Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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22
66
Neutrinos
f
rom
S
tar
s
log
(
L/L
)
log
(
M
/
M
)
Fig. 6.2 The observational data on the mass-luminosity relation for the main-
sequence stars
(
Carroll and Ostlie, 2007. With permission from Pearson Education,
Inc.
)
these stars is most likely to be the
f
usion o
ff
our hydro
g
en nuclei into on
e
4
He nucleus. To be s
p
ecific, we write out the reactio
n
4
p
4
+
2
e
−
→
4
H
e+2
ν
e
+
Q
,
(
6.36
)
where
Q
≈
4
m
p
+
2
m
e
−
m
4
He
≈
2
6
.
7
3MeVis
j
ust
0
.
7%
of the sum of initia
l
proton masses. If the Sun were composed entirely of hydrogen and all th
e
h
ydro
g
en were converted into helium via the above reaction, the time
f
or this
p
rocess to be com
p
leted would be
τ
∼
0
.
7% ×
m
p
×
(
M
/m
p
)
/
L
≈
10
1
1
y
r
,
where the present solar luminosity is assumed. If only 10
%
of the hydrogen i
n
the
S
un is consumed, the solar li
f
etime will be o
f
the ri
g
ht order o
f
ma
g
nitude
τ
∼
10
10
y
r, consistent with the nuclear timescal
e
τ
nuc
ττ
given in Eq.
(
6.9
).
The hydrogen burning proceeds in two different ways: the proton-proton
(
pp
)
c
hain and the carbon-nitrogen-oxygen
(
CNO
)
circle, which are also the mai
n
s
ources of stellar neutrinos. The advanced nuclear fusions
,
such as the carbo
n
a
n
d
si
l
icon
b
urnin
g
,requiremuc
hh
i
gh
er temperatures
∼
10
9
K
.
He
n
ce we
s
hall only consider the hydro
g
en-burnin
g
sta
g
e, which is relevant to the
S
un
a
nd thus the production of solar neutrinos
(
Bahcall, 1989
).
6.1.4 The Mass-Lum
i
nos
i
t
y
Relat
i
on
The color-ma
g
nitude dia
g
ram has been shown in Fi
g
.6.1,
f
rom which one ca
n
s
ee a clear correlation between the effective temperature and luminosity. Fo
r
6
.2 Neutrinos
f
rom the
S
un 22
7
t
h
e main-sequence stars, t
h
ere exists an empirica
l
mass-
l
uminosit
y
re
l
ation
s
hown in Fig. 6.2. More explicitly
(
Cox and Giuli, 1968; Prialnik, 2000
),
L
L
∝
M
M
ν
,
(
6.37
)
w
h
e
r
e
ν
=
3
∼
5 denotes the
p
ower index
f
or the main-se
q
uence stars.
G
iven
a
chemical com
p
osition similar to the Sun, it has been found that ν
≈
3
.
6
h
olds
f
or the stars with their masses lyin
g
in the ran
g
e20
M
>
M
>
2
M
,
ν
≈
4
.
5
for 2
M
>
M
>
0
.
5M
a
nd
ν
≈
2
.
6
for 0
.
5
M
>
M
>
0
.
2
M
(
Salaris and Cassisi, 2005
)
. Let us use the basic equations to understand suc
h
a
nele
g
ant relation. With the help o
f
the condition o
f
hydrostatic equilibriu
m
i
nEq.
(
6.11
)
, we approximately obtain
P
∼M
2
/
R
4
by equatin
g
the left
-
h
and side of Eq.
(
6.11
)
with
P/R
a
n
d
ta
k
ing t
h
e average
d
ensit
y
ρ
∼
M
/R
3
.
A
ssuming the equation of state of the ideal gas
(
i.e.
,
P
∝
ρT
)
,wegetth
e
t
em
p
era
t
ur
e
T
∝
P/ρ
∼M
/R
. These relations, together with Eq.
(
6.24
),
a
llow us to derive an empirical mass-luminosity relation fo
r
ν
=
3:
L
∝
R
4
M
T
4
∝
R
4
M
M
R
4
=
M
3
.
(
6.38
)
F
or massive stars wit
h
M
>
10
M
,
the mass-luminosit
y
relation becomes
l
ess steep. The reason for this difference is simply that the radiation pressur
e
i
s dominant in the massive stars, so the pressure is
g
iven b
y
P
∝
T
4
a
n
d
T
∝M
1
/
2
/
R arises from
P
∼M
2
/
R
4
.
