Neutrino Signal of Collapse-Induced Thermonuclear Supernovae: The Case for Prompt Black Hole Formation in SN1987A

0.89 MB·

by  Kfir Blum

Draft version January 15, 2016
PreprinttypesetusingLATEXstyleemulateapjv.01/23/15
NEUTRINO SIGNAL OF COLLAPSE-INDUCED THERMONUCLEAR SUPERNOVAE: THE CASE FOR
PROMPT BLACK HOLE FORMATION IN SN1987A
Kfir Blum1,2,3, Doron Kushnir2,3
1DepartmentofParticlePhysicsandAstrophysics,WeizmannInstituteofScience,Rehovot,760000Israel
2SchoolofNaturalSciences,InstituteforAdvancedStudy,Princeton,NewJersey,08540USAand
3JohnN.BahcallFellow
Draft version January 15, 2016
ABSTRACT
6 Collapse-induced thermonuclear explosion (CITE) may explain core-collapse supernovae (CCSNe).
1 We present a first, preliminary analysis of the neutrino signal predicted by CITE and compare it
0 to the neutrino burst of SN1987A. For strong ((cid:38)1051 erg) CCSNe, as SN1987A, CITE predicts a
2 proto-neutron star (PNS) accretion phase, accompanied by the corresponding neutrino luminosity,

n that can last up to a few seconds and that is cut-off abruptly by a black hole (BH) formation. The
a neutrino luminosity can later be revived by accretion disc emission after a dead time of few to a few
J ten seconds. In contrast, the competing neutrino mechanism for CCSNe predicts a short ((cid:46) sec)

3 PNS accretion phase, followed, upon the explosion, by a slowly declining PNS cooling luminosity.
1 We repeat statistical analyses used in the literature to interpret the neutrino mechanism, and apply
themtoCITE.Thefirst1-2secoftheneutrinoburstareequallycompatiblewithCITEandwiththe

] neutrinomechanism. However, thedatahintstowardsaluminositydropatt=2-3sec, thatisinsome
E
tensionwiththeneutrinomechanismwhilebeingnaturallyattributedtoBHformationinCITE.The
H occurrence of neutrino signal events at 5 sec in SN1987A places a constraint on CITE, suggesting
. that the accretion disc formed by that time. We perform 2D numerical simulations, showing that
h CITE may be able to accommodate this disc formation time while reproducing the ejected 56Ni mass
p
and ejecta kinetic energy within factors 2-3 of observations. We estimate the accretion disc neutrino
-
o luminosity and show that it can roughly match the data. This suggests that direct BH formation is
r compatible with the neutrino burst of SN1987A. With current neutrino detectors, the neutrino burst
t of the next strong Galactic CCSN may give us front-row seats to the formation of an event horizon in
s
a real time. Access to phenomena near the event horizon motivates the construction of a few Megaton
[ neutrino detector that should observe extragalactic CCSNe on a yearly basis.

1
v
1. INTRODUCTION et al. 2015; Melson et al. 2015a,b).
2
Burbidgeetal.(1957)suggestedadifferentmechanism
2 Thereisstrongevidencethattype-IIsupernovae(SNe)
for the explosion that does not involve the emitted neu-
4 are explosions of massive stars, initiated by the gravita-
trinos. They suggested that the adiabatic heating of the
3 tional collapse of the stars’ iron core (Burbidge et al.
outer stellar shells as they collapse triggers a thermonu-
0 1957; Hirata et al. 1987; Smartt 2009). It is widely
clear explosion (see also Hoyle & Fowler 1960; Fowler
. thought that the explosion is obtained due to the de-
1 & Hoyle 1964). This collapse-induced thermonuclear ex-
position in the envelope of a small fraction (∼1%) of
0 plosion(CITE)hastheadvantageofnaturallyproducing
thegravitationalenergy(∼1053erg)releasedinneutrinos
6 E ∼1051erg from the thermonuclear burning of ∼M
1 from the core, leading to the ∼1051erg observed kinetic kin (cid:12)
of light elements, with a gain of ∼MeV per nucleon. A
v: energy (Ekin) of the ejected material (see Bethe 1990; few 1D studies suggested that this mechanism does not
Janka2012,forreviews). However,one-dimensional(1D)
i lead to an explosion because the detonation wave is ig-
X simulations indicate that the neutrinos do not deposit
nited in a supersonic in-falling flow (Colgate & White
sufficient energy in the envelope to produce the typical
r 1966; Woosley & Weaver 1982; Bodenheimer & Woosley
a Ekin∼1051erg. While some two-dimensional (2D) stud- 1983), and the idea was subsequently abandoned. While
ies indicate successful explosions (Bruenn et al. 2013;
the results of these studies are discouraging, they only
Bruenn et al. 2014; Nakamura et al. 2015; Suwa et al.
demonstrate that some specific initial stellar profiles do
2016), others indicate failures or weak explosions (Taki-
not lead to CITE, and they do not prove that CITE is
waki et al. 2014; Dolence et al. 2015), and these studies
impossible for all profiles.
are affected by the assumption of rotational symmetry
Recently, Kushnir & Katz (2015) have shown that
and by an inverse turbulent energy cascade that, unlike
CITE is possible in some (tuned) 1D initial profiles,
many physical systems, appears to amplify energy on
that include shells of mixed helium and oxygen, but re-
large scales. Therefore, three-dimensional (3D) studies
sulting in weak explosions, E (cid:46)1050erg, and negligible
are necessary to satisfactorily demonstrate the neutrino kin
amounts of ejected 56Ni. Subsequently, Kushnir (2015a)
mechanism, but so far 3D studies have resulted in either
used 2D simulations of rotating massive stars to explore
failures or weak explosions (Takiwaki et al. 2014; Lentz
theconditionsrequiredforCITEtooperatesuccessfully.
It was found that for stellar cores that include slowly (a
akfi[email protected]
few percent of breakup) rotating shells of mixed He-O
[email protected]
2
with densities of few×103gcm−3, a thermonuclear det- neutrinos at later times (Burrows 1988; Loredo &
onation that unbinds the stars’ outer layers is obtained. Lamb 2002): as we review below, neutrino signal
With a series of simulations that cover a wide range of events were detected 5-10 sec after core-collapse.
progenitor masses and profiles, it was shown that CITE Our first question is: does this argument rule out
is insensitive to the assumed profiles and thus a robust CITE?
process that leads to supernova explosions for rotating
We show here that the answer is negative. CITE,
massive stars. The resulting explosions have E in the
kin and more generally direct BH formation, can be
rangeof1049−1052ergandejected56Nimasses(M )of
Ni reconciledwiththeSN1987Aneutrinosignal. Even
up to ∼1M , both of which cover the observed ranges
(cid:12) though BH formation should indeed temporarily
of core-collapse supernovae (CCSNe, including types II
quench the neutrino burst, the subsequent forma-
and Ibc).
tion of an accretion disc around the BH can pro-
It is difficult to test observationally if the initial con-
duce a neutrino luminosity consistent with obser-
ditions required for CITE exist in nature. Nevertheless,
vations.
