Solar Physics

, Volume 291, Issue 12, pp 3467–3484

Comparative Analyses of Brookhaven National Laboratory Nuclear Decay Measurements and Super-Kamiokande Solar Neutrino Measurements: Neutrinos and Neutrino-Induced Beta-Decays as Probes of the Deep Solar Interior

Article

DOI: 10.1007/s11207-016-1008-9

Cite this article as:
Sturrock, P.A., Fischbach, E. & Scargle, J.D. Sol Phys (2016) 291: 3467. doi:10.1007/s11207-016-1008-9

Abstract

An experiment carried out at the Brookhaven National Laboratory over a period of almost 8 years acquired 364 measurements of the beta-decay rates of a sample of \({}^{32}\mbox{Si}\) and, for comparison, of a sample of \({}^{36}\mbox{Cl}\). The experimenters reported finding “small periodic annual deviations of the data points from an exponential decay … of uncertain origin”. We find that power-spectrum and spectrogram analyses of these datasets show evidence not only of the annual oscillations, but also of transient oscillations with frequencies near 11 year−1 and 12.5 year−1. Similar analyses of 358 measurements of the solar neutrino flux acquired by the Super-Kamiokande neutrino observatory over a period of about 5 years yield evidence of an oscillation near 12.5 year−1 and another near 9.5 year−1. An oscillation near 12.5 year−1 is compatible with the influence of rotation of the radiative zone. We suggest that an oscillation near 9.5 year−1 may be indicative of rotation of the solar core, and that an oscillation near 11 year−1 may have its origin in a tachocline between the core and the radiative zone. Modulation of the solar neutrino flux may be attributed to an influence of the Sun’s internal magnetic field by the Resonant Spin Flavor Precession (RSFP) mechanism, suggesting that neutrinos and neutrino-induced beta decays can provide information about the deep solar interior.

Keywords

Nuclear physics Solar structure 

1 Introduction and Context

We now have extensive information concerning the structure and variation of the solar photosphere, chromosphere, corona, and solar wind. We have obtained from helioseismology detailed information concerning the time-averaged structure of the convection zone, the radiative zone, and the tachocline that separates the convection and radiative zones (see, for instance, Turck-Chièze and Couvidat, 2011). However, by comparison, we have little information concerning the dynamics of the solar core. It has been recognized for some time that one way to probe the deepest layers of the Sun is by means of neutrinos (Bahcall, 1989). Several neutrino observatories have been constructed and put into operation, beginning with the Homestake radiochemical experiment (Davis, 1987), and including notably the Super-Kamiokande (Fukuda, 2003) and Sudbury Neutrino Observatory (SNO; Boger et al., 2000) Cerenkov experiments. The results of these neutrino experiments have confirmed the basic model of nuclear processes.

Almost all of the data analyses carried out by the neutrino consortia have been based on the assumption that the neutrino composition and flux are essentially constant. An early exception was the research of Davis and his collaborators concerning possible variability of measurements made by the Homestake experiment (Davis, 1996). The other neutrino consortia have carried out very limited inquiries concerning variability. However, a number of investigators who are not associated with the consortia have examined neutrino measurements and claimed to find evidence of variability (among them Sakurai, 1981; Bieber et al., 1990; Haubold and Gerth, 1990; Grandpierre, 1996).

An important development in recent years has been the study of evidence that beta decays are variable, and that the Sun could be in part responsible for the variation (Falkenberg, 2001). The currently favored mechanism is that (as suggested by Falkenberg) neutrinos can stimulate beta decays (Fischbach et al., 2009). The cross section for this process appears to be enormously larger than the cross sections for the processes responsible for neutrino detections in the current neutrino detection experiments (Sturrock and Fischbach, 2015). This development opens the possibility of a new generation of neutrino observatories that are far more sensitive and far less expensive than current observatories.

One of the early experiments to yield evidence of variability of the beta-decay process was one installed at the Brookhaven National Laboratory (BNL). It was designed to determine the half-life of 32Si (Alburger, Harbottle, and Norton, 1986). For almost eight years, the experimenters tracked the (beta) decay rate of a specimen of 32Si and, for comparison, the decay rate of a specimen of the long-lived nuclide 36Cl (half-life of approximately 300,000 years). Measurements beginning in February 1982 were made at approximately four-week intervals, each consisting of a total of 40 hours of counting for each specimen. For each nuclide, the daily decay rate was formed from an average of 20 separate measurements in the course of the day, the 36Cl and 32Si measurements being interwoven. The authors reported that “small periodic annual deviations of the data points from an exponential decay were observed, but were of uncertain origin”. These oscillations were found to have an amplitude (depth of modulation) of \(6 \times 10^{-5}\), with a maximum on February 9.

