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A&A
Volume 573, January 2015
Article Number A81
Number of page(s) 20
Section Astrophysical processes
DOI https://doi.org/10.1051/0004-6361/201424858
Published online 23 December 2014

© ESO, 2014

1. Introduction

Table 1

The names, GBM trigger numbers, durations, fluence, detectors used, and optimal S/N for the eight bursts studied in this paper.

Gamma-ray bursts (GRBs) are the most luminous explosions in the Universe known to date. The first GRB was discovered in 1967 (Klebesadel et al. 1973), and after over 45 years of research efforts, it is now believed that GRBs originate from highly relativistic outflows from central compact sources at cosmological distances with bulk Lorentz factors Γ > 100 (e.g. Lithwick & Sari 2001; Hascoët et al. 2012). This is often understood in terms of the “fireball model” (Goodman 1986; Paczynski 1986; Rees & Meszaros 1992, 1994; Piran 1999), where the GRB itself is produced by dissipation of kinetic energy from the relativistic flow. However, the shape of GRB spectra does not naturally fit the synchrotron spectra predicted by this model. Even after many dedicated missions to explore GRBs, such as the Burst And Transient Source Explorer (BATSE, Fishman et al. 1989; Meegan et al. 1992) onboard the Compton Gamma-Ray Observatory (CGRO), the BeppoSAX satellite (Boella et al. 1997), the Swift satellite (Gehrels et al. 2004), and the Fermi Gamma-Ray Space Telescope (Atwood et al. 2009), no single consensus theory has emerged to explain all the features of the prompt emission, although various possibilities in addition to the basic fireball model have been suggested (see, e.g., Zhang 2014, for a recent overview).

To study the physical properties of GRB prompt emission, the observed γ-ray spectrum is usually fitted to a chosen model (either physical or empirical). Then the best-fit parameters can be compared with the physical parameters used in theoretical models and computer simulations. Over the past 20 years, the preferred fitting model has been the empirical Band function (Band et al. 1993), which consists of a smoothly joined broken power-law with low-energy power-law index α, high-energy power-law index β, and a characteristic energy Ep parameterized as the peak energy in the observed νFν spectrum.

Since the observed spectral behaviour varies from burst to burst and over time within a single burst, it is crucial to study the fitted parameters from a carefully selected sample of GRBs in a systematic way. Well-constrained spectral parameters are also important to distinguish among various theoretical models. However, because of the observed high-energy cutoff nature of the spectrum and the fact that it is harder to detect high-energy γ-ray photons, the high-energy power-law index is often poorly constrained for most bursts. Thanks to the broad spectral coverage of the Gamma-Ray Burst Monitor (GBM, Bissaldi et al. 2009; Meegan et al. 2009) onboard Fermi, we are now able to obtain the spectral indices with good precision.

Motivated by the fact that most catalogue studies of large GRB samples do not consider the quality of high-energy photon statistics (e.g., Kaneko et al. 2006; Nava et al. 2011; Goldstein et al. 2012, 2013; Gruber et al. 2014; Yu et al., in prep.), we present time-resolved spectroscopy for eight energetic GRBs with good high-energy statistics in the GBM GRB zoo (Bissaldi et al. 2011) to obtain an accurate measurement of β. We describe the selection criteria, analysis procedures, and empirical fitting models in Sect. 2. The observational results are presented in Sect. 3. We present the fitting results from the standard Band function in Sect. 3.1, and a test synchrotron model in Sect. 3.2. In Sect. 4 we discuss the theoretical implications of the observed parameter distributions in the context of different models. The conclusion is given in Sect. 5. Unless otherwise stated, all errors are given at the 1σ confidence level.

2. GBM data analysis

2.1. Instrumentation

The GBM is a sensitive scintillation array onboard the Fermi satellite. It consists of twelve thallium activated sodium iodide (NaI(Tl)) detectors covering the energy range from 8 keV to 1 MeV and two bismuth germanate (BGO) detectors covering the energy range from 200 keV to 40 MeV. This provides spectral coverage over three orders of magnitude, which makes the GBM a powerful observing instrument for GRB prompt emission.

2.2. Burst, detector, and data selection

Our sample consists of bursts that are among the most energetic bursts observed by Fermi GBM until 21 August 2013. They were selected according to two criteria: (1) the total fluence in 1 keV–1 MeV, f> 1.0 × 10-4 erg cm-2; and (2) the signal-to-noise ratio level, S/N ≥ 10.0 above 900 keV (i.e. the NaI limit) in the BGO. The advantage of analysing bursts with significant photon statistics above 900 keV is that the high-energy power-law index can be better constrained. Moreover, high fluence provides more statistics for time-resolved spectral analysis. Table 1 lists the eight long GRBs (time in which 90% of burst fluence observed, T90> 2 s) that satisify these selection criteria. There are no short bursts in the sample because they do not satisfy our fluence criterion. GRB 130427A is the brightest burst observed by GBM. This brightness caused a pulse pile-up effect in the detectors in its complex-shaped main pulse after t = T0 + 2.4 s. However, it also has a bright first pulse that is well suited for testing the synchrotron model (Preece et al. 2014) and that satisfies our selection criteria by itself. Therefore, this first pulse (t<T0 + 2.4 s) is included in our analysis.

