6.4 Re-acceleration of Intermediate- and High-energy Electrons
6.4.2 Dependence on the rigidity profile of diffusion
The form of the HPS is shown in the previous subsection to influence the re-acceleration of spectra at the TS, because it is largely preserved during modulation as a result of rigidity- independent diffusion. This is also the case at high energies, since very large MFPs phase out modulation. If the rigidity dependence of the diffusion coefficients is however altered such that modulated spectra do not necessarily reflect the characteristics of the original energy distribution, the properties of this new dependence would dictate the form of the distributions
10−2 10−1 100 101 10−5
10−4 10−3 10−2 10−1 100 101 102
Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o 1 AU 94 AU 122 AU
Pk = 0.34 GV
10−2 10−1 100 101
10−5 10−4 10−3 10−2 10−1 100 101 102
Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o 1 AU 94 AU 122 AU
Pk = 0.45 GV
10−2 10−1 100 101
10−5 10−4 10−3 10−2 10−1 100 101 102
Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o 1 AU 94 AU 122 AU
Pk = 0.60 GV
Figure 6.11:Modelled spectra at the TS (94 AU,θ= 55◦) and Earth (1 AU,θ= 90◦), respectively shown in red and blue, modulated from the HPS at 122 AU shown in black. Solid and dashed lines respectively represent solutions with and without DSA effects. Each panel corresponds to a configuration in Figure 5.8 for the indicated values ofPk(of Eq. 3.24). Drifts are neglected for these solutions.
incident at the TS and hence the extent by which they are modified during DSA. It is therefore worthwhile investigating the dependence of electron re-acceleration on the rigidity profile of diffusion. Section 5.4.1 demonstrates how varyingPk (of Eq. 3.24) alters the rigidity profile of the diffusion coefficients, and how this in turn affects the modulation of electron energy spectra;
recall thatPkis the rigidity above which the energy range of the turbulence power spectrum becomes more prevalent in defining diffusion than the inertial range. It is shown in particular that increasingPkreduces the segment of the rigidity profile along which MFPs increase with rigidity according toP1.23, yielding rigidity-independent diffusion up to higher energies and larger MFPs at lower energies. This is furthermore limits the hardening of TS spectra in the region of 0.1 to 1 GeV, since the power law at lower energies transitions more directly into the softer power law retained from the HPS at high energies. This is shown in Figure 5.9.
The resultant spectra of the diffusion configurations explored in Section 5.4.1 are illustrated
10−2 10−1 100 101 1
1.5 2 2.5 3
Kinetic Energy (GeV)
Ratio of differential intensities
1 AU, 0.34 GV 1 AU, 0.45 GV 1 AU, 0.60 GV 94 AU, 0.34 GV 94 AU, 0.45 GV 94 AU, 0.60 GV
α = 10o
Figure 6.12: Ratios of the solutions in Figure 6.11 with DSA effects to those without, and shown for different values ofPkin different line styles as indicated in the legend. Ratios at Earth (1 AU,θ= 90◦) and the TS (94 AU,θ= 55◦) are shown in blue and red respectively.
again in Figure 6.11, but with the inclusion of shock-accelerated spectra. The accelerated in- tensities are higher by similar margins for all three diffusion configurations considered, but in the energy region of the power-law transition, where DSA effects seem to bottleneck, shock- accelerated intensities appear more pronounced for larger values ofPk. The TS spectrum being accelerated in case ofPk=0.60 GV in Figure 6.11, for instance, is essentially distributed accord- ing to spectral indices of−1.35and−3.18at intermediate and high energies respectively, with somewhat harder spectra prevailing over a narrow energy region at the transition between these power laws. For smaller values ofPkthis hardened spectral region is wider and the aver- age spectral index thereof larger. DSA effects become more prominent in the transition region for larger values ofPk, because the hardening of spectra at these energies is largely circum- vented by the diminishing of the segment of the rigidity profile increasing withP, and because the softer spectra that emerge are easier to accelerate. This ties in once again to the theme in- troduced in Section 6.2 that DSA raises the intensities of softer spectra by greater amounts than those of hard spectra. These amounts for each diffusion configuration are illustrated in Figure 6.12 by ratios of solutions with shock-acceleration to those without. The ratios demonstrate that the re-accelerated contribution at the TS declines less rapidly across the power-law transi- tion from 100 MeV to just above 1 GeV for larger values ofPk, and that the large depression of acceleration effects visible forPk =0.34 GV (also pointed out in Section 6.3) at a few hundred MeV disappears. Similar effects are shown in Figure 6.12 for the contribution of re-accelerated electrons at Earth, although these extend to lower energies.
At the low-energy end of Figure 6.12, where ratios appear to remain constant with energy, the re-accelerated contribution is higher for the configurations wherePk is smaller - at the TS, Pk =0.34 GV yields a factor increase below 100 MeV of at least 2.6 as opposed to a factor just short of 2.4 forPk =0.60 GV. This effect, which is hardly noticeable on the scale of Figure 6.11, follows likely as a result of more effective particle confinement; note that below 100 MeV the
distributions incident at the TS as well the rigidity profiles of diffusion have the same form for the configurations considered, while only the values of MFPs differ for each. Recall from Chapter 4, and also Section 6.2, that impaired diffusion enhances acceleration. The larger MFPs ensuing at low energies whenPkis scaled up therefore causes DSA to raise spectral intensities by smaller amounts. Note however that above 100 MeV the re-accelerated contributions are greater for larger values ofPk (for reasons already explained above) despite the MFPs being larger in those cases. This suggests that the manner in which diffusion modifies energy spectra at the TS (or equivalently, the form of the rigidity dependence of diffusion) is more important than the level of diffusion in determining the amount by which DSA raises intensities. Hence, aside from the compression ratio, the form of energy distributions incident at the TS remains the predominant role player influencing the acceleration of these distributions. Finally, the ratios in Figure 6.12 converge toward high energies; since both the incident spectra and MFPs are identical there for each configuration, the associated acceleration effects are also the same.