TOWARD A BETTER UNDERSTANDING OF THE GRB PHENOMENON: A NEW MODEL FOR GRB PROMPT EMISSION AND ITS EFFECTS ON THE NEW L_i^NT– E_peak,i^rest,NT RELATION

The fireball model remains the most popular scenario for the gamma-ray burst (GRB) phenomenon (Cavallo & Rees 1978; Goodman 1986; Paczynski 1986; Shemi & Piran 1990; Rees & Mészáros 1992, 1994; Mészáros & Rees 1993). In this model, the GRB central engine is a stellar-mass black hole or a rapidly spinning and highly magnetized neutron star formed by either the collapse of a supermassive star (collapsar; Woosley 1993; MacFadyen & Woosley 1999; Woosley & Heger 2006) or the merger of two compact objects (Paczynski 1986; Fryer et al. 1999; Rosswog 2003). In both cases, the original explosion creates a bipolar collimated jet composed mainly of photons, electrons, positrons, and a small fraction of baryons. The relativistic explosion ejecta within the jet are not homogeneous—they form multiple high density layers, which propagate at various velocities. When the fastest layers catch up with the slowest, the charged particles contained in the layers are accelerated through mildly relativistic collisionless shocks (internal shocks; Rees & Mészáros 1994; Kobayashi et al. 1997; Daigne & Mochkovitch 1998). The particles subsequently cool via emission processes such as synchrotron, Synchrotron Self Compton (SSC), and Inverse Compton (IC). The internal shock phase is usually associated with the so-called GRB prompt emission,¹⁷ mainly observed in the keV−MeV energy range (see, e.g., the spectral catalogs by Gruber et al. 2014; von Kienlin et al. 2014) and usually lasting from a few ms up to several tens to hundreds of seconds. As the ejecta interact with the interstellar medium they slow down via relativistic collisionless shocks (external shocks; Rees & Mészáros 1992; Mészáros & Rees 1993), accelerating charged particles, which then emit non-thermal synchrotron photons. This external shock phase is usually associated with the so-called GRB afterglow emission observed at radio wavelengths up to X-rays and in some cases even up to the GeV regime hours after the prompt phase, and days and even years for the lowest frequencies. The detailed origin of the gamma-ray emission, however, is not fully understood and many theoretical difficulties remain, such as the composition of the jet, the energy dissipation mechanisms, as well as the radiation mechanisms (e.g., Zhang 2011).

Another prediction of the fireball model is the existence of intense, thermal-like emission from the jet photosphere, expected to be observed simultaneously with the non-thermal prompt emission (Goodman 1986; Mészáros 2002; Daigne & Mochkovitch 2002; Rees & Mészáros 2005). Indeed, the high density of the outflow makes it optically thick to Thomson scattering close to the source, but as the jet expands its density decreases and radiation can escape, resulting in a thermal-like component more or less affected by sub-photospheric processes and jet-curvature effects.

GRB prompt emission spectra in the keV–MeV energy regime were predominantly fitted by the so-called Band function (Band et al. 1993; Greiner et al. 1995) prior to the launch of the Fermi Gamma-ray Space Telescope (hereafter Fermi) in 2008. This empirical function is a smoothly broken power law (PL) whose shape is defined with four parameters: two indices α and β corresponding to the spectral slopes of the low- and high-energy PLs, respectively; a break energy parametrized to correspond to the maximum of the νF_ν spectrum E_peak (Gehrels 1997); and a normalization factor. The derived Band spectra usually suggested a non-thermal origin of the radiation leading to the natural proposition that they were produced by synchrotron mechanisms. However, the high values obtained for α were often in conflict with the synchrotron scenario predictions in both the slow and fast electron cooling regimes (Cohen et al. 1997; Crider et al. 1997; Preece et al. 1998; Ghisellini et al. 2000). With the aim to identify emission from the fireball model's jet photosphere, Ghirlanda et al. (2003) and Ryde (2004) fitted a thermal spectral shape—using a pure blackbody (BB) component, and a combination of a BB and a PL, respectively—to the GRB prompt emission observed with the Burst And Transient Source Experiment (BATSE) on board the Compton Gamma Ray Observatory (CGRO). Despite the good fits obtained in a few cases, it was often difficult to assess whether thermal spectra were better than the non-thermal Band ones. At the same time, González et al. (2003) simultaneously fitted BATSE and CGRO/EGRET data of GRB 941017, finding a significant deviation from the Band function at high energies ($\gt 1$ MeV); these fits improved with the addition of a PL to account for the high-energy deviation.

