Cosmic Ray Variability: Century of Its ObservationsView this Special Issue
Latitudinal Dependence of Cosmic Rays Modulation at 1 AU and Interplanetary Magnetic Field Polar Correction
The cosmic rays differential intensity inside the heliosphere, for energy below 30 GeV/nuc, depends on solar activity and interplanetary magnetic field polarity. This variation, termed solar modulation, is described using a 2D (radius and colatitude) Monte Carlo approach for solving the Parker transport equation that includes diffusion, convection, magnetic drift, and adiabatic energy loss. Since the whole transport is strongly related to the interplanetary magnetic field (IMF) structure, a better understanding of his description is needed in order to reproduce the cosmic rays intensity at the Earth, as well as outside the ecliptic plane. In this work an interplanetary magnetic field model including the standard description on ecliptic region and a polar correction is presented. This treatment of the IMF, implemented in the HelMod Monte Carlo code (version 2.0), was used to determine the effects on the differential intensity of Proton at 1 AU and allowed one to investigate how latitudinal gradients of proton intensities, observed in the inner heliosphere with the Ulysses spacecraft during 1995, can be affected by the modification of the IMF in the polar regions.
The Solar Modulation, due to the solar activity, affects the Local Interstellar Spectrum (LIS) of Galactic Cosmic Rays (GCR) typically at energies lower than 30 GeV/nucl. This process, described by means of the Parker equation (e.g., see [1, 2] and Chapter 4 of ), is originated from the interaction of GCRs with the interplanetary magnetic field (IMF) and its irregularities. The IMF is the magnetic field that is carried outwards during the solar wind expansion. The interplanetary conditions vary as a function of the solar cycle which approximately lasts eleven years. In a solar cycle, when the maximum activity occurs, the IMF reverse his polarity. Thus, similar solar polarity conditions are found almost every 22 years . In the HelMod Monte Carlo code version 1.5 (e.g., see ), the “classical” description of IMF, as proposed by Parker , was implemented together with the polar corrections of the solar magnetic field suggested subsequently in [6, 7]. This IMF was used inside the HelMod  code to investigate the solar modulation observed at Earth and to partially account for GCR latitudinal gradients, that is, those observed with the Ulysses spacecraft [8, 9]. In order to fully account for both the latitudinal gradients and latitudinal position of the proton-intensity minimum observed during the Ulysses fast scan in 1995, the HelMod Code was updated to the version 2.0 to include a new treatment of the parallel and diffusion coefficients following that one described in . In the present formulation, the parallel component of the diffusion tensor depends only on the radial distance from the Sun, while it is independent of solar latitude.
2. The Interplanetary Magnetic Field
Nowadays, we know that there is a Solar Wind plasma (SW) that permeates the interplanetary space and constitutes the interplanetary medium. In IMF models the magnetic-field lines are supposed to be embedded in the nonrelativistic streaming particles of the SW, which carries the field with them into interplanetary space, producing the large scale structure of the IMF and the heliosphere. The “classical” description of the IMF was proposed originally by Parker (e.g., see [2, 5, 11–14] and Chapter 4 of ). He assumed (i) a constant solar rotation with angular velocity (), (ii) a simple spherically symmetric emission of the SW, and (iii) a constant (or approaching an almost constant) SW speed () at larger radial distances (), for example, for (where is the Solar radius), since beyond the wind speed varies slowly with the distance. The “classical” IMF can be analytically expressed as  where is a coefficient that determines the IMF polarity and allows to be equal to , that is, the value of the IMF at Earth's orbit as extracted from NASA/GSFC's OMNI data set through OMNIWeb [16, 17]; and are unit vector components in the radial and azimuthal directions, respectively, is the colatitude (polar angle); is the polar angle determining the position of the Heliospheric Current Sheet (HCS) ; is the Heaviside function: thus, allows to change sign in the two regions above and below the HCS  of the heliosphere; finally, with the spiral angle. In the present model is assumed to be independent of the heliographic latitude and equal to the sidereal rotation at the Sun's equator. The magnitude of Parker field is thus
In 1989 , Jokipii and Kota have argued that the solar surface, where the feet of the field lines lie, is not a smooth surface, but a granular turbulent surface that keeps changing with time, especially in the polar regions. This turbulence may cause the footpoints of the polar field lines to wander randomly, creating transverse components in the field, thus causing temporal deviations from the smooth Parker geometry. The net effect of this is a highly irregular and compressed field line. In other words, the magnitude of the mean magnetic field at the poles is greater than in the case of the smooth magnetic field of a pure Parker spiral. Jokipii and Kota  have therefore suggested that the Parker spiral field may be generalized by the introduction of a perturbation parameter which amplifies the field strength at large radial distances. With this modification the magnitude of IMF, (3), becomes  The difference of the IMF obtained from (3) and (4) is less than ~1% for colatitudes (e.g, see Figure 1(a)) and increases for colatitudes approaching the polar regions (e.g., see Figure 2 of ).
