Magnetically Aligned Velocity Anisotropy in the Taurus Molecular Cloud

MolecularCloud

MarkHeyer1,HaoGong1,2,EveOstriker2,ChristopherBrunt1,3

ABSTRACT

VelocityanisotropyinducedbyMHDturbulenceisinvestigatedusingcom-putationalsimulationsandmolecularlineobservationsoftheTaurusmolecularcloud.Anewanalysismethodispresentedtoevaluatethedegreeandangleofvelocityanisotropyusingspectroscopicimagingdataofinterstellarclouds.TheefficacyofthismethodisdemonstratedonmodelobservationsderivedfromthreedimensionalvelocityanddensityfieldsfromthesetofnumericalMHDsimula-tionsthatspanarangeofmagneticfieldstrengths.Theanalysisisappliedto12

COJ=1-0imagingofasub-fieldwithintheTaurusmolecularcloud.Velocityanisotropyisidentifiedthatisalignedwithin∼10◦ofthemeanlocalmagneticfielddirectionderivedfromopticalpolarizationmeasurements.Estimatedval-uesofthefieldstrengthbasedonvelocityanisotropyareconsistentwithresultsfromothermethods.WhencombinedwithnewcolumndensitymeasurementsforTaurus,ourmagneticfieldstrengthestimateindicatesthattheenvelopeofthecloudismagneticallysubcritical.TheseobservationsfavorstrongMHDturbu-lencewithinthelowdensity,sub-critical,moleculargassubstrateoftheTauruscloud.

Subjectheadings:ISM:clouds–ISM:magneticfields–ISM:kinematicsanddynamics–ISM:individual(TaurusMolecularCloud)–physicaldataandpro-cesses:MHD;methods:dataanalysis

1.Introduction

Dense,interstellarmolecularcloudsofferauniqueandvaluablelaboratorytoinvestigatemagneto-turbulentphenomena.Thesecloudsareexpectedtobefullyturbulentsystemswith

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averylargedynamicrangebetweendrivinganddissipationscales.Thedegreeofmagneticcouplingtotheturbulentflowshasimportantimplicationsforthenatureofgasdynamicsandstarformationwithinmolecularclouds.Astrong,wellcoupledfieldcanaffectthestarformationefficiencyinacloudbyreducingtheamountofmaterialthatissusceptibletogravitationalcollapseandstarformation,andalsoaffectthescaleatwhichcollapseoccurs(Mouschovias1976;Vazquez-Semadenietal2005).Magneticfieldsalsostronglyaffectthedegreeofgasdensitycompressioninshocks.Suchshock-generateddensityperturbationsmayprovidetheseedsofprotostellarcoresandprotoclusters.Giventhepotentialimpactofthemagneticfieldonthegasdynamicsofmolecularclouds,itisimperativetomeasure(orestimate)magneticfieldstrengthsandtodevelopaccuratedescriptionsofmagnetohydrody-namic(MHD)turbulenceunderconditionsapplicableinstar-formingclouds.

GoldreichandSridhar(1995,hereafterGS95)developedatheoryforstrong,incompress-ible,MHDturbulencethatprovidesdefinitivepredictionsofthespectrumandanisotropyofvelocityfields.Wave-waveinteractionsareexpectedtosheartheAlfv´enwavepacketintheplaneperpendiculartothemeanfield.Correspondingly,waveenergyismoreefficientlyredistributedtosmallerscalesinthedirectionperpendiculartothefieldthanthroughthecascadeparalleltothefield.GS95proposethatacriticalbalanceisachievedbetweennon-linearinteractionsandwavepropagation,suchthatthetimescalestotransferenergyalongthetwodirectionsarecomparable,

λ󰀄/vA∼λ⊥/v

(1)

whereλ󰀄andλ⊥arethewavelengthsparallelandperpendiculartothemeanfieldandvisthemeanvelocityfluctuationatthescaleofthecorrespondingcomponent.Foranenergy-conservingcascade,v∝λ⊥1/3,soequation(1)implies

λ󰀄∝λ⊥2/3

(2)

Thecorrespondingvelocityscalinglawalongthemagneticfieldisv∝λ󰀄1/2.AcriticallybalancedAlfv´eniccascadeleadstoascale-dependentanisotropyofthevelocityfield.ThisanisotropyhasbeendemonstratedwithcomputationalsimulationsforbothincompressibleandcompressibleMHDturbulence(e.g.Maron&Goldreich2001;Cho,Lazarian,&Vishniac2002;Vestuto,Ostriker,&Stone2003).

