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Supernova neutrino spectra and applications to flavor oscillations [Elektronische Ressource] / Mathias Thorsten Keil

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DissertationSupernova Neutrino Spectraand Applications toFlavor OscillationsbyMathias Thorsten KeilInstitut fu¨r Theoretische Physik T30Univ.-Prof. Dr. Manfred LindnerMax-Planck-Institut fu¨r Physik(Werner-Heisenberg-Institut)Advisor: Dr. habil. Georg G. Raffeltsupported by the grant SFB 375Institut fu¨r Theoretische Physik T30, Univ.-Prof. Dr. Manfred LindnerSupernova Neutrino Spectraand Applications toFlavor OscillationsMathias Thorsten KeilVollsta¨ndiger Abdruck der von der Fakult¨at fu¨r Physik der Technischen Universit¨atMu¨nchen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften(Dr. rer. nat.) genehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr. F. v. FeilitzschPru¨fer der Dissertation: 1. Univ.-Prof. Dr. M. Lindner2. Univ.-Prof. Dr. M. DreesDie Dissertation wurde am 27.05.2003 bei der Technischen Universit¨at Mu¨ncheneingereicht und durch die Fakulta¨t fu¨r Physik am 25.06.2003 angenommen.SummaryWe study the flavor-dependent neutrino spectra formation in the core ofa supernova (SN) by means of Monte Carlo simulations. Several neutrinodetectors around the world are able to detect a high-statistics signal from agalactic SN. From such a signal one may extract information that severelyconstrains the parameter space for neutrino oscillations. Therefore, reliablepredictions for flavor-dependent fluxes and spectra are urgently needed.

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Dissertation

SupernovaNeutrinoSpectra
andApplicationsto
FlavorOscillations

ybMathiasThorstenKeil

Institutfu¨rTheoretischePhysikT30
Univ.-Prof.Dr.ManfredLindner
Max-Planck-Institutfu¨rPhysik
(Werner-Heisenberg-Institut)
Advisor:Dr.habil.GeorgG.Raelt
supportedbythegrantSFB375

Institutfu¨rTheoretischePhysikT30,Univ.-Prof.Dr.ManfredLindner

SupernovaNeutrinoSpectra
andApplicationsto
FlavorOscillations

MathiasThorstenKeil

Vollsta¨ndigerAbdruckdervonderFakulta¨tfu¨rPhysikderTechnischenUniversita¨t
Mu¨nchenzurErlangungdesakademischenGradeseinesDoktorsderNaturwissenschaften
(Dr.rer.nat.)genehmigtenDissertation.

Vorsitzender:Univ.-Prof.Dr.F.v.Feilitzsch
Pru¨ferderDissertation:1.Univ.-Prof.Dr.M.Lindner
2.Univ.-Prof.Dr.M.Drees

DieDissertationwurdeam27.05.2003beiderTechnischenUniversita¨tMu¨nchen
eingereichtunddurchdieFakulta¨tfu¨rPhysikam25.06.2003angenommen.

SummaryWestudytheflavor-dependentneutrinospectraformationinthecoreof
asupernova(SN)bymeansofMonteCarlosimulations.Severalneutrino
detectorsaroundtheworldareabletodetectahigh-statisticssignalfroma
galacticSN.Fromsuchasignalonemayextractinformationthatseverely
constrainstheparameterspaceforneutrinooscillations.Therefore,reliable
predictionsforflavor-dependentfluxesandspectraareurgentlyneeded.
Inallhydrodynamicsimulationsthetreatmentofνµ,τandν¯µ,τisrather
schematic,withtheexceptionofthemostrecentGarching-Groupsimulation,
wheretheinteractionprocesseswereupdatedpartlybasedonourresults.The
interactionscommonlyincludedintraditionalsimulationsareiso-energetic
scatteringonnucleonsNν→νN,scatteringonelectronsandpositrons
e±ν→νe±,andelectron-positronpairannihilatione+e−→νµ,τν¯µ,τ.Inour
MonteCarlosimulationswevarythestaticstellarbackgroundmodelsinad-
ditiontosystematicallyswitchingonandoffoursetofinteractionprocesses,
i.e.,recoilandweakmagnetisminNν→νN,scatteringone±andνe/¯νe,
e+e−andνeν¯eannihilationintoνµ,τν¯µ,τpairs,andneutrinobremsstrahlung
offnucleonsNN→NNνν¯.Asνµ,τandν¯µ,τsources,NN→NNνν¯and
νeν¯e→νµ,τν¯µ,τdominate.Thelatterprocesshasneverbeenstudiedbefore
inthecontextofSNeandturnsouttobealwaysmoreimportantthanthe
traditionale+e−annihilationprocessbyafactorof2–3.Forenergytransfer,
themostimportantreactionsareNν→νNwithrecoil,andscatteringon
e±.Weakmagnetismhasaverysmalleffectandscatteringonνeandν¯eis
negligible.Thechargedcurrentreactionse−p→νenande+n→ν¯epdominateforνe
andν¯e.Incomparingournumericalresultsforallflavorswefindthestandard
<hierarchyofmeanenergiesνe<ν¯e∼νµ,τ,with,however,verysimilar
valuesforνµ,τandν¯e.Theluminositiesofνµ,τandν¯ecandifferbyupto
afactorof2fromLν¯e≈Lνe.TheGarchingGroupobtainssimilarresults
fromtheirself-consistentsimulationwiththefullsetofinteractions.These
resultsarealmostorthogonaltothepreviousstandardpictureofexactly
equalluminositiesofallflavorsanddifferencesinmeanenergiesofuptoa
2.ofortfacExistingconceptsforidentifyingoscillationeffectsinaSNneutrinosignal
needtoberevised.Wepresenttwomethodsfordetectingtheearth-matter
effectthatareratherindependentofpredictionsfromSNsimulations.An
earth-inducedfluxdifferencecanbemeasuredbythefutureIceCubedetector
inAntarcticaandaco-detectorlikeSuper-orHyper-Kamiokande.Atasingle
detectorwithhighenergyresolutiontheFouriertransformoftheinverse-
energyspectrumcanrevealthemodulationsofthespectrum.Bothmethods
aresensitivetothesmallmixingangleθ13andtheneutrinomasshierarchy.

i

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urF¨

meine

Eltern

iv

Acknowledgments
Mantience,ypeandoplekconindness.tributedIetoxpressthismydissertagratitudetiontowitheveryotheirneawhodvice,helpesuppdmort,akingpa-
thisworkcometrue.
ForemostIamindebtedtomyadvisor,GeorgRaffelt,notonlyforpropos-
ingtheprojectandtherebygettingmeintoAstroparticlePhysics,buteven
moreforhisgreatsupportandadviceinsomanyways.Ibenefitedverymuch
fromourcloseandveryinspiringcollaboration.Alsofortheproofreadingand
uncountablevaluablesuggestionsonthemanuscriptIamverygrateful.
Thenumericalpartofthisworkwasonlypossiblewiththecodethat
Hans-ThomasJankahaddevelopedandkindlyintroducedmeto.Ienjoyed
thecollaborationwithThomasandappreciatehispatiencewithmyques-
tionsandhisexhaustiveanswersonallkindsofcode-andsupernova-related
questions.ThomasandhisGarchingSupernovaGroup,i.e.,RobertBuras
andMarkusRampp,contributedveryvaluableresultsfromtheirhydrody-
namicsimulations.IalsothankRobertforourmanylongdiscussionsand
hisvaluablesuggestionsonpartsofthemanuscript.
WithAmolDigheIhadawonderfulcollaborationontheneutrino-
oscillationpart.ButalsoforallotherquestionsanddiscussionsIfound
manAmol’suscriptdoorbywidehisopproen.oInafreadingddition,andIstuhankggestions.AmolAforlsoifmproorvingdiscussionspartsofwtheith
MichaelKachelrieß,SergioPastor,DimitrySemikoz,andRicardTom´asIam
verygrateful.
myAsfamilyimpowhortanstuppastheortedsauppndortatencouragedthemeinstitutetwehroughoutreallatllhosetheyfriendsears,eveandn
mywhendeepIwestastfarhanksawatyo.Fmoyrptarenheirtslovaseweandllsasuppmyortibrothernanyapndossiblesister.wCayIhristophowe
evenfittedsomeproofreadingintohisbusyconferenceschedule,thankyou.
andFeovreyourrythinglovyeo,usuppdidort,formande,weoncrdsouragecannotmenet,yxpressourpmyatiencegratitude,yourandsmlileovs,e
foryou,Meike.

v

Partsofthisdissertationhavealreadybeenpublishedas:

•M.“MonT.teCKeil,arloG.sG.tudyofRaffelt,supaendrnovH.anT.eutrinoJanka,spectraformation,”
Astrophys.J.inpress,
tro-ph/0208035.as

•R.“Electron-Buras,H.neutrinoT.Jankpaira,M.T.annihilatiKeil,on:G.AG.newRaffelt,sourceandforM.muonRampp,andtau
Asneutrinostrophys.inJ.sup587ernov(2003)ae,”320[astro-ph/0205006].

•M.T.Keil,G.G.Raffelt,andH.T.Janka,
“SupernovaNeutrinoSpectraFormation,”
Nucl.Phys.BProc.Suppl.118(2003)506.

•G.G.Raffelt,M.T.Keil,R.Buras,H.T.Janka,andM.Rampp,
“Supernovaneutrinos:Flavor-dependentfluxesandspectra,”
tro-ph/0303226.as

•A.S.Dighe,M.T.Keil,andG.G.Raffelt,
“DetectingtheneutrinomasshierarchywithasupernovaatIceCube,”
JCAP,inpress(2003),hep-ph/0303210.

•A.“IdenS.tDigheifying,M.earthT.Kematteril,andeGffects.Go.nRsaffeuplt,ernovaneutrinosatasingle
detector,”p-ph/0304150.he

vi

tsentCon

1Introduction

2TheCore-CollapseParadigm
2.1TheLifeofaStar:BalancingForces.............
2.2TheDeathofaMassiveStar:CoreCollapse........
2.3Reincarnation:AProto-NeutronStarEmerges.......
2.4Self-ConsistentSimulations..................
2.5Flavor-DependentNeutrinoEmission............

3NeutrinoInteractions
3.1Beta-Processes........................
3.2Neutrino-NucleonScattering.................
3.3WeakMagnetism.......................
3.4Bremsstrahlung........................
3.5PairAnnihilation.......................
3.6ScatteringonElectronsandElectronNeutrinos......
3.7OtherReactions........................

4SettingtheStageforNeutrinoTransport
4.1CharacterizingProto-NeutronStars.............
4.2Proto-Neutron-StarProfiles.................
4.3OpticalDepthvs.ThermalizationDepth..........
4.4ThermalizationDepthsinOurStellarModels.......

5CharacterizingNon-ThermalNeutrinoSpectra
5.1GlobalParameters......................
5.2AnalyticFits.........................

6RelativeImportanceofNeutral-CurrentReactions
6.1MonteCarloSetup......................
6.2Accretion-PhaseModelI...................

vii

..........

..............

........

....

....

1

7701113161

1212326272282343

3553639324

949415

757595

6.3SteepPowerLaw.........................
6.4ShallowPowerLaw........................
6.5Summary.............................
7ComparisonofAllFlavors
7.1ResultsfromOurMonteCarloStudy..............
7.2PreviousLiterature........................
7.3SpectralShape..........................
7.4Summary.............................
8DetectingOscillationsofSNNeutrinos
8.1OscillationsofSNNeutrinos...................
8.2DetectingOscillationsWithTwoDistantDetectors......
8.2.1TheBasicPrinciple....................
8.2.2TheSNSignalatIceCube................
8.2.3TheOscillationSignalatIceCube...........
8.2.4Super-orHyper-KamiokandeandIceCube.......
8.3DetectingtheEarth-MatterEffectataSingleDetector....
8.4Summary.............................
9DiscussionandSummary

tionsbbreviaAA

BMonteCarloCode
B.1GeneralConcept.........
B.2StructureoftheCode.......

viii

......................

..........

264646767647970885589898983999101201105

109

111111131

1Chapter

ductiontroIn

Thephysicsofneutrinooscillationshasreceivedtremendousattentiondur-
ingthepastyearsandtodayoscillationsarethoughttobeestablished.A
largenumberofexperimentshascontributedtoourpresentknowledgeof
thetheamllowixingedrscangeshemesareandknown.parameters.WithinForthesenveextralfewofthedecadesmixtingheparametersmagnitude
ofthesmallmixingangleθ13andthequestionwhethertheneutrinomass
hierarchyisnormalorinvertedwillremainopenandcanbesettledonlyby
futureprecisionmeasurementsatdedicatedlong-baselineoscillationexper-
iments(Bargeretal.2001,Cerveraetal.2000,Freund,Huber,&Lindner
2001)ortheobservationofafuturegalacticsupernova(SN).
Alongwiththeinterestinneutrinooscillationsdevelopedthebranch
ofneutrinoastronomy.FortheearlyexperimentsRaymondDavisJr.and
MasatoshiKoshibareceivedlastyear’sPhysicsNobelPrize“forpioneering
contributionstoastrophysics,inparticularforthedetectionofcosmicneu-
trinos”(Nobele-Museum,2002).Currentandplannedexperimentswillbe
abletorecordsome10,000–100,000neutrinosfromafuturegalacticSNand
thereforewouldprovidevaluableinformationonSNphysicsandneutrino
properties(Barger,Marfatia,&Wood2001,Minakataetal.2002).
Thereexistsalargenumberofpublicationsaddressingthequestion,what
theneutrinosignalofagalacticSNwilltellusaboutneutrinooscillations.
Withdifferenthetiatembetweasuremeneenexistoftingsucoshacillationneutrinoscenariossignal(Cithiuwo&uldKuobep2000,ossibleDigheto
&Smirnov2000,Duttaetal.2000,Fuller,Haxton,&McLaughlin1999,
&SLunardiniato2002).&SAtmirnothevsame2001b,time,2003,anMinakundersatatanding&Nofunokawneutrinoa2001,osTacillationskahasihsi
crucialforinterpretingarecordedsignal.
In1987neutrinosfromaSNweredetectedforthefirstandonlytime
(Biontaetal.1987,Hirataetal.1987).ThissignalfromSN1987Ainthe

1

2

Chapter1.Introduction

LargeMagellanicCloudwasanalyzedtakingneutrinooscillationsintoac-
count(Jegerlehner,Neubig,&Raffelt1996,Kachelriessetal.2002,Lunardini
&Smirnov2001a,Smirnov,Spergel,&Bahcall1994).However,theneutrino
detectorsthatwereoperationalatthattimemeasuredonlyabout20events
connectedtoSN1987A.Withsuchlowstatistics,determiningSNandneu-
trinoparameterswasdifficult,butsomelimitscouldbeobtained.
FortheanalysisofaSNneutrinosignalitwouldbehelpfultohaveafirm
predictionfortheneutrinospectraemittedbythecoreoftheSN.Oscillation
effectsinaneutrinosignaldependonthespectraldifferencesbetweenthe
variousflavors.Suchspectraldifferencesareobtainedbyneutrino-transport
simulationscoupledtoselfconsistenthydrodynamiccalculations.Inorder
togetreliableneutrinospectrathewayneutrinosaretransportedinthese
simulationsplaysacrucialrole.Inthepast,thetransportofνµwasvery
schematic(hereandinthefollowingνµstandsforeithermuonortauneu-
trinos,oranti-neutrinos,unlessstateddifferently).Spectraobtainedforνµ
inthesesimulationsthereforeinheritlargeuncertainties.However,theseun-
certaintiesbarelyaffectthedynamicsoftheexplosion.FromtheSNmodel
builderspointofview,detailsofνµspectraformationdothereforenotplay
acrucialrole.Thegreatestchallengefornumericalmodelsstillistounder-
standhowSNeexplode.Themostelaboratesimulationsagreeratherwellon
theearlystagesoftheSNmechanism,butdonotshowexplosions.
Accordingtothecommonlyaccepted“delayed-explosionscenario”amas-
sivestarwithamassgreaterthanabout8Meventuallybecomesacore-
collapseSN.Attheendofitslife,suchastarhasaccumulatedanironcore
ofabout1.5M,closetotheChandrasekharlimit.Oncethecoremassis
greaterthanthestabilitylimititcollapseswithinsomehundredmilliseconds
toaproto-neutronstar.Duetothiscollapsealargeamountofgravitational
bindingenergyisliberated,butparticlescannotescapethedensemedium.
Ashockwavethatformedattheedgeoftheproto-neutronstartravelsout-
wardsandplowsthroughthestillinfallingouterlayersoftheironcore.The
disintegrationoftheseinfallingironnucleiconsumesthekineticenergyofthe
shockwaveandfinallycausesittostallafterafew100ms.Neutrinoemission
drainsenergyfromtheproto-neutronstarandatthesametimeheatsupthe
regionbehindthestalledshockwave.Thisenergytransfercausestheshock
wavetoregainvelocityandfinallyblowofftheenvelopeofthestar.
Inacrudeapproximationtheproto-neutronstarisablackbodysource
forneutrinosofallflavors.Theflavordependentdifferences,thataremost
importantforoscillations,ariseduetotheflavordependenceoftheneutrino-
matterinteractions.Forthecaseofelectronneutrinos(νe)andelectronanti-
neutrinos(ν¯e)thedominantreactionsarethecharged-currentinteractions
withnucleonse−p↔νenande+n↔ν¯ep.Sincetherearemoreneutrons

3

thanprotonsinaproto-neutronstar,theinteractionsaremoreefficientfor
νeandkeepνeinlocalthermalequilibrium(LTE)uptolargerradiithan¯νe.
Thereforetheemergingν¯earehotterthantheνe.Theotherflavorsundergo
varioustypesofneutral-currentinteractionsthateachareofcomparable
importance.ThedetailedinterplayofthesereactionsinthecontextofSNe
hasbeenstudiedforthefirsttimeinthepresentwork.
Traditionally,theνµ-matterinteractionsthatareimplementedinnumer-
icalsimulationsareiso-energeticscatteringwithnucleons,scatteringonelec-
tronsandpositrons,andelectron-positronpairannihilationintoνµ¯νµpairs.
Forseveralyearstherehavebeensuggestionsonimprovingtheserates.Bur-
rows&Sawyer(1998)andReddyetal.(1999)calculatedtheeffectofmany-
bodycorrelationsthatchangetheneutrino-nucleoninteractions.Weakmag-
netismevencausesdifferentratesforνµandν¯µ(Horowitz&Li2000,Horowitz
2002,Vogel&Beacom1999).Astandardsimplificationinnumericalsimula-
tionswastotakethenucleonmasstobeinfiniteforneutrino-nucleonscatter-
ing.However,ithasbeenknownforsometimethattheinfluenceofnucleon
recoilsinthesescatteringreactionswasnotnegligible(Jankaetal.1996,
Raffelt2001).Animportantsourcetermforνµisnucleonbremsstrahlung
NN→NNνµν¯µthatevendominatesforlowenergies(Hannestad&Raffelt
1998,Suzuki1991,1993,Thompson,Burrows,&Horvath2000).Thetradi-
tionallyincludedsourceforνµinhydrodynamicsimulations,e+e−→νµν¯µ,
turnsouttoberatherunimportant.Forthefirsttimeweshowedthat
electron-neutrinopairannihilationνeν¯e→νµν¯µisalwaysafactorof2–3
moreimportantthanthistraditionalprocess.
Inordertostudytherelativeimportanceofallpossibleinteractionpro-
cessesandtheirdependenceonareasonablerangeofstellarbackground
modelswehaveadaptedtheMonteCarlocodeofJanka(1987,1991)and
addednewmicrophysicstoit.WegobeyondtheworkofJanka&Hillebrandt
(1989a,b)inthatweincludethebremsstrahlungprocess,nucleonrecoilsand
weakmagnetism,νeν¯epairannihilationintoνµν¯µ,andscatteringofνµonνe
andν¯e.Withtheseextensionsweinvestigatetheneutrinotransportsystem-
aticallyforavarietyofmediumprofilesthatarerepresentativefordifferent
SNphases.Raffelt(2001)hasrecentlystudiedtheνµspectra-formationprob-
lemwiththelimitationtonucleonicprocesses(elasticandinelasticscatter-
ing,recoils,bremsstrahlung),toMaxwell-Boltzmannstatisticsfortheneutri-
nos,andplane-parallelgeometry.Ourpresentstudycomplementsthismore
schematicworkbyincludingtheleptonicprocesses,Fermi-Diracstatistics,
andsphericalgeometry.InadditionweapplyourMonteCarlocodetothe
transportofνeandν¯eandthusareabletocomparetheflavor-dependent
fluxesandspectra.
Withthecompletesetofneutrinointeractionswefindasituationthat

4

Chapter1.Introduction

isalmostorthogonaltowhathasbeenassumedsofar.Thestandardpicture
concerningSNneutrinofluxesandspectrawasanexactequipartitionof
theneutrinoluminosityamongallflavorsandastronghierarchyofthemean
energiesofthedifferentneutrinoflavorsνµ>ν¯e>νewithadifference
ofuptoafactorof2betweenν¯eandνµ.Theseassumptionsinturnlead
tostrongoscillationeffectsinthepredictedsignalofagalacticSN.Our
findingssuggestthatthedifferencebetweenν¯eandνµisabout10%
orless,buttheluminositiescandifferbyafactorof2ineitherdirection,
dependingontheexplosionphase.
Thisnewpictureisalsosupportedbyallhydrodynamicsimulationsin-
cludingBoltzmannsolversforthetransportofneutrinos.Ifthesesimulations
usethetraditionalsetofneutrinointeractionstheyfinddifferencesbetween
ν¯eandνµaround20–30%.VeryrecentlytheGarchinggroupimple-
mentedwhatwefoundtobetherelevantsetofinteractionsintheirtwo-
dimensionalhydrodynamicsimulation(Burasetal.2003b).Theirfindings
areequivalenttotheresultsofourMonteCarlosimulation.
Forstudiesofneutrino-oscillationeffectsinSNneutrinospectrathese
findingsmeanafundamentalchangeofparadigm,becausethemainfocusof
thesestudiesaretheeffectsonν¯eandν¯µ.Availabledetectorswillrecordonly
ν¯ewithhighstatistics.Thebasicideawastoidentifyanoscillationeffect
bycomparingcharacteristicsoftheobservedandthepredictedν¯espectrum.
Ifoscillationswerepresent,onewouldobservedeviations,becausethe¯νe
andν¯µspectrawouldmix.Inadditiontothefactthatthepredictionshave
largeuncertainties,theseconceptsfailoncethenewfindingsaretakeninto
account.Thereforedifferentmethodsareneeded.
Accordingtoourfindingsandthemostrecentresultsofself-consistent
SNsimulations,aconceptforinferringneutrino-oscillationparametersfrom
thesignalofagalacticSNshouldbebasedonthefluxdifferencesbetween
ν¯eandν¯µratherthandifferencesinmeanenergies.WithaSNrateinour
galaxyofafewpercenturytheconceptneedstoinvolvedetectorsthatwill
beoperationalforseveraldecades.
Withsuchmethodsonemightbeabletodetermineoscillationparameters
thataredifficulttoobtainbyearth-basedexperiments.Ifthecurrentlydis-
cussed“neutrinofactories”and“superbeams”arebuilttheywillbeableto
determineneutrinoparametersingreatdetail(Apollonioetal.2002,Barger
etal.2001,Cerveraetal.2000,Freund,Huber,&Lindner,2001,Huberet
al.2003,Huber,Lindner,&Winter2002).Today,itisnotclearwhethersuch
experimentswillbecomereality.Bothconceptscanbenefitfromoneanother,
becausedetectorsbeingbuiltforearthbasesexperimentscanalsoserveas
SNdetectorsandresultsobtainedfromoneconceptcanbeusedtoimprove
ther.othe

5

Inviewofourfindings,wedevelopedtwomethodsforidentifyingtheearth
simmatterulationeffectresultsinarSNathernteutrinohandsignaletailedpthatonlyredictionsdep.eThndefionrstmrobustethofdineaturesvolveosf
twoneutrinodetectorsatdifferentsites,suchthatforonetheneutrinopath
goesthroughtheearthandtheotherseesthesignal“fromabove.”Theenergy
depositedinonedetectordiffersfromthatintheotheriftheearth-matter
effectispresent,duetodifferentoriginalν¯eandν¯µfluxes.Surprisingly,the
futurehigh-energyneutrinotelescopeIceCubeattheSouthPole,thatwill
beKeil,gin&takRingaffeltdata2003a).fromDue2005,towitsillyieluniquedthelomocation,stacwcuithrateasignalco-detector(Digheon,
thenAlternativorthernely,hwithaemisphere,singlee.g.,detectorinJapaonen,canthesidenetuptifycotvheersmomostdulationsoftheofstkhey.
energyspectrumthatareinducedbytheearth-mattereffectwiththehelp
ofwithaFtheouriesrizeatransndformenergy(Dighe,resolutionKeil,o&fRavaffealtilable2003b).detectors.ThisisHoawediffivecr,ultwithtaska
detectorlikeHyper-Kamiokandeoralargescintillationdetectorthisisa
poweThisrfulprodissertationcedure.isstructuredasfollows.InChapter2weexplainthe
discurrencusstthenexplosioneutrino-matterparadigmitnthateractiwebonsetlievhateisarerealizedrelevanintinsidenature.theWeptroto-hen
hasneutronneversbtareeninCstudiedhapterb3.eforeWeinputthespcoecialntextofemphasisSNe.oInntChehapterνeν¯e4prowepcessresenthatt
thestellarprofilesthatweuseinournumericalsimulations.Weintroduce
theconceptofa“thermalizationdepth”andapplyittoourstellarprofiles
inordertoestimatetherelativeimportanceoftheinteractionprocesses.
Fordescribingthespectrathatweobtainfromourcalculationsweusethe
characteristicquantitiesandfitsthatweexplicateinChapter5.Withallthe
necessarytoolsinhandwethenfirstdescribeournumericalresultsforνµin
Chwhereaptewre6alsoandgtivheenanmoovveeorviewntooftthehecoprmpeviousarisonofliterature.allflavAsorsiannChapteapplicationr7,
inofaourdetectedfindingsSNweshoneutrinowinCsignal.hapterA8ndhowfinallytoidenwetifytsummarizeheearth-ourmatterfindingseffectin
Chapter9.

6

hCapter1.nItroduction

2Chapter

aradigmPore-CollapseCThe

Massivestarsaccumulateironintheircenterafterseveralstagesofnuclear
burning.Theironcorebecomesunstableonceitreachesacertainmasslimit
andcollapses.Accordingtothesocalled“delayed-explosionmechanism,”
thecollapseisinvertedintoanexplosionandashockwavetravelsoutwards
poweredbytheenergydepositionoftheneutrinoflux.Theneutrinosare
emittedbytheproto-neutronstarthatformedinthecenter.
rateThereamongetxistheseansimumberulations,ofnhowumericalever,simfailtoulations.produceEveenthexplosions.mostIteislabano-
unsettledissuewhatthemissingingredientsare.
theeWithxplosion.afocFuslavonorthedepnendeneutrinotdifferencesemission,winethediscussemittedthenvariouseutrinospstagesectraof
ariseduetoneutrino-matterinteractions.Electronneutrinosandelectron
aneralti-nneutraleutrinoscaurrenretprodominancessetslygoproverndutcedhebytranschpaorgrted-ofthecurrenottherproflavcesses.ors.Sev-

2.1TheLifeofaStar:BalancingForces

Duringitswholelifeastarhastobalancetwopossiblyfataleffectsthat
workinoppositedirections.Ononeside,thereisthegravitationalpull,that
triestocollapsethestar,ontheothersidethethermalpressurethatexpands
thestar.Theconnectionofthermalandgravitationalenergyisgivenbythe
virialtheorem.Forthetime-averagedkineticenergyEkinandgravitational
energyEGofanatominsidethestarthevirialtheoremis
1Ekin=−2EG.(2.1)
Bymakingthestarmorecompact,EGbecomesmorenegative.Sincethe
temperatureisproportionaltoEkin,itrisesasthestar’sradiusshrinks.
7

8

Chapter2.TheCore-CollapseParadigm

Shrinkingthestarbyheatingitup,orviceversa,correspondstoanega-
tiveheatcapacityandstabilizesthestar.Foracertainstellarradiusthe
gravitationalpullisbalancedbythethermalpressure.
Astarradiatesphotonsandthereforelosesenergy.Byburningjustenough
ofitsfuelinordertoproducetheenergyradiatedaway,thestarbecomesa
stableobject.Initially,thisfuelishydrogen,thatproducesenergyinnuclear-
fusionreactionstohelium.Thestrongerthegravitationalpull,thehotterthe
star’sinteriorandthemoreenergyisradiatedaway.Thereforemoremassive
starsburnfuelatahigherrate.
Duetothenuclear-fusionreactionsmoreandmoreheliumcollectsin
thecentralregion.Whenhydrogenisdepletedatthecenter,thehelium
eventuallyreacheshighenoughdensityandtemperatureinordertoignite.
Duetothestrongtemperaturedependenceofnuclearfusionreactions,helium
burninginthecentralregionisspatiallywellseparatedfromthehydrogen
burningthatcanstillcontinueattheedgeoftheheliumcore.Thehelium
fusionprocessproducesagainheavierelementsthatcanalsostartfusion
reactionsatalaterstage.Sincethestarhastobedenserandhottertoignite
theburningofheavierelementsitbecomesbrighterandlosesmoreenergyat
thisstage.Additionally,theseburningprocessesliberatelessenergyineach
reaction.Thereactionraterisesdrastically.Whilehydrogenburningcanlast
millionsorevenbillionsofyearsthesiliconburningstageusuallytakesless
thanaday(seee.g.Herantetal.1997).
Afterseveralstagesofnuclearburningtheinitialcompositionofhelium
andhydrogentransformedintoanonion-shellstructurewithheavierelements
uptoiron-groupnucleiformingthestar’score(Figure2.1).Forstarswith
amassgreaterthanabout8Mthecoremainlyconsistsof56Feand56Ni.
Duetosiliconburningtheironcoreeventuallyreachesamassclosetoits
Chandrasekharlimitof1.2–1.5M.Ironcannotigniteandfurtherfuseto
heavierelementssinceitisalreadythemosttightlyboundelement.
TheChandrasekharlimitisastabilitycriterionforcompactobjectslike
whitedwarfsortheironcoreofamassivestar.Suchobjectsarestabilized
byelectrondegeneracypressure.Thefollowingverybasicexplanationofthe
ChandrasekharlimitcanbefoundinShapiro&Teukolsky(1983).Thegrav-
itationalenergyofaparticularelectronatradiusRisEG∝−NR−1forN
gravitatingparticles.Bymovingtothecentertheelectrongetslowergravi-
tationalenergyandthereforeoccupiesanenergeticallyfavoredstate.Aslong
astheelectronisnon-relativisticitsFermienergyEFisproportionaltothe
FermimomentumpFsquared.Therefore,whentheelectronmovestowards
thecenter,EF∝pF2∝N2/3R−2risesfasterthanthepotentialenergyfalls.
ForsomeradiusthetotalenergyEG+EFassumesaminimumvalueand
thestarisinastablestate.Whenthemassrisesandthusthegravitational

2.1.TheLifeofaStar:BalancingForces

9

Fig.2.1.—Schematicpictureofatypicalprogenitor’sonionstructure(not
drawntoscale).

potentialgetsdeeper,alsopFhastoriseandmoreandmoreelectronsbe-
comerelativistic.Forrelativisticelectrons,however,EF∝pF∝N1/3R−1.
Theminimuminthetotalenergydisappears.Dependingonitsmassbeing
aboveorbelowtheChandrasekharlimitatthattimethestarcanhavetwo
estinies.ddistinct

•Ifthethteotalmass,energyM∝ispNo,sioftivet.heTstarhereforeisbethelowsttarhelowersiChandrasekhartsenergylimitby
expanding,whichinturnlowerspF,electronsbecomenon-relativistic
andthestarisstableagain.

