OSSOS. VIII. The Transition between Two Size Distribution Slopes in the Scattering Disk

The populations of small bodies in our solar system are incrementally grinding themselves into dust through mutual collisions. On short timescales, collisions are infrequent, though on occasion, the aftermath can be directly observed (e.g., Jewitt et al. 2010). Over the age of the solar system, collisions may be the main force that shaped the observed size distribution of all but the largest trans-Neptunian objects (TNOs; Schlichting et al. 2013), or the size distribution may be a result of formation conditions (Fraser et al. 2014). As dynamical evolution is not size-dependant for these small TNOs, we do not expect the size distribution to be affected by removal of TNOs from the scattering population due to interactions with the giant planets. The size distribution of populations that are shaped by collisions can be described by a power law of the form $\tfrac{{dN}}{{dD}}\propto {D}^{-q}$, where an idealized infinite collisional cascade will produce an exponent of q = 3.5 (Dohnanyi 1969).

In the outer solar system, the luminosity distribution must be used as a proxy for the size distribution, because TNOs are unresolved. Luminosity is measured as an apparent magnitude, which can be directly converted to an absolute magnitude H when combined with a measured distance. The H-magnitude can then be directly mapped to diameter, as long as an albedo is measured (or assumed). A handful of small (H ∼ 9–14) TNOs and Centaurs have had their albedos measured observationally, and they range from 4% to 16% (Duffard et al. 2014). The size distribution can be written in terms of absolute magnitude H as $\tfrac{{dN}}{{dH}}\propto {10}^{\alpha H}$, where the size distribution exponent q is related to the H-magnitude exponent α by q = 5α + 1 (assuming albedo is size-independent; Irwin et al. 1995; Fraser et al. 2008; Petit et al. 2008).

Measuring the size distribution of a small-body population tells us about their composition, collisional processes that shape them, and may also provide information on their formation. Collisional simulations of the asteroid belt (e.g., Bottke et al. 2005; Pan & Schlichting 2012) have found that the sizes of the largest asteroids are set by the initial formation sizes, which in combination with mass depletion of the asteroid belt (caused by Jupiter's migration), sets any structure in the size distribution. The size distribution of the asteroids can be measured to much smaller sizes (larger H-magnitudes) than the TNOs due to the fact that it is much closer, and thus smaller objects will be above survey detection limits. The asteroid size distribution at smaller sizes shows intriguing structure, which collisional simulations have shown to likely be caused by a combination of formation size and the initial number density of the asteroid belt; the transition between primordial and collisionally evolved populations happens at ∼10–100 km (Bottke et al. 2005; Morbidelli et al. 2009a). By measuring the size distribution of the Kuiper Belt across several orders of magnitude in size, as has been done in the asteroid belt, we may gain an additional constraint on the timing and manner of Neptune's migration, which severely depleted the mass of the Kuiper Belt (Malhotra 1995; Gomes et al. 2004; Nesvorný 2015).

The magnitude distribution of the Kuiper Belt has long been modeled as a single slope at large sizes (e.g., Jewitt et al. 1996). Gladman et al. (2001) found that the smallest TNOs had a size distribution inconsistent with a single power law. Later, Bernstein et al. (2004) measured a rollover, proving that a single power law was not adequate to describe the observed Kuiper Belt. Surveys are now reaching deep enough and detecting enough TNOs that additional structure in the size distribution is required to match observations (Fuentes & Holman 2008; Shankman et al. 2013, 2016; Fraser et al. 2014; Alexandersen et al. 2016).

Here, we focus our analysis on the scattering TNOs and Centaurs. Because they come closer to the Sun than most TNOs, we can observe smaller TNOs within this population than any other in the Kuiper Belt. Scattering TNOs and Centaurs are part of the dynamically "hot" population. TNOs in the dynamically hot population have had their orbits excited to higher inclinations and eccentricities by scattering off Neptune or past/current entanglement with mean-motion resonances (Gladman 2005). Previous work has demonstrated that the hot population, due to its different collisional and formation history, has a different size distribution than the dynamically cold population of the main classical Kuiper Belt (Petit et al. 2011; Fraser et al. 2014). We specifically exclude those TNOs that are currently resonant from the analysis presented in this manuscript, as they are likely to have experienced a different pathway to dynamical excitation than the scattering TNOs and Centaurs (i.e., Gladman et al. 2012).

Shankman et al. (2016) used scattering TNOs detected in four well-characterized surveys to measure the scattering TNO H-distribution to great precision. In this work, we provide an update for the measurement of the scattering TNO H-distribution with the inclusion of the full discovery data set of the Outer Solar System Origins Survey (OSSOS; Bannister et al. 2016; the full data set is in Bannister et al. 2018). OSSOS has completed its observing, more than tripling the sample of scattering TNOs and Centaurs since the analysis of Shankman et al. (2016).