For each burst, up to three NaI detectors with viewing angle smaller than 60 degrees and the BGO without blockage by either the Large Area Telescope (Atwood et al. 2009) or the solar panels were included to maximize signals and reduce the level of background noise. We used the time-tagged event (TTE) data, which provide high temporal (continuous temporal coverage with 2 μs time tags) and spectral resolution (128 pseudo-logarithmically scaled energy channels). The channels with energy lower than 8 keV for NaIs and 245 keV for BGOs, together with the overflow channels, were excluded. As a result, an effective spectral range from 8 keV to 40 MeV was used in the analysis. Moreover, effective area corrections were applied to each pair of NaI and BGO detectors.

2.3. Time-resolved spectral analysis

The light curves were binned using a fixed S/N for each burst (but varying across bursts, see last column of Table 1) to avoid artificial binning bias while preserving the general shape of the light curve by avoiding merging peaks and valleys (e.g. Guiriec et al. 2010), resulting in a total of 299 spectra. The binned light curves are shown in Figs. A.1 and A.2 with time relative to the GBM trigger time T0.

Time-resolved spectroscopy was then performed with the GBM official spectral analysis software RMFIT1 v4.3BA and the GBM response matrices v2.0. To account for the change in orientation of the source with respect to the detectors caused by the slew of the spacecraft, RSP2 files containing the detector response matrices (DRM) for every two degrees on the sky were used. For each burst a low-order polynomial (order 2–4) was fitted to every energy channel according to a user-defined background interval before and after the prompt emission phase and interpolated across the emission interval.

Bhat (2013) reported that the typical minimum variability timescales (MVT) for short and long GRBs are 24 ms and 0.25 s. The average temporal resolution of the time bins (Tbin) used in this paper is 2.18 s, which is longer than the MVT. The pulse duration (Tpulse) ranges from seconds to tens of seconds (see Figs. A.1 and A.2), which is, of course, by definition shorter than or equal to the burst duration T90. So we have the typical values of MVT <Tbin<Tpulse<T90.

The variable temporal S/N binning technique can avoid the resulting statistics being dominated by the brightest few bursts. This is because the optimal S/N for each burst is different, which leads to similar number of bins for every bursts (see Tables A.1–A.8). The fitting results will be given in Sect. 3 and discussed in Sect. 4. GRB 100724B is discussed separately because of its ambiguous parameter distributions. We checked the statistics contributed by individual bursts and found that our conclusions are not affected if any one burst (even for GRB 100724B, see Sect. 3.1) is removed from the overall sample.

2.4. Empirical fitting models

2.4.1. Band function

The Band function (Band et al. 1993) was fitted to every spectrum: (1)where (2)In these equations, A is the normalization factor at 100 keV in units of photons s-1 cm-2 keV-1, α is the low-energy power-law index, β is the high-energy power-law index, and Ep is the peak energy in units of keV in the observed νFν spectrum. The energy Ec is where the low-energy power-law with an exponential cut-off ends and the pure high-energy power-law starts.

2.4.2. Synchrotron model

thumbnail Fig. 1

Schematic spectra for the SSM cooling scenarios. The left, middle, and right panels show the “slow”, “both”, and “fast” cases in the energy flux space. The shaded region represents the possible location of νboth (i.e. Ep) when fitting the observed spectrum using a model with smoothly jointed power-laws. The photon distribution slopes are also indicated for each different case.

The optically thin synchrotron shock model (SSM) predicts two different spectra, “fast cooling” and “slow cooling” (e.g. Sari et al. 1998; Preece et al. 2002), depending on the injection and evolution of the relativistic electron population. Both of them consist of a lower and a higher frequency break, fixed by the values of the cooling frequency νcool and the lowest injection frequency νmin for the relativistic electrons. The electrons in the shock are accelerated to a lowest energy γmin. Assuming a power-law behaviour for the electron energy distribution , where γe ≥ γmin is the electron energy, the emission spectrum also has a power-law shape. As long as p> 2, the distribution is characterized by its lower cut-off at γmin, and the integrated energy of the population does not diverge at high electron energies.

There is a critical energy γcool such that electrons with energies above γcool emit a significant amount of their energy via synchrotron cooling. The values of γcool and γmin correspond to νcool and νmin, and the slow-cooling spectrum is given by (3)while the fast-cooling spectrum is given by (4)Subtracting 1 from the spectral indices will give the photon indices (i.e. α and β), which we obtain in Sect. 3, and which lead to a synchrotron “line-of-death” α = −2/3 for both scenarios and a second line-of-death α = −3/2 (Preece et al. 1998) for the fast-cooling scenario. Figure 1 shows the schematic spectra for the slow- and fast-cooling scenario as well as the so-called “both” case where νcool/νmin (slow-cooling) or νmin/νcool (fast-cooling) is close to unity. The both case can be considered to describe an intermediate case of moderately fast-cooling.

