Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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7
N
eutrinos
f
rom Supernovae
The discovery of neutrinos emitted from the Supernova 1987A
(
SN 1987A
)
exp
l
osionisanoutstan
d
ing mi
l
estone in
b
ot
h
neutrino p
h
ysics an
d
neutrino
a
stronom
y
.
O
n the one hand, this
f
ortunate observation o
f
supernova neutri-
n
os provides a stron
g
support for the modern theory of supernova explosions.
On the other hand, it implies that there exists another class of astrophysi-
c
al neutrino sources or astroph
y
sical laboratories.
A
part of this chapter is
to introduce the standard
p
icture of core-colla
p
se su
p
ernovae and
p
roduction
m
echanisms of supernova neutrinos. After a brief account of the experimenta
l
d
etection of the neutrino burst from the SN 1987
A
, we shall explore its impli-
c
ations on neutrino masses and neutrino lifetimes. The flavor conversions of
s
upernova neutrinos, including the effects of collective neutrino oscillations,
w
ill
be d
i
scussed
in
deta
il
.
7
.1
S
tellar
C
ore
C
ollapses and
S
upernova Neutrino
s
The evolution and fate of stars depend crucially on their initial masses. Th
e
reason is simply that the sel
f
-
g
ravity o
f
stars should be balanced by the pres
-
s
ure force to maintain h
y
drostatic equilibrium. For main-sequence stars, ther
-
m
al nuclear reactions serve as the energy source and offer the desired pressure
f
orce. In this section we shall consider the thermal pressure
f
rom de
g
enerat
e
electrons or neutrons. This is the case for white dwarfs and neutron stars
,
which have burnt out nuclear fuels at the final stage of stellar evolution. After
d
iscussin
g
t
h
ee
l
ectron
d
e
g
eneracy pressure, we s
h
a
ll
s
h
ow t
h
at t
h
e
d
e
g
en
-
erate stellar core becomes unstable and colla
p
ses when its mass exceeds the
C
handrasekhar limi
t
M
C
h
∼
1
.
4
M
(
Chandrasekhar, 1931a, 1931b, 1935
)
.
We shall pa
y
particular attention to the core-collapse supernovae which
fi
rs
t
experience the collapse and then rebounce, ejecting the stellar mantle an
d
enve
l
ope an
dl
eavin
g
neutron stars or
bl
ac
kh
o
l
es at t
h
e center. T
h
ero
le
played by neutrinos in the core-collapse supernovae, to
g
ether with possibl
e
m
echanisms of su
p
ernova ex
p
losions, will also be discussed.
Z.-Z. Xing et al., Neutrinos in Particle Physics, Astronomy and Cosmology
© Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2011
250 7 Neutrinos
f
rom
S
upernovae
7.1.1 Degenerate
S
tars
Fo
r
a
m
ass
i
ve sta
r
w
i
t
h
M
8
M
,
nuclear
f
usion reactions in the cor
e
will continue until the iron-group nuclei
(
e.g.,
56
Fe an
d
56
N
i
)
are abundantly
produced. At this moment the massive star takes on the onion-like structur
e
with an iron core surrounded by the shells of silicon, sulfur, oxy
g
en, neon,
c
arbon and helium from inner layers to outer layers. Since iron is the most
ti
g
htly bound nucleus, no
f
usion reactions can take place.
S
othetherma
l
p
ressure from fusion reactions is not available and the core will contract under
i
ts self-gravity. This contraction ceases if the iron core becomes degenerate
a
nd the de
g
eneracy pressure o
f
electrons is hi
g
henou
g
h to support the wei
g
ht
of the core. White dwarfs result from the stars that are not sufficientl
y
massiv
e
to ignite car
b
on an
d
oxygen in t
h
e core, an
d
t
h
e gravity is
b
a
l
ance
db
y
the pressure from degenerate electrons
(
Liebert, 1980; Chandrasekhar, 1984
;
K
oester and Chanmugan, 1990; Hansen and Liebert, 2003
)
.
Now let us consider a degenerate gas of electrons at the temperature of
a
bsolute zero
(
i.e.,
T
=
0
)
. Due to the Pauli exclusion principle, all the state
s
with momenta ranging from zero to the Fermi momentu
m
p
F
a
re occu
p
ied
.