The analysis here builds on the work of Shankman et al. (2016), using the same methodologies. We first discuss the OSSOS survey, summarizing the mechanics of the Survey Simulator, which allows us to forward-bias our model to allow statistical comparison with our observational sample of TNOs. In Section 3, we summarize the statistical analysis that we use to find the range of acceptable H-magnitude distributions allowed by our observed sample. Section 4 includes our population measurements, and in Section 5, we discuss how our measurements of the scattering disk fit into the larger context of the solar system.

Because scattering TNOs and Centaurs have high eccentricities, and their pericenter distances can range from nearly Jupiter-crossing to >40 au, the observing biases are extreme and must be accounted for carefully; e.g., small Centaurs and TNOs with closer pericenters are far more likely to be detected in magnitude-limited surveys (as is visible in Figure 1). By using only TNOs detected by well-characterized surveys in this analysis, where the magnitude limits, pointings, and tracking efficiencies are known and published,¹⁰ we are able to forward-bias models of the scattering disk and statistically compare the resulting biased simulated detections with the real TNO discoveries.

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OSSOS is a large program on the Canada–France–Hawaii Telescope over five years to discover TNOs while carefully characterizing tracking fractions, detection efficiencies, and pointing directions, allowing the survey biases to be fully quantified (Bannister et al. 2016). This methodology has been followed for three other large Kuiper Belt surveys: The Canada–France Ecliptic Plane Survey (CFEPS; Petit et al. 2011), the CFEPS high latitude component (HiLat; Petit et al. 2017), and the survey of (Alexandersen et al. 2016, hereafter referred to as MA). Combining these three surveys with OSSOS gives a well-characterized set of surveys (which we refer to throughout this paper as the "OSSOS ensemble"), whose combined detected TNOs provide powerful constraints on the intrinsic TNO orbital distributions and populations when used in combination with the Survey Simulator. This statistical reproduction of the survey biases is discussed extensively in other works, (e.g., Kavelaars et al. 2009; Petit et al. 2011; Bannister et al. 2016; Shankman et al. 2016; Lawler et al. 2018).

Shankman et al. (2016) analyzed a scattering TNO sample of 22 objects from CFEPS, HiLat, MA, and the first two (of eight) observing blocks of OSSOS. OSSOS has since detected dozens of new scattering TNOs and Centaurs, bringing the full sample available for analysis to 68 TNOs (17 Centaurs, 51 scattering). The orbital elements of the full sample analyzed here are shown in Figure 1 and Table 3 in the Appendix, and further detail is available in Table 3 (ensemble catalog) in Bannister et al. (2018).

Here, we use the dynamical classification scheme of Gladman et al. (2008) to determine membership in the scattering and Centaur classes. These two classes are both unstable on timescales much shorter than the age of the solar system. The distinction between them is semimajor axis a relative to Neptune's orbit; Centaurs have smaller a and scattering TNOs have larger (shown by different symbols in Figure 1). Their changes in a over time are usually due to close encounters with one of the giant planets, but can also be due to dynamical diffusion for the more weakly bound TNOs (Bannister et al. 2017): those that have largest pericenter distances q > 37 au tend to also have the largest semimajor axes of the sample (see Figure 1). The Centaurs show similar evolution in semimajor axis and represent the low-a tail of the scattering population (e.g., Gomes et al. 2008), thus it is expected that they will share the same H-distribution.

Previous work has shown that there is a sharp transition in the H-distribution of the TNOs, though the form of the transition is unclear (Shankman et al. 2013, 2016; Fraser et al. 2014; Alexandersen et al. 2016). We parameterize this transition using a bright-end slope α_b, a faint-end slope α_f, a break magnitude H_b, and a differential contrast c. We use the terminology that c = 1 is a knee, c > 1 is a divot. We refer the reader to Figure 9 in Shankman et al. (2016) for a graphical demonstration of the effect of these two different transitions on the cumulative and differential number distributions in H.

The slope at the bright end of the TNO H-distribution α_b, a range of H ≃ 4–7, is well-probed by previous work (e.g., Fraser & Kavelaars 2009; Petit et al. 2011). Our OSSOS ensemble detections range from H_r values of 6 to 14.5 because of the very close pericenter distances of some of these TNOs, and thus this analysis is sensitive to a much fainter H_r range than previous work. We note that several of the scattering TNOs included in this sample were not observed in r-band because some blocks of CFEPS observed only in g. These have had their g-band and H_g magnitudes transposed to r by assuming that g − r = 0.7, which is at the neutral end of the observed color range of dynamically excited TNOs (Tegler et al. 2016). Shankman et al. (2016) used g − r = 0.7, and also demonstrated that using g − r values ranging from 0.5 to 0.9 makes no difference to the statistical analysis performed below (see Figure 8 in Shankman et al. 2016).

