THE INFLUENCE OF COSMIC-RAY MODULATION AT HIGH HELIOSPHERIC LATITUDES ON THE SOLAR DIURNAL VARIATION OBSERVED AT EARTH

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1 The Astrophysical Journal, 629: , 2005 August 10 # The American Astronomical Society. All rights reserved. Printed in U.S.A. THE INFLUENCE OF COSMIC-RAY MODULATION AT HIGH HELIOSPHERIC LATITUDES ON THE SOLAR DIURNAL VARIATION OBSERVED AT EARTH H. Moraal School of Physics, North-West University, Private Bag X6001, 1 Hoffman Street, Potchefstroom 2520, South Africa; fskhm@puknet.puk.ac.za R. A. Caballero-Lopez Instituto de Geofísica, Universidad Nacional Autónoma de México, 04510, México K. G. McCracken Institute for Physical Science and Technology, University of Maryland, College Park, MD and J. E. Humble School of Mathematics and Physics, University of Tasmania, Hobart, Tasmania 7001, Australia Receivved 2005 March 14; accepted 2005 April 29 ABSTRACT During the solar minimum period of 1954 the cosmic-ray diurnal variation as observed by neutron monitors and muon telescopes underwent a dramatic swing in its direction of maximum intensity, from the normal value between 16:00 and 18:00 local time to as early as 08:00. It is shown that this swing can be explained as being due to a negative radial density gradient of cosmic rays in the inner heliosphere and that this negative gradient is caused by large radial and latitudinal diffusion mean free paths that bring in particles from high latitudes. In principle, such large diffusion mean free paths should simultaneously cause high intensities, as were observed in Subject headingg: cosmic rays 1. INTRODUCTION The solar time diurnal variation since 1932 in Galactic cosmic radiation has been studied by various investigators, and reliable data are now available for three heliomagnetic cycles from 1933 to Using data from eight ionization chambers that were in operation prior to 1953, Thambyahpillai & Elliot (1953) demonstrated that the phase exhibited a 22 yr periodicity, being several hours earlier at the sunspot minima of 1933 and 1954 than at the 1944 minimum. A number of investigators, e.g., Forbush (1954) and Parsons (1957, 1959) confirmed that result, as summarized by Venkatesan & Dattner (1959), who also showed that the phase during the solar minimum of 1954 was 6 8 hr earlier than that observed throughout the interval Subsequent analysis of both ionization chamber and neutron monitor data by Bieber & Chen (1991) showed that the diurnal phase was earlier in 1954 than at any other time in the interval Figure 1 plots the annual diurnal phase, based on the results reported in all three of the above papers, together with recent neutron monitor data from Climax and Haleakala (Hawaii). We note that the diurnal expansion and contraction of Earth s atmosphere introduces a diurnal variation into the muon component of the cosmic radiation (Duldig 2000) with a maximum at about 04:00 local time. This small diurnal variation, a direct consequence of the diurnal heating of the atmosphere by the Sun, remains essentially invariant from year to year. For the purposes of this paper we note that (1) it does not affect the neutron data (i.e., in Fig. 1) and (2) its removal from the ionization chamber data would shift the free space phases depicted in Figure 1 to a later time by 1 2 hr for observed phases <12:00 hr, to a later time by 0 1 hr for an observed phase between 12:00 and 16:00 hr, and to earlier times by hr for 556 phases later than 16:00 hr. For this paper we have not corrected for this effect, but we note that the time of maximum in 1954 might be up to 2 hr early as a result of it. Therefore, all discussions are in terms of the time of maximum being 10:00 hr in All phases have been corrected for the effects of the deflection of the cosmic radiation in the geomagnetic field. Consequently, they indicate the direction of the maximum cosmic-ray intensity in interplanetary space. Assuming a 10:00 hr phase for 1954 indicates that the predominant flow of cosmic rays at Earth was approximately aligned with the outward direction of the heliospheric magnetic field (HMF). This