A TEST OF THE MILLISECOND MAGNETAR CENTRAL ENGINE MODEL OF GAMMA-RAY BURSTS WITH SWIFT DATA

The central engine of gamma-ray bursts (GRBs) remains an open question in GRB physics (Zhang 2011). Observations of GRB prompt emission and early afterglow pose the following constraints on a successful central engine model: (1) The central engine must be able to power an outflow with an extremely high energy and luminosity (e.g., Zhang & Mészáros, 2004; Mészáros 2006); (2) The ejecta must have a low baryon loading, with energy per baryon exceeding 100 (e.g., Lithwick & Sari, 2001; Liang et al. 2010); (3) The central engine should be intermittent in nature to account for the observed light curves with rapid variability (Fishman & Meegan 1995); (4) The engine should last for an extended period of time to power delayed erratic X-ray flares (Burrows et al. 2005; Zhang et al. 2006) or long-lasting X-ray emission followed by a sudden drop (i.e., "internal plateau," Troja et al. 2007; Liang et al. 2007; Lyons et al. 2010); (5) Finally, Fermi observations require that the central engine should be strongly magnetized to launch a magnetically dominated outflow at least for some GRBs (Zhang & Pe'er 2009).

Two types of GRB central engine models have been discussed in the literature (e.g., Kumar & Zhang 2014 for a review). The leading type of models invokes a hyper-accreting stellar-mass black hole (e.g., Popham et al. 1999; Narayan et al. 2001; Lei et al. 2013), from which a relativistic jet is launched via neutrino-anti-neutrino annihilation (Ruffert et al. 1997; Popham et al. 1999; Chen & Beloborodov 2007; Lei et al. 2009), Blandford-Znajek mechanism (Blandford & Znajek 1977; Lee et al. 2000; Li 2000), or episodic magnetic bubble ejection from the disk (Yuan & Zhang 2012).

The second type of models invokes a rapidly spinning, strongly magnetized neutron star dubbed a "millisecond magnetar" (Usov 1992; Thompson 1994; Dai & Lu 1998a; Wheeler et al. 2000; Zhang & Mészáros 2001; Metzger et al. 2008, 2011; Bucciantini et al. 2012). Within this scenario, the energy reservoir is the total rotation energy of the millisecond magnetar, which reads

$$\begin{equation} E_{\rm rot} = \frac{1}{2} I \Omega _{0}^{2} \simeq 2 \times 10^{52}\, {\rm erg}\, M_{1.4} R_6^2 P_{0,-3}^{-2}, \end{equation} \tag{ 1 }$$

where I is the moment of inertia, Ω₀ = 2π/P₀ is the initial angular frequency of the neutron star, M_1.4 = M/1.4 M_☉, and the convention Q = 10^xQ_x is adopted in cgs units for all other parameters throughout the paper.

Assuming that the magnetar with initial spin period P₀ is being spun down by a magnetic dipole with surface polar cap magnetic field B_p, the spindown luminosity would evolve with time as (Zhang & Mészáros 2001)

$$\begin{eqnarray} L(t) & = & L_0 \frac{1}{(1+t/\tau)^2} \simeq \left\lbrace \begin{array}{@{}ll}L_0, & t \ll \tau, \\ L_0 (t/\tau)^{-2}, & t \gg \tau. \end{array} \right. \end{eqnarray} \tag{ 2 }$$

where

$$\begin{equation} L_0 = 1.0 \times 10^{49} {\rm erg\; s^{-1}} (B_{p,15}^2 P_{0,-3}^{-4} R_6^6) \end{equation} \tag{ 3 }$$

is the characteristic spindown luminosity, and

$$\begin{equation} \tau = 2.05 \times 10^3 {\rm s} \left(I_{45} B_{p,15}^{-2} P_{0,-3}^2 R_6^{-6} \right) \end{equation} \tag{ 4 }$$

is the characteristic spindown time scale.

The spin-down behavior of the magnetar can leave characteristic imprints in the observed GRB emission. Dai & Lu (1998a) first proposed an energy injection model of millisecond pulsars to interpret a rebrightening feature of the first optical afterglow detected in GRB 970228. The required B_p is ∼10¹³ G, not quite a magnetar strength. The prompt GRB emission has to be attributed to additional physical processes, e.g., magnetic dissipation in a differentially rotating neutron star (Kluźniak & Ruderman 1998) or strange quark star (Dai & Lu 1998b). Zhang & Mészáros (2001) studied energy injection from a central engine with a general luminosity law L(t) = L₀(t/t₀)^−q (the magnetar injection corresponds to q = 0 for t < τ and q = 2 for t > τ), and pointed out that besides the rebrightening feature discussed by Dai & Lu (1998a, 1998b), for more typical magnetar parameters, one can have a shallow decay phase followed by a normal decay phase in the early afterglow of a GRB. Such a shallow decay phase (or plateau) was later commonly observed in Swift early X-Ray Telescope (XRT) light curves (Zhang et al. 2006; Nousek et al. 2006; O'Brien et al. 2006; Liang et al. 2007). It can be readily interpreted as energy injection from a millisecond magnetar central engine (Zhang et al. 2006). An alternative energy injection model invokes a short-duration central engine, which ejects materials with a stratified Lorentz factor (Γ) profile. Energy is gradually added to the blastwave as the blastwave is gradually decelerated to progressively lower Γ (Rees & Meszaros 1998; Sari & Mészáros 2000; Uhm et al. 2012). Both models can interpret the shallow decay phase of most X-ray light curves.

A tie-breaker GRB was discovered in early 2007. GRB 070110 (Troja et al. 2007) showed an extended plateau with a near flat light curve extending to over 10⁴ s before rapidly falling off with a decay index α ∼ 9 (throughout the paper the convention F_ν∝t^−αν^−β is adopted). Such a rapid decay cannot be accommodated in any external shock model, so that the entire X-ray plateau emission has to be attributed to internal dissipation of a central engine wind. Such an "internal plateau" was later discovered in several more GRBs (Liang et al. 2007; Lyons et al. 2010). The near steady X-ray emission observed in GRB 070110 may not be easy to interpret within a black hole central engine model, but is a natural prediction of the magnetar central engine model (Equation (2) when t ≪ τ). The rapid t⁻⁹ decay near the end is not predicted in the magnetic dipole radiation model. Troja et al. (2007) interpreted it as being due to collapse of the magnetar to a black hole after losing centrifugal support.¹ Interestingly, internal plateaus are also discovered in a good fraction of short GRBs (Rowlinson et al. 2010, 2013). Modeling various afterglow features for both long and short GRBs within the framework of the millisecond magnetar (or pulsar with weaker magnetic field) central engine model has gained growing attention (Dai et al. 2006; Gao & Fan 2006; Fan & Xu 2006; Metzger et al. 2008, 2011; Dall'Osso et al. 2011; Fan et al. 2011; Bucciantini et al. 2012; Bernardini et al. 2013; Gompertz et al. 2013, 2014). Numerical simulations of binary neutron star mergers indeed show that a stable magnetar can survive if the initial masses of the two neutron stars are small enough, which would power a short GRB (Giacomazzo & Perna 2013).

