solar model parameters and direct measurements of solar neutrino fluxes

Solar model parameters and direct measurements of solar neutrino fluxes

Abhijit Bandyopadhyay,1 Sandhya Choubey,2,3 Srubabati Goswami,2,4 and S. T. Petcov5,*1Tata Institute of Fundamental Research, Mumbai 400005, India2Harish-Chandra Research Institute, Allahabad 211 019, India

3Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom4Physik-Department T30d, Technische Universitaet Muenchen, D-85748 Garching, Germany

5Scuola Internazionale Superiore di Studi Avanzati and Instituto Nazionale di Fisica Nucleare, I-34014 Trieste, Italy(Received 2 January 2007; published 29 May 2007)

We explore a novel possibility of determining the solar model parameters, which serve as input in thecalculations of the solar neutrino fluxes, by exploiting the data from direct measurements of the fluxes.More specifically, we use the rather precise value of the 8B neutrino flux, �B obtained from the globalanalysis of the solar neutrino and KamLAND data, to derive constraints on each of the solar modelparameters on which �B depends. We also use more precise values of 7Be and pp fluxes as can beobtained from future prospective data and discuss whether such measurements can help in reducing theuncertainties of one or more input parameters of the standard solar model.

DOI: 10.1103/PhysRevD.75.093007 PACS numbers: 14.60.Pq, 26.65.+t, 96.60.Jw

I. INTRODUCTION

There has been remarkable progress in the studies ofsolar neutrinos in the last several years. The evidences ofsolar neutrino (��) oscillations, obtained first in theHomestake experiment and strengthened by the results ofKamiokande, SAGE, and GALLEX/GNO collaborations[1], were made subsequently compelling by the data ofSuper-Kamiokande (SK), SNO, and KamLAND (KL) ex-periments [2–4].1 The combined charged current (CC) andneutral current (NC) data from SNO and the �� e� elasticscattering data from SK experiment showed that the solar�e undergo flavor conversion on their way from the centralpart of the Sun, where they are produced, to the Earth.Under the plausible assumption of CPT-invariance, theresults of the KL reactor neutrino experiment [4] estab-lished the large mixing angle (LMA) Mikheyev-Smirnov-Wolfenstein (MSW) oscillations/transitions as the domi-nant mechanism at the origin of the observed solar �edeficit. The existing global neutrino oscillation data allowus to conclude that the solar �e undergo transitions (pre-dominantly) into almost an equal mixture of �� and ��neutrinos. The ratio of the CC and NC event rates observedat SNO provided a measure of the solar �e transitionprobability at energies of E� �5–10� MeV, while theSNO NC data permitted to determine with rather good

precision the 8B component of the solar �e flux [9]. Theglobal solar neutrino data allowed to obtain information onthe other important components of the solar neutrinoflux—the fluxes of pp, pep, and 7Be neutrinos, and toconstrain the flux of CNO neutrinos [10]. The combinedsolar neutrino and KamLAND data lead to a determinationof the neutrino oscillation parameters which drive the solar�e oscillations—the neutrino mass squared difference�m2

21 and the mixing angle �12, with an unexpectedlyhigh precision (see, e.g., [11,12]).

The latest SNO result on the 8B neutrino flux, �B, asreported in [9], reads

�NCB � 4:94�1� 0:088� � 106 cm�2 s�1: (1)

This is in good agreement with the standard solar model(SSM) prediction [13]

�SSMB � 5:79�1� 0:23� � 106 cm�2 s�1: (2)

The global oscillation analysis of the solar neutrino andKamLAND data performed in [11], in which �B is treatedas a free parameter, yields the following value of the flux:

�GlobalB � 4:88�1� 0:036� � 106 cm�2 s�1: (3)

The value of �B thus obtained corresponds to fB � 0:84,where fB is defined as

fB ��B

�SSMB

: (4)

The values of the pp and 7Be neutrino fluxes can also bedetermined similarly from a global analysis of the solarneutrino and KamLAND data, in which the solar luminos-ity constraint is used [10]. This analysis showed that withthe current data sets the pp neutrino flux can be determinedwith an uncertainty which is the same as the estimateduncertainty in the SSM prediction for the flux [13]. The

*Also at: Institute of Nuclear Research and Nuclear Energy,Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria.

1We would like to recall that the hypothesis of neutrinooscillations was formulated in [5]. In [6] it was suggested thatthe solar �e can take part in oscillations involving another activeor sterile neutrino. The article [6] appeared before the first dataof the Homestake experiment were reported. For a detaileddiscussion of the evolution of the solar neutrino problem since1970 and of the variety of different ‘‘solutions’’ proposed overthe years, see, e.g., [7,8].

PHYSICAL REVIEW D 75, 093007 (2007)

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http://dx.doi.org/10.1103/PhysRevD.75.093007

uncertainty obtained for the 7Be flux in the same analysis islarger than the estimated one in the SSM prediction for thesame flux [13]. Future high precision measurements of the7Be neutrino flux by Borexino and KamLAND can lead toa reduction of the uncertainties in both 7Be and pp fluxesdetermined from the solar neutrino data.

In the present article we explore the possibility of usingthe precision data (current and prospective) on the (i) 8B,(ii) 8B and 7Be, and (iii) 8B, 7Be, and pp, solar neutrinofluxes in order to obtain ‘‘direct’’ information (i.e., toconstrain or determine) on at least some of the solar modelparameters—nuclear reaction rates, opacity, diffusion,heavy element surface abundance, etc., which enter intothe calculations of the fluxes in the SSM. It is important toestablish whether high precision measurements of the 8B,7Be, and pp neutrino fluxes can provide also significantconstraints on the indicated SSM parameters because noneof the latter can be determined directly experimentally. Therelevant nuclear reaction rates are measured in direct ex-periments. The LUNA collaboration working at the Gran-Sasso National Laboratory in Italy has measured theS-factor S33 at center-of-mass (c.m.) energy of Ec:m: 16 keV [14], corresponding to the energy at which thereaction proceeds in the central part of the Sun. However,all the other nuclear reaction S-factors of interest have beenmeasured at energies which are considerably higher thanthose the nuclei have in the Sun: data on S1;14 and S34, forinstance, have been obtained recently [15,16] at Ec:m: *

70 keV and Ec:m: * 130 keV, respectively. As a conse-quence, one has to employ an extrapolation procedure(based on nuclear theory) in order to obtain the values ofthe rates at the energy of interest. The solar luminosity L�is measured directly with very high accuracy. However, itis still important to determine this fundamental solar ob-servable from solar neutrino flux measurements. The latterprovide ‘‘real time’’ information on the rates of the nuclearfusion reactions in the central region of the Sun. Both thephotons, observed in the form of solar luminosity, and theneutrinos, emitted by the Sun, are simultaneously producedin these reactions. It takes neutrinos approximately 8 mi-nutes to reach the Earth. In contrast, the ‘‘conventionally’’measured luminosity of the Sun is determined by photonsemitted by the solar surface, which, however, were pro-duced in the central region of the Sun �4� 104 yearsearlier—the time it takes for these photons to reach thesurface of the Sun (see, e.g., [17]). Thus, a comparison ofthe experimentally measured solar luminosity with thatobtained from neutrino flux measurements allows, in par-ticular, to test the thermo-nuclear fusion theory of energygeneration in the Sun and the hypothesis that the Sun, inwhat concerns the energy generation, taking place in itsinterior, and the energy radiation from its surface, is in anapproximate steady state.

