198 9MNRAS.239. .571K 3 There aretworelatedyetdifferentmeasuresofthedistribution ofmassinthegalacticdiscnear the Oortlimitmaybedetermined bysummingalllocalobservedmatter-an observationally the sameasvolumemass densityatthegalacticplane.Thisquantityhasunits ofMpc', the Sun.Themostwidelyusedandcommonlydetermined measureisthelocalvolumemass difficult task, whichleadstoconsiderable uncertainties.Theuncertainties ariseinpartdue to and itslocalvalueisoften calledthe‘Oortlimit’.Thecontributionofidentified materialto density -i.e.theamountof massperunitvolumeneartheSun,whichforpractical purposesis 1 Introduction 0 © Royal Astronomical Society • Provided by theNASA Astrophysics Data System A techniquetodeterminetheintegralsurfacemass The massdistributioninthegalacticdisc-1. Accepted 1989February3.Received3;inoriginalform The requisiteobservationaldataareanobserveddistributionfunctionof velocities anddistancesforasampleoftracerstarsextending^1kpcfromthe Summary. Wepresentanewtechniqueforthedeterminationofintegral function withavarietyofmodeldistributionfunctions,calculatedforwide population. Theanalysisinvolvescomparisonoftheobserveddistribution galactic plane,andthespatialdensitydistributioncorrespondingtothattracer surface massdensityofthegalacticdiscatsolargalactocentricdistance. Konrad KuijkenandGerardGilmoreInstituteofAstronomy, density ofthediscnearSun Madingley Road,CambridgeCB3OHA range ofassumedgalacticpotentialsandcorrespondingforcelawsK{z\to it utilizesthefullobserveddistributionfunction,and soavoidstheneedfor effects duetothelikelychangeinorientationofstellarvelocityellipsoid calculated inasufficientlygeneralwaythatonecanincludethequitelarge determine thebestdescriptionofdata.Themodeldistributionsare binning dataandtheuseofafewvelocitymomentsto representalargenumber Mon. Not.R.astr.Soc.(1989)239,571-603 observed rotationcurve.Animportantfeatureofthe analysistechniqueisthat extended darkhalogeneratingalargepartoftheradialaccelerationin is constrainedtobeconsistentdynamically,inthatthelocalmassdensityof as onemovesfurtherfromthegalacticplane.ThederivedbestfitT^-forcelaw 1988 November1 Galaxy isalsodetermined,insuchawayastoensureconsistencywiththe of observations. z 198 9MNRAS.239. .571K -2 limit hasanextensivehistory,andwillbediscussedfurtherinPaperIIIofthisseries(Kuijken& important quantity(fordynamicalpurposes)isthemeanvolumedensityofpatchily interstellar medium(ISM)toadopt.Thislatteruncertaintyisexacerbatedsincethephysically Hewett 1985),butmostlyfromuncertaintiesintheappropriatevolumedensityof difficulties indetectingverylowluminositystars,evenneartheSun{cf.Gilmore,Reid& Gilmore 1989b). distributed ISMatthesolargalactocentricdistance.ThedynamicaldeterminationofOort mass density.ThisquantityhasunitsofMpc,andistheamountdiscinacolumn perpendicular tothegalacticplane,extendingsufficientlyfarthatitincludesallmass 572 K.KuijkenandG.Gilmore rotation curvesandthelarge-scaledistributionofmassingalaxies. associated withthegalacticdisc.Itisthisquantitywhichofrelevancetointerpretation which iseffectivelyfullyexplainedbyobservedmass.Thentheunidentifiedcontributionto was unidentified(i.e.alocal‘missingmass’problem),butalsodeterminesurfacedensity its nature.Forexample,onemightdeducethatsomefractionofthelocalvolumemassdensity unidentified inthevolumemassdensitynearSunwasnotsame‘missingmass’which local volumedensitywouldhavetoasmallscaleheight,inorderthatitscontribution contribution tothelocalvolumemassdensitywhichwasnotidentified,evenwithoutknowing galactic disc,onecouldimmediatelydetermine(oratleastconstrain)thescaleheightofany dominates theextendedouterpartsofgalaxies. the surfacedensitybesmall.Itwouldthenplausibletodeducethatany‘local’missingmass K^(z), atthesolargalactocentricdistanceR=R. which, whilenotbeingaproblemfordeterminationoflocaldynamicalquantitiessuchasthe There areseveralinherentlimitationsinpublishedtechniquesforastudyofkinematicdata, require similarobservationaldata,namelydistancesandvelocitiesforasampleofsuitable from theplane.Itthusremovesneedtodescribeanarrayofdistance-velocitydataby tracer stars(thoughonmoredistanttracersinthelattercase),butratherdifferentanalyses. important physicaleffects(theorientationofthestellar velocityellipsoidfarfromthegalactic moments, suchasther.m.s.velocitydispersion. It alsoprovidesthefreedomtoinclude observed andpredicteddistributionfunctionsofstellarvelociitesasafunctiondistance limit -neartheSun,anddiscussconsistencyof theselocalandmoreglobaldynamical plane) andconstraints(consistencywiththegalacticrotation curve)inthemodelling. density ofthegalacticdiscnearSun.Itinvolvesmaximumlikelihoodcomparison as theintegralsurfacedensity.Inthispaperwepresentanewtechniqueforanalysisof Oort limit,areamajorlimitationindeterminationofmoregeneraldynamicalquantitiessuch 0 quantities. stellar kinematicdata,whichisalsoappropriatefordeterminationoftheintegralsurfacemass point alongitstrajectory, and,afterafewrevolutionsandverticaloscillations, deducethe be tolaunchatest-particle, andwatchitsorbit.Thenwecouldcalculatetheforce itfeelsatany How canwemeasurethegravitational fieldinastellarsystem?Theobviousthing todowould Gilmore 1989b)wereaddressthedeterminationof thelocalvolumemassdensity-Oort set, anddeterminetheintegralsurfacemassdensityof thegalacticdisc.InPaperHI(Kuijken& 2 Themeasurementofgalactic potentials 0 The secondmeasureofthedistributionmassinsolarvicinityisintegralsurface If oneknewboththelocalvolumemassdensityandintegralsurfaceof Both thesequantitiesarederivedfromameasurementoftheverticalgalacticforcefield, Determination ofthevolumemassdensityandintegralsurfacenearSun In PaperIIofthisseries(Kuijken&Gilmore1989a) weapplythistechniquetoanewdata- © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 198 9MNRAS.239. .571K z3 3 gravitational potentialnearthisorbit.Whilesuchaprocedureisfeasibleforstudyingthe gravitational fieldoftheEarth(afteraccountistakenatmosphericdragandothernon-gravi- more thanafewauacrossthetime-scalesinvolvedmakeinvestigationsofthistypeimpractical. tational effects),aswellforstudyofthesolarsystemandsomebinarystars,systems tions andvelocitiesmeasurableatasingletimetostudythegravitationalpotential.Excellent We mustthereforeturntothetheoryofstellarstatistics,anduseensembleposi- reviews ofthissubjectcanbefoundinOort(1965),andBinney&Tremaine(1987). The dynamicsofanylargestellarsystemaregovernedbythecollisionlessBoltzmannequation not possibletodefinesuchan/(x,¿T),butforthemomentwewillignorecomplications,as infinitely smallvolumes.Sometimes,thestructureoforbitissocomplicated(fractal)that where fisthephasespacedensityatpoint(x,v)in[i.e.thereare/(*,dxv is appropriateforsystemscomprisedpredominantlyofregularorbits.Adiscussionthe occurrence ofirregularorbitsintheMilkyWayGalaxywillbedeferreduntilAppendixB. a distributionfunctioniscalled‘fine-grained’,asitsconstructionrequiresaveragingover stars inavolumeofsizedxcentredonwithvelocitythevaboutv\Such 2.1 THECOLLISIONLESSBOLTZMANNEQUATION gravity forces,andthosearebeingdescribedthroughasmoothbackgroundpotential. Dt dtdxdv Consequently, /doesnothavetodescribetheentireGalaxy;onecanconcentrateonany stars arepresentornot.Thisarisesbecausedonotinteractexceptthroughlong-range the Galaxy,irrespectiveofwhatcausesthispotential. subsample