197 9ApJ. . .232 . .259G The AstrophysicalJournal,232:259-276,1979August15 © 1979.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. panion (Lamb1977;Lightman,Rees,andShapiro neutron starsaccretingmatterfromabinarycom- sources (Pounds1977;Formanetal.1978)andthe strong evidencethatthesearerotatingmagnetic periods oftheseX-raysourcesduetothetorque larly significantbecausetheyprovidethemostcom- exerted onthestarbyaccretingmatterareparticu- of suchstars.Thechangesobservedinthepulsation pelling evidencethatthesourcesareindeedneutron the accretionflownearneutronstar(Eisnerand stars (PringleandRees1972;Lamb,Pethick, at theUniversityofIllinoisand NASAcontractNAS5-23315 Pines 1973;RappaportandJoss1977;Mason1977), magnetic momentsofthesestarscanbesubstantially and becausetheyfurnishcluesaboutthepatternof improved, accuratemeasurementsofperiodchanges Indeed, ifourknowledgeofaccretionflowsandthe Lamb 1976;Lamb,Pines,andShaham1976,1978a,b). 1978) havegeneratedgreatinterestintheproperties at Caltech. 2 1 The discoveryofmorethanadozenpulsatingX-ray AlfredP.SloanFoundationResearch Fellow. Researchsupportedinpartby NSFgrantPHY78-04404 © American Astronomical Society • Provided by theNASA Astrophysics Data System field penetratestheinnerpartofdiskviaKelvin-Helmholtzinstability,turbulentdiffusion, The transitionzoneiscomposedoftwoqualitativelydifferentregions,abroadouterwhere the startomagnetosphericflownearstar.Usingtwo-dimensionalhydromagnetic and reconnection,producingabroadtransitionzonejoiningtheunperturbeddiskflowfarfrom magnetospheric boundaryofaneutronstaraccretingmatterfromdisk.Thestellarmagnetic the angularvelocityisKeplerianandanarrowinnerzoneorboundarylayerwhereitdeparts equations, wecalculatetheinnerandouterradiiofthiszoneitsradialverticalstructure. the boundarylayerbutbecomesincreasinglydistortedatlargerradii. significantly fromtheKeplerianvalue.Thestar’smagneticfieldisonlyslightlydeformedwithin to thesurfaceofstar,resultingaccretiontorque,andpatternfallingplasmaat plasma channeledbythemagnetosphericfieldfallsinacircularringatmagneticpoles. accreting X-raysources,consideringinturntheflowofmatterfrominneredgedisk stellar surface.Because^20%ofthestar’smagneticfluxthreadsdiskoutsideitsinneredge, Subject headings:hydromagnetics—stars:accretionmagneticneutron We investigatetheinteractionbetweenstellarmagneticfieldandaccretingplasmaat We discusstheimplicationsofflowsolutionsfoundhereforneutron-starmodels I. INTRODUCTION ACCRETION BYROTATINGMAGNETICNEUTRONSTARS. Department ofPhysics,UniversityIllinoisatUrbana-Champaign;and II. RADIALANDVERTICALSTRUCTUREOFTHE X-rays :binaries Department ofPhysics,UniversityIllinoisatUrbana-Champaign 1 TRANSITION ZONEINDISKACCRETION Received 1978August3;accepted1979February13 California InstituteofTechnology 2 ABSTRACT F. K.Lamb P. Ghosh AND 259 could providenewconstraintsontheequationofstate the configurationofmagneticfieldinside Lamb, andPethick1977,hereafter,PaperI),wepre- of matteratveryhighdensities(Lamb1977). influenced bytheangularmomentumtransportedto of accretionbyrotatingmagneticneutronstars(Ghosh, torque onthestarisdeterminedbymatchingflows the starbyaccretingmatter.Pointingoutthat magnetosphere, andshowedthatbotharestrongly sented solutionsfortheflowofaccretingmatterand inside andoutsidethemagnetosphereinaphysically used toderiveinterestingboundsontheaccretion acceptable way,weshowedthatthesesolutionscanbe in thetransitionzonebetweenexteriorflowand torque evenwithoutknowingthedetailsofflow laws. Theseboundsdemonstratethattheearlydimen- magnetosphere, byapplyinggeneralconservation and Pines1973)ofthesignmagnitudetorque sional estimate(PringleandRees1972;Lamb,Pethick, is correctifthetransitionzone isnarrow,andthatin this casesameestimate isalsoapproximately on slowlyrotatingneutronstars accretingfromadisk correct evenforfastrotators. Weconsideredavariety of exteriorflows,anddiscussed theimplicationsofour In thefirstpaperreportingresultsofourstudy 197 9ApJ. . .232 . .259G results forsecularperiodchangesinpulsatingX-ray 260 innermost radiusofthediskandaccretiontorque model ofsteadyaxisymmetricdiskaccretionbyan detailed descriptionoftheflowintransitionzone. actual valueoftheaccretiontorquedoesrequirea sources. Finally,wenotedthatdeterminationofthe zone andthereforeenablesustocomputeboththe aligned rotatorwhichincludesadetaileddescription of theradialandverticalstructuretransition The resultingdissipationisroughlyconsistentwith that thisflowissteady,anduseassumptionto Keplerian regionandthestellarsurface.Weassume at aconsistentdescriptionoftheflowbetween acting onthestar.Thismodelrepresentsafirstattempt determine theeffectiveconductivityofdiskplasma. that expectedfrommagnetic-fluxreconnection.The term periodfluctuationsandspin-downepisodes been measured,includingHerX-l,whiletheshort- accretion torquepredictedbythemodelisingood all thepulsatingX-raysourcesinwhichthisratehas from themodelasconsequencesoffluctuationsin agreement withtheobservedsecularspin-upratesof observed inHerX-landCenX-3follownaturally torque mayalsoexplaintheparadoxicalexistence(see episodes issomewhatreduced.Thissamebraking a brakingtorquesufficienttoaccountfortheobserved mass-accretion rate.Inparticular,themodelpredicts spin-down episodesiftheaccretionrateduring Paper I)ofalargenumberlong-periodX-raysources. (Ghosh andLamb1979)wecalculatetheaccretion processes thatoccurthere,whileinasubsequentpaper and structureofthetransitionzonephysical tions describedhere,andcomparetheimpliedchanges torque predictedbyourmodel,usingtheflowsolu- of theaccretionflowinthismodel,includingsize transition zoneisnotnarrow,butinsteadrather the mainconclusionsreachedinpresentpaperare in stellar-rotationrateswiththoseobserved.Among beyond itsinneredge,and(2)thatasaresult,the (1) thatthestellarmagneticfieldthreadsdiskwell broad. Abriefsummaryofourprincipalresultshas been givenpreviously(GhoshandLamb1978). calculate thestructureofaccretionflow.Wefirst field linesislimitedbyreconnection,andthereby We assumethatinasteadystatethisdistortionof the Kelvin-Helmholtzinstability,turbulentdiffusion, disk overasizableareaastheresultofdevelopment show thatthestellarmagneticfieldwillpenetrate and magnetic-fluxreconnection.Thestellarfieldlines in determiningtheflow.The transitionzoneiscom- is unaffectedbythestellar magneticfield,butina is determinedbytheeffectiveviscosityindiskand arrive atthefollowingpictureofaccretionflow. and radialdirectionsbythemotionofdiskplasma. that threadthediskaredistortedinbothazimuthal and themagnetosphere, stressassociatedwiththe broad transitionzonebetween theunperturbeddisk stellar magneticfieldbecomes increasinglyimportant Far fromthestar,motionofaccretingmatter Recently, wehavedevelopedatwo-dimensional In thepresentpaperweprovideadetailedaccount In §IIweoutlinethemodelandmethodusedto © American Astronomical Society • Provided by theNASA Astrophysics Data System GHOSH ANDLAMB 21/ posed oftwoparts,abroadouterpart,wherethe value. part whereitdepartssignificantlyfromtheKeplerian azimuthal velocityisKeplerian,andanarrowinner velocity andthemagneticfieldchangeonalength behaves likeaboundarylayer,inthatboththeflow equations intheinnertransitionzone.Thiszone matter leavesthediskvertically,fallingtoward magnetospheric fieldtypicallybyafactorof~5,and scale $«r.Herescreeningcurrentsreducethe plasma acrosstheresidualmagnetosphericfieldadds