Astronomy&Astrophysicsmanuscriptno.article˙lveltz c ESO2008 (cid:13) February2,2008 Galactic kinematics with RAVE data I. The distribution of stars towards the Galactic poles L.Veltz1,2,13,O.Bienayme´1,K.C.Freeman2,J.Binney3,J.Bland-Hawthorn4,B.K.Gibson5,G.Gilmore6,E.K. Grebel7,8,A.Helmi9,U.Munari10,J.F.Navarro11,Q.A.Parker4,12,G.M.Seabroke6,A.Siebert1,13,M.Steinmetz13, F.G.Watson4,M.Williams2,13,R.F.G.Wyse14,andT.Zwitter15 8 0 (Affiliationscanbefoundafterthereferences) 0 2 ReceivedFebruary2,2008 n a ABSTRACT J 4 WeanalyzethedistributionofGandKtypestarstowardstheGalacticpolesusingRAVEandELODIEradialvelocities,2MASSphotometricstar 1 counts,andUCAC2propermotions.Thecombinationofphotometricand3Dkinematicdataallowsustodisentangleanddescribethevertical distributionofdwarfs,sub-giantsandgiantsandtheirkinematics. ] We identify discontinuities within the kinematics and magnitude counts that separate the thin disk, thick disk and a hotter component. The h respectivescaleheightsofthethindiskandthickdiskare225 10pcand1048 36pc.Wealsoconstraintheluminosityfunctionandthekinematic p ± ± distributionfunction.Theexistenceofakinematicgapbetweenthethinandthickdisksisincompatiblewiththethickdiskhavingformedfrom - o thethindiskbyacontinuousprocess,suchasscatteringofstarsbyspiralarmsormolecularclouds.Othermechanismsofformationofthethick r disksuchas‘createdonthespot’orsmoothly‘accreted’remaincompatiblewithourfindings. t s Keywords.Stars:kinematics–Galaxy:disk–Galaxy:fundamentalparameters–Galaxy:kinematicsanddynamics–Galaxy:structure– a [ 1 1. Introduction thatthe [α/Fe] versus[Fe/H]distributionis relatedto the kine- v matics(Fuhrmann1998;Feltzingetal.2003;Soubiran&Girard 0 Itisnowwidelyacceptedthatthestellardensitydistributionper- 2005;Brewer&Carney2006;Reddyetal.2006)andprovides 2 pendicular to the Galactic disk traces at least two stellar com- an effective way to separate stars from the thin and thick disk 1 ponents, the thin and the thick disks. The change of slope in 2 components.Agesandabundancesareimportanttodescribethe the logarithm of the vertical density distributions at 700pc . ∼ variousdiskcomponentsandtodepictthemechanismsoftheir 1 (Cabrera-Lavers et al. 2005) or 1500 pc (Gilmore & Reid 0 1983)abovetheGalacticplaneisu∼suallyexplainedasthesigna- formation. A further complication comes from the recent indi- cations of the presence of at least two thick disk components 8 ture of a transitionbetween these two distinct components:the with differentdensity distributions,kinematicsand abundances 0 thinandthethickdisks.Thethickdiskisanintermediatestellar : (Gilmoreetal.2002;Soubiranetal.2003;Wyseetal.2006). v population between the thin disk and the stellar halo, and was i initiallydefinedwiththeotherstellarpopulationsbycombining X Manyoftherecentworksfavorthepresentlyprevailingsce- spatial, kinematicandabundanceproperties(see a summaryof nariosofthickdiskformationbytheaccretionofsmallsatellites, r the Vatican conference of 1957 by Blaauw 1995 and Gilmore a puffinguptheearlystellarGalacticdiskortidallydisruptingthe & Wyse 1989). Its properties are described in a long series of stellardisk(seeforexampleSteinmetz&Navarro2002;Abadi publicationswithoftendivergingcharacteristics(seetheanaly- et al. 2003; Brook et al. 2004). We note however that chemo- sisbyGilmore1985,Ojha2001, Robinetal.2003andalsoby dynamical models of secular Galactic formation including ex- Cabrera-Laversetal.2005,thatgiveanoverviewofrecentim- tended ingredients of stellar formation and gas dynamics can provements).Majewski(1993)comparedanearlyexhaustivelist alsoexplaintheformationofathickdiskdistinctfromthethin ofscenariosthatdescribemanypossibleformationmechanisms disk(Samland&Gerhard2003;Samland2004). forthethickdisk. In this paper, we attempt to give an answer to the simple but still open questions:are the thin and thick disks really two Inthispaper,weusetherecentRAVEobservationsofstellar distinctcomponents?Isthereanycontinuoustransitionbetween radialvelocities,combinedwithstarcountsandpropermotions, them?Thesequestionswerenotfullysettledbyanalysisofstar torecoverandmodelthefull3Ddistributionsofkinematicsand countsbyGilmore&Reid (1983) andlaterworkers.Otherim- densitiesfornearbystellarpopulations.Inaforthcomingstudy, portant signatures of the thick disk followed from kinematics: metallicitiesmeasuredfromRAVEobservationswillbeincluded the age–velocity dispersion relation and also the metallicity– todescribethegalacticstellarpopulationsandtheirhistory.The velocitydispersionrelation.Howevertheidentificationofathin– descriptionofdataisgiveninSect.2,themodelinSect.3,and thick discontinuity depends on the authors, due to the serious the interpretation and results in Sect. 4. Among these results, difficultyofassigningaccurateagestostars(seeEdvardssonet we identify discontinuities visible both within the density dis- al.1993andNordstro¨metal.2004).Morerecentlyitwasfound tributionsand the kinematicdistributions.Theyallow to define more precisely the transition between the thin and thick stellar Sendoffprintrequeststo:[email protected] Galacticdisks. 