PHOTOSPHERE EMISSION IN THE X-RAY FLARES OF SWIFT GAMMA-RAY BURSTS AND IMPLICATIONS FOR THE FIREBALL PROPERTIES

The radiation physics of prompt gamma-ray emission is still a mystery. It is related to the unknown composition and radiation mechanism of the gamma-ray burst (GRB) jets (e.g., Zhang 2014). The standard fireball-shock model predicts a GRB prompt emission spectrum as superposition of a quasi-thermal photosphere emission component and a nonthermal component from internal shocks (e.g., Mészáros & Rees 2000; Zhang & Mészáros 2002; Daigne & Mochkovitch 2002; Pe'er et al. 2006, 2012; Toma et al. 2011). GRB observations with CGRO/BATSE in the 25–2000 keV band revealed that a typical GRB spectrum in the BATSE band is empirically fitted with a smooth broken power-law (PL) function, the so-called "Band function" (Band et al. 1993; Preece et al. 2000). Possible superposition of a thermal component on the nonthermal spectrum was claimed in some BATSE GRBs (e.g., Ghirlanda et al. 2003; Ryde 2004, 2005; Bosnjak et al. 2006; Ryde & Pe'er 2009). With the gamma-ray burst monitor (GBM) and Large Area Telescope (LAT) on board the Fermi mission, GRBs are now observed in a broad spectral band from several keV to 300 GeV. Most Fermi GRB spectra are still well fitted with the Band function (Abdo et al. 2009; Zhang et al. 2011; Lu et al. 2012). This has led to the suggestion that the composition of GRB jets is magnetically dominated (e.g., Zhang & Pe'er 2009). However, evidence for a blackbody (BB) component is also found. The most prominent case is GRB 090902B, whose time-integrated broadband spectrum can be decomposed into a dominating multicolor BB comment and an extra PL component in the GBM and LAT band (Ryde et al. 2010). For small enough time bins, the time-resolved spectrum can be decomposed into a BB component plus a PL (Zhang et al. 2011). Another interesting case with detection of a BB component is GRB 081221. The spectrum of GRB 081221 shows a bimodal feature, which is well fit with the Band function plus a BB component with a typical kT ∼ 20 keV (S.-J. Hou et al. 2014, in preparation). A weak BB component may also contribute to the total flux with a fraction of less than 10% in GRBs 110721A (Axelsson et al. 2012), 100724B (Guiriec et al. 2011), and GRB 120323A (Guiriec et al. 2013). These observations indicate that the intrinsic GRB spectra of some GRBs are composed of a thermal (or quasi-thermal) component and a nonthermal component (the broad Band function or the exponential cutoff PL function), as expected in the standard fireball model.

The Swift mission plays a critical role in revealing the nature of the GRBs with its rapidly slewing capacity and multiwavelength observations (Gehrels et al. 2004). Bright X-ray flares have been observed with the X-Ray Telescopes (XRTs; Burrows et al. 2005b) on board Swift from tens to 10⁵ s post GRB triggers with the Burst Alert Telescope (BAT; Barthelmy et al. 2005) for half of the GRBs (Burrows et al. 2005a; Falcone et al. 2006; Chincarini et al. 2007, 2010). The majority of flares happened at t < 1000 s, and some flares may occur at ∼10⁵ s after the triggers. Both spectral and temporary properties of X-ray flares have been extensively studied (e.g., O'Brien et al. 2006; Liang et al. 2006; Falcone et al. 2007; Chincarini et al. 2010; Margutti et al. 2011; Qin et al. 2013; Wang & Dai 2013; Hu et al. 2014). It is generally believed that X-ray flares are of an internal origin and may signal the restart of the GRB central engine post the prompt gamma-ray phase (e.g., Burrows et al. 2005a; Zhang et al. 2006; Liang et al. 2006; Dai et al. 2006; see review by Zhang 2007). The spectra of most X-ray flares in the XRT band is adequate to be fitted by an absorbed single PL model, although the GRB Band function or the single PL plus a BB component may improve the fits for some flares (Falcone et al. 2007; Page et al. 2011). Chincarini et al. (2010) confirmed that the X-ray flares are tightly linked to the prompt emission by analyzing the width evolution with energy, the ratio between rising and decaying timescales, as well as the spectral energy distribution of X-ray flares. Margutti et al. (2010) found that the peak energies (E_p) of the early flares observed in GRBs 060904B and 060418 are marginally consistent with the E_p  −  L_iso relation derived from prompt gamma-rays (Yonetoku et al. 2004; Liang et al. 2004), but deviate from the E_p  −  E_{γ, iso} relation for typical GRBs (Amati et al. 2002).

This paper is dedicated to the search for a BB component in the joint XRT+BAT band (0.3–150 keV) for the observed X-ray flares with the Swift mission in a sample as described in Section 2. We present a joint spectral analysis of the X-ray flares that were simultaneously observed with BAT and XRT and find a thermal emission component embedded the joint spectra of a fraction of flares (Section 3). By applying the standard fireball photosphere theory (Section 4.1), we constrained the fireball properties of these flares, and found tight Γ_ph  −  L_BB, E_p  −  L_BB correlations for the thermal component of these flares (Section 4.2). We draw conclusions in Section 5, and discuss the implications for understanding the physics of X-ray flares. The concordance cosmology with parameters H₀ = 71 km s⁻¹Mpc⁻¹, Ω_M = 0.3, and Ω_Λ = 0.7 is adopted.

