jsons_folder/2014MNRAS.439..623G.json

{
  "Robert J.J. Grand ⋆ , Daisuke Kawata, Mark Cropper ": [],
  "Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT ": [],
  "ABSTRACT ": [
    "Recent numerical N-body simulations of spiral galaxies have shown that spiral arms in N-body simulations do not rotate rigidly as expected in classic density wave theory, but instead seem to rotate at a similar speed to the local rotation speed of the stellar disc material. This in turn yields winding, transient and recurrent spiral structure, whose co-rotating nature gives rise to changes in the angular momentum (radial migration) of star particles close to the spiral arm at many radii. From high resolution N-body simulations, we highlight the evolution of strongly migrating star particles (migrators) and star particles that do not migrate (non-migrators) around a spiral arm. We investigate the individual orbit histories of migrators and non-migrators and find that there are several types of migrator and non-migrator, each with unique radial evolution. We find the important quantities that a˙ect the orbital evolution to be the radial and tangential velocity components in combination with the azimuthal distance to the spiral arm at the time the star particle begins to feel tangential force. We contrast each type of orbit to compare how these factors combine for migrators and non-migrators. We find that the positive (negative) migrators sustain a position behind (in front of) the spiral arm, and feel continuous tangential force as long as the spiral arm persists. This is because the positive (negative) migrators are close to the apocentre (pericentre) epicycle phase during their migration, and rotate slower (faster) than the co-rotating spiral arm. On the other hand, non-migrators stay close to the spiral arm, and pass or are passed by the spiral arm one or two times. Although they gain or lose the angular momentum when they are behind or in front of the spiral arm, their net angular momentum change becomes close to zero. We discuss also the long term e˙ects of radial migration on the radial metallicity distribution and radial angular momentum and mass profiles. ",
    "Key words: galaxies: evolution -galaxies: kinematics and dynamics -galaxies: spiral -galaxies: structure "
  ],
  "1 INTRODUCTION ": [
    "Radialmigrationreferstoachangeintheguidingcentreofastarasitorbitsthegalacticcentre.Theprincipalphenomenonthatcausesradialmigrationistheinteractionwithnon-axisymmetricstructuresuchasspiralarms,especiallyaroundtheco-rotationradius(e.g.(<>)Goldreich&Lynden-Bell(<>)1965;(<>)Lynden-Bell&Kalnajs(<>)1972;(<>)Sellwood&Binney(<>)2002).Unlikeothermechanismssuchasmolecularcloudscattering((<>)Spitzer&Schwarzschild(<>)1951,(<>)1953)anddiffusionthroughphasespace((<>)Binney&Lacey(<>)1988;(<>)Brunettietal.(<>)2011),radialmigrationbyspiralarmsdoesnotnecessarilyincreasetheorbitaleccentricityofthestars((<>)Lynden-Bell&Kalnajs(<>)1972).",
    "Thephenomenonwasfirstdiscoveredby(<>)Goldreich&Lynden-Bell((<>)1965)and(<>)Lynden-Bell&Kalnajs((<>)1972).Forastanding,long-liveddensitywavepattern,theangularrotationspeedofthespiralarms,thepatternspeedp,isconstantovertheentireradialrangeofthespiralstructure.Thereisasingleradialpointinthediscwherethespiralarmsrotateatthesamespeedasthediscmaterial.Astarnearthespiralarmatthisradiusmaythereforebeacceleratedordeceleratedsuchthatthestarguidingcentremovesawayfromortowardsthegalacticcentrerespectively.Thisco-rotationradiuswasfoundtobetheonlyradiusinthediscaroundwhichradialmigrationfromspiralarmswithoutheatingcanoccur((<>)Lynden-Bell&Kalnajs(<>)1972;(<>)Lépineetal.(<>)2003).",
    "(<>)Sellwood&Binney((<>)2002)showedthattheradialmi-",
    "grationprocesscanhavelongtermeffectsontheentirestellardisc(whichtheytermradialmixing)ifthediscdevelopssuccessivetransientspiralarms.Transientspiralarmsareseeninallsimulationsandtheirexistencerepresentsadeparturefromclassicdensitywavetheory((<>)Lin&Shu(<>)1964),inwhichthespiralarmsarelong-livedandstandingwavefeatures.Instead,whilethereductionofnumericalnoiseasaresultofimprovedresolution(N > 106 ,asdescribedin(<>)Fujiietal.(<>)2011)enablesthespiralmorphologyofgalaxiestopersistformanyrotationperiods,simulationsrepeatedlyshowthatindividualspiralarmsdisappearandreappearonthetimescaleofagalacticrotation.",
    "Radialmigrationproducedfromtransientspiralarmshasbeenhighlightedinmanyrecentnumericalstudies((<>)Roškaretal.(<>)2008;(<>)Minchev&Famaey(<>)2010;(<>)Fujiietal.(<>)2011;(<>)Grandetal.(<>)2012a,(<>)b;(<>)Minchevetal.(<>)2011,(<>)2012;(<>)Fujii&Baba(<>)2012;(<>)Roškaretal.(<>)2012;(<>)Solwayetal.(<>)2012),althoughtheprecisemechanismoftheradialmigrationprocessisstilldebated,whichstemsmainlyfromuncertaintiesinourunderstandingofthespiralarmnature.Forexample,(<>)Minchev&Quillen((<>)2006);(<>)Minchev&Famaey((<>)2010);(<>)Comparetta&Quillen((<>)2012);(<>)Chamandyetal.((<>)2013)explainthetransientspiralarmsasasuperpositionofmultipledensitywavemodepatternsthatspanseparateradialrangesthatoverlap.Becauseinnerpatternsrotatefasterthanouterpatterns,theyaresaidtoconstructivelyinterfereperiodicallywhichcausesthegrowthanddecayofaspiralarmonthetimescaleofaninterference.Ontheotherhand,otherstudies(e.g(<>)Wadaetal.(<>)2011;(<>)Grandetal.(<>)2012a,(<>)b;(<>)Fujiietal.(<>)2011;(<>)Fujii&Baba(<>)2012)reportthespiralarmtobeanamplifiedover-densitywhoserotationspeedmatchesthatofthediscmaterialatallradii.Suchspiralarmsarenaturallywindingandtransientasshownby(<>)Wadaetal.((<>)2011)whoreportstransientspiralarmsthatexhibitaverysmoothlydecreasingpatternspeedasafunctionofradius.Theco-rotatingspiralarmissupportedby(<>)Grandetal.((<>)2012a,(<>)2013),whoperformedN-bodysimulationsofnon-barredgalaxiesembeddedinastaticdarkmatterhalopotentialandtracedthespiralarmpeakdensitydirectlyfromthedensitydistribution.Thisisfurtherbacked-upby(<>)Roca-Fàbregaetal.((<>)2013),whoperformedhighresolutionN-bodysimulationswithalivedarkmatterhalo.",
    "Theorbitalevolutionofparticlesthatradiallymigratehastodatebeendiscussedinfewstudies.(<>)Roškaretal.((<>)2012)explainthatinaspiraldiscofmultipleco-existingdensitywavepatterns,radialmigrationoccursonlyattheco-rotationradiusofeachpattern.Fromasampleofstarparticleschosenfromthetop10% ofmigrators,theyinterpretradialmigrationoveralargeradialrangeastwosuccessivediscreteparticle-patterninteractions,whereaparticlemaybetransportedfromtheco-rotationradiusofonepatternclosetothelocationoftheotherpattern.Inourpreviouswork,(<>)Grandetal.((<>)2012a,(<>)b),weshowedthatradialmigrationcanoccurcontinuouslyoveralargeradialrangeuntilthespiralarmdisappears.WeexaminedindividualN-bodystarparticlesandfoundthattheorbitaleccentricitywaslargelyconserved.(<>)Babaetal.((<>)2013)studytheorbitevolutionofstarparticlesintheirhighresolutionN-bodysimulation.However,theyfocusonarandomsampleofstarparticlesassociatedwiththe“non-steady”spiralarmsinordertolinktheformationanddisruptionofspiralarmsto",
    "themotionsofitsconstituentstars,withlessfocusonradialmigration.",
    "Inthispaper,wecomplementthesestudiesandbuilduponourownworkbyrunningahighresolutionsimulationofagalacticdisctoexploreindetailtheinteractionofstarparticleswiththespiralarm.Inparticularwefocusonstarparticlesthatshowsignificantradialmigration(migra-tors),andstarparticlesthatshowalmostnomigration(non-migrators)betweenthebirthanddeathofaspiralarm.Wepresentdetailedstep-by-stepevolutionofeachgroupofparticle,whichrevealsseveraltypesofmigratorsaswellasnon-migratorseachwithdifferentorbitalcharacteristics,noneofwhich(toourknowledge)havebeenreportedinliterature(includingourpreviousworks,(<>)Grandetal.(<>)2012a,(<>)b).Individualorbitsaretrackedextensivelytocoverthetimebefore,duringandafterasinglespiral-particleinteraction.Ourspiralarmpeaktracingmethod((<>)Grandetal.(<>)2012b)enablesustofollowtheevolutionoftheparticlepositionwithrespecttothespiralarm,whichisaquantitycurrentlyunexploredintheliterature.Thisisanimportantdiagnosticthatallowsustoidentifyandexplainthepropertiesofthemigrating/non-migratingstarparticlesinourMilkyWaysizedsimulation.Therefore,thesetypesoforbitsmaybeobservablefortheGalaxyinGalacticsurveyssuchasRAVE(e.g.(<>)Steinmetzetal.(<>)2006;(<>)Pasettoetal.(<>)2012a,(<>)b;(<>)Siebertetal.(<>)2012;(<>)Williamsetal.(<>)2013),Gaia (e.g.(<>)Lindegrenetal.(<>)2008),Gaia-ESO((<>)Gilmoreetal.(<>)2012),APOGEE(e.g.(<>)AllendePrietoetal.(<>)2008;(<>)Bovyetal.(<>)2012),SEGUE(e.g.(<>)Yannyetal.(<>)2009;(<>)Leeetal.(<>)2011),LAMOST(e.g.(<>)Chenetal.(<>)2012)and4MOST((<>)deJong(<>)2012).",
    "Themainresultsofthispapercomefromthedetailedanalysisofmanyindividualstarparticleorbits,henceforbrevityweshowtheresultsfromonehighresolutionsimulation.Howeverwebrieflydiscusstheapplicabilityoftheseresultstoothersimulationsofdifferentspiralstructure,anddiscusstheeffectsofradialmigrationontheglobalproperties,suchasthemetallicityandangularmomentumdistribution.",
    "Thispaperisorganisedasfollows.InSection2,wedescribethesimulation.InSection3,wedescribeourparticleselection.InSection4weanalysethegroupevolutionofthesamplesinvariousphasespaceprojectionsanddescribeoverallmacroscopicbehaviour.Wethenexaminetheorbitsofindividualparticlesandcategoriseseveraltypesofmigra-torsandnon-migratorsinSection5.Theorbitalcharacteristicsofeachtyperevealdeterminingfactors,whichwhencombinedtogetherdistinguishthemigratorsfromthenon-migrators.Insection6,wediscusstheapplicabilityoftheresultstoothersimulations,andbrieflyshowtheevolutionoftheglobalmassandmetallicitydistributionsasaconsequenceoftheradialmigration.Insection7,wesummariseourconclusions."
