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A&A 389, 641­664 (2002) DOI: 10.1051/0004-6361:20020431
c ESO 2002

Astronomy & Astrophysics

Colors of Minor Bodies in the Outer Solar System
A statistical analysis
O. R. Hainaut1 and A. C. Delsanti1,
1 2

,

2

Europ ean Southern Observatory, Casilla 19001, Santiago, Chile Observatoire de Paris-Meudon, 5 place Jules Janssen, 92195 Meudon Cedex, France

Received 12 Octob er 2001 / Accepted 14 February 2002 Abstract. We present a compilation of all available colors for 104 Minor Bodies in the Outer Solar System (MBOSSes); for each ob ject, the original references are listed. The measurements were combined in a way that does not introduce rotational color artifacts. We then derive the slop e, or reddening gradient, of the low resolution reflectance sp ectra obtained from the broad-band color for each ob ject. A set of color-color diagrams, histograms and cumulative probability functions are presented as a reference for further studies, and are discussed. In the color-color diagrams, most of the ob jects are located very close to the "reddening line" (corresp onding to linear reflectivity sp ectra). A small but systematic deviation is observed toward the I band indicating a flattening of the reflectivity at longer wavelengths, as exp ected from lab oratory sp ectra. A deviation from linear sp ectra is noticed toward the B for the bluer ob jects; this is not matched by lab oratory sp ectra of fresh ices, p ossibly suggesting that these ob jects could b e covered with extremely evolved/irradiated ices. Five ob jects (1995 SM55 , 1996 TL66 , 1999 OY3 , 1996 TO66 and (2060) Chiron) have almost p erfectly solar colors; as two of these are known or susp ected to harb our cometary activity, the others should b e searched for activity or fresh ice signatures. In the color-color diagrams, 1994 ES2 , 1994 EV3 , 1995 DA2 and 1998 HK151 are located very far from the main group of ob jects; it is susp ected that this corresp onds to inaccurate measurements and not intrinsically strange ob jects. The color distributions were analyzed as functions of the orbital parameters of the ob jects and of their absolute magnitude. No significant correlation is observed, with the following exceptions: Cub ewanos with low orbital excitation (low i, e and/or E = e2 + sin2 i), and therefore exp eriencing on average fewer and less violent collisions have significantly redder colors; Cub ewanos with faint absolute magnitude M (1, 1) tend to b e redder than the others, while Plutinos present the opp osite trend. The color distribution of the various MBOSS classes are analyzed and compared using generic statistic tools. The comets were found to b e significantly bluer than the other MBOSSes. Finally, we compare the various 1D and 2D color distributions to simple models, in order to throw some light on the question of the bimodality of MBOSS color distributions. It is found that with the current data set, all color distributions are compatible with simple, continuous distribution models, while some color distributions are not compatible with simple bimodal distribution models. Table 1 is also available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/389/641, and the tables and complete set of figures corresp onding the up-to-date database are available on the web at http://www.sc.eso.org/~ohainaut/MBOSS. Key words. comets: general ­ Kuip er Belt ­ solar system: general ­ methods: statistical

1. Intro duction
As soon as colors were available for a few Transneptunian Ob jects (TNOs), it was very tempting to see families and groups in the various color and color-color diagrams.
Send offprint requests to : O. R. Hainaut, e-mail: ohainaut@eso.org Table 1 is also available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/389/641 Tables 3, 5, 6 and the list of pap ers are only available in electronic form at http://www.edpsciences.org

Immediately after the discovery of 1992 QB1 , D. Jewitt presented a thorough analysis of one data point (Jewitt 1992), with the un-attackable argument that for years, people had done similarly detailed analysis on zero data points. The first detailed analysis was published by the Hawaii group (Luu & Jewitt 1996a). In the following years, with the continuous increase of the data set available, more analyses were published, but they are usually over-exploiting the data, without much consideration for the statistical significance of their (otherwise interesting) claims. The physical properties of TNOs are difficult to assess; indeed, their faint magnitudes prevent them from


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being studied in detail using the arsenal of observation techniques that could be used were they brighter. Only a handful of them were observed spectroscopically; the spectra reveal continuous, featureless gradients in the visible (e.g. Boehnhardt et al. 2001; Davies 2000), and a fairly flat spectrum in the near-IR (e.g. Davies 2000; Brown et al. 1998; McBride et al. 1999; Brown et al. 1999), in some cases displaying water absorption bands (e.g. Brown et al. 1999). The bulk of physical studies comes from Visible and/or IR colors.

1.1. Dynamical families and their inter-connections
This section gives a broad overview of the connections between the different classes of MBOSSes. We keep this introduction to the minimum: it is not meant as a review of this quickly evolving field. It is now broadly accepted that the TNOs constitute the largest members of the Edgeworth-Kuiper belt, which is in turn a remnant of the proto-planetary nebula extending beyond the region where planets formed. In that region, the small quantity of material available and the fairly quiet dynamical environment prevented the formation of planet-sized bodies. The "Main Belt" TNOs, also called Cubewanos (for 1992 QB1 , the first discovered, Jewitt & Luu 1992) or "Classical TNOs", are found on orbits of low/moderate eccentricity and inclinations with semi-ma jor axis >40 AU. Their orbital parameters are not primordial, as they are still in the influence area of the outer planets. Beyond 45 AU, i.e. where the gravitational field of the planets has no effects anymore, it is hypothesized that there could lie a very thin belt of smaller ob jects with a dynamically very cold ­and primordial­ orbit distribution, forming the "Cold Disk" (Hahn 2000). This class has not been identified observationally yet. Other dynamicists (e.g. Morbidelli 2001) can explain the observed distribution of MBOSSes without the need for such a Cold Disk. During the latest stages of the planetary accretion, the proto-Uranus and Neptune scattered a significant number of proto-planetesimals (Malhotra 1995). According to some dynamicists (Malhotra 1996; Hahn & Malhotra 1999), this caused their orbit semi-ma jor axis to increase, the so-called planet migration. In that process, the orbital resonances associated with Neptune swept the inner part of the Edgeworth-Kuiper Belt, trapping the ob jects whose orbit passed through the resonance. The orbit of these ob jects were excited in e and i, eccentricity and inclination, resp. (Malhotra 1995; Malhotra 1996; Hahn & Malhotra 1999). Ob jects from this class are known as "Resonant TNOs", or Plutinos, after the most famous of their members, Pluto. For other dynamicists (Morbidelli 2001), planet migration is not needed to explain the population of the orbital resonances. Some of the TNOs from the inner Kuiper Belt are on orbits that are unstable over the age of the Solar System because of interactions with Uranus and Neptune. Because of these instabilities, there is a continuous flow of such

ob jects toward the Jupiter-Saturn region, where they can stay for a few million years (Kowal et al. 1979; Asher & Steel 1993). Thanks to their proximity to the Sun, they can develop a significant cometary activity, as it was the case for the first ob ject discovered in this class: Chiron (Kowal 1977; Kowal et al. 1979). In his history of outer Solar System astronomy, Davies (2001) explains that "Kowal looked for a group of mythological characters unrepresented amongst the asteroids. He found the Centaurs, strange creatures, half human and half horses". His choice was a very good one, as it also fits their dual appearance, half comet, half asteroid. Through the interactions with Uranus and Neptune, some TNOs are also ejected on very eccentric, very elongated orbits with large semi-ma jor axis a; these are called "Scattered TNOs". The dynamical distinction between Centaurs and Scattered TNOs is not very clear; the classification is based on their orbit semi-ma jor axis, the limit being loosely defined. In this paper, we define ob jects from this class with a < 35 as Centaurs, those with larger a, as Scattered TNOs. Finally, as their orbits are not stable, some of the Centaurs can fall further toward the inner solar system, where they will appear as Short Period (SP) Comets (Kowal et al. 1979); for instance, simulation by Asher & Steel (1993) showed that 20% of the test particles originally on Pholus-like orbit would end up on comet-like orbits. Nevertheless, the main source of short period comets is believed to be directly the inner Kuiper Belt; a very efficient way to transfer ob jects from that region to cometlike orbit is through collisions. It is estimated that 90% of the SP comets originating from the Kuiper Belt correspond to collisional fragments that were directly ejected from the Belt (Farinella & Davis 1996).

1.2. Physical prop erties
From a physical point of view, Cubewanos, Plutinos, Scattered TNOs, Centaurs and SP Comets (i.e. Minor Bodies in the Outer Solar System, MBOSS) are closely related, and are all believed to have the Edgeworth-Kuiper Belt as a common origin. It is therefore quite natural to consider that they have the same intrinsic physical nature. Nevertheless, their current location and past history may have affected them in different ways. The MBOSSes cover a broad range of colors, from neutral (solar colors), possibly slightly bluish to very red (see, for instance, Jewitt & Luu 1998; Boehnhardt et al. 2001; Delsanti et al. 2001) for the TNOs, Meech et al. (2002) for the SP comets. Three main phenomena are suspected to contribute to this color diversity: ­ Aging: considering that their surface is likely to be covered with organic-rich water ice, irradiation of the surface layers by high-energy particles (cosmic rays, hard UV...) will cause the organic molecules to lose Hydrogen atoms, and a progressive polymerization. This results in a slow, progressive reddening of


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643

the ob ject, with a time-scale of 107 yrs, the blue albedo decreasing down to a few percent (Strazzula & Johnhson 1991; Strazzula 1998). With further irradiation, the icy surface is expected to become dark grey (uniform albedo of a few percent), with a time-scale ten times longer (Thompson et al. 1987); ­ Impact on the surface will also alter the color of the ob ject by exposing underlying, non-irradiated (and neutral-bluish) material. The frequency of these impacts will differ considerably depending on the region of the MBOSS being considered. The collisions in the Edgeworth-Kuiper Belt have been studied in detail (Stern 1996; Davis & Farinella 1997); ­ Cometary activity is also expected to alter the ob ject color, as it removes the upper layer of the active regions and deposits fresh dust on the surface, which is likely to make it neutral to blue. While cometary activity obviously has an effect on comets, it must also be considered for Centaurs, as exemplified by (2060) Chiron. Recent observations also suggest that the cometary activity could play a role for TNOs (Hainaut et al. 2000; Sekiguchi et al. 2002). It is also interesting to note that the laboratory experiments simulating the ob ject surface for different irradiation levels (i.e. different ages) all present fairly linear spectra over the visible wavelengths, but with changing slope (Thompson et al. 1987). The three phenomena described above will produce a variegated ob ject, with regions of different ages, each of them presenting a different spectrum, hence a different color. However, the average spectrum over the surface being the average of linear spectra with different slopes, this total spectrum is expected also to be linear over the visible wavelength range. While the reddening by irradiation will affect all MBOSSes in a similar way (to some extent, see Thompson et al. 1987), the time scale for collision re-surfacing will be very different for the different classes. Similarly, the importance of the cometary activity will be a function of the heliocentric distance of the ob ject. The thickness of the irradiation crust and/or of a dust mantle will also determine whether an ob ject is active or not; the impact rate will therefore be linked to the cometary activity. The different MBOSS populations will therefore be affected ­ at least ­ by these three phenomena, but the equilibrium between them will be different, resulting in different color distributions. The balance between collisions and reddening has been studied numerically in the case of TNOs (Luu & Jewitt 1996a). The model used was fairly simple ­ in particular, it did not take into account the darkening of the surface that follows the reddening, but could reproduce the color diversity observed. One can hope that it will become possible to use the observed color distributions to further constrain the relative importance of the evolution phenomena for the different MBOSSes. Recent papers have been published discussing fairly large samples of TNOs; while some authors see these ob jects evenly spread over the whole color range

(Barucci et al. 2000; Boehnhardt et al. 2001; Delsanti et al. 2001; Davies 2000), others have reported that the TNO population is distributed into two well separated color classes: one of solar color, the other very red (Tegler & Romanishin 1998). It is intriguing that different groups obtain such different results. We will investigate whether this can be a random, selection effect, or if other conclusions have to be reached.

1.3. Structure of this pap er
The purposes of this paper are the following: ­ In Sect. 2, we will first give a description of the individual ob jects that is as complete as possible, based on all the published photometric information. For each ob ject, we will compute the "reddening", or spectral gradient, which describes the global slope of the reflectance spectrum; ­ In Sect. 3, we will look for correlations between the objects' surface characteristics and size, orbital parameters, etc.; ­ Section 4 is devoted to the description of individual classes of ob jects; ­ In Sect. 5, we will compare the data of the different MBOSS classes, in order to cast some light on their similarities and differences. ­ Finally, in Sect. 6, we will compare the colors of the various MBOSS classes with simple models, in order to investigate the reality of their possible bimodal or continuous distribution. In these sections, we will each time highlight the results of the individual tests and give the conclusions that can be drawn from them taken individually. We will then discuss the results from a more general point of view and summarize them in Sect. 7. Most of the tables and plots shown in this paper are directly generated from the measurement database (described later on). We intend to keep this database up-to-date (contributions from measurers are welcome), to make it available as a web page and, if needed, publish updates of this analysis every time the size of the studied population is multiplied by 2­3, i.e. when we can expect a ma jor step in significance of the results described. Finally, in Appendix A, we give the complete list of reference for each ob jects. We also give there a fairly detailed description of the statistical tests used in the paper and the numerical results of these tests. It is recommended that the reader who is not familiar with these techniques read the Appendix first. The present paper represent a snapshot of the color database, which is continuously growing. Up-to-date versions of the tables and figures, as well as many additional figures, are available on our web site at http://www.sc.eso.org/~ohainaut/MBOSS.


