calculate, especially for weak surfaces, are much smaller than the dynamical lifetimes of NEAs82, which agrees with our initial choice of a mixed impactor-flux model for Dimorphos.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors analyze disk-resolved images of Didymos binary system collected by the DART and\nLICIAcube spacecraft in order to access its geology and evolution. In general, the study is well-\npresented and scientifically relevant.\nI provide some concerns below that might be relevant or could stem from my limited\nunderstanding of the subtopic, thus misunderstanding.\nI have issues with several claims in the abstract.\nFirst, \"Didymos has a non-circular equatorial perimeter \". This is inferred from image 2a, if I\nunderstand that correctly. However, given the orientation indicated by the axes, the viewing\ngeometry is far from being pole-on, perhaps by ~30 degrees. If so, then any interpretation\nconcerning the equatorial perimeter is invalid. Also, part of the projected disk is not illuminated,\nabout half of the contour is thus not defined. Some parts of the work use this claim as input for\nfurther interpretation. I am not sure if these interpretations are valid as well.\nSecond, \"At high elevations, its surface is undulating and contains large boulders and craters; at\nlow elevations, its surface is smoother and possesses fewer large boulders and craters.\" No\nboulder counts and statistics on Didymos are provided, so this is not supported by the data.\nThird, \"Dimorphos possesses a uniform surface covered by boulders.\" This is probably caused by\ninconvenient wording. This statement can also be understood in the sense that boulders are\ndistributed uniformly in size, which is not correct according to Fig. 5. Moreover, I have some\nconcerns about the analysis of the boulder statistics. The larger boulders (orange) are much more\nuniform than the smaller ones. Also, the histogram is not symmetric wrt the equator, but rather\nwrt the sub-spacecraft point. I wonder if viewing effects could corrupt/bias the boulder counts for\nlarger observing latitudes. The histogram looks suspiciously similar to the cosine function.\nFinally, \"If part of the Bauptistina family, Didymos likely represents the latest of multiple asteroid\ngenerations stemming from the original parent.\" This scenario is presented in the text as an\nalternative to a different scenario. Both scenarios are still valid and the selection of just one for the\nabstract is incorrect.\nLine 22 and Fig. 2a: The triangular shape is a local feature as it is not visible in Fig. 2d. Also, such\na feature is prone to viewing geometry (it is not equator-on). It would be useful to indicate the\norientation by the axes as it is done in Fig. 2b (instead of the arrow z+ which is not sufficiently\nindicative).\nFig. 2c: Three types of terrains. Could illumination/viewing geometry affect what you see? The\nsurface seems similar if going from north to south (or top to bottom) along the terminator. Clearly,\nthe three surface types are not preserved to the right (along the terminator), where shadowing\nreveals the roughness.\nI have downloaded the global digital terrain model of Dimorphos from\nhttps://lib.jhuapl.edu/papers/dart-an-autonomous-kinetic-impact-into-a-near-eart/ and I cannot\nmatch your figure 4f. I see a surface full of crater-like features in Fig. 4f, which is surprising and\nlooks unrealistic given the disk-resolved images (and the model I have downloaded).\nLine 317: Do we need a mechanism to flatten the shape of the primary? Could be already\nprimordial, right? Asteroids are not born spheres.\nLine 320: It is a speculation that is not addressed in the previous text.\nLine 324: it is not clear to me what \"are likely >15-m-diameter \" means.\nI found some typos in lines: 19, 50 (Fig. 2a?), 54 (missing comma?), 61 (missing space between\nvalue and unit, multiple times in the text), 155 (input), 267.\nInconsistency between near-Earth and Near-Earth.\nWeird numbering of sections (Acknowledgements and further).", "reviewer_2": "Referee comments on “The geology and evolution of a the Near-Earth binary asteroid system\n(65803) Didymos” written by Olivier Barnouin and others\nThis paper describes the first detail results of the geology on Didymos and Dimorphos studied by\nthe images taken from DART and LICIA cube spacecraft. The authors studied the shape and\nsurface geology of both asteroids and determined the crater size frequency distributions and the\nboulder surface distributions. They also conducted numerical studies for the factor of safely on\nDidymos, and found that the surface cohesion could be less than 1 Pa to reproduce the boulder\ntracks found on Didymos. Furthermore, according to the previous studies for the shape of asteroid\nBennu, they suggested that the interior cohesion of ~10 Pa was necessary for surface sliding, mass\nshedding, and some interior deformation. Similar to the discussion of Didymos, they also\nestimated the surface cohesion of Dimorphos and it was less than 0.9 Pa. Based on the surface\ncohesion of both bodies, the authors applied a crater scaling relationship for a crater chronology\non both asteroids. Then, they estimated the surface age of both asteroids from the crater size\nfrequency distribution. Finally, they found that the surface ages of Didymos and Dimorphos were\nvery young compared to the Bauptistina family, and they speculated that this means that the both\nasteroids could be the latest of multiple asteroid generations stemming from the original planet.\nThis paper is quite worthwhile for the advance of the study of rubble-pile bodies and binary-\nasteroids although most of the results may be confirmed or improved by the related mission of\nHERA, if it is succeeded. Moreover, this paper describes the geological features on both bodies\nbefore DART impact, so we can obtain quite important information by comparing pre- and post-\nimpact geology on these bodies after HERA mission is completed. I think that this paper newly\nclarified other aspects of rubble bodies compared to Bennu, Ryugu, and Itokawa. Therefore, I\nstrongly recommend the editor to publish this paper without delay. My major and minor concerns\nto this paper are described as follows. I appreciate if the authors answer to these comments.