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REVIEW 5 major objections 4 minor 92 references

Signature of Triaxially Precessing Magnetars in Gamma-ray Burst X-Ray Afterglows

T0 review · 5 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Gamma-ray burst X-ray plateaus and their quasi-periodic oscillations can be produced by a triaxially precessing newborn millisecond magnetar, and the model fits four observed bursts.

desk verdict Solid formalism and a nice Gold-sample consistency check, but the Silver/Bronze evidence rests on an unexamined instantaneous-tracking assumption that external-shock physics likely violates. read the letter →

arxiv 2411.15883 v1 pith:Z7NRY5YX submitted 2024-11-24 astro-ph.HE astro-ph.SR

Biao Zhang , Shu-Qing Zhong , Long Li , Zi-Gao Dai This is my paper · ORCID
classification astro-ph.HEastro-ph.SR PACS 98.70.Rz97.60.Gb
keywords gamma-rayburstsmagnetarstriaxialfreeprecessionX-rayafterglowsquasi-periodicoscillationsplateausmagneticdipoleradiationneutronstarcentralengines
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that newborn millisecond magnetars are triaxially deformed and freely precessing, so their magnetic inclination angle varies periodically and imprints quasi-periodic oscillations on gamma-ray burst X-ray afterglow plateaus. The authors identify four bursts from Swift/XRT data with regular flux variations on their plateaus and show that a magnetic-dipole radiation model of a triaxially precessing magnetar reproduces the light curves, including three observed QPO periods of 157 s, 650 s, and 246 s. For the two 'Gold' bursts with internal plateaus, the predicted collapse times of 431 s and 3549 s match the observed plateau ends of 435 s and 3570 s. If the claim holds, these afterglow oscillations are direct evidence that newborn millisecond magnetars precess during the first hours after the burst.

What carries the argument

The central machinery is the closed-form solution for triaxial free precession of a rigid neutron star: the unit angular-momentum components in the body frame are Jacobi elliptic functions, $\hat{L}_1 = \sin\theta_0\,\mathrm{cn}(\Omega_P t, m)$, $\hat{L}_2 = \sin\theta_0\sqrt{1+\delta}\,\mathrm{sn}(\Omega_P t, m)$, and $\hat{L}_3 = \cos\theta_0\,\mathrm{dn}(\Omega_P t, m)$, with precession frequency $\Omega_P$ and triaxiality parameter $m = \delta\tan^2\theta_0$. These motions change the magnetic inclination angle $\alpha$ between the dipole moment and the angular-momentum axis, and the dipole luminosity is modulated as $L_{\rm iso,X}(t) \propto (1+t/((1+z)\tau_{\rm sd}))^{-2}\,[1+k\sin^2\alpha]$, where the factor $1+k\sin^2\alpha$ comes from plasma-filled magnetosphere simulations. A second piece of machinery is the collapse-time formula $T_{\rm col}$ for a supra-massive magnetar, which links the best-fit mass, magnetic field, and spin period to the observed end of an internal plateau. The argument works because the precession frequency stays nearly constant before the spin-down timescale $\tau_{\rm sd}$, so a clean period appears in the plateau window.

What would settle it

A forward-shock simulation that takes a periodically modulated energy-injection rate with a period near 200 s and asks whether the oscillation survives in the external-plateau phase; if the shock smooths the variation, the Silver (GRB 050730) and Bronze (GRB 210610A) evidence would disappear. A second check would be a new internal-plateau burst whose fitted parameters predict a collapse time that disagrees with the observed plateau end.

Watch

Extended reading notes

Core claim

The paper establishes that the X-ray afterglow plateaus and the periodic flux variations on them can be produced by the magnetic dipole radiation of a triaxially freely precessing magnetar whose spin-down is electromagnetic. Treating the neutron star as a rigid triaxial body and solving the Euler equations with Jacobi elliptic functions, the magnetic inclination angle $\alpha$ oscillates as the angular-momentum axis precesses and nutates in the body frame, modulating the dipole luminosity through the magnetospheric factor $\lambda(\alpha) = 1 + k\sin^2\alpha$. The model fits the Swift/XRT light curves of GRB 060202, GRB 180620A, GRB 050730, and GRB 210610A, reproducing the regular flux variations and, for the first three, the QPO periods (model 153 s, 621 s, 239 s versus observed 157 s, $650 \pm 50$ s, 246 s). For the two Gold bursts, the best-fit magnetar masses, fields, and spin periods predict collapse times of 431 s and 3549 s, consistent with the observed internal-plateau ends of 435 s and 3570 s.

