Axisymmetric stability criteria for a composite system of st

1、axisymmetric stability criteria for a composite system of st 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 dephysiquedesparticulesdemarseille(cppm)/cent

2、renationaldelarecherchescienti que(cnrs)/institutnationaldephysiquenucleaireetdephysiquedesparticules(in2p3)etuniversitedelamediterraneeaix-marseilleii,163,avenuedeluminycase902f-13288marseille,cedex09,france;3departmentofastronomyandastrophysics,theuniversityofchicago,5640ellisave,chicago,il60637,u

3、sa;4nationalastronomicalobservatories,chineseacademyofscience,a20,datunroad,beijing100012,china. accepted2021.received2021.;inoriginalform2021 abstract usingthe uid-magneto uidformalism,weobtainaxisymmetricstabilitycriteriaforacompositediscsystemconsistingofstellarandgaseousmagnetizedsingularisother

4、maldiscs(msids).both(m)sidsarepresumedtoberazor-thinandaregravitationallycoupledinaself-consistentaxisymmetricbackgroundequilibriumwithpower-lawsur-facemassdensitiesand atrotationcurves.thegaseousmsidisembeddedwithanon-force-freecoplanarazimuthalmagnetic eldb(r)ofradialscalingr 1/2.louzourecentlycon

5、structedexactglobalstationarycon gurationsforbothaxisym-metricandnonaxisymmetriccoplanarmagnetohydrodynamic(mhd)perturbationsinsuchacompositemsidsystemandproposedthemhdds criteriaforaxisym-metricstabilitybythehydrodynamicanalogy.inadi erentperspective,wederiveandanalyzeherethetime-dependentwkbjdispe

6、rsionrelationinthelow-frequencyandtight-windingregimetoexamineaxisymmetricstabilityproperties.byintroducingarotationalmachnumberdsfortheratioofthestellarrotationspeedvstothestellar 2 numericallytovelocitydispersionas,onereadilydeterminesthestablerangeofds establishtheds criteriaforaxisymmetricmsidst

7、ability.thosemsidsystemsro-tatingeithertoofast(ringfragmentation)ortooslow(jeanscollapse)areunstable. 2 thestablerangeofdsdependsonthreedimensionlessparameters:theratioforthealfvenspeedtothesoundspeedinthegaseousmsid,theratioforthesquareofthestellarvelocitydispersiontothegassoundspeedandtheratiofort

8、hesurfacemassdensitiesofthetwo(m)sids.ourwkbjresultsof(m)sidinstabilityprovidephys-icallycompellingexplanationsforthestationaryanalysisoflouzou.wefurtherintroduceane ectivemhdqparameterforacompositemsidsystemandcomparewiththeearlierworkofelmegreen,jogandshenlou.asexpected,anaxisym-metricdarkmatterha

9、loenhancesthestabilityagainstaxisymmetricdisturbancesinacompositepartialmsidsystem.intermsoftheglobalstarformationrateinadiscgalaxysystem,itwouldappearphysicallymoresensibletoexaminethemhdqmstabilitycriterionagainstgalacticobservations.relevancetolarge-scalestructuresindiscgalaxiesarealsodiscussed.

10、keywords:mhdwavesism:magnetic eldsgalaxies:kinematicsanddynam-icsgalaxies:spiralstar:formationgalaxies:structure. 1introduction incontextsofgalacticstructures,discstabilitiesandglobalstarformationratesinspiralgalaxies,wederiveherein-stabilitycriteriaforaxisymmetriccoplanarmagnetohydro-dynamic(mhd)pe

11、rturbationsinacompositediscsystem withanazimuthalmagnetic eldinthegasdisccomponentandestablishageneralizedde nitionofane ectivemhdqmparameterappropriatetosuchamagnetizedgravita-tionalsystem.formulatedassuch,thisisanidealizedandlimitedtheoreticalmhddiscproblemyetwithseveralkeyconceptualelementsinclud

12、ed.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 tremaine1987andbertinlin1996andextensiverefer-encestherein).theoriginalstudi