  ],
  "2 SIMULATION ": [
    "ThesimulationinthispaperisperformedwithaTreeN-bodycode((<>)Kawata&Gibson(<>)2003;(<>)Kawataetal.(<>)2013),andsimulatesagalaxycomprisedofasphericalstaticdarkmatterhaloandalivestellardisconly(foramoredetaileddescriptionofthesimulationset-up,see(<>)Grandetal.(<>)2012a).",
    "Alivedarkmatterhalocanrespondtothe",
    "self-gravitatingstellardiscbyexchangingangularmomentum.Thisisprominentonlongtimescalesforlong-livednon-axisymmetricstructuressuchasabar((<>)Debattista&Sellwood(<>)2000;(<>)Athanassoula(<>)2002,(<>)2012).However,theeffectofthelivedarkmatterhaloisexpectedtobesmallfortransientspiralarms,whichjustifiestheuseofastaticdarkmatterhaloforourinvestigation.Furthermore,forpracticalreasonsalivedarkmatterhaloisoftenmod-elledwithparticlesmoremassivethandiscparticles,whichmayintroducesomescatteringandheatingthatdependsonthescaleofthemassdifferencebetweentheparticlespecies.Therefore,intheinterestofcomputationalspeedandamorecontrolledexperiment,weuseastaticdarkmatterhalo.",
    "Thedarkmatterhalodensityprofilefollowsthatof(<>)Navarroetal.((<>)1997)withtheadditionofanexponentialtruncationterm((<>)Rodionov&Athanassoula(<>)2011):",
    "(1)",
    "whereˆc isthecharacteristicdensitydescribedby(<>)Navarroetal.((<>)1997),theconcentrationparameter,c = r200/rs = 20,andx = r/r200.Thetruncationterm,exp(−x2),isneededtoensurethatthetotalmassisconvergent.Itdoesnotchangethedarkmatterdensityprofileintheinnerregionofthedisc,whichisthefocusofthispaper.Thescalelengthisrs,andr200 istheradiusinsidewhichthemeandensityofthedarkmattersphereisequalto200ˆcrit (whereˆcrit= 3H02/8ˇG;thecriticaldensityforclosure):",
    "(2)",
    "whereM200 = 1.5 × 1012 M⊙,h = H0/100 km s−1 Mpc−1 .Weassume0 = 0.266,b= 0.0044 andH0 = 71 km s−1 Mpc−1 .",
    "Thestellardiscisassumedtofollowanexponentialsurfacedensityprofile:",
    "(3)",
    "whereMd,∗ = 5 × 1010 M⊙ isthediscmass,Rd,∗ = 3.5 kpcisthediscscalelengthandzd,∗ = 350 pcisthediscscaleheight.ThesimulationpresentedinthispaperhasN = 1×107 particlesinthestellardisc,sothateachstarparticleis5000 M⊙.ThisissufficienttominimisenumericalheatingfromPoissonnoise((<>)Fujiietal.(<>)2011;(<>)Sellwood(<>)2013).Weapplyafixedsofteninglengthof160 pc(Plummerequivalentsofteninglengthof53 pc)forstarparticleswiththesplinesofteningsuggestedby(<>)Price&Monaghan((<>)2007)."
  ],
  "3 PARTICLE SELECTION OF STRONG MIGRATORS AND NON-MIGRATORS ": [
    "Inthisstudy,wefocusonsinglespiralarm-starparticleinteractionsonthetimescaleofthespiralarmlifetime,whichwefindtobe˘ 100 Myr.Wescrutinizetheevolutionofasampleofstarparticlesinordertoobtaindetailedinformationontheseinteractions.",
    "Thefirststepistoidentifycoherentspiralarmsforwhichwecanreliablytracethespiralarmpeakposition.Areliabletraceisdefinedasaradialrangeforwhicheachradialpointshowsasmoothsingledensitypeakinazimuth.Asnapshotinwhichthespiralarmexhibitsadoublepeakstructureanywherewithinthedefinedradialrangefortracingisrejected.Doublepeakfeaturesareusuallyassociatedwiththeformationanddestructionstagesofspiralarmevolution,andarenotsuitableforanunambiguoustrace(see(<>)Grandetal.(<>)2012b,formoredetails).Fig.(<>)1showstheface-ondensitysnapshotsofourselectedspiralarm.Thereisatimewindowoverwhichthespiralarmcanbereliablytraced,andawidertimewindowwhichwillbeusedtoexaminethepriorandsubsequentstarparticlebehaviour.Wecouldtracethespiralarmreliablyattheradiibetween6and10kpcinthetraceabletimewindowspanningt = 1.968 to2.024 Gyr(highlightedwiththewhitelineinFig.(<>)1).Outsideofthistimewindow,weextrapolatethespiralarmpositionbyrotatingtheR = 7, 8 and9 kpcpeakpositionsofthet = 2.0 Gyrsnapshotwiththecircularvelocity(anchorsmarkedwithwhitecrosses)toguidetheeyetothespiralarmwhenitformsanddisrupts.Thepatternspeediscalculatedbysimplesubtractionofthepeaklinebetweensnapshots.Fig.(<>)2showstheradialprofileofthetime-averagedpatternspeed(dashedredline)andtheangularrotationalvelocityofthedisc(solidblackline).Thelatteriscalculatedfromtheradialforceaveragedoverazimuthforeachradius.Fig.(<>)2confirmsthatthespiralarmisco-rotatingwiththerotationalvelocityofthedisc(asshownin(<>)Grandetal.(<>)2012a;(<>)Roca-Fàbregaetal.(<>)2013).Fortherestoftheanalysis,wefocusonparticleinteractionswiththisspiralarm.",
    "Wechooseasnapshotatatimewhenthespiralarmisfullyformedandadoptt = 2.0 Gyr(Fig.(<>)1)asthetimeofparticleselection.Fromthetracedspiralarmatthistime,wedefinearegionwithinarangeof4 kpceithersideofthedensitypeakofthespiralarmintheazimuthaldirection.Theregionisboundedbyaradialrangeof6 -10 kpcandaverticalheightof|z| < 0.1 kpc.Thesampleisconstitutedofparticlesfoundinsidethisspatiallydefinedregionatt = 2.0 Gyr.Thisprobesawiderangeinradiusandazimuthalpositionwithrespecttothespiralarm,andrestrictsthemajorityofparticlestobeintheplaneofthediscbecausemoremigrationtakesplaceintheplaneofthedisc.Notethatovertime,theverticaloscillationscausesomeparticlestomovetoheightsthatexceedtheselectioncutof|z| < 0.1 kpc.Wefoundthatabout15% ofthestarparticlesselectedreachzmax> 0.35 kpc(oneinitialgalacticscaleheight).However,wefindthatthereisnosignificantdifferencebetweenthetrendsdiscussedinthispaperforparticlesofdifferentmaximumheights.",
    "Thetimeatwhichthestarparticlesareselectedisde-finedasthecentraltimestep,Tc= 2 Gyr,andatimewindowisthendefinedasT = Tfin−Tini,whereTini= Tc−48 MyrandTfin= Tc+ 48 Myr.Thistimewindowspansthe˘ 100 Myrlifetimeofthespiralarm.Notethatthistimewindowislongerthanthetimewindowforwhichwecouldtracethespiralarm.However,asseeninFig.(<>)1,thespiralarmbeginstoformataroundt = 1.952 Gyranddisruptsataroundt = 2.048 Gyr.Themotionofnearbystarparticlescanbeaffectedatthesetimes.Infact,wewilldemonstrateinSection.5thatthemotionofsomestarparticlescanbeaffectedasearlyast ˘ 1.9 Gyrandlastuntilt ˘ 2.2 Gyr.",
    "Figure 1. The face-on density snapshots showing the sequence of the traceable spiral arm. The white line indicates the traced spiral arm of interest. The spiral peak position for R = 7, 8 and 9 kpc radii at the t = 2.0 Gyr snapshot are rotated with the corresponding circular velocity and marked as anchors (white crosses) on the t = 1.952, 2.032 and 2.048 Gyr snapshots, to guide the eye to the forming and disrupting stages of the spiral arm respectively. ",
    "Hencethistimewindowissetbyconvenienceandnotastrictdefinitionoftheformationanddestructiontime.ThestarparticlesampleisthenplottedintheLz,ini− Lzplane,whereLz,iniisthez−componentofangularmomentumofthestarparticlesatthebeginningofthetimewindow,Tini,andLzisthechangeinangularmomentumfromtheinitialtime,Tini,tothefinaltime,Tfin.ThisisplottedinFig.(<>)3.FromFig.(<>)3,weseethatthereisawiderangeofinitialangularmomentumvaluesoverwhichtheangularmomentumi.e.theguidingcentre,ischanged,whichisconsistentwithourpreviousstudies((<>)Grandetal.(<>)2012a,(<>)b).",
    "FromthestarparticledistributioninFig.(<>)3,weselectasampleintherange1.86 × 103 < Lz,ini< 1.97 × 103 kpc km s−1 (particlesamplesatotherLz,iniexhibitsimilarbe-haviour,sowefocusononesample),whichcorrespondstoaguidingcentreradiusofabout8kpc.Thesampleisfurthercutintosubgroups:strongmigrators(bothnegativeandpositive)andnon-migrators.Thestrongpositive(negative)migratorsconsistofthoseparticlesthathavethelargestpositive(negative)Lz,andthenon-migratorsarethosethathavethelowestchangesinangularmomentum.Thesestarparticlesareselectedsuchthatthereare˘ 200 − 300 particlesineachstarparticlegroup,andarehighlightedinFig.(<>)3."
