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arxiv: 2604.22598 · v1 · submitted 2026-04-24 · 🌌 astro-ph.HE

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Synchrotron polarization of anisotropic electron distribution in GRB prompt emission

Kang-Fa Cheng , Kai-Xian Luo , Xiao-Hong Zhao , Jirong Mao , Hong-Bang Liu , Yu-Hang Mo , Jin-Rong Huang , Rong-Li Weng
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Wen-Jie Xie Gao-Jin Yu
Authors on Pith no claims yet

Pith reviewed 2026-05-08 10:24 UTC · model grok-4.3

classification 🌌 astro-ph.HE
keywords gamma-ray burstssynchrotron polarizationanisotropic electronspitch angle distributionmagnetic reconnectionGRB prompt emissionpolarization degreetoroidal magnetic field
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The pith

Anisotropic electron pitch-angle distributions lower synchrotron polarization in GRB gamma-ray and X-ray bands relative to isotropic cases.

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

The paper calculates synchrotron polarization for GRB prompt emission assuming electrons follow an energy-dependent anisotropic pitch-angle distribution at low energies, taken from magnetic reconnection simulations, while high-energy electrons remain isotropic. This produces systematically lower polarization degrees in gamma-ray and X-ray bands than the usual isotropic assumption, with optical polarization either lower or higher depending on the slope parameter m. A sympathetic reader would care because polarization data across bands can distinguish the underlying electron acceleration physics in these explosions, where spectra alone are often ambiguous. The authors compare their results to existing GRB observations and conclude the anisotropic model offers a possible explanation for the polarization and spectral properties of some bursts.

Core claim

Within a globally toroidal magnetic field, the synchrotron polarization degrees produced by anisotropically distributed electrons are systematically lower in the gamma-ray and X-ray bands than those from isotropically distributed electrons. In the optical band the polarization degree can be either lower or higher depending on the value of the energy slope m. Numerical comparisons with data indicate that an anisotropic distribution may explain the polarization and spectral data of some gamma-ray bursts.

What carries the argument

The energy-dependent anisotropic electron pitch-angle distribution, where the mean value of sin squared alpha follows a power-law proportionality to gamma to the power m for Lorentz factors below gamma_iso while remaining isotropic above it, which modifies the synchrotron emissivity and Stokes parameters.

If this is right

  • Polarization degrees in gamma-ray and X-ray bands are reduced compared to the isotropic electron case.
  • Optical polarization can be either reduced or increased relative to isotropic, depending on the value of the slope m.
  • The anisotropic model supplies a possible match to both polarization and spectral observations in some GRBs.
  • Multi-band polarization measurements can be used to constrain the electron pitch-angle distribution in the emission region.

Where Pith is reading between the lines

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

  • If the model holds, it would favor magnetic reconnection over shock acceleration as the dominant process setting the electron distribution in GRB jets.
  • Simultaneous gamma-ray, X-ray, and optical polarimetry of future GRBs could provide a direct test of the predicted band-dependent polarization differences.
  • The same anisotropic prescription might be applied to polarization calculations in other reconnection-powered sources such as pulsar wind nebulae or blazar jets.

Load-bearing premise

The specific anisotropic form for low-energy electrons, with mean sin squared alpha proportional to gamma to the m, that is taken from magnetic reconnection simulations applies directly to the GRB prompt emission region.

What would settle it

Observation of gamma-ray polarization degrees in GRBs that are not systematically lower than isotropic-model predictions, or optical polarization values that contradict the m-dependent range, would contradict the central results.

Figures

Figures reproduced from arXiv: 2604.22598 by Gao-Jin Yu, Hong-Bang Liu, Jin-Rong Huang, Jirong Mao, Kai-Xian Luo, Kang-Fa Cheng, Rong-Li Weng, Wen-Jie Xie, Xiao-Hong Zhao, Yu-Hang Mo.

