6.5 Electron Re-acceleration at Very Low Energies
6.5.2 The role of dissipation-range turbulence in electron acceleration
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.
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 6.19: Modelled energy spectra at Earth (1 AU,θ = 90◦) and the TS (94 AU, θ = 55◦) with respect to the HPS (black lines) at 122 AU. The reference HPS is employed as input spectrum on the left, while Eq. 5.3 is applied on the right withh0 =1.0. Solutions in the top and bottom panels respectively correspond to the diffusion configurations described by Eq. 5.5 withg0 = −0.5 and−1.0. Solid and dashed lines respectively represent solutions with and without shock acceleration. Drifts are neglected.
From the results above, the largest acceleration consistently occurs in proximity to where the rigidity dependence of diffusion changes and thus where energy distributions change in re- sponse. Particularly, the most pronounced re-accelerated contributions occur at the transitions where softer distributions are succeeded by hard distributions. This effect is visible near the energy equivalent toPk1 (of Eq. 5.5) for all solutions with up-turns in their low-energy diffu- sion coefficients. Larger values ofg0 lead to greater intensity increases at the transition near 10 MeV, because larger up-turns in low-energy MFPs yield softer spectra at energies preceding the transition. When the reference input spectrum of Eq. 5.1 is applied (see the top panel of Figure 6.20), the aforementioned effect also surfaces for rigidity-independent diffusion (g0 = 0) at∼0.5 MeV, whereβbegins to decline appreciably toward lower energies, causing the in- tensities there to increase more rapidly and the spectrum to soften. When a low-energy MFP up-turn is present, however, the acceleration effects associated with theβ-induced bend be- come progressively smaller for larger values ofg0and intensities are instead increased more at the diffusion-induced transition near 10 MeV. This is because for larger values ofg0 the spec-
10−4 10−3 10−2 10−1 100
101
Kinetic Energy (GeV)
Ratio of differential intensities
1 AU, 0 1 AU, 0.5 1 AU, 1.0 94 AU, 0 94 AU, 0.5 94 AU, 1.0 α = 10o
HPS: Eq. 5.1
10−4 10−3 10−2 10−1
100 101
Kinetic Energy (GeV)
Ratio of differential intensities
1 AU, 0 1 AU, 0.5 1 AU, 1.0 94 AU, 0 94 AU, 0.5 94 AU, 1.0
α = 10o HPS:
Eq. 5.3, h
0 = 1.0
Figure 6.20:Ratios of the solutions in Figure 6.19 with shock acceleration to those without for the input spectra as indicated in each panel. Line styles correspond to the values forg0(of Eq. 5.5) indicated in the legend;g0 = 0refers to the rigidity-independent cases presented for the same input spectra in Figure 6.17. The ratios are shown in blue at Earth (1 AU,θ= 90◦) and in red at the TS (94 AU,θ= 55◦).
trum above 0.5 MeV softens more, which in physical terms implies that electrons accelerated up until energies at the bend can more easily be accelerated further, and consequently the peak in the re-accelerated contribution moves to larger energies (or in this case, to the next spec- tral bend at 10 MeV). A hard distribution following a transition, by contrast, impairs further acceleration of electrons; the re-accelerated contribution is therefore not distributed to higher energies, but instead peaks at the transition itself.
In summary, the results of this section not only reaffirms the expected result of larger accelera- tion where modulated spectra soften, but also reveals insight pertaining to transitions between different energy distributions: Acceleration is more pronounced at spectral transitions where the distribution preceding the transition is softer and the distribution succeeding it is harder.
Moreover, by virtue of the aforementioned mechanisms, it is shown that a low-energy MFP up- turn can indeed facilitate greater acceleration effects in cases where the distribution supplied by the HPS is too hard to accelerate. Comment on what combinations of input spectra and diffusion configurations are most realistic is reserved for Chapter 7.
0 50 100
−100
−50 0 50 100
Radial distance (AU)
Radial distance (AU)
α = 10o A > 0
16 MeV
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
0 50 100
−100
−50 0 50 100
Radial distance (AU)
Radial distance (AU)
α = 10o A < 0
16 MeV
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
Figure 6.21: Contour plots for 16 MeV electron intensity ratios, illustrating the factor by which solu- tions with shock acceleration at the TS increases the intensities of solutions where shock acceleration is suppressed. The plots are shown in the meridional plane of the heliosphere for A>0 (left) and A<0 (right). The dashed white halfcircle indicates the TS position at 94 AU, the dash-dotted lines indicate the polar extent of the HCS forα= 10◦, and the dashed black line shows the trajectory alongθ= 55◦at which the corresponding ratios in Figures 6.6 and 6.8 are shown. The Sun is located at the origin.