With the promptly slewing capacity of the Swift mission, the prompt emission of some GRBs was simultaneously observed with BAT and XRT. We search for the flares that are also bright in the BAT band to make joint spectral fits. We take the XRT light curves of the GRBs observed with Swift from http://www.swift.ac.uk/xrt_spectra/ (Evans et al. 2009). The XRT light curves are binned dynamically with a minimum signal-to-noise ratio of three. The XRT light curves are usually composed of an underlying component with multiple PL-decaying segments and some flares (Zhang et al. 2006; Nousek et al. 2006). We select only those flares that do not significantly overlap with adjacent flares. We empirically fit the light curves around the selected flares with a model of a broken PL plus a single PL to obtain the profiles of the flares (Margutti et al. 2010, 2011; Chincarini et al. 2010.), similar to what was usually done when decomposing the prompt gamma-ray pulses (e.g., Norris et al. 1996). We extract the background-subtracted BAT light curves with a bin size of 1.024 s, and use the Bayesian black method to analyze the profile of the BAT light curves. For the details of our analysis on the BAT light curves, please refer to Hu et al. (2014). We adopt the following criteria to select the flares for our analysis. First, the flares are bright, with F_p/F_u > 5, where F_p is the flux at the peak time (t_p) and F_u is the flux of the underlying PL component at the peak time derived from our empirical fits. Since X-ray flares are asymmetric with a slow decay and relatively fast rising wing (Chincarini et al. 2010), we normally select a time interval [t_p − 10s, t_p + 20s] for our analysis, but sometimes the time interval may vary based on the detailed light curve behavior. Second, the BAT light curves in the corresponding time intervals have a signal-to-noise ratio that is greater than 4σ, where σ is the standard deviation of the background. With these criteria, the extracted spectra are dominated by the flares/pulses in the XRT–BAT band. Based on these criteria, we identified a sample of 32 bright flares for our analysis by the end of 2014 April.⁸ These flares belong to different GRBs, thus we finally obtained a sample of 32 GRBs. Their joint BAT and XRT light curves and the selected time intervals are illustrated in Figure 1. Associated peaks in the BAT and XRT bands are found in our selected flares, although the corresponding X-ray flares are usually much broader than the pulses in the BAT band.

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We extract the BAT spectrum with the HEAsoft package (ver.6.15). The BAT event data are downloaded from the Swift archive.⁹ We use the tool batbinevt to extract the spectrum and apply corrections to the spectra with tools batupdatephakw and batphasyserr. A response matrix is then generated with batdrmgen. The XRT spectra are extracted using an interacting tool available from the XRT team.¹⁰ We bin the XRT spectrum at least 20 counts per bin with a tool grppha.

The derived BAT and XRT spectra are jointly fitted with the spectral analysis package Xspec (version: 12.8.1). The Galactic absorption is fixed (Kalberla et al. 2005) and the intrinsic absorption is adopted as a free parameter. Three spectral models are considered, i.e.,

1. the single power-law model (PL model),

$$\begin{equation} { N(E)={ K_{\rm PL}} E^{-\Gamma }}; \end{equation} \tag{ 1 }$$

2. the blackbody radiation model (BB model),

$$\begin{equation} { A(E)=\frac{8.0525 \times K_{\rm BB} E^2 d E}{(kT)^4 (e^{(E/kT)}-1)}}; \end{equation} \tag{ 2 }$$

and

3. the Band function,

$$\begin{equation} A(E)= \left\lbrace \begin{array}{@{}l@{}l@{}} K(E/100)^{\alpha }e^{-E/E_0} \quad {{\rm if}\; E<(\alpha -\beta)E_0 }&\\ K[(\alpha -\beta)E_c/100]^{(\alpha -\beta)}(E/100)^{\beta }e^{-(\alpha -\beta)} & \\ {{\rm if}\; E > (\alpha -\beta)E_0} \end{array}\right.. \end{equation} \tag{ 3 }$$

The strategy and procedure of our spectral fits are described as the following.

• 1.  
  Since the observed spectra in the 0.3–2 keV band highly suffer absorption by metals (denoted as the corresponding neutral hydrogen column density N_H) along the line of sight, we first make a preliminary fit to a joint XRT+BAT spectrum in the 2–150 keV band with the PL model in order to judge the preferred models that we use.
• 2.  
  We inspect the deviation between the observed spectrum and the absorbed PL fit to select preferred models. In the case in which the model is adequate to fit the spectrum in the 2–150 keV band, with a reduced χ² < 1.1, the preferred model remains as the absorbed PL model. In the case in which both the high-energy (>40 keV) end and the low-energy (2  −  5 keV) end of the spectrum are consistent with the fitting line, but the data show a bump feature in the middle range, the preferred model to improve the fit is the absorbed single PL plus a BB component (the PL+BB model). In the case in which the observed spectrum is curved, with both the high-energy (>40 keV) end and low-energy (∼2  −  5 keV) end of the spectrum deviating from the single PL fitting line, the preferred model to improve the fit is the absorbed Band function.
• 3.  
  We make global fits to the observed spectrum in the 0.3–150 keV band with the preferred models and compare the results to the fit with the absorbed PL model. If the parameters of the preferred models are well constrained and the reduced χ² is ∼1.0, we adopt the fitting results of the preferred models.

We finally obtain the following results about the joint spectra of the X-ray flares. Eight flares are adequate to be fitted with the absorbed single PL model and 11 are well fitted with the absorbed Band function, whereas 13 are better fitted with the absorbed PL+BB model. The observed spectra with our fitting curves are shown in Figures 2–4 and the results are reported in Tables 1–3. The quoted errors of spectral fitting parameters have a confidence level of 90%. To access the adopted model fits, we compare the $\chi ^2_r$ of the adopted model fits to that of the single PL fits in Figure 5. One can find that the absorbed PL+BB model and the absorbed Band function model significantly improve the fits over that with the absorbed PL model for two-thirds of the spectra in our sample. Note that the $\chi ^2_{r}$ of the finally adopted single PL fits of two GRBs (06060607A and 130606A) are larger than 1.1, since the fits with the PL+BB models and the Band function cannot reduce the χ².

