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REVIEW 4 major objections 6 minor 154 references

Constrain magnetar parameters by taking into account the evolutionary effects of radius and moment of inertia with \emph{Swift}/XRT data

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Neglecting how a neutron star's radius and inertia shrink as it spins down biases gamma-ray-burst magnetar parameters by 20–50%.

desk verdict Careful and transparent, but the central 'R/I' claim collapses to I-only under the paper's own mu-conservation assumption; the numbers may survive a rewrite, the framing should not. read the letter →

arxiv 2507.11110 v2 pith:I5JWPE53 submitted 2025-07-15 astro-ph.HE

Lin Lan , He Gao , Shunke Ai , Wen-Jin Xie , Yong Yuan , Long Li , Li-Ping Xin , Jian-Yan Wei This is my paper · ORCID
classification astro-ph.HE
keywords gamma-rayburstsmagnetarsX-rayplateausneutronstarradiusevolutionmomentofinertiaequationstategravitationalwavesSwift/XRT
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 tries to establish that the radius and moment of inertia of a newborn millisecond magnetar change appreciably as it spins down, and that ignoring this evolution has systematically distorted every previous estimate of magnetar parameters derived from gamma-ray-burst X-ray plateaus. It re-analyzes 105 long GRBs observed by Swift/XRT before 2023 December, fitting each plateau with a spin-down model that lets radius and moment of inertia evolve according to neutron-star equations of state. The central result is that constant-radius, constant-inertia fits overestimate the surface dipole field by roughly a factor of 1.5, underestimate the initial spin period by roughly a factor of 0.7, and leave the ellipticity nearly unchanged, with case-dependent biases of 20% to 50%. If true, previously published magnetar parameters for plateau GRBs need revision, and the new corrected sample reveals universal relations connecting spin, field, and ellipticity to the energy of the GRB jet.

What carries the argument

The load-bearing object is the piecewise power-law ansatz for neutron-star radius and moment of inertia during spin-down, $R\simeq R_0(\Omega/\Omega_k)^m$ and $I\simeq I_0(\Omega/\Omega_k)^k$ for $\Omega_1<\Omega<\Omega_k$, with the indices, critical velocities, and reference values imported from general-relativistic rotating-star models. Combined with the assumption that the magnetic dipole moment $\mu\equiv B_p R^3$ is conserved, this converts the standard dipole-plus-gravitational-wave spin-down law into a modified law whose plateau luminosity decays with new temporal slopes controlled by the index $k$. This machinery is what turns the observed plateau break time and luminosity into corrected values of $B_p$, $P_0$, and $\epsilon$, and it is what generates the new light-curve segments seen in the model fits.

What would settle it

Simulate plateau light curves with a constant radius and moment of inertia, fit them with the evolving R/I model, and check whether the recovered B_p, P_0, and ϵ show the claimed systematic 20–50% offsets; if the recovery is unbiased, the reported bias is an artifact of the model choice rather than a real evolutionary effect.

Watch

Extended reading notes

Core claim

The paper argues that the radius R and moment of inertia I of a newborn millisecond magnetar shrink appreciably as it spins down, and that analyses which take R and I as fixed constants misestimate the dipole field B_p, initial spin period P_0, and ellipticity ϵ inferred from GRB X-ray plateaus. Averaging over four equations of state, two baryon masses, and two radiative efficiencies, constant-R/I fits overestimate log B_p by roughly 0.17 dex (a factor near 1.5), underestimate P_0 by roughly 0.26 ms (a factor near 0.7), and leave ϵ essentially unchanged; individual scenarios show systematic biases of 20% to 50%. The paper also reports sample-wide power-law correlations, $\epsilon\propto P_0^{1.57\pm0.22}$, $\epsilon\propto B_p^{0.97\pm0.13}$, and $B_p\propto P_0^{1.30\pm0.16}$, together with anticorrelations of $P_0$, $B_p$, and $\epsilon$ with jet energy, and interprets these as indicating that the ellipticity originates from magnetically induced distortion while the observed spin period is an equilibrium period set by magnetar–disk interaction rather than the birth spin.