CanMHDinducedvelocityanisotropy,aspredictedbyGS95,bemeasuredininterstellarclouds?Watsonetal(2004)andWiebe&Watson(2007)haveattributedthepolarizationpropertiesofbothOHmasersandthermalmolecularlineemissiontodirectionallydependentopticaldepthsinducedbyMHDturbulence.Morepanoramicobservationalviewsofthegasdynamicsrelyonspectroscopicimagingdataofatomicormolecularlineemission,mostnotably,theHI21cmline,andthelowrotationaltransitionsof12COanditsisotopomers,

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CO,andC18O.Inprinciple,thespatialvariationoflineshapesandvelocitydisplacementsofferaproxyviewoftheprevailingclouddynamics.Recoveringtheformofthevelocitypowerspectrumoritsequivalentstructurefunctionfromthespectroscopicdatacubes,T(x,y,v),ischallenging,owingtothecomplexintegrationofthevelocityanddensityfieldsalongthelineofsightandtheeffectsoflineexcitationandopacitythatmayfilterormaskdynamicalinformationfromsomefractionofthevolume(Brunt&MacLow2004;Ossenkopfetal2006).PrincipalComponentAnalysis(PCA)isapowerfulmethodtoexaminespectroscopicimagingdataofinterstellarclouds.Itreordersthedataontoasetofeigenfunctionsandeigenimages(Heyer&Schloerb1997;Brunt&Heyer2002).Characteristicvelocitydif-ferences,δv,andspatialscales,τ,arederivedforeachprincipalcomponentforwhichthesignalvarianceisdistinguishedfromthestatisticalnoiseofthedata.Thesetofδv,τpointscanbeempiricallylinkedtothetruevelocitystructurefunctionusingmodelvelocityanddensityfields(Brunt&Heyer2002;Bruntetal2003).Themethodhasbeenappliedtoalargesetof12COand13COimagingobservationsofgiantmolecularcloudslocatedwithin4kpcoftheSuntoestablishtheuniversalityofturbulencewithinthemolecularinterstel-larmedium(Brunt2003;Heyer&Brunt2004).However,thesestudiesdidnotconsidervelocityanisotropy.Theeigenvectorswerederivedfromthecovariancematrixthatwasac-cumulatedfromallspectrawithinthedatacubewithnoorientationconstraints.Therefore,anydynamicalsignatureofanisotropyalongagivenaxiswasnecessarilydilutedbyisotropiccontributionstothecovariancematrix.Thecorrespondingeigenimagesidentifiedlocationswithintheprojectedplanewherevelocitydifferencescanoccurbutatanyangle.

Inthispaper,wedescribeamodifiedapplicationofPCAonspectroscopicimagingdatatorecoverstructurefunctionsalongperpendicularaxes(§2).In§3,theutilityofthisanalysisisdemonstratedonspectroscopicdatacubesderivedfrommodelvelocityanddensityfieldsfromdecayingMHDsimulationscoveringarangeofmagneticfieldstrengths.In§4,weapplythisanalysisto12COJ=1-0observationsofasub-fieldwithintheTaurusMolecularCloudtoshowthatsuchanisotropyispresentandthatthedegreeofanisotropyprovidesacoarseestimatetothestrengthofthemagneticfieldinthisregion.

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2.DescriptionofAnalysisMethod:AxisConstrainedCovarianceMatrix

Toexaminethedegreeofvelocityanisotropyininterstellarclouds,wehavemadeasimplemodificationtotheapplicationofPrincipalComponentAnalysis.Adirectionalcon-straintisimposedontheeigenvectorsbycalculatingthecovariancematrixfromthesequenceofspectraalongonespatialaxis(position-velocityslicesofthedatacube).Theposition-velocityimagecanbeextractedonesliceatatimetopreservespatialresolutionorcanbe

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generatedbyaveragingcontiguousslicestoincreasethesignaltonoiseratio.Foragivendatacube,T(x,y,v),withdimensionsnx,ny,nv,theposition-velocityslicealongthexdirection,averagedoverthickness∆intheydirection,is

Wy(x,v)=

1

nx

nx󰀁i=1

W(xi,vk)W(xi,vl),(4)

(suppressingtheysubscriptonW).Theeigenvalueequationissolvedforthisaxis-constrainedcovariancematrix,

Cxux=λxux(5)toproducethesetofnveigenvectors,ux(v),thatdescribevelocitydifferencesexclusively

alongthisparticularposition-velocityslice.Tospatiallyisolatewherethesedifferencesoccurforeachcomponent,thespectraareprojectedontothecorrespondingeigenvector,

Ix(xi)=

nv󰀁k=1

W(xi,vk)ux(vk)(6)

Theeigenprojection,Ix(x),hasdimensionsnx×1.Thecharacteristicvelocitydifference,δvx,andscale,τx,aredeterminedfromthescalelengthofthenormalizedautocorrelationfunctionsofux(v)andIx(x)respectively(Brunt&Heyer2002).Typically,only4to5(δv,τ)pairscanbeextractedfromtheaxisconstrainedeigenvectorsandprojectionsforagivenposition-velocitysliceowingtothelimitedspatialdynamicrange.Thesestepsarerepeatedforallposition-velocityslices(j1=1,1+∆,1+2∆,...,ny−∆)inthedatacubetoproduceacompositesetof(δvx,τx)pairsderivedfromny/∆setsofeigenvectorsandeigenprojections.Similarly,toexaminestructurealongtheyaxis,thesestepsareappliedtoposition-velocityslicesalongtheydirectionaveragedoverx−thickness∆=i2−i1+1,

