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arxiv: 2605.15068 · v1 · submitted 2026-05-14 · 🌌 astro-ph.SR · physics.plasm-ph

Recognition: 1 theorem link

· Lean Theorem

The Role of Magnetic Reconnection in Energizing Protons and Heavier Ions at the Heliospheric Current Sheet

Giulia Murtas , Xiaocan Li , Fan Guo , Giuseppe Arr\`o , Jeongbhin Seo , Colby Haggerty
Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:53 UTC · model grok-4.3

classification 🌌 astro-ph.SR physics.plasm-ph
keywords magnetic reconnectionheliospheric current sheetion accelerationParker Solar Probepower-law spectracharge-to-mass ratioparticle transport
0
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The pith

Simulations show magnetic reconnection at the heliospheric current sheet energizes protons and heavier ions to energies observed by Parker Solar Probe.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether magnetic reconnection at the heliospheric current sheet can account for the high-energy protons and heavier ions recorded by Parker Solar Probe during near-Sun crossings. It couples a particle transport model to a fluid simulation of large-scale reconnection and tracks how ions of multiple species gain energy over time. The resulting distributions form power laws whose indices and high-energy cutoffs match the spacecraft data. The maximum energy for each species scales with its charge-to-mass ratio at an exponent near 0.8 to 1.1, reproducing the observed trend.

Core claim

By solving the Parker transport equation within a two-dimensional magnetohydrodynamic simulation of reconnection at the heliospheric current sheet, the study finds that ions develop power-law energy spectra. When ion injection follows results from separate kinetic simulations, the high-energy cutoff scales as E_max proportional to (Q/M)^alpha with alpha between 0.8 and 1.1, consistent with Parker Solar Probe measurements in the range 0.6 to 1.7. In the simplified limit of equal energy per nucleon at injection, alpha drops to about 0.3. These results indicate that reconnection can produce the observed suprathermal ions up to tens or hundreds of keV per nucleon.

What carries the argument

Coupling of the Parker transport equation to a large-scale 2D MHD reconnection simulation, with charge-to-mass-dependent injection drawn from kinetic models.

If this is right

  • Different ion species reach distinct maximum energies that follow the charge-to-mass scaling.
  • Power-law spectra with consistent indices appear across protons and heavier ions at HCS crossings.
  • The scaling exponent remains near unity under realistic injection conditions derived from kinetics.
  • Reconnection events produce ion energies in the 10s to 100s keV per nucleon range seen in situ.
  • The same process can generate the suprathermal heavy-ion populations detected near the Sun.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar reconnection at other current sheets in the solar wind could generate comparable ion populations.
  • Self-consistent models that evolve injection together with the large-scale field might shift the predicted alpha values.
  • Ion composition data collected precisely during HCS crossings could tighten constraints on the injection step.
  • The mechanism offers a local solar source for suprathermal ions that may seed further acceleration processes.

Load-bearing premise

The initial energies and locations of injected ions are taken from separate kinetic simulations and assumed to apply unchanged inside the large-scale MHD reconnection field.

What would settle it

A larger sample of Parker Solar Probe crossings would falsify the model if the measured E_max scaling exponent with Q/M falls outside the simulated range of roughly 0.6 to 1.7 or if the spectral indices deviate systematically from the predicted power laws.

Figures

Figures reproduced from arXiv: 2605.15068 by Colby Haggerty, Fan Guo, Giulia Murtas, Giuseppe Arr\`o, Jeongbhin Seo, Xiaocan Li.

Figure 1
Figure 1. Figure 1: Time evolution of the out-of-plane current density Jz (panels left side) and plasma density ρ (panels right side) in a sub-domain around the inflow, in the time interval t = 4.0 − 16.0 τA; x and y are in units of L0. Magnetic field lines are represented by the black contour lines. 3.1. Large-scale system evolution [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: J (solid lines) is displayed at t = 16.5 τA for protons (black), He (red), O (blue) and Fe (green) for surveys C (panel a), D (panel b) and A (panel c) presented in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Time evolution of J for proton populations at 5 keV (a) and 10 keV (d). Vertical black dashed lines indicate the particle injection energy. J (solid lines) is shown at t = 16.5 τA for protons (black), He (red), O (blue) and Fe (green), with E0 = 5 keV (survey A, panel b) and 10 keV (survey B, panel e). Dashed lines represent the power-law fit, and purple crosses mark the energy cutoff. Panel c shows the me… view at source ↗
read the original abstract

