THE ASTROPHYSICAL JOURNAL, 565:1366È1375, 2002 February 1 ( The American Astronomical Society. All rights reserved. Printed in U.S.A.

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1 THE ASTROPHYSICAL JOURNAL, 565:366È375, 22 February ( 22. The American Astronomical Society. All rights reserved. Printed in U.S.A. COMPARATIVE ANALYSIS OF GALLEX-GNO SOLAR NEUTRINO DATA AND SOHO/MDI HELIOSEISMOLOGY DATA: FURTHER EVIDENCE FOR ROTATIONAL MODULATION OF THE SOLAR NEUTRINO FLUX PETER A. STURROCK AND MARK A. WEBER Center for Space Science and Astrophysics, Stanford University, Stanford, CA 9435; sturrock=ñare.stanford.edu, weber=plage.stanford.edu Received 2 February 26; accepted 2 July 5 ABSTRACT Recent histogram analysis of GALLEX-GNO and SAGE data indicates that the solar neutrino Ñux, in the energy range of gallium experiments, varies on a timescale of weeks. Such variability could be caused by modulation of the neutrino Ñux by an inhomogeneous magnetic Ðeld in the solar interior if neutrinos have a nonzero magnetic moment. We may then expect the detected neutrino Ñux to oscillate with a frequency set by the synodic rotation frequency in the region of the solar interior that contains the magnetic structure. We investigate this possibility by carrying out a comparative analysis of the GALLEX- GNO solar neutrino data and estimates of the solar internal rotation rate derived from the MDI helioseismology experiment on the SOHO spacecraft. We Ðnd that while the Lomb-Scargle spectrum does not show a signiðcant peak in the band appropriate to the radiative zone, it does show two closely spaced peaks in the band appropriate to the convection zone. In order to explore the relationship of these features to the SunÏs internal rotation, we introduce a resonance statistic ÏÏ that is a measure of the degree of resonance ÏÏ of oscillations in the neutrino Ñux and the solar rotation as a function of radius and latitude. A two-dimensional map of the resonance statistic indicates that the modulation is occurring in the convection zone, near the equator. In order to derive a signiðcance estimate for this result, we next evaluate the integral of this statistic over selected equatorial sections corresponding to the convection zone and the radiative zone. This statistic yields strong evidence that modulation is occurring in the convection zone and no evidence that modulation is occurring in the radiative zone. Subject headings: neutrinos È Sun: interior È Sun: particle emission È Sun: rotation. INTRODUCTION There has for some time been great interest in the possibility that the solar neutrino Ñux is variable. Sakurai (979, 98) suggested that the neutrino Ñux may vary with an approximate 2 yr period, but most early tests focused on the possibility that the Ñux varies with the solar cycle. This period ( yr) is so long that power spectrum analysis is not feasible. Hence, the preferred test for variability was an indirect one, namely, the search for a correlation between the solar neutrino Ñux and an index of solar variability such as the Wolf sunspot number (Bahcall, Field, & Press 987; Bahcall & Press 99; Bieber et al. 99; Dorman & Wolfendale 99; Wilson 987), the surface magnetic Ðeld strength (Massetti & Storini 993; Oakley et al. 994), the intensity of the green line corona (Massetti & Storini 996), or the solar wind Ñux (McNutt 995). Walther (997) has criticized such claims, speciðcally the claim that the neutrino Ñux is anticorrelated with the sunspot number, on the grounds that the authors used smoothed data and then applied tests that are appropriate only if data points are independent. Snodgrass & Oakley (999) have in turn criticized WaltherÏs claim, and Walther (999b) has responded to that criticism (see also Walther 999a). In recent articles, Boger, Hahn, & Cumming (2) and Wilson (2) also conclude that there is no statistically signiðcant association between Homestake data and the usual solar activity indices. Homestake data (Cleveland et al. 998) comprise 8 runs (for the time period 97È994). However, the GALLEX (Anselmann et al. 993, 995; Hampel et al. 996, 997) and GNO (Altmann et al. 2) consortia, working with the same gallium experiment, have now acquired 84 runs (for the period 99È2). Since the half-life of the 366 gallium capture product 7Ge (.43 days) is appreciably shorter than that of the chlorine capture product 37Ar (35. days), the former is better suited to a search for variations on a timescale of weeks. It is also important to note that, if one of two signals is known to be periodic, search for the same periodicity in the other signal may be a more sensitive discriminator of correlation than a simple correlation test. For instance, a scatter diagram of daily rainfall measurements for Austria and for Arizona would not yield an impressive case for an association, but spectrum analysis would show that each has an unmistakable annual variation. It is possible to take advantage of a known periodicity to pull a signal out of the noise.ïï A radio receiver is designed to take full advantage of the fact that each transmission has a well-deðned carrier frequency. Although the overall duration of solar neutrino data (24 yr for the Homestake experiment) is not long enough to permit a deðnitive spectrum analysis for oscillations as slow as the solar cycle, the duration is adequate for a search for oscillations of shorter period. Haubold & Gerth (99) have used spectrum analysis to search for evidence of oscillations with period of order yr. More recently, Haubold (997) has used wavelet analysis for this purpose. It is well known that indices such as sunspot number, surface magnetic Ðeld strength, coronal brightness, etc., are strongly inñuenced by solar rotation. They display oscillations with a basic synodic period of about 27 days and harmonics of this frequency. These oscillations arise from the fact that the solar magnetic Ðeld is highly inhomogeneous and that some components of this Ðeld, which may have their origin deep in the convection zone, last for several or many solar rotations. (Neugebauer et al. 2