Load-bearing premise

For the two external-plateau bursts, the model assumes the observed X-ray luminosity tracks the instantaneous magnetar-wind luminosity with no smoothing by the external forward shock, so that roughly 200 s flux variations survive into the light curve.

Editorial extensions

If this is right

  • If the model is correct, the Gold bursts (GRB 060202 and GRB 180620A) are direct evidence that newborn millisecond magnetars precess within the first few thousand seconds after formation.
  • The consistency between predicted and observed collapse times supports the standard picture that a supra-massive magnetar collapses into a black hole at the end of an internal plateau.
  • The fitted ellipticities, of order $10^{-6}$ to $10^{-5}$, give concrete targets for gravitational-wave searches for newborn magnetars and for neutron-star equations of state.
  • Adding two long GRBs with QPOs to the previously reported short-GRB sample strengthens the statistical case that precessing magnetars power a substantial fraction of GRB plateaus.
  • The Bronze burst shows that even without a detectable periodicity, the same precession mechanism can leave regular flux variations on an external plateau, widening the predicted observational signature.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A straightforward test the paper does not perform: simulate the external forward shock's response to a periodically modulated injection luminosity; if oscillations of about 200 s are smeared by the shock, the Silver and Bronze classifications lose their force.
  • The triaxial model predicts that the QPO period should drift after the spin-down timescale as the magnetar spins down; searching for such a frequency chirp in long-lived plateaus would distinguish triaxial precession from a strictly biaxial wobble.
  • If precession is generic in newborn magnetars, stacking many plateau light curves should reveal low-amplitude oscillations at a characteristic period set by the ellipticity distribution, a population-level signature testable with current X-ray archives.
  • The same varying inclination angle that modulates X-ray luminosity should also modulate the beam geometry, so prompt gamma-ray emission or early radio and optical afterglows might show correlated periodicity; the paper does not explore these windows.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper searches Swift/XRT afterglows for GRBs whose X-ray plateaus show regular flux variations, identifies four bursts (GRB 060202, GRB 180620A, GRB 050730, GRB 210610A), and classifies them as Gold, Silver, and Bronze samples according to whether the post-plateau decay is steep and whether a QPO is detected. It constructs a model of magnetic dipole radiation from a triaxially freely precessing magnetar whose spin-down is electromagnetically dominated, fits the four light curves with ten free parameters per burst using MCMC, and reports good agreement between observed and model QPO periods for the three periodic cases. For the two Gold bursts it also derives collapse times from the fitted mass, magnetic field, and spin period and compares them with the observed plateau-end times.

Significance. If the central claim holds, the paper would add four long GRBs to the small sample supporting newborn-magnetar precession, and it would be the first application of a triaxial rather than biaxial precession model to GRB afterglow plateaus. The paper has real strengths: Appendix B gives a complete and explicit proof that the two standard solution forms for triaxial free precession are equivalent; the reported period ratios (153/157, 621/650, 239/246 s) are close; and the collapse-time agreement for the two Gold bursts is a nontrivial internal consistency check. However, the evidential value of the Silver and Bronze samples depends on an unverified instantaneous-luminosity assumption for external-shock emission, and the paper does not establish that the ten-parameter precessing model is preferred over simpler alternatives. The Gold-sample result may survive these concerns, but the broader claim about all four bursts currently needs substantial additional work.