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Table 1. Average magnitudes and colors; online version at CDS, and an up dated version is available on-line at http://www.sc.eso.org/~ohainaut/MBOSS.
Ob ject 2060 Chiron 5145 Pholus 7066 Nessus 8405 Asb olus 10199 Chariklo 10370 Hylonome 2P/Encke 6P/d'Arrest 10P/Temp el 2 22P/Kopff 28P/Neujmin 1 46P/Wirtanen 53P/VanBiesbro ek 86P/Wild3 87P/Bus 93K2P/Helin-Law. 96P/Machholz 1 107P/Wilson-Harr. 143P/Kowal-Mrkos 1992 QB1 1993 FW 1993 RO 1993 SB 1993 SC 1994 ES2 1994 EV3 1994 GV9 1994 JQ1 1994 JR1 1994 JS 1994 JV 1994 TA 1994 TB 1994 VK8 1995 DA2 1995 DB2 1995 DC2 1995 FB21 1995 HM5 1995 QY9 1995 QZ9 1995 SM55 1995 TL8 1995 WY2 1996 RQ20 1996 RR20 1996 SZ4 1996 TC68 1996 TK66 1996 TL66 1996 TO66 1996 TP66 1996 TQ66 1996 TS66 1997 CQ29 1997 CR29 1997 CS29 1997 CT29 1997 CU29 1997 GA45 1997 QH4 1997 QJ4 1997 RL13 1997 RT5 1997 RX9 1997 SZ10 1998 BU48 1998 FS144 1998 HK151 1998 KG62 1998 QM107 1998 SG35 1998 SM165 1998 SN165 1998 TF35 1998 UR43 1998 VG44 1998 WH24 1998 WV24 1998 WV31 1998 WX24 1998 WX31 1998 XY95 1999 CC158 1999 CD158 (1)/(2) M11 ± Cent/27 6.398 ± 0.049 Cent/36 7.158 ± 0.097 Cent/17 -- Cent/34 8.966 ± 0.057 Cent/38 6.486 ± 0.033 Cent/7 -- SPC/4 -- SPC/2 -- SPC/2 -- SPC/1 -- SPC/6 -- SPC/1 -- SPC/1 -- SPC/1 -- SPC/1 -- SPC/1 -- SPC/1 -- SPC/2 -- SPC/2 -- QB1/8 6.864 ± 0.121 QB1/8 6.533 ± 0.151 Plut/6 8.488 ± 0.113 Plut/4 8.024 ± 0.143 Plut/14 6.711 ± 0.054 QB1/2 7.525 ± 0.115 QB1/4 7.108 ± 0.089 QB1/1 6.815 ± 0.091 QB1/5 6.603 ± 0.127 Plut/7 6.844 ± 0.071 QB1/2 7.255 ± 0.062 QB1/2 7.195 ± 0.058 Cent/2 11.413 ± 0.126 Plut/10 7.505 ± 0.080 QB1/2 7.025 ± 0.144 QB1/7 7.964 ± 0.118 QB1/2 8.112 ± 0.085 QB1/6 6.848 ± 0.148 QB1/4 7.017 ± 0.099 Plut/7 7.881 ± 0.111 Plut/6 7.487 ± 0.126 Plut/2 7.886 ± 0.400 QB1/3 4.333 ± 0.053 Scat/1 4.585 ± 0.056 QB1/3 6.861 ± 0.110 QB1/5 6.890 ± 0.104 Plut/2 6.586 ± 0.133 Plut/2 8.181 ± 0.159 QB1/1 6.734 ± 0.073 QB1/2 6.281 ± 0.074 Scat/10 5.227 ± 0.133 QB1/16 4.544 ± 0.049 QB1/9 6.958 ± 0.063 Plut/6 7.137 ± 0.078 QB1/9 5.986 ± 0.112 QB1/5 6.763 ± 0.183 QB1/2 7.076 ± 0.135 QB1/10 5.065 ± 0.085 QB1/2 6.498 ± 0.230 QB1/4 6.206 ± 0.108 QB1/1 7.744 ± 0.500 QB1/3 6.983 ± 0.133 Plut/3 7.424 ± 0.124 QB1/1 9.361 ± 0.300 QB1/1 6.736 ± 0.030 QB1/1 7.800 ± 0.100 QB1/1 8.145 ± 0.060 Cent/1 7.033 ± 0.057 QB1/1 -- Plut/3 6.879 ± 0.039 QB1/2 6.065 ± 0.078 Cent/1 10.226 ± 0.060 Cent/2 10.828 ± 0.023 QB1/2 5.799 ± 0.190 QB1/4 5.736 ± 0.410 Cent/2 8.683 ± 0.193 Plut/3 8.090 ± 0.130 Plut/2 6.349 ± 0.057 QB1/7 4.512 ± 0.108 Plut/1 7.112 ± 0.040 Plut/1 7.643 ± 0.070 QB1/1 6.232 ± 0.090 QB1/1 6.225 ± 0.075 Scat/1 6.492 ± 0.167 Scat/1 5.430 ± 0.074 QB1/1 4.903 ± 0.066 G rt ± 0.642 ± 1.629 52.054 ± 2.105 45.727 ± 2.567 15.075 ± 2.981 13.677 ± 1.548 10.667 ± 4.090 3.770 ± 3.070 15.138 ± 2.886 -- -- 11.687 ± 3.947 -- -- -- -- -- -- -- 20.983 ± 1.167 37.328 ± 6.651 12.172 ± 5.517 19.363 ± 7.579 12.253 ± 4.554 36.763 ± 3.488 80.403 ± 7.434 27.511 ± 7.555 -- -- 24.825 ± 5.805 -- 37.024 ± 5.331 35.801 ± 6.104 39.035 ± 4.615 32.582 ± 6.345 17.189 ± 7.292 -- 36.530 ± 7.927 -- 6.761 ± 4.993 10.588 ± 4.022 15.709 ± 5.234 1.269 ± 2.875 33.942 ± 3.051 21.766 ± 9.753 21.258 ± 5.329 40.209 ± 5.038 19.062 ± 4.832 -- 27.932 ± 3.716 3.355 ± 3.011 5.371 ± 2.467 32.326 ± 3.690 35.809 ± 4.398 28.922 ± 5.214 34.308 ± 5.942 20.636 ± 8.257 28.988 ± 2.815 -- 28.730 ± 3.680 -- 28.694 ± 6.173 9.307 ± 6.115 -- -- -- 31.431 ± 3.246 26.985 ± 3.102 20.767 ± 6.964 8.017 ± 3.404 23.450 ± 3.122 16.299 ± 3.246 12.259 ± 2.768 33.103 ± 3.836 7.311 ± 4.410 34.880 ± 4.193 9.494 ± 5.465 24.105 ± 3.881 23.435 ± 3.338 14.117 ± 3.243 10.197 ± 4.283 37.747 ± 5.234 26.201 ± 4.606 36.230 ± 7.184 20.293 ± 3.657 13.430 ± 3.734 B- 0.679 1.299 1.090 0.750 0.802 0.643 0.770 V ± ± ± ± ± ± -- ± -- -- -- -- -- -- -- -- -- -- ± ± ± ± ± ± ± ± -- -- ± -- -- ± ± ± ± -- -- -- ± ± ± ± ± ± ± ± ± -- ± ± ± ± ± ± ± ± ± -- ± -- ± ± -- -- -- ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± -- ± ± ± ± 0. 0. 0. 0. 0. 0. 039 099 040 040 049 082 V- 0.359 0.794 0.793 0.513 0.479 0.464 0.388 0.565 0.575 0.533 0.508 0.355 0.328 0.116 0.543 0.267 0.429 0.406 0.580 0.713 0.517 0.576 0.475 0.673 0.940 0.516 0.740 0.945 0.656 0.850 0.771 0.672 0.706 0.659 0.547 0.770 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 102 121 040 034 072 206 141 094 124 060 056 060 109 118 082 127 152 082 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 463 520 515 394 695 648 553 707 531 600 640 334 377 654 655 635 728 538 667 744 634 R ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± -- ± -- ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± -- ± ± -- -- -- ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 027 032 041 068 029 059 062 067 048 022 079 073 081 144 020 075 027 017 020 098 101 128 077 065 150 124 099 097 115 070 091 080 083 061 131 R- 0.356 0.814 0.695 0.523 0.542 0.490 0.408 0.450 I ± ± ± ± ± ± ± ± -- -- ± -- -- -- -- -- -- -- ± ± ± ± ± ± ± ± -- -- ± -- ± ± ± -- ± -- ± -- ± ± -- ± ± ± ± ± ± -- ± ± ± ± ± ± ± ± ± -- ± -- ± ± -- -- -- -- ± -- ± ± -- ± ± ± ± ± ± ± -- ± -- ± ± ± ± ± 0. 0. 0. 0. 0. 0. 0. 0. 037 056 066 045 030 122 060 040 I- 0.472 1.040 0.790 0.690 0.730 0.390 J ± ± ± ± ± ± -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- ± 0. 0. 0. 0. 0. 0. 132 051 122 056 040 173 H± ± 0.082 ± 0.046 ± 0.268 ± 0.142 ± 0.044 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 0.400 ± 0.203 -- -- -- -- -- -- -- -- -- -- -- -- -- -- 1.200 ± 0.470 -- -- -- -- -- -- -- -- -- -- 0.350 ± 0.117 -0.210 ± 0.170 0.170 ± 0.078 -- 0.650 ± 0.071 -- -- 0.300 ± 0.156 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- J- 0.290 0.375 0.309 0.315 0.411 H-K ± 0.064 ± 0.099 -0.037 ± 0.047 -0.089 ± 0.385 0.095 ± 0.253 0.093 ± 0.046 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -0.040 ± 0.197 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -0.040 ± 0.112 0.810 ± 0.158 0.020 ± 0.092 -- -- -- -- -0.100 ± 0.233 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --

0.040

0.440

0.079

0. 0. 0. 0. 0. 1. 0. 1.

820 836 932 933 802 012 710 500

0. 0. 0. 0. 0. 0. 0. 0.

028 145 089 162 071 105 150 150

0. 0. 0. 0. 0. 0. 0. 0.

560 672 431 515 514 738 970 840

0. 0. 0. 0. 0. 0. 0. 0.

022 197 127 192 114 077 150 199

1.010

0.180

0.520 0.563 0.740 0.727 0.515

0.120 0.133 0.210 0.108 0.172 0.160 0.108 0.060 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 066 076 208 120 160 170 120 079 060 075 100 097 120 182 061

1. 1. 1. 1.

261 080 010 310

0. 0. 0. 0.

139 132 060 270

0.160 0.580 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 096 093 050 052 051 190 099 070 062 078 050 052 047 073 081 110 120 157 053 090 058 0.370 0.400 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 310 641 458 609 760 620 590 428 375 683 750 645 605 620 592

0. 0. 0. 0. 1. 1. 0. 1. 0. 1. 0. 0. 0. 1. 1. 0. 0. 1.

649 696 880 645 045 004 935 150 783 002 694 666 984 186 010 990 750 049

1.157 1.039 0.700

0.145

0.638

0.098 0.159 0.130

0.131 0.628 0.120 0.511

0.103 0.649 0.119 0.362

1. 1. 0. 0. 1. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 1.

140 105 910 510 000 730 725 966 712 085 784 951 924 770 834 090

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0.939 0.962 0.871

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.238 0. 0.098 0. 0.077 0.

080 074 076 090 060 060 089 091 095 111 101 055 063 010 089 050

650 648 560 469 561 520 456 687 446 697 565 567 602 500 513 700 602 645 571 477

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

030 052 067 065 074 030 050 079 089 064 106 056 043 030 069 050 080 140 063 065

0.570 0.398 0.640 0. 0. 0. 0. 0. 0. 0. 546 648 419 651 268 668 547

0.078 0.073 0.040 0. 0. 0. 0. 0. 0. 0. 063 073 088 119 117 116 111

0.357 0. 0. 0. 0. 640 772 552 543

0.114 0. 0. 0. 0. 112 153 088 089


O. R. Hainaut and A. C. Delsanti: MBOSS colors, a statistical analysis Table 1. Continued.
Ob ject 1999 CF119 1999 DE9 1999 HB12 1999 HR11 1999 HS11 1999 KR16 1999 OX3 1999 OY3 1999 RY215 1999 RZ253 1999 TC36 1999 TD10 1999 TR11 1999 UG5 2000 EB173 2000 OK67 2000 PE30 2000 QC243 2000 WR106 (1)/(2) Scat/1 Scat/2 Scat/1 QB1/1 QB1/1 QB1/1 Cent/3 QB1/1 QB1/1 QB1/2 Plut/5 Scat/2 Plut/1 Cent/5 Plut/17 QB1/2 Scat/1 Cent/1 QB1/1 M11 ± 7.031 ± 0.077 4.804 ± 0.056 -- -- -- 5.505 ± 0.020 7.272 ± 0.196 6.303 ± 0.040 -- 5.428 ± 0.056 4.920 ± 0.070 8.706 ± 0.022 8.058 ± 0.140 10.483 ± 0.134 4.657 ± 0.110 6.138 ± 0.063 -- 7.949 ± 0.049 3.048 ± 0.059 G rt ± 13.450 ± 4.603 20.506 ± 2.281 8.150 ± 3.096 29.372 ± 4.428 30.142 ± 4.784 44.581 ± 1.577 28.215 ± 3.746 0.952 ± 2.294 -- 29.962 ± 3.002 32.331 ± 2.382 11.893 ± 1.908 44.369 ± 7.259 25.886 ± 2.677 22.884 ± 3.969 15.972 ± 7.056 4.713 ± 2.049 6.961 ± 2.724 39.611 ± 3.536 B-V -- 0.915 ± 0.870 ± 0.920 ± 1.010 ± 1.100 ± 1.072 ± 0.710 ± 0.800 ± 0.820 ± 1.008 ± 0.770 ± 1.020 ± 0.964 ± 0.954 ± 0.727 ± 0.710 ± 0.724 ± 1.017 ± ± 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 058 060 120 160 050 117 010 100 170 050 050 080 085 050 108 050 062 071 V- 0.557 0.572 0.500 0.530 0.680 0.740 0.692 0.370 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 646 687 495 750 607 565 517 380 448 711 R ± ± ± ± ± ± ± ± -- ± ± ± ± ± ± ± ± ± ± ± 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 083 042 050 100 100 030 055 020 058 041 040 070 060 090 068 040 044 071 R- 0.391 0.559 0.320 0.800 0.600 0.770 0.475 0. 0. 0. 0. 780 647 625 470 I ± ± ± ± ± ± ± -- ± ± ± ± -- ± ± -- ± ± ± ± 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 107 049 080 070 090 030 109 080 062 056 032 I -J ± -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- J -H ± -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- H-K ± -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --

645

0.625 0.623 0.450 0.397 0.730

0.042 0.061 0.040 0.069 0.071

(1) Class: QB1 = Cub ewano, Plut = Plutino, Cent = Centaur, SPC = Short Perio d Comet, LPC = Long Perio d Comet. (2) Numb er of ep o chs. Grt is the sp ectral gradient S (%/100 nm). M11 is the absolute R magnitude.