In the present treatment, the heliosphere is divided into polar regions and a equatorial region where different description of IMF are applied. In the equatorial region the Parker's IMF, Equation (3) is used, while in the polar regions we used a modified IMF that allows a magnitude as in (4) where equatorial regions are those with colatitude . The symbol indicates the corresponding polar regions.
In order to have a divergence-free magnetic-field we require that the perturbation factor  has to be where is the minimum perturbation factor of the field. The perturbation parameter is left to grow with decreasing of the colatitude. However, in their original work, Jokipii and Kota  estimated the value of the parameter between and .
Since the polar field is only a perturbation of the Parker field, it is a reasonable assumption that stream lines of the magnetic field do not cross the equatorial plane, thus, remaining completely contained in the solar hemisphere of injection. This allows one to estimate an upper limit on the possible values of (see, Figure 1(b)). Currently, we use by comparing simulations with observations at Earth orbit during Solar Cycle 23 (see, Section 5).
3. The Propagation Model
Parker in 1965  (see also, , Chapter 4 of  and references therein) treated the propagation of GCRs trough the interplanetary space. He accounted for the so-called adiabatic energy losses, outward convection due to the SW and drift effects. In the heliocentric system the Parker equation is then expressed (e.g., see [2, 19]): where is the number density of particles per unit of particle kinetic energy , at the time . is the solar wind velocity along the axis , is the symmetric part of diffusion tensor , is the drift velocity that takes into account the drift of the particles due to the large scale structure of the magnetic field [21–23], and, finally, where is the rest mass of the GCR particle. The last term of (7) accounts for adiabatic energy losses [1, 24]. The number density is related to the differential intensity as ([2, 25], Chapter 4 of  and references therein): where is the speed of the GCR particle.
Equation (7) was solved using the HelMod code (see the discussion in ). This treatment (i) follows that introduced in [10, 26–30] and (ii) determines the differential intensity of GCRs using a set of approximated stochastic differential equations (SDEs) which provides a solution equivalent to that from (7). The equivalence between the Parker equation, that is a Fokker-Planck type equation, and the SDEs is demonstrated in [28, 31]. In the present work, we use a 2D (radius and colatitude) approximation for the particle transport. The model includes the effects of solar activity during the propagation from the effective boundary of the heliosphere down to Earth's position.
The set of SDEs for the 2D approximation of (7) in heliocentric spherical coordinates iswhere and is a random number following a Gaussian distribution with a mean of zero and a standard deviation of one. The procedure for determining the SDEs can be found in .
In a coordinate system with one axis parallel to the average magnetic field and the other two perpendicular to this the symmetric part of the diffusion tensor is (see e.g., ): with the diffusion coefficient describing the diffusion parallel to the average magnetic field and and are the diffusion coefficients describing the diffusion perpendicular to the average magnetic field in the radial and polar directions, respectively. In this work is that one proposed by Strauss and collaborators in  (see also [33–35]): where is the diffusion parameter—described in Section 2.1 of —which depends on solar activity and polarity, is the particle speed in unit of speed of light, is the particle rigidity expressed in GV, and, finally, is the heliocentric distance from the Sun in AU.
