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Global MHD simulations reach the asymptotic monster-shock regime and confirm that on the equator the peak Lorentz factor scales as sigma_x c/(omega R_x), while a wrinkled magnetosphere fragments the shock front, lowers the effective magneti

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2026-08-02 21:06 UTC pith:OLZIKUOJ

load-bearing objection First global MHD study to reach the asymptotic monster-shock regime, confirming the equatorial scaling and adding genuinely new off-equator and fragmentation results; the single-fluid MHD validity margin is thinner than one would like, but the paper is honest about it and deserves full refereeing.

arxiv 2602.21290 v3 pith:OLZIKUOJ submitted 2026-02-24 astro-ph.HE physics.plasm-ph

Global Magnetohydrodynamic Simulations of Monster Shocks in Neutron Star Magnetospheres

Michael P. Grehan , Bart Ripperda , Andrei M. Beloborodov , Christopher Thompson , Elias R. Most This is my paper
classification astro-ph.HE physics.plasm-ph
keywords magnetarsmonster shocksrelativistic MHDfast magnetosonic wavesneutron star magnetospheresshock fragmentationplasma astrophysicshigh-energy transients
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims that a global, two-dimensional relativistic MHD simulation can follow a fast magnetosonic wave launched in a neutron-star dipole magnetosphere all the way into the asymptotic "monster shock" regime, and that on the equator the peak upstream Lorentz factor follows the analytic scaling Gamma = sigma_x c/(omega R_x), with fitted slope near unity. It claims that off the equator the shock becomes oblique and disappears at finite latitude, matching the predicted drift-velocity pattern, and that when the background dipole is wrinkled by additional modes, the shock front fragments, the effective magnetization drops to roughly (B_d/B')^2, and secondary shocks appear intermittently along a line of sight. A reader should care because monster shocks are a leading candidate for powering magnetar X-ray bursts and possibly fast radio bursts, and this is the first global MHD study to reach the high magnetizations needed to test the theory.

Core claim

In ideal single-fluid MHD, a small-amplitude fast wave launched from the star grows relative to the dipole background roughly as r^2, and where E^2 approaches B^2 inertial effects turn it into an ultra-relativistic shock. The paper's central result is that global 2D MHD reproduces the analytic scaling Gamma = sigma_x c/(omega R_x) on the equator, with fitted slope near unity and a nonzero intercept, and confirms the approximate radial decay of the upstream Lorentz factor. Off the equator the shock becomes oblique and vanishes at finite latitude, consistent with the drift-velocity profile predicted analytically. In a cylindrical test problem the same mechanism reaches sigma_x = 5e4 and still

What carries the argument

The mechanism is the fast magnetosonic wave's relative growth in a dipole background: the wave field decays as 1/r while the dipole decays as 1/r^3, so the ratio grows as r^2 until E^2 -> B^2, at which point the force-free description fails and inertial plasma flows form an ultra-relativistic shock. The analytic identity Gamma = sigma_x c/(omega R_x) is the load-bearing object being tested; the simulations verify it by fitting Gamma_max against magnetization, amplitude, and frequency. A second element is a harmonic wrinkle perturbation of the vector potential (Equation 8), which provides zones of constructive and destructive interference that fragment the shock and introduce secondary maxima

Load-bearing premise

The single-fluid MHD description assumes the electron-positron plasma stays magnetized enough that the ideal jump conditions hold; the paper's own validity condition is satisfied only by a factor of about 2.5 for fiducial magnetar parameters, so if pair loading or kinetic precursors alter the jump conditions, the confirmed scalings would shift.

