Ann. Henri Poincaré 4, Suppl. 1 (2003) S303 – S317
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/01S303-15
DOI 10.1007/s00023-003-0924-z
Annales Henri Poincaré
Solar Neutrino Production
Douglas Gough
Abstract. It is commonly and quite plausibly presumed that neutrino flavour transitions are responsible for essentially all of the discrepancy between the observed
flux of neutrinos from the Sun and the theoretically calculated neutrino production
rate. Indeed, the comparison between the detection rates by SuperKamiokande and
by the Sudbury Neutrino Observatory are consistent with this presumption. Helioseismological analysis has set quite tight constraints on the conditions in the Sun’s
core, where the neutrino-emitting nuclear reactions take place. These constraints
have been obtained subject to certain assumptions which need to be investigated
before secure precise conclusions concerning neutrino production can be drawn.
1 The standard solar model
The so-called standard solar model is not a true standard. It is a standard only
so far as the general principles of its construction are concerned – and even those
vary with time. Therefore, when discussing differences between measured neutrino
fluxes and a standard solar model, one must always be meticulous in recording
which standard model is being referred to.
A standard solar model is spherically symmetrical, in hydrostatic equilibrium.
It is evolved either from some early stage of its existence as a star, such as the
base of the probably fully convective Hayashi track, or, more frequently, from only
the putative zero-age main sequence: a homogeneous state in thermal balance, the
rate Ln of generation of thermal energy by nuclear reactions being balanced by
the luminosity L radiated by photons at the surface r = R. The only process that
modifies the structure (on the main sequence) after the zero age is the change in
chemical composition. This is brought about principally by nuclear transmutation
in the core, and, to a much lesser extent, by the gravitational settling of heavy
elements. The gravitational energy liberated by these processes is tiny compared
with the nuclear energy. No internal motion is explicitly considered, with the exception that in the convection zone (which occupies the outer 29 per cent by radius
of the model) the chemical composition is taken to be homogenized, and a simple
mixing-length formula is adopted to relate the flux of heat and, rarely, momentum
to the superadiabatic temperature gradient. Thus there is no macroscopic mixing
of the products of the nuclear reactions in the core. The mass of the model is conserved – there is neither mass loss nor accretion – and rotation and the magnetic
field are ignored. It seems that all of these simplifications are quite reasonable.
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Moreover, they render it possible, in principle, for anyone to reproduce the results
(within numerical error).
Given the mass M of the star, the model is evolved for a time t to its
current state, defined according to its radius R and its luminosity L, and the
surface abundance ratio Zs /Xs , which is determined spectroscopically; here X
is the relative hydrogen abundance (by mass) and Z is the abundance of heavy
elements (i.e. all elements other than H and 4 He), the abundance of 4 He being
Y = 1 − X − Z. That current state is achieved by choosing appropriate values for
the initial composition (X0 , Y0 , Z0 ), and also for a parameter α which relates the
mixing length in the prescription for convection to the pressure scale height. The
age t is believed to be about 4.55 Gy (e.g. von Hippel, Simpson and Manset,
2001), but the precise value adopted for modeling differs from model to model.
In order to reduce predictions of different models to a common age it is adequate
to extrapolate linearly, using partial derivatives that have been quoted in the
literature (e.g. by Bahcall and Ulrich, 1988), some of which I reproduce in §4.
Standard solar models have been computed with a variety of equations of
state, opacity formulae and nuclear reaction rates. To assess the dependence of
the neutrino fluxes on the values of the cross-sections that have been adopted, one
can again use linear extrapolation with published partial derivatives; to assess the
dependence on equation of state and opacity is more difficult, and usually requires
numerical computation. However, Bahcall and his collaborators in several publications (e.g. Bahcall and Ulrich, 1988) have explicitly addressed the sensitivity of
the neutrino fluxes to a host of the uncertainties in the theory, and ChristensenDalsgaard (1996) has considered the sensitivity of other properties of the models.
As is well known, the fluxes of solar neutrinos measured at the Earth are
substantially less than the corresponding theoretical values computed from standard solar models under the assumption of no neutrino flavour transitions. The
discrepancies have triggered an enormous amount of research into solar modeling, resulting in keenly honed models which can be relied upon to reproduce the
consequences of the physics that has been put into them.
