# Blocks Size Frequency Distribution in the Enceladus Tiger Stripes Area: Implications on Their Formative Processes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dataset and Methodology

^{2}, we computed the corresponding area of the studied surface.

_{min}, which is the threshold value above which the power-law exists. The estimation of x

_{min}is done through the Kolmogorov-Smirnoff (KS) statistic and allows to find the value minimizing it. Afterwards, the parameter ${\alpha}_{Cl}$ is determined through the maximum likelihood estimator (MLE). The uncertainty for both ${\alpha}_{Cl}$ and x

_{min}is then derived through a non-parametric bootstrap procedure that generates a large number of synthetic datasets from a power-law random generator and performs a number of KS tests to verify if the generated and observed data come from the same distribution. This technique returns a p-value that can be used to quantify the plausibility of the hypothesis. Considering the significance level of 0.10 [43], if the p-value is $\ge $0.1, then it is possible to conclude that any difference between the empirical data and the model can be explained with statistical fluctuations. On the contrary, if the p-value is <0.1, then the data set does not come from a power-law distribution, but instead, from a different one.

## 3. Results

^{2}wide: we can therefore plot the unbinned cumulative number of blocks per km

^{2}(Figure 5B) deriving values that span from 4$\xb7$10

^{−3}/km

^{2}at a diameter of 300 m, up to 5$\xb7$10

^{−1}/km

^{2}for sizes ~125 m.

_{min}for all counts is 149.8 m (with a total number of 688 blocks larger than x

_{min}), while the value of the power-law index $\alpha $ is −5.4 (Figure 6A). To evaluate the uncertainty for x

_{min}and $\alpha $ we generated 5000 synthetic datasets using the non-parametric bootstrap procedure. The scatter-plot of x

_{min}against $\alpha $ is presented in Figure 6B in order to evaluate how a change in x

_{min}results in a different value for $\alpha $, while the frequency histograms of x

_{min}and $\alpha $ are indicated in Figure 6C,D. The resulting error of x

_{min}is 16.2 m, while the one for $\alpha $ is 0.37. As indicated in Figure 6A, the p-value obtained is 0.185. Since this value is $\ge $0.1, it is possible to affirm that (i) any difference between the empirical data and the model can be explained with statistical fluctuations, (ii) we cannot reject the power-law model for these data, and (iii) the data are consistent with a power-law.

^{2}or power-law indices, we decided to split the full dataset of blocks into three main groups (Figure 7): (i) one belonging to the Damascus Sulcus, (ii) one characterized by the blocks situated in close proximity to the Baghdad Sulcus, while (iii) the last one surrounding the Cairo Sulcus. Even if the Baghdad Sulcus branches out in two parts on the studied area, we decided to consider the corresponding blocks as belonging to the same group since the tiger stripe has the same origin.

^{2}in Figure 8B,D,F. The Damascus area (1208 km

^{2}) has a total number of 1838 counted blocks (the lowest spatial scale of the images used for the Damascus block identification is 31.9 m/pixel) and it is the one that shows the largest diameter range, between ~40 and 366 m (Figure 8A). The mode of the skewed distribution for all blocks $\ge $127 m (4 pixels) lies at 132.4 m, the median at 150.7 m (with a mad of 23.5 m), the mean is 162.0 m, and the standard deviation is 38.2 m. The cumulative number of blocks per km

^{2}ranges from a minimum of 8.3$\xb7$10

^{−4}/km

^{2}at a size of 366 m to 5.1$\xb7$10

^{−1}/km

^{2}at 127 m (Figure 8B).

^{2}. Out of the 11,883 blocks counted (the lowest spatial scale of the images used for the Baghdad block identification is 25.3 m/pixel), the largest one has a diameter of 343 m and it is located 1150 m far from the Baghdad left branch. The mode of the distribution of blocks $\ge $101 m is 102.3 m (Figure 8C), the median is 118.5 m with a mad of 19 m, while the mean lies at 127.5 m, with a standard deviation of 30.1 m. The cumulative number of blocks per km

^{2}at the maximum identified size (~340 m) is 6.0$\xb7$10

^{−4}/km

^{2}and it is comparable to the Damascus one, while at 101 m size the cumulative number of blocks per km

^{2}is 1.30/km

^{2}(Figure 8D).