The Gamma-ray Burst Monitor (GBM) and the Large Area Telescope (LAT) on board Fermi have significantly improved GRB prompt emission observations after 2008. Although the Band function remains a good model to describe GBM spectra (e.g., Gruber et al. 2014), the joint spectral analysis of some GRBs using both GBM and LAT confirmed that in some cases an additional PL is required to improve the fit quality (e.g., Abdo et al. 2009a; Guiriec et al. 2010; Ackermann et al. 2010, 2011, 2013). Moreover, Fermi results show that this additional PL can also account for deviations from the Band function below a few tens of keV. Indeed, Ackermann et al. (2011) reported the existence of an intense peak in the prompt emission light curves (LCs) of GRB 090926A, from the lowest to the highest energies, associated with this additional PL. Similar associations of sharp structures in GRB LCs with an additional PL were also reported in Guiriec (2011a) and González et al. (2012), the latter using CGRO data. In GRB 090926A, the additional PL exhibited a spectral break around 1.4 GeV, which was interpreted as resulting from γ–γ opacity making possible an estimate of the bulk Lorentz factor, Γ, between 200 and 700. Guiriec et al. (2010) reported that a similar PL deviation from the Band function could also be identified in GBM-only data of some GRBs, suggesting a spectral turnover well below the 1.4 GeV of GRB 090926A, since little or no emission was observed in the LAT data for these events. The origin of this spectral component remains challenging.

The quest for GRB prompt photospheric emission continued in the Fermi Era. Within the framework of the fireball model, the expected photosphere component should outshine the non-thermal Band component (Zhang & Pe'er 2009), but this photosphere component remained undetected. Such a surprising observational result triggered a heated debate on the GRB prompt emission mechanism. A highly magnetized jet was suggested to suppress the photosphere emission component (Daigne & Mochkovitch 2002; Nakar et al. 2005; Zhang & Pe'er 2009). Alternatively, it was suggested that the Band component is produced from a dissipative photosphere (e.g., Rees & Mészáros 2005; Thompson 2006; Beloborodov 2010; Lazzati et al. 2013; Chhotray & Lazzati 2015). Ryde et al. (2010) and Pe'er et al. (2012) simultaneously fitted a thermal component with a PL to the data of GRB 090902B and suggested that the main keV–MeV prompt emission could be of thermal origin and, therefore, the signature of the jet's photosphere. Then, Guiriec et al. (2011b) reported for the first time that the prompt emission νF_ν spectra of GRB 100724B were best fitted with a double curvature model (BB+Band) rather than the Band function alone. Contrary to the conclusions of Ryde et al. (2010) and Pe'er et al. (2012), the photospheric component (BB) identified in Guiriec et al. (2011b) was subdominant compared to the non-thermal component (Band function). Such a subdominant BB is not expected from the pure fireball model, but is more consistent with a highly magnetized outflow, as suggested by Daigne & Mochkovitch (2002), Nakar et al. (2005), and Zhang & Pe'er (2009). Guiriec et al. (2011b) suggested that to produce such a subdominant BB component, the outflow must be highly magnetized close to the source but the jet must have a low magnetization at large radii for the internal shocks to be efficient. Alternatively, Zhang & Yan (2011b) suggested that efficient energy dissipation is still possible, even if the magnetization parameter σ is still greater than unity. They envisaged a collision-induced magnetic reconnection and turbulence (ICMART) to effectively dissipate the magnetic energy in the outflow (see also Zhang & Zhang 2014 for a simulation of GRB lightcurves within such a model.) A similar component has now been reported in a handful of other GRBs (Burgess et al. 2011; Guiriec 2011a; Guiriec et al. 2011b, 2013; Axelsson et al. 2012) observed with Fermi. Interestingly, as initially proposed in Guiriec et al. (2011b) and confirmed in Guiriec et al. (2013), fits using a Band function with a BB result in Band function shapes that are much more compatible with the predictions of the synchrotron emission origin. Moreover, as shown by Guiriec et al. (2013), in those GRBs in which an intense but still subdominant thermal-like emission is detected, a strong correlation appears between the energy fluxes and the νF_ν spectral peak energy of the non-thermal component only (hereafter ${F}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{NT}}$ relation¹⁸ ). For GRBs in which the thermal-like contribution affects very slightly the shape of the non-thermal one, the fit to a Band function alone to the data leads to a similar F–E_peak relation (Golenetskii et al. 1983; Borgonovo & Ryde 2001; Liang et al. 2004; Ghirlanda et al. 2010b, 2011a, 2011b; Guiriec et al. 2010, 2013; Lu et al. 2012). The ${F}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{NT}}$ relations have similar slopes for all GRBs, indicating a universal-like mechanism for the non-thermal prompt emission, and, when corrected for the redshift, the fits between the luminosity and ${E}_{\mathrm{peak}}^{\mathrm{rest}}$ of the non-thermal component (hereafter ${L}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{rest},\mathrm{NT}}$) align perfectly for all tested GRBs (Guiriec et al. 2013).