2 The principal nuclear reactions
To appreciate how neutrino production depends on conditions in the solar core it
is necessary to consider the balance of the thermonuclear reactions. I shall concentrate on the principal neutrino-emitting reactions of the pp chain:
3
He(p, e+ ν)4 He
hep (2 × 10−5 %)
|
p(p, e+ ν)2 H(p, γ) 3 He(3 He, 2p) 4 He
ppI (85%)
|
|
3
He(4 He, γ) 7 Be(e− , ν)7 Li(p,4 He)4 He
ppII (15%)
p(pe− , ν)2 H
|
7
Be(p, γ)8 B(e+ ν)8 Be∗ (4 He)4 He ppIII (0.02%) .
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The figures in parentheses are approximately the percentage terminations via the
corresponding branch of the chain at the centre of a standard model; about 0.25%
of the deuterium is produced by the pep reaction.
Because the temperature does not vary by a very large amount through the
energy-generating core, it is adequate to approximate the temperature dependence
of the reaction rates by power laws. All but the pep reaction and the spontaneous
decays of 8 B and 8 Be are two-body reactions, between species i and j, say, whose
rates per unit mass may be written rij Rij Xi Xj ρT ηij , where Xk is the abundance of species k, ρ is density, and Rij and ηij are constants. For the reactions
of interest I shall use atomic number unambiguously for the nuclear-species label k, for I shall not need to consider explicitly the details of the relatively rapid
terminating branches of the chain. I shall use the label e for electrons. Since the
abundant elements are almost fully ionized, the electron abundance is approximately 12 (1 + X). The three-body pep reaction rate can therefore be written
rpep Rpep (1 + X)X 2 ρ2 T ηpep .
Except in the outermost reaches of the core, where the reactions are slow,
all the components of the chain of reactions are in balance, the overall reaction
rate being determined by the slow pp and pep reactions which gradually reduce
the abundance X of hydrogen fuel. Since the reaction rates decrease with radius,
r, through the core, as also do ρ and T , hydrogen exhaustion decreases outwards,
leaving a positive gradient of X (Figure 1). The proton capture by deuterium is
essentially instantaneous, so the entire process creating 3 He from H is controlled
by the pp reaction.
Figure 1: Neutrino production in a standard solar
model. The 7 Be and 8 B neutrino fluxes, Φi
with i = 7 and 8 respectively, are proportional to 0R φi dr, where r is a radius variable and R is
the radius of the Sun. The pp neutrino production rate and, approximately, the thermal energy
generation rate are proportional to φpp ; the pep production rate varies similarly (although not
identically). The functions that are plotted are the normalized production rates fνx = Φ−1
x φx .
The temperature T and the hydrogen abundance X (whose ordinate scale is on the right of the
diagram) are included for comparison.
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For a good simple first approximation to the sensitivities of the neutrinoproducing reactions to conditions in the core it is adequate to balance the 3 Heproducing reaction against 3 He destruction by only the ppI chain, and similarly
to balance the 7 Be-producing reaction against 7 Be destruction by only the ppII
chain. Thus
A11 X 2 ρT η11 A33 X32 ρT η33
(1)
A34 (1 − X)X3 ρT η34 A7e (1 + X)X7 T η7e ,
(2)
and, because Z 1,
from which the abundance X7 of 7 Be can be calculated. Finally, the abundance
X8 of 8 B can be calculated from the p,γ reaction with 7 Be:
r8 = A8 X8 ρ = A71 X7 XρT η71 .
(3)
In the solar core, ηpep η11 4, η31 8, η33 16, η34 17, η7e − 12 and
η71 13. Therefore the neutrino-producing reaction rates are given approximately
by
rpep Rpep (1 + X)X 2 ρ2 T 4 ,
(4)
r11 R11 X 2 ρT 4 ,
r13 R13 R11 /2R33 X 2 ρT 2 ,
r7e R34 R11 /2R33 (1 − X)XρT 11 ,
R11 1 − X 2 24.5
R34 R71
X ρT
.
r8 R7e
2R33 1 + X
(5)
(6)
(7)
(8)
The exact functional forms of the neutrino production rates per unit radius r in
the Sun, φpp = 4πr2 ρr11 , φhep = 4πr2 ρr13 , φ7 = 4πr2 ρr7e and φ8 = 4πr2 ρr8 ,
are plotted in Figure 1. Because most of the terminations are via ppI, the thermal
energy generation rate is essentially, although not exactly, proportional to φpp Q,
where Q is the total energy released per p-p reaction in converting 4 protons and 2
electrons to 4 He (it is not exactly proportional because there are losses associated
with neutrino emission, and because the branching ratios between ppI and ppII
and, much less importantly, between ppII and ppIII vary with temperature); the
functional form of φpep is similar, but not identical, to that of φpp .
