Distribution Inference for Physical and Orbital Properties of Jupiter's Moons

Home , Ananke (moon), Carme (moon), Pasiphae (moon), Themisto (moon)

Hindawi Advances in Astronomy Volume 2018, Article ID 1894850, 15 pages https://doi.org/10.1155/2018/1894850

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

F. B. Gao ,1 X. H. Zhu,2 X. Liu,1 andR.F.Wang1

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

Correspondence should be addressed to F. B. Gao; [email protected]

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. Tis 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.

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 diferences 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 specifc example is illustrated. Terefore, 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 the asteroid’s power and put it into orbit. So, part of the irregular moons might be created by the captured asteroids Tere are 69 (the number has been refreshed to 79 by a team and then collided with other moons [5, 6], thus forming the from Carnegie Institution for Science in July 2018. https:// various groups we see today. Te identifcation of satellite sites.google.com/carnegiescience.edu/sheppard/moons/jupi- families is tentative (please see [7, 8] for more details), and termoons) confrmed moons of Jupiter, around 65 of which these families bear the names of their largest members. have been well investigated [1, 2]. Considering the formation Te most detailed modelling of the collisional origin of the of Jupiter’s moons is infuenced by diverse factors, which families was reported in [9, 10]. results in their physical characteristics difering greatly According to this identifcation scheme [11], 60 moons [3], Jupiter’s moons are divided into two basic categories: were classifed into 8 diferent groups, including Small Inner regular and irregular. Te regular satellites are so named Regulars and Rings, Galileans, Temisto group, Carpo group, becausetheyhaveprogradeandnear-circularorbitsoflow Himalia group, Carme group, and Ananke group as well as inclination, and they are in turn split into two groups: Inner Pasiphae group, in addition to 9 satellites that do not belong satellites and Galilean [4]. Te irregular satellites are actually to any of previous groups. Te detailed information about all the objects whose orbits are far more distant and eccentric. the groups of Jupiter’s moons can be found in Appendix A. Tey form families that share similar orbits (semi-major In recent years, many scientists have paid considerable axis, inclination, and eccentricity) and composition. Tese attention to astronomical observation, physical research, and families, which are considered to be part of collisions, arise deep space exploration of small bodies, including asteroids, when the larger parent bodies were shattered by impacts comets, and satellites, and so on. For example, planetary from asteroids captured by Jupiter’s gravitational feld. Tat scientist Carry collected mass and volume estimates of 17 is to say, at the early time of moons’ formation of the Jupiter, near-Earth asteroids, 230 main-belt and Trojan asteroids, 12 mass of the original moon’s ring was still sufcient to absorb comets, and 28 trans-Neptunian objects from the known 2 Advances in Astronomy literature [12]. Te accuracy and biases afecting the meth- According to Glivenko–Cantelli theorem [18], if the ods used to estimate these quantities were discussed and sample comes from distribution �(�),then�� will almost best-estimates were strictly selected. For the asteroids in surely converge to zero when ��→∞. Terefore, we retrograde orbit, there are at least 50 known moons of only focus on three satellite groups that have more than ten Jupiter’s that are retrograde, some of which are thought satellites in their groups, respectively. Of all these groups, to be asteroids or comets that originally formed near the Temisto group and Carpo group only contain one satellite, gas giant and were captured when they got too close. respectively, and only 4 moons were found separately in Small Kankiewicz and Włodarczyk selected the 25 asteroids with Inner Regulars and Rings, as well as in Galileans group. the best-determined orbital elements and then estimated Moreover, there are 5 moons in Himalia group. For these 5 their dynamical lifetimes by using the latest observational groups, there is no sufcient data for distribution inference, data, including astrometry and physical properties [13]. so we will focus on the Carme group, the Ananke group, and However, few researchers have tried to extrapolate the dis- thePasiphaegroup. tribution of Jupiter’s satellites through statistical methods as In the following sections, in order to use the one-sample yet. K-S test, three sets of observed data from Jupiter’s moons In this paper, distributions of physical and orbital prop- will be tested against some commonly used distributions erties for the moons of Jupiter will be conducted by using in statistics. Te list of these distributions is shown in one-sample Kolmogorov-Smirnov (K-S) test and maximum Table 1. likelihood estimation [14–16]. Based on the analysis of satel- For the 9 continuous distributions in Table 1, the one- lites’ data, it is found surprisingly that the physical and orbital sample K-S test will be used to select the distribution with characteristics obey some distribution, such as Birnbaum- the highest confdence level. In order to characterize these Saunders distribution [17], logistic distribution, Weibull dis- distributions with well-defned parameter values, maximum tribution, and t location-scale distribution. Furthermore, the likelihood estimation is also used. In addition, the parameter probability density curves of the data distribution are gener- values of these distributions can be calculated from the ated, and the diferences of physical and orbital characteristics observed data. However, when confdence level (typically set inthethreegroupsarepresented.Inaddition,theresultsof to 0.05) decreases, the rejection domain of the test becomes theoretical inference results are then proved to be feasible smaller, so the observed values that initially fall into the through one concrete example. Terefore, the results may be rejection domain may eventually fall into the acceptance helpful to astronomers to discover new moons of Jupiter in domain. Tis situation will bring some trouble in practical the future. application. To this end, we adopt p value, which represents the obtained confdence level by using the one-sample K- 2. Method of Distribution Inference S test. In addition, the use of p value not only avoids determining the level of signifcance in advance, but also In statistics, the K-S test, one type of nonparametric test, is makes it easy to draw conclusions about the test by comparing used to determine whether a sample comes from a population the p value and signifcance level of the test. If the p value is with a specifc distribution. Te null hypothesis of one- greater than 0.05, we declare that the null hypothesis can be sample K-S test is that the Cumulative Distribution Function accepted. Furthermore, if the p values of several distributions (CDF) of the data follows the adopted CDF. For one-sample are all greater than 0.05, the distribution with the largest p case, null distribution of statistic can be obtained from the value should be selected, and the corresponding distribution null hypothesis that the sample is extracted from a reference will be the most appropriate one to ft the observed data. Our distribution. Te two-sided test for “unequal” CDF tests the results can be found in the tables in Appendix B. null hypothesis against the alternative that the CDF of the data is diferent from the adopted CDF. Te test statistic is 3. Distribution Inference of Satellite Groups the maximum absolute diference between the empirical CDF calculated by x and the hypothetical CDF: In this section, distributions of several diverse physical and orbital properties for the Carme group, Ananke group, and � � �� = sup �� (�) −�(�)� , Pasiphae group are inferred sequentially. � (1) where �(�) is a given CDF and 3.1. Carme Group. Tereare15moonsintheCarmegroup (please see Appendix A.). Due to the lack of enough data 1 � from S/2010 J1, the number of adopted moons is 14, which � (�) = ∑� (� ) � � (−∞,�] � (2) means the length of each data set is 14 from the mathematical �=1 perspective. In addition, considering all the mean densities 3 � have been calculated at being 2.60 g/cm , the surface gravity is the empirical distribution function of the observations �. 2 � (� ) of Carme and other moons in this group is 0.017 m/s and Here (−∞,�] � is the indicator function with the following 2 form 0.001 m/s , respectively [19]. Terefore, these data obviously do not obey the distributions. {1, �� ≤�, Based on the previous method of statistical distribution �(−∞,�] (��)={ (3) 0, ��ℎ��. inference and MATLAB 2016a (Intel Core i5-3230 M, CPU { 2.60 GHz), distribution inferences in the Carme group can Advances in Astronomy 3

