Constraining the Type of Central Engine of GRBs with Swift Data
Liang Li Xue-Feng Wu Wei-Hua Lei Zi-Gao Dai En-Wei Liang Felix Ryde
astro-ph.HE



Liang Li
liang.li@fysik.su.se, xfwu@pmo.ac.cn, leiwh@hust.edu.cn, dzg@nju.edu.cn

0000-0002-0786-7307]Liang Li
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China
Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden
Department of Physics, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden


Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China



School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China




School of Astronomy and Space Science, Nanjing University, Nanjing, Jiangsu 210093, China



GXU-NAOC Center for Astrophysics and Space Sciences, Department of Physics, Guangxi University, Nanning 530004, China



Department of Physics, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden




The central engine of gamma-ray bursts (GRBs) is poorly constrained. There exist two main candidates: a fast-rotating black hole and a rapidly spinning magnetar.
Furthermore, X-ray plateaus are widely accepted to be the energy injection into the external shock.
In this paper, we systematically analyze the Swift/XRT light curves of 101 GRBs having plateau phases and known redshifts (before 2017 May).
Since a maximum energy budget (∼2×10^52 erg) exists for magnetars but not for black holes, this provides a good clue to identifying the type of GRB central engine.
We calculate the isotropic kinetic energy E_ K,iso and the isotropic X-ray energy release E_ X,iso for individual GRBs.
We identify three categories based on how likely a black hole harbors a central engine: 'Gold' (9 out of 101; both E_ X,iso and E_ K,iso exceed the energy budget), 'Silver' (69 out of 101; E_ X,iso less than the limit but E_ K,iso is greater than the limit), and 'Bronze' (23 out of 101, the energies are not above the limit).
We then derive and test the black hole parameters with the Blandford-Znajek mechanism, and find that the observations of the black hole candidate ('Gold'+'Silver') samples are consistent with the expectations of the black hole model.
Furthermore, we also test the magnetar candidate ('Bronze') sample with the magnetar model, and find that the magnetar surface magnetic field (B_p) and initial spin period (P_0) fall into reasonable ranges.
Our analysis indicates that if the magnetar wind is isotropic, a magnetar central engine is possible for 20% of the analyzed GRBs.
For most GRBs, a black hole is most likely operating.






§ INTRODUCTION


One of the most fundamental, yet unanswered, questions in gamma-ray burst (GRB) physics is the nature of the central engine. Observationally, they have to fulfill several requirements. First, they should be able to release a huge amount of energy (∼10^49-10^55 erg). Second, indirect estimates of the jet Lorentz factor ∼100, together with the compactness problem, require the central engine to be made up of baryons as much as possible. Finally, the rapid variability hints towards an intermittent central engine <cit.>.

There are two main progenitor models for the central engine. On the one hand, the magnetar model involves a rapidly spinning and highly magnetized neutron star. In this scenario, the rotational energy of the neutron star powers the GRB <cit.>. The limited energy budget provided by the magnetar model might, however, be at odds with observed GRB energies <cit.>.
On the other hand, a newly formed black hole (hereinafter BH) surrounded by an accretion disk <cit.> can produce bipolar jets through the Blandford-Znajek (BZ) mechanism <cit.> or through the production and polar annihilation of neutrino-antineutrino pairs <cit.>.
The rapid rotation of the central engine required by GRB models might, however, prevent the collapse to a BH <cit.>.
Distinguishing between a BH and a neutron star as the central engine is of the utmost importance as it would help to better constrain the progenitor of GRBs and their emission mechanisms.

The long time activity and decay of the central engine can leave footprints in the X-ray and optical afterglows and can thus help distinguish between the proposed progenitors. For example, the optical rebrightening of GRB 970228 was interpreted to be the result of energy injection by a millisecond pulsar <cit.>. In addition, <cit.> showed that a shallow decay followed by a normal decay phase is naturally obtained for the energy ejection by a millisecond magnetar.
The widely accepted scenario for the X-ray plateaus (also called the shallow decay) is the energy injection, either matter dominated or Poynting-flux (electron-positron wind) dominated, into the external shock. The simplest way to deal with the energy injection process is by assuming that the central engine is a millisecond magnetar, and that the injected energy is carried out by a Poynting-flux/electron-positron wind extracting the spin energy of the magnetar through the magnetic dipole radiation (MDR).
Of particular importance is the interpretation of the X-ray plateau as energy injection into the external forward shock, which is an evidence for extended central engine activity. The energy can be provided either in terms of Poynting flux from the central magnetar or by the deposition of kinetic energy into the forward shock by a stratified jet.
Both models are successful in explaining the normal decay phase at the end of the plateau.
However, the plateau is followed by a sharp flux drop as steep as F ∼ t^-9. Such a sharp decay is very difficult to explain using the forward shock model, since it requires the radiation to be released from a small radius. In this case, the plateau has to be of internal origin (as opposed to an external one, when it can be explained by energy injection). Such plateaus are easily explained in the magnetar framework as residual rotational and magnetic energy that can be extracted after the main extraction event. However, no model for the emission mechanism exists to date.
The spin-down luminosity of the energy injection from a central engine would evolve with time as (; the magnetar injection corresponds to q=0 for t<τ and q=2 for t>τ)

    L(t)≃ L_0(t/t_0)^-q={[         L_0,        , t≪τ; L_0(t/τ)^-2,        , t≫τ;              ].

Here, q is the luminosity injection index. For BHs, they can also inject energy, but q will depend on density profiles of the star <cit.>.

An additional criterion that could be used to differentiate between the magnetar and the BH model is the total energy emitted during the bursts. Indeed, heuristic arguments lead to a maximum energy that can be released by a magnetar, corresponding to the maximum rotational energy of 2×10^52 erg (). Such a limit does not exist for BHs.
The emitted radiation energy will only be some fraction of the rotational energy depending on the radiative efficiency (see, e.g., ).
More importantly, such a limit requires an assumption to be made on whether or not the energy emitted from a magnetar is isotropic. The reason is that the inferred total energy is given by integrating the observed flux over all solid angles. The angular distribution of the flux must therefore be assumed.
For a millisecond magnetar, the energy is carried away by a Poynting-flux/electron-positron wind extracting the spin energy through the MDR.  The wind expected to be launched is nearly isotropic wind (e.g. ). For instance, for the pulsar wind nebular of Crab, the MDR of a neutron star is nearly isotropic.
Moreover, in the case of a neutron star merger as the origin of the magnetar central engine in short GRBs, the total energy is also expected to be emitted quasi-spherically (e.g. ). However, if the magnetar is born at the center of a collapsar, the progenitor star could cause a collimation of the emitted energy, both during the prompt phase () and the afterglow phase (). However, <cit.> argue for the magnetar wind to be close to isotropic within a collapsar based on the energetics of the associated supernovae (SNe). Only a small fraction of the total energy is then channeled into the GRB jet, while most of the total energy is responsible for the SN explosion.
Therefore, the isotropic injection has a low efficiency.
However, the energy injection from a fast-rotating BH through the BZ mechanism is naturally considered to be collimated <cit.>.
The collimated injection has a high efficiency, since a large fraction of the energy can be injected into the GRB jet.

In this paper, we assume that the magnetar wind is quasi-spherical during the energy injection phase of the X-ray afterglow. We analyze the plateau afterglow of GRBs in order to investigate the plausibility of having a magnetar as a central engine. We extensively search the Swift/XRT data for GRBs whose light curves have a plateau phase and calculate the X-ray and kinetic energy released during this phase. We then investigate whether or not the data are consistent with the BH central engine model.

The paper is organized as follows.
The XRT data reduction and the sample selection are presented in 2.
The physical parameters of the GRBs and the hypothetical BH/magnetar parameters are derived in 3.
The statistical analyses of the external shock model parameters are presented in 4.
The discussions and conclusions are presented in 5.
Throughout this paper, a concordance cosmology with parameters H_0= 71 km s^-1 Mpc^-1, Ω_M=0.30, and Ω_Λ=0.70 is adopted.



§ DATA ANALYSIS


We restricted our study to GRBs observed by Swift/XRT from 2004 December to 2017 May, which show a transition from a plateau to either a normal decay phase (for the case of canonical light curves) or a very steep decay (for the case of "internal plateaus").
We first identify such bursts by visual inspection of the X-ray light curves. To derive the properties of the GRBs in the rest frame, we only select the bursts with known redshifts.



 §.§ Sample Definition


Two possible forms of light curves might confuse our sample selections performed through visual inspection. First, we might take an intrinsic single power-law (SPL) decay as a broken power-law decay. Second, we might also take a normal decay followed by a postjet break phase as a plateau followed by a normal decay phase. 

Therefore, to characterize a plateau, we perform a temporal fit to the X-ray light curves within a time interval (t_s, t_e) for each burst.
Here, t_s is the time when the plateau begins and t_e when the segment after the plateau break ends.
We employ either a smoothly broken power law (BKPL),

    F =  F_0   [(t/t_ b)^α_1ω+(t/t_ b)^α_2ω]^-1/ω,

or an SPL[Note that our basic model is the BKPL model; in order to exclude an intrinsic SPL decay cases, we also apply the SPL model for some cases.] to fit the light curves,

    F = F_0  t^-α.

In the above equations, α, α_1, and α_2 are the temporal slopes; t_ b is the break time; F_ b = F_0  2^-1/ω is the flux of the break time; and ω describes the sharpness of the break[The smaller the ω, the smoother the break; we often fixed it to be 3.].
We perform the fits to the data with a nonlinear, least-squares fitting using the Levenberg-Marquardt algorithm, and an IDL routine called mpfitfun.pro[<http://purl.com/net/mpfit>] <cit.>.

To exclude an intrinsic SPL decay, we first apply the above two models. We then choose the best model according to the following two principles.
(i) We compare the χ^2_ r value for both models and choose the model having the χ^2_ r closest to 1[We excluded GRB 050416A, GRB 080928, GRB 081118, and GRB 101219B from our samples since we confirmed that SPL is the best-fit model. We retained GRB 060418, GRB 061021, GRB 070529, GRB 081221, and GRB 131030A in our samples since we confirmed that BKPL is the best-fit model.].
For example, the value of the χ^2_ r of GRB 070529 for the BKPL model fitting is given 29/30 (close to 1), and for the SPL model fitting it is 68/33 (much larger than 1). We then adopt the BKPL as the best model for GRB 070529.
(ii) In case both models have similar χ^2_ r values (close to 1), we always chose the simplest model (SPL).
For instance, GRB 101219B has χ^2_ r 16/18 (close to 1) for the BKPL model and 21/21 (close to 1) for the SPL model. In this case, the SPL model is the best model for our choice.
We then remove those cases that have the SPL model as the best fit[We excluded GRB 080210, GRB 130514A, and GRB 140430A from our samples.].

To properly select only plateau phases and remove the normal decay followed by a jet-break phase, we only keep the temporal decays that are too fast to be explained by an external shock. For this, we use the closure relations to obtain an estimate of what should be the spectral index α(β) in the case of different spectral regimes (ISM/wind),

    α^s_1<α(β_1)≡{[                    3β_1-1/2, ν>max(ν_m,ν_c)(ISM and wind);                      3β_1/2,               ν_m<ν<ν_c(ISM);                    3β_1+1/2,              ν_m<ν<ν_c(wind);                              ].

Here, α_1 is the temporal decay index before the break time, Γ_1 is the photon spectral index, β_1 is the spectral index (β=Γ-1), ν_m is the minimum frequency, and ν_c is the cooling frequency.
Comparing the theoretical spectral indexes to the measured ones, we only keep the bursts that satisfy[With this selection, the bursts removed from the study are GRB 081208, GRB 081203A, GRB 080413B, GRB 090313, GRB 111123A, GRB 130427B, and GRB 140213A.] α^s_1<α(β).

A total of 101 GRBs are selected. These are shown in Figs. <ref>, <ref> and <ref>.



 §.§ Type of Plateau


In order to identify the type of the plateau (internal or external), we compare the temporal (α_1 and α_2) and spectral (Γ_1 and Γ_2) indices[Throughout the paper, the convention F_ν∝ν^-βt^-α is adopted.] before and after the plateau break, and combine them through the closure relations. Since the external plateau is purely dynamical, no change in the spectral index is expected. As such, we require Γ_1=Γ_2 (within errors), which is associated with the equality between the measured decay index and that obtained by closure equations from Γ_1 and Γ_2.
The prebreak slope α_1 (corresponding to q=0) is given by

    α_1= {[             2α_2-3/3,  ν_m<ν_<ν_c     (ISM);             2α_2-1/3, ν_m<ν_<ν_c     (Wind);             2α_2-2/3, ν_>ν_c     (ISM/Wind) ].

while the postbreak slope α_2 (corresponding to q=1; ), according to Zhang et al. (2006), is

    α_2= {[      3β_2/2=3(p-1)/4,    ν_m<ν_<ν_c   (ISM);      3β_2+1/2=3p-1/4,   ν_m<ν_<ν_c   (Wind);      3β_2-1/2=3p-2/4, ν_>ν_c     (ISM/Wind) ].

where β_2 is the spectral index during the normal decay phase, and p is the electron's spectral distribution index. These criteria gave 89 bursts in this sample.
Fig.<ref> shows all GRBs' external plateaus in the α_1-α_2 plane, with three theoretically favored lines (Equation <ref>) plotted.
Fig.<ref> shows the β-α plane based on either the ISM or the wind model.
The GRBs satisfying the "closure relations" are identified as our sample GRBs with colored data points.