In addition to the reactions I have discussed, there are also those of the CNO
cycle. Like the 8 B rate, the neutrino production rates are quite sensitive to temperature – they are roughly proportional to XZρT 20 – but the total neutrino flux
is quite small, so I shall not discuss them explicitly. They must, of course, be taken
into account when making detailed comparison between theory and measurement.
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3 Helioseismology
Some aspects of the structure of the Sun have been determined seismologically.
Observable seismic modes are resonant acoustic waves, which sense, above all, the
sound speed c. In addition, but to a lesser extent, they sense the density, ρ, through
both buoyancy and the self-gravity of the waves. From accurate measurements of
thousands of resonant frequencies it has been possible to determine certain properties of both c(r) and ρ(r) with remarkable precision. (Deviations from spherical
symmetry are tiny, and are hardly detectable; in any case they appear to be confined to the near-surface layers of the Sun, and are therefore unlikely to have a
substantial impact on studies of neutrino production.)
Given a stellar model (usually referred to as a reference model) whose structure is pointwise close to that of the Sun, it is adequate to estimate the small
deviation of the model from the Sun by linearizing the frequency differences δνi of
modes i in the differences δc and δρ between the sound speed and density of the
Sun and the corresponding values in the model. Thus δνi is the sum of weighted
integrals of δc and δρ, the weight functions Kci , Kρi being different
for different modes. By taking an appropriate linear combination of the data, αi (r0 )δνi ,
it is possibleto design corresponding weight functions Ac (r; r0 ) =
αi Kci and
Aρ (r; r0 ) =
αi Kρi which make the frequency differences much easier to interpret. For example, Ac (r; r0 ) can be designed to be localized about r = r0 with Aρ
everywhere small; if the coefficients
αi are normalized to make Ac unimodular (i.e.
having an integral of unity), then αi δνi , considered as a functional of the model,
represents a localized average of δc. Similarly one can construct a localized average
of δρ, but that is harder because the density kernels are smaller than thesoundspeed kernels (in the sense that they project, without the combinations
αi δνi
being too seriously contaminated by data errors, into an accessible function space
of lower dimension than that effectively spanned by {Kci }). Alternatively, one can
seek combinations of the data that generate functions δc, δρ which, when added
to the sound speed cm and density ρm of the model, reproduce the observations
precisely (or, more usefully, merely reproduce the observations to within the observational errors; functions that satisfy the real, necessarily erroneous data precisely
tend to have highly oscillatory components which are extremely sensitive to the
data, and therefore to their errors, and which therefore have no useful physical
meaning. It is more prudent to try to select functions that are robust to data errors yet which reproduce the data adequately. How that selection is made, formally,
and often really, depends on prejudice). The resulting representations of the Sun
constitute an extremely valuable data set with which to compare the theoretical
solar models.
To illustrate how closely a modern standard model represents the Sun, I
illustrate in Figure 2 the relative deviations δc2 /c2 = 2δc/c and δρ/ρ between the
Sun and the standard model S of Christensen-Dalsgaard et al. (1996). The reason
for considering c2 and not c is that for a perfect gas, which roughly represents
the solar material, c2 is proportional to temperature (and is inversely proportional
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Figure 2: Relative differences (a) δc2 /c2 and (b) δρ/ρ between the spherically averaged squared
sound speed and density in the Sun (inferred seismologically) and the squared sound speed and
density in the standard model of Christensen-Dalsgaard et al. (1996). The seismic data are spatial
averages, weighted by localized kernels whose characteristic widths (approximately full widths
at half maxima) are denoted by the horizontal bars; the vertical bars represent formal standard
errors (which are correlated). The theoretical model was computed with Z0 /X0 = 0.0245, and
Y0 0.27 (from Takata and Gough, 2001).