2 L.Veltzetal.:GalactickinematicswithRAVEdata 2. Observationaldata otherGalacticparametersthatoccursinotherGalacticdirections (densityandkinematicscalelengths,Oort’sconstants,R ,V ...). 0 0 ThreetypesofdataareusedtoconstrainourGalacticmodelfor The selected J K=[0.5-0.7] color interval corresponds to the stellar kinematics and star counts (the modeldescription is − K3-K7 dwarfs and G3-K1 giants (Koornneef 1983; Ducati et giveninSect.3):theTwo-MicronAll-SkySurvey(2MASSPSC; al. 2001). They may be G or K giants within the red clump Cutrietal.2003)magnitudes,theRAVE(Steinmetzetal.2006) region (the part of the HR diagram populated by high metal- andELODIEradialvelocities,andtheUCAC2(Zachariasetal. licity He-burning core stars). The absolute magnitudes of red 2004)propermotions.Eachsampleofstarsisselectedindepen- clump stars are well defined:nearby HIPPARCOS clump stars dently of the other,with its own magnitudelimit and coverage have a mean absolute magnitude M = 1.61 with a disper- K ofskyduetothedifferentsource(catalogue)characteristics. − sion of 0.22 (Alves 2000, see Cannon 1970 for the first (1)Weselect220502MASSstarswithinan8-degreeradius proposed∼use of clump stars as distance indicators, see also oftheSouthandNorthGalacticPoles,withmK magnitudesbe- Salaris&Girardi2002;Girardietal.1998andotherreferences tween5-15.4.StarcounthistogramsforbothGalacticpolesare in Cabrera-Lavers et al. 2005). This mean absolute magnitude usedtoconstraintheGalacticmodel. doesnot vary significantly with [Fe/H] in the abundancerange (2) We select 105170 UCAC2 stars within a radius of 16 [ 0.6,0] (Alves 2000). Studying nearby stars in 13 open clus- − degreesoftheGalacticpoles,withm 2MASSmagnitudesbe- tersand2 globularclusters, Grocholski& Sarajedini2002find K tween6-14.Weadjustthemodeltofithistogramsoftheµ and that the mean absolute magnitude of clump stars is not depen- U µ proper motion marginal distributions; the histograms com- denton metallicity when the [Fe/H] abundanceremainswithin V binestarsin1.0magnitudeintervalsform =6to9and0.2mag- the interval [ 0.5,0.1]. Sarajedini 2004 finds that, at metallic- K nitudesintervalsform =9to14. ity [Fe/H]= −0.76, the mean absolute magnitude of red clump K − (3) We select 543 RAVE stars ( with mK 2MASS magni- starsdropsto MK = −1.28,ashiftof0.33mag.Mostofthegi- tudesfrom8.5to11.5)withinaradiusof15degreesoftheSGP. ants with metallicity [Fe/H] lower than -0.8 dex are excluded Wegrouptheminthreehistogramsaccordingtom magnitudes. by our color selection from our sample. Hence, we did not K We completethisradialvelocitysamplewith 392othersimilar model giants of the metal-weak thick disk, first identified by stars:TYCHO-IIstarsselectedtowardstheNGPwithinanarea Norris1985(seealso,Morrison,Flynn&Freeman1990).This of 720 squaredegrees,with B-V colorsbetween 0.9-1.1.Their represents however only a minor componentof the thick disk. magnitudesarebrighterthanm =8.5,theywereobservedwith Although, Chiba & Beers (2000) find that 30 % of the stars K ∼ the ELODIE spectrographand were initially used to probe the with 1 > [Fe/H] > 1.7 are thick disk stars, but stars with − − verticalGalacticpotential(Bienayme´ etal. 2006).Allthesera- [Fe/H] < 1 represent only 1 per cent of the local thick disk − dialvelocitysamplesplayakeyroleinconstrainingthevertical stars(Martin&Morrison1998). velocitydistributionsofstarsandtheshapeofthevelocityellip- K dwarfs within the J K=[0.5-0.7] color interval also − soid. have well defined absolute magnitudes that depend slightly on metallicity and color. We determine their mean ab- solute magnitude, M =4.15, from nearby HIPPARCOS K MK −2024−0lllll.ll2lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll0llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll.llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll0llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll0lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll.lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll2lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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J−K pole directions: there, the kinematic data are simply related to thecardinalvelocitiesrelativetothelocalstandardofrest(LSR). Fig.1. MK/J KHRDiagramfromHipparcosstarswithσπ/π UCAC2 proper motions are nearly parallel to the U and V ve- − ≤ 0.1 cross-matched with the 2MASS catalogue. Vertical dashed locities, and RAVE radialvelocitiesare close to the verticalW linesrepresentourcolorselectionJ K=[0.5-0.7] velocitycomponent. − 2.2.Howaccurateistheavailabledata? Thestarmagnitudesaretakenfromthe2MASSsurveywhichis 2.1.Dataselection presentlythemostaccuratephotometricallskysurveyforprob- In