In this analysis, as in previous work (Shankman et al. 2013, 2016), we seek to measure the slope of the faint end of the H-distribution α_f, the contrast of the transition c, and the H-magnitude where the break occurs H_b. We test H-distributions from a grid covering α_f from 0.1 to 0.9, and c from 1 to 100, with two different break magnitudes, H_b = 8.3 (preferred break magnitude from Shankman et al. 2016) and H_b = 7.7 (preferred break magnitude from Fraser et al. 2014).

3.1. The Survey Simulator and Statistical Analysis

Our method of forward-biasing a model distribution with different H-distributions is discussed in detail in Shankman et al. (2013) and Shankman et al. (2016). Briefly, we start with a version of the scattering distribution modeled by the emplacement simulation of Kaib et al. (2011), with the dynamically hotter inclination distribution used in Shankman et al. (2016). We then draw orbits from this simulation. Orbits are randomly oriented (random ω and Ω), and objects are placed with a random mean anomaly on these orbits (which sets the distance), and are given an H-magnitude from within a chosen H-distribution, and then an r-magnitude is calculated. The Survey Simulator then determines if that r-magnitude, rate of motion, and on-sky position was detectable in the OSSOS survey ensemble. This process is continued until a large number (hundreds) of simulated detections are created. The cumulative distributions of simulated detections are then statistically compared with the cumulative distributions of the 68 real Centaurs and scattering TNOs in semimajor axis a, inclination i, r-magnitude m_r, pericenter distance q, distance at detection d, and H-magnitude in r-band H_r. These six cumulative distributions are shown in Figure 2 for the real TNOs as well as simulated detections using three different H-distributions.

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The statistical analysis is described in detail in Shankman et al. (2016), and we summarize below. We first calculate the Anderson–Darling (AD) statistic (Anderson & Darling 1954), comparing the observed TNOs and the simulated detections for a given H-distribution. An AD statistic is computed for each parameter. From previous work, we found that the most powerful lever arms for this analysis (because they vary most for different modeled H-distributions) come from using the parameters q, d, and H_r, so we sum the AD statistics calculated for each of these three distributions (following the analysis method of Parker 2015). This summed AD statistic is bootstrapped by selecting at random 68 objects from the distribution of simulated detections, calculating the AD statistic between this random sample and the simulated detections in each parameter, and summing them. This random selection and AD statistic calculation is repeated hundreds of times. The distribution of summed AD statistics for random samples of the simulated distribution is then compared to the summed AD statistic for the real TNOs. If that AD statistic or larger occurs for <5% of the random distributions, we can reject that distribution with >2σ (>95%) confidence. To explain in another way, if <95% of random subsets of the model are farther from the parent model than the observations are, then the model cannot be rejected.

3.1.1. The Scattering Inclination Distribution

Figure 2 shows a good match between the observations and the preferred model for five of the six parameters measured; the inclination distribution has a rather poor fit at high inclinations (this is true for all H-distributions tested). The paucity of high-inclination objects in the model as compared with observations was noted and discussed in Shankman et al. (2016). The difficulty of generating high-inclination objects in emplacement models is a well-noted problem (e.g., Kaib et al. 2011), and suggests that a small fraction of scattering TNOs may require a different emplacement pathway in order to match the real Kuiper Belt. Suggested mechanisms in the literature include diffusion from the Oort Cloud (Kaib et al. 2009; Brasser et al. 2012), interaction with a distant massive planet (Gomes et al. 2015), and interaction with a rouge planet that was later ejected from the solar system (Gladman & Chan 2006). Creating dynamical emplacement models of the Kuiper Belt that obtain a realistic inclination distribution is currently an area of active research.

We perform a simple experiment to make sure that the inclination distribution does not severely affect the three parameters we test (H, d, and q) by doubling and halving all of the inclinations in the model and re-running our statistical test. We find that the bootstrapped AD values only vary by 1%–2% for these two very different inclination distributions, and so we conclude that while the inclination distribution shown in Figure 2 does not provide a great match to observations, the other properties of the model still provide an excellent fit to the real scattering TNOs.