The internal plateau <cit.> cases are selected based on Γ_1, different Γ_2, and α_2>2+β. This sample is made up of only 12 bursts. The bursts that cannot be placed in either of these two categories are discarded from further analysis[This is the case for GRB 050730, GRB 060526, GRB 061021, GRB 060607A, GRB 070110, GRB 081029, GRB 100219A, GRB 100302A, GRB 100902A, GRB 111209A, GRB 120521C, GRB 130408A.].



 §.§ X-Ray Plateau and Afterglow Kinetic Energies


We derive the X-ray isotropic energy released (E_ X,iso) during the plateau phase by integrating the fluence from t_ s  to t_ b,

    E_ X,iso=4π k D_ L^2S_ X/1+z,

where k=(1+z)^β-1, β is the spectral index, S_ X is the X-ray fluence[To obtain more precise X-ray fluence, we corrected them from absorbed to unabsorbed by multiplying the ratio of the observed and unabsorbed flux, which are collected from the Swift archive.] integrated over the plateau phase, D_L is the luminosity distance, and z is the redshift.

Another important parameter is the isotropic kinetic energy (E_ K,iso), which is measured from the afterglow flux during the normal decay.
Here, E_ K,iso is calculated following the method discussed in <cit.>.
Since E_ K,iso depends on the afterglow models (we note it here again briefly for completeness), it is important to identify the afterglow spectral regimes. We first use the observed quantity (α,β) in the normal decay phase to constrain/determine the afterglow spectral regimes of each individual burst (). We find that 18 of the 101 GRBs belong to spectral regime I (ν_ x>max(ν_m,ν_c)), and 83 out of the 101 GRBs belong to spectral regime II (ν_m<ν_ x<ν_c).
Here, ν_x is the frequency of the X-ray band. In addition, 42 out of the 83 bursts satisfy the ISM model, and 31 out of 83 satisfy the wind model.

We consider the relations obtained by <cit.> to compute the kinetic energy of the external plateau sample using the normal decay segment. Since the normal decay phase is always observed at late times (t∼ 10^4 s) in our samples, we consider the slow cooling scenario (ν_m<ν_ x<ν_c; see also ). In the case (ν_ x>max(ν_m,ν_c)), the same express expression can be applied to both the wind-like and the constant interstellar medium since the kinetic energy does not depend on the medium profile in this spectral regime.

    E_ K,iso,52=[ν F_ν(ν=10^18Hz)/5.2×10^-14 erg s^-1cm^-2]^4/(p+2)
    × D_28^8/(p+2)(1+z)^-1t_d^(3p-2)/(p+2)
    × (1+Y)^4/(p+2)f_p^-4/(p+2)ε_B,-2^(2-p)/(p+2)
    ×ε_e,-1^4(1-p)/(p+2)ν_18^2(p-3)/(p+1),

where ν F_ν (ν=10^18) Hz is the energy flux[Note that since ν F_ν is not an observed quantity, and it can be converted to the observed flux through integral frequencies over the X-ray energy range (0.3-10 keV).
We note that for a few cases, the photon index Γ_ X,2=2.0 during the normal decay phase, we adopt Γ_ X,2≃ 2.01 (within the errors) make a approximate treatment in order to avoid the 1-β=0 (β=Γ-1) for the calculations.] at frequency 10^18 Hz in units of erg s^-1 cm^-2, t_ d is the time in the observer frame in units of days, D_28=D/10^28, is the luminosity distance in units[The convention Q=10^xQ_x is adopted in cgs units for all other parameters throughout the paper.] of 10^28 cm, n is the number density in the constant ambient medium, A_* is the parameter of the stellar wind, Y is the Compton parameter, and ϵ_e and ϵ_B are the energy fraction assigned to the electron and magnetic field, respectively. f_p is a function of the power-law electron distribution p index[f_p is derived from the "external plateau" sample with p>2. Only one case, GRB 100302A, has p<2, and it was discarded.] <cit.>,

    f_p=6.73 (p-2/p-1)^p-1(3.3×10^-6)^(p-2.3)/2.


We adopted standard values of the microphysical parameters of the shock derived from observations <cit.>, namely n=1, Y=1, and (ϵ_e,ϵ_B)= (0.1, 0.01).

When ν_m<ν<ν_c, the kinetic energy depends on the type of external medium.
For a wind medium, one has

    E_ K,iso,52=[ν F_ν(ν=10^18Hz)/7.4×10^-14 erg s^-1cm^-2]^4/(p+1)
    × D_28^8/(p+1)(1+z)^-(p+5)/(p+1)t_d^(3p+1)/(p+1)
    × f_p^-4/(p+1)ε_B,-2^-1ε_e,-1^4(1+p)/(p+1)
    × A_*,-1^-4/(p+1)ν_18^2(p-3)/(p+1),

while for a constant ISM, one has

    E_ K,iso,52=[ν F_ν(ν=10^18Hz)/6.5×10^-14 erg s^-1cm^-2]^4/(p+3)
    × D_28^8/(p+3)(1+z)^-1t_d^3(p-1)/(p+3)
    × f_p^-4/(p+3)ε_B,-2^-(p+1)/(p+3)ε_e,-1^4(1-p)/(p+3)
    × n^-2/(p+3)ν_18^2(p-3)/(p+3).


After deriving E_ K,iso at the break time t_ b, we next estimate the upper limit of E_ K,iso(t_0) by E_ K,iso(t_ b). If t_s>t_0, the upper limit of E_ K,iso(t_0) can be estimated by

    E_ K,iso(t_0)<E_ K,iso(t_b)/(t_ b/t_ s)^(1-q).


The error on E_ K,iso (t_ b) is derived by  bootstrapping. For other parameters with simple mathematical expressions, the errors are calculated by error propagation.

After determining the bursts with an external plateau, we further split them into three groups. The classification is made depending on whether the X-ray isotropic energy E_ X,iso and or the kinetic isotropic energy E_ K,iso is larger than 2×10^52 erg, corresponding to the maximum energy released by a magnetar. This is illustrated in Fig.<ref>.




  * Gold Sample: bursts for which the E_ X,iso of the plateau phase is greater than 2×10^52 erg.
The sample is made up of nine GRBs and the power-law fits to the light curves are shown in Fig.<ref>. This corresponds to 9% of all the bursts.


  * Silver Sample: bursts for which the E_ X,iso
 of the plateau phase is less than 2×10^52 erg, but  E_ K,iso is greater than 2×10^52 erg. It is also consistent with the theoretical prediction of the BH central engine model.
We note that because the computation of E_ K,iso relies on assumed values for the microphysical parameters, it is not possible to completely rule out the magnetar progenitor for the bursts. This sample is made up of 69 bursts, and their light curves are presented in Fig.<ref>. This corresponds to 68% of all the bursts.


  * Bronze Sample: bursts for which neither E_ X,iso nor E_ K,iso are above 2×10^52 erg. For the 23 bursts in this sample, it is not possible to completely rule out the BH progenitor. Fig.<ref> shows the fitted light curves.
This corresponds to 23% of all the bursts.


Under the assumption that a magnetar emits its energy isotropically, this classification indicates that 77% of the bursts have a BH central engine; see further discussion in 5.1.



 §.§ γ-Ray Energetics


The k-corrected isotropic energy released in the gamma-ray band (E_γ, iso) is retrieved from the published paper.
For bursts that did not have published values, we derived them from the observed fluence (S_γ) in the detector band and k-correct to the rest-frame (1-10^4) keV using the spectral parameters.

We obtained the fluence and spectral parameters (including the α and β spectral indexes and the peak energy E_ p) of the Band function from published papers <cit.> or the GCN Circulars Archive[<https://gcn.gsfc.nasa.gov>].
First, if a burst was detected by Swift/BAT and also by Fermi/GBM and/or Konus/Wind, we considered the spectral parameters from either the Fermi/GBM or Konus/Wind instrument.
Second, if a burst was only detected by the Swift/BAT instrument and fit with a power-law or a cutoff power-law model, we proceeded with the following steps:
(1) for the cutoff power-law model, the fit parameters include α and E_ p <cit.>, so we adopt the typical value of β=-2.3.
(2) for the SPL model, we derive E_ p from the empirical correlation log E_ p=(2.76± 0.07)-(3.61±0.26) logΓ^ BAT <cit.>, where Γ is the photon index of the Swift/BAT band, which was found from the Swift data archive, and adopted the typical β value of -2.3.
(3). We finally test the results against the Amati relation <cit.>, as shown in Fig.<ref>. Our results are consistent with the Amati relation, which indicates that our method is reasonable.

We derive the k-correction <cit.> factor in the rest-frame 1-10^4 keV band, and derive the isotropic energy E_γ,  iso as

    E_γ, iso=4π k D_L^2S_(γ)/1+z.

The isotropic luminosity at the break (L_ X) can be derived as

    L_ X, iso=4π k D_L^2F_ X,b,

where F_ X,b is the flux at the break time (t_ b), which was obtained from the light curve fit.

The luminosity at the break time (L_ X, iso) and the isotropic release energy of the plateau phase (E_ X,iso), together with the time intervals (t_s, t_e) for the light curve fitting, the temporal indices of the plateau phase (α_1) and the normal decay or steeper decay phase (α_2), the break time (t_ b), the X-ray fluence for the plateau phase (S_ X), which is the integrated observed flux during the time interval (t_s, t_b), and the photon indices[The photon indices of the plateau phase were taken from the table published in <cit.>, and the rest of the values were computed from the Swift BA+XRT repository: <http://www.swift.ac.uk/burst_analyser/>] for the shallow decay/plateau phase (Γ_1) and a normal or steeper decay phase (Γ_2), are obtained from the Swift archive,[We applied the same photon model for the both pre- and postbreak phases, either from the PC model to the PC model, or from the WT model to the WT model, since the data for the late time observed is usually a WT model.], which are reported in Table 1.
In addition, the table also gives the spectral indices of the plateau phase and of the following segment, both obtained from either the Swift archive or from the closure relation.



 §.§ Jet-opening angle


In this section, we study the beaming correction with three methods and compare their results. The beaming correction is defined as

    f_ b=1-cosθ_j≃(1/2)θ_j^2.

It is used to correct the γ-ray and kinetic energies.
We obtain the geometrically corrected gamma-ray energy through E_γ=f_ b E_γ,  iso， and E_ K=f_ b E_ K,iso.  The latter  quantity is used for collimated outflows from a BH central engine (See 3.2).

The jet-opening angle can be derived from the observational data <cit.>,

    θ_j=
    0.124 (t_j/1 day)^3/8(1+z/2)^-3/8(E_ k,iso/10^52 erg)^-1/8(n/1 cm^-3)^1/8,

where θ_j is the jet-break time, E_ K,iso is the isotropic kinetic energy, and n is the medium density.
If the light curve shows a jet break, we measure θ_j using Equation (<ref>). In our sample, 18 GRBs show a jet break. For the rest of the GRBs without a jet break, we use the observed time of the last data point as the jet-break time (t_j) to estimate θ_j, and get an upper limit.

Another method to estimate the jet-opening angle is by empirical relations <cit.>, θ_j≈5.0/Γ_0, where Γ_0 is the initial Lorentz factor, also derived using the empirical relation as

    Γ_0=27 E_γ, iso,52^0.26,

where E_γ, iso, 52 is the isotropic energy, in units of 10^52 erg.

Fig.<ref> shows the distributions of the beaming factors f_ b.
We compare the distributions of these two methods.
These distributions can be fitted with Gaussian functions; we have logf_ b=-1.68±0.61 from using the late data point method and logf^'_ b=-2.14±0.47 from the empirical relation method. We find that the f_ b values from the first method is greater than those from the second method, and the f_ b of θ_j=5^o lies between the two distributions.
In <cit.>, a coefficient 5 was adopted for the estimation.
In turn, if it is initially defined as θ_j≡ a_j/Γ_0, we then can estimate a_j. Fig.<ref>b shows the distribution of a_j, with the best Gaussian fit a_j= 5.11 ± 7.21. This result is consistent with the finding in <cit.>.
The results, which include θ_j, f_ b, a_j, θ^'_j, and f^'_b, are presented in Table 4.

