Advances in Astronomy

Advances in Astronomy / 2018 / Article

Research Article | Open Access

Volume 2018 |Article ID 1894850 | https://doi.org/10.1155/2018/1894850

F. B. Gao, X. H. Zhu, X. Liu, R. F. Wang, "Distribution Inference for Physical and Orbital Properties of Jupiter’s Moons", Advances in Astronomy, vol. 2018, Article ID 1894850, 15 pages, 2018. https://doi.org/10.1155/2018/1894850

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

Academic Editor: Geza Kovacs
Received04 Jun 2018
Revised01 Sep 2018
Accepted30 Sep 2018
Published01 Nov 2018

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 [1416]. 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.


Name of Selected DistributionsParametersMeaning

Betafirst shape parameter
second shape parameter

Birnbaum-Saundersscale parameter
shape parameter

Gammashape parameter
scale parameter

Logisticmean
scale parameter

Nakagamishape parameter
scale parameter

Normalmean
standard deviation

Ricenoncentrality parameter
scale parameter

location-scalelocation parameter
scale parameter
shape parameter

Weibullscale parameter
shape parameter

For the 9 continuous distributions in Table 1, the one-sample K-S test will be used to select the distribution with the highest confidence level. In order to characterize these distributions with well-defined parameter values, maximum likelihood estimation is also used. In addition, the parameter values of these distributions can be calculated from the observed data. However, when confidence level (typically set to 0.05) decreases, the rejection domain of the test becomes smaller, so the observed values that initially fall into the rejection domain may eventually fall into the acceptance domain. This situation will bring some trouble in practical application. To this end, we adopt p value, which represents the obtained confidence level by using the one-sample K-S test. In addition, the use of p value not only avoids determining the level of significance in advance, but also makes it easy to draw conclusions about the test by comparing the p value and significance level of the test. If the p value is greater than 0.05, we declare that the null hypothesis can be accepted. Furthermore, if the p values of several distributions are all greater than 0.05, the distribution with the largest p value should be selected, and the corresponding distribution will be the most appropriate one to fit the observed data. Our results can be found in the tables in Appendix B.

3. Distribution Inference of Satellite Groups

In this section, distributions of several diverse physical and orbital properties for the Carme group, Ananke group, and Pasiphae group are inferred sequentially.

3.1. Carme Group

There are 15 moons in the Carme group (please see Appendix A.). Due to the lack of enough data from S/2010 J1, the number of adopted moons is 14, which means the length of each data set is 14 from the mathematical perspective. In addition, considering all the mean densities have been calculated at being 2.60 g/cm3, the surface gravity of Carme and other moons in this group is 0.017 m/s2 and 0.001 m/s2, respectively [19]. Therefore, these data obviously do not obey the distributions.

Based on the previous method of statistical distribution inference and MATLAB 2016a (Intel Core i5-3230 M, CPU 2.60 GHz), distribution inferences in the Carme group can be found in Table 2, and the last column represents the p value. The smaller the p value, the greater the significance because it tells us that hypothesis under consideration may not be sufficient to explain the observations. The hypothesis will be rejected if any of these probabilities is less than or equal to a small, fixed but arbitrarily predefined threshold value. The null hypothesis here refers to data obeying a particular distribution, and the alternative hypothesis assumes that the data does not obey the distribution.


CharacteristicsDistribution inferenceParameter estimatesp-value

Semi-major axis (107 km)logistic distribution = 2.3326, = 0.006513460.9988
Mean orbit velocity (103 km/h)logistic distribution = 8.21989, = 0.01128050.6607
Orbit eccentricity (10−1)Birnbaum-Saunders distribution = 2.54254, = 0.03308880.6245
Inclination of orbit (102°)t location-scale distribution= 1.6511, = 0.0017, = 0.87510.6662
Equatorial radius (km)t location-scale distribution= 1.65708, = 0.440683, = 1.145010.7119
Escape velocity (km/h)t location-scale distribution = 7.32997, = 1.58419, = 1.068210.6619

From Table 2, semi-major axis and mean orbit velocity obey the logistic distribution, of which parameter denotes the average orbital eccentricity and parameter plays a key role in representing the variance of the data set, through the variance formula , so the variances of these logistic distributions are 1.3957E-04 and 4.1863E-04, respectively. As the parameter increases, it indicates that the average semi-major axis and mean orbit velocity of the Carme group increase. As the parameter increases, the data in these two characteristics gradually disperse, and the discrepancy between the data and the average value increases. The meaning of decreasing the parameter and shares the same principle as that of increasing the parameters. The orbit eccentricity follows the Birnbaum–Saunders distribution, which is unimodal with a median of . The mean value and the variance of the distribution can be calculated by the following relationships:

Therefore, the mean value of orbital eccentricity can be calculated to be 0.2543 and the variance is 7.08E-9.

