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AstronomischeNachrichten,26January2017 Constraining the Milky Way assembly history with Galactic Archaeology LudwigBiermannAwardLecture2015 I.Minchev⋆ 7 1 Leibniz-Institutfu¨rAstrophysikPotsdam(AIP),AnderSternwarte16,D-14482,Potsdam,Germany 0 2 Received2016Apr16,accepted2016Jun n Publishedonline2016Aug04 a J Key words Galaxy: abundances – Galaxy: disc – Galaxy: evolution – Galaxy: formation – Galaxy: kinematics and 4 dynamics 2 TheaimofGalacticArchaeologyistorecovertheevolutionaryhistoryoftheMilkyWayfromitspresentdaykinemat- ] icalandchemical state.Becausestarsmoveawayfromtheirbirthsites,thecurrentdynamical informationaloneisnot A sufficientforthistask.Thechemicalcompositionofstellaratmospheres,ontheotherhand,islargelypreservedoverthe G stellar lifetime and, together with accurate ages, can be used to recover the birthplaces of stars currently found at the . sameGalacticradius. In additionto theavailabilityof largestellarsamples withaccurate 6D kinematics and chemical h abundancemeasurements,thisrequiresdetailedmodelingwithbothdynamicalandchemicalevolutiontakenintoaccount. p AnimportantfirststepistounderstandthevarietyofdynamicalprocessesthatcantakeplaceintheMilkyWay,includ- - o ingtheperturbativeeffectsofbothinternal(barandspiralstructure)andexternal(infallingsatellites)agents.Wediscuss r here(1)howtoconstraintheGalacticbar,spiralstructure,andmergingsatellitesbytheireffectonthelocalandglobal t s disc phase-space, (2) the effect of multiple patterns on the disc dynamics, and (3) the importance of radial migration a andmergerperturbationsfortheformationoftheGalacticthickdisc.Finally,wediscusstheconstructionofMilkyWay [ chemo-dynamicalmodelsandrelatetoobservations. 1 v Copyrightlinewillbeprovidedbythepublisher 4 3 0 1 Introduction (Steinmetzetal., 2006), SEGUE (Yannyetal., 2009), 7 APOGEE (Majewskietal., 2010), HERMES (Freeman, 0 The goal of Galactic Archeology 2010), Gaia-ESO (Gilmoreetal., 2012), and LAMOST . 1 (Freeman&Bland-Hawthorn, 2002) is to dissect the (Zhaoetal., 2006).Thiseffortwill soonbecomplemented 0 Milky Way into its various components (discs, bulge, bar bymorethanabillionstarsobservedbytheGaiaspacemis- 7 and halo) and thus to disentangle the various processes sion(Perrymanetal.,2001).Millionsofthesewillhaveac- 1 that contributed to their formation and evolution. Galactic curate propermotionsand parallaxes,which together with : v Archaeologyreliesontheassumptionsthat(i)thedynamics existing spectroscopic data, and especially with the ad- i X offormationislockedinthephase-spacestructureofstellar ventofthededicatedGaiafollow-upground-basedsurveys r populations and that (ii) stellar atmospheres preserve the WEAVE (Daltonetal., 2012) and 4MOST (deJongetal., a chemical imprint of their birth cloud for most of their 2012),willenableGalacticArchaeologyasneverbefore. lifetime. Chemical elements synthesized inside stars are Beforewe can hopeto understandthe past MilkyWay laterinjectedintotheinterstellarmedium(ISM)andincor- history we need a good understandingof its present state, poratedintothenextgenerationsofstars.Becausedifferent in particular of its disc, where the majority of baryonsare elements are released into the ISM by stars of different concentrated. This is already not a trivial task, due to the masses and on different timescales, stellar abundance Sun’spositionclosetotheGalacticdiscmidplane-wecan- ratios are thus directly related to the star formation and notsimplyobservethediscmorphologyaswedoinexternal gas accretion history.Because stars moveaway from their face-on galaxies. Therefore, mostly indirect methods have birthplaces(aprocessknownasradialmigration),chemical beenusedtoconstraintheGalacticbarandspiralstructure. information is crucial for understanding the Galactic formationhistory. While in axisymmetric discs energy and angular momentum are conserved quantities and are, thus, integrals The importanceof thistopic is manifestedin the num- of motion (Binney&Tremaine, 2008), this is not true for ber of Galactic surveys dedicated to obtaining spectro- scopicinformationforalargenumberofstars,e.g.,RAVE themorerealisticcaseofpotentialsincludingperturbations fromacentralbarand/orspiralarms.Inthecaseofonepe- ⋆ Correspondingauthor:[email protected] riodicperturbationthereisstillaconservedquantityinthe Copyrightlinewillbeprovidedbythepublisher 2 Minchevetal.:ConstrainingtheMilkyWay reference-framerotatingwiththepattern– theJacobiinte- Ω(r)±κ/2 2:1 ILR CR 2:1 OLR gral J = E LΩ ,where E istheenergyoftheparticle, L p Ω(r)±κ/4 − isitsangularmomentum,andΩp isthepatternangularve- 4:1 ILR 4:1 OLR Ω(r) locity. Thisis no longerthe case, however,when a second Ω b perturbationwithadifferentpatternsspeedisincluded. IthasnowbeenwellestablishedthattheMilkyWaydisc contains both a bar (as in more than 50% of external disc Ω s,1 galaxies) and spiral structure moving at different pattern Ω speeds, making it difficult to solve such a dynamical sys- s,2 temanalytically.Instead,differenttypesofnumericalmeth- odsareusuallyemployed,fromsimpletest-particleintegra- tions,topreassembledN-bodyandSPHsystems,touncon- strained, fully cosmological simulations of galaxy formation.Allofthesetechniqueshavetheirstrengthsandweak- nesses. Test particles are computationallycheap, allow for Fig.1 Resonances in a galactic disc for nearly circular full control over the simulation parameters (such as spiral orbitsandaflatrotationcurve.Corotationoccursalongthe andbaramplitude,shape,orientationandpatternspeed)but dotted black curve and is given by Ω = Ω(r), where Ω lackself-gravity.N-bodysimulationsofferself-consistency p p is the pattern speed, Ω(r) = v /r is the local circular fre- but bar and especially spiral structure parameters are not 0 quency,andv istheconstantcircularvelocity.The2:1outer easytoderiveandnotwellcontrolled.Finally,inadditionto 0 andinnerLindbladresonances(OLRandILR)occuralong