Is the Grand Tack model compatible with the orbital distribution of main belt asteroids?

4.6 MB·

by  Rogerio Deienno

Is the Grand Tack model compatible with the orbital distribution
of main belt asteroids?
RogerioDeiennoa,RodneyS.Gomesb,KevinJ.Walshc,AlessandroMorbidellid,David
Nesvorny´c
aInstitutoNacionaldePesquisasEspaciais,AvenidadosAstronautas1758,CEP12227-010Sa˜oJose´dosCampos,SP,
7
Brazil
1
bObservato´rioNacional,RuaGeneralJose´Cristino77,CEP20921-400RiodeJaneiro,RJ,Brazil
0
cDepartmentofSpaceStudies,SouthwestSpaceResearchInstitute,1050WalnutSt.,Boulder,CO80302,USA
2
dLaboratoireLagrange,UMR7293,Universite´Coˆted’Azur,CNRS,ObservatoiredelaCoˆted’Azur,Boulevardde

n l’Observatoire,06304NiceCedex4,France
a
J

0
1
Abstract

]
P The Asteroid Belt is characterized by the radial mixing of bodies with different physical
E
properties,averylowmasscomparedtoMinimumMassSolarNebulaexpectationsandhasan
.
h excitedorbitaldistribution,witheccentricitiesandinclinationscoveringtheentirerangeofvalues
p allowedbytheconstraintsofdynamicalstability. ModelsoftheevolutionoftheAsteroidBelt
- showthattheoriginofitsstructureisstronglylinkedtotheprocessofterrestrialplanetformation.
o
r The GrandTack modelpresentsa possible solution to the conundrumof reconcilingthe small
t
s massofMarswiththepropertiesoftheAsteroidBelt,includingthemassdepletion,radialmixing
a and orbitalexcitation. However,while the inclinationdistributionproducedin the GrandTack
[
modelisingoodagreementwiththeoneobserved,theeccentricitydistributionisskewedtowards

1 values larger than those found today. Here, we evaluate the evolution of the orbitalproperties
v oftheAsteroidBeltfromthe endoftheGrandTackmodel(attheendofthegasnebulaphase
5
whenplanetsemergefromthedispersinggasdisk),throughoutthesubsequentevolutionofthe
7
Solar Systemincludingan instabilityofthe GiantPlanetsapproximately400Mylater. Before
7
2 theinstability,theterrestrialplanetsweremodeledondynamicallycoldorbitswithJupiterand
0 Saturn locked in a 3:2 mean motionresonance. The modelcontinuesfor an additional4.1 Gy
1. afterthegiantplanetinstability.Ourresultsshowthattheeccentricitydistributionobtainedinthe
0 GrandTackmodelevolvestowardsoneverysimilartothatcurrentlyobserved,andthesemimajor
7 axis distribution does the same. The inclination distribution remains nearly unchangedwith a
1 slightpreferencefordepletionatlowinclination;thisleadstotheconclusionthattheinclination
:
v distributionattheendoftheGrandTackisabitover-excited.Also,weconstraintheprimordial
i eccentricitiesofJupiterandSaturn,whichhaveamajorinfluenceonthedynamicalevolutionof
X
theAsteroidBeltanditsfinalorbitalstructure.
r
a
Keywords: Asteroids,dynamics,Origin,SolarSystem,Planetarydynamics,Planets,migration
Emailaddress:[email protected](RogerioDeienno)
PreprintsubmittedtoIcarus January12,2017
1. Introduction
TheAsteroidBeltischallengingtounderstandbutiscriticalforstudiesoftheformationand
earlyevolutionoftheSolarSystem. TheorbitalconfigurationoftheAsteroidBeltisbelievedto
havebeenestablishedintwophases. Thefirstphasedatesbacktothefirstfewmillionyearsof
Solar System’sformationandshouldbestudiedin conjunctionwith theformationofthe inner
andouterplanets,especiallyJupiterandSaturn. ThesecondphaseoccurredwhentheAsteroid
Belt witnessed a Giant Planet instability, long after the damping effects of the gaseous Solar
Nebulahaddissipated
Ingeneral,simulationsofthedynamicalre-shapingoftheAsteroidBeltaremadeinconjunc-
tionwiththeformationoftheinnerplanets. Thefirstsimulationsofterrestrialplanetformation
(ChambersandWetherill,1998)includedasetofplanetaryembryosuniformlydistributedinthe
innerregionoftheSolarSystemwithorbitsinitiallydynamicallycold(loweccentricityandin-
clination). Throughnumericalintegrationsoftheequationsofmotionoftheseembryos,adding
amodelofaccretionbycollisions,thesystemevolvestoformplanetsintheinnerregionofthe
SolarSystemonstableorbits. Whileearlyresultsabouttheformationofterrestrialplanetswere
promising,oneoftheproblemsfoundintheseintegrationswasrelatedwiththefinaleccentric-
ities of the planets, which weresystematically largerthanthe realones. Themodelsproduced
morepromisingresultswhenthepresenceofasubstantialpopulationofplanetesimalswasalso
accountedfor;infact,thedynamicalfrictionexertedbytheplanetesimalsactedtodecreasethe
excitationoftheplanet’sfinalorbits(Chambers,2001;O’Brienetal.,2006).
