Neutrino spectra from stellar electron capture

6 Pages·0.26 MB·English

by  K. Langanke

Neutrino spectra from stellar electron capture
K. Langanke1, G. Mart´ınez-Pinedo1,2 and J. M. Sampaio1
1Institut for Fysik og Astronomi, ˚Arhus Universitet, DK-8000 ˚Arhus C, Denmark
2 Departement fu¨r Physik und Astronomie, Universit¨at Basel, Basel, Switzerland
(February 8, 2008)
Using the recent shell model evaluation of stellar weak interaction rates we have calculated the
neutrino spectra arising from electron capture on pf-shell nuclei under presupernova conditions.
We present a simple parametrization of the spectra which allows for an easy implementation into
collapse simulations. We discuss that the explicit consideration of thermal ensembles in the parent
nucleus broadens the neutrino spectra and results in larger average neutrino energies. The capture
rates and neutrino spectra can be easily modified to account for phase space blocking by neutrinos
which becomes increasingly important during thefinal stellar collapse.
1
PACS numbers: 26.50.+x, 23.40.-s, 21.60.Cs
0
0 Weak interactions play a central role in the final evo- tions, but to our knowledge neutrino spectra have not
2
lution of massive stars [1]. These processes, mainly elec- been derived on the basis of the FFN compilation. As

n troncaptureandbeta decays,createneutrinos whichfor mentioned above, such spectra are also not required for
a all densities until the collapse becomes truly hydrody- presupernova studies as the neutrinos leave the star un-
J namic (i.e. ρ 1011 g cm−3) escape the star carrying hindered.

8 away energy a∼nd reducing the entropy. Thus, in simu- Due to progress in nuclear many-body modeling and
1
lations which follow the star’s evolution until the iron in computer hardware and guided by experimental data

core reaches central densities of order a few 109 g cm−3 it has recently been possible to treat the nuclear struc-
1
it is sufficient to know the average neutrino energies of ture problem involved in the calculation of stellar weak
v
9 the various weak processes to determine the energy loss interaction rates in a reliable way [10,11]. The calcu-
3 rate along the stellar trajectory, e.g. [2]. Such simula- lations have been performed on the basis of large-scale
0 tionsdefinethe‘presupernovamodels’,e.g.[3],whichare shell model studies which reproduce all experimentally
1 then being used as input to detailed studies of the col- available relevant data quite accurately [12]. These shell
0
lapseandexplosionmechanism,e.g.[4]. Inrecentyearsit model rates show some marked differences to the FFN
1
has become apparentthatthe various neutrino reactions estimates,leading to significantchangesinthe presuper-
0
/ play an essential role in these simulations and, as neu- nova evolution of massive stars [2,13]. It appears there-
h
trinoreactionsdependsensitivelyonenergy,thesestudies fore reasonable that this compilation [11] should also be
t
- require a detailed bookkeeping of the neutrino spectra. used in collapse and explosion studies which build on
l
c Thisgoalisachievedwithinhydrodynamicalmodelswith the presupernova models. To make such use possible we
u explicit neutrino Boltzmann transport [5–7]. The neces- willherestudytheneutrinospectracorrespondingtothe
n sary input into such simulations are the weak rates and shell model rates and show a way how these spectra can
:
v the associated neutrino spectra. beeasilyandconsistentlyimplementedincollapsecodes.
i Sofar,electroncaptureonnucleihasonlybeentreated It is important to note that under the conditions of the
X
quite schematicallyincollapsesimulationswith neutrino presupernova models and in the subsequent stellar evo-
r Boltzmann transport, e.g. [5–7]. The ensemble of nuclei lution beta decay is strongly blocked by the appreciable
a
in the stellar composition, given by nuclear statistical electronchemicalpotentialandthetotalelectroncapture
equilibrium,isrepresentedbyan‘averagenucleus’,whose rates are orders of magnitude larger [10,2,13]. Thus it is
capture rate then is derived on the basis of the indepen- quite sufficient to focus on the neutrino spectra arising
dentparticlemodel(evenreducedtoamodelwhichonly fromelectroncapturesforpost-presupernovasimulations
considers f and f orbitals [5]). The correspond- where detailed neutrino transport is important.