calculated in§IV.Herethemagneticstresscarriesa to theheatgeneratedindiskbyviscousdissipation. star alongstellarfieldlines. length scale~r. and Sunyaev1973).Currentsflowinginthiszone is verysimilartothatinastandarda-disk(Shakura Even so,thestructureofaccretionflowinthiszone dissipation associatedwiththemotionofdisk significant fluxofangularmomentumwhileenergy screen theresidualmagnetosphericfieldtozeroona namely, anarrowinnerzonewheremostofthescreen- that thestructureoftransitionzonefoundhere, ing occurstogetherwithabroadouterzonewherethe residual stellarfluxthreadsthedisk,islikelytobea existence ofamaximumstellar-rotationrateconsistent general featureofdiskaccretion.Next,wediscussthe transition zone,thestabilityofflowfrom with steadyaccretion,theunusualnatureofinner 0 inner edgeofthedisktoneutronstar,and new featurestobeexpectedinaccretionbyoblique in pulsatingX-raysources.Wealsonotesomeofthe emission fromthestellarsurfaceandperiodchanges implications ofoursolutionsforthepatternX-ray temporaneous workofScharlemann(1978)and with theworkofotherauthors,includingcon- Ichimaru (1978). rotators. Finally,wecomparethepresentcalculations assume thatthestarisrotatingwithangularvelocity star withadipolarmagneticfieldofmomentWe perpendicular totheplaneofdisk.Weassume everywhere. Inthemathematicalanalysiswhichfollows, further thattheflowissteadyandhasaxialsymmetry whose originisatthecenterofneutronstarand we usethecylindricalcoordinatesystem(¿ü,,z) the cylindricalradius,andr=(œ+z)is whose polarz-axisliesalongthestellar-rotationaxis. Q aboutthemagneticfieldaxis,andthatthisaxisis In thiscoordinatesystem,istheazimuthangle,cö (semithickness h«r),then inside itr^¿ó. distance fromthecenterof the star.Ifdiskisthin plasma isnotpossibleandthen introduceamodelfor screening ofthestellarmagnetic fieldfromthedisk s In §IIIwesetupandsolvethehydromagnetic The structureofthebroadoutertransitionzoneis In §Vwediscussourflowsolutions.First,argue Our mainconclusionsaresummarizedin§VI. In thisstudyweconsiderdiskaccretionbyaneutron In thepresentsectionwefirst showthatcomplete II. THEMODEL Vol. 232 197 9ApJ. . .232 . .259G 3 2 1 12 2 that threadthedisk.Finally,wedescribeoverall the steady-stateconfigurationofstellarfieldlines No. 1,1979 analysis. picture oftheaccretionflowthatemergesfromour Arons andLea1976;EisnerLamb1977;Michel the magnetosphereandoutsideplasmacompletely star iswellknown(seeLamb,Pethick,andPines1973; of roughlysphericalaccretionbyamagneticneutron keep thefieldconfinedtointeriorofmagneto- screen thestellarmagneticfieldfromplasmaand case ofaccretionfromathindisk,however.Tosee this, supposethatcurrentsonthesurfacesofdisk spheric cavity.Thispictureisnotself-consistentinthe 1911 a,b,c):currentsinthetransitionregionbetween field, andwouldbecompletelyexcludedfromthedisk. B ~pr~isthestrengthofunscreenedstellar the magnetosphericfieldthreadsdiskplasma,ona oppositely directedandofmagnitude~B,where spheric fieldaboveandbelowthediskwouldbe were toscreenthediskcompletely.Thenmagneto- Here k=f+k^isthewavevectorofmode, time scalemuchshorterthantheradialdriftof But thisconfigurationwillevolvetowardoneinwhich discuss here: result ofavarietyprocesses,threewhichwe disk plasma.Thustheoriginalassumptionthat magnetospheric fieldisexcludedfromthedisknot region andthediskdrivesKelvin-Helmholtz continuity betweenthelow-densitymagneticfield g =(GMIr)(hlr)ofthegravitationalacceleration v =r(^—4*)<0.1cisthevelocitydiscontinuity self-consistent. Thefieldpenetratesthediskasa in termsoftheKeplerianangularvelocityQand instability withagrowthrate field regionandthediskplasma,v=(Bf/top) between thediskplasmaandstellarmagneticfield (compare Northrop1956andScharlemann1978). teristic wavenumberdefinedintermsofthemagnitude 0 perpendicular totheinterfacebetweenmagnetic angular velocity£lofthestar,k=gJvisacharac- 0 Northrop’s expansionschemetothenexthigherorder;for unaffected bycompressibility.Onecanseethistaking satisfy k«k#.Inthisregime,thegrowthrateisalmost is anAlfvénvelocitydefinedintermsoftheradial r unstable modes,thegrowthrateisonlyincreasedbyanamount component oftheexternalmagneticfieldand external magneticfieldandp is thegaspressureindisk. times theleadingterm,where pisthepressureof plasma densitypinthedisk.Thisexpressionfor/kh ing effectsofmagnetictensionandgravityifk#> z QK shows thattheinterfaceisunstabledespitestabiliz- K A sgA r e&a mag a) FailureofCompleteScreeninginDiskAccretion 1 The pictureofmagneticfieldscreeninginthecase 1. Kelvin-Helmholtzinstability.—Thevelocitydis- Inthepresentcase,v«candmodesofinterest 0 © American Astronomical Society • Provided by theNASA Astrophysics Data System 2ô01 ~ i(Wc)(/>mag/p)(/WP) 2l Vkh ~(k^Vçflc-Kkk)v gA -49/2 -1 < 2X10ar(dInpl^hi />) 8e&s ACCRETION BYMAGNETICNEUTRONSTARS 2 -74/526/53/512/51 231 303 8 _1 17_8 8 21 2 10 kclv »andk{^kf)islargerthanthecritical wavenumber k=\clv^). and B~2?,thecharacteristicwavenumberkmay k ä1.1x10a/x3o(Af/Afo)^i7r8cm, be expressedas where aistheviscosityparameterofdisk,plju- Thus thecriticalwavenumberkis~10r" in unitsof10gausscm,Misthemassneutron radius. Assumingthatkcanbeassmall27r/r,the star, Misthemass-accretionratetilinunitsof at r~10cmandincreasesrapidlywithincreasing r0 the instabilitytis in thedisk,r~|p|,tolinear-growthtimeof (those withk>),theratioofradial-drifttime of magnetictension.Then,fortheunstablemodes stabilizing effectofgravitycompletelydominatesthat cg 10 gs,andristheradiusinunitsofcm. Thus, theKelvin-Helmholtzinstabilitygrowsinitially r0g time. SincefcA/27r~2atr10cm,themarginally g interface tobecomecomparableh,andwetherefore the nonlinearregimeforperturbation8Aof unstable modesonlyneedtodevelopashortwayinto thorough mixingofthediskplasmaandstellar expect theKelvin-Helmholtzinstabilitytolead on atimescalemuchshorterthantheradial-drift 30 magnetic fieldinlessthantheradial-drifttime. cg vection, asinsomediskmodels(Liang1977;Shakura, where uisthermsvelocityofturbulence,exter- gradient inthediskleadstovigorous,large-scalecon- r 17 in atimeT~h^'.Here7jistheturbulentdiffusivity, nal magneticfieldwilldiffuseverticallythroughthedisk Sunyaev, andZilitinkevich1978),pu>BIStt, which maybeestimatedas0.15w*/*(Parker1971), the a-diskrelationul~ach,wherecissound where listhelengthscaleoflargesteddies.Using kh dr c velocity, onefinds 8 Thus forreasonablevaluesofa(>10'),theexternal c in lessthantheradial-drifttime. magnetic fieldwilldiffusetothemidplaneofdisk netic fieldwithinthediskplasma,neartopand below thediskwillreconnecttosmall-scalemag- to oneanother,drivenbydifferentialrotationand reconnection ofthesmall-scalemagneticfieldloops bottom surfacesofthedisk.Subsequentshearingand t circulation inconvectivecells, willcausethemagneto- t 0 spheric fieldtopenetrateinside thedisk.Theefficiency ts t of thisprocessdependson the strengthofsmall- scale magneticfieldwithin the diskandrateof 21/5/5710 For astandarda-disk(ShakuraandSunyaev1973) 2. Turbulentdiffusion.—Iftheverticaltemperature Tlr Ä8.6Xl0a^r-(M/M).(3) 3. Reconnection.—Themagneticfieldaboveand dt17o8 55/5/8 T/T -1.4X10a~V-30“\M¡M©) dKH 7,f12 x (1-njQ)-vm)'.