The Progenitor Dependence of the Preexplosion Neutrino Emission in Core-Collapse Supernovae

2.1 MB·English

by  Evan O'Connor

SUBMITTEDTOAPJ.JULY4,2012
PreprinttypesetusingLATEXstyleemulateapjv.12/16/11
THEPROGENITORDEPENDENCEOFTHEPREEXPLOSIONNEUTRINOEMISSION
INCORE-COLLAPSESUPERNOVAE
EVANO’CONNOR1 ANDCHRISTIAND.OTT1,23
SubmittedtoApJ.July4,2012
ABSTRACT
Weperformspherically-symmetricgeneral-relativisticsimulationsofcorecollapseandthepostbouncepre-
explosionphasein32presupernovastellarmodelsofsolarmetallicitywithzero-age-main-sequencemassesof
2 12M to120M . Usingenergy-dependentthree-speciesneutrinotransportinthetwo-momentapproximation
(cid:12) (cid:12)
1
with an analytic closure, we show that the emitted neutrino luminosities and spectra follow very systematic
0
trendsthatarecorrelatedwiththecompactness(∼M/R)oftheprogenitorstar’sinnerregionsviatheaccretion
2
rateinthepreexplosionphase. Wefindthatthesequalitativetrendsdependonlyweaklyonthenuclearequa-

l tionofstate,butquantitativeobservationalstatementswillrequireindependentconstraintsontheequationof
u
state and the rotation rate of the core as well as a more complete understanding of neutrino oscillations. We
J
investigate the simulated response of water Cherenkov detectors to the electron antineutrino fluxes from our
4
modelsandfindthatthelargestatisticsofagalacticcorecollapseeventmayallowrobustconclusionsonthe

innerstructureoftheprogenitorstar.
]
E
Subject headings: equation of state - hydrodynamics - neutrinos - stars: evolution - stars: neutron - stars:
H
supernovae: general
.
h
p
1. INTRODUCTION the region behind the stalled shock to revive the shock and
-
o maketypical∼1Bcore-collapsesupernovaexplosion(Janka
r The radial instability of the electron-degenerate, et al. 2007; Müller et al. 2012 and references therein). Only
t Chandrasekhar-mass core marks the beginning of the fi- hyper-energetic(i.e., O(10)B)explosionsmayrequireadif-
s
a nal episode in the life of a massive star with zero-age ferent mechanism (Ugliano et al. 2012), e.g., rapid rotation
[ main-sequence(ZAMS)massintherange∼8M(cid:12)−130M(cid:12). combinedwithstrongmagneticfields,whichmayleadtoen-
Collapseensuesand,oncefullydynamical,separatesthecore ergeticjet-drivenexplosions(e.g.,Burrowsetal.2007).
1
intoasubsonicallyhomologouslycontractinginnercoreand
v For a galactic or near-extragalactic core-collapse super-
asupersonicallycollapsingoutercore. Atnucleardensity,the
0 nova, neutrinos offer the unique possibility of directly ob-
repulsivecomponentofthenuclearforceleadstoastiffening
0 serving the dynamics and thermodynamic conditions preva-
oftheequationofstate(EOS).Thisstabilizestheinnercore,
1 lentinthesupernovacore. Togetherwithgravitationalwaves
which overshoots its new equilibrium, then rebounds into
1 (see, e.g., Ott 2009; Kotake 2011) they will herald the next
7. tfhroemouittesrecdogree,.laTunhcishiinngstaansttroinngtihmyedriosdyrenfaemrriecdshtoocakswcaovree nearby supernova possibly hours before any telescope sensi-
0 bounce.Theinnercorehasamassof∼0.5M atbounceand tive to electromagnetic waves will notice the event. In the
2 (cid:12) probable case that the next galactic supernova occurs in a
thismaterialbecomestheunshockedcoreoftheprotoneutron
1 dust-enshrouded region and/or close or behind the galactic
star. The shock formed at core bounce propagates into the
: center, thesupernovamaybeimpossibletoobserveinbroad
v outer core, but dissociation of accreting nuclei into neutrons
bandsoftheelectromagneticspectrum, makingneutrinoand
i andprotonsandelectroncaptureonfreeprotonsintheregion
X gravitational-waveobservationsevenmoreimportant.
behind the shock (the postshock region) soon sap its might,
r driving it into submission to the ram pressure of accretion. TheobservationofneutrinosfromSN1987AintheLarge
a
The shock stalls and turns into a standing accretion shock Magellanic Cloud (Hirata et al. 1987; Bionta et al. 1987;
thatmustberevivedtounbindthestellarenvelopeanddrive Alekseev et al. 1987) confirmed the basic picture of core
acore-collapsesupernovaexplosion. collapse and early protoneutron star evolution (e.g., Sato &
Suzuki 1987; Bruenn 1987; Burrows & Lattimer 1986; Bur-
Neutrinos play a pivotal and dominant role in stellar col-
rows1987,1988;Arnettetal.1989;Jegerlehneretal.1996;
lapse and core-collapse supernovae. Neutrinos and antineu-
Loredo & Lamb 2002; Yüksel & Beacom 2007; Pagliaroli
trinos of all flavors carry away the ∼300B (=3×1053ergs)
etal.2009bandreferencestherein),butthesmallnumberand
of gravitational binding energy of the remnant neutron star
poortimingoftheobservedeventsdidnotallowfar-reaching
over tens of seconds after core bounce. Aided by multi-
androbustconclusionsoncore-collapsesupernovadynamics
dimensional fluid instabilities, they probably deposit, within
andtheinvolvedneutrinophysics,nuclearphysicsandastro-
afewhundredmillisecondsafterbounce,sufficientenergyin
physics.
1TAPIR, Mailcode 350-17, California Institute of Technology, The situation will be completely different when the
Pasadena,CA91125,[email protected],[email protected] neutrino burst from a galactic core-collapse supernova
2KavliIPMU,UniversityofTokyo,Kashiwa,Japan reaches current and near-future neutrino detectors on Earth.
3AlfredP.SloanResearchFellow Super-Kamiokande (Fukuda et al. 2003; Ikeda et al.
2
[Super-Kamiokande Collaboration] 2007), IceCube (Ab- 2009; Dasgupta & Dighe 2008; Duan et al. 2010 and refer-
basi et al. [IceCube Collaboration] 2011), LVD (Aglietta encestherein). However, anumberofrecentstudiessuggest
et al. 1992; Vigorito [LVD Collaboration] 2011), Borexino thatcollectiveoscillationsmaybecompletelyoratleastpar-
(Alimonti et al. [BOREXINO Collaboration] 2009; Cado- tially suppressed in the preexplosion accretion phase of or-
nati et al. 2002), KamLAND (Piepke 2001), SNO+ (Kraus dinarycore-collapsesupernovae(Chakrabortyetal.2011b,a;
& Peeters 2010), Noνa (Davies, [for the NOνA Collabo- Sarikasetal.2012),butseeCherryetal.(2012)andDasgupta
ration] 2011) and others will together see many thousands etal.(2012)fordiscrepantresults.
of neutrinos from a core collapse event at 10kpc (Scholberg
Provided that collective oscillations can be ignored in the
2012).Distanceestimatesbasedontheobservedneutrinoflux
preexplosion phaseand thatthe θ mixing angleindeed has
(Kachelrieß et al. 2005), sky localization (Beacom & Vogel 13
the large value suggested by recent measurements (An et al.
1999;Tomasetal.2003),andtriggeringofgravitational-wave
2012), the neutrino mass hierarchy may be inferred from
searches by the reconstruction of the time of core bounce
the qualitative shape of the early postbounce neutrino signal
(Pagliaroli et al. 2009a; Halzen & Raffelt 2009) will likely
(Kachelrießetal.2005;Serpicoetal.2012).
allbepossible.
Apreexplosionaccretionphasewithsuppressedcollective
Well-timedhigh-statisticscoincidentneutrinoobservations
oscillations would also offer the opportunity to probe the
willallowtoprobeindetailabroadrangeofsupernovaastro-
structure of the progenitor star on the basis of the observed
physics, nuclear physics, and neutrino physics (see Burrows
neutrinosignal.Thedetailsofthepreexplosionneutrinoemis-
et al. 1992; Wurm et al. 2012; Raffelt 2010 for overviews).
sion have been discussed carefully, e.g., by Thompson et al.