major comments (5)
  1. [§3.2.3 and Eq. (4)] The two external-plateau bursts (GRB 050730 and GRB 210610A) are fitted with Eq. (4), which sets the observed X-ray luminosity proportional to the instantaneous magnetar wind luminosity. The paper states in §3.3.1 that external plateaus are powered by the magnetar injecting rotational energy into the external forward shock. Forward-shock X-ray emission at observer time t receives contributions from electrons energized over a window whose width is comparable to the shock age because of equal-arrival-time and angular-spreading effects, so the shock acts as a low-pass filter on the wind luminosity. For GRB 050730 the claimed 246 s period is compared with data in the interval (187, 790) s, where the smoothing window is comparable to the period; a proper convolution may erase the oscillation. The authors should either compute the convolved external-shock light curve and refit, or explicitly restrict the model claim to the two Gold bursts, where internal dissipation of the wind makes Eq. (4) more plausible.
  2. [§3.2 and Table 2] The model has ten free parameters per burst, and no comparison is made with a biaxially precessing magnetar or with a non-precessing smooth plateau model. Because the precession period is controlled by freely fitted parameters (ϵ2, ϵ3, θ0 and the spin period P0), while the overall normalization and decay are controlled by ηX, Bp, P0 and M, the reported period agreements are expectations from fitting rather than independent predictions. The paper should report AIC/BIC or a likelihood-ratio test against the smooth broken power law already fitted in Eq. (34) and against a biaxial precession model with fewer parameters. Without such a model-selection step, the abstract's phrase 'provides further evidence' is not supported by the statistical analysis presented.
  3. [§3.3.1 and Eq. (37)] The collapse-time checks for GRB 060202 and GRB 180620A are internal consistency tests, not independent predictions: the values of M, Bp and P0 entering Eq. (37) are the same parameters fitted to the pre-break light curve, and the observed collapse time is the plateau-end time tb from the smooth broken power law. The comparison is also very sensitive to the GM1 EOS parameters and to the fitted mass, which sits close to MTOV for GRB 060202 (M ≈ 2.41 M⊙). The authors should propagate the posterior uncertainties of M, Bp and P0 into a posterior distribution for Tcol, and should test at least one other EOS, to show that the agreement is not a consequence of the chosen EOS and the freedom in M.
  4. [§3.3.2 and Eq. (7)] The spin evolution assumed in Eq. (7) is inconsistent with the precession-modulated torque in Eq. (2): since λ(α) depends on precession phase, Ω̇ is modulated, so the exact solution is not Ω0[1 + t/((1+z)τsd)]^(−1/2) with a constant τsd. For GRB 180620A, the fitted parameters give τsd ≈ 9.4×10^2 s while the plateau extends to tb/(1+z) ≈ 3.57×10^3 s, contrary to the text's claim in §3.3.2 that for the Gold samples the collapse time is less than τsd and that Ω and ΩP remain nearly constant during the plateau. The resulting period drift should be included in the model, or the paper should quantify why it is negligible for the fitted time windows.
  5. [§3.1 and Table 1] The sample is selected by visual inspection from Swift/XRT data spanning May 2005 to November 2023, and the QPO significances are reported as LSP false-alarm probabilities that do not account for the effective number of bursts inspected or the number of trial periods scanned. The quoted 0.01% FAP levels are therefore overconfident, especially for the new detections in GRB 060202 and GRB 050730. The authors should provide a trial-corrected significance estimate or an injection/recovery analysis. This is not fatal for the Gold-sample evidence, but it affects the strength of the claim for the newly identified QPOs.
minor comments (4)
  1. [§2.1] There is a typo in 'The obeserved isotropic X-ray Luminosity' immediately before Eq. (8); it should read 'observed'.
  2. [§3.2] The text says the magnetar mass is set as a free parameter for the Silver and Bronze samples, but for the Gold samples it is also a free parameter; the distinction is not about whether M is free but about whether M is required to be below MTOV. This could be stated more clearly.
  3. [§3.3.2] The discussion describes the spin-frequency evolution as 'a broken power law as described by equation (7)', but Eq. (7) is a single power-law-like expression with no break; the language should be made consistent.
  4. [§2.1, Eq. (8)] The K-correction exponent in Eq. (8) should be defined explicitly; the text does not state the convention for the X-ray spectral index Γ used in the luminosity-flux conversion.

Circularity Check

2 steps flagged · score 6.0 of 10

QPO periods and Gold collapse times are in-sample consistency checks: both are derived from the same free parameters (epsilon2, epsilon3, theta0, P0 or M, Bp, P0) that were MCMC-fitted to the very light curves whose QPO/break they are said to reproduce.