(2) K 261 a) 197 9ApJ. . .232 . .259G 3724/51/40 is everywherestrongenoughtosupplymostofthe reconnection betweenmagneticfieldsofunequal field isgreaterthanthatofthestellarmagnetic effective “viscous”stress(EardleyandLightman ratio oftimescales strength. Ifthesmall-scalemagneticfieldindisk 262 relative efficiencyofthisprocessisdescribedbythe reconnection ratebetweenmagneticfieldsofunequal above andbelowthediskatr>1.Further,if than T. field willpenetratethediskinatimemuchshorter strength isdeterminedbytheweakerfield,then standard diskthatthermsstrengthofthissmall-scale now theshapeassumedbymagneticfieldina thread thediskwellbeyonditsinneredge.Consider the diskplasmainatimeshortcomparedtoradial- halves ofthediskreconnect.Thus,actualvalue long beforeitbecomesthislarge,theoppositely plasma hadtheusualCoulomborBohmvalue, steady state. drift time.Asaresult,thestellarmagneticfieldwill cause themagnetosphericfieldtoinvadeandmixwith Once again,forreasonablevaluesof£theexternal depends ontheflowpattern(seeYasyliunas1975). reconnecting fieldandfisanumericalfactorwhich 2h(tv)-\ wherevistheAlfvénvelocityinweaker directed toroidalmagneticfieldsintheupperandlower equilibrium valueofB#wouldbeextremelylarge.But disk plasma.Iftheeffectiveconductivityof through thediskduetofiniteconductivityof TlT ^5xl0-/V30^i7~(M/M)r. exactly balancedbythediffusionofmagneticfield the increaseinB#duetotheseprocessesmustbe focusing oftheflowcompressesanyazimuthalfield Here wehaveusedthereconnectiontimescaletæ 1975), thenitcanbeshownfromthetheoryof between thestaranddisk,sinceifthisregionwere plasma onthosepartsofthefieldlineswhichlie that ispresent,therebyamplifyingit.Inasteadystate, muthal direction,therebygeneratingatoroidalfield disk requiresthepresenceofasmallamount to thestarstretchesstellarfieldlinesinazi- reconnect thereandnotoroidalfieldwoulddevelop. by thedifferentangularvelocitiesofstarand However, evenaverysmall quantityofplasma a perfectvacuum,thefieldlineswouldcontinually by theplasmaflowandreconnection. this regionfromthatofavacuum tothatcorrespond- ing toourpicture.Weillustrate thisbyestimatingthe suffices tochangetheelectrodynamic propertiesof 8 d number density«ofparticles aboveandbelowthedisk A dRo8 r In summary,thereareavarietyofprocesseswhich The angularmotionofthediskplasmawithrespect Our pictureofstellarfieldlinesstretchedazimuthally is determinedbyabalancebetweenamplification © American Astronomical Society • Provided by theNASA Astrophysics Data System from thepoloidalfieldB.Similarly,radial p b) Steady-StateConfiguration GHOSH ANDLAMB (4) -21 2 2 field linethreadingthediskdecreasestowardstar thus theelectron-ionrelativevelocitycaneasilybe the reasonableassumptionthatplasmainthis required tosupportthecurrentsimpliedbyazi- comparable to,butcannotexceed,thelocalsound current J=(c/47t)|VXB^lisaconductioncurrent; region isnotcharge-separated,therequiredpoloidal muthal pitchgivenbyequation(37)below.Making the stellarsurface,weobtain disk andassumethattheazimuthalpitchofastellar speed. Ifweestimatethesoundspeedbythatin vacuum electrodynamicproperties. is avacuum,dynamicallyspeaking,ithasverynon- force: althoughtheregionaboveandbelowdisk (1975) haspointedout,thestrengthofCoulomb how smallamassintheformofchargedparticles on alengthscaler,reducingtoverysmallvalueat field componentBfromthez-component.Again, a typicalplasmafield,andunderscores,asMestel matter fromabinarycompanion.Thisestimateshows certain tobepresentaroundaneutronstaraccreting flow andreconnection. the diskplane,therebygeneratingaradialmagnetic the diskplasmapinchesstellarfieldlinesinwardin in theazimuthaldirection,inwardradialdriftof suffices tochangeavacuumelectromagneticfieldinto Such asmalldensity(~10thatinthedisk)is that ispresent,andtheactualvalueofBdetermined the radialfocusingofflowamplifiesanyfield fine thestellarmagneticfieldinsideascreeningradius by abalancebetweenamplificationtheplasma everywhere betweenthediskplane andthesurfaceofstar perturbed accretiondiskandthemagnetosphere.The the diskconstitutesatransitionzonebetweenun- in followingthisdevelopment,tohavemindthe effect onthestellarmagneticfield.Itwillbehelpful, tions whichdescribeindetailtheaccretionflowandits corotate withthestar,Alfvénradiusr,insidewhich the star.Thustransitionzone,whichextendsfrom r insidewhichtheplasmaisforcedtocorotatewith motion ofthediskplasmaacrossstellarfieldlines stellar magneticfieldcontrolsthe accretionflow(seeLamb, the definitionofAlfvénradiuscanbemademathe- the flowissub-Alfvénicinstellarmagneticfield.Although r, whichmaybe10-10timeslargerthantheradius in thistransitionzonegeneratescurrentswhichcon- overall picturethatemergesfromouranalysis. do notyetknowwhethertheaccretion flowissub-Alfvénic meaningful basisforestimating theradiusinsidewhich matically precise(seePaperI,§ l\b) andprovidesaphysically the corotationradiusr,insidewhichplasmaisforcedto Lamb 1978),wefollowedwidespreadconventioninidentifying Pethick, andPines1973;Eisner and Lamb1977;PaperI),we p rz r A co s co 2 In amanneranalogoustothestretchingoffieldlines The regionwherethestellarmagneticfieldthreads In thenexttwosectionswesetupandsolveequa- Inourbriefsummaryofthepresentmodel(Ghoshand 23 n >0.2(yJ£)(1-Ü/Q)rcm".(5) sK8 c) ASelf-consistentPicture Yol. 232 197 9ApJ. . .232 . .259G No. 1,1979 the dominantstresstransportingangularmomen- r outto,isquitebroad.Withinthetransitionzone ficantly fromtheKeplerianvalue.Theboundary viscous stresstotheassociatedwithstellar tum oftheaccretingmatterchangesfromeffective parts, anouterpart,wheretheangularvelocityis magnetic field.Thiszonedividesnaturallyintotwo we arguethatthistwo-partstructureofthetransition between thesetwopartsdefinestheradiusr.In§V the magneticfieldchangethereonalengthscale like aboundarylayer,inthattheflowvelocityand zone islikelytobeageneralfeatureofdiskaccretion. Keplerian, andaninnerpart,whereitdepartssigni- flow-alignment radiusroneveryfieldlineoftheaccre- the angularvelocityofplasmaisreducedfrom netospheric field,typicallybyafactorof~5.The netic stress,andcirculatingcurrentsscreenthemag- Keplerian valuetothecorotationalbymag- cos lines threadingtheboundarylayerthereforedefines the diskandaccretingmatterbeginstofalltoward inner transitionzoneorboundarylayerisalsothe region wherethedominantmagneticstressdisrupts 8 «rç.Hencethezonehasathickness=r—~ the staralongstellarfieldlines.Thesetof field linesinthisregionareonlyapproximatestream- the accretionbundle,namely,thosefieldlinesalong 8 whichissmallcomparedtor.Inthiszone which matteraccretesontothestar.Theplasmain below ithassignificantcross-fieldmotion.Thusthe accretion bundleatthediskplaneandjustabove 0 lines oftheflow.Asmatterfallsclosertostar,its cross-field motionisdeceleratedbytheincreasing force ofthestarandapproachesfreefall.Thereisa aligned motionisacceleratedbythegravitational tion bundle,insidewhichthecross-fieldmotionof strength ofthestellarmagneticfieldwhileitsfield- f field-aligned motion.Thisradius,whichmustbeless than orequaltotheAlfvénradiusr(seePaperI),is in generaldifferentonfieldlines.Solutions accreting matterbecomesnegligiblecomparedwithits for thefield-alignedaccretionflowinsiderhavebeen 0co given inPaperI. 