•ForamassabovetheChandrasekharlimitthetotalenergybecomes
moreandmorenegativebyloweringtheradiusandthestarstartsto
collapse.

Thenumberofnucleonsdeterminesthemassoftheironcore.Uptodetails
oflimittheiscthenhemicalcobtainedomposbiystionolvingoftheEFs=tar,EGMfor=rNmnuelativisticcleon.eThelectronsandChandrasekharyields
1.5M.

10

Chapter2.TheCore-CollapseParadigm

2.2TheDeathofaMassiveStar:Core
Collapse

AssoonastheironcoreofaSNprogenitorreachesitsChandrasekharlimit
thegravitationalpullwinsovertheinternalpressureandthecorestartsto
collapse.Theriseindensityandtemperatureduetothecollapsesetsthe
stageforprocessesspeedinguptheinfall.Electroncapturesonnucleiand
,protons

e−(Z,A)→νe(Z−1,A)
e−p→νen,(2.2)
becomemoreandmoreefficientanddonotonlytakeawaydegeneracypres-
sure,butalsodrainenergyduetotheemittedneutrinosthatescapetheiron
coreunhindered.Anotherenergysinkisphotodisintegrationofironnuclei.
Duetotherisingtemperatureinsidethecorephotonsstarttodisintegrate
ironnuclei.Withinmillisecondsthephotodisintegrationundoespartsofthe
workofthenuclear-fusionprocesses,andthereforetakesawayenergyand
thermalpressure.
Asthecollapseoftheironcorespeedsup,twodistinctregionsdevelop
(Figure2.2left,Figure2.3Ricinphase1):

•thehomologouslyinfallinginnerpartwherethespeedissub-sonicand
proportionaltoradius(Bruenn1985).Thisregioncontracts,keeping
itsrelativedensityprofile,untilitreachesnucleardensitywherethe
equationofstatestiffensdrastically.Withitsenormousinertiathecore
contractsevenbeyondthenewequilibriumandslightlyexpandsagain,
usuallyreferredtoasthe“corebounce.”

•thesuper-sonicallyinfallingouterpart.Duetothelowersoundspeed
atlargerradii,thematterassumessuper-sonicspeeds.Theinfallcor-
respondsthereforebasicallytoafreefall.

Ashockwaveformsattheinterfaceofthetworegionsduetotheslight
reexpansionoftheinnercore.Thecorebouncetriggerstheexplosion.
Thecollapsehappensonatimescaleofsome100ms(firstphasein
Figure2.3).Therefore,theouterpartoftheironcoreisstillinfallingandthe
star’senvelopehas“notevennoticed”thatthecorecollapsed.

2.3.Reincarnation:AProto-NeutronStarEmerges

11

Fig.innerh2.2.—omologousCartoonandoftheacoollapsutersingupestr-sar.oLenicallyft:Theinfallinginitialsregion.tage,Miwddithle:theA
hardcorehasformed,theshock(white)hasbouncedandtravelsoutwards.
Theouterlayersarestillinfalling,passingthroughtheshockwave,andthen
droppingataslowerrate.Right:Theshockwavehasblownofftheouterlay-
ersofthestar;thenakedproto-neutronstarcontractsandcoolsbyneutrino
.nemissio

2.3Reincarnation:AProto-NeutronStar
Emerges

Theshockwavemovesoutwardsthroughtheinfallingouterlayersofthe
ironcoreandlosesenergybydisintegratingthenucleiitplowsthrough.The
nuclearbindingenergyof0.1Mofironisabout1.7×1051erg,comparable
totheexplosionenergy,i.e.,approximately1%ofthereleased1053ergof
gravitationalbindingenergy.Thedissociationofironalsoliberatesprotons
thatcaptureelectrons,andinturnemitelectronneutrinos.Inthecentral
regionofthecore,theseneutrinosaretrappedbythesurroundingmaterial.
Diffusionisslowcomparedtothedynamictimescale.AsshowninFigure2.3,
theregionofneutrinotrappingbelowRνbuildsupduringcollapse.Aslongas
theshockwavehasnotpassedRνneutrinotrappingisveryefficient,because
therearestillironnucleiaroundandduetocoherenceeffectsthecrosssection
forneutrinoscatteringonnucleiislargecomparedtothatforscatteringon
ns.cleounThenweenterthesecondphaseofFigure2.3.TheshockpassesRνand
disintegratesthestillinfallingironnucleiinthevicinityofRν.Suddenly,there
arenolongerironnucleiaround.Therefore,insteadofneutrinosscattering
onnuclei,onlyscatteringonfreenucleonstakesplacewithamuchlower
crosssection.Atthesametime,thefreeprotonscaptureelectronsandemit
electronneutrinos.Theseneutrinosinthevicinityoftheshockcanescape

12

Chapter2.TheCore-CollapseParadigm

theFig.radius2.3.—Sofchtheematicoriginalplotiofronthecore.timeTheveeolutionvoflutionorviasriousdividedradialinzfouroneswithinphases:
1.Kelvin-CollapseHelmholtzofthecooironlingcore,ofthe2.pproto-romptnνeeutronburst,star.3.Asalongccretiondashesphase,therandadius4.
ofinnerthecioreron(cRoreic),(RandFe)disotsshotown,theshortradiusdashesbelowtrepresenhatnttheeutrinosinaterfacerettrappotheed
(shoRνc).k(RTheshosck).olidline,AdaptesdftartingromJankshortlyaa(1993).fter0.1sshowsthepositionofthe

freelyandcauseasuddenpeakintheelectron-neutrinoflux.Theluminosity
brieflyrisestoaround1054ergs−1,referredtoasthepromptneutrinoburst.
Thisalsoweakenstheshock.
Leptonnumberisapproximatelyconservedinsidethecorebecauseneu-
trinosaretrappedinthedensemattermainlyduetoneutrino-nucleonscat-
tering.TheprocessesofEquations(2.2)causeaβequilibriumtodevelopand
aFermiseaofνebuildsup.Mostofthegravitationalbindingenergyisnow
storedintheFermiseasofelectronsandelectronneutrinos.
DuringthethirdstageinFigure2.3,ataradiusofafew100km,only
aboutafew100msafterbounce,theshockwavestallsduetoitsenergy
losscausedbythedisintegrationofironnucleiandneutrinoemission.Still
infallingmatterfromouterlayersoftheironcorepassesthroughthestanding
shockontothenewlyformedproto-neutronstar,alsoshowninthemiddle

2.4.Self-ConsistentSimulations

13

whpanelatousfeFdtoigurebe2.2.theTinheteredgefaceofbetthewepenroto-thenhomeutronsologoustarilsyaatndRsicupiner-soFigurenically2.3,
infallingregionsoftheironcore.
causeConthetractiontempoferaturethetoproto-riseandneutronthesntaraeutrinondthealuminositccretionyofincreases.matterInsttillhe
regionneutrinosbetwceoeonltRheνandsurfacetheoftstandingheprotshockwo-neutronavecsonvtar.ectionInttheaklesastsplace.tageTheof
ofFiguretheo2.3tutgoinghestalledneutrinosshocdkepwaositveisinthereviverdegionbytjheustbenergyehindthattheasfhockraction.If
53onlyabsorbafedewbpyerthecenmtoatterftheblibehinderattheeds10tandingerggshorackwvitationalave,thesbindinghockwillenergymovaree
ofoutthewsardstar.Tagain.henakFinallyed,thispproto-neutronowerfulstarrexplosionemainsbloinwstheoffcentheterwaholendcoenvolselopbye
neutrinoemission.TheKelvin-Helmholtz-coolingphaseisdisplayedasthe
lastDuephasetotinheFigureinitially2.3astalledndtheandrightafterwpanelardsofFreviveigureds2ho.2.ckwave,thismech-
andanismWiilssroneferred(1985).toaOsveartdelaheyyeedarsemxplosion.oreandItmwoasrerefirstfinednsuggestedumericbyalsiBethemu-
reslationsulththatavenobeeenxplosdevieonslopead.reTheobtainemostd(elabBurasorateetal.among2003a,themLiebeagreend¨inorfertheiret
al.2001,Mezzacappaetal.2001,Rampp&Janka2000,Thompson,Burrows,
&Pinto2002).IthasbeenstressedbyBurasetal.(2003a)thatsomecrucial
pieceinthegamemightbemissing.
neutronEvensiftarithesbsorntandardbytheSNpictureisexplosion.correct,Iftthehereareprogenitoralsocstaraseshaswhenalargeno
mfindassa,ssayuddengreeatend.rDthanuet20oaM,ccretionthetevheolutionmassoofftthehepproto-roto-nneutroneutronsstartarccanan
endsexceedupitsasaChablacndrakshole,ekhatrlimit.erminatingTheneprotoutrino-eneutromisnsisontaralmtohenstcatollaanpsesinsatanndt.
Thedetailsdependonthenuclearequationofstateandmatteraccretion,
issuesthatarenotyetsettled.AdiscussioncanbefoundinBeacom,Boyd,
&Mezzacappa(2001)andreferencestherein.

2.4Self-ConsistentSimulations

TherehavebeennumericalsimulationsofSNeshowingdelayedexplosions
foralmosttwodecadesnow(Bethe&Wilson1985,Wilson1985).Thesepi-
oneeringworksobtainedrobustexplosions.Withincreasingcomputerpower
andrefinedinputphysicsthepreviousexplosionscouldnotbeconfirmed
byothergroups.Uptonow,onlytheLivermoregroupobtainsrobustexplo-

14

Chapter2.TheCore-CollapseParadigm

sions(Totanietal.1998).However,theyhavetoincludeconvectioninsidethe
proto-neutronstarjustbelowtheneutrinospheresbyaparameterizationof
neutron-fingerinstabilitiesinordertogetexplosions(Wilson&Mayle1988,
1993).Thistreatmentofconvectionhastheeffectofenhancingtheearly
neutrinoluminosities,whichinturncausesmoreenergytobedepositedbe-
hindtheshock.Byneutron-fingerconvectiontheneutronrichouterpartof
theproto-neutronstarmixeswiththeinnerpartinanalogytosalt-finger
convectioninhotsaltwaterandcoldfreshwater.Howeveritiscontroversial
whetherconvectionisrealizedinanactualSN(Burrows1987,Bruenn,Mez-
zacappa,&Dineva1995,Keil,Janka,&M¨uller1996,Mezzacappaetal.1998,
Ponsetal.1999).Intheirtwo-dimensionalsimulations,Burasetal.(2003b)
findconvectionofLedouxtype.
AllgroupsperformingcomputersimulationsofthehydrodynamicsofSNe
withastate-of-the-arttreatmentoftheneutrinotransport,i.e.,Boltzmann
solvers,donotgetexplosionsatall(Liebend¨orferetal.2001,Mezzacappaet
al.2001,Rampp&Janka2000,Thompson,Burrows,&Pinto2002).Even
themostelaborateSNsimulationswithmulti-dimensionalhydrodynamics
andallrelevantmicrophysicsintheneutrinointeractionsincludedfailto
explode(Burasetal.2003b).Thereasonforthemodelsnottoexplodeis
.urecobsThedelayed-explosionmechanismdescribedintheprevioussectionisin
principlefoundbynumericalsimulations.Thecollapsesetsin,ashockwave
forms,bouncesandmovesoutwards.Onitswayitlosesenergymainlydue
tothedisintegrationofnucleiandfinallystalls.Thesimulationsalsoshow
theappearanceofacoolingregionabovetheneutrinospherewhereenergy
islostduetoneutrinoradiation.Behindtheshocktheseneutrinosheatthe
matter,asdescribedabove.Inthesimulations,however,theheatingisnot
sufficientforrevivingtheshock.Insteaditfallsbackandnoexplosiontakes
place.Quitenaturally,themainfocusofmodelbuildersistofindthemissing
inputphysicsinordertoobtainanexplosion.
Inspiteofthenumericalfailureofobtainingexplosions,therearegood
reasonstobelieveinthedelayed-explosionscenario.Observationalhintssug-
gestthatthecorecollapseparadigmshould,atleasttosomeextent,be
realizedinnature.WeknowthatSNeoccurafterstarshavereachedtheir
finalstageofnuclearburning.Forexample,SN1987Awasabluesupergiant
beforeitexploded.TherewereneutrinosobservedfromSN1987A(Biontaet
al.1987,Hirataetal.1987)aboutthreehoursbeforethefirstphotonswere
seen,whichisaclearevidenceforacorecollapse.Althoughtherehasnot
beenaneutronstaridentifiedatthesiteofSN1987A,forotherSNremnants
neutronstarshavebeenobserved.Whilethedetailsoftheexplosionmech-
anismarestillunderdebate,basicfeaturesappeartoberobust.Ifacore

2.4.Self-ConsistentSimulations

30 20 25 15 [MeV] 20 [MeV] 15 10〈 ε 〉〈 ε 〉 10 5 5-1] 50-1] 6ν−e
ν 40e erg s 30 erg s 4νµ
5252 20 2L [10L [10 10 0 0 0 0.02 0.04 0.06 0 0.5 1 1.5 2
Time [s]Time [s]

νe−νeνµ

15

Fig.2.4.—Numericalresultsofthefirst2saftercorebouncetakenfrom
Totanietal.(1998).Topleft:Onlythemeanenergyofνeisaffectedbythe
promptburst.Topright:Theevolutionofmeanenergiesshowsthetraditional
hierarchy.Bottomleft:Thepromptνeburstlastingforafewmilliseconds.
Bottomright:Timeevolutionoftheneutrinoluminosities,scaledonboth
axesascomparedtotheleftpanel.

collapseoccurstheriseofneutrinoluminosityandalsothepromptneutrino
burstareunavoidableaslongasthecollapsedoesnotcontinuetoforma
theblacpkroto-holeneutronimmediatelystar.isAnaolsonngoingotneudoubted.trinoTemissiohroughoutnfortshiseveralthesissecowendswillby
assumethatthedelayed-explosionmechanismisrealizedinnature.
Allnumericalsimulationsagreeonthisqualitativebehavior.Anexample
thatwetookfromTotanietal.(1998)isgiveninFigure2.4.Neutrinomean
energiesandluminositiesareshownasfunctionsoftimepostbounce.Solid
alinesprrominenepresenttνfeature,e,dashenamelydν¯e,athendpdottedromptννµe.Inburstthebrighotattom-fterleftthebpanelounce.weshoThew
meanenergyofνeinthetop-leftpanelshowsashortriseduringtheprompt
burst.Theleftpanelsshowthemeanenergiesandluminositiesduringthe
firstaftert60hemspromptandtνheerighbursttptheanelsaccretionduringphasethefitakrste2sosvaer,ftervisiblecorebinotheunce.riseaRighndt
fallofluminosities.Aftertheendoftheaccretionphasedeleptonizationand
coolingoftheproto-neutronstarpowerstheneutrinoemission.Thecooling
ofphastheesntextartssatectio0.5n.s.Howtheflavor-dependentdifferencesariseisthetopic

16

Chapter2.TheCore-CollapseParadigm

2.5Flavor-DependentNeutrinoEmission
Assoonasthecollapsestartstheneutrinoluminosityincreases.Inthisearly
stageonlyelectronneutrinosareproducedinlargeamountsduetoelectron
capturesonfreeprotons.Theprotonsoriginatefromphoto-disintegratediron
nuclei.Sincetheseprocesseshappenmostfrequentlyinthedenseinnercore
themajorpartofthereleasedelectronneutrinosistrapped.Thistrappingis
thereasonfortheFermiseaofνetodevelop.Whentheshockwavepassesthe
regionwherethestarbecomestransparenttoneutrinos,i.e.,the“neutrino
sphere,”thetrappedνecanescape.Additionally,theinfallingironnucleiare
disintegratedandtheelectroncapturerateinthevicinityoftheneutrino
spheregoesup.Altogether,thisresultsinthepromptνeburst(leftpanels
2.4).FigureNeutrinosbelowtheneutrinospherearestilltrapped.Duringtheaccre-
tionphase,lastingforafew100ms,infallingmatterdepositsenergyonthe
surfaceoftheproto-neutronstar,increasingtheneutrinoluminosities(bot-
tomrightpanelFigure2.4).Afterwards,theproto-neutronstarstillgainsen-
ergybycontraction.TheFermiseasofelectronsandνehaveahugeamount
ofenergystored.Thestarreleasesthisenergyoveratimescaleofseveral
seconds.ForthecalculationoftheLivermoregroup(Totanietal.1998)we
plottedthedeleptonizationinFigure2.5.Thelowerpanelshowsthenumber
ofleptonspernucleon(YL)asafunctionofradius.Fromtoptobottomthe
linescorrespondto0.5,1,3,5,10,and15safterbounce.Accordingly,the
toppanelshowsthetemperatureprofileatthesetimes.Themaximummoves
tolowerradiiastimeincreases.Atthesametimetheinnermostpartheats
up.Eventhoughneutrinosaretrappedbelowtheirneutrinospheres,due
tothesteepgradientinYLtheydiffuseoutwards.Sinceelectronsarevery
degenerateinthisregion,evenmoredegeneratethanνe,neutrinoswillscat-
tertowardsregionsoflowerelectrondegeneracy.Additionally,whenscat-
teringonelectrons,neutrinoswillalwaysloseenergy,sincetheelectronhas
toendupwithanenergyabovethesurfaceoftheelectrons’Fermisea.By
thisdownscatteringtheelectron-neutrino-andelectron-degeneracyenergy
ismainlyreleasedasheat.Thereforethetemperaturemaximumfollowsthe
deleptonization(seee.g.Raffelt1996).
Thereasonwhyneutrinosaretrappedistheirfrequentinteractionwith
thesurroundingmatterviavariousinteractionprocesses.Themediumbasi-
callyconsistsofneutrons,protons,electrons,positrons,andofcourseneutri-
nos.Duetothepresenceofelectronsandpositrons,νeandν¯ecanundergo
chargedcurrentreactionsthatareabsentforallotherflavors.Qualitatively
thetransportofνeandν¯eisthereforedifferentfromthetransportofνµand

2.5.Flavor-DependentNeutrinoEmission

50 40T [MeV] 30 20 0.5 0.4 0.3LY 0.2 0.1 0 0 2 4 6 8 10
Radius [km]

17

Fig.2.5.—Deleptonizationofaproto-neutronstarasobtainedbytheLiver-
moregroup(Totanietal.1998).Top:Weshowthetemperatureasafunction
ofradiusat0.5,1,3,5,10,and15safterbounce.Withincreasingtimethe
peakmovestolowerradii.Bottom:Forthesametimeswegivethelepton
numberpernucleon.Astimeincreasesleptonnumbergetslower.

ντ.presenTot.beInexact,centratlrhereegioisns,alsowaheresmalltheptoemppulationeratureofmisuonsmaxiandmal,antheti-mmuonsuon
massistwicethetemperature.Sinceforanon-degenerateparticlethemean
energyisroughlythreetimesthetemperature,≈3T,thereisathermal
distributionofmuonspresentintheinnercore.However,inthevicinityof
nethenutrinoeutrinospectra,spheretheofteνmµpeandratureν¯µ,iwsahereboutmuons1/10wofouldthemaffectuonthemases.Tmerginghus
muonsarestronglysuppressedanddonotaffecttheneutrinoemission.
cenSctralregionhematicallyof,ttheheproto-emissionneutronofνesandtar,ν¯eiscorrespdisplaondingyeditonFtheigureblac2k.6.Inshadedthe
areaonthelefthandsideofFigure2.6,β-processeskeepνeandν¯einLTE.
Duetothedecreasingdensitythisreactionbecomesinefficientatsomera-
dius,definingtheneutrinosphere.Fromthatradiusneutrinosstartstreaming
pofreelysi.tionSinceofthethenβ-ceutrinorossspheresectionisdepenerendsgyondepeneundenttrino.Tenergyhereforealsoitisthenotrpadialos-

18

Chapter2.TheCore-CollapseParadigm

Fig.2.6.—Schematicpictureoftheνeandν¯espheresinaproto-neutronstar.

Fig.2.7.—SameasFigure2.6butforνµ.

sibletodefineauniqueneutrino-sphereradius.Dependingontheirenergy
neutrinosdecouplefromtheproto-neutronstaratdifferentradiiandthere-
foredifferentlocaltemperatures.Forthisreasonthespectrumofneutrinos
leavingthestardoesnotrepresentathermaldistributionwithatempera-
turecorrespondingtothemediumtemperatureattheneutrinospherethat
is,anyway,notwelldefined.
Theenergydependenceoftheinteractionprocessesofνeandν¯eisexactly
thesame.Theproto-neutronstar,however,containsmoreneutronsthan
protons,whichleadstoahigherabsorptionrateforνethanforν¯e.Therefore,
theνesphereliesatalargerradiusthantheν¯esphere.Sincethisistruefor
allneutrinoenergiesthemeanenergyofemittedνeisalwayslessthanthe
meanenergyofν¯e.
Forroughestimates,theneutrinospherecanbeconsideredablackbody
radiatingneutrinos.Thesurfaceareaisthusproportionaltothenumber
ofemittedneutrinos.Again,theenergydependenceoftheneutrino-sphere
radiusmakesamorereliableestimateverydifficult.
InFigure2.7weshowaschematicviewoftheνµtransport.Comparing
Figures2.7and2.6tellsusthatthetransportofνµismorecomplicated.
Therearenochargedcurrentinteractionsofνµwiththemedium.Instead,

2.5.Flavor-DependentNeutrinoEmission

19

thereisavarietyofneutralcurrentinteractionsthatqualitativelydifferfrom
oneanotherand,atthesametime,areofcomparableimportanceforthe
transportprocess.Naively,onewouldexpectthatνµdecoupledeeperinside
thestarthanν¯e,becausetheneutral-currentinteractionratesarelower.
Then,themeanenergyofemergingνµwouldbehighercomparedtoν¯eand
thefluxlower.Inreality,thesituationismoresubtle.
Intheinner-mostregiontheνµarekeptinLTEbyenergyexchanging
scatteringprocessesandthefollowingpairprocesses:
•BremsstrahlungNN↔NNνµν¯µ
•Neutrino-pairannihilationνeν¯e↔νµν¯µ
•Electron-positron-pairannihilatione+e−↔νµν¯µ
Theradiuswherethesereactionsbecomeinefficientdefinesthenumber
sphere.Outsidethisspherenoparticlecreationorannihilationtakesplace
andνµcannolongerbeinLTE.
Duetothescatteringreactions
•Nνµ→Nνµ
•e±νµ→e±νµ
thereisstillenergyexchangewiththemedium.However,thetwoprocesses
arequalitativelyverydifferent.Scatteringone±islessfrequentsincethereare
lesse±thannucleons.Ontheotherhandtheamountofenergyexchangedin
eachinteractionwithe±isverylargecomparedtothesmallrecoilofnucleons.
Attheradius,wherescatteringone±freezesoutliestheenergysphere.A
diffusiveregimestarts,whereneutrinosonlyscatteronnucleonsandtherefore
exchangelittleenergyineachreaction.Thisregimeisterminatedbythe
transportsphere,definedbytheradiusatwhichalsoscatteringonnucleons
becomesineffectiveandtheνµstartstreamingfreely.
Evenifweassumethatnucleonscatteringhappensiso-energetically,i.e.,
noenergycanbeexchangedoutsidethethermalizationsphere,themean
energyofneutrinosescapingthestarisonlyabout50–60%ofthoseclose
tothethermalizationsphere.Duetoitsdependenceonthesquareofneu-
trinoenergythenucleonscatteringcrosssectionhasafiltereffect,becauseit
tendstoscatterhighenergyneutrinosmorefrequently(Raffelt2001).Ifone
weretospecifyaneutrinosphereaccordingtothecrudeapproximationof
ablackbodyradiatingneutrinos,itwouldcoincidewiththenumbersphere.
NeutrinosleaveLTEatthenumbersphere.Thepositionofthenumbersphere
determinestheflux,becauseneutrinocreationisnoteffectivebeyondthat

20

Chapter2.heTCore-ollapseCaradigmPradius.Theνµfluxthatpassesthenumbersphereisconserved.Ontheother
hand,themeanenergyofνµinthisareaisstillsignificantlylowereddue
toscatteringprocessesbeforetheνµleavethestar.Themeanenergyofνµ
emergingthestarisusuallyfoundtobelargerthanthatof¯νe.Simulations
thatincludetheνµinteractionsonlyveryapproximateobtainalargehierar-
chyinthemeanenergies(Figure2.4).Thesystematicstudyofallinteraction
processesisamajorpartofthiswork.Ourfindingsshowthatthetransport
ofνµismoresubtlethanpreviouslyassumed.Onceallinteractionprocesses
aretakenintoaccount,themeanenergiesofνµandν¯eareverysimilarand
mightevencrossover.Thenumberfluxescandifferbylargeamounts.

3Chapter

teractionsInNeutrino

Wediscussneutrinointeractionswiththestellarmediumthatweassume
toconsistofprotons,neutrons,electrons,positrons,andneutrinosofallfla-
vors.Themostimportantreactionsforelectronneutrinosandelectronanti-
neutrinosarechargedcurrentprocessesonnucleons(β-processes).Allother
flavorsundergoneutral-currentinteractionsonly.
Theneutral-currentprocessesfallintothreequalitativelydifferentgroups.
TrueLTEisonlyobtainedbythepaircreationandannihilationprocesses,
i.e.,bremsstrahlung,theνeν¯e,ande+e−pairprocesses.Scatteringone±
andνe,¯νeexchangeslargeamountsofenergyinasinglereaction.Neutrino-
nucleonscatteringqualitativelydiffersfromtheotherscatteringreactionsin
beingmorefrequentbutatthesametimeexchangingonlysmallamountsof
energyineachreaction.Goingbeyondthetraditionaliso-energeticnucleon
scatteringweintroducerecoilandweakmagnetism.
Forthefirsttime,wehaveinvestigatedtheνeν¯epairprocessindetailand
wereabletoshowthatitisfarmoreimportantthanthetraditionale+e−pro-
cess.Bycrossingsymmetrytheνeν¯epairprocessisrelatedtoνµνescattering.
Thisscatteringprocess,however,isnegligiblecomparedtoe±scattering.We
havepresentedtheseresultsinasimilarforminourpublication:
R.Buras,H.T.Janka,M.T.Keil,G.G.Raffelt,andM.Rampp,“Electron-
neutrinopairannihilation:Anewsourceformuonandtauneutrinosinsu-
pernovae,”Astrophys.J.587(2003)320.

