THE ASTROPHYSICAL JOURNAL, 565 : 1366È1375, 2002 February 1
( 2002. 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 94305 ; sturrock=Ñare.stanford.edu, weber=plage.stanford.edu
Received 2001 February 26 ; accepted 2001 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 GALLEXGNO 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
1.
INTRODUCTION
gallium capture product 71Ge (11.43 days) is appreciably
shorter than that of the chlorine capture product 37Ar (35.0
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 (1990) have
used spectrum analysis to search for evidence of oscillations
with period of order 1 yr. More recently, Haubold (1997)
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. 2000
There has for some time been great interest in the possibility that the solar neutrino Ñux is variable. Sakurai (1979,
1981) 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 (11 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 1987 ;
Bahcall & Press 1991 ; Bieber et al. 1990 ; Dorman & Wolfendale 1991 ; Wilson 1987), the surface magnetic Ðeld
strength (Massetti & Storini 1993 ; Oakley et al. 1994), the
intensity of the green line corona (Massetti & Storini 1996),
or the solar wind Ñux (McNutt 1995). Walther (1997) 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 (1999) have in turn criticized WaltherÏs claim, and Walther (1999b) has responded
to that criticism (see also Walther 1999a). In recent articles,
Boger, Hahn, & Cumming (2000) and Wilson (2000) also
conclude that there is no statistically signiÐcant association
between Homestake data and the usual solar activity
indices.
Homestake data (Cleveland et al. 1998) comprise 108
runs (for the time period 1970È1994). However, the
GALLEX (Anselmann et al. 1993, 1995 ; Hampel et al. 1996,
1997) and GNO (Altmann et al. 2000) consortia, working
with the same gallium experiment, have now acquired 84
runs (for the period 1991È2000). Since the half-life of the
1366
ANALYSIS OF GALLEX-GNO AND SOHO/MDI DATA
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 1996). The
currently favored candidate for the solution of the solar
neutrino deÐcit (see, e.g., Bahcall, Krastev, & Smirnov 1998)
is the MSW e†ect (Mikhevev & Smirnov 1986a, 1986b,
1985c ; Wolfenstein 1978, 1979), 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 (1971), who considered propagation through
magnetic Ðeld in the core of the Sun. This leads to spin
precession (see also Fujikawa & Schrock 1980), 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 (1986a, 1986b) and Barbieri & Fiorentini (1988) 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 (1981) 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 (1988a, 1988b) and Lim & Marciano (1988)
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 (1997) and Pulido &
Akhmedov (2000).
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. (1986a, 1986b) 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 \ 0.7. (Here and elsewhere the radius is normalized to that of the photosphere.)
Nuclear burning occurs near the center : about 80% of the
burning occurs within r \ 0.15 (see, e.g., Bahcall 1989).
1367
Between the nuclear-burning core and the tachocline is the
radiative zone. According to data derived from helioseismology (Schou et al. 1998), the core and radiative zone
are in substantially rigid rotation with a sidereal rotation
rate in the range 13.75 ^ 0.25 yr~1. 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) [ 1 .
(1)
One may convert from these units to degrees per day or
nHz as follows :
l(yr~1) \ 0.98526l(degrees per day) ,
l(yr~1) \ 31.6875l(nHz) .
(2)
At the tachocline, the sidereal rotation rate jumps from
about 13.75 yr~1 at r \ 0.66 to about 14.53 yr~1 at
r \ 0.74. 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~1 at r \ 1 at the poles to a maximum of
about 14.84 yr~1 at r \ 0.93 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 5È31 weeks, with a median of 10
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 1997) and
found evidence for an oscillation at 12.85 yr~1, comparable
with the synodic rotation frequency of the radiative zone
(12.75 ^ 0.25 yr~1). 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 1998).
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. 1999), based on a least-squares procedure
(Knight, Schatten, & Sturrock 1979), 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 GALLEXGNO data (Sturrock & Scargle 2001). 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.