What would settle it

A particle-in-cell simulation in a dipolar background with sigma_x = 100 and the wave parameters of the paper's Table 1: if the fitted slope of Gamma_max versus c sigma_x/(omega R_x) departs clearly from unity, or if shocks appear at latitudes where the analytic drift profile predicts none, the single-fluid claim is falsified.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The equatorial scaling Gamma = sigma_x c/(omega R_x) is confirmed in global MHD for the first time, with fitted slope near unity across variations of magnetization, amplitude, and frequency.
  • Off the equator, the shock weakens and vanishes at finite latitude, so the geometry of emission from a monster shock depends strongly on viewing angle and the drift funnel near the equator.
  • A wrinkled background with comparable-amplitude modes fragments the shock front and reduces the effective magnetization to sigma_eff ~ (B_d/B')^2, implying that pre-existing turbulence changes the shock's dissipative power.
  • Secondary shocks can appear intermittently along a given line of sight, which would produce time-variable, multiple-peaked emission from a single event.
  • The same mechanism operates in a cylindrical geometry with sigma_x up to 5e4, indicating the scaling is not an artifact of the dipole setup and may apply to winds of rapidly rotating compact objects.

Where Pith is reading between the lines

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

  • If the fragmentation seen here is generic in real magnetar magnetospheres, a single outburst could produce several closely spaced sub-bursts in X-ray or radio light curves rather than one smooth flash; this is an observable consequence not spelled out in the paper.
  • The effective magnetization reduction sigma_eff ~ (B_d/B')^2 suggests that pre-existing wrinkles could suppress the efficiency of maser precursor radio emission by making the shock dissipate at lower Lorentz factor; a kinetic simulation with a wrinkled background would test this directly.
  • The single-fluid validity condition is satisfied only marginally for fiducial magnetar parameters, so pair loading or kinetic precursor emission could shift the absolute value of Gamma even if the linear scaling remains intact; higher-multiplicity regimes deserve separate study.
  • The paper treats only axisymmetric harmonic wrinkles; if non-axisymmetric Alfven waves are added in 3D, the fragmentation pattern and the intermittency along a line of sight could be substantially different.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. This paper presents axisymmetric 2D ideal-GRMHD simulations (using BHAC) of ultra-relativistic magnetized 'monster shocks' in a magnetar magnetosphere. Three setups are considered: (i) a spherical fast magnetosonic (FMS) wave launched into a dipolar field; (ii) FMS generation by collision of Alfvén waves launched through localized surface twists; and (iii) FMS propagation through a 'wrinkled' dipole with a harmonic standing perturbation. The central quantitative claim is that, on the equator, the peak upstream Lorentz factor follows Γ_max ∝ σ_x c/(ω R_x) with a fitted slope consistent with the analytic prediction of Beloborodov (2023), and that off the equator the shock becomes oblique and disappears at finite latitude according to the drift-velocity profile. In the wrinkled background, the shock front fragments, the effective magnetization drops to σ_eff ∼ (B_d/B')^2, and secondary shocks can appear along a line of sight. The paper also presents a quasi-1D cylindrical testbed reaching σ_x = 5×10^4 and an Alfvén-to-FMS conversion study.

Significance. The paper makes a substantial contribution. It appears to be the first global MHD demonstration of the asymptotic monster-shock regime in a dipolar field, with convergence tests (Fig. 15) and a control run without the FMS wave (WigNoFMS) that support the numerical results. The cylindrical suite at σ_x up to 5×10^4 provides a strong, independently checkable test of the analytic scaling in a different geometry, and the explicit derivation in Appendix E is a useful extension of B23. The oblique-shock analysis and the fragmentation/secondary-shock phenomenology are new and observationally relevant for burst light curves and precursor emission. The authors are transparent about the model's limitations: they state that pair production, cooling, and kinetic precursor emission are neglected, and they quote the single-fluid MHD validity condition (Sec. 5). If the confirmed scalings hold, this will be a valuable reference for interpreting kinetic simulations and future observations. The main caveat is that the single-fluid MHD condition is only marginally satisfied for fiducial magnetar parameters (factor ∼2.5), but this is a limitation of the physical model rather than an internal inconsist