Fast characteristic temporal variations in the preexplosion
(2003)andLiebendörferetal.(2004)andweshallnotrepeat
neutrinofluxeswouldbe tell-tale signsofmulti-dimensional
themhere. Itis,however,necessarytooutlineitsmostsalient
fluid instabilities in the postshock region (Ott et al. 2008;
features. For simplicity, we neglect neutrino oscillations in
Marek et al. 2009; Lund et al. 2010; Brandt et al. 2011)
thefollowing.
and/or early postbounce ring-down oscillations of a rapidly
spinning protoneutron star (Ott et al. 2012). A sudden deep In core collapse and in the subsequent postbounce evolu-
drop of the accretion-driven component of the neutrino lu- tion, emission of νe and ν¯e occurs via charged and neutral
minosity (primarily in νe and ν¯e) within a few hundred mil- currents, while heavy-lepton neutrinos νx = {νµ,ν¯µ,ντ,ν¯τ}
lisecondswouldindicatetheonsetofexplosion(e.g.,Burrows are created exclusively via thermal neutral-current pair pro-
et al. 1992; Hüdepohl et al. 2010; Fischer et al. 2012, 2010) cesses. Before core bounce, only νe are emitted from elec-
and an abrupt cut-off of the entire neutrino flux within sec- tron capture in the collapsing core. Milliseconds after core
ondsofitsonsetwouldindicatetheformationofablackhole bounce,theshockbreaksoutoftheνe neutrinosphere(where
(Burrows1986,1988;Beacometal.2001;Liebendörferetal. the optical depth is τνe ≈ 2/3) and a strong burst of νe is
2003,2004;Sumiyoshietal.2006;Fischeretal.2009). The emittedfor∼20msfromrapidelectroncaptureonthefreshly
spectralcharacteristicsandlong-termspectralevolutionofthe abundantfreeprotonsbehindtheshock. νxarecopiouslycre-
neutrino flux could provide important constraints on the nu- ated in the hot interior of the protoneutron star after bounce
clearEOS(Robertsetal.2012;Mareketal.2009)and/orthe and begin to diffuse out, leading to a steep rise, quick lev-
spin of the progenitor core (Ott et al. 2008; Marek & Janka eling and subsequent slow decay of the νx luminosity (Lνx).
2009). The ability of some detectors to distinguish interac- ν¯e productionviacharged-currentpositroncaptureisinitially
tionsofdifferentneutrinoflavorswouldleadtoconstraintson suppressed due to the high degeneracy of the electrons. The
theneutron-to-protonratiointheneutrino-drivenwindphase, latter is partially lifted after bounce at the moderate-density,
allowinganobservationaltestofcore-collapsesupernovaeas hot edge of the protoneutron star and Lν¯e rises, reaching or
potentialsitesforr-processnucleosynthesis(Hüdepohletal. surpassingthevalueatwhichLνe levelsoffaftertheneutron-
2010;Fischeretal.2010;Wurmetal.2012). izationburstdecays. Thesubsequentpreexplosionluminosity
canroughlybesplitintoadiffusivecomponentfromthecore
The neutrino signature of core collapse, of the subsequent and accretion luminosity (∝GM[R ]M˙/R , where R is an
core-collapse supernova evolution and of the protoneutron ν ν ν
approximate neutrinosphere radius) from or from above the
starcoolingphase,isinvariablyintertwinedwithneutrinoos-
neutrinosphere (Burrows 1988). L is primarily diffusive,
cillation physics. The robustness of all of the above men- νx
while L and L are dominated by accretion. In general,
tioned observational conclusions will depend on our under- νe ν¯e
L ≈L >L ,but4L =L +L +L +L >L +L .
standingoftheimpactofneutrinoflavoroscillations. Neutri- νe ν¯e νx νx νµ ν¯µ ντ ν¯τ νe ν¯e
ν havethelowestopacity,sincetheyinteractonlyvianeutral
nospropagatingfromtheiremissionsitetodetectorsonEarth x
currents. Theydecouplefrommatteratthesmallestradiiand
may experience (i) so-called vacuum oscillations driven by
highesttemperaturesandthushavethehighestaverageener-
neutrinomassdifferences(Pontecorvo1968),(ii)oscillations
gies(cid:104)(cid:15) (cid:105). ν¯ haveaslightlyloweropacitythanν ,leadingto
mediated by a resonance in ν–e− scattering (the Mikheyev- ν e e
the well established neutrino energy hierarchy in the preex-
Smirnov-Wolfenstein [MSW] effect; Mikheev & Smirnov
plosion phase (cid:104)(cid:15) (cid:105)>(cid:104)(cid:15) (cid:105)>(cid:104)(cid:15) (cid:105). The mean energy of all
1985; Wolfenstein 1978), and (iii) oscillations due to ν–ν νx ν¯e νe
speciesgrowswithincreasingpostbouncetime,reflectingthe
scattering (Pantaleone 1992; see Duan et al. 2010 for a re-
recessionoftheneutrinospheresduetothecontractionofthe
view). Vacuum and MSW oscillations are well understood
protoneutronstar.
andtheiroutcomesdependessentiallyonlyonneutrinomix-
ingparameters,inparticulartheneutrinomasshierarchyand Considering that the accretion luminosity will scale with
themixingangles. Theν–ν-scatteringdrivenoscillations,on thepostbounceaccretionrateM˙, onewouldnaturallyexpect
the other hand, have a non-linear Hamiltonian that may lead an increase of the detected neutrino events with increasing
to so-called collective oscillations with very complex spatial mass of the stellar core. Since higher accretion rates corre-
and temporal outcome that remains to be fully understood spondtomorematerialcompressingandsettlingmorerapidly
(see,e.g.,Hannestadetal.2006;Duanetal.2007;Foglietal. ontheprotoneutronstar,thelatter’souterregionswillbehot-
3
ter. Thus the thermal neutral-current emission will be en- preservingtheoverallsystematicswithaccretionrate. Buras
hanced,leadingtohigherluminositiesandhighermeanneu- et al. (2006) also were the only authors to suggest that the
trinoenergies. accretion-ratedependenceofluminosityandtotalemitteden-
ergyinthepreexplosionphasecouldbeusedtoinferthestruc-
Thevariationofthepreexplosionneutrinosignalwithpro-
ture of the progenitor. The other studies, being focused on
genitorstarZAMSmasswasfirstdiscussedbyWoosleyetal.
aspectssuchasneutrinooscillations,blackholeformation,or
(1986)basedonthepioneeringsimulationsofWilson(1985)
thelate-timepost-explosionevolution,didnotconsiderobser-
andWilsonetal.(1986). Theseauthorsprovidedtotalemis-
vationalconsequences.
sion characteristics and spectra that show a systematic in-
crease of total energy emitted in neutrinos and mean ν¯e en- Thompson et al. (2003), using a limited set of three pro-
ergywithZAMSmassintherangefrom10−25M(cid:12). Mayle genitor models ({11, 15, 20}M(cid:12) at ZAMS), found similar
et al. (1987), before SN 1987A, carried out simulations of systematics as the aforementioned studies, but also carried
a range of progenitor stars with ZAMS mass in the range outananalysisoftheexpectedsignalinvariousneutrinode-
12−100M(cid:12). They found that the νe neutronization burst tectorsinthefirst250msafterbounce. TheycomputedIBD
shows little dependence on the progenitor, due to the rather event rates for their 20-M and 11-M and found a factor
(cid:12) (cid:12)
universal homologous collapse and bounce dynamics. Fur- of two more IBD events for the former, which would allow
thermore, they mentioned, though did not discuss in detail, a high-confidence distinction between these progenitors for
thattheluminositiesandmeanneutrinoenergiesincreaseasa a galactic core collapse event. However, their 15-M model
(cid:12)
functionofironcoremass(andnotZAMSmass). Amorede- yielded a postbounce neutrino signal very similar to that of
tailedandclearphysicaldiscussionwasprovidedbyBruenn their 11-M model and would be indistinguishable by neu-
(cid:12)
(1987), who contrasted the predicted neutrino signal from trino observations alone. This suggests that ZAMS mass is
the early postbounce phase in two different progenitor core not a good parameter to describe presupernova stellar struc-
models with neutrino observations of SN 1987A. He noted ture(cf.,Bruenn1987).