2. Dataset and general description of individual MBOSSes 2.1. Dataset ­ Average magnitudes and colors
In order to get the most significant results, the statistical analysis presented in this paper were based on a complete compilation of all the TNO and Centaur colors that have been reported in the "Distant EKO" web page (Parker 2001), as of 2001. Several additional papers, preprint and private communications about TNOs and Comets were also added. We realize that such a compilation can never be complete and up-to-date; the current database is frozen in its current state, and we plan to add new and missing papers in future versions. Refer to Appendix A for the references that were used for each ob ject. Authors are encouraged to send us their measurements electronically (ohainaut@eso.org), so that we can include them in this database. When available, the individual magnitudes were used, so that non-standard color indexes (i.e. not the traditional B - V , V - R...) can be computed (we hereby encourage the authors to publish these individual magnitudes). Where the magnitudes were not available, we used the published color indexes. In this compilation, no correction has been made for the different photometric systems used. Only the name of the filter is taken into account, so that RBessel = RKC , K = K = K s, etc. We assume that the errors introduced by these assumptions are small compared to the measurement errors. As all the TNO measurements were obtained after 1992, a large fraction of them were calibrated using the standard stars by Landolt (1992). If the authors computed the color term of their system and applied them, the magnitude they published are de facto in the Bessel system as described by Landolt, further reducing possible color discrepancies between the different filter system used. For a given epoch (loosely defined as "within a few hours"), we computed all the possible colors and magnitudes based on the available colors and magnitudes. It is important to note that no additional color indexes were

computed at that stage (i.e. if V is available at one epoch, and R at another, the V - R index is not computed mixing these epochs). Some publications list colors obtained by combining magnitudes obtained at different epochs. These were not entered in the database. We also checked for and removed multiple entries for the same measurements that appeared in different papers. The magnitudes and colors from different epochs (and different authors) were combined in order to obtain one average magnitude and color set per ob ject. However, no new color indexes are computed even if we now have enough data (e.g. if an author reported a R - I and another I - J , we do not compute nor use the resulting R - J ), as these would not be obtained from simultaneous data. In this way, even if the ob ject presents some intrinsic magnitude variability, we do not introduce any additional color artifacts. The average magnitude that we publish here corresponds to the average of the (possibly varying) magnitudes, and the average colors is the average of the measured colors. The variations of magnitude will not contribute to the color error. For this combination, x, the average magnitude or ¯ color is obtained by weighted average of the individual magnitudes and colors. We did not a priori reject any published measurement, nor give a stronger or lighter weight to the measurements from a given author or team. We did not give a larger weight to measurements obtained on a larger telescope. For this study, we fully trust and rely on the published error bars: the weight of a measurement is set to 1/ :
N i=1 xi /i N i=1 1/i

x= ¯ Using small imate 1... N puted

·

(1)

this weight, very good measurements (trusting their ) will be given a strong weight compared to approxvalues. In case of multiple measurements xi , i = of an item (magnitude or color), the error is comas a combination of the individual errors i and of


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O. R. Hainaut and A. C. Delsanti: MBOSS colors, a statistical analysis Table 2. Solar colors used in this pap er, from Hardorp (1980), Campins et al. (1985) and Allen's Astrophysical Quantities, Cox (2000). Color U -B U -V V -R V -I Value 0.204 0.845 0.36 0.69 Color V -K J -H H -K Value 1.486 0.23 0.06

the dispersion of the measurements around their mean x, ¯ using =
N i=1 i N i=1 1/i

+

N i=1

(N - 1)

(xi - x)2 /i ¯
N i=1

1/i

·

(2)

The first term in the root corresponds to the increase of the Signal-to-Noise ratio resulting from the multiple measurements, while the second term is the variance of the measurements (the N - 1 is because we don't have a priori knowledge of the mean). With this combination, a measurement with a large error will have a small contribution to the final average and error. On the other hand, two equally good measurements having different values will have a resulting error that is larger than the individual ones, reflecting a possible variation and a definite uncertainty on the value. In case only one measurement is available, it is reported with its error bar in the final table. Some ob jects were measured several times. The dispersion of these measurements is similar to the error bars, suggesting that no dramatic systematic effects affect the different teams, and therefore indicating that the combined data are of better quality than the individual ones. The averaging program also includes a warning system checking for very different values of given color of an ob ject (the limit corresponds to an incompatibility at the 3 level taking into account the sum of the considered error bars). The current database did not trigger this warning. The classical color indexes are reported in Table 1. In this table, the un-named ob jects are identified by their temporary MPC designation (e.g. 1992 QB1 ), while the named ob jects are identified with their number and name. For uniformity, we don't use the number of numbered but still un-named ob ject. In the case of the numbered comets, their IAU designation is used. The table also lists the number of independent epochs that were combined for each ob ject.

The results are also listed in Table 1. Assuming a value for the surface albedo p, these absolute magnitude can be converted into the radius RN of the ob ject [km] using the formula from Russell (1916)
2 pRN = 2.235 â 10 22

â 10

0.4(M -M (1,1))

,

(4)

where M is the R magnitude of the Sun and M (1, 1) is the absolute R magnitude from Eq. (3). As the albedo p is not known (except for a couple of ob jects), and in particular because neutral-grey ob jects could either correspond to extremely old surfaces (with p as low as 0.02, Thompson et al. 1987) or to ob jects covered with fresh ice (therefore having a higher albedo, possibly as high as that of Pluto, 0.3, or Chiron, 0.1), we do not give a generic conversion of M (1, 1) into a radius.

2.3. Sp ectral gradient
The information contained in the color indexes can be converted into a very low resolution reflectivity spectrum R() (Jewitt & Meech 1986), using R() = 10
-0.4(m()-m ())

,

(5)

2.2. Absolute magnitude
For each epoch, we attempted to compute an absolute R magnitude: for this purpose, we used either the measured R magnitude, when available, or another magnitude and the corresponding color index with R. The helio- and geo-centric distances (r and , resp., [AU]) were computed using a two-body ephemerides program with the orbital elements available at MPC, and the absolute magnitude M (1, 1) was computed using M (1, 1) = R - 5 log(r). (3)

where m and m are the magnitude of the ob ject and of the Sun at the considered wavelength. Normalizing the reflectivity to 1 at a given wavelength (in our case, the V central wavelength), we have R() = 10
-0.4((m()-m(V ))-(m()-m(V )) )

.

(6)

Because i) the phase angle is usually small for MBOSS observations, ii) this angle does not change significantly from epoch to epoch, and iii) the phase function is unknown for most MBOSSes, we neglect the phase correction. Because of i) and ii), this correction would in any case not change significantly the result. The M (1, 1) for all available epochs were averaged using the same procedure as described above; the error was also computed.

The solar colors used are listed in Table 2. The reflectivity spectra are given in Fig. 1. Boehnhardt et al. (2001) have compared such magnitude-based reflectivity spectra with real spectra (i.e. obtained with a spectrograph) for 10 ob jects observed quasi-simultaneously with a large telescope (one of ESO's 8 m VLTs) through broad-band filters and with a low resolution spectrograph. He found a excellent agreement between real and magnitude-based spectra. We can introduce a description of the reflectivity spectrum: the reddening S , also called slope parameter or spectral index, which is expressed in percent of reddening per 100 nm: S (1 , 2 ) = 100. R(2 ) - R(1 ) · (2 - 1 )/100 (7)


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Table 3. Reddening Index S and its error, and deviation d from a linear sp ectrum for the ob jects in the database, available electronically at http://www.edpsciences.org.

ferent and worth a more detailed study. In both cases, observers should take a closer look at the ob jects identified by a large deviation |d| in Table 3. These ob jects are (7066) Nessus, 1991 QB1 , 1993 SC, 1994 ES2 , 1994 TB, 1996 RR20 and 1999 KR16 . It is interesting to note that the coordinates (S , d) of an ob ject are very similar to the "principal components" (PC1,PC2) that Barucci et al. (2001) have obtained from an analysis of the colors of 22 ob jects: the position of a MBOSS in a multi-dimensional color diagram is determined primarily by PC1 (which can physically be associated to S ) and to a much lesser extent by PC2 (which would be related to d). The additional dimension of the multi-dimensional color diagram contain little information. We intent to apply a similar analysis to this dataset.
Fig. 1. Examples of reflectivity sp ectra, sorted by increasing gradient. The reflectivity is normalized to 1 for the V filter; the sp ectra have b een arbitrarily shifted for clarity. For each ob ject, the dotted line is the linear regression over the V , R, I range, corresp onding to the gradient S . Similar reflectivity sp ectra are available for all the ob jects of the database on our MBOSS web site.

2.4. Color-color diagrams
Figure 2 shows a selection of color-color diagrams; the whole collection, for all possible color indexes, is available on the MBOSS web site. To guide the eye, the reddening line is drawn on each diagram. This line is constructed computing the colors for an ob ject of a given reddening S using Eq. (7), and then connecting all the points for -10 < S < +70%/100 nm (a tick is placed every 10%). An ob ject located directly on this line has a perfectly linear reflectivity spectrum, and its slope S can be estimated using the tick-marks on the line. Ob jects above the line have a concave spectrum (positive d), while ob jects below the line have a convex spectrum (negative d) over the spectral range considered. As it was noted in Sect. 1.2, the three physical processes that are suspected to effect the color of a MBOSS surface independently produce linear reflectivity spectra (in first approximation, over the visible wavelength range). The average over the complete surface of an ob ject will therefore also be a linear spectrum. Within that hypothesis, if no other physical processes plays an important role, and if the MBOSSes have the same original intrinsic composition, the ob jects should all lie on the reddening line. A young-surfaced ob ject would have solar-like colors, and the aging will move the ob ject up the reddening line, while collision and activity will move it back down. Similarly, an ob ject left undisturbed long enough would evolve moving up the reddening line till it reaches the maximum possible reddening, then, the continued irradiation of its surface would cause it to further darken (Thompson et al. 1987), possibly moving back down on the reddening line. In that case, one could expect to find among the neutral ob jects some MBOSSes covered with fresh ice, together with objects with ancient ice, with a very dark albedo. This is tested later (cf. Sect. 3.4).

Boehnhardt et al. (2001) realized that all the ob jects observed with a high S/N display a linear reflectivity spectrum over the V - R - I range. We can therefore introduce a global value for S , describing the spectrum over the V - R - I range. We obtained the value of S , together with its uncertainty, by linear regression of R as given in Eq. (6). We restricted this fit to the ob jects having at least two color indexes measured. The values of S and its error are listed in Table 3. We also restricted the fit to the V , R and I filters, excluding B , because the B reflectivity shows a systematic trend, as described below. The error on S is a combination of the error on each R() (obtained by propagation of the errors on the colors) and on the linear regression. In order to further characterize the shape of the spectrum, we introduced d, the total deviation of the reflectivity with respect to the linear regression: d=
=B ,V ,R,I

(R() - Rl ()),

(8)

where Rl is the reflectivity expected from the linear fit. Positive values of d correspond to spectra with a global concavity, while negative values correspond to a convexity. The concavity d can be interpreted in two ways: either i) one considers that the result published by Boehnhardt et al. (2001) can be generalized to all MBOSSes; in that case, ob jects with a large |d| suffer from large uncertainties and should be re-measured with a better S/N , or ii) it is considered as real, and large values of |d| denote ob jects whose spectral characteristic are intrinsically dif-


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Fig. 2. MBOSS color-color diagrams. The meaning of the different symb ols is given in Fig. 4. The reddening line ranges gradients from -10 to 70%/100 nm; a tick mark is placed every 10 units. The outliers ob jects in the B - V , R - I are 1994 ES2 (top left), 1994 EV3 (top right), 1998 HK151 (b ottom left), and 1995 DA2 (middle right). All other combinations of colors are available on the MBOSS web site.