Even though evidence of a magnetar central engine is mounting, it remains unclear whether the rich GRB data accumulated over the years with the GRB mission Swift indeed statistically requires the existence of (presumably) two types of central engines. If indeed magnetars are operating in some GRBs while hyper-accreting black holes are operating in others, do the data show statistically significant differences between the two samples? Do those GRBs that seem to have a magnetar signature have physical parameters that are consistent with the predictions of the magnetar central engine model?

This paper is to address these interesting questions through a systematic analysis of the Swift XRT data. The XRT data reduction details and criteria for sample selection are presented in Section 2. In Section 3, physical parameters of the GRBs and the hypothetical magnetars are derived for all the samples. A statistical comparison of the physical properties between the magnetar samples and the non-magnetar sample are presented in Section 4, and conclusions are drawn in Section 5 with some discussion. Throughout the paper, a concordance cosmology with parameters H₀ = 71 km s⁻¹ Mpc⁻¹, Ω_M = 0.30, and Ω_Λ = 0.70 is adopted.

The XRT data are downloaded from the Swift data archive.² We developed a script to automatically download and maintain all the XRT data on the local UNLV machine. The HEAsoft packages version 6.10, including Xspec, Xselect, Ximage, and the Swift data analysis tools, are used for the data reduction. An IDL code was developed by the former group member B.-B. Zhang to automatically process the XRT data for a given burst in any user-specified time interval (see Zhang et al. 2007c for details). We adopt this code with slight modifications to solve the problem designed for this paper. The same IDL code was used in several previous papers (Zhang et al. 2007c; Liang et al. 2007, 2008, 2009) of our group. More details about the data reduction procedures can be found in Zhang et al. (2007c) and Evans et al. (2009).

Our entire sample includes more than 750 GRBs observed between 2005 January and 2013 August, whose XRT data are all processed with our data reduction tool. Since the magnetar signature typically invokes a shallow decay phase (or plateau) followed by a steeper decay segment (a normal decay for canonical light curves, or a very steep decay for internal plateaus), our attention is on those GRBs that show such a transition in the X-ray light curves. We first identify such bursts by inspecting their light curves. In order to grade their magnetar candidacy, we next perform a temporal fit to the plateau behavior within a time interval (t₁, t₂), where t₁ is the beginning of the plateau, while t₂ is the end of the segment after the plateau break (either last observed data point if there is no further break in the lightcurve, or the break time if a second break appears). Since we are mostly interested in the behavior around the break time t_b, the exact positions of t₁ and t₂ do not matter much, so we pick them through visual inspection of the light curves. We then fit the light curves with a smooth broken power law

$$\begin{equation} F = F_{0} \left[\left(\frac{t}{t_b}\right)^{\omega \alpha _1}+ \left(\frac{t}{t_b}\right)^{\omega \alpha _2}\right]^{-1/\omega }, \end{equation} \tag{ 5 }$$

where t_b is the break time, F_b = F₀ · 2^−1/ω is the flux at the break time t_b, α₁ and α₂ are decay indices before and after the break, respectively, and ω describes the sharpness of the break. The larger the ω parameter, the sharper the break.

An IDL routine named "mpfitfun.pro" is employed for our fitting (Moré 1977; Markwardt 2009). This routine performs a Levenberg–Marquardt least-square fit to the data for a given model to optimize the model parameters. After processing all the data, we grade all long GRBs in our sample into four groups ("Gold," "Silver," "Aluminum," and "non-magnetar") according to their likelihood of being powered by a magnetar central engine.

1. Gold. This sample is defined by those bursts that display an "internal plateau." These plateaus are followed by a decay slope steeper than 3, which is essentially impossible to interpret within the external shock models (Gao et al. 2013b).³ It demands a long-lasting central engine, and a near steady flux is consistent with emission from a spinning down magnetar. The rapid decay at the end of plateau may mark the implosion of the magnetar into a black hole (Troja et al. 2007; Zhang 2014). There are altogether only 9 robust cases identified in this Gold sample, 3 of which have redshift measurements. The light curves of these 9 GRBs together with the broken power-law fittings (red curves) are shown in Figure 1, and the fitting parameters are summarized in Table 1.

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2. Silver. This sample includes GRBs with a shallow decay phase followed by a normal decay phase, and the pre- and post-break temporal and spectral properties are well consistent with the external forward shock model with energy injection of a magnetar as defined in Equation (2). Specifically, one requires two independent criteria to define this sample. First, the temporal and spectral properties of the afterglow after the break (the normal decay phase) should satisfy the "closure relation" of the external shock model (e.g., Zhang & Mészáros 2004; Gao et al. 2013b), i.e.,

$$\begin{equation} \alpha _{2}= \left\lbrace \begin{array}{@{}ll}\displaystyle\frac{3\beta }{2}=\frac{3(p-1)}{4}, & \nu _{m}\;{<}\;\nu \;{<}\;\nu _{c} \hbox{ (ISM)} \\ \displaystyle\frac{3\beta +1}{2}=\frac{3p-1}{4}, & \nu _{m}\;{<}\;\nu \;{<}\;\nu _{c} \hbox{ (Wind) }\\ \displaystyle\frac{3\beta -1}{2}=\frac{3p-2}{4}, & \nu \;{>}\;\nu _{c} \hbox{ (ISM or Wind)} \end{array} \right. \end{equation} \tag{ 6 }$$

Here β is the spectral index of the normal decay segment (which is X-ray photon index minus 1), and p is the electron's spectral distribution index. Second, the pre-break slope α₁ should correspond to q = 0, while the post-break slope α₂ should correspond to q = 1 (for a constant energy fireball, the scaling law is the same as q = 1, Zhang & Mészáros 2001), so according to Zhang et al. (2006) and Gao et al. (2013b), one should have

$$\begin{equation} \alpha _{1}= \left\lbrace \begin{array}{@{}ll}\displaystyle\frac{2\alpha _{2}-3}{3}, & \nu _{m}\;{<}\;\nu \;{<}\;\nu _{c} \hbox{ (ISM)} \\ \displaystyle\frac{2\alpha _{2}-1}{3}, & \nu _{m}\;{<}\;\nu \;{<}\;\nu _{c} \hbox{ (Wind)} \\ \displaystyle\frac{2\alpha _{2}-2}{3}, & \nu \;{>}\;\nu _{c} \hbox{ (ISM or Wind)} \end{array} \right. \end{equation} \tag{ 7 }$$

In our entire sample, 69 GRBs can be grouped into this Silver sample, with 33 having measured redshifts. The light curves with fitting curves are presented online at http://grb.physics.unlv.edu/~lhj/Silver/, and the fitting results are reported in Table 1. Two examples (GRBs 060729, see also Grupe et al. 2007, and 070306) are shown in Figure 2. Figure 3 shows all the GRBs in the α₁–α₂ plane, with three theoretically favored lines of the magnetar models (Equation (7)) plotted. Those GRBs falling onto these lines (within error bars) and also satisfy the closure relations are identified as Silver sample GRBs (colored data points). In Figure 4 we present the distribution of electron spectral index p derived from the Silver sample. It has a Gaussian distribution with a center value p_c = 2.51 ± 0.04. Figure 5 shows the distribution of Silver sample in the (α, β)-plane combined with the closure relations for the models (ISM and wind medium).

