K/ = KoBf1(R)f2 (R) K_r = aKlh KW = bKh KA = (kA\ — • (2-39-3)
Here f is the ratio of the speed of the CR particles to the speed of light; f[(R) gives the rigidity dependence in GV; Ko is a constant in units of 6.0x1020 cm2 s-1 with Ko =25; a = 0.05 is a constant which determines the value of k_r which contributes to perpendicular diffusion in the radial direction, and b is a constant determining the value of k_q which contributes to perpendicular diffusion in the polar direction. Diffusion perpendicular to the HMF was therefore enhanced in the polar direction by assuming Kqq = k_q = bK// with b = 0.05 and 0.15 respectively. (see also Kota and Jokipii, 1995; Potgieter, 1996). The coefficient (ka)o specifies the amount of drifts allowed, with (ka )o = 1.0 a maximum. The effective radial diffusion coefficient is given by
with y the angle between the radial direction and the averaged HMF direction. Note that y ^ 90° when r > 10 AU with the polar angle 9 ^ 90°, and y ^ 0° when 9 ^ 0°, which means that K// dominates
Krr in the inner and polar regions and k_ dominates in the outer equatorial regions of the Heliosphere. Differential intensities, J rc R2 f, are calculated as particles m-2sr"1s"1MeV-1.
Solutions were computed in Ferreira et al. (1999b) with a simple rigidity dependence for K// and k__ (meaning both K_r and k_q ) given by f(R) = fR/Ro at R > °-4 GF, 1 ' ]f (0.4GV))Ro at R < 0.4 GV, where Ro= 1 GV. This simple approach has proven to be most useful (Potgieter, 1996). For the spatial dependence, f2 (9, r ) = 1 + r/r (2.39.6)
was assumed, with r\ = 1 AU. Note that K// and Kj_ have the same rigidity dependence that becomes flat and constant below 0.4 GV. This feature causes the electron modulation at a given position in the Heliosphere to become almost constant at energies < 50 MeV. At energies < 10 MeV Jovian electrons may contribute to the computed spectra (Haasbroek et al., 1996) but is neglected in Ferreira et al. (1999b) study. Different assumptions for f1(R) may change the slope of the spectra at low energies as was illustrated in detail by Potgieter (1996) but this is not important for the results and conclusions of the Ferreira et al. (1999b) study.
Modeling the modulation of electrons in the Heliosphere, a 2D model with an emulated wavy current sheet was used as developed by Hattingh and Burger (1995a). Obviously, the 2D model differs from a 3D model in the way the HCS is handled. However, using a 2D model is well justified and for a comparison between this 2D model and the 3D model developed by Hattingh (1998) - see also Hattingh and Burger (1995b); it will be considered below in detail in Section 2.41 on the basis of paper Ferreira et al. (1999a). For an additional description of the 'tilt angle' dependence of the model see Section 2.37 on the basis of paper Burger and Potgieter (1999).
The electron differential intensities as a function of tilt angle a are shown in Fig. 2.39.1 for 1.94 GeV electrons at 9 = 90° (equatorial plane) for both polarity cycles. Solutions are shown at three radial distances and for two different values of k±9 . Panels (a) and (b) show solutions at 1 AU; panels (c) and (d) for 5 AU and panels (e) and (f) for 80 AU for b = 0.05 and b = 0.15 respectively.
From Fig. 2.39.1 follows that at 1 AU the intensity for the A < 0 polarity cycle is higher than for the A > 0 cycle. As Kj_9 was enhanced by increasing b from 0.05 to 0.15, a reduction occurs in the differences between the two epochs. The intensities for both epochs are lower for the increased value of K9 and do not have such a strong a dependence as for a smaller K9 . For A > 0 this diminished dependence on a is especially evident for a < 40°. At 5 AU the intensities for the A > 0 and A < 0 cycles cross at a ~ 15° with the A < 0 intensities lower than those for the A > 0 for a < 15°. As for 1 AU, the increase in K9 led to a decrease of the a dependence, especially with a < 40°. The spectra shown no longer cross, but it still occurs at a slightly larger radial distance. For 80 AU, the intensities for the A > 0 are consistently higher than for the A < 0 epoch and the increase in K9 had little or no effect on the differential intensities as a function of a. This indicates that the increase in Kj_9 is more important in the inner and middle Heliosphere (compare also Ferreira and Potgieter, 1999).
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