After insertin
g
T
∝M
1
/
2
/
R
i
nt
o
E
q.
(
6.38
)
, we can arrive at a linear mass-luminosity relation
L
∝M
.
T
hese
s
imple ar
g
uments help us understand the observed correlation between stella
r
m
asses and luminosities, at least to some extent. It is worth mentioning tha
t
the mass-luminosity relation is not applicable to the other branches o
f
stars
s
hown in Fi
g
. 6.1. Those kinds of stars may have the cores with de
g
enerat
e
e
l
ectrons, an
dh
ence t
h
ea
b
ove simp
l
eargumentsareinva
l
i
d.
6
.2 Neutrinos from the
S
un
A
lthough how stars formed in the first place remains an open question, it
i
s believed that they ori
g
inated
f
rom a
g
ravitationally bound system o
f
the
primordial gas. The theory of Big Bang nucleosynthesis tells us that the mass
f
raction of helium in the primordial
g
as is about 25
%
, and the rest is almost
h
ydro
g
en. The bound
g
as contracts due to the ener
g
y loss via electroma
g-
n
etic radiation, but the temperature of this system increases because of th
e
n
e
g
ative speci
fi
c heat. When the temperature is hi
g
henou
g
htoi
g
nite hydro-
g
en, the ener
g
y released from hydro
g
en fusion makes the
g
as expandin
g
.
A
sa
result, the gravity is balanced by the pressure force and the system achieves
hy
drostatic equilibrium. This is the case
f
or the main-sequence stars, like th
e
22
86
Neutrinos
f
rom
S
tar
s
S
un. When all the hydro
g
enatthestellarcoreisconsumed,
f
urther evolution
of the star depends on its initial mass. For a mass below
(2
∼
3
)
M
,
th
e
h
elium as a product of hydrogen fusion accumulates at the center and ther
e
i
sa
h
y
d
ro
g
en-
b
urnin
g
s
h
e
ll
aroun
d
t
h
e
d
ense
h
e
l
ium core. Meanw
h
i
l
e, matter
a
roundthecorestartstoex
p
and and hence the tem
p
erature of the surfac
e
d
ecreases. In ot
h
er wor
d
s, t
h
estar
b
ecomes re
dd
er an
d
is ca
ll
e
d
are
d
giant
.
Thecoreo
f
ared
g
iant is so hot and dense that helium
f
usion becomes pos
-
s
ible. The stage with a helium-burning core and a hydrogen-burning shell i
s
known as a horizontal-branch star. For a mass up to
(
6
∼
8
)
M
,
car
b
on an
d
oxy
g
en can be produced in the stellar core, but
f
urther
f
usion processes ar
e
n
ot allowed. Since all the fuels are burnt out, these stars with a degenerate
c
ore wi
ll
coo
ld
own
b
y emittin
g
neutrinos an
d
p
h
otons unti
l
t
h
ey
d
isappear
f
rom view.
S
uch a star is re
f
erred to as a white dwar
f
. For more massiv
e
s
tars wit
h
M
>
(
6
∼
8)
M
,
nuclear fusion at the center can continue t
o
the
f
ormation o
f
the most stable element — iron. When the de
g
enerate iro
n
c
ore reaches the
C
handrasekhar limit and there are no more ener
g
ies release
d
f
rom nuclear reactions
,
the core becomes unstable. Further contraction allows
p
h
otons to
d
isassociate t
h
enuc
l
ei, an
d
t
h
is process wi
ll
consume ener
g
ies an
d
f
urther reduce the pressure. In addition, electrons are captured b
y
the nuclei
a
n
d
converte
d
into e
l
ectron neutrinos. T
h
e
l
atter wi
ll
escape an
d
carry away
m
ore ener
g
ies. So the unstable core must continue to collapse.
A
collapse
d
s
tar and its subsequent evolution ma
y
lead to a t
y
pe-II supernova. Neutrinos
emitted from the type-II supernova explosion will be discussed in Chapter 7
.
W
e shall subsequentl
yf
ocus our interest on the
S
un, a t
y
pical main-
s
equence star, by introducing the standard solar model
(
SSM
)
.Weshallalso
pay particular attention to the production of solar neutrinos.
