Accepted to the Astronomical Journal: January 21, 2017 PreprinttypesetusingLATEXstyleAASTeX6v.1.0 THE STRUCTURE OF THE DISTANT KUIPER BELT IN A NICE MODEL SCENARIO R. E. Pike1,2, S. Lawler3, R. Brasser4, C. J. Shankman1, M. Alexandersen2, and J. J. Kavelaars1,3 1DepartmentofPhysicsandAstronomy,UniversityofVictoria,Victoria,BC,Canada 2InstituteofAstronomyandAstrophysics,AcademiaSinica,Taipei10617,Taiwan 3NationalResearchCouncilofCanada,Victoria,BC,Canada 4EarthLifeScienceInstitute,TokyoInstituteofTechnology,Meguro,Tokyo152-8550,Japan 7 1 ABSTRACT 0 2 This work explores the orbital distribution of minor bodies in the outer Solar System emplaced as a n result of a Nice model migration from the simulations of Brasser & Morbidelli (2013). This planetary a migration scatters a planetesimal disk from between 29-34 AU and emplaces a population of objects J intotheKuiperbeltregion. Fromthe2:1Neptuneresonanceandoutward,thetestparticlesanalyzed 4 populatetheouterresonanceswithorbitaldistributionsconsistentwithtrans-Neptunianobject(TNO) 2 detections in semi-major axis, inclination, and eccentricity, while capture into the closest resonances ] is too efficient. The relative populations of the simulated scattering objects and resonant objects P in the 3:1 and 4:1 resonances are also consistent with observed populations based on debiased TNO E surveys, but the 5:1 resonance is severely underpopulated compared to population estimates from . h survey results. Scattering emplacement results in the expected orbital distribution for the majority p of the TNO populations, however the origin of the large observed population in the 5:1 resonance - o remains unexplained. r t s a 1. INTRODUCTION Nesvorny´ 2015b). Some regions of the Kuiper belt have [ different physical properties; for example, the cold clas- Thetrans-Neptunianobjects(TNOs)populatethere- sicalobjects,withalessexcitedinclinationdistribution, 1 gion beyond Neptune, and the specifics of their forma- v have a steeper size distribution than the hot classical tion location and evolutionary history are the subject 1 objects (Bernstein et al. 2004). The surface colors of of much study. Early ideas of a quiescent belt, surviv- 4 TNOs are also correlated with their dynamics; differ- 0 ing beyond the giant planets, ware clearly incomplete ent color distributions correspond to different dynam- 7 based on the orbital characteristics of the early TNO 0 ical sub-populations (e.g. Tegler et al. 2003). Brown discoveries (Levison et al. 2008). Even the first known . et al. (2012) suggests that the surface colors of TNOs 1 TNO, Pluto, has large eccentricity and inclination; the 0 could be produced by forming these objects interior to TNOs must have been dynamically evolved in the past 7 theirpresentlocationandmovingthemoutwardtotheir (Malhotra 1995). The TNO population has dynami- 1 currentorbits. TheouterSolarSystempopulationchar- : cally excited eccentricity and inclination distributions, v acteristics are complex and provide clues about the for- andtheobjectsextendoutto largesemi-majoraxes. In i X addition,theresonantpopulationsaremuchlargerthan mation and evolution of the Solar System. An acceptable model of Solar System evolution will r expected for a Kuiper belt which experienced no dy- a reproduce these aspects of the TNO population. A dynamical sweeping or scattering. The over-population of namicalinstabilityofthegiantplanetscanincreaseboth objectsinresonanceisanindicatorofapreviousdynam- theeccentricityandinclinationofsmallbodiesandscat- ical instability affecting the outer Solar System (Malho- ter a large number of TNOs into resonance (Levison tra 1993, 1995; Hahn & Malhotra 2005; Gomes et al. etal.2008). AslowmigrationofNeptuneoutwardwould 2005; Levison et al. 2008). capture TNOs into mean motion resonances (Malhotra The relative sizes and physical properties of the sub- 1995). This slow sweeping pumps up the eccentricity of populationsintheKuiperbeltholdcluestotheregion’s capturedTNOswithoutsignificantlyalteringtheirperi- evolutionary history. The sizes of the 3:2 and 2:1 reso- centerdistances. Thelargebinaryfractionofsomesub- nant populations compared to the classical Kuiper belt populationsofTNOsfavorsasloworminimalmigration andtherelativesizesofthecoldandhotclassicalpopu- because these binaries are likely to have been disrupted lations are dependent on the specifics of the dynamical bymoreviolentscatteringinteractions(Parker&Kave- evolution (e.g. Malhotra 1995; Chiang & Jordan 2002; 2 laars2010). Smoothmigrationscenariostypicallyresult 2. MIGRATION MODEL in TNO populations which are not sufficiently dynami- The model from Brasser & Morbidelli (2013)1 is ex- cally excited in inclination, and the possibility of more amined here, referred to as the B&M simulation. The granular and less smooth migration models have been B&M simulation is considered as an example of a Nice explored (Hahn & Malhotra 2005; Nesvorny´ 2015b). model type scenario and used to assess the accuracy Thommesetal.