Wx(y,v)=

1

ny

ny󰀁j=1

W(yj,vk)W(yj,vl)(8)

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Acorrespondingsetof(δvy,τy)pairsarederivedfromnx/∆setsofeigenvectors,uy(v),andeigenprojections,Iy(y).Foreachaxis,weconsolidatetheτvaluesintoonepixelwidebinsandcalculatethemeanandstandarddeviationofδvvaluesforeachbin.Powerlawsarefittoeachsettoderivearelationshipbetweenthemagnitudeofvelocitydifferencesinthelineprofilesandscaleoverwhichthesedifferencesoccur,whenconstrainedtoeachaxis,

αx

<δvx>=v◦,xτxαy

<δvy>=v◦,yτy.

(9a)(9b)

ThePCAscalingexponents,αx,αy,areempiricallylinkedtothescalingexponentsofthe

firstordervelocitystructurefunction

γ=1.69α−0.α≤0.67γ=0.93α−0.03α>0.67

(10a)(10b)

γ

γx

(Bruntetal2003).Thefirst-orderstructurefunctions,δvx=v◦,xτxandδvy=v◦,yτyyprovideequivalentinformationtothepowerspectrumofthevelocityfieldalongthekxandkyaxes,respectively,forkz=0.Averagingover∆iny(orinx)isequivalenttointegratingalongky(orkx).

Thismethodoffersatooltoderivevelocitystructurefunctionsalonganytwoperpen-dicular(spatial)axesofaspectroscopicdatacube.Withaprioriknowledgeofthelocalmagneticfielddirection,onecouldsimplyrotatethedatacubetoalignthex-axisalongthisdirectionanddeterminetheparallelandperpendicularstructurefunctions.However,thisorientationmaynotnecessarilycorrespondtotheangleatwhichvelocityanisotropyislargest.AmorerigoroustestofMHD-inducedanisotropyisthedemonstrationthatvelocityanisotropyismaximizedwhenoneofthetwoorthogonalaxesliesalongthelocalmagneticfielddirection.Todeterminetheangleofmaximumanisotropy,θMAX,thespectroscopicdatacubeisrotatedthroughasequenceofangles,θ,intheplaneoftheskyfromwhichthexandy-axisstructurefunctionsarecalculatedforeachangle.Tocomparewithpolarizationobservationsthatmeasurepositionangleseastofnorth,wedefineθastheanglemeasuredcounter-clockwisefromtheyaxis.Toquantifythedifferencebetweenthexandy-axisstruc-turefunctionsforeachangle,weconsidertwoseparatemeasuresofanisotropy.Thefirstanisotropyindex,Ψ1,ismotivatedbyGS95whopredictdifferencesinthescalingexponents,γx,γy,

γx−γy

Ψ1=

v◦,y+v◦,x

(12)

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Foranisotropicvelocityfield,Ψ1≈0andΨ2≈0.

ThemodulationofΨbypositionangleenablesamoreaccuratedeterminationoftheamplitudeandangleatwhichthevelocityanisotropyismaximized.Thismodulationinvolvesrotationaboutanaxissothereisdegeneracyforanglesθandθ+180.Wefindthatthefunction

Ψ(θ)=Ψ◦cos[2(θ−θMAX)](13)providesareasonablefittothevariationoftheanisotropyindex.ThecoefficientΨ◦gives

theamplitudeoftheanisotropyandthephase,θMAX,istheangleofmaximumanisotropythatcanbecomparedtothelocalfielddirection,󰀍θB󰀎.

3.

MHDSimulations

Todemonstratethattheanalysisdescribedin§2canindeedrecoverthespatialstatisticsofthevelocityfield,wehaveanalyzedasetofcomputationalsimulationsofdecayingMHDturbulencefromOstriker,Stone,&Gammie(2001).Themodelsspanarangeofmagneticfieldstrengthsparameterizedbytheratioofthermaltomean-fieldmagneticenergydensities,

2

β=c2envelocitybasedonthes/vAwherecsisthesoundspeedforH2andvAistheAlfv´√meanfield,󰀍B󰀎/

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3.1.

DirectMeasurementofVelocityAnisotropy

TheMHDvelocityanisotropyismanifestbythespectralpropertiesofthevelocityfieldalongaxesparallelandperpendiculartothemagneticfield.Cho,Lazarian,&Vishniac(2002)andVestuto,Ostriker,&Stone(2003)examinedthespectralslopesofdirectionalpowerspectraorequivalently,the2ndorderstructurefunction,ofvelocityfieldsfromcomputationalsimulations.Bothstudiesfoundsteeperspectralslopesandsmallernormalizationconstantsforstructurefunctionsextractedalongthemagneticfielddirectionrelativetothosealonganaxisperpendiculartothefield.Thatis,thevelocityfieldcontainsmorepowerwhenk⊥≈kandk󰀄≈0thanwhenk󰀄≈kandk⊥≈0,foragivenk.

