Magnetic field amplification in collapsing, non-rotating stellar cores

0.75 MB·English

by  Martin Obergaulinger

Astronomy&Astrophysicsmanuscriptno.b˙nonrot˙SN c ESO2011
(cid:13)
January7,2011
Magnetic field amplification in collapsing,
non-rotating stellar cores
M.Obergaulinger, andH.-Th.Janka
1 Max-Planck-Institutfu¨rAstrophysik,Karl-Schwarzschild-Str.1,D-85748Garching,Germany
2 RacahInstituteofPhysics,TheHebrewUniversity,Jerusalem91904,Israel
Receivedxxmonthxxxx/Acceptedxxmonthxxxx
ABSTRACT
1
1
Context.Theinfluenceofmagneticfieldsonstellarcorecollapseandexplosionisnotwellexplored.Itdependsonthepossibilityto
0
amplifytheinitialpre-collapsefields.Intheabsenceofrotationthiscanhappenbycompression,convection,thestandingaccretion
2
shockinstability,andtheaccumulationandgrowthofAlfve´nwavesintheaccretionflows.

n Aims.Weinvestigatesuchamplificationmechanismsofthemagneticfieldduringthecollapseandpost-bounceevolutionofthecore
a ofanon-rotating15M starwithvariedinitialfieldstrengths,takingintoaccountthemicrophysicalequationofstateandneutrino
J physicsthatplayacruc⊙ialroleinsupernovacores.
6 Methods.Weperformsimulationsofidealmagnetohydrodynamicswithneutrinotransportinaxisymmetry.Thetransportofelectron
neutrinos andantineutrinos istreatedwithanew scheme that solvestheenergy-dependent set of radiationenergy andmomentum
] equationsintwodimensionsbyusingananalyticclosurerelation.
R Results.The magnetic field undergoes kinematic amplification by turbulent flows. We also find indications for amplification by
S interacting waves travelling upwards and downwards inside accretion streams. The fields can reach up to equipartition with the
velocityfield.Veryhighmagneticfieldstrengthsrequireverystrongpre-collapsefieldsandareabletoshapethepost-bounceflow,
.
h leadingtoapatterndominatedbylow-ordermultipoles.Suchmodelsareclosesttoasuccessfulexplosion.
p Conclusions.Magneticfieldscanbuilduptointerestingstrengthseveninnon-rotatingcollapsingstellarcores.Startingwithfieldsin
- thepre-collapsecoreaspredictedbypresentstellarevolutionmodels(109–1010G),typicalneutronstarfieldsof1012–1013Gemerge,
o
whereasprogenitorfieldsofsome1011–1012Gleadtofieldsofmagnetar strengthofabout1014–1015G.Onlyinthelattercasethe
r
t magnetic fieldshavesignificant dynamical effectson theflowsinthesupernova coreand mayhave aninfluence ontheexplosion
s mechanism.However,innoneofour2Dsimulations,wefindanexplosionuntil500millisecondspost-bounce.
a
[ Keywords.Magnetohydrodynamics(MHD)-Supernovae:general-Stars:magneticfields-Stars:magnetars

1
v
8 1. Introduction cally affected explosions very much dependent on the amount
9 of field amplification happening during and after collapse.
1 Mostscenariosfortheexplosionmechanismofcore-collapsesu- Rapid rotation may amplify a weak seed field quite gener-
1 pernovae (SNe) involve a combination of energy deposition in ically to dynamically relevant values, e.g., by winding up
1. themattersurroundingthenascentproto-neutronstar(PNS)and a poloidal field (a process linear in time) or, exponen-
0 multi-dimensionalhydrodynamicflows.Examplesformeansof tially in time, by the magneto-rotational instability (MRI,
1 energy transfer to the SN ejecta are the prompt bounce shock, Akiyamaetal. 2003). Magneto-rotational explosions, theoret-
1 neutrinos, magnetic fields and acoustic waves. In the neutrino- ically discussed by Meieretal. (1976), have been studied
: heatingmechanismneutrinostap thegravitationalpotentialen- in various approximations, e.g., by Bisnovatyi-Koganetal.
v
i ergy released during collapse and deposit a part of it behind (1976);Symbalisty(1984);Akiyamaetal.(2003);Kotakeetal.
X the stalled shock. This processis enhancedand thus supported (2004); Obergaulingeretal. (2006); Cerda´-Dura´netal. (2007);
r by non-radial fluid flows triggered by hydrodynamic instabili- Obergaulingeretal. (2009); recent simulations employing de-
a tieslikeconvectionandthestandingaccretionshockinstability tailed microphysics have been performed by Burrowsetal.
(SASI;Blondinetal.2003;Foglizzo2001,2002) (2007);Scheideggeretal.(2008).
Thoughitissafetoassumethattheexplodingstarwillpos- However, according to stellar-evolution models, the major-
sessamagneticfieldofsome(unknown)strengthandtopology, ity of progenitors is expected to rotate slowly. Although many
its possible influence on the explosion is less clear. The main stars on the upper main sequence show a surface rotation pe-
reason for the small number of conclusive investigations into riodclosetothecriticalvalueformassshedding,theywillmost
this topic is that important effects are expected to occur only likelylosemostoftheirangularmomentumduringtheirsubse-
when the magnetic field is roughly in (energetic) equipartition quentevolution,e.g.,by strongstellar windsor magneticbrak-
withthegasflow,aconditioncorrespondingtoextremelystrong ing(Hegeretal.2005;Meynetetal.2011).Non-rotatingcores,
fields similar to those observed in magnetars. This makes full albeitlackingtheabove-mentionedefficientchannelsforampli-
MHD simulations including a treatment of the important neu- fication,mayexperiencefieldgrowthbyseveraleffects:
trinophysicsintheSNcoreindispensable.