An importantingredientwas the presenceof Jupiter, which should have completed its for-
mation much earlier than the inner planets (ChambersandWetherill, 1998; Chambers, 2001;
Petitetal.,2001). Primarily,theinfluenceofJupiterontheAsteroidBeltistopromotedestruc-
tivecollisions(fragmentation)ratherthanconstructivecollisions(accretion)(Petitetal.,2002).
However,Jupiteralonecannotexcitetheeccentricityofplanetesimalssomuchastoexplainthe
currentexcitedorbitsofasteroids(Petitetal.,2002). Inaddition,thereissignificantdiversityin
the physicalpropertiesof asteroidsfoundin the MainAsteroid Belt, buttheir main taxonomic
classesarefoundinroughlyoverlappingdistributions–althoughS-classbodiespredominatein
the inner regions and C-class bodies in the outer regions (see DeMeoandCarry, 2014). The
solutionoftheseissueshavebeenattributedtotheoriginalpresenceofplanetaryembryosinthe
AsteroidBelt(Petitetal.,1999;O’Brienetal.,2007). Theseembryos,onceexcitedbyJupiter,
would havescattered the orbitsof the planetesimals. In the end, the Asteroid Belt wouldhave
beendepletedofplanetesimalsandtotallydevoidofembryos.
Despite the many successes in the modelingof the terrestrial planets and Asteroid Belt by
the simulations described above, systematic problemspersisted. The planet formed in the ap-
proximate region of Mars systematically showed a much larger mass than the real Mars (see
Raymondetal.,2009). AnexperimentbyHansen(2009)foundthatifthereissharpouteredge
intheinitialmassdistributionofsolidsatabout1.0AU,thenthemodelsconsistentlyreproduce
themassofMars.
Walshetal.(2011)proposedamechanismtomodifytheoriginalmassdistributionofsolids
andproducethetruncateddiskexploredbyHansen(2009),byaccountingfortheearlymigration
of Jupiter and Saturn when they were still embedded in the gaseous proto-planetarydisk. An
outcomefoundinmanyhydrodynamicalmodels(MassetandSnellgrove,2001;Morbidellietal.,
2007;PierensandNelson,2008;PierensandRaymond,2011;D’AngeloandMarzari,2012)of
the interaction between giant planetsand gaseousdisks is that the type-IIinward migrationof
aJupiter-massplanetishaltedandevenreversedwhenasecond,lessmassiveplanet,isformed
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externalto the first one. Thisprovidesthe explanationfor whyJupiterdid notmigrateto very
close the Sun, as is seen for giant planets in many other planetary systems (UdryandSantos,
2007;Cummingetal.,2008). Instead,Jupiterwouldhavemigratedfirstinwards,thenoutwards.
BecauseofthechangeindirectionoftheorbitalmotionofJupiter(a“tack”insailor’sjargon),
theWalshetal.(2011)modelisnamedthe“GrandTack”.ThetimingoftheformationofSaturn
isconstrainedbythemassdistributionoftheterrestrialplanets,whicharebestreproducedwhen
Jupiterreversesmigrationat1.5AUandtruncatesthediskat1AU.