early phase was observed in 1954 by three ionization chambers (Huancayo, Cheltenham/Fredericksburg, and Christchurch) and by a large threefold coincidence telescope at Hobart (Parsons 1959) and can be regarded as well established. Figure 1 shows that the while the phase of the diurnal variation was substantially earlier in the vicinity of each of the sunspot minima of 1933, 1954, 1976, and 1996, the actual phases attained varied between 08:00 and 14:00 hr (10:00 and 15:00 hr after temperature correction). This rather wide range is to be contrasted with the consistent phase of 18:00 hr attained for the majority of the interval Recently, McCracken et al. (2004) reported that the concentration of the cosmogenic isotope 10 Be (responding to cosmicray energies in the vicinity of 2 GeV nucleon 1 ) and the balloon measurements of Neher (1967 and references therein) indicate that there was an unusual influx of cosmic radiation into the vicinity of Earth in The 10 Be data for the period subsequent to 1954 and the MeV data obtained by the Interplanetary Monitoring Platform (IMP) and other satellites indicate that this has not occurred during subsequent minima. McCracken et al. (2004) speculated that the influx and the

2 COSMIC-RAY MODULATION AND SOLAR DIURNAL VARIATION 557 field structure. These transport processes are described by the cosmic-ray transport equation, which can be written in the form of an equation of continuity for the omnidirectional part of the distribution function f, þ :=S þ p p2 hṗif ¼ Q; ð1þ S ¼ 4p 2 ðcvf K =:f Þ ð2þ Fig. 1. Phase (time of maximum) of the diurnal variation in free space from 1932 to The Hafelekar data are from Thambyahpillai & Elliot (1953) and for Cheltenham and Huancayo from Bieber & Chen (1991), and from 1985 onward the data are from NeutronMonitor/neutron_ mon.html. The corrections for geomagnetic deflection are as given by Bieber & Chen (1991). The inset shows directional conventions. 08:00 09:00 hr diurnal maximum in 1954 were both due to enhanced cosmic-ray diffusion from high heliolatitudes. They noted that the mean sunspot number for 1954 was one of the lowest on record and suggested that this may indicate a more ordered heliomagnetic field than those during subsequent solar minima. The development of the cosmic-ray transport equation by Parker (1965) provided a physical basis that allowed the 22 yr periodicity of the diurnal variation phase to be understood. Thus, Bieber & Chen (1991) demonstrated that the observed 22 yr periodicity in phase and 11 yr periodicity in amplitude were consistent with the drift and gradient terms in the cosmicray transport equation. They demonstrated that the product of the parallel mean free path and the radial gradient of GeV cosmic rays was a factor of 2 greater at the sunspot minima of 1944 and 1965 than at that of However, they did not examine the cause of this difference or the very early diurnal phase in Lemmer & Moraal (1991) analyzed the Bieber & Chen results analytically and showed that the predominantly 11 yr amplitude and 22 yr phase variations are a simple consequence of the geometry of the Parker spiral magnetic field and are only indirectly due to the drift terms in the cosmic-ray anisotropy. Using these previous results, together with those from the numerical solution of the cosmic-ray transport equation, it is shown in this paper that the early diurnal phase of 1954 can be explained in terms of a negative radial density gradient that is a consequence of large diffusion mean free paths during solar minimum conditions. The 08:00 14:00 hr range (10:00 15:00 hr after temperature correction) for the phase during the four socalled qa > 0 sunspot minima 1933, 1954, 1976, and 1996 is then explained in terms of different radial gradients at these times. 