Axisymmetric stability criteria for a composite system of stellar and magnetized gaseous si

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

Mon.Not.R.Astron.Soc.000,000–000(0000)Printed5February2008

A(MNLTEXstyle lev1.4)

Axisymmetricstabilitycriteriaforacompositesystemof

stellarandmagnetizedgaseoussingularisothermaldiscs

Yu-QingLou1,2,3,4andYueZou1

arXiv:astro-ph/0511348v1 11 Nov 2005

1Physics2Centre

DepartmentandTsinghuaCenterforAstrophysics(THCA),TsinghuaUniversity,Beijing100084,China;dePhysiquedesParticulesdeMarseille(CPPM)/CentreNationaldelaRechercheScienti que(CNRS)/InstitutNationaldePhysiqueNucl´eaireetdePhysiquedesParticules(IN2P3)etUniversit´edelaM´editerran´eeAix-MarseilleII,163,AvenuedeLuminyCase902F-13288Marseille,Cedex09,France;3DepartmentofAstronomyandAstrophysics,TheUniversityofChicago,5640EllisAve,Chicago,IL60637,USA;4NationalAstronomicalObservatories,ChineseAcademyofScience,A20,DatunRoad,Beijing100012,China.

Accepted2004...Received2004...;inoriginalform2004

ABSTRACT

Usingthe uid-magneto uidformalism,weobtainaxisymmetricstabilitycriteriaforacompositediscsystemconsistingofstellarandgaseousmagnetizedsingularisothermaldiscs(MSIDs).Both(M)SIDsarepresumedtoberazor-thinandaregravitationallycoupledinaself-consistentaxisymmetricbackgroundequilibriumwithpower-lawsur-facemassdensitiesand atrotationcurves.ThegaseousMSIDisembeddedwithanon-force-freecoplanarazimuthalmagnetic eldBθ(r)ofradialscalingr 1/2.Lou&Zourecentlyconstructedexactglobalstationarycon gurationsforbothaxisym-metricandnonaxisymmetriccoplanarmagnetohydrodynamic(MHD)perturbationsinsuchacompositeMSIDsystemandproposedtheMHDDs criteriaforaxisym-metricstabilitybythehydrodynamicanalogy.Inadi erentperspective,wederiveandanalyzeherethetime-dependentWKBJdispersionrelationinthelow-frequencyandtight-windingregimetoexamineaxisymmetricstabilityproperties.ByintroducingarotationalMachnumberDsfortheratioofthestellarrotationspeedVstothestellar

2

numericallytovelocitydispersionas,onereadilydeterminesthestablerangeofDs

establishtheDs criteriaforaxisymmetricMSIDstability.ThoseMSIDsystemsro-tatingeithertoofast(ringfragmentation)ortooslow(Jeanscollapse)areunstable.

2

ThestablerangeofDsdependsonthreedimensionlessparameters:theratioλfortheAlfv´enspeedtothesoundspeedinthegaseousMSID,theratioβforthesquareofthestellarvelocitydispersiontothegassoundspeedandtheratioδforthesurfacemassdensitiesofthetwo(M)SIDs.OurWKBJresultsof(M)SIDinstabilityprovidephys-icallycompellingexplanationsforthestationaryanalysisofLou&Zou.Wefurtherintroduceane ectiveMHDQparameterforacompositeMSIDsystemandcomparewiththeearlierworkofElmegreen,JogandShen&Lou.Asexpected,anaxisym-metricdarkmatterhaloenhancesthestabilityagainstaxisymmetricdisturbancesinacompositepartialMSIDsystem.Intermsoftheglobalstarformationrateinadiscgalaxysystem,itwouldappearphysicallymoresensibletoexaminetheMHDQMstabilitycriterionagainstgalacticobservations.Relevancetolarge-scalestructuresindiscgalaxiesarealsodiscussed.

Keywords:MHDwaves—ISM:magnetic elds—galaxies:kinematicsanddynam-ics—galaxies:spiral—star:formation—galaxies:structure.

1INTRODUCTION

Incontextsofgalacticstructures,discstabilitiesandglobalstarformationratesinspiralgalaxies,wederiveherein-stabilitycriteriaforaxisymmetriccoplanarmagnetohydro-dynamic(MHD)perturbationsinacompositediscsystem

withanazimuthalmagnetic eldinthegasdisccomponentandestablishageneralizedde nitionofane ectiveMHDQMparameterappropriatetosuchamagnetizedgravita-tionalsystem.Formulatedassuch,thisisanidealizedandlimitedtheoreticalMHDdiscproblemyetwithseveralkeyconceptualelementsincluded.Thesimplephysicalrationale

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

2Y.-Q.LouandY.Zou

isthatzonalregionsofhighergasdensityandmagnetic eldarevulnerabletoactiveformationofmassivestarswithvar-iousscalesinvolveddi eringbymanyordersofmagnitudes.Onthesameground,non-axisymmetricstabilitycriteriaareequallyimportantbutaremorechallengingtoestablish(seee.g.Shuetal.2000forrelevantissues).Overfourdecades,importantdevelopmenthavebeenmadeforinstabilitycrite-riarelevanttogalacticdiscdynamics(seeLin1987,Binney&Tremaine1987andBertin&Lin1996andextensiverefer-encestherein).Theoriginalstudiesofaxisymmetricinstabil-itieswereconductedbySafronov(1960)andToomre(1964)whointroducedthedimensionlessQparameterforthelocalstability(i.e.Q>1)againstaxisymmetricring-likedistur-bances.Fordiscgalaxies,itwouldbemorerealisticandsen-sibletoinvestigateacompositediscsystemconsistingofastellardisc,amagnetizedgasdiscandamassivedarkmat-terhalo.Therehavebeenextensivetheoreticalstudiesonthistypeofcompositetwo-componentdiscsystemsofvar-ioussub-combinations(Lin&Shu1966,1968;Kato1972;Jog&Solomon1984a;Bertin&Romeo1988;Romeo1992;Elmegreen1995;Jog1996;Lou&Fan1998b,2000a,b;Lou&Shen2003;Shen&Lou2003,2004a,b;Lou&Zou2004;Lou&Wu2005).Inparticular,therehavebeenseveralstud-iestryingtode neapropere ectiveQparameterforlocalaxisymmetricinstabilitycriterioninacompositediscsys-tem(Elmegreen1995;Jog1996;Lou&Fan1998b,2000a,b;Shen&Lou2003,2004a,b).Fromdi erentperspectives,theseanalyseso erinsightsforinstabilitypropertiesofacompositediscsystemandprovideatheoreticalbasisforunderstandingthelarge-scaledynamicsinsuchasystem(e.g.Lou&Fan2000a,b;Lou&Shen2003;Shen&Lou2003,2004a,b;Lou&Zou2004;Lou&Wu2005).