to the mean molecular mass), so c2 is perhaps a more natural variable to think
about. The quantities plotted are averages of the pointwise differences, weighted by
localized kernels each of whose characteristic spreads (roughly 1.04 of the full width
at half maximum) is denoted by a horizontal bar. Each vertical bar represents ±
one standard error arising from the random errors (assumed to be independent) in
the frequency data. Care must be taken in interpreting small-scale undulations in
any curve one might draw by eye through the data plotted in the figure, because
the errors in those data are correlated. If one is interested in trends and does
wish to imagine such a curve, then as a rule of thumb one might first multiply
the errors by a factor 3, but even that should be taken with a large pinch of salt
because the only published analysis of such error correlation (Gough, Sekii and
Stark, 1996) was carried out on a somewhat different problem with a different (and
rather smaller) frequency data set.
A remarkable property of the functionals plotted in Figure 2 is that their values are so small. That almost certainly implies that the reference model is quite
a good representation of the Sun. However, the model differs from the Sun by
a significant amount, because the deviations δc2 /c2 and δρ/ρ exceed the formal
standard errors by a large factor, more than 10 in places. The situation is qualitatively similar if other standard models are used as the reference, such as the model
of Bahcall and Pinsonneault (1992), the model of Brun, Turck-Chièze and Zahn
(1999), or that of Bahcall, Pinsonneault and Basu (2001). It is important that in
the fullness of time the cause of these deviations be understood.
The functionals plotted in Figure 2 each appear to have a component that
varies gradually with r, and a prominent localized feature (in the values of δc2 /c2
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immediately beneath the base of the convection zone at r/R 0.71, and also
in the apparent derivative of δρ/ρ in the same location if one does not heed my
warning about error correlation). The large-scale component is probably a product
of small errors that have a global impact on the model, such as in one or more of the
nuclear reaction rates, or the chemical composition; these have a direct influence
on the temperature or pressure, which are directly constrained by the equations of
stellar structure through their spatial derivatives. The somewhat larger-amplitude
localized feature is the product of an error in a local variable, whose derivative is
not directly so constrained. It is probably a result of rotationally induced mixing
in a thin layer beneath the convection zone called the tachocline. Material that
would otherwise have been richer in helium than the convection zone as a result
of gravitational settling is homogenized with the convection zone; consequently
the mean molecular mass, and also the density, is decreased in the tachocline, and
the sound speed is thereby increased (Elliott and Gough, 1999). Such a localized
anomaly far out in the envelope has a negligible direct effect on the neutrino fluxes.
The smoothly varying component, however, is likely to have some repercussions,
albeit perhaps small, on our inferences about conditions in the core.
4 Neutrino fluxes
In the days when modelers were trying to adjust their models to reproduce the
measured neutrino fluxes without admitting neutrino transitions, it was essential
to appreciate how the theoretical fluxes respond to perturbations to a model. That
requires one to know where in the core neutrinos of different energy are produced,
and how the fluxes vary with ρ, T and X. Much of that information is contained
in Figure 1 and equations (4)–(8), together with the energy information contained
in Figure 3. At present, however, with the many new parameters characterizing
neutrino flavour transitions that need to be determined, reliable fine tuning of the
models against neutrino data is hardly possible. Only the high-energy 8 B production rate, inferred from SuperKamiokande and the Sudbury Neutrino Observatory
(SNO), is currently available (Ahmad et al., 2001, 2002). Fortunately, that is consistent with the models, as I explain below. The lower-energy neutrinos contribute
somewhat to the radiochemical detections by 37 Cl at Homestake (Cleveland et al.,
1998) and, more importantly, to the 71 Ga detections by SAGE (Abdurashitov et
al., 1999) and GALLEX (Hampel et al. 1999). By combining all the measurements,
estimates of values of neutrino parameters such as those indicated in Figure 4 have
been constructed on the basis of transitions between electron neutrinos and neutrinos of some one other flavour, but we must await new neutrino data of different
kinds before it is possible to determine all the parameters.