that there are significant uncertainties in connecting a given
ZAMS mass to precollapse structure. Instead of a progeni- Inthisarticle,wepresentafreshlookattheprogenitorde-
torwithanassociatedZAMSmass, heconsideredamassive pendenceoftheneutrinosignatureinthepreexplosionaccre-
(and high-entropy) 2.05-M iron core model and a lower- tionphaseofcore-collapsesupernovae. Weperform1Dgen-
(cid:12)
mass (and lower-entropy) 1.35-M iron core model in his eral relativistic radiation-hydrodynamics core collapse sim-
(cid:12)
spherically-symmetric(1D)neutrinoradiation-hydrodynamic ulations of 32 progenitor models from the single-star solar-
simulations. Heshowedthatthemoremassivecoreleadstoa metallicity presupernova model suite of Woosley & Heger
consistentlyhigherν¯ luminosityinboththeaccretionanddif- (2007)andfollowthepostbouncepreexplosionevolutionfor
e
fusionsectors. ThewaterCherenkovdetectorsthatobserved 450ms. In ZAMS mass, these models range from 12M(cid:12)
neutrinos from SN 1987A are most sensitive to the inverse to 120M(cid:12), but guided by the previous results discussed in
betadecay(IBD)reactionν¯ +p→n+e+. Bruenn(1987)pre- theabove,wechoosenottoparameterizeoursimulationsby
e
dicted a factor of two difference in the integrated early IBD ZAMSmass. Insteadweemploythecompactnessparameter
events between the massive and the low-mass core in these ξM ∼M/R(M)(forarelevantmassscaleM,measuredatthe
detectors. Heconcludedthattheneutrinosignalobservedby time of bounce). As shown in O’Connor & Ott (2011), ξM
thesedetectorsfromSN1987Awasmostconsistentwiththe is the most important stellar structure parameter governing
low-masscore. Burrows(1988),whocarriedoutaparameter the postbounce accretion evolution to a remnant mass scale
study of quasi-hydrostatic protoneutron star cooling, consid- M.Wedemonstratethatthepreexplosionneutrinoemissionis
ering various initial masses, ad-hoc accretion rates, and dif- very well parameterized by the compactness. The preexplo-
ferent nuclear EOS, found a similar trend. He showed that sion luminosities and mean energies of all neutrino species
more massive cores, higher accretion rates, and softer EOS increase essentially monotonically with increasing ξM. We
lead to stronger, higher-energy neutrino emission. Some of computepredictedintegratedIBDeventsforagalacticcore-
hisstrongestemitterswerecasesinwhicheventuallyablack collapse supernova in the Super-Kamiokande detector and
holewasformed. show that the clear systematics governed by ξM carries over
to observation, even when standard MSW neutrino oscilla-
Liebendörfer and collaborators carried out a sequence of
tions are taken into account. Our results thus indicate that
studiesoftheprogenitordependenceoftheneutrinosignalus-
– in the absence of complicated collective neutrino oscilla-
ing modern general relativistic 1D radiation-hydrodynamics
tions – a high-statistics detection of neutrinos from the pre-
simulations (Liebendörfer et al. 2001, 2002, 2003, 2004).
explosion phase will allow, in principle, a tight constraint of
They showed that the ν neutronization burst is indeed al-
e thecompactnessoftheprogenitorstar’score. This,however,
most independent of progenitor structure (as first suggested
willrequireknowledgeofthenuclearEOSandoftherotation
by Mayle et al. 1987). They also qualitatively and quantita-
rateofthecollapsedcore,since,asweshow,bothcandilute
tivelyconnectedtheevolutionofthepostbouncepreexplosion
theotherwiseclearcompactness-dependentneutrinoemission
luminosity to the postbounce accretion rate, but did not dis-
systematics.
cuss observational implications. Their results were corrobo-
ratedbysimilarlysophisticatedsubsequentstudiesofKachel- This paper is structured as follows. In §2, we discuss our
rieß et al. (2005); Buras et al. (2006); Fischer et al. (2009); generalrelativistichydrodynamicscodeGR1Dandintroduce
Sumiyoshi et al. (2008); Fischer et al. (2010, 2012); Serpico itsextensiontoneutrinoradiation-hydrodynamicsinthetwo-
etal.(2012). Ofthese,Burasetal.(2006)presentedthemost momentapproximation,nuGR1D.Theinitialmodelsandthe
comprehensive analysis and also compared between 1D and employed EOS are discussed in §3. In §4, we present re-
axisymmetric(2D)results. Theyfoundthatin2D,convection sults from a benchmark collapse and postbounce simulation
intheprotoneutronstaraltersthestructureofthelatter,affect- thatallowsustocomparewiththepreviouslypublishedcode
ingtheneutrinoemissionstarting∼100msafterbounce, but comparisonofLiebendörferetal.(2005)toassessnuGR1D’s
4
abilitytoreproduceresultsoffullBoltzmannneutrinotrans- tensorareP =E X2andP =g E /3,respec-
rr,(ν),thin (ν) ii,(ν),thick ii (ν)
port. Wepresenttheresultsofoursimulationsin§5,analyze tively.
thedependenceoftheneutrinosignalonprogenitorcompact-
Wesetouttosolvethesystemofequationsviastandardhy-
ness, discusspredictedIBDsignalsfromagalacticcorecol-
perbolicmethodsborrowedfromconservativehydrodynamic
lapse event in the Super-Kamiokande detector, and explore
schemes (Pons et al. 2000). Transport variables live at cell
potential degeneracies introduced by EOS and rotation. Fi-
centers and we employ piece-wise linear reconstruction to
nally, in §6, we critically summarize our work and conclude
cellinterfaceswithvanLeer’slimiter(vanLeer1977)andthe
by contrasting our results with the early neutrino signal ob-
HLLEapproximateRiemannsolver(Einfeldt1988)forcalcu-
servedfromSN1987A.
lating the intercell fluxes. A complication arises when solv-
ingtheneutrinomomentequationsinthehigh-opacitylimit.
2. METHODS In this case, the standard fluxes returned from the Riemann
solveraredominatedbyanumericaldiffusionterm. Wefol-
We make use of the open-source 1D general relativistic
low Audit et al. (2002) and modify the fluxes to correct for
hydrodynamics code GR1D (O’Connor & Ott 2010; avail-
this. Essentially,themodifiedfluxesreturnthediffusion-limit
able at http://www.stellarcollapse.org) outfit-
fluxinthehighopacitylimit. Themodifiedformoftheneu-
ted with an energy-dependent multi-species M1 neutrino
trinoenergydensityfluxthroughthei+1/2interfaceisgiven
transport scheme in which the zeroth and first moments of
by
the neutrino distribution function are evolved. We refer the
reader to O’Connor & Ott (2010) for details on GR1D and a˜+Fi,R −a˜−Fi+1,L+(cid:15) a˜+a˜−(Ei+1,L−Ei,R)
describe in the following our current implementation of the Fi+1/2= r,(ν) r,(ν) (ν) (ν) (ν) . (5)
transport scheme. For this first application, we neglect the r,(ν) a˜+−a˜−
computationally-expensiveenergy-couplingneutrinointerac- The corresponding modified interface flux for the neutrino
tions and transport terms – these terms are undoubtedly im- fluxevolutionequationis,
portantformakinghighlyaccuratepredictionsoftheneutrino
signature(see, e.g., Lentzetal.2012a,b), butareunlikelyto Pi+1/2=(cid:15) P˜i+1/2+(1−(cid:15)2 )(Pi+1,L+Pi,R)/2, (6)
rr,(ν) (ν) rr,(ν) (ν) (ν) (ν)
affect the general trends we observe. We will address them
infuturework,butprovideadiscussionontheconsequences with
ofneglectingthesetermsviaacomparisontofullBoltzmann (cid:15) (a˜+Pi,R −a˜−Pi+1,L)+a˜+a˜−(Fi+1,L−Fi,R)
neutrinotransportsimulationsin§4. P˜i+1/2= (ν) rr,(ν) rr,(ν) (ν) (ν) . (7)
rr,(ν) a˜+−a˜−
OurM1schemecloselyfollowsShibataetal.(2011),who
formulate the M1 evolution equations in a closed covariant
In these equations, (cid:15) controls the modification to the
(ν)
form. The scheme is simplified greatly by neglecting the
fluxestoaccountforthehighopacity. FollowingAuditetal.
energy-coupling terms. This further requires that the veloc-
(2002),wetake
ity dependent terms are also ignored. In this limit, and us-
(cid:18) (cid:19)
ing the Schwarzschild-like metric and radial-polar slicing of 1
(cid:15) =min 1, , (8)
GR1DandsettingG=c=M(cid:12)=1,theevolutionequationsfor (ν) κ(ν)∆r
theneutrinoenergydensity,E ,andtheneutrinofluxvector,
(ν)
Fr,(ν),simplifyfromEquations3.37and3.38ofShibataetal. where κ(ν) is the sum of the scattering and absorptive opac-
(2011)to ities. These opacities are strong functions of energy and are
alsospeciesdependent. Wenotethatwhen(cid:15) is1,theinter-
∂E + 1∂ (cid:18)αr2F (cid:19)=α2St , (1) cellfluxesreducetothestandardHLLEapp(rνo)ximation. The
t (ν) r2 r X2 r,(ν) (ν) characteristicspeedsneededfortheHLLEschemearecalcu-
latedinthesamespiritastheneutrinopressuretensor(Shibata
and
etal.2011;Kurodaetal.2012),
1 (cid:18)αr2 (cid:19) E (1−p )
∂tFr,(ν)+r2∂r X2 Prr,(ν) =αX2S(rν)+α (ν) r (ν) , (2) λ(ν)= 3p(ν2)−1λ(ν),thin+3(1−2p(ν))λ(ν),thick, (9)
whereSα istheneutrinointeractionsourceterm(seebelow), where in the zero velocity limit, λ = ±α/X and
αisthelapsefunctionandX =(1−2M(r)/r)−1/2. P isthe √ (ν),thin
rr,(ν) λ =±α/( 3X). a˜+ anda˜− arethemaximumandmini-
neutrinopressuretensorandistakentobeaninterpolationbe- (ν),thick
mumvalues,respectively,ofthesecharacteristicspeedseval-
tweenthetwolimitingcasesoffreestreaminganddiffusion.
uatedfromboththerightandleftreconstructedvariables.