^{2}wide and the total number of counted blocks is 3349 (the lowest spatial scale of the images used for the Cairo block identification is 18.4 m/pixel). The biggest block here identified has a diameter of 206 m (Figure 8E). The mode of the distribution of blocks larger than 4 pixels (~74 m) lies at 77.1 m, the median at 89.0 m (with a mad of 13.8 m), while the mean is 93.3 m, and the standard deviation is 18.3 m. The cumulative number of blocks per km

^{2}is 2.5$\xb7$10

^{−3}/km

^{2}at the size of 206 m while it is 2.9/km

^{2}at 74 m (Figure 8F).

_{min}and $\alpha $ generating 5000 synthetic datasets using the non-parametric bootstrap procedure. The results are presented in Figure 9, Figure 10 and Figure 11, outlined with the same fashion of Figure 6.

_{min}value of 149.6 $\pm $ 14.2 m and a power-law index of −5.11 $\pm $ 0.43. The resulting p-value is 0.21. The Clauset et al. [43] methodology applied on the Baghdad counts, instead, returned a x

_{min}value of 117.3 $\pm $ 16.5 m, a power-law index of −5.13 $\pm $ 0.34 and a p-value of 0.14. Finally, for the Cairo case we obtained a x

_{min}value of 87.8 $\pm $ 5.1 m, a $\alpha $ value of −6.32 $\pm $ 0.34, and a p-value of 0.11.

## 4. Discussion

^{2}), Martens et al. [1] evidenced that the ejection process occurring during cryovolcanic eruptions from the TS could be a supported block formative mechanism. It is difficult to conceive that wide enough cracks could let ~100 m size blocks pass, since to form such large nozzles both high stresses and strain rates would be needed [44]. However, Hansen et al. [45] indicated that Enceladus collimated jets could reach speeds of 1 km/s or higher, and assuming plume temperatures higher than 296 K, blocks with sizes ~100 m could be lifted and ejected [1].

^{2}wide scarp located in 67P small lobe [51]. Besides gravity and tidal stresses, this location can be considered somehow geologically similar to the Enceladus SPT due to its ubiquitous fracture pattern and the active jets, hence allowing the comparison between the two bodies. By studying the Hathor non-dislodged blocks SFD in the 7–40 size range, Pajola et al. [33] derived a power-law index of −5.2 + 0.5/−0.6 (see Figure 11 and Figure 12 of Pajola et al. [33]), which has been attributed to both sublimation and fracturing processes. This value is similar to the one we obtained for the SPT full counts (Figure 6, $\alpha $ = −5.4 $\pm $ 0.4), as well as for the Damascus (Figure 9, $\alpha $ = −5.1 $\pm $ 0.4) and Baghdad case (Figure 10, $\alpha $ = −5.1 $\pm $ 0.3). Despite a larger block size range for Enceladus, similar power-law indices may suggest that sublimation and surface stresses favor similar fractures development in the icy matrix, hence resulting in comparable block disaggregation.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The eight projected ISS-NAC images covering the region of interest between Damascus, Baghdad and Cairo Sulci (highlighted in light blue, following Crow-Willard and Pappalardo [7]). The yellow outline shows the full area covered by the dataset. On the left side of the saw-tooth yellow boundary, a bad signal-to-noise part of the image has been eliminated. The red rectangle shows the location of Figure 3, while the orange circles show the jets’ sources identified by Porco et al. [10].

**Figure 3.**Methodology used to identify the blocks located on Enceladus tiger stripes. (

**A**) Subframe of the ISS-NAC N1597182500 image. The yellow arrow shows the direction of the solar irradiation. (

**B**) The identified blocks are grouped in size (m).

**Figure 5.**(

**A**) Frequency histogram of all blocks identified in the yellow area of Figure 4. The grey shadowed area shows the blocks that have been identified, but their diameter is smaller than 4 pixels. The main statistical properties of the right-skewed distribution are computed only for values $\ge $125 m. (

**B**) Log-log plot showing the cumulative unbinned number of blocks per km

^{2}. As for A, the grey shadowed area shows all blocks <125 m.