Table 1: A list of common distributions. Name of Selected Distributions Parameters Meaning � frst shape parameter Beta � second shape parameter � scale parameter Birnbaum-Saunders � shape parameter � shape parameter Gamma � scale parameter � mean Logistic � scale parameter � shape parameter Nakagami � scale parameter � mean Normal � standard deviation � noncentrality parameter Rice � scale parameter � location parameter � location-scale � scale parameter � shape parameter � scale parameter Weibull � shape parameter

Table 2: Te distribution inference in each physical and orbital characteristic (Carme group).

Distribution Parameter Characteristics p-value inference estimates Semi-major axis logistic � = 2.3326, � = 7 0.9988 (10 km) distribution 0.00651346 Mean orbit velocity logistic � =8.21989,� = 3 0.6607 (10 km/h) distribution 0.0112805 Birnbaum- Orbit eccentricity � =2.54254,� = −1 Saunders 0.6245 (10 ) 0.0330888 distribution Inclination of orbit t location-scale � = 1.6511, � = 2 0.6662 (10 ∘) distribution 0.0017, � =0.8751 � =1.65708,� = Equatorial radius t location-scale 0.440683, � = 0.7119 (km) distribution 1.14501 Escape velocity t location-scale � =7.32997,� = 0.6619 (km/h) distribution 1.58419, �=1.06821 be found in Table 2, and the last column represents the distributions are 1.3957E-04 and 4.1863E-04, respectively. p value. Te smaller the p value, the greater the signif- As the parameter � increases, it indicates that the average cance because it tells us that hypothesis under consider- semi-major axis and mean orbit velocity of the Carme group ation may not be sufcient to explain the observations. increase. As the parameter � increases, the data in these Te hypothesis will be rejected if any of these probabili- two characteristics gradually disperse, and the discrepancy ties is less than or equal to a small, fxed but arbitrarily between the data and the average value increases. Te predefned threshold value. Te null hypothesis here refers meaning of decreasing the parameter � and � shares the to data obeying a particular distribution, and the alterna- same principle as that of increasing the parameters. Te orbit tive hypothesis assumes that the data does not obey the eccentricity follows the Birnbaum–Saunders distribution, distribution. which is unimodal with a median of �.Temeanvalue From Table 2, semi-major axis and mean orbit velocity and the variance of the distribution can be calculated by the obey the logistic distribution, of which parameter � denotes following relationships: the average orbital eccentricity and parameter � plays a key 2 role in representing the variance of the data set, through the � 2 2 �=�(1+ ), variance formula � � /3, so the variances of these logistic 2 4 Advances in Astronomy

Table 3: Te distribution inference in each orbital or physical characteristic (Ananke group).

Distribution Parameter Characteristics p-value inference estimates � = 2.11665, � = Semi-major axis t location-scale 7 0.00915557, � = 0.9304 (10 km) distribution 1.52402 Mean orbit velocity Weibull � =2.29923,� = 3 0.9788 (10 km/h) distribution 21.9834 � =8.72481,� = Orbit eccentricity t location-scale −1 0.0203158, � = 0.6017 (10 ) distribution 1.36328 Inclination of orbit t location-scale � =1.4879,� = 2 0.9075 (10 ∘) distribution 0.0127, � =1.4538 � =1.80724,� = Equatorial radius t location-scale 0.660297, � = 0.7231 (km) distribution 1.19615 13 Weibull � =38.3573,� = Mass (10 kg) 0.4304 distribution 0.368827 � =0.167044,� = Surface gravity t location-scale 2 3 0.0636087, � = 0.4210 ((m/s )/10 ) distribution 1.40815 Escape velocity t location-scale � =8.24903,� = 0.7262 (km/h) distribution 2.61085, � =1.15837