The diagrams from Fig. simple interpretation of the are clustered along that line colors, the deviations from

2 are in agreement with this reddening line: the MBOSSes . For the (B - V ) and (V - R) the line are compatible with

the error bars of the individual points, indicating that the spectra are linear over this wavelength range. The plots involving the (R - I ) color, however, show a systematic deviation from the line for the reddest ob jects (particularly


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649

M(1,1)

a [AU]

e

i

E

M(1,1)

a [AU]

e

i

E

M(1,1)

a [AU]

e

i

E

M(1,1)

a [AU]

e

i

E

M(1,1)

a [AU]

e

i

E

M(1,1)

a [AU]

e

i

E

Fig. 3. Visible color distributions as functions of the absolute magnitude M (1, 1), the orbit semi-ma jor axis a [AU], the eccentricity e, the inclination i and the orbit excitation E (see text for definition). The meaning of the different symb ols is given in Fig. 4. Other colors are available electronically and on line at the MBOSS web site.

visible in the (B - R) vs. (R - I ) diagram). This corresponds to the fact that the spectrum of the reddest ob jects flattens toward the IR, where it is typically flat/neutral (Davies 2000; McBride et al. 1999). One also notes a systematic deviation from the reddening line of the neutral to neutral-red points in the B - V vs. V - R diagram: the bulk of these points are significantly above the line; this corresponds to the bend observed around the B wavelength in many reflection spectra from Fig. 1. This bend is not observed in the "fresh ice" laboratory spectra published by Thompson et al. (1987), suggesting that in spite of their low reddening, the surface of these ob jects could be significantly processed. It is also interesting to note a small group of 5 ob jects clustered very near the solar colors in the B - V vs. V - R diagram, i.e. in the range corresponding to fresh ice surface. Two of these ob jects are either known or suspected to be cometary active, i.e. (2060) Chiron (Tholen et al. 1988; Meech & Belton 1989) and 1996 TO66 (Hainaut et al. 2000). A detailed study of

the others (1995 SM55 , 1996 TL deserved.

66

and 1999 OY3 ) is well

In addition to the simple "reddening line", it would be interesting to produce an evolution track of the color of laboratory ices, for increasing irradiation doses. Such work will be presented in another paper by the same authors. The diagrams show notable outliers (i.e. isolated points, far from the reddening line and the general cluster of ob jects): ­ 1994 ES2 , which has only one fairly old set of color measurements; ­ 1994 EV3 , whose V - R index is well established (and reasonable); other colors have only one measurement available; ­ 1998 HK151 , whose B - V has only one fairly old measurement while other colors are well established, and ­ 1995 DA2 , whose B measurements is also fairly old; ­ 1996 TO66 is an outlier only in the IR color diagram.


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O. R. Hainaut and A. C. Delsanti: MBOSS colors, a statistical analysis

For all these ob jects, it seems that less accurate magnitudes from the early years of TNO photometry (before the VLT/Keck era) are the reason of the strange colors. We therefore encourage observers to acquire new colors for these ob jects. For instance, preliminary measurements of VLT data for 1994 ES2 and 1994 EV3 (Delsanti, priv. comm.) put these ob jects back in the main group.

no way to quantify the significance of that correlation. The correlation coefficients were computed for the complete MBOSS population (i.e. al l ob jects) and for the Plutinos, Cubewanos and Centaurs/Scattered ob jects separately. The numerical results of the tests are listed in Table C.1. R e s u l ts :

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-ma jor axis, e, the eccentricity and i, the inclination. We also consider the "excitation" E of an ob ject's orbit, defined as E= e2 + sin2 (i), (9)

sin(i) is related to the ob ject's velocity perpendicular to the ecliptic, and e to its radial velocity. Therefore, E is an estimate of the velocity of the ob ject with respect to another ob ject 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. 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 S as a function of the orbital parameters and absolute magnitude. In each figure, each ob ject 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. R e s u l ts : ­ a plots: the extreme redness of ob jects 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 ob jects.

­ For the complete MBOSS population: the colors and gradient are not correlated with any of a, e, i, E nor M (1, 1) (i.e. all the correlation coefficients are extremely low in absolute value). ­ i, e, E : for the Cubewanos, there is a systematic anticorrelation between the colors (and gradient) and the e, i and E parameters (i.e. ob jects 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 (0.2), both effects canceling each-other for E . ­ M (1, 1): for the Cubewanos, there is a correlation between the colors and the absolute magnitude, indicating that the ob jects 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 ob ject 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 (P rob 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.

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


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651

R e s u l ts : t - te s t ­ 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-ma jor axis a. ­ The t-test applied to the color distribution vs. i, e and E reveals a strong, highly significant trend for the Cubewanos: ob jects with higher orbital excitations are bluer. The effect is systematic in all colors (and gradient), and its significance is in the P rob = 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 E ob jects are less affected by (collisional) re-surfacing than ob jects 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 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 ob jects can be discovered. As a consequence, the test on the Plutinos explores intrinsically smaller ob jects. Setting the cut-off magnitude for the Plutinos equal to that of the Cubewanos produces unbalanced samples, but the result remains the same. R e s u l ts : f - te s t ­ The color distributions of ob jects as a function of the absolute magnitude: in earlier versions of this database, the B - and - I ( indicating any other filter) of ob jects with fainter M (1, 1) had significantly broader color distributions than the ob jects 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. Ob jects 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 8­10 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, e and/or 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).

3.4. Broadening of the M (1 , 1 ) distributions?
As mentioned earlier, neutral-bluish ob jects 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 ob jects are expected to be covered with highly irradiated ice (corresponding to the laboratory samples that received doses of 109 erg cm-2 Thompson et al. 1987), and would have a much narrower range of albedo. If we assume that all these ob jects have a similar radius distribution, the resulting M (1, 1) distribution should be significantly broader for the neutral ob jects 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. R e s u l ts : ­ Restricting the test to the Plutinos, it appears that the redder ob jects 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 ob jects have a broader M (1, 1)


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Table 4. Average colors of the various classes of ob jects. For each color, the table lists the numb er of ob jects included in the statistics, as well as the average color and the square root of the corresp onding variance (which is undefined and set to 0 in case only one ob ject is available).
Color U -B U -V U -R U -I B B B B B B V V V V V R R R R - - - - - - - - - - - - - - - V R I J H K R I J H K I J H K 0 0 0 0 20 20 17 0 0 0 20 17 4 0 0 17 0 0 0 0 0 0 2 1 1 20 P l u ti n -- -- -- -- os -- -- -- -- 1 1 1 1 33 30 25 0 0 0 40 30 5 0 0 30 0 0 0 0 0 0 4 3 3 35 Cub ewan 1.000 ± 1.720 ± 2.120 ± 2.500 ± os 0. 0. 0. 0. 000 000 000 000 0 0 0 0 15 15 14 5 4 4 15 13 6 5 5 14 6 5 5 6 5 5 5 5 5 15 Centau -- -- -- -- 0.930 1.513 2.119 2.800 3.395 3.428 0.590 1.169 1.801 2.245 2.299 0.582 1.243 1.658 1.695 rs -- -- -- -- 0.219 0.349 0.489 0.807 0.679 0.623 0.136 0.270 0.552 0.555 0.506 0.134 0.382 0.400 0.367 1 1 1 1 8 8 7 0 0 0 9 9 1 0 0 9 0 0 0 0 0 0 1 1 1 9 Scatt 0.970 1.710 2.150 2.410 ered ± 0. ± 0. ± 0. ± 0. 000 000 000 000 0 0 0 0 2 2 2 0 0 0 13 4 0 0 0 4 0 0 0 0 0 0 0 0 0 4 Comets ---- ---- ---- ---- 0.795 ± 0.035 1.355 ± 0.064 1.860 ± 0.141 ---- ---- ---- 0.430 ± 0.140 0.964 ± 0.136 ---- ---- ---- 0.465 ± 0.066 ---- ---- ---- ---- ---- ---- ---- ---- ---- 12.894 ± 7.192

0.886 ± 0.176 1.464 ± 0.261 1.977 ± 0.412 ---- ---- ---- 0.580 ± 0.091 1.118 ± 0.239 2.345 ± 0.213 ---- ---- 0.542 ± 0.161 ---- ---- ---- ---- ---- ---- 0.800 ± 0.566 0.360 ± 0.000 -0.040 ± 0.000 21.760 ± 12.305

0.946 ± 0.185 1.561 ± 0.249 2.131 ± 0.329 ---- ---- ---- 0.629 ± 0.132 1.206 ± 0.225 1.795 ± 0.495 ---- ---- 0.613 ± 0.137 ---- ---- ---- ---- ---- ---- 0.228 ± 0.355 0.330 ± 0.234 0.243 ± 0.494 26.537 ± 14.100

± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.863 ± 0.126 1.376 ± 0.262 1.914 ± 0.412 ---- ---- ---- 0.528 ± 0.116 1.031 ± 0.242 1.452 ± 0.000 ---- ---- 0.509 ± 0.138 ---- ---- ---- ---- ---- ---- 0.350 ± 0.000 0.310 ± 0.000 -0.040 ± 0.000 16.948 ± 11.906

I-J I-H I-K J -H J -K H-K Grt (%/100 nm)

0.685 ± 0.233 1.087 ± 0.234 1.124 ± 0.209 0.340 ± 0.051 0.383 ± 0.076 0.025 ± 0.083 24.316 ± 15.086

distribution) but although it is visible in all indexes, it is not statistically significant. ­ The reversed effect (bluer ob jects 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 5­6% 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 ob jects 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 ob jects 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.

Fig. 4. Color-color diagrams of the average p opulations

4. Individual p opulations 4.1. Average and Variances
For each class of MBOSS, we compute the average color indexes. Table 4 lists the average colors of the various classes of ob jects together with the square root of their variances (which will become equivalent to the standard deviation for large samples with a normal distribution). These values are of practical interest, for instance when preparing observations of an ob ject whose colors are not known.

Figure 4 displays the average populations' colors in a set of color-color diagrams.

4.2. Histograms and cumulative probability functions
It is customary to visually compare distributions of objects using their histograms, i.e. the number of ob jects if given bins. Such histograms are displayed in Fig. 5. However, one has to be extremely careful in working with such plots: the size of the bin has a strong influence on the shape of the final histogram. Indeed, binning the data is equivalent to smoothing the data with a window equal


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Fig. 5. CPF and histograms for the color indexes for all the classes of ob ject.

to the bin size. Structures in the distribution that have a size similar to or smaller than the bin will be masked in the histogram, an effect that can create dangerously convincing ­ but wrong ­ artifacts. A better way to represent a distribution is its Cumulative Probability Function (CPF). If one of the sample is x1 , x2 , ..., xn (e.g. the V - R color indexes of

n Centaurs), the corresponding CPF F (x) is the fraction of the sample whose value is smaller than x. The CPF always has a typical "S" shape, with F (-) = 0, and F increases by step at each xi till it reaches a value of 1 when x is larger than all the xi . The advantage of the CPF is that no information is lost with respect to the original distribution. While its use is not as instinctive as that of


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Fig. 5. Continued. CPF and histograms for the Gradient S .

the histogram, it is a worthwhile exercise to train ones eye to use them. The CPFs are also displayed in Fig. 5

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 KolmogorovSmirnov (KS) test, which are described in more detail in Appendix B; each of them produce a probability P rob. Low values of P rob 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 ob jects. They are defined as following: ­ Odd: ob jects with odd database; ­ Even: ob jects with even database; ­ 1999: ob jects discovered in ­ non-99: ob jects not discove numbers in the internal numbers in the internal 1999, and red in 1999.

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. R e s u l ts ­ Shift: apart from the comet V - R that appears to be on average bluer than that of the other ob jects, 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 .

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 P rob 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 ob jects 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 ob jects, that would have been present for the earlier years.

5.2.1. t -test: Are the mean colors compatible?
Table 4 lists the mean colors of the different classes. The color of an ob ject 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 P rob are listed in Table C.3; the results for the artificial classes are displayed in Table C.4. R e s u l ts ­ The 1999/non-1999 test classes do not show incompatibilities at the 9­10% level, except for very small

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 ob jects.


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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 ob jects; ­ 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. 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 P rob are listed in Table C.5. R e s u l ts : ­ 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.
Fig. 6. Examples of CPF of the TNO color distributions, compared with the CPF of bimodal and continuous model distributions. The model distributions have b een adjusted for a match of the observed distribution.

R e s u l ts : ­ 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.

6. Distribution bimo dality
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 ob jects 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

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 P rob are listed in Table C.6 for the real classes of ob jects. Those for the test classes are available only electronically.


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O. R. Hainaut and A. C. Delsanti: MBOSS colors, a statistical analysis Table 6. Parameters of the models maximizing the KS statistics for the 2D distribution of TNO colors, and corresp onding d and P r ob. The table is available in electronic form at http://www.edpsciences.org.

Table 5. 1D color distribution models and corresp onding KS statistics for various color indexes. The first column indicates which p opulation is taken into consideration (Plutinos, Cub ewanos or b oth), the second gives the color distribution used. The parameters listed corresp onds to the b est fitting models. The table is available in electronic form at http://www.edpsciences.org.

this compilation are often combining the result of various groups (a few ob jects 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 colorcolor diagrams.

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 ob jects is created at random. The observed distribution is then compared to the model population. We verified that the resulting P are 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 P rob 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

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 P rob. 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. R e s u l ts : ­ Plutinos and Cubewanos have color distributions that are compatible with a simple, uniform, continuous distribution (the worse probability is 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


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7.2. Individual objects
nPoints: 10000

Distribution: ( 0.68 0.40) -> ( 1.10 0.75) Type: Cont. nPoints: 10000

Distribution: ( 0.75 0.50) -> ( 1.05 0.65) Type: Bimod.