ofstars,andapplythecollisionlessBoltzmannequationtoit.Wewillrefersuch subsamples astracerpopulations,sincewemayusetheirkinematicstotracethepotentialof that itisconvenienttowriteoutthecollisionless Boltzmann equationincylindricalpolar For presentpurposes,theGalaxyisadequatelydescribed asbeingrotationallysymmetric,so scale fieldintheMilkyWaypresumablyis,wecanset components (t^,íl,): coordinates {R,(f>,z)inwhichz=0isthediscplane of symmetry,withcorrespondingvelocity where theaccelerationsv,v^,explicitinequation (1)havebeenequatedtotheforces(real general bemadetosatisfy anequationsuchas(3)whichcontainsonlytwo functions oftwo and K:fhasfiveindependent variables(wesuppressedany^-dependence)and socannotin and Ktobederived.Note, though,thatageneralfunction/willnotallow solutionforK K{R,z) isthegravityforce. Thenclearlyknowledgeoif{x,v)allowstheforce components K and fictitious)thatcausethem, and^-gradientsin/andinthepotentialsetto zero. Thevector Rz z z R R The collisionlessBoltzmannequationissatisfiedbyanystellarpopulation,whetherother If wehaveasteady-statetracerpopulation,andtime-independentpotential,asthelarge- © Royal Astronomical Society • Provided by theNASA Astrophysics Data System hV. dR * àf R dv^dv z Mass distributioninthegalacticdisc-1573 (2) (3) 198 9MNRAS.239. .571K 1 v: 2 vK+1Rv 2=v fz(z,v)=f(E, (7) have toextractthepotentialïp{z)fromthese;thereare manywaysofdoingthis. be derivedfromstarcounts,andthe^-distribution asafunctionofz,f(v;z).Wesomehow much asageneralfunction oftwovariables.Thisbecomesclearifweplotthe contours offin for whichthisistrue;f{z, v) onlyhastwoone-variablefunctionaldegreesof freedom,notas require twoobservablespertainingtoasuitabletracer population:thedensityv(z),whichcan where E=ip{z)+\vanddip(z)/dz—~K.Inother words,theenergyEisonlyintegral f(^yRy^z,v)=fR ^(R,R,v^)xf(zv.InthiscasethecollisionlessBoltzmannequation(3) the (z,v)phasespace.Byequation (7),thesearecurvesofconstantE=ip{z)+¿v , fromwhich The mostdirectmethodof recoveringip(z)isprobablybyfindingthecurves of constantfin of motioninthisidealizedone-dimensionalstellarsystem. with assolution 3.1 DIRECTCONTOURINGOFf{z,V) z Boltzmann equationadistributionfunctionwhichseparatesintotwofactors, virtually independent.Thismeansthatitispossibletohaveasasolutionofthecollisionless VHz) canbederived.Note thatafunctionf{zv)ingeneralwillnotadmitsolution for'ip(z) vz separates aswell,givingusthez-terms the z-forceK(R,z)changeslittlewithduringanorbit,andR-z-motionsare have quiteloweccentricity,sothattheyconfinethemselvestoasmallrangeofR.Therefore, return toitstreatmentlateron(seeSection7below).Moststellarorbitsinthediscpopulations z z z For adiscgalaxy,thedominantgradientinJeans’equationisz-gradient,providedwe z z z9 stay reasonablyclosetotheplane.Therefore,wewillforpresentignoreoterm,and z z z z 3 Applicationtodiscgalaxies =Í^rÍ¿<‘ z z9 z to itas‘the’Jeans’equationinfuture. vK of astellartracerpopulation.Wewillmainlybeinterestedinthe^-equation(4),andrefer derive thembothfrommeasurementsofthemomentsvelocitydistributionsanddensity Rz where v{R,z)isthespacedensityofstars,ando(R,z)theirvelocitydispersiontensor(i.e. integrating overallvelocityspaceproducesJeans’equations: variables. Sinceequation(3)cannoteasilybesolvedingeneralwithrealdata,onesimplifiesthe analysis andproceedsbytakingvelocitymoments.Multiplyingthroughv 574 K.KuijkenandG.Gilmore °ij(ij))- Inthiswaywehaveseparatedthetwoforcecomponents,andcaninprinciple r zR So howcanweobtainKinthecaseofadistribution functionwhichobeysequation(7)?We z © Royal Astronomical Society • Provided by theNASA Astrophysics Data System dv 7 = 0 (5) (4) 198 9MNRAS.239. .571K 2 fr =E-it>{z),(8) have tosatisfy the (z,t^)-plane:inthesecoordinates,allcontoursmusthavesameshape,sincethey being curvesofconstantenergy(seeFig.1).Sincewearedealingwithrealdataratherthan completely well-knownfunctions,wehavetotryinsomewayfindthe/thatsatisfiesthese potential problems: constraints andismostconsistentwiththedata. bins neededtoobtainreasonablestatisticshavebelarge.Sincethequantityweare velocity distributionwhereitissteepest,i.e.nearv=(||);butthismostofthestars interested inisthederivativeofpotential,coarsebinsmakeithardtoobtainagood data points.Forthenumberofvelocityobservationsthatistypicallyavailable(c/.paperII), z measured, andmoreoverthefractionofstarswhichdohavevelocitiesmaydependon distance. Thisdistancebiashastobetakenintoaccount. are. Alotofweightendsupbeingputonthepoorlydefinedwingsdistribution. derivative. to takeaccountofthephysicalconstraintsonpotential,observationalerrorsand hundred starsisclearlynotfeasible.However,inSection3.4wewilldevelopthisideafurther which makesfulluseofthedataset,formsunderlyingideabehindourmethodanalysis. uneven samphngofthestarswithradialvelocitydata.Afittoentiredistributionfunction, z We canofcourseusethevelocitymomentsequation(6)instead,analogoustoJeans’ Figure 1.Contoursofconstant phase-spacedensityinthedistance-velocity{z,v)~and (z,t^)-planes.The equation (4).Thezeroeth,firstandsecondmomentsare(integratingovertherange0to°oso z shape ofthecontoursyields potentialrp{z),andtheirseparationyieldsthedistributionfunction, f(E). 3.2 MOMENTEQUATIONS z z If wetrytomeasuret/j{z)fromthecontoursoffforarealdataset,runintoseveral (i) Thedatahavetobebinnedinsomeway,aswearedealingwithonlyafinitenumberof (in) Notallthestarsfromwhichdensitylawsarecalculatedhavetheirradialvelocity (ii) Measurementerrorsinthevelocitiesaredifficulttodealwith.Theywillmainlyaffect Making rehabledeductionsfromdirectcontouringofadatasetcontainingonlyfew (iv) Theerroranalysisiscomplicated. z © Royal Astronomical Society • Provided by theNASA Astrophysics Data System f-contours in(z,v)plane 2 z (pc) z (pc) Mass distributioninthegalacticdisc-1575 * o N O o o o to 0 200400600800 1000 2 f-contours in(z,v)plane z 198 9MNRAS.239. .571K 3 2 /(z,0)^=iv| (9) that wecanuseoddvelocitymomentsofthedistributiontoo) 576 K.KuijkenandG.Gilmore the generalterminthisseriesis(for«>0) Z (v(Zj)independently,itmakesmoresensetouseaparameterizationofplausible isothermal components. the potential,ratherthansolvingforthemandpotentialself-consistently.Heusedthree v(z) = obtain asensiblepotential(ifthereisonethatconsistentwiththedata),evenifchoiceof potentials instead;thiswaywehavemanyfewerparameterstoplaywithandareguaranteed supply then,hecalculatedthefractionsofvariouscomponentsusingafirst-passguessat be gainedfromnotfittingacontinuumof‘isothermalcomponents’.Thatisthecoursewehave model thephasespacedensityoftracerintermsisothermalcomponents,thereislittleto depends ontheindividualoJs,though,sothismethodstillleavesalottobedesired. dispersions forthevariouscomponentsisnotquiteoptimal.Theresultingpotentialstill v(yj) =2 Instead offixingtheparameterslatter,andthenusingthesetomodeldensity In most^-studies,thedensityv(z)isknowntobetterprecisionthanvelocitydistribution. which isjustamomentoff: gradient, itisthereforepreferabletoworkintheotherdirection,andpredictvelocity taken, whichisdescribedinthefollowingsection. z{so tion modelscanthenbecomparedwiththeobservedvelocitydata. distribution ofatracerindifferentmodelpotentials,givenitsdensity.Thesevelocitydistribu- Zzxso Reparameterizing thez-heightintermsofpotential 'ip,wehave 