OSSOS. XVIII. Constraining Migration Models with the
2:1 Resonance Using the Outer Solar System Origins
Survey

Ying-Tung Chen, Brett Gladman, Kathryn Volk, Ruth Murray-Clay, Matthew

Lehner, J. Kavelaars, Shiang-Yu Wang, Hsing-Wen Lin, Patryk Sofia

Lykawka, Mike Alexandersen, et al.

► To cite this version:

Ying-Tung Chen, Brett Gladman, Kathryn Volk, Ruth Murray-Clay, Matthew Lehner, et al. OS-SOS. XVIII. Constraining Migration Models with the 2:1 Resonance Using the Outer Solar System Origins Survey. The Astronomical Journal, 2019, 158 (5), pp.214. 10.3847/1538-3881/ab480b hal-02415216

HAL Id: hal-02415216

https://hal.science/hal-02415216v1

Submitted on 5 Sep 2025

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# OSSOS. XVIII. Constraining Migration Models with the 2:1 Resonance Using the Outer Solar System Origins Survey

Ying-Tung Chen (陳英同)<fnref n="1" />, Brett Gladman<fnref n="2" />, Kathryn Volk<fnref n="3" />, Ruth Murray-Clay<fnref n="4" />, Matthew J. Lehner<fnref n="1" />,<fnref n="5" />,<fnref n="6" />, J. J. Kavelaars<fnref n="7" />,<fnref n="8" />, Shiang-Yu Wang (王祥宇)<fnref n="1" />, Hsing-Wen Lin (林省文)<fnref n="9" />,<fnref n="10" />, Patryk Sofia Lykawka<fnref n="11" />, Mike Alexandersen<fnref n="1" />, Michele T. Bannister<fnref n="12" />, Samantha M. Lawler<fnref n="7" />, Rebekah I. Dawson<fnref n="13" />, Sarah Greenstreet<fnref n="14" />,<fnref n="15" />, Stephen D. J. Gwyn<fnref n="7" />, and Jean-Marc Petit<fnref n="16" />

## Abstract

Resonant dynamics plays a significant role in the past evolution and current state of our outer solar system. The population ratios and spatial distribution of Neptune's resonant populations are direct clues to understanding the history of our planetary system. The orbital structure of the objects in Neptune's 2:1 mean-motion resonance ("twotinos") has the potential to be a tracer of planetary migration processes. Different migration processes produce distinct architectures, recognizable by well-characterized surveys. However, previous characterized surveys only discovered a few twotinos, making it impossible to model the intrinsic twotino population. With a well-designed cadence and nearly 100% tracking success, the Outer Solar System Origins Survey (OSSOS) discovered 838 trans-Neptunian objects, of which 34 are securely twotinos with well-constrained libration angles and amplitudes. We use the OSSOS twotinos and the survey characterization parameters via the OSSOS survey simulator to inspect the intrinsic population and orbital distributions of twotinos. The estimated twotino population, $4400_{-1100}^{+1500}$ with $H_r < 8.66$ (diameter ~100 km) at 95% confidence, is consistent with the previous low-precision estimate. We also constrain the width of the inclination distribution to a relatively narrow value of $\sigma_i = 6_{-1}^{+1}$ and find that the eccentricity distribution is consistent with a Gaussian centered on $e_c = 0.275$ with a width $e_w = 0.06$. We find a single-slope exponential luminosity function with $\alpha = 0.6$ for the twotinos. Finally, for the first time, we meaningfully constrain the fraction of symmetric twotinos and the ratio of the leading asymmetric islands; both fractions are in the range of 0.2-0.6. These measurements rule out certain theoretical models of Neptune's migration history.

**Key words:** celestial mechanics – Kuiper Belt: general – surveys

## 1. Introduction

The small celestial bodies currently in the trans-Neptunian region are believed to be the remaining planetesimals that formed early in the history of the solar system. Based on these surviving bodies, many theoretical models of the outer solar system's early dynamical history, including the migration of Neptune to its current orbit, have been constructed in attempts to reproduce the current structure and populations of trans-Neptunian objects (TNOs) in the outer solar system (e.g., Malhotra 1995; Levison & Morbidelli 2003; Brasser & Morbidelli 2013; Nesvorný et al. 2016). Different modes for planetary migration can produce different architectures in the present outer solar system. In particular, differences in the premigration structure of the planetesimal disk, the speed and smoothness of Neptune's migration, and the dynamical

excitation of Neptune's orbit during migration can all leave signatures in the current TNO population (e.g., Hahn & Malhotra 2005; Murray-Clay & Chiang 2005; Levison et al. 2008; Nesvorný 2015a, 2015b; Nesvorný et al. 2016).

During planetary migration, two of Neptune's strongest resonances, the 3:2 resonance (whose members are referred to as "plutinos") and the 2:1 resonance (whose members are referred to as "twotinos"), would have captured a large number of planetesimals; these two resonances contain large numbers of observed resonant objects. The 2:1 resonance with Neptune, which is roughly located at semimajor axis $a = 47.8$ au, lies near the outer edge of the main classical belt (Gladman et al. 2008). The spatial and orbital distribution of the current twotino population provides clues for understanding the premigration planetesimal disk and the history of Neptune's migration. In addition to the usual eccentricity and inclination

<footnote n="1">¹Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 10617, Taiwan; ytchen@asiaa.sinica.edu.tw</footnote>

<footnote n="2">²Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada</footnote>

<footnote n="3">³Lunar and Planetary Laboratory, The University of Arizona, 1629 E. University Boulevard, Tucson, AZ 85721, USA</footnote>

<footnote n="4">⁴Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA</footnote>

<footnote n="5">⁵Department of Physics and Astronomy, University of Pennsylvania, 209 S. 33rd Street, Philadelphia, PA 19125, USA</footnote>

<footnote n="6">⁶Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA</footnote>

<footnote n="7">⁷Herzberg Astronomy and Astrophysics Research Centre, National Research Council of Canada, 5071 W. Saanich Road, Victoria, BC V9E 2E7, Canada</footnote>

<footnote n="8">⁸Department of Physics and Astronomy, University of Victoria, Elliott Building, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada</footnote>

<footnote n="9">⁹Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="10">¹⁰Institute of Astronomy, National Central University, 32001, Taiwan</footnote>

<footnote n="11">¹¹School of Interdisciplinary Social and Human Sciences, Kindai University, Japan</footnote>

<footnote n="12">¹²Astrophysics Research Centre, School of Mathematics and Physics, Queen's University Belfast, Belfast BT7 1NN, UK</footnote>

<footnote n="13">¹³Department of Astronomy & Astrophysics, Center for Exoplanets and Habitable Worlds, The Pennsylvania State University, University Park, PA 16802, USA</footnote>

<footnote n="14">¹⁴B612 Asteroid Institute, 20 Sunnyside Avenue, Suite 427, Mill Valley, CA 94941, USA</footnote>

<footnote n="15">¹⁵DIRAC Center, Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195, USA</footnote>

<footnote n="16">¹⁶Institut UTINAM UMR6213, CNRS, Univ. Bourgogne Franche-Comté, OSU Theta F-25000 Besançon, France</footnote>

<footnote>Received 2019 June 1; revised 2019 September 23; accepted 2019 September 24; published 2019 November 1</footnote>
distributions, the 2:1 resonance's multiple libration islands provide an additional constraint on the solar system's dynamical history. Like other *n*:1 mean-motion resonances, the 2:1 resonance has three stable libration centers (see, e.g., Beauge 1994; Malhotra 1996): the "symmetric" libration center, where objects' perihelion location librates around a point centered 180° away from Neptune, and two "asymmetric" libration centers, where objects' perihelion locations librate around points that lead or trail Neptune by ~60°–100°. In simulations of resonant capture during planetary migration, the orbital distribution and relative population ratios of the two asymmetric islands (the leading and trailing islands) within the 2:1 resonance can differ depending on migration parameters (Chiang & Jordan 2002; Murray-Clay & Chiang 2005). In particular, the distribution of the twotino population can provide an important constraint on the speed and total distance of Neptune's migration if this population is sufficiently constrained by outer solar system surveys.

Past observational surveys have suggested that the twotino population does differ between the libration islands, but only with rough estimates. The Deep Ecliptic Survey (DES) was a pioneering project to systematically survey the ecliptic plane in the early 2000s. This project discovered five twotinos (Elliot et al. 2005). Combining the DES discoveries and other known twotinos, the observationally biased results showed that two twotinos lie at longitudes behind that of Neptune, and seven are located ahead of it, but interpretation of the asymmetry is difficult due to incomplete knowledge of survey bias. A rough estimate of the simple debiased trailing-to-leading ratio (3:6) excludes a very rapid migration history (migration timescale $\tau \le 10^6$ yr) for Neptune (Murray-Clay & Chiang 2005). A subsequent characterized<fnref n="17" /> survey for outer solar system objects was the Canada-France Ecliptic Plane Survey (CFEPS; Kavelaars et al. 2009; Petit et al. 2011). Combining the discovery of five twotinos and the survey characterization, CFEPS was only able to make a rough estimate of the twotino population and a simple comparison with DES results (Gladman et al. 2012). It is impractical to estimate the fraction of twotinos in the symmetric island and also the ratio of the leading and trailing islands with such a small number of detections. Hence, wider and deeper surveys that detect more twotinos were necessary for understanding the intrinsic population and structure.

The Outer Solar System Origins Survey (OSSOS) was a Large Program with the 3.6 m Canada-France-Hawaii Telescope, operating 2013–2017, that searched for TNOs in 155 deg² of sky on and near the ecliptic plane (Bannister et al. 2016, 2018). The detailed characterization of OSSOS provides the capability to accurately constrain the intrinsic distributions of specific TNO subpopulations (Lawler et al. 2018a). Volk et al. (2016) used the twotino detections in the first quarter of OSSOS and in CFEPS (total: 4+5) by utilizing the OSSOS survey simulator to place constraints on the intrinsic number of twotinos and their distribution in the three libration islands. However, these constraints were relatively weak due to the small number of observed twotinos. The full sample of 34 OSSOS twotinos provides more precise constraints, which are extremely valuable for testing outer solar system evolution models. These observed twotinos have very

precisely determined orbital parameters, including accurate resonant libration centers and amplitudes, due to the well-designed cadence of the OSSOS observations, and these parameters are key to estimating the resonance's intrinsic population and distribution for comparison to the end states of theoretical models.

