Gas motions in the plane of the spiral galaxy NGC 3631

A(MNLTEXstylefilev1.4)

GasmotionsintheplaneofthespiralgalaxyNGC3631

A.M.Fridman1,2,O.V.Khoruzhii1,3,E.V.Polyachenko1,A.V.Zasov2,

O.K.Sil’chenko2,A.V.Moiseev2,A.N.Burlak1,2,V.L.Afanasiev4,S.N.Dodonov4,J.H.Knapen5,6

ofAstronomyoftheRussianAcademyofScience,48,PyatnitskayaSt.,Moscow,109017,RussiaAstronomicalInstitute,MoscowStateUniversity,Universityprospect,13,Moscow,1199,Russia

3NationalResearchCenter”TroitskInstituteforInnovationandThermonuclearResearches”,Troitsk,Moscowreg.,142092,Russia

4SpecialAstrophysicalObservatoryoftheRussianAcademyofSciences,Zelenchukskaya,377140,Russia

5IsaacNewtonGroupofTelescopes,Apartado321,SantaCruzdeLaPalma,E-38700Spain6DepartmentofPhysicalSciences,UniversityofHertfordshire,Hatfield,HertsAL109AB

2Sternberg1Institute

arXiv:astro-ph/0012116v1 5 Dec 2000ABSTRACT

Thevelocityfieldofthenearlyface-ongalaxyNGC3631,derivedfromobservationsintheHαlineandHiradioline,isanalysedtostudyperturbationsrelatedtothespiralstructureofthegalaxy.Weconfirmourpreviousconclusionthattheline-of-sightvelocityfieldgivesevidenceofthewavenatureoftheobservedtwo-armedspiralstructure.Fourieranalysisoftheobservedvelocityfieldisusedtodeterminethelocationofcorotationofthespiralstructureofthisgalaxy,andtheradiusofcorotationRcisfoundtobeabout42′′,or3.2kpc.Thevectorvelocityfieldofthegasintheplaneofthediscisrestored,andtakingintoaccountthatwepreviouslyinvestigatedverticalmotions,wenowhaveafull3Dgaseousvelocityfieldofthegalaxy.Weshowclearevidenceoftheexistenceoftwoanticyclonicandfourcyclonicvorticesnearcorotationinaframeofreferencerotatingwiththespiralpattern.Thecentresoftheanticyclonesliebetweentheobservedspiralarms.Thecycloneslieclosetotheobservedspirals,buttheircentresareshiftedfromthemaximainbrightness.

Keywords:galaxies:individual:NGC3631–galaxies:ISM–galaxies:kinematicsanddynamics–galaxies:spiral–galaxies:structure–Hαline,21cmline:galaxies.

1INTRODUCTION

Thebranchofastronomyknownasdynamicsofgalacticdiscshasacquired,throughlongyearsofdevelopment,anunquestionablyclassicalstatus,buthassofarevolvedasapartoftheoreticalastrophysics(e.g.,Fridman&Poly-achenko1984;Binney&Tremaine1987).Observationaldata,whicharethebasisfordynamicalinvestigations,havebeenuptonowmostlyone-dimensional:amassdistribu-tioninadiscisusuallyreconstructedfromasurfacebright-nessprofileandfromalong-slitmajor-axisvelocityprofileorrotationcurve.Suchanapproachleadsdynamiciststosupposeastrictaxisymmetryofgalacticdiscs.However,ev-identlythelatterarenotaxisymmetricingeneral.Particu-larly,barsandspiralarmsareaclearmanifestationofsuchnon-axisymmetry.Thus,inordertomakedynamicalanaly-sesmorereliable,oneneedstwo-dimensional(2D)data.WiththeadventofCCDdetectors,2Dphotometricstudiesbegantoappear.Inparticular,Kent(1984,1985)hasundertaken2DdecompositionofCCDimagesofgalax-cRAS󰀁

iesandhasdeterminedalotofexponentialdiscparameters.

InaseriesofworksAthanassoulaandco-workers(Consid`ere&Athanassoula1988;Garc´ıaG´omez&Athanassoula1993)haveusedanazimuthalFourieranalysisofimagesofgalac-ticdiscstorevealpropertiesoftheirspiralstructure,suchasthenumberofarmsandtheirpitchangle.Two-dimensionalvelocityfields,however,arerarelyincludedintostate-of-artdynamicalinvestigations.Themaximumyieldobtainedfromsuchdataisusuallyarotationcurvecalculatedinzero-orderapproximationofcircularrotation,i.e.againundertheassumptionofaxisymmetry.However,2Dvelocityfieldscontainmuchmoreinformation.

Theobservedline-of-sightvelocityofgasinspiralgalax-iescontainsacontributionnotonlyoftheregularrotation,butalsoofthevelocityperturbationsduetothespiralden-sitywaves.Inprinciple,ananalysisofthevelocityfielden-ablesonetoseparateallthesecomponentsofgasmotion,butthistaskisfarfromsimple:theexpectedamplitudeofthemainharmonicsrelatedtowavemotionisaboutoneorderofmagnitudelowerthanthemaximalvelocityofrotationof

2A.M.Fridmanetal.

agalaxy.Inaddition,thepresenceofbothnonplanaroscilla-tionsofthegasalongtherotationaxisandlocalnon-circular

motionsmakestheobservedvelocityfieldverycomplicatedanddifficulttointerpret.Althoughspiral-relatedperturba-tionsofthegasmotionweredetectedbothinourGalaxy(Yuan1969andreferencestherein)andinmanyexternalgalaxiesbeginningwiththeclassicalworkofRotsonM81(1975),theamplitudeofperturbedvelocitiesandpatternangularvelocityarebadlyknownevenforthebestobservedgalaxies.