2 ANALYSIS OF GALLEX-GNO AND SOHO/MDI DATA 367 have recently presented evidence of a rotational pattern in the solar wind that lasts for more than three solar cycles.) The solar rotation period is short enough, compared with the duration of neutrino experiments, that it is quite reasonable to search for the inñuence of rotation by means of time series spectrum analysis. According to standard physics,ïï one would not expect the solar neutrino Ñux to vary. However, it has been realized for some time that neutrino processes may be governed by nonstandard physics ÏÏ (see, e.g., Ra elt 996). The currently favored candidate for the solution of the solar neutrino deðcit (see, e.g., Bahcall, Krastev, & Smirnov 998) is the MSW e ect (Mikhevev & Smirnov 986a, 986b, 985c; Wolfenstein 978, 979), whereby electron neutrinos may be converted into either muon or tau neutrinos as they propagate through matter in the solar interior. The MSW e ect involves a nonzero neutrino mass but no nonzero magnetic moment. The possibility that neutrino magnetic moment may have something to do with the neutrino deðcit was Ðrst advanced by Cisneros (97), who considered propagation through magnetic Ðeld in the core of the Sun. This leads to spin precession (see also Fujikawa & Schrock 98), which converts some of the left-hand electron neutrinos, produced by nuclear reactions in the core, into sterile right-hand neutrinos which are not detectable. At a later date, Voloshin, Vysotskii, & Okun (986a, 986b) and Barbieri & Fiorentini (988) examined the possible variation of the solar neutrino Ñux due to propagation through the solar convection zone, taking account of the e ect of matter as well as magnetic Ðeld. They found that matter tends to suppress spin precession, although spin precession is still possible if the product of the magnetic moment times the magnetic Ðeld strength is large enough. This is now known as the VVO ÏÏ e ect. Schechter & Valle (98) considered the e ect of a possible ÑavorÈo -diagonal (transition) magnetic moment and found that this could lead to the simultaneous precession of both spin and Ñavor ( spin Ñavor precession ÏÏ or SFP ÏÏ) if neutrinos propagate through a magnetic Ðeld. Akhmedov (988a, 988b) and Lim & Marciano (988) analyzed the propagation of neutrinos through matter permeated by magnetic Ðeld, with application to propagation through the solar convection zone. They found that, for a given neutrino energy, there is a certain density at which a resonant process occurs, enhancing SFP. This is now known as resonant spin Ñavor precession ÏÏ (RSFP). The implications of RSFP for solar neutrinos have been reviewed more recently by Akhmedov (997) and Pulido & Akhmedov (2). Much of the above analysis was stimulated by the possibility that the solar neutrino Ñux may vary in the course of the solar cycle, although Voloshin et al. (986a, 986b) also drew attention to the possibility that the Ñux may vary in the course of a year, as a result of a possible latitude dependence of the internal magnetic Ðeld and the inclination of the SunÏs rotation axis to the ecliptic. These issues remain important, but, for reasons given above, we have become more interested in the possibility that the Ñux may vary with the solar rotation frequency. The solar interior is divided into two di erent regions by the tachocline,ïï located at about r \.7. (Here and elsewhere the radius is normalized to that of the photosphere.) Nuclear burning occurs near the center: about 8% of the burning occurs within r \.5 (see, e.g., Bahcall 989). Between the nuclear-burning core and the tachocline is the radiative zone. According to data derived from helioseismology (Schou et al. 998), the core and radiative zone are in substantially rigid rotation with a sidereal rotation rate in the range 3.75 ^.25 yr~. We Ðnd it convenient to measure frequencies in cycles per year, since this leads to a simple relationship between the sidereal and synodic values: l(synodic) \ l(sidereal) [. () One may convert from these units to degrees per day or nhz as follows: l(yr~) \.98526l(degrees per day), l(yr~) \ l(nHz). (2) At the tachocline, the sidereal rotation rate jumps from about 3.75 yr~ at r \.66 to about 4.53 yr~ at r \.74. Above the tachocline, in the convection zone which extends to the photosphere, the rotation rate varies markedly with radius and latitude, from a minimum of about 8.68 yr~ at r \ at the poles to a maximum of about 4.84 yr~ at r \.93 at the equator. Early in our research, these considerations led us to focus our attention on the radiative zone since this is larger than the convection zone, contains gas of higher pressure that could contain stronger magnetic Ðeld, and is believed to rotate e ectively like a rigid body. In searching for evidence for rotational modulation of the solar neutrino Ñux by examining radiochemical data, one is faced with the serious difficulty that the timing of the data acquisition is comparable with the period of oscillations one is looking for. For the Homestake experiment, the run duration was in the range 5È3 weeks, with a median of weeks. For the GALLEX-GNO experiment, the run duration was in the range 2È6 weeks, with a median of 4 weeks. It has nevertheless proved possible to search for oscillations in the frequency band of the rotational frequencies. We analyzed the Homestake data using a maximum likelihood procedure (Sturrock, Walther, & Wheatland 997) and found evidence for an oscillation at 2.85 yr~, comparable with the synodic rotation frequency of the radiative zone (2.75 ^.25 yr~). We have also found some evidence that the solar neutrino Ñux, as measured by the Homestake experiment, exhibits modulation related to heliographic latitude (Sturrock, Walther, & Wheatland 998). However, since there has been so much controversy concerning the analysis of Homestake data, and since (as pointed out earlier) data derived from gallium experiments are better suited to a search for oscillations with a period of weeks, we have turned to the study of GALLEX-GNO data. An early spectrum analysis of GALLEX data alone (Sturrock et al. 999), based on a least-squares procedure (Knight, Schatten, & Sturrock 979), revealed some evidence of a periodicity that may be related to solar rotation. As an independent investigation of the question of variability, we have recently studied the histogram of GALLEX- GNO data (Sturrock & Scargle 2). One expects the histogram of a stationary time series to be unimodal. We Ðnd that the histogram formed from either GALLEX-GNO data or SAGE data (or GALLEX, GNO, and SAGE data combined) is bimodal, indicating that the Ñux measured by gallium experiments varies and that it does so on a timescale of weeks.