We finally adopted the following criterion to calculate the beaming factor: if a burst has an observed light curve jet break, we then use it to estimate θ_j; otherwise, we adopt θ_j=5^o.



 §.§ γ-Ray Radiative efficiency

Another point we want to investigate is whether the γ-ray radiative efficiency is different among the samples.
The radiative efficiency is defined as <cit.>

    η_γ=E_γ, iso/E_γ, iso+E_ K,iso=E_γ/E_γ+E_ K.

Since the kinetic energy (E_ K,iso) increases with time during the plateau phase but remains the same during the normal decay phase, so the radiation efficiency η_γ thus depends on which E_ K epoch is adopted.
Here we focus on the radiation efficiency at the end of the shallow decay phase, η_γ(t_ b).
The physical meaning for this is the efficiency of converting the spin-down energy of a BH/magnetar to γ-ray radiation.

The X-ray efficiency can also be defined as the conversion of the BH/magnetar spin-down energy to radiation, i.e.

    η_ X=L_ X,iso/L_ X,iso+L_ K,iso=L_ X/L_ X+L_ K.


In Fig.<ref>, we compare E_γ, iso and E_ K,iso (Fig.<ref>a) and L_ X,iso and L_ K,iso (Fig.<ref>b) for different samples.
The same distribution is also shown in Figs. <ref>a and <ref>b. The typical value is log η_γ(t_ b)= -0.53±0.56 for η_γ and is log η_x(t_ b)=-0.54±0.58 for η_x. The radiative efficiencies in the γ- and X-ray energy bands, in general, present the same distribution.



§ CENTRAL ENGINE PROPERTIES 


In this section, we derive the properties of the central engine based on our observations, assuming two models.



 §.§ BH Central Engine


One of the leading models for the GRB central engine is a stellar-mass BH surrounded by a hyper-accreting disk <cit.>, with a typical accretion rate of 0.01-1 M_ s^-1. There are two main energy reservoirs that provide the jet power, namely the accretion energy in the disk that is carried by neutrinos and antineutrinos, which annihilate each other and power a bipolar outflow <cit.>, and the spin energy of the BH, which can be tapped through magnetic fields via the BZ <cit.> mechanism <cit.>.

As suggested by <cit.>, the jet might be dominated by the BZ power especially at late times <cit.>. The rotational energy of a BH with angular momentum J_∙ is a fraction of the BH mass M_∙ (or dimensionless mass m_∙=M_∙/M_⊙),

    E_ rot=f_ rot(a_∙)M_∙/M_ c^2 erg= 1.8 × 10^54 f_ rot(a_∙) M_∙/M_ erg,

where a_∙ = J_∙ c/(GM_∙^2) is the BH spin parameter and

    f_ rot(a_∙)=1-√((1+√(1-a_∙^2))/2 ).


We then connect the kinetic energy of the jet to the spin energy of the BH through the relation

    η E_ rot=f_ b E_ K,iso,

where η is the efficiency of converting the spin energy to the kinetic energy of the jet through the BZ mechanism, and f_ b is the beaming factor (see <ref> for a detailed discussion).
For a maximally rotating BH, about 31% of the rotational energy can be taken out to power a GRB <cit.>. This efficiency is not sensitive to the BH spin a_∙.
Therefore, we take η≃ 0.3 in the following.

From Equations (<ref>) and (<ref>), one can constrain the BH spin parameter once we know the BH mass M_∙, the jet-opening angle θ_ j, and the kinetic energy of the jet E_ K,iso. Following <cit.>, the typical value for the BH mass in a long GRB is 6 M_. In this paper, we adopt M_∙ =3 M_ for short GRBs[Our samples include five short GRBs, and four of them belong to the Bronze sample.] and 6 M_ for long GRBs.

For a_∙ >0.1, the bimodal distribution of the index q can be explained by the spin evolution of the BH. The BH evolves with time during a GRB, since it would be spun up by accretion and spun down by the BZ mechanism <cit.>.
Due to and the different initial parameter sets for different GRBs, the evolution of the characteristics varies from burst to burst. Therefore, we have q>0 for the spin-down evolution and q<0 for the spin-up case.

For a_∙ <0.1, the BH spin does not necessarily need to be very low. Instead, the value we obtained based on Eq.(<ref>) should be the change in the BH spin Δ a_∙ during a GRB. The very small value of Δ a_∙ means that the BH spin does not evolve. This is possible when the BH spin is at a critical value where either the spin-up due to accretion is comparable to the spin-down due to the BZ mechanism or the accretion rate is quite low. A clear evidence comes from the distribution of q when a_∙ <0.1, for which q is close to zero.

The correlation of a_∙-q for our different samples is presented in Fig.<ref>.
We find that the Gold sample bursts cluster into the region where a_∙>0.1. The results are consistent with the prediction of the BH central engine model.
Here we note that since the Bronze sample bursts could still have a BH central engine, we also calculate a_∙ for them. Interestingly, we found all Bronze sample bursts have  a_∙< 0.1.

The BH parameters (a_∙), the duration of the prompt emission T_ 90, the isotropic energy release in the gamma-ray band E_γ, iso, the isotropic kinetic energy E_ K,iso, the electron spectral index p, the luminosity inject index q, and a redshift z (searched from the published papers or the GCN Circulars Archive) are reported in Table 2 for the Gold and the Silver samples.

For a magnetized BH-accretion disk system, there is another magnetic mechanism: the magnetic coupling effect between the BH and the disk through closed magnetic field lines <cit.>. Similar to the BZ mechanism, the magnetic coupling effect also extracts rotational energy from the spinning BH. Only if the BH spin is initially small would the magnetic coupling would act as an additional spin-up process. A similar discussion of this aspect was made by <cit.> within the context of the magnetar central engine model. In more general cases, the magnetic coupling effect would not significantly affect the BH spin evolution <cit.>.



 §.§ Magnetar Central Engine


A rapidly spinning magnetar model invokes a rapidly spinning, strongly magnetized neutron star called a "magnetar". 
There are two critical magnetar parameters for a magnetar undergoing bipolar spin-down: the initial spin period P_0 and the surface polar cap magnetic field B_p, which can be derived using the characteristic luminosity L_0 and the spin-down time scale τ. We derive the magnetar parameters (P_0 and B_ P) using Eqs. (<ref>)-(<ref>), assuming isotropic emission from the magnetar.

For a magnetar, the characteristic spin-down luminosity is:

    L_0=1.0×10^49 ergs^-1 (B_p,15^2P_0,-3^-4R_6^6).

where R is the stellar radius[We assume the typical value for R_6=1, M=M_1.4 and I_45∼2.].
In Eguation (<ref>), t is a measure of time in the rest frame, and τ is the initial spin-down timescale, which is defined by

    τ=P/2Ṗ_0=2.05×10^3s(I_45B_p,15^-2P_0,-3^2R_6^-6),

where Ṗ is the rate of period increase due to MDR, and I_45 is the stellar moment of inertia in units of 10^45g cm^-2 <cit.>.
The initial spin-down timescale, τ, in general, can be identified as the observed break time, t_ b,

    τ=t_b.

and the characteristic spin-down luminosity L_0, in general, should include two terms,

    L_0=L_ X+L_ K=(L_ X,iso+L_ K,iso),

where L_ X,iso is the X-ray isotropic luminosity due to the internal dissipation of the magnetar wind, which is the observed X-ray luminosity of the internal plateau and measured at the observed break time (one can only derive an upper limit for external plateaus), and

    L_ K,iso=E_ K,iso(1+z)/t_b,

is the kinetic isotropic luminosity which is injected into the blast wave during the energy injection phase.
It depends on the isotropic kinetic energy E_ K,iso after the injection phase is over and the observed break time t_ b.
The isotropic kinetic energy E_ K,iso can be derived from afterglow modeling, as discussed above.

Since it is not possible to completely rule out the magnetar progenitor for the Silver sample, we investigate the magnetar parameters for both the Silver and Bronze samples.

The derived values of P_ 0 and B_ P are presented in Table 3 and Fig.<ref>.
We found that all bursts in the Bronze sample fall in a reasonable range in the (P_ 0-B_ P) plot (Fig. <ref>). B_ P has a typical value of ∼ 10^15G, and is in the range [10^14G,10^16G]; no case have B_ P below 10^14G, and only one case is greater than 10^16G. We find that most derived P_ 0 are close to 1 ms; this is consistent with the breakup limit of a neutron star, which is about 0.96 ms <cit.>.[We note that four of the five short GRBs are obtained in the Bronze sample, while the rest of the 19 GRBs are long GRBs.].

However, we also found that approximately half of the Silver sample bursts have B_ P < 10^14G. Moreover, most of the bursts have P_ 0 that is far away below to 1 ms. This result implies that the Silver sample bursts are more likely to have a BH central engine.

Using the derived value of P_0, the total rotation energy of the millisecond magnetar within this scenario is

    E_ rot=1/2IΩ^2≃ 2×10^52 erg M_1.4R_6^2P_0,-3^-2,

where I is the moment of inertia, Ω=2Π/P_0 is the initial angular frequency of the neutron star, M_1.4=M/1.4M_, and E_ rot is the maximal total energy available and can be compared to the observed and inferred energies (see <ref>).

In conclusion, the properties of the bursts in the Bronze sample are consistent with the predictions of the magnetar central engine model.



 §.§ Total Energy and Correlations
 

In Fig.<ref>, we compare the inferred E_γ, iso+E_ K,iso with the total rotation energy E_ rot of the BH (Silver; <ref>) and the millisecond magnetar (Bronze; <ref>).
It is found that the GRBs are generally above and slightly above the E_ rot=E_γ, iso+E_ K,iso line. This is consistent with the hypothesis, namely, that all of the emission energy ultimately comes from the spin energy of the BH or magnetar.

In Fig.<ref>, we show the T_90 (rest frame) as a function of various energies for both the BH and magnetar candidate samples.
All isotropic energies for most cases are above 2×10^52 erg while all other energies for most cases with a beam correction are below 2×10^52 erg.

In Fig.<ref>, we compare the histograms of the isotropic energies (E_γ,iso, E_ K,iso, E_ X,iso) of the BH and magnetar candidate samples. The typical values of all kinds of energies (E_γ, iso, E_K, iso, E_X, iso) for the BH candidate samples (Gold+Silver) are systemically larger than those for the magnetar sample (Bronze).

Fig.<ref> displays the T_90 (rest frame) as a function of redshift for the samples. Two interesting findings: first, four out of the five short GRBs are in the magnetar candidate sample (Bronze); second, the redshifts of all five short bursts are below 1.

Fig.<ref> shows the correlations of L_ X,iso-E_γ, iso and L_ X,iso-t_ b/(1+z) for the entire sample (BH candidate Gold+Silver samples and magnetar candidate Bronze sample).
We find that a higher isotropic γ-ray energy implies a higher isotropic X-ray break luminosity, and the bursts of the Gold sample cluster into the region of high isotropic γ-ray energy and isotropic X-ray break luminosity. The best fit to the correlation is log L_ X, iso, 49=(-1.33± 0.10)+(1.06±0.09)log E_γ,  iso,52, with a Spearman correlation coefficient of 0.78 and a chance probability of p<10^-4.
For the L_ X,iso-t_ b/(1+z) correlations, the best fit is log L_ X, iso, 49=(2.74± 0.40)+(-1.09±0.09)log t_ b/(1+z), with a Spearman correlation coefficient of 0.79 and a chance probability of p<10^-4. The anticorrelation between the luminosity and the rest-frame duration of the plateau phase is consistent with the previous findings <cit.>.

In Fig.<ref>, we show the scatter plots of η_ X versus η_γ(t_ b), E_γ,  iso, E_ K,iso(t_ b), and E_ rot (Bronze). One interesting finding is the η_ X and E_ K,iso(t_ b) presenting an anticorrelation for the BH candidate samples (Gold+Silver) while presenting a positive correlation for the magnetar candidate sample (Bronze).



§ COMPARISON OF STATISTICAL CHARACTERS


Since most of the bursts in our sample have an external plateau, this indicates that our GRB samples conform well with the external shock models, and they offer an excellent sample to study external shock model parameters. Here we investigate whether there are noticeable differences between the BH candidate (Gold+Silver) and magnetar candidate (Bronze) bursts for the observation properties and model parameters.

In addition, we also investigate the external plateau and the internal plateaus in our bursts.



 §.§ Temporal indices α


Fig.<ref> displays the distributions of the temporal indices α in different subsamples and different temporal segments. They are all well fitted with Gaussian distributions for each sample.
For the α_ x,1 distributions, we have α_ x,1=0.55±0.30 for the BH candidate samples and α_ x,1=0.32±0.48 for the magnetar candidate sample.
The same analysis based on the type of plateau gives α_ x,1=0.60±0.47 for the internal plateau sample, α_ x,1=0.50±0.31 for the external plateau ones (Fig.<ref>a).
For the α_ x,2 distributions, we have logα_ x,2=0.18±0.11 for the BH candidate samples and logα_ x,2=0.07±0.13 for the magnetar candidate samples.
For the internal/external plateau samples, one has logα_ x,2=0.46±0.33 for the internal plateau sample and logα_ x,2=0.17±0.11 for the external plateau sample (Fig.<ref>b).

