3 Correlation with orbital parameters and size

In this section, we search for correlation between the color or reddening distributions and the orbital parameters, i.e. a, the semi-major axis, e, the eccentricity and i, the inclination. We also consider the "excitation'' $\cal E$ of an object's orbit, defined as

$${\cal E} = \sqrt{e^2 + \sin^2(i)},$$ (9)

$\sin(i)$ is related to the object's velocity perpendicular to the ecliptic, and e to its radial velocity. Therefore, $\cal E$ is an estimate of the velocity of the object with respect to another object that would be at the same distance on a circular ecliptic orbit, it is therefore also related to the probability of collision, as well as the velocity of the impacts. The color vs. $\cal E$ plots therefore explore possible effect of collisions. The colors are also plotted as functions of M(1,1), the absolute magnitude (cf. Sect. 2.2 and Eq. (3)).

3.1 Plots

Figure 3 displays some of the color indexes and the reddening slope $\cal S$ as a function of the orbital parameters and absolute magnitude. In each figure, each object is represented using the symbol of its class (cf. Fig. 2). The complete set of diagrams is available on the MBOSS web site; only some examples are reproduced here.

Results:

• a plots: the extreme redness of objects with a>40 AU reported by Tegler & Romanishin (2000) is supported by the appearance of the B-V.
• No striking bimodality appears in any of the plots.
• None of the plot do show any convincing trend. In particular, the V-J vs. M(1,1) plot does not show any convincing trend: we do not confirm the correlation that Jewitt & Luu (1998) had observed on only 5 objects.

  
3.2 Correlation

In order to quantify possible correlations between the colors (and gradient) and the various orbital elements and absolute magnitude, we computed the correlation coefficient for each "Color'' versus "orbital element'' distribution. The test itself is described in Appendix B. As a reminder, while the correlation coefficient indicates how strong the correlation is (large absolute values), there is no way to quantify the significance of that correlation. The correlation coefficients were computed for the complete MBOSS population (i.e. all objects) and for the Plutinos, Cubewanos and Centaurs/Scattered objects separately. The numerical results of the tests are listed in Table C.1.

Results:

• For the complete MBOSS population: the colors and gradient are not correlated with any of a, e, i, $\cal E$ nor M(1,1)(i.e. all the correlation coefficients are extremely low in absolute value).
• i, e, $\cal E$: for the Cubewanos, there is a systematic anti-correlation between the colors (and gradient) and the e, iand $\cal E$ parameters (i.e. objects with higher orbital excitation being bluer). This effect is visible for all colors, and is the strongest for the B-R index. This effect completely disappears for the Centaurs/Scattered TNOs (coefficients close to zero). In the case of the Plutinos, the colors present a weak anti-correlation with the eccentricity (up to -0.2), and a weak correlation with the inclination ($\sim$0.2), both effects canceling each-other for $\cal E$.
• M(1,1): for the Cubewanos, there is a correlation between the colors and the absolute magnitude, indicating that the objects with fainter M(1,1) are redder. This effect is completely nonexistent for the Centaurs/Scattered TNOs, but appears reversed (i.e. bright M(1,1) redder) for the Plutinos, with slightly weaker correlations.

3.3 Are there some more subtle effects

In order to test for more subtle effects than a simple correlation between the colors (or gradient) and the orbital elements (and M(1,1)), each population is divided in two sub-samples, i.e. the object having the considered element smaller than a given value, and those having that element larger. The boundary value is chosen as the median of the sample, i.e. to split the population in sub-samples of similar sizes. The median value is probably not the best choice on physical bases, but a physically better choice might lead to samples of fairly different sizes, which could cause asymmetry artifacts. The cut-off values are listed in Table C.1 with the results of the test described below.

The two samples are then compared using Student's t-test and f-test, which are described in Appendix B. In summary, small values of the probability associated to the t-test, indicate that both sub-populations have significantly different mean of the considered color (Prob is the probability that both subsumes are randomly drawn from a similar population), while small values of the probability associated to the f-test reveal that the sub-samples have different variances. Each test was performed on the whole MBOSS population and on the Plutinos and Cubewanos only. The numeric results of the tests and the cut values are listed in Appendix, in Table C.1.

Results: $\mathsfsl{t}$-test

• For the complete MBOSS population as well as for the individual families, the t-test does not reveal any significant trend with respect to the semi-major axis a.

• The t-test applied to the color distribution vs. i, e and $\cal E$ reveals a strong, highly significant trend for the Cubewanos: objects with higher orbital excitations are bluer. The effect is systematic in all colors (and gradient), and its significance is in the Prob=10^-3 range. This support the result presented by Trujillo et al. (2001) (i.e. a correlation between color and eccentricity, supported by similar t-tests) on a smaller sample. They interpreted this as an evidence that low $\cal E$ objects are less affected by (collisional) re-surfacing than objects with a high excitation (because the collisions are on average less violent, and/or less numerous). Stern (oral communication at the Meudon 2001 meeting) also mentioned the correlation with $\cal E$.

