There are several instances of particle acceleration within the bounds of the heliosphere, rang- ing from particles accelerated during transient events on the Sun to acceleration in planetary magnetospheres. Of recent interest is the acceleration of CRs associated with magnetic recon-
nection, which in the outer heliosphere is thought to occur near the HCS and regions where SW flow has stagnated near the HP [e.g.Lazarian and Opher, 2009]. It has also been proposed as a mechanism for the acceleration of ACRs. As discussed in Section 2.7.1, the original paradigm of TS-accelerated ACRs has become further challenged by the concept of stochastic acceleration, which was demonstrated byStrauss et al.[2010b] to be able to account for ACR observations in the heliosheath. Stochastic acceleration may be modelled in the TPE with the inclusion of the momentum diffusion term
1 p2
∂
∂p
p2κP
∂f
∂p
, (3.43)
withκP the momentum diffusion coefficient. This process, its incorporation in a CR modula- tion model, and the form ofκP are discussed in detail byStrauss[2010]. This is not the focus of the present study, which aims to investigate the re-acceleration of galactic electrons at the TS for which DSA is still considered the prevalent mechanism. Tautz et al.[2013] found that the spectral effects of stochastic acceleration become negligible if particle speeds significantly exceed the local Alfv´en speeds. Using this as a guideline, it would be unlikely for CR electrons, which remain relativistic down to very low energies (Figure 3.2), to be accelerated stochasti- cally at any region within the heliosphere. This requires further investigation. It assumed for the purposes of this study thatκP = 0.
As stated before, this study endeavours to illustrate DSA as a mechanism for electron re- acceleration, and is also illustrated by surveying the features that emerge from the acceleration of ACRs (or more technically, TSPs). The DSA process is not explicitly contained in the TPE, but is simulated through the SW velocity divergence in the term for adiabatic energy changes, and is further reproduced by specifying the appropriate continuity and streaming conditions for the CR flux across the TS. This section discusses these modelling aspects. See alsoAxford et al.[1977],Axford[1981],Lagage and Cesarsky[1983],Drury[1983], andJones and Ellison[1991]
for more on shock acceleration.
3.6.1 Solar wind velocity divergence
The migration of CRs in energy is controlled in the TPE through the term for adiabatic energy changes, which contains the divergence of the SW velocity. It hence depends on the value of
∇ ·V~swwhether particles gain or lose energy in this region. In the inner heliosphere, the radial SW speed remains largely constant, but is rapidly reduced near and at the TS, and continues to decrease asr−2in the heliosheath under the incompressibility assumption. In terms of∇ ·V~sw this translates to the following:
∇ ·V~sw≡:
>0 r < rT S−L upstream region
<0 rT S−L≤r ≤rT S shock region
= 0 r > rT S downstream region
(3.44)
Recall thatrT Sdenotes the TS position andLthe width of the precursor. Figure 3.11 illustrates the SW profile near the TS and the corresponding divergences of the SW velocity. Note that
∇ ·V~sw > 0, which occurs upstream of the TS, results in adiabatic cooling. This gives rise to characteristic CR energy spectra in the form of positively distributed power laws, and is dis-
91 91.5 92 92.5 93 93.5 94 94.5 95 95.5 96 10−3
10−2 10−1 100
Radial distance (AU)
|div V sw| (400 km.s−1 .AU−1 ) 100 200 300 400 500
Solar wind speed (km.s−1 )
div V
sw = 0 div V
sw > 0 div V
sw < 0
Figure 3.11: The SW speed (Vsw; top panel) and the corresponding divergence of the SW velocity (∇ ·V~sw; bottom panel) as functions of radial distance in the equatorial plane (θ= 90◦). Note that∇ ·V~sw
is written asdiv Vswin this figure, and that it is the absolute value of this quantity plotted in the bottom panel. Regions of different signs of∇ ·V~sw, as indicated, are separated by vertical dash-dotted lines.