\nMajor comments:\nOne of the most important results of this paper is that the authors estimated the surface cohesion\nof the two rubble-pile bodies, and they are very small <1Pa. So, I think that Fig. 8 and Fig. 9 are\nimportant figures to determine them in this paper. Especially, the numerical calculation of Fig.8\nis very impressive for me to predict the boulder tracks. But there are some assumptions such as\ndensity, friction angle, upper layer thickness and rotation rate in this calculation. I am wondering\nhow the FS is sensitive to these parameters, e.g. how much dose the FS change when the density\nchanges from 3080 to 2520 kg/m3 or the friction angle changes from 20 to 40 degree? I think that\nthis is closely related to the estimation of the surface cohesion. In addition to Fig.8, Fig.9 is also\nimpressive for me. But I cannot follow how to derive several curves on Fig.9 (a). Ref.41 is cited\nin this part, but I don’t understand why 10 Pa cohesion is necessary in this model.\nMinor comments:\n1. Figure 3: There is no explanation for Fig. 3(a). Only (b) is described.\n2. Line 42: “gsd” is not so common word. Please explain what “gsd” is.\n3. Line 47-48: The following sentence is obscure, “The visual identification … on both\nasteroids”. Could you please rewrite clearly?\n4. Line 68-71 and Fig.5: These sentences indicate that the boulder number density decreases\nwith the increase of the latitude. I am wondering what materials occupy the inter space\namong boulders at high latitude. Is the space among boulders occupied by grains smaller\nthan 0.5m?\n5. Fig.6: I cannot imagine how to prepare Fig.6(c). What is the definition of slope? Is it an\naverage of some area?\n6. Line 89: Could you please explain “local debris aprons”? I don’t see any apron in the photo.\n7. Figure 7(a): What is µ in the caption? There is no definition of µ.\n8. Line 155: “nputs” may be typo. It may be “inputs”.\n9. Figure 8: There is no number “a” in the top figure of Fig.8. The caption “C=0.5 Pa” of Fig.8f\nmay be “C=1 Pa”.\n10. Line 177-178: What are the mass concentrations which induce the protrude? I cannot image\nit. Is it large blocks without porosity?\n11. Line 184-185: Theses sentences may have a contradiction because it says that the strength is\nneeded for mass-movements and mass shedding but the interior deformation is needed for\nthe flattening. Could you please explain it more easily?\n12. Line 196-198: I cannot understand the following sentences, “The deformation calculations\n… and enter orbit”. Please make clear. What is the deformation calculation and so on?\n13. Figure 9(b): This figure is so impressive to explain the assumption that produce the triangular\nridge and mass shedding. I recommend the authors to discuss this figure in the main text.\n14. Line 234: “low C (le 0.9 Pa)” may be typo. It may be “low C (<0.9 Pa)”.\n15. Line 274: What is the origin of the width of a surface age, 0.09-0.3Myr? Do you use the\ndifferent strength to obtain 0.09 and 0.3 Myr?\n16. Line 280: Why does the author adopt 5m for the limit of armoring effect?\n17. Line 394-395: The following sentence is obscure, so please rewrite it clearly, “Intermediate\nto small craters … below others”. What is “others”?\n18. Line 417: “global datasaet” may be typo. It may be “global dataset”", "reviewer_3": "This is an interesting paper detailing primary characteristics of the Didymos-Dimorphos binary\nsystem prior to the DART impact. Results here will be of broad relevance to NEO and asteroid\nstudies, in particular, estimates of material strength and surface/formation ages here provide key\nconstrains on system origin mechanisms and the overall (extremely young) system age. Methods\nand approaches utilized are generally state-of-the-art and well explained. I recommend its\npublication after addressing the minor issues indicated below.\nlines 17-18 For the non-specialists, please clarify the definition of intermediate, major, minor, and\nz axes. It is notable that both objects have equatorial circumferences that are much closer to\ncircular than most small bodies, i.e., they are much less ellipsoidal and are closer to oblate\nspheroids with a b/a near unity. This seems inconsistent with the tone of the first sentence in the\nabstract that instead emphasizes the equator of Didymos as being non-circular, when it instead\nappears to be remarkably near-circular.\nline 17 Text is missing in parentheses after Fig. 1 reference\nFig. 1 caption: Are the terms “flattened” and “oblate” meant to convey different things? If yes,\nplease clarify, and if not, use one term rather than both. Dimorphos seems similarly flattened, i.e.,\nit has a similar c/a, to the secondaries in the plot, which seems to contradict the statement in the\ncaption. What is notable is its high b/a, i.e., its more circular equator compared to other\nsecondaries.\nFig. 3 caption: Please insert a size scale bar. What is the white region on the right side of (a)?\nLine 61-63: 100-6 is an odd description; does this mean 6 to 100 cm in length? Please justify why\nis a 16 m boulder is “too big to have been formed by local impacts”.\nLine 155: nputs ◊ input\nLine 161: This is a surprisingly high density for a rubble pile, please comment.\nLine 175 “non-circular perimeter at Didymos”: Didymos is quite nearly circular at its equator (b/a\n= 0.96). Can the needed strength to account for b/a = 0.96 be quantified, and is this consistent\nwith the estimates based on surface sliding and mass shedding cited on line 190?\nLine 205: an near ◊ a near\nLine 293-294: Is it possible to use the inferred age differences to place any meaningful limit on the\nradiation torques that led to the spin up of the primary?