3.4 CONSTRUCTING/(£JFROMv(y>) zz This equationisanAbel transform, whichhasthewell-knowninversion(see, e.g.Binney& Tremaine 1987): z Z Rather thanputnoconstraintsonthepotentialsthatcancomeoutofthissolution,andsolve If weneedtoparameterizethepotentialsinanycase,andrequireatleastsixparameters Given adistributionfunction/(£Jandpotentialt^(z),wecancalculatethedensityv{z), © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Z = 2 7t vix) J2[Ez-V(z)] f(z,v)d z WE-xp) fz(E z K. KuijkenandG.Gilmore — dv/dip fÁE z dE. 7 dip, dE,. (18) (20) (19) (21) 198 9MNRAS.239. .571K 7 lwl/2iM,£l f{z,v) into(E).Therefore,beforebeingabletomakeaninversionsuchasthatgivenin that thetracerisisothermal(/^e”^^)anexampleoflatter,leadingtoatrivial high z,arerequiredtomeasurethepotentialatlowz(andhencelocalvolumedensityof instead postulatingarangeofpotentialsyj(z),andforeachthemcalculatingf{z,v)from inversion toequation(13). equation (21),wehavetomakesomeAnsatzaboutthepotential,orf{E).Assuming potential exceedsE,i.e.beyondthepointz=V~Therefore,wecanderive technique. Itisimportanttonotethatf{E)dependsonthedensityonlyatpointswhere so thatthereisauniquerelationbetweenv(\p)andf(E).Becauseofthisequivalencev(tp) potential. quantities v(z),f{E)andyj{z)isknown,asoneneedsthepotentialtobeableconvert matter p). Therefore eitherthetailofvelocitydistributionatlowz,ordensityandpotential hand, becausehigh-energystarsarepresentatallheights,measurementsofthepotentialvery some sense‘half-derivatives’/ and f(E),thereisatriangularmathematicalrelationshipbetweenthethreefunctions'ip(z), close totheplanedorequireknowledgeofhigh-energytaildistributionfunction. of Katanyheightrelatesdirectlytothetotalsurfacedensityintegratedthatheight,thisis somewhat unstable,butnotasunstabletakingadirectderivativeofthedata;theyarein about theshapeofpotentialnearertoplane.Since,aswewillseebelow,ameasurement at largedistancesfromtheplanehigh-zdataalone,withouthavingtoworryindetail extremely useful,allowingustoobtainmeaningfulresultsfromhigh-zdataalone.Ontheother z v{z) andf{E):anyoneofthemcanbededucedfromtheothertwo.Abelinversionsare important tomakesurethatthemodelsaresufficientlygeneral.Onotherhand,constraints v(z). Wethencomparethesedistributionstothevelocitydata,andselectbest-fittingmodel force lawscanberuledoutonphysicalgrounds,andsoshouldnotincludedamongthe can bebuiltintothemodelpotentialswhichdirectderivationsfromdatahavemoredifficulty possible solutions(provided,ofcourse,thatacceptable fitstothedatacanbeobtainedfrom consistent withaI£-lawwhichdidnotshowsuchturnover.Thisillustratesthefactthatsome started todecreaseaboveafewhundredparsecs,whichleadsonededucenegative coping with.Forexample,theanalysisbyHill(1960)ofaKgiantsamplefound|{z)which z z z z z massive haloing. of thelocalcircularspeedbyitsgravity,wewould notexpectalsotofindevidencefora the potentials:ifweweretofindaveryheavydiscmass, forexample,whichcangeneratemost the restrictedsolutionset).Consistencywithgalactic rotationcurvecanalsobebuiltinto dynamical masses-aphysicalabsurdity.ThesamedatawereshownbyOort(1960)tobe z 0 z z *In thatanyFouriercomponent e*ofv(vOcorrespondstoone«wmf{E). z density pofgravitatingmatter followsfromPoisson’sequation: V'K=-AnGp. (22) Given ameasurementof the gravitationalfieldR(R,z)inanaxisymmetric galaxy, thetotal 4 FromKtodeterminationofthediscsurfacedensity z z z z The essentialfeatureofouranalysisisthatweavoidtheassumptionisothermality,by It isworthexaminingequation(21)inmoredetail,asitformsthebasisofouranalysis When startingfromasetof{z,v)dataforthetracerpopulation,onlyfirstthree This isnotadirectmeasurementofthepotential,butrathermodellingit,soit z © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Mass distributioninthegalacticdisc-1579 198 9MNRAS.239. .571K == 2 2 m]z lz Rlhu]lz 2 Az)= 580 In thecaseofadiscgalaxywecanexpress/^-gradientinV-RtermsOort constants ofgalacticrotationAand#(see,e.g.Mihalas&Binney1981): between heights-zandrelativetothediscplane0is Assuming that(A-B)varieslittleoverthez-rangeofinterest,totalcolumndensity2(z) As discgalaxiestypicallyhaveflatrotationcurvesfarfromthecentre,{A-B)~0.Thereis This relationshipcanbeusedinbothdirections,eithertodeducethewaysurfacedensity varies withz(thisrequiresveryhighaccuracydata),ortoconstrainthepotentialabovemostof thus astrongcorrespondencebetweenthesurfacedensitytoheightzandverticalforceK. we investigatethiscorrespondenceinmoredetail,andconsidertheaccuracyofneglecting the discmaterialtobephysicallyplausible.Itismostlywithlatterpossibilityinmindthat nz)=2pze-%l-e- °). We willexamineinturnthez-forceandsurfacedensityofmasscomponentswhichmakeup using equation(24)toderiveasurfacedensityfrommeasurementofK. p{R,z)=pe-*e-\ The firstmasscomponentwewillconsiderisadouble-exponentialdisc: realistic massmodelsoftheMilkyWay,inordertoevaluateerrorsthataremadewhen 4.1 THEACCURACYOFKTOSURFACEMASSDENSITYCONVERSIONS z-gradient of(A-B)nearthediscplane. The Poissonequationforsuchamassdistributionis solvedinAppendixAtermsofHankel transforms: therewefindthepotential(equationA10) z 4.1.1 Double-exponentialdiscs d0 The surfacedensitytoheight zis and thez-force(equationA12) z K(R)=- d z ■>p{R,z)= - ziZ + Royal Astronomical Society •Provided bythe NASA Astrophysics Data System AjcG [dzRdR AnG Ul -|z| ZJTCr 1 1 \dKd z | p(z)dz = AjiGpt AjzGp d K. KuijkenandG.Gilmore h R h 1 R + 2{A-B) 2 K\ (A—B' z 2,12 ’ -1-¿U|_T-Izl/zo 23l2 l dkJ{kR)(h~ +k)~ 0R dkJ(kR)(h +k)z sign(z) QR0 (RK l jzG 2 z-k 0 _/cz l _“z|/zo e 2 zñ ~k (23) (24) (25) (26) (27) (28) 198 9MNRAS.239. .571K 2 2 parameters. Whiletheseagreequitewellatlowz,asexpected,thereisclearlyasystematic, Fig. 2showsacomparisonbetweenK(z)and2jrG2(z),forplausiblesolarneighbourhood with theOortconstantsasbefore.Notehoweverthatthesenowreferonlyto roughly linear,deviationbetweenthetwo,whicharisesfromneglecteddKldR-iormof potential duetothedouble-exponentialmassdistributionabove,andnotthatofentire equation (23).(Ratherthanwriteoutthisterminfullwhatfollows,wecontinuetodescribeit Figure 2.Thesurfacedensity2(z)ofadoubleexponentialdisc(radialscalelengthh=4.5kpc,vertical curves areexplainedinthetext. scaleheight z=300pc)atradialdistanceR7.8kpc,andapproximationstoit,derivedfromK.The Galaxy. ToavoidconfusionwiththerealOortconstants,whendescribingpotentialduetoa which theyrefer.)Ontheassumptionthat(A-B^varieslittlewithz,wecanthencalculate subsystem oftheGalaxy,welabelAandBwithsufficesdenotingmasscomponentto for aninfinitelythinexponentialdiscofthesamesurfacedensityasdouble-exponential much betterfit.Forcomparison,thedottedlineshowswhathappensifwecalculateAandB as thenextapproximationto2jrG2.Thisisshowndash-dottedlineinFig.2,clearlya fractional errorA(2)/2atz=1and2kpcoverthe plausiblerangeofvalueshandRis disc usedhere.Thepotentialvariesmorerapidlyneartheplaneinthincase,and force duetoadouble-exponentialdiscby the radialvariationsinipwhichcausecorrection terms.Theerrorsareclearlyoftheorder plotted. Changingzdoesnotaffectthissignificantly, aslongitremainssmallcomparedto z surface density. consequently the{A-B^-itrmdoestoo,yieldingaslightlyworseapproximationto R of afewpercent.Wethereforeconcludethattosufficient accuracywecanapproximatethez- R QzR |/q-2(A2-B^k| In thecaseofsphericalcomponents, wedonotexpectsuchaclosecorrespondence