Toquantifythevelocityanisotropyinthesimulationsusedinthisstudyandtocomparewithsimulatedobservationsshownin§3.2,thetrue2ndorderstructurefunction,S2(τ),iscalculateddirectlyfromeachmodelvelocityfield,v2,

S2(τ󰀄,τ⊥)=󰀍[v2(x)−v2(x+τ)]2󰀎

(14)

whereτ=τ󰀄e󰀄+τ⊥e⊥;e󰀄,e⊥areunitvectorsparallelandperpendicularrespectivelytothelocalmeanmagneticfielddirection,andtheanglebracketsdenoteaspatialaverageoverthevolume(Cho,Lazarian,&Vishniac2002).Here,werestrictouranalysistotheprojectedplaneappropriateforthevelocityfieldtofacilitatecomparisonwithmodelobservationsin§3.2.Inthiscase,thev2componentprojectsintotheplane1definedbyaxes1and3.Figure1showsthe2ndorderstructurefunctions,S2(0,τ⊥)andS2(τ󰀄,0).Powerlawsarefitoverthepixelrange5-15toexcludethesteepcomponentatsmallscalesthatresultsfromgrid-scalenumericaldissipationofthesimulation.Theamplitudesandspectralindicesofanequivalent,firstorderstructurefunction,(S2)1/2,arelistedinTable2.VelocityanisotropyisclearlyidentifiedintheB2andB3simulationsnapshotsastheslopeandamplitudeoftheorthogonalstructurefunctionsaredifferent.Fortheintermediate(snapshotsC2,C3)andweak(snapshotsD2,D3)B-fieldcases,thestructurefunctionsarestatisticallyequivalent,indicativeofgloballyisotropicvelocityfieldswithslopes(∼0.5)thataretypicalofstronglysupersonic,super-Alfvenicturbulentflows.Theabsenceofvelocityanisotropyresultsfromthelocaldistortionsofthemagneticfieldthatdiluteanysignaturetolargescaleanisotropy.

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3.2.

ModelSpectroscopicDataCubes

Observersdonotdirectlyrecoverthe3dimensionalvelocityfields.Widefieldspectro-scopicimagingmeasureslineintensityasafunctionofpositionontheskyandvelocityalonganaxis.Thepreciseshapeofalineprofileisdependentondensity,theprojectedveloc-itycomponent,temperature,andchemicalabundancethatareintegratedalongthelineofsightandaffectedbylineexcitationandopacity.Toplacethemodelvelocityanddensityfieldsfromthecomputationalsimulationsinthesamedomainasobservations,wegeneratesyntheticlineprofilesof12COand13COJ=1-0emission.DetailsofthelineexcitationandradiativetransfercalculationsaredescribedbyBrunt&Heyer(2002).Theassumedabun-dancevaluesof12COand13COrelativetoH2are1.0×10−4and1.25×10−6respectively.Weadoptauniformkinetictemperatureof15K,whichcorrespondstoaonedimensionalsoundspeedof0.22kms−1.TheadoptedmeanvolumedensityofH2isn=1000cm−3.Thechoiceofconstructingsyntheticprofilesofthehighopacity12COemissionismo-tivatedbytwofactors.First,the12COJ=1-0lineisthemostcommontracerofcloudstructuresotherearemanyobservationaldatasetsavailabletocomparewiththesemodels.Tobesure,12COdoesnoteffectivelyprobethehighdensitycoresofmolecularcloudswherestarformationtakesplace.However,theseregionscompriseasmallfractionofthecloudmassandvolume(Heyer,Ladd,&Carpenter1996;Goldsmithetal2008).Brunt&Heyer(2002)examinedtheeffectsoflineopacityonthegasdynamicsperceivedbyobservations.Withtheexceptionofmicro-turbulentvelocityfields,theyfoundthat12COmeasurementsreliablyrecoverthevelocityfieldstatistics.Althoughthelocalopticaldepthcanbelargewithinavolume,themacro-turbulentvelocityfieldsprovideaneffectivelargevelocitygra-dientconditionthatallowsmostphotonsfromthesurfaceofthelocalvolumetoescape.Inaddition,owingtoradiativetrapping,12COisdetectedoverabroaderareathantheloweropacitylinessotherearesimplymoremeasurementsandinformationonthelargestscales.Nevertheless,tore-examinetheeffectsoflineopacity,wealsogenerateandanalyzesyntheticprofilesofthe13COJ=1-0transition.

3.3.Axis-ConstrainedPCAAppliedtoModelDataCubes

Theutilityoftheanalysisdescribedin§2isassessedbyitsapplicationtothesynthetic12

COand13COdatacubesconstructedfromtheMHDmodeldensityandvelocityfields.Doestheanalysisrecovervelocityanisotropywhenthisispresentintherawvelocityfield,forthecaseofstrongmagneticfields?Doesthemethodverifyisotropicvelocityfieldsintheintermediateandweakfieldcases?

