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Astronomy&Astrophysicsmanuscriptno.cheminm99 cESO2016 (cid:13) January20,2016 Asymmetric mass models of disk galaxies - I. Messier 99 LaurentChemin1,2,Jean-Marc Huré1,2,CarolineSoubiran1,2,StefanoZibetti3,StéphaneCharlot4,andDaisuke Kawata5 1 Univ.Bordeaux,LAB,UMR5804,F-33270,Floirac,Francee-mail:[email protected] 2 CNRS,LAB,UMR5804,F-33270,Floirac,France 3 INAF-OsservatorioAstrofisicodiArcetri,LargoEnricoFermi5,I-50125Firenze,Italy 4 Institutd’AstrophysiquedeParis,CNRS&UniversitéPierre&MarieCurie(UMR7095),98bisBdArago,F-75014Paris,France 6 5 MullardSpaceScienceLaboratory,UniversityCollegeLondon.Dorking,Surrey,RH56NT,UnitedKingdom 1 ReceivedJanuary20,2016;acceptedXX 0 2 n ABSTRACT a J Massmodelsof galacticdiskstraditionallyrelyonaxisymmetricdensityand rotationcurves, paradoxically actingasiftheirmost remarkableasymmetricfeatures,suchaslopsidednessorspiralarms,werenotimportant.Inthisarticle,werelaxtheaxisymmetry 9 approximationandintroduceamethodology thatderives3Dgravitationalpotentialsofdisk-likeobjectsandrobustlyestimatesthe 1 impactsofasymmetriesoncircularvelocitiesinthediskmidplane.Massdistributionmodelscanthenbedirectlyfittedtoasymmetric line-of-sightvelocityfields.Appliedtothegrand-designspiralM99,thenewstrategyshowsthatcircularvelocitiesarehighlynonuni- ] A form,particularlyintheinnerdiskofthegalaxy,asanaturalresponsetotheperturbedgravitationalpotentialofluminousmatter.A cuspyinnerdensityprofileofdarkmatterisfoundinM99,intheusualcasewhereluminousanddarkmattersharethesamecenter. G Theimpactofthevelocitynonuniformityistomaketheinnerprofilelesssteep,althoughthedensityremainscuspy.Onanotherhand, . amodelwherethehaloiscoredominatedandshiftedby2.2-2.5kpcfromtheluminousmasscenterismoreappropriatetoexplain h mostofthekinematicallopsidednessevidencedinthevelocityfieldofM99.However,thegravitationalpotentialofluminousbaryons p isnotasymmetricenoughtoexplainthekinematicallopsidednessoftheinnermostregions,irrespectiveofthedensityshapeofdark - o matter.Thisdiscrepancypointsoutthenecessityofanadditionaldynamicalprocessintheseregions:possiblyalopsideddistribution r ofdarkmatter. t s Keywords. galaxies:kinematicsanddynamics–galaxies:spiral–galaxies:structure–galaxies:individual(Messier99,NGC4254) a [ 21. Introduction case, Franxetal. (1994) and Schoenmakersetal. (1997) initi- v atedthederivationofhigh-orderharmonicswithfirst-orderkine- 1Rotationcurvesandsurfacedensityprofilesofgalacticdisksare matical componentsfrom gaseous velocity fields. They argued 0 the observational pillars most models of extragalactic dynam- that kinematicalFourier coefficientsare useful to constrain de- 6 ics are based on. Rotation curves are needed to constrain the viationsfromaxisymmetryandthe natureofdynamicalpertur- 1 0total mass distribution, the parameters of dark matter haloes, bations. Using that technique, Gentileetal. (2005) concluded, .or the characteristics of modified Newtonian dynamics, while for instance, that the kinematical asymmetries in the Hi veloc- 1surface density profiles are helpful to constrain the structural ityfield ofa dwarfdiskpresentinga core-dominateddarkmat- 0 parameters of disks and bulges, and generate the velocity con- terhalo(DDO47)couldlikelyoriginatefromaspiralstructure. 6 tributions of luminous matter essential to mass models. As the However,theiramplitudeswerenothighenoughtoaccountfor 1 :density and rotationvelocity profilesare axisymmetricby con- thevelocitydifferenceexpectedbetweentheCDMcuspandthe vstruction,massmodelsimplicitlyassumethattherotationalve- cored halo preferred by the rotation curve fittings. In the sec- Xilocityisonlymadeofuniformcircularmotions.Thoughattrac- ondcase,Spekkens&Sellwood(2007)proposedtofitabisym- tiveforitssimplicity,thisapproachremainsareductiveexploita- metricmodelofbar-like/ovaldistortiontotheHαvelocityfield r ationofvelocityfieldsandmultiwavelengthimagesofstellarand ofanotherlow-massspiralgalaxy(NGC2976)andarguedthat gaseousdisks, which are informationrich. In particular,it pre- negligible high-order Fourier motions in velocity fields do not vents one from measuring the rotational support through per- necessarily imply that the bisymmetric perturbation is negligi- turbations(spiralarms,lopsidedness,etc.),whichareobviously ble, and that the rotation curve should be similar to the under- themoststrikingfeaturesofgalacticdisks.Inaneraofconflict lyingcircularmotionsonlyifthedeparturesfromcircularityre- between observationsand expectationsfrom Cold Dark Matter main small. These authors also showed that the inner slope of (CDM) simulations, the cusp-core controversy (see the review therotationcurveofNGC2976islikelyaffectedbythebar-like ofdeBlok2010,andreferencestherein,butseeGovernatoetal. perturbation.Numericalsimulationsofbarreddisksarrivedata (2010)), it appeared fundamental to assess the impact of such similarconclusionabouttheimpactofthebarontheinnershape perturbationson the shape of rotation curves, and more gener- of rotation curves (Valenzuelaetal. 2007; Dicaireetal. 2008). allyonmassmodelsanddensityprofilesofdarkmatter. Randriamampandryetal. (2015) performed numerical simula- This is the reason why efforts have been made to deter- tionstodeterminea correctedrotationcurveforanotherbarred mine the kinematical asymmetries inferred by perturbations or galaxy (NGC 3319), free from the perturbing motions induced to model the effects of dynamical perturbations. In the former by the bar. While these simulations demonstrate it is possible Articlenumber,page1of20 A&Aproofs:manuscriptno.cheminm99 disksofM99.Themeaningoftheseasymmetricoutputsisdis- cussed in Section 5. We then perform asymmetric mass distri- bution models of M99 (Section 6), and compare them to the axisymmetry-based models of Section 3. The conclusions for M99andtheprospectsofournewstrategyforgalacticdynamics arefinallygiveninSection7. 2. Thegrand-designspiralgalaxyMessier99 2.1.Abriefpresentation We selected the galaxy Messier 99 (M99 hereafter) because it is a SAc type disk harboring a very prominent spiral struc- ture (Fig. 1). Located in the Virgo Cluster (adopted distance of 17.1 Mpc, Freedmanetal. 1994), observing M99 is a good Fig. 1. Composite SDSS gri-image of the grand-design spiral galaxy opportunitytobenefitfromhigh-sensitivityandhigh-resolution M99.Northisup;eastisleft.Theimagesizeis7 6.3. multiwavelengthobservationsofthestellardiskandinterstellar ′ ′ × medium. The integrated Hα profile from data of Cheminetal. (2006) has a width at 20% of the maximum Hα peak of to hide cuspier DM distributions into artificial cored distribu- 216 km s 1. Combined with a small disk inclination (20 , − ◦ tions under the effect of stellar bars, they perfectly illustrate Makarovetal. 2014), this implies a massive galaxy with most the difficulty of performing mass models and constraining the ofrotationvelocitiesgreaterthan245kms 1.Thismakesitan − shapeofdarkmatterdensityprofilesfromobservationsofbarred idealtargetto study the structure and kinematicsof the disk in galaxies. Other numerical models based on closed-loop orbits detailandtotestournewmassmodelingstrategy. showed that shallow kinematics and core-like haloes could ac- The spiral structure is asymmetric. A one-arm mode domi- tually be explainedby cuspy triaxial distributionsof dark mat- natestheHidisk(Phookunetal.1993;Chungetal.2009),while terviewedwithparticularprojectionangles(Hayashi&Navarro thestellardistributionexhibitsmorethanonesinglearm(Fig.1). 2006).Finally,otherstudiesfocusedonextractinginHispectra Phookunetal. (1993) proposeda scenario where the Hi arm is the line-of-sight (l.o.s) velocity components supposed to trace triggeredbygasinfallingandwindingonthedisk,resultingfrom the axisymmetric rotational velocities better than the compo- atidalencounter.Basedonnumericalsimulations,Vollmeretal. nents based on more usual intensity-weightedmeans (Ohetal. (2005)mimictheasymmetricdiskandperturbedHikinematics 2008).Appliedtotwodarkmatterdominateddisks,NGC2366 by a flyby of a massive companion,coupledwith ram pressure andIC2574,whichareprototypesofgalaxieswhosedarkmatter strippingfromtheVirgointraclustermedium(ICM).Theseau- densityconflictswiththecosmologicalcusp,thismethodyielded thorsarguedthatM99 is entering the Virgo cluster for the first steeper rotation curves in the inner disk regions. However, the time.FurthernumericalmodelsofDuc&Bournaud(2008)also velocity differences with the intensity-weighted mean velocity explainedtheoriginofthelarge-scaleHitail,whichapparently curve were not sufficient to reconcile the observation with the connects M99 to VIRGOHI21, a 108 M Hi cloud wandering CDMcusp. in the Virgo ICM (Minchinetal. 2007),⊙by a tidal interaction In this context, in this article we propose a new approach thatwaswithanothercandidatecompanionthaninVollmeretal. to model the mass distribution of disk galaxies. Our strategy (2005). In principle, mass distribution models should only be goesbeyondthedecompositionofrotationcurvesandfullyex- carried out with targets supposedly in dynamical equilibrium. ploitsthebidimensionaldistributionofluminousmatter,thusthe The possibly infalling Hi mass of 108 M only represents asymmetric nature of stellar and gaseous disks. Our approach 2%ofthetotalneutralgasmassof∼M99,and⊙ about0.2%ofth∼e determinesthe 3D gravitationalpotentialof anydisk-like mass total luminousmass (Section 3). This should be not enough to componentthroughhyperpotentialstheorizedbyHuré(2013).It affectthelarge-scaledynamicsofM99.Furthermore,rampres- thenderivesthecorrespondingcircularvelocitymapinthedisk sure stripping predominantly affects the outskirts of the neu- midplane,whichallowsustodeterminewhereandtowhichex- tralatomicgascomponent,nottheinner,densestregionsofthe tent the circular motions should deviate from axisymmetry. A gaseousdiskofM99. 2Dmassdistributionmodelcanthenbedirectlyfittedtoa l.o.s velocityfield,byaddingthe2Dvelocitycontributionsfromlu- 2.2.High-resolutionHαkinematics minous baryons to that from the missing matter. The impacts ofthevelocityasymmetriesonthemassmodelsandstructureof Optical long-slit and resolved observations also revealed per- darkmatterhaloescanthenbeeninvestigatedbycomparingwith turbedkinematicsoftheionizedgasdiskofM99(Phookunetal. resultsobtainedwiththeaxisymmetricmassmodels. 1993; Kranzetal. 2001; Cheminetal. 2006). In particular, Weapplythatmethodologytoaprototypeofunbarred,spi- Kranzetal. (2001) arguesfor the presenceof a possible stellar ralgalaxyMessier99,whosegeneralpropertiesarepresentedin bar that would impact the Hα kinematics in the inner 1.5 kpc. Section2.Theaxisymmetricmassmodelofthehigh-resolution Asymmetricmotionshavealsobeenclearlyevidencedalongthe rotationcurveofM99isdetailedinSection3.ThisSectionalso Hα spiral arms (Phookunetal. 1993; Cheminetal. 2006), as presentsamoreelaboratedaxisymmetricmassmodel,fitteddi- wellasalopsidedHαvelocityfield(Cheminetal.2005b).The rectlytoahigh-resolutionvelocityfieldofM99.Thebasisofthe kinematicaldatawe usetomodelthemassdistributionofM99 derivation of the 3D gravitational potentials for luminous mat- are from the 3D spectroscopy survey of Virgo cluster galaxies ter is described in Section 4, which also presents the inferred byCheminetal. (2005a,2006). Intheir catalogofHαvelocity gravitationalpotentials,accelerations,andcircularvelocitiesfor fieldsofbrightVirgospiralandirregulardisks,theFabry-Perot the contributions from the stellar, atomic , and molecular gas observationsofM99haveangularandspectralsamplingsof1.6 ′′ Articlenumber,page2of20 LaurentCheminetal.:Asymmetricmassmodelsofdiskgalaxies-I.Messier99 Fig.2.Hαintegratedemission,velocityfield,andvelocitydispersionmapofM99(fromlefttoright,respectively). (130 pc at the adopted distance) and 8.2 km s 1. The Hα This Section is thereforeorganizedas a traditionalaxisym- − ∼ velocityfieldofCheminetal.(2006)resultedfromanadaptive metric mass model of a galactic disk. We first derive the rota- binningand a spatial interpolationof the Hα datacube,follow- tion curve of M99 and estimate the asymmetric drift contribu- ing prescriptions given in Daigleetal. (2006). We did not use tion,andthenwepresentmultiwavelengthobservationsneeded these interpolated data but an improved version of the binned to infer the mass surface density profiles and the velocity con- datacubeofCheminetal.(2006),wherenewbinsarenowmade tributionsofthestellarandinterstellarmatter.Theresultsofthe ofauniquepixellocatedatthebarycenterofinitialbins.Every massmodelingarethendiscussedtoinvestigatewhichdarkmat- binhasthusanequalangularsize,inaccordancewiththekine- terhaloshapeismoreappropriate.Inaddition,wediscussthe2D matical analysis of many other Hα Fabry-Perot velocity fields axisymmetricmodeling. (Epinatetal. 2008a,b). Figure 2 shows the revisited integrated Hα emission, velocity, and dispersion velocity fields of M99. 