Figure 1
Figure 1. Figure 1: The curves of logγiso versus m2 with different γmin,α. where F(x) and G(x) are written as (Rybicki & Lightman 1979)    F(x) = x R ∞ x K5/3(ξ)dξ G(x) = xK2/3(x), (7) where K5/3(ξ) and K2/3(x) are Bessel functions, x = ν ′/ν′ c , and ν ′ c = 3qeB′ sin α 4πmec γ 2 . Note that B′ and ν ′ are the MF strength and the frequency in the jet comoving frame. If the electron spectral index is p, the analytical expr… view at source ↗
Figure 2
Figure 2. Figure 2: The flux density spectra (upper panels), the negative spectral index of flux density spectra (middle row panels), and the energy-resolved PDs (bottom panels) with different energy slope of m1. We fix the energy slope of m2 = 0.3 in this figure, and take γmin,α = 10, 100, and 500 in the left, middle, and right column panels, respectively. For comparison, the black solid line corresponds to the case of isotr… view at source ↗
Figure 3
Figure 3. Figure 3: Similar to view at source ↗
Figure 4
Figure 4. Figure 4: The distribution of the PDs as a function of the energy slopes of m1 (bottom panels) and m2 (upper panels) in the γ-ray (400 keV), X-ray (10 keV), and optical (6 × 1014 Hz) bands of GRB prompt emission. We fix m1 = −0.3 in the upper panels and fix m2 = 0.3 in the bottom panels. We take γmin,α = 10, 100, and 500 in the left, middle, and right column panels, respectively. with a PD of 27+11 −11% and was foun… view at source ↗
Figure 5
Figure 5. Figure 5: The distribution of the PDs as a function of normalized viewing angles (q) for various m2 in the γ-ray (the left panels) and X-ray (the bottom panels) bands. We take γmin,α = 10 and 100 in the upper and bottom panels, respectively. The black solid line represents the case of isotropic electron distribution. We fix the energy slope of m1 = −0.3 in this figure. 2. The energy slope of m1 typically affects onl… view at source ↗
Figure 6
Figure 6. Figure 6: The numerical results of the anisotropic electron distribution model compared to the PD and spectral data of GRBs 180914B, 190530A, and 100826A. The top-left panel shows the energy spectrum, and the bottom-left panel presents the corresponding spectral indices. The curves in the top-right panel display the PDs as a function of m2, with data points representing the observed PDs of the bursts. The bottom-rig… view at source ↗
read the original abstract

In gamma-ray bursts (GRBs), the electron pitch angle ($\alpha$) is usually assumed to be isotropically distributed. However, recent numerical simulations indicate that only the high-energy electrons (with Lorentz factors $\gamma>\gamma_{iso}$) are distributed isotropically, whereas the low-energy electrons (with $\gamma<\gamma_{iso}$) follow an energy-dependent anisotropic distribution during magnetic reconnection. The mean value of $\sin^2 \alpha$ approximately follows the relation $\langle \sin^2 \alpha \rangle \propto \gamma^{m}$ for $\gamma<\gamma_{iso}$. In principle, polarization measurements may help us constrain the pitch-angle distribution of electrons in GRBs, since different pitch-angle distributions produce distinct synchrotron polarization signatures. The polarization of GRBs produced by isotropically distributed electrons has been extensively studied. In this paper, we investigate synchrotron polarization produced by anisotropically distributed electrons within a globally toroidal magnetic field in GRB prompt emission. Our results show that the synchrotron PDs in the $\gamma$-ray and X-ray bands produced by anisotropically distributed electrons are systematically lower than those produced by isotropically distributed electrons, while the PD in the optical band could be either lower or higher than that of isotropically distributed electrons, depending primarily on the value of the energy slope $m$. In addition, we compared our numerical results with observational data, and the comparison suggests that an anisotropic distribution of electrons may offer a potential explanation for the PD and spectral data of some GRBs.