Spread: 0.070 0.070 KS: d=0.173 Prob=.640

Spread: 0.070 0.070 KS: d=0.180 Prob=.589

Cubewanos Distribution: ( 0.64 0.32) -> ( 1.15 0.72) Type: Cont. nPoints: 10000

Cubewanos Distribution: ( 0.70 0.38) -> ( 1.00 0.60) Type: Bimod. nPoints: 10000

Spread: 0.100 0.100 KS: d=0.299 Prob=.426E-01

Spread: 0.100 0.100 KS: d=0.190 Prob=.417

A small group of ob jects ­ 1995 SM55 , 1996 TL66 , 1999 OY3 , 1996 TO66 and (2060) Chiron ­ have almost perfectly solar colors, suggesting they are covered with neutrally colored fresh ice. Chiron is known to be cometary active, and 1996 TO66 is suspected to be so too. The other ob jects from this group therefore deserve a closer study to look for activity and/or fresh ice spectral signature. Four ob jects appear as outliers from the general population: 1994 ES2 , 1994 EV3 , 1998 HK151 and 1995 DA2 . In all cases, we suspect that they do not correspond to physically distinct ob jects, but that the colors reported are not accurate.

7.3. Gradient and colors
In the color-color diagram, the ob jects follow closely the "reddening line" (which is the locus of ob jects having a linear reflectivity spectrum). This confirms that most MBOSSes have globally linear reflectivity spectra in the visible. Nevertheless, some systematic effects are visible:
Plutinos & Cubewanos Distribution: ( 0.70 0.40) -> ( 1.10 0.70) Type: Cont. nPoints: 10000 Plutinos & Cubewanos Distribution: ( 0.70 0.45) -> ( 1.00 0.65) Type: Bimod. nPoints: 10000 Spread: 0.100 0.100 KS: d=0.210 Prob=.545E-01 Spread: 0.100 0.100 KS: d=0.266 Prob=.623E-02

Fig. 7. Examples of color-color diagrams of the TNOs, sup erimp osed to the model distributions used for the KS analysis describ ed in the text. Left column corresp onds to continuous distributions, right to bimodal. Top row is for Plutinos, middle for Cub ewanos, b ottom for b oth together.

­ The diagrams involving the I band indicate that the spectrum of many ob jects becomes flatter toward the near-IR, which is expected if one considers the laboratory spectra. ­ Also, many ob jects present a bend in the B region. The latter is not matching the laboratory spectra for fresh ices. Maybe they correspond to ob jects in an evolved state (e.g. covered with extremely irradiated ices, which can have reflectivity spectra bend in the blue region), as opposed to having recently been re-surfaced.

7.4. Correlations with orbital elements 7.4.1. Semi-major axis a
There is no correlation between the color (and spectral gradient) of the ob jects and their orbit semi-ma jor axis. This stands for the whole MBOSS population as well as for the individual families. Therefore, the traditional increasing reddening of asteroids with a that is observed for Main-Belt asteroids and Tro jans stands for the MBOSSes as a whole (i.e. they are on average considerably redder than ob jects closer to the Sun, cf. Table 4), but not within the MBOSSes themselves.

bimodal distributions, except the 2 diagrams mentioned above, which are not compatible with (simple) bimodal distributions.

7. Discussion and summary 7.1. Dataset
We compiled the colors of Minor Bodies in the Outer Solar System from 40 references, totaling measurements during 486 epochs of 104 ob jects, i.e. 13 SP Comets, 14 Centaurs, 9 Scattered TNOs, 20 Plutinos and 48 Cubewanos. For each ob ject, these measurements have been carefully combined, taking care not to introduce rotational artifacts in the colors, and weighting each measurement with its error bar. The final error bar reflects the combined signal/noise ratio and the dispersion between the measurements. The absolute R magnitudes (M (1, 1)) and the mean reflectivity slope S have been computed for each ob ject, together with the deviation from a linear spectrum. The color-color diagrams are presented. The mean of each color (and its error) is presented for all the MBOSS classes.

7.4.2. Orbit excitation i , e and E
For the Plutinos, Centaurs and Scattered TNOS, the correlation between colors (including spectral gradient) and the other orbital parameters is either nonexistent or very weak: no trend is apparent in the different color vs. parameter plots, and this is confirmed by weak correlation coefficients and statistical tests. One notable exception: the color of the Cubewanos presents a very strong, very significant correlation with the eccentricity, inclination and "excitation" (E obtained


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by combining quadratically i and e): Cubewanos with a small excitation are systematically and significantly redder than those with a higher excitation. This confirms the results presented by Trujillo et al. (2001) and Stern (oral comm. at Meudon 2001 workshop), who obtained similar correlations on smaller samples. In addition, the Cubewanos with large orbital excitation have significantly broader color distributions than the others. In a more general way, the difference is greater between Cubewanos with a small and large orbital excitation than the difference between the different classes of MBOSSes. This suggests either that ­ ob jects with higher excitation suffer a more efficient resurfacing, because of more numerous and stronger collisions than those with smaller excitation, or that ­ ob jects with low excitation constitute a different population, possibly covered with a more primordial surface, or of a different nature.

7.5. Absolute magnitude
The tests involving the absolute magnitude of the object (M (1, 1), neglecting the solar phase correction, which is unknown but expected to be small) deserve special attention. Cubewanos with faint M (1, 1) tend to be redder than the others (this effect is visible through the correlation coefficients and the t-tests). Plutinos present the opposite trend (faint M (1, 1) tend to be bluer), with about the same significance. It is difficult to explain this through a selection effect at discovery. However, the Plutinos extend to fainter M (1, 1) than the Cubewanos (a effect of the latter being on average further away from the Sun, therefore fainter than Plutinos of the same absolute magnitude). At this point, these opposite trends are not explained. The width of the color distributions of the ob jects with faint M (1, 1) is never significantly different than those of the larger ob jects. The models of collisional resurfacing balancing the reddening (Luu & Jewitt 1996a; Jewitt & Luu 2001) predict that the smaller ob jects will have a broader range of colors, which is not observed. Therefore, the current database does not support this model. Jewitt & Luu (2001) discuss also that, for that model, the colors of a given ob ject should vary with same amplitude as the variation of colors between ob jects of the same diameters, which is not the case. However, as this database does not explore the rotational variations, this cannot be further explored.

would indicate that they are on average exposed to a narrower range of resurfacing effects ­ for instance, less collisions because they spend a significant fraction of their time far out of the densely populated regions, or strictly no cometary activity because they are the most distant ob jects from the Sun. This could give constraints on the conditions to which they are exposed. We performed a series of statistical tests on the color distributions (f -test, t-test and KS). These tests indicate that the comets' colors are significantly bluer than those of the other MBOSSes. This result is very strong for the Cubewanos and Plutinos (with a probability that the comet are actually similar to these ob jects of 10-3 on individual color indexes), and weaker for the Centaurs and Scattered TNOs (probability of the order of a %). There is no evidence that the Plutinos, Cubewanos, Centaurs and Scattered TNOs have significantly different color distributions. Non-physical, arbitrary populations (in which the objects are distributed according to their designation) were used to test the statistical methods; they indicate that probabilities larger than 5­10% should not be considered as reliable.

7.7. Bimodality of the color distributions
Visually comparing the color CPFs of the various classes, it appears that the Cubewanos and Centaurs tend to have "bimodal" (2 well separated steps) or "broken" (2 well separated slopes) distributions, while the Plutinos tend to have very continuous distributions (i.e. uniform CPF slope over the whole range). However, statistical comparison of the observations with simple 1D and 2D model distributions indicate that, in no case we have enough data to rule out the validity of simple, continuous distributions to represent the data. This does not mean that the distributions are continuous, but that we have to be extremely careful if saying that they are not. Jewitt & Luu (2001) have performed some statistical tests (bin, dip and interval distribution tests) on a smaller sample; these tests do not provide evidence that the B - V and V - R of their sample are distributed bimodally.

7.8. Prosp ects
We plan to maintain and update the observations database and keep it available on the web (at http://www.sc.eso.org/~ohainaut/MBOSS). We encourage the observers to send us the tables of their publications electronically. We intend to update this paper when the number ob jects in the database will have doubled or when the conclusions will have significantly changed.

7.6. Comparison b etween p opulations
The color distribution of the Scattered TNOs systematically cover a narrower range than those of the other classes; this is not substantiated by the statistical tests, but possibly because the distributions have fairly different shapes. Nevertheless, if confirmed in the future, this

App endix A: Color measurement references
The list of papers that were used for each ob ject to build this database is available in the electronic form at


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http://www.edpsciences.org, and maintained on our web site http://www.sc.eso.org/~ohainaut/MBOSS

App endix B: Statistical tests
In this appendix, we describe in detail the statistical tests used thorough this paper.

raise 10 such flags, we can expect that one of them will be a random effect. The statistic tests are described in more detail, together with their original references and with the algorithms we used in Press et al. (1992).

B.2.1. Student's t test
This test checks whether the means of two distributions are significantly different. The basic implementation of this test implies that the variance of both distributions are equal. For the MBOSSes colors, this cannot be guaranteed (we deal with that question with the next section). We therefore used a modified version of the t test that deals with unequal variances: t= xA - xB , (Var(xA )/NA + Var(xB )/NB )1/2 (B.2)

B.1. Correlation coefficient
Pearson's correlation coefficient r evaluates the association between two continuous variables x and y (such as the orbit semi-ma jor axis a and the V - R color). r is given by r=
i

(xi - x)(yi - y) ¯ ¯ 2 (xi - x) ¯ ¯2 i (yi - y )
i

(B.1)

where x and y are the mean of x and y . r is in the -1, 1 ¯ ¯ range. Large values (positive or negative) indicate a strong correlation between the two variables, while a value close to 0 indicates that they are uncorrelated. Unfortunately, there is no reliable way to quantify the significance of that correlation for small samples (less than 500 elements).

B.2. Comparing two distributions
The tests described in this section aim at comparing two continuous, 1D distributions (such as the V - R colors of two MBOSS families). These three tests estimate the validity of the null hypothesis "the two samples are extracted from the same population." This is performed by computing an estimator (f , t and d resp., defined below), whose direct interest is limited. From the estimator, a much more interesting value is derived: P rob, the probability that the statistical estimator is as large as measured by chance. P rob is the probability to get a statistical estimator as large as or larger than the value measured while the two samples compared being actually random sub-samples of a same distribution. Large values of P rob indicate that it is very probable to get the measured estimator by chance, or in other words, that we have no reason to claim (on statistical bases) that the two samples come from different distributions. Remember, however, that this does not allow us to say that the samples are identical, only that they are not statistically incompatible. On the other hand, small values of P rob indicate that the chances of getting the observed estimator by chance while extracting the two samples from the same distributions are small, or in other words, that the two samples are not statistically compatible. The size of the sub-samples is taken into account in the computation of P rob. While it is definitely safer to work on "large" samples, the advantage of these methods is that they start to give fairly reliable results with fairly small samples; in this study, we set the threshold as 7. The probability at which one can conclude that samples are different depends on the certainty level required. Traditional values are 0.05 and 0.003, corresponding to the usual 2 and 3 levels. For this study, we will start raising the warning flags at P rob 0.1. Of course, if we

where xA and xB are the two color distributions considered, x and Var(x) their means and variances, and N the number of ob jects. The statistic P rob of t is distributed approximately as the original Student's t, and is given by the Student's distribution probability function A, which is related to the incomplete beta function (see Press et al. 1992, for details). Small values of P rob indicate that the distributions are different.

B.2.2. f test
The f -test evaluates whether two distributions have significantly different variances. The statistic f is simply the ratio of the largest variance to the smaller one: f= Var(xA ) · Var(xB ) (B.3)

Very large values of f indicate that the difference is significant. P rob, the statistics of F , is obtained by the f -distribution probability function, which is related to the incomplete beta function.

B.2.3. Kollmogorov-Smirnov test
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 Kolmogorov-Smirnov (KS) test. The distributions are compared through their Cumulative Probability Function (CPF) S (x), which is defined as the fraction of the sample whose value is smaller or equal to x. f starts at 0 and increases till it reaches 1 for the x corresponding to largest element of the distribution. d, the KS test, is the maximum (vertical) distance between the CPFs S1 and S2 of the samples to be compared, i.e. d=
-
max

|S1 (x) - S2 (x)|.