between 2(z) andK{z),astheradial andthez-gradientsofpotentialarecomparable size. d simplified ifwecanexpressthedifference\K\—2jrG2(z) asalinearterminz. K takingtheoppositesigntoz.Aswillbeseen later,thecalculationsareconsiderably d 4.1.2 Sphericalmasscomponents R |^| =27rG2(z)+2(^-R^|z, (29) 0 d =0 z Z z =0 Rather thanshowaseriesofsimilarplotsfordifferentvaluesR,handz,inFig.3the © Royal Astronomical Society • Provided by theNASA Astrophysics Data System R() Mass distributioninthegalacticdisc-I581 K-E CONVERSION z 198 9MNRAS.239. .571K r 582 2 Ps~p() M{r) from theuseofequation(29).Theverticalscaleheightdiscis300pc. Figure 3.Fractionalerrorsindiscsurfacedensitymeasurementsatz=1kpc(left)and2(right),arising have amasscomponentofdensity Evaluating thecorrectiontermsisthereforeofmoreimportanceforsuchcomponents.Ifwe and massprofile The correctionterm2{A]-isgivenby [rk] 2(z)=2 and where /•isthesphericalradius■Jr+z,then ÍTr’ © Royal Astronomical Society • Provided by theNASA Astrophysics Data System A -^3 x „ _GM(r) 1 r \nrp{r)dr, p(jR+7)dz. Error indynamicalsurfacedensity K. KuijkenandG.Gilmore Z“0 GM(R) ^ -firKr) A -4jiGp{R). r3 1 dGM(r) r dr R [kpc] 0 r-R r-R (30) (31) (33) (32) (34) 198 9MNRAS.239. .571K 2 2 2 2 2 2 2 -2 1- Therefore, ifweapplythesamecorrectionasfordiscabove(seeequation29),error make is A(Z) =IK(z)\-[2jrG2(z)+2(A-B)o121] Here pistheaveragedensitywithinr=R.Asphericalcoronawithaflatrotationcurvehas which, expandinginzaboutthepoint{R,=0)gives (36) is-%(z/R).Inpractice,weexpectacoronatohavesomekindofcoreradius,which density p(r) ; 587 198 9MNRAS.239. .571K 2 -2 which determinesthecoronalmass.Sinceinourmodelswehaveextrainformationabout is thenfittedbyallowingthecoronaandspheroidparameters(scalelength,solarneighbour- numerical valueforthefiducialmodelderivedfromresultsofPaperII.Therotationcurve disc density,thecoronaisalreadywell-constrainedfromdatatoaradiusof1.6Æ. hood density)tovary.Asthespheroidscalelengthdoesnotaffectrotationcurve their probableerrors.Thediscpotentialiscompletelyspecifiedineachmodel,withthe 588 K.KuijkenandG.Gilmore by non-circularmotionsneartheGalacticCentre,perhapsassociatedwithasmallbar inside R=0.1,ishereincludedinthefits,spiteofsuggestionsthatitmaywellbecaused not fitted,andthespheroidfixedtobecorrespondinglylessmassive,didappreciably accordance withthebest-fitvalueofCOSI.Notethatinnerpeakinrotationcurve, significantly outsidetheinnerregionofrotationcurve,wehavefixedittobe80pc,in model isillustratedinFig.5.InordertoobtainanestimateoftheerrorF,furthermodels (Gerhard &Vietri1986).Othermodelfits,inwhichtheinnerpartofrotationcurvewas were calculatedinwhicheachinputparameterTable2wasvariedbyonestandard corona +spheroiddensityandduetothe(A-Z?)-terminPoisson’sequation(23).This change thepotentialnearSun. of 37Mpcand55. deviation, withoutalteringtheotherones.Finally,inordertoevaluatecorrelationbetween 0 K andFinthebest-fitpotential,calculationswererepeatedforlocaldiscsurfacedensities r, togetherwiththevalueofFandcontributions2-)dueto and Kwhichdescribethegravitationalpotentialinequation(39)(inourunits): F= 0.041—0.0094K±0.008,(43) 0 0 cp{AB The inputparametersforthemassmodelsaregiveninTable2,togetherwithanestimateof Results aregiveninTable3.Thetoplineshowsthefinalmodelparametersp,and From theseresults,wederivethefollowingrotationcurveconstraintonparametersF s 0c © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Table 3.Rotationcurvefitsforvariouschoicesofgalactic structure parameters. ThesymbolsaredefinedinthetextandTable1. 2i4R=205km/s 2AK=239km/s o 0 2 2 r S0=55X/pc S0=37X/pc /i#=5.5kpc /iR=3.5kpc R=8.5kpc Ä=7.1kpc 0 0 V=205km/s 0 0 V=235km/s o 0 Fiducial Model _1 -2 -1 Table 2.Theinputparametersforthemassmodel. 2AR 222±17kms R 7.8±0.7kpc S 46At0pc V 220±15kms hji 4.5±1.0kpc 0 0 0 0 Parameter -3 0.00119 1695 0.00120 2019 0.00157 1917 0.00121 2346 0.00096 2155 A^opc pc 0.00120 1865 0.00120 2025 0.00120 2013 0.00120 1790 0.00121 2299 0.00120 2544 PsfQ r c Description Fiducial diskdensity Exterior rotationcurve Local circularspeed Disk exponentialscalelength Galactocentric distance -32 AÍqpc (I0km/s)(l00pc)“ 0.0109 0.0329 0.0147 0.0432 0.0077 0.0241 0.0139 0.0418 0.0087 0.0262 0.0114 0.0342 0.0103 0.0311 0.0100 0.0305 0.0118 0.0354 0.0101 0.0306 0.0119 0.0356 Pc,Q FpF(A2_B2}Fiot -0.0006 0.0235 -0.0039 0.0290 -0.0081 0.0351 -0.0055 0.0363 -0.0055 0.0301 -0.0006 0.0256 -0.0037 0.0267 -0.0040 0.0314 -0.0025 0.0280 -0.0049 0.0293 -0.0019 0.0292 198 9MNRAS.239. .571K which translatesinto Figure 5.TherotationcurveoftheGalaxy,asmeasuredfromHi(R<0.45R),CO(0.45Z?).Thefittedmodelisdescribedinthetext. the planeofdisc,andhencethato=0.Thisconsiderablysimplifiesdynamical So farwehaveassumedthatEisanintegralofthemotionforourtracerpopulation.As required tomodelthem. dimensional system.Unfortunately,realgalaxiesaremorecomplex,andsoisthemathematics saw before,itthenfollowsthatoneoftheprincipalaxesvelocityellipsoidisalignedwith 7 Orientationofthevelocityellipsoid velocity ellipsoiddoesoncetheplane-parallelapproximation breaksdown.However,thereare particular densityinapotential.Consequently wedonotknowexactlywhatthe analysis, asitmeansthatwecanstudythez-motionsinisolation,iftheymadeupaone- ()() Eddington potentials)analyticsolutionsdoexist. In thesepotentials,theHamilton-Jacobi 0 equation (whichyieldstheequationsofmotion in generalizedcoordinates)separates a fewtheoreticalresultswhichareuseful.InStäckel potentials(sometimesalsoknownas particular Stäckelpotentialautomaticallyyieldsthe alignmentofthevelocityellipsoid.A confocal ellipsoidalcoordinates,sothatknowingthe coordinatesystemcorrespondingtoa tre. Non-separablepotentials donothavethisniceproperty:thevelocityellipsoids ofdifferent coordinates, sothatoneaxisofthevelocityellipsoid alwayspointstowardstheGalacticCen- special caseisasphericalgalaxy,inwhichthe equationsseparateinsphericalpolar Rz solutions tothecollisionless Boltzmannequationdonothavealignedaxes. z described byCarlberg& Innanen (1987),usingsoftwarewrittenbyR.G.Carlberg (private shows theprojectionsin (Æ,z)-planeofthreeorbitscomputedinthedisc-halo potential There isnogeneralmethodforconstructingdistributionfunctionswhichreproducea To illustratethesignificance ofthisnon-separabilityindeterminingtheo^-term, Fig.6 © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 88 2019C8S O19c3OOO12OM 2 ^coronp-P R«)||800po^.„=4500.1)02AR=222.km/s 'oorn.0=VP P,ph.0=0/Pdi.k.0=46.0M/pcV=220.km/s 0 Oao ROTATION CURVEFIT Mass distributioninthegalacticdisc-1 R [pc] 589 (44) 198 9MNRAS.239. .571K upper boundontheerror that canbemadewhenignoringit(cfFig.6above). is small.Theseareextreme cases,andsotheeffectofasphericaltilttermshould beagood will contentourselveshere withtwolimitingcases:auniformdisc(notilt)asdescribed before, function oftwovariables,but/three.