One can estimate the neutrino production rates in the Sun by adjusting the
fluxes of a standard model, using differences δc2 and δρ, such as those illustrated
in Figure 2, and the scaling laws (4)–(8). The results depend, of course, on the
equation of state and the putative nuclear reaction rates that are adopted. To close
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Figure 3: Spectrum of neutrinos from a standard solar model. The energy ranges in which the
various solar neutrino detectors are sensitive are indicated above the graph (from J.N. Bahcall).
the scaling it is necessary also to estimate X and T , which are related to c2 and
ρ by the equation of state but cannot be inferred directly in a radiative zone by
seismology alone. How one estimates X depends on what is assumed about the
state of the radiative zone. If only the seismically more precisely determined c2
is used, then perhaps the closest approximation to the standard-model assumptions is to presume that the zone is in hydrostatic and thermal balance, with no
macroscopic material motion to transport any of the products of the nuclear reactions. These assumptions were used for this purpose by Gough and Kosovichev
(1990): they are prescribed by the energy equation r−2 ∂(r2 F )/∂r = 4πr2 ρ
, where
F = −(4ac̃T 3 /3κρ)∂T /∂r is the radiative heat flux; here is the nuclear heatgeneration rate, κ is the opacity, and a and c̃ are respectively the radiation density
constant and the speed of light. These constraints, together with the equation of
state relating c2 , ρ, T and X, enable one to determine X and T . Similar procedures have been adopted by Shibahashi (1993, 1999) and Shibahashi and Takata
(1996). (It is important to realize that in determining the opacity κ it is necessary to specify the distribution of the heavy elements in the Sun; it is perhaps
most natural to adopt the distribution of the standard reference model, although
taking a uniform distribution has been more common.) If, on the other hand, the
seismological determination of both ρ and c2 are used, then it is more robust to
estimate X by adding to the reference abundance, after due homogenization of
the tachocline, a (small) constant whose value is determined by demanding that
the luminosity be unchanged; that procedure also leaves essentially unchanged
the amount of hydrogen consumed by nuclear reactions during the main-sequence
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Flux
Capture rates (snu)
Detector
37 Cl
71 Ga
SK+SNO
BPB
BTCZ
JCD
Sun
Obs.
BPB
BTCZ
JCD
Sun
Obs.
Obs.
pep
0.22
hep
0.04
pp
0
7 Be
1.15
8B
5.76
CNO
0.42
0.22
0.22
0.04
0.04
0
0
1.13
1.11
5.56
5.43
0.41
0.40
2.8
0.1
69.7
34.2
12.1
9.0
2.8
2.8
0.1
0.1
69.8
69.7
33.8
33.2
11.7
11.4
8.8
8.6
(106 cm−2 s−1 )
total
7.6±1.2
7.0
7.4
7.2
2.56±0.23
128±8
127
127
126
74.6±5
8B
5.05
4.99
4.87
4.82
5.05
4.99
4.87
4.82
5.44±0.99
Table 1: Neutrino fluxes produced by various representations of the Sun, and the measured
fluxes on Earth. BPB, BTCZ and JCD are the standard solar models of Bahcall, Pinsonneault
and Basu (2001), Brun, Turck-Chièze and Zahn (1999) and Christensen-Dalsgaard et al. (1996);
‘Sun’ represents the predictions of the seismic representation of the Sun presented by Gough and
Scherrer (2002), with T and X determined from c2 and ρ as described in the text, using the
nuclear reaction rates adopted by Bahcall, Pinsonneault and Basu (2001). The 37 Cl observations
are from Cleveland et al. (1998), SAGE and GALLEX from Abdurashitov et al. (1999) and
Hampel et al. (1999) respectively, and SuperKamiokande and SNO from Fukuda et al. (2001)
and Ahmad et al. (2001). The ages of the standard models from the zero-age main sequence (onset
of hydrogen burning) are 4.57 Gy (BPB), probably 4.55 Gy (BTCZ) and 4.52 Gy (JCD). The
fluxes can be reduced to a common age with the help of the logarithmic derivatives ∂lnΦ/∂lnt
quoted by Bahcall and Ulrich (1988): 0.00, -0.11, -0.07, 0.69, 1.28 and 1.13 for pep, hep, pp, 7 Be,
8 B and CNO neutrinos respectively.
evolution. The neutrino fluxes that result when the reaction rates of Bahcall, Pinsonneault and Basu (2001) are adopted are listed in Table 1. Included in the table,
for comparison, are the fluxes of the standard models of Bahcall, Pinsonneault and
Basu (2001), Brun, Turck-Chièze and Zahn (2001) and Christensen-Dalsgaard et
al. (1996). The differences are, at present, less than the uncertainties in the inference that can be drawn from the neutrino flux measurements.