6.2.1 The
S
tandard
S
olar Mode
l
The answer to why the Sun shines was first
g
iven by Hans Bethe, who worke
d
out the solar nuclear fusion chains in 1939
(
Bethe, 1939
)
. Neutrinos produced
f
rom these nuclear reactions were completely i
g
nored, because the existence
of neutrinos had not been experimentally established by that time
(
Beth
e
a
nd Peierls, 1934
)
. In 1940, George Gamow and Mario Schoenberg pointed
out that neutrinos could take away a lar
g
eamounto
f
ener
g
ies and should be
very important in a collapsing star
(
Gamow and Schoenberg, 1940, 1941
).
N
eutrinos were first discovered by Clyde Cowan and Frederick Reines i
n
1
956
(
Cowan
et al.
,
1956
)
. A theoretical calculation of solar neutrino fluxe
s
was carried out by John Bahcall in 1964, and a proposal for detecting solar
n
eutrinos was put forward by Raymond Davis in the same year
(
Bahcall,
1
964; Davis, 1964
)
. The SSM was originally built and continuously refined b
y
Bahcall and his collaborators
(
Bahcal
l
e
tal
.
,
1982
,
2001
,
2006
;
Bahcall an
d
U
lrich, 1988; Bahcall and Pinsonneault, 1995
)
.
Letussummarizethemaina
pp
roximations made in the
SS
M, or mor
e
generally in a theory of the main-sequence stars
(
Bahcall, 1989
)
.
(
1
)
Th
e
6
.2 Neutrinos
f
rom the
S
un 22
9
S
un is sphericall
y
s
y
mmetric and its intrinsic properties onl
y
depend on it
s
radius.
A
s a consequence, the rotation and ma
g
netic fields can be ne
g
lected
.
(
2
)
Hydrostatic equilibrium is always assumed in the SSM. Any significant
d
eviation
f
rom h
y
drostatic equilibrium would have alread
y
led the
S
un t
o
a
global contraction or expansion in about one hour
(
i.e., the dynamica
l
t
im
escale
τ
dyn
τ
∼
10
3
s)
.
(
3
)
The energy transport is induced by photons o
r
c
onvection. T
h
era
d
iative ener
g
y transport is
d
ominant in t
h
eso
l
ar core, w
h
i
l
e
the convective energy transport becomes important near the solar surface
.
(
4
)
The energy production is attributed to thermal nuclear reactions, and
t
h
eener
g
y
g
ain or
l
oss in
d
uce
db
yt
h
e
l
oca
l
contraction or expansion s
h
ou
ld
be taken into account.
(
5
)
The chemical composition is changed only throug
h
n
uclear reactions, and thermal or
g
ravitational di
ff
usion o
f
nuclear element
s
s
hould be included. Based on these approximations, the basic equations
g
iven
i
nSection6.1canthenbea
pp
lied to the Sun
.
W
e proceed to outline the input parameters in the
SS
M. They include
the primordial chemical abundances, radiative opacit
y
, equation o
f
state and
n
uc
l
ear reaction rates. T
h
eprimor
d
ia
l
c
h
emica
l
composition is usua
ll
ygive
n
b
y
the helium mass
f
raction
Y
a
n
d
t
h
emeta
l
-to-
h
y
d
ro
g
en rati
o
Z/
X
.
T
he
l
atter can be extracted from a measurement of the chemical abundances of
the solar surface. Note that the metal here stands for the elements heavie
r
than helium.
S
ince the sur
f
ace temperature is relativel
y
low, nuclear reactions
c
annot be initiated and thus the abundance at the surface can re
p
resent the
primordial one. We are able to measure the chemical abundance of the sola
r
s
ur
f
ace in two di
ff
erent wa
y
s. The
fi
rst is the mass spectroscop
y
o
f
meteorite
s
a
t terrestrial laboratories. However, the volatile elements including hydrogen,
h
e
l
ium, car
b
on, nitrogen, oxygen an
d
neon mig
h
t
h
ave
b
een
d
ep
l
ete
d
in t
h
e
pristine meteorites. In t
h
is case si
l
icon is usua
lly
c
h
osen to measure t
h
ere
l
a-
tive abundances of other elements. Hence some prior knowledge of silicon a
t
the solar sur
f
ace is needed to determine the chemical composition. The othe
r
wa
y
is to anal
y
ze the solar spectrum and atomic lines. It depends on th
e
m
odel of the solar atmos
p
here and s
p
ectrum formation. These two methods
a
re consistent wit
h
an
d
comp
l
ementary to eac
h
ot
h
er, an
d
t
h
e
l
atest resu
l
ts
with widely-used compilations are listed in Table 6.4
(
Asplund
et al.