ThemigrationofJupiterwouldhavestronglyaffectedanyplanetesimalsformedinthepresent-
day Asteroid Belt, with a primary consequence of substantially depleting the entire region of
small bodies. The inward migration phase primarily pushes the asteroids originally inside of
Jupiter’sorbit(named“S-class”inWalshetal.(2011))downtolowersemimajoraxes(insideof
1 AU), thoughWalshetal. (2011) foundthatabout10%of these bodiesare scattered outward
onto orbits with semimajor axis a between 4-10 AU. During the outward migration of Jupiter
and Saturn, these bodies are encounteredagain, and about 1% are scattered back into the As-
teroid Belt. Meanwhile Jupiter and Saturn eventuallyencounterprimitiveplanetesimals(titled
“C-class”inWalshetal.(2011)),andafractionofapercentofthesearealsoscatteredintothe
AsteroidBelt. Thisprovides,atthetimewhenthegasnebulahasdispersed,afinalbeltwhichis
depletedinmassbyafactorofabout1,000,thatcontainstwodifferentclassesofbodiespartially
mixedinheliocentricdistanceandwithorbitsexcitedineccentricitiesandinclinations(although
thefinaleccentricitydistributiondoesnotmatchwellthecurrentone,asdiscussedbelow).
Numerousconstraints,suchastheagesofthelastimpactbasinsontheMoon(Bottkeetal.,
2007), the impact age distribution of HED meteorites (Marchietal., 2013), and the small to-
tal chondriticmassaccretedbythe Moonsince its formation(Morbidellietal., 2012), pointto
anepochofincreasedbombardmentintheinnerSolarSystemabout∼400−700Myafterthere-
movalofgasfromtheproto-planetarydisk(whereastheGrandTackhappenedbeforetheremoval
ofthegas).Thisperiodofincreasedbombardmentisusuallycalled“TerminalLunarCataclysm”
or“LateHeavyBombardment”(LHB)(seeHartmannetal.,2000;Chapmanetal.,2007,forre-
views),andwewilladopttheLHBnomenclaturehere. TheoriginoftheLHBhasbeenlinked
to a dynamicalupheavalin the outer Solar System frequentlyreferred to as the “Nice model”
(Gomesetal., 2005; Levisonetal., 2011;Bottkeetal., 2012). Duringthisdynamicalupheaval
thegiantplanetswouldhavesufferedaninstabilityandaperiodofmutualcloseencountersthat
radicallychangedtheirorbits.Inturn,theorbitalchangeofthegiantplanetswouldhaveseverely
affectedthedistributionoftheasteroidsinthemainbelt(Morbidellietal.,2010). Thebestguess
on when this instability occurred, from variousconstraints, is 4.1 Gy ago (Bottkeetal., 2012;
Morbidellietal.,2012).
This is important because the final Asteroid Belt in Walshetal. (2011) lacks objects with
small eccentricities. Indeed, according to the Grand Tack model, the eccentricity distribution
expectedfortheAsteroidBeltatthetime whenthegasnebuladispersed,some3-10Myrafter
theemergenceoffirstsolidsandroughly4.5Gyrago,peaksaround0.4. Ontheotherhand,the
currentdistributionoftheAsteroidBeltpeaksaround0.1(Morbidellietal.,2015). Ithasnever
been studied whetherthe Grand Tack finaldistributioncould evolveto onesimilar to what we
seetodayduetotheperturbationscausedbythegiantplanetinstabilityduringtheNiceModel.
Thegoalofthispaperistopresentsuchastudy.
Inthiswork,wewillstudyindetailtheevolutionoftheAsteroidBeltorbitalstructure,from
the end of the Grand Tack model to the giant planet orbital instability, through the instability
phase,andfinallyduringthelast∼4Gyuntiltoday.
Thisworkwillthereforeunfoldasfollows:section2explainsourSolarSystem’sconfigura-
3
Case 1 Case 2
0.03 0.08
0.07
0.06
ntricity 0.02 ntricity 00..0045
Ecce 0.01 Ecce 0.03
0.02
0.01
0 0
0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700
Time (My) Time (My)
Figure 1: Eccentricity behavior ofJupiter (black) andSaturn (gray) before the planetary instability as afunction of
time. Thesetwoplanetsareincoplanarorbits,andlockedintheir3:2MMRwithJupiterat5.4AU.Left: Case1isa
initialconfigurationfromMorbidellietal.(2007)whereplanetsevolveintoaresonantconfigurationinahydrodynamical
simulation. Right: Case2isanarbitrarycasewhereplanetsareplacedinresonantorbits,allowinglargevariationsin
eccentricity.
tionandthemethodusedforthenumericalsimulations. Insection3wediscussthedynamical
evolutionoftheAsteroidBeltthroughoutthevariousphasesoftheSolarSystem’sevolution,as
wellastheinfluenceoftheprimordialeccentricitiesofJupiterandSaturnonthefinalstructure
oftheAsteroidBelt. Finally,section4summarizesthemainconclusionsofthispaper.