  ],
  "4 EVOLUTION OF SAMPLE IN PHASE SPACE ": [
    "Weploteachparticleinthesampleinvariousprojectionsinphasespace,andhighlightthepositivemigrator(blue",
    "Figure 2. The pattern speed (red dashed line) of the traced spiral arm highlighted above. The angular speed of the stellar disc is also plotted (solid black line). The pattern speed matches the angular speed of the disc material well. ",
    "circles),negativemigrator(redsquares)andnon-migrator(blackdiamonds)groupsinFig.(<>)4.Eachcolumncorrespondstoadifferenttime:theleftcolumnshowsthesampleattheearliesttimethatthespiralarmcouldbereliablytraced(followingcriteriain(<>)Grandetal.(<>)2012b),themiddlecolumn",
    "Figure 4. Top row: The negative migrators (red squares), positive migrators (blue circles) and non-migrators (black diamonds) of the sample in vθ− vRspace. Middle row: Shows the azimuthal distance between the star particles and the spiral arm peak position, R, as a function of azimuthal velocity, vθ. Bottom row: Plots the radius of the same star particles with vθ. Each column shows these projections at three time epochs, increasing from left to right. The circular velocity at the 8 kpc radius, vc= 243.5 km s−1 , is marked in each panel by the vertical dashed green line. ",
    "isthetimeofselection,Tc,andtherightcolumnisthelatesttimethatthespiralarmcouldbereliablytraced.",
    "ThetoprowofFig.(<>)4showsthesampleplottedintheradialvelocity,vR-azimuthalvelocity,vθ,plane.Positiveradialvelocity,vR> 0,isinthedirectionofthegalacticcen-tre.ThecircularvelocityatR = 8 kpcisvc(R = 8) = 243.5 kms−1 ,whichismarkedbythedashedgreenlinesinFig.(<>)4.Thepositiveandnegativemigratorsappeartooccupyseparateregionsofvelocityspace,whereasthenon-migrators(blackdiamonds)aremoreevenlydistributedandoverlapthemigratorgroups.Thepositivemigratorshaveoutwardfacingvelocities(vR< 0)duringtheiroutwardmigration.ItisinterestingtoseethattheirazimuthalvelocitytendstobeslowerthanthecircularvelocityatR = 8 kpc.Theoppositeisappliedtothenegativemigrators,whichmoveinward(vR> 0)androtatefasterthanthecircularvelocityatR = 8 kpc.",
    "Theazimuthaldistanceofastarparticlewithrespecttothespiralarm,R,isdefinedasthelengthofanarcthatjoinsthestarparticleazimuthpositiontothespiralarmpeakazimuthpositionatthatradius.ThesecondrowofFig.(<>)4showsthisquantityasafunctionofazimuthalvelocityfor",
    "thesample.Inthisplane,eachgroupofparticles(includingnon-migrators)isveryclearlyseparated.Thepositivemi-gratorsalwaysstaybehindthespiralarm(R < 0),andthenegativemigratorsalwaysstayinfrontofthespiralarm(R > 0),throughoutthetraceablespiralarmlifetime.Thenon-migratorsareclusteredaroundthespiralarm.Atthet = 1.968 Gyr(leftpanel),thearrangementofeachparticlegroupishighlyordered.Thenon-migratorgroupinparticularisspreadoveralargerangeofazimuthalvelocity,whichappearstightlycorrelatedwiththeazimuthaldistancebetweenstarparticleandspiralarm.Forexample,att = 1.968 Gyr,atvθ= 220 km s−1 thepositivemigratorshaveanegativeR,i.e.theyarebehindthespiralarm,whilethenon-migratorshaveapositiveR,i.e.theyareinthefrontofthespiralarm.Conversely,atvθ˘ 265 km s−1 ,thenegativemigratorshaveapositiveR,i.e.theyareinthefrontofthespiralarm,whilethenon-migratorshaveanegativeR,i.e.theyarebehindthespiralarm.InSection.5.3,wedemonstratethatnon-migratorsmustpassorbepassedbythespiralarmatsomepointduringthespiralarmlifetime.Roughlyspeaking,thestarparticlewillcrossthespiralarmif:|R| < | R tt01 vθ,subdt|,wheret1 −t0 < T ,",
    "Figure 3. The change in angular momentum of the sample of particles over the time window Tfin− Tini, as a function of their initial angular momentum. Over-plotted are the strong positive migrators (white with black outline symbols), strong negative mi-grators (white symbols) and non-migrator particles (black symbols) selected. The units are kpc km s−1 . ",
    "andvθ,sub= vθ− vc,wherevθistheazimuthalvelocityofthestarparticleandvcisthecircularvelocityattheparticleradius.Althoughthesestarparticleswereselectedatt = 2.0 Gyr,theyaremoreorderedatt = 1.968 Gyr.ThisindicatesthattheR−vθphasespacecanbediagnosticattheearlystagesofspiralarmformation(middle-leftpanelofFig.(<>)4)inpre-determiningwhetherastarparticlewillbeamigratororanon-migrator.Atlatertimes,thegroupsbecomelessclearlyseparated,butstillmaintainthetrend.",
    "ThebottomrowofFig.(<>)4showsthesampleintheR − vθplane.Att = 1.968 Gyr(leftpanel),migratorandnon-migratorparticlegroupsoccupythesameregionofthisspace,andbecomemoreseparatedatthelatertimes(middleandrightpanelsofFig.(<>)4)owingtothemigrationtakingplace.Thedistributionofnon-migratorsinthisplanehighlightstheepicyclicmotionofthestarparticles.Starparticlesthatareataradiusgreaterthantheguidingcentreofthesample,R > Rg˘ 8 kpc,possessazimuthalvelocitieslowerthanthecircularvelocityattheguidingcentre,vθ< vc(Rg),whereasstarparticlesataradiussmallerthantheguidingcentrepossessazimuthalvelocitieslargerthanthecircularvelocityattheguidingcentre.Positivemigra-torsobviouslymovetowardlargerradii,andareoutsideoftheirguidingcentreowingtotheirrelativelylowrotationvelocitywithrespecttothecircularvelocityatthatradius.Inotherwords,thepositivemigratorsarealwaysclosetotheapocentrephaseduringtheirmigration,andthenegativemigratorsarealwaysclosetothepericentrephase.",
    "Fig.(<>)5showstheevolutionofbothpositivemigrators(bluestars)andnegativemigrators(redstars)plottedontheface-ondensitymapsofthediscintheR − fsplane.Theevolutionisshowninarotatingframethatco-rotateswiththecircularvelocityatR = 8 kpc.The coordinateofallstarparticleshasbeensubtractedbyanamountcorrespondingtotherotatingframesuchthatthespiralarmandsampleparticlesremainwithinthe120-280degreeazimuthwindowi.e.fs= true− frt,wheret = Tini−t.Herefristheangularrotationspeedoftheframe.Thedirectionofmotionisfromrighttoleft.Eachmigratorparticleisoutlinedinwhite(black)toindicatetheoutward,",
    "Figure 5. Time sequence of a close up of the density map in the R−fsplane. The rotation is from right to left in a rotating frame that co-rotates with the circular velocity at R = 8 kpc. Positive migrators are marked as blue stars, and negative migrators are marked as red stars. The radial velocity direction is indicated by the white and black outline of the symbols, which represent outward and inward moving radial velocities respectively. The white line (present in some panels) highlights the peak position of spiral arm at each radius. ",
    "vR< 0 (inward,vR> 0)directionoftheradialvelocityvector,inordertoindicatetheepicyclephasei.e.vR< 0 meansthatthestarparticleismovingfrompericentretoapocentre,whilevR> 0 indicatesthatthestarparticleismovingfromapocentretopericentre.Bothgroupsexhibitarangeofradialvelocitiesateachsnapshot,whichindicatesthereissomespreadintheepicyclephasewithinthegroups.Forthepositivemigrators,particlesclosertothespiralarmatt = 1.928 and2.024 Gyrhavenegativeradialvelocities",
    "Figure 6. The same as Fig. (<>)5 but showing non-migrator particles. The red line indicates the spiral arm peak position. ",
    "(approachingapocentre)andparticlesfurtherfromthespiralarmhavepositiveradialvelocities(movingawayfromapocentre).Theoppositetrendisseeninthenegativemi-gratorgroup.Despitethedifferentepicyclephasesofthesemigratorgroups,allstarparticlesinthepositiveandnegativemigratorgroupsradiallymigrateeventually,asweshowbelow.",
    "Fig.(<>)6showstheevolutionofthenon-migratorsintheR − fsdensityplaneintherotatingframedescribedaboveforFig.(<>)5.Thesymbolsareoutlinedinblackandwhitecorrespondingtoinwardandoutwardradialvelocityunitvectorsrespectively.Mostofthenon-migratorsareclusteredclosearoundthespiralarm(peakpositionmarkedinred),andappearspatiallyseparatedaccordingtothedirectionofradialmotion.Forexample,att = 1.992 and2.008 Gyr,most",
    "starparticlesbehind(infrontof)thespiralarmaremovingtowardspericentre(apocentre).Thisisaclearcontrastfromthemigrators.Att = 1.976 Gyr,thepositivemigratorsbehindthespiralarmaremovingtowardapocentre,i.e.outward,whilethenegativemigratorsinthefrontofthespiralarmaremovingtowardpericentre,i.e.inward."
