Latitudinal Dependence of Cosmic Rays Modulation at 1 AU and Interplanetary Magnetic Field Polar Correction

Abstract

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.

1. Introduction

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 [3]), 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 [4]. In the HelMod Monte Carlo code version 1.5 (e.g., see [2]), the “classical” description of IMF, as proposed by Parker [5], was implemented together with the polar corrections of the solar magnetic field suggested subsequently in [6, 7]. This IMF was used inside the HelMod [2] 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 [10]. 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 [3]). He assumed (i) a constant solar rotation with angular velocity (𝜔), (ii) a simple spherically symmetric emission of the SW, and (iii) a constant (or approaching an almost constant) SW speed (𝑉_sw) at larger radial distances (𝑟), for example, for 𝑟>𝑟_𝑏≈10𝑅_⊙ (where 𝑅_⊙ is the Solar radius), since beyond 𝑟_𝑏 the wind speed varies slowly with the distance. The “classical” IMF can be analytically expressed as [15] 𝐁_Par=𝐴𝑟²𝐞_𝑟−Γ𝐞_𝜑1−2𝐻𝜃−𝜃^′, (1) where 𝐴 is a coefficient that determines the IMF polarity and allows |𝐁_Par| to be equal to 𝐵_⊕, that is, the value of the IMF at Earth's orbit as extracted from NASA/GSFC's OMNI data set through OMNIWeb [16, 17]; 𝐞_𝑟 and 𝐞_𝜑 are unit vector components in the radial and azimuthal directions, respectively, 𝜃 is the colatitude (polar angle); 𝜃^′ is the polar angle determining the position of the Heliospheric Current Sheet (HCS) [18]; 𝐻 is the Heaviside function: thus, [1−2𝐻(𝜃−𝜃^′)] allows 𝐁_Par to change sign in the two regions above and below the HCS [18] of the heliosphere; finally, 𝜔Γ=tanΨ=𝑟−𝑟_𝑏sin𝜃𝑉_sw, (2) with Ψ the spiral angle. In the present model 𝜔 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 𝐵_Par=𝐴𝑟²1+Γ². (3)

In 1989 [6], 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 [6] have therefore suggested that the Parker spiral field may be generalized by the introduction of a perturbation parameter [𝛿(𝜃)] which amplifies the field strength at large radial distances. With this modification the magnitude of IMF, (3), becomes [6] 𝐵_Pol=𝐴𝑟²1+Γ²+𝑟𝑟_𝑏²𝛿(𝜃)². (4) The difference of the IMF obtained from (3) and (4) is less than ~1% for colatitudes 20^∘≤𝜃≤160^∘ (e.g, see Figure 1(a)) and increases for colatitudes approaching the polar regions (e.g., see Figure 2 of [2]).

(a)

(b)

Figure 1 

(a) Maximum percentage difference between 𝐁Pol and 𝐁Par as a function of the solar distance inside the colatitude regions 20∘≤𝜃≤160∘ (𝜃20∘), 25∘≤𝜃≤155∘ (𝜃25∘), and 30∘≤𝜃≤150∘ (𝜃30∘). (b) Maximum value allowed of 𝛿𝑚 as a function of colatitude: values in the “allowed” region guarantee that the stream line originated from the polar magnetic field do not cross the equatorial plane. Since (5) is symmetric with respect to the solar equatorial plane, values greater than 90∘ of colatitude lead to same results of those presented.

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) 𝐁_Pol=𝐴𝑟²𝐞_𝑟+𝑟𝑟_𝑏𝛿(𝜃)𝐞_𝜃−𝜔𝑟−𝑟_𝑏sin𝜃𝑉_sw𝐞_𝜑×1−2𝐻𝜃−𝜃^′𝐁Polarregions,_Par=𝐴𝑟²𝐞_𝑟−𝜔𝑟−𝑟_𝑏sin𝜃𝑉_sw𝐞_𝜑×1−2𝐻𝜃−𝜃^′Equatorialregion, (5) where equatorial regions are those with colatitude 𝑋^∘≤𝜃≤(180^∘−𝑋^∘). The symbol 𝜃_𝑋^∘ indicates the corresponding polar regions.

In order to have a divergence-free magnetic-field we require that the perturbation factor [𝛿(𝜃)] has to be 𝛿(𝜃)=𝛿_𝑚1−2𝐻𝜃−𝜃^′,sin𝜃 (6) where 𝛿_𝑚 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 [6] estimated the value of the parameter 𝛿 between 10⁻³ and 3×10⁻³.