Another self-consistency check is to compare the observed change of the decay slope (Δα_ x=α_ x,2-α_ x,1).
We find log Δα_ x=0.01±0.18 for the BH candidate samples and log Δα_ x=0.05±0.39 for the magnetar candidate, and logΔα_ x=0.51±0.60 for the internal plateau sample and logΔα_ x=-0.01±0.17 for the external plateau sample (Fig.<ref>c).

The temporal indices (both α_1 and α_2) for the BH candidate samples (Gold+Silver), in general, are steeper than those for the magnetar sample (Bronze). For the internal and external examples, the prebreak temporal indices α_1 are consistent with each other, but the postbreak temporal indices α_2 for the internal sample is much steeper than those for the external sample, consistent with an internal origin. The degrees of the break Δα for the BH candidate samples (Gold+Silver) is significantly less than those of the Silver and the magnetar samples (Bronze).



 §.§ Photon spectral indices Γ


Fig.<ref> shows the distributions of the photon spectral indices Γ in different subsamples and different plateau types. They are all well-fitted with Gaussian distributions for each sample.

For the Γ_ x,1 distributions, we have Γ_ x,1=1.91±0.20 for the BH candidate samples and Γ_ x,1=1.99±0.23 for the magnetar candidate sample (Fig.<ref>a).
For the Γ_ x,2 distributions, one has Γ_ x,2=1.94±0.20 for the BH candidate samples and Γ_ x,2=1.88±0.25 for the magnetar candidate sample (Fig.<ref>b).
For the ΔΓ_ x distributions, one has ΔΓ_ x=0.03±0.19 for the BH candidate samples and log ΔΓ_ x=-0.08±0.24 for the magnetar candidate sample (Fig.<ref>c).

We also investigate the Γ distributions in different plateau types.
For the internal plateau samples, one has Γ_ x,1=2.00±0.44, Γ_ x,2=1.81±0.53, and ΔΓ_ x=0.13±0.26, respectively.
For the external plateau samples, one has Γ_ x,1=1.92±0.20, Γ_ x,2=1.95±0.17, and ΔΓ_ x=0.01±0.20, respectively.

The spectral indices (both Γ_1 and Γ_2), and the degrees of the break ΔΓ for the different subsamples (the BH candidate samples and the magnetar candidate sample), in general, present the same distributions.
For the internal and external samples, a change is not found in the external sample, while a significant change in the internal sample is found.

The results indicate that there is no spectral evolution for the subsamples and the external plateau sample, while the internal plateau sample has a significant spectral change between a plateau phase and the following steeper decay phase. This is consistent with the internal plateau having an internal physical origin.



 §.§ Break time t_ b


Fig.<ref> shows the distributions of the observed break times (t_ b). A Gaussian fit to the distributions gives log(t_ b/s)=(4.05±0.68) and log(t_ b/s)=(4.38±0.63) for the BH candidate samples and the magnetar candidate sample, respectively. Separating the "internal" plateau and "external" plateau samples, we have log(t_ b/s)=(4.22±0.54) for the "internal" plateau sample and log(t_ b/s)=(4.07±0.72) for the "external" plateau sample. So, both external and internal plateau samples are consistent in terms of t_ b.

The typical break time for the  BH candidate samples (Gold+Silver) are statistically earlier than that of the magnetar candidate sample (Bronze). Similar results can be found between the internal (earlier) and external (later) samples. The results indicate that the end energy injection for the "external plateau" is in general earlier than that for the "internal plateau" on average.



 §.§ Electron spectral index p


We derive the electron spectral indices p, derived using the temporal indices:

    p={[                    4α_2+2/3, ν>max(ν_m,ν_c)(ISM and wind);                    4α_2+3/3,               ν_m<ν<ν_c(ISM);                    4α_2+1/3,              ν_m<ν<ν_c(wind);                              ].


The p distributions for our subsamples are p=2.63±0.63
for the BH candidate samples and p=2.49±0.15 for the magnetar candidate sample (Fig.<ref>).
The global p value is within the range [1.89, 3.70], and it has a Gaussian distribution with a centre value p=2.69±0.05. Except for 11 GRBs that have internal plateaus, all the rest have p>2.0, except for one case (GRB 100302A) which had p<2.0 (Fig.<ref>).

The results are consistent with the typical value of p for relativistic shocks due to first-order Fermi acceleration <cit.>.



 §.§ Energy Injection Parameter q


Similarly, the luminosity injection index q (see Equation. <ref>) can be determined from the temporal index α and the spectral index β for different afterglow models; here, α and β are measured in the shallow decay segment preceding the break:

    q={[           2α_1-2β_1+2/1+β_1, ν>max(ν_m,ν_c)(ISM and wind);           2α_1-2β_1+2/2+β_1,               ν_m<ν<ν_c(ISM);             2α_1-2β_1/1+β_1,              ν_m<ν<ν_c(wind);                              ].


The q distributions for our subsamples are q=0.24±0.50 for the BH candidate samples, and q=0.38±0.35 for the magnetar candidate sample (Fig.<ref>).
Also, for the internal plateau, we have q=0.39±0.35, and for the external plateau, we have q=0.22±0.35.

The different subsamples present different central values; the typical value for the BH candidate samples (Gold+Silver) is slightly less than for the magnetar candidate sample (Bronze). The typical q value for the internal sample is greater than the external sample. Totally, the central q value for subsamples is around 0.3, which is consistent with the predictions of an energy injection model <cit.>.



§ DISCUSSIONS AND CONCLUSIONS


In the analysis above, several assumptions have been made, which, if relaxed, make the magnetar energy budget even more strained. For instance, the radiative efficiency is unknown but could indeed be low (e.g. ). Therefore, the limiting radiative energy for a magnetar burst could be only a fraction of the maximum allowed rotational energy. Moreover, the energy estimated from the X-ray afterglows could underestimate the true energy if they are strongly suppressed by Compton cooling <cit.>.
On the other hand, the total energy from a magnetar GRB should come from the rotational energy of the magnetar itself. Except for the kinetic energy E_ K,iso (t_ b), which is measured at the end of the plateau phase that is considered in this paper, the energy of the GRB itself (E_γ,iso) including GeV emission should also be considered and should be added to the total energy; therefore, it could cause some GRBs in our magnetar sample to exceed the maximum energy allowed by the magnetar.
The main assumption made is, however, on the sphericity of the magnetar wind, which we discuss in the next section.

Furthermore, the related detailed studies have been done by many authors <cit.>. For instance, <cit.> considered long GRBs to be formed by rapidly spinning, newly born magnetars, and claimed that the high spin-down luminosity caused by MRD losses represents a natural mechanism for prolonged energy injection in the external shock in the first hours after the formation of the magnetars.
The individual light curves of the different cases can be fitted well by their model, and the initial spin period P_0 and magnetic dipole field B_ p, which are derived from their fitting, also match well the values expected from the magnetar model. The results are consistent with our findings.
The simulation results in <cit.> suggest that the if a rapidly spinning magnetar powered the GRB X-ray plateau, then magnetars are required to have two different progenitors and formation paths and different magnetic field formation efficiencies.
<cit.> show evidence of energy injection for many short GRBs. In addition, they suggest that the remnant of neutron-star-neutron-star mergers could produce an unstable magnetar rather than collapse directly and immediately to a BH.
The unstable magnetar powers the X-ray plateau.
The results show that nearly half of events forming magnetars could collapse to a BH after a few hundred seconds.
Moreover, in <cit.>, they studied GRB magnetar central engines using the observed plateau luminosity and duration correlation based on both long and short GRBs, and they found that the magnetar emission most likely is narrowly beamed (<20^o) and has a lower efficiency (≲ 20%) in converting rotational energy into the observed X-ray plateau emission.




 §.§ Sphericity of the magnetar wind


One important assumption in the analysis above is whether or not the emitted energy is isotropic, since this affects the derived energetics of the bursts. In the framework of a BH central engine, the rotational energy can naturally be assumed to be extracted through the jet and is deposited into the X-ray plateau. In this case, a beaming correction needs to be made: E_ K = f_ b E_ K, iso.
On the other hand, for a magnetar central engine, the rotational energy is initially emitted in an isotropic wind. Indeed, for short GRBs, the total energy is typically expected to be emitted quasi-spherically (e.g. ). Here, we have also assumed the energy to be emitted quasi-isotropically even for long GRBs during their X-ray afterglow phase (see, e.g. ). Then only a small part of the rotational energy of a magnetar is injected into the external shock of the jet. This means that for the magnetar sample, using this assumption, we should not use the beaming correction, so that f_ b = 1.

We note, however, that it is still unclear whether the energy release for long GRBs during their X-ray afterglow phase is spherical or collimated <cit.>, and there needs to be further theoretical and numerical modeling, as well as observational constraints, to understand the late-time energy release from a magnetar.
Indeed, there are arguments for a collimated outflow from a magnetar central engine. For instance, the progenitor star of long GRBs is expected to cause a collimation of the emitted energy, at least during the prompt phase <cit.>, and this would change the derived total energetics. However, for very late times, such as during the X-ray plateau phase, which is a hundred seconds to a day following the trigger of the GRB, there should be less material surrounding the magnetar. Therefore, the magnetar emission could be expected to come out more isotropically.

Similarly, <cit.> found, by using Swift data of GRB afterglows, that while an isotropic assumption for the emitted energy is reasonable for short GRBs, for most long GRBs, this assumption leads to unrealistic values of the derived rotation period of the neutron star. <cit.> therefore argue that this means that the assumption of isotropic energy release during the afterglow phase is not correct. We note, however, that it could also point to the fact that the progenitor is not a magnetar.

By contrast, an indirect argument for quasi-isotropic magnetar winds is given in <cit.> who studied the energetics of GRBs associated with SNe. They noted that the SN energy is narrowly distributed around the characteristic magnetar energy of ∼ 10^52 erg and is always larger than the GRB energy. On the other hand, the GRB energy varies a lot. They therefore suggest that the central engine for the SN-associated GRBs is a magnetar and that the SN is driven by a close-to-spherical magnetar wind. A consequence is that the jet does not drive the stellar explosion; indeed, it is not clear how the jet would transfer energy to the star after it has escaped.
<cit.> had to make an assumption, though, that the magnetar wind is not fully isotropic based on the fact that SNe, in general, are observed to have some degree of asphericity (e.g. ). The asphericity is, however, assumed to be much smaller than the highly collimated jetted flows from BH central engines, with only f_ b < ∼ 0.5-0.2.
Moreover, <cit.> argued for a magnetar central engine in GRB 111209A, since the associated SN 2011kl did not show any evidence for Ni production. However, its properties are peculiar compared to other GRB/SN cases and might thus not be representative (). In the case of GRB 111209A, a correction for collimation of f_b < 1/50 was required for E_γ, which would suggest an initial jetted outflow.  On the other hand, considering the gamma-ray radiative efficiency found in <ref> (Eq.<ref>, Fig.<ref> and Fig.<ref>) and the derived E_ X, iso the total energy budget could be strongly strained for the simple estimates of a magnetar central engine, in particular if E_ X is assumed to be  isotropic. Indeed, <cit.> argues that additional energy sources, such as radioactive heating, have to be considered apart from the magnetar itself for a consistent scenario.

Following these lines of argument, the assumption made in this paper of f_ b = 1 might be too high. Instead, by employing a factor of f_ b <∼ 0.5-0.2, as argued by <cit.>, the energies become smaller, and more bursts are consistent with being magnetar-bound. For instance, using f_ b = 0.5 (0.2), there are 2 (18) bursts in the Silver sample that would be classified as possibly harboring a magnetar central engine and thus belonging to the Bronze sample (see Table 5). We also calculate the beaming correction, f_ b, that individual bursts need in order to reach the magnetar limit (green line in Fig.<ref>a). These f_ b values are significantly larger than the f_ b values obtained from the jet break. On the other hand, it shows that only a slight anisotropy is needed in order for the total energy to get below the magnetar limit. In such cases, the central engine could still be a magnetar.

This points toward the fact that there could be a continuum of degrees of asphericity (or collimation) of the magnetar wind. By assuming a fully isotropic wind as we did in this paper, then most bursts are inconsistent with the magnetar central engine (our Gold and Silver samples). Only a small fraction of bursts, both short GRBs and long GRBs in our Bronze sample, are consistent with a magnetar central engine assumption. By contrast, <cit.> assume a high degree of collimation and find a large fraction of all bursts to be consistent with the magnetar central engine.
As argued above, the degree of asphericity could lie in between the assumption of <cit.> (high degree of collimation) and this paper (isotropic wind) and could, for instance, be determined by varying the mass of the envelope surrounding the magnetar, which would have a different impact from the focusing of the wind.