3 368 STURROCK & WEBER Vol. 565 In 2 we determine the spectrum of the combined GALLEX-GNO data using the Lomb-Scargle technique, which, as we explain, is superior to the technique used in our analysis of GALLEX data (Sturrock et al. 999). In 3 we compare the spectrum with information concerning the SunÏs internal rotation, with a view toward determining the region in which modulation of the neutrino Ñux takes place. The results of these investigations are discussed in SPECTRUM ANALYSIS In this section we analyze the combined GALLEX-GNO data by a method developed by Lomb (976) and Scargle (982) for the analysis of irregular time series. Such a procedure is necessary, since the timing of radiochemical experiments is not regular. Bretthorst (988) has shown by Bayesian analysis that the Lomb-Scargle technique is the optimum procedure for Ðnding a sinusoid in irregularly spaced data. He has further shown that irregular timing, far from being a drawback, is in fact a great asset in that it is substantially una ected by the window function ÏÏ representing the sampling times (and therefore much less subject to aliasing), and it allows one to extend the spectrum to higher frequencies than would be possible for regular spacing. The timing and Ñux measurements of the GALLEX- GNO data are presented in Table. Most of the runs last about 4 weeks, but, on the other hand, the half-life of the capture product 7Ge is only.43 days (Bahcall 989), so that the survival probability of a germanium nucleus produced on the Ðrst day of a run is of order 2%. Hence, the measurement of each run is weighted toward captures late in the run, so that we can obtain a satisfactory estimate of the spectrum of the neutrino Ñux by assigning the measurement made during each run to the end time of that run. TABLE GALLEX-GNO DATA EXPOSURE TIMES EXPOSURE TIMES EXPERIMENT RUN Start End FLUX ESTIMATE EXPERIMENT RUN Start End FLUX ESTIMATE GALLEX GALLEX GALLEX GALLEX [22 GALLEX GALLEX GALLEX GALLEX GALLEX [2 GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX [78 GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GALLEX GNO GALLEX GNO GALLEX GNO GALLEX GNO GALLEX GNO [46 GALLEX GNO GALLEX GNO GALLEX GNO [5 GALLEX [9 GNO GALLEX GNO GALLEX GNO GALLEX GNO GALLEX GNO GALLEX GNO GALLEX GNO GALLEX GNO GALLEX GNO GALLEX GNO GALLEX [89 GNO

4 No. 2, 22 ANALYSIS OF GALLEX-GNO AND SOHO/MDI DATA 369 The Ñux estimates are shown as a function of end time in Figure. Later in this section we discuss another choice and show that the spectrum changes very little. Figure may look like a scatter diagram, but we will show later in this section that appearances can be deceptive and that what looks like a scatter diagram may in fact hide a strong oscillation. We note that some of the Ñux estimates (e.g., those for GALLEX runs 3, 62, and 64) are quite high compared with the overall distribution of estimates. These are referred to later as outliers,ïï and we shall investigate the implications of moderating the inñuence of these runs. We also note that some of the estimates are negative; such negative estimates can be a quite appropriate result of the routine operation of the data reduction procedure, since this necessarily involves statistical assessments. We denote by t the time of each data point (here chosen to be the end time i of each run), where i( \,..., N) enumerates the runs. We denote by g the Ñux measurements and deðne the mean h and the standard i deviation p by the usual expressions: h \ N ; N gi (3) and p2\ N [ ; N (gi [ h)2. (4) Following Lomb (976) and Scargle (982), the power spectrum is given, as a function of angular frequency u,by S(u) \ AM; (g [ h) cos [u(t [ q)]n2 i i 2p2 ; cos2 [u(t [ q)] i B ] M; (g i [ h) sin [u(t i [ q)]n2, (5) ; sin2 [u(t [ q)] i where q is deðned by the relation tan (2uq) \ ; sin (2ut i ) ; cos (2ut ). (6) i Since we are here concerned with possible rotational modulation of the neutrino Ñux, we show in Figure 2 the result of Flux End Time FIG..ÈPlot of Ñux estimates vs. end time for GALLEX-GNO experiments. Power FIG. 2.