Another SSM parameter on which the solar neutrinofluxes depend is the ratio of the surface abundance in

mass of the elements heavier than helium and of thesurface abundance (in mass) of hydrogen (surface heavyelement composition), Z=X. At the present there is ratherlarge uncertainty in the SSM estimated value of this pa-rameter (see, e.g., [13,18]), as will be discussed in some-what greater detail further. Moreover, the estimateduncertainty in the value of Z=X is obtained [13,18] assum-ing that the total spread of all modern determinations ofZ=X is equal to the 3� uncertainty in Z=X. Clearly, adetermination of the surface element composition parame-ter Z=X from neutrino flux measurements could be veryhelpful for resolving the indicated problems [19,20]. Itcould also be very useful for solar model building.

In the analysis that follows we will use the SSM byBahcall and Pinsonneault from 2004 [13] as a ‘‘bench-mark’’ solar model for the predictions of the solar neutrinofluxes and the estimated uncertainties in these predictions,originating from the different SSM parameters.

II. PRELIMINARY OBSERVATIONS

There are six principal nuclear reactions and decays inwhich neutrinos are produced in the Sun (see, e.g., [21]).Four of them generate neutrinos with continuous energyspectrum. These are the fusion of two protons (pp �’s), andthe decays of the nuclei 8B (8B �’s), 13N, and 15O (CNO�’s). The other two, the fusion of two protons and anelectron and the capture of an electron by a 7Be nucleusproduce neutrino lines (the so-called pep and 7Be �’s).The shapes of the energy spectra of the pp, 8B, and CNOneutrinos are determined by nuclear physics and are wellknown. However, the SSM predictions for the total valuesof the pp, pep, 7Be, 8B, and the CNO neutrino fluxesdepend on several SSM input parameters. The uncertain-ties associated with these parameters lead to (normaliza-tion) uncertainties in the predicted fluxes. There arealtogether 11 input SSM parameters on which the SSMpredictions for the 8B, 7Be, pp, and the CNO solar neu-trino fluxes in general depend [13]. These are first of allthe S-factors (see, e.g., [22]) of the nuclear reactions1H�p; e�e�2H, 3He�3He; 2p�4He, 3He�4He; ��7Be,14N�p; ��15O, 7Be�p; ��8B, and of e� capture on 7Be.The standard notations for these are S11, S33, S34, S1;14,S17, and Se�7, respectively. The additional parameters aredirectly related to the physics of the Sun: they are [13] thesolar luminosity L�, age ��, opacity O�, diffusionD�, andthe ratio of the mass fractions of the elements heavier thanhelium and of hydrogen at the surface of the Sun (surfacecomposition), Z=X. However, the formal definition of pa-rameters like diffusion and opacity involves complicatedphysics and they are not real physical parameters in thesense the S-factors, luminosity, age of sun, etc., are. Thetype of dependence of a given solar neutrino flux (8B, 7Be,pp, pep, and CNO) on a specific SSM parameter varieswith the flux. For instance, for the three fluxes of interestfor our subsequent discussion, we have [22]:

BANDYOPADHYAY, CHOUBEY, GOSWAMI, AND PETCOV PHYSICAL REVIEW D 75, 093007 (2007)

093007-2

�B � CB�S11��2:59�S33�

�0:40�S34�0:81�S1;14�

0:01�S17�1:0

� �Se�7��1:0 � �L��6:76��1:28�O��2:93

� �D��2:20�Z=X�1:36; (5)

�Be � CBe�S11��0:97�S33�

�0:43�S34�0:86�L��

3:40��0:69

� �O��1:49�D��

�0:96�Z=X�0:62; (6)

�pp � Cpp�S11�0:14�S33�

0:03�S34��0:06�S1;14�

�0:02

� �L��0:73��0:07�O��0:14�D��0:13�Z=X��0:08;

(7)

where CB, CBe, and Cpp are constants.Most of the nuclear reaction S-factors of interest are

measured typically at c.m. energies exceeding consider-ably the energies at which the reactions take place in thecentral region of the Sun. As a consequence, the results onthe S-factors obtained experimentally have to be extrapo-lated to the lower energy of �20 keV, which is of interestfor the solar neutrino flux calculations. The extrapolationprocedure brings additional theoretical uncertainty in theS-factor values. This uncertainty can be substantial for agiven S-factor.

A program of systematic high precision measurements,at lowest possible energies, of the nuclear reaction rates ofinterest for astrophysics is being realized by the LUNAcollaboration at the Gran-Sasso National Laboratory inItaly (see, e.g., [14]). The LUNA collaboration has mea-sured the S-factors S33, S1;14, and S34 at c.m. energiesEc:m: 16:5 keV [14], Ec:m: * 70 keV [15], and Ec:m: 168:9; 147.7; 126.5 keV [16], respectively. The S33-factorwas for the first time measured in laboratory at energyoccurring in a star, more specifically, in the region of theGammow peak for the reaction 3He�3He; 2p�4He (Ec:m: �15–27� keV). The results obtained for the S1;14 read

S1;14�0� � 1:61� 0:08� 0:16 keV b; (8)

which implies a 1� error of approximately 12%. The S34

factor has been determined with a 4% uncertainty [16]:S34�0� � 0:547� 0:017 keV b. Recently it has also beenmeasured in [23] at four values of the c.m. energy from theinterval (400–950) keV. Combining earlier results withthose obtained in [23] and in [16], and using an extrapola-tion to an energy of �20 keV, the authors of [16] obtainS34�0� � 0:550� 0:012 keV b.

Because of the importance of the S17-factor for theprediction of �B and the interpretation of the data of theHomestake, SK, and SNO experiments, considerable ef-forts have been made to determine it with a relatively highprecision. This was done using data (i) from a directmeasurement of the cross section of the reaction p7Be! 8B � [24], and (ii) of indirect studies of thesame reaction (via the Coulomb dissociation process �8B! p 7Be [25], and heavy-ion transfer and breakup

processes [26,27]). In the SSM calculations of the solarneutrino fluxes [13] the most precise result reported in [24]is used. The value recommended in [24] reads:

S17�0� � 21:4� 0:5�expt� � 0:6�theo� eV b; (9)

the quoted 1� error being smaller than 5%.2 However,from more recent data, obtained with radioactive ionbeams, the following value was found in [27]: S17�0� �18:2� 1:7 eV b. One of the theoretical uncertainties in thevalue of S17 quoted above in Eq. (9), is associated with theextrapolation method used to obtain this result. In [29] it isargued that a larger extrapolation error than is usuallytaken into account should be assigned in the evaluationof the uncertainties in S17�0�.

In what concerns the uncertainties in the predictions ofthe solar neutrino fluxes due to the other non-nuclearphysics parameters, the largest, according to the SSMBP04 [13], is a consequence of the lack of sufficientlyprecise knowledge of the surface element composition ofthe Sun. New values for the abundances in mass of theelements C,N,O,Ne, and Ar have been derived [30] usinga three-dimensional rather than one-dimensional model ofthe solar atmosphere, including hydrodynamical effects,etc. The new abundance estimates together with the bestestimates for other solar abundances [31] imply Z=X �0:0176, which is considerably smaller than the earlierresult [31] Z=X � 0:0229. The new values of the solarsurface abundances of C, N,O, Ne, and Ar, when incorpo-rated into solar models, lead to serious discrepancies withhelioseismological data [13,18,32].3 The estimated uncer-tainty in the value of Z=X, according to [13], is approxi-mately 15% (see Table II). The latter is obtained [13,18]4

assuming that the total spread of all modern determinationsof Z=X is equal to the 3� uncertainty in Z=X. It is pertinentto point out that the knowledge of the solar neutrinooscillation parameters �m2

21 and sin2�12 with relativelysmall uncertainties can be crucial for a successful highprecision determination of the fluxes of solar 8B, 7Be,and pp neutrinos. The existing data allow a determinationof �m2

21 and sin2�12 at 3� with an error of approximately11% and 25%, respectively. Much higher precision can(and most likely will) be achieved in the future. The datafrom phase-III of the SNO experiment [9] using 3Heproportional counters for the neutral current rate measure-

2The earlier recommended value [28], S17�0� � 194�2 eV b,

had a 1� uncertainty of approximately 15%.3For this reason the authors of our benchmark SSM BP04 did

not include the new most recent estimate of the parameter Z=Xin the calculations of the solar neutrino fluxes. The possibleeffects of this new result on the SSM predictions of the solarneutrino fluxes were considered in [18,32].