The synchrotron fitting model (SYNC) that we apply is a modified triple power-law with sharp breaks defined as: (5)where A is the normalization factor at 100 keV in units of photons s-1 cm-2 keV-1, α, β, and γ are the power-law indices of the three segments (from low to high energies), and Eb,1 and Eb,2 are the two break energies in units of keV. Here we fixed α = −2/3 and β − γ = 1/2 to create a SYNC-slow model (Eq. (3)). This makes it a four-parameter model with freely varying A, Eb,1, Eb,2, and β (or equivalently, γ). We also tried to fit the SYNC model with fixed α = −2/3 and β = −3/2 to create a SYNC-fast model (Eq. (4)). This also makes a four-parameter model with freely varying A, Eb,1, Eb,2, and γ.

2.4.3. Blackbody model

We also added a blackbody model (BB) to the SYNC fits. It is a Planck function defined as (6)where A is the normalization factor at 1 keV in units of photons s-1 cm-2 keV-1 and kT is the temperature of the blackbody in units of keV.

3. Fitting results

3.1. Band fits

The Band function has long been known to provide a good fit to prompt emission spectra (Band et al. 1993), where the typical reduced-χ2 ≈ 1 (there is a caveat that the χ2 statistics may not be suitable for non-Gaussian data) and the Castor C-statistics values (CSTAT, Cash 1979) are low (often a few hundred to a thousand for GBM fits depending on the data quality of individual burst) among the simplest models (e.g., Goldstein et al. 2012; Gruber et al. 2014; Yu et al., in prep.). If, in addition to a low CSTAT value corresponding to a low reduced-χ2 value (≈1), all parameters in an individual spectral fit have 1σ relative error σparameter/ (parametervalue) < 1.0 (for power-law indices we use absolute error σparameter< 1.0), we define the fit as a constrained fit. For all these good fits, we verified that the data points are within ≈99.73% confidence level to the model curves. Although we found that in some extreme cases the asymmetric errors of β may be unconstrained on the negative side, our selection criteria can filter most of these cases by ensuring the symmetric error (which is the mean of the asymmetric errors) to be well behaved. As a result, 216 of the total 299 spectra (≈72%) were constrained. Figure 2 shows the distributions of the constrained parameters for the Band model: the low-energy power-law index α, the high-energy power-law index β, the peak energy in the observed νFν spectrum Ep, and the difference between the low- and high-energy power-law indices Δs ≡ α − β.

The distributions of α, β, Ep, and Δs are clustered around values of , , keV (), and , respectively. The asymmetric distribution errors were determined by taking the difference between the median values of the cumulative distribution function (CDF) and the 68% quantiles. Note that shows that the overall sample distribution is consistent with the synchrotron line-of-death (see Sect. 2.4.2). About a third of the individual spectra are consistent with the value α = −2/3 within 1σ. The slope is consistent with typically observed values. The average errors of α and β are σα ~ 0.1 and σβ ~ 0.2. So in Fig. 2 a bin width equal to 0.2 was chosen to display the histograms. This implies that the observed dispersions in the power-law index distributions cannot be explained solely by statistical uncertainties. The dispersion is also observed within bursts, indicating that spectral evolution has a non-negligible effect on the parameter distribution. Moreover, it is observed that σEp ~ 0.1Ep.

The distribution of Ep peaks at keV and is only slightly higher than those found in the GBM time-averaged spectral catalogues (Goldstein et al. 2013; Gruber et al. 2014) and the BATSE spectral catalogues (e.g. Kaneko et al. 2006). According to Fig. 2, 91% of all Ep ≤ 1 MeV. The remaining 9% has the highest Ep = 2.1 MeV (GRB 130504C, see Table A.7). Nava et al. (2011) presented a time-averaged spectral analysis on 44 short GBM GRBs and found that the distribution peaks at keV. This suggests that our long bursts could be as hard as short bursts, which is expected since we selected the bursts with relatively better statistics in the BGO channels. Our bursts lie at the high Ep-long T90 end in the “long and soft vs. short and hard” classification of GRBs (Kouveliotou et al. 1993). However, the brightest three short GBM GRBs show an Ep as high as 6 MeV (Guiriec et al. 2010). This shows that the Ep dispersion within long or short bursts can also be huge. In addition, Ep is observed to be decreasing throughout a burst, with intensity-tracking behaviour during sub-pulses within a single burst (see Sect. 4.1).

As shown in Fig. 2, 50% of the hard β> − 2 are from GRB 100724B. We show in Sect. 4.2 that this burst is consistent with both the slow- and fast-cooling scenario, and that the general conclusion is not affected beacuse removing this burst will only make the distribution peak narrower.

thumbnail Fig. 2

Distributions of the constrained parameters obtained from the Band model. The upper left panel shows the distributions of the values of Ep. The lower left panel shows the distributions of the values of α. The upper right panel shows the distributions of the values of β. The lower right panel shows the distributions of the values of Δs = α − β. The blue lines show the distributions of GRB 100724B.