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Table 1. Results of Our Fits with the Single Power-law Model to the Joint Spectra Observed with BAT and XRT for the Eight Flares

GRB	Interval	Γ	$N_{H,22}^{\rm host}$	χ2/Bins
	(s)			($\rm \chi ^2_{red}$)
060607A	82–112	$1.60_{-0.05}^{+0.05}$	$1.39_{-0.38}^{+0.44}$	176/143(1.26)
100212A	70–90	$1.75_{-0.05}^{+0.06}$	$0.14_{-0.03}^{+0.03}$	117/136(0.88)
100728A	107–137	$1.38_{-0.01}^{+0.01}$	$0.36_{-0.04}^{+0.04}$	209/195(1.09)
100901A	381–411	$1.34_{-0.03}^{+0.03}$	$0.54_{-0.15}^{+0.19}$	139/136(1.05)
111103B	100–130	$1.68_{-0.03}^{+0.03}$	$0.33_{-0.04}^{+0.05}$	143/156(0.94)
120514A	135–165	$1.88_{-0.05}^{+0.05}$	$1.47_{-0.15}^{+0.16}$	151/169(0.91)
130606A	145–160	$1.03^{+0.03}_{-0.03}$	$3.89^{+3.88}_{-2.70}$	115/95(1.26)
140108A	81–96	$1.40^{+0.02}_{-0.02}$	$0.08_{-0.08}^{+0.20}$	100/111(0.93)

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Table 2. Results of Our Fits with the Band Function Model^a to the Joint Spectra Observed with BAT and XRT for the 11 Flares

GRB	Interval	α	β	Ep	$N^{\rm host}_{H,22}$	χ2/Bins
	(s)			(keV)		($\rm \chi ^2_{red}$)
090715B	61–91	$-0.86_{-0.13}^{+0.32}$	$-1.81_{-0.12}^{+0.14}$	$16.65_{-9.59}^{+9.59}$	$2.04_{-0.83}^{+0.80}$	211/185(1.17)
100619A	85–100	−1.00	$-1.94_{-0.05}^{+0.05}$	$12.47_{-1.87}^{+1.86}$	$0.55_{-0.09}^{+0.10}$	116/116(1.03)
100704A	160–190	$-1.09_{-0.17}^{+0.38}$	$-2.64_{-0.13}^{+0.12}$	$6.24_{-3.20}^{+3.20}$	$0.39_{-0.10}^{+0.07}$	203/204(1.03)
100814A	135–165	−1.00	$-1.77_{-0.10}^{+0.08}$	$13.94_{-3.25}^{+3.35}$	$0.23_{-0.10}^{+0.11}$	159/162(1.01)
100906A	100–130	−1.00	$-2.64_{-0.07}^{+0.07}$	$6.00_{-0.54}^{+0.52}$	$4.40_{-0.89}^{+1.01}$	198/172(1.18)
110102A	198–213	−0.49 ± 0.06	−1.33 ± 0.03	10.00	0.12	129/123(1.08)
110119A	190–215	−1.00	$-1.78_{-0.08}^{+0.06}$	$13.41_{-2.10}^{+ 2.44}$	$0.18_{-0.03}^{+0.03}$	223/180(1.27)
111123A	135–165	−1.00	$-1.65_{-0.07}^{+0.06}$	$17.12_{-3.42}^{+3.60}$	$0.20_{-0.03}^{+0.03}$	192/195(1.00)
121123A	230–260	−0.90 ± 0.03	⋅⋅⋅	$57.54_{-4.04}^{+4.59}$	$0.10_{-0.03}^{+0.04}$	173/183(0.97)
121211A	158–188	$-1.12^{+0.07}_{-0.04}$	$-2.39^{+0.13}_{-0.16}$	10.00	$1.15_{-0.22}^{+0.26}$	172/173(1.03)
140206A	53–68	$-0.93^{+0.05}_{-0.04}$	⋅⋅⋅	$99.32^{+10.56}_{-8.52}$	$11.16_{-3.49}^{+4.38}$	142/135(1.09)

Notes. ^aParameters without error are not well constrained by the data. GRB 121123A and GRB 140206A favor cutoff power-law model, which is a good approximation for Band function.

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Table 3. Results of Our Fits with the Single Power-law Plus the BB Model to the Joint Spectra Observed with BAT and XRT for the 13 Flares

