6.4 Re-acceleration of Intermediate- and High-energy Electrons
6.4.3 The effects of electron drifts
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.
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 A > 0
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 A < 0
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 A > 0
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 A < 0
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 A > 0
1 AU 94 AU 122 AU
Pk = 0.60 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 A < 0
1 AU 94 AU 122 AU
Pk = 0.60 GV
Figure 6.13: Similar to Figure 6.11, but with full drifts and shown for both the A>0 (left) and A<
0 (right) cycles. Solid and dashed lines respectively represent solutions with and without DSA effects.
The panels from top to bottom correspond to the different configurations of Eq. 3.24 shown in Figure 5.8 forPk =0.34 GV, 0.45 GV and 0.60 GV as indicated.
Considering Figure 6.13, which shows drift-enabled model solutions for both polarities and for each of the earlier considered diffusion configurations, the acceleration effects attributed to changes in the rigidity profile of diffusion are more obvious than those following from polarity- dependent changes. In each instance, the intensities are raised by a larger margin during the A
<0 cycle (yielding harder spectra) than during the A>0 cycle, so that larger DSA effects are expected for the latter polarity. This is easier to discern from Figure 6.14. The re-accelerated contribution is shown to be consistently larger during the positive polarity cycle than during the negative. An exception is observed at the TS over an energy region between roughly 300 and 700 MeV wherein the opposite holds forPk=0.34 GV, and where the contributions during the two cycles are at least similar for the other diffusion configurations. Note from Figure 5.16 that drift effects diminish rapidly at the TS across this narrow energy interval so that the aforementioned features are to be expected. No such exception is visible in Figure 6.14 at Earth.
10−2 10−1 100 101 1
1.5 2 2.5 3
Kinetic Energy (GeV) Ratio of differential intensities
no drifts A > 0 A < 0 Pk = 0.34 GV
α = 10o θ = 90o r = 1 AU
10−2 10−1 100 101
1 1.5 2 2.5 3
Kinetic Energy (GeV) Ratio of differential intensities
no drifts A > 0 A < 0
α = 10o θ = 55o r = 94 AU
10−2 10−1 100 101
1 1.5 2 2.5 3
Kinetic Energy (GeV) Ratio of differential intensities
no drifts A > 0 A < 0 α = 10o θ = 90o r = 1 AU Pk = 0.45 GV
10−2 10−1 100 101
1 1.5 2 2.5 3
Kinetic Energy (GeV) Ratio of differential intensities
no drifts A > 0 A < 0
α = 10o θ = 55o r = 94 AU
10−2 10−1 100 101
1 1.5 2 2.5 3
Kinetic Energy (GeV) Ratio of differential intensities
no drifts A > 0 A < 0 α = 10o θ = 90o r = 1 AU Pk = 0.60 GV
10−2 10−1 100 101
1 1.5 2 2.5 3
Kinetic Energy (GeV) Ratio of differential intensities
no drifts A > 0 A < 0
α = 10o θ = 55o r = 94 AU
Figure 6.14: Ratios of the full-drift solutions in Figure 6.13 with DSA effects to those without for both polarities, along with the corresponding no-drift ratios of Figure 6.12. Panels from top to bottom corre- spond to the indicated values ofPk(of Eq. 3.24), with ratios shown at Earth (left) and the TS (right).
While drifts clearly influence the acceleration process of energy spectra at the TS by altering their form, the possible reciprocation of this interaction must also be considered. Section 5.5 demonstrates the magnitude of drift effects and the contribution of drifts to intensities dur- ing opposite magnetic polarities in conjunction with the effects of different rigidity profiles of diffusion. The discussion to follow explains how these effects and contributions differ when DSA is also present. First, it is considered how DSA affects the contribution of drifts to electron intensities: Figure 6.15 shows the ratios of shock-accelerated spectral intensities with drifts to those without; the ratios also presented in Figure 5.17 where DSA is suppressed are shown for comparison. Similar to the unshocked solutions, the drift contributions associated with shock- accelerated spectra are distributed over a wider range of energies at Earth than at the TS, and are larger during the A<0 cycle than during the A>0 cycle. As before, these contributions are also larger for the cases where diffusion levels are smaller, that is, for smaller values ofPk.