13、esofaxisymmetricinstabil-itieswereconductedbysafronov(1960)andtoomre(1964)whointroducedthedimensionlessqparameterforthelocalstability(i.e.q1)againstaxisymmetricring-likedistur-bances.fordiscgalaxies,itwouldbemorerealisticandsen-sibletoinvestigateacompositediscsystemconsistingofastellardisc,amagnetiz

14、edgasdiscandamassivedarkmat-terhalo.therehavebeenextensivetheoreticalstudiesonthistypeofcompositetwo-componentdiscsystemsofvar-ioussub-combinations(linshu1966,1968;kato1972;jogsolomon1984a;bertinromeo1988;romeo1992;elmegreen1995;jog1996;loufan1998b,2000a,b;loushen2021;shenlou2021,2021a,b;louzou2021;

15、louwu2021).inparticular,therehavebeenseveralstud-iestryingtode neapropere ectiveqparameterforlocalaxisymmetricinstabilitycriterioninacompositediscsys-tem(elmegreen1995;jog1996;loufan1998b,2000a,b;shenlou2021,2021a,b).fromdi erentperspectives,theseanalyseso erinsightsforinstabilitypropertiesofacompos

16、itediscsystemandprovideatheoreticalbasisforunderstandingthelarge-scaledynamicsinsuchasystem(e.g.loufan2000a,b;loushen2021;shenlou2021,2021a,b;louzou2021;louwu2021). themainmotivationhereistoexplorebasicproper-tiesthemsidmodelinacompositesystemandobtaincon-ceptualinsightsforastrophysicalapplicationsi

17、nmagnetizedspiralgalaxiesandinestimatingglobalstarformationratesindiscgalaxies(e.g.,loubian2021).intheoreticalstud-iesofmodelingdiscgalaxies,theclassofsidmodelshasadistinguishedhistorysincethepioneeringworkofmestel(1963)(zang1976;toomre1977;lemos,kalnajslynden-bell1991;lynden-belllemos1993;syertrema

18、ine1996;goodmanevans1999;charkrabarti,laughlinshu2021)andhasgainedconsiderableattentionandinter-estsrecentlybyconsideringacompositediscsystemandbyincorporatinge ectsofmagnetic eld(shuetal.2000;lou2021;loufan2021;loushen2021;shenlou2021,2021a,b;shen,liulou2021;louzou2021;louwu2021).speci cally,shueta

19、l.(2000;seealsogallietal.2021)studiedglobalstationary(i.e.,zeropatternspeed)perturba-tioncon gurationsinanisopedicallymagnetizedsidwith-outinvokingtheusualwkbjortight-windingapproxima-tion.theyobtainedexactglobalsolutionsforbothalignedandunalignedaxisymmetricandnon-axisymmetriclogarith-micspiralcon

20、gurationsandinterpretedtheaxisymmetricsolutionforperturbationswithradialpropagationsasde-marcatingtheboundariesbetweenthestableandunstableregimes.bytheseaxisymmetricinstabilities,asidwitha su cientlyslowrotationspeedwouldjeanscollapseinducedbyperturbationsoflargerradialscales,whileasidwithasu cientl

21、yfastrotationspeedmaysu ertheringfragmen-tationinstabilityinducedbyperturbationsofsmallerradialscales(see g.2ofshuetal.2000).byintroducingaro-tationalmachnumberd,de nedastheratioofthesidrotationspeedvtotheisothermalsoundspeeda,thecrit-icalvaluesofthehighestandlowestdforanaxisymmetricstabilitycanbede

22、termineddirectlyfromthemarginalsta-bilitycurve.tosupporttheirphysicalinterpretations,theyinvokedthewell-knowntoomreqparameterandfoundthatthehighestd,namelytheminimumoftheringfragmenta-tioncurve,correspondstoaqvalueveryclosetounity,thusheuristicallysuggestingthecorrespondencebetweenthed criterionandt

23、heq criterion. di erentfromyetcomplementarytotheanalysisofshuetal.(2000)onasingleisopedicallymagnetizeddisc,lou(2021)studiedglobalcoplanarmhdperturbationsinasin-glemsidembeddedwithanazimuthalmagnetic eldandrevealedthattheminimumofthemhdringfragmenta-tioncurveinthismsidmodelistightlyassociatedwiththe