\nLine 364-365: The accretion time of boulders in orbit around Didymos onto Dimorphos would seem\nto be extremely short. Is the tidal locking timescale short enough to explain why boulders are\nenhanced at low latitudes on Dimorphos, as suggested here? Or is it necessary to assume multiple\nmass shedding events from Didymos, with tidal locking occurring on much longer timescales\nbetween such events, as is described in hypothesis # 2 in lines 389-395?"}, "author_response": "Please find below our comments to three reviewers. This paper was greatly improved by their\ncomments and we really appreciate their helpful inputs. Our responses are in blue. Text that are\nblue and italic show the updates that we made to the text. We have also provided a revised\nmanuscript, plus a manuscript showing our changes relative to the original manuscript. Many of\nthe changes made that do not directly address the reviewers’ comments were undertaken because\nwe identified additional typos, and improved the text for clarity while replying to the reviewers.\nREVIEWER COMMENTS\nReviewer #1 (Remarks to the Author):\nThe authors analyze disk-resolved images of Didymos binary system collected by the DART and\nLICIAcube spacecraft in order to access its geology and evolution. In general, the study is well-\npresented and scientifically relevant.\nI provide some concerns below that might be relevant or could stem from my limited\nunderstanding of the subtopic, thus misunderstanding.\nI have issues with several claims in the abstract.\nFirst, \"Didymos has a non-circular equatorial perimeter \". This is inferred from image 2a, if I\nunderstand that correctly. However, given the orientation indicated by the axes, the viewing\ngeometry is far from being pole-on, perhaps by ~30 degrees. If so, then any\ninterpretationconcerning the equatorial perimeter is invalid. Also, part of the projected disk is not\nilluminated, about half of the contour is thus not defined. Some parts of the work use this claim\nas input for further interpretation. I am not sure if these interpretations are valid as well.\nResponse: Thanks for this comment, which must refer to original Figure 2b (not a). We have\nmodified Figure 2 to clarify our observation and avoid any confusion. Although we have kept\nthe arrows which show interpreted shape corners in Figure 2b, we removed the hashed circles in\nFigure 2b and any references to the off-axis observation. Instead, we have added a circular dotted\nline to Figure 2f, where we are looking directly at Didymos’ GDTM (shape model) from the -Z\npole. Both 2b and 2e images continue to show that Didymos is not circular (like Bennu and\nunlike Ryugu); but has plausible corners or edges like Bennu (due to its longitudinal, N-S\nridges). The Didymos model is not sufficiently good to do a spherical harmonics assessment as\nwas done for Bennu, to quantitatively confirm their presence, so this is best we can do until Hera\narrives. Because these features are plausible, we believe the interpretation is also plausible and\nworth mentioning.\nWe have edited the main text also. See lines 22-26:\nUnlike Ryugu or KW4 [5, and reference therein], Didymos' perimeter when viewed somewhat\nobliquely (Fig. 2b) or along its south pole (Fig. 2f), is not particularly circular, a trait that is\nreminiscent of Bennu, which has longitudinal (N-S) ridges that are apparent when Bennu is\nviewed from the same direction [6].\nSecond, \"At high elevations, its surface is undulating and contains large boulders and craters; at\nlow elevations, its surface is smoother and possesses fewer large boulders and craters.\" No\nboulder counts and statistics on Didymos are provided, so this is not supported by the data.\nResponse: Thanks for this comment as well: We forgot to add references to the submitted Pajola\net al. (2023), which is now also added (line 39 – to be revised for the same special issue in\nNature Comms that we submit this paper to; they provide their boulder data there). Their\nfindings (see figure above from their manuscript) are consistent with the visual inspection\nprovided here that we felt were fairly evident that regions at higher elevation possess more\nboulders than lows . There is an obvious band of smoothness at the low equator with no boulders\nvisible in the Didymos images that is lacking at the high poles (where many large boulders are\nvisible). The roughness assessments by Vincent et al. (2023; also submitted to the Nature\nComms special issue), which were indicated in the original manuscript (line 50), also indicate\nroughness correspondence with elevation.\nThird, \"Dimorphos possesses a uniform surface covered by boulders.\" This is probably caused\nby inconvenient wording. This statement can also be understood in the sense that boulders are\ndistributed uniformly in size, which is not correct according to Fig. 5. Moreover, I have some\nconcerns about the analysis of the boulder statistics. The larger boulders (orange) are much more\nuniform than the smaller ones. Also, the histogram is not symmetric wrt the equator, but rather\nwrt the sub-spacecraft point. I wonder if viewing effects could corrupt/bias the boulder counts\nfor larger observing latitudes. The histogram looks suspiciously similar to the cosine function.\nResponse: We have edited the wording “Dimorphos possesses a uniform surface covered by\nboulders” in the abstract with “Dimorphos possesses an intimate mixture of boulders”. Since\nsubmitting this paper, Pajola et al 2023 finalized his submitted paper on the boulder size\nfrequency distribution and noted that boulders about >2m are complete in the region mapped\nusing a statistical methodology (see figure below). For this size of boulder, we stop seeing any\nlatitudinal dependency on the size of boulders. We have updated Fig. 5 to reflect this change and\nhave updated the wording in the text (see lines 68-71). We also indicate in the discussion, that\nthis distribution of boulders better satisfies what would be expected for the formation of a\nsecondary (line 370-374): Their random distribution, regardless of latitude or longitude (Fig. 5),\nis consistent with models of secondary formation using large discrete element particles [40]\nwhere the satellite often forms with a non-principal axis rotation, and can accrete material at all\nlatitudes and longitudes.