The inclination of orbit, equatorial radius, and escape velocity are subject to the t location-scale distribution, which contains the scale parameter , the location parameter , and the shape parameter . Without loss of generality, we assume that the data vector obeys the t location-scale distribution, and then we have , which obeys Student’s t-distribution; here represents the degrees of freedom. As can also be seen in Table 2, the inclination of orbit obeys the t location-scale distribution with parameters (1.6511, 0.0017, 0.8751) and the mean inclination is 165.11°. The equatorial radius follows the t location-scale distribution with parameters (1.65708, 0.440683, 1.14501). Therefore, the moons in Carme group have an average equatorial radius of 1.65708 km. In addition, the escape velocity characteristic obeys the t location-scale distribution with parameters (7.32997, 1.58419, 1.06821) and the average escape velocity is 7.32997 km/h. As the parameter changes, the average equatorial radius and escape velocity of the group also change accordingly. In the t location-scale distribution, the variance is , and when the shape parameter is greater than two, the variance of the distribution is defined. Therefore, the variances of these two specific t location-scale distributions cannot be defined.

3.2. Ananke Group

There are 10 moons in the Ananke group, of which we do not have enough data about S/2010 J2. Thus, the length of each characteristic of the remaining moons in this group will be 9.

From our discussion on the t location-scale distribution in Carme group, it is easy to understand the distribution inference of semi-major axis, mean orbit velocity, equatorial radius, surface gravity, and escape velocity in the Ananke group (see Table 3). However, compared with Table 2, it is noted that the orbit eccentricity and mass properties in the Ananke group are subject to the Weibull distribution. Parameters and represent the scale and shape parameters of the distribution [20], respectively, which together determine the mean and variance of the distribution.


CharacteristicsDistribution inferenceParameter estimatesp-value

Semi-major axis (107 km)t location-scale distribution = 2.11665, = 0.00915557, = 1.524020.9304
Mean orbit velocity (103 km/h)Weibull distribution = 2.29923, = 21.98340.9788
Orbit eccentricity (10−1)t location-scale distribution = 8.72481, = 0.0203158, = 1.363280.6017
Inclination of orbit (102°)t location-scale distribution= 1.4879, = 0.0127, = 1.45380.9075
Equatorial radius (km)t location-scale distribution = 1.80724, = 0.660297, = 1.196150.7231
Mass (1013 kg)Weibull distribution = 38.3573, = 0.3688270.4304
Surface gravity ((m/s2)/103)t location-scale distribution = 0.167044, = 0.0636087, = 1.408150.4210
Escape velocity (km/h)t location-scale distribution = 8.24903, = 2.61085, = 1.158370.7262

3.3. Pasiphae Group

There are 19 moons in the Pasiphae group; due to the lack of data about three moons, S/2011 J2, S/2017 J1, and S/2016 J1, the remaining 16 will be studied in this subsection.

The distributions in Table 4 are inferred to be similar to the distribution in Tables 2 and 3. Therefore, we can easily understand the parameters of these distributions.


CharacteristicsDistribution inferenceParameter estimatesp-value

Semi-major axis (107 km)t location-scale distribution = 2.38241, = 0.0273627, = 0.7303970.3730
Mean orbit velocity (103 km/h)logistic distribution = 8.22803, = 0.2260460.4015
Orbit eccentricity (10−1)logistic distribution = 2.95251, = 0.5993730.9550
Inclination of orbit (102°)logistic distribution = 1.5136, = 0.03390.8987
Equatorial radius (km)Weibull distribution = 3.69781, = 0.7816480.0859
Mass (1013 kg)Weibull distribution = 72.5121, = 0.2665850.1096
Surface gravity ((m/s2)/103)Weibull distribution = 3.09876, = 0.8437940.0594
Escape velocity (km/h)Weibull distribution = 17.0302, = 0.8072360.0722

4. Comparison of Data Properties

Table 5 is given according to the previous distribution inference.