Load-bearing premise

The entire size and even the sign of the claimed bias rests on the assumed piecewise power-law curves for radius and moment of inertia versus spin rate, and on the assumption that the magnetic dipole moment $B_pR^{3}$ stays fixed while the neutron star spins down.

Editorial extensions

If this is right

  • All previously published magnetar parameters for plateau GRBs that assumed constant R and I should be revised, with systematically lower B_p and higher P_0.
  • The reported universal correlations connect the central engine's spin, field, and ellipticity to jet energetics: faster-spinning, lower-field, less-deformed magnetars tend to power more energetic jets.
  • The inferred ellipticity-scale with both B_p and P_0 implies that gravitational-wave emission is significant only for the fastest-spinning remnants, sharpening the target list for aLIGO and the Einstein Telescope.
  • On the current sample, magnetar dipole radiation would be detected by EP/WXT in about 30% of cases, by EP/FXT in about 81%, and by SVOM/MXT in about 40%.
  • Gravitational waves from the GW-dominated remnants with measured redshifts are below advanced-LIGO sensitivity, and only GRBs 150323A and 170607A reach Einstein Telescope sensitivity.
  • The correlations imply that the measured P_0 may not be the true birth spin but an equilibrium period from magnetar–disk interaction, which changes how P_0 should be interpreted in progenitor models.

Reading between the lines

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

  • Because the fitted parameters are adjusted to reproduce the same plateau luminosity and break time, part of the reported 20–50% offset could be absorbed by the fitting degeneracy between B_p, P_0, and ϵ; a posterior predictive check on simulated light curves would separate the physical bias from the model-induced shift.
  • If future observations show that B_p, rather than the magnetic dipole moment B_pR^3, stays fixed while the magnetar spins down, the size and direction of the derived parameter biases would change, potentially weakening the claimed universal correlations.
  • Since 62 of the 105 GRBs lack a measured redshift and are assigned z = 1, distance errors could smear the jet-energy correlations; a redshift-complete sample from upcoming missions would provide a sharper test of the reported scaling laws.
  • The two events predicted to be detectable by the Einstein Telescope offer a concrete observational route: a non-detection of GW emission from GRBs 150323A or 170607A at ET sensitivity would place an upper bound on the ellipticity that either supports or conflicts with the high-ϵ required by the magnetar interpretation.
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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

4 major / 6 minor

Summary. This paper presents a systematic re-analysis of 105 long GRBs with X-ray plateau emission from the Swift/XRT archive (detected before 2023 December). The authors fit each light curve with a smooth triple power law to separate the jet-emission and magnetar-wind phases, then run MCMC fits of a magnetar spin-down model that includes power-law-evolving radius and moment of inertia (R ∝ Ω^m, I ∝ Ω^k, with indices taken from Lan et al. 2021) under the assumption that the magnetic dipole moment μ = B_p R^3 is conserved, in order to infer B_p, P_0, and ε. The paper compares these results with the conventional constant-R/I fits, claims that neglecting R/I evolution biases B_p, P_0, and ε by 20–50%, reports 'universal' correlations (ε ∝ P_0^1.57±0.22, ε ∝ B_p^0.97±0.13, B_p ∝ P_0^1.30±0.16, and several jet–wind energy scaling relations), estimates EP/SVOM detectability of magnetar wind radiation, and evaluates aLIGO/ET detectability of GW emission, finding that only GRBs 150323A and 170607A could reach the ET sensitivity threshold.

