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Mon.Not.R.Astron.Soc.000,1–10(0000) Printed30January2012 (MNLATEXstylefilev2.2) On the magnetic flux problem in star formation ⋆ Jonathan Braithwaite ArgelanderInstitutfu¨rAstronomie,AufdemHu¨gel71,53121Bonn,Germany 30January2012 2 1 0 ABSTRACT 2 Strongmagneticfieldsplayacrucialroleintheremovalofangularmomentumfromcollaps- n ingcloudsandprotostellardiscsandarenecessaryfortheformationofdiscwindsaswellas a jetsfromtheinnerdiscandindeed,stronglarge-scalepoloidalmagneticfieldsareobserved J inprotostellardiscsatallradiidownto∼ 10R⊙.Nevertheless,bythetimethestarisvisible 6 virtuallyall of the originalmagneticflux has vanished.I exploremechanismsforremoving 2 thisfluxduringtheformationoftheprotostaronceitismagneticallydisconnectedfromthe parentcloud, lookingat both radiativeand convectiveprotostars.This includesa numerical ] R investigationofbuoyantmagneticfield removalfromconvectivestars. Itisfoundthatif the S stargoesthroughafullyconvectivephaseallremainingfluxcaneasilyberemovedfromthe . protostar,essentially onan Alfveńtimescale.If onthe otherhandthe protostarhasnofully h convectivephasethensomefluxcanberetained,thequantitydependingonthenetmagnetic p helicity,whichisprobablyquitesmall.Onlysomefractionofthisfluxisvisibleatthestellar - o surface.Ialsolookathowthesamemechanismscouldpreventfluxfromaccretingontothe r staratall,meaningthatmasswouldonlyaccreteasfastasitisabletoslippasttheflux. t s a Keywords: (magnetohydrodynamics)MHD –stars:magneticfields– ISM:clouds– ISM: [ magneticfields–stars:formation–stars:magneticfields 1 v 0 4 1 INTRODUCTION is conserved if flux freezing is valid. It is related to the gravita- 6 tional and magnetic energies by (ignoring factors of order unity) .5 Santadrsmfaogrmnetfircomencelroguiedssainrewgheincehraglrlayvictoamtiopnaaral,btlheertomaoln,eroatnatoitohnearl. λ2 = |Egrav|/Emag.Acloudwithλ & 1issaidtobe‘magneti- 1 callysupercritical’andwillcollapse,intheabsenceofsignificant Onceastarhasformed,thethermalandgravitationalenergiesare 0 thermal or rotational energy. Conversely a cloud with λ . 1 is 2 stillcomparabletoeachotherbutthemagneticenergy(asinferred ‘magneticallysubcritical’andthemagneticfieldsupportsthecloud 1 from Zeeman measurements) is at least six orders of magnitude againstgravity. : lowerineventhemoststronglymagnetisedstars. v This is not what one prima facie expects if a cloud retains ObservationsofcloudcoresandtheISM(e.g.Crutcheretal. i X itsmagnetic fluxasit collapses, since gravitational and magnetic 1999; Heiles&Troland 2005; Girartetal. 2006, 2009) show that energyhavethesamescalingwithradius,i.e.E ∝ 1/R.Incon- clouds docontain strong magnetic fieldsand appear tobemildly r a trast,thermalandrotationalenergyrisefaster,andmusttherefore magnetically subcritical on scales above ∼ 1000 AU. The rota- becontinually extractedfromthecloud toallow itscollapse. Ex- tionalandthermalenergiesappear toberelativelysmallonthese tractionofthermalenergyhappensviaradiationandrepresentsno large scales. The so-called “magnetic flux problem” can be di- barrier to star formation in theory or otherwise (at least for stars vided into two parts, the first of which is: how issufficient mag- underabout10M⊙);extractionofrotationalenergyislesswellun- netic flux lost so that the cloud becomes magnetically supercriti- derstoodbuttheoreticalmechanismsexist,usingmagneticfieldsor cal?Gravitationalcontractiondowntothatscalecouldproceedvia gravitational instability, and there is plenty of evidence for mag- ambipolar diffusion(Mestel&Spitzer1956) orsomeother diffu- neticbraking(Gillisetal.1974,1979;Mouschovias&Paleologou sive process, and dynamical collapse begins once supercriticality 1979;Stahler&Palla2005andrefs.therein). isreached.Recentobservationalsupportforthismodelisgivenby Therelativestrengthofthegravitationalandmagneticfields Davidsonetal.(2011). isoftenexpressedasadimensionlessmass-to-fluxratio,definedas The second part of the magnetic flux problem – that ad- λ ≡ 2πG1/2M/Φ, where M and Φ are the mass and magnetic dressedhere–ishowtoremovetheremainingfluxoncethecloud flux, or locally in a disc context as 2πG1/2Σ/B where B and z z hasbecomesupercritical,toexplaintheincrediblyweakmagnetic Σ are the field normal to the disc, and surface density; this ratio fieldsseeninstars(e.g.Mestel&Spitzer1956;Nakano1984;Galli 2009). The supercritical collapse might take place too quickly to allow ambipolar diffusion to remove the rest of the flux before ⋆ E-mail:[email protected] fluxfreezingresumesowingtorisingionisationfraction(although (cid:13)c 0000RAS 2 JonathanBraithwaite opinionsdifferonthispoint,seeLi(1998);Desch&Mouschovias AUand alocalmass-to-flux ratioof λ ∼ 1.7. Thefieldislarge- (2001);Banerjee&Pudritz(2006)amongstothers). scaleandhasadirectionsimilartothatinthesurrounding cloud. InthispaperIlookatasolutiontothisproblem,namelythat Using meteorites, Levy&Sonett (1978) measured 3 G at radius thecollapsecanproceedwithaslittleorasmuchlossoffluxasdis- 1 AU in the proto-solar system, translating to λ ∼ 5, using ob- sipativeprocessesallow,andthattheexcessfluxisdestroyedonce servations(e.g.Wilner&Lay2000)givingsurfacedensitiesofor- theprotostarforms,orrather,onceithasbecomedisconnectedin der 104 g cm−2 at this radius. Finally, inthe FU Orionis system somesensefromitsparentcloud.InthenextsectionIreviewrel- Donatietal. (2005) used the Zeeman effect tomeasure avertical evant observational and theoretical results of collapse from 1000 fieldcomponentof1kGwithafillingfactorof20%ataradiusof AUtoprotostarformation.Insection3,Iexplorethemechanisms 0.05AU,i.e.10R⊙.Withasurfacedensityof∼4×104gcm−2, fordestroying fluxinstarsviamagnetohydrodynamic (MHD)in- thisgivesameanmass-to-fluxratioλ ≈ 0.1.1 Thispoloidalfield stabilityinnon-convectivestarsandbuoyancyinconvectivestars, is