(1999)suggestedthattheearlyconfig- of this model in producing the scattering and resonant uration of giant planets was dynamically unstable. The TNO populations. The TNO comparison population is early incarnations of rapid planetary migration models fromtheCanada-FranceEclipticPlaneSurvey(CFEPS; are known as the ‘Nice model’ (Tsiganis et al. 2005; Petit et al. 2011, 2016) and the Alexandersen et al. Morbidelli et al. 2005; Gomes et al. 2005). In these sce- (2016)survey. Usingadynamicalsimulatedmodelfrom narios, there is a large dynamical instability, such as asourceexternaltotheobservationalsurveyresultspro- Saturn and Jupiter crossing their 2:1 mean motion res- vides a useful test of the orbital distribution models onance, scattering Uranus and Neptune, which subse- created by the survey team. This work focuses on the quently scatter small bodies, emplacing the TNOs and ‘scattereddisk’portionoftheB&Msimulation–allpar- Oort cloud and causing the Late Heavy Bombardment. ticles beyond Neptune and interior to the Oort cloud ThisscenarioalsoresultsincaptureoftheJovianTrojan (a<3,000 AU). asteroids(Morbidellietal.2005). Thismorerapidplan- Brasser & Morbidelli (2013) use a Nice model frame- etary scattering event results in different characteristics worktopopulatethescattereddiskandOortcloudand of captured objects as compared to a smooth migration determine the relative sizes of these populations. This model. Nice model migration includes a dynamical instability A detailed comparison of TNO detections and nu- following the removal of the gas in the Solar System’s merical simulation results requires a carefully observed protoplanetary disk, after the last encounter between Kuiper belt in addition to a well-sampled simulation. the ice giants. The planetary evolution track from Lev- SeveralrecentsurveysoftheKuiperbelthaveattempted ison et al. (2008) ‘Run A’ was repeated, which starts toprovideTNOdiscoverieswithknowndiscoverybiases withNeptuneat27.5AUwithaneccentricityof0.3and (Schwamb et al. 2010; Petit et al. 2011; Adams et al. Uranus at 17.5 AU and an eccentricity of 0.2. Uranus 2014; Alexandersen et al. 2016; Bannister et al. 2016; migrates outward to ∼19 AU and Neptune migrates to Petit et al. 2016). These surveys characterize their dis- ∼31 AU over ∼ 100 Myr. At the end of the simula- covery biases to facilitate comparison with population tion, Neptune is slightly beyond its true position, at models. 30.8 AU instead of 30.1 AU, however this was left un- This work explores the specific effects of the ‘Nice’ changed during the simulation to avoid disrupting res- modelscenarioonthescatteringandresonantTNOpop- onant objects and all objects were rescaled after com- ulationsindetail. AsagenericexplorationofouterSolar pletion. The 30,000 test particle initial locations were System evolution, the predicted TNO populations from 29 < a < 34 AU (based on the final position of Nep- the Nice migration simulation by Brasser & Morbidelli tune, this rescales to 28.2 < a < 33.2 AU), i < 1◦, (2013) are tested against the real TNO detections from and e = 0.15. The eccentricity of the test particle ini- Petit et al. (2011, 2016) and Alexandersen et al. (2016). tial conditions results from the eccentricity of the outer Thecombinationofrealdetectionsandcharacterizations planets (Levison et al. 2008). Each test particle was fromthesurveysandasurveysimulatorprovidesapow- initially given unique position and velocity vectors. For erfultoolforcomparinganexternalmodeltothesurvey thescattereddiskcomponent,aftertheplanetmigration detections. The simulation and test particle classifica- was completed, all test particles beyond 3,000 AU were tion are discussed in Section 2. An explanation of the removed from the simulation. The particles and giant