A complementary study, approaching the problem from a different perspective, has been
recentlypresentedinRoigandNesvorny´ (2015). Wewillcompareourresultswiththeirsatthe
endofsection3.2.
2. Methods
Ournumericalsimulationscontainfiveplanets,Venus,Earth,Mars,Jupiter,andSaturn(all
withtheircurrentmasses),plus10,000masslessparticlesrepresentingthefinaloutcomeofthe
Asteroid Belt in the Grand Tack simulations(describedbelow). Uranus, Neptune, and the pu-
tativeextraicegiantinvokedinNesvorny´ andMorbidelli(2012)arenotincludedinanyofour
simulations (the same applies for the planet Mercury). These planets are indeed too far from
theAsteroidBelttohaveanyimportantdirecteffectonitsstructure. Acaveat,however,isthat
in somesimulationstheextraicegiantissenttemporarilyontoanAsteroidBelt-crossingorbit
before being ultimately ejected from the Solar System. In the case the Asteroid Belt-crossing
phaselastforsufficientlylongtime,theeffectsofthisbodyontheAsteroidBeltshouldbetaken
into account(see Brasiletal., 2016). Nevertheless, because we have limited understandingon
how exactlythe evolutionof the planetsoccurred(i.e. how deepa planetcrossedthe Asteroid
Beltandforhowlong),weprefertoneglectthisputativeAsteroidBelt-crossingeventandfocus
onlyontheeffectsinducedbyJupiter,Saturnandtheterrestrialplanets,whoseevolutionisbetter
constrained.
Theorbitalconfigurationofthefiveplanetswaschosenasfollows:
• Beforetheplanetaryinstability:Theterrestrialplanetsaremodeledtobeonsimpleplanar
orbits(Brasseretal.,2013). JupiterandSaturn,alsoonplanarorbits,werelockedintheir
mutual3:2 MMRwith Jupiterat 5.4AU. Two cases, differingforthe eccentricityof the
giant planets (Fig. 1) are considered. Case 1 (left) is an initial configuration from the
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hydrodynamicalsimulations of Morbidellietal. (2007), where the planets migrated into
theresonanceandpreservedorbitswithverysmalleccentricitiesduetothetidaldamping
effect exerted by the gas disk. Case 2 (right) is an arbitrary case where the planets are
placed in resonant orbits, but with larger eccentricities (although still smaller than their
currentvalues).Forthisphase(forbothcases),thenumericalintegrationsarecontinuedfor
400My(i.e. until4.1Gyago),thebestestimateofthetimeofthegiantplanetinstability
(Gomesetal.,2005;Bottkeetal.,2012;Morbidellietal.,2012).
• After the planetaryinstability: The initial asteroid’sorbits are those recordedat the end
of the simulations of the previous phase, but the planets are placed onto their present-
dayorbits. Bydoingthis,wemodeltheinstantaneoustransitionoftheplanetsfromtheir
pre-instability orbits to the current orbits. In other words, we simulate the response of
the Asteroid Belt to the change of the planetary orbits, but we assume that the actual
dynamicalpath that the planets followed in this transition had no importanteffects. We
takethissimpleapproachbecausepreciselywhathappenedduringtheplanetaryinstabil-
ity is unknown, and many different paths could have been followed by the giant planets
(Nesvorny´ andMorbidelli,2012). Certainly,noinstabilitysimulationcanhavethepreten-
siontoreproduceexactlywhathappened. Weknowfrompreviousstudies(Brasseretal.,
2009; Morbidellietal., 2010) that the change of the giant planets’ orbits had to be fast
andof the “jumping-Jupiter”type, i.e. dominatedbya few impulsiveeventsdueto mu-
tualplanetaryencounters. Thus,webelievethatsubstitutingtheoriginalplanetaryorbits
with the current orbits is the best approximation of the real planetary evolution we can
implementwithoutthe risk of incurringinto arbitrarysimulationartefacts. In particular,
byusingthecurrentplanetarysystem,thelocationsandstrengthsofallresonancesexactly
matchtherealones,whereasnosingularinstabilitysimulationwouldfinishwithprecisely
a replica of the current-day planetary system. For the same reasons, this approach has
alreadybeenadoptedbyBottkeetal.(2012)intheirstudyoftheprojectilefluxfromthe
AsteroidBelttotheterrestrialplanets. Theoppositeapproach(i.e. simulatingthedynam-
icsofthegiantplanetsduringtheinstabilityphase)hasbeentakenbyRoigandNesvorny´
(2015).Thesecondphaseofsimulationswereconductedfor4.1Gy,untilatotaltimescale
of4.5GywascoveredbythecombinationofthetwophasesoftheSolarSystemevolution.
