Essential transport coefficients are given and the inclusion of diffusive shock acceleration effects is discussed. The general characteristics of diffusive shock acceleration are further illustrated in Chapter 4 by applying the model to the anomalous oxygen acceleration of cosmic rays.
Introduction
Solar Variability
The convention followed in this study is that the polarity of the Sun is positive (A>0) when the field lines point outward in the northern hemisphere of the Sun and inward in the south, while the opposite applies for a negative polarity (A<0). It is during this erratic phase of high solar activity that the Sun's polarity is reversed; since about 11 years elapse between successive solar maxima, the SMF polarity returns to its previous state after about 22 years.
The Solar Wind
The SW velocity lattice profiles are modeled for FLS1 and FLS3 in Figure 2.3 using Eq. There is no consensus yet on the speed profile of the SW in the heliosheath, although convincing.
The Heliospheric Magnetic Field
Furthermore, Figure 2.6 illustrates that the HMF magnitude also exhibits 11-year solar activity dependence as evidenced by its correlation with the number of sunspots. Note from Figure 2.7 that the radial HMF magnitude profile predicted by the Parker model shows characteristics similar to the observations.
The Heliospheric Current Sheet
Note, however, that the HCS does not necessarily extend equally on either side of the equatorial plane [e.g. These results are shown in the resting image of the sun, which is at the origin.
Global Features of the Heliosphere
The termination shock
In this study it is not the cause of the precursor of SW but its consequences. The inner and outer boundaries shown as black lines represent the TS and HP profiles, respectively.
The heliosheath
Scherer and Fichtner [2014], on the other hand, show that by including a helium component in the ISM simulations, the scenario of a BS is still valid. Depending on the outcome of the above dispute, the outer heliosphere can be defined either as the region between the HP and the BS or more generally as the turbulent region of the ISM beyond the HP.
A dynamic and irregular heliosphere
Cosmic Rays
Anomalous cosmic rays
Garcia-Munoz et al [1973] and Hovestadt et al [1973] reported an increase in the CR fluxes of helium and oxygen. These improvements were attributed to the contribution of ACRs, named for their relatively high abundance [McDonald et al., 1974].
Galactic electrons
For more details on the possible acceleration mechanisms involved in the formation of ACR, see the review by Giacalone et al [2012]. Jupiter's electrons dominate the inner heliosphere, with an intensity that peaks at Jupiter's position at 5 AU.
![Figure 2.17: Radial intensities of low-energy electrons as observed by Voyager 1 [Webber, 2014]](https://thumb-ap.123doks.com/thumbv2/pubpdfnet/10414813.0/37.892.145.780.130.444/figure-radial-intensities-energy-electrons-observed-voyager-webber.webp)
Summary
Further discussions elaborate on the mathematical representation of these processes in the transport equation and its transformation to a spherical coordinate system. The details of the numerical model implemented in this study to solve the transport equation are briefly reviewed.
The Parker Transport Equation
Due to the complexity of the transport equation and its constituent parts, analytical solutions cannot be obtained without simplifying assumptions and therefore numerical solutions are sought instead. The electron and proton rest mass energies, i.e. e0=0.511 MeV and E0 =938.3 MeV, are shown as horizontal dotted lines.
Particle Diffusion
Turbulence
It is now believed that δBslab,x2 ∼B2 in the heliosheath [see, e.g., Roux et al., 1999, for a similar assumption on Earth]. Finally, when the vortices (or length scales of fluctuations) become too small, energy is quickly lost to the environment, resulting in the steeper decay we see above kID, where the inertial range transitions into the dissipation range.
Parallel diffusion
Note from Figure 3.5 that the proton and electron MFPs are similar at 1 GV, and so these radial profiles are representative of both species. According to this configuration, VA remains roughly constant at about 35 km.s−1 up to TS, above which it increases by a factor of s1/2 and continues to increase proportionally to B in the heliosheath; recall that due to the incompressibility assumption discussed in Section 2.3 the SW velocity asr−2 decreases in the heliosheath, implying that ρm remains constant in this region, as expected.
Perpendicular diffusion
The perpendicular diffusion coefficients most compatible with the numerical modeling in this study are those of Burger et al. [2000]. for improved latitudinal transport. See Burger et al [2000], Potgieter [2000] and Heber and Potgieter [2006] for a physical justification of this improvement, and Potgieter [1996] and Ferreira et al [2000]. on the effects of anisotropic orthogonal diffusion.