24、generalizedmhdqmparameteroriginallyintroducedbyloufan(1998a)indevelopingthegalacticmhdden-sitywavetheory(fanlou1996).foracompositesystemoftwocoupledunmagnetizedhydrodynamicsids,loushen(2021)constructedstationaryglobalperturbationcon- gurationsandshenlou(2021)suggestedastraightfor-wardd criterionfort

25、heaxisymmetricringfragmentationinstabilityinsuchasystemonthebasisofalow-frequencywkbjanalysis;theyrevealedthattheminimumoftheringfragmentationintheircompositesidmodelisagaincloselyrelatedtoapropere ectiveqparameter(elmegreen1995;jog1996).furthermore,foracompositesidsystemwithanisopedicallymagnetized

26、gaseoussidandastellarsidinthe uiddescription,louwu(2021)haveconstructedglobalstationarymhdperturbationstructuresandexam-inedstabilitypropertiestoanticipateasimilard criterioninparalleltothecaseofshenlou(2021).meanwhile,shenlou(2021b)havefurthergeneralizedbothworkofsyertremaine(1996)andloushen(2021)t

27、othesitu-ationofacompositesystemfortwogravitationallycoupledscale-freediscs;theyalsostudiedtheaxisymmetricstabilityofsuchacompositesystemintermsofthemarginalstabilitycurvesandproposedadslou(2021). criterionbytheanalogyofshenwehaverecentlyexaminedtwo-dimensionalcoplanarmhdperturbationsinacompositesys

28、temconsistingofastellarsidandagaseousmsid.bothsidsareexpedientlyapproximatedasrazor-thinandthegasdiscisembeddedwithanon-force-freecoplanarmagnetic eld(louzou2021).inthis uid-magneto uidmsidmodelapproxima-tion,weobtainedexactglobalstationarymhdsolutionsforalignedandunalignedlogarithmicspiralperturbat

29、ioncon- gurationsinsuchacompositemsidsystem,expressedintermsofthestellarrotationalmachnumberds.inrefer-encetotheresultsofasinglesid(shuetal.2000;gallietal.2021;lou2021),itwouldbenaturaltosuggestthat using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite dis

30、c system consisting of stellar and gaseous magnetized singular isothermal discs (msids). lou zou2021).incomparisonwith thesinglesidcase,thestablerangeofd2 sisreducedasaresultofthemutualgravitationalcouplingbetweenthetwosids.however,incomparisonwiththecaseofacompositeunmagnetizedsidsystem(loushen2021

31、;shenlou 2021),thestablerangeofd2 sexpandsconsiderablyduetothepresenceofacoplanarmagnetic eld. tocon rmheuristicargumentsfortheaboveanalogyandourintuitivephysicalinterpretations,weconductinthispaperalow-frequencytime-dependentstabilityanalysisinthewkbjortight-windingapproximationforthecompos-itemsid

32、system(shenlou2021,2021a;louzou2021).weshalldemonstrateunambiguouslythevalidityofdemar-catingthestableandunstableregimesbythestellarrota-tionalmachnumberds.toplaceouranalysisinpropercontexts,wealsodiscussspeci callyhowthetwoe ectiveqparametersofelmegreen(1995)andofjog(1996)arere-latedtoourdsparamete

33、rwhenotherrelevantparametersarespeci ed,andshowthatthetwoe ectiveqparame-tersarepertinenttotheringfragmentationinstabilityinacompositemsidsystem. insection2,wederivethetime-dependentdisper-sionrelationusingthewkbjapproximationforpertur-bationsinacompositemsidsystemandintroduceafewkeydimensionlesspar

34、ameters.insection3,wepresentthedsite msidcriterionsystemandbeingtwoe ectivestableagainstqparametersarbitraryforaxisymmet-acompos-ricperturbationswithradialpropagations.mainresultsanddiscussionsaresummarizedinsection4. 2afluid-magnetofluidformalism weconsiderbelowacompositesystemconsistingoftwogravit