In the current treatment, has a radial dependence proportional to , but no latitudinal dependence. Mc Donald and collaborators (see, ) remarked that (i) a spatial dependence of —like the one proposed here—can affect the latitudinal gradients at high latitude and (ii) it is consistent with that originally suggested in . Furthermore, the perpendicular diffusion coefficient is taken to be proportional to with a ratio for both and -coordinates. The latter value is discussed in Section 5.
In addition, the practical relationship between and monthly Smoothed Sunspot Numbers (SSN)  values—discussed in Section 2.1 of —is currently updated using the most recent data from . As in , the data are subdivided into four sets, that is, ascending and descending phases for both negative and positive solar magnetic-field polarities (Table 1). It has to be remarked that after each maximum the sign of the magnetic field (i.e., the parameter in (5)) is reversed. The updated practical relationships between in and SSN values for for the four periods (from I up to IV, listed in Table 1) are The rms (root mean square) values of the percentage difference between values obtained with (13a), (13b), (13c), and (13d) from those determined using the procedure discussed in Section 2.1 of  applied to the data from  were found to be 6.0%, 10.1%, 7.0%, and 13.2% for the period I (ascending phase with ), II (descending phase with ), III (ascending phase with ), and IV (descending phase with ), respectively.
4. The Magnetic Field in the Polar Regions
Section 2 describes an IMF following the Parker Field with a small region around the poles in which such a field is modified. As already mentioned (see e.g., [6, 39–43]), the correction is needed to better reproduce the complexity of the magnetic field in those regions.
Moreover, Ulysses spacecraft (see e.g., [44–46]) explored the heliosphere outside the ecliptic plane up to of solar latitude at a solar distance from ~1 up to ~5 AU. Using these observations, the presence of latitudinal gradient in the proton intensity could be determined (e.g., see Figure 2 of  and Figure 5 of ). The data collected during the latitudinal fast scan (from September 1994 up to August 1995) show (a) a nearly symmetric latitudinal gradient with the minimum near ecliptic plane, (b) a southward shift of the minimum, and (c) an intensity in the North polar region at exceeding the South polar intensity. In  a latitudinal gradient of ~0.3%/degree for proton with kinetic energy >0.1 GeV was estimated. While in  the analysis to higher energy was extended estimating a gradient of ~0.22%/degree for proton with kinetic energy >2 GeV. The minimum in the charged particle intensity separating the two hemispheres of the heliosphere occurs ~10° South of the heliographic equator . In addition, an independent analysis that takes into account the latitudinal motion of the Earth and IMP8 confirms a significant (~) southward offset of the intensity minimum  for MeV proton. Furthermore in , a southward offset of about ≈7° is evaluated; this offset of the intensity minimum results to be independent of the particle energy up to 2 GeV. Finally, in , the intensity in the North polar region at is observed to exceed the South polar intensity of ~6% for protons with MeV.
Using the present HelMod code (version 2.0), we could investigate (i) the latitudinal gradient of GCR intensities resulting from solar modulation and (ii) how the magnetic-field structure of the polar regions, as defined in Section 2, is able to influence the GCR spectra on the ecliptic plane. As previously defined, we denote with a polar region of amplitude from polar axis, that is, and . Three regions with , and were investigated. Outside any of these regions, the ratio between and in (5) is less than ~1% (see Figure 1) and, thus, it ensures a smooth transition between polar and equatorial regions. For the purpose or this study we consider an energy binning closer to those presented in . The KET instrument  collects proton data in three “channels” one with energies ranging from 0.038 GeV up to 2.0 GeV and two for particle with kinetic energy GeV and GeV, respectively. A successive reanalysis of the collected data allowed the authors to subdivide the 0.25–2 GeV “channel” in three “subchannels” of intermediate energies. Since the Present Model is optimized—as discussed in —for particles with rigidity greater than 1 GV (i.e., ≈0.444 GeV), the present results are compared only with the corresponding “channel” or “subchannels” suited for the corresponding energy range (see Table 2).