this paper,we restrict our analysisto stars near the Galactic ingtheGalacticstellarpopulations.Nevertheless,sinceourcolor poleswithJ Kcolorsbetween0.5-0.7(seefig.1).Thisallowsus rangisnarrow,wehavetotakecarethatthephotometricerrors − torecoversomeGalacticproperties,avoidingthecouplingwith onJandKdonotbiasouranalysis. L.Veltzetal.:GalactickinematicswithRAVEdata 3 The meanphotometricaccuracyrangesfrom0.02in K and only noticeable difference between the histograms at the NGP J at magnitudes m =5.0, to 0.15 in K and 0.08 in J at magni- and SGP is that the peak of the proper motion distribution is K tudes m =15.4. The error in J K is not small considering the slightlymoreflattenedattheSGP,formagnitudesm >13(see K K − size(∆(J–K)=0.2)oftheanalyzedJ Kinterval,0.5to0.7.We fig. 6). Thisdifferenceis relatedto the differenterrordistribu- − do not expect, however, that it substantially biases our analy- tionstowardstheNGPandSGP. sis. For m brighterthan 10,the peakof giantsis clearly iden- The analyzed stars are located at distances from 200pc to K tifiedintheJ KdistributionwithintheJ K=[0.5-0.7]interval 1kpcfordwarfsandto1.5kpcforgiants.A2masyr 1errorrep- − (see Fig.2 or−Figure 6 from Cabrera-Lav−ers et al., 2005). This resents10kms 1at1kpc,and6masyr 1,anerrorof30kms 1. − − − peakvanishesonlybeyondm =11.AtfainterKmagnitudes,the Thiscanbecomparedtotheσ valuesfortheisothermalcom- K U dwarfsdominateandtheJ Khistogramofcolorshasaconstant ponents, for instance 60kms 1 for the thick disk that is the − − ∼ slope. This implies that the error in color at faint magnitudes dominantstellar population1.5kpcfromtheplane.Addingthe doesnotaffecttofirstorderthestarcounts. errors in quadrature to the velocity dispersion would modify a We find fromtheshape ofthe counthistogramsthat, in the real proper motion dispersion of 60kms 1 to an apparent dis- − directionoftheGalacticPoleandwithourcolorselectionJ K= persion of 67kms 1. The apparent dispersion would be only − [0.5-0.7]thelimitofcompletenessism 15.5-15.6.More−over, 60.8kms 1 if the stars have a 2masyr 1 accuracy. Therefore, K − − ∼ the contamination by galaxies must be low within the 2MASS we overestimatethe σ dispersionof the thick disk by 5 to 10 U PSC.Itisalsounlikelythatcompactorunresolvedgalaxiesare percent. This effect is lower for the thin disk components (the present: according to recent deep J and K photometric counts stars are closer and their apparent proper motion distributions (see Figure 15 of Iovino et al. 2005), with our color selection, arebroader).Wehavenotyetincludedtheeffectofpropermo- galaxiescontributeonlybeyondm 16.We concludethatwe tionerrorswithinourmodel.Thiserrorhasjustanimpactofthe K ∼ haveacompletesampleofstarsformagnitudesfrom5.0to15.4 determinationonthevelocitydispersionsσ andσ andonthe U V inK,towardstheGalacticpoles. ellipsoid axis ratio σ /σ of each stellar disk component,but U W The UCAC2 and RAVE catalogues however are not com- does not change the determination of vertical velocity disper- plete. Making it necessary to scale the proper motions and ra- sions σ which are mainly constrained by the magnitude star W dialvelocitiesdistributionspredictedbyourmodelforcomplete countandtheradialvelocities.Hence,itisnotsignificantinour samples. The total number of stars given by the model for the kinematicdecompositionoftheGalacticdisk. distribution of proper motions (or radial velocity) in a magni- Theaccuracyofpropermotionscanalsobegaugedfromthe tude interval is multiplied by the ratio between the number of stabilityofthepeaksofpropermotiondistributions:comparing starsobservedinUCAC2(orRAVE)dividedbythenumberof 112µ andµ histogramsfordifferentmagnitudeintervals,we U V starsobservedin2MASS. findnofluctuationslargerthan3-5masyr 1. − A more complete test is performed by comparing the UCAC2 proper motions (with our J K color selection) to the − recentPM2000catalogue(Ducourantetal. 2006) in an areaof 8 16 degrees around α =12h50m, δ = 14deg close to 2000 2000 × the NGP. PM2000 proper motions are more accurate, with er- rorsfrom1to4masyr 1.Themeandifferencesbetweenproper − 6 motions from both catalogues versus magnitudes and equato- rial coordinatesdo notshow significant shifts, just fluctuations of the order of 0.2 masyr 1. We also find that the disper- − 8 sionsofproperm∼otiondifferencesare 2masyr 1 form <10, − K 4masyr 1 with m =10-13, and 6ma∼syr 1 with m =13-14. − K − K K ThesedispersionsaredominatedbytheUCAC2errors. 