accompanied by a somewhat weaker toroidal component, con- beforeinvestigatingthelatternumericallyinsection3.2.2.Idiscuss trastingwithlocaldynamomodelswhereapredominantlytoroidal theresultsinsection4andconcludeinsection5. fieldisgeneratedinsitu.Althoughtheseobservational resultsare approximateandrelatetodifferentradiiindifferentobjects,wesee thatdiscsareclosetomagneticallycriticalevenveryclosetothe protostar–thereisnoevidenceforλanywherenearashighasin 2 ACCRETIONANDPROTOSTARFORMATION stars(λ=103−108,seebelow). It is important to note that a small λ(r) = I now summarise relevant results and evidence that at least part 2πG1/2Σ(r)/B (r) in the disc does not necessarily mean ofthesolutiontothemagneticfluxproblemmustliein,orinthe z that mass and flux are being advected inwards in that same immediatevicinityof,theprotostar. ratio (and if the aforementioned value of λ ≈ 0.1 is true, this is impossible). The mass might be slipping inwards past the magneticfieldlines;allwecansay withcertaintyisthat theflux 2.1 Inwardsadvectionofmagneticfluxinadisc present in the disc was advected inwards at some point in the Onceacloudhasbecomesupercritical,itcancollapsedynamically. past. In theturbulent-disc scenario, vanBallegooijen (1989) used Magneticbraking becomesineffectiveonce thecollapseissuper- geometricalargumentstoshowthattheturbulentdiffusionshould Alfveńic,sotherotationalenergybecomeslargerinrelationtothe causethemagneticfieldtodiffuseoutwardsrelativetothegasat other energies. Normally this leads to formation of a disc of ra- the roughly same speed at which the gas moves inwards. There dius 100 − 1000 AU, although some systems might lack a disc maybewaysaroundthis:forinstance,Spruit&Uzdensky(2005) (Stassunetal.1999,2001;Rebulletal.2006).However, weshall proposed amechanismtoadvect fluxinwardsindiscreteclumps, lookhereatthemagneticfluxaccretedviaadisc. whichisperhapssupportedbytheaforementionedobservationsof Gas in an accretion disc can lose its angular momentum Donatietal. (2005). In contrast, in the disc-wind/jet model, one and spiral inwards by passing it either vertically to a disc wind expectsprimafaciethatfluxisadvectedinwardssinceaccretionis or jet (Blandford&Payne 1982), or radially outwards to disc fastandthereisnoturbulentdiffusion. gas exterior to itself via some instability-driven turbulence. The Ifinthesteadystatethestarisaccretingmassandfluxinthe instability could be gravitational (e.g. Jappsen&Klessen 2004; ratio λ∗, then mass and flux must be passing through each sur- Vorobyov&Basu2006;Zhuetal.2009),magnetorotational(MRI, faceofconstantradiusinthediscinthesameratio(ignoringout- Balbus&Hawley 1991; Pessah 2010 and refs. therein) perhaps flow), even though the local ratio will in general be much lower with buoyancy (e.g. Keppensetal. 2002) or purely hydrodynam- λ(r) ≪ λ∗, requiring almost perfect slippage at all radii. Since ical(Lithwick2009). thepropertiesofthediscvarysignificantlyoverthelargerangein Jetsareobservedinmanyobjects,arelaunchedfromradii∼1 radii,itisnaturaltoinferthatthereissomefeedbackmechanism AU(Bacciottietal.2002;Andersonetal.2003)andoftenappear whichpreventsfluxfrommovinginwardsfasterthanitcanbeab- togetherwithaslower,lesscollimatedwind.Aroundatenthofthe sorbedintothestar,whichinturnwilllimitthemassaccretionrate accretedmassisejected(Kurosawaetal.2006;Podioetal.2006). to that at which the mass can effectively slip past the field lines. Thereisevidenceinfavourofboththeories;forinstancetheout- Thismechanismcouldworklocally,‘feeling’somequantitysuch flowmodelissupportedbymeasurementsoftheangularmomen- astheradialgradientinfieldstrength,oritcouldbecyclic.Since tumofjets(Rayetal.2007),butnotbythefactthatdiscsareself- observations show that λ(10R⊙) ≪ λ∗ it seems likely that this luminous,sinceanoutflowwhichcarriesalloftheangularmomen- feedbackoriginatesfromabottleneckinthecentralregionwhere tumalsoremovesalloftheaccretionenergy.Itislikelytherefore it is noteworthy that a high ionisation fraction renders ambipolar thatdiscsareoutflow-orturbulence-dominatedindifferentzones, andOhmicdiffusionineffective.Finally,notethatwedonotknow atdifferent timesandindifferent objects(seeCombetetal.2010 thevalueofλ∗ whileanembeddedstarisaccreting–itmaywell andrefs.thereinforarecentcomparisonofthetwo). beverymuchlowerthanthemass-to-fluxratioofstarswhichhave Thereisstrongevidencethatdiscsdocontainstrong,ordered, finishedaccreting.Itispossiblethatmuchofthefluxislostonce netpoloidalflux.Thisiscrucialfortheoutflowmodel(Ouyedetal. themainaccretionphaseisoverandthestarhasbecomedetached 1997; Banerjee&Pudritz 2006, 2007; Beckwithetal. 2008) but fromitssurroundings. possiblyalsohelpfulinincreasingtheefficiencyoftheMRI,bring- Thepurposeofthispaperistoshowthatanyexcessfluxcan ingtheShakura-Sunyaevαparameterproducedinsimulationsup bedestroyedoncetheprotostarforms.Excessfluxistakentomean totheobservationally-inferredvalue(Kingetal.2007;Pessahetal. thedifferencebetweenthefluxaccretedontotheprotostarandthe 2007). Moreover, there are direct measurements of the magnetic field in protostellar discs at various radii. Vlemmingsetal. (2010) used methanol masers in the massive-protostar system 1 Note that β ∼ λ2(h/r)M∗/Mdisc where β = 8πP/B¯2; in a disc CepheusAHW2,findingafieldof23mGatadiscradiusof∼1000 thereforeitispossibletohavebothβ>1andλ<1. (cid:13)c 0000RAS,MNRAS000,1–10 On themagneticfluxprobleminstarformation 3 fluxlaterobservedonthestar.Theeasiestwaytoimaginehowthis a b canhappenistoassumefirstthatallfluxwhichsurvivesdiffusive processesduringthesupercriticalcorecollapseisaccretedontothe protostar, which therefore at least initially contains the entire accreted flux. The star then becomes magnetically ‘detached’ from itsparentcloud,andthenafteraccretionhasfinishedthefluxisde- stroyed according tothemechanisms described insection3.One might actually expect these mechanisms to work continuously as accretionisongoing,sothatthefluxoftheinfallingmaterialislost assoon asit arrivesat the protostar; this scenario isdiscussed in section3.3. c d 2.2 Conditionsintheproto-andmain-sequencestar Of crucial importance for the following discussion is the appear- ance of convection, which appears in protostars when atempera- tureof106 Kisreachedanddeuteriumfusionbegins.Ifthemass remains below 0.4M⊙, the star remains on the fully-convective Hayashitrackallthewayontothemain-sequence(MS)whenhy- drogen ignites. Above 0.4M⊙ a radiative interior eventually de- velops,eitherafteraccretionhasceased,whereuponthestarenters Figure1.Themagneticdisconnectionofacollapsingcore/protostarfrom the Henyey track (Henyeyetal. 