At 1 AU and as a function of the solar colatitude, the GCR intensities for protons are shown in Figure 2. For a comparison with Ulysses observations, the modulated intensities of protons—resulting from HelMod code—were investigated from (North) and down to (South). They were obtained using the HelMod code and selected using the energy bins reported in Table 2. In Figure 2, the latitudinal intensity distribution is normalized to the corresponding South Pole intensity. The quoted errors include statistical and systematic errors. The distributions were interpolated using a parabolic function expressed as where is the normalized intensity, is the latitudinal angle (the latitudinal angle is ), and , , and are parameters determined from the fitting procedure. The so obtained fitted curves are shown as continuous lines in Figure 2. Furthermore, the latitudinal positions of minimum intensity (), percentages of North-South asymmetry of intensities (), and differences in percentage between the maximum and minimum intensities () were also determined from the fitting procedure and are listed in Tables 3, 4, and 5, respectively. The quoted errors—following a procedure discussed in —are derived varying the fitted parameters in order to obtain a value of the parameter two times larger than the one resulting from the best fit (). is defined as with where is the central value of the th latitudinal bin of the differential intensity distribution and are the errors including the experimental and Monte Carlo uncertainties. For , that is, assuming a modified polar magnetic-field for and , we found a general agreement with Ulysses observations. The position of is compatible within the errors with one observed in , as well as the values of and .
The HelMod code allows one to investigate the relevance of the treatment of the polar region magnetic-field with respect to the resulting modulation of GCR. Thus, the latitudinal normalized intensities were obtained excluding a few (small) regions nearby the poles. This was determined from reducing the latitudinal spatial phase-space admissible for pseudo-particles (see ), that is, the latitudinal extension of GCR particles taken into account—to (), () and (). The so obtained latitudinal gradients are compared with the full latitudinal extension, (), in Figure 3. By an inspection of Figure 3, one may lead to the conclusion that the GCR diffusion nearby the polar axis has a large impact on the latitudinal gradients in the inner heliosphere. As a consequence, the IMF description in the polar regions is relevant in order to reproduce the observed modulated GCR spectra.
5. Comparison with Observations during Solar Cycle 23
The agreement of HelMod simulated spectra with observations during solar cycle 23 is investigated via quantitative comparisons using (17) and (18). However, since the structure of the heliosphere is different in high and low solar activity the two periods are separately analyzed.
The HelMod Code  (version 2.0) allowed us to investigate how the modulated (simulated) differential intensities are affected by the (1) particle drift effect, (2) polar enhancement of the diffusion tensor along the polar direction (, e.g., see ), and, finally, (3) values of the tilt angle () calculated following the approach of the “R” and “L” models . This analysis also allows us to estimate the values of IMF parameters that better describe the modulation along the entire solar cycle. The effects related to particle drift were investigated via the suppression of the drift velocity (No Drift approximation), this accounts for the hypothesis that magnetic drift convection is almost completely suppressed during solar maxima. The differential intensities were calculated for with values of of 1, 8, and 10, that is, no enhancement, that suggested in  and that suggested in [2, 48] (and reference there in), respectively. Furthermore, the modulated proton spectra were derived from a LIS whose normalization constant depends on the experimental set of data and were already discussed in ). In addition, the differential intensities were calculated accounting for particles inside heliospheric regions where solar latitudes are lower than .
During the period of high solar activity for the solar cycle 23, the BESS collaboration took data in the years 1999, 2000, and 2002 (see sets of data in ). For period not dominated by high solar activity in solar cycle 23, BESS, AMS, and PAMELA collaborations took data, that is, BESS-1997 , AMS-1998 , and PAMELA-2006/08 .