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3. Aluminum. Other GRBs with a shallow decay segment transiting to a steeper decay are included in the Aluminum sample. They either do not satisfy external shock closure relations in the post-break phase, or do not satisfy the α₁–α₂ relations predicted in the magnetar external shock models. These are marked as gray points in Figure 3. Those GRBs that fall onto the three magnetar model lines but are still denoted as Aluminum are the ones that do not satisfy the closure relations in the post-break phase. On the other hand, since early magnetar spindown may not fully follow the simple dipole spindown law (e.g., Metzger et al. 2011), and since the observed X-ray emission may not come from the external forward shock emission (e.g., can be from external reverse shock, Dai 2004; Yu & Dai 2007, or from internal dissipation of the magnetar wind, Yu et al. 2010), these GRBs could be still powered by magnetars. We therefore still assign them as magnetar candidates, but with a lower grade. There are 135 solid cases in the sample, 67 of which have redshift measurements. The light curves with fitting curves are presented online at http://grb.physics.unlv.edu/~lhj/Aluminum/. Two examples (GRBs 070420 and 080430) are presented in Figure 2.

4. Non-magnetar. All the other long GRBs we have analyzed are included in the non-magnetar sample. They either have a single power-law decay, or have erratic flares that prevent identifying a clear shallow decay phase, or present a rebrightening behavior, or the data are too poor to reach a robust conclusion. There are more than 400 GRBs in this group, 111 of which have redshift measurements. Strictly speaking, some of these GRBs may still host a magnetar central engine. We define these GRBs as "non-magnetar," simply because they do not present a clear magnetar signature. Two examples (GRBs 061007, see also Schady et al. 2007, Mundell et al. 2007, and 081028) are presented in the Figure 2.

Finally, we also independently processed the X-ray data of short GRBs that may harbor a magnetar central engine (cf. Rowlinson et al. 2013). We select the short GRBs that have measured redshifts and high-quality X-ray data. The light curves with fitting curves are presented online at http://grb.physics.unlv.edu/~lhj/SGRB/.

Our purpose is to analyze and compare the physical properties of GRBs with or without a magnetar signature. In this section, we use data to derive relevant physical parameters. Redshift measurements are crucial to derive the intrinsic parameters (energy, luminosity, etc), so in the following we focus on those GRBs with z measurements only.

3.1. Energetics, Luminosity, and Radiation Efficiency

The isotropic prompt γ-ray emission energy E_{γ, iso} is usually derived from the observed fluence S_γ in the detector's energy band, and extrapolated to the rest-frame 1–10⁴ keV using spectral parameters (the low- and high-energy spectral indices $\hat{\alpha }$, $\hat{\beta }$, and the peak energy E_p for a standard "Band-function" fit, Band et al. 1993) and through k-correction. However, since the BAT energy band is narrow (15–150 keV), for most GRBs the spectra can be only fit by a cutoff power law or a single power law (Sakamoto et al. 2008, 2011). We therefore apply the following procedure to estimate the Band spectral parameters: (1) If a burst was also detected by Fermi GBM or Konus Wind, we adopt the spectral parameters measured by those instruments. (2) For those bursts that are not detected by other instruments but can be fit with a cutoff power law model, we adopt the derived $\hat{\alpha }$ and E_p parameters,⁴ and assume a typical value of $\hat{\beta }= -2.3$. (3) For those GRBs that can be only fit with a single power law, we have to a derive E_p using an empirical correlation between the BAT-band photon index Γ^BAT and E_p (e.g., Sakamoto et al. 2009; Zhang et al. 2007b; Virgili et al. 2012; Lü et al. 2012). The typical parameters $\hat{\alpha }= -1$, $\hat{\beta }=-2.3$ are adopted to perform the simulations. We can then calculate the E_{γ, iso} according to

$$\begin{eqnarray} E_{\rm \gamma,iso}&=&4\pi k D^{2}_{L} S_{\gamma } (1+z)^{-1}\nonumber \\ &=&1.3\times 10^{51}\; {\rm erg}\; k D^{2}_{28} (1+z)^{-1} S_{\gamma,-6} \end{eqnarray} \tag{ 8 }$$

where z is the redshift, D = 10²⁸cmD₂₈ is the luminosity distance, and k is the k-correction factor from the observed band to 1–10⁴ keV in the burst rest frame (e.g., Bloom et al. 2001).

Another important parameter is the isotropic kinetic energy E_{K, iso} measured from the afterglow flux. This value is increasing during the shallow decay phase, but becomes constant during the normal decay phase (Zhang et al. 2007a). We follow the method discussed in Zhang et al. (2007a) to calculate E_{K, iso} during the normal decay phase using the X-ray data. Noticing that fast-cooling is disfavored at this late epoch, we derive several relevant cases. For ν > max(ν_m, ν_c), the afterglow flux expression does not depend on the medium density, so the following expression (Zhang et al. 2007a) applies to both interstellar medium (ISM) and wind models⁵

$$\begin{eqnarray} E_{\rm K,iso,52 } & = & \left[\frac{\nu F_\nu (\nu =10^{18} {\rm Hz})}{5.2\times 10^{-14} \,{\rm erg\, s^{-1}\, cm^{-2}} }\right]^{4/(p+2)} \nonumber \\ &&\times D_{28}^{8/(p+2)}(1+z)^{-1} t_d^{(3p-2)/(p+2)}\nonumber \\ &&\times (1+Y)^{4/(p+2)} f_p^{-4/(p+2)}\epsilon _{B,-2}^{(2-p)/(p+2)} \nonumber \\ &&\times \epsilon _{e,-1}^{4(1-p)/(p+2)} \nu _{18}{^{2(p-2)/(p+2)}}. \end{eqnarray} \tag{ 9 }$$

For the ν_m < ν < ν_c ISM model, one has (Zhang et al. 2007a)

$$\begin{eqnarray} E_{\rm K,iso,52} & = & \left[\frac{\nu F_\nu (\nu =10^{18} {\rm Hz})}{6.5\times 10^{-13} \,{\rm erg\, s^{-1}\, cm^{-2}} }\right]^{4/(p+3)} \nonumber \\ &&\times D_{28}^{8/(p+3)}(1+z)^{-1} t_d^{3(p-1)/(p+3)}\nonumber \\ &&\times f_p^{-4/(p+3)} \epsilon _{B,-2}^{-(p+1)/(p+3)} \epsilon _{e,-1}^{4(1-p)/(p+3)} \nonumber \\ &&\times n^{-2/(p+3)} \nu _{18}{^{2(p-3)/(p+3)}}. \end{eqnarray} \tag{ 10 }$$

