5 Population comparisons

5.1 Using the CPFs

The eye is extremely good at finding patterns and comparing shapes. In this first paragraph, we shall analyze visually the color histograms and CPFs. Of course, this analysis is only qualitative, and no claim is made with respect to the significance of these descriptions. They are meant to attract the attention of the reader to features that might eventually become significant - or may disappear when more data become available. In the next section, we will reconsider these comparisons with the cold (and less imaginative) eye of statistical tests.

Results

• Shift: apart from the comet V-R that appears to be on average bluer than that of the other objects, no systematic difference of color is apparent;
• Broadening: defining the width of a color distribution by the interval over which the CPF is strictly between 0 and 1 (excluded), no class is systematically the broadest of all (the broadest is always either the Cubewanos or the Centaurs). The population with the narrowest color distribution is in almost all cases the Scattered TNOs;
• Several distributions show some discontinuities; in particular the Cubewanos and Centaurs' B-V, B-R and B-I as well as the Cubewanos' V-I display a broken CPF, with first a sharp then a shallower increases (corresponding to a narrow then a broad peak in the histograms). Other distributions are very smooth with a constant increase rate, such as the Plutinos' B-V, B-R, B-I, V-I and R-I.

5.2 Statistical tests

In this section, we will apply statistic tools to the available dataset in order to cast some light on the question of similarities and differences between the different classes of objects.

The problem at hand is to compare samples of 1D continuous distributions (colors, e.g. V-R), in order to decide whether they are statistically compatible. We will consider the MBOSS classes two by two. For that purpose, we shall use the t-test, the f-test, and the Kolmogorov-Smirnov (KS) test, which are described in more detail in Appendix B; each of them produce a probability Prob. Low values of Prob indicate that the distributions are statistically incompatible, but larger values can only be interpreted as stating that the distributions are not incompatible, not that they are equal; this is also discussed in more details in Appendix B.

In order to get a known comparison when studying the real MBOSS populations, we introduced two pairs of artificial subsets of the objects. They are defined as following:

• Odd: objects with odd numbers in the internal database;
• Even: objects with even numbers in the internal database;
• 1999: objects discovered in 1999, and
• non-99: objects not discovered in 1999.

Odd and Even are two populations of about the same size, while 1999 is much smaller than non-99. As the members of these populations are chosen using non-physical properties from the whole sample, we expect them to be equivalent, and that the statistical tests will give large values of Prob when comparing them. We performed all the tests on the "Odd/Even'' and "1999/non-99'' pairs, and report the results together with the tests on real classes. This allows the reader to get an idea of the statistical tests' calibration. 1999 was chosen as opposed to earlier years, because of the fairly large number of objects in the database (19) and because in that year the survey techniques were already quite advanced; in that way, we should not have a bias against small objects, that would have been present for the earlier years.

5.2.1 $\mathsfsl{t}$-test: Are the mean colors compatible?

Table 3 lists the mean colors of the different classes. The color of an object is function of the nature of its surface and of the reddening and resurfacing it experienced. For a given population, the mean color will therefore give an information on the equilibrium reached between the aging reddening and the different re-surfacing processes.

The question we address in this section is whether the mean color of different classes are significantly different. The traditional way to compare the means of distributions is to use Student's t test; the implementation used for this work is described in Appendix B.2.1. The values of t and Prob are listed in Table C.3; the results for the artificial classes are displayed in Table C.4.

Results

• The 1999/non-1999 test classes do not show incompatibilities at the 9-10% level, except for very small samples, indicating that one should consider these levels with suspicion when comparing small samples;
• The mean V-R of the Comets is incompatible with those of the Cubewanos (with a significance of 10^-3), of the Plutinos, and the Centaurs, and marginally incompatible with those of the Scattered TNOs. This validates the visual impression that one has looking at the CPFs: the Comets are significantly bluer than the other objects;
• There is no significant incompatibility between the Plutinos, Cubewanos and Centaurs. Although the low- and high orbital excitation Cubewanos have incompatible color distributions, once considered all together, are indistinguishable (with the t-test) from the other classes.

5.2.2 $\mathsfsl{f}$-test: Are the variances compatible?

The variance of the color distribution contains some information on the diversity of the population, and on the range covered by the reddening and resurfacing processes. For instance, one could expect that - although reaching a different mean equilibrium - the aging, the collisions and the cometary activity broaden the color distribution in a similar way, ranging from bluish, fresh ice, to deep red, undisturbed, aged surface.

In this section, we will determine whether the variances of the color distributions are significantly different (independently of their mean, that can be either similar or different). This is quantified using the f-test, described in Appendix B.2.2. The values of F and Prob are listed in Table C.5.

Results:

• None of the variances show incompatibilities with a high level of significance.

The f-test does not give any significance to the fact that the color distributions of the Scattered TNOs cover systematically a narrower range than that of the other classes. This can be either because it is not significant, but also because the distributions have fairly different shapes.

5.2.3 KS test: Are the distributions compatible?

Obviously, the whole information from a distribution is not contained in its two first moments (mean and variance). A more complete comparison of the color distributions is therefore interesting. The ideal statistics tool for this purpose is the KS test (described in Appendix B.2.3), in which the two samples are compared through their complete Cumulative Probability Function (CPF). The values of d and the associated probability Prob are listed in Table C.6 for the real classes of objects. Those for the test classes are available only electronically.

Results:

• The Comets' V-R are incompatible (10^-2-10^-3) with the Cubewanos, Plutinos and Centaurs, but the incompatibility with the scattered TNOs that was noticed with the t-test is not apparent here;
• The other classes do not present significant incompatibilities.