10 From the internal and external error analysis, RAVE radial velocities show a mean accuracy of 2.3kms 1 (Steinmetz et − al. 2006).Radialvelocitiesof starsobservedwith the ELODIE 12 e´chelle spectrographare an order of magnitudemore accurate. Theseerrorshavenoimpactonthedeterminationofthevertical velocitydispersionofstellarcomponentsthatrangesfrom10to 14 50kms 1,butthereducedsizeofourradialvelocitysamplesto- − wardsthepoles(about1000stars)limitstheaccuracyachieved −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 inmodelingtheverticalvelocitydispersions. J−K Fig.2. K / J K Color Magnitude Diagram obtained with 3. ModelofthestellarGalacticdisks − 2MASS stars within a 8 degrees radius around the North Galactic pole. Dashed lines representthe limit of our color se- ThebasicingredientsofourGalacticmodelaretakenfromtradi- lection:J K=[0.5,0.7]. tionalworksonstarcountandkinematicmodeling,forinstance − seePritchet(1983);Bahcall(1984);Robin&Cre´ze´ (1986).Itis alsosimilar totherecentdevelopmentsbyGirardietal.(2005) Towards the North Galactic Pole (NGP), the error on orbyVallenarietal.(2006). the UCAC2 proper motions used in our analysis varies from The kinematic modeling is entirely taken from Ratnatunga 1masyr 1 for the brightest stars to 6masyr 1 at m =14. etal.(1989)andisalsosimilartoGould’s(2003)analysis.Both − − K Towards the South Galactic Pole (SGP), the error distribution propose closed-form expressions for velocity projections; the lookssimilar,withtheexceptionofasmallfractionofstarswith dynamicalconsistencyissimilar toBienayme´ etal.(1987) and m from 11 to 14 having errors around 8 or 13 masyr 1. The Robinetal.(2003,2004). K − 4 L.Veltzetal.:GalactickinematicswithRAVEdata Ouranalysis,limitedtotheGalacticpoles,isbasedonaset Weadoptanasymmetricdriftproportionaltothesquareofσ : RR of20stellardiskcomponents.Thedistributionfunctionofeach V = σ2 /k ,wherethecoefficientk isalsoamodelparame- lag RR a a component or stellar disk is built from three elementary func- ter.We assumenullstreammotionsfortheothervelocitycom- tions describing the vertical density ρ (dynamically self con- ponents,thus U = U and W = W . i sistent with the vertical gravitational potential), the kinematic Forsimplihcityi,we−ha⊙veasshumiedth−att⊙heσ /σ ratioisthe RR φφ distribution f (3D-gaussians)andtheluminosityfunctionφ . sameforallthecomponents.Itiswellknownthatthe assump- i ik Wedefine (z,V ,V ,V ;M)tobethedensityofstarsinthe tions of a constant σ /σ ratio, of a linear asymmetric drift R φ z RR φφ N Galacticposition-velocity-(absolutemagnitude)space andof2DgaussianUandVvelocitydistributionsholdonlyfor cold stellar populations(see for instance Bienayme´ & Se´chaud = ρ(z)f(V ,V ,V )φ (M) 1997).Thesesimpleassumptionsallowadirectcomparisonwith i i R φ z ik N Xik similarstudies.Itallowsalsoanexactintegrationofcountequa- tionsalongthelineofsight.Thustheconvergenceofparameters the index i differentiates the stellar disk components and the foranysinglemodelisachievedinareasonableamountoftime index k the absolute magnitudes used to model the luminosity (oneweek).Themodelincludes20isothermalcomponentswith function. σ from3.5to70kms 1.Wechooseastepof3.5kms 1which Fromthismodel,weapplythegeneralizedequationofstellar zz − − issufficienttogivearealistickinematicdecompositionandper- statistics: mit calculation in a reasonable time. The two first components σ = 3.5 and 7kms 1 were suppressed since they do not con- A(m,µ,µ ,V )= N(z,V ,V ,V ;M)z2ωdz zz − l b r Z R φ z tributesignificantlytocountsformK>6andarenotconstrained byouradjustments.Thecomponentsbetween10and60 kms 1 − to determine the A(m) apparent magnitude star count equation areconstrainedbystarcounts,propermotionshistogramsupto aswellasthemarginaldistributionsofbothcomponentsµl and magnitude 14 in K and radial velocity histograms for magni- µbofpropermotionsandthedistributionsofradialvelocitiesfor tudes mK=[5.5-11.5]). The model includes isothermal compo- any direction and apparentmagnitudes.For the Galactic poles, nentsfrom60to 70kms 1 to properlyfit the star countsatthe − wedefineµU andµV asthepropermotioncomponentsparallelto faintest apparent magnitudesmK > 15.0. All the values of the thecardinaldirectionsofU andV velocities.Foramoregeneral kinematiccomponentsdependontheadoptedgalacticpotential. inversemethodoftheequationofthestellarstatistic,seePichon ThevelocityellipsoidsareinclinedalongtheGalacticmerid- etal.(2002). ianplane.Themainaxisofvelocityellipsoidsaresetparallelto con-focalhyperboloidsasinSta¨ckelpotentials.Wesetthefocus atz =6kpconthemainaxisgivingthemrealisticorientations 3.1.Theverticaldensity hyp (seeBienayme´ 1999).Thenon-zeroinclinationimpliesthatthe Eachstellardiskismodeledwithanisothermalvelocitydistribu- vertical density distributions of each isothermal component is tion,assumingthattheverticaldensitydistribution(normalized notfullydynamicallyconsistentwiththepotential.Sincethez- atz=0)isgivenbytherelation: distancesare below 1.5kpc for the majority of stars with kine- maticdata,andsincethemaintopicofthispaperisnotthede- ρi(z)=exp −Φ(z)/σ2zz,i (1) terminationoftheGalacticpotential,wedonotdevelopamore (cid:16) (cid:17) consistentdynamicalmodel. where Φ(z) is the vertical gravitational potential at the solar Galactic position and σ is the vertical velocity dispersion of zz,i