CharacteristicsCarme GroupAnanke GroupPasiphae Group

Semi-major axislogistic distributiont location-scale distributiont location-scale distribution
Mean orbit velocitylogistic distributiont location-scale distributionlogistic distribution
Orbit eccentricityBirnbaum–Saunders distributionWeibull distributionlogistic distribution
Inclination of orbitt location-scale distributiont location-scale distributionlogistic distribution
Equatorial radiust location-scale distributiont location-scale distributionWeibull distribution
MassNoneWeibull distributionWeibull distribution
Surface gravityNonet location-scale distributionWeibull distribution
Escape velocityt location-scale distributiont location-scale distributionWeibull distribution

Based on the previous distribution inference, the properties of the moons’ data can be compared more specifically and conveniently. According to the distribution of the specific parameters, we get the following probability density function (PDF) diagram.

As shown in Figure 1, the semi-major axis in the Ananke group is the smallest, followed by the Carme group and the Pasiphae group. In addition, the PDF of the Pasiphae group is relatively flat, indicating a large dispersion in semi-major axis around Jupiter, while the data in the Carme and Ananke groups differ slightly. As can be seen from Figure 2, the trend of mean orbital velocity distribution in Carme group is the closest to each other, with the mean value being the smallest among the three groups, followed with the Pasiphae and Ananke groups. Obviously, the curve of the Pasiphae group is the flattest, which shows the mean orbit velocity in the Pasiphae group differing greatly. Figure 3 shows that the mean value in the Ananke group is the smallest, followed by the Carme and Pasiphae groups. It is obvious that the PDF curve of the Pasiphae group is flatter compared to the others, which means the orbital eccentricity of the Pasiphae group has a relatively large dispersion. In Figure 4, although the inclinations of orbit in the Ananke group and Pasiphae group are relatively close in value compared with the Carme group, and the data looks more dispersed than those in Carme group, the inclination of orbit in the Ananke group has the same distribution as the Carme group.

Figure 5 shows the PDF curves of equatorial radii of these three groups. Data attributes are similar; most of the moons’ 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 are of relatively small mass.

Figure 7 illustrates the surface gravity PDF curves in the Ananke and Pasiphae groups. The difference in density between the two groups indicates that the surface gravity of the Ananke moons is higher than that of the Pasiphae group. The PDF plots corresponding to the escape velocity are shown in Figure 8, where it is clear that the Pasiphae group is flatter than the other two groups. The Carme group also has similar escape velocity density to the Ananke group, but the former average escape velocity is smaller.

5. Verification of Rationality of Theoretical Results

In this section, we take the semi-major axis and mean orbit velocity of the moons in the Carme group as concrete examples to illustrate the rationality of the statistical inferences in the preceding sections. As can be seen in Table 2, semi-major axis and mean orbit velocity obey the logistic distribution with parameters (2.3326, 0.00651346) and (8.21989, 0.0112805), respectively. So the corresponding predicted PDFs can be written as and

Note that the orbital and physical properties are not independent of each other. For instance, the semi-major axis is related to the mean orbit velocity through the relation , where is the mass parameter. Then the PDF of mean orbit velocity can also be derived analytically as followsAlthough the PDF of mean orbit velocity obtained by different methods has different mathematical representations, from Figure 9, we can find that (PDF curve represented by red circles) obtained by the analytical method is in good agreement with (PDF curve represented by blue circles) obtained by statistical inference.

However, what we need to pay attention to here is that some physical features and orbital elements are mixed and can be linked by some mathematical formulas similar to the above. Theoretically, the distribution of another variable in the formula can be solved by a known defined distribution; although this may be a complex process because the probability density function may contain some transcendental functions and gamma functions, it can be achieved. Yet the known distribution becomes uncertain now; that is, the distribution exists with a certain probability. So, there will be a certain risk when we calculate the distribution of the linked variable based on this uncertain distribution, especially when the possibility of the inferred distribution is not very high.

In addition, in order to further show that the results of the KS test agree well with the actual observed results, we also compared the best-fit CDFs (the CDFs of inferred statistically) and the observed CDFs. From Figures 10 and 11, it can be seen that the best-fit CDF and the observed CDF of semi-major axis agree better than the case of mean orbital velocity. This 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.