WefollowShibataetal.(2011),whoexpressP as
ii,(ν)
Finally, the source terms in Equations 1 and 2 are taken
3p −1 3(1−p )
P = (ν) P + (ν) P , (3) fromShibataetal.(2011). Inthezerovelocitylimit,
ii,(ν) 2 ii,(ν),thin 2 ii,(ν),thick
St=(η −κ E )/α, (10)
where p istheEddingtonfactor,takenheretobethemax- (ν) a,(ν) (ν)
(ν)
imum entropy closure in a closed, analytic form (Minerbo Sr=−(κa,(ν)+κs,(ν))Fr,(ν)/X2, (11)
1978;Cernohorsky&Bludman1994),
where η , κ , and κ are the neutrino emissivity, neu-
(ν) a,(ν) s,(ν)
1 f2 trino absorption opacity, and the neutrino scattering opac-
p = + (ν)(6−2f +6f2 ). (4) ity, respectively. We precompute the neutrino interaction
(ν) 3 15 (ν) (ν)
terms for each neutrinos species (we treat ν , ν¯ and ν =
e e x
Intheno-velocitylimitforGR1D, f =|F /(E X)|. The {ν ,ν¯ ,ν ,ν¯ }) and neutrino energy group in dense tabular
(ν) r,(ν) (ν) µ µ τ τ
free streaming and diffusion limits of the neutrino pressure form as a function of density ρ, temperature T, and electron
5
fractionY . Wethenuselinearinterpolationforefficienton-
e ✶✵✹
the-fly interpolation. We include all standard iso-energetic
scattering processes, charged-current absorption and emis-
sion, and thermal pair-production processes (Burrows et al. ✶✵✷ ☎✆✝✞✟
2006; Bruenn 1985) in the calculation of the neutrino in- ☎✠✝✟
✶
teraction terms. Since the neutrino–matter interactions for
heavy-lepton neutrinos and antineutrinos are slightly differ-
ent, NuLib averages the two values of the emissivities and (cid:0)✵✽
▼
opacities. Ourlibraryofneutrinointeractionroutines(which ①
we call NuLib) is open source and available as a GitHub (cid:0)✵✻
repositoryathttp://www.nulib.org. NuLibrequires
an EOS for the evaluation of the emissivities and opacities. (cid:0)✵✹
Our treatment of thermal pair processes in GR1D warrants
some comments. Since we do not currently consider energy (cid:0)✵✷
(orspecies)couplingforthermalemissionprocessessuchas
electron–positronannihilationtoaneutrino–antineutrinopair,
(cid:0)
wecomputeanemissivitybasedonthethermalcontentofthe ✶✷ ✶✺ ✷(cid:0) ✷✺ ✸(cid:0) ✹(cid:0) ✺(cid:0) ✻(cid:0) ✼(cid:0)✽(cid:0) ✶(cid:0)(cid:0)✶✷(cid:0)
matterignoringanyfinalstateneutrinoblocking.Welimitthe ❩✁✂✄ (cid:210)
neutrino energy density to the blackbody occupation density FIG.1.— Compactness parameters for the 32 considered presupernova
modelsofWoosley&Heger(2007)versusZAMSmassasevaluatedfrom
byusingKirchhoff’slawtodetermineaneffectiveopacityfor collapse simulations with the LS220 EOS. We show both ξ1.75 and ξ2.5.
neutrino–antineutrinoannihilationfromthethermalemissiv- ThemappingbetweenZAMSmassandprecollapsestructureishighlynon-
ity. Asweshallsee,thismethodperformswellatpredicting monotonic,makingtheformeranill-suitedparameterfordescribingprogen-
itorstructureincorecollapsesimulations.
thethermalneutrinofluxoftheheavy-leptonneutrinosduring
nuclearequationofstate.Theempiricalparameterintroduced
thepreexplosionphase.
inO’Connor&Ott(2011)isthecompactnessoftheprogen-
InnuGR1D,wefirstupdatethehydrodynamicvariablesto itor, measured at the time of core bounce. It is an inverse
then+1-thtimestep. Wethencomputetheneutrinoopacities measureoftheradialextentofagivenmasscoordinateatthe
and emissivities associated with the updated hydrodynamic timeofbounce,
variables. We update the radiation field operator-split. The
fluxtermissolvedexplicitly,usingtheradiationmomentsof ξ = M/M(cid:12) (cid:12)(cid:12) , (12)
the n-th timestep. We calculate the neutrino–matter interac- M R(Mbary=M)/1000km(cid:12)t=tbounce
tion terms using the n+1 radiation moments via a local im-
where R(M =M) is the radial coordinate that encloses a
plicit update. With the n+1 radiation energy density source bary
baryonicmassofM atthetimeofcorebounce. InO’Connor
term,wethenupdatetheenergydensityandelectronfraction
& Ott (2011), we chose M =2.5M , since this is the rele-
of the matter. We use 24 energy groups, with lowest-energy (cid:12)
vantmassscaleforblackholeformation,i.e.,atypicalmax-
groupcentersat0.5MeVand1.5MeV,andthenspacedloga-
imumbaryonicmassatwhicharangeofEOScannolonger
rithmicallyupto200MeVforν ,ν¯ andν . Wenotethatfor
e e x
support a neutron star against gravity. In this study, we pri-
thehighestenergybinsitoccasionallyoccursthattheevolved
marily use ξ . The motivation for this is that during the
neutrinofluxvectorexceedstheevolvedneutrinoenergyden- 1.75
postbouncepreexplosionphase,therelevantmassscale,espe-
sity. This tends to occur in the most dynamic phases of our
ciallyformodelswithrelativelysmallcompactness,ismuch
simulations and where the opacities vary significantly from
less than 2.5M . In this study, we choose 1.75M because
onezonetothenext. Whenthisisthecasewelimittheneu- (cid:12) (cid:12)
thisisclosetotheaveragebaryonicmassinsidetheshockat
trinofluxtotheneutrinoenergydensity.
200–300ms after bounce for all models: in the two extreme
3. INITIALMODELSANDEQUATIONSOFSTATE modelsthatspanthespaceincompactnessparameter(model
s12WH07, [ξ =0.24 and ξ =0.022], on the lower end;
1.75 2.5
We employ the most recent non-rotating solar-metallicity model s40WH07 [ξ =1.33 and ξ =0.59] on the upper
1.75 2.5
single-starmodelsetfromthestellarevolutioncodeKEPLER end),thebaryonicmassaccretedthroughtheshockat250ms
(Woosley & Heger 2007). This model set contains the pre- afterbounceis1.45Mand2.05M,respectively. Wefurther
supernova configuration of 32 stars ranging in ZAMS mass justifyourmotivationofusingξ overξ in§5.1.InFig.1,
1.75 2.5
from 12M(cid:12) to 120M(cid:12). We denote individual models by weplotbothξ1.75 andξ2.5 versusZAMSmassforall32con-
sXXWH07,whereXXcorrespondstotheintegerZAMSmass sideredmodels. ξ isprovidedinTable1forallmodels.
1.75
of the model, e.g., s12WH07 is the 12-M model of this
(cid:12) ForFig.1,onenotesthatwhileξ andξ differquanti-
model set. In O’Connor & Ott (2011), we investigated this 1.75 2.5
tatively,thereisnosignificantqualitativedifferencebetween
and other model sets in the context of black hole formation.
them. The overall trends transcending individual models re-
Undertheassumptionofafailedcore-collapsesupernova,we
main, including the two regions of high compactness near
found a strong empirical relation between the properties of
22–25M and35–45M . ξ simplyprovidesamorefine-
thepresupernovastructureandtheevolutionofthefailingsu- (cid:12) (cid:12) 1.75
grained parameterization at the lower mass scale relevant in
pernova, e.g., the time to black hole formation. This led to
the first few hundred milliseconds after bounce. Note, how-
a clear prediction: If we observe black hole formation in a
ever, that there are a few models that have similar ξ , but
failed core-collapse supernova via neutrinos, the lifetime of 2.5
rather different density structure at small enclosed masses
the protoneutron star (and thus of the neutrino signal) relays
andradiiand,hence,adifferentξ . Modelss14WH07and
direct information about the presupernova structure. How- 1.75
s16WH07areexamples.
ever,suchaprediction,(i)requiresafailedsupernova,which
maynotbethenorm,and(ii)hasastrongdependenceonthe In this study we perform core collapse simulations with
6
eachprogenitorandtwoEOS.WeusetheEOSofLattimer&
✼✵
Swesty (1991) with a nuclear incompressibility of 220MeV.