The BNL experiment attracted the attention of Jenkins et al. (2009) and Fischbach et al. (2009), who speculated that the annual oscillation might be related to the annual variation of the Earth–Sun distance (as had been proposed by Falkenberg, 2001). Examining the ratio of the 32Si and 36Cl decay rates, Jenkins et al. confirmed that the maximum occurs in early February, not far from the date (January 3) of closest approach of Earth to the Sun. They noted that long-term measurements of the decay rate of 226Ra, carried out at the Physikalisch Technische Bundesanstalt (Siegert, Schrader, and Schotzig, 1998), appeared to show a similar annual oscillation. Jenkins et al. suggested (as had Falkenberg) that solar neutrinos may be responsible for variations in beta-decay rates. This suggestion was of course viewed with skepticism, since the cross section for collisions between neutrinos and known particles is believed to be of order \(10^{-43}~\mbox{cm}^{2}\) (Bahcall, 1989).

An annual oscillation does not in itself provide strong evidence for beta-decay variability, since it may be caused by some environmental influence on the detection system itself. However, a power-spectrum analysis of BNL data gives evidence also of frequencies in the range 11 – 13 year−1 (periods in the range 25 – 35 days; Javorsek et al., 2010; Sturrock et al., 2010). In his studies of the decay rate of several other nuclides, Parkhomov (2011) has independently identified oscillations of decay data with frequencies in this range.

According to standard theory, neutrinos can travel freely through the Sun (Bahcall, 1989). If there is any non-standard interaction between neutrinos and the particles or field of any region of the solar interior, the neutrino flux as measured at Earth may therefore be influenced by that interaction. One of the known nonstandard processes is that the neutrino flux can be influenced by magnetic field through Resonant Spin-Flavor Precession (RSFP) (Akhmedov, 1988). An internal magnetic field (which may be primordial or produced by a dynamo process) is likely to be non-axially symmetric. Hence the RSFP process must be expected to lead to periodic fluctuations of the solar neutrino flux, the period being the rotation period as seen from Earth.

There are no doubt other processes in the Sun that can influence the neutrino flux. Fluctuations that have their origin in the deep solar interior may arise from stochastic processes such as random fluctuations in nuclear burning, as well as from rotation. However, these two types of fluctuation are in principle distinguishable by a power-spectrum analysis: measurements derived from rotation may be expected to exhibit recognizable high-Q, narrow-band oscillations, whereas stochastic fluctuations are expected to lead to broad-band fluctuations. The bandwidth of rotational modulation depends on the coherence and stability of the rotation and on the duration of the measurements. At the photosphere, the rotation rate is latitude-dependent and extends over a wide frequency range. However, according to helioseismology data, the rotation rate in the radiative zone does not vary significantly with latitude, and does not vary greatly with radius (Couvidat et al., 2003; Schou et al., 2002). The corresponding rotation is expected to lead to oscillations in or near to the frequency range 12 – 14 year−1 as measured on Earth. For the nuclear core, the situation is more complex, and different publications suggest different estimates.

The question of whether some nuclear decay rates are variable remains controversial primarily because the known cross sections for collisions of neutrinos with other particles (electrons, protons, etc.) are far too small to be compatible with the apparent influence of neutrinos on beta decays (Sturrock and Fischbach, 2015). However, Singleton, Inan, and Chiao (2015) have suggested that beta-decay variability may be related to a possible neutrino-induced decoherence process. Another possibility is that the role of the \(\mathrm{W}^{-}\) boson is modified by neutrinos, as suggested by the schematic Feynman diagram shown in Figure 1, which may explain the curious fact that (as we show below) nuclides with very different half-lives exhibit beta decays of similar amplitude.
Figure 1

Conjectured schematic Feynman diagram to represent the influence of ambient neutrinos on the beta-decay process.

Another reason that the issue remains controversial is that some experiments fail to yield evidence of variability, or are claimed not to show such evidence. The Jenkins–Fischbach articles led to critical articles by Cooper (2009), Norman et al. (2009), and Semkow et al. (2009), which led to responses by Krause et al. (2012), O’Keefe et al. (2013), and Jenkins, Mundy, and Fischbach (2010). More recent examples are Bellotti et al. (2012) and Kossert and Nahle (2014, 2015). However, it is important to bear in mind that the study of beta-decay variability is at a pre-theoretical stage, so we must be cautious about assuming that the results of any one experiment can invalidate the results of a quite different experiment. Concerning the Bellotti et al. results, we should bear in mind that the measurements extended over only 217 days and examined only one nuclide. Concerning the Kossert and Nahle results, we should note that their measurements were made with a triple to double coincidence ratio (TDCR) system that involves assumptions that may not be valid for stimulated decays, and that there was an error in their power-spectrum analysis (Sturrock et al., 2016).