Significance. If the results hold, the quantitative recalibration of magnetar parameters (initial period, dipole field, ellipticity) from X-ray plateaus, the uniform 105-GRB sample, and the explicit prospective-detector predictions would be a useful contribution to GRB central-engine studies. The paper's strengths include a transparent MCMC setup with stated uniform priors, per-GRB fit tables, K-S tests between the constant and evolving scenarios, closed-form spin-down solutions, and concrete falsifiable predictions (two ET-reachable sources and EP/FXT detection-rate numbers). The main impediments to accepting the claims as stated are internal: the radius-evolution mechanism contradicts the μ-conservation assumption adopted in Eq. (12), and the headline correlations are susceptible to the z=1 assignment for 62 of 105 events and to fitting degeneracies in ε, so the results as currently framed require substantial revision rather than minor polishing.

major comments (4)
  1. [Section 3, Eqs. (12)–(18)] The central 20–50% claim is framed in the abstract and Section 1 as an effect of 'R/I evolutionary effects,' with the Section 1 motivation L_dip ∝ R^6. But under the assumption stated beside Eq. (12) that μ ≡ B_p R^3 is conserved during spin-down, Eq. (2) gives L_dip = η_X μ^2 Ω^4/(6c^3), which is independent of R at fixed Ω and fixed μ; the R^6 scaling is only realized if B_p, not μ, is held fixed. Consistently, the analytic solutions in Eqs. (15)–(18) contain the I-evolution index k but never the R-evolution index m. Thus, within the paper's own framework, radius evolution has no independent effect on the dipole luminosity, and the claimed 20–50% bias, if real, must arise from I evolution alone. The authors must state which quantity (B_p or μ) is held fixed in the numerical integrations behind Figure 4 and the MCMC fits; if μ is conserved, the title, abstract, and Section 1 overstate the role of R and must be revised, and if B_p is held fixed, the analytic solutions are missing the m-dependence they should contain. Either way, the mechanism behind the headline bias needs to be restated.
  2. [Section 2 and Section 4.2 (Tables 13–14)] A redshift z = 1 is imposed for 62 of the 105 GRBs (Section 2), and every derived quantity that feeds the correlations — L_p, E_wind, E_jet,iso, and hence the fitted P_0, B_p, and ε — scales with D_L^2(z). The 'universal' correlations in Figures 11–13, 16–18 and Tables 13–14, as well as the E_wind–E_jet,iso relation of Eqs. (24)–(26), therefore mix a 43-event sample with measured redshifts with a 62-event sample whose luminosities are set by the arbitrary z=1 distance. The paper should rerun the correlation analysis using only the 43 GRBs with measured redshift (the EP/SVOM section already restricts to these 43) and report whether the slopes, signs, and significances survive; at minimum, a sensitivity test with an alternative assigned redshift (e.g., the median redshift of the measured subsample) is needed to show that the claimed correlations are not distance-assumption artifacts.
  3. [Section 4.1, Figures 11–13 and Eq. (19)] The near-unity slope of the reported ε ∝ B_p^0.97±0.13 correlation is close to what the fitting model itself enforces rather than an independent physical relation: in the EM+GW co-dominated regime the spectral break occurs near L_GW ≈ L_dip, which gives ε ∝ B_p R^3/(I Ω) = μ/(I Ω), i.e., a slope of unity in log-log space at fixed Ω. In the EM-dominated regime, by contrast, the dipole light curve is essentially independent of ε, so the narrow Gaussian-style ε posteriors shown in the Appendix (e.g., ε_3 = 0.05 ± 0.03 for GRB 050922B in Figure 24) need to be shown to be likelihood-driven rather than prior- or upper-limit-driven. Please provide likelihood slices or posterior profiles in ε for representative EM-dominated, GW-dominated, and co-dominated fits, and check whether the ε–P_0 and ε–B_p correlations persist when the analysis is restricted to GRBs for which ε is genuinely constrained by the data.
  4. [Section 4.1, Table 10 (and the abstract)] The abstract's blanket statement that neglecting R/I evolution biases B_p, P_0, and ε by 20–50% is not supported uniformly across the scenarios. In Table 10, the K-S test for B_p gives p < 10^-1 for both Mb = 2.5 M⊙ scenarios (η_X = 0.1 and 0.5), which does not reject the null at the conventional 5% level, and the corresponding B_p deviations in Section 4.1 are factors of 1.1 and 1.0, respectively. The averaged factor of 1.5 quoted in the conclusion conceals this range (factors from 1.0 to 2.0 across the four scenarios). The text following Table 10 states that the null hypothesis is rejected without qualifying these marginal cases; the abstract and conclusion should present the bias as scenario-dependent (quantified per EoS, M_b, and η_X) rather than as a uniform 20–50% systematic.
minor comments (6)
  1. [Section 1 heading] The heading 'INTRODUTION' should be 'INTRODUCTION'; the title header also contains the spacing error 'Swift/XR T data'.
  2. [Eqs. (15) and (17)] In Eqs. (15) and (17), B_p appears without a subscript; under the μ-conservation assumption of Eq. (12) the prefactor should be written with B_{p,0} (or directly with μ) so that the R-independence of the prefactor is explicit.
  3. [Section 4.3 and Conclusion] The FXT sensitivity threshold is 1×10^-11 erg s^-1 cm^-2 in Section 4.3 but 3×10^-11 in the conclusion bullet; these values should be made consistent.
  4. [Figure 18 caption] Figure 18's caption says 'correlations between the P0 and Ejet,iso' but the panels plot ε against E_jet; the caption should be corrected.
  5. [Section 4.1 and Conclusion] In Section 4.1, deviations such as '△(log Bp) ∼ 0.3 G' mix units: log B_p is dimensionless, so the shift should be quoted in dex; likewise, 'I ∼ 2.5×10^45 g cm^−2' in the conclusion should read g cm^2.
  6. [Table 10] The K-S p-values in Table 10 are reported only as upper bounds (p < 10^-n); reporting the actual p-values or the test statistics would allow readers to assess the marginal cases (p < 10^-1) that are central to the scenario-dependence discussion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivation is a self-contained data fit with external numerical input, and the apparent R-evolution inconsistency is a correctness issue rather than a circular step.