4The authors of [18] made a comment worth quoting concern-ing the uncertainties in the element abundances under discus-sion: ‘‘Estimating the uncertainty in an abundance determinationis even more difficult than arriving at a best-estimateabundance.’’

SOLAR MODEL PARAMETERS AND DIRECT . . . PHYSICAL REVIEW D 75, 093007 (2007)

093007-3

ment could lead to a reduction of the error in sin2�12 to21% [33,34]. If instead of 766.3 t yr one uses simulated 3 ktyr KamLAND data in the same global solar and reactorneutrino data analysis, the 3� errors in �m2

21 and sin2�12

diminish to 7% and 18% [34]. The most precise measure-ment of �m2

21, could be achieved [33] using Super-Kamiokande doped with 0.1% of Gadolinium (SK-Gd)for the detection of reactor ��e [35]: the SK detector getsthe same flux of reactor ��e as KamLAND and after 3 yearsof data taking, �m2

21 could be determined with an error of3.5% at 3� [33]. A dedicated reactor ��e experiment with abaseline L� 60 km, tuned to the minimum of the ��esurvival probability, could provide the most precise deter-mination of sin2�12 [36]: with statistics of �60 GW kt yrand a systematic error of 2% (5%), sin2�12 could bemeasured with an accuracy of 6% (9%) at 3� [34]. Theinclusion of the uncertainty in �13 (sin2�13 < 0:05) in theanalyzes increases the quoted errors by (1–3)% to approxi-mately 9% (12%) [34]. Even higher precision in the mea-surement of �m2

21 and sin2�12 can be reached with onemodule (of 147 kt fiducial mass) of the water Cerenkovdetector MEMPHYS doped with 0.1% of gadolinium(MEMPHYS-Gd), and with a 50 kt scale liquid scintillatordetector (LENA), installed in the Frejus underground labo-ratory [37]. The improved determination of �m2

21 and �12

with KamLAND or dedicated post-KamLAND reactorneutrino experiments has been studied also, e.g.,in refs. [38– 40], where the potential improvements ofthe precision on these parameters from future solarneutrino experiments have been investigated, e.g., inRefs. [10,34,36].

III. PRESENT KNOWLEDGE AND FUTUREMEASUREMENTS OF SOLAR NEUTRINO

FLUXES

Precise knowledge of solar neutrino fluxes is a keyingredient of our analysis. In this section, we summarizethe presently existing data and information on solar neu-trino fluxes and the improvements in the determination ofthe fluxes that are possible in the future.

Among all the eight solar neutrino fluxes, at the presentwe only have a direct experimental determination of thetotal 8B neutrino flux produced inside the Sun, through themeasurement of the rate of the neutral current reaction ondeuterium in the SNO experiment (cf. Eq. (1)). The 1�uncertainty in the value of the 8B neutrino flux determinedfrom the NC SNO data is approximately 8.8% (see Eq. (1)).Global oscillation analyses of solar and KamLAND data,in which the 8B neutrino flux is treated as a free parameter,allow to determine �B with even higher precision, asEq. (3) shows. In Fig. 1 we present the range of allowedvalues of fB (defined in Eq. (4)), obtained in analyses of thecurrent data and of prospective data from future experi-ments. The corresponding 1� uncertainties in fB are givenin Table I. Combining the results from other solar neutrino

experiments with the NC data from SNO improves theprecision of determination of the solar neutrino oscillationsparameters �m2

21 and sin2�12 and hence reduces the 1�error on the 8B neutrino flux to 4.4%. This is furtherreduced to 3.6% by addition of the KamLAND results inthe global data analysis. Also shown in Fig. 1 and Table Iare the expected uncertainty on fB with the inclusion ofprospective data from future planned/proposed experi-ments. The phase-III of SNO is expected to provide direct(uncorrelated) measurement of the NC event rate with aprecision greater than that achieved in the earlier saltphase. For the third phase data from SNO we have assumedthe same central values of the CC and NC event rates asthose observed during the salt phase, but smaller uncer-tainties in the measured CC and NC rates, namely, 4.0%and 6.4%, respectively [41]. With the inclusion of theindicated prospective results from this phase (referred toas SNO-III in Fig. 1 and Table I), the 1� uncertainty in fBcould be reduced to 3.2%. This uncertainty would furtherdiminish to 2.5% and 1.7% if we added successively theprospective data from a ‘‘generic pp’’ and from ‘‘SPMIN’’

0.7 0.75 0.8 0.85 0.9 0.95fB

0

1

2

3

4

5

6

7

8

9

10

11

12

∆χ2

SolarSolar+KLSolar + SNO III + KLSolar +SNO III + pp + KLSolar + SNO III + pp + SPMIN

1σ

2σ

3σ

FIG. 1 (color online). The dependence of ��2 on fB, showingthe range of allowed values of fB, determined using the currentlyexisting data and prospective data from future experiments.

TABLE I. 1� uncertainties of fB.

Data set used1� uncertaintyin fB (in %)

Solar 4.4Solar KamLAND 3.6Solar SNO� III KamLAND 3.2Solar SNO� III pp KamLAND 2.5Solar SNO� III pp SPMIN 1.7


093007-4

experiments to the analysis. The ‘‘generic pp’’ experimentin Fig. 1 and Table I refers to a high precision �� e�

elastic scattering experiment5 with assumed 1% error in themeasured reaction rate. The latter is simulated at the best-fit values of �m2

21 � 8� 10�5 eV2 and sin2�12 � 0:31.The SPMIN refers to a reactor experiment with a baselineof 60 km, tuned to the Survival Probability MINimum [36].The results given in Fig. 1 and Table I for this experimentcorrespond to statistics of 3 kt yr of data and a systematicerror of 2%. We refer the reader to [34,36] for furtherdetails of our method used for the analysis of projecteddata from future experiments.

The SSM prediction for 7Be neutrino flux reads:

�SSMBe � 4:86�1� 0:12� � 109 cm�2 sec�1; (10)

the estimated uncertainty being 12%. The flux of lowenergy pp neutrinos is calculated very precisely withinthe SSM and the estimated uncertainty is 1%:

�SSMpp � 5:94�1� 0:01� � 1010 cm�2 sec�1: (11)

The 7Be and pp neutrino fluxes can be determined in asolar model independent way together with the 8B neutrinoflux by treating all three fluxes as free parameters in globaloscillation analyses [10]. Below we summarize the resultsobtained for the 7Be and pp neutrino fluxes using presentand future prospective data from solar neutrino experi-ments [10]. The results for �pp depend on whether theluminosity constraint [44] is included in the analysis or not;the determination of �Be is essentially independent of theluminosity constraint [10]. Without employing the lumi-nosity constraint, both �Be and �pp are determined fromthe existing data with a precision which is much worse thanthe estimated precision of the BP04 SSM predictions forthe two fluxes. The inclusion of the luminosity constraint inthe analysis brings about a drastic improvement in thedetermination of �pp: even with the present solar andKamLAND data, �pp is determined with an uncertaintyof approximately 2% at 1�.