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Here,weexaminethesyntheticspectroscopicdatacubesderivedfromthev2velocityfieldforpositionangleθ=90◦,i.e.correspondingtoalignmentofthex-axiswiththemeanmagneticfielddirectionintheplaneofthesky.Theresultsoftheaxis-constrainedPCAmethod(with∆=2),asappliedtoallmodelsnapshots,areshowninFigure2.MagneticallyalignedanisotropyisclearlyidentifiedfortheB2andB3simulationsnapshotsasaseparationofthesetofpoints(<δv>,τ)derivedrespectivelyalongthexandyaxesofthemodeldatacubes.ThisseparationofpointsisqualitativelysimilartothecorrespondingtruestructurefunctionscalculateddirectlyfromthevelocityfieldsthatareshowninFigure1.Fortheintermediate-(C2,C3)andweak-(D2,D3)magneticfieldsnapshots,thereisastrongoverlapofpoints(<δv>,τ)derivedfortheorthogonalaxes.Thisindicatesvelocityisotropywithrespecttothemeanmagneticfield,andisinagreementwiththetruevelocitystructurefunctionsforthemodels.

Toassessthemethodquantitatively,bisectorfitsofpowerlawswithparameters,α,v◦,arefittoeachsetofpointsforeachaxisovertherange3≤τ≤30pixels.Thescalingexponents,γ󰀄andγ⊥,ofthestructurefunctionarederivedfromthefittedparameters,α󰀄andα⊥,accordingtoequation10.Theresultsforthe12COand13COmodeldatacubesaresummarizedinTable3.WiththeexceptionoftheC3modeldatacube,therearenosignificantdifferencesbetweenthepowerlawparametersderivedfrom12COand13COmodelcubes,demonstratingthatopacityeffectsdonotsignificantlyskewthederivedvelocityfieldstatistics.

Forthestrongfieldsimulations,theseparationofpointsinFigure2isduetoacombi-nationofalargernormalizationconstantandshallowerindexfortheperpendicularstructurefunction.Moreover,theanisotropyisstrongerinthelaterstagesimulation(comparingB3withB2).Thereare,however,discrepanciesbetweenthevaluesofγdetermineddirectlyfromthevelocityfieldinTable2andthosedeterminedbyPCAthatarelistedinTable3.Theroot-mean-squaredifferencebetweenpowerlawindicesis0.12(23%).Thisdiscrepancyisdueinpart,tothedifficultyinmeasuringapowerlawindexofstructurefunctionsofvelocityfieldsproducedbythecomputationalsimulationsthathavelimitedinertialrange(Vestuto,Ostriker,&Stone2003).Inaddition,thePCAeigenprojectionalongasingleaxistendstolimitthedynamicrangeofspatialscalesoverwhichthepowerlawsparametersarederived.Despitethisdiscrepancyofthescalingexponents,theseresultsdemonstratetheabilityoftheaxisconstrainedPCAeigenfunctionstoshowaclearsignatureofvelocityanisotropyinducedbyMHDturbulence.

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4.

TheTaurusMolecularCloud

TheTaurusMolecularCloudprovidesavaluableplatformtoinvestigateinterstellargasdynamicsandthestarformationprocess,owingtoitsproximity(140pc)andthewealthofcomplementarydata.Narayananetal(2008)presentnewwide-fieldimagingobservationsof12

COand13COJ=1-0emissionfromthecentral100deg2oftheTauruscloudcomplex,ob-tainedwiththeFCRAO14mtelescope.Theimagesidentifyalowcolumndensitysubstrateofgasthatcontainsubtlestreaksofelevated12COemissionalignedalongthelocalmagneticfielddirectionasdeterminedfromstellarpolarizationmeasurements(Heiles2000).Imagesof12COJ=1-0integratedintensityandcentroidvelocitywithmeasuredpolarizationvectorsfromthissubfieldareshowninFigure3.Theseshowaconnectionbetweenthedensityandvelocityfields.Whiletheoriginofthesestreaksisunknown,theirrigorousalignmentwiththepolarizationvectorsstronglysuggeststhattheinterstellarmagneticfieldplaysaprominentroleinthegasdynamicsofthislowdensitymaterial.