Themainmotivationhereistoexplorebasicproper-tiestheMSIDmodelinacompositesystemandobtaincon-ceptualinsightsforastrophysicalapplicationsinmagnetizedspiralgalaxiesandinestimatingglobalstarformationratesindiscgalaxies(e.g.,Lou&Bian2005).Intheoreticalstud-iesofmodelingdiscgalaxies,theclassofSIDmodelshasadistinguishedhistorysincethepioneeringworkofMestel(1963)(Zang1976;Toomre1977;Lemos,Kalnajs&Lynden-Bell1991;Lynden-Bell&Lemos1993;Syer&Tremaine1996;Goodman&Evans1999;Charkrabarti,Laughlin&Shu2003)andhasgainedconsiderableattentionandinter-estsrecentlybyconsideringacompositediscsystemandbyincorporatinge ectsofmagnetic eld(Shuetal.2000;Lou2002;Lou&Fan2002;Lou&Shen2003;Shen&Lou2003,2004a,b;Shen,Liu&Lou2004;Lou&Zou2004;Lou&Wu2005).Speci cally,Shuetal.(2000;seealsoGallietal.2001)studiedglobalstationary(i.e.,zeropatternspeed)perturba-tioncon gurationsinanisopedicallymagnetizedSIDwith-outinvokingtheusualWKBJortight-windingapproxima-tion.Theyobtainedexactglobalsolutionsforbothalignedandunalignedaxisymmetricandnon-axisymmetriclogarith-micspiralcon gurationsandinterpretedtheaxisymmetricsolutionforperturbationswithradialpropagationsasde-marcatingtheboundariesbetweenthestableandunstableregimes.Bytheseaxisymmetricinstabilities,aSIDwitha

su cientlyslowrotationspeedwouldJeanscollapseinducedbyperturbationsoflargerradialscales,whileaSIDwithasu cientlyfastrotationspeedmaysu ertheringfragmen-tationinstabilityinducedbyperturbationsofsmallerradialscales(see g.2ofShuetal.2000).Byintroducingaro-tationalMachnumberD,de nedastheratiooftheSIDrotationspeedVtotheisothermalsoundspeeda,thecrit-icalvaluesofthehighestandlowestDforanaxisymmetricstabilitycanbedetermineddirectlyfromthemarginalsta-bilitycurve.Tosupporttheirphysicalinterpretations,theyinvokedthewell-knownToomreQparameterandfoundthatthehighestD,namelytheminimumoftheringfragmenta-tioncurve,correspondstoaQvalueveryclosetounity,thusheuristicallysuggestingthecorrespondencebetweentheD criterionandtheQ criterion.

Di erentfromyetcomplementarytotheanalysisofShuetal.(2000)onasingleisopedicallymagnetizeddisc,Lou(2002)studiedglobalcoplanarMHDperturbationsinasin-gleMSIDembeddedwithanazimuthalmagnetic eldandrevealedthattheminimumoftheMHDringfragmenta-tioncurveinthisMSIDmodelistightlyassociatedwiththegeneralizedMHDQMparameteroriginallyintroducedbyLou&Fan(1998a)indevelopingthegalacticMHDden-sitywavetheory(Fan&Lou1996).ForacompositesystemoftwocoupledunmagnetizedhydrodynamicSIDs,Lou&Shen(2003)constructedstationaryglobalperturbationcon- gurationsandShen&Lou(2003)suggestedastraightfor-wardD criterionfortheaxisymmetricringfragmentationinstabilityinsuchasystemonthebasisofalow-frequencyWKBJanalysis;theyrevealedthattheminimumoftheringfragmentationintheircompositeSIDmodelisagaincloselyrelatedtoapropere ectiveQparameter(Elmegreen1995;Jog1996).Furthermore,foracompositeSIDsystemwithanisopedicallymagnetizedgaseousSIDandastellarSIDinthe uiddescription,Lou&Wu(2005)haveconstructedglobalstationaryMHDperturbationstructuresandexam-inedstabilitypropertiestoanticipateasimilarD criterioninparalleltothecaseofShen&Lou(2003).Meanwhile,Shen&Lou(2004b)havefurthergeneralizedbothworkofSyer&Tremaine(1996)andLou&Shen(2003)tothesitu-ationofacompositesystemfortwogravitationallycoupledscale-freediscs;theyalsostudiedtheaxisymmetricstabilityofsuchacompositesystemintermsofthemarginalstabilitycurvesandproposedaDs&Lou(2003).

criterionbytheanalogyofShenWehaverecentlyexaminedtwo-dimensionalcoplanarMHDperturbationsinacompositesystemconsistingofastellarSIDandagaseousMSID.BothSIDsareexpedientlyapproximatedasrazor-thinandthegasdiscisembeddedwithanon-force-freecoplanarmagnetic eld(Lou&Zou2004).Inthis uid-magneto uidMSIDmodelapproxima-tion,weobtainedexactglobalstationaryMHDsolutionsforalignedandunalignedlogarithmicspiralperturbationcon- gurationsinsuchacompositeMSIDsystem,expressedintermsofthestellarrotationalMachnumberDs.Inrefer-encetotheresultsofasingleSID(Shuetal.2000;Gallietal.2001;Lou2002),itwouldbenaturaltosuggestthat

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

thestationaryaxisymmetricsolutionswithradialpropaga-tionsgiverisetomarginalstabilitycurves(seeFigure2inthispaperlaterandLou&Zou2004).Incomparisonwith

thesingleSIDcase,thestablerangeofD2

sisreducedasaresultofthemutualgravitationalcouplingbetweenthetwoSIDs.However,incomparisonwiththecaseofacompositeunmagnetizedSIDsystem(Lou&Shen2003;Shen&Lou

2003),thestablerangeofD2

sexpandsconsiderablyduetothepresenceofacoplanarmagnetic eld.

Tocon rmheuristicargumentsfortheaboveanalogyandourintuitivephysicalinterpretations,weconductinthispaperalow-frequencytime-dependentstabilityanalysisintheWKBJortight-windingapproximationforthecompos-iteMSIDsystem(Shen&Lou2003,2004a;Lou&Zou2004).Weshalldemonstrateunambiguouslythevalidityofdemar-catingthestableandunstableregimesbythestellarrota-tionalMachnumberDs.Toplaceouranalysisinpropercontexts,wealsodiscussspeci callyhowthetwoe ectiveQparametersofElmegreen(1995)andofJog(1996)arere-latedtoourDsparameterwhenotherrelevantparametersarespeci ed,andshowthatthetwoe ectiveQparame-tersarepertinenttotheringfragmentationinstabilityinacompositeMSIDsystem.

InSection2,wederivethetime-dependentdisper-sionrelationusingtheWKBJapproximationforpertur-bationsinacompositeMSIDsystemandintroduceafewkeydimensionlessparameters.InSection3,wepresenttheDsite MSIDcriterionsystemandbeingtwoe ectivestableagainstQparametersarbitraryforaxisymmet-acompos-ricperturbationswithradialpropagations.MainresultsanddiscussionsaresummarizedinSection4.

2AFLUID-MAGNETOFLUIDFORMALISM

Weconsiderbelowacompositesystemconsistingoftwogravitationallycoupled(M)SIDswithoneoftheSIDsbeingmagnetizedandthusreferredtoasMSID.Forphysicalvari-ables,weuseeithersuperscriptorsubscriptstoindicateanassociationwiththestellarSIDandeithersuperscriptorsubscriptgtoindicateanassociationwiththegaseousMSID.ThestellarSIDandthegaseousMSIDcanhavecon-stantyetdi erentrotationalspeedsVsandVg(relatedtothephenomenonofasymmetricdriftinthegalacticcontext);wethuswritethebackgroundangularrotationspeeds softhestellarSIDand gofthegaseousMSIDas

s=Vs/r=asDs/r

(1)

and

g=Vg/r=agDg/r,

(2)separately,whereasandagaretheconstantvelocitydis-persionofthestellarSIDandtheisothermalsoundspeedofthegaseousMSID,respectively;DsandDgarethecor-respondingrotationalMachnumbers.Therelevantepicyclicfrequenciesintermsof sand garegivenby

κ22 s

s≡

dr

(r2 s)=2 2s

(3)

MHDstabilitycriteriaforcompositeMSIDs

3

and

κ2g

g≡

2 dr

(r2 g)=2 2g,

(4)

respectively.SimilartoasingleMSID,wetaketheback-groundazimuthalmagnetic eldtobeintheformof

B1/2θ(r)=Fr ,

(5)

whereFisaconstant(Lou2002;Lou&Fan2002)and

Br=Bz=0.

(6)

Foramoregeneralpower-lawradialvariationoftheaz-imuthalmagnetic eldandthoseofotherrelatedback-groundvariables,theinterestedreaderisreferredtoarecentanalysisofShen,Liu&Lou(2005).