6. Application of the Distribution Inference

As mentioned previously, identification of irregular satellite families is tentative. However, after the distribution was inferred, it is found that these features of moons in the three different groups obey some selected distributions. Furthermore, the obtained population distribution can also be used to predict the characteristic data. Considering the rationality of proving this prediction method, suppose some characteristic data of a moon is unknown in a given irregular moon group. One specific feature of the moon can be predicted by using other moons’ data and one-sample K-S method. Here is an example:

Assume that some characteristic data of the moon Erinome (S/2000 J4) in Carme group is poorly known. Now, we try to predict the equatorial radius of the Erinome. First, we use one-sample K-S method to find the most appropriate continuous distribution for these data. The inference results are as shown in Table 6.


p-valuehParameter Values

Beta4.02E-051 = 0.202265
= 0.173517

Birnbaum-Saunders0.03071 = 2.37288
= 0.943409

Gamma0.03081 = 1.05985
= 3.11366

Logistic0.04631 = 1.95282
= 1.72324

Nakagami0.00391 = 0.282272
= 43.5177

Normal0.00371 = 3.3
= 5.94531

Rice5.73E-081 = 0.138623
= 4.66385

t location-scale0.68060 = 1.67257
= 0.487941
= 1.15623

Weibull0.06980 = 3.06061
= 0.897697

Note: h = 0 and h = 1 indicate acceptance of the null hypothesis and rejection of the null hypothesis, respectively.

From the p values displayed in Table 6, it becomes clear that the corresponding p value 0.6806 is the largest and , so the best distribution for the remaining characteristic data in the Carme group is t location-scale distribution with the PDFwhere is the Gamma function.

The distribution implies that average equatorial radius of the moons in Carme group is 1.67257 km with confidence interval (we believe that the confidence interval will be smaller and shorter with continuous progress of observation technology) under the level of significance 0.05. For the true value of Erinome’s equatorial radius being 1.6 km, it is easy to find that we can use the estimated value to predict the equatorial radius or as a reference to study other relevant physical and orbital characteristics.

7. Conclusions

By using the one-sample K-S nonparametric test method of statistical inference, the distribution laws of the physical and orbital properties of Jupiter’s moons are investigated statistically in this paper. The physical and orbital characteristics of Jupiter’s moons are found to obey the Birnbaum–Saunders distribution, the logistic distribution, the Weibull distribution, and the t location-scale distribution.

In addition, the probability density curves of the data distributions are also generated, and the differences in the physical and orbital characteristics of the three groups are explained in more detail.

Through a specific example, we find that some moons’ missing data can be inferred by using the aforementioned distributional model and probability density function. More importantly, with the help of the distribution, it can be even helpful to predict the physical or orbital features of the undiscovered moon.

If future observations will allow for the expansion of the number of Jupiter’s moons, we believe that the distribution laws will be slightly modified as potential newly discovered distribution functions fit the increased sample better, and the distributions will probably tend to be more uniform; i.e., some of the different properties follow the same distribution obviously, but these will not change dramatically over a long period of time.

Appendix

A.

See Table 7.


NumberNameDesignationThe Year of Discovery

Small Inner Regulars and Rings

XVIMetis1979
XVAdrastea1979
VAmalthea1892
XIVThebe1979

Galileans

IIo1610
IIEuropa1610
IIIGanymede1610
IVCallisto1610

Themisto Prograde Irregular Group

XVIIIThemistoS/2000 J12000

Himalia Prograde Irregular Group

XIIILeda1974
VIHimalia1904
XLysithea1938
VIIElara1905
LIIIDiaS/2000 J112000

Carpo Prograde Irregular Group

XLVICarpoS/2003 J202003

Retrograde Irregular Groups

---

Ananke Retrograde Irregular Group

LIIS/2010 J22010
XXXIIIEuantheS/2001 J72001
XXIXThyoneS/2001 J22001
XLMnemeS/2003 J212003
XXIIHarpalykeS/2000 J52000
XXXHermippeS/2001 J32001
XXVIIPraxidikeS/2000 J72000
XLIIThelxinoeS/2003 J222003
LXS/2003 J32003
XXIVIocasteS/2000 J32000
XIIAnanke1951

Carme Retrograde Irregular Group

XLIIIArcheS/2002 J12002
XXXVIIIPasitheeS/2001 J62001
LHerseS/2003 J172003
XXIChaldeneS/2000 J102000
XXXVIIKaleS/2001 J82001
XXVIIsonoeS/2000 J62000
XXXIAitneS/2001 J112001
XXVErinomeS/2000 J42000
LIS/2010 J12010
XXTaygeteS/2000 J92000
XICarme1938
XXIIIKalykeS/2000 J22000
XLVIIEukeladeS/2003 J12003
LVIIS/2003 J52003
XLIVKallichoreS/2003 J112003

