A countless number of objects orbit the sun, including asteroids, comets, and several large planets. Until recently, most objects known to orbit the sun were either very large (planets) or very small (asteroids or comets). In recent decades, the discoveries of many large bodies similar in size to Pluto and orbiting the sun beyond Neptune have challenged the clear distinction between planets and non-planets. These discoveries culminated with the revocation of Pluto’s planetary status by the International Astronomical Union (IAU) in 2006 and the creation of a new class of bodies known as dwarf planets: spherical objects orbiting the sun which have not cleared their orbital neighborhood of other large objects and debris. If you’re not already familiar with the controversy, here’s a good video to catch you up.

The current IAU definition of a planet stipulates that, to be a planet, a solar system body must (1) orbit the sun (and not another planet), (2) have achieved hydrostatic equilibrium, i.e., have sufficient mass to pull itself into a spherical shape, and (3) clear its orbital neighborhood of debris and other objects (Parsons & Wand, 2008). Solar systems bodies such as Pluto which meet criterion 1 and criterion 2, but not criterion 3, are classified as dwarf planets. Criterion 3 is controversial, as it is rather ambiguous and open to interpretation. Some are dissatisfied with the classification of Pluto as a dwarf planet. Is there a better taxonomy?

A recent multivariate statistics class gave the me opportunity to cluster large bodies orbiting the sun to shed light on questions such as “is Pluto a planet?” I approached this problem by taking a dataset of 11 physical parameters for 14 large solar system bodies and creating a model of these 11 parameters (observed variables) as linear combinations of only 2 latent variables plus error terms. This allows for the 14 solar system objects to be described as points in 2-dimensional (rather than 11-dimensional) space, which makes grouping these points into clusters easy. Such clustering allows for a new taxonomy of large solar system object orbiting the sun.

The reduction from 11 dimensions to 2 dimensions was performed using maximum-likelihood factor analysis (FA). The FA model explains the variance of the dataset using only 2 factors or latent variables. This is a bit like the Genius playlists which iTunes creates from your music library: different songs load more or less strongly onto different playlists such as “West Coast Jazz” or “Classic Hard Rock” which attempt to explain the variety of music in your library using relatively few factors. In FA, each observed variable is assigned loadings which tell how strongly each factor contributes to that observed variable. The factor loadings of the FA model, bounded between -1 and 1, mostly result in the observed variables loading strongly onto one latent variable, with many loadings > 0.9. Subsequent clustering of points in factor space was performed using k-means clustering. The observed variables included 6 parameters specifying the size, tilt, shape, and other properties of an object’s orbit, the mass of the object, the length of a “day” on the object, its number of moons, how massive the object is relative to the debris in its orbital neighborhood, and, finally, the ability of the object to gravitationally clear its orbital neighborhood. For more technical details of my model and an explanation of all parameters in my dataset, please scroll to the appendix at the end of this article!

For this analysis, 14 large solar system bodies were selected: (1) Mercury, (2) Venus, (3) Earth, (4) Mars, (5) Jupiter, (6) Saturn, (7) Uranus, (8) Neptune, (9) Pluto, (10) Ceres, (11) Eris, (12) Vesta, (13) Haumea, and (14) Pallas. Bodies 1 – 8 include all planets currently recognized by the IAU. Bodies 9, 10, 11, and 13 include 4 out of 5 dwarf planets currently recognized by the IAU. A fifth dwarf planet, Makemake, is excluded due to the fact that its mass has not been accurately measured. Ceres, one of the dwarf planets included in this analysis, is also a large asteroid, along with bodies 12 and 14. All dwarf planets except for Ceres orbit the sun in a region known as the Kuiper belt, a collection of icy objects orbiting the sun beyond the orbit of Neptune (Soter, 2006).

Factor Analysis

For factor 1, the observed variables with the strongest loadings (absolute value) were the Stern-Levison parameter (0.98) which describes the ability of a body to gravitationally influence its neigborhood, mass (0.98), and Soter’s planetary discriminant (0.98), which gives the ratio of a body’s mass over the mass of all other debris in the object’s orbit. For factor 2, the observed variables with the strongest loadings were semimajor axis (0.98), which describes the size of an orbit, and average heliocentric velocity (-0.98), or the speed of the object. Based on the factor loadings, the first factor can be interpreted as a latent variable corresponding to gravitationally influence. The second factor can be interpreted as a latent variable describing the orbital properties of an object. The 2-dimensional plane of factor space shows several easily identifiable clusters. A 2-cluster model of the factor space identified clusters congruent with the current IAU classification of planets, i.e., one cluster contained the 8 planets while the other cluster contained the 6 non-planets. However, the fit of this cluster model was poor (47.1%). A much better fit was achieved using 4 clusters based on the number of visually identifiable clusters. Indeed, this clustering had a much better fit (94.2%). One cluster was composed of the 4 rocky planets orbiting inside the asteroid belt (Mercury, Venus, Earth, and Mars). A second cluster was composed of the gas giant planets orbiting outside the asteroid belt (Jupiter, Saturn, Uranus, and Neptune). A third cluster consisted of those objects orbiting within the asteroid belt (Ceres, Vesta, and Pallas). The fourth and final cluster consisted of Kuiper belt bodies orbiting beyond the orbit of Neptune (Pluto, Eris, and Haumea).

