7.3 Model Results for Electron Modulation and Acceleration
7.3.3 Reproducing radial electron intensity profiles
For the sake of discernibility, only the solutions with shock-acceleration effects are shown in Figure 7.3 in the heliosheath, while the transport of re-accelerated electrons into this region is
Radial distance (AU) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 )
85 90 95 100 105 110 115 120
100 101 102 103
a, < 97 AU b, 97−103 AU c, 103−109 AU d, 109−115 AU e, > 115 AU rTS
A < 0 θ = 55o 6 MeV e−
a b c d e
Radial distance (AU)
Ratio of Differential Intensities
85 90 95 100 105 110 115 120
1 2 3 4 5 6 7 8 9 10
a, < 97 AU b, 97 − 103 AU c, 103 − 109 AU d, 109 − 115 AU e, > 115 AU rTS
A < 0 θ = 55o 6 MeV e−
a b c d e
Figure 7.6:TOP: Modelled radial intensity profiles for 6 MeV electrons for the configurations (a) to (e) as defined in Table 7.1. The profiles are estimated to be valid only in the accordingly labelled regions specified in the legend. Profiles with and without DSA effects are shown in solid and dashed lines respectively. The TS position is indicated with a vertical dash-dotted line. BOTTOM: Similar to the top panel, but illustrating the ratios of profiles with DSA effects to those without for each configuration.
instead illustrated using radial intensity profiles. Distributions of 6 MeV electron intensities in the heliosheath are presented in Figure 7.6 for each of the configurations presented in Table 7.1. Note however that each configuration, labelled (a) to (e), is technically only estimated to be valid roughly within the regions described for each profile in the figure legend, because they are meant to emulate different instances in time (or equivalently, in the solar activity cycle).
These regions are shown as consecutively shaded and unshaded areas. From these profiles it is possible to compare the modulation and acceleration effects following from the different con- figurations. The total heliosheath modulation is greater for configurations (a) and (e) than for (b) to (d), which follows largely as a result of the tilt angle for which they are solved; consider e.g. that the levels of the profiles for (b) to (c), attained using similar tilt angles, are alike, while configurations (b) and (e) with similar magnetic fields at Earth yield noticeably different inten- sity levels. The tilt angle, as a proxy for solar activity, is hence the most prevalent parameter of
those listed in Table 7.1 in determining modulation effects in the model: it controls the scaling constants of the diffusion coefficients as well as the latitude dependence of the SW speed (see e.g. Figure 7.2), which in turn affects both the diffusion length scales and global HMF mag- nitudes. Consequently, the smaller diffusion length scales following from the fast SW stream atθ= 55o enhances the acceleration of electrons at the TS, and hence the ratios in the bottom panel of Figure 7.6 are generally larger for solutions associated with minimum solar activity. It is important not to confuse the tilt angle’s role as proxy for solar activity with its physical role of defining the HCS inclination; the aforementioned effects follow as a result of the former.
In order to reproduce the observed intensities for 5 to 12 MeV electrons in the heliosheath, the appropriate conditions must be simulated in the model throughout the region. Since the dis- tribution of observed spectra with radial distance has already been successfully reproduced in Figure 7.3, the configurations of Table 7.1 that were used to do so provide a convenient starting point. If each configuration is assumed to be roughly valid in a limited region around the radial distance it is associated with, the segments of the profiles of Figure 7.6 within these regions can be combined to construct realistic radial intensity profiles (both with and without DSA effects) for the entire heliosheath; the transitions of profiles between different regions are smoothed by means of taking a moving average. These profiles are shown in Figure 7.7 at a representative energy of 6 MeV alongside intensities measured by Voyager 1 in the 5 to 12 MeV interval. The model reproduces the observed radial distribution of intensities quite accurately, especially in the respective proximities of the TS and HP: From the TS out to at least 100 AU the effects of DSA are essential to account for the shape of the first large incline in intensities, while modelled and observed intensities are also similar forr >109 AU, where acceleration effects have mostly diminished; see also the bottom panel of Figure 7.7. At intermediate distances (100–109 AU), the observations are better represented without DSA effects, since the re-accelerated contribu- tion causes the model to overestimate intensities in this region. Atr < rT S, the model predicts intensities at roughly the same level of the highest points of the observed intensity peaks. The factor of about 6 by which these intensities are raised from levels without shock effects is much larger than the magnitudes of the peaks shown in Figure 7.7, however these might in fact be larger if the background intensities (shown in red) are indeed artificially raised as discussed in Section 7.3.2. As also concluded in the previous subsection, DSA as implemented in the model is efficient enough to produce intensity increases of the magnitude of these peaks and larger.
A final issue to consider is what causes the peak-like shapes of the observed intensity increases near the TS if DSA is indeed assumed to be an involved mechanism. A simple explanation involves a magnetic connection between the TS and the spacecraft at the time of observation, which would allow particles re-accelerated at the shock to be transported along field lines to the spacecraft for as long as the two are connected. If this is assumed not to be the cause, the following mechanisms are considered: In the model, diffusion levels are too high upstream of the shock to limit the passage of re-accelerated electrons to such narrow spatial intervals.
In this context one might expect any event creating local low-diffusion conditions to be able to account for their narrow shape. Its cause may however be spatial as well as temporal, e.g.
the relative movement of the spacecraft and TS, or the movement of either of these in and out of different magnetic sectors [Hill et al., 2014]. It is suggested by Stone et al. [2005] that
Radial distance (AU) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 )
85 90 95 100 105 110 115 120
100 101 102 103
5−12 MeV, > 94 AU 5−12 MeV, < 94 AU 6 MeV (no DSA) 6 MeV
rTS
b c e
a d
A < 0 θ = 55o
Radial distance (AU) Ratio of Differential Intensities
85 90 95 100 105 110 115 120
1 2 3 4 5 6 7 8
6 MeV rTS
e
a b c d
A < 0 θ = 55o
Figure 7.7: Similar to Figure 7.6, this figure shows 6 MeV electron intensities modelled by combining the segments of profiles valid in each labelled region. Computed profiles with and without DSA are respectively shown in solid and dashed black lines. Triangular and circular markers denote Voyager 1 observations [Webber, 2014] forr < rT S andr > rT S, while the red markers represent background intensities. The bottom panel shows the ratio of the computed profile with DSA to that without.
these peaks may result because of the interaction of particles accelerated at the TS with large- scale disturbances, e.g. merged interaction regions, forming in interplanetary space due to the coalescence of coronal mass ejections, that are convected outward by high SW speeds. Such temporal effects may be studied more effectively using time-dependent models [e.g. Ferreira and Potgieter, 2004;Manuel et al., 2011, 2014].