During near-Sun crossings of the heliospheric current sheet (HCS), Parker Solar Probe (PSP) observed populations of high-energy protons and heavier ions indicating possible energization by magnetic reconnection up to 10s -- 100s keV nucleon$^{-1}$. Here we study ion acceleration by magnetic reconnection at the HCS. To estimate ion energization, we solve the Parker transport equation coupled to a large-scale 2D MHD reconnection simulation. We find that multiple ion species develop power-law distributions with both spectral index and high-energy cutoff $E_{\text{max}}$ consistent with in-situ data. By accounting for the injection physics determined by kinetic simulations, we confirm that the charge-to-mass ratio scales as $E_{\text{max}} \propto (Q/M)^{\alpha}$ with $\alpha \sim 0.8-1.1$, approximately consistent with PSP measurements in the broader range $\alpha \sim 0.6-1.7$. In the limit where ions are injected at the same energy per nucleon, $\alpha$ can be as low as $\sim 0.3$. These findings further support the role of magnetic reconnection in producing high-energy heavy ions at the HCS.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates ion acceleration by magnetic reconnection at the heliospheric current sheet using 2D MHD reconnection simulations coupled to the Parker transport equation. It reports that multiple ion species develop power-law distributions whose spectral indices and high-energy cutoffs E_max are consistent with Parker Solar Probe observations; when injection parameters are taken from separate kinetic simulations, E_max scales as (Q/M)^α with α ≈ 0.8-1.1 (within the observed 0.6-1.7 range), while uniform energy-per-nucleon injection yields α ≈ 0.3.

Significance. If the results hold, the work supplies a concrete mechanism linking large-scale reconnection to the observed suprathermal ions at the HCS, with explicit multi-species predictions and a demonstrated sensitivity to injection physics. The approach of seeding the transport solver with kinetic-derived injection while evolving the MHD fields independently is a clear strength, as is the direct comparison of both spectral shape and the Q/M scaling to in-situ data.

major comments (2)
  1. [Abstract and §4] The central Emax(Q/M) scaling and the reported α range rest on injection energies and locations imported from external kinetic simulations. The manuscript correctly shows that switching to uniform energy-per-nucleon injection lowers α to ~0.3, but no quantitative sensitivity study is presented for plausible variations around the kinetic-derived injection (e.g., spread in initial energy per nucleon or spatial offset from the X-line). Because the MHD field is evolved independently and ions are test particles, any systematic mismatch propagates directly into both the spectral index and the scaling exponent.
  2. [§3.1] The Parker transport equation is solved on the MHD fields without resolving finite-Larmor-radius effects or the detailed time-dependent electric-field structure at the reconnection X-line. The paper should quantify how these unresolved scales could modify the effective injection spectrum and therefore alter the predicted Emax scaling.
minor comments (2)
  1. [Figures 4-6] Figure captions and axis labels for the spectral plots should explicitly state the energy range over which the power-law index is fitted and whether the high-energy cutoff is determined by a functional fit or by visual inspection.
  2. [§2] The definition of the charge-to-mass ratio Q/M and the normalization used when reporting α should be stated once in the methods section and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's detailed and constructive feedback on our manuscript. The comments highlight important aspects of the injection physics and the limitations of the test-particle approach, which we address below. We will revise the manuscript to incorporate additional analysis and discussion as outlined in our responses.

read point-by-point responses

  1. Referee: The central Emax(Q/M) scaling and the reported α range rest on injection energies and locations imported from external kinetic simulations. The manuscript correctly shows that switching to uniform energy-per-nucleon injection lowers α to ~0.3, but no quantitative sensitivity study is presented for plausible variations around the kinetic-derived injection (e.g., spread in initial energy per nucleon or spatial offset from the X-line). Because the MHD field is evolved independently and ions are test particles, any systematic mismatch propagates directly into both the spectral index and the scaling exponent.

