5.3 Modelling the Electron Input Spectrum
5.3.2 Electron intensities at very low energies
Recall from Figure 2.17 the peak-like increases in the radial intensity profiles recorded by Voy- ager 1 near the TS for electron energies of 2.5 to 14 MeV. In the accompanying text DSA is men- tioned as a mechanism that might be at least partially responsible for these features. Should this indeed be the case, it must be borne in mind that the particles that constitute these narrow increases must have migrated from lower energies during acceleration. Section 6.5.1 demon- strates that the efficiency of electron re-acceleration at Voyager-observed energies is in fact sen- sitive to the features of the HPS below 1 MeV. The intensity levels and spectral shape of the very local interstellar spectrum are however unknown below 1 MeV from both an observa- tional and modelling perspective. It is hence necessary that the form of the HPS is investigated at these very low energies. Figure 5.6 demonstrates that electron intensities continue to grow should Eq. 5.1 be extended below 1 MeV. It also happens more rapidly with decreasing energy because of the decrease ofβ from unity. This behaviour is however unlikely to be accurate.
Consider firstly that most studies on turbulence in the heliosphere predict an up-turn in the MFPs of electrons at low energies (Section 5.4.3), which would imply that should the HPS in- tensities be allowed to increase indefinitely toward lower energies, the intensities registered at Earth would be largely unaltered from that. This contribution, which would be in the order of105 electrons m−2 s−1sr−1MeV−1 at 0.1 MeV if Eq. 5.1 is assumed valid there, exceeds the intensities recorded near Earth for electrons at these low energies; see e.g. Figure 2.18. The greatest contribution to these measured intensities is furthermore attributed to particles ema- nating from the Sun or planetary magnetospheres [e.g.Ferreira et al., 2001a, b], which constrains the galactic electron contribution even further. The differential flux profile reported for 50 keV electrons in the heliosheath byHill et al.[2014] suggests that intensities may be as low as3·10−2 electrons m−2 s−1sr−1MeV−1 near the HP along Voyager 1’s direction of travel. This implies a HPS with diminishing intensity toward lower energies. However, since published data cov-
10−4 10−3 10−2 10−1 100 10−2
10−1 100 101 102 103 104 105 106 107 108 109
Kinetic Energy (GeV) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 )
Langner et al. (2001) GALPROP
Eq. 5.1, h1=−1.55
Figure 5.6: Plausible scenarios from galactic CR propagation modelling for electron intensities at the HP (122 AU) for low (E <100 MeV ) to very low energies (E <1 MeV). The dash-dotted line represents a ‘polar approach’ model solution byLangner et al. [2001] and the dashed line a GALPROP solution [Bisschoff, 2014,private communication], while the HPS of Eq. 5.1 (in solid red) is included for comparison.
erage for electrons at these very low energies remains quite sparse, the predictions of galactic propagation models are also considered again. For example, the interstellar spectra calculated using the polar approach of Langner et al. [2001] yield 103 electrons m−2 s−1 sr−1 MeV−1 at 0.1 MeV, signifying that the intensities flatten out and diminish with decreasing energy. The spectra byStrong et al.[1994, 2000] appear similar [see alsoStrong et al., 2011], while GALPROP yields flattened spectra from 1 MeV to as low as 0.01 MeV for rigidity-independent diffusion within this energy range [Bisschoff, 2014,private communication]. See alsoBisschoff and Potgieter [2014] for more details on the instance of the GALPROP model referenced here.
With model predictions as illustrated in Figure 5.6 and available data considered, a number of mostly speculative scenarios are explored. It can be conjectured that the HPS flattens toward energies below 1 MeV, or at least decreases slow enough to appear so. This may be simulated by adapting Eq. 5.1 to reflect a convenient triple power-law form according to
jHPS = 0.16 β2
E
EN
1.5
+ Eb2
EN
1.5
1 + Eb2
EN
1.5
h2−h1 1.5
E
EN
1.5
+ Eb1
EN
1.5
1 + Eb1
EN
1.5
h1−h0 1.5
E EN
h0
, (5.3)
in units of electrons m−2 s−1 sr−1 MeV−1. Hereh0represents the spectral index of the lowest energy segment of the HPS, andEb1 andEb2 respectively represent the energies about which power laws transition fromh0 toh1 andh1 toh2. The value ofEb2 is retained fromEb in Eq.
5.1 as 0.67 GeV, while it is assumed that Eb1 = 10−3Eb2. The roles and values of earlier spec- ified quantities remain unchanged. The spectra resulting from Eq. 5.3 for different values of h0 are illustrated in the left panel of Figure 5.7. The first and second of these, with h0 =1.0 and 1.5, are in reasonable agreement with GALPROP andLangner et al.[2001] estimates respec-
10−5 10−4 10−3 10−2 10−6
10−4 10−2 100 102 104 106 108
Kinetic Energy (GeV) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 )
1.0 1.5 3.0 6.0 Eq. 5.1
10−5 10−4 10−3 10−2
10−6 10−4 10−2 100 102 104 106 108
Kinetic Energy (GeV) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 )
1.1 1.4 1.8 Eq. 5.1
Figure 5.7: Modifications to the HPS described by Eq. 5.1 (solid red line) forE .1 MeV. In the left panel, the legend values correspond to different values ofh0, representing the power-law indices of the lowest-energy segment of the HPS as described by Eq. 5.3. Simulating an exponential decrease of intensities toward lower energies as described by Eq. 5.4, the spectra on the right are presented for different values ofΦas shown in the legend.
tively, while the spectrum forh0 = 6.0 intercepts the 50-keV observed intensity ofHill et al.
[2014]. While none can be ruled out given the very limited available data, the forms of the last two spectra, withh0 =3.0 and 6.0, are less preferable due to their more pronounced deviation from the relatively well-established power law above 1 MeV. This deviation may be avoided by shiftingEb1to lower energies, but this would require larger values ofh0to yield the above correspondences. There may also be more suitable forms than power laws to describe the HPS at these very low energies, since these types of distributions may be erroneously associated with the spectral imprints of energy changes that are not necessarily involved here, e.g. adi- abatic cooling. Hence, a second scenario to consider is a HPS with an exponential intensity decrease toward lower energies, which is more reasonable than a power law. This is modelled by modifying Eq. 5.1 as follows:
jHPS= 0.16 β2
E
EN
1.5
+
Eb2
EN
1.5
1 +
Eb2
EN
1.5
h2−h1 1.5
E EN
h1
exp − Ec
E Φ!
, (5.4)
in units of electrons m−2 s−1 sr−1 MeV−1, with the energy at which the low-energy intensity decrease ensues taken asEc = 0.4 MeV, Φdenoting the strength of this decrease, and other quantities as already defined. The values ofΦare varied and the resultant spectra shown in the right panel of Figure 5.7. This HPS intercepts the observed 50-keV electron profile forΦ = 1.4, while deviations from the Voyager-observed power law above 1 MeV are small for allΦ-values considered. The scenarios presented in Figure 5.7 for the HPS are utilised in subsequent sec- tions, where their influence will be considered in conjunction with that of the properties gov- erning modulation processes and DSA as discussed in the current and next chapters respec-
10−2 10−1 100 101 10−1
100 101
Rigidity (GV) Mean free path λ || (AU)
0.34 GV 0.45 GV 0.60 GV
Figure 5.8:The rigidity profile of the parallel MFP for electrons at Earth for different values ofPkin Eq.
3.24 as indicated in the legend.
tively. At any rate, the findings presented above, revealing the form of the very low-energy HPS, are mostly speculative and as such not conclusive. These features may only be inferred upon further investigation.