7/2 5/2
ing neutrino spectrumis then approximatedinthe spirit Under the stellar conditions we are concerned with
of this model, assuming a nucleus-independent energy electron capture is dominated by Gamow-Teller (GT)
splitting of 3 MeV between the f and f orbitals. transitions[14]. The appropriateformalismhasbeende-
7/2 5/2
We note, that the standard for the stellar weak interac- rived in [8,9]:
tion rates in presupernova evolutions, however, was set
by the work of Fuller, Fowler and Newman (FFN) [8,9] λ= ln2 gA 2 (2Ji+1)e−Ei/(kT) |hj|| kσktk+||ii|2
who estimated the rates for the nuclei with mass num- K (cid:18)g (cid:19) G(Z,A,T) P2J +1
V Xi,j i
bers A = 21 60 on the basis of the independent par-
∞
−
ticle model and experimental data, whenever available. wp(Q +w)2F(Z,w)S (w)dw, (1)
ij e
Presupernovaevolutionstudiesthenconsideredthe FFN ×Zwl
rate tables for a proper nuclear composition. FFN list
where the sums in i and j run over states in the parent
the average neutrino energies for the various weak reac-
and daughter nuclei, respectively. For the constant K
1
we used K = 6146 6 s [15] and g ,g are the vec- the phase space integralof the electron capture formula,
V A
±
tor and axialvector coupling constants. G(Z,A,T) = Eq. (1). The neutrino distribution S (E ) depends on
ν ν
exp( E /(kT))isthepartitionfunctionoftheparent positionandtimeandcanbecalculatedwithintheBoltz-
i − i
nPucleus. The sum in the GT matrix element runs over mann transport formalism. Once Sν(Eν) is known, the
all nucleons. In the phase space integral w is the total present neutrino spectra and the capture rates of [10]
energyoftheelectroninunitsofm c2,andp=√w2 1 can be easily corrected for neutrino blocking. For the
e
−
isits momentuminunits ofm c. Finallythe Q-valuefor neutrino spectra this is achieved by folding the uncor-
e
a transition between two nuclear states i,f is defined in rected spectra n(E ), as presented in this paper, with
ν
units of m c2 as the blocking factor (1 S (E )). The corrected capture
e ν ν
−
ratesareobtainedbymultiplyingthetabulatedrates[11]
1
Qij = mec2(Mp−Md+Ei−Ej), (2) witFhorRtnh(eEfo)(ll1ow−inSgν(dEis)c)duEss.ion it is useful to realize that
theneutrinospectradependbasicallyonthreequantities
where M ,M are the nuclear masses of the parent and
p d
daughter nucleus, respectively, while Ei,Ej are the exci- the electron chemical potential
tation energies of the initial and final states. The lower •
integral limit in (1) is wl = 1 if Qij > −1 or wl = |Qij| • the ‘effective’ Q-value, Qeff =Mp−Md+Ei
if Q < 1. For the stellar conditions we are interested
ij − the GT strength distribution.
in, the electrons are well described by Fermi-Dirac (FD) •
distributions,withtemperatureT andchemicalpotential Generally one expects that neutrino energies are in-
µe: creased for larger electron chemical potentials, favor-
ablemassdifferencesbetweendaughterandparentnuclei
1
S (E )= , (3) (M M ), and from thermally excited states. Further-
e e p d
exp Ee−µe +1 more−, large neutrino energies are favored if the strong
kT
(cid:16) (cid:17) GT transition resides at low excitation energies in the
with E = wm c2. The remaining factor appearing in daughter nucleus. Due to nuclear pairing structure ar-
e e
the phase space integrals is the Fermi function, F(Z,w), guments [10]this is the casefor odd-odddaughternuclei
that corrects the phase space integral for the Coulomb (captureoneven-evenparents),whilethebulkoftheGT
distortionoftheelectronwavefunctionnearthenucleus. strength is somewhat higher ( 2 3 MeV) for odd-A
∼ −
Applyingtheformalismdescribedaboveanddetermin- nuclei and is shifted by additional 2-3 MeV in even-even
ing the nuclear matrix elements within large-scale shell daughter (capture on odd-odd parents) [12]. We note
modelcalculations,stellarelectroncapturerateshavere- that these differences in the GT distributions result in
cently been calculated for pf-shell nuclei. These nuclei the fact that low-lying rather weak GT transitions con-
dominate the weak processes in the presupernova evolu- tribute relatively more to the electron capture rates on
tion of massive stars and details of the rate evaluations odd-oddnucleiandodd-Anuclei,whileforeven-evenpar-
canbefoundin[10]. Herewewillusethesameapproach ents the GT bulk often resides at such low energies (e.g.