full rationale

The paper's empirical chain is largely self-contained: the spin-down solutions (Eqs. 15-18) are derived from the stated spin-down law and the piecewise power-law R/I forms (Eqs. 12-13), the magnetar parameters are MCMC fits to Swift/XRT light curves, and the 20-50% bias claims are model-comparison outputs on the same data rather than predictions constructed from the fitted values. The 'universal correlations' are least-squares summaries of posterior estimates, so they are empirical outputs that may be affected by statistical degeneracies, but they are not definitional reductions. The main self-citation is to Lan et al. (2021), which supplies RNS-based numerical R/I behavior as external input; it is load-bearing but is not an unverified uniqueness or ansatz claim smuggled in through citation. One genuine internal issue is that the stated assumption mu = BpR^3 conserved makes Bp^2R^6 = mu^2 in Eq. (2), so the analytical dipole luminosity in Eqs. (15)-(18) contains only the I-evolution index k and not the R-evolution index m; this means the 'radius evolutionary effect' is not actually realized in the paper's own analytic framework. That is a correctness and interpretation concern, not circularity, and the paper itself partially acknowledges model sensitivity in the Section 5 caveat that the R/I evolution effect 'would be reduced or even completely suppressed' under other physical conditions.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central claim rests on no new entities: no new particle, force, or conserved quantity is postulated. The 'equilibrium spin period' and 'magnetically induced distortion' explanations are existing concepts from the cited literature, not entities invented here. The free parameters and axioms above are the inputs the paper actually depends on, and the most consequential are the imported R/I power-law indices and the mu-conservation assumption, which together set the magnitude of the claimed biases.