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 𝛿_𝑚 (see, Figure 1(b)). Currently, we use 𝛿_𝑚=1×10⁻⁵ by comparing simulations with observations at Earth orbit during Solar Cycle 23 (see, Section 5).

3. The Propagation Model

Parker in 1965 [1] (see also, [2], Chapter 4 of [3] 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]): 𝜕U=𝜕𝜕𝑡𝜕𝑥_𝑖𝐾^𝑆_𝑖𝑗𝜕U𝜕𝑥_𝑗−𝜕𝜕𝑥_𝑖𝑉_sw,𝑖+𝑣_𝑑,𝑖U+13𝜕𝑉_sw,𝑖𝜕𝑥_𝑖𝜕𝛼𝜕𝑇^rel𝑇U, (7) where U is the number density of particles per unit of particle kinetic energy 𝑇, at the time 𝑡. 𝑉_sw,𝑖 is the solar wind velocity along the axis 𝑥_𝑖 [20], 𝐾^𝑆_𝑖𝑗 is the symmetric part of diffusion tensor [1], 𝑣_𝑑 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, 𝛼_rel=𝑇+2𝑚_𝑟𝑐²𝑇+𝑚_𝑟𝑐², (8) where 𝑚_𝑟 is the rest mass of the GCR particle. The last term of (7) accounts for adiabatic energy losses [1, 24]. The number density U is related to the differential intensity 𝐽 as ([2, 25], Chapter 4 of [3] and references therein): 𝐽=𝑣U,4𝜋 (9) where 𝑣 is the speed of the GCR particle.

Equation (7) was solved using the HelMod code (see the discussion in [2]). 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 is1Δ𝑟=𝑟²𝜕𝑟𝜕𝑟²𝐾^𝑆_𝑟𝑟𝜕Δ𝑡−(𝜃)𝐾𝜕𝜇^𝑆𝑟𝜇1−𝜇²(𝜃)𝑟+𝑉Δ𝑡_sw+𝑣_𝑑,𝑟Δ𝑡+2𝐾^𝑆_𝑟𝑟^1/2𝜔_𝑟√(𝜃)1Δ𝑡,Δ𝜇=−𝑟²𝜕𝜕𝑟𝑟𝐾^𝑆_𝜇𝑟1−𝜇²(𝜃+𝜕)Δ𝑡(𝜃)𝐾𝜕𝜇^𝑆𝜇𝜇1−𝜇²(𝜃)𝑟²−1Δ𝑡𝑟𝑣_𝑑,𝜇1−𝜇²(𝜃)−Δ𝑡2𝐾^𝑆_𝑟𝜇𝑟1−𝜇²(𝜃)2𝐾^𝑆_𝑟𝑟^1/2𝜔_𝑟√+1Δ𝑡𝑟1−𝜇²(𝜃𝐾)^𝑆𝜇𝜇𝐾^𝑆_𝑟𝑟−𝐾^𝑆_𝑟𝜇²0.5𝐾^𝑆_𝑟𝑟^1/2𝜔_𝜇√𝛼Δ𝑡,Δ𝑇=−^rel𝑇3𝑟²𝜕𝑉_sw𝑟²𝜕𝑟Δ𝑡, (10a)(10b)(10c)where 𝜇(𝜃)=cos𝜃 and 𝜔_𝑖 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 [2].

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 𝐾^𝑆_𝑖𝑗 is (see e.g., [32]): 𝐾^𝑆_𝑖𝑗=𝐾_||000𝐾_⟂,𝑟000𝐾_⟂,𝜃, (11) with 𝐾_|| the diffusion coefficient describing the diffusion parallel to the average magnetic field and 𝐾_⟂,𝑟 and 𝐾_⟂,𝜃 are the diffusion coefficients describing the diffusion perpendicular to the average magnetic field in the radial and polar directions, respectively. In this work 𝐾_|| is that one proposed by Strauss and collaborators in [10] (see also [33–35]): 𝐾_||=𝛽3𝐾₀𝑃𝑟1GV1+,1AU (12) where 𝐾₀ is the diffusion parameter—described in Section 2.1 of [2]—which depends on solar activity and polarity, 𝛽 is the particle speed in unit of speed of light, 𝑃=𝑝𝑐/|𝑍|𝑒 is the particle rigidity expressed in GV, and, finally, 𝑟 is the heliocentric distance from the Sun in AU.