3.1Beta-Processes

Forelectronneutrinos,themostimportantneutrino-matterinteractionsare
theβ-processesνen↔e−pandν¯ep↔e+p.Inprinciple,alsoνµcanun-
dergochargedcurrentreactions.ThecentralregionoftheSNcorecertainly

21

22

Chapter3.NeutrinoInteractions

containsmuons,buttheirpresenceiscommonlyneglectedinhydrodynamic
simulations.Inthevicinityoftheneutrinospherethetemperatureistoo
lowforathermaldistributionofmuons.Foroursimulationsitistherefore
justifiedtoneglectthechargedcurrentreactionsofνµ.
Thesquaredspin-summedmatrixelementforneutrinoabsorptionbyneu-
tronsisgivenby(Yueh&Buchler1976b)
|M|2=32GF2(α+1)2(k2·k1)(k4·k3)
sinsp+(α−1)2(k2·k3)(k4·k1)+(α2−1)mnmp(k1·k3).
(3.1)Themomentak1,k2,k3,andk4areassignedtothecorrespondingparticlesas
showninFigure3.1;GFistheFermicouplingconstant,α=CA/CV=1.26,
andmnandmpthemassesoftheneutronandproton,respectively.Together
withphase-spaceblockingandthephase-spaceintegrations,Equation(3.1)
yieldstheabsorptionrateorequivalentlytheinverseofthemeanfreepath
(mfp)

=λ−1nnd3k2d3k4d3k32
|M|(2π)521222423spins
4×fn(2)(1−fp(4))(1−fe(3))δ(k1+k2−k3−k4),(3.2)
wherennistheneutrondensity,jtheenergyofparticlej,andfn(2),fp(4),
andfe(3)aretheoccupationnumbersofneutrons,protons,andelectrons,
.elyectivrespForanti-neutrinoswehavetointerchangeneutronsandprotonsandre-
placeelectronsbypositrons.Forthereverserateonehastoreplaceblocking
factorsandtheoccupationnumbersappropriately.

νek1k2n

−ek3k4p

Fig.3.1.—Feynmandiagramforνeabsorption.

3.2.ScatteringucleonNNeutrino-

23

SincetherearemoreneutronsthanprotonspresentintheSNcore,theβ-
ratesforνearealwayshigherthanforν¯e.Allinteractionprocessespresented
intheremainingsectionsofthischapterareneutral-currentinteractions.
Dueneutrinos.totheThesmallerfollocwingouplingsectionsconstanaretsmtoreheyraelevreantsub-toνdominan,btecaforuseechalectronrged
µcurrentreactionsdonotcontributetothespectraformationofνµ.(Ifnot
stateddifferently,νµalwaysreferstoνµ,τandν¯µ,τ.)

3.2Neutrino-NucleonScattering
Themostfrequentneutral-currentprocessisneutrino-nucleonscatteringand
thereforethedominantopacitysourceforνµ.Foraneutrinowithinitial
energy1andfinalenergy2,thedifferentialcrosssectionisgivenby
dσ=CA2(3−cosθ)GF222S(ω,k),(3.3)
d2dcosθ2π2π
withω=1−2,kthemodulusofthemomentumtransfertothemedium,and
θthescatteringangle.S(ω,k)representsthedynamicalstructurefunction
thatparameterizestheresponseofthenuclearmedium.
Thedynamicalstructurefunctionincludesalleffectsthatchangethebe-
haviorofthenucleoninthemedium,likefluctuations,correlations,andde-
generacyeffects.Thefulldynamicstructurefunctioncoversthewhole(ω,k)
plane.Thereforeitdoesnotonlydescribeneutrinoscatteringonnucleons
butalsoneutrinobremsstrahlungoffnucleons.Bothprocessesarerelated
bycrossingsymmetryasshowninFigure3.2,wherethegreyshadedblob
representsthenuclearmedium.However,thefullstructurefunctionisun-
known.Commonapproximationstreatthespace-like(ω2≤k2)andtime-like
(ω2≥k2)domainsonadifferentfooting.

Fig.3.2.—Left:Diagramofneutrinonucleonscattering.Thegreyshaded
blobrepresentsthenuclearmedium.Right:Bremsstrahlungisobtainedby
leg.eutrinononecrossing

24

Chapter3.NeutrinoInteractions

Thecommonsimplificationoftheneutrino-nucleonscatteringcrosssec-
tionintraditionalSNsimulationswastoassumethatthereactionisiso-
energetic,i.e.,thenucleonshavean“infinitemass”andthereforecannot
absorbenergybutarbitraryamountsofmomentum.Inthisapproximation
thestructurefunctionis

Sno−recoil(ω,k)=2πδ(ω).(3.4)

Theonlymulti-particleeffectthatisusuallytakenintoaccountisfinal-state
blockingduetothenucleondegeneracy.
Moreelaboraterecentapproximationsincludecorrelatednucleonsand
afinitenucleonmass.(BurrowsandSawyer1998,Reddy,Prakash,and
Lattimer1998,Reddyetal.1999).Theseresultswerecalculatedinthe
random-phaseapproximation,thatincludesnucleon-spincorrelationsbutnot
nucleon-spinfluctuations.Therefore,bremsstrahlungisnotallowed,because
neutrinoscoupletothenucleonspinandtheemissionofaneutrinopairre-
quiresaspinflip.IntherecentsimulationsbytheGarchinggroup(Rampp
andJanka2002),thecorrelationeffectsweretakenintoaccounttogether
recoil.thiwSomedifferentaspectsofmulti-particleeffectsaretakenintoaccount
bynucleonexcitations.ThediagrammaticstructureisgiveninFigure3.3.
Raffelt(2001)developedaschematicstructurefunctioninthisspiritfor
bremsstrahlung.Bycrossinganeutrinolegoneobtainsinelasticscattering,
thatallowsforanenergytransferunrelatedtonucleonrecoil.Eventhough,
thisenergytransferisofthesamesizeasnucleonrecoil,thequadraticden-
sitydependenceoftheinelasticscatteringprocessyieldslowerratesinmore
diluteregions,whererecoilisstilleffective.InadetailedstudyRaffelt(2001)
showsthatoncerecoilisincludedinaSNsimulationtheinelasticscattering
contributioncanbeneglected.Fortheimplementationofνµ-nucleonscatter-

NN

ν

νν− νN

N

NNNNFig.3.3.—Left:Diagramofneutrinobremsstrahlungoffnucleons.Thispro-
cessissubdominantforenergyexchangecomparedtorecoil(Raffelt2001).
Right:The“crossedprocess”correspondstoinelasticneutrino-nucleonscat-
tering.

ucleonNNeutrino-3.2.Scattering

25

ingRaffelt(2001)providesastructurefunctionforrecoilingnucleonsthat
weadaptedtoourcode.
Wedonotdistinguishbetweenprotonsandneutrons.Sincefornon-
relativisticnucleonsthescattering1crosssectionisp1roportional2toCV2+3CA2,
thevectorcurrent(CV=−forneutronsand−2sinθWforprotons)
issmallcomparedtotheaxia2lcomponent,where2weuse|CA|=1.26/2.
Neglectingthevectorpartsimplifiesthecalculationssignificantly,sinceoth-
erwise,therearedifferentstructurefunctionsfortheaxialcurrent,thevector
current,andthemixedterm.
Ignoringnucleondegeneracyeffects,thestructurefunctionthatincorpo-
ratesnucleonrecoilsis(Raffelt2001)
πω−ωk
Srecoil(ω,k)=ωkTexp−4Tωk,(3.5)
withωk=k2/2m,mthenucleonmass,andTthemediumtemperature.
MultiplyingEquation(3.3)withthedensityofnucleonsandignoring
phase-spaceblockingoftheessentiallynon-degeneratenucleonsyieldsthe
differentialratesthatcanbeintegratedforobtainingtherequiredenergy
andangulardifferentialrates.Inthecaseofrecoilthenumericalintegra-
tionsarerathertrickybecauseEquation(3.3)isstronglyforwardpeaked,cf.
Figure3.4.Inourcodeweemploythe“rejectionmethod”forobtainingthe
integratedrates(Pressetal.1992).

µ0.50.505ε2[MeV]
-1-1-0.5-0.5001015
200251teRa1015500

Fig.3.4.—ThedifferentialinteractionrateforνNscatteringaccordingto
Equations(3.3)and(3.5)forinitialenergy1=10MeVasafunctionofthe
outgoingenergy2andscatteringangleµ=cosθ.

26

Chapter3.NeutrinoInteractions

Inthecaseofνeandν¯e,scatteringonnucleonsislessimportantthanthe
β-processes.Thereforewelefttheoriginalimplementationofourcodeinthe
electron-neutrinosectorunchanged.TheimplementationfollowsTubbs&
Schramm(1975)andcanbefoundinJanka(1987).Itbasicallycorresponds
tousingthestructurefunctiongiveninEquation(3.4).

3.3WeakMagnetism
Weakmagnetismarisesasacorrectiontonucleonscatteringduetoour
schematictreatmentofthedynamicalstructurefunction.Thecompletestruc-
turefunctionwould,ofcourse,containweakmagnetism.Theeffectisabsent
ifweusethe“traditional”approximationofinfinitenucleonmass.
Asmentionedearlier,thereareessentiallynomuonsortauspresentinthe
vicinityoftheneutrinosphere.Therefore,νµandν¯µareusuallytreatedinthe
sameway.However,theneutrino-nucleoncrosssectionisdifferentforneutri-
nosandanti-neutrinosonceweakmagnetismisincluded.Thecontributions
tothecrosssectionfromtheanomalousmagneticmomentofnucleonsand
theinterferencetermoftheaxialandvectorcurrenthavedifferentsignsfor
neutrinosandanti-neutrinosduetoparityviolationinthestandardmodel
(Horowitz2002).Theinteractionrateishigherforneutrinosthanforanti-
neutrinos.Neutrinosintherelevantregimeoftheproto-neutronstarhaveenergies
ontheorderofseveral10MeV.Tofirstorderin/m,whereistheneutrino
energyandmthenucleonmass,thecorrectionfactortotheneutrino-nucleon
crosssectionduetoweakmagnetismis(Horowitz2002)
1±4CA(CV+F2)k,(3.6)
CA2(3−cosθ)m
wherekisthemomentumtransfer,F2astructurefunctionparameterizing
thestructureofnucleons.Forneutrino-protonscatteringF2=21(µp−µn)−
2sin2(θw)µp≈1.019,whereµp,narethemagneticmomentsofneutronand
proton,respectively,andθwistheweakmixingangle.Forneutrino-neutron
scattering,theindicesnandphavetobeinterchangedyieldingF2=−0.963.
TheuppersigninEquation(3.6)istobeusedforneutrinosandthelower
signforanti-neutrinos.
InordertobeconsistentwithEquation(3.3)wekeepterms∝CA2and
neglectterms∝CV2.Inaddition,wesubstituted1cosθ,takenattherest
frameofthenucleon,byourmomentumtransferk.Thisiscorrectforfor-
wardandbackwardscatteringbutonlyanapproximationforotherangles.
Itdoesnotchangetheangulardependencebymuch,butinordertokeep

Bremsstrahlung3.4.

27

ourstructure-functionprescriptionofthecrosssection,wecannotmultiply
factorsthatdependonk.
Thefirst-ordercorrectionfactorhasthedisadvantagethatitbecomes
negativeforlargemomentumtransfer.Inordertoavoidnumericalproblems
weuse2
1±4C2A(CV+F2)k(3.7)
CA(3−cosθ)2m
instead,whichcorrespondstoEquation(3.6)uptofirstorderink/m.For
neutrinomomentaintheregionoftheneutrinospherethisapproximation
inducesanerroroflessthan1%.

remsstrahlungB3.4Neutrinobremsstrahlungoffnucleonsinsidethestellarmediumisrelated
toneutrinonucleonscatteringbycrossingsymmetry(Section3.2).Raffelt
tive(2001)lysdephoewndedonthatthedtheeetailedmerrgingate,neafactutrinothatspewcetraconfirmfromainSoNurdosimnotsulations.ensi-
Aschematicapproachisthereforewelljustified.Sincebremsstrahlungonly
playsacrucialroleforνwedonotconsiderνandν¯.Weadopttheap-
proachgiveninRaffeltµ(2001).Theratefortheeabsorptioneofaνµbyinverse
bremsstrahlungNNνµν¯µ→NNisgivenby
¯322λ−1=CA2GFnB212¯d(2kπ)3f(¯)24¯S(ω),(3.8)
wherscatteringeω=case.+¯T,henBoviser-btheanrredqucleonuantitiesdensitby,elongandt|oCtAhe|ν¯=1µ.that26/is2asabsionrbtheed
togetherwiththeprimaryν.
Theenergy-differentialrateµforemissionofbremsstrahlungisobtainedby
adjustingthephase-spaceblocking:
d˙nC2G22d3k¯
d=A2FnB2π2[1−f()]2¯(2π)3[1−f(¯)]24¯S(−ω)(3.9)
TheheuristicansatzforS(ω)givenbyRaffelt(2001)hastheformofa
ziantLoren2Γ2√πα2nBT
S(ω)=ω2+Γ21+exp(−ω/T),Γ=m2mTπ+m2,(3.10)
πwiththeconstantαπ≈(2m/mπ)2/4π,wheremisthenucleonmassandmπ
thepionmass.ForcarryingouttheintegrationinEquations(3.8)and(3.9)
theΓ2inthedenominatorofScanbeneglected.Averydetailedanalysisof
thisbremsstrahlungratecanbefoundinRaffelt(2001).

28

Chapter3.NeutrinoInteractions

3.5PairAnnihilation
Wenowturntotheleptonicsourcereactionsforνµν¯µpairs,e+e−→νµν¯µ
andνeν¯e→νµν¯µ.Traditionally,onlye+e−pairannihilationwasimplemented
inSNsimulations.Asonepartofthisdissertationwehaveforthefirsttime
infacvtesortofigated2–3mtheoreimpimpoortancertantofν(Buraseν¯e,Jpairanka,Keannihilationil,Raffethatlt,&turnsRamopputto2003a).bea
stantsThemwhileatrixtheelemenphase-tssfpaceorbinothtegratproionscessesaonlyreidendiffertbicalyupthetochemicalcouplingpoten-con-
tials.Aftersummingoverallspinsandneglectingtherestmasses,thesquared
matrixelementis
|M|2=8GF2(CV+CA)2u2+(CV−CA)2t2(3.11)
sinspwiththeMandelstamvariablest=−2k1·k3andu=−2k1·k4.Themomenta
areassignedto+the−particlesasindicatedinFigure3.5.Theweakinteraction
constantsforeeannihilationare
CV=−1+2sin2θW,CA=−1(3.12)
22whileforνeν¯eannihilationtheyare
CA=CV=21.(3.13)
Fortheinteractionrateswehavetoperformthephase-spaceintegrations,
usingblockingfactorsforthefinalstatesandoccupationnumbersforinitial-
stateparticles(Hannestad&Madsen1995,Yueh&Buchler1976a).Three
integrationsremainthatcannotbecarriedoutanalytically.
icalInpotenordertialstopforaerformllintheteractingphase-spartipacecleins.Mtegralsu-andwethaau-leveptotonsspecifyareachlmosem-t

−ν,eek1k2e+,νe−

νµk3k4 −νµ

Fig.3.5.—Pairannihilationprocessesproducingνµν¯µpairs.

AnnihilationiraP3.5.

29

absentinproto-neutronstaratmospheres,therefore,chemicalpotentialsof
thecorrespondingneutrinoscanonlyariseduetodifferencesintheneutrino
andanti-neutrinointeractions,i.e.,weakmagnetism.Thischemicalpotential
buildsupdynamically,becauseanti-neutrinosescapemoreeasilythanneu-
trinos.Forthee+e−reactionsthelocalvalueofµecanbeobtainedfromρ,
T,andYebyinverting

ne(µe)−ne+(−µe)=Ye,(3.14)
nonsybarpowhersitroenns,er(µespe)aectivndelyn.e+(The−µeb)arayorenntheumbnerumdberensitdyensisitiesofelectronsand
ρnbaryons=mn(1−Ye)+mpYe.(3.15)
Forνeandν¯ethechemicalpotentialisobtainedbytherelation
µνe=µe+µp−µn,(3.16)
withthechemicalpotentialsµpandµnofprotonsandneutrons,respectively.
pairsTheistreahatsobnelowwhythetheνµνenν¯eumbpairerprosphere,cessiνseanandiν¯mpeareortanbtassicoallyurceaforpartνµνo¯fµ
thestellarmediumlikee±.Theenergysphereofνliesalwaysdeeperinside
thestarthantheνeandν¯espheres(Section4.4).µThusνeandν¯eareinLTE
andarepartofthemedium+−asfarasthetransportofνµisconcerned.
portanUnttilveryleptonicrecensourcetlyeforeν.Inannihilationmostnwasumericalconsideredsimtoulationsbetitheissmosttillim-the
µSNonlyconditionsincludedcitisreationobmviousecthathanismνefν¯eorpairstheνaµreflavmoreors.Cimpoomparingrtant.IntheFrigureates3for.6
weshowbothleptonicpairratesasafunctionoftheirrespectivedegeneracy
parameters.Thechemicalpotentialofνµwassettozeroandνe,¯νe,ande±
areassumedtobeinLTE.Intheregionbelowtheneutrinospheresthede-
pairgeneracyprocessisparametersalwaysobmeyoreηiνemp=ηeortan+tηp−thanηne+<ηe−eandtannihilation.hereforetheneutrino
Anotherqualitativedifferenceofbothratesbecomesapparentbylooking
ofate±theeandνnergy-e,¯νethedifferentialdifferenprotialductionproductionrate.Inratethed2casen/dofdvtaisnishingequalforbodegeneracythνµ
andν¯µ,asshowninFigure3.7.Ingeneral,however,electronsandelectron
neutrinoswillbesignificantlydegenerate.Fortheextremecaseηe=ηνe=10
weshowthedifferentialproductionratesinFigure3+.8.−Intheupperpanel
borepresthνenµtsandtheν¯fµacttcreationhattheratesprefactorsareveury2sanditmilar2inforeEquatione(3.11)annihilation.aresTimilar,his

30

Chapter3.NeutrinoInteractions

2]-31.5 cm-1 s71dn / dt [100.500246810
ηFig.3.6.—Pairproductionratesbytheprocessνeν¯e→νµν¯µasafunctionof
ηνe(upperline)ande+e−→νµν¯µasafunctionofηe(lowerline).Weused
T=12MeVandηνµ=0.

namely(CV+CA)2≈0.542and(CV−CA)2≈0.462.Comparingtheratesfor
νµandν¯µ,whichcorrespondstoexchanginguandt,thenhasnobigeffect.
Fortheνeν¯e2processthesituation2isdifferent.Inthiscase(CV+CA)2=1
and(CV−CA)=0andonlyucontributes.Interchangingν¯µwithνµ
correspondstoreplacingu2byt2.Therefore,thekinematicsforparticlesand
anti-particlesaredifferent.Sinceinthecaseofην=0alsothedistribution
functionsofparticlesandanti-particlesaredifferenetthisleadstoadifference
intheenergydependenceoftheνµandν¯µproductionrate.Thelowerpanel
ofFigure3.8displaysbothrates.Thenumbersofproducedνµand¯νµare
course.ofequal,Thereasonfortheaverageenergyofνµtobelargerthanofν¯µcanbe
understoodbylookingatthereactionνeν¯e→νµν¯µinthecenterofmomentum
(CM)frame.Thedifferentialcrosssectionis
2Gσddcosθ=4πF2(1+cosθ)2,(3.17)
whereθistheanglebetweentheingoingνeandtheoutgoingνµ,orequiva-
lently,betweentheingoingν¯eandtheoutgoingν¯µ.Putanotherway,forward
scatteringisfavoredandbackwardscatteringforbidden.Thisisduetoangu-
larmomentumconservation.Theingoingνeandν¯ehaveoppositehelicities
and,intheCMframe,oppositemomenta,sothattheircombinedspinsadd
upto1.Thesameistruefortheoutgoingparticlessothatbackwardscat-
teringwouldviolateangularmomentumconservation.Intherestframeof
themediumtheingoingνetendstohaveenergiesoftheorderofitsFermi
energy,whiletheingoingν¯etendstohaveenergiesoforderT.Becausefor-

-+ee−ννee

31

3.5.PairAnnihilation31
]-31600-1 cm1400e+e-
1200− MeV1000νeνe
-1 s80042600400n / dEdt [1020002d020406080100120140
Energy [MeV]Fig.3.7.—Differentialνµandν¯µproductionratesd2n/ddtvs.neutrino
energyforηe=ηνe=0andT=12MeV.

νµ,τ−νµ,τ

νµ,τ−νµ,τ

1210e+e-νµ,τ
−νµ,τ8642]0-3−-1 cm25νeνeνµ,τ
−νµ,τ MeV20-1 s154210n / dEdt [1052d0020406080100120140160180
Energy [MeV]Fig.3.8.—Differentialνµandν¯µproductionratesd2n/ddtforηe=ηνe=10
andT=12MeV.Upperpanelfore+e−→νµν¯µ,lowerpanelforνeν¯e→νµν¯µ.

32

Chapter3.NeutrinoInteractions

wardscatteringisfavored,theoutgoingνµtendstoinheritthelargerenergy
oftheingoingνe.
Thedifferencesofthesourcespectra,however,donottranslateintosig-
nificantspectraldifferencesoftheνµandν¯µfluxesemittedfromtheSN
core.Whilepairannihilationsandnucleonbremsstrahlungareresponsible
forproducingorabsorbingneutrinopairsandthustheirequilibrationwith
thestellarmediumbelowthe“neutrino-energysphere,”otherprocesses,no-
tablyνµe±scatteringandnucleonrecoils,aremoreefficientfortheexchange
ofenergybetweenneutrinosandthemediumbetweentheequilibrationand
transportspheres.Inournumericalrunswewillfindinfactthataddingthe
newprocesstoaSNsimulationprimarilymodifiesthefluxwithonlyminor
modificationsofthespectrum.
Thenumericalimplementationisvastlysimplifiedbyexploitingthefact
thatνeandν¯eareinLTEintheregionwherethepairprocessesareeffec-
intive.Itegrationnsteadweofcanusingusethetheactualequilibriumneutrinodistributiondistributionswithinthethelocalphase-msediumpace
temperature.Thisapproximationbreaksdownatlargerradiiwheretheνeν¯e
processisunimportantanyway.
Inordertofurtherreducecomputationtimeoneoftheremainingthree
phase-spaceintegrationscanbeapproximatedbytheanalyticexpressions
giveninTakahashi,ElEid,&Hillebrandt(1978).Thisalsorequiressim-
plifyingtheblockingfactors.Withµe=−µe+≥0wecanapproximate
thepositronoccupationnumberbyaMaxwell-Boltzmanndistribution.For
µe/T∼>2thisholdstoverygoodaccuracy.Thegreatestdeviationisat
µe/T=0andyieldsblockingfactorstoolowbyabout10%.However,e−
andνearealwaysdegenerateintherelevantregions.

3.6ScatteringonElectronsandElectron
NeutrinosEventhoughthescatteringreactionsone±andonνe,¯νearecloselyrelated
tothepairannihilationsdiscussedintheprevioussection,scatteringone±
ismoreimportantthanonνeorν¯e.Thematrixelementsforthesereactions
arejustthecrossedversionsoftheleptonicpairprocesses,
|M|2=8GF2(CV+CA)2s2+(CV−CA)2u2(3.18)
sinspwiththeweakinteractioncoefficientsofEquations(3.12)or(3.13)forscat-
teringone−oronνe,respectively.Fors=2k1·k2andu=−2k1·k4the
momentaareassignedtotheparticlesaccordingtoFigure3.9.Crossingthe

3.6.ScatteringonElectronsandElectronNeutrinos

ννµµkk13kk42e−,νee−,νe

Fig.3.9.—Leptonicscatteringprocesses.

33

matrixelementEquation(3.18)againbyinterchangingu↔t,weobtain
scatteringone+orν¯e.Thisisalsotrueforscatteringofν¯µone−orνe;
scatteringofν¯µone+orν¯ebringsusbacktoEquation(3.18).Thenumerical
procedureforcalculatingtheratesisthesameasforthepair-annihilation
processes,givenintheprevioussection.
Incontrasttothepairratesoftheprevioussection,thedifferentialscat-
teringratesaremonotonicallyrisingfunctionsofthedegeneracy.InFig-
ure3.10weshowtheratesforνµscatteringone±(lowerline)andtherates
forνµscatteringonνeorν¯e,normalizedtothelatterrate.Scatteringonνe
±andν¯eisalwaysmorefrequentthanscatteringoneincaseofηνe=ηe.How-
ever,forrealisticsituationswithηνe<ηeweexpectthatscatteringone±has
1–2timestherateofscatteringonνeandν¯e.Therefore,neutrino-neutrino
scatteringisexpectedtobearelativelyminorcorrection.Inournumerical

18 scattering on16η=014ν−e and νe
/dt)e12e+ and e-
108(dn/dt) / (dn642000.511.522.533.54
ηFig.3.10.—Thermallyaveragedscatteringrateforνµone±asafunction
ofηe(lowerline)andforνµonνeandν¯e(upperline)asafunctionofηνe.
Theratesarenormalizedtothescatteringrateone±atηe=0.Weused
T=12MeVandηνµ=0.

34

hCapter3.eutrinoNteractionsInstudieswewillindeedfindthatthisprocesshasonlyasmalleffectonthe
fluxes.andectraspneutrino

3.7OtherReactions

InteractionsthatalsotakeplaceinsideaSN,butareunimportantforour
purpose,areabsorptionandemissionofνeandν¯ebynucleiaswellasscatter-
ingofneutrinosonnuclei.Inthevicinityoftheneutrinosphereallnucleiare
disintegrated.Reactionsonnucleitakeplaceatlargerradiiandaretherefore
importantfortheheatingprocessbehindthestalledshockwave.
Throughoutthepastliterature,oftentimestheplasmonprocesswastaken
intoaccountasasourceforneutrinopairs.Theplasmonisanexcitationof
themediumthatcanemitaνν¯paironceitdecays.Intherelevantarea
closetotheneutrinospherethisprocessisseveralordersofmagnitudeless
frequentthantheotherpairprocesses.

4Chapter

TraSettingnsporthetStageforNeutrino

Proto-neutronstaratmospheresarecharacterizedbytheirradialprofilesof
temperature,nucleondensity,andthenumberofelectronspernucleon.In
ordertoprobeavarietyofpossibleatmosphericstructuresweusetwoself-
consistentprofilesobtainedbyhydrodynamicsimulationsand,inaddition,
power-lawparameterizationswithconstantelectronfractions.Afterdefining
theopticaldepthandaneffectivemeanfreepathforthermalizingprocesses
wecalculatethethermalizationdepths,i.e.,thelastpointsofinteractionfor
allthethermalizingprocessesineachofthebackgroundmodels.Thissimple
approachisapowerfultestoftherelativeimportanceofvariousneutrino-
matterinteractionsandtheirdependenceonstellarparameters.

4.1CharacterizingProto-NeutronStars

Alldensittheyofsneutrinocatterersinandteractiontheirratestempwiththeerature.bacBothkgrounddensitymediumρanddepteempndeonraturethe
Tareformationfwunctionseassuofmethethatradius.thepIntheroto-neutronregionstarrelevcaonsntistforonlynofeutrinospectra

protons•neutrons•nselectro••positrons
•neutrinosofallflavors.

35

36Chapter4.SettingtheStageforNeutrinoTransport
Indensitorderyatndocthealculatechemicaleverypintotentialeractionforerateachforofatllheseradii,wmattereneedctheonstituennumtsbears
afunctionofradiusinadditiontothetemperature.Asmentionedearlierno
muonsandtausarepresent.Therefore,usuallywesetthechemicalpotential
ofνµtozero.Onlyinthecaseofweakmagnetismasmallνµdegeneracy
developsandweobtainachemicalpotentialforνµ.
Ourneutron-staratmospheresarecharacterizedbyT(r),ρ(r),andthe
electronfractionperbaryonYe(r).Sincememn,p,thedensityisgivenby
thenumberdensitiesofneutronsandprotonstimestheirmasses
ρ(r)=nnmn+npmp.(4.1)
Inadditionweusen
Ye(r)=nn+pnp(4.2)
toobtainthedensitiesofneutronsandprotons.Bysolving
2nn,p(r)=dfn,p(,µn,p)=d1+exp(−µn,p),(4.3)
Tweobtainthechemicalpotentialsofprotonsandneutronsµnandµp.
Similarly,wedetermineµeby
nBYe=ne−ne+=d(f(,µe)−f(,−µe)).(4.4)
Avanishingelectronfractiondoesnotmeanthattherearenoelectrons
present,butratherthatthereisanequalthermaldistributionofelectrons
.itronsospandAfterallotherchemicalpotentialsareknown,µνeisgivenbyEqua-
(3.16)tionµνe=µe+µp−µn.
Thiscompletesthesetofthermodynamicquantitiesneededforthetransport
ofneutrinosinaSNcore.
4.2Proto-Neutron-StarProfiles
Onepossibilityforourstudyoftheinfluenceofneutrinointeractionsandthe
stellarneutronbsactarakgroundtmospheresmodeltohatntwheereoemergbtainginedbyneutrinoself-spconsistenectratishytodrousedpynamicroto-

4.2.Proto-Neutron-StarProfiles

37

simtionulations.phase;hAsaenceforthfirstewexamplewillwrefereusetoaitamostdelherepresen“Accretion-tativePhasefortheModaelccre-I”
(Fig.4.1).ItwasprovidedtousbyO.E.B.Messerandwasalreadyusedin
Raffelt(2001)foramoreschematicstudy.BasedontheWoosley&Weaver
w15asMpprogeerformednitorwithmodethellSabNelceoddesd15sev7b,eloptedhebyNewtonianMezzacappacollapseetsal.im(ulation2001).
Thesnapshotistakenat324msafterbouncewhentheshockisatabout
120km,i.e.,theproto-neutronstarstillaccretesmatter.Inthissimulation
thescatteringtraditionalonnumcleons,icrophe+ysicse−forνµannihilationtranspoandrtwνase−scincluded,attering.i.e.,iso-energetic
µAsanotherself-consistentexample(Accretion-PhaseModelII,Fig.4.2)
weobtaineda150mspostbouncemodelfromM.Rampp(personalcom-
mincludesunication)anathatpproxusesimateavgeryeneralsimilarrelativipsticrogenitortreatment(s15s7b2).inTsphericalhesimsymme-ulation

1]3 0.1 g/cm 0.0113 0.001 [10ρ 0.00011412108T [MeV]6420.50.4e0.3Y0.20.10.0 20 30 40 50 60 70 80 90 100
Radius [km]

Fig.4.1.—Accretion-PhaseModelI,aSNmodel324msafterbouncefrom
aNewtoniancalculation(O.E.B.Messer,personalcommunication).