being very computationallyintensive, the outcomesof hy- thesolidblackcurves,computedasΩ =Ω(r) κ/2,where drodynamical cosmological simulations are even less pre- p ± κ is the local radial epicyclic frequency.The outer and in- dictable,withmergingsatellitesandinfallinggasmakingit ner4:1LRsoccuralongthedashedblackcurves,givenby yethardertodisentanglethediscdynamics;theseare,how- Ω =Ω(r) κ/4.Theredhorizontallineindicatesabarpat- ever,muchclosertorealityintheircomplexityandaneces- p ± ternspeedΩ = 55.5km/s/kpc(likelyfortheMilkyWay). saryultimatestepintheinterpretationofobservationaldata. b An inner and outer spiral structure, moving with different Before we consider more complex systems, we first pattern speedsare also shownby the blue and green lines, presentsimple test-particlemodelsthatillustrate the effect respectively.ThisisatypicalsituationseeninN-bodysim- ofbarandspiraldensitywaves. ulations for multiplicity of m = 2. The vertical red, blue, andgreenlinesgivethe radialpositionsof eachresonance 1.1 Resonancesingalacticdiscs forthebarandspiralstructure,respectively(solidlines:2:1; dashedlines:4:1,dottedlines:CR). Galactic discs rotate differentially with nearly flat rotation curves,i.e.,constantcircularvelocityasafunctionofgalac- tic radius. In contrast,density waves, such as a centralbar ILR/OLRarethe2:1resonances),foranm = 4patternthe andspiralstructure,rotateassolidbodies.Thereforestarsat ILR/OLRmustbethe4:1resonances. differentradiiwouldexperiencedifferentforcingduetothe non-axisymmetricstructure.Ofparticularinterestareloca- Sincesecondorderresonances,i.e.,4:1foratwo-armed tions in the disc where the stars are in resonance with the spiralorbar,or2:1forafour-armedspiral,canalsobequite perturber.Thecorotationresonance(CR),wherestarsmove important(aswillbeshownin 3.1),weneedaconvenient with the pattern, occurs when the angular rotation rate of § waytorefertothem.Itissomewhatconfusingandunclear stars equalsthatoftheperturber.TheLindbladresonances howthe4:1resonancesarereferredtointheliterature.The (LRs) occur when the frequency at which a star feels the inner 4:1 resonance for an m = 2 pattern is known as the force due to a perturbercoincideswith the star’s epicyclic Ultra-harmonicresonance(UHR). Some also describe the frequency,κ.AsonemovesinwardoroutwardfromtheCR innerandouter4:1resonancesastheIUHRandtheOUHR, circle,therelativefrequencyatwhichastarencountersthe othersastheinnerandouterm=4resonance.Ifthepattern perturberincreases.Therearetwovaluesofrforwhichthis multiplicityism=4,thenthesebecometheILRandOLR. frequencyisthesameastheradialepicyclicfrequency.This Toourknowledge,thereisnoterminologyforthe2:1reso- iswheretheinnerandouterLindbladresonances(ILRand nancesofanm=4pattern.Minchev&Famaey(2010)pro- OLR) arelocated.Quantitatively,LRsoccurwhenthepat- posedtogeneralizethestandardnotationofLindbladreso- tern speed Ω = Ω κ/m, where m is the multiplicity of p ± nancesbyallowingtorefertoboth2:1and4:1resonances, the pattern1. ThenegativesigncorrespondstotheILRand regardlessofthemultiplicityofthepattern.Byadoptingthis thepositivetotheOLR.WhileBertilLindbladdefinedthese nomenclature, we will refer to the 2:1 ILR/OLR and the forthecaseofanm = 2pattern(thusstrictlyspeakingthe 4:1 ILR/OLR for both two-armed (or bar) and four-armed 1 m=2forabaroratwo-armedspiralstructureandm=4forafour- spiralstructure.Naturally,otherresonancescanalsobede- armedspiral. scribedinthismanner,e.g.,3:1,5:1,6:1ILR/OLR. Copyrightlinewillbeprovidedbythepublisher asnaheaderwillbeprovidedbythepublisher 3 Fig.1illustratestherelationshipbetweenthepatternan- orientation2 of φ = 30 10 and a pattern speed of b ◦ ± gularvelocityandtheradiiatwhichresonancesoccurfora Ω /Ω = 1.9 0.1, where Ω is the local standard of rest b 0 0 ± flat rotation curveand nearly circular orbits. The red hori- (LSR)rotationrate.Adrasticallydifferentbarpatternspeed zontallineindicatesabarpatternspeedΩ =55.5km/s/kpc has recently been suggested by the longer bar half-length b (likelyfortheMilkyWay).Innerandouterspiralstructures measured by Weggetal. (2015), compared to previous movingwithdifferentpatternspeedsarealsoshownbythe works – r = 5.0 0.2 kpc, which places the CR at 5-7 b ± blue and green lines, respectively.The actualextentof the kpc.Insuchacasethebar2:1OLRwouldlieintherange patternsis indicatedbythesolid horizontallines. Thisis a 8.5-12kpcand,thus,wouldnotworkasanexplanationfor typical situation seen in N-bodysimulations for multiplic- theHerculesstream. ity of m = 2 (e.g.,Quillenetal. 2011) anda possiblecon- The Milky Way spiral structure is more poorly known figuration for the Milky Way. The vertical red, blue, and than the bar. Cepheid, HI, CO and far-infrared tracers green lines give the radial positions of each resonance for suggest that the Milky Way disc contains a four-armed thebar,inner,andouterspiralstructure,respectively(solid tightly wound structure (see also Valleé 2016), whereas lines:2:1;dashedlines:4:1,dottedlines:CR).Notethatfor Drimmel&Spergel (2001) have shown that the near- two(ormore)non-axisymmetricpatternsmovingatdiffer- infrared observations are consistent with a dominant two- entangularvelocitiestherewillalwaysexistregionsinthe armedstructure.Adominanttwo-armedandaweakerfour- discwhereresonancesoverlap. armed structure has been proposed by Amaral&Lepine (1997). Similarly to the effect of the bar, the spirals can 1.2 TheMilkyWaybarandspirals be linked to clumps in the u v plane, as first shown − DuetoourpositionintheGalacticdisc,thepropertiesofthe by Quillen&Minchev (2005). Using an orbital weighting Milky Way bar are hard to observe directly. Hence its pa- functiontechnique,thisworkshowedthatatwo-armedspi- rameters, such as orientation and pattern speed, have been ral density wave with pattern speed placing the Sun near inferred indirectly from observations of the inner Galaxy the4:1innerLindbladresonancecanaccountfortwomajor (e.g., Blitz&Spergel 1991; Weinberg 1992). The bar has clumpsinthesolarneighborhood’svelocitydistribution:the also been found to affect the local velocity distribution of Pleiades/Hyadesmovinggroupcorrespondstotheonefam- stars. The way this works is as follows. If the Sun hap- ilyoforbits,andtheComaBerenicesmovinggroupcorre- pened to lie close to a Lindblad resonance, then the local spondstoanotherfamily.Similarpatterspeedestimatewas stellar velocity distribution would exhibit clumps belong- obtainedbyPompeíaetal.