, 2009
)
.
It is worth remarking that the initial chemical composition can significantly
aff
ect the radiative opacity o
f
the solar core, which has a
g
reat impact on th
e
c
entral tem
p
erature and thus solar neutrino fluxes
.
B
uilding the SSM may begin with the so-called zero-age-main-sequenc
e
sta
r
of
1
M
(
Prialnik, 2000
)
.Sincetheag
e
τ
ττ
,l
uminosit
y
L
a
n
d
r
ad
i
us
R
of the Sun have been precisel
y
measured, the
y
must be reproduced b
y
the SSM. These important parameters of the Sun are listed in Table 6.5
.
The densit
y
and sound velocit
y
pro
fi
les o
f
the
S
un,
f
rom its sur
f
ace to the
d
eep core, can be determined from the stud
y
of the solar acoustic oscillations
(
i.e., the so-called helioseismology
)
. They also provide strict constraints o
n
the SSM
(
Basu and Antia, 2008
)
.
2
30 6
Neutrinos
f
rom
S
tar
s
Table 6.4 The latest and widely-used mass fractions of hydrogen
(
X
)
, helium
(
Y
)
and metals
(
Z
)
in the present-day solar photosphere
(
Asplund
et al
.
,
2009. Wit
h
permission from Annual Reviews
)
R
e
f
erence
s
X
YZZ
/X
A
nders and Grevesse, 1989
0
.
731
40
.
2
4
85
0
.
0
201 0
.
0
27
4
G
revesse and Noels, 1993
0
.
7336
0
.
2
4
85
0
.
0
179 0
.
0
2
44
G
revesse and
S
auval, 1998
0
.
73
4
5
0
.
2
4
85
0
.
0
169 0
.
0
23
1
L
odders, 2003
0
.
7
4
91
0
.
2377
0
.
0
133 0
.
0
17
7
A
s
p
lun
d
et al.
,
2005
0
.
7392
0
.
2
4
85
0
.
0
122 0
.
0
16
5
Lodde
r
s
et al.
,
2009
0
.
7390
0
.
2
4
69
0
.
0
1
4
10
.
0
19
1
A
s
p
lun
d
e
tal
.
,
2009
0
.
7381
0
.
2485
0
.
0
134 0
.
0
18
1
Table 6.5 Some important parameters of the Sun
(
Bahcall, 1989. With permission
from the Cambridge University Press
)
P
arameter
V
alue
L
uminosity
(
L
)
3
.
86
×
10
33
er
g
s
−
1
Mass
(
M
)1
.
99
×
10
33
g
R
adius
(
R
)
6
.
96
×
10
10
c
m
A
ge
(
τ
ττ
)4
.
57
×
10
9
yr
Central density
(
ρ
C
)
148 g cm
−
3
Central hydrogen abundance
(
X
C
)0
.
34
6.2.2 Proton-proton
C
hain and
C
N
OC
ycl
e
A
s shown in Eq.
(
6.36
)
, the main energy source in the Sun is the nuclear fusion
reaction which converts four
p
rotons into a helium nucleus and releases th
e
energy
Q
≈
26
.
73
Me
V
. Two neutrinos are emitted from this reaction an
d
c
arry away only 2
%
of the total ener
g
y. The burnin
g
of hydro
g
en proceed
s
via th
e
pp
chain and CNO c
y
cle
.
(
1
)
Th
eppc
h
ai
n
.
The first process is
p
+
p
→
D
+
e
+
+
ν
e
.
T
h
e positron
m
a
y
imme
d
iate
ly
anni
h
i
l
ate wit
h
an am
b
ient e
l
ectron an
d
t
h
us it contri
b
ute
s
to the energy production. The electron neutrinos produced from this reaction
a
r
e
r
e
f
e
rr
ed to as t
h
e
pp
neutrinos. T
h
ey
h
ave a continuous energy spectrum
a
nd a maximal ener
gy
E
max
ν
=0
.
4
20 MeV. The rate o
f
pp
react
i
o
n
sca
n
be
c
alculated by using the standard theory of weak interactions together wit
h
the experimental data on the proton-proton scatterin
g
and properties o
f
the
d
euteron
(
Bahcall
et al
., 1982
)
. The second process is the electron-capture
reaction
p
+
e
−
+
p
→
D
+
ν
e
,
where the energy of the electron neutrino is
6
.2 Neutrinos
f
rom the
S
un 2
3
1
E
ν
=1
.