The purpose of this article is to examine the possibility that the decay rates of some nuclides may be oscillatory with a frequency compatible with that of solar rotation, bearing in mind the possibility that oscillations may be intermittent or vary in some other way. We investigate this possibility by means of power-spectrum and spectrogram analyses of the BNL dataset, and compare the results with a similar analysis of available Super-Kamiokande solar neutrino measurements.

2 BNL Power Spectrum Analysis

The BNL dataset comprises 364 daily decay-rate measurements of each of 36Cl and 32Si acquired over the interval 1982.106 to 1989.929 (duration 7.823 years; Alburger, Harbottle, and Norton, 1986). For each nuclide, the daily decay rate was formed from an average of 20 separate measurements in the course of the day, the 36Cl and 32Si measurements being interwoven.

For convenience of power spectrum analysis, we adopt a date format (one that does not involve leap years) that has proved useful in the analysis of solar neutrino data (Sturrock, 2006). We first count dates in “neutrino days”, for which January 1, 1970, is designated “neutrino day 1” (\(t(\mbox{ND}) = 1\)). We then convert dates to “neutrino years”, denoted by \(t(\mbox{NY})\), as follows:
$$ t(\mbox{NY}) = 1970 + t(\mbox{ND})/365.2564. $$
(1)
Dates in neutrino years differ from true dates by less than one day. We next detrend (to remove the exponential decay) and normalize the data for each nuclide as follows. From the times \(t_{r}\) and count-rate measurements \(y_{r}\), we form
$$ V=\sum_{n} \bigl(\log(y_{n})+ \kappa t_{n}-K\bigr)^{2}, $$
(2)
and determine the values of \(\kappa\) and \(K\) that minimize \(V\). The normalized (and detrended) values are then given by
$$ x_{n} = y_{n}/z_{n}\quad \mbox{where}\ z_{n} =\exp(K-\kappa t_{n}). $$
(3)
Plots formed from the normalized decay-rates for 36Cl and 32Si are shown in Figures 2 and 3, respectively. It is interesting that for 36Cl, the standard deviation of measurements for 1986 to 1990 is notably less than that of measurements from 1982 to 1986. For 32Si, measurements for 1985 to 1987 are a little lower than measurements for the rest of the dataset; measurements for 36Cl do not show a similar feature. These comparisons suggest that 36Cl and 32Si were not subject to exactly the same influences even though their environmental conditions were identical.
Figure 2

A plot of detrended and normalized 36Cl data.

Figure 3

A plot of detrended and normalized 32Si data.

We have carried out power-spectrum analyses for the frequency range \(0\,\mbox{--}\,16~\mbox{year}^{ - 1}\) of each dataset, using a likelihood procedure (Sturrock, 2003). The power, as a function of frequency, is given by
$$ S( \nu) = \frac{1}{2 \sigma^{2}}\sum_{r = 1}^{R} x_{r}^{2} - \frac{1}{2 \sigma^{2}}\sum_{r = 1}^{R} \bigl( x_{r} - A\mathrm{e}^{\mathrm{i}2\pi \nu t_{r}} - A^{*}\mathrm{e}^{ - \mathrm{i}2\pi \nu t_{r}} \bigr)^{2}, $$
(4)
where for each frequency, the complex amplitude \(A\) is adjusted to maximize \(S\).
The results are shown in Figures 4 and 5, and the top 20 peaks are shown in Tables 1 and 2. We see that the power spectrum for 36Cl shows a very strong annual modulation. The power at exactly 1.00 year−1 is 14.61. For an expected exponential power distribution, the probability of finding this value or more by chance is \(\mathrm{e}^{-S}\), i.e.\(4.5 \times 10^{-7}\) (Scargle, 1982).
Figure 4

Power spectrum formed from normalized 36Cl measurements.

Figure 5

Power spectrum formed from normalized 32Si measurements.

Table 1

Top 20 peaks in the frequency band 0 – 16 year−1 in the power spectrum formed from the decay-rate measurements for 36Cl.