GRBa	zref	Interval	ΓPL	KPL	kT	KBB	$N_{\rm H,22}^{\rm host}$	χ2/Bins
		(s)			(keV)			($\rm \chi ^2_{r}$)
060510B	4.91	290–320	1.50 ± 0.04	0.66 ± 0.08	$1.41_{-0.20}^{+0.32}$	$0.03^{+0.007}_{-0.006}$	$3.37_{-1.31}^{+1.52}$	158/168(0.97)
060526	3.212	236–266	$1.71_{-0.05}^{+0.06}$	$1.68_{-0.27}^{+0.34}$	$1.56_{-0.22}^{+0.64}$	$0.11_{-0.02}^{+0.04}$	$5.33_{-1.32}^{+1.72}$	145/163(0.92)
060814	0.843	121–151	$1.63_{-0.04}^{+0.04}$	$2.02_{-0.26}^{+0.26}$	$1.90_{-0.39}^{+0.43}$	$0.07_{-0.02}^{+0.03}$	$1.28_{-0.21}^{+0.24}$	147/146(1.04)
061121	1.314	65–90	$1.32_{-0.02}^{+0.02}$	$7.86_{-0.65}^{+0.70}$	$2.74_{-0.38}^{+0.31}$	$0.53_{-0.18}^{+0.18}$	$2.23_{-0.39}^{+0.47}$	126/135(0.97)
070129	⋅⋅⋅	300–330	$1.65_{-0.05}^{+0.05}$	$1.16_{-0.16}^{+0.16}$	$1.53_{-0.26}^{+0.49}$	$0.03_{-0.01}^{+0.01}$	$0.25_{-0.04}^{+0.04}$	199/193(1.06)
080810	3.355	90–120	$1.48_{-0.05}^{+0.05}$	$0.50_{-0.08}^{+0.08}$	$1.22_{-0.18}^{+0.30}$	$0.03_{-0.01}^{+0.01}$	$1.63_{-0.72}^{+0.82}$	169/156(1.12)
080928	1.696	193–223	$1.60_{-0.04}^{+0.04}$	$2.06_{-0.27}^{+0.29}$	$1.38_{-0.15}^{+0.19}$	$0.12_{-0.02}^{+0.02}$	$1.14_{-0.30}^{+0.35}$	162/181(0.92)
090709A	⋅⋅⋅	74–104	1.32 ± 0.03	$1.44_{-0.14}^{+0.16}$	$2.27_{ -0.21}^{+0.18}$	0.15 ± 0.03	$2.28_{-0.57}^{+0.70}$	196/173(1.17)
100725B	⋅⋅⋅	200–230	$2.40_{-0.08}^{+0.08}$	$9.63_{-2,00}^{+2.18}$	$1.10_{-0.14}^{+0.24}$	$0.13_{-0.03}^{+0.03}$	$0.64_{-0.08}^{+0.09}$	176/176(1.03)
110801A	1.867	344–374	1.63 ± 0.05	$1.67_{-0.27}^{+0.31}$	$1.24_{-0.11}^{+ 0.15}$	0.11 ± 0.01	$1.51_{-0.47}^{+0.62}$	226/203(1.15)
121217A	⋅⋅⋅	715–745	$1.44_{-0.03}^{+0.04}$	$1.43_{-0.13}^{+0.25}$	$2.61_{-0.21}^{+0.20}$	$0.14_{-0.03}^{+0.02}$	$0.01_{-0.01}^{+0.12}$	193/162(1.23)
130514A	3.68	95–125	1.67 ± 0.03	$2.48_{-0.24}^{+0.26}$	$2.47_{-0.24}^{+0.21}$	0.16 ± 0.03	0.09 ± 0.05	205/199(1.06)
130609B	⋅⋅⋅	178–208	1.79 ± 0.03	$7.29_{-0.69}^{+0.75}$	$2.07_{-0.16}^{+0.14}$	0.34 ± 0.06	0.16 ± 0.04	215/195(1.13)

References. (1) Price 2006; (2) Berger & Gladders 2006; (3) Thoene et al. 2007; (4) Bloom et al. 2006; (5) Prochaska et al. 2008; (6) Vreeswijk et al. 2008; (7) Cabrera Lavers et al. 2011;(8) Schmidl et al. 2013.

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The selected X-ray flares in our sample are also detected by BAT; therefore, they also belong to the prompt emission phase even though most of them were detected at the late episode of the prompt emission phase. We compare the spectral parameters between the X-ray flares and the earlier prompt gamma-ray emission before X-ray flares are detected. As is well known, the dominant component in the prompt gamma-ray emission spectra is the Band function component (Band et al. 1993; Zhang et al. 2011). Owing to the narrow energy coverage of BAT, the prompt gamma-ray spectra observed with BAT are usually adequately fitted with a single PL model. The spectra of the selected flares in 11 GRBs in our sample can be fitted with the Band function. Ten out of these 11 GRBs were also observed with the Fermi GBM or Konus–Wind during the earlier prompt emission phase. The spectra of these GRBs are well fitted with the Band function or cutoff PL models. We collect the spectral parameters of these GRBs from the Fermi/GBM Catalog (Goldstein et al. 2012; Paciesas et al. 2012.)¹¹ or from GCN Circulars (Golenetskii et al. 2009). Figure 6 shows the comparisons of E_p and α between the prompt gamma-ray emission (time-integrated spectra including both earlier gamma-rays and later gamma-rays overlapping with X-ray flares) and the X-ray flares.¹² One can observe that the E_ps of the flares are usually much lower than those of prompt gamma-rays, except for GRBs 121123A and 140206 since the flares in these two GRBs were detected during the peak time of prompt gamma-ray emission. The α values of the flares are typically in the range of 0.8–1.2, whereas it scatters in the range of 0–1.6 for the prompt gamma-rays. Note that GRB spectra usually show significant evolution with time and the evolution feature among GRBs is diverse (e.g., Liang & Kargatis 1996; Lu et al. 2010, 2012). Thus, revealing time-resolved spectra with different E_p using overlapping data between BAT and XRT, which is essential to unveil radiation physics (e.g., Lyu et al. 2014).

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4.1. Baryonic Photosphere Model

As shown in Mészáros & Rees (2000), a thermal component should be embedded in the observed GRB spectrum, if the GRB outflow is matter dominated. For a continuous outflow, the photosphere emission should be also continuous as thermal photons entrained in the outflow progressively become transparent. As a result, early prompt gamma-ray emission and later X-ray flares should have their own photosphere emission components. The thermal component in the BAT–XRT joint spectra would reveal the photosphere properties of the X-ray flares. The same theoretical framework can be applied to study the properties of the X-ray flare photosphere if the outflow is matter dominated.