Improvement in the determination of �Be will be pos-sible from a direct measurement of the 7Be neutrino flux, asis envisaged in the forthcoming Borexino solar neutrino[45] and KamLAND experiments [46]. Data from theBorexino experiment with a total 1� error of 10% couldlead to a determination of the 7Be neutrino flux with a 1�uncertainty of �10%, which is somewhat smaller than theestimated uncertainty in the BP04 SSM prediction for �Be

(see Eq. (10)). With a total uncertainty of 5% in themeasured event rate in Borexino, �Be is expected to bedetermined with 1� error of 5.5%, while a measurement ofthe Borexino event rate with an error of 3% could lead to afactor of 3 improvement in the precision on �Be withrespect to the currently estimated 12% uncertainty in theSSM prediction for �Be.

Precision data from the Borexino experiment is alsoexpected to bring a significant improvement in the preci-sion of the determination of�pp. As long as the luminosityconstraint is imposed, a measurement of the Borexinoevent rate with a 5% error could lead to a determinationof�pp with a 1� uncertainty of 0.5%. The addition of highprecision data from a ‘‘generic pp’’ experiment is notexpected to lead to any significant reduction of the uncer-tainty in the value of �pp as long as the luminosity con-straint is taken into account [10]. However, the data fromthese experiments could certainly improve noticeably theprecision of the pp flux determination, which can beachieved without using the luminosity constraint.Moreover, it should be possible to test the applicabilityof the photon luminosity constraint itself and, more gen-erally, the thermo-nuclear theory of the energy generationin the Sun, by comparing the measured value of the solarphoton luminosity with the value obtained using the resultsfrom the LowNu and the other solar neutrino experiments[10].

IV. DETERMINING THE SSM INPUTPARAMETERS FROM DIRECT SOLAR NEUTRINO

FLUX MEASUREMENTS

In the theoretical framework of the SSM, the depen-dence of the solar neutrino flux from the ith nuclearprocess, �i, on the input parameters of the SSM is given,as we have already seen on the examples of the 8B, 7Be,and pp fluxes, by power laws:

�i � Ci �Yall j

xijj : (12)

Here Ci is a constant and ij in the exponent is thelogarithmic derivative of �i with respect to the SSM inputparameter xj,

ij �@ ln�i

@ lnxj: (13)

The values of the different logarithmic derivatives for theBP04 SSM [13] are given in Table II. We have, in general,eight such equations for the eight different solar neutrinofluxes.6

5There are a number of planned sub-MeV solar neutrinoexperiments (LowNu experiments) aiming to observe and mea-sure directly the pp neutrino flux using either charged currentreactions (LENS, MOON, SIREN [42]) or the �� e� elasticscattering process (XMASS, CLEAN, HERON, MUNU,GENIUS [42]). It was recently shown in [43] that it may bepossible to probe the temperature profile of the energy produc-tion in the Sun by observation of low energy neutrinos in theLENS experiment.

6In this counting we have included also the fluxes of hep and17F solar neutrinos [13], which, however, are predicted to beexceedingly small. These two fluxes do not play any significantrole in the analyses of the presently existing solar neutrino data.


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Let us first consider a general formulation of the prob-lem where we assume that we have a total of N inputparameters in the solar model calculations and that out ofthe total eight fluxes, K different fluxes along with their 1�uncertainties can be determined from direct measurements.We can then pick up the set of K equations from the aboveset (12) and can solve them for any subset xj1

; xj2; . . . ; xjK

of K different input parameters of the solar model, usingthe values of rest other (N-K) parameters xj as the valuesx0j found in SSM. We can express the constants Ci’s in

terms �SSMi ’s and x0

j ’s using the equation

�SSMi � Ci �

Yall j

�x0j �ij : (14)

Thus, we write the set of K equations for the fluxes�i1 ; �i2 ; . . . ; �iK as

�i

�SSMi

�

Qj�j1;j2;...;jK

�xj�ijQ

j�j1;j2;...;jK�x0j �ijQ

all j �x0j �ij

�Y

j�j1;j2;...;jK

�xjx0j

�ij; i � i1; i2; . . . ; iK: (15)

Taking logarithm on both sides of Eq. (15) we get

ln��i

�SSMi

��

Xj�j1;j2;...;jK

ij ln�xjx0j

�; i � i1; i2; . . . ; iK:

(16)

The above set of K equations can be written in a matrixform as

i1j1i1j2

:: i1jKi2j1

i2j2:: i2jK

: : : :: : : :

iKj1iKj2

:: iKjK

0BBBBB@

1CCCCCA

lnx1

x01

lnx2

x02::

lnxKx0K

0BBBBBBB@

1CCCCCCCA�

ln �1

�SSM1

ln �2

�SSM2::

ln �K

�SSMK

0BBBBBBB@

1CCCCCCCA: (17)

A general solution of Eq. (17) is given by

ln�xrx0r

��

DetXrDetA

; r � j1; j2; . . . ; jK; DetA � 0;

(18)

where

A �

i1j1i1j2

:: i1jKi2j1

i2j2:: i2jK

: : : :: : : :

iKj1iKj2

:: iKjK

0BBBBB@

1CCCCCA (19)

and Xr is the matrix obtained by replacing the rth columnof the matrix A by the column matrix in the right-hand sideof Eq. (17). Thus, we can write the solution for the set of Kparameters of the solar model fxrg, r � j1; j2; . . . ; jK, inthe form

xr � x0r exp

�DetXrDetA

�; r � j1; j2; . . . ; jK: (20)

To evaluate the uncertainties in each of the K SSMparameters of the set fxjg, j � j1; j2; . . . ; jK, determinedusing the data on the solar neutrino fluxes, we take thederivative of the logarithm of both sides of Eq. (12) for i �i1; i2; . . . ; iK:

ln�i �Xall j

ij lnxj; i � i1; i2; . . . ; iK: (21)

This set of K equations can also be written in matrix formas

TABLE II. Values of the logarithmic derivatives ij �@ ln�i@ lnxj

, corresponding to different solarneutrino fluxes�i (pp, pep, hep, 7Be, j, 8B, 13N, 15O, 17F) and the input parameters of the solarmodel (from Ref. [13]).

j pp;j pep;j hep;j Be;j B;j N;j O;j F;j

S11 0:14 �0:17 �0:08 �0:97 �2:59 �2:53 �2:93 �2:94S33 0:03 0:05 �0:45 �0:43 �0:40 0:02 0:02 0:02S34 �0:06 �0:09 �0:08 0:86 0:81 �0:05 �0:05 �0:05S1;14 �0:02 �0:02 �0:01 0 0:01 0:85 1:00 0:01S17 0 0 0 0 1:00 0 0 0L� 0:73 0:87 0:12 3:40 6:76 5:16 5:94 6:25Z=X �0:08 �0:17 �0:24 0:62 1:36 1:99 2:06 2:17�� 0:07 0 �0:11 0:69 1:28 1:01 1:27 1:29O� 0:14 0:24 0:54 �1:49 �2:93 �1:81 �2:25 �2:35D� 0:13 0:22 0:38 �0:96 �2:20 �2:86 �3:10 �3:22Se�7 0 0 0 0 �1:00 0 0 0