to study the neutrino spectra emerging in the electron Ni isotopes) that a distiction between low-lying strength
capture reactions. Energyconservationrequiresthat the and GT bulk is not very meaningful.
neutrinoemittedaftercaptureofanelectronwithenergy To illustrate the discussion we plot in Fig. 1 the
w on an initial state i leading to a final state j is: normalized neutrino spectra for electron capture on the
A=56 isobars. The calculation has been performed for
Eν =mec2(w+Qij). (4) typicalconditionsduringsiliconshellburningofa15M⊙
star [2] (T =4 109 K, ρ=3 108 g cm−3, Y =0.45).
e
× ×
The neutrino spectra for a specific nuclear transition Theresultingelectronchemicalpotentialthenisµ =2.5
e
i j is given by the respective partial rate per energy MeV.
→
interval. The total spectrum is then the sum over all We note that 56V, 56Cr, 56Mn and 56Fe have Q =
0
possible transitions. The normalized neutrino spectrum M M <0, thus making electroncapture in the labo-
p d
n(Eν) is obtained by dividing by the total electron cap- rato−ryimpossible. Theothertwonuclei56Co(Q0 =4.06
ture rate. MeV) and 56Ni (Q = 1.62 MeV) decay dominantly by
0
Inthederivationabovewehaveexplicitlyassumedthat electron capture. With the exception of 56Co the neu-
the neutrinos produced by the electron capture process trino spectra for the other 5 nuclei are very similar:
leavethestar;i.e. thereisnoblockingofthephasespace they are peaked around rather small neutrino energies
due to the presence of neutrinos in the stellar environ- E 1 2MeVwithawidthof1.4-1.8MeV.Thereason
ν
≈ −
ment. The spectra, which we will discuss below, are all for this quite similar structure is twofold. For the nuclei
derived based on this assumption. In the collapse phase withnegativeQ -values,electroncaptureishinderedand
0
following the presupernova evolution neutrinos are get- requires electrons from the exponentially decreasing tail
ting increasingly trapped in the core. This will require oftheFDdistribution. Obviouslytoachieveanapprecia-
theinclusionofaneutrinoblockingfactor(1 Sν(Eν))in ble rate it is advantegeous to keep the neutrino energies
−
2
forthegroundstate. Thisreflectsthestrongangularmo-
0.8
56V 56Cr 56Mn mentum mismatch for the ground state which does not
0.6 connectto statesin56Fe below2 MeVandhasonlyvery
weak transitions to states below 3 MeV. This is differ-
0.4
ent for the excited state which can reach the lower-lying
-1V)0.2 J = 2 states in 56Fe and in that way produces neutri-
e
M 0 nos with larger energies. Interestingly the excited J =1
n ( 56Fe 56Co 56Ni state at 1.7 MeV excitation energy in 56Co produces a
000 ... 4260 0 2 4 6 8 1 0 0 2 4 Eν6 (M8eV1)0 0 2 4 6 8 10 12 MmsernialedeateucehtVttrrerri.irinoxnWngloeailerctesgasmpnepfeeoatecntuvntetorreeurrttagomhobiaentlthsew.etetihtWffh5hi6eisshFcastieslitetvapgaetertteohaQeukeghnveeeadanxnslecuseritraetaagatotyrteefaisoQatanhnrneoeeffderuun∼nthesrdetri5rgnno.Eyco8nesνig,Mnwc=GeotiVhtnTh8e-,
parent increases the effective Q-value, and thus the av-
FIG. 1. Normalized neutrino spectra for stellar electron
erage neutrino energy, it reduces the contribution to the
capture on selected A = 56 isobars. The spectra have been
rate due to the Boltzmann weight. However,the excited
calculated for stellar conditions which are typical for silicon
1+ state yields the clue to the higher-energy neutrino
shell burningin a 15 M⊙ star [2].