ÈLomb-Scargle spectrum of the GALLEX-GNO solar neutrino data for the frequency range È2 yr~. our Lomb-Scargle analysis of the GALLEX-GNO data only for frequencies in the range È2 yr~. (There is an interesting peak at 6.8 yr~, but analysis of this peak is complicated by the fact that this does not correspond to any known solar oscillation.) As a guideline to the signiðcance of the peaks in the spectrum, we may note that the probability of Ðnding a peak of power S or more is e~s (Scargle 982). Hence, for a speciðed frequency, a peak with S [ 3. is signiðcant at the 5% level, and a peak with S [ 4.6 is signiðcant at the % level. However, we will need to make signiðcance estimates not for a speciðed frequency but for a range of possible frequencies (see 3). If modulation were occurring in the radiative zone, we might expect a fairly well deðned peak, since that region is believed to be in rigid rotation. According to helioseismology data (Schou et al. 998), the rotation rate of the radiative zone is in the range 2.4È2.9 yr~. (Further information is given later; see Fig..) We see from Figure 2 that there is no signiðcant peak in this range. There is an interesting peak with S \ 4.5 at l \ 3.4 yr~ which, if real, may be related to the tachocline. The rotation rate of the convection zone varies with both radius and latitude. However, neutrinos detected on Earth have traversed the solar interior within a small fraction of a solar radius of the equatorial section, for which the rotation rate is in the range 4.3È4.8 yr~ sidereal or 3.3È3.8 yr~ synodic. This band represents a combination of uncertainty in the measurements and a real variation of the rotation rate with radius. We see, in Figure 2, that there is a prominent double peak in this band, one at l \ 3.59 with power S \ 4.82 and the other at l \ 3.67 with power S \ 4.4. We now demonstrate that the Lomb-Scargle procedure can indeed detect an oscillation in a time series that looks like a scatter diagram, even if the period of oscillation is less than the mean interval between samples. From the end times, we form u \ sin (2nl t), for l \ 5, and then reorder the end times by sorting the corresponding u-values. We also reorder the Ñux estimates by sorting the list. We then form a simulated data set by combining the reordered times and the reordered Ñux values. The time Ñux diagram for the reorganized data is shown in Figure 3. The authors

5 37 STURROCK & WEBER Vol Flux 5 Power End Time FIG. 3.ÈPlot of Ñux vs. time for GALLEX-GNO data, reordered as explained in text. can see no obvious di erence between this Ðgure and Figure. On the other hand, the spectrum of the reorganized data is that shown in Figure 4. It reveals a very strong periodicity at l \ 5. The minor peak at l \ is a weak alias of the imposed oscillation, arising from a strong periodicity in the window function at about l \ 3. (This peak is located at w 2l [ l. There are similar weak peaks [not shown] at w l [ l, l ] l, etc.) It is a strong point of the Lomb- w w Scargle method that we see no peak in the spectrum at the frequency l. We may note that the period of the imposed w (and detected) oscillation is 3.5 weeks, less than the mean time between runs (4 weeks). This demonstrates that it is indeed possible to detect oscillations of frequency higher than the mean sampling frequency, provided that the data acquisition is somewhat irregular. In forming spectra, we have e ectively assigned each Ñux measurement to the end time of the run (a point we shall discuss further in Appendix A). However, we have also formed the spectrum by assigning Ñux measurements to the mean time of each run. The resulting spectrum is shown in Figure 5. As we see, the spectrum is not particularly sensi FIG. 5.ÈLomb-Scargle spectrum of the GALLEX-GNO Ñux estimates, here referred to the mean of the start time and end time of each run. tive to the precise choice of time chosen to represent each run. Finally, we consider another modiðcation of our spectrum analysis. The Lomb-Scargle analysis is equivalent to forming a least-squares Ðt of the data (suitably normalized) to a sine wave. In forming least-squares Ðts, one may be concerned about overemphasizing a few data points with extreme values of the dependent variable. We can check to see whether this consideration is signiðcant for our problem by adopting a procedure that de-emphasizes the outliers.ïï We have, therefore, as an experiment, formed the following variable: B y \ arctan Ag [ h. (7) p When we form the Lomb-Scargle spectrum of this quantity, we obtain the spectrum shown in Figure 6. We see that the height of the peak at 3.6 is increased rather than decreased, Power FIG. 4.ÈLomb-Scargle spectrum of reordered data Power FIG. 6.ÈLomb-Scargle spectrum of the GALLEX-GNO Ñux estimates, modiðed by an arctan operation to moderate the e ect of outliers.