What is a Planet?

There are several questions I sought to address using clustering of points in factor space. (1) What is the best taxonomy of solar system objects based on their physical properties? (2) Does Ceres belong in the same category as the other IAU dwarf planets? (3) Is Pluto a planet or non-planet? The first question is addressed by describing physical properties of large solar system objects in a low dimensional space and clustering objects to create new categories for conceptualizing solar system objects. The FA model yeilds 4 clusters readily identifiable upon visual inspection and confirmed with k-means clustering. An interesting interpretation of this finding is that solar system objects can be best conceptualized using 4 distinct categories: inner planets, outer planets, asteroids, and Kuiper belt objects. Furthermore, the 8 planets currently recognized by the IAU might be better understood as 2 separate categories of worlds. The outer planets (Jupiter, Saturn, Uranus, and Neptune) are far more massive than the inner planets (Mercury, Venus, Earth, and Mars). The mass of Jupiter, for instance, is 318 times greater than that of the most massive inner planet, Earth. The outer planets all have larger factor 1 scores than the inner planets. Since the inner planets and outer planets belong to distinct clusters, it is likely that the best taxonomy would separate these worlds into “hyperplanets” (Jupiter, Saturn, Uranus, and Neptune) and “hypoplanets” (Mercury, Venus, Earth, and Mars). Because we humans live on Earth, we like to think that Earth is just as qualified to be a planet as Jupiter. However, from the point of view of an alien living on Jupiter, Earth might be seen as a dwarf planet owing to its small mass and humble moon count.

Since the ideal taxonomy should group solar system objects into 4 categories, it no longer seems appropriate to count Ceres as a dwarf planet. Ceres has a very different factor 2 score than the other dwarf planets (Pluto, Eris, and Haumea) recognized by the IAU, reflecting the fact that Ceres orbits much closer to the sun than the other dwarf planets. In fact, judging from the factor space plot, Ceres might just as easily be part of the inner planet cluster as the dwarf planet cluster if the asteroids Vesta and Pallas are removed. Because the current IAU criteria for dwarf planethood requires the body in question to be spherical due to its own gravity, Ceres (a spherical object) is seen as more planet-like than Vesta and Pallas (ellipsoidal objects). The state of an object which has rounded itself out into a sphere by its own gravity is known as hydrostatic equilibrium. Because achievement of hydrostatic equilibrium is an abrupt change best described as a binary variable, this trait was not taken into account in the current analysis, as FA works best with continuous variables. Were it feasible to include hydrostatic equilibrium in the dataset of physical parameters, it is possible that Ceres would cluster together with other dwarf planets.

I conclude from this analysis that an ideal taxonomy of large solar system objects includes 4 groupings: hyperplanets (the massive outer planets: Jupiter, Saturn, Uranus, and Neptune), hypoplanets (the smaller inner planets, Mercury, Venus, Earth, and Mars), Kuiper belt objects (Pluto, Eris, and Haumea), and large asteroids (Ceres, Vesta, and Pallas). A limitation of this analysis is that many physical parameters which might be useful for describing and clustering large solar system objects are best expressed as binary variables and therefore excluded from this analysis. For instance, presence of a magnetic field, presence of a substantial atmosphere, and achievement of hydrostatic equilibrium are arguably important physical parameters but difficult to quantify on a continuous scale. Additionally, large solar system objects which lack moons and have not been visited by spacecraft have masses which are not well characterized. For this reason, objects such as the dwarf planet Makemake and the large asteroid Juno were excluded from this analysis. Inclusion of more solar system objects would perhaps increase the fit of the k-means clustering. Moreover, future analyses might also include other objects such as comets and small asteroids to see if these objects fit with existing clusters or form new clusters of their own. Finally, an ultimate taxonomy should be applicable to other star systems in addition to the solar system. Planets orbiting other stars (exoplanets) might be included in future analyses to ensure a universal taxonomy applicable to other star systems.

Appendix (for nerds)

Dataset

The observed variables were the following properties: (1) semimajor axis, (2) orbital eccentricity, (3) orbital inclination, (4) mean anomaly*, (5) orbital period, (6) average heliocentric velocity, (7) mass, (8) equatorial angular velocity, (9) number of moons, (10) Soter’s planetary discriminant (mu), and (11) the Stern-Levison parameter (lambda). All parameter values were obtained from the WolframAlpha computational knowledge engine unless otherwise noted (WolframAlpha, 2015). Parameters 1 – 6 specify the orbit of a body, including its ellipsoidal shape and orientation relative to the plane of the solar system, as well as the speed with which the object orbits the sun. Parameters 7 – 9 describe physical properties intrinsic to the solar system body. Equatorial angular velocity is the rotational rate of the body in rotations per Earth day. Parameters 10 – 11 are measures of how well a solar system body fulfills criterion 3 of the IAU’s definition of a planet. Mu, Soter’s planetary discriminant, is the ratio of a body’s mass over the mass of all other debris in the object’s orbit. Values for mu were taken from (Soter, 2006). Lambda, or the Stern-Levison parameter, is a measure of a body’s potential to gravitationally influence its orbital neighborhood and is computed as the mass of the body squared over its orbital period (Soter, 2006). This quotient is usually multiplied by a constant k such that a body with lambda > 1 is a planet. Because lambda values were normalized prior to dimensionality reduction, the constant factor k is ignored herein. You can download my dataset here: DATASET

