The Brittle Boulders of Dwarf Planet Ceres

Boulders on Ceres can only be distinguished in framing camera images acquired in lamo at an altitude of around 400 km and lower orbits of the extended mission (Russell et al. 2007). The framing camera is a narrow-angle camera with a field of view of 5.5° × 5.5° (Sierks et al. 2011). lamo coverage of the illuminated surface was near-complete for the camera's clear filter, but color imaging was sparse. The clear filter (F1) is a polychromatic filter with 98% transmission in the 450–920 nm wavelength range (Sierks et al. 2011). lamo images were acquired between 2015 December 16 and 2016 August 27 and have a typical scale ⁴ of 35 m per pixel^–1 (Roatsch et al. 2017). The average scale of lamo images that we used in our analysis (at least one for each crater with boulders) is 35.8 ± 1.3 m pixel^–1. The boulder-finding procedure was identical to that followed for Vesta boulders (Schröder et al. 2020). In summary, the second author browsed the entire data set of lamo clear filter images and identified, measured, and mapped all boulders using the J-Ceres GIS program, which is a version of jmars ⁵ (Christensen et al. 2009), after which the first author reviewed the results for accuracy and completeness. Boulders were identified as positive relief features in projected images at a zoom level of 1024 pixels deg^–1. The lamo resolution is about 230 pixels deg^–1 at the equator, so this represents a zoom factor of about 4. Boulder size was determined using the J-Ceres crater measuring tool, which draws a circle around a boulder fitted to three points that are selected by the user on the visible boulder outline. The measurement uncertainty is about a single pixel. The limited accuracy of the pointing information for lamo images leads to mismatches between projected images. We used small craters inside and outside the crater as tie points to align the projected images to a Ceres background mosaic and relative to each other. All of this leads to uncertainty in the location of the boulders on the order of 500 m. We are confident that we could reliably identify boulders with a size of at least 3 pixels (105 m), although a criterion of 4 pixels (140 m) is more likely to ensure that mapping is complete (Schröder et al. 2020; Pajola et al. 2021). We did not distinguish between boulders located either inside or outside the crater rim, a choice that we justify in Section 3.3.

The illumination conditions at the time of imaging affect the visibility of a boulder, mainly by the strength of its shadow. The photometric angles at the center of the lamo images, calculated for an ellipsoid Ceres, are plotted as a function of latitude L in Figure 1. We see that the illumination conditions for imaging, and thereby boulder visibility, systematically changed according to latitude. The spacecraft looked at nadir most of the time (low emission angle), but the incidence and phase angle increased with L. If we distinguish three latitudinal zones as "low" (∣L∣ < 30°), "mid" (30° < ∣L∣ < 60°), and "high" (∣L∣ > 60°), the average incidence angle at the image center is ι = 48° ± 4° for low latitudes, ι = 62° ± 5° for mid-latitudes, and ι = 75° ± 5° for high latitudes. For a spherical boulder that is half-buried in a plane surface, the maximum length of the shadow is $l=r({\cos }^{-1}\iota -\cos \iota )$, with boulder radius r and incidence angle ι. A boulder at low latitudes will cast the shortest shadow, with l = 0.83r. A boulder with a diameter of 3 and 4 pixels will cast a shadow of 1.2 and 1.6 pixels, respectively. Especially for the larger diameter, the shadow is long enough to be well visible. Thus, although boulders will be more easily recognized in high-latitude images, we are confident that all boulders with a diameter of 4 pixels can be recognized in low-latitude images. Nevertheless, it is likely that we missed boulders with a 3 pixel diameter in low-latitude images due to their short shadows. In Figure 2, we investigate how the increase of ι with latitude affects our mapping. The figure shows lamo images of two craters with abundant boulders: high-latitude crater Jacheongbi (69°S, ι = 78°) and low-latitude crater Unnamed17 (10°S, ι = 42°). Both craters appear fresh, with boulders that are easily recognized by the shadows they cast on their ejecta blankets. At lamo resolution, the blankets appear equally smooth at either incidence angle. The rightmost panels show the distribution of boulders as mapped using J-Ceres. The stronger shadows of the Jacheongbi boulders did not lead us to recognize them in higher numbers compared to Unnamed17, which confirms that differences in visibility may only be consequential at the most extreme incidence angles (Wilcox et al. 2005). The figure also shows that half of Jacheongbi's interior is in shadow. Boulders are abundant on the sunlit half of the crater floor, which suggests that boulder numbers in high-latitude craters are severely underestimated, with potentially important consequences for the SFD.