\nFigure showing boulder size frequency distribution on Dimorphos. Xmin provides a statistical\nestimate of the size of boulder above which the population is considered complete using a\nmethodolofy described in DeSouza, I., et al. 2015. Improved techniques for size-frequency\ndistribution analysis in the planetary sciences: Application to blocks on 25143 Itokawa. Icarus\n247, 77–80. https://doi.org/10.1016/j.icarus.2014.10.009\nFinally, \"If part of the Bauptistina family, Didymos likely represents the latest of multiple\nasteroid generations stemming from the original parent.\" This scenario is presented in the text as\nan alternative to a different scenario. Both scenarios are still valid and the selection of just one for the\nabstract is incorrect.\nResponse: Agreed. But this latter possibility is the most intriguing and plausible given the\ninferred weak material strength of the asteroid. We have removed this sentence in the abstract,\nbut clarify why we think this is a very likely possibility in the main text. See new text at line\n354-358.\nHowever, if the interior bulk cratering strength remains small, near the 10-20 Pa value inferred\nby the possibility that Didymos flattened, Didymos may be an Nth-generation rubble pile [68],\nmeaning that it did not directly originate from the catastrophic disruption of (298) Baptistina.\nRather, Didymos may represent the latest of multiple asteroid generations stemming from the\noriginal parent [68].\nLine 22 and Fig. 2a: The triangular shape is a local feature as it is not visible in Fig. 2d. Also,\nsuch a feature is prone to viewing geometry (it is not equator-on). It would be useful to indicate\nthe orientation by the axes as it is done in Fig. 2b (instead of the arrow z+ which is not\nsufficiently indicative).\nResponse: We have addressed this in the methods section now (line 656-682). The limbs from\nthe DRACO images are not included in the low fidelity Didymos shape model; this results in a\nlack of triangular shape in the GDTM as seen in Fig. 2d and e. The images do show this\ntriangular shape (Fig. 2a-c)_ and should be trusted; they are the real data. This addition of the\nlimb data would likely enhance the square appearance of Didymos when viewed from -Z pole.\nFig. 2c: Three types of terrains. Could illumination/viewing geometry affect what you see? The\nsurface seems similar if going from north to south (or top to bottom) along the terminator.\nClearly, the three surface types are not preserved to the right (along the terminator), where\nshadowing reveals the roughness.\nResponse: The three terrain types do not extend towards the terminator as shown in Fig. 2c (now\nupdated), but we do not think that it is a viewing issue. The incidence angle for much of the\nvisible portion of the asteroid seen by DRACO is at very suitable angles for identifying\nmorphology (see Method 1).\nI have downloaded the global digital terrain model of Dimorphos\nfrom https://lib.jhuapl.edu/papers/dart-an-autonomous-kinetic-impact-into-a-near-eart/ and I\ncannot match your figure 4f. I see a surface full of crater-like features in Fig. 4f, which is\nsurprising and looks unrealistic given the disk-resolved images (and the model I have\ndownloaded).\nResponse: The shape model you have downloaded is not the latest. The shape model used here is\nnow available via the SBMT and the PDS (see data availability section). This model is also\nexplained in detail in a separate paper now accepted at PSJ (Daly et al. 2024). This new model\nshows the distinct trough depression identified in the Fig 4f. It is resolved through\nphotoclinometry in the disk-resolved images, where variations in brightness illuminate the\ntrough. It you look closely at the images the troughs and lineaments are visible as well. We have\nnow added a star indicating the location of DART’s impact point, so it is easier for a reader to\nlink the image Fig. 4e with the GDTM view shown in Fig. 4f.\nLine 317: Do we need a mechanism to flatten the shape of the primary? Could be already\nprimordial, right? Asteroids are not born spheres.\nResponse: Agreed. And we have made it clear in earlier parts of the text that Didymos could\nhave been born flattened. To address this comment, we have re-affirmed this possibility in the\nbracketed text (line 328-329): “(to explain Didymos' squashed appearance if it were not born\nso)”.\nLine 320: It is a speculation that is not addressed in the previous text.\nResponse: In response to your comment, and one made by reviewer 2, we have now updated the\nsentence pointing out the observation of the triangle limb-shape of the asteroid discussed in Fig.\n4 and linking it to Fig. 9b to make it clear what we are discussing here (line 331-332): The\ntriangular shape of the Didymos ridge (Fig. 2a) may have been created by the mass shedding\nevent (Fig. 9b).\nLine 324: it is not clear to me what \"are likely >15-m-diameter \" means.\nResponse: Thanks for noticing this. With your above comment, we clarified this section by\nadding the following text (line 335-345):\nNumerical simulations [40] show that mass shedding events are relatively efficient at forming\nasteroid satellites, requiring only ∼ 2–3% of the Didymos volume to be shed to form Dimorphos.\nSuch a volume of material would be equivalent to a 3–5 m deep global layer (this layer would be\ndeeper if mass was shed from a localized area). This is insufficiently thick to wipe away the\nlargest craters on Didymos with depth near ∼15 m that define the age of Didymos. As a\nconsequence, we would expect, if the bulk shape is maintained, that the largest craters on\nDidymos could be relatively well preserved following a mass shedding event. While we cannot\nrule out effects of image resolution, the difficulty in clearly identifying craters <100 m-diameter\non Didymos may be the result of surface modification caused by the deformation, landsliding,\nand mass-shedding.\nI found some typos in lines: 19, 50 (Fig. 2a?), 54 (missing comma?), 61 (missing space between\nvalue and unit, multiple times in the text), 155 (input), 267.