Fig. 2.39.3. As in Fig. 2.39.2, but at a polar angle of 0 = 5°. From Ferreira et al. (1999b).

The a dependence is clearly negligible compared to the equatorial regions as follows from comparing panels (a) to (f) in Fig. 2.39.3 with those in Fig. 2.39.2. The only significant a dependence is when a > 60°. This is understandable, because when the tilt angles become very large the modulation of intensities must respond to the presence of the wavy HCS in these high latitude regions of the simulated Heliosphere. The increase in K9 from 5% to 15% of K// gives the largest effect on the level of modulation in the polar regions as a comparison of panels (b), (d) and (f) in Fig. 2.39.3 with those in Fig. 2.39.2 illustrates. This is due to the fact that the enhancement of K9 in the polar directions becomes more effective with decreasing polar angles. The significant reduction of the latitude dependence of the electron intensities due to the enhancement of K9 also follows clearly from comparing panels (b),(d) and (f) in Fig. 2.39.3 with those in Fig. 2.39.2.

According to Ferreira et al. (1999b) studying the effects on electron modulation of enhancing K9 in the polar directions, from 5% to 15% of K//, it was found that this increase reduced the differences between the modulated intensities as a function of tilt angle a for the two magnetic polarity cycles. This is especially strong for the inner Heliosphere in the equatorial regions and most of the Heliosphere in the polar regions. The increase in K9 also led to a decrease in the a dependence of the differential intensities for a < 40° for the inner Heliosphere in the equatorial regions as shown in Fig. 2.39.1 and Fig. 2.39.2. For the polar regions, shown in Fig. 2.39.3, the increase in K9 had little or no change in the a dependence of the intensities for a < 60°, but it caused a significant reduction in the global latitude dependence of electron modulation.

2.40. Rigidity dependence of the perpendicular diffusion coefficient and the Heliospheric modulation of CR electrons

Potgieter et al. (1999) note that the diffusion perpendicular to the Heliospheric magnetic field (HMF) plays an important role in the modeling of the Heliospheric propagation and modulation of galactic CR. This followed directly from the simulation of latitude dependent modulation, first studied about 30 years ago with a two-dimensional model by Fisk (1976). Even with the introduction of global and neutral sheet drifts in models of increasing complexity (Kota and Jokipii, 1983; Potgieter and Moraal, 1985; le Roux and Potgieter, 1991; Burger and Hattingh, 1998) the importance of the perpendicular diffusion coefficient has remained and is arguably the most important element of the diffusion tensor. But because no comprehensive theory exists for it, the best that can be done at this stage is to make reasonable assumptions about its value, spatial and rigidity dependence. Fortunately, the modulation of CR electrons in the Heliosphere provides a useful tool in understanding and in determining the diffusion coefficients. Computed electron modulation responds directly to what is assumed for the energy dependence of the diffusion coefficients below 500 MeV, in contrast to protons which experience large adiabatic energy changes below this energy and which consequently obscure the effects of changing the energy dependence of any of the diffusion coefficients. Another aspect is that drifts become progressively less important with decreasing electron energy, to have almost no effect on electron modulation below 100-200 MeV. For the work of Potgieter et al. (1999), electron modulation was used to illustrate how important perpendicular diffusion is, in particular its rigidity dependence, to the Heliospheric modulation of CR electrons.

According to Potgieter et al. (1999), the modulation of CR electrons in the Heliosphere provides a useful tool in understanding and estimating the diffusion tensor applicable to Heliospheric modulation. Using a comprehensive modulation model including all major mechanisms to study electron modulation, especially at energies below 500 MeV, Potgieter et al. (1999) found that perpendicular diffusion is very important to electron modulation at these energies. Electrons respond directly to the energy dependence of the diffusion coefficients below 500 MeV, in contrast to protons which experience large adiabatic energy losses below this energy. As a result of this and because drifts become unimportant for electrons at these low energies, important conclusions can be made about the absolute values, spatial and especially the rigidity dependence of the diffusion coefficients.