3.2. SYNC fits

Various studies have shown that a thermal component of about a few times 10 keV may generally exist (e.g. Mészáros et al. 2002; Ryde 2005; Guiriec et al. 2011; Axelsson et al. 2012; Guiriec et al. 2013; Burgess et al. 2014a,b). In addition, Burgess et al. (2014a) showed that a SYNC-type model alone cannot be reconciled with the flatness of α. We found that in most of our spectra adding a blackbody component can greatly improve the fit. Therefore, all the spectra were fitted again to include a blackbody component in the SYNC model. The theoretical implications for the SYNC+BB and Band model are discussed in Sect. 4.

thumbnail Fig. 3

Distributions of the constrained parameters obtained from the SYNC+BB model with slow-cooling constraints (i.e. α = −2/3 and β − γ = 1/2). The upper left panel shows the break energies Eb,1 and Eb,2. The lower left panel shows the kT distribution. The upper right panel shows the photon indices β of the middle power-law segment. The lower right panel shows the ratio between the two breaks, Eb,2/Eb,1. The blue lines show the distributions of GRB 100724B. Values that are out of the plotting region are accumulated in the boundary bins.

thumbnail Fig. 4

νFν spectral evolution of the SYNC-slow model for GRB 130427A. The evolution of the SYNC component evolves from cyan to blue, while the BB component evolves from yellow to red. No clear correlation is found between the two components.

Two SYNC+BB models (i.e. SYNC-slow+BB and SYNC-fast+BB) were fitted to all spectra using a customized version of RMFIT. We validated that these are good fits to the data by various goodness-of-fit measures: (1) reduced-χ2 values are close to unity; (2) CSTAT values are comparable to, often lower than, those for the Band fits (e.g. Gruber et al. 2014; Burgess et al. 2014a); and (3) quantile-quantile plots for the cumulative observed vs. model count rates lie very close to x = y, thus confirming that a SYNC+BB model description is consistent with the data. For reference, the CSTAT values for all spectra are listed together with the degrees of freedom (DOF) and the fitted parameters in Tables A.1–A.8. It is found that both the SYNC-slow+BB and -fast+BB models provide constrained fits in more than 65% of all spectra. We show in Sect. 4 that such a test model can provide constraints on various prompt emission mechanism theories. The distributions for the SYNC-slow+BB constrained parameters are plotted in Fig. 3. The time-resolved spectral evolution for GRB 130427A is shown in Fig. 4. There is no clear correlation found between the fluxes of the SYNC and BB components.

The upper left panel of Fig. 3 shows the distributions of Eb,1 and Eb,2. We found that there are two clear peaks for the breaks around keV and keV for Eb,1 and Eb,2, respectively. The asymmetric distribution errors were obtained by the same procedure by constructing CDFs as described in Sect. 3.1. Comparing to the Band fits, it is observed in most of the cases that Eb,1<Ep ≈ Eb,2. We found that 100% of Eb,1< 1 MeV and 97% of Eb,2< 3 MeV.

The lower left panel shows the kT distribution. The parameter distribution of keV creates a bump at ~30 keV. This kT distribution is consistent with most of the sub-dominant thermal bursts observed (e.g. Ryde 2005; Guiriec et al. 2011, 2013; Axelsson et al. 2012; Burgess et al. 2014a,b). When the Planck-to-SYNC flux ratio is high, the Planck function dominates the curvature of the lowest end of the spectrum.

The upper right panel shows the distribution of β, where β − γ = 1/2. The parameter distribution of translates into the electron distribution index . A synchrotron spectrum with p> 2 (i.e. β< − 1.5) requires no upper cut-off for the total energy of the electrons to remain finite (Sect. 2.4.2). Therefore, the measured high-energy slopes for the SYNC model do not require such a cut-off. In addition, this is also consistent with afterglow-deduced distributions of p ~ 2.3 (e.g., Curran et al. 2010; Ryan et al. 2014). GRB 100724B provided most of the cases where β> − 1.5, which matches the fast-cooling index value.

The lower right panel shows the distribution of the ratio between the two breaks, Eb,2/Eb,1. It is observed that Eb,2 and Eb,1 have a peak ratio at , and over 90% are below 10. If we assume Eb,1 and Eb,2 are related to Emin = hνmin and Ecool = hνcool, then a ratio of Eb,2/Eb,1< 10 poses a very tight constraint on the theoretical models (see Sect. 4.3).

The parameter distributions for the SYNC-fast model are nearly identical to those of the SYNC-slow model (which is expected because the value of β = −3/2 is only 0.1 away from the SYNC-slow β distribution peak). The only difference observed is that γ extends to much steeper values (from − 1.75 to − 4.50 with a peak around − 2.0–− 2.5, not a normally distributed population), which reflects the fact that since the power-law segments are no longer connected, γ can go much steeper in the time bins that contain mostly upper limits in the high-energy channels.

In brief, the following features are observed in the SYNC fits: (1) over 90% of Eb,2/Eb,1< 10; (2) a bump or flattening feature at ~ 30 keV; and (3) a general hard-to-soft evolution for the peak or break energy is observed. We discuss the theoretical implications of these observational results in the next section.