Frequency (year−1)

Power

Order

0.82

9.28

6

1.03

15.71

1

1.29

6.89

13

1.93

10.37

5

2.57

7.46

11

5.03

8.09

10

6.97

6.01

17

8.17

7.41

12

8.64

5.79

18

9.05

5.73

20

9.65

6.13

16

10.14

8.47

9

11.10

10.44

4

11.39

8.75

8

11.82

5.74

19

12.00

6.17

15

12.66

13.29

2

12.81

10.68

3

14.10

9.14

7

14.45

6.74

14

Table 2

Top 20 peaks in the frequency band 0 – 16 year−1 in the power spectrum formed from the decay-rate measurements for 32Si.

Frequency (year−1)

Power

Order

0.17

22.84

1

1.06

9.45

3

1.29

8.79

4

1.95

5.64

19

2.59

8.70

5

4.08

5.89

17

5.05

11.13

2

6.51

6.14

16

6.79

6.23

14

8.63

6.56

12

9.20

6.68

11

10.11

7.34

8

10.22

6.76

10

10.41

6.17

15

10.57

6.38

13

11.07

8.63

7

11.98

5.67

18

12.76

8.69

6

14.49

7.20

9

15.28

5.64

20

We have carried out a shuffle test (Bahcall and Press, 1991) of the 36Cl data, randomly reassigning values \(x_{r}\) to values \(t_{r}\). We find from 10 000 shuffles that the probability of finding the power 14.61 or more by chance is \(2.7 \times 10^{-7}\). Hence both tests show that the probability of finding the annual modulation shown in Figure 4 by chance is lower than \(5 \times 10^{-7}\).

It is interesting that the power-spectrum analysis of the 32Si data shown in Figure 5 does not show such a strong annual modulation. We see from Table 2 that there is a peak with power 9.45 at frequency 1.06 year−1. The power at exactly 1.00 year−1 is 3.51, for which \(\mathrm{e}^{-S}\) has the value 0.02. It is notable that the 36Cl and 32Si power spectra are quite different even though the instrumental and environmental conditions were absolutely identical.

It is also interesting to note from Tables 1 and 2 that there is some evidence of harmonics of the annual modulation, notably oscillations with frequencies close to 2 year−1 and 5 year−1. These harmonics, if real, would not be compatible with the early suggestion that an annual modulation is due to the varying Earth–Sun distance (Falkenberg, 2001; Jenkins et al., 2009).

Table 3 shows the power, amplitude (depth of modulation), and phase (phase of maximum always intended) of the annual oscillation (at exactly 1.00 year−1) for each nuclide, as derived from our count-rate analysis. We obtain an estimate of 171.5 years for the half-life of 32Si that is consistent with that derived by Alburger, Harbottle, and Norton (1986). The half-life of 36Cl is too long to be determined by this procedure in just a few years.
Table 3

Inferred half-life, power, amplitude, and phase of maximum of the annual modulation (\({\nu=1}~\mbox{year}^{-}1\)) of the normalized 36Cl and 32Si data (the 36Cl half-life is too long to determine accurately).

Nuclide

Half-life (y)

Power

Amplitude

Phase of maximum

36Cl

 

14.46

5.29e−04

0.70

32Si

171.5

3.52

2.72e−04

0.90

3 BNL Spectrogram and Phasegram Analyses of the Annual Modulations

We see from Figure 2 that the 36Cl dataset is distinctly non-uniform, and there is some evidence in Figure 3 that the same is true of the 32Si dataset. It is therefore helpful to examine these datasets by means of spectrograms and phasegrams. Figures 6 and 7 show spectrograms formed from 36Cl data and 32Si data, respectively, for the frequency band 0 – 8 year−1. Each spectrogram is formed by sequentially selecting 200 consecutive measurements and forming the power of that selection, using the same likelihood procedure as in Section 2 (Sturrock, 2003). The power is indicated by the color.
Figure 6

Spectrogram formed from 36Cl data for the frequency band 0 – 8 year−1.

Figure 7

Spectrogram formed from 32Si data for the frequency band 0 – 8 year−1.

We see from Figure 6 that 36Cl data show a strong annual modulation for the time interval 1984 to 1986. We see from Figure 7 that the 32Si data show intermittent oscillations (but no stable oscillation) near the annual frequency.

We form phasegrams by a procedure similar to that used to form spectrograms. Each phasegram is formed by sequentially selecting 200 consecutive measurements and forming the power of that selection as a function of phase of year (over the range 0 to 1) at the set frequency 1 year−1. Figures 8 and 9 show phasegrams formed from the normalized 36Cl and 32Si data, respectively.
Figure 8

Phasegram formed from 36Cl data for the annual modulation at frequency 1 year−1.