theconsideredstellarcomponenti.TheSun’spositionz above 3.3.Theluminosityfunctions the Galactic plane is also used as a modelparameter. S⊙uch ex- Theluminosityfunctionofeachstellardiskcomponentismod- pressionswereintroducedbyOort(1922),assumingthestation- eled with n different kinds of stars according to their absolute arity of the density distributions. They ensure the consistency magnitude: between the vertical velocity and density distributions. For the ovebrttaiicnaeldgrbayviBtaiteionnayalmpe´oteetnatila.l(w20e0u6s)ebtahseedrecoenntthdeetaenrmalyinsaistioonf φi(M)= φik(M)= 1 cike−12(cid:18)Mσ−MMk(cid:19)2 HIPPARCOSandTYCHO-IIredclumpgiants.Theverticalpo- kX=1,n √2πσM kX=1,n tentialisdefinedatthesolarpositionby: where c is the density for each type of star (index k) of each ik stellardiskcomponent(indexi). Φ(z)=4πG Σ z2+D2 D +ρ z2 0 eff We use four types of stars to model the local luminosity (cid:18) (cid:18)p − (cid:19) (cid:19) function (see Fig. 3). More details on the way that we have withΣ0 =48M pc−2,D=800pcandρeff =0.07M pc−3. determined it is given in section 4.4. Stars with a mean abso- Itis quitesi⊙milar to the potentialdeterminedby⊙Kuijken& lute magnitude M = 1.61 are identified to be the red clump K Gilmore(1989)andHolmberg&Flynn(2004). giants (k = 1) that w−e will call ‘giants’, with M = 0.89 K − and M = 0.17 for first ascent giants that we categorize as K − ‘sub-giants’ (k = 2 3) and M = 4.15 are labelled dwarfs 3.2.Thekinematicdistributions − K (k = 4) (see fig. 1). We neglected ’sub-giant’ populationshav- The kinematical model is given by shifted 3D gaussian veloc- ingabsolutemagnitude M between0.2and2.Theirpresences K ityellipsoids.Thethreecomponentsofmeanstreamingmotion marginallychangetheratioofgiantstodwarfs,sincetheirmag- ( U , V , W )andvelocitydispersions(σ ,σ ,σ ),referred nitudesarelower,andtheirtotalnumberinthemagnitudecounts RR φφ zz h i h i h i tothecardinaldirectionsoftheGalacticcoordinateframe,pro- appearssignificantlysmallerthantheothercomponents.Infact, videasetofsixkinematicquantities.Themeanstreammotionis we initially tried to introduce 10 types of stars (spaced by 0.7 relativetotheLSR.TheSun’svelocityU andW aremodelpa- absolute magnitude intervals). This still improvesthe fit to the rameters.Wedefinethe V streammotio⊙nas: V⊙ = V V . data.Howeverdueto the smallcontributionof the ‘sub-giants’ lag h i h i − ⊙− L.Veltzetal.:GalactickinematicswithRAVEdata 5 componentswithM =[0.2 2],theywerenotdeterminedwith thickdiskcomponentsofourGalaxy.Thisdiscontinuitymustbe K − ausefulaccuracy.Weadoptσ = 0.25,justifiedbythenarrow theconsequenceofsomespecificprocessofformationforthese M rangeofabsolutemagnitudesbothforredclumpgiantsandfor Galacticcomponents. dwarfsontheluminosityfunction. Fittingamulti-parametermodeltoalargedata-setraisesthe The4x20coefficientsc areparametersofthemodel.Inor- questionoftheuniquenessofthebestfitmodel,andtherobust- ik dertoobtainarealisticluminosityfunction,wehaveaddedcon- nessofoursolutionandconclusions.Forthispurpose,wehave straintstotheminimizationprocedure.Foreachkinematiccom- exploredthestrengthofthebestGalacticmodel,byfittingvari- ponenti,weimposeconditionsontheproportionofdwarfs,gi- oussubsetsofdata,bymodifyingvariousmodelparametersand antsandsub-giantsfollowingthelocalluminosityfunction.We adjusting the others. This is a simple, but we expect efficient, havemodeledourdeterminationofthelocalluminosityfunction way to understand the impact of parameter correlations and to ofnearbystars(seeFig.3).Weobtained: seewhatisreallyconstrainedbymodelorbydata.Asummary ofthemainoutcomesisgivenbelow. – aratioofthedensityofdwarfs(k=4)tothedensityofgiants From these explorations, we choose to fix or bound some (k=1)of12.0,soweimpose: ci,4 >10 ci,1 importantGalacticmodelparameterswhichwouldotherwisebe – a ratio of the density of giants (k=1) to the density of sub- poorlyconstrained:i)wefixtheverticalGalacticpotential(ad- giants(k=2)of2.3,soweimpose: ci,1 >2 justingtheK forcedoesnotgivemoreaccurateresultsthanfor ci,2 z – andthedensityofsub-giants(k=2)isgreaterthanthedensity instance in Bienayme´ et al., 2006, since we only increase by a ofsub-giants(k=3),soweimpose:c > c . factor 2 the number of stars with measured radial velocities), i,1 i,2 ii)theasymmetricdriftsofallkinematiccomponentsarelinked If we do not include these constraints, the variouscomponents througha unique linear asymmetric drift relation with just one arepopulatedeitheronlywithdwarfsoronlywithgiants. freeparameter;thesolarvelocitycomponentV isalsofixed,iii) theaxisratioofthevelocityellipsoidsisboun⊙ded;forthindisk components(σ 25kms 1) we set σ /σ > 1.5, for thick −210 ˚ ¸ (cid:204) ˝ disks(σW >30Wkm≤s−1,σU/σ−W >1.1). U W The agreement between our fitted model and the observed countsisillustratedbythevariousmagnitude,propermotionand −1) radialvelocitydistributions(Figs.4,5,6,7,8).Wecanconsider ag