peak. We notice that there are many more states in the
excitationenergyrange 2–4MeV whichareconnected
small. Thus captureto low-lyingstates occurswith elec- to the low-lying states i∼n the daughter nucleus 56Fe by
tronswith lowerenergiesthancapture tothe bulk ofthe strongGT transitions. FFN have coinedthe term‘back-
GT strength, but both are accompanied by low-energy resonances’ for these states [8,9] as they are part of the
neutrinos. For 56Ni the Q -value allows capture of elec- bulk of the GT strength built on the low-lying states in
0
trons with all energies. However, the GT distribution the inverse direction (Ref. [10] explains how these states
in the daughter 56Co is well concentrated at low excita- are considered in the rate evaluation.) Electron capture
tionenergies,resultingagaininarathernarrowneutrino onthese backresonancesoccurswith favorableQeff-value
spectrum. and hence allows the emission of neutrinos with rather
high energies. Despite the Boltzmann suppression the
gain in phase space combined with the large matrix ele-
0.4 ments ensure that these states combined contribute no-
ticeably to the total rate and produce the second peak
ground state in the total neutrino spectrum. Noting that the inte-
0.3
first exc. state gral over the spectrum reflects the relative contribution
1+ exc. state
1) backresonances to the total rate, we remark in passing that Fig. 2 also
-V total impliesthatatthe chosenconditionselectroncaptureon
e0.2
M 56Co is dominated by the one on the ground state. Al-
(
n though similar in nuclear structure, the electron capture
on the odd-odd nuclei 56Mn and 56V does not produce
0.1
a double-bump structure due to the negative Q value
0
which favors emission of low-energy neutrinos.
0 At the conditions of silicon shell burning, depicted in
0 2 4 6 8 10
Fig. 2, the electron chemical potential is yet not large
Eν (MeV)
enough (µ = 2.5 MeV) to allow significant capture
e
FIG. 2. Partial contributions of individual states in the on the odd-odd nucleus 56Co from low-lying states to
parent nucleus to the neutrino spectrum for stellar electron the bulk of the GT+ distribution in the daughter nu-
captureon56Co. Thecalculation hasbeenperformed forthe cleus which resides at around 7–9 MeV in 56Fe [16].
∼
same conditions as in Fig. 1. The spectra are multiplied by However,inthe subsequentstellarevolutionµe increases
their relative weight to thetotal capturerate. rather fast. Thus, capture to the bulk, e.g. with signif-
icantly larger GT strength, becomes easier for the low-
The56Cospectrumisquitedifferent,showingadouble- lying states. This increases their relative weight com-
bump structure. The reason is explained in Fig. 2 paredwith the one of the backresonancesandis alsonot
which shows the partial neutrino spectra contributed by compensatedbytherelativegainintheBoltzmannfactor
selected states in the parent nucleus. The spectra for of the latter. This behavior is demonstrated in Fig. 3,
both, the ground state (J = 4) and the first excited again for 56Co. The temperatures have been chosen ac-
state (J = 3) show the single-peak structure, as gen- cordingly,usingthe stellartrajectoriesasgivenbyHeger
erally observed for the other nuclei. We also find the et al. [2]. We also observe that, once capture to the
spectra for the first excited state somewhat wider than bulk of the GT distribution dominates the capture rate,
3
0.5 0.8
0.4 µTe9==02..925 µTe9==13..339 Tµe9==13..582 0.6 Tµ9e==31..339 Tµ9e==31..852 Tµe9==24..513
0.3
0.4
0.2
-1V)0.1 -1V)0.2
e e
M0.0 M0.0
n ( 0 000 ....4 2031 0 2 4 6 8µT e 9= = 241 .. 051 3 0 2 4Eν (6Me8µTVe9==)441..0241 0 2 4 6 8µTe9==871..012512 n ( 0 00 ...4 260 0 2 4 6 8µT e 9== 441 .. 024 1 0 2 4Eν (6Me8µTVe9==)871..01250 2 4 6 8Tµe9==111100..6012
FIG. 3. Normalized neutrino spectra for stellar electron FIG. 4. Normalized neutrino spectra for stellar electron
capture on 56Co at several different phases of the final evo- capture on 56Fe at several different phases of thefinal evolu-
lution of a 15 M⊙ star. The stellar parameters have been tionofa15M⊙ star. Thestellarparametershavebeentaken
taken from Table 1 of [2]. The chemical potentials are given fromTable1of[2]. Thelastpanelcorrespondstotypicalcon-
in MeV, while T9 definesthe temperaturein 109 K. ditions during the collapse phase (T = 1010 K, ρ = 3×1010
g/cm3, and Ye =0.42). The chemical potentials are in MeV,
while T9 defines thetemperature in 109 K.