3.1.TangentialandradialvelocitiesinM99 We restrictedtheHαkinematicsto R = 11.5kpc,whichcorre- spondstoadistributionof9002velocitypixels.Thisallowedus As the mass distribution modeling first needs a rotation curve, toderiveaccuratevelocityandvelocitydispersionprofiles.Be- the3Dvelocityspace(v ,v ,v )isdeducedfromfittingtotheHα yondtheopticalsizeofthestellardisk,theHikinematicsistoo velocityfieldofM99theRfoθllozwingmodel: scatteredtoinferusefulkinematicalinformation. v =v +(v cosθ+v sinθ)sini+v cosi, (1) l.o.s sys θ R z 3. AxisymmetricmassmodelingofM99 where v is the l.o.s velocity, v and i are the systemic l.o.s sys Atraditionalaxisymmetricmassmodelconsistsindecomposing velocity and the inclination of the M99 disk, and θ is the az- arotationcurveintocontributionsfromluminousbaryons(stel- imuthalangle in the deprojectedorbit.In the axisymmetricap- lar and gaseous disks, bulge, etc.) and dark matter. The fitting proach, these velocities are assumed uniform, i.e. only depen- procedure yields fundamental scale parameters for the hidden denton R. Theazimuthalvelocityis the rotationcurve.The v R masscomponent,andeventuallyafactorthatenablesthescaling and v components are usually omitted in kinematical studies z ofluminositiesintosurfacedensitiesforstars(themass-to-light becausetheyaregenerallyassumedtobenegligible.Asseenbe- ratio, M/L). This section details the rotation curve decomposi- low,itisthecaseforv butsinceoneofourgoalsistostudythe R tion,whichwerefertoasthe1Daxisymmetriccasehereafter,as impactoftheradialmotionsonthemassmodeling,wedecided the rotation velocity only dependson one angular scale, which to fit them. Derivingv with or withoutv doesnotimpactthe θ R isthegalactocentricradius. shape or amplitude of the tangentialcomponent.Then, a prob- ThisSectionalsoenvisagesasecondstrategy,wheretheax- lemin leavingv free inEq. 1is thatitshouldexhibitartificial z isymmetric modeling is carried out directly from the velocity variationsifthegalaxyhasakinematicallopsidedness.Therea- field. The motivation for this is, first, that circular motions are son for this is that the systemic velocity must be naturally im- expectedto dominatethe kinematicsand, since rotation curves pactedbyalopsidedness(seeAppendixB),butsincevariations stemfromvelocityfields,onemaylogicallyperformmassmod- ofv arenotallowedhere,thev cositermabsorbsthekinemat- sys z elswith2Dresolveddatainsteadof1Dcurves.Thesecondmoti- icalsignatureoflopsidedness.Asboththegravitationalpotential vationisthatthedevelopmentofapipelinethatmakesfittingsin ofluminousmatterandtheHαkinematicsofM99arelopsided 2Dismandatoryforthenewasymmetricmethodologydiscussed (Section 4.2 and Appendix B), the best solution to avoid such fromSections4to6.Themostnaturalwaytounderstandtheim- artificialv variationsis thusto assume v = 0 hereafter. Other z z pactoftheasymmetriesonthemassmodelsis,thus,tofittothe face-on,grand-designspiralsofsimilarstarformationactivityto HαvelocityfieldofM992Dvelocitymodelsbuiltunderaxisym- M99 are knownto have negligibleverticalmotions(e.g.,NGC metryassumptions.Inshort,onecannotdirectlycomparethere- 628,Kamphuis&Briggs1992). sults from the decomposition of the rotation curve with those Since theHα gasiswell confinedinthe opticaldisk thatis presented in Sect. 6 for the asymmetric fittings of the velocity notwarped,wehavenotallowedv ,i,thepositionangleofthe sys field. One needs an intermediate step between them, which we diskmajoraxis(Γ)andthecoordinatesofthedynamicalcenter refertoasthe2Daxisymmetriccasehereafter.Thoughthedata tovarywithradius.Wechoseθ=0 alignedwiththesemimajor ◦ to be fitted are not totally axisymmetric because of the evident axis of the receding side of the galaxy disk. We used an incli- signaturesof,forexample,spiralarms,wenonethelesscallthis nation of 20 , which is the one of the stellar disk, i.e. the pho- ◦ 2Dcaseaxisymmetricbecausetheindividualcontributionsfrom tometricvalue.Thisvaluediffersfromthekinematicalvalueof darkmatter,stars,andgasaredistributedaxisymmetrically. 31 6 derivedinCheminetal.(2006).Thephotometricvalue ◦ ◦ ± Articlenumber,page3of20 A&Aproofs:manuscriptno.cheminm99 Fig.3.Leftandmiddle:ProfilesoftangentialandradialvelocityofM99.Shadedareaindicatesthe1σr.m.s.fromthefittings.Agreensolidline representsthefittingforthewholedisk,whilebluedottedandredcrossedcirclesarerespectivelytheresultsfortheapproaching andreceding sidesfittedseparately.Rightpanel:HαvelocitydispersionofM99.Symbolsrepresenttheobserveddispersion;thedashedlineisasmoothmodel oftheobservationusedtoderivetheasymmetricdrift. is more appropriate for the mass modeling than larger inclina- Theformalerrorfromthefittingsaresmallattheseradiiandre- tions(seeSection3.3).NonlinearLevenberg-Marquardtfittings mainnegligiblerelativetotheamplitudeofv . R were performed with uniformly weighted velocities. We found It is expected that the observed tangential velocity of v = (2398 0.5)kms 1andΓ = (67 0.5) .Thekinematical the kinematical tracer differs from its circular velocity be- sys − ◦ ± ± centerappearsslightlyoffsetfromthephotometriccenter(peak cause of asymmetric drift. Starting from Eq. (4.227) of of the stellar density). However that difference is not signifi- Binney&Tremaine(2008),andassumingthatrandommotions cantowingtothemeasureduncertainties(seealsoCheminetal. drivesthegaspressure,anisotropicdispersionellipsoid,andthat 2006). For simplicity, we adopt the photometric center as the the productv v is independentof z, the circular velocities, v , R z c kinematical center. These results are in very good agreement arededucedfromthetangentialmotionsby withCheminetal.(2006).TheHαsystemicvelocityagreeswell with the centroid of the integrated Hα profile from our dataset dlnρ dlnσ2 (2392 km s−1) or the value found by Chungetal. (2009) from v2c =v2θ −σ2l.o.sR dR + dRl.o.s, (2) rthiveedinvteRgarantdedvθHwiipthroafillleoothfeMrp9a9ra(m23e9te5rskfimxesd−1a)t.tThheesen,awdoepdteed- where σl.o.s is the observed l.o.s dispersion and ρ is the total values.We choseanadaptiveangularsamplingwitharingsize (atomic+molecular) mass volume density that can be replaced ofatleast2.5′′ (welllargerthantheseeingoftheobservations) bythesurfacedensityΣifwealsoassumeaconstantdiskthick- andwithaminimumnumberof30pixelsperringtoensuregood ness. The atomic and molecular gas densities are those pre- qualityfits.Withtheserules,radialbinsarefullyuncorrelated. sented in Section 3.2. The Hα velocity dispersion profile, cor- Figure3showstheresultingprofilesofv andv ,whoseval- rected from instrumental broadening, is shown in Fig. 3 and θ R uesare reportedinTab.C.1ofAppendixC.Therotationcurve listed in Appendix C. The mean dispersion is 20 km s 1 for a − is regular. It shows a peak at R 3 kpc, which roughlycorre- standarddeviationof 2kms 1,whichiscomparabletovalues − ∼ ∼ spondstotheradiusequivalentto2.2timesthestellardiskscale- foundformanyotherstar-forminggalaxiesofsimilarmorphol- length(seeSection3.2),andaflatpartatlargerradii(v 270 ogy and mass (Epinatetal. 2010; Kametal. 2015). The pro- θ kms 1).ThisrotationcurveisconsistentwiththeHαme∼asure- file variations of the density and dispersion remain too small − ments presented in Phookunetal. (1993), Kranzetal. (2001), to imply a significant gas asymmetric drift. Equation 2 yields and Cheminetal. (2006) after scaling at similar inclinations. an almost constant pressure support, well represented by an WealsoverifiedthegoodconsistencywithCOkinematicsfrom average value v v 1 km s 1. Because of this minor c θ − h − i ∼ Leroyetal.(2009).Therotationcurvesfortheapproachingand correction relatively to v , asymmetric drift is neglected here- θ recedinghalvesoftheHαdiskhavealsobeenfittedseparately. after. Such values are fully consistent with results found for Theyare asymmetricin the inner2 kpc,and around3-3.5kpc. other dwarf or massive disk galaxies (deBlok&Bosma 2002; Thisisoneofthesignaturesofthekinematicallopsidednessof Gentileetal.2007;Swatersetal.2009;Dalcanton&Stilp2010; M99. The Very Large Array (VLA) data did not allow us to Westfalletal.2011;Martinssonetal.2013). perform a reliable tilted-ring model of the Hi velocity field for R > 11.5 kpc. The scatter of the Hi kinematics is indeed too large to consider an outer Hi rotation curve, and constrain the 3.2.Stellarandgaseoussurfacedensities parameters of a disk warping, if any warp exists in M99. This Themassmodelsneedvelocitycontributionfromluminousmat- has nonetheless no consequence on the results presented here- ter.We haveconsideredindividualcontributionsfroma molec- afterbecauseHidensitiesaresmallandthegravitationalimpact ular gas disk (mol), an atomic gas disk (atom), a stellar disk oftheatomicgasdiskisnegligibleintheoverallmassbudget. (⋆,D), and a stellar bulge (⋆,B). The corresponding velocity The rotation of M99 is clockwise, assuming trailing spiral componentsarededucedfrommasssurfacedensities. arms. With this rule, positive v (negative, respectively) corre- The molecular gas disk surface densities are from CO 1–0 R spondstomotionsradiallyorientedinward(outward).Asacon- mm observations of Rahmanetal. (2011) from the Combined sequence, the fitted profile presents globally inward radial mo- Array for Research in Millimeter-wave Astronomy (CARMA) tionsinM99,exceptintheinnerR = 2.7kpc,betweenR = 4.9 array,atanangularresolutionof4.3 .TheCO0-thmomentmap ′′ kpcandR=5.9kpcandlocallyatR=3.7kpcandR=6.9kpc, has been translated to H surface densities using a conversion 2 whichareconsistentwithv directedoutward.BeyondR 10 factor of 1.8 1020 cm 2 (K km s 1) 1. The H gas mass is R − − − 2 kpc,thescatteroftheradialcomponentbecomeslarger,th∼ough 5 109 M .×The atomic gas disk surface densities come from∼ thisscatterismainlyconsistentwithavelocitydirectedoutward. th×e Very L⊙arge Array Hi survey of Virgo Cluster galaxies by Articlenumber,page4of20 LaurentCheminetal.:Asymmetricmassmodelsofdiskgalaxies-I.Messier99 Fig.4.Surfacedensitymapsfortheatomicgas,stellarandmoleculargasdisksofM99(fromlefttoright,respectively).Densities(inM pc 2) − areshowninadecimallogarithmicscale. ⊙ Chungetal.(2009),originallyfromPhookunetal.(1993),atan is1.6 .ThestellarmassofM99is 4.2 1010 M .Intotal,it ′′ angularresolutionof 25 .Usingtheadopteddistance,theHi yieldsaluminousmassof5.2 1010∼M .× ⊙ ′′ gas mass is 5 10∼9 M , which is thus the same as the H The images have been dep×rojected⊙with constant kinemati- 2 mass.Thetot∼alm×assdensi⊙tymapsfortheatomicandmolecular calparametersasfunctionofradius,usingi=20 andaposition ◦ contributionsarefinallyobtainedbymultiplyingthe HiandH angle of 67 . We set the x > 0 axis of our deprojected frame 2 ◦ densities by the usualfactor of 1.37 to take the contributionof tobecoincidentwithtothesemimajoraxisoftherecedinghalf elementsthatareheavierthanhydrogenintoaccount. of the disk. We have also assumed thatall luminousmass con- tributionsshare the same dynamicalcenter, that is, fixed at the The stellar mass surface densities are estimated using the positionofthephotometriccenter.Figure4showstheresulting methodofZibettietal.(2009),basedonthepixel-by-pixelcom- surfacedensitymaps.TheprominentspiralstructureofM99is parison of optical and near-infrared (NIR) colors with a suite observedatallangularscaleswithmoreobviousspiralpatterns of stellar populationsynthesis models. Zibettietal. (2009) and for the moleculargas and stellar components.The outer stellar Zibetti(2009)providedetailsoftheimagereductionandsignal- spiralarm(x<0)coincideswellwiththeouterspiralarmofthe to-noise enhancement via adaptive smoothing, which allows atomicgasdisk. one to extract accurate (error < 0.05 mag) surface brightness at each pixel in g, i, and H-bands, respectively, and, in turn, g i, i H colors. We compute the same colors for a set of 3.3.Luminousanddarkmattervelocitycontributions − − 150,000 composite stellar population synthesis (SPS) models with variable star formation histories (exponentially declining Thefollowingmodelisfittedtoakinematicalobservable plus random bursts), metallicities, and two-componentdust at- v2 =v2 +v2 , (3) tenuations as in Charlot&Fall (2000), following the prescrip- θ,mod DM lum tions of daCunhaetal. (2008). The models are binned in the where v the circular velocity contribution from the missing g i,i HspaceandthemedianM/LinH bandiscomputedfor DM ea−chb−in.Foranygivenpixel,themeasure−dg iandi Hcolors mass(darkmatter,orDM),andvlumthatfromthetotalluminous − − mass,givenby selectthebininthemodellibrariesandthecorrespondingM/Lis assignedtothepixel.BymultiplyingtheH-bandsurfacebright- v2 =v2 +v2 +v2 +v2 . (4) lum atom mol ⋆,D ⋆,B ness with this M/L, we obtain the stellar mass surface density. With respectto the SPS libraryadoptedin Zibettietal. (2009), Equations 3 and 4 assume tracers at centrifugal balance, theonlydifferenceisthatthebaseofsimplestellarpopulations whichisprobablynottotallyverifiedbecauseofthediskasym- (SSP)isnottheso-calledCB07versionoftheBruzual&Charlot metries.Inour1Daxisymmetricapproach,azimuthallyaveraged (2003,BC03)models,butthe2012updatedversionoftheorig- surfacedensityprofileshavebeenderivedfromtheresolvedden- inal BC03 SSPs1. In fact, in recent years, many observations sitymapsofSect.3.2.Thestellarmasssurfacedensityprofileis have shown that models with a very strong contribution by showninFig.5.Thebulgecomponenthasamassof 4.7109 ∼ TP-AGB stars (e.g., Maraston 2005, CB07) fail to reproduce M andfollowsanexponentiallawofcentraldensityandscale- the optical-NIR spectral energy distribution of galaxies in the len⊙gth(Σ ,h) (7450M pc 2,0.3kpc).Themainstellarcon- 0 − low- to intermediate-redshift Universe (e.g., Krieketal. 