 §.§ Conclusions


In this paper, we systematically analyzed the Swift/XRT light curves having a plateau phase in 101 GRBs which were detected before 2017 May. Assuming an isotropic magnetar wind, we draw the following conclusions:




  * A Gold sample, which includes 9 out of the 101 GRBs. Both E_ X,iso and E_ K,iso exceed the energy budget of a magnetar.
Therefore, these most likely harbor a BH central engine.


  * A Silver sample, which includes 69 out of the 101 GRBs. Estimates of their kinetic energy E_ K,iso are larger than the energy budget of a magnetar. Likewise, these most likely have a BH central engine.


  * A Bronze sample, which includes 23 out of the 101 GRBs. These bursts have energy output lower than the maximal energy budget of magnetars, and as such can be considered candidates for the magnetar central engine.


  * We further test the data with the BZ mechanism based on the BH model, and find that the observations of the "Gold" and "Silver" samples are consistent with expectations.
We also test the magnetar model for the Silver and Bronze samples and find that the magnetar surface magnetic filed (B_ p) and initial spin period (P_0) fall into a reasonable range for the Bronze sample but not for the Silver sample. This implies that the Bronze sample is consistent with having a magnetar central engine, but the Silver sample, in general, is inconsistent with the magnetar model.


We also compare the properties of the two central engine candidate samples, and the two types of plateau bursts.




  * BH candidate (Gold+Silver) and Magnetar candidate (Bronze) samples.

 We found that the observational properties (α_1, α_2, Δα, Γ_1, Γ_2, and ΔΓ) and the model parameters (q), in general, do not differ significantly. This indicates that it could be insensitive to the central engine, or in the case where these parameters were to be sensitive to the central engine, it would argue for a common central engine type. However, one interesting result is that the p value distribution for the magnetar candidate sample is significantly narrower than that for the BH candidate sample.
A significant difference is also found for t_ b, z, T_90, L_ X,iso, energies (E_ X,iso, E_ K,iso, and E_γ, iso), and p, which could imply a different physical reason for the different samples.


  * Internal plateau and External plateau sample.

For the internal and the external plateaus, we found that α_2, Δα, Γ_2, ΔΓ, t_ b, and q in general, present different distributions.
The results are consistent with the expectation for different physical progenitors for the two types of plateaus, while the α_1 and Γ_1 generally show similar distributions.



We conclude that, under the assumption of isotropic energy release from a magnetar, most GRBs should harbor a BH central engine.

We acknowledge the use of the public data from the Swift data archive and the UK Swift Science Data Center.
We thank Damien Bégué, Yu Wang, David Yu, Hüsne Dereli, Jin-Jun Geng, Yun-Feng Liang, and Christoffer Lundman for helpful discussions.
We appreciate the valuable comments by the referee and thank Bing Zhang for useful discussions that greatly improved the paper.
This work is supported by the National Basic Research Program (“973" Program) of China (grant No. 2014CB845800), the National Key Research and Development Program of China (grant No. 2017YFA0402600), the National Natural Science Foundation of China (grant Nos. 11725314, 11573014, 11673068, and 11773010), the Youth Innovation Promotion Association (2011231), the Key Research Program of Frontier Sciences (QYZDB-SSW-SYS005), the Strategic Priority Research Program “Multi-waveband gravitational wave Universe” (grant No. XDB23000000) of the Chinese Academy of Sciences, and the Swedish National Space Board, the Swedish Research Council.
L.L. acknowledges the support from the Erasmus Mundus Joint Doctorate Program through grant No. 2013-1471 from the EACEA of the European Commission.
F.R. is supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine. Part of this work made use of our private Python library.



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The X-Ray Observation Properties and Fitting Results of Our Samples
1

GRB
   T_ s-T_ ea
   t_ ba
   α_1
   α_2
   Γ_ x,1b
   Γ_ x,2b
   S_ Xc
   L_ X,isod
   E_ X,isod


   
(ks)   
(ks)   
   
   
   
   
(10^-7 erg cm^-2)   
(10^47 erg s^-1)   
(10^50 erg)
   Gold

050401   0.13-548   5.1±0.6   0.55±0.01   1.47±0.06   1.78±0.21   1.79±0.15   10.88±0.57   55.73±5.10   208.70±10.91   

050730   4.01-408   9.7±0.5   0.72±0.12   2.69±0.04   1.43±0.06   1.63±0.05   11.57±0.64   111.72±12.58   368.39±20.39   

060607A   0.46-171   12.6±0.3   0.44±0.02   3.60±0.09   1.51±0.10   1.56±0.08   12.91±0.17   25.64±1.27   273.82±3.52   

061222A   0.21-1443   47.6±3.1   0.70±0.01   1.74±0.03   1.83±0.06   2.07±0.09   20.18±0.39   4.70±0.34   220.53±4.25   

080721   0.11-1396   3.1±0.2   0.80±0.01   1.65±0.01   1.81±0.02   1.96±0.06   40.29±0.68   239.60±12.48   644.10±10.93   

100902A   1.19-1489   839.9±48.9   0.73±0.02   4.53±0.00   2.28±0.16   2.38±0.54   6.93±0.10   0.88±0.11   268.32±3.79   

111209A   0.43-49   18.4±0.1   0.81±0.00   5.64±0.05   1.24±0.01   1.54±0.05   288.62±0.33   7.79±0.05   364.77±0.41   

140206A   0.42-1316   13.6±1.9   0.73±0.02   1.41±0.02   1.68±0.09   1.81±0.06   15.34±0.72   21.01±3.23   265.97±12.53   

150403A   0.08-2834   2.5±0.1   0.50±0.01   1.45±0.01   1.67±0.01   1.75±0.04   55.11±0.58   279.55±5.79   588.12±6.20   

   Silver

050315   5.37-888   231.4±29.4   0.68±0.03   1.94±0.16   1.89±0.07   2.16±0.21   6.64±0.29   0.37±0.05   64.21±2.83   

050319   0.44-1389   36.8±10.8   0.49±0.04   1.56±0.20   1.92±0.08   2.17±0.23   3.40±0.41   5.43±0.88   78.17±9.38   

050505   2.89-1127   26.1±3.2   0.77±0.05   1.80±0.06   1.98±0.08   2.00±0.10   3.09±0.17   10.91±1.72   111.58±6.13   

050802   0.32-830   6.6±0.7   0.66±0.03   1.66±0.04   1.76±0.09   1.90±0.09   3.46±0.16   4.32±0.60   26.44±1.21   

050803   0.42-1329   16.3±1.3   0.39±0.03   1.82±0.06   1.81±0.10   2.56±0.24   3.58±0.16   0.10±0.01   1.71±0.08   

060115   5.37-418   39.9±27.0   0.60±0.18   1.46±0.22   2.16±0.19   2.15±0.29   0.42±0.13   0.97±0.68   11.19±3.42   

060210   3.83-1581   28.4±10.4   0.83±0.07   1.41±0.05   2.02±0.06   1.97±0.09   5.31±0.82   13.82±5.64   165.07±25.61   

060418   0.36-537   1.3±0.4   0.87±0.13   1.57±0.05   1.94±0.21   1.91±0.23   1.85±0.36   13.46±5.73   10.94±2.15   

060502A   0.33-1581   30.0±9.2   0.55±0.04   1.19±0.06   1.91±0.14   1.84±0.17   2.29±0.27   0.44±0.11   13.88±1.65   

060604   1.24-767   22.3±5.5   0.40±0.09   1.27±0.06   2.07±0.16   2.07±0.17   0.73±0.10   0.77±0.17   8.28±1.13   

060605   0.20-127   8.7±0.6   0.46±0.03   2.19±0.08   1.97±0.12   2.08±0.15   1.45±0.05   13.47±1.18   42.73±1.43   

060614   1.00-2770   50.2±2.6   0.12±0.04   1.97±0.04   1.73±0.10   1.83±0.12   3.02±0.12   (2.12±0.14)e-3   (11.71±0.48)e-2   

060714   0.30-1245   4.6±1.6   0.43±0.11   1.31±0.05   1.81±0.15   2.00±0.18   1.03±0.20   9.04±3.17   17.58±3.36   

060729   0.40-11845   65.8±2.2   0.15±0.02   1.42±0.01   2.04±0.04   2.03±0.05   13.08±0.31   0.18±0.01   10.41±0.25   

060814   1.00-1325   13.2±1.8   0.51±0.05   1.43±0.03   2.00±0.11   2.15±0.09   3.35±0.24   0.57±0.08   6.55±0.47   

060906   0.33-258   12.6±1.9   0.29±0.06   1.82±0.17   2.07±0.21   1.77±0.25   0.54±0.05   2.52±0.38   15.27±1.33   

060908   0.08-450   0.7±0.2   0.44±0.10   1.56±0.07   2.03±0.29   1.97±0.22   1.41±0.21   34.19±9.41   12.86±1.95   

061121   0.30-2097   6.7±0.5   0.41±0.03   1.48±0.02   1.95±0.12   1.82±0.06   9.55±0.43   7.68±0.72   44.70±2.02   

070129   1.15-1490   21.9±3.6   0.24±0.06   1.20±0.04   2.28±0.19   2.13±0.18   1.06±0.12   1.80±0.23   14.11±1.55   

070306   0.50-1036   30.3±1.7   0.13±0.03   1.88±0.05   1.79±0.09   2.11±0.18   6.77±0.27   2.72±0.14   40.47±1.64   

070508   0.08-646   0.9±0.1   0.48±0.02   1.43±0.01   1.77±0.04   1.74±0.07   9.47±0.33   19.66±1.08   17.62±0.62   

070529   0.17-445   1.7±0.8   0.64±0.13   1.29±0.05   1.75±0.21   2.10±0.30   0.70±0.14   12.95±6.48   10.52±2.16   

080310   1.35-378   10.7±0.9   0.13±0.09   1.63±0.06   1.96±0.17   1.82±0.14   0.87±0.06   2.77±0.26   12.39±0.91   

080430   0.30-2879   29.4±4.7   0.39±0.03   1.15±0.03   2.05±0.11   2.06±0.13   1.94±0.16   0.11±0.01   3.16±0.26   

080516   0.14-47   4.2±2.1   0.32±0.10   1.10±0.12   2.40±0.57   2.21±0.40   0.82±0.22   16.04±6.10   18.51±4.97   

080605   0.10-230   0.5±0.0   0.52±0.04   1.37±0.02   1.67±0.04   1.71±0.07   4.61±0.37   113.13±11.75   32.62±2.64   

080905B   0.20-895   4.2±1.1   0.28±0.11   1.46±0.04   1.53±0.16   2.11±0.12   3.80±0.62   32.05±10.09   51.92±8.47   

081008   0.52-261   18.5±4.7   0.86±0.04   1.85±0.14   2.05±0.13   1.83±0.32   3.88±0.24   1.48±0.46   38.19±2.39   

081221   0.26-506   0.6±0.1   0.25±0.12   1.32±0.02   1.99±0.07   2.04±0.11   1.98±0.35   219.92±22.88   24.91±4.36   

090407   5.47-978   120.7±17.4   0.59±0.04   1.95±0.14   2.24±0.13   1.84±0.37   3.05±0.19   0.14±0.02   17.12±1.06   

090418A   0.14-216   3.1±0.5   0.49±0.06   1.57±0.04   2.02±0.22   1.96±0.10   3.92±0.31   11.59±2.50   26.74±2.14   

090510   0.10-67   1.4±0.2   0.62±0.04   2.15±0.09   1.67±0.10   2.05±0.20   3.28±0.19   5.01±0.77   7.39±0.42   

090516   3.89-521   16.9±2.8   0.77±0.09   1.86±0.08   2.05±0.09   2.16±0.12   1.68±0.17   14.99±3.36   56.37±5.73   

090529   6.85-774   234.1±118.0   0.51±0.10   1.74±0.54   1.90±0.44   1.66±0.58   0.41±0.08   0.03±0.02   6.67±1.27   

090530   0.38-791   41.6±15.6   0.54±0.04   1.30±0.13   2.03±0.15   1.93±0.32   1.15±0.16   0.11±0.03   5.02±0.69   

090618   0.40-2841   8.4±0.5   0.72±0.01   1.51±0.01   1.93±0.04   1.86±0.05   24.04±0.53   1.34±0.08   19.13±0.42   

091018   0.07-520   0.5±0.1   0.31±0.06   1.24±0.02   1.84±0.15   2.04±0.09   1.55±0.17   11.60±1.57   4.05±0.45   

100418A   0.40-2091   125.0±17.1   -0.01±0.05   1.66±0.10   1.89±0.29   1.84±0.17   2.09±0.23   (2.26±0.31)e-2   2.24±0.24   