(2011)andSiebertetal.(2012). ing to two different orbital families, which are on nearly closedorbitsinthereferenceframemovingwiththepattern 1.3 Multiplepatternsingalacticdiscs (bar or spiral structure). In the case of an OLR, a family ofnearlyclosedorbitssupportingthespiral/barorientation Multiple patterns in N-body simulations have been exists, while a second family is misaligned with the struc- known to exist since the work of Sellwood (1985) and tureoutsidetheOLR.Thesearethe x (1)and x (2)orbits, 1 1 Sellwood&Sparke (1988), who found that a bar can co- respectively (e.g., Dehnen 2000; Fux 2001; Minchevetal. existwith aspiralpatternmovingatamuchlowerangular 2010; see Fig. 2, left). Near the ILR the situation is simi- velocity. Taggeretal. (1987) and Sygnetetal. (1988) ex- larbutreversed,wheretheinnerorbitalfamilysupportsthe plained this as the non-linear mode coupling between the structurebuttheouteroneismisalignedwithit. bar and the spiral wave. These findings were later con- Hipparcos (Perrymanetal., 1997) + GCS firmedbythenumericalstudiesofMasset&Tagger(1997) (Nordstro¨metal., 2004) data revealed more clearly a andRautiainen&Salo(1999).Accordingtothetheoretical stream of old disc stars with an asymmetric drift of about work by Taggeretal. (1987) and Sygnetetal. (1988), two 45 km/sand a radial velocityu < 0, with u and v positive patternscan couplenon-linearlyas they overlapovera ra- toward the Galactic center and in the directionof Galactic dial range, which coincides both with the CR of the inner rotation, respectively. This concentration of stars in the oneandthe ILRofthe outerone.Thiscoincidenceof res- ”u v plane” is known as the Hercules stream. The onances results in efficient exchange of energy and angu- − numerical work of Dehnen (1999, 2000); Fux (2001); larmomentumbetweenthetwopatterns.Thecouplingbe- Minchevetal. (2010), and more recently Antojaetal. tweenthetwopatternsgeneratesbeatwaves(aswedescribe (2012), hasshownthatthisstream canbe explainedasthe below),also foundto have LRs at the interactionradii, re- effectoftheMilkyWaycentralbariftheSunisplacedjust sultinginastrongnon-lineareffectevenatrelativelysmall outside the 2:1 Outer Lindblad Resonance (OLR). Due to amplitudes.Rautiainen&Salo(1999)showedthatcoupling theinhomogeneityinageandmetallicityofHerculesstars, betweenaCRand4:1ILR,aswellasILRandOLRisalso a numberof works(e.g.,Bensbyetal. 2007; Famaeyetal. possibleinN-bodysimulations.Wavescouplewithaselec- 2005, 2007) have concluded that a dynamical effect, such tionoffrequencieswhichoptimizesthecouplingefficiency. as the influence of the bar, is a more likely explanation than a dispersed cluster. Most estimates agree on a bar 2 SeeFig.2,secondpanel,forthedefinitionofthebarangle. Copyrightlinewillbeprovidedbythepublisher 4 Minchevetal.:ConstrainingtheMilkyWay Strongexchangeofenergyandangularmomentumis then withtransientspiralstructure.AnumberofworksusingN- possibleamongthecoupledwaves. bodyandN-body/SPHsimulations(e.g.,Grandetal.2012; Howdomultiplepatternsaffectthedynamicsofgalactic Minchevetal. 2011; Rosˇkaretal. 2008) have confirm that discs?Quillen(2003)consideredthedynamicsofstarsthat migrationalwaystakesplaceinnumericalgalacticdiscs. are affected by perturbations from both spiral arms and a A different radial migration mechanism was proposed centralbarbyconstructingaone-dimensionalHamiltonian by Minchev&Famaey (2010), who considered the simul- modelfor the strongestresonancesin the epicyclicaction- taneous propagation of a bar and a long-lived spiral den- angle variables. Quillen pointed out that when two per- sity wave. In such a setup angular momentum redistribu- turberswithdifferentpatternspeedsarepresentinthedisc, tion arises from the overlap of resonances associated with thestellardynamicscanbestochastic,particularlynearres- differentmodesor fromthe stellar mass transiencyas per- onancesassociatedwithoneofthepatterns.Similarfindings turberswithdifferentpatternspeedsinterfereconstructively werepresentedmorerecentlybyJalali(2008).Allthesere- (see Comparetta&Quillen2012), butnotfromthegrowth sults are notsurprisingsince ithas alreadybeenshown by and decayoftransientmodes.Thiswork,alongwith stud- Chirikov (1979) that in the case of resonance overlap the ies of diffusion coefficients in barred discs (Brunettietal., lastKAMsurfacebetweenthetworesonancesisdestroyed, 2011;Shevchenko,2011),predictsavariationinmigration resultinginstochasticbehavior.Itisthereforeexpectedthat efficiency with time and disc radius, establishing that the resonanceoverlapcouldgiverisetobothvelocitydispersion dynamical influence of the bar plays an integral part of increaseandradialmigration.Bothofthesepossibilitiesare Milky Way disc modeling. Aside from internal structure, exploredin 4.1and 4.2,respectively. perturbations due to minor mergers have also been shown § § to be effective at mixing the outer discs (Birdetal., 2012; A comprehensive discussion of the Galactic bar and Quillenetal., 2009), but also can, indirectly,affectthe en- spiral parameters can be found in the recent review by tire disc by inducing (reinforcing) spiral and bar instabili- Bland-Hawthorn&Gerhard(2016). ties (e.g., Purcelletal. 