Simulation results

0 2 4 6 8 10 12 14 16 Time (hour)

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Electron Differential Intensity [e /(s sr cm 2 M eV )]

300 keV

Helios Data

r

= 0.05 AU

r

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Figure 5.6: Simulated temporal evolution of the differential intensity for a range of radial mean free path (λrr) values as indicated in the legend for300keV electrons.

Figure 5.7 shows a collection of contour plots that illustrate the evolution of the omni-directional differential intensities for the third SEP event on28May1980. The dashed white line represents the magnetic field line that is the best magnetically connected field line to the Earth at 1AU, which is indicated as observer C. Observers A and B, at0.3 AU, are located on the best con- nected magnetic field line and a field line+45^◦westward, respectively. Observers D, E, and F, all located at1AU, are+45^◦westward,+90^◦ westward, and−90^◦eastward, respectively. An injection broadness ofσ0 = 20^◦ and a radial mean free path ofλrr = 0.25 AU were used in this simulation for the300keV electrons. The570minute mark indicates the start of the third SEP event and the intensity evolves over the next30minutes. The important effect of two di- mensional scattering is seen where the SEPs start spreading longitudinally already at a radial distance of0.3AU and significantly when the particles reach1AU.

Figure 5.8a shows the differential SEP intensity and anisotropy for an injection broadness of σ0= 5^◦and a radial mean free path ofλrr = 0.1AU at the position of the six virtual observers.

Figure 5.8b is for the case ofσ₀ = 20^◦ and the sameλ_rr. For these, and all subsequent simula- tions, the injection function as inferred by using the Neupert effect is used, while the simulated differential intensity is normalized to observations. The location of the observers is the same as in figure 5.7. Helios-1 (at∼0.3AU and close to best magnetic connection) and IMP-8 (at∼1 AU and∼50^◦westward of best magnetic connectivity) observations are shown for comparison (see figures10and11inAgueda and Lario[2016])

88 5.5. SIMULATION RESULTS

(a) (b)

(c) (d)

Figure 5.7: Contour plots showing the temporal evolution of the omni-directional differential intensity, in the equatorial plane, for the third event of figure 5.6. The labelled symbols indicate the position of the six virtual observers assumed in the model. (a) shows the peak intensity reaching0.3AU while (b) and (c), both6minutes apart, show the effect of perpendicular diffu- sion. (d) which is∼11minutes later shows the decay phase at1AU.

Time (hour)

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Differential Intensity [e/(s sr MeV)]

0.3 AU Helios Data 1.0 AU IMP8 Data A - 0.3 AU = 0 B - 0.3 AU = 45 C - 1.0 AU = 0 D - 1.0 AU = 45 E - 1.0 AU = 90 F - 1.0 AU = 90

0 2 4 6 8 10 12 14 16

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Anisotropy

(a)

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Differential Intensity [e/(s sr MeV)]

0.3 AU Helios Data 1.0 AU IMP8 Data A - 0.3 AU = 0 B - 0.3 AU = 45 C - 1.0 AU = 0 D - 1.0 AU = 45 E - 1.0 AU = 90 F - 1.0 AU = 90

0 2 4 6 8 10 12 14 16

Time (hour) 0

1 2

Anisotropy

(b)

Figure 5.8: Simulated omni-directional differential intensities and anisotropies for λ_rr = 0.10 AU and an injection broadness of (a)σ0 = 5^◦and (b)σ0 = 20^◦.

90 5.5. SIMULATION RESULTS

Time (hour)

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Differential Intensity [e/(s sr MeV)]

0.3 AU Helios Data 1.0 AU IMP8 Data A- 0.3 AU = 0 B - 0.3 AU = 45 C - 1.0 AU = 0 D - 1.0 AU = 45 E - 1.0 AU = 90 F - 1.0 AU = 90

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Anisotropy

(a)

Time (hour)

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Differential Intensity [e/(s sr MeV)]

0.3 AU Helios Data 1.0 AU IMP8 Data A - 0.3 AU = 0 B - 0.3 AU = 45 C - 1.0 AU = 0 D - 1.0 AU = 45 E - 1.0 AU = 90 F - 1.0 AU = 90

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Time (hour) 0

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Anisotropy

(b)

Figure 5.9: Simulated omni-directional differential intensities and anisotropies forλ_rr = 0.25 AU and an injection broadness of (a)σ0 = 5^◦and (b)σ0 = 20^◦.

The timings of the peaks from the injections correspond well with the observed intensity peaks from data, specifically the Helios-1 data at0.3AU. The onset phase also corresponds well, with the decay phase again not completely captured by the model. This is, most likely, again related to the extended release of flare particles not captured by the Neupert effect. The simulation results at1AU also compare relatively well with the IMP-8 measurements, given that the pa- rameters used here were not tuned to reproduce any observations. The gradual rise of the IMP-8 intensity measurements (which is about an order of magnitude smaller than the antici- pated intensities), is attributed to the large connection angle (∼50^◦) between the spacecraft and the events. The interplay between the effectiveness of perpendicular diffusion and the size of the source region is also clearly visible: Virtual observers magnetically connected to the source directly see the injected time profile, while observers not magnetically connected see a more gradual rise in intensity due to the presence of perpendicular diffusion.

The results presented in figure 5.8 therefore illustrate the effects of perpendicular diffusion and magnetic connectivity on the simulation results: in figure 5.8a, observer B is not mag- netically connected to the (small/narrow) source, while it is magnetically connected to the (broad/wide) source in figure 5.8b. At1AU the effect of perpendicular diffusion becomes ap- parent when comparing e.g. observers C and E, where SEPs reaching observer E must do so via perpendicular diffusion, and simulations hence show a smooth onset, and associated small anisotropies.Strauss et al.[2017a] also confirms the importance of perpendicular diffusion dur- ing these events (see figures13,14and15ofStrauss et al.[2017a]).

Figure 5.9 shows the same simulation from figure 5.8, but now with simulated omni-directional differential intensities and anisotropies forλrr= 0.25AU and an injection broadness ofσ0 = 5^◦ (figure 5.9a) andσ₀ = 20^◦(figure 5.9b). This configuration clearly shows the Neupert effect’s inability to correctly simulate the decay phase of the flare as it underestimates the decay phase with several orders of magnitude. Also, the differential intensities at Earth (observer C) are lower than expected when comparing to the IMP-8 data for a narrow injection (5^◦), while the broader injection (20^◦) compares closer to the spacecraft data, but the underestimation of the decay phase still remains in both cases.