1955; Palla&Stahler 1993) and therestofthecloud(forclarity,verysimplisticandnottoscale).(a)Acol- movestotheleftontheHRdiagram,orinamoremassiveprotostar lapsingoverdenseregiondrawsfieldlinestogether,theclassical‘hourglass’ whileaccretionisstillongoing–deuteriumburnsintheconvective picture.(b)Furthercollapse.(c)Reconnectionoccursoutsidetheprotostar, which becomes disconnected from thecloud. (d)A super-Alfveńic wind shell at the same rate at which it is accreted. Above about 4M⊙ ensuresthatthestariscompletelycausallydisconnectedfromitssurround- convectionretreatsrighttothesurfaceofthestar(Palla&Stahler ings,andthefluxinthestarcannowbedestroyed. Thewindexpels the 1993) and may dissappear entirelyor havelittleeffect on theac- surroundingcloudmaterial. creting gas. This marks a fundamental difference between high- mass and low-mass stars, withthe all the material in stars below 4M⊙ having experienced convection, but only theinner 4M⊙ of on stars which are still accreting; in both convective and non- moremassivestarshavingexperienced convection. Inaddition, it convectivestarsthefieldsmaynotbeverydifferentfromthoseon ispossiblethatverymassiveprotostarssomehowbypassthefully themain-sequence(seee.g.Alecianetal.2009). convective phase during growth up to2M⊙, retaining aradiative Thisagreeswithwhatoneexpectstheoretically.Inaconvec- interioratalltimes. tivestar,themagnetic fieldreaches asteady stateindependent of OntheMS, starsbelow 0.4M⊙,whicharefullyconvective, theinitialconditions.Inaradiativestarthemagneticfieldevolves andstarsbetween0.4and1.5M⊙,whichhaveaconvectiveenve- onadynamictimescaleuntilitfindssomestableequilibrium,the lope,displayfluctuatingdynamofields.Itisnotobviousfromfirst geometryandstrengthofwhichdoesdependontheinitialcondi- principles whether a magnetic-convective steady state should de- tions. pendontheinitialconditions;allwecansayisthatnosuchdepen- denceisapparent fromtheobservations, whichdohowever show a clear positive correlation between rotation speed and magnetic 3 DESTRUCTIONOFFLUXINASTAR activity(Pizzolatoetal.2003;seealsoMorinetal.2010). Incontrast,inMSstarswitharadiativeenvelope(>1.5M⊙) Once accretion has stopped and the star has developed a super- the observations of magnetic fields arethe following. Of theless Alfveńic wind, the star is ‘detached’ from the parent cloud and massivestars(upto6M⊙)somefraction(∼ 10%,seee.g.Power itsflux no longer needs tobe conserved. In fact, thisdetachment 2007) display a weak, large-scale field of 0.2 − 30 kG; the so- mayhappenearlierthanthecompletecessationofaccretion,when called Ap and Bp stars. At higher masses (O and early B) it theaccretionratedecreases,andthestarcouldpossiblydetachand seems that a similar fraction is magnetic (Grunhutetal. 2011). reattachmanytimesbeforefinaldetachment,inresponsetoFUOri- The fields strengths measured correspond to a mass-to-flux ratio onisaccretionepisodes.Apossiblesequenceofeventsisillustrated ofλ.10−3 assumingsimilarinteriorandsurfacefieldstrengths. infig.1,andalthoughtheactualpathbetweenstages(a)and(d)is Thesefieldsarepresumablyinequilibrium,sincethereisnopossi- uncertain,thedetailisunimportant.Inowlookattwomechanisms bilityofacontemporarydynamo.Therestofthepopulation,how- todestroyfluxintheidealisedcaseofafullydetachedstar,which ever,seemtohavefieldsatthegauss-levelorlower(Lignie`resetal. work respectively via MHD instability and reconnection to equi- 2009; Petitetal. 2011), which may be for instance produced by libriuminsidearadiativestar(section3.1)andbuoyantexpulsion asubsurface-convective-dynamo mechanism (Cantiello&Braith- fromtheinteriorofaconvectivestar(section3.2).Afterwards,in waite,inprep.)orbeevolvingdecayingdynamicallyonatimescale section3.3,Ilookatthemorerealisticscenariowherethesemech- givenbyτA2/P whereτA andP aretheAlfveńtimescaleandro- anismsproceedwhilethestarisstillaccreting. tation period (Braithwaite & Cantiello, in prep.) In any case, the important point is that in contrast to convective stars, the mag- 3.1 Non-convectivestars netic properties of radiative stars do depend on the initial condi- tions. For a recent review of magnetism in main-sequence stars In the simplest picture, a stably-stratified star initially contains a seeDonati&Landstreet2009.Fieldscanalsobeobserveddirectly symmetric, dipole-like purely-poloidal field which was inherited (cid:13)c 0000RAS,MNRAS000,1–10 4 JonathanBraithwaite contextsfromthelaboratory(Chui&Moffatt1995;Hsu&Bellan 2002)tothesolarcorona(Zhang&Low2003).Therefore: Hf ≈Hi. (1) Considerationofdimensionsgivesus |Hf|=ψfR∗Ef, (2) whereψ isadimensionlessparameteroftheequilibrium;wecan f solveforE andH onceweknowitsvalue.Inalarge-scaleequi- f f libriumψ shouldbeoforderunity;inasmall-scaleequilibriumit f willbesmallerbutnotethatduringarelaxationtoequilibriumat constanthelicityanequilibriumwithhigherψ isalwaysfavoured f sinceitrepresentsalowerenergystate. So,theenergyandstrengthofthemagneticfieldonthemain sequenceisdeterminedbythehelicityoftheparentcloud, which canbeexpressedas|H |=ψ R E whereψ isthedegreeof cl cl cl cl cl Figure2.Formoftheinstabilityinapoloidalfield.Ontheleftisshownthe twistinthecloud,whichcanhaveanyvaluefrom0toorderplusor equilibriumfield-lineloops;ontherighttheformoftheunstablemotions minusunity(helicityiseitherpositiveornegativedependingonthe atsomeparticularazimuthalwavenumber. senseofthetwist).Thislogicwasrecentlyconfirmednumerically in the context of intergalactic bubbles (Braithwaite 2010), where a relaxation to equilibrium was found approximately to conserve from the cloud. Any purely poloidal field is subject to an MHD helicity,uponwhichthefinalequilibriumenergythereforedepends. instability (Markey&Tayler 1973, 1974; Flowers&Ruderman Finally,notethatinthestandardhourglassmodel,helicityis 1977;Braithwaite2007;Marchantetal.2010)whichissimilarto zero and so one expects the field to disappear completely after theinstabilityinapairofinitiallyalignedbarmagnetswhichrotate disconnection from the cloud. A non-zero helicity requires some untiltheyliewithopposite polesnexttoeachother.Thefreeen- asymmetrybetweenthehemispheres. ergyisthevolumeintegralofB2/8π outsidethemagnets,which dropsasaresultoftherealignment.Afluidstarcanbethoughtof asacollectionofbarmagnets;themagnetsdonotstayalignedand 3.1.2 Stratification theinstabilitygrowsonthedynamical(Alfveń)timescale,around Assuming that the magnetic energy is much less than the gravi- 10yearsinasolar-typestarwithafieldstrengthof1kG–inany tational, a positive radial entropy gradient prevents the gas from casemuchshorter thananyformationtimescale.Seefig2.Even- movingsignificantlyintheradialdirection(wespeakofastably- tuallylengthscalesbecomesmallenoughsothatmagneticfluxis stratifiedstar),sothatduringtherelaxationtoequilibriumthegas destroyed inside the star, resulting in the complete destruction of motionisconfinedtosphericalshells.2Thismeansthattheabsolute themagneticfield.Notethatevenifthefieldisnotpurelypoloidal fluxthroughasphericalshellcannotincreaseduringrelaxationto but is ‘wound-up’ to some degree, there is nothing to stop it un- equilibrium.Thiscanbeseenbyconsideringtwoormoreregions windingfirstandthendevelopingthisinstability. on a spherical shell of positive and negative radial component of Ifthefieldisnotsymmetricalasabovebuthassomenet‘twist’ magnetic field B ; see fig. 3. As the fluid moves around on the then MHD instability will not destroy it entirely. Instead, it will r sphericalshelldiscontinuities(currentsheets)developbetweenre- evolveontheAlfveńtimescaleintosomestableequilibrium.The gions of different B (not just between positive and negative B main factors which determine the nature of the resulting equilib- r r butbetweenanydifferingvalues)andtheresultofthesesheetsis rium are a) the net twist, or more formally the magnetic helicity that gas from either side with initially different B ends up with andb)theinitialradialdistributionofmagneticflux. r some B which is an average of the two. Indeed in general, dis- r sipativeprocessescanonlyleadtoadropinthesphericalsurface integral |B |dS at any given radius. Therefore, if the star ini- r 3.1.1 Helicity tiallyconHtainsastrongmagneticfieldinthecentreandaweakfield at the surface, which is what one might expect if each fluid ele- Letusimaginetherelaxationtomagnetohydrodynamicequilibrium ment conserves itsflux during accretion and compression so that in a star of radius R∗ which contains a magnetic field of energy B ∝ ρ2/3, then the resulting equilibrium will also have only a E = (4π/3)R3B2/8π, where B is the r.m.s. magnetic field. In weakfieldatthesurface.Itispossibleforastarto‘hide’alarge the following, quantities initially and at (finally) equilibrium are magneticenergyinitsinteriorbutshownothingatthesurface.In marked with the subscripts i and f respectively. We can say the fact,ifB∝ρ2/3thoughoutthestarthenApstars,whichhavemag- followingabouttheequilibriumstate. netictothermalenergydensity ratiosatthesurfaceof between1 The reconnection destroys magnetic energy on small length and100(i.e.theplasma-βis0.01to1),haveglobalEmag/|Egrav| scalesbuthaslittleeffectonthemagnetichelicity,aglobalquan- ratios of the about same value. Of course, a global ratio of unity titydefinedasthevolumeintegralofthescalarproductofthemag- orgreaterisimpossiblesincethestarwouldbegravitationallyun- netic field with its vector potential H ≡ (1/8π) A·BdV. It bound.However,thelowerconductivitynearthesurfacemeansthat can be shown that in the case of infinite conductivRity, helicity is astarwhichinitiallyhasB ∝ ρ2/3 willundergodiffusionwhich conserved (Woltjer 1958). Helicity has units of energy × length andsoispresentmoreinthelargerstructuresthanistheenergy– anditisapproximatelyconservedduringreconnectiontakingplace 2 Inaddition,themotionisapproximatelyincompressible,sothevelocity on small scales, a property which has been very useful in many fieldhasjustonedegreeoffreedom. (cid:13)c 0000RAS,MNRAS000,1–10 On themagneticfluxprobleminstarformation 5 current sheet agravitationalpotentialwell.Thegascontainsanon-equilibrium magnetic field of r.m.s.magnitude B with acharacteristic length scale l, which is initially much smaller than the size of the sys- spherical shell temR(below,thecasewithinitiall ∼ Rislookedat).Magnetic reconnectionoccursataspeed vrec =αvA (3) wherethereconnectionspeedparameterαisoforderunity(inthe solar corona context a value 0.1 is often assumed) and vA is the Alfveńspeed.Asreconnectionproceedsonatimescaleofl/(αvA), intheabsenceofothereffectsthelengthscalelincreasesuntilsome Figure3.Reconnectiononasphericalshellwhereadiscontinuityappears global equilibrium is reached with l of the order the size of the inthemagneticfield.Thegreyarrowsrepresentthevelocityfield;notethat system.However,whilethisreconnectionisproceeding,thereare themotionsareessentiallytwo-dimensional. buoyant motions