Differentmethodswereproposedtodeterminekinemat-icalparametersofdensitywavesfromtheobservedline-of-sightvelocityfields(seeSakhibov&Smirnov1987,19,1990;Bonnareletal.1988;Canzian,1993;Sempereetal.1995;Schoenmakersetal.1997;Westpfahl1998andrefer-encestherein).However,allthesemethodshaveoneortwoprincipalshortcomings.First,theyarebaseduponanal-legedpossibilitytorestoretheequilibriumrotationvelocitywithouttheanalysisoftheresidualvelocities.Suchapossi-bilityexistsifwedealwiththeresultsofamodelexperimentandknowtheformofthegravitationalpotential(exactlythecasewithCanzian’s1993investigation).However,whenweanalysetheline-of-sightvelocityfieldofarealgalaxy,independentrestorationoftherotationcurvebecomesim-possible(Lyakhovichetal.1997;Fridmanetal.1997).Arotationcurvedeterminedintheframeofamodelofpurecircularmotionhassystematicerrorsoftheorderoftheresidualvelocities,andthustheresidualfieldbuiltonthebasisofthiscurvedoesnotrepresenttherealfieldofveloc-ityperturbationscausedbythedensitywave(Lyakhovichetal.1997;Fridmanetal.1997).Second,allapproachesmen-tionedabovearebasedontheassumptionofa2Dcharacterofthegalacticmotioninadisc,whereasanyrealgalacticdiscisa3Dobjectandregularmotionsinducedbyaden-sitywavearealsothreedimensionalinprinciple(Fridmanetal.1997;Fridmanetal.,1999).Thustheonlydirectap-proachtoanalysetheobservedvelocityfieldistoseekforself–consistentsolutionsforthefullvectorvelocityfield.Inotherwords,therotationvelocityandallthreecomponentsoftheperturbedvelocityshouldbedeterminedsimultane-ouslyfromtheanalysisofobservationaldata,takingintoaccountthe3Dnatureofthegalacticdiscs.

Arecentattempttorestorethecomplete(threecom-ponent)vectorvelocityfieldinthegaseousdiscsofgrand-designgalaxiesfromtheobservedfieldofline-of-sightve-locities(Lyakhovichetal.1997;Fridmanetal.1997),gaveusahopetobuild,inthefuture,anobservationally-basedfoundationofthedynamicsofthegalacticdiscs.Theknowl-edgeofthecompletevelocityfieldgivesusatonce(1)rota-tioncurve,(2)allthebasicresonances:Lindbladandcoro-tational,and(3)knowledgeoftheresidualvelocityfield,containingrecentlydiscoveredstructuressuchasgiantanti-cyclones(Fridmanetal.1997)andcyclones(Fridmanetal.1999),the“constituentparts”ofthespiraldensitywaves.Finally,knowledgeofthecompletevelocityfieldhelpstodeterminethecollectiveprocess—akindofinstability—whichisresponsibleforthespiral–vortexstructureofagivendisc.Thus,wecanstatewithoutriskofexaggerationthattheobservedvelocityfieldprovidesthenecessaryobserva-tionalbasefortheconstructionofthedynamicalportraitofagalaxyunderconsideration.

Theaimofthepresentarticleistorestoreandanal-

ysethevelocityfieldofthegaseousdiscofthegrand–designgalaxyNGC3631,forwhichtwotypesofline-of-sightveloc-itydatawereobtained,wellcomplementingeachother:intheradioHiandopticalHαlines(seebelow).TheHiobser-vationsusedforthisstudywereobtainedbyKnapen(1997)withtheWesterborkSynthesisRadioTelescope,andtheHαobservationswerecarriedoutattheSpecialAstrophysicalObservatory(SAO)withits6-mreflectorequippedwithanF/2.4focalreducerandascanningFabry–Perotinterferom-eter.

NGC3631isaratherbrightnon-barredgalaxywithwell-definedspiralstructure.Itsopticalaxialratioisclosetounity:accordingtotheRC3catalogue(Vaucouleursetal.1991),loga/b=0.02±0.07,sothisgalaxylooksnearlyface-on.Suchanorientationisveryfavourableforstudyinggasmotionsperpendiculartotheplaneofthegalaxy,whichwasthemaintopicofourpreviouspaper(Fridmanetal.,1998,referredhereafterasPaperI).

InPaperIweshowedthatnon-circulargasmotionsinNGC3631havearegularcharacter,andthattheyarerelatedtotheobservedtwo-armedspiralstructure,whichhasawavenature.Thevertical(thatisperpendiculartotheplaneofthedisc)componentofthegasmotionsasrevealedbyaFourieranalysismethod(Fridmanetal.1997),wasalsofoundtobeinducedbythespiraldensitywave.TheinclinationangleofthediscofNGC3631wasfoundtobeabout170,whichenables,usingthesameobservationaldata,therestorationofthevectorvelocityfieldintheplaneofthisgalaxy,whichisthemainobjectiveofthepresentpaper.

NGC3631isagrand-designspiralgalaxyoftypeSAc,atadistanceof15.4Mpc,asestimatedfromitsrecessionvelocityusingaHubbleconstantof75kms−1Mpc−1,whichgivesanangularscaleof75pcperarcsec.Interestingly,thegalaxyhasbeenincludedinArp’s(1966)atlasofpeculiargalaxies,thankstoits“straightarms”,and“absorptiontubecrossingfrominsidetooutsideofsouthernarm”.Thesefea-turescanberecognisedintheR-imageshowninFig.1,butthegalaxyasawholelookstousrathernormal.TheatomichydrogendistributionhasbeendescribedbyKnapen(1997andreferencesthereintoearlierwork),andtheionizedhy-drogenhasbeenstudiedthroughemissionintheHαlineby,amongothers,Boeshaar&Hodge(1977),Hodge(1982)andRozas,Beckman&Knapen(1996).

