Inverse Velocity Dispersion in solar energetic particle events: Solar Orbiter’s Case Study

(Solar Orbiter Nugget #68 by Z. Ding¹, R. F. Wimmer Schweingruber¹, A. Kollhoff¹, P. Kühl¹, L. Yang¹, L. Berger¹, A. Kouloumvakos², N. Wijsen³, J. Guo⁴, D. Pacheco⁴, Y. Li⁴, M. Temmer⁵, J. Rodriguez Pacheco⁶, R. C. Allen⁷, G. C. Ho⁷, G. M. Mason², Z. Xu⁸, S. Gunaseelan¹)

1. Introduction

Solar energetic particles (SEPs) usually reach an observer in order of speed, with the fastest and highest energy particles arriving first, creating the classic velocity dispersion (VD) signature. On 5 September 2022, Parker Solar Probe, only 15 solar radii from the Sun, observed an inverse velocity dispersion (IVD) in which high energy protons arrived after lower energy ones [1]. While one obvious explanation is that higher energy particles simply require more time to be accelerated considering the diffusive shock acceleration mechanism, recent Solar Orbiter observations and a re-analysis of STEREO data show that IVDs are not rare: several events persist for more than 10 h for the IVD period, with peak proton energies delayed by tens of MeV. Are these long duration IVDs merely a consequence of acceleration times, or do evolving shocks, shifting magnetic connectivity and transport processes together reshape SEP arrival patterns? Unravelling the interplay among these processes is crucial for pinpointing the true origin of IVDs.

2. Observations

Fig. 1 summarizes the 7 June 2022 SEP event recorded by Solar Orbiter. In the upper panel, below 1 MeV the classical velocity dispersion appears, but from 1 MeV to 20 MeV the arrival order reverses and an inverse velocity dispersion persists for almost ten hours. The transition (“nose”) energy lies near 1.1 MeV. The lower panel plots five representative energy channels. Each rises gradually and reaches a plateau near 22:00 UT, while their onset times increase with energy, confirming the long duration inverse dispersion. This clear event provides an excellent test case for uncovering the physics behind inverse velocity dispersion in SEPs.

Figure 1: Proton dynamic spectrum (upper panel) and time-intensity profiles (lower panel) for the 2022 June 7 event observed by STEP, EPT, and HET on board Solar Orbiter. The curved dashed line in the upper panel represents the onset times fit. The colored vertical lines in the lower panel indicate the onset times for each energy channel. The vertical solid line marks the CME eruption time, and the vertical dashed black line denotes the release time derived from VDA.

3. Model

Fig.  2 (a) shows a snapshot of solar wind and coronal mass ejection (CME) from EUHFORIA simulations [2]. In the snapshot, the propagation speed in the western portion of the CME is faster than that in the eastern portion. This asymmetric expansion of the CME is due to variations in the upstream solar wind conditions, that is, the CME expands faster in fast streams. Two observers at 0.96 au are indicated: Solar Orbiter (SC1, 163° ) first connects to the western flank, while a virtual spacecraft (SC2, 115° ) connects to the nose. Fig. 2 (b) compares modeled and measured plasma time series at Solar Orbiter. The spacecraft crossed a stream interaction region and entered a fast wind before the eruption. The simulated shock arrival time, density, and flow speed match the measurements, providing confidence that EUHFORIA simulations reasonably capture the shock propagation.

Figure 2. Panel (a): equatorial (left) and meridional (right) snapshots of the radial solar wind speed from EUHFORIA. Panel (b): comparison of in situ plasma and magnetic field between observation and EUHFORIA simulation for the 2022 June 7 event. The panels from top to bottom present the solar wind proton number density, the solar wind speed, and the magnetic field magnitude. The blue and black lines show the EUHFORIA simulation results and measurements from Solar Orbiter, respectively.

We utilized the Heliopheric Energetic Particle Acceleration and Transport (HEPAT) model to simulate this SEP event [3,4]. Fig. 3 compares shock parameters along magnetic field lines to Solar Orbiter (SC1) and to SC2. SC1 first connects to the shock’s slow, oblique western flank (θ_BN ≈ 50°,  E_max ≈ 2 MeV) and then drifts toward the nose, where the shock speed rises to about 1200 km s⁻¹, the geometry turns quasi parallel, and E_max reaches ≈15  MeV before dropping again. SC2 starts at the nose, so it sees that high shock speed and an initial E_max ≈ 30 MeV. After ~0.6 au its field line shifts to the slower flank and E_max falls. Thus, as a foot point migrates from flank to nose it samples ever stronger shock regions, raising the local maximum particle energy with time. This evolving connectivity, not acceleration time alone, explains the delayed arrival of high energy particles and the observed inverse velocity dispersion.

Figure 3. Radial evolution of the shock parameters along the field line connecting to SC1 (Solar Orbiter) and SC2. From top to bottom, we show the shock compression ratio, shock speed, shock obliquity angle, and maximum proton energy. 

Fig. 4 contrasts simulated proton dynamic spectra for Solar Orbiter (SC1) and SC2 during the first 24 h after the CME eruption. Without cross field diffusion (upper panels), SC1 shows a clear inverse dispersion pattern with a 2 MeV nose energy because its field line drifts from the flank into faster, stronger shock regions. SC2, connected at the nose early on, displays normal velocity dispersion. Considering limited cross field diffusion (lower panels), SC1 samples a wider shock area, raising the nose energy and shortening the IVD phase, while the spectra of well connected SC2 remain unchanged. Thus, a large cross field diffusion may blur the IVD signature by letting high energy particles cross field lines quickly.

Figure 4. Modeled proton dynamic spectra observed at SC1 (a) and SC2 (b). The upper panels show the case without cross-field diffusion, and the lower panels show the case with cross-field diffusion.

4. Conclusion

The delayed rise in maximum particle energy measured by Solar Orbiter is not merely the time a young shock needs to accelerate particles; instead, it reflects how the spacecraft’s magnetic footpoint progressively samples stronger regions of an evolving, non‑uniform shock. Our results predict an east–west asymmetry: IVD is most likely to occur when the observer first connects to the shock’s weak western flank and later migrates to the nose, so the local maximum energy grows and high‑energy particles arrive later. We further suggest that the IVD signature persists only while cross‑field diffusion remains weak.

This work has been published in Ding et al., 2025, A&A, https://doi.org/10.1051/0004-6361/202553806

Affiliations

1 Institute of Experimental and Applied Physics, Kiel University, D-24118 Kiel, Germany

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

3 Centre for mathematical Plasma Astrophysics, KU Leuven Campus Kulak, 8500 Kortrijk, Belgium

4 National Key Laboratory of Deep Space Exploration/School of Earth and Space Sciences, University of Science and Technology of China, Hefei 230026, China

5 Institute of Physics, University of Graz, Graz, Austria

6 Universidad de Alcalá, Alcalá de Henares 28805, Spain

7 Southwest Research Institute, San Antonio, TX 78228, USA

8 California Institute of Technology, MC 290-17, Pasadena, CA 91125, USA

References

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[2] Pomoell, J. & Poedts, S. 2018, JSWSC, 8, A35 https://doi.org/10.1051/swsc/2018020

[3] Ding, Z., Li, G., Mason, G., et al. 2024, A&A, 681, A92

https://doi.org/10.1051/0004-6361/202347506

[4] Ding, Z., Wijsen, N., Li, G., & Poedts, S. 2022, A&A, 668, A71

https://doi.org/10.1051/0004-6361/202244732