In the current treatment, 𝐾_|| has a radial dependence proportional to 𝑟, but no latitudinal dependence. Mc Donald and collaborators (see, [36]) remarked that (i) a spatial dependence of 𝐾_||—like the one proposed here—can affect the latitudinal gradients at high latitude and (ii) it is consistent with that originally suggested in [6]. Furthermore, the perpendicular diffusion coefficient is taken to be proportional to 𝐾_|| with a ratio 𝐾_⟂,𝑖/𝐾_||=0.13 for both 𝑟 and 𝜃𝑖-coordinates. The latter value is discussed in Section 5.

In addition, the practical relationship between 𝐾₀ and monthly Smoothed Sunspot Numbers (SSN) [37] values—discussed in Section 2.1 of [2]—is currently updated using the most recent data from [38]. As in [2], the 𝐾₀ 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 𝐴 parameter in (5)) is reversed. The updated practical relationships between 𝐾₀ in AU²GV⁻¹s⁻¹ and SSN values for 1.4≤SSN≤165 for the four periods (from I up to IV, listed in Table 1) are (I)𝐾₀=0.000297−2.9⋅10⁻⁶SSN+8.1⋅10⁻⁹SSN²+1.46⋅10⁻¹⁰SSN³−8.4⋅10⁻¹³SSN⁴,(II)𝐾₀=0.000304846−5.8⋅10⁻⁶SSN,ifSSN≤20,0.00195SSN−2.3⋅10⁻¹⁰SSN²+9.1⋅10⁻⁵()𝐾,ifSSN>20,III₀=0.0002391−8.453⋅10⁻⁷()𝐾SSNIV₀=0.000247−1.175⋅10⁻⁶SSN. (13a)(13b)(13c)(13d)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 [2] applied to the data from [38] were found to be 6.0%, 10.1%, 7.0%, and 13.2% for the period I (ascending phase with 𝐴<0), II (descending phase with 𝐴<0), III (ascending phase with 𝐴>0), and IV (descending phase with 𝐴>0), respectively.

Table 1 

Definition of ascending and descending periods.

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 ±80^∘ 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 [9] and Figure 5 of [8]). 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 80^∘ exceeding the South polar intensity. In [9] a latitudinal gradient of ~0.3%/degree for proton with kinetic energy >0.1 GeV was estimated. While in [8] 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 [9]. In addition, an independent analysis that takes into account the latitudinal motion of the Earth and IMP8 confirms a significant (~8^∘±2^∘) southward offset of the intensity minimum [9] for 𝑇>100 MeV proton. Furthermore in [8], 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 [9], the intensity in the North polar region at 80 is observed to exceed the South polar intensity of ~6% for protons with 𝑇>100 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 𝜃_𝑋^∘ a polar region of amplitude 𝑋^∘ from polar axis, that is, 𝜃<𝑋^∘ and 𝜃>180^∘−𝑋^∘. Three regions with 𝑋^∘=20^∘,25^∘, and 30^∘ were investigated. Outside any of these regions, the ratio between 𝐁_Pol and 𝐁_Par 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 [8]. The KET instrument [47] 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 𝑇>0.1 GeV and 𝑇>2 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 [2]—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).

Table 2 

Kinetic energy bins (in GeV) selected with HelMod Code and the corresponding proton energy channel for KET instruments on board the Ulysses spacecraft [8].

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 80^∘ (North) and down to −80^∘ (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 𝐼𝜃_lat=𝑎+𝑐𝜃_lat+𝑑², (14) where 𝐼(𝜃_lat) is the normalized intensity, 𝜃_lat is the latitudinal angle (the latitudinal angle is 𝜃_lat=90^∘−𝜃), and 𝑎, 𝑏, and 𝑐 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 (𝜃_lat,min), percentages of North-South asymmetry of intensities (Δ_N-S), and differences in percentage between the maximum and minimum intensities (Δ_max) 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 [2]—are derived varying the fitted parameters in order to obtain a value of the parameter 𝜂_rms,lat two times larger than the one resulting from the best fit (𝐼_best). 𝜂_rms,lat is defined as 𝜂_rms.lat=∑_𝑖𝜂_𝑖,lat/𝜎_{𝜂,𝑖,lat}²∑_𝑖1/𝜎²_{𝜂,𝑖,lat} (15) with 𝜂_𝑖,lat=𝐼𝜃_𝑖−𝐼_best𝜃_𝑖𝐼_best𝜃_𝑖, (16) where 𝜃_𝑖 is the central value of the 𝑖th latitudinal bin of the differential intensity distribution and 𝜎_{𝜂,𝑖,lat} are the errors including the experimental and Monte Carlo uncertainties. For 𝜃₃₀^∘, that is, assuming a modified polar magnetic-field for 𝜃<30^∘ and 𝜃>150^∘, we found a general agreement with Ulysses observations. The position of 𝜃_lat,min is compatible within the errors with one observed in [8], as well as the values of Δ_N-S and Δ_max.