InSection2theresultsofaFourieranalysisoftheobserveddistributionsofoptical(HαandR-band)surfacebrightnessandofHisurfacedensityaregiven,andcomparedwithaFourieranalysisofazimuthaldistributionsoftheob-servedline-of-sightvelocities.Section3presentsamodeloflarge-scalegasmotion,inwhichweassumethatthegasro-tatesinthegalacticplane,andsimultaneouslyparticipatesintheperturbed3D–motionscausedbythetwo-armedden-sitywave.Thelatterfactallowsustorestricttheexpansionoftheline-of-sightvelocityfieldtothefirstthreeFourierharmonics(mobs=1−3)(Fridmanetal.1997).Twoin-dependentmethodsareusedtodeterminethepositionsofcorotationandotherresonances,basedontherelationshipsbetweenthephasesofazimuthalFϕandradialFroscil-lationsoftheperturbedvelocityandthephaseFσoftheperturbedsurfacedensity.InSection4wethenproceedtorestorethevectorvelocityfieldofthegasintheplaneofthedisc.Weshowthatinaframeofreferencecorotatingwiththespiraldensitywavepattern,giantcyclonesexistalong-󰀁

cRAS,MNRAS000,sideanticyclonesnearthecorotationradius.Anticyclonesweredescribedinourearlierwork(Fridmanetal.1997),whereascycloneswerepredicted(Fridmanetal.2000)inthosegaseousgalacticdiscsinwhichthegradientoftheaz-imuthalresidualvelocityexceedsthegradientoftherotationvelocityinthereferenceframecorotatingwithspirals.WebrieflysummarizeourmainconclusionsinSection5.

WerefertoPaperIforadescriptionoftheobservationswhichareusedinthiswork.

2SPIRALSTRUCTUREOFNGC3631

Althoughthetwo-armedspiralstructureofNGC3631iswelldefinedinopticallight,ithasanumberofirregulari-ties,especiallyintheouterpartsofthedisc.TheHimaphasmuchlowerangularresolutionthanopticalimages,yetitalsoclearlyshowsevidenceofthetwo-armedspiralstruc-ture(Knapen1997).Forthepurposeofourstudyweshouldbeconvincedthatthesecondharmonicofthebrightness,mb=2,ofthespiralstructureexceedsallotherharmon-ics.Tocheckthis,wedividedthegalacticdiscintoellipticalrings,correspondingtocircularringsafterdeprojection,andcarriedoutaFourieranalysisoftheazimuthalbrightnessdistribution,usinganHαimageofNGC3631obtainedattheSAO6-mtelescopethroughinterferometricobservations(PaperI),anR-bandimageofthegalaxy,asobtainedfromtheIsaacNewtonGroup(ING)archive,andtheneutralhy-drogendistribution,asobtainedfrom21cmobservations(Knapen1997).WeshowtheR-bandimage,takenwiththe1-mJacobusKapteynTelescope,inFig.1,whichoutlinesthemainspiralarmstructureinthisgalaxy.Technicalin-formationonthisimagecanbefoundinKnapen(1997).TheseriesofhistogramsinFig.2showsthecontributionsofthedifferentFourierharmonicstothedeviationfromanaxi-allysymmetricaldistributionofbrightness,ordispersion,forthethreeimagesmentionedabove.Throughoutthepaper,werestrictourconsiderationtotheregionofthegalacticdiscR<80′′whichcorrespondstotheextentoftheopticalspirals.

Fig.2showsclearlythatthesecondharmonic,whichcorrespondstotheobservedtwo-armedstructure,indeeddominatesthespectrum.ThehighlevelofthefirstharmonicintheHαimageiscausedbythenon-symmetricaldistribu-tionofstar-formingregionsinthespiralarms,anddoesnotreflectthetruecontributionofthefirstFourierharmonictothemassdistributioninthegalaxy.ThisassumptionissupportedbythelowlevelofthefirstharmonicintheR-bandimageandtheHimap.Todemonstratethisyetmoreclearly,weshowinFig.3theexistenceofatightcorrelationbetweenlinesofmaximumvaluesofthesecondharmonicintheR-andHαimages(toppanel),andaveryweakcorre-lationbetweenpointsandlinesofmaximumvaluesofthefirstharmonics(bottompanel).

Asshownearlier(Sakhibov&Smirnov19;Canzian1993;Fridmanetal.1997),ifthecircularvelocityofgasinagalaxyisperturbedbyatwo-armedspiralpattern,thismustleadtotheappearanceofthefirstandthirdFourierharmonics(mobs=1and3)intheazimuthaldistributionoftheobservedline-of-sightvelocity.Inaddition,thesecondharmonic(mobs=2)mayalsoappearifthedensitywaveinducesverticaloscillationsofthegas(Fridmanetal.1997;

󰀁

cRAS,MNRAS000,GasmotionsinNGC3631

3

Figure2.ContributionofindividualFourierharmonicstothe

deviation,ordispersion,ofthebrightnessdistributionfromaxialsymmetry,asderivedfrom(a)ourHαimage(PaperI),(b)ourR-bandimage(INGarchive),and(c)21cmmap(Knapen1997).

Fridmanetal.1998).Thepredominanceofthefirstthreeharmonicsintheline-of-sightvelocityfieldofthegalaxyisclear,asfirstdemonstratedinpaperI.Here,weshowthisresultforbothopticalandradiovelocitymeasurementsinFig.4and5,usingPA=336◦,whichgivesminimumdis-persioninamodelofpurecircularmotionfortheradioline-of-sightvelocitydatainthepartofthegalaxyunderconsideration.TheopticaldataarepracticallyinsensitivetothechangeofPAbylessthan10◦.Fig.4showsthecontri-

4A.M.Fridmanetal.