1. After bounce, convection develops in the PNS and the
Stellar evolution calculations, predicting only weak pre- surrounding hot-bubble region. Breaking down into three-
collapse magnetic fields, render the prospects for magneti- dimensionalturbulence,convectionmayprovidetheαeffect
2 M.ObergaulingerandH.-Th.Janka:Magneticfieldamplificationincollapsing,non-rotatingstellarcores
responsible for a small-scale dynamo amplifying the field Table1.VariablesandsymbolsusedintheMHDequationsand
on length scales comparable to the scale of turbulent forc- theneutrinotransportformulationpresentedinthisarticle.
ing,i.e.,thesizeofconvectiveeddies(Thompson&Duncan
1993). However, the generation of a large-scale field, e.g., Symbol Variable
on the scale of the PNS, probablyrequiresa non-vanishing ρ gasdensity
kinetic helicity of the turbulent flow, which can most nat- Ye electronfraction
v gasvelocity
urally be accounted for by (differential) rotation (see, e.g.,
b magneticfield
Brandenburg&Subramanian2005).
ε internalenergydensity
2. Endeveetal. (2010) have demonstrated that the standing-
e totalenergydensity
⋆
accretion-shock instability (SASI Blondinetal. 2003), ex-
P gaspressure
citedbyafeedbackcycleofacousticandadvectiveperturba- P totalpressure
⋆
tionstravellingbetweentheshockwaveandthedeceleration Φ gravitationalpotential
regionabovethePNS,hasthepotentialtoamplifythemag- S0 neutrino-matterenergyexchange
ν
netic field by up to four orders of magnitude.Dynamically S0 neutrino-matterlepton-numberexchange
ν;n
relevant field strengths (in which case the field reaches at Si neutrino-mattermomentumexchange
ν
least10%oftheequipartitionstrength)canbereachedonly Eν neutrinoenergydensity
when the pre-collapsefield in the stellar coreis sufficiently Nν neutrinonumberdensity
Fi neutrinoenergyflux
strong. ν
Vi neutrinonumberflux
3. Non-radialfluidmotionstriggeredbytheseinstabilitiescan ν
p scalarEddingtonfactor
excite perturbations of the magnetic field propagating as ν
Pij neutrinoenergypressuretensor
Alfve´nwavesalongthefieldlines.Theoutwardpropagation ν
Rij neutrinonumberpressuretensor
of Alfve´n waves excited close to the PNS has to compete ν
χ0 0th-momentopacity
with theaccretionofgastowardsthecentre.Assumingthat
χ1 1st-momentopacity
the accretion flow decelerates in this region continuously,
Guiletetal.(2010)arguethattheremustbeanAlfve´npoint
atwhichtheAlfve´nspeedequalstheaccretionvelocity,and
inthispaper,theschemeprovidesareasonablygoodrepresenta-
the propagation of the wave (measured in the lab frame)
tionoftheneutrinoeffectsthatplayaroleduringcorecollapse,
comestoarest.TheyshowthatAlfve´nwavesareamplified
bounce,shockpropagationandintheaccretionlayerbehindthe
exponentially at such a stagnation point. At the conditions
stalled supernovashock.Moreover,itis acomputationallyeffi-
of a supernova core, the amplification should be most effi-
cient treatment of the neutrino transport, and two-dimensional
cientforamagneticfieldstrengthofafew1013Gandcould
simulationsuptoseveralhundredmillisecondsafterbounceare
yield final fields of the order of 1015 G. Dissipation of the
easilypossible.
waveenergycanincreasetheentropyofthegas,thusmodi-
Itisthegoalofthisworktoinvestigatetherelevanceoffield
fyingthedynamicsintheaccretionregion.Forevenstronger
amplification mechanismslike those in points 1.-3. concerning
fields, the Alfve´n point can be close to the shock wave. In
their importance for the evolution and potential revival of the
this case, Suzukietal. (2008) find that the explosion can
stalled shock by core-collapse simulations including a reason-
be drivensolely by the energydeposited by the dissipation
ablygoodtreatmentofthe relevantmicrophysics.Inparticular,
ofAlfve´nwaves.Thisprocesstransmittingenergyfromthe
wewillstudythefieldgrowthconnectedtoconvectiveandSASI
(convectivelyactive)PNStothemuchlessdensesurround-
activityinthepost-shocklayerandtheroleofenergytransport
ingmediumbearsastrongsimilaritytotheproposedmecha-
anddissipationbyAlfve´nwaves.Tothisend,weperformMHD
nismforheatingthesolarcoronabyAlfve´nwavesemerging
simulations of collapse and post-bounce evolution of the core
fromthesolarsurface.
of a star of 15 solar masses (Woosleyetal. 2002). Varying the
strength of the initial field, assumed to be purely poloidal, we
Previous simulations of magnetohydrodynamicstellar core candeterminedifferentsitesandmechanismsoffieldamplifica-
collapse have used a wide variety of methods to treat the tionandidentifytheback-reactionofthefieldontotheflow.
effects of neutrinos: in the simplest models, they were ei-
This article is organised as follows: Sect.2 describes our
ther ignored (e.g., Obergaulingeretal. 2006; Mikamietal. physical model and the numerical methods; Sect.3 introduces
2008) or treated by simple parameterizations or local source the initial conditions; Sect.4 presents the results of our simu-
terms (e.g., Cerda´-Dura´netal. 2008); more complex ap-
lations; Sect.5 gives a summary of the study and draws some
proaches include trapping/leakage schemes (e.g., Kotakeetal. conclusions;appendixAgivesabriefoverviewofourtreatment
2004), multi-dimensional,energy-dependentflux-limited diffu- oftheneutrinotransport.
sion (Dessartetal. 2006) or the isotropic-diffusion source ap-
proximation which discriminates between trapped and free-
streaming components (Liebendo¨rferetal. 2009). In our ap- 2. Physicalmodelandnumericalmethods
proach to radiation-magnetohydrodynamics (RMHD), we em-
ploy a new multi-dimensional and energy-dependent scheme 2.1.Magnetohydrodynamics
for the neutrino transport in supernova cores, namely a
We assume thatthe evolutionof the gasandthe magneticfield
two-moment solver for the neutrino energy (lepton number)
isdescribedbytheequationsofNewtonianidealmagnetohydro-
and momentum equations with an analytic closure relation
dynamics(MHD),
(Cernohorsky&vanWeert 1992; Ponsetal. 2000; Auditetal.