TheLS220EOSisbasedonacompressibleliquid-dropmodel ✲✁ ✻✵ ✢❡ ✢❡
of the nucleus. Of the publicly available nuclear EOS, the ✺✵
LS220 EOS best matches the constraints from nuclear the- ✹✵
✁
ory and astrophysical observations (see Fig. 1 of Ott et al. (cid:0) ✸✵ ✜✚✚
2011 and Demorest et al. 2010; Hebeler et al. 2010; Steiner
et al. 2010; Özel et al. 2010). We also employ the relativis- ♥ ✷✵ ✛✚✚ ✢①
ticmeanfieldEOSofShenetal.(2011)thatisbasedonthe ✶✵ ✚✌✙ ✚ ✙ ✗✚ ✗✙
TM1parameterset. ItisverydifferentfromtheLS220EOS. ✵
✵ ✺✵ ✶✵✵ ✶✺✵ ✷✵✵ ✷✺✵ ✵ ✺✵ ✶✵✵ ✶✺✵ ✷✵✵ ✷✺✵
The maximum neutrino-less β-equilibrium cold neutron star
gravitationalmassesare2.04M(cid:12) and2.24M(cid:12) fortheLS220 ✷✹ ❆✠✡☛☞✌✍✎☛✏✑✏✒❆✓
and HShen EOS, respectively. The radius of a neutrino-less ❱☞✒✏☞✔
β-equilibrium cold neutron star with a gravitational mass of ✷✵ ✕✖✠✒✗✘ ✢①
1.4M usingtheLS220EOSis12.7km.FortheHShenEOS,
(cid:12)
the corresponding radius is 14.6km. For details on our par- ✟✞ ✶✻
✁
ticular implementation and the treatment of the low-density ✞ ✢❡
EOS, we refer the reader to O’Connor & Ott (2010, 2011). ✶✷ ✢❡
The EOS tables, reader and interpolation routines are avail-
ablefromhttp://www.stellarcollapse.org. ✽
✵ ✺✵ ✶✵✵ ✶✺✵ ✷✵✵ ✷✺✵ ✵ ✺✵ ✶✵✵ ✶✺✵ ✷✵✵ ✷✺✵
❜✂✄☎✆✝ ❜✂✄☎✆✝
4. COMPARISONOFNUGR1DTOBOLTZMANNTRANSPORT FIG.2.—Neutrinoluminositiesandrootmeansquaredenergiesplottedasa
functionofpostbouncetimeforthes15WW95progenitor.Theseluminosities
(toppanels)andenergies(bottompanels)correspondtothecomparisonstudy
Sinceourimplementationofneutrinotransportisnewand
ofLiebendörferetal.(2005).Theleftpanelscontainresultsforνe,theright
approximate, a comparison with published results of full panelsshowν¯e(thicklines)andνx(thinlines)results. Theinsetplotinthe
Boltzmann neutrino transport is warranted. This will allow upperleftpanelshowstheνeluminosityaroundcorebounce.Showninsolid
blacklinesareluminositiesandrootmeansquaredenergiesobtainedwith
ustotesttheabilityofourcodetoreproducetheneutrinolu-
nuGR1D.Thebluedashedlinesandreddashed-dottedlinesaretheresults
minositiesandspectralpropertiesinthepreexplosionphase. fromLiebendörferetal.(2005)usingtheVERTEXcode(Rampp&Janka
2002)andAgile-BOLTZTRANcode(Liebendörferetal.2004),respectively.
WecomparenuGR1DwiththeresultsofLiebendörferetal.
Adetaileddiscussionofthedifferencesisprovidedinthetext.
(2005), a comparison study between two Boltzmann neu-
species: ν (left panels), and ν¯ and ν (right panels). Both
trino transport codes4. The two codes, Agile-BOLTZTRAN e e x
theluminosityandrootmeansquaredenergiesaredefinedin
(Liebendörfer et al. 2004), and VERTEX (Rampp & Janka
Liebendörfer et al. (2005), Section 4. The black solid lines
2002), approach the neutrino transport problem in very dif- aretheresultsobtainedwithnuGR1D,thereddashed-dotted
ferent ways. Their results compare very well in the New- lines are the predictions of Agile-BOLTZTRAN, and the blue
tonian limit, but show significant quantitative differences in dashedlinesaretheVERTEXresults. Overall,theagreement
thegeneralrelativisticcase. Subsequentmodificationstothe
isgood,howeverthereareareseveralsystematicdifferences:
approximate general relativistic potential used in VERTEX
(Marek et al. 2005) have since removed many of the quan- (i)Themagnitudesoftheνeandν¯eluminositiesintheearly
titativedifferencesbetweenthecodes. Thegeneralrelativis- postbounce phase predicted by nuGR1D agree well with the
tic test case of Liebendörfer et al. (2005) was the collapse Agile-BOLTZTRAN results but they are systematically lower
andearlypostbounceevolutionofa15-M (atZAMS)solar- than the VERTEX results. This discrepancy, which also ex-
(cid:12)
metallicityprogenitorofWoosley&Weaver(1995),referred ists between VERTEX and the Agile-BOLTZTRAN, has been
to as model s15WW95 in the following. They employed furtherinvestigatedinMareketal.(2005)andhassincebeen
the LS180 EOS (Lattimer & Swesty 1991) and a baseline resolved. The updated VERTEX code employs an improved
setofneutrino-matterinteractions, includingcouplingofen- generalrelativisticpotentialandgivescomparableamplitudes
ergy groups via inelastic scattering processes (Bruenn 1985; toAgile-BOLTZTRAN,andhencenuGR1D.
Liebendörferetal.2005). Werepeattheirtesthere,usingthe (ii)Thetimewhenthesilicon–oxygeninterface(locatedat
same initial conditions and EOS, with our current approxi- abaryonicmasscoordinateof1.43M inmodels15WW95)
(cid:12)
mations and compare the neutrino observables. We empha- accretesthroughtheshock,whichismarkedbyasharpdrop
sizeagainthatthecurrentversionofnuGR1Dlacksinelastic in the ν and ν¯ luminosities, is earlier (at ∼140ms) in our
e e
neutrino–electronscatteringandvelocitydependenttransport simulations than in the simulations of Liebendörfer et al.
terms. Both are included in the simulations of Liebendörfer (2005)(∼180msintheVERTEXsimulations5). Wehavecar-
et al. (2005). Our transport scheme evolves only the zeroth riedoutanumberofteststotrytounderstandthisdifference.
and first moment of the neutrino distribution function, using We have varied the zero point of the internal energy, which
ananalyticclosuretotruncatetheseriesofmomentequations, affects the relativistic enthalpy entering the momentum and
whereas Liebendörfer et al. (2005) solve the full Boltzmann energy equations and alters the collapse time. We have at-
equationforneutrinotransport. tempteddifferentmappingsofthepresupernovastellarstruc-
In Fig. 2, we show the luminosities (top panels) and root
mean squared energies (bottom panels) of three neutrino 5 Such a sharp drop is not seen in the Agile-BOLTZTRAN results.
Liebendörfer et al. (2005) attribute the lack of a sharp drop in the
Agile-BOLTZTRANresultstotheuseofanadaptivegrid,whichintroduces
4 The numerical data from this study are available online at artificialdiffusionandsmearsoutsharpdensityfeatures. Nevertheless,the
\unhbox\voidb@x\hbox{http://iopscience.iop.org/ slowdeclineoftheνeandν¯eluminositiesbeginsaroundthesametimeasin
0004-637X/620/2/840/fulltext/datafiles.tar.gz} theVERTEXsimulations.