Table 3 

Latitudinal positions of minimum proton intensity (𝜃_lat,min) (in degrees) as a function of the kinetic energy, using three values for the extension of the polar regions.

Table 4 

Percentages of North-South asymmetry of proton intensities (Δ_N-S) as a function of the kinetic energy, using three values for the extension of the polar regions.

Table 5 

Differences in percentage between the maximum and minimum proton intensities (Δ_max) as a function of the kinetic energy, using three values for the extension of the polar regions.

Figure 2 

Latitudinal relative intensity at 𝑟=1AU, obtained at different solar colatitudes for protons in the energy range defined in the Table 2 and using three definitions of polar regions: 𝜃20∘, 𝜃25∘, and 𝜃30∘.

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 [2]), that is, the latitudinal extension of GCR particles taken into account—to 1^∘<𝜃<179^∘ (𝜃_𝑅,1), 2^∘<𝜃<178^∘ (𝜃_𝑅,2) and 10^∘<𝜃<170^∘ (𝜃_𝑅,10). The so obtained latitudinal gradients are compared with the full latitudinal extension, 𝜃_𝑅,0 (0^∘<𝜃<180^∘), 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.

Figure 3 

For the polar regions 𝜃<30∘ and 𝜃>150∘, latitudinal relative intensity at 𝑟=1AU, accounting for GCR particles with 0∘<𝜃<180∘ (𝜃𝑅,0), 1∘<𝜃<179∘ (𝜃𝑅,1), and 2∘<𝜃<178∘ (𝜃𝑅,2), respectively, are shown as a function of the proton kinetic energy and solar colatitude. Results with 10∘<𝜃<170∘ (𝜃𝑅,10) are comparable with those obtained with 𝜃𝑅,2.

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 [2] (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 (𝐾_⟂,𝜃, e.g., see [48]), and, finally, (3) values of the tilt angle (𝛼_𝑡) calculated following the approach of the “R” and “L” models [49]. 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 𝐾_⟂,𝜇=𝜌_𝐸𝐾_⟂,𝑟 with values of 𝜌_𝐸 of 1, 8, and 10, that is, no enhancement, that suggested in [50] 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 [2]). In addition, the differential intensities were calculated accounting for particles inside heliospheric regions where solar latitudes are lower than |5.7^∘|.

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 [51]). For period not dominated by high solar activity in solar cycle 23, BESS, AMS, and PAMELA collaborations took data, that is, BESS-1997 [51], AMS-1998 [52], and PAMELA-2006/08 [53].

Following the procedure described in [2], the observation data were compared with those obtained from HelMod code using the error-weighted root mean square (𝜂_rms) of the relative difference (𝜂) between experimental data (𝑓_exp) and those resulting from simulated differential intensities (𝑓_sim). For each set of experimental data and with the approximations and/or models described above, we determined the quantity 𝜂_rms=∑_𝑖𝜂_𝑖/𝜎_𝜂,𝑖²∑_𝑖1/𝜎²_𝜂,𝑖, (17) with 𝜂_𝑖=𝑓_sim𝑇_𝑖−𝑓_exp𝑇_𝑖𝑓_ref𝑇_𝑖, (18) where 𝑇_𝑖 is the average energy of the 𝑖th energy bin of the differential intensity distribution and 𝜎_𝜂,𝑖 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 𝛿_𝑚—from 0, that is, the nonmodified Parker IMF, up to 3×10⁻⁵, see Figure 1(b)—,  𝐾_⟂,𝑟/𝐾_||=𝜌_𝑘 (from 0.10 up to 0.14) and 𝐾_⟂,𝜃/𝐾_⟂,𝑟=𝜌_𝐸 (from 1 up to 10) seeking a set of parameters set that minimize 𝜂_rms. In Table 6, the average values of 𝜂_rms (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 𝛼_𝑡 and for the No Drift approximation and without any enhancement of the diffusion tensor along the polar direction (𝐾_⟂,𝜇). The results derived with the enhancement of the diffusion tensor along the polar direction indicate that for 𝜌_𝐸=8 and 10 one obtains a value of 𝜂_rms that is from 1.5 up to 3 times larger with respect the case without enhancement (For a comparison, the scalar approximation presented in [2], that is, assuming that the diffusion propagation is independent of magnetic structure, leads to and average 𝜂_rms 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 [2]). The minimum difference with respect to the experimental data occurs when 𝜌_𝑘=0.11–0.13 and 𝛿_𝑚=1.0×10⁻⁵ for both “R” and “L” models, with the “L” model slightly preferred to “R”.