In this study, we extend the work of Volk et al. (2016) to the full OSSOS twotino sample, and we build an empirical model of the intrinsic twotino population based on the observed OSSOS twotinos and models of the twotino population from theoretical simulations. This model is fine-tuned to match the orbital distributions of OSSOS twotinos using the survey simulator. We are thus able to use the OSSOS detections to provide the first quantitative constraints on the twotino orbital distribution, including the fraction of the population belonging to each libration island.

Section 2 of this paper describes the observations and characterization of the OSSOS twotino sample. Section 3 describes the basic dynamics of the twotino population and details the empirical model. In Section 4, we examine how well this empirical model matches the OSSOS observations using a statistical analysis of the population and constrain the fraction of the twotino population in each of the libration islands. In Section 5, we discuss the connections between our observations and theoretical models. A summary of this study is provided in Section 6.

## 2. Observations

Bannister et al. (2016, 2018) provided a detailed description of the survey design and implementation for OSSOS. The major characteristics of this survey are (1) a total coverage area of 155 deg² over eight blocks on the sky (each ≈20 deg²) within 10° of the ecliptic plane, (2) limiting magnitudes in the range 24.1–25.2 in the *r* band, (3) precovery and follow-up observations implemented with the same telescope before and after the discovery observations, and (4) precise characterization of the survey's detection efficiency. The 5 yr survey, with its fully characterized detection efficiency and complete tracking observations, obviates ephemeris bias and provides a unique approach to understanding intrinsic TNO populations. In particular, one of the major science goals of OSSOS was to reveal the intrinsic resonant populations in the outer solar system, so the survey block locations were chosen to optimize the detection of a large number of 3:2 and 2:1 objects, i.e., areas leading and trailing Neptune by 30°–150° (see Figure 1). In the full survey, OSSOS discovered 838 TNOs, including 34 twotinos. Each detected TNO was dynamically classified using numerical integrations according to the scheme outlined in Gladman et al. (2008) and Bannister et al. (2018); the best-fit orbit is used for an object's nominal dynamical classification, and the minimum and maximum semimajor axis orbits consistent with the observations are used to determine whether that classification is secure. Each OSSOS twotino was observed 24–47 times over total arc lengths of 1.1–4.1 yr, which typically resulted in well-constrained orbits with only 0.1% uncertainty at the most in semimajor axis; thus, all of the OSSOS twotinos have secure classifications in the 2:1 resonance. The procedure to determine each twotino's libration center and amplitude (with uncertainty) is fully described in Appendix B of Volk et al. (2016). Briefly, these parameters are determined by integrating 250 clones of each twotino that span the uncertainties associated with the best-fit orbit and analyzing

<footnote n="17">¹⁷ A "characterized" survey is one that provides a measured detection efficiency versus magnitude and rate for all of its survey fields, along with a fully tracked sample (or one for which the tracking efficiency can be modeled).</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** The spatial distribution of the OSSOS observational blocks (blue shaded areas) and detected TNOs (blue dots) are shown projected onto the ecliptic plane. The detected twotinos are shown as red dots projected onto the ecliptic plane, and the gray dots are the empirical model of the intrinsic twotino population in this study (see Section 3). The gray shaded areas indicate the location of the Galactic plane along the ecliptic where TNO detection is very difficult due to the high background star density. Neptune's indicated location is at the rough midpoint of the full survey (2014 July 1). Note that the upper right quadrant is (northern) "fall."</figcaption></figure>

their libration characteristics. The orbital elements and resonance parameters of all OSSOS twotinos are listed in Table 1.

### 3. Twotino Model

#### 3.1. Dynamics in the 2:1 Resonance

Occupancy of the 2:1 mean-motion resonance is diagnosed by a libration (oscillation) of the resonant argument,

$$ \phi_{21} = 2\lambda - \lambda_N - \varpi, \quad (1) $$

as time evolves. Here the two mean longitudes ($\lambda$ and $\lambda_N$) are that of the TNO and Neptune, and $\varpi$ is the TNO's longitude of perihelion. We define the amplitude of the libration of this angle to be $\Delta\phi_{21} = (\phi_{21max} - \phi_{21min})/2$. For the 2:1 resonance, there are three possible centers, or "islands" (see Figure 2), about which $\phi_{21}$ librates.

1. Symmetric libration around a central (roughly "average") $\langle\phi_{21}\rangle = 180^{\circ}$, usually with an amplitude $\Delta\phi_{21} = 140^{\circ}-170^{\circ}$. While resonant libration is typically modeled as a sinusoidal process (see, e.g., Murray & Dermott 1999), high-amplitude libration such as that of the

symmetric twotinos is more of a "triangular wave" (an example of this type of libration is shown in Appendix C of Volk et al. 2016). This means that all values of $\phi_{21}$ in the range $180^{\circ} \pm \Delta\phi_{21}$ are equally likely, so twotinos spend equal time at all allowed values of $\phi_{21}$.

2. Leading asymmetric libration around a libration center $\langle\phi_{21}\rangle$ between $\sim 60^{\circ}$ and $130^{\circ}$. The exact libration center depends on the heliocentric orbital eccentricity $e$ (Nesvorný & Roig 2001). The libration amplitude around this central value depends on the location of the libration center and thus also indirectly on $e$.

3. Trailing asymmetric libration around a libration center $\langle\phi_{21}\rangle$ between $\sim 230^{\circ}$ and $300^{\circ}$ but otherwise identical to the leading asymmetric island.

The goal of this study is to model the distributions of these quantities in a way that agrees with the observations by OSSOS. All surveys have detection biases (see Gladman et al. 2012 for a detailed discussion), but in the case of OSSOS, we can analyze those biases and compare them with the measurements of the 2:1 population, showing that OSSOS has strong but well-understood biases in the detection of twotinos (see Figure 2). The comparison with observations requires subjecting a model of these distributions to the
<table><caption>**Table 1**