Figure 9

Phasegram formed from 32Si data for the annual modulation at frequency 1 year−1.

We see from Figures 6 and 8 that for 36Cl, the annual modulation is most evident for the time interval 1984 to 1986, with a phase of 0.71. We see from Figures 7 and 9 that for 32Si, the annual modulation is most evident for the same time interval, but with a phase that drifts from 0.95 to 0.85.

It is interesting that the two phasegrams are not the same. It is also notable that for each nuclide, the indicated phase of the decay rate is not close to 0 (or 1), the value expected if the oscillations were due to the varying Earth–Sun distance alone.

It appears that regardless of what is the cause (or causes) of the annual variations in the beta-decay rates, this cause is not steady and does not have exactly the same influence at all times and for all nuclides. This suggests that the variability in the beta-decay rates is not to be understood purely in terms of environmental influences on the detection system or in terms of the varying Earth–Sun distance.

4 BNL Spectrogram Analysis of Possible Solar Rotational Modulations

As we point out in a recent article (Sturrock et al., 2015), an annual oscillation of a beta-decay rate may be due to a solar influence, but it may also be due (entirely or in part) to some other influence, such as an annual variation in the environmental conditions or to a cosmic influence. Hence, to test for a solar influence, we should look for other evidence. We here search for an oscillation or oscillations that may be associated with solar rotation. If the influence is due to neutrinos, modulation of the neutrino flux may in principle occur anywhere inside the Sun. Theoretically, the neutrino flux could be influenced by either the Mikheyev, Smirnov, Wolfenstein (MSW) effect (Mihkeyev and Smirnov, 1986; Wolfenstein, 1978) that is determined by the density structure, or by the Resonant Spin Flavor Precession (RSFP) effect (Akhmedov, 1988), which depends on both the density and the magnetic field. Either process provides a mechanism for converting an electron neutrino (the type produced by nuclear reactions in the core) into either a muon neutrino or a tau neutrino, neither of which would be detected by a neutrino experiment such as Super-Kamiokande (Bahcall, 1989). The fact that neutrino experiments detect fewer neutrinos than expected is attributed to these processes (Ianni, 2014). It seems reasonable to assume that if beta decays are influenced by neutrinos, then this influence is due primarily – perhaps exclusively – to electron neutrinos.

The Sun is sufficiently stable for the MSW effect not to lead to any detectable time variation. On the other hand, the solar magnetic field, as it is observed at the photosphere, is highly asymmetric and highly variable. The same is likely to be true throughout the convection zone (normalized radius 0.7 to 1), and may also be true in the radiative zone (normalized radius 0.3 or 0.4 to 0.7). If the RSFP process is operative in a region where the magnetic field is sufficiently strong and sufficiently inhomogeneous, we may therefore expect that the solar neutrino flux may exhibit modulation in a band of frequencies appropriate to the Sun’s internal rotation. As determined by helioseismology, the equatorial sidereal rotation rate is in the range 13.5 – 15.0 year−1, which converts into a synodic rate (as seen from Earth) of 12.5 – 14.0 year−1 (Schou et al., 2002; Couvidat et al., 2003). However, the rotation rate in the deep interior is quite uncertain, and there are indications from analyses of Super-Kamiokande measurements that some part of the solar interior may rotate as slowly as 10.4 year−1 (sidereal) or 9.4 year−1 (synodic; Sturrock and Scargle, 2006). A reasonable search band for the rotational modulation of beta-decay rates (as seen from Earth) would therefore seem to be 9 – 14 year−1, corresponding to periods in the range 26 – 41 days.

We show spectrograms (again formed from 200 consecutive measurements) for 36Cl and 32Si data for the frequency band 8 – 16 year−1 in Figures 10 and 11, respectively. We see that both spectrograms give evidence of an oscillation with frequency of about 12.7 year−1, which is compatible with a source in the solar radiative zone. For both spectrograms, this oscillation is evident over the time interval 1983.5 to 1985.5. However, the two spectrograms differ in that there is clear evidence of an oscillation with frequency about 11.2 year−1 in the 36Cl spectrogram (Figure 10), but only slight evidence of a similar oscillation in the 32Si spectrogram (Figure 11). Hence, regardless of the source of these oscillations, it does not have exactly the same influence on the 36Cl measurements and on the 32Si measurements. We comment further on this difference in Section 6. The depth of modulation of the oscillation at 12.7 year−1 is \(8.4 \times 10^{-4}\) for 36Cl and \(7.6 \times 10^{-4}\) for 32Si.
Figure 10

Spectrogram formed from 36Cl data for the frequency band 8 – 16 year−1.

