Spectral Properties of X-Ray Flares in Gamma-Ray Bursts

Introduction

X-ray flares are a common phenomenon observed in the X-ray afterglows of gamma-ray bursts (GRBs; [13, 12, 4]). In the sample of GRBs observed by the Swift satellite, approximately one-third of the bursts exhibited X-ray flares during the afterglow phase ([12]). From the manifestations of the flares, the X-ray flares of different GRBs exhibit diverse characteristics: some GRBs show only a single flare, while others display multiple successive flares, and sometimes with multiple flare components overlapping with each other (e.g., [12, 19, 6, 34, 48, 31]).

Corresponding author: Zejun Jiang

[email protected]

Corresponding author: Jirong Mao

[email protected]

The physical origin of X-ray flares remains debated. Due to the similar curves of light variation and characteristics of the spectral evolution of X-ray flares to prompt emission pulses, many researchers generally believe that X-ray flares and prompt emission may have the same physical origin, and both are related to prolonged activity of the central engine of GRBs (e.g., [6, 49, 38, 44, 13, 12, 15] (Ma & Gao 2025)). Therefore, X-ray flares are regarded as an important probe for exploring the physical properties of the central engine of GRBs.

Observationally, X-ray flares exhibit the following characteristics. (1) The profile of light variation can usually be described by an asymmetric structure that has a rapid rise and rapid decline, and the duration of the decline is approximately twice that of the rise ([12, 23]). (2) The duration of the flare is related to its isotropic equivalent X-ray energy $E_{\mathrm{X,iso}}$ and peak time; that is, flares with higher $E_{\mathrm{X,iso}}$ have shorter durations, whereas later-peaking flares have longer durations ([12, 48]). (3) The peak luminosity $L_{\mathrm{p}}$ of flares declines with increasing peak time, indicating that later-peaking flares are less energetic (i.e., lower $E_{\mathrm{X,iso}}$; [12, 48]). The spectral evolution of X-ray flares is observed to exhibit diverse features, and their physical origin remains unclear. Much research has studied the spectral characteristics of X-ray flares ([6, 40, 13, 12, 19, 36, 1, 31] (Lü et al. 2022)).

Some theoretical models have been proposed, such as the fragmentation model of collapsing stars ([27]), the accretion disk fragmentation model ([42]), the intermittent accretion model driven by magnetic fields ([43]), the magnetic reconnection model after the merger of millisecond pulsars ([14]), the magnetic dissipation model of decelerating shells ([21]), and the internal shock models ([6, 20, 28]). X-ray flares undoubtedly carry rich physical information about the central engine, providing us with a unique perspective for understanding the physical processes of GRBs.

Spectral analysis of GRBs is one of the most important methods for understanding their physical origins. In this paper, based on the X-ray observation data from Swift/XRT, we use a simple power-law model to fit the spectra of X-ray flares and construct a comprehensive spectral data sample of these flares. By comparing and analyzing the spectral properties of X-ray flares with afterglow emission and with prompt emission, we can test and restrict some physical models of GRBs. This paper is organized as follows: Section 2 is the sample selection and spectral fitting of X-ray flares. Section 3 is the statistical analysis of the spectra of X-ray flares. Section 4 is a discussion. Section 5 is the summary of our work.

Sample Selection and Spectral Fitting

Sample Selection

In this paper, the selection of samples of X-ray flares is mainly based on the data of X-ray flares that have been verified by [12] and [48]. According to the temporal parameters of X-ray flares provided by those authors, we conducted time-sliced spectral analyses for the three phases: the flare phase, the preflare phase, and the postflare phase.

Our final sample comprises 521 X-ray flares originating from 206 GRBs observed between 2005 April 6 and 2015 March 23. Within this GRB sample, 98 bursts exhibit a single flare, 57 bursts display two flares, and the remaining 51 bursts show three or more flares. The sample includes 99 short-duration GRBs (S-GRBs) and 202 long-duration GRBs (L-GRBs), among which a total of 99 GRBs have known redshifts.

In addition, we referred to the third Swift/BAT Gamma-Ray Burst Catalog published by Lien et al. [30] and collected the prompt emission spectral parameters of these 206 GRBs, including the power-law photon index ($\alpha_{\mathrm{PL}}$) and the cutoff power-law photon index ($\alpha_{\mathrm{CPL}}$). According to the classification criteria for GRB subclasses proposed by Bi et al. (2018), our sample contains 17 X-ray flashes (XRFs), 122 X-ray-rich GRBs (XRRs), and 67 classical GRBs (C-GRBs). The relevant data of the GRB X-ray samples are summarized in Table 1.