the spectrum becomes single-peaked (approximately for
µ =4.2). Afurther increaseinµ thensimply increases
e e
the average neutrino energy (e.g. for µ =8.1). electron chemical potential makes the capture energeti-
e
Wesummarizethattherelativeheightofthetwopeaks cally easier, the neutrino peak energy moves to higher
reflects the competition between electron chemical po- energies. We mention that the last panel of the figure
tential (its decrease reduces the capture from low-lying already corresponds to a phase of the contraction after
states to the bulk of the GT distribution) and tem- the presupernovamodelwhichwehaveapproximatedby
perature (its decrease reduces the Boltzmann weight of T = 1010 K, ρ = 3 1010 g/cm3 and Ye = 0.42. It
×
the backresonances). If one follows the stellar evolution demonstrates that the simple structure of the spectrum
backwards in time, i.e. to smaller temperatures, den- remainsalsoduringthatstellarevolutionstagewhichre-
sities and electron chemical potential, the spectrum ul- quires detailed neutrino transport.
timatively becomes single-peaked as it is dominated by As stated above detailed neutrino transport becomes
the temperature-favored ground state contribution. The important in the final evolution of massive stars, follow-
spectrumalsodevelops‘discontinuities’(seee.g. forµ = ing the presupernova models. Due to electron captures
e
0.9)whichreflectthefactthatthespectrumrepresentsa the matter in the final presupernova models is neutron-
sum over several initial and final states which all have a rich and the nuclei present have negative Q0-values. On
definite minimal neutrino energy Emin =m c2(Q +1). theotherhand,theelectronchemicalpotentialhasgrown
ν e ij
At Emin, the electron momentum vanishes (p = 0), but stronglyenoughuntilthis pointthus allowingcaptureto
ν
pF(z,ω) is finite resulting in a finite value for n(Emin). the bulk of the GT strength. Fig. 5 shows the neu-
ν
Forlargerµ valuesthesediscontinuitiesaresmearedout trino spectra for the 6 nuclei which dominate electron
e
as the individual spectra noticeably overlap. capture in the presupernova models of a 15 M⊙ star [2].
As examplified for 56Fe in Fig. 4 the situation is quite Due to Heger et al. the core density and temperature
different if one studies the neutrino spectrum emerging are ρ = 9.1 109 g/cm3 and T = 7.2 109 K, while
× ×
fromcaptureonaneven-evennucleusasfunctionofelec- the Ye value is 0.432 [2]. The chemical potential then
tron chemical potential. To understand the reason we is µe = 8.1 MeV. We note that all neutrino spectra are
note two facts. First, as the GT distribution in the single-peaked. The average neutrino energy released by
daughteris quite concentratedat low excitationenergies nucleiisabout3MeV,whileitis6.25MeVforcaptureon
adouble-peakstructurerelatedtothedistinctcaptureto free protons which under these presupernova conditions
low-lying states and the GT bulk does not emerge. Sec- become abundant enough to significantly contribute to
ond,ineven-evennucleithebackresonancesareathigher electron capture.
excitationenergiesthaninodd-oddnuclei[10]. Thecon- As neutrino cross sections scale with E2, high-energy
ν
sequencesfortheneutrinospectrumareobvious. Ifcom- neutrinos are more easily trapped. Capture on free pro-
pared to 56Co, the relative contribution of the low-lying tonshasa morefavorableQ0 value thancaptureonneu-
states to the electron capture on 56Fe is significantly en- tronrich nuclei present in the presupernova matter com-
hanced with respect to the backresonances and no high- position. Asaconsequencetheneutrinoaverageenergyis
energy neutrino peak emerges. As an increase of the higherforcaptureonfreeprotons. Neverthelesscaptures
onnucleistillproduces the largeramountofhigh-energy
4
0.4 0.1 0.4
1H 52V 62Co
ground state total
0.3 1st exc. state without backresonances
0.08 2nd exc. state backresonances
0.2 3rd exc. state 0.3
4th exc. state
-1V)0.1 -1V) 0.06 5th exc. state
Me 0 Me 0.2
n ( 63Ni 59Fe 65Ni n (0.04
000 .. .2031 0 2 4 6 8 1 0 0 2 Eν4 (M6eV)8 10 0 2 4 6 8 10 12 0.0020 2 E4ν (Me6V) 8 10 0.010 2 E4ν (Me6V) 8 10
FIG. 6. Partial contributions of individual states in the
FIG. 5. Normalized neutrino spectra for stellar electron
parent nucleus to the Neutrino spectrum for stellar electron
capture on the six most important ‘electron-capturing nu- capture on 59Fe. The calculation has been performed for the
clei’ in the presupernova model of a 15 M⊙ star, as iden- presupernovaconditionsofFig. 5. Thespectraaremultiplied
tified in [2]. The stellar parameters are T = 7.2×109 K,
by their relative weight to the total capture rate. The left
ρ = 9.1×109 g/cm3, and Ye = 0.43. The solid lines repre-
panel shows the neutrino spectra calculated for the ground
sentthespectraderivedfromtheshellmodelelectroncapture state(solid line) and the5lowest excited states in 59Fe. The
rates. Thedashed lineshowsthefittothespectra, usingthe
right panel compares thetotal neutrinospectrum (solid line)
parametrization of Eq. (6) and adjusting the parameter q to
with the one obtained only from the backresonances (dotted
theaverageneutrinoenergyoftheshellmodelspectrum. The
line,foradefinitionseetextand[10]). Thedashedlineshows
dashed-dotted spectrum corresponds to the parametrization
the contributions from the low-lying individual states which
recommended in [5].