0 terminology basedontheproperties oftheflowthatare be stableundertheconditionsthat obtainthere(see§V).We in ourmodel,orevenwhetherthis isrequiredfortheflowto calculated here. have thereforepreferred,inthe presentaccount,tousea A f The innertransitionzonebetweenrandbehaves co0 © American Astronomical Society • Provided by theNASA Astrophysics Data System Inner transitionzoner8«Magneticstress Unperturbed diskrLargeEffectiveviscousstress Outer transitionzonerEffectiveviscousstress Magnetosphere RrMagneticstress co0 s Qs co Region RadiusVariablesTransport ACCRETION BYMAGNETICNEUTRONSTARS Distinct RegionsoftheAccretionFlow Inner OuterinPhysicalofAngularMomentum TABLE 1 has astructureverysimilartothatofstandarddisk field linestransportsangularmomentumbetweenthe the magneticstressassociatedwithtwistedstellar residual stellarmagneticfieldthreadingthedisk.First, except forthefollowingmodificationscausedby energy associatedwiththeeffectiveviscousstressis disk andthestar.Second,viscousdissipationof Weak screeningcurrentsaregeneratedinthiszoneby partial screeningbythecurrentflowinginbound- the residualstellarmagneticfieldwhichisleftafter the radialcross-fielddriftofdiskplasma;thus associated withthecross-fieldmotionofplasma. augmented bytheresistivedissipationofenergy kept confinedwithinthescreeningradiusrwhich marks theouteredgeoftransitionzone. ary layerisfurtherscreenedonalengthscale~rand accretion flowaresummarizedinTable1. flow inthetransitionzonebycalculatingradial tions fortheboundary-layervariablesandidentify boundary layer.Wefirstdevelopaclosedsetofequa- and verticalstructureoftheinnertransitionzoneor the relevantboundaryconditions.Next,weshowthat figuration intheboundarylayeraredescribedby function oftheboundary-layerconstantsand to calculateitsstructurenumerically.Finally,we a boundarylayerexistsanddescribethemethodused discuss thecharacteristicsofboundarylayerasa angular velocityofthestar. s following equations. the equationexpressingangular-momentumconserva- tion, whichmaybewrittenintheform where tit(r)=ATrrh\v\pis theradialmass-flowrate scribed byfivebasicequations.Thefirstoftheseis in thediskplaneandÙis angularvelocityofthe dr The broadoutertransitionzonebetweenrand The characteristicsofthedifferentregions 0s In thissectionwebeginourdetailedanalysisofthe The accretionflowandthemagneticfieldcon- The radialstructureoftheboundarylayerisde- 2 2 Length Scale d{MrQ)ldr =a{dKildr) +B^B.r,(6) d of ChangeDominantMechanism III. THEBOUNDARYLAYER i) Radial-StructureEquations a) Equations 263 197 9ApJ. . .232 . .259G _1 264 angular momentumistransportedintheboundary plasma. Thetwotermsontheright-handsideofthis carried awaybythematterleavingboundary layer; thefirstdescribesangularmomentum equation correspondtothetwomechanismsbywhich momentum transportedbythemagneticstress.The layer verticallywhilethesecondrepresentsangular momentum conservation,is viscous stresscanbeneglected,asexplainedin§V. equation, in theradialdirection. Here isthethermalpressureandraunitvector law, namely, and theappropriateversionofgeneralizedOhm’s field, andcristheeffectiveelectricalconductivityof the plasma. boundary layer.Assumingthattheflowissteady,this effective conductivitytotheflowpatternwithin Here Jistheelectric-currentdensity,Eelectric azimuthal pitch, relationship canbeexpressedintermsoftheaverage (8) and(9),is which followsfromtheradialcomponentsofequations at theupperandlowersurfacesofdisk.Theresult, In writingequation(10),wehaveusedthefactsthat lower halvesofthedisk,andthatradialcomponent that whentheangularvelocityofplasmaexactly of theelectricfieldEcanbeexpressedas= eff equals £2,themagneticfieldhasnotoroidalcom- the rateofslippagefieldlinesthroughdisk ponent andJiszero.Thevalueoforameasure plasma. Weemphasizethatthevalueofcrdoesnot only ontheassumptionofasteadyflow:anydis- depend onthedetailsofdissipativeprocess,but ai). an effectiveconductivityinagreementwithequation sipative process,ifitleadstoasteadyflow,mustgive boundary layer.Wedescribe thislossbytheequation r — Qrj9c.Thelatterfollowsfromtheobservation S reff eff s2 The secondbasicequation,whichexpressesradial- The thirdandfourthequationsareaMaxwell The fifthbasicequationistheonewhichrelates Now considerthevertical masslossfromthe © American Astronomical Society • Provided by theNASA Astrophysics Data System reverses onalengthscalehbetweentheupperand 2 v(dvldr) =—GMjr+Q,r—p'Kdp/dr) r 2 == *eff =(cMh-'r-\a-D)-V*.(11) Yo ^(,BplB)=(B,(10) s (zs=h2 = ii) MassFlowOutoftheDisk J= a(E+C-HXB).(9) ef{ dtifaldr =47rrpcg(r), (12) s 1 VxB =4itJIc(8) + (47rp)-{(yxB)XB}-f.(7) 9 GHOSH ANDLAMB 2 poloidal magneticfieldjustaboveandbelowthe profile ofmasslossfromtheboundarylayer.Although ing a“gate”functiong(r)whichdescribestheradial layer intermsofthelocalsoundspeedcandintroduc- scaling theverticalflowvelocityoutofboundary threading ofthediskplasmabystellarfield,we reconnection ofthestellarmagneticfieldwithin boundary layercannotbyitselfproducecomplete further thattheflowfromupperandlowersurfaces assume thatwhenitiscombinedwithturbulentdiffu- toward thestar.However,inwardradialmotion loss profileislargelydeterminedbytheshapeof sufficiently completetochanneltheflow.Weassume sion andotherdissipativeprocesses,threadingis flow unimpededalongthemagneticfieldfromdisk boundary layer.Werethedipolarfieldofstarun- of thedisktowardstarisstable.Thenmass- field inward,distortingeachlineintoashapethat present caseofanalignedrotator,andmattercould line wouldbehighestatthemagneticequator,which coincides withthemidplane(z=0)ofdiskin distorted, thegravitationalpotentialalongeachfield has alocalminimumofthegravitationalpotentialat of theboundary-layerplasmapinchespoloidal the midplaneofboundarylayerandtwolocal s these maximafromthemidplane,sincefieldlines mass-loss rateissensitivetotheverticaldistancez^of maxima aboveandbelowit(Scharlemann1978).The below thediskandhenceaccretingmattermust are approximatestreamlinesoftheflowaboveand flow “over”themaximaofgravitationalpotentialin from themidplanelikeexp(—zlh)]. the midplane[inastandarddisk,e.g.,pdropsaway order tofalltowardthestar,andsincedensityof matter variesrapidlywiththeverticaldistancefrom the abbreviatedform mined bythreeequations:theverticallyaveraged where Cisanumber~1,yettobedetermined,and momentum andthermalenergy,theequationof equations expressingtheconservationofvertical state. Wewritethevertical-momentumequationin is ~A.ThechoiceC=lwouldcorrespondtohydro- we haveassumedthattheverticalpressurescaleheight present case,hydrostaticequilibriumdoesnotholdin the verticaldirection,sincethereismassflow, static equilibriumintheverticaldirection.In but thevertical-flowvelocityofmatteris,T,andhfrom teristic radius,whichwedenotebyr.Wetheneliminate equations exhibitboundary-layerbehavioratacharac- zone asfollows.First,weshowthatthehydromagnetic in theoutertransitionzone. all continuousatr,joiningsmoothlytotheirvalues outer transition-zonesolutionsatthisradius,while the radial-structureequationsandsolvelatter p B numerically, subjecttotheboundaryconditionsgiven above. Oncetheradialstructureisknown,values enhance themagneticfieldwithin themagnetosphere, eight variablesü,v.B,M,h,T,/>,andp.Fromthese radius followimmediately.Weturnnowtothedetails. of thevertical-structurevariablesasfunctions SK0r eight equationsweeliminatethefourvariablesthat d0 true valueofBatrissomewhat largerthanthis;however, co0 the correctionfactorisnotlarge (see§llldbelow). 