Col-OSSOS: Color and Inclination Are Correlated throughout the Kuiper Belt

Four main tests were used to assess the orbital inclination versus spectral slope distribution of our data set. The one-dimensional two-sample (2-s) Kolmogorov–Smirnov (KS; Chakravarti et al. 1967) and Anderson–Darling (AD; Stephens 1974) tests were used to determine whether the inclination distributions of the two TNO color groups defined in Section 3.1 come from a single parent distribution. These two tests are broadly similar and also complementary. The KS test is more sensitive to the center of the distributions, while the AD test gives more weight to the tails. The AD test is generally considered more powerful than the KS test. These two tests are nonparametric with no reference to a Gaussian. We therefore report 1 minus the p-value of the test (in percentage) as the level of confidence that the two distributions are different, rather than a number of σ deviation from a mean. We also used the student's T-test of means (Snedecor & Cochran 1989) to determine whether the mean inclinations of the gray and red TNOs are equal. Finally, we derived the Spearman coefficient (Zwillinger & Kokoska 2000) of our data set to search for a correlation between orbital inclination and spectral slope. Unlike the other three tests, the Spearman test does not require dividing our data set into two color groups. All of the results discussed in the following sections are summarized in Table 3. p-values were determined by computing each statistic for sets of inclination/spectral slope pairs constructed by independently randomly sampling the observed distribution of each parameter (Efron & Tibshirani 1993). We sampled with replacement so the same value of inclination or spectral slope could be sampled several times. The reported CL then indicates the probability that a random sample produces a larger test statistic than the observed population.

We first consider the full data set as a whole. All statistical tests indicate a strong (CL > 99.7%) discrepancy between the inclination distributions of the gray and red TNOs, as well as between their mean inclinations. The Spearman test indicates that the spectral slope anti-correlates with inclination at the same significance level (Table 3). The color versus inclination trend is particularly obvious when comparing the ratios of gray-to-red objects between low and high inclinations. From Figure 1, it appears that the inclination transition from where gray and red TNOs are roughly equally numbered to where red TNOs are almost nonexistent is located around i = 21°. We therefore chose to compare the ratios of red-to-gray objects above and below this value. Only five TNOs—i.e., 11% of the objects—can be unambiguously classified as red at i > 21° (Figure 1). In contrast, red objects account for 42% of the low-inclination (5° < i < 21°) population. To assess the likelihood of producing such an observation from a random population of objects, we bootstrap with replacement from the observed TNO samples of N_g and N_r objects, where N_g and N_r are the observed number of gray and red TNOs. We find that less than 0.1% of the simulations produce a higher or equal ratio of gray-to-red objects than in the observed sample of high-i (i > 21°) TNOs.

While the above tests confirm a trend in color versus inclination, they cannot determine whether this trend results from the two color classes having different inclination distributions or from a direct correlation between color and inclination. To assess the possibility of a correlation, we consider only objects with 5° < i < 21°, where both color classes are thoroughly sampled by surveys (Figure 1). Using the Spearman test, we find no evidence for a correlation between color and inclination for these objects (Table 3). This indicates that the Spearman test finds a correlation for all (low-i and high-i) TNOs because the number of red TNOs drops off near 21° inclination. It follows that spectral slope and inclination of the dynamically excited TNOs are not directly correlated. Rather, red and gray objects have different inclination distributions. This opens interesting prospects for our understanding of the origin of TNO colors (see further discussion in Section 4).

