Journal of Modern Physics, 2013, 4, 4956 http://dx.doi.org/10.4236/jmp.2013.44A008 Published Online April 2013 (http://www.scirp.org/journal/jmp) QuasiBiennial Modulation of the Solar Neutrino Flux: A “Telescope” for the Solar Interior Loris D’Alessi1, Antonio Vecchio1, Vincenzo Carbone1,2*, Monica Laurenza3, Marisa Storini3 1Dipartimento di Fisica, Università della Calabria, Cosenza, Italy 2Liquid Crystal Laboratory (INFM), Cosenza, Italy 3INAF/IAPSRoma, Via del Fosso del Cavaliere, Roma, Italy Email: *vincenzo.carbone@fis.unical.it Received February 18, 2013; revised March 21, 2013; accepted March 31, 2013 Copyright © 2013 Loris D’Alessi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT An oscillating magnetic field deep within the solar radiative region can significantly alter the helioseismic gmodes. The presence of density gradients along gmodes, can excite Alfvén waves resonantly, the resulting waveforms show sharp spikes in the density profile at radii comparable with the neutrino’s resonant oscillation length. This process should ex plain the observed quasibiennial modulation of the solar neutrino flux. If confirmed, the coupling between solar neu trino flux and gmodes should be used as a “telescope” for the solar interior. Keywords: Solar Neutrinos; Solar Cycle; Solar Interior 1. Introduction Sakurai claimed that fluctuations of the core temperature, which is responsible of the pp chain efficiency, should be at the origin of this quasibiennial modulation [6]. How ever the analysis by Lanzerotti [7] carried on a set of data which cover a longer time period, exclude any connection between events in the core with the ones which occur in the photosphere. The results of analysis of Kamiokande data over the cycle 22 of the 11yr solar cycle [8] showed that there is no correlation of the solar neutrino flux with sunspot numbers on 11yr time scale. Time variability of solar neutrino flux has been studied since the appearance of the first results of Homestake experiment [1]. In an early attempt to interpret the dis crepancy between theoretical and observed flux, Sheldon [2] suggested a dependence of neutrino flux with solar activity, due to a time variable production rate of the neutrinos in the core of the Sun. The most famous evi dence of the solar cycle is the time variation of sunspots number, with a characteristic period of about 11 years, extensively investigated in the past (see e.g., Ref. [3]). The occurrence of the solar cycle is related to the dynamo effect that generates the magnetic field of the Sun through the αω process (the usual αeffect coupled with the dif ferential rotation) [4]. The spatial behaviour of the solar cycle is related to the latitudinal migration of magnetic structures toward the solar equator as the 11year cycle goes on, thus generating the characteristic “butterfly dia gram” in both space and time domain. Superimposed on these largescale effects, the presence of smallscale ap parently stochastic fluctuations is observed [5]. Apart from the 11year cycle, intracycle periodicities have been discovered in many solar activity proxies. The most prominently recognized periods are in the so called quasibiennial oscillations (QBOs) range on time scales from 1.5 to 3.5 years [912]. This periodicity is better detected in correspondence of main cycle maxima and it suffers, as the 11year cycle, of period length modulation [13]. Quite interestingly, corresponding QBOs have been found also in other contexts related to solar variability, as in solar wind fluctuations, interplanetary magnetic field intensity, galactic cosmic ray (CR) flux [1417] energetic proton fluxes recorded in the interplanetary space [18] and in the solar rotation rate [19]. In they early work, Sakurai [6] invoked the presence of the quasibiennial modula tion for solar neutrino flux, in an attempt to solve the puzzle of missing neutrinos [20]. To date the puzzle has been solved in favour of neutrino flavour transformation However, through the analysis of Homestake data Sa kurai [6] showed the existence of a quasibiennial perio dicity both in the solar neutrino flux and in the sunspot number. In order to make a connection between the pe riodicity observed both in solar neutrino and sunspot data, *Corresponding author. C opyright © 2013 SciRes. JMP