5�2 �2 = (��)2 (1+ ) . From our discussion on the t location-scale distribution 4 in Carme group, it is easy to understand the distribution inference of semi-major axis, mean orbit velocity, equatorial (4) radius, surface gravity, and escape velocity in the Ananke Terefore, the mean value of orbital eccentricity can be group (see Table 3). However, compared with Table 2, it is calculated to be 0.2543 and the variance is 7.08E-9. noted that the orbit eccentricity and mass properties in the Te inclination of orbit, equatorial radius, and escape Ananke group are subject to the Weibull distribution. Param- velocity are subject to the t location-scale distribution, which eters � and � represent the scale and shape parameters of the contains the scale parameter �, the location parameter �, distribution [20], respectively, which together determine the and the shape parameter �. Without loss of generality, we mean and variance of the distribution. assume that the data vector � obeys the t location-scale distribution, and then we have (� − �)/�∼ �(�) ,which 3.3. Pasiphae Group. Tere are 19 moons in the Pasiphae obeys Student’s t-distribution; here � represents the degrees group; due to the lack of data about three moons, S/2011 J2, of freedom. As can also be seen in Table 2, the inclination of S/2017 J1, and S/2016 J1, the remaining 16 will be studied in orbit obeys the t location-scale distribution with parameters this subsection. ∘ (1.6511, 0.0017, 0.8751) and the mean inclination is 165.11 .Te Te distributions in Table 4 are inferred to be similar to equatorial radius follows the t location-scale distribution with the distribution in Tables 2 and 3. Terefore, we can easily parameters (1.65708, 0.440683, 1.14501). Terefore, the moons understand the parameters of these distributions. in Carme group have an average equatorial radius of 1.65708 km. In addition, the escape velocity characteristic obeys the t 4. Comparison of Data Properties location-scale distribution with parameters (7.32997, 1.58419, 1.06821) and the average escape velocity is 7.32997 km/h. As Table 5 is given according to the previous distribution infer- the parameter � changes, the average equatorial radius and ence. escape velocity of the group also change accordingly. In the Based on the previous distribution inference, the prop- 2 t location-scale distribution, the variance is � �/(� − 2),and erties of the moons’ data can be compared more specifcally when the shape parameter � is greater than two, the variance and conveniently. According to the distribution of the specifc of the distribution is defned. Terefore, the variances of parameters, we get the following probability density function these two specifc t location-scale distributions cannot be (PDF) diagram. defned. As shown in Figure 1, the semi-major axis in the Ananke group is the smallest, followed by the Carme group and the 3.2. Ananke Group. Tere are 10 moons in the Ananke group, Pasiphae group. In addition, the PDF of the Pasiphae group of which we do not have enough data about S/2010 J2. Tus, is relatively fat, indicating a large dispersion in semi-major the length of each characteristic of the remaining moons in axis around Jupiter, while the data in the Carme and Ananke this group will be 9. groups difer slightly. As can be seen from Figure 2, the Advances in Astronomy 5

Table 4: Te distribution inference in each orbital or physical characteristic (Pasiphae group).

Characteristics Distribution inference Parameter estimates p-value 7 Semi-major axis (10 km) t location-scale distribution � =2.38241,� = 0.0273627, � = 0.730397 0.3730 3 Mean orbit velocity (10 km/h) logistic distribution � = 8.22803, � = 0.226046 0.4015 −1 Orbit eccentricity (10 ) logistic distribution � =2.95251,� = 0.599373 0.9550 2 Inclination of orbit (10 ∘) logistic distribution � =1.5136,� = 0.0339 0.8987 Equatorial radius (km) Weibull distribution � =3.69781,� = 0.781648 0.0859 13 Mass (10 kg) Weibull distribution � = 72.5121, � = 0.266585 0.1096 2 3 Surface gravity ((m/s )/10 ) Weibull distribution � =3.09876,� = 0.843794 0.0594 Escape velocity (km/h) Weibull distribution � = 17.0302, � = 0.807236 0.0722

Table 5: Distribution inference summary.

Characteristics Carme Group Ananke Group Pasiphae Group Semi-major axis logistic distribution t location-scale distribution t location-scale distribution Mean orbit velocity logistic distribution t location-scale distribution logistic distribution Orbit eccentricity Birnbaum–Saunders distribution Weibull distribution logistic distribution Inclination of orbit t location-scale distribution t location-scale distribution logistic distribution Equatorial radius t location-scale distribution t location-scale distribution Weibull distribution Mass None Weibull distribution Weibull distribution Surface gravity None t location-scale distribution Weibull distribution Escape velocity t location-scale distribution t location-scale distribution Weibull distribution

Semi-major Axis Around Jupiter Mean Orbit Velocity 40 25

35 20 30

25 15 20 PDF PDF 10 15

10 5 5

0 0 1.5 2 2.5 3 7.4 7.6 7.8 8 8.2 8.4 8.6 8.8 9 9.2 7 3 Semi-major Axis Around Jupiter (10 km) Mean Orbit Velocity (10 km/h)

Carme Group Carme Group Ananke Group Ananke Group Pasiphae Group Pasiphae Group

Figure 1: Te PDF curves of semi-major axis around Jupiter. Figure 2: Te PDF curves of mean orbit velocity. trend of mean orbital velocity distribution in Carme group which means the orbital eccentricity of the Pasiphae group is the closest to each other, with the mean value being the has a relatively large dispersion. In Figure 4, although the smallest among the three groups, followed with the Pasiphae inclinations of orbit in the Ananke group and Pasiphae group and Ananke groups. Obviously, the curve of the Pasiphae are relatively close in value compared with the Carme group, group is the fattest, which shows the mean orbit velocity in and the data looks more dispersed than those in Carme the Pasiphae group difering greatly. Figure 3 shows that the group, the inclination of orbit in the Ananke group has the mean value in the Ananke group is the smallest, followed by same distribution as the Carme group. the Carme and Pasiphae groups. It is obvious that the PDF Figure 5 shows the PDF curves of equatorial radii of these curve of the Pasiphae group is fatter compared to the others, three groups. Data attributes are similar; most of the moons’ 6 Advances in Astronomy

Orbit Eccentricity Equatorial Radius 5 0.8 4.5 0.7 4 0.6 3.5 3 0.5

2.5 0.4 PDF PDF 2 0.3 1.5 0.2 1 0.5 0.1

0 0 1 1.5 2 2.5 3 3.5 4 4.5 5 0 12345678910 -1 Orbit Eccentricity(10 ) Equatorial Radius (km)

Carme Group Carme Group Ananke Group Ananke Group Pasiphae Group Pasiphae Group

Figure 3: Te PDF curves of orbit eccentricity. Figure 5: Te PDF curves of equatorial radius.