Following the procedure described in , the observation data were compared with those obtained from HelMod code using the error-weighted root mean square () of the relative difference () between experimental data () and those resulting from simulated differential intensities (). For each set of experimental data and with the approximations and/or models described above, we determined the quantity with where is the average energy of the th energy bin of the differential intensity distribution and are the errors including the experimental and Monte Carlo uncertainties; the latter account for the Poisson error of each energy bin. The simulated differential intensities are interpolated with a cubic spline function. The modulation results are studied varying the parameters —from 0, that is, the nonmodified Parker IMF, up to , see Figure 1(b)—, (from 0.10 up to 0.14) and (from 1 up to 10) seeking a set of parameters set that minimize . In Table 6, the average values of (in percentage, %) for low solar activity periods are listed. They were obtained in the energy range (Above 30 GeV, the differential intensity is marginally (if at all) affected by modulation.) from 444 MeV up to 30 GeV using the “L” and “R” models for the tilt angle and for the No Drift approximation and without any enhancement of the diffusion tensor along the polar direction (). The results derived with the enhancement of the diffusion tensor along the polar direction indicate that for and 10 one obtains a value of that is from 1.5 up to 3 times larger with respect the case without enhancement (For a comparison, the scalar approximation presented in , that is, assuming that the diffusion propagation is independent of magnetic structure, leads to and average of ~15%). From inspection of Table 6, one can remark that the drift mechanism leads to a better agreement with experimental data. Furthermore the “R” and “L” models for tilt angles are comparable within the precision of the method (discussed in ). The minimum difference with respect to the experimental data occurs when –0.13 and for both “R” and “L” models, with the “L” model slightly preferred to “R”.
In Figures 4, 5, and 6, the differential intensities determined with the HelMod code are shown and compared to the experimental data of BESS-1997, AMS-1998, and PAMELA-2006/08, respectively; in these figures, the dashed lines are the LIS as discussed in . These modulated intensities are the ones calculated for a heliospheric region where solar latitudes are lower than , using , and with the “L” model. Finally, one can remark that the present code, combining diffusion and drift mechanisms, is also suited to describe the modulation effect in periods when the solar activity is no longer at the maximum.
In Table 7 we present the averages (in percentage, %) during the periods dominated by high solar activity. The simulated differential intensities were obtained for a heliospheric region where solar latitudes are lower than without any enhancement of the diffusion tensor along the polar direction (). The simulations with 8 and 10 lead to comparable with those presented in Table 7. However, using provides a better agreement to experimental data at lower energy. From inspection of Table 7, one can note that “R” and “L” models for tilt angles yield comparable results within the precision of the method. Furthermore the minimum difference with the experimental data occurs when and with the “L” model slightly preferred to “R”. The HelMod parameter configuration, which minimizes the difference to the experimental data, are reported in Table 8: one may remark that the No Drift approximation is (almost) comparable to a drift treatment for both BESS-2000 and BESS-2002 with data collected during and after the maximum of the solar activity. Apparently, the drift treatment is needed in order to describe BESS-1999 with data taken during a period approaching the solar maximum.
In Figures 7, 8, and 9, the differential intensities determined with the HelMod code are shown and compared with the experimental data of BESS-1999, BESS-2000, and BESS-2002, respectively; in these figures, the dashed lines are the LIS as discussed in . These modulated intensities are the ones calculated for a heliospheric region where solar latitudes are lower than , using independently of the latitude and including particle drift effects with the values of the tilt angle from the “L” model. Finally, it is concluded that the present code combining diffusion and drift mechanisms is suited to describe the modulation effect in periods with high solar activity [2, 50, 54].
In this work an IMF, which combines the Parker Field and its polar modification, is presented. In the polar regions, the Parker IMF was modified with an additional latitudinal components according to those proposed by Jokipii and Kota in . We found the maximum perturbed value with this component yielding, as a physical result, streaming lines completely confined in the solar hemisphere of injection.
The proposed IMF is, then, used within the HelMod Monte Carlo code to determine the effects on the differential intensity of protons at 1 AU as a function of the extension of polar region, in which the modified magnetic-field is employed. We found that a polar region contained within of colatitude is that one ensuring a very smooth transition to the equatorial region and allows to reproduce qualitatively and quantitatively the latitudinal profile of the GCR intensity, and the latitudinal dip shift with respect to the ecliptic plane. Finally we determined how the polar region diffusion is mostly responsible of the proton intensity latitudinal gradient observed in the inner heliosphere with the Ulysses spacecraft during 1995.
K. Kudela wishes to acknowledge VEGA Grant agency project 2/0081/10 for support. Finally, the authors acknowledge the use of NASA/GSFC's Space Physics Data Facility's OMNIWeb service and OMNI data.