Concerningtheasteroiddistributionweconsidertwocases. Inournominalcaseweusethe
asteroid distribution resulting from the simplest of all Grand Tack scenarios, namely the one
that only includedJupiter and Saturn (Figure 2. of SupplementaryInformationof Walshetal.
(2011),referencedas“2-planetGrandTacksimulation”hereafter).However,forcomparisonwe
alsoconsiderthedistributiongeneratedintheGrandTackscenariothatalsoincludeUranusand
Neptune(presentedinthemainpaperofWalshetal.(2011),referencedas“4-planetGrandTack
simulation”hereafter).
BecausetheoriginalGrandTacksimulationshadonly∼1,000particlesinthefinalsamplein
theAsteroidBelt,inordertoimprovestatisticswegenerated10,000masslessparticlesmatching
thebasictraitsofthea,e,idistributionobtainedinthe2-planetGrandTacksimulation,withthe
followingproperties:
• Eccentricitydistributionisanormaldistributionwithmeanof0.38andsigmaof0.17.
• InclinationdistributionisaRayleighdistributionwithasigmaof10degrees.
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• Semimajoraxisdistributionisnearlyanuniformdistributionspreadovertherange1.8–3.6
AU.
Whilethesefunctionalfitsreproduceeachdistributionindividually,combinedtheyproduced
an excessof objectswith high-eccentricityand low-inclinationcomparedto the distributionof
asteroidsattheendoftheGrandTacksimulations. Toremovethisexcess,the10,000particles
werere-sampledinbinsofwidth0.05ineccentricityand5degreesininclination. Ineachbin,
weallowedthetotalnumberofparticlestobeatmost4timesmorenumerousthantheasteroids
attheendoftheGrandTacksimulationsinthesamebin. IftherewerenooriginalGrandTack
asteroids in the bin, a maximum of 4 particles from the functional fits was allowed. This re-
samplinglimitedtheexcesspopulationofhigh-eccentricityandlow-inclinationbodiesfoundin
the10,000particledistributions,andresultedinafinalpopulationof6,424bodies.Theadvantage
ofthisprocedure,relativetousingdirectlytheGrandTackasteroidsasinitialconditions,isthat
the functional fits allow us to generate more particles (but this could have been achieved also
bycloningtheGrandTackasteroids)andalsoplacesomeparticlesinbinsoriginallywithzero
GrandTackasteroids,thusmakingthedistributionsmoother.
Forthe4-planetGrandTacksimulationwesimplyresampledtheabovedistributionattribut-
ing to each particle a “weight”, representingthe probabilitythat said particle exists at the end
ofthe4-planetGrandTackrun. Becausethesemimajoraxisandinclinationdistributionsinthe
2-planetand4-planetGrandTacksimulationsarebasicallythesame,theseweightsarecomputed
fromtheeccentricitydistributionsonly,whicharesignificantlydifferent.Theseweightsarethen
usedtobuildanewfinaldistributionfromthatobtainedinournominalcase.
Inallsimulationsallplanetsinteractwitheachotherwhilealsoperturbingthetestparticles.
Test particles interact with the planets but not among themselves. The integrations have been
conductedusing Mercury (Chambers, 1999), in the Hybrid option with a time step of 10 days
(morethan20stepsperorbitalperiodofVenus).
3. Results
Figure 2 shows the currentstructure of the Asteroid Belt plotting all objects with absolute
magnitude H<10 from the Minor Planet Center catalog1, where the Main Asteroid Belt is as-
sumedtobeobservationallycompleteatthissize(Jedickeetal.,2002). Ourgoalistoverifyifit
ispossiblethattheorbitalconfigurationoftheAsteroidBeltattheendoftheGrandTackmodel
couldevolvetoonesimilartothatshowninthisfigure. Todoso,weconsideredthetwophases
oftheSolarSystemevolutionpreviouslydiscussedinsection2.