Inthe uidapproximationofastellarSID,themassconservation,theradialcomponentofthemomentumequa-tionandtheazimuthalcomponentofthemomentumequa-tionaregivenbelowinorder,namely

Σs (rΣsus) (Σsjs)

rr2

+u

s u

s

us1 t

r2

r3=

r

s j

s

js

Πs

t+ur2Σs θ

,(9)whereusistheradialcomponentofthebulk owvelocityofthestellarSID,js≡rvsisthestellarspeci cangularmomentumalongthez direction,vsistheazimuthalcom-ponentofthestellarbulk owvelocity, isthetotalgrav-itationalpotential,Πsistheverticallyintegratedpressure,andΣsisthesurfacemassdensityofthestellarSID.

Inthemagneto uidapproximationforthegaseousMSID,themassconservation,theradialcomponentofthemomentumequationandtheazimuthalcomponentofthemomentumequationaregivenbelowinorder,namely

Σg

(rΣgug) (Σgjg)rr2

+ug

ug

ug

1

t

r2

r3=

r

dzBθ

Σg

r

Br

g

Πg t

+ug

j jgr2Σg θ

+1 (rBθ)

4π

θ

(12),

whereugistheradialcomponentofthegasbulk owveloc-ity,jg≡rvgisthegasspeci cangularmomentuminthez

direction,vg

istheazimuthalcomponentofthegasbulk owvelocity,Πgisthetwo-dimensionalgaspressure,ΣgisthegassurfacemassdensityandBrandBθaretheradialandazimuthalcomponentsofthemagnetic eldB.Thelasttwotermsontheright-handsidesofequations(11)and(12)aretheradialandazimuthalcomponentsoftheLorentzforceduetothecoplanarmagnetic eld.Thetwosetsof uidandmagneto uidequations(7) (9)and(10) (12)aredynam-icallycoupledbythetotalgravitationalpotential through

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

4Y.-Q.LouandY.Zou

thePoissonintegral

∞s

(r,θ,t)= dψ

G(Σg+Σ)ζdζ

Bθ

r

+

t=

1 θ

(ugBθ vgBr),(15)

Bθ

r

(ugBθ vgBr).

(16)

UsingPoissonintegral(13),onereadilyderivesthefollowingexpressionsforthebackgroundsurfacemassdensities

Σsa22

0=

s(1+Ds)

2πGr(1+δ)

,(18)

whereδ≡Σgs

0/Σ0isthesurfacemassdensityratioofthetwodynamicallycoupledbackground(M)SIDsandCAistheconstantAlfv´enwavespeedintheMSIDde nedC2A≡ by

dzB2θ/(4πΣg0).

(19)Apparently,equations(17)and(18)requires

a2D2222

s(1+s)=ag(1+Dg) CA/2.

(20)

Wenowintroducetwomoreusefuldimensionlessparameters

here.The rstparameterβ≡a22

s/agstandsforthesquareoftheratioofthestellarvelocitydispersiontotheisothermalsoundspeedintheMSIDandthesecondparameterλ2C22≡A/agstandsforthesquareoftheratiooftheAlfv´

enwavespeedtotheisothermalsoundspeedintheMSID.Inlate-typediscgalaxies,thestellarvelocitydispersionasisusuallyhigherthanthegassoundspeedag,wethusfocusonthecaseofβ≥1(Jog&Solomon1984a,b;Bertin&Romeo1988;Jog1996;Elmegreen1995;Lou&Fan1998b;Lou&Shen2003;Shen&Lou2003,2004a,b;Lou&Wu2005).

Beforegoingfurther,wenotethatatypicaldiscgalaxysysteminvolvesamassivedarkmatterhalo,astellardiscandagaseousdiscofinterstellarmedium(ISM)onlargescales,wheretheISMdiscismagnetizedwiththemagneticenergydensity(～1eV/cm3)beingcomparabletotheen-ergydensitiesofthermalISMandofrelativisticcosmic-raygas(e.g.Lou&Fan2003).Tocomprehensivelyun-derstandmulti-wavelengthobservationsoflarge-scalespi-ralstructuresofdiscgalaxiesandtodeveloppotentiallypowerfulobservationaldiagnostics(Lou&Fan2000a,b),itwouldbemorerealisticandnecessarytotakeintoaccountofmagnetic elde ectsinacompositemagnetizeddisc-halomodel. Whilethereareexceptions,galacticmagnetic elds

Thecosmic-raygasissetasideheremerelyforthesakeofsim-

typicallytendtobecoplanarwiththediscplaneofaspiral

galaxyonlargescales.Onsmallerscales,regionsofclosedandopenmagnetic eldsaremostlikelyintermingledbythesolaranalogy(e.g.Lou&Wu2005).Asa rststep,Lou(2002)carriedoutacoplanarMHDperturbationanal-ysisforstationaryalignedandunalignedlogarithmicspiralstructuresinasingleMSIDembeddedwithanazimuthalmagnetic eldanddemonstratedthattheminimumoftheringfragmentationcurveinthisMSIDmodelisclearlyre-latedtothegeneralizedMHDQMparameter(Lou&Fan1998a).SincethebackgroundMHDrotationalequilibriumadoptedbyLou&Fan(1998a)isnotanMSIDmodel,itwouldbemoresatisfyingtojustifythestatementandin-terpretationofLou(2002)inadynamicallyself-consistentmanner.Weshallde neaQMparametersimilartothatofLou(2002)andshowthatthisQMisequivalenttothatofLou&Fan(1998a).

ForasingleMSID,wereadilyderivelinearequations(bysettingrelevantparametersforthestellarSIDtovanish)forcoplanaraxisymmetricMHDperturbationswithFourierharmonicdependenceexp(ikr+iωt),wherekistheradialwavenumberandωistheangularfrequency.IntheusualWKBJortight-windingapproximationofkr 1,weobtainthelocalWKBJdispersionrelationforMHDdensitywavespropagatinginanMSIDintheformof

ω2=κ2222g+k(ag+CA) 2πG|k|Σg

0,

(21)

whichistheMHDgeneralizationoftheWKBJdispersionrelationforcoplanarperturbationsinanunmagnetizedSID.

Toderiveane ectiveQMparameterfortheaxisym-metricstability(i.e.,ω2withanarbitraryk,thedeterminant≥0)againstofMHDtheright-handperturbationssideofequation(21)shouldbenegativeforallk.Thisrequires

κg(C2

21/2QM≡

A+ag)

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

mutualgravitationalcouplingbetweenthetwoSIDsbythetermontheright-handside.The rstfactorontheleft-handsideisforperturbationsinthestellarSIDapproximatedasa uid,whilethesecondfactorisforcoplanarMHDpertur-bationsinthegaseousMSID[seeexpression(21)].Itshouldbeemphasizedthatwhiledispersionrelation(23)isalo-calone,thebackgroundphysicalvariablesareinrotationalMHDequilibriuminaconsistentmanner.Inparticular,Σg0givenbyexpression(18)containstheinformationofback-groundmagnetic eldviaC2

A.

AsnotedbyLou&Shen(2003),thedispersionrelationderivedhereforcoplanarMHDperturbationsinacomposite(M)SIDapproachisqualitativelysimilartothosepreviouslyobtainedbyJog&Solomon(1984),Elmegreen(1995)andJog(1996)inspirit,butonemajordistinctionisthatintheformulationofourcomposite(M)SID,therotationspeedsofthetwoSIDsVsandVgaredi erentingeneral.Wethushaveinequalityκs=κg,whileinthoseearlieranalyses,κs=κgwaspresumedapriorifollowingtheassumptionofVs=Vg.

Thisisconceptuallyrelatedtothephenomenonofasymmetricdrift(e.g.Binney&Tremaine1987).Physically,stellarvelocitydispersionsmimicapressure-likeforceforthestellarcomponent,whilethethermalISMgasandmagneticpressureforcestogetheractonthemagnetizedgascompo-nent.Inthesamegravitationalpotentialwelldeterminedbythetotalmassdistribution,thedi erenceinthestellarpressure-likeforceandthesumofthegaseousandmagneticpressureforceswouldleadtodi erentVsandVgandthustheasymmetricdrift.TheraresituationofVsandVgbeingequalmayhappenunderveryspecialcircumstances.