For the ν_m < ν < ν_c wind model, one has (Gao et al. 2013b)

$$\begin{eqnarray} \nu _{m}&=&5.5\times 10^{11}Hz \left(\frac{p-2}{p-1}\right)^{2} \nonumber\\ && \times (1+z)^{1/2}\epsilon ^{1/2}_{B,-2} \epsilon ^{2}_{e,-1}E^{1/2}_{\rm K,iso,52}t^{-3/2}_{d}, \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} &&\nu _{c}=4.7\times 10^{18}Hz(1+z)^{-3/2}A^{-2}_{\ast,-1} \epsilon ^{-3/2}_{B,-2}E^{1/2}_{\rm K,iso,52}t^{1/2}_{d}, \nonumber \\ \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} &&F_{\nu,max}=5.7\times 10^{2}\mu Jy(1+z)^{3/2}A_{\ast,-1} \epsilon ^{1/2}_{B,-2}D^{-2}_{28}E^{1/2}_{\rm K,iso,52}t^{-1/2}_{d}, \nonumber \\ \end{eqnarray} \tag{ 13 }$$

so that

$$\begin{eqnarray} \nu F_{\nu }(\nu =10^{18}Hz)&=&\nu F_{\nu,max}\left(\frac{\nu }{\nu _{m}}\right)^{-(p-1)/2} \nonumber \\ &=& F_{\nu,max}\nu ^{(3-p)/2}\nu ^{(p-1)/2}_{m} \nonumber \\ &=& 7.4\times 10^{-14}\; {\rm erg\; cm^{-2}\, s^{-1}} \nonumber \\ &\times & D^{-2}_{28} (1+z)^{(p+5)/4} A_{\ast,-1}f_{p}\epsilon ^{(p+1)/4}_{B,-2}\epsilon ^{p-1}_{e,-1}\nonumber \\ &\times &E^{(p+1)/4}_{\rm K,iso,52}t^{(1-3p)/4}_{d} \nu ^{(3-p)/2}_{18}, \end{eqnarray} \tag{ 14 }$$

and

$$\begin{eqnarray} E_{\rm K,iso,52} & = & \left[\frac{\nu F_\nu (\nu =10^{18}\, {\rm Hz})}{7.4\times 10^{-14}\, {\rm erg\, s^{-1}\, cm^{-2}} }\right]^{4/(p+1)} \nonumber \\ & \times & D_{28}^{8/(p+1)}(1+z)^{-(p+5)/(p+1)} t_d^{(3p-1)/(p+1)}\nonumber \\ & \times &f_p^{-4/(p+1)} \epsilon _{B,-2}^{-1} \epsilon _{e,-1}^{4(1-p)/(p+1)} \nonumber \\ & \times & A_{\ast,-1}^{-4/(p+1)} \nu _{18}{^{2(p-3)/(p+1)}}. \end{eqnarray} \tag{ 15 }$$

Here νf_ν(ν = 10¹⁸ Hz) is the energy flux at 10¹⁸ Hz (in units of erg s⁻¹ cm⁻²), n is the density of the constant ambient medium, A_* is the stellar wind parameter, t_d is the time in the observer frame in days, and Y is the Compton parameter. The electron spectral index p and the spectral index β are connected through

$$\begin{equation} p= \left\lbrace \begin{array}{@{}ll}2\beta +1, & \nu _{m}\;{<}\;\nu \;{<}\;\nu _{c} \\ 2\beta, & \nu \;{>}\;\nu _{c}, \end{array} \right. \end{equation} \tag{ 16 }$$

and f_p is a function of the power law distribution index p (Zhang et al. 2007a)

$$\begin{equation} f_{p}\sim 6.73\left(\frac{p-2}{p-1}\right)^{p-1} (3.3\times 10^{-6})^{(p-2.3)/2} \end{equation} \tag{ 17 }$$

In our calculations, the microphysics parameters of the shock are assigned to standard values derived from observations (e.g., Panaitescu & Kumar 2002; Yost et al. 2003): _e = 0.1 and _B = 0.01. The Compton parameter is assigned to a typical value Y = 1.

After deriving the break time t_b through light curve fitting, we derive the break time luminosity as

$$\begin{equation} L_b = 4\pi D^2 F_b, \end{equation} \tag{ 18 }$$

where F_b is the X-ray flux at t_b. Since the XRT band is narrow, no k-correction is possible to calculate L_b.

A jet break was detected in some GRBs in our sample. For these GRBs, we correct all the isotropic values to the beaming-corrected values by multiplying the values by the beaming correction factor (Frail et al. 2001)

$$\begin{equation} f_b = 1-\cos \theta _j \simeq (1/2) \theta _j^2, \end{equation} \tag{ 19 }$$

i.e., E_γ = E_{γ, iso}f_b, and E_K = E_{K, iso}f_b. The jet angle information was searched from the literature (e.g., Liang et al. 2008; Racusin et al. 2009; Lu et al. 2012; Nemmen et al. 2012), which is collected in Table 2.

Table 2. The Properties of GRBs with Known Redshifts in our "Gold," "Silver," and Short GRB Samples