10−2 10−1 100 101 0
1 2 3 4 5 6 7 8 9
Kinetic Energy (GeV)
Ratio of differential intensities
0.34 GV 0.45 GV 0.60 GV
A > 0 α = 10o θ = 90o r = 1 AU
10−2 10−1 100 101
0 2 4 6 8 10 12 14 16
Kinetic Energy (GeV)
Ratio of differential intensities
0.34 GV 0.45 GV 0.60 GV
A < 0 α = 10o θ = 90o r = 1 AU
10−2 10−1 100 101
0 1 2 3 4 5 6 7 8
Kinetic Energy (GeV)
Ratio of differential intensities
0.34 GV 0.45 GV 0.60 GV A > 0
α = 10o θ = 55o r = 94 AU
10−2 10−1 100 101
0 2 4 6 8 10 12 14 16 18
Kinetic Energy (GeV)
Ratio of differential intensities
0.34 GV 0.45 GV 0.60 GV A < 0
α = 10o θ = 55o r = 94 AU
Figure 6.15: Ratios of the full-drift shock-accelerated solutions of Figure 6.13 to the no-drift shock- accelerated solutions of Figure 6.11. These are shown in solid lines for both A > 0 (left) and A <
0 (right) alongside the corresponding ratios in dashed lines (also presented in Figure 5.17) for which shock-acceleration is disabled. The ratios are displayed both at Earth and at the TS in the top and bot- tom panels respectively, with line colours representing the values ofPkindicated in the legend.
Aside from displaying the same qualitative features, however, the drift contributions associ- ated to shock-accelerated solutions are shown to be notably smaller for both polarities. The same holds for drift effects as conventionally defined by ratios of intensities for drift-enabled solutions during the A<0 cycle to those of the A>0 cycle: These ratios, shown in Figure 6.16, also display qualitative features for shock-accelerated solutions that are similar to the ratios of solutions with no acceleration, although the ratios are again smaller where DSA is involved.
The explanation for these effects is relatively simple: Just as larger diffusion coefficients de- crease modulation, resulting in larger intensities and smaller intensity gradients, DSA simi- larly raises intensities at the TS and thereby all intensities at smaller radial distances as well.
The smaller intensity gradients that follow reduce the imprint of drifts on the distribution func- tion in the TPE. Therefore, where DSA effects are present, drift effects and the contribution of drifts to intensities are diminished in the same way that larger MFPs would diminish them.
While it appears from the above that particle drifts and DSA inhibit one another in terms of
10−2 10−1 100 101 0.5
1 1.5 2 2.5 3
Kinetic Energy (GeV)
Ratio of differential intensities
0.34 GV 0.45 GV 0.60 GV α = 10o
θ = 90o r = 1 AU
10−2 10−1 100 101
0.5 1 1.5 2 2.5 3 3.5 4
Kinetic Energy (GeV)
Ratio of differential intensities
0.34 GV 0.45 GV 0.60 GV α = 10o
θ = 55o r = 94 AU
Figure 6.16: Ratios of the drift-enabled shock-accelerated solutions in Figure 6.13 with A<0 to that for which A>0. These are shown in solid lines, whereas the accompanying dashed lines represent the corresponding ratios (also shown in Figure 5.16) without DSA effects. The ratios are displayed both at Earth and at the TS in the panels on the left and right respectively, with line colours representing the values ofPkindicated in the legend.
global modulation, this does not imply (in physical terms, at least) that drifts are not involved in the acceleration process. When CRs drift along a shock and intermittently scatter across or off of it they gain energy in what is referred to as shock drift acceleration. These energy gains are similar to what is achieved through DSA and can be accounted for in the matching condi- tions at the TS [Jokipii, 1982], while the Parker TPE contains all the relevant physics to describe this process. This concept is yet to be developed further in the context of the current model and offers an intriguing subject for further research. See also the discussions byBall and Melrose [2001] andMoraal[2001] for more on this topic.