E. N. Parker, “The passage of energetic charged particles through interplanetary space,” Planetary and Space Science, vol. 13, no. 1, pp. 9–49, 1965.View at: Google Scholar
P. Bobik, G. Boella, M. J. Boschini et al., “Systematic investigation of solar modulation of galactic protons for solar cycle 23 using a monte carlo approach with particle drift effects and latitudinal dependencehe,” The Astrophysical Journal, vol. 745, p. 132, 2012.View at: Google Scholar
C. Leroy and P. G. Rancoita, Principles of Radiation Interaction in Matter and Detection, World Scientific Publishing, River Edge, NJ, USA, 3rd edition, 2011.
R. D. Strauss, M. S. Potgieter, and S. E. S. Ferreira, “Modeling ground and space based cosmic ray observations,” Advances in Space Research, vol. 49, pp. 392–407, 2012.View at: Google Scholar
E. N. Parker, “Dynamics of the interplanetary gas and magnetic fields,” The Astrophysical Journal, vol. 128, p. 664, 1958.View at: Google Scholar
J. R. Jokipii and J. Kota, “The polar heliospheric magnetic field,” Geophysical Research Letters, vol. 16, pp. 1–4, 1989.View at: Google Scholar
U. W. Langner, Effect of termination shock acceleraion on cosmic ray in the helisphere [Ph.D. thesis], Potchestroom University, Potchestroom, South Africa, 2004.
B. Heber, W. Dröge, P. Ferrando et al., “Spatial variation of >40 MeV/n nuclei fluxes observed during the Ulysses rapid latitude scan,” Astronomy and Astrophysics, vol. 316, no. 2, pp. 538–546, 1996.View at: Google Scholar
J. A. Simpson, “Ulysses cosmic-ray investigations extending from the south to the north polar regions of the Sun and heliosphere,” Nuovo Cimento della Societa Italiana di Fisica C, vol. 19, no. 6, pp. 935–943, 1996.View at: Google Scholar
R. D. Strauss, M. S. Potgieter, I. Büsching, and A. Kopp, “Modeling the modulation of galactic and jovian electrons by stochastic processes,” The Astrophysical Journal Letters, vol. 735, no. 2, article 83, 2011.View at: Publisher Site | Google Scholar
E. N. Parker, “Newtonian development of the dynamical properties of ionized gases of low density,” Physical Review, vol. 107, pp. 924–933, 1957.View at: Google Scholar
E. N. Parker, “The hydrodynamic theory of solar corpuscular radiation and stellar winds,” The Astrophysical Journal, vol. 132, p. 821, 1960.View at: Google Scholar
E. N. Parker, “Sudden expansion of the corona following a large solar flare and the attendant magnetic field and cosmic-ray effects,” The Astrophysical Journal, vol. 133, p. 1014, 1961.View at: Google Scholar
E. N. Parker, Interplanetary Dynamical Processes, Wiley-Interscience, New York, NY, USA, 1963.
M. Hattingh and R. A. Burger, “A new simulated wavy neutral sheet drift model,” Advances in Space Research, vol. 16, no. 9, pp. 213–216, 1995.View at: Google Scholar
J. H. King and N. E. Papitashvili, “Solar wind spatial scales in and comparisons of hourly Wind and ACE plasma and magnetic field data,” Journal of Geophysical Research A, vol. 110, no. 2, Article ID A02104, 2005.View at: Publisher Site | Google Scholar
NASA-OMNIweb, online database, 2012, http://omniweb.gsfc.nasa.gov/form/dx1.html.