2. THE COSMIC-RAY TRANSPORT MODEL AND ANISOTROPY EXPRESSIONS Cosmic-ray particles scatter off the irregularities in the HMF. Since these irregularities are frozen into the solar wind, the cosmic rays are also convected outward with the wind speed, and as they are carried in the wind they cool adiabatically due to the expansion of the wind with radial distance. The particles also experience gradient and curvature drifts in the background is the particle flux, Q is a source function, hṗi ¼ ðp/3þv =:f /f is the adiabatic loss rate of momentum p, V is the solar wind velocity, K(r, P, t) is the diffusion tensor, and C ¼ (1/3)@ ln f /@ ln p is the Compton-Getting coefficient. Momentum may be changed to rigidity P, kinetic energy per nucleon T, or particle speed ¼ v/c with the relationship P ¼ pc/q ¼ A/Z½T(T þ 2E 0 ) Š 1 =2 ¼ (A/Z )(T þ E 0 ), where A and Z are mass and charge number, respectively, and E 0 ¼ 938 MeV is the rest-mass energy of a proton. When S and hṗi are put into the equation, it results in the more familiar form, written by Parker þ V =:f := ð K =:f Þ 1 3 ln p ¼ Q: The diffusion tensor K contains the elements k (r, P, t) and? (r, P, t) for scattering along and perpendicular to the HMF B, together with an antisymmetric coefficient T ¼ P/(3B) that describes gradient, curvature, neutral sheet, and shock drift effects (e.g., Isenberg & Jokipii 1979; Moraal 2001). The HMF structure is described by the standard Parker spiral magnetic field, given in spherical polar coordinates (r,, ) byb ¼ B e (r e /r) 2 (e r tan e ), with ð3þ tan ¼ ðr r 0 Þsin =V; ð4þ where is the angular frequency of solar rotation and r 0 is the Alfven radius of several solar radii. The observed values of the field at Earth B e vary from 5 nt at solar minimum to 10 nt at solar maximum conditions. The value of /V ¼ 1AU 1 if V ¼ 400 km s 1 and ¼ 1 solar rotation per days. The streaming vector (2) is written in the usual fashion in terms of an anisotropy vector x ¼ 3S/(4p 2 vf ), diffusion mean free paths k i ¼ 3 i /v, and gyroradius ¼ 3 T /v ¼ P/Bc as x ¼ kgþ3cv=v ¼ k k g k k? g? e B < g þ 3CV=v; where e B ¼ B/B and g ¼ :f /f is the density gradient. When the cosmic-ray intensity is azimuthally symmetric (@/@ ¼ 0), this vector has radial, latitudinal, and azimuthal components given by r ¼ 3CV=v k rr g r þ ðsign BÞ sin ð Þg ; ¼ ðsign BÞ sin ð Þg r k g ; ¼ k r g r þ ðsign BÞ cos ð Þg ; where k rr ¼ k k cos 2 þ k? sin 2, k ¼ k?,andk r ¼ (k? k k )sin cos ¼ (k k rr )tan. The quantity sign B ¼þ1 ( 1) in the qa > 0(<0) drift cycles. ð5þ ð6þ ð7þ ð8þ

3 558 MORAAL ET AL. Vol. 629 The radial and azimuthal components of this anisotropy constitute the diurnal variation, with magnitude and phase 2 ¼ r 2 þ 2 ; ¼ 180 þ tan 1 = r =15 hr; ð9þ where the phase is expressed as the time of maximum intensity in hours. For a typical power-law intensity spectrum of the form P 2:5 ( f / P 4:5 ), the Compton-Getting coefficient C ¼ 1:5, so that in a solar wind V ¼ 400 km s 1, and for relativistic particles (v ¼ c ¼ 3 ; 10 8 ms 1 ), the convective part of the radial anisotropy is c ¼ 3CV/v ¼ 0:6%. In the most classical interpretation, summarized by Moraal (1976), there are no drifts, k? Tk k, and there is a balance between radially inward diffusion and outward convection, so that r ¼ 0, and it therefore follows from equation (6) that g r ¼ 3CV /(vk k cos 2 ). Putting this into equation (8) then gives ¼ ¼ 3CV tan /v ¼ (3CV /v)(r r 0 ) ¼ 0:6% at r ¼ 1 AU, and ¼ 18:00 hr. Because of the (r r 0 ) term, this is sometimes called the corotation anisotropy because its value is the same as it would be if particles corotated with the Sun (which they do not). The directional conventions are shown on Figure 1. There are significant deviations from this baseline value, because the observations stated above indicate an 11 yr wave in the amplitude, with minimum values at solar minimum (Bieber & Chen 1991), and a 22 yr wave in phase, with the time of maximum near to 18:00 hr in the qa < 0 solar minima of 1944, 1965, and 1987, while it is nearer 12:00 in the qa > 0 minima of 1933, 1954, and 1976, with the 1954 value, according to some of the observations, perhaps as early as 08:00. We follow the notation of Bieber & Chen (1991) by letting ¼ k? /k k and concentrate on the position of Earth where (r r 0 )sin ¼ 400 km s 1. Thus, from equation (4) it follows that tan ¼ 1/W,whereW ¼ V(km s 1 )/400 is the dimensionless solar wind speed. Then the radial and azimuthal components (eqs. [6] and [8]) can be