Inclination Mass 200 0.08

180 0.07 160 0.06 140 0.05 120 100 0.04 PDF PDF 80 0.03 60 0.02 40 0.01 20 0 0 0 51015 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 13 2∘ 10 Inclination (10 ) Mass ( kg)

Carme Group Ananke Group Ananke Group Pasiphae Group Pasiphae Group Figure 6: Te PDF curves of mass. Figure 4: Te PDF curves of inclination.

escape velocity density to the Ananke group, but the former averageescapevelocityissmaller. radiuses are less than 4 km. Figure 6 indicates similarities between the Ananke and Pasiphae groups. As can be seen from these curves, most of the moons in these two groups 5. Verification of Rationality of are of relatively small mass. Theoretical Results Figure 7 illustrates the surface gravity PDF curves in the Ananke and Pasiphae groups. Te diference in density In this section, we take the semi-major axis and mean orbit between the two groups indicates that the surface gravity of velocity of the moons in the Carme group as concrete exam- the Ananke moons is higher than that of the Pasiphae group. ples to illustrate the rationality of the statistical inferences Te PDF plots corresponding to the escape velocity are shown in the preceding sections. As can be seen in Table 2, semi- in Figure 8, where it is clear that the Pasiphae group is fatter major axis (��) and mean orbit velocity (��V) obey the than the other two groups. Te Carme group also has similar logistic distribution with parameters (2.3326, 0.00651346) Advances in Astronomy 7

Surface Gravity 6 20

5 15 4

3 PDF 10 PDF

2 5

1 0 0 8.15 8.20 8.25 8.30 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Mean Orbit Velocity (10 km/h) 2 3 Surface Gravity ((m/Ｍ )/10 ) statistically predicted distribution Ananke Group analytically derived distribution Pasiphae Group Figure 9: Comparison of statistical prediction distribution and Figure 7: Te PDF curves of surface gravity. analytical derivation distribution.

Escape Velocity 0.25 Note that the orbital and physical properties are not independent of each other. For instance, the semi-major axis 0.2 � is related to the mean orbit velocity V through the relation V = √��/�,where�� is the mass parameter. Ten the PDF of mean orbit velocity can also be derived analytically 0.15 as follows

PDF ��,��V (V;�,�) 0.1 �� =2 � ( ; 2.3326, 0.00651346) V3 ��,�� V2 0.05 (7) 2 2 �� −(��−2.3326V )/0.00651346V =2 3 2 V −(��−2.3326V2)/0.00651346V2 0 0.00651346 (1 + � ) 0 2468101214161820 Escape Velocity (km/h) Although the PDF of mean orbit velocity obtained by dif- Carme Group ferent methods has diferent mathematical representations, � (V;�,�) Ananke Group from Figure 9, we can fnd that ��,��V (PDF curve Pasiphae Group represented by red circles) obtained by the analytical method is in good agreement with ��,��V(V;�,�)(PDF curve repre- Figure 8: Te PDF curves of escape velocity. sented by blue circles) obtained by statistical inference. However, what we need to pay attention to here is that and (8.21989, 0.0112805), respectively. So the corresponding some physical features and orbital elements are mixed and predicted PDFs can be written as can be linked by some mathematical formulas similar to the above. Teoretically, the distribution of another variable in ��,�� (V;�,�) the formula can be solved by a known defned distribution; although this may be a complex process because the prob- �−(V−2.3326)/0.00651346 (5) = ability density function may contain some transcendental 0.00651346 (1 + �−(V−2.3326)/0.00651346)2 functions and gamma functions, it can be achieved. Yet the known distribution becomes uncertain now; that is, the and distribution exists with a certain probability. So, there will be a certain risk when we calculate the distribution of the linked ��,��V (V;�,�) variable based on this uncertain distribution, especially when �−(V−8.21989)/0.00112805 (6) the possibility of the inferred distribution is not very high. = . In addition, in order to further show that the results 0.00112805 (1 + �−(V−8.21989)/0.00112805)2 of the KS test agree well with the actual observed results, 8 Advances in Astronomy

Table 6: Distribution inference of Carme group without Erinome (S/2000 J4).

p-value h Parameter Values � = 0.202265 Beta 4.02E-05 1 � =0.173517 � =2.37288 Birnbaum-Saunders 0.0307 1 � =0.943409 � =1.05985 Gamma 0.0308 1 � = 3.11366 � =1.95282 Logistic 0.0463 1 � =1.72324 � = 0.282272 Nakagami 0.0039 1 � =43.5177 � =3.3 Normal 0.0037 1 � =5.94531 � =0.138623 Rice 5.73E-08 1 � =4.66385 � =1.67257 t location-scale 0.6806 0 � =0.487941 � =1.15623 � =3.06061 Weibull 0.0698 0 � =0.897697 Note: h =0andh = 1 indicate acceptance of the null hypothesis and rejection of the null hypothesis, respectively.

1 6. Application of the Distribution Inference 0.9 As mentioned previously, identifcation of irregular satellite 0.8 families is tentative. However, afer the distribution was inferred, it is found that these features of moons in the 0.7 three diferent groups obey some selected distributions. 0.6 Furthermore, the obtained population distribution can also 0.5 be used to predict the characteristic data. Considering the rationality of proving this prediction method, suppose some (x) or F(x) (x) or

