Advances in Astronomy

Volume 2018, Article ID 1894850, 15 pages

https://doi.org/10.1155/2018/1894850

## Distribution Inference for Physical and Orbital Properties of Jupiter’s Moons

^{1}School of Mathematical Science, Yangzhou University, Yangzhou 225002, China^{2}Department of Mathematics, Shanghai University, Shanghai 200444, China

Correspondence should be addressed to F. B. Gao; moc.anis@oabafoag

Received 4 June 2018; Revised 1 September 2018; Accepted 30 September 2018; Published 1 November 2018

Academic Editor: Geza Kovacs

Copyright © 2018 F. B. Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

According to the physical and orbital characteristics in Carme group, Ananke group, and Pasiphae group of Jupiter’s moons, the distributions of physical and orbital properties in these three groups are investigated by using one-sample Kolmogorov–Smirnov nonparametric test. Eight key characteristics of the moons are found to mainly obey the Birnbaum–Saunders distribution, logistic distribution, Weibull distribution, and* t* location-scale distribution. Furthermore, for the moons’ physical and orbital properties, the probability density curves of data distributions are generated; the differences of three groups are also demonstrated. Based on the inferred results, one can predict some physical or orbital features of moons with missing data or even new possible moons within a reasonable range. In order to better explain the feasibility of the theory, a specific example is illustrated. Therefore, it is helpful to predict some of the properties of Jupiter’s moons that have not yet been discovered with the obtained theoretical distribution inference.

#### 1. Introduction

There are 69 (the number has been refreshed to 79 by a team from Carnegie Institution for Science in July 2018. https://sites.google.com/carnegiescience.edu/sheppard/moons/jupitermoons) confirmed moons of Jupiter, around 65 of which have been well investigated [1, 2]. Considering the formation of Jupiter’s moons is influenced by diverse factors, which results in their physical characteristics differing greatly [3], Jupiter’s moons are divided into two basic categories: regular and irregular. The regular satellites are so named because they have prograde and near-circular orbits of low inclination, and they are in turn split into two groups: Inner satellites and Galilean [4]. The irregular satellites are actually the objects whose orbits are far more distant and eccentric. They form families that share similar orbits (semi-major axis, inclination, and eccentricity) and composition. These families, which are considered to be part of collisions, arise when the larger parent bodies were shattered by impacts from asteroids captured by Jupiter’s gravitational field. That is to say, at the early time of moons’ formation of the Jupiter, mass of the original moon’s ring was still sufficient to absorb the asteroid’s power and put it into orbit. So, part of the irregular moons might be created by the captured asteroids and then collided with other moons [5, 6], thus forming the various groups we see today. The identification of satellite families is tentative (please see [7, 8] for more details), and these families bear the names of their largest members. The most detailed modelling of the collisional origin of the families was reported in [9, 10].

According to this identification scheme [11], 60 moons were classified into 8 different groups, including Small Inner Regulars and Rings, Galileans, Themisto group, Carpo group, Himalia group, Carme group, and Ananke group as well as Pasiphae group, in addition to 9 satellites that do not belong to any of previous groups. The detailed information about all the groups of Jupiter’s moons can be found in Appendix A.

In recent years, many scientists have paid considerable attention to astronomical observation, physical research, and deep space exploration of small bodies, including asteroids, comets, and satellites, and so on. For example, planetary scientist Carry collected mass and volume estimates of 17 near-Earth asteroids, 230 main-belt and Trojan asteroids, 12 comets, and 28 trans-Neptunian objects from the known literature [12]. The accuracy and biases affecting the methods used to estimate these quantities were discussed and best-estimates were strictly selected. For the asteroids in retrograde orbit, there are at least 50 known moons of Jupiter’s that are retrograde, some of which are thought to be asteroids or comets that originally formed near the gas giant and were captured when they got too close. Kankiewicz and Włodarczyk selected the 25 asteroids with the best-determined orbital elements and then estimated their dynamical lifetimes by using the latest observational data, including astrometry and physical properties [13]. However, few researchers have tried to extrapolate the distribution of Jupiter’s satellites through statistical methods as yet.

In this paper, distributions of physical and orbital properties for the moons of Jupiter will be conducted by using one-sample Kolmogorov-Smirnov (K-S) test and maximum likelihood estimation [14–16]. Based on the analysis of satellites’ data, it is found surprisingly that the physical and orbital characteristics obey some distribution, such as Birnbaum-Saunders distribution [17], logistic distribution, Weibull distribution, and* t* location-scale distribution. Furthermore, the probability density curves of the data distribution are generated, and the differences of physical and orbital characteristics in the three groups are presented. In addition, the results of theoretical inference results are then proved to be feasible through one concrete example. Therefore, the results may be helpful to astronomers to discover new moons of Jupiter in the future.

#### 2. Method of Distribution Inference

In statistics, the K-S test, one type of nonparametric test, is used to determine whether a sample comes from a population with a specific distribution. The null hypothesis of one-sample K-S test is that the Cumulative Distribution Function (CDF) of the data follows the adopted CDF. For one-sample case, null distribution of statistic can be obtained from the null hypothesis that the sample is extracted from a reference distribution. The two-sided test for “unequal” CDF tests the null hypothesis against the alternative that the CDF of the data is different from the adopted CDF. The test statistic is the maximum absolute difference between the empirical CDF calculated by* x* and the hypothetical CDF:where is a given CDF andis the empirical distribution function of the observations . Here is the indicator function with the following form

According to Glivenko–Cantelli theorem [18], if the sample comes from distribution , then will almost surely converge to zero when . Therefore, we only focus on three satellite groups that have more than ten satellites in their groups, respectively. Of all these groups, Themisto group and Carpo group only contain one satellite, respectively, and only 4 moons were found separately in Small Inner Regulars and Rings, as well as in Galileans group. Moreover, there are 5 moons in Himalia group. For these 5 groups, there is no sufficient data for distribution inference, so we will focus on the Carme group, the Ananke group, and the Pasiphae group.

In the following sections, in order to use the one-sample K-S test, three sets of observed data from Jupiter’s moons will be tested against some commonly used distributions in statistics. The list of these distributions is shown in Table 1.