100615A   0.20-163   19.8±4.5   0.39±0.03   1.26±0.10   2.24±0.19   1.92±0.25   12.82±1.44   4.10±0.60   67.44±7.57   

100621A   0.40-1769   99.3±20.6   0.86±0.03   1.66±0.08   2.33±0.10   3.06±0.31   25.66±0.99   0.11±0.03   20.58±0.80   

100704A   0.50-1051   33.3±6.4   0.68±0.03   1.38±0.05   2.04±0.10   1.92±0.17   5.10±0.32   6.15±1.14   139.15±8.85   

100814A   1.00-4848   148.9±7.3   0.50±0.02   2.09±0.06   1.90±0.05   2.08±0.13   12.25±0.28   0.58±0.03   68.13±1.58   

100906A   0.30-202   13.2±1.3   0.76±0.03   2.12±0.80   1.96±0.10   2.05±0.15   11.20±0.35   3.73±0.53   59.73±1.86   

111008A   0.34-1014   7.4±0.9   0.29±0.04   1.34±0.03   1.92±0.09   1.88±0.13   2.57±0.22   52.83±6.48   116.40±9.81   

111123A   4.22-183   42.9±12.3   0.81±0.08   2.02±0.29   2.23±0.20   2.31±0.47   0.99±0.10   1.44±0.52   21.80±2.28   

111228A   0.50-2572   6.5±0.8   0.22±0.05   1.23±0.02   2.01±0.15   2.15±0.10   3.29±0.30   1.00±0.11   4.63±0.42   

120327A   0.30-146   3.0±0.6   0.60±0.06   1.51±0.06   1.69±0.14   1.82±0.17   1.97±0.20   20.09±4.75   35.78±3.65   

120712A   0.10-67   4.6±1.5   0.89±0.04   1.64±0.10   2.01±0.15   2.25±0.20   1.12±0.08   15.03±6.22   36.00±2.55   

120811C   0.30-81   1.7±0.8   0.36±0.15   1.16±0.09   1.78±0.15   2.24±0.21   0.86±0.28   34.08±11.77   14.41±4.75   

120922A   0.74-751   3.4±1.6   0.43±0.19   1.09±0.04   2.20±0.24   1.98±0.15   0.87±0.31   18.93±8.11   18.53±6.56   

121024A   3.90-236   29.7±12.7   0.75±0.11   1.67±0.18   1.93±0.14   1.58±0.27   1.07±0.19   0.49±0.23   13.87±2.48   

121128A   0.20-70   1.6±0.2   0.52±0.07   1.68±0.04   1.91±0.11   1.97±0.13   2.91±0.26   43.37±7.23   34.86±3.13   

121211A   3.94-362   36.1±16.6   0.71±0.09   1.51±0.19   1.94±0.12   2.27±0.44   1.46±0.27   0.15±0.07   4.21±0.79   

130420A   0.76-1295   65.4±29.2   0.72±0.03   1.27±0.09   2.13±0.10   1.96±0.23   1.98±0.24   0.10±0.04   9.03±1.08   

130606A   5.32-238   13.0±3.6   0.44±0.30   1.82±0.14   1.79±0.17   1.85±0.21   0.57±0.16   15.88±5.53   32.93±9.38   

130612A   0.76-44   3.0±2.1   0.38±0.36   1.24±0.16   2.19±0.40   2.11±0.43   0.09±0.04   0.95±0.60   0.91±0.46   

131030A   0.36-1661   3.4±1.1   0.81±0.05   1.28±0.02   1.71±0.04   1.95±0.09   6.88±0.91   9.39±3.26   31.22±4.13   

131105A   0.41-734   4.8±1.3   0.23±0.13   1.20±0.05   1.90±0.17   1.99±0.18   1.04±0.20   3.32±0.92   7.74±1.50   

140512A   0.23-264   17.7±1.7   0.79±0.01   1.68±0.05   1.76±0.07   1.90±0.09   24.66±0.64   0.96±0.10   35.81±0.94   

140518A   0.29-19   2.6±0.6   0.19±0.10   1.53±0.18   1.99±0.15   1.94±0.34   0.43±0.07   28.04±4.55   17.74±2.88   

140703A   3.81-79   14.5±1.6   0.60±0.10   2.28±0.10   1.78±0.10   1.94±0.15   3.66±0.30   16.93±2.78   80.03±6.66   

141121A   56.28-1393   358.5±51.9   0.44±0.12   2.47±0.26   1.79±0.23   1.72±0.21   2.58±0.24   0.06±0.01   14.93±1.39   

150910A   0.20-224   7.6±0.4   0.57±0.01   2.36±0.06   1.83±0.20   1.75±0.08   16.07±0.32   8.86±0.57   80.16±1.59   

151027A   0.40-908   4.4±0.2   0.22±0.02   1.68±0.02   2.03±0.04   2.01±0.06   17.07±0.50   9.88±0.47   30.99±0.91   

160121A   0.20-46   20.8±4.9   0.30±0.05   2.14±0.77   1.97±0.23   1.85±0.66   1.20±0.15   0.84±0.14   11.76±1.42   

160227A   1.00-1647   21.2±3.7   0.24±0.09   1.22±0.03   1.73±0.09   1.73±0.10   3.06±0.36   3.43±0.56   42.01±4.91   

160327A   0.30-42   4.8±1.9   0.40±0.15   1.60±0.14   1.71±0.22   2.03±0.23   0.45±0.09   16.51±7.14   20.33±4.04   

161117A   1.05-1365   7.5±1.4   0.36±0.11   1.17±0.03   2.11±0.86   1.95±0.08   2.17±0.29   3.42±0.57   13.82±1.86   

170113A   0.20-877   6.0±1.2   0.57±0.03   1.26±0.03   1.83±0.11   1.81±0.09   3.77±0.32   7.02±1.36   37.13±3.17   

   Bronze

051109B   0.18-158   3.3±0.8   0.26±0.10   1.31±0.08   2.22±0.33   1.86±0.34   0.23±0.04   (8.38±1.73)e-4   (3.63±0.57)e-3   

051221A   3.00-848   26.7±5.2   0.18±0.15   1.43±0.06   1.82±0.22   1.96±0.15   0.39±0.07   (1.77±0.34)e-2   0.32±0.06   

060526   1.00-314   97.1±20.7   0.63±0.60   2.89±0.48   1.91±0.16   1.91±0.45   1.22±0.08   0.38±0.14   27.66±1.77   

060708   0.20-1228   8.8±2.2   0.59±0.04   1.32±0.05   2.10±0.18   2.08±0.16   0.88±0.08   (8.73±2.07)e-3   0.14±0.01   

061021   4.20-4236   29.7±13.9   0.75±0.07   1.19±0.03   1.94±0.08   1.87±0.08   2.81±0.61   0.02±0.01   0.89±0.20   

061110A   3.00-756   73.2±51.6   0.19±0.27   1.16±0.25   2.11±0.72   1.65±0.46   0.14±0.06   (3.56±2.06)e-3   0.23±0.10   

061201   0.11-120   2.6±0.5   0.53±0.09   2.01±0.11   1.33±0.18   2.01±0.40   1.36±0.11   (8.83±2.57)e-3   (4.13±0.34)e-2   

070110   4.08-29   20.4±0.2   0.03±0.05   9.39±0.69   2.07±0.08   2.09±0.19   3.23±0.04   2.75±0.11   20.55±0.24   

080707   0.11-279   14.4±4.3   0.36±0.05   1.29±0.10   2.05±0.30   1.83±0.23   0.47±0.07   0.15±0.04   1.95±0.29   

081007   0.30-1534   68.9±28.9   0.75±0.03   1.35±0.10   2.01±0.13   2.03±0.21   2.66±0.26   0.01±0.01   2.03±0.20   

081029   2.76-268   18.2±1.2   0.43±0.06   2.70±0.15   1.92±0.10   2.04±0.21   1.03±0.05   6.24±0.61   31.11±1.42   

100219A   5.00-121   26.4±3.6   0.24±0.31   2.96±0.83   1.66±0.30   1.48±0.39   1.70±0.18   5.26±1.28   69.43±7.38   

100302A   1.29-1062   17.0±11.5   0.30±0.14   0.92±0.08   1.88±0.35   1.95±0.22   0.17±0.07   1.61±0.72   7.12±2.92   

100425A   0.34-498   25.3±13.7   0.48±0.06   1.16±0.14   2.22±0.26   1.91±0.39   0.43±0.09   0.17±0.07   3.43±0.74   

110715A   0.10-847   0.2±0.0   0.20±0.12   1.00±0.01   1.87±0.10   1.83±0.09   1.84±0.40   35.25±3.79   3.42±0.74   

110808A   4.00-617   40.9±37.2   0.33±0.20   1.07±0.19   2.54±0.49   1.61±0.39   0.35±0.18   0.05±0.03   1.71±0.87   

120422A   0.30-1703   164.6±69.5   0.27±0.07   1.27±0.21   2.09±0.26   1.61±0.29   0.22±0.05   (1.98±0.61)e-4   0.05±0.01   

120521C   0.57-42   21.2±5.2   0.31±0.10   2.55±0.80   1.82±0.32   2.19±0.77   0.25±0.03   4.35±1.06   14.92±1.86   

130408A   9.55-79   27.2±2.7   0.55±0.21   3.87±0.42   2.26±0.19   2.53±0.25   1.04±0.09   11.83±2.66   30.26±2.63   

130603B   0.07-118   7.6±1.3   0.68±0.05   1.90±0.12   1.94±0.16   1.97±0.28   2.23±0.12   0.04±0.01   0.75±0.04   

140903A   0.21-111   8.9±1.7   0.10±0.07   1.28±0.09   1.70±0.21   1.56±0.20   0.93±0.13   (3.15±0.46)e-2   0.30±0.04   

151215A   0.30-175   1.9±2.0   0.54±0.20   1.17±0.14   2.23±0.22   1.91±0.55   0.22±0.11   3.92±3.47   3.42±1.76   

161108A   0.65-645   41.6±23.3   0.24±0.09   0.94±0.13   1.78±0.25   1.76±0.32   0.72±0.24   0.08±0.02   2.65±0.87   

aThe start, end, and the break times for the light curve fits.
bThe X-ray photo indices before and after the break time.
cThe fluence during the plateau phase, calculated by integrating the fitting light curve from start to break time.
dThe luminosity at the break time, in units of 10^47 erg s^-1. The X-ray release energy during plateau phases, which is calculated by integrating the luminosity during the time interval of the plateau phase, in units of 10^50 erg.




lccccccccccccccccccccc


0pt


The Properties of the BH Candidate (Gold+Silver) Samples
2

GRB
   T_90
   E_γ isoa
   E_ K,iso(t_b)b
   E_ K,iso(t_0)b
   a_0c
   p
   q
   z

   
(s)   
(10^52 erg)   
(10^52 erg)   
(10^52 erg)   
(10^-1)   
   
   
(redshift)   

   Gold

050401   33.3   42.23±8.95   71.53±29.27   0.75±0.50   1.39±0.29   2.29±0.06   -0.26±0.14   2.90   

050730   60.0   0.01±0.00   ...d   ...   ...   2.50±0.10   1.06±0.12   3.967   

060607A   102.2   7.14±0.34   ...   ...   ...   2.50±0.10   0.74±0.12   3.0749   

061222A   71.4   35.64±14.53   158.72±17.52   19.66±6.52   2.06±0.12   3.33±0.03   0.61±0.06   2.088   

080721   16.2   69.31±4.40   78.49±6.24   30.10±2.92   1.45±0.06   3.20±0.01   0.71±0.02   2.591   

100902A   428.8   24.09±1.96   ...   ...   ...   2.50±0.10   0.27±0.11   4.5   

111209A   14400.0   82.27±10.88   ...   ...   ...   2.50±0.10   1.40±0.01   0.677   

140206A   93.6   30.68±1.12   46.28±8.34   21.92±8.46   1.12±0.10   2.88±0.02   0.79±0.10   2.73   

150403A   40.9   1.51±0.17   248.80±15.64   4.02±0.31   2.57±0.08   2.26±0.01   -0.21±0.01   2.06   

   Silver

050315   95.6   5.59±0.58   388.63±97.31   70.55±24.44   3.19±0.40   3.58±0.16   0.55±0.06   1.949   

050319   152.5   8.02±1.70   41.83±19.26   2.86±1.66   1.06±0.25   3.08±0.20   0.39±0.07   3.240   

050505   58.9   21.69±1.78   224.36±77.07   15.65±6.21   2.44±0.43   2.73±0.06   -0.21±0.06   4.27   

050802   19.0   3.23±0.47   16.33±4.93   0.55±0.21   0.67±0.11   2.55±0.04   -0.12±0.07   1.71   

050803   87.9   0.25±0.02   37.50±12.04   10.09±6.43   1.01±0.17   3.09±0.06   0.64±0.15   0.42   