surveysimulatordebiasingprocedureisprovidedinSec- planets were integrated for an additional 3.8 Gyr with tion 3. Section 4 presents the results of the classifica- SWIFT RMVS3 (Levison & Duncan 1994). Because of tion and analysis; the B&M model of the Kuiper belt, significant scattering loss, after 1 Gyr and 3.5 Gyr, the as a proxy for the scattering that likely occurs in a Nice remaining test particles were cloned three times to en- model type scenario, does a reasonable job of populat- sure a sufficiently well-sampled Kuiper belt (effectively ing the outer Solar System scattering components and 270,000testparticles). Theresultsfromtheendstateof resonancesbeyonda∼45. Thenotableexceptionisthe the3.8GyrB&Mscattereddisksimulationsareutilized 5:1population, whichisnotwellreproducedinthissce- nario,independentoftheresultsfromPikeetal.(2015). The discussion and conclusions are presented in Section 1 The B&M model end state including the particle classifica- 5. tionsforalltestparticlesfrom200.24 that are not scattering or resonant. sified test particles. The primary goal is to describe the behavior of the test particles at the start of the addi- tionalintegrations,tocharacterizetheB&Mmodel‘end state.’ A classification of resonance requires an oscillation of theresonantangle,φ ,overtime,wherepandq arein- pq tegers, and φ describes the p:q mean-motion resonant pq angle. Each particle whose semi-major axis is within 1.5 AU of a Neptune resonance location had the rele- vant resonant angle computed: φ =pλ−qλ −(p−q)(cid:36). (1) pq N 4 60 1:1 2:1 3:1 4:1 5:1 6:1 7:1 8:1 9:1 10:1 11:1 3:2 5:2 7:2 9:2 11:2 13:2 15:2 17:2 19:2 21:2 4:3 8:3 10:3 14:3 16:3 20:3 50 ) 5:3 7:3 13:3 17:3 19:3 23:3 s 5:4 7:4 9:4 11:4 13:4 15:4 21:4 e e 7:58:59:5 13:5 19:5 gr40 13:6 e 17:8 19:7 D ( n 30 o i t a n20 i l c n I 10 0 1:1 2:1 3:1 4:1 5:1 6:1 7:1 8:1 9:1 10:1 11:1 3:2 5:2 7:2 9:2 11:2 13:2 15:2 17:2 19:2 21:2 4:3 8:3 10:3 14:3 16:3 20:3 0.8 5:3 7:3 13:3 17:3 19:3 23:3 5:4 7:4 9:4 11:4 13:4 15:4 21:4 7:58:59:5 13:5 19:5 y 13:6 t0.6 i 17:8 19:7 c i r t n e c0.4 Centaur c E Scattering Detached Resonant 0.2 Inner Classical Main Classical Outer Classical 0.0 20 40 60 80 100 120 140 Semi-major axis [AU] Figure 1. Test particle inclination, i, and eccentricity, e, distributionswithsemi-majoraxis,a,oftheendstateofthe B&Mmodel. Thedashedlinesmarkresonanceswheremore thantwotestparticlesarefound. Theopacityofthedashed lines scales with the number of particles in the resonance. The large number of inner and main classical objects is ap- parent. The outer classical objects are consistent with em- placementthroughresonancedropout,similartotheslightly larger e detached objects. The solid lines indicate specific pericenterlocations,q of35and40. Neptuneisindicatedby the large dark blue circle. 5 3. COMPARING THE B&M MODEL TO THE randomizing the position angles (ω, Ω, and M). This SOLAR SYSTEM cloning ensures a sufficient number of simulated detections; objects are randomly drawn until a larger sample 3.1. Direct Model Comparison than the known TNOs is ‘detected’ to ensure the pa- The subcomponents of the B&M simulation results rameter space is sufficiently well sampled. are directly compared to the CFEPS L7 model, the or- In order to determine the detectability of the B&M bitalelementdistributionsandabsolutepopulationsizes orbital model, an absolute magnitude, H, distribution of different TNO components based on the CFEPS de- for the particles is assumed. This is a proxy for object tections and detection biases (Petit et al. 2011; Glad- size,whichisnotdirectlymeasurableforunresolvedob- man et al. 2012) and other models based on CFEPS jects. H distributions are typically parameterized as an (Shankmanetal.2013;Pikeetal.2015;Shankmanetal. exponential, equivalent to a single power law in diame- 2016). The L7 model is consistent with the CFEPS deter; indifferentialformthesinglepowerlawhasasingle tections, but is not uniquely consistent; for example, an slope of α: equallyconsistentmodelcouldcontainadditionalpopu- lationsatverylargepericentersentirelyundetectableby dN/dH ∝10αH. (2) the surveys. As a result, agreement between the B&M simulationandtheL7modelimpliesthattheB&Msim- Shankman et al. (2013, 2016) describes a broken H- ulations must be consistent with the detections, while distribution by joining two different differential sin- disagreement with the L7 model does not necessary gle