1368
STURROCK & WEBER
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. 1999). 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 ° 4.
2.
SPECTRUM ANALYSIS
In this section we analyze the combined GALLEX-GNO
data by a method developed by Lomb (1976) and Scargle
(1982) for the analysis of irregular time series. Such a procedure is necessary, since the timing of radiochemical
experiments is not regular. Bretthorst (1988) has shown by
Bayesian analysis that the Lomb-Scargle technique is the
optimum procedure for Ðnding a sinusoid in irregularly
Vol. 565
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 GALLEXGNO data are presented in Table 1. Most of the runs last
about 4 weeks, but, on the other hand, the half-life of the
capture product 71Ge is only 11.43 days (Bahcall 1989), so
that the survival probability of a germanium nucleus produced on the Ðrst day of a run is of order 20%. 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 1
GALLEX-GNO DATA
EXPOSURE TIMES
EXPOSURE TIMES
EXPERIMENT
RUN
Start
End
FLUX ESTIMATE
EXPERIMENT
RUN
Start
End
FLUX ESTIMATE
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 . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
1991.367
1991.427
1991.485
1991.543
1991.600
1991.658
1991.718
1991.775
1991.830
1991.890
1991.945
1992.079
1992.139
1992.197
1992.249
1992.634
1992.713
1992.790
1992.866
1992.943
1993.019
1993.096
1993.173
1993.255
1993.326
1993.403
1993.480
1993.556
1993.633
1993.710
1993.786
1993.863
1993.940
1994.016
1994.093
1994.170
1994.247
1994.323
1994.400
1994.781
1994.838
1994.896
1991.425
1991.485
1991.543
1991.600
1991.658
1991.718
1991.773
1991.830
1991.888
1991.945
1992.022
1992.137
1992.194
1992.249
1992.328
1992.710
1992.787
1992.863
1992.940
1993.016
1993.093
1993.170
1993.252
1993.323
1993.400
1993.477
1993.553
1993.630
1993.707
1993.784
1993.860
1993.937
1994.014
1994.090
1994.167
1994.244
1994.321
1994.397
1994.474
1994.838
1994.896
1994.953
64
42
295
20
[20
125
106
129
105
36
80
55
144
2
96
103
123
144
32
100
58
70
115
138
36
62
59
56
64
44
71
[9
39
52
85
51
107
112
90
140
104
[89
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GALLEX . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
GNO . . . . . . . . . .
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
1994.956
1995.030
1995.107
1995.186
1995.266
1995.337
1995.416
1995.490
1995.567
1995.647
1995.701
1996.123
1996.183
1996.243
1996.298
1996.489
1996.544
1996.661
1996.716
1996.776
1996.891
1996.945
1997.025
1998.384
1998.460
1998.556
1998.652
1998.729
1998.806
1998.882
1998.959
1999.036
1999.189
1999.285
1999.381
1999.458
1999.573
1999.649
1999.726
1999.830
1999.880
1999.953
1995.030
1995.107
1995.184
1995.266
1995.337
1995.414
1995.490
1995.567
1995.644
1995.701
1995.759
1996.180
1996.243
1996.295
1996.353
1996.544
1996.601
1996.716
1996.776
1996.888
1996.945
1997.025
1997.060
1998.460
1998.556
1998.652
1998.729
1998.806
1998.882
1998.959
1999.036
1999.112
1999.285
1999.381
1999.458
1999.573
1999.649
1999.726
1999.803
1999.880
1999.953
2000.033
43
[22
44
56
49
24
119
51
50
22
102
140
57
118
155
108
151
[78
106
186
70
184
101
71
48
97
69
[46
45
116
[51
126
123
53
26
97
114
46
49
33
67
79
ANALYSIS OF GALLEX-GNO AND SOHO/MDI DATA
The Ñux estimates are shown as a function of end time in
Figure 1. Later in this section we discuss another choice and
show that the spectrum changes very little. Figure 1 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
i
to be the end time of each run), where i( \ 1, . . . , N) enumerates the runs. We denote by g the Ñux measurements and
i
deÐne the mean h and the standard deviation p by the usual
expressions :
h\
1 N
;g
i
N
1
(3)
and
1
N
; (g [ h)2 .