4
42 MeV. The ener
g
y spectrum o
f
such
pep
neutrinos is in princip
le
d
iscrete, but the broadenin
g
effects due to the thermal motion actually mak
e
t
h
e spectrum continuous. Note t
h
at t
h
enuc
l
ear matrix e
l
ements re
l
evant t
o
the
pp
a
n
d
pep
r
eactions are the same, so the ratio o
f
these two reactio
n
rates is inde
p
endent of the SSM. The
p
e
p
reaction rate can be accuratel
y
c
alculated by means of the electroweak theory and with the help of the
pp
reaction rate. T
h
enextstepint
he
pp
c
hain is the production o
f
3
H
e
in
the
p
roton-ca
p
ture reaction D
+
p
→
3
He
+
γ
.
This is the onl
y
reactio
n
f
or the deuteron burning in the Sun, and there are no neutrinos produce
d
f
rom this process. The termination o
f
th
e
pp
c
h
ain is main
ly
via t
h
e proces
s
3
H
e+
3
He
→
4
H
e+
2
p
(
i.e., the
pp
-I chain
)
. There is also another termination
wit
h
t
h
epro
b
a
b
i
l
ity
2
×
10
−
7
v
i
athe
r
eact
i
o
n
p
+
3
He
→
4
H
e+
e
+
+
ν
e
,
where neutrinos with ener
g
ies up to
E
m
a
x
ν
=
18
.
7
7 MeV are
g
enerated
.
A
lthough thes
e
h
e
p
n
eutrinos have sufficiently high energies, their flux i
s
u
n
f
ortunately too small to be detected. The remainin
g
two possibilities
f
or
t
h
ete
rmin
at
i
o
n
of t
h
e
pp
c
hain are related to the production o
f
ber
y
lliu
m
7
B
et
h
roug
h
3
H
e
+
4
He
→
7
B
e+
γ
a
nd its subsequent interactions:
(
a
)
the
pp
-II c
h
ain c
h
aracterize
dby
7
Be
+
e
−
→
7
L
i
+
ν
e
a
n
d
7
Li +
p
→
2
4
He
with a branchin
g
fraction 9
9
.
8
7%, producin
g
7
B
e neutrinos with eithe
r
E
ν
=
0
.
8
62 MeV
(
with the probability 8
9
.
7%)
o
r
E
ν
=0
.
38
4Me
Vd
ue to t
he
excited state of lithium
(
with the probability 10
.
3%)
;
(
b
)
th
e
pp
-
III
cha
in
c
haracterized b
y
7
Be
+
p
→
8
B+
γ
,
8
B
→
8
B
e
∗
+
e
+
+
ν
e
a
nd
8
Be
∗
→
2
4
He
with a branching fraction
0
.
13
%
, producin
g
8
Bn
eut
rin
os w
i
th a
m
a
xim
al
ener
g
y
E
m
a
x
ν
=15MeV.
A
lthou
g
h the reaction rate o
f
7
B
e
+
p
→
8
B
+
γ
i
s highly suppressed, it is practically easier to detect
8
B
neutrinos because
their energies are much higher than those of the more abundant
pp
,
pep
a
n
d
7
B
e neutrinos. In
f
act, all the solar neutrino experiments are sensitive t
o
8
B
n
eutrinos.
(
2
)
T
he
C
N
O
cycle
.S
ince carbon is most abundant amon
g
the heavy
elements in stars, the
C
N
O
c
y
cle o
ff
ers another important wa
y
to realize
h
ydrogen fusion. More explicitly, we have
1
2
C
+
p
→
1
3
N
+
γ
,
1
3
N
→
1
3
C
+
e
+
+
ν
e
a
n
d
13
C+
p
→
14
N+
γ
, where electron neutrinos arise
f
rom th
e
β
+
d
eca
y
so
f
1
3
Na
n
dt
h
us a
r
e
kn
ow
n
as
13
N
neutrinos. The
f
urther
p
roton-
c
ap
t
ure process
1
4
N+
p
→
1
5
O+
γ
r
are
l
y occurs,
b
ut it gives
b
irt
h
t
o
1
5
O
n
eutrinos and terminates one o
f
the
C
N
O
c
y
cles via
15
O
→
15
N+
e
+
+
ν
e
a
n
d
15
N+
p
→
1
2
C
+
4
He. There exists a slim chance
(
∼
0
.