093007-6

i1j1i1j2

:: i1jKi2j1

i2j2:: i2jK

: : : :: : : :

iKj1iKj2

:: iKjK

0BBBBB@

1CCCCCA

lnxj1

lnxj2

::

lnxjK

0BBBBB@

1CCCCCA

�

ln�i1 �P

j�j1;...;jKi1j lnxj

ln�i2 �P

j�j1;...;jKi2j lnxj

::

ln�iK �P

j�j1;...;jKiKj lnxj

0BBBBBBBB@

1CCCCCCCCA: (22)

A general solution of Eq. (22) is then given by

lnxr �DetDr

DetA; r � j1; j2; . . . ; jK; DetA � 0;

(23)

where A is the matrix given by Eq. (19) and Dr is thematrix obtained by replacing the rth column of the matrixA by the column matrix appearing in the right-hand side ofEq. (22). Obviously, the right-hand side of Eq. (23)should be a linear combination of the quantities �i

�i, i �

i1; i2; . . . ; iK, and xjxj

, j � j1; j2; . . . ; jK, and therefore

Eq. (23) can be written as

lnxr �X

i�i1;...;iK

Pir ln�i X

j�j1;...;jK

Qjr lnxj;

r � j1; j2; . . . ; jK;(24)

where the coefficients Pir and Qjr involve the differentlogarithmic derivatives ij. It follows from the precedingequation that the 1� relative uncertainty in xr is given by

� lnxr ��Xi�i1;...;iK

P2ir�� ln�i�

2 X

j�j1;...;jK

Q2jr�� lnx0

j �2

s; or

ln�1

�xrxr

��

��Xi�i1;...;iK

P2ir

�ln�1

��i

�i

��2

Xj�j1;...;jK

Q2jr

�ln�1

�x0j

x0j

��2

vuut ; or

�xrxr�

��1� exp� ��X

i�i1;...;iK

P2ir

�ln�1

��i

�i

��2

Xj�j1;...;jK

Q2jr

�ln�1

�x0j

x0j

��2

vuut ��;

(25)

where ��i=�i is the 1� relative uncertainty in the directly measured flux�i and �x0j=x

0j is the 1� relative uncertainties of

the parameter x0j as estimated in the SSM.

V. DETERMINING ONE SSM INPUT PARAMETER USING MEASURED 8B NEUTRINO FLUX

Following the technique described in Sec. IV, we can use the measured value of the 8B neutrino flux, �B, and its

uncertainty, as given in Eq. (3), to determine one of the parameters xj1of the solar model together with its uncertainty,

�xj1xj1

.

They are given, respectively, by Eqs. (20) and (25) for K � 1:

xj1� x0

j1

��B

�SSMB

�1=B;j1 ; (26)

�xj1

xj1

�

��1� exp� ��

1

B;j1

�2�

ln�1

��B

�B

��2

Xj�j1

�B;jB;j1

�2�

ln�1

�x0j

x0j

��2

vuut ��: (27)

We show results for the central value of the input parame-ters with respect to their values used in BP04 SSM andtheir corresponding uncertainties in Table III. The extremeright column of Table III gives the factor by which thecentral values of different parameters will change withrespect to their values as used in the BP04 SSM [13] ifwe use the measured 8B neutrino flux, given in Eq. (3), fortheir calculations. Note that if the measured mean value of�B differs from the value predicted by the SSM,�SSM

B , thevalue of the parameter xj1

obtained using Eq. (26) would

differ from its SSM predicted value x0j1

by a factor con-trolled by the corresponding logarithmic derivative B;j1

�@ ln�B@ lnxj1

.We present in Table III the uncertainty of each of the

SSM input parameters, evaluated from Eq. (27), for threedifferent benchmark values of the uncertainty in the mea-sured 8B neutrino flux. For comparison we have alsoincluded in column 5 of Table III the estimated uncertain-ties in the SSM parameters in the BP04 SSM. It followsfrom Eq. (27) that the relative uncertainty in the parameter


093007-7

xj1depends on the inverse power of the magnitude of the

corresponding logarithmic derivative B;j1. It should be

clear from Eq. (5) and Table II that since the relevantlogarithmic derivatives in the cases of the S-factors S33

and S1;14 are relatively small, these quantities cannot be

determined with smaller uncertainties by using even a highprecision measurement of the 8B neutrino flux. We there-fore do not show the results for these cases in Table III. Itfollows from the results reported in Table III, in particular,that the SSM parameters like S11, Z=X, L�, and O� can bedetermined with uncertainties less than 10% owing to therelatively large values of their corresponding logarithmicderivatives. For the uncertainty in diffusion parameter D�we get approximately 10.6%. The table shows also that useof a direct high precision measurement of the 8B neutrinoflux can allow to determine the parameter Z=X with anuncertainty which is smaller than its currently estimateduncertainty in the BP04 SSM [13]. At the same time, theparameter S17 is determined much less precisely—withuncertainty of 25%, in spite of the fact that �B / S17.

We see from Table III that the uncertainties of most ofthe SSM parameters under discussion are essentially stablewhen the 1� error in the measured 8B neutrino fluxchanges from 2% to 4%. The reason for such a behavioris that for very small values of ��B=�B, the first term inthe right-hand side of Eq. (27) is much smaller than thesecond term which controls the uncertainty �xj1

=xj1. In

Fig. 2 we have plotted the fractional uncertainty in each ofthe SSM parameters as a function of the 1� error in themeasured value of the 8B neutrino flux.7 Figure 2 showsthat if the 8B flux uncertainty is larger than �5%, the firstterm in Eq. (27) would dominate over the second and�xj=xj can exhibit a stronger dependence on the uncer-tainty ��B=�B. The degree of this dependence is con-trolled by the corresponding Bj1

value. However, as wehave shown before (cf. Fig. 1 and Table I), the uncertaintyin the value of fB, determined from the current data, isalready approximately 4%. The second term in Eq. (27) isdominant and we do not expect any significant improve-

TABLE III. The uncertainties for each of the SSM input parameters, which are expected to be obtained if a high precision directmeasurement of the 8B neutrino flux is used to determine the corresponding SSM parameter. Given are also the uncertainties in theSSM input parameters [13], used in the SSM calculations of the solar neutrino fluxes.

Name of model parameters (xj1) �xj1

xj1(%)

�x0j1

x0j1

(%)xj1x0j1

��B�B� 2% ��B

�B� 3% ��B

�B� 4%

S11 9.05 9.10 9.16 0.4 1.07S34 30.01 30.19 30.44 9.4 0.81S17 24.80 24.94 25.12 3.8 0.84Luminosity L� 3.35 3.37 3.39 0.4 0.97Z=X 9.11 9.27 9.48 15.0 0.88Age of sun 19.18 19.28 19.42 0.4 0.87Opacity O� 7.69 7.73 7.79 2.0 1.06Diffusion D� 10.54 10.59 10.66 2.0 1.08Se�7 25.08 25.22 25.40 2.0 1.19

0 5 10 158.5

9

9.5

10

10.5

11

0 5 10 1570

75

80

85

90

95

100

0 5 10 15

30

32

34

36

38

0 5 10 1524

26

28

30

0 5 10 15

3.4

3.6

3.8

4

0 5 10 158

10

12

14

0 5 10 15

19

20

21

22

23

0 5 10 15

8

8.5

9

0 5 10 15

10.5

11

11.5

12

12.5

0 5 10 1524

26

28

30

S11

S33 S

34

.

S17

LO

τOO

O D

Se-7

Z/X

x axis = ∆φB/φ

B (%)

y axis = ∆xj1/x

j1(%)

.

. O.