  ],
  "5 INDIVIDUAL PARTICLE ORBITS ": [
    "Inthissection,weanalysetheorbitsofthepositive,negativeandnon-migratorparticlesofthesampleindividually.Wetook˘ 100 randomsamplesofeachstarparticlegroup,andfollowedtheevolutionofeachorbitindividually.Wescru-tinisedtheorbitsoverthecourseoftheparticle-spiralarminteraction,andcategoriseseveraltypesofmigratorsandnon-migrators.Belowweshowanexampleofeachtypeandoutlinetheirdefiningfeatures.Werefertopositivemigra-torswithasuffix‘g’becausetheygainangularmomentum,negativemigratorswithasuffix‘l’becausetheyloseangularmomentumandnon-migratorswithasuffix‘n’.InTable.(<>)1,welisttheorbitalpropertiesofeachorbitaltypementionedbelow."
  ],
  "5.1 Orbits of positive migrators ": [
    "Fig.(<>)7showsanexampleofthreedifferenttypesofpositivemigrator.Intheleftpanels,weshowaselectionofsnapshotsfromthetimesequenceevolutionoftheseparticlesinthesamerotatingframeadoptedinFig.(<>)5.Thesymbolineachsnapshotindicatesthepositionoftheparticleatthetimegiveninthebottom-rightcornerofthepanel,andthelinesshowthehistoryoftheorbitintheco-rotatingframe.Becausethespiralarmrotateswiththecircularvelocity(Fig.2),thespiralarmintheleftpanelsofFig.(<>)7movesrelativetotherotatingframeintime:forR < 8 kpc,thespiralarmmovestolowerfs,whereasforR > 8 kpcthespiralarmmovestohigherfs.Weremovetherelativemotionbetweenthespiralarmandstarparticleorbithistoriesbymakingafurtheradjustmenttotheorbithistory.Wecalculatethedifferencebetweenthevelocityoftherotatingframeandthecircularvelocityattheradiusofthelinepoint.Ateachsubsequenttimestepafterthelinepointappearsthepositionofthelinepointisshiftedbytheamountcorrespondingtothisvelocitydifference.Thepurposeofthisadjustmentistogiveanideaofwherethepastparticlepositionswererelative to the spiral arm atprevioustimes,althoughitisnotexactbecauseitisimpossibletoplacethepastorbitaroundthedynamicallychangingspiralarm.",
    "The1st-and2nd-rightpanelsofFig.(<>)7showstheevolutionofparticleangularmomentum,Lz,andthetangentialforceperunitmass,Fθ,wherethepositivedirectionisoppositetothedirectionofrotation,i.e.fromlefttorightintheleftpanelsofFig.(<>)7.Bluesolid,blackdashedandgreendot-dashedlinescorrespondtotheblue,blackandgreenlinesintheleftpanelsofFig.(<>)7.",
    "Wedefineradialmigrationtobeachangeinangularmomentumovertime,suchasthechangeseeninthetop-rightpanelofFig.(<>)7,e.g.theblackdashedlinefromt ˘ 1.89 to2.08 GyrwhenLzincreases.NotethatthetangentialforceisalwaysnegativeduringtheincreaseinLz.Themagnitude",
    "Figure 7. The evolution of three types of positive migrator. Left panels: The time evolution of the particles in the close up R − fsdensity map in a rotating frame equal to the circular velocity at R = 8 kpc. Rotation is from right to left. The symbols depict the current position of each star particle, and the lines show the history of each star particle orbit relative to the spiral arm (see text for more details). Top-right panel: The evolution of the angular momentum of the star particles. Second-right panel: The evolution of the tangential force per unit mass, Fθ, acting upon the star particles. Third-right panel: The radial evolution of the star particles. Bottom-right panel: The azimuthal angle of the star particles in the rotating frame of the left panels, fs, (lines) and spiral arm azimuthal angle at the particle radius (symbols). The latter can only be calculated in the traceable spiral arm time window defined in Section. 3. In all right-hand panels, the time at which each particle first feels the tangential force is indicated by a vertical line of corresponding line style. ",
    "ofthetangentialforceindicatestherateofchangeofangularmomentum,whichallowsustoseemoreclearlywhenandwherethestarparticlesmigrateintheleftpanelsofFig.(<>)7.Thethird-rightandbottom-rightpanelsofFig.(<>)7showtheevolutionoftheparticleradius,R,andazimuthangleintherotatingframe,fs,respectively.Inthebottompanel,thebluestar,blacktriangleandgreensquareindicatetheazimuthangleofthespiralarmthatweidentifiedinFig.(<>)1atthestarparticleradiusatthecorrespondingtime.Notethatwecouldtracethespiralarmforonlypartoftheperiodwhenthespiralarmisclearlyseen.However,thisdemonstratesthatthespiralarmaffectstheorbitofthestarparticleswellbeforethearmisclearlyseenandevenafteritbeginstodisrupt.Atleastinthisshortperiodwhenwecanclearlytracethespiralarm,wecanshowtheparticle",
    "positionwithrespecttothespiralarmasseeninthebottompanelofFig.(<>)7.TheevolutionofeachquantityshownintherightpanelsofFig.(<>)7willdeterminethetypeofeachpositivemigrator."
  ],
  "5.1.1 Type 1g positive migrator ": [
    "Thefirsttype(Type1g)ofpositivemigrator(bluestarandsolidbluelineintheleftpanelsandbluesolidlineintherightpanelsofFig.(<>)7)isquiteclosetothespiralarmwhenitbeginstofeelanegativetangentialforceataroundt ˘ 1.9 Gyr(2nd-rightpanelofFig.(<>)7).ThetopleftpanelofFig.(<>)7showsthatthespiralarmbeginstobuildupatt ˘ 1.9 Gyr.Itappearsthatthedensityenhancementsaroundfs= 180 andfs= 220 atR ˘ 8 kpcatt = 1.912 Gyrmergeandform",
    "thesinglespiralarmaroundt ˘ 1.976 Gyr.Att = 1.912 Gyrthebluestarislocatedbehindthedensityenhancementatfs= 220,andthereforethedirectionoftangentialforceisnegative(Fθ< 0)andthestarparticleisaccelerated.",
    "Att ˘ 1.9 Gyr,theparticleisapproachingtheapocen-trephaseoforbit(3rd-rightpanelofFig.(<>)7).However,asaresultofstrongnegativetangentialforceatthisphase(2ndrightpanelofFig.(<>)7),theradiusofthestarparticledoesnotdecreaseagainaccordingtonormalepicyclemotion.Thisisbecauseofthecompetitionbetweenmainlytheradialgravitationalforceandtheincreaseincentrifugalforcecausedbythegaininangularmomentum.Inthiscase,theradialforceisbalancedbytheincreasedcentrifugalforce.Asaresult,thestarparticlepausesataradiusofR ˘ 7.9 kpcfor˘ 20 − 30 Myrataroundt ˘ 1.94 Gyr,thenincreasesagainoncetheangularmomentumhasincreasedsuchthatthecentrifugalforceislargeenoughtoovercometheradialgravitationalforce.",
    "Notethatirrespectiveoftheevolutionofradius,theincreaseinangularmomentumissustainedastheparticlecontinuestobeacceleratedbythenegativetangentialforce,becausetheparticleisalwaysbehindthespiralarm.Thebottom-rightpanelofFig.(<>)7showsthattheazimuthangleoftheparticleisalwayslargerthanthatofthespiralarm(starparticleisbehindthespiralarm)duringtheepochatwhichthespiralarmisclearlytraced.Thestrongmigratorsareabletostayononesideofthespiralarm,becausethespiralarmco-rotateswiththediscmaterial((<>)Grandetal.(<>)2012a).Asaresult,positivemigratorsmaintaintheirpositionbehindthespiralarmandcontinuetobeacceleratedandmigratealongthespiralarm.Ataroundt ˘ 2.08 Gyrwhenthespiralarmisdisrupted,theparticlestopsgainingangularmomentumandthestarparticleresumesepicyclemotion."