CITEmakesafewpredictionsthataredifferentfromthe
The fact that accretion disc during stellar collapse
predictions of the neutrino mechanism, and that can be
can produce the required late-time neutrino lumi-
compared to observations. For example, CITE predicts
nosity should come as no surprise. Similar sce-
thatstrongerexplosions(i.e.,largerE andhigherM )
kin Ni
narios have been investigated in the literature in
are obtained from progenitors with higher pre-collapse
the context of the collapsar model for gamma-ray
masses. Kushnir (2015b) showed that the observed cor-
bursts (GRBs; MacFadyen & Woosley 1999), and
relationbetweenM andtheluminositiesoftheprogen-
Ni
theresultingdiscshavebeenshowntoexhibitcopi-
itors for type II SNe is in agreement with the prediction
ous neutrino emission2 (Popham et al. 1999; Mac-
of CITE and in possible contradiction with the neutrino
Fadyen & Woosley 1999). It is interesting to note
mechanism. Another prediction of CITE is that neu-
tron stars (NSs) are produced in weak (E (cid:46)1051erg) that Loredo & Lamb (2002) in their analysis of
kin
explosions,whilestrong(E (cid:38)1051erg)explosionsleave SN1987AfoundthatdirectBHformationisfavored
kin by the neutrino data, but they set a prior against
a black hole (BH) remnant. This prediction suggests
thispossibility. Werepeathereasimilarlikelihood
that a BH was formed in SN1987A (E ≈1.5·1051erg;
kin analysis of SN1987A and show that CITE can in-
Utrobin&Chugai2011)duringthefirstfewsecondsafter
deed give a somewhat better fit to the data.
core collapse (direct BH formation, to be distinguished
from BH formation from fallback, which lasts hours to 2. A key to CITE is the formation of a rotationally-
days). In contrast, simulations based on artificially trig- inducedaccretionshock(RIAS)duringthecollapse
gered explosions within the neutrino mechanism predict ofthestellarenvelopebelowtheHe-Olayer(Kush-
that the compact object in SN1987A is a NS (see, e.g., nir2015a). TheRIASprovidesthematchforther-
Peregoetal.2015). Atthetimeofwriting, aNShasnot monuclear explosion. Importantly for us here, the
yet been detected in the cite of SN1987A (Graves et al. RIAS formation time is precisely the formation
2005; Larsson et al. 2011), but see Zanardo et al. (2014) time of the accretion disc that is needed to restart
for a possible recent hint. the neutrino luminosity after BH formation. As
In this paper we continue to explore the observational mentioned above, SN1987A data implies that the
consequences of CITE. We focus on the neutrino signal accretiondiscneutrinoluminosityshouldbeopera-
characterizing core-collapse, and derive constraints from tivebyt∼5sec. Oursecondquestionis: canCITE
the neutrino burst that accompanied SN1987A (Bionta operate successfully with RIAS formation time as
et al. 1987; Hirata et al. 1987). Specifically, we ask, and early as a few seconds?
begin to answer, the following two questions.
Kushnir (2015a) made preliminary studies of the
dependence of CITE on the pre-collapse stellar
1. Asmentionedabove,CITEpredictsthataBHwas
profile, but for profiles which resulted in strong
formed directly during the event of SN1987A. The
explosions (E > 1051erg) the RIAS formation
reason for this expectation is that a strong explo- kin
sion (E (cid:38)1051erg) requires a high mass for the times considered there were significantly larger
kin
He-Oshell((cid:38)1M ), whichinturnrequiresamas- than 5sec. Here we extend the analysis of Kushnir
(cid:12)
sive core ((cid:38)4M ) below the shell. By the time (2015a) by further numerical simulations. Guided
(cid:12)
by the neutrino data of SN1987A we tailor the
CITE operates (on the order of the free-fall time
initial profile to initiate neutrino emission from a
of the He-O layer ∼30sec) the mass below the He-
disc at t ≈ 5 sec. With this t constraint
O shell accrets onto the central proto-neutron star disc disc
we are able to find a profile in which CITE op-
(PNS),toppingthecriticalmassandturningitinto
erates and yields values of M ∼ 0.035 M and
a BH (within 1-3 sec; O’Connor & Ott 2011). Ni (cid:12)
E ∼ 0.6·1051 erg, in the ballpark of, though a
kin
Direct BH formation in SN1987A has been pre- factor2-3belowobservationsforSN1987A(Hamuy
viously considered unlikely, as it was argued to 2003; Utrobin & Chugai 2011).
abruptly terminate the PNS neutrino emission1.
Wefurtherestimatetheneutrinoemissionfromthe
This would be incompatible with the detection of
accretiondiscatthebaseoftheRIAS,findingaν¯
e
luminosity of about L ∼ 0.5·1051 erg/sec and
1 Anotherfrequentargumentintheliterature(see,e.g.,Mirizzi ν¯e
mean neutrino energy ∼ 10 MeV. These results
etal.2015)isthatBHformationwouldnotbecompatiblewiththe
neutrinomechanismfortheexplosion. Thisargumentisofcourse
irrelevanttoouranalysishere. 2 SeealsoLiuetal.(2015)forarecentanalysis.
3
are on the low side, but not inconsistent with the
50
rangeallowedbythedata. Whilemoresimulations
45 Kamiokande
are needed for conclusive results, our preliminary IMB
findings indicate that CITE can operate in rough 40 Baksan
35
agreement with the neutrino data of SN1987A.
V]30
In Section 2 we review the neutrino light curve from Me25
SN1987A, recalling the signal events at 5−10 sec that E [20
placeanimportantconstraintonCITE.Wefurthernote 15
a hint for a drop in ν¯e luminosity around t ∼ 2 sec, 10
and examine it in Section 2.1. While the luminosity 5
drop is not very statistically significant, we find it in-
0
teresting to repeat a likelihood analysis as of Loredo & -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
t [sec]
Lamb (2002) and Pagliaroli et al. (2009) for the neu-
Fig. 1.— Time series of the neutrino burst of SN1987A. Blue
trino mechanism, but focusing on interpretation within
(diamond), Red (circle), and black (cross) markers denote recon-
CITE. In Section 2.2 we show that the neutrino mecha- structedeventenergiesfortheKamiokande,IMB,andBaksande-
nism is in some tension with the luminosity drop, of or- tectors, respectively, with 1σ error bars. Horizontal blue line de-
dertwostandarddeviations. InSection2.3wegiveatoy notes the traditional 7.5 MeV threshold imposed in analyses of
Kamiokande data. Note that the three time series are offset by
model parameterization of the neutrino burst expected
unknownrelativedelays, likelyoforder100ms; herewesetthese
in CITE, with the same number of free parameters as delaystozero.
used in Loredo & Lamb (2002) and in Pagliaroli et al.