S
ince the density o
f
quantum states in the momentum space is d
3
p
/
(
2
π
)
3
,
t
h
e tota
ln
u
m
be
r
of e
l
ect
r
o
n
s
in
t
h
ese states
r
eads
N
e
N
=
2
V
p
F
0
4
π
p
π
2
d
p
d
(2
π
)
3
=
Vp
3
F
3
π
2
,
(
7.1
)
where the
f
actor 2 takes account o
f
the spin states an
d
V
i
st
h
evo
l
u
m
eof
the s
y
stem. Hence
p
F
is determined b
y
the number densit
y
of electron
s
n
e
≡
N
e
N
/
V ; i.e.
,
p
F
=(3
π
2
n
e
)
1
/
3
.T
h
e correspon
d
ing Fermi energy
ε
F
≡
p
2
F
/
(
2
m
e
)
f
or non-relativistic electrons is
g
iven b
y
ε
F
=(3
π
2
n
e
)
2
/
3
/
(
2
m
e
)
. Multiplying
the integrand in Eq.
(
7.1
)
b
y
p
2
/
(
2m
e
)
and performing the integration ove
r
t
h
e momentum, we can o
b
tain t
h
e tota
l
energ
y
E
o
f this system. The latter
i
s related to the pressure as follows
(
Landau and Lifshitz, 1980
)
:
P
=
2
E
3
V
=
(
3
π
2
)
2
/
3
5
m
e
n
5
/
3
e
.
(
7.2
)
This e
q
uation of state is also valid for nonzero tem
p
eratures, but the conditio
n
f
or strong degeneracy T
ε
F
s
hould be satisfied. As the gas of degenerate
electrons is compressed, the density increases, so does the mean ener
g
y. In
this case one should consider the relativistic effects i
f
ε
F
becomes much larger
than
m
e
.
The Fermi ener
g
yo
f
relativistic electrons is
g
iven as
ε
F
=
p
F
=
(
3
π
2
n
e
)
1
/
3
.
Multiplying the integrand in Eq.
(
7.1
)
b
y
ε
=
p
and inte
g
ratin
g
over the momentum, we arrive at
(
Landau and Lifshitz, 1980
)
P
=
E
3
V
=
(3
π
)
2
/
3
4
n
4
/
3
e
.
(
7.3
)
The system consistin
g
o
f
only de
g
enerate electrons is actually unstable, so th
e
positively charged nuclei must be present to balance the negative charges of
7
.1
S
tellar
C
ore
C
ollapses and
S
upernova Neutrinos 25
1
electrons. However, the de
g
eneracy pressure
f
rom electrons is dominant ove
r
the thermal
p
ressure of the nuclei. The latter is henceforth assumed to hav
e
n
o impact on the equation of state of the whole system. The number densit
y
o
f
electrons is
g
iven b
y
n
e
=
ρ
/
(˜
μ
e
m
H
), where ˜
μ
e
a
n
d
m
H
s
tan
d
respective
ly
f
or the mean molecular wei
g
ht per electron and the atomic mass unit. Wit
h
the help of Eqs.
(
7.2
)
and
(
7.3
)
, we may rewrite the equation of state a
s
P
=
K
i
K
ρ
γ
(
fo
r
i
=1
,
2
)
,
(
7.4
)
i
n
wh
i
ch
K
1
=(
3
π
2
)
2
/
3
/
[
5
m
e
(˜
μ
e
m
H
)
5
/
3
]
with
γ
=5
/
3
f
or non-relativistic
electrons
,
and
K
2
=(3
π
2
)
2
/
3
/
[4(˜
μ
e
m
H
)
4
/
3
]
with γ
=
4
/
3
for extremel
y
rel
-
ativistic electrons. One always has ˜
μ
e
≈ 2 for heavy nuclei, and thus the
c
orrespon
d
in
g
K
1
a
n
d
K
2
a
re independent of the density. Note that Eq.
(
7.4
)
i
s
j
ust the equation of state for the pol
y
tropic process with the index
n
de
-
fi
ned b
y
γ
=
1
+
1
/n
.
So one may solve Eq.
(
6.12
)
for hydrostatic equilibrium
with the help of Eqs.