  ],
  "5.1.2 Type 2g positive migrator ": [
    "Thesecondtype(Type2g)ofpositivemigrator(blacktriangleandsolidblacklineintheleftpanelsandblackdashedlineintherightpanelsofFig.(<>)7)beginstofeelanegativetangentialforceataboutthesametimeastheType1gpositivemigratormentionedabove(t ˘ 1.89 Gyr),andisaccelerated.Again,thereisacompetitionbetweentheradialgravitationalforceandtheincreaseinangularmomentum.However,atthistimetheparticlehasjustpassedtheapoc-entrephaseoftheepicyclemotion(3rd-rightpanelofFig.(<>)7)whentheangularmomentumbeginstoincrease.Therefore,thestarparticlebeginstomoveinwardforawhile,untiltheangularmomentumincreasessufficientlytoovercomethegravitationalforce.Asaresult,theamplitudeoftheepicyclemotionisshortened.Thisshorteninginamplitudeisshownbythechangedpericentreradiibetweent ˘ 1.83 and1.93 Gyrinthe3rd-rightpanelofFig.(<>)7.Thepericen-treradiusislargeratthelatertimeowingtotheincreaseinguidingcentre.NotethatalthoughtheradialevolutionoftheorbitlooksdifferenttothatoftheType1gpositivemigratorshownabove,theangularmomentumsteadilyincreasesirrespectiveoftheirradialevolutionbecausethestarparticleisalwayslocatedbehindthespiralarmandaccelerated(bottom-rightpanelofFig.(<>)7).",
    "Thisisthemostcommontypeofpositivemigratorinthissample.Theshortenedepicyclemotionpresentin",
    "theradialevolutionoftheType2gpositivemigratorisalsoreportedin(<>)Roškaretal.((<>)2012).Thestrongestmigra-torsintheirsimulationareshowntoexhibitseveralshortenedepicyclemotions,whichtheyinterpretaseffectsofco-rotationresonancesoftwospiralwaves:oneinner,fasterrotatingspiralpatternandoneouter,slowerrotatingspiralpattern.Forexample,inthetoppanelsofFig.11of(<>)Roškaretal.((<>)2012),theorbitofthemigratorshows˘ 6 epicyclemotionsfromt ˘ 5.4 to5.9 Gyr.Inourstudy,wefocuson1-2epicycleperiodswhichcorrespondstothelifetimeofthespiralarminoursimulation.TheType2gmigratordemonstratesthatduringtheshortenedepicyclemotiontheguidingcentreofthestarparticlecontinuouslyincreasesowingtotheangularmomentumincreaseateveryradius.Therefore,wethinkthatitisdifficulttoattribute6shortenedepicyclemotionsseenin(<>)Roškaretal.((<>)2012)toonlytwoco-rotationresonances.Instead,wethinkthatthespiralarmfeatureco-rotateswiththediscmaterialateveryradius((<>)Grandetal.(<>)2012a),andinducesacontinuousgaininangularmomentumofmigratorsaslongasthefeaturepersists.Wesuspectthattheirspiralarmisshort-lived,andthatthe6epicyclemotionsseenover˘ 0.5 Gyrareaffectedbyseveraltransientspiralarmfeaturesthatco-rotateandacceleratestarparticlesbehindthespiralarmatallradii."
  ],
  "5.1.3 Type 3g positive migrators ": [
    "Thethirdtype(Type3g)ofpositivemigrator(greensquareandgreensolidlineintheleftpanelsandgreendot-dashedlineintherightpanelsofFig.(<>)7)originatesmuchfartherfromthespiralarmasitbeginstobuildupataroundt ˘ 1.9 Gyr(top-leftpanelofFig.(<>)7),andconsequentlybeginstofeelanegativetangentialforceatthelatertimeoft ˘ 1.93 GyrwithrespecttoTypes1gand2g.Atthistime,thestarparticleapproachesthepericentrephaseoftheepicycle(3rdrightpanelofFig.(<>)7),andthereforeitmovesclosertothespiralarmatt ˘ 1.976 Gyrintheleftpanelsandbottom-rightpanelofFig.(<>)7.Inthiscase,boththeoutwardepicyclemotionandtheangularmomentumincreasefacilitatetheoutwardmotionofthestarparticle.Asaresult,theparticleradiusincreasesveryrapidly(third-rightpanelofFig.(<>)7).",
    "Notethatthestarparticlereachestheapocentreatt ˘ 2.02 Gyr,wellbeforetheangularmomentumgainendsatt ˘ 2.06 Gyr.Thismeansthataroundt ˘ 2.02 Gyr,theradialgravitationalforcebecomesgreaterthantheenhancedcentrifugalforceprovidedbytheboostinangularmomentum.Theradialevolutionofthestarparticlethenproceedsinwardswhiletheparticlecontinuestogainangularmomentumuntilepicyclemotionresumesatt ˘ 2.06 Gyr."
  ],
  "5.2 Orbits of negative migrators ": [
    "Fig.(<>)8showstheevolutionofanexampleofeachtypeofnegativemigratorfoundintheparticlesample.EachpanelisthesameasinFig.(<>)7."
  ],
  "5.2.1 Type 1l negative migrator ": [
    "Thefirsttype(Type1l)ofnegativemigrator(bluestarandsolidbluelineintheleftpanelsandbluesolidlineinthe",
    "Figure 8. The same as Fig. (<>)7 but for types of negative migrator. ",
    "rightpanelsofFig.(<>)8)isthecounterpartoftheType1gpositivemigratorshowninFig.(<>)7.The2nd-rightpanelofFig.(<>)8showsthatthestarparticlebeginstofeelapositivetangentialforceatatimeofaroundt ˘ 1.91 Gyr,whentheparticleismovingtowardsthepericentrephaseoforbit(3rdrightand1st-leftpanelofFig.(<>)8).Asthespiralarmgrowsindensity,e.g.t = 1.944 and1.976 Gyr(2nd-and3rd-leftpanelsofFig.(<>)8),thestarparticlecontinuestofeelastrongpositivetangentialforceaccompaniedbyasteepnegativeslopeintheangularmomentumevolution(1st-rightpanelofFig.(<>)8).Att ˘ 1.98 Gyr,theradialgravitationalforceandcentrifugalforcearetemporarilybalancedandthestarparticlestaysatR ˘ 7.2 kpcfor˘ 30 − 40 Myr-asimilarradialpausetothatoftheType1gmigratordescribedintheprevioussection.Att ˘ 2.02 Gyr,theradialgravitationalforceplusthelossinangularmomentumovercomethecentrifugalforce,hencethestarparticlemovesradiallyinwardagain.Asthespiralarmbeginstofadeataroundt ˘ 2.06 Gyr,thetangentialforcediminishes,andthestarparticleresumesnormalepicyclemotion.Again,thestarparticlehas",
    "remainedinfrontofthespiralarmandcontinuallymigratedalongthespiralarm."
  ],
  "5.2.2 Type 3l negative migrator ": [
    "TheotherexamplestarparticleshowninFig.(<>)8isthenegativemigratorcounterpartoftheType3gpositivemigrator,andthereforewedesignateitasType3l(blacktriangleandblacksolidlineintheleftpanelsandblackdashedlineintherightpanelsofFig.(<>)8).ThisstarparticleoriginatesfartherfromthespiralarmthanType1lasthespiralarmbeginstobuildupatt = 1.912 Gyr(top-leftpanelofFig.(<>)8).Therefore,thestarparticlebeginstofeelthepositivetangentialforce(deceleration)atatimeoft ˘ 1.93 Gyr,whichislaterthanthetimeatwhichtheType1lnegativemigratorbeginstoloseangularmomentum(2nd-leftand2nd-rightpanelsofFig.(<>)8).Atthistime,thestarparticleisclosetotheapocentrephaseoforbit(3rd-rightpanelofFig.(<>)8),andconsequentlymovesclosertothespiralarmatt ˘ 1.976 Gyr(3rd-leftpanelofFig.(<>)8).SimilartotheType3gpositivemigrator,theinwarddirectionoftheangular",
    "momentumchangeandepicyclemotionmeansthatthestarparticlemovesrapidlytowardsthecentreofthegalaxy(leftpanelsofFig.(<>)8).Again,thestarparticlereachestheperi-centreatt ˘ 2.01 Gyr,beforetheangularmomentumlosshasceased.Thismeansthatthecentrifugalforcebecomesgreaterthantheradialgravitationalpotentialeventhoughthestarparticlecontinuestoloseangularmomentum.Thetangentialforcediminishesatt ˘ 2.07 Gyr,whentheparticleresumesepicyclemotion."