have been considered to calculate the electron capture rate
[10].
neutrinosinlighterstars(M<20M⊙)asthey dominate
therateinthe presupernovamodels. Thisisdifferentfor
in collapse simulations, they have to be represented by
heavier stars where the capture on free protons in the
parametrizationswhichareaccurate,fastandcanbeeas-
presupernova model corresponds already to 30–50% [2].
ily implemented. Our proposal for the parametrization
Forpresupernovaconditionstheshellmodelevaluation
is based on the following approximations. Suppose that
predictsslightlylargeraverageneutrinoenergiesthanthe
i) the electron capture on a state in the parent nucleus,
FFN rates [2]. A possible explanation for this difference
described by GT transitions, leads to a single state in
is given by the fact that the shell model rates explicitly the daughter nucleus at energy E∗ and that ii) Brink’s
consider capture from thermally excited states. The Q- hypothesis is valid, i.e. this statefis at E∗ +E if the
valuefor59FeisQ = 5.696MeV.However,theexcited f i
0 − capture is onan excitedstate in the parentat excitation
states have more favorable Q values than the ground
eff energy E . If we equal electron energy and momentum,
i
statethusmakingelectroncapturemoreeasyandinsev-
which is a valid approximation for the conditions we are
eral cases supporting larger neutrino energies. Fig. 6
interested in, the neutrino spectrum has the form [5]
shows the neutrino spectrum calculated for the 6 lowest
states in 59Fe. (Here the distributions are multiplied by N
their relative weight in the total rate). One finds that n(Eν)=Eν2(Eν −q)21+exp (E q µ )/kT (5)
ν e
indeed neutrino spectra from excited states often have a { − − }
wider tail. This results from capture to low-lying states with q = Q E∗ and a constant N which normalizes
0 − f
in the daughter. In contrast to the low-lying transitions the neutrino spectrum to unity. This form is obviously
the GT bulk approximately obeys Brink’s hypothesis, valid for capture on free protons where the parameter q
e.g. [17]. This states that the GT strength of excited is the reaction Q value, q = 1.29 MeV. For finite nu-
0
−
statesisthesameasforthegroundstate,onlyshiftedby clei, q should be considered a fit parameter. It can be
the excitation energy of the parent state. In particular, adjusted to the average neutrino energy which is listed
Brink’s hypothesis implies that the relevant energy dif- in the shell modelrate tabulations [11]for a gridoftem-
ference Qeff Ej is the same for capture to the GT bulk perature/density points and can be easily interpolated
−
for all parent states. Another observation has already in-between. We have tested this proposal and gener-
been mentioned above. Although the total capture rate allyfindthattheparametrizationapproximatestheshell
is still dominated by the transition to low-lying states, model spectra rather well, as can be seen by the dashed
the main source for high-energy neutrinos, however, are curves in Fig. 5. Of course, our parametrization fails if
the ‘backresonances’[8,9]. the spectrum is double-peaked as observed for capture
If the shell model neutrino spectra are to be used on odd-odd nuclei under special conditions (e.g. see Fig.