Figure1.R-bandimageofNGC3631asobtainedfromtheINGarchive

butionsofdifferentFourierharmonicstothedispersioninthemodelofpurecircularmotion,averagedovertheinnerpartofthedisc(R<40′′),whereasFig.5showssimilarhistogramsfortheouterpartoftheopticaldisc(40′′ComparisonoftheHα(Figs.4aand5a)withtheradiodata(Figs.4cand5c)showsthattheamplitudeoftheharmonicsismuchsmallerinthelatterthanintheformercase.Suchadifferencecouldbeareasontodistrusttheopticalorradiodata,ifthesedatawererelatedtothesamevelocityfield,butthisis,infact,notthecase.

AspointedoutinPaperI,reasonsfordifferencesbe-tweenradioandopticalestimatesoftheamplitudesofthesecondFourierharmonicsontheonehand,andofthefirstandthethirdharmonicsontheotherhand,shouldbediffer-ent.Inthelattercase,thedifferencemaybecausedbythelowresolutionoftheradiodata.ThisiswellillustratedinFigs.4band5b,wheretheamplitudeswerecalculatedaftersmoothingoftheopticalvelocityfieldtoaresolutionof14′′,closetothatoftheradiodata.Thesquaresoftheampli-tudesofthethirdharmonicinFig.4bandofthefirstandthirdharmonicsinFig.5bareone-third/one-fifthasmanyasthoseintheoriginalHαdata.ItthusfollowsfromFigs.4and5thatthehistogramsofthesmoothedHαdataoccupyanintermediatepositionbetweenhistogramsoforiginalHαandHidata—amplitudesinthehistogramsb)isclosertothoseinthehistogramsc)thanthoseinhistogramsa).ThesecondharmonicofthesmoothedHαline-of-sightvelocityfieldisnaturallymuchhigherthanthesecondharmonicoftheHidata.ThiscanbeexplainedbythedifferentopticaldepthofthegaseousdiscintheHαand21cmlines,ifonetakesintoaccountthatthesecondharmoniciscausedbytheverticalmotionsinthedensitywavewhichareantisym-metricalwithrespecttothecentralplaneofthedisc(seediscussioninPaperI).

ItfollowsfromFigs.4and5thatthecontributionofharmonicswithmobs>3intotheobservedline-of-sightve-locityfieldisnotsignificantforthisgalaxy.Notethatthisconclusionisnotauniversalrule-ourpreliminaryanaly-sisofdataforotherobjectsshowsthatitmightbewrong

cRAS,MNRAS000,󰀁

GasmotionsinNGC36315

Figure3.SuperpositionofthepositionsofthemaximumvalueoftheharmonicsintheR(asterisks)andHα(squares)images:(a)toppanel,secondharmonics;(b)lowerpanel,firstharmonics.

forsomeothergalaxies,eventhoughtheyhaveatwo-armedgrand-designspiralpattern.Theanomalouslylargevalueoftheamplitudeofthe5thharmonicinFig.5cisperhapscausedbythecoincidenceofthetypicalspatialscalesofthe5thharmonicintheouterpartofthediscwiththeresolu-tionoftheradiodata.ThishypothesisissupportedbytheappearanceofanartificialanomalousamplitudeofthefifthharmonicinthesmoothedHαdata(Fig.5b).

Theextractionofthefirstthreeharmonicsfromtheline-of-sightvelocityfieldrevealsthattheresidualshaveanon-regularnoisy-likecharacter.Fig.6showsahistogramofthedistributionofthevelocityresidualswithm>3alloverthediscwithinthedeprojectedradiusR<80′′.ItcanbesatisfactorilyapproximatedbyaGaussianwithadispersion√

6A.M.Fridmanetal.

Figure5.AsFig.4,nowfortheregion40′′3

PHASERELATIONSHIPSANDTHE

POSITIONOFCOROTATION

ThemethodusedtorestorethevectorvelocityfieldandtodeterminethecorotationradiusRcfromobservationsofline-of-sightvelocitydistributionswasdescribedindetailbyLyakhovichetal.(1997)andappliedtosomegalaxiesbyFridmanetal.(1997)andFridmanetal.(2000).ThemethodisbasedonthecomparisonofFouriercoefficientsofazimuthaldistributionsoftheobservedline-of-sightve-locity,withthoseexpectedforamodelwheretheperturbed

Figure6.Histogramofresiduals(differencesbetweentheob-servedandthemodelline-of-sightvelocityfields)inthemodeloftheline-of-sightvelocityfieldwhichtakesintoaccountonlythefirstthreeFourierharmonics.Thelineshowstheχ-distributionwithadispersionequaltothatofthemodel.(a)Fortheorigi-nal,unsmoothed,Hαline-of-sightvelocityfield,(b)forthe21cmvelocityfield.

velocityV

˜componentsattheparticularmomentoftimeV

˜r,ϕ,V˜zarecausedbyatwo-armedspiraldensitywave:V

˜r(R,ϕ)=Cr(R)cos[2ϕ−Fr(R)],(1)V˜ϕ(R,ϕ)=Cϕ(R)cos[2ϕ−Fϕ(R)],and(2)V

˜z(R,ϕ)=Cz(R)cos[2ϕ−Fz(R)],(3)

whereCi(R)andFi(R)areanamplitudeandphaseofi-componentofvelocity.