6 No. 2, 22 ANALYSIS OF GALLEX-GNO AND SOHO/MDI DATA 37 Power FIG. 7.ÈLomb-Scargle spectrum of the GALLEX-GNO Ñux estimates, modiðed by a square root operation to moderate the e ect of outliers. so it would appear that this peak is not due to the undue inñuence of a few outliers. As another experiment, we have carried out the following procedure to moderate the inñuence of outliers: y \ g o g (8) o g o The resulting Lomb-Scargle spectrum is shown in Figure 7. Once again, we see that the peak at 3.6 is enhanced as a result of our moderating the inñuence of outliers. Thus, the overall result of our analyses is that there exists a signiðcant peak in the power spectrum near l \ RESONANCE STATISTIC Our purpose here is to examine the relationship between the variability of the neutrino Ñux and internal rotation. It is convenient to begin by introducing a visual display of this relationship. Schou et al. (998) have tabulated the rotation rate l (r, j) and the error estimate p (r, j) for values of h h the radius r and 25 values of the latitude j, deriving their estimates from the MDI helioseismology experiment on the SOHO spacecraft. These data determine a probability distribution function (PDF) of the rotation frequency for each pair of values r, j: C D P(l o r, j) \ (2n)~@2p~ exp [ (l [ l h )2. (9) h 2p2 h Since we wish to compare the rotation frequencies with the neutrino Ñux variability as measured on Earth, it is appropriate to inspect the synodic rotation rates rather than the sidereal rates. We can deðne a measure of the degree of resonance of the neutrino Ñux with internal rotation by forming the resonance statistic,ïï P lb $(r, j) \ dl S(l)P(l o r, j), () la where S is the spectrum computed in 2 and displayed in Figure 2. In this integral, we need to select limits of integration that are sufficiently wide to cover all signiðcant contributions from the PDF. We have adopted l \ and l \ 2, but a much smaller range would have been a satisfactory. b A color map of $ as a function of radius and latitude is presented in Figure 8. This Ðgure is essentially a mapping of the power spectrum of the solar neutrino time series onto the solar interior. Where the map is colored yellow or red, $ is large compared with its average value of unity. This denotes a resonance ÏÏ between the neutrino Ñux and the local solar rotation in the sense that the two oscillations ÏÏ (Ñux variability and rotation) have the same frequency. When the color is blue, there is no such resonance. If two regions of the solar interior have the same rotation rate, they will tend to have the same value of the statistic $. Another way of looking at this Ðgure, which is perhaps physically more signiðcant, is the following. Let us assume that there is a well-deðned oscillation in the neutrino Ñux, and let us assume that this oscillation is due to modulation of the Ñux by a structure (such as a magnetic structure) in the solar interior. Then we can attempt to locate that structure by Ðnding the location (or locations) where the rotation rate has just the correct value to account for the dominant oscillation of the neutrino Ñux. The map shown in Figure 8 may now be viewed as a PDF for the location of the modulating structure. We see from the map that, if the neutrino Ñux is variable, and if the variability is due to modulation by some internal solar structure, then that structure is probably in the lower part of the convection zone and (as we would expect) near the equator. The map o ers no evidence that such modulation is occurring in the core or in the radiative zone. It is to be expected that the modulation will occur near the equator, since neutrinos are produced within a radius of.5 of the center of the Sun (see, e.g., Bahcall 989). Even allowing for the tilt of the SunÏs axis with respect to the ecliptic, we Ðnd that most of the neutrinos detected by GALLEX-GNO have passed well within 6 of the solar equator in penetrating the surface of the Sun. Over the maximum range of 6, the rotation rate varies by less than % within the convection zone and less than that in the radiative zone. We now seek a statistical evaluation of the signiðcance of the modulation of the neutrino Ñux by an internal structure or structures in either the radiative zone or the convection zone. We begin with the convection zone, and, for reasons just stated, we now restrict our attention to possible modulation near the equatorial section of that zone. We introduce a PDF characterizing the distribution of rotation rates over the entire equatorial section of the convection zone (CZ). This is formed from the PDF of equation (9) as follows: P P(l o CZ) \ ru dr P(l o r, ). () r [ r u l rl We may verify that P lb dl P(l o CZ) \. (2) la We denote by r and r the lower and upper limits of the convection zone. l The former u is set by the upper limit of the tachocline, so r \.74, and the latter by the photosphere, l so r \. Over this range the minimum sidereal frequency u

7 372 STURROCK & WEBER Vol. 565 FIG. 8.ÈMap of the resonance statistic $, deðned by eq. (), upon a meridional section of the solar interior. Red or yellow denotes a resonance ÏÏ between the neutrino Ñux and the local solar rotation, indicating that the two oscillations (Ñux variability and rotation) have the same frequency. is 4.27 yr~, and the maximum is 4.83 yr~, a range of almost 4%. The resulting PDF is shown in Figure 9. We now form the integral resonance statistic ÏÏ P lb! \ dl S(l)P(l o CZ), (3) la where S(l) is the power spectrum computed by the Lomb- Scargle process in 2. This procedure is known in Bayesian parlance (see, e.g., Bretthorst 988) as an integration over a nuisance parameter.ïï We denote by! the actual value of! derived from the data, and we Ðnd that d! \ We now need to assess the signiðcance of this value. d We adopt the method introduced by Bahcall & Press (99) in their study of the apparent anticorrelation between Homestake measurements and the sunspot number. The procedure is to shuffle ÏÏ the data many times, reassigning the Ñux measurements among runs. We have carried out, shuffles of the data, and the results are shown in Figure. We Ðnd that of the, random simulations, only 6 have values![!. From this analysis, we infer that there is a probability d of approximately.6% of obtaining the actual value of! by chance. We conclude that there is a resonance between the modulation of the neutrino Ñux, as measured by the GALLEX- GNO experiment, and the rotation of the equatorial section of the solar convection zone, as measured by the MDI experiment, and that this correspondence is signiðcant at the.2% level. PDF FIG. 9.ÈThis probability distribution function is representative of the range of (synodic) rotation frequencies in the convection zone.

