The delayed arrival of faster solar energetic particles as a probe into the shock acceleration process

(Solar Orbiter Nugget #70 by Y. Li^1,2, J. Guo^1,3, D. Pacheco^1,3, Y. Wang^1,3, M. Temmer², Z.. Ding⁴, and R. F. Wimmer-Schweingruber⁴)

1. Introduction

Solar energetic particles (SEPs) are accelerated by solar flares and/or CME-driven shocks and pose a significant radiation risk to spacecraft and crewed missions. The long-standing picture is that more energetic (faster) particles arrive earlier, producing the well-known velocity dispersion (VD) pattern in dynamic spectra and supporting the assumption that particles are released at approximately the same time irrespective of their energy and propagate along similar trajectories. Recent measurements by Solar Orbiter (SO) and Parker Solar Probe (PSP) [1-4], however, reveal cases where higher-energy particles arrive later, i.e., an inverse velocity dispersion (IVD).

This study reports 10 IVD proton events detected by SO and presents a detailed analysis of three events to probe the underlying physics. We find that the IVD features can be well interpreted by the diffusive shock acceleration (DSA) mechanism through which particles need longer time to obtain higher energies before being released from the acceleration site. This finding suggests that DSA is probably a dominant mechanism in such events, with respect to magnetic reconnection, to accelerate protons to tens of MeV, which could be potentially radiation damaging to instruments and humans in space. We further determine, innovatively, the physical conditions and time scales during the actual acceleration process of these particles at the shock that cannot be observed directly.

Results

From early 2022 to mid-2024, SO observed many SEP events. Among them, ten events at SO heliocentric distances of ~0.49–0.95 au with clear IVD signatures were identified; three with complete remote-sensing observations (2023-11-09, 2023-12-24, 2023-12-31) covered by Earth-view and STEREO-A spacecraft were investigated in depth here.

We take the 2023-11-09 SEP event as an example: two C-class flares (C1.3 from AR13481; C2.6 from AR13480) and a halo CME were associated with the event. White-light graduated cylindrical shell model (GCS) fits give a shock speed ~870 km s⁻¹. In-situ particle data measured by SO at 0.66 au show the event onset at 13:32 UT with the first-arrival energy at ~3 MeV. Below this energy, there is a classical velocity dispersion (VD) pattern, while above it, there is a clear inverse velocity dispersion (IVD) feature. A shock arrived at SO about two days later and produced energetical storm particles (ESPs) at low energies.

For the VD part (E<~2 MeV), a standard velocity dispersion analysis (VDA)[5] was applied. This assumes that the first-arriving particles are released simultaneously and travel along the same path length, and particles with higher energies would arrive earlier than those with lower energies. To determine the release time (t₀) and path length (L₀) of particles with the velocity dispersion feature, the following simple function is often applied:

t_onset(E)=t₀+L₀/v(E)

where t_onset(E) is the observed SEP onset time for particles with kinetic energy E. VDA gives the release time t₀ = 11:39 ± 7 UT, broadly consistent with the flare hard-X-ray peak and Type II radio burst start (~11:05 UT), indicating that these particles were likely accelerated by the flare process or the initial phase of the shock.

For the IVD part (E'>~7 MeV), we consider a procedure where particles of different energies were released at a different time t_release(E') (as the shock propagates outward) and they also arrived at the observer with different path lengths L(E'), as described in the equation below:

t_onset(E')=t_release(E')+L(E')/v(E')

The above equation cannot be solved directly, but we used an iterative approach to derive the best-fit t_release(E') and L(E') accounting for the outward motion of the CME shock (shortening the particle path) using a drag-based model (DBM [6]). This yields energy-dependent release times that increase with energy, i.e., the higher-energy protons were released later, when the shock was already at ~0.05–0.2 au. This indicates that these particles were likely accelerated by the shock as it traveled outward.

Using the first three hours after the onset in each energy band (to minimize transport effects which are amplified at later stages of SEP events), the proton energy-spectra show distinct power-law indices: about –2.14 for the low-energy VD part and –3.41 for the high-energy IVD part (2023-11-09), implying different acceleration phases or mechanisms, consistent with the previous evidence.

Fig1: Panel a: dynamic spectrum of the 2023-11-09 event. The onset times of the VD particles are marked with blue circles, while the IVD particles are marked with orange triangles. The transition energy is around 3 MeV, marked by green triangles. Panel b uses the onset times of the VD particles in panel a (blue dots), to derive the release time and path length of protons below about 2 MeV (shown in the legend). Orange triangles are the onset times of higher-energy IVD particles versus energy (represented by 1/beta, which is c/v on the x-axis), and the orange circles are the derived release time which increase with energy. The DSA-fitted release time is plotted as the dotted line. Panel c: energy spectra of VD (blue) and IVD (orange) particles.