In summary, until now GRB prompt emission spectra have been found to be composed of: (1) a Band function alone; (2) a combination of a Band function and a PL; (3) a combination of a BB and a PL; or (4) a combination of a Band function and a BB (Abdo et al. 2009a, 2009b; Ackermann et al. 2010, 2011; Guiriec 2011a; Guiriec et al. 2010, 2011b, 2013; Ryde et al. 2010; Zhang et al. 2011c; Pe'er et al. 2012). The non-thermal Band function usually overpowers the spectra, the quasi-thermal photosphere component usually carries less than a few tens of percent of the total energy, and the additional PL component extends from low to high energies.

Here we reanalyze two of the brightest Fermi GRBs, GRB 080916C and GRB 090926A, in the context of the multiple components described above and discuss the possible simultaneous existence of such separate components by comparing spectral analyses at various timescales from time-integrated to very fine-time intervals. Through the evolution of these components with time, we trace back the emission mechanisms which produced them, following the jet physical mechanisms back to the central engine energy reservoir. In Section 2, we describe the data selection as well as the observations, and in Section 3 we present the procedures we followed for the analysis. Sections 4 and 5 are dedicated to the results of the time-integrated and the coarse-time analyses, respectively. Section 6 reports the fine-time spectral analysis results, which are the main topics of this article. In Section 7, we investigate the impact of the multiple spectral components on the relations between the energy flux of the non-thermal component and its observed νF_ν spectral-peak energy (namely ${F}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{NT}}$) and between the luminosity of the non-thermal component and its intrinsic νF_ν spectral-peak energy (namely ${L}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{rest},\mathrm{NT}}$). Section 8 introduces our suggested model for GRB prompt emission. Section 9 discusses a possible interpretation of our observational results; however, this section does not aim to be a comprehensive or a detailed review of all the models that exist in the literature and that may support our observational results. Finally, in Section 10, we summarize the most important observational results and comment on their future impact.

The GBM consists of 12 sodium lodide (NaI) detectors covering energies between 8 keV and 1 MeV. GBM also includes two bismuth germanate (BGO) detectors (b0, b1) covering energies between 200 keV and 40 MeV, located on opposite sides of the spacecraft. The LAT is a pair-conversion telescope sensitive to photons with energies from 20 MeV to >300 GeV. When a photon enters the LAT, it is converted into an electron–positron pair when interacting with the conversion tungsten foils located between the tracker silicon strip detector planes. Detailed information about the GBM and the LAT can be found in Meegan et al. (2009) and Atwood et al. (2009), respectively.¹⁹

GBM detected GRB 080916C and GRB 090926A on 2008 September 8 at T₀ = 00:12:45 UT (Goldstein & van der Horst 2008) and on 2009 September 26 at T₀ = 04:20:26.99 UT (Bissaldi 2009), respectively. These bursts were also detected with the LAT (Tajima et al. 2008; Abdo et al. 2009b for GRB 080916C, and Uehara et al. 2009; Ackermann et al. 2011 for GRB 090926A). The properties of the two GRBs are reported in Table 1. We show the LCs of GRB 080916C and GRB 090926A in several energy ranges in Figures 1 and 2, respectively.

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Table 1.  List of GRB Properties

GRB Name	Locationa	Redshift z	${{\rm{T}}}_{90}(s)$ b	Observed Highest Energy Photon (GeV)
GRB 080916C	(R.A., Decl.) = (11984717,−5663833) (±0 5)c	4.24 ± 0.26d	63 ± 1	33
GRB 090926A	(R.A., Decl.) = (35340070,−6632390) (±1 5)e	2.1062f	20 ± 2	19.6

Notes.

^aSee GBM GRB catalog at http://fermi.gsfc.nasa.gov/ssc. ^bDuration computed between 50 and 300 keV (Kouveliotou et al. 1993). ^cThe Gamma-ray burst Optical/Near-infrared Detector (GROND—Clemens et al. 2008). ^dGreiner et al. (2009). ^eSwift/UVOT (Vetere et al. 2009). ^fVery Large Telescope (VLT—Malesani et al. 2009).