* Inclusion of mean anomaly was a mistake, as it is a transient measure reflecting the current position of the object in its orbit and not appropriate for considering whether an object is a planet. More appropriate would have been mean anomaly rate. However, since mean anomaly did not load strongly onto either factor, I do not think its inclusion tremedenously changed the results of the analysis. I acknowledge fully that its inclusion was a source of noise.

Factor Analysis

Prior to normalization, the values in the parameter space of the dataset span many orders of magnitude. Because some columns of the dataset include zero values, the log transform is not defined for all data. Accordingly, I have used the inverse hyperbolic sine (IHS) transform, which is defined for zero values, as an alternative (Burbidge, Magee, & Robb, 1988). An exploratory maximum-likelihood FA model was implemented in R 3.1.0 using the fa() function from the psychology toolbox (Revelle, 2015). Because the most parsimonious description possible is desired of the solar system bodies in a very low dimensional space, only 2 factors were extracted. The estimated loadings were rotated using the oblimin rotation method, which is an oblique rotation. Use of an oblique rotation allows for the rotated axes to align with clusters of points in factor space more easily than an orthogonal rotation such as varimax. The FA model was generated using the correlation matrix (rather than covariance matrix) of the dataset, as this corrects for the vastly different scalings of observaed variables. Factor scores were computed according to Thurstone’s regression method.

The maximum-likelihood FA model yielded 2 factors accounting for 58% and 42% of the total variance. Although an oblique rotation method was used, the factor scores are minimally correlated (r = -0.03). The root mean square of the residuals (RMSR) of the correlation matrix was 0.12 and the off-diagonal fit of the model had a value of 0.93. The observed variables which are best accounted for by the FA factors (i.e., variables with communalities > 0.9) are average heliocentric velocity, semimajor axis, lambda, mass, and mu. The median communality for all observed variables was 0.87, and the minimum communality was 0.13 (equilateral angular velocity). Most observed variables loaded strongly onto one factor, as evidenced by the median complexity value of 1.1. Notable exceptions were equilateral angular velocity, with a complexity value of 1.9, and number of moons, with a complexity value of 2.0. Both of these observed variables loaded about equally onto both factors. A table of factor loadings, communalities, specific variances, and complexities is available here: Factors

k-means clustering

The k-means clustering algorithm was applied to identify clusters of points in factor space which may potentially map onto semantic categories such as planet, dwarf planet, or asteroid. Cluster analysis was performed using the kmeans() function in R. Two clustering analyses were performed in factor space: an initial cluster analysis searching for k = 2 clusters, ideally mapping onto planets and non-planets, and a second clustering analysis with the number k of predetermined clusters driven by visual inspection of scatter plots of the factor space. The results of the k = 2 search identified a planet cluster and a non-planet cluster. The within clusters sum of squares for the planet and non-planet clusters were SS = 8.6 and SS = 5.2, respectively. The ratio of the between SS to the total SS gave a rather inadequate clustering fit of 47.1%. Subsequently, visual inspection of the factor space scatter plot identified 4 clusters, and the k-means algorithm was repeated for k = 4. The results of the clustering showed one cluster corresponding to the inner planets (Mercury, Venus, Earth, and Mars; SS = 0.50), another cluster corresponding to the outer planets (Jupiter, Saturn, Uranus, and Neptune; SS = 0.85), a third cluster corresponding to dwarf planets in the Kuiper belt (Pluto, Eris, and Haumea; SS = 0.098), and a fourth cluster corresponding to asteroids (Ceres, Vesta, Pallas; SS = 0.047). The ratio of the between clusters SS to the total SS gave a clustering fit of 94.2%.

References

Burbidge, J. B., Magee, L., & Robb, A. L. (1988). Alternative transformations to handle extreme values of the dependent variable. Journal of the American Statistical Association, 83(401), 123–127.

Parsons, J., & Wand, Y. (2008). A question of class. Nature, 455(7216), 1040–1041.

Revelle, W. (2015) psych: Procedures for Personality and Psychological Research, Northwestern University,

Evanston, Illinois, USA, http://CRAN.R-project.org/package=psych Version = 1.5.4.

Soter, S. (2006). What is a Planet? The Astronomical Journal, 132(6), 2513.

Wolfram Alpha LLC. 2015. Wolfram|Alpha. Accessed May 6, 2015.