In the following, we apply the standard baryonic photosphere model (Mészáros & Rees 2000) to constrain the physical properties of the X-ray flare outflows with the detections of a BB component.

Let us assume that a total luminosity of L₀ is released at an initial radius R₀, the initial dimensionless entropy is defined as $\eta =L_0/\dot{M}c^2$, where $\dot{M}$ is the mass rate of the outflow, and c is the speed of light. The initial temperature of the fireball is

$$\begin{eqnarray} T_0 &=&\left(L_0/4\pi R_0^2 a c\Gamma _i^2\right)^{1/4}\left/\right. (1+z)\nonumber\\ &=& 1.1\ {\rm MeV}\ L_{0,52}^{1/4}R_{0,7}^{-1/2}\Gamma _i^{-1/2}\left/\right. (1+z), \end{eqnarray} \tag{ 4 }$$

where Γ_i is the initial Lorentz factor at R₀, a is the radiation constant a = 7.57 × 10⁻¹⁵ erg cm⁻³ K⁻⁴, and the notation Q_n is defined as Q/10ⁿ in units of cgs. The scattering optical depth for the fireball with radius R and Lorentz factor Γ is given by

$$\begin{equation} \tau =\sigma _T n^{\prime } \delta R=\sigma _{\rm T} L_0/(4\pi R^2 m_p c^3 \eta \Gamma)(R/2\Gamma), \end{equation} \tag{ 5 }$$

where σ_T is the Thompson scattering cross section. Taking τ = 1, one obtains the photosphere radius as

$$\begin{equation} R_{\rm ph}=\frac{\sigma _T L_0}{8\pi m_p c^3 \eta \Gamma _{\rm ph}^2}, \end{equation} \tag{ 6 }$$

where Γ_ph is the Lorentz factor of the photosphere and m_p is the proton mass. Since the photosphere of a baryonic fireball is usually above the acceleration saturation radius, we have Γ_ph ≃ η, then

$$\begin{equation} R_{\rm ph}=\frac{\sigma _{\rm T} L_0}{8\pi m_p c^3 \Gamma _{\rm ph}^3}. \end{equation} \tag{ 7 }$$

The luminosity of the BB component is given by

$$\begin{equation} L_{\rm BB}=4\pi R^2 \Gamma _{\rm ph}^2 a T^{\prime 4} c=4\pi R^2\frac{(1+z)^4}{\Gamma _{\rm ph}^2}\sigma T^4, \end{equation} \tag{ 8 }$$

where T' and T = Γ_phT'/(1 + z) are the photosphere temperature in the comoving frame and the observer frame, respectively. The BB luminosity L_BB can be calculated with the observed flux, i.e., $L_{{\rm BB}}=4\pi D_L^2 F_{\rm BB}$. Therefore, we obtain

$$\begin{equation} { \Gamma _{\rm ph}=\left[\frac{(1+z)^2}{D_{\rm L}}\frac{\sigma _T L_{0}}{8\pi m_p c^3}\left(\frac{\sigma T^4}{F_{{\rm BB}}}\right)^{1/2}\right]^{1/4}} \end{equation} \tag{ 9 }$$

and

$$\begin{equation} { R_{\rm ph}=\left[\frac{D_L^3}{(1+z)^6}\frac{\sigma _T L_{0}}{8\pi m_p c^3}\left(\frac{F_{{\rm BB}}}{\sigma T^4}\right)^{3/2}\right]^{1/4}}. \end{equation} \tag{ 10 }$$

Assuming a radiation efficiency of 0.2 and the k correction of a factor of two for the nonthermal component with Band function parameters (α = −1, β = −2.3, and E_p = 300 keV), we may estimate L₀ ∼ 10 L_r, with L_r dominated by the nonthermal component. Eight out of the 13 GRBs with the BB component detection have redshift information. Using the observed flux and temperature of the BB component, one can obtain Γ_ph and R_ph for the eight GRBs, which are presented in Table 4. The derived Γ_ph is in the range of 50–150 and the photosphere radius is about R_ph ∼ 10¹³ cm. Since the photosphere radius is above the acceleration saturation radius, the Lorentz factor of the emission region producing X-ray flares should be nearly equal to Γ_ph. This provides a unique approach to estimate the Lorentz factor of the X-ray flares (Pe'er et al. 2007), which is poorly known thus far. Since these are bright X-ray flares, which occur at the late episode of the prompt emission, it is not unreasonable to have the inferred Lorentz factor ≳ 100.

Table 4. Derived Fireball Properties of the Eight Flares that have Detection of a Thermal Component and Redshift Measurement in Our Sample

GRB	LBB, 50	Ep'	Γph	Rph, 13	Rsat, 10	R0, 7a
060510B	6.23 ± 1.45	23.46 ± 5.32	141.04	1.42	1.25	8.85
060526	8.36 ± 3.35	18.52 ± 7.60	104.9	1.96	6.26	59.69
060814	0.19 ± 0.07	9.86 ± 2.23	57.97	0.57	0.56	9.61
061121	4.55 ± 1.54	17.88 ± 2.02	124.83	1.85	1.18	9.48
080810	2.05 ± 0.42	14.97 ± 3.68	98.01	1.39	1.22	12.48
080928	1.92 ± 0.29	10.47 ± 1.44	71.77	2.01	3.74	52.16
110801A	2.22 ± 0.20	10.00 ± 1.21	68.34	2.26	5.50	80.46
130514A	16.01 ± 3.00	32.04 ± 2.72	153.5	1.33	3.63	23.67

Note. ^aDerived from Equation (14).