Themainideabehindthisisthefollowing.Theline-of-sightvelocityisconnectedwiththevelocitycomponentsofthegasbytherelationship(Lyakhovichetal.1997;Fridmanetal.1997)

Vobs(R,ϕ)=Vs+Vϕ(R,ϕ)cosϕsini++Vr(R,ϕ)sinϕsini+Vz(R,ϕ)cosi,

(4)

whereVsisthesystemicaccountthatVr=V

˜velocityofthegalaxy.Takinginto

r,Vϕ=Vrot+V˜ϕ,Vz=V˜z,whereVrotistherotationvelocity,andsubstituting(1)–(3)in(4)weobtainthemodelrepresentationoftheline-of-sightvelocity:Vmod(R,ϕ)=Vs+sini[a1(R)cosϕ+b1(R)sinϕ+

󰀁

cRAS,MNRAS000,+a2(R)cos2ϕ+b2(R)sin2ϕ++a3(R)cos3ϕ+b3(R)sin3ϕ],

(5)

whereFouriercoefficientsrelatedtophasesandamplitudesofthevelocitycomponentsare:a1=Vrot+

CrsinFr+CϕcosFϕ

2

,

(7)a2=CzcosFzcoti,(8)b2=CzsinFzcoti,(9)

a3=−

CrsinFr−CϕcosFϕ

2

.(11)

CalculatingtheFouriercoefficientsoftheobservedline-of-sightvelocityfield(aobsi,bobs

i,withi=1,2,3)andequat-ingthemtothemodelones(Eqs.6-11),weobtainthebasetodeterminetheamplitudesandphasesofallthreevelocitycomponents.

TheFouriercoefficientsaobs1,bobs1,aobs2,bobs2,aobs

3andbobs

3aswellasthebestfitparametersofthegalacticdisc(PA,inclination,andcentreposition)maybefoundfromtheobservedline-of-sightvelocitydistributionVjobs(rj,ϕj)byminimisingineachellipticalringthequantityχ2(Rk),asdeterminedbythefollowingequation(seeLyakhovichetal.1997;Fridmanetal.1997fordetails):χ2

(Rk)=

󰀂

(Vj(Rj,ϕj)−Vsj

−

n−

wherek󰀂=3

aobsn(Rk)cosnϕj+bobs

n(Rk)sinnϕjn=1

designates󰀁thenumberoftheringand󰀃

sini

󰀅2

,(12)

summingisperformedoverallpixelsbelongingtothering.

Thefirstharmonicoftheobservedvelocityfieldcon-tainscontributionfromboththerotationvelocityandtheperturbedmotion,whichcannotbeseparatedwithouttak-ingsomeadditionalproposition,butthethirdharmonicshouldbeunambiguouslyrelatedtothevelocityperturba-tionconnectedwiththeobservedspiralarms.Toverifythis,Fridmanetal.(1997)proposedtousethe“modifiedthirdharmonic”,whichhasaformcos(2ϕ−F3+π/2),whereF3isthephaseoftheoriginalthirdharmonicoftheobservedline-of-sightvelocity.Inthecaseofatightlywoundspiral,itwasshownthatthemaximaofthis”modifiedthirdhar-monic”followthemaximaoftheperturbedsurfacedensityofthediscoutsidecorotation(thatisF3=Fσ+π/2).In-sidecorotation,theycancoincidewithmaximaorminimaoftheperturbedsurfacedensity,dependingontherelationbetweentheamplitudesoftheradialandazimuthalresidualvelocities.IfCr󰀁

cRAS,MNRAS000,GasmotionsinNGC3631

7

Figure7.Comparisonofthepositionsofthemaxima(a)and

minima(b)ofthe“modifiedthirdharmonic”oftheHαline-of-sightvelocityfield(triangles)withtheformofthespiralarmscharacterizedbypositionsofmaximaofthesecondFourierhar-monicofRbrightnessmap(asterisk).Themodifiedthirdhar-moniciscos(2ϕ−F3+π/2),whereF3isthephaseoftheoriginalthirdharmonic.

armsintheradialregion25′′Toobtainamoreaccurateestimateofthepositionofcorotationwepreviouslyproposedtwomethods(Lyakhovichetal.1997;Fridmanetal.,1997).Thefirstmethodisbasedonthecomparisonoftheradialbehaviourofthesinuscom-ponentsofthefirst(bobs1)andthird(bobs

3)Fourierharmonicsoftheline-of-sightvelocityfield.Inthecaseoftightlywoundspirals,thefollowingrelationsarefulfilled(fordetailsseeLyakhovichetal.1997andFridmanetal.,1997):

8A.M.Fridmanetal.

Figure8.Behaviourof|bobs3(R)|−|bobs1(R)|asafunctionof

galactocentricradiusRinNGC3631,asderivedfrom(a)origi-nalHαline-of-sightvelocityfield,(b)smoothedHαvelocityfield,(c)21cmvelocityfield.Errorbarscorrespondto3σlevel.Ac-cordingtoresultsofthedensitywavetheoryintheapproxima-tionoftightlywoundspirals,thedifferenceshouldbenegativeinsidecorotationandpositiveoutside.Thusthedatashowthatthecorotationradiusisat40′′±7′′.

|bobs3(R)|−|bobs

1(R)|≤0,

for

R1(R)|≥0,

forR>Rc.

(13)

Theseinequalitiesenablethedeterminationofthelo-cationofthecorotationradiusfromobservationaldata.Ac-cordingtoEq.(13),corotationislocatedintheregionwhere

thedifference|bobs3(R)|−|bobs

1(R)|changessign,fromminustoplus.