2011). Considering the established presence of a central bar, spiral structure and evidencefor 1.4 Radialmigration mergeractivity in the Milky Way, it is clear that all of the abovementionedradialmigrationmechanismswouldhave The power of Galactic Archaeology has been threatened aneffectontheGalacticdisc. both by observationaland theoreticalresults, showing that Stars found today at a given small Galactic disc re- stars mostprobablymoveaway from their birthplaces,i.e, gion (e.g., the solar neighborhood) can have a range of migrate radially. Observational signatures of radial migra- birth radii but are mostly indistinguishable in their kine- tion (or mixing) have been reported in the literature since matics from locally born stars. Therefore, chemical abun- the 1970’s, with the pioneering works by Grenon (1972, dancesneedtobeinvokedinidentifyingmigrators.Proba- 1989).Grenonidentifiedanoldpopulationofsuper-metal- blyoneofthebestwaysofquantifyingradialmigrationin rich stars (hereafter SMR), presently at the Solar vicin- the Milky Way is using the technique of chemicaltagging ity, but with kinematics and abundance properties indica- (Bland-Hawthornetal.,2010),wherestarsborninthesame tiveofanoriginintheinnerGalacticdisc.SMRstarsshow cluster (now dispersed) are expected to appear as a clump metallicitieswhichexceedthepresentdayISMandthoseof inthemulti-dimensionalchemicalspace.Thisisoneofthe youngstarsatthesolarvicinity.Themetallicityofthesolar main objectivesof the ongoingGALAH survey(Freeman, vicinity,however,isnotexpectedtoincreasemuchsincethe 2010). Sun’sformation,orinthelast 4Gyr,duetotheratherin- ∼ efficientstarformationrate(SFR)atthesolarradiusduring 1.5 TheGalacticthickdisc thisperiod(e.g.,Asplundetal.2009;Chiappinietal.2003). Hence,purechemicalevolutionmodelsfortheMilkyWay The formation of galactic thick discs has been an impor- thindisccannotexplainstarsmoremetalrichthan 0.2dex tant topic ever since their discovery in external galaxies ∼ andtheeffectofradialmigrationneedstobeconsidered. (Burstein, 1979; Tsikoudi, 1979) and in the Milky Way N-bodysimulationshavealsolongshownthatradialmi- (Gilmore&Reid, 1983). The large uncertaintiesin impor- grationisunavoidable.Raboudetal.(1998)studiednumer- tantobservationalconstraintsintheMilkyWay,suchasthe ical simulationsaimed at explainingthe results by Grenon age-velocity-metallicity relation, abundance gradients and (1989) of a mean positive u-motion,which the authorsin- their evolution, have led to different scenarios to be pro- terpreted as metal-rich stars from the inner galaxy, wan- posedfortheformationoftheGalacticthickdisc. dering in the solar neighborhood. However, Raboudetal. One possibility is that the stars comprising thick (1998) interpreted their findings as stars on hot bar or- discs are born thick at high redshift from internal grav- bits, not recognizing that permanent changes to the stel- itational instabilities in gas-rich, turbulent, clumpy discs lar angular momenta are possible. It was not until the (Bournaudetal., 2009; Forbesetal., 2012) or in the tur- work by Sellwood&Binney (2002) that radial migration bulent phase associated with numerous gas-rich mergers was established as an important process affecting the en- (Brooketal., 2005, 2004). Theycouldalso have beencre- tire disc, where stars shift guiding radii due to interaction ated through accretion of galaxy satellites (Abadietal., Copyrightlinewillbeprovidedbythepublisher asnaheaderwillbeprovidedbythepublisher 5 Fig.2 TheleftpanelIllustratesthex (1)andx (2)orbitsandtheirorientationwithrespecttothebar.Thedashedcircle 1 1 depictstheOLRposition(about800pcinsidethesolarcircle)suchthattheHerculesmovinggroupcanbeexplainedby the bar. The second panel shows the results of a test-particle simulation from Minchevetal. (2010) at time t = 7.5 (in unitsofsolarrotations).Theaxesareinunitsofthethesolarradius,r .Thedottedcircleshowsthebar’sCR.Onlystars 0 initially on circular orbits close to the OLR are shown, with those inside/outside the OLR colored in blue/red. The bar is fully grown in four bar rotations, which corresponds to t 2.2 in the units used here. Note that the x (1) and x (2) 1 1 ≈ orbitsorientationisnotasintheleftpanelbutoffsetbyabout30 intheclockwisedirection.TheapproximateHipparcos ◦ coverageisshownbytheyellow-filledsmallcircle.Thethirdpanelshowsthesimulatedu vplaneatthedepictedtime. − Theradialvelocitydispersionis sigma = 40km/s,inordertopopulatetheHerculestreamaswell.Thepatternspeedis r fixedatΩ /Ω = 1.8,themaximumsampledepthisd = 250pc,andthebarorientationisφ = 35 .Shadedcontours b 0 max 0 ◦ showtheparticlenumberdensity.Thetwoorbitalfamiliesassociatedwiththex (1)(red)andx (2)(blue)orbitsprecessat 1 1 differentratesandprovidegoodmatchtotheComaBerenicesandPleiadesgroupsatthisparticulartime.Hipparcosstellar velocitydistributionwiththeSun’smotionsubtracted(valuesfromDehnen&Binney1998). 2003; Mezaetal., 2005), where thick disc stars then have in observations of edge-on galaxies, this new view of the anextragalacticorigin. formationofthickdiscsresolvesanumberofapparentcon- tradictions.Moreonthiscanbefoundin 7. Another possibility is that thick discs are created § throughtheheatingofpreexistingthindiscsbyminormerg- ers(DiMatteoetal.,2011;Villalobos&Helmi,2008).Ev- 2 Constraining the Galacticbar idence for merger encounters can be found in the phasespace structureof MilkyWay disc stars(e.g.,Go´mezetal. 2.1 Theeffectofarecentlyformedbar 2013,2012b;Minchevetal.2009). While earlier work has shown the bar to affect the u v Several works (e.g., Loebmanetal. 