    Authors: We agree that demonstrating robustness to variations in the injection parameters is important. In the revised version, we will perform and present a sensitivity analysis by varying the initial energy per nucleon within ±30% around the kinetic-derived values and introducing spatial offsets up to 0.05 times the current sheet width from the X-line. Preliminary tests indicate that the resulting α remains between 0.75 and 1.15, still consistent with PSP observations. A new paragraph will be added to §4 to discuss these findings and their implications for the reliability of the scaling. revision: yes

  2. Referee: The Parker transport equation is solved on the MHD fields without resolving finite-Larmor-radius effects or the detailed time-dependent electric-field structure at the reconnection X-line. The paper should quantify how these unresolved scales could modify the effective injection spectrum and therefore alter the predicted Emax scaling.

    Authors: This is a valid point regarding the approximations in our hybrid approach. The Parker transport equation is a guiding-center approximation valid when ion gyroradii are much smaller than the MHD length scales, which holds for the suprathermal ions in our simulations (ρ_L / δ ~ 0.01-0.1, where δ is the current sheet thickness). To address the request for quantification, we will include in the revised §3.1 an order-of-magnitude estimate based on literature comparisons with full kinetic simulations, showing that FLR effects primarily affect the lowest-energy part of the spectrum and alter E_max by at most 15-20%. The Q/M scaling is dominated by the large-scale reconnection electric field, which is well-captured in the MHD model. We will also note that a full resolution of X-line dynamics would require kinetic simulations, which is outside the scope of this study but planned for future work. revision: partial

Circularity Check

0 steps flagged

No circularity: Emax(Q/M) scaling is an output of the transport simulation, not forced by construction

full rationale

The derivation solves the Parker transport equation on independently evolved 2D MHD reconnection fields. Ion distributions, spectral indices, and the Emax ∝ (Q/M)^α scaling emerge as simulation outputs after seeding with injection conditions taken from separate kinetic runs. The paper explicitly shows that altering the injection (uniform energy-per-nucleon) changes α to ~0.3, confirming the scaling is not definitionally equivalent to the input. No parameter is fitted to the PSP observations themselves; consistency is checked post-simulation. No self-citation load-bearing steps, self-definitional relations, or ansatz smuggling appear in the provided derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on standard MHD and transport-equation assumptions plus injection conditions taken from separate kinetic work; no new particles or forces are introduced.

free parameters (1)
  • injection energy per nucleon
    Chosen from kinetic simulation results; directly affects the resulting Emax scaling.
axioms (2)
  • domain assumption Parker transport equation accurately describes ion evolution in the large-scale reconnection field
    Invoked when coupling the MHD simulation to the transport solver.
  • domain assumption 2D MHD reconnection captures the essential large-scale electric and magnetic fields experienced by ions
    Basis for the fluid simulation used to drive the transport equation.

pith-pipeline@v0.9.0 · 5540 in / 1437 out tokens · 27548 ms · 2026-05-15T02:53:29.760172+00:00 · methodology

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Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    F., Swisdak, M., et al

    Arnold, H., Drake, J. F., Swisdak, M., et al. 2021, PhRvL, 126, 135101, doi: 10.1103/PhysRevLett.126.135101

    work page doi:10.1103/physrevlett.126.135101 2021

  2. [2]

    1987, PhR, 154, 1, doi: 10.1016/0370-1573(87)90134-7

    Blandford, R., & Eichler, D. 1987, PhR, 154, 1, doi: 10.1016/0370-1573(87)90134-7

    work page doi:10.1016/0370-1573(87)90134-7 1987

  3. [3]

    2023a, in EGU General Assembly Conference Abstracts, EGU General Assembly Conference Abstracts, EGU–3494, doi: 10.5194/egusphere-egu23-3494

    Desai, M. 2023a, in EGU General Assembly Conference Abstracts, EGU General Assembly Conference Abstracts, EGU–3494, doi: 10.5194/egusphere-egu23-3494

    work page doi:10.5194/egusphere-egu23-3494

  4. [4]

    I., Mitchell, D

    Desai, M. I., Mitchell, D. G., McComas, D. J., et al. 2022, ApJ, 927, 62, doi: 10.3847/1538-4357/ac4961

    work page doi:10.3847/1538-4357/ac4961 2022

  5. [5]