(2009) to describe the neutrino mechanism. The CITE accretion luminosity for t > tBH. CITE can fulfil this
model is fitted to data with reasonable parameters and requirement via accretion disc luminosity. Nevertheless,
naturally addresses the luminosity drop at t∼2 sec. beforegoingintomoredetailwecanconcludethatCITE
In Section 3 we use 2D numerical simulations to modelsthatpredictRIASformationontimetdisc >5sec
demonstrate that CITE can operate successfully with areexcludedbytheSN1987AneutrinodataofIMB.Itis
early RIAS formation time t ∼ 5 sec, yielding E thusimportanttoinvestigateifreasonablestellarprofiles
disc kin
and M in rough agreement with SN1987A. We show can be found in which CITE operates successfully with
Ni
that the BH accretion disc at the base of the RIAS can an RIAS launch time tdisc ≈ 5 sec. We will tackle this
revive the neutrino emission with luminosity in the ball- task in Section 3.
park seen in the data. We conclude in Section 4. In Please look again at Figure 1. Our main point in the
AppendixAwerecapsomedetailsofthephenomenolog- rest of this section, is that the neutrino light curve is
ical modelling of the neutrino mechanism and add some compatible with two different physical mechanisms ac-
more statistical analyses. countingfortheinitialdensesequenceofeventsontimes
t (cid:46) 2 − 3 sec, and the subsequent reduced luminosity
on times t > 5 sec. In fact, there is a time gap be-
2. SN1987A NEUTRINO DATA
tween the Kamiokande event at t=1.9 sec and the next
Figure 1 depicts the time series of the SN1987A neu- Kamiokande events at t > 9 sec. A comparably signif-
trino burst. Blue (diamond), Red (circle), and black icant gap (the statement of significance requires mod-
(cross) markers denote reconstructed event energies for elling, that we provide later on) exists between the IMB
the Kamiokande (Hirata et al. 1987), IMB (Bionta et event at t=2.7 sec and the next pair of events at t=5
al. 1987), and Baksan (Alekseev et al. 1987) detectors, and 5.6 sec.
respectively, with 1σ error bars. Horizontal blue line The time gaps in the neutrino data were noted in,
denotes the traditional 7.5 MeV threshold imposed in e.g., Spergel et al. (1987), Lattimer & Yahil (1989), and
analyses of Kamiokande data. Note that the three time Suzuki & Sato (1988). Spergel et al. (1987) commented
series are offset by unknown relative delays, likely of or- on the possible hint for a discontinuity, but fitted a con-
der 100ms. Here we set thesedelaysto zero; this has no tinuous exponential PNS cooling model to the neutrino
impact on our results3. light-curve, finding a reasonable global fit. We have re-
Figure 1 shows signal events from IMB (that had, re- done the analysis of Spergel et al. (1987) and we agree
portedly, no background) at t>5 sec. The Kamiokande with their numbers. Indeed, the SN1987A neutrino data
event at t = 10.4 sec, with Eν > 10 MeV, also has only is too sparse for conclusive detailed modeling. How-
a small probability of being due to background (Loredo ever, it is important to note that Spergel et al. (1987),
&Lamb2002). Forourpurposeinthispaper, theimpli- and other analyses (such as, e.g., Loredo & Lamb 2002;
cation is that neutrino emission should last for at least Pagliaroli et al. 2009; Ianni et al. 2009), did not have
5-10 sec after core-collapse. Models for CITE that pre- a theoretical model contender to the PNS accretion and
dict BH formation on time tBH ∼1−3 sec, must invoke cooling luminosity predicted within the neutrino mech-
another neutrino source to replace the PNS cooling and anism. The situation for us is different. A time gap
in the neutrino data, with intense PNS luminosity for
3WealsonotethatKamiokandeobservedfouradditionalevents t ≤ t ∼ 1−3 sec, silence for a few seconds, and re-
attimes17.6,20.3,21.4,and23.8sec,withenergies6.5,5.4,4.6,and BH
newedaccretion discluminosity, ispreciselywhat weex-
6.5 MeV, respectively. These late-time events were below thresh-
oldfortheoriginalKamiokandeanalysis. Nevertheless,theywere pect from CITE. In the next subsections we explore this
included(togetherwithproperbackgroundtreatment)inthelike- point further with some statistical analyses.
lihoodanalysisofLoredo&Lamb(2002),Pagliarolietal.(2009),
and Ianni et al. (2009), and though we do not show them in Fig-
2.1. A luminosity drop at t∼2 sec?
ure1weincludetheseeventsinouranalysistoo.
4
Toobtainabasicassessmentoftheneutrinosourcelu-
minosity, we make two simplifying assumptions:
(i) We assume that the neutrino distribution function at 101
the source is a modified Fermi-Dirac spectrum with in- c]
stantaneous temperature T(t)and ν¯e luminosity Lν¯e(t), g/se
er
ddNEν¯(de0t)(t)= c (Lαν¯)eT(t2)(t)ex(pE(E/T/(Tt)()t)2)++α 1, (1) 52n [10 100
L bi
where c (α)=(cid:0)1−2−3−α(cid:1)Γ(4+α)ζ(4+α). The mean per
L y
ν¯e energyforthisspectrumis4 (cid:104)Eν(cid:105)(t)=cT(α)T(t)with osit10-1
cT(α)=cL(α)/cL(α−1). Wesetα=2. Thesuperscript min
on dN(0)/dEdt denotes the spectrum at the source, be- Lu
ν¯e
fore neutrino flavor mixing.
(ii) We neglect the contribution of ν¯µ and ν¯τ at the 10-2
source. UsingEq.(1)wecomputetheν¯ differentialflux 10-1 100 101
at the detector, e t [sec]
Fig. 2.— Inferred ν¯e luminosity. Blue markers: Poisson likeli-
P dN(0) hoodanalysisbasedonthesimplifiedformulaeEqs.(1-2),usingthe
Φ (t)= ee ν¯e (t), (2) combined Kamiokande, IMB, and Baksan data, with thick (thin)
ν¯e 4πDS2N dEdt error bars showing 1σ (2σ) ranges. Black circles: Rν/Eν binned
luminosity estimator from data; see Eq. (A2) and text around it.
withtheelectronantineutrinosurvivalprobabilityP = Notethat(i)theluminosityestimator(black)doesnotaccountfor
ee
0.67 and with D =50 kpc the distance to SN1987A. detectorbackground, whilethePoissonfit(blue)subtractsit; (ii)
SN omittingBaksandatadoesnotaffecttheresultssignificantly.
We perform a Poisson likelihood analysis for the
Kamiokande, IMB, and Baksan neutrino data of
SN1987A, including background and detector effi- have α = 0. Our choice of α = 2 applies if the dom-
ciency effects. We implement the analysis suggested inant ν¯e source is e+n → pν¯e from a plasma contain-
in Pagliaroli et al. (2009) and in Ianni et al. (2009) that ing free nucleons and e± pairs, a reasonable scenario if
modifies the method of Loredo & Lamb (2002) in the the luminosity is dominated by accreting matter (see,
treatment of detector efficiency. We include only the e.g., Perego et al. 2015). Contributions due to ν¯µ and
dominant inverse-beta decay (IBD) reaction (Strumia & ν¯τ at the source are straightforward to include and do
Vissani 2003). For detector efficiency and backgrounds, not affect our conclusions. Last, our parametrization of
we use the updates given by Vissani (2015). neutrino mixing ignores possibly important matter and
Our first analysis of the data is as follows. We neutrino self-induced oscillation effects (see, e.g., Kar-
split the full neutrino event time series (16 events in tavtsev et al. 2015; Mirizzi et al. 2015). Our choice of
Kamiokande, 8 events in IMB, and 5 events in Baksan) P =0.67≈cos2θ applies for normal mass hierarchy,
ee 12
into eight time bins of equal log-space duration delim- with a strong matter effect causing adiabatic alignment
itedby[0,0.25,0.5,1,2,4,8,16,32]sec. Ineachtimebin, of the ν¯ flavor state with the mass eigenstate ν¯ (Fogli
e 1
centred around the time t , we fix the source parameters et al. 2002). We chose this treatment of flavor conver-
i
L (t) and T(t) to constant values L , T , independent sion mainly to facilitate comparison with previous work
ν¯e i i
frombintobin. Wethenperformabinbybinlikelihood that used the same prescription (Loredo & Lamb 2002;
analysis in the two source parameters L and T . Pagliaroli et al. 2009; Ianni et al. 2009).