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Various practices for displaying boulder SFDs seen in the literature were discussed by Schröder et al. (2020). In this paper, we display the SFD in both cumulative and differential format, using the incremental (binned, histogram) version for the latter (Colwell 1993). The cumulative distribution of boulders on solar system bodies is often assumed to follow a power law, for which the number of boulders with a size larger than d is

$$\begin{eqnarray}&&N(\gt d)={N}_{\mathrm{tot}}{\left(\displaystyle \frac{d}{{d}_{\min }}\right)}^{\alpha },\end{eqnarray} \tag{ 1 }$$

with α < 0 the power-law exponent and N_tot the total number of boulders larger than ${d}_{\min }$. The exponent of a cumulative distribution of a quantity that follows a power law is identical to that of the associated incremental differential distribution with a constant bin size on a logarithmic scale, if the logarithmic bins are chosen to be wide enough (Hartmann 1969; Colwell 1993). The power-law exponent is best estimated from the SFD by means of the maximum-likelihood (ML) method (Newman 2005; Clauset et al. 2009). The ML power-law exponent (α < 0) is estimated directly from the boulder size measurements as

$$\begin{eqnarray}&&\hat{\alpha }=-N{\left(\sum _{i=1}^{N}\mathrm{ln}\displaystyle \frac{{d}_{i}}{{d}_{\min }}\right)}^{-1},\end{eqnarray} \tag{ 2 }$$

with d_i the size of the boulder i and N the total number of boulders with a size larger than ${d}_{\min }$. The standard error of $\hat{\alpha }$ is

$$\begin{eqnarray}&&\sigma =-\hat{\alpha }/\sqrt{N}\end{eqnarray} \tag{ 3 }$$

plus higher-order terms, which we ignore. The estimator in Equation (2) is unbiased only for a sufficiently large sample size. Clauset et al. (2009) also provided details of a statistical test that evaluates whether a power law is an appropriate model for the data. ⁶ The test randomly generates a large number of synthetic data sets according to the best-fit power-law model (specified by $\hat{\alpha }$ and ${d}_{\min }$) and calculates for each the Kolmogorov–Smirnov statistic, which is a measure of how well the synthetic data agree with the model. A p-value, defined as the fraction of synthetic data sets that have a larger statistic than the real data set, quantifies how well the power law performs. The authors adopted p < 0.1 to mean rejection of the power-law model.

We fitted power laws to the global boulder population but also to the populations associated with individual craters to investigate possible variations of the exponent over the surface. To evaluate whether any such variations found are meaningful, we simulate the Ceres boulder population on the basis of the power law, using the observed population sizes of individual craters as input. For all craters, we adopt the same exponent, namely that of the power law that best fits the global population. The continuous power-law probability distribution is also known as the Pareto distribution (Newman 2005). To simulate a size distribution of boulders associated with impact craters, we draw a random variate U from a uniform distribution on (0, 1) using the randomu routine in IDL with an undefined seed. Then the boulder diameter

$$\begin{eqnarray}&&d={d}_{\min }{U}^{1/\alpha }\end{eqnarray} \tag{ 4 }$$

follows a Pareto distribution, with α the power-law index (associated with the cumulative distribution function, with α < 0). We adopted a minimum boulder diameter of ${d}_{\min }=140$ m (4 pixels). We simulated populations for all craters and estimated the power-law exponent for each using the ML method. We then compared the resulting distribution of exponents with the observed one.