8 No. 2, 22 ANALYSIS OF GALLEX-GNO AND SOHO/MDI DATA 373 Fraction 2 Fraction PDF Statistic FIG..ÈFor analysis of the convection zone, the ordinate denotes the fraction of, simulations that have values of the statistic!, deðned by eq. (3), larger than the value indicated by the abscissa. The vertical line denotes the actual value of the statistic,!, derived from the data. Less than.2% of the simulations have![!. d d We have carried out the same calculations for the radiative zone (RZ). We use equation (9) to calculate P(l o RZ), except that we now adopt r \.3 and r \.66. The l u resulting PDF is shown in Figure. When we calculate the resonance statistic!, using equation (3), we obtain the value! \.4. We have carried out, shuffles of the d data, and the results are shown in Figure 2. We see that, of the, random simulations, almost 3 have values![!. From this analysis, we infer that there is no signið- d cant resonance between oscillations of the solar neutrino Ñux and rotation of the radiative zone DISCUSSION The purpose of this investigation has been to seek further evidence concerning possible rotational modulation of the solar neutrino Ñux. Earlier evidence has come from spectrum analysis of Homestake data (Sturrock et al. 997, FIG..ÈThis probability distribution function is representative of the range of (synodic) rotation frequencies in the radiative zone Statistic FIG. 2.ÈFor analysis of the radiative zone, the ordinate denotes the fraction of, simulations that have values of the statistic!, deðned by eq. (3), larger than the value indicated by the abscissa. The vertical line denotes the actual value of the statistic,!, derived from the data. By contrast with the situation shown in Fig., almost d 3% of the simulations have![!. d 998) and from histogram analysis of data from the gallium experiments (Sturrock & Scargle 2). We have found that a Lomb-Scargle spectrum of the GALLEX-GNO data yields a double peak in the frequency range corresponding to the solar internal rotation at low latitudes. Bahcall (999) has pointed out that rotational modulation would be very surprising. One would not expect the SunÏs magnetic Ðeld to have a coherent e ect on the neutrino Ñux, since the magnetic Ðeld at the visible surface of the Sun is spotty and variable, so that the e ect should be washed out. However, modulation of the neutrino Ñux would be caused by the deep, strong Ðeld, not by the shallow, weak Ðeld. At this time, we have no direct information concerning the strength, structure, or stability of the SunÏs internal Ðeld. According to Fan, Fisher, & McClymont (994), a free Ñux tube near the base of the convection zone will rise to the surface in a few weeks only if its Ðeld strength exceeds 5 G. It is also possible that part of the magnetic Ñux is anchored ÏÏ in the radiative zone, in which case it may retain its structure for years or more. Another reason that the result would be surprising is that it points toward the VVO, SFP, and RSFP processes (discussed in ), each of which requires a magnetic moment signiðcantly greater than that attributed to the neutrino in standard particle physics. By forming a statistic representing the degree of correspondence between the neutrino Ñux oscillation and the internal rotation as a function of radius and latitude, we have found evidence that the rotational modulation occurs deep in the convection zone at low latitudes. By integrating this statistic over radius, we have determined that the correspondence between the neutrino Ñux oscillation and rotation of the convection zone is signiðcant at the.2% level. On the other hand, there is no statistically signiðcant evidence for a correspondence between the neutrino Ñux oscillation, as determined by GALLEX-GNO, and rotation of the radiative zone.