  ],
  "5.2.3 No Type 2l negative migrator ": [
    "WecouldnotfindanegativemigratorcounterparttotheType2gpositivemigrator.Thismaybeattributedtothepresenceofanotherspiralarme.g.theoneseenonthefrontsideofthemainspiralarmataround(R kpc,fs)= (8, 130) at1.976 Gyr(3rd-leftpanelofFig.(<>)8)relativetothemainspiralarmonwhichwefocus(R, fs)= (8, 185).Thisiscloserthanthespiralarmbehindthemainspiralarmat(R, fs)= (8, 265).Thecloserproximitytothemainspiralofthedensityenhancementonthefront-sideofthespiralarmincomparisontothedensityenhancementonthebacksideofthespiralmaycausethemotionsofstarparticlesinfrontofthespiraltobemoreinfluencedbyneighbour-ingdensityenhancementsthanthosebehindthemainspiralarm.",
    "TomirrortheType2gpositivemigrator,theType2lnegativemigratorswouldhavebeguntoloseangularmomentumjustaftertheypassedpericentre.Inthisepicyclephase,theType2lstarparticlewouldtemporarilymoveawayfromthespiralarm(becauseithasahigherrotationvelocityinthisphasethanthespiralarm),anditispossiblethattheover-densityinfrontofthespiralarm“moppedup”thesestarparticles,whichwouldeitherreduceorchangesignofthepositivetangentialforceactinguponthestarparticle.Thiswouldmeanthatthesestarparticlesgainedsomeangularmomentumfromthisover-densityduringthetimewindowexamined.Therefore,thesestarparticleswouldnotbestrongmigrators,becausetheywouldpopulatearegionoflower|Lz| inFig.(<>)3.Inthiscase,theywillnotbeselectedinourstrongmigratorsample."
  ],
  "5.3 Orbits of non-migrators ": [
    "Inthissection,weanalysetheorbitalevolutionofthosestarparticlesthatexperienceverylittleornonetchangeinangularmomentumoverthetimewindow(non-migrators).Fig.(<>)9showsanexampleofeachofthethreetypesofnon-migratorinthesample."
  ],
  "5.3.1 Type 1n non-migrator ": [
    "Thefirsttypeofnon-migrator(Type1n,representedbythebluestarandsolidbluelineintheleftpanelsandthesolidbluelineintherightpanelsofFig.(<>)9),isveryclosetowherethespiralarmbeginstobuildupatt ˘ 1.912 Gyr(1st-leftpanelofFig.(<>)9),anditbeginstofeelanegativetangentialforceearlyatt ˘ 1.86 Gyr(2nd-rightpanelofFig.(<>)9).Att = 1.944 Gyr(2nd-leftpanelFig.(<>)9),thestarparticleislocatedinbetweenthemainspiralarmat(R, fs)= (8, 185) andtheweakerspiralarmbehinditat(R, fs)= (8, 205) -",
    "incontrasttothepositivemigratorswhichwerebehindbothdensityenhancementsatthistime.Thestarparticlemovestowardspericentre(3rd-rightpanelofFig.(<>)9),whileitgainsangularmomentum(top-rightpanelofFig.(<>)9).Becausethestarparticleisaroundpericentreandlocatedclosebehindthearm(3rd-leftpanelofFig.(<>)9),thestarparticlemovestowardsthespiralarmuntilt ˘ 1.99 Gyrwhenitpassesthespiralarm,asseeninthebottom-rightpanelandthe3rdand4th-leftpanelsofFig.(<>)9.Theparticleisnowlocatedonthefrontsideofthespiralandasaresultthetangentialforceactinguponthestarparticlehasbecomepositive,whichcausesthestarparticletobegintoloseangularmomentum.Thestarparticlethenresumesepicyclemotionasthespiralarmfadesatt ˘ 2.05 Gyr.",
    "Thegeneralpropertiesofthesenon-migratorsisthattheyoriginateveryclosetothespiralarm,andorbitthespiralarmuntilitdisappears,continuallygainingandlosingangularmomentumsuchthatthenetangularmomentumgainisLz˘ 0,attheendofthetimewindow.Itisworthnotingthatwhenthestarparticleiscirclingaroundthespiralarm,theamplitudeoftheepicyclemotionissmaller.Thisisbecausethetangentialforcefromthespiralarmactsalwaystochangetheguidingcentreinthedirectionoppositetothatofepicyclemotion.Therefore,asthestarparticlemovestowardspericentre,itisaccelerated(Fθ< 0),whichincreasestheguidingcentreandinturnincreasestheperi-centreradius.Forexample,thepericentreradiusislargeratt ˘ 1.96 Gyrthanatt = 1.84 Gyr,andatt ˘ 2.09 Gyrthepericentreisreturnedtothatatt ˘ 1.84 Gyr.Thisindicatestheamplitudeoftheepicyclemotionisshortened,becausethetangentialforcefromthespiralarmactsagainsttheepicyclemotion.TheType1norbitisfoundtooriginatealsoinfrontofthespiralarmwhenitfirstfeelsthetangentialforce."
  ],
  "5.3.2 Type 2n non-migrator ": [
    "Thesecondtype(Type2n)ofnon-migrator(blacktriangleandsolidblacklineinleftpanelsanddashedblacklineinrightpanelsofFig.(<>)9)islocatedinfrontofthespiralarmasitformsatt ˘ 1.944 Gyr(1st-and2nd-leftpanelsofFig.(<>)9).Itbeginstofeelapositivetangentialforceandmigratesinwardatt ˘ 1.91 Gyr.Thisoccursjustbeforethestarparticlereachestheapocentreoftheorbit,andatt ˘ 1.95 − 1.97 Gyrthelossofangularmomentumandtheinwardepicyclemotion(1st-and3rd-rightpanelsofFig.(<>)9)causetheradiusofthestarparticletodecreaserapidlyinamannersimilartoType3lnegativemigrator.Thiscausesadecreaseinpericentreradius(seetheradiusatt ˘ 1.87 andt ˘ 2.0 Gyrin3rd-rightpanelofFig.(<>)9).However,unliketheType3lnegativemigrator,thisnon-migratoristooclosetothespiralarmwhenitreachesapocentreatt = 1.94 Gyrandasaresultispassedbythespiralarmatt ˘ 1.98 Gyr(bottom-rightpanelofFig.(<>)9).Thiscausesthechangeoftangentialforcefrompositivetonegative,andthestarparticlethengainsbacktheangularmomentumlostpreviously(top-rightpanelofFig.(<>)9).Thisonceagainensuresthatthisnon-migratorhasanetangularmomentumchangeofLz˘ 0 attheendofthetimewindow.",
    "ThedifferenceinradialevolutionbetweenthisType2nandtheType1nnon-migratorsisthattheType2nnon-migratorshowsanincreaseinepicycleamplitudebecause",
    "Figure 9. The same as Fig. (<>)7 but for types of non-migrator. ",
    "thepericentreatt ˘ 2.0 Gyrislowerwithrespecttotheearliertime,t ˘ 1.86 Gyr,i.e.theamplitudeofepicyclemotionisincreasedinsteadofdecreasedasinthecaseoftheType1nnon-migrator.",
    "Itisinterestingtonotethatthistypeofnon-migratorisnotfoundtooriginatefrombehindthespiralarm.Thismayberelatedtothedifferencesbetweenthefront-andback-sideofthespiralarmasdiscussedinSection.5.2.3.Inaddition,asdiscussedinSection5.1.1and5.3.1,thereisasmalldensityenhancementthatmergeswiththeback-sideofthespiralarmatt ˘ 1.976 Gyr.Thereforetheback-sideofthespiralarmseemstobuilduplaterthanthefront-side,andthislikelycausesthedifferenceinthevarietyintheorbitsofstarparticlesthatoriginatefromthefront-andback-sideofthespiralarm."
  ],
  "5.3.3 Type 3n non-migrator ": [
    "Thethirdtype(Type3n)ofnon-migratorparticlefoundinthissample(thegreensquareandsolidgreenlineinleftpanelsandgreendot-dashedlineintherightpanelsofFig.",
    "9",
    "(<>)9)beginstofeelanegativetangentialforceatt ˘ 1.88 Gyr(2nd-rightpanelofFig.(<>)9)justasitreachestheapocen-treoftheorbitatt ˘ 1.912 Gyr(top-leftandthird-rightpanelsofFig.(<>)9).Thisisaccompaniedbyanincreaseinangularmomentum(top-rightpanelofFig.(<>)9)fromaboutt ˘ 1.88 − 1.99 Gyr.Duringthistime,theradiusofthestarparticleproceedstoapocentreandexhibitsasmalldipinradiusatt ˘ 1.96 Gyr(3rd-rightpanelofFig.(<>)9).Thisisobviouslythefeatureofanincreasedpericentreradiusowingtotheimbalancebetweentheradialgravitationalforceandcentrifugalforceboostedbyagaininangularmomentum.Thepericentre-likefeatureisalsoseeninthebottom-rightpanelofFig.,whichshowsadecreasingfsofthestarparticle(line)thatbringsthestarparticleclosertothespiralarminazimuthangle(squaresymbols)att ˘ 1.96 Gyr.",
    "Att ˘ 1.99 Gyr,thestarparticlepassesthespiralarmandbeginstoloseangularmomentum(rightpanelsinFig.(<>)9).Uptot ˘ 1.99 Gyr,thisnon-migratorexhibitedsimilarevolutiontoType1gpositivemigrators,butitdifferedafterthistimebycrossingthespiralarminsteadofremainingbehindit.Thisisentirelybecausethisnon-migratoristooclose",
    "Table 1. Summary of orbital characteristics for each orbital type. Columns show 1) orbital type name 2) the time at which the particle first feels the tangential force, tcapture(measured by eye to provide an indication) 3) initial particle position with respect to the spiral arm at tcapture4) the epicycle phase of the particle at tcapture5) the e˙ect on their epicycle amplitude 6) whether or not they remain on the same side of the spiral throughout the spiral lifetime. ",
    "Figure 10. Close up of the density map in the R − fsplane at t = 1.976 Gyr, with contours of the tangential force over-plotted in white. Solid lines indicate positive tangential force (acting in the direction ˆ) and dashed lines indicate negative tangential force (acting in the direction −ˆ). The position of the Type 1l negative migrator and Type 2n non-migrator are represented by a blue star and black square respectively, and their orbit histories shown by the solid lines of the same colour. ",
    "tothespiralarm,suchthatevenavθthatisslightlyfasterthanthespiralarmrotationvelocitywilltaketheparticlepastthespiralarm.Againthishighlightstheimportanceofazimuthangleofthestarparticlewithrespecttothespiralarm.Thistypeofnon-migratorexhibitsshortenedepicycleamplitudewhileitremainsonthesamesideofthespiralarm.ThisisdifferentfromType2n,whichshowsextendedepicycleamplitude,anditisalsodifferentfromType1n,whichshowsshortenedepicycleamplitudeasitcirclesthespiralarm.ThisType3nnon-migratorisfoundbothinfrontofandbehindthespiralarm,unlikeType2n."