2010; tributioncome∼sfromadi⊙skwithascalelengthofh = 1.7kpc ⋆ Zibettietal. 2013; Melnick&DePropris 2014), while models andacentraldensityof 1800M pc 2.Thedensityprofileis − with a more moderate TP-AGB contribution (e.g. BC03) work better modeled with the∼addition o⊙f an outer truncated compo- better. This motivates our decision to opt for BC03 SSPs. The nenthavingascalelengthof 20kpcforacentraldensityof M/Lin H bandestimatedwithTP-AGBlightmodelsaretypi- 40M pc 2. ∼ ∼ − cally0.1 −0.2upto0.3dexhigherthanestimatedwithTP-AGB Th⊙ecircularvelocityderivesfromtheradialaccelerationg , R heavymo−dels(seeFig.3ofZibettietal.2009).The2012update as v2 = Rg , assuming a pressureless mass component. The c − R ofBC03introducessomeimprovementsinthetreatmentofthe circularvelocitycontributionv ofthebulgehasbeenderived ⋆,B stellarremnant,whichresultsinlargerM/Lbyroughly10%for from the bulge density assuming a spherical bulge. The circu- old stellar populations.The pixelscale of the stellar mass map lar velocity contribution of the stellar disk v has been de- ⋆,D duced from a residual density profile obtained by subtracting the bulge density to the total stellar density profile. The main 1 bruzual.org/~gbruzual/bc03/Updated_version_2012 disk and outer components are thus contained in this residual Articlenumber,page5of20 A&Aproofs:manuscriptno.cheminm99 Fig.5.Left:MasssurfacedensityprofileofstarsinM99.Abulge-diskdecompositionmodel(greensolidline)totheobservedprofile(symbols) isseen, aswellasbulge anddisk components (reddashed lines).Right: Axisymmetric massdistributionmodel of M99. Therotationcurve is representedwithfilledsymbols,anditsmodelwithanorangesolidline.Contributionsfromthestellarbulgeanddiskareshownwithreddashed lines,fromtheatomicandmoleculargasdiskswithbluedash-dottedlines,andfromthedarkmattercomponent withgreenopensymbols.The darkmattermodelisthebest-fitNFWhalowhoseparametersaregiveninTab.D.1. profile,andwedonotdifferentiatebetweenbothdiskpartshere- The velocity contribution of dark matter is assumed to be after.Theverticaldensitylawofthestellardiskcannotbemea- thatofasphericalhalo(r=R)whosecentercoincideswiththat sured directly for an almost face-ongalaxylike M99. We have of the disk of luminous matter (hereafter centered-halo case). assumed it follows a sech-squared law (vanderKruit&Searle TheDMhalomodelswefittedaretheEinastomodel(EINhere- 1981)withaconstantscaleheightof 0.35kpc.Thisvaluecor- after),thecuspymodelinferredfromcosmologicalsimulations ∼ respondsto20%oftheM99diskscalelength,followingresults (theNavarro-Frenk-Whitemodel;NFWhereafter),andthecore- foundforedge-ondisks(e.g.,Yoachim&Dalcanton2006).The dominatedmodel(pseudoisothermalsphere;PIShereafter). vertical structure of gaseous disks is less observationally con- ThemassdensityprofileoftheEinastomodel(Navarroetal. strained than for stellar disks. We considered that the gaseous 2004)isdefinedas disksarethinstructuresofnegligiblescaleheight. Figure5showstheindividualvelocityprofiles.Oneseesthat r 1/n ρ (r)=ρ exp 2n 1 . (5) tcheeptstienlltahredinisnkerdokmpcinwahteesrethietiostchoemrlpuamrainboleuwscitohntthriebubutilognes,coexn-- EIN −2 −  r−2! −  tribution.We emphasizethatusing an inclinationof,for exam- Here r is the characteristic scale radius at which the density 2 ple, 30 asinCheminetal.(2006)wouldsystematicallymake profile−hasalogarithmicslopeof 2,ρ isthescaledensityat ◦ 2 thes∼tellarcontributionoverestimatetherotationcurveinthein- thatradius,andnisadimensionles−sinde−xthatshapestheprofile. ner disk regions. Unless the stellar mass of M99 has been sig- ThecircularvelocityimpliedbytheEinastomodelis nificantly overestimated or the asymmetric drift has been sig- nificantlyunderestimated,usingalowerkinematicalinclination v2EIN(r)=4πGnr32ρ 2e2n(2n)−3nγ(3n,ξ) r−1, (6) − − than those given by Phookunetal. (1993), Kranzetal. (2001), or Cheminetal. (2006)is the bestsolutionto reconcilethe Hα where γ(3n,ξ) = 0ξe−tt3n−1dt is the incomplete gamma func- kinematicswiththephotometricinclinationandtheresultsfrom tionandξ=2n(r/rR )1/n(Cardoneetal.2005;Mamon&Łokas 2 modelsofstellarpopulationssynthesis. 2005; Retana-Mon−tenegroetal. 2012). This three-parameter The larger concentrationof molecular gas in the inner disk modelhas the flexibility to choose between a steep, intermedi- causes v to dominate over v ; the latter velocity steadily ate, or shallow density profile, depending on the value of the mol atom increases toward the outer regions and peaks at a radius larger index. At fixed characteristic density and radius, models with than the last point of the Hα rotation curve. The radius of the small(large)indicescorrespondtoshallow(steep,respectively) peakofthestellardiskcontributionR = 2.2h = 3.7kpccoin- innerdensityprofiles(Cheminetal.2011). ⋆ cideswellwiththatoftheinnerpeakoftherotationcurve.This ThedensityoftheNFWprofile(Navarroetal.1997)is confirms our finding of a small scalelength for the stellar disk ofM99.Atthisradius,ifthevelocitycontributionofthestellar r r 2 disk dominates the rotation curve at a level of more than 75% ρNFW(r)=4ρ−2 −r2 r+−r2 2! , (7) (v /v > 75%) for galaxies similar in morphologicaltype to − ⋆,D θ M99,thenthestellardiskissaidtobemaximum(Sackett1997). correspondingtoacircularvelocityprofile We measure a stellar disk velocity fraction of 67% at R = 3.7 kpc.Also, takingthebulgecontributionintoaccountleadsto a v2 (r)=v2 log(1+η) η/(1+η)/ x(log(1+c) c/(1+c)) , NFW 200 − − stellar fraction of 72% and all luminous matter gives a veloc- (cid:0) (cid:1) (cid:0) (cid:1)(8) ity fraction of 78%. The stellar disk of M99 is thus not maxi- mum,andthe luminousbaryonsarebarelymaximumwith this where x = r/r . The parameter r is the virial radius 200 200 definition. This confirms that dark matter is a significant con- derived where the density equals 200 times the critical density tribution to the total mass budgetof M99, as often observedin 3H2/(8πG) for closure of the Universe; and η = cx, where c 0 nearby spirals (e.g., Bershadyetal. 2011; Westfalletal. 2011; is the concentrationof the DM halo.We fixed the Hubblecon- Martinssonetal.2013). stant at the value found by the Planck Collaboration H = 68 0 Articlenumber,page6of20 LaurentCheminetal.:Asymmetricmassmodelsofdiskgalaxies-I.Messier99 ATOMICGAS STARS MOLECULARGAS Fig.6.Gravitationalpotential(Φ),tangential,andradialacceleration(g andg ,respectively)mapsofthedisksoftheatomic,moleculargas,and θ R stellarcomponentsofM99.Thesemapsareforthez=0kpcmidplane.Theaxisymmetriccontributionfromthebulgepotentialisnotincludedin thestellardiskcomponent. km s 1 Mpc 1 (PlanckCollaborationetal. 2015). The two pa- practice, it is the bidimensionaltangential velocity only that is − − rameters are the scale velocity v and the halo concentration modified in Eq 1, as the parameters of dark matter vary while 200 c.TheNFWhaloissaidtobecuspybecausethedensityscales fitting the observation. However, we have built two l.o.s pro- asr 1 inthecenter,whichissteepercomparedwiththedensity jections: with and without fixed radial motions. The modeling − profileofthepseudoisothermalsphere without v assumes v = 0, while that with fixed v uses the R R R axisymmetricprofilederivedinSection3.1.Modelingwithnon- r2 circularmotionsisreferredasthe2D+v casehereafter.Though ρ (r)=ρ c , (9) R PIS 0 r2+rc2 not perfect, as vR is not dynamically motivated unlike vθ, the 2D+v caseremainsasimplemethodtoestimatetheglobalim- R whichthustendstotheconstantvalueρ0inthecoreregionofthe pactofnoncircularmotionsin thedynamicalmodelingandthe halo. Thismodelis thereforereferredas a core.The parameter innershapeofthedarkmatterdensity. r isthecharacteristiccoreradiusofthehalo.Thedensityprofile c impliesacircularvelocity The SPS modeling of Zibettietal. (2009) already provide thescalingofthestellarmass.Ourmodelsonlyhavethoseofthe v2 (r)=4πGρ r2(1 r /ratan(r/r )). (10) DMcomponentasfreeparameters.Fittingswerethuscarriedout PIS 0 c − c c with52d.o.fs(9000,respectively)fortheNFWandPISmodels, EachofvEIN,vNFW,orvPIS replacesvDMinEq.3. and 51 d.o.fs (8999) for the EIN model in the 1D axisymmet- riccase (2D).As inSect. 3.1.,we useduniformweightingsfor the2Dmodels.Rotationcurvedecompositionoftenusesnormal 3.4.Massdistributionmodeling weightings that are the inverse of the squared uncertainties on Therotationcurveistheobservableforthe1Dmodeling,while rotationvelocities, ∆ . We defined∆2 as the quadraticsum of vθ vθ theHαvelocityfieldofM99hasbeenfittedinthe2Dcase.For the formal error from the rotation curve fitting with a system- that,amodeledl.o.svelocitymapisbuiltfollowingEqs1and3 atic error(halfthevelocitydifferencebetweentheapproaching withfixedkinematicalparameters,usingaxisymmetricvelocity andrecedingdiskhalves).TheerrordistributionisnotGaussian, contributionsfrom dark matter, molecules, atoms, and stars. In whichpreventsusfromusingnormalweightings.Wethusused Articlenumber,page7of20 A&Aproofs:manuscriptno.cheminm99 thenumberofpointsperradialbinastheweightingfunctionof velocities.Thoughthisisnothomogeneoustothe2Dweighting function,itturnedouttobethemostappropriatewaytoaccount fora distributionofvelocitiesin the rotationcurvecomparable tothepixeldistributioninthevelocityfield. Appendix D (Tab. D.1) reports the fitted parameters of the different halo models. The quoted parameter errors correspond to the formal 1σ error from the fittings. Both 1D and 2D fit- tingsarecorrectandyieldparametersingoodagreementwithin the errors. The 2D modeling yields more constrained parame- ters than for the 1D case. We also note the degeneracy for the halo parameters of the Einasto model. This degeneracy is par- tially raised in the 2D case, as the Einasto index is about 2.5 Fig. 7. Fraction (in %) of the perturbed potential to the unperturbed times more constrained.We base the analysis upon the Akaike potentialforluminousmatterinM99.Theredsolid(dotted)lineisfor InformationCriterion(AIC;Akaike1974)tocomparethediffer- thestellardisk(stellarbulge+disk)potential,greendashedlineforthe enthalomodels,followingCheminetal.(2011).Thecriterionis moleculargasdisk,andbluedash-dottedlinefortheatomicgasdisk. definedbyAIC=2N+χ2,whereNisthenumberofparameters tobefittedbythemodeling(N =2forNFWandPIS,N =3for EIN).SincetheAICinvolvesboththeχ2andthenumberofpa- 4.1.Methodology rameters,anAICtestisthenappropriatetocomparemodelsthat arenotnestedanddonothaveasimilarnumbersofparameters, The major difference between the axisymmetric case and the suchastheNFW,PIS,andEINforms.AnAICtestcannotbein- asymmetric approach is that we need to derive the 3D, asym- vokedtoruleoutaspecificmodel,butinsteadhelpsustodecide metric gravitational potential of luminous baryons beforehand. whichmodelismorelikelythanothers.Cheminetal.(2011)ap- WethuscomputedthepotentialinCartesiancoordinates(x,y,z), pliedthiscriteriontoanalyzerotationcurvesofdisksfromtheHi which enables us to derive both radial and azimuthal forces at NearbyGalaxy Survey(Walteretal. 2008; deBloketal. 