written as r ¼ 0:004CW = d þ W 2 = 1 þ W 2 dr ; ð10þ ¼ d ð1 ÞW= 1 þ W 2 dr ; ð11þ where d ¼ k k g r and dr ¼ g W /(W 2 þ 1) 1 =2.Fromthemeasurement of r and and assuming c ¼ 0:6% as above, Bieber & Chen (1991) calculated the two unknown quantities d and dr for a series of neutron monitor observations and for the Cheltenham ionization chamber over several solar cycles from 1933 to (They noted that g is bidirectional in nature and therefore introduced and calculated a bidirectional latitudinal gradient, but this is not strictly necessary.) The basic result was that d in the qa > 0 cycles is considerably smaller than in the qa < 0 cycles, while dr alternates in sign between these two cycles due to the well-known switch in g. Lemmer & Moraal (1991) pointed out that the dr drift terms are not the primary cause of the observed 11 yr periodicity in amplitude and 22 yr periodicity in phase because (1) they are too small and (2) they are always in the same direction, regardless of the drift state of the heliosphere and the side of the wavy neutral sheet near the ecliptic plane. First, from ¼ P/Bc it follows that ¼ 0:0044P(GV) AU in a 5 nt field. Thus, with ¼ 45, the magnitude of the two drift terms in equations (6) and (8) is less than c if jg j < 200% AU 1. Typical modulation calculations (Caballero-Lopez et al. 2004), however, indicate that jg j10=p(gv)% AU 1. Thus, for all rigidities of interest, these drift terms are typically only 5% of the convective term. Second, modulation models also confirm that when sign B ¼þ1 ( 1) north (south) of the neutral sheet in the qa > 0 drift cycles , , and , there is an intensity minimum on the sheet. Conversely, when sign B ¼ 1 (+1) north (south) of the sheet (as in the qa < 0 cycles , , and ), there is an intensity maximum on the sheet, Thus, g switches sign together with sign B, and the small drift terms in equations (6) (8), or equations (10) and (11), are always 0 for both positions relative to the neutral sheet and in both magnetic cycles. This does not mean that drifts are not important for changes in the diurnal variation. In x 3 it is shown that these drifts completely rearrange the spatial distribution of cosmic rays in the heliosphere, thereby altering the gradients. Thus, drifts are important in determining the anisotropy, but the mechanism is through the gradient-driven diffusive term. Thus, given that C is fairly fixed at 1.5 at energies >1 GeV, the diurnal variation is a function of just three parameters. They are the diffusive parameter d ¼ k k g r, the ratio ¼ k? /k k,andthe solar wind speed W ¼ V/400. In x 3 we explore this parameter dependence. 3. ORIGIN OF THE AMPLITUDE AND PHASE SWINGS The primary reason for the 11 and 22 yr variations in the amplitude and phase of the diurnal variation is the change in the radial gradient g r from qa > 0to<0cycles, while the very early time of maximum in 1954 is due to this radial gradient becoming negative. This can be demonstrated by starting with a numerical solution of the cosmic-ray transport equation (3). Details of such a solution can be found in Caballero-Lopez et al. (2004). In the present case it is sufficient to use a drastically simplified version of that solution, noting that temporal, termination shock, and heliosheath effects play no role of any consequence in the diurnal variation in the inner heliosphere. Consequently, the model is azimuthally symmetric (@/@ ¼ 0); the outer boundary is at r b ¼ 100 AU, where a proton input spectrum of the form given by Caballero-Lopez et al. (2004) is placed; the drift in the wavy neutral sheet (assumed tilt angle is 10 ) is simulated as in method (3) of Caballero-Lopez & Moraal (2003); the solar wind speed is 400 km s 1 in the ecliptic plane, increasing to 800 km s 1 at the poles, as described in Caballero-Lopez et al. (2004); while the diffusion mean free paths were chosen as k rr ¼ 0:5P(GV) (cos 2 þ 2) AU, and k ¼ 0:05P(GV) AU. Figure 2 shows the 10 GeV intensity calculated from these solutions as a function of radial distance. The four panels are for an effective one-dimensional case in which both the drift and latitudinal diffusion are