4. Theoretical implications

4.1. Hard-to-soft evolution and intensity-tracking behaviour

We show the light curves overlaid on the evolutions of Ep, Eb,1, Eb,2, and kT for every burst in Figs. A.1 and A.2. Hard-to-soft evolution over the whole bursting period is observed in every burst with in-pulse intensity-tracking behaviour. These two modes of evolutionary trend have been observed in many GRBs (e.g., Ford et al. 1995; Liang & Kargatis 1996; Kaneko et al. 2006; Preece et al. 2000, 2014; Guiriec et al. 2010; Lu et al. 2010; Peng et al. 2010; Ghirlanda et al. 2011; Burgess et al. 2014a). Hard-to-soft evolution is a natural prediction from the SSM (Daigne & Mochkovitch 1998), in which the relative Lorentz factors of the colliding shells become lower and the spectra become softer. For instance, Lu et al. (2012) reported a time-resolved spectral analysis for 62 Fermi bursts (51 long + 11 short) with a detailed study of the Ep evolution. They found that the two modes for Ep evolution are present in different pulses and in different bursts. Despite the complexity of the problem, they suggested that the intensity-tracking behaviour could be at least partially attributed to the superposition of hard-to-soft pulses in a highly superimposed light curve. As all bursts in our sample are multi-pulsed (although for GRB 130427A only the first pulse is analysed, see Sect. 2.2), this possibility cannot be excluded. We also observed that the Ep in later pulses is never as high as in the first pulse, even if a later pulse has a higher peak flux. This suggests that the hard-to-soft evolution dominates the intensity-tracking behaviour, and that the hard-to-soft evolution is an intrinsic property of GRBs with intensity-tracking behaviour added on top.

4.2. Synchrotron emission and the Band function fits

Table 2

Electron distribution index p for different cases.

The values of Δs and β obtained from the Band fits can be used to compute the electron distribution power-law index p and to distinguish among different cooling scenarios (Preece et al. 2002). Preece et al. (2002) performed time-resolved spectroscopy on 156 BATSE GRBs and found that the results are consistent with the “slow, low”, “both”, or “fast, high” cases (with “low” and “high” referring to just the lower or higher spectral break).

The relative rate of electron cooling against energy injection into the electron population marks the difference between slow- and fast-cooling. To obtain the synchrotron cooling and energy injection timescale requires knowledge of the physical parameters of the ejecta, such as magnetic field strength and electron Lorentz factor, as well as precise modelling of the energy output from the central engine. This makes accurate measurements of these timescales difficult. In the internal shock model, the relative Lorentz factors between colliding shells are only mildly relativistic (Daigne & Mochkovitch 1998), and the synchrotron cooling timescale of the relativistic electrons in the ejecta frame is (7)where Γe is the Lorentz factor of the electrons relative to the ejecta and B is the magnetic field in the shock. One could compare, for instance, tsyn with the MVT observed in the light curve (as experienced in the ejecta frame), which is then taken to represent the rate of energy injection into the synchrotron electron population. However, the inferred values of the physical parameters, such as magnetic field strength (see the discussion below), vary in a wide range among bursts and sub-pulses within a single bursts. Taken together with the uncertainty in the spatial and temporal profile of the particle acceleration sites (e.g. extended turbulent regions vs. shock acceleration, or intermittent vs. continuous injection), it becomes hard to predict a clear preference for a given cooling regime because of the difficulty of unambiguously interpreting the observable timescales. We show in the following that a mix of both the slow- and fast-cooling is implied by the GBM data.

Table 2 shows the values of p obtained from the Δs and β distributions (see Eqs. (8)–(12) in Preece et al. 2002). Column 1 shows the three cases where p depends on both Δs and β. Column 2 shows the respective value of α in each case. Columns 3 and 6 show the formulae for p as a function of Δs and β respectively. Columns 4 and 7 give the ranges of possible values of p calculated from the distributions of Δs and β for all eight bursts, and Cols. 5 and 8 give the same for GRB 100724B alone.

The values of p in Col. 4 are inconsistent with the “fast, high” case in Col. 7. The “fast, low” case predicts Δs = 5/6, which is clearly rejected, as shown in Fig. 2. The distribution of fast-cooling γSYNC as mentioned in Sect. 3.2 indicates that the electron distribution index above γmin can take any value from p = 1.5–6.0. Theories of electron shock-acceleration typically predict p values between 2 and 3, which makes these very steep values for p suggestive of a cut-off or deviation from a power-law slope in the accelerated particle distribution, rather than a single very steep slope. A steep electron distribution index can also occur when the shock normal is at an angle to the magnetic field, allowing electrons to escape the acceleration region early (Ellison & Double 2004; Baring 2006; Summerlin & Baring 2012; Burgess et al. 2014a). The “slow, high” case, which refers to the higher energy break in the left panel of Fig. 1, predicts Δs = 1/2 and is clearly rejected, as shown in Fig. 2. The average values of Δs and β for GRB 100724B are 1.0 and −1.7 respectively, which are also consistent with the “slow, low” and “both” cases (Cols. 5 and 8), at the same time consistent with the “fast, low” case, which predicts Δs = 5/6.