slopes, α and α , at a specific magnitude, bright faint require that the B&M simulation results are inconsis- H . In addition to the change of slope, they transition tent with real detections. To mitigate this issue, the also propose a sudden drop (a ‘divot’) in number den- B&MmodelistestedbothagainsttheL7modelanddi- sity after the transition, parameterized as the contrast, rectlyagainstthesurveydetectionsusingtheAnderson- c. (A contrast of one has no drop and is referred to Darling (AD) statistical test2. as a ‘knee’ in the literature.) The survey detections used in this work are primarily in g-band, so H , ab- g 3.2. Comparing the B&M model to Real TNO solute magnitude in g band is used. The B&M simula- Detections tion particles are assigned an absolute brightness, H , g from three different size distributions from the litera- Tofacilitatedirectcomparisonoftheorbitaldistribu- ture. Theseare: asingleslopeofα=0.9asinGladman tionsfromtheendstateofB&MtorealTNOdetections, et al. (2012); a knee distribution with α = 0.87, an observational bias is applied to the B&M simulation bright α = 0.2, and H = 8.35 as in Fraser et al. particles using a survey simulator. The survey simula- faint g−transition (2014),convertedtog usingg−r=0.65fromPetitetal. tor uses survey detection characteristics (pointings, de- (2011); and a divot distribution with α = 0.8, tection efficiencies, tracking efficiencies, etc.) to deter- bright α = 0.5, H = 9.0, and a contrast c = 5.6 mine which input model objects would have been de- faint g−transition as in Shankman et al. (2013). For the majority of the tected by the survey (Kavelaars et al. 2009). The B&M B&Mpopulationsthechoiceofsizedistributionshadno “observed”testparticlesarecomparedtothedetections impactontheconclusions,soforthesepopulationsonly fromtheCFEPSecliptic(Petitetal.2011)andhighlat- the knee distribution is presented. itude(Petitetal.2016)surveysaswellastheAlexander- The survey simulator biased B&M model reflects the sen et al. (2016) survey. In both plots where B&M sim- detectability of different populations, see Figure 2. The ulation particles are compared to real TNOs, the TNOs selectionbiaseswhichcomplicateTNOpopulationstud- are the characterized detections from these surveys. To iesareapparentwhencomparingFigure1andFigure2. determine the acceptability of the B&M particles as a The closest objects dominate the simulated detections. model of the populations, the survey simulator-biased Thefractionofdetachedobjectsdetectedissignificantly test particles are compared to the real TNO detections. smaller than other populations, because of their large TheB&Msimulationresultswereusedastheinputor- pericenters. Thechoiceofsizedistributionmodelaffects bital model for the CFEPS survey simulator. Conduct- theexpectednumberofdetectionsroughlyasafunction ing a survey simulator analysis of a simulation output of perihelion distance. The single slope produces more requires a large number of orbit samples. The simula- small object detections per each large object, so for the tion particles were cloned, preserving a, e, i, and in the samenumberofsimulateddetections, asingleslopedis- case of resonant objects, the resonant angle (φ), while tribution results in more low-a and large-H detections, while a knee or divot distribution is more likely to have 2SeeJonesetal.(2006)foranexplanationofcomparingcumu- larger-a and low-H detections. The survey simulator lativeparametersandhowtheADstatisticisusedforrejection. biased results for several TNO subpopulations are pre- 6 sented in Section 4. 60 1:1 2:1 3:1 4:1 5:1 6:1 7:1 8:1 9:1 10:1 11:1 3:2 5:2 7:2 9:2 11:2 13:2 15:2 17:2 19:2 21:2 4:3 8:3 10:3 14:3 16:3 20:3 50 ) 5:3 7:3 13:3 17:3 19:3 23:3 s 5:4 7:4 9:4 11:4 13:4 15:4 21:4 e e 7:58:59:5 13:5 19:5 gr40 13:6 e 17:8 19:7 D ( n 30 o i t a n20 i l c n I 10 0 1:1 2:1 3:1 4:1 5:1 6:1 7:1 8:1 9:1 10:1 11:1 3:2 5:2 7:2 9:2 11:2 13:2 15:2 17:2 19:2 21:2 4:3 8:3 10:3 14:3 16:3 20:3 0.8 5:3 7:3 13:3 17:3 19:3 23:3 5:4 7:4 9:4 11:4 13:4 15:4 21:4 7:58:59:5 13:5 19:5 y 13:6 t0.6 i 17:8 19:7 c i r t n e c0.4 Centaur c E Scattering Detached Resonant 0.2 Inner Classical Main Classical Outer Classical 0.0 20 40 60 80 100 120 140 Semi-major axis [AU] Figure 2. SimilartoFigure1,inclination,i,andeccentricity, e, distribution with semi-major axis, a, of the B&M simulation end state biased using a survey simulator. The 30,000 particlesshownwere‘detected’bythesurveysimulatorusing the Petit et al. (2011, 2016) and Alexandersen et al. (2016) surveypointingswithHmagnitudesrandomlyassignedfrom a single slope H-magnitude distribution with α=0.9. This plot includes only detections with H < 8, which roughly g corresponds to >170 km in diameter. The significant selec- tioneffectsofTNOsurveysareapparent; theinnerclassical and close resonances are much easier to detect compared to more distant populations. The knee and divot distributions show qualitatively similar detection biases. 