The physics of strong magnetic fields of magnetars.docx

The physics of strong magnetic fields of magnetars.docx

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Thephysicsofstrongmagneticfieldsofmagnetars

Thephysicsofstrongmagneticfieldsofmagnetars

DepartmentofAstronomy,NanjingUniversity,Nanjing,210093,China.

Abstract：

Aphasetransitionfromparamagnetismtoferromagnetisminneutronstarinteriorisexplored.Sincethereis3P2neutronsuperfluidinneutronstarinterior,itcanbetreatedasasystemofmagneticdipoles.Underthepresenceofbackgroundmagneticfield,themagneticdipolestendtoaligninthesamedirection.Belowacriticaltemperature,thereisaphasetransitionfromparamagnetismtoferromagnetism.Andthisgivesaconvenientexplanationofthestrongmagneticfieldofmagnetars.Inourpointofview,thereisanupperlimitforthemagneticfieldstrengthofmagnetars.Themaximumfieldstrengthofmagnetarsisabout（3.0-4.0）1015gauss.Thiscanbetesteddirectlybyfurtherinvestigations.

Keywords:

magneticfields—stars:

neutron—pulsar:

general

I.Introduction

Itisgenerallyacceptedthatthereareverystrongmagneticfields,B>1012gauss,formostradiopulsars（e.g.ShapiroandTeukolski1983）.Thereprobablyaremagnetarswithmagneticfieldstrengthexceedingthequantumcriticalthreshold,Hcr=4.4141013gauss（Duncan&Thompson1992;Paczynski1992;Usov1992;Thompson&Duncan1995,1996）.AnomalousX-rayPulsars（AXPs）andSoftGammaRepeaters（SGRs）areidentifiedasclassesofmagnetarcandidates（e.g.Kouvelliotouetal.1998,1999;Hurleyetal.1999;Mereghetti&Stella1995;Wilsonetal.1999;Kaspietal.1999）.Themagneticfieldsofthemagnetars,whichcanbededucedfromtheirlongspinperiods（P~5-12s）andspindownrate（

）basedonthestandardmagneticdipolarradiation,aresostrongastoreachtwoordersofmagnitudethequantumcriticalthresholdHcr.Moreover,thesameestimateofthemagneticfieldisderivedfromtheobservationofanabsorptionlineintheX-rayspectrumofSGR1806-20（Ibrahimetal.2002）.

Apuzzleiswhatistheoriginofstrongmagneticfieldofmagnetars.Apossibleexplanationisthatthestrongmagneticfieldofneutronstarsoriginatesfromthecollapseofthecoreofasupernovawiththeconservationofmagneticflux（e.g.Ostriker&Gunn,1969）（thefossilfieldhypothesis）;thereforemagneticfieldof

oftheprogenitorsisrequired.Braithwaite&Spruit（2004）analysedthestructureofthemagneticfieldsinAstarsandwhitedwarfs,anddemonstratedtheformationofstablefieldconfigurationsaccordingtothefossilfieldhypothesis.Ferrario&Wickramasinghe（2006）studiedthefielddistributionofthewhitedwarfprogenitors,andtheysuggestedthatmagnetarsaredescendedfromthehighmagneticfieldmassiveprogenitors.However,thedetailedcalculationaboutthemagneticfieldofmagnetarshavenotbeencarriedoutdirectly.Moreover,theprogenitorsofmagnetarshavenotbeenidentifiedinobservations.

AnotherpopularmechanismispresentedbyDuncanandThompson（Duncan&Thompson1992;Thompson&Duncan1993）.Theypointoutthatthereareviolentconvectionintheprocessofpro-neutronstarsformation,andanefficientα-Ωdynamocanamplifythemagneticfieldtothefieldstrengthofmagnetarsinseveralsecondsiftheinitialspinperiodofthemagnetarislessthan3msandtheproto-neutronstarsrotatedifferentially.Nevertheless,aspointedoutbyBraithwaite&Spruit（2004）,thedynamohypothesismaybedifficulttoproduceenoughhighfieldandexplainthecorrelationwithrotation.Recently,Vink&Kuiper（2006）studiedtheremnantenergeticofmagnetars.Theydetectedthatthesupernovaexplosionenergyofthemagnetarsissimilartothatofnormalradiopulsars.However,ifthefieldsofthemagnetarsoriginatefromthedynamohypothesis,millisecondinitialspinperiodshouldbeexpected,theexplosionshouldbeasenergeticasthehypernovasincetherotationalenergyofthemillisecondproto-neutronislargeenoughnottobeneglectedinpoweringsupernova.

Thefossilfieldhypothesisandthedynamohypothesisarerelativelymaturemodelsastotheoriginofthehighmagneticfield.Bothofthemhavebeenemployedtointerpretsomeobservationssuccessfully,whereassomenewobservationalresultsmaychallengetothem.Altogether,theoriginofthehighmagneticfieldsofmagnetarsisanactiveandabsorbingissuewhichneedfurtherwork.Severalmodelshavealreadybeenproposedbasedontheferromagnetismofthenetronstarmatter,e.g.Iwazaki（2005）arguedthatthehugemagneticfieldofmagnetarsissomecolorferromagnetismofquarkmatter.

Inthispaperweproposeanewideafortheoriginofthemagnetars.Thestrongmagneticfieldsofthemagnetarsmayoriginatefromtheinducedmagneticmomentofthe3P2neutronCooperpairsintheanisotropicneutronsuperfluid.

