The Structure of the Distant Kuiper Belt in a Nice Model Scenario

6.9 MB·

by  Rosemary E. Pike

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 dy-
namical 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. tionsforalltestparticlesfrom20<a<160AU,asgeneratedin
thiswork,isavailableat: doi:10.11570/16.0009.
3
in this work; see Brasser & Morbidelli (2013) for more The particle mean longitude is λ = Ω+ω +M, Ω is
details on the migration simulation. the longitude of the ascending node, ω is the argument
of pericenter, and the longitude of perihelion is (cid:36) =
2.1. Additional Integrations Ω+ω. λ refers to the mean longitude of Neptune. If
N
φ oscillatesinsteadofcirculates,thenthetestparticle
In this work, the end state planet and test particle pq
is classified as resonant.
positionsfromtheB&Msimulationwereintegratedfor-
Diagnosing resonance based on a visual inspection is
ward 30 Myr in order to determine dynamical classifi-
straightforward, but an automated detection method is
cations. Additional integrations were necessary because
required for large numbers of particles. A spectrogram
resonance classification requires more frequent output
analysis was used on a series of windows to identify os-
thanthepreviousfullsimulationinordertoconclusively
cillation in φ over time (Shankman et al. 2017). The
classify the test particles. The Sun, Jupiter, Saturn, pq
behavior of φ in overlapping windows of 5 Myr was
Uranus, Neptune, and test particle end states from the pq
analyzedusingafastFouriertransform(FFT).Ifapar-
previous simulation were provided as input for SWIFT
ticle’s φ was resonant for all windows it was classified
RMVS4 (Levison & Duncan 1994). The particle posi-
as stable; unstable particles only displayed oscillations
tions were recorded every 300 years to ensure sufficient
in a subset of windows. If an object was resonant, the
sampling of the resonant angle. During the 30 Myr in-
libration amplitude (maximum to minimum φ oscilla-
tegration, the particles between 20 < a < 160 AU were pq
tion) and libration center (median value of φ ) of the
recorded, to focus on the scattering TNOs as well as pq
particle were calculated.
some of the distant resonances.
Inthissimulationalltestparticlesbeyond34AU,the
Neptune’s final semi-major axis from the B&M simu-
original extent of the implanted disk, must have been
lationswas∼0.8AUfartherfromtheSunthanthetrue
scattered by Neptune. ‘Scattering objects’ specifically
position of Neptune. After the simulations completed,
refers to the dynamically unstable objects in the simu-
the semi-major axes of all objects in the B&M simula-
lation, particles that experience a change in a≥1.5 AU
tion were adjusted to correspond with their locations if
in the first 10 Myr of the additional integration (Glad-
Neptune’ssemi-majoraxiswerethecurrentactualvalue,
man et al. 2008). This is intended to refer to objects
usingascalefactorof30.047/30.8(ascalingof<2.5%).
that are currently scattering, instead of the ‘scattered’
Because everything in the simulation is shifted by the
or ‘scattered disk’ classification which refers to objects
same factor, the dynamics of all planets and test par-
thatwerescatteredinthepast,adifficultcriteriontoas-
ticles remain the same. This adjustment was done to
sessinrealTNOs. Objectsthatexhibitsemi-majoraxis
facilitate direct comparison with the Solar System, and
evolutionandhavea<30AUareclassifiedasCentaurs.
for the remainder of this work, the positions discussed
If a particle is not currently scattering or resonant,
for Neptune, the test particles, and the mean-motion
then the particle is classified based on its current a, e,
resonances in the simulations are the scale of the Solar
and i values. Objects are classified as main classical if
System today.
they are between the 3:2 and 2:1 resonance, 39.4 AU –
47.7 AU, and have eccentricities e < 0.24. Inner classi-
2.2. Particle Classifications
calobjectsarefoundbetweenNeptuneandthe3:2reso-
The subpopulations determined here are compared nance, 30.04 AU – 39.4 AU with eccentricities e<0.24.
with survey detections from Petit et al. (2011, 2016) Testparticlesbeyondthe2:1resonanceat47.7AUinthe
andAlexandersenetal.(2016),soaclassificationscheme same eccentricity range are classified as outer classical
consistentwiththosesurveyclassificationsisused(from objects. DetachedobjectsareparticlesbeyondNeptune
Gladman et al. 2008). Figure 1 shows a plot of all clas- with e>0.24 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
(
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i
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a
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i
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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 detec-
tions; 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 de-
ter; 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
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gr40 13:6
e 17:8 19:7
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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
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t
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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 simula-
tion 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 scatter-
ing, 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 res-
onance 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).