GRB	θja	Eγ, iso, 52b	Lb, 49c	τ3c	Bp, 15d	P0, −3d	Bp, θ, 15e	P0, θ, −3e	Erot, 50f
Gold
060522	5	0.71 ± 0.71	1.38 ± 0.22	0.06 ± 0.01	2.34 ± 0.71	1.19 ± 0.59	37.93 ± 11.42	19.28 ± 9.51	0.54 ± 0.29
060607A	5	9.08 ± 7.11	0.58 ± 0.07	0.13 ± 0.02	0.18 ± 0.04	0.44 ± 0.10	2.91 ± 0.67	7.15 ± 1.58	3.91 ± 1.29
070110	5	3.09 ± 2.51	0.07 ± 0.03	3.68 ± 0.06	0.23 ± 0.11	0.74 ± 0.36	3.79 ± 1.78	11.97 ± 5.84	1.39 ± 0.76
Silver
050401	5	$32^{+26}_{-7}$	0.47 ± 0.01	1.51 ± 0.21	0.09 ± 0.02	0.15 ± 0.02	1.44 ± 0.32	2.43 ± 0.34	34.20 ± 8.11
050505	1.67 ± 0.35	$16^{+13}_{-3}$	0.41 ± 0.01	1.49 ± 0.30	0.07 ± 0.02	0.13 ± 0.03	3.21 ± 1.20	6.28 ± 1.45	5.12 ± 1.78
050803	5	$0.24^{+0.24}_{-0.08}$	(8.92 ± 0.31)e-4	11.24 ± 0.13	0.07 ± 0.01	0.21 ± 0.01	1.22 ± 0.04	3.39 ± 0.08	17.37 ± 7.88
060108	5	$0.59^{+0.84}_{-0.08}$	(5.96 ± 0.48)e-3	4.70 ± 2.44	0.21 ± 0.36	0.53 ± 0.50	3.28 ± 0.58	8.65 ± 2.43	2.68 ± 1.79
060526	3.61 ± 0.57	$5.2^{+5.6}_{-0.4}$	(3.91 ± 0.27)e-2	2.38 ± 1.08	0.10 ± 0.09	0.21 ± 0.18	2.14 ± 1.59	4.74 ± 1.98	8.93 ± 2.69
060604	5	$0.5^{+0.12}_{-0.1}$	(1.15 ± 0.05)e-2	3.63 ± 2.17	0.21 ± 0.08	0.49 ± 0.36	3.39 ± 1.87	7.98 ± 3.88	3.16 ± 2.45
060605	1.55 ± 0.57	$2.5^{+3.1}_{-0.6}$	0.13 ± 0.01	1.56 ± 0.11	0.03 ± 0.01	0.06 ± 0.01	1.68 ± 0.23	3.19 ± 0.31	19.57 ± 3.35
060614	7.57 ± 2.29	$0.24^{+0.04}_{-0.04}$	(2.49 ± 0.08)e-5	44.31 ± 3.22	0.06 ± 0.01	0.32 ± 0.03	0.69 ± 0.10	3.42 ± 0.34	17.06 ± 2.92
060729	18 ± 1.61	$0.33^{+0.29}_{-0.06}$	(1.56 ± 0.02)e-3	47.38 ± 1.96	0.06 ± 0.01	0.33 ± 0.02	0.25 ± 0.02	1.48 ± 0.07	91.23 ± 7.84
060906	1.15 ± 0.12	$13^{+12}_{-1}$	(3.08 ± 0.21)e-2	2.73 ± 0.70	0.04 ± 0.02	0.09 ± 0.04	2.55 ± 1.70	6.36 ± 2.78	4.94 ± 2.55
060908	0.46 ± 0.29	$7^{+4}_{-1}$	0.26 ± 0.07	0.25 ± 0.06	0.57 ± 0.41	0.33 ± 0.17	100.6 ± 73.24	59.08 ± 30.06	0.06 ± 0.03
061110A	5	$0.28^{+0.28}_{-0.06}$	(3.83 ± 0.81)e-5	41.62 ± 3.23	0.39 ± 0.09	2.32 ± 0.41	6.31 ± 1.43	37.68 ± 6.71	0.14 ± 0.04
070306	3.38 ± 1.72	$6^{+5}_{-1}$	(2.06 ± 0.06)e-2	11.89 ± 0.69	0.02 ± 0.01	0.06 ± 0.01	0.36 ± 0.04	1.39 ± 0.11	104.2 ± 14.5
070529	5	$9^{+9}_{-3}$	0.12 ± 0.01	0.47 ± 0.24	0.44 ± 0.17	0.39 ± 0.33	7.06 ± 1.58	6.33 ± 3.63	5.02 ± 3.55
080605	5	$21^{+9}_{-4}$	1.49 ± 0.16	0.17 ± 0.02	0.47 ± 0.11	0.22 ± 0.03	7.63 ± 1.75	3.52 ± 0.56	16.26 ± 4.22
080607	5	$280^{+130}_{-90}$	1.36 ± 0.27	0.34 ± 0.05	0.13 ± 0.05	0.11 ± 0.03	2.10 ± 0.77	1.72 ± 0.47	67.85 ± 26.14
080721	5	$110^{+110}_{-50}$	2.50 ± 0.13	0.86 ± 0.04	0.07 ± 0.01	0.09 ± 0.01	1.18 ± 0.12	1.45 ± 0.10	96.86 ± 13.41
080905B	5	$3.4^{+3.1}_{-0.6}$	0.31 ± 0.01	1.20 ± 0.36	0.11 ± 0.08	0.16 ± 0.07	1.84 ± 1.30	2.58 ± 1.10	30.32 ± 15.55
081008	5	$6^{+3}_{-1}$	(1.19 ± 0.06)e-2	5.36 ± 2.22	0.08 ± 0.10	0.22 ± 0.16	1.29 ± 0.63	3.60 ± 1.63	15.45 ± 10.29
081203A	5	$17^{+13}_{-4}$	(3.23 ± 0.11)e-2	3.62 ± 2.80	0.06 ± 0.01	0.13 ± 0.07	0.93 ± 0.39	2.17 ± 0.73	42.65 ± 4.11
081221	5	282.29 ± 4.68	1.72 ± 0.20	0.18 ± 0.02	0.38 ± 0.11	0.21 ± 0.04	6.19 ± 1.79	3.35 ± 0.67	18.01 ± 5.61
090423	>12	$8^{+1}_{-1}$	0.82 ± 0.04	0.47 ± 0.08	0.19 ± 0.07	0.28 ± 0.07	1.30 ± 0.47	1.88 ± 0.44	59.46 ± 22.42
090618	6.7∓1.08	$15^{+1}_{-1}$	(1.62 ± 0.02)e-2	4.73 ± 0.93	0.25 ± 0.09	0.48 ± 0.13	3.08 ± 1.09	5.79 ± 1.23	5.98 ± 1.94
090927	5	0.43 ± 0.06	(5.16 ± 0.27)e-3	55.36 ± 32.68	0.05 ± 0.02	0.36 ± 0.25	0.73 ± 0.21	5.82 ± 0.86	6.01 ± 1.04
091208B	7.3 ± 1.42	4.88 ± 0.30	0.05 ± 0.01	0.56 ± 0.10	0.96 ± 0.60	0.72 ± 0.34	10.71 ± 6.65	8.02 ± 3.77	3.11 ± 1.67
100418A	5	0.14 ± 0.02	(1.16 ± 0.11)e-4	53.48 ± 13.64	0.05 ± 0.03	0.32 ± 0.13	0.80 ± 0.51	5.23 ± 2.16	7.31 ± 3.65
111008A	5	85.23 ± 4.82	0.72 ± 0.02	1.25 ± 0.38	0.05 ± 0.04	0.11 ± 0.04	0.88 ± 0.59	1.67 ± 0.67	72.35 ± 35.86
111228A	5	5.45 ± 0.13	(8.48 ± 0.24)e-3	3.81 ± 1.23	0.36 ± 0.25	0.64 ± 0.26	5.78 ± 1.19	10.32 ± 4.22	1.89 ± 0.94
120422A	5	0.13 ± 0.02	(2.31 ± 0.26)e-6	129.5 ± 17.4	0.42 ± 0.12	3.80 ± 0.75	6.85 ± 1.96	61.66 ± 12.09	0.05 ± 0.02
121024A	5	10.78 ± 0.98	(6.30 ± 0.45)e-3	10.01 ± 2.49	0.06 ± 0.02	0.26 ± 0.09	1.05 ± 0.58	4.19 ± 1.48	11.38 ± 5.16
121027A	5	6.61 ± 0.33	(3.38 ± 0.16)e-3	52.19 ± 16.18	0.02 ± 0.01	0.15 ± 0.07	0.29 ± 0.15	2.41 ± 1.09	34.58 ± 18.26
121128A	5	78.91 ± 4.57	0.38 ± 0.06	0.50 ± 0.08	0.21 ± 0.08	0.18 ± 0.05	3.39 ± 1.29	2.98 ± 0.82	22.61 ± 8.72
121229A	5	6.64 ± 1.88	(1.81 ± 0.25)e-3	15.21 ± 2.25	0.07 ± 0.02	0.34 ± 0.08	1.06 ± 0.36	5.55 ± 1.32	6.52 ± 2.27
SGRBs
051221A	...	$0.28^{+0.21}_{-0.11}$	24.71 ± 4.97	(8.8 ± 0.23)e-3	0.57 ± 0.01	2.47 ± 0.16	...	...	22.88 ± 4.41
060801	...	$0.17^{+0.02}_{-0.02}$	0.03 ± 0.02	8.7 ± 4.1	11.21 ± 4.21	1.95 ± 0.34	...	...	52.62 ± 24.26
061201	...	$0.018^{+0.002}_{-0.001}$	1.08 ± 0.23	0.08 ± 0.01	6.01 ± 0.12	4.59 ± 0.05	...	...	9.48 ± 1.88
070809	...	$0.001^{+0.001}_{-0.001}$	12.14 ± 5.33	(4.5 ± 2.5)e-3	2.06 ± 1.03	5.55 ± 1.25	...	...	6.49 ± 4.31
090426	...	$0.42^{+0.5}_{-0.04}$	0.09 ± 0.05	1.9 ± 1.2	4.79 ± 3.11	1.87 ± 0.53	...	...	57.46 ± 5.44
090510	...	$0.3^{+0.5}_{-0.2}$	0.15 ± 0.02	2.1 ± 0.2	5.05 ± 0.26	1.87 ± 0.05	...	...	57.36 ± 3.02
100724A	...	$0.07^{+0.01}_{-0.01}$	0.23 ± 0.07	0.23 ± 0.03	8.22 ± 0.59	4.14 ± 0.15	...	...	11.67 ± 1.87
101219A	...	$0.48^{+0.03}_{-0.03}$	0.13 ± 0.06	9.7 ± 3.8	2.86 ± 0.81	0.96 ± 0.13	...	...	217.7 ± 71.5
130603B	...	$0.22^{+0.02}_{-0.02}$	2.22 ± 0.49	0.11 ± 0.01	2.16 ± 0.12	2.61 ± 0.07	...	...	29.32 ± 1.63