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

3 major / 2 minor

Summary. The paper claims that synchrotron polarization degrees (PDs) in GRB prompt emission, computed via numerical integration of the emissivity over an energy-dependent anisotropic electron pitch-angle distribution (⟨sin²α⟩ ∝ γ^m for γ < γ_iso) taken from magnetic reconnection simulations and embedded in a globally toroidal magnetic field, are systematically lower than the isotropic case in the γ-ray and X-ray bands; the optical PD can be lower or higher depending on m. A direct comparison of these numerical PDs with selected GRB observations is presented as suggesting that the anisotropy may help explain both PD and spectral data for some bursts.

Significance. If the imported pitch-angle distribution applies, the work supplies a concrete mechanism for the lower-than-expected PDs often seen in GRB prompt emission and generates an m-dependent prediction for the optical band that is in principle falsifiable with multi-wavelength polarimetry. The direct numerical integration over the assumed distribution is reproducible and avoids analytic approximations that might hide the anisotropy effects.

major comments (3)
  1. [Introduction / model setup] The functional form ⟨sin²α⟩ ∝ γ^m (for γ < γ_iso) is imported directly from reconnection simulations and inserted into the GRB prompt-emission calculation without derivation from GRB-relevant processes (shock acceleration, Weibel instability, or prompt-region turbulence) or robustness tests against other plausible anisotropies; this assumption is load-bearing for every reported PD offset relative to the isotropic case.
  2. [Results / data comparison] The comparison with observational PD and spectral data is described only as 'suggestive' and lacks quantitative fit statistics, error budgets, or explicit parameter choices for the GRB sample; without these it is impossible to judge whether the anisotropic model provides a statistically meaningful improvement over the isotropic baseline.
  3. [Numerical method / field geometry] All calculations assume a globally toroidal magnetic field geometry; the systematic PD reduction in the γ-ray/X-ray bands is not shown to survive under other plausible field configurations (e.g., tangled or radially dominated) that are also discussed in the GRB literature.
minor comments (2)
  1. The notation for the anisotropic distribution and the precise definition of γ_iso could be stated more explicitly in the equations to facilitate reproduction.
  2. A brief statement of the numerical integration method (quadrature scheme, energy and angle grids) would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us improve the clarity and scope of the manuscript. We address each major comment point by point below, indicating revisions where made.

read point-by-point responses

  1. Referee: [Introduction / model setup] The functional form ⟨sin²α⟩ ∝ γ^m (for γ < γ_iso) is imported directly from reconnection simulations and inserted into the GRB prompt-emission calculation without derivation from GRB-relevant processes (shock acceleration, Weibel instability, or prompt-region turbulence) or robustness tests against other plausible anisotropies; this assumption is load-bearing for every reported PD offset relative to the isotropic case.

    Authors: We acknowledge that the adopted functional form for the pitch-angle anisotropy is taken directly from magnetic reconnection simulations without a first-principles derivation tailored to GRB prompt-emission regions. Our study focuses on the observational implications of this distribution as reported in the simulation literature. We have added a dedicated paragraph in the Introduction discussing the potential applicability to GRB shocks (including references to related acceleration mechanisms) and explicitly noting the assumption's limitations. Robustness tests against alternative anisotropy prescriptions are beyond the scope of the present work but are highlighted as an important avenue for future investigation. revision: partial

  2. Referee: [Results / data comparison] The comparison with observational PD and spectral data is described only as 'suggestive' and lacks quantitative fit statistics, error budgets, or explicit parameter choices for the GRB sample; without these it is impossible to judge whether the anisotropic model provides a statistically meaningful improvement over the isotropic baseline.

    Authors: We agree that the data comparison remains qualitative and is appropriately labeled 'suggestive'. The model depends on multiple parameters (including m, γ_iso, and the bulk Lorentz factor), which precludes a straightforward statistical fit without additional priors. We have revised the relevant section to provide more explicit documentation of the parameter values adopted for each burst in the sample and to include a brief discussion of associated uncertainties. A full quantitative analysis (e.g., χ² minimization or Bayesian model comparison) would require an extensive parameter survey that we consider outside the scope of this initial exploration. revision: partial

  3. Referee: [Numerical method / field geometry] All calculations assume a globally toroidal magnetic field geometry; the systematic PD reduction in the γ-ray/X-ray bands is not shown to survive under other plausible field configurations (e.g., tangled or radially dominated) that are also discussed in the GRB literature.