minor comments (6)
  1. [Sec. 3, Fig. 2 and Appendix D] The confirmation of Eq. (9) relies on linear fits with a free y-intercept of about 1.2–1.5, while the analytic prediction has zero intercept. For the lowest magnetization σ_x=25, the intercept is roughly 45% of the predicted Γ_max. The text calls this a 'small constant offset,' which understates the effect at the edge of the simulated range. I recommend explicitly stating that what is confirmed is the scaling slope, not the absolute normalization of Eq. (9), and noting that the dipole-case offset is not separately diagnosed as in Appendix E.
  2. [Abstract; Sec. 5, Eqs. (18)–(19)] The abstract states that monster shocks 'are described by relativistic magnetohydrodynamics (MHD).' This is stronger than the paper's own validity condition, which is satisfied only by a factor of ∼2.5 for fiducial parameters. Since the stress-test concern about pair-physics corrections is real, I suggest qualifying the abstract and conclusion with 'approximately' or 'in the regime 1/Γ_u ≫ 2(ω/ω_x)^{1/2},' as already implied by Sec. 5.
  3. [Eq. (2)] The notation E_w is used both for the vector electric field at the surface and for its scalar amplitude. Please disambiguate, e.g., E_w(t) φ̂ and E_w0.
  4. [Sec. 3.1, Fig. 6] The term 'explosive configuration' is used without definition. Please clarify that it refers to the transition from inflow-dominated to outflow-dominated internal shocks in the star frame.
  5. [Appendix B, Fig. 14] The numerical diffusion coefficient D is set to 10 for magnetization-varied runs and 5 for amplitude/frequency-varied runs. Since Fig. 14 shows that D affects the peak Lorentz factor and the y-intercept, a one-sentence justification in the main text (or a note in Table 1) would help the reader assess the systematic uncertainty in the fitted offsets.
  6. [Sec. 4.2, Eq. (17)] The secondary-shock condition k'_r E' > 2ω_F E0/c is derived under the assumption of vanishing B'_θ and a purely radial wrinkle wavenumber. This is a heuristic estimate; please state these restrictions more explicitly and note that the full invariant contains additional terms that may be of the same order for a generic wrinkle.

Circularity Check

0 steps flagged

No circular reduction: the simulations independently test B23 analytic scalings with free fits, and the wrinkle phenomenology is new simulation output.

full rationale

The paper's derivation chain is to construct ideal-MHD initial conditions with specified wave amplitude, frequency, and magnetization profile; evolve them with BHAC; and then measure the equatorial upstream Lorentz factor, drift-velocity pattern, and shock geometry, comparing with analytic predictions from B23 (Eqs. 9, 10, 12-15). B23 is a published, parameter-free analytic theory with stated assumptions; it does not take the simulation results as inputs. The main claimed confirmation is not obtained by forcing the predicted slope: Appendix D and Fig. 16 report linear fits with the slope left free, and the fitted slopes match the predicted values (11.93 vs 11.25 for amplitude variation, 22.93 vs 22.36 for frequency variation, 0.070 vs 0.071 for magnetization variation). Thus the scaling Gamma_max ∝ c sigma_x/(omega R_x) is measured, not imposed. The off-equator drift profile and shock disappearance angle are compared to the independent formulas Eqs. (12)-(13). The cylindrical testbed (Appendix E) re-derives the B23 analysis in a different geometry and reports a slope that does not fully agree with its own analytic prediction (0.2 vs 0.125), showing the comparison is falsifiable rather than constructed. The wrinkled-background results (Sec. 4.2) are new simulation outcomes interpreted with a local perturbation argument (Eqs. 16-17), not predictions manufactured from fitted values. The only load-bearing caveat is the validity of single-fluid ideal MHD, assessed in Sec. 5 via a B23 criterion and satisfied only marginally (Eqs. 18-19); this is a physical-assumption risk about pair loading and kinetic effects, not a circular reduction of the simulation results to their inputs. Self-citations to B23 are central but independent: the analytic predictions predate the simulations and are externally falsifiable, so they do not constitute load-bearing self-citation under the stated rules.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The central claims rest on standard relativistic MHD plus three paper-specific choices: (1) the ideal single-fluid description with a stated validity condition that is only marginally satisfied for fiducial magnetar parameters; (2) an assumed outer density profile sigma_bg ~ (R_x/r)^3; (3) an axisymmetric harmonic wrinkle representation. No new physical entities are introduced. The main free parameters are numerical diffusion D, fitted intercepts in the scaling confirmation, and the wrinkle/twist amplitude-wavenumber scans that control the new phenomenology. The B23 analytics are treated as input benchmarks rather than re-derived here.