i i
Figure 2 shows the result for the fitted source lumi- We now wish to compare different theoretical models
nosity L . In each bin, the blue marker denotes the best for the supernova. In Section 2.2 we follow Loredo &
i
fit luminosity, with thick (thin) vertical error bars ob- Lamb (2002) and Pagliaroli et al. (2009) and perform
tained by fixing T to its best fit point and letting L a likelihood analysis of the neutrino mechanism using
i i
vary within ∆χ2 < 1 (∆χ2 < 4). (The black circles are phenomenological models. In Section 2.3 we devise ana-
explained in Appendix A, and are not important for the logues models, with the same number of free parame-
discussion in this section.) The horizontal bars denote ters, for the neutrino luminosity expected in CITE. We
the time bin duration5. The time gap in Figure 1 is re- compare the performance of CITE models to that of the
produced in Figure 2 as an order of magnitude drop in neutrino mechanism.
the ν¯ luminosity around t∼2 sec.
e
Regarding our parameterization in Eqs. (1-2). Other 2.2. Neutrino mechanism
assumptions about neutrino flavor mixing or source fla- Calculations within the neutrino mechanism suggest
vorcompositioncanbemadebyreinterpretingtheprod- that the supernova explosion, if it is to occur at all,
uct PeeLν¯e. Our results are not affected significantly should occur within a few hundred milliseconds after
by the choice of α. A plain thermal spectrum would core collapse (see, e.g., O’Connor & Ott 2011; Pejcha
& Thompson 2014). Before the explosion, accretion of
4 For reference, cL(0) ≈ 5.68, cL(2) ≈ 118.26, cT(0) ≈ 3.15, the stellar envelope onto the PNS produces accretion lu-
cT(2)≈5.07. minosity with nontrivial time dependence, that (for ν
5 We find it more informative for our current purpose to plot e
and ν¯ ) can dominate over the cooling luminosity of the
theinstantaneousmeanluminosityLi ineachbin,ratherthanthe e
energy per bin tiLi, despite the logarithmic bin assignment. We PNS. However, after the explosion has cleared away the
thankJohnBeacomfordiscussiononthispoint. accreting matter, for post-bounce times t > 1 sec, the
5
neutrino mechanism has a robust prediction of contin- 2.3. CITE: accretion for ∼2−3 sec, then black hole
uous PNS cooling luminosity that is slowly decreasing formation
with a characteristic time scale of a few seconds.
TheneutrinoluminosityinCITE,foraprogenitorstar
Pagliaroli et al. (2009) used phenomenological mod-
relevant to SN1987A and skipping the abrupt initial ν
els of the neutrino flux to represent the predictions of e
deleptonization burst that is unimportant for our pur-
the neutrino mechanism. The main models discussed
pose, follows three main phases.
there were: (i) simple exponential cooling of the PNS
(EC model, details in Appendix A), and (ii) exponential 1. PNS forms on core collapse, followed by ac-
cooling + truncated accretion model (ECTA model, de- cretion through a quasi-static accretion shock.
tails in Appendix A), with more free parameters added This is the usual stalled shock of the core
to describe the early accretion phase, that is truncated bounce. Electron-flavor neutrino luminosity is
by hand at t =0.55 sec. dominated by nucleon conversion reactions with
accretion
anWdeEhCavemroedperlos,duicnedagthreeembeesnttfitwpitohinPtsagfolirartohlei EeCtTaAl. Lνe (cid:16)≈ Lν¯(cid:17)e,(cid:16) where Lν(cid:17)¯e(cid:16) ∼ G(cid:17)M2PRNPSNM˙SPNS ∼
(2009). We also reproduce the Poisson likelihood dif- 1052 MPNS M˙PNS 25 km erg/sec. PNS
2 M(cid:12) 0.1 M(cid:12)/sec RPNS
ference ∆χ2 ≈−10 in favor of the ECTA model best-fit coolingproducesadditionalluminosityL ∼(0.3−
x
point as compared with that of the EC model. This 0.6)L , with L ≈ L ≈ L ≈ L ≡ L .
statistical preference for the ECTA model led Loredo & State-ν¯oef-the art νeµxampleν¯sµare prνeτsentedν¯iτn Peregxo
Lamb (2002) and Pagliaroli et al. (2009) to argue that etal.(2015)andinMirizzietal.(2015). Theaccre-
the data provides evidence for an early accretion phase tion phase lasts for 1-3 sec after bounce, until the
on time t(cid:46)0.5 sec. PNS accumulates baryonic mass ∼2−3 M (de-
(cid:12)
Loredo & Lamb (2002) and Pagliaroli et al. (2009) did pending on details of the EOS (O’Connor & Ott
not have a model that could address a sharp luminosity 2011)) leading to BH formation
dropattimet>1sec. Itisinterestingtore-inspecttheir
results, keeping in mind CITE as an alternative theory. 2. BH forms, absorbing the matter downstream to
The neutrino time series as shown in Figure 1 has no the accretion shock on a time scale of miliseconds.
events in either Kamiokande or IMB6 during the time Spherical accretion directly onto the BH produces
t = 2.7−5 sec. We can use the best-fit EC and ECTA small neutrino luminosity, Lν¯e (cid:46) 1047 erg/sec, be-
modelsofPagliarolietal.(2009)tocalculatethePoisson causetheaccretingmatterintheabsenceofashock
probabilityforobservingnoeventsduringthistime. The doesnothavetimetoradiateitsgravitationalbind-
results are given in Table 1. For the EC (ECTA) model, ing energy before it goes through the horizon (or,
this probability is 0.4% (2.2%). fromtheperspectiveofanobserveratinfinity,gets
redshifted to nothing). Thus, BH formation leads
to an abrupt cut-off in the neutrino luminosity
TABLE 1 3. A quiescent phase, corresponding to quasi-
Poisson probability for the neutrino data during sphericalaccretionontheBH,shouldbeginatt
t=2.7−5 sec. For each theoretical model we calculate BH
and last for ∼1−10 seconds. However, for CITE
the number of signal events expected in each detector
during this time. The Poisson probability for observing no to work (Kushnir 2015a), angular momentum in
events is given in the last column. IMB is assumed to have the envelope must produce a centrifugal barrier,
zero background. We take the threshold energy
leading to an accretion disc and to the launch of
Eth=4.5 MeV for Kamiokande, with which we expect 0.43 the RIAS that propagates outward and eventually
background events during t=2.7−5 sec.
triggers the explosion. Matter in the disc at the
baseoftheRIASheatsupandre-initiatesneutrino
Model Kamiokande IMB Poissonprobability emission, dominated again by nucleon conversion
EC(ν mechanism) 3.65 1.56 0.4% reactions and potentially reaching ∼1051 erg/sec.