L. D’ALESSI ET AL. 50 [21], also implying a rest mass for neutrinos. Neverthe less, the origin of the biennial modulation of the solar neutrino flux and its interaction with the solar magnetic field are still debated [12,2227]. These modulations should be induced by direct cou pling of neutrino flux with solar magnetic field through neutrino magnetic moment. However, based on Kam LAND data analysis constraint [28], assuming an upper bound for neutrino magnetic moment 12 10 2 2.810 1 0 m jm j (1) a neutrino oscillation length λosc ≈ 100  200 km and magnetic field fluctuation with amplitude δB ≈ 50  100 kG, the deviation from the mean rate for SFP mechanism results to be of the order of % [28]. This means that the coupling of neutrino’s magnetic moment and magnetic field gives negligible effects. On these basis, the most reliable mechanism seems to be the modulation of the production rate of the nuclear reactions or the variation of physical parameters, mainly the density, at the solar core. In order to affect appreciably the neutrino flux, the density fluctuations have to satisfy both the following requests at the position of the MSW oscillation [29,30]: 1) The correlation length of these fluctuations has to be of the same order of neutrino oscillation length; 2) The fluctuations amplitude have to be at least of ~1%. The most plausible mechanism, which in principle could originate fluctuations in matter density with the required properties, is the Alfvén/gmodes resonance [31]. The presence of density gradients along gmodes, can excite Alfvén waves resonantly, the resulting waveforms show sharp spikes in the density profile at radii compa rable with the neutrino’s resonant oscillation length. Hence, the study of shortterm periodicities of the solar cycle should lead to improve knowledge of the global properties of the Sun, with particular regard to solar neu trinos and energetic particle emission. In particular, the possible coupling neutrinosolar activity can help to un derstand the physical processes occurring in the solar deeper layers not accessible to helioseismic probing. In the present paper we resume the study of the quasi biennial solar cycle (see [12,3233]) by investigating the time evolution of two different datasets, through the em pirical mode decomposition (EMD), with particular at tention to the statistical significance of the analysis. We claim that the modulation can be the manifestation of the interaction of solar neutrino flux with Alfvén/gmode resonance modulated by an oscillating magnetic field deep within the solar radiative region. 2. The Neutrino Datasets In order to investigate the relationship between solar neutrinos and magnetic activity, we report the results of EMD analysis carried out solar neutrino flux data re corded from the Homestake experiment (dataset νH) (a total of 108 records from 1970 to 1994 [34]) and from the SAGE experiment (dataset νS) (a total of 168 records from 1990 to 2008 [35]). The data from these two experiments cover a time window of ~20 yr, passing through the maxima of two solar cycles (Cycles 21 and 22 for Home stake data and Cycles 22 and 23 for SAGE data). The EMD results for the two solar neutrino datasets have been compared with that obtained from the data of several solar cycle indicators: sunspot number (SN) and area (SA), flux of interplanetary protons in the energy range 0.50  0.96 MeV/nucleon measured by the charged particles meas urements experiment (channel P2) aboard the IMP8 spacecraft (P2) and cosmic ray intensity measured by the Rome neutron monitor with cutoff rigidity of 6 GV (NM)1. 