\nInconsistency between near-Earth and Near-Earth.\nWeird numbering of sections (Acknowledgements and further).\nResponse: We address all three of the above minor editorial issues. Thanks for finding them.\nReviewer #2 (Remarks to the Author):\nThis paper describes the first detail results of the geology on Didymos and Dimorphos studied by\nthe images taken from DART and LICIA cube spacecraft. The authors studied the shape and\nsurface geology of both asteroids and determined the crater size frequency distributions and the\nboulder surface distributions. They also conducted numerical studies for the factor of safely on\nDidymos, and found that the surface cohesion could be less than 1 Pa to reproduce the boulder\ntracks found on Didymos. Furthermore, according to the previous studies for the shape of\nasteroid Bennu, they suggested that the interior cohesion of ~10 Pa was necessary for surface\nsliding, mass shedding, and some interior deformation. Similar to the discussion of Didymos,\nthey also estimated the surface cohesion of Dimorphos and it was less than 0.9 Pa. Based on the\nsurface cohesion of both bodies, the authors applied a crater scaling relationship for a crater\nchronology on both asteroids. Then, they estimated the surface age of both asteroids from the\ncrater size frequency distribution. Finally, they found that the surface ages of Didymos and\nDimorphos were very young compared to the Bauptistina family, and they speculated that this\nmeans that the both asteroids could be the latest of multiple asteroid generations stemming from\nthe original planet. This paper is quite worthwhile for the advance of the study of rubble-pile\nbodies and binary asteroids although most of the results may be confirmed or improved by the\nrelated mission of HERA, if it is succeeded. Moreover, this paper describes the geological\nfeatures on both bodies before DART impact, so we can obtain quite important information by\ncomparing pre- and postimpact geology on these bodies after HERA mission is completed. I\nthink that this paper newly clarified other aspects of rubble bodies compared to Bennu, Ryugu,\nand Itokawa. Therefore, I strongly recommend the editor to publish this paper without delay. My\nmajor and minor concerns to this paper are described as follows. I appreciate if the authors\nanswer to these comments.\nOne of the most important results of this paper is that the authors estimated the surface cohesion\nof the two rubble-pile bodies, and they are very small <1Pa. So, I think that Fig. 8 and Fig. 9 are\nimportant figures to determine them in this paper. Especially, the numerical calculation of Fig.8\nis very impressive for me to predict the boulder tracks. But there are some assumptions such as\ndensity, friction angle, upper layer thickness and rotation rate in this calculation. I am wondering\nhow the FS is sensitive to these parameters, e.g. how much dose the FS change when the density\nchanges from 3080 to 2520 kg/m3 or the friction angle changes from 20 to 40 degree?\nResponse: Increasing the density to 3080kg/m3 has a fairly minor consequence to the findings\nhere. The surface slopes decrease and the cohesion needed to prevent surface failure changes\nfrom 1 to 0.5 Pa. A decrease in density to 2520kg/m3 would increase slope angles, and the\ncohesions needed from 1 to 2 Pa. Friction angles of 25 degrees would call for more cohesion\n(We don’t think 20 degrees is reasonable for any natural occurring soil). If you look at Barnouin\net al. 2022 in JGR Planets you can understand the effects of friction angle: steeper angles require\nless cohesion but these values of cohesion do not change dramatically with friction angle.\nRegolith thickness is probably the biggest factor. A 3m layer of regolith would require a\ncohesion of 0.3 Pa. To first order, our finding that the surface of this asteroid must have a\ncohesion of ~ 1Pa does not change much with all the variables mentioned. We have added the\nfollowing phrase in italic to the previous sentence at lines 169-173.\n“Thinner layers and a lower bulk asteroid density would require less cohesion [35], while\nthicker layers and larger densities (within ±280 kg/m3) would require more cohesion but always\n<2 Pa. For our nominal assumptions (density of 2800 kg/m3and 10m thick regolith layer),\nsurface C >1 Pa would shut down surface displacements completely.”\nIn addition to Fig.8, Fig.9 is also impressive for me. But I cannot follow how to derive several\ncurves on Fig.9 (a). Ref.41 is cited in this part, but I don’t understand why 10 Pa cohesion is\nnecessary in this model.\nResponse: We derive the failure curves in Fig.9(a) using the method described in Zhang et al.\n(2022; the original Ref. 41). Each failure behavior is associated with a specific failure condition\nand is represented by an independent failure curve. For the surface landslides (the orange curve)\nand mass shedding (the blue curves), we use surface slope distributions and evaluate failure\nconditions for different near-surface cohesions. For the interior deformation (the red curve), we\nuse the interior structural stress state distribution derived from a semi-analytical stress model to\nevaluate what cohesion failure (deformation) occurs. The structure of Didymos would undergo\nthe corresponding failure behavior if its surface/interior cohesion is smaller than the value given\nby each curve. Therefore, at Didymos' current spin period of 2.26 hr., as indicated by the cross-\nover of the red curve and the green vertical line in Fig.9(a), maintaining ~10 Pa of interior bulk\ncohesion is necessary to prevent its internal structure from deformation. ~1Pa at the surface leads\nto landsliding.\nWe have updated the caption to Figure 9 (new text in Italic):\nFailure and reaccumulation of Didymos. (a) Failure-mode diagram of Didymos for a density of\n2800 kg/m3. These curves mark the minimum cohesion required to prevent structural failure in\nDidymos across different spin periods. The analysis considers three potential failure modes\ncommon to rubble-pile objects. Each curve is derived based on the corresponding failure\ncriteria. Interior deformation that affects the entire asteroid occurs if C is below the red curve,\nwhich for Didymos…\nMinor comments:\n1. Figure 3: There is no explanation for Fig. 3(a). Only (b) is described.