Notes. ^aThe jet opening angle (in units of degree (°)) measured from afterglow observations (Racusin et al. 2009; Lu et al. 2012; Nemmen et al. 2012), or assumed as θ_j = 5° if no observation is available. SGRBs are assumed to be isotropic. ^bE_{γ, iso} is calculated using fluence and redshift extrapolated into 1–10000 keV (rest frame) with a spectral model and a k-correction, in units of 10⁵² erg. ^cIsotropic luminosity of break time (in units of 10⁴⁹ erg s⁻¹), and the spin-down time (in units of 10³ s). ^dDipolar magnetic field strength at the polar cap in units of 10¹⁵G, and the initial spin period of the magnetar in units of milliseconds, with an assumption of an isotropic wind. ^eThe same as d, but with beaming correction made. ^fThe rotational energy (in units of 10⁵⁰ erg) of the magnetar assuming R₆ = 1 and M = 1.4 M_☉.

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The GRB radiation efficiency is defined as (Lloyd-Ronning & Zhang 2004)

$$\begin{equation} \eta _{\gamma } = \frac{E_{\rm \gamma,iso}}{E_{\rm \gamma,iso}+E_{\rm K,iso}} = \frac{E_\gamma }{E_\gamma +E_{\rm K}}. \end{equation} \tag{ 20 }$$

Since E_{K, iso} (and E_K) are increasing functions of time during the shallow decay phase, η_γ is different when E_{K, iso} (E_K) at different epochs are adopted. Following Zhang et al. (2007a), we take a typical blastwave deceleration t_dec and the end of the shallow decay phase t_b to calculate the radiative efficiencies. Within the framework of the magnetar central engine model, the two efficiencies carry different physical meanings: η_γ(t_dec) denotes the efficiency of dissipating the magnetar wind energy during the prompt emission phase, while η_γ(t_b) denotes the total efficiency of converting the spindown energy of a magnetar to γ-ray radiation.

3.2. Magnetar Parameters

For a magnetar undergoing dipolar spindown, two important magnetar parameters, i.e., the initial spin period P₀ and the surface polar cap magnetic field B_p, can be solved by the characteristic luminosity L₀ (Equation (3)) and the spindown time scale τ (Equation (4)).

The spindown time scale can be generally identified as the observed break time, i.e.,

$$\begin{equation} \tau = t_b/(1+z). \end{equation} \tag{ 21 }$$

One caution is that τ can be shorter than t_b/(1 + z) if the magnetar is supra-massive, and collapses to a black hole before it is significantly spun down. On the other hand, the angular velocity of the magnetar does not change significantly until reaching the characteristic spindown time scale, so that the collapse of the supra-massive magnetar, if indeed happens, would likely happen at or after τ. In our analysis, we will adopt Equation (21) throughout.

The characteristic spindown luminosity should generally include two terms:

$$\begin{equation} L_0 = L_{\rm X} + L_{\rm K} = (L_{\rm X,iso} + L_{\rm K,iso}) f_b, \end{equation} \tag{ 22 }$$

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

$$\begin{equation} L_{\rm K,iso} = E_{\rm K,iso} (1+z) / t_b \end{equation} \tag{ 23 }$$

is the kinetic luminosity that is injected into the blastwave during the energy injection phase. It depends on the isotropic kinetic energy E_{K, iso} after the injection phase is over, which can be derived from afterglow modeling discussed above. For the Gold sample, the L_{X, iso} component dominates, while for Silver and Aluminum samples, the L_{K, iso} component dominates. In any case, both components should exist and contribute to the observed flux (Zhang 2014). One can also define an X-ray efficiency to define the radiative efficiency for a magnetar to convert its spindown energy to radiation, i.e.,

$$\begin{equation} \eta _{\rm X} = \frac{L_{\rm X}}{L_{\rm X}+L_{\rm K}} = \frac{L_{\rm X,iso}}{L_{\rm X,iso}+L_{\rm K,iso}}. \end{equation} \tag{ 24 }$$

In our analysis, we try to calculate both L_{X, iso} and L_{K, iso} from the data. For the Gold sample GRBs that show internal plateaus, L_{X, iso} can be readily measured. For the cases where the internal plateau lands on an external shock component (e.g., Troja et al. 2007), L_{K, iso} can be also derived by modeling the late X-ray afterglow in the normal decay phase. For the Gold sample cases where no late external shock component is available, one can only set up an upper limit on L_{K, iso}. For Silver and Aluminum samples, the internal plateau component is not detectable. Through simulations, we find that the external shock component would not be significantly modified if the internal plateau flux is below 50% of the observed external shock flux. Therefore for all the Silver and Aluminum sample GRBs, we place an upper limit of L_{X, iso} as 50% of the observed X-ray flux.

4.1. Magnetar Parameters and Collimation

We derive magnetar parameters (P₀ and B_p) of the Gold, Silver and Aluminum samples using Equations (3), (4), and (22).⁶ First, we assume that the magnetar wind is isotropic, so that f_b = 1. The derived P₀, B_p are presented in Table 2 and Figure 6(a). Most "magnetars" have B_p below 10¹⁵ G, some even have B_p below 10¹³ G, which are not considered as magnetars. More problematically, most derived P₀'s are much shorter than 1 ms. This directly conflicts with the break-up limit of a neutron star, which is about 0.96 ms (Lattimer & Prakash 2004). This suggests that the isotropic assumption for these long GRB magnetar winds is not correct. We then introduce the beaming factor f_b for each GRB. If θ_j is measured, we simply adopt the value. Otherwise, we choose θ_j = 5^o, a typical jet opening angle for bright long GRBs (Frail et al. 2001; Liang et al. 2008). Very interestingly, after such a correction, all the data points of Gold and Silver sample GRBs fall into the expected region in the P₀–B_p plot (Figure 6(b)). Also the additional conditions imposed by the causality argument (i.e., that the speed of sound on the neutron star cannot exceed the speed of light, Lattimer et al. 1990, and Equations (9) and (10) of Rowlinson et al. 2010) are satisfied for all GRBs in all three (Gold, Silver and Aluminum) magnetar samples, if one assumes M = 1.4 M_☉. All these suggest that the long GRB magnetar winds are likely collimated. Some Aluminum sample GRBs are still to the left of the allowed region (with P₀ shorter than the break-up limit). This may suggest that those Aluminum sample bursts are not powered by magnetars, or are powered by magnetars with even narrower jets.