3. The Empirical Mode Decomposition The periodicities and their relative amplitudes have been identified through the EMD, a technique developed to process nonstationary data [36] and successfully applied in different contexts, e.g. [37,38]. In the EMD framework, a time series X(t) is decomposed into a finite number of oscillating intrinsic mode functions (IMF) as tCtrt (2) The IMFs Cj(t) represent a set of basis functions ob tained from the dataset under analysis by following the “sifting” procedure described by Huang et al. [36]. This procedure starts by identifying local minima and local maxima of the raw signal X(t). The envelopes of maxima and minima are then obtained through cubic splines and the mean between them, namely m1(t), is calculated. The differences between the raw time series and the mean series htXt mt 11 , represents an IMF only if it satisfies two criteria: 1) the number of extremes and zero crossings does not differ by more than one; 2) at any point, the mean value of the envelopes defined by the local maxima and the local minima is zero. 4. Results and Discussion The EMD represents a powerful tool to study the solar QBOs and highly nonstationary signal. Since these oscil lations are high during the activity maxima [9,11,13] and their frequency is not constant from a cycle to another [13] the EMD is more suitable than the classical Fourier and wavelet analysis, to properly identify the QBOs. In fact it is well known that, in presence of nonstationary 1SN and SA data at: http://solarscience.msfc.nasa.gov/SunspotCycle. shtml; P2 data at: http://sdwww.jhuapl.edu/IMP/imp_cpme_data.html; CR data at: http://www.fis.uniroma3.it/svirco/. Copyright © 2013 SciRes. JMP
L. D’ALESSI ET AL. CopyrigJMP 51 signals, the Fourier power spectrum, as well as the time integrated Wavelet spectrum, detect broader and lowered peaks. Since the Fourier transform looks for a global frequency and does not take into account possible period modulations, an underestimation of the contribution of the QBOs could occur. For each dataset, the QBO con tribution to the original signal has been isolated through partial sum of IMFs oscillating with time scales in the range 1.4 yr ≤ τi ≤4 yr, where τi denotes a typical average period for the ith IMF. The QBO contribution from the Homestake is shown in Figure 1 toghether with the quasibiennial signals of P2 and NM, while in Figure 2 the QBO signal extracted from SAGE data is shown toghether with the QBOs of sunspot data. As a reference, the time history of the sunspot area for the period of ref erence is reported in the lower panel. After properly identifying the QBO components through the EMD from the different indicators, we com pare them by evaluating Pearson’s correlation coefficient. For each correlation coefficient, a confidence level of 95% is derived both through Fisher’s transformation (ΔrF) and bootstrap methods (Δrboot). Finally an estimation of the pvalue (i.e. the probability to obtain by chance a correlation coefficient greater then that observed) is given by random phases method (PRP) [39]. Results de monstrate that the correlation is stronger around the solar cycle maxima where the QBO amplitudes are higher. In particular, the QBOs isolated from Homestake data are in phase with particles data around the maxima of cycle 21 and 22, while QBOs isolated from SAGE data seem to be correlated with those of sunspot data near the maxima of cycle 22 and 23. This correlation is significant even ex tending the time window to 11yr starting from mid1991. In Table 1 are shown the results of the correlative analysis. Figure 1. Upper panel: QBO isolated from Homestake data (solid line), P2 proton flux (dotted line) and galactic CRs (dashdotted line). Lower panel: Time history of the sunspot areas for the period of reference (in unit of millionths of a solar hemisphere). Dashed vertical lines correspond to maxima of solar cycles. Figure 2. Upper panel: QBO isolated from SAGE data (solid line), sunspot area (dotted line) and sunspot number (dashdotted line). Lower panel: Time history of the sunspot areas for the period of reference (in unit of millionths of a solar hemisphere). Dashed vertical lines correspond to maxima of solar cycles. ht © 2013 SciRes.