0 0 rd s0 7 3 The equationwhichdescribestheverticalradiative 3 Finally, weadoptanideal-gasequationofstatefor The variablesÜ,B,andtifaresubjecttothe The vertical-structurevariables/>,/?,T,andhare We solveforthestructureofinnertransition zd Equations (6)-(15)provideeightequationsforthe Sincethescreeningcurrents within theboundarylayer takes thevalue^r"atr. © American Astronomical Society • Provided by theNASA Astrophysics Data System co i) ExistenceofaBoundaryLayer 2 icaTVphK =(Jl)/2 and where termsoffirstandhigherorderin8/rhave been neglected.Inequations(22)and(23), 0 co0 is thefastnessparameterof therotator(Eisnerand r neutron starinseconds,R isthestellarradiusin Lamb 1977;PaperI),Pis the rotationperiodof units of10cm,andLis theaccretionluminosity r 0 6 37 First, letusassumethattheKeplerianmotionends Using definitions(16)-(20),thedifferentialequations 3l9l8l89/ 812 db/dx =-Cb%F-(a>-o>)-,(23) bs o> =Ql(GMr-y s0 1/6363258/63 2 ^ 0.82(C-") u^ -vl[(2GMIry>\Slry>];(18) n0wp r0 21 85n + C(y,l)ubF-i(a>^a>)-,(22) x p-^-^pi\MIMr(24) wrs 1730Q 312 1 a> =Q>l(GMr~);(17) 0 dœjdx =C^F-,(21) x=(r -r)lS.(16) co0 b^BJBfrJ; (19) F=Kf¡Kl. (20) d 265 197 9ApJ. . .232 . .259G 37-1 1110 17- 8 a2/27 an4/189 266 the dimensionlessparameters in unitsof10ergss.Equations(21)-(23)contain and which arenotdeterminedbythestructureequations in theboundary-layerapproximation.Here is thecharacteristicAlfvénradiusforspherical accretion (EisnerandLamb1977) in termsoftheheighthandviscosityparametera This expressionfor0canbeconvenientlyevaluated the dimensionlessquantitiesC,yandi/j.For momentum flux,withtheresultÿ=(hlr)aoc of astandarddisktransportingnonetangular- example, ifoneneglectstheveryweakr-dependence ” 0 - 1/20 M =1,andtil10gsoneobtains length scale8withintheboundarylayerintermsof and of 0andevaluatesitatr=10cmwitha1, 82727195 result form usingequations(13)-(20)and(29),withthe s Thus, theboundary-layerapproximationisself-con- the diskplane,cannowberecastinnondimensional œ9bp0 s0 sistent. Since^sdrcca,bothare quite insensitivetotheeffectiveviscosityindisk. the boundary-layerconstantsC,C^,andare 0 the azimuthalpitchy^(x),gatefunctiong(x),and This equationandequations(21)-(23)formacom- 0 0 the variables known. Withtheresultingknowledgeofo>,u,b,andF, plete setfortheradial-structurevariablesandcanbe 16/2716/272/278l solved numericallyforarotatorofgivenfastness,if 1/276/275/278(0) 0s which specifytheverticalsemithickness, temperature, Qs bp r 2ll210 1/87/82(O)7654: dFIdx xO.nSiy^/^)^-. 8 =0.031{y~Cp“)r«r(29) r *0.41(y/C-).(30) 0öco 0&wpA e =hlh,6EET¡T£ pips >andyeepjp, Equations (25)and(26)givetheradiusrradial i/j EE[(9tifl327T)(Klac)(GMr)(2kIGMmy]. C =6y,p(ro/r)(8o/^o)0"'(26) sS s 0 0Bp b(iA Equation (12),whichdescribesthemasslossfrom © American Astronomical Society • Provided by theNASA Astrophysics Data System ii) InternalStructureoftheBoundaryLayer 1,2<0)7/2r (0)min2 C =2y(ro/r)"(S/o)(25) Mi>A0 r =fjL(2GM)~]\^~(27) A GHOSH ANDLAMB (28) (31) (32) 4 the equationofstate,completingboundary-layer (denoted byasubscripts)atthesameradius,canbe density, andpressureintheboundarylayerterms calculated fromthevertical-structureequationsand of thecorrespondingquantitiesinastandarddisk the boundarylayerisrelativelyinsensitiveto the constantsCC^,and.First,wetakey^x)= following choiceforthefunctionsy^ix)andg(x) solution. precise valueofyasmaybeseenfromthesolutions const. =yintheboundarylayer.Thestructureof g(;t) satisfiestheconditions(1)thatg#0atandout- figuration foranassumedg(x)iscalculatedand for severalvaluesofyintherange0.5-5whichare function g(x)canbedeterminedaccuratelybyan described below.Second,wenotethatthegate iterative procedureinwhichthemagneticfieldcon- in theregionbetweenrandflow-alignmentradius g =1wellinsidetheboundarylayer.Theseconditions insensitive tothepreciseformofg(x),providedthat accretion flowandthemagneticfieldconfiguration procedure mustawaitaquantitativedescriptionofthe checked forself-consistency.However,useofthis r. Nevertheless,theotherpropertiesofmodelare g(x) =1,0, radialvelocityu,andmass-flowrateF,inappropriatedimensionlessunits(seetext)forthefollowingvaluesof We incorporateconstraint(34)intooursolutionsby method. First,wefixtheconstantsC,andy. adjusting Caccordingtothefollowingiterative culate Ccfromequation(34),andchecktheinitial solve equations(21)-(23)and(31)toobtainè,cal- Next wechooseaninitialvaluefortheconstant choose anewvalueofC,iteratinguntilself- and finalvaluesofforself-consistency.Wethen consistent valueisobtained. large numberofexamplesthatwehavecalculated. values oftheboundary-layerconstants,C,and y, andthevalueoffastnessparameterœ. Each caseislabeledbyfournumbers,namely,the boundary layerinsometypicalcasesselectedfromthe various valuesoftheboundary-layerconstants.This tive totheparticularvaluesofboundary-layer 0 the boundarylayerpresentedhereisquitegeneral,and insensitivity demonstratesthatthebasicstructureof constants, C,andyapropertycharacteristicof 0bpö that theactualvaluesofboundary-layerconstants r are notcrucialtothetheory.Henceforthweshall boundary layersgenerally.ThisisillustratedinFigure bp0 w adopt thevaluesC=2.5,2,andy1. 0 circulating intheboundarylayerscreenmagneto- layer forthreedifferentvaluesofcü.Thecurrents w reduced bythemagneticstressesincrossinglayer, and reachescorotationattheinneredge.Theradial spheric fieldbyafactorb~0.2.Theangularvelocity characteristic oftheouter transition zoneatro,as velocity ofthematterequalsslowradialdrift of thematter,whichisKeplerianatouteredge, b9p required bythevelocityboundary condition;within 0s the boundarylayeritfirstincreases inwardascentri- 1, whichshowstheradialstructureforœ=0.3and fugal supportfailsandthenpasses throughamaximum öp0 and decreasesinwardasthe opposingmagneticpres- bp0 s 0 s Fig. 1.—Radialstructureoftheboundarylayerforastarfastnesswhereisanumericalfactor limited byreconnection.Asafirstapproximation,we model thiscomplexprocessbyassuminganamplifica- Assuming theflowtobesteady,thispitchdetermines the effectiveconductivityofdiskplasmafor IFeff =op,wheretheparameteraislessthanorof ef{ field. (9) thencompletesthedescriptionofmagnetic the outertransitionzone.Addingequations(8)and poloidal currents,viaequation(11).Wefurther currents, andthattheratioyJÇisconstantthroughout plasma fortoroidalcurrentsisthesameaspoloidal assume thattheeffectiveconductivityofdisk a the residualpoloidalmagneticfield, ¿>ut=BliLbr~.Thepointymarkstheouteredge ofthetransitionzone. ~1, andareconnectionrateÇvjlh,wherev= 6, anddensity£,inappropriate dimensionlessunits(seetext)asafunctionofthe radiusy=r/r.Alsoshownis 0zs A 0 Finally, weneedtodeterminethemagneticfield Fig. 4.