  ],
  "5.4 The tangential force ": [
    "Wehaveshownabovethattimeperiodsofsustainedtangentialforce,whichcausestrongradialmigration,depend",
    "onthestarparticlepositionwithrespecttothespiralarm.Forexample,thenegativemigratorsthatfeelstrongpositivetangentialforcesloseangularmomentumwhiletheyfeelthatforce,whichispossiblebecausetheparticlealwaysremainsinfrontofthespiralarm.However,thereexistsomenon-migratorsthatspendtimeinsimilarregionsaroundthespiralasmigrators,butuponwhichalmostnotangentialforceacts.",
    "Toexplorethis,wecomputeamapofthetangentialforce.Fig.(<>)10showsthedensitymapofthespiralarmwithcontoursoftangentialforceover-plottedinwhiteatt = 1.976 Gyr(solidforpositivevaluesanddashedfornegativevalues).Asexpected,thetangentialforceiszeroatthedensitypeakofthespiralarm.Thetangentialforcebecomesstrongerwithincreasingazimuthaldistancefromthespiralarm,untilitreachesamaximum,˘ 10-20 degreeawayfromthedensitypeakoneitherside.Atfartherdistancesfromthepeakofthespiralarm,thetangentialforcedecreasesandeventuallychangesthesign,owingtotheforcecontributionsfromneighbouringspiralarms.Itisinterestingtoseethedifferencebetweenthefront-andback-sidesofthespiralarm.Onthefront-sideofthearmthepointofthemaximumtangentialforceisclosertothedensitypeakofthespiralarmthanthatontheback-sideofthespiralarm.Thisisbecausetheneighbouringspiralarmonthefront-sideiscloserthantheoneontheback-side.Ontheback-side,thesmalldensityenhancementmergestothemainspiralarmaroundR = 8.5 kpc,whichpushesthepointofthemaximumtangentialforcefartherfromthedensitypeakofthemainspiralarm.",
    "WeshowtheorbithistoryoftheType1lnegativemi-grator(bluestarandsolidbluelineinFig.(<>)10)showninSection.5.2.1andtheType2nnon-migrator(blacksquareandsolidblacklineinFig.(<>)10)showninSection5.3.2.Thefig-ureshowsthatthenegativemigratoriscapturedinaregionofstrongtangentialforceinfrontofthespiralarm,whichcontinuesoverthespiralarmlifetime(see2nd-rightpanelofFig.(<>)8).However,thenon-migrator,despitebeinglocatedinfrontofthespiralarmandhavingaverysimilarorbittothenegativemigrator,feelsalmostnotangentialforceoverthistimeperiod(see2nd-rightpanelofFig.(<>)9)becausethestarparticleistooclosetothepeakofthespiralarm.Thisimpliesthatitisnotenoughthatthestarparticleislocated",
    "Figure 11. The change in angular momentum, Lz, between t = 1.0 and 2.0 Gyr of the disc particles as a function of their angular momentum at t = 1.0 Gyr. Red colours indicate regions of high number density. ",
    "Figure 12. The cumulative profiles of total disc mass fraction (upper curve) and angular momentum (lower curve). The solid red and dashed black curves show the profiles at t = 1.0 and t = 2.0 Gyr, respectively. ",
    "ononesideofthespiralarminordertobecomeamigra-tor:themigratormustbefarenoughawayfromthespiralarmpeakline-yetnottoofar!-inordertobecapturedbyregionsofstrongtangentialforcesuchasthoseshowninFig.(<>)10.Wethereforerefinetheconditionforastrongpositive(negative)migratortobeastarparticleabletoremainatasuitable distancebehind(infrontof)thespiralarm.Thisre-emphasizestheimportanceoftheR parameterindistinguishingstrongmigratorsfromnon-migrators."
  ],
  "6 BROADER IMPLICATIONS OF RADIAL MIGRATION ": [
    "Themainresultsofthispaperfocusonthecomplicatedinteractionsbetweenthespiralarmandavarietyoforbitaltypes,ontimescalesroughlyequaltothespiralarmlifetime.Eachtypeofmigratorparticleexperiencesachangeinangularmomentumonthesetimescales(Fig.(<>)3).Onlongertimescales,i.e.1Gyr,astarparticlemayinteractwithaspiralseveraltimes.Thecumulativeeffectofconsecutivespiralarm-starparticlesinteractionsisreflectedintheevolutionofglobalpropertiesofthedisc,suchasthemassandmetaldistribution.Inthissection,webrieflydiscusstheeffectsofradialmigrationinthissimulationonlongertimescales.",
    "Co-rotatingspiralarmshavebeenshowntocauseradialmigrationatmanyradii,andthisworkconfirmsthatmigrationwillcontinueaslongasthestarparticleremainsonthesamesideofthespiralarminaregionofstrongtangentialforce.Thismeansthatradialmigrationinthissimulationisefficientattransportingstarparticlestodifferentguidingradii.ThisisshowninFig.(<>)11,whichshowsthechangeinangularmomentumofdiscparticlesbetweenthetimesof1.0 and2.0 Gyrasafunctionoftheirangularmomenumatt = 1.0 Gyr.ThisshowsmuchmoreradialmigrationthanfortheshortertimescaleshowninFig.(<>)3(˘ 5 timesasmuch).Althoughthisradialmigrationredistributestheindividualangularmomentaofstarparticles(seealsoFigs.(<>)7,(<>)8and(<>)9),",
    "theoverallangularmomentumdistributionofthediscandthecumulativemassprofileremainalmostunchanged(seeFig.(<>)12).",
    "Themovementofstarparticlesfromoneguidingradiustoanotheraffectsthedistributionofmetalsinthedisc.Toindicatetheeffectonthemetaldistributionoftheradialmigrationinthissimulation,weartificiallyassignmetallic-ityvaluestoeachstarparticleinthesimulationatatimeoft = 1.0 Gyr.Werandomlyassigneachstarparticleametalvaluebydrawingfromthegaussianmetallicitydistributionfunctionateachradiuswithadispersionof0.05 dex.Themeanonwhichagaussianiscentredatagivenradiusisdefinedbyametallicitygradientof−0.05 dex/kpc,and[Fe/H](R = 0)=0.25dex.ThetoprowofFig.(<>)13showsthesmootheddistributionofstarsinmetallicityasafunctionofradiusatt = 1.0 Gyr(leftpanel)andt = 2.0 Gyr(rightpanel).Themetallicitygradientdoesnotchangemuchatallbetween1.0and2.0Gyr.Thisisbecausetheradialmigrationisnotstrongenoughtomixstarsatsucharateastoaffecttheslope.However,themetallicitydistributionatanygivenradiusbroadenswithtime.ThebottomrowofFig.(<>)13showsthesameasthetoprowbutforaninitialradialmetallicitygradientof−0.1 dex/kpc.Att = 2.0 Gyr,althoughtheradialmetallicitygradientremainsunchanged,themetallicitydistributionshowsamuchlargerbroadeningowingtothethesteepergradient,asexpected.Despiteourcrudemetallicityanalysis,asimilareffectisseeninFig.16of(<>)Casagrandeetal.((<>)2011),whoshowthatinthesolarneigh-bourhood,themetallicitydistributionofstarsagedbetween1 − 5 Gyriscomparativelybroadincomparisonwiththatofstarsyoungerthan1 Gyr,andbothpopulationshavethesamepeakmetallicityvalue(seealso(<>)Haywood(<>)2008).Westressthattheseresultsarefromthissimulationonly,andshouldnotbetakenasthegeneralcaseforsimulatedspiralgalaxies.Forexample,parameterssuchasspiralarmstrengthandpitchanglelikelyplayaroleintheamountofradialmigrationandthereforethedegreeofmixingthatoc-",
    "cursinaspiraldisc.Moreover,thisresultisderivedfromaN-bodysimulationinwhichmetallicitieshavebeenassignedartificiallyatanarbitrarytime.Forarobustanalysisontheeffectofspiralmorphologyonmetaldistributions,manysimulationsthatincludethegascomponentandrecipesforstarformationandchemicalevolutionneedtobeanalysedandcomparedwitheachotherandobservation.Thistopicdeservesathoroughnumericalstudy,andwereservefurtherdiscussionuntilthen.",
    "Alloftheanalysispresentedinthispaperhasbeenwithregardtoonesimulation,whichhasaflocculentspiralstructurewithnobulgeorbaratthecentre.However,wehaveseensomeevidencethatthegroupcharacteristicsofthepositive,negativeandnon-migratorsarepresentalsoinsimulatedgalaxiesofdifferentspiralstructure,andshowsimilarseparationsinthephasespaceplanesshowninFig.(<>)4.Thesesimulationsincludea3-4armedgalaxywithastaticbulgeinthecentre(simulationFin(<>)Grandetal.(<>)2013),andthebarred-spiralgalaxyoftwoarmspresentedin(<>)Grandetal.((<>)2012b)."