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Very interestingly, the magnetar properties of short GRBs derived under the isotropic assumption actually lie reasonably in the allowed region (Figure 6(a), blue dots). After jet correction for long GRB magnetars (but keep short GRB magnetar wind isotropic), the derived magnetar parameters are well mixed in the same region. This suggests that the isotropic assumption for short GRBs is reasonably good. This is understandable within the framework of the progenitor models of GRBs. Short GRBs are believed to be powered by mergers of NS–NS or NS–BH systems (Paczynski 1986; Eichler et al. 1989; Paczynski 1991; Narayan et al. 1992). During the merger process, only a small amount of materials are launched (Freiburghaus et al. 1999; Rezzolla et al. 2010; Hotokezaka et al. 2013). A millisecond magnetar is expected to launch a near isotropic wind. This wind, instead of being collimated by the ejecta (e.g., Bucciantini et al. 2012), would simply push the ejecta behind and accelerate the ejecta and make a bright electromagnetic signal in the equatorial directions (Fan & Xu 2006; Zhang 2013; Gao et al. 2013a; Yu et al. 2013; Metzger & Piro 2013). In the jet direction, the magnetar wind emission is not enhanced by the beaming effect, so that one can infer correct magnetar parameters assuming an isotropic wind. For long GRBs, on the other hand, jets are believed to be launched from collapsing massive stars (Woosley 1993; MacFadyen & Woosley 1999). The initially near isotropic magnetar wind is expected to be soon collimated by the stellar envelope to a small solid angle (Bucciantini et al. 2008).

4.2. Statistical Properties and Correlations of other Parameters

Figure 7 shows the correlations of L_b–E_{γ, iso} and L_b–t_b for the entire sample. As shown in Figure 7(a), a higher isotropic γ-ray energy generally has a higher X-ray break luminosity. For the Gold and Silver samples, a Spearman correlation analysis gives a dependence

$$\begin{equation} \log L_{b,49}=(1.48 \pm 0.17) \log E_{\rm \gamma,iso,52}+(2.56 \pm 0.75), \end{equation} \tag{ 25 }$$

with a correlation coefficient r = 0.83, and a chance probability p < 0.001. Adding the Aluminum sample only slightly worsens the correlation (log L_{b, 49} = (1.02 ± 0.10)log E_{γ, iso, 52} + (2.64 ± 2.04), with r = 0.72 and p < 0.001). Such a correlation is expected, which may be caused by a combination of intrinsic (a more energetic magnetar gives more significant contribution to both prompt emission and afterglow) and geometric effects (a narrower jet would enhance both prompt emission and afterglow).

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Figure 7(b) presents an anti-correlation between L_b and t_b (Dainotti et al. 2010). Our Gold + Silver sample gives

$$\begin{equation} \log L_{b,49}=(-1.83 \pm 0.20) \log t_{b,3}+(0.2 \pm 0.18) \end{equation} \tag{ 26 }$$

with r = 0.84 and p < 0.001. Adding the Aluminum sample only slightly worsens the correlation (log L_{b, 49} = (− 1.29 ± 0.15)log t_{b, 3} − (0.43 ± 0.14) with r = 0.66 and p < 0.001). Such an anti-correlation is consistent with the prediction of the magnetar model: Given a quasi-universal magnetar total spin energy, a higher magnetic field would power a brighter plateau with a shorter duration, or vice versa (see also Xu & Huang 2012).

In Figure 8, we compare the inferred E_γ + E_K with the total rotation energy E_rot (Equation (1)) of the millisecond magnetar. It is found that the GRBs are generally above and not too far above the E_rot = E_γ + E_K line. This is consistent with the magnetar hypothesis, namely, all the emission energy ultimately comes from the spin energy of the magnetar. Figure 8(a) includes all the GRBs in the Gold/Silver/Aluminum samples, with θ_j = 5^o assumed if the jet angle is not measured. Figure 8(b) presents those GRBs with jet measurements only. Essentially the same conclusion is reached.

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A very interesting question is whether there are noticeable differences between the magnetar and non-magnetar samples. One potential discriminator would be the total energetics of the GRBs. While the magnetar model predicts a maximum value of the total energy (Equation (1)), the black hole model is not subject to such a limit. In Figure 9 we make some comparisons. The first three panels compare the histograms of the isotropic energies (E_{γ, iso}, E_{K, iso}, and E_{γ, iso} + E_{K, iso}) of the magnetar and non-magnetar samples. For the magnetar sample, we in one case includes the most secure (Gold + Silver) sample only (blue hatched), and in another case includes all magnetar candidates (Gold + Silver + Aluminum) (red solid). The non-magnetar sample is marked in gray. The best Gaussian fits to the three samples are presented as blue, red, and black dotted curves, respectively. The center values of all the fits are presented in Table 3. It is found that without jet correction, the isotropic values of the magnetar and non-magnetar samples are not significantly different.

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Table 3. The Center Value of Gaussian Fitting of the Distributions

	Gold+Silver	Gold+Silver+Aluminum	Non-magnetar
Eγ, iso	(52.87 ± 0.33) erg	(52.89 ± 0.09) erg	(53.20 ± 0.04) erg
EK, iso	(53.11 ± 0.09) erg	(53.99 ± 0.06) erg	(53.94 ± 0.02) erg
Etotal, iso	(53.31 ± 0.05) erg	(54.05 ± 0.05) erg	(54.01 ± 0.05) erg
	Silver	Silver+Aluminum	Non-magnetar
Eγ	(48.55 ± 0.11) erg	(49.06 ± 0.13) erg	(50.11 ± 0.12) erg
EK	(50.55 ± 0.17) erg	(51.13 ± 0.12) erg	(51.54 ± 0.18) erg
Etotal	(50.62 ± 0.07) erg	(51.06 ± 0.09) erg	(51.81 ± 0.11) erg

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Next, we introduce beaming correction, and replot the histograms of the jet-corrected energies of the magnetar and non-magnetar samples. The results are presented in the later three panels in Figure 9. One can see a clear distinction between the robust magnetar sample (Gold + Silver) and the non-magnetar sample. For the total energy (E_γ + E_K), while the former peaks around 50.62 erg, the latter peaks around 51.81 erg. More interestingly, all the Gold+Silver magnetar sample GRBs have a total energy smaller than the limit set by the spin energy (Equation (1)), while for some non-magnetar sample GRBs, this upper limit is exceeded. The results are generally consistent with the hypothesis that two types of GRB central engines can both power GRBs.