2008) any desired z. This can be derived independentlyfor each stel- andfoundthatinmostofconfigurationstheEinastomodelisan larorgaseouscontribution.ThegravitationtalpotentialΦofthe improvementwith respectto the NFW andPIS models. As the massdistributionisinprinciplededucedfromtheconvolutionof smaller theAIC, themorelikelyonemodelwithrespectto an- thevolumemassdensitybytheGreenfunction,namely other,wethencomparedtwomodelsbyderivingthedifference betweentheir respectiveAIC (Tab.D.2).ReportingAICdiffer- Φ= G dρ′ , (11) encesexplainswhyTab.D.1doesnotlisttheχ2ofeachfitting. − $ r r | − ′| TheEinastoandNFWcuspsarefoundtobethemostlikely whereGisthegravitationalconstant.However,theGreenfunc- models, compared with the pseudoisothermal sphere. We esti- tionwrittenby1/r r =[(x x)2+(y y)2+(z z)2] 1/2,iswell ′ ′ ′ ′ − mate a DM density slope atthe firstpointofthe rotationcurve knowntodiverge|a−tea|chpoi−ntwhere−x = x,y−= y andz = z. ′ ′ ′ R = 0.27 kpc of 0.88 0.37 for the 1D case ( 0.87 0.33 This function renders any direct estimate of Φ inaccurate and − ± − ± for the 2D case) for the Einasto model, thus slightly shallower generallyencouragesmodelerstoincorporateasofteninglength thanfortheNFWcusp( 1).TheAICdifferencebetweenthe tobypassthe divergence.Here, we usethe newformalismpre- ∼ − NFWandEinastocuspsimpliesthatthethree-parametermodel sentedinHuré(2013)whoshowedthattheNewtonianpotential ismorelikely.However,itisdifficulttojudgetherealpertinence is exactly reproduced by using an intermediate scalar function of this result because of the degeneracy of the Einasto model ,namelyΦ=∂2 .In3D,thishyperpotentialiswrittenas parameters. The best-fit 1D mass model with the NFW halo is H xyH shown in Fig 5. Results for the 2D model that take the radial motionsinto accountfollow the same trends,and an innerDM H(x,y,z)=−G$ ρ(x′,y′,z′)κxy(X,Y,Z)dx′dy′dz′, (12) slopeof 0.84 0.31isdeducedfortheEINmodel.Thisslope Ω′ − ± isonlymarginallyshallowerthanforthemodelwithoutvR.The with X = x x ,Y = y y andZ = z z.Theκfunctionisa ′ ′ ′ noncircularradialmotionshavea negligibleimpacton shaping − − − hyperkerneldefinedby theDMdensityprofileofM99,atleastwithinthisaxisymmetry approach. XY X+ r r κxy(X,Y,Z)= Zatan +Yln | − ′| . (13) − Z|r−r′| √Y2+Z2 Thisapproachis particularlysimple and efficientfor 2D or 3D distributionssince is,incontrasttoΦ,theconvolutionofthe 4. DynamicalasymmetriesofM99luminousmatter surfaceorvolumedHensitywitharegular,finiteamplitudekernel. Themethodologythusdoesnotmakeuse ofa softeninglength Thebenefitfroma2Danalysisshouldbecomemoreinteresting inthederivationofthepotential.Inpractice,thisconvolutionis ifthevelocitycontributionfromluminousmattercouldstickto performedusingthesecond-ordertrapezoidalruleandthemixed theasymmetricrealityoftheluminousmatter.Theobjectiveof derivativesare estimatedat the same orderfromcenteredfinite this section is thus to describe the methodology and products differences.Furthermore,thevolumedensityofthetracerarede- oftheasymmetricapproach(gravitationalpotentials,radial,and ducedfroma surfacedensitymap,consideringthatthe vertical tangentialforces),andinparticulartheresolvedcircularvelocity density follows a sech-squared or exponential law of constant contributionofluminousmattertobeusedbythe2Dasymmetric scaleheightwithradius.Theprecisionoftheseschemesissuffi- modelingpresentedinSection6. cientforthepresentpurpose. Articlenumber,page8of20 LaurentCheminetal.:Asymmetricmassmodelsofdiskgalaxies-I.Messier99 Fig.8.Left:Circularvelocityfieldforthecontributionofluminousmatteronly.Middle:Zoomintheinner4kpcwithcontoursrepresentingthe emissionfrommoleculargas.AdashedcircleshowsthelocationofR=2.2h =3.7kpc,whereh isthestellardiskscalelength.Right:Variation ⋆ ⋆ ofvelocitywithazimuthatR=3.7kpc.Opencirclesrepresentthevelocitiesoffourpointsinside,down-andupstreamofthespiralarms,indicated withwhitefilledcirclesinthemiddlepanel.Thedashedlineistheaxisymmetricvalue.Thevelocityscaleoftheleftaxisisforluminousmatter only,whilethatoftherightaxistakesanadditionalcontributionfromdarkmatterintoaccount. Thevolumedensityofgasandstarswerederivedusingtheir inregionswhereperturbationsofthemoleculargaspotentialare respectivesurface densities(Section 3.2and Fig. 4). Similar to stronger. On overall average, we find that the gravitational po- theaxisymmetriccase,weconsideredasech-squaredlaw,using tentialisdisturbedatthelevelof 10%fortheatomicgascom- ∼ ascaleheightof0.35kpcfortheverticalvariationofthedensity ponent, 5% for molecules, and 2% for stars. The addition ∼ ∼ of the stellar disk, and that the molecular and atomic gasdisks of the bulge potential to the axisymmetric part of the potential havenegligiblescaleheights.Oncethe3Dgravitationalpotential ofthestellardiskonlymarginallymodifiestheratioofperturba- of a tracer is derived, the azimuthal and radial components of tionsforthestellarcomponent. thegravitationalaccelerationareobtainedfromgθ = θΦand Figure6alsoshowsthetangentialandradialaccelerationsin −∇ gR = −∇RΦ,respectively.Fromthe3Dproducts,wecanextract themidplane.Themostimportantresultsarethatgθ isfarfrom thegravitationalpotential,radial,andtangentialaccelerationsin negligibleandthenthatg andg arestronglyasymmetric.The R θ the galaxy midplane (z = 0 kpc). This is necessary to fit the morphologyoftheg mapsisintrinsicallylinkedtospiralstruc- R observedkinematicsofionizedgasthatisassumedtolieinthat tures in the density and potential maps, while structures in the plane. g maps differ markedly from the other density, potential, and θ radial acceleration fields. The sign of g is almost exclusively R negativeexcepton the trailing sides of the gaseousarms, prin- 4.2.Asymmetricgravitationalpotentialsandaccelerations cipally at R = 1 2.5 kpc and R = 4.5 5 kpc. Interestingly, − − these radii coincideor are very close to regionswhere the ion- Figure 6 shows the midplane potentialmaps for the stellar and izedgashasitsradialmotionoutwardlydirected(Section3.1and gaseousdiskcomponentsofM99.Asweareprimarilyinterested Fig.3).The(absolute)radialforceincreasesradiallyinthetrail- intheasymmetriccomponents,theaxisymmetricpotentialofthe ingsidesofthe(density)spiralsandreacheslocalmaxima.Once bulgeisnotrepresentedhere(seeFig.A.3foritsradialprofile). the peak of density has been met, it decreases on the leading Weonlyshowpixelsforwhichthedensityisstrictlypositivefor sides to reach local minima. As for the azimuthalacceleration, eachcomponent,asthesearethemostimportantregionsforthe itsstructureisgovernedbystrikingalternationsofpositiveand analysis.Beyondtheobservableextentofeachdisk,the poten- negativepatterns.Basically,g variesrapidlythroughthespiral tial well decreases smoothlywith radiusand hasno interesting θ arms, admitting a minimum where the density is maximum. It featuresthatdeservetobeshown. also presents a large gradient in the center of the stellar disk. Theabsolutestrengthofthepotentialisobservedtoincrease We estimate that the mean tangential acceleration is about 70, fromtheatomicgas,tothemoleculargas,andfinallytothestel- 300, and 3950 times smaller than the mean radial acceleration larcomponent.Asexpected,noneofthepotentialsshowpureax- fortheatomicgas,moleculargas,andstellardisk(respectively). ialsymmetry,astheyexhibitlopsidedandspiral-likefeatures.To Thetangentialforceisthuscompletelynegligibleovertheentire quantify the nonaxisymmetric perturbations, we have modeled diskonaverage,butthisis onlydueto the alternatingpatterns. thegravitationalpotentialsviaaseriesofharmonics,asdetailed Indeed,thisisnotthecaseanymoreonsmallscalessinceg and θ in Appendix A. We find that the stellar and gaseous potentials g havecomparableamplitudes. areindeedlopsided(m = 1mode),andexhibitspiralstructures R (m = 2 modes) and other less prominentm = 3 perturbations. We also find thatthe stellar potentialis notbarred and that the 4.3.Nonuniformcircularvelocity lopsidednessdominatestheamplitudeofthestellarperturbations in the inner disk region,while it totally dominatesthose in the On a global scale only, circular motions can almost be consid- disk of atomic gas. We derivedthe azimuthallyaveragedratios eredaspurelyaxisymmetric.However,theimplicationthatboth of total perturbed against unperturbed potentials to summarize g , 0andg areasymmetricisthatthecircularvelocitymust θ R the importanceof the perturbations.These ratios are written as be nonuniform.To quantifythe velocitynonuniformity,we de- Φ (R)cos(m(θ θ (R))/Φ (R) = Φ(R,θ) /Φ (R) 1, rivedthecircularvelocityfieldv forthetotalluminousmatter h m,0 m − m 0 iθ h iθ 0 − lum followingEq. A.1 andare shown in Fig. 7. Theyshowthat the following Eq. 4. The major difference from the axisymmetric P degree of perturbation significantly increases with decreasing approachisthateachcircularvelocitycontributionnowdepends massdensity.Thestellarpotentialispreferentiallylessdisturbed on(x,y).Thatvelocityfieldisbuiltona(x,y)-gridsimilartothe Articlenumber,page9of20 A&Aproofs:manuscriptno.cheminm99 Fig.9.Radialprofileofthenonuniformityfactorν,definedasthemaxi- mumvelocityvariationwithazimuthrelativelytotheaxisymmetricve- locity. The solid line is for total (luminous+dark) matter, blue dotted lineforstellaranddarkmatter(nogas),andreddashedlineforthetotal luminousmatter(nodarkmatter). Fig.10.AzimuthalvelocityprofilesatR=5.4and10.8kpcinthesim- ulationoftheMW-likegalaxyofKawataetal.(2014b).Filledsymbols arethetangentialvelocityofthegascomponentofthesimulatedgalaxy. Thegreensolidlineisthenonuniformcircularvelocitypredictedfrom stellar component. Velocity contributions of the gaseous disks theasymmetricmethodology,thevioletdashedlineistheuniformcir- havethusbeeninterpolatedatthenodesofthestellargrid.Fig- cularmotion. ure8showsthev fieldforthewholediskaswellasinaregion lum focusingintheinnerR=4kpc. The velocity field presents many spiral patterns, which are responds to the ν factor when the total gaseous component is mostly caused by the perturbed stellar and molecular contri- omitted.We estimate thatgasis responsibleforabouthalf ofν butions. The map admits extrema that depend on the location and the scatter of circular velocity at R = 2.2h , although the ⋆ withrespecttothespiralarm.Basically,thevelocitysharplyin- stellar disk velocity contribution is larger by 100 km s 1 than − creases on the trailing sides of the spiral arms where the den- thatoftotalgasatthisradius.IntheR=9 10kpcspiralarms, sity of stars and gas increases, then peaks at higher densities, atomicgascontributesbyupto 20%toν.−Self-gravityofgasis tosharplydecreaseontheleadingsidesofthearms.Streaming thereforenotnegligibleinM99,∼inparticular,inthedensestdisk motions observed along spiral arms of galaxies, usually iden- regions.Theimpactofgascouldhavebeenevenmoreimportant tified by wiggles in contours of l.o.s velocities, naturally find witha higherangularresolutionforthe atomicgascomponent. theirorigininthenonuniformityofcircularmotions.Examples Indeed,thelowangularresolutionofthecurrentHiobservations ofmodeledl.o.svelocityfieldwithapparentstreamingmotions hassmearedoutthesurfacedensityoftheatomicgas,whichhas are presented in Section 6. An example of highly nonuniform thenlikelypreventedusfromderivinghighervelocitycontrasts circularvelocitiesisshownintherightpanelofFig.8withthe through arm-interarm regions, such as those evidenced within azimuth-velocity diagram at R = 2.2h⋆ = 3.7 kpc. At this ra- thestellarandmolecularcomponents. dius,thestellarcontributionismaximumandtheaxisymmetric circularvelocityoftotalluminousmatteris v 207kms 1 lum − (or259kms−1 whenacontributionfrom,e.hg.,thie∼best-fitNFW 5. Comparisonwithpreviousworksandnumerical haloofthe2DaxisymmetriccaseofSection3.4isincluded).The simulations overallvariationofvelocity,65kms 1 (52kms 1 withDM),is − − verysignificant;thestandarddeviation,whichis 13kms−1(10 It is important to clarify that nonuniform is not equivalent to kms−1),issignificantaswell.Thesharpgradien∼tsinthetrailing noncircular,althoughbothphenomenahaveconnectionsasthey sidesof the spiralarmsfor azimuths101◦ to 127◦, and 294◦ to are causedby perturbedpotentials.In mostkinematicalstudies 311◦ are of 54 and 59 km s−1, respectively (44 and 47 km s−1 of disk galaxies, noncircularmotions are often only associated with dark matter included). Such nonuniformity is remarkable withasymmetries.Thisishoweveronlypartoftherealitysince consideringtheveryproximityofthepoints(e.g.,θ = 294◦ and asymmetry/nonuniformityappliesbothtononcircularandcircu- 311◦areseparatedby1.2kpconly). larmotions.Ourmethodcanonlyestimatethedegreeofnonuni- We define a nonuniformityfactor, ν, as the maximumvari- formityofcircularmotionsforM99,however.Itshouldalsobe ationofcircularvelocityrelativelytotheaxisymmetriccircular possibletoestimatethisdegreeforthenoncircular(radial)com- velocity(Fig.9).Thenonuniformityfactorisimportantinthein- ponentwithmoreadvanceddynamicalmodeling,butthisisbe- nerdiskregions.Asruleofthumb,itexceeds10%forR.2.2h . yondthescopeofthearticle. ⋆ withamaximumof 30%( 40%withoutDM)atR=2.5kpc. Also, the literature usually refers to circular as the axisym- ∼ ∼ ThisfactorislessstrongintheouterspiralarmatR 10kpc,up metricvalueoftherotationalmotiononly,whiledeparturesfrom ∼ toalevelof 9%(25%withoutDM).ForR > 12kpc,bumps that mean value are often called noncircular. This circularity- ∼ can be identified as caused by the m = 1,2,3 perturbationsin axisymmetryassociationturnsouttobeinadequatesincedepar- the gravitational potential of the atomic gas. Here, however, ν turesfromtheaxisymmetricvelocities,suchasthose identified smoothlydecreasesbecauseofthedominantaxisymmetriccon- inSection4.3,directlystemfromtheradialforceasanaturalre- tribution of dark matter. A close inspection of the v , v , sponseto perturbedpotentials.Thedesignationnonuniformfor atom mol and v maps reveals the important contribution of gas in the such departuresis thus more appropriate.The interesting point ⋆,D nonuniformity,thus in generating velocity wiggles in the M99 for circular motions in M99 is that nonuniformity is the rule spiral arms. This effect is shown with the dotted line that cor- rather than exception, whereas axisymmetry is rarely expected Articlenumber,page10of20