38Chapter4.SettingtheStageforNeutrinoTransport

trytranspasodertedscribweithdbayllRamrelevppant&inJankateractions(2002).exceptTheνethreν¯ee→neuνµν¯trinoµflav(Sectionors7are.1
andRamppetal.2002).
Whenweusetheseself-consistentmodelswiththesamemicrophysics
thatwasusedintheoriginalmodelsbyMesserandRamppwefindgood
agreementwiththeirresults.Changingthemicrophysicalinputthenenables
ustostudytheinfluenceofthevariousinteractionprocesses(Chapter3)on
thedoneebemergingfore.Sinceneutrinoweaspreectrasiminulatingasasystematictaticpwaroto-y,nwhiceutronhhasstarnevaerndbalsoeen
varytheinputphysics,oursimulationsarenotselfconsistent.However,once
ourfindingsareimplementedintheself-consistenthydrodynamicsimulations
theyqualitativelyfindthesameresultsaswedo(Burasetal.2003b).
Theself-consistentlyobtainedstellarbackgroundmodelsareintherele-
vofanttemprangeeraturetoagaondoddensitapproyx.Asimationajdifferenustptoawerpproaclawshwforectanhethusradialdepparameterizeendence

10]3 1 g/cm 0.113 0.01 [10ρ 0.0012015T [MeV]1050.400.350.30e0.25Y0.200.150.100.05 20 30 40 50 60 70 80 90 100
Radius [km]

Fig.4.2.—Accretion-PhaseModelII,aSNcoreat150mspostbouncefrom
ageneral-relativisticsimulation.(M.Rampp,personalcommunication).

4.3.OpticalDepthvs.ThermalizationDepth

39

theprofilesinasimplewayandtherebystudytheinfluenceofthestellarpro-
files.Especiallysteepprofilesthatarecharacteristicforthelatephaseofthe
proto-neutronstararenotavailabletousfromself-consistentsimulations.
Weusetwopower-lawprofilesoftheform
ρ=ρ0r0p,T=T0r0q,(4.5)
rrwthatithac≈onstan20–25teMlectroneVfforractiontheepmerergingbaryonneYe.utrinosWetaoodjustbtainpmoarametersdelsatmucoh-
spheresintheballparkofresultsfromproto-neutronstarevolutioncalcula-
bytions.RaffeWeltdefine(2001),aa“ndsteep”a“shpoallower-w”lawone;mothedel,characcorrespterisondingticsaretogtheiveonneinTuseda-
ble4.1.TheshallowmodelcouldbecharacteristicofaSNcoreduringthe
accretionphasewhilethesteepmodelismorecharacteristicfortheneutron-
thatstarcoallowsolingustophase.inveThestigateconsthetantrelativelectroneimpforactionrtanceoYfetisheanotheleptonicrpproaramecessester
asafunctionoftheassumedYe.

Table4.1.Characteristicsofpower-lawmodels.

wShalloSteep510pqq/p2.50.2510.2
ρ0[1014gcm−3]2.00.2
rT00[k[Mme]V]1031.661020.0

4.3OpticalDepthvs.ThermalizationDepth
Bycomparingtheopticaldepthsduetodifferentneutrinointeractionswe
canassesstherelativeimportanceoftheneutrinoprocessesinthestellar
backgroundmodel.Inprincipletheopticaldepthτisdefinedastheintegral
oftheinversemfpλ−1(seee.g.Shapiro&Teukolsky1983,Suzuki1989):
∞∞1τi(r,)=rdrλ(r,)=rdrσi()ni(r),(4.6)
i

40Chapter4.SettingtheStageforNeutrinoTransport

whereistandsforanyinteractionchannel,σiisthecorrespondingcross
section,andnithenumberdensity.Theopticaldepthspecifieshowfrequently
aspecificprocessoccursonaneutrino’swayoutoftheSNcore.
Sincewewanttoknowhowefficientacertainprocessisatexchanging
energywehavetotakeintoaccountthequalitativelydifferentnatureof
theneutrino-matterinteractions.Therelevantinteractionprocessesfallinto
threeclasses:

1.Creationprocesses,thatareabletokeepneutrinosinLTE.They
cancreateandabsorbneutrinosandthereforealsocontributetothe
energyexchangebetweenneutrinosandthebackgroundmedium.

2.Scatteringwithlargeenergytransfer.Inthiscasetheneutrinos
scatteronparticleswithalowmasscomparedtotheneutrino’smo-
mentum(i.e.,e±andneutrinos).

3.Scatteringonnucleons.Heretheenergytransferisverysmallina
singlescattering,Thefactthatallowsfortheeffectivemfpdescription
discussedintheprevioussection.

Allthreeclassescontributetotheopacity,butonlythefirsttwoclassesof
reactionscanexchangelargeamountsofenergyinsingleinteractions.
Makinguseofthisqualitativedifferencebetweennucleonscatteringand
theotherinteractionswecandefinethelocation(thermalizationdepth)
whereaneutrinoofgivenenergylastexchangedenergyefficientlywiththe
mediumbyareactionsuchasνµe−→e−νµ.Lookingattheexampleofνµe−
scatteringwehavetwodifferentmfps,namelyonefornucleonscatteringλT
andonefortheenergyexchangingνµe−processλE.Thetotalmfpisthen
givenbyλtot=1/(λT−1+λE−1).Betweentwoenergyexchangingscatterings
theneutrinoundergoesmanyisoenergeticscatterings.Thisyieldsaneffective
mfpfortheenergyexchangereactionthatislargerthanλE.
Thewayofaneutrinothroughthemediumcanbeconsideredarandom
walk.Therefore√thetraveleddistancescalesasthesquarerootoftimethatis
proportionaltoN,whereNrepresentsthenumberofinteractions.Foreach
reactiontheprobabilityofenergyexchangeisλtot/λE,sotheinverseisthe
numberofreactionsuntilthenextenergyexchange.Altogetherthetraveled
distancefromoneenergy-exchangereactiontotheothercorrespondstothe
effectivemfpoftheenergyexchangingprocess:
λλEEeffλE=λtotλtot=λT−1+λE−1(4.7)

4.3.OpticalDepthvs.ThermalizationDepth

41

Withtheeffectivemfpwecanthencalculatetheopticaldepthforthermal-
ization

τtherm(r,)=rdrλE(r,)λT(r,)+λE(r,).(4.8)
∞111
Whentheopticaldepthbecomeslessthanoforderunity,thecorresponding
processisnolongerrelevant.Thecorrespondingradiusisreferredtoasthe
epth:dthermalization2τtherm(Rtherm)=3,(4.9)
whereRthermdependsontheneutrinoenergy.
Withthisconceptwecandeterminetheenergy-dependentspherewhere
acertainclassofreactionsbecomesinefficient.Foraspecificneutrinoenergy
thecorrespondingradiusisgivenbythelargestRthermofanyreactioninthe
class.Forthefirstclassthisspherecorrespondstothenumbersphere,where
particlecreationfreezesout,asdefinedinSection2.5.Thesecondclassof
reactionsfreezesoutattheradiusthatdeterminestheenergysphere.
Forthethirdclass,i.e.,neutrino-nucleonscattering,wearenotableto
obtainthethermalizationdepth.Ineachreactiontheenergytransferisvery
small,thusmanyreactionsarenecessarytothermalizetheneutrino.Wecan
thereforenotdeterminethelocationofthetransportsphere,i.e.,theradius
atthatneutrino-nucleonscatteringbecomesinefficient.
Asweshowinthenextsection,duetotheenergydependenceitisnot
possibletodefineauniqueneutrinosphere.Evenforνeandν¯e,wherethe
β-processesdominateandthereforetheonlyrelevantsphereistheirnum-
bersphere,itslocationisstronglyenergydependent.Theotherflavorsare
emittedattheirnumbersphere,loseenergyinelasticscatteringuptotheir
transportsphere,andthendiffusebeforetheystartstreamingfreelyfrom
theirtransportsphere.Comparedtoνeandν¯eonewouldrefertothenum-
bersphereasneutrinosphereofνµ.Thespectrumemittedatthisneutrino
sphere,however,isverydifferentfromtheemittedspectrum.
Inordertoillustratetheνµtransportletusconsiderbremsstrahlungand
iso-energeticscatteringonnucleonsonly.Fornucleonbremsstrahlungthe
energydependenceisratherweakandatthesametimethedensitydepen-
denceisverystrong.Therefore,toagoodapproximationonemaypicture
thebremsstrahlung-numbersphereasablackbodysurfacethatinjectsneutri-
nosintothescatteringatmosphereandabsorbsthosescatteredback(Raffelt
2001).Withiso-energeticscatteringonnucleons,theneutrinofluxandspec-
trumemergingfromthetransportsphereistheneasilyunderstoodinterms
oftheenergy-dependenttransmissionprobabilityoftheblackbodyspectrum

42Chapter4.SettingtheStageforNeutrinoTransport

launchedatthenumbersphere,thatisinthissimplepictureequaltothe
energysphere.Thetransportcrosssectionscalesas2,implyingthatthe
transmittedfluxspectrumisshiftedtolowerenergiesrelativetothetemper-
atureattheenergysphere.Thissimple“filtereffect”accountssurprisingly
wellfortheemergingfluxspectrum(Raffelt2001).Fortypicalconditions
themeanfluxenergiesare50–60%ofthosecorrespondingtotheblackbody
conditionsattheenergysphere.
Moreover,itisstraightforwardtounderstandthattheeffectivetempera-
tureoftheemergingfluxspectrumisnotoverlysensitivetotheexactlocation
ofthenumbersphere.Ifthepairprocessesaresomewhatmoreeffective,the
numbersphereisatalargerradiuswithalowermediumtemperature.How-
ever,thescatteringatmospherehasasmalleropticaldepthsothatthehigher-
energyneutrinosarelesssuppressedbythefiltereffect,partlycompensating
thesmallerenergy-spheretemperature.FortypicalsituationsRaffelt(2001)
foundthatchangingthebremsstrahlungratebyafactorof3wouldchange
theemergingneutrinoenergiesonlybysome10%.Thisfindingsuggeststhat
theemittedaverageneutrinoenergyisnotoverlysensitivetothedetailsof
thepairprocesses.
Withthecompletesetofinteractionsneutrinosundergoenergyexchang-
ingscatteringreactionsbelowtheenergyspherebeforetheyenterthedif-
fusiveregime.Thereforethemeanenergyoftheemergingfluxisevenless
sensitivetotheexactlocationofthenumberspherethanfoundbyRaffelt
(2001).InSection6.3weshowthatchangingthebremsstrahlungratebya
factorofthreeupordownhasalmostnoeffectonthemeanenergyofthe
flux.neutrinoemerging

4.4ThermalizationDepthsinOurStellar
delsMo

Wenowturntoapplyingtheconceptofthermalizationdepthstotheneutron-
staratmospheresdescribedabove(Section4.2).Weconsidertheneutrinomfp
fornucleonbremsstrahlungNN→NNνµν¯µ,pairannihilatione+e−→νµν¯µ
andνe¯νe→νµν¯µ,andscatteringonchargedleptonsνµe±→e±νµ.The
reactionratesusedweredescribedinChapter3.
InFigures4.3and4.4wegivethethermalizationdepthRthermasa
functionofneutrinoenergyforthetwohydrodynamicallyself-consistent
accretion-phasemodels.Fromtoptobottomthepanelsshowtheresults
forνe,¯νe,andνµ,respectively.Thestep-likecurvesrepresentthetempera-
tureprofilesintermsofthemeanneutrinoenergy,ν=3.15Tfornon-

4.4.ThermalizationDepthsinOurStellarModels

43

degenerateneutrinosatthelocalmediumtemperature;thestepscorrespond
totheradialzonesofournumericalsimulation.Slightlyabovethesestepsthe
thinlinerepresentsthemeanenergyofneutrinosflowinginradialdirection
obtainedbyournumericalsimulation.TheothercurvesrepresentRthermfor
bremsstrahlung(b),e+e−annihilation(p),νeν¯eannihilation(n),andscat-
teringone±(s).Inthecaseofνeandν¯ewedonotincludebremsstrahlung
andνeν¯eannihilation.Particlecreationisdominatedbythechargedcurrent
reactionsonnucleons(urca).
Forthepower-lawmodelsweshowRthermforνµinFigures4.5and4.6.
Thedifferentpanelscorrespondtotheindicatedvaluesoftheelectronfraction
Ye.NotethatYerepresentsthenetelectrondensityperbaryon,i.e.,thee−
densityminusthatofe+sothatYe=0impliesthatthereisanequalthermal
populationofe−ande+.
Theνµabsorptionrateforthebremsstrahlungprocessvariesapprox-
imatelyas−1,theνµNtransportcrosssectionas2sothattheinverse
mfpforthermalizationvariesonlyas1/2.ThisexplainswhyRthermfor
bremsstrahlungisindeedquiteindependentof.Therefore,bremsstrahlung
aloneallowsonetospecifyaratherwell-definednumbersphere.Theother
processesdependmuchmoresensitivelyonsothatameanenergysphere
ismuchlesswelldefined.
Bothelectronscatteringandtheleptonicpairprocessesaresoineffective
atlowenergiesthattrueLTEcannotbeestablishedevenforastonishingly
deeplocations.Bremsstrahlungeasily“plugs”thislow-energyholesothat
onecanindeedexpectLTEforallrelevantneutrinoenergiesbelowacer-
tainradius.Forhigherenergies,theleptonicprocessesdominateandshift
theenergyspheretolargerradiithanbremsstrahlungalone.Therelative
importanceofthevariousprocessesdependsonthedensityandtemperature
profilesaswellasYe.
Toassesstheroleofthevariousprocessesfortheoverallspectraforma-
tiononeneedstospecifysometypicalneutrinoenergy.Onepossibilitywould
beforneutrinosinLTE.Anotherpossibilityisthemeanenergyofthe
neutrinoflux,inparticularthemeanenergyofthoseneutrinoswhichactu-
allyleavethestar.InFigures4.3–4.6thethinlinerepresentsthisfluxmean
energy.Inthecaseofνµitisclearlyvisiblethatevenafterthelastenergy
exchangingprocess,i.e.,e±scattering,becameineffectivethemeanenergy
ofthefluxstilldrops.Thisdropisduetoscatteringonnucleons.Amongthe
pairprocessesitisalwaystheνeν¯eannihilationprocessthathasthelargest
Rthermintherangeoffluxenergy.However,especiallyinthesteeppower-law
profiletheintersectionofthebremsstrahlung’sandνeν¯eannihilation’sther-
malizationdepthsarerathercloseatthemeanfluxenergy.Thee+e−process
isalwayslessimportantthanνeν¯e,aswasalreadydiscussedinSection3.5.

44

hCapter4.ettingSthetageSforeutrinoNrTpansortInournumericalsimulationswefindthesameresults(Chapter6).
Thefinalshapeoftheνµspectraandthereforealsotheirmeanenergy
cannotdependmuchonanyoftheνµ-creationprocesses.Theenergysphere
representedbyRthermofe±scatteringstretchesinallprofilesfarbeyondthe
otherthermalizationdepths.Additionally,alsothescatteringonnucleons
hasasignificantimpactonthemeanenergies.Itsimportancecomparedto
electronscatteringisverydifficulttoguess.Recall,itisnotpossibletodefine
athermalizationdepthfornucleonrecoils.Bothprocessesarequalitatively
verydifferentinthatneutrinostransferonlyasmallfractionoftheirenergy
pernucleonscattering.Numericallywehavecomparedtheimportanceofall
processesandpresentourresultsinChapter6.

4.4.ThermalizationDepthsinOurStellarModels

νe

−νe

νµ,τ

45

60νe 50 40 [MeV] 30ε 20 10 0 60−νe 50 40 [MeV] 30ε 20 10 0b 60νpµ,τ 50n 40 [MeV]s 30urcaε 20 10 0 20 30 40 50 60 70
[km]RthermFig.4.3.—RthermasafunctionofneutrinoenergyforourAccretion-Phase
νMo.EdelnergyI.Feromxctophangingtobproottomcesses:thepanelsbremsstrahlungshowthe(solidresultsline),feor+eν−e,¯νeannihila-,and
µtion(dashed),νeν¯eannihilation(dotted),andscatteringone±(dash-dotted).
steps“Urca”rdeepresennotetsthe=3ch.15argeT;d-ctuherrenthintrelineactioncloseoftoνe3.15andTν¯eisonthenumcleoeanns.eTnergyhe
ofthefluxflux,cf.Chapter5.

bpnsurca

46

Chapter4.SettingtheStageforNeutrinoTransport

νe

−νe

60νe 50 40 [MeV] 30ε 20 10 0 60−νe 50 40 [MeV] 30ε 20 10 0b 60νpµ,τ 50n 40 [MeV]s 30urcaε 20 10 0 20 30 40 50 60 70 80 90
[km]RthermFig.4.4.—SameasFigure4.3fortheAccretion-PhaseModelII.

bpnsurca

νµ,τ

4.4.ThermalizationDepthsinOurStellarModels

= 0.00Ye

= 0.05Ye

47

60 = 0.00Ye 50 40 [MeV] 30ε 20 10 0 60 = 0.05Ye 50 40 [MeV] 30ε 20 10 0 60 = 0.50Ye 50 40 [MeV] 30ε 20 10 0 12 13 14 15 16 17 18 19 20
[km]RthermvaFig.lueso4.5.—fYe.RTtherhismforfigureνµcinorresthepsondsteeptopothewebr-lawottommopdaneellofwithFigurethe4.3.indicated

= 0.50Ye

48

[MeV]ε

[MeV]ε

[MeV]ε

Chapter4.SettingtheStageforNeutrinoTransport

60 50 40 30 20 10 0 60 50 40 30 20 10 0 60 50 40 30 20 10 0 10

Y = 0.00e

= 0.05Ye

= 0.50Ye

0 30 25 20 15 10 [km]RthermFig.4.6.—SameasFig.4.5fortheshallowpower-lawmodel.

5Chapter

Non-ThermalCharacterizingSpNeutrinoectra

TheneutrinofluxesemergingfromaSNcorearenotthermal.Therearenu-
merouswaystodescribespectralcharacteristicswithafewparameters,for
exampleenergymoments.Formanyapplicationsitisusefultohaveanana-
lyticdescriptionofthespectra.Throughouttheliteratureitiscommontouse
annominalFermi-Diracdistribution.Wepresentasimplertwo-parameterfit
thatdescribesourMonteCarlodatabetter,andcomparebothdescriptions.

5.1GlobalParameters

WhileaSNcorecanbecrudelyunderstoodasablackbodysourceforneu-
trinosofallflavors,theescapingfluxesarenotstrictlythermal.Anumber
ofobservableshavebeendefinedintheliteratureinordertodescribethe
propertiesofthesespectra.
Letusassumef(,µ)isthedistributionofneutrinosasafunctionof
energyandµ=cosθ,whereθistheanglebetweentheradialdirectionand
thedirectionofmotion.FarawayfromtheSNcoreneutrinosstreamfreely
inradialdirection,thereforeµ=1.
Asthemostintuitivequantitycharacterizingthespectrumthemeanen-
used,commonlyisergy

∞+10d−1dµf(,µ)
=0∞d+1−1dµf(,µ).(5.1)
Inthisratiothenumeratoristheenergydensityandthedenominatoristhe
numberdensity.

49

50Chapter5.CharacterizingNon-ThermalNeutrinoSpectra

WithinthenumbersphereneutrinosareinLTE,butintheouterregions
somenetfluxofneutrinosdevelops.Therefore,itisoftenusefultoextract
spectralcharacteristicsforthoseneutrinoswhichareactuallyflowingbyre-
movingtheisotropicpartofthedistribution.Specifically,wedefinethemean
fluxenergyby∞+1
0d−1dµµf(,µ)
flux=0∞d+1−1dµµf(,µ).(5.2)
Outsidethetransportsphereallneutrinoswillflowessentiallyintheradial
direction,implyingthattheangulardistributionbecomesadelta-functionin
theforwarddirectionsothatflux=.Inthetrappingregionsthetwo
averagesareverydifferentbecausethedistributionfunctionisdominatedby
itsisotropicterm.However,formostdiscussionsonlytheemergingspectra
arerelevantandbothdefinitionsofthemeanenergy,Equations(5.1)and
(5.2)areequivalent.TherighthandsideofEquation(5.2)thencorresponds
totheratioofluminositytonumberflux.
Deepinsidethecore,belowthenumbersphereneutrinosareinLTE.
Withavanishingchemicalpotentialtheirdistributionfunctionisgivenby
2f(,µ)=1+exp(/T)(5.3)

hereforetand4=7πT≈3.1514T.(5.4)
ζ1803Weusethisrelationtocomparethenumericallyobtainedmeanenergyto
themediumtemperature,cf.e.g.Figure4.3.
Sincetheemergingspectraarenotthermal,additionalcharacteristics
areuseful.Arathergeneralwaytocharacterizethespectrumbeyondmean
valuesisbyitsenergymoments(Janka&Hillebrandt1989a)
+1∞n0d−1dµf(,µ)
n=0∞d+1−1dµf(,µ).(5.5)
Usually,valuesaregivenonlyupton=2intheliterature,i.e.,themean
energyandthesecondmoment.
Withthefirsttwomomentsaneasywaytoseebyhowmuchaspectrum
deviatesfrombeingthermalisthepinchingparameter,definedbyRaffelt
(2001).Itisbasicallytheratioofthefirsttwomoments
21p≡a2,(5.6)

FitsictAnaly5.2.

51

normalizedsuchthataFermi-Diracdistributionwithvanishingchemicalpo-
tentialyieldsp=1.Therefore,
a≡22FD=4860008ζ3ζ5≈1.3029.(5.7)
FD49π
ForaMaxwell-Boltzmanndistributionthisquantitywouldbe4/3.Aspec-
trumthatisthermaluptoitssecondmomenthasp=1,whilep<1signi-
fiesapinchedspectrum(high-energytailsuppressed),p>1ananti-pinched
spectrum(high-energytailenhanced).
Insomepublicationstheroot-mean-squareenergyrmsisgiveninstead
oftheaverageenergy.Therms-energyinvolvestheratioofthirdandfirst
tmenmoyenerg

∞+1
rms=0∞d−+11dµ3f(,µ)=3.(5.8)
0d−1dµf(,µ)
Tfromhischneutrinosaracteristtoictshepectralstellaremnergyediumisuinserefulforactionseswithtimatingcrosstheesectionsnergyproptransor-fer
2andtionalthetorms-.Howenergyever,iscdifficult,omparingbtheecausemeantheenergyrelationdofefinedbointhEquanquationtities(5.1)de-
pendsonthespectralshape.

itsFnalyticA5.2Forapplicationslikeneutrinooscillationsitisusefultohaveananalyticfitto
theSNneutrinospectra.Wepresentatwo-parameterfitthatreproducesthe
firstfiveormoremomentsofournumericallyobtainedspectratoaverygood
accuracy(Section7.3).Inadditiontotheoverallnormalizationweintroduce
theparametersαand¯bydefining
α1fα()=e−(α+1)/¯,(5.9)
¯c

(5.9)

ormalizationnthewithc=∞dfα()=(α+1)−(α+1)Γ(α+1)¯.(5.10)
0Theaverageenergyisalways=¯,whileαcontrolsthepinchingofthe
function.Justlikethemeanenergyallobservablesdiscussedintheprevious
sectioncanbeexpressedintermsofαand¯bysimpleanalyticrelations.

52Chapter5.CharacterizingNon-ThermalNeutrinoSpectra

1.5>ε 1.4 / <rms 1.3>ε< 1.2 2 3 4 5 0 1 2 3 4 5
ηαFig.5.1.—Theratioofrmsenergyandmeanenergy.Theleftpanelshows
thedependenceonαasgiveninEquation(5.14);intherightpanelweplot
thedependenceonη.

Thepinchingparameterpiscloselyrelatedtothewidth
w=2−2=ap−1=1¯,(5.11)
α1+witha≈1.3029asdefinedinEquation(5.7).Therelationforthepinching
parametercanbereadoffEquation(5.11),
22+α
ap=2=1+α.(5.12)
Forthemoregeneralcaseoftheratiosofarbitraryneighboringmoments
wefindnn+α
n−1=1+α,¯(5.13)
thatcanbeeasilygeneralizedtoarbitraryratiosofmomentswithEqua-
(5.10).tionTakingtheratioofthirdandfirstenergymomentyieldstherms-energy
2rms=(2+α)(3+α).¯(5.14)
)α+(1AsshowninintheleftpanelofFigure5.1,rmsisalwayssome25–45%
largerthanintherangeofα=2–5,thatturnsouttobetheparameter
rangeobtainedbynumericalsimulations.Therefore,thetwodifferentenergy
averagescontaininformationonthespectralshape.

5.2.FitsictAnaly

53

Fromanythreemomentsitispossibletoobtainthenormalization,α,
and¯.Fromallourstudieswefoundthatthespectraareverywelldescribed
bythesethreeparameters.
IntheliteratureonefrequentlyencountersanominalFermi-Diracdis-
tributioncharacterizedbyatemperatureTandadegeneracyparameterη
toaccording2fη()=1+exp−η(5.15)
TasarepresentationoftheSNneutrinospectrum(Janka&Hillebrandt1989a).
Thefunctionalformismotivatedbytheequilibriumdistributionofneutrinos
toinsideintheterpret,star.becausTheevthealuesofemergingthespparametersectraareTnotandη,thermal.however,arenoteasy
Thefunctionalformoffηismorecomplicatedthanfα.Mostcalculations
invFigureolving5.2fηwehasvhoewtob/eTcarriedandpoutasnafuncumericallytionofηand/or.Uptoapprosexcondimatelyor.deInr,

1.0210.98p0.960.940.920.94.003.80 / T >3.60ε3.40< 3.203.00-2-10123
η

Fig.5.2.—Meanenergyandpinchingparameterasafunctionofthedegen-
eracyparameterforaFermi-Diracdistribution.Asdashedlinesweshowthe
expansionsgiveninEquation(5.16).

54Chapter5.CharacterizingNon-ThermalNeutrinoSpectra

expansionsare

/T≈3.1514+0.1250η+0.0429η2
p≈1−0.0174η−0.0046η2.(5.16)
TheseexpansionsareshowninFigure5.2asdashedlines.
Therearenoanalyticexpressionsfortheenergymomentsandtherefore
alsotherms-energycanonlybeobtainednumerically.Forthespecialcaseof
η=0theexpressionis
930rms=441πT≈4.5622T.(5.17)
WithEquation(5.4)thiscorrespondsto≈0.691rms,butrecallthat
SNneutrinospectraarenotthermalspectraandη=0.Thenumerically
obtaineddependenceoftheratioofrmsandmeanenergyisdisplayedinthe
rightpanelofFigure5.1.Itisrathersimilartothecaseoftheαfit.
ofboInthviewoffunctionsanalyticarerathersimplicitsiymilar.theTαhfitenisumericallycertainlysupedeterminedrior,butsptheectrashaparees
similarlywelldescribedbybothfηandfα.Asweshowinthefollowingfor
areasonablerangefortheparametersbothfunctionsdifferbynomorethan
.10%outabIntheupperpanelofFigure5.3weshowfα(),theintegralnormalized
tounity,forseveralvaluesofα.Thebroadestcurveisforα=2whilefor
thedecremennarrotswerasonesshotwnheinwidthTablegiven5.1.inTheEquationmiddle(5.11)panelwaosfdFigureecreased5.3insho10%ws
thecorrespondingcurvesfη()withtheη-valuesgiveninTable5.1.The
broadestcurvesineach√panelareidenticalandcorrespondto2exp(−3/¯)
withawidthw0=/3.

Table5.1.Parametersforfit-functionsofFigure5.3.

αηWidthw0=/√32.−∞
0.9w02.70371.1340
00..87ww003.68755.12254.40142.7054
00..56ww0011.7.333313.8926.9691

5.2.tAnalyicFits55

Fig.5.3.—Normalizedfitfunctions.Upperpanel:α-fitaccordingtoEqua-
tion(5.9).Middlepanel:Fermi-DiracfitaccordingtoEquation(5.15).In
bothpanelsthebroadestcurvecorrespondstof()=2exp(−3/¯),i.e.,to
α=2andη=−∞,respectively.Fortheothercurvesthewidthwasde-
creasedindecrementsof10%,seeTable5.1.Bottompanel:Ratioofthefits
fα/fηforthefirstfourcases.

