Boulders on asteroid Toutatis as observed by Chang’e-2

Boulders are ubiquitously found on the surfaces of small rocky bodies in the inner solar system and their spatial and size distributions give insight into the geological evolution and collisional history of the parent bodies. Using images acquired by the Chang’e-2 spacecraft, more than 200 boulders have been identified over the imaged area of the near-Earth asteroid Toutatis. The cumulative boulder size frequency distribution (SFD) shows a steep slope of −4.4 ± 0.1, which is indicative of a high degree of fragmentation. Similar to Itokawa, Toutatis probably has a rubble-pile structure, as most boulders on its surface cannot solely be explained by impact cratering. The significantly steeper slope for Toutatis’ boulder SFD compared to Itokawa may imply a different preservation state or diverse formation scenarios. In addition, the cumulative crater SFD has been used to estimate a surface crater retention age of approximately 1.6 ± 0.3 Gyr.


Spatial and Size Distribution of Boulders
Considering the appearance and structure of Toutatis, we herein refer to the "head", "body" and "neck" to indicate the small lobe, large lobe and conjunction between them, respectively, following the descriptions of ref. 1. The spatial distribution of boulders on the imaged surface of Toutatis is shown in Fig. 1. A total of 222 boulders were identified over the imaged area (~6.68 km 2 ). They have sizes ranging from 10 to 61 m, with an average size of 22 m. The two largest boulders (> 50 m) are located in the neck region ( Fig. 1). Approximately 90% of boulders are no larger than 30 m in diameter.
The cumulative size-frequency distribution (SFD) of boulders per unit area is shown in a log-log plot in Fig. 2. We limit our analysis to boulders larger than 20 m in size, which exhibits a slope (power-index) of − 4.4 ± 0.1. In addition, we divide all boulders into two populations for the purpose of assessing the boulder distributions on the head and the body for comparison. By using the sharply perpendicular silhouette about the neck region as a boundary, over 50 boulders are counted on the head, and 160 rocks are identified on the body. Their separate distributions of boulders are normalized by the imaged areas of the head and body respectively, which give similar distribution trends with previous statistics 6 . Figure 3 shows that the slopes of size distribution of boulders above 20 m are − 3.9 ± 0.2 and − 4.5 ± 0.1, respectively.
The number density of boulders larger than 20 m over the Toutatis' surface is ~17/km 2 , which is a bit smaller than that of Itokawa (~30/km 2 ), but much larger than that of Eros (~2/km 2 ) 13,18 . The stark difference between Eros and Toutatis may result from the diverse formation process of boulders as discussed  -c)). The left side of Toutatis is sheltered by the solar panel (a,b), whereas c is the first panoramic image acquired by Chang' e-2. The resolution (R) and imaging distance (D) are shown. The Sun-Toutatis-Chang' e-2 phase angle for outbound imaging is ~6.4°. (d) The sketch map shows the outline of Toutatis, and the approximate positions of 222 boulders identified in this work. The two largest boulders with geometric mean sizes larger than 50 m are marked by black arrows around the neck region.
below. The number density of boulders larger than 20 m on the head (~15/km 2 ) is roughly the same as that on the body (~18/km 2 ), which bears a resemblance to that on the entire imaged side of Toutatis Toutatis' cumulative boulder SFD shows a slope of − 4.4 ± 0.1 for boulder sizes between 20 m and 60 m, which is significantly steeper than the slopes of − 3.2 for Eros boulders with sizes ranging from 15 to 80 m, and of − 3.3 ± 0.1 for Itokawa boulders with sizes ranging from 6 to 38 m 13,18 . Other small objects like the Martian moon Phobos also reports a similarly shallower slope 19 . The slope of cumulative boulder SFD steeper than − 2 is suggestive of very fragmented material, and a steeper slope is further indicative of greater degree of fragmentation 13,20 . Boulders could be comminuted by later impacts, which will steepen their size distribution because the largest sizes may be destroyed but not replenished 15 . The steep slope obtained for Toutatis implies that the boulders may have experienced much processing in the geological  evolution, including breaking, sorting, and transporting 13,17 . As most boulders above 20 m are probably surviving fragments from Toutatis' parent body as discussed below, relatively steeper slope may suggest that the body (− 4.5 ± 0.1) has suffered more energetic impacts than the head (− 3.9 ± 0.2) does. If this is the case, the larger fracturing of the body may be one of reasons leading to less dense internal structure than the head, consistent with differences in the observed ratios of moments of inertia 21 .