The power-law exponent estimated according to Equation (2) is biased for low boulder numbers (Clauset et al. 2009), which was illustrated for simulated boulder populations by Schröder et al. (2020). Another potential source of bias is measurement error of boulder size. While the boulders described in this paper are large in absolute terms, they typically measure only a few pixels across, and measurement errors on the order of a pixel can be expected. Here we investigate the consequences of measurement errors using boulder populations simulated according to Equation (4). First, we consider a group of "craters," each with a simulated boulder population of different size. The SFD of all boulder populations follows a power law with exponent −4, with a minimum boulder size of 40 m. We modified the boulder sizes according to three different definitions of measurement error, where we distinguish between systematic and random errors. Measurement errors are sized 1 pixel of 35 m, equal to the spatial resolution of the Ceres lamo images. We estimated the power-law exponent for each crater, including only boulders larger than 4 pixels (140 m) in the fit. Note that we can only assess how many boulders have met this requirement after performing the simulation. Figure 3 shows the results. In panel (a), the boulder sizes are all measured correctly, and the power-law exponent is retrieved reliably for craters with large boulder numbers. The negative bias at small boulder numbers inherent in Equation (2) can be recognized clearly. In panel (b), boulder sizes are affected by measurement errors with a random character; sizes are either decreased by 1 pixel (size overestimated), increased by 1 pixel (size underestimated), or left unchanged, each with an equal probability of 1/3. Panels (c) and (d) explore the consequences of systematic measurement errors by either under- or overestimating all boulder sizes by 1 pixel. Systematic errors can be introduced by the method of measuring. Figure 3 shows that all types of measurement error lead to biased power-law exponents. Underestimating the boulder sizes increases the exponent by a little less than unity (shallower power law), while overestimating the sizes decreases the exponent by unity (steeper power law). The bias is stronger for overestimating the sizes than for underestimating, which causes the random measurement errors in panel (b) to decrease the exponent by a little less than unity (steeper power law). Another consequence of random measurement errors is that the exponent converges only at larger boulder numbers compared with the situation without measurement errors, which is not reflected in the formal uncertainty of the exponent (Equation (3)).

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Another aspect of this problem is the shape of the cumulative SFD. Figure 4 investigates how the shape changes for the same three cases of random and systematic measurement errors. For each case, we generated four populations of the same size, again with a power-law exponent of −4. In panel (a), the boulder sizes are all measured correctly, and the SFD follows the straight line of the power law up to a diameter of about 200 m. Beyond this size, the simulated curves diverge considerably due to chance. In panel (b), the boulder sizes are affected by measurement errors with a random character, which steepens the SFD slightly. In panel (c), the boulder sizes are systematically underestimated, which makes the SFD shallower and introduces a slightly convex curvature. In panel (d), the boulder sizes are systematically overestimated, which steepens the SFD considerably and introduces a slightly concave curvature. Our simulations demonstrate that bias due to measurement error is unavoidable when typical boulder sizes are on the order of a few image pixels. The results in Figures 3 and 4 may allow us to identify or predict such bias for the Ceres boulder population.

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The SFD of the Vesta boulder population is better described by a Weibull distribution than a power law (Schröder et al. 2020). We investigate whether the same holds true for the Ceres boulder population. The Weibull distribution was initially derived empirically and is often used to describe the particle distribution resulting from grinding experiments (Rosin & Rammler 1933). Where the power law follows naturally from a single-event fragmentation that leads to a branching tree of cracks that have a fractal character, the Weibull distribution results from sequential fragmentation (Brown & Wohletz 1995). Because we only include boulders larger than a certain size in the fit, we employ a left-truncated Weibull distribution with the cumulative form (Wingo 1989)

$$\begin{eqnarray}&&N(\gt d)=N\exp \left[-\alpha \left({d}_{i}^{\beta }-{d}_{\min }^{\beta }\right)\right],\end{eqnarray} \tag{ 5 }$$

where N is the number of boulders larger than ${d}_{\min }$. We estimate the Weibull parameters α and β = 3(γ + 1) from the boulder sizes ${d}_{i}\gt {d}_{\min }$ using the ML method. To maximize the log-likelihood function, these two equations must be satisfied:

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{N}{\sum ({d}_{i}^{\beta }-{d}_{\min }^{\beta })}\end{eqnarray} \tag{ 6 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{N}{\beta }+\sum \mathrm{ln}{d}_{i}-N\displaystyle \frac{\sum ({d}_{i}^{\beta }\mathrm{ln}{d}_{i}-{d}_{\min }^{\beta }\mathrm{ln}{d}_{\min })}{\sum ({d}_{i}^{\beta }-{d}_{\min }^{\beta })}=0.\end{eqnarray} \tag{ 7 }$$

We find $\hat{\beta }$ from a simple grid search and $\hat{\alpha }$ by inserting $\hat{\beta }$.