Spectral Fitting

The X-ray observations of GRBs in our sample were obtained from the public website of the Neil Gehrels Swift Observatory¹, which provides data from both the Burst Alert Telescope (BAT) and the X-ray Telescope (XRT). Detailed descriptions of the Swift/XRT data reduction procedures, light-curve fitting, spectral fitting, example results, and the usage of the online database are presented by Evans et al. [16, 17]. The Swift/XRT team also maintains an X-ray spectral repository² for GRBs, offering time-averaged spectra and their best-fit parameters for each burst. This repository website further provides an automated tool for time-sliced spectral fitting, allowing users to select specific time intervals for analysis. In this work, we primarily employed this tool to perform time-sliced spectral fits for the X-ray flares and their preflare and postflare phases.

For the definition of the flare duration, we follow the method of Chincarini et al. [12] and Yi et al. [48]. The start time ($T_{\mathrm{start}}$) and end time ($T_{\mathrm{end}}$) of an X-ray flare are taken as the intersections of power-law fits to the rising and decaying phases of the flare with the underlying continuum, respectively. The peak time $T_{\mathrm{peak}}$ is the time of the maximum flux, and the duration is then $T_{\mathrm{end}} - T_{\mathrm{start}}$. This approach cleanly separates the flare emission from the underlying afterglow (e.g., [19, 12, 48]).

We provide a schematic diagram to explain the spectral fitting of the X-ray flare in Figure 1.

Figure 1 illustrates our strategy for spectral fitting of X-ray flares. Taking an X-ray light curve containing multiple flares as an example, the figure marks three consecutive and overlapping flares (flare 1, flare 2, and flare 3). For such multiflare events, we performed individual spectral fits for each flare as well as a combined fit for the entire flare sequence (total-flare fitting), with the results labeled as “total-flare” in Table 1. Time-sliced spectral analysis was also conducted for the preflare and postflare intervals characterized by relatively smooth and continuous afterglow emission (regions marked as preflare and postflare in the figure). Based on the observational modes of the X-ray data, we carried out fits separately for data acquired in windowed timing (WT) mode and photon counting (PC) mode, which are shown in the figure in blue and red, respectively.

Figure 2 presents two representative X-ray light curves illustrating the selection of time intervals for time-sliced spectral analysis of different types of flares. The left panel shows the case of GRB 070704, a burst exhibiting only a single flare, where the time intervals for the flare, preflare, and postflare phases are marked. The right panel displays the case of GRB 090715B, a burst exhibiting multiple consecutive flares. In this case, the time intervals for the eight consecutive overlapping flares are individually marked.

Our time-resolved spectral analysis followed the strategy shown in Figures 1 and 2. For each GRB, we examined three distinct light-curve intervals: preflare, flare and postflare phases. Spectral extractions were performed via the

UKSSDC online tool, which implements the standard XRT reduction routines ([16, 17]. We simply repeat the treatment described by Evans et al. [16, 17] below.

We defined a circular extraction region for source events, centered on the afterglow position. In PC mode, this radius depended on the count rate, ranging between 5 and 30 pixels (i.e. $2.^{\prime\prime}36$ per pixel) – see Table 1 of Evans et al. [16]. For WT data, we instead used a rectangular box of $20 \times40$ pixels. Background sampling came from a nearby source-free circle, with any interloping sources flagged and removed via XIMAGE. For PC observations, our default background annulus had radii of 60 pixels (inner, $142^{\prime\prime}$) and 110 pixels (outer, $260^{\prime\prime}$); we adjusted its location whenever the CCD edge posed a problem.

Pileup corrections are built into the UKSSDC pipeline, relying on a rate-dependent point-spread function (PSF) masking routine that discards the bright central core. In PC mode, the pipeline compares the observed PSF with the standard PSF (Moretti et al. [37]) to identify pileup, and then applies an annular source region (inner radius $R_{\mathrm{p}}$) when required. In WT mode, such an annular box is triggered above 150 counts $\mathrm{s}^{-1}$. To minimize residual systematics from pileup, we enforced the recommended event-grade selections: we kept grades 0–4 for PC mode and retained only grade 0 for WT mode.