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Astronomy&Astrophysicsmanuscriptno.cheminm99 cESO2016 (cid:13) January20,2016 Asymmetric mass models of disk galaxies - I. Messier 99 LaurentChemin1,2,Jean-Marc Huré1,2,CarolineSoubiran1,2,StefanoZibetti3,StéphaneCharlot4,andDaisuke Kawata5 1 Univ.Bordeaux,LAB,UMR5804,F-33270,Floirac,Francee-mail:[email protected] 2 CNRS,LAB,UMR5804,F-33270,Floirac,France 3 INAF-OsservatorioAstrofisicodiArcetri,LargoEnricoFermi5,I-50125Firenze,Italy 4 Institutd’AstrophysiquedeParis,CNRS&UniversitéPierre&MarieCurie(UMR7095),98bisBdArago,F-75014Paris,France 6 5 MullardSpaceScienceLaboratory,UniversityCollegeLondon.Dorking,Surrey,RH56NT,UnitedKingdom 1 ReceivedJanuary20,2016;acceptedXX 0 2 n ABSTRACT a J Massmodelsof galacticdiskstraditionallyrelyonaxisymmetricdensityand rotationcurves, paradoxically actingasiftheirmost remarkableasymmetricfeatures,suchaslopsidednessorspiralarms,werenotimportant.Inthisarticle,werelaxtheaxisymmetry 9 approximationandintroduceamethodology thatderives3Dgravitationalpotentialsofdisk-likeobjectsandrobustlyestimatesthe 1 impactsofasymmetriesoncircularvelocitiesinthediskmidplane.Massdistributionmodelscanthenbedirectlyfittedtoasymmetric line-of-sightvelocityfields.Appliedtothegrand-designspiralM99,thenewstrategyshowsthatcircularvelocitiesarehighlynonuni- ] A form,particularlyintheinnerdiskofthegalaxy,asanaturalresponsetotheperturbedgravitationalpotentialofluminousmatter.A cuspyinnerdensityprofileofdarkmatterisfoundinM99,intheusualcasewhereluminousanddarkmattersharethesamecenter. G Theimpactofthevelocitynonuniformityistomaketheinnerprofilelesssteep,althoughthedensityremainscuspy.Onanotherhand, . amodelwherethehaloiscoredominatedandshiftedby2.2-2.5kpcfromtheluminousmasscenterismoreappropriatetoexplain h mostofthekinematicallopsidednessevidencedinthevelocityfieldofM99.However,thegravitationalpotentialofluminousbaryons p isnotasymmetricenoughtoexplainthekinematicallopsidednessoftheinnermostregions,irrespectiveofthedensityshapeofdark - o matter.Thisdiscrepancypointsoutthenecessityofanadditionaldynamicalprocessintheseregions:possiblyalopsideddistribution r ofdarkmatter. t s Keywords. galaxies:kinematicsanddynamics–galaxies:spiral–galaxies:structure–galaxies:individual(Messier99,NGC4254) a [ 21. Introduction case, Franxetal. (1994) and Schoenmakersetal. (1997) initi- v atedthederivationofhigh-orderharmonicswithfirst-orderkine- 1Rotationcurvesandsurfacedensityprofilesofgalacticdisksare matical componentsfrom gaseous velocity fields. They argued 0 the observational pillars most models of extragalactic dynam- that kinematicalFourier coefficientsare useful to constrain de- 6 ics are based on. Rotation curves are needed to constrain the viationsfromaxisymmetryandthe natureofdynamicalpertur- 1 0total mass distribution, the parameters of dark matter haloes, bations. Using that technique, Gentileetal. (2005) concluded, .or the characteristics of modified Newtonian dynamics, while for instance, that the kinematical asymmetries in the Hi veloc- 1surface density profiles are helpful to constrain the structural ityfield ofa dwarfdiskpresentinga core-dominateddarkmat- 0 parameters of disks and bulges, and generate the velocity con- terhalo(DDO47)couldlikelyoriginatefromaspiralstructure. 6 tributions of luminous matter essential to mass models. As the However,theiramplitudeswerenothighenoughtoaccountfor 1 :density and rotationvelocity profilesare axisymmetricby con- thevelocitydifferenceexpectedbetweentheCDMcuspandthe vstruction,massmodelsimplicitlyassumethattherotationalve- cored halo preferred by the rotation curve fittings. In the sec- Xilocityisonlymadeofuniformcircularmotions.Thoughattrac- ondcase,Spekkens&Sellwood(2007)proposedtofitabisym- tiveforitssimplicity,thisapproachremainsareductiveexploita- metricmodelofbar-like/ovaldistortiontotheHαvelocityfield r ationofvelocityfieldsandmultiwavelengthimagesofstellarand ofanotherlow-massspiralgalaxy(NGC2976)andarguedthat gaseousdisks, which are informationrich. In particular,it pre- negligible high-order Fourier motions in velocity fields do not vents one from measuring the rotational support through per- necessarily imply that the bisymmetric perturbation is negligi- turbations(spiralarms,lopsidedness,etc.),whichareobviously ble, and that the rotation curve should be similar to the under- themoststrikingfeaturesofgalacticdisks.Inaneraofconflict lyingcircularmotionsonlyifthedeparturesfromcircularityre- between observationsand expectationsfrom Cold Dark Matter main small. These authors also showed that the inner slope of (CDM) simulations, the cusp-core controversy (see the review therotationcurveofNGC2976islikelyaffectedbythebar-like ofdeBlok2010,andreferencestherein,butseeGovernatoetal. perturbation.Numericalsimulationsofbarreddisksarrivedata (2010)), it appeared fundamental to assess the impact of such similarconclusionabouttheimpactofthebarontheinnershape perturbationson the shape of rotation curves, and more gener- of rotation curves (Valenzuelaetal. 2007; Dicaireetal. 2008). allyonmassmodelsanddensityprofilesofdarkmatter. Randriamampandryetal. (2015) performed numerical simula- This is the reason why efforts have been made to deter- tionstodeterminea correctedrotationcurveforanotherbarred mine the kinematical asymmetries inferred by perturbations or galaxy (NGC 3319), free from the perturbing motions induced to model the effects of dynamical perturbations. In the former by the bar. While these simulations demonstrate it is possible Articlenumber,page1of20 A&Aproofs:manuscriptno.cheminm99 disksofM99.Themeaningoftheseasymmetricoutputsisdis- cussed in Section 5. We then perform asymmetric mass distri- bution models of M99 (Section 6), and compare them to the axisymmetry-based models of Section 3. The conclusions for M99andtheprospectsofournewstrategyforgalacticdynamics arefinallygiveninSection7. 2. Thegrand-designspiralgalaxyMessier99 2.1.Abriefpresentation We selected the galaxy Messier 99 (M99 hereafter) because it is a SAc type disk harboring a very prominent spiral struc- ture (Fig. 1). Located in the Virgo Cluster (adopted distance of 17.1 Mpc, Freedmanetal. 1994), observing M99 is a good Fig. 1. Composite SDSS gri-image of the grand-design spiral galaxy opportunitytobenefitfromhigh-sensitivityandhigh-resolution M99.Northisup;eastisleft.Theimagesizeis7 6.3. multiwavelengthobservationsofthestellardiskandinterstellar ′ ′ × medium. The integrated Hα profile from data of Cheminetal. (2006) has a width at 20% of the maximum Hα peak of to hide cuspier DM distributions into artificial cored distribu- 216 km s 1. Combined with a small disk inclination (20 , − ◦ tions under the effect of stellar bars, they perfectly illustrate Makarovetal. 2014), this implies a massive galaxy with most the difficulty of performing mass models and constraining the ofrotationvelocitiesgreaterthan245kms 1.Thismakesitan − shapeofdarkmatterdensityprofilesfromobservationsofbarred idealtargetto study the structure and kinematicsof the disk in galaxies. Other numerical models based on closed-loop orbits detailandtotestournewmassmodelingstrategy. showed that shallow kinematics and core-like haloes could ac- The spiral structure is asymmetric. A one-arm mode domi- tually be explainedby cuspy triaxial distributionsof dark mat- natestheHidisk(Phookunetal.1993;Chungetal.2009),while terviewedwithparticularprojectionangles(Hayashi&Navarro thestellardistributionexhibitsmorethanonesinglearm(Fig.1). 2006).Finally,otherstudiesfocusedonextractinginHispectra Phookunetal. (1993) proposeda scenario where the Hi arm is the line-of-sight (l.o.s) velocity components supposed to trace triggeredbygasinfallingandwindingonthedisk,resultingfrom the axisymmetric rotational velocities better than the compo- atidalencounter.Basedonnumericalsimulations,Vollmeretal. nents based on more usual intensity-weightedmeans (Ohetal. (2005)mimictheasymmetricdiskandperturbedHikinematics 2008).Appliedtotwodarkmatterdominateddisks,NGC2366 by a flyby of a massive companion,coupledwith ram pressure andIC2574,whichareprototypesofgalaxieswhosedarkmatter strippingfromtheVirgointraclustermedium(ICM).Theseau- densityconflictswiththecosmologicalcusp,thismethodyielded thorsarguedthatM99 is entering the Virgo cluster for the first steeper rotation curves in the inner disk regions. However, the time.FurthernumericalmodelsofDuc&Bournaud(2008)also velocity differences with the intensity-weighted mean velocity explainedtheoriginofthelarge-scaleHitail,whichapparently curve were not sufficient to reconcile the observation with the connects M99 to VIRGOHI21, a 108 M Hi cloud wandering CDMcusp. in the Virgo ICM (Minchinetal. 2007),⊙by a tidal interaction In this context, in this article we propose a new approach thatwaswithanothercandidatecompanionthaninVollmeretal. to model the mass distribution of disk galaxies. Our strategy (2005). In principle, mass distribution models should only be goesbeyondthedecompositionofrotationcurvesandfullyex- carried out with targets supposedly in dynamical equilibrium. ploitsthebidimensionaldistributionofluminousmatter,thusthe The possibly infalling Hi mass of 108 M only represents asymmetric nature of stellar and gaseous disks. Our approach 2%ofthetotalneutralgasmassof∼M99,and⊙ about0.2%ofth∼e determinesthe 3D gravitationalpotentialof anydisk-like mass total luminousmass (Section 3). This should be not enough to componentthroughhyperpotentialstheorizedbyHuré(2013).It affectthelarge-scaledynamicsofM99.Furthermore,rampres- thenderivesthecorrespondingcircularvelocitymapinthedisk sure stripping predominantly affects the outskirts of the neu- midplane,whichallowsustodeterminewhereandtowhichex- tralatomicgascomponent,nottheinner,densestregionsofthe tent the circular motions should deviate from axisymmetry. A gaseousdiskofM99. 2Dmassdistributionmodelcanthenbedirectlyfittedtoa l.o.s velocityfield,byaddingthe2Dvelocitycontributionsfromlu- 2.2.High-resolutionHαkinematics minous baryons to that from the missing matter. The impacts ofthevelocityasymmetriesonthemassmodelsandstructureof Optical long-slit and resolved observations also revealed per- darkmatterhaloescanthenbeeninvestigatedbycomparingwith turbedkinematicsoftheionizedgasdiskofM99(Phookunetal. resultsobtainedwiththeaxisymmetricmassmodels. 1993; Kranzetal. 2001; Cheminetal. 2006). In particular, Weapplythatmethodologytoaprototypeofunbarred,spi- Kranzetal. (2001) arguesfor the presenceof a possible stellar ralgalaxyMessier99,whosegeneralpropertiesarepresentedin bar that would impact the Hα kinematics in the inner 1.5 kpc. Section2.Theaxisymmetricmassmodelofthehigh-resolution Asymmetricmotionshavealsobeenclearlyevidencedalongthe rotationcurveofM99isdetailedinSection3.ThisSectionalso Hα spiral arms (Phookunetal. 1993; Cheminetal. 2006), as presentsamoreelaboratedaxisymmetricmassmodel,fitteddi- wellasalopsidedHαvelocityfield(Cheminetal.2005b).The rectlytoahigh-resolutionvelocityfieldofM99.Thebasisofthe kinematicaldatawe usetomodelthemassdistributionofM99 derivation of the 3D gravitational potentials for luminous mat- are from the 3D spectroscopy survey of Virgo cluster galaxies ter is described in Section 4, which also presents the inferred byCheminetal. (2005a,2006). Intheir catalogofHαvelocity gravitationalpotentials,accelerations,andcircularvelocitiesfor fieldsofbrightVirgospiralandirregulardisks,theFabry-Perot the contributions from the stellar, atomic , and molecular gas observationsofM99haveangularandspectralsamplingsof1.6 ′′ Articlenumber,page2of20 LaurentCheminetal.:Asymmetricmassmodelsofdiskgalaxies-I.Messier99 Fig.2.Hαintegratedemission,velocityfield,andvelocitydispersionmapofM99(fromlefttoright,respectively). (130 pc at the adopted distance) and 8.2 km s 1. The Hα This Section is thereforeorganizedas a traditionalaxisym- − ∼ velocityfieldofCheminetal.(2006)resultedfromanadaptive metric mass model of a galactic disk. We first derive the rota- binningand a spatial interpolationof the Hα datacube,follow- tion curve of M99 and estimate the asymmetric drift contribu- ing prescriptions given in Daigleetal. (2006). We did not use tion,andthenwepresentmultiwavelengthobservationsneeded these interpolated data but an improved version of the binned to infer the mass surface density profiles and the velocity con- datacubeofCheminetal.(2006),wherenewbinsarenowmade tributionsofthestellarandinterstellarmatter.Theresultsofthe ofauniquepixellocatedatthebarycenterofinitialbins.Every massmodelingarethendiscussedtoinvestigatewhichdarkmat- binhasthusanequalangularsize,inaccordancewiththekine- terhaloshapeismoreappropriate.Inaddition,wediscussthe2D matical analysis of many other Hα Fabry-Perot velocity fields axisymmetricmodeling. (Epinatetal. 2008a,b). Figure 2 shows the revisited integrated Hα emission, velocity, and dispersion velocity fields of M99. 