Here, we test whether the observed inclination distribution discrepancy between gray and red TNOs holds for the individual dynamical subpopulations of objects. We consider three distinct groups of dynamically excited TNOs. The first two groups, the hot classicals and the resonant objects, are each considered separately as they comprise enough objects to allow a statistically robust analysis. The remaining objects, including centaurs, scattering disk, and detached objects, are considered as a single group. We emphasize that considering those objects together does not mean we assume they have a common origin. We adopt this approach because these populations, especially the scattering disk and detached objects, are too sparsely sampled in our data set to be analyzed individually. Our goal here is to check whether the inclination discrepancy between gray and red TNOs holds for the remaining TNOs in our data set. Nevertheless, it is believed that the scattering disk directly feeds the centaur population (Duncan & Levison 1997). Since objects from the Kuiper Belt roughly preserve their inclinations as they become centaurs (Volk & Malhotra 2013), this seems the most sensible class merger. Figure 3 shows the inclination versus color distribution for the individual populations of TNOs.

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Applying previous statistical tests to the individual groups, we find that the discrepancy between the mean inclinations of the gray and red TNOs holds with CL = 96.1% on average for the hot classicals (Table 3). The Spearman correlation between inclination and spectral slope is less significant (CL = 93.2%). The KS and AD test return somewhat weaker results but this slightly depends on the magnitude and inclination cuts used to define our sample (see Appendix B). Our results therefore are in agreement with previous works that reported an inclination versus color correlation for classical TNOs (Tegler & Romanishin 2000; Hainaut & Delsanti 2002; Trujillo & Brown 2002; Doressoundiram et al. 2005; Peixinho et al. 2008). These works however could have been contaminated by the presence in the sample of objects belonging to peculiar compositional groups, including cold classical objects. Here, we show that the observed discrepancy between gray and red hot classicals holds when interlopers are removed from this population.

A similar trend is found in the resonant group and in the group of centaurs/scattering/detached objects at a comparable significance level (Table 3). We also observe that out of the 102 resonant TNOs in our data set, where 49 are gray and 31 are red (the rest being unclassified), the 14 highest-inclination objects all belong to the gray class. When bootstrapping two random samples of 49 and 31 objects from the resonant population, we find that this occurs in less than 0.1% of the simulations, thus strongly supporting the conclusion that the gray and red resonant TNOs have different inclination distributions. Finding that resonant objects follow the same trend as other dynamical populations of TNOs is interesting. It implies that the initial orbital inclinations these objects had before being captured into orbital resonances with Neptune were not completely randomized by the capture process and the dynamical mechanisms, such as Kozai oscillation (e.g., Gomes et al. 2005), that take place in those resonances.

We stress that the group of centaurs/scattering/detached TNOs is strongly biased by the presence of centaurs in the sample. Excluding those objects completely removes the trend of gray and red TNOs having different orbital inclinations in that group. We note however that the scattering disk population does not contain any red objects at i > 21°, similar to what is observed in the resonant population and, to a lesser extent, the hot classical (1 interloper) and the centaur (2 interlopers) populations (Figure 3). Scattering disk objects might therefore follow the same trend as other TNO populations, but the current small number of scattering TNOs with well-measured colors prevents us from drawing any firm conclusion. Detached objects seem to constitute the only population that does not follow the trend, but their sample is also too small for conclusive analysis.

To summarize, we divided our data set into two color classes at 20.6%/(10³ Å) and found a highly significant difference between the orbital inclination distributions of the gray and red TNOs. This discrepancy cannot be detected at the CL > 99.7% significance level in the individual dynamical population of TNOs, but it does show up in each of these populations at lower statistical significance. This indicates that (1) this trend is common to every population of dynamically excited TNOs and (2) that our overall data set is not biased by one particular dynamical population. We emphasize that the results presented here slightly vary depending on the adopted magnitude and inclination cuts used to define the analyzed sample, as well as the spectral slope value of 20.6%/(10³ Å) we adopt to differentiate between gray and red TNOs. We find however that a reasonable modification of those values does not affect the conclusions of our study in any significant way (see Appendix B).