Note that in the heliosheath the divergence is zero, which is not visible on a logarithmic scale. The TS precursor length scale is specified asL=1.2 AU and the compression ratio iss= 2.5.
cussed further in Section 5.4.2. In the shock region, where∇ ·V~sw<0, the absolute value of the divergence, denoted|∇ ·V~sw|, is particularly large. In the TPE this translates to a large transfer- ral of energy to particles over a relatively narrow radial interval, making the TS a very effective site for CR acceleration. Indeed, Voyager observations near the TS confirmed that the bulk of the SW energy is dissipated into the heating of non-thermal particles [Richardson et al., 2008;
Decker et al., 2012]. The SW divergence can hence be exploited to model DSA in the TPE [Jokipii, 1992]. Consider finally the SW flow and corresponding velocity divergence in the downstream region of the TS. Section 2.3 explained the implications of assuming incompressible flow in the heliosheath, which included that∇ ·V~sw = 0. Hence, particles are neither heated nor do they incur energy losses in the heliosheath.
Note that should a steeper SW profile than r−2 be assumed in the heliosheath [e.g. Langner et al., 2006a, b;Ferreira et al., 2007a, b], particles gain energy through adiabatic heating in this region. These steeper profiles are not substantiated as deviations from the incompressibility assumption, but are modelled instead to account for a hydrodynamically shrunken heliosheath resulting from the loss of SW protons due to charge exchange with interstellar neutrals [e.g.
Fahr et al., 2000]. At any rate, incompressibility is assumed in this study withVsw ∝ r−2 in
the heliosheath, not (necessarily) because of its physical viability, but because other forms of heating must be suppressed to study the effects of DSA at the TS alone. Similarly, particle heating at the TS can be suppressed for a control scenario by imposing ∇ · V~sw = 0 when
∇ ·V~sw <0in the region ofrT S−L≤r ≤rT S; this is especially useful to emphasise the effects of DSA by comparing TPE solutions with heating in the shock region to solutions without. The next subsection addresses the continuity conditions at the TS that allows the acceleration to reflect the theoretically expected results of DSA.
3.6.2 Features of diffusive shock acceleration
The hallmark of shock-accelerated CRs is that their energy spectra are power-law distributed, which follows from two conditions respectively concerning the continuity of CR densities and their streaming across the TS. The first of these conditions specifically entails that the differen- tial CR number density,Up, related to the CR distribution function according toUp= 4πp2f, is continuous across the TS, which also implies that the distribution function is continuous across this boundary. Hence,
f−=f+, (3.45)
where f− = limr→rT Sf(r) and f+ = limrT S←rf(r) respectively represent downstream and upstream values off with the superscripts−and+as introduced in Section 2.6.1. Secondly, the streaming of particles across the TS from the reference frame of a stationary observer must obey the condition
∇ ·S~ =Q, (3.46)
with Q the source of particles (not to be confused with Qs in Eq. 3.1), and where S~ is the differential particle current density, given by
S~ = 4π
C ~Vswj−Kt· ∇j
= −4πp2 V~sw 3
∂f
∂lnp +Kt· ∇f
!
, (3.47)
withCthe Compton-Getting factor [Gleeson and Axford, 1968] specified as C = 1− 1
3j
∂
∂p(pj) =−1 3
∂lnf
∂lnp. (3.48)
Eq. 3.46 essentially states that the CR flux that diverges from the shock must have its origin at a source on the shock. See also the treatment of the above byGleeson and Axford[1967]. In the case where the particle flux is directed perpendicular to the TS, Eq. 3.46 reduces to
S++S−= lim
→0
Z rT S+ rT S−
Qdr. (3.49)
Note that solving Eq. 3.49 is simplified by invoking Eq. 3.45, since terms containing the dif- ference betweenf− andf+ may be eliminated, while the product Kt · ∇f in Eq. 3.47 may furthermore be expanded in spherical coordinates using the transformed tensor of Eq. 3.39. It
10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103
−3.2
−3
−2.8
−2.6
−2.4
−2.2
−2
−1.8
−1.6
−1.4
Kinetic energy (GeV)
Spectral Index
Electrons Ions
Figure 3.12:The spectral indices,γ(s), of Eq. 3.54 as a function of kinetic energy, of CR electron and ion distributions ensuing from the DSA of a monoenergetic source at the TS. A fixed compression ratio of s= 2.5is used so that it follows from Eq. 3.54 thatγ(s)varies between−1.5below the rest-mass energy and−3.0above; the rest-mass energies for electrons and protons are indicated using vertical lines.