3.1.TangentialandradialvelocitiesinM99 We restrictedtheHαkinematicsto R = 11.5kpc,whichcorre- spondstoadistributionof9002velocitypixels.Thisallowedus As the mass distribution modeling first needs a rotation curve, toderiveaccuratevelocityandvelocitydispersionprofiles.Be- the3Dvelocityspace(v ,v ,v )isdeducedfromfittingtotheHα yondtheopticalsizeofthestellardisk,theHikinematicsistoo velocityfieldofM99theRfoθllozwingmodel: scatteredtoinferusefulkinematicalinformation. v =v +(v cosθ+v sinθ)sini+v cosi, (1) l.o.s sys θ R z 3. AxisymmetricmassmodelingofM99 where v is the l.o.s velocity, v and i are the systemic l.o.s sys Atraditionalaxisymmetricmassmodelconsistsindecomposing velocity and the inclination of the M99 disk, and θ is the az- arotationcurveintocontributionsfromluminousbaryons(stel- imuthalangle in the deprojectedorbit.In the axisymmetricap- lar and gaseous disks, bulge, etc.) and dark matter. The fitting proach, these velocities are assumed uniform, i.e. only depen- procedure yields fundamental scale parameters for the hidden denton R. Theazimuthalvelocityis the rotationcurve.The v R masscomponent,andeventuallyafactorthatenablesthescaling and v components are usually omitted in kinematical studies z ofluminositiesintosurfacedensitiesforstars(themass-to-light becausetheyaregenerallyassumedtobenegligible.Asseenbe- ratio, M/L). This section details the rotation curve decomposi- low,itisthecaseforv butsinceoneofourgoalsistostudythe R tion,whichwerefertoasthe1Daxisymmetriccasehereafter,as impactoftheradialmotionsonthemassmodeling,wedecided the rotation velocity only dependson one angular scale, which to fit them. Derivingv with or withoutv doesnotimpactthe θ R isthegalactocentricradius. shape or amplitude of the tangentialcomponent.Then, a prob- ThisSectionalsoenvisagesasecondstrategy,wheretheax- lemin leavingv free inEq. 1is thatitshouldexhibitartificial z isymmetric modeling is carried out directly from the velocity variationsifthegalaxyhasakinematicallopsidedness.Therea- field. The motivation for this is, first, that circular motions are son for this is that the systemic velocity must be naturally im- expectedto dominatethe kinematicsand, since rotation curves pactedbyalopsidedness(seeAppendixB),butsincevariations stemfromvelocityfields,onemaylogicallyperformmassmod- ofv arenotallowedhere,thev cositermabsorbsthekinemat- sys z elswith2Dresolveddatainsteadof1Dcurves.Thesecondmoti- icalsignatureoflopsidedness.Asboththegravitationalpotential vationisthatthedevelopmentofapipelinethatmakesfittingsin ofluminousmatterandtheHαkinematicsofM99arelopsided 2Dismandatoryforthenewasymmetricmethodologydiscussed (Section 4.2 and Appendix B), the best solution to avoid such fromSections4to6.Themostnaturalwaytounderstandtheim- artificialv variationsis thusto assume v = 0 hereafter. Other z z pactoftheasymmetriesonthemassmodelsis,thus,tofittothe face-on,grand-designspiralsofsimilarstarformationactivityto HαvelocityfieldofM992Dvelocitymodelsbuiltunderaxisym- M99 are knownto have negligibleverticalmotions(e.g.,NGC metryassumptions.Inshort,onecannotdirectlycomparethere- 628,Kamphuis&Briggs1992). sults from the decomposition of the rotation curve with those Since theHα gasiswell confinedinthe opticaldisk thatis presented in Sect. 6 for the asymmetric fittings of the velocity notwarped,wehavenotallowedv ,i,thepositionangleofthe sys field. One needs an intermediate step between them, which we diskmajoraxis(Γ)andthecoordinatesofthedynamicalcenter refertoasthe2Daxisymmetriccasehereafter.Thoughthedata tovarywithradius.Wechoseθ=0 alignedwiththesemimajor ◦ to be fitted are not totally axisymmetric because of the evident axis of the receding side of the galaxy disk. We used an incli- signaturesof,forexample,spiralarms,wenonethelesscallthis nation of 20 , which is the one of the stellar disk, i.e. the pho- ◦ 2Dcaseaxisymmetricbecausetheindividualcontributionsfrom tometricvalue.Thisvaluediffersfromthekinematicalvalueof darkmatter,stars,andgasaredistributedaxisymmetrically. 31 6 derivedinCheminetal.(2006).Thephotometricvalue ◦ ◦ ± Articlenumber,page3of20 A&Aproofs:manuscriptno.cheminm99 Fig.3.Leftandmiddle:ProfilesoftangentialandradialvelocityofM99.Shadedareaindicatesthe1σr.m.s.fromthefittings.Agreensolidline representsthefittingforthewholedisk,whilebluedottedandredcrossedcirclesarerespectivelytheresultsfortheapproaching andreceding sidesfittedseparately.Rightpanel:HαvelocitydispersionofM99.Symbolsrepresenttheobserveddispersion;thedashedlineisasmoothmodel oftheobservationusedtoderivetheasymmetricdrift. is more appropriate for the mass modeling than larger inclina- Theformalerrorfromthefittingsaresmallattheseradiiandre- tions(seeSection3.3).NonlinearLevenberg-Marquardtfittings mainnegligiblerelativetotheamplitudeofv . R were performed with uniformly weighted velocities. We found It is expected that the observed tangential velocity of v = (2398 0.5)kms 1andΓ = (67 0.5) .Thekinematical the kinematical tracer differs from its circular velocity be- sys − ◦ ± ± centerappearsslightlyoffsetfromthephotometriccenter(peak cause of asymmetric drift. Starting from Eq. (4.227) of of the stellar density). However that difference is not signifi- Binney&Tremaine(2008),andassumingthatrandommotions cantowingtothemeasureduncertainties(seealsoCheminetal. drivesthegaspressure,anisotropicdispersionellipsoid,andthat 2006). For simplicity, we adopt the photometric center as the the productv v is independentof z, the circular velocities, v , R z c kinematical center. These results are in very good agreement arededucedfromthetangentialmotionsby withCheminetal.(2006).TheHαsystemicvelocityagreeswell with the centroid of the integrated Hα profile from our dataset dlnρ dlnσ2 (2392 km s−1) or the value found by Chungetal. (2009) from v2c =v2θ −σ2l.o.sR dR + dRl.o.s, (2) rthiveedinvteRgarantdedvθHwiipthroafillleoothfeMrp9a9ra(m23e9te5rskfimxesd−1a)t.tThheesen,awdoepdteed- where σl.o.s is the observed l.o.s dispersion and ρ is the total values.We choseanadaptiveangularsamplingwitharingsize (atomic+molecular) mass volume density that can be replaced ofatleast2.5′′ (welllargerthantheseeingoftheobservations) bythesurfacedensityΣifwealsoassumeaconstantdiskthick- andwithaminimumnumberof30pixelsperringtoensuregood ness. The atomic and molecular gas densities are those pre- qualityfits.Withtheserules,radialbinsarefullyuncorrelated. sented in Section 3.2. The Hα velocity dispersion profile, cor- Figure3showstheresultingprofilesofv andv ,whoseval- rected from instrumental broadening, is shown in Fig. 3 and θ R uesare reportedinTab.C.1ofAppendixC.Therotationcurve listed in Appendix C. The mean dispersion is 20 km s 1 for a − is regular. It shows a peak at R 3 kpc, which roughlycorre- standarddeviationof 2kms 1,whichiscomparabletovalues − ∼ ∼ spondstotheradiusequivalentto2.2timesthestellardiskscale- foundformanyotherstar-forminggalaxiesofsimilarmorphol- length(seeSection3.2),andaflatpartatlargerradii(v 270 ogy and mass (Epinatetal. 2010; Kametal. 2015). The pro- θ kms 1).ThisrotationcurveisconsistentwiththeHαme∼asure- file variations of the density and dispersion remain too small − ments presented in Phookunetal. (1993), Kranzetal. (2001), to imply a significant gas asymmetric drift. Equation 2 yields and Cheminetal. (2006) after scaling at similar inclinations. an almost constant pressure support, well represented by an WealsoverifiedthegoodconsistencywithCOkinematicsfrom average value v v 1 km s 1. Because of this minor c θ − h − i ∼ Leroyetal.(2009).Therotationcurvesfortheapproachingand correction relatively to v , asymmetric drift is neglected here- θ recedinghalvesoftheHαdiskhavealsobeenfittedseparately. after. Such values are fully consistent with results found for Theyare asymmetricin the inner2 kpc,and around3-3.5kpc. other dwarf or massive disk galaxies (deBlok&Bosma 2002; Thisisoneofthesignaturesofthekinematicallopsidednessof Gentileetal.2007;Swatersetal.2009;Dalcanton&Stilp2010; M99. The Very Large Array (VLA) data did not allow us to Westfalletal.2011;Martinssonetal.2013). perform a reliable tilted-ring model of the Hi velocity field for R > 11.5 kpc. The scatter of the Hi kinematics is indeed too large to consider an outer Hi rotation curve, and constrain the 3.2.Stellarandgaseoussurfacedensities parameters of a disk warping, if any warp exists in M99. This Themassmodelsneedvelocitycontributionfromluminousmat- has nonetheless no consequence on the results presented here- ter.We haveconsideredindividualcontributionsfroma molec- afterbecauseHidensitiesaresmallandthegravitationalimpact ular gas disk (mol), an atomic gas disk (atom), a stellar disk oftheatomicgasdiskisnegligibleintheoverallmassbudget. (⋆,D), and a stellar bulge (⋆,B). The corresponding velocity The rotation of M99 is clockwise, assuming trailing spiral componentsarededucedfrommasssurfacedensities. arms. With this rule, positive v (negative, respectively) corre- The molecular gas disk surface densities are from CO 1–0 R spondstomotionsradiallyorientedinward(outward).Asacon- mm observations of Rahmanetal. (2011) from the Combined sequence, the fitted profile presents globally inward radial mo- Array for Research in Millimeter-wave Astronomy (CARMA) tionsinM99,exceptintheinnerR = 2.7kpc,betweenR = 4.9 array,atanangularresolutionof4.3 .TheCO0-thmomentmap ′′ kpcandR=5.9kpcandlocallyatR=3.7kpcandR=6.9kpc, has been translated to H surface densities using a conversion 2 whichareconsistentwithv directedoutward.BeyondR 10 factor of 1.8 1020 cm 2 (K km s 1) 1. The H gas mass is R − − − 2 kpc,thescatteroftheradialcomponentbecomeslarger,th∼ough 5 109 M .×The atomic gas disk surface densities come from∼ thisscatterismainlyconsistentwithavelocitydirectedoutward. th×e Very L⊙arge Array Hi survey of Virgo Cluster galaxies by Articlenumber,page4of20 LaurentCheminetal.:Asymmetricmassmodelsofdiskgalaxies-I.Messier99 Fig.4.Surfacedensitymapsfortheatomicgas,stellarandmoleculargasdisksofM99(fromlefttoright,respectively).Densities(inM pc 2) − areshowninadecimallogarithmicscale. ⊙ Chungetal.(2009),originallyfromPhookunetal.(1993),atan is1.6 .ThestellarmassofM99is 4.2 1010 M .Intotal,it ′′ angularresolutionof 25 .Usingtheadopteddistance,theHi yieldsaluminousmassof5.2 1010∼M .× ⊙ ′′ gas mass is 5 10∼9 M , which is thus the same as the H The images have been dep×rojected⊙with constant kinemati- 2 mass.Thetot∼alm×assdensi⊙tymapsfortheatomicandmolecular calparametersasfunctionofradius,usingi=20 andaposition ◦ contributionsarefinallyobtainedbymultiplyingthe HiandH angle of 67 . We set the x > 0 axis of our deprojected frame 2 ◦ densities by the usualfactor of 1.37 to take the contributionof tobecoincidentwithtothesemimajoraxisoftherecedinghalf elementsthatareheavierthanhydrogenintoaccount. of the disk. We have also assumed thatall luminousmass con- tributionsshare the same dynamicalcenter, that is, fixed at the The stellar mass surface densities are estimated using the positionofthephotometriccenter.Figure4showstheresulting methodofZibettietal.(2009),basedonthepixel-by-pixelcom- surfacedensitymaps.TheprominentspiralstructureofM99is parison of optical and near-infrared (NIR) colors with a suite observedatallangularscaleswithmoreobviousspiralpatterns of stellar populationsynthesis models. Zibettietal. (2009) and for the moleculargas and stellar components.The outer stellar Zibetti(2009)providedetailsoftheimagereductionandsignal- spiralarm(x<0)coincideswellwiththeouterspiralarmofthe to-noise enhancement via adaptive smoothing, which allows atomicgasdisk. one to extract accurate (error < 0.05 mag) surface brightness at each pixel in g, i, and H-bands, respectively, and, in turn, g i, i H colors. We compute the same colors for a set of 3.3.Luminousanddarkmattervelocitycontributions − − 150,000 composite stellar population synthesis (SPS) models with variable star formation histories (exponentially declining Thefollowingmodelisfittedtoakinematicalobservable plus random bursts), metallicities, and two-componentdust at- v2 =v2 +v2 , (3) tenuations as in Charlot&Fall (2000), following the prescrip- θ,mod DM lum tions of daCunhaetal. (2008). The models are binned in the where v the circular velocity contribution from the missing g i,i HspaceandthemedianM/LinH bandiscomputedfor DM ea−chb−in.Foranygivenpixel,themeasure−dg iandi Hcolors mass(darkmatter,orDM),andvlumthatfromthetotalluminous − − mass,givenby selectthebininthemodellibrariesandthecorrespondingM/Lis assignedtothepixel.BymultiplyingtheH-bandsurfacebright- v2 =v2 +v2 +v2 +v2 . (4) lum atom mol ⋆,D ⋆,B ness with this M/L, we obtain the stellar mass surface density. With respectto the SPS libraryadoptedin Zibettietal. (2009), Equations 3 and 4 assume tracers at centrifugal balance, theonlydifferenceisthatthebaseofsimplestellarpopulations whichisprobablynottotallyverifiedbecauseofthediskasym- (SSP)isnottheso-calledCB07versionoftheBruzual&Charlot metries.Inour1Daxisymmetricapproach,azimuthallyaveraged (2003,BC03)models,butthe2012updatedversionoftheorig- surfacedensityprofileshavebeenderivedfromtheresolvedden- inal BC03 SSPs1. In fact, in recent years, many observations sitymapsofSect.3.2.Thestellarmasssurfacedensityprofileis have shown that models with a very strong contribution by showninFig.5.Thebulgecomponenthasamassof 4.7109 ∼ TP-AGB stars (e.g., Maraston 2005, CB07) fail to reproduce M andfollowsanexponentiallawofcentraldensityandscale- the optical-NIR spectral energy distribution of galaxies in the len⊙gth(Σ ,h) (7450M pc 2,0.3kpc).Themainstellarcon- 0 − low- to intermediate-redshift Universe (e.g., Krieketal. 