3.1. Beforetheplanetaryinstability
Figure 3 showsthe currentAsteroid Belt orbitaldistribution(grayline) comparedwith the
initialconditionsusedinthisexperimentthatweregeneratedfromtheGrandTacksimulations(as
describedinsection2);thesolidlineisfortheresultofthe2-planetGrandTacksimulationand
thedashedlineisforthe4-planetsimulation.Withtheexceptionoftheinclinationdistributions,
whichmatchfairlywell,theGrandTackandtherealdistributionsareclearlydifferent,i.e.,the
eccentricitydistributionsintheGrandTacksimulationsareskewedtowardslargevalues,andthe
semimajoraxisdistributionsarenearlyuniforminsteadofhavingthecleargapsfoundintoday’s
1http://www.minorplanetcenter.org/iau/MPCORB.html
6
0.6 MMRs with Jupiter 7:2 3:1 5:2 7:3 2:1 N(cid:176) of Ast. = 668
0.5
ity 0.4 Earth
c
ntri 0.3 Jupiter
e
c 0.2
c
E 0.1
Mars
0
n
) 40 6
(cid:176)(
n 30
o
i
t
a 20
n
i
l
c 10
n
I
0
1.6 2 2.4 2.8 3.2 3.6 4
Semimajor axis (AU)
Figure2:CurrentstructureoftheAsteroidBeltwithallH<10objectsfromtheMinorPlanetCentercatalog. Graydots
representtheasteroids. Curveddashedlinesinthetoppanelrepresenttheboundaries forEarth-,Mars-,andJupiter-
crossingorbits(fromlefttoright). Inthebottompanel,thecurveddashedlinerepresentsthecurrentlocationoftheν6
secularresonance,whichoccurswhentheprecessionrateofanasteroid’slongitudeofperihelionisequaltothemean
precessionrateoftheperihelionofSaturn.VerticaldashedlinesshowMMRsbetweenasteroidsandJupiter.
population.The2-planetand4-planetGrandTacksimulationsalsogiveeccentricitydistributions
quite differentfrom each other, with the distribution obtained in the 4-planetsimulation being
evenmoreskewedtolargevalues.ManyparticlesinthefinalGrandTackdistributions,however,
mustclearlybeunstable,becausetheirsmallsemimajoraxisand/orlargeeccentricitymakethem
planet-crossing.
Figure4, showssnapshotsof theevolutionofthe AsteroidBeltin oursimulations,starting
from the final 2-planet Grand Tack configuration. The left panel illustrates the initial system,
correspondingtothesolidblacklinesofFig. 3. Consideringtheevolutiondescribedinsection
2(withplanetsonpre-instabilityorbits),andusingCase1(low-eccentricitygiantplanetsorbits
from Morbidellietal. (2007)), after 400 My of evolution we obtain the distribution depicted
in the right panel of Figure 4. The colors of the particles in the left panels correspond to the
particles’lifetimesinthissimulation.ResultsrelatedtoCase2(giantplanetswithmoreeccentric
orbits), will be considered only for measuring the influence of the primordialeccentricities of
JupiterandSaturnandpresentedinsection3.3.
Thedistributionafter400Myisalreadyclosertothecurrentone(seeFig.2)becausethevery
high eccentricity objects have been removed due to interactions with the planets whose orbits
they crossed. The main differencewith the currentmain beltdistribution is that oursimulated
beltextendsinwardsof2.2AUinsemimajoraxis. Thisisbecausethepowerfulν resonanceis
6
notinplaceyet,giventhatJupiterandSaturnarestillinresonanceandonquasi-circularorbits.
7
Time = 0.000 Gy Time = 0.000 Gy
1 1
ber 0.9 ber 0.9
um 0.8 um 0.8
N N
ve 0.7 ve 0.7
mulati 00..56 mulati 00..56
u u
C 0.4 C 0.4
d d
malize 00..23 malize 00..23
Nor 0.1 Nor 0.1
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 45
Eccentricity Inclination ((cid:176))
Time = 0.000 Gy Time = 0.000 Gy
1 45
mber 0.8 3450
u
ormalized Relative N 000...246 (cid:176)Inclination () 1122305050
N 5
0 0
1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Semimajor axis (AU) Eccentricity
Figure3: Normalized cumulative numberofasteroids asafunction oftheeccentricity (topleft)andinclination (top
right). Normalizedrelativenumberofasteroidsasafunctionofsemimajoraxis(bottomleft). Thewidth∆aofthebins
ofthehistogramis0.02AUinsemimajoraxis. Inallthesepanelsthegraylinereferstotherealdistribution,thesolid
anddashedblacklinesrefertothefinaldistributionsinthe2-planetand4-planetGrandTacksimulationsrespectively.
