Cosmic Ray Transport and Acceleration Model
3.1 Introduction
Chapter 2 gave an introduction to CRs and the major defining properties of the heliosphere.
As CRs are transported through the heliosphere they are subjected to a number of modulating physical processes that cause their distribution function to change as a function of position, time, and energy. It is the purpose of this chapter to characterise these processes mathemati- cally and model the responding CR behaviour by solving a charged-particle transport equation.
The relevant transport equation is firstly introduced and the processes accounted for therein are discussed. Further discussions elaborate on the mathematical representation of these processes in the transport equation and its transformation to a spherical coordinate system. As a result of the complexity of the transport equation and its constituent parts, analytical solutions cannot be attained without simplifying assumptions and hence numerical solutions are sought instead.
The details of the numerical model implemented in this study to solve the transport equation are reviewed concisely.
10−5 10−4 10−3 10−2 10−1 100 101 102 103 10−14
10−12 10−10 10−8 10−6 10−4 10−2 100 102 104
Rigidity (GV)
Kinetic Energy (GeV)
Electrons Protons
Anomalous Oxygen
Figure 3.1:The kinetic energy of a particle or nucleon as a function of its rigidity (according to Eq. 3.4) for the CR species indicated in the legend. The electron and proton rest-mass energies, that is,e0=0.511 MeV andE0 =938.3 MeV, are shown as dotted horizontal lines. The electrons and protons both have Θ-values of 1, whereasΘ =16 for ACR Oxygen.
distribution function, the terms from left to right describe time-dependent changes, convection due to the SW flow at a velocity ofV~sw, CR drifts in terms of the pitch-angle averaged guiding centre drift velocity ofh~vDi, spatial diffusion as described by the diffusion tensor,Ks, adiabatic energy changes through the divergence of the SW velocity (∇ ·V~sw), and the contribution of CR sources inside the heliosphere. This study only formally considers the transport of GCRs and ACRs, for which the initial distributions are specified as boundary conditions at the HP and TS respectively. The source term denotedQs=Qs(~r, p, t)in Eq. 3.1 is hence not regarded.
Note that while the TPE is written in terms of momentum, it is solved in terms of rigidity,P, which is related to the former according to
P = pc
|q|, (3.2)
where p is the particle’s relativistic momentum and |q| = Ze its charge, with Z the atomic number, ethe elementary charge, andcthe speed of light. The relation between a particle’s rigidity and its total kinetic energy, E, can furthermore be derived from its total relativistic energy as
P = A
Ze
pE(E+ 2E0), (3.3)
or conversely
E= s
P2 Ze
A 2
+E20−E0, (3.4)
withAthe mass number andE0 the rest-mass energy of the particle. Note that when consid- ering CR ions,E denotes the total kinetic energy per nucleon. For electrons,E0 =e0 =0.511 MeV (the electron rest-mass energy) while for ions the proton rest-mass energy of∼938.3 MeV
10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.1
Kinetic Energy (GeV)
Ratio of particle to light speed, β
Electrons Ions
Figure 3.2:The ratio of particle speed to light speed for CR ions and electrons. The electron and proton rest-mass energies are shown as vertical dashed lines ate0 =0.511 MeV andE0=938.3 MeV, at which each profile respectively attains a speed of∼0.87c. Note that light speed is never fully attained and that βprogresses asymptotically to unity above the relevant rest-mass energy.
is applied. Furthermore, the ratio of mass number to charge can used to distinguish between different CR species:
Θ = A
Ze, (3.5)
which is 2 for most GCRs (e.g. Helium, Carbon, Oxygen, ...), being fully ionised, except for galactic protons, for whichΘ =1. This quantity is also taken as 1 for electrons. Of course, for singly-charged ACRs it follows thatΘ = A. The relation in Eq. 3.4 is demonstrated in Figure 3.1 for different CR species. Note that for each presented species,E is proportional toP2 and P respectively below and above their rest-mass energies. This may also be inferred from Eq.
3.3 or 3.4 in the limits of E E0 and E E0. Figure 3.4 shows that this proportionality changes at the same kinetic energy (E ≈ E0) for protons and ACR Oxygen, since the energy is measured per nucleon, although the rigidities at which this occurs are separated by a factor given by the ratio of theirΘ-values. AtE > E0, both protons and electrons are relativistic and display identical relationships betweenEandP, becauseΘ =1 for both. Generally, any range of relativistic energies has a corresponding rigidity range encompassing the same values. The difference in the relation between E andP for relativistic and non-relativistic CRs is further reflected in their shock-accelerated energy spectra; see Section 3.6.2.
Another important quantity that can be specified in terms of those introduced above is the ratio of particle speed,v, to light speed, that is,
β(≡v/c) = P q
P2+ ZeA2
E02
=
pE(E+ 2E0) E+E0
. (3.6)
This quantity is illustrated for CR ions and electrons in Figure 3.2 as a function of E. Ions
Figure 3.3: A schematic representation of the HMF-aligned diffusion configuration. The directions in which the coefficientsκ||,κ⊥randκ⊥θgovern CR diffusion are indicated with respect to a Parker field line, using black, red and blue arrows respectively. The radial SW outflow is indicated with the bold black arrows labelledV. Figure adapted fromHeber and Potgieter[2006].
and electrons attain speeds near the speed of light above the proton and electron rest-mass energies respectively. Note that different species of CR ions do not show differentβ-profiles, because Eq. 3.6 in terms ofE does not contain an explicit dependence onΘ. The parameter β is an important quantity in CR modulation, since it is contained in the expressions for both the diffusion coefficients and GCR input spectra. Note finally that while the TPE is solved for the distribution function, f = f(~r, p, t), CR intensities are presented in this study in terms of differential intensity,j=j(~r, p, t), which is related tof according to
j(~r, p, t) =p2f(~r, p, t), (3.7) with units of particles / unit area / unit time/ unit solid angle / unit kinetic energy (per nu- cleon, in the case of CR ions). See e.g.Strauss[2010] for a full derivation of this quantity.