In Figures 10(a) and (b), we compare E_{γ, iso} and E_{K, iso} for the magnetar and non-magnetar samples. The kinetic energy of the blastwave E_{K, iso} is evaluated at t_b for Figure 10(a), and at t_dec for Figure 10(b) (similar to Zhang et al. 2007a). It is interesting to see that at t_dec, the magnetar central engine tends to power more efficient GRBs (due to the initial small E_K value) than the black hole central engine. It is interesting to see after the energy injection phase (at t_b), the γ-ray efficiencies of magnetar and non-magnetar samples are no longer significantly different. The same conclusion is also manifested in Figures 11(a) and (b), where we plot the histograms of η_γ for different samples.

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If one accepts that millisecond magnetars are powering some GRBs, it would be interesting to constrain the internal energy dissipation efficiency η_X (Equation (24)) from the data. In both Figures 7(a) and (b), it is found that the Gold sample GRBs have a relatively large L_b value. This is generally consistent with the expectation that a larger η_X would give rise to an internal plateau (Zhang 2014). In Figure 10(c), we compare L_{K, iso} and L_{X, iso}. It indeed shows that the Gold sample GRBs have a much higher η_X than other GRBs. On the other hand, it is curious to ask why there is a gap in this phase space. It appears that some magnetars are particularly efficient to dissipate the magnetar wind energy, while most magnetars are not. Plotting the histograms of η_X (Figure 11(c)), it looks indeed like a bimodal distribution of η_X, even though this second high η_X component is not significant enough. In Figure 12, we present the scatter plots of η_X against other parameters, including η_γ(t_b), E_{γ, iso}, E_{K, iso}, and E_rot. In all cases, the Gold sample (the ones with very high η_X) tend to stick out and emerge as a separate population.

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In order to address whether (at least) some GRBs might have a magnetar central engine, we have systematically analyzed the X-ray data of all the Swift GRBs (∼750) detected before 2013 August. By applying some criteria to judge how likely a GRB might harbor a millisecond magnetar central engine, we characterized long GRBs into several samples: Gold, Silver, and Aluminum magnetar samples, as well as the non-magnetar samples. For comparison, we also independently processed the data of short GRBs that might have a magnetar central engine (Rowlinson et al. 2010, 2013). By deriving the basic magnetar parameters P₀ and B_p from the data, we are able to reach two interesting conclusions.

First, it seems that at least for the Gold and Silver sample GRBs, the derived properties seem to be consistent with the expectations of the magnetar central engine model. The consistency includes the following: 1. After beaming correction, the derived P₀ and B_p seem to fall into the reasonable range expected in the magnetar central engine model; 2. The L_b–t_b anti-correlation seems to be consistent with the hypothesis that there is a quasi-universal energy budget defined by the spin energy of the magnetars (Equation (1)); 3. The sum of E_γ and E_K is generally smaller than E_rot, the total energy budget of a magnetar; 4. Most importantly, it seems that the magnetar and non-magnetar samples are different. The robust magnetar sample (Gold + Silver) GRBs all have a beaming-corrected energy smaller than the maximum energy allowed by a magnetar, i.e., E_{rot, max} ∼ 2 × 10⁵² erg. The non-magnetar sample, on the other hand, can exceed this limit. The two samples have two distinct distributions in E_γ, E_K, and (E_γ + E_K), suggesting that they may be powered by different central engines.

Second, both long and short GRBs can be powered by a millisecond magnetar central engine. The characteristic magnetar signature, an internal plateau, is found in both long and short GRBs, suggesting that different progenitors (both massive star core collapses and compact star mergers) can produce a millisecond, probably supra-massive magnetar as the central engine. The data is consistent that a long GRB magnetar wind is collimated, while a short GRB magnetar wind is essentially isotropic. All these have profound implications in several related fields in high-energy, transient astronomy. For example, if the recently discovered fast radio bursts (FRBs; Lorimer et al. 2007; Thornton et al. 2013) are indeed produced when a supra-massive neutron star collapses into a black hole (Falcke & Rezzolla 2014; Zhang 2014), our analysis suggests that such supra-massive neutron stars very likely do exist in GRBs, and that the FRB/GRB association suggested by Zhang (2014) should be quite common, probably up to near half of the entire GRB population. This is higher than the rate of plausible detections made by Bannister et al. (2012), but that low detection rate (2 out of 9 GRBs, Bannister et al. 2012) may be due to the sensitivity limit of the Parkes 12 m telescope they have used. A rapid-slewing larger radio telescope would be able to detect more FRB/GRB associations, which would open a new window to study cosmology (Deng & Zhang 2014) and conduct cosmography (Gao et al. 2014). For another example, the conclusion that short GRBs can be powered by a millisecond magnetar with a near isotropic magnetar wind would give rise to relatively bright, early electromagnetic counterparts of gravitational wave bursts due to NS–NS mergers (Zhang 2013; Gao et al. 2013a; Yu et al. 2013; Metzger & Piro 2013; Fan et al. 2013), which gives promising prospects of detecting electromagnetic counterparts of gravitational wave signals in the Advanced LIGO/Virgo era.

Our analysis also poses some curious questions. One is regarding the magnetar dissipation efficiency η_X. The results seem to suggest that some magnetars are efficient in dissipating their magnetar wind energy to X-ray radiation, while most others are not. A straightforward inference would be that there might be a dichotomy within the magnetar central engines. A more plausible scenario would be that some (or probably) most normal plateaus (those followed by normal decays) could be also dominated by internal dissipation emission (e.g., Ghisellini et al. 2007; Kumar et al. 2008a, Kumar et al. 2008b). They are not identified as internal plateaus because their post-break decay is not steep enough. Physically they may be stable magnetars or supra-massive magnetars with a much later collapsing time, so that the collapsing signature (very steep decay) is not detected. If so, the η_X distribution may be more spread out, without a clear bimodal distribution. This possibility is worth exploring in the future.

Another mystery is regarding collimation of magnetar wind in short GRBs. Our analysis suggests that at late times the magnetar wind is essentially isotropic. On the other hand, during the prompt emission phase, at least some short GRBs show evidence of collimation (e.g., Burrows et al. 2006; Soderberg et al. 2006; Berger 2013 for a review). There is no well studied short GRB prompt emission model within the magnetar central engine scenario. Suggested scenarios invoke an early brief accretion phase (Metzger et al. 2008), an early brief differential rotation phase (Fan et al. 2013), or an early brief phase-transition phase (e.g., Cheng & Dai 1996; Chen & Labun 2013). The short GRB could be collimated by the torus within the accretion scenario (Bucciantini et al. 2012).

Upon finishing this paper, we were drawn attention to Yi et al. (2014), who performed an independent analysis on a sub-sample of GRB magnetar candidates (essentially our Gold sample). They assumed that the long GRB magnetar winds are isotropic and used the data to constrain magnetar wind dissipation efficiencies. Through a comparison with X-ray emission efficiency of spin-down powered pulsars, they offer support to the millisecond magnetar central engine model from a different point of view.

We acknowledge the use of the public data from the Swift data archive and the UK Swift Science Data Center. We thank an anonymous referee for helpful comments, He Gao and En-Wei Liang for helpful discussion, and Zi-Gao Dai, Xue-Feng Wu, and Shuang-Xi Yi for sharing their paper with us and related discussion. This work is supported by the NASA ADAP program under grant NNX10AD48G.