060115   139.6   6.05±1.16   85.68±119.93   26.30±40.87   1.52±0.67   2.61±0.22   0.41±0.27   3.53   

060210   255.0   16.04±0.89   42.78±20.56   16.78±8.86   1.08±0.28   2.88±0.05   0.53±0.07   3.91   

060418   103.1   12.75±1.53   6.50±3.16   4.02±2.29   0.42±0.10   3.10±0.05   0.63±0.21   1.489   

060502A   28.4   1.76±0.10   4.05±1.39   0.32±0.21   0.33±0.06   2.59±0.06   0.44±0.12   1.51   

060604   95.0   0.26±0.06   29.57±11.70   4.19±3.01   0.89±0.19   2.36±0.06   0.32±0.20   2.1357   

060605   79.1   1.67±0.16   145.59±51.59   0.48±0.22   1.97±0.38   3.25±0.08   -0.52±0.07   3.78   

060614   108.7   0.04±0.01   8.33±1.26   0.14±0.03   0.48±0.04   2.96±0.04   -0.70±0.06   0.125   

060714   115.0   7.95±1.50   5.39±2.21   1.38±0.81   0.38±0.09   2.75±0.05   0.44±0.15   2.711   

060729   115.3   0.53±0.05   15.56±0.76   0.19±0.03   0.65±0.02   2.90±0.01   0.07±0.03   0.54   

060814   145.3   6.09±0.42   32.66±7.73   9.20±4.16   0.94±0.12   2.57±0.03   0.51±0.15   1.9229   

060906   43.5   9.13±1.42   28.23±24.05   0.05±0.05   0.87±0.31   2.76±0.17   -0.75±0.11   3.686   

060908   19.3   3.39±0.15   4.86±2.02   1.01±0.67   0.36±0.08   3.08±0.07   0.27±0.23   1.8836   

061121   81.3   37.89±3.43   28.81±5.64   0.23±0.07   0.88±0.09   2.31±0.02   -0.55±0.07   1.314   

070129   460.6   8.06±0.83   42.92±12.30   2.04±1.24   1.08±0.16   2.27±0.04   -0.03±0.17   2.3384   

070306   209.5   4.96±0.41   126.93±17.89   5.76±2.00   1.85±0.14   3.51±0.05   0.25±0.08   1.4959   

070508   20.9   9.29±0.69   4.09±0.63   0.18±0.03   0.33±0.02   2.24±0.01   -0.33±0.03   0.82   

070529   109.2   4.70±0.50   17.06±15.08   17.91±22.88   0.68±0.24   2.39±0.05   1.02±0.39   2.4996   

080310   365.0   7.20±1.86   15.82±4.29   0.34±0.13   0.65±0.09   2.50±0.06   -0.85±0.10   2.42   

080430   16.2   0.45±0.04   8.37±2.10   0.39±0.25   0.48±0.06   2.20±0.03   0.33±0.13   0.767   

080516   5.8   0.98±0.18   35.50±37.61   0.92±1.82   0.98±0.36   2.14±0.12   -0.07±0.47   3.2   

080605   20.0   11.88±1.01   6.90±1.77   1.12±0.33   0.43±0.06   2.16±0.02   -0.18±0.04   1.6398   

080905B   128.0   5.76±0.83   16.44±6.21   4.69±3.24   0.67±0.14   2.95±0.04   0.59±0.19   2.374   

081008   185.5   8.30±1.19   79.29±72.06   1.16±1.17   1.46±0.51   2.80±0.14   -0.18±0.08   1.9685   

081221   34.0   34.30±1.55   8.03±1.35   4.16±0.88   0.47±0.04   2.75±0.02   0.18±0.10   2.26   

090407   310.0   0.94±0.20   (6.90±3.49)e2   5.13±3.04   4.20±1.04   2.94±0.14   -0.58±0.07   1.4485   

090418A   56.0   17.06±2.31   12.57±2.99   1.45±0.87   0.58±0.07   3.10±0.04   0.31±0.17   1.608   

090510   0.3   0.34±0.07   8.24±4.58   0.51±0.31   0.47±0.13   3.20±0.09   -0.05±0.08   0.903   

090516   210.0   69.02±7.54   75.01±21.53   34.50±11.42   1.42±0.23   3.48±0.08   0.47±0.09   4.109   

090529   100.0   1.41±0.46   91.37±155.70   0.62±1.29   1.57±0.77   2.65±0.54   -0.41±0.27   2.625   

090530   48.0   0.68±0.08   3.19±1.40   0.14±0.11   0.29±0.07   2.73±0.13   0.34±0.12   1.266   

090618   113.2   13.39±1.09   12.72±0.98   3.08±0.43   0.59±0.02   3.01±0.01   0.53±0.04   0.54   

091018   4.4   0.56±0.06   7.96±1.96   2.91±1.49   0.46±0.06   2.33±0.02   0.51±0.22   0.971   

100418A   7.0   0.10±0.06   33.95±11.78   (4.53±4.21)e-4   0.96±0.18   2.55±0.10   -0.95±0.14   0.6235   

100615A   39.0   5.82±0.24   16.66±5.28   0.27±0.18   0.67±0.11   2.67±0.10   0.10±0.12   1.398   

100621A   63.6   2.75±0.29   91.35±62.62   4.43±4.08   1.57±0.48   2.88±0.08   0.45±0.11   0.542   

100704A   197.5   17.75±1.07   29.45±7.24   2.55±1.12   0.89±0.12   2.84±0.05   0.42±0.08   3.6   

100814A   174.5   15.01±0.97   (6.16±1.25)e3   35.06±8.51   9.68±0.41   3.12±0.06   -0.42±0.03   1.44   

100906A   114.4   29.48±0.98   (3.33±0.97)e2   3.49±1.40   2.96±0.42   3.16±0.80   -0.20±0.06   1.727   

111008A   63.46   42.11±3.66   23.77±3.92   3.15±0.88   0.80±0.07   2.78±0.03   0.25±0.08   4.9898   

111123A   290.0   23.78±1.47   89.24±71.83   20.07±18.06   1.55±0.53   3.70±0.29   0.36±0.15   3.1516   

111228A   101.2   4.17±0.45   13.89±2.91   1.82±0.90   0.61±0.07   2.31±0.02   0.21±0.17   0.714   

120327A   62.9   8.78±0.34   8.45±5.07   0.67±0.46   0.48±0.14   2.35±0.06   -0.10±0.11   2.813   

120712A   14.7   18.57±0.27   8.69±4.74   1.77±1.34   0.49±0.14   3.19±0.10   0.59±0.13   4.1745   

120811C   26.8   6.96±1.04   27.80±24.19   15.01±15.21   0.87±0.30   2.21±0.09   0.65±0.28   2.671   

120922A   173.0   21.38±3.36   59.62±41.06   18.00±16.35   1.27±0.39   2.12±0.04   0.21±0.30   3.1   

121024A   69.0   1.84±0.22   32.61±45.48   2.92±4.39   0.94±0.44   2.56±0.18   -0.19±0.12   2.298   

121128A   23.3   1.05±0.11   14.73±3.61   4.40±1.49   0.63±0.08   3.24±0.04   0.42±0.10   2.2   

121211A   182.0   0.17±0.02   37.76±37.17   24.19±26.08   1.01±0.37   2.68±0.19   0.80±0.19   1.023   

130420A   123.5   0.23±0.28   3.72±1.86   0.24±0.16   0.32±0.08   2.69±0.09   0.38±0.08   1.297   

130606A   276.58   21.67±2.04   86.86±92.64   25.01±28.81   1.53±0.59   2.76±0.14   -0.39±0.24   5.913   

130612A   4.0   0.76±0.11   3.19±4.66   1.02±1.75   0.29±0.13   2.33±0.16   0.17±0.51   2.006   

131030A   41.1   17.22±0.60   6.73±2.61   4.38±1.80   0.43±0.09   2.70±0.02   0.81±0.06   1.295   

131105A   112.3   34.31±2.63   22.24±10.29   4.48±3.53   0.78±0.19   2.26±0.05   0.35±0.25   1.686   

140512A   154.8   9.07±0.41   51.44±14.15   0.76±0.28   1.18±0.17   2.57±0.05   0.03±0.05   0.725   

140518A   60.5   4.99±0.92   8.41±5.03   1.25±0.86   0.48±0.14   3.04±0.18   0.14±0.13   4.707   

140703A   67.1   1.95±0.11   (1.26±0.58)e3   (2.53±1.26)e2   5.53±1.21   3.37±0.10   -0.20±0.10   3.14   

141121A   549.9   0.02±0.00   (5.42±3.90)e4   (4.10±3.31)e3   9.14±2.98   3.63±0.26   -0.39±0.17   1.47   

150910A   112.2   2.06±0.35   (8.75±2.03)e2   8.10±4.27   4.69±0.50   3.48±0.06   -0.29±0.13   1.359   

151027A   129.69   3.38±0.32   28.71±2.04   3.51±0.38   0.88±0.03   3.24±0.02   0.12±0.03   0.81   

160121A   12.0   0.77±0.25   (2.08±0.98)e2   8.52±4.08   0.24±0.07   3.18±0.77   -0.68±0.12   1.960   

160227A   316.5   4.06±1.70   12.76±2.62   1.92±0.77   0.59±0.06   2.63±0.03   0.38±0.11   2.38   

160327A   28.0   8.90±5.88   10.38±6.74   2.67±2.49   0.53±0.16   3.13±0.14   0.51±0.23   4.99   

161117A   125.7   23.29±0.76   42.15±11.73   9.31±17.07   1.07±0.15   2.23±0.03   0.23±0.92   1.549   

170113A   20.66   0.63±0.38   8.07±1.98   1.61±0.70   0.47±0.06   2.68±0.03   0.53±0.10   1.968   

aWe searched for E_γ,  iso in published papers or is using the fluence and redshift extrapolated into 1-10^4keV (rest frame) with a spectral model and a k-correction, in units of 10^52 erg.
bThe kinetic energy, which is derived from the normal decay phase. The initial kinetic energy (upper limit) is estimated from the kinetic energy at break time (t_ b).
cThe BH spin parameter a_∙, in units of 10^-1 s.
dWe did not calculate the kinetic energy for the bursts that have an internal plateau.




lccccccccccccccccccccc


0pt


The Properties of the Magnetar Candidate (Bronze) Sample
3

GRB
   T_90
   E_γ,iso
   q
   E_ K,iso(t_b)a
   B_pb
   P_0b
   E_ rotc
   z

   
(s)   
(10^52 erg)   
   
(10^52 erg)   
(10^15 G)   
(10^-3 s)   
(10^50 erg)   
(redshift)   

051109B   14.3   (6.79±1.06)e-4   0.02±0.22   0.02±0.01   14.44±3.54   12.87±3.05   (4.02±0.10)e-2   0.08

051221A   1.4   0.36±0.20   0.25±0.21   1.50±0.37   0.52±0.11   1.33±0.25   3.78±1.05   0.5465

060526   296.0   1.62±0.26   0.50±0.43   ...d   0.68±0.15   3.31±0.71   0.61±0.08   3.221

060708   10.2   0.96±0.10   0.47±0.22   0.52±0.22   1.71±0.47   2.51±0.67   1.06±0.23   1.92

061021   46.2   0.50±0.21   0.55±0.08   ...   9.27±4.33   24.95±8.20   0.01±0.00   0.3463

061110A   40.7   0.44±0.14   0.05±0.51   0.38±0.26   0.59±0.42   2.47±1.49   1.09±0.53   0.758

061201   0.76   0.03±0.03   0.30±0.19   0.10±0.07   7.68±1.82   6.10±2.21   0.18±0.02   0.111

070110   88.4   2.97±0.26   -0.03±0.06   ...   1.21±0.02   2.70±0.06   0.91±0.01   2.352

080707   27.1   0.33±0.05   0.20±0.22   1.64±0.62   0.56±0.22   1.06±0.30   5.97±3.18   1.23

081007   10.0   0.06±0.01   0.49±0.11   1.95±0.98   0.29±0.13   1.17±0.45   4.86±3.21   0.5295

081029   270.0   12.55±1.93   0.35±0.09   ...   0.90±0.07   1.90±0.11   1.85±0.12   3.8479

100219A   18.8   1.78±0.35   0.44±0.36   ...   0.68±0.10   1.72±0.24   2.26±0.37   4.6667

100302A   17.9   2.03±0.36   0.29±0.30   ...   1.90±1.29   3.86±1.57   0.45±0.09   4.813

100425A   37.0   0.49±0.12   0.16±0.18   1.32±0.76   0.42±0.26   1.06±0.50   5.98±5.37   1.755

110715A   13.0   2.79±0.13   0.23±0.11   1.81±0.27   4.63±3.79   1.10±0.14   5.48±1.30   0.82

110808A   48.0   17.86±3.03   -0.12±0.28   0.88±0.69   0.44±0.42   1.40±1.05   3.41±3.65   1.35

120422A   5.35   (6.31±1.18)e-3   -0.79±0.13   0.07±0.08   1.04±0.44   6.56±4.00   0.15±0.03   0.283