We identified a total of 4423 boulders on the surface of Ceres with a diameter larger than 3 image pixels (105 m), of which 1092 were larger than 4 pixels (140 m). All boulders are associated with impact craters. The details of all craters with at least one boulder larger than 4 pixels (n = 58) are listed in Table 1. First, we summarize some general statistics of the global boulder population. Figure 5(a) shows a (weak) correlation between the number of boulders per crater and crater size. The largest craters in the sample have a number of boulders that is much smaller than expected. For example, Occator is the largest crater with a diameter of 90 km, and it has 26 boulders larger than 4 pixels. From the general trend in the figure, we can expect to find a number at least an order of magnitude larger for a crater of its size. But many of its boulders may have been destroyed or hidden from view by the large-scale flows that are present in and around the crater (Schenk et al. 2019). In fact, three out of the largest four craters in our sample show evidence of such flows: Azacca, Ikapati, and Occator (Buczkowski et al. 2018b; Krohn et al. 2018; Schenk et al. 2019). Boulders around these craters are also difficult to distinguish from features resulting from partly submerged topography. The fourth crater in this group, Gaue, is very old (Schmedemann et al. 2016; Pasckert et al. 2018). Age must affect the number of boulders, as they are destroyed over time. The figure also shows the number of boulders associated with craters on Vesta (Schröder et al. 2020), where we adopted the same minimum size (140 m) as for the Ceres boulders. Clearly, the number of boulders produced on average by impacts on Ceres is larger than on Vesta for craters of the same size.

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Table 1. All Craters on Ceres with at Least One Boulder Larger than 4 pixels (d > 140 m)

Name	Longitude	Latitude	D	${d}_{\max }$	nd>3 pixels	nd>4 pixels	α	σα
	(°E)	(deg)	(km)	(m)
Azacca	218.4	−6.6	49.0	171	32	4
Braciaca	84.3	−22.7	7.7	179	17	1
Cacaguat	143.6	−1.2	13.6	353	115	26	−4.9	1.0
Cozobi	287.3	+45.3	22.4	182	28	6	−8.5	3.5
Emesh	158.2	+11.1	18.0	214	43	9	−7.1	2.4
Gaue	86.2	+30.8	79.0	254	29	8	−3.9	1.4
Haulani	10.9	+5.7	32.8	279	164	20	−7.5	1.7
Ialonus	168.5	+48.2	15.9	205	29	10	−7.7	2.4
Ikapati	45.6	+33.8	47.6	155	31	5
Jacheongbi	2.3	−69.2	26.8	498	473	160	−4.5	0.4
Juling	168.5	−35.9	17.3	202	85	22	−6.8	1.5
Kokopelli	124.5	+18.3	31.2	256	264	93	−4.6	0.5
Kupalo	173.5	−39.4	25.1	231	81	21	−5.7	1.2
Ninsar	263.3	+30.3	39.0	190	33	8	−7.7	2.7
Nunghui	272.3	−54.0	21.9	287	284	84	−5.1	0.6
Occator	239.4	+19.7	90.0	229	195	26	−7.0	1.4
Oxo	359.6	+42.2	9.1	172	5	2
Rao	119.0	+8.1	11.1	338	76	25	−5.2	1.0
Ratumaibulu	77.7	−67.3	18.2	219	98	23	−6.7	1.4
Sekhet	255.8	−66.4	38.0	259	167	59	−5.9	0.8
Shennong	28.1	+69.0	28.0	187	61	14	−6.5	1.7
Tawals	238.0	−39.1	7.9	160	15	2
Thrud	31.1	−71.3	6.5	140	18	1
Tupo	88.4	−32.4	31.8	218	97	27	−6.5	1.3
Unnamed02	10.8	−2.7	6.6	219	52	14	−6.3	1.7
Unnamed03	37.1	+34.8	6.5	221	10	2
Unnamed04	58.2	+58.6	10.1	214	21	2
Unnamed06	86.4	−10.7	5.3	225	18	3
Unnamed09	270.1	+50.2	8.7	174	16	3
Unnamed11	279.1	−23.0	15.0	386	209	74	−4.7	0.5
Unnamed12	307.0	−35.7	7.0	188	15	7	−7.5	2.9
Unnamed14	344.1	+18.1	9.3	248	35	11	−6.9	2.1
Unnamed15	244.4	−67.2	9.3	180	33	7	−8.0	3.0
Unnamed16	266.4	−34.3	13.3	267	181	46	−6.8	1.0
Unnamed17	21.1	−10.0	18.0	213	248	45	−7.9	1.2
Unnamed18	355.7	+33.6	8.0	174	26	6	−7.7	3.1
Unnamed19	153.1	−68.5	11.7	150	30	7	−18.2	6.9
Unnamed21	348.6	−3.5	7.4	179	24	8	−8.3	3.0
Unnamed22	62.4	+17.0	8.8	167	22	4
Unnamed23	232.0	−64.7	6.0	156	11	1
Unnamed24	312.7	−45.0	11.5	247	33	7	−5.0	1.9
Unnamed26	252.6	+62.9	21.5	237	44	12	−8.1	2.3
Unnamed27	191.6	−25.8	10.9	191	46	7	−7.5	2.8
Unnamed28	46.7	+79.7	13.9	176	42	10	−9.9	3.1
Unnamed29	199.1	−11.4	15.3	179	57	15	−8.7	2.2
Unnamed30	238.5	+50.0	20.3	237	159	45	−8.2	1.2
Unnamed31	276.5	−5.0	5.0	153	20	1
Unnamed32	300.8	+68.6	6.8	150	20	4
Unnamed33	19.6	−47.9	7.6	179	27	4
Unnamed34	111.0	−39.4	18.6	248	112	33	−6.8	1.2
Unnamed36	86.5	+58.8	9.6	215	22	4
Unnamed37	332.3	−52.9	10.3	265	35	7	−4.5	1.7
Unnamed42	131.7	+65.7	19.4	188	35	8	−6.0	2.1
Unnamed43	134.1	+68.0	15.1	266	68	18	−5.8	1.4
Unnamed44	350.0	+68.2	26.0	226	14	5
Unnamed48	162.1	−18.0	8.9	143	2	1
Victa	301.0	+36.2	28.9	184	39	12	−8.9	2.6
Xevioso	310.6	+0.7	8.3	216	16	3