Figure9.Variationof(bobs3−bobs1)cosFσwithgalactocentricra-diusRinNGC3631,asderivedfrom(a)originalHαline-of-sight

velocityfield,(b)smoothedHαvelocityfield,(c)21cmvelocityfield.Errorbarscorrespondto3σlevel.Accordingtoresultsofthedensitywavetheoryintheapproximationoftightlywoundspirals,thedifferenceisnegativeinsideandpositiveoutsidecoro-tation.Thusthedatashowthecorotationradiustobeabout43′′±4′′.

Fig.8showstheradialbehaviourof|bobs3(R)|−|bobs

1(R)|inNGC3631,asobtainedfromtheoriginalHα(a),smoothedHα(b),and21-cm(c)line-of-sightvelocityfields.Fig.8showsthatthisfunctionderivedfromHαline-of-sightvelocityfieldisnegativewithintheerrorsintheinnerpartofthegalaxy,andpositiveintheouterregion,inaccordancewiththeexpectationsfrom(13).FromthesedataitfollowsthatthecorotationradiusisatR=40′′dataisrelativelyhigher,due±7′′.Theleveloferrorsin21-cmtolowerresolu-

󰀁

cRAS,MNRAS000,tion.Nevertheless,behaviourof|bobs3(R)|−|bobs

1(R)|derivedfrom21-cmline-of-sightvelocityisingoodagreementwithHαdatawithinoneσlevelofconfidence.

Inaddition,yetanothermethodtoestimateRcwaspro-posed(Lyakhovichetal.1997;Fridmanetal.1997),basedontherelationbetweenthephasesoftheperturbedsurfacedensityandtheradialperturbedvelocityofagas,whichisfulfilledfortrailing󰀄

tightlywoundspirals:

Fσ−Fr=

πatR0atR>Rc

=(1−signˆω)π/2.

(14)

Thisrelationshowsthatgasshouldmovealongspiralarmsinwardsinsideofcorotation,andoutwardsoutsideofit.Asshowninthecitedpapers,itmayalsobewrittenastherelationshipbetweenthephaseoftheperturbedsurfacedensityandtheFouriercoefficientsbobs3andbobs1:

(bobs3(R)−bobs1(R))cosFσ(R)≤0,

forR1(R))cosFσ(R)≥0,

forR>Rc.

(15)

Fig.9showsthedependenceof(bobs3−bobs1)cosFσongalactocentricradiusR.Thesituationisquitesimilartopresentedinthepreviousfigure.ThecurrentapproachleadstotheestimateofRcof43′′theresultsfrom±4′′.

Combiningthesemethods,whichagreewell,wemayconcludethatthecorotationradiusRc5′′,or3.2kpc±0.38kpc.

≈42′′±4

RESTOREDVELOCITYFIELDOFGASINTHEGALACTICPLANE

AsshowninSection2,loweringtheangularresolutionleadstoanunderestimateoftheamplitudesoftheFourierhar-monics.Therefore,torestoreatwocomponentvectorveloc-ityfieldofagasinthediscofNGC3631,weusethefullresolutionHαline-of-sightvelocityfield.

Torestorethevelocityfieldintheplaneofagalactic

disc,i.e.,itsradialV

˜randazimuthalVϕ=Vrot+V˜ϕvelocitycomponents,itisnecessarytodeterminefiveunknownfunc-tions:Vrot(r),Cr(r),Cϕ(r),Fr(r),Fϕ(r)(seeEqs.1and2).ThesefivefunctionsareconnectedwiththeFouriercoeffi-cientsoftheobservedline-of-sightvelocityfieldbythefourrelations(6),(7),(10),and(11).Anadditionalcondition,re-quiredtoclosethesystem,shouldhaveatheoreticalorigin.Unfortunately,uptonow,areliablecondition,validforanydensitywaveamplitude,isnotavailable.Severalpossibilitiesdiscussedintheliterature(Sakhibov&Smirnov,19;Frid-manetal.1997)arebasedonsomekindofapproximationandhavelimitedapplicability.Toovercomethisdifficulty,weproposethefollowingapproach.

Amongthefunctionslistedabove,Vrot(r)canbemostreliablyestimatedintwoindependentwaysfromobserva-tionaldata.ThefirstoneusestheequilibriumconditionofagaseousdiscrotatinginagravitationalpotentialΨ

V2rot/r=∂Ψ/∂r.

(16)

Theright-handsideoftheEq.(16)isdeterminedfromthemassdistributioninagalaxy,oritssurfacebright-nessmaps,assumingthemass-to-lightratioisknownandconstantwithradius.Forthispurpose,weuseathree-componentdynamicalmodelofaspiralgalaxysimilartotheoneusedbySumin,Fridman,&Haud(1991).Althoughthe

󰀁

cRAS,MNRAS000,GasmotionsinNGC3631

9

Figure10.Theradialdependenceofbobs3−bobs=CrcosFrandbobs3+bobs1

1=CϕsinFϕobservedinthespiralgalaxyNGC3631.Anestimateoftheamplitudesofthevelocitycomponentsfromtheextremesofthesefunctionsgivesmax(Cr)≃max(Cϕ)≃60km/s.

modelisrathercrude,theresultingrotationcurveVrot(R)

correspondstotheregionapproximatelybetween(a1)minand(a1)max.

Thesameresultcanbeobtainedinanotherway.Fromequation(6)itfollowsthatthedifference|a1notexceedtheamplitudesCrandCϕ,which,−inVrotturn,|can-areconnectedbyequations(7),(10),and(11)withtheFourier

coefficientsbobs1,aobs3,andbobs

3,determinedfromourobser-vations.InFig.10,weshowtheradialbehaviourofbobs3obs=CrcosFrandbobs3+bobs1

=CϕsinFϕ.Theextremes−b1ofthesefunctionsallowestimatesoftheamplitudesCrandCϕ.FromFig.10weconcludethatinNGC3631amaxi-mumvalueoftheamplitudeoftheresidualvelocitiesoccursat60km/s,i.e.

|a1−Vrot|max≤60km/s.