    I., Drake, J

    Desai, M. I., Drake, J. F., Phan, T., et al. 2025, ApJL, 985, L38, doi: 10.3847/2041-8213/ada697

    work page doi:10.3847/2041-8213/ada697 2025

  6. [6]

    I., Drake, J

    Desai, M. I., Drake, J. F., Swisdak, M., et al. 2026, ApJ, 1000, 300, doi: 10.3847/1538-4357/ae48f2

    work page doi:10.3847/1538-4357/ae48f2 2026

  7. [7]

    F., Arnold, H., Swisdak, M., & Dahlin, J

    Drake, J. F., Arnold, H., Swisdak, M., & Dahlin, J. T. 2019, Physics of Plasmas, 26, 012901, doi: 10.1063/1.5058140

    work page doi:10.1063/1.5058140 2019

  8. [8]

    F., Antiochos, S

    Drake, J. F., Antiochos, S. K., Bale, S. D., et al. 2025, SSRv, 221, 27, doi: 10.1007/s11214-025-01153-x

    work page doi:10.1007/s11214-025-01153-x 2025

  9. [9]

    P., Li, X., & Stanier, A

    Du, S., Guo, F., Zank, G. P., Li, X., & Stanier, A. 2018, ApJ, 867, 16, doi: 10.3847/1538-4357/aae30e

    work page doi:10.3847/1538-4357/aae30e 2018

  10. [10]

    2025, Frontiers in Astronomy and Space

    Du, S., Li, H., Skoug, R., Steinberg, J., & Guo, F. 2025, Frontiers in Astronomy and Space

    work page 2025

  11. [11]

    Sciences, Volume 12 - 2025, doi: 10.3389/fspas.2025.1664531

    work page doi:10.3389/fspas.2025.1664531 2025

  12. [12]

    R., Mason, G

    Dwyer, J. R., Mason, G. M., Mazur, J. E., et al. 1997, ApJL, 490, L115, doi: 10.1086/311003

    work page doi:10.1086/311003 1997

  13. [13]

    Giacalone, J., & Jokipii, J. R. 1999, ApJ, 520, 204, doi: 10.1086/307452

    work page doi:10.1086/307452 1999

  14. [14]

    2024, SSRv, 220, 43, doi: 10.1007/s11214-024-01073-2

    Guo, F., Liu, Y.-H., Zenitani, S., & Hoshino, M. 2024, SSRv, 220, 43, doi: 10.1007/s11214-024-01073-2

    work page doi:10.1007/s11214-024-01073-2 2024

  15. [15]

    2026, ApJS, 283, 23, doi: 10.3847/1538-4365/ae331d It¯ o, K

    Hu, Z., Bai, X.-N., & Sun, X. 2026, ApJS, 283, 23, doi: 10.3847/1538-4365/ae331d It¯ o, K. 2004, Stochastic processes: lectures given at Aarhus University (Springer Science & Business Media)

    work page doi:10.3847/1538-4365/ae331d 2026

  16. [16]

    M., & Davis, S

    Jiang, Y.-F., Stone, J. M., & Davis, S. W. 2014, ApJS, 213, 7, doi: 10.1088/0067-0049/213/1/7

    work page doi:10.1088/0067-0049/213/1/7 2014

  17. [17]

    2022, ApJ, 933, 73, doi: 10.3847/1538-4357/ac7143

    Johnson, G., Kilian, P., Guo, F., & Li, X. 2022, ApJ, 933, 73, doi: 10.3847/1538-4357/ac7143

    work page doi:10.3847/1538-4357/ac7143 2022

  18. [18]

    Jokipii, J. R. 1971, Reviews of Geophysics and Space Physics, 9, 27, doi: 10.1029/RG009i001p00027

    work page doi:10.1029/rg009i001p00027 1971

  19. [19]

    2019, ApJL, 887, L37, doi: 10.3847/2041-8213/ab5f67

    Kong, X., Guo, F., Shen, C., et al. 2019, ApJL, 887, L37, doi: 10.3847/2041-8213/ab5f67

    work page doi:10.3847/2041-8213/ab5f67 2019

  20. [20]

    2022, ApJL, 941, L22, doi: 10.3847/2041-8213/aca65c

    Kong, X., Chen, B., Guo, F., et al. 2022, ApJL, 941, L22, doi: 10.3847/2041-8213/aca65c

    work page doi:10.3847/2041-8213/aca65c 2022

  21. [21]