3

AXISYMMETRICSTABILITYANALYSIS

FORACOMPOSITEMSIDSYSTEM

WedescribebelowcoplanarMHDperturbationanalysisfortheaxisymmetricstabilityofacompositeMSIDbasedonalow-frequencytime-dependentWKBJapproach.Generaliz-ingthenotationsofShen&Lou(2003)yetwiththee ectofmagnetic eldincluded,weherede ne

H1≡κ22a2s+ks 2πG|k|Σs

0,

(24)

H2≡κ2222g+k(ag+CA) 2πG|k|Σg0,

(25)

G1≡2πG|k|Σs0,(26)

G2≡2πG|k|Σg0,

(27)

whereΣs0andΣg

0aregivenbyequations(17)and(18).In

additiontotheappearanceofC2

Ainequation(25),Σg0givenbyexpression(18)alsocontainsthemagnetic elde ect.Dispersionrelation(23)canbecastintotheformof

ω4 (H1+H2)ω2+(H1H2 G1G2)=0,

(28)

MHDstabilitycriteriaforcompositeMSIDs

5

withthetworootsω22

+andω givenbyω2±(k)=(H1+H2)/2

H2)2Similartothe±[(proofH1+ofShen 4(&H1H2Lou(2003, G1G2)]1/2/2.

(29)

2004a),theω2+

rootremainsalwayspositive.Incontrast,ω2

maybecomenegative,leadingtoaxisymmetricMSIDinstabilities.Sub-stitutionsofexpressionsH1,H2,G1,G2andde nitions(17)and(18)andexpression(20)intoequation(29)for

theminus-signsolutionwouldgiveω2

intermsof vedi-mensionlessparametersD2

s,K≡|k|r,δ,βandλ2,namely

ω2 (k)

=

a2s

β

,(33) ≡B4K4+B3K3+B2K2+B1K+B0,

(34)B4≡[1 (1+λ2)/β]2,

(35)B3≡2(1+y)(δ 1)[1 (1+λ2)/β]/(1+δ),

(36)

B2≡[y2+2y 3+(8β 4 2λ2+2λ4+2βλ2)/β2],(37)

B1≡4(1+y)(1 δ)[1 1/β+λ2/(2β)]/(1+δ),

(38)B0D≡4[1 1/β+λ2/(2β)]2,

(39)

wherey≡2

s.TheanalysishereparallelsthatofShen&Lou(2003);thenovelmagnetic elde ecttobeexplorediscontainedinthedimensionlessparameterλ2.

Asaresultoftheone-to-onecorrespondencebetweenD22sandDgdictatedbyexpression(20),itisstraightforwardtoderiveanequivalentformofexpression(30)intermsofD2ginsteadofD2s.MathematicalsolutionsofD2gandD2s

becomeunphysicalforeitherD22

g<0orDs<0orbothbe-ingnegative.Onecanreadilyshowfromcondition(20)thatD22s<Dgforβ≥1(Lou&Zou2004).Therefore,itsu ces

torequireD2

s>0.Inthesubsequentanalysis,wemainly

useD2

s–thesquareoftherotationalMachnumber–toexaminetheaxisymmetricstabilitypropertyinacompositeMSIDsystem.

Bysettingλ2=0forzeromagnetic eldinexpressions(30) (39),theyallreducetothecorrespondingexpressionsforacompositesystemoftwocoupledunmagnetizedSIDsanalyzedbyShen&Lou(2003).Forscale-freediscsmoregeneralthanSIDs,thereaderisreferredtotheworkofSyer

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

6Y.-Q.LouandY.Zou

&Tremaine(1996),Shen&Lou(2004a,b)andShen,Liu&Lou(2005).Toderiveane ectiveMHDQMparameterforacompositediscsystemofoneSIDandonegaseousMSID,

wemustdeterminethevalueofKminatwhichω2

reachestheminimumvalue.

3.1The

D2s Criterion

intheWKBJRegime

Beforede ningane ectiveQMparameter,we rstshow

unambiguouslytheDsandcon rmourearlier interpretationscriterionforaxisymmetricforthemarginalstabilitysta-bilitycurvesinacompositeMSIDsystem(Lou&Zou2004).

Accordingtosolution

(30),ω2

isafunctionofKandD2.Bysettingω2

s =0andassigningvaluesofparametersδ,βandλ2insolution(30),weendupwithanequation

forD2.Contoursofω22

sandK inDsandKwithvarious

combinationsofδ,βandλ2

aregivenbelowtocomparewithourmarginalstabilitycurvesobtainedearlier(Lou&Zou2004;seealsoShen&Lou2003,2004a,b,Shen,Liu&Lou2005andLou&Wu2005).

Asanexampleofillustration,weset|m|=0,δ=0.2,β=1.5andλ2=1anddeterminenumericallycontour

curvesofω22

intermsofDsandKasdisplayedinFig.1.

Physically,thetworegionslabelledω2

<0inthelower-leftandupper-rightcornersareunstable,whiletheregionla-belledbyω2

>0isstableagainstaxisymmetriccoplanarMHDperturbations.Forcomparison,weshowtheglobalmarginalstabilitycurveinacompositeMSIDsystemwiththesameparametervaluesinFig.2( gure11ofLou&Zou2004),whereαisadimensionlesse ectiveradialwavenum-ber(seeShuetal.2000;Lou2002;Lou&Shen2003;Lou&Zou2004).IntheWKBJapproximationoflargeKandα,thetwoupper-rightsolidcurvesinFigs.1and2showgoodmutualcorrespondencefortheringfragmentationinstabil-ity.Thusourpreviousinterpretationfortheglobalstation-aryaxisymmetricMHDperturbationcon gurationasthemarginalstabilitycurveiscon rmedbytheWKBJanaly-sishere.Incomparison,thecorrespondenceinthesmallKregimeisqualitativewithapparentdeviations;theWKBJapproximationworksbetterforalocalanalysis,whiletheresultsofFig.2areglobalandexactwithouttheWKBJapproximation.Itisclearthatthiscomparisonrevealsthephysicalnatureofthedemarcationcurvesastheaxisym-metricstabilityboundaries.

Byanω2 contourplotforD2

sversusK,thestablerangeofD2

sintheWKBJapproximationcanbereadilyidenti ed.Forexample,inFig.1with|m|=0,δ=0.2,β=1.5andλ2=1,thecompositeMSIDsystemhasastablerangefromD20.1205atK=0.5075toD2s=s=6.3428atK=3.5149.IntheWKBJapproximation,weexplorenumericallyandshowsomequalitativetrendsofvariationsforthemarginalstabilitycurveswithparametersδ,βandλ2inFigs.3 5.Ingeneral,theincreaseofδandβtendstomakeacompositeMSIDsystemmorevulnerabletoinstability(compareFigs.1,5and6),whiletheincreaseofthemagnetic eldstrengthλ2expandsthestablerangeandreducesthedangerofin-stabilities(compareFigs.3and4).

Figure1.Acontourplotofω2 asafunctionofKandD2s

with|m|=0,δ=0.2,β=1.5andλ2=1.Thetwoseparated

regionslabelledω2

<0inthelower-leftandupper-rightcornersareunstable.Thetwosolidcurvesmarkω2

=0.Figure2.TheglobalmarginalstabilitycurveofD2s

versusef-fectivedimensionlessradialwavenumberαfor|m|=0,δ=0.2,β=1.5andλ2=1[see gure11ofLou&Zou(2004)].

Figure3.Acontourplotofω2 asafunctionofKandD2s

with|m|=0,δ=0.2,β=1andλ2=0.09.Thetwoseparated

domainslabelledbyω2<0areunstable.Thetwosolidcurvesmarkω2

=0.

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

Figure4.Acontourplotofω2 asafunctionofKandD2with|m|=0,δ=0.2,β=1andλ2s=3.61.Thetwodomainslabelled

byω2 <0areunstable.Thetwosolidcurvesmarkω2 =0.