Pasiphae or No Strong Clustering Retrograde Irregular Group

XXXIVEuporieS/2001 J102001
LVS/2003 J182003
LIVS/2016 J12016
XXXVOrthosieS/2001 J92001
XLVHelikeS/2003 J62003
LVIIIS/2003 J152003
XXXIIEurydomeS/2001 J42001
XXVIIIAutonoeS/2001 J12001
LVIS/2011 J22011
XXXVISpondeS/2001 J52001
LIXS/2017 J12017
VIIIPasiphae1908
XIXMegacliteS/2000 J82000
IXSinope1914
XXXIXHegemoneS/2003 J82003
XLIAoedeS/2003 J72003
XVIICallirrhoeS/1999 J11999
XLVIIICylleneS/2003 J132003
XLIXKoreS/2003 J142003

The new Jupiter satellites discovered but yet to be numbered or named.
All numbered or named satellites are in the table above

S/2003 J22003
S/2003 J42003
S/2003 J92003
S/2003 J102003
S/2003 J122003
S/2003 J162003
S/2003 J192003
S/2003 J232003
S/2011 J12011

B. Distribution Inference Results

See Tables 8, 9, and 10


BetaBirnbaum-SaundersGammaLogisticNakagamiNormalRicet location-scaleWeibull

Semi-major axis (107 km)p-value3.55E-040.9898null0.99870.01360.98910.9882null0.7911
h10null0100null0
parameters = 27.2522
= 0.272087
= 2.33268
= 0.00500346
Matrix is close to singular. = 2.3326
= 0.00651346
= 9985.19
= 5.44166
= 2.33271
= 0.0121133
= 2.33268
= 0.0116728
The calculated Hessian is not positive definite. = 2.33856
= 201.843

Mean orbit velocity (103 km/h)p-value8.80E-030.5488null0.66077.2972E-50.55980.5461The calculated Hessian is not positive definite.0.2983
h10null0000null0
parameters = 68.2281
= 0.245874
= 8.22086
= 0.00232688
Matrix is close to singular. = 8.21989
= 0.0112805
= 46161.3
= 67.5832
= 8.22088
= 0.0198551
= 8.22086
= 0.0191329
MLE (Maximum Likelihood Estimation) did not converge. = 8.23058
= 452.354

Orbit eccentricity (10−1)p-value4.80E-030.62454.0418E-70.56950.61280.61180.6002null0.3726
h1000000null0
parameters = 4.79853
= 0.247005
= 2.54254
= 0.0330888
= 909.093
= 0.00279831
= 2.53584
= 0.0506215
= 226.315
= 6.47876
= 2.54393
= 0.0880083
= 2.54251
= 0.0848305
The calculated Hessian is not positive definite. = 2.58664
= 30.5827

Inclination of orbit (102°)p-value0.09940.09840.09910.26970.09980.10050.10050.66620.2271
h000000000
parameters = 32931.5
= 166729
= 1.6494
= 0.005040
= 39419.4189
= 0.0000418
= 1.6502
= 0.004132
= 9866.0175
= 2.7205
= 1.6494
= 0.008298
= 1.6494
= 0.008298
= 1.6511
= 0.0017
= 0.8751
= 1.6533
= 212.8357

Equatorial radius (km)p-value1.78E-050.02540.02490.03740.00240.00242.01E-080.71190.0574
h111111100
parameters = 0.206976
= 0.185628
= 2.32292
= 0.913758
= 1.10481
= 2.87704
= 1.90719
= 1.606
= 0.286619
= 40.5921
= 3.17857
= 5.73011
= 0.116687
= 4.50454
= 1.65708
= 0.440683
= 1.14501
= 2.98203
= 0.911426

Mass (1013 kg)p-value6.83E-058.58E-059.31E-040.00286.02E-043.10E-046.68E-12null0.0478
h1111111null1
parameters = 0.0841809
= 0.128804
= 153.808
= 6.88668
= 0.160966
= 5891.52
= 151.943
= 941.547
= 0.0699297
= 1.44943e+7
= 948.336
= 3522.9
= 113.882
= 2491.35
MLE did not converge. = 31.522
= 0.309664