(17)

Theconditions(16)and(17)donotallowanexactcal-culationofthefunctionVrot(r).Nevertheless,theysetlim-itsonthevariationsofboththeamplitudeandtheformof

Vrot(r).Withintheselimits,varyingthemass-to-lightratio(±40%)andassumingitdoesnotdependingonradiusweobtainasetoftrialcurves(Fig.11)andanalysethevelocityfieldsrestoredfromEqs.(6)–(11)foragivenVrot(r).

Fig.12showsfourexamplesoftherestoredvelocityfieldofNGC3631,intheplaneofthedisc,andinaref-erenceframerotatingwiththespiralpattern.PositionsofthemaximaofthesecondFourierharmonicoftheR-bandsurfacebrightnessmapareoverlaidtoindicatetherelativelocationofspiralarmsandstructuresinthevelocityfield.

Itisclearlyseenthat,inspiteofbroadvariationsinVrot(r),thegeneralstructureofthevelocityfieldchangesonlyslightly.Inallcasesthisstructuredemonstratesthepresenceoftwoanticyclones,locatedonthecorotationcir-cle,andbetweenspiralarms.Variationsoftherotationcurveonlyleadtosmallchangesinthebasicquantitativefeaturesoftheanticyclones.Inallcases,themaximumnoncircularvelocityinthevorticesisabout60km/s,theirradialwidthisabout20′′,andthevariationoftheazimuthalpositionofthecentresofanticyclonesislessthan10◦.Thisprovesthattheexactshapeoftherotationcurvedoesnotimpactonthebasicconclusionoftheexistenceofgiantanticyclonesin

10A.M.Fridmanetal.

Figure12.RestoredvelocityfieldofNGC3631intheplaneofthediscinthereferenceframerotatingwiththepatternspeed.OverlaidasterisksshowthelocationsofmaximaofthesecondFourierharmonicoftheR-bandbrightnessmapofthegalaxy.Thecirclemarksthepositionofthecorotation.Solidlinesdemonstratevortexseparatricesornearlyclosedstreamlinesintheabsenceofaseparatrix(theonesgreaterinsizecorrespondtoanticyclones,andthesmaller–tocyclones).(a)Thecurvemarkedbysymbol”a”inFig.11usedasVrot(r).

thegaseousdiscofNGC3631.Theseanticyclonesaresimi-lartothoserevealedearlierinthevelocityfieldofNGC157(Fridmanetal.1997).

AnothernewresultisthatregionsofcyclonicshearareseeninFig.12.Asaconsequenceoftherelativelyhigham-plitudeofthedensitywaveinNGC3631,acyclonicshearisproducedbythedensitywaveinsomeregions,whichdomi-natestheanticyclonicshearcausedbydifferentialrotation.TheappearanceofcyclonesingaseousgalacticdiscswithastrongdensitywavewaspredictedearlierbyFridmanetal.(2000).

TochoosebetweenthevelocityfieldspresentedinFig.12thefieldwhichisclosesttotherealvelocityfieldofthegalacticdiskofNGC3631,weusethefollowingcri-terion.Inthecourseofmanyrevolutionsthegrowthofthedensitywaveamplitudeisstoppedbythesaturationofacorrespondinginstability.Thusinthereferenceframecoro-tatingwithspiralarmsthevelocityfieldofthegalacticdiskshouldbestationary.Insuchafieldthevorticesshouldhaveclosedseparatrices(linesdividingtwofamiliesoftrajecto-ries:trappedandtransitones).Accordingtothecriterionstatedabove,thevelocityfieldpresentedinFig.12cisdis-cRAS,MNRAS000,󰀁

GasmotionsinNGC363111

Figure12.(b)Thecurvemarkedbysymbol”b”inFig.11isusedasVrot(r).

tinctinthatsensethatitisonlycasewherecycloneshavecloseseparatrices.ItisanargumentinfavourofchoosingjustthisexampleastheclosesttotherealvelocityfieldoftheNGC3631gaseousgalacticdisk.Atthesametimethatmeansthatthecurve”c”inFig.11isclosetotherealrota-tioncurveofthedisk.TherelativepositionofthevorticesandspiralarmsinFig.12cisinagoodagreementwiththeoreticalpredictions(Fridmanetal.1999).

5CONCLUSIONS

Wecanbrieflysummarizeourmainconclusionsfromthispaperasfollows:

cRAS,MNRAS000,󰀁

(i)Basedonananalysisofline-of-sightvelocityfieldsof

gaseousemissionlinesforthespiralgalaxyNGC3631weconfirmobservationallythetheoreticalconclusionsfromourpreviousworkaboutthewavenatureofitstwo-armedspiralstructure,anddiscussitsoriginsandpropertiesinlightofthistheoreticalframework.

(ii)Usingtwoindependentmethods,wefindthatthecorotationradiusinthisgalaxyisatabout42′′or3.2kpc.(iii)Theprojectionoftherestoredthree-dimensionalvec-torvelocityfieldofgasintheplaneofthegalaxy,andinareferenceframecorotatingwiththespiralpattern,revealsthepresenceoftwoanticyclonicvorticesnearcorotation.Wethusconfirmthetheoreticalpredictionforeseenearlieronthebasisofastudyofthegeneralprinciplesofthewavenatureofspiralstructureingalaxies.

12A.M.Fridmanetal.

Figure12.(c)Thecurvemarkedbysymbol”c”inFig.11isusedasVrot(r).