Figure5.Acontourplotofω2 asafunctionofKandD2s

with|m|=0,δ=0.2,β=10andλ2=1.Thetwodomainslabelled

byω2 <0areunstable.Thetwosolidcurvesmarkω2

=0.

Figure6.Acontourplotofω2 asafunctionofKandD2s

with|m|=0,δ=1,β=10andλ2=1.Thetworegionslabelledbyω2 <0areunstable.Thetwosolidcurvesmarkω2

=0.MHDstabilitycriteriaforcompositeMSIDs

7

Figure7.TheglobalmarginalstabilitycurveofD2s

versusef-fectivedimensionlessradialwavenumberαfor|m|=0,δ=0.2,β=10andλ2=1[see gure13ofLou&Zou(2004)].

Theroleofcoplanarringmagnetic eldinstabilizing

acompositeMSIDsystemcanbephysicallyunderstoodasfollows.Accordingtodispersionrelation(23),weseethat,intheMSIDfactor,thegasandmagneticpressuretermsareexplicitlyassociatedwiththesquareoftheradialwavenum-ber|k|,whilethebackgroundsurfacemassdensitytermislinearin|k|.Fortheringfragmentationinstabilitythatoc-cursatrelativelylargeradialwavenumbers,thetwopressuretermsdominateandtheincreaseofmagneticpressuretendstoenhancetheaxisymmetricstabilityofacompositeMSIDsystem.FortheJeanscollapseinstabilitythatoccursatrela-tivelysmallwavenumbers,thebackgroundgassurfacemassdensitytermintheMSIDfactorbecomesdominantoverthetwopressureterms.BytherotationalMHDradialforcebal-ancecondition(18),thebackgroundgassurfacemassden-sitytendstobereducedbytheincreaseofmagnetic eldstrengthandthustheJeanscollapseinstabilitytendstobesuppressed.Inaddition,theright-handsideofdispersionre-lation(23)representsthemutualgravitationalcouplinginthepresenceofcoplanarMHDperturbations.Areductionofbackgroundgassurfacemassdensitywillweakenthiscou-plingandthusincreasetheaxisymmetricstability.

Forafurthercomparison,wereproducetheglobalmarginalstabilityresultsof gure13inLou&Zou(2004)hereasFig.7,whichhasthesamesetofparametersasFig.5forthelocalWKBJsolutionresults.Wenoteagainthatfortheunstableregion(upperright)oflargeradialwavenum-ber,labelledastheringfragmentationinLou&Zou(2004),Figs.5and7showverygoodcorrespondenceasexpected.Asreference,Table1containsseverallistsfortheoverall

stablerangeofD2

swithm=0anddi erentsetsofpa-rametersδ,βandλ2

,includingtheresultsbothfromhereusingtheWKBJapproximationandfromthoseofLou&Zou(2004)forexactglobalMHDperturbationcalculationsinacompositeMSIDmodel.

Asshownabove,axisymmetricstabilitypropertiesofacompositeMSIDsystemcanbequalitativelyunderstoodusingtheWKBJanalysis.Nevertheless,theWKBJapproxi-

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

8Y.-Q.LouandY.Zou

Table1.ThestablerangeofD2s

foraxisymmetriccoplanarMHDperturbationsofanywavelengthsfordi erentvaluesofδ,βandλ2intheWKBJapproximation.ThevaluesinparenthesesarethosedeterminedbyLou&Zou(2004)forglobalmarginalsta-bilitycurvesandaremoreaccurate,especiallyfordescribingtheJeanscollapseregimeinvolvinglargeradialspatialscales.

0.2

1.510.1205(0.7695)6.3428(6.1554)0.210.090.1594(0.9063)5.9573(5.7561)0.213.610.1111(0.6783)7.9494(7.7905)0.21010.1184(0.7534)4.5499(4.4310)11010.0396(0.4063)2.2787(2.2434)

[A2K2

2r2

min+A1Kmin+A0 1/2]

=a2s

( 1/2and de nedbyequation

A2K2

min(34)takes A1K(41)

min)

,onthevalueatKmin.AlthoughtheformsofthesemathematicalexpressionsarestrikinglysimilartothoseofElmegreen(1995),allrelevantcoe cientsandvariablescontainthee ectofmagnetic eldthrougnλ2.Byde nition(41)ofQ2Eabove,itisclearthat

forQ22

E>1,theminimumofω >0andthusthecom-positeMSIDsystemwouldbestableagainstaxisymmet-riccoplanarMHDperturbationsofarbitraryradialwave-lengths.ThisgeneralizedMHDparameterQ2Ecorresponds

tothestablerangeofD2

swhereDsistherotationalMachnumberofthestellarSIDmodelledasa uid.

Theformidableappearanceofω2

givenbyequation(40)wouldpreventusfromderivingastraightforwardana-lyticalexpressionofKmin.Nonetheless,insteadofminimiz-ingω2

directly,itismuchsimplertodeterminethecriticalvalueKcforKcorrespondingtotheminimumofvariable

W≡ω22

+ω .Byequation(28),wehaveW≡ω22

+ω =H1H2 G1G2.

(42)

ForpossibleextremaofW,therelevantcubicequationthatKcshouldsatisfyis

dW

dK

=0,

(43)

orequivalently

dK3+aK2+bK+c=0,

(44)withthefourcoe cientsexplicitlyde nedd≡4 1+λ2

by

(δ+1),

(45)a≡ 3 βδ+λ2+1

(y+1),

(46)b≡2(δ+1) 2βy 2+λ2+2β+2yλ2+2y

,(47)c≡ (y+1) 2βy+2βyδ 2+λ2+2β

.

(48)

Formostparameterregimesunderconsideration,thereis

onlyonerealsolutionforthecubicequation(44).ThisrealsolutionKctakesthelengthybutstraightforwardformofKc=(x q/2)1/3+( x q/2)1/3 a/(3d),

(49)

wherex≡(q2/4+p3/27)1/2,p≡(b/d) (a/d)2/3andq≡2(a/d)3/27 ab/(3d2)+c/d.WethenusethisKctoestimateKminandtodeterminethee ectiveMHDQEparameterasthegeneralizationofQE′.RelevantcurvesofMHDQ2E

versusD2

scorrespondingtodi erentvaluesofδ,βandλ2aredisplayedinFigures8 10.

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

Figure8.SeveralcurvesofQ22EversusDsfordi erentβvalues

withspeci edparameters|m|=0,δ=0.2andλ2=1.Foreach

Q2twointersectionpointsatQ2stableEcurve,therangeofDE=1bracketthe

2s.ThisstableD2s

rangeshrinksasβincreases.

Figure9.SeveralcurvesofMHDQ22relevantparametersEversusDsfordi erentδ

valuesasindicated.Other|m|=0,β=10andλ2=1are xed.ForeachMHDQ2tionpointsatQ2stableEcurve,thetwointersec-rangeofD2E=1bracketthes.ThisstableD2srangeshrinksasδincreases.

Byvaryingparametersδ,βandλ2,weobserveseveral

trendsofvariationinthepro leofMHDQ22

EversusDsasdisplayedinFigs.8 10.Whenβincreaseswith xedδandλ2values,orδincreaseswith xedβandλ2values,the

stablerangeofD2

sshrinksingeneral,whiletheincreaseofλ2tendstoexpandthestablerangeofD2s.SimilarvariationtrendshavebeennoticedearlierfortheDsexactglobalperturbationsolutionsinour compositecriterionandMSIDthemodel(Lou&Zou2004).