Figure 11

Spectrogram formed from 32Si data for the frequency band 8 – 16 year−1.

In order to assess the significance of the features shown in Figures 10 and 11, it is necessary to estimate the probability of finding such a feature in a specified band.

We proceed by first estimating the probability \(P_{1}\) of finding a feature with the actual power or more in a band of unit width (1 year−1). Since we plan to shuffle the data, the choice of band is unimportant; only the width matters. We find that the actual peak power in the band 12 – 13 year−1 is 13.29 for the 36Cl power spectrum, and 8.69 for the 32Si power spectrum. By applying the shuffle test to 36Cl data, we find that there is a probability of only \(1.5 \times 10^{-5}\) of finding by chance a feature of power 15.71 or more in a band of unit width. By applying the shuffle test to 32Si data, we find that there is a probability of only \(1.5 \times 10^{-3}\) of finding by chance a feature of power 9.45 or more in a band of unit width.

We may infer from these results the probabilities of finding such features in bands other than those of unit width. The probability of finding such a feature in a band of width \(B\) is given by
$$ P_{B} = 1 - \left( 1 - P_{1} \right)^{B}. $$
(5)
If we choose to adopt a search band of 9 – 14 year−1 (of width 5 year−1) for the 36Cl data, we arrive at a probability of \(7.5 \times 10^{-5}\) of finding in that band a feature of power as large as or larger than the actual power (15.71). For the 32Si data, we would arrive at a probability of 0.0075 of finding in that band a feature of power as large as or larger than the actual power (9.45).

We find that the depths of modulation of the rotational signals shown in Figures 10 and 11 are \(7 \times 10^{-4}\) for 36Cl and \(6 \times 10^{-4}\) for 32Si.

5 Super-Kamiokande Power Spectrum Analysis

We now examine the Super-Kamiokande dataset (Super-Kamiokande I) as published in Yoo et al. (2003). This comprises 358 measurements of the B8 neutrino flux, of energy 5 – 20 MeV, acquired over the time interval June 2, 1996, to July 15, 2001. The mean flux over that time interval was \(2.35 \times 10^{6}~\mbox{cm}^{-2}\,\mbox{s}^{-1}\). For each of the 358 measurements, we are given the start time and end time, the best estimate of the flux, and upper and lower error estimates. When we examine the flux estimates shown in Figure 5 of Yoo et al. (2003) by eye, we form the impression that the neutrino flux is essentially constant. However, it is known that power-spectrum analyses can reveal an oscillation in a time series that appears by eye to be featureless (see, for instance, Press et al., 1992, Section 13.8). Not surprisingly, evidence of an oscillation becomes more obvious the more information is taken into account.

We have carried out an analysis of the Super-Kamiokande measurements by a likelihood procedure that is similar to the procedure described in Section 4, but takes the start and end times of each measurement into account as well as the count-rate measurement and the lower and upper error estimates (Sturrock and Scargle, 2006). We adjusted the offset for each frequency; as a result, the power calculation is a nonlinear process.

The resulting power spectrum is shown in Figure 12, and the top 20 peaks in the frequency range 0 – 16 year−1 are shown in Table 4. We find the strongest peak at 9.43 year−1 with power 13.24, and the second-strongest peak at 12.31 year−1 with power 6.24. These are the strongest peaks in the power spectrum over the frequency range 0 – 30 year−1 (Sturrock and Scargle, 2006). We have assessed the significance of these oscillations by means of the shuffle test (Sturrock and Scargle, 2006). We find that there is a probability of only 0.0004 of finding a peak of power 13.24 or more in unit bandwidth. This would lead to a probability of only 0.002 of finding a peak of power 13.24 or more in a rotational band extending from 9 to 14 year−1.
Figure 12

Power spectrum formed from Super-Kamiokande measurements using a likelihood procedure that takes the start and end time of each run and the error estimates into account.

Table 4

Top 20 peaks in the frequency band 0 – 16 year−1 in the power spectrum formed from Super-Kamiokande B8 flux measurements.