due to magnetically-induced density variations, which also have a length scale l: imagine the buoyant motion of aparcelofgasofsizelanddensityρthroughitssurroundingsof leadstothefieldinalayerbelowthesurfacerelaxingtoapotential field,meaningthatthefieldstrengthatthesurfacewillbesimilarto densityρ0.Matchingthebuoyancyforcetotheaerodynamicdrag, wehave thatatthebottomofthelayer,wheretheplasma-βismuchgreater thanunity;theB ∝ ρ2/3 relationcontinuestoholddeeperinthe ρ0l2vt2 ∼g|ρ0−ρ|l3 (4) interior.Forinstance,ina2M⊙ZAMSstarthepotential-fieldlayer shouldgrowtoadepthof∼0.05R∗afteratime107yr(calculated wherevt istheterminalvelocity(upwardsordownwardsdepend- by equating t to L2/η where L isdepth and η the magnetic dif- ingonthesignofρ0−ρ)andg isthegravitationalacceleration. fusivityatthatdepth);atthatdepththedensityis∼ 104 timesits Itcaneasilybeshownthattheterminalvelocityisachievedafter photosphericvalue;ifafieldstrengthof3kGseenonthesurface theregionhasrisenadistancel.Fromtheequationofhydrostatic isroughlyconstantdowntothatdepthandthenB ∝ ρ2/3 below equilibriumwecanreorganise(4)togive wthoaut,ldthbeeghloidbdalenrawtioithEinmtahge/|sEtagrr.aAv|lt∼ern5at×ive1l0y−th4e.Mfieolsdtsotfretnhgethfluinx vt ∼ l 1/2 |ρ0−ρ| 1/2 c0, (5) theinteriormaydependmuchlesssteeplyondensity;infactifone (cid:18)Hp(cid:19) (cid:18) ρ0 (cid:19) imaginesbuildinga(radiative)starbyaccretion,addingspherical where Hp is the pressure scale height and c0 is the sound speed shellsfrominsidetooutside(asseemslikelyfromentropyconsid- intheexternalmedium(recallthatc20 ∼ P0/ρ0).Wecanassume erations),itisdifficulttoseehowthefluidelementscouldpossibly ρ ≈ ρ0, so that the motion is very subsonic3. Pressure balance retaintheiroriginalfluxwithoutaveryunlikelyandunstablemag- gives4 neticfieldconfiguration.Inanycase,thereisstillpotentialforthe B2 B2 fieldstrengthdeepintheinteriortobemuchgreaterthanthatseen P0+ 0 =P + . (6) 24π 24π onthesurface. Thisignorestheanisotropicnatureofmagneticpressure;inreality neighbouringblobswillpushandpulloneachotherinsomecom- 3.2 Convectivestars plexway,butwelimitourselvesheretolookingjustatthegeneral tendencyforbuoyantrise.InanisentropicstarwehaveP =Kργ Alternatively,ifatsometimethestarbecomesconvectivethenthe so magneticfieldwilltendtorisetothesurface,itsenergybeingdis- sipatedbyreconnectionintheatmosphere.Whilethemagneticen- P0−P ≈γρ0−ρ, (7) ergy density is greater than the convective, as might be expected P0 ρ0 in a protostar at least initially, the important condition is not the incontrast toastar withan extra thermodynamic degree of free- convectivemotion,butthenear-isentropicstatewhichthismotion dom where buoyant balance ρ = ρ0 can be achieved despite the maintains. Deviationsfromuniformentropy arealsounimportant pressure difference: for instance in a star with an ideal-gas EOS when the field is above equipartition with the convective kinetic the temperature can be adjusted. Using (6) and (7), (5) becomes energy–buoyancyeffectsaredominatedbythemagneticfield. (droppingfactorsoforderunityandassuming|B−B0|∼B) l 1/2 3.2.1 Buoyantlossofmagneticflux vt ∼(cid:18)H (cid:19) vA. (8) p Amagneticfieldinanisentropicstarhasatendencytorisetowards ThiscorrespondstotheresultofParker(1975)fortheriseofaflux thesurface(Reisenegger2009,andrefs.therein).Anymagnetised tubeinaconvectivezonewherethehydrodynamicforceassociated regionmustbeinpressureequilibriumwithitssurroundings,which withtheconvectivemotioncanbeignored.Comparingthisto(3) meansthatitsgaspressuremustbelowerthanthatinthesurround- ings.Sincetheentropyisconstantwehaveρ = ρ(P),sothatthe regionmustalsohavelowerdensity.Thereisnowaytostopbuoy- 3 InthecaseofX-raycavitiesingalaxyclusters,wehaveρ≪ρ0andl≈ ant rise to the surface unless reconnection can occur fast enough Hp,sothatvt∼c0.Fromtheapparentabsenceofshocksinthesesystems we infer that the motion is in fact mildly subsonic and cannot therefore that a global magnetic equilibrium is reached where buoyant re- solvethecooling-flowproblemwithshockheating. gionsaresomehowhelddownbymagnetictension.Itistherefore 4 Itisappropriateheretousean‘isotropic’magneticpressureequaltoone informativetoestimatetherelevanttimescales. thirdoftheenergydensity,aswithanyrelativisticfluid(seee.g.Braithwaite Consider a body of gas sitting in hydrostatic equilibrium in 2010). (cid:13)c 0000RAS,MNRAS000,1–10 6 JonathanBraithwaite we see that the condition for reconnection to proceed faster than (thirdorder)increaseefficiencybygivingaloweffectivediffusiv- buoyantmotionis ityatmodestresolution(1923here).ThecodeincludesOhmicand wellasthermalandkineticdiffusion.UsingCartesiancoordinates l .α2. (9) avoidsproblemswithsingularitiesandsimplifiestheboundarycon- H p ditions:periodicboundariesareusedhere. Itseemsthereforethattherelaxationofanarbitrarymagneticfield The simulations model the star as a self-gravitating ball of withsmallinitiall toequilibriumasseeninsimulationsofasta- ideal gas (γ = 5/3) of radius R in hydrostatic equilibrium with blystratifiedstarcannotoccurinastarwithρ=ρ(P)beyondthe radialdensityandpressureprofilesobeyingthepolytroperelation pointwherethelengthscaleissomefractionofthescaleheight.If P ∝ ρ1+1/n, with the index n set to 3/2 here to give constant ontheotherhandtheinitialconditionshavealargerlengthscalel entropy, as opposed to the radiative-star approximation n = 3 thanthatin(9),thebuoyantmotionisalwaysfasterthantherecon- in Braithwaite&Nordlund 2006. Surrounding the star is a hot, nectionspeedandsoitisimpossibleforthemagneticfieldtoreor- poorly-conductingatmosphere,whichhastheeffectsbothofforc- ganiseitself.Regionsofstrongfieldwillrisetothesurfacewhile ingthemagneticfieldtorelaxintoapotential(curl-free)statewhile regionsofweakfieldwillsinktothecentre;atorabovethesurface keepingtheAlfveńspeedoutsidethestarnumericallyconvenient, ofthestar(inthelow-plasma-β,force-freeregime)therecouldbe