2002).Presentlyitincludesonlyelectronneutrinosandantineu- ∂ρ+∇ (ρv) = 0, (1)
t
trinosandarelativelysimpletreatmentonlyofthemostrelevant ∂(ρY )+∇ (ρ·Y v) = S0 , (2)
interaction rates of these neutrinoswith neutrons, protons, and t e · e ν;n
nuclei.Nevertheless,inthecontextofthequestionsfocussedon ∂(ρvi)+ P δij+ρvivj bibj = ρ iΦ+Si, (3)
t ∇j ⋆ − ∇ ν
(cid:16) (cid:17)
M.ObergaulingerandH.-Th.Janka:Magneticfieldamplificationincollapsing,non-rotatingstellarcores 3
∂e +∇ ((e +P )v (v b)b) = ρv ∇Φ+S0, (4) Table2.Averagemagneticfieldstrengthsandnormalisedmag-
t ⋆ · ⋆ ⋆ − · · ν
∂ b ∇ (v b) = 0, (5) neticenergiesforourcomputedmodelswithmagneticfield(ex-
t
− × × cludingmodels15-B0).Seetextfordetails.
describingthe conservationof mass, electronfraction,gas mo-
mentum, total energyof the matter, and magnetic flux, respec- model s15-B10 s15-B11 s15-B11.5 s15-B12
tively.Inadditiontotheseevolutionaryequations,themagnetic b 1010 1011 3.16 1011 1012
0 ×
fieldhastofulfilthedivergenceconstraint, b [G] 1.2 1013 1.4 1014 4.2 1014 1.4 1015
β1i4 5.9×10 9 7.8×10 7 7.5×10 6 5.9×10 5
∇ b=0. (6) βk14 0×.20 − ×24 − ×82 − 7×00 −
· 14
b [G] 9.1 1012 1.2 1014 2.1 1014 4.7 1014
Thesymbolsusedinthissystemhavestandardmeaning;theyare βcinv 1.1×10 7 2.9×10 5 9.8×10 5 4.8×10 4
listedinTab.1.Thetotalenergydensityandpressurearedefined βkcnv 0.0×0041− 0.×055− 0×.16 − 1×.6 −
asthesumofthecontributionsofthegasandthemagneticfield, cnv
b [G] 3.8 1012 3.0 1013 6.3 1013 1.4 1014
e⋆ =ε+1/2ρv2+1/2b2andP⋆ = P+1/2b2,respectively. βsitb 1.1×10 6 7.8×10 5 3.3×10 4 1.4×10 3
The MHD part of the system is closed by the equation of βsktb 0.0×020− 0×.16 − 0×.80 − 3×.7 −
state of Shenetal. (1998), relating gas density, internal-energy bstb [G] 3.6 1011 6.0 1012 1.0 1013 1.2 1013
density,andelectronfractiontotemperature,T,gaspressureP, βgiain 2.2×10 6 5.7×10 4 2.7×10 3 1.0×10 2
entropy,S,thecompositionoffourspecies(neutrons,protons,α βkgain 0.00×008−9 0.×016− 0.×071− 0×.26 −
gain
particles,andarepresentativeheavynucleus),andthechemical
potentialsofneutrons,protons,andelectrons.
To include some key features of general relativity into our
Newtonianmodel,weusethemodifiedgravitationalpotentialof specifythetensorofthesecondmoment,Piνj,forwhichweuse
Mareketal. (2006). Following the lines proposed by these au- asimpleanalyticclosurerelationfromMinerbo(1978).Usinga
thors,weincludethegasandneutrinoenergydensityandpres- tensorialgeneralisationofa one-dimensionalEddingtonfactor,
sureaswellasanapproximateLorentzfactoroftheflowinthe our method is generically multi-dimensional. A more detailed
sourcetermofthePoissonequationforΦ.Overall,thisleadsto discussioncanbefoundinAppendixA.
asomewhatdeeperpotentialwellwhenthecoreapproachesnu- We note that our set of equations is a generalisation of the
clear matter density and servesas a good approximationto the usualdiffusionansatz,inwhichthesystemofmomentequations
strongerrelativisticgravityofaPNS. is truncated at the level of the energy equation and closed by
Our code is based on an implementation of the MHD expressingthefluxintermsofthegradientoftheenergydensity,
equations on a fixed Eulerian grid in the constrained- mostlyconnectingdiffusionandfree-streaminglimitsbyaflux
transport discretisation (Evans&Hawley 1988). It uses high- limiter.Retainingthefirsttwomoments,itleadstoahyperbolic
resolution shock-capturing (HRSC) methods with approx- system,whichcanbesolvedbycommonmethodssuchashigh-
imate Riemann solvers (Lax-Friedrichs, HLL, or HLLD resolutionshock-capturingmethods.Moredetailsonthiscanbe
(Miyoshi&Kusano 2005)) in the multi-stage framework foundinAppendixA.
(Titarev&Toro 2005). We reconstruct the conserved vari-
ables using high-order monotonicity-preserving (MP) methods
(Suresh&Huynh 1997). Time stepping is done by explicit 3. Modelsandinitialconditions
Runge-Kuttaschemesof2nd,3rd,or4thorder.