OSSOS TNOs in Neptune's 2:1 Resonance<sup>a</sup></caption><thead><tr><th colspan="2">Designations</th><th>a<br>(au)</th><th>e</th><th>i<br>(deg)</th><th>d<sup>b</sup><br>(au)</th><th>⟨φ<sub>21</sub>⟩<br>(deg)</th><th>A<sub>φ21</sub><br>(deg)</th><th>Mag<br>(r)</th><th>H<br>(r)</th><th>Libration Island</th></tr></thead><tbody><tr><td>OSSOS</td><td>MPC</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>o5c025</td><td>2015 VA<sub>166</sub></td><td>47.761</td><td>0.41828</td><td>13.153</td><td>35.2</td><td>65.78</td><td>8<sup>+3</sup><sub>-3</sub></td><td>24.30</td><td>8.8</td><td>Leading</td></tr><tr><td>o5c016</td><td>2015 VZ<sub>165</sub></td><td>47.689</td><td>0.36811</td><td>4.115</td><td>32.3</td><td>67.48</td><td>8<sup>+2</sup><sub>-2</sub></td><td>22.97</td><td>7.9</td><td>Leading</td></tr><tr><td>o5d008</td><td>2015 VD<sub>166</sub></td><td>47.671</td><td>0.35856</td><td>1.911</td><td>32.2</td><td>68.29</td><td>11.0<sup>+0.9</sup><sub>-0.7</sub></td><td>24.37</td><td>9.3</td><td>Leading</td></tr><tr><td>o5d014</td><td>2015 VH<sub>166</sub></td><td>47.750</td><td>0.31534</td><td>3.390</td><td>33.6</td><td>70.39</td><td>3.9<sup>+0.7</sup><sub>-0.8</sub></td><td>24.09</td><td>8.8</td><td>Leading</td></tr><tr><td>o5d026</td><td>2015 VF<sub>166</sub></td><td>47.701</td><td>0.31562</td><td>1.569</td><td>36.5</td><td>70.89</td><td>8<sup>+2</sup><sub>-3</sub></td><td>24.39</td><td>8.8</td><td>Leading</td></tr><tr><td>o5d021</td><td>2015 VD<sub>157</sub></td><td>47.702</td><td>0.31024</td><td>0.892</td><td>34.7</td><td>71.15</td><td>7.6<sup>+1.4</sup><sub>-1.6</sub></td><td>24.34</td><td>8.9</td><td>Leading</td></tr><tr><td>o5c031</td><td>2015 VY<sub>165</sub></td><td>47.733</td><td>0.30280</td><td>1.288</td><td>36.7</td><td>71.28</td><td>5.5<sup>+0.2</sup><sub>-0.1</sub></td><td>23.20</td><td>7.5</td><td>Leading</td></tr><tr><td>o5d010</td><td>2015 VL<sub>166</sub></td><td>47.796</td><td>0.32197</td><td>5.457</td><td>33.1</td><td>71.29</td><td>12.1<sup>+0.5</sup><sub>-1.0</sub></td><td>24.93</td><td>9.7</td><td>Leading</td></tr><tr><td>o5c014</td><td>2015 VB<sub>166</sub></td><td>47.647</td><td>0.32663</td><td>1.045</td><td>32.2</td><td>72.62</td><td>19.5<sup>+0.6</sup><sub>-2.2</sub></td><td>23.61</td><td>8.5</td><td>Leading</td></tr><tr><td>o3l80</td><td>2013 SA<sub>101</sub></td><td>47.752</td><td>0.27033</td><td>3.689</td><td>50.7</td><td>77.32</td><td>20.3<sup>+1.4</sup><sub>-0.3</sub></td><td>23.82</td><td>6.8</td><td>Leading</td></tr><tr><td>o4h15</td><td>2014 UH<sub>228</sub></td><td>47.627</td><td>0.32585</td><td>3.110</td><td>32.7</td><td>77.34</td><td>30.3<sup>+1.6</sup><sub>-0.6</sub></td><td>23.94</td><td>8.8</td><td>Leading</td></tr><tr><td>o5d019</td><td>2015 VJ<sub>166</sub></td><td>47.595</td><td>0.29551</td><td>3.653</td><td>34.2</td><td>78.17</td><td>28.6<sup>+0.7</sup><sub>-0.9</sub></td><td>24.40</td><td>9.1</td><td>Leading</td></tr><tr><td>o5d039</td><td>2015 VG<sub>166</sub></td><td>47.805</td><td>0.20982</td><td>17.175</td><td>38.5</td><td>78.27</td><td>18.6<sup>+1.0</sup><sub>-0.8</sub></td><td>24.86</td><td>9.0</td><td>Leading</td></tr><tr><td>o3l34</td><td>2013 SQ<sub>100</sub></td><td>47.847</td><td>0.25342</td><td>12.957</td><td>42.3</td><td>81.07</td><td>30.5<sup>+1.6</sup><sub>-2.1</sub></td><td>23.64</td><td>7.4</td><td>Leading</td></tr><tr><td>o5d035</td><td>2015 VE<sub>166</sub></td><td>47.836</td><td>0.21248</td><td>4.344</td><td>38.3</td><td>85.83</td><td>27.6<sup>+1.8</sup><sub>-1.8</sub></td><td>24.96</td><td>9.1</td><td>Leading</td></tr><tr><td>o4h43</td><td>2014 US<sub>228</sub></td><td>47.941</td><td>0.32533</td><td>7.165</td><td>44.3</td><td>90.8</td><td>47.6<sup>+2.2</sup><sub>-6.0</sub></td><td>24.07</td><td>7.6</td><td>Leading</td></tr><tr><td>o3l08</td><td>2013 UH<sub>17</sub></td><td>47.694</td><td>0.30232</td><td>9.332</td><td>34.1</td><td>91.43</td><td>46.6<sup>+1.4</sup><sub>-3.6</sub></td><td>24.35</td><td>8.9</td><td>Leading<sup>c</sup></td></tr><tr><td>o3o33</td><td>2013 JJ<sub>64</sub></td><td>47.768</td><td>0.08252</td><td>7.650</td><td>46.5</td><td>175.36</td><td>113.2<sup>+2.2</sup><sub>-18.3</sub></td><td>23.99</td><td>7.3</td><td>Symmetric<sup>d</sup></td></tr><tr><td>o5m57</td><td>2015 KX<sub>173</sub></td><td>47.911</td><td>0.23826</td><td>15.529</td><td>41.6</td><td>179.72</td><td>148.9<sup>+0.9</sup><sub>-1.5</sub></td><td>24.58</td><td>8.4</td><td>Symmetric</td></tr><tr><td>o3e55</td><td>2013 GX<sub>136</sub></td><td>48.003</td><td>0.25156</td><td>1.100</td><td>37.0</td><td>179.87</td><td>155.0<sup>+0.4</sup><sub>-0.6</sub></td><td>23.41</td><td>7.7</td><td>Symmetric</td></tr><tr><td>o5d055</td><td>2015 VK<sub>166</sub></td><td>48.039</td><td>0.20252</td><td>2.323</td><td>39.9</td><td>179.87</td><td>160.2<sup>+1.1</sup><sub>-1.1</sub></td><td>23.20</td><td>7.2</td><td>Symmetric</td></tr><tr><td>o3l17</td><td>2013 UZ<sub>17</sub></td><td>47.532</td><td>0.17602</td><td>7.179</td><td>39.6</td><td>179.88</td><td>152.6<sup>+0.6</sup><sub>-1.5</sub></td><td>23.98</td><td>7.9</td><td>Symmetric</td></tr><tr><td>o5m60</td><td>2015 KL<sub>167</sub></td><td>47.440</td><td>0.25558</td><td>10.304</td><td>42.1</td><td>179.91</td><td>153.5<sup>+2.0</sup><sub>-2.6</sub></td><td>24.47</td><td>8.2</td><td>Symmetric</td></tr><tr><td>o5d027</td><td>2015 VC<sub>166</sub></td><td>48.105</td><td>0.24046</td><td>2.917</td><td>37.4</td><td>179.92</td><td>164.4<sup>+11.1</sup><sub>-1.7</sub></td><td>24.65</td><td>8.9</td><td>Symmetric</td></tr><tr><td>o5t47</td><td>2015 RX<sub>277</sub></td><td>47.642</td><td>0.20629</td><td>0.073</td><td>47.8</td><td>179.93</td><td>154.6<sup>+1.4</sup><sub>-0.6</sub></td><td>24.37</td><td>7.5</td><td>Symmetric</td></tr><tr><td>o5s27</td><td>2015 RY<sub>277</sub></td><td>47.457</td><td>0.27731</td><td>3.457</td><td>38.8</td><td>180.08</td><td>153.5<sup>+3.3</sup><sub>-0.4</sub></td><td>23.79</td><td>7.8</td><td>Symmetric</td></tr><tr><td>o5m49</td><td>2015 KN<sub>167</sub></td><td>47.896</td><td>0.21359</td><td>6.029</td><td>40.9</td><td>257.05</td><td>50.7<sup>+6.4</sup><sub>-5.1</sub></td><td>24.34</td><td>8.2</td><td>Trailing</td></tr><tr><td>o5p121</td><td>2015 GY<sub>51</sub></td><td>47.549</td><td>0.30996</td><td>8.955</td><td>44.9</td><td>263.15</td><td>53.8<sup>+46.0</sup><sub>-1.4</sub></td><td>23.83</td><td>7.3</td><td>Trailing<sup>e</sup></td></tr><tr><td>o4h54</td><td>2014 UY<sub>229</sub></td><td>47.629</td><td>0.21375</td><td>3.403</td><td>56.3</td><td>275.89</td><td>27.4<sup>+1.1</sup><sub>-1.0</sub></td><td>24.63</td><td>7.1</td><td>Trailing</td></tr><tr><td>o4h53</td><td>2014 UW<sub>229</sub></td><td>47.725</td><td>0.17033</td><td>14.607</td><td>53.3</td><td>275.95</td><td>19.0<sup>+0.5</sup><sub>-0.4</sub></td><td>24.31</td><td>7.0</td><td>Trailing</td></tr><tr><td>o3e05</td><td>2013 GW<sub>136</sub></td><td>47.744</td><td>0.34410</td><td>6.660</td><td>33.0</td><td>277.25</td><td>39.1<sup>+0.9</sup><sub>-0.3</sub></td><td>22.69</td><td>7.4</td><td>Trailing</td></tr><tr><td>o5m26</td><td>2015 KM<sub>167</sub></td><td>47.763</td><td>0.33369</td><td>8.490</td><td>34.5</td><td>283.19</td><td>30.0<sup>+1.4</sup><sub>-4.3</sub></td><td>24.44</td><td>9.0</td><td>Trailing</td></tr><tr><td>o5p141</td><td>2015 GZ<sub>54</sub></td><td>47.756</td><td>0.27653</td><td>1.957</td><td>48.3</td><td>283.6</td><td>19.1<sup>+1.2</sup><sub>-1.4</sub></td><td>24.57</td><td>7.7</td><td>Trailing</td></tr><tr><td>o3o18</td><td>2013 JE<sub>64</sub></td><td>47.789</td><td>0.28524</td><td>8.335</td><td>36.1</td><td>284.12</td><td>22.5<sup>+0.4</sup><sub>-0.5</sub></td><td>23.56</td><td>7.9</td><td>Trailing</td></tr></tbody></table>

**Notes.**

<sup>a</sup> All digits given are significant, although *d* and *H* are more accurate than listed here.

<sup>b</sup> Object distance at discovery.

<sup>c</sup> Island switching after 5 Myr.

<sup>d</sup> Island switching.

<sup>e</sup> A few symmetric clones exist.

detection biases via a simulation of which types of objects are most likely to be detected by OSSOS. So comparing a model of the intrinsic twotino distribution with observations requires exposing the model to the same biases as the observations. We employ the OSSOS survey simulator (version 11) to simulate observational biases so that different models can be tested against the real OSSOS detections. This simulator uses a synthetic model to generate objects in our model twotino population and then determines whether the characterized survey could have detected each object based on detection efficiency and other parameters. See Petit et al. (2011) and Lawler et al. (2018a) for an introduction and discussion of this kind of survey simulator.

Part of the reason this approach is so important is that both the location of the blocks relative to Neptune and their relative depths have a very strong effect on the orbital elements of the detections. This produces very nonuniform sensitivity to the current resonant angle and the body's long-term libration amplitude, an effect that is quantified below. For the present moment, we just point out that because $\lambda = \bar{\omega} + M$, where *M* is the mean longitude,

$$ \phi_{21} = 2M + \bar{\omega} - \lambda_N. \qquad (2) $$

Because there is a strong bias toward detection near perihelion where $M \approx 0$, most of the time, the resonant particle's resonant amplitude at detection is roughly
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Illustration of the 2:1 resonance's three libration islands using numerical simulations of three real OSSOS twotinos in a reference frame that corotates with Neptune (the single light blue dot). The red (o3e55/2013 GX<sub>136</sub>), yellow (o4h43/2014 US<sub>228</sub>), and blue (o3e05/2013 GW<sub>136</sub>) paths indicate libration in the symmetric, leading, and trailing islands, respectively. The step size between dots is about 1 yr, and the total time span is about 29,000 yr. The observational bias for detecting twotinos in the different libration islands as a function of longitude relative to Neptune is evident in this figure, as the objects are easier to detect near perihelion.</figcaption></figure>

$\phi_{21} \approx \varpi - \lambda_N$, meaning that particles are usually detected with resonant arguments whose value is just the longitude difference ahead of Neptune of the observing location. A characterized survey provides observing locations, depths, and a list of detections, but, nearly as importantly, the nondetection (or rare detection) of resonant objects in particular directions also provides valuable constraints on the intrinsic distribution of the resonance angles for the population.

