Rigidity dependence of 11 year cosmic ray modulation: Implication for theories

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1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009ja014798, 2010 Rigidity dependence of 11 year cosmic ray modulation: Implication for theories H. S. Ahluwalia, 1 M. M. Fikani, 1 and R. C. Ygbuhay 1 Received 17 August 2009; revised 19 March 2010; accepted 26 April 2010; published 3 July [1] We use data obtained with the global network of detectors, with median rigidity (Rm) of response covering a wide range (1 to 200 GV), to derive the rigidity dependence of the 11 year modulation of galactic cosmic rays (GCR) for 4 sunspot cycles (20 to 23). We find that observed rigidity dependence is represented by a power law with negative exponents; the exponents do not depend on the solar magnetic polarity. However, the dependence on GCR rigidity is significantly flatter than is expected from the Quasi linear Theory of modulation formulated by Jokipii. Citation: Ahluwalia, H. S., M. M. Fikani, and R. C. Ygbuhay (2010), Rigidity dependence of 11 year cosmic ray modulation: Implication for theories, J. Geophys. Res., 115,, doi: /2009ja Introduction [2] The study of the time variations of galactic cosmic ray (GCR) intensity at earth orbit began in earnest after Forbush [1966] established a network of shielded ion chambers (ICs) at global sites in nineteen thirties. They blossomed into a distinct field of research with the advent of the International Geophysical Year (IGY) in when a global network of neutron monitors [Simpson, 2000] and muon telescopes [Elliot, 1952] was set up (on the surface and underground) and the contributions from the atmospheric and environmental effects to detector counting rates came to be understood [Dorman, 1957]. The task of integrating the data obtained with diverse detectors at worldwide locations eased with the derivation of the coupling functions relating secondary species (leptons, mesons, baryons) observed on the ground to GCR spectrum incident at the top of the atmosphere [Dorman, 1957]. However, an understanding of the response of detectors to time variations of GCR rigidity spectrum came very slowly [Ahluwalia and Ericksen, 1971; Ahluwalia and Fikani, 2007, and references therein]. [3] In the meantime, several interesting theoretical models have been advanced to understand the observed modulations without measurable lasting success in discriminating among them. The advent of the Space Age (coming on the heels of the IGY) made it possible to make in situ measurements of physical quantities in space leading to a clarification of the basic concepts underlying the mechanisms needed to understand the observations. [4] Morrison [1956] proposed that certain features of the Forbush decreases could be explained in terms of diffusion in the tangled magnetic fields pervading the interplanetary medium, moving (radially) away from the sun. The discovery of the magnetized solar wind [Neugebauer and 1 Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico, USA. Copyright 2010 by the American Geophysical Union /10/2009JA Snyder, 1962; Coleman et al., 1962] impelled Ahluwalia and Dessler [1962] to propose a physical process for the convection of GCRs away from the sun via an electric drift (E B, E = B V) in the Parker interplanetary magnetic field (IMF) spiral, leading to a diurnal anisotropy (in solar time) observed by a detector on the spinning earth. Parker [1964] and Krymsky [1964] independently pointed to the need for including diffusive inward flow, in near balance with the convective radial outward flow to account for the small ( 0.5%) amplitude of the diurnal anisotropy. These insights led to the development of the Parker [1965] equation; a corollary of it is a successful diffusion convection model in which convection is the driver of observed GCR modulations. When details of IMF structure and its evolution with time became better understood [Forman and Gleeson, 1975] it came to be appreciated that Parker equation contains information about all features of GCR modulations. For example, it includes contributions from other drifts in an inhomogeneous IMF; an appreciation of their role for helio latitudinal transport of GCRs came much later [Kota and Jokipii, 1983]. [5] Jokipii [1971] presents an alternate coherent exposition of a comprehensive theory of cosmic ray transport in the solar wind containing random magnetic irregularities proposed by Morrison; GCRs are scattered as they diffuse into the heliosphere from the interstellar medium, causing modulations. Jokipii succeeds in deriving the rigidity (R) dependence of the diffusion coefficient (k) over a range of rigidities from 0.1 to 10 GV; the diffusion is mainly along the mean magnetic field. He finds that there exists a sharp transition at about 2 GV; below it k / R 0.5 while above it k / R 2. According to the diffusion convection theory of GCR modulation, the modulation function (M) varies as the reciprocal of the diffusion coefficient (k); see Gleeson and Axford [1968]. The rigidity dependence of the observed modulations is then key to testing the elegant Jokipii Quasilinear diffusion theory (QLT). We use data from the global network of detectors to carry out this test, over a range of rigidities (1 to 200 GV), for the 11 year modulation for four 1of7