Interestedinthecontributionofmagneticfieldstothedynamics
of supernovae, we simulate the evolution of a star of 15.0 so-
2.2.Neutrinotransportandneutrino-matterinteractions larmasseswithsolarmetalicity(Woosleyetal.2002).We map
the pre-collapse model to a grid of n = 320 radial zones dis-
We treat the transport of neutrinos by solving the energy- r
tributedlogarithmicallyoveraradialdomainof[r =0;r
dependent, multi-dimensional moment equations for neutrino min max ≈
2800 km] and n = 64 lateral zones distributed uniformly be-
energy,E ,andneutrinomomentum,Fi, θ
ν ν tweenthenorthandsouthpole.Theresolutionatthegridcentre
∂E +∇ (E v+F ) ω v ∂ Pjk = S0, (7) is(δr)ctr =400m.Tenenergybinsbetweenǫmin =0.1MeVand
t ν · ν ν − ∇j k ω ν ν ǫ =280MeVareusedfortheneutrinos.
∂tFνi +∇j(Fνivj+Piνj)+Fνi∇jvj = Sν1;i. (8) maxThe topology of the magnetic field at the onset of collapse
is highly uncertain. On the main sequence, field amplification
Presently only electron neutrinos, ν , and antineutrinos, ν¯ are
e e by,e.g.,gradualcontractionof the star or convectioncompetes
considered. The source terms, S0,1, include the interaction of
ν with the loss of magnetic energy in stellar winds and in work
neutrinos with the gas by emission, absorption, and scattering
the magnetic field does by exerting torques on the stellar mat-
reactions;weuseareducedsetofprocesses,viz.
ter.Intheabsenceofrotation,starslackanimportantingredient
ofmostlarge-scaledynamos.Insuchacase,theymaybedom-
1. emissionandabsorptionofelectronneutrinosbyneutrons,
inated by a small-scale turbulent field rather than a large-scale
2. emission and absorption of electron anti-neutrinos by pro-
field, e.g., a lower-order multipole. Nevertheless, we assume a
tons,
3. elasticscatteringofallneutrinosoffnucleons, simple initial field, viz. a modified dipole (Fig.1). While this
isprobablynota verygoodapproximationtorealstellarcores,
4. emissionandabsorptionofelectronneutrinosbyheavynu-
it representsan optimalconfigurationfor the Alfve´n-waveam-
clei,
5. coherentelasticscatteringofallneutrinosoffheavynuclei. plificationmechanismwe areinterestedin becauseofthe large
coherent radial component favouring the radial propagation of
Furthermore, fluid-velocity dependenteffects such as the PdV Alfve´nwaves.Furthermore,weobservethatthefieldisreplaced
work associated with diverging flows appear in the equations byamorecomplexsmall-scalefieldintheregionsofthecoreaf-
as well. To close the system of moment equations, we have to fectedbyhydrodynamicinstabilitiessuchasconvectionandthe
4 M.ObergaulingerandH.-Th.Janka:Magneticfieldamplificationincollapsing,non-rotatingstellarcores
Core bounce occurs after 168 milliseconds of collapse; the
shockformsatamasscoordinateof M 0.45 M andstarts
sh
propagating outwards. A strong negative≈gradient⊙of the elec-
tronfractiontriggerspromptPNSconvectioninalayerbetween
log |b| [G] an inner radius of about 15 and an outer radius that is initially
at about50 km and decreasesto about40 km at the end of the
7.92 8.63 9.33 10.04
simulation. In Fig.2, we show the upper boundary of the PNS
convectionzonebya solidblackline.Convectivevelocitiesin-
2.5 side the PNS reach maximum values of 5 108 cms 1 and
−
angle-averagedr.m.s.valuesof 2.5 10≈8cm×s 1.
−
≈ ×
Afterroughly50mspost-bounce,theshockstartstooscillate
2.0
as the standing accretionshock instability (SASI Blondinetal.
] 2003) develops in the post-shock region. Perturbations created
m
k1.5 attheshockwave areadvectedwith theflowtowardsthePNS,
0
0 leading to the parallel alignment of the patterns of lateral ve-
0
1 locity and of the (grey) mass-shell lines lines in Fig.2. From
z [1.0 theseperturbations,theSASIdevelopsinthepost-shockregion.
TypicaltransversevelocitiesassociatedwiththeSASIreachval-
0.5 ues about 3 times as large as the convective velocities deeper
insidethePNS.Withincreasingtime,theSASIbegins,starting
intheimmediatepost-shockregion,toaffectalargerfractionof
0.0
0.0 0.5 1.0 1.5 2.0 2.5 thepost-bouncevolume,untilatt&100msthePNSconvection
x [1000 km] zone is separated from the SASI region by a thin stable layer
locatedroughlybetweendensitiesof1011 and1012 gcm 3.The
−
Fig.1. Initial (pre-collapse) field configuration of model borderbetweenthestablelayerandtheSASIregionisshownin
s15-B10:field strength (colour scale) and field lines. For sim- Fig.2byasolidblackline.
plicity,onlyonequadrantisshown.
Forlaterreference,welistthedefinitionsweuseforthefour
distinctregionsofthecoreandtheirrespectiveboundaries:
SASI.Hence,wedeemtheinfluenceofsuchanartificialchoice 1. The inner hydrodynamically stable PNS extends from the
fortheinitialfieldonourresultsonlyaminorone. origin to roughly the radius where the density of the gas
ThefieldisthesameastheoneusedbySuwaetal.(2007), dropsbelow1014gcm 3.
−
definedbyavectorpotentialoftheform 2. This inner core is surroundedby the PNS convectionzone,
characterisedbyanegativegradientoftheelectronfraction.
Aφ = b0 r03 rsinθ. (9) Theouterboundaryofthislayerisassociatedwiththeloca-
2 r3+r3 tionoftheminimumoftheYe profile.
0
3. AstablelayersurroundsthePNSconvectionzoneandsep-
We set the normalisationradius r denotingthe location of the aratesthelatterfromthe
0
dipole to r = 1000 km and vary the parameter b setting the 4. gainregion wherehot-bubbleconvectionand the SASI op-
0 0
field at the centre of the core between 1010 and 1012 G, i.e., in erate. The boundary between the stable and the gain layer
a range somewhat above that found by (Hegeretal. 2005) for is approximately given by the gain radius that defines the
rotating progenitors.See Fig.1 for a visualisation of the initial transition from neutrino cooling below to neutrino heating
fieldofmodels15-B10. above.