ECTA(ν mechanism) 2.23 1.17 2.2%
We construct a toy phenomenological parametrization
CITE 0.03 0.01 62%
for the neutrino luminosity of SN1987A in CITE, in the
spirit of the neutrino mechanism analysis of Loredo &
Lamb (2002) and Pagliaroli et al. (2009). We use six
We learn that the time gap is somewhat unlikely from free parameters, the same number of parameters as used
the point of view of the neutrino mechanism. In fact, in Loredo & Lamb (2002) and in Pagliaroli et al. (2009)
we suspect that the improved global likelihood of the to define the ECTA model of the neutrino mechanism.
ECTA model, interpreted in Loredo & Lamb (2002) and Two of our parameters define the basic CITE time
in Pagliaroli et al. (2009) as evidence for an early ac- scales: t denoting BH formation, and t denot-
BH disc
cretion phase, may actually be driven to some extent by ing accretion disc formation (and launch of the RIAS).
the need to not over-shoot the late time luminosity gap To model the PNS accretion phase (phase 1 above),
at t > 2 sec. To clarify this point, in Appendix A we we build on the simulations of Perego et al. (2015)
constructabinnedMonte-Carlo(MC)analysisthatpro- for their HC19.2 pre-supernova progenitor model, when
vides time-dependent information on the performance of their artificial trigger (denoted ”PUSH” in Perego et
the fit. al. 2015) is not used to start an explosion. We de-
fine two parameters, f and f . During the inter-
L E
val 0 < t < 0.8 sec, where Perego et al. (2015) pro-
6 TherewerenoeventsinBaksan,either,forthistimeperiod. vidednumericalresults,ourmodelluminosityisL (t)=
ν¯e
6
f ×LHC19.2(t),L (t)=f ×LHC19.2(t),withmeanneu- ularly un-natural with t of a few seconds, as
L ν¯e x L x disc
trino energy (cid:104)E (cid:105)(t) = f ×(cid:104)E (cid:105)HC19.2(t), (cid:104)E (cid:105)(t) = long as CITE can operate successfully. However,
f ×(cid:104)E (cid:105)HC19.2ν¯(et). HereE, LHC1ν9¯e.2 is the ν¯ lumxinosity we should stress that while the formation of the
E x ν¯e e disc, by itself, is a built-in ingredient in CITE, the
reported by Perego et al. (2015), etc. For 0.8 sec <
precise timing t = 5 sec we deduce here from
t < t (where Perego et al. 2015, did not provide disc
BH the neutrino data is not a generic prediction of the
numerical results), we let the luminosity decrease as
model: asseeninKushnir(2015a),CITEcouldop-
L (t), L (t) ∝ 1/t, while the energies are set to rise
linν¯eearly (cid:104)xE (cid:105)(t), (cid:104)E (cid:105)(t) ∝ t. For t > t , we set erate just as well with tdisc >10 sec. This is to be
Lx(t) = 0,ν¯aend Lν¯e(tx) = 1+2Lt/dtidscisc (cid:16)1−e−(t/tBdiHsc)k(cid:17) with cinontthreasptreedviwouitshittehme.more robust prediction of tBH
k = 100. This form gives a fast rise for the accretion
disc ν¯e luminosity, consistent with what we find in our • The values of fL and fE we find correspond to
numericalsimulationsinthenextsection. Themeanen- moderate modulation of the results of Perego et
ergy during the disc phase is (cid:104)Eν¯e(cid:105)(t)= 1+2Et/dtidsicsc. afrlo.m(20va1r5y).ingMtuhcehinlaprugterprme-ocdolulalaptsieonpsrocfiolueldwiatrhisine
Tosummarize, oursixfreeparametersare(1)t and
BH observational constraints.
(2) t , denoting BH and subsequent disc formation
disc
times; (3) f and (4) f , constant factors by which we
L E • Last, the best fit disc neutrino energy E is
modulatethenumericalresultsfortheHC19.2SN1987A disc
higher by about a factor of 2, and the best fit
progenitor model of Perego et al. (2015), to obtain the
disc neutrino luminosity L is higher by about
luminosity before BH formation; and (5) L and (6) disc
disc an order of a magnitude, than our estimate of
E , characterizing the late neutrino emission of the
disc the disc emission in the next section. However,
accretion disc around the BH.
these parameters are not tightly constrained by
Calculating the likelihood for our CITE parametriza-
the data. For example, keeping the other pa-
tion, we find several configurations with Poisson likeli-
rameters at the same value as in (3), but reduc-
hood superior to the best fit ECTA model of Pagliaroli
ing L to 1051 erg/sec, gives Poisson likelihood
et al. (2009). For instance, the following model disc
for CITE that is worse by ∆χ2 ≈ 6.2 compared
CITE: (3) to the L = 4 × 1051 erg/sec of (3), but still
disc
t =2.7 sec, t =5 sec, f =0.67, f =0.56, improved by ∆χ2 ≈ 0.6 compared to the ana-
BH disc L E
L =4×1051 erg/sec, E =15 MeV, logues ECTA model of the neutrino mechanism.
disc disc Varying both L and E within ∆χ2 = 4
disc disc
has ∆χ2 smaller by 6.8 compared to the ECTA model around the reference values in (3) we find values
of the neutrino mechanism. For comparison, within the in the range L ∼ (1−10)×1051 erg/sec and
disc
neutrino mechanism, the ECTA model has ∆χ2 smaller E ∼10−20 MeV, with higher L correlated
disc disc
by 9.8 than that of the EC model, the latter having 3 with lower E and vice-verse.
disc
parameters less; this was considered in Loredo & Lamb
(2002) and in Pagliaroli et al. (2009) as evidence for an 3. NUMERICAL SIMULATIONS
accretion phase.
Inthissectionweperform2Dnumericalsimulationsof
OurCITEmodelisobviouslyconsistentwithnoevents
CITE. We have two goals:
during the time t = 2.7−5 sec. We give the Poisson
(i) to verify that CITE can operate with RIAS launch
probabilityinTable1. Comparingtotheneutrinomech-
time t ≈5 sec, reproducing E and M in the ball-
anism, the 2.7−5 sec time gap is the source for the im- disc kin Ni
park of observations for SN1987A.
provedlikelihoodofCITE.InAppendixAwerepeatour
(ii) to study the accretion disc neutrino luminosity rele-
binned MC analysis for the model in (3). Incidentally,
vant for CITE on times t>t .
as we have based the first second of our CITE model disc
With respect to item (ii), we stress that our calcula-
light-curve on the non-exploding numerical simulation
tions are preliminary. Our code is Newtonian and our
of Perego et al. (2015), it is safe to say that the early
treatment of neutrino transport is simplistic. More so-
timeneutrinodatadoesnotrequireatransitionbetween
phisticatedcodesexistintheliterature(see, e.g.,Perego
accretion luminosity to PNS cooling luminosity. Contin-
et al. 2015; Mirizzi et al. 2015; O’Connor & Ott 2011);
ued accretion is consistent with the data.