free parameters (6)
  • X-ray radiation efficiency eta_X = 0.1 and 0.5
    Chosen, not measured; all derived Bp, P0, and epsilon scale with it. The paper acknowledges eta_X is uncertain (Xiao & Dai 2019; Rowlinson et al. 2014; Gao et al. 2016) and uses two values without marginalizing.
  • Baryonic mass Mb = 2.0 and 2.5 M_sun (SLy only 2.0)
    Two fixed values chosen per EoS. The R/I evolution curves depend strongly on mass, and the mass is not constrained by the data; SLy is run only at 2.0 M_sun since 2.5 M_sun is above its maximum mass.
  • Redshift substitution z=1 = z = 1 for 62 of 105 GRBs
    Luminosities and energies are derived from redshift; assuming z=1 for GRBs without measurement changes Lp, Ejet, Ewind and hence every derived magnetar parameter and every reported correlation.
  • Jet opening angle theta_j = 5 degrees when not measured
    Used to correct Ejet via Ejet = Ejet,iso(1 - cos theta_j); most GRBs in the sample receive the default value, which sets the Ejet scale for the correlations.
  • Uniform prior bounds for MCMC = P0 in [0.3, 40 ms], Bp in [1e13, 1e16 G], epsilon in [1e-6, 1e-2]
    The priors shape the posteriors for weakly constrained parameters, especially epsilon in EM-dominated fits, where the quoted median epsilon is partly prior-driven rather than data-driven.
  • Power-law indices m, k and velocities Omega_k, Omega_1 = values from Lan et al. (2021) per EoS and mass
    The R(Omega) and I(Omega) evolution is parameterized by these imported numbers; they set the size of the claimed 20-50% biases and are not tabulated in this paper.
assumptions (7)
  • domain assumption The X-ray plateau is powered by isotropic EM dipole spin-down of a newborn millisecond magnetar, with the spin-down law of Eq. (2).
    Central assumption of the entire inference; standard in the field but not independently verified in this paper.
  • domain assumption Vacuum dipole formula and quadrupole GW formula with a constant radiation efficiency eta_X.
    Used in Eq. (2); the standard magnetar spin-down model with a single efficiency parameter.
  • ad hoc to paper The magnetic dipole moment mu = Bp R^3 is conserved during spin-down.
    Stated near Eq. (12). If Bp instead of mu is held fixed, the R^6 evolution strongly suppresses L_dip at fixed Omega, so the derived corrections depend critically on this choice.
  • domain assumption R(Omega) and I(Omega) follow the piecewise power laws of Eqs. (12)-(13) with indices m, k from Lan et al. (2021).
    The R/I evolutionary effects are imported wholesale from the authors' prior RNS study; the validity of m, k is not re-established here.
  • domain assumption A post-plateau decay slope in [-1, -2] identifies magnetar spin-down with GW- or EM-dominated losses.
    Used both as a sample selection criterion in Section 2 and as the interpretation of the fitted slopes.
  • domain assumption Differential rotation and monopolar wind emission are negligible at plateau times (t > hundreds of seconds).
    Argued in the Section 5 caveats; the inferred P0 may not be the true birth spin period.
  • standard math Flat LCDM cosmology with Planck 2020 parameters.
    Used for luminosity distances and k-corrections; standard background input.

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Cite this review

Pith. "Pith review of Constrain magnetar parameters by taking into account the evolutionary effects of radius and moment of inertia with \emph{Swift}/XRT data." pith.science (2026). https://pith.science/paper/I5JWPE53

@misc{pith2026250711110,
  author       = {Pith},
  title        = {Pith review of: Constrain magnetar parameters by taking into account the evolutionary effects of radius and moment of inertia with \emphSwift/XRT data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I5JWPE53}},
  note         = {Machine review of arXiv:2507.11110}
}
abstract

A newly born millisecond magnetar has been proposed as one possible central engine of some GRBs with X-ray plateau emission. In this work, we systematically analyzed the Swift/XRT data of long GRBs with plateau emission that were detected before 2023 December, and estimated the physical parameters by considering the $R/I$ evolutionary effects. We found that neglecting the $R/I$ evolutionary effects can lead to systematic overestimation or underestimation of magnetar parameters such as $B_p$, $P_0$, and $\epsilon$ from 20\% to 50\%. We also found that some tight correlations, which can be approximately expressed as $\epsilon\propto P_0^{1.57\pm0.22}$, $\epsilon\propto B_p^{0.97\pm0.13}$, $B_p\propto P_0^{1.30\pm0.16}$, $E_{\rm wind}\propto E_{\rm jet,iso}^{0.83\pm0.07}(E_{\rm jet}^{0.76\pm0.06})$, $P_0\propto E_{\rm jet,iso}^{-0.29\pm0.03}(E_{\rm jet}^{-0.26\pm0.02})$, $B_p\propto E_{\rm jet,iso}^{-0.58\pm0.06}(E_{\rm jet}^{-0.55\pm0.05})$, and $\epsilon\propto E_{\rm jet,iso}^{-0.55\pm0.07}(E_{\rm jet}^{-0.52\pm0.06})$ for our selected EoSs. The universal correlations suggest that a nascent magnetar with the faster $P_0$, lower $B_p$, and lower $\epsilon$ are more inclined to power a more energetic GRB jet, and the $\epsilon$ and $P_0$ of newborn magnetar are likely to originate from the magnetically induced distortion and correspond to the equilibrium spin period as a result of interaction between the magnetar and its accretion disk, respectively. Finally, we found that the GW signals from the remnants of those GW-dominated GRBs with redshift measurements cannot reach aLIGO sensitivity threshold, and only two cases (GRBs 150323A and 170607A) can reach ET sensitivity threshold. Future GW observations could not only offer the first smoking gun that a protomagnetar can serve as the central engine of GRBs but also play a crucial role in precisely constraining the neutron star EoS.