(Thisisagood reasonwhylittleisgainedfromusinga rotation ofthetracerstars, butthentheanalysisbecomesmoremodel-dependent. Rather,we and asphericalpotential,which mightbeexpectedifthedisccontributionto rotationcurve on EorLcanbemade.Forexample,onemight specifytheToomreQ-parameter,or is known.Inanycase,evenif/wereknowntherewould stillbenouniquesolution,sincevisa three integralsofmotion:E,Landtheempirical‘third integral’/,forwhichnoanalyticform Stäckel potentialapproximationinanalysesofthistype.) SomeAnsatzaboutthedependence a similarinversionfromdensitypluspotentialto distributionfunction,butnowthereare continue thisphilosophyinthecaseofanaxisymmetric potential.Todothis,wehavetomake distribution functionratherthanjustthesecond moment. Itwouldbenicetoable Much emphasiswasplacedintheprevioussections ontheimportanceoflookingat 7.1 EVALUATIONOFTHETILTINGTERMINJEANS’EQUATION the Jeans’equation. may usenumericalorbitintegrationstoestimatethelikelyrangeofvaluesfora^-termin potential (inwhichcasetheywouldhavebeenalignedwiththedashedlinesinFig.6).Thusone these orbitstilttowardstheGalacticCentre,butnotasstronglytheywouldinaspherical in theplane,butdivergeslightlyatlargez.Consequentlydifferentorbitsfilloutboxeswhich communication) andJ.R.Lewis(privatecommunication).Theenvelopesoftheseorbitsagree separable distributionfunctioncangiveaverygoodapproximationtotheseorbits.Notethat are notpartofasinglecoordinategrid,andthispotentialisseparable.Nonetheless, z 3 z 3 590 K.KuijkenandG.Gilmore © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Figure 6.ThreeorbitsinaplausibleGalaxypotential.ThedashedlinespassthroughtheGalacticCentre. T [kpc] 198 9MNRAS.239. .571K 2 Rlhlt 2 2 2s the‘effective force’ The tiltingtermintheJeans’equationis(l/Rv)(d/dR)(Rvo);wethereforeneedR- We willmodelthistermaccordingtotheassumptionsmadeabouttiltingofvelocity gradients ofo(andhence),andv.ThenifthediscGalaxyisradially Thus weobtain balance implies(fordiscsurfacedensityfi)oandhence present purposes. moments onelooksat;weareimplicitlyassumingthatthereisnosignificantchangeinthe ellipsoid, andthenuseKintheinversionfromdensitytodistributionfunction.Thisisnot exponential, andhasaconstantverticalscaleheight,asisseeninexternaldiscgalaxies, obtained inthiswaywillhavethecorrectsecondmoment,shouldbequitereasonablefor shape ofthedistributionfunctionasaresulttilting.Since strictly accurate,asthetiltingtermtakesadifferentformdependingonwhichvelocity J__d_ is easytoshowthat S(R,z) = coordinates, andhasthevalue,incylindrical-polarcoordinatesofz=0,o:a:1.Thenit assume thattheaxisratioofstellarvelocityellipsoidremainsconstantinspherical-polar co betheforcethatwouldmeasuredbyanapplicationof‘no-tilt’Jeans’equation(10). RvdR Rz velocity ellipsoidofconstantaxisratioawhichpointsattheGalacticCentre.T(r,z)is ol^distributionsasdescribed above,wehavetocomparethem distribution functionisquiteGaussian.Themaximumlikelihoodfittingweusetoevaluatethe zef way bytheprobabilitydensity(asweareonlyinterested intherelativelikelihoodofmodels, (Edwards 1972):thissimplymeansthatweselectthe modelthathasthehighestprobability data inordertoselectthebestfittingone.Inthis work wehaveusedmaximumlikelihood 8 Comparisonofmodelswithobservations:maximumlikelihood function ofz,sincewehave notcompletelysampledavolumewithourspectroscopic survey. this causesnoproblemwithnormalizations).Themain advantagesofthemaximumlikelihood of reproducingthedataset.Forcontinuousdistributions theprobabilityisreplacedinusual in thatitdoesnotrelyonmeasuring justasingledispersionparameter,butrather onfittingthe entire probabilitydistribution function. approach arethatitdoes not requireanybinning(unlikexmethods),andis non-parametric, z eff L{m) =0Prob(datapoint/undermodelm) (53) z What weobserveisnotin fact thephasespacedistribution,butvelocitydistribution asa © Royal Astronomical Society • Provided by theNASA Astrophysics Data System data i K. KuijkenandG.Gilmore V s vKc dz.(52) z R=7.8kpc h=4.5kpca=2 0r 198 9MNRAS.239. .571K Err{v,v) />;)=/>Jz)=^V’ (54) limited sample.ThusthedistributionT(z,^)ofobservedheightsandvelocitiesina to weightbytheobservingvolume,sincethischangeswithdistancefromplaneforanarea- cone ofsolidangleQtowardsthegalacticpoleis Errors inthedatawillaffectdistributionofobservedstarsdistanceandvelocity.To It istothemodeldistributionsF(v\z)thatdata aretobecompared. velocity asafunctionofdistance fromthegalacticplane.Aseparatedistribution functionis The techniquedescribedin thispaperpredictsarangeofobservabledistribution functionsof distribution ofobservedpositionandvelocity.Notethatforthedistanceconvolutionwehave predicted foreachadopted setofnumericalvaluesparameterizingthepotential. Maximum The errordistributionsareapproximatelyGaussianinvelocityandlog(distance),so account forthese,weconvolvedthemodelswitherrordistributionstoobtain likelihood isthenusedto findwhichofthepredicteddistributionfunctions isthebest 8.1 THEEFFECTOFMEASUREMENTERRORS so thelog(likelihood)ofourvelocitydatapoints(zv¡)formodelfisgivenby The velocitydistributionatheightz,f(v;z),isjusttheconditionalofvgiven -^(^obs I^obs) description oftheobserved distributionfunctiondescribedinPaperH where Errandarethedistributionsofzt>abouttruevalues.Thedistribution vohs zZ 8.2 PRESENTATIONOFMODELFITTINGRESULTS Err¿z,z) of observedvelocitiesthestarswhichweobservetobeatheightzistherefore which wemaximizebyfittingappropriatemodels. log L=Xlogf(v¡;Zj),(55) •^(^obs? ^obs> obs obs /?z vz zvobs obs obs v © Royal Astronomical Society • Provided by theNASA Astrophysics Data System i 2 -^(^»obs?^obs) J dvE(z,v) \Q>zdzErr(z,)v{ ' ohsobs zohs 2 Qzdz dvf(,v)Err{v zobsv -^(-^obs? ^obs) 1 2 v(z) ^ —{v—Vote)/2of, 2 ^-(lnz-lnzobs)/2a? nz Mass distributioninthegalacticdisc-1593 (59) (57) (58) (56) 198 9MNRAS.239. .571K unhelpful, duetothecombinationofnoiseindataandcomplexityinterpreting the firstandsecondvelocitymomentsofmodelobserveddistributionfunctions. tion function.Hence,inPaperII,weshow,additiontosuchlikelihoodplots,acomparisonof model distributionfunctionwithcontoursoftheobservedisquite observations issomewhatproblematic.Visualcomparisonofacontourplotthebest-fit combinations ofnumericalparameters,whileuseful,aredifficulttorelatedirectlyadistribu- maximum likelihoodvalueforthevelocitydispersionisrmsvelocity.Foradistribution maximum likelihoodsolutionfinds.IfallthemodeldistributionfunctionsareGaussian, significance ofanydiscrepancies.Similarly,plotstherelativelikelihooddifferent velocity moment.Forillustrativepurposes,toprovideanestimateoftheresultscomparing more extendedthanaGaussian,thermsisthereforenotasgoodstatistictouselower but fitsthefulldistribution.Themomentsareplottedasavisualaidforassistanceof our datawiththemodels,wewillplotfirstandsecondvelocitymomentsinPapern.We reader. emphasize, however,thatthemodelfittingdoesnotusemomentsofdistributionfunction, dominated bythewings.Asobservedvelocitydistributionateachzdistanceisindeed {-\v\ ¡VqXthemaximumlikelihoodstatisticismeanmodulus\v,whichnotsostrongly with moreextendedwingsthisisnolongertrue-e.g.foranexponentialdistribution,f°cexp 594 Determination ofthedistributionmassingalacticdiscnearSunrequirestwo in PaperHIweconsiderthelocalvolumemassdensity neartheSun.Theessentialfeature mass densityofthegalacticdisc.InPaperII(Kuijken &Gilmore1989a)weapplythis further inPaperIIIofthisseries(Kuijken&Gilmore1989b). variety ofdeterminationsthelocalvolumemassdensityhavebeenmade,andarediscussed complementary andalmostdistinctexperiments.Thefirstoftheseistomeasurethe‘Oort 9 Summary technique toanewdatasetmeasuretheintegralsurface massdensityofthegalacticdisc,and the massofdisc,andpotentiallyconstrainsitsnature. Hencetheinterestintheseresults. The