ourestimatesheremotivatetheapplicationofthesetools
Finally we comment on the fit results in (3).
tothescenarioofCITE.Beyondthetechnicallimitations
• The value of t = 2.7 sec is in good agreement ofthesimulation,theproblemoftheneutrinoluminosity
BH
with progenitor models as in Perego et al. (2015). of BH accretion discs suffers from theoretical uncertain-
We believe that t ∼ 1−3 sec is a robust pre- ties due to the implementation of viscosity. Here we set
BH
diction of CITE for strong explosions, as can be theviscositytozero. Estimatesfordifferentassumptions
seenformanalyticalestimatesaswellasnumerical oftheviscositycanbefoundinPophametal.(1999)and
simulations (O’Connor & Ott 2011). in MacFadyen & Woosley (1999). Because of these lim-
itations, we do not attempt to reproduce the neutrino
• We view the requirement t = 5 sec as an ob- lightcurveinanydetailbesidesfromtheroughluminos-
disc
servational constraint on CITE, at least when at- ity and time scales.
tempting to interpret SN1987A IMB data. We an- We aim to simulate the process of the accretion disc
alyze the implications of this constraint further in at times t(cid:38)3 sec and the subsequent CITE, and we do
the next section. We do not see anything partic- not attempt to reproduce the early phase of PNS and
7
BH formation (again see, e.g., Perego et al. 2015; Mi- 1.6M⊙ Si O He- He
rizzi et al. 2015; O’Connor & Ott 2011, for details of O
this early phase). We assume that once sufficient mass, 10
M >2−3M ,hasaccretedthroughtheinnerboundary
(cid:12)
of our simulation, r (to be specified below), the cen-
inner
tral object forms a BH. Before this time, the flow below 8
r ∼107 cm in our simulation does not capture correctly
the standing shock above the PNS; however, for t>t
BH
the shocked material is quickly absorbed in the BH and 6
by t > 4 sec – still many dynamical times prior to disc
formation in our simulation – we expect that our calcu-
lation provides a reasonable approximation of the flow 4
down to r near the last stable orbit. log (ρ[gcm−3])
10
T[109K]
3.1. Pre-collapse profile 2 M[M⊙]
We use the same methods from Kushnir (2015a), so jz=0[1017cm2s−1]
here we only highlight a few aspects of the simulations. MESA
We do not simulate the collapse at r <r and details 0
inner 107 108 109 1010
of the progenitor on r < r are unimportant for the
inner r[cm]
results. Onr >r thepre-collapseprofileiscomposed
inner
ofshellswithconstantentropyperunitmass,s,constant Fig. 3.— Pre-collapse stellar profile (density, temperature, en-
composition, and in hydrostatic equilibrium. We place closed mass, and specific angular momentum on the equatorial
1.6M within r < 2·108cm, representing a degenerate plane, jz=0) used in CITE simulation (the profile below r =
(cid:12) 2·108cmhasnegligibleeffectontheresults, seetextfordetails).
iron core. This choice roughly reproduces the PNS mass
Thedensity,temperature,andenclosedmassprofilesaresimilarto
inPeregoetal.(2015)fortheirHC19.2progenitormodel pre-collapse profiles of a 20M(cid:12) star (dashed gray), calculated by
when PUSH is not used to trigger an explosion. The RoniWaldmanwithMESA(Paxtonetal.2011).
regionbetweenr andr =2·108cmisfilledwiths=
inner
1kB iron(inhydrostaticequilibrium),whichisprevented Heretff(r)isthefree-falltimeatpre-collapseradialcoor-
from burning. This prescription is chosen for simplicity, dinate r, M(r) is the enclosed mass, and rf is the radial
andwedefermoredetailedanalysistofuturework. Note coordinateonthez =0planewherethecentrifugalforce
that the region inwards of r = 2·108 cm falls through fraction f first becomes greater than zero. The factor of
(cid:112) 2 in Eq. (4) sums (i) the (almost) free-fall trajectory of
r in time t ≈ π r3/2GM(r) ≈ 0.4 sec after core
inner the mass element initially at r down to the disc for-
collapse,sothecompositioninthisregionhasanegligible f
mation radius r ≈ (f/2)r (cid:28) r , and (ii) the time
effect on the results of the simulation. disc f f
it takes the rarefaction wave starting at core-collapse to
The base of the He-O shell is placed at a mass coordi-
propagate out from r =0 to r (Because the initial pro-
nate of 6M , a radius of 4.25·109cm, and a density of f
(cid:12) file is in hydrostatic equilibrium, this sound travel time
104gcm−3. Theshelliscomposedofequalmassfraction
is again roughly equal to the free-fall time at r ; Kush-
of helium and oxygen. The local burning time at the f
nir & Katz 2015). In Figure 3, r = 1.2·109 cm and
base of the shell is ≈700s, which is 100 times the free- f
M(r )=2.34 M , so we estimate t ≈5.2 sec.
fall time at this position. The total mass of the He-O f (cid:12) disc
The disc neutrino luminosity can be estimated by the
shell is ≈2.7M . Pure oxygen (helium) is placed below
(cid:12)
gravitationalbindingenergyaccretingthroughthedisc,
(above) the He-O shell. Oxygen is replaced with silicon
where T >2·109K, to prevent fast initial burning. The GM M˙
angularmomentumisinitiallydistributedsuchthatf , L ∼ disc disc (5)
rot ν¯e 2r
theratioofthecentrifugalforcetothecomponentofthe disc
gravitational force perpendicular to the rotation axis, is (cid:20)(f/2)r (cid:21)−1(cid:20)M(r )(cid:21)(cid:34) M˙ (cid:35)
constant f = 0.02 throughout the profile, except for ≈1051 f f disc erg/sec.
rot,0 107 cm 2M 0.05M /sec
the following: (cid:12) (cid:12)
• f =0 at r <1.2·109cm. This estimate assumes that half the disc emission is in
rot
ν¯ . We scaled the mass accretion rate through the disc
e
• f = 0 at large radii, and increases linearly with by a typical value.
rot
decreasing radius between r = 2 · 1010cm and Note that Eqs. (4-5) are only used to tune the initial
r = 1010cm to f . This is done for numerical stellar profile before running the numerical simulations.
rot,0
stability, and has a small effect on the results. We do not use these estimates in the numerical calcula-
tions described next.
The stellar profile used in the analysis is shown in Fig-
ure 3.
3.2. Simulations and results
The stellar profile defined here is designed to achieve
t ≈ 5 sec and disc neutrino luminosity L ∼ The problem considered is axisymmetric, allowing the
1d0i5sc1 erg/sec. An estimate of the disc formationν¯etime use of two-dimensional numerical simulations with high
resolution. We employ the FLASH4.0 code with ther-
is given by
monuclear burning (Eulerian, adaptive mesh refinement
(cid:115)
r3 Fryxell et al. 2000) using cylindrical coordinates (R,z)
t ≈2t (r )=π f . (4) to calculate one quadrant, with angular momentum im-
disc ff f 2GM(r )
f
8
plementation as in Kushnir (2015a). Layers below the forviscosityeffects, findingaccretiondiscneutrinolumi-
inner boundary, r , are assumed to have already col- nosity with a range encompassing our result here.
inner
lapsed,andthepressurewithinthisradiusisheldatzero
throughout the simulation. We assume that neutrinos
escape freely through the outer layers.