Toassessthedegreeofvelocityanisotropywithinthissub-regionoftheTaurusmolecularcloud,wehaveappliedtheaxisconstrainedPCAmethodtothe12COdatafromthisimagingsurvey.TheprecisefieldisdescribedbythesolidboxinFigure3.Wedonotconsiderthe13

COJ=1-0datasincethesignalisweakfromthislowcolumndensitysectorofthecloud.Themean,localpolarizationangle,derivedfrom16measurementswithinthefieldis52◦±10◦.Assumingthepolarizationisinducedbyselectiveabsorptionofbackgroundstarlightbymagneticallyaligned,elongateddustgrains,thisanglecorrespondstothelocalmagneticfielddirection(Purcell1979;Draine2003).Figure4showsthevariationoftheanisotropyindices,Ψ1andΨ2,withpositionangle(measuredeastofnorth)for12COdatawithinthissubfieldoftheTauruscloud.ForΨ1,whichconsidersthedifferencesinscalingexponents,thefittedparametersareΨ◦=0.49±0.03andθMAX=41◦±2◦.ForΨ2,whichmeasuresanisotropybasedonthedifferencesofthenormalizationconstants,Ψ◦=0.56±0.03andθMAX=46◦±2◦.Theangleofmaximumanisotropyiswithin6-11◦ofthelocalmagneticfielddirectionandthemeanpositionangleoftheemissionstreaksof12COemission.Thexandy-axisstructurefunctionsderivedatθMAX=46◦areshowninFigure5.Thesedistributionsshowthesamepatternofoffsetsbetweentheparallelandperpendicularstructurefunctionsmeasuredinthestrongfieldsimulationsnapshots(B2,B3)showninFigure2.FortheTaurusfield,thepowerlawindexofthestructurefunctionderivedfrom12COalongthex-axis(i.e.thedirectionalignedwiththepolarization)issteeper(0.81±0.05)thantheindexofthey-axisstructurefunction(0.34±0.06).Thesteeperpowerlawalongthex-axisisindicativeofavelocityfieldmoredominatedbylargescales.Similartothemodelstructurefunctionsinthestrongmagneticfieldcases,thenormalizationofthey-axisstructurefunction,v◦,yis0.08kms−1andlargerthanthevalueofthex-axisstructurefunction(v◦,x=0.02kms−1).Thus,thesmoothvariationofdensityalongthepresumedmagneticfieldismirroredbyasmooth

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variationinthevelocity,andthestrongervariationindensityintheperpendiculardirection(streakiness)ismirroredbyastrongervariationinthevelocity.Indeed,preliminaryanalysisshowsthatinthedirectionperpendiculartotheprojectedmagneticfield,displacementsbetweenthepeaksinintegratedintensityandvelocitycentroidsaresimilarwithtypicalvalues0.2to0.4pc.

TheresultsshowninFigures3,4,5aresuggestiveofvelocityanisotropyinducedbystrongMHDturbulence,asdescribedbyGS95andverifiedbycomputationalsimulations(Cho,Lazarian,&Vishniac2002;Vestuto,Ostriker,&Stone2003).Wenotethattheobservedspectralslopeparalleltothefield,γ󰀄,issteeperthanthevaluepredictedforincompressibleMHDturbulencebyGS95butissimilartovaluesderivedforthestrongfield(B2,B3)simulations.VelocityanisotropycouldbeproducedbyprocessesotherthanMHDturbulence.Asystematicflowofmaterialthatis“channeled”bythemagneticfieldmayalsogeneratedifferencesintheparallelandperpendicularstructurefunctions.Suchlargescalegradientswouldproducesteepspectralindices(γ≥1).However,theobservedhighfrequencyvariationofvelocitiesperpendiculartothefieldarenotcharacteristicofsuchlargescaleshearflows.Regardlessofitsorigin,thenearalignmentofthevelocityanisotropywiththelocalmagneticfielddirectiondemonstratestheimportanceoftheinterstellarmagneticfieldonthegasdynamicswithinthislowdensitycomponentoftheTaurusmolecularcloud.

4.1.TheMagneticFieldStrengthintheTaurusCloudEnvelope

Sinceanisotropyisonlyevidentinmodelswithstrongmagneticfields,theidentificationofsuchanisotropywithinobservationaldataoffersaproxymeasureofthemagneticfieldanditseffectupontheneutralgas(Vesuto,Ostriker,&Stone2003).Specifically,theamplitude󰀂ofthemeanmagneticfield,B◦=|󰀍B󰀎|=cs

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constrainedbynon-LTEexcitationmodelsthatmatchtheobserved12COand13COJ=1-0intensitiesfromthesub-thermallyexcitedcomponentoftheTauruscloud(Goldsmithetal2008).Themagneticfieldstrengthcorrespondingtothesevaluesofβ,kinetictemperature,andgasdensityis14µG.Asnotedabove,thisisalowerlimitonthetotalmagneticfieldstrengthsincethevelocityanisotropyisnotsensitivetotheline-of-sightcomponentofthemagneticfield.

ZeemanmeasurementsoftheOHlineemissionfromtheL14darkcloud,located∼4degreestothesouth-westofthesubfieldinTaurus,identifyalineofsightfieldstrengthof11µG(Crutcher&Troland2000).Whilethisvalueiscomparabletoourcoarseestimateofthefield,theseOHZeemanobservationsaretowardhighercolumndensitymaterial(N(H2)∼2×1022)thanislikelypresentintheTaurussubfield.Ifthishighercolumndensityreflectsalargervolumedensityandifthemagneticfieldiscorrespondinglycompressed,thefieldinthediffusepartsoftheTauruscloudmaybesmaller.

