Electrons - negatively charged, less massive and generally faster than their ionic CR counter- parts - are modelled with a ratio of mass number to charge of Θ = 1 and have a rest-mass energy ofe0 = 5.11·10−4 GeV. Indeed, much of the desired electron behaviour follows natu- rally in the model if the rest-mass energy is correctly specified. For example, electrons become
77
10−3 10−2 10−1 100 101 10−6
10−5 10−4 10−3 10−2 10−1 100 101 102 103 104
Kinetic energy (GeV) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o
1 AU, A > 0 1 AU, A < 0 94 AU, A > 0 94 AU, A < 0 122 AU
Figure 5.1: Modelled energy spectra for galactic electrons at the TS (94 AU, θ = 55◦) and at Earth (1 AU,θ = 90◦) as modulated from the reference HPS (black line) at 122 AU. Line styles and colours respectively represent the magnetic polarities and radial distances as indicated in the legend. These solutions are shown without the effects of DSA and for solar minimum conditions (α= 10◦).
relativistic with particle speeds attaining significant fractions of light speed at an energy of roughly a factor ofE0/e0 lower than for nuclei (Figure 3.2), withE0 the rest-mass energy for protons. These same attributes result in electrons having smaller radii of gyration than other heavier CR species at most energies, while displaying the same gyroradius and length scales of drift as protons atE > E0; see Figure 3.9. Their negative charge naturally implies that the drift patterns expected for positive particles are reversed for electrons during both solar polarity cycles. Furthermore, the relationship between a particle’s kinetic energy and its rigidity is also shown in Figure 3.1 to differ above and below the rest-mass energy, which is demonstrated in Section 3.6.2 to affect the form of accelerated energy spectra. This is further explored in Section 6.2. See alsoMoraal and Potgieter[1982],Langner and Potgieter[2004], andCaballero-Lopez et al.
[2010] for more on the species-specific differences that are of interest in CR modulation studies.
Further revisions to the model follow from the revelations of the Voyager 1 spacecraft, and pertains mostly to the form of electron energy spectra in the heliosheath. The measurements presented in Figure 2.18 suggest a power-law distributed energy spectrum for 4 to 40 MeV elec- trons at the HP, with a yet steeper distribution following atE &1 GeV as inferred from data collected by the PAMELA detector at Earth. The heliopause spectrum (HPS) ofPotgieter et al.
[2014b], which is discussed further in Section 5.3, displays these features and is implemented in this study as the reference input spectrum for galactic electrons. Although CRs are likely to undergo modulation in the outer heliosheath as well [Scherer et al., 2011; Strauss et al., 2013b;
Luo et al., 2015], the HP is the outermost boundary considered in the current model so that an input spectrum specified here suffices. The Voyager observations also prompt revision of the coefficients describing electron diffusion. Note from Figure 2.18 that the power-law distribu- tion of the measured intensities is preserved at radial distances throughout most of the inner
1 10 20 30 40 50 60 70 80 90 100 110 120 10−3
10−2 10−1 100 101 102
Radial distance (AU) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o θ = 55o 16 MeV, A > 0
16 MeV, A < 0 200 MeV, A > 0 200 MeV, A < 0 1000 MeV, A > 0 1000 MeV, A < 0 rTS
Figure 5.2: Modelled radial intensity profiles for galactic electrons at sample energies of 16 MeV, 200 MeV and 1 GeV, represented by different line colours as indicated in the legend. Solutions are shown alongθ = 55◦ for solar minimum conditions (α= 10◦) and both magnetic polarities as also indicated in the legend. The position of the TS is marked with the vertical dashed line at 94 AU and the HP is situated at 122 AU.
heliosheath, which suggests rigidity-independent diffusion across the energy range where this occurs [see also Potgieter and Nndanganeni, 2013a]. The intensities also decrease from the HP to the TS with a factor of a few hundred at these energies. To model these features, the use of the parallel diffusion coefficient byBurger et al.[2008] is discontinued in favour of the phe- nomenologically informed coefficient described by Eqs. 3.23 to 3.25. The full motivation for this is discussed in Section 3.3.2. The perpendicular diffusion coefficients ofBurger et al.[2000]
are however still implemented, but are adapted to retain the rigidity dependence of the paral- lel coefficient (see Section 3.3.3). The rigidity profiles of the selected coefficients are revisited in Section 5.4. Finally, and more generally, the heliosheath is assumed to be less vast in the current and subsequent chapters than in Chapter 4, with the TS and HP positions respectively specified asrT S = 94AU andrHP = 122AU to reflect the Voyager 1 detection of these bound- aries. SeeFerreira et al.[2004b] for a study on electron modulation where the positions of these boundaries are varied. While the TS compression ratio is retained ass= 2.5, the effects of DSA are suppressed in this chapter by setting∇ ·V~sw= 0in the shock region (See Section 3.6.1). No source functions are specified at the TS or elsewhere; all solutions are merely modulated forms of the HPS. For illustrative purposes, full drift efficiency is assumed (i.e. κD,0 = 1.0). Other modulation parameters that are not mentioned here remain unchanged from those specified in Section 4.2, while any departures are pointed out explicitly in the following sections.