combinationofboththelocalvolumedensityandintegralcolumncourse through thegalacticdiscatsolargalactocentricdistance.Thisdeterminationrequiresdata determines (oratleastseverelyconstrains)thescaleheight ofanyunidentifiedcontributionto distribution dataforatracerpopulationwhichextendsabovemostofthetotalmassdisc. with theobserveddistributionfunction,soastodetermine thebestavailabledescriptionof uses maximumlikelihoodtechniquestocomparethe setofpredicteddistributionfunctions of thetechniquedescribedhereisthatonecalculates thedistributionfunctionofvelocities for distantstars,withthefundamentalrequirementbeingthatonehavevelocityandspace limit’, orvolumemassdensityintheplaneofGalaxynearSun.Thisexperiment z v integrals ofthemotionconstraining theorbitsofstarsinarealisticgalacticpotential. Thusone observed asafunctionofdistance,forrangeplausible gravitationalpotentials.Onethen requires thecombinationofhighprecisionlocaldatawithsomesupplementarydistantdata.A does notknowapriorithe orientationofthestellarvelocityellipsoidsfarfrom thegalactic ‘true’ galacticgravitational potential. Presentation ofthequalitymaximumlikelihoodmodelasadescription When consideringthesemoments,itisimportanttoconsiderthetypesofmodelswhich In thispaperwehavederivedanewtechniquefor thedeterminationoftotalsurface The secondexperimentisthedeterminationofintegralsurfacemassdensityinacolumn The analysisiscomphcated alittlebytheabsenceofananalyticdescription thefullsetof © Royal Astronomical Society • Provided by theNASA Astrophysics Data System K. KuijkenandG.Gilmore 198 9MNRAS.239. .571K plane. Thiscomplicationishandledherebyconsidering avarietyoflimitingcasesinthe modelling procedure. the constructionofdistributionfunctionare approximate, sinceEisnotanexact computations usedtocalculatethemodeldistribution functions.Theinitialcalculationofthe second momentsisexact,whilethedensityprofile determinedbyobservation.Laterstepsin integral ofthemotion. Bahcall, J.N.,1984a.Astrophys. Z.,276,156. Bahcall, J.N.,1984c.Astrophys. J.,287,926. Bahcall, J.N.,1984b.Astrophys. J.,276,169. Armandroff, T.E.,1989.Astr.J., 97,375. References z efi To summarizeourtechnique,wepresentinFig.8a flowchart,illustratingthesequenceof © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Figure 8.Asummaryofthetechniquedescribedinthispaper. Mass distributioninthegalacticdisc-1 595 198 9MNRAS.239. .571K aR 1 596 Binney, J.,1982.In:MorphologyandDynamicsofGalaxies,edsMartinet,L.&Mayor,M.,p.1,Geneva Burton, W.B.&Gordon,M.A.,1978.Astr.Astrophys.,63,7. Binney, J.&Tremaine,S.,1987.GalacticDynamics,PrincetonUniversityPress,Princeton. Clemens, D.R,1985.Astrophys.J.,295,422. Carney, B.W.,Latham,D.W.&Laird,J.B.,1989.Astr.J.,91,423. Caldwell, J.A.P.&Ostriker,R,1981.Astrophys.7.,251,61. Appendix A:Thepotentialduetoafînite-thickness radially exponentialdisc Gerhard, O.E.&Vietri,M.,1986.Mon.Not.R.astr.Soc.,223,311. Edwards, A.W.F.,1972.Likelihood,CambridgeUniversityPress. Casertano, S.,1983.Mo«.Aoí.R.asir.Soc.,203,735. Carlberg, R.G.&Innanen,K.A.,1987.Astr.J.,94,666. Gilmore, G.,Reid,I.N.&Hewett,P.C,1985.Mon.Not.R.astr.Soc.,213,257. Gunn, J.E.,1987.DarkMatterintheUniverse,IAUSymp.No.117,p.557,edsKormendy,&Knapp, Kuijken, K.&Gilmore,G.,1989b.Mo«.Afoi.R.asir.Soc.,239,651. Kuijken, K.&Gilmore,G.,1989a.Mon.Not.R.astr.Soc.,239,605. Gilmore, G.,1984.Mon.Not.R.astr.Soc.,207,223. In thisappendixwederivethepotentialduetoadisc ofdensity King, I.R.,1965.Asir.7,70,296. Kerr, F.J.&Lynden-Bell,D.,1986.Mon.Not.R.astr.Soc.,221,1023. Hill, E.R.,1960.Bull.astr.Inst.Neth.,15,1. Hénon, M.&Heiles,C,1964.Astr.J.,69,73. Hanson, R.B.,1987.Astr.J.,94,409. Gradshteyn, I.S.&Ryzhik,M.,1980.TableofIntegrals,SeriesandProducts,AcademicPress,London. Gilmore, G.&Wyse,R.F.G.,1987.In:TheGalaxy,p.247,edsCarswell,R.,Reidel,Dordrecht. Oort, J.H.,1960.Bull.astr.Inst.Neth.,15,45. Lewis, J.R.,1989.Mon.Not.R.astr.Soc.,submitted. p(R,z)=pe p(z) Ostriker, J.P.&Caldwell,A.R,1979.In:TheLargeScaleCharacteristicsoftheGalaxy,p.459,ed.Burton,W., Oort, J.H.,1965.In:GalacticStructure,p.455,edsBlaauw,A.&Schmidt,M.,UniversityofChicagoPress. Mihalas, D.&Binney,J.,1981.GalacticAstronomy,Freeman,SanFrancisco. Martinet, L.,1974.Asir.AsiropAys.,32,329. Lin, C.C.,Yuan,&Roberts,W.Jr.,1978.Astr.Astrophys.,69,181. Miyamoto, M.&Nagai,R.,1975.Pubisastr.Soc.Japan,27,533. for thespecialcase Zinn, R.,1985.Asirop/rys.293,424. Woolley R.v.d.&Stewart,J.M.,1961.Mon.Not.astr.Soc., 136,329. Woolley,R.v.d.R., 1957.M0«.Not.R.astr.Soc.,117,198. Whitmore, B.,McElroy,D.&Schweizer,F.,1987.Astrophys.J., 314,439. van derKruit,P.C.,1988.Astr.Astrophys.,192,117. Schmidt, M.,1985.TheMilkyWayGalaxy,IAUSymp.No.1Ó6,p.75,edsvanWoerden,H.,Allen,R.J.& Ostriker, J.P.&Caldwell,A.R,1983.In:Kinematics,DynamicsandStructureoftheMilkyWay,p.249,ed. Sandage, A.R.&Fouts,G.,1987.Asir./.,92,74. Murray, C.A.,1986.Mon.Not.R.astr.Soc.,223,649. yOjUHe-^. Schweizer, F.,Whitmore,B.&Rubin,V.C,1983.Astrophys.!.,288,909. Schechter, P.L.,1988.TalkpresentedatCITAworkshopTheMassoftheGalaxy, dz © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Observatory, Sauvemy. Reidel, Dordrecht. Burton, W.B.,Reidel,Dordrecht. G. R.,Reidel,Dordrecht. Shuter, W.,Reidel,Dordrecht. K. KuijkenandG.Gilmore (Al) (A2) 198 9MNRAS.239. .571K aR f(k) = For ageneralaxisymmetricdensityp(R,z\Poisson’sequationcanbesolvedintermsof Hankel transforms where JisaBesselfunction,toyieldthegravitationalpotential profile p(Æ,z)/p0)isindependentofradius,thiscanberearrangedto tp(R,z) =-2jiG {) I(k) = density factorizesintoaradialandverticalpartp{R,z)=p{R)p{z\i.e.ifthe (see, e.g.Casertano1983).p{kz)istheHankeltransformofp{R,z)inRvariable.If If theradialdependenceisexponential,p(R)=pe~,asgivenbyequation(Al),thenu- integral canbeevaluatedanalytically: This canagainbecalculatedanalyticallyifpisexponential, p(z)=e(anynormalization where equation(A6)isobtainedfromGradshteyn&Ryzhik(1980)(6.611). can beabsorbedintop):inthatcase ip(R,z) =-2jiG R Rz f I(k)=2 Rd z z d The apparentsingularityfor ß=kisnotreal,ascanreadilybeverifiedwithFHopital’s rule. z The Ç-integralofequation(A5)canberewrittenas © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Thus thepotentialdueto adouble-exponentialdisc,ofradialscalelength1/a andvertical -|z = 2e* Jo oo klzß xdxJ{kx)f(x), au 0 ße~-kc- uduJ(ku)pQ~ = 0d 2 ß-k Jo d £cosh(kÇ)p(+2kz) JO J-oo z dkJ{kR) 0 dkJ{ kR) 0 23/2 = pa(a+k), d uduJ(ku)p{u] kz çl QR da da dÇp{k,Ç)e ^. d_ 12 [p^+kr] Jo oo au duJ(ku)pe~ 0d Mass distributioninthegalacticdisc-1597 Ul kl; dÇe p{£). z d£p{£)e z -k\z-^\ (A3) (A4) (A6) (A7) (A5) (A8) (A9) 198 9MNRAS.239. .571K 1+ 71 ç 598 K.KuijkenandG.Gilmore respectively. Theseare scaleheight 1/ß,is The radialandverticalforces,K,aregivenbythenegativez-/^-derivatives, ip(R,z)= -AnGap, and K(R,z)= ~AnGap These integralshaveoscillatingintegrands,sosomecareneedstobetakenwhenevaluating proposed byvanderKruit(1988),thisintegral,whilenotanalytic,canneverthelessbe them numerically;ifthisisdoneproperly,though,theresultingconvergencequiterapid. K(R,z)= -4jiGap infinite integralisrequiredtoevaluatethepotential,whichverycostlycomputationally. peaked asanexponential,theIintegralwillchange.Ingeneral,thismeanthatadouble evaluated relativelyefficientlyasasummedseries. However, ifthedensityprofileistakentobeofform Only forlargeß(smallscaleheight)dowehavetogokbeforetheintegralsconverge. are suchthat(l+2m/^)ß=kforsomevalueofm, the denominatorwillbezero.Theseterms where themthbinomialcoefficientofpower-1 has beendenotedby((,f)-Ifk,andß p(z)=sechí(/3z/|) (Al3) corresponding situationin equation(A9).Theresultingtermisthen are stillregular,however;toevaluatethemwecan useTHopitaPsrule,aswasdoneforthe 4 =2íZ with thebinomialexpansionandintegrateresulting series.Thisyields,aftersomeworking, As alltheintegralsarewithininterval[0,1],we can expandthetermsinvolving(t+1^ ¿ =e'*,weobtain zR rd zd z 2¿ í k If weinsertthez-density(A13)intoequation(A8),setrj=ß/k^andmakesubstitution If wehaveadifferentverticaldensityprofile,forexampleonewhichisnotasstrongly © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 2V m I j(l +fc|z|)e m =0 J-*ui e 1 + dt t+2^cosh(fe) 2 t|;(1 +2,,/l)ul (l +2ml%)ß-k /?