Note. Crater and boulder diameters are D and d, respectively, and α is the power-law exponent of the (cumulative) boulder SFD as derived with the ML method (only for craters with n_{d>4 pixels} > 5).

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We investigate the relation between the sizes of the largest boulder (L) and of the crater (D) in Figure 5(b). Being a single measurement, the largest boulder size is a poor statistic but is nevertheless often used to characterize boulder populations (Thomas et al. 2001; Küppers et al. 2012; Jiang et al. 2015; Schulzeck et al. 2018b). We compare the Ceres distribution with the relation provided by Lee et al. (1996) for craters formed in rocky targets (L = 0.25D^0.7 with L and D in meters) and the empirical range established by Moore (1971) for a selection of lunar and terrestrial craters (L = 0.01^1/3 KD^2/3, with K ranging from 0.5 to 1.5). The former relation represents more or less the upper limit of the latter range. The largest boulders on Ceres do not agree well with either relation from the literature, with sizes that are almost independent of the size of the craters. Again, it is mostly the largest craters that break the trend, probably because of the aforementioned flow features and old age. The largest boulder we found on Ceres is a 500 m large block on the rim of Jacheongbi (figure inset), a relatively large crater (27 km). The figure also shows the largest boulders of the Vesta craters. While the Vesta data also do not perfectly agree with either relation from the literature, most fall within the range of Moore (1971). On average, the largest boulders on Vesta are somewhat smaller than those on Ceres.

We aggregate all boulders counted on the surface of Ceres to find the cumulative power-law exponent of the global boulder population. We note that the resulting global SFD is biased, as the boulder populations of the largest craters were almost certainly decimated by large-scale flows (see Section 3.1). Figure 6 shows the SFD in both cumulative and differential representation. At the top of the differential plot, we show the uncertainty in size resulting from a 1 pixel measurement error. We chose a logarithmic bin size of 0.07 with the boulder size in meters, ensuring that the size is on the order of the measurement error at the larger end of the scale. As the error is larger than the bin size at the smaller end of the scale, we can expect boulders to end up in adjacent bins merely by chance. We recognize the characteristic rollover of the distributions toward smaller diameters, caused by the limited spatial resolution and the measurement error. We fit two power laws to the data with the ML method, one with the minimum boulder size (${d}_{\min }$) fixed and the other with ${d}_{\min }$ estimated by the ML algorithm. When fixing ${d}_{\min }$ to 4 pixels (140 m), we find a power-law exponent of α = −5.8 ± 0.2 (n = 1092; black dashed lines in Figure 6). By extrapolating the power law to smaller diameters, we find that the number of boulders with a diameter around 3 pixels may be severely underestimated; the observed number in the bin closest to the 3 pixel limit is 2328, but the extrapolated, expected number is about 4000. The counts for boulders larger than 4 pixels are probably close to complete. We note that the counts at the largest diameters do not match well with the power law in both the cumulative and differential representation. The statistical test provided by Clauset et al. (2009) confirms that this power law is not a good model for the data (p = 0). When we let the ML algorithm itself choose the minimum boulder size, we find a larger ${d}_{\min }=169$ m and a steeper power law with α = −6.7 ± 0.3 (n = 400; red dashed lines in Figure 6). The statistical test indicates that this power law is a good model for the data (p = 0.37).