thatgloballytheagreementisgood,ifwenotethesmallχ2 val- K-m −310 uesobtained.We justcommentthe maindisagreementsvisible −3 withinthesedistributions.Theycanbecomparedtorecentsim- pc ber (cid:228) ilarstudies(Girardietal.2005,Vallenarietal.2006). m Theagreementfortheapparentmagnitudedistributionlooks u n ( satisfyinginFig.4. φFL −410 (cid:228) ThecomparisonofobservedandmodeledµU propermotion (cid:228) distributionsdoes not show satisfactory agreementclose to the maximaofhistogramsatapparentmagnitudem <10(NGPor K SGP, see Fig. 5). We have not been able to determine if this is −3 −2 −1 0 1 2 3 4 5 duetotheinabilityofourmodeltodescribetheobserveddata, MK forinstance due to simplifyingassumptions(gaussianityof the Fig.3. Local luminosity function:The histogram is our deter- velocitydistribution,asymmetricdriftrelation,constantratioof mination of the local luminosity functionfor nearby stars with velocitydispersions,etc...).Wenotethatthisdisagreementmay error bars. The red (or dark grey) dashed line is a fit of the lu- just result from an underestimate of the impact of the proper minosity function with four gaussians (blue or light grey line) motionerrors. correspondingtothedwarfs,thegiantsandthetwotypesofsub- Some possible substructuresare seen in propermotion his- giants. tograms for the brightest bins (mK <7, Fig. 5); they are close to the level of Poissonian fluctuations and marginally signifi- cant. One of the possible structures corresponds to the known Herculesstream(U¯ = 42 kms 1andV¯ = 52kms 1,Famaey − − − − etal2005). 4. Resultsanddiscussion Forfaintmagnitude(m >11)bins(Fig.6),smallshifts( 3- K The181freemodelparametersareadjustedthroughsimulations. 5masyr 1)ofµ explainmostofthedifferencesbetweenN∼orth − U Each simulation is compared to histograms of counts, proper andSouthandthelargerχ2. motionsandradialvelocities(see Sect. 2 forthe descriptionof At m within 10-13 (Fig. 6), the wings of µ histograms K U data histograms and see Figures 4, 5, 6, 7, 8) for the compari- look slightly different between North and South directions; it son of the best fit model with data. The adjustment is done by apparentlyresultsfromshiftsofNorthhistogramsversusSouth minimizing a χ2 function using the MINUIT software (James ones. 2004). Equal weight is given to each of the four types of data A disagreement of the model versus observations also ap- (magnitudecounts,µ propermotions,µ propermotions,and pears within the wings of µ distributions, (m within 10-13, U V V K radial velocities). This gives relatively more weight to the ra- Fig. 6). This may introduce some doubtconcerningour ability dialvelocitydatawhosecontributioninnumberistwoordersof to correctly recover the asymmetric drift, because the negative magnitudesmallerthanforthephotometryandpropermotions. propermotiontailofµ distributionsdirectlyreflectstheasym- V By adjusting our Galactic model, we derive the respective metricdriftoftheV velocitycomponent.However,weestimate contributionsof dwarfsand giants, andof thin and thick disks. thatourdeterminationofthe asymmetricdriftcoefficientis ro- Onenoticeableresultisthekinematicgapbetweenthethinand bustandmarginallycorrelatedtotheothermodelparameters. 6 L.Veltzetal.:GalactickinematicswithRAVEdata Fig.4.MagnitudecounthistogramtowardstheNorthGalacticPole.Left:modelprediction(dashedline)issplitaccordingtostar types:giants(redorblackline),sub-giants(dot-dashedanddotted)anddwarfs(greenorgreyline).Therightfigurehighlightsthe contributionsofthinandthickdisks(respectivelythinandthicklines),fordwarfs(greenorgrey)andgiants(redorblack). Fig.5. µ and µ histograms towards the North Galactic Pole (right) and the South Galactic Pole (left) for magnitudes6 to 10: U V model(dashedline) andcontributionsfromthedifferenttypesofstars:giants(redordarkthinlines),sub-giants(dot-dashedand dottedlines)anddwarfs(greenorgreythicklines). L.Veltzetal.:GalactickinematicswithRAVEdata 7 Fig.6.SameasFig.5formagnitudes10to14. Fig.7.RadialvelocityhistogramstowardstheNorthGalacticPoleformagnitudes5.5to8.5forELODIEdata:model(dashedline) andcontributionsofthedifferenttypeofstars:giants(redordarklines),sub-giants(dot-dashedanddotted)anddwarfs(greenor greyline). Thesecomparisonsofobservedandmodeldistributionssug- motiondata,andalso forpossibleinconsistencyinourgalactic gestnewdirectionstoanalyzedata.Inthefuture,weplantouse modeling. the present galactic model to simultaneously fit the RAVE ra- dialvelocitydistributioninallavailablegalacticdirections.This result will be compared to a fit of our model to proper motion 4.1.Thetransitionfromdwarfstogiants distributionsoverall galactic directions.This will give a better WithintheJ K=[0.5-0.7]interval,thepropermotionisanexcel- insightintotheinconsistencybetweenradialvelocityandproper − lentdistanceindicator:thereisafactorof14betweentheproper 8 L.Veltzetal.:GalactickinematicswithRAVEdata