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In the baryonic jet scenario, the Lorentz factor of the outflow evolves as

$$\begin{equation} \Gamma (R)=\left\lbrace \begin{array}{@{}ll}R/R_0,& {\rm if}\,\, R<R_{\rm sat}, \\ {\eta }, & {\rm if}\,\, R>R_{\rm sat}, \end{array} \right. \end{equation} \tag{ 11 }$$

where R_sat = ηR₀ is the saturation radius for acceleration. The observer-frame temperature is given by

$$\begin{equation} T=\left\lbrace \begin{array}{@{}ll}T_0(R_{\rm ph}/R_{\rm sat})^{-2/3},& {\rm if}\,\, R_{\rm ph}>R_{\rm sat}, \\ T_0, & {\rm if}\,\, R_{\rm ph}<R_{\rm sat}, \end{array} \right. \end{equation} \tag{ 12 }$$

where T₀ is the initial temperature of the fireball at R₀. The photosphere luminosity is expected to be

$$\begin{equation} L_{{\rm BB}}=\left\lbrace \begin{array}{@{}l@{\quad}l}L_0(R_{\rm ph}/R_{\rm sat})^{-2/3},& {\rm if}\,\, R_{\rm ph}>R_{\rm sat} \\ L_0, & {\rm if}\,\, R_{\rm ph}<R_{\rm sat} \end{array}. \right. \end{equation} \tag{ 13 }$$

Thus, R_sat, R₀, and T₀ can be estimated with

$$\begin{equation} \left\lbrace \begin{array}{@{}ll}R_{\rm sat}=R_{\rm ph}(L_{\rm BB}/L_0)^{3/2},\\ R_0=R_{\rm sat}/\Gamma _{\rm ph},\\ T_0=T(R_{\rm ph}/R_{\rm sat})^{2/3}=T_{\rm ph}(L_0/L_{\rm BB}). \end{array} \right. \end{equation} \tag{ 14 }$$

The results are reported in Table 4. The typical values of R₀ and R_s are ∼10⁸–10⁹ cm and a few ∼10¹⁰ cm, respectively. Interestingly, the value of the initial fireball radius R₀ is comparable to that derived from some GRBs using the prompt emission thermal emission data (e.g., in GRB090902B; Ryde et al. 2010). These results suggest that at least for some X-ray flares, emission may be from a baryon-dominated fireball.

4.2. Γ_ph  −  L_BB and E_p  −  L_BB Correlations for the Photosphere Radiation

Liang et al. (2010) and Lü et al. (2012) found tight correlations between the initial Lorentz factor and the isotropic gamma-ray energy/luminosity. With the Γ_ph values derived above for the eight flares, we show Γ_ph as a function of L_BB in comparison with the Γ₀  −  L_{γ, iso} in Figure 7. Interestingly, a tight Γ_ph  −  L_BB relation is found, i.e., log Γ_ph = (2.30 ± 0.08) + (0.20 ± 0.05)log L_{BB, 52}, with a Spearman correlation coefficient of 0.69 and a chance probability of 0.006 (N = 8). Adding the nonthermal component to the luminosity, the relation becomes log Γ_ph = (2.15 ± 0.03) + (0.25 ± 0.03)log L_{BB + PL, 52}, with a Spearman correlation coefficient of 0.88 and a chance probability p ∼ 10⁻⁴ (N = 8). We also over-plot the Γ₀  −  L_{γ, iso} relation (Lü et al. 2012) for comparison. Note that the Γ₀ reported in Lü et al. (2012) were estimated using other methods. In particular, for most GRBs, Γ₀ is estimated using the fireball deceleration timescales measured in the early afterglow light curves, and hence stands for the Lorentz factor before the deceleration time. On the other hand, as mentioned above, the baryonic photosphere radius is larger than the saturation radius R_sat. Therefore, the Γ_ph is essentially comparable to Γ₀. It is found that the Γ_ph  −  L_BB relation is consistent with the Γ₀  −  L_{γ, iso} relation.

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The physical origin of Γ₀  −  L_{γ, iso} is unknown. The observed Γ_ph  −  L_BB relation is a natural consequence of the baryonic photosphere model. Within this model, one expects $\Gamma _{\rm ph} \propto L_{\rm BB}^{0.27} R_0^{-0.18}$. If R₀ does not vary significantly within a burst, the correlation between Γ_ph and L_BB is naturally expected. Fan et al. (2012) started from radiation physics and suggested that emission from a baryonic photosphere can naturally produce a Γ_ph  −  L_BB relation and suggested that the observed Γ₀  −  L_{γ, iso} can be attributed to this, suggesting that the bulk of GRB emission comes from the baryonic photosphere (see also Lazzati et al. 2013).