7 4. RESULTS: POPULATIONS OF THE OUTER 1.0 SOLAR SYSTEM n o Using the methods described in Section 2.2, the end- cti0.8 a stateorbitsofthetestparticlesintheB&Msimulations Fr0.6 were placed into orbital classes. The full population e v statistics for the B&M model are summarized in Table ti0.4 a 1andthemodelobjects’orbitaldistributionisplottedin ul m 0.2 Figure 1. Several subpopulations are discussed in detail u C here. 0.0 20 40 60 80 100 120 140 160 Table 1. Test Particle Classifications Semi-major Axis (AU) 1.0 n o Classification Number of Particles Fraction of Total ti0.8 c a Resonant 3,910 42% r F0.6 Inner Classical 871 9% e v Main Classical 1,943 21% ati0.4 ul Outer Classical 181 2% m 0.2 Scattering 535 6% Cu Detached 1,921 21% 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Centaurs 5 0.05% Eccentricity 1.0 Total Particles 9,366 100% n o cti0.8 Unbiased Simulation a Fr0.6 Shankman 2013 4.1. Scattering Objects e v Biased Simulation (SS) The B&M scattering objects populate the considered ti0.4 a Biased Simulation (Knee) region from 24–155 AU at a nearly constant rate over ul m Biased Simulation (Divot) 0.2 semi-major axis. These objects are shown in Figure u C Detections 3, and their distribution in a, e, and i is statistically 0.0 consistent with the model of scattering objects used by 0 20 40 60 80 100 120 140 160 180 Shankman et al. (2013, 2016). The Shankman et al. Inclination (Degrees) (2013, 2016) scattering object model used orbital pa- Figure 3. The cumulative fraction of scattering objects in rametersbasedonthesimulationresultsfromKaibetal. the B&M simulation with the a, e, and i values are indi- (2011). The scattering objects from Kaib et al. (2011) catedwiththeblue(dashed)line. Thegreen(dash-dot)line were produced by a significantly different planetary mi- indicatesthemodelusedtorepresenttheunbiasedmodeldis- tributionfromShankmanetal.(2013,2016),whichisconsis- gration and evolution scenario, however the signatures tentwiththeblue(dashed)B&Msimulationendstate. The in the orbital structure produced by the specific mi- B&M particles were assigned three different H-magnitude gration in the scattering objects are not statistically distributions: the single slope (SS), knee, and divot. The particles were then biased using the survey simulator. The distinguishable. Based on comparing these scattering magenta ‘x’ marks indicate actual detections from the sur- object orbital element distributions, we conclude, as in veyssimulated,forcomparisonwiththeB&Mbiasedsimula- Shankman et al. (2013), that the specifics of planetary tion results. The observed orbital element distributions are better matched by the knee or divot size distributions than migration do not strongly affect the scattering popula- the single slope. tion. Of the populations considered in this work, the scat- teringobjectsarethemostsensitivetothechoiceofsize distribution. Thedetectabilityofscatteringobjectswith The biased B&M simulation with the single slope, the divot, knee, and single slope size distributions ap- knee, and divot H-distributions all provide a statisti- pliedtotheB&Msimulationscatteringobjectsisshown cally acceptable semi-major axis distribution match for in Figure 3 (see Equation 2 for details). Previous work the real detections. The single slope results in a signif- from Shankman et al. (2013, 2016) finds that the sin- icantly worse eccentricity distribution, nearly rejectable gleslopeisrejectableforthescatteringobjects, andthe by the AD statistic. The knee and divot e-distributions divot is the preferred model (although several knee dis- are non-rejectable. The B&M model does not include tributions are acceptable). any retrograde objects; these retrograde objects are re- 8 turning Oort cloud objects (Brasser et al. 2012) which particlesareresonantforaminimumof5Myr. Toassess were excluded