free parameters (4)
  • TVDLF extra diffusion coefficient D = 5 or 10 (default 1)
    Chosen by hand for numerical stability at sigma_x = 100 (Appendix B). Figure 14 shows D = 10 systematically lowers the peak Lorentz factor, and the paper states this 'reduces the y-intercept' of the Gamma-sigma_x relation, i.e., it directly shapes the fitted offset used in the confirmation.
  • Y-intercept of Gamma_max(c sigma_x / (omega R_x)) linear fits = +1.225 to +1.488 (spherical); +2.55 (cylindrical)
    Free intercepts in the scaling fits (Fig. 2 right; Fig. 16) absorb numerical diffusion and pre-acceleration; the abstract's 'confirm analytical predictions' is calibrated against the slope after removing this fitted offset.
  • Wrinkle amplitude a~ and wavenumber m = a~ = 0.3-0.5; m = 32 pi - 64 pi
    Scanned by hand (Table 3). The claimed fragmentation, secondary-shock, and Lorentz-factor-enhancement effects occur only in this 'comparable amplitude' window; results are not mapped outside it.
  • Surface twist parameters (kappa, Delta, theta_0, omega_0, omega_A) = kappa = 50, Delta = 0.05 pi, theta_0 = pi/4; omega_0 and omega_A varied
    Initial conditions of Eq. (6) chosen by hand (Table 2). The claim that larger amplitude and lower frequency twisting gives stronger shocks is stated qualitatively ('Exact scaling relationships ... require further investigation').
axioms (5)
  • domain assumption Ideal single-fluid MHD describes the monster shock, valid when 1/Gamma_u >> 2(omega/omega_x)^1/2 (B23)
    Stated in Section 5 with fiducial estimates (Eqs. 18-19) giving only a factor ~2.5 margin (5e-5 vs 2e-5). Pair production, cooling, and kinetic precursor emission are neglected by design (stated in Sec. 1 and Sec. 5); if these alter jump conditions the central scalings shift.
  • domain assumption External magnetization profile sigma_bg = (R_x/r)^3 for r > R_x (Eq. 5)
    Assumed density shape used in the forward-shock radius predictions (Eqs. 14-15). Only one uniform-profile run (mw3opp) tests the alternative formula; the profile is an initial-condition choice, not constrained by data.
  • domain assumption The wrinkle perturbation (Eq. 8) represents realistic magnetospheric perturbations
    Motivated by force-free simulations (Burnaz et al. 2025) and crust-quake models (Qu & Bransgrove 2026), but the specific axisymmetric harmonic standing-wave form with omega' ~ m/(pi r) is chosen for this paper; non-axisymmetric Alfven waves are excluded (Sec. 2.3).
  • standard math B23 analytical predictions for equatorial and forward shocks (Eqs. 9, 10, 12-15)
    Used as the benchmarks the simulations are compared against. B23 is a published parameter-free analytic derivation by a co-author; the present paper treats it as unproved background and tests it numerically, but the agreement statistics (fitted intercepts) are part of this paper's choices.
  • domain assumption Zero-temperature, zero-velocity background with density floors applied wherever E < 0.7B or T < 1e-3
    Initial and flooring conditions in Sec. 2 and Appendix B; the floors and entropy switch discard ~0.4% of total energy (stated), which is small but not propagated through the scaling fits.

pith-pipeline@v1.3.0-alltime-deepseek · 22481 in / 21979 out tokens · 199701 ms · 2026-08-02T21:06:12.709355+00:00 · methodology