7
turetoourEuleriangrid,andwehavetestedvariationsinpre-
collapse electron capture by parameterizingY as a function TABLE1
ofdensityinthecollapsephase(Liebendörfer2e005). Noneof KEYNEUTRINOQUANTITIES
thesetestsledtoachangeofthepostbouncetimetosilicon–
oxygeninterfaceaccretionbymorethan∼10ms. Model ξ1.75 E4ν0e0ms/E4ν¯0e0ms/E4ν0x0ms N2ib0d0ms N4ib0d0ms
LS220/HShen LS220/HShen
Withoutcurrentlyhavingthemodelingtechnologytotestit, [B] [103] [103]
wesuspectthattheapparentdifferencemaybeduetotheonly
s12 0.235 19.24/14.19/ 7.73 1.02/0.92 2.13/1.78
otherobviousdifferencebetweennuGR1DandVERTEX:our
s13 0.383 22.18/16.58/ 8.76 1.25/1.09 2.53/2.07
simplistic treatment of the EOS at low densities. Below a s14 0.537 25.19/19.35/ 9.24 1.36/1.20 3.06/2.49
density of 6×107gcm−3, VERTEX replaces the nuclear sta- s15 0.580 25.51/19.59/ 9.17 1.30/1.16 3.13/2.59
s16 0.338 18.91/13.72/ 8.20 1.11/0.95 2.00/1.68
tisticalequilibrium(NSE)EOSwithanEOSthatspecifically
s17 0.383 19.93/14.54/ 8.57 1.20/1.02 2.13/1.78
dependsonthecompositionofthematter.Agile-BOLTZTRAN s18 0.738 28.66/22.26/10.30 1.55/1.36 3.62/2.92
assumesallmatterbelowthisdensityissiliconandburnsitto s19 0.544 23.36/17.49/ 9.37 1.43/1.23 2.67/2.18
NSE when the temperature reaches 0.44MeV (Liebendörfer s20 0.944 29.39/22.70/11.08 1.79/1.55 3.64/2.91
s21 0.325 18.48/13.41/ 8.09 1.04/0.91 1.95/1.65
etal.2005).InGR1D,asdiscussedinO’Connor&Ott(2010),
s22 0.972 30.06/23.29/11.26 1.82/1.58 3.76/3.00
weassumeNSEcompositionsfromthenuclearEOS.Atden- s23 1.256 42.00/33.88/14.68 2.29/2.00 5.99/4.67
sities below the validity regime of the nuclear EOS, we take s24 1.167 39.14/31.35/13.66 2.16/1.89 5.44/4.24
thecompositionsatthelowestdensitypointfromthenuclear s25 1.040 32.30/25.28/11.87 1.93/1.68 4.15/3.28
s26 0.727 24.97/18.84/10.10 1.59/1.38 2.88/2.33
EOSandusetheTimmesEOS(Timmes&Arnett1999). For
s27 0.783 25.67/19.45/10.21 1.61/1.40 3.01/2.42
reference,theinitialdensityofthesilicon–oxygeninterfaceis s28 0.640 25.86/19.83/ 9.54 1.36/1.21 3.16/2.62
0.4–1.0×107gcm−3,wheretherangerepresentstheextentin s29 0.556 23.24/17.38/ 9.43 1.45/1.22 2.64/2.17
s30 0.760 26.61/20.30/10.19 1.60/1.40 3.20/2.58
densityspace.
s31 0.687 25.23/19.09/ 9.95 1.56/1.35 2.96/2.40
(iii) The ν and ν¯ root mean squared energies, predicted s32 0.883 28.10/21.58/10.70 1.71/1.49 3.43/2.75
e e s33 0.965 30.36/23.57/11.28 1.82/1.58 3.82/3.04
bynuGR1DagreeverywellwiththeBoltzmanntransportre-
s35 1.129 35.93/28.50/12.77 2.06/1.79 4.83/3.79
sultsduringthepostbouncephase. Thedifferenceseeninthe s40 1.328 48.32/39.45/17.51 2.49/2.21 7.23/5.75
ν root mean squared energy is similar to that observed by s45 1.300 46.42/37.81/16.56 2.42/2.14 6.86/5.44
x
Thompson (2002) and Lentz et al. (2012a) when investigat- s50 0.701 25.60/19.42/10.00 1.57/1.32 3.02/2.45
s55 0.577 23.56/17.65/ 9.52 1.48/1.25 2.69/2.20
ingtheeffectsofinelasticneutrino–electronscattering. Inthe
s60 0.461 21.52/15.88/ 9.07 1.33/1.12 2.36/1.95
postbounceevolution,thisinteractionisexpectedtopredom- s70 0.755 25.83/19.58/10.17 1.60/1.39 3.04/2.46
inately affect the ν neutrino. We currently ignore this pro- s80 0.591 23.75/17.80/ 9.63 1.48/1.26 2.71/2.22
x
cessinnuGR1Dandnotethattheν luminositypredictedby s100 0.792 29.90/23.33/10.59 1.61/1.42 3.82/3.07
x s120 0.474 22.31/16.64/ 9.02 1.33/1.15 2.52/2.07
nuGR1DstillagreeswellwiththefullBoltzmannresults.
(iv) Another difference between the evolution in nuGR1D NOTE. —ForeachmodelintheWoosley&Heger(2007)modelset
andthefullBoltzmanntransportresultsarisesinthecollapse weshowξ1.75,thecumulativeemittedneutrinoenergyinνe,ν¯e anda
phase. The lack of velocity terms and inelastic ν −e− scat- singleνxat400msafterbounce.Thenumberscorrespondtothemodels
e runwiththeLS220EOS.Wealsopresent,foreachmodel,theestimated
teringsignificantlyeffectsthecompositionoftheinnercore.
numberofIBDeventsinaSuper-Kamiokande-likedetectorat200and
Insimulationswithinelasticνe−e−scattering,neutrinosfrom 400ms after bounce for a supernova at a fiducial galactic distance of
electroncaptureonfreeprotonsdown-scatteroffofelectrons 10kpcforbothEOS.
tolowerenergies. Sincetheopticaldepthislower,theseneu-
effectcanbecapturedwithGR1D’sleakageschemeinwhich
trinos can then escape, deleptonizing the core. In our simu-
itistrivialtoincluderedshiftterms(O’Connor&Ott2010).
lations, thesehigh-energyneutrinoscannotdown-scatterand
Capturing redshift in an energy-dependent transport scheme
therefore cannot escape. Deleptonization is suppressed until
requiresenergy-groupcoupling. Wehavealsocomparedour
laterphases,whencentraldensityandtemperaturearehigher.
M1schemetotheresultsofFischeretal.(2009)forthe40M
Thelackofvelocity-dependenttermsdelaysfulltrappingthat (cid:12)
modelfromWoosley&Weaver(1995)usingtheLS180EOS.
would normally begin to occur at ρ(cid:38)1×1012gcm−3 until
Inthismodelablackholeformswithin500msofbounce.We
nuclear densities, allowing for further deleptonization. At
finddifferencesintheν andν¯ luminositiesof∼10–20%in
bouncethecentralvalueofY are∼0.22comparedto∼0.29 e e
e thelast∼50ms,butgoodagreementintheearlypostbounce
inVERTEXandAgile-BOLTZTRAN. Therootmeansquared
phase.
ν energies predicted by nuGR1D are higher during the pre-
e
bounce phase, because the neutrinos do not experience the
down-scatteringviainelasticneutrino–electronscattering. 5. RESULTS
Wefindthatdifferencesintheinnercorestructureandcom- 5.1. TrendsintheNeutrinoObservables
position at bounce do not manifest themselves in the neu-
trinosignalafterthecollapsephase. Therefore,forthisstudy, Weperformcorecollapseandearlypostbounceevolutions
we find the current version of nuGR1D to be acceptable, of all 32 models introduced in §3 using both the LS220 and
since our primary focus is the neutrino signal of the preex- the HShen EOS. In Fig. 3 we present the three neutrino lu-
plosion accretion phase. We do note, however, that the lack minositiesandaverageenergiesforeachmodelandEOSasa
of energy-coupling terms in our transport can cause quali- functionofpostbouncetime.Wedonotexpectacleartrendin
tative differences near black hole formation. When energy- theneutrinoobservableswithZAMSmass. However, wedo
coupling terms, such as gravitational redshift, are included, expecttrendsbasedonthepresupernovastructureofthestar,
thentheν andν¯ luminositiesdropoffattimesverycloseto whichiswellencapsulatedbythecompactnessparameterin-
e e
black hole formation, as seen in Fischer et al. (2009). This troducedin§3. Inthetopsetofpanelsweshowsimulations
8
runusingtheLS220EOS.Inthebottompanels, simulations which the high accretion rates lead to the accumulation of
performedwiththeHShenEOSareshown. Tohighlightthat ∼2M ofmaterialinsidetheshockwithin∼200–300msof
(cid:12)
thereisindeedatrendwithpresupernovastructure,wecolor- bounce. InthecaseoftheLS220EOS,thisleadstoveryhigh
code individual models according to their compactness pa- temperatures throughout the protoneutron star as it becomes
rameter.Themappingbetweenlinecolorandξ isprovided more and more compact and closer to gravitational collapse
1.75
ontheright. TomoredirectlyhighlighttheEOSdependence, toablackhole. Inoursimulations, themostcompactmodel
weincludetheluminosityandaverageenergiesoftwomodels s40WH07(ξ =1.33,ξ =0.59)formsablackhole503ms
1.75 2.5
runwiththeLS220EOSintheHShenEOSpanelswiththick after bounce. The slightly less compact model s45WH07
dashed lines. These models, s12WH07 and s40WH07, have (ξ = 1.30, ξ = 0.55) forms a black hole 563ms after
1.75 2.5
the lowest and highest compactness parameter in our model bounce. The high temperatures present in the LS220 simu-
set,respectively. lations at these times will not be obtained until postbounce
times(cid:38)1sinmodelsusingtheHShenEOS6.