Byde nition(41)fortheMHDQEparameter,onede-terminesthestablerangeofD2

sintheWKBJapproxima-tionasshowninFigures8 10.Forexample,givenδ=0.2,β=1.5andλ2=1inFig.8,wehaveQ2E>1for

0.12<D2

s<6.34;thecompositeMSIDsystemisthusstableagainstaxisymmetriccoplanarMHDperturbations

withinthisrangeofD2

s.Morenumericalresultsforthesta-blerangesofD2

sfordi erentparametersaresummarizedin

MHDstabilitycriteriaforcompositeMSIDs

9

Figure10.SeveralcurvesofMHDQ222Otherrelevantparameters|mEversusDsfordi erentλ

values.|=0,δ=0.2andβ=1.5are xed.Foreachcurve,thetwointersectionpointsatQ2bracketthestablerangeofDThisstableDE=1

2s.2srangeexpandson

bothendsasλ2increases.

Table2.ApproximatestablerangesofD2determinedbythecriterionQ2s

≥1inacompositesystemof(M)SIDs.TheD2s

val-uesoutsidetheparenthesesaregivenbytheMHDQEparameter

generalizingthatofElmegreen(1995),whiletheD2s

valuesin-sidetheparenthesesarederivedfromtheMHDQJparameter

generalizingthatofJog(1996).ThesetwosetsofD2s

valuescor-respondingtoMHDQEandMHDQJparametersarenearlythe

sameandthestableD2s

rangesareveryclosetothevaluesgivenbyourexactglobalDs criterionusingtheMHDperturbationprocedureofLou&Zou(2004;seealsoTable1here).

0.2

1.510.12(0.12)6.34(6.34)0.210.090.16(0.16)5.96(5.96)0.213.610.11(0.11)7.95(7.95)0.21010.12(0.12)4.55(4.55)11010.04(0.04)2.28(2.28)

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

10Y.-Q.LouandY.Zou

Table3.NumericalvaluesfortheminimaofD2s

ringfragmen-tationcurve(Lou&Zou2004)andthecorrespondingvaluesofe ectiveMHDQparameters,includingboththeMHDQEpa-rametergeneralizingQE′ofElmegreen(1995)andtheMHDQJparametergeneralizingQJ′ofJog(1996).

0.21.516.15541.02161.02230.210.095.75611.02481.02470.213.617.79051.01441.01590.21014.43101.01841.018811012.24341.01151.0102

[κ2g+k2(a22+

2πGkΣs0

g+CA)]

2πGkΣs0

[κ2g+k2(a2g+C2A)]

+

2[β(1+y) 1+λ2/2]+K2(1+λ2)+

K(1+y)/(1+δ)Kmin(1+y)/(1+δ)

(1+

Q2J)

≡F=

(52)

2[β(1+y) 1+λ2/2]+K2

min(1+λ2)

[seeequations(5)and(6)ofJog(1996)andequation(30)

ofShen&Lou(2003)].ItfollowsimmediatelythatQ2J>1andQ2J<1correspondtoaxisymmetricMHDstabilityandinstability,respectively.GivenQ2J>1,itindeedfollowsthat

H1H2 G1G2>0fortheω2

correspondingtoKmin.That

is,theminimumofω2

ispositiveforarbitraryKandcon-sequently,thecomposite(M)SIDsystemisstableagainstaxisymmetriccoplanarMHDdisturbances.TheprocedureofobtainingtheMHDQJparametercanbesummarizedas

follows.Foragivensetofδ,β,λ2andD2

s,one rstdeter-minesthevalueofKminnumericallyusingequation

(30).ByinsertingthisKminintoequation(52),onethenobtainsthenumericalvalueofMHDQ2JinacompositeMSIDsystem

forthegivensetofδ,β,λ2andD2

s.

WehaveexploredrelevantparameterregimesofinterestandrevealedseveralqualitativevariationtrendsofMHDQ2JparameterasshowninFigures11 13.Similartothe

MHDQ2parameter,therangeofD22

EsforMHDQJ>1correspondstotheaxisymmetricstabilityofacompositeMSIDsystem.CorrespondingtoMHDQ2J>1,wehave

calculatedstablerangesofD2

sgivenseveralsetsofδ,βandλ2andthedetailedresultsaresummarizedinTable2along

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

MHDstabilitycriteriaforcompositeMSIDs

11

Figure12.SeveralcurvesofMHDQ22|m|JversusDsfordi erentδ

values.Otherrelevantparameters=0,β=10andλ2=1are xed.ForeachMHDQ2atQ2bracketthestableJcurve,thetwointersectionpoints

rangeofD2s.ThisstableD2J=1srangeshrinksasδincreases.

Figure13.SeveralcurvesofMHDQ222Otherrelevantparameters|mJversusDsfordi erentλ

values.|=0,δ=0.2andβ=1.5

are xed.ForeachQ2curve,thetwointersectionpointsatQ21bracketthestableJrangeofD2.ThisstableD2rangeexpandsJ=

asλ2ss

increases.

withthosecorrespondingtotheMHDQ2Eparameter.Itis

apparentthatthestablerangeofMHDD2

sgivenbyMHDQ2

JMeanwhile,≥1isalmostinFiguresthesame11 13,asthatwerevealgiventhebyMHDsamevariationQ2E≥1.trendsasthoseobtainedbyMHDQ2EandbyexactglobalMHDperturbationprocedureofLou&Zou(2004).Whenβincreaseswith xedδandλ2values(seeFig.11),orδincreaseswith xedβandλ2values(seeFig.12),thestable

rangeofD2

stendstoshrink,whiletheincreaseofλ2tends

toexpandthestablerangeofD2

s(seeFig.13).

Onceagain,wefollowthesimilarprocedureofLou&Zou(2004)torevealthecloserelationshipbetweenthemin-imumoftheD2

smarginalstabilitycurveandthee ectiveMHDQparameter.Forspeci edvaluesofparametersδ,βandλ2,wecomputetheminimumoftheringfragmen-tationD2

scurveusingtheexactglobalMHDperturbation

procedureofLou&Zou(2004),inserttheresultingD2

sinto

expression(52)andobtainthevalueofMHDQ2Jparame-tercorrespondingtotheminimumoftheringfragmentationcurve.Forexample,givenδ=0.2,β=1.5andλ2=1in gure11ofLou&Zou(2004)(orFigure2here),themin-imumofD2

s

intheringfragmentationcurveis～ingde nition(52),weobtainthecorrespondingMHDQ2J=1.02223.Moredetailednumericalresultsaresumma-rizedinTable3forreference.

Bythesenumericalexperiments,wedemonstratethat

thevaluesofMHDQ22

JandMHDQEcorrespondingtothe

minimaofD2

s

ringfragmentationcurvesarenearlythesame,withtherelevantvaluesofMHDQ2Jbeingveryclosetounity.Therefore,thee ectiveMHDQ2JparameterisalsopertinenttotheMHDringfragmentationinstabilityforaxisymmetriccoplanarMHDperturbationsinacompositeMSIDsystem.

IncomparisontothedeterminationofMHDQ2E,thesearchfortheMHDQJvaluerequiresnumericalexplo-rationsforeachgivenD2

s.Themajoradvantageisthatthede nition(52)forMHDQ2Jremainsvalidfortheentirepa-rameterregimeandavoidsimpropersituationsofunusualsetsofδ,βandλ2.