FIG. 2 (color online). The 1� fractional uncertainty (�xj1xj1

) ofthe various SSM input parameters as a function of the uncer-tainty in the measured 8B neutrino flux (��B=�B).

7Obviously, the uncertainty in the value of a given SSM inputparameter determined exploiting a solar neutrino flux measure-ment depends also on the uncertainties in the remaining SSMinput parameters used in the evaluation.


093007-8

ment of the precision of determination of the SSM inputparameters with more precise measurement of �B alone.Small improvements can nonetheless be expected, espe-cially in what concerns the precision of determination ofZ=X.

VI. DETERMINING TWO SSM INPUTPARAMETERS USING MEASURED 8B AND 7Be

NEUTRINO FLUXES

As we have discussed in Sec. III, a relatively highprecision measurement of the 7Be neutrino flux can beperformed by the Borexino experiment. The KamLANDexperiment can also provide valuable data on �Be. Onecould use the experimentally measured values of the 8Band 7Be neutrino fluxes to determine any two of the SSMinput parameters.8 Equations (20) and (25) for K � 2 give

the central values and the uncertainties for any two pa-rameters xj1

and xj2as

xj1� x0

j1�

��B

�SSMB

�Be;j2

�

��Be

�SSMBe

��B;j2

��1=B;j1Be;j2�Be;j1B;j2 �; (28)

xj2� x0

j2�

��B

�SSMB

�Be;j1

�

��Be

�SSMBe

��B;j1

��1=B;j2Be;j1�Be;j2B;j1 �; (29)

and

��xj1

xj1

��

��1� exp

264� 1

B;j2�2��ln�B�

2 � 1Be;j2�2��ln�Be�

2 P

j�j1;j2�B;jB;j2�

Be;jBe;j2�2��lnx0

j �2

�B;j1B;j2�

Be;j1Be;j2�2

375

1=2��; (30)

��xj2

xj2

��

��1� exp

264� 1

B;j1�2��ln�B�

2 � 1Be;j1�2��ln�Be�

2 P

j�j1;j2�B;jB;j1�

Be;jBe;j1�2��lnx0

j �2

�B;j2B;j1�

Be;j2Be;j1�2

375

1=2��: (31)

Equations (28) and (29) imply, in particular, that thevalues of the parameters xj1

and xj2could differ from their

respective SSM values by factors determined by the fourlogarithmic derivatives—B;j1

, B;j2, Be;j1

, and Be;j2.

For calculating the SSM parameter uncertainties, we takeall possible combinations of fxj1

; xj2g and use Eqs. (30) and

(31) to get the corresponding errors on these sets of twoparameters, assuming a measurement of 8B and 7Be neu-trino fluxes, respectively, with 4% and 6% uncertainty.9

The uncertainties of xj1and xj2

, as given in Eqs. (30) and(31), are controlled in a rather complicated way by both themagnitude and the relative signs of the different logarith-mic derivatives. However, it follows from Eqs. (30) and

(31) that if for a certain pair of SSM parameters xj1 and xj2the relation B;j2

=B;j1 Be;j2

=Be;j1holds, these pa-

rameters would be determined with poor accuracy even ifone uses high precision data on the 8B and 7Be neutrinofluxes. Table II suggests that such pairs can be, for instance,fS33; S34g, fL�; O�g, and fZ=X;D�g.

Among the solar physics parameters the opacity O� canbe determined with a 9% uncertainty in pair with S34, orS17, or Se�7, for which we get at the same time uncertaintyof approximately 16%. For the diffusion D� we find anuncertainty of 11% when determined in combination withS33 or S34. In these cases, however, S33 and S34 are foundwithin 36% and 14%, respectively.

In Table IV we present results only for those combina-tions of two SSM parameters which are determined with an

TABLE IV. The list of different combinations of two SSMparameters which are determined with an uncertainty smallerthan 15% using data of prospective direct measurements of 8Band 7Be neutrino fluxes with 1� errors of 4% and 6%, respec-tively. The relative uncertainties on the SSM parameters thusdetermined are also given.

Combinations�xj1xj2

(%)�xj2xj2

(%)j1 j2

S34 S11 12.71 8.81S34 Z=X 14.25 12.26S34 D� 13.79 11.35

8We have also examined the uncertainties of the SSM parame-ters one obtains if only high precision prospective data on the7Be neutrino flux is used to determine the parameters. We havefound, in particular, that S33 and S34 can be determined with aprecision of 43.3% (33%) and 17.2% (12%), respectively, if the7Be neutrino flux is measured with 1� error of 10% (2%). Thisshould be compared with the uncertainties of 76% and 30% inS33 and S34 we have obtained in the previous section, using 8Bneutrino flux measurement with 4% uncertainty. Even if we takethe 1� error in �Be to be 2%, all the other SSM parameters aredetermined with uncertainties which are larger than those wefound in Sec. V when the same parameters are determined fromthe measured �B with 1� error of 4%.

9While �B is already known with a 4% uncertainty, �Be couldbe determined, as we have discussed above, with a 6% error bythe Borexino experiment (see Ref. [10] for details).


093007-9

uncertainty smaller than 15% using directly measuredvalues of �B and �Be. It is interesting to note fromTable IV that the set fS34; Z=Xg can be determined with arelatively good precision: the predicted uncertainty of Z=Xis smaller than the estimated one within the BP04 SSM[13], while the uncertainty in S34 is approximately 14.3%.The parameter S34 can be determined with 12.7% uncer-tainty in the combination fS34; S11g. As can be seen bycomparing Table III with Table IV, one gets a somewhatbetter precision on Z=X if the value of Z=X is obtainedfrom the experimental information on the 8B neutrino fluxonly.10 However, the uncertainty in the determination ofS34 can be much smaller when in addition to the data on the7Be neutrino flux one uses the data on the 8B neutrino fluxas well.

In Table V we present results on the uncertainties of theparameter combination fZ=X; S34g, determined from dataon the 8B and 7Be neutrino fluxes. The results correspondto 1� errors of 2%, 3%, and 4% in the measured value ofthe 8B neutrino flux, and of 2%, 4%, 6%, 8%, and 10% inthe measured value of�Be. We observe that the uncertaintyin the determination of Z=X would be smaller than itscurrently SSM estimated one of 15% if �Be is measuredwith an error not exceeding approximately 10% at 1�.

In Fig. 3 we show the expected 1� uncertainties ofthe pair of parameters fS34; Z=Xg, �S34=S34, and��Z=X�=�Z=X�, as continuous functions of the uncertainty

in the measured 7Be neutrino flux, ��Be=�Be, for threedifferent values for the uncertainty in the measured 8Bneutrino flux, ��B=�B. The figure shows that the depen-dence of �S34=S34 on ��Be=�Be is stronger than that on��B=�B, while ��Z=X�=�Z=X� depends on the accuracyof measurement of both fluxes. This feature is due to thefact that Be;Z=X ’ 2:2� B;Z=X, while Be;S34

’ B;S34.