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Waves launched from the neutron star surface or inner magnetosphere propagate through the magnetosphere as small perturbations, but can grow relative to the background magnetic field and steepen into ``monster shocks'' -- ultra-relativistic magnetized shocks which can power high-energy emission. Such shocks can develop around isolated magnetars, merging binaries, and collapsing neutron stars. They occur in magnetically dominated plasma and are described by relativistic magnetohydrodynamics (MHD). We present global relativistic MHD simulations of monster shocks in unperturbed and perturbed (``wrinkled'') backgrounds with a global dipolar geometry. Our simulations confirm analytical predictions for equatorial shocks and provide new insight into the behavior of oblique shocks off the equator. Simulations where the shock is formed through Alfv\'{e}n mode to fast mode conversion are also presented, demonstrating the generic nature of the monster shock mechanism. We explore how the presence of additional modes in the magnetosphere modifies the shock behavior. Modes of comparable amplitude can fragment the shock front, substantially reduce the magnetization, produce localized enhancements in the Lorentz factor relative to an unperturbed dipole background, and intermittently generate additional shocks along a line of sight.

discussion (0)

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Reference graph

Works this paper leans on

37 extracted references · 7 canonical work pages · cited by 3 Pith papers

  1. [1]

    1988, Physics of Fluids, 31, 839, doi: 10.1063/1.866765

    Alsop, D., & Arons, J. 1988, Physics of Fluids, 31, 839, doi: 10.1063/1.866765

    doi:10.1063/1.866765 1988

  2. [2]

    C., et al

    Andersen, B. C., et al. 2020, Nature, 587, 54, doi: 10.1038/s41586-020-2863-y

    doi:10.1038/s41586-020-2863-y 2020

  3. [3]

    Arnowitt, R., Deser, S., & Misner, C. W. 1962, in in Gravitation: An Introduction to Current Research (Chap. 7). Edited by Louis Witten. John Wiley & Sons Inc, 227

    1962

  4. [4]

    Beloborodov, A. M. 2023, ApJ, 959, 34, doi: 10.3847/1538-4357/acf659 —. 2024, ApJ, 975, 223, doi: 10.3847/1538-4357/ad698c

    doi:10.3847/1538-4357/acf659 2023

  5. [5]

    M., & Thompson, C

    Beloborodov, A. M., & Thompson, C. 2007, ApJ, 657, 967, doi: 10.1086/508917

    doi:10.1086/508917 2007

  6. [6]

    Bernardi, D., Yuan, Y., & Chen, A. Y. 2025a, ApJ, 980, 222, doi: 10.3847/1538-4357/adabe5 —. 2025b, PhRvL, 135, 265201, doi: 10.1103/y9p7-1zms

    doi:10.3847/1538-4357/adabe5

  7. [7]

    D., Ravi, V., Belov, K

    Bochenek, C. D., Ravi, V., Belov, K. V., et al. 2020, Nature, 587, 59, doi: 10.1038/s41586-020-2872-x

    doi:10.1038/s41586-020-2872-x 2020

  8. [8]

    R., & Bransgrove, A

    Burnaz, L., Most, E. R., & Bransgrove, A. 2025, arXiv e-prints. https://arxiv.org/abs/2508.18033

    Pith/arXiv arXiv 2025

  9. [9]

    Y., Yuan, Y., Li, X., & Mahlmann, J

    Chen, A. Y., Yuan, Y., Li, X., & Mahlmann, J. F. 2022, arXiv e-prints, arXiv:2210.13506, doi: 10.48550/arXiv.2210.13506

    Pith/arXiv arXiv doi:10.48550/arxiv.2210.13506 2022

  10. [10]

    C., & Thompson, C

    Duncan, R. C., & Thompson, C. 1992, ApJL, 392, L9, doi: 10.1086/186413

    doi:10.1086/186413 1992

  11. [11]

