Research paperMultidimensional scaling analysis of the solar system objects
Introduction
A very wide range of observed objects orbiting the Sun, characterized by their masses, densities, orbits and several other characteristics, provide valuable insight on the solar system’s (SS) formation and evolution [1]. The richness of data from the SS surveys is an influential tool for statistical studies as it provides a window into the structure of the SS [2], [3]. According to the International Astronomical Union (IAU) definitions, the SS contain three categories of dynamically and physically classified objects orbiting the Sun: planets, dwarf planets (DPs), and small SS bodies [4], [5]. The five IAU-recognized DPs in the SS include the asteroid Ceres and the trans-Neptunian objects Pluto, Haumea, Makemake, and Eris. Ceres is the only DP in the asteroid belt. The rest of the DPs are located in the trans-Neptunian region that includes the bodies within the Kuiper belt area, the members of the scattered disc and any potential Hills cloud or Oort cloud objects. The cohort population of Kuiper belt consists of three categories namely, comets (between 1 and 20 km in diameter), planetesimals (between 50 and 300 km in diameter), and small planets (between 300 and 2400 km in diameter). In the last cohort of the population, Pluto is the largest. The Oort cloud, which is a thousand times more distant than the Kuiper belt, is a theoretical cloud of mostly icy planetesimals suggested to encircle the Sun at distances ranging from 2000 to 200,000 au. This region lies beyond the bubble-like region in the interstellar medium (the heliosphere). It may be the source for long-period comets and many of the centaurs entering the inner SS.
In 2014, Trujillo and Sheppard [6] pointed out that there is a set of Kuiper belt objects in the distant SS and in the inner Oort cloud (e.g., Sedna and Sedna-like objects like 2012 VP113) that yield an unexplained clustering in orbital parameters. In fact, the trans-Neptunian area shows a complex dynamical structure that cannot be justified by a driven instability of the orbital history corresponding to giant planets alone. For instance, the origins of a highly inclined cohort of trans-Neptunian objects (particularly for the retrograde cases) are still ambiguous under the current evolutionary framework. Recently, Batygin and Brown [7], [8] suggested the existence of an additional Neptune-like planet residing in a distant, eccentric and mildly inclined orbit. In this model, the distant eccentric perturber plays an important role in the intricate orbital behavior of distant Kuiper belt objects. The so-called ‘Planet Nine’ hypothesis explains the existence of dynamical phenomena such as the anomalous orbits of long-period Kuiper belt objects that cannot be inferred from the interactions with the known SS planets (for a review, see [9]). It is indeed the first speculation about a new SS member established upon some anomalous orbital structure of known objects [10], like the high perihelia of the sednoid population (i.e., Sedna, 2012 VP113 and 2015 TG387 which are all possible DPs), in the inner Oort cloud. Although the inner Oort cloud initially lies beyond 2000 au, the authors in Refs. [6], [11] have argued that the sednoid objects are located in the inner Oort cloud, wherein the existence of a large planet in the trans-Neptunian region is predicted.
The physical characteristics of larger Kuiper belt populations, specifically DPs, may shed some light on the diversity of this cohort population [12]. DPs are large enough to be rounded by their own gravity. In other words, the masses of these objects and consequent gravity pulled them into a partly spherical shape. The distinction between planets and DPs is recognized by the neighborhood around their orbit which has not cleared for the so-called DPs. In fact, a DP is an object which is too large to drop within a smaller category, but too small to be taken into account a mature planet. Removing DPs out of the list of planets was proposed by Brown [13] who discovered Eris and many of the new DPs. However, this change of classification was avoided by Stern by means of the term ‘Dwarf Planet’ for the first time in 1991 [14]. Nonetheless, in 1990, Zappalà et al. [15] showed that a totally automatic process applied to a large set of suitable objects and considering their finite preciseness can yield a credible group classification. The data processing method that groups similar objects is called cluster analysis [16]. This mathematical tool helps finding relationships embedded in high dimensional datasets with the most similar objects in the same cluster and the most dissimilar between distant clusters. The distance/similarity measures are essential to the classification and clustering [17]. The hierarchical clustering (HC) is a technique to reveal similarities by means of distance/similarity measures. One important computational method for clustering and visualizing data is the multidimensional scaling (MDS). In 1995, Zappala et al. [18] first used the HC algorithm in asteroid studies. Masiero et al. [19], identified new asteroid families and improved the lists of previously known families, using the HC method and WISE/NEOWISE physical properties. Banda and Anrgyk [20] adopted dissimilarity measures and MDS for large scale solar data analysis. They compared different classifiers in order to determine the amount of dimensionality reduction. The data analysis on the extra-solar planets using the robust clustering was also investigated in [21]. To explore the properties of the statistical distributions of exoplanets, Jiang et al. considered clustering techniques to examine possible set of exoplanets. These techniques were also used in meteoroid stream searches (e.g., see [22], [23]).