Tab.2 list important parameters of our models: apart from
the initialfieldstrength,b , we presenttimeaverages(overt Despite growing SASI activity and increasing entropy in
0
[280ms,300ms])ofthemeanfield strengthinthePNSatden∈- the post-shock region and despite a slowly growing radius of
sitiesabove1014 gcm 3,intheconvectionzoneinsidethePNS the stalled shock, no explosion takes place until 550 ms after
−
(b ),inthestablelayersurroundingthePNS(b ),andinthe bounce. This is in agreement with simulations for 15M stars
cnv stb
gainregionwhereneutrinosdepositenergybehindtheSNshock (butdifferentprogenitormodels)byMarek&Janka(200⊙9)and
(b ).Moreover,time-averaged(overt [280ms,300ms])ra- Burrowsetal.(2006).
gain
∈
tiosofmagneticenergytointernalenergyandtokineticenergy
in the same regions,βi and βk, respectively,are listed. We will
4.2.Modelswithweakinitialfields
definetheseregionsmorequantitativelyinSect.4.1.
Models s15-B10and s15-B11 (b = 1010,11 G, respectively)
0
exhibitverysimilardynamicscomparedbothtoeachotherand
4. Results tothenon-magnetisedreferencemodel.Thestructureofthecore
isduringseveralhundredmillisecondsofpost-bounceevolution
4.1.Anon-magnetisedreferencemodel
thesameasthatofmodels15-B0:aconvectivelayerinsidethe
The model with b = 0, model s15-B0, serves as a reference proto-neutronstar at ρ & 1012 gcm 3 and a SASI region, sep-
0 −
casetowhichwecancomparethemagnetisedmodels;further- aratedbyathinstable decelerationlayer;hence,theprofilesof
more, it allows us to compare our results with previousresults entropyandv areinqualitativeagreementwiththeonesshown
θ
forsimilarcores.Theevolutionofthemeanentropyprofileand inFig.2.
of the root-mean-squaredlateralvelocityof themodelasfunc- We discuss the evolution of the magnetic field of model
tionoftimeisshowninFig.2. s15-B10in the following. A space-time diagram of the model
M.ObergaulingerandH.-Th.Janka:Magneticfieldamplificationincollapsing,non-rotatingstellarcores 5
Fig.2.Summaryoftheevolutionofmodels15-B0.Theupperandlowerpanelsshowangularaveragesoftheentropyperbaryon
andtheofroot-mean-squareofthelateralvelocity,respectively(colourcoding)asafunctionofradiusandpost-bouncetime.The
greylinesindicatethepositionsofdifferentmassshells;theintervalsbetweensolidanddashedlinesare0.1and0.025solarmasses,
respectively. The thick black lines show (from small to large radii) the outer boundariesof the PNS convection zone and of the
stablelayerbeneaththeSASIregionandtheoutermostpositionoftheshockwave.Thethindashedblacklinesdepictiso-density
surfacesof10ngcm 3,n=14,13,....
−
(Fig.3) shows angular averages of the current density1 j = sityasρ2/3,leadingtoagainofafactorof 500and 25inthe
(∇ b)φ and the field strength. We use the current density to maximummaxfieldstrengthandtotalmagnetic≈energy,re≈spectively.
×
detectspatialvariationsofthemagneticfield.Whilenotallpat-
ternsof jarenecessarilycausedbyAlfve´nwaves,Alfve´nwaves FurtheramplificationoccursinthePNSconvectionzone(see
areassociatedwithanon-zerocurrentdensity.Hence,wecantry Fig.4).Withinroughly10msaftertheonsetofPNSconvection,
toidentifythesewavesinthecurrentdistribution. the mean field is amplified by a factor of 3 between radiiof
∼
Withoutchangingthefieldgeometry,compressionamplifies 20and 60km.Fieldgrowthisdrivenbytheconvectiveover-
∼ ∼
themagneticfieldduringcollapsealongwiththemaximumden- turnmotionsofthegas(indicatedinthedeviationsofYefromits
angularmean value, see right half of Fig.4) and leads to a fil-
1 Forsimplicity,weomitthefactor 4π inthedefinitionofthecurrent amentary field topology (same figure, left half). This strongly
density and consider only the φ-compconent, neglecting the vanishing non-uniformstructureisreflectedinthelargefluctuationsofthe
components jrand jθ. currentdensityvisibleinthetoppanelofFig.3(first 100ms
∼
6 M.ObergaulingerandH.-Th.Janka:Magneticfieldamplificationincollapsing,non-rotatingstellarcores
Fig.3.Evolutionofmodels15-B10.Thepanelsshowincolourcodingthelogarithmofthelateralaverageofthecurrentdensity
(top panel; the intensity of the colour encodes the absolute value and the blue and red colours distinguish between positive and
negativesigns)andthelogarithmofthemeanmagneticfieldstrength(bottompanel).ThelineshavethesamemeaningasinFig.2.
at radii between 15 and 60 km). After about 100 ms, the everfluctuationstheremaybeduetoconvectivecellsatthesame
∼ ∼
amplitudeof variationsof the currentdensity in the convective radii.
layer decreases.While j still varieson shorttemporaland spa-
tialscales,largercoherentfluctuationsaremostlyabsentduring As soon as the SASI develops, we find appreciable lateral
thisphase:comparethe largerred (jφ < 0) patchesin the PNS velocitiesalsointheregionabovethePNSconvectionzone.An
convection zone at t . 100 ms to the convection zone at later earlystageofthedevelopmentoftheSASIcanbeseenalreadyin
timesstandingoutasabroadblue(jφ > 0)bandwithlittlecon- theYe variationclosetotheshockinFig.4:smalldeformations
tributionofnegativecurrents.Inthisphase,theangularaverage of the shock excite perturbationsadvected inwards. These per-
ofthecurrentdensityinthePNSconvectionzoneisveryregular turbationsare associatedwith kinksinthe magneticfield lines.