2 Figure 1. (a). Annual mean hourly rates are plotted for NMs at CL, HU as well as VELA, IC, and ions for cycle 20; data are normalized to 100% in May (b) The rigidity dependence. sunspot cycles (20 to 23); the diffusion approximation is shown to be valid over the stated range of GCR rigidities [Ahluwalia and Fikani, 2007]. 2. Detector Rm Values [6] One uses parameterization of latitude survey data obtained with neutron monitors (NMs) to determine their response functions. For details on methodology, the reader is referred to Dorman [1957], Webber and Quenby [1959], Lockwood and Webber [1967], to name a few. The response function for a NM is a product of the integral multiplicity and GCR differential spectrum; the former is a constant but the latter undergoes changes with the sunspot activity. The NM differential response curves have broad (as opposed to peaked, inverted V shape) maxima at 7 GV for sea level (1033 g/cm 2 ) sites and 5 GV for the mountain altitudes (700 g/cm 2 ). [7] Nagashima et al. [1989] carried out an extensive analysis of the latitude surveys for (three solar cycles) and used them to derive response functions for NMs at different atmospheric depths at sunspot activity maximum and minimum; they are used now by the heliophysics community for different purposes [Clem and Dorman, 2000; Bieber et al., 2007]. Ahluwalia and Fikani [2007] use them to compute the values of the median rigidity of response (Rm) for NMs at sea level and mountain altitude sites worldwide, for solar activity maximum and minimum (see their Figures 1 and 2); 50% of a NM counting rate lies below this value of GCR rigidity spectrum. [8] The values of Rm are used by the heliophysics community as a proxy for the study of the rigidity dependence of GCR modulations; see discussion by Lockwood and Webber [1996], pp. 21, , 577]. In other words, Rm can be substituted for R in the theoretical formulas. However, the researchers define Rm in different ways. The values computed by Lockwood and Webber [1996] make two assumptions. First, GCR modulation ceases above 100 GV. Second, Rm value depends on the modulation function assumed by them (see their Figure 2). The observations obtained with a variety of detectors at global sites show that modulation extends to far higher GCR rigidities [Peacock and Thambyahpillai, 1967; Ahluwalia and Ericksen, 1970; Elliot et al., 1972; Speller et al., 1972; Thambyahpillai, and Speller, 1975; Nagashima et al., 1987; Ahluwalia, 1992]. [9] Ahluwalia and Fikani [2007] compute Rm values in a direct manner; in particular, no assumption is made as to the form of the modulation function or the value of the limiting Figure 2. (a) Same as in Figure 1a for cycle 21. (b) The rigidity dependence for cycle 21. 2of7

3 GCR rigidity for a given epoch. For cycles 21/22 Lockwood and Webber [1996] value of Rm = 5.4/7.0 GV for Mt. Washington NM may be compared to Ahluwalia and Fikani value of 10 GV. Later, Lockwood and Webber [1997] give Rm = 10 GV (for cycles 21/22) for Mt. Washington NM in agreement with our value. Later still, Lockwood et al. [2001] give Rm = 14 GV for cycles 20, 21, and 22 for Mt. Washington NM which exceeds our value. In fact, Rm values given by Lockwood et al. [2001] are more than a factor of two higher than those given by Lockwood and Webber [1996] for NMs at different global sites but they are closer to the values computed by Ahluwalia and Fikani [2007], see Table 2 in their paper for a comparison of the two sets of Rm values. [10] A similar situation exists with respect to detectors on space probes and IMP 8 satellite. Van Allen [1979] computes 1.3 GeV as the mean energy for > 80 MeV GCR protons detected on board Pioneers 10 and 11. Lockwood and Webber [1992] give Rm = 1.5 GV for > 70 MeV protons on Voyagers 1 and 2. Later, Lockwood and Webber [1997] compute Rm = 1.8 GV for > 70 MeV protons recorded on IMP 8 compared to Lopate and Simpson [1991] value of Rm = 2.3 GV for the penetrating IMP 8 protons (>90 MeV). [11] Ahluwalia and Lopate [2008] compute the mean energy of response to GCR protons for for Climax neutron monitor (CL/NM) and IMP 8 cosmic ray nuclear composition instrument, covering cycles 21 and 22. They find that for penetrating proton channel