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We also estimated the power-law exponent for individual craters that have at least six boulders larger than 4 pixels (Table 1) and plot these as a function of the number of boulders in the population in Figure 7. The figure also includes three simulations of the power-law exponent distribution of individual craters. The simulation uses the observed population sizes and adopts the best-fit power-law exponent for the global boulder population (α = −5.8). The observations and simulations agree in showing a large scatter in the exponents for smaller population sizes and their expected negative bias (Clauset et al. 2009; Schröder et al. 2020). The degree of scatter is similar for both simulated and observed data, indicating that the observed variety in power-law exponents is merely due to differences in population size and not some physical property of boulders or craters. However, the observed exponents of craters with small boulder populations are typically more negative than in the simulations. Additionally, the power-law exponent of the crater with the largest number of boulders, Jacheongbi, is further from −5.8 than that of the simulated "Jacheongbi." It is almost as if the observed distribution of exponents is skewed with respect to the simulated distribution. This suggests that the power-law model does not correctly describe the boulder SFD.

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Schulzeck et al. (2018b) independently counted boulders around several Ceres craters and fitted power laws to the SFD, using ML to estimate both the exponent and the minimum diameter. We compare their exponents with ours in Figure 8. The exponents agree within the error bars for the craters with more than 70 boulders (Jacheongbi, Nunghui, and Unnamed11). The exponents for the other three craters (Juling, Ratumaibulu, and Unnamed17) agree less well, which is not surprising given the small size of their boulder populations. The exponents for the crater with the largest number of boulders (Jacheongbi with 160 boulders) match closely (−4.5 ± 0.4 versus −4.4 ± 0.7). This suggests that our counts are consistent with those of Schulzeck et al. (2018b). In Section 2.2, we uncovered evidence for bias resulting from measurement errors in the boulder sizes, which were probably similar in magnitude for Schulzeck et al. (2018b). Can this bias be responsible for the unusual steepness and convex shape of the SFD of the global boulder population? Given that the boulder size measurements were almost certainly subject to errors of at least 1 image pixel, the "true" SFD is probably less steep (see Figures 3 and 4). The power-law exponent may be smaller by about unity, but with a value somewhere between −4.8 and −5.7, the SFD would still be unusually steep. Measurement errors also affect the shape of the SFD (Figure 4). Random errors would actually lead to a slightly more concave shape of the SFD. Only systematically underestimating the boulder sizes would lead to a convex SFD, but this would also tend to make it shallower. Thus, we cannot attribute the downturn of the SFD toward large sizes to measurement error, and it must be an intrinsic property of the Ceres boulder population.

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We conclude that the power law is not a good model for the SFD of boulders larger than 4 pixels. The ML algorithm was able to find an acceptable power law for a larger minimum size (${d}_{\min }=169$ m), but there is no reason to exclude the well-resolved boulders larger than 4 pixels but smaller than 169 m (more than half of the total) from the fit. This is the same situation as for the Vesta boulder SFD, for which the Weibull distribution proved to be a better model than the power law (Schröder et al. 2020). The best-fit Weibull distribution for the Ceres global boulder population has N = 1092, α = 1.32, and β = 0.45 (Figure 9). The fractal dimension D_f = 3 − β for the cracks in the rock is 2.5. The Weibull distribution fits the SFD better than the power law, and, contrary to the latter, it does not predict that the number of boulders with a size of 3 pixels is massively underestimated. Figure 9 also shows Weibull distributions for the Vesta global boulder population (Schröder et al. 2020). There are two best-fit curves, one including and the other excluding the boulders of Marcia, the largest crater on Vesta. Just like the largest Ceres craters, Marcia shows evidence of flows, which may have destroyed many of its boulders. As Vesta is smaller than Ceres, its Weibull distributions plot below the Ceres distribution. Given the uncertainty surrounding the boulder populations of the largest craters on both worlds, a detailed comparison of the Weibull parameters provides little insight.