1%
)
for the production
of
16
O
in the proton-capture reaction o
f
15
N
(
i.e.,
15
N+
p
→
16
O+
γ
)
,whic
h
i
nitiates another
C
N
O
c
y
cle and produces
1
7
Fn
eut
rin
os v
i
a
16
O
+
p
→
1
7
F+
γ
a
nd
1
7
F
→
17
O
+
e
+
+
ν
e
.
This c
y
cle is terminated b
y
the production of
hel
i
u
m
v
i
a
1
7
O
+
p
→
14
N+
4
H
e. All the
C
N
O
neutrinos result from the
β
+
d
ecays, so t
h
e correspon
d
in
g
neutrino ener
g
y spectra are continuous. T
he
experimental detection of CNO neutrinos seems ver
y
difficult because thei
r
energies and fluxes are not sufficiently high
.
2
3
2
6
Neutrinos
f
rom
S
tar
s
6
.2.
3S
olar Neutrino Fluxe
s
N
uclear ener
g
ies and neutrinos are released
f
rom the
pp
c
hain and
C
N
O
c
ycle. The energies are mostly carried away by photons, which diffuse fro
m
t
h
eso
l
ar core unti
l
t
h
ey reac
h
t
h
ep
h
otosp
h
ere w
h
ere t
h
eme
d
ium
b
ecomes
trans
p
arent. The di
ff
usion time o
fp
hotons is about 1
0
4
y
r. In comparison
,
n
eutrinos almost freely escape from the Sun after they are produced. Henc
e
t
h
e detect
i
o
n
of so
l
a
rn
eut
rin
os ca
n
offe
r
d
ir
ect ev
i
de
n
ce fo
r
t
h
e
rm
o
n
uc
l
ea
r
reactions in the center of the Sun. We summarize various sources of sola
r
n
eutrinos toget
h
er wit
h
t
h
eir average an
d
maxima
l
energies in Ta
bl
e6.6
.
Note that
7
B
ea
n
d
pep
n
eutrinos
h
ave
d
iscrete
l
ine spectra
.
Table 6.6 Various sources o
f
solar neutrinos to
g
ether with their avera
g
e ener
g
ies
E
ν
a
nd maximal energies
E
ma
x
ν
a
s compared with the average thermal energies
E
(
in units of MeV
)(
Giunti and Kim, 2007. With permission from the Oxfor
d
University Press
)
S
ource Reactio
n
E
ν
E
m
a
x
ν
E
pp p
+
p
→
D
+
e
+
+
ν
e
0
.
2668 0
.
423 13.
0
98
7
p
e
pp
+
e
−
+
p
→
D+
ν
e
1
.
4
55
1
.
455 11.919
3
h
e
p
3
H
e
+
p
→
4
H+
e
+
+
ν
e
9
.
628 1
8
.
778 3
.
7
370
7
Be
7
B
e+
e
−
→
7
L
i
+
ν
e
0
.
863
1
0
.
863
112.
6008
7
Be
7
B
e
+
e
−
→
7
L
i
∗
+
ν
e
0
.
3855 0
.
3855
12.
6008
8
B
8
B
→
8
B
e
∗
+
e
+
+
ν
e
6
.
735
∼
1
56
.
6305
1
3
N
1
3
N
→
1
3
C
+
e
+
+
ν
e
0
.
7063 1
.1
98
2
3
.
4577
15
O
15
O
→
1
5
N
+
e
+
+
ν
e
0
.
996
41.
73
1
7
21.
5706
1
7
F
1
7
F
→
17
O
+
e
+
+
ν
e
0
.
9977 1
.
736
42.
363
S
ince nuclear reactions in the
pp
c
hain and
C
N
O
c
y
cle are alread
y
known
,
one ma
y
directl
y
calculate solar neutrino fluxes for all the sources liste
d
i
n Table 6.6. The most relevant quantity for the reaction rat
e
R
i
j
i
sthe
a
stroph
y
sical
f
acto
r
S
ij
defined in Eq.
(
6.33
)
. A neutrino flux is actually
c
omputed by integrating the local production rate over the volume of the
Sun
,
Φ
i
=
M
0
R
i
ρ
(
M
)
,
T
(
M
)
,
{
X
j
X
}
d
M
,
(
6.39
)
where the subscri
p
t
i
denotes the neutrino source type, and
R
i
d
escribes the
c
orrespon
d
in
g
pro
d
uction rate per unit mass. T
h
e
l
atter apparent
l
y
d
epen
ds
on the matter densit
y
, local temperature and chemical composition. No
w
that both the thermal energy and neutrinos are generated from the hydrogen
6.3 Experimental Detection o
fS
olar Neutrinos 23
3
fusion reaction 4
p
4
+2
e
−
→
4
H
e+2
ν
e
+
Q
w
i
th
Q
=26
.