on IMP 8 the mean energy changes by 2 (3 to 6.5 GeV) whereas for CL/NM the change is 23% (9.5 to 11.7 GeV). However, the corresponding change for the computed modulation function is a factor 3.8 (500 to 1740 MeV). These findings lead us to question the results of the rigidity dependence of the transport parameters, reported in the literature; experimental observations cannot be used to test a modulation theory without knowing the correct Rm value for the detector. [12] For underground muon telescopes (MTs) we use Rm values computed by Fujimoto et al. [1984] from the theoretical calculations of Murakami et al. [1979]. They obtain the MT response functions at different atmospheric depths by solving numerically the equations of hadronic cascades in the atmosphere; Feynman scaling is invoked for the hadronic interactions of GCRs incident at the top of the atmosphere. Their definition of Rm value for a detector is the same as ours. Gaisser [1974] performed a similar calculation, using accelerator data on proton proton interactions for >1500 GeV in lab frame; the results are in essential agreement with Murakami et al. [1979] (detailed) computations, where they overlap. Ahluwalia and Ericksen [1971] show that MTs are much more sensitive to rigidity dependent changes in modulation than are NMs. [13] Ahluwalia and Sabbah [1993] study the characteristics of the diurnal anisotropy for a solar magnetic cycle ( ) at GCR rigidities: 16 GV Rm 331 GV. They show that diurnal anisotropy has east west and radial components as expected from the diffusion convection model [Riker and Ahluwalia, 1987; Ahluwalia, 1988, 1994a] derived from the Parker equation. Furthermore, Ahluwalia and Sabbah show that diurnal anisotropy data are consistent with the ratio: k?/kk = 0.1, independent of GCR rigidity; k?, kk are the components of the diffusion coefficient normal and parallel to mean IMF. They also confirm that at high GCR rigidities the contribution of drift terms (in the Parker equation) to diurnal anisotropy is small but finite. Bieber and Chen [1991] reach similar conclusions about diurnal anisotropy derived from the data for a narrower range (17 GV Rm 67 GV) of GCR rigidities but a longer period ( ). Earlier, Riker and Ahluwalia [1987] showed that if one assumes that kk/r 2, the value of the heliospheric radial gradient comes out to be extremely small compared to what is observed. They conclude that rigidity dependence of kk must be considerably flatter. [14] Ahluwalia and Fikani [2007] and Ahluwalia et al. [2009] have used the computed Rm values for the detectors in a detailed study of the rigidity dependence of the large amplitude Forbush decreases observed during the declining phase of four sunspot cycles (19 to 22); GCR rigidity range covered: 1 GV Rm 300 GV. 3. Eleven Year Modulation [15] Forbush [1954] showed that 11 year modulation observed by four ICs (Rm = 67 GV) of his worldwide network bear an inverse correlation with sunspot numbers (SSNs) for , covering sunspot cycles 17 and 18. The relationship continues to be valid for an extended data set available today [Ahluwalia, 2005]. Also, comprehensive analyses by Ahluwalia [1980, 1994a, 1994b, 1996, 2000a, 2000b] have established several new features, not foreseen by Forbush. [16] Figure 1a is a plot of the annual mean hourly rates for NMs at Climax (Rm = 11 GV) and Huancayo (Rm = 33 GV), IC, neutrons produced in the skin of VELA satellites by protons with an energy > 25 MeV (data supplied by late John R. Asbridge, Los Alamos National Labs), and ions (>0.1 GeV) observed by Bazilevskaya et al. [1991] at a high latitude balloon altitudes (Rm = 3.9 GV) for ; the data are normalized to 100% in May The epochs of sunspot maximum (M) and minima (m) are indicated on the bottom scale; vertical dashed line marks the epoch of the polar field reversal in the solar northern hemisphere for cycle 20 [Howard, 1974]. [17] The general behavior is similar for all detectors; GCR intensity (I) is high in 1965, 1 year after m epoch (a typical behavior) and is most depressed by the time solar polar field reversal starts in 1969, a significant additional decrease occurs for >0.1 GeV ions in We define the modulation amplitude as follows: Amplitude ð%þ ¼ I1965 I1969 I1965 ð100þ [18] Ahluwalia [2003] discusses the modulation observed after 1972 (solar minicycle). The inverse relationship between the amplitude of modulation and SSNs as well as with Rm values for the detector are features to be noted; the amplitude is small for IC (Rm = 67 GV) and large for >0.1 GeV ions (Rm = 3.9 GV); rigidity dependence of the modulation amplitude (%) for (negative) cycle 20 is shown by the diagonal line in Figure 1b, the plot includes data from NMs at Deep River (DR, Rm = 16 GV), Hermanus ð1þ 3of7