    Authors: The globally toroidal field is chosen because it permits high polarization degrees in the isotropic-electron case, thereby isolating the effect of the pitch-angle anisotropy. We recognize that other geometries (tangled or radial) are also discussed in the GRB literature and would generally suppress overall PDs. We have added a short paragraph in the Conclusions noting that the reported anisotropy-induced reduction is most relevant for ordered-field configurations and that the interplay with disordered fields remains to be quantified in future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; results follow from numerical integration over imported distribution

full rationale

The central results are obtained by direct numerical integration of the synchrotron emissivity and polarization over the assumed anisotropic pitch-angle distribution ⟨sin²α⟩ ∝ γ^m (γ < γ_iso) taken from external reconnection simulations, inserted into a toroidal B-field geometry. This produces PD values in γ-ray, X-ray, and optical bands that are outputs of the integration, not equivalent by construction to the input distribution or to any fitted GRB data. The comparison with observational data is a post-hoc suggestion of possible explanation and does not involve parameter fitting that would render the predictions tautological. No self-citation load-bearing steps, self-definitional relations, or ansatzes smuggled via citation are present in the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the functional form of anisotropy taken from external simulations and the assumption of a globally toroidal field; m acts as a tunable parameter whose value controls the optical-band outcome.

free parameters (1)
  • m
    Energy slope in ⟨sin²α⟩ ∝ γ^m; its value determines whether optical PD is lower or higher than the isotropic case and is varied to explore outcomes.
axioms (2)
  • domain assumption Low-energy electrons follow an energy-dependent anisotropic pitch-angle distribution with ⟨sin²α⟩ ∝ γ^m for γ < γ_iso.
    Directly adopted from recent numerical simulations of magnetic reconnection.
  • domain assumption The magnetic field in the GRB prompt emission region is globally toroidal.
    Standard assumption for ordered field geometry in the emitting region.

pith-pipeline@v0.9.0 · 5612 in / 1477 out tokens · 75365 ms · 2026-05-08T10:24:46.504061+00:00 · methodology

discussion (0)

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Works this paper leans on

45 extracted references · 43 canonical work pages

  1. [1]

    1993, ApJ, 413, 281, doi: 10.1086/172995

    Band, D., Matteson, J., Ford, L., et al. 1993, ApJ, 413, 281, doi: 10.1086/172995

    work page doi:10.1086/172995 1993

  2. [2]

    Hard X-Ray Polarization Catalog for a Five-year Sample of Gamma-Ray Bursts Using AstroSat CZT Imager

    Chattopadhyay, T., Gupta, S., Iyyani, S., et al. 2022, ApJ, 936, 12, doi: 10.3847/1538-4357/ac82ef 12Cheng et al

    work page doi:10.3847/1538-4357/ac82ef 2022

  3. [3]

    2025, ApJ, 985, 112, doi: 10.3847/1538-4357/add1d7 —

    Cheng, K., Mao, J., Zhao, X., et al. 2025, ApJ, 985, 112, doi: 10.3847/1538-4357/add1d7 —. 2024a, ApJ, 975, 277, doi: 10.3847/1538-4357/ad8234

    work page doi:10.3847/1538-4357/add1d7 2025

  4. [4]

    2024b, A&A, 687, A128, doi: 10.1051/0004-6361/202348050

    Cheng, K., Zhao, X., Mao, J., & Chen, Z. 2024b, A&A, 687, A128, doi: 10.1051/0004-6361/202348050

    work page doi:10.1051/0004-6361/202348050

  5. [5]

    , keywords =

    Cheng, K. F., Zhao, X. H., & Bai, J. M. 2020, MNRAS, 498, 3492, doi: 10.1093/mnras/staa2595

    work page doi:10.1093/mnras/staa2595 2020

  6. [6]