Surface gravity ((m/s2)/102)p-value1.64E-050.01710.02320.01620.00230.00243.21E-08null0.0346
h1111111null1
parameters = 0.211325
= 0.186944
= 0.180961
= 0.87834
= 1.16534
= 0.208399
= 0.149513
= 0.120199
= 0.298776
= 0.224286
= 0.242857
= 0.421927
= 0.001165
= 0.334886
The calculated Hessian is not positive definite. = 0.232103
= 0.933741

Escape velocity (km/h)p-value1.40E-050.01260.01410.02870.00150.00181.37E-080.66190.0358
h111111101
parameters = 0.210438
= 0.186748
= 10.5296
= 0.871682
= 1.16971
= 12.0298
= 8.93572
= 7.99281
= 0.296761
= 769.929
= 14.0714
= 24.8177
= 0.703387
= 19.6157
= 7.32997
= 1.58419
= 1.06821
= 13.4252
= 0.932058


BetaBirnbaum-SaundersGammaLogisticNakagamiNormalRicetlocation-scaleWeibull

Semi-major axis (107 km)p-value0.03280.13470.13710.67850.13950.13640.14210.93040.684
h100000000
parameters = 17.098
= 0.178068
= 2.1074
= 0.0142595
Matrix is close to singular. = 2.1139
= 0.0123376
= 2.10761
= 0.0311751
= 2.10761
= 0.0311751
= 2.1074
= 0.0295767
= 2.11665
= 0.00915557
= 1.52402
= 2.11821
= 141.939

Mean orbit velocity (103 km/h)p-value4.30E-30.910.92110.96520.93120.94850.94080.70320.9788
h100000000
parameters = 2.619
= 0.185662
= 2.24183
= 0.052298
= 368.072
= 0.006.09908
= 2.24867
= 0.0683413
= 92.6977
= 5.05313
= 2.2449
= 0.1227135
= 2.24187
= 0.116491
MLE did not converge. = 2.29923
= 21.9834

Orbit eccentricity (10−1) p-value8.99E-050.16340.16160.65570.15980.1510.15810.60170.0821
h100000000
parameters = 8.26353
= 0.198125
= 8.75391
= 0.0087315
Matrix is close to singular. = 8.73706
= 0.0313777
= 3235.86
= 76.6427
= 8.75424
= 0.0813835
= 8.7539
= 0.0772086
= 8.72481
= 0.0203158
= 1.36328
= 8.79917
= 82.6918

Inclination of orbit (102°)p-value0.50270.49040.49910.71140.50770.51670.51670.90750.6297
h000000000
parameters = 1951.25
= 11193.4
= 1.4841
= 0.0210
= 2286.47
= 0.000649
= 1.4863
= 0.0154
= 575.3654
= 2.2045
= 1.4844
= 0.0309
= 1.4841
= 0.0308
= 1.4879
= 0.0127
= 1.4538
= 1.4986
= 52.7307

Equatorial radius (km)p-value7.43E-040.50450.41180.26890.1580.10362.40E-030.72310.4889
h100000100
parameters = 0.225054
= 0.144156
= 2.32358
= 0.827948
= 1.45368
= 2.11188
= 2.18108
= 1.39766
= 0.395632
= 23.241
= 3.07
= 3.91806
= 0.126333
= 3.40798
= 1.80724
= 0.660297
= 1.19615
= 3.18574
= 1.07908

Mass (1013 kg)p-value0.00250.03410.06680.030.0376.00E-032.74E-08null0.4304
h1101111null0
parameters = 0.0964716
= 0.106068
= 60.0159
= 4.33548
= 0.219599
= 1411.32
= 78.0193
= 298.807
= 0.0927262
= 898663
= 309.925
= 944.346
= 22.4209
= 670.178
Maximum likelihood estimation did not converge. = 38.3573
= 0.368827

Surface gravity ((m/s2)/102)p-value7.52E-040.28020.22270.30420.09820.09333.50E-030.4210.2732
h100000100
parameters = 0.246544
= 0.14788
= 0.198988
= 0.732471
= 1.82168
= 0.137236
= 0.190652
= 0.101607
= 0.482908
= 0.129
= 0.25
= 0.271825
= 0.0041522
= 0.253955
= 0.167044
= 0.0636087
= 1.40815
= 0.270122
= 1.21203