4 Figure 3. (a) Same as in Figure 1a for cycle 22. (b) The rigidity dependence for cycle 22. (HE, Rm = 20 GV) and Rome (RO, Rm = 23 GV). The straight line fit is a power law in rigidity, with an exponent 1.26 and a correlation coefficient (cc) 0.99 in log log space. [19] Figure 2a is a plot of the annual mean hourly rates for NMs at Climax, Huancayo, Kiel (Rm = 17 GV), Deep River, IC, and ions (>0.1 GeV) measured at a high latitude balloon altitudes by Bazilevskaya et al. [1991] for ; data are normalized to 100% in May 1965, and epochs of sunspot maximum (M) and minima (m) are indicated on the bottom scale, vertical dashed line marks the epoch of solar polar field reversal in the northern hemisphere for cycle 21. [20] Ahluwalia [1994b] noted that GCR intensity is most depressed nearly 3 years after the M epoch for cycle 21; later Ahluwalia [2000a] showed that IMF (B) reached a maximum value in In Figure 1a, one notes that recovery for (positive) cycle 21 takes a longer time compared to that for (negative) cycle 20, a pattern first noted by Ahluwalia [1980]. These findings were not anticipated by the Forbush analysis. The rigidity dependence of amplitude (%) for (positive) cycle 21 is shown in Figure 2b, data are also included for NM at Rome. The straight line fit to data is a power law in rigidity, with an exponent 1.36 and a correlation coefficient (cc) 0.99 in log log space. In spite of the significant differences in the timelines of the two cycles, the rigidity dependence does not seem to be all that different for the positive and negative cycles. [21] Figure 3a is a plot of the annual mean hourly rates for NMs at Climax, Huancayo, Kiel, Deep River, IC, and ions (>0.1 GeV) measured at a high latitude balloon altitudes by Bazilevskaya et al. [1991] for ; data are normalized to 100% in May 1965, and epochs of sunspot maximum (M) and minima (m) are indicated on the bottom scale, vertical dashed line marks the epoch of solar polar field reversal in the northern hemisphere for cycle 22. There are no data for IC after 1993; DR/NM operations were stopped after Like cycle 20 before it, (negative cycle) cycle 22 recovers more rapidly than (positive) cycle 21. [22] The rigidity dependence of the amplitude (%) for cycle 22 is shown in Figure 3b, data are also included for NMs at Hermanus and Rome. The straight line fit to data is a power law in rigidity, with an exponent 1.11 and a correlation coefficient (cc) 0.99 in log log space; the rigidity dependence for cycle 22 is a little flatter than the preceding cycles. [23] Figure 4a is a plot of the annual mean hourly rates (%) for NMs at Climax, Haleakala, Hermanus, Kiel, Oulu Figure 4. (a) Same as in Figure 1a for cycle 23. (b) The rigidity dependence for cycle 23. 4of7

5 (Rm = 16 GV), Rome, as well as the penetrating protons and the Helium channels on IMP 8 for (cycle 23); the data are normalized to 100% in January 1996, and epochs of sunspot maximum (M) and minima (m) are indicated on the bottom scale, vertical dashed line marks the epoch of solar polar field reversal in northern hemisphere. The data are not yet processed for NMs at Climax and Haleakala after 2006 (for lack of support to PI); IMP 8 stopped operating after October One notes that all NMs show recovery above the 1996 level; they have all recovered to the highest level ever since continuous monitoring by NMs began in 1951 [Ahluwalia et al., 2010]. [24] The rigidity dependence of amplitude (%) for (positive) cycle 23 is shown in Figure 4b, data are also included for MT at Mawson, Australia, at 37 m water equivalent depth of rock within the cone of viewing of the telescope (Rm = 164 GV). This is the first time that data from a MT at an underground site has been used for the study of the rigidity dependence of 11 year modulation. The straight line fit to data is a power law in rigidity, with an exponent 1.22 and a correlation coefficient (cc) 0.99 in log log space, over a wide range of GCR rigidity spectrum (1 to 200 GV). It is very gratifying that a single power law fits the observations over 2 decades of GCR rigidity spectrum. 