Figures

Figures reproduced from arXiv: 2507.11110 by the authors.

Figure 1
Figure 1. Joint BAT+XRT light curve of GRB 111008A. The black dots are BAT data extrapolated to the XRT band (0.3–10 keV), and the blue dots are XRT data. The solid red line is the fit with a smooth triple power-law function, and the vertical dashed line is the separation of the jet emission phase and magnetar wind emission phase. 0 1 2 3 4 5 6 log tjet, z or log twind, z (s) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Probabilit… view at source ↗
Figure 2
Figure 2. The distributions of jet emission duration (tjet,z) and the magnetar wind emission duration (twind,z) in the rest frame (left panel), as well as the distribution of the plateau luminosity Lp (right panel). The red dashed lines are the best Gaussian fits. We adopt z = 1 for the LGRBs without redshift measurement in our sample. log Lp/ergs−1 = 47.78 ± 0.87, respectively. In [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Lp as a function of tb/(1 + z) (left panel) or tjet/(1 + z) (right panel) for the LGRBs in our sample. The red solid and dashed lines are the least-squares linear fit and its 95% confidence level, respectively. of inertia should be related to the rotational speed (Lattimer 2012; Gao et al. 2020). Lan et al. (2021) studied in detail how R and I evolve as the magnetar’s rotational speed by using the numerical methods … view at source ↗
Figures from the paper (27 more)
Figure 4
Figure 4. Figure 4: Numerical results for dipole radiation luminosity by considering the R and I evolution effects in choice of different NS EoSs, baryonic masses, radiation efficiencies, as well as the dominant energy loss term [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Comparisons of the derived magnetar physical parameter Bp histograms between the R and I evolution correction and the constant R and I in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5, respectively [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: The cumulative distributions for the derived magnetar physical parameter Bp between the R and I evolution correction and the constant R and I in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5, respectively [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Comparisons of the derived magnetar physical parameter P0 histograms between the R and I evolution correction and the constant R and I in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5, respectively [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: The cumulative distributions for the derived magnetar physical parameter P0 between the R and I evolution correction and the constant R and I in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5, respectively [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Comparisons of the derived magnetar physical parameter ϵ histograms between the R and I evolution correction and the constant R and I in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5, respectively [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: The cumulative distributions for the derived magnetar physical parameter ϵ between the R and I evolution correction and the constant R and I in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5, respectively [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: The correlations between the ϵ and P0 in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5. The red solid and red dashed lines are the best-fitting results and the 95% confidence level, respectively [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: The correlations between the ϵ and Bp in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5. The red solid and red dashed lines are the best-fitting results and the 95% confidence level, respectively [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: The correlations between the Bp and P0 in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5. The red solid and red dashed lines are the best-fitting results and the 95% confidence level, respectively [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]