Frequency (year−1)

Power

Order

0.36

4.07

6

0.80

2.68

12

2.31

2.32

17

2.96

2.54

15

4.43

3.71

7

5.90

2.72

11

8.30

5.26

3

8.75

2.49

16

8.98

3.27

9

9.43

13.24

1

10.69

3.45

8

11.29

3.18

10

11.57

2.66

13

12.31

6.01

2

12.69

4.44

4

13.17

2.25

18

13.83

2.20

20

14.66

2.24

19

14.87

2.64

14

15.72

4.11

5

For comparison of the peak at 12.31 year−1 with the peaks near 12.7 year−1 found in the BNL power spectra (Tables 1 and 2), we note from the shuffle test that there is a probability of 0.09 of finding a peak of power 6.24 or more in unit bandwidth. This would convert into a probability of 0.03 of finding a peak of power 6.24 or more in a band of width 0.3 year−1, which is the separation of the BNL and Super-Kamiokande peaks.

For comparison with the spectrograms formed from BNL data (Figures 6, 7, 10, 11), we now show the corresponding spectrograms formed from Super-Kamiokande data. Figure 13 shows the spectrogram formed (from measurements corrected for the varying Earth–Sun distance) for the frequency range 0 – 8 year−1. We see that there is no evidence of an annual oscillation. Figure 14 shows the spectrogram formed for the frequency range 8 – 16 year−1. We see evidence of a fairly steady feature at frequency 9.4 year−1 and an intermittent feature centered approximately at 12.5 year−1. It appears that the reason the 9.4 peak in the power spectrum is more significant than the 12.3 year−1 peak is that the former oscillation remains fairly steady over the duration of the measurement, whereas the latter is intermittent.
Figure 13

Spectrogram formed from Super-Kamiokande data (corrected for the varying Earth–Sun distance) for the frequency band 0 – 8 year−1.

Figure 14

Spectrogram formed from Super-Kamiokande data for the frequency band 8 – 16 year−1.

For comparison with the depth of modulation of the rotational modulation found in BNL measurements, we note that the depth of modulation of the rotational modulation at 12.31 year−1 in Super-Kamiokande data is 0.052.

We noted in Section 4 that the spectrogram formed from BNL 36Cl measurements shows a feature at 11 year−1. There is weak evidence of a similar feature in Figure 14. These results raise the question what pattern of internal rotation of the Sun might explain evidence of rotation features at 9.4 year−1, 11 year−1, and approximately 12.5 year−1? We discuss this question in the next section.

6 Discussion

Since an annual oscillation may be due to environmental influences on an experiment, the main focus of this article is whether beta-decay rates show evidence of an influence related to solar rotation, However, we first review the results of our search for an annual oscillation.

We found in Sections 2 and 3 that there is strong evidence of an annual oscillation in 36Cl measurements, but little evidence of such an oscillation in 32Si measurements. We estimated a probability of only \(4.5 \times 10^{-7}\) that the annual oscillation in 36Cl data is due to random fluctuations in the measurements. We found slight evidence of an annual oscillation in 32Si data. This is an interesting result. If the annual oscillation in the 36Cl data were due to an environmental influence such as temperature fluctuations, then we would expect to see very similar evidence in both datasets, since both nuclides were subject to exactly the same environmental influences. That we see strong evidence of an annual oscillation in 36Cl data but not in 32Si data therefore suggests that the annual oscillation evident in 36Cl data is not due to an environmental influence such as temperature.

Moreover, the spectrogram shown in Figure 6 shows that the annual oscillation in 36Cl data is intermittent rather than steady, and we find (Table 3 and Figure 8) that the annual oscillation has its maximum at phase 0.7, i.e. in mid-September. These results argue against the early suggestion (Falkenberg, 2001; Jenkins et al., 2009) that an annual oscillation in a decay rate may be due to the annual oscillation of a solar flux caused by the annual variation of the Earth–Sun distance, since this influence would be steady and would lead to a phase of maximum near January 3.

These results therefore seem to be incompatible with the hypothesis that the annual oscillations are due to the influence of a steady flux of solar neutrinos. However, they appear to be compatible with the hypothesis that the annual oscillations are due to the influence of cosmic neutrinos (in addition to solar neutrinos) if we allow for the possibility that different nuclides are responsive to neutrinos of different energy and that the spatial structure and energy composition of cosmic neutrinos are variable. This hypothesized variability in the structure and composition of the cosmic-neutrino distribution may be the result of gravitational interaction with the Sun and with stochastic deflections and stochastic acceleration due to gravitational encounters with stars, galactic nuclei, black holes, etc., which processes would depend on the speed (and therefore the energy) of the particles. The possible role of cosmic neutrinos in influencing beta decays will be discussed in more detail in a future article.