35、ationallycoupled(m)sidswithoneofthesidsbeingmagnetizedandthusreferredtoasmsid.forphysicalvari-ables,weuseeithersuperscriptorsubscriptstoindicateanassociationwiththestellarsidandeithersuperscriptorsubscriptgtoindicateanassociationwiththegaseousmsid.thestellarsidandthegaseousmsidcanhavecon-stantyetdi

36、erentrotationalspeedsvsandvg(relatedtothephenomenonofasymmetricdriftinthegalacticcontext);wethuswritethebackgroundangularrotationspeeds softhestellarsidand gofthegaseousmsidas s=vs/r=asds/r (1) and g=vg/r=agdg/r, (2)separately,whereasandagaretheconstantvelocitydis-persionofthestellarsidandtheisother

37、malsoundspeedofthegaseousmsid,respectively;dsanddgarethecor-respondingrotationalmachnumbers.therelevantepicyclicfrequenciesintermsof sand garegivenby 22 s s dr (r2 s)=2 2s (3) mhdstabilitycriteriaforcompositemsids 3 and 2g g 2 dr (r2 g)=2 2g, (4) respectively.similartoasinglemsid,wetaketheback-groun

38、dazimuthalmagnetic eldtobeintheformof b1/2(r)=fr , (5) wherefisaconstant(lou2021;loufan2021)and br=bz=0. (6) foramoregeneralpower-lawradialvariationoftheaz-imuthalmagnetic eldandthoseofotherrelatedback-groundvariables,theinterestedreaderisreferredtoarecentanalysisofshen,liulou(2021). inthe uidapprox

39、imationofastellarsid,themassconservation,theradialcomponentofthemomentumequa-tionandtheazimuthalcomponentofthemomentumequa-tionaregivenbelowinorder,namely s (rsus) (sjs) rr2 +u s u s us1 t r2 r3= r s j s js s t+ur2s ,(9)whereusistheradialcomponentofthebulk owvelocityofthestellarsid,jsrvsisthestellar

40、speci cangularmomentumalongthez direction,vsistheazimuthalcom-ponentofthestellarbulk owvelocity, isthetotalgrav-itationalpotential,sistheverticallyintegratedpressure,andsisthesurfacemassdensityofthestellarsid. inthemagneto uidapproximationforthegaseousmsid,themassconservation,theradialcomponentofthe

41、momentumequationandtheazimuthalcomponentofthemomentumequationaregivenbelowinorder,namely g (rgug) (gjg)rr2 +ug ug ug 1 t r2 r3= r dzb g r br g g t +ug j jgr2g +1 (rb) 4 (12), whereugistheradialcomponentofthegasbulk owveloc-ity,jgrvgisthegasspeci cangularmomentuminthez direction,vg istheazimuthalcomp

42、onentofthegasbulk owvelocity,gisthetwo-dimensionalgaspressure,gisthegassurfacemassdensityandbrandbaretheradialandazimuthalcomponentsofthemagnetic eldb.thelasttwotermsontheright-handsidesofequations(11)and(12)aretheradialandazimuthalcomponentsofthelorentzforceduetothecoplanarmagnetic eld.thetwosetsof

43、 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). l

44、ou enwavespeedinthemsidde nedc2a by dzb2/(4g0). (19)apparently,equations(17)and(18)requires a2d2222 s(1+s)=ag(1+dg) ca/2. (20) wenowintroducetwomoreusefuldimensionlessparameters here.the rstparametera22 s/agstandsforthesquareoftheratioofthestellarvelocitydispersiontotheisothermalsoundspeedinthemsida

45、ndthesecondparameter2c22a/agstandsforthesquareoftheratioofthealfv enwavespeedtotheisothermalsoundspeedinthemsid.inlate-typediscgalaxies,thestellarvelocitydispersionasisusuallyhigherthanthegassoundspeedag,wethusfocusonthecaseof1(jogsolomon1984a,b;bertinromeo1988;jog1996;elmegreen1995;loufan1998b;lous

46、hen2021;shenlou2021,2021a,b;louwu2021). beforegoingfurther,wenotethatatypicaldiscgalaxysysteminvolvesamassivedarkmatterhalo,astellardiscandagaseousdiscofinterstellarmedium(ism)onlargescales,wheretheismdiscismagnetizedwiththemagneticenergydensity(1ev/cm3)beingcomparabletotheen-ergydensitiesofthermali