from the simulation, so the lack of ret- the B&M simulation, these resonant B&M populations rograde objects does not invalidate the model. When are compared to debiased survey results from the lit- the detected retrograde TNO (from Petit et al. (2016) erature, as well as biased using a survey simulator for ati>90◦ inFigure3)isexcludedfromtheanalysis,the comparison with real resonant object detections. inclination distribution of the three H-distributions all provideanacceptablematch. BasedontheB&Mmodel Table 2. Resonance Occupation ofthea,e,andidistributionofthescatteringobjects,a knee(Fraseretal.2014)ordivot(Shankmanetal.2013) Resonance Semi-Major Axis Number of Fraction size distribution provides a significantly better repre- p:q (AU) Test Particles Stablea sentation of the real detections than a single slope H- distribution. The orbital parameters of the B&M scat- 1:1 30.05 4 100% tering objects are consistent with the parameters from 5:4 34.87 64 100% Kaibetal.(2011),confirmingthatpropertiesofthescat- 4:3 36.40 588 100% tering population are not particularly dependent on the 3:2 39.37 1640 99% specificsofthescatteringevent(Shankmanetal.2013). 5:3 42.24 217 100% The scattering objects in the B&M simulation with a 7:4 43.63 67 100% knee or divot size distribution provide a good model of 2:1 47.70 111 97% this population. 7:3 52.86 39 100% 4.2. Resonant Test Particles 5:2 55.35 337 99% The B&M simulation contains 42% resonant test par- 3:1 62.50 77 85% ticles. The B&M model inner resonances are overpopu- 4:1 75.71 70 80% latedrelativetotheSolarSystem. ThisanalysisofB&M 5:1 87.86 19 84% resonanttestparticlesfocusesonresonancesbeyondthe main classical belt, including the 2:1 resonance. aParticles are considered stable when resonant for the 30 Myr integration. TheorbitaldistributionoftheB&Mresonantparticles atsmallersemi-majoraxes,suchasthe3:2,4:3,and5:4, areunlikelytorepresenttherealobjectdistributionsbe- 4.3. Comparing Population Sizes: Resonant and causeoftheinitialB&Msimulationdesign. Thisislikely Scattering Objects a result of the extended disk of particles before scattering, which results in some unrealistic sweeping capture. A successful model of planetary migration should re- When the test particles with initial semi-major axes in- produce the relative population sizes of the Kuiper belt terior and exterior to the final position of Neptune are resonances. The B&M simulation population ratios are considered separately, the a>30 AU particles are cap- presented in two ways in this section. In Table 3, the tured into these closer resonances twice as efficiently as end state B&M populations are compared to published thea<30AUparticles. However,thecaptureefficiency debiased literature population estimates. The ‘L-Scale beyondthe2:1resonancedoesnotdependoninitialpar- Factor’, literature scale factor, in this table shows the ticlea,soallparticlesareusefulinanalyzingthedistant factor the simulation would need to be increased by in populations. Manyofthemoredistantresonantpopula- ordertomatchtheliteratureestimate, andthe95%un- tionsintheB&Msimulationprovideanexcellentmatch certainty in the literature population estimate is trans- to observations, as these resonances are populated by lated into the scale factor 95% uncertainty. For Table captured scattering objects which are well represented 4, the B&M model is biased using the survey simula- bytheB&Mmodel, inpartduetothegenericnatureof tor and compared to the number of real TNO detec- scattering (see Section 4.1). tions from the surveys. The ‘B-Scale Factor’, or biased TheresonanceoccupationattheendstateoftheB&M scale factor, is the median number of times the model simulationsispresentedinTable2. Theresonanceswith of that dynamical sub-class must be sampled in order population estimates from the comparison surveys are to produce the number of real detections listed, with included here; resonances populated in the B&M sim- 95% confidence intervals from the population estimate ulation but not identified in the surveys are excluded. distributions. These scale factors indicate the factor by The number of test particles in each resonance is re- which the starting disk would have to be increased in ported, as well as the fraction classified as ‘stable’; this order to result in the appropriate population size. In requires that the test particle be resonant for the entire each table, when