Irregular Proton Injection to High Energies at Interplanetary Shocks

Figure 1 shows a 30 minute overview across the shock transition. Panels (a)–(b) reveal the presence of shock-accelerated particles at energies of up to 100 keV, while particle fluxes at higher energies do not respond to the shock passage. At these high energies, the fluxes were enhanced following a large solar energetic particle (SEP) event (see Klein et al. 2022).

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The most striking feature of the period prior to the shock arrival at SolO is the irregular energetic particle enhancements particularly evident at 10–30 keV energies (Figure 1(b), black box), found in the time interval ∼15 minutes before the shock crossing, corresponding to 2 × 10⁵ km or 2500 ion inertial lengths, d_i . These particle enhancements have the novel feature of being dispersive in energy and are the focus of this work. The typical timescales at which the irregularities are observed are of 10–20 s, corresponding to spatial scales of about 50 d_i . Such signatures were previously inaccessible to observations, as shown in Figure 1(c), where the time profile of ion differential flux in the 0.012–0.015 MeV channel, rising exponentially up to the shock (Giacalone 2012), is shown at full resolution (blue) and averaged using a ∼1 minute window, typical of previous IP shock measurements. Figure 1(d) shows pitch-angle intensities for 0.011–0.019 MeV ions (i.e., energies at which the irregular enhancements are observed). Pitch angles are computed in the plasma rest frame assuming that all ions are protons and performing a Compton–Getting correction (Compton & Getting 1935), thereby combining magnetic field data from the magnetometer (MAG; Horbury et al. 2020), solar wind plasma data from the Proton and Alpha Particle Sensor (PAS) on the Solar Wind Analyser (SWA) instrument suite (Owen et al. 2020), and particle data from EPD/STEP (Yang et al. 2023). For the interval studied, low pitch angles are in the 30° field of view of STEP, relevant for shock-reflected particles. The irregular enhancements of energetic particles are field aligned, as is evident for the strongest signal close to the shock transition. The flux enhancement visible in PAS (Figure 1(e)) at lower energies starting immediately before the shock (22:00 UT) also reveals a field-aligned population. The study of the PAS low-energy population and the behavior very close to the shock transition is the object of another investigation (Dimmock et al. 2023).

The magnetic field reveals a wave foreshock ∼2 minutes upstream of the shock, in conjunction with a population of low-energy (∼4 keV) reflected particles seen by SWA/PAS, visible as the light blue enhancement in Figure 1(e) around 22:00 UT. Interestingly, the magnetic field is quieter where signals of irregular injection are found, indicating that efficient particle scattering may be reduced in this region (Lario et al. 2022). In this "quiet" shock upstream, we found two structures compatible with shocklets in the process of steepening (∼21:57 UT), very rarely observed at IP shocks (Wilson et al. 2009; Trotta et al. 2023a).

The shock parameters were estimated using upstream/downstream averaging windows varied systematically between 1 and 8 minutes (Trotta et al. 2022a). The shock was oblique, with a normal angle θ_Bn = 44 ± 15 (obtained with the Mixed Mode 3 technique, MX3; Paschmann & Schwartz 2000), compatible with MX1,2 and magnetic coplanarity). The shock speed in the spacecraft frame and along the shock normal is V_shock = 400 ± 5 km s⁻¹. The shock Alfvénic and fast magnetosonic Mach numbers are M_A ∼ 7.6 and M_fms ∼ 4.6, respectively. Thus, the event provides us with the opportunity to study a shock with particularly high Mach number in comparison with other IP shocks, while the shock speed is moderate with respect to typical IP shocks (Kilpua et al. 2015). The shock is supercritical and therefore expected to have a corrugated, rippled front (Trotta & Burgess 2019; Kajdic et al. 2021). The presence of reflected particles, enhanced wave activity in close proximity (1 minute) to the shock transition, and upstream shocklets in the process of steepening is consistent with the local shock parameters (Blanco-Cano et al. 2016).

To further elucidate the dispersive nature of the suprathermal particles, we show the STEP energy spectrogram in 1/v versus t space (Figure 2). Here, particle speeds are referred to the center of the relative energy bin and computed in the spacecraft rest frame, assuming that all particles detected are protons (see Wimmer-Schweingruber & Pacheco 2021 for further details). During the period of irregular particle enhancements, we also combined magnetic field and plasma data to compute the particle pitch angles in the solar wind frame (Compton & Getting 1935), revealing that the particles detected by STEP are closely aligned with the field (not shown here). Interestingly, by visual inspection, it can be seen that these dispersive signals are shallower going far upstream, consistent with the fact that they are injected from more distant regions of the shock.

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The dispersive flux enhancements are associated with irregular acceleration of protons along the shock front. Indeed, due to their dispersive nature, the particles detected by STEP cannot be continuously produced at the shock and propagated upstream, but they must come from a source that is only temporarily magnetically connected to the spacecraft due to time and/or space irregularities. Then, the fastest particles produced at the irregular source are detected first by the spacecraft, followed by the slower ones, yielding the observed dispersive behavior. Given the short timescales at which energetic particle enhancements are observed with respect to the shock and the quiet behavior of upstream magnetic field in the 10 minutes upstream of the shock, we assume that particles do not undergo significant scattering from their (irregular) production to the detection at SolO. It is then natural to investigate the connection with the shock. The bottom left panel of Figure 2 shows the local ${\theta }_{\mathrm{Bn}}(t)\equiv {\cos }^{-1}\left({\bf{B}}(t)\cdot {\hat{{\bf{n}}}}_{\mathrm{shock}}/| {\bf{B}}(t)| \right)$ changing significantly when the dispersive signals are observed, indicating that the spacecraft was indeed connected to different portions of the (corrugated) shock front, which in turn is expected to respond rapidly to upstream changes, as recent simulation work elucidated (e.g., Trotta et al. 2023b). Note that, given the single-spacecraft nature of the observations, the average shock normal computed with MX3 for both local and average θ_Bn estimation was used.