For the actual fitting, we used XSPEC v12.15.1 (Arnaud et al. [2]) and adopted an absorbed power-law form: phabs * zphabs * powerlaw (Evans et al. [17]). The Galactic column (phabs) was not fitted; instead, we fixed it to the 21 cm value from Kalberla et al. [26] and Willingale et al. [46]. The host-galaxy absorption (zphabs) was also held fixed, using the known spectroscopic redshift for each burst. The remaining parameter, the intrinsic neutral column density $N_{\mathrm{H,int}}$, was left free and fitted independently for each time-resolved spectrum.

We chose the Cash statistic ($C$-stat; Cash [10]) in XSPEC for our fitting work. Including a separate background spectrum triggers automatic switching to the $W$-statistic in XSPEC. That is actually the recommended route for Poisson-limited, low-count X-ray data (Arnaud et al. [2]). Before we could compute the $C$-stat, we had to group each spectrum channel by channel. We used the FTOOL grppha with the group min 1 setting (Evans et al. [17]) to make sure every bin had at least one count. This mild grouping does two things: it preserves the energy resolution as much as possible, and it clears out all those empty zero-count bins. A good number of our time-resolved spectra are pretty faint, especially at late times or during those dull intervals. There, many channels only register a few counts each. That is exactly why we turn to the $C$-statistic (Cash [10]) — it is a maximum-likelihood estimator built for Poisson data, and it gives us unbiased parameters even when the counts are really low.

We checked the quality of our spectral fits in two ways. First, we visually compared the data with the best-fit model by examining the residuals. Second, we looked at the empirical ratio of the $C$-statistic to the degrees of freedom (dof). Admittedly, the Cash statistic does not behave exactly like $\chi^2$ in terms of a strict probability distribution. Still, both Humphrey et al. [24] and Kaastra [25] have shown that it can serve as a reliable guide for judging the goodness of fit in X-ray work. Combining the residual behavior with the $C$-stat/dof value gives us a workable, practical criterion for judging model adequacy. Over the full 0.3–10 keV band, the residuals (data/model) fluctuate randomly around unity. We see no persistent deviations or organized structures. At the same time, the $C$-stat/dof ratios all come out close to 1.0. This, empirically, tells us that the absorbed power-law model describes our observed X-ray spectra reasonably well.

In Figure 3, we present the histogram of the $C$-stat/dof distributions derived from the X-ray spectral fits for the flare, preflare, and postflare phases of our selected sample. The vertical dashed line at 1.0 marks the ideal value for a statistically acceptable fit according to the empirical $C$-stat/dof criterion. The individual $C$-stat/dof values are listed in Table 1.

We illustrate the time-resolved spectral fitting procedure using the Swift/XRT observation of GRB 070704 as an example. Spectra were extracted for three epochs: preflare (159–270 s), flare (280–500 s), and postflare (550–1078 s). Each spectrum was fitted in the 0.3–10 keV band with an absorbed power-law model, where the Galactic absorption was fixed at $N_{\mathrm{H,Gal}} = 8.54 \times10^{21}\ \mathrm{cm}^{-2}$ and the intrinsic neutral absorption $N_{\mathrm{H,int}}$ was left free. The best-fit parameters and fit statistics are summarized in Table 1. The derived $N_{\mathrm{H,int}}$ values are $(1.00^{+0.43}_{-0.37}) \times10^{22}\ \mathrm{cm}^{-2}$ for the preflare phase, $(8.8^{+1.1}_{-1.0}) \times10^{21}\ \mathrm{cm}^{-2}$ for the flare phase, and $(4.7^{+5.3}_{-4.3}) \times10^{21}\ \mathrm{cm}^{-2}$ for the postflare phase. The corresponding photon indices are $1.98 \pm0.28$, $1.76 \pm0.07$, and $1.47 \pm0.35$, respectively. The absorbed power-law model provides acceptable fits for all three phases: $C$-stat/dof = 0.84 (preflare, dof = 344), $C$-stat/dof = 0.95 (flare, dof = 788), and $C$-stat/dof = 0.78 (postflare, dof = 194). Figure 4 presents the unfolded spectra and the data-to-model ratio for the three phases. In all cases, the residuals are randomly scattered around 1.0 with no systematic trends.

Spectral Fitting Results

We performed spectral fitting for all 521 X-ray flares, as well as for the preflare and postflare phases. The best-fit parameters and related physical quantities are summarized in Table 1.