(4)
i
N[1
1
Following Lomb (1976) and Scargle (1982), the power spectrum is given, as a function of angular frequency u, by
p2 \
A
1 M; (g [ h) cos [u(t [ q)]N2
i
i
; cos2 [u(t [ q)]
2p2
i
M; (g [ h) sin [u(t [ q)]N2
i
i
,
]
; sin2 [u(t [ q)]
i
where q is deÐned by the relation
S(u) \
B
(5)
; sin (2ut )
i .
(6)
; cos (2ut )
i
Since we are here concerned with possible rotational modulation of the neutrino Ñux, we show in Figure 2 the result of
tan (2uq) \
300
250
200
Flux
150
100
50
0
−50
−100
1990
1992
1994
1996
End Time
1998
2000
2002
FIG. 1.ÈPlot of Ñux estimates vs. end time for GALLEX-GNO
experiments.
1369
5
4.5
4
3.5
3
Power
No. 2, 2002
2.5
2
1.5
1
0.5
0
10
11
12
13
14
15
16
Frequency
17
18
19
20
FIG. 2.ÈLomb-Scargle spectrum of the GALLEX-GNO solar neutrino
data for the frequency range 10È20 yr~1.
our Lomb-Scargle analysis of the GALLEX-GNO data
only for frequencies in the range 10È20 yr~1. (There is an
interesting peak at 6.08 yr~1, 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 1982). Hence, for a
speciÐed frequency, a peak with S [ 3.0 is signiÐcant at the
5% level, and a peak with S [ 4.6 is signiÐcant at the 1%
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. 1998), the rotation rate of the
radiative zone is in the range 12.4È12.9 yr~1. (Further information is given later ; see Fig. 11.) 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 \ 13.04 yr~1 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 14.3È14.8 yr~1 sidereal or 13.3È13.8 yr~1
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 \ 13.59 with power
S \ 4.82 and the other at l \ 13.67 with power S \ 4.40.
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 \ 15, and then
0 the corresponding
0
reorder the end times by sorting
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
1370
STURROCK & WEBER
300
Vol. 565
6
250
5
200
4
Power
Flux
150
100
50
3
2
0
−50
1
−100
1990
1992
1994
1996
End Time
1998
2000
2002
0
10
11
12
13
FIG. 3.ÈPlot of Ñux vs. time for GALLEX-GNO data, reordered as
explained in text.
14
15
16
Frequency
17
18
19
20
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.
can see no obvious di†erence between this Ðgure and Figure
1. On the other hand, the spectrum of the reorganized data
is that shown in Figure 4. It reveals a very strong periodicity
at l \ 15. The minor peak at l \ 11 is a weak alias of the
0
imposed
oscillation, arising from a strong periodicity in the
window function at about l \ 13. (This peak is located at
2l [ l . There are similarw weak peaks [not shown] at
l w[ l 0, l ] l , etc.) It is a strong point of the Lombw
0 method
w
0that we see no peak in the spectrum at the
Scargle
frequency l . We may note that the period of the imposed
w oscillation is 3.5 weeks, less than the mean
(and detected)
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-
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 :
y \ arctan
A B
g[h
.
p
(7)
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 13.6 is increased rather than decreased,
6
35
5
30
4
Power
Power
25
20
15
3
2
10
1
5
0
10
0
10
11
12
13
14
15
16
Frequency
17
18
FIG. 4.ÈLomb-Scargle spectrum of reordered data
19
20
11
12
13
14
15
16
Frequency
17
18
19
20
FIG. 6.ÈLomb-Scargle spectrum of the GALLEX-GNO Ñux estimates,
modiÐed by an arctan operation to moderate the e†ect of outliers.