switched off (Fig. 2a), the previous case but with latitudinal diffusion switched on (Fig. 2b), and the addition of drift in the qa > 0(Fig.2c) andqa < 0(Fig.2d) cases. The solid line is the intensity in the ecliptic plane, while the dashed, dotted, and dot-dashed lines are for polar angles (colatitudes) of 60,30,and0 (pole), respectively. The effective one-dimensional intensities are lower in the ecliptic plane than above the poles because with the chosen parameters, the modulation parameter Vr/ rr is 1.5 times larger in the ecliptic than at the pole. This latitudinal gradient is much smaller in case b due to the latitudinal diffusive transport being switched on. The same behavior is observed in the qa > 0 solution of case c, but in the qa < 0 case d there is a small negative latitudinal gradient with the intensity maximum in the ecliptic plane. This behavior is well understood as due to the switch in drift direction, which is basically equatorward in the qa > 0 and poleward in the qa < 0cycles. The central argument for our interpretation of the diurnal variation is that in the ecliptic plane of the inner heliosphere

4 No. 1, 2005 COSMIC-RAY MODULATION AND SOLAR DIURNAL VARIATION 559 Fig. 4. Time of the maximum of the diurnal variation. Fig. 2. Solutions of the cosmic-ray transport equation showing 10 GeV intensities as a function of radial distance for (a) a one-dimensional model without drift or latitudinal diffusion, (b) a no-drift model with latitudinal diffusion, (c)modelb with qa > 0 drift added, and (d )modelb with qa < 0drift added. (r < 10 AU), the radial gradient is negative in the no-drift and qa > 0 cases b and c but positive in the qa < 0cased. This behavior happens because the latitudinal gradient g ¼ f 1 /(r@) ¼ f 1 /@s becomes very large as r! R. Consequently, the latitudinal diffusive transport term (@/@s) ½sin ð /@sþš becomes increasingly important with decreasing r, short-circuiting the latitudinal gradient in the inner heliosphere. In the qa < 0casethe radial gradient remains positive because this effect is counteracted by the poleward drift, which increases /r 2 in the inner heliosphere (Caballero-Lopez & Moraal 2003). It is also noteworthy that the intensity in the ecliptic plane goes through a broad extreme value, leading to values of g that are of the same order of magnitude as g r. Thus, since Tk k,the dr drift terms in equations (10) and (11) are negligible. Based on this insight, it is a simple matter to parameterize the magnitude and time of maximum of the diurnal variation in equation (9) with the values given by equations (10) and (11), with dr ¼ 0. The results are shown in Figures 3 and 4, where the independent variable is d ¼ k k g r,forsixvaluesof ¼ k? /k k and W ¼ 1(V ¼ 400 km s 1 ). Figure 3 shows that the magnitude goes through a minimum value at d 0:6%, which corresponds to a time of maximum of 15:00 hr. Figure 4 shows, however, that the time of maximum is a monotonically increasing function of d, limited between 09:00 hr (if d 3 c in eqs. [9] [11]) and 24:00 hr if d 3 c (or if ¼ 1). Based on the numerical solutions above, we demarcate two regions for reasonable values of d for the qa > 0 and <0 cycles. This clearly shows that the magnitude in the two cycles is of the same order but that the time of maximum is much earlier in the qa > 0 cycle than in the qa < 0 cycle. Figures 3 and 4 favor the smaller values of for two reasons. First, observations indicate that the time of maximum is near 18:00 hr in the qa < 0 cycle, which is progressively violated for larger values of, and second, the smaller the value of, the earlier the time of maximum will be in the qa > 0 cycle. As discussed above, the observed early phase of 08:00 in 1954 in Figure 1 may indicate a diurnal maximum at about 10:00 after allowance for the effects of muon decay in the atmosphere. Figure 4 shows that such an early time of maximum can only be reached in the limiting case for small and large d ¼ k k g r. We note, however, that the solar wind speed will Fig. 3. Magnitude of the diurnal variation. Fig. 5. Magnitude of the diurnal variation as a function of solar wind speed.