andshowslittletemporalvariation.Itisdominatedbyonelarge- Geometrically,they take the formof sheets orientedparallelto
scalepattern,viz.columnsofstrongmagneticfieldgeneratedby theshock,i.e.,roughlyatconstantradius,containingamagnetic
flowsconvergingattheaxisduetotheconstraintofaxisymme- fieldwithastrongθ-component.Thistranslatesintoanon-zero
try(Fig.4).Theseverypersistentfeaturesareassociatedwitha derivative ∂rbθ, i.e., a non-zero current density. Consequently,
strongpositivecurrentdensitywhosevalueisdominatingwhat- theperturbationscanbeidentifiedeasilyinthetoppanelofFig.3
as the alternating blue and red bands starting at the shock and
M.ObergaulingerandH.-Th.Janka:Magneticfieldamplificationincollapsing,non-rotatingstellarcores 7
14
13
]
G
b [12
g
o
l
11
10
Fig.4. Snapshot of model s15-B10 shortly after the onset of
10 100
PNS convectionat r . 50 km. The left and right panels show r [km]
thelogarithmofthemagneticfieldstrengthandthedeviationof
the electron fraction from its angular average. Vectors indicate Fig.6. Average radial profiles of the field strength of model
thevelocityfield. s15-B10 in the post-bounce phase (time averages over t
∈
[100;200] ms (black), t [200;300] ms (green), t
∈ ∈
[300;400] ms (red) post-bounce). The thin straight lines show
profiles r 2(r<50km)and r 3/2(r>80km).
− −
∝ ∝
4.2.1. Sub-andsuperalfve´nicregions
Inthefollowing,wediscussthepropagationofMHDwavesin
thecore.We willnottouchuponfastmodesandfocusonlyon
slowandAlfve´nwaves.WhereasAlfve´nandslowmodesprop-
agatingacrossthe field linesare clearlydistinctsolutions(they
have,e.g.,differentpolarisationstatesanddifferentpropagation
speeds),theydegeneratewhenpropagatingalongthefieldlines.
In this case, they have identical propagation speeds, viz. the
Alfve´n speed. Because we expect the dynamics to be domi-
natedbythe modespropagatingfastest, i.e.,theonestravelling
(anti-)parallel to the field lines, we will consider these modes
only. Because of the degeneracy of slow and Alfve´n modes in
such a case, we will refer to these modes as Alfve´n waves for
short,postponinganinvestigationofthepossibledifferencesbe-
tween the two classes of modes. It is expected (J. Guilet, pri-
vate communication)that slow modesbehavevery similarly to
Alfve´nwaves.
Fig.5.Snapshotofmodels15-B10att 120mspost-bounce.
An essential requisite for this evolution is that the flow is
≈
Thepanelsshow,apartfromthevelocityfield,thelogarithmof
superalfve´nic,i.e.,theflowvelocityislargerthantheAlfve´nve-
themagneticfieldstrength(left)andthecurrentdensity(right).
locityofthemagneticfield
√b2 b 1012gcm 3 1/2
cA = √ρ =106cms−1 10|12|G! ρ − ! . (10)
fallingtowardsthedecelerationlayer.Propagatingattheveloc-
A magneticfield b allowsforthe propagationof Alfve´nwaves
ityoffluidaccretion,thesebandsareparalleltothemassshells
inandagainstthedirectionofthefieldatthesamepropagation
(greylines).
speed, c relative to the flow of the gas carrying the magnetic
A
WhentheaccretionflowdeceleratesabovethePNSconvec- field,i.e.,withavelocity c = b/b c .Thetotalvelocityof
A A
± ± | |
tionzone,thesesheetspileupatthedecelerationlayer(cf.Fig.5 suchapairofwavesinthelabframe,wherethegasmoveswith
atr&50km),leadingtoalocalincreaseofthemagneticenergy. avelocityv,isv c .Hence,anAlfve´npointdevelopswherethe
A
±
The bottom panel of Fig.3 indicates that this process happens componentoftheAlfve´nspeeddirectedalongthefluidvelocity
throughoutthe entire evolution and magnetic energy continues hasthesameabsolutevalueasv.Atthatpoint,oneofthepairof
toaccumulate. Alfve´nwaves, viz.the one with anAlfve´nvelocityantiparallel
8 M.ObergaulingerandH.-Th.Janka:Magneticfieldamplificationincollapsing,non-rotatingstellarcores
Fig.7. Space-time diagram of the magnetic field strength (top panel) and the logarithmof the Alfve´n number(bottom panel) of
models15-B11.5alongthenorthpole.
tov,istrapped,i.e.,hasvanishingtotalvelocity,whiletheother – A regulates the total (lab-frame) propagation speed of the
onepassestheAlfve´npointatatotalspeed2v. wkave,composedofthe(comoving)Alfve´nvelocityandthe
| |
Toquantifytheimportanceofthemagneticfield,wedefine flowspeed, c +v; A =1indicatesanAlfve´npointwhere
A
thetotal,parallel,andperpendicularAlfve´nnumbers, thewavesst±all. | k|
– A isaparameterrelevanttotheexcitationofperturbations
v or⊥thogonaltothemagneticfield,comparingthevelocitydis-
A = | | , (11)
tot c placingmagneticfieldlineswiththetensionofthefieldlines.
A
v| c| Alfve´n waves rely on magnetic tension as restoring force;
Ak = c·2A, (12) hence, their excitation occurs for A < 1; otherwise, the
A field can be distorted (e.g., amplified⊥in a stretch-twist-fold
v c
A = | × A|. (13) dynamo),butthedistortionsarebeyondthelinearregimeof
⊥ c2 Alfve´nwaves.