The above model configuration yields the solutions shown in Figures 5.1, 5.2 and 5.3, which serve as standard references for comparison for subsequent solutions. The first of these shows electron energy spectra at the TS along the approximate Voyager 1 trajectory at a polar angle of θ= 55o, and at Earth in the equatorial plane atθ= 90◦, for solar minimum conditions and both
0 50 100
−100
−50 0 50 100
Radial distance (AU)
Radial distance (AU)
α = 10o A > 0
200 MeV
−0.5 0 0.5 1 1.5 2
0 50 100
−100
−50 0 50 100
Radial distance (AU)
Radial distance (AU)
α = 10o A < 0
200 MeV
−0.5 0 0.5 1 1.5 2
Figure 5.3: Contour plots illustrating 200 MeV galactic electron intensities in the meridional plane of the heliosphere for A>0 (left) and A<0 (right). The colour bars show intensities as percentages of the maximum on a logarithmic scale. The dashed white halfcircle indicates the TS position at 94 AU, while the dash-dotted lines indicate the polar extent of the HCS forα= 10◦. The Voyager 1 trajectory is approximated by the dashed black line alongθ= 55◦, while the Sun is located at the origin.
magnetic polarities. Note how the TS spectrum retains the power law that the HPS exhibits at lower energies. This follows because of the rigidity-independent MFP chosen for electrons forP < 0.34GV (see Section 3.3.2). Caballero-Lopez et al.[2010] andPotgieter and Nndanganeni [2013a] show similar results. From the form of the spectrum at Earth, it appears that adiabatic energy losses, which force ion spectra into characteristic E+1-distributed power laws (as in Section 4.2), are absent for electrons in the energy range considered here. This, too, is shown to be a result of rigidity-independent diffusion; see Section 5.4. Furthermore, both the energy spectra in Figure 5.1 and the spatial distributions of Figures 5.2 and 5.3 demonstrate substantial decreases in intensities from the HP to the TS, creating a prominent modulation barrier. This effect, particularly visible toward lower energies, is the result of the impaired diffusion brought about by the small MFPs specified for particles in this region and had been anticipated before its detection by the Voyager spacecraft [Potgieter and Le Roux, 1989]. See also the modelling results ofFerreira and Potgieter[2002],Ferreira et al.[2004b],Nkosi et al.[2011] andPotgieter and Nndanganeni[2013a] for similarly large electron modulation in the heliosheath. This barrier is also modelled byLangner et al.[2004] for protons. The more complicated problem, however, is predicting the contribution of galactic electrons to the intensity at Earth. See e.g. Potgieter and Nndanganeni[2013a] andNndanganeni[2015] for comment on this matter.
While electron transport appears to be generally sensitive to diffusion properties, the effects of electron drifts are confined to a narrow domain in energy under the current model configu-
ration. Figure 5.1 shows that polarity-dependent modulation is most prominent for electrons of 0.1 to 1 GeV. Their spatial distributions (as seen in Figures 5.2 and 5.3) may be explained at the hand of the expected electron drift patterns: Recall that their movement along the HCS is outward during the A<0 polarity cycle, poleward in the outer heliosphere, and back down- ward from the poles toward the equatorial regions in the heliospheric interior. Of course, these directions are reversed during the A>0 cycle. Figure 5.3 illustrates these patterns for 200 MeV electrons, with contours assuming a convex (or outward bulging) shape in their direction of travel. During the A<0 cycle, enhanced polar diffusion at high latitudes facilitates poleward movement of electrons through the heliosheath. Likewise, this also encourages transport from these electron-rich polar regions to the HCS in the interior, from where electrons drift outward.
This collectively serves to supplement intensities up to the TS and a short way beyond. This can also be discerned from Figure 5.2. For A>0, the drift patterns in the heliosheath tend to carry electrons toward the equatorial regions, from where the passage of electrons inward along the HCS is stifled by the low-diffusion barrier. The inward drifts along the HCS still manage to carry electrons across the TS and hence supplement intensities in the interior, however not to the extent achieved during the negative polarity cycle. This discussion is expanded in Section 5.5. As for the rest of the chapter, the features and processes earmarked in the text above for further comment form the basis of discussion in following sections.