On the other hand, the BATSE β and Δs distributions suggested that the “slow, low”, “fast, low”, and “both” cases are all viable processes (see, e.g., Fig. 2 of Preece et al. 2002; Kaneko et al. 2006). Gruber et al. (2014) also showed similar conclusions in the GBM time-averaged spectra. Burgess et al. (2014a) performed a Bayesian time-resolved spectral analysis using physical synchrotron and thermal models instead of the Band function to several GBM GRBs and found that the slow-cooling scenario better explains the observed data, and their results suggest continuous energy injection is important. Uhm & Zhang (2014) predicted that using a decaying magnetic field as a function of radius, with a decay index b, it is possible for most GRBs to cool via the fast-cooling scenario with α ~ − 1.0. They predicted that the asymptotic value of the low-energy electron distribution should be p = (6b − 4)/(6b − 1) instead of p = 2 for a constant magnetic field (e.g. Preece et al. 2002), and the spectral index s = (−p + 1)/2 = 3/(12b − 2) = α + 1. We found that in more than 77% of the constrained fits b has values between 0.6 and 2.6. There is no clear evolutionary trend of b. The variability of b within bursts is difficult to reconcile with a large-scale power-law dependence on radius of the magnetic field. However, this can still be the case, but just not as clearly manifested in the data as predicted by Uhm & Zhang (2014).

In brief, our results are consistent with slow-cooling with the low-frequency break seen (or in the “both” case, undistinguished between slow- and fast-cooling). For GRB 100724B, fast-cooling is also consistent with the low-frequency break seen. This implies that the second line-of-death, α = −3/2, could also be avoided.

4.3. Synchrotron models fits

The SYNC-slow model is basically a three-segment broken power-law, with the middle- and high-energy segment connected (i.e. βSYNC − γSYNC = 1/2). It is essentially an extended version of the Band model, in which the curvature of Band is replaced by two breaks and the power-law segment in between. This implies that when we are comparing the results from Band and SYNC fits, it should be kept in mind that either βSYNC or γSYNC could be picking up βBAND. This is discussed below in this subsection. Moreover, a sharply joined broken power-law is intrinsically non-physical. The actual spectrum should always be smooth, so sharp power-law fits run the risk of covering a single smooth transition with multiple sharp breaks. However, a smoothly joined triple power-law would contain too many parameters, and fitting such a complicated empirical model is statistically unsound. The constrained power-law indices in our SYNC-slow and -fast models mitigate the problem by assuming a synchrotron origin of the observed spectrum a priori, thereby limiting the possible shapes of fitted spectra.

Theoretically, the SYNC model has excluded the synchrotron emission from Maxwellian electrons. This is because we wished the synchrotron emission to occur at the correct frequencies (i.e. γ-rays), which requires the energy per emitting electron to be higher than that obtained by simply averaging (as demonstrated by Daigne & Mochkovitch 1998). This implies a small subset of electrons at very high energies, far away from the thermal pool from which they were drawn. Alternatively, if the Maxwellian electron distribution peak and the minimum injection Lorentz factor (i.e. γmin) remain close, the effect of adding the Maxwellian electrons will be a smoothing of the synchrotron function that we could not model with our Band or SYNC models. Moreover, Burgess et al. (2011) has shown that the Maxwellian electron population is sub-dominant. To avoid further complication of the fitting model, we therefore assumed the synchrotron emission is just from the population of shock-accelerated electrons (see Sect. 3.2). However, the Maxwellian does not just have to exist as left-over thermal pool from the thermal parts of the jet. It can also be created in the shock region by thermalization of electrons crossing the shock (see, e.g., Spitkovsky 2008).

According to the SYNC-slow fitting results, there are two cases to consider: (1) the γSYNC is the high-energy segment in the slow-cooling scenario, that is, νmin and νcool are the predicted break values; or (2) γSYNC is the middle-energy segment in the slow-cooling scenario; in this case, the triple power-law is just mimicking the slowly varying Band model. If (1) is true, then we can take γSYNC = −2.5–− 2.0, and we will have p = 2.0–3.0. Table 2 shows that the SYNC-slow model is consistent with the “both” case; if (2) is true, then instead of comparing with γSYNC, we should compare with βSYNC in Eq. (3), and we will have p = 3.0–4.0. Table 2 shows that the SYNC-slow model is also consistent with the “slow, low” case.

thumbnail Fig. 5

Selected spectrum from GRB 130606B plotted in νFν space. The black, red, and blue solid curve show the fitted spectrum for the Band, SYNC-slow+BB, and SYNC-fast+BB model, respectively, while the dash-dotted curves show individual SYNC or BB components. The vertical dash-dotted black, red, and blue line show the Ep and break energies for the Band, SYNC-slow+BB, and SYNC-fast+BB model, respectively.