9 374 STURROCK & WEBER Vol. 565 The above result is consistent with the common view that the SunÏs magnetic Ðeld is generated by a dynamo process and that this process occurs near the tachocline and in the convection zone. According to this view, the magnetic Ðeld in the radiative zone is probably weak compared with that in the convection zone. Indeed, there is observational evidence supporting the conjecture that there are long-lasting magnetic structures deep inside the Sun. Bai (988) and others have presented evidence that solar activity exhibits longitudinal structure that persists for more than a solar cycle. We referred, in, to the recent article by Neugebauer et al. (2), presenting evidence that some structure within the Sun, which inñuences the structure of the solar wind, persists for more than three solar cycles. It is also well known that the corona exhibits rigid rotation, rather than di erential rotation (see, e.g., Hoeksema & Scherrer 987). StenÑo (977) and Zirker (977) have proposed that this can best be understood as the e ect of magnetic structures deep in the convection zone. It will be interesting to examine whether these longlived structures in the corona (i.e., coronal holes) and solar wind exhibit rotation rates consistent with the rotation rates characteristic of neutrino modulation. The rotation rate that one associates with low-latitude solar activity is the Carrington frequency (4.39 yr~ sidereal or 3.39 yr~ synodic), not too far from the value (3.6 yr~ synodic) found here. However, some long-lived features such as polar coronal holes exhibit a slower rotation, nearer 2.9 yr~ synodic (Timothy, Krieger, & Vaiana 975; Bohlin 977). It is interesting that our earlier analysis of Homestake data (Sturrock et al. 997) pointed to modulation at this lower frequency. We plan to reexamine the Homestake data, using the Lomb-Scargle procedure of spectrum analysis, and to compare the properties of Homestake measurements with those of the GALLEX-GNO measurements. As we mentioned in, it seems most likely that rotational modulation, if real, is due to the RSFP e ect. It is signiðcant that this process occurs in a resonant layer where the density has the value o \ 3.*m2E~, (4) where E is the neutrino energy in ev and *m2 is the di erence between m2 for electron neutrinos and for neutrinos of a di erent Ñavor (see, e.g., Pulido & Akhmedov 2). It is notable that, of experiments now in operation, the gallium experiments detect the lowest energy neutrinos (down to kev). Hence, the resonant layer relevant to the gallium experiment will be deeper than that for chlorine experiments (of order MeV) and much deeper than that for Super-Kamiokande, SNO, etc. (of order MeV). It follows that if the modulation of the neutrinos detected by GALLEX-GNO occurs at r \.8, neutrinos detected by Homestake will be modulated at about r \.9, and those detected by Super-Kamiokande would be modulated at about r \.95. Hence, the magnetic Ðeld will be weaker in the regions relevant to the Homestake and Super- Kamiokande experiments than in the region relevant to the gallium experiments. It may also prove signiðcant that about 26% of the neutrino Ñux detected by the gallium experiments comes from the 7Be lines, whereas only 4% of the Ñux detected by Homestake comes from lines (Bahcall 989), since modulation of the continuum contributions may tend to be washed out by phase mixing. For these reasons, it may turn out that rotational modulation is more readily detectable in gallium experiments than in other experiments. Rotational modulation, if real, should lead to new information concerning neutrinos and the solar interior. This article is based on work supported in part by NASA and NSF. It is a pleasure to acknowledge the interest of and helpful suggestions from Evgeni Akhmedov, Blas Cabrera, Sasha Kosovichev, John Leibacher, Eugene Parker, Joao Pulido, Je Scargle, and Guenther Walther. We are indebted to P. Anselmann and W. Hampel and their collaborators of the GALLEX consortium, as well as to M. Altmann and his collaborators of the GNO consortium, for generously making their data publicly available. We also extend our thanks to an anonymous referee for critical comments and helpful suggestions. APPENDIX A EFFECTIVE TIMING OF RUNS IN RESPONSE TO SINUSOIDAL MODULATION Allowing for the decay of the capture products, the count at the end of a run is given by If P te C \ dt f (t)e~k(te~t). te~d f \ Aeiut`ia ] cc, (A) (A2) C \ A [ [ e~(i`iu)d]eipte`ia ] cc. i ] iu (A3) The experimenters assign a Ñux value g based on the assumption that the Ñux is constant, i.e., g \ ic( [ e~id)~. We wish to determine an o set time * and a conversion factor K such that Kg agrees with the actual Ñux at time t [ *. On e (A4)