Source of Boulders
As discussed, boulders are usually produced on asteroids by impact cratering or/and catastrophic disruption of the parent body. In either case, only those boulders ejected at velocities smaller than the escape velocity of the final body can be re-accreted on the asteroid's surface. As for the impact cratering process, ejected fragments are somewhat associated with the source crater 13 .
Previous investigations revealed the correlations between the largest boulder size (L) and the source crater diameter (D) for the Moon, the Martian moons Phobos and Deimos, as well as the asteroid Ida 16,22 . One of the empirical relations is expressed as L ~ 0.25 D 0.7 , with L and D in meters, which was applied for Eros and Itokawa 13,23 . For Toutatis, the largest boulder is ~61 m in geometric mean diameter. According to the above empirical relationship, the corresponding source crater is ~2570 m, which is roughly four times larger than the largest impact crater (~530 m) on the surface of Toutatis 1 . Even for the depression at the big end of Toutatis, its size is only ~800 m. Due to the limited dimension of Toutatis, the probability of the existence of km-scale impact craters on the side not imaged by Chang' e-2 is very small. Moreover, the largest depression (possibly an impact crater) identified based on global radar-derived model was 750 m across 10 . Therefore, we infer that the largest boulder is not produced by the impact cratering. As a matter of fact, the maximum diameter of boulders generated by the largest impact crater (~530 m) on the imaged side of Toutatis is ~20 m. Also, there exist about 30 boulders with sizes ranging from 30 to 61 m discerned on the surface of Toutatis. For other small bodies, such as Phobos and Ida, boulders of this size range correspond to craters with 1-10 km diameter 22 , which are involved in the impact cratering. These observations provide strong evidence that most boulders (larger than 20 m) are probably surviving fragments from the parent body of Toutatis, re-accreted after the initial disruption.
In addition to the analysis above, we quantitatively examine the origin of boulders on the surface of Toutatis by estimating a total volume of ejecta represented by the boulders and a total volume of ejecta represented by the craters on Toutatis. In this work, 222 boulders with sizes from 10 to 61 m give a total volume ~1.8 × 10 6 m 3 , on the basis of the assumption 23 that the height of a boulder is half of its geometric mean dimension. There are fifty craters with diameters from 40 to 530 m identified from the flyby images of Toutatis, as mentioned by ref. 1. According to similar estimation method for asteroids Eros and Itokawa 13,23 , the volume of materials produced in the formation of these craters can be estimated to be ~1.9 × 10 7 m 3 , if the excavated volume is half that of the crater volume which is expressed as V ~ 0.07 D 3 , where D is crater diameter in meters. Thus, the boulders are about 10% the excavated volume of these craters, which would become higher if we take all boulders < 20 m into account which are counted incompletely in this work. The percentage on Toutatis (10%) is much larger than that on Eros (0.4%) where most boulders are believed to originate from the Shoemaker crater, but smaller than that on Itokawa (25%) where most boulders are of endogenic origin, i.e., fragments from the parent body 13,14 . The high ratio of the total volume of the boulders to the total excavated volume of the craters on Toutatis may also imply that most boulders cannot solely be created as products of cratering on the surface of Toutatis.

Surface Age
Although optical images of Toutatis obtained by the Chang' e-2 spacecraft have a significantly better resolution than that of the previously published radar-derived model 10,21,24 , the overall quality is relatively poor because they were acquired by the engineering camera, making the detection of degraded craters a herculean task. As a result, fifty craters with diameters from 40 to 530 m have been discerned and catalogued as impact craters with various degree of confidence 1 . Toutatis has an overall shallow slope of the cumulative crater SFD (Fig. 4), with a distinct shallower slope for diameter smaller than ~0.15 km. Note that this size is well above the spatial resolution, therefore the slope turnover is unlikely due to limited resolution. The observed roll over of the crater SFD at smaller sizes shown by Toutatis qualitatively resembles that of Stein crater SFD, also shown in Fig. 4 for a comparison. As suggested for Steins 25 , the rollover of the crater SFD at small diameter may indicate the presence of some crater obliteration mechanisms. Small craters can be erased by regolith displacement due to the cumulative effects of seismic shaking of small impacts or a large impulse of erasure triggered by the formation of large craters. Similar processes have been demonstrated to be common on small bodies like Steins, Ida, Eros, and Itokawa [25][26][27][28] .
In order to evaluate the crater retention age of Toutatis we estimate the impactor flux and convert impactor size to crater size by adopting a crater scaling law 25,29 . Given the likely highly fractured nature of Toutatis, we adopt a rubbly material crater scaling law, with the impact flux following the model for the dynamical evolution of asteroid belt 30 . As for the impact rate, we choose average impact condition for main belt asteroids. This is mainly because the time spent by Toutatis in the main belt is typically 100 times larger than the time spent on the near-Earth orbit, hence most of the observed craters on Toutatis are formed via collisions within the main belt. In order to estimate the crater retention age we fit the crater SFD at sizes > 0.15 km (Fig. 4), the resulting surface age of Toutatis is estimated to be ~1.6 Gyr, for rubbly material and a strength of 2 × 10 6 Pa, supporting the validity of our assumption 31 . The nominal age is relatively close to an age estimate for Steins (using a similar crater scaling law). We warn, however, that low number statistics (particularly at large crater sizes) preclude a robust age assessment for both asteroids. Moreover, the cratering age for a small asteroid is a function of its material strength, and weaker materials would result in younger ages. The formal error estimate for our age estimate is ± 0.3 Gyr, but in reality it is larger due to the uncertain factors described above.