2010; tributioncome∼sfromadi⊙skwithascalelengthofh = 1.7kpc ⋆ Zibettietal. 2013; Melnick&DePropris 2014), while models andacentraldensityof 1800M pc 2.Thedensityprofileis − with a more moderate TP-AGB contribution (e.g. BC03) work better modeled with the∼addition o⊙f an outer truncated compo- better. This motivates our decision to opt for BC03 SSPs. The nenthavingascalelengthof 20kpcforacentraldensityof M/Lin H bandestimatedwithTP-AGBlightmodelsaretypi- 40M pc 2. ∼ ∼ − cally0.1 −0.2upto0.3dexhigherthanestimatedwithTP-AGB Th⊙ecircularvelocityderivesfromtheradialaccelerationg , R heavymo−dels(seeFig.3ofZibettietal.2009).The2012update as v2 = Rg , assuming a pressureless mass component. The c − R ofBC03introducessomeimprovementsinthetreatmentofthe circularvelocitycontributionv ofthebulgehasbeenderived ⋆,B stellarremnant,whichresultsinlargerM/Lbyroughly10%for from the bulge density assuming a spherical bulge. The circu- old stellar populations.The pixelscale of the stellar mass map lar velocity contribution of the stellar disk v has been de- ⋆,D duced from a residual density profile obtained by subtracting the bulge density to the total stellar density profile. The main 1 bruzual.org/~gbruzual/bc03/Updated_version_2012 disk and outer components are thus contained in this residual Articlenumber,page5of20 A&Aproofs:manuscriptno.cheminm99 Fig.5.Left:MasssurfacedensityprofileofstarsinM99.Abulge-diskdecompositionmodel(greensolidline)totheobservedprofile(symbols) isseen, aswellasbulge anddisk components (reddashed lines).Right: Axisymmetric massdistributionmodel of M99. Therotationcurve is representedwithfilledsymbols,anditsmodelwithanorangesolidline.Contributionsfromthestellarbulgeanddiskareshownwithreddashed lines,fromtheatomicandmoleculargasdiskswithbluedash-dottedlines,andfromthedarkmattercomponent withgreenopensymbols.The darkmattermodelisthebest-fitNFWhalowhoseparametersaregiveninTab.D.1. profile,andwedonotdifferentiatebetweenbothdiskpartshere- The velocity contribution of dark matter is assumed to be after.Theverticaldensitylawofthestellardiskcannotbemea- thatofasphericalhalo(r=R)whosecentercoincideswiththat sured directly for an almost face-ongalaxylike M99. We have of the disk of luminous matter (hereafter centered-halo case). assumed it follows a sech-squared law (vanderKruit&Searle TheDMhalomodelswefittedaretheEinastomodel(EINhere- 1981)withaconstantscaleheightof 0.35kpc.Thisvaluecor- after),thecuspymodelinferredfromcosmologicalsimulations ∼ respondsto20%oftheM99diskscalelength,followingresults (theNavarro-Frenk-Whitemodel;NFWhereafter),andthecore- foundforedge-ondisks(e.g.,Yoachim&Dalcanton2006).The dominatedmodel(pseudoisothermalsphere;PIShereafter). vertical structure of gaseous disks is less observationally con- ThemassdensityprofileoftheEinastomodel(Navarroetal. strained than for stellar disks. We considered that the gaseous 2004)isdefinedas disksarethinstructuresofnegligiblescaleheight. Figure5showstheindividualvelocityprofiles.Oneseesthat r 1/n ρ (r)=ρ exp 2n 1 . (5) tcheeptstienlltahredinisnkerdokmpcinwahteesrethietiostchoemrlpuamrainboleuwscitohntthriebubutilognes,coexn-- EIN −2 −  r−2! −  tribution.We emphasizethatusing an inclinationof,for exam- Here r is the characteristic scale radius at which the density 2 ple, 30 asinCheminetal.(2006)wouldsystematicallymake profile−hasalogarithmicslopeof 2,ρ isthescaledensityat ◦ 2 thes∼tellarcontributionoverestimatetherotationcurveinthein- thatradius,andnisadimensionles−sinde−xthatshapestheprofile. ner disk regions. Unless the stellar mass of M99 has been sig- ThecircularvelocityimpliedbytheEinastomodelis nificantly overestimated or the asymmetric drift has been sig- nificantlyunderestimated,usingalowerkinematicalinclination v2EIN(r)=4πGnr32ρ 2e2n(2n)−3nγ(3n,ξ) r−1, (6) − − than those given by Phookunetal. (1993), Kranzetal. (2001), or Cheminetal. (2006)is the bestsolutionto reconcilethe Hα where γ(3n,ξ) = 0ξe−tt3n−1dt is the incomplete gamma func- kinematicswiththephotometricinclinationandtheresultsfrom tionandξ=2n(r/rR )1/n(Cardoneetal.2005;Mamon&Łokas 2 modelsofstellarpopulationssynthesis. 2005; Retana-Mon−tenegroetal. 2012). This three-parameter The larger concentrationof molecular gas in the inner disk modelhas the flexibility to choose between a steep, intermedi- causes v to dominate over v ; the latter velocity steadily ate, or shallow density profile, depending on the value of the mol atom increases toward the outer regions and peaks at a radius larger index. At fixed characteristic density and radius, models with than the last point of the Hα rotation curve. The radius of the small(large)indicescorrespondtoshallow(steep,respectively) peakofthestellardiskcontributionR = 2.2h = 3.7kpccoin- innerdensityprofiles(Cheminetal.2011). ⋆ cideswellwiththatoftheinnerpeakoftherotationcurve.This ThedensityoftheNFWprofile(Navarroetal.1997)is confirms our finding of a small scalelength for the stellar disk ofM99.Atthisradius,ifthevelocitycontributionofthestellar r r 2 disk dominates the rotation curve at a level of more than 75% ρNFW(r)=4ρ−2 −r2 r+−r2 2! , (7) (v /v > 75%) for galaxies similar in morphologicaltype to − ⋆,D θ M99,thenthestellardiskissaidtobemaximum(Sackett1997). correspondingtoacircularvelocityprofile We measure a stellar disk velocity fraction of 67% at R = 3.7 kpc.Also, takingthebulgecontributionintoaccountleadsto a v2 (r)=v2 log(1+η) η/(1+η)/ x(log(1+c) c/(1+c)) , NFW 200 − − stellar fraction of 72% and all luminous matter gives a veloc- (cid:0) (cid:1) (cid:0) (cid:1)(8) ity fraction of 78%. The stellar disk of M99 is thus not maxi- mum,andthe luminousbaryonsarebarelymaximumwith this where x = r/r . The parameter r is the virial radius 200 200 definition. This confirms that dark matter is a significant con- derived where the density equals 200 times the critical density tribution to the total mass budgetof M99, as often observedin 3H2/(8πG) for closure of the Universe; and η = cx, where c 0 nearby spirals (e.g., Bershadyetal. 2011; Westfalletal. 2011; is the concentrationof the DM halo.We fixed the Hubblecon- Martinssonetal.2013). stant at the value found by the Planck Collaboration H = 68 0 Articlenumber,page6of20 LaurentCheminetal.:Asymmetricmassmodelsofdiskgalaxies-I.Messier99 ATOMICGAS STARS MOLECULARGAS Fig.6.Gravitationalpotential(Φ),tangential,andradialacceleration(g andg ,respectively)mapsofthedisksoftheatomic,moleculargas,and θ R stellarcomponentsofM99.Thesemapsareforthez=0kpcmidplane.Theaxisymmetriccontributionfromthebulgepotentialisnotincludedin thestellardiskcomponent. km s 1 Mpc 1 (PlanckCollaborationetal. 2015). The two pa- practice, it is the bidimensionaltangential velocity only that is − − rameters are the scale velocity v and the halo concentration modified in Eq 1, as the parameters of dark matter vary while 200 c.TheNFWhaloissaidtobecuspybecausethedensityscales fitting the observation. However, we have built two l.o.s pro- asr 1 inthecenter,whichissteepercomparedwiththedensity jections: with and without fixed radial motions. The modeling − profileofthepseudoisothermalsphere without v assumes v = 0, while that with fixed v uses the R R R axisymmetricprofilederivedinSection3.1.Modelingwithnon- r2 circularmotionsisreferredasthe2D+v casehereafter.Though ρ (r)=ρ c , (9) R PIS 0 r2+rc2 not perfect, as vR is not dynamically motivated unlike vθ, the 2D+v caseremainsasimplemethodtoestimatetheglobalim- R whichthustendstotheconstantvalueρ0inthecoreregionofthe pactofnoncircularmotionsin thedynamicalmodelingandthe halo. Thismodelis thereforereferredas a core.The parameter innershapeofthedarkmatterdensity. r isthecharacteristiccoreradiusofthehalo.Thedensityprofile c impliesacircularvelocity The SPS modeling of Zibettietal. (2009) already provide thescalingofthestellarmass.Ourmodelsonlyhavethoseofthe v2 (r)=4πGρ r2(1 r /ratan(r/r )). (10) DMcomponentasfreeparameters.Fittingswerethuscarriedout PIS 0 c − c c with52d.o.fs(9000,respectively)fortheNFWandPISmodels, EachofvEIN,vNFW,orvPIS replacesvDMinEq.3. and 51 d.o.fs (8999) for the EIN model in the 1D axisymmet- riccase (2D).As inSect. 3.1.,we useduniformweightingsfor the2Dmodels.Rotationcurvedecompositionoftenusesnormal 3.4.Massdistributionmodeling weightings that are the inverse of the squared uncertainties on Therotationcurveistheobservableforthe1Dmodeling,while rotationvelocities, ∆ . We defined∆2 as the quadraticsum of vθ vθ theHαvelocityfieldofM99hasbeenfittedinthe2Dcase.For the formal error from the rotation curve fitting with a system- that,amodeledl.o.svelocitymapisbuiltfollowingEqs1and3 atic error(halfthevelocitydifferencebetweentheapproaching withfixedkinematicalparameters,usingaxisymmetricvelocity andrecedingdiskhalves).TheerrordistributionisnotGaussian, contributionsfrom dark matter, molecules, atoms, and stars. In whichpreventsusfromusingnormalweightings.Wethusused Articlenumber,page7of20 A&Aproofs:manuscriptno.cheminm99 thenumberofpointsperradialbinastheweightingfunctionof velocities.Thoughthisisnothomogeneoustothe2Dweighting function,itturnedouttobethemostappropriatewaytoaccount fora distributionofvelocitiesin the rotationcurvecomparable tothepixeldistributioninthevelocityfield. Appendix D (Tab. D.1) reports the fitted parameters of the different halo models. The quoted parameter errors correspond to the formal 1σ error from the fittings. Both 1D and 2D fit- tingsarecorrectandyieldparametersingoodagreementwithin the errors. The 2D modeling yields more constrained parame- ters than for the 1D case. We also note the degeneracy for the halo parameters of the Einasto model. This degeneracy is par- tially raised in the 2D case, as the Einasto index is about 2.5 Fig. 7. Fraction (in %) of the perturbed potential to the unperturbed times more constrained.We base the analysis upon the Akaike potentialforluminousmatterinM99.Theredsolid(dotted)lineisfor InformationCriterion(AIC;Akaike1974)tocomparethediffer- thestellardisk(stellarbulge+disk)potential,greendashedlineforthe enthalomodels,followingCheminetal.(2011).Thecriterionis moleculargasdisk,andbluedash-dottedlinefortheatomicgasdisk. definedbyAIC=2N+χ2,whereNisthenumberofparameters tobefittedbythemodeling(N =2forNFWandPIS,N =3for EIN).SincetheAICinvolvesboththeχ2andthenumberofpa- 4.1.Methodology rameters,anAICtestisthenappropriatetocomparemodelsthat arenotnestedanddonothaveasimilarnumbersofparameters, The major difference between the axisymmetric case and the suchastheNFW,PIS,andEINforms.AnAICtestcannotbein- asymmetric approach is that we need to derive the 3D, asym- vokedtoruleoutaspecificmodel,butinsteadhelpsustodecide metric gravitational potential of luminous baryons beforehand. whichmodelismorelikelythanothers.Cheminetal.(2011)ap- WethuscomputedthepotentialinCartesiancoordinates(x,y,z), pliedthiscriteriontoanalyzerotationcurvesofdisksfromtheHi which enables us to derive both radial and azimuthal forces at NearbyGalaxy Survey(Walteretal. 2008; deBloketal. 