Fig.8.NumberofgiantsanddwarfsinRAVEdatacomparedtomodelprediction.Leftcolumn:Radialvelocityhistogramstowards theSouthGalactic Pole formagnitudes8.5to 11.5forRAVE data,model(dashedline) andcontributionsofthe differenttypeof stars: giants (red or dark lines), sub-giants (dot-dashedand dotted lines) and dwarfs (green or grey line). Center column: Radial velocityhistogramsforallstars (black)andforgiants(redor grey):modelfor allstars (blackdashedline) andforgiants(redor greydashedline).Rightcolumn:Radialvelocityhistogramsforallstars(black)andfordwarfs(greenorlightgrey):modelforall stars(blackdashedline)andfordwarfs(greenorlightgreydashedline). motionofadwarfandthepropermotionofagiantwiththesame tweengiants-sub-giantsanddwarfsoccursatm 10.1.Thisis K ∼ apparentmagnitudesandvelocities.Combiningpropermotions arobustresultfromourstudythatdependsslightlyontheabso- andapparentmagnitudes,ourbest-fitGalacticmodelallowsus lute magnitude adoptedfor dwarf and giant stars. We have not toseparatethecontributionsofdwarfsandgiants(Fig.4). triedtochangeourcolorrange.Ifwetakeabroadercolorinter- We deduce that, towards the Galactic poles, most of the val,thedispersionaroundtheabsolutemagnitudeofdwarfswill brightstars are giants. At m = 7.2, only 10% are dwarfs and belarger,butourresultsarenotexpectedtochange.Foranother K at m = 9.6only 50%are giants. We havecheckedif the con- colorinterval,wecanexpectthisresulttobedifferent,sincewe K tributionofsub-giantswithabsolutemagnitudeM = [0.2 2] wouldbelookingatadifferentspectraltypeofstar. K canchangethecontributionof dwarfsandgiants.Atm <−10, A confirmation of the dwarf-giant separation between magni- K othredecronotfrimbuatginointuodfesulobw-geari.nStowitthheMraKtio=o[f0.g2ia−n2ts]aisndatdlewaasrtfosnies tpurdeelismminKar=yd[e5t.e5rm−in1a1t.i5o]ncoofmtheessftreolmlarRpAarVamEestpeersct(rTa.efWf,iltohgt(hge) unchanged.Furthermore,theRAVEdataconfirmourmodelpre- and[Fe/H])ofRAVEstars,wechoosetodefinegiantstarswith diction.ThisisincontradictionwithCabrera-Laversetal.(2005) log(g) < 3 anddwarfswith log(g) > 4.The comparisonof the statementbasedontheWainscoatetal.(1992)modelwhichesti- numberofgiantsanddwarfspredictedbyourbestmodeltothe matesthat,atmagnitudem <10,giantsrepresentmorethan90 observedoneisingoodagreement(seefig.8). K %ofthestars.TheWainscoatmodelassumesonlyonediskwith ascaleheightof270pcforthegiantsand325pcforthedwarfs. 4.2.Thescaleheightsofstellarcomponents Inourmodel,wefindascaleheightof225pcbothforthegiants andthedwarfs.Thisexplainswhywefindmoredwarfsatbright Ourdynamicalmodelingofstarcountsallowsustorecoverthe magnitudes(mK <10). verticaldensitydistributionofeachkinematiccomponentρi(z), Faint stars are mainly dwarfs, 80% at m = 11.6 while at with theexactshapesdependingontheadoptedverticalpoten- K m = 11.9, only 10% are giants. The 50%-50% transition be- tialΦ(z).We recoverthewell-knowndouble-exponentialshape K L.Veltzetal.:GalactickinematicswithRAVEdata 9 which is in relativelygoodagreementwith the valuesobtained fromourmodel. For dwarfs that dominate the counts at faint apparentmag- nitudes m > 11 (distances larger than 240pc), we use the K ∼ photometricdistance: zphot =10(mK−MK−5)/5 (2) whereM isequalto4.15(thevalueforthedwarfs). K Doingso,weobtainthenumberdensityn(z )ofstarsseen phot along the line of sight at the SGP and NGP (Fig. 10). These plotsshowawell-definedfirstmaximumatz =500pc(SGP) phot or 700pc (NGP) related to the distribution of thin disk dwarfs. At 0.9-1.1kpc, n(z ) has a minimum and then rises again at phot largerdistances,indicatingthethickdiskdwarfcontribution. However, the use of photometric distances can introduce a systematic error for thick disk dwarfs that have lower metal- Fig.9. Model of the vertical stellar density ρ(z) towards the licities . The mean metallicity of the thick disk populationat 1 the North Galactic Pole (dashed line) and its thin and thick kpcis [Fe/H] -0.6(Gilmoreetal.1995;Carraroetal.1998; h i≃ disk decomposition (respectively thin and thick lines). The Soubiranetal.2003). thin disk includes the isothermal kinematic components with Themetallicityvariationfrom[Fe/H]=0.0forthethindiskto σ <25kms 1,thethickdisksincludecomponentswithσ > [Fe/H]=-0.6forthethickdiskmeansthattheabsolutemagnitude W − W 25kms−1. MKchangesfrom4.15to4.5.So,wesmoothlyvarytheabsolute magnitudewiththemetallicityfromthethintothethickdisk,in thisway: of the total verticalnumberdensity distribution ρtot(z) (Fig. 9). M ([Fe/H])= M +0.035m (3) K K,0 K Since we estimate that the kinematicdecompositionin isother- malcomponentsisclosertotheidealizedconceptofstellarpop- whereM isequalto4.15. K,0 ulationsanddisks, we identifythe thindisk asthe components Thecountscontinuetoshowtwomaxima(Fig.11),evenif with vertical velocity dispersions σ smaller than 25kms 1 theminimumislessdeep.Theminimumdelineatesadiscontin- W − and the thick disk with σ from 30 to 45.5kms 1 (Fig. 12). uoustransitionbetweenthethinandthickcomponents. W − Following this identification, we can fit an exponential on the thin and thick disk vertical density component (thin line and The superposition of the model on the number density thick lines respectively of Fig. 9). The scale height of the thin n(z ) showsonlyapproximateagreement(Fig.10).We think phot diskis225 10pcwithin200-800pc.Forthe thickdisk,within that is due to non-isothermalityof the real stellar components. ± 0.2-1.5kpc,thescaleheightis1048 36pc.Ifweconsiderallthe Anyway,thefactthatthemodeldoesnotreproduceexactlythe ± kinematic componentswithout distinguishing between the thin observationdoesnotweakentheconclusionaboutthekinematic andthickdisk,wecanfitadoubleexponentialwithascalelength separation of the thin and thick disk. It reinforces the need ofthethindisk217 15pcandofthethickdisk1064 38pc.We for a clear kinematic separation between the two disks in the ± ± calculate the error of the scale length from the error on the in- kinematicdecomposition(Fig.12). dividualkinematicdiskcomponentsφ (seeTab.1).Wehave kin,i performedaMonte-Carlosimulationonthevalueofthecompo- Wealsonotice,inFig.10,thedifferenceincountsbetween nentsandobtainedtheerrorbarsforthescalelengthofthethin the North and the South. This difference allows us to deter- andthickdiskbothindependentlyandtogether. mine the distance of the Sun above the Galactic plane, z = We note that our density distribution is not exponentialfor +20.0 2.0pc, assuming symmetry between North and So⊙uth. ± z<200pc:thismainlyresultsfromthefactthatwedonotmodel We also note thatthe transitionbetweenthin andthick disksis components with small velocity dispersions σ <8kms 1. morevisibletowardstheSGPthantowardstheNGP. W − Thusourestimateddensityatz=0cannotbedirectlycompared, for instance, to Cabrera-Lavers et al. (2005) results. With this 4.3.Thethin–thickdisktransition,andthekinematic proviso,thestarnumberdensityratioofthicktothindiskstars distributionfunction atz=0pcis8.7%forthedwarfs. One candidate to trace the thin and thick disk are the red The minimum at z 1kpc in the n(z) distribution (Fig. 10) ∼ clump giants. In fact, at z-distances larger than 500pc (i.e. provides very direct evidence of the discontinuity between ∼ m larger than 7.0, see Fig. 4, there are more thick disk gi- stellar components with small velocity dispersions (σ =10- K W antsthanthindi∼skgiants.Cabrera-Laversetal.(2005)havean- 25kms 1) and those with intermediate velocity dispersions − alyzedthemusing2MASSdata.Todothis,theyselectallstars (σ 45.5kms 1)(leftpanelFig.12). W − ∼ withcolorJ-K=[0.5-0.7]andmagnitudem < 10.But,beyond Anothermanifestationofthistransitioniswellknownfrom K magnitude9,theproportionofgiantsrelativetosub-giantsand thelogρ(z)densitydistribution(Fig.9)whichshowsachangeof dwarfsdecreasesquickly.Atm =9.6,giantsjustrepresenthalf slopeatz=500-700pc.Thisfeaturecanbesuccessfullymodeled K ofthestars,andtheirdistanceisabout1.7kpc.Thus,wemustbe with two (thin and thick) components (e.g. Reid and Gilmore cautiouswhenprobingthethickdiskwithclumpgiantsandwe 1983),whichisanindicationofadiscontinuitybetweenthethin havefirsttodeterminetherespectivesub-giantanddwarfcontri- andthickdisksofourGalaxy. butions.However,Cabrera-Laversetal.(2005)obtainedascale Itisconclusiveevidence,onlyifweshowthatwecannotfit heightof267 13pcand1062 52pcforthethinandthickdisks accuratelythestarcountsorverticaldensitydistributionswitha ± ± 10 L.Veltzetal.:GalactickinematicswithRAVEdata 2500 SGP NGP 2000 2000 ofstars 1500 ofstars 1500 er er Numb 1000 Numb 1000 500 500 500 1000 1500 500 1000 1500 Photometricdistance(pc) Photometricdistance(pc) Fig.10.Data (histogramwith errorbars) andmodel(dashedline) forthe NGP(left) andSGP(right)verticaldensitydistribution usingphotometricdistancesn (z)fordwarfstars.Thetransitionbetweenthinandthickcomponentsisrevealedbyaminimumat phot z 1kpc.Themaincontributingcomponentsareplotted,forthethindisk(thincontinuousline)σ =10.5(dot-dashed),14&17.5 W (tr∼ipledot-dashed),21&24.5kms 1(dotted)andforthethickdisk(thickcontinuousline)σ =45.5kms 1. − W − 3000 SGP NGP 2500 stars 2000 stars 2000 erof erof 1500 b b m m Nu 1000 Nu 1000 500 500 1000 1500 500 1000 1500 Photometricdistance(pc) Photometricdistance(pc) Fig.11.HistogramsoftheverticaldensitydistributionfortheNGP(left)andSGP(right)usingphotometricdistancesn (z)for phot dwarfstarswithasmoothvariationinthe[Fe/H]fromthethintothethickdisk. Fig.12. Left:Thelocalσ kinematicdistributionfunction.Thecontributingcomponentstostarcountscanbeputtogetherinathin W diskcomponent(σ <25kms 1),athickdisk(isothermalwithσ =45.5kms 1)andahottercomponentwithσ 65kms 1.The W − W − W − twofirstcomponentswithσ =3.5and7kms 1 aresettozerobyconstruction.Right:AKinematicDistribution∼Function(KDF) W − thattriestoreproducethemagnitudestar countsandthekinematicdata:thismodelhasbeenobtainedrequiringthecontinuityof theKDFfromσ =10to48kms 1. w −