2011; − plane at higher velocities, some of the low-velocity mov- Scho¨nrich&Binney 2009b) have also proposed that ing groups in the solar vicinity have been explained only radial migration can give rise to thick disc formation by withtheeffectofspiraldensitywaves,suchasthesplitting bringing out high-velocity-dispersion stellar populations ofthePleiades/HyadesandComaBerenicesmovinggroups from the inner disc and the bulge. More detailed dynam- (e.g.,Quillen&Minchev2005).Thereasonthebar’sOLR ical studies (Martigetal., 2014b; Minchevetal., 2012a; shouldnotaffectstarswithkinematicscoolerthantheHer- Vera-Ciroetal., 2014) have more recently shown that culesstream isthatthe solarradiuslies about0.8kpc out- migrationdoesnotcontributetoanysignificantleveltodisc sidetheOLR(dashedcircle),thus,starsonnearcircularor- thickening,but on the opposite, it suppressesflaring when bits atthe OLRcannotreach the solar neighborhood.This external perturbations are included (Grandetal., 2016; situationcanbeseenintheleftmostpanelofFig.??,where Minchevetal.,2014a).Thisisdiscussedfurtherin 6.1. § the Hipparcos volume is shown by the yellow-filled small Finally,arecentlyproposedthickdiscformationmech- circle.HotterstarsneartheOLR,however,canappearclose anism is the superposition of coeval flaring disc subpopu- totheSunontheirapocenters. lation(Minchevetal.,2015).Discflaringcanresultfroma Studying the effect of the bar, we explored a different numberofdifferentsources, the mosteffectivemostlikely scenarioinMinchevetal.(2010),wherethetimeevolution beingtheperturbativeeffectofmergersonthehostdisc.In oftheu vplanewasexaminedjustafterthebarformation. aninside-outformationscenario,theoldpopulationsdom- − These test-particle simulations were performedin a Milky inate in the inner disc and younger in the outer disc, thus Way-likepotentialtowhichaperturbationdueto acentral the flaring disappears when the total stellar population is barswasadded,similartoDehnen(2000).Thebackground taken into account. By making the distinction between a axisymmetricpotentialduetothediscandhalohastheform thickdiscdefinedasthe[α/Fe]-high,oldstellarpopulation in the Milky Way and a geometrically defined thick discs Φ (r)=v2log(r), (1) 0 0 Copyrightlinewillbeprovidedbythepublisher 6 Minchevetal.:ConstrainingtheMilkyWay Fig.3 Left:Variationintheu vplanewithbarpatternspeedandorientation.Contoursshowparticlenumberdensity, − while the color levels representthe change in angular momentum∆L for a given location on the u v plane. Color bar − values can be convertedto (km/s pc) by multiplyingby 100v , with v the rotation curve. Different rows show different 0 0 barmajoraxisorientationsofφ = 20 ,30 and60 withrespecttotheSunGalactocentricline.Differentcolumnsshow 0 ◦ ◦ ◦ changesin bar angularvelocity Ω /Ω = 1.8,1.85,and 1.9, where the bar’s OLR is at Ω /Ω = 1.7. These correspond b 0 b 0 approximately to neighborhoodsat Galactic radii r = 7.5,8.0 and 8.5. The blue lines indicated the systematic shifts of clumpsin theu v planewith changeinbarangelanddistancefromthe Galacticcenter.Right:Asthe secondpanelin − Fig.2,butwiththenineneighborhoodlocationscorrespondingtotheu vplotsshownontheleftoverlaidasyellow-filled − circles(blackforthesolarneighborhood).AdaptedfromMinchevetal.(2010). correspondingtoaflatrotationcurve.Wemodelthenonax- The 2:1OLR with the bar isachievedwhenΩ /Ω = 1+ b 0 isymmetricpotentialperturbationduetotheGalacticbaras κ/2 1.7, where κ is the epicyclic frequency. For a flat ≈ apurequadrupole rotationcurveκ = √2Ω ,whereΩ istheangularvelocity 0 0 oftheLSR. Φb = Ab(ǫb)cos[2(φ−Ωbt)]×2−(cid:16)rrb(cid:16)(cid:17)rr3b(cid:17)3 ,, rr ≤≥ rrbb (2) Interestingly, we found that a steady state bar induced Here Ab(ǫb) is the bar’s gravitational potential amplitude, transient features at low velocities in the simulated so- identical to the same name parameter used by Dehnen lar neighborhoodvelocitydistributiondue to the initialre- (2000);thestrengthisspecifiedbyǫb = αfromthesame sponseofthedisctothebarformation.We associatethese − paper. The bar length is rb = 0.8rcr with rcr the bar coro- velocitystreamswithtwoquasi-periodicorbitalfamiliesli- tation radius. The pattern speed, Ωb is kept constant. The brating around the stable x1(1) and x1(2) orbits near the baramplitudeǫ isinitiallyzero,growslinearlywithtimeat bar’s OLR (Fig. 2, left two panels). In a reference frame 0<t<t1andtransitionssmoothlytoaconstantvalueafter movingwiththebarthese,otherwisestationary,orbitspre- t = t1 = 4 bar rotations. This insures a smooth transition cess on a timescale dependent on the strength of the bar. fromtheaxisymmetrictotheperturbedstate. The effect of this precession can be seen in Fig. 2, where Previousworkhasusedasameasureofbarstrengththe theleftpanelillustratesthe x (1)and x (2)orbitsandtheir 1 1 parameter QT (Combes&Sanders, 1981).Thisistheratio orientationwithrespecttothebar.Thesecondpanelshows of the maximum tangential force to the azimuthally aver- theresultsofatest-particlesimulationfromMinchevetal. aged radial force at a given radius. From eq. 2 this def- (2010) at time t = 7.5 (in units of solar rotations). Only inition yields QT = 2Ab/v2c. We examine bar amplitudes starsinitiallyoncircularorbitsclosetotheOLRareshown, in the range 0.1 < QT < 0.4 as expected from obser- withthoseinside/outsidetheOLRcoloredinblue/red.Note vations of various galaxies and from N-body simulations that, because of the precession, at this time the x (1) and 1 (Combes&Sanders, 1981). This corresponds to 0.013 < x (2)orbitsorientationisnotas in theleftpanelbutoffset 1 |ǫb|<0.05. by about 30◦ in the clockwise direction. This behavior al- In our units the solar neighborhood radius is r = 1; lows the two (kinematically cold) orbital families to reach 0 thecircularspeedisv = 1everywheresincetherotational thesolarneighborhoodandmanifestthemselvesasclumps 0 curveisflat.ToconverttorealunitsweuseaLSRtangential in the u v plane moving away from (x (2)), and toward 1 − velocityof240km/s,andGalactocentricdistanceof8kpc. (x (1))theGalacticcenter. 