56

Chapter5.haracterizingCNon-ThermaleutrinoNSpectralimitingThewidthlimitingisbwe0ha/√vior5≈of0f.α(44721)forw0l.arEgevidenαistlyδ(the−¯c)urvwehilesfαfor()fηc(an)tach-e
commoWefinddateathatmucthehbnroadereutrinosprangeectraofwareidthsalwathanysfitthewcithurvpesfηarameters().inthe
<<<<0.range75w02.∼Inαt∼he5obor0ttom∼pη∼anel4(ofSectionFigure75.3.1),wi.e.eswhoithwathewidthratiosaboofvetheabfioutt
functionsforthewidthsdownto0.7w0.Fromthisplotwefindthatexceptfor
tothebleowtteresttehannergies10%.aTndveherefore,ryhightheetnergwotiesyptesheoftwofitsfitafreunctionslargelyeaqreuivequivalentalenfort
mostOnprathectibcalasispurofpaofses.ewhigh-statisticsMonteCarlorunsweshowinSec-
ationb7.3roaderthatrangetheofnuenergiesmericalbsyptheectra“pareowaer-lactuallyw”fitbeftterunctionsapprofxα().imatedInovaddi-er
thetion,sptheseectrumtfunctionshatisamreostrmoreelevanflexibletforatstudyingrepresenthetingEthearthhigh-effectinenergyneutrinotailof
oscillations.

6Chapter

RelativeImportanceof
ReactionstNeutral-Curren

WebrieflyintroduceourMonteCarlocode,thatwasdevelopedtostudy
neutrinotransportinproto-neutron-staratmospheres.Fortheproto-neutron-
staratmospherespresentedinSection4.2weshowourresultsthatwereob-
tainedbyswitchingonandoffvariousneutral-currentinteractionprocesses.
Thetraditionalsetofprocessesincludedinνµtransportturnsouttobe
aratherpoorapproximation.Contrarytowhathadbeenassumedinthe
past,nucleon-nucleonbremsstrahlungandtheνeν¯epairprocessarethemain
sourcesforνµν¯µproduction.Abovethenumberspherethespectraareshaped
byelectronscatteringandscatteringonrecoilingnucleons.Thetraditional
e+e−→νeν¯eprocessandscatteringonνe/¯νeturnouttoberathernegligible.
TheseresultsandthoseofChapter7representamajorpartofthisdis-
sertationandwerealreadypublishedinasimilarformas:
M.T.Keil,G.G.Raffelt,andH.T.Janka,“MonteCarlostudyofsupernova
neutrinospectraformation,”Astrophys.J.inpress,astro-ph/0208035.

6.1MonteCarloSetup

FourortMheontestellarCarlobcacodekgroundusingvmoadelsrioussinetstroofducednineutrino-matterSection4i.2ntweeractionshave.runIn
isApptoendixassessBtheweigivmpacteaoftdetailedhevadriousescriptionneutrinooftinhecoteractionsde.Ouronthemainfluxinterestand
itspisectrumsufficienfttoormation.simulateSincethenneutrinoseutrinoaretrainnspLoTrtEbelostartingwtheslighntumlybbereloswtphere,hat
radiusAtthatwherelowweerhbavetoundaryospofecifyourabproto-oundaryneutron-condition.staratmospherewealways

57

58Chapter6.RelativeImportanceofNeutral-CurrentReactions

imposeablackbodyboundarycondition.WeassumeneutrinostobeinLTE,
i.e.,atthelocaltemperatureandtheappropriatechemicalpotential.Fornu-
onlymericalνµrareunsinvolvediscusseddandinwetakhischmagnetismaptertheislatterdiscussedistakinenthetovnextacnish,hbapter.ecauseAs
aconsequenceofthisinnerboundarycondition,theluminosityemergingat
thesurfaceisgeneratedwithinthecomputationaldomainandcalculatedby
ourMonteCarlotransport.Asmallfluxacrosstheinnerboundarydevelops
becauseofthenegativegradientsoftemperatureanddensityintheatmo-
sphere.Itsmagnitudedependsontheradialresolutionoftheproto-neutron-
sHotarwevatmer,osforpheoreurandsetupsisthethisreforefluxanisartismallfactcofomparedthentotumericalheluminositimplemenytatation.the
surfaceandthereforetheemergingneutrinospectradonotdependonthe
lowerboundarycondition.
Theshallowenergydependenceofthethermalizationdepthofthenu-
cleonbremsstrahlungimpliesthatwheneverweincludethisprocessitisnot
latterdifficulttootocdeephooinseatherstareasonableisveryloCcatiPU-exonforpenstheiveloasweronebspeoundarynds.Tmostakingofthethe
simulationforcalculatingfrequentscatteringsofneutrinosthatareessen-
E.TLintiallyTheouterboundaryisdeterminedbytherequirementthatneutrinos
streamfreely.IntheouterregionsthetransportisnotCPU-expensive.There-
wefore,alwtaheysusexactealocgridationofof30tehequallyouterspbacedoundaryradialisznotonevseryandccanrucial.thusHowevobtainer,
betterspatialresolutionbychoosingtheouterboundaryfairlylow.
WealwaysincludeνµNscatteringasthemainopacitysource.Forenergy
exchange,weswitchonorofftheprocessesgiveninTable6.1.Wenever
includeinelasticnucleonscatteringνµNN→NNνµasthisprocessisnever

Table6.1.Neutral-currentprocessesincludedinourνµtranport.

ProcessLabel
bbremsstrahlungnucleonrecoilr
scatteringone+ande−s
e+e−pairannihilationp
νeν¯eannihilationn
scatteringonνeandν¯esn

6.2.Accretion-PhaseModelI

59

importantrelativetorecoil(Raffelt2001).Forconfirmingthatscatteringon
νeandν¯eisreallyunimportantifνµe±isincluded(Section3.6)weusedthe
Accretion-PhaseModelI.Inallothermodelsweignorethisprocess.Wealso
neglectνµνµorνµν¯µscatteringeventhoughsuchprocessesmayhavealarger
ratethansomeoftheincludedleptonicprocesses.Processesofthistypedo
notexchangeenergybetweentheneutrinosandthebackgroundmedium.
Theyarethereforenotexpectedtoaffecttheemergingfluxesandshould
alsohaveaminoreffectontheemittedspectra.Westudyweakmagnetism
.7Chapterin

6.2Accretion-PhaseModelI

Ourfirstgoalistoassesstherelativeimportanceofdifferentenergy-exchange
processesfortheνµtransport.AsafirstexamplewebeginwithourAccretion-
PhaseModelI.TheresultsfromournumericalrunsaresummarizedinTa-
ble6.2,whereforeachrunwegiveflux,2flux,ourfitparameterαdeter-
minedbyEquation(5.12),andthepinchingparameterpfluxfortheemerging
fluxspectrum,definedinEquation(5.6).Wefittedthetemperatureand
degeneracyparameterofaneffectiveFermi-Diracspectrumproducingthe
samefirsttwoenergymoments,andgivetheluminosity.Onlyflux,2flux,
andtheluminositywereobtainedfromthenumericalsimulations.Allother
characteristicswerecalculatedfromthesefirsttwoenergymoments.
inalBThefioltzmannrstrowcontransptoainsrtthemcalculationuonnbyeutrinoMesser.fluxcThomakaracteristicseacoftonnectionheorig-to
theseresults+we−ranourcodewiththesameinputphysics,i.e.,νµe±scatter-
ing(s)andeeannihilation(p).Thereremainsmalldifferencesbetweenthe
originalspectralcharacteristicsandours.Thesecanbecausedbydifferences
intheimplementationoftheneutrinoprocesses,bythelimitednumberof
energyandangularbinsintheBoltzmannsolver,thecoarserresolutionof
btheorundaryadialgridcondition.inourWeMoninteCterpretarlotheruns,firstatndwobyrowoursosfTimpleableblac6.2kabsodyagreeinglower
sufficientlywellwitheachotherthatadetailedunderstandingofthediffer-
encesisnotwarranted.Henceforthwewillonlydiscussdifferentialeffects
withinourownimplementation.
Inthenextrow(bsp)weincludenucleonbremsstrahlungwhichhasthe
effectofincreasingtheluminositybyasizableamountwithoutaffecting
muchthespectralshape.Thissuggeststhatbremsstrahlungisimportantas
asourceforνν¯pairs,butthatthespectrumisthenshapedbytheenergy-
exchangeinscµµatteringwithe±.Inthenextrow(bp)weswitchoffe±scatter-
ingsothatnoenergyisexchangedexceptbypair-producingprocesses.The

60Chapter6.RelativeImportanceofNeutral-CurrentReactions

spectralenergyindeedincreasessignificantly.However,thebiggestenergy-
roexcwshawnegeeffincludeectinrecoilthes(brp)caatteringndthenregimeisadditionallynucleoen±rscecoil.atteringInthe(brsnp),extbtwotho
loweringthespectralenergiesandalsotheluminosities.
Thepictureofallrelevantprocessesi+s−completedbyaddingνeν¯epair
annihilation(brspn),whichissimilartoeepairannihilation,butafactor
ofeffect2–3mwhicohreisimpounderstortantod(Secintionterms3.5).ofoTurheblaclumkbinosoidytyispictureagainforincthereasneudm,baern
moandveestonergylargersrpheres.adiiInoncethe“n”lowiserswitcpanelhedofoFn,igurether4.3adiatingweseetsurfacehatoRfttherhem
“blackbody”increasesandmorepairsareemitted.Forboth“p”and“n”
Rthermisstronglyenergydependentandthereforeitisimpossibletodefinea
sharpthermalizationradius.Switchingoff“r”again(bspn)showsthatalso
with“n”included,“r”reallydominatesthemeanenergyandshapingofthe
ectrum.ps

Table6.2.MonteCarloresultsforAccretion-PhaseModelI.

Energyexchangeflux2fluxαpfluxTηLν
originalrun17.5388.2.70.975.21.114.4
––sp–16.6362.2.21.015.3−0.315.8
b–sp–16.3351.2.11.025.4−2.219.1
b––p–17.8419.2.11.025.9−1.920.1
br–p–15.1285.3.00.964.31.618.6
bbrrssppn–14.214.4255.264.2.82.700.98.9744.2.31.11.214.817.6
–b––ssppnn16.916.6369.358.2.42.301.99.0055.3.20.40.220.221.7
bbrrss–––n14.014.4251.263.2.62.700.99.9744.3.31.20.617.013.1
–rsp–14.5265.2.80.974.31.213.0
b–rrssnppnn14.714.3269.260.3.12.700.96.9744.2.31.71.216.817.9

Note.—TheEnergyexchangeisspecifiedinTable6.1.Wegive
fluxandTinMeV,2fluxinMeV2,andLνin1051ergs−1.

6.2.Accretion-PhaseModelI

61

Tostudytherelativeimportanceofthedifferentpairprocesses,weswitch
offtheleptonicones(row“brs”)andcomparethistoonlytheleptonicpro-
cesses(row“rspn”).Inthisstellarmodelbothtypescontributesignificantly.
Comparingthen“brsp”with“brsn”showsthatamongtheleptonicprocesses
“n”isclearlymoreimportantthan“p”.
Thelastrow(brsnpn)includesinadditiontoallotherprocessesscattering
onνeandν¯e.AsalreadyshowninSection3.5,thisprocessisabouthalfas
importantasscatteringone±anditsinfluenceontheneutrinofluxand
spectraisnegligible.Weshowthiscaseforcompletenessbutdonotinclude
scatteringonνeandν¯eforanyofourfurthermodels.
InordertoillustratesomeofthecasesofTable6.2weshowintheupper
panelofFigure6.1severalfluxspectrafromhigh-statisticsMonteCarlo
runs.Startingagainwiththeinputphysicsoftheoriginalhydrodynamic

spbspbrspbrspn

] 25-1sp s-1 20bspbrsp MeV 15brspn53 10 5Flux [10 0 0 10 20 30 40 50 60
sp0.03bspbrsp)0.02εbrspnf(0.01

0.00 0 10 20 30 40 50 60
[MeV]ε

Fig.6.1.—High-statisticsspectraforAccretion-PhaseModelIwithdifferent
inputphysicsasinTable6.2.Upperpanel:Differentialparticlefluxes.Lower
panel:Spectranormalizedtoequalparticlefluxes.

62Chapter6.RelativeImportanceofNeutral-CurrentReactions

simannihilationulation(sp)(n).weEachaddofbthesepremsstrahlungrocessesh(b),asasrecoilignifican(r),tandandfinallyclearlyνeν¯veisiblepair
influenceonthecurves.Thepair-creationprocesses(“b”and“n”)hardly
chstronglyangethemospdifiesectralthessphapeectralbutshapeincreas.Inetthehelonwuermpberanelflux,ofwFigurehereas6.1werecoilshow(r)
thesamecurves,normalizedtoequalparticlefluxes.Inthisrepresentation
itisparticularlyobviousthatthepairprocessesdonotaffectthespectral
e.pshaTheverydifferentimpactofpairprocessesandnucleonrecoilshasa
simpleexplanation.Thethermalizationdepthforthepairprocessesisdeeper
thanthatoftheenergy-exchangingreactions,i.e.,thenumbersphereisbelow
theenergysphere.Therefore,theparticlefluxisfixedmoredeeplyinthe
starwhilethespectraarestillmodifiedbyenergy-exchangingreactionsin
phere.atmoscatteringsthe

6.3SteepPowerLaw
Asanotherexamplewestudythesteeppower-lawmodeldefinedinEqua-
tion(4.5)andTable4.1.Thismodelissupposedtorepresenttheouter
layersofalate-timeproto-neutronstarbutwithoutbeinghydrostatically
self-consistent.ItconnectsdirectlywithRaffelt(2001),wherethesamepro-
filewasusedinaplaneparallelsetup,studyingbremsstrahlungandnucleon
recoil.TheresultsofourrunsaredisplayedinTable6.3andagreevery
nicelywiththoseobtainedbyRaffelt(2001),correspondingtoourcases“b”
br”.“andForinvestigatingtheimportanceofleptonicprocesses,werunourcode
withavarietyofneutrinointeractionsandinadditionassumeaconstant
electronfractionYethroughoutthewholestellaratmosphere.Thisassump-
tionissomewhatartificial,butgivesustheopportunitytostudyextreme
casesinacontrolledway.IntherelevantregionYe=0.5yieldsthehighest
possibleelectrondensity.Inadditionwestudytheelectronfractionbeing
oneorderofmagnitudesmaller,Ye=0.05,andfinallytheextremecasewith
anequalnumberofelectronsandpositrons,Ye=0.
Thefirstleptonicprocessweconsiderise+e−pairannihilation.Compar-
ingtherows“bp”withtherow“b”showsanegligibleeffectonthespectrum,
butariseinluminosity.IncreasingYebringstheluminosityalmostbackto
the“b”case,becausetheelectrondegeneracyrisesandthepositrondensity
decreasessothatthepair±processbecomeslessimportant.
Addingscatteringoneforcesthetransportedneutrinostostaycloser
tothemediumtemperature,i.e.,reducestheirmeanenergy.Ofcourse,the

6.3.SteepPowerLaw

63

sci.e.,foratteringhigherrateYeincreaswegeteswloiwtherthespnectralumbeerdnergies.ensityFofortheelectronsluminositandpyostheitronssit-,
weuationswitischomorene±scatteringcomplicated.wewSinceouldtheexpectneutrinoalowerfluxluminositenergiesy.Hodecreasewever,twhenhe
opacityofthemediumtoneutrinosisstronglyenergydependentandlow
theenergnyumberneutrinoflux.sOcannbescaalance,pemothere“eabsp”silythanluminositieshigh-areenergylargerones,comparedincreasingto
ones.bp”“the±againToandcompareinsteadtheswitchscatteringonreconoile(r).withQualitativthatonelyn,uthecleonsene,wrgyeetxcurnohangeff“iss”
±veexcryhangedifferenaltargefromamounthetofearlierenergycas,e.wInhiletheforsscatteringcatteringononneuacleonsntheeutrinoenergycan
exchangeissmall.Butsinceneutrino-nucleonscatteringisthedominant
verysourceoffrequentopacit.Tyhisthatleadskeepstoathestrongerneutrinosinssuppressionidetheinstar,thethehigh-senergycatteringstailaofre
theneutrinospectrumandthereforetoavisiblysmallermeanfluxenergy

Table6.3.MonteCarloresultsforthesteeppower-lawmodel.

EnergyexchangeYeflux2fluxαpfluxTηLν
b––––—25.8962.1.21.11——21.0
br–––—19.5487.2.60.986.00.714.5
b––p–025.4890.1.61.06——23.8
b––p–0.0525.6908.1.61.06——23.2
b––p–0.525.5917.1.41.08——21.6
b–sp–024.2787.1.91.03——24.5
b–sp–0.0523.8753.2.01.02——24.5
b–sp–0.521.3591.2.31.006.8−0.323.1
br–p–0.520.0507.2.70.986.01.016.8
brsp–020.3518.2.90.975.91.419.7
brsp–0.0520.3518.2.90.975.91.419.5
brsp–0.519.6488.2.70.985.91.118.7
brspn020.7535.3.00.965.81.823.9
b×3rspn0.0520.3522.2.70.976.01.324.2
brspn0.0520.6530.3.00.965.91.723.8
b×0.3rspn0.0520.7534.3.10.965.81.823.4
brspn0.519.8499.2.70.975.91.221.4

64Chapter6.RelativeImportanceofNeutral-CurrentReactions

andlowereffectivespectraltemperature,buthigherαandhighereffective
high-degeneracyenergy.Manneutrinosyn(ucleondifferentscatteringfromse,±hoscwever,attering).arenTehededereforetondowneutrinosgradesttahey
oflongerbackathighscattering.eTnergieshisandsuppresseexpestriencehealneutrinoargerofluxpacitysignificanandatly.largeramount
Intherunsincludingbothscatteringreactions(brsp),wefindamixture
ofmeathenefffluxectsenergofey.±andnucleonscatteringsandanenhancedreductionofthe
ergyandFinally,pincaddinghing,thebutnaneutrinoincreasedpairproluminositcessyyieldsasexpalmostectednofromchangetheainnalo-en-
gouscaseinSection6.2.Althoughthisprofileisrathersteep,leptonicpair
processesarestillimportant(Figure4.5).
Inbremsstrahlungordertoewestimatehavetpeherformedsensitivitoneytorunthewithexactthetreatmenbremsstrahlungtofnuclerateon
artificiallyenhancedbyafactorof3,andonewhereitwasdecreasedbya
factor0.3.Allotherprocesseswereincluded.Theemergingfluxesandspectra
arguedindeedindonotSectiondep4end.3.sensitivelyontheexactstrengthofbremsstrahlungas

6.4ShallowPowerLaw
Fosteeprthecaseshalloapplies.wpowAserwelawcan(Taablelready6.4)inferalmostfromtheFiguresame4.6,discussionleptonicasproforcessesthe
areoncemore“p”iompr“n”ortanaret.Thisincluded,leadstandoatomsuchtrongerhigherspectralincreaseopincfthehingneutrinowhene+fluxe−
annihilationisswitchedon.Scatteringone±downgradesthetransported
neutrinofluxbyalargeramount.

6.5Summary
Wefindthattheνµspectraarereasonablywelldescribedbythesimplepic-
tureofablackbodyspheredeterminedbythethermalizationdepthofthe
nucleonicbremsstrahlungprocess,the“filtereffect”ofthescatteringatmo-
sphere,andenergytransfersbynucleonrecoils.Thisisalsotruefortheνµ
fluxincaseofsteepneutron-staratmospheres.Formoreshallowatmospheres
pairannihilation(e+e−andνeν¯e),however,yieldsalargecontributiontothe
emittedνµfluxande±scatteringreducesthemeanfluxenergysignificantly.
Itisthereforeimportantforstate-of-thearttransportcalculationstoinclude
theseleptonicprocesses.Thetraditionalprocesse+e−→νµν¯µisalwayssub-

amSum6.5.ry

65

dominantcomparedtoνeν¯e→νµν¯µ.Therelativeimportanceofthevarious
reactionsdependsonthestellarprofile.
ingNeutrinosatmosphereemittedwithoutfrromecoilsablacandklbodyeptonicsuprorfaceacessesndhavfilteredeanbayntai-spinccatter-hed
spectrum(Raffelt2001).However,afterallenergy-exchangingreactionshave
beenincludedwefindthatthespectraarealwayspinched.Whendescribed
byourα-fit,αrangesfromabout2.5to3.5.ForaneffectiveFermi-Dirac
distribution,thenominaldegeneracyparameterηistypicallyintherange
1–2,dependingontheprofileandelectronconcentration.
Thetraditionalsetofneutrinointeractionsthatiscommonlyincluded
intrahydroformation.dynamicNucleonSNsimrecoilsulationsissignificnotantlysufficienlowtertothemaccouneantfeornergyνµ,spandec-
bremsstrahlungaswellastheνeν¯epairprocesscausehigherluminosities
comparedtoresultsobtainedwiththetraditionalinteractions.

Table6.4.MonteCarloresultsfortheshallowpower-lawmodel.

EnergyexchangeYeflux2fluxαpfluxTηLν
b––––—27.71120.1.21.12——20.3
bb–r––p–––0—27.720.1974.521.2.72.500.98.9986.3.31.00.443.113.4
bb––––pp––00.5.0527.928.31019.990.2.72.700.98.9888.3.51.11.043.338.3
bb––sspp––00.0525.425.5815.830.2.82.600.97.9877.5.61.21.146.346.2
b–sp–0.523.5706.2.60.987.11.044.8
br–p–0.522.5624.3.30.956.12.233.1
brsp–022.3612.3.30.956.12.139.6
brsp–0.0522.2609.3.20.956.12.139.1
bbrrssppn–00.522.221.7608.585.3.33.100.94.9566.0.12.21.954.739.2
bbrrssppnn00.5.0521.822.4587.615.3.33.400.95.9466.1.11.92.351.354.9

66

hCrapte6.RleeativmIportanceofeutral-CNrrenutReactions

7Chapter

ComparisonofAllFlavors

Thenewenergy-exchangechannelsstudiedintheprevioussectionlowerthe
averageνµenergies.Inordertocomparetheνµfluxesandspectrawiththose
ofνeandν¯eweperformanewseriesofrunswhereweincludethefullsetof
relevantmicrophysicsforνµandalsosimulatethetransportofνeandν¯e.
Ourfindingsforνeandν¯eagreewiththepreviousliterature,whereasour
resultsforνµshowlowermeanenergies.Themeanenergiesof¯νeandνµare
almostequal.Theνµluminositycandifferfromthatofνeandν¯ebyuptoa
factorof2.Weakmagnetismcausesadifferencebetweenthemeanenergies
ofνµandν¯µofafewpercent.
Wepresentanoverviewoftheresultsobtainedbyothergroupsandfind
thatthemorerecentsimulationsagreeratherwellonthedifferentialeffectsin
energiesandluminositiesofthedifferentflavors.Thesefindingsaswellasour
resultssuggestthatassumptionscommonlymadeforcalculatingoscillation
effects,i.e.,equalluminositiesandadifferenceinthemeanenergiesofν¯eand
ν¯µofuptoafactorof2,needtobechanged.
Weshowresultsofthedetailedspectralshapesobtainedinoursimula-
tions.OurαfitdescribesthespectraslightlybetterthananominalFermi-
distribution.raciDPartsofthischapterwerealreadypublishedinasimilarformas:
M.T.Keil,G.G.Raffelt,andH.T.Janka,“MonteCarlostudyofsupernova
neutrinospectraformation,”Astrophys.J.inpress,astro-ph/0208035.

7.1ResultsfromOurMonteCarloStudy

Forourνeand¯νetransportweemploythesamemicrophysicsasinJanka&
Hillebrandt(1989a,b),i.e.,charged-currentreactionsofe±withnucleons,iso-
energeticscatteringonnucleons,scatteringone±,ande+e−pairannihilation.

67

68

Chapter7.ComparisonofAllFlavors

Inprincipleoneshouldalsoincludenucleonbremsstrahlungandtheeffectof
nucleonrecoilsforthetransportofνeandν¯e,buttheireffectswillbeminimal.
Therefore,wepreferredtoleavetheoriginalworkingcodeunmodifiedfor
theseflavors.
InthefirstthreerowsofTable7.1wegivethespectralcharacteristics
fortheAccretion-PhaseModelIfromtheoriginalsimulationofMesser.The
usualhierarchyofaverageneutrinoenergiesisfound,i.e.,νe:ν¯e:νµ=
0.86:1:1.20.Theluminositiesareessentiallyequalbetween¯νeandνewhile
νµ,¯νµ,ντ,andν¯τeachprovideabouthalfoftheν¯eluminosity.
OurMonteCarlorunsofthisprofileestablishthesamepictureforthe
sameinputphysics(label“sp”).Althoughourmeanenergiesareslightly
offsettolowervaluesforallflavorsrelativetotheoriginalrun,ourenergies
relativetoeachotherareνe:ν¯e:νµ=0.84:1:1.19andthusvery
similar.However,onceweincludeallenergyexchangingprocesses(brspn)
wefind0.84:1:1.02instead.Therefore,νµnolongerexceedsν¯eby
much.Theluminosityofνµisabouthalfthatofνeorν¯ewhichareapproxi-
matelyequal,inroughagreementwiththeoriginalresults.Eventhoughthe
additionalprocesseslowerthemeanenergyofνµtheyyieldamorethan10%
higherνµluminosity,mainlyduetoνeν¯eannihilation.
Differentratesforneutrinosandanti-neutrinosariseduetotheweak
magnetismcorrectionoftheneutrino-nucleonscatteringcrosssection(Sec-
tion3.3).Thenexttwolines(wm)showresultsfromrunswhereweincluded
weakmagnetisminourνµtransport.Weakmagnetismcausesasignificant
correctiontotheneutrino-nucleoncrosssectionthatmainlyarisesdueto
thelargeanomalousmagneticmomentsofprotonsandneutrons(Vogel&
Beacom1999,Horowitz2002).Itincreasestheneutrinointeractionratebut
lowerstherateforanti-neutrinos.Itisexpectedtobeasmallcorrectionin
theSNcontext,buthasneverbeenimplementedforνµandν¯µsofar.Fol-
lowingHorowitz(2002)weaddweakmagnetismtoournucleon-recoilrate
asgiveninSection3.3.
OurMonteCarlocodetransportsonlyonespeciesofneutrinosatatime.
Inordertotesttheimpactofweakmagnetismweassumedthatachemical
potentialforνµwouldbuildup,andassumedafixedvaluefortheνµdegen-
eracyparameterthroughoutourstellarmodel.Wetheniteratedseveralruns
forνµandν¯µwithdifferentdegeneracyparametersuntiltheirparticlefluxes
wereequalbecauseinastationarystatetherewillbenonetfluxofµ-lepton
.rembnuWeperformedthisprocedureforourAccretion-PhaseModelI.Themean
energyofνµgoesdownby1%andgoesupforν¯µby4%.Themeanlumi-
nositieschangeinoppositedirectionsbyabout2–3%.Sinceweadjustedthe
particlefluxesbyhandinordertofixthechemicalpotential,wecannotclaim

7.1.ResultsfromOurMonteCarloStudy

Table7.1.ComparingMonteCarloresultsfordifferentflavors.
Accretion-PhaseModels

69

Flavorflux2fluxfluxαpfluxTηLν
ν¯eflux
Accretion-PhaseModelI
Originalνµ,¯νµ17.5388.1.202.70.975.21.114.4
ν¯e14.6253.14.40.913.53.429.2
νe12.5190.0.863.60.933.22.830.8
unsrOurνµ,¯νµ(sp)16.6362.1.192.21.015.3−0.315.8
νµ,¯νµ(brspn)14.3260.1.022.70.974.31.217.9
νµ(wm)14.2254.1.012.90.964.11.617.4
ν¯µ(wm)14.9281.1.062.80.974.31.518.3
¯νe14.0237.13.80.933.62.731.7
νe11.8175.0.842.90.973.41.431.9
Accretion-PhaseModelII
Originalνµ,¯νµ17.2380.1.092.50.985.20.832.4
ν¯e15.8300.14.00.924.03.068.1
νe12.9207.0.823.10.963.71.765.6
unsrOurνµ,¯νµ15.7317.1.022.50.984.80.827.8
ν¯e15.4283.14.20.923.83.273.5
νe13.0207.0.843.40.953.62.173.9

Note.—WegivefluxandTinMeV,2fluxinMeV2,Lνin1051ergs−1.