Discussion
Boulders are ubiquitous on the surfaces of asteroids. For a relatively large asteroid like Lutetia, gravity is strong enough that boulders ejected from impacts are concentrated around the source crater. For small asteroids like Itokawa, most of ejected boulders must have been lost due to the low escape velocity, and those re-accreted are likely to be scattered globally on the surface. As far as the present investigation is concerned, most of boulders on the surface of asteroids appear to be connected to source craters, like the Shoemaker crater on Eros, the Stickney crater on Phobos and the central crater on Lutetia, whereas the distribution of observed boulders on Itokawa likely results from processes involved in Itokawa's re-accretion following the disruption of the parent body 13,15,23,32 .
Toutatis can be regarded as an intermediate body between Eros and Itokawa in many respects, for instance, body size, porosity, as well as the number density of boulders > 20 m and the ratio of the total volume of the boulders to the total excavated volume of the craters as mentioned above. As small-sized objects, Toutatis and Itokawa show a relatively uniform distribution of boulders, without local concentration around craters. The recent research shows that the size distribution and segregation of boulders may be correlated with the gravity field 33 . Boulders could also have been disintegrated by subsequent micrometeoroid impacts and thermal fatigue processes 12,34 . Thermal fragmentation induced by the diurnal temperature variations is considered to be a dominant mechanism of rock weathering and fragmentation on kilometer-sized asteroids 12 . Toutatis' boulders exhibit a clearly steeper slope than Itokawa, probably implying a different preservation state of boulders. If so, the preservation state of boulders may have something to do with the past orbital evolution. As discussed by ref. 35, near-Earth objects have a rather complex orbital evolution, with significant oscillations of the perihelion distance. The latter is important because the closer to the Sun the object is, the higher the thermal effects. Interestingly, numerical simulations 35 show that Toutatis minimum perihelion could have been as low as 0.02 AU with respect to 0.06 AU of Itokawa, suggesting a higher relevance of thermal effects for Toutatis. The best fit of cumulative crater SFD infers a cratering age of ~1.6 ± 0.3 Gyr, for rubbly material and a target strength of 2 × 10 6 Pa. Other parameters adopted for the age estimates are: target density (2.1 g/cm 3 ), impactor density (2 g/cm 3 ), and intrinsic collisional probability 2.86 × 10 −18 km −2 yr −1 corresponding to the average main belt value. The rubbly material crater scaling law is derived from ref. 36

Methods
We define boulders as rocks and features with (1) a clearly identifiable brightness variation and (2) a bright positive relief with shadow next to it, following the criteria in the identification of boulders on the Lutetia surface 15 . In order to investigate the size-frequency statistics of boulders on Toutatis, measurements were made mainly based on seven optical images with resolutions of ~3.6-8.3 m/pixel and imaging distances of 29.0-67.7 km 1 . When calculating the size of boulders, different approaches have been used. Ref. 17 took the maximum size as the geometrical diameter of a boulder. Afterwards, ref. 23 took the geometric mean of the long and short axes of a boulder. Recently, ref. 18 used a boulder diameter more representative of its mass or volume. The difference among these methods is owing to the deviations in definition of boulder size.
We cannot simply apply any of above methods to our measurements, as we have not connected the shape model of Toutatis with the images acquired by Chang' e-2. Furthermore, some facets of Toutatis imaged by Chang' e-2 are not perpendicular to camera shooting direction. In our work, we choose to use the maximum length of the pixel line as the size for boulders resembling spherical shapes. For irregularly shaped boulders, we determine their size as geometric mean of maximum size and minimum size. The real size of boulders is estimated by evaluating width of pixels defining each boulder. It should be noted that the largest boulder has a smaller size than that previously measured 1,6 . This is due to the difference in the measurements of boulder size. For the same boulder, the geometric mean in this work is smaller than the maximum in ref. 1.