the populations have similar scale fac- classification integration of 30 Myr. Unstable resonant tors (with overlapping uncertainty) this indicates that 9 these populations are produced consistently within the As discussed in the previous section, the B&M simu- B&Mmodelandscalefactoragreementbetweentheta- lation was highly efficient at populating resonances in- bles indicates that the population size is consistent by terior to the classical Kuiper belt compared to those both methods. beyond the 3:2, causing an overpopulation in the 5:4, The population sizes of the B&M model sub- 4:3, and 3:2 resonances (see Table 3). These low-a res- componentsarecomparedtopopulationestimatesofthe onances include a component captured by resonance resonances from carefully characterized surveys (Petit sweeping. The initial planetesimal disk extended to et al. 2011; Alexandersen et al. 2016; Bannister et al. 33.2 AU, contrary to expectations about the real proto- 2016; Petit et al. 2016). The population estimates planetesimal disk. The starting conditions for Neptune (Gladman et al. 2012; Pike et al. 2015; Alexandersen in the B&M simulation placed the 5:4 resonance and etal.2016;Volketal.2016)werecalculatedbycreating the 4:3 resonance within the initial disk before plane- aparametricmodeloftheresonance,thenusingthesur- tary migration. The 3:2 resonance was just beyond the vey simulator to forward bias the model to statistically original extent of the implanted disk, but the test par- compare it to the observations. (See Gladman et al. ticles had sufficient eccentricity to reach an apocenter (2012)orVolketal.(2016)foradetailedexplanationof crossing this resonance, so sweeping may still be effec- theparamaterization.) Thesizeoftheunderlyingpopu- tive. TheB&M5:4,4:3,and3:2resonancesaretheonly lationnecessarytogeneratethenumberofdetectionsin resonancesthatincludeasignificantnumberofparticles the survey is the estimated size of the population. The swept into resonances, resulting in too high a capture population estimates for many resonances explored by efficiency, so these populations are not representative of these surveys are summarized in Table 3. the real TNOs. Table3. LiteratureEstimatesofResonantPopulationsfrom Surveys Resonance a B&M Simulation Population Estimate Survey; Source p:q (AU) L-Scale Factora N(H <8) g 1:1 30.05 5.0+20 10+40 Alexandersen et al. (2016)b −4 −9 5:4 34.87 0.4+2.4 10+60 CFEPS; Gladman et al. (2012) −0.36 −9 4:3 36.40 0.4+0.4 70+100 CFEPS; Gladman et al. (2012) −0.32 −50 1.6+0.6 1200+500 CFEPS; Gladman et al. (2012) −0.6 −400 3:2 39.37 1.4+0.6 1100+400 Alexandersen et al. (2016) −0.4 −300 1.2+0.4 900+330 CFEPS+OSSOSc; Volk et al. (2016) −0.4 −270 5:3 42.24 12+12 450+470 CFEPS; Gladman et al. (2012) −6 −280 7:4 43.63 40+60 300+400 CFEPS; Gladman et al. (2012) −20 −200 20+10 340+200 CFEPS; Gladman et al. (2012) 2:1 47.70 −10 −220 20+10 360+230 CFEPS+OSSOSc; Volk et al. (2016) −10 −180 7:3 52.86 20+40 320+760 CFEPS; Gladman et al. (2012) −10 −270 6+8 1100+1400 CFEPS; Gladman et al. (2012) 5:2 55.35 −4 −700 4+4 770+680 CFEPS+OSSOSc; Volk et al. (2016) −2 −420 10+20 340+800 CFEPS; Gladman et al. (2012) 3:1 62.50 −8 −290 6+8 220+270 Alexandersen et al. (2016) −4 −150 4:1 75.71 3+10 80+360 Alexandersen et al. (2016) −3 −80 5:1 87.89 240+420 1900+3300 CFEPS; Pike et al. (2015) −180 −1400 Note that the 5:3, 4:3, and 3:2 resonances are overpopulated in theB&Msimulation. aThe‘L-ScaleFactor’(literaturescalefactor)indicateshowmuch theB&Msimulation(withHg <8.35)mustbescaleduptomatch the population estimates (Population Estimate scaled to 8.35 ÷ NumberofB&Mmodelobjects). Theuncertaintiesaretheprop- agated95%uncertaintiesfromthepopulationestimates. bPopulation estimate is for the stable Neptune Trojans, as all of theB&Mtestparticlesinthisresonancearestable. cOSSOS:the OuterSolarSystemOriginsSurvey(Bannisteretal.2016) 10 The number of objects in many B&M resonant pop- of the scattering object population and some of the n:1 ulations (2:1, 7:3, 5:2, 3:1, 4:1) and the population es- resonancepopulationsinTable4areconsistentwithob- timates from Gladman et al. (2012), Volk et al. (2016), servations. Fortherealsurveydetections,thescattering and Alexandersen et al. (2016) are consistent. The L- / 3:1 / 4:1 populations have a ratio of 12 / 4 / 1; the scale factors for these populations in Table 3 are con- biased B&M simulation gives a consistent ratio of 20 sistent with each other and notably agree with the ob- / 5 / 1 before conversion to the scale factor. The B- servational survey results from Gladman et al. (2012) and L-scale factors (∼5-10) between the detections and andVolketal.