73 MeV, we can
rou
g
hly estimate the total neutrino flux:
Φ
ν
≡
i
Φ
i
≈
2
×
L
4
π
(
1AU
)
2
·
1
Q
≈
6
.
4
×
10
10
c
m
−
2
s
−
1
,
(
6.40
)
where the
f
actor 2 takes account o
f
the
f
act that two neutrinos are produce
d
f
rom hydro
g
en fusion, and
L
i
s the solar luminosit
y
.
A
recent revision of th
e
a
bundances of heavy elements at the solar surface yields
(
Z/
X
)
=
0
.
0165
(
Asplund
et al
.
,
2005
)
as compared with the previous result
(
Z/
X
)
=
0
.
0229
(
Grevesse and Sauval, 1998
)
. This new determination of solar metallicity is
i
n contradiction with the helioseismological measurements
(
Chapli
n
et al
.
,
2
007; Bas
u
et al
.
,
2007
)
. In contrast, the earlier and larger value of metallicity
a
grees very well with the solar density and pressure profiles extracted fro
m
the helioseismological observations
(
Serenelli
et al
.
,
2009
)
. Two SSMs hav
e
been constructed by using the previous
(
larger
)
and recent
(
smaller
)
results
of heavy-element abundances, so they are referred to as BPS
(
GS98
)
an
d
BPS
(
AGS05
)
, respectively
(
Pena-Garay and Serenelli, 2008
).
Fi
g
. 6.3 illustrates the distribution o
f
solar neutrino
fl
uxes as a
f
unction
of the solar radius fractio
n
r/
R
.
Note that solar neutrinos are actually
pro
d
uce
d
in t
h
ecorere
g
ion wit
h
r<
0
.
35
R
.Inparticu
l
ar, a
l
most a
ll
t
h
e
8
B
neutrinos are produced from the core
(
r
0
.
1
R
)
, so their total
flux depends sensitively on the central temperature
T
.
It has been found
that
Φ
8
B
∝
T
18
,
Φ
7
Be
∝
T
8
a
n
d
Φ
pp
∝
T
−
1
.
2
h
o
ld
approximate
ly
.T
he
pp
n
eutrinos are
g
enerated in the re
g
ion from which most of the solar luminosity
a
rises. Hence the above temperature dependence of
pp
n
eutrinos is main
ly
a
scribed to the constraint coming from the solar luminosity
(
Bahcall, 1989
).
Theoretical predictions for the ener
g
y spectra of solar neutrinos from th
e
SSM of BPS
(
GS98
)
2008 are shown in Fig. 6.4. In addition, solar neutrin
o
fluxes predicted by the SSMs of BPS
(
GS98
)
2008 and BPS
(
AGS05
)
2008 are
l
isted in Table 6.7.
A
difference between the
p
redictions of these two models
f
or every neutrino flux is given in the last column of Table 6.7
.
6
.3 Ex
p
erimental Detection o
f
Solar Neutrino
s
Solar neutrinos can be detected by means of radiochemical reactions, wate
r
Cherenkov detectors and
(
or
)
liquid scintillators. Some typical and important
experiments on solar neutrinos will be briefly introduced in this section, from
w
h
ic
h
one can see w
h
yt
h
ere was t
h
ewe
ll
-
k
nown so
l
ar neutrino pro
bl
em
.
6.3.1 Radiochemical Method
s
It was Bruno Pontecorvo who
fi
rst su
gg
ested detectin
g
electron neutrinos via
the inverse beta decay
(
Pontecorvo, 1946; Alvarez, 1948
)
2
3
4
6
Neutrinos
f
rom
S
tar
s
0
5
10
1
5
2
0
0
0
.
05
0
.
1
0
.1
5
0
.2
0
.2
5
0
.
3
0
.
35
dΦ
i
d
x
8
B
7
B
e
p
e
p
pp
h
e
p
x
B
PS
(
GS98
)
200
8
0
5
1
0
1
5
20
0
0
.
05
0
.
1
0
.
15
0
.
2
0
.
2
5
0
.
3
0
.