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In Figure 10, we plot the distribution of all boulders with a size of at least 3 pixels (105 m) on a color-composite map of Ceres. In the same figure and on the same scale, we also show the distribution of boulders larger than 105 m on Vesta using counts from Schröder et al. (2020). With a surface area 3.2 times that of Vesta, the Ceres map is much larger. On both bodies, boulders are confined to craters. On Vesta, boulders are mostly absent from a large area of low albedo that appears to be enriched with carbonaceous chondrite material (Denevi et al. 2016; Schröder et al. 2020). The situation is different on Ceres. Several distinctly blue craters (Occator, Haulani, and Kupalo) have boulders, which is consistent with blue being a marker of youth (Schmedemann et al. 2014). Large areas are devoid of boulders, especially at lower latitudes, but these do not stand out in terms of color or albedo like on Vesta. Even though the poles were partly in shadow during lamo, we find many boulders there.

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We quantify the boulder abundance in three latitude zones (low, mid, and high) on both Ceres and Vesta in Figure 11. We calculated the density of both boulders and craters with at least one boulder larger than 105 m by dividing the total number of boulders/craters in a latitudinal zone by the total surface area of the zone (including shadowed terrain), calculated under the assumption that Ceres and Vesta are spheres with radii of 469 and 263 km, respectively. Adopting Poisson error bars allows one to assess whether any differences are the result of chance. The boulder density graph (Figure 11(a)) confirms our visual impression that the density is higher at the high latitudes of Ceres, despite the fact that the polar terrain was partly in shadow. The small (Poisson) error bars indicate that this is very likely not due to chance. The Ceres boulder density at mid-latitudes is also a little higher than that at low latitudes. The boulder density at low latitudes is more uncertain than the error bar indicates. First, boulder numbers may be underestimated because of the limited visibility of the shadows cast by smaller boulders (see Section 2.1). Second, boulder counts for Occator and Haulani, with their large-scale flows, are uncertain (see Section 3.1). In contrast to Ceres, the Vesta boulder density does not vary with latitude within the Poisson error margins. It is a little lower than the Ceres boulder density at low and mid-latitudes and much lower at high latitudes. The density of the craters with boulders (Figure 11(b)) also increases on Ceres toward high latitudes, although the correlation is not as strong due to the larger error bars. The density of craters with boulders on Vesta does not vary with latitude within the error margins. The crater density may be similar at high latitudes on both bodies, although the error bars are large. Interestingly, the density of craters with boulders on Vesta is significantly higher than that on Ceres for low and mid-latitudes, despite the boulder density being lower. This implies that, on average, craters on Ceres have more boulders than on Vesta, consistent with Figure 5(a).

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Examples ⁷ of the spatial distribution of boulders around individual craters are shown in Figure 12. Boulders are located both inside the crater and outside the rim, typically, all within one crater radius. The figure distinguishes between boulders in two different size classes, but sorting of boulders according to their size is not evident, consistent with the findings of Schulzeck et al. (2018b). There are no accumulations of boulders with the size range considered in our global study at the foot of the steep slopes, which argues against a formation of such boulders by postimpact weathering. High-resolution images (scales of 3–10 m pixel⁻¹) acquired in the Dawn extended mission show evidence for boulder transport on crater walls, such as bounce marks on unconsolidated talus and boulders at the downslope end of tracks. Figure 13(a) shows an example of boulders collected at the foot of a crater wall. Such boulders are consistently smaller than those we consider here. As very high-resolution images are only available for a very small fraction of the surface, we do not consider them for our global study. We therefore decided to group boulders inside and outside the craters together and treat them as a single population. Another image from the extended mission, in Figure 13(b), shows clusters of boulders that appear to derive from former, larger boulders. Such fields of debris may originate from either the impact of the larger boulder on the surface or weathering and/or erosion in place, demonstrating that boulders disintegrate by fracturing.