\nResponse: This is now addressed in response to this comment and reviewer 3. See new Figure 3\ncaption reads:\nComposite images of Dimorphos (DRACO image collected on Sept 26, at 23:14:09 UTC; a) and\nDidymos (LUKE images collected on Sept 26, at 23:17:27; b). The images are composites of two\nstretches to show the full outline of the bodies. Regions of Dimorphos illuminated by Didymos\nshine appear light-grey in (a). Arrows in (b) point to the dark limb of Didymos back-lit by DART\nejecta. In (b), the +Z (N) direction is tilted by a few 10s of degrees into the page. Dimorphos and\nDART ejecta can be seen in (b) at roughly 8:30 clock position from the center of Didymos.\n2. Line 42: “gsd” is not so common word. Please explain what “gsd” is.\nResponse: Thanks for catching this. We rephrased to refer to the pixel scale of images, rather\nthan GSD.\n3. Line 47-48: The following sentence is obscure, “The visual identification … on both\nasteroids”. Could you please rewrite clearly?\nResponse: Clarified now with (line 48-50) “The terrains identified are consistent with the\nspatial distribution of surface roughness measured on Didymos [14].”\n4. Line 68-71 and Fig.5: These sentences indicate that the boulder number density decreases\nwith the increase of the latitude. I am wondering what materials occupy the inter space\namong boulders at high latitude. Is the space among boulders occupied by grains smaller\nthan 0.5m?\nResponse: We no longer think this is true. The total number of boulders are identical at all\nlatitudes. See our response to reviewer 1 for more details. For boulders that a statistical approach\nshows we identify completely, we do not see a latitudinal or longitudinal dependence on their\nnumbers (see updated Fig. 5).\n5. Fig.6: I cannot imagine how to prepare Fig.6(c). What is the definition of slope? Is it an\naverage of some area?\nResponse: We compute the slope of each facet by using the dot product of the normal vector and\nthe local gravity vector. We weight the slope of each facet by their surface area in order to show\na slope distribution representative of what any region on the asteroid has. The peak in the\ndistribution shows the average area weighted slope of the oblate ellipsoid comprising\nDimorphos. The tail beyond 10 degrees is primarily due to the steep slopes of the boulders that\nare captured in the shape.\nWe have written some clarifications in the Fig. 6 caption: (c) The global distribution of slopes\nweighted by surface area.\n6. Line 89: Could you please explain “local debris aprons”? I don’t see any apron in the photo.\nResponse: This is an accumulation of rocks leaning or laying against a large boulder. They are\npointed out by the arrows. We clarified the text now (line 87-90):\nOur preferred interpretation, therefore, is that many of these boulders leaning or laying on other\nboulders are a type of local debris aprons, where rocks have just slid off larger boulders and\naccumulated by their side with no preferred orientation. These rock accumulations are not an\nindication of widespread regional regolith transport.\n7. Figure 7(a): What is μ in the caption? There is no definition of μ.\nResponse: μ is now removed. We define dashed red line (a) in a new caption for Figure 7:\nGeometric crater depth to diameter ratio (a; see [26] for definition) measured on Dimorphos,\nand crater size frequency distribution on Didymos (b) and Dimorphos (c). Dotted red-line shows\n(a) d/D results for Dimorphos from DTMs using the latest GDTM [10]. Didymos’ GDTM was of\ninsufficient quality to measure reliable d/D.\n8. Line 155: “nputs” may be typo. It may be “inputs”.\nResponse: Fixed\n9. Figure 8: There is no number “a” in the top figure of Fig.8. The caption “C=0.5 Pa” of Fig.8f\nmay be “C=1 Pa”.\nResponse: Both these issues are now fixed. Thanks for noticing.\n10. Line 177-178: What are the mass concentrations which induce the protrude? I cannot image\nit. Is it large blocks without porosity?\nResponse: Yes. That is basically what we think these are. We have added this to the main text to\nclarify (line 180): large distinct mass concentrations (large blocks)\n11. Line 184-185: Theses sentences may have a contradiction because it says that the strength is\nneeded for mass-movements and mass shedding but the interior deformation is needed for\nthe flattening. Could you please explain it more easily?\nResponse: Spinning asteroids require some strength to allow mass shedding – if they didn’t have\nthat strength, the asteroid would deform into a pancake and then eventually disrupt. However,\nthis strength can still be small enough to allow some deformation.\nWe clarified the paragraph where this line is located to address this confusion (line 188-206):\nThe presence of 10 Pa strength within Didymos is consistent with the evidence for surface mass-\nmovements and mass shedding. Several studies show that asteroids that asteroid spin-up\nfacilitates mass movement and mass shedding, especially for lower porosity objects like Didymos, if\nthe interior possesses some strength [e.g., 43, 46]. The interior strength of Didymos, however, must\nnot be so large that it prevents the asteroid from flattening (unless the asteroid were formed in a flattened\nstate to begin with). To permit such internal deformation while also shedding, asteroid spin-up\ncalculations first used at Bennu [43], but adapted for Didymos show that an interior average\nbulk C of ∼10 Pa (shear stress level at zero pressure) is required near the current spin-rate for a\nbulk density of 2800 kg/m3 and a friction angle of 35◦ (Fig. 9). The deformation would permit\nthe observed flattening of Didymos and the non-circular equatorial circumference, as any\ninterior mass concentration shift outwards. Anything stronger than the C of >20 Pa would\nprevent any interior deformation until Didymos spins up to <2.21 hours. Interior strengths\nsimilar to the surface C≤1 Pa would see the entire asteroid deform into a near pancake (which is\nnot observed), before mass shedding begins. It should be noted that at the center of Didymos, the\noverburden stresses would exceed 100 Pa, and friction effects, rather than cohesion, dominate.