Modelling Two Consecutive Energetic Storm Particle Events observed by Solar Orbiter 

Fig. 1 presents the proton time intensity profiles and 2-hour time-interval proton spectra observed by SolO/EPD for the August 30 and the September 5 events. The key observations from the study include: 1) Diverse Event Durations: The two ESP events differed significantly in duration - the August event lasted approximately 7 hours, while the September event extended over 16 hours. These durations are linked to the passage of shock-sheath structures associated with each event. 2) Unique 'Crossover' Feature: A rare 'crossover' phenomenon was observed in the September event, where low-energy protons near the shock had lower intensities than high-energy protons. This contrasts with typical SEP events. 3) Distinct Proton Spectra: Analysis of the proton spectra revealed that the August event displayed a typical negative power-law, while the September event showed a positive power-law-like behavior upstream of the shock, transitioning to a typical negative power-law across the shock. This difference poses challenges to conventional explanations based solely on  particle transport effects.

As demonstrated in Fig. 2(a), the power spectral density (PSD) of the total magnetic field was computed to discern the resonant behavior with different particle energies upstream of the shock, revealing distinct differences between the two events. For the August event, the PSD decayed rapidly within about two hours, whereas for the September event, it showed a slower decay rate, taking approximately six hours. This suggests notable differences in the turbulence dynamics of the events. Further analysis of the spectral slopes, derived from power-law fitting of the spectra, indicated that the September event had a steeper spectral slope, signaling stronger and more developed turbulence ahead of the shock.

From this PSD analysis, we explain how accelerated particles are distributed at the shock and their escape mechanisms as illustrated in Fig. 2(b). Particles within certain distances upstream the shock are trapped by self-excited waves, while those beyond can escape into the ambient solar wind. The different escape lengths observed in the two events, influenced by the decay rate of turbulence strength, played a significant role in the spatial distribution of particles. In scenarios where the escape boundary extends further, the spectral index of the escaped particle intensity can turn positive, leading to a crossover phenomenon. This combined effect of trapping and escape, controlled by different escape boundaries, explains the distinct particle distributions observed in the two events.

Figure 2: (a) Total magnetic field Power Spectral Density (PSD) for August 30, 2022 (upper) and September 05, 2022 (lower) events. The left panels represent the 2-hour interval of PSD as a function of frequency upstream of the shock. 0 represents the time of shock. The dashed vertical lines indicate the corresponding resonant frequencies of different energies by Taylor’s hypothesis. The three energies are labelled in the middle panel. The middle panels show the ratio of PSD as a function of time upstream of the shock, compared to the PSD closest to the shock. The right panels show the fitted power law indices in the frequency range [0.003,0.06] Hz as a function of time. (b) Two schematics of the distribution of accelerated particles at an interplanetary shock for two momenta, adapted from [7]. Both schematics illustrate the momentum-dependent scale length of the exponential decay upstream of the shock and the corresponding trapping and escape of particles. The only difference between the schematics is the different escape length scales LP1 and LP2. The subpanel shows the intensity of the escaped particles for P1 and P2.

Model

We first modeled the solar wind and CME propagation using EUHFORIA [3]. Movie 1 shows the radial speed of the solar wind in EUHFORIA and the following shell structure. The shell structures in the iPATH model are constructed via the realistic 3D shock fronts and time-dependent downstream flow speed from EUHFORIA, thereby capturing the spatial extension of the downstream region of the shock. The radial width of the shell structure for the August event is much narrower than that of the September event, a difference attributed to the distinct deceleration histories of the plasma downstream of the shock. There is strong deceleration of CME for the September event, causing the shell structure to be stretched more by the rapid deceleration of the downstream flow. The radial width of the shell structure is essential for understanding the duration of the ESP event. The arrival of the shock front (the first shell) at the observer marks the onset of the ESP event, while the passage of the last shell marks its end. This demonstrates that the shell module of the iPATH model can serve as a powerful tool to study ESP events.

In this study we examined two consecutive but very different ESP events observed by the Solar Orbiter, we effectively modelled these two events in terms of their intensity profiles, spectral slopes and  ESP duration using combined EUHFORIA and iPATH models. The unusual crossover of time profiles and positive spectral indices in the September event are attributed to the long duration of enhanced turbulence upstream of the shock, which effectively hinders the escape of lower-energy particles from the shock. Our simulation shows that this duration is a consequence of the deceleration history of the CME and the shock it drives. Examination of these two events underscore the need for comprehensive simulations that incorporate realistic solar wind, CME dynamics, and turbulence near the shock to fully grasp the nature of SEP events.

This work is supported in part by NASA grants 80NSSC19K0075, 80NSSC22K0268, 80NSSC21K1327, and NSF ANSWERS 2149771 at UAH-USA. GL also acknowledges support through ISSI team 469. Solar Orbiter is a mission of international cooperation between ESA and NASA, operated by ESA. The Suprathermal Ion Spectrograph (SIS) is a European facility instrument funded by ESA under contract number SOL.ASTR.CON.00004.  We thank ESA and NASA for their support of the Solar Orbiter instruments whose data were used in this paper.  Solar Orbiter post-launch work at JHU/APL is supported by NASA contract NNN06AA01C and at CAU by German Space Agency (DLR) grant \# 50OT2002. The UAH-Spain team acknowledges the financial support by the Spanish Ministerio de Ciencia, Innovación y Universidades FEDER/MCIU/AEI Projects ESP2017-88436-R and PID2019-104863RB-I00/AEI/10.13039/501100011033. SP acknowledges support from the projects C14/19/089  (C1 project Internal Funds KU Leuven), G.0B58.23N and G.0025.23N (WEAVE)   (FWO-Vlaanderen), 4000134474 (SIDC Data Exploitation, ESA Prodex-12), and Belspo project B2/191/P1/SWiM.

¹ Centre for mathematical Plasma Astrophysics, KU Leuven, 3001 Leuven, Belgium

² Department of Space Science and CSPAR, University of Alabama in Huntsville, Huntsville, AL 35899, USA

³ Johns Hopkins Applied Physics Lab, Laurel, MD 20723, USA

⁴ Institute of Physics, University of Maria Curie-Skłodowska, Pl. M. Curie-Skłodowska, 20-031 Lublin, Poland

⁵ Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

⁶ Department of Astronomy, University of Maryland College Park, MD 20742, USA

⁷ Institute of Experimental and Applied Physics, University of Kiel, Leibnizstrasse 11, D-24118 Kiel, Germany

⁸ Universidad de Alcalá, Alcalá de Henares 28805, Spain

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