Wefindthatthereislittlevariationinthepeakluminosityof
theν neutronizationburstsignal.Forall32modelssimulated Most neutrino detectors are most sensitive to the electron
e
usingtheLS220(HShen)EOS,thepeakamplitudevariesby antineutrino luminosity through the dominant IBD interac-
lessthan3%(5%)fromtheaverage. Thisreflectstheuniver- tion, ν¯ +p→n+e+. IntheleftpanelofFig.4, weconsider
e
salnatureofthecollapseoftheinnercore(Liebendörferetal. the cumulative emitted ν¯ energy for each model using the
e
2002). After the neutronization burst, the postbounce lumi- LS220 EOS. We color code the models based on their com-
nositiesofallspeciesincreasesystematicallywithincreasing pactnessparameterandincludetworeferencemodelsthatuse
compactnessparameter. Modelswithhigherξ havehigher theHShenEOS,models12WH07andmodels40WH07. The
1.75
temperatures throughout the protoneutron star (O’Connor & graphsshowninthispanelaretheintegralofthegraphsshown
Ott 2011). This increases the diffusive neutrino luminosity inthetopcenterpanelofFig.3. Itisobviousthatthecumu-
and is best seen in the ν luminosities. The postbounce ac- lative amount of ν¯ energy emitted during the preexplosion
x e
cretion rate also increases with the compactness parameter. phasestronglycorrelateswiththecompactnessoftheprogen-
Higher accretion rates, and the deeper gravitational poten- itor model. For example, the amount of emitted ν¯ energy
e
tialduetothehigherprotoneutronstarmass,increasetheac- from model s40WH07 (ξ =1.33) is always between two
1.75
cretion luminosity, which is most directly reflected in the ν andthreetimesofthatofmodels12WH07(ξ =0.24). We
e 1.75
and ν¯ signals. After roughly 100ms, the average energy of makethispointmorequantitativeinthecenterandrightpan-
e
theemittedneutrinosalsoshowsanincreasingtrendwiththe els of Fig. 4. In the center (right) panel we plot the cumu-
compactnessparameter. Themattertemperatureattheneutri- lative emitted ν¯ energy at 100, 200, 300, and 400ms after
e
nosphereishigherinmodelswithlargercompactness,there- bounce for both EOS as a function of ξ (ξ ). For refer-
1.75 2.5
foreahigheraverageneutrinoenergyisobservedatinfinity. ence, we present a subset of these numbers in Table 1. We
see a very clear correlation that depends only weakly on the
The neutrino luminosities and average energies from sim-
chosenEOS.Note,however,thatformodelswithsmallcom-
ulations using the HShen EOS are systematically lower than pactness parameter (ξ (cid:46)0.8), the correlation between the
theluminositiesandaverageenergiesfromsimulationsofthe 1.75
totalemittedν¯ energyandthecompactnessparameterisnot
samemodelrunwiththeLS220EOS.Thisisclearlyseenin e
as strong after 400ms of postbounce evolution. Comparing
the bottom set of panels in Fig. 3: models s12WH07-LS220
the center and right panels justifies our choice of ξ over
ands40WH07-LS220haveluminositiesandaverageenergies 1.75
ξ asexplainedin§3.
that are comparable to or larger than in the corresponding 2.5
HShen models. For a fixed accretion rate (or fixed progeni- The onset of an explosion will break the correlation ob-
tormodel),thelocationoftheneutrinosphereofeachspecies served in Fig. 4. Once it is launched, the accretion lumi-
influences the emitted luminosities and spectra. In models nosity effectively turns off and only the diffusion luminosity
evolved with the stiff HShen EOS, the neutrinospheres are remains. One also expects this diffusion luminosity to show
located systematically at larger radii and lower matter tem- a correlation with the compactness of the progenitor, since
peraturesthaninmodelsrunwiththesofterLS220EOS.For the remnant protoneutron star’s thermodynamic conditions,
example, in the s12WH07 simulations, the Rosseland-mean such as the central entropy and its mass, are essentially set
ν neutrinospheresat200msafterbouncehaveradiiandtem- by the presupernova structure. However, it is currently un-
e
peratures of ∼35.3km and ∼4.86MeV; and ∼39.5km and clearwhetheroneshouldobtainacorrelationbetweenexplo-
∼4.46MeVfortheLS220andtheHShenEOS,respectively. sion time and the compactness parameter. Clarification will
The larger neutrinosphere radii are responsible for the lower require a more complete understanding of the core-collapse
accretion luminosity since the latter is set essentially by the supernova explosion mechanism and may require extensive
product of the mass accretion rate and the gravitational po- parameterstudieswithfullyself-consistentthree-dimensional
tential at the protoneutron star surface (Liebendörfer et al. simulations.
2002). The latter is located at larger radii in simulations us-
ing the HShen EOS. The differences in the neutrinosphere 5.2. Detectability
radii and temperatures between the LS220 and HShen EOS
also give an explanation for the systematically lower aver- WeusethepubliclyavailablesoftwareSNOwGLoBES7 to
age neutrino energies seen in the HShen simulations. Mat- predict the neutrino signal observed in Earth-based neutrino
ter at larger radii has been compressed less and therefore is
cooler. This leads to average neutrino energies that can be 6 While we do not follow these models to black hole formation in our
currentstudy,inO’Connor&Ott(2011)wefoundthattheblackholefor-
up to 5MeV lower for the HShen EOS than for the LS220
mationtimesofthes40WH07ands45WH07modelsare∼1.3sand∼1.4s,
EOS for the same progenitor model (see Fig. 3). The dif- respectively,whenusingtheHShenEOS.
ferenceintheneutrinoluminositiesandaverageenergiesbe- 7 available at http://www.phy.duke.edu/~schol/
tween the two EOS is largest for models with ξ (cid:38)1.2, in snowglobes
1.75
9
✶✹✵
✲(cid:0) ✶✷✵ ▲▲✞✞✟✟✟✟✠✠✡✡☛☞✟✠
✶✵✵
✽✵ ✕✖✗✘✙
✺(cid:0) ✻✵ ✒✎✑
✹✵ ✒✎✏
♥
✷✵ ✌❡ ✌❡ ✌① ✒✎✍
✵
✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵ ✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵ ✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵ ✍✎✓
✸✵ ✸✵
✌❡ ✌❡ ✸✵ ✌① ✍✎✔
✷✝ ✷✝
✍✎✑
✷✝
✷✵ ✷✵ ✍✎✏
✷✵
✶✝ ✶✝
✶✵ ✶✵ ✶✝
✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵ ✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵ ✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵
❜✁✂✄☎✆ ❜✁✂✄☎✆ ❜✁✂✄☎✆
✶✹✵
✲(cid:0) ✶✷✵ ❍❍✞✞✟✟✠✠✡✡☛☛☞✍✌✎
✶✵✵
✺(cid:0) ✻✽✵✵ ▲▲✢✢✣✣✣✣✤✤✥✥✦✧✤✣ ✕✒✔✘✙✚✛✜
✹✵ ✕✒✓
♥
✷✵ ✏❡ ✏❡ ✏① ✕✒✑
✵
✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵ ✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵ ✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵ ✑✒✖
✸✵ ✸✵
✏❡ ✏❡ ✸✵ ✏① ✑✒✗
✷✝ ✷✝
✑✒✔
✷✝
✷✵ ✷✵ ✑✒✓
✷✵
✶✝ ✶✝
✶✵ ✶✵ ✶✝
✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵ ✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵ ✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵
❜✁✂✄☎✆ ❜✁✂✄☎✆ ❜✁✂✄☎✆
FIG.3.—Neutrinoluminosities(toppanels)andaverageenergies(bottompanels)plottedasafunctionofpostbouncetimeforall32modelsofWoosley
&Heger(2007). ThetopsetofpanelsshowsresultsobtainedwiththeLS220EOS.ThebottompanelshowsthesamefortheHShenEOS,butincludes,for
reference,twoLS220models: s12WH07ands40WH07. Theleft,center,andrightpanelsshowresultsforνe,ν¯e,andνx,respectively. Thecurvesarecolor-
andline-weight-codedwithincreasingcompactness(ξ1.75),themappingfromcolortocompactnessparameterisshownontheright. Thereisacleartrendin
allluminositiesandaverageenergieswithcompactnessparameter. Theprogenitorwiththehighestcompactness,s40WH07,formsablackholeat502msafter
bounce. Noneofthesemodelsexplode,buttheonsetofanexplosioninanyofthesemodelswouldleadtoasuddendeepdrop(strongestforνeandν¯e)inthe
luminositiesandaverageenergies(Fischeretal.2010). Thesmallerdropobservedformostmodelsmodelshereisduetothesuddendecreaseoftheaccretion
ratewhenthesilicon–oxygeninterfacereachesthestalledshock.