VII. DETERMINING THREE SSM INPUTPARAMETERS USING DATA ON THE 8B, 7Be, AND

pp NEUTRINO FLUXES

As we have already discussed in Sec. III, the pp neutrinoflux, �pp, is determined with an uncertainty of 2% fromthe present solar and reactor neutrino data if one employsthe luminosity constraint. Future solar neutrino experi-ments can provide a remarkably precise measurement of�pp [10]. In this section we use high precision prospective

0 1 2 3 4 5 6 7 8 9 10

∆φBe / φBe (in %)

8

9

10

11

12

13

14

15

16

[ ∆(

Z/X

) / (

Z/X

) ] (

in %

) ∆φB/φB = 2%∆φB/φB = 3%∆φB/φB = 4%

0 1 2 3 4 5 6 7 8 9 10

∆φBe / φBe (in %)

4

6

8

10

12

14

16

18

20

22

∆S34

/ S

34 (

in %

)

∆φB/φB = 2%∆φB/φB = 3%∆φB/φB = 4%

FIG. 3 (color online). The 1� fractional uncertainties of theSSM input parameters S34 and Z=X, determined from data on the8B and 7Be neutrino fluxes, as a function of the uncertainty in themeasured 7Be neutrino flux, for three different values of the 1�error in the measured 8B neutrino flux.

TABLE V. Relative uncertainties of the set of two parame-ters—Z=X and S34, determined using prospective high precisiondata on the 8B and 7Be neutrino fluxes. The results correspond todifferent sets of assumed 1� errors in the measured values of thetwo fluxes.

��B�B�%� ��Be

�Be(%) ��Z=X�

�Z=X� (%) �S34S34

(%)

2 2 8.81 6.854 9.83 10.026 11.32 13.838 13.11 17.93

10 15.10 22.21

3 2 9.30 7.164 10.29 10.246 11.72 14.008 13.47 18.07

10 15.42 22.34

4 2 9.96 7.604 10.88 10.566 12.26 14.258 13.95 18.27

10 15.85 22.51

10The reason for this behavior can be traced to the complicatednature of Eqs. (30) and (31).


093007-10

data on 8B, 7Be, and pp neutrino fluxes to simultaneously determine any three of the SSM input parameters.11

Equations. (20) and (25) for K � 3 give the central values and uncertainties for any three parameters xj1, xj2

, and xj3as

xr �� pp

�SSMpp

��pp;r

��Be

�SSMBe

��Be;r

��B

�SSMB

��B;r

�1=DetA

� x0r ; �r � j1; j2; j3� (32)

and

�ln�

1�xrxr

��2�

�1

DetA

�2�

��2pp;r

�ln�1

��pp

�pp

��2 �2

Be;r

�ln�1

��Be

�Be

��2 �2

B;r

�ln�1

��B

�B

��2

X

j�j1;j2;j3

��pp;rpp;j �Be;rBe;j �B;rB;j�2

�ln�1

�xjxj

��2�; �r � j1; j2; j3�; (33)

where A is a 3� 3 matrix given by

A �pp;j1

pp;j2pp;j3

Be;j1Be;j2

Be;j3

B;j1B;j2

B;j3

0B@

1CA: (34)

�pp;r, �Be;r, and �B;r are the cofactors of the matrixelements pp;r, Be;r, and B;r respectively (r �j1; j2; j3). We take different combinations of three SSMparameters and calculate their uncertainties using 3%, 4%,and 1% as illustrative 1� errors in the measured values of8B, 7Be, and pp neutrino fluxes, respectively. We find thatfor almost all solar model parameters, the uncertaintiesreduce when using the combined information on the 8B,7Be, and pp neutrino fluxes, compared to what we haveobtained by using only prospective data on 8B and/or 7Befluxes. In Table VI we present results only for those sets ofthree SSM input parameters which are determined withuncertainties smaller than 15% each.

As Table VI shows, the most precise determination ofS34 occurs in the combination fS34; L�; Se�7g, while Z=X isbest determined in the sets fS17; L�; Z=Xg andfSe�7; L�; Z=Xg. For Z=X, the uncertainties we get aresmaller than that in the BP04 SSM predictions.

Note also that we get a rather accurate determination ofS17 from the combination fS34; S17; L�g, and of L� fromfS34; L�; Z=Xg. Although the uncertainty on S17 thus ob-tained of��8%–10%� is larger than the currently estimateduncertainty in the value of S17 found from relevant nuclearreaction data (see Sec. II), our results on S17 can be used, inparticular, as a consistency check, e.g., of the extrapolationprocedure employed to get S17 from the data. In whatconcerns the other parameters, we get the best determina-tion of S11 from fS11; ; L�; Se�7g, ofD� from fS17; L�; D�g,and of �� from fS17; L�; ��g. Introducing the pp flux in theanalysis a larger number of combinations of input parame-

ters can be determined within 15% uncertainty as com-pared to the case of using the 8B and 7Be fluxes. Thishappens due to a complicated interplay between the vari-ous terms in Eq. (33). However, note that in all thesecombinations the luminosity, L� is present as this parame-ter is controlled extremely well by the pp flux.

We stress that even though the precision we get for L� isworse than the precision achieved in the direct measure-ment of L�, the method we used to determine L� allows toperform a fundamental test of the thermo-nuclear fusiontheory of energy generation in the Sun, as well as to test thehypothesis that the Sun is in an approximate steady state in

TABLE VI. The fractional uncertainties in different combina-tions of the SSM input parameters for which we get uncertaintysmaller than 15%, assuming 3%, 4%, and 1% errors in the‘‘measured’’ �B, �Be, and �pp neutrino fluxes, respectively.

Combinations �xj1xj2

(%)�xj2xj2

(%)�xj3xj3

(%)xj1

xj2xj3

S11 S34 L� 6.16 7.98 1.62S11 L� O� 13.67 1.72 12.25S34 L� Z=X 8.87 1.00 8.63S34 L� O� 10.45 1.69 5.31S34 L� D� 9.03 1.57 5.77S34 L� �� 11.40 1.66 12.86S11 S17 L� 4.73 11.05 1.73S11 L� Se�7 4.53 1.73 11.05S33 S17 L� 12.40 8.47 1.95S33 L� Se�7 13.43 1.95 8.47S34 S17 L� 8.61 8.98 1.86S34 L� Se�7 6.33 1.85 8.98S17 L� Z=X 9.26 1.21 6.65S17 L� �� 9.05 1.85 7.34S17 L� O� 9.21 1.84 3.40S17 L� D� 9.74 1.74 4.68L� Z=X Se�7 1.21 6.65 9.88L� �� Se�7 1.85 7.42 9.29L� O� Se�7 1.84 3.40 9.48L� D� Se�7 1.73 4.68 10.12

11If we use only pp flux with 1% error then L� gets determinedwith 2.36% uncertainty. For other parameters using only the ppflux does not cause any improvement over what we get with the8B flux with 4% uncertainty.


093007-11

what regards the energy produced in its central region andthe energy emitted from its surface.

VIII. CONCLUSIONS

In the present article we have studied the possibility ofusing the precision data (current and prospective) on the(i) 8B, (ii) 8B and 7Be, and iii) 8B and 7Be and pp, solarneutrino fluxes in order to obtain direct information (i.e., toconstrain or determine) on at least some of the 11 solarmodel parameters—opacity, diffusion, heavy element sur-face abundance, nuclear reaction S-factors, etc., whichenter into the calculations of the fluxes in the SSM. Ourwork was inspired by the remarkable progress made in thestudies of solar neutrinos in the last several years, whichled to an unexpectedly precise determination of the solarneutrino oscillation parameters and of the 8B neutrino flux,as well as by the prospects for high precision measure-ments of the 7Be and pp neutrino fluxes. It was stimulatedalso by the realization that the solar physics parameterslike the opacity (O�), diffusion (D�), and heavy elementsurface abundance (Z=X), can never be measured in directexperiments. The solar photon luminosity L� is measureddirectly with very high accuracy. However, the convention-ally measured luminosity of the Sun is determined byphotons produced in the central region of the Sun, whichtook �4� 104 years to reach the surface of the Sun fromwhich they are emitted (see, e.g., [17]). The luminositydetermined from solar neutrino flux measurements pro-vides ‘‘real time’’ information on the rates of nuclearfusion reactions in the central region of the Sun, in whichthe solar energy is generated: neutrinos are simultaneouslyproduced in these reactions with the photons observed inthe form of solar luminosity, but it takes solar neutrinosapproximately only 8 minutes to reach the Earth. Similarconsiderations apply, perhaps to a somewhat less extent, tothe S-factors S11, S33, S34, S1;14, and S17, directly related tothe rates of the nuclear fusion reactions, on which the SSMpredictions for the solar neutrino fluxes depend and inwhich the solar energy is generated. They can be and aremeasured in direct experiments on Earth. However, this isdone at energies which are, in most cases, considerablyhigher than the energies at which the reactions take place inthe central part of the Sun. As a consequence, one has toemploy an extrapolation procedure (based on nuclear the-ory) in order to obtain the values of the rates at the energyof interest, corresponding to the physical conditions in thecentral part of the Sun.