The data set built for this work uses photometric measurements from a large variety of discovery surveys, each with its own detection biases and discovery efficiency. This raises the question as to whether these observing biases could be responsible for the observed inclination discrepancy between gray and red TNOs reported in this work. A discovery bias in favor of red TNOs comes from the combination of their higher optical albedos (Fraser et al. 2014; Lacerda et al. 2014) and redder colors with respect to gray objects. The observed ratio of red-to-gray objects depends on the filter used for the observations. Observations performed in broadband filters at longer wavelengths will discover a higher fraction of red TNOs than observations made at shorter wavelengths. This can introduce a color-inclination correlation if low-latitude discovery surveys (which are biased toward lower inclination TNOs) were performed at longer wavelengths than high-latitude surveys. To test whether this effect can be responsible for the observed distribution, let us consider the worst-case scenario, which would be if off-ecliptic TNO surveys were made in the V-band (550 nm), while on-ecliptic surveys were made in the R-band (660 nm).

The number of objects with albedos between a and a + da, between radii R and R + dR, and heliocentric distances between r and r + dr detected by the survey is

$$\begin{eqnarray}&&n(R,r,a)=A\,f(R)g(r)h(a){da}\,{dr}\,{dR},\end{eqnarray} \tag{ 1 }$$

where f(R) is the TNO size distribution, g(r) is the Kuiper Belt radial distribution, h(a) is the TNO albedo distribution, and A is a constant. Assuming a power-law distribution in R, $f(R)\propto {R}^{-q}$, and recasting the expression for R in terms of apparent magnitude m, it can be shown (Schwamb et al. 2018) that the number of objects with albedo a₀ ≤ a ≤ a₁ and magnitude m₀ ≤ m ≤ m₁ observed within the boundaries r₀ and r₁ of the Kuiper Belt is given by

$$\begin{eqnarray}\begin{array}{rcl}N({m}_{0}\lt m\lt {m}_{1}) & = & C{\int }_{{a}_{o}}^{{a}_{1}}h(a){a}^{\tfrac{5\alpha }{2}}{da}\\ & & \times \,{\int }_{{r}_{o}}^{{r}_{1}}g(r){r}^{-5\alpha }{{\rm{\Delta }}}^{-5\alpha }{dr}{10}^{\alpha m},\end{array}\end{eqnarray} \tag{ 2 }$$

where C is a constant, Δ is the geocentric distance to the object, and $\alpha =(q-1)/5\approx 1.0$ (Fraser et al. 2014) is the power-law slope of the apparent magnitude distribution. Here, we have assumed that the albedo, radial, and size distributions are independent. For simplicity, this expression does not reflect the apparent latitudinal and longitudinal variation of TNO density of the resonant populations. We are only interested in the ratio of gray-to-red objects observed within a given survey. Assuming that within each orbital class, separate color populations share the same spatial distribution, the latitudinal and longitudinal variations will be reflected in both color populations equally. Thus, for our derivation such variations can be ignored. In the case of a population composed of two color groups, where each group has its own albedo distribution, but similar size and radial distributions, and where the intrinsic number of objects in the two groups is given by A₂ = γ A₁, where γ is a constant, the observed ratio of populations 1 and 2 is

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{1}}{{N}_{2}}({m}_{0}\lt m\lt {m}_{1})=\gamma \displaystyle \frac{{\int }_{{a}_{o}}^{{a}_{1}}{h}_{1}(a){a}^{\tfrac{5\alpha }{2}}{da}}{{\int }_{{a}_{o}}^{{a}_{1}}{h}_{2}(a){a}^{\tfrac{5\alpha }{2}}{da}}.\end{eqnarray} \tag{ 3 }$$

Here, we have assumed that the fraction of objects in the two groups is not a function of the discovery survey's observing depth. This is true as long as the size distribution can be modeled as a single power law ($f(R)\propto {R}^{-q}$), i.e., as long as the limiting magnitude of the surveys is not sensitive to the break (or divot) magnitude (see, e.g., Shankman et al. 2013, 2016; Fraser et al. 2014; Lawler et al. 2018b). A fraction of objects in our data set, mainly centaurs and scattering disk objects, are fainter than the magnitude break measured for these objects (H = 7.7 or 8.3; Lawler et al. 2018b). The presence of a break in the size distribution is explored numerically below in Section 3.3.2.