A
While A compareskinetic and magnetic energiesdirectly,A The gas flows triggered by PNS convection as well as by
tot
and A measure two effects important to Alfve´n waves propak- the SASI are highly superalfve´nic. Though there is, in princi-
gating⊥alongthefield: ple,asmallstableregionbetweentheconvectionandtheSASI
M.ObergaulingerandH.-Th.Janka:Magneticfieldamplificationincollapsing,non-rotatingstellarcores 9
regions, where the accretion is decelerated and only small lat-
eralvelocitiesarepresent,theflowinthisregionremainssuper- 6
alfve´nicbecauseofovershootingfromthePNSconvectionand
the SASI regions. Travelling along a field line connecting the
convection zone with the SASI region, one observes a drop of
allthreeAlfve´nnumbersinthedecelerationregion,buttheystill
remainconsiderablyaboveunity.Hence,thereisneitheranysig-
nificantexcitationofAlfve´nwavesnoristhereanAlfve´npoint 4
where amplification as proposed by Guiletetal. (2010) could ]
s
happen. Magnetic perturbations are unable to travel upwards m/
c
againsttheaccretionflow;furthermore,therelativeweaknessof 8
0
thefieldprohibitsdynamicback-reaction.Thus,thisisapurely 1
[
kinematiceffect. | θ
v
The combination of these effects leads to the time evolu- |
2
tionoftheaveragefieldprofileshowninFig.6.AcrossthePNS
convectionzone,theaveragefieldstrengthdropsbyanorderof
magnitude.At its upper boundary,the pile-up of field accreted
fromtheouterlayersleadstoastronglocalincreaseofthefield
strength,whileintheSASIregion,theradialdecreaseofthefield
continuesfollowingaslightlyflatterlaw(b r 3/2)thaninthe
PNSconvectionzone(b r 2). Thesteepen∝ing−ofthe slopeat 0
∝ − 10 100
r 200kmmarkstheminimumradiusoftheshockwave.
∼ r [ km]
For model s15-B10, the flow is superalfve´nic almost
everywhere. Sizeable subalfve´nic regions develop only for
significantly stronger fields. The convective flows of model Fig.8. Root-mean-square averages of the θ-component of the
s15-B11.5, with an initial field strength given by b = velocity as a function of radius for models s15-B10 (black),
0
1011.5G 3.16 1011G,areonlymoderatelysuperalfve´nic.In s15-B11(red),ands15-B11.5(blue).Timeaveragesoversev-
particular≈,along×thepole,wherethemagneticfieldisenhanced eral tens of milliseconds in the late post-bounce phase (at t
≈
artificiallyforsymmetryreasons,theAlfve´nnumbersarebelow 300ms)areshown.
unity(Fig.7).ThethinlayerseparatingPNSconvectionandhot
bubbleconsistsofbothsub-andsuperalfve´nicregions,whilethe
accretionflowismostlysuperalfve´nic. Accretionproceedsmostlythroughanarrowstreamatthepoles,
Inanalogytothemodeldiscussedabove,perturbationsofthe and mass conservation leads to an increase of the flow speed
magnetic field created at the shock wave are advected towards as the gas is accreted through a small solid angle rather than
the PNS. The field piling up above the PNS, dominated by a uniformly over all angles. Eventually, the flow reaches sound
stronglateralcomponentgeneratedbythecomplexoverturnsof speed. Analogously, flux conservation increases the magnetic
thegasflow,formsshellsaroundthePNS.Sincetheradialcom- field strength in the accretion funnels. Depending on the ini-
ponentisweak,Alfve´nwavesareboundtotravelatroughlycon- tial field strength,the Alfve´nnumberof the accretionflow and
stantradius.Consequently,thelocationofanAlfve´npointalong β= P/P maybeoforderunityormuchgreater.
mag
afieldlineisdeterminednotbytheradialflowalonebutmostly DuetothehighMachnumber,pressureperturbationsofthe
bythe lateralvelocity.Whereasthe formertendsto gothrough accretionflow maysteepen,insomecases yieldingdiscontinu-
zerointhedecelerationregion,thelattercan,inprinciple,have ities. In particular,weak shock waves may mark the lower end
astochasticandchaoticbehaviouranddynamicalevolution. of the column,providinga veryabruptdecelerationof the gas.
Thus,whiletheremaybeAlfve´npointsinthisregion,their Duetothedissipationofkineticintointernalenergy,theentropy
location is highly variable; they may even, depending on the of the matter above the shock is higher than that of the gas in
flow, cease to exist for a certain time when the flow becomes the deceleration and PNS convection layers where S is nearly
superalfve´nicorsubalfve´nicintheentireregion.Thismayhave uniform.
adverse effects on the efficiency of the amplification process In the presence of a magnetic field, these features are not
proposed by Guiletetal. (2010). Despite the stochastic nature purelyhydrodynamicbutratherMHDstructuresassociatedwith
of these variations,we find a systematic trend:the strongerthe abendingofthefieldlines,i.e.acurrentdensity:thefield,pre-
magneticfield is, the fartheroutstretchesthe layerof accumu- dominantlyradialinthe accretioncolumn,isfoldedbytherel-
lated magnetic field on average. Since this layer corresponds atively high lateral velocity in the deceleration region and is
tothehydrodynamicallystableregionbetweenPNSconvection mostlymostly lateralin this layer.Thiseffectisweakestin the
andtheSASIregion,thetrendofagrowingradiusoftheregion polaraccretionfunnelsbecausetheaxialsymmetryrestrictsthe
wherethefieldaccumulatescanbeseenmosteasilyintheaver- developmentof a non-radialfield considerably.In all accretion
ageprofilesofthelateralvelocityshowninFig.8.Thiseffectis columns,bothvelocityandmagneticfieldaredominatedbytheir
an,albeitsmall,indicationofdynamicback-reactionofthefield respective radial components;for the polar accretion flow, this
ontotheflow. dominance of the radial componentsholds even at radii below
thelowerendoftheaccretionstream,i.e.,forr .70 100km.