• This effect completely vanishes for the Plutinos and Centaurs/Scattered TNOs: the various sub-samples are statistically equivalent. It appears diluted when considering all MBOSSes together.

• Considering the Plutinos colors as a function of their absolute magnitude, a marginally significant trend appears, indicating that those with a fainter M(1,1) are slightly bluer. For the Cubewanos, the opposite trend exists: those with a fainter M(1,1) are slightly redder. It is also worth noting that the median M(1,1) (used as cut-off between the sub-samples) is fainter for the Plutinos (7.4) than for the Cubewanos (6.6). This is a natural discovery selection effect: as the Plutinos are on average at smaller heliocentric distances than the Cubewanos, smaller objects can be discovered. As a consequence, the test on the Plutinos explores intrinsically smaller objects. Setting the cut-off magnitude for the Plutinos equal to that of the Cubewanos produces unbalanced samples, but the result remains the same.

Results: $\mathsfsl{f}$-test

• The color distributions of objects as a function of the absolute magnitude: in earlier versions of this database, the B-* and *-I(* indicating any other filter) of objects with fainter M(1,1) had significantly broader color distributions than the objects with brighter absolute magnitudes. This was tracked down to an instrumental effect: the B and I magnitudes are typically affected by larger error bars because of the lower quantum efficiency of the detectors in these bands. Objects with fainter absolute magnitude are on average fainter than those with a brighter M(1,1), and therefore have larger error bars. Recent addition to the database of a large number of observations obtained with 8-10 m-class telescopes diluted that effect out. In the current version of the database, the MBOSSes with faint M(1,1) (i.e. probably the smaller ones) do not present a significantly different color distribution width than the bright ones. This is true for the MBOSSes as a whole and for the different classes. This is contrary to the predictions of a collisional resurfacing model (Luu & Jewitt 1996a; Jewitt & Luu 2001).

• Cubewanos with higher orbital excitation (i.e. larger i, eand/or $\cal E$) have significantly broader color distributions than the others. The effect is strongly significant except for B-V and *-I, where the effect is partially diluted by larger error bars on B and I magnitudes. For the Plutinos and Centaurs/Scattered TNOs, the effect disappears, except for the inclination (but it is not significant).

 

 

Table 4: Average colors of the various classes of objects. For each color, the table lists the number of objects included in the statistics, as well as the average color and the square root of the corresponding variance (which is undefined and set to 0 in case only one object is available).