For simplicity, we consider the simple scenario where the two color groups each have a unique albedo, a₁ and a₂. Then, the albedo distribution is given by the Dirac delta function ${h}_{n}(a)=\delta ({a}_{n}-a)$, and Equation (3) becomes

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{1}}{{N}_{2}}({m}_{0}\lt m\lt {m}_{1})=\gamma \,{a}_{1}^{\tfrac{5\alpha }{2}}{a}_{2}^{-\tfrac{5\alpha }{2}}.\end{eqnarray} \tag{ 4 }$$

In the V-band (550 nm), gray and red TNOs have mean albedos of 6% and 12%, respectively (Fraser et al. 2014; Lacerda et al. 2014), implying a red-to-gray ratio of $\tfrac{{N}_{\mathrm{red}}}{{N}_{\mathrm{gray}}}\,=5.66\gamma$. In this work, we derived mean spectral slopes of 11.4%/(10³ Å) and 29.1%/(10³ Å) for the gray and red color classes of TNOs, respectively. This implies albedos of 6.7% and 15.7% in the R-band (660 nm), and $\tfrac{{N}_{\mathrm{red}}}{{N}_{\mathrm{gray}}}=8.41\gamma$. Therefore, in the V spectral band the observed red-to-gray ratio is 67% that in the R spectral band.

In our data set, we measure a red-to-gray ratio of 0.821 for 5° < i < 10° (23 red objects, 28 gray, and 17 unclassified). Assuming the worst-case scenario, where all on-ecliptic discovery surveys were performed in R and all off-ecliptic surveys were performed in V, the expected red-to-gray ratio at high inclination would be 0.550. Yet, the observed ratio for i > 21° is 0.119 (5 red objects, 42 gray, and 6 unclassified). If considering g- (480 nm) and r-band (620 nm) filters instead of V and R, then the observed ratio in g is 56% that in r, and the expected ratio in our data set would be 0.450 at i > 21°. Choosing different inclination cuts for the two subsamples slightly varies these numbers but does not affect the conclusion. Therefore, it appears that even the observationally heavily biased scenario considered here cannot account for the observed relationship between color and inclination.

We further test the effects of discovery biases on the color versus inclination distribution of TNOs using the OSSOS survey simulator (Bannister et al. 2016, 2018; Lawler et al. 2018a). This simulator replicates the detection characteristics (pointings, detection and tracking efficiencies) of a given survey to determine which input model objects would be detected by the survey. We use this simulator to model a heavily biased survey composed of a single on-ecliptic block performed in the r-band and two high-latitude blocks, with mean inclinations of 12° and 22°, performed in g. All of the blocks are equally sensitive to the input objects in our simulation. Considering TNOs' mean (g–r) = 0.78 color, the observational depth was set to m_r = 24.0 for the on-ecliptic block, and m_g = 24.75 for the off-ecliptic ones. We use as input populations the Canadian–France Ecliptic Plane Survey (CFEPS) L7 model of the Kuiper Belt (Petit et al. 2011) and the improved verison of the model of Kaib et al. (2011, hereafter K11) by Shankman et al. (2016) of the scattering disk and centaur populations.