## 3.2. Twotino Distribution in Resonant Phase Space

The construction of our synthetic model is based in part on the real detections and in part on the parameter space found to be occupied in previous studies of the dynamics of the 2:1 resonance. In modeling the CFEPS (Gladman et al. 2012) and the first quarter of the OSSOS twotino sample (Volk et al. 2016), an empirical model was adopted that uniformly filled the stable phase space in eccentricity and libration amplitude identified in models of the current 2:1 resonance (Nesvorný & Roig 2001; Tiscareno & Malhotra 2009); this model was sufficient to describe the initially small sample of OSSOS twotinos. With the full sample of 34 twotinos, this simplified, uniform model of the three libration islands can be statistically

rejected. To construct a more detailed model of the twotino population, we look to other models in the literature. Chiang & Jordan (2002) explored the distribution of libration centers $\langle\phi_{21}\rangle$ and amplitudes $\Delta\phi_{21}$ at the end of planetary migration simulations; these distributions are shown in Figure 7 of Chiang & Jordan (2002) and guide the modeling described in the following sections. In a similar, independent theoretical study that was more focused on the resonant trapping efficiency during migration as a function of initial inclination, Li et al. (2014) recovered similar distributions, although they showed that the regions of the $\langle\langle\phi_{21}\rangle, \Delta\phi_{21}\rangle$ phase space inhabited today should depend on the inclinations of the initial TNOs before resonant trapping. Restricting the simulations of Li et al. (2014) to near-planar TNO initial conditions results in a very similar expected phase space distribution to that of Chiang & Jordan (2002). In addition, Malhotra et al. (2018) provided the boundaries of twotino stability, which could be a guide to exploring $(a, e)$ phase space. Here we detail the distributions of intrinsic orbital parameters in our model, as guided by these theoretical models. Our chosen range of parameters and the distribution of the twotinos inside those ranges are statistically not rejectable by the real observed twotino population when biased by the survey simulator (see Figures 3 and 4).
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Distribution of libration amplitude $\Delta\phi_{21}$ (left panel) and eccentricity (right panel) vs. current resonant argument $\phi_{21}$. The yellow dots indicate the OSSOS twotino detections (with libration amplitude uncertainties indicated by vertical lines in the left panel). The gray, red, and blue dots show the model of the intrinsic twotino population for the symmetric, leading, and trailing islands, respectively. The object at $\phi_{21} \sim 206^\circ$, $\Delta\phi_{21} \sim 113^\circ$, and $e \simeq 0.08$ is o3o33, which is currently transitioning from the symmetric island to the asymmetric island. The object at $\phi_{21} \sim 249^\circ$, $\Delta\phi_{21} \sim 54^\circ$, and $e \simeq 0.31$ is o5p121, which has larger uncertainties due to the existence of a few symmetric clones.</figcaption></figure>

### 3.2.1. Symmetric Twotino Model

The symmetric twotino population is relatively simple to model. All such objects have a libration center ($\phi_{21}$) = 180°, and we assign a libration amplitude $\Delta\phi_{21}$ randomly from a uniform distribution over the range 140°–167°; the amplitude range is based on the range showing 4 Gyr orbital stability in Li et al. (2014). Because we impose a sawtooth wave time evolution for $\phi_{21}$ (see Section 3.1, based on this being a better representation of the forward time evolution of detected OSSOS symmetric TNOs than a sinusoidal time history), this generates the uniform distribution in $\phi_{21}$ shown in the left panel of Figure 3. The orbital eccentricity $e$ for symmetric librators is modeled as uniform from $e = 0.07$ to 0.3. These choices, when biased by the survey simulator, yield an acceptable match to the symmetric twotinos detected by OSSOS (Figure 4).

We note that one of the nine symmetric librators detected by OSSOS is not described well by this model, because it has a rather low libration amplitude (o3o33/2013 JJ64 with $\Delta\phi_{21} \approx 113^\circ$). In the known phase space structure of the resonance, such a small libration amplitude can only exist if the orbital eccentricity is small, and indeed, with $e = 0.08$, this is the lowest-eccentricity object in the sample. It is also the case that the libration behavior of o3o33 is not strictly symmetric on timescales longer than a few libration cycles. In the 10 Myr integrations performed for the dynamical classification, clones of o3o33 switch between the small-amplitude symmetric libration described above and even smaller-amplitude

asymmetric libration around both the leading and trailing islands. Its behavior on timescales longer than ~1 Myr is very chaotic, which is expected for very low eccentricity objects in the twotino population (see, e.g., Nesvorný & Roig 2001). Given that only one such object was found by OSSOS, we have elected not to do the complicated modeling of this particularly chaotic portion of the symmetric resonant phase space. Thus, all of our symmetric population estimates concern the $0.07 < e < 0.3$ and $140^\circ < \Delta\phi_{21} < 167^\circ$ portion of the phase space that essentially all other known symmetric twotinos inhabit, including the eight remaining OSSOS detections (Figure 3, left panel).

### 3.2.2. Asymmetric Twotino Population Model

Modeling the twotinos in the asymmetric islands is much more involved. The structure of the resonance's asymmetric islands is not simple to represent in an analytic fashion, so we instead base our model on empirical relationships in the literature derived from numerical studies of these islands. Note that we follow the same procedure for both asymmetric islands, except for the determination of the island at the last step. Our algorithm for generating an asymmetric twotino in our model is as follows.

1. Choose an eccentricity.

The $e$ distribution turns out to be satisfactorily modeled as a Gaussian with a mean ($e_c$) at $e = 0.275$ and
<figure><img src="image_6.png" /><figcaption>Figure 4. Same as Figure 3, but small dots now represent the twotinos from the model that are recovered (i.e., “observed”) after running them through the OSSOS survey simulator. The gray (symmetric) simulated detections show the survey’s nonuniform sensitivity to twotinos with different values of $\phi_{21}$. The dense cluster of leading (red) dots near $\phi_{21} = 70^{\circ}$ is mostly made up of model twotinos detected in the C/D observing block; this block had the greatest sensitivity to fainter objects and was located at the nearly optimal longitude for detecting leading twotinos.</figcaption></figure>

a width ($e_w$) of 0.06. We truncate the outlying tails of this Gaussian distribution to force values of $e$ to be in the range 0–0.53; the upper limit represents the eccentricity where an object’s perihelion distance allows strong interactions with Uranus and would thus be unstable. This intrinsic $e$ distribution can be seen in the right panel of Figure 3, with the expected tendency to detect the higher-$e$ members evident in Figure 4’s right panel.

2. Choose the libration center $\langle\phi_{21}\rangle$.

To get an acceptable match, we converged on the following empirical algorithm. The libration center $\langle\phi_{21}\rangle$ is highly correlated with $e$ (see Figure 10(c) of Chiang & Jordan 2002). To generate values of $\langle\phi_{21}\rangle$ from their model distribution, we use the following polynomial functions to fit Figure 10(c) of Chiang & Jordan (2002). We first calculate a minimum value from the polynomial function:

$$ \langle\phi_{21}\rangle_{\text{low}} = 133.2 - 315.3e + 377.1e^2 - 0.4916/e. \quad (3) $$

To account for the fact that there is some variation in $\langle\phi_{21}\rangle$ even for a single value of $e$, we then pick a random number $f$ on (0, 1) and weight the libration center toward the low-amplitude edge via

$$ \langle\phi_{21}\rangle = \langle\phi_{21}\rangle_{\text{low}} + 20 \times \left[ 1 - \sin\left(\frac{f\pi}{2}\right) \right]. \quad (4) $$

This tendency for asymmetric twotinos to be concentrated toward the lowest libration center allowed for a given $e$

was required to get a satisfactory match and is thus an aspect of the real distribution.

3. Choose the libration amplitude $\Delta\phi_{21}$.

We bound the lower and upper limits of the libration amplitude distribution as

$$ \Delta\phi_{21\text{min}} = 79.03 \times \exp\left[-\frac{(\langle\phi_{21}\rangle - 121.3)^2}{2 \times 15.51^2}\right], \quad (5) $$

$$ \Delta\phi_{21\text{max}} = 9.10\langle\phi_{21}\rangle - 0.04425\langle\phi_{21}\rangle^2 - 0.0884/\langle\phi_{21}\rangle - 404 \quad (6) $$

based on the boundaries of the asymmetric islands in Figure 7 of Chiang & Jordan (2002). We then generated the amplitude $\Delta\phi_{21}$ between these limits, weighting lower amplitudes more heavily using a linear distribution with a slope of -0.75.