Bottomright: inclinationasafunctionoftheeccentricity. LargegraydotsrepresenttherealasteroidswithH<10from
theMinorPlanetCentercatalog. Smallblackdotsrepresentourinitialdistributiongeneratedasdescribedinsection2
(thedistributionattheendofthe2-planetGrandTacksimulation,i,e.,afterthedispersionofthegasnebula).
ThisextensionofthemainbelttowardstheterrestrialplanetsispreciselytheE-beltadvocatedin
Bottkeetal.(2012).
Althoughnot shown, the distribution after 700 My of evolution is nearly indistinguishable
fromthatshownintherightpanelofFig. 4. Thisdemonstratesthattheexacttimingofthegiant
planetinstabilityisnotimportantforthisstudy.
Inordertobetterunderstandtheevolutionduringthefirst400My,bylookinginmoredetail
atFig. 42,weseethataverystrongdepletion(∼71%)hasoccurred,approximatelyhalfinthe
first100My. Thisisduetothelargecollectionofparticlesonclearlyunstableorbits,asnoted
above. In fact, this happened mostly to objects with e > 0.4, due to either collision with the
terrestrialplanets(10%oftheremovedparticles)orJupiter(1.5%)orejectionsontohyperbolic
orbits (88% of the removed particles). Only 0.5% of the removed particles collided with the
Sun. ThisnumberismuchsmallerthanthatforthecurrentNear-Earthasteroids(Farinellaetal.,
1994;Gladmanetal.,1997). Thisisbecausethegiantplanetsareonmorecircularorbits,which
makesthemeanmotionresonancesmuchlesseffectiveinrisingeccentricitiesthaninthecurrent
2An animation of the entire evolution, from 0 to 4.5 Gy, can be found electronically at ex-
tranet.on.br/rodney/rogerio/asteroids.mp4
8
Time = 0.000 Gy Time = 0.400 Gy
1 MMRs with Jupiter 7:2 3:1 5:27:3 2:1 N(cid:176) of Ast. = 9888 0.4 1 MMRs with Jupiter 7:2 3:1 5:27:3 2:1 N(cid:176) of Ast. = 2897 4.5
Eccentricity 0000....2468 Earth Jupiter 000...23355 Gy) Eccentricity 0000....2468 Earth Jupiter 334.5 Gy)
0 Mars 0.2 me ( 0 Mars 2.5 me (
(cid:176)nation () 234000 00..115 Lifeti (cid:176)nation () 234000 12.5 Lifeti
ncli 10 0.05 ncli 10 1
I 0 0 I 0 0.5
1.6 2 2.4 2.8 3.2 3.6 4 1.6 2 2.4 2.8 3.2 3.6 4
Semimajor axis (AU) Semimajor axis (AU)
Figure4: Left: Distribution oftheparticles resultingfromtheGrandTack2-planet simulation, whichconsitutes the
initialconditionforourfirstsimulationfrom0to400My. Right: distributionoftheparticlessurvivingat400My,just
beforetheplanetaryinstability,whichconstitutestheinitialconditionforoursecondsimulation,from400Myto4.5Gy.
Particlesarecoloredaccordingtotheirlifetimeinthesimulationforwhichtheyrepresenttheinitialconditions. Thus,
onlyredparticlessurviveuntiltheendofthesimulation.Curveddashedlinesinthetoppanelsrepresenttheboundaries
forEarth-,Mars-,andJupiter-crossingorbits(fromlefttoright). VerticaldashedlinesshowthemainMMRsbetween
asteroidsandJupiter.
SolarSystem(meanmotionresonancesarestableifJupiterisonacircularorbitandwouldnot
opengaps, Morbidelli(2002)). Also, theorbitsofthegiantplanetsareclosertoeachother,so
theν secularresonanceisfurtheroutinthebelt. Consequently,neitherthemeanmotionnorthe
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secularresonancesareabletolifttheparticles’eccentricitiestounity.
Althoughthedistributionof particlesattheendofthe GrandTack evolutioncoversa wide
rangeofeccentricities,semimajoraxes,andinclinations(asshownintheleftpanelofFig. 4),in
ordertoestimatetheAsteroidBeltmassWalshetal.(2011)consideronlyparticlesinthe“belt
region”, with q > 1.9 AU and a < 3.2 AU. In this region, ourdepletionratio is ∼ 24% during
thefirst400Myand∼16%withinthefirst100My. TheGrandTackmodelplaces∼ 1.3×10−3
M ofS-typesasteroidsinthe“beltregion”,andthreetimesasmuchofC-typesasteroids,which
⊕
impliesatotalmassof∼5.2×10−3M forthepost-GrandTackmainbelt.