  1. fitted input called prediction [Section 3.2 and Sections 3.2.1-3.2.3; Eqs. (17), (25), (26); Table 2]
    "The free parameters in our model include the mass of the magnetar M, the radiation efficiency ηX, the surface polar cap magnetic field Bp, the initial spin period P0, the ellipticity ϵ2, the parameter ξ related to ϵ3, the initial wobble angle θ0, the azimuthal angle of the dipole moment η, the polar angle of the dipole moment χ, and the parameter k. ... The PDS of the model points in the time interval (312, 2305) s peaks at 621 s. Our model's period of 621 s is in good agreement with the observed period of 650 ± 50 s reported by Zou & Liang (2022)."

    The model's precession period is fixed by ϵ2, ϵ3, θ0, and P0 through Eqs. (17), (25), and (26), but these are exactly the free parameters MCMC-fitted to the same XRT plateau points that carry the QPO. Sampling the fitted periodic curve and taking its LSP peak is an in-sample operation: a periodic function fitted to a periodic signal is expected to peak near the signal frequency. The reported agreements (153 vs 157 s, 621 vs 650 s, 239 vs 246 s) therefore do not constitute an out-of-sample prediction of the period; no external benchmark (e.g., a pre-fit period, an independent band, or a period derived from the collapse time) is used.

  2. fitted input called prediction [Section 3.3.1; Eq. (37); Table 2]
    "By substituting the values of M, Bp, and P0 (see Table 2) obtained from fitting the X-ray afterglow light curves into equation (37), the theoretical collapse time for the supra-massive magnetars in the Gold samples GRB 060202 and GRB 180620A can be derived. For GRB 060202, the theoretical collapse time Tcol = 431 s is consistent with the observed collapse time tcol = 435 s. Similarly, for GRB 180620A, the theoretical collapse time Tcol = 3549 s is also consistent with the observed collapse time tcol = 3570 s."

    Eq. (37) takes M, Bp, and P0 from the MCMC fit to the same afterglow whose break time is being matched, and the paper states 'the magnetar mass is set as a free parameter' for the Gold samples. Moreover, the fit is restricted to data ending at the observed break (e.g., 'from 312 s to 7853 s, corresponding to ... the time of the break'), so tcol is used to delimit the fitted interval. With at least one free parameter (M) per burst and the formula's strong sensitivity to M-MTOV, Tcol ≈ tcol is a consistency check of the fitted parameter combination rather than a first-principles prediction; it is not tested against an independent mass or EOS measurement outside the fitted values.

full rationale

Most of the modeling chain is mathematically self-contained and not circular: the triaxial free-precession solution, Eqs. (25)-(29), is explicitly proved equivalent to the Landau-Lifshitz solution in Appendix B, and the dipole radiation formula is a standard parametrization with λ(α) = 1 + k sin^2 α. Citations to Zou & Liang (2022), Suvorov & Kokkotas (2020/2021), and Lasky et al. (2014) are not by the present authors and are used as ordinary literature support; there is no load-bearing self-citation chain and no uniqueness theorem imported from the authors' own prior work. The circularity that is present is narrower: the headline quantitative successes (QPO periods and Gold collapse times) are computed from free parameters fitted to the same data whose periodicities and breaks they are said to reproduce. That makes them in-sample consistency checks rather than independent predictions; the model's flexibility (about ten free parameters for four light curves) is what permits the agreement. A further non-circular but important physical risk is Eq. (4), which identifies the observed X-ray luminosity with the instantaneous wind luminosity for the external-plateau bursts even though the paper states those plateaus are powered by external-shock injection; a forward shock would smooth the injected luminosity on timescales comparable to the shock age, potentially erasing the claimed 246 s QPO and the Bronze variations. For those two bursts the evidential weight is therefore reduced. Overall the central claim is not fully forced by definition, but the claimed confirmatory period and collapse-time matches are partly circular because they are generated by the fitted parameters themselves. Score 6.