The columns of Table 1 are organized as follows: Column (1) gives the GRB name; Column (2) indicates the GRB subtype (XRF, XRR, or C-GRB); Column (3) lists the redshift $z$; Column (4) presents $T_{90}$, the time interval over which 5% to 95% of the total fluence is accumulated; Column (5) presents the power-law photon index $\alpha_{\rm PL}$ from the simple power-law model for the prompt emission; Column (6) presents the photon index $\alpha_{\rm CPL}$ from the cutoff power-law model. The subsequent columns report the fitting results for the flare, preflare, and postflare phases. For each phase, the following parameters are provided: the observing mode (Obs., either WT mode or PC mode), the time interval used in the fit, the photon index ($\Gamma$), the intrinsic hydrogen column density $N_{\rm H}$(intrinsic) in units of $10^{20}\ \mathrm{cm}^{-2}$, and the $C$-statistic value along with the degrees of freedom ($C$-stat/dof). The flare number (No.) is indicated, where “tot.” denotes a combined fit for multiple consecutive flares (total-flare), and an asterisk (*) marks flares for which data from both WT and PC modes overlap and are spectrally fitted separately.

Statistical Analysis

Spectral Statistics of X-ray Flares

Our X-ray flare sample is mainly based on the flares identified by Chincarini et al. [12] and Yi et al. [48]. According to their criteria, a flare is recognized by its temporal morphology: a well-defined rise, peak, and decay, clearly separable from the underlying afterglow continuum. Minor fluctuations or unresolved features are not considered as valid flares. The light-curve profile must be well fitted by an empirical function, the Norris 05 function Chincarini et al. [12], or a smooth broken power law Yi et al. [48]. Overlapping flares are included only when individual pulses can be reliably separated.

For GRBs with multiple consecutive flares, we perform spectral fits for each individual flare and a global fit for the whole flare sequence, following the schemes shown in Figure 1 and Figure 2. The start and end times of individual flares are determined using the flare boundaries from Chincarini et al. [12] and Yi et al. [48], combined with GRB light curves; all flare time intervals are nonoverlapping.

We select GRB 090715B, a GRB exhibiting multiple consecutive flares, as an example; its light curve is displayed in Figure 2. This source shows eight separate X-ray flares. We carry out spectral fits for each individual flare and a global fit covering all eight consecutive flares (total-flare). The fitting results are shown in Figure 5.

In our sample, we perform joint spectral fits on 74 light-curve segments containing consecutive flares from 73 GRB sources. For GRB 141031A, the WT-mode and PC-mode data overlapped during its consecutive flares, so we carried out two separate joint spectral analyses using WT-mode and PC-mode data, respectively. The 73 GRBs that exhibited multiple consecutive flares contained a total of 315 individual flares. We conducted statistical analyses on the fitting results from both the joint spectra and individual-flare spectra, and the distributions of $C$-stat/dof are shown in Figure 6. Figure 6(a) presents the $C$-stat/dof distribution for the 74 joint spectral fits of multiflare sequences, spanning 0.75–1.45. Figure 6(b) shows the $C$-stat/dof distribution for the 315 individual-flare spectral fits, covering 0.65–1.45. As shown in the figure, we obtained spectral fits of reasonable quality for both individual flares and consecutive multiflare sequences.

We conducted a statistical analysis of the spectra of X-ray flares. In our sample, we obtained 521 X-ray flares from 206 GRBs. Among the sample, 98 GRBs exhibited a single flare, 57 GRBs showed two flares, and 51 GRBs displayed three or more flares. We performed joint spectral fits on 74 light-curve segments with consecutive X-ray flares from 73 GRB sources. In our sample, the total number of X-ray flares observed in WT and PC modes is 521, with 406 in WT mode and 115 in PC mode. We analyzed the photon index distributions of X-ray flares for the combined WT+PC mode, WT mode, and PC mode. The resulting histograms are shown in Figure 7.