—Structureoftheoutertransition zoneforastaroffastnessw=0.3.Shownarethevertical semithicknessc,temperature s © American Astronomical Society • Provided by theNASA Astrophysics Data System r Then, balancingthegrowthanddecayof 2A(47t/o) 1/2 ACCRETION BYMAGNETICNEUTRONSTARS Q k b 2 (37) 3 1 The ratioyJÇisfixedbyrequiringthatjoin the outertransitionzoneisassumedtobeKeplerian. The lastdeterminesthenewverticalstructure,which the energy-dissipationrateduetocross-fieldmotion. equations (9)and(11),determinesthescreening disk structure,thisboundarycondition,togetherwith vertical diskstructureandresidualmagneticfieldis procedure canthenbeiterateduntilaself-consistent can becomparedwiththetrialstructure,and current /,theresidualpoloidalmagneticfieldB,and found. smoothly toitsvalueyatr.Givenatrialvertical temperature, density,andverticalthicknessinthis the boundary-layercurrents,isfurtherreducedby nbr~ —B,indicatingthatthestellardipolefield, this zone.Thepoloidalfieldisscreenedonalength disk. region tothecorrespondingquantitiesinastandard 2 as maybeseenfromFigure4,whichcomparesthe similar tothatofastandarddiskatthesameradius, techniques, onefinds We canexpressthisscreeningbywritingB~ generated bytheradialcross-fielddriftofplasma. screening fieldB.Solvingforusingstandard screened toafractionbofitsunscreenedvalueby scale bytheazimuthalcurrentJ#=—^uC~vB z 0 0s z s 0 er2 B =y*o-M«(l-^sY^-^y-^Bár,), s0 As inthestandardmodel,angularvelocity The structureoftheoutertransitionzoneisvery Consider nowthemagneticfieldconfigurationin b) BoundaryConditionsandMethodofSolution c) CharacteroftheSolutions 269 (38) 197 9ApJ. . .232 . .259G 3 3 in termsofthehypergeometricfunctionF(t)= where e=>=1,decreases the factthatf(y)reachesitsasymptoticvalueat its asymptoticvalue0.78fory>3.Thez-component F(3I2, 1/2,2,t),andy=r/risthedimensionless field flowbetweentheboundarylayerandflow- The formoftheright-handexpressionemphasizes conveniently describedbythedimensionlessvariable 270 fibr~ —Bithasbeenassumedthatthescreening the solution(40).First,inwritingBform monotonically withincreasingy,closelyapproaching calculation ofthesehighermultipolesisanotherrefine- (1) B(ro)=^br~,and(2)thetotalscreening This doesnotfollowfromtheboundary-layersolu- the startoexactlythatofadipolemoment/x6. of themagneticfieldinoutertransitionzoneis the accretionflowandmagneticfieldconfigura- tions presentedabove.However,thefactsthat currents and(//)thegeneratedbycross- currents insider,consistingof(/)theboundary-layer flow-alignment radiusr. multipole componentsinBwhichmaybesignificant the dipolarapproximationisgoodexceptforhigher to roughlythatofadipolemoment/xè,mean currents insiderreducethestellarfieldatradii» alignment radiusr,reducethebasicdipolarfieldof tion intheregionbetweenboundarylayerand ment whichmustawaitaquantitativedescriptionof at radiilessthanoroftheorderseveraltimesr.The the screeningradius where Bfirstvanishes,and i?=0outside.This tion whicharenotincludedinequation(37).Thusa limited bydissipativeprocessesotherthanreconnec- very largevalueofcjatradii.Inreality,o^ffis things, thisimplies<7ocwhichleadstoa tion (11)withygivenbyequation(37).Amongother 0 better approximationistousetheresult(40)inside solution (40)wastoassumethat(jisgivenbyequa- calculated fromequation(37) andtheknownvertical approximation isshownin Figure 3. 210 0s z zQ 0 0 f z 0 0 f 0 z 2 eff eff 0 ei{ bout =B^bor-*1-[f(y)IO.n](ylyy°'*°.(40) s Two approximationshavebeenmadeinobtaining The secondapproximationmadeinobtainingthe The azimuthalpitchy#ofthe magneticfieldcanbe © American Astronomical Society • Provided by theNASA Astrophysics Data System 32/7340/73 ^ r/r*103«“yo“(l-,(41) s0 n32 + j-F(j)j,n=40/47(39) GHOSH ANDLAMB 213 5 1 5/2 1/2 _3 increasing radiusforr>,scalingroughlyas corotation point=(GM/Q.),whereÜQ mum value~ythendecreases,changessignatthe from r,y#increasesypassesthroughamaxi- y .increaseswith structure ofthiszone.Theresultisthatasrincreases (Ghosh andLamb1979). exert aspin-downtorqueonthestar.Mostof neutron-star X-raysources,andbrieflycompareour these solutions,describesomeoftheimplicationsfor the transitionzonebetweencorotatingmagneto- radial andverticalstructureoftheaccretionflowin used ittoconstructdetailedsolutionsdescribingthe for diskaccretionbymagneticneutronstarsandhave spin-down torquecomesfromtheregionwhere results withotherwork. sphere andthedisk.Inpresentsectionwediscuss Thus theinnerradiusrofdiskisdetermined implicitly bytherelation and IV,namely,anarrowinnerboundarylayerwhere brake theazimuthalmotioninaradialdistance8«r. which triestoenforcecorotation,islargeenough disk plasmabythestressofstellarmagneticfield, value. Thisoccurswherethetorqueexertedon flow velocityisdeterminedbytheeffectiveviscous disk, appearstobeageneralfeatureofsteady,axi- most ofthescreeningoccurs,togetherwithabroad be suchapointmayseenasfollows.Wherethe which alsofollowsfromequation(6).Thattheremust begins todepartsignificantlyfromtheKeplerian and centrifugalforcecloselybalance,theradial- outer zonewheretheresidualstellarfluxthreads motion, oneexpectsB#~Bsothatintheabsenceof as r~.Thediskendswheretheazimuthalvelocity shear stress,whichisrelativelyweakandscalesroughly symmetric diskaccretion,aswenowdiscuss. ence ofthemagneticfield and onlystrengthensthis roughly asr.Screeningsteepenstheradialdepend- as (8/r)r,whereasfordiskflowtherightsidescales screening theleftsideofequation(42)scalesroughly stellar magneticfieldbeginstocontroltheazimuthal c SKs 09 009 the magneticstressbecomes largeenoughtodisrupt (42) becausetheyaretoo weak, inathindisk,to conclusion. Viscousstresses do notappearinequation 0O cs c c disrupt theKeplerianmotion; thus,atthepointwhere 0 Z9 In theprevioussectionswehaveoutlinedamodel The structureofthetransitionzonefoundin§§III In theaccretiondisk,Ù^Q,forceofgravity K (rÆÆ/47r)277r2S ^{rpv^v^lirrflh,8«r 20o a) GeneralCharacterofthePresentSolutions v. DISCUSSION Vol. 232 (42) 197 9ApJ. . .232 . .259G 2 m 2 2 2 transition zoneorboundarylayer.Eveniftherewere boundary layer,asshownbelow). this motion,theviscousstressis,ofnecessity,negli- the radiuswhereitoccurs,becauseofsteepradial no screening,thiswidthwouldbeonlyafractionof gible (theviscousstressisnegligibleevenwithinthe No. 1,1979 plasma isperpendiculartothepoloidalmagneticfield dependence ofB.Buttheradialmotiondisk width ofthetransitionzonetoelectromagnetic poloidal magneticfield,confiningitandreducingthe drives toroidalcurrents.Thesecurrentsscreenthe of thestar,creatinganazimuthalelectricfieldwhich screening length, extent thatthesameconductivitycanbeusedto layer, whereB#æBandtheflowissteady,thiscon- component ofthemagneticfield;inboundary plasma forpoloidalcurrentsisrelatedtothetoroidal currents. Nowtheeffectiveconductivityofdisk Here <7istheeffectiveconductivityfortoroidal evaluate equation(43),oneobtainstheresult ductivity satisfiesequation(11)with~1.Tothe Thus 8~risnotself-consistent. (44) ,thiswouldrequirev~(A/r)(l—w)%.Since zone isnarrow(8«r),asfollows. where cu=Q/iîandv(GM/r).Thisis is notcompletelybalancedbythecentrifugalforce (42), dB¡dræ(A/r)(l—<*>pg,whichistoosmall. there, andwillleadtoatypicalinwardvelocityof the typicalvalueofQinboundarylayer,although greater thanQ,islessthegravitationalforce sufficient toshowthattheangular-velocitytransition the magneticpressuregradient,equations(44)and the netradialforce~(1—co)pg.Butfromequation pressure gradient,dB/dr,werelargeenoughtobalance be possibleiftheradialcomponentofmagnetic equation (44).Aself-consistentsolutionmightstill which for8~rismuchlargerthanallowedby magnitude (45) canbesolvedfortheself-consistentwidthof That themagneticpressuregradientdoesprovidesome boundary layer,withtheresult, decreases neartheinneredge oftheboundarylayer where thisgradientisgreatest. Thevalueoftheratio also evidentinFigures1and 2,whichshowthatv z eff y/| thatemergesfromthe numericalsolutionsisin substituting theresult(46) into equation(42)andis support intheself-consistentsolutionmaybeseenby 0 r0s 0 sK0 0s sK s 0 r a 1/32 Consider nowthewidth8ofangular-velocity Ignoring forthemomentradialcomponentof First, suppose8wereinstead~r.Fromequation 0 8/r -(1<0^(1+a>)-(Ä/ro)«1.