The consistency between the Γ_ph  −  L_BB relation and the Γ₀  −  L_{γ, iso} relation (Lü et al. 2012) is intriguing. A direct implication would be that the prompt emission of GRBs is photosphere-dominated, as suggested by Fan et al. (2012). Indeed, some GRBs (e.g., GRB 090902B and GRB 081221) have been shown to be thermal-dominated (Ryde et al. 2010; Zhang et al. 2011; Hou et al. 2014). These GRBs indeed follow the Γ₀  −  L_{γ, iso} relations. On the other hand, the dominant emission component of most GRBs, namely, the traditional "Band" component, is likely not of a thermal origin. The arguments against the thermal origin of the Band function peaks include the following. (1) The observed spectral index below E_p is too soft to be interpreted by the photosphere model (Deng & Zhang 2014). (2) The hard-to-soft evolution across broad pulses as observed in many GRBs (Lu et al. 2010, 2012) is difficult to interpret within the photosphere model (Deng & Zhang 2014), but is straightforward to interpret within the synchrotron emission model (Uhm & Zhang 2014; Bošnjak & Daigne 2014). (3) The E_p of the Band component is sometimes above the "death line" defined by the photosphere model (Zhang et al. 2012). (4) Most importantly, several GRBs (100724B, 110721A, and 120323A; Guiriec et al. 2011, 2013; Axelsson et al. 2012) have both the photosphere component and the Band component identified, with the photosphere component being the sub-dominant component. The X-ray flares in our sample, even if they include a thermal component in the spectrum, are also nonthermal-dominated. Adding the nonthermal component to the luminosity, these flares also roughly align with the Γ₀  −  L_{γ, iso} relation, even though they introduce some scatter to the correlation. It was also proposed that the observed Γ₀  −  L_{γ, iso} correlation in GRBs may stem from an intrinsic physical reason related to the central engine by Lei et al. (2013), who showed that, for a hyper-accreting black hole central engine, both a neutrino–antineutrino annihilation model (which produces a fireball) and a Blandford–Znajek model (which produces a magnetically dominated outflow) can give rise to a Γ₀  −  L_{γ, iso} correlation similar to what is observed. We note the Γ_ph  −  L_{γ, iso} relation of X-ray flares (blue triangles) in Figure 7 lines below the main correlation of prompt gamma-rays. This may be a hint that the outflow is more magnetically dominated. Indeed, a generalized photosphere model invoking an arbitrary magnetization of the outflow (Gao & Zhang 2014, see also Veres & Mészáros 2012) suggests that the photosphere usually occurs during the phase when the outflow is still being accelerated. Also, the thermal to nonthermal flux ratio is low due to the early suppression of photosphere emission from a magnetized outflow and later efficient release of magnetic energy due to ICMART processes (Zhang & Yan 2011).

We further explore the E_p  −  L_BB relation for the BB component. We show L_BB as a function of E_p for the flares in comparison with the same correlation within GRBs 081221 (S.-J. Hou et al. 2014, in preparation), 090618 (Page et al. 2011),¹³ and 090902B (Zhang et al. 2011); the three GRBs with time-resolved thermal emission identified. Such an E_p  −  L_BB correlation is found among flares and also within individual flares. Generally, the E_p  −  L_BB correlation within a GRB is tighter than that among the flares. Our Spearman correlation analysis yields $\log L_{\rm BB}=(46.43\,{\pm}\, 0.95)+(3.32\,{\pm}\, 0.78)\log E^{^{\prime }}_{\rm p}$ with a scatter 0.37 for the flares (r = 0.71, p = 0.005), $\log L_{\rm BB}=(47.72\,{\pm}\, 0.52)+(2.13\,{\pm}\, 0.25)\log E^{^{\prime }}_{\rm p}$ with a scatter 0.16 for GRB 081221 (r = 0.81, p  <  10⁻⁴), $\log L_{\rm BB}=(47.73\,{\pm}\, 0.45)+(1.87\,{\pm}\, 0.16)\log E^{^{\prime }}_{\rm p}$ with a scatter 0.21 for GRB 090902B (r = 0.86, p  <  10⁻⁴), and $\log L_{\rm BB}=(48.21\,{\pm}\, 0.04)+(2.70\,{\pm}\, 0.09)\log E^{^{\prime }}_{\rm p}$ with a scatter 0.03 for GRB 090618 (r = 0.99, p = 0.001). This is probably because r₀ has a small scatter within a same GRB. The slope (ρ) of the E_p  −  L_BB correlations (i.e., $L_{\rm BB}\propto E_{\rm p}^{\rho }$) within individual GRBs varies slightly among GRBs. Globally, one can observe an E_p  −  L_BB correlation for all data in Figure 8 in a broad E_p and L_BB range. Our best fit gives $\log L_{\rm BB}=(48.51\,{\pm}\, 0.10)+(1.68\,{\pm}\, 0.05)\log E^{^{\prime }}_{\rm p}$ (r = 0.95, p < 10⁻⁴), with a scatter of 0.28.

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When over-plotting the E_p  −  L_{γ, iso} relation (Yonetoku et al. 2004; Liang et al. 2004) in Figure 8, we find that the E_p  −  L_BB relations also align with the observed E_p  −  L_{γ, iso} relation. A straightforward inference would be that GRB emission is photosphere dominated (Fan et al. 2012). However, due to the reasons discussed above, one may not draw this conclusion directly. Indeed, it was found that there is a correlation between the peak energy of the thermal and nonthermal components in GRBs (Burgess et al. 2014). As a result, the nonthermal component would also follow the Yonetoku relation. Indeed, E_p∝L^1/2 is naturally predicted in synchrotron radiation models (Zhang & Mészáros 2002), especially if the emission radius is not sensitively dependent on the bulk Lorentz factor (e.g., Zhang & Yan 2011). The observed Yonetoku relation may be a consequence of both thermal emission and synchrotron nonthermal emission producing a positive relation between E_p and L.