We now consider the evidence of an influence of solar rotational modulation on the beta-decay rates analyzed in Section 4. We found evidence of a transient oscillation of frequency about 12.7 year−1 in both 36Cl and 32Si spectrograms (Figures 10 and 11). This frequency is compatible with an influence on the solar neutrino flux that occurs in the radiative zone. We found from the shuffle test that these features could have occurred by chance in a band of unit width with probabilities \(1.5 \times 10^{-5}\) and \(1.5\times 10^{-3}\), respectively. The probability of finding both features within a band of 5 year−1 is \(5.6\times 10^{-7}\).

Figure 10 shows a feature with frequency about 11.0 year−1 early in the spectrogram formed from 36Cl data. There is a hint of a similar feature late in the spectrogram formed from 32Si data (Figure 11). A synodic frequency of 11 year−1 corresponds to a sidereal frequency of 12 year−1, which is the rotation frequency of a region (presumed to be a tachocline) that appears to be the site of r-mode oscillations in the Sun (Sturrock et al., 2012; Sturrock and Bertello, 2010; Sturrock, Fischbach, and Jenkins, 2011; Sturrock et al., 2012; Sturrock et al., 2013; Sturrock et al., 2015).

Figure 11 also shows evidence of a transient feature with a frequency of about 10 year−1. We have speculated that the oscillation with a frequency of about 9.4 year−1 found in Super-Kamiokande data (evident in Figures 12 and 13) may have its origin in the solar core (Sturrock et al., 2012). Hence an oscillation with a frequency of about 10 year−1 may have its origin in a region between the core and the inner tachocline.

Figure 15 gives a schematic representation of the sidereal internal rotation of the Sun, as inferred from studies of rotational and r-mode oscillations (we note that the synodic rotation rate is lower than the sidereal rate by 1 year−1). The proposed value of the rotation rate of the radiative zone (13.5 year−1), based on the results of Section 4, is consistent with estimates derived from helioseismology (Couvidat et al., 2003; Schou et al., 2002). The proposed rotation rate of the inner tachocline (12 year−1) is based on analyses of r-mode oscillations. The proposed rotation rate of the core (10.5 year−1) is based on our analysis of Super-Kamiokande data. This value is compatible with estimates (\({\le}\, 11.6~\mbox{year}^{ - 1}\)) cited by Elsworth et al. (1995), based on data acquired by the Birmingham Solar Oscillations Network. However, Garcia et al. (2007) derived a range 13.0 to 13.6 year−1 from their analysis of solar gravity-mode oscillations detected by the Global Oscillations at Low Frequency instrument as part of the SOHO Mission. This discrepancy remains to be resolved. The estimates shown in Figure 15 are compatible with the results of Benomar et al. (2015), who conclude from their analysis of asteroseismology and spectroscopic measurements that the rotation rates of the convective and radiative zones of main-sequence stars differ by less than a factor of 2.
Figure 15

Equatorial cut of the conjectured internal rotation of the Sun.

It is interesting to note that these estimates of the internal rotation rate all fall within the range of the rotation rate of the photosphere (8.7 year−1 to 14.8 year−1), but this probably has no physical significance, since this strong differential rotation is confined to the outermost layer of the convection zone.

If further analyses of further experiments support the proposition that beta decays can be induced by neutrinos, it will be necessary to seek a theoretical understanding of this process, examining options such as the possibility that the role of the \(\mathrm{W}^{-}\) boson is modified by neutrinos, as suggested by the schematic Feynman diagram shown in Figure 1.

Acknowledgements

We acknowledge with thanks the generous cooperation of David Alburger in providing the BNL data and relevant experimental information. We thank the Super-Kamiokande Observatory for making available their solar neutrino measurements. We thank our colleague Guenther Walther for advice concerning the statistical analyses in Section 2 and Section 5. We thank Alexander Parkhomov of the Lomonosov Moscow State University, Heinrich Schrader of the Physikalisch-Technische Bundesanstalt, and Gideon Steinitz of the Geological Survey of Israel, for sharing data with us, which, while not addressed specifically in this article, have provided crucial background information. We thank David Gilbert for assistance in preparing the figures, and we thank an anonymous referee for comments and suggestions that significantly improved the article.

Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Kavli Institute for Particle Astrophysics and Cosmology and the Center for Space Science and AstrophysicsStanford UniversityStanfordUSA
  2. 2.Department of Physics and AstronomyPurdue UniversityWest LafayetteUSA
  3. 3.NASA/Ames Research Center, MS 245-3Moffett FieldUSA