(2016)thatthe5:2resonancehasalarge themodelarewithinuncertaintiesforthesepopulations. population. The B&M model for the scattering objects, 3:1, and 4:1 resonance is self-consistent. Table 4. Populations from the B&M model biased using 4.4. The Large Population of 5:1 Resonators a survey simulator. The knee H-distribution is presented because the effects of different H-distributions are minimal. Pike et al. (2015) investigated the 5:1 Neptune resonance using both real and simulated detections. The threerealTNOdetectionswerefoundbyCFEPS(Petit Sub- a B&M/Survey Survey et al. 2011, 2016) and a population model was created population (AU) B-Scale Detectionsb basedontheconstraintsprovidedbythedetectionsand (p:q) Factora the survey characterization. The parametric model of 1:1 30.05 4 +20 1 the 5:1 resonance from Pike et al. (2015) is consistent −4 5:4 34.87 0.2 +0.8 1 with a minimum population estimate based on the or- −0.2 4:3 36.40 0.2 +0.3 6 bitaldistributionof5:1resonatorsfromtheB&Mmodel. −0.1 3:2 39.37 0.8 +0.3 42 IntheB&Mevolution, the5:1resonanceispopulated −0.2 by the same mechanisms as the 3:1 and 4:1 resonances, 5:3 42.24 2.5 +1.8 12 −1.2 but both the L- and B-scale factor comparisons show 7:4 43.63 5 +6 5 −3 that the B&M model significantly under-predicts this 2:1 47.70 10 +8 9 −5 population compared to survey estimates and discover- 7:3 52.86 3 +6 2 −2 ies (Pike et al. 2015). If the B&M model is correct, 5:2 55.35 1.4 +−10..47 8 the efficiency of detection for the 5:1 resonance would 3:1 62.50 11 +15 4 be extremely low; producing three 5:1 detections in the −7 4:1 75.71 6.6 +31 1 surveyswouldresultin∼1,000scatteringobjectdetec- −6.4 5:1 87.89 180 +330 3 tions. InordertobeconsistentwiththeB&Mscattering −140 Scattering 20–155 5 +4 12 / 3:1 / 4:1 populations, the 5:1 B&M population would −2 need to be ∼ 20−100× larger, requiring an unreason- aThe ‘B-Scale Factor’ (biased scale factor) is the number of ably large starting planetesimal disk. The detections times the B&M model of the selected population with the knee in the 5:1, and thus the large population in that reso- size distribution must be sampled by the survey simulator to generate the number of detections found by the surveys. The nance, requires a different population source. This con- 95%uncertaintyquotediscalculatedbyrandomlyresamplingthe firms that the extremely large population estimate for population. ThesimulateddetectionswerecountedtoHg <8.35, thekneeinthesizedistribution. the5:1foundbyPikeetal.(2015)isunexplainedbythe bTotal number of detections as found by Gladman et al. (2012), currently explored models. Petitetal.(2016),andAlexandersenetal.(2016)surveys. The 5:1 resonance is significantly underpopulated in theB&Msimulationcomparedtoexpectationsfromsur- To avoid relying on the specific orbital distributions vey results. Pike et al. (2015) predict an enhancement from the survey population models, the B&M model of a factor of ∼50−100 compared to the local scatter- is biased using the survey simulator (see Section 3.2). ing objects, but this is not confirmed in typical Kuiper Someofthepublishedsurveypopulationestimateshave beltformationmodels(Hahn&Malhotra2005;Levison large 2σ uncertainties. The biased B&M model output et al. 2008). The non-resonant populations in the B&M ispresentedinTable4. ThescalefactorsforthekneeH- model within ±5 AU of the 5:1 resonance (89 AU) in- magnitude distribution are presented; the choice of size clude54scatteringand159detachedtestparticles. This distribution made no statistically significant difference isalineardensityof5scatteringand16detachedobjects in the results. per AU. The B&M 5:1 resonance has 16 particles, with Then:1resonancesarelikelypopulatedbythecapture an approximate width of 1 AU it is ∼ 3 times denser ofscatteringobjects,soiftheB&Mmodelaccuratelyre- than the scattering object population and comparable produces the scattering population it should reproduce indensitytothedetachedpopulation, inconsistentwith the n:1 population statistics as well. The relative sizes the enhancement predicted by (Pike et al. 2015).

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