3
5
d
Φ
i
d
x
1
7
F
15
O
13
N
1
3
N
x
BPS(GS98) 2008
Fig. 6.3 The distribution of solar neutrino fluxe
s
d
Φ
i
d
x
w
ith respect to the sola
r
r
ad
i
us f
r
act
i
on
x
≡
r/
R
, where the SSM of BPS
(
GS98
)
2008 has been assumed
(
Pena-Garay and Serenelli, 2008
)
ν
e
+
37
C
l
→
37
Ar
+
e
−
.
(
6.41
)
T
h
is met
h
o
d
was
l
ater use
db
y Davis in t
h
e pioneerin
g
Homesta
k
eso
l
ar neu
-
trino experiment
(
Davis, 1964
)
. The energy threshold for the Cl-Ar reactio
n
i
nEq.
(
6.41
)
i
s
E
th
ν
=0
.
8
14 MeV, so t
h
e Homesta
k
e experiment can on
ly
d
etect t
h
ose so
l
ar neutrinos wit
h
me
d
ium an
dh
i
gh
ener
g
ies. T
h
is experimen
t
was located in the Homestake Gold Mine at Lead in South Dakota. Its de
-
tector was a single horizontal steel tank and contained a volume of 615 tons
o
f
liquid perchloroeth
y
lene
C
2
Cl
4
,
an
dh
ence t
h
eneutrinotar
g
et was a
b
ou
t
2
.
1
6
×
1
0
30
atoms
(
130 tons
)
of
37
C
l. The ke
y
problem of a chlorine experi
-
m
ent is to extract an
d
count t
h
e resu
l
tant argon atoms. First, a sma
ll
amoun
t
o
f
isotopicall
y
pure
36
A
r
o
r
38
A
r carrier
g
as is placed in the tank of liqui
d
perchloroethylene. Then the detector is exposed to solar neutrinos for abou
t
two
m
o
n
t
h
s
.Thi
st
im
e
in
te
r
va
li
sc
h
ose
n
suc
h
t
h
at t
h
e
n
u
m
be
r
of
3
7
A
r
ato
m
s
c
an nearly grow to the saturation value
(
i.e., the production rate is equal to
the decay rate
)
. After the exposure, the argon atoms are extracted by purgin
g
6.3 Experimental Detection o
fS
olar Neutrinos 235
whe
r
ethe
1
σ
t
heoretical uncertainties are given
(
Pena-Garay and Serenelli, 2008
)
Table 6.7 Solar neutrino fluxes predicted by the SSMs of BPS
(
GS98
)
2008 and
BPS
(
AGS05
)
2008. Note that the fluxes are given in units of 10
10
(
pp
)
,10
9
(
7
Be
)
,
10
8
(
pe
p
,
13
Na
n
d
15
O
)
,10
6
(
8
B
a
n
d
1
7
F
)
and 10
3
(
h
e
p
)
c
m
−
2
s
−
1
,
respectively
(
Pena-Garay and Serenelli, 2008
)
Source BPS
(
GS98
)
2008 BPS
(
AGS05
)
2008 Difference
pp
5
.
97
(1
±
0
.
0
06
)
6
.
04
(
1
±
0
.
005
)
1.
2
%
p
e
p
1
.
4
1
(1
±
0
.
0
11
)
1
.
45
(
1
±
0
.
010
)
2.
8
%
he
p
7
.90
(1
±
0
.
1
5
)8
.
22
(1
±
0
.15
)4
.
1
%
7
B
e
5
.
0
7
(1
±
0
.
0
6
)
4
.
55
(1
±
0
.
0
6
)
10%
8
B5
.94
(1
±
0
.
1
1
)
4
.
7
2
(1
±
0
.11
)
21%
13
N2
.88
(1
±
0
.
1
5
)1
.
8
9
(
1
+0
.
14
−
0
.
13
)
34
%
15
O
2
.
1
5
(
1
+
0
.
1
7
−
0
.
16
)1
.
3
4
(
1
+0
.
16
−
0
.
15
)
31
%
17
F5
.
8
2
(
1
+
0
.
19
−
0
.
17
)3
.
2
5
(
1
+0
.
16
−
0
.
15
)
44
%
with the helium gas. The recovery efficiency of
37
A
r is determined b
y
compar-
i
ng the measured isotopic composition of the extracted argon atoms with th
e
a
mount o
f
isotopically pure carrier
g
as at the be
g
innin
g
o
f
each run. Finally
,
the extracted
37
A
r atoms are counted by observing the
2
.
8
2keV
A
uger elec
-