To further support this idea, similarly to velocity dispersion analyses (VDAs) used to determine the injection time of SEP events (e.g., Lintunen 2004; Dresing et al. 2023), we chose the clearest dispersive signal (∼100 s upstream of the shock), and we superimpose the following relation (indicated by the magenta line in Figure 2):

$$\begin{eqnarray}&&{t}_{{\rm{O}}}(v)={t}_{i}+\displaystyle \frac{s}{v},\end{eqnarray} \tag{ 1 }$$

where t_O represents the time at which the flux enhancement is observed for a certain speed v, t_i is the time of injection at the source, and s is the distance traveled by the particles from the source to the spacecraft. Thus, the argument is that the dispersive signals are due to accelerated particles produced by different portions of the shock front temporarily connected with the spacecraft, as sketched in Figure 2 (right). We note that, due to the very high energy-time resolution of STEP, it was possible to perform the VDA on such small (∼seconds) timescales. Determining t_i based on the time when the highest energy particles are observed (t_i ∼ − 130 s), the source distance that we obtain through Equation (1) is s ≈ 4 × 10⁴ km (∼500d_i ), compatible with their generation at the approaching shock, for which we would expect $s\sim {V}_{\mathrm{shock}}{\rm{\Delta }}t/\sin ({\theta }_{\mathrm{Bn}})$, where V_shock is the average shock speed and Δt is the time delay between the observation of the dispersive signal and the shock passage. This is also compatible with the fact that the other dispersive signals observed farther upstream, such as the one before 21:54, about 500 s upstream of the shock (see Figure 2), show a shallower inclination though a more precise, quantitative analysis of this behavior is complicated by the high noise levels of the observation and will be the object of later statistical investigation employing more shock candidates (Yang et al. 2023).

Further insights about shock front irregularities are limited by the single-spacecraft nature of these observations. Therefore, we employ 2.5-dimensional kinetic simulations, with parameters compatible with the observed ones, to model the details of the shock transition, where proton injection to suprathermal energies takes place, relevant to our interpretation of the dispersive signals and enabling us to see how the shock surface and normal behave at small scales (see Figure 2). In the simulations, protons are modeled as macroparticles and advanced with the particle-in-cell method, while the electrons are modeled as a massless, charge-neutralizing fluid (Trotta et al. 2020).

In the model, distances are normalized to the ion inertial length d_i , times to the upstream inverse cyclotron frequency ${{{\rm{\Omega }}}_{{ci}}}^{-1}$, velocity to the Alfvén speed v_A , and the magnetic field and density to their upstream values B₀ and n₀. The shock is launched with the injection method (Quest 1985) where an upstream flow speed V_in = 4.5v_A was chosen, corresponding to M_A ∼ 6. The shock nominal θ_Bn is 45°. The simulation domain is 512 d_i ×512 d_i , with resolution Δx = Δy = 0.5 d_i and a particle time step Δt_pa =0.01 ${{\rm{\Omega }}}_{{ci}}^{-1}$. The number of particles per cell used is always greater than 300. This choice of parameters is compatible with the local properties of the IP shock as estimated from the SolO measurements. However, inherent variability routinely found in the simulations at small scales and in the observations at larger scales must be considered when comparing numerical and observational results. We note that these simulations are initialized with a laminar upstream, and therefore, the fluctuations that impact the shock are self-generated (due to particle reflection and subsequent upstream propagation). An exhaustive characterization of these self-induced fluctuations is discussed in Kajdic et al. (2021).

Simulation results are shown in Figure 3. In the top panel, we present the proton density for a simulation snapshot where the shock transition is well developed, showing the strongly perturbed character of the shock front. In such an irregular shock transition, particle dynamics become extremely complex (e.g., Lembege & Savoini 1992). To further elucidate the irregularities of the shock front, we computed the shock position in the simulation domain (with the criterion B > 3B₀, as in Trotta et al. 2023b) and evaluated the local θ_Bn along it (Figure 3(a), inset), showing high variability (see the cartoon in Figure 2).

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In the bottom panel of Figure 3, we study the self-consistently shock-accelerated protons. The upstream energy spectrum is shown in the inset, with a peak at the inflow population energies and a suprathermal tail due to the accelerated protons. To address particle injection, we analyze the upstream spatial distribution of such suprathermal protons (Figure 3(b)) at the energies highlighted in the inset, which are a factor of 10 larger than the typical energies of particles in the upstream inflow population, in a similar fashion as the energy separation between the STEP energies at which the irregular enhancements are observed (∼10 keV) and the solar wind population energies measured by PAS (∼1 keV). It can be seen that suprathermal particles are not distributed uniformly, and their spatial distribution varies with their locations along the shock front, another indication of irregular injection. Furthermore, we observed that the length scale of the irregularities is of 50 d_i , directly comparable with the irregularities seen in the STEP fluxes (see Figure 1). Higher energy particles also show irregularities.