    2024, ApJ, 972, 9, doi: 10.3847/1538-4357/ad51fe

    Comisso, L. 2024, ApJ, 972, 9, doi: 10.3847/1538-4357/ad51fe

    work page doi:10.3847/1538-4357/ad51fe 2024

  7. [7]

    2023, ApJ, 959, 137, doi: 10.3847/1538-4357/ad1241

    Comisso, L., & Jiang, B. 2023, ApJ, 959, 137, doi: 10.3847/1538-4357/ad1241

    work page doi:10.3847/1538-4357/ad1241 2023

  8. [8]

    2019, ApJ, 886, 122, doi: 10.3847/1538-4357/ab4c33 —

    Comisso, L., & Sironi, L. 2019, ApJ, 886, 122, doi: 10.3847/1538-4357/ab4c33 —. 2021, PhRvL, 127, 255102, doi: 10.1103/PhysRevLett.127.255102 —. 2022, ApJL, 936, L27, doi: 10.3847/2041-8213/ac8422

    work page doi:10.3847/1538-4357/ab4c33 2019

  9. [9]

    2020, ApJL, 895, L40, doi: 10.3847/2041-8213/ab93dc Di Gesu, L., Donnarumma, I., Tavecchio, F., et al

    Comisso, L., Sobacchi, E., & Sironi, L. 2020, ApJL, 895, L40, doi: 10.3847/2041-8213/ab93dc Di Gesu, L., Donnarumma, I., Tavecchio, F., et al. 2022, ApJL, 938, L7, doi: 10.3847/2041-8213/ac913a

    work page doi:10.3847/2041-8213/ab93dc 2020

  10. [10]

    Eichler, D., Livio, M., Piran, T., & Schramm, D. N. 1989, Nature, 340, 126, doi: 10.1038/340126a0

    work page doi:10.1038/340126a0 1989

  11. [11]

    Detection of X-Ray Polarization from the Blazar 1ES 1959+650 with the Imaging X-Ray Polarimetry Explorer

    Errando, M., Liodakis, I., Marscher, A. P., et al. 2024, ApJ, 963, 5, doi: 10.3847/1538-4357/ad1ce4

    work page doi:10.3847/1538-4357/ad1ce4 2024

  12. [12]

    The Astrophysical Journal960(1), 87 (2024) https://doi.org/10.3847/1538-4357/acfc43

    Feng, Z.-K., Liu, H.-B., Xie, F., et al. 2024, ApJ, 960, 87, doi: 10.3847/1538-4357/acfc43

    work page doi:10.3847/1538-4357/acfc43 2024

  13. [13]

    R., Pols, O

    Ghisellini, G., Celotti, A., & Lazzati, D. 2000, MNRAS, 313, L1, doi: 10.1046/j.1365-8711.2000.03354.x

    work page doi:10.1046/j.1365-8711.2000.03354.x 2000

  14. [14]

    2021, MNRAS, 504, 1939, doi: 10.1093/mnras/stab1013

    Gill, R., & Granot, J. 2021, MNRAS, 504, 1939, doi: 10.1093/mnras/stab1013 —. 2024a, MNRAS, 527, 12178, doi: 10.1093/mnras/stad3991 —. 2024b, MNRAS, 527, 12178, doi: 10.1093/mnras/stad3991

    work page doi:10.1093/mnras/stab1013 2021

  15. [15]

    , keywords =

    Gill, R., Granot, J., & Kumar, P. 2020, MNRAS, 491, 3343, doi: 10.1093/mnras/stz2976

    work page doi:10.1093/mnras/stz2976 2020

  16. [16]

    2026, MNRAS, doi: 10.1093/mnras/stag452

    Gill, R., He, J., Granot, J., et al. 2026, MNRAS, doi: 10.1093/mnras/stag452

    work page doi:10.1093/mnras/stag452 2026

  17. [17]

    2022, ApJ, 933, 18, doi: 10.3847/1538-4357/ac67d5

    Goto, R., & Asano, K. 2022, ApJ, 933, 18, doi: 10.3847/1538-4357/ac67d5

    work page doi:10.3847/1538-4357/ac67d5 2022

  18. [18]