\nThe spin-up calculations also indicate that 10 m-radius boulders would need to travel more than\n100 m along the surface to possess enough momentum to be ejected and enter orbit.\nLine 196-198: I cannot understand the following sentences, “The deformation calculations\n… and enter orbit”. Please make clear. What is the deformation calculation and so on?\nResponse: To address this comment, we have clarified what we meant by “deformation\ncalculation by using instead “spin-up” calculation in this line (now line 204). We define what\nwe mean by ‘spin-up’ calculation further up in the text (line 194): To permit such internal\ndeformation while also shedding, asteroid spin-up calculations first used at Bennu [43], but\nadapted for Didymos show that an interior average bulk C of ∼10 Pa (shear stress level at zero\npressure) is required near the current spin-rate for a bulk density of 2800 kg/m3 and a friction\nangle of 35◦ (Fig. 9).\nFigure 9(b): This figure is so impressive to explain the assumption that produce the triangular\nridge and mass shedding. I recommend the authors to discuss this figure in the main text.\nResponse: In response to this and reviewer 1 comment, we have now added a reference to Fig 9b\nin the main text and added a reference to Fig 4a where we show the triangular shape limb of\nDidymos. See line 331-332 The triangular shape of the Didymos ridge (Fig. 2a) may have been\ncreated by the mass shedding event (Fig. 9b).\nLine 234: “low C (le 0.9 Pa)” may be typo. It may be “low C (<0.9 Pa)”.\nResponse: We fixed the text. It was indeed a typo (now line 241).\n15. Line 274: What is the origin of the width of a surface age, 0.09-0.3Myr? Do you use the\ndifferent strength to obtain 0.09 and 0.3 Myr?\nResponse: We have clarified what we did and referenced Figure 7c: We used the lower range of\nsurface strength inferred from our analysis and initial beta estimates because the craters we have\ndo not penetrate deeply in Dimorphos, and more likely encounter a weaker surface. See line 278-\n285:\nIn the case of Dimorphos, the craters are small (≤ 10 m) in diameter suggesting that their impact\nenergy is not vcery high. The impactors for these craters probably never penetrated below a few meters\nin depth, and so are less likely to have encountereda stronger substrate inferred by the lineaments. . The\nlower-end of strength (0–0.1 kPa) estimates determined by numerical simulations of the DART\nimpact [32–34] and inferences from Ryugu are, therefore, preferred. Consequently, the surface\nage of Dimorphos must also be well-modeled by impacts in a weak (near the gravity regime)\ntarget, with a surface age of 0.09–0.3 Myr (Fig. 7c)\nLine 280: Why does the author adopt 5m for the limit of armoring effect?\nResponse: We have now clarified the text now. See line 289-296: As the number density of\nboulders on Dimorphos is larger than on Bennu [11], we might expect that the transition into the\narmoring regime occurs at a larger diameter, maybe even at ∼5 m where we see the break in\nslope of the Dimorphos CSFD (Fig. 7). But this transition could be offset by the greater yield\nstrength of individual S-type boulders relative to more crushable C-type ones that may reduce the\nefficiency of armoring [58]. The exact consequence on the true surface age of Dimorphos is\ntherefore not exactly known and is likely a second-order effect compared to the poorly\nunderstood cratering strength.\nLine 394-395: The following sentence is obscure, so please rewrite it clearly, “Intermediate\nto small craters … below others”. What is “others”?\nResponse: This was revised (line 406-407): Intermediate to small craters on Didymos should be\npresent below some steep slopes, but heavily modified below others.\nLine 417: “global datasaet” may be typo. It may be “global dataset”\nResponse: We fixed this typo (line 429).\nReviewer #3 (Remarks to the Author):\nThis is an interesting paper detailing primary characteristics of the Didymos-Dimorphos binary\nsystem prior to the DART impact. Results here will be of broad relevance to NEO and asteroid\nstudies, in particular, estimates of material strength and surface/formation ages here provide key\nconstrains on system origin mechanisms and the overall (extremely young) system age. Methods\nand approaches utilized are generally state-of-the-art and well explained. I recommend its\npublication after addressing the minor issues indicated below.\nlines 17-18 For the non-specialists, please clarify the definition of intermediate, major, minor,\nand z axes. It is notable that both objects have equatorial circumferences that are much closer to\ncircular than most small bodies, i.e., they are much less ellipsoidal and are closer to oblate\nspheroids with a b/a near unity. This seems inconsistent with the tone of the first sentence in the\nabstract that instead emphasizes the equator of Didymos as being non-circular, when it instead\nappears to be remarkably near-circular.\nResponse: We see what look like N-S longitudinal ridges in some of the images of Didymos. We\nhave modified the abstract to clarify what we mean, both in response to this reviewer’s comment\nand reveiwer1. The abstract now reads: “These images reveal that the primary asteroid,\nDidymos, is flattened and has plausible undulations along its equatorial perimeter.” This\naddress both reviewer 1 and 2 concerns. We have also added a schematic in Figure 1 to make\nclear what we mean by the three axes, a, b and c, and clarify in the main text that these are the\naxis to the best fit ellipsoids: “Using best fit ellipsoids, Didymos has an intermediate to major\naxis ratio b/a = 0.96, and a minor to major axis ratio c/a = 0.73.”\nline 17 Text is missing in parentheses after Fig. 1 reference\nResponse: This is now fixed (see line 17).