L. D’ALESSI ET AL. 52 Notes on tables contents. In Tables 13 are reported, for different couples of QBOs extracted from datasets X and Y, the corresponding Pearson correlation coefficient r evaluated in the period indicated in the caption. ΔrF and Δrboot represent the 95% confidence intervals for the cor relation coefficient from Fisher’s and bootstrap tests, respectively. PRP indicates the probability, calculated through the random phases test, to obtain values greater than rXY due to chance. namely the continuity equation for mass density 5. The MagnetoGravity Modes The observed correlation between solar quasibiennial cycle and solar neutrino flux fluctuations on quasibien nial time scales could represent a direct observation of instabilities induced by quasibiennial dynamo effects in the deeper regions of solar radiative zone. The theory of coupling between large scale magnetic fields and solar matter has been investigated by Burgess et al. [31]. In particular, in the presence of backgroud magnetic fields of reasonable intensity, density gradients allows gmodes to excite Alfvén waves resonantly, causing mode energy to be funnelled along magnetic field lines away from the solar equatorial plane. Magnetogravity waves are described by the usual compressible, ideal magnetohydrodynamic equations, Table 1. Results of correlative analysis for Homestake, en ergetic proton and cosmic ray data for 5 yr around maxima of cycle 21 (1980.25). XY r ΔrF Δrboot PRP νHP2 0.96 [0.90, 0.98] [0.91, 0.98] 0.01 νHNM –0.90 [–0.96, –0.75] [–0.95, –0.78] 0.06 P2NM –0.98 [–0.99, –0.95] [–0.99, –0.97] 0.01 Table 2. Results of correlative analysis for Homestake, en ergetic proton and cosmic ray data for 5 yr around maxima of cycle 22 (1990.75). XY r ΔrF Δrboot PRP νHP2 –0.92 [0.82, 0.97] [0.82, 0.97] 0.03 νHNM 0.93 [–0.99, –0.97] [–0.99, –0.98] 0.01 P2NM –0.99 [–0.97, –0.80] [–0.96, –0.85] 0.03 Table 3. Results of correlative analysis for SAGE, sunspot number and area for 11 yr starting from mid1990. XY r ΔrF Δrboot PRP νSSA 0.58 [0.46, 0.69] [0.45, 0.69] <0.01 νSSN 0.67 [0.56, 0.75] [0.53, 0.78] <0.01 SASN 0.89 [0.85, 0.92] [0.86, 0.92] <0.01 0 t v (3) the momentum equation with the gravity term 1 4π P t vvvg BB (4) and the magnetic field induction equation t BvB (5) The system of equations can be closed by relating the pressure P to the mass density through an energy equa tion dd 1 dd PP Q tt (6) where d/dt is the total time derivative, γ is the ratio of heat capacities, and Q is the sum of all energy density sources and losses, such as heat conductivity, viscosity and ohmic dissipation. Assuming an equilibrium situation where the velocity field and current density are both zero, Equations (3)(6) are linearized by using lowfrequency approximation, in order to filter out the pressure pmodes, and an exponent tial density profile. A plane geometry with a local gravity directed along the zaxis and the background magnetic field along the xaxis is used. Background quantities de pend on z, and a standard massdensity profile coming from solar models exp zH ,,,exp xy 0c is assumed (here the density at the solar centre ρc and the density heightscale H are constant) [29]. All fluctuating quanti ties depends on space and time through xyztAzikx kyt If we consider a slowly varying background magnetic field, we expect that the system, which varies on times of the order of the helioseismic characteristic periods, has enough time to adapt the configuration corresponding to the instantaneous amplitude of the background magnetic field (adiabatic hypothesis). Under this assumption, by using a background magnetic field which varies in time according to 00 0 Bf Be, where 0 , we ob tain two equations for the Fourier coefficients of magnetic field fluctuations 00 Bf t bB and the velocity fluctuations v 2 00 xx x Axzx iik b P ivtikb z bv ve g ve (7) (“primed” quantities are fluctuations). The above set of equations is formally identical to that founded by Burgess Copyright © 2013 SciRes. JMP