    , keywords =

    Granot, J. 2003, ApJL, 596, L17, doi: 10.1086/379110

    work page doi:10.1086/379110 2003

  19. [19]

    , keywords =

    Granot, J., & K¨ onigl, A. 2003, ApJL, 594, L83, doi: 10.1086/378733

    work page doi:10.1086/378733 2003

  20. [20]

    Granot, J., & Taylor, G. B. 2005, ApJ, 625, 263, doi: 10.1086/429536

    work page doi:10.1086/429536 2005

  21. [21]

    D., Briggs, M

    Kaneko, Y., Preece, R. D., Briggs, M. S., et al. 2006, ApJS, 166, 298, doi: 10.1086/505911

    work page doi:10.1086/505911 2006

  22. [22]

    2024, arXiv e-prints, arXiv:2406.05783, doi: 10.48550/arXiv.2406.05783

    Kole, M., de Angelis, N., Bacelj, A., et al. 2024, arXiv e-prints, arXiv:2406.05783, doi: 10.48550/arXiv.2406.05783

    work page doi:10.48550/arxiv.2406.05783 2024

  23. [23]

    M., Liodakis , I., Middei , R., et al

    Kouch, P. M., Liodakis, I., Middei, R., et al. 2024, A&A, 689, A119, doi: 10.1051/0004-6361/202449166

    work page doi:10.1051/0004-6361/202449166 2024

  24. [24]

    The Physics of Gamma-Ray Bursts and Relativistic Jets

    Kumar, P., & Zhang, B. 2015, PhR, 561, 1, doi: 10.1016/j.physrep.2014.09.008

    work page Pith review doi:10.1016/j.physrep.2014.09.008 2015

  25. [25]

    2019, ApJ, 870, 96, doi: 10.3847/1538-4357/aaf41d

    Lan, M.-X., Geng, J.-J., Wu, X.-F., & Dai, Z.-G. 2019, ApJ, 870, 96, doi: 10.3847/1538-4357/aaf41d

    work page doi:10.3847/1538-4357/aaf41d 2019

  26. [26]

    2021, ApJ, 909, 184, doi: 10.3847/1538-4357/abe3fb

    Lan, M.-X., Wang, H.-B., Xu, S., Liu, S., & Wu, X.-F. 2021, ApJ, 909, 184, doi: 10.3847/1538-4357/abe3fb

    work page doi:10.3847/1538-4357/abe3fb 2021

  27. [27]

    2025, ApJ, 988, 202, doi: 10.3847/1538-4357/ade4c4 M´ esz´ aros, P

    Mai, Q., Liu, H., Tuo, J., et al. 2025, ApJ, 988, 202, doi: 10.3847/1538-4357/ade4c4 M´ esz´ aros, P. 2006, Reports on Progress in Physics, 69, 2259, doi: 10.1088/0034-4885/69/8/R01

    work page doi:10.3847/1538-4357/ade4c4 2025

  28. [28]

    2003, JCAP, 2003, 005, doi: 10.1088/1475-7516/2003/10/005

    Nakar, E., Piran, T., & Waxman, E. 2003, JCAP, 2003, 005, doi: 10.1088/1475-7516/2003/10/005

    work page doi:10.1088/1475-7516/2003/10/005 2003

  29. [29]

    2004, Reviews of Modern Physics, 76, 1143, doi: 10.1103/RevModPhys.76.1143

    Piran, T. 2004, Reviews of Modern Physics, 76, 1143, doi: 10.1103/RevModPhys.76.1143

    work page doi:10.1103/revmodphys.76.1143 2004

  30. [30]

    D., Briggs, M

    Preece, R. D., Briggs, M. S., Mallozzi, R. S., et al. 2000, ApJS, 126, 19, doi: 10.1086/313289

    work page doi:10.1086/313289 2000

  31. [31]

    , keywords =

    Rossi, E. M., Lazzati, D., Salmonson, J. D., & Ghisellini, G. 2004, MNRAS, 354, 86, doi: 10.1111/j.1365-2966.2004.08165.x

    work page doi:10.1111/j.1365-2966.2004.08165.x 2004

  32. [32]