We have derived the basic equations for determining thecentral values of the SSM input parameters and their un-certainties using results of direct measurements of solarneutrino fluxes (Sec. IV, Eqs. (20) and (25)). If we have rmeasured solar neutrino fluxes, at most r SSM input pa-rameters can be determined using the data on the solarneutrino fluxes. For the remaining SSM parameters wehave to use the SSM values and estimated uncertainties.

All our numerical results are based on the predictions ofthe SSM of Bahcall and Pinsonneault from 2004 [13].These include the dependence of different solar neutrinofluxes (8B, 7Be, pp) on the SSM input parameters, and,whenever necessary, the predicted values of the SSM inputparameters and their uncertainties.

We used first the precise value of the 8B neutrino flux,�B, obtained from global analysis of solar neutrino andKamLAND data, to determine each of the SSM parameterson which �B depends. If the measured mean value of �Bdiffers from the value predicted by the SSM, �SSM

B , thevalue of the parameter xj1

obtained using the data on �B

would differ from its SSM predicted value x0j1

by a factor

controlled by the logarithmic derivative B;j1� @ ln�B

@ lnxj1(see

Eq. (26)). The relative uncertainty in the parameter xj1thus

found depends on the inverse power of the magnitude ofB;j1

(Eq. (27)). Since, according to the BP04 SSM, therelevant logarithmic derivatives in the cases of the nuclearreaction S-factors S33 and S1;14 are relatively small (seeTable II), these quantities cannot be determined with suffi-ciently good accuracy even by using a high precisionmeasurement of the 8B neutrino flux. The results of thispart of our analysis are summarized in Table III. We havefound, in particular, that the SSM parameters like S11,Z=X, L�, and O� can be determined with uncertaintiesless than 10% owing to the relatively large values of theircorresponding logarithmic derivatives. For the uncertaintyin diffusion parameter D� we get approximately 10.6%.Our results show that the parameter Z=X can be determinedwith an uncertainty which is smaller than its currentlyestimated uncertainty in the BP04 SSM [13]. We havefound also that the uncertainties of most of the SSMparameters under discussion practically do not changewhen the 1� error in the measured 8B neutrino flux isreduced from 4% to 2%.

We have performed a similar analysis by combining aprospective high precision measurement of the 7Be neu-trino flux with the 8B neutrino flux measurement. In thiscase it is possible to determine simultaneously two SSMinput parameters, xj1

and xj2, including their uncertainties,

using the data on the 8B and 7Be neutrino fluxes. Ourresults show, in particular, that the values of the parametersxj1

and xj2thus determined could differ from their respec-

tive SSM values by factors determined by the four loga-rithmic derivatives—B;j1

, B;j2, Be;j1

, and Be;j2

(Eqs. (28) and (29)). We have calculated the SSM parame-ter uncertainties for all possible combinations of two SSMparameters fxj1

; xj2g, assuming that 8B and 7Be neutrino

fluxes are measured with 1� errors of 4% and 6%, respec-tively. Such precision on �Be can be reached in theBorexino experiment. We have found that the uncertaintiesof xj1

and xj2are controlled in a rather complicated way by

both the magnitude and the relative signs of the differentlogarithmic derivatives (Eqs. (30) and (31). If for a certain


093007-12

pair of SSM parameters xj1 and xj2 the relationB;j2

=B;j1 Be;j2

=Be;j1holds, these parameters would

be determined with poor accuracy even if one uses highprecision data on the 8B and 7Be neutrino fluxes. Thelogarithmic derivatives taken from the BP04 SSM(Table II) suggest that such pairs can be, for instance,fS33; S34g, fL�; O�g, and fZ=X;D�g. We have found alsothat among the solar physics parameters the opacity O�can be determined with a 9% uncertainty in pair with S34,or S17, or Se�7, for which we get at the same time uncer-tainty of approximately 16%. For the diffusion D� we findan uncertainty of 11% when determined in combinationwith S33 or S34. In these cases the latter are found withuncertainties of 36% and 14%, respectively. Most of theresults from this part of our study are summarized inTable IV.

We have obtained also rather detailed results on theuncertainties of the combination fZ=X; S34g, determinedfrom data on the 8B and 7Be neutrino fluxes (Table V andFig. 3). We have found, in particular, that the uncertainty inthe determination of Z=X would be smaller than its cur-rently SSM estimated one of 15% if �Be is measured withan error not exceeding approximately 10% at 1�.

Finally, we have analyzed the possibility to use highprecision prospective measurements of the 8B, 7Be, and ppsolar neutrino fluxes to simultaneously determine any threeof the SSM input parameters. We have taken differentcombinations of three SSM parameters and calculate theiruncertainties using 3%, 4%, and 1% as illustrative 1�errors in the measured values of �B, �Be, and �pp, re-spectively. We have found that for almost all solar modelparameters, the uncertainties reduce when using the com-bined information on the 8B, 7Be, and pp neutrino fluxes,compared to what we have obtained by using only pro-

spective data on 8B and/or 7Be fluxes. Results for those setsof three SSM input parameters which are determined withuncertainties smaller than 15% each are collected inTable VI. Our results show, in particular, that the mostprecise determination of S34, using our method, occurs inthe combination fS34; L�; Se�7g, while Z=X is best deter-mined in the set fS17; L�; Z=Xg. For Z=X the uncertaintieswe get are smaller than that in the respective BP04 SSMpredictions. The best determination of S11 is found to befrom the set fS11; ; L�; Se�7g, ofD� from fS17; L�; D�g, andof �� from fS17; L�; ��g. Even though the precision weobtained for L� is worse than the precision achieved in thedirect measurement of L�, the method used to determineL� allows to perform a fundamental test of the thermo-nuclear fusion theory of energy generation in the Sun, aswell as to test the hypothesis that the Sun is in an approxi-mate steady state in what regards the energy produced in itscentral region and the energy emitted from its surface.

The results obtained in the present article underline theimportance of performing high precision measurements of7Be and pp solar neutrino fluxes.

ACKNOWLEDGMENTS

S. C. and S. T. P. would like to thank S. M. Bilenky foruseful discussions. The collaboration of T. Schwetz at theinitial stage of this study is acknowledged with grateful-ness. This work was supported in part by the Italian MIURand INFN under the programs ‘‘Fisica Astroparticellare’’(S. T. P.). The work of S. G. was supported by theAlexander-von-Humboldt-Foundation. The work of A. B.is partly supported by the Partner Group program betweenthe Max Planck Institute for Physics and Tata Institute ofFundamental Research.

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solar model parameters and direct measurements of solar neutrino fluxes

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