\nFig. 1 caption: Are the terms “flattened” and “oblate” meant to convey different things? If yes,\nplease clarify, and if not, use one term rather than both. Dimorphos seems similarly flattened,\ni.e., it has a similar c/a, to the secondaries in the plot, which seems to contradict the statement in\nthe caption. What is notable is its high b/a, i.e., its more circular equator compared to other\nsecondaries.\nResponse: We have removed the use of “flattened” and “oblate”, and just highlighted the\ndifferences in the axis ratio of Didymos and Dimorphos to other primaries and secondaries.\nFig. 3 caption: Please insert a size scale bar. What is the white region on the right side of (a)?\nResponse: We have added a white bar and clarified what the light-grey/white region is in Fig 1a.\nSee updated figure caption.\nLine 61-63: 100-6 is an odd description; does this mean 6 to 100 cm in length? Please justify\nwhy is a 16 m boulder is “too big to have been formed by local impacts”.\nResponse: We have clarified this section by adding the following sentence (line 63-65): The\nlargest of the observed boulders is 16 m in length, too big to have been formed by the craters\nobserved on Dimorphos, which are either similar or smaller than this length.\nLine 155: nputs à input\nResponse: Now line 153. We edited the line as recommended.\nLine 161: This is a surprisingly high density for a rubble pile, please comment.\nResponse: We comment on this in the discussion section (see lines 310-323). This is the current\nbest estimate for the density for Didymos, with the caveat that the shape of Didymos may be too\nsmall, and there are ongoing efforts to refine the mass of Didymos. Such a density (low porosity)\ncould be justified if the asteroid has large interior aggregates, and or fines that accumulate\nbetween voids at depth.\nLine 175 “non-circular perimeter at Didymos”: Didymos is quite nearly circular at its equator\n(b/a = 0.96). Can the needed strength to account for b/a = 0.96 be quantified, and is this\nconsistent with the estimates based on surface sliding and mass shedding cited on line 190?\nResponse: We have now clarified that protrusions in Didymos’ perimeter, that may be N-S\nridges are likely indicators of some form of mass concentration within the asteroid. These mass\nconcentrations are typically stronger (see lines 177-182).\nLine 205: an near à a near\nResponse: Replaced.\nLine 293-294: Is it possible to use the inferred age differences to place any meaningful limit on\nthe radiation torques that led to the spin up of the primary?\nResponse: Given the hypothesis given at the end of the manuscript, we do not think we can\nprovide constraints on the radiation torques until Hera arrives. Further, there are just too many\ndegrees of freedom since we also have to think about tides and BYORP which can add/subtract\nangular momentum. But we appreciate the thought.\nLine 364-365: The accretion time of boulders in orbit around Didymos onto Dimorphos would\nseem to be extremely short. Is the tidal locking timescale short enough to explain why boulders\nare enhanced at low latitudes on Dimorphos, as suggested here? Or is it necessary to assume\nmultiple mass shedding events from Didymos, with tidal locking occurring on much longer\ntimescales between such events, as is described in hypothesis # 2 in lines 389-395?\nResponse: Since submitting this manuscript, we determined that only boulders >=2m were\ncompletely observed, and that our inclusion of boulders <2m could have been biased by\nobservation conditions. Using this new constraint, we find no dependency of boulder totals on\nlongitude and latitude, which actually works much better with numerical models done by co-\nauthor Agrusa, which show that non-principal axis rotation is more likely than not when the\nsecondary first forms. This implies that multiple mass shedding events may not be necessary,\nalthough cannot be eliminated because it remains difficult to generate Dimorphos’ oblate shape.\nThese implications are now located in lines 363-376.\nFurther, the lack of any preference in the total number or size of boulders with latitude and\nlongitude, the presence of an intimate mixture of boulders and cobbles, and the lack of obvious\nfine regolith are all consistent with Dimorphos being the result of mass shedding from Didymos\nand gravitational re-accumulation. Material shed from Didymos is expected to form a ring of\nmaterial that includes all sizes around Didymos [43, 45, 69], except for the very finest (<100\nμm) particles. These finest particles get quickly swept away by solar radiation forces [e.g., 70],\nwithin hours as exemplified by observation of the DART tail [54]. The rest of the rocks\naccumulate quickly, in a handful of days. Their random distribution, regardless of latitude or\nlongitude (Fig. 5) is consistent with models of secondary formation using large discrete element\nparticles [40] where the satellite often forms with a non-principal axis rotation, and can accrete\nmaterial at all latitudes and longitudes. These large particle models, however, also usually\ngenerate an elongated secondary, which Dimorphos was not. Some additional accumulation\nprocess, therefore, may be required to explain its oblate shape.\nREVIEWERS' COMMENTS\nReviewer #1 (Remarks to the Author):\nI would like to express my gratitude to the authors for their thorough revision of the manuscript\nbased on the feedback provided by the referees. Addressing the comments and incorporating\nsuggestions has significantly strengthened the quality and clarity of the manuscript.\nI do not have additional comments.\nReviewer #2 (Remarks to the Author):\nI appreciate the authors to improve their manuscript according to my comments, and I am\nsatisfied with their answers to my comments and their revision. Therefore, I recommend the editor\nto publish this paper.\nI don’t have any comments and questions on this revised manuscript.\nReviewer #3 (Remarks to the Author):\nThe authors have responded to all of my concerns and I support publication of the current\nmanuscript."}]

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