Color	Plutinos		Cubewanos		Centaurs		Scattered		Comets
U-B	0	-- --	1	1.000 $\pm$ 0.000	0	-- --	1	0.970 $\pm$ 0.000	0	-- --
U-V	0	-- --	1	1.720 $\pm$ 0.000	0	-- --	1	1.710 $\pm$ 0.000	0	-- --
U-R	0	-- --	1	2.120 $\pm$ 0.000	0	-- --	1	2.150 $\pm$ 0.000	0	-- --
U-I	0	-- --	1	2.500 $\pm$ 0.000	0	-- --	1	2.410 $\pm$ 0.000	0	-- --
B-V	20	0.886 $\pm$ 0.176	33	0.946 $\pm$ 0.185	15	0.930 $\pm$ 0.219	8	0.863 $\pm$ 0.126	2	0.795 $\pm$ 0.035
B-R	20	1.464 $\pm$ 0.261	30	1.561 $\pm$ 0.249	15	1.513 $\pm$ 0.349	8	1.376 $\pm$ 0.262	2	1.355 $\pm$ 0.064
B-I	17	1.977 $\pm$ 0.412	25	2.131 $\pm$ 0.329	14	2.119 $\pm$ 0.489	7	1.914 $\pm$ 0.412	2	1.860 $\pm$ 0.141
B-J	0	-- --	0	-- --	5	2.800 $\pm$ 0.807	0	-- --	0	-- --
B-H	0	-- --	0	-- --	4	3.395 $\pm$ 0.679	0	-- --	0	-- --
B-K	0	-- --	0	-- --	4	3.428 $\pm$ 0.623	0	-- --	0	-- --
V-R	20	0.580 $\pm$ 0.091	40	0.629 $\pm$ 0.132	15	0.590 $\pm$ 0.136	9	0.528 $\pm$ 0.116	13	0.430 $\pm$ 0.140
V-I	17	1.118 $\pm$ 0.239	30	1.206 $\pm$ 0.225	13	1.169 $\pm$ 0.270	9	1.031 $\pm$ 0.242	4	0.964 $\pm$ 0.136
V-J	4	2.345 $\pm$ 0.213	5	1.795 $\pm$ 0.495	6	1.801 $\pm$ 0.552	1	1.452 $\pm$ 0.000	0	-- --
V-H	0	-- --	0	-- --	5	2.245 $\pm$ 0.555	0	-- --	0	-- --
V-K	0	-- --	0	-- --	5	2.299 $\pm$ 0.506	0	-- --	0	-- --
R-I	17	0.542 $\pm$ 0.161	30	0.613 $\pm$ 0.137	14	0.582 $\pm$ 0.134	9	0.509 $\pm$ 0.138	4	0.465 $\pm$ 0.066
R-J	0	-- --	0	-- --	6	1.243 $\pm$ 0.382	0	-- --	0	-- --
R-H	0	-- --	0	-- --	5	1.658 $\pm$ 0.400	0	-- --	0	-- --
R-K	0	-- --	0	-- --	5	1.695 $\pm$ 0.367	0	-- --	0	-- --
I-J	0	-- --	0	-- --	6	0.685 $\pm$ 0.233	0	-- --	0	-- --
I-H	0	-- --	0	-- --	5	1.087 $\pm$ 0.234	0	-- --	0	-- --
I-K	0	-- --	0	-- --	5	1.124 $\pm$ 0.209	0	-- --	0	-- --
J-H	2	0.800 $\pm$ 0.566	4	0.228 $\pm$ 0.355	5	0.340 $\pm$ 0.051	1	0.350 $\pm$ 0.000	0	-- --
J-K	1	0.360 $\pm$ 0.000	3	0.330 $\pm$ 0.234	5	0.383 $\pm$ 0.076	1	0.310 $\pm$ 0.000	0	-- --
H-K	1	-0.040 $\pm$ 0.000	3	0.243 $\pm$ 0.494	5	0.025 $\pm$ 0.083	1	-0.040 $\pm$ 0.000	0	-- --
Grt (%/100 nm)	20	21.760 $\pm$ 12.305	35	26.537 $\pm$ 14.100	15	24.316 $\pm$ 15.086	9	16.948 $\pm$ 11.906	4	12.894 $\pm$ 7.192

  
3.4 Broadening of the $\mathsfsl{M(1,1)}$distributions?

As mentioned earlier, neutral-bluish objects could have their surface covered with fresh ice (resulting from a recent re-surfacing), or, on the contrary, with extremely ancient, extremely irradiated ice (with doses of 10¹⁰ erg cm^-2), whose color is also expected to be neutral (cf. laboratory spectra published by Thompson et al. 1987). The albedo of the ancient ice is expected to be significantly lower than that of the fresh ice. On the other hand, very red objects are expected to be covered with highly irradiated ice (corresponding to the laboratory samples that received doses of 10⁹ erg cm^-2Thompson et al. 1987), and would have a much narrower range of albedo. If we assume that all these objects have a similar radius distribution, the resulting M(1,1) distribution should be significantly broader for the neutral objects than for the red ones (cf. Eq. (4)).

In order to test this hypothesis, we consider the M(1,1)distribution as function of the colors (i.e. the reverse of the previous section). We split the observed sample in two, i.e. those with colors redder than a limit, and the others. The cut-off value is set at the mid-point between the minimum and maximum values of the considered color. The average values of M(1,1) and their variances are computed for each sub-samples, and are compared using the f-test (cf. Sect. B), which evaluates whether the two variances are compatible. This test was performed for all the colors and the gradient distributions, for the complete MBOSS population, and for the Plutinos, Cubewanos and Centaurs/Scattered TNOs only. The numerical results of these tests are listed in Table C.2.

Results:

• Restricting the test to the Plutinos, it appears that the redder objects have a broader M(1,1) distribution (strongly significant effect) than the bluer ones. This effect is visible and significant (<1%) in all color indexes.
• This effect is also visible for the Centaurs/Scattered TNOs (redder objects have a broader M(1,1) distribution) but although it is visible in all indexes, it is not statistically significant.
• The reversed effect (bluer objects have a broader M(1,1)distribution) for the Cubewanos. The significance is not as strong as for the Plutinos, but still to be considered (all of the indexes are at the $\sim$5-6% level or stronger).
• No effect is visible for the MBOSSes as a whole.

Combining this with the results of correlations (Sect. 3.2), it implies that the objects with a bighter M(1,1), i.e. the redder Plutinos and the bluer Cubewanos, cover a broader M(1,1) range than the fainter ones. This is an effect of the steep luminosity function: there are much fewer objects per unit magnitude brighter than the cut-off magnitude (therefore the sample covers a broad range of M), while the number of objects per unit magnitude is much larger for those fainter than the cut-off. As the samples have roughly the same size, the faint ones cover a smaller magnitude range than the bright ones.