Ｈ 0.4 ＆ characteristic data of a moon is unknown in a given irregular 0.3 moon group. One specifc feature of the moon can be predicted by using other moons’ data and one-sample K-S 0.2 method. Here is an example: 0.1 Assume that some characteristic data of the moon Eri- nome (S/2000 J4) in Carme group is poorly known. Now, 0 2.3 2.31 2.32 2.33 2.34 2.35 we try to predict the equatorial radius of the Erinome. First, 7 Semi−major Axis (10 km) we use one-sample K-S method to fnd the most appropriate continuous distribution for these data. Te inference results observed CDF are as shown in Table 6. best−ft CDF From the p values displayed in Table 6, it becomes clear Figure 10: Comparison of the observed CDF and the best-ft CDF that the corresponding p value 0.6806 is the largest and ℎ=0, of semi-major axis. so the best distribution for the remaining characteristic data in the Carme group is t location-scale distribution with the PDF �(�|�,�,�) we also compared the best-ft CDFs (the CDFs of inferred −(�+1)/2 statistically) and the observed CDFs. From Figures 10 and 11, Γ ((�+1) /2) �+((�−�)/�)2 it can be seen that the best-ft CDF and the observed CDF = [ ] , (8) �√��Γ (�/2) � of semi-major axis agree better than the case of mean orbital � velocity. Tis should be due to the fact that the value of the −∞<�<+∞, former is 0.9988, which is obviously larger than that of the latter 0.6607. where Γ(⋅) is the Gamma function. Advances in Astronomy 9 3 0 0 1951 1610 1610 1610 1610 1938 1974 1892 1938 1979 1979 1979 2010 2010 1905 2001 2001 2001 2001 2001 2001 1904 2003 2003 2003 2003 2003 2003 2003 2002 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 Te Year of Discovery pard/satellites/jupsatdata.html). 7 2 1 J 3 0 0 2 / S/2010 J1 S/2003 J1 S/2001 J3 S/2001 J2 S/2001 J8 S/2001 J7 S/2010 J2 S/2002 J1 S/2001 J6 S/2000 J1 S/2003 J3 S/2003 J5 S/2000 J7 S/2000 J3 S/2000 J2 S/2000 J5 S/2000 J6 S/2000 J9 S/2001 J11 S/2000 J4 S/2003 J11 S/2000 J11 S/2003 J21 S/2000 J10 S/2003 J22 S/2003 J20 Designation — Galileans Retrograde Irregular Groups Small Inner Regulars and Rings Carpo Prograde Irregular Group Himalia Prograde Irregular Group Carme Retrograde Irregular Group Temisto Prograde Irregular Group Ananke Retrograde Irregular Group a a e e eS h s t h l r t o i a e s Dia Kale Leda Elara Aitne Metis y Tebe Arche Name m Carpo Isonoe Kalyke Carme Iocaste Europa Tyone Taygete Callisto Mneme Ananke Himalia Euanthe Pasithee Erinome Adrastea Eukelade Temisto Praxidike Chaldene Telxinoe Hermippe Kallichore Harpalyke Ganymede Table 7: Information of Jupiter’s moons (source: the Jupiter satellite and moon page. Carnegie Institution. http://home.dtm.ciw.edu/users/shep XXVII XLII LX XXIV XII XXII XXX XL LVII XLIV XXI XXXVII XXVI LI XX XI XXIII XLVII XXIX XXXI XXV XXXIII Number XVI II XVIII XIII XLVI LII XLIII XV VA II III IV VI XIV XL XXXVIII LH VII LIII 10 Advances in Astronomy 2011 2011 1914 2017 2016 2001 2001 1999 1908 2001 2001 2001 2003 2003 2003 2003 2003 2003 2003 2003 2003 2003 2003 2003 2003 2003 2003 2000 Te Year of Discovery S/2011 J1 S/2017 J1 S/2011 J2 S/2016 J1 S/1999 J1 S/2001 J1 S/2001 J5 S/2001 J9 S/2001 J4 S/2003 J2 S/2003 J8 S/2003 J7 S/2003 J4 S/2003 J9 S/2003 J6 S/2000 J8 S/2001 J10 S/2003 J15 S/2003 J13 S/2003 J14 S/2003 J18 S/2003 J10 S/2003 J12 S/2003 J16 S/2003 J19 S/2003 J23 Designation Table 7: Continued. All numbered or named satellites are in the table above Pasiphae or No Strong Clustering Retrograde Irregular Group Te new Jupiter satellites discovered but yet to be numbered or named. Kore Name Aoede Helike Sinope Sponde Cyllene Euporie Autonoe Pasiphae Orthosie Megaclite Callirrhoe Eurydome Hegemone XXXV XLV LV LIV Number XXXIV LVIII XXXVI LIX VIII XIX IX XXXII XXVIII LVI XXXIX XLI XLVIII XVII XLIX Advances in Astronomy 11 0 0 =1.6533 =31.522 0.2983 =13.4252 =2.33856 =2.98203 =201.843 =8.23058 =2.58664 =30.5827 =452.354 =0.232103 Weibull = 0.933741 =212.8357 =0.911426 =0.932058 =0.309664 � � � � � � � � � � � � � � � � l0 l1 l1 l l l t u u u null null MLE =1.6511 =0.0017 =0.8751 =1.14501 =1.58419 did not =7.32997 =1.06821 =1.65708 positive positive positive positive defnite. defnite. defnite. defnite. = 0.440683 � converge. converge. � � � Likelihood � � � � (Maximum