We have presented a joint spectral analysis of 32 GRB X-ray flares that were simultaneously observed with BAT and XRT. Our results show a diverse range of flare spectra. The joint spectra of eight flares are adequate to fit with an absorbed single PL model. The derived photon indices are Γ_PL < 2, which suggests that their E_p of the νF_ν spectra are beyond the XRT + BAT band. The joint spectra of 11 flares are well fitted with an absorbed Band function, with an E_p in the 3 ∼ 100 keV range. More interestingly, the joint spectra of 13 X-ray flares are fitted with the absorbed BB+PL model. The derived temperature is around 1 ∼ 3 keV. Phenomenally, the observed spectra of the 13 GRBs are analogous to several GRBs whose thermal component was identified in their spectra, but only with a much lower temperature. Assuming that the thermal emission is the photosphere emission of a relativistic fireball, we derive the physical properties of the eight flares that have redshift measurements. The derived Γ_ph is in the range of 50  −  150, and the typical R₀, R_s, and R_ph are ∼10⁸, ∼10¹⁰, and ∼10¹³ cm, respectively. We also find tight correlations for Γ_ph  −  L_BB and E_p  −  L_BB in the sample, which are straightforward within the photosphere model.

Combining our sample of X-ray flares with GRBs that have detected thermal components, one may derive the following overall picture of GRB thermal emission. Some GRBs, exemplified by GRB 090902B, have a dominant photosphere emission component with a temperature around 200 keV. It has an extra nonthermal PL component in the GBM and LAT band, with a peak and turnover energy beyond the LAT band. The high-energy component is likely of an inverse Compton scattering origin (Pe'er et al. 2012; Beloborodov et al. 2014). A similar high-energy component was also observed in GRB 090926A, with the high-energy peak identified (Ackermann et al. 2011). Next, in the case of GRB 081221, the time-resolved spectra show clear two-component features, with the lower and higher peaks having thermal and nonthermal origins, respectively. More common cases are those that have a sub-dominant thermal emission component, such as GRBs 090618 (Page et al. 2011), 110721A (Axelsson et al. 2012), 100724B (Guiriec et al. 2011), and GRB 120323A (Guiriec et al. 2013), in which the BB component contributes to the total flux with a fraction below 10%. The BB component in the 13 flares studied here may contribute to an even lower fraction of total flux. The PL indices of the nonthermal component in these flares are typically Γ < 2, suggesting that the peak of this component may be beyond the BAT band. All of these make a sequence of thermal-component dominance in the GRB (or X-ray flare) spectra, which may suggest that the GRB central engine may have a distribution of the initial magnetization σ, which causes the diversity of the observed spectra (e.g., Zhang et al. 2011). The results presented here also suggest that the nondetection of thermal component from GBM data in some GRBs may be due to the fact that the thermal component is weak and soft, which can be detected only in the X-ray band. For 13 bursts with PL plus BB spectral, BAT observation is dominated by nonthermal emission. Meanwhile, the mechanism to produce nonthermal emission and its efficiency are not clear, thus our results are reasonable in jets with different initial dimensionless entropy η and a magnetization factor σ₀ (Zhang 2014). A more general theory of the photosphere model invoking arbitrary magnetization (Gao & Zhang 2014) should be applied to fully diagnose the outflow parameters.

Our analysis suggests that the Γ_ph  −  L_BB and E_p  −  L_BB relations discovered for the X-ray flare photosphere component seem to be consistent with the observed Γ₀  −  L_{γ, iso} and E_p  −  L_{γ, iso} correlations derived from prompt gamma-ray emission (Lü et al. 2012; Yonetoku et al. 2004). These correlations are expected, since they are inherited from the baryonic photosphere model itself (Fan et al. 2012). On the other hand, the consistency with the correlations found in GRBs, in general, is intriguing. A straightforward inference would be that the GRB prompt emission is thermal-dominated, and the Band function is modified thermal emission from the photosphere (Fan et al. 2012; Lazzati et al. 2013). However, in view of the difficulties that the photosphere model encounters when interpreting the Band function (e.g., Deng & Zhang 2014; Zhang et al. 2012) and the fact that a dominant nonthermal component has been discovered in many GRBs (Guiriec et al. 2011, 2013; Axelsson et al. 2012), one must attribute those apparent consistencies to more profound physical reasons. For the Γ₀  −  L_{γ, iso} relation, it has been found that a hyper-accreting black hole central engine can naturally produce a similar correlation for both a thermal fireball and a magnetically dominated jet (Lei et al. 2013). For the E_p  −  L_{γ, iso} relation, both photosphere emission (Thompson et al. 2007; Fan et al. 2012; Lazzati et al. 2013) and synchrotron radiation (Zhang & Mészáros 2002; Zhang & Yan 2011) can reproduce a similar correlation. It is possible that different energy dissipation mechanisms and radiation mechanisms, by coincidence, can produce similar correlations as those observed. The combined results above, including the temperature and flux of BB component, the dynamic quantities, the relationship Γ_ph  −  L_BB, and E_p  −  L_BB, show that the GRB central engine is a hybrid system with both thermal energy and magnetic energy. Maybe for some X-ray flares, emission comes from a baryon-dominated fireball. More data and more detailed analyses of these correlations, by separating the thermal and nonthermal components with emission in the GeV band (e.g., Wang et al. 2006; Murase et al. 2011), are needed to better understand the underlying physics of GRB prompt emission and X-ray flares.

We appreciate valuable comments from the referee, and we thank Bin-Bin Zhang, Kim Page, and He Gao for helpful discussions. This work made use of data supplied by the UK Swift Science Data Center at the University of Leicester. It is supported by the National Basic Research Program (973 Programme) of China (grant 2014CB845800), the National Natural Science Foundation of China (grants 11025313, 11163001, 11363002), the Guangxi Science Foundation (2013GXNSFFA019001), the Key Laboratory for the Structure and Evolution of Celestial Objects of the Chinese Academy of Sciences, and the Strategic Priority Research Program "The Emergence of Cosmological Structures" of the Chinese Academy of Sciences, grant No. XDB09000000.