Assumptions & free parameters 10 free parameters · 8 assumptions · 0 invented entities

The model is a phenomenological fit: ten parameters per burst are fitted to each light curve, and the two quoted consistency checks, precession period and collapse time, are computed from those same fitted parameters rather than from external calibration. Several physical assumptions, such as undamped free precession and direct luminosity tracking for external plateaus, are adopted without quantitative justification.

free parameters (10)
  • M (magnetar mass) = GRB 060202: 2.4083 M_sun; GRB 180620A: 2.3845 M_sun; GRB 050730: 1.5475 M_sun; GRB 210610A: 1.4654 M_sun
    Controls the moment of inertia and the collapse time through Eq. (37); fitted by MCMC.
  • ηX (X-ray radiation efficiency) = GRB 060202: 2.795e-3; GRB 180620A: 0.407e-3; GRB 050730: 0.1706; GRB 210610A: 0.0493
    Normalizes the conversion of magnetar luminosity to observed X-ray luminosity in Eq. (4).
  • Bp (surface polar cap magnetic field) = GRB 060202: 2.027e15 G; GRB 180620A: 1.106e15 G; GRB 050730: 8.489e15 G; GRB 210610A: 8.747e15 G
    Sets the dipole luminosity scale and spin-down timescale in Eqs. (5) and (6).
  • P0 (initial spin period) = GRB 060202: 1.353 ms; GRB 180620A: 1.844 ms; GRB 050730: 3.976 ms; GRB 210610A: 6.007 ms
    Determines the initial rotation energy and the base precession frequency through Ω0.
  • ϵ2 (triaxial ellipticity parameter) = GRB 060202: 8.712e-6; GRB 180620A: 1.538e-6; GRB 050730: 18.124e-6; GRB 210610A: 21.539e-6
    One of the two deformation parameters controlling the free-precession solution and period.
  • ξ (excess of ϵ3 over ϵ2) = GRB 060202: 7.472e-6; GRB 180620A: 2.894e-6; GRB 050730: 16.510e-6; GRB 210610A: 25.818e-6
    With ϵ3 = ϵ2 + ξ, this sets the second deformation parameter and the triaxiality measure δ.
  • θ0 (initial wobble angle) = GRB 060202: 0.252 rad; GRB 180620A: 0.458 rad; GRB 050730: 0.500 rad; GRB 210610A: 0.536 rad
    Controls the amplitude of precession and the amplitude of the flux variations.
  • η (azimuthal angle of the dipole moment) = GRB 060202: 1.956 rad; GRB 180620A: 0.440 rad; GRB 050730: 0.004 rad; GRB 210610A: 0.162 rad
    Phase parameter in Eq. (30) that, with χ, sets the time-dependent inclination angle α.
  • χ (polar angle of the dipole moment) = GRB 060202: 0.514 rad; GRB 180620A: 0.550 rad; GRB 050730: 0.811 rad; GRB 210610A: 0.741 rad
    Polar angle of the dipole moment in the body frame, one of the angles determining α.
  • k (magnetospheric factor amplitude) = GRB 060202: 0.652; GRB 180620A: 0.821; GRB 050730: -0.635; GRB 210610A: -0.423
    Amplitude in the magnetospheric factor λ = 1 + k sin²α; fitted freely within |k| ≤ 1.
assumptions (8)
  • standard math Euler equations for a rigid triaxial body describe the magnetar's free precession.
    Section 2.2, Eq. (9); standard classical mechanics for torque-free rigid-body motion.
  • standard math The Jacobi elliptic function solution and the Landau-Lifshitz parametrization are valid.
    Section 2.2, Eqs. (12)-(16); Appendix B proves equivalence with the Gao et al. (2023) form.
  • domain assumption The magnetospheric factor is λ = 1 + k sin²α with |k| ≤ 1.
    Eq. (3), adopted from Spitkovsky (2006) and Suvorov & Kokkotas (2020); a hybrid magnetospheric model, not derived here.
  • domain assumption Spin-down is dominated by electromagnetic dipole radiation, with gravitational-wave spin-down negligible.
    Section 4 checks LGW/LEM << 1 using fitted parameters, but this is not imposed as a prior in the fit.
  • domain assumption The angular velocity vector is approximately parallel to the angular momentum vector when computing α.
    Appendix C justifies this for the small fitted ellipticities, but it is an approximation used in Eq. (33).
  • domain assumption For external-plateau bursts, the observed X-ray luminosity directly follows the instantaneous magnetar dipole luminosity without external-shock smoothing.
    Eq. (4) is applied to GRB 050730 and GRB 210610A in Section 3.2 despite their classification as external plateaus, where energy injection into an external shock is the standard picture.
  • domain assumption Free precession is not damped over the hundreds-of-seconds plateaus.
    No damping timescale, such as internal dissipation or superfluid friction, is included; the model assumes several coherent precession cycles.
  • domain assumption The GM1 equation of state with MTOV = 2.37 M_sun, α = 1.58e-10 s^-β, and β = -2.84 governs the collapse time.
    Section 3.3.1, Eq. (37), citing Lasky et al. (2014) and Lü et al. (2015); an EOS assumption inherited from the literature.