    B., & Lightman, A

    Rybicki, G. B., & Lightman, A. P. 1979, Radiative processes in astrophysics

    1979

  33. [33]

    1998, ApJL, 497, L17, doi: 10.1086/311269

    Sari, R., Piran, T., & Narayan, R. 1998, ApJL, 497, L17, doi: 10.1086/311269

    work page doi:10.1086/311269 1998

  34. [34]

    Sazonov, V. N. 1969, Soviet Ast., 13, 396

    1969

  35. [35]

    2009, ApJ, 698, 1042, doi: 10.1088/0004-637X/698/2/1042

    Toma, K., Sakamoto, T., Zhang, B., et al. 2009, ApJ, 698, 1042, doi: 10.1088/0004-637X/698/2/1042

    work page doi:10.1088/0004-637x/698/2/1042 2009

  36. [36]

    L., & Zhang, B

    Uhm, Z. L., & Zhang, B. 2014, Nature Physics, 10, 351, doi: 10.1038/nphys2932

    work page doi:10.1038/nphys2932 2014

  37. [37]

    2025, ApJ, 994, 194, doi: 10.3847/1538-4357/ae12ee

    Wang, X.-Y., Li, J.-S., & Lan, M.-X. 2025, ApJ, 994, 194, doi: 10.3847/1538-4357/ae12ee

    work page doi:10.3847/1538-4357/ae12ee 2025

  38. [38]

    Woosley, S. E. 1993, ApJ, 405, 273, doi: 10.1086/172359

    work page doi:10.1086/172359 1993

  39. [39]

    2018, ApJL, 864, L16, doi: 10.3847/2041-8213/aada4f

    Yang, Y.-P., & Zhang, B. 2018, ApJL, 864, L16, doi: 10.3847/2041-8213/aada4f

    work page doi:10.3847/2041-8213/aada4f 2018

  40. [40]

    2011, ApJL, 743, L30, doi: 10.1088/2041-8205/743/2/L30

    Yonetoku, D., Murakami, T., Gunji, S., et al. 2011, ApJL, 743, L30, doi: 10.1088/2041-8205/743/2/L30

    work page doi:10.1088/2041-8205/743/2/l30 2011

  41. [41]

    2002, ApJ, 571, 876, doi: 10.1086/339981

    Zhang, B., & M´ esz´ aros, P. 2002, ApJ, 571, 876, doi: 10.1086/339981

    work page doi:10.1086/339981 2002

  42. [42]

    2019, Science China Physics, Mechanics, and Astronomy, 62, 29502, doi: 10.1007/s11433-018-9309-2

    Zhang, S., Santangelo, A., Feroci, M., et al. 2019, Science China Physics, Mechanics, and Astronomy, 62, 29502, doi: 10.1007/s11433-018-9309-2

    work page doi:10.1007/s11433-018-9309-2 2019

  43. [43]

    2025, Science China Physics, Mechanics, and Astronomy, 68, 119502, doi: 10.1007/s11433-025-2786-6

    Zhang, S.-N., Santangelo, A., Xu, Y., et al. 2025, Science China Physics, Mechanics, and Astronomy, 68, 119502, doi: 10.1007/s11433-025-2786-6

    work page doi:10.1007/s11433-025-2786-6 2025

  44. [44]

    2014, ApJ, 780, 12, doi: 10.1088/0004-637X/780/1/12

    Zhao, X., Li, Z., Liu, X., et al. 2014, ApJ, 780, 12, doi: 10.1088/0004-637X/780/1/12

    work page doi:10.1088/0004-637x/780/1/12 2014

  45. [45]

    2025, ApJ, 986, 32, doi: 10.3847/1538-4357/add0ac

    Zhong, Q., Liu, H.-B., Cheng, K., et al. 2025, ApJ, 986, 32, doi: 10.3847/1538-4357/add0ac

    work page doi:10.3847/1538-4357/add0ac 2025