Particle Drifts
The weak scattering limit is reached at P > PD, where the drift scales predicted by Eq. The drift directions of CRs can therefore be resolved from d~l× h~vDi = 0, which would yield streamlines similar to those shown in Figure 3.8.
The Transport Equation in Spherical Coordinates
This follows because the curl of a vector can be rewritten as the divergence of a second-order tensor, leading to . This identity allows the TPE to be rewritten in a more compact form and facilitates the transformation to spherical coordinates.
Modelling the Acceleration of Cosmic Rays
Solar wind velocity divergence
In the shock region, where∇ ·V~sw<0, the absolute value of the divergence, denoted |∇ ·V~sw|, is particularly large. Finally, consider the SW flow and corresponding velocity divergence in the downstream region of the TS.
Features of diffusive shock acceleration
In such a case, the particles sense the curvature of the shock and escape, causing a break in the accelerated potential distribution. The establishment of an accelerated spectrum is also time-dependent, since the power law, e.g.
The Numerical Modulation Model
The progenitor of the 2-D model used in this study was originally developed by le Roux et al.[1996]. This is for better numerical fitting of the discontinuous SW velocity drop at TS.
Summary
Because of their formation by the acceleration processes of interest in this study, ACRs reveal a great deal of information about the features of these processes and do so more clearly than their GCR counterparts. It is intended that the knowledge gathered in this chapter will inform further implementation of the CR electron re-acceleration model in the TS.
Model Configuration and Reference Solutions
In the energy region just before this discontinuity, the TS spectrum is slightly inflated for A. Furthermore, note from Figure 4.2 that the modeled radial intensity profiles reach a maximum at TS in the equatorial plane (also visible in Figure 4.3). .
Shock Strength and Solar Wind Compression
At E > Ecut, the power law distribution is no longer limited by time, but by the curvature of the shock (see section 4.4). It appears that while the power law is arguably the most notable feature of the TS spectrum, varying compression ratios also have some influence on how far the particles are accelerated.
Dependence on Elements of the Diffusion Tensor
Effective radial diffusion
Because κ⊥r/θ ∝ κ||, the effects of varying κ|| as shown in the left panel also implicitly reflects the effects of changing the perpendicular coefficients. However, when studying the TS spectrum at low heliographic latitudes, varying κ⊥ is sufficient to illustrate the effects of the variation of the effective radial diffusion.
Polar diffusion
Note also that the modeled spectra fall well below the power law in Figure 4.5 for lower values of κ⊥r,0, although they still parallel it at high energies. Another interesting result is that varying κ||,0, as shown on the left side of Figure 4.6, produces effects similar to that of Figure 4.5.
The Effects of Cosmic-ray Drifts
From Figure 4.7, the intensity enhancements and depressions at the poles appear much less pronounced than at the equator. Note that the drift-free solutions for the two inclination angles shown in Figure 4.8 do not coincide.
A Latitude-dependent Termination Shock
The compression ratio
As expected, the spectrum atθ= 90◦ is similar to the spectrum shown in Figure 4.4 for a fixed, global compression ratio of = 2.5. These displacement patterns are illustrated for both polarities, with s = s(θ), using the contour plots in Figure 4.12 and are consistent with the patterns described in Section 4.5.
The pick-up ion source strength
The dashed white semicircle indicates the TS position (at 90 AU), while the dashed white lines indicate the polar extent of the HCS for α= 10◦ . For similar reasons as given to forego the use of the latitude-dependent compression ratio, the injection efficiency is considered fixed for all polar angles in subsequent chapters.
Low-energy Features of the ACR Spectrum
4.15, which is also approximately where the transition between the power laws shown in Figure 4.14 occurs. This explains why solutions injected above this energy in Figure 4.13 follow the expected γ =−1.5 power law.
Summary and Conclusions
This is found to localize ACR acceleration at the TS to the equatorial regions of the heliosphere. The insights gained from the characteristics of accelerated ACRs are carried over to subsequent chapters to inform further application of the model to the reacceleration of galactic electrons at the TS.
Configuring the Model for Electron Transport
Further revisions to the model follow from the Voyager 1 spacecraft revelations, and are mainly concerned with the shape of electron energy spectra in the heliosheath. No source functions are specified with the TS or elsewhere; all solutions are just modulated forms of the HPS.