TheChandrasekhar&Fermi(1953)methodoffersanadditionalmeasureofthemagneticfieldstrengthininterstellarclouds.Itattributesdeviationsofthelocalmagneticfieldfromthemeanfielddirectiontolinear-amplitudetransverseMHDwavessuchthat

(δB/Bp)=|δv|/vA

(15)

whereBpistheprojectionofthemeanmagneticfieldontheplaneofthesky,δBand|δv|arecomponentsofthemagneticandvelocityperturbationstransversetoBp,andvAistheAlfv´envelocity.Assumingpolarizationvectorsaccuratelytrackthelocalmagneticfielddirectionandtransversevelocityperturbationsinthetwodirectionsperpendiculartoˆarecomparable,theChandrasekhar-FermimethodisrewrittenintermsofobservationalB

measures,

σpol=f(4πρ◦)1/2σv/Bp(16)whereσpolisthedispersionofpolarizationanglesmeasuredinradians,σvisthelineof

sightvelocitydispersion,ρ◦isthemeandensityofthegas,andthefactor,f,accountsfordensityinhomogeneityandlineofsightintegration.Ostriker,Stone,&Gammie(2001)andPadoanetal(2001)determinef≈0.4-0.5fromcomputationalsimulations.Thedispersionofmeasuredopticalpolarizationangleswithinthetargetfieldis0.17radians.Thelineofsightvelocitydispersiondeterminedfromthe13COdatais0.38kms−1.Assumingameandensityof250cm−3andf=0.5,thederivedmeanfieldstrengthis14µG.Thus,ourPCA-basedestimateofthemagneticfieldstrengthinTaurusalsocomparesfavorablytothevaluederivedbytheChandrasekhar-Fermimethod.

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4.2.

TheMagneticSupportoftheTaurusCloudEnvelope

Thedegreetowhichthemagneticfieldcansupportavolumeagainstself-gravitationalcollapseisparameterizedbythemasstofluxratiowithrespecttothecriticalvalue,(M/Φ)crit=0.16/G1/2(Nakano&Nakamura1978).Themagneticcriticalindex,µ,istheratioofthemasstofluxratioofavolumetothiscriticalvalue,

µ=(M/Φ)/(M/Φ)crit=7.6×10−21N(H2)/B

(17)

whereN(H2)isthegascolumndensityincm−2alongfieldlinesandBisthemagneticfieldstrengthexpressedinµG.Owingtoprojectionsofthemagneticfieldandthemassdistributionalongfieldlines,theobservedindex,µobs,overestimatesthetruemagneticindex.Assumingrandomorientationsofthemagneticfieldandflattenedgasdistributionwithrespecttotheobserver,onecanderiveastatisticalcorrectiontotheobservedvalue,<µ>=µobs/3toassesswhetheravolumeissuper-critical(<µ>greaterthan1)orsub-critical(<µ>lessthan1)(Heiles&Crutcher2005).

Goldsmithetal(2008)derivethedistributionofmolecularhydrogenover100deg2oftheTaurusMolecularCloudusingthe12COand13COJ=1-0dataofNarayananetal(2008).FromtheGoldsmithetal(2008)image,themeancolumndensitywithintheTaurussubfieldanalyzedinthisstudyis1.5×1021cm−2.Foramagneticfieldwithstrength14µG,thiscolumndensitycorrespondstoanobservedmagneticindexofµobs=0.81.Applyingthestatisticalcorrectionforprojections,<µ>=0.27.ThislowcolumndensitysubfieldwithintheTauruscloudismagneticallysub-criticalindicativeofamagneticallysupportedcloudenvelope.Suchsub-critical,lowcolumndensityenvelopesareexpectedgiventheexposuretotheambientUVradiationfieldthatmaintainsasufficientdegreeofionizationtocoupletheneutralmaterialtoions.Theambipolardiffusiontimescaleislongwithrespecttothedynamicaltimeoftheenvelope.WhilestarformationwithinthehighdensitycoresandfilamentsoftheTauruscloudattesttothegravitationalcollapseandlackofmagneticsupportwithinlocalizedregions,theseoccupyasmallfractionofthemassandareaofthecloud.Goldsmithetal(2008)reportthat50%ofthemassand75%oftheareaofTaurushavemolecularcolumndensitieslessthan2×1021cm−2.Ifthiscolumndensityregimeissimilartothesubfieldanalyzedinthisstudy,thentheTaurusmolecularcloudenveloperemainsmagneticallysupported.