Estimation) MLE did not � Hessian is not Hessian is not Hessian is not Hessian is not Te calculated Te calculated Te calculated Te calculated location-scale 0 0 =1.6494 =19.6157 =2491.35 = 113.882 =2.54251 =4.50454 = 2.33268 = 8.22086 = 0.001165 = 0.116687 = 0.334886 =0.008298 =0.703387 =0.0191329 = 0.0116728 � =0.0848305 � � � � � � � � � � � � � � � 0 0 = 3522.9 =1.6494 =5.73011 =14.0714 = 2.33271 =3.17857 =24.8177 =2.54393 = 8.22088 =948.336 =0.421927 =0.242857 = 0.0121133 =0.008298 = 0.0198551 = 0.0880083 � � � � � � � � � � � � � � � � 1 0 =2.7205 0.0998 0.1005 0.1005 0.6662 0.2271 =67.5832 = 46161.3 =40.5921 = 226.315 =9985.19 =5.44166 =769.929 =6.47876 =0.296761 =0.286619 =0.224286 =0.298776 =9866.0175 =0.0699297 � � = 1.44943e+7 � � � � � � � � � � � � � � 0 0 =1.606 =1.6502 = 2.3326 =151.943 =7.99281 =1.90719 =8.21989 = 941.547 =8.93572 =2.53584 =0.149513 =0.120199 = 0.004132 � = 0.0112805 =0.0506215 � � � � � = 0.00651346 � � � � � � � � � � Table 8: Carme group. null null =1.16971 =1.10481 =1.16534 =5891.52 =12.0298 =909.093 =2.87704 Gamma Logistic Nakagami Normal Rice =0.160966 =0.208399 = 0.0000418 � � =39419.4189 � � � to singular. to singular. =0.00279831 � � � � � � Matrix is close Matrix is close � 0 0 =1.6494 =153.808 =10.5296 = 2.32292 = 2.33268 =2.54254 = 8.22086 =0.87834 = 6.88668 =0.180961 =0.913758 =0.871682 =0.005040 Saunders = 0.0330888 � � � Birnbaum- � =0.00232688 � � � =0.00500346 � � � � � � � � � 11 1 1 1 1 1 0 1 1 1000000n 11 1 1 1 1 1n 11 1 1 1 1 1n 11 1 1 1 1 1 00 1 0000 0 0 000 Beta =166729 =32931.5 =27.2522 = 68.2281 =4.79853 = 0.211325 =0.210438 =0.185628 =0.186748 = 0.128804 =0.245874 =0.186944 =0.272087 =0.206976 =0.247005 =0.0841809 � � � � � � � � � � � � � � � � h h h h h h h h value 1.64E-05 0.0171 0.0232 0.0162 0.0023 0.0024 3.21E-08 null 0.0346 value 1.78E-05 0.0254 0.0249 0.0374 0.0024 0.0024 2.01E-08 0.7119 0.0574 -value 0.0994 0.0984 0.0991 0.2697 -value 1.40E-05 0.0126 0.0141 0.0287 0.0015 0.0018 1.37E-08 0.6619 0.0358 -value 3.55E-04 0.9898-value null 8.80E-03 0.5488 0.9987 null 0.0136 0.6607 0.9891 7.2972E-5 0.9882 0.5598 null 0.5461 0.7911 -value 4.80E-03 0.6245 4.0418E-7 0.5695 0.6128 0.6118 0.6002 null 0.3726 -value 6.83E-05 8.58E-05 9.31E-04 0.0028 6.02E-04 3.10E-04 6.68E-12 null 0.0478 p- p- p p p p p p parameters parameters parameters parameters parameters parameters parameters parameters ) kg) 2 13 )/10 2 ) ) km) km/h) ∘ 7 3 −1 2 Semi-major axis (10 Mean orbit velocity (10 Orbit eccentricity (10 Inclination of orbit (10 Equatorial radius (km) Mass (10 Surface gravity ((m/s Escape velocity (km/h) 12 Advances in Astronomy = 14.419 =1.4986 = 1.11631 = 2.11821 =3.18574 =1.21203 =141.939 =8.79917 =38.3573 =82.6918 =2.29923 =21.9834 =52.7307 =1.07908 Weibull =0.270122 =0.368827 � � � � � � � � � � � � � � � � l0 l u scale =0.0127 =1.4879 =1.4538 =1.19615 = 2.11665 =1.15837 =1.40815 =8.72481 =1.80724 =2.61085 =1.52402 =1.36328 =8.24903 =0.167044 =0.660297 converge. =0.0203158 � likelihood � � tlocation- =0.0636087 � Maximum � � = 0.00915557 � � � � � � � MLE did not � � not converge. � � estimation did � 0.1581 0.6017 0.0821 =1.4841 =2.1074 =0.0308 =8.7539 =14.9178 =670.178 =2.24187 =3.40798 = 0.116491 = 22.4209 =0.253955 = 0.531109 = 0.126333 � � � =0.0295767 =0.0772086 = 0.0041522 � � � � � � � � � � � � � =13.7 =0.25 =3.07 =1.4844 =0.0309 =2.2449 =2.10761 =16.9184 =3.91806 � =309.925 =8.75424 = 944.346 � � =0.271825 = 0.0311751 = 0.1227135 =0.0813835 � � � � � � � � � � � � � =0.129 =445.3 =23.241 =2.2045 =5.05313 =2.10761 =898663 =3235.86 =76.6427 =92.6977 = 0.0311751 =575.3654 =0.418467 =0.395632 =0.482908 � � =0.0927262 � � � � � � � � � � � � � � = 2.1139 =1.4863 =0.0154 =2.18108 =9.82501 =78.0193 =1.39766 =8.73706 =5.97056 =298.807 =2.24867 =0.101607 =0.190652 = 0.0123376 = 0.0313777 � =0.0683413 � � � � � � � � � � � � � � � Table 9: Ananke group. = � = 1411.32 = 2.11188 =1.82168 =1.45368 =1.58042 = 2286.47 =368.072 = 8.66859 Gamma Logistic Nakagami Normal Rice =0.137236 =0.219599 = 0.000649 � � � to singular. to singular. � � � � � � 0.006.09908 � � Matrix is close Matrix is close =1.4841 =2.1074 =0.0210 =8.75391 =2.24183 =2.32358 = 60.0159 = 10.6647 =4.33548 = 0.198988 =0.732471 =0.778473 = 0.052298 =0.827948 Saunders =0.0087315 =0.0142595 � � � � � � � � Birnbaum- � � � � � � � � 1000 0 0 000 10 0 0 0 0 1 0 0 10 0 0 0 0 1 0 0 10 0 0 0 0 1 0 0 1000 0 0 000 1000 0 0 000 11 0 1 1 1 1n 0000 0 0 000 Beta =2.619 =17.098 = 11193.4 =1951.25 =0.14788 =8.26353 =0.198125 =0.144156 =0.145225 � =0.185662 =0.178068 =0.230441 =0.225054 = 0.106068 =0.246544 � =0.0964716 � � � � � � � � � � � � � � h h h h h h h h value 7.52E-04 0.2802 0.2227 0.3042 0.0982 0.0933 3.50E-03 0.421 0.2732 value 7.43E-04 0.5045 0.4118 0.2689value 0.158 6.67E-04 0.1036 0.3295 2.40E-03 0.2788 0.7231 0.2429 0.4889 0.1138 0.0989 1.70E-03 0.7262 0.3567 value 0.0328 0.1347 0.1371 0.6785 0.1395 0.1364 0.1421 0.9304 0.684 value 8.99E-05 0.1634 0.1616 0.6557 0.1598 0.151 value 0.0025 0.0341 0.0668 0.03 0.037 6.00E-03 2.74E-08 null 0.4304 -value 0.5027 0.4904 0.4991 0.7114 0.5077 0.5167 0.5167 0.9075 0.6297 -value 4.30E-3 0.91 0.9211 0.9652 0.9312 0.9485 0.9408 0.7032 0.9788 p- p- p- p- p- p- p p parameters parameters parameters parameters parameters parameters parameters parameters ) kg) 2 13 ) ) km) km/h) ∘ 7 3 −1 2 Semi-major axis (10 Mean orbit velocity (10 Orbit eccentricity (10 Inclination of orbit (10 Equatorial radius (km) Mass (10 Surface gravity ((m/s2)/10 Escape velocity (km/h) Advances in Astronomy 13 =1.5508 = 72.5121 = 17.0302 =3.69781 =8.48321 =2.35522 =19.8563 =3.24837 =23.9875 =23.6908 =3.30468 Weibull =0.781648 =0.309876 =0.266585 =0.807236 =0.843794 � � � � � � � � � � � � � � � � l0 l0 l0 l0 l0 l0 l l l l l l t u u u u u u =1.5203 =0.0599 =2.38241 positive positive positive positive positive defnite. defnite. defnite. defnite. defnite. =0.730397 converge. � = 0.0273627 � =5657176.32 � � MLE did not Hessian is not Hessian is not Hessian is not Hessian is not Hessian is not � Te calculated Te calculated Te calculated Te calculated Te calculated � location-scale =1.5191 =0.0599 =1.06913 =27.9843 =0.14977 =5639.23 =6.39782 =215.387 =2.68061 =8.27394 =2.28844 =0.472316 =0.393632 � = 0.192009 =0.946989 = 0.0105316 � � � � � � � � � � � � � � � =4.5 =0.35 =1.5203 =0.0599 � = 8.28331 =7838.61 =8.10876 =2507.97 =1.04493 =35.2429 =2.29334 =20.0625 =2.90656 � =0.154515 =0.406311 =0.587651 � � � � � � � � � � � � � � =2.3148 = 5.28181 =9.47173 =81.8925 = 112.484 =1566.94 =57.3364 =68.7679 = 0.44625 =2.02049 =162.6092 = 0.253336 = 0.270001 = 0.242429 � = 0.0664969 � � � � � =6.36374E+7 � � � � � � � � � � =2.32 =1.5136 = 0.0339 = 11.3591 = 2.9666 =2437.85 =2.95251 = 722.941 =12.8644 =2.50052 = 8.22803 � =0.214293 =0.599373 = 0.205059 = 0.226046 � =0.0823946 � � � � � � � � � � � � � � Table 10: Pasiphae group. =0.0023 =17059.2 = 222.433 = 455.023 =5.83678 =24.4022 =6.69584 = 655.7110 Gamma Logistic Nakagami Normal Rice =0.147015 = 0.822159 =0.896179 =0.770973 =0.390547 =0.434085 =0.0103103 =0.0182041 � � � � � � � � � � � � � � � � =1.5191 =0.0389 =1.14416 = 12.9337 =1.08145 =185.784 =1.20376 =8.27432 =8.45355 = 2.66229 = 2.28804 =2.80607 =0.232932 =0.068061 = 0.426688 � Saunders � =0.0465993 � � � � � � � Birnbaum- � � � � � � � = 11 1 1 1 1 1n 11 1 1 1 1 1n 11 1 1 1 1 1n 1000 0 0 000 1000000n 1000000n 11 1 1 1 1 1n 0000 0 0 000 � Beta =554.177 =3.19406 = 3091.06 = 4.22446 = 0.133182 =0.203821 =0.216072 =0.201972 =0.237408 =0.262587 =0.208766 = 0.766602 = 0.286862 =0.203804 =0.206344 0.0820299 � � � � � � � � � � � � � � � h h h h h h h h value 9.34E-5 0.0186 0.0238 0.0139 0.0062 0.0055 2.15E-08 null 0.0594 value 1.85E-04 0.0224 0.0377 0.0153 0.0139value 0.0079 1.55E-04 0.0157 2.09E-08 0.0296 null 0.0142 0.0859 0.011 0.0074 2.39E-08 null 0.0722 value 0.0012value 0.1039 9.38E-04 0.1069 0.2085 0.2952 0.2025 0.1092 0.4015 0.1238 0.1961 0.2048 0.1125 0.1906 0.373 0.1848 null 0.1794 value 0.0132 0.3639 0.5771 0.955 0.7272 0.9138 0.8914 null 0.9072 value 0.0012 0.0051 0.0111 0.0047 0.0091 8.16E-04 4.36E-11 null 0.1096 -value 0.5345 0.5633 0.5423 0.8987 0.5206 0.5002 0.5002 0.5004 0.2133 p- p- p- p- p- p- p- p parameters parameters parameters parameters parameters parameters parameters parameters ) kg) 2 13 )/10 2 ) ) km) km/h) ∘ 7 3 −1 2 Semi-major axis (10 Mean orbit velocity (10 Orbit eccentricity (10 Inclination of orbit (10 Equatorial radius (km) Mass (10 Surface gravity ((m/s Escape velocity (km/h) 14 Advances in Astronomy