120521C   26.7   10.05±1.30   0.35±0.28   ...   0.93±0.23   2.11±0.37   1.50±0.25   6.0

130408A   28.0   8.21±1.63   0.18±0.18   ...   0.44±0.06   1.13±0.14   5.24±1.14   3.757

130603B   0.18   0.36±0.05   -0.27±0.11   1.73±1.02   0.97±0.37   1.32±0.42   3.82±1.85   0.3564

140903A   0.3   0.06±0.01   -0.70±0.13   0.34±0.15   2.04±0.45   3.00±0.78   0.74±0.13   0.351

151215A   17.8   0.55±0.48   0.19±0.19   1.16±1.46   1.43±3.74   0.98±0.95   6.91±13.57   2.59

161108A   105.1   0.54±0.13   0.33±0.22   0.66±0.32   0.53±0.30   1.68±0.77   2.37±1.30   1.159

aThe kinetic energy, which is calculated from the normal decay segment; the kinetic energy of the afterglow. 
bDipolar magnetic field strength at the polar cap in units of 10^15 G, and the initial spin period of the magnetar in units of milliseconds, assuming an isotropic magnetar wind.
cThe total rotation energy of the millisecond magnetar.
dWe did not calculate the kinetic energy for the bursts that have an internal plateau.




lccccccccccccccccccccc


0pt


The Results of the Estimation Jet-opening Angle
4

GRB
   θ_ja
   f_ba
   a_jb
   θ^'_jc
   f^'_bc

   
(10^-2)   
(10^-2)   
   
(10^-2)   
(10^-2)   

   Gold

050401   15.09±0.62   1.14±0.09   10.79   6.99±0.39   0.24±0.03   

050730   ...   ...   ...   59.76±1.62   17.36±0.91   

060607A   ...   ...   ...   11.10±0.14   0.62±0.02   

061222A   21.42±0.22   2.29±0.05   14.64   7.31±0.77   0.27±0.11   

080721   21.85±0.16   2.38±0.03   17.75   6.15±0.10   0.19±0.01   

100902A   ...   ...   ...   8.09±0.17   0.33±0.01   

111209A   ...   ...   ...   5.88±0.20   0.17±0.01   

140206A   22.53±0.38   2.53±0.09   14.81   7.60±0.07   0.29±0.01   

150403A   19.21±0.11   1.84±0.02   5.77   16.62±0.49   1.38±0.09   

   Silver

050315   16.26±0.38   1.32±0.06   6.87   11.83±0.32   0.70±0.04   

050319   22.17±0.96   2.45±0.21   10.29   10.77±0.59   0.58±0.07   

050505   15.14±0.50   1.14±0.08   9.10   8.31±0.18   0.35±0.01   

050802   24.31±0.71   2.94±0.17   8.90   13.64±0.51   0.93±0.08   

050803   33.30±1.00   5.49±0.33   6.26   26.57±0.62   3.52±0.17   

060115   12.60±1.69   0.79±0.21   5.43   11.59±0.58   0.67±0.07   

060210   21.96±0.95   2.40±0.21   12.20   8.99±0.13   0.40±0.01   

060418   23.92±1.07   2.85±0.25   12.52   9.54±0.30   0.46±0.03   

060502A   37.92±1.21   7.10±0.45   11.85   15.98±0.24   1.28±0.04   

060604   20.74±0.76   2.14±0.16   3.95   26.25±1.56   3.43±0.50   

060605   7.39±0.25   0.27±0.02   2.28   16.18±0.40   1.31±0.07   

060614   57.78±0.84   16.23±0.46   6.93   41.67±3.32   8.57±2.38   

060714   28.93±1.14   4.16±0.33   13.39   10.79±0.53   0.58±0.06   

060729   81.80±0.38   31.63±0.28   18.69   21.86±0.51   2.38±0.12   

060814   30.50±0.66   4.61±0.20   13.17   11.57±0.21   0.67±0.02   

060906   11.94±0.88   0.71±0.10   5.73   10.41±0.42   0.54±0.05   

060908   21.96±0.83   2.40±0.18   8.14   13.47±0.16   0.91±0.02   

061121   34.03±0.64   5.73±0.21   23.64   7.19±0.17   0.26±0.01   

070129   24.81±0.69   3.06±0.17   11.52   10.75±0.29   0.58±0.03   

070306   21.11±0.28   2.22±0.06   8.65   12.20±0.26   0.74±0.04   

070508   30.55±0.42   4.63±0.13   14.73   10.36±0.20   0.54±0.02   

070529   17.39±1.41   1.51±0.24   7.02   12.37±0.34   0.77±0.04   

080310   16.64±0.47   1.38±0.08   7.51   11.07±0.75   0.61±0.11   

080430   49.60±1.17   12.05±0.56   10.86   22.81±0.50   2.60±0.12   

080516   6.36±0.57   0.20±0.04   1.71   18.60±0.87   1.73±0.17   

080605   16.90±0.41   1.43±0.07   8.69   9.72±0.21   0.47±0.02   

080905B   23.01±0.79   2.64±0.18   9.80   11.73±0.44   0.69±0.05   

081008   12.50±1.05   0.78±0.13   5.85   10.67±0.40   0.57±0.05   

081221   20.59±0.31   2.11±0.06   13.94   7.38±0.09   0.27±0.01   

090407   16.82±0.78   1.41±0.13   4.47   18.80±1.04   1.77±0.23   

090418A   15.38±0.37   1.18±0.06   8.68   8.85±0.31   0.39±0.03   

090510   11.78±0.60   0.69±0.07   2.40   24.56±1.28   3.01±0.35   

090516   13.30±0.37   0.88±0.05   10.80   6.15±0.17   0.19±0.01   

090529   17.12±2.75   1.46±0.47   5.06   16.90±1.42   1.43±0.32   

090530   31.32±1.35   4.86±0.41   7.64   20.48±0.64   2.09±0.14   

090618   49.17±0.34   11.85±0.16   26.06   9.42±0.20   0.44±0.02   

091018   25.15±0.57   3.14±0.14   5.84   21.49±0.64   2.31±0.14   

100418A   38.01±1.29   7.14±0.48   5.64   33.66±5.25   5.62±3.41   

100615A   13.80±0.38   0.95±0.05   5.89   11.70±0.13   0.69±0.02   

100621A   32.17±1.89   5.13±0.60   11.30   14.22±0.39   1.01±0.06   

100704A   20.22±0.48   2.04±0.10   11.54   8.76±0.14   0.38±0.01   

100814A   23.35±0.46   2.71±0.11   12.75   9.15±0.15   0.42±0.01   

100906A   10.25±0.29   0.53±0.03   6.67   7.68±0.07   0.29±0.00   

111008A   18.48±0.29   1.70±0.05   13.19   7.00±0.16   0.25±0.01   

111123A   9.50±0.62   0.45±0.06   5.84   8.12±0.13   0.33±0.01   

111228A   45.20±0.88   10.04±0.38   17.69   12.76±0.35   0.81±0.05   

120327A   12.11±0.65   0.73±0.08   5.75   10.52±0.11   0.55±0.01   

120712A   8.15±0.40   0.33±0.03   4.70   8.66±0.03   0.38±0.00   

120811C   8.48±0.72   0.36±0.06   3.79   11.17±0.44   0.62±0.05   

120922A   17.05±1.19   1.45±0.20   10.21   8.34±0.34   0.35±0.03   

121024A   12.93±1.65   0.83±0.21   4.09   15.79±0.49   1.25±0.08   

121128A   9.15±0.21   0.42±0.02   2.50   18.27±0.51   1.67±0.09   

121211A   17.91±1.84   1.60±0.33   3.03   29.53±0.89   4.34±0.26   

130420A   36.71±1.83   6.66±0.66   6.74   27.22±8.91   3.69±6.18   

130606A   8.70±0.88   0.38±0.08   5.23   8.31±0.20   0.35±0.02   

130612A   9.55±1.24   0.46±0.12   2.40   19.89±0.75   1.98±0.15   

131030A   37.50±1.36   6.95±0.50   21.22   8.83±0.08   0.39±0.01   

131105A   22.41±1.02   2.50±0.23   15.17   7.38±0.15   0.27±0.01   

140512A   16.23±0.43   1.31±0.07   7.78   10.43±0.12   0.54±0.01   

140518A   4.83±0.27   0.12±0.01   1.98   12.18±0.58   0.74±0.08   

140703A   4.98±0.23   0.12±0.01   1.60   15.55±0.23   1.21±0.04   

141121A   11.08±0.74   0.61±0.08   1.08   51.16±2.00   12.83±1.06   

150910A   9.53±0.20   0.45±0.02   3.10   15.33±0.67   1.18±0.11   

151027A   27.26±0.19   3.69±0.05   10.10   13.47±0.33   0.91±0.05   

160121A   5.78±1.13   0.17±0.06   1.46   19.82±1.65   1.96±0.53   

160227A   29.86±0.58   4.42±0.17   11.61   12.85±1.40   0.83±0.78   

160327A   6.23±0.35   0.19±0.02   2.97   10.48±1.80   0.55±0.65   

161117A   26.62±0.72   3.52±0.19   16.30   8.16±0.07   0.33±0.01   

170113A   26.20±0.58   3.41±0.15   6.26   20.89±3.28   2.18±2.39   

   Bronze

051109B   41.94±1.24   8.67±0.51   1.70   123.29±5.00   66.97±5.04   

051221A   40.77±1.01   8.19±0.40   8.45   24.10±3.51   2.90±3.66   

060526   ...   ...   ...   16.32±0.67   1.33±0.13   

060708   57.94±2.16   16.32±1.18   15.47   18.71±0.53   1.75±0.10   

061021   ...   ...   ...   22.15±2.45   2.45±1.30   

061110A   44.19±2.72   9.60±1.16   9.65   22.88±1.82   2.61±0.63   

061201   31.17±2.04   4.82±0.63   3.44   45.28±10.32   10.10±11.27   

070110   ...   ...   ...   13.94±0.32   0.97±0.04   

080707   23.18±0.81   2.67±0.19   4.70   24.61±0.91   3.02±0.24   

081007   49.07±2.29   11.80±1.08   6.50   37.69±2.13   7.03±0.89   

081029   ...   ...   ...   9.58±0.38   0.46±0.04   

100219A   ...   ...   ...   15.92±0.80   1.27±0.14   

100302A   ...   ...   ...   15.39±0.71   1.18±0.12   

100425A   27.32±1.49   3.71±0.40   6.12   22.29±1.41   2.48±0.39   

110715A   37.45±0.52   6.93±0.19   13.20   14.17±0.17   1.00±0.02   

110808A   33.05±2.64   5.41±0.86   18.88   8.74±0.39   0.38±0.04   

120422A   82.70±7.70   32.29±5.66   5.98   69.05±3.36   22.95±2.30   

120521C   ...   ...   ...   10.15±0.34   0.52±0.04   

130408A   ...   ...   ...   10.70±0.55   0.57±0.06   

130603B   20.06±1.08   2.00±0.21   4.15   24.12±0.81   2.90±0.20   

140903A   24.16±1.01   2.90±0.24   3.14   38.45±1.67   7.31±0.68   

151215A   16.97±1.96   1.44±0.33   3.92   21.59±4.87   2.33±2.59   

161108A   35.96±1.65   6.40±0.58   8.29   21.68±1.33   2.35±0.32   

aEstimate of the jet-opening angle from the last data point of the Swift/XRT observation using Eq.(<ref>).
ba_j value was estimated by setting the ratio of the θ_j,0 equal to θ_j.
cEstimate of the jet-opening angle from the empirical relation of θ_j,0≈5.0/Γ_0 <cit.>.



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0pt


The relationship between the f_ b value and the sample size
5

f_ b
   Silver to Bronzea
   GRB-SN Burstsb
   Fractions
   Fractions

   
(Numbers)   
(Numbers)   
(for Silver to Bronze)   
(for GRB-SN Bursts)   

f_ b=0.7   0   0   0.0%   0.0%   

f_ b=0.6   1   0   1.6%   0.0%   

f_ b=0.5   2   0   3.2%   0.0%   

f_ b=0.4   5   0   8.1%   0.0%   

f_ b=0.3   6   0   11.3%   0.0%   

f_ b=0.2   18   0   29.0%   0.0%   

f_ b=0.1   62   3   100.0%   100.0%   

aNumber of Silver bursts that become Bronze bursts after applying a beaming correction (E_ K,iso).
bNumber of GRB-SN bursts from the Silver burst sample that become Bronze bursts after applying a beaming correction (E_ K,iso).







































































Swift/XRT
Python