(iv)WeshowtheexistenceofcyclonicvorticesinNGC3631,apartfromthatofanticyclones,asmentionedabove.Suchcyclonicvorticesareaconsequenceofahighamplitudeofthedensitywaveinthisgalaxy.

REFERENCES

Arp,H.,1966,ApJS,14,1

Binney,J.,Tremaine,S.,1987,GalacticDynamics,PrincetonUni-versityPress,Princeton

Boeshaar,G.,Hodge,P.W.,1977,ApJ,213,361

Bonnarel,F.,Boulesteix,J.,Georgelin,Y.P.,Lecoarer,E.,

Marcelin,M.,Bacon,R.,Monnet,G.,1988,A&A,1,59Canzian,B.,1993,ApJ,414,487Consid`ere,S.,Athanassoula,E.,1988,Astron.Astrophys.Suppl.

Ser,76,365

Fridman,A.M.,Polyachenko,V.L.,1984,Physicsofgravitating

systems,v.1,2,Springer-Verlag,NewYork,Berlin,Heidelberg,Tokyo

Fridman,A.M.,Khoruzhii,O.V.,Lyakhovich,V.V.,Avedis-ova,V.S.,Sil’chenko,O.K.,Zasov,A.V.,Rastorguev,A.S.,

cRAS,MNRAS000,󰀁

AcknowledgementsThisworkwasperformedwithpar-tialfinancialsupportofRFBRgrantN99-02-18432,“Lead-ingScientificSchools”grantN00-15-96528,and“Funda-mentalSpaceResearches.Astronomy”grants:N1.2.3.1,N

1.7.4.3.TheJacobusKapteynTelescopeisoperatedontheislandofLaPalmabytheIsaacNewtonGroupintheSpan-ishObservatoriodelRoquedelosMuchachosoftheInsti-tutodeAstrof´ısicadeCanarias.DatawereretrievedfromtheINGarchive.

GasmotionsinNGC363113

Figure12.(d)Thecurvemarkedbysymbol”d”inFig.11isusedasVrot(r).

Afanasiev,V.L.,Dodonov,S.N.,Boulesteix,J.,1997,Astroph.andSpaceSci.,252,115

Fridman,A.M.,Khoruzhii,O.V.,Zasov,A.V.,Sil’chenko,O.K.,

Moiseev,A.V.,Burlak,A.N.,Afanasiev,V.L.,Dodonov,S.N.,Knapen,J.H.,1998,Astron.Lett.,24,7(PaperI)

Fridman,A.M.,Khoruzhii,O.V.,Polyachenko,E.V.,Zasov,A.V.,

Sil’chenko,O.K.,Afanasiev,V.L.,Dodonov,S.N.,Moiseev,A.V.,1999,PhysicsLettersA,2,85

Fridman,A.M.,Khoruzhii,O.V.,Lyakhovich,V.V.,Sil’chenko,

O.K.,Zasov,A.V.,Afanasiev,V.L.,Dodonov,S.N.,Boulesteix,J.,2000,A&A,submittedGarc´ıaG´omez,C.,Athanassoula,E.,1993,Astron.Astrophys.

Suppl.Ser,100,431

Hodge,P.W.,1982,ApJ,256,447

Kent,S.M.,1984,Astrophys.J.Suppl.Ser.,56,105Kent,S.M.,1985,Astrophys.J.Suppl.Ser.,59,115cRAS,MNRAS000,󰀁

Knapen,J.H.,1997,MNRAS,286,403

Lyakhovich,V.V.,Fridman,A.M.,Khoruzhii,O.V.,Pavlov,A.I.,

1997,AstronomyReport,41,447

Rozas,M.,Beckman,J.E.,Knapen,J.H.,1996,A&A,307,735Sakhibov,F.Kh.,Smirnov,M.A.,1987,Astron.Zh.,,255Sakhibov,F.Kh.,Smirnov,M.A.,19,Astron.Zh.,66,921Sakhibov,F.Kh.,Smirnov,M.A.1990,Astron.Zh.,67,690

Schoenmakers,R.H.M.,Franx,M.,deZeeuw,P.T.,1997,MN-RAS,292,349

Sempere,M.J.,Garc´ıa-Burillo,S.,Combes,F.,Knapen,J.H.,

1995,A&A246,45

Sumin,A.A.,Fridman,A.M.,Haud,V.A.,Pis’maAstron.Zh.,

1991,17,698

Vaucouleurs,G.de,Vaucouleurs,A.de,Corwin,H.G.,et.al.,Buta,

R.J.,Paturel,G.,Fouque,P.,1991.ThirdReferenceCata-logueofBrightGalaxies.Springer,NewYork(RC3)

14A.M.Fridmanetal.󰀕󰀓󰀓󰀔󰀛󰀓/.󰀔󰀙󰀓-󰀔󰀗󰀓,󰀔󰀕󰀓󰀔󰀓󰀓󰀛󰀓󰀙󰀓󰀓󰀔󰀓󰀕󰀓󰀖󰀓󰀗󰀓󰀘󰀓󰀙󰀓󰀚󰀓󰀛󰀓󰀜󰀓Figure11.Examplesoftrialcurvesusedtorepresentthero-tationcurve(Vrot(r))inNGC3631,shownbysolidlines,alongwiththeobservedbehaviourofaobs1(r)(triangles).Abscissaistheradiusinarcsecandordinate—rotationvelocityinkm/s.Ac-cordingtotheanalysispresentedbelow,thethickestlinemarksthecurvecorrespondingtotherealrotationcurveofNGC3631gaseousdisk.

Westpfahl,D.J.,1998,ApJS115,203Yuan,C.,1969,Ap.J.,158,871

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cRAS,MNRAS000,