consequently follows that ∂f
∂r −
= κ+rr κ−rr
∂f
∂r +
− Vsw− −Vsw+ 3κ−rr
∂f
∂lnp −κ−rθ−κ+rθ rT Sκ−rr
∂f
∂θ + Q κ−rr
, (3.50)
which relates the particle flux up- and downstream of the TS, and is referred to as the match- ing condition [Langner, 2004]. This equation and the TPE as written in Eq. 3.42 are solved simultaneously as discussed in Section 3.7.
The source in Eq. 3.46 can be specified as a monoenergetic delta function, representing a seed particle population injected at the TS, according to
Q=Q∗δ
r−rT S r0
δ
P−Pmin P0
, (3.51)
with P0 =1 GV andr0 =1 AU. Here, Q∗ is of arbitrary magnitude with the units off, and Pminis the rigidity at which the seed population is injected. When modelling the acceleration of ACRs,Pminessentially represents the rigidity at which PUIs are injected. In a 1-D case, for a source specified as in Eq. 3.51, an analytical solution can be obtained for Eq. 3.46 [Steenkamp, 1995;Langner, 2004] that yields
f ∝P−3s/(s−1), (3.52)
wheresis the compression ratio. In differential intensity, this proportionality becomes j=p2f ∝P(s+2)/(1−s)
. (3.53)
To furthermore express this proportionality in terms of kinetic energy, the relation betweenE
andP, as discussed in Section 3.2, must be considered. At energies respectively much higher and much lower than the rest-mass energy, it follows thatP ∝EandP ∝E1/2, so that Eq. 3.53 becomes
j ∝Eγ(s)withγ(s)∈
s+ 2
1−s, s+ 2 2−2s
. (3.54)
To demonstrate,γ(s)is shown in Figure 3.12 as function of kinetic energy for CR electrons and ions. As a result of the smaller rest-mass energy of electrons, the relationship of their kinetic en- ergy with rigidity becomes linear at lower energies, and hence the transition of their accelerated spectra to the smaller high-energy spectral index also occurs at lower energies than for their heavier CR counterparts. The form of Eq. 3.54, where the spectral index of shock-accelerated spectra arising from a monoenergetic source depends only ons, is encountered repeatedly in further chapters, although the nature of the dependence onsmay differ depending on other involved modulation parameters, e.g. those contained inKt. The idealised solution of Eq. 3.46 given by Eq. 3.54 for an infinite plane shock hence provides a guideline for the forms one can expect numerically computed shock-accelerated spectra to have. A notable deviation (in a 2-D geometry) occurs when the diffusion length scale,κrr/Vsw, of a particle becomes comparable to the shock radius, that is, when
κrr Vsw
−
+ κrr
Vsw
+
&rT S. (3.55)
In such an event particles detect the curvature of the shock and escape, which results in the termination of the accelerated power-law distribution. The establishment of the accelerated spectrum is also time-dependent, since the power law of e.g. Eq. 3.54 can only be achieved up to such an energy as there is time for particles to be accelerated to. Such an acceleration time may be expressed in terms of momentum,p, as
τa= 3 Vsw− −Vsw+
Z p p0
"
κrr
Vsw
−
+ κrr
Vsw
+# dp0
p0 , (3.56)
wherep0 (≡ Pmin(Ze)/c) denotes the initial momentum. More on these quantities follow in Chapter 4. See also the related discussions byJones and Ellison[1991].