detectors. SNOwGLoBES(Scholberg2012),whichinturnre- of events for each model run with the LS220 EOS and, for
lies on GLoBES (Huber et al. 2005, 2007), is a set of rou- reference,twomodelsrunwiththeHShenEOS.Thelinesare
tines that compute the interaction rates of supernova neutri- color-coded according to compactness parameter. Note that
nos in user-specified detector configurations. Variables in- theverticalscaleinthisfigureisinthousandsofevents. We
clude detector material (e.g., water, scintillator, argon, and notethatouranswersagreewiththetotalnumberofexpected
lead), detector volume, detector response functions, and a eventsinSuper-Kamiokandefromagalacticcore-collapsesu-
host of relevant neutrino interactions. For this investigation, pernovaat10kpc,whichisestimatedtobe∼7000(Scholberg
we consider only IBD events in a water Cherenkov detec- 2012). To arrive at this number from our results, consider
tor. We choose a detector mass of 32kT, the mass of wa- the lowest ZAMS mass progenitor in our model set, model
terinSuper-Kamiokandethatissensitivetocore-collapsesu- s12WH07. After 450ms of evolution, 15B of ν¯ energy has
e
pernova neutrinos (Scholberg 2012). For reference, we use beenradiated(Fig.4),whichcorrespondsto2000IBDevents
the wc100kt30prct smearing rates and efficiencies pro- (Fig. 5). For 50B of released energy (∼1/6 of 300B, the
vided with SNOwGLoBES. We construct SNOwGLoBES ini- fiducial energy released in neutrinos over the entire cooling
tial fluence data from our simulations binned in 5ms inter- phase),onewouldthenexpect∼7000events. However,asis
vals. Weprovidetheseenergy-dependentfluencesathttp: clear from Fig. 5, the number of events from the next galac-
//www.stellarcollapse.org/M1prog for all mod- ticsupernovamaybehigherthanthisfiducialnumber. More
els, neutrino species, and both EOS in 5ms intervals up to importantly,therateofeventsinthepreexplosionphasewill
450msafterbounce.Weassumeafiducialgalacticsupernova giveusdetailedinformationontheprogenitorcorestructure.
distanceof10kpc. InFig.5,weshowthecumulativenumber
10
✺✵ ✺✵ ✺✵
▲✌✍✍✎✏✑✍ ✑✎✎✘✏ ▲✌✍✍✎
✹✵ ▲✌✍✍✎✏✒✎ ✹✵ ✍✎✎✘✏ ✹✵ ❍✌✢✣✤
✙✎✎✘✏
✦✧★✩✪✫✬✭ ✒✎✎✘✏
✠✡✡❡☛ ✸✵ ✠✡✡❡☛ ✸✵ ✠✡✡❡☛ ✸✵
❡✟❴ ♥☞✷✵ ❡✟❴ ♥☞✷✵ ❡✟❴ ♥☞✷✵
❙ ❙ ❙
✶✵ ✶✵ ✶✵
✦✧★✩✪✫✮✯
✵ ✵ ✵
✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵ ✵✓✷ ✵✓✹ ✵✓✔ ✵✓✚ ✶✓✵ ✶✓✷ ✶✓✹ ✵✓✵ ✵✓✶ ✵✓✷ ✵✓✸ ✵✓✹ ✵✓✺ ✵✓✔
t(cid:0)t❜✁✂✄☎✆ ✥✝✞ ✛✖✜✗ ✕✖✗
FIG.4.—Cumulativeemittedelectronantineutrinoenergy. Weshowthecumulativetotalsviaseveralmethods. Intheleftpanelweshowtimeseriesdatafor
eachofthe32modelsofWoosley&Heger(2007)runwiththeLS220EOS.Thecolorcodingcorrespondstothevalueofξ1.75 andisgiveninFig.3. This
cumulativeenergyisthetimeintegralofthetop-centerpanelofFig.3. Thedashedblacklinescorrespondtomodelss12WH07ands40WH07runwiththe
HShenEOS.Inthecenterandrightpanelswepresenttheemittedenergyatselectpostbouncetimes,foreachmodelandEOS,plottedversusthecompactness
ξ1.75(centerpanel)andξ2.5(rightpanel)ofthemodel.
✮ onthedetector’sresponsefunctionandefficiency.
✬✭
✶✵
An additional independent path to experimentally probing
✾ ✖✩✪✫ the inner structure of the progenitor is via the total neutrino
✣ ✤✌✌✥✦ ▲☛☞☞✌ energy emitted in all species over the first 10s of seconds
✞✟✠✡ ✽ ✕ ✧☞✌✌✌✌✥✥✦✦ ❍☛✍✎✏ after the initial collapse. This method requires a measure-
■✆✝ ✼ ✔✢ ★✌✌✥✦ ✯✰✱✲✳✴✿✭ ment of the total fluence of neutrinos of all species, not just
✻ ✜ electron antineutrinos. An example of a neutrino interaction
✓ capable of relaying such information is the mono-energetic
✺
✗ de-excitation of a neutral-current neutrino-excitation of 12C
✑
✹ ✑✒✓ ✑✒✔ ✑✒✕ ✑✒✖ ✗✒✑ ✗✒✓ ✗✒✔ (Scholberg 2012). Such a measurement would require good
✸ ①✘✙✚✛ energy resolution and a significant source of carbon, for ex-
ample, a liquid scintillator neutrino detector. We note that
✷ evenwithaliquidscintillatordetector,thedominantneutrino
interactionisstillIBD(Scholberg2012).
✶ ✯✰✱✲✳✴✬❀
If such a measurement was made, and there is not a sig-
✵
✵ ✶✵✵ ✷✵✵ ✸✵✵ ✹✵✵ nificantamountofrotation(seethediscussiononrotationin
❜(cid:0)✁✂✄☎ §5.3),onecanimmediatelyinferthegravitationalbindingen-
FIG.5.— Cumulative IBD events in a Super-Kamiokande-like water
ergyoftheremnant,sinceneutrinoscarryawaythevastma-
Cherenkovdetectoratafiducialgalacticdistanceof10kpcversuspostbounce
time. WeusetheSNOwGLoBESpackagetodeterminetheintegratedIBD jority(∼99%8)ofthegravitationalbindingenergy. Fortyp-
eventrateina32kTwaterCherenkovdetectorat10kpc. Thecolorcoding ical nuclear EOS like the ones considered here, this results
correspondstothevalueofξ1.75andisprovidedinFig.3. Thedashedlines inaone-to-onemappingofthereleasedgravitationalbinding
areresultsformodelss12WH07ands40WH07runwiththeHShenEOS.
energy to the baryonic mass of the remnant, and, hence, the
IntheinsetweshowthecumulativeIBDhitsasafunctionofξ1.75foreach
modelandEOSatfourpostbouncetimes:100,200,300,and400ms. gravitationalmassoftheremnant. Thisismosteasilyseenby
fittingthegravitationalbindingenergyofacold(T=0.1MeV),
In order to more directly quantify the differences between
neutrino-less β-equilibrium, non-rotating neutron star to its
variations in progenitor compactness and variations in the
baryonic mass. From cold neutron star TOV solutions using
nuclear EOS, we plot in the inset of Fig. 5 the number
theLS220EOSonecanobtainanempiricalfittobetterthan
of expected IBD events in a Super-Kamiokande-like water
3%aboveabaryonicmassof1.15M ,
Cherenkovdetectoratvariouspostbouncetimesversusξ . (cid:12)
1.75
Thereisawelldefinedtrend: ThenumberofIBDeventsde- E ∼ 1.12×1053(M /M )2ergs. (13)
binding bary (cid:12)
tected in the first 100, 200, 300, and 400ms increases with
thecompactnessparameterofthemodels. Forreference, we AsimilarfitfortheHShenEOSgives,
includetheexpectednumberofeventsat200and400msfor
E ∼9.78×1052(M /M )2ergs, (14)
both EOS in Table 1. We find that the EOS dependence of binding bary (cid:12)
the expected number of events is similar to the EOS depen- andisaccurateto5%abovebaryonicmassesof1.15M . Be-
(cid:12)
dence of the total emitted ν¯e energy: the HShen EOS leads lowMbary=1.15M(cid:12), theempiricalquadraticfitisnotasac-
to a lower number of events (compare the inset of Fig. 5 to curate. However,allmodelsconsideredherereachabaryonic
thecenterpanelofFig.4). ThedependenceonEOSissome- protoneutronstarmassof1.15M within∼10msofbounce.
(cid:12)
whatstrongerhere,sincethetheloweraverageν¯eenergypre- Hence, we believe that the above fits are acceptable for the
dictedfromstifferEOStranslatesintoareducedcrosssection iron-corecorecollapseeventsconsideredhere.
in Earth-based detectors. In addition to the total number of
events,awaterCherenkovdetectormeasuresindividualener- 8 Theremaining∼1%oftheenergyispredominantlysharedamongthe
gies,andthus,allowsforthereconstructionofthecumulative kineticenergyoftheexplosion,theoriginalbindingenergyoftheunbound
emittedν¯ energyovertime. Thisreconstructionwilldepend stellarmantle,andthebindingenergyoftheironcoreattheonsetofcollapse.
e