2.40.2. The propagation and modulation model, main results, and discussion

The propagation and modulation model that was used in Potgieter et al. (1999) is the same which was used in Ferreira et al. (1999b) and was described above, in Section 2.39.2 (see Eq. 2.39.1 up to Eq. 2.39.6). The results are shown in Fig. 2.40.1 and illustrate in general that when the rigidity dependence of k__ (that is both K_r and k__q in Eq. 2.39.3) is taken independently from that for k// at energies below ~ 500 MeV, it clearly dominates modulation at these lower energies. Potgieter et al. (1999) assumed for these results that function f (d, r) in Eq. 2.39.3 is described by Eq. 2.39.6, with r1 = 1 AU, Ko = 25, a = 0.05 and b = 0.15. The HMF magnetic cycle was chosen to be A > 0 (e.g. the solar polarity cycle in 1999). The different rigidity dependencies for K// and k_ are shown in the inserted graph where k_ was multiplied by 10 for illustrative purposes. In this case the rigidity dependence for k// was according to the damping model - composite slab - 2D geometry of Bieber et al. (1994); see also Potgieter (1996). It is evident that the computed spectra in the inner Heliosphere are still compatible to data at higher energies but below ~ 100 MeV the modulation becomes unreasonably large which is apparently not supported by measurements. However, these results illustrate that although k_r and k_q is only 5% and 15% of the value of k// respectively, perpendicular diffusion, especially in the polar direction, dominates electron modulation below ~ 100 MeV and that it is as such a very important parameter that should be studied in detail. If the increase in the low energy part of observed electron spectra with decreasing energy was taken as a characteristic of modulated electron spectra then k__ ^ pR, as shown in Fig. 2.40.1, is not a workable option.

As it was argued by Kota and Jokipii (1995) that k_q plays a crucial role in CR modulation which is the assumption for work of Potgieter et al. (1999), because perpendicular diffusion enhanced in the polar direction seems a necessity for getting computed latitude dependencies compatible to the Ulysses observations. Because k__q may be considerably larger than k__r in the Heliospheric polar regions, Fig.

2.40.2 illustrates whether a further enhancement of k_q may change the features shown in Fig. 2.40.1 by taking b = 0.40 instead of 0.15 in Eq. 2.39.3. This increase caused additional low energy electrons to reach the equatorial plane, compared to Fig. 2.40.1. A further increase in b had little additional effect at these low energies, so that there is clearly a limit to what his approach can do.

To extend the study on the modulation aspects shown in Fig. 2.40.1 and Fig. 2.40.2, Ferreira (1999) constructed an analytical expression for k// , applicable to electrons, using the theoretical work of Hattingh (1998) and Burger and Hattingh (1998) where they on their part used the formalism of Bieber et al. (1994), especially the random sweeping model for dynamical turbulence with pure slab geometry (see also Zank et al., 1998). This expression is depicted in Fig. 2.40.3 as a function of kinetic energy for 1 AU, 10 AU, 50 AU and 100 AU in the equatorial plane. No explicit latitude dependence was assumed.

It is evident from Fig. 2.40.3 that the radial dependence of K// is much more sophisticated with main feature the changing slopes of the function and that K// is much larger in the outer Heliosphere at high energies than at low energies, with the opposite at 1 AU.

The corresponding computed spectra are shown in Fig. 2.40.4. In this case a = 0.05 and b = 0.15 in Eq. 2.39.3. The electron data from the Ulysses/KET experiment for 1997 are shown to provide a reference for inner Heliospheric electron intensities during minimum modulation. The compatibility between the data and the model is reasonable at energies > 400 MeV, but not at energies < 100

MeV. Although Potgieter et al. (1999) used a very sophisticated function for K// and k_ it does not give electron modulation compatible to Ulysses data at low energies. In this case the only way to assure compatibility is to make k__ almost independent of kinetic energy at low energies because it dominates electron modulation at these low energies - see also Ferreira (1999), Ferreira and Potgieter (1999). It should be kept in mind, however, that measured low energy electrons might contain a Jovian contribution.

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