⊕
Thus,after400My,westillhave∼76%ofthemassremainingintheprimordialmainbelt,
which is ∼3.9×10−3 M , or about6-7 the currentmass of the currentAsteroid Belt (∼6×10−4
⊕
M ).
⊕
3.2. Aftertheplanetaryinstability
Thesecondphaseofthenumericalsimulationsbeginaftertheplanetaryinstability. Here,as
explainedinsection2,theplanetsareinstantaneouslyplacedontotheircurrentorbits. Withthe
new orbits, the secular resonances(particularlythe ν resonanceat2.05AU) appearwith their
6
fullpowerandseveralmeanmotionresonanceswithJupiterbecomeunstabledueto thelarger
eccentricities of Jupiter and Saturn. The lifetimes of the particles that constituted the initial
conditionsforthissecondsimulationareillustratedviaacolorscaleintherightpanelsofFig.4.
Figure 5 shows the final orbitaldistributionof the particlessurvivingat the end of the full
simulation, at4.5 Gy. One may note thatthereis a nice qualitativematch betweenthe current
AsteroidBeltdistributionandthatproducedinthesimulation(comparewithFig. 2). Therange
ofdistributioninsemimajoraxes,eccentricities,inclinations,andthestructureoftheKirkwood
andsecularresonantgapslookverysimilarinthetwocases.
9
Time = 4.500 Gy
0.6 MMRs with Jupiter 7:2 3:1 5:2 7:3 2:1 N(cid:176) of Ast. = 669
0.5
ity 0.4 Earth
c
ntri 0.3 Jupiter
e
c 0.2
c
E 0.1
Mars
0
n
) 40 6
(cid:176)(
n 30
o
i
t
a 20
n
i
l
c 10
n
I
0
1.6 2 2.4 2.8 3.2 3.6 4
Semimajor axis (AU)
Figure5:Configurationofthesystem4.1Gyaftertheplanetaryinstability(i.e.atthecurrenttime).Blackdotsrepresent
theasteroidsthesurvivedinthesimulation. CurveddashedlinesinthetoppanelrepresenttheboundariesforEarth-,
Mars-,andJupiter-crossingorbits(fromlefttoright).Inthebottompanel,thecurveddashedlinerepresentsthecurrent
locationoftheν6secularresonance.VerticaldashedlinesshowMMRsbetweenasteroidsandJupiter.
Fig. 6 comparesthe final distribution of the surviving simulated particles with the current
Asteroid Belt orbital distribution, in a manner similar to what we did in Fig. 3 for the initial
distribution.
Fig.6(bottomleftpanel)comparesthefinalsemimajoraxisdistributions. Thevisualcom-
parison is very satisfactory, with the exception of the zone between the 7:3 and 2:1 MMRs,
whichismorerelativelypopulatedbyrealasteroidsthaninourmodel. Tobemorequantitative,
thefractionofthetotalasteroidpopulationorbitingbetweenmajorKirkwoodgapsisreportedin
Table1forboththeobservedandthemodelpopulations.Again,weseeafairlygoodagreement.
Theinclinationdistribution(Fig. 6topright)doesnotchangeverymuch(comparewithFig.
3)fromthebeginningtotheendofoursimulations,movingjustslightlytowardslargerinclina-
tionvalues. Indeedlowinclinationasteroidsareremovedsomewhatmorefrequentlythanhigh
Kirkwoodgapratios Zone1 Zone2 Zone3 Zone4
MajorBelt 0.106 0.373 0.093 0.427
Simulation 0.147 0.394 0.141 0.318
Table1:FractionoftheasteroidpopulationorbitingbetweenmajorKirkwoodgaps.Allfractionsarecaulculateddividing
thenumberofasteroidsinbetweentwoadjacentKirkwoodgapsbythetotalnumberofasteroidsbetweenthe7:2and
the2:1MMRs. Thus: Zone1representsthefractionofasteroidsbetweentheMMRs7:2and3:1. Zone2thefraction
betweenthe3:1and5:2MMRs.Zone3betweenthe5:2and7:3MMRs,andZone4betweenthe7:3and2:1MMRs.
10