5
1.). However,we do notexpect that these conditions oc- [1] T.A. Weaver, S.E. Woosley and G.M. Fuller, in Numer-
cur for pf-shell nuclei during the collapse phase where ical Astrophysics, eds. J. Centrella, J. LeBlanc and R.
the electronchemical potential is high enough compared Bowers (Jones and Bartlett, 1984) p. 374
to the reaction Q value to allow appreciable electron [2] A.Heger,S.E.Woosley,G.Mart´ınez-PinedoandK.Lan-
0
capturetotheGTbulk. Finallywenotethatfor52Vthe ganke, submitted to Astrophys.J. , astro-ph/0011507
shell model spectrum is wider than the parametrization. [3] S.E.WoosleyandT.A.Weaver,Astrophys.J.Suppl.Ser.
This is caused by the fact that the spectrum for the in- 101, 181 (1995)
dividualstates showsnoticeabledifferences anddoesnot [4] A. Mezzacappa, Nuclei in the Cosmos, eds. J.
Christensen-Dalsgaard andK.Langanke,Nucl.Phys.A,
strictly follow the Brink hypothesis. For example, the
excited state of 52V at 22 keV has J =5 and thus there to be published
are no low-lying states in the daughter 52Ti which can [5] S.W. Bruenn, Astrophys.J. Suppl.Ser. 58, 771 (1985)
[6] A.MezzacappaandS.W.Bruenn,Astrophys.J.410,740
be reached by GT transitions. This is different for the
(1993)
excited J = 1 state at 141 keV which connects strongly
[7] H.-T.JankaandE.Mu¨ller, Astron.Astrophys.306,167
to the 52Ti ground state.
(1996)
Bruenn suggested a similar parametrization for the
[8] G.M.Fuller,W.A.FowlerandM.J.Newman,Astrophys.
neutrino spectra emerging from electron capture on pf- J. Suppl.Ser. 42, 447 (1980)
shell nuclei, however simply setting q =Q 3 MeV [5].
0 [9] G.M.Fuller,W.A.FowlerandM.J.Newman,Astrophys.
−
The resulting spectra are compared to the shell model J. Suppl.Ser. 48, 279 (1982)
spectra in Fig. 5. Despite the simple guess for the pa- [10] K. Langanke and G. Mart´ınez-Pinedo, Nucl. Phys. A
rameter, the agreement is quite acceptable. 673, 481 (2000)
In summary, the knowledge of the neutrino energy [11] K. Langanke and G. Mart´ınez-Pinedo, At. Data and
spectra at every point and time in the core is quite rel- Nucl. Data Tables, in print
evant for simulations of the final collapse and explosion [12] E. Caurier, K. Langanke, G. Mart´ınez-Pinedo, and F.
phase of a massive star. In the collapse phase, neutri- Nowacki, Nucl.Phys. A 653, 439 (1999)
nos are mainly produced by electron capture on nuclei [13] A. Heger, K. Langanke, G. Mart´ınez-Pinedo and S.E.
and protons and their emerging energy spectra are an Woosley, Phys.Rev.Lett. in print,astro-ph/0007412
important ingredient in the simulations. In this paper [14] H.A. Bethe, G.E. Brown, J. Applegate and J.M. Lat-
we have presented neutrino spectra for stellar electron timer, Nucl.Phys. A 324, 487 (1979)
capture during the final presupernova evolution stage of [15] I.S. Towner and J.C. Hardy, in: Symmetries and Fun-
massivestars. Thespectrahavebeenconsistentlyderived damental Interactions in Nuclei, eds. W.C. Haxton and
E.M. Henley (World Scientific, Singapore, 1995) p. 183
in the framework of the recently evaluated capture rates
[16] K.LangankeandG.Mart´ınez-Pinedo,Phys.Lett.B453,
which have been calculated on the basis of state-of-the-
187 (1999)
art large-scaleshell model studies. Furthermore we have
[17] M.B. Aufderheide, I. Fushiki, S.E. Woosley, D.H. Hart-
calculated the spectra for stellar conditions which have
mann, Astrophys.J. Suppl.Ser.91, 389 (1994)
beenobtainedinpresupernovaevolutionofmassivestars,
using the same shell-model weak interaction rates. The
calculated presupernova neutrino spectra show a rather
simple structure which is easily parametrizable. This
parametrization is easily implementable into the simu-
lation codes and allows for a derivation of the neutrino
spectra consistent with the shell-model weak-interaction
rates.
ACKNOWLEDGMENTS
Our workhas been supported by the Danish Research
Council. GMP thanks the Carlsberg Foundation for
a fellowship. JMS acknowledges a scholarship of the
Fundac¸a˜o para a Ciˆencia e Tecnologia.
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