4. Discussion and Summary [25] We have studied GCR 11 year modulations for four sunspot cycles for ; two are positive and two negative (two Hale cycles). We use data from a variety of detectors of the global network, covering a wide range of GCR rigidity spectrum (1 to 200 GV). The observed modulations all show a power law dependence on rigidity R, with no systematic difference between exponents for the positive and negative cycles, around a mean value 1.24 ± [26] Parker equation can only be solved numerically. Besides it points to no preferred choice of transport parameters or the configuration of IMF. A spherical symmetry solution of the Parker equation [Gleeson and Urch, 1973] is often invoked to understand modulation in terms of a force field parameter ; it represents a GCR rigidity loss in the heliosphere. The appeal of this approach lies in the fact that observed modulation (over a range of rigidities) can be described in terms of a single parameter ( ); it is a chargeless heliospheric potential (volts), GCR species behave as if they are all positively charged, see Ahluwalia [2005] for a discussion of this approach. The model retains much of the physics of the Parker equation e.g., diffusion, convection, and adiabatic energy loss, only drifts are left out. [27] For the force field solution at R > 1 GV, 3 =RM i.e., M / 1/R, where M (dimensionless) is the modulation function [Gleeson and Axford, 1968; Caballero Lopez and Moraal, 2004, equation (17)], under the assumption that diffusion coefficient (k) is separable into the form: k(r, R) = k1(r)k2(r), where k2 (R) / R and k1 / 1/B [Ahluwalia, 2005]. It follows that the slope of lnm versus R curve gives the rigidity dependence of the diffusion coefficient at these GCR rigidities. The observations indicate that a simple power law fits the data quite well down to 3 GV. At lower rigidities, McDonald et al. [1992] show that k2(r) / R s, the power law index changes from s 0.6 to 0.9 in the inner heliosphere, in keeping with our result at the earth orbit at higher rigidities. Other colleagues have used this approach to fit observations for a variety of GCR species at earth orbit, over an extended time period [Gleeson and Urch, 1972; Evenson et al., 1983; Perko and Burlaga, 1992; McCracken and McDonald, 2001]. McCracken et al. [2004] used this approach in a comprehensive study of modulation of galactic cosmic radiation over the past 1100 years ( ) using 10 Be data from Greenland and South Pole. Similarly, Usoskin et al. [2005] employed the same technique to compute the monthly values of for from the data obtained with the global network of NMs; the computed values of correspond to fragmentary estimates of GCR spectrum from balloon/spacecraft measurements for the overlapping time periods. [28] It may be useful to recall that Lockwood and Webber [1979] tried to explain cycle 20 modulation for Mt. Washington NM and other global detectors with the idea of a transition rigidity advocated by Jokipii [1971]. They assume k / R from 3 GV to 25 GV and k / R 2 at higher rigidities, up to 100 GV. They expect hysteresis effect of 1.5% for the plots of Mt. Washington NM monthly rate versus the rates of Hermanus and Pic du midi NMs; it is not observed. Their experience is then similar to Riker and Ahluwalia [1987] experience with the diurnal anisotropy data, for an overlapping time period. We chose to include Mawson MT data in this analysis hoping to extend the rigidity range of the Lockwood Webber analysis, expecting that datum point for MT (Rm = 164 GV) may clearly require a two region power law fit that Lockwood and Webber were looking for. Instead, we are surprised to see that power law fit to cycle 23 data is mainly defined by three points, two at the lowest and one at the highest GCR rigidities with some scatter for NM data; Ahluwalia and Ericksen [1970, 1971] note that MTs are much more sensitive to rigidity dependent changes in modulation than are NMs. So, if a transition rigidity exists it must have a value > 164 GV, a far higher value than Jokipii s suggested value. [29] The empirical evidence presented in this paper suggests that QLT developed by Jokipii [1971] needs a revision to fit the observations at higher GCR rigidities. We note that several improved theories have been developed following Jokipii s work, presenting a detailed and deep analysis of the perpendicular diffusion coefficient for GCRs [Shalchi, 2009]. These efforts are driven by attempts to explain data collected by the detectors on board Voyagers 1 and 2 [Burlaga et al., 1993]; they respond to lower GCR rigidities as discussed in section 2. These spacecrafts are now in the heliosheath region, observing some unusual and interesting modulations that cannot be explained by the force field solution of the Parker equation. In the last decade, most of the refinements in cosmic ray transport theory and modeling at lower GCR rigidities are aimed at understanding the Voyager data. Ahluwalia [2005] argues that heliosheath related modulations contribute minimally to cosmic ray detector counting rate at earth orbit, at R > 3 GV. [30] The results presented in this paper should alert theorists and modelers to pay more attention to the physical significance of results obtained at higher GCR rigidities. The lack of an agreement between the observations presented and discussed in this paper and the Jokipii quasilinear theory for parallel diffusion probably points to 5of7

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