In Figure 7, we performed Gaussian fits to the histograms. For the WT+PC mode, the photon indices range from 0.63 to 6.30, with a mean value of 1.94 and a Gaussian peak centered at $x_{\rm c}=1.85\pm0.01$. For the WT mode, the photon indices range from 0.67 to 4.36, with a mean of 1.92 and a Gaussian peak at $x_{\rm c}=1.85\pm0.02$. For the PC mode, the photon indices range from 0.73 to 6.30, with a mean of 2.02 and a Gaussian peak at $x_{\rm c}=1.86\pm0.02$.³ We applied the Kolmogorov–Smirnov test (K-S test) to compare the photon index distributions between the WT and PC modes, yielding a $p$-value of $p=0.1956$. The K-S test indicates that there is no significant difference between the photon index distributions obtained in the two observing modes.

Temporal Evolution of the Photon Index in X-Ray Flares

To investigate the temporal evolution of the photon index in X-ray flares, we selected GRBs exhibiting multiple flares. The sample composition is as follows: 14 GRBs exhibit four X-ray flares (GRB 050712, GRB 050904, GRB 060124, GRB 070721B, GRB 071031, GRB 081028, GRB 090417B, GRB 090429A, GRB 100614A, GRB 111016A, GRB 111209A, GRB 140817A, GRB 141031A, GRB 150323C); seven GRBs exhibit five X-ray flares (GRB 070107, GRB 081210, GRB 100725B, GRB 100901A, GRB 110709B, GRB 130606A, GRB 130925A); one GRB exhibits six X-ray flares (GRB 080310); two GRBs exhibit seven X-ray flares (GRB 070129 and GRB 070616); one GRB exhibits eight X-ray flares (GRB 090715B); two GRBs exhibit nine X-ray flares (GRB 100728A and GRB 110119A); one GRB exhibits ten X-ray flares (GRB 051117A); and one GRB exhibits twelve X-ray flares (GRB 100212A).

Figure 8 presents the correlation between the photon index and the peak time ($t_{\mathrm{peak}}$) of multiple X-ray flares occurring in the same GRB. Figures 8(a) and 8(b) show the cases in which each GRB has four X-ray flares. Figures 8(c) and 8(d) show the cases in which each GRB has five X-ray flares. Figure 8(e) displays the cases that each GRB has six, seven, and eight X-ray flares, respectively. Figure 8(f) displays the cases that each GRB has nine, ten, and twelve X-ray flares, respectively. As illustrated, the photon index of X-ray flares increases progressively with peak time, indicating a spectral softening trend as the peak time increases. Our analysis further demonstrates that, for GRBs with multiple flare episodes, the photon index of X-ray flare exhibits a distinct temporal evolution pattern: later flares tend to have softer spectra than earlier ones, suggesting a progressive spectral softening over time.

Statistical Analysis of Photon Indices in Preflare, Flare, and Postflare Phases

To investigate the spectral evolution characteristics of X-ray flares, we conducted a comparative statistical analysis of the photon indices in the preflare, flare, and postflare phases. For GRBs with multiple consecutive flares in our sample, we adopted two approaches for statistical analysis. First, we analyzed the overall emission of multiple consecutive flares (total-flare sample). Second, we analyzed all individual flares (all-flare sample). The K-S test and the t-test were employed to statistically examine the variations in photon index distributions among the preflare, flare, and postflare phases.

In the histogram distributions, Gaussian fits were applied separately to the photon index distributions of the preflare, flare, and postflare phases, and the differences in the Gaussian peak centers were compared. Additionally, K-S tests were performed on the photon index distributions among these three phases. In this statistical analysis, we combined all observation data from both WT and PC modes for an integrated analysis.

Figure 9 presents the histograms of spectral index distributions along with Gaussian fits for the total-flare sample, which includes bursts exhibiting multiple flares. The central values derived from the Gaussian fits are: $x_c = 1.82 \pm0.02$ for flare; $x_c = 1.88 \pm0.02$ for preflare; and $x_c = 1.94 \pm0.03$ for postflare. The K-S test yields the following $p$-values: $p = 0.452$ for flare versus preflare; $p = 0.002$ for flare versus postflare; $p = 0.063$ for preflare versus postflare. At a significance level of 0.05, no significant differences are found between the flare and preflare distributions, nor between the preflare and postflare distributions. However, a statistically significant difference exists between the flare and postflare distributions, manifested as a notably larger spectral index in the postflare phase than that during the flare, indicating a trend toward spectral softening after the flare.

To further investigate the evolution of the spectral indices across the preflare, flare, and postflare phases, we performed one-to-one matching of the sample data among the three phases. The paired scatter plots for flare versus preflare, flare versus postflare, and preflare versus postflare are presented in Figure 10.