3.3

ACompositePartialMSIDSystem

Inmostdiscorspiralgalaxies,thereareoverwhelmingob-servationalevidencefortheexistenceofmassivedarkmatter

haloesingeneral.Toincludethelarge-scalegravitationalef-fectofamassivedarkmatterhalo,weaddagravitationalpotentialΦtermassociatedwiththedarkmatterhaloinourbasicMHDequations(8),(9),(11)and(12),whereΦispresumedtobeaxisymmetricforsimplicityandforthelackofinformation.BasedonN bodynumericalsimulationsforgalaxyformation,typicalvelocitydispersionsofdarkmatter‘particles’arefairlyhigh(morethanafewhundredkilome-terspersecond).Hence,anothermajorsimpli cationofouranalysisistoignoreperturbationresponsesofthemassivedarkmatterhalotocoplanarMHDperturbationsinacom-positeMSIDsystem(e.g.,Syer&Tremaine1996;Shuetal.2000;Lou2002;Lou&Fan2002;Lou&Shen2003;Shen&Lou2004a,b).Asbefore,weintroduceadimensionlessratioF≡ /( +Φ)forthefractionofthediscpotentialrelativetothetotalpotentialallinabackgroundequilib-riumstate(e.g.,Syer&Tremaine1996;Shuetal.2000;Lou2002;Lou&Shen2003;Lou&Zou2004).Theback-groundrotationalMHDequilibriumofacompositeMSIDisthusstronglymodi edbythisadditionalΦterm.fore,wewrite s=asDs/r, g=agDg/r,κs=

√

Asbe-2 g.Itfollowsfromtheradialforcebalanceinthe

(M)SIDsystemthatthebackgroundsurfacemassdensitiesnowbecome

Σsa20

=F

s(1+D2

s)

2πGr(1+δ)

,(54)

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

12Y.-Q.LouandY.Zou

Figure14.Anω2 contourplotasafunctionofKandD2s

withm=0,δ=0.2,β=1.5,λ2=1andF=0.1.Theregionlabelled

byω2

<0isunstableandtheJeanscollapseregimedisappearscompletely.IncomparisonwithFig.1,itisapparentthatthe

stablerangeofD2s

isenlargedasFdecreases.where0≤F<1forapartialcompositeMSIDandF=1

forafullcompositeMSIDthathasbeenstudiedindetailintheprevioussubsections.PerformingthestandardMHDperturbationanalysis,welinearizedependentphysicalvari-ablesbutignoredynamicalfeedbacksfromthemassivedarkmatterhalotocoplanarMHDperturbationsintheMSIDsystem.IntheWKBJapproximation,itisthenstraightfor-wardtoderiveastrikinglysimilardispersionrelationintheformofequation(23)butwithmodi edbackgroundequilib-riumproperties(53)and(54).FollowingthesameprocedureofDstheω 2

criterionanalysisdescribedinsection4.1,weobtain

contourplotasafunctionofstellarrotationalMach

numberD2

sandradialwavenumberK≡|k|rwiththepo-tentialratioFasanadditionalparameter.TypicalresultsaredisplayedinFig.14asanexampleofillustration.

TheexampleofFig.1withF=1correspondstoafullcompositeMSIDsystemwherenobackgrounddarkmatterhaloisinvolved,ashasbeenstudiedinsubsection3.1.AsFbecomeslessthan1correspondingtoanincreaseofthepotentialfractionofthemassivedarkmatterhalo,thestable

rangeofD2

sbecomesenlargedasclearlyshowninFigure14

forthecaseofF=0.1.Fromtheseω2

<0contourplots,itisapparentthattheintroductionofamassivedarkmatterhalotendstostabilizeacompositeMSIDsystemasexpected(Ostriker&Peebles1973;Binney&Tremaine1987;Lou&Shen2003).Forlate-typespiralgalaxies,onemaytakeF=0.1orsmaller.SuchacompositepartialMSIDsystemarestableagainstaxisymmetriccoplanarMHDdisturbances

inawiderangeofD2

s.3.4

QuantitativeEstimatesand

GalacticApplications

Forournumericalexamples,thediscmassdensityratiopa-rameterδ≡ΣgΣs

0/0hasbeentakentobe0.2and0.02(seeFig.12).Forlate-typespiralgalaxies,thisratiorangesfrom0.05to0.15.Forrelativelyyoungandgas-richspiralgalax-

ies,thisratioδcanreach0.2andevenhigher.Innearbyspiralgalaxies,thestrengthofmagnetic eldistypicallyin-ferredtobeafewto10µG;bytheequipartitionargument,magnetic eldstrengthmayreachafewtensofµGincir-cumnuclearregionsandwithintowardsthecentre(e.g.,Louetal.2001).Weshalltaketheratioλ≡CA/agtobeoftheorderof1.Asestimates,wehavetakenβ≡(as/ag)2tobe1,1.5,10,and30inFigure11.Forspiralgalaxies,theratioas/agcanbeoftheorderoforgreaterthan5or6.WiththeseestimatesinFigs.7 12,weseeclearlythatwithoutamassivedarkmatterhalo,atypicaldiscgalaxysystemwithasu cientlyfastrotation(e.g.,aVsring-fragmentationinstability.～150 Although250kms 1)wouldsu erthethepresenceofmagnetic eldo ersthestabilizinge ect(seeFig.9)againstthering-fragmentationinstability,thedevel-opmentofsuchinstabilitywouldbeunavoidablefortypi-callyinferredgalacticmagnetic eldstrenghs.ByFig.14,theinclusionofamassivedarkmatterhaloholdsthekeytopreventsuchring-fragmentationinstability.

IntheobservationalstudiesofKennicutt(1989)onglobalstarformationratesinspiralgalaxies,thetheoret-icalrationaleisthereforenotsu cientlystrong.Firstly,theapplicationoftheToomorestabilitycriterionforasinglediscistoosimpletobephysicallysensible.Secondly,foracompositediscsystemwithoutmagnetic eld,thegeneral-izedcriteria(Elmegreen1995;Jog1996;Lou&Fan1998)fortheToomreinstabilitycannotbereadilyappliedtoarealgalacticdisc.Thirdly,evenforacompositediscsystemwithacoplanarmagnetic eldwhichshowsclearstabilizingef-fects,theMHDring-fragmentationwouldoccurfortypicallyinferredparametersofaspiralgalaxywithoutasu cientlymassivedarkmatterhalo.Finally,ourconclusionisthere-forethatinrelatingToomre-typeinstabilitieswithglobalstarformationratesinspiralgalaxies,oneshouldconceivenewphysicalrationalesbyincorporatingthedynamicalin-terplaybetweenamassivedarkmatterhaloandamagne-tizedcompositediscsystem(Wang&Silk1994;Silk1997;Lou&Fan2002a,b).

4SUMMARYANDDISCUSSION

Inthispaper,weexaminedtheaxisymmetricMHDlinearstabilitypropertiesofacompositesystemconsistingofastellarSIDandagaseousMSIDcoupledbythemutualgravityandamassivedarkmatterhalo,usingtheWKBJapproximation.Ourmainpurposeistocon rmthephysicalinterpretationfortheglobalmarginalstabilitycurve(Lou&Zou2004)andtoestablishtheMHDgeneralizationoftheQparameter(Safronov1960;Toomre1964)foracom-positeMSIDsysteminreferencetotheearlierwork(e.g.,Elmegreen1995;Jog1996;Lou&Fan1998a,b;Lou2002;Lou&Shen2003;Lou&Zou2004).Wenowsummarizethemaintheoreticalresultsbelow.

WehaverecentlyconstructedexactglobalsolutionsforstationarycoplanarMHDperturbationsinacompositesys-temofastellarSIDandagaseousMSIDforbothaligned

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

MHDstabilitycriteriaforcompositeMSIDs

13

andunalignedlogarithmicspiralcases(Lou&Zou2004).Lou&Zou(2004)haveextendedtheanalysesofShuetal.(2000)onanisopedicallymagnetizedSID,ofLou(2002)onasinglecoplanarlymagnetizedSID,andofLou&Shen(2003)andShen&Lou(2003)onanunmagnetizedcompos-iteSIDsystem.Inabroaderperspective,acompositeMSIDsystemisonlyaspecialcasebelongingtoamoregeneralclassofcompositescale-freemagnetizeddiscsystems(Syer&Tremaine1996;Shen&Lou2004a,b;Shenetal.2005).InanalogyofShuetal.(2000),Lou&Shen(2003)andShen

&Lou(2003),Lou&Zou(2004)naturallyinterpretedD2

smarginalcurvesforstationaryaxisymmetriccoplanarMHDperturbationswithradialpropagationsasthemarginalsta-bilitycurves.Basedonthelow-frequencytime-dependentWKBJanalysishere(Shen&Lou2003,2004a),weestab-lishthephysicalscenarioforthepresenceofthetwounstableregimesreferredtoasthe‘MHDringfragmentationinsta-bility’andthe‘MHDJeanscollapse’inacompositeMSIDsystem.Consequently,itisintuitivelyappealingandphys-icallyreliabletoapplyourexactglobalDscompositeMSIDsysteminordertoexamine itscriterionaxisymmet-foraricstabilityandobtainthestablerangeofD2

s

,whereDsistherotationalMachnumberofthestellarSID.