−
Accretion streams at intermediate latitudes tend to punch into
4.2.2. Accretioncolumns
gasthreadedbystronglateralfield linesandmovingatconsid-
Theaccretioncolumnsdisplaymuchstrongerradialcomponents erablespeedalongconstantradius.
ofthefieldandthevelocity,whilenon-radialfieldsofaconsider- Forasufficientlystrongmagneticfield,anAlfve´npointmay
ablestrengthmaybepresentinthefluxsheetssurroundingthem. be located in the accretionstream;for a steep gradientof v or
r
10 M.ObergaulingerandH.-Th.Janka:Magneticfieldamplificationincollapsing,non-rotatingstellarcores
evenadiscontinuityatthelowerendofthestream,thisisamost
14
naturalplacefortheAlfve´npoint.
Conditionsaremostfavourablefortheformationofastable
Alfve´npointinradialpolaraccretionstreams.Weshowthetime
evolutionofthemagneticfieldandtheAlfve´nnumberalongthe
northpoleof models15-B11.5inFig.7. Initially,accretionis 13
roughlyspherical, but after t 80 ms, polar accretion streams
≈
beginto form;as a consequence,the radialvelocitybehindthe
shockincreasesabruptly.Thisstreamremainsactivefortherest G]
oonf tthimeesimscuallaetsioonf.aIbtoius,th1o0wmevilelirs,escuobnjdecs.tTtohesstreonogscflilulacttiuoantsioanfs- g |b| [12
fecttheouterandinnerterminationradiusoftheaccretionflow, o
l
the velocity,and the magneticfield. Theyare caused by lateral
motionsduetoconvectionandtheSASIinthesurroundinggas
that lead to an episodically varying cross section of the fun- 11
nel.Followingthevariationsoftheaccretionvelocity,theMach
numberof the flow oscillates aroundunity.For this model, the
sameholdsfortheAlfve´nnumber.
Theaccretionflowterminatesinasteepgradienthostingan
10
Alfve´n point. This point is first, at t 100 ms post-bounce,
10 100
≈
situated at a radius of 80 kilometres, i.e., about 30 kilome- r [km]
tres above the PNS convection layer; later it retreats to lower
radii as the core contracts. The intermediate layer between the Fig.9. Space-time averages of the magnetic field strength of
termination point of the accretion column and the PNS con- modelss15-B10(black),s15-B11(green),ands15-B11.5(or-
vection layer is, though threaded by a weaker magnetic field, ange line). We average over times between t = 200 ms and
filled by gas at subalfve´nic velocities. Hence, at these radii, t = 300 ms post-bounce. The fields of models s15-B11 and
Alfve´n waves could propagate from below towards the accre- s15-B11.5arescaledaccordingtotheirinitialstrength.
tion flow. Given a typicalAlfve´n velocityin this layer of c
A
2 108 cms 1(b /1011 G), they should traverse the intermed∼i-
− 0
ate×layerwithinτ 10ms/(b /1011 G),i.e.,onthetimescale
A 0
∼
oftheoscillationsoftheaccretioncolumn.Thishastwoconse-
quences:first,itcomplicatestheidentificationofAlfve´nwaves, 15
and,second,itmaylimittheapplicabilityofthescenarioofam-
plificationofAlfve´nwavesinthiscontext.
The fluctuations of the accretion velocity and the position
(radius,possiblyalsolatitude)oftheterminationpointoftheac- 14
cretioncolumninduceperturbationspropagatinginthelayerbe-
tweentheaccretioncolumnandthePNSconvectionregion,i.e., G]
outsidearadiusofabout50kilometresandinsidea radiusthat b| [
recedesgraduallyfromabout80 100kilometresatt 100ms og |13
toroughly60kmatt 300ms.P−artially,theseperturba≈tionscan l
≈
be entrainedin matter flowingupwardsandthus sustainingthe
magneticfieldinthehot-bubbleregion.Additionally,overshoot
fromtheconvectivelayer belowcontributesto a certaindegree 12
tomaintainingtheturbulenceinthedecelerationregion.
4.2.3. Fieldamplification 11
10 100
Though,inprinciple,thedynamicsinthislayercanbeexpressed r [km]
in terms of basic MHD waves, it is not trivial to identify these
simplesolutionsinthehighlyvariablefields,preventingdeeper Fig.10. Left panel: Temporally averaged (over post-bounce
insightsintothefieldamplificationmechanisms.Therefore,we times of 120ms . t . 200 ms) radial profiles of the magnetic
musttrytofindindirectevidencefortheamplificationprocesses fieldstrengthalongthenorthpoleformodelss15-B10(black),
atwork.Tothisend,wedistinguishbetweenkinematicanddy- s15-B11(green),s15-B11.5(orange).Thefieldstrengthisrep-
namic amplification, i.e., processes that do not depend on the resentedbysolidlines;thedashedlinesshowthefieldstrength
seed magnetic field (at least during an initial phase when it is scaledbytheratioofinitialfieldstrength.
stillsufficientlyweak)andprocessesforwhichtheamplification
issomefunctionoftheseedfield,respectively.
Convection and the SASI amplify the field kinematically. filesofallthreemodelsareremarkablysimilar.Themaindiffer-
Hence,theyshouldamplifythefieldbyagivenfactorindepen- encesappearinthedecelerationregion,whereneithertheradial
dent of the seed field. This is demonstratedby the comparison collapsenorconvectionortheSASI arethemainamplification
of space-time averages of radial profiles of the magnetic field mechanisms.
strength of models with different initial fields shown in Fig.9. The Alfve´n mechanism due to Guiletetal. (2010), on the
Correctedforthescalingwiththeinitialfield strength,thepro- other hand, depends on the presence and the location of