3.2. Preferred H-distribution

Using the 68 detected scattering TNOs and Centaurs from the OSSOS ensemble, we find that the least-rejectable H-distribution is for α_f = 0.5 and c = 3.2, using α_b = 0.9. This H-distribution is shown as a blue solid line in Figure 2, and by a blue star in Figure 3.

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We are unable to statistically reject a knee distribution. A transition to a faint slope α_f = 0.4 at H_b = 7.7 is non-rejectable at 3σ significance in our analysis; this preferred knee H-distribution is shown by a purple dashed–dotted line in Figure 2 and by a purple star in Figure 3. For comparison, the best-fit knee distribution from Fraser et al. (2014) is shown with a green star, including 1σ error bars.

The preferred divot H-distribution from Shankman et al. (2016) remains a viable explanation for the scattering TNO H-distribution (white star in Figure 3), but the analysis here increases the number of rejectable models, more tightly constraining the acceptable parameter space of α_f and c. As in Shankman et al. (2013, 2016), a single power law (c = 1, α_f = α_b = 0.9) is rejectable at >3σ significance (shown with a black star in both plots in Figure 3).

Interestingly, we are not able to rule out either break magnitude H_b we tested. We tested two different values of H_b: 8.3 and 7.7, based on predictions from previous work (Fraser et al. 2014; Shankman et al. 2016). The yellow contours in Figure 3 highlight the H-distributions, which are rejectable by our analysis at the lowest significance (i.e., least-rejectable distributions). Contours of <1σ rejectability occur for both H_b values that we tested, and (α_f = 0.5, c = 3.2) are the least-rejectable H-distributions for both values of H_b.

We use the Survey Simulator to determine the number of scattering TNOs and Centaurs brighter than a given H-magnitude that must be drawn from the Kaib et al. (2011) scattering TNO model to allow 68 detections (Table 1: scattering TNOs), or 17 detections from the a < 30 au subset of the Kaib et al. (2011) scattering TNO model (Table 2: Centaurs). Error bars on these intrinsic populations are calculated by running this experiment many times and finding the populations that bracket 95% of the estimates; the error bars given are thus 95% confidence intervals on the intrinsic population.

Table 1.  Population Estimates for Scattering TNOs

			Hr < 8.66	Hr < 10	Hr < 12
Hb	αf	c	Population	Population	Population	Comment
8.3	0.5	3.2	(0.9 ± 0.2) × 105	(2.9 ± 0.7) × 105	(2.7 ± 0.7) × 106	preferred divot, this work
7.7	0.4	1	(0.8 ± 0.2) × 105	$({3.5}_{-0.6}^{+0.9})\times {10}^{5}$	$({2.4}_{-0.4}^{+0.6})\times {10}^{6}$	preferred knee, this work
8.3	0.5	5.6	$({1.0}_{-0.2}^{+0.3})\times {10}^{5}$	$({2.6}_{-0.5}^{+0.7})\times {10}^{5}$	$({2.1}_{-0.4}^{+0.6})\times {10}^{6}$	preferred, Shankman et al. (2016)
8.3	0.4	1	(0.8 ± 0.2) × 105	(4.0 ± 0.9) × 105	$({2.8}_{-0.7}^{+0.6})\times {10}^{6}$	least-rejectable knee, Hb = 8.3
7.7	0.5	3.2	(0.7 ± 0.2) × 105	$({2.8}_{-0.6}^{+0.7})\times {10}^{5}$	$({2.7}_{-0.6}^{+0.7})\times {10}^{6}$	least-rejectable divot, Hb = 7.7
Previously Published Population Estimates
8.3	0.5	5.6			∼1 × 106	estimate from Shankman et al. (2013)
8.3	0.5	5.6			(2.4–8.3) × 105	estimate from Shankman et al. (2016)
−	0.8	−	$({5}_{-3}^{+5})\times {10}^{3}$		$({4}_{-3}^{+4})\times {10}^{6}$	CFEPS estimate (Petit et al. 2011)

Note. Error bars on population estimates are 95% confidence intervals.

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Table 2.  Population Estimates for Centaurs

			Hr < 8.66	Hr < 10	Hr < 12
Hb	αf	c	Population	Population	Population	Comment
8.3	0.5	3.2	${110}_{-40}^{+60}$	${390}_{-150}^{+200}$	${3500}_{-1400}^{+1800}$	preferred divot, this work
7.7	0.4	1	${130}_{-70}^{+80}$	${550}_{-290}^{+340}$	${3700}_{-2000}^{+2300}$	preferred knee, this work
Previously Published Population Estimates
7.7	0.2	1	≤75,000	$({2.8}_{-2.5}^{+10.0})\times {10}^{4}$		calculated from Uranian co-orbitalsa
8.5	0.5	6	≤75,000	$({2.8}_{-2.5}^{+13.0})\times {10}^{4}$		calculated from Uranian co-orbitalsa
7.7	0.2	1	${2500}_{-2100}^{+11,000}$	${7100}_{-6800}^{+32,000}$		calculated from Neptunian co-orbitalsa
8.5	0.5	6	${2900}_{-2500}^{+11,000}$	${7500}_{-7100}^{+32,000}$		calculated from Neptunian co-orbitalsa

Note. Error bars on population estimates are 95% confidence intervals.