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Pith. "Pith review of Signature of Triaxially Precessing Magnetars in Gamma-ray Burst X-Ray Afterglows." pith.science (2026). https://pith.science/paper/Z7NRY5YX

@misc{pith2026241115883,
  author       = {Pith},
  title        = {Pith review of: Signature of Triaxially Precessing Magnetars in Gamma-ray Burst X-Ray Afterglows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z7NRY5YX}},
  note         = {Machine review of arXiv:2411.15883}
}
read the original abstract

The X-ray afterglows of some gamma-ray bursts (GRBs) exhibit plateaus, which can be explained by the internal dissipation of a newborn millisecond magnetar wind. In the early phase of these newborn magnetars, the magnetic inclination angle undergoes periodic changes due to precession, leading to periodic modulation of the injection luminosity due to magnetic dipole radiation. This may result in quasi-periodic oscillations (QPOs) on the plateaus. In this paper, we identify four GRBs with regular flux variations on their X-ray afterglow plateaus from Swift/XRT data before November 2023, three of which exhibit periodicity. Based on the likelihood of supporting a precessing magnetar as the central engine, we classify them into three categories: Gold (GRB 060202 and GRB 180620A), Silver (GRB 050730), and Bronze (GRB 210610A). We invoke a model of magnetic dipole radiation emitted by a triaxially freely precessing magnetar whose spin-down is dominated by electromagnetic radiation, to fit the light curves. Our model successfully reproduces the light curves of these four GRBs, including the regular flux variations on the plateaus and their periodicity (if present). Our work provides further evidence for early precession in newborn millisecond magnetars in GRBs.

Figures

Figures reproduced from arXiv: 2411.15883 by the authors.

Figure 1
Figure 1. The geometry of NS precession in the body frame. The unit angular momentum Lˆ precesses around eˆ3, and the angle θ between them undergoes nutation during precession. The unit dipole moment µˆ remains constant in the body frame, with its polar angle and azimuthal angle denoted as χ and η, respectively. The angle between the unit angular momentum Lˆ and the unit dipole moment µˆ is represented by α. The obeserved iso… view at source ↗
Figure 2
Figure 2. The X-ray light curves of GRBs in our sample. GRB 060202 and GRB 180620A are the Gold samples, GRB 050730 is the Silver sample, and GRB 210610A is the Bronze sample. The black data points represent the XRT data of the afterglow, and the red solid curves are the smooth broken power-law model fitted to the XRT data. 2. Silver Sample: this category comprises bursts with a normal decay phase following the plateau, indic… view at source ↗
Figure 3
Figure 3. Left panel: the sampled theoretical model points (red dots) prior to the break time obtained from the best-fitting model light curve in comparison with the XRT afterglow data (black dots) of GRB 180620A. Right panel: the PDS (red curve) of the theoretical sampled points before 2305 s, obtained from LSP. The vertical red solid lines represent the peak of the PDS, corresponding to P = 621 s. The vertical black solid l… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Left panel: the sampled points (red dots) from the best-fit model curve during the plateau phase of GRB 060202, and the XRT data (black dots) in time interval (149, 1399) s. Right panel: the PDSs of the observed points (black curve) and the sampled model points (red cu…
Figure 5
Figure 5. Figure 5: Left panel: the sampled X-ray lightcurve (red dots) derived from our best model fit and the XRT data (black dots) of GRB 050730. Right panel: same as the right panel of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Left panel: sampled points (red dots) from the fitted curve and observed afterglow (black dots) of GRB 210610A. Right panel: zoomed-in view of the plateau phase (data before 858 seconds), with the note that the horizontal axis is linear. parameter ξ related to ϵ3 obtai…
Figure 7
Figure 7. Figure 7: The corner plot of the posterior probability distribution of model parameters obtained by MCMC fitting the X-ray afterglow of GRB 180620A with our model. The best-fitting parameters and the corresponding 1σ uncertainties are depicted in the diagonal histograms with bla…
Figure 8
Figure 8. Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Same as [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Same as [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

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