(B.4)

The distribution of d's statistic can be calculated: the probability to get a d larger than the observed one, the


660

O. R. Hainaut and A. C. Delsanti: MBOSS colors, a statistical analysis

Table C.1. Size, mean and square-root of the variance of the sub-samples used for statistical tests (for a description of the subsamples, see Sect. 3), whose results are listed in the right-most columns. The correlation coefficient (last column) is computed on the whole sample. This table refers to the whole MBOSS p opulation (left) and the Cub ewanos only (right).
All MBOSS p opulations Col. Pop. 1 Pop. 2 (small) (large) n1 x / ¯ n2 x± ¯ vs. a: semi-ma jor axis. acut = 42.00AU B - V 39 0.91/0.20 36 0.93/0.17 B - R 39 1.49/0.29 33 1.53/0.26 B - I 34 2.04/0.44 29 2.08/0.35 V - R 42 0.59/0.12 40 0.61/0.14 V - I 34 1.14/0.25 33 1.18/0.24 5 1.62/0.41 V - J 11 2.05/0.49 R - I 35 0.56/0.14 33 0.60/0.14 6 0.69/0.23 0 -- I-J 8 0.43/0.32 4 0.27/0.36 J -H H - K 7 0.02/0.07 3 0.22/0.51 40 22.86/13.09 37 25.44/14.58 S vs. e: eccentricity. ecut = 0.18 B - V 34 0.94/0.19 42 0.90/0.18 B - R 32 1.54/0.26 41 1.48/0.29 B - I 25 2.14/0.34 38 2.01/0.43 V - R 40 0.61/0.13 44 0.59/0.12 V - I 29 1.19/0.23 40 1.12/0.25 6 1.81/0.47 11 1.96/0.50 V -J R - I 28 0.60/0.14 42 0.56/0.15 I-J 1 0.73/-- 6 0.69/0.24 4 0.29/0.36 9 0.42/0.30 J -H 8 0.00/0.06 H - K 3 0.27/0.48 36 24.73/14.03 42 22.71/13.60 S vs. i: inclination. icut = 6.94 B - V 38 0.93/0.20 38 0.91/0.17 B - R 36 1.53/0.27 37 1.48/0.29 B - I 32 2.09/0.38 31 2.03/0.42 V - R 43 0.62/0.12 42 0.58/0.13 V - I 36 1.18/0.23 34 1.12/0.25 6 1.89/0.52 11 1.91/0.49 V -J R - I 36 0.59/0.14 35 0.56/0.14 2 0.43/0.06 5 0.80/0.14 I-J 5 0.47/0.42 8 0.32/0.24 J -H H - K 4 -0.01/0.07 7 0.12/0.31 39 26.02/13.31 40 21.67/13.98 S vs. E = B-V B-R B-I V -R V -I V -J R-I I-J J -H H-K S vs. M ( B-V B-R B-I V -R V -I V -J R-I I-J J -H H-K S 36 34 29 41 33 5 32 1 4 2 38 1, 1) 34 32 31 40 34 8 33 2 7 6 36 Statistics t /P r o b -0.40/0.691 -0.72/0.476 -0.40/0.690 -0.67/0.508 -0.70/0.486 1.77/0.109 -1.29/0.200 --/-- --/-- --/-- -0.81/0.418 0.99/0.324 0.93/0.358 1.26/0.214 0.75/0.455 1.25/0.216 -0.63/0.543 1.35/0.183 --/-- --/-- --/-- 0.64/0.522 0.51/0.614 0.65/0.517 0.57/0.568 1.29/0.201 1.11/0.270 -0.09/0.932 1.04/0.303 --/-- 0.73/0.497 --/-- 1.42/0.161 f /P r o b 1. 1. 1. 1. 1. 1. 1. 33/0.396 30/0.451 62/0.194 40/0.292 03/0.932 43/0.783 02/0.941 --/-- --/-- --/-- 1.24/0.510 -0. 0. -0. 0. -0. -0. 0. r 01 00 06 02 02 05 00 -- -- -- 0.01 Cub ewanos p opulation Col. Pop. 1 Pop. 2 (small) (large) n1 x± ¯ n2 x± ¯ vs. a: semi-ma jor axis. acut = 43.80AU B - V 13 0.97/0.23 14 0.89/0.12 1.52/0.31 11 1.57/0.14 B - R 13 8 2.04/0.43 12 2.16/0.20 B-I 0.59/0.13 16 0.67/0.13 V - R 17 V -I 9 1.09/0.24 15 1.27/0.20 1 1.00 ± -- 3 1.89/0.16 V -J 9 0.59/0.18 15 0.63/0.12 R-I 0 -- 0 -- I-J 1 -0.21 ± -- 2 0.47/0.25 J-H H-K 1 0.81 ± -- 1 -0.10 ± -- 13 21.27 ± 12.29 15 29.95 ± 15.63 S vs. e: eccentricity. ecut = 0.07 B - V 12 1.05/0.16 15 0.83/0.12 1.67/0.16 12 1.41/0.25 B - R 12 8 2.27/0.11 12 2.00/0.36 B-I V - R 17 0.65/0.11 16 0.60/0.16 1.23/0.11 14 1.18/0.28 V - I 10 1 2.07 ± -- 3 1.53/0.46 V -J 0.64/0.12 14 0.60/0.16 R - I 10 I-J 0 -- 0 -- 1 0.30 ± -- 2 0.22/0.61 J-H 1 0.81 ± -- H - K 1 -0.10 ± -- 13 28.05/6.59 15 24.08 ± 19.13 S vs. i: inclination. icut = 3.75 B - V 14 1.01/0.18 13 0.84/0.13 1.68/0.14 11 1.38/0.24 B - R 13 B - I 10 2.30/0.14 10 1.92/0.32 0.69/0.11 14 0.54/0.12 V - R 19 1.30/0.19 10 1.06/0.20 V - I 14 1 2.07 ± -- 3 1.53/0.46 V -J R - I 14 0.66/0.13 10 0.55/0.14 0 -- 0 -- I-J 1 0.30 ± -- 2 0.22/0.61 J-H 1 0.81 ± -- H - K 1 -0.10 ± -- 16 33.31 ± 13.38 12 16.07/9.69 S vs. E = 1. 1. 1. 1. 1. 1. 1. 18/0.606 53/0.214 66/0.177 29/0.423 48/0.265 63/0.676 08/0.834 --/-- --/-- --/-- 1.20/0.580 - - - - - - - 17 21 15 25 21 10 20 -- -- -- -0.19 0. -0. 0. 0. -0. 0. -0. 02 01 03 00 01 45 01 -- -- -- -0.01 0. 0. 0. 0. 0. 0. 0. B-V B-R B-I V -R V -I V -J R-I I-J J-H H-K S vs. M ( B-V B-R B-I V -R V -I V -J R-I I-J J-H H-K S 14 13 10 19 14 1 14 0 1 1 16 1, 1) 14 12 11 18 13 4 12 0 3 2 15 Statistics t /P r o b 1.14/0.270 -0.48/0.639 -0.75/0.471 -1.66/0.108 -1.86 /0.083 --/-- -0.71/0.494 --/-- --/-- --/-- -1.64/0.112 5.72 /0.000 3.94 /0.001 3.75 /0.002 1.38/0.178 1.66/0.112 --/-- 1.38/0.180 --/-- --/-- --/-- 0.75/0.461 2. 3. 4. 3. 4. 41 /0.0 86 /0.0 15 /0.0 12 /0.0 15 /0.0 --/-- 3.00 /0.0 --/-- --/-- --/-- 3.95 /0.0 2 0 0 0 0 f /P r o b r

3.78 /0.024 -0.29 5.23 /0.013 0.02 4.50 /0.027 0.11 1.00/1.000 0.07 1.47/0.504 0.30 --/-- -- 2.43/0.140 0.17 --/-- -- --/-- -- --/-- -- 1.62/0.410 0.17 1.76/0.274 5.01 /0.003 8.19 /0.001 2.69 /0.039 7.32 /0.001 --/-- 2.62 /0.079 --/-- --/-- --/-- 8.43 /0.001 58 57 45 19 15 -- 0.01 -- -- -- -0.26 - - - - - -0.51 -0 . 7 6 -0 . 7 1 -0.58 -0.59 -- -0.41 -- -- -- -0 . 6 1 0. 0. 0. 0. 0.

1. 1. 1. 1. 1. 1. 1.

10/0.772 30/0.450 55/0.258 14/0.678 14/0.718 12/0.954 08/0.851 --/-- --/-- --/-- 1.06/0.843 50/0.225 11/0.760 20/0.618 08/0.800 22/0.570 11/0.830 01/0.982 --/-- 2.96/0.200 --/-- 1.10/0.766

0. 0. 0. 0. 0. 0. -0. - - - - -

08 09 04 12 12 01 15 -- -- -- -0.07 17 22 18 28 22 18 15 -- -0.38 -- -0.22 0. 0. 0. 0. 0. 0. 0.

1. 1. 1. 1. 1. 1. 1.

- - - - - - -

3 1.72/0.283 1 7.09 /0.000 1 7.30 /0.001 4 2.46 /0.052 0 1.89/0.214 --/-- 06 1.39/0.516 --/-- --/-- --/-- 01 1.91/0.284

e2 + i2 : orbit excitation. Ecut = 0.28 0.93/0.19 1.54/0.24 2.10/0.34 0.62/0.11 1.20/0.21 1.88/0.41 0.60/0.14 0.39/-- 0.64/0.40 -0.07/0.04 25.88/13.01 : absolute m 0.92/0.14 1.51/0.26 2.08/0.36 0.61/0.13 1.17/0.22 1.69/0.46 0.58/0.12 0.60/0.18 0.31/0.26 0.13/0.34 24.13/12.37 40 39 34 43 36 12 38 6 9 9 40 agn 34 34 26 37 29 6 30 2 4 3 36 0.90/0.18 1.47/0.30 2.03/0.44 0.57/0.13 1.11/0.26 1.92/0.53 0.56/0.15 0.75/0.18 0.27/0.19 0.11/0.27 21.52/14.25 itude. M (1, 0.93/0.23 1.51/0.30 2.06/0.44 0.60/0.12 1.15/0.26 2.25/0.35 0.56/0.16 0.87/0.25 0.52/0.46 0.03/0.07 23.97/15.15 0.72/0.473 1.17/0.246 0.74/0.462 1.98 /0.051 1.52/0.133 -0.14/0.889 1.16/0.251 --/-- --/-- --/-- 1.41/0.162 1)cut = 6.86 -0.09/0.928 0.01/0.991 0.21/0.835 0.33/0.745 0.41/0.687 -2.60 /0.023 0.53/0.599 --/-- --/-- --/-- 0.05/0.960

e2 + i2 : orbit excitation. Ecut = 0.13 1.01/0.18 13 0.84/0.13 1.68/0.14 11 1.38/0.24 2.30/0.14 10 1.92/0.32 0.69/0.11 14 0.54/0.12 1.30/0.19 10 1.06/0.20 2.07 ± -- 3 1.53/0.46 0.66/0.13 10 0.55/0.14 -- 0 -- 0.30 ± -- 2 0.22/0.61 -0.10 ± -- 1 0.81 ± -- 33.31 ± 13.38 12 16.07/9.69 : absolute magnitude. M (1, 1)cut 0.90/0.16 9 0.97/0.23 1.47/0.28 9 1.64/0.19 2.01/0.36 7 2.23/0.21 0.59/0.14 12 0.69/0.13 1.12/0.20 9 1.30/0.24 1.67/0.47 0 -- 0.55/0.12 9 0.67/0.15 -- 0 -- 0.25/0.43 0 -- 0.35/0.64 0 -- 20.30 ± 11.41 10 34.10 ± 17.43 95 /0.006 24 /0.000 55 /0.000 63 /0.001 99 /0.000 --/-- 3.13 /0.004 --/-- --/-- --/-- 3.95 /0.001 = 6.73 -0.84/0.414 -1.61/0.125 -1.62/0.125 -2.02 /0.054 -1.85 /0.084 --/-- -1.94 /0.072 --/-- --/-- --/-- -2.21 /0.044 2. 4. 4. 3. 3. 1.13/0.808 6.16 /0.001 6.62 /0.002 1.64/0.289 1.80/0.260 --/-- 1.24/0.671 --/-- --/-- --/-- 1.91/0.284 1. 2. 2. 1. 1. 96/0.270 18/0.277 86/0.211 22/0.758 41/0.570 --/-- 1.70/0.408 --/-- --/-- --/-- 2.33/0.150 -0.59 -0 . 7 7 -0 . 7 2 -0.55 -0.54 -- -0.36 -- -- -- -0.58 0.31 0.53 0.66 0.53 0.62 -- 0.62 -- -- -- 0.53

2.45 /0.012 1.28/0.498 1.53/0.267 1.22/0.550 1.38/0.375 1.72/0.569 1.88 /0.085 --/-- --/-- --/-- 1.50/0.235

two data sets being drawn from the same distribution, is given by P rob(d > observed) =Q
KS

and the function Q
in f

KS

is defined as e . (B.7)

Q ( Ne + 0.12 + 0.11/ Ne ) d , (B.5)

KS

() = 2
j =1

(-1)j

-1 -2j 2 2

App endix C: Results of the statistical tests
where Ne is the effective number of data points, Ne = N1 N2 , N1 + N2 (B.6) This appendix presents the detailed results of the statistical tests described in the paper. Tables C.3, C.5 and C.6 concern the comparison between the colors of the various classes of MBOSSes.


O. R. Hainaut and A. C. Delsanti: MBOSS colors, a statistical analysis Table C.1. Continued, for Plutinos only (left) and Centaurs/Scattered TNOs only (right).
Plutinos p opulation Col. n1 Pop. 1 (small) x± ¯ Pop. 2 (large) x± ¯ - - - - - Statistics t /P r o b 0.59/0.572 0.58/0.578 0.59/0.573 0.97/0.351 0.79/0.452 --/-- -0.70/0.507 --/-- --/-- --/-- -0.75/0.475 0. 0. 1. 0. 0. 99/0.337 71/0.486 05/0.312 99/0.336 83/0.422 --/-- 0.70/0.498 --/-- --/-- --/-- 0.68/0.504 -0.25/0.808 -0.55/0.591 0.00/0.996 -0.65/0.521 0.38/0.711 --/-- 0.66/0.521 --/-- --/-- --/-- -0.34/0.739 f /P r o b 1. 1. 1. 1. 2. 31/0.636 53/0.493 88/0.370 42/0.740 45/0.214 --/-- 2.18/0.276 --/-- --/-- --/-- 1.32/0.631 - - - - - 0. 0. 0. 0. 0. r 16 20 10 21 15 -- 0.10 -- -- -- 0.19 19 14 34 09 25 -- -0.21 -- -- -- -0.09 0. 0. 0. 0. 0. 24 32 24 26 15 -- 0.09 -- -- -- 0.26 0. 0. 0. 0. 0. Centaur and Scattered TNO p opulations Col. n1 Pop. 1 (small) x± ¯ Pop. 2 (large) x± ¯ Statistics t /P r o b 0. 0. 0. 0. 0. 00/1.000 03/0.978 46/0.648 23/0.817 70/0.492 --/-- 0.99/0.337 --/-- --/-- --/-- 0.52/0.610 1.09/0.287 0.97/0.344 0.89/0.383 1.10/0.282 0.46/0.647 --/-- 0.01/0.994 --/-- --/-- --/-- -0.83/0.418 - - - - - 0. 0. 0. 0. 0. 20/0.845 31/0.760 56/0.583 36/0.722 27/0.790 --/-- 0.02/0.984 --/-- --/-- --/-- 0.13/0.901 f /P r o b 1. 1. 1. 1. 1. 86/0.343 46/0.564 46/0.560 31/0.677 34/0.672 --/-- 1.49/0.544 --/-- --/-- --/-- 1.87/0.336 1. 1. 1. 1. 1. - - - - - 0. 0. 0. 0. 0.