Discussion

We infer the physical origin of the IVD using the diffusive shock acceleration (DSA) theory. In the DSA picture, particles gain energy by repeated shock crossings while scattering off converging magnetic irregularities; energy gain and the resulting spectrum depend on the upstream/downstream flows and turbulence [7]. Using the IVD onset spectral index gamma  and the measured upstream speed u_u in the shock frame, the shock compression ratio r and downstream speed u_d are derived following previous theories [8-9].

With the parameters of the shock acceleration conditions retrieved from the DSA model, we further determine energy-dependent acceleration time scales under different shock conditions. In DSA, the ability of the shock to confine particles for continuous acceleration is parameterized by the particle mean free path lambda₀: smaller values of lambda₀ lead to better confinement and more efficient acceleration of particles. Observationally derived energy-dependent acceleration time is then used to retrieve the theoretical mean free path of particles at the acceleration shock close to the Sun. Fitting the observed IVD release-time curves for the three events yields lambda₀ ~ 10^-4 au, implying strong turbulence and efficient confinement near the acceleration site. Although more statistics are needed for a more general conclusion, this indicates common characteristics of the DSA process, e.g.,  lambda₀ is determined by the excitation of Alfvén waves near the shock front by streaming protons [11]. 

The analysis further compares above-derived near-Sun mean free paths with in-situ values at SO inferred from ESP profiles upon the in-situ shock arrival [12]. For the 2023-11-09 and 2023-12-24 events, the mean free path is obtained to increase with solar distance following a power law, which are broadly consistent with inner-heliosphere turbulence constraints and support strong confinement near the shock when the IVD particles were released [14].

Fig 2: Panel a: Shock crossing times versus final proton energy derived from different events with different shock properties (shown as different deltau in the legend). Panel b: acceleration time matrix for the 2023-11-09 event. Color scale represents the acceleration time required to accelerate a particle from an initial energy (x-axis) to a final energy (y-axis). Panel c:  acceleration time versus the final proton energy for three different events. Each colored band corresponds to the possible solutions of the acceleration time with lambda₀  ranging from 10⁻⁵ to 10⁻³ au. The markers with error bars are results derived from the observations based on the IVD analysis.

Conclusion

Our alternative explanation of IVD is based on the evolving magnetic connection between the observer and shock front. As the shock propagates outward, the observer’s magnetic connection can connect to regions with increasing acceleration efficiencies, which leads to a later release at high energies as shown in Fig. 3.

Fig. 3: Sketch depicting the connectivity scenario which could potentially explain the later incremental arrival of higher energy protons to the observer. The blue star marks the cobpoint, which is a point at the shock that is magnetically connected to the observer

This mechanism has been investigated by several studies using Magnetohydrodynamics (MHD) models of shock evolution embedded with particle acceleration and transport models by Kouloumvakos et al.[2]. for the event PSP observed on 2022-09-05 [1]. and by Ding et al.[4] who successfully reproduced the observed IVD signatures by SO on 2022-06-07. 

In summary, this study shows a new type of solar energetic proton events observed by SO that has an unusual inverse velocity dispersion structure (at energies above a few MeV) during the event onset in addition to the typical velocity dispersion at lower energies. We analyzed the travel path and release time of particles for both VD and IVD components and have quantified the IVD release time as a function of proton energies. We found that the VD proton release time is consistent with the flare X-ray bursts and initial Type II radio emission, which indicates that these particles are likely associated with the flare process or early-shock formation close to the Sun. For IVD protons, the release time is much later when the shock is at a solar distance between about 0.05 and 0.2 au and the release time increases with particle energy. These observations could be explained by the longer time required to accelerate higher-energy particles in the diffusive shock acceleration mechanism, based on which we derived the shock parameters during the acceleration process. Therefore, this new trait in the high-energy range of solar proton events helps us to innovatively retrieve physical parameters during the acceleration process directly from observations, in particular the actual acceleration time scales that cannot be observed directly.

For further details, see doi: https://doi.org/10.1093/nsr/nwaf348

Affiliations

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

2 Institute of Physics, University of Graz, Austria

3 CAS Center for Excellence in Comparative Planetology, University of Science and Technology of China, China

4 Institute of Experimental and Applied Physics, Kiel University, Germany

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