J. R. Jokipii and B. Thomas, “Effects of drift on the transport of cosmic rays. IV—modulation by a wavy interplanetary current sheet,” The Astrophysical Journal, vol. 243, pp. 1115–1122, 1981.View at: Google Scholar
J. R. Jokipii, E. H. Levy, and W. B. Hubbard, “Effect of particle drift on cosmic-ray transport. I. general properties, application to solar modulation,” The Astrophysical Journal Letters, vol. 213, pp. L85–L88, 1977.View at: Google Scholar
E. Marsch, W. I. Axford, and J. F. McKenzie, “Solar wind,” in Dynamic Sun, B. N. Dwivedi, Ed., pp. 374–402, 2003.View at: Google Scholar
J. R. Jokipii and E. H. Levy, “Effects of particle drifts on the solar modulation of galactic cosmic rays,” The Astrophysical Journal Letters, vol. 213, pp. L85–L88, 1977.View at: Publisher Site | Google Scholar
M. S. Potgieter and H. Moraal, “A drift model for the modulation of galactic cosmic rays,” The Astrophysical Journal, vol. 294, part 1, pp. 425–440, 1985.View at: Google Scholar
R. A. Burger and M. Hattingh, “Steady-state drift-dominated modulation models for galactic cosmic rays,” Astrophysics and Space Science, vol. 230, no. 1-2, pp. 375–382, 1995.View at: Publisher Site | Google Scholar
J. R. Jokipii and E. N. Parker., “On the convection, diffusion, and adiabatic deceleration of cosmic rays in the solar wind,” The Astrophysical Journal, vol. 160, p. 735, 1970.View at: Google Scholar
L. J. Gleeson and W. I. Axford, “Solar modulation of galactic cosmic rays,” The Astrophysical Journal, vol. 154, p. 1011, 1968.View at: Google Scholar
Y. Yamada, S. Yanagita, and T. Yoshida, “A stochastic view of the solar modulation phenomena of cosmic rays,” Geophysical Research Letters, vol. 25, pp. 2353–2356, 1998.View at: Google Scholar
M. Gervasi, P. G. Rancoita, I. G. Usoskin, and G. A. Kovaltsov, “Monte-Carlo approach to Galactic cosmic ray propagation in the heliosphere,” Nuclear Physics B, vol. 78, no. 1–3, pp. 26–31, 1999.View at: Google Scholar
M. Zhang, “A Markov stochastic process theory of cosmic-ray modulation,” The Astrophysical Journal, vol. 513, pp. 409–420, 1999.View at: Google Scholar
K. Alanko-Huotari, I. G. Usoskin, K. Mursula, and G. A. Kovaltsov, “Stochastic simulation of cosmic ray modulation including a wavy heliospheric current sheet,” Journal of Geophysical Research A, vol. 112, no. 8, Article ID A08101, 2007.View at: Publisher Site | Google Scholar
C. Pei, J. W. Bieber, R. A. Burger, and J. Clem, “A general time-dependent stochastic method for solving Parker's transport equation in spherical coordinates,” Journal of Geophysical Research A, vol. 115, no. 12, Article ID A12107, 2010.View at: Publisher Site | Google Scholar
C. W. Gardiner, Handbook of Stochastic Methods: For Physics, Chemistry and Natural Sciences, Springer, New York, NY, USA, 1985.
J. R. Jokipii, “Propagation of cosmic rays in the solar wind,” Reviews of Geophysics and Space Physics, vol. 9, pp. 27–87, 1971.View at: Google Scholar
I. D. Palmer, “Transport coefficients of low-energy cosmic rays in interplanetary,” Reviews of Geophysics and Space Physics, vol. 20, pp. 335–351, 1982.View at: Publisher Site | Google Scholar
M. S. Potgieter and S. E. S. Ferreira, “Effects of the solar wind termination shock on the modulation of Jovian and galactic electrons in the heliosphere,” Journal of Geophysical Research A, vol. 107, article 1089, 9 pages, 2002.View at: Publisher Site | Google Scholar
W. Dröge, “Probing heliospheric diffusion coefficients with solar energetic particles,” Advances in Space Research, vol. 35, no. 4, pp. 532–542, 2005.View at: Publisher Site | Google Scholar
F. B. McDonald, P. Ferrando, B. Heber et al., “A comparative study of cosmic ray radial and latitudinal gradients in the inner and outer heliosphere,” Journal of Geophysical Research A, vol. 102, no. 3, pp. 4643–4651, 1997.View at: Google Scholar