5 560 MORAAL ET AL. region of 400 km s 1 or higher and only weakly dependent on solar cycle (e.g., Bieber & Chen 1991). On the basis of the modeling reported herein, it is possible that the diurnal variation provides a direct means of investigating the solar wind speed in the past. To this end, it will be important that the effects of muon decay must be removed from the data as accurately as possible. Fig. 6. Time of the maximum of the diurnal variation as a function of solar wind speed. also affect the phase through its influence on the spiral angle, as in equation (4). In Figures 5 and 6 we calculate equations (10) and (11) for other values of V, with ¼ 0. Figure 5 shows that the magnitude in the qa > 0 cycle ( d ¼ 1:2 and 0.6) is a strong function of V, while in the qa < 0 cycle ( d ¼ 1:2 and 0.6) it is weakly V-dependent. The time of maximum in Figure 6 depends only moderately on V, with the strongest dependence in the qa > 0 cycle ( d ¼ 1:2 and 0.6) for low solar wind speeds. This is precisely the region of interest for the anomalously early time of maximum observed in 1954, and it suggests that the solar wind speed may have decreased below 400 km s 1. Thus, our interpretation of the early times of maximum observed in 1954 is a scenario with a strong negative radial gradient and low solar wind speed. Direct measurements of the solar wind speed do not exist for this epoch. The measured solar wind speeds at Earth since about 1964 have always been in the 4. SUMMARY AND CONCLUSIONS We have demonstrated that a diurnal variation with a time of maximum in the morning sector (<12:00 hr) can be explained as due to a negative radial gradient and that such a negative radial gradient is plausible in qa > 0 drift cycles. This is in agreement with observations that the time of maximum in these cycles is earlier than that in qa < 0 cycles. However, the extreme observations in 1954, when the time of maximum shifted to about 10:00 after allowance for muon decay in the atmosphere, suggests that the solar wind speed may have been <400 km s 1 for that epoch. Since no wind speed observations exist for this epoch, it poses an interesting challenge to try to infer them in an indirect way. In a follow-up paper we will attempt a detailed year-by-year simultaneous modeling of the cosmic-ray intensity and the diurnal variation with a series of full numerical solutions of the cosmic transport equation (3). The specific feature that we need to extract from these solutions is that early times of maximum of the diurnal variation are accompanied by high intensities or low levels of modulation, as implied by the observations of Neher (1967). This work was supported by the South African National Research Foundation grant GUN and NSF grant ATM R. C. L. was supported by UNAM-DGAPA grant IN The Haleakala data were supplied by the University of New Hampshire, supported by NSF grant ATM Bieber, J. W., & Chen, J. 1991, ApJ, 372, 301 Caballero-Lopez, R. A., & Moraal, H. 2003, Proc. 28th Int. Cosmic Ray Conf. ( Tsukuba), 7, 3871 Caballero-Lopez, R. A., Moraal, H., & McDonald, F. B. 2004, J. Geophys. Res., 109, 5105 Duldig, M. L. 2000, Space Sci. Rev., 93, 207 Forbush, S. E. 1954, J. Geophys. Res., 59, 525 Isenberg, P. A., & Jokipii, J. R. 1979, ApJ, 234, 746 Lemmer, M., & Moraal, H. 1991, Proc. 22nd Int. Cosmic Ray Conf. (Dublin), 3, 521 REFERENCES McCracken, K. G., Beer, J., & McDonald, F. B. 2004, Adv. Space Res., 34, 397 Moraal, H. 1976, Space Sci. Rev., 19, , in COSPAR Colloq. 11, The Outer Heliosphere: The Next Frontiers, ed. K. Scherer et al. (Amsterdam: Pergamon), 147 Neher, H. V. 1967, J. Geophys. Res., 72, 1527 Parker, E. N. 1965, Planet. Space Sci., 13, 9 Parsons, N. R. 1957, Australian J. Phys., 10, , Ph.D. thesis, Univ. Tasmania Thambyahpillai, T., & Elliot, H. 1953, Nature, 171, 918 Venkatesan, D., & Dattner, H. 1959, Tellus, 11, 116