L. D’ALESSI ET AL. 53 et al. [31], apart for the fact that in our case 2 22 0 0 4π Bf t A vt is a timedependent Alfvén speed. After some algebraic calculations we finally obtain an equation for the fluctuating magnetic field, whose solution determines all other fluctuating quantities 22 2 22 22 2 1d xA 2 2 22 dd d 10 z z N z bb xA kv t z N z kb 222 z kv t (8) where the perpendicular wavevector is y kkk and we defined the BruntVäisälä frequency 2 Nz gz Pz 00 00 dd 11 dd P z 0B ,cosht (9) which represents the characteristic frequency of the sys tem. Equation (8) describes magnetogravity waves. In the limit 0 it leads to the standard helioseismic gmodes. In absence of gravity and B0 = cost. Equation (8) describes Alfvén waves with frequency ω = ΩA = kx vA. By retaining both gravitation and magnetic field a new singular point occurs when the coefficient of the second derivative term in Equation (8) vanishes. Since this happens at ω = ΩA, it can be viewed as being due to resonance between gmodes and Alfvén waves [31]. Let us come back to the Sun. Since ΩA varies with the dis tance from the centre of the Sun, while, according to usual helioseismology the gmodes frequency is inde pendent on position, the resonance occurs at a particular radius inside the Sun, namely when ΩA crosses the fre quency of one of the gmodes. The occurrence of the resonance depends on the value of B0. This means that, in our case, the existence of the resonance is modulated in time by the term f(ω0t), that is the resonance is timede pendent. Solutions of Equations (7) gives the eigenvalue spec trum as roots of the trascendental equation [31] An 0cosht (10) where x kC where C0(t) represents the timedependent Alfvén veloc ity at the solar centre, and Accordingly, the instantaneous resonant position is given by 0π ,2πln tan 4 x kC t Antn i Nk H 1 π ,2lnRecosh, r znt Hnt (11) The time dependence of solution of Equation (10) re sults in a modulation of the distance between neighbor ing resonant layers with the same period of the back ground magnetic field. This is shown in Figure 3, where we report the time evolution of the distance as a function of the position of the resonance d,1, , rr r zntznt znt (12) The background magnetic field is assumed to have a sinusoidal variation, with a profile defined by 22 00 0,cos sin 22 tt ft 1 π (13) where 0r 3 10 and the function f is defined in the interval [ε, 1] (we used ). As noted by Burgess et al. [31], for reasonably values of the background magnetic field intensity, the distance between resonant layers, at the neutrino’s resonant region, are of the order of the neutrino’s oscillation length. In particular the spikes which occur in density profiles, as a consequence of the resonance, could increase the prob ability of interactions between neutrino flux and solar matter [29]. In Figure 4 we report the time evolution of the lagran gian density perturbation 1 exp,1 2,1 r C zznt Hidnt (14) where d(n, t) denotes the growing factor of the eigenfre quency 11id , and C is defined as follows 2 122 2 1 ,1 41 1 ph s vntkH Cci (15) 1 where ,, hx . As it is evident, the reso nance oscillates in time with a frequency ω0. vnt ntk 6. Conclusions Recent analysis carried on BiSON and GOLF data [40] show that quasibiennial signal has the same amplitude for pmodes at all frequencies. On the other hand the 11yr modulation affects predominantly high frequency pmodes occuring on shallow regions close to the solar surface. This suggests that the dynamo mechanism re sponsible of the mean cycle has its origin at shallow re gions of the solar interior (resonably located near the bot tom of the shear layer extending 5% below the surface), Copyright © 2013 SciRes. JMP
L. D’ALESSI ET AL. Copyright © 2013 SciRes. JMP 54 Figure 3. Time evolution of the distance between resonant Alfvén layers. In the yz plane are reproduced respectively the po sition of the resonances (in solar radius units) and the distance between the resonant layers (in km). The x axis represents the time (in yr). Figure 4. Time evolution of neighbouring density profiles in the region zr ~0.3 Rsun. while another separated quasibiennial dynamo mecha nism could be originated in deeper layers. In this sce nario, the quasibiennial dynamo located in the inner layers of the Sun, is more likely to induce a fluctuating background magnetic field. The latter is the key ingredi ent of the model since allows that correlation length be tween density spikes variates in time. This mechanism could thus produce the observed variations, at the quasi biennial scale, of the solar neutrino flux. with magnetogravity modes is of great interest for solar physics. This coupling could represent a new way to in vestigate the physical properties in the very inner layers of the Sun thus playing the role of a “telescope” for the solar interior. 7. Acknowledgements This work was partially supported by the ASI/INAF Contract No. I/022/10/0, by the European Social Fund— The modulation of solar neutrinos and the coupling
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