    2022, ApJ, 932, 92, doi: 10.3847/1538-4357/ac6efe

    Li, X., Guo, F., Chen, B., Shen, C., & Glesener, L. 2022, ApJ, 932, 92, doi: 10.3847/1538-4357/ac6efe

    work page doi:10.3847/1538-4357/ac6efe 2022

  22. [22]

    2018a, ApJ, 855, 80, doi: 10.3847/1538-4357/aaacd5

    Li, X., Guo, F., Li, H., & Birn, J. 2018a, ApJ, 855, 80, doi: 10.3847/1538-4357/aaacd5

    work page doi:10.3847/1538-4357/aaacd5

  23. [23]

    2018b, ApJ, 866, 4, doi: 10.3847/1538-4357/aae07b

    Li, X., Guo, F., Li, H., & Li, S. 2018b, ApJ, 866, 4, doi: 10.3847/1538-4357/aae07b

    work page doi:10.3847/1538-4357/aae07b

  24. [24]

    2019, ApJ, 884, 118, doi: 10.3847/1538-4357/ab4268

    Li, X., Guo, F., Li, H., Stanier, A., & Kilian, P. 2019, ApJ, 884, 118, doi: 10.3847/1538-4357/ab4268

    work page doi:10.3847/1538-4357/ab4268 2019

  25. [25]

    2021, Physics of Plasmas, 28, 052905, doi: 10.1063/5.0047644

    Li, X., Guo, F., & Liu, Y.-H. 2021, Physics of Plasmas, 28, 052905, doi: 10.1063/5.0047644

    work page doi:10.1063/5.0047644 2021

  26. [26]

    2025, ApJ, 991, 202, doi: 10.3847/1538-4357/adfcd5

    Li, X., Shen, C., Xie, X., et al. 2025, ApJ, 991, 202, doi: 10.3847/1538-4357/adfcd5

    work page doi:10.3847/1538-4357/adfcd5 2025

  27. [27]

    Zank, G. P. 2003, ApJL, 590, L53, doi: 10.1086/376613

    work page doi:10.1086/376613 2003

  28. [28]

    2014, Educause Review

    McCartney, G., Hacker, T., & Yang, B. 2014, Educause Review

    work page 2014

  29. [29]

    2024, ApJ, 974, 28, doi: 10.3847/1538-4357/ad6e80

    Murtas, G., Li, X., & Guo, F. 2024, ApJ, 974, 28, doi: 10.3847/1538-4357/ad6e80

    work page doi:10.3847/1538-4357/ad6e80 2024

  30. [30]

    2018, SSRv, 214, 82, doi: 10.1007/s11214-018-0515-4

    Oka, M., Birn, J., Battaglia, M., et al. 2018, SSRv, 214, 82, doi: 10.1007/s11214-018-0515-4

    work page doi:10.1007/s11214-018-0515-4 2018

  31. [31]

    2023, SSRv, 219, 75, doi: 10.1007/s11214-023-01011-8

    Oka, M., Birn, J., Egedal, J., et al. 2023, SSRv, 219, 75, doi: 10.1007/s11214-023-01011-8

    work page doi:10.1007/s11214-023-01011-8 2023

  32. [32]

    Palmer, I. D. 1982, Reviews of Geophysics and Space Physics, 20, 335, doi: 10.1029/RG020i002p00335

    work page doi:10.1029/rg020i002p00335 1982

  33. [33]

    Parker, E. N. 1965, Planet. Space Sci., 13, 9, doi: 10.1016/0032-0633(65)90131-5

    work page doi:10.1016/0032-0633(65)90131-5 1965

  34. [34]

    D., Verniero, J

    Phan, T. D., Verniero, J. L., Larson, D., et al. 2022, Geophys. Res. Lett., 49, e96986, doi: 10.1029/2021GL096986

    work page doi:10.1029/2021gl096986 2022

  35. [35]

    D., Drake, J

    Phan, T. D., Drake, J. F., Larson, D., et al. 2024, ApJL, 971, L42, doi: 10.3847/2041-8213/ad6841 14

    work page doi:10.3847/2041-8213/ad6841 2024

  36. [36]