4. Choose the current value of the resonant argument.

We choose $\phi_{21}$ from a uniform distribution between its limits by choosing a random phase $R$ in the interval (-1, 1) and then setting

$$ \phi_{21} = \langle\phi_{21}\rangle + R \times \Delta\phi_{21}. \quad (7) $$

5. Choose the leading or trailing island.

The model defines a fraction, $0 < f_L < 1$, of the asymmetric twotinos that are in the leading island. To assign a libration island, a uniform random number in this range is drawn, and if it is larger than $f_L$, the object is converted to a
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Distribution of eccentricity and libration amplitude for the real OSSOS twotino detections (yellow dots) and the simulated detections from the twotino model (small purple and gray dots) after running it through the OSSOS survey simulator. As elsewhere, the island-changing o3o33 with $\Delta\phi_{21} \sim 113^\circ$ is not part of our modeling.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Estimated visibility map for the phase space of the 2:1 resonance in OSSOS. To generate this map, we assume a uniform underlying distribution in *e*, *i*, and $\Delta\phi_{21}$ within the allowed resonant phase space for each libration island and a slope of $\alpha = 0.6$ for the *H* distribution. The color scale is log(normalized visibility), where normalized visibility = 1 occurs for the (*e*, $\Delta\phi_{21}$) bin with the most simulated detections across all three libration islands when the model is passed through the OSSOS survey simulator in the left-hand plots and for the (*i*, $\Delta\phi_{21}$) bin with the most modeled detections in the right-hand plots. Note that the white regions in the middle left and top left panels are just the phase spaces that exceed the maximum $\Delta\phi_{21}$ in Nesvorný & Roig (2001) and thus have no generated twotinos.</figcaption></figure>
trailing twotino by the simple transformation

$$ \langle \phi_{21} \rangle \rightarrow 360^{\circ} - \langle \phi_{21} \rangle. \qquad (8) $$

We treat $f_L$ as a free parameter, and we vary it in order to match the observed debiased distribution (see Section 4.2).

Biasing the model twotino distribution described above using the OSSOS survey simulator yields the distribution shown in Figure 5, which is compared statistically to the real detections in Section 4. OSSOS detected 17 leading and eight trailing asymmetric twotinos. The dominant reason for the roughly factor of 2 enhancement in leading island detections is not that they are intrinsically more numerous. As a comparison of Figures 3 and 4 already shows, the OSSOS blocks that pointed in the direction of the leading asymmetric librators achieved fainter limiting magnitudes than those pointed toward the trailing librators and thus yield the majority of the expected detections, even for a population with equal numbers of leading and trailing twotinos. This is most clearly illustrated in Figure 6, which shows how sensitive OSSOS was to the phase space within each libration island. This “visibility map” is constructed by uniformly filling the phase space of the resonance in $e, i$, and $\Delta\phi_{21}$ within the limits shown in the figure and plotting the relative distribution of the resulting simulated detections.

The resulting Figure 6 nicely illustrates the strong biases induced by the survey’s pointing and the flux bias. For example, there is a very strong bias favoring the detection of $e > 0.3$ symmetric twotinos that has only a weak dependence on libration amplitude (because the latter are all large and thus high-amplitude twotinos could be present in any OSSOS block); the lack of such detections by OSSOS proves they are unstable or scarcely populated and motivated us to place an upper limit. Figure 6 also shows the clear bias toward the leading asymmetric island (top panels) compared to the trailing island (middle panels) and required an upper eccentricity cutoff of 0.3 in the model. The difference in the biases for asymmetric twotinos is striking and caused by the location of Neptune relative to the galactic plane during the OSSOS epochs. For leading asymmetric twotinos, the blocks from R.A. = 0h to 3h mean that eccentric objects at perihelion will be close and bright, and the C/D block (Figure 1) is perfectly placed for the detection of very low libration amplitude objects. There is thus a strong concentration of objects where bias is high but also a need to have the intrinsic $e$ distribution decline sharply above 0.4. In contrast, for the trailing blocks, low libration amplitude twotinos would have perihelia longitudes in the galactic plane, and if they have large $e$, they will be at large $d$ when in the blocks between R.A. = 13h and 16h. For larger libration amplitudes, some better detectability develops because the perihelia for some of the objects can librate first over to the M and O blocks and then to E and P. The trailing twotinos again suggest that $e > 0.4$ is rare or absent. Lastly, the right column shows the expected general bias toward the detection of lower-$i$ twotinos; the fact that there are detections with $i > 10^{\circ}$ implies that the true inclination distribution has a certain number of large-$i$ members. In addition, the leading (and symmetric) panels show that placing the M block to cover latitudes from 6° to 10° has an important effect in allowing the detection of $i > 5^{\circ}$ to better constrain the inclination distribution. Figure 6 was intended to illustrate the general biases present and how complicated they can be; the underlying distributions are not uniform in these quantities, and below, we will validate our

parameterizations of the nonuniform orbital element distribution. Only with the ability to precisely measure the detection biases can one reliably try to extract the true orbital element distributions. OSSOS was designed precisely to do this.

For a final perspective of the complexity, we implemented a numerical simulation using MERCURY (Chambers 1999) to obtain a stability map of the boundaries in the libration center and width space. We randomly generated 40,000 synthetic particles with $47.2 \text{ au} \le a \le 48.5 \text{ au}$, $0^{\circ} \le i \le 20^{\circ}$, and $0 \le e \le 0.41$. In order to generate orientation angles that gave in initially librating asymmetric twotino orbit, the resonant arguments and amplitudes are randomly chosen within the ranges described in Section 3.1. Each libration island had an equal number of particles. Using Equation (1), the argument of perihelion is calculated with the random ascending node, random true anomaly, and resonant arguments/amplitudes chosen above. We identified 8134 surviving particles librating as twotinos at the end of a 1 Gyr simulation. (The surviving particles were identified as twotinos if the 2:1 resonant argument librated over the last 5 Myr.) We note that a few of these particles are in the phase of island switching, like o3o33 described in Section 3.2.1.

The libration centers and amplitudes of the surviving particles in this simulation are bounded by the contours in Figure 7. This stability map of the synthetic particles is relatively consistent with the measured distribution of the OSSOS twotinos, our unbiased model, and the simulations of Chiang & Jordan (2002) and Li et al. (2014; assuming initial conditions of low inclination). The biggest exception is the abundance of low-amplitude leading librators. This discrepancy likely results from the uniform initial distribution of orbital elements in the numerical study; initial conditions that give $\Delta\phi_{21} < 10^{\circ}$ are confined to a small portion of phase space and underrepresented, whereas the detection biases greatly favor their discovery in OSSOS, as just discussed. A second, smaller discrepancy is that the numerical study retains more moderate to large libration amplitude twotinos. Most of integrated particles in this region frequently transit between the symmetric island and an asymmetric island, which may artificially inflate the measurement of their asymmetric libration amplitude. This discrepancy may also be due to our initial distribution of particles but in the opposite way (overrepresenting large-amplitude librators compared to how the solar system populated the resonances).

## 4. Validation of the Model

The validity of our empirical twotino model is established by comparing the simulated detections with the real OSSOS twotino detections. In our comparison, we consider the given orbital elements separately as a one-dimensional cumulative distributions. We use the bootstrapped Anderson-Darling (AD) test to compare the simulated 1D distributions to the observed distributions. The AD statistic for the $N$ real twotinos is computed by randomly drawing subsamples of $N$ synthetic detections from the biased detections and bootstrapping the AD statistic for these subsamples; we choose to reject models if the AD statistic for $N$ real twotinos is below the 5% confidence level. The verification includes $e$ ($e_c$ and $e_w$), $i, d, H, \phi_{21}, \langle\phi_{21}\rangle$, and $\Delta\phi_{21}$, to identify the rejectability of model parameters.

The validation of the models was done in the following steps.
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Distribution of libration centers and amplitudes for the leading (left panel) and trailing (right panel) islands. Yellow points are the OSSOS twotino detections (which are a biased set). The black points show our favored model distribution (unbiased), based on the work of Chiang & Jordan (2002) and adjusted to match the OSSOS sample in Section 4. The magenta and blue points show the results of a capture study (Li et al. 2014) for initial orbital inclinations of *i* < 35° and *i* > 35°, respectively; the low-*i* results are a better match to the OSSOS twotino population (all of which have *i* < 35°) and consistent with the distribution simulated in Chiang & Jordan (2002), as expected. The contours indicate the stable phase space of the twotino population based on the surviving particles after a 10 Myr numerical simulation (see text for discussion).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Bootstrapped AD test distribution vs. width of the inclination distribution $\sigma_i$ for the leading asymmetric (dashed line), trailing asymmetric (dotted line), all asymmetric (red line), symmetric (blue line), and all twotino (yellow line). A value of $\sigma_i = 6^\circ$ has the lowest rejectability for each set of twotinos independently and for the entire ensemble. The horizontal lines at 5% and 1% show models rejected at moderate and high confidence.</figcaption></figure>
1. Libration centers and amplitudes for asymmetric twotinos. The range of the current resonance angles, $\phi_{21}$, and libration amplitudes, $\Delta\phi_{21}$, generated within the model was first confirmed to encompass the values of the known detections. After that, the boundaries for these distributions were fixed. Figure 3–5 show that the model covers the known detections, except for the one unstable low-$e$ symmetric object mentioned in Section 3.2.1 (o3o33) that we discard for the purposes of this study.

2. Inclination. The inclination distribution was then adjusted. The TNO inclination distributions are acceptably represented by a functional form of $\sin(i)$ times a Gaussian distribution of width $\sigma_i$ (Brown 2001; Gulbis et al. 2010; Gladman et al. 2012). We varied this width freely to determine what range is not rejectable using the bootstrapped AD test. Figure 8 shows the results of this statistical analysis of the twotino inclination distributions in the three libration islands. In all cases, the OSSOS detections can be acceptably modeled by an intrinsic inclination distribution with a width of $\sigma_i = 6^\circ$, with the 95% confidence interval being roughly $4^\circ-8^\circ$ for different libration islands. Note that, although there are fewer detections, the symmetric population independently supports this same inclination width range, as does each asymmetric island when tested independently. As a possible source of twotinos, the scattering TNOs could be transiently stuck in 2:1 resonance and occupy a significant ratio of symmetric twotinos (Yu et al. 2018). The peak of the inclination distribution of the transient-sticking twotinos only slightly shifts up by $\sim 1^\circ$; i.e., the transient-sticking twotinos have a similar hot inclination distribution as the scattering TNOs. However, the OSSOS symmetric twotinos prefer a colder inclination distribution, which indicates that the twotinos probably originate from the primordial disk rather than the scattering disk. For comparison, the inclination distribution width of the plutinos was estimated by Volk et al. (2016) to have a width of $8^\circ \le \sigma_i \le 21^\circ$ at a 95% confidence level and Alexandersen et al. (2016) to have a width of $11^\circ \le \sigma_i \le 21^\circ$ at a 95% confidence level; this range of widths is essentially nonoverlapping for the twotino inclination distribution in this study. The intrinsic inclination distributions of plutinos and twotinos are thus different, which must be a result of how their resonances were populated and eroded. We thus fix $\sigma_i = 6^\circ$ for our nominal twotino model that will be used to estimate the total population and ratio of asymmetric islands. The quality of the match between simulated and real OSSOS detections for this modeled inclination distribution is shown in Figure 9. The cosmogonic implications of this narrow width are discussed in Section 5.