In this paper, we consider different options of distances for processing the characteristics of the SS objects, such as the Canberra, Jaccard, Lorentzian and Manhattan metrics. For distinguishing the properties of the SS objects, the information is visualized using the HC and MDS techniques. The HC and MDS generate maps of points in two- and three-dimensional spaces representing the SS objects according with their characteristics. The relative significance of the points and the emerging patterns provide a direct interpretation of the results. Here we are mostly interested in massive bodies like planets, DPs and possible DPs. The reason for choosing these specific objects is due to the fact that they are all potentially massive enough to be in the hydrostatic equilibrium which is a significant criterion in the distinction between the DPs and small SS objects.
The paper is organized as follows. Section 2 presents the dataset and tables containing a partial list of observational data prepared by several reliable web sites. Section 3 introduces the mathematical background including the distance metrics and clustering methods that are essential for processing the data. Section 4 processes the data and analyses the results. Section 5 takes the hypothesis of the existence of a distant, Neptune-like planet into account and carries out a series of numerical experiments using clustering techniques. Finally, Section 6 outlines the main conclusions.
Section snippets
The dataset
We consider the dataset corresponding to bodies with indices in the SS as listed in Tables 1 and 2. Due to its size and for the sake of presentation, the information is divided in two parts and each table lists the data for 8 indices. In Table 1, we find the indices Aphelion, Perihelion, Semi-major axis, Orbital Eccentricity, Orbital Period, Orbital Velocity, Inclination of Orbit to Ecliptic, and Longitude of Ascending Node, that we shall abbreviate by writing A, P, S-m A, OE, OP, OV,
Distances and clustering methods
The HC and MDS techniques process data distributed in a high-dimensional space. The high dimension of the original data represents a problem for its visualization and the HC and MDS try to represent it a lower dimensional space. The HC follows a graphical representation by some kind of 2-dim ‘tree’ where the objects under analysis are the ‘leafs’. The MDS represents the objects by points and allows representations in 2- and 3-dim spaces.
Once the data is collected, we have a set of n-dim objects
Data analysis and results
In the data analysis we construct the maps, using metrics (1a)–(1d), generated by HC trees and MDS 3-dim for visualization. Before any clustering calculations we introduce a data pre-processing, or normalization, that is, the conversion to a numerical set of values with a similar significance. For each index we subtract the mean and divide by the standard deviation for each characteristic, . This is a common technique in signal processing. The idea of this normalization is to avoid
Analysis of adding planet nine to the SS
In this section, we consider the effect of adding Planet Nine to our numerical experiments. According to the Planet Nine hypothesis, this distant and eccentric planetary member of the SS is wholly responsible for the anomalous component of the trans-Neptunian orbital distribution. In other words, this proposed object modulates the perihelia of bodies in the anti-aligned cluster and explains the high perihelion of Kuiper belt objects. The appearance of Planet Nine, as yet, explains some
Conclusions
The number of discovered objects, particularly DPs, is increasing steadily. To investigate the distributions of their distances from the sun, orbital elements (like, eccentricity and inclination), sizes, masses, periods and other characteristics becomes relevant. An exact visualization of such distribution can have importance for the theories addressing the formation and evolution of the SS. In this study, we selected 27 SS bodies, including 8 planets, 5 DPs and 14 candidates of larger DPs,
Acknowledgments
This research has made use of data and/or services provided by the International Astronomical Union’s Minor Planet Center.
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