5.4 The Rigidity Dependence of Electron Diffusion
5.4.2 Implications of rigidity-independent diffusion
As already discussed in Section 5.2, spectra at the TS inherit the power-law distributed form of the HPS at low energies up to a few hundred MeV. This follows as an obvious result of the rigid- ity independence of the MFPs specified at these energies. However, when spectra are consid- ered at energies as low as the electron rest-mass energy (and lower), more intricate conditions apply. Since the particle speed of electrons become less comparable with light speed at these low energies, β (implicitly contained within Eq. 3.9) forces diffusion coefficients to deviate from the flat rigidity profile assumed for MFPs belowPk. Hence, under such a configuration, modulated spectra do not retain the form of the HPS at very low energies. To study the im- plications of rigidity-independent diffusion however, the governing coefficients should remain
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Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o
E+1 1 AU 122 AU
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Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
α = 10o
E+1 1 AU 122 AU
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Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
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E+1 1 AU 122 AU
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Kinetic energy (GeV) Differential Intensity (part.m−2 .s−1 .sr−1 .MeV−1 )
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Figure 5.11:Modelled energy spectra at Earth (1 AU,θ= 90◦) for various diffusion configurations and 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 correspond to the profiles with the same line styles in Figure 5.10. The spectra at Earth (all in blue) under the most rapidly declining diffusion scenario (dash-dotted lines) invariantly follow aE+1-distributed power law (dashed magenta line). Drifts are disabled for these solutions.
constant down to the lowest energy considered. This may be achieved by incorporating a1/β factor into the diffusion coefficient to suppress the existing dependence on the particle-to-light- speed ratio. The diminishing of low-energy diffusion may also be exaggerated for illustrative purposes by strengthening the dependence onβ. Figure 5.10 illustrates these scenarios, which respectively represent a rigidity-independent diffusion configuration withκ∝β0, the standard configuration of Section 5.2 withκ ∝β1, and an exaggerated configuration whereκ ∝ β2. It should be stressed however that these cases are presented purely for illustrative purposes, and that they are by no means proposed to be physically valid. Since the parallel and perpendicular diffusion coefficients are assumed to share the same rigidity dependence,κis used here to de- note diffusion coefficients in general and is thus interchangeable with bothκ||andκ⊥,r/θ. Note that while it is the convention to illustrate diffusion properties in terms of MFP versus rigidity (e.g. Figure 5.8), it is more beneficial to the end of demonstrating the dependences onβto show
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Kinetic energy (GeV) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 )
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Figure 5.12: Modelled energy spectra at the radial distances indicated in the legend and shown in different colours. The spectra are displayed atθ = 55◦, except for that at Earth (1 AU), which is shown at θ = 90◦. The input spectrum at 122 AU is the reference HPS of Eq. 5.1. Spectra in solid lines are shown for the κ ∝ β2 configuration, while dashed-dotted lines represent spectra resulting from rigidity-independent diffusion. The dashed red line shows the characteristicE+1power law followed by adiabatically cooled distributions. Drifts are disabled.
the corresponding diffusion coefficient. The coefficient is furthermore rather expressed as func- tion of kinetic energy so that corresponding features in the accompanying energy spectra are more readily identifiable.
Selecting a representative sample of the input spectra introduced in Section 5.3.2, the resultant modulated spectra at Earth are shown in Figure 5.11 for each of the aforementioned diffusion configurations. For diffusion coefficients with a flat rigidity profile the modulated spectra, shown as solid lines in Figure 5.11, mostly retain the form of the HPS . Also, not surprisingly, forκ∝β1andκ∝β2modulation increases toward lower energies, causing greater deviations from the form and intensity of the HPS. However, for the latter configurations where diffusion is allowed to decrease toward lower energies, the modulated spectra at Earth follow a E+1- distributed power law regardless of the form of the input spectrum employed. This spectral characteristic is indicative of the involvement of adiabatic energy losses. This process plays a well-established role in the modulation of ions [see e.g.Moraal and Potgieter, 1982;Potgieter and Moraal, 1988;Caballero-Lopez et al., 2010;Strauss et al., 2011b], where thisE+1power law is also shown to emerge for non-relativistic particles. Indeed, spectra are only aligned to this char- acteristic distribution below the electron rest-mass energy in Figure 5.9. In most modulation studies, however, electrons are relativistic in the typical energy ranges considered, and had diffusion been allowed to decline sufficiently above the rest-mass energy, they would instead displayE+2-distributed spectra at these energies [seeCaballero-Lopez et al., 2010;Strauss et al., 2011b]. Unlike for protons though, electron MFPs are not known to decline so rapidly until much lower energies [Teufel and Schlickeiser, 2003] that are typically smaller than that consid-
ered in this study; electrons are therefore not naturally observed to attain the adiabatic cooling limit. Aside from requiring diminished diffusion, these energy losses are incurred only when
∇ ·V~sw > 0, which is only achieved at r < rT S. The spectral effects of this process in the inner heliosphere are demonstrated in Figure 5.12: Note that when diffusion declines sharply toward lower energies (according to e.g. κ ∝ β2), spectra indeed follow theE+1 power law, while the form of the HPS is retained for rigidity-independent diffusion. Moreover, it appears from Figure 5.12 that radial gradients are extinguished in the adiabatic limit so that cooled spectra throughout the inner heliosphere eventually converge at low-enough energies; recall thatκalso grows withr so that spectra at larger radial distances will attain the adiabatic limit at lower energies.
In summary, the assumption of rigidity-independent diffusion firstly allows the intensities of modulated spectra to be reduced in such a manner that the general form of the HPS is retained.
It also follows that since diffusion coefficients are not permitted under this assumption to attain sufficiently small values, other modulating processes such as adiabatic energy losses do not manifest in the inner heliosphere. While it is reasonable to assume that the diffusion coefficients of electrons do not diminish toward lower energies (or at least not too drastically), there may nevertheless be more likely scenarios than rigidity-independent diffusion at these very low energies. This is investigated further in the next subsection.