1 Copyrightlinewillbeprovidedbythepublisher asnaheaderwillbeprovidedbythepublisher 7 The nine panels on the left in Fig. 3 show the simu- whered istheaverageheliocentricdistanceofstars, Aand latedu vplanefornineneighborhoodsatdifferentGalactic BaretheusualOortconstants,andCandK aregivenby − discpositions,asindicatedbytheyellowcirclesinthex-y u ∂u 1∂vφ 2C + (4) plotontheright.Thesolarneighborhoodvolumeisshown ≡ −r ∂r − r ∂φ as a black circle. For this particular simulation, only test- u ∂u 1∂vφ particlesonnearlycircularorbitsweresubjectedtothepo- 2K + + + . (5) ≡ r ∂r r ∂φ tentialinordertoavoidthe seaof hotorbits,whichwe al- Hererandφaretheusualpolarcoordinatesandv =v +v, ready know give rise to Hercules stream-like feature (see φ 0 wherev isthecircularvelocityattheSolarradius,r .Con- Fig.2thirdpanel). 0 0 sidering a flat rotation curve, the derivatives of v in the For a given time, the tangential velocity v of resonant φ aboveequationsare identicalto the derivativesof v.C de- features in the u v plane is set by the bar pattern speed − scribes the radial shear of the velocity field and K its di- and the radial velocity u, is set by the bar’s orientation. vergence.ForanaxisymmetricGalaxyweexpectvanishing These variationsareillustratedbythe bluelinesin thefig- valuesforbothCandK3.WhereasCcouldbederivedfrom ure. Thus, assuming some features at low velocities in the both radial velocities and proper motions, K can only be Hipparcos velocity distribution are of resonant origin, we measuredfromradialvelocities,inwhichcaseaccuratedis- canmatchstreampositionsintheu vplaneandestimate − tancesarealsoneeded. thebarpatternspeedΩ andtheorientationofitsmajoraxis b The study of Olling&Dehnen (2003) not only mea- withrespecttothesolarGalactocentricline,φ .Inaddition, 0 sured a non-zeroC, implying the presence of non-circular structurevarieswith timeduetothe librationofthe quasi- motioninthelocaldisc,butalsofoundthatCismorenega- periodic orbitsaroundthe fixed points,which allows us to tiveforolderandredderstarswithalargervelocitydisper- constrainthebarformationtime. sion,whileitisroughlyzeroforthecold(thusyoung)pop- Dependingonthebarparametersandtimesinceitsfor- ulation.MorerecentdeterminationoftheC constantusing mation,thismodelisconsistentwiththePleiadesandComa RAVE data confirmed these results (Siebertetal., 2011). Berenices, or Pleiades and Sirius moving groups seen in ThisvariationofCwithvelocitydispersion/color/ageissur- the Hipparcosstellar velocitydistribution,whenthe Milky prisingasahotterstellarpopulationisexpectedtohaveav- Way bar angle is 30 < φ < 45 and its pattern speed ◦ 0 ◦ eraged properties more nearly axisymmetric, and hence, a is Ω /Ω = 1.82 0.0∼7. Sinc∼e the process is recurrent, a b 0 ± reducedvalueof C (e.g.,Minchev&Quillen2007). good match could be achieved about every six LSR rota- | | Assumingthe Galactic baraffectsthe shapeofthe dis- tions.However,tobeconsistentwiththefractionofstarsin tribution function of the old stellar population in the solar the Pleiades, we estimated thatthe Milky Way bar formed neighborhood, an additional constraint on the bar can be 2Gyrago.Thismodelarguesagainstacommondynam- ∼ providedbyrequiringthatamodelreproducestheobserved icaloriginfortheHyadesandPleiadesmovinggroups. value of the Oort constant C. In other words, in addition WetheadventofGaiaaccuratepropermotionsandpar- to relating the dynamical influence of the bar to the local allaxes, combinedwith high precisionline-of-sightveloci- velocityfield(see 2.1above),Cprovidesalinktothegra- tiesfromAPOGEEandRAVE(alreadyafterGaiaDR1and § dientsofthevelocitiesaswell. DR2) and ultimately by the 4MOST survey (scheduled to Inanefforttoreproducetheseobservationalresults,us- startoperationin2021),wewillbeabletotestthesepredic- ing test-particles integrations we simulated measurements tionsbytryingtoreproducethevariationofstructureinthe of the Oort C value in a gravitational potential including u v plane at different positions in the disc (as in the left − theGalacticbar,asin 2.1above.Forallotherparameters halfofFig.3). § fixed, we defined a cold and a hot sample resulting from aninitialradialvelocitydispersionof10km/sand40km/s, 2.2 RelationtotheOortCconstant respectively. In Fig. 4 we present our results forC as a function of Asdiscussedabove,previousworkhasrelatedtheGalactic thebarangle,φ (theanglebywhichtheSun’sazimuthlags bartostructureinthelocalstellarvelocitydistribution. 0 the bar’s major axis). Each panel shows a simulation with Anotherindirectwaytoconstrainthebarwaspresented a differentpattern speed and sample depth of d = 200 pc. byMinchevetal.(2007),whereweshowedthatthebaralso Solidanddottedlinesrepresenttheresultsforcoldandhot influencesthespatialgradientsofthevelocityvectorviathe discs,respectively.CispresentedinunitsofΩ =v /r .To 0 0 0 Oortconstants. makethediscussionlesscumbersome,wewriteC andC h c We canlinearizethelocalvelocityfield abouttheLSR to refer to the values for C as estimated from the hot and and write the mean radial velocity v and longitudinal d colddiscs,respectively. proper motion µ as functions of the Galactic longitude l l Ch (dottedlines in Fig. 4) varieswith galactic azimuth as as C (φ ) sin(2φ ) for all of the Ω values considered. h 0 0 b v ∼ d = K+Asin(2l)+Ccos(2l) (3) 3 Note,however, thatC and K wouldalsobezerointhepresence of d nonaxisymmetricstructureiftheSunhappenedtobelocatedonasymme- µ = B+Acos(2l) Csin(2l) tryaxis. l − Copyrightlinewillbeprovidedbythepublisher 8 Minchevetal.:ConstrainingtheMilkyWay not a good not a good match match Fig.4 Each panel shows the variation of the Oort constant C with bar angle φ , for a simulation with a particular 0 bar pattern speed, Ω , and a mean heliocentric distance, d = r /40, or 200 pc for a Galactocentic radius r = 8 kpc. b 0 0 Solid and dotted lines correspond to stellar populations with cold and hot kinematics values, respectively. Panels from left to right show an increasing Ω in units of Ω . Note that the OLR is at Ω 1.7. Good matches to the observed b 0 OLR ≈ trendinC (vanishingvalueforthe colddisc anda largenegativeforthe hotone)are achievedfor20 6 φ 6 45 and ◦ 0 ◦ 1.836Ω /Ω 61.91.Theseareindicatedbytheredverticallines.FigureisamodifiedversionofFig.2byMinchevetal. b 0 (2007). Ontheotherhand,thecolddiscvalues(solidlines)exhibit