5.Summary

Wehavedevelopedananalysismethodtoassessvelocityanisotropywithininterstellarmolecularcloudsfromspectroscopicimagingobservations.Suchanisotropyispredictedfrom

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theoryofstrongMHDturbulence(GS95).TheutilityofourmethodisdemonstratedusingMHDsimulationswithvaryingmagneticfieldstrengths.Velocityanisotropyisrecoveredinmodelswithstrongmagneticfields(β=0.01)orientedperpendiculartotheline-of-sight.Noanisotropyismeasuredinsimulationswiththemagneticfieldpressuremorecomparabletothelocalthermalpressure,orafewtimeslarger(β=1.0,0.1).Theanalysisisappliedto12COJ=1-0emissionfromalowdensitysub-regionwithintheTaurusmolecularcloud.Wedetectvelocityanisotropythatisalignedwithin∼10degreesofthelocalmagneticfielddirection.Thiscoincidenceofthefielddirectionwithmeasuredanisotropyinsmall-scalevelocityvariationsdemonstratesastrongcouplingoftheinterstellarfieldwiththeneutralgasthatmayresultfromMHDturbulentflows.Ourestimateoftheplane-of-skymagneticfieldstrengthbasedonourvelocityanisotropyanalysisisinagreementwiththevaluederivedusingtheChandrasekhar-Fermimethod.Basedonourestimatedmagneticfieldstrengthcombinedwithcolumndensitymeasurements,wefindthatthelow-densityenvelopeofTaurus,whichcomprisesthebulkofthecloud’smass,ismagneticallysubcritical.ThisworkwassupportedbyNSFgrantAST00852totheFiveCollegeRadioAs-tronomyObservatory.ECOissupportedbyNSFgrantAST0507315.

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Table1:MHDSimulationParameters

B2C2D2B3C3D30.010.101.000.010.101.000.070.040.050.190.090.097.47.67.24.94.94.9

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Table2:StructureFunctionParameters:(S2(τ))1/2=v◦τγ

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Table3:StructureFunctionParametersDerivedfromModelDataCubes

12

CO

γ󰀄0.550.570.530.620.340.61

ModelB2C2D2B3C3D3

COγ⊥v◦,󰀄0.320.440.590.230.530.41

0.080.100.100.040.100.07

13

v◦,⊥0.150.120.100.130.080.09

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Fig.1.—Thesecondordervelocitystructurefunction,S2(τ),alongaxesparallel(redtri-angles)andperpendicular(bluecircles)tothemeanmagneticfielddirectionforvelocityfieldsfromturbulentsimulations.Velocityanisotropyisevidentinthestrongfieldmodelcases(B2,B3)asalargerscalingamplitudeandshallowerindexforthestructurefunctionperpendiculartothemeanfielddirection.

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Fig.2.—Theaxis-constrainedPCA<δv>,τrelationshipsderivedfromsyntheticspectro-scopicdatacubesof12COJ=1-0emissionforthexaxis(alongthemeanmagneticfield;redtriangles)andyaxis(perpendiculartothemeanmagneticfield;bluecircles).Theerrorbarsreflectthestandarddeviationofvalueswithineach1pixelwidebinofτ.ThemethodrecoverstheanisotropyintrinsictotheB2andB3modelvelocityfieldsandverifiestheisotropicvelocityfieldsoftheintermediate(C2,C3)andweakfield(D2,D3)models.

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Fig.3.—(left)Imageof12COJ=1-0emissionofasub-fieldwithintheTaurusmolecularcloudintegratedoverthevelocityinterval5.5to7.5kms−1and(right)imageof12COvelocitycentroid(Narayananetal2008),withoverlayofopticalpolarizationvectorsfromthecompilationbyHeiles(2000).Themolecularlineemissionandvelocitiesexhibitstreaksthatarealignedalongthelocalmagneticfielddirection.ThesolidlineboxoutlinestheareauponwhichtheaxisconstrainedPCAmethodisapplied.Thedottedlineboxshowstheareawithinwhichthepolarizationanglesareaveragedtoesimatethemeanmagneticfielddirection.

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Fig.4.—Thevariationoftheanisotropyindices,Ψ1(left)andΨ2(right),withpositionangle,θ.Foreachindex,thesolidlineshowsthefitofequation13tothesetofpoints.Thedashedverticallinesshow󰀍θB󰀎±1σinferredfromopticalpolarizationmeasurementsofbackgroundstarswithinthesubfield.Theangleofmaximumanisotropyisnearlyalignedwiththelocalmagneticfielddirection,whichsuggestsarelationshipbetweenvelocityanisotropyandtheinterstellarmagneticfieldinducedbystrongMHDturbulence.

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Fig.5.—ThePCAderivedδv,τrelationshipderivedfromthe12COspectroscopicdatacubeoftheTaurussub-fieldrotatedtoalignthex-axiswiththeangleofmaximumanisotropy,θMAX=46◦.Theredtrianglesarepointsderivedalongtherotatedx-axisandthebluecirclesarepointsderivedalongtherotatedy-axis.Theerrorbarsreflectthestandarddeviationofvaluesineachbin.ThepatternissimilartothatfoundfromtheB2,B3simulationsnapshotsinFigure2andsuggestsanimportantroleofthemagneticfieldonthelocalgasdynamics.