Weperformtwodifferentsimulationrunsbasedonthe 101
same stellar profile.
1. First, the thermonuclear explosion was calculated
with r = 60km, a resolution (i.e. minimal al-
inner
lowed cell size within the most resolved regions)
of ≈14km and a 13-isotope α-chain reaction net-
work (similar to the APPROX13 network supplied 100
with FLASH with slightly updated rates for spe-
cificreactions, especiallyfixingatypoforthereac-
tion 28Si(α,γ)32S, which reduced the reaction rate
by a factor ≈4). This setup is sufficient for cal-
culating the disc formation, RIAS launch, and the Lν¯ [1051ergs−1]
resultingthermonuclearexplosion. Anignitionofa e
detonation wave was obtained at t≈25sec, which hEν¯ei [MeV]
resultedinanexplosionwithEkin ≈6·1050ergand 10-15 5.5 6 6.5 7
MNi ≈ 0.035M(cid:12). Both of these values are in the t[s]
ballpark of, though smaller by a factor of ≈2−3
than the observed values of SN1987A (Utrobin & Fig. 4.—Meanenergy(red)andluminosity(black)forν¯e taken
fromthenumericalsimulation.
Chugai 2011).
2. Second, to calculate the neutrino light curve we
used r =30km, a resolution of ≈2km and the
inner T [K]
APPROX19reactionnetwork(toallowheliumdis-
integration to nucleons). The required high reso- Outgoing
shock wave
lution and small value of rinner allowed us to con- ν emission
tinue the calculation for only a few seconds after
thediscformed. Thenucleonconversionrateswere
estimated by ≈9·1023(cid:0)T/1011K(cid:1)6X ergs−1g−1
n
(Qian&Woosley1996),whereXn isthemassfrac- m]
tion of neutrons. The baryonic mass below r c
treach≈ed52sMec(cid:12), iantcr≈e2a.s5insgecLandttohe≈R5IA·S10f5o0rmeregidnsn−ae1tr 6z [10
disc ν¯e
where the mean energy of the neutrinos7 is esti-
mated by (cid:104)E (cid:105)≈10MeV (see Figure 4).
ν¯e
A snapshot of the disc and RIAS at time 5.5sec is
showninFigure5. Theneutrinoemissionoriginatesfrom
radii30−100km,butmostlydominatedfrom30−40km
where the typical densities are few×109gcm−3.
Increasing the resolution to ≈1km changes the results R [106 cm]
by less than 10%, but increasing rinner to 40km leads to Fig. 5.— BH accretion disc neutrino luminosity in CITE. Log-
a reduced luminosity by 30−40% and reduced energies arithmic temperature map at 5.5s since collapse with neutrino
by 10%. We conservatively estimate that our results are emission contours (black, 10 contours logarithmically distributed
accurate to only a factor of a few, as our simulations are between3·1019−3·1020ergs−1g−1). Theinsetshowsazoomed
maparoundtheneutrinoemissionregion.
Newtonian and velocities of ∼0.5c are achieved near the
emission region. Furthermore, the Schwartzchild radius
ofthecentralBHatthistimeisRs ≈10km,soourdisc, 4. CONCLUSIONS
that ignores general relativistic effects, is located not far
The neutrino burst of SN1987A has been traditionally
above the last stable circular orbit. Nevertheless, our
results demonstrate that L ∼ 1051ergs−1 with E ∼ used to advocate for the neutrino mechanism operating
ν¯e ν¯e in exploding CCSNe. Our goal in this paper was to give
10 MeV is possible for t>t .
disc a first analysis of the neutrino signal expected in CITE,
OurresultscanbecomparedtothoseofPophametal.
asacompetingmechanismofCCSNe, andtocompareit
(1999) and of MacFadyen & Woosley (1999) in the con-
to the SN1987A signal. The questions we addressed and
text of the collapsar model, that shared a similar setup
our results were as follows.
to ours. The latter included a free parameter to account
There is a common claim in the literature that di-
7Meanneutrinoenergywasapproximatedfromthemattertem- rectBHformationwouldbeincompatiblewithSN1987A.
perature, averaged by neutrino emissivity and assuming α=2 in This claim is usually based on two arguments: (i) the
Eq.(1). neutrino mechanism predicts a NS remnant, and/or (ii)
9
directBHformationwouldcut-offtheneutrinoemission, We close with comments on further work.
leaving the signal events at t>5 sec unexplained.
We find that this claim is, at least currently, unjus- • We are eager to see independent simulations of
tified. First, the neutrino mechanism has not yet been CITE, to compare with the work of Kushnir
shown to operate successfully and reproduce the obser- (2015a). In particular, it is important to investi-
vations of SN1987A. Therefore its failure is not a good gate whether the pre-collapse initial conditions re-
cause to exclude BH formation. CITE provides one po- quired for CITE can be obtained with stellar evo-
tential counter example. Second, if the progenitor of lution models.
SN1987Apossessedarotatingenvelopethenanaccretion
• Many particle physics analyses used the neutrino
discwouldformaroundtheBH.Suchaccretiondiscsare
burst of SN1987A to constrain new physics be-
knowntobecopiousneutrinoemittersandcouldexplain
yond the Standard Model, such as axions or ster-
the late-time neutrino events of SN1987A.
ile neutrinos (see, e.g., Raffelt (1996)). Most of
In Section 2 we gave a statistical analysis of the neu-
these works assumed PNS cooling luminosity, as
trino emission in CITE, along the lines used by Loredo
suggested within the neutrino mechanism. Our re-
&Lamb(2002)tostudytheneutrinomechanism. While
sults here imply that these analyses may need to
the statistical significance of such analysis is limited by
be revisited.
the sparse data, we find that: (i) there is a hint in the
data for a luminosity drop around t ∼ 2 sec, right in
• IfCITEworksinnature,thentheneutrinoburstof
theballparkwhereCITEpredictsBHformation; (ii)the
the next strong Galactic CCSN may give us front-
neutrino mechanism is in some tension with this lumi-
row seats to the formation of an event horizon in
nosity drop, while CITE could address it naturally.
real time with current neutrino detectors. This
The neutrino events at t > 5 sec imply that CITE
may have already happened, albeit with limited
should be operative with RIAS formation as early as
statistics, with SN1987A. Access to phenomena
that. This is a nontrivial constraint that was not con-
near the event-horizon motivates construction of a
sidered in Kushnir (2015a). It can be summarized by
fewMegatonneutrinodetectorthatwillobserveex-
Eqs. (4-5) with tdisc ≈ 5 sec and Lν¯e ≈ 1051 erg/sec. In tragalactic CCSNe on a yearly basis (Kistler et al.