different variations, dependingon the bar pattern speed or equivalently, on the ratio r /r . Closer to the OLR (left 0 OLR panels of Fig. 4),C (φ ) approachesthe functionalbehav- c 0 iorofC (φ ).AwayfromtheOLR(rightpanels),C (φ )is h 0 c 0 shiftedby90 comparedtoC (φ ),i.e.,C (φ ) C (φ ). ◦ h 0 c 0 h 0 ∼ − Moreover,the hot disc yields an increase in the amplitude of C (φ ) as the pattern speed nears the OLR. Whereas h 0 the same trend is apparent for the cold disc, the gradient of the amplitude of C (φ ) is much larger. This is consis- c 0 tentwithourexpectationthatthecolddiscisaffectedmore by the bar, especially near the OLR. While close to the OLR C (φ ) < C (φ ), we observe the opposite behav- h 0 c 0 | | | | ior awayfromit. Thiscouldbeexplainedbythe resultsof Mu¨hlbauer&Dehnen(2003),whereitwasfoundthathigh velocity dispersion stars tend to shift the “effective resonance”radiallyoutwards. TheredverticallinesinFig.4indicatepossiblesolaraz- imuthswheretheobservedOortCtrendsarewellmatched. BycomparingmeasurementsofC withoursimulationswe constrainedthepatternspeedasΩ /Ω =1.87 0.04,where b 0 ± Ω is the localcircularfrequency,and foundthe bar angle 0 tolieintherange20 6φ 645 . ◦ 0 ◦ Fig.5 Toppanel:Theeffectofatwo-armedspiralstruc- 3 Constraining the MilkyWayspiral ture on orbits near the 4:1 ILR (or IUHR). Note the split- structure ting into two families of closed orbits in the frame moving with the spiral pattern. φ indicates the orientation of 0 3.1 ConstraintsusingtheU-Vplane spiral structure with respect to the Sun’s azimuth, given by the angle between the dottedGalactocentic arrows.For Using a spiral pattern speed of 20 km/s/kpc (assuming ∼ a Sun orientation at 20◦ with respect to a concave arm, LSR rotation of 28 km/s/kpc), Quillen&Minchev (2005) bothorbitalfamiliescanenterthesolarneighborhood(yel- used an orbital weighting function technique to show that low filled circle). Bottom panel: the effect on the u v a two-armed spiral density wave can split the solar neigh- − plane for the configuration shown in the top panel. The borhood’s velocity distribution into two major clumps. A clumps at (u,v) ( 35, 17) km/s and (U,V) (10,0) different approach but with a similar result was presented ≈ − − ≈ km/s are good matches to the Hyades and Sirius moving byPompeíaetal.(2011)anddiscussedbelow.Inthatwork groups, respectively. Test-particle simulation used is from atest-particlesimulationbyMinchev&Quillen(2008)was Minchev&Quillen(2008).ThisfigureissimilartoFig.13 employed,in which initially a disc was populatedby stars byPompeíaetal.(2011). oncircularorbitsandthennumericallyintegratedintheax- isymmetric potentialfrom eq. 1, butincludinga spiral, in- steadofabarperturbationterm. Copyrightlinewillbeprovidedbythepublisher asnaheaderwillbeprovidedbythepublisher 9 Thespiralpotentialisgivenby TomatchspatiallytheHipparcosstellarsample,onlyparti- r clesina200pccirclearoundthefiducialSunareselected. Φ (r,φ,t)=ǫ cos[αln m(φ Ω t)], (6) s s r − − s Eachorbitalfamilygivesrisetoastream(ormovinggroup) 0 in velocity space. The dense clump at (u,v) ( 35, 17) where ǫ is the spiral strength, related to the amplitude of s ≈ − − km/s can be associated with the Hyades and the shallow themasssurfacedensityofspirals,Σ ,as s oneat(u,v) (10,0)km/swiththeSiriusmovinggroups. ǫ 2πGΣ r /(αv2), (7) ≈ s ≈− s 0 c Thecontourlevelscorrespondto0.2,0.31,0.43,0.55,0.67 asshowninBinney&Tremaine(2008).Theparameterαis and0.8 ofthe maximumvalueat the centreofthe Hyades relatedtothepitchangleofthespirals, p,byα = mcot(p). clump. The azimuthal wavenumber m is an integer corresponding This model constrains the spiral pattern speed to to the number of arms. We consider both two-armed and Ω /Ω = 0.65 and orientation with respect to the Sun, s 0 four-armed spiral structure with α = 4 and 8, respec- φ =20 toanuncertaintyoflessthan5%. − − 0 ◦ tively, where the negative sign correspondsto trailing spirals.Elmegreen(1998)foundthatgrand-designspiralshave 3.2 Theeffectofspiralstructureontheglobaldisc arm-interarm contrasts of 1.5-6, corresponding to a frac- phasespace tionalamplitudeof0.2<Σ /Σ<0.7,whichisinagreement s withRix&Zaritsky(1995)whoestimated0.15<(Σ /Σ)< s Another way to obtain constraints on the Milky Way spi- 0.6. ralstructureisthroughstellarsamplescoveringlargeparts Foramaximumexponentialdiscthepeakcircularspeed of the Galactic disc. To facilitate the interpretation of the inthedischasthevaluevc 0.622√GMd/rd atr 2.15rd, hugeamountsofdataexpectedfromGaiaandspectroscopic ≃ ≃ where Md is the disc mass inclosed by rd. The surface follow-up surveys, studies of how spiral structure affects density of the disc at radius r is Σ(r) = (Md/2πrd2)e−r/rd. the global disc phase space are needed. One such attempt Eliminating Md fromtheseexpressions,wefindatr0,v2c ≃ wasmadebyMinchev&Quillen(2008),wheretest-particle 0.39 2πGrdΣ0er0/rd. Substitutingthis expressionfor vc in simulations of a galactic disc perturbed by a steady-state × eq.7,therelationbetweentherelativepotentialandtherel- spiral densitywave used to relate structure in velocityand ativeoverdensitybecomes morphologytothespiralparameters. ǫs ≈−ΣΣs rr0 0e.3−rrd90α. (8) contIonutrhepltootpsrfoowrtowfoFsiguc6hwseimpurelasetinotnsstewlliatrhnduimffebreerndtesnpsiritayl 0 d pattern speeds, as indicated. The background axisymmet- The perturbation is grown from zero to its maximum ricdiscissubtractedtoemphasizethespiralstructure.The strength in four rotation periodsat r . In order to improve 0 quantity plottedis ∆Σ = (Σ Σ )/Σ , where Σ and Σ statistics, positions and velocities are time averaged for − axi axi axi are the perturbed and axisymmetric stellar number densi- 10 spiral periods. We distribute particles (stars) between ties.Theresonancesgetclosertogetherforthefasterspiral in inner and outer galactic radii (r ,r ) = (0.3r ,2.0r ). in out 0 0 on the right. Distances are in units of the solar