sincethedensityisnottoolow;inotherwordsitisasimpleway reconnectionandlossofmagnetichelicity,eventuallyresultingin of modelling a vacuum. Since the field strength drops by a large completelossofthemagneticfieldofthestarviaacontinuouspro- factorduringtheprocessbeingmodelled,afield-uppingroutineis cessofmagneticbuoyantconvection.Inprincipleitseemspossible employed, which artificially amplifies the field to keep the mag- thatintheabsenceofheat-drivenconvection,somefieldmightbe netic energy constant (see Braithwaite&Nordlund 2006 for de- retainedandformaglobalequilibrium5;however, anequilibrium tails),which greatly reduces the computational demand and, per- notonlyrequiresnon-zerohelicityasinthecaseofstablestratifi- haps more importantly, keeps the Alfveń timescale much shorter cationexploredinsection3.1,butshouldeventuallybedestroyed thanthediffusive(Ohmic)timescale.Atthebeginning,thestaris bytheconvectivemotions. given an initially random magnetic field containing energy at all However,howcanonejustifyignoringtheconvectivemotion lengthscalesdowntoalimitofafewgrid-spacings,andtheMHD intheaboveanalysis?Tosuppressconvection,afieldmustnotonly equationsareintegratedintimetofollowtheevolutionofthefield. becoherentonscaleslargerthantheconvectivecells(∼ Hp)but Infig.4wecanseeanexampleoftheevolutionofafieldin alsoatleastatequipartitionwiththethermalenergy(Mestel1970), thisnumericalmodel.Atthebeginningthefieldissomewhatcon- whichwecanassumeisnotthecaseforHayashi-trackstars.How- centratedintotheinteriorofthestar,andrisestowardsthesurface ever, while the energy in the field is greater than the kinetic en- on an Alfveń timescale. Another example of this, starting with a ergyoftheconvectiontheconvectionissub-Alfveńicandtherefore differentrandomisationoftheinitialfield,isshowninfig.5. makes littledifference tothemagnetic buoyant motions explored Taking a closer look, at some point in time some recognis- above. able structure becomes visible, in the form of flux tubes lying Inanycase,oncethemagneticfieldhasdecayedtoequipar- quasi-horizontally around the surface of the star. These resem- titionwiththeconvection,theconvectivemotioncannolongerbe blethestructureofnon-axisymmetricequilibriafoundinprevious ignored, the analysis above becoming invalid. Thedomain of the simulations(Braithwaite2008)whenasimilarlyrandom, butless convectivedynamohasbeenentered,andsomesteady-stateispre- centrally-concentratedinitialfield(i.e.withsignificantfluxthread- sumablyreachedwiththemagneticenergyatmostcomparableto ingthesurfaceatt=0)islefttoevolveinastably-stratifiedstar– theconvectionenergy.Any‘memory’oftheoriginal(large)mag- horizontally-lyingfluxtubesarrangedinsomemeanderingpattern neticflux ispresumably erased, and themain-sequence magnetic below the surface of the star6; these tubes evolve on a timescale fielddoesnotdependontheconditionsintheparentcloud. givenbytheOhmictimescaleand/oralargemultipleofthethermal timescale (whichever is shorter) under the stellar surface, which canof coursebesignificant compared to,or longer than, amain- 3.2.2 Numericalmodelofanisentropicstar sequence lifetime. In contrast, the similar structures found in the current study are evolving on a shorter timescale related to the I now describe a numerical model of flux loss from an isen- Alfveń timescale in the stellar interior, due to the buoyant force tropic star. The stellar model and computational setup are simi- pushing them upwards into the atmosphere of the star, and the lar to those described in Braithwaite&Nordlund 2006, to where Ohmictimescaleabovethestellarsurface,asthetubesdecaydue the reader is referred for a fuller account of the setup of the to(relativelylow)finiteconductivity.Itisnotimmediatelyobvious model;abriefoutlineisgivenhere.ThecodeusedistheSTAGGER what this timescale should be, except that it will be rather small CODE(Nordlund&Galsgaard1995,Gudiksen&Nordlund2005), comparedtotheevolutiontimescaleofaprotostar.Inthesimula- a high-order finite-difference Cartesian MHD code which uses a tions,theexteriorhasamuchlowerconductivitythantheinterior, ‘hyper-diffusion’scheme,asystemwherebydiffusivitiesarescaled whichforcestheexteriorfieldtorelaxtoapotential(curl-free)con- withthelengthscalespresentsothatbadlyresolvedstructurenear figuration,butitisimpossibletomakethiseffectquantitativelyre- the Nyquist spatial frequency is damped whilst preserving well- alisticinthenumericalmodelwhiledynamic-timescaleprocesses resolvedstructureonlongerlengthscales.This,andthehigh-order are ongoing. In all the simulations the decay timescale is a few spatialinterpolationandderivatives(sixthorder)andtime-stepping Alfveń timescales. In reality the decay above the surface (in the 5 Therangeofavailable equilibria inisentropic stars,oralternatively in stars with a barotropic EOS ρ = ρ(P), is more restricted than in non- 6 Notethatinthesimulations itisdifficulttodefinethepreciselocation barotropicstarssincethereisonlyonedegreeoffreedominadjustingthe ofthestellarsurface,asthetransition fromisentropicandhighelectrical thermodynamicstatetobalancetheLorentzforce,buttheirexistencecannot conductivitytoisothermalandlowelectricalconductivitytakesplaceover beruledout. severalgridspacings (cid:13)c 0000RAS,MNRAS000,1–10 On themagneticfluxprobleminstarformation 7 Figure4.Theevolutionofaninitiallyturbulentmagneticfieldinanon-stablystratifiedstar.Thethreesnapshotsareatt/τA=0,3.1and7.8whereτAisthe Alfveńtimescale.AftersomeAlfveńtimes,thefieldhasbubbleduptothesurfaceandshowsnosignsofreachinganequilbrium.Theblueshadingrepresents thestellarsurface. 