IntheWKBJapproximation,werelateourMHDDsthe MHDcriterionregime,tothenamely,twoe ectivetheMHDQparametersQEextendedtoizingthatofElmegreen(1995)andtheMHD criterionQJgeneral-generalizingthatofJog(1996).However,ourprocedures criteriondi erfromthoseofElmegreen(1995)andofJog(1996),be-causeourMHDbackgroundofrotationalequilibriumisdy-namicallyself-consistentwithκs=κgingeneral.WeshowthatMHDgeneralizationsofbothQEleadtonearlythesamestablerangefor D2

andQJ criteria

sextendedtotheMHDrealm.Thiscon rmsthecloserelationbetweenourMHDDsComplementarily, criterionweandshowthethatMHDtheQEvalues andoftheQJ MHDcriteria.QE

andQJcorrespondingtotheminimaoftheD2

sringfrag-mentationcurvesintheexactglobalMHDperturbationpro-cedureofLou&Zou(2004)areallclosetounity.Ourin-terpretationoftheaxisymmetricmarginalstabilitycurveasthedemarcationofstableandunstableregimesisphys-icallysensible,andtheMHDQEandQJparametersareassociatedwiththeMHDringfragmentationinstabilityinacompositeMSID.

Finally,wehaveshowntheaxisymmetricMHDstabilitypropertyofacompositepartialMSIDbyincludingthegrav-itationale ectfromanaxisymmetricmassivedarkmatterhalo.Itisapparentthatthedarkmatterhalohasastrongstabilizinge ectforacompositeMSIDsystem.

Inadditiontotheoreticalinterestofdiscinstabili-tiesforforminglarge-scalegalacticstructures(n.b.,non-axisymmetriconesarenotstudiedhere),therehasbeenakeendesiretosomehowrelatesuchinstabilitiestoglobalstarformationratesindiscgalaxiesandtheirevolution(e.g.,Jog&Solomon1984a,b;Kennicutt1989;Wang&Silk1994;Silk1997).Theoverallchainofstarformationpro-cessesfromlarge-scalediscinstabilities,togiantmolecular

clouds,tocloudcollapses,toclustersofstars,todiscac-cretionontoindividualstarsandsoonisquitecomplicatedandinvolvesmanyscalesofdi erentordersofmagnitudes.Conceptually,large-scaleaxisymmetricringstructuresinadiscmustbefurtherbrokendownnon-axisymmetricallyintosmallerpiecesinordertoinitiatethisconceivedchainofcol-lapses.Whilevariousstagesofthis‘chain’havebeeninten-sivelystudiedseparately,theultimaterelationorconnectionbetweenthelarge-scaleaxisymmetricinstabilitiesandtheglobalstarformationrateremainsunclearandneedstobeestablished(e.g.Elmegreen1995;Lou&Bian2005).

Observationally,Kennicutt(1989)attemptedtoinferanempiricalrelationbetweentheQparameterofthestellardiscaloneandtheglobalstarformationrate.Wang&Silk(1994)pursuedasimilarideawithanestimateofQparam-eterforacompositediscsystemoftwo uiddiscs;however,theirapproximationforQparametermaybeo toomuchundervariousrelevantsituations(Lou&Fan1998b,2000a).Shouldthislineofreasoningindeedcontainanelementoftruthforaddressingtheissueoftheglobalstarformationrate,thentheQparameteradoptedshouldreallycorrespondtothatofacompositediscsystemwithamagnetizedgasdisccomponentandinthepresenceofamassivedarkmat-terhalo.Thebasicphysicalreasonbehindthissuggestionisthatstarsformdirectlyinthemagnetizedgasdiscunderthejointgravitationalin uenceofthedarkmatterhalo,thestellardiscandthemagnetizedgasdiscitself.Ifthislineofreasoningdoesindeedmakephysicalsense,thenaninterest-ingpossibilityarises.Thatis,thedarkmatterhalomayplayanimportantroleofregulatingglobalstarformationratesindiscgalaxiesandthusgalacticevolution.Forexample,ifthemassofadarkmatterhaloisverymuchgreaterthanthemassofacompositediscsystem,thenstarformationactivitiesbecomeweaker.Ontheotherhand,ifthedarkmatterhaloisnotsu cientlymassive,thenthediscsystemrapidlyevolvesintoabarsystem.Itisalsopossiblethatthedarkmatterhaloisonlymarginaltomaintainastabilityofacompositedisc.Inthiscase,theglobalstarformationactivitiesinthediscsystemproceedinaregulatedmanner.

OurMHDmodelanalysisinthispaper,highlyidealizedinmanyways,doescontainseveralrequisiteelementsfores-tablishinganMHDgeneralizationoftheQparameterandthecorrespondingcriterionforaxisymmetricstabilityorin-stability.Observationally,itwouldbeextremelyinterestingtoexaminetherelationbetweentheMHDQMparameteringalacticsystemsandglobalstarformationrates.Thisisnotexpectedtobeatrivialexercisegivenvarioussourcesofuncertainties.

Bypresumingthatsuchaxisymmetricdiscinstabilities

characterizedbyeitherD2

sorQMparametersmightsome-howhintatorconnecttotheglobalstarformationrate,thereareafewmodelproblemssimilartothecurrentonethatcanbeexploredfurther.Forexample,themodelsofShen&Lou(2004b)andShen,Liu&Lou(2005)canbecombinedtoconstructacompositediscsystemconsistingoftwoscale-freediscswiththegaseousonebeingcoplanarlymagnetizedinthepresenceofadarkmatterhalo.Likewise,

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurati

14Y.-Q.LouandY.Zou

theworkofLou&Wu(2005)canbegeneralizedtotwocoupledscale-freediscwiththegaseousonebeingisopedi-callymagnetizedinthepresenceofdarkmatterhalo(Lou&Wu2006inpreparation).Therealsituationismorecom-plicated.Wehopetheseanalysesmayo ercertainhintsandinsightsfordi erentaspectsoftheproblem(Lou&Bai2005inpreparation).

ACKNOWLEDGEMENTS

ThisresearchhasbeensupportedinpartbytheASCICen-terforAstrophysicalThermonuclearFlashesattheUni-versityofChicagounderDepartmentofEnergycontractB341495,bytheSpecialFundsforMajorStateBasicSci-enceResearchProjectsofChina,bytheTsinghuaCen-terforAstrophysics,bytheCollaborativeResearchFundfromtheNationalNaturalScienceFoundationofChina(NSFC)forYoungOutstandingOverseasChineseSchol-ars(NSFC10028306)attheNationalAstronomicalOb-servatory,ChineseAcademyofSciences,byNSFCgrants10373009and10533020(YQL)attheTsinghuaUniversity,andbythespecialfund20050003088andtheYangtzeEn-dowmentfromtheMinistryofEducationthroughtheTs-inghuaUniversity.YQLacknowledgessupportedvisitsbyTheoreticalInstituteforAdvancedResearchinAstrophysics(TIARA)ofAcademiaSinicaandNationalTsinghuaUni-versityinTaiwan.ThehospitalityandsupportofSchoolofPhysicsandAstronomy,UniversityofStAndrews,Scotland,U.K.,andofCentredePhysiquedesParticulesdeMarseille(CPPM/IN2P3/CNRS)etUniversit´edelaM´editerran´eeAix-MarseilleII,Francearealsogratefullyacknowledged.A liatedinstitutionsofYQLsharethiscontribution.

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