On the magnetic flux problem in star formation

0.39 MB·English

by  Jonathan Braithwaite

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
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0
ABSTRACT
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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 Alfve´ntimescale.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
Alfve´nic,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 mag-
netic 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 ac-
creted 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-Alfve´nic 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 Alfve´nic 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 aretheAlfve´ntimescaleandro- 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(Alfve´n)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-
evolveontheAlfve´ntimescaleintosomestableequilibrium.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
Alfve´nspeed.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 keepingtheAlfve´nspeedoutsidethestarnumericallyconvenient,
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 Alfve´n 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 Alfve´n timescale. Another example of this, starting with a
ergyoftheconvectiontheconvectionissub-Alfve´nicandtherefore 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-
Alfve´n 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
Alfve´n 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
Alfve´ntimescale.AftersomeAlfve´ntimes,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 Alfve´ntimescaleonwhichthesemechanismsworkismuchshorter
colouredaccordingtothevalueoftheradialfieldcomponent(blue&vio- thantheaccretiontimescale.However,sincewedonotobservethe
letarenegative,green&yellow/orangearepositive).Thisrunissimilarto staruntilthemainaccretionphaseisover, itisdifficulttodistin-
theoneshowninfig.4,butwithadifferentrandominitialisation;thissnap- guishobservationallybetweenongoingfluxdestructionduringthe
shotistakenafter12.2Alfve´ntimescaleshavepassed(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.
theAlfve´ntimescalebecomeslessthanthecollapsetimescale,i.e. 2008);alsothereistheissueofthelackofuniversalmagneticfields
whenthefirstcoreformsandthecollapsebecomessub-Alfve´nic. 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. (anAlfve´ntimescale,∼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. An equilibrium can then survive essentially
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