(! +2m/g)e~-fc~'^ e 2 o *00 0 2 dkJ(kR)(a +k) 0 2 dkJ{kR){a + 0 kdkJ{kR){a t 23/ k)~ kßsign(z) }2 3/2^-^ 2 3/2 ß—k ßc 2 ß- k dt 1 t (t'+fŸ' -k\z_ß cQ z ß-k< (A10) (All) (A12) (Al 5) (A14) (A16) 198 9MNRAS.239. .571K The cáse¿=0correspondstotheexponentialdensityofequation(A2),andsoshouldgiveus the sameresultasequation(A8).Thisisindeedcase:for£=0onlyfirstbinomial Appendix B:Regularityoforbitsatthesolargalactocentricradius The resultsdescribedinthispaperhavebeenobtainedassumingthatwearedealingwith coefficient isnon-zero,andthistermidenticaltoequation(A9). More realistically,thedistributionfunctionwillquicklybecomemoreandlumpyif now discusstheseassumptionsinsomemoredetail. function) and(ii)haveanintegralofmotionakintotheenergyverticaloscillations.We in whichwedonottakethelimittoarbitrarilysmallvolumes;butthenthereisnoguarantee which ergodicallyfillafour-dimensionalregionofphasespace(i.e.theypassarbitrarilyclose examined athighresolution.Inmoreextremecases,somepotentialsallow‘chaotic’orbits, ject toadditionalconstraintsduetheneedsatisfymorethanonesetofintegrals.(We Boltzmann equationasaprobabilitydensity(cf.Bwnsy&Tremaine1987,section4.1). that thisfunctionwillsatisfythecollisionlessBoltzmannequation(Binney1982).Thisproblem orbit). Insomecases,itisstillpossibletodefinean‘almostfine-grained’distributionfunction, never reaches(whichwouldleadustodeduceaphasespacedensityofzeroforpointsonthe through whichsuchanorbitpassescontainsacontinuoussubvolumeinsidethe to eachpointinthisregion).Suchorbitsmaypossess‘strangeattractorsYforwhichitis function asawell-behavedoftheisolatingintegralsmotion,sothatthereareno is resolvedbytheinterpretationofdistributionfunctionwhichsatisfiescollisionless and smallersubvolumesconverges-instead,anyvolumeelement,nomatterhowsmall, orbits which(i)obeythecollisionlessBoltzmannequation(i.e.haveafine-graineddistribution impossible tofindsmallvolumesinsidewhichthephasespacedensityaveragedoversmaller integrals. Inthiscasetherewilltypicallybemorethanonesetofapproximateisolating probably alsoacceptabletouseadistributionfunctionwhichisoftheclassical difficulties intheanalysis.Ifthereismildstochasticity,i.e.somestarsareonchaoticorbits,it thank JamesBinneyforemphasizingthispoint,andallayingourapprehensionsaboutthe like thatofourGalaxy.Mostthediscstarsmove alongroughlycircularorbits,withonly the orbitsofrelevanceinsolarneighbourhoodthennoanalysisinvolvingsolutions possible effectsofchaoticorbits.)Nevertheless,ifthereismorethanmildstochasticityamong collisionless Boltzmannequationismeaningful. integrals, sothatonecandescribeameaningfuldistributionfunctionprobabilitydensity,sub- small excursionsaboutanexactcircle.Forsuch orbits wecanlinearizetheequationsof Innanen (1987),amassconcentration neartheGalacticCentrewillcausesuch plungingorbits higher oroflowerangular momentumthanthatofacircularorbitthesame energymake integral ofthemotion,whichapproachesenergy ofthez-motionsinlimitsmall larger excursionsinthe r-direction. Iftheazimuthalvelocityofthesestars isquitesmall oscillations. Theradialextentoftheboxdependson theangularmomentum:starsonorbitsof box orbitinthecomovingmeridionalplane[the(R, z)-plane]. Inadditiontotheenergyand orbit, andgiverisetoindependentharmonicoscillations inther-andz-directions.Thisisa motion: thegravitationalaccelerationsarethenproportional tothedeviationsfromcircular Galaxy. Aswasshownby Martinet(1974),andmostrecentlyinvestigated byCarlberg& compared tothelocalcircular velocity,theywillplungedeepintotheinner regionsofthe component oftheangularmomentumaboutaxis ofsymmetry,theseorbitsrespectathird A physicalfine-graineddistributionfunctioncanbedefinedonlyinextremesituations. If allorbitsareregular,thestrongJeans’theoremallowsonetoconsiderdistribution It isthereforecrucialtoknowhowimportantchaotic orbitsareinaxisymmetricpotentials, © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Mass distributioninthegalacticdisc-1599 198 9MNRAS.239. .571K 21/ 1 1 600 K.KuijkenandG.Gilmore ones. Ifastarmovesonanorbitwiththreeintegrals,theoccupiesthree-dimensional its orbitdeterminethetotalandazimuthalspeeds,onlyfreedomthatisleftdirection box orbits,forexample).Thisgivessuchorbitsaveryregularappearance,asthedirectionof manifold inphasespace.Specifyingthepositioncoordinatespaceusesupthesethree this isnolongertrue,becausethenthevelocityvectorhasonerealdegreeoffreedomleft. motion ateverypointisalwaysoneofonlyafinitesetpossibilities.Ifnothirdintegralexists fill atwo-dimensionalregiononthesesurfaces. Binney 1982)ofaregularorbitareclosedcurves;chaoticorbits,ontheotherhand,ergodically large z-excursioninanorbitwithlowz-velocityatz=0.Finally,thesurfacesofsection(see,e.g. range ofvalues.Chaoticorbitscanthereforebeidentifiedbythemuchlessregularappearance degrees offreedom,andhencefixesthevelocityuptofinitedegeneracy(fourfoldincase to ‘lose’theirthirdintegrals.Ineffect,whathappensisthatthecorrelation(expressedby of theirmotioninthemeridionalplane.Aswesawabove,anothermanifestationchaosisa Since thespatialcoordinatesofastartogetherwithenergyandangularmomentum the onepresentedinthispaperisflawed,astheserelyonexistenceofathirdintegral:it radial kineticenergyintothez-directioninaquasi-randomfashion,andhenceleadstogreatz- third integral)betweentheverticalspeedatapogalacticonandmaximumheightwhich precisely therelationbetweenmaximumheightreachedbyastaranditsverticalspeedat excursions fororbitswhichinitiallyhadsmallverticalspeeds.Ifasignificantnumberofthe large extentwhetheraparticularorbithasthirdintegralornot.Toseehowimportantsuch orbits areinthepresentstudy,wethereforeneedtoinvestigateangularmomentum of thevelocityvectorinmeridionalplane,whichmustthereforebeallowedaninfinite stars foundneartheSunareonsuchchaoticorbits,thenentirebasisof^-studiesas Table Bl.Themodelcontainsanucleusandmore extendedbulgeinthecentreaswella the KlawatRdeducedfromourK-dwarfsample.Theresultingparametersareshownin Miyamoto &Nagai(1975)‘disc-spheroids’,wasadaptedtofittherotationcurvedescribedin potential usedbyCarlberg&Innanen(1987),whichcomprisesasuperpositionofmodified centric radius.Iftheorbitsareconstrainedtoremainneargalacticplane,thenaKanaly- distribution ofthestarswhicharefoundwithinafewkpcdiscplaneatsolargalacto- the meridionalplaneofaseriesstarswhichwerelaunched fromthesolarcirclewithvertical in thispotential,usingsoftwareduetoR.G.Carlberg (privatecommunication)andJ.R.Lewis outer potentialareinanycaseofnoconsequencefor thisinvestigation.Orbitswerecalculated disc, andhasarotationcurvethatisapproximatelyflat toaradiusof15kpc.Thedetailsthe Section 6,aswellaradialexponentialscalelengthforthedisc[givenbyc.Ka+6)]and z =0thattellsusthestrengthofgravitationalfieldneardiscplane. star attainsduringitsorbitislostbytheencounterwithgalacticnucleus.Thisscatters the exactmodelparametersofpotential,except that ifweremovethenucleusatcentre momentum, v^>100kms"atthesolarcircle,are regular boxorbits-thisisnotsensitiveto (private communication),forstarswhichcrossthesolar position.Fig.Blshowstheorbitsin sis ispossible. regular andremainconfined tothediscplane. orbits withzeroangular momentum, whichpassrightthroughtheaxisof the Galaxy,are of theGalaxy(andhence ignorethepeakofrotationcurveinsider=1 kpc),theneven speeds of40kms“,andvariousazimuthalspeeds. Asexpected,starswithhighangular such astheorbitwithinitial azimuthalvelocity70kms~^Theseorbitsariseas librationsabout z0 z There aremanydifferentwaysinwhichchaoticorbitscanbedistinguishedfromregular As wehaveseen,thedistanceofclosestapproachtocentreGalaxydeterminesa In ordertoseehowlowtheangularmomentumofanorbithasbeforitchaotic, Apart fromthenormalbox orbits,wealsofindsometubeorbitsinthemeridional plane, © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 198 9MNRAS.239. .571K 1 penultimate oneisaresonantbox orbitandthefinalisashellorbit. s .Thefirstorbitischaotic,the secondnearlyresonant,thethirdatubeorbit,following three areboxesthe the solarpositionwithvelocities (0,^,40).Fromtoptobottom,10,40,70,100,130, 160,190,220km Figure Bl.Meridionalsections oforbitsinthegalacticpotentialdescribedtext.Stars werelaunchedat © Royal Astronomical Society • Provided by theNASA Astrophysics Data System galactic potential(fordefinitionsofthesymbols see theirpaper). to Carlberg&Innanen’s(1987)fitthe Table Bl.Theparametersofourmodification b (kpc) a (kpc) /¿ (kpc) h (kpc) Ä hi (kpc) A ßi Mass {Mq Parameter 3 2 1 1.45 10 Mass distributioninthegalacticdisc-1 0.125 0.090 0.325 disk- halo 5.5 2.4 0.1 0.5 0.4 910 nucleus bulge 9.3 101.0 0.25 1.5 601 198 9MNRAS.239. .571K -1 1 1 -1 1 1 1 1 1 -1 1 602 K.KuijkenandG.Gilmore more complicatedthanthoseoftheboxorbits-neverthelessstarsontuberemain a resonant,closedorbitinthemeridionalplane.Theyareregular,buttheirthirdintegrals the cutoffofangularmomentumregularorbitstobeatanazimuthalvelocitysolar tube orbits,thenthereareusuallyalsosome,andsometimesindeedmany,chaoticorbitsofthe circle oflOOkms,beyondwhichtherearenomoretubeorbits.Theorbitwithinitial same energyandangularmomentum(Hénon&Heiles1964).Hencethefactthatweimpose confined totheplane,andsocanbeusedforaKstudy.However,ifpotentialallowssome point markedwithanasteriskonthetubeorbitinFig.B1)meridionalvelocitiesallowedto function ismostlikelysmooth,atanypointincoordinatespaceonewouldexpecttoseestars would bedominatedbystarswithsuchastrongorbitaltilt.Sincethestellardistribution However, itseemsunlikelythatanobservedsampleoftracerstarsatsomepointintheGalaxy which crossthemselvesmoreorlesssymmetricallywithrespecttothe(Æ,z)-coordinates. Thus, aslongthestarswhichcompriseoursampleofKdwarfs(PaperII)haveasymmetric on boxorbits,andtubeorbitswithlocaltiltsofoppositesign.Themeantiltshouldbemuch regular forevenlowerangularmomentum,sincetheydonotapproachthenucleussoclosely. stars onsuchanorbitareallmoreorlessinthesamedirection,unlikeboxorbits, drift ofc.10kms"(Mihalas&Binney1981).Theirradialexcursionsarethereforeafewkpc disc dominate.Thesestarsarewell-knowntohavehighangularmomenta,withanasymmetric drifts ofatmost120kms~orso,theirorbitswillbealmostentirelyregular. smaller thanthemostextremeallowedpossibihty. thus allhaveawell-behavedthirdintegral. instead allorbitsnearitarenormal,non-resonantboxorbits. by Armandroff(1989)showsthatreviseddistances tosomeoftheseclustersyielda kinematically hotterthantheolddisc,hadangularmomentaroughlyhalfwaybetween at most,andtheorbitsareallboxesinmeridionalplane.Theclassicaldiscpopulationstars azimuthal velocityv^=190kms,whichisalsoresonant,doesnotsireanytubeorbits, clusters’ (Zinn1985),whichshowaspatialandmetallicity distributionsimilartothethickdisc over theasymmetricdriftofthispopulation,atonly around30kms'.The‘discglobular stellar mass.Earlyinvestigationsofthispopulationsuggestedthatthesestars,whichare direction ofgalacticrotation(Gilmore&Wyse1987) alsoyieldsahighcircularspeedforthese stars, werethoughttohavealargerasymmetricdrift, of68±29kms'-however,recentwork for thesestars).However,therecentstudiesofkinematicsnearbyhigh-propermotion might thusbeprimecandidatesforbeingonchaoticorbits,asunlikethestarsofextreme galactic poleyieldsashear intherotationspeedof36±5kms'kpc' . Thusallthe sample results.AradialvelocitysurveyofGdwarfs insituafewkpcabovetheplane smaller valueof27kms'fortheasymmetricdrift, in agreementwiththelocalpropermotion intermediate PopulationIIstarshaverelativelysmallverticalcrossingspeeds{o~40kms Population II,whichhavelargeverticalspeedskeepthemfromtheGalacticCentre, z these starswillindeedhave sufficientangularmomentumtokeepthemawayfrom troublenear stars. Finally,apropermotionsurvey(Murray1986) ofstarsupto-1kpctowardsthesouth ‘classical’ discpopulationsandthoseoftheextremePopulationII(Gilmore1984).Suchstars observations seemtoindicate thattheasymmetricdriftofthickdiscstarsis c.30kms'. stars bySandage&Fouts(1987)andCarney,Latham &Laird(1989)areallinagreement the GalacticCentre.Thus itappearsthatthethickdiscstars,likethoseof thindisc,move zz 1 The velocityellipsoidsoftubeorbitscanbestronglytilted.Insomeregions(e.g.nearthe At lowheights,within~1kpcofthediscplane,stellarpopulationsyoungandold At heightsofac.1-5kpc,theintermediatePopulationII(or‘thickdisc’)starsdominate Orbits withhigherverticalcrossingspeedsintheplanethanthoseshownFig.B1remain © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Since theazimuthalvelocity dispersionofthethickdiscstarsisc.40km s',mostof 198 9MNRAS.239. .571K 1 1 present analysis. predominantly alongregularorbits.Wearethereforejustifiedinusingthesestarsforthe momenta onaverage,andafractionofthemwillcomeveryclosetothegalacticnucleusduring have velocitydispersionsofonlyc.45kms~orless.ThusextremePopulationIIstarsarenot their orbits,andbescattered.AsshownbyCarlberg&Innanen(1987),someofthesestarsdo distant starsshouldreflectthat.AswewillseeinPaperIIhowever,evenourmost indeed appeartomoveonchaoticorbits,inthatstarsofverylowangularmomentumare present inoursamplesignificantnumbers-thereforetheorbitsweare if suchstarsarepresentingreatnumbersoursample,thevelocitydispersionsformost statistically significantlydepletedinthesolarneighbourhood,asistobeexpectedifthesestars predominantly regular,validatingthefundamentalassumptionsunderlyingouranalysis. asymmetric drift(t^<100kms~)isaround80kms"inthesolarneighbourhood,andhence, are scatteredtogreaterheights.However,theverticalvelocitydispersionofstarswithlarge The sameisnottrueofthestarsextremepopulationII:theyhaveverylowangular © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Mass distributioninthegalacticdisc-1603 198 9MNRAS.239. .571K © Royal Astronomical Society Provided bythe NASA Astrophysics Data System