1 obviously, but these will not change dramatically over a long 0.9 period of time.

0.8 Appendix 0.7

0.6 A. 0.5 See Table 7. (x) or F(x) (x) or

Ｈ 0.4 ＆ 0.3 B. Distribution Inference Results 0.2 SeeTables8,9,and10 0.1

0 Data Availability 8.18 8.19 8.2 8.21 8.22 8.23 8.24 8.25 8.26 3 Mean Orbit Velocity (10 km/h) Te data used to support the fndings of this study are available from the corresponding author upon request. observed CDF best−ft CDF Conflicts of Interest Figure 11: Comparison of the observed CDF and best-ft CDF of mean orbit velocity. Te authors declare that they have no conficts of interest.

Te distribution implies that average equatorial radius of Acknowledgments themoonsinCarmegroupis1.67257kmwithconfdence Te authors acknowledge the support of National Natural [1.23204, 2.1131] interval (we believe that the confdence Science Foundation of China (NSFC) through grant Nos. interval will be smaller and shorter with continuous progress 11672259, 11302187, and 11571301; the Ministry of Land and of observation technology) under the level of signifcance Resources Research of China in the Public Interest through 0.05. For the true value of Erinome’s equatorial radius being grant No. 201411007; and Top-notch Academic Programs 1.6 km, it is easy to fnd that we can use the estimated value to Project of Jiangsu Higher Education Institutions (TAPP) predict the equatorial radius or as a reference to study other through grant No. PPZY2015B109. relevant physical and orbital characteristics.

7. Conclusions References [1] S. S. Sheppard, Te Jupiter satellite and moon page,Carnegie By using the one-sample K-S nonparametric test method of Institution, 2017. statistical inference, the distribution laws of the physical and orbital properties of Jupiter’s moons are investigated statis- [2]D.B.Jupiter,“moons,”NASA Planetary Science Division,2017, tically in this paper. Te physical and orbital characteristics https://solarsystem.nasa.gov/planets/jupiter/moons. of Jupiter’s moons are found to obey the Birnbaum–Saunders [3] D. Jewitt and N. Haghighipour, “Irregular satellites of the distribution, the logistic distribution, the Weibull distribu- planets: Products of capture in the early solar system,” Annual tion, and the t location-scale distribution. Review of Astronomy and Astrophysics,vol.45,pp.261–295,2007. In addition, the probability density curves of the data [4] S. S. Sheppard, “Outer irregular satellites of the planets and their distributions are also generated, and the diferences in the relationship with asteroids, comets and Kuiper Belt objects,” physical and orbital characteristics of the three groups are Proceedings of the International Astronomical Union,vol.1,no. explained in more detail. 229, pp. 319–334, 2005. Trough a specifc example, we fnd that some moons’ [5]B.W.CarrollandD.A.Ostlie,An introduction to modern missing data can be inferred by using the aforementioned astrophysics, Pearson, 2nd edition, 2006. distributional model and probability density function. More [6] Elkins-Tanton LT, “Jupiter and Saturn,” Facts on File, 2010. importantly, with the help of the distribution, it can be [7] D. Jewitt, S. Sheppard, and C. Porco, “Jupiter’s outer satellites even helpful to predict the physical or orbital features of the and Trojans,” in Jupiter. Te planet, satellites and magnetosphere, undiscovered moon. F. Bagenal, T. E. Dowling, and W. B. McKinnon, Eds., vol. 1, pp. If future observations will allow for the expansion of the 263–280, Cambridge planetary science, Cambridge, UK, 2004. number of Jupiter’s moons, we believe that the distribution [8] Wikipedians, “Moons of Jupiter,” in Jupiter,pp.78–94,Pedi- laws will be slightly modifed as potential newly discovered aPress GmbH, 2018. distribution functions ft the increased sample better, and [9] D. Nesvorn´y, J. L. A. Alvarellos, L. Dones, and H. F. Levison, the distributions will probably tend to be more uniform; i.e., “Orbital and collisional evolution of the irregular satellites,” Te some of the diferent properties follow the same distribution Astronomical Journal,vol.126,no.1,pp.398–429,2003. Advances in Astronomy 15

[10] D. Nesvorn´y, C. Beaug´e, and L. Dones, “Collisional origin of families of irregular satellites,” Te Astronomical Journal,vol. 127, no. 3, pp. 1768–1783, 2004. [11] S. S. Sheppard and D. C. Jewitt, “An abundant population of small irregular satellites around Jupiter,” Nature,vol.423,no. 6937, pp. 261–263, 2003. [12] B. Carry, “Density of asteroids,” Planetary and Space Science,vol. 73, no. 1, pp. 98–118, 2012. [13] P. Kankiewicz and I. Włodarczyk, “Dynamical lifetimes of asteroids in retrograde orbits,” Monthly Notices of the Royal Astronomical Society,vol.468,no.4,pp.4143–4150,2017. [14]R.V.HoggandA.T.Craig,Introduction to mathematical statistics, Pearson Education Asia Limited and Higher Education Press, 2004. [15] H. Xie, X. Cui, B. Wan, and J. Zhang, “Statistical analysis of radio interference of 1000 kV UHV AC double-circuit transmission lines in foul weather,” CSEE Journal of Power and Energy Systems,vol.2,no.2,pp.47–55,2016. [16] R. R. Wilcox, “Some practical reasons for reconsidering the Kolmogorov-Smirnov test,” British Journal of Mathematical and Statistical Psychology,vol.50,no.1,pp.9–20,1997. [17] Z. W. Birnbaum and S. C. Saunders, “A new family of life distributions,” Journal of Applied Probability,vol.6,no.2,pp. 319–327, 1969. [18] K. B. Athreya and S. N. Lahiri, Measure Teory and Probability Teory, Springer Texts in Statistics, Springer, New York, NY, USA, 2006. [19] List of moons in orbit around Jupiter, 2018, https://www .universeguide.com/planet/jupiter#themoons. [20] R. Jiang and D. N. P. Murthy, “A study of Weibull shape parameter: Properties and signifcance,” Reliability Engineering &SystemSafety, vol. 96, no. 12, pp. 1619–1626, 2011. Journal of International Journal of Advances in The Scientifc Applied Bionics Chemistry Engineering World Journal Geophysics and Biomechanics Hindawi Hindawi Hindawi Publishing Corporation Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 http://www.hindawi.comwww.hindawi.com Volume 20182013 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

Active and Passive Shock and Vibration Electronic Components

Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

Submit your manuscripts at www.hindawi.com

Advances in Advances in Mathematical Physics Astronomy Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

International Journal of Rotating Machinery Advances in Optical Advances in Technologies OptoElectronics

Advances in Advances in Physical Chemistry Condensed Matter Physics Hindawi Hindawi Hindawi Hindawi Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 Hindawi Volume 2018 www.hindawi.com www.hindawi.com

International Journal of Journal of Antennas and Advances in Advances in International Journal of Chemistry Propagation High Energy Physics Acoustics and Vibration Optics Hindawi Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018