70

40 30 [MeV] 20ε 10 40 30 [MeV] 20ε 10 40 30 [MeV] 20ε 10

Chapter7.ComparisonofAllFlavors

‹›ε›‹εflux

νe

−νe

νµ,τ

100 80 60 40 20Radius [km]tinFig.uous7.1.—linesCsomhowparisouronMonoftteheCAcarlocretrunsion-PwhilehaseMcrossesodelrIcepresenalctulationstheo.Criginalon-
sithemlowulationerbpanelyMresseepresenr.Ttheourstepsfindingscorresapfterondtoincluding=3the.f15ullT.setTohinfrelevlinesanitn
interactionprocesses.

aneffectontheluminosities.
InFigure7.1wecompareourcalculationsfortheAccretion-PhaseModelI
withthoseoftheoriginalsimulation.Thestep-likecurverepresentsthemean
energyofneutrinosinLTEforzerochemicalpotential.Thesmoothsolidline
isthemeanenergyfromourruns,thedottedlinegivesflux.Thecrosses
arethecorrespondingresultsfromtheoriginalruns.Inthecaseofνµ(lower
panel)weshowtheresultsofour“sp”runasthicklinesandtheresultsofour
“brspn”run,i.e.,thecompletesetofinteractionsincluded,asthinlines.The
boundariesofoursimulationdomainwerethesameforallflavors,namely
Rin=20kmandRout=100km.
Asanotherexampleofanaccretingproto-neutronstarweusethe

7.1.ResultsfromOurMonteCarloStudy

71

Accretion-PhaseModelII.Theneutrinointeractionsincludedinthismodel
werenucleonbremsstrahlung,scatteringone±,ande+e−annihilation.Nu-
cleoncorrelations,effectivemass,andrecoilweretakenintoaccount,fol-
lowingBurrows&Sawyer(1998,1999),aswellasweakmagnetismeffects
(Horowitz2002)andquenchingofgAathighdensities(Carter&Prakash
2002).Alltheseimprovementstothetraditionalmicrophysicsaffectmainly
νµandtosomedegreealsoν¯e.Weakmagnetismtermsdecreasethenucleon
scatteringcrosssectionsforν¯µmorestronglythantheymodifyνµscatterings.
Inthishydrodynamiccalculation,however,νµandν¯µweretreatedidentically
byusingtheaverageofthecorrespondingreactioncrosssections.Theeffects
ofweakmagnetismonthetransportofνµandν¯µarethereforenotincluded
toveryhighaccuracy.Note,moreover,thattheoriginaldatacomefroma
generalrelativistichydrodynamicsimulationwiththesolutionoftheBoltz-
mannequationforneutrinotransportcalculatedinthecomovingframeof
thestellarfluid.Thereforetheneutrinoresultsareaffectedbygravitational
redshiftand,dependingonwheretheyaremeasured,mayalsobeblueshifted
byDopplereffectsduetotheaccretionflowtothenascentneutronstar.
OurMonteCarlosimulationincontrastwasperformedonastaticback-
groundwithoutgeneralrelativisticcorrections.Itincludesbremsstrahlung,
recoil,e+e−pairannihilation,scatteringone±,andνeν¯eannihilation,i.e.,our
microphysicsissimilarbutnotidenticalwiththatusedintheoriginalrun.
Asanouterradiuswetook100km;allfluxparametersaremeasuredatthis
radiusbecausefartheroutDopplereffectsoftheoriginalmodelwouldmake
itdifficulttocomparetheresults.Keepinginmindthatweuseverydifferent
numericalapproachesandsomewhatdifferentinputphysics,theagreement
inparticularforνeandν¯eisremarkablygood.Thisagreementshowsonce
morethatourMonteCarloapproachlikelycapturesatleastthedifferential
effectsofthenewmicrophysicsinasatisfactorymanner.
FortheAccretion-PhaseModelIIweshowthecomparisonwithourruns
inFigure7.2.Inthiscase,forthetransportofνµourinnerboundaryis
Rin=16km,whileforνeandν¯eweuseRin=24km.Forνeandν¯ethe
charged-currentprocesses(urca)keeptheseneutrinosinLTEuptolarger
radiithanpairprocessesinthecaseofνµ.WithourchoiceofRintheneutrinos
areinLTEwithintheinnermostradialzones.Inthisprofileourchoiceof
boundariesreducestheCPUtimeneededanddoesnotaffecttheresults.As
statedbeforeouterboundarieswerechosentobeatRout=100km.
TheresultsaresimilartotheAccretion-PhaseModelI.Theluminosities
arenotequipartitionedbutinsteadfollowroughlyLνe≈Lν¯e≈2Lνµ.The
ratiosofmeanenergiesareνe:ν¯e:νµ=0.82:1:1.09intheoriginal
runand0.84:1:1.02inourrun.
Insummary,bothaccretion-phasemodelsagreereasonablywellinthe

72

40 30 [MeV] 20ε 10 40 30 [MeV] 20ε 10 40 30 [MeV] 20ε 10

Chapter7.ComparisonofAllFlavors

‹›ε›‹εflux

νe

−νe

νµ,τ

100 80 60 40 20Radius [km]Fig.7.2.—SameasFigure7.1fortheAccretion-PhaseModelII.

findsνe:ν¯esomethingratiolikforeallν¯r:uns.νMoreo=1:1ver,.20.usingDepetraditionalndingoninputtheiphmplemenysicsotnea-
µetionofthenewinputphysicsanddependingonthemodelonefindsresults
betweenν¯e:νµ=1:1.02and1:1.09.
Inordertoestimatethecorrespondingresultsforlaterstagesoftheproto-
powneutronerqsoftartheevtoemplutionewratureeemploprofileyowurithinsteeparpoweeasonabler-lawmorangedel.soWtehatvaryq/pthe=
0.roughly25–0.35,equalwithnqumbanderpfluxesdefinedforνinEandν¯quationb(ecause4.5).aYefewissfixedecondsbyafterbdemandingounce
eedeleptonizationshouldbeessentiallycomplete.Thefluxesoftheseneutrinos
dependverysensitivelyonYesothatthisconstraintisonlyreachedtowithin
about30%withouttuningYetothreedecimalplaces.However,themean
energiesareratherinsensitivetotheexactvalueofYe.Thisisillustrated
inwherethewfierstshoswectionresultsoffTorableYe7.2=0.by15theand0steep.20.pTohweenr-umlawbermodfluxeleswithofνeqand=2ν¯.e5

7.1.ResultsfromOurMonteCarloStudy

Table7.2.ComparingPoMonwert-eLaCwarloModeresultslsfordifferentflavors.

73

FlavorYeflux2fluxν¯fluxfluxαpfluxTηLν
eSteepPowerLawp=10
5.=2qνµ,¯νµ0.1520.4525.1.102.80.965.91.523.5
ν¯e0.1518.5413.13.80.924.63.023.5
νe0.1512.7198.0.693.40.943.42.412.8
νµ,¯νµ0.220.4521.1.143.00.975.91.523.3
ν¯e0.217.9383.14.10.924.43.111.7
νe0.213.4218.0.753.70.933.42.924.4
0.=3qνµ,¯νµ0.117.7393.1.142.90.965.01.812.7
¯νe0.115.5289.13.90.934.02.88.8
νe0.110.5132.0.684.10.922.63.06.6
5.=3qνµ,¯νµ0.0715.8310.1.223.10.954.42.17.9
νµ(wm)0.0715.5296.1.193.30.944.22.37.7
ν¯µ(wm)0.0716.5337.1.273.20.954.52.18.3
ν¯e0.0713.0207.13.40.943.52.34.3
νe0.079.4103.0.725.00.902.13.94.1
ShallowPowerLawp=5,q=1
νµ,¯νµ0.322.0596.1.143.30.946.02.253.9
ν¯e0.319.3440.14.50.914.53.785.7
νe0.314.7262.0.763.70.933.82.756.5

74

Chapter7.ComparisonofAllFlavors

Atdifferthebysamelesstthanime,the30%avforeYragees=0p.15,ectralbutenergiesdifferbbyaarelyfcactorhange.of3forYe=0.2.
Theratiosofmeanenergiesarenotverydifferentfromthoseofthe
calmaccretion-eaningphasebecausemodels.weaOfcdjustedourse,thethesteallarbsoluteprofilefluxinorderenergiestoohavbtainenorphealisticysi-
values.FortheluminositieswefindLνe<Lνµ,differentfromtheaccretion
phase.Thesteeppowerlawimpliesthattheradiatingsurfacesaresimilarfor
allflavorssothatitisnotsurprisingthattheflavorwiththelargestenergies
alsoproducesthelargestluminosity.
bFecauseoroturhissteepshouldestpprobrofileehwoewmalsoucphespecrformedtrarareunsaffectedincludinginwlate-eaktimepmagnetism,rofiles.
AsalreadyfoundfortheAccretion-PhaseModelI,themeanenergyofν¯µgoes
upby4%,whereasinthecaseofνµitdecreasesby1%.Themeanluminosities
arealmostunaffected.Weconcludethatweak-magnetismcorrectionsare
small.Transportingνµandν¯µseparatelyinaself-consistenthydrodynamic
simulationisprobablynotworththecostincomputertime.
Wefindthatνµalwaysexceedsν¯ebyasmallamount,theexactvalue
dependingonthestellarmodel.Duringtheaccretionphasetheenergiesseem
tobealmostidentical,latertheymaydifferbyupto20%.Wehavenotfound
amodelwheretheenergiesdifferbythelargeamountswhicharesometimes
assumedintheliterature.AtlatetimeswhenYeissmallthemicrophysics
governingν¯etransportisclosertothatforνµthanatearlytimes.Therefore,
oneexpectsthatatlatetimesthebehaviorof¯νeismoresimilartoνµthanat
ofeaerlynetrgieimes.satWelatedontimotesseeforasneylf-cargonsuismentetntfosrteexpllarmectingodelas.nextremehierarchy
Weneverfindexactequipartitionoftheflavor-dependentluminosities.
Dependingonthestellarprofilethefluxescanmutuallydifferbyuptoa
factorof2ineitherdirection.

7.2PreviousLiterature
StudiesofoscillationeffectsinSNneutrinospectrausuallyassumeexactly
equalneutrinoluminositiesforallflavorsandastronghierarchyforthemean
energies.Oftentimestheyassumethedifferencebetweenthemeanenergies
ofν¯eandν¯µtobeaboutafactorof2.Wefindanalmostorthogonalpicture
inoursimulations.Wheredoesthiscomefrom?
Tothebestofourknowledge,exceptintheveryrecentmodelsbyBuraset
al.(2003b),themicrophysicsemployedforνµtransportwasroughlythesame
inallpublishedsimulations.Itincludediso-energeticscatteringonnucleons,
e+e−annihilationandνµe±scattering.Ofcourse,thetransportmethodand

LiteratureiousPrev7.2.

75

thenumericalimplementationoftheneutrinoprocessesdifferinthecodes
ofdifferentgroups.Thenewresultsofself-consistentcalculationspointin
thesamedirectionasthefindingsfromourMonteCarlostudiespresented
inthepreviouschapter.Thereforethesefindingsarealmostorthogonalto
thepreviousassumptionsoscillationstudiesarebasedon.Ourrepresentative
sampleofpertinentresultsissummarizedinTable7.3.Wehavealsoincluded
ourAccretion-PhaseModelsIandIIandaveryrecentmodelbytheGarching
groupincludingallrelevantprocesses.Notethatthesimulationsdiscussed
belowdidnotinallcasesusethesamestellarmodelsandequationsofstate
forthedensematterintheSNcore.
WebeginwiththesimulationsoftheLivermoregroupwhofindrobust
explosionsbyvirtueoftheneutron-fingerconvectionphenomenon.Neutrino
transportistreatedinthehydrodynamicmodelswithamultigroupflux-
limiteddiffusionscheme.Mayle,Wilson,&Schramm(1987)gavedetailed
resultsfortheirSNsimulationofa25Mstar.Forhalfasecondafter
bouncetheyobtainedasomewhatoscillatorybehavioroftheneutrinolu-
minosities.Afterthepromptpeakoftheelectronneutrinoluminosity,they
gotLνe≈Lν¯e≈2Lνµ≈50–130×1051ergs−1.Afteraboutonesecondthe
valuesstabilize.Thiscalculationdidnotproducethe“standard”hierarchy
ofenergies.However,thereisclearlyatendencythat¯νebehavemoresimilar
toνµatlatetimes.
ThemostrecentpublishedLivermoresimulationisa20Mstar(Totani
etal.1998).Itshowsanastonishingdegreeofluminosityequipartitionfrom
theaccretionphasethroughouttheearlyKelvin-Helmholtzcoolingphase.
Abouttwosecondsafterbouncetheνµfluxfallsoffmoreslowlythanthe
otherflavors.InTable7.3weshowrepresentativeresultsforanearlyanda
latetime.Themeanenergiesandtheirratiosareconsistentwithwhatwe
wouldhaveexpectedonthebasisofourstudy.
Withadifferentnumericalcode,Bruenn(1987)foundfora25Mpro-
genitorqualitativelydifferentresultsforluminositiesandenergies.Atabout
0.5safterbouncetheluminositiesandenergiesbecamestableatthevalues
giveninTable7.3.Thissimulationisanexampleforanextremehierarchy
ofmeanenergies.
InBurrows(1988)allluminositiesaresaidtobeequal.Inadditionitis
statedthatforthefirst5secondsνµ≈24MeVandtherelationtothe
otherflavorsisνe:ν¯e:νµ=0.9:1:1.8.Detailedresultsareonly
givenforν¯e,sowearenotabletoaddthisreferencetoourtable.Thelarge
varietyofmodelsinvestigatedbyBurrows(1988)andthedetailedresults
forν¯egobeyondthescopeofourbriefdescription.InalaterpaperMyra
&Burrows(1990)studieda13Mprogenitormodelandfoundtheextreme
hierarchyofenergiesshowninourtable.

76

literature.hetromfaracteristicshcfluxtndenedeporvFla7.3.bleaT

Chapter7.ComparisonofAllFlavors

µνLeν¯L20202092.20202026.1.500.525.165308.16303085.652203014.34415.0.07.300.390.33315.0.3.400.406.33308.0.3.300.300.
eνLν¯νe
:0:105.:1:108.:1:155.:2:138.:1:158.:1:175.:1:137.:0:108.:1:137.:1:198.:1:157.:1:188.
00000000659213151524µ1212ν425412131313e1112ν¯tpb025201110090182009.59815088715
922.5.58011e111ν99.30.130.50.31.01110

(1987)l.ateelyaMetanioT1998)(al.t(1987)nnBrue(1990)swurroB&rayM(1989b)Hillebrandt&aJank1989)(iukSuz1991)(iukSuz1993)(iukSuz

14293120.18323202.32686609.28747402.444302.32
:1:168.:1:148.:1:128.:1:148.:1:158.
01815130.32original)(IledoMeashion-PtercAcashion-PtercAc01414120.32un)rour(IledoMe01716130.15(original)IIledoMeashion-PtercAc01615130.15run)(ourIIledoMeashion-PtercAc016.816.514.1.250).momcnalosre(pl.ateBuras

iousPrev7.2.Literature

uedintCon7.3—bleaT

µνLeν¯LeνL

µνLeν¯L8252526.10303014.
eνLeνν¯:1:148.:1:109.
µνeν¯νefoeadinst
rms910.5(2001)l.aterorfe04212
tpbwsholineswingollofThe04291610.52001)(al.etMezzacappaend¨Lieb

.1−serg5110inνLandMeV,ins,in(tpb)ouncebostpimettheegiveW—Note.

77

78

Chapter7.ComparisonofAllFlavors

WiththeoriginalversionofourcodeJanka&Hillebrandt(1989b)per-
formedtheiranalysesfora20Mprogenitorfromacore-collapsecalculation
byHillebrandt(1987).Ofcourse,likeourpresentstudy,thesewereMonte
Carlosimulationsonafixedbackgroundmodel,notself-consistentsimula-
tions.Takingintoaccountthedifferentmicrophysicsthemeanenergiesare
consistentwithourpresentwork.Themeanenergiesofνeweresomewhaton
thelowsiderelativetoν¯eandtheν¯eluminositywasoverestimated.Bothcan
beunderstoodbythefactthatthestellarbackgroundcontainedanoverly
largeabundanceofneutrons,becausethemodelresultedfromapost-bounce
calculationwhichonlyincludedelectronneutrinotransport.
Suzuki(1989)studiedmodelswithinitialtemperatureanddensitypro-
filestypicalofproto-neutronstarsatthebeginningoftheKelvin-Helmholtz
coolingphaseabouthalfasecondafterbounce.Heusedtherelativelystiff
nuclearequationofstatedevelopedbyHillebrandt&Wolff(1985).Inour
tableweshowtheresultsofthemodelC12.FromSuzuki(1991)wetook
themodellabeledC20whichincludesbremsstrahlung.ThemodelC48from
Suzuki(1993)includesmultiple-scatteringsuppressionofbremsstrahlung.
Suzuki’smodelsaretheonlyonesfromthepreviousliteraturewhichgobe-
yondthetraditionalmicrophysicsforνµtransport.Itisreassuringthathis
ratiosofmeanenergiescomeclosesttotheoneswefind.
Overthepastfewyears,firstresultsfromBoltzmannsolverscoupled
withhydrodynamicsimulationshavebecomeavailable,forexampletheun-
publishedonesthatweusedasourAccretion-PhaseModelsIandII.For
convenienceweincludetheminTable7.3.Further,weincludeaveryrecent
accretion-phasemodeloftheGarchinggroup(Burasetal.,personalcommu-
nication)thatincludesthefullsetofmicrophysicalinput.Finally,weinclude
twosimulationssimilartotheAccretion-PhaseModelI,onebyMezzacappa
etal.(2001)andtheotherbyLiebend¨orferetal.(2001).Theselatterpapers
showrmsenergiesinsteadofmeanenergies.Recallingthattheformertendto
beabout45%largerthanthelattertheseresultsareentirelyconsistentwith
ouraccretion-phasemodels.Moreover,theratiosofrmstendtoexaggerate
thespreadbetweentheflavor-dependentmeanenergiesbecauseofdifferent
amountsofspectralpinching,i.e.,differenteffectivedegeneracyparameters.
ToillustratethispointwetakethefirsttworowsfromTable7.1asanex-
ample.Theratioofthemeanenergies1=17.5MeVand2=14.6MeV
is1/2=1.19.UsingEquation5.14forα1=2.7andα2=4.4theratio
ofrmsenergiesequals1.31.
Tosummarize,thefrequentlyassumedexactequipartitionoftheemitted
energyamongallflavorsappearsonlyinsomesimulationsoftheLivermore
groupandsomebySuzuki.Wenotethattheflavor-dependentluminosities
tendtobequitesensitivetothedetailedatmosphericstructureandchemical

7.3.eShapectralSp

79

mecompanenosition.ergiesOnwastheonlyotherfoundhand,inthetheoearlyften-asimssumedulationseoxtremefBruenhierarcn(hy1987)of
andofMyra&Burrows(1990),possiblyaconsequenceoftheneutron-star
equationofstateusedinthesecalculations.
Ifweignoreresultswhichappeartobe“outliers”,thepictureemerging
fromTable7.3isquiteconsistentwithourownfindings.Fortheluminosi-
ties,direction,typicallydepeLνending≈Lonν¯etheandeavofactlutionaorofry2–3bphase.etweFeornthethismandeanLeνµinnergieseitherwe
reTheadtmyorpicealrecenratiostsinimulathetiornsangeinvofolvingνe:aBν¯oel:tzmaνµnnso=0lv.ers8–0.s9:how1:a1.co0–1.3.nsis-
tentbehaviorandwillinfutureprovidereliableinformationaboutneutrino
fluxesandspectra.Thereisacleartendencythattheinclusionofallrelevant
microphysicsdecreasesthehierarchyofmeanenergies.
AnalyzingdetectedneutrinospectrafromafuturegalacticSNwillde-
mandessaryveforryboth,accurateapossiblepredictionsfmeasuremenromseltf-ofnconsisteneutrinotsimoscillationulations.Thisparametersisnec-
andalsofortestingthecurrentSNparadigm.

7.3SpectralShape
Thusfarwehavecharacterizedtheneutrinospectrabyafewsimpleparame-
ters.However,itisextremelyusefultohaveasimpleanalyticfittotheoverall
spectrumthatcanbeused,forexample,tosimulatetheresponseofaneu-
trinodetectortoaSNsignal.Tostudythequalityofdifferentfitfunctions
wehaveperformedafewhigh-statisticsMonteCarlorunsfortheAccretion-
PhaseModelIandtwoofthesteeppower-lawmodels(p=10,q=2.5and
3.5)includingallinteractionprocesses.Moreover,wehaveperformedthese
runsfortheflavorsνe,¯νe,andνµ.
Inordertogetsmoothspectralcurveswehaveaveragedtheoutputof
70,000timesteps.Inadditionwehaverefinedtheenergygridoftheneutrino
interactionrates.Bothmeasuresleavethepreviousresultsunaffectedbut
increasecomputingtimeanddemandformemorysignificantly.
InFigure7.3weshowourhigh-statisticsMonteCarlospectra(MC)for
theAccretion-PhaseModelItogetherwiththeα-fitfunctionfα()definedin
Equation(5.9)andtheη-fitfunctionfη()ofEquation(5.15).Theanalytic
functionscanonlyfitthespectrumwelloveracertainrangeofenergies.We
havechosentooptimizethefitfortheeventspectruminadetector,assuming
thecrosssectionscaleswith2.Therefore,weactuallyshowtheneutrinoflux
spectramultipliedwith2.Accordingly,theparametersαand¯,aswellasη
andTandthenormalizationsaredeterminedsuchthattheenergymoments

80

Chapter7.ComparisonofAllFlavors

432,Below,aenacdhspareectrumreproweshoducedwtbheytheratiofitso.fourMonteCarloresultswith
thefitfunctions.Intheenergyrangewherethestatisticsinadetectorwould
bboethretasypesonableoffifortsarepresengalacticttSheN,saMonyfteromCarlo5–10MresultseVupnicelyto.aHoroundweve40r,inMeallV,
casestheα-fitworkssomewhatbetterthantheη-fit.
Wehaverepeatedthisexerciseforthesteeppower-lawmodelswithq=
2.resp5andectivelythe.oTheneqwithualitqyo=3ft.5heafitsndisshocwtheomparableresultstointheFpiguresrevious7.5eandxample.7.4,

7.4Summary
Inofalltcomparingheconsideredthespmectraodofels.allnOnceeutrinoallrelevflavoanrstiwnetfinderactionsacareonsistentincludedbehavtiorhe
differencesbetweenν¯andνspectrabecomerathersmall.Concerningmean
energiesfromourtweoreaµlisticprofilesthatrepresenttheaccretionphase,
wealwaysfindνe:ν¯e:νµ=0.8–0.9:1:1.0–1.09.Thesefindings
arecompletelyconsistentwithfullhydrodynamicsimulationsthatinclude
thefullsetofinteractions,i.e.,therecentGarching-groupmodels.Forthe
luminositieswefindgoodagreement,too.TherunscommonlyshowLνe≈
Lν¯e≈2Lνµ.
Forourpower-lawmodelswefindsomewhatstrongerhierarchiesinthe
meanenergies.Itisnotclearifthiswillstillholdinself-consistentsimula-
tions.Therearecurrentlynoself-consistentmodelswithanaccuratetreat-
mentoftheneutrinotransportavailablethatcontinuedthecalculationsto
thelatephaseswheresteepproto-neutron-starprofilesarefound.
Inourcollectionofpreviousliteraturewefoundavarietyofdifferent
assumedpredictions.intheThesliteraturetrongforhierarccalculatinghiesinmeanneutrino-eonergiesscillationthatweffectserecareommonlyrather
outliers.Alsotheexactequipartitionofluminositiesisnotfoundinelaborate
neutrinotransportmodels.
Thestandardpictureofflavor-dependentSNneutrinospectraneedstobe
changed.Oscillationstudiesshouldallowforlargedifferencesintheν¯eand
ν¯µluminosities,andforalmostequalmeanenergiesoftheseflavors.Results
ofdifferentself-consistentsimulationsshowlargevariationsintheirabsolute
values.Therefore,onlyconceptsthatrelyondifferentialeffectsofflavor-
dependentSNneutrinospectracanhavepredictivepowerforidentifying
signatures.oscillation

Sum7.4.ryam

fitαMC

fitηMC

νe

81

8 7 6αMC fitηMC fitνe
) 5ε f( 42ε 3 2 1 01.051.00MC / fit0.950.90 0 10 20 30 40 0 10 20 30 40 50
8 7 6αMC fitηMC fitν−e
) 5ε f( 42ε 3 2 1 01.051.00MC / fit0.950.90 0 10 20 30 40 0 10 20 30 40 50
8 7 6αMC fitηMC fitνµ,τ
) 5ε f( 42ε 3 2 1 01.051.00MC / fit0.950.90 0 10 20 30 40 0 10 20 30 40 50
[MeV]ε [MeV]εFig.7.3.—High-statisticsspectraforAccretion-PhaseModelIincludingall
interactionprocesses.TheMonteCarlo(MC)resultsareshownascrosses,
theanalyticfitfunctionsassmoothlines.Theleft-handpanelsuseasfits
fα()accordingtoEquation(5.9),theright-handpanelsfη()accordingto
Equation(5.15).BelowthespectraweshowtheratiobetweenMonteCarlo
t.fiand

82

fitαMC

Chapter7.ComparisonofAllFlavors

fitηMC

10 fitη fitαMCMC 8)ε 6 f(2ε 4 2 01.05MC / fit0.950.85 0 10 20 30 40 0 10 20 30 40
10αMC fitηMC fit
8)ε 6 f(2ε 4 2 01.05MC / fit0.950.85 0 10 20 30 40 0 10 20 30 40
10αMC fitηMC fitν
8)ε 6 f(2ε 4 2 01.05MC / fit0.950.85 0 10 20 30 40 0 10 20 30 40
[MeV]ε [MeV]εFig.7.4.—SameasFigure7.3forthep=10,q=3.5powerlaw.

νe

−νe

νµ,τ

ryamSum7.4.

fitαMC

fitηMC

νe

6α fitη fitνe
MCMC)ε 4 f(2ε 2 01.05MC / fit0.950.85 0 10 20 30 40 50 0 10 20 30 40 50
6αMC fitηMC fit−νe
)ε 4 f(2ε 2 01.05MC / fit0.950.85 0 10 20 30 40 50 0 10 20 30 40 50
6αMC fitηMC fitνµ,τ
)ε 4 f(2ε 2 01.05MC / fit0.950.85 0 10 20 30 40 50 0 10 20 30 40 50
[MeV]ε [MeV]εFig.7.5.—SameasFigure7.3forthep=10,q=2.5powerlaw.

83

84

hCrapte7.omCoparisnofllAvFlaors

Chapter8

DetectingOscillationsofSN
Neutrinos

Bynow,neutrinooscillationsarefirmlyestablishedandthenextgoalistopin
downtheoscillationparametersthatarestillunknown,liketheneutrinomass
hierarchyandthesmallmixingangleθ13.Takingourfindingsonneutrino
spectraintoaccountrequiresnewmethodsforextractinginformation,but
enableWesuspresentotuasemethoratiosdofforobsidenervtableifyingstheinsteadeffectofoftheirearth-ambsoluteattermoagnitude.scillations
inenergyaddepetectedositedSNintwneutrinoodetectors.signalbyByidencomparingtifyingthesptectralimedmepoendencedulationsofwtithhe
abutFourier-demandsbtransformetteremethonergydrtheesolutiondetectthaniontahettawo-singledetectordetectormethoisd.pFoindingssible,
oscillationsignaturesinthesignalcanpindownthemasshierarchyorrestrict
angle.mixingmallsthePartsofthischapterwerepublishedinasimilarforminourpublications
A.S.Dighe,M.T.Keil,andG.G.Raffelt,
“DetectingtheneutrinomasshierarchywithasupernovaatIceCube,”JCAP,
sinupeprernossva(ne2003),utrinoshepatas-ph/0303210,ingledetecandtor,”“Idehentifyingp-ph/0304150.earthmattereffectson

8.1OscillationsofSNNeutrinos

Moreandmoreaccuratemeasurementshavefirmlyestablishedneutrinoos-
cillations.Aftermanyyearsofatmospheric-andsolar-neutrinoexperiments
andalsolong-baselineexperiments(KamLANDandK2K)wehaveagood
understandingofhowneutrinososcillate(Bahcall,Gonzales-Garcia,&Pe˜na-
Garay2003,Foglietal.2002,Gonzales-Garcia&Nir2002,deHolanda&

85

86

Chapter8.DetectingOscillationsofSNNeutrinos

Smirnov2003).Theflavorsνe,νµ,andντareinweakinteractioneigenstates
andthereforerelevantwhenwetalkaboutinteractionsofparticles.Eachof
theseweakeigenstatesisanon-trivialsuperpositionofthreemasseigenstates
ννν1,ν2,andν3,
1eνµ=Uν2,(8.1)
νν3τwhereUistheleptonicmixingmatrixthatcanbewritteninthecanonical
form100c130eiδs13c12s120

U=0c23s23010−s12c120.(8.2)
0−s23c23−e−iδs130c13001
Herec12=cosθ12ands12=sinθ12etc.,andδisaphasethatcanleadto
CP-violatingeffects,thatare,however,irrelevantforSNneutrinos.Thethree
matricesthenjustcorrespondtotherotationmatricesinthe2-3,1-3,and
1-2planes,respectively.
Themasseigenstatesaremostconvenientfordescribingneutrinoprop-
agationinspace.Iftherearethreedifferentmasseigenstatestherecanbe
threedifferentmasses.Theoscillation,i.e.,theinterferenceofthethreeeigen-
states,dependsonthedifferenceofthesquaresoftheirmasses.Usually,the
mass-squareddifferencesaregivenas∆m2ij≡mi2−mj2.Ofcourse,onlytwo
ofthemareindependent.Comparedtothecurrentlimitontheabsolute
neutrinomass,experimentsfindverysmallmass-squareddifferences.Twoof
thethreemassesappeartoberatherdegenerate.Thethirdmasseigenstate
canthenliefaraboveorfarbelowthisdoublet.Whetherthethirdmass
eigenstateliesaboveorbelowthealmostdegeneratedoubletisreferredto
asnormalorinvertedhierarchy,respectively.Whichhierarchyisrealizedin
natureisanopenquestion.
Themasssquareddifferencesrelevantfortheatmosphericandsolarneu-
trinooscillationsshowthestronghierarchy∆m2atm∆m2.Thishierarchy,
combinedwiththeobservedsmallnessoftheangleθ13atCHOOZ(Apollonio
etal.1999)impliesthattheatmosphericneutrinooscillationsessentiallyde-
couplefromthesolaronesandeachoftheseisdominatedbyonlyoneofthe
mixingangles.Theatmosphericneutrinooscillationsarecontrolledbyθ23
thatmaywellbemaximal(45◦).Thesolarcaseisdominatedbyθ12,thatis
largebutnotmaximal.Toareasonablygoodaccuracy∆m2atm≈∆m232and
∆m2≈∆m221andtherefore∆m231≈∆m232.Fromaglobal3-flavoranalysis
ofalldataonefindsthe3σrangesforthemassdifferencesandmixingangles
summarizedinTable8.1.