There are three kinds of neutrino flux measurements that have been undertaken to date, two radiochemical and two scattering. The first of the radiochemical measurements (Homestake) involved neutrino capture by 37 Cl, the first results
from which were announced nearly four decades ago (Davis, 1964); the others
(SAGE and GALLEX) involve capture by 71 Ga. The scattering measurements use
water: ordinary water in the case of Kamiokande (Fukuda et al., 1996), and subsequently the larger SuperKamiokande (Fukuda et al.,1998), and heavy water in the
case of SNO. All the detectors are more sensitive to neutrinos of higher energy, but
the thresholds differ (Figure 3). SAGE and GALLEX have the lowest threshold;
they are the only detectors in operation that are sensitive to the pp neutrinos, and,
because there are so many pp neutrinos, these neutrinos dominate the counts. The
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Homestake threshold is too high to detect pp neutrinos; its counts are a combination of principally pep, 7 Be and 8 B neutrino captures, together with a small
contribution from the CNO cycle, with 8 B neutrinos dominating. The scattering
detectors are sensitive essentially to only the 8 B neutrinos (there is only a very
small contribution from hep), and therefore provide a cleaner, and more sensitive,
diagnostic.
The only observational inference that can be compared directly with the
inferred solar fluxes at present is that of Ahmad et al. (2000, 2001). It concerns only
the high-energy 8 B neutrinos, and was drawn initially by combining the electronscattering data from SuperKamiokande (Fukuda et al. 2001), which provide a
nonuniformly weighted average Φes of the fluxes of neutrinos of all flavours: Φes =
Φe + βΦx (where Φe is the actual electron-neutrino flux at the detector, Φx is
the combined flux of µ and, presumably, τ neutrinos which have been created by
transitions from electron neutrinos, and β = 0.154 was calculated theoretically)
with the charge-current data from SNO, which are sensitive to only Φe . The data
are plotted on a Φe − Φx diagram in Figure 5; they intersect at Φe = 1.75 ±
0.14cm−2 s−1 and Φx = 3.69 ± 1.13 × 106 cm−2 s−1 , from which one obtains a total
neutrino flux Φ = Φe + Φx = 5.44 ± 0.99cm−2 s−1 at the Earth (actually at a
distance of 1 AU from the Sun). This result has been confirmed recently by Ahmad
et al. in their second paper, in which they report neutral-current data from SNO
which are also sensitive to all neutrino flavours, although with a relative weighting
which is somewhat different from the electron-scattering value. The newer SNO
measurement is not as precise as the SuperKamiokande measurement because there
has not yet been time enough to accumulate the data.
The total neutrino flux Φ corresponds to the production rate of electron neutrinos in the Sun. The agreement with the theoretical predictions recorded in Table
1 is a remarkable scientific achievement. It suggests very strongly that the solar
models actually provide a good representation of the Sun. This is especially the
case because the 8 B-neutrino production rate is so very sensitive to temperature.
One anticipates, therefore, that the theoretical predictions of the less-temperaturesensitive 7 Be and particularly the pp and pep neutrinos are much more robust.
5 Concluding remarks
The remarkable agreement between the neutrino production rates predicted by
the solar models and the direct neutrino-flux measurements is partly fortuitous.
Although, as I have explained, the models are in quite good agreement with helioseismological inference, there are significant discrepancies. These need to be
understood, and not merely explained away, before one can have confidence in our
view of the state of the solar interior.
It is not uncommon for solar modelers to compare their models with just the
seismologically determined sound speed through the Sun, for that is the quantity
we know the most accurately; sometimes only a particular simple feature of the
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Figure 4: An MSW neutrino-oscillation solution for ∆m2 (difference between the (principal)
squared mass eigenvalues associated with the electron-neutrino transitions) and the mixing angle
δ, obtained from the standard solar model of Bahcall, Pinsonneault and Basu (2001) and the
neutrino flux measurements (from Bahcall, Gonzalez-Garcia and Pẽna-Garay, 2002).
sound speed is considered, such as the inferred depth of the convection zone.