661

n2

n2

r 06 11 16 15 18 -- -0.16 -- -- -- -0.13

vs. a: semi-ma jor axis. acut = 38.70AU B - V 14 0.87/0.17 6 0.92/0.20 1.44/0.25 6 1.52/0.31 B - R 14 B - I 11 1.93/0.37 6 2.07/0.51 0.57/0.10 6 0.61/0.08 V - R 14 1.08/0.20 6 1.19/0.31 V - I 11 -- 4 2.35/0.21 V -J 0 0.52/0.14 6 0.58/0.20 R - I 11 I-J 0 -- 0 -- 1 0.40 ± -- J - H 1 1.20 ± -- -- 1 -0.04 ± -- H-K 0 14 20.32 ± 11.91 6 25.13 ± 13.68 S vs. e: eccentricity. ecut = 0.23 B-V B-R B-I V -R V -I V -J R-I I-J J -H H-K S B-V B-R B-I V -R V -I V -J R-I I-J J -H H-K S vs. E = B-V B-R B-I V -R V -I V -J R-I I-J J -H H-K S 10 0.92/0.20 10 1.51/0.30 8 2.09/0.48 10 0.60/0.09 8 1.17/0.27 2 2.43/0.00 8 0.57/0.18 0 -- 1 0.40 ± -- 1 -0.04 ± -- 10 23.66 ± 11.96 10 0.88/0.16 10 1.43/0.23 9 1.98/0.35 10 0.57/0.09 9 1.14/0.21 2 2.23/0.29 9 0.57/0.14 0 -- 2 0.80/0.57 1 -0.04 ± -- 10 20.80 ± 10.97 10 0.85/0.15 10 1.42/0.23 9 1.88/0.33 10 0.56/0.10 9 1.07/0.21 2 2.26/0.32 9 0.52/0.15 0 -- 1 1.20 ± -- 0 -- 10 19.86 ± 12.98 10 0.90/0.20 10 1.50/0.29 8 1.98/0.50 10 0.59/0.10 8 1.09/0.28 2 2.46/0.03 8 0.51/0.19 0 -- 0 -- 0 -- 10 22.72 ± 14.05

vs. a: semi-ma jor axis. acut = 24.40AU B - V 11 0.91/0.23 11 0.91/0.17 1.48/0.36 11 1.47/0.30 B - R 11 B - I 10 2.10/0.52 11 2.00/0.43 0.58/0.15 11 0.56/0.13 V - R 11 1.17/0.29 10 1.08/0.25 V - I 10 5 1.90/0.56 2 1.38/0.10 V -J 0.60/0.15 11 0.54/0.12 R - I 10 I-J 5 0.74/0.20 1 0.39 ± -- 5 0.34/0.05 1 0.35 ± -- J -H 0.03/0.08 1 -0.04 ± -- H-K 5 11 24.00 ± 16.93 11 20.72 ± 12.36 S vs. e: eccentricity. ecut = 0.38 B B B V V V R I J H -V -R -I -R -I -J -I -J -H -K S -V -R -I -R -I -J -I -J -H -K S 12 0.87/0.20 12 1.40/0.32 11 1.96/0.47 12 0.54/0.13 11 1.09/0.25 3 1.42/0.29 11 0.55/0.13 3 0.53/0.18 2 0.35/0.09 2 0.08/0.02 12 19.15 ± 13.55 12 0.91/0.19 12 1.49/0.29 10 2.11/0.45 12 0.58/0.11 10 1.13/0.25 2 1.25/0.08 11 0.55/0.15 2 0.43/0.06 1 0.29 ± -- 1 0.06 ± -- 12 21.93 ± 12.32 11 0.95/0.18 11 1.53/0.33 10 2.15/0.46 12 0.60/0.13 11 1.14/0.28 4 2.00/0.54 12 0.55/0.15 3 0.84/0.18 4 0.34/0.03 4 -0.02/0.08 12 23.96 ± 14.97 11 0.90/0.21 11 1.44/0.37 11 2.00/0.49 12 0.56/0.15 12 1.10/0.28 5 1.95/0.48 12 0.55/0.13 4 0.81/0.16 5 0.35/0.04 5 0.00/0.08 12 21.18 ± 16.39

1. 1. 2. 1. 1.

83/0.382 70/0.443 10/0.319 22/0.774 67/0.485 --/-- 1.49/0.587 --/-- --/-- --/-- 1.18/0.812 1. 1. 2. 1. 1. 54/0.532 57/0.515 03/0.342 30/0.701 84/0.411 --/-- 1.83/0.415 --/-- --/-- --/-- 1.64/0.473

23/0.756 0.11 07/0.902 0.07 02/0.983 0.16 05/0.935 0.10 27/0.709 0.00 --/-- -- 1.41/0.593 -0.07 --/-- -- --/-- -- --/-- -- 1.22/0.747 0.07 1. 1. 1. 1. 1. 24/0.729 60/0.449 21/0.785 92/0.293 27/0.736 --/-- 1.20/0.768 --/-- --/-- --/-- 1.77/0.358 08 13 18 16 13 -- -0.06 -- -- -- -0.09 - - - - - 0. 0. 0. 0. 0.

vs. i: inclination. icut = 5.30

vs. i: inclination. icut = 13.20 B B B V V V R I J H

e2 + i2 : orbit excitation. Ecut = 0.27 10 0.88/0.20 10 1.45/0.28 9 1.98/0.43 10 0.59/0.09 9 1.12/0.25 1 2.43 ± -- 9 0.54/0.17 0 -- 2 0.80/0.57 1 -0.04 ± -- 10 21.60 ± 12.11 10 0.89/0.16 10 1.48/0.25 8 1.97/0.42 10 0.57/0.09 8 1.12/0.25 3 2.32/0.25 8 0.54/0.16 0 -- 0 -- 0 -- 10 21.92 ± 13.15 -0.19/0.849 -0.29/0.772 0.07/0.942 0.29/0.773 0.04/0.965 --/-- -0.05/0.962 --/-- --/-- --/-- -0.06/0.955 1. 1. 1. 1. 1. 42/0.606 0.03 28/0.721 0.10 04/0.976 -0.08 09/0.902 0.06 02/0.992 -0.08 --/-- -- 1.16/0.858 -0.08 --/-- -- --/-- -- --/-- -- 1.18/0.811 0.08 36/0.217 64/0.471 81/0.450 20/0.792 03/0.956 --/-- 1.04/0.944 --/-- --/-- --/-- 1.24/0.752 29 30 44 27 41 -- -0.42 -- -- -- -0.30 - - - - - 0. 0. 0. 0. 0.

vs. E = B B B V V V R I J H -V -R -I -R -I -J -I -J -H -K S

e2 + i2 : orbit excitation. Ecut = 0.44 12 0.88/0.19 12 1.42/0.30 11 2.01/0.45 12 0.55/0.12 11 1.12/0.24 3 1.42/0.29 11 0.58/0.12 3 0.53/0.18 2 0.35/0.09 2 0.08/0.02 12 20.44 ± 12.69 11 0.94/0.19 11 1.52/0.35 10 2.09/0.51 12 0.59/0.14 11 1.10/0.30 4 2.00/0.54 12 0.53/0.15 3 0.84/0.18 4 0.34/0.03 4 -0.02/0.08 12 22.67 ± 16.03 -0.72/0.479 -0.71/0.488 -0.40/0.692 -0.80/0.434 0.14/0.891 --/-- 0.83/0.416 --/-- --/-- --/-- -0.38/0.710 = 7.03 0.36/0.726 0.38/0.706 0.81/0.432 0.58/0.569 0.53/0.604 --/-- -0.45/0.658 --/-- --/-- --/-- -0.50/0.624 - - - - - 2. 1. 1. 1. 1. 30/0.256 37/0.671 30/0.719 20/0.790 08/0.939 --/-- 1.01/0.999 --/-- --/-- --/-- 1.10/0.890 -0.03 -0.06 0.06 -0.08 -0.02 -- 0.02 -- -- -- -0.09 1. 1. 1. 1. 1. 02/0.967 0.03 38/0.601 -0.01 29/0.694 0.02 40/0.585 0.00 55/0.504 -0.08 --/-- -- 1.49/0.539 -0.11 --/-- -- --/-- -- --/-- -- 1.60/0.450 0.00

vs. M (1, 1): absolute magnitude. M (1, 1)cut = 7.42 B - V 10 0.93/0.21 10 0.85/0.14 1.01/0.330 1.53/0.29 10 1.40/0.22 1.19/0.252 B - R 10 9 2.14/0.43 8 1.79/0.32 1.94 /0.073 B-I 0.60/0.09 10 0.56/0.09 0.93/0.364 V - R 10 V -I 9 1.21/0.22 8 1.01/0.23 1.82 /0.089 2.43/0.00 2 2.26/0.32 --/-- V -J 2 9 0.60/0.15 8 0.47/0.15 1.83 /0.088 R-I 0 -- 0 -- --/-- I-J 1 1.20 ± -- --/-- J - H 1 0.40 ± -- H - K 1 -0.04 ± -- 0 -- --/-- 10 24.84 ± 11.54 10 18.68 ± 12.86 1.13/0.275 S

2. 1. 1. 1. 1.

vs. M (1, 1): absolute magnitude. M (1, 1)cut B-V 9 0.91/0.15 10 0.94/0.23 1.46/0.30 10 1.51/0.35 B-R 9 9 2.02/0.43 8 2.20/0.49 B-I 0.56/0.13 10 0.59/0.12 V - R 10 V - I 10 1.10/0.26 8 1.17/0.25 3 1.46/0.27 2 2.13/0.68 V -J 0.55/0.14 9 0.58/0.14 R - I 10 2 0.60/0.18 2 0.87/0.25 I-J 3 0.35/0.06 2 0.35/0.04 J -H H-K 3 0.04/0.07 2 0.03/0.09 10 20.87 ± 13.41 10 23.93 ± 14.06 S

Acknow ledgements. We are very grateful to the authors of pap ers containing large tables who send us their measurements electronically, and to M. Vair, who compiled a significant fraction of the first versions of the database from the original pap ers. We also want to thank John Davies for accepting to

undertake the task of reviewing this pap er and for his numerous valuable comments. This research has made use of NASA's Astrophysics Data System Bibliographic Service.


662

O. R. Hainaut and A. C. Delsanti: MBOSS colors, a statistical analysis

Table C.2. f test: broadening of color and gradient distributions as a function of the absolute magnitude M (1, 1).
All MBOSS p opulation Color Cut Blue Pop n1 x ¯ B-V 0.94 34 7.00 1.56 33 7.03 B-R 2.13 28 6.73 B-I 0.60 38 6.92 V -R 1.18 31 6.80 V -I V -J 1.82 7 6.02 0.58 31 6.91 R-I 0.69 2 0.00 I-J 0.32 5 6.39 J -H -0.04 4 6.04 H-K S 23.45 36 6.93 Plutinos p opulation only Color Cut Blue Pop n1 x ¯ B-V 0.88 10 7.66 1.40 10 7.66 B-R 1.78 8 7.70 B-I 0.56 10 7.66 V -R 1.02 8 7.74 V -I V -J 2.43 2 0.00 0.51 8 7.74 R-I -- 0 0.00 I-J 0.40 1 6.71 J -H -0.04 1 6.71 H-K S 19.06 10 7.66 . 1. 1. 1. 1. 1. 1. 1. 0. 1. 1. 1. . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 43 43 43 43 49 00 49 00 00 00 43 n2 10 10 9 10 9 2 9 0 1 0 10 55 74 58 53 54 53 48 00 74 05 57 n2 34 33 29 39 32 7 32 2 6 5 36 Red Pop x ¯ 6.73 6.77 6.77 6.79 6.72 6.86 6.64 0.00 6.57 6.67 6.79 Red Pop x ¯ 6.73 6.73 6.58 6.73 6.54 0.00 6.54 0.00 7.88 0.00 6.73 . 1. 1. 1. 1. 1. 0. 1. 0. 0. 1. 1. . 1. 1. 1. 1. 1. 0. 1. 0. 0. 0. 1. 22 22 19 22 14 00 14 00 00 00 22 57 38 60 42 55 84 59 00 92 58 47 Statistics f pr ob 1.02 0.947 1.58 0.202 1.02 0.957 1.17 0.639 1.01 0.984 3.34 0.168 1.15 0.706 14.61 0.326 3.58 0.195 2.26 0.529 1.13 0.718 Statistics f pr ob 7.94 0.005 7.94 0.005 7.56 0.015 7.94 0.005 5.32 0.040 4.46 0.563 5.32 0.040 -- -- -- -- -- -- 7.94 0.005 Cub ewanos p opulation only Color Cut Blue Pop. Red Pop. n1 x ¯ n2 x ¯ B-V 0.93 11 5.83 1.14 12 6.46 0.60 1.59 10 5.80 1.11 11 6.55 0.68 B-R 2.16 9 5.65 1.09 9 6.52 0.71 B-I 0.63 15 6.04 0.98 15 6.54 0.66 V -R 1.21 11 5.82 1.07 11 6.46 0.72 V -I V -J 1.77 2 0.00 0.00 2 0.00 0.00 0.61 10 5.66 1.08 11 6.58 0.63 R-I -- 0 0.00 0.00 0 0.00 0.00 I-J -0.21 1 4.54 0.00 2 0.00 0.00 J -H -0.10 1 5.07 0.00 1 4.54 0.00 H-K S 26.20 12 5.87 1.01 13 6.49 0.70 Centaurs and Scattered p opulation only Color Cut Blue Pop Red Pop. n1 x ¯ n2 x ¯ B-V 0.91 9 7.73 2.14 10 7.16 2.55 1.48 9 7.73 2.14 10 7.16 2.55 B-R 2.08 8 7.01 2.09 9 7.35 2.63 B-I 0.57 10 7.72 1.94 10 7.10 2.60 V -R 1.12 9 7.45 1.83 9 7.08 2.76 V -I V -J 1.45 2 0.00 0.00 3 7.54 1.28 0.55 9 7.65 1.67 10 6.91 2.65 R-I 0.69 2 0.00 0.00 2 0.00 0.00 I-J 0.32 2 0.00 0.00 3 6.29 0.98 J -H -0.04 2 0.00 0.00 3 7.28 1.46 H-K S 20.29 10 7.72 1.94 10 7.10 2.60 Statistics f pr ob 3.63 0.045 2.64 0.146 2.35 0.248 2.23 0.145 2.21 0.227 807.68 0.045 2.93 0.109 -- -- -- -- -- -- 2.07 0.227 Statistics f pr ob 1.42 0.630 1.42 0.630 1.58 0.559 1.81 0.391 2.28 0.266 2.40 0.831 2.52 0.208 14.61 0.326 3.43 0.410 1.14 0.896 1.81 0.391

