6.5 Electron Re-acceleration at Very Low Energies
6.5.1 Consequences of the form of input spectra
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.
10−4 10−3 10−2 10−1 10−5
10−4 10−3 10−2 10−1 100 101 102 103 104 105 106
Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o
1 AU 94 AU 122 AU
10−4 10−3 10−2 10−1
10−5 10−4 10−3 10−2 10−1 100 101 102 103 104 105 106
Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o
1 AU 94 AU 122 AU
10−4 10−3 10−2 10−1
10−5 10−4 10−3 10−2 10−1 100 101 102 103 104 105 106
Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o
1 AU 94 AU 122 AU
10−4 10−3 10−2 10−1
10−5 10−4 10−3 10−2 10−1 100 101 102 103 104 105 106
Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o
1 AU 94 AU 122 AU
Figure 6.17: Modelled energy spectra at Earth (1 AU,θ= 90◦) and the TS (94 AU,θ= 55◦) with respect to different input spectra (black lines) at 122 AU. In the top-left panel the reference HPS is employed as input, in the bottom left Eq. 5.4 withΦ =1.4 is applied, while Eq. 5.3 is applied on the right withh0= 1.0 and 6.0 for the top and bottom panels respectively. Solutions are shown forβ-compensated rigidity- independent diffusion (as explained in Section 5.4.2). Solid and dashed lines respectively represent solutions with and without shock acceleration. Drifts are neglected for these solutions.
These scenarios are at best speculative, although the energy distribution of electrons at these low energies can possibly be inferred from the magnitude of observed intensity increases mea- sured near the TS along the Voyager 1 trajectory [see McDonald et al., 2003;Decker et al., 2005;
Stone et al., 2005]. If these increases can indeed be attributed to the acceleration of electrons at the TS, the low-energy HPS can be approximated by the energy distribution that reproduces a similar magnitude intensity increase through DSA. Each of the aforementioned scenarios are investigated in this regard. In order to have the spectra incident at the TS directly reflect the features of the HPS, rigidity-independent diffusion is assumed for the following illustrations.
Figure 6.17 shows how spectra at Earth and the TS, as modulated from various forms of the HPS, are accelerated. Note that for each case the features of the HPS are mostly retained by the modulated spectra. Immediately noticeable is the large intensity increase in the case of the ref- erence HPS; the softer spectrum induced by the deviation ofβfrom unity in Eq. 5.1 provides a distribution that is easy to accelerate. Since the distribution incident at the TS has a spec-
10−4 10−3 10−2 10−1 100
101
Kinetic Energy (GeV)
Ratio of differential intensities
Eq. 5.1, h1 = −1.35 Eq. 5.3, h0 = 1.0 Eq. 5.3, h0 = 6.0 Eq. 5.4, Φ = 1.4
α = 10o θ = 90o r = 1 AU
10−4 10−3 10−2 10−1
100 101
Kinetic Energy (GeV)
Ratio of differential intensities
Eq. 5.1, h1 = −1.35 Eq. 5.3, h0 = 1.0 Eq. 5.3, h0 = 6.0 Eq. 5.4, Φ = 1.4
α = 10o θ = 55o r = 94 AU
Figure 6.18:Ratios of the model solutions in Figure 6.17 with shock acceleration to those without. Each set of ratios are shown for one of the input spectra specified in the legend, and are shown both at Earth and the TS in the top and bottom panels respectively.
tral index that is similar to (or even slightly smaller than) that producible by the shock itself, DSA raises the intensities appreciably and even some hardening is observed in the shocked solutions. Figure 6.18 shows that the intensities are increased at the TS with an average factor larger than 10 below 1 MeV, which translates to a re-accelerated contribution of up to a factor of 9 at Earth for the same energies. This is a remarkably large effect, but in fact very similar to the magnitude of the peak-like intensity increases reported by Decker et al.[2005] near the shock for 0.35 to 1.5 MeV electrons. The acceleration effects in the cases where the other input spectra are employed are less spectacular, since the form of the spectra hardens toward lower energies;
the factor by which the acceleration increases intensities both at the TS and Earth actually de- creases from the factor achieved above 10 MeV. Of these cases, the HPS described by Eq. 5.4 andΦ =1.4 yields an intensity increase at the TS that declines from a factor of 2.5 at 10 MeV to about 2 at 1 MeV and is mostly diminished at 0.1 MeV. The input spectrum described by Eq. 5.3 withh0 =1.0 yields an intensity increase at the TS maintained at a factor of approximately 1.5 below 1 MeV, whereas, at the same energies, the intensities of the TS spectrum associated with theh0 =6.0 case are not raised at all. At Earth the same features are visible, but are translated to lower energies (likely as a result of adiabatic cooling). The findings presented here are again consistent with earlier results in that the hardness of spectra arriving at the TS determines the magnitude of the acceleration effects that follow.
While the reference HPS of Eq. 5.1 yields TS spectra that produce favourable acceleration fea- tures in the correct range of energies, its form is unlikely to be accurate. Recall from Section 5.3.2 that the intensities of galactic electrons arriving at the heliosphere are not expected to increase indefinitely toward low energies, but instead to eventually fall away. Note that the so- lutions in this subsection are shown only for rigidity-independent diffusion. Hence, if the other (possibly more realistic) forms suggested for the HPS are employed in conjunction with more appropriate low-energy diffusion properties, spectral forms at the TS that are more conducive to efficient acceleration may be attained.