3. Parameters for symmetric twotinos. The validation of the symmetric twotinos is relatively straightforward. As mentioned in Section 3.2.1, we adopt a uniform distribution in an appropriate range for $\phi_{21}$, $\Delta\phi_{21}$, and $e$. This simple model is a statistically acceptable match to the OSSOS detections. The validation of e, i, d, H, $\phi_{21}$, $\langle\phi_{21}\rangle$, and $\Delta\phi_{21}$ distribution is nonrejectable on the condition of a 95% confidence level. Based on only eight symmetric twotinos in the OSSOS sample, we avoid overinterpretation of details and are mostly interested in the population size.

4. Size distribution and eccentricity for asymmetric twotinos. We adopt a single exponential H-magnitude distribution because it turns out to not be rejectable, although the feature near $H = 8.1-8.5$ in the H distribution of Figure 9 could be a hint of a knee or divot H distribution (Lawler et al. 2018b). We assume the magnitude distribution is the same for twotinos in all three islands. We then vary the logarithmic slope, $\alpha$, of the cumulative absolute magnitude exponential distribution $N(<H_r) \propto 10^{\alpha H_r}$, and, for the asymmetric twotinos, we vary the Gaussian center $e_c$ and width $e_w$ of the eccentricity distribution. These three parameters are strongly coupled because of the observational biases, and we thus consider them together in a single statistical analysis. (In contrast, the symmetric twotinos were acceptably modeled with a simple, uniform eccentricity distribution.) We then repeat the analysis of Volk et al. (2016; their Figure 4), computing the bootstrapped probability that the real sample's AD statistic would be more discrepant than an equally sized subsample from a model defined by the ($\alpha, e_c, e_w$) triplet. We assign a model a rejectability that is the worst of the three AD probabilities for the e, d, and H cumulative distributions; we found that this choice did a better job of ruling out a model that is clearly discrepant in one variable than the “summed AD statistic” used previously. Rejectability maps of these three parameters are shown in Figure 10. The least rejectable trio of values is $\alpha = 0.6, e_c = 0.275$, and $e_w = 0.06$, and we adopt these parameters as our nominal model. The analysis shows that the data suggest an asymmetric eccentricity distribution that is strongly peaked near $e_c \approx 0.28$ with a narrow width of only about 0.06, and that such a model provides an entirely acceptable match to the data (with a worst-of-three AD probability as high as 0.58). To 95% confidence, we cannot rule out the case of lower $e_c$ and large $e_w$ (that is, a wide Gaussian centered at low $e$) due to the detection bias against low-$e$ twotinos, but such models produce worse matches to the eccentricity distribution. An exponential with a single value of $\alpha$ (i.e., a size distribution without a break or divot) satisfactorily represents the data but is the worst of the matches (Figure 9). We also individually tested the size distribution with a few published size distributions: (1) the knees in Fraser et al. (2014), (2) the knee in Lawler et al. (2018b), and (3) the divot in Lawler et al. (2018b). The knee and divot in Lawler et al. (2018b; based on the scattering population) are not rejectable and provide higher AD values than that in this study, whereas both knees for the hot population in Fraser et al. (2014) are nearly rejectable (at a 93% confidence level). The H-magnitude distribution is likely the first thing that will be modified if a future well-characterized survey can more than double the OSSOS sample, and it will likely then formally reject the single exponential.

5. Fraction of all twotinos in the symmetric island and asymmetric twotinos in the leading island. Finally, we estimate the parameters $f_S$ (the fraction of all twotinos that are symmetric librators) and $f_L$ (the fraction of the asymmetric twotinos that are in the leading asymmetric island). Thus, the fraction of all twotinos in the leading island is $(1-f_S)^*f_L$. The OSSOS sample constrains that $0.2 < f_S < 0.55$ and $0.2 < f_L < 0.65$ at the 95%
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Cumulative plots of orbital parameters for discovery distance (*d*), orbital inclination (*i*), orbital eccentricity (*e*), and absolute magnitude *H* distribution for the full model of all OSSOS twotinos (symmetric and asymmetric twotinos). The gray dotted line shows the intrinsic (model) distribution for the parameter; the black line indicates the biased results produced by the OSSOS survey simulator. The red dots give the distribution of the real twotinos detected by OSSOS. The real detections are statistically consistent with being drawn from the biased distribution.</figcaption></figure>

confidence level. This analysis is described in detail in
Section 4.2.

Figures 9 and 11 show cumulative plots of various orbital parameters for our nominal model before and after being subjected to the OSSOS survey simulator compared to the real OSSOS twotinos. The real and modeled detections are nonrejectable matches. These certainly suffice to provide reasonable population estimates of the various twotino islands, which is our main goal.

4.1. *Population Estimates*

To estimate the total twotino population in each libration
island, we generate simulated twotinos from the models
described above using the exponential absolute magnitude
distribution with slope α = 0.6 to a magnitude cutoff of
H_r < 10.0. This cutoff value covers the full range of OSSOS
twotino H_r magnitudes. For each libration island, model

twotinos are generated and run through the survey simulator
until the number of tracked objects matches the number of
detected twotinos. This test is repeated 10⁴ times for each
island. The median number of simulated twotinos that must be
generated to match the real number of detections is our estimate
for the relevant twotino population; the distribution median of
number of simulated twotinos is used to calculate the
uncertainty in our estimates.

Table 2 lists the median number of twotinos consistent with
the real OSSOS detections in each libration island based on our
nominal model, as well as the 95% confidence ranges for these
population estimates. Note that the population estimate for the
trailing island is more uncertain than that for the leading island
due to the survey being less sensitive to twotinos in the trailing
island (Figure 6). For comparison with the earlier results of
Gladman et al. (2012) and Volk et al. (2016), these population
estimates down to H<sub>r</sub> = 8.66 are also listed in Table 2.
Gladman et al. (2012) and Volk et al. (2016) made use of
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Bootstrapped AD probability maps for a range of values of $\alpha$, $e_c$, and $e_w$ in the intrinsic asymmetric twotino model. The area outside of the white contours indicates a rejected value of the worst AD statistic at a >95% confidence level (solid) or 68% confidence level (dashed). The most nonrejectable model parameters are shown by the black dots and stated at the top right. Each panel is a two-dimensional cut of our three-dimensional parameter space search. For each panel, we fix one parameter to its best value and show the AD rejectability map for the other two parameters.</figcaption></figure>

CFEPS data that was acquired in the g band and set a faintest magnitude of $H_g = 9.16$. To convert to r, we assume $g - r = 0.5$ (thus, the last three columns are comparable), which was used in Volk et al. (2016) and approximately consistent with the mean twotino color in Marsset et al. (2019). Although the small number of twotinos in the two earlier studies made it impossible to usefully constrain the populations of the individual libration islands, their reported estimates of the total twotino population are in statistical agreement with our results, with OSSOS roughly halving the estimated uncertainty.

## 4.2. Relative Populations of the Different Islands

The population ratio of the asymmetric islands in the current epoch could provide clues to Neptune's migration rate. The fraction of symmetric twotinos will also constrain theoretical models but has little discussion in the literature. For a statistical analysis of these populations, we repeated the work shown in Figure 10 of Volk et al. (2016), which only used nine twotinos (four from the 13AO and 13AE blocks of OSSOS and five from CFEPS). For each candidate value of $f_S$ and $f_L$ on a grid from 0 to 1 using steps of 0.025, we generate twotinos (using

the favored empirical model described above) until the survey simulator produces 10,000 tracked twotino detections. Then we randomly draw a sample of 34 from these simulated twotino detections and check whether that sample matches the OSSOS observed symmetric fraction of 9/34 and observed leading asymmetric fraction of 17/25. This process was repeated 1000 times to calculate a probability distribution across the tested range of $f_S$ and $f_L$ (see Figure 12).

OSSOS provides the first meaningful quantitative bounds on the relative island fractions. The OSSOS sample demands with high certainty that $0.2 < f_S < 0.6$ and $0.2 < f_L < 0.7$ at the 99% confidence level and that $0.2 < f_S < 0.55$ and $0.2 < f_L < 0.65$ at the 95% confidence level. The figure shows a mild preference for both $f_S$ and $f_L$ at 40%, but a 50% value for $f_L$ is perfectly acceptable.