Figure 14
Figure 14. Figure 14: The correlations between the isotropic jet energy Ejet,iso and the magnetar wind energy Ewind (left panel), as well as between the corrected jet energy Ejet and the magnetar wind energy Ewind (right panel). The red solid line and dashed lines are the least-square fit …
Figure 15
Figure 15. Figure 15: Left panel: The distributions of isotropic jet prompt gamma-ray/X-ray energy (Ejet,iso), corrected jet prompt gamma-ray/X-ray energy (Ejet), and X-ray energy release of the magnetar wind (Ewind) for the LGRBs in our sample. Right panel: The distribution of the energy …
Figure 16
Figure 16. Figure 16: The correlations between the P0 and Ejet,iso (black circles), and Ejet (gray circles) in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5. The red solid and red dashed lines are the best-fitting results and the 95% confidence level for P0 − Ejet,iso, re…
Figure 17
Figure 17. Figure 17: The correlations between the Bp and Ejet,iso (black circles), and Ejet (gray circles) in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5. The red solid and red dashed lines are the best-fitting results and the 95% confidence level for P0 − Ejet,iso, re…
Figure 18
Figure 18. Figure 18: The correlations between the P0 and Ejet,iso (black circles), and Ejet (gray circles) in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5. The red solid and red dashed lines are the best-fitting results and the 95% confidence level for P0 − Ejet,iso, re…
Figure 19
Figure 19. Figure 19: The GRB Luminosity thresholds of Swift/XRT, EP/WXT, EP/FXT, SVOM /MXT in the flux limit F XRT th = 2 × 10−12 erg s−1 cm−2 , F WXT th = 1 × 10−10 erg s−1 cm−2 , F FXT th = 1 × 10−11 erg s−1 cm−2 , F MXT th = 7 × 10−11 erg s−1 cm−2 , respectively. The data of the X-ray …
Figure 20
Figure 20. Figure 20: Left panel: the X-ray luminosity light curves of GW-dominated GRBs with z measurement in our sample. Right panel: the X-ray luminosity light curves of GW-dominated GRBs without z measurement in our sample, and we adopt z = 1 to estimate the luminosity of these GRBs. f…
Figure 21
Figure 21. Figure 21: The GW characteristic amplitude of GRB 170607A in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5, respectively. of magnetar physical parameters can be approximately expressed as ϵ ∝ P 1.57±0.22 0 , ϵ ∝ B0.97±13 p , Bp ∝ P 1.30±0.16 0 , with the 1σ dev…
Figure 22
Figure 22. Figure 22: Detectability test of GW signals with aLIGO and ET detectors for our GW-dominated GRBs with z measurement in four samples of EoSs with Mb = 2.0M⊙ and ηX = 0.5. The black solid line is the sensitivity limits for ET, and the red and green solid lines are the sensitivity…
Figure 23
Figure 23. Figure 23: Similar to [PITH_FULL_IMAGE:figures/full_fig_p034_23.png]
Figure 24
Figure 24. Figure 24: EM-dominated case: the corner plots and best-fitting results of GRB 050922B in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5, respectively [PITH_FULL_IMAGE:figures/full_fig_p036_24.png]
Figure 25
Figure 25. Figure 25: GW-dominated case: the corner plots and best-fitting results of GRB 091208B in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5, respectively [PITH_FULL_IMAGE:figures/full_fig_p037_25.png]
Figure 26
Figure 26. Figure 26: EM+GW-dominated case: the corner plots and best-fitting results of GRB 180626A in four samples of EoSs with Mb = 2.0 M⊙, 2.5 M⊙ and ηX = 0.1, 0.5, respectively [PITH_FULL_IMAGE:figures/full_fig_p038_26.png]
Figure 27
Figure 27. Figure 27: The best-fitting results of our entire GRB sample in AP3 EoS with Mb = 2.0 M⊙ and ηX = 0.5 [PITH_FULL_IMAGE:figures/full_fig_p039_27.png]
Figure 27
Figure 27. Figure 27: — continued [PITH_FULL_IMAGE:figures/full_fig_p040_27.png]
Figure 27
Figure 27. Figure 27: — continued [PITH_FULL_IMAGE:figures/full_fig_p041_27.png]
Figure 27
Figure 27. Figure 27: — continued [PITH_FULL_IMAGE:figures/full_fig_p042_27.png]

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