  ],
  "7 CONCLUSIONS ": [
    "WehaveperformedhighresolutionN-bodysimulationsofaspiraldiscembeddedinastaticdarkmatterhalopotential.Wefocusonasampleofstarparticlestakenfromaroundthespiralarmanddividethissampleintothreegroupsaccordingtotheamountofchangeinangularmomentum,i.e.radialmigration,ofthestarparticlesoveragiventimeperiodwhenthespiralarmisstrong.Thesegroupsare:starparticlesthatgainthemostangularmomentum(positivemigrators),starparticlesthatlosethemostangularmomentum(negativemigrators)andstarparticlesthatshowalmostnochangeinangularmomentum(non-migrators).",
    "Wefollowtheevolutionofthesegroupsinvθ− vR,vθ− R andvθ− R planes,andcometothefollowingconclusions:",
    "Theslopeinthevθ− R spaceofthedistributionofnon-migratorsindicatesthatnon-migratorslocatedbehindthearmhavehigherazimuthalvelocitiesthefartherbehindthearmtheyare.Similarly,thenon-migratorsinfrontofthespiralarmhavelowerazimuthalvelocitiesthefartherinfrontofthespiralarmtheyarewhenthespiralisforming.",
    "Wethenexaminedtheorbitalevolutionofindividualstarparticlesineachparticlegroupindetail.Wecategorisedandcontrastedseveralorbitaltypesofeachgroup,eachof",
    "whichisanewtypeoforbitshownforthefirsttimeinthispaper.",
    "Therearethreetypesofpositivemigrators,eachchar-acterisedbythetimetheyfeelthetangentialforce,theirphaseofepicyclemotionwhenthisforceisintroducedandtheirazimuthaldistancefromthespiralarm:",
    "Type3goriginatesfartherfromthearmthanTypes1gand2g,andbeginstomigrateatlatertimesthanTypes1gand2g,aroundthepericentrephase,andasaresultexhibitsrapidchangesinradius.",
    "Thisvarietyoforbitsshowsthatpositivemigratorscanbeinanyepicyclephaseatthetimethetangentialforceisintroduced.However,migratorsthatoriginatefarfromthespiralarmmustbeinthepericentrephaseoforbitinordertocatchthespiralarmandundergolargeangularmomentumchanges.Migratorsthatarelocatedclosertothespiralarmbegintomigratearoundapocentrebecauseiftheyareinpericentre,theywillpassthespiralarmandanyincreaseinangularmomentumwillbearrested,asweshowinSection.5.3.Hence,thekeyforstrongpositivemigratorsistostaybehindthespiralarm,whichisverifiedinthebottom-rightpanelofFig.(<>)7foreachpositivemigratorexample.Inthisway,irrespectiveofthephaseofepicyclemotion,thestarparticlescontinuetogainangularmomentumfromthespiralarm.Wefindsimilartypesoforbitsforthenegativemigrators.However,wedidnotfindanyType2negativemigratoranaloguetoType2gpositivemigrator.Wethinkthatthisisbecausethespiralarmonwhichwefocushasdifferentpropertiesandformationprocessesonthefront-andback-sideofthespiralarm.",
    "Wefoundthreetypesofnon-migrators:",
    "Figure 13. Metallicity distribution of all disc stars as a function of radius at t = 1.0 Gyr (left panel) and t = 2.0 Gyr (right panel). Top: The initial assigned metallicity gradient is −0.05 dex/kpc, and the dispersion of the gaussian metallicity distribution function is 0.05 dex. Bottom: The initial assigned metallicity gradient is −0.1 dex/kpc, and the dispersion of the gaussian metallicity distribution function is 0.05 dex. In all panels the initial metallicity gradient is shown by the dashed black line. Contours and red colours indicate regions of high number density. ",
    "Eachtypeofnon-migratorshowsthatpositionrelativetothespiralarmisveryimportant,andhasconsequencesforthestarparticleevolution.ItisagainimportanttonotethatalthoughtheseparticleswereselectedasshowingLz˘ 0 withintheselectedmigrationtimewindow,theymaygoontomigratearoundotherspiralarmsatlatertimes,andmayhaveundergonemigrationaroundpastspiralarmsatearliertimes.",
    "Theimportanceoftheproximityofthestarparticletothespiralarmandtheepicyclephaseofmotionoftheparticlefoundintheorbitalanaylsisisconsistentwiththe",
    "trendseeninthesampledistributionintheR − vθplaneshowninthemiddlerowofFig.(<>)4.Itshowsthatstrongmi-gratorsarealwayslocatedonthesamesideofthespiralarmthroughoutthelifetimeofthespiralarm,whilenon-migratorstendtopass/bepassedbythespiralarmowingtothefaster/slowerrotationvelocitiesofthenon-migratorsrelativetothatofthespiralarm,givencloseenoughproximity.Thisunderscoresthecriterionforstarparticlestobecomestrongpositive(negative)migratorsastheabilityofastarparticletoremainbehind(infrontof)andstayclosetothespiralarmaslongasitispresent.",
    "Wehavefoundthattherearecertaindistancesfromthespiralarmatwhichexistregionsofstrongtangentialforce.Migratorstendtobecapturedbytheseregions,whereasnon-migratorsmaymissthestrongforceregionsbymovingtooclosetothespiralarm.Inthatcase,thestarparticledoesnotfeelmuchtangentialforceanddoesnotchangeangularmomentum.Thisemphasizestheimportanceofazimuthaldistancefromthespiralarm.Indeedwefindthatstrongmigratorsremainatasuitabledistancefromthespiralarm.",
    "Thedetailedinformationofparticleorbithistoriesoflargestarsamplesmayprovidecluestohowthespiralarmisformedanddestroyed.Thisinformationislikelytobefoundinnotjuststrongmigratorsandnon-migrators,butfrommuchlargerstarparticlesamplesthatexhibitmanydifferentdegreesofradialmigration.Fromthesekindofdata,itmaybepossibletoseehowthespiralarmisconstructedastheinfluxofstarparticlesfromregionsaroundthediscintotheformingspiralarmincreasesthedensityandeventuallyleadstothefullyformedspiralfeatures.Inthesamespirit,thefuturetrajectoriesofmanystarparticlesthatmakeupthespiralfeaturesmaygiveusaninsightintohowthesefeaturesaredisrupted.Thenon-linearnatureoftheparticlemotionthatwehaveglimpsedmakesthismethodofanalysisapromisingdirectiontopursue,andweleavethiscomplicatedinvestigationtofuturestudies.",
    "Theapplicabilityoftheseresultstotransientspiralsislimitedtotheco-rotatingspiral,giventhattheresultsholdformanyradii.Therefore,thisprecludestheclassicdensitywavetheory,whichisbasedonasingleco-rotationradius.Furthermore,therapidgrowthofspiralarmsfromsmallseedsandthepersistenceofspiralstructureintheabsenceoflargeperturbingmasses(seealsoD’Onghiaetal.2013)indicatestheimportanceofhighlynon-linearprocessesinthediscthatareatleastnotfullycapturedinlinearapproximationssuchasclassicswingamplificationtheory.Itmaybethatthespiralstructurearisesfromswingamplifi-cationatmanyradiithatoriginatesatoneormoreradii,andamplifiesandspreadstootherregionsofthediscwhenthepreviousamplificationspillsoverintoneighbouringradiiandpropagatesthegrowthmechanism(seealso(<>)Grandetal.(<>)2012a).Westressthatthisisourspeculation,andnotethatadetailedstudyisrequiredtoexploreotherpossiblemechanismsofspiralarmformation.",
    "Finally,weremarkupontheapplicabilityoftheseresultstorealspiralgalaxies.Weexpectthattheorbitalbe-haviourfoundinthispaperappliestoanyco-rotatingspiralarm,andassuch,allisolatedgalaxiesincludingthosewithabar(Grandetal.2012ab).However,itispossiblethatgalaxieswithcompanionsatellitesareexperiencingdifferentevolution((<>)Dobbsetal.(<>)2010;(<>)Purcelletal.(<>)2011).Therefore,forthemomentwerestrictourfindingstobeimportantforisolatedgalaxies."
  ],
  "ACKNOWLEDGEMENTS ": [
    "Theauthorsthanktherefereeforthoughtfulandconstructivecommentsthatledtotheimprovementofthemanuscript.ThecalculationsforthispaperwereperformedontheUCLLegion,theCrayXT4atCenterforComputationalastrophysics(CfCA)ofNationalAstronomicalObservatoryofJapanandtheDiRACFacilities(through",
    "theCOSMOSconsortium)jointlyfundedbySTFCandtheLargeFacilitiesCapitalFundofBIS.WealsoacknowledgePRACEforawardingusaccesstoresourceCartesiusbasedintheNetherlandsatSURFsara.Thisworkwascarriedout,inpart,throughtheGaia ResearchforEuropeanAstronomyTraining(GREAT-ITN)network.TheresearchleadingtotheseresultshasreceivedfundingfromtheEuropeanUnionSeventhFrameworkProgramme([FP7/2007-2013]undergrantagreementnumber264895."
  ]
}