**Figure 6.**(

**A**) Normalized cumulative distribution function showing the obtained sizes, the power-law fitting curve and the x

_{min}limit (the grey shadowed area is not considered by the fit). The total number of blocks larger than x

_{min}and used for the fit is 688. (

**B**) Scatter-plot of x

_{min}against $\alpha $ resulting from 5000 synthetic datasets. (

**C**) Frequency histogram of x

_{min}. (

**D**) Frequency histogram of the power-law index $\alpha $.

**Figure 7.**The division of the study area into three different groups: the Damascus, Baghdad and Cairo blocks.

**Figure 8.**(

**A**,

**C**,

**E**) Frequency histogram of the sizes identified in the Damascus, Baghdad and Cairo areas. The grey shadowed area shows the blocks that have been identified, but their diameter is smaller than 4 pixels. The main statistical properties of the right-skewed distributions are computed only for values $\ge $4 pixels. (

**B**,

**D**,

**F**) Log-log plots showing the cumulative unbinned number of blocks per km

^{2}for the Damascus, Baghdad and Cairo area, respectively. As for (

**A**,

**C**,

**E**), the grey shadowed area shows all blocks smaller than 127 m (Damascus), 101 m (Baghdad) and 74 m (Cairo).

**Figure 9.**The Damascus case study. (

**A**) Normalized cumulative distribution function showing the obtained sizes, the power-law fitting curve and the x

_{min}limit (the grey shadowed area is not considered by the fit). The total number of blocks larger than x

_{min}and used for the fit is 327. (

**B**) Scatter-plot of x

_{min}against $\alpha $ resulting from 5000 synthetic datasets. (

**C**) Frequency histogram of x

_{min}. (

**D**) Frequency histogram of the power-law index $\alpha $.

**Figure 10.**The Baghdad case study. (

**A**) Normalized cumulative distribution function showing the obtained sizes, the power-law fitting curve and the x

_{min}limit (the grey shadowed area is not considered by the fit). The total number of blocks larger than x

_{min}and used for the fit is 1156. (

**B**) Scatter-plot of x

_{min}against $\alpha $ resulting from 5000 synthetic datasets. (

**C**) Frequency histogram of x

_{min}. (

**D**) Frequency histogram of the power-law index $\alpha $.

**Figure 11.**The Cairo case study. (

**A**) Normalized cumulative distribution function showing the obtained sizes, the power-law fitting curve and the x

_{min}limit (the grey shadowed area is not considered by the fit). The total number of blocks larger than x

_{min}and used for the fit is 659. (

**B**) Scatter-plot of x

_{min}against $\alpha $ resulting from 5000 synthetic datasets. (

**C**) Frequency histogram of x

_{min}. (

**D**) Frequency histogram of the power-law index $\alpha $.

**Figure 12.**Diameter versus TS distance plots obtained for all counts (

**A**), for the Damascus blocks (

**B**), for the Baghdad counts (

**C**) and for the Cairo blocks (

**D**). The vertical green dashed lines are the four pixels limit derived for each case, see Figure 5 and Figure 8. On all plots, the counts located inside the grey shadowed area are not considered because they are below the four pixels identification limit.

**Table 1.**Image ID, acquisition date and spatial scale of the Cassini ISS-NAC images used in this analysis.

Image ID | Acquisition Date (DD-MMM-YYYY) | Spatial Scale (m/pixel) |
---|---|---|

N1597182401 | 11 August 2008 | 11.5 |

N1597182434 | 11 August 2008 | 15.0 |

N1597182467 | 11 August 2008 | 18.4 |

N1597182500 | 11 August 2008 | 21.8 |

N1597182533 | 11 August 2008 | 25.3 |

N1604166970 | 31 October 2008 | 12.1 |

N1604167158 | 31 October 2008 | 31.9 |

N1637462964 | 21 November 2009 | 14.4 |

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**MDPI and ACS Style**

Pajola, M.; Lucchetti, A.; Senter, L.; Cremonese, G. Blocks Size Frequency Distribution in the Enceladus Tiger Stripes Area: Implications on Their Formative Processes. *Universe* **2021**, *7*, 82.
https://doi.org/10.3390/universe7040082

**AMA Style**

Pajola M, Lucchetti A, Senter L, Cremonese G. Blocks Size Frequency Distribution in the Enceladus Tiger Stripes Area: Implications on Their Formative Processes. *Universe*. 2021; 7(4):82.
https://doi.org/10.3390/universe7040082

**Chicago/Turabian Style**

Pajola, Maurizio, Alice Lucchetti, Lara Senter, and Gabriele Cremonese. 2021. "Blocks Size Frequency Distribution in the Enceladus Tiger Stripes Area: Implications on Their Formative Processes" *Universe* 7, no. 4: 82.
https://doi.org/10.3390/universe7040082