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Boulders degrade over time and eventually disappear from the surface. Basilevsky et al. (2015) predicted that the survival time of meter-sized boulders on Ceres is very similar to that on Vesta, based on estimates of the potential impactor flux and the expected impact velocities. Schröder et al. (2020) determined a survival time of about 300 Ma for Vesta boulders, much larger than the ∼10 Ma predicted by Basilevsky et al. (2015). The authors attributed this apparent discrepancy to the fact that the boulders in their sample were 1–2 orders of magnitude larger than the meter-scale associated with the prediction. The boulders in our sample are even larger than the Vesta boulders studied by Schröder et al. (2020) because of the lower lamo image resolution at Ceres, but typically by a factor of 2 rather than an order of magnitude. It should therefore be possible to test the Basilevsky et al. (2015) prediction of similar survival times on Vesta and Ceres, if accurate age estimates are available for our Ceres craters.

Estimating the age of a crater is typically done by counting smaller craters in a selected area on the ejecta blanket and modeling the resulting SFD. Just as for Vesta, two alternative chronologies have been used to model crater SFDs for Ceres: the lunar-derived model (LDM) adapts the lunar production and chronology functions to impact conditions on Ceres, whereas the asteroid-derived model (ADM) derives a production function by scaling the observed SFD from the main asteroid belt to the SFD of the Ceres craters (Hiesinger et al. 2016). Most papers on the topic of Ceres dating employ both chronologies and provide two age estimates for a particular crater. Table 2 lists craters for which age estimates are available. The uncertainty associated with the age is typically large, as the two chronologies can yield widely different values. Additional sources of uncertainty are the choice of counting area and the assumed strength of the target surface. The tabulated ages were estimated assuming an impact into hard rock. Williams et al. (2018) found that the ADM (and, presumably, the LDM) ages are much larger for a rubble surface (e.g., Cacaguat: 14.4 instead of 3.3 Ma; Rao: 133 instead of 30.4 Ma).

We define "areal density" as the total number of boulders identified in and around a crater divided by the crater equivalent area, calculated as the area of a circle with the diameter of that crater (Table 1). We determined the areal density of boulders larger than 105 m in and around the craters in Table 2. Some areal densities are unreliable. Boulder numbers are underestimated for craters at high latitudes, which were partly in the shadow during lamo (Shennong and Unnamed4/26/28/36/44). Boulder numbers are uncertain for craters that show postimpact modifications in the form of flows (Haulani, Ikapati, and Occator). The distribution of boulders around craters with a reliable density was shown in Figure 12. We relate the boulder density to crater age in Figure 14(a), where the age ranges are spanned by the LDM and ADM estimates. The two variables are anticorrelated. There is a large degree of scatter in the data, which may be due to the fact that the boulder density also depends on latitude (see Section 3.3). The data suggest that the maximum boulder survival time is around 150 Ma, where we note that the age of the oldest crater in the figure (Gaue) is very uncertain. Support for this maximum age comes from craters of this age for which we did not find boulders; one such unnamed crater at (162°E, +78°) has an estimated age of 89–252 Ma (Ruesch et al. 2018). Two others, Messor at (234°E, +50°) and an unnamed crater at (186°E, +23°), have estimated ages of 96–192 and 88–205 Ma, respectively (Scully et al. 2018). All craters estimated to be younger than 150 Ma in the papers referenced in Table 2 have boulders. Given the uncertainty in the crater age estimates due to the different models, we adopt an uncertainty of 50 Ma for the boulder survival time. Whereas the correlation with crater age is expected, we also find that the boulder density is correlated with crater size (Figure 14(b)). This may at least partly be explained by the fact that large craters are, on average, older than small craters. Another explanation might be that larger craters sample different, deeper crustal layers in a stratified crust, a concept that we discuss in the next section.

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Basilevsky et al. (2015) estimated the survival time of meter-sized boulders on Vesta, Ceres, and the Moon by predicting the impactor velocity and density distributions. Boulders on Vesta and Ceres should live equally long but 30 times shorter than on the Moon. Schröder et al. (2020) found that the maximum boulder survival time on Vesta is 300 Ma, the same as that of lunar boulders (Basilevsky et al. 2013). The authors attributed this apparent contradiction to the fact that the Vesta boulders in their study are larger than 60 m, rather than meter-sized. In other words, large boulders live longer than small ones. At 150 ± 50 Ma, the survival time of the Ceres boulders in our sample is only half that of the Vesta boulders, despite being larger (>105 m). Thus, the lifetime of Ceres boulders may be less than half that of Vesta boulders of the same size.