tribution (or part thereof) of CRs incident on the TS. This becomes especially relevant when interpreting the acceleration features for spectra arriving at the TS after being modulated from an initial spectrum, such as the HPS, specified further downstream. The dependence of accel- eration effects on the hardness of spectra is a recurring concept in following sections.
10−3 10−2 10−1 100 101 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 ) A > 0
α = 10o
1 AU 94 AU 122 AU
10−3 10−2 10−1 100 101
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 )
A < 0 α = 10o
1 AU 94 AU 122 AU
Figure 6.5:Modelled energy spectra for galactic electrons at the TS (94 AU,θ= 55◦) and Earth (1 AU, θ = 90◦) as modulated from the reference HPS (black line) at 122 AU. Line colours represent radial distances as indicated in the legend, while solutions with and without shock acceleration are shown in solid and dashed lines respectively. These solutions are shown for A>0 (top panel) and A<0 (bottom panel) during solar minimum conditions (α= 10◦).
than the index associated with the shock compression ratio, and uniformly raises the intensi- ties instead. The factors by which the intensities are raised also depend on how similar the aforementioned indices are. Hence, the distribution up to nearly 100 MeV arriving at the TS, after having retained the power-law index of−1.35during its modulation in the heliosheath, is too hard to be altered significantly by a shock withs= 2.5, which can at the very most yield an accelerated index of−3.0. As a result its intensity is raised uniformly with a factor of∼2.5 as mentioned before. The succeeding depression in acceleration effects, centred around 100 to 200 MeV, similarly follows because the spectrum incident at the TS becomes very hard due to decreasing modulation (as result of increasing MFPs) with increasing energy. Consequently, the intensities are raised by a barely noticeable factor on the scale of Figure 6.5. However, as modulation eventually diminishes toward 1 GeV due to very large MFPs, and the TS spectrum inherits the soft−3.18power-law index from the HPS, acceleration effects are restored. Due to the similarity of the incident spectrum to that producible by the TS, intensities are raised by
10−3 10−2 10−1 100 101 1
1.5 2 2.5 3
Kinetic energy (GeV)
Ratio of Differential Intensities
α = 10o
1 AU, A < 0 1 AU, A > 0 94 AU, A < 0 94 AU, A > 0
Figure 6.6: Ratios of the spectra shown in Figure 6.5 at the TS (94 AU,θ = 55◦) and at Earth (1 AU, θ = 90◦) with shock acceleration to those without. Line styles and colours respectively represent the magnetic polarities and radial distances as indicated in the legend.
a surprisingly large factor at such high energies of nearly 1.6. Figures 6.7 and 6.8 show radial profiles that are representative of electrons in each of the aforementioned spectral regions and reflect similar acceleration features at the TS position. Ultimately though, as in Section 6.2, acceleration effects wane due to the TS curvature limit (Eq. 3.55) at E & 1 GeV. The higher energies obtained by re-accelerated electrons during the positive polarity cycle as shown in Figure 6.6 follows because the contribution of drifts to intensities manifests at higher energies during this cycle (see Figure 5.17). Drifts harden spectra between∼100 and 400 MeV by rais- ing intensities at the TS sharply between these energies; by the insights garnered earlier, DSA cannot raise the intensities of such hard distributions appreciably, and hence the re-accelerated contribution is diminished. From the ratios of Figure 6.6 it follows that the re-accelerated con- tribution is generally larger for A>0 than for A<0 (see also Figure 6.8), except for a narrow region between 300 and 700 MeV where the opposite is true. The discussion on drifts and their influence on the re-acceleration of electrons, and vice versa, is continued in Section 6.4.3.
6.3.2 Radial distributions of re-accelerated electrons
Though the spectral changes incurred due to DSA are revealed to be subtle, a further con- sideration to gauge the influence of this mechanism is the distribution of re-accelerated elec- trons throughout the heliosphere. Its largest contribution to electron intensities is of course expected at the shock itself, but subject to the various transport processes involved, intensities elsewhere may also be bolstered. Figure 6.7 shows the radial profiles of electrons at different energies both with and without the contribution of DSA. It is revealed, as in Figures 6.5 and 6.6, that re-acceleration is most prominent at lower energies, and that the contribution of these re-accelerated electrons is appreciable even in the deep heliospheric interior. Considering only
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 )
A > 0 α = 10o θ = 55o 16 MeV
200 MeV 1000 MeV rTS
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 )
A < 0 α = 10o θ = 55o 16 MeV
200 MeV 1000 MeV rTS
Figure 6.7:Modelled radial intensity profiles with DSA at the TS (solid lines) and without (dashed lines) 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 A>0 and A <0 as shown in the top and bottom panels respectively. The TS is marked at 94 AU with the vertical dashed line and the HP is at 122 AU.
these intensity profiles, the effects of DSA are not obvious aside from a general increase in intensities. Earlier studies [e.g.Ferreira et al., 2004b;Langner and Potgieter, 2004; Potgieter and Langner, 2004] on the DSA of galactic electrons showed a more definitive intensity peak at the TS position, while the profiles in Figure 6.7 appear in contrast smooth across this boundary.
The re-accelerated contribution does however peak at the TS, as illustrated by the ratios in Figure 6.8, and constitutes a factor increase in intensities there of at least 2.5 for 16 MeV elec- trons. By the same low-diffusion properties responsible for the massive decrease in galactic electron intensities across the heliosheath, the electrons re-accelerated at the TS do not diffuse far into the heliosheath either. In fact, under the current configuration, the model predicts that no re-accelerated electrons (forE <1 GeV, at least) reach the HP. By contrast, the contribution inward from the TS is quite noticeable; the diffusion coefficients increase towards the TS inside the heliosphere so that re-accelerated electrons may permeate further into the heliosphere. The re-accelerated contribution in fact raises intensities throughout most of the interior by a factor
1 10 20 30 40 50 60 70 80 90 100 110 120 1
1.5 2 2.5 3 3.5
Radial distance (AU)
Ratio of Differential Intensities
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 6.8: Ratios of the profiles shown in Figure 6.7 with shock acceleration to those without. Line styles and colours respectively represent the magnetic polarities and electron energies as indicated in the legend. The TS is marked at 94 AU with the vertical dashed line and the HP is at 122 AU.
of up to 2. This spread-out intensity profile upstream of the TS and the steep increase of in- tensities downstream in the heliosheath collectively serve to obscure intensity increases arising due to local acceleration, and hence a definitive peak-like structure is not observed at the shock in Figure 6.7. These profiles are shown at a polar angle of θ = 55◦; Section 6.6 illustrates the spatial distributions of re-accelerated electrons more globally and in greater detail.