^aCalculated from observations and models of Alexandersen et al. (2013, 2016). Note that populations here are actually for the a < 34 au scattering population, a large fraction of which will be Centaurs; see the text.

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4.1. The Size of the Scattering TNO Population

4.2. The Size of the Centaur Population

Although we are unable to formally reject either a knee or divot distribution, the power of forward-biasing combined with statistical analysis of the full OSSOS data set has vastly reduced the allowed parameter space compared to previous analyses (Shankman et al. 2013, 2016). But even with the earlier much smaller number of detections, this analysis technique is powerful. While the range of parameter space that was non-rejectable in Shankman et al. (2013) was many times larger than in our analysis here, the preferred divot from the analysis in Shankman et al. (2013) still provides a good agreement to the fit obtained here, even though that analysis only included 11 TNOs, while the analysis here contains over six times as many TNOs.

5.1. Knee or Divot?

5.2. Comparison with Other TNO Populations

5.3. Comparison with the Cratering Record

This work is an exploration of the scattering TNO H-distribution with the full OSSOS sample, expanding on the analysis of Shankman et al. (2016) with a threefold larger set of detections (68 rather than 22 TNOs) and including fainter H_r magnitudes than previous work. We have demonstrated that existing models (H-distributions with either a divot or knee transition from bright- to faint-end slopes) provide acceptable matches for the H-distribution observed for scattering TNOs, but we have greatly constrained the allowed parameter space of possible faint slopes α_f and contrasts of the transition. Our preferred H-distribution has a bright-end slope α_b = 0.9, a faint slope α_f = 0.5, and a divot of contrast c = 3.2, though a knee distribution with α_f = 0.4 is also acceptable. The H-magnitude at the break is not important to our fit, and we find equally statistically acceptable H-distributions for H_b = 7.7 or 8.3, both of which were proposed by previous analyses. Large surveys such as Pan-STARRS and LSST will detect many new TNOs, especially at relatively bright H-magnitudes, and that will likely provide more statistical constraint on exactly where the break magnitude is, providing more information on the initial planetesimal formation size and collisional history of the Kuiper Belt.

We find that the shallower slope at faint magnitudes makes populations that are consistent with both the cratering record on Pluto and the population required to be the source of the Jupiter Family Comets.

A full exploration of possible size distributions would be best done in the context of a formation and evolutionary model of the solar system. The current degeneracy across potential break locations and divot or knee distributions may be addressed through additional constraints from formation theories. In order to explore this, one must understand the conditions under which accretion takes place, e.g., born big (Morbidelli et al. 2009a) or pebble accretion (Shannon et al. 2016), and must also understand the dynamical excitation process, e.g., whether Neptune's migration was smooth (Hahn & Malhotra 2005), grainy (Nesvorný & Vokrouhlický 2016), or chaotic (Tsiganis et al. 2005). By using these dynamical constraints, we can understand the process that emplaced the hot TNOs and shut off collisional grinding, leaving the Kuiper Belt with the size distribution we observe today.

The authors acknowledge the sacred nature of Maunakea and appreciate the opportunity to observe from the mountain. CFHT is operated by the National Research Council (NRC) of Canada, the Institute National des Sciences de l'Universe of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii, with OSSOS receiving additional access due to contributions from the Institute of Astronomy and Astrophysics, Academia Sinica, Taiwan. Data are produced and hosted at the Canadian Astronomy Data Centre; processing and analysis were performed using computing and storage capacity provided by the Canadian Advanced Network For Astronomy Research (CANFAR).

S.M.L. gratefully acknowledges support from the NRC-Canada Plaskett Fellowship. M.T.B. appreciates support from UK STFC grant ST/L000709/1. W.F. acknowledges support from Science and Technology Facilities Council grant ST/P0003094/1. This project was funded by the National Science and Engineering Research Council and the National Research Council of Canada.

Facility: CFHT (MegaPrime). -

Software: matplotlib (Hunter 2007), scipy (Jones et al. 2001).

The Appendix comprises Table 3.