6 Distribution bimodality

Figure 6: Examples of CPF of the TNO color distributions, compared with the CPF of bimodal and continuous model distributions. The model distributions have been adjusted for a match of the observed distribution.

In this section, we will tackle the question of the bimodality of the TNO color distributions. Tegler and Romanishin have repeatedly reported that their observations lead to a classification of the objects in 2 separate groups in the color-color diagrams (Tegler & Romanishin 1998; Tegler & Romanishin 2000), one being of neutral-blue colors, while the other is very red. While this bimodality appears evident to the eye on their color-color diagrams, other authors (Barucci et al. 2000; Davies 2000; Delsanti et al. 2001) do not confirm it: their color-color diagrams show continuous distributions. Is Tegler and Romanishin's bimodality a selection artifact, or is it real? Since their original report, they have refined their claim, indicating that the bimodality affects only the most distant MBOSSes, i.e. the Cubewanos (Tegler & Romanishin 2000).

One of the reason invoked by Tegler and Romanishin to explain that they see this bi-modality while others don't, is that their own photometry is more accurate than that of other groups. While it is true that measuring faint MBOSSes is tricky, this claim cannot be valid anymore: i) many measurements (by other groups) have been performed on VLT-class telescopes, ensuring very good S/N ratios, and ii) the measurements presented in this compilation are often combining the result of various groups (a few objects combine >10 different measurements, many >5). The small resulting errors (which takes into account the dispersion between these measurements) indicate that the dispersion is rather small.

We will now compare the observed color distributions to simple models - continuous and bimodal ones, and try to decide whether the data are incompatible with one or the other. We will first consider the 1D distributions (e.g. B-V), then 2D distributions, corresponding to color-color diagrams.

  
6.1 1D distributions

The MBOSS color distributions will now be compared individually with a continuous distribution model, and with a bimodal distribution. The model distributions which are chosen are extremely simple; indeed, the idea is not to find a physical model that reproduces the data, but just to decide if the observed sample is compatible or not with a type of distribution. The model parameters are the following:

• Continuous distribution: the colors are uniformly distributed between C₁ and C₂;
• Bimodal distributions: the colors are uniformly distributed between B₁ and B₂ (group 1), and between B₃ and B₄ (if B₁ = B₂ and B₃ = B₄, the two "blobs'' have no internal spread). An additional parameter f gives the fraction of TNOs in the first blob. We impose a requirement that B₃ - B₂ > 0.2 mag, i.e. that there is a clear separation between the two blobs of a bimodal distribution. We adjust the parameters to maximize Prob.

For each color index, we considered the 2 models for the Plutinos, the Cubewanos and both Plutinos and Cubewanos together. The parameters of the models, as well as the corresponding probabilities, are available in Table 5. Figure 6 displays examples of the CPF for the TNO color indexes and the corresponding models.

Results:

• Plutinos and Cubewanos have color distributions that are compatible with a simple, uniform, continuous distribution (the worse probability is $\sim$4% for the Cubewanos' and Plutinos+Cubewanos' B-I);
• Some of the color distributions are not compatible with a bimodal distribution as defined (i.e. no adjustment of the distribution parameter can give a probability of compatibility larger than a few percent). These are the Plutinos' B-V and V-R, and the Cubewanos' V-R and R-I. When considering both classes together, the V-R, V-I and R-I are not compatible with a bimodal distribution.

  
6.2 2D distributions

The traditional color indexes ( B-V, V-R, R-I, etc., but also B-I, B-K, R-J, etc) are based on the standard photometric systems. There is no reason to believe that this system is specially adequate for TNO or Centaur work. It is possible that groups would appear in 2D (or >2D) diagrams, that would not appear in the 1D distributions. An illustration of this is the clustering of the MBOSSes around the reddening line, an effect that would not be visible in the 1D distributions. In this section, we will re-do a similar KS analysis in various 2-dimension space. Ideally, we could extend this work to a N-dimension space. Unfortunately, the KS tool does not exist for D > 2.

As in the 1D case, we will compare the observed distributions with model distributions. We will also use a bimodal model, in which the colors are spread around 2 individual points in the color-color diagram, and a continuous model, in which the colors are spread around a line joining 2 points in the color-color diagram. In order to simulate these model distributions, a large number (10 000) of test objects is created at random. The observed distribution is then compared to the model population. We verified that the resulting Pare not significantly varying for larger model sample, nor for one random population to the next.

In addition to the coordinates of the center of both blobs in the color-color diagram being considered, the parameters of the models are the spread of the distribution in x and y and, in case of a bimodal distribution, the fraction of the population in the first blob. The parameters were adjusted iteratively in order to maximize the KS probability.

Table 6 lists the parameters of the models giving the higher P and the corresponding values of d and P. Examples of the random populations simulating the models have been plotted on the color-color diagrams displayed in Fig. 7.

Results:

Adjusting the parameters of the model distributions, we could obtain fairly high values of Prob in all cases except for the B-V/R-I and B-V/V-R diagrams for Plutinos and Cubewanos, that cannot be reproduced by a bimodal distribution. In other words, all the 2D distributions considered are compatible with both continuous and bimodal distributions, except the 2 diagrams mentioned above, which are not compatible with (simple) bimodal distributions.

Figure 7: Examples of color-color diagrams of the TNOs, superimposed to the model distributions used for the KS analysis described in the text. Left column corresponds to continuous distributions, right to bimodal. Top row is for Plutinos, middle for Cubewanos, bottom for both together.