47、smandofrelativisticcosmic-raygas(e.g.loufan2021).tocomprehensivelyun-derstandmulti-wavelengthobservationsoflarge-scalespi-ralstructuresofdiscgalaxiesandtodeveloppotentiallypowerfulobservationaldiagnostics(loufan2000a,b),itwouldbemorerealisticandnecessarytotakeintoaccountofmagnetic elde ectsinacompos

48、itemagnetizeddisc-halomodel. whilethereareexceptions,galacticmagnetic elds thecosmic-raygasissetasideheremerelyforthesakeofsim- typicallytendtobecoplanarwiththediscplaneofaspiral galaxyonlargescales.onsmallerscales,regionsofclosedandopenmagnetic eldsaremostlikelyintermingledbythesolaranalogy(e.g.lou

49、wu2021).asa rststep,lou(2021)carriedoutacoplanarmhdperturbationanal-ysisforstationaryalignedandunalignedlogarithmicspiralstructuresinasinglemsidembeddedwithanazimuthalmagnetic eldanddemonstratedthattheminimumoftheringfragmentationcurveinthismsidmodelisclearlyre-latedtothegeneralizedmhdqmparameter(lo

50、ufan1998a).sincethebackgroundmhdrotationalequilibriumadoptedbyloufan(1998a)isnotanmsidmodel,itwouldbemoresatisfyingtojustifythestatementandin-terpretationoflou(2021)inadynamicallyself-consistentmanner.weshallde neaqmparametersimilartothatoflou(2021)andshowthatthisqmisequivalenttothatofloufan(1998a).

51、 forasinglemsid,wereadilyderivelinearequations(bysettingrelevantparametersforthestellarsidtovanish)forcoplanaraxisymmetricmhdperturbationswithfourierharmonicdependenceexp(ikr+it),wherekistheradialwavenumberandistheangularfrequency.intheusualwkbjortight-windingapproximationofkr 1,weobtainthelocalwkbj

52、dispersionrelationformhddensitywavespropagatinginanmsidintheformof 2=2222g+k(ag+ca) 2g|k|g 0, (21) whichisthemhdgeneralizationofthewkbjdispersionrelationforcoplanarperturbationsinanunmagnetizedsid. toderiveane ectiveqmparameterfortheaxisym-metricstability(i.e.,2withanarbitraryk,thedeterminant0)again

53、stofmhdtheright-handperturbationssideofequation(21)shouldbenegativeforallk.thisrequires g(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

54、shen(2021),thedispersionrelationderivedhereforcoplanarmhdperturbationsinacomposite(m)sidapproachisqualitativelysimilartothosepreviouslyobtainedbyjogsolomon(1984),elmegreen(1995)andjog(1996)inspirit,butonemajordistinctionisthatintheformulationofourcomposite(m)sid,therotationspeedsofthetwosidsvsandvga

55、redi erentingeneral.wethushaveinequalitys=g,whileinthoseearlieranalyses,s=gwaspresumedapriorifollowingtheassumptionofvs=vg. thisisconceptuallyrelatedtothephenomenonofasymmetricdrift(e.g.binneytremaine1987).physically,stellarvelocitydispersionsmimicapressure-likeforceforthestellarcomponent,whiletheth

56、ermalismgasandmagneticpressureforcestogetheractonthemagnetizedgascompo-nent.inthesamegravitationalpotentialwelldeterminedbythetotalmassdistribution,thedi erenceinthestellarpressure-likeforceandthesumofthegaseousandmagneticpressureforceswouldleadtodi erentvsandvgandthustheasymmetricdrift.theraresitua

57、tionofvsandvgbeingequalmayhappenunderveryspecialcircumstances. 3 axisymmetricstabilityanalysis foracompositemsidsystem wedescribebelowcoplanarmhdperturbationanalysisfortheaxisymmetricstabilityofacompositemsidbasedonalow-frequencytime-dependentwkbjapproach.generaliz-ingthenotationsofshenlou(2021)yetwiththee ectofmagnetic eldincluded,weherede ne h122a2s+ks 2g|k|s 0, (24) h22222g+k(a