10−6 10−5 10−4 10−3 10−2 10−1 10−4
10−3 10−2 10−1 100 101 102 103 104 105 106
Kinetic energy (GeV.nuc−1) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 .nuc)
α = 10o θ = 90o
0.005 0.010 0.020 0.005 0.010 0.020 Pmin = 0.03 GV P
min = 0.3 GV
Figure 4.13: Modelled TS spectra for ACR Oxygen in the equatorial plane for varying values ofκ⊥r,0 and source functions representing the PUI seed population injected atPmin = 0.03GV (in blue) and Pmin = 0.3GV (in red) respectively. Here, drifts are disabled (κD,0= 0), and the expected power laws (withγ=−1.5fors= 2.5) for each set of solutions indicated with dotted lines.
Physically, this means that all the PUIs injected at the TS in the equatorial plane are accelerated to ACRs, while ten times less are injected at the poles.Strauss[2010] reports that this generally results in solutions that are a factor of ∼ I(θ) lower than those shown in Figure 4.11, with slight spectral changes occurring at high energies due to a modified intensity gradient,∇f. For similar reasons to that stated for refraining from the use of the latitude-dependent compression ratio, the injection efficiency is considered fixed for all polar angles in subsequent chapters.
10−6 10−5 10−4 10−3 10−2 10−1 10−4
10−3 10−2 10−1 100 101 102 103 104 105 106
Kinetic energy (GeV.nuc−1) Differential Intensity (particles.m−2 .s−1 .sr−1 .MeV−1 .nuc)
α = 10o θ = 90o
0.005 0.010 0.020
γ = −1.87
γ = −1.5 Pmin = 0.03 GV
Figure 4.14: Similar to Figure 4.13, but with only the solutions withPmin = 0.03 GV shown. Also illustrated here are the two power laws with γ = −1.87and γ = −1.5 fitted to theκ⊥r,0 = 0.005 spectrum respectively below and above∼0.3 MeV. nuc−1.
andPmin = 0.3GV (≡E =190 keV. nuc−1). Note that the deviations from the power law are pronounced for the lower injection energy (similar to Figure 4.5), but suppressed for the higher- energy injection. The latter solutions thus display accelerated spectra following a power law withγ =−1.5as expected fors = 2.5, while the former appear not to. It is necessary to com- ment though that an injection energy of 190 keV exceeds what may reasonably be expected from PUIs, since their maximum energies are predicted to be no more than four times the en- ergy of SW ions [Moebius et al., 1988], i.e. ∼6 to 40 keV. These solutions are presented though merely as a case for comparison. At any rate, the solutions with the more realistic lower injec- tion energy do not immediately follow the expected−1.15power law at low energies. Upon closer inspection however, the lowered intensities turn out not to be mere deviations from the expected power law, but a structured consequence of the effective compression ratio observed by low-energy particles.
Figure 4.14 reveals that toward high energies TS spectra for all diffusion levels are parallel to the expected power law, albeit at a lower intensity. Below ∼ 0.3 MeV. nuc−1 the spectra become softer. The bottommost spectrum, for instance, with κ⊥r,0 = 0.005, displays a power law with γ ∼ −1.87. This spectral index corresponds to a SW compression at the TS ofs = 2.1, which implies that ifV2 = 0.4V, the maximum SW flow speed observed by ACRs in this energy region is2.1·V2 = 0.84V, withV2 andV = 400 km.s−1 respectively representing the downstream and unshocked upstream SW speeds. One may infer from either Figure 3.11 or 4.16 thatVsw = 0.84V∗is attained within the precursor structure atr∼89.8 AU, i.e. 0.2 AU from the (sub)shock. One thus expects that particles with diffusion length scales at the TS of roughly less than 0.2 AU will experience an effective compression ratio of no more thans= 2.1. Indeed, such length scales correspond to energies lower than∼0.3 MeV. nuc−1, as indicated in Figure
10−5 10−4 10−3 10−2 10−1 100 10−2
10−1 100 101 102 103 104
Kinetic energy (GeV.nuc−1) Diffusion length scale, κ rr / V sw(AU)
0.020 0.010 0.005
Figure 4.15: The diffusion length scale (κrr/Vsw) of ACR Oxygen at the TS (90 AU, θ = 90◦) as function of kinetic energy for varying values of κ⊥r,0 as shown in the legend. Here κrr/Vsw = (κrr/Vsw)− + (κrr/Vsw)+ where the superscripts − and +respectively denote up- and downstream values as introduced in Section 2.6.1. The red lines indicate the length scale observed for the energy at which the power laws indicated in Figure 4.14 intersect. The solid and dashed horizontal blue lines respectively represent the width of the TS precursor and the length scale at which the ACR cut-off is expected to occur.
4.15, which is also roughly where the transition between the power laws shown in Figure 4.14 occurs. Above this energy, length scales increase so as to eventually exceed 1 AU (the width of the modelled TS-precursor structure) and particles experience the full compression ratio of s = 2.5. This explains why the solutions injected above this energy in Figure 4.13 follow the expectedγ =−1.5power law. A lower limit on the compression ratio also exists: Since the SW speed decreases discontinuously across the subshock with a factor ofs= 0.7V∗/0.4V∗ = 1.75, the softest spectral index one might expect isγ =−2.5. A more refined study of these features may therefore include fitting TS spectra with multiple power laws corresponding to the range ofs= 1.75→2.5constrained above.
The above analysis demonstrates that should a precursor as shown in e.g. Figure 4.16 be present, CRs with small diffusion length scales, typically at low energies, will experience only a fraction of the total compression ratio and be accelerated accordingly. Similar effects are re- ported byKruells and Achterberg[1994]. Should the precursor be widened as in Figure 4.16, and the energy gains from SW compression hence distributed over a larger interval, the TS spec- trum will only display the expected power-law structure at larger energies, i.e from∼10 MeV.
nuc−1 for a precursor width of 2.0 AU. However, if the precursor is diminished, Figure 4.16 shows that the expected power law is already obtained at lower energies, although obscured by the intensity enhancement seen in earlier sections preceding the cut-off. It is therefore the expectation that should the precursor width tend toward zero so as to obtain a fully discontin- uous shock, no deviations from the theoretically expected power law will be visible. Indeed, studies where a TS precursor is absent [Potgieter and Moraal, 1988; Steenkamp, 1995; Steenberg
88 88.5 89 89.5 90 90.5 91 0.3
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
Radial distance (AU) Solar Wind speed (400 km.s−1 )
0.6 AU 1.0 AU 2.0 AU
10−4 10−3 10−2 10−1 10−1
100
Kinetic energy (GeV.nuc−1)
Differential Intensity X E1.5
0.6 AU 1.0 AU 2.0 AU
Figure 4.16: The radial profile of the SW speed in the TS region is shown on the left for varying pre- cursor widths as indicated in the legend. The discontinuous jump at 90 AU represents the subshock.
On the right are the modelled ACR Oxygen TS spectra for varying precursor widths corresponding to those shown on the left, withκ⊥r,0 = 0.005andκD,0 = 0. These solutions are multiplied byE1.5and normalised to unity at0.5·10−6GeV. nuc−1. This scale is such that flattening out of spectra to a constant value signifies alignment with the expected power law.
and Moraal, 1999] report no such deviations, and hence their solutions show no sensitivity at low energies to the choice of the injection energy either.