ANALYSIS OF GALLEX-GNO AND SOHO/MDI DATA
No. 2, 2002
7
6
Power
5
4
3
2
1
0
10
11
12
13
14
15
16
Frequency
17
18
19
20
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 o 1@2 .
ogo
(8)
The resulting Lomb-Scargle spectrum is shown in Figure 7.
Once again, we see that the peak at 13.6 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 \ 13.6.
3.
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. (1998) have tabulated the rotation
rate l (r, j) and the error estimate p (r, j) for 101 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
(l [ l )2
h
.
(9)
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(l o r, j) \ (2n)~1@2p~1 exp [
h
$(r, j) \
P
lb
dl S(l)P(l o r, j) ,
(10)
la
where S is the spectrum computed in ° 2 and displayed in
Figure 2. In this integral, we need to select limits of integra-
1371
tion that are sufficiently wide to cover all signiÐcant contributions from the PDF. We have adopted l \ 0 and l \
a
b
20, but a much smaller range would have been satisfactory.
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
0.15 of the center of the Sun (see, e.g., Bahcall 1989). 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 16¡ of the solar
equator in penetrating the surface of the Sun. Over the
maximum range of 16¡, the rotation rate varies by less than
1% 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(l o CZ) \
We may verify that
P
lb
P
1
ru
dr P(l o r, 0) .
r [r
u
l rl
(11)
dl P(l o CZ) \ 1 .
(12)
la
We denote by r and r the lower and upper limits of the
u
convection zone.l The former
is set by the upper limit of the
tachocline, so r \ 0.74, and the latter by the photosphere,
so r \ 1. Over l this range the minimum sidereal frequency
u
1372
STURROCK & WEBER
Vol. 565
FIG. 8.ÈMap of the resonance statistic $, deÐned by eq. (10), 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.
!\
P
lb
dl S(l)P(l o CZ) ,
(13)
la
where S(l) is the power spectrum computed by the LombScargle process in ° 2. This procedure is known in Bayesian
parlance (see, e.g., Bretthorst 1988) as an integration over a
““ nuisance parameter.ÏÏ We denote by ! the actual value of
d
! derived from the data, and we Ðnd that
! \ 2.762. We
d
now need to assess the signiÐcance of this value.
We adopt the method introduced by Bahcall & Press
(1991) 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 10,000 shuffles of the data, and the results are
shown in Figure 10. We Ðnd that of the 10,000 random
simulations, only 16 have values ! [ ! . From this
d of approxanalysis, we infer that there is a probability
imately 0.16% 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 0.2% level.
0.035
0.03
0.025
0.02
PDF
is 14.27 yr~1, and the maximum is 14.83 yr~1, a range of
almost 4%. The resulting PDF is shown in Figure 9.
We now form the ““ integral resonance statistic ÏÏ
0.015
0.01
0.005
0
10
11
12
13
Frequency
14
15
FIG. 9.ÈThis probability distribution function is representative of the
range of (synodic) rotation frequencies in the convection zone.
ANALYSIS OF GALLEX-GNO AND SOHO/MDI DATA
No. 2, 2002
0
0
10
10
−1
−1
10
Fraction
Fraction
10
−2
10
−3
10
−4
0
−4
0.5
1
1.5
2
Statistic
2.5
3
3.5
4
FIG. 10.ÈFor analysis of the convection zone, the ordinate denotes the
fraction of 10,000 simulations that have values of the statistic !, deÐned by
eq. (13), larger than the value indicated by the abscissa. The vertical line
denotes the actual value of the statistic, ! , derived from the data. Less
than 0.2% 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 \ 0.30 and r \ 0.66. The
l 11. When weu calculate the
resulting PDF is shown in Figure
resonance statistic !, using equation (13), we obtain the
value ! \ 1.140. We have carried out 10,000 shuffles of the
d the results are shown in Figure 12. We see that, of
data, and
the 10,000 random simulations, almost 3000 have values
! [ ! . From this analysis, we infer that there is no signiÐd
cant resonance
between oscillations of the solar neutrino
Ñux and rotation of the radiative zone.