In comparison with our estimates of the island fractions, the two migration simulations in Chiang & Jordan (2002) predict values of $f_S = 0.16$ and $0.13$ and $f_L = 0.48$ and $0.23$ for their slow migration (Ib) and fast migration (Iib) models, respectively. Both of these results are rejected at the 99% confidence level by the OSSOS population estimates. A prediction of $f_S = 0.42$ and $f_L = 0.34$ for the low-inclination ($i < 5°$)
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Cumulative plots of the libration parameters for the two asymmetric islands (combined). The red dots are the real detections, the gray dotted curves show the intrinsic empirical model, and the solid black curves indicate the simulated detections from the empirical model. The left and middle panels illustrate the heavy detection biases present in the OSSOS twotino sample. In particular, the middle panel shows that the intrinsic 60/40 split between the trailing and leading island populations is heavily biased in the detected sample toward more detected leading twotinos. This is due to the greater depth and favorable location of the fall blocks (see Figure 1); nearly 70% of the detected asymmetric sample is thus found in the leading island.</figcaption></figure>

twotinos from Figure 9 of Li et al. (2014) is in agreement with our results. Pike & Lawler (2017) provided an estimate of $f_S = 0.37 \pm 0.07$ and $f_L = 0.69 \pm 0.13$ (the uncertainties were estimated assuming Poisson errors given the number of particles studied) from a detailed Nice model simulation based on Brasser & Morbidelli (2013), which is rejected at more than a 95% confidence level by our analysis. Lawler et al. (2019) included dynamical classifications of the four Neptune migration simulations in Kaib & Sheppard (2016), which used different migration timescales and either grainy or slow migration. This analysis provides a measure of $f_S/f_L$ for the twotinos in these four simulations: grainy slow = $0.42 \pm 0.16/0.36 \pm 0.19$, grainy fast = $0.31 \pm 0.05/0.54 \pm 0.08$, smooth slow = $0.13 \pm 0.08/0.52 \pm 0.19$, and smooth fast = $0.29 \pm 0.05/0.56 \pm 0.09$ (again, uncertainties estimated assuming Poisson errors). Note that the slow migration simulations from Lawler et al. (2019) have fewer particles in resonances, so the island fraction numbers are more uncertain than for the fast migration simulations. Of these four studied cases, grainy slow migration provides the best match with our study, and the smooth slow migration simulation is rejected at a >99% confidence level.

As discussed in Pike & Lawler (2017), the specifics of Neptune's migration, as well as the test particle initial conditions, have a significant effect on the relative populations of the twotino asymmetric islands. The dynamical processes to create the enhancement of the libration island are still not comprehensively studied, and there is probably considerable uncertainty/variation in the model results themselves. The range of $f_S$ and $f_L$ allowed by OSSOS will provide important constraints for future comprehensive numerical simulations of twotino production.

Outstanding issues that will hopefully be addressed in future Kuiper Belt surveys are as follows. (1) Is there any difference in the population of the leading and trailing islands? Our results give no significant indication that there is (and a sample of hundreds of twotinos from a survey with very well understood biases will be needed to convincingly demonstrate it), but if shown to exist, this property might strongly constrain the style of planetary migration. (2) Why is today's symmetric fraction so high? There is no a priori reason to have thought that the

population is essentially equally split between symmetric, trailing, and leading librators, but our results indicate that this is roughly true. Smooth slow migrations tend to produce a twotino population poor in symmetric librators, so this may be a clue. Resonant sticking (see below) might serve to provide a steady-state input of temporary resonators that are more likely to be symmetric librators, but they would typically have inclinations more typical of scattering objects (15°) than the cold distribution, limiting the allowed contribution of temporary twotinos.

## 5. Discussion

The OSSOS survey's block R.A.s were chosen to optimize the detection of resonant objects, especially twotinos. The R.A. of the C/D block is near the center of the leading island, and the S, T, P, and E blocks lean lightly against the lower boundaries of the leading and trailing islands (see Figure 1). This survey design is very useful for probing the detailed structure of the twotino libration islands. Many OSSOS twotinos with $\phi_{21} \sim 70^\circ$ and low $\Delta\phi_{21}$ ($<10^\circ$) were discovered in the C/D block, which has excellent placement with deep discovery observations (longer exposure time), and the D block also benefited from better seeing conditions than the other blocks (see Figure 4). These discoveries significantly constrain the distribution in phase space occupied by the current twotino population, especially their ranges of $\phi_{21}$ and $\Delta\phi_{21}$. The previous lack of low libration amplitude twotinos was thus a selection effect. In the C/D block, OSSOS discovered six twotinos with $\Delta\phi_{21} < 10^\circ$, the lowest being $\Delta\phi_{21} = 3^\circ 9$. However, the simulations of Chiang & Jordan (2002) and Pike & Lawler (2017) predict a much smaller population of objects at such small libration amplitudes than that estimated from the OSSOS detections. The OSSOS results thus indicate that the initial conditions or migration parameters used in these migration simulations will need to be adjusted to produce more low libration amplitude twotinos.

OSSOS also revealed interesting regions of resonant parameter space that have not been sufficiently probed by observations. In the OSSOS sample, $\phi_{21}$ is a very peculiar object. It has very low eccentricity ($0.0825$) and a unique
<table><caption>**Table 2**
Twotino Population Estimates</caption><thead><tr><th>Libration Island</th><th>e Distribution</th><th>Median Pop.<br>H<sub>r</sub> &lt; 10</th><th>Median Pop.<br>H<sub>r</sub> &lt; 8.66<sup>a</sup></th><th>Gladman et al. (2012)<br>H<sub>g</sub> &lt; 9.16</th><th>Volk et al. (2016)<br>H<sub>g</sub> &lt; 9.16</th></tr></thead><tbody><tr><td>Symmetric</td><td>Uniform e = 0.07–0.3</td><td>6500<sup>+2900</sup><sub>+4300</sub></td><td>1300<sup>+600</sup><sub>+900</sub></td><td>...</td><td>...</td></tr><tr><td>Leading</td><td>e<sub>c</sub> = 0.275, e<sub>w</sub> = 0.06</td><td>4800<sup>+1700</sup><sub>+2200</sub></td><td>1600<sup>+600</sup><sub>+600</sub></td><td>...</td><td>...</td></tr><tr><td>Trailing</td><td>e<sub>c</sub> = 0.275, e<sub>w</sub> = 0.06</td><td>7100<sup>+3400</sup><sub>+5200</sub></td><td>1500<sup>+700</sup><sub>+1000</sub></td><td>...</td><td>...</td></tr><tr><td>Total</td><td>...</td><td>18400<sup>+4800</sup><sub>+7100</sub></td><td>4400<sup>+1100</sup><sub>+1500</sub></td><td>3700<sup>+2400</sup><sub>+4400</sub></td><td>4000<sup>+2000</sup><sub>+2500</sub></td></tr></tbody></table>

**Note.**

intermediate libration amplitude of 113°. When we integrate the orbital evolution of o3o33, this object switches its libration center many times between the symmetric and asymmetric islands over the course of 10 Myr. We note that 2001 UP18, discovered by DES (Elliot et al. 2005), also has a small eccentricity (0.0815) and is in the same process of switching islands. This behavior appears more common in this small low-e portion of the twotino phase space. Note that the low eccentricity also implies a larger observational bias against detecting such objects since the perihelion distance is larger. The stability and net flux between the symmetric and asymmetric islands are not clear, but they must have reached a steady state at the current time, given how rapidly the switching occurs.

The results of our global statistical analysis are somewhat limited by having only 34 OSSOS twotinos available. While this larger sample improves the uncertainties in the twotino population estimates by more than a factor of 2 compared to previous studies, the small number of detections still dominates the population uncertainties given in Section 4.1. Nevertheless, our results allow an improved comparison of the OSSOS twotino population estimates to the plutino population estimate from Volk et al. (2016). The estimated number of $10,000_{-3000}^{+3600}$ plutinos brighter than $H_r < 8.66$ is somewhat more than double the number of twotinos with the same $H_r$ limit, confirming the result of Gladman et al. (2012). This low current ratio of the twotino to plutino populations must reflect a combination of their primordial population ratio (which depends on both the migration process and the primordial TNO distribution at the start of large-scale planetary migration) and their differential dynamical loss rate over the age of the solar system. Hahn & Malhotra (2005) demonstrated that the capture efficiency of the 2:1 resonance in Neptune’s migration (initial $\langle e \rangle = 0.001$, $\tau = 10^7$ yr) is significantly higher than that of the 3:2 resonance. Tiscareno & Malhotra (2009) presented a simulation of this erosion process for both resonances, showing how the ancient ratio of the twotino and plutino populations can be related to the present ratio; the erosion rate of twotinos is about two times larger than that of plutinos over 4 Gyr. Future numerical studies incorporating the density/space distribution of the planetesimal disk, migration processes, erosion rate, and estimates of the current plutino and twotino populations will provide a cosmogonic picture to understand the history of the outer solar system.

The inclination distribution in this study is consistent with the estimates of Gladman et al. (2012) and Volk et al. (2016); all studies prefer a colder inclination distribution for the twotinos than that of either the 3:2 or 5:2 resonances. This feature is very different from the slow migration simulations of Nesvorný (2015b), which indicates a similar hotter inclination

distribution for both the 3:2 and 2:1 resonances. The fact that OSSOS detected very few high $i > 10^\circ$ twotinos, despite considerable sensitivity to them, indicates that the twotino population is intrinsically cold. It is worth noting that Tiscareno & Malhotra (2009) identified inclination dependence for long-term twotino stability, which would mean that a $\sin(i)$-Gaussian distribution would not the best representation of the twotino inclination distribution even if the initially trapped population was (see the bootstrapped AD test and the cumulative distribution in Figures 8 and 9, which support this idea). Although the inclination-dependent stability of twotinos in a 1 Gyr timescale could contribute to a relatively cold inclination distribution (Tiscareno & Malhotra 2009), this effect is not noticeable in the results of Nesvorný (2015b). Further studies of how today’s twotinos ($a \sim 47$ au) could be significantly dynamically colder in inclination than the two other largest-resonance populations ($3:2 a \sim 39$ au, $5:2 a \sim 55$ au) are needed, especially given that the twotino resonance is located between them.

Although we find that the existing detections do rule out many parameterizations of the intrinsic resonance distributions, there is still freedom related to how the libration centers and amplitudes can be modeled/adjusted. We are convinced, however, that this will not greatly alter the population estimates, considering the comparable population estimations with a relatively simple twotino model in Gladman et al. (2012) and Volk et al. (2016). More cosmogonic numerical studies of early solar system dynamics are required in order to determine how much variation there is in the current epoch’s distribution of $\langle \phi_{21} \rangle$ and $\Delta\phi_{21}$ and how these distributions depend on migration rates, TNO initial conditions, and long-term dynamical erosion. With only the existing numerical studies, we are unable to precisely determine if variations in the early solar system (for example, the rate and extent of planetary migration or the e and i distribution of small TNOs at the start of planet migration) result in different distributions today.