    2023, ApJ, 957, 102, doi: 10.3847/1538-4357/acfa78

    Golbraikh, E., & Lyubarsky, Y. 2023, ApJ, 957, 102, doi: 10.3847/1538-4357/acfa78

    doi:10.3847/1538-4357/acfa78 2023

  12. [12]

    P., Ghosal, T., Beattie, J

    Grehan, M. P., Ghosal, T., Beattie, J. R., et al. 2025, PhRvD, 112, 063046, doi: 10.1103/8xf2-x2nq

    work page doi:10.1103/8xf2-x2nq 2025

  13. [13]

    1991, Physics of Fluids B, 3, 818, doi: 10.1063/1.859877

    Hoshino, M., & Arons, J. 1991, Physics of Fluids B, 3, 818, doi: 10.1063/1.859877

    work page doi:10.1063/1.859877 1991

  14. [14]

    M., & Beloborodov, A

    Kaspi, V. M., & Beloborodov, A. M. 2017, ARA&A, 55, 261, doi: 10.1146/annurev-astro-081915-023329

    doi:10.1146/annurev-astro-081915-023329 2017

  15. [15]

    V., & Ciolfi, R

    Kastaun, W., Kalinani, J. V., & Ciolfi, R. 2021, PhRvD, 103, 023018, doi: 10.1103/PhysRevD.103.023018

    doi:10.1103/physrevd.103.023018 2021

  16. [16]

    R., Beloborodov, A

    Kim, Y., Most, E. R., Beloborodov, A. M., & Ripperda, B. 2025, ApJL, 982, L54, doi: 10.3847/2041-8213/adbff9

    work page doi:10.3847/2041-8213/adbff9 2025

  17. [17]

    L., Thompson, C., & Hanna, C

    Lehner, L., Palenzuela, C., Liebling, S. L., Thompson, C., & Hanna, C. 2012, PhRvD, 86, 104035, doi: 10.1103/PhysRevD.86.104035

    work page doi:10.1103/physrevd.86.104035 2012

  18. [18]

    F., Aloy, M

    Mahlmann, J. F., Aloy, M. Á., & Li, X. 2024, ApJ, 972, 139, doi: 10.3847/1538-4357/ad60c4

    doi:10.3847/1538-4357/ad60c4 2024

  19. [19]

    F., & Beloborodov, A

    Mahlmann, J. F., & Beloborodov, A. M. 2025, ApJL, 981, L17, doi: 10.3847/2041-8213/adb5fd

    doi:10.3847/2041-8213/adb5fd 2025

  20. [20]

    2020, ApJL, 898, L29, doi: 10.3847/2041-8213/aba2cf

    Mereghetti, S., Savchenko, V., Ferrigno, C., et al. 2020, ApJL, 898, L29, doi: 10.3847/2041-8213/aba2cf

    doi:10.3847/2041-8213/aba2cf 2020

  21. [21]

    R., Beloborodov, A

    Most, E. R., Beloborodov, A. M., & Ripperda, B. 2024, ApJL, 974, L12, doi: 10.3847/2041-8213/ad7e1f

    work page doi:10.3847/2041-8213/ad7e1f 2024

  22. [22]

    2006, ApJ, 641, 626, doi: 10.1086/500349

    Zanna, L. 2006, ApJ, 641, 626, doi: 10.1086/500349

    doi:10.1086/500349 2006

  23. [23]

    C., Krolik, J

    Noble, S. C., Krolik, J. H., & Hawley, J. F. 2009, ApJ, 692, 411, doi: 10.1088/0004-637X/692/1/411

    doi:10.1088/0004-637x/692/1/411 2009

  24. [24]

    2019, A&A, 629, A61, doi: 10.1051/0004-6361/201935559

    Olivares, H., Porth, O., Davelaar, J., et al. 2019, A&A, 629, A61, doi: 10.1051/0004-6361/201935559

    doi:10.1051/0004-6361/201935559 2019

  25. [25]