4.
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. 1997,
0.025
0.02
PDF
0.015
0.01
0.005
0
10
−2
10
−3
10
10
1373
11
12
13
Frequency
14
15
FIG. 11.ÈThis probability distribution function is representative of the
range of (synodic) rotation frequencies in the radiative zone.
10
0
0.5
1
1.5
2
Statistic
2.5
3
3.5
4
FIG. 12.ÈFor analysis of the radiative zone, the ordinate denotes the
fraction of 10,000 simulations that have values of the statistic !, deÐned by
eq. (13), larger than the value indicated by the abscissa. The vertical line
denotes the actual value of the statistic, ! , derived from the data. By
d
contrast with the situation shown in Fig. 10, almost
30% of the simulations
have ! [ ! .
d
1998) and from histogram analysis of data from the gallium
experiments (Sturrock & Scargle 2001). 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 (1999) 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 (1994), 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 105 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
° 1), 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 0.2% 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.
1374
STURROCK & WEBER
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 (1988) and others have presented
evidence that solar activity exhibits longitudinal structure
that persists for more than a solar cycle. We referred, in ° 1,
to the recent article by Neugebauer et al. (2000), 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 1987). StenÑo (1977) and Zirker
(1977) 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 (14.39 yr~1 sidereal or 13.39 yr~1 synodic), not too far from the value (13.6
yr~1 synodic) found here. However, some long-lived features such as polar coronal holes exhibit a slower rotation,
nearer 12.9 yr~1 synodic (Timothy, Krieger, & Vaiana
1975 ; Bohlin 1977). It is interesting that our earlier analysis
of Homestake data (Sturrock et al. 1997) 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 ° 1, 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 \ 1013.0*m2E~1 ,
(14)
Vol. 565
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 2000). It is
notable that, of experiments now in operation, the gallium
experiments detect the lowest energy neutrinos (down to
100 keV). Hence, the resonant layer relevant to the gallium
experiment will be deeper than that for chlorine experiments (of order 1 MeV) and much deeper than that for
Super-Kamiokande, SNO, etc. (of order 10 MeV). It follows
that if the modulation of the neutrinos detected by
GALLEX-GNO occurs at r \ 0.8, neutrinos detected by
Homestake will be modulated at about r \ 0.9, and those
detected by Super-Kamiokande would be modulated at
about r \ 0.95. Hence, the magnetic Ðeld will be weaker in
the regions relevant to the Homestake and SuperKamiokande 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 14% of the Ñux detected by
Homestake comes from lines (Bahcall 1989), 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
C\
P
te
dt f (t)e~k(te~t) .
(A1)
te~D
If
f \ Aeiut`ia ] cc ,
C\
A
[1 [ e~(i`iu)D]eipte`ia ] cc .
i ] iu
(A2)
(A3)
The experimenters assign a Ñux value g based on the assumption that the Ñux is constant, i.e.,
g \ iC(1 [ e~iD)~1 .
(A4)
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
No. 2, 2002
ANALYSIS OF GALLEX-GNO AND SOHO/MDI DATA
1375
combining equations (A2), (A3), and (A4), we Ðnd that this requirement is satisÐed if
Keiu* \
A
BC
i ] iu
i
D
1 [ e~iD
.
1 [ e~(i`iu)D
(A5)
We have computed the o†set time * and the conversion factor K for all runs, for the frequency of interest, l \ 13.6. We Ðnd
that * ranges from a minimum of 5.17 days to a maximum of 6.86 days ; the mean is 5.93 days and the standard deviation 0.66
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 80¡ 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 1.47 to a maximum of 3.98 ; the mean is 3.50 and the standard deviation is 0.60.
This translates into a standard deviation of only 17% 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. (1997), which returns the best estimate of the amplitude.
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