Burgess et al. (2014a) used a physical non-thermal plus thermal synchrotron kernel to fit a few GBM GRBs and found that slow-cooling is physically possible. Since the typically observed value of α ~ −1.0, the fast-cooling model has been disfavoured because it predicts α should be as steep as − 3/2 below νmin (Sari et al. 1998). A blackbody contribution to the lower part of the spectrum would render it even more difficult to reconcile the α slope with the “fast, high” case. On the one hand, our fit results for the SYNC-slow model yield p values closer to the expected range between 2 and 3. On the other hand, a SYNC-fast model, implying that most of the energy of the electrons is radiated away, has the advantage of allowing for a lower efficiency. The total energy in γ-rays is typically similar to the inferred kinetic energy of the ejecta. Therefore, if the efficiency in converting accelerated electron energies to radiation is low, the efficiency in extracting energy from the ejecta to the non-thermal electron population has to become extremely high to compensate for this (see e.g. Nousek et al. 2006; Granot et al. 2006, for detailed discussions). The fact that in the SYNC fits, both spectral breaks consistently occur fairly close to one another does alleviate the problem in that it provides essentially a “moderately fast-cooling” scenario, regardless of the precise order of the breaks. This, however, raises the question of how the universal break ratio between νmin and νcool inferred from our sample can be understood, as the positions of these breaks are not theoretically expected to be related.

The fast-cooling model with a decaying magnetic field (Uhm & Zhang 2014) predicts a Band function spectral shape with b ~ 1.0–1.5 (see their Fig. 4), in which the curved Band shape is a summed effect for the emissions of electrons at different times. A decay index b ≲ 2.6 (see Sect. 4.2) implies stronger magnetic dissipation, and the electrons at later time could be cooled by slow-cooling, thus the positions of νcool and νmin could reverse and move closer to each other, so that the “both” case is possible. Uhm & Zhang (2014) showed that this is possible on a timescale ~ 1.0 s, consistent with the typical Tbin used in this paper (see Sect. 2.3). We found that the Band and SYNC model have very similar shapes (Fig. 5) that are consistent with this interpretation and thus provide further support for the “both” case.

4.4. Thermal origin of prompt emission

Recently, Beloborodov (2013) suggested that the evolution of Ep could be a manifestation of thermal emission. As shown in Fig. 2, more than 90% of Ep values remain below 1 MeV. The observed clustering of Ep ~ few hundred keV, instead of a wide distribution, is hard to explain in the SSM. The observed spectral width in the νFν space is log (E1/E2) ≈ 1.0–1.5 decades in photon energy (Beloborodov 2013), where E2 − E1 is the width at half-maximum. This is narrower than a synchrotron model would predict (Daigne et al. 2011).

Early photospheric models assumed a freely expanding radiation-dominated outflow with no baryonic loading or magnetic field (Goodman 1986; Paczynski 1986). This predicts a sharply defined peak with a Planck spectrum (Beloborodov 2011), which contradicts the observed non-thermal spectra in most GRBs. Detailed radiation transfer simulations have shown that a thermal origin of the Band function is possible (Pe’er et al. 2006; Giannios 2008; Beloborodov 2010; Vurm et al. 2011), and Beloborodov (2013) computed that the maximum Ep of a spectrum from thermal plasma is given by 30ΓP keV under high radiation efficiency, where ΓP is the Lorentz factor of the Planckian photospheric shell. With the typical values of the Lorentz factor of GRBs to be ~ 100, Ep,max ~ 3 MeV in the rest frame. This value is consistent with most of our observed Ep ≲ 500 keV, but only when assuming a redshift z ≲ 0.83. Deng & Zhang (2014) also found that α ~ − 1.0 could be achieved if the radiating photosphere has a constant or increasing luminosity. However, they stated that it is difficult to reproduce the observed hard-to-soft evolution under natural conditions.

5. Conclusions

We performed time-resolved spectroscopy for eight energetic, long GRBs observed by Fermi GBM during the first five years of its mission. We obtained well-constrained Band spectral parameters and studied their theoretical implications. We showed that even in the bursts with good high-energy statistics above 900 keV, most observed properties can be explained using the synchrotron shock model. We also tested the observed spectra with a synchrotron plus blackbody model using slow- and fast-cooling parametric constraints, and found that the “both” case is consistent with the data, which requires a narrow distribution of the break ratio Eb,2/Eb,1< 10 with a peak at . The population of p is found to be 2–3, in accordance with the expected range. The picture of a “moderately fast-cooling” scenario can also explain the narrow distribution of the break ratio and relax the efficiency problem for the slow-cooling scenario.

Recently, Frontera et al. (2013) reported the result of the time-resolved spectral analysis of four GRBs observed by BATSE and BeppoSAX. They found that a specially devised empirical Comptonized model is the best-fit model to most of their time-resolved spectra. They also found that using a simple power-law plus blackbody model (PL+BB) does not give fitting results better than the conventional Band function. This is consistent with the results from the time-resolved GBM GRB catalogue (Yu et al., in prep.) that most of the time-resolved spectra are best fitted by a Comptonized model, and only very few spectra are best fitted by PL+BB, although they are generally acceptable fits. We showed that the spectral shape ≳1 MeV could be harder than a Comptonized model or simple power-law.