10 No. 2, 22 ANALYSIS OF GALLEX-GNO AND SOHO/MDI DATA 375 combining equations (A2), (A3), and (A4), we Ðnd that this requirement is satisðed if Ai ] iubc [ e~id Keiu* \ i [ e~(i`iu)dd. (A5) We have computed the o set time * and the conversion factor K for all runs, for the frequency of interest, l \ 3.6. We Ðnd that * ranges from a minimum of 5.7 days to a maximum of 6.86 days; the mean is 5.93 days and the standard deviation.66 days. We may also express this as an equivalent phase error brought about by adopting the end time as the reference time of each run. We Ðnd that this phase error ranges from 69 to 92, with a mean of 8 and a standard deviation of 9. Such a small variation in the phase error is unimportant in forming power spectra. We Ðnd that K ranges from a minimum of.47 to a maximum of 3.98; the mean is 3.5 and the standard deviation is.6. This translates into a standard deviation of only 7% of the mean value of K. Such a small variation in the scaling is unimportant. We see that if we are interested only in the spectrum of the neutrino Ñux, we may assign each Ñux measurement to the end time since an almost constant phase discrepancy is unimportant. However, if one wishes to extract from the data the actual amplitude and phase of the neutrino Ñux variation, then it is preferable to adopt a maximum likelihood method, such as that of Sturrock et al. (997), which returns the best estimate of the amplitude. Akhmedov, E. Kh. 988a, Phys. Lett. B, 23, 64 ÈÈÈ. 988b, Soviet J. Nucl. Phys., 48, 382 ÈÈÈ. 997, in Fourth International Solar Neutrino Conference, ed. W. Hampel (Heidelberg: MPI), 388 Altmann, M., et al. 2, Phys. Lett. B, 49, 6 Anselmann, P., et al. 993, Phys. Lett. B, 34, 445 ÈÈÈ. 995, Phys. Lett. B, 357, 237 Bahcall, J. N. 989, Neutrino Astrophysics (Cambridge: Cambridge University Press) ÈÈÈ. 999, Russell Lecture, AAS Meeting, Chicago Bahcall, J. N., Field, G. B., & Press, W. H. 987, ApJ, 32, L69 Bahcall, J. N., Krastev, P. I., & Smirnov, A. Yu. 998, Phys. Rev. D, 58, 966- Bahcall, J. N., & Press, W. H. 99, ApJ, 37, 73 Bai, T. 988, ApJ, 328, 86 Barbieri, R., & Fiorentini, G. 988, Nucl. Phys. B, 34, 99 Bieber, J. W., Seckel, D., Stanev, T., & Steigman, G. 99, Nature, 348, 47 Boger, J., Hahn, R. L., & Cumming, J. B. 2, ApJ, 537, 8 Bohlin, J. D. 977, Sol. Phys., 5, 377 Bretthorst, G. L. 988, Bayesian Spectrum Analysis and Parameter Estimation, Vol. 48 of Lecture Notes in Statistics, ed. J. Berger et al. (Berlin: Springer) Cisneros, A. 97, Ap&SS,, 87 Cleveland, B. T., et al. 998, ApJ, 496, 55 Dorman, L. I., & Wolfendale, A. W. 99, J. Phys. G., 7, 789 Fan, Y., Fisher, G. H., & McClymont, A. N. 994, ApJ, 436, 97 Fujikawa, K., & Schrock, R. E. 98, Phys. Rev. Lett., 45, 963 Hampel, W., et al. 996, Phys. Lett. B, 388, 384 ÈÈÈ. 997, Phys. Lett. B, 447, 27 Haubold, H. J. 997, Ap&SS, 258, 2 Haubold, H. J., & Gerth, E. 99, Sol. Phys., 27, 347 Hoeksema, J. T., & Scherrer, P. H. 987, ApJ, 38, 428 Knight, J. W., Schatten, K. H., & Sturrock, P. A. 979, ApJ, 227, L53 Lim, C.-S., & Marciano, W. J. 988, Phys. Rev. D, 37, 368 Lomb, N. 976, Ap&SS, 39, 447 Massetti, S., & Storini, M. 993, Sol. Phys., 48, 73 ÈÈÈ. 996, ApJ, 472, 827 REFERENCES McNutt, R. L., Jr. 995, Science, 27, 635 Mikhevev, S. P., & Smirnov, A. Yu. 986a, Nuovo Cimento, 9, 7 ÈÈÈ. 986b, Soviet J. Phys., 42, 93 ÈÈÈ. 986c, Soviet Phys.ÈJETP, 64, 4 Neugebauer, M., Smith, E. J., Ruzmaikin, A., Feynman, J., & Vaughan, A. H. 2, J. Geophys. Res., 5, 235 Oakley, D. S., Snodgrass, H. B., Ulrich, R. K., & Vandekop, T. L. 994, ApJ, 437, L63 Pulido, J., & Akhmedov, E. Kh. 2, Astropart. Phys., 3, 227 Ra elt, G. R. 996, Stars as Laboratories for Fundamental Physics (Chicago: Univ. Chicago Press) Sakurai, K. 979, Nature, 278, 46 ÈÈÈ. 98, Sol. Phys., 74, 35 Scargle, J. D. 982, ApJ, 263, 835 Schechter, J., & Valle, J. W. F. 98, Phys. Rev. D, 24, 883 (erratum 25, 283 [982]) Schou, J., et al. 998, ApJ, 55, 39 Snodgrass, H. B., & Oakley, D. S. 999, Phys. Rev. Lett., 83, 894 StenÑo, J. O. 977, A&A, 6, 797 Sturrock P. A., & Scargle, J. N. 2, ApJ, 55, L Sturrock, P. A., Scargle, J. D., Walther, G., & Wheatland, M. S. 999, ApJ, 523, L77 Sturrock, P. A., Walther, G., & Wheatland, M. S. 997, ApJ, 49, 49 ÈÈÈ. 998, ApJ, 57, 978 Timothy, A. F., Krieger, A. S., & Vaiana, G. S. 975, Sol. Phys., 42, 35 Voloshin, M. B., Vysotskii, M. I., & Okun, L. B. 986a, Soviet J. Nucl. Phys., 44, 44 ÈÈÈ. 986b, Soviet Phys.ÈJETP, 64, 446 Walther, G. 997, Phys. Rev. Lett., 79, 4522 ÈÈÈ. 999a, ApJ, 53, 99 ÈÈÈ. 999b, Phys. Rev. Lett., 83, 895 Wilson, R. M. 987, Sol. Phys., 2, ÈÈÈ. 2, ApJ, 545, 532 Wolfenstein, L. 978, Phys. Rev. D, 7, 2369 ÈÈÈ. 979, Phys. Rev. D, 2, 2634 Zirker, J. B. 977, Rev. Geophys. Space Phys., 5, 257