Such scant comparison is dangerous, for it can give one a false impression of
the faithfulness of the models. This was highlighted recently by Watanabe and
Shibahashi (2001). By artificially varying the heavy-element abundance Z(r) in
the core, they constructed a range of low-flux solar models, with ΦCl varying from
7.2 to 2.45 snu, ΦGa from 126 to 101 snu, and Φ8 from 4.8 to 1.3 ×106 cm−2 s−1 ;
models with higher fluxes could equally easily have been constructed. These were
not standard models, evolved from zero age, but were seismic models in hydrostatic
and thermal balance. Unlike a similar set of standard models, their sound speeds
all agree with those inferred for the Sun. But their densities do not agree, so many
of the models can be ruled out. However, there is some leeway, because density is
not determined with high accuracy. Although Watanabe and Shibahashi did not
consider how their Z profiles might have come about, their results provide a stark
warning that processes currently ignored in standard models, such as putative weak
macroscopic circulatory flow in the radiative zone, or material and possibly heat
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Figure 5: Flux Φ8 of 8 B neutrinos in the form of µ and τ flavours, plotted against the flux of 8 B
electron neutrinos. The charge-current data ΦSNO
from SNO and the electron-scattering data
cc
ΦSK
es from SuperKamiokande are depicted by bands of width two standard errors (±σ from the
most likely value); their centres intersect at the spot. The diagonal bands bounded by solid lines
inclined at -45◦ represent the total flux to within ±σ; the dashed lines represent the predictions
of the standard solar model of Bahcall, Pinsonneault and Basu (2001) (after Ahmad et al., 2001).
transport by waves, could be present; such processes might influence the neutrino
production rates. Additionally, some of the parameters influencing microscopic
processes, such as nuclear reaction rates or opacity, could also be in error.
As a concluding example of the latter, it is interesting to note that recent work
by Grevesse and Sauval (2002) and Prieto, Lambert and Asplund (2002) has led to
substantial decreases in the spectroscopically estimated photospheric abundances
of several heavy elements. This effectively reduces Zs /Xs . What are the implications? One cannot determine that without a detailed calculation, but one can make
an estimate. Although the abundance revisions modify the relative abundances of
the heavy elements, one can, as a first approximation, ignore that. From the standpoint of stellar structure, the effective change in opacity is roughly equivalent to
reducing Zs /Xs by some 16 per cent. It is interesting that the change is broadly consistent with the spectroscopic error estimates, but it is rather greater than the presumably overoptimistic estimates of Basu (1998), which are based on a helioseismological analysis in which the OPAL opacity was accepted, with due warning, as also
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was the OPAL equation of state, without comment, despite evidence that it is deficient at the helioseismological level of precision (Kosovichev et al., 1992; Baturin
et al., 2000). With the help of the partial derivatives ∂lnΦpep /∂ln(Zs /Xs ) −0.1,
∂lnΦhep /∂ln(Zs /Xs ) −0.2, ∂lnΦ7 /∂ln(Zs /Xs ) 0.6, ∂lnΦ8 /∂ln(Zs /Xs ) 1.3
and ∂lnΦCNO /∂ln(Zs /Xs ) 1.9, which were computed from the grid of theoretical solar models of Gough and Novotny (1990), one can estimate that the fluxes
ΦCl , ΦGa and Φ8 from standard solar models are reduced by 1.4 snu, 8.3 snu,
and 1.0 ×106 cm−2 s−1 , respectively. Seismic models with given c2 (r), on the other
hand, suggest that the reductions are 0.8 snu, 3.3 snu and 0.62 ×106 cm−2 s−1 . In
either case, the deviation from the values inferred by Ahmad et al. (2001, 2002)
are increased. Evidently, there remains more work to do to increase the reliability
of the predictions of the neutrino production rates for the purpose of constraining
neutrino-transition parameters more tightly.
6 Acknowledgements
I am very grateful to Günter Houdek for stimulating conversations, to Di Sword
for preparing the LATEX file, and to Richard Sword for helping with the diagrams.
References
[1] J.N. Abdurashitov, et al., Phys. Rev. C (Nucl. Phys.), 60, 055801 (1999).
[2] Q.R. Ahmad, et al., Phys. Rev. Lett., 87, 071301–071306 (2001).