Modelling the Electron Input Spectrum
A heliopause spectrum for galactic electrons
As mentioned before, it is characterized by two power laws – the first, at lower energies, shows a larger spectral index than the second, which follows roughly at energies above 1 GeV. The latter is taken aske2 =-3.18 to reflect the reported value of the PAMELA experiment. The HPS with the former serves as the reference input spectrum.
Electron intensities at very low energies
In the left panel, the legend values correspond to different values of h0, representing the power law indices of the lowest energy segment of the HPS as described by Eq. 5.4, the spectra on the right are shown for different values of Φas indicated in the legend.
The Rigidity Dependence of Electron Diffusion
Electron diffusion at high energies
The solutions correspond to the MFP profiles shown in the same line styles in Figure 5.8 and Pk and 0.60 GV according to that figure's legend. But just as the profiles presented in Figure 5.8 converge above Pk, the energy spectra of Figure 5.9 also converge above the corresponding energy at both the TS and Earth.
Implications of rigidity-independent diffusion
For diffusion coefficients with a flat stiffness profile, the modulated spectra, shown as solid lines in Figure 5.11, largely retain the shape of the HPS. The spectral effects of this process in the inner heliosphere are demonstrated in Figure 5.12: Note that when diffusion decreases sharply to lower energies (according, for example, to κ ∝ β2), spectra indeed follow the E+1 power law, while the shape of the HPS is conserved for stiffness-independent diffusion.
Modelling dissipation-range turbulence effects
As seen before, stiffness-independent diffusion (g0 = 0; κ ∝ β0) preserves the shape of the input spectra throughout the heliosphere. Note that the resulting modulated spectra in Figure 5.14 have been softened from the shape of the HPS in this energy range.
Drift-related Electron Transport and Modulation
Solutions are shown for different values of Pk corresponding to the configurations illustrated in Figure 5.8 and as indicated in the legend. Solutions are shown for both polarities and different values of Pk (corresponding to the configurations illustrated in Figure 5.8) as indicated in the legend.
Summary and Conclusions
The insights from the two previous chapters are therefore combined, first to study how the acceleration of electrons at the TS is affected by larger modulating processes in the heliosphere, and also to establish the dependence of the DSA mechanism on the spectral properties of incident electrons. It is further aimed at illustrating the contribution of reaccelerated electrons to global intensities and how this affects other processes accounted for in TPE.
Spectral Properties of Shock-accelerated Electrons
Insights from accelerating a monoenergetic electron population
The observed power-law spectral indices for each compression ratio in both energy regions are listed in Table 6.1. However, at the considered energies, the diffusion length scales remain constant (due to the stiffness independence of the MFP) and over the TS width so that the electrons observe the full compression ratio.
The imprint of DSA on electron distributions
The multiplied E1.5 spectra remain constant at most of the energies considered, implying that there is no hardening from the original spectrum, except for the spectrum for s = 4.0. Therefore, to summarize, the effect that DSA has on the energy distribution of particles injected into the TS is twofold: First, if the initial distribution will have a spectral index, χ, that is smaller than that expected to produce shot for monoenergetic injection; that is, γ(s) according to Eq.
Standard Features of Re-accelerated Galactic Electrons
Energy distributions of re-accelerated electrons
The subsequent drop in acceleration effects, centered around 100 to 200 MeV, similarly follows as spectrum ingress at the TS becomes very difficult due to decreasing modulation (as a result of increasing MFPs) with increasing energy. But in the end, as in section 6.2, the acceleration effects decrease due to the TS curvature limit (equation 3.55) at E & 1 GeV.
Radial distributions of re-accelerated electrons
However, the reaccelerated contribution peaks at the TS, as illustrated by the ratios in Figure 6.8, and constitutes a factor increase in intensities there of at least 2.5 for 16 MeV electrons. By the same low-diffusion properties that account for the massive decrease in galactic electron intensities across the heliosheath, the electrons reaccelerated at the TS also do not diffuse far into the heliosheath.
Re-acceleration of Intermediate- and High-energy Electrons
Shock acceleration and the heliopause spectrum
Indeed, Fig. 6.9 shows that the intensities of the TS spectra increase uniformly, so the power-law distributions are preserved not only during the modulation but also during the re-acceleration process. Under the standard model configuration and with negligible offsets, Figure 6.10 shows that DSA increases the spectral intensities at TS by factors of up to 1.6 at energies exceeding 1 GeV.
