The transport model of this study can be used to answer some interesting questions concerning the interplay between the effect of perpendicular diffusion and the broadness of the injection region as a proxy for the broadness of the acceleration region. Considering the four panels of figure 4.8 where the perpendicular diffusion is kept constant for each panel, it is possible to confuse the effect of the broadness of the injection with the effect of perpendicular diffusion since both cases will lead to a wider SEP distribution at e.g. 1AU. To disentangle these effects, the following method is proposed. The left column of figure 4.10 shows five panels (r = 0.2, 0.4, 0.6, 0.8, and1.0AU) with the same level of perpendicular diffusion,a = 0.01, where the maximum particle intensities for7MeV electrons are shown as a function of azimuthal angle for each panel. The model ofStrauss et al. [2017a] is used where it quantifies the amount of perpendicular diffusion present in the model. The second column shows the results fora= 0.1 and the results fora= 0.3are shown in the third column (in order of increasing perpendicular diffusion). This methodology is similar to that presented inStrauss et al.[2017a]. The injection broadness is kept constant atσ0= 5◦across all panels of figure 4.10. Standard Gaussian curves (blue) are fitted to each simulation result using
I(φ) =I0exp
−(φ−φ0)2 2σ2
, (4.44)
whereI0is the maximum particle intensity at a specific azimuthal angleφandφ0is the center of the Gaussian.
The effects of the change in perpendicular diffusion is immediately observed when considering the change in broadness of the fitted Gaussian curves,σ. Now,σ is calculated for each radial distance together with each choice of perpendicular diffusion. This is shown in figure 4.13a for the case where the initial injection broadness isσ0= 5◦, starting at the inner radial boundary of r0= 0.05AU. For weak perpendicular diffusion, e.g.a= 0.01, the particles stick to the narrow injection where the azimuthal extent only grows from the initial5◦ at0.05AU to10◦ at1AU.
For the efficient perpendicular diffusion case, e.g.a= 0.3, a significant broadening is observed from the initial 5◦ at0.05 AU to ∼ 35◦ at1AU. The same method is followed for σ0 = 20◦, shown in figures 4.11 and 4.13b, and forσ0= 40◦shown in figures 4.12 and 4.13c.
66 4.3. GAUSSIAN MODELLING OF SEPS
a = 0.01 a = 0.1 a = 0.3
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.2 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.4 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.6 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.8 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 1.0 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.2 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.4 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.6 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.8 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 1.0 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.2 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.4 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.6 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.8 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 1.0 AU Gaussian fit
Simulation result
Figure 4.10: Gaussian curves fitted to the maximum particle intensities for7MeV electrons with a constant injection broadnessσ0 = 5◦. The first column represents a perpendicular diffusion ofa = 0.01, the second column a= 0.1, and third columna = 0.3. See text for details about a. The black solid line is the results from the transport model and the blue-dotted line is the fitted Gaussian curve. The bottom axes,∆φ, represent the difference (due to the spiral HMF) between the injection azimuth angle at the Sun and the observation azimuth angle a radial distance away.
a = 0.01 a = 0.1 a = 0.3
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.2 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.4 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.6 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.8 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 1.0 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.2 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.4 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.6 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.8 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 1.0 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.2 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.4 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.6 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.8 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
10 5 10 3 10 1 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 1.0 AU
Gaussian fit Simulation result
Figure 4.11: The same as figure 4.10 but now forσ0 = 20◦.
68 4.3. GAUSSIAN MODELLING OF SEPS
a = 0.01 a = 0.1 a = 0.3
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.2 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.4 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.6 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.8 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 1.0 AU Gaussian fit
Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.2 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.4 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.6 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.8 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 1.0 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.2 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.4 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.6 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 0.8 AU
Gaussian fit Simulation result
180 120 60 0 60 120 180 (Degrees)
105 103 101 101 103
Maximum omni-directional intensity [e/(s sr cm1) MeV] r = 1.0 AU
Gaussian fit Simulation result
Figure 4.12: The same as figure 4.10 but now forσ0 = 40◦.
0.0 0.2 0.4 0.6 0.8 1.0 Radial distance (AU)
0 5 10 15 20 25 30 35 40
Injection broadness (Degrees) a = 0.01 a = 0.1 a = 0.3
(a)σ0= 5◦
0.0 0.2 0.4 0.6 0.8 1.0
Radial distance (AU) 15
20 25 30 35 40 45
Injection broadness (Degrees) a = 0.01 a = 0.1 a = 0.3
(b)σ0= 20◦
0.0 0.2 0.4 0.6 0.8 1.0
Radial Distance (AU) 35
40 45 50 55
Injection broadness (Degrees) a = 0.01 a = 0.1 a = 0.3
(c)σ0= 40◦
Figure 4.13: The broadness of the fitted Gaussian functions for different levels of perpendicular diffusion and injection broadnesses starting at5◦(panel (a)),20◦(panel (b)), and40◦(panel (c)).
70 4.3. GAUSSIAN MODELLING OF SEPS
0.0 0.2 0.4 0.6 0.8 1.0
Parker spiral length (AU) 0
5 10 15 20 25 30 35 40 45 50 55
Inj ec tio n br oa dn es s (D eg re es )
a = 0.01 a = 0.1 a = 0.3
Figure 4.14: Combined results of all panels from figure 4.13. The shaded region indicates the area of degeneracy where the effects of the injection broadness and transport mechanisms can- not be distinguished from each other.
The combined results of figure 4.13 are shown in figure 4.14 and give a possible guide to an- swer the initial question of the interplay between the perpendicular diffusion and the injection broadness. The grey shaded area indicates the area of degeneracy. In this area an injection of σ0 = 5◦ anda = 0.3cannot be distinguished from an injection of σ0 = 20◦ anda = 0.01 as measured at1AU. Similarly, an injection ofσ0 = 5◦anda= 0.1cannot be distinguished from an injection ofσ0 = 20◦anda= 0.01as measured at1AU. Due to these intersections between the broadnesses there are several grey shaded areas (not shown) from∼ 0.4AU and further radially outward. This result shows that to disentangle these effects, observations much closer to the Sun are needed. For example, measuring particle intensities between0.1and0.3AU will significantly improve our ability to distinguish between these effects. Both the Parker Solar Probe and the Solar Orbiter will provide us with particle intensity data to do just that.
Dresing et al. [2014] also address the degeneracy problem by investigating energetic electron anisotropies to distinguish between the broadness of an injection source and the effect of per- pendicular diffusion that leads to widespread events (events with a longitudinal separation of at least 80◦ between AR and spacecraft). A strong anisotropy suggests the particles trav- elled relatively scatter-free and were magnetically well-connected to the injection (source) re- gion, while a weak anisotropy suggests the particles were extensively scattered to become more isotropic, and the directionality vanishes. The authors conclude that several mechanisms are responsible for the widespread events, but that the effect of perpendicular transport and the broadness of the initial injections play the main roles.