Future studies of the 2:1 resonance are probably not warranted until at least a factor of 2 more objects are discovered. This will likely happen once the Large Synoptic Survey Telescope (LSST; Ivezić et al. 2019) begins operations; it is the only project on the foreseeable horizon with the scope to make a major leap forward in the number of detections. Although OSSOS was a deeper survey than LSST, the roughly 2 orders of magnitude more sky area to be covered by LSST will more than compensate for this, and the survey should detect several hundred twotinos. It will be critically important for the LSST survey to ensure that the completeness of the TNO detections is known, as OSSOS makes it clear that the number of detections is a very strong function of depth at various sky locations (e.g., Figure 1 shows that the

<footnote><sup>a</sup> The limit of $H_r = 8.66$ is equivalent to the limit of $H_g = 9.16$ used in Gladman et al. (2012) and Volk et al. (2016), assuming $g - r = 0.5$.</footnote>
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Probability map of the fraction of symmetric island twotinos ($f_S$) and asymmetric twotinos in the leading island ($f_L$), constrained by the observed population fractions of the symmetric and leading asymmetric islands ($f_{S, OSSOS}$, $f_{L, OSSOS}$; i.e., the number of detections in each island should be the same as the ratio in the OSSOS sample). The rejected regions for this probability distribution are indicated with solid (99% confidence level) and dashed (95% confidence) white curves. Note that the absolute probabilities (color map) are low only because the cells have very high resolution; the contours correctly enclose the normalized probabilities. The $f_S$ and $f_L$ from Chiang & Jordan (2002), Li et al. (2014), Pike & Lawler (2017), and Lawler et al. (2019) are overplotted with dots and labeled.</figcaption></figure>

observations of the deepest OSSOS block, labeled C/D, were
very good at finding twotinos, especially leading asymmetric
librators). The LSST project would thus need to invest
significant effort into understanding if there is any longitudinal
sensitivity to its detection probability due to seasonal weather
trends and background stellar density.

6. Summary

The empirical model developed for this work acceptably
reproduces the distribution of OSSOS twotino detections in all
of the parameters studied. This match allows us to confidently
derive population estimates for the 2:1 populations in each
libration island. Note that the AD statistic provides the
nonrejectable range of a parameter, rather than assessing its
best value. We summarize our main findings as follows.

1. Although the model presented in Volk et al. (2016) is rejected after the inclusion of the full OSSOS twotino sample, it proves we can probe the dynamical structure within the islands. Our model parameters and population estimates are fully validated with the OSSOS survey simulator to match with the detected OSSOS twotino sample. These are the first results from a fully characterized survey with a large enough sample population to investigate the full twotino parameter space and the

differences in the distributions between the different
libration islands.

2. Our analysis strongly supports previous claims (Gladman et al. 2012; Volk et al. 2016) that the width $\sigma_i$ of the twotino population inclination distribution is significantly colder than other resonances. Volk et al. (2016) showed a best estimate of $\sigma_i = 12^\circ$ for the 3:2 resonance width, while our estimate for the twotino population is $\sigma_i = 6^\circ$. These distributions are different at more than the 95% confidence level.

3. The combination of the empirical model and the OSSOS survey simulator provides an excellent match to the observed twotino population. However, more detections, especially of fainter and more distant twotinos, would be very helpful to further constrain the parameter space for comparison to migration models.

4. Our estimates of the overall distributions of libration centers and amplitudes are in good agreement with the simulations of Chiang & Jordan (2002) and Li et al. (2014), with the exception of the low libration amplitude objects. Our biased results indicate that the current populations of the symmetric, leading, and trailing islands are all equal given current uncertainties. Our estimates of the libration island fractions appear discrepant with some models of outward Neptune migration, especially slow and smooth ones.
The authors acknowledge the staff of the Canada-France-Hawaii Telescope for their amazing effort in this project. We are grateful to Jian Li for kindly providing the data from his study of the twotino inclination distribution. Y.-T.C. gratefully acknowledges support from NRC-Canada and UBC, enabling a scholarly visit in the early phase of this study. Y.-T.C. also acknowledge Renu Malhotra for helpful conversations. K.V. and R.M.C. acknowledge support from NASA grant NNX15AH59G. K.V. acknowledges additional support from NASA grant NNX14AG93G and NSF grant AST-1824869. B.G. acknowledges NSERC Canada for research support. M.T.B. acknowledges support from UK Science and Technology Facilities Council grant ST/P0003094/1. The Center for Exoplanets and Habitable Worlds is supported by the Pennsylvania State University, the Eberly College of Science, and the Pennsylvania Space Grant Consortium. S.M.L. gratefully acknowledges support from the NRC-Canada Plaskett Fellowship. R.I.D. is supported in part by Alfred P. Sloan Foundation's Research Fellowship. Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii, with this project receiving additional access due to contributions from the Institute of Astronomy and Astrophysics, Academia Sinica, National Tsing Hua University, and Ministry of Science and Technology, Taiwan.

Facility: CFHT (MegaCam).

Software: orbfit (Bernstein & Khushalani 2000), MERCURY (Chambers 1999), OSSOS survey simulator (Bannister et al. 2016; Lawler et al. 2018a).

## ORCID iDs

Ying-Tung Chen (陳英同) https://orcid.org/0000-0001-7244-6069

Kathryn Volk https://orcid.org/0000-0001-8736-236X

Ruth Murray-Clay https://orcid.org/0000-0001-5061-0462

Matthew J. Lehner https://orcid.org/0000-0003-4077-0985

J. J. Kavelaars https://orcid.org/0000-0001-7032-5255

Patryk Sofia Lykawka https://orcid.org/0000-0003-0926-2448

Mike Alexandersen https://orcid.org/0000-0003-4143-8589

Michele T. Bannister https://orcid.org/0000-0003-3257-4490

Samantha M. Lawler https://orcid.org/0000-0001-5368-386X

Rebekah I. Dawson https://orcid.org/0000-0001-9677-1296

Sarah Greenstreet https://orcid.org/0000-0002-4439-1539

Jean-Marc Petit https://orcid.org/0000-0003-0407-2266

## References

Alexandersen, M., Gladman, B., Kavelaars, J. J., et al. 2016, AJ, 152, 111

Bannister, M. T., Gladman, B. J., Kavelaars, J. J., et al. 2018, ApJS, 236, 18

Bannister, M. T., Kavelaars, J. J., Petit, J.-M., et al. 2016, AJ, 152, 70

Beauge, C. 1994, CeMDA, 60, 225

Bernstein, G., & Khushalani, B. 2000, AJ, 120, 3323

Brasser, R., & Morbidelli, A. 2013, Icar, 225, 40

Brown, M. E. 2001, AJ, 121, 2804

Chambers, J. E. 1999, MNRAS, 304, 793

Chiang, E. I., & Jordan, A. B. 2002, AJ, 124, 3430

Elliot, J. L., Kern, S. D., Clancy, K. B., et al. 2005, AJ, 129, 1117

Fraser, W. C., Brown, M. E., Morbidelli, A. r., Parker, A., & Batygin, K. 2014, ApJ, 782, 100

Gladman, B., Lawler, S. M., Petit, J.-M., et al. 2012, AJ, 144, 23

Gladman, B., Marsden, B. G., & Vanlaerhoven, C. 2008, in The Solar System Beyond Neptune, ed. M. A. Barucci et al. (Tucson, AZ: Univ. of Arizona Press), 43

Gulbis, A. A. S., Elliot, J. L., Adams, E. R., et al. 2010, AJ, 140, 350

Hahn, J. M., & Malhotra, R. 2005, AJ, 130, 2392

Ivezić, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111

Kaib, N. A., & Sheppard, S. S. 2016, AJ, 152, 133

Kavelaars, J. J., Jones, R. L., Gladman, B. J., et al. 2009, AJ, 137, 4917

Lawler, S. M., Kavelaars, J. J., Alexandersen, M., et al. 2018a, FrASS, 5, 14

Lawler, S. M., Pike, R. E., Kaib, N., et al. 2019, AJ, 157, 253

Lawler, S. M., Shankman, C., Kavelaars, J. J., et al. 2018b, AJ, 155, 197

Levison, H. F., & Morbidelli, A. 2003, Natur, 426, 419

Levison, H. F., Morbidelli, A., Van Laerhoven, C., Gomes, R., & Tsiganis, K. 2008, Icar, 196, 258

Li, J., Zhou, L.-Y., & Sun, Y.-S. 2014, MNRAS, 443, 1346

Malhotra, R. 1995, AJ, 110, 420

Malhotra, R. 1996, AJ, 111, 504

Malhotra, R., Lan, L., Volk, K., & Wang, X. 2018, AJ, 156, 55

Marset, M., Fraser, W. C., Pike, R. E., et al. 2019, AJ, 157, 94

Murray, C. D., & Dermott, S. F. 1999, Solar System Dynamics (Cambridge: Cambridge Univ. Press)

Murray-Clay, R. A., & Chiang, E. I. 2005, ApJ, 619, 623

Nesvorný, D. 2015a, AJ, 150, 68

Nesvorný, D. 2015b, AJ, 150, 73

Nesvorný, D., & Roig, F. 2001, Icar, 150, 104

Nesvorný, D., Vokrouhlický, D., & Roig, F. 2016, ApJL, 827, L35

Petit, J.-M., Kavelaars, J. J., Gladman, B. J., et al. 2011, AJ, 142, 131

Pike, R. E., & Lawler, S. M. 2017, AJ, 154, 171

Tiscareno, M. S., & Malhotra, R. 2009, AJ, 138, 827

Volk, K., Murray-Clay, R., Gladman, B., et al. 2016, AJ, 152, 23

Yu, T. Y. M., Murray-Clay, R., & Volk, K. 2018, AJ, 156, 33