Section3weperformed2Dnumericalsimulationsguided 2011).
by these constraints. Without yet attempting a system-
atic survey of possible profiles, we were able to find such
profile that yields an explosion in the rough ballpark of
We thank John Beacom, Avishay Gal-Yam, Juna
theobservations(E aboutafactorof3andM about
kin Ni Kollmeier, and Fukugita Masataka for discussions, Ko-
a factor of 2 below that of SN1987A). Further study of
hta Murase for early collaboration, and Boaz Katz, Eli
different initial profiles is needed to derive more conclu-
WaxmanandYosefNirforcommentsonthemanuscript.
sive results. We also gave order of magnitude estimates
D. K. gratefully acknowledges support from the Friends
oftheluminosityandtypicalenergyoftheneutrinoemis-
oftheInstituteforAdvancedStudy. FLASHwasinpart
sionproducedbytheBHaccretiondiscatthebaseofthe
developed by the DOE NNSA-ASC OASCR Flash Cen-
RIAS, finding results on the low side, but not inconsis-
ter at the University of Chicago.
tent with the data.
APPENDIX
MODELLING OF THE NEUTRINO MECHANISM, AND MONTE CARLO STUDY
We recap here some details on the EC and ECTA models of Loredo & Lamb (2002) and Pagliaroli et al. (2009). We
also describe a Monte Carlo (MC) analysis designed to clarify the time dependence in the different models.
The EC model has 3 free parameters for the neutrino source: (i) NS initial temperature T , (ii) NS cooling time
c
τ , and (iii) NS neutrinosphere radius R . The ν¯ luminosity is assumed to scale as L (t) ∝ R2T4(t) with T (t) =
c c ν¯e c c c
Tce−4τtc. In addition to the source parameters, three unknown time shifts between the zero of time in the three
detectors Kamiokande, IMB, and Baksan are also marginalized over, with the best-fit model of Pagliaroli et al. (2009)
corresponding to these time shifts being zero. In addition to direct ν¯ emission, equal luminosity is assumed to
e
be emitted in µ and τ flavors, L = L = L ≡ L . The x-flavor temperature is set by hand to 1.2 times the ν¯
ν¯e ν¯µ ν¯τ x e
temperature. ToaccountforL intermsoftheeffectiveL ofEq.(2),weconvertLeffective =Lmodel+(1/P −1)Lmodel,
x ν¯e ν¯e ν¯e ee ν¯x
with L =L =L .
ν¯x ν¯µ ν¯τ
The ECTA model adds, on top of the 3 parameters of EC, 3 more free parameters intended to describe an early
accretionphaseprecedingtheexplosion: (iv)accretiontemperatureT ,(v)accretiontimescaleτ ,and(vi)aparameter
a a
µ proportional to the over-all accretion luminosity. Pagliaroli et al. (2009) assumes that the accretion luminosity
consists purely of e flavor, setting L =0 during the accretion phase and turning it back on once accretion is stopped
x
and replaced by NS cooling as above. In addition to the free parameters (iv-vi), time dependence for the accretion
luminosity, L ∼ L /(1+t/0.5 sec), is introduced in Loredo & Lamb (2002) and in Pagliaroli et al. (2009) without
x x,0
counting the functional form or the time scale of 0.5 sec as another free parameter (instead, the 0.5 sec time scale is
argued to arise in numerical simulations).
We move on to describe our binned MC procedure. In the limit that energy-dependent detector efficiency and
background are not important, a good proxy for the source luminosity during some time interval ∆t is given by the
10
sum of event inverse-energy,
Rν ≡ 1 (cid:88) 1 , (A1)
E ∆t E
ν k
k
where the sum goes over the neutrino events detected during ∆t. To see this, note that the detection cross section
at the relevant neutrino energies (8 MeV < E < 45 MeV) can be approximated by σ(E ) ≈ σ¯(E /MeV)2, with
ν ν ν
σ¯ = 6.8×10−44 cm2. Ignoring background and energy-dependent detector efficiency, we can compute the expected
value of R /E given source luminosity L (constant in time during ∆t),
ν ν ν¯e
(cid:28)R (cid:29) N P (cid:90) σ(E) dN(0) (cid:18)N P σ¯(cid:19)
ν ≈ p ee dE ν¯e = p ee L , (A2)
E 4πD2 E dEdt 4πD2 ν¯e
ν SN SN
where N is the effective number of target protons in the detector. For an ideal detector we have L ≈
(cid:16) (cid:17)(cid:16)p (cid:17) ν¯e
1032 (cid:104)Rν/Eν(cid:105) ×1053 erg/sec. Inpracticeenergy-dependentefficiencyintroducesaneffectivelow-energythresh-
Np MeV−1sec−1
old that lowers the proportionality coefficient on the RHS of Eq. (A2) in a detector-dependent way. In addition,
a small correction is introduced due to the small mismatch between the incoming neutrino energy and the recon-
structed positron energy in the IBD detection event. Combining all three detectors, we find that the replacement
N →0.29(N +N +N )=1.8×1032 in Eq. (A2) for the luminosity estimator
p p,Kam p,IMB p,Bak
(cid:18) (cid:19)
R /E
LRν/Eν ≈ ν ν ×5.6·1052 erg/sec, (A3)
ν¯e MeV−1sec−1
reproduces the data well for the source parameter α = 2 adopted in most of this section8. The result of applying
Eq. (A3) to the data is shown by black markers in Figure 2. Note that the luminosity estimator in Eq. (A3) does not
account for detector background, while the Poisson fit (blue in Figure 2) automatically subtracts it.
Armed with our quick-to-compute luminosity estimator LRν/Eν from Eq. (A3), we generate mock data samples and
ν¯e
compute the distribution of LRν/Eν in different time bins. In Figure 6 we show the Monte Carlo (MC) results for
ν¯e
LRν/Eν (red markers), computed for the best fit EC (left) and ECTA (right) models of Pagliaroli et al. (2009). The
ν¯e
MC results we show are converged to a few percent with 5·104 mock samples.
Figure6suggeststhatmuchofthestatisticaltensionassociatedwiththesimpleECPNScoolingmodelisdrivenby
the luminosity drop at t∼2 sec. In order to not overshoot the event rate during this time, the EC model is forced to
low luminosity on earlier times, leading to tension in the t∼0.25−0.5 sec time bin. The ECTA model can fix some
of this tension, raising the luminosity at t (cid:46) 0.5 sec while using extra free parameters to keep the late time “cooling
PNS” luminosity not too high.
Neutrinomechanism,EC Neutrinomechanism,ECTA
101 Data 101 Data
Model Model
1] 1]
− −
gs 100 gs 100
r r
e e
2 2
5 5
0 0
1 1
[ [
Eν Eν
R/ν¯νe10-1 R/ν¯νe10-1
L L
10-2 10-2
10-1 100 101 10-1 100 101
t[s] t[s]
Fig. 6.—MCdistributionofthebinnedLRν/Eν sourceluminosityestimator. Left: ECmodel. Right: ECTAmodel.
ν¯e
In Figure 7 we repeat our MC procedure for the LRν/Eν binned luminosity estimator in CITE, using the model of
ν¯e
Eq. (3). Compared with the ECTA model, we find somewhat improved consistency with the data.
REFERENCES
8 Changingtoα=0wefindforthescalingfactor0.29→0.25.
Noneofourresultsisaffectedsignificantly.