(46) 0s © American Astronomical Society • Provided by theNASA Astrophysics Data System 1,2l Vr ~(S//o)(l-.Thepresentsolutionsforthe and anisotropicconductivitytensor,indicatethatthe of thediskplasmainthisregiongeneratestoroidal radii, wheretheydependsensitivelyonthemodelof 09 magnetic fieldconfigurationareunreliableatlarger magnetic fielddissipationadoptedandpredictvalues 0 havior oftheaccretiontorquewillbeunaffectedby of whicharequitelarge.Nevertheless,thenature 0 improvements inthetreatmentofthisregionsolong of theoutertransitionzoneandqualitativebe- the innerradiusofdiskandwidth zone inthepresentsolutionsseemslikelytobea as thescreeninglengthremains>r,aresultwhich found onlyforvaluesofthe fastnessparameterœ where themagneticfieldbeginstocontrolflow. existence ofasteadyflow,theisotropycon- the magneticfielddissipationprocessbutonlyon general featureofsteady,axisymmetricdiskaccretion. seems relativelysecure. 0 that thenetradialforceatr , whichhasgravitational, reason forthisisthatwhen œ >wthecentrifugal less thanacertainmaximum fastnessa>^1.The boundary layerdonotdependonthedetailsof 0 force ontheplasmainboundary layerissogreat ductivity, andthereasonableassumptionthatä1, Ftirthermore, theargumentspresentedhereshowthat r 0 s 0 smax max Next, considertheoveralleffectofazimuthal Finally, considertheextentofthisouterpart In summary,thetwo-partstructureoftransition In ourmodel,stationaryinflowsolutionscanbe b) ExistenceofaMaximumFastness 271 197 9ApJ. . .232 . .259G 2/31/ 2/31 regions inreverseorder. flow-alignment radiusrand (2)theflowbetweenr and thesurfaceofstar. Weconsiderthesetwo parts: (1)theflowbetween thediskplaneand the diskplaneandstar divides naturallyintotwo field. Athoroughdiscussionofthisassumptionwould centrifugal, andmagneticcomponents,isoutward nature oftheproblem. the flowinthisregion.Herewesimplyoutline require adetailedknowledgeofthemagneticfieldand neutron starischanneledbythepoloidalmagnetic d) FlowfromtheBoundaryLayertoNeutronStar found onlyifPexceedsacertaincriticalvalue. (24). Given/x,M,andAT,steady-flowsolutionscanbe tion mayceasealtogether(compareDavidsonand comparable tow,whilefor»)20.Werethe layer (§III)isvalid. cous stress~r¡rdQ¡dr\s>(r/A)(l+w)-)-. the ratioofmagneticstress~BBI4tttovis- the dominantstressesaremagnetic,ratherthanviscous features : (~ 100eV)ofastandarddiskatr(seeFig.3).Thus, to therateofenergygenerationthroughoutouter the resultofresistivedissipationelectricalcurrents occurs atarate~GMtálr.Thisiscomparable as inordinaryshearboundarylayers.Thisfollows structure oftheboundarylayerdeterminedby of thelayer.Thisfollowsfromequations(13)and(45), f f min maxs s 0 s 0 min rs t0s z(}) 0 0 The accretionflowinthemagnetospherebetween The rangeofvalues/¿,Af,Aï,andspinperiodP In constructingthepresentflowsolutionswehave The boundarylayerhasseveralnovelandinteresting 2. Theradialinflowissupersonicthroughoutmost 3. Energygenerationwithintheboundarylayeris 1. Itisanelectromagneticboundarylayer,inthat © American Astronomical Society • Provided by theNASA Astrophysics Data System c) NatureoftheBoundaryLayer GHOSH ANDLAMB _31/2 to theextentthatcross-sectional areaofthe this pointtheflowissub-Alfvénic overtheentire field motionhasnotbeenstudiedindetail.Itseems above andbelowthediskis givenbyequation(48), accretion bundle.Theaverage Machnumberwell and isshowninFigure5 for thecaseco=0.3.At the resultis using theboundary-layersolutionsobtainedin§III. can becalculatedovertheentireaccretionbundle flow inthisregiontobestableisJÍ<1.TheAlfvén likely, however,thatasufficientconditionforthe In termsofthedimensionlessboundarylayervariables, the flow,andhencethatflowinsideflow- Mach numberoftheverticalflowatdiskplane of finiteelectricalconductivityandappreciablecross- stability ofmagnetohydrodynamicflowsinthepresence between thediskplaneandrisnotknown,since alignment surfaceisstable. expect that<1overmostofthecrosssection flow interiortoristhereforethattheAlfvénicMach A necessaryconditionforthestabilityofaccretion flow whichpassesfromsuper-Alfvénictosub-Alfvénic. nature oftheinstabilitythatoccursinafield-aligned then (Williams1975;Adam1978)haveclarifiedthe this conditionthereforelimitstheextentofstable where B=/xr[4—3(r/r)]isthetypicalstrength region. Thenthecross-sectionalareaofaccretion number, Jtp,=v^v,notcrossunityfromaboveany- magnetosphere; detailedcalculationsperformedsince (1974) firstpointedoutthatwithinthemagnetosphere frame ofthestar(seePaperI).LambandPethick since .Given<^1,we bundle. Clearly,<^>islessthanunityforr< constants chosenin§III,onefinds continuity, andthevaluesofboundary-layer bundle is assuming thatthemagneticfieldisdipolarinthis without suchknowledgeonecanestimatetheAlfvén tion flowattheflow-alignmentsurface.However,even tion bundlerequiresadetailedknowledgeoftheaccre- poloidal magneticfield.Adeterminationofthefield the magneticfieldandvisAlfvénvelocityin where inthisregion.Hereistheflowvelocityalong definition, completelyfield-alignedinthecorotating of theunscreenedpoloidalfieldalongaccretion Using equations(29),(30),and(47),theequationof Mach numberaveragedovertheaccretionbundle, such aflowisstableonlyifitsub-Alfvénic,andthat strength, plasmadensity,andvelocityovertheaccre- s A f f 0 A flp0A f0 K 1/254 <^a> ~(^i,/%)Wro)[4-3(rlr)]-^(BIB) The correspondingstabilityconditionfortheflow The flowbetweenrandthestellarsurfaceis,by 0p9 f 21 a{r) ^47rr(8/r)(r/ro)[4-2>{rlr)]~.(47) 0 -MrrKsr] « Vol. 232 (48) 197 9ApJ. . .232 . .259G 3/2 5/4 1/2 92 l2 54 71/2583/28 1/4 (47) andnomassislostfromtheaccretionbundledue, for example,toRayleigh-Taylorinstability(thelatter accretion bundleinthisregionisgivenbyequation tion bundlejustaboveandbelowthediskplaneasafunction 0.5(i?/r)S ~0.12i?,whereRisthestellarradius. proaches unity.Itthereforeseemslikely,thoughnot the plane,1^/%increasesslowly,while(r/r)x would lowertheMachnumber).Thequantitiesthat of theboundary-layercoordinateforastarfastness No. 1,1979 and theflow-alignmentsurface. certain, that<1alsoholdsbetweenthediskplane appear inequation(48)takethevalues(rn/%)~0.1, )Nwherevistheradialvelocityof while ifvisevaluatedusingtheradialvelocityin for S,oneobtainstheestimater~0.20C\ (8/r)% intheboundarylayerandresult(29) disk outsidetheboundarylayer,resultingestimate what underestimatesrpinthecaseofaxisymmetric equation (30),weconcludethatthisexpressionsome- of risstillsmaller.Oncomparingtheseresultswith field firstdominatestheaccretionflowwhere to thosegivenabovein§Ya,thatthestellarmagnetic disk accretionbyanalignedrotator. magnetic stressesarecomparabletothematerial r r namely pVrV#~ismoreappropriatetoan eq. [42]ifB~atrand8h).Rees(1974) stresses; thiscriterionisequivalenttoequation(42)if r 0œA 0 B# ^Batranddetermines8tobe~4h(thelatter at rand8~h.Sincethepresentmodelassumes suggested asimilarcondition,pvv~B/47r,todeter- oblique rotator,butgivesthesamevalueofras mine r;thisisequivalenttoequation(42)ifB#~B 8 ~h(thealgebraicexpressiongivenbytheseauthors, follows fromeq.[46]),thevaluesofrgivenbythese dimensional estimatesroughlyagreewiththevalue 0 here. field balancesthegaspressureindisk,thatis, the ideaofpressurebalance.PringleandRees(1972) obtained fromthemoredetailedcalculationsdescribed be locatedwherethepressureofstellarmagnetic where B¡8tt~/?.Thiscriteriongivestheestimate suggested thatthemagnetosphericboundarywould these authors,orr~0.26a,ifoneusesthe rz0 choose asolutionareactuallythesameequation three boundaryconditionsusedbyScharlemannto the inneredgeofdisk.Hismodifiedcondition, the stellarmagneticfieldBbyscreeningcurrentsat mann (1978)modifiedthePringle-Reesconditionby disk modelofShakuraandSunyaev(1973).Scharle- r ~0.57xy'V,ifoneusesthediskmodelof centrifugal forcesactingontheplasmainboundary conditions isnotlogicallyconsistent.Ichimaru(1978) radius ofthediskdeterminedusingtheseboundary approximated intwodifferentways.Thus,theinner gave r~0.86a.However,thefirsttwoof {B +A2?)