Dependence on the rigidity profile of diffusion
These quantities for each diffusion configuration are illustrated in Figure 6.12 by ratios of solutions with shock acceleration to those without. Similar effects are shown in Figure 6.12 for the contribution from reaccelerated electrons on Earth, although these extend to lower energies.
The effects of electron drifts
Note from Figure 5.16 that drift effects decay rapidly at TS across this narrow energy range, so the aforementioned properties can be expected. 0 (right) next to the corresponding conditions in dashed lines (also shown in Figure 5.17) for which shock acceleration is disabled.
Electron Re-acceleration at Very Low Energies
Consequences of the form of input spectra
In order for the spectra incident on the TS to directly reflect the characteristics of the HPS, rigidity-independent diffusion is assumed for the following illustrations. Each set of ratios is shown for one of the input spectra specified in the legend, and is shown at the Earth and the TS in the top and bottom panels, respectively.
The role of dissipation-range turbulence in electron acceleration
In particular, the most pronounced reaccelerated contributions occur at transitions where softer distributions are followed by hard distributions. Larger values of g0 lead to larger intensity increases at the transition near 10 MeV, because larger ramp-up in low-energy MFPs gives softer spectra at energies prior to the transition.
Global Spatial Distributions of Re-accelerated Electrons
The graphs are shown in the meridional plane of the heliosphere for A>0 (left) and A<0 (right). Note also that the large contributions of re-accelerated 200 MeV electrons to the intensities in the interior of the heliosphere are a result of the spectral features around those energies; the ratios in Figure 6-6 are actually smaller on the TS than on Earth for energies near 200 MeV.
Summary and Conclusions
The implications of these outcomes are far-reaching in the context of the acceleration of galactic electrons. The collective insights gained in the present and preceding chapters are sufficient to inform the application of the model to reproduce electron observations in the outer heliosphere.
Configuration to Reproduce Electron Observations
This time lag only increases if the decrease in SW radial velocity in the heliosheath is assumed to be steeper than r−2, as some studies suggest [e.g. Florinski et al., 2004; Ferreira et al., 2007a]. See also Richardson and Burlaga's overview[2013] on the SW and related properties in the outer heliosphere and heliosheath.
Model Results for Electron Modulation and Acceleration
Reproduction of electron energy distributions
Moreover, the Voyager measurements may also suggest that the underlying turbulence exhibits a radial dependence, as the MFP upward spin becomes less pronounced with increasing distance from the TS. Note also that the parameters used to reproduce the spectrum at Earth are similar to those used to repro-.
![Figure 7.3: Modelled energy spectra for galactic electrons, with accompanying observations from Voy- Voy-ager 1 [Webber, 2014] and PAMELA [Adriani et al., 2011] respectively presented using circular and tri-angular markers](https://thumb-ap.123doks.com/thumbv2/pubpdfnet/10414813.0/150.892.107.748.129.574/modelled-galactic-electrons-accompanying-observations-adriani-respectively-presented.webp)
The features of electron re-acceleration
For the lower energy range, however, it is desirable for the model to achieve factors as large as the observed magnitude of acceleration effects (or larger). Model (2) estimates for acceleration effects are shown in Table 7.2 to be much larger than the corresponding model (1) estimates for the range 0.35 to 1.5 MeV.
Reproducing radial electron intensity profiles
Remarkably, both configurations in Figure 7.5 yield nearly indistinguishable shocked spectra and reproduce the measurements observed by Voyager at the TS, while the effects of the different diffusion stiffness profiles are clearly visible for the spectra without DSA. Distributions of the electron intensities of 6 MeV in the heliosheath are shown in Figure 7.6 for each of the configurations shown in Table 7.1.
Summary and Conclusions
Borrmann, Effects of different solar wind velocity profiles in the heliosheath on the modulation of cosmic ray protons, Astrophy. Zhang, M., Galactic and anomalous cosmic ray acceleration in the heliosheath, in Physics of the Inner Heliosheath, AIP Conf.
![Figure 2.2: A schematic representation [adapted from Heber and Potgieter, 2006] of the form of the SMF and the distribution of regions of opposite polarity during different phases of the solar activity cycle.](https://thumb-ap.123doks.com/thumbv2/pubpdfnet/10414813.0/17.892.209.726.123.481/schematic-representation-potgieter-distribution-opposite-polarity-different-activity.webp)