(
6.13
)
and
(
7.4
)
. The resultant second-order differentia
l
equation for the density is
(
Prialnik, 2000
)
(
n
+
1
)
K
4
π
G
N
n
·
1
r
2
·
d
d
r
r
2
ρ
(
n
−
1
)
/n
·
d
ρ
d
r
=
−
ρ,
(
7.5
)
whe
r
e
K
=
K
1
for
n
=3
/
2
a
n
d
K
=
K
2
for
n
=
3. T
h
e
d
ensit
yd
istri
b
ution
ρ
(
r
)
for 0
r
R should fulfill the initial condition
s
ρ
(
R
)
=0
(
wit
h
R
being the radius of the star
)
and
d
ρ
d
r
=
0 at the center. These conditions
f
ollow from the equation of state and the pressure profile wit
h
P
(
R
)
=0an
d
d
P
d
r
=0
at
r
=
0. After a change of variable
s
ρ
=
ρ
c
Θ
n
a
n
d
r
=
α
ξ
,
w
h
er
e
α
≡
{
(
n
+1
)
K
/
4
π
G
N
ρ
(
n
−
1)
/
n
c
}
1
/
2
a
n
d
ρ
c
i
st
h
e centra
ld
ensity, we o
b
tai
n
the Lane-Emden e
q
uation o
f
inde
x
n
(
Chandrasekhar, 1938; Prialnik, 2000
):
1
ξ
2
·
d
d
ξ
ξ
2
d
Θ
d
ξ
=
−
Θ
n
(
7.6
)
wit
h
t
h
e correspon
d
in
g
initia
l
con
d
ition
s
Θ
=
1
a
n
d
d
Θ
d
ξ
=0
at
ξ
=
0
.
In general, Eq.
(
7.6
)
can be numerically solved. It has been found tha
t
Θ
d
ecreases monotonically to zero at a finit
e
ξ
=
ξ
R
f
or
n
<
5
(
Shapiro and
Teukolsky, 1983
)
. Note tha
t
Θ
(
ξ
R
)
= 0 corresponds to
ρ
=
0atthesur
f
ac
e
of the star. Hence the radius and the mass of the star are given by
(
Shapiro
a
nd Teukolsky, 1983
)
R
=
αξ
R
=
ρ
(1
−
n
)
/
2
n
c
ξ
R
(
n
+1
)
K
4
π
G
N
1
/
2
,
(
7.7
)
a
nd
252 7 Neutrinos
f
rom
S
upernovae
M
=
R
0
4
π
r
2
ρ
(
r
)
dr
=4
π
ρ
(
3
−
n
)
/
2
n
c
ξ
2
R
(
n
+
1
)
K
4
π
G
N
3
/
2
d
Θ
d
ξ
ξ
=
ξ
R
,
(
7.8
)
where Eq.
(
7.6
)
has been used to evaluate the integration. Eliminatin
g
ρ
c
in
E
qs.
(
7.7
)
and
(
7.8
)
, we get an interesting mass-radius relatio
n
M∝
R
(3
−
n
)
/
(1
−
n
)
,
(
7.9
)
i
mplyin
g
that
M
∝ R
−
3
f
or
n
=3
/
2. More interestin
g
ly, the mas
s
M
i
s
i
ndependent o
f
R
an
d
ρ
c
fo
r
n
=
3
.Int
h
is case a mass
l
imit must exis
t
as
ρ
c
→∞
a
n
d
R
→
0.
S
ubstitutin
g
n
=
3,
K
=
K
2
,
ξ
R
≈
6
.
897
an
d
d
Θ
d
ξ
ξ
=
ξ
R
=0
.
0
424 into Eq.
(
7.8
)
, one gets the Chandrasekhar mass limit
M
Ch
≈
5
.
7˜
μ
−
2
e
M
≈
1
.
4
M
for ˜
μ
e
=2
(
Chandrasekhar, 1931a, 1931b
,
1
935
)
. The existence of such a mass limit can be understood in an intuitiv
e
way, which was first presented by Lev Landau
(
Landau, 1932; Shapiro and
Teukolsky, 1983
)
. For a star with the total baryon numbe
r
N
b
N
N
,
i
ts
g
rav
i
ta
-
tiona
l
potentia
l
energy is given
b
y
E
g
∝
−
G
N
N
b
N
N
R
−
1
.
On the other hand
,
the Fermi ener
g
yo
f
relativistic de
g
enerate electrons is E
F
∝
N
1
/
3
b
NN
R
−
1
.
I
f
N
b
NN
i
s very large
(
i.e., the star is very massive
)
, the total energy
E
=
E
g
+
E
F
c
an be negative and thus decrease without bound as the radiu
s
R
d
ecreases.
A
s a result, a limit o
f
N
b
NN
or
M
=
N
b
NN
m
p
e
xi
sts for
E
=0
(
i.e., the balanc
e
between the gravity and the degeneracy pressure of electrons
).