/87tä/?,whereAi?isthescreeningfield, attempting totakeintoaccounttheenhancementof z0 0 r0 0 layer. However,Ichimaru’smodelisdefectivein by attemptingtotakeintoaccountgravitationaland modified thePringle-Reescriterioninadifferentway 0z Q general questionwhichariseswithallcriteriabased important respects,someofwhicharenotedbelow.A these criteriaignorebothinertial forces(seeEisner penetrate themagnetosphericcavity.Atveryleast, static criteriontodynamicaccretionflowswhich on theideaofpressurebalanceisrelevancethis which resultsfromtheKelvin-Helmholtz instability and Lamb1977)thecoupling betweenthestellar and otherprocesses(see§II above). magnetic fieldandtheorbital motionoftheplasma disk 0A 0A 0A disk Lamb andPethick(1974)argued,forreasonssimilar Finally, considertheestimatesthatwerebasedon Vol. 232 197 9ApJ. . .232 . .259G tions andshowquitedifferentscalingsintermsofthe estimating rhavequitedifferentphysicalinterpreta- physical parametersoftheproblemfromthat No. 1,1979 Despite theseimportantdifferences,theearliercriteria criterion foundfromthepresentdetailedcalculations. factor of~2thevaluefoundhere,aconsequence typically giveestimatesforrwhicharewithina field andtheresultinginsensitivityofrtoparticular of thesteepradialdependencestellarmagnetic (1978) andIchimaru(1978).Scharlemannstressedthe has alsobeenconsideredrecentlybyScharlemann criterion used. field andsuggestedamodelofplasmaflowfromthe in mixingthediskplasmaandmagnetospheric likely importanceoftheKelvin-Helmholtzinstability inner edgeofthedisktowardneutronstarwhichis present work.Ichimaruproposedamodeloftheflow very similartothemodelwehaveadoptedin we mentionfirst,thathisequation(7)implicitly 0 assumes thattheangularvelocityofplasmachanges several deficienciesandinconsistencies.Amongthem in whichplasmapenetratesintoaboundarylayerby angular velocityjustinside,eventhoughthereareno discontinuously fromtheKeplerianvalueoutside anomalous diffusion,buthistreatmentsuffersfrom velocity ofthestarandthatfluxangular the magnetosphererotatesrigidlywithangular outer surfaceoftheboundarylayertostellar 0 momentum towardthestarisalwayszero.Both equation (29)assumesthattheinflowingplasmawithin stresses inthemodeltoeffectthischange.Second,his 0 Third, Ichimaru’sequation(31)implicitlyassumes that itspinsdownasaccretes(seePaperI,§IV.) implies thatthetorqueonstariszeroandhence and [11]).Inparticular,theformerrequiresthat eq. [27]). tional termdescribingthechangeinmaterialstress magnetosphere isthesameasthatindiskoutside, that theangularvelocityofplasmainside different, hisequation(31)shouldincludeanaddi- contrary totheassumptionimplicitinhisequation(7). stellar fieldlinesbeinfinitelyrigid,whilethelatter tion zoneatthemagnetosphericboundaryofan tion oftheradialandverticalstructuretransi- across themagnetosphericboundary(seeLamb1975, Since thesetwoangularvelocitiesareingeneral assumptions aretoorestrictive(cf.PaperI,eqs.[10] the stellarmagneticfieldthreads thedisknearits magnetic stressactingon the diskplasmabecomes aligned rotatingneutronstaraccretingmatterfroma inner edgeofthediskislocatedwhereintegrated inner edgeviatheKelvin-Helmholtz instability,tur- comparable totheintegrated materialstressassociated Keplerian disk.Thecalculationindicates(1)thatthe with itsinwardradialdriftand orbitalmotion;(2)that In summary,thecriteriapreviouslyproposedfor The flowofplasmafromthedisktowardstar In thepresentpaperwehavedescribedafirstcalcula- © American Astronomical Society • Provided by theNASA Astrophysics Data System VI. CONCLUDINGREMARKS ACCRETION BYMAGNETICNEUTRONSTARS the corotatingmagnetosphere;(3)thattransition transition zonebetweentheunperturbeddiskflowand bulent diffusion,andreconnection,producingabroad field islargelybutnotentirelyscreenedbycurrents zone iscomposedoftwoqualitativelydifferentregions, flowing intheboundarylayer;and(5)thatforsuffi- boundary layerwherethedeparturefromKeplerian the magnetic-fluxreconnectionprocessinboundary motion issubstantial;(4)thatthestellarmagnetic Keplerian, andanarrowinnertransitionzoneor a broadoutertransitionzonewherethemotionis fluctuations intheflow,atleastonsufficientlysmall layer butonlyontheexistenceofasteadyflowand ciently faststellarrotationtherearenosteady-flow magnetic fluxreconnectionislikelytoproduce crudely isotropiceffectiveconductivity.Although of theboundarylayerdonotdependondetails solutions. Theinnerradiusofthediskandwidth There weshow(1)thatthemagneticfieldwhich use theresultsobtainedheretocalculatetorque exerted ontheneutronstarbyaccretingplasma. stationary flowmodelmayprovideanadequate size andstructureoftheboundarylayer,present spatial andtemporalscales,hencevariationsinthe changes sofarmeasuredinpulsatingX-raysources, can dominatethetorqueduetomatterinboundary threads thediskinoutertransitionzoneexertsan appreciable torqueonthestar;(2)thatthis including HerX-l,areconsistentwithdiskaccretion; emission, continues;(3)thatallthesecularperiod layer, ifthestarisrotatingfastenough,braking and 4U0900—40maybeexplainednaturallyas and (4)thattheshort-termperiodfluctuations star’s rotationevenwhileaccretion,andhenceX-ray description oftheaverageflowtowardstar. the flowbetweendiskplaneandflow-alignment mass-accretion rate. consequences ofrelativelysmallfluctuationsinthe spin-down episodesobservedinHerX-l,CenX-3, the boundary-layerapproximation,therebyeliminating to solvefortheflowindiskplanewithoutmaking radius, and(3)tousetheunderstandinggainedhere and dissipationinthetransitionzone,(2)toobtain problems thatmeritstudy.Theseincludethemathe- layer. tion oftheazimuthalpitchthroughboundary sophisticated treatmentofmagneticfieldamplification all undeterminedconstantsanddeterminingthevaria- solutions tothehydromagneticequationsthatdescribe metric flowsofthetypeconsideredhere,andtime- rotators, thestabilitypropertiesofsteadyaxisym- matical descriptionoftheaccretionflowinoblique light ontheoriginsoffluctuations intheluminosity, understanding ofthelasttwoproblemswouldshed dependence ofreconnectioninthetransitionzone.An ing thesteady-flowassumption madeinthepresent X-ray sources,andwouldprovide abasisforevaluat- pulse period,shape,and spectrumofpulsating work. Finally,inviewof the qualitativedifference In asubsequentpaper(GhoshandLamb1979),we Work iscurrentlyunderway(1)todevelopamore The presentresultsalsosuggestseveralother 275 197 9ApJ. . .232 . .259G between thetransitionzoneinnearlysphericalaccre- 276 tion andthatinaccretionfromaKepleriandisk,it would beinterestingtostudyhowthetransitionzone ing accretionflowsinwhichtheinfallingmatterhas changes fromtheoneformtoother,byinvestigat- flattening oftheflowbutnotenoughtoformathin Adam, J.A.1978,PlasmaPhys.19,77. sufficient angularmomentumtocauseappreciable Basko, M.M.,andSunyaev,R.A.1976,M.N.R.A.S.,175, Arons, J.,andLea,S.M.1976,Ap.207,914. Keplerian disk. .1977,Ap.J.,215,897. —.1979,submittedtoAp.J. Davidson, K.,andOstriker,J.P.1973,Ap.J.,179,585. Cole, J.D.,andHuth,H.1959,Phys.Fluids,2,624. Eardley, D.M.,andLightman,A.P.1975,Ap.J.,200,187. Eisner, R.F.,andLamb,F.K.1976,Nature,262,356. Ichimaru, S.1978,Ap.J.,224,198. 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