8.1.OscillationsofSNNeutrinos

87

Obviously,inthecontextofSNeneutrinooscillationscanonlybefoundif
flavor-dependentdifferencesintheSNneutrinofluxesandspectraarepresent.
Wedenotethefluxesofν¯eandν¯µatearththatwouldbeobservablein
theabsenceofoscillationsbyFe¯0andFµ¯0,respectively.Inthepresenceof
oscillationsaν¯edetectoractuallyobserves
Fe¯D()=p¯D()Fe¯0()+1−p¯D()Fµ¯0,(8.3)
wherep¯D()istheν¯esurvivalprobabilityafterpropagationthroughtheSN
mantleandperhapspartoftheearthbeforereachingthedetector.Water
Cherenkovdetectorsaswellasscintillationdetectorscanonlydetectν¯ewith
agoodefficiency.Thereforewegivetheν¯efluxatthedetector.
Whenneutrinostravelinmattertheiroscillatorybehaviorcanchange
leadssignificanthroughtly.Depematterndingsonurroundingtheneutrinothesource,sourceliktheeipnathstarstoworardsinaadSNetectorand
mightalsogothroughtheearth.Thismatterhasasignificantpopulationof
electrons,butotherleptonsareabsent.Bycharged-currentinteractionsthe
mediumdistinguishesbetweenelectronflavorandotherflavors,whichimplies
apotentialforelectronneutrinosandanti-neutrinos.Inotherwords,the
mediumcanchangeitsrefractiveindexforelectron(anti-)neutrinos.Sincethe
otherflavorsareunaffected,theoscillatorybehaviorcanchangesignificantly,
dependingontheelectrondensity.
Asignificantmodificationofthesurvivalprobabilityduetothepropaga-
tionthroughtheearthappearsonlyforthosecombinationsofneutrinomixing
parametersshowninTable8.2.Theearthmattereffectdependsstronglyon
twoparameters,thesignof∆m232andthevalueof|θ13|(Dighe&Smirnov

Table8.1.Neutrinomixingparameters

ObservationMixingangle∆m2[meV2]
Sun,KamLANDθ12=27◦–42◦∆m221=55–190
Atmosphere,K2Kθ23=32◦–60◦|∆m232|=1400–6000
CHOOZθ13<14◦∆m231≈∆m232

Note.—Neutrinomixingparametersfromaglobalanal-
ysisofallexperiments(3σranges)byGonzalez-Garcia&
2002).(Nir

88

Chapter8.DetectingOscillationsofSNNeutrinos

2000,Dighe2001).Thenormalhierarchycorrespondstom1<m2<m3,i.e.,
∆m232>20,whereastheinvertedhierarchycorrespondstom3<m1<m2,
i.e.,∆m32<0.Notethatthepresenceorabsenceoftheeartheffectdis-
criminatesbetweenvaluesofsin2θ13lessorgreaterthan10−3,i.e.,θ13less
orlargerthanabout1.8◦.Thus,theeartheffectissensitivetovaluesofθ13
thataremuchsmallerthanthecurrentlimit(Table8.1).
Letusconsiderthosescenarioswherethemasshierarchyandthevalueof
θ13aresuchthattheeartheffectappearsforν¯e.Insuchcasestheν¯esurvival
probabilityp¯D(E)isgivenby
p¯D≈cos2θ12−sin2θ¯e2⊕sin(2θ¯e2⊕−2θ12)
×sin2∆m2⊕L12.5MeV,(8.4)
10−5eV21000km
wheretheenergydependenceofallquantitieswillalwaysbeimplicit.Here
θ¯e2⊕isthemixinganglebetweenν¯eandν¯2inearthmatterwhile∆m2⊕isthe
masssquareddifferencebetweenthetwoanti-neutrinomasseigenstates¯ν1
andν¯2,Listhedistancetraveledthroughtheearth,andistheneutrino
energy.Wehaveassumedaconstantmatterdensityinsidetheearth,which
isagoodapproximationforL<10,500km,i.e.,aslongastheneutrinosdo
notpassthroughthecoreoftheearth.

Table8.2.TheeartheffectinaSNsignal.

13-MixingNormalHierarchyInvertedHierarchy
sin2θ13∼<10−3νeandν¯eνeandν¯e
sin2θ13∼>10−3ν¯eνe

Note.—Theeartheffectappearsfortheindicatedfla-
vorsinaSNsignal.

8.2.DetectingOscillationsWithTwoDistantDetectors

8.2DetectingOscillationsWithTwo
etectorsDtnDista

89

8.2.1TheBasicPrinciple
HowsignificanttheoscillationsignalinadetectedSNneutrinospectrum
is,basicallydependsonthedifferencebetweenthe¯νeandν¯µfluxesandthe
distancetraveledthroughtheearth.Thereisnoexactpredictionforthespec-
traemittedbyaSN,butsomefeaturesareestablished.Themeanenergies
ofν¯eandν¯µwillbeverysimilar,adifferenceoftypically0–20%shouldbe
expected.Atthesametimethenumberfluxeswillberatherdifferentby
uptoafactor2inanydirection.Theratiooffluxeswillcertainlychange
withtime,becausetherearetwodistinctphasesresponsiblefortheneu-
trinosignal,namelytheaccretionphaseandKelvin-Helmholtzcoolingofthe
proto-neutronstar(Chapter2).
Withthetimedependenceofthefluxdifferenceitispossibletofindan
oscillationsignaturewithoutadetailedknowledgeoftheoriginalSNsignal.
Oscillationsinsidetheearthdependonthefluxdifferenceandwilltherefore
alsochangewithtime.Comparingthetimedependenceofaneutrinosignal
oftwodetectorsyieldsagoodchancefordetectingtheeartheffect.Thesetup
shouldbeasfollows:onedetectorseestheSNfromabove,i.e.,withoutearth
effect,andanotherdetectorthatneutrinosreachaftertravelingthroughthe
earth.Thechanceforencounteringthissetupincreaseswiththeseparation
ofthedetectors.Detectorsthatareseparatedbyalongdistanceandwillbe
operationalformanydecadesarethefutureIceCubedetectorinAntarctica
(Ahrensetal.2002b)and,forexample,theSuper-Kamiokandedetectorin
Japan.

8.2.2TheSNSignalatIceCube
Ithasbeenrecognizedforalongtimethatneutrinotelescopesdetecting
CherenkovlightinicecandetectaSNneutrinoburstbecausetheCherenkov
glowoftheicecanbeidentifiedastime-correlatednoiseamongallphototubes
(Halzen,Jacobsen,&Zas1994,1996).Thisapproachhasbeenusedbythe
neutrinotelescopeAMANDAtoexcludetheoccurrenceofagalacticSNover
arecentobservationperiod(Ahrensetal.2002a).
TheSNneutrinosstreamingthroughtheantarcticiceinteractaccording
toν¯ep→ne+andsomeotherlessimportantreactions.Thepositrons,in
turn,emitCherenkovlightproducingahomogeneousandisotropicglowof
theice.Theopticalmodules(OMs)thatarefrozenintotheiceareimmersed
inthisdiffusebathofphotonsandpickupanumbercorrespondingtotheir

90Chapter8.DetectingOscillationsofSNNeutrinos

angularacceptanceandquantume+fficiency.Estimatingtheeventrateofa
SNsignalinducedbytheν¯ep→nereactionyields(Digheetal.2003a)
2Γevents=62s−152Lν¯e−110kpcffluxfdet,(8.5)
Dserg10whereLν¯eistheν¯eluminosityafterflavoroscillations,andDthedistance
betweenSNanddetector.Thefactorsffluxandfdetparameterizetheflux
anddetectorcharacteristics,respectively.Bothfactorsareoforder1.Forthe
fluxwehave
15MeV8ν¯3e
fflux=ν¯e15(15MeV)3
=(3+α)(2+α)8ν¯e2,(8.6)
(1+α)215(15MeV)2
whereweusedEquation(5.13)inthelastline,i.e.,weappliedourαparam-
eterization.Forα=3andν¯e=15MeVthefactoris1.
Thefactorfdetcontainstheefficiencyforconvertingtheneutrinosignal
intophotonsbytheiceaswellastheefficiencyfordetectingthesephotons.
Forourpurposeweassumefdet=1.
Inthe4800OMsofIceCubeaSNinourgalaxywouldyieldatotalevent
numberofabout1.5×106photonsifweassumetheSNradiates5×1052erg.
Thisrateneedstobecomparedtothebackgroundcountingrateof300Hz
perOMandthusforadurationof10sweexpect1.44×107photonsintotal.
AssumingPoissonfluctuations,theuncertaintyofthisnumberis3.8×103,
i.e.,0.1%oftheSNsignal.Therefore,onecandeterminetheSNsignalwith
astatisticalsub-percentprecision,ignoringfornowproblemsofabsolute
calibration.detectorInordertoillustratethestatisticalpowerofIceCubetoobserveaSN
signalweusetwodifferentnumericalSNsimulations.Thefirstwasperformed
bytheLivermoregroup(Totanietal.1998)thatinvolvestraditionalinput
physicsforνµinteractionsandaflux-limiteddiffusionschemefortreating
neutrinotransport.Thegreatadvantageofthissimulationisthatitcovers
thefullevolutionfrominfallovertheexplosiontotheKelvin-Helmholtz
coolingphaseoftheproto-neutronstar.WeshowtheLivermoreν¯eandν¯µ
lightcurvesinFigure8.1(leftpanels).Forallflavorsthesecurvesarealso
displayedinFigure2.4.
OursecondsimulationwasperformedwiththeGarchingcode(Rampp
&Janka2002).Itincludesallrelevantneutrinointeractionrates,including
nucleonbremsstrahlung,neutrinopairprocesses,weakmagnetism,nucleon
recoils,andnuclearcorrelationeffects.Theneutrinotransportpartisbased

8.2.DetectingOscillationsWithTwoDistantDetectors

25 25 20 20〈ε〉 15 15 10 10 0 1 2 3 4 0 250 500 750
-1] 6−−νe 6
5 4νµ 5 4
erg s 3 352 2 2L [10 1 1 0 0 1 2 3 4 0 0 250 500 750
Time [s]Time [ms]

91

Fig.8.1.—Supernovaν¯eandν¯µlightcurvesandaverageenergies.Left:Liv-
ermoresimulation(Totanietal.1999).Right:Garchingsimulation(Raffeltet
2003).al.

onaBoltzmannsolver.Theneutrino-radiationhydrodynamicsprogramal-
lowsonetoperformsphericallysymmetricaswellasmulti-dimensionalsim-
ulations.Theprogenitormodelisa15Mstarwitha1.28Mironcore.
Theperiodfromshockformationto468msafterbouncewasevolvedintwo
dimensions.Thesubsequentevolutionofthemodelissimulatedinspherical
symmetry.At150mstheexplosionsetsin,althoughasmallmodification
oftheBoltzmanntransportwasnecessarytoallowthistohappen(Jankaet
al.2003).Recallthatunmanipulatedfull-scalemodelswithanaccuratetreat-
mentofthemicrophysicscurrentlydonotobtainexplosions(Section2.4).
Thisrunwillbecontinuedbeyondthecurrentepochof750mspostbounce;
wehereusethepreliminaryresultscurrentlyavailable(Raffeltetal.2003).
WeshowtheGarchingν¯eandν¯µlightcurvesinFigure8.1(rightpanels).
WetaketheLivermoresimulationtorepresenttraditionalpredictionsfor
flavor-dependentSNneutrinofluxesandspectrathatwereusedinmany
previousdiscussionsofSNneutrinooscillations.TheGarchingsimulationis
takentorepresentasituationwhentheν¯µinteractionsaremoresystem-
aticallyincludedsothattheflavor-dependentspectraandfluxesaremore
similarthanhadbeenassumedpreviously.Wethinkitisusefultojuxtapose
theIceCuberesponseforbothcases.
AnotherdifferenceisthatinLivermoretheaccretionphaselastslonger.
Sincetheexplosionmechanismisnotfinallysettled,itisnotobviouswhich

92

Chapter8.DetectingOscillationsofSNNeutrinos

] 1.5 1.5-1 s6 1 1 0.5 0.5Count Rate [10 0 0 0 1 2 3 4 0 250 500 750
]4 6 6 4 4Counts / Bin [10 2 2 0 0 0 1 2 3 4 0 250 500 750
Time [ms]Time [s]

onFig.theL8.2.—ivSupermoreernosimvaulationsignalin(left)IceCubandteheaGassumingrchingaonedistance(righot),f1in0kbpocth,bcasesased
andignoringhaveflaavddeordonoisescillations.fromaInbacthekgbroundottomprateanelsof300wehHzavpeeruOsedM.50msbins

caseismorerealistic.Moreover,therecouldbedifferencesbetweendifferent
SNe.Theoverallfeaturesarecertainlycomparablebetweenthetwosimula-
.tionsInFigure8.2weshowtheexpectedcountingratesinIceCubeasgiven
inEquation(8.5)foranassumeddistanceof10kpcand4800OMsforthe
Livermore(left)andGarching(right)simulations.Wealsoshowthissignal
in50msbinswherewehaveaddednoisefromabackgroundof300Hzper
OM.Thebaselineisattheaveragebackgroundratesothatnegativecounts
correspondtodownwardbackgroundfluctuations.
Onecouldeasilyidentifytheexistenceanddurationoftheaccretionphase
andthustestthestandarddelayed-explosionscenario.Onecouldalsomea-
suretheoveralldurationofthecoolingphaseandthusexcludethepresenceof
significantexoticenergylosses.Therefore,manyoftheparticle-physicslimits
basedontheSN1987Aneutrinos(Raffelt1999)couldbesupportedwitha
statisticallyserioussignal.IftheSNcoreweretocollapsetoablackholeafter
sometime,thesuddenturn-offoftheneutrinofluxcouldbeidentified.In
short,whenagalacticSNoccurs,IceCubeisapowerfulstand-aloneneutrino
detector,providinguswithaplethoraofinformationthatisoffundamental
astrophysicalandparticle-physicsinterest.

8.2.DetectingOscillationsWithTwoDistantDetectors

93

8.2.3TheOscillationSignalatIceCube
Inordertocalculatetheextentoftheeartheffectf2orIceCube,−w5ewill2as-
sumethattherelevantmixingparametersare∆m12=6×10eVand
sin2(2θ12)=0.9.Wefurtherassumethatthesourcespectraaregivenbythe
functionalformofourαfitEquation(5.9).Thevaluesoftheparametersα
andforboththeν¯eandν¯µspectraareingeneraltimedependent.
InFigure8.3weshowthevariationoftheexpectedIceCubesignalwith
earth-crossinglengthLforthetwosetsofparametersdetailedinTable8.3.
Thefirstcouldberepresentativeoftheaccretionphase,thesecondofthe
coolingsignal.Weusethetwo-densityapproximationfortheearthdensity
profile,wherethecorehasadensityof11.5gcm−3andaradiusof3,500km,
whilethedensityoftheearthmantlewastakentobe4.5gcm−3.Weobserve
thatforshortdistances,correspondingtonear-horizontalneutrinotrajecto-
ries,thesignalvariesstronglywithL.Betweenabout3,000and10,500km
itreachesanasymptoticvaluethatwecallthe“asymptoticmantlevalue.”
ForCase(a),thisvaluecorrespondstoabout1.5%depletionofthesignal,
whereasfor(b)itcorrespondstoabout6.5%depletion.
Beyondanearth-crossinglengthofabout10,500km,theneutrinoshave
tocrosstheearthcorewithanotherlargejumpindensity.Thecoreef-

Fig.8.3.—VariationoftheexpectedIceCubesignalwithneutrinoearth
crossinglengthLfortheassumedfluxandmixingparametersofTable8.3.
Thesignalisnormalizedto1whennoeartheffectispresent,i.e.forL=0.
Thedashedlineisforthecaserepresentingtheaccretionphase,thesolidline
forthecoolingphase.

94

Chapter8.DetectingOscillationsofSNNeutrinos

IceCube Signal(a)5040301401201008060402002010(b)
1.91.91.91401.91401.9
1.71.71.71201201.71.7
−νµ1.51001.51.5−νµ1.51001.5
00_νe1.31.31.380_νe1.31.380
Φ /Φ1.11.11.16000Φ /Φ1.11.160
0.90.90.9400.90.940
0.70.70.7200.70.720
0015101620173018401950201510162017301840192500
ενµενµ
Fig.8.4.—AsymptoticIceCubesignalmodificationbytheeartheffect.The
fixedfluxparametersare(a)ν¯e=15MeV,αν¯e=4.0,andαν¯µ=3.0
and(b)ν¯e=15MeV,αν¯e=αν¯µ=3.0.Thecontoursareequallyspaced
startingfrom1.02(light)in0.02decrementstosmallervalues(darker).

fectschangetheasymptoticmantlevaluebyroughly1%ascanbeseenin
Figure8.3.Weneglectthecoreeffectsinthefollowinganalysis,andthe
“asymptoticvalue”alwaysreferstotheasymptoticmantlevalue.
Forthelargestpartoftheskytheeartheffecteitherappearswiththis
asymptoticvalue(“neutrinoscomingfrombelow”),oritdoesnotappear
atall(“neutrinosfromabove”).Therefore,wenowfocusontheasymptotic
valueandstudyhowthesignalmodificationdependsontheassumedflux
parameters.InTable8.4weshowthesignalmodificationforν¯e=15MeV,
αν¯e=4.0,andαν¯µ=3.0asafunctionofν¯µandthefluxratioΦν¯0e/Φν¯0µ.In
Table8.5weshowthesamewithαν¯e=αν¯µ=3.0.Theresultsareshownin
theformofcontourplotsinFigure8.4.
Evenformildlydifferentfluxesorspectrathesignalmodificationisseveral

Table8.3.Fluxparametersfortworepresentativecases.

ExamplePhaseν¯eν¯µαν¯eαν¯µΦν¯0e/Φν¯0µAsymptotic
[MeV][MeV]EarthEffect
(a)Accretion1517431.5−1.5%
(b)Cooling1518330.8−6.5%

8.2.DetectingOscillationsWithTwoDistantDetectors

Table8.4.AsymptoticIceCubesignalmodificationbytheeartheffect

Fluxratioν¯µ[MeV]
Φν¯0e/Φν¯0µ151617181920
2.01.0261.0141.0020.9880.9740.960
1.91.0231.0110.9990.9850.9710.956
1.81.0211.0090.9950.9820.9670.952
1.71.0181.0050.9920.9780.9630.948
1.61.0151.0020.9880.9740.9590.944
1.51.0120.9980.9840.9690.9540.939
1.41.0080.9940.9800.9650.9490.934
1.31.0040.9900.9750.9600.9440.928
1.21.0000.9850.9700.9540.9380.922
1.10.9950.9800.9640.9480.9320.915
1.00.9890.9740.9570.9410.9250.908
0.90.9830.9670.9500.9340.9170.901
0.80.9760.9590.9420.9250.9090.892
0.70.9670.9500.9330.9160.8990.883
0.60.9580.9400.9230.9060.8890.873
0.50.9460.9280.9110.8940.8770.862

Note.—Thefixedfluxparametersareν¯e=15MeV,
αν¯e=4.0,andαν¯µ=3.0.

95

96

Chapter8.DetectingOscillationsofNSTable8.5.SameasTable8.4withαν¯e=αν¯µ=3.0.

ratioFluxΦν¯0e/Φν¯0µ

2.01.91.81.71.61.51.41.31.21.11.00.90.80.70.60.5

15

.0361.0331.0311.0281.0251.0221.0191.0151.01011.006.0001.9940.9860.9780.9680.9560

16

1.0241.0221.0191.0161.0131.0091.0051.0010.9960.9910.9850.9780.9700.9610.9500.938

¯νµ[MeV]
1817

0121.0101.0061.0031.9990.9950.9910.9860.9810.9750.9690.9610.9530.9440.9330.9200.

1.0000.9960.9930.9890.9850.9810.9760.9710.9650.9590.9520.9450.9360.9260.9160.903

19

0.9860.9830.9790.9750.9710.9660.9610.9550.9490.9430.9360.9280.9190.9100.8990.886

20

0.9720.9680.9640.9600.9550.9510.9450.9400.9330.9270.9190.9110.9030.8930.8820.870

eutrinosN

8.2.DetectingOscillationsWithTwoDistantDetectors97

percent,byfarexceedingthestatisticaluncertaintyoftheIceCubesignal,
althoughtheabsolutecalibrationofIceCubemayremainuncertaintowithin
severalpercent.However,thesignalmodificationwillvarywithtimeduring
theSNburst.Duringtheearlyaccretionphasethatisexpectedtolastfor
afew100msandcorrespondstoasignificantfractionoftheoverallsignal,
theν¯µfluxmaybealmostafactorof2smallerthantheν¯eflux,butitwill
beslightlyhotterandlesspinched(Raffeltetal.2003).Thiscorresponds
toCase(a)ofTable8.3;itisevidentfromFigure8.3andTable8.4that
thisimpliesthattheeartheffectisverysmall.DuringtheKelvin-Helmholtz
coolingphasethefluxratioisreversedwithmore¯νµbeingemittedthanν¯e,
butstillwiththesamehierarchyofenergies.ThiscorrespondstoCase(b);in
thiscasetheeartheffectcouldbeabout6%.Thistimedependencemayallow
onetodetecttheeartheffectwithoutapreciseabsolutedetectorcalibration.
Inordertoillustratethetimedependenceoftheeartheffectweshow
inFigure8.5theexpectedcountingrateinIceCubeforboththeLivermore
(leftpanels)andGarching(rightpanels)simulations.Intheupperpanelswe
showtheexpectedcountingratewithflavoroscillationsintheSNmantle,
butnoeartheffect(solidlines),orwiththeasymptoticeartheffect(dashed
lines)thatobtainsforalargeearth-crossingpath.Naturallythedifferences
areverysmallsothatweshowinthelowerpanelstheratioofthesecurves,
i.e.,theexpectedcountingratewith/withouteartheffectasafunctionof
timeforbothLivermoreandGarching.WhilefortheLivermoresimulation
thereisalargeeartheffectevenatearlytimes,thechangefromearlytolate
timesinbothcasesisaround4–5%.Therefore,themostmodel-independent
signatureisatimevariationoftheeartheffectduringtheSNneutrinosignal.
Inordertodemonstratethestatisticalsignificanceoftheseeffectswein-
tegratetheexpectedsignalforbothsimulationsseparatelyfortheaccretion
phaseandthesubsequentcoolingphase;theresultsareshowninTable8.6.
Forbothsimulationstheeartheffectitselfanditschangewithtimeisstatis-
ticallyhighlysignificant.DuetomoreschematicinputphysicstheLivermore
modeloverestimatesthedifferencesbetween¯νeandν¯µspectra.However,the
relativechangeoftheeartheffectduringaccretionandcoolingisnotvastly
differentbetweenthetwosimulations.Recallingthattheabsolutedetector
calibrationmaybeveryuncertainsothatonehastorelyonthetemporal
variationoftheeartheffect,thedifferencebetweenLivermoreandGarching
becomesmuchsmaller.Weexpectthatitisquitegenericthatthetemporal
changeoftheeartheffectisafewpercentoftheoverallcountingrate.

98Chapter8.DetectingOscillationsofSNNeutrinos
]1.50-11.506 sL ≠L = 0 0
1.001.000.500.50Count Rate [100.000.00 0 1 2 3 4 0 250 500 750
1.001.000.960.960) / (L=0)0.92≠0.92(L0.880.88 0 1 2 3 4 0 250 500 750
Time [ms]Time [s]Fig.8.5.—EartheffectinIceCube.Theupperpanelsshowtheexpected
countingratebasedontheLivermore(left)andGarching(right)models,
includingflavoroscillations.Thesolidlineiswithouteartheffect(L=0),
thedashedlinewithasymptoticeartheffect(L=0).Thelowerpanelsshow
theratiobetweenthesecurves,i.e.,theratioofcountingrateswith/without
ect.effrtheaTable8.6.IceCubeCherenkovcountsforthenumericalSNmodels.
LivermoreGarching
AccretionCoolingAccretionCooling
Integrationtime[s]0–0.5000.500–30–0.2500.250–0.700
ts]Coun[SignalSNNoEarthEffect519,080818,043173,085407,715
AsymptoticEarthEffect488,093751,137171,310390,252
Difference30,98766,9061,77517,463
FractionalDifference−5.97%−8.18%−1.03%−4.28%
√Background[Counts]720,0004,320,000360,000648,000
Background/Signal0.16%0.25%0.35%0.20%

8.2.DetectingOscillationsWithTwoDistantDetectors99

8.2.4Super-orHyper-KamiokandeandIceCube
OnecanmeasuretheeartheffectinIceCubeonlyinconjunctionwithanother
high-statisticsdetector.WedonotattempttosimulateindetailtheSNsig-
nalinthisotherdetectorbutsimplyassumethatitcanbemeasuredwitha
precisionaboutasgoodasinIceCube.OnecandidateisSuper-Kamiokande,
awaterCherenkovdetectorthatwouldmeasurearound104eventsfroma
galacticSNatadistanceof10kpc.Therefore,thestatisticalprecisionforthe
totalneutrinoenergydepositioninthewaterisaround1%andthusworse
thaninIceCube.EventhoughSuper-Kamiokandewillmeasurealargernum-
berofCherenkovphotonsthanIceCube,asingleneutrinoeventwillcause
anentireCherenkovringtobemeasured,i.e.,thephotonsarehighlycor-
related.Therefore,intheestimatedstatistical√Nfluctuationofthesignal,
thefluctuatingnumberNisthatofthedetectedneutrinos.Ifthefuture
Hyper-Kamiokandeisbuilt,itsfiducialvolumewouldbeabout30timesthat
ofSuper-Kamiokande.Inthiscasethestatisticalsignalprecisionexceedsthat
ofIceCubefortheequivalentobservable.
WedenotetheequivalentIceCubesignalmeasuredbySuper-orHyper-
KamiokandeasNSKandtheIceCubesignalasNIC.Ifthedistancestraveled
bytheneutrinosbeforereachingthesetwodetectorsaredifferent,theearth
effectontheneutrinospectramaybedifferent,whichwillreflectintheratio
NSK/NIC.Ofcourse,intheabsenceoftheeartheffectthisratioequalsunity
definition.ybThegeographicalpositionofIceCubewithrespecttoSuper-orHyper-
Kamiokandeatanorthernlatitudeof36.4◦iswell-suitedforthedetection
oftheeartheffectthroughacombinationofthesignals.UsingFigure8.3
wecanalreadydrawsomequalitativeconclusionsabouttheratioNSK/NIC.
Clearly,NSK/NIC=1ifneutrinosdonottravelthroughtheearthbefore
reachingeitherdetector.Ifthedistancetraveledbyneutrinosthroughthe
earthismorethat3,000kmforbothdetectors,theeartheffectsonbothNSK
andNICarenearlyequalandtheirratiostaysaroundunity.Iftheneutrinos
come“fromabove”forSKand“frombelow”forIceCube,orviceversa,the
earthmattereffectwillshiftthisratiofromunity.
InFigure8.6,weshowcontoursofNSK/NICfortheSNpositioninterms
ofthelocationonearthwheretheSNisatthezenith.Themapisanarea
preservingHammer-Aitoffprojectionsothatthesizesofdifferentregionsin
thefiguregivesarealisticideaofthe“good”and“bad”regionsofthesky.In
ordertogeneratethecontoursweusetheparametersofCase(b)inTable8.3
sothattheasymptoticsuppressionofthesignalisabout6.5%.Theskyfalls
intofourdistinctregionsdependingonthedirectionoftheneutrinosrelative
toeitherdetectorasdescribedinTable8.7.Whentheneutrinoscomefrom

100

Chapter8.DetectingOscillationsofSNNeutrinos

aboveforbothdetectors(RegionD)thereisnoeartheffect.Iftheycome
frombelowinboth(RegionC),theeartheffectislargeinboth.Depending
ontheexactdistancetraveledthroughtheearth,theeventratiocanbelarge,
butgenerallyfluctuatesaround1.Intheotherregionswheretheneutrinos
comefromaboveforonedetectorandfrombelowfortheother(RegionsA
andB)therelativeeffectislarge.

eaFig.rth.T8.6.—hereCongionstoursA,oBf,NC,SKD/NareICdonescrtheibedminapTofablethe8.7.skyprojectedonthe

Table8.7.RegionsinFigure8.6fortheeartheffectinIceCubeand
nde.aKamioker-Sup

RegionSkyfractionNeutrinoscomefromNSK/NIC
Ker-SupeIceCubBA00.35.35abbeloovwebabeloovwe01.935.070
C0.15belowbelowFluctuationsaround1
D0.15aboveabove1

8.3.DetectingtheEarth-MatterEffectataSingleDetector101

8.3DetectingtheEarth-MatterEffectata
DetectorSingleAccordingtoEquation(8.4)themodulationsimprintedonthedetectedspec-
traareproportionaltosin2(#/).Thereforetheprefactor#shouldappear
asadominantfrequencyintheFouriertransformoftheinverseenergyspec-
trum(Digheetal.2003b),becauseininverse-energyspacethemodulations
oftheSNneutrinospectrumarenearlyequispaced.
Theequidistantpeaksinthemodulationoftheinverse-energyspectrum
areanecessaryfeatureoftheeartheffects.Indeed,thenet¯νefluxatthe
detectormaybewrittenusing(8.3)and(8.4)intheform
Fe¯D=sin2θ12Fµ¯0+cos2θ12Fe¯0+∆F0A¯⊕sin2(∆m2⊕L/),(8.7)
where∆F0≡(Fe¯0−Fµ¯0)dependsonlyontheprimaryneutrinospectra,
whereasA¯⊕≡−sin2θ¯e2⊕sin(2θ¯e2⊕−2θ12)dependsonlyonthemixingpa-
rametersandisindependentoftheprimaryspectra.Thelasttermin(8.7)
istheearthoscillationtermthatcontainsafrequencyk⊕≡2∆m2⊕Lin−1,
withthecoefficient∆F0A¯⊕beingacomparativelyslowlyvaryingfunctionof
−1.Thefirsttwotermsin(8.7)arealsoslowlyvaryingfunctionsof−1,and
hencecontainfrequenciesin−1thataremuchsmallerthank⊕.
Thefrequencyk⊕iscompletelyindependentoftheprimaryneutrinospec-
tra,andindeedcanbedeterminedtoagoodaccuracyfromtheknowledgeof
thesolaroscillationparameters,theearthmatterdensity,andthedirection
oftheSN.Ifthisfrequencycomponentisisolatedfromtheinverse-energy
spectrumofν¯e,theeartheffectswouldbeidentified.
Duetotheenergydependenceoftheprefactorsoftheoscillationterm
∆F0A¯⊕andof∆m2⊕inEquation(8.7)thepeakintheFourier-transformed
spectrumgetsacertainwidtharoundk⊕.Thispeakcanbeclearlyidentified
iftheeartheffectispresent.
Inanexperimenttherearetwoeffectsobscuringthesignal.Oneisthe
statisticalfluctuationsofthesignalandtheotheristhesmearingofthe
modulationsignalbytheenergyresolutionofthedetector.Higherstatistics
aswellasbetterenergyresolutionenhancetheoscillationsignature.Apre-
scriptionforidentifyingtheprominentpeakontopofthebackgroundcan
befoundinDigheetal.(2003b).IfthedirectionoftheSNisknownwecan
estimatek⊕andbythatsimplifythepeakidentification.
Oncethepeakisidentifiedthisisaclearsignalofearth-mattereffects.
AsstatedinSection8.2.4anidentificationoftheearth-mattereffectwill
severelyrestricttheneutrinomixingparameterspace,sincetheeffectsare
presentonlywiththecombinationsoftheneutrinomasshierarchyandthe

102Chapter8.DetectingOscillationsofSNNeutrinos

mixingangleθ13giveninTable8.2.Inparticular,ifsin2θ13ismeasuredata
laboratoryexperimenttobegreaterthan10−3,thentheeartheffectsonthe
ν¯espectrumimplythenormalmasshierarchy.However,iftheeartheffects
arenotdetected,itdoesnotruleoutanyneutrinomixingparameters,owing
tothecurrentuncertaintiesintheprimaryfluxes.
Withthepositionofthepeakitisevenpossibletodetermine∆m2toan
accuracyofa2fewpercentand−theref5ore2muchbetterthanthecurrentlimits,
thatare∆m=(5.5–19)×10eV.TheFourier-transformmethodwill
measure∆m2toaprecisionof10%,comparabletowhatcanbereachedby
KamLAND(deGouvea&Pe˜na-Garay,2001).