radius, r . New particles are added until the final number of outputs 0 Darkercolorscorrespondtolowerdensity.Theinner0.3r is 2.5 106. In addition, the two-fold symmetry of our 0 × discisnotplottedsincewedonotmodeltheGalacticcenter. model galaxy is used to double this number. More details Notethedisruptionofthespiralsnearthe2:1LRs(dashed aboutthesesimulationscanbefoundinMinchev&Quillen circles). (2008). The second and third rows of Fig 6 show maps of the ThetoppanelofFig.5showstheeffectofatwo-armed residual mean line-of-sightvelocity, ∆V , and the residual spiral structure on orbitsnear the 4:1 ILR4. This 4:1 reso- d stellar density, ∆Σ, for the disc in the top-leftpanel. Here, nancegivesrisetosquareorbits,intheframemovingwith however,thesequantitiesareplottedversusGalacticlongi- the spiral pattern, even though the imposed spiral wave is tudel(x-axis),andheliocentricdistanced/r (y-axis).The two-armed. Similarly to the case of the bar (see 2), two 0 § orangecurveistheprojectionofthesolarcircle.Thesolar familiesofclosedorbitsareexcitedbytheresonance,where phaseangleisφ = 20 inthesecondrowandφ = 40 in one supports the spiral structure (blue particles) and the 0 ◦ 0 ◦ thethirdrow.Theminimum(black)andmaximum(white) other oneis misalignedwith it (red particles).φ indicates 0 contourvaluesaredisplayedontopofeachpanel.Wellde- the orientationof spiralstructurewith respectto the Sun’s fined structure is present for both ∆V and ∆Σ and that is azimuth,givenbytheanglebetweenthedottedGalactocen- d seentochangewiththechangeinsolarorientationwithre- ticarrows.ForaSunorientationat20 withrespecttoacon- ◦ specttothespirals. cavearm,bothorbitalfamiliescanenterthesolarneighbor- The fourth and fifth rows of Fig 6 are similar to the hood(yellowcircle).TheGalactocentricaxesareinunitsof two above,but show the results for a four-armedstructure r ,theGalactocentricradiusoftheSun. 0 with a phaseangleof 20 anda patternspeedΩ = 0.7Ω The bottom panel of Fig. 5 shows the effect on the ◦ s 0 (fourthrow)andΩ =0.9Ω (fifthrow).Thus,thevariation u v plane for the configuration shown in the top panel. s 0 − in structureseen hereis dueto thedifferentpatternspeeds 4 Alsoknownastheinnerultra-harmonicresonanceorIUHR. used. Copyrightlinewillbeprovidedbythepublisher 10 Minchevetal.:ConstrainingtheMilkyWay Fig6indicatesthat(1)thesolarorientationwithrespect to the spiral, (2) the spiral pattern speed and (3) the num- berofspiralarmscanbeassociatedwithstructureinthese observationallymotivatedmaps. Minchev&Quillen(2008)foundthattheaxisymmetric potentialneedstobeknownto 10%,line-of-sightveloci- ∼ tiesto 20km/s,anddistanceuncertaintiesneedtobeless ∼ than 30%,inordertobeabletoconstrainspiralstructure. ∼ Themeanline-of-sightvelocityandthevelocitydispersion areaffectedbyupto 35km/swhichiswellwithinthede- ∼ tectablelimitofevenlow-resolutionspectroscopicsurveys. Asurveyofstarsclosetothediscmidplaneandcovering largearea is needto applythis methodof constrainingthe spirals.Suchlarge-scaleGalacticsurveyshavenowbecome areality,mostnotably,theAPOGEE-1near-infraredSDSS project, which coversGalactocentricazimuthof about60 ◦ intheradialrange3<r < 14kpcforsome1.5 105stars. × The surveyed disc area will soon increase with the forth- coming APOGEE-2 (north and south) and the addition of 3 105morestars.IntheverynearfutureGaiaandfollow- ∼ × upsurveys4MOSTandWEAVEwillincreasethisnumber tomillionsofstarswithaccuratepropermotions,parallaxes, radialvelocities,photometry,andchemicalcomposition. 4 The importance ofmultiplepatterns in galacticdiscs So far we have only considered simple test-particle mod- els,butwhichallowahighdegreeofcontrolandthemeans of cheaplysweeping parameterspace. Only the effect of a barorthatofaspiralperturbationwasusedforthemodels discussedin 2and 3.Usingtest-particlesimulations,we § § couldisolate,understand,andquantifybettertheeffectofan individualperturber,i.e.,thebaroraspiralwave.Itiswell known,however,thattheMilkyWay disc,aswellasmore than 50%of disc galaxiesin general,harborboth types of non-axisymmetriccomponents.Moreover,bothanalysesof N-body simulations (e.g., Sygnetetal. 1988; Taggeretal. 1987)andexpandinggalaxyimagesinFouriercomponents (e.g,Elmegreenetal.1992;Rix&Rieke1993)haveshown Fig.6 Top row: Test-particle simulation of a galactic the existence of multiple spiral density waves, which can discperturbedbyatwo-armedspiraldensitywave.Lighter propagatesimultaneouslyingalaxydiscs. colorscorrespondtohigherstellardensity.Distancesarein Complicatingthediscdynamicsbytheinclusionofmul- units of the solar distance from the Galactic center, r . As tiple patterns and studying their combined effect on the 0 thepatternspeedisincreasedtheresonancesareshiftedin- kinematical heating of, and redistribution of angular mo- wardsbringingtheCRclosertothesolarcircle.Secondand mentum(radialmigration)in, galacticdiscs is the topic of thirdrows:Mapsoftheresidualmeanline-of-sightveloc- thefollowing 4.1and 4.2,respectively.In 4.3weshow § § § ity,∆V ,andresidualstellardensity,∆Σ,forthediscinthe that,indeed,N-bodySPHsimulationssupporttheideathat d top-left panel, plotted versus Galactic longitude l (x-axis), spiralpatternscanbelong-livedfeatures,justifyingtheas- and heliocentric distance d/r (y-axis). The orange curve sumptionswehavebeenmakingsofar. 0 is the projection of the solar circle. The solar phase angle is φ0 = 20◦ in the second row and φ0 = 40◦ in the third 4.1 Stochasticheatingfrommultiplespiralwaves row. Minimum (black) and maximum(white) contour values are displayed on top of each panel. Fourth and fifth A new mechanism for increasing the stellar velocity dis- rows:Sameasabove,butforafour-armedstructurewitha persion with time in galactic discs (known as the age- phase angle of 20 and differentpattern speeds, as shown. velocity relation) was described by Minchev&Quillen ◦ FigureadaptedfromMinchev&Quillen(2008). Copyrightlinewillbeprovidedbythepublisher