7.1.2
C
ore-colla
p
se
S
u
p
ernovae
Supernovae are exploding stars that emit a large amount of thermal energy i
n
a
relatively short time
(
e.g., from one year to several days
)
and overshine al
l
the other stars in the host galaxies. They are in general classified as type-I an
d
type-II supernovae, which can be further divided into subclasses
(
e.g., type-Ia,
b, c
)
according to the existence of hydrogen, helium or silicon spectral lines
(
Bethe, 1990
)
. Although the explosion mechanism for type-Ia supernova
e
remains an open question, it is commonly assumed that the ener
g
y
f
ro
m
n
uclear
f
usion reactions accounts
f
or the ex
p
losion. In this case neutrinos
a
re not as important as in the core-collapse type-Ib, type-Ic and type-I
I
s
upernovae, w
h
ere t
h
eexp
l
osion is powere
db
yt
h
e
g
ravitationa
l
potentia
l
ener
g
y and nearly all the ener
g
y is released in the form of neutrinos. We shal
l
c
oncentrate on type-II supernovae in the following
.
As mentioned before, a massive star
(
M
8
M
)
will develop a degener-
a
te iron core at the final sta
g
e of its evolution. Moreover, the silicon-burnin
g
sh
e
ll
wi
ll
continuous
l
y contri
b
ute mass to t
h
e core an
d
eventua
ll
y
l
ea
d
to a
n
excess over the
C
handrasekhar mass limit and thus the
g
ravitational collapse
.
Two other microsco
p
ic
p
rocesses at this moment make the situation worse.
A
s the iron core contracts, the temperature will slightly increase. Hence th
e
7
.1
S
tellar
C
ore
C
ollapses and
S
upernova Neutrinos 253
photodissociation of heavy nuclei
(
e.g.,
γ
+
56
Fe
→
13
4
He + 4
n
−
1
24 MeV
)
becomes more efficient. This endothermic reaction consumes thermal ener-
gies and thus reduces the thermal pressure. On the other hand, the de
-
g
enerate electrons will be captured by heavy nuclei or
f
ree nucleons via
e
−
+
A
(
Z
,
N
)
→
A
(
Z
−
1
,N
+1
)+
ν
e
.
Since neutrinos are weakly interacting
with matter, they will escape from the core immediately after production an
d
take away enormous thermal ener
g
ies. Furthermore, the number o
f
electrons
h
as been diminished such that the degeneracy pressure is accordingly re
-
d
uced. Both the photodissociation of nuclei and the electron-capture process
c
an further reduce the thermal pressure and aggravate the collapse
(
Bethe
,
1
990
;
Janka
,
e
tal
.
,
2007
).
Th
eco
ll
apse is o
b
vious
l
y
d
etermine
db
yt
h
e
g
ravity an
dh
y
d
ro
d
ynamics.
Interestin
g
ly, it has been discovered that the inner part o
f
the iron core col
-
l
apses in a homologous way; i.e., the distribution of temperature and density
i
s similar and only scales with respect to time
(
Goldreich and Weber, 1980
).
F
or the inner core, the velocity o
f
in
f
allin
g
matter is proportional to the ra-
d
ius but smaller than the velocity of sound, while the outer core falls inward
s
upersonically
(
Yahil and Lattimer, 1982; Yahil, 1983
)
. The critical
(
or sonic
)
point can be taken as the position where the infall velocit
y
equals the sound
velocity. Thus the inner core is in good contact with itself, whereas the outer
one is not.
A
s the core collapses, the matter densit
y
in the center reaches
a
nd exceeds the nuclear densit
y
ρ
0
∼
3
×
10
14
g
cm
−
3
.
The nuclear matter i
s
l
ess compressi
bl
ean
d
t
h
e pressure
b
ui
ld
supgra
d
ua
ll
y, so t
h
e gravitationa
l
c
ollapse o
f
the inner core will be hindered. However, the
f
reely-
f
allin
g
outer
part cannot immediatel
y
feel the pressure, so the matter still falls inward. If
the infall velocity ultimately goes to zero, the pressure wave will become
a
s
hock at the large radius just beyond the sonic point
(
Cooperstein and Baron,
1
990; Bethe, 1990
)
. It is expected that such a shock wave will traverse th
e
w
h
o
l
estaran
db
ea
bl
etoexpe
l
t
h
e mant
l
ean
d
enve
l
ope,
g
ivin
g
rise to a
s
upernova
(
Colgate and Johnson, 1960
)
. Meanwhile, a neutron star or blac
k
h
ole remains at the center. But most of the detailed numerical simulations
h
ave not con
fi
rmed this prompt-shock explosion picture. When the shock
wave propa
g
ates outward, it will disassociate heavy nuclei into nucleons a
t
the energy expense of 9 MeV per nucleon. It turns out that the shock wave
s
ta
ll
st
y
pica
lly
at a ra
d
ius a
b
out 400
k
m. W
h
et
h
er t
h
eprompts
h
oc
k
ca
n
s
ucceed or not depends sensitivel
y
on the pre-supernova evolution, such as
the equation of state and the mass of the core
(
Baron
e
ta
l.