Our results confirmed that while most properties of energetic GRBs can be explained in the conventional theoretical models, the radiative process in GRB prompt emission is complicated and cannot be fully explained by a single distribution of electrons (e.g. due to anisotropic distribution of electron energies or continuous acceleration or photospheric emission). The possibility of a decaying magnetic field that modifies the fast-cooling spectrum was also explored, yielding a magnetic field decay index 0.6 <b< 2.6 for 77% of the constrained fits. However, it is difficult to reconcile the variability of b within bursts with a mechanism where the spectra are shaped by a single large-scale decaying magnetic field. Nevertheless, such a field might still exist, but its impact may be obscured by more local conditions in the flow.

1

The public version of the RMFIT software is available at http://fermi.gsfc.nasa.gov/ssc/data/analysis/rmfit/

Acknowledgments

The authors wish to thank Frédéric Daigne, Alexander van der Horst, Re’em Sari, Bing Zhang, and the anonymous referee for insightful suggestions. H.F.Y. and J.G. acknowledge support by the DFG cluster of excellence “Origin and Structure of the Universe” (www.universe-cluster.de). The GBM project is supported by the German Bundesministeriums für Wirtschaft und Technologie (BMWi) via the Deutsches Zentrum für Luft und Raumfahrt (DLR) under the contract numbers 50 QV 0301 and 50 OG 0502.

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Online material

Appendix A: Time-resolved fitting results

Table A.1

Band parameters for GRB 090902B.

Table A.2

Band parameters for GRB 100724B.

Table A.3

Band parameters for GRB 100826A.

Table A.4

Band parameters for GRB 101123A.

Table A.5

Band parameters for GRB 120526A.

Table A.6

Band parameters for GRB 130427A.

Table A.7

Band parameters for GRB 130504C.

Table A.8

Band parameters for GRB 130606B.

thumbnail Fig. A.1

From top to bottom: light curves of GRB 090902B, GRB 100724B, GRB 100826A, and GRB 101123A with the evolutions of constrained Ep, Eb,1, Eb,2, and kT overlaid.

thumbnail Fig. A.2

From top to bottom: light curves of GRB 120526A, GRB 130427A, GRB 130504C, and GRB 130606B with the evolutions of constrained Ep, Eb,1, Eb,2, and kT overlaid.

All Tables

Table 1

The names, GBM trigger numbers, durations, fluence, detectors used, and optimal S/N for the eight bursts studied in this paper.

In the text
Table 2

Electron distribution index p for different cases.

In the text
Table A.1

Band parameters for GRB 090902B.

In the text
Table A.2

Band parameters for GRB 100724B.

In the text
Table A.3

Band parameters for GRB 100826A.

In the text
Table A.4

Band parameters for GRB 101123A.

In the text
Table A.5

Band parameters for GRB 120526A.

In the text
Table A.6

Band parameters for GRB 130427A.

In the text
Table A.7

Band parameters for GRB 130504C.

In the text
Table A.8

Band parameters for GRB 130606B.

In the text

All Figures

thumbnail Fig. 1

Schematic spectra for the SSM cooling scenarios. The left, middle, and right panels show the “slow”, “both”, and “fast” cases in the energy flux space. The shaded region represents the possible location of νboth (i.e. Ep) when fitting the observed spectrum using a model with smoothly jointed power-laws. The photon distribution slopes are also indicated for each different case.
In the text
thumbnail Fig. 2

Distributions of the constrained parameters obtained from the Band model. The upper left panel shows the distributions of the values of Ep. The lower left panel shows the distributions of the values of α. The upper right panel shows the distributions of the values of β. The lower right panel shows the distributions of the values of Δs = α − β. The blue lines show the distributions of GRB 100724B.
In the text
thumbnail Fig. 3

Distributions of the constrained parameters obtained from the SYNC+BB model with slow-cooling constraints (i.e. α = −2/3 and β − γ = 1/2). The upper left panel shows the break energies Eb,1 and Eb,2. The lower left panel shows the kT distribution. The upper right panel shows the photon indices β of the middle power-law segment. The lower right panel shows the ratio between the two breaks, Eb,2/Eb,1. The blue lines show the distributions of GRB 100724B. Values that are out of the plotting region are accumulated in the boundary bins.
In the text
thumbnail Fig. 4

νFν spectral evolution of the SYNC-slow model for GRB 130427A. The evolution of the SYNC component evolves from cyan to blue, while the BB component evolves from yellow to red. No clear correlation is found between the two components.
In the text
thumbnail Fig. 5

Selected spectrum from GRB 130606B plotted in νFν space. The black, red, and blue solid curve show the fitted spectrum for the Band, SYNC-slow+BB, and SYNC-fast+BB model, respectively, while the dash-dotted curves show individual SYNC or BB components. The vertical dash-dotted black, red, and blue line show the Ep and break energies for the Band, SYNC-slow+BB, and SYNC-fast+BB model, respectively.
In the text
thumbnail Fig. A.1

From top to bottom: light curves of GRB 090902B, GRB 100724B, GRB 100826A, and GRB 101123A with the evolutions of constrained Ep, Eb,1, Eb,2, and kT overlaid.
In the text
thumbnail Fig. A.2

From top to bottom: light curves of GRB 120526A, GRB 130427A, GRB 130504C, and GRB 130606B with the evolutions of constrained Ep, Eb,1, Eb,2, and kT overlaid.
In the text

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