II.Anisotropic3P2neutronsuperfluid

Therearetworelevantregimesofneutronsuperfluidintheneutronstarinterior:

oneistheisotropic1S0neutronsuperfluidwiththecriticaltemperatureT（1S0）11010Kinadensityrange11010<（g/cm3）<1.61014.Theotheristheanisotropic3P2neutronsuperfluidwithawidedensityrange1.31014<（g/cm3）<7.21014.Thecriticaltemperature

is

.

（1）

Wenotethattheenergygap（3P2）isalmostaconstantaboutthemaximumwithanerrorlessthan3%inaratherwidedensityregion3.31014<（g/cm3）<5.21014（seeFig.2ofElgagøyetal.1996,butweneglecttheFstateofneutronCooperpairhere）.

A3P2neutronCooperpairhasaspinwithquantumnumber,=1.Themagneticmomentofthe3P2neutronCooperpairistwicethatoftheabnormalmagneticmomentofaneutron,2ninmagnitude,anditsprojectionontheexternalmagneticfield（z-direction）isz×（2n）,z=1,0,-1,where

.

Itisinterestingtonotethatthebehaviorofthe3P2neutronsuperfluidisverysimilartothatoftheliquid3Heatverylowtemperature（Leggett1975）:

1）Theprojectiondistributionforthemagneticmomentofthe3P2neutronCooperpairsintheabsenceofexternalmagneticfieldisstochastic,or“EqualSpinPair”（ESP）phasesimilartotheA-phaseoftheliquid3Heatverylowtemperature（Leggett1975）.The3P2neutronsuperfluidisbasicallyisotropicwithoutnetmagneticmomentintheabsenceoftheexternalmagneticfield.WenameitastheA-phaseofthe3P2neutronsuperfluidsimilartotheAphaseoftheliquid3Heatverylowtemperature（Leggett1975）.

2）However,theprojectiondistributionforthemagneticmomentofthe3P2neutronCooperpairsinthepresenceofexternalmagneticfieldisnotstochastic.Thenumberof3P2neutronCooperpairwithparamagneticmomentismorethantheoneswithdiamagneticmoment.Therefore,the3P2neutronsuperfluidhasanetinducedparamagneticmomentanditsbehaviorisanisotropicinthepresenceofexternalmagneticfield.WenameitastheB-phaseofthe3P2neutronsuperfluidsimilartotheBphaseoftheliquid3Heatverylowtemperature（Leggett1975）.

Thefullstoryofanisotropic3P2neutronsuperfluidismorecomplicatedwhenmagneticinteractionisintroduced.Weconsiderthisproblemfromtheeffectivefieldtheorypointofview（Feng&Jin2005）.Itispossiblethattheinducedmagneticfieldofthe3P2superfluidwillboosttheneutronstarmagneticfieldtothatofmagnetars,wheneverthestellarinternaltemperatureisbelowthephasetransitiontemperature.WewilldiscussthisinsectionV.

III.Inducedparamagneticmomentofthe3P2neutronsuperfluidintheB-phase

Theinducedparamagneticmomentofthe3P2neutronsuperfluidintheB-phasemaybesimplyestimatedasfollows:

thesystemofthe3P2neutronCooperpairswithspinquantumnumber,

maybetreatedasaBose-Einsteinsystem.ItobeystheBose-Einsteinstatistics.

Amagneticdipoletendstoaligninthedirectionoftheexternalmagneticfield.The3P2neutronCooperpairhasenergyZ2nB（Z=1,0,-1）intheappliedmagneticfieldduetotheabnormalmagneticmomentoftheneutrons.Herenistheabsolutevalueofthemagneticmomentofaneutron,weusethisconventionfromnowon.Bisthebackgroundmagneticfield.Whileintheeffectivefieldframe,Bisthetotalmagneticfield.DetaileddiscussionwillbepresentedinsectionV.Wedenotethenumberdensityof3P2neutronCooperpairswithspinprojectionz=1,0,-1byn1,n0andn-1respectively.Theirrelativeratiosare

（2）

.（3）

Thedifferenceofthenumberdensityof3P2neutronCooperpairswithparamagneticanddiamagneticmomentis

（4）

.（5）

TheBrillouinfunction,f（nB/kT）,isintroducedtotakeintoaccounttheeffectofthermalmotionWenotethatfItisanincreasingfunction,inparticular,

and

.f（nB/kT）increasewithdecreasingtemperature.AndthisisthemathematicalformulafortheB-phaseofthe3P2superfluid.

Arelevantquestionishowmanyneutronshavebeencombinedintothe3P2Cooperpairs?

Thetotalnumberofneutronsisgivenby

（Herekisawavevector,

pisthemomentum）.Theneutronscombinedintothe3P2CooperpairsarejustinathinlayerintheFermisurfacewiththickness

.Thenwehave

（k<

.（6）

Here,wewouldliketoemphasizethattheenergygap,,isthebindingenergyoftheCooperpairratherthanavariationoftheFermienergyduetoadisturbanceorduetothevariationofparticlenumberdensity（Inthelattercase,itiseasytomisunderstandtoget

or

）.Therefore,wethinktheeq.（6）isaproperestimation.

Forthenon–relativisticneutrongas（

）,thefractionoftheneutronsthatcombinedintothe3P2Cooperpairsis

（7）

.

TheFermienergyoftheneutronsystemmaybecalculatedbytheformula

.（8）

Theenergygapoftheanisotropicneutronsuperfluidis（3P2）~0.05MeV（Elgagøyetal.