, 1985a, 1985b
)
.
U
sually the prompt shock will
f
ail i
f
the mass o
f
the iron core is
g
reater tha
n
1
.
2
5M
(
Arnett, 1983; Hillebrandt, 1982a, 1982b, 1984; Cooperstein
e
tal.
,
1
984; Baron and Cooperstein, 1990
)
.
B
ecause o
f
the
f
ailure o
f
the prompt-shock model, the neutrino transpor
t
m
odel was proposed
(
Colgate and White, 1966
)
and the delayed-shock mode
l
was developed
(
Bowers and Wilson, 1982a, 1982b; Wilson, 1985; Bethe an
d
Wilson, 1985
)
. Neutrinos play a crucial role in this mechanism. When the
254 7 Neutrinos
f
rom
S
upernovae
m
atter densit
y
o
f
the inner core increases to about 1
0
12
g
c
m
−
3
,e
l
ectron
n
eutrinos
(
with energies around 10 MeV
)
resulting from the electron-captur
e
processes will be trapped due to coherent scattering of neutrinos on heav
y
n
uc
l
ei via t
h
eneutra
l
-current interactions. T
h
etrappe
d
e
l
ectron neutrino
s
f
orm a degenerate Fermi sea through the beta equilibrium reactio
n
e
−
+
p
↔
n
+
ν
e
.
Hence t
h
e gravitationa
l
potentia
l
energy
d
uring t
h
eco
ll
apse
h
as
b
ee
n
c
onverted into the chemical potentials o
f
electrons and neutrinos. Neutrinos
d
iffuse out of the core and heat the materials in the region below the stalle
d
s
hock, and ultimately revive the shock and lead to a successful explosio
n
(
Colgate and White, 1966; Bethe and Wilson, 1985
)
. A simulation includin
g
the neutrino transport actually demonstrates that a proper fraction of the
n
eutrino energy could successfully revive the stalled shock
(
Janka and M¨uller,
1
993
)
. However, it is still unclear how neutrinos deposit the right amount o
f
energies and whether the convection and rotation effects are inevitable for a
s
uccessful explosion
(
Janka
et al.
,
2007
).
7.1.3
S
u
p
ernova Neutrinos
A
s pointed out by George Gamow and Mario Schoenberg, neutrinos may
play a crucial role in a collapsing star
(
Gamow and Schoenberg, 1940, 1941
).
In the su
p
ernova context neutrinos can be tra
pp
ed in the core via coherent
n
eutrino-nucleon scattering
(
Freedman, 1974; Mazurek, 1975; Sato, 1975
).
The interactions o
f
neutrinos with the back
g
round nucleons and electron
s
h
ave been systematically studied
(
Tubbs and Schramm, 1975; Lamb an
d
P
et
h
ic
k
, 1976; Bet
h
e
et al
., 1979
)
. The mean free path of neutrinos for elasti
c
s
cattering is given by
(
Bethe
et al
., 1979
)
λ
ν
=1.
0
×
1
0
8
cm
ρ
10
12
g
c
m
−
3
M
e
V
E
ν
2
N
2
6
A
X
h
+
X
n
−
1
,
(
7.10
)
w
h
ere ρ
i
st
h
e matter
d
ensity
,
E
ν
d
enotes t
h
eneutrinoenergy,
X
h
an
d
X
n
represent the mass
f
ractions o
f
heav
y
nuclei and nucleons,
N
a
n
d
A
a
r
ethe
n
umbers of neutrons and nucleons in an avera
g
e nucleus. We have taken
s
i
n
2
θ
w
=1
/
4 in obtaining Eq.