2008) any desired z. This can be derived independentlyfor each stel- andfoundthatinmostofconfigurationstheEinastomodelisan larorgaseouscontribution.ThegravitationtalpotentialΦofthe improvementwith respectto the NFW andPIS models. As the massdistributionisinprinciplededucedfromtheconvolutionof smaller theAIC, themorelikelyonemodelwithrespectto an- thevolumemassdensitybytheGreenfunction,namely other,wethencomparedtwomodelsbyderivingthedifference betweentheir respectiveAIC (Tab.D.2).ReportingAICdiffer- Φ= G dρ′ , (11) encesexplainswhyTab.D.1doesnotlisttheχ2ofeachfitting. − $ r r | − ′| TheEinastoandNFWcuspsarefoundtobethemostlikely whereGisthegravitationalconstant.However,theGreenfunc- models, compared with the pseudoisothermal sphere. We esti- tionwrittenby1/r r =[(x x)2+(y y)2+(z z)2] 1/2,iswell ′ ′ ′ ′ − mate a DM density slope atthe firstpointofthe rotationcurve knowntodiverge|a−tea|chpoi−ntwhere−x = x,y−= y andz = z. ′ ′ ′ R = 0.27 kpc of 0.88 0.37 for the 1D case ( 0.87 0.33 This function renders any direct estimate of Φ inaccurate and − ± − ± for the 2D case) for the Einasto model, thus slightly shallower generallyencouragesmodelerstoincorporateasofteninglength thanfortheNFWcusp( 1).TheAICdifferencebetweenthe tobypassthe divergence.Here, we usethe newformalismpre- ∼ − NFWandEinastocuspsimpliesthatthethree-parametermodel sentedinHuré(2013)whoshowedthattheNewtonianpotential ismorelikely.However,itisdifficulttojudgetherealpertinence is exactly reproduced by using an intermediate scalar function of this result because of the degeneracy of the Einasto model ,namelyΦ=∂2 .In3D,thishyperpotentialiswrittenas parameters. The best-fit 1D mass model with the NFW halo is H xyH shown in Fig 5. Results for the 2D model that take the radial motionsinto accountfollow the same trends,and an innerDM H(x,y,z)=−G$ ρ(x′,y′,z′)κxy(X,Y,Z)dx′dy′dz′, (12) slopeof 0.84 0.31isdeducedfortheEINmodel.Thisslope Ω′ − ± isonlymarginallyshallowerthanforthemodelwithoutvR.The with X = x x ,Y = y y andZ = z z.Theκfunctionisa ′ ′ ′ noncircularradialmotionshavea negligibleimpacton shaping − − − hyperkerneldefinedby theDMdensityprofileofM99,atleastwithinthisaxisymmetry approach. XY X+ r r κxy(X,Y,Z)= Zatan +Yln | − ′| . (13) − Z|r−r′| √Y2+Z2 Thisapproachis particularlysimple and efficientfor 2D or 3D distributionssince is,incontrasttoΦ,theconvolutionofthe 4. DynamicalasymmetriesofM99luminousmatter surfaceorvolumedHensitywitharegular,finiteamplitudekernel. Themethodologythusdoesnotmakeuse ofa softeninglength Thebenefitfroma2Danalysisshouldbecomemoreinteresting inthederivationofthepotential.Inpractice,thisconvolutionis ifthevelocitycontributionfromluminousmattercouldstickto performedusingthesecond-ordertrapezoidalruleandthemixed theasymmetricrealityoftheluminousmatter.Theobjectiveof derivativesare estimatedat the same orderfromcenteredfinite this section is thus to describe the methodology and products differences.Furthermore,thevolumedensityofthetracerarede- oftheasymmetricapproach(gravitationalpotentials,radial,and ducedfroma surfacedensitymap,consideringthatthe vertical tangentialforces),andinparticulartheresolvedcircularvelocity density follows a sech-squared or exponential law of constant contributionofluminousmattertobeusedbythe2Dasymmetric scaleheightwithradius.Theprecisionoftheseschemesissuffi- modelingpresentedinSection6. cientforthepresentpurpose. Articlenumber,page8of20 LaurentCheminetal.:Asymmetricmassmodelsofdiskgalaxies-I.Messier99 Fig.8.Left:Circularvelocityfieldforthecontributionofluminousmatteronly.Middle:Zoomintheinner4kpcwithcontoursrepresentingthe emissionfrommoleculargas.AdashedcircleshowsthelocationofR=2.2h =3.7kpc,whereh isthestellardiskscalelength.Right:Variation ⋆ ⋆ ofvelocitywithazimuthatR=3.7kpc.Opencirclesrepresentthevelocitiesoffourpointsinside,down-andupstreamofthespiralarms,indicated withwhitefilledcirclesinthemiddlepanel.Thedashedlineistheaxisymmetricvalue.Thevelocityscaleoftheleftaxisisforluminousmatter only,whilethatoftherightaxistakesanadditionalcontributionfromdarkmatterintoaccount. Thevolumedensityofgasandstarswerederivedusingtheir inregionswhereperturbationsofthemoleculargaspotentialare respectivesurface densities(Section 3.2and Fig. 4). Similar to stronger. On overall average, we find that the gravitational po- theaxisymmetriccase,weconsideredasech-squaredlaw,using tentialisdisturbedatthelevelof 10%fortheatomicgascom- ∼ ascaleheightof0.35kpcfortheverticalvariationofthedensity ponent, 5% for molecules, and 2% for stars. The addition ∼ ∼ of the stellar disk, and that the molecular and atomic gasdisks of the bulge potential to the axisymmetric part of the potential havenegligiblescaleheights.Oncethe3Dgravitationalpotential ofthestellardiskonlymarginallymodifiestheratioofperturba- of a tracer is derived, the azimuthal and radial components of tionsforthestellarcomponent. thegravitationalaccelerationareobtainedfromgθ = θΦand Figure6alsoshowsthetangentialandradialaccelerationsin −∇ gR = −∇RΦ,respectively.Fromthe3Dproducts,wecanextract themidplane.Themostimportantresultsarethatgθ isfarfrom thegravitationalpotential,radial,andtangentialaccelerationsin negligibleandthenthatg andg arestronglyasymmetric.The R θ the galaxy midplane (z = 0 kpc). This is necessary to fit the morphologyoftheg mapsisintrinsicallylinkedtospiralstruc- R observedkinematicsofionizedgasthatisassumedtolieinthat tures in the density and potential maps, while structures in the plane. g maps differ markedly from the other density, potential, and θ radial acceleration fields. The sign of g is almost exclusively R negativeexcepton the trailing sides of the gaseousarms, prin- 4.2.Asymmetricgravitationalpotentialsandaccelerations cipally at R = 1 2.5 kpc and R = 4.5 5 kpc. Interestingly, − − these radii coincideor are very close to regionswhere the ion- Figure 6 shows the midplane potentialmaps for the stellar and izedgashasitsradialmotionoutwardlydirected(Section3.1and gaseousdiskcomponentsofM99.Asweareprimarilyinterested Fig.3).The(absolute)radialforceincreasesradiallyinthetrail- intheasymmetriccomponents,theaxisymmetricpotentialofthe ingsidesofthe(density)spiralsandreacheslocalmaxima.Once bulgeisnotrepresentedhere(seeFig.A.3foritsradialprofile). the peak of density has been met, it decreases on the leading Weonlyshowpixelsforwhichthedensityisstrictlypositivefor sides to reach local minima. As for the azimuthalacceleration, eachcomponent,asthesearethemostimportantregionsforthe itsstructureisgovernedbystrikingalternationsofpositiveand analysis.Beyondtheobservableextentofeachdisk,the poten- negativepatterns.Basically,g variesrapidlythroughthespiral tial well decreases smoothlywith radiusand hasno interesting θ arms, admitting a minimum where the density is maximum. It featuresthatdeservetobeshown. also presents a large gradient in the center of the stellar disk. Theabsolutestrengthofthepotentialisobservedtoincrease We estimate that the mean tangential acceleration is about 70, fromtheatomicgas,tothemoleculargas,andfinallytothestel- 300, and 3950 times smaller than the mean radial acceleration larcomponent.Asexpected,noneofthepotentialsshowpureax- fortheatomicgas,moleculargas,andstellardisk(respectively). ialsymmetry,astheyexhibitlopsidedandspiral-likefeatures.To Thetangentialforceisthuscompletelynegligibleovertheentire quantify the nonaxisymmetric perturbations, we have modeled diskonaverage,butthisis onlydueto the alternatingpatterns. thegravitationalpotentialsviaaseriesofharmonics,asdetailed Indeed,thisisnotthecaseanymoreonsmallscalessinceg and θ in Appendix A. We find that the stellar and gaseous potentials g havecomparableamplitudes. areindeedlopsided(m = 1mode),andexhibitspiralstructures R (m = 2 modes) and other less prominentm = 3 perturbations. We also find thatthe stellar potentialis notbarred and that the 4.3.Nonuniformcircularvelocity lopsidednessdominatestheamplitudeofthestellarperturbations in the inner disk region,while it totally dominatesthose in the On a global scale only, circular motions can almost be consid- disk of atomic gas. We derivedthe azimuthallyaveragedratios eredaspurelyaxisymmetric.However,theimplicationthatboth of total perturbed against unperturbed potentials to summarize g , 0andg areasymmetricisthatthecircularvelocitymust θ R the importanceof the perturbations.These ratios are written as be nonuniform.To quantifythe velocitynonuniformity,we de- Φ (R)cos(m(θ θ (R))/Φ (R) = Φ(R,θ) /Φ (R) 1, rivedthecircularvelocityfieldv forthetotalluminousmatter h m,0 m − m 0 iθ h iθ 0 − lum followingEq. A.1 andare shown in Fig. 7. Theyshowthat the following Eq. 4. The major difference from the axisymmetric P degree of perturbation significantly increases with decreasing approachisthateachcircularvelocitycontributionnowdepends massdensity.Thestellarpotentialispreferentiallylessdisturbed on(x,y).Thatvelocityfieldisbuiltona(x,y)-gridsimilartothe Articlenumber,page9of20 A&Aproofs:manuscriptno.cheminm99 Fig.9.Radialprofileofthenonuniformityfactorν,definedasthemaxi- mumvelocityvariationwithazimuthrelativelytotheaxisymmetricve- locity. The solid line is for total (luminous+dark) matter, blue dotted lineforstellaranddarkmatter(nogas),andreddashedlineforthetotal luminousmatter(nodarkmatter). Fig.10.AzimuthalvelocityprofilesatR=5.4and10.8kpcinthesim- ulationoftheMW-likegalaxyofKawataetal.(2014b).Filledsymbols arethetangentialvelocityofthegascomponentofthesimulatedgalaxy. Thegreensolidlineisthenonuniformcircularvelocitypredictedfrom stellar component. Velocity contributions of the gaseous disks theasymmetricmethodology,thevioletdashedlineistheuniformcir- havethusbeeninterpolatedatthenodesofthestellargrid.Fig- cularmotion. ure8showsthev fieldforthewholediskaswellasinaregion lum focusingintheinnerR=4kpc. The velocity field presents many spiral patterns, which are responds to the ν factor when the total gaseous component is mostly caused by the perturbed stellar and molecular contri- omitted.We estimate thatgasis responsibleforabouthalf ofν butions. The map admits extrema that depend on the location and the scatter of circular velocity at R = 2.2h , although the ⋆ withrespecttothespiralarm.Basically,thevelocitysharplyin- stellar disk velocity contribution is larger by 100 km s 1 than − creases on the trailing sides of the spiral arms where the den- thatoftotalgasatthisradius.IntheR=9 10kpcspiralarms, sity of stars and gas increases, then peaks at higher densities, atomicgascontributesbyupto 20%toν.−Self-gravityofgasis tosharplydecreaseontheleadingsidesofthearms.Streaming thereforenotnegligibleinM99,∼inparticular,inthedensestdisk motions observed along spiral arms of galaxies, usually iden- regions.Theimpactofgascouldhavebeenevenmoreimportant tified by wiggles in contours of l.o.s velocities, naturally find witha higherangularresolutionforthe atomicgascomponent. theirorigininthenonuniformityofcircularmotions.Examples Indeed,thelowangularresolutionofthecurrentHiobservations ofmodeledl.o.svelocityfieldwithapparentstreamingmotions hassmearedoutthesurfacedensityoftheatomicgas,whichhas are presented in Section 6. An example of highly nonuniform thenlikelypreventedusfromderivinghighervelocitycontrasts circularvelocitiesisshownintherightpanelofFig.8withthe through arm-interarm regions, such as those evidenced within azimuth-velocity diagram at R = 2.2h⋆ = 3.7 kpc. At this ra- thestellarandmolecularcomponents. dius,thestellarcontributionismaximumandtheaxisymmetric circularvelocityoftotalluminousmatteris v 207kms 1 lum − (or259kms−1 whenacontributionfrom,e.hg.,thie∼best-fitNFW 5. Comparisonwithpreviousworksandnumerical haloofthe2DaxisymmetriccaseofSection3.4isincluded).The simulations overallvariationofvelocity,65kms 1 (52kms 1 withDM),is − − verysignificant;thestandarddeviation,whichis 13kms−1(10 It is important to clarify that nonuniform is not equivalent to kms−1),issignificantaswell.Thesharpgradien∼tsinthetrailing noncircular,althoughbothphenomenahaveconnectionsasthey sidesof the spiralarmsfor azimuths101◦ to 127◦, and 294◦ to are causedby perturbedpotentials.In mostkinematicalstudies 311◦ are of 54 and 59 km s−1, respectively (44 and 47 km s−1 of disk galaxies, noncircularmotions are often only associated with dark matter included). Such nonuniformity is remarkable withasymmetries.Thisishoweveronlypartoftherealitysince consideringtheveryproximityofthepoints(e.g.,θ = 294◦ and asymmetry/nonuniformityappliesbothtononcircularandcircu- 311◦areseparatedby1.2kpconly). larmotions.Ourmethodcanonlyestimatethedegreeofnonuni- We define a nonuniformityfactor, ν, as the maximumvari- formityofcircularmotionsforM99,however.Itshouldalsobe ationofcircularvelocityrelativelytotheaxisymmetriccircular possibletoestimatethisdegreeforthenoncircular(radial)com- velocity(Fig.9).Thenonuniformityfactorisimportantinthein- ponentwithmoreadvanceddynamicalmodeling,butthisisbe- nerdiskregions.Asruleofthumb,itexceeds10%forR.2.2h . yondthescopeofthearticle. ⋆ withamaximumof 30%( 40%withoutDM)atR=2.5kpc. Also, the literature usually refers to circular as the axisym- ∼ ∼ ThisfactorislessstrongintheouterspiralarmatR 10kpc,up metricvalueoftherotationalmotiononly,whiledeparturesfrom ∼ toalevelof 9%(25%withoutDM).ForR > 12kpc,bumps that mean value are often called noncircular. This circularity- ∼ can be identified as caused by the m = 1,2,3 perturbationsin axisymmetryassociationturnsouttobeinadequatesincedepar- the gravitational potential of the atomic gas. Here, however, ν turesfromtheaxisymmetricvelocities,suchasthose identified smoothlydecreasesbecauseofthedominantaxisymmetriccon- inSection4.3,directlystemfromtheradialforceasanaturalre- tribution of dark matter. A close inspection of the v , v , sponseto perturbedpotentials.Thedesignationnonuniformfor atom mol and v maps reveals the important contribution of gas in the such departuresis thus more appropriate.The interesting point ⋆,D nonuniformity,thus in generating velocity wiggles in the M99 for circular motions in M99 is that nonuniformity is the rule spiral arms. This effect is shown with the dotted line that cor- rather than exception, whereas axisymmetry is rarely expected Articlenumber,page10of20

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