    M., & Hui, L

    Parfrey, K., Beloborodov, A. M., & Hui, L. 2013, ApJ, 774, 92, doi: 10.1088/0004-637X/774/2/92

    doi:10.1088/0004-637x/774/2/92 2013

  26. [26]

    2017, Computational Astrophysics and Cosmology, 4, 1, doi: 10.1186/s40668-017-0020-2

    Porth, O., Olivares, H., Mizuno, Y., et al. 2017, Computational Astrophysics and Cosmology, 4, 1, doi: 10.1186/s40668-017-0020-2

    doi:10.1186/s40668-017-0020-2 2017

  27. [27]

    2026, ApJ, 998, 190, doi: 10.3847/1538-4357/ae3a9d

    Qu, Y., & Bransgrove, A. 2026, ApJ, 998, 190, doi: 10.3847/1538-4357/ae3a9d

    doi:10.3847/1538-4357/ae3a9d 2026

  28. [28]

    2019, Astrophys

    Ripperda, B., Bacchini, F., Porth, O., et al. 2019, Astrophys. J. Suppl., 244, 10, doi: 10.3847/1538-4365/ab3922

    work page doi:10.3847/1538-4365/ab3922 2019

  29. [29]

    M., Mösta, P., Desai, D., & Wu, S

    Siegel, D. M., Mösta, P., Desai, D., & Wu, S. 2018, ApJ, 859, 71, doi: 10.3847/1538-4357/aabcc5

    doi:10.3847/1538-4357/aabcc5 2018

  30. [30]

    Sironi, L., Plotnikov, I., Nättilä, J., & Beloborodov, A. M. 2021, PhRvL, 127, 035101, doi: 10.1103/PhysRevLett.127.035101 Sądowski, A., Narayan, R., Tchekhovskoy, A., & Zhu, Y. 2013, MNRAS, 429, 3533, doi: 10.1093/mnras/sts632

    doi:10.1103/physrevlett.127.035101 2021

  31. [31]

    M., Ripperda, B., Chernoglazov, A., et al

    TenBarge, J. M., Ripperda, B., Chernoglazov, A., et al. 2021, Journal of Plasma Physics, 87, 905870614, doi: 10.1017/S002237782100115X

    doi:10.1017/s002237782100115x 2021

  32. [32]

    Thompson, C., Lyutikov, M., & Kulkarni, S. R. 2002, Astrophys. J., 574, 332, doi: 10.1086/340586

    doi:10.1086/340586 2002

  33. [33]

    2004, PhRvD, 70, 124030, doi: 10.1103/PhysRevD.70.124030

    Troischt, P., & Thompson, C. 2004, PhRvD, 70, 124030, doi: 10.1103/PhysRevD.70.124030

    work page doi:10.1103/physrevd.70.124030 2004

  34. [34]

    2025, PhRvL, 134, 035201, doi: 10.1103/PhysRevLett.134.035201

    Vanthieghem, A., & Levinson, A. 2025, PhRvL, 134, 035201, doi: 10.1103/PhysRevLett.134.035201

    doi:10.1103/physrevlett.134.035201 2025

  35. [35]

    2026, arXiv e-prints, arXiv:2602.10895, doi: 10.48550/arXiv.2602.10895

    Wang, C.-W., Xiong, S.-L., Wang, Y., et al. 2026, arXiv e-prints, arXiv:2602.10895, doi: 10.48550/arXiv.2602.10895

    doi:10.48550/arxiv.2602.10895 2026

  36. [36]

    M., Chen, A

    Yuan, Y., Beloborodov, A. M., Chen, A. Y., et al. 2022, ApJ, 933, 174, doi: 10.3847/1538-4357/ac7529

    doi:10.3847/1538-4357/ac7529 2022

  37. [37]

    2021, ApJ, 908, 176, doi: 10.3847/1538-4357/abd405

    Yuan, Y., Levin, Y., Bransgrove, A., & Philippov, A. 2021, ApJ, 908, 176, doi: 10.3847/1538-4357/abd405

    doi:10.3847/1538-4357/abd405 2021