[3] Q.R. Ahmad, et al., Phys. Rev. Lett., 89, 011301-1–5 (2002).
[4] J.N. Bahcall, M.C. Gonzalez-Garcia, and C. Peña-Garay, 2002,
JHEP 04(2002)007, hep-ph/0111150.
[5] J.N. Bahcall and M.H. Pinsonneault, Rev. Mod. Phys., 64, 885–926 (1992).
[6] J.N. Bahcall, M.H. Pinsonneault and S. Basu, Ap J., 555, 990–1012 (2001).
[7] J.N. Bahcall and R.K. Ulrich, Rev. Mod. Phys., 60, 297–372 (1988).
[8] S. Basu, MNRAS, 298, 719–728 (1998).
[9] V. A. Baturin, W. Däppen, D. O. Gough, S. V. Vorontsov, MNRAS, 316,
71–83 (2000).
[10] A.S. Brun, S. Turck-Chièze and J-P. Zahn, Ap J., 525, 1032–1041 (1999).
[11] J. Christensen-Dalsgaard, Proc. VI IAC Winter School: ‘The structure of the
Sun’, (ed. T. Roca Cortés & F. Sánchez, Cambridge University Press), 47–139
(1996).
S316
D. Gough
Ann. Henri Poincaré
[12] J. Christensen-Dalsgaard, et al., Science, 272, 1286–1292 (1996).
[13] B. T. Cleveland, et al., Ap J., 496, 505–526 (1998).
[14] R. Davis, Jr, Phys. Rev. Lett., 12, 303–305 (1964).
[15] J.R. Elliott and D.O. Gough, Ap J., 516, 475–481 (1999).
[16] S. Fukuda, et al., Phys. Rev. Lett., 86, 5651–5655 (2001).
[17] Y. Fukuda, et al., Phys. Rev. Lett., 77, 1683–1686 (1996).
[18] Y. Fukuda, et al., Phys. Rev. Lett., 81, 1158–1162 (1998).
[19] D.O. Gough, T. Sekii and P.B. Stark, Ap J., 459, 779–791 (1996).
[20] D.O. Gough and A.G. Kosovichev, Inside the Sun, (ed. G. Berthomieu & M.
Cribier, Kluwer, Dordrecht), 327–340 (1990).
[21] D.O. Gough and P.H. Scherrer, The Century of Space Science, (ed. M. Huber,
J. Geiss, J. Bleeker & A. Russo, Kluwer, Dordrecht), 1035–1063 (2002).
[22] D.O. Gough and E. Novotny, Solar Phys., 128, 143–160 (1990).
[23] N. Grevesse and A.J. Sauval, Adv. Space Res., 30, 3–11 (2002).
[24] W. Hampel et al., Phys. Lett. B, 447, 127–133 (1999).
[25] T. von Hippel, C. Simpson and N. Manset, (eds), Astrophysical Ages and
Timescales, Ast. Soc. Pac. Conf. Ser., 245 (2001).
[26] A. G. Kosovichev et al., MNRAS, 259, 536–558 (1992).
[27] C.A. Prieto, D.L. Lambert and M. Asplund, Ap J., 573, L137–L140 (2002).
[28] H. Shibahashi, Frontiers of Neutrino Astrophysics (ed. Y. Suzuki & K. Nakamura, Universal Academy Press, Tokyo) 93 (1993).
[29] H. Shibahashi, Adv. Space Res., 24, 137–145 (1999).
[30] H. Shibahashi and M. Takata, PASJ, 46, 377–387 (1996).
[31] M. Takata and D.O. Gough, Proc SOHO 10 / GONG 2000 Workshop: Helioand asteroseismology at the dawn of the millennium, (ed. A. Wilson, ESA
SP-464, Noordwijk) 543–546 (2001).
[32] S. Watanabe and H. Shibahashi, PASJ, 53, 565–575 (2001).
Vol. 4, 2003
Solar Neutrino Production
Douglas Gough
Institute of Astronomy
Madingley Road
Cambridge, CB3 0HA
UK
and
Department of Applied Mathematics and Theoretical Physics
Silver Street
Cambridge, CB3 9EW
UK
and
Physics Department
Stanford University
California, 94305-4085
USA
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