Escape velocity (km/h)p-value6.67E-040.32950.27880.24290.11380.09891.70E-030.72620.3567
h100000100
parameters = 0.230441
= 0.145225
= 10.6647
= 0.778473
= 1.58042
= 8.66859
= 9.82501
= 5.97056
= 0.418467
= 445.3
= 13.7
= 16.9184
= 0.531109
= 14.9178
= 8.24903
= 2.61085
= 1.15837
= 14.419
= 1.11631


BetaBirnbaum-SaundersGammaLogisticNakagamiNormalRicet location-scaleWeibull

Semi-major axis (107 km)p-value0.00120.10390.10690.29520.10920.12380.11250.3730.1848
h100000000
parameters = 4.22446
= 0.262587
= 2.28804
= 0.068061
= 222.433
= 0.0103103
= 2.32
= 0.0823946
= 57.3364
= 5.28181
= 2.29334
= 0.154515
= 2.28844
= 0.14977
= 2.38241
= 0.0273627
= 0.730397
= 2.35522
= 23.6908

Mean orbit velocity (103 km/h)p-value9.38E-040.20850.20250.40150.19610.20480.1906null0.1794
h1000000null0
parameters = 3.19406
= 0.286862
= 8.27432
= 0.0465993
= 455.023
= 0.0182041
= 8.22803
= 0.226046
= 112.484
= 68.7679
= 8.28331
= 0.406311
= 8.27394
= 0.393632
The calculated Hessian is not positive definite. = 8.48321
= 19.8563

Orbit eccentricity (10−1)p-value0.01320.36390.57710.9550.72720.91380.8914null0.9072
h1000000null0
parameters = 0.766602
= 0.237408
= 2.66229
= 0.426688
= 6.69584
= 0.434085
= 2.95251
= 0.599373
= 2.02049
= 9.47173
= 2.90656
= 1.04493
= 2.68061
= 1.06913
MLE did not converge. = 3.24837
= 3.30468

Inclination of orbit (102°)p-value0.53450.56330.54230.89870.52060.50020.50020.50040.2133
h000000000
parameters = 554.177
= 3091.06
= 1.5191
= 0.0389
= 655.7110
= 0.0023
= 1.5136
= 0.0339
= 162.6092
= 2.3148
= 1.5203
= 0.0599
= 1.5191
= 0.0599
= 1.5203
= 0.0599
= 5657176.32
= 1.5508
= 23.9875

Equatorial radius (km)p-value1.85E-040.02240.03770.01530.01390.00792.09E-08null0.0859
h1111111null0
parameters = 0.203821
= 0.201972
= 2.80607
= 1.20376
= 0.770973
= 5.83678
= 2.50052
= 2.9666
= 0.242429
= 81.8925
= 4.5
= 8.10876
= 0.192009
= 6.39782
The calculated Hessian is not positive definite. = 3.69781
= 0.781648

Mass (1013 kg)p-value0.00120.00510.01110.00470.00918.16E-044.36E-11null0.1096
h1111111null0
parameters = 0.0820299
= 0.133182
= 185.784
= 8.45355
= 0.147015
= 17059.2
= 722.941
= 2437.85
= 0.0664969
= 6.36374E+7
= 2507.97
= 7838.61
= 215.387
= 5639.23
The calculated Hessian is not positive definite. = 72.5121
= 0.266585

Surface gravity ((m/s2)/102)p-value9.34E-50.01860.02380.01390.00620.00552.15E-08null0.0594
h1111111null0
parameters = 0.216072
= 0.206344
= 0.232932
= 1.08145
= 0.896179
= 0.390547
= 0.205059
= 0.214293
= 0.270001
= 0.44625
= 0.35
= 0.587651
= 0.0105316
= 0.472316
The calculated Hessian is not positive definite. = 0.309876
= 0.843794

Escape velocity (km/h)p-value1.55E-040.01570.02960.01420.0110.00742.39E-08null0.0722
h1111111null0
parameters = 0.208766
= 0.203804
= 12.9337
= 1.14416
= 0.822159
= 24.4022
= 11.3591
= 12.8644
= 0.253336
= 1566.94
= 20.0625
= 35.2429
= 0.946989
= 27.9843
The calculated Hessian is not positive definite. = 17.0302
= 0.807236

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors acknowledge the support of National Natural Science Foundation of China (NSFC) through grant Nos. 11672259, 11302187, and 11571301; the Ministry of Land and Resources Research of China in the Public Interest through grant No. 201411007; and Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (TAPP) through grant No. PPZY2015B109.

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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.

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