SIDC-team, “The International Sunspot Number,” Monthly Report on the International Sunspot Number, online catalogue, 1964–2010.View at: Google Scholar
I. G. Usoskin, G. A. Bazilevskaya, and G. A. Kovaltsov, “Solar modulation parameter for cosmic rays since 1936 reconstructed from ground-based neutron monitors and ionization chambers,” Journal of Geophysical Research A, vol. 116, no. 2, Article ID A02104, 2011.View at: Publisher Site | Google Scholar
A. Balogh, E. J. Smith, B. T. Tsurutani, D. J. Southwood, R. J. Forsyth, and T. S. Horbury, “The heliospheric magnetic field over the south polar region of the sun,” Science, vol. 268, no. 5213, pp. 1007–1010, 1995.View at: Google Scholar
H. Moraal, “Proton modulation near solar minimim periods in consecutive solar cycles,” in Proceedings of the International Cosmic Ray Conference, vol. 6, p. 140, 1990.View at: Google Scholar
C. W. Smith and J. W. Bieber, “Solar cycle variation of the interplanetary magnetic field spiral,” The Astrophysical Journal Letters, vol. 370, no. 1, pp. 435–441, 1991.View at: Google Scholar
L. A. Fisk, “Motion of the footpoints of heliospheric magnetic field lines at the sun: implications for recurrent energetic particle events at high heliographic latitudes,” Journal of Geophysical Research, vol. 101, pp. 15547–15554, 1996.View at: Google Scholar
M. Hitge and R. A. Burger, “Cosmic ray modulation with a Fisk-type heliospheric magnetic field and a latitude-dependent solar wind speed,” Advances in Space Research, vol. 45, no. 1, pp. 18–27, 2010.View at: Publisher Site | Google Scholar
T. R. Sanderson, R. G. Marsden, K. P. Wenzel, A. Balogh, R. J. Forsyth, and B. E. Goldstein, “High-latitude observations of energetic ions during the first Ulysses polar pass,” Space Science Reviews, vol. 72, no. 1-2, pp. 291–296, 1995.View at: Publisher Site | Google Scholar
R. G. Marsden, “The 3-D heliosphere at solar maximum,” The Publications of the Astronomical Society of the Pacific, vol. 113, pp. 129–130, 2001.View at: Google Scholar
A. Balogh, R. G. Marsden, and E. J. Smith, The Heliosphere Near Solar Minimum. The Ulysses Perspective, Springer, New York, NY, USA, 2001.
B. Heber, M. S. Potgieter, and P. Ferrando, “Solar modulation of galactic cosmic rays: the 3D heliosphere,” Advances in Space Research, vol. 19, no. 5, pp. 795–804, 1997.View at: Google Scholar
M. S. Potgieter, “Heliospheric modulation of cosmic ray protons: role of enhanced perpendicular diffusion during periods of minimum solar modulation,” Journal of Geophysical Research A, vol. 105, no. 8, pp. 18295–18303, 2000.View at: Google Scholar
J. T. Hoeksema, “The large-scale structure of the heliospheric current sheet during the Ulysses epoch,” Space Science Reviews, vol. 72, no. 1-2, pp. 137–148, 1995.View at: Publisher Site | Google Scholar
S. E. S. Ferreira and M. S. Potgieter, “Long-term cosmic-ray modulation in the heliosphere,” The Astrophysical Journal, vol. 603, pp. 744–752, 2004.View at: Google Scholar
Y. Shikaze, S. Orito, S. Haino et al., “Measurements of 0.2–20 GeV/n cosmic-ray proton and helium spectra from 1997 through 2002 with the BESS spectrometer,” Astroparticle Physics, vol. 28, no. 1, pp. 154–167, 2007.View at: Publisher Site | Google Scholar
M. Aguilar, J. Alcaraz, and J. Allaby, “The Alpha Magnetic Spectrometer (AMS) on the International Space Station: part I—results from the test flight on the space shuttle,” Physics Reports, vol. 366, pp. 331–405, 2002.View at: Google Scholar
O. Adriani, G. C. Barbarino, G. A. Bazilevskaya et al., “PAMELA measurements of cosmic-ray proton and helium spectra,” Science, vol. 332, no. 6025, pp. 69–72, 2011.View at: Publisher Site | Google Scholar
D. C. Ndiitwani, S. E. S. Ferreira, M. S. Potgieter, and B. Heber, “Modelling cosmic ray intensities along the Ulysses trajectory,” Annales Geophysicae, vol. 23, no. 3, pp. 1061–1070, 2005.View at: Google Scholar