Table 6 

Average values (last three columns) of 𝜂_rms (in percentage, %) as a function of 𝛿_𝑚 (×10⁻⁵) and 𝜌_𝑘, for BESS-1997, AMS-1998, PAMELA-2006/08, obtained from (17) without enhancement of the diffusion tensor along the polar direction (𝜌_𝐸=1), using “R” and “L” models for the tilt angle and No Drift approximation. The differential intensities were calculated accounting for particles inside the heliospheric regions for which solar latitudes are lower than |5.7^∘|.

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 [2]. These modulated intensities are the ones calculated for a heliospheric region where solar latitudes are lower than |5.7^∘|, using 𝜌_𝑘=0.13, 𝛿_𝑚=1.0×10⁻⁵ and 𝜌_𝐸=1 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.

Figure 4 

Proton differential intensity determined with the HelMod code (continuous line) compared to the experimental data of BESS-1997; the dashed line is the LIS (see the text).

Figure 5 

Proton differential intensity determined with the HelMod code (continuous line) compared to the experimental data of AMS-1998; the dashed line is the LIS (see the text).

Figure 6 

Proton differential intensity determined with the HelMod code (continuous line) compared to the experimental data of PAMELA 2006/08; the dashed line is the LIS (see the text).

In Table 7 we present the averages 𝜂_rms (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 |5.7^∘| without any enhancement of the diffusion tensor along the polar direction (𝐾_⟂,𝜇). The simulations with 𝜌_𝐸= 8 and 10 lead to 𝜂_rms comparable with those presented in Table 7. However, using 𝜌_𝐸=1 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 𝜌_𝑘=0.10 and 𝛿_𝑚=(2.0−3.0)×10⁻⁵ 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.

Table 7 

Average values (last three columns) of 𝜂_rms (in percentage, %) as a function of 𝛿_𝑚(×10⁻⁵) and 𝜌_𝑘, for BESS-1999, BESS-2000, BESS-2002 obtained from (17) without enhancement of the diffusion tensor along the polar direction (𝜌_𝐸=1), using “R” and “L” models for the tilt angle and No Drift approximation. The differential intensities were calculated accounting for particles inside the heliospheric regions for which solar latitudes are lower than |5.7^∘|.

Table 8 

Average 𝜂_rms (in percentage, %), for BESS-1999, BESS-2000, and BESS-2002 obtained from (17) without enhancement of the diffusion tensor along the polar direction (𝜌_𝐸=1), 𝛿_𝑚=2.0×10⁻⁵, 𝜌_𝑘=0.10, and using “R” and “L” models for the tilt angle and No Drift approximation. The differential intensities were calculated accounting for particles inside the heliospheric regions for which solar latitudes are lower than |5.7^∘|.

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 [2]. These modulated intensities are the ones calculated for a heliospheric region where solar latitudes are lower than |5.7^∘|, using 𝐾_⟂,𝜇=𝐾_⟂,𝑟 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].

Figure 7 

Proton differential intensity determined with the HelMod code (continuous line) compared to the experimental data of BESS-1999; the dashed line is the LIS (see the text).

Figure 8 

Proton differential intensity determined with the HelMod code (continuous line) compared to the experimental data of BESS-2000; the dashed line is the LIS (see the text).

Figure 9 

Proton differential intensity determined with the HelMod code (continuous line) compared to the experimental data of BESS-2002; the dashed line is the LIS (see the text).

6. Conclusion

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 [6]. 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 30^∘ 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.

Acknowledgments

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.