2004; Braithwaite 2008) and consequently no transport of radial fluxfromtheinteriortothesurface.ItseemsthereforethatanMHD equilibrium cannot be reached from these initialconditions in an isentropicstar;howeveritisimpossibletoruleoutatthisstagethat differentinitialconditions, perhapswithlargelengthscalesanda largemagnetichelicitymightleadtoanequilibrium.Therangeof equilibriainanisentropicstarishowevermorerestrictedthanthat availableinastably-stratifiedstar,andthepossibilitythatanequi- libriumcouldbereachedfromanyrealisticinitialconditionsseems ratherunlikely.Inaddition,theconvectivemotionsdrivenbyheat flux, whichare not included inthese simulations, would presum- ablypreventequilibriumformation.Thesequestionswillbestudied morefullyinaforthcomingpublication. 3.3 Ongoingfluxlossduringaccretion We have seen how flux can be destroyed once the protostar has stoppedaccretingandisfullydetachedfromitssurroundings,but itseemsplausiblethatsomethingsimilartothesetwoprocessesre- movefluxinaccretedmaterialas,orsoonafter,itisaccreted.This Figure5.Whenthefieldreachesthesurfaceofthestar,familarstructures ispossiblebecauseunlessthefieldisalreadyextremelyweakthe becomevisible.Here,fieldlinesareplottedandthesurfaceofthestaris Alfveńtimescaleonwhichthesemechanismsworkismuchshorter colouredaccordingtothevalueoftheradialfieldcomponent(blue&vio- thantheaccretiontimescale.However,sincewedonotobservethe letarenegative,green&yellow/orangearepositive).Thisrunissimilarto staruntilthemainaccretionphaseisover, itisdifficulttodistin- theoneshowninfig.4,butwithadifferentrandominitialisation;thissnap- guishobservationallybetweenongoingfluxdestructionduringthe shotistakenafter12.2Alfveńtimescaleshavepassed(correspondingtothe embedded accretionphaseandfluxdestructiononlyattheendof lastframeoffig.4.Fluxtubesliealongthesurfaceinapparentlyrandom thatphasewhenthestarbecomesdetached. patterns. In the magnetospheric accretion model (Koenigl 1991; Longetal.2008),thereisagap(withlowplasmaβ)betweenthe low plasma-β regime) should (helped by convective motions ab- starandtheinneredgeofthedisc;gasischannelledfromtheinner sent from these simulations) proceed on a dynamic timescale in edge along fieldlinesanchored around the magnetic poles of the localisedreconnectionzones,asitdoesinthesolarcorona. star.Thebulkofthestellarfluxself-connectswithinthemagneto- Infig.6wecanseetheprocessinalittlemoredetail–inatime sphere,sothestellarfieldisessentiallydetachedfromthesurround- animation from a third run with a different random initialisation. ings–itisimplicitlyassumedinthismodelthatsomemechanism Several runs were done, with different random initialisations and existstobreakfieldlinesbetweenthestarandtheparentcloud. withdifferentinitialradialprofilesofmagneticenergy,butineach In addition, note that the fieldstrengthdrops off sufficiently runtheresultwasessentiallythesame. quickly with increasing distance from the star that the magnetic Broadlyspeaking,thesenumericalresultsconfirmthepredic- energyinthevicinityofthestarismuchgreaterthanthatfurther tion made in section 3.2. This behaviour is distinct from that of away. [More precisely, the magnetic energy per unit radius as a thesamemagneticfieldinastably-stratifiedstar,wherethereisno functionofradius(Emag(r) =B2r2/2)mustdropofffasterthan transport of material in the radial direction (Braithwaite&Spruit log normal, meaning that dlnB/dlnr < −3/2; the values for (cid:13)c 0000RAS,MNRAS000,1–10 8 JonathanBraithwaite Figure6.Atimesequenceatt/τA =0,1.6,4.2,9.4,13.1and19.3(toprowlefttorightandthenbottomrowlefttoright).Thefieldquicklyreachesthe surface,butonlyinsomeplaces,sincetheremustbeadownwardsflowinotherplaces.Discretefluxtubesbecomerecognisable,asinthefourthframe.In thelastframe,theentirefieldcanbethoughtofasdiscretefluxtubes.Asthesefluxtubesarepushedupwardsbybuoyancytheylosetheiraxialcomponent intotheatmospherewhichcausesthemtobecomelongerandthinner.Thisissimilartoprocessinstablystratifiedstars,exceptthatithappensonadynamic timescaleratherthanadiffusivetimescale. dipoleandsplit-monopolefieldsare−3and−2respectively.]This reduced by mixing and reconnection. Again, the accreted flux is means that energy liberated by reorganisation of the field inside not relevant, but here also the accreted helicity isnot necessarily andimmediatelyoutsidethestaroutweighsanyenergyrequiredto conserved,dependingonthepropertiesofthedynamo. changethegeometryofthefieldfurtheraway. Ifthesemechanismscanpreventsignificantfluxfrombuilding Inthegeneralcasewhereanetnon-zero(butprobablysmall) upinthestar,theaccretiondiscwillnotbeabletodragmagnetic helicityhasbeenaccreted,aradiativestarmustcontainanequilib- flux inwards; the magnetic field lines necessarily find some way riumwhichevolvesquasistaticallyasaresultoftheadditionofnew of slipping outwards relativeto the inspiralling gas. This may be mass,fluxandhelicity.Imaginethatthestarhasaweakdipolefield facilitatedbyanoutwards-directedLorentzforceresultingfromthe andaccretesmaterialwithmagneticfieldalignedtothatdipole:the buildupoffluxtowardsthecentre.Whenthediscfirstformsitmust materialcansimplyflipoveruntilitsfieldisantiparalleltothatof initiallybeabletodragfluxinwards,accountingforthelargeflux the star (similarly to the case of two bar magnets in section 3.1) observedintheinnerdisc;atsomepointthatinwardsdragswitches whereuponitsfieldisannihilatedagainstthatofthestar;thefield offorismuchreducedasmagneticpressurebuildsupinthecentre. lines originally connecting the blob to the outside world will re- connectsoastobypassthestar.Inthiswayalowerenergystateis reached,magneticenergybeingconvertedintoheat.Infact,since 4 DISCUSSION thefluxoftheaccretedmaterialisnotconserved,onlytheaccreted massandhelicityarerelevant fortheevolutionofthestarandits Let us assume for the moment that a significant fraction of the magneticfield. original flux survives transport into the protostar. In low-mass Ifontheotherhandthestarisconvectiveorhasasignificant (<0.4M⊙)protostarswhichhavenotyetbecomefullyconvective, convectiveenvelope,itsfieldwillbeinadynamo-poweredsteady itispossible that theflux isdestroyed whileaccretion continues, state.Highlymagnetisedmaterialarrivingatthesurfacewillstay butattheverylatestwhenitceases.Ifitispossibletomeasurethe atthesurfaceuntilbothitsentropyandmagneticbuoyancycanbe magneticfieldonalow-masspre-MSstaraftersignificantaccretion (cid:13)c 0000RAS,MNRAS000,1–10 On themagneticfluxprobleminstarformation 9 has ended but before deuterium ignition, one should see a global been suggested (Zinnecker&Yorke 2007; Maitzenetal. 