Each input object was cloned several times, allowing its orbital parameters a, e, and i to vary by 10% with respect to their nominal values, while randomizing its position angles (ω, Ω, and M). This cloning ensures that a sufficient number of simulated TNOs are randomly produced. We ran the simulator until a sample of 10,000 objects sampling the full parameter space was detected. Simulations were run independently for the on-ecliptic block and the high-latitude ones to ensure the same number of objects (5000) were detected by the blocks. Optical (g–r) colors were drawn from the double normal distribution measured in Section 3.1 and randomly attributed to the simulated objects. The input color ratio is irrelevant here; we are interested only in the relative color ratios between low-i and high-i objects in the output survey population. In order to determine the detectability of the simulated objects, an absolute magnitude distribution was assumed in the magnitude range of interest (5 < H_r < 11). We used two different distributions: a single-slope distribution with an exponent of α = 1.0 and a broken-slope distribution with slope exponents of α_b = 1.0 and α_f = 0.2 for the bright and faint end of the distribution, respectively (Fraser et al. 2014). The break magnitude of the broken-slope distribution was set to ${H}_{{\rm{r}}}=7.7$ for the gray TNOs and ${H}_{{\rm{r}}}=6.9$ for the red ones in order to account for their respective mean albedos of 6% and 12%.

The population of detected objects in our simulated survey reveals a trend of increasing color with decreasing inclination (Figure 4), in agreement with our survey setup. We first consider the population drawn from the single-slope distribution. The red-to-gray ratio of the objects found in the on-ecliptic block is 54% that in the off-ecliptic blocks for the L7 model and 62% for the K11 model, in good agreement with the predicted value of 56% derived in Section 3.3.1. However, a fraction of the high-i TNOs in our simulation are detected by the low-latitude block. Because of this, the color discrepancy between the low-i and high-i objects in our simulation is weaker than predicted analytically (Section 3.3.1). The red-to-gray ratio of the high-inclination (>21°) objects is 0.689 that of the low-inclination (i < 10°) ones for the L7 model, and 0.854 for the K11 model. This strengthens our proposal that survey biases cannot account for the observed color versus inclination distribution. Finally, we find that the presence of a break in the magnitude distribution increases the red-to-gray ratio of the population detected in the high-latitude blocks relative to the on-ecliptic one; the on-ecliptic block ratio is 63% that in the off-ecliptic blocks for the L7 model, and 69% for the K11 model. This reinforces our conclusion that the observed color versus inclination correlation cannot be a result of the survey detection biases.

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We assess a subset of past surveys to further show that the correlation we report of color versus inclination distribution of the dynamically excited TNO populations is not the result of a discovery bias. Indeed, real surveys are far from being as biased as in the worst-case scenario we explored in Sections 3.3.1 and 3.3.2. We select the surveys with the most discoveries and an obtainable pointing history, including Chiang & Brown (1999), Gladman et al. (1998, 2001), Trujillo et al. (2001), Allen et al. (2002), Bernstein et al. (2004), Petit et al. (2006), Fuentes & Holman (2008), Fuentes et al. (2009), Fraser & Kavelaars (2009), Fraser et al. (2008, 2010), Alexandersen et al. (2016), the Deep Ecliptic Survey (DES; Millis et al. 2002; Elliot et al. 2005), CFEPS (Petit et al. 2011, 2017), and OSSOS (Bannister et al. 2016, 2018). This list is by no means exhaustive, but we ensure it covers well the range of ecliptic latitudes and of filters sampled by past surveys, and it includes the surveys that have reported the most detections of TNOs with H_r > 5.0 (the DES, CFEPS, and OSSOS). We omit the two surveys of Schwamb et al. (2010) and Rabinowitz et al. (2012), which were up to large ecliptic latitudes, in red-band and in broadband red+blue filters, respectively, as we could not retrieve their precise pointing history.

Figure 5 highlights that these surveys did not favor the detection of red TNOs on highly inclined orbits. On the contrary, most of the surveys performed far from the ecliptic plane used red-band filters, whereas the majority of surveys using blue-band and broadband blue+red filters were conducted close to the ecliptic. We should therefore expect objects detected on highly inclined orbits to be preferentially red compared to the low-inclination ones, which is contrary to the correlation we report. We also note that only a small fraction of the known TNOs were detected in the high-latitude fields, in agreement with their narrow orbital inclination distribution. This implies that even if a discovery bias exists in our data set, it probably only has a marginal effect on its color-inclination distribution.

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