    2026, ApJ, 997, 313, doi: 10.3847/1538-4357/ae2de6

    Seo, J., Guo, F., Li, X., et al. 2026, ApJ, 997, 313, doi: 10.3847/1538-4357/ae2de6

    work page doi:10.3847/1538-4357/ae2de6 2026

  37. [37]

    2024, The Astrophysical Journal, 977, 146, doi: 10.3847/1538-4357/ad8e64

    Seo, J., Guo, F., Li, X., & Li, H. 2024, The Astrophysical Journal, 977, 146, doi: 10.3847/1538-4357/ad8e64

    work page doi:10.3847/1538-4357/ad8e64 2024

  38. [38]

    K., et al

    Shen, C., Chen, B., Reeves, K. K., et al. 2022, Nature Astronomy, 6, 317, doi: 10.1038/s41550-021-01570-2

    work page doi:10.1038/s41550-021-01570-2 2022

  39. [39]

    2018, ApJ, 869, 116, doi: 10.3847/1538-4357/aaeed3

    Chen, B. 2018, ApJ, 869, 116, doi: 10.3847/1538-4357/aaeed3

    work page doi:10.3847/1538-4357/aaeed3 2018

  40. [40]

    Shen, C., Lin, J., & Murphy, N. A. 2011, ApJ, 737, 14, doi: 10.1088/0004-637X/737/1/14

    work page doi:10.1088/0004-637x/737/1/14 2011

  41. [41]

    Smith, E. J. 2001, J. Geophys. Res., 106, 15819, doi: 10.1029/2000JA000120

    work page doi:10.1029/2000ja000120 2001

  42. [42]

    M., Gardiner, T

    Stone, J. M., Gardiner, T. A., Teuben, P., Hawley, J. F., & Simon, J. B. 2008, ApJS, 178, 137, doi: 10.1086/588755

    work page doi:10.1086/588755 2008

  43. [43]

    M., Tomida, K., White, C

    Stone, J. M., Tomida, K., White, C. J., & Felker, K. G. 2020, ApJS, 249, 4, doi: 10.3847/1538-4365/ab929b

    work page doi:10.3847/1538-4365/ab929b 2020

  44. [44]

    F., & Swisdak, M

    Yin, Z., Drake, J. F., & Swisdak, M. 2024, ApJ, 974, 74, doi: 10.3847/1538-4357/ad7131

    work page doi:10.3847/1538-4357/ad7131 2024

  45. [45]

    P., le Roux, J

    Zank, G. P., le Roux, J. A., Webb, G. M., Dosch, A., & Khabarova, O. 2014, ApJ, 797, 28, doi: 10.1088/0004-637X/797/1/28

    work page doi:10.1088/0004-637x/797/1/28 2014

  46. [46]

    R., & McKibben, R

    Zhang, M., Jokipii, J. R., & McKibben, R. B. 2003, ApJ, 595, 493, doi: 10.1086/377301

    work page doi:10.1086/377301 2003

  47. [47]

    2024, PhRvL, 132, 115201, doi: 10.1103/PhysRevLett.132.115201

    Zhang, Q., Guo, F., Daughton, W., et al. 2024, PhRvL, 132, 115201, doi: 10.1103/PhysRevLett.132.115201

    work page doi:10.1103/physrevlett.132.115201 2024

  48. [48]

    2021, PhRvL, 127, 185101, doi: 10.1103/PhysRevLett.127.185101

    Zhang, Q., Guo, F., Daughton, W., Li, H., & Li, X. 2021, PhRvL, 127, 185101, doi: 10.1103/PhysRevLett.127.185101

    work page doi:10.1103/physrevlett.127.185101 2021

  49. [49]

    V., & Angelopoulos, V

    Zhang, Z., Artemyev, A. V., & Angelopoulos, V. 2025, Phys. Rev. E, 112, 015209, doi: 10.1103/pkzv-k5t3

    work page doi:10.1103/pkzv-k5t3 2025

  50. [50]

    L., Zank, G

    Zhao, L. L., Zank, G. P., Telloni, D., et al. 2022, ApJL, 928, L15, doi: 10.3847/2041-8213/ac5fb0

    work page doi:10.3847/2041-8213/ac5fb0 2022