2008; MHD equilibrium rather like those seen in radiative stars on the Bogomazov&Tutukov 2009; Ferrarioetal. 2009) that magnetic MS.Itisprobablydifficultorimpossibletomeasurethemagnetic starsaremergerproducts,themergerresultingindifferentialrota- fieldonlow-massprotostarswhicharestillaccretingsignificantly, tionandsomekindofdynamoactivity.Thisscenariowouldexplain butthismeasurement coulddistinguishbetweenongoing fluxde- thelackofmagneticstarsinshort-periodbinaries(Abt&Snowden structionanddestructiononlyaftercompletedetachmentfromthe 1973). However, this could also be explained by magnetic inhi- surroundings. Inprinciplefluxdestructioncouldbeginassoonas bition of fragmentation, as is seen in simulations (Machidaetal. theAlfveńtimescalebecomeslessthanthecollapsetimescale,i.e. 2008);alsothereistheissueofthelackofuniversalmagneticfields whenthefirstcoreformsandthecollapsebecomessub-Alfveńic. inbluestragglers.Inanycase,itseemslikelythatthemagneticstars Oncedeuterium igniteshowever, andthestar becomes fullycon- areunusualinsomesenseandthatthenormalstateofaffairsisfor vective–whichitremainsthroughoutthemain-sequence–thestar thefluxtodroptothatcorrespondingtolessthan1gauss. isexpectedtoloseallofitsoriginalfluxandhelicityandwithitthe ‘magneticinitialconditions’ fromtheparent cloud. Thisexplains whythemagneticfieldsoflow-massstarsappeartodependonlyon 5 CONCLUSIONS rotationspeed.Solar-typestars(eventualmass0.4−1.5M⊙)also passthroughafully-convectivephasebeforearadiativecoredevel- I have examined mechanisms to destroy flux in a protostar, with ops;inthisenvelopeadynamosimilartothatinlow-massstarsis a view to shedding some light on the observational fact that the expected,andobserved.Insidetheradiativecore,onemightexpect magnetic flux in main-sequence stars is very small compared to tofindtheremnantofthepreviousconvectivedynamoinanequi- that inmolecular cloud coresinstar-formingregions. Itisuncer- librium whose strength depends on the helicity generated by (or tain how much of the original flux is already removed from the surviving) thisdynamo. Notethatforadynamo togenerateanet gas by various diffusive processes during collapse and accretion, helicityfromzeroinitialhelicitytheremustbesymmetry-breaking andthemechanismsexploredherecanremovefluxwhichsurvives or positive feedback; alternatively an initial net helicity inherited theseprocessesandisaccretedontotheprotostar.Themechanisms fromthemolecularcloudmaypersistthroughthedynamophaseif canbecategorisedaccordingtowhethertheyoperateinconvective thephase doesnot lasttoolong,or mayperhapsbeamplified.In ornon-convectiveprotostars.Inconvectivestars,themagneticfield anycase,theresidualfieldisunlikelytobestrongerthanequipar- risesbuoyantlytothesurfaceofthestaronadynamicaltimescale titionwiththepreviousconvectiveenergy. (anAlfveńtimescale,∼10yrinasolar-typestarwitha1kGfield Intermediate-mass stars (> 1.5M⊙) also have a fully- or somewhat longer in a protostar with larger radius) and its en- convectiveprotostellarphase,butconvectionintheenvelopeeven- ergy is destroyed by reconnection in the atmosphere, much as is tuallydiesaway.Intheradiativeenvelopewearethereforeseeing observedinthesolarcorona,untilthefieldstrengthhasdroppedto eitherformerlyconvectivematerial,whichhaslostall‘memory’of thatwhichcanbemaintainedbyaconvectivedynamo.Inaradia- itsinitialmagneticfield,asinthesolar-typestars,ormaterialwhich tivestar(ortheradiativezoneofastar),anarbitrarymagneticfield accreted onto the star after significant convection disappeared, in evolvesonthesamedynamical timescaleintoanequilibrium,the whichcasetheaccretedhelicitymaysurvive.Thusthedifference strengthofwhichdependsnotontheoriginalfluxbutonthemag- between magnetic field strengths in intermediate-mass MS stars netichelicity.Anon-zerohelicityrequiressomeasymmetryinthe andthebimodalitybetweenmagneticandnon-magneticstarscould magnetic field of the accreted material; in thestandard hourglass beatleastpartiallyduetodifferencesintheaccretionhistory;the modelthehelicityiszero. non-magneticstarswouldhaveaccretedalloftheirmaterialwhile Insummaryanyexcess‘unwanted’fluxcanbedestroyedonce still convective, while the magnetic stars would have become ra- thestarbecomesmagneticallyindependent fromitssurroundings, diativewhileaccretioncontinuedandwereablefromthatpointon- via buoyancy and/or MHD instability and reconnection on a dy- wardstoaccumulatemagnetichelicityfromtheaccretedgas.This namical timescale. This should happen either while the main ac- ispossiblyconnectedtotheobservationalresultthatthemagnetic cretionphaseisongoingorasitcomestoanend–inanycase,by fraction increases with mass, from ∼ 1% at 1.5M⊙ to ∼ 20% thetimewecanobservethestaritwillalreadyhavelostitsoriginal at 6M⊙ (Power 2007). In more massive stars deuterium fusion flux. contributes relatively little to the energy and convection may be Acknowledgements.TheauthorwouldliketothankLeonMes- unimportant,inwhichcasewedoexpecttoseemagneticequilibria tel, A˚ke Nordlund and Wouter Vlemmings for useful discussions whichdependoninitialconditions.However,ifastarformsfroma andassistance. relativelysymmetricalcloudthenhelicityshouldbesmallandwe expectalmostallofthemagneticenergytobelostduringtherelax- ation to equilibrium. 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