(
7.10
)
. The contribution from neutral-curren
t
i
nteractions of neutrinos with protons, which is proportional to
(1
−
4s
i
n
2
θ
w
),
i
s therefore vanishing. Taking
X
h
≈
1
and
X
n
≈
0 during the infall phase, one
obta
in
s
λ
ν
≈
2
km f
o
r E
ν
=
10
M
e
V
,
ρ
=10
12
gc
m
−
3
,
N
≈
50
an
d
N
/
A
=
0
.
6(
Bethe, 1990
)
. The radius of the core corresponding t
o
ρ
=10
12
g
c
m
−
3
i
sabou
t
R
=
30 km, and hence neutrinos are essentiall
y
trapped in the core
a
nd can only get out of it by diffusion
(
Cooperstein, 1988a, 1988b
)
.
A
fter bein
g
trapped, the electron neutrinos come into chemical equilib
-
rium with the degenerate electrons and nucleons through the beta process
e
−
+
p
↔
ν
e
+
n
.There
f
ore, a de
g
enerate Fermi sea o
f
neutrinos builds up
,
a
nd the correspondin
g
chemical potential is determined b
y
μ
ν
e
=
μ
e
−
ˆ
μ
w
i
th
ˆ
μ
≡
μ
n
−
μ
p
≈
100 MeV. The chemical potential of degenerate electron
s
7
.1
S
tellar
C
ore
C
ollapses and
S
upernova Neutrinos 255
Fi
g
. 7.1 Luminosities o
f
neutrino
s
ν
α
an
d
antineutrinos
ν
α
(
for
α
=
e
,
μ
,
τ
)
versus
the time after the core bounce. Four stages of the delayed-explosion scenario are
shown
(
Raffelt, 1996. With permission from the University of Chicago Press
)
:
(
1
)
the collapse and bounce;
(
2
)
the shock propagation and prompt neutrino burst;
(
3) the matter accretion and mantle cooling while the shock stalls; (4) the Kelvin-
Helmholtz cooling o
f
the neutron star a
f
ter explosio
n
is
μ
e
≈
111 MeV
(
ρ
13
Y
e
YY
)
1
/
3
,
where
ρ
13
denotes the matter densit
y
in units
of 1
0
13
g
cm
−
3
a
n
d
Y
e
YY
is the number fraction of electrons. For the nuclea
r
d
ensit
y
ρ
0
=
3
×
10
14
g
c
m
−
3
and a t
y
pical electron
f
raction
Y
e
YY
=0
.
4
,w
e
h
av
e
μ
e
≈
2
50 MeV an
d
μ
ν
e
≈
1
50 MeV
(
Bethe, 1990
)
. Hence the gravi
-
tationa
l
potentia
l
energy re
l
ease
dd
uring t
h
eco
ll
apse
h
as
b
een
l
arge
l
ycon
-
verted into the chemical potentials o
f
de
g
enerate electrons and neutrinos
.
Shortly after the bounce, the shock wave disassociates the infalling heavy
n
uclei into free nucleons, on which the electron-capture rate is much larger
.
O
n the other hand, the matter density in the re
g
ion just below the shoc
k
wave has been dramatically reduced, allowing the electron neutrinos to freel
y
escape. T
h
is is t
h
eso-ca
ll
e
d
prompt neutronization or
d
e
l
eptonization
b
urs
t
o
f
neutrinos. The electron ca
p
ture and the neutrino burst lead to a moderate
reduction of the le
p
ton number in the iron core. As a conse
q
uence, the neutri-
n
os o
f
all
fl
avors can now be produced via the electron-positron annihilation
e
+
+
e
−
→
ν
α
+
ν
α
,
the plasmon deca
ys
γ
∗
→
ν
α
+
ν
α
and the nucleonic
b
remsstra
hl
ung
N
+
N
→
N
+
N
+
ν
α
+
ν
α
(
Suzuki, 1991, 1993
)
. For illustra-
tion, the neutrino luminosit
yf
rom the standard dela
y
ed-explosion scenario i
s
s
hown in Fi
g
. 7.1, where one can see the hi
g
hly-peaked neutrino burst and
the approximate equilibration of neutrino luminosities at later times. Note
t
h
at t
h
eneutrino
l
uminosities are
d
epen
d
ent on t
h
etime.
S
imilar to electron neutrinos, muon and tau neutrinos are also tra
pp
ed
a
nd keep in local thermal equilibrium with electrons and nucleons. They dif-
f
use outward to their respective neutrino spheres,
f
rom which the
y
ma
yf
reel
y
escape. Hence the energy spectra of neutrinos can be well described by th
e