8.4Summary
WithournewfindingsfortheSNneutrinospectramethodsforidentifying
oscillationsignaturesinanobservedneutrinospectrumneedtobechanged.
fWect.eFopresenrthetedItwoceCubveerydproetectomrisingincoAntanceptsrcticaforeadrthetectingmatter-theeffeaectsrth-amreatterpresenef-t
inthesignalofafuturegalacticSNonthelevelofafewpercent.IftheIce-
Cubesignalcanbecomparedwithanotherhigh-statisticssignal,notablyin
Supvisibleer-asKamiokaadifferencendeorbeHyptwer-eentheKamiokdande,etectors.theAsearthoneiseffectloobkingecomesforacsignallearly
modificationintherangeofafewpercent,theabsolutedetectorcalibration
maynotbegoodenoughinoneorbothoftheinstruments.However,fortyp-
icalnumericalSNsimulationstheeffectistimedependentandmostnotably
codiffolingersbetphase.weenTtheherefore,earlyoaneccretiowouldnhaphavesetoandsearcthehforasubsequentemptoralneutronvariationstar
oftherelativedetectorsignalsofafewpercent.Thelargenumberofoptical
modulesinIceCuberendersthistaskstatisticallypossible.Infactdepend-
ingonthedifferencesinflavor-dependentfluxes,thestatisticalaccuracyof
Super-Kamiokandemayturnouttobethelimitingfactor.Thislimitationis
notsignificantforHyper-Kamiokande.
TheuniquelocationofIceCubeinAntarcticaimpliesthatforabout70%
oftheskythisdetectorseestheSNthroughtheearthwhenSuper-andHyper-
Kamiokandeseeitfromabove,ortheotherwayround,i.e.,thechancesof
awerreelattoivseeseigthenalSNdifffromerenceabboevteworeenboththetdhroughetectorstheareelaarth,rge.tIfhebcoomthparisdetectoonrofs
thesignalswouldwouldnotrevealtheeartheffect.
OncefuturedetectorslikeHyper-Kamiokandeorlargescintillationde-
tectothroughrsbtecoheFmeoavurierailabletransapformowoferfuthelwinavyersofeiden-energytifyingspectrum.earth-mInattertheeinvffectsersei-s

ryamSum8.4.

103

energyspectrummodulationsduetoearthmatterwillbeapproximatelyeq-
uispacedandthereforeappearasasinglepeakintheFouriertransformed
spearth-ectrum.matterWithothisscillationsmethoanddevoneencanmeasureunam∆m2biguouslytoaaproccuravetcyheofafexistenceewpeor-f
cent.Identifyingtheearth-mattereffectsintheneutrinosignalofafuture
galacticSNwouldrestricttheparameterspaceforneutrinooscillations
sevlishederelyt.oIbf,elinargeinaddition,thestheensemofsinagnitude2θ13of∼>10the−3mbyixingalong-angleθbaseline13canexpbeeerimenstabt-
(Bargeretal.2001,Cerveraetal.2000,Freundetal.2001),itimpliesthe
normalmasshierarchy.
Ontheotherhand,ifsin2θ13∼<10−3hasbeenestablished,theearth
smaeffectllθis13wunaovuldoidableimplywhatevthatertheptherimaryhierarcSNhyis.neutrinoNotfluxesobservinganditspforectrasucharea
moresimilarthanindicatedbystate-of-the-artnumericalsimulations.For
sin2θ13∼>10−3notobservingtheearth-mattereffectsdoesnotallowoneto
ordrawduetotheconclusions,neutrinobecausemasistcanhierarcbehy.duetheoriginalSNspectraandfluxes,

104

Chapter8.DetectingOscillationsofNSeutrinosN

9Chapter

SummaryandDiscussion

Ithasbeenknownforsometimethatthe“traditional”setofνµ-matterin-
teractionsemployedbyhydrodynamicSNsimulationswasincomplete.In
additiontotheingredientsthatwereknowntobemissing,i.e.,nucleon
bremsstrahlungandnucleonrecoil,weshowedthatνeν¯eannihilationinto
νµν¯µisalwaysmoreimportantthanthetraditionale+e−annihilationpro-
cessbyafactorof2–3.
Inasystematicapproachwestudiedtheformationofneutrinospectra
andfluxesinaSNcore.UsingaMonteCarlocodeforneutrinotransport,
wevariedthemicroscopicinputphysicsaswellastheunderlyingstatic
proto-neutronstaratmosphere.Weusedtwobackgroundmodelsfromself-
consistenthydrodynamicsimulations,andseveralpower-lawmodelswith
varyingpower-lawindicesforthedensityandtemperatureanddifferentcon-
stantvaluesfortheelectronfractionYe.
Theνµtransportopacityisdominatedbyneutral-currentscattering
onnucleons.Inaddition,therearenumber-changingprocesses(nucleon
bremsstrahlung,leptonicpairannihilation)andenergy-changingprocesses
(νµe±andνµνe,νµν¯escattering).Recoilinnucleonscatteringallowsfora
smallenergyexchangeineachcollision.Theνµspectraandfluxesareroughly
accountedforifoneincludesonesignificantchannelofpairproductionand
oneforenergyexchangeinadditiontoνµNscattering.Forexample,thetra-
ditionalsetofmicrophysics(iso-energeticνµNscattering,e+e−annihilation,
andνµe±scattering)yieldscrudelycomparablespectraandfluxestoacal-
culationwherepairsareproducedbynucleonbremsstrahlungandenergy
isexchangedbynucleonrecoil.Theoverallresultisrobusttowithin30%
againstthedetailedchoiceofmicrophysics.
However,inviewofneutrinooscillations,whereflavor-dependentflux
differencesareimportant,state-of-the-artsimulationsshouldaimatahigh
precisionforthefluxesandspectralenergies.Therefore,oneneedstoinclude

105

106

Chapter9.DiscussionandSummary

bremsstrahlung,leptonicpairannihilation,neutrino-electronscattering,and
energytransferinneutrino-nucleoncollisions.Asexpected,thetraditional
e+e−annihilationprocessisalwaysmuchlessimportantthanνeν¯eannihila-
tion.Noneofthereactionsstudiedherecanbeneglectedexceptperhapsthe
traditionale+e−annihilationprocess,andνµνeandνµν¯escattering.
Theexistingtreatmentsofthenuclear-physicsaspectsoftheNN→NNνν¯
bremsstrahlungprocessareratherschematic.Wefind,however,thatthe
νµfluxesandspectradonotdependsensitivelyontheexactstrengthof
thebremsstrahlungrate.Therefore,whileamoreadequatetreatmentof
bremsstrahlungremainsdesirable,thefinalresultsareunlikelytobemuch
ected.ffaThetransportofνµandν¯µisusuallytreatedidentically.However,weak-
magnetismeffectsrendertheνµNandν¯µNscatteringcrosssectionssomewhat
different(Horowitz2002),causingasmallνµchemicalpotentialtobuildup.
Wefindthatthedifferencesbetweentheaverageenergiesofνµandν¯µare
onlyafewpercentandcanthusbeneglectedformostpurposes.
Includingallprocessesworksinthedirectionofmakingthefluxesand
spectraofνµmoresimilartothoseofν¯ecomparedtoacalculationwiththe
traditionalsetofinputphysics.Duringtheaccretionphasetheneutron-star
atmosphereisrelativelyexpanded,i.e.,thedensityandtemperaturegradients
arerelativelyshallow.Ourinvestigationsuggeststhatduringthisphaseνµ
isonlyslightlylargerthanν¯e,perhapsbyafewpercentor10%atmost.
Thisresultagreeswiththefirsthydrodynamicsimulationincludingallof
therelevantmicrophysicsexceptνeν¯eannihilation(Accretion-PhaseModel
II)providedtousbyM.Rampp.Fortheluminositiesofthedifferentneutrino
speciesonefindsLν¯e≈Lνe≈2Lνµ.ThesmallnessofLνµisnotsurprising
becausetheeffectiveradiatingsurfaceismuchsmallerthanforν¯e.
DuringtheKelvin-Helmholtzcoolingphasetheneutron-staratmosphere
willbemorecompact,thedensityandtemperaturegradientswillbesteeper.
Therefore,theradiatingsurfacesforallspecieswillbecomemoresimilar.
InthissituationLνµmaywellbecomelargerthanLν¯e.However,therela-
tiveluminositiesdependsensitivelyontheelectronconcentration.Therefore,
withoutaself-consistenthydrostaticlate-timemodelitisdifficulttoclaim
thisluminositycross-overwithconfidence.
Theratioofthespectralenergiesismostsensitivetothetemperature
gradientrelativetothedensitygradient.Inourpower-lawmodelsweused
ρ∝r−pandT∝r−q.Varyingq/pbetween0.25and0.35wefindthatν¯e:
νµvariesbetween1:1.10and1:1.22.Notingthattheupperrangeforq/p
seemsunrealisticallylargeweconcludethatevenatlatetimesthespectral
differencesshouldbesmall;20%soundslikeasafeupperlimit.However,the
power-lawmodelsmightoverestimatethespreadofmeanenergies,ascan

107

beinferredfromcomparingtheshallowpower-lawmodelwiththerealistic
accretion-phasemodels.Westressthatthereisnophysicalreasonforthe
inequalityν¯e<νµ.Itmaywellbethatatsomepointthemeanenergies
areequalorevencrossover.
Thestatementsabouttherelativeneutrinoenergiesinthepreviouslit-
eraturefallintotwoclasses.Onegroupofworkers,usingthetraditionalset
ofmicrophysics,foundspectraldifferencesbetween¯νeandνµonthe25%
level,arangewhichlargelyagreeswithourfindingsinviewofthedifferent
microphysics.Otherpapersclaimratiosaslargeasν¯e:νµ=1:1.8or
evenexceeding1:2.Wehavenoexplanationfortheselatterresults.Atleast
withintheframeworkofoursimplepower-lawmodelswedonotunderstand
whichparametercouldbereasonablyadjustedtoreachsuchextremespectral
erences.diffInahigh-statisticsneutrinoobservationofafuturegalacticSNonemay
wellbeabletodiscoversignaturesofflavoroscillations.However,whenstudy-
ingthesequestionsonehastoallowforthepossibilityofverysmallspectral
differences,andconversely,forthepossibilityoflargefluxdifferences.This
situationisalmostorthogonaltowhathasoftenbeenassumedinpapers
studyingpossibleoscillationsignatures.Arealisticassessmentofthepoten-
tialofafuturegalacticSNtodisentangledifferentneutrinomixingscenarios
shouldallowforthepossibilityofverysmallspectraldifferencesamongthe
differentflavorsofanti-neutrinos.Thespectraldifferencesbetweenνeand
νµ,τarealwaysmuchlarger,butalargeSNneutrino(asopposedtoanti-
neutrino)detectordoesnotexist.
PreviousanalysesonoscillationeffectsintheneutrinofluxofaSNnot
onlyassumedequipartitionoftheluminositiesandalargehierarchyinmean
energies,butalsoheavilyreliedontheabsolutemagnitudeofmodelpre-
dictions.Comparingsuchpredictionswithresultsfromanobservationhas
littlepredictivepowerduetouncertaintiesinthemodels,e.g.,thenuclear
equationofstate.Reliablemethodsinvolveonlyrelativefluxdifferencesand
smallrelativedifferencesinenergies.Withsuchrathermodelindependent
assumptionsonecanunambiguouslyidentifyoscillationeffects.
Weproposedamethodforidentifyingearth-mattereffectsbycompar-
ingtheSNneutrinosignaloftwodetectors.Onedetectorrecordsthesignal
fromabove,theotherthroughtheearth.Iftheenergydepositedperunit
volumeinbothdetectorsdiffers,theearth-mattereffectisdetected.Sur-
prisingly,thefutureIceCubedetectorattheSouthPole,willbeableto
determinethedepositedenergywithaprecisioncomparabletothediscussed
Hyper-Kamiokandedetector.Theuniquelocationhastheadvantagethat
withSuper-orHyper-Kamiokandeasaco-detectorthelikelihoodforen-
counteringtherequiredsetup(exactlyonelineofsightthroughtheearth)

108

Chapter9.DiscussionandSummary

is70%.Iftheearth-mattereffectis2de>tected−w3ecaninferthehierarchyof
neutrinomassestobenormalifsinθ13∼10isestablishedbyotherex-
pecarise,meanptos.sitFiovresisign2nθal13a∼<llo10ws−3forthesteraorthngeffectconclusionswillalwayswhereasbethepresent.reasonsInafnory
anotobservinganeffectcaneitherbeduetosmallerfluxdifferencesfrom
theSOurNosroecondscillationproposedparameters.mechanism,namelyidentifyingtheeartheffectat
asingledetectorwould,togetherwiththefirstmethod,enhancethecovered
arefutureaoflargethesskycintot85illation%.AdedetectortectorwillliketrecordheproptheosnedeutrinosHypefromr-KamaiokagalacticndeorSNa
wspithectrumsuchcaanhighreveaalmoccuracydulationsthattheFinducedourierbytranstheeformarthofeffect.theiIntvwerseould-energyeven
bepossibletodeterminethesolarmasssplittingatthelevelofaccuracythat
willbereachedbyKamLAND.
originalThesefluxesmethoanddssphaveectra.theWeajdvustanatagessumeofbethatingaindepdifferenceendenbtetofweentheethexact¯ν
eandtime¯νvµafluxesriationisofthepresenfluxt.Fortdifferencehetwo-asthedetectorobservsetupablewineporderroposetotogetuseridtohef
depesystematicndenceisfuncertainoundintiesallinntheumericalabsosimluteuladtionetectors.Thephynormalization.sicalreasonThisisttimehat
laterneutrinoonbypemissionroto-niseutronfirstpsotarwecredoobling.ytheaccretionofinfallingmaterialand
actionsWithwethiswerefirstabletosystematicsignificanstudytlyiofmproallvreelevtheantnunderstandingeutrino-maoftterflainvtoer-r-
dependentSNneutrinoemission.Ourfindingsessentiallymarkachangeof
paradigmforoscillationstudiesinvolvingSNneutrinos.Themeanenergies
ofνµandν¯earealmostequalwhereastheluminositiesdifferbyuptoafac-
tordetectof2,cneutrino-ontraryotoscillatitheonprevieffects,ouspforearadigm.xampleThewithntewwopicturedistantallowsdetectorsonebtoy
bycomparingidentifyingthettheimemvoariationdulationsofofthethedepspositedectrumenergywith,otrheinahelpsingleofaFdetectorourier
transform.TheseconceptsrelyonrobustfeaturesofSNsimulationsinstead
ofcomparingpredictionswithmeasurements,likepreviousmethods.Thede-
cantectionaofadditionallySNbyconitselftributwilletboepavarticleeryvphyaluablesics.astrophysicsobservationand

AndixeApp

Abbreviations

νµν¯µCMETLMpmfSNYeYL

—————————

unlessspecifieddifferently,νµstandsforνµ,ντ,¯νµ,andν¯τ
unlessspecifieddifferently,ν¯µstandsforν¯µandν¯τ
centerofmomentum
equilibriumhermaltcallosolarmass
meanfreepath
avernoupselectronfractionperbaryon
leptonfractionperbaryon

109

110

AppndixeA.Abbreviations

ndixeAppB

MonteCarloCode

eneraGB.1Conceptl

OurMonteCarlocodeisbasedonthatdevelopedbyJanka(1987)where
adetaileddescriptionofthenumericalaspectscanbefound.Thecodewas
firstappliedtocalculationsofneutrinotransportinSNebyJanka&Hille-
brandt(1989a,b)andJanka(1991).ItusesMonteCarlomethodstofollow
theindividualdestiniesofsampleneutrinos(particle“packages”withsuit-
ablyattributedweightstorepresentanumberofrealneutrinos)ontheirway
throughthestarfromthemomentofcreationorinflowtotheirabsorptionor
escapethroughtheinnerorouterboundaries.Theconsideredstellarback-
groundisassumedtobesphericallysymmetricandstatic,andthesample
neutrinosarecharacterizedbytheirweightfactorsandbycontinuousvalues
ofenergy,radialposition,anddirectionofmotion,representedbythecosine
oftheanglerelativetotheradialdirection.Theratesofneutrinointeractions
withparticlesofthestellarmediumcanbeevaluatedbytakingintoaccount
Fermionblockingeffectsaccordingtothelocalphase-spacedistributionsof
neutrinos(Janka&Hillebrandt1989b).
AsbackgroundstellarmodelsweusetheonesdescribedinSec.4.2.They
aredefinedbyradialprofilesofthedensityρ,temperatureT,andelectron
fractionYe,i.e.thenumberofelectronsperbaryon.Thecalculationsspanthe
rangebetweensomeinnerradiusRinandouterradiusRout.Theseboundthe
computationaldomainwhichisdividedinto30equallyspacedradialzones.
Ineachzoneρ,T,andYearetakentobeconstant.Rinischosenatsuchhigh
densityandtemperaturethattheneutrinosareinLTEinatleastthefirst
radialzone.Routisplacedinaregionwheretheneutrinosessentiallystream
freely.AtRinneutrinosareinjectedisotropicallyaccordingtoLTE.While
asmallnetfluxacrosstheinnerboundarydevelops,theneutrinosemerging

111

112

AppendixB.MonteCarloCode

fromthestararegeneratedalmostexclusivelywithinourcomputational
domain.IfRinischosensodeepthattheneutrinosareinLTE,theassumed
boundaryconditionforthefluxwillnotaffecttheresults.
Thestellarmediumisassumedtobeinthermodynamicequilibriumwith
nucleibeingcompletelydisintegratedintofreenucleons.Basedonρ,T,and
Yewecalculatealltherequiredthermodynamicquantities,notablythenum-
berdensities,chemicalpotentials,andtemperaturesofprotons,neutrons,
electrons,positrons,andtherelevantneutrinos.Exceptforrunsthatinclude
weakmagnetism,thechemicalpotentialsforνµandντaretakentobezero.
Nextwecomputetheinteractionratesineachradialzoneforallincludedpro-
cesses.Inthesimulationsdiscussedinthepresentwork,fermionphase-space
blockingiscalculatedfromtheneutrinoequilibriumdistributionsinsteadof
thecomputedphase-spacedistributions.Thissimplificationsavesalotof
CPUtimebecauseotherwisetherateshavetobere-evaluatedwheneverthe
distributionofneutrinoshaschangedafteratransporttimestep.Theap-
proximationisjustifiedbecausephase-spaceblockingismostimportantin
regionswhereneutrinosfrequentlyinteractandthusareclosetoLTE.Test
runswithoutthisapproximationshowthattheresultsarenotaffectedwithin
ournumericalaccuracy.
AtthestartofaMonteCarlorun,800,000testneutrinosarerandomly
distributedinthemodelaccordingtothelocalequilibriumdistributions.
Eachtestneutrinorepresentsacertainnumberofrealneutrinos.Inthisinitial
setupthenumberofrealneutrinosisdeterminedbyLTE.Thentransportis
started.Thetimestepisfixedat∆t=10−7s;recallthattheinteractionrates
donotchange.Atthebeginningofeachstepneutrinocreationtakesplace.
Thenumberoftestparticlesthatcanbecreatedisgivenbythenumber
ofneutrinosthatwerelostthroughtheinnerandouterboundariesplus
thoseabsorbedbythemedium.Basedon∆t,theproductionrates,andthe
factthattheinnerboundaryradiatesneutrinos,wecalculatethenumberof
neutrinosthatareproducedinonetimestepanddistributethemamongthe
availabletestneutrinosbyattributingsuitableweightfactors.Thesample
particlesarecreatedwithinthemediumorinjectedattheinnerboundaryin
appropriateproportions.
Duringatimestepthepathofeachtestparticlethroughthestellar
atmosphereisfollowedbyMonteCarlosampling.Withrandomnumbersand
storedtablesofthetotalinteractionrateswedecidewhetheritfliesfreelyor
interacts.Ifitinteractsitcanscatteroritcanbeabsorbed;inthelastcasewe
turntothenextparticle.Forscatteringwedeterminethenewmomentum
andpositionandthetimeofflightbeforetheinteractiontookplaceand
continuewiththeprocessuntilthetimestepisusedup.Particlesleaving
throughthelowerorupperboundariesareeliminatedfromthetransport.

B.2.StructureoftheCode

113

Afteracertainnumberoftimesteps(typicallyaround15,000)theneu-
trinodistributionreachesastationarystateandfurtherchangesoccuronly
duetostatisticalfluctuations.Atthatstagewestartaveragingtheoutput
quantitiesoverthenext500timesteps.

B.2StructureoftheCode
numThermoberofbarydynamiconsPisropcertomputediesofbythethegivMediumenmediumForeachdensitry.adialzUsingonetthehe
givandenthetempfactethatratureelTecandtronsYeandptogetherositronswitharetheinnLumTbEerwedensiobtaintyofthebarycheonsmicnalB
potentialofelectronsandpositronsbynumericallysolvingEquation(4.4).
InWiththeµsameandTwaythethenumpbroperedrtiesensioftiesofprotonselectronsandandneutronsposareitronsdarecetermined.alculated.

sistMainsofPaarlot:opTthheatproTime-StcesseseponeLotoipmestThepeamfteainrtpheartotofhert.heEachprogramtimestcoepn-
correspondsto10−7sofneutrinotransport.Sinceweuseblockingfactorsof
onlyneutrinoscalculateinLTEtheiasntaneractionapproxratesimationinttheotfirshetrealtimestepneutrinoandostoreccupation,themwine
arratime-ys.stepFlromooptspheseecificratesratesthearecumobtaulainedtedfpromtrobabilitieshearraaysrebyobtained.linearinInterptheo-
lation.

InjectsampleionaneutrinosndtCreathatiwonereofaNbsorbeutedorinosrleftIntheeacmohdeltimetstephroughthethenuminnerberoorf
outerboundaryduringtheprecedingtimestepisnowavailableforinjection
atliktheelihoodinnerofbeachoundaryeventandtheseemissionsampleeventsneutrinosintheamreedium.dividedAintoccordingonetogroupthe
thatisinjectedattheinnerboundaryandvariousothersthatcorrespondto
ntheumcberreationofrealmecneutrinoshanismsthatinsidewtheouldmbeedium.proWducedithinthethetsameimelikstepelihoisodcstalcu-he
latedandequallydistributedamongthesampleneutrinosastheirweights.
randomNeutrinosenergyinjectedaatccordingthelotowertheboLTEundarydisgettributionRinasinttheirherfirstadialcell,position,andana
Theanglecareationccordingmectothehanismsdistributioninsidetheofmnediumeutrinosareftreelyheschtreamingargedcoffurrenatsinphere.ter-
−++−acandtionsNNe→pNN→ννµeν¯µna,νnedν¯ee→nν→µν¯µν¯epfforoνrµ.νeByandMonν¯ete,eCaerlo→mνν¯ethofordsawlleflavobtainors,
theradialposition,energy,andangleversustheradialdirection.

114

AppndixeB.tMonearloCCodebTreraonfspnorteutrinosAftertoabllenewtranspnortedeutrinoseaqualsrethecreatednumbtranserpofortstneutrinosarts.Tthehatnurm-e-
mainedinthestarplustheinjectedandcreatedneutrinos.Thetransport
loisopusedthenuptakorestheyoneleaveneutrinothemoafterdel.theNeutrinosothertthathroughrtheemainedstaruninsidetilthetheirmtodimeel
fromtheprecedingtimestephavethewholetime∆t=10−7s.Sinceinjected
andlastsconlyreatedr∆tneutrinoswhereraisppaearatuniformlysomerandomdistributedtimetrandomheirntranspumbeorrtbettimeweenstep0
and1.Thenthetimeuntilthefirstinteraction(TIA)isobtainedwiththehelp
oftrino.theIfThighesIAtliestotaloiutsident∆teractionweratestoreforthetheneutrinoenergyofproptheertiestransafterpoirtedtfnreelyeu-
traveleduntiltheendof∆tandjumptothenextone.Iftheinteraction
takesplacewithintheavailabletimewehavetodecidewhichinteraction
takesplace.Theinteractionratesarealsodensitydependent,i.e.formost
locationsinsidethestarweunderestimatedTIAbyusingtheupperlimitof
thetotalrate.Therefore,whendecidingwhichinteractiontakesplaceoneof
the“interactions”leavestheneutrinopropertiesunchangedandinfactisno
interactionBeforewaetagoll.toThistheicompnteractionensatesfworethefirsttoolocalculatewestimatethenTewIApolosicallytion.of
theneutrinoatthepointofinteraction.Ifitleavesthemodelthroughthe
innerjumpotorotheuternexbtoone.undaryInoncaseitsitwasytaytoswithatthinpointhetiatreaiseofintliminatederestawenduwsee
theinteractionratesattheinteractionpointtodecidewhichprocesstakes
place.directionThenoftheaccordingoutgoingtothatneutrinoprocessarethedneweterminedenergyandandstorangleed.vIfersus∆tirsnadialot
usedupwedetermineanewTIAandgoonuntil∆tisusedup.
Afterallneutrinosuseduptheirtimeorleftthestarwegoovertothe
step.timenext

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