Table C.3. T-Test (t and P r ob) for the colors of the various classes: are the mean colors compatible? Note: index, the first line lists the numb er of ob jects from b oth classes considered that are used in the statistics, and gives d and P r ob. The values of d and P r ob that are based on sufficiently large samples to b e reliable (i.e. more in each class) are printed in b oldface. Only the color indexes for which some computations could b e p erformed
Color B-V - B-R - B-I - V -R - V -I - V -J R-I - J -H J -K H-K G rt - Pl-QB1 20 33 1.2 0.238 20 30 1.3 0.196 17 25 1.3 0.209 20 40 1.7 0.097 17 30 1.2 0.225 45 2.2 0.069 17 30 1.5 0.138 24 1.3 0.364 13 ---- 13 ---- 20 35 1.3 0.196 Pl-Cent 20 15 0.6 0.530 20 15 0.5 0.652 17 14 0.9 0.397 20 15 0.2 0.809 17 13 0.5 0.598 46 2.2 0.066 17 14 0.8 0.451 25 1.1 0.455 15 ---- 15 ---- 20 15 0.5 0.596 Pl-Scat 20 8 4 0.708 20 8 8 0.435 17 7 3 0.738 20 9 2 0.251 17 9 9 0.391 41 ---- 17 9 5 0.593 21 ---- 11 ---- 11 ---- 20 9 0 0.334 Pl-Com 20 2 1.9 0.083 20 2 1.5 0.188 17 2 0.8 0.456 20 13 3.4 0.003 17 4 1.7 0.123 40 ---- 17 4 1.5 0.154 20 ---- 10 ---- 10 ---- 20 4 2.0 0.090 QB1-Cent 33 15 0.3 0.799 30 15 0.5 0.638 25 14 0.1 0.934 40 15 1.0 0.348 30 13 0.4 0.668 56 0.0 0.984 30 14 0.7 0.494 45 -0.6 0.573 35 -0.4 0.738 35 0.8 0.526 35 15 0.5 0.631 QB1-Scat 33 8 1.5 0.151 30 8 1.8 0.101 25 7 1.3 0.233 40 9 2.3 0.037 30 9 1.9 0.076 51 ---- 30 9 2.0 0.070 41 ---- 31 ---- 31 ---- 35 9 2.1 0.057 QB1-Com 33 2 3.7 0.009 30 2 3.2 0.033 25 2 2.3 0.149 40 13 4.5 0.000 30 4 3.0 0.026 50 ---- 30 4 3.6 0.009 40 ---- 30 ---- 30 ---- 35 4 3.2 0.019 Cent-Scat 15 8 0.9 0.367 15 8 1.1 0.302 14 7 1.0 0.330 15 9 1.2 0.246 13 9 1.3 0.225 61 ---- 14 9 1.3 0.226 51 ---- 51 ---- 51 ---- 15 9 1.3 0.200

For each color the second line than 7 ob jects are listed.
Scat-Com 82 1.3 0.225 82 0.2 0.844 72 0.3 0.781 9 13 1.8 0.091 94 0.6 0.543 10 ---- 94 0.8 0.447 10 ---- 10 ---- 10 ---- 94 0.8 0.468

- - - - -

0. 0. 0. 1. 0.

-

0.

-

1.

Cent-Com 15 2 2.2 0.049 15 2 1.6 0.143 14 2 1.6 0.167 15 13 3.1 0.005 13 4 2.0 0.069 60 ---- 14 4 2.4 0.034 50 ---- 50 ---- 50 ---- 15 4 2.2 0.054

Table C.4. t-test applied to the non-physical test classes. Similar tables were obtained for the f and KS tests. Color B-V B-R B-I V -R V -I V -J R-I J -H J -K H -K Grt Odd-Even Nr p er sample t P r ob 27/26 0.1 0.948 25/25 -0.3 0.758 21/21 1.0 0.343 30/30 -0.5 0.605 24/23 -0.3 0.750 5/4 0.2 0.848 24/23 -0.1 0.960 3/3 -0.7 0.541 2/2 -0.6 0.646 2/2 -0.8 0.569 28/27 0.5 0.636 99-non99 Nr p er sample t P r ob 9/44 -0.1 0.894 7/43 0.3 0.749 5/37 1.8 0.123 8/52 -0.1 0.948 6/41 1.8 0.108 0/9 ---- 7/40 2.4 0.033 0/6 ---- 0/4 ---- 0/4 ---- 8/47 0.7 0.499


O. R. Hainaut and A. C. Delsanti: MBOSS colors, a statistical analysis Table C.5. F-Test for the colors of the various classes: are the color variances compatible?
Color B-V B-R B-I V -R V -I V -J R-I J -H J -K H-K G rt Pl-QB1 20 33 1.1 0.836 20 30 1.1 0.792 17 25 1.6 0.305 20 40 2.1 0.083 17 30 1.1 0.756 45 5.4 0.198 17 30 1.4 0.451 24 2.5 0.419 13 ---- 13 ---- 20 35 1.3 0.537 Pl-Cent 20 15 1.6 0.360 20 15 1.8 0.237 17 14 1.4 0.515 20 15 2.3 0.098 17 13 1.3 0.634 46 6.7 0.148 17 14 1.4 0.508 25 **** 0.001 15 ---- 15 ---- 20 15 1.5 0.403 Pl-Scat 20 8 9 0.381 20 8 0 0.916 17 7 0 0.920 20 9 6 0.361 17 9 0 0.913 41 ---- 17 9 4 0.681 21 ---- 11 ---- 11 ---- 20 9 1 0.979 Pl-Com 20 2 24.6 0.315 20 2 16.8 0.380 17 2 8.5 0.528 20 13 2.4 0.088 17 4 3.1 0.387 40 ---- 17 4 5.9 0.169 20 ---- 10 ---- 10 ---- 20 4 2.9 0.409 QB1-Cent 33 15 1.4 0.406 30 15 2.0 0.120 25 14 2.2 0.089 40 15 1.1 0.822 30 13 1.4 0.407 56 1.2 0.856 30 14 1.1 0.954 45 48.6 0.003 35 9.5 0.060 35 35.1 0.006 35 15 1.1 0.716 QB1-Scat 33 8 2.1 0.301 30 8 1.1 0.768 25 7 1.6 0.397 40 9 1.3 0.743 30 9 1.2 0.717 51 ---- 30 9 1.0 0.906 41 ---- 31 ---- 31 ---- 35 9 1.4 0.643 QB1-Com 33 2 27.2 0.301 30 2 15.3 0.401 25 2 5.4 0.659 40 13 1.1 0.736 30 4 2.7 0.446 50 ---- 30 4 4.3 0.252 40 ---- 30 ---- 30 ---- 35 4 3.8 0.293 Cent-Scat 15 8 3.0 0.147 15 8 1.8 0.452 14 7 1.4 0.706 15 9 1.4 0.658 13 9 1.2 0.774 61 ---- 14 9 1.1 0.883 51 ---- 51 ---- 51 ---- 15 9 1.6 0.508 Cent-Com 15 2 38.5 0.251 15 2 30.1 0.284 14 2 11.9 0.446 15 13 1.1 0.919 13 4 3.9 0.287 60 ---- 14 4 4.1 0.273 50 ---- 50 ---- 50 ---- 15 4 4.4 0.248 Scat-Com 82 12.8 0.424 82 16.9 0.370 72 8.5 0.514 9 13 1.5 0.606 94 3.1 0.375 10 ---- 94 4.4 0.254 10 ---- 10 ---- 10 ---- 94 2.7 0.439

663

1. 1. 1. 1. 1.

1.

1.

Table C.6. KS Test for the colors of the various classes: are the distribution compatible?
Color U -B U -V U -R U -I B-V B-R B-I V -R V -I V -J R-I J -H J -K H-K G rt Pl-QB1 01 ---- 01 ---- 01 ---- 01 ---- 20 33 0.2 0.788 20 30 0.4 0.081 17 25 0.4 0.048 20 40 0.3 0.148 17 30 0.3 0.222 45 0.8 0.082 17 30 0.3 0.152 24 0.8 0.242 13 0.7 0.641 13 0.7 0.641 20 35 0.3 0.115 Pl-Cent 00 ---- 00 ---- 00 ---- 00 ---- 20 15 0.3 0.508 20 15 0.3 0.508 17 14 0.3 0.584 20 15 0.2 0.763 17 13 0.2 0.975 46 0.7 0.135 17 14 0.2 0.688 25 0.8 0.158 15 0.6 0.724 15 0.8 0.362 20 15 0.2 0.678 Pl-Scat 01 ---- 01 ---- 01 ---- 01 ---- 20 8 0.2 0.895 20 8 0.2 0.895 17 7 0.2 0.989 20 9 0.2 0.793 17 9 0.3 0.692 41 1.0 0.150 17 9 0.2 0.802 21 1.0 0.201 11 1.0 0.289 11 0.1 1.000 20 9 0.2 0.837 Pl-Com 00 ---- 00 ---- 00 ---- 00 ---- 20 2 0.6 0.353 20 2 0.5 0.585 17 2 0.5 0.670 20 13 0.5 0.012 17 4 0.5 0.350 40 ---- 17 4 0.5 0.350 20 ---- 10 ---- 10 ---- 20 4 0.4 0.389 QB1-Cent 10 ---- 10 ---- 10 ---- 10 ---- 33 15 0.2 0.619 30 15 0.3 0.173 25 14 0.3 0.300 40 15 0.3 0.327 30 13 0.3 0.230 56 0.5 0.454 30 14 0.3 0.420 45 0.6 0.357 35 0.7 0.224 35 0.3 0.947 35 15 0.3 0.386 QB1-Scat 11 1.0 0.289 11 1.0 0.289 11 1.0 0.289 11 1.0 0.289 33 8 0.4 0.217 30 8 0.5 0.079 25 7 0.4 0.283 40 9 0.4 0.098 30 9 0.5 0.034 51 0.8 0.362 30 9 0.5 0.034 41 0.8 0.461 31 0.7 0.641 31 0.7 0.641 35 9 0.5 0.016 QB1-Com 10 ---- 10 ---- 10 ---- 10 ---- 33 2 0.7 0.146 30 2 0.8 0.085 25 2 0.8 0.120 40 13 0.6 0.000 30 4 0.7 0.021 50 ---- 30 4 0.7 0.021 40 ---- 30 ---- 30 ---- 35 4 0.7 0.025 Cent-Scat 01 ---- 01 ---- 01 ---- 01 ---- 15 8 0.4 0.269 15 8 0.4 0.291 14 7 0.4 0.271 15 9 0.4 0.389 13 9 0.3 0.539 61 0.7 0.583 14 9 0.4 0.234 51 0.6 0.724 51 0.8 0.362 51 0.8 0.362 15 9 0.3 0.561 Cent-Com 00 ---- 00 ---- 00 ---- 00 ---- 15 2 0.5 0.517 15 2 0.5 0.517 14 2 0.6 0.433 15 13 0.5 0.023 13 4 0.5 0.229 60 ---- 14 4 0.6 0.123 50 ---- 50 ---- 50 ---- 15 4 0.5 0.225 Scat-Com 10 ---- 10 ---- 10 ---- 10 ---- 82 0.6 0.366 82 0.5 0.651 72 0.6 0.494 9 13 0.3 0.539 94 0.2 0.996 10 ---- 94 0.3 0.907 10 ---- 10 ---- 10 ---- 94 0.2 0.996

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