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For the analysis we only selected the optimal NaI detectors with angles to the source smaller than $30^\circ$ based on the best source locations and with no blockage by another part of the spacecraft (e.g., LAT, radiators, solar panels) as well as with no shadowing by another GBM module. According to these criteria, we selected detectors n0, n3, and n4 for GRB 080916C and n3, n6, and n7 for GRB 090926A.²⁰

Usually, the source is only in the field of view of one BGO detector, and only this BGO detector is retained for the analyses. Therefore, for GRB 080916C, we used b0. However, in the case of GRB 090926A, the burst occurred in the median plane of the spacecraft, which is parallel to the sides to which the BGO detectors are attached. Therefore, the two BGO detectors can be simultaneously used in this specific case. Because of the peculiar location of GRB 090926A in the spacecraft coordinates, the reconstructed signal may be distorted due to the inaccurate modeling of the absorption in the back of the BGO modules (photomultipliers for instance), especially at low energy. Our analysis revealed that although the signals reconstructed from the various NaI detectors match perfectly, discrepancies between the NaI and BGO modules are observed in the overlapping energy region between 200 and ∼750 keV for b0 and between 200 and ∼600 keV for b1 for GRB 090926A. Beyond ∼750 and ∼600 keV for b0 and b1, respectively, NaI and BGO data are in agreement.

We used in our analysis time tagged event (TTE) data, which have the finest time (i.e., 2 μs) and energy resolution that can be achieved with the GBM. We used the NaI data from 8 to ∼900 keV cutting out the overflow high-energy channels as well as the K-edge from ∼30 to ∼40 keV. For GRB 080916C, we used BGO data from 200 keV to ∼39 MeV cutting out the overflow energy channels. For GRB 090926A, we also excluded the low-energy channels <750 and <600 keV for b0 and b1, respectively, since they may be affected by calibration problems. We then generated the response files for each GBM detector based on the best source location.

To optimize the analysis according to the instrument performance, we study the photons converted in the front and in the back of the LAT tracker as two separate data sets (hereafter FRONT and BACK, respectively) using the appropriate instrument response functions. We analyzed the FRONT and BACK data passing the Pass 7 transient cuts (Ackermann et al. 2012) within a ${10}^{\circ }$ region of interest centered on the best source location and with energy above 100 MeV. To perform spectral analysis of the LAT data between 20 and 100 MeV, we used the LAT low-energy (LLE) data, which are designed to increase the LAT sensitivity below 100 MeV for transient sources as well as to make spectral analysis below 100 MeV possible (for instance, see the Appendix of Ajello et al. 2014).

The background in each of the GBM detectors as well as in the LLE data was estimated by fitting polynomial functions to the LCs in various energy ranges before and after the source active time period. The background was then measured by interpolating the functions to the source active time period. For GBM data, the background was fitted to the CSPEC data files, which have the same energy resolution as the TTE data but with a coarser time resolution. CSPEC data cover a much longer time range, making the estimation of the background more reliable especially for long GRBs. We then exported the background estimated from the CSPEC data to our analysis of the TTE data. The background for the LAT FRONT and BACK data was estimated by averaging the data over several orbits (during which the GRB was not present) when Fermi was located at the same position in the orbit as at the trigger time and when the LAT was pointing in the same direction as at the trigger time (Vasileiou 2013).

To perform the spectral analysis, we used the XSpec and Rmfit tool kits. Starting from a cutoff PL (C) or a Band function (Band), we built twelve increasingly complex spectral models with two or three components adding a PL, a BB, C, or Band. Sketches of the various models are presented in Figure 3. The best spectral parameter values and their 1σ uncertainties were estimated by optimizing the Castor C-statistic (hereafter Cstat), which is a likelihood technique that converges to ${\chi }^{2}$ for a specific data set when there are enough counts. We performed the spectral analysis over three timescales—we refer to these later in the text as: time-integrated, coarse-time bins, and fine-time bins.

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In order to check the consistency of the NaI and BGO detectors through all tested models, we fitted both detector types simultaneously and we applied a free effective area correction (EAC) factor between b0 and each of the NaI detectors selected for the analysis. For each model fitted we then fixed these EAC values. To evaluate the consistency of the GBM data (NaI and b0) with the LAT-LLE, LAT-FRONT, and LAT-BACK data, we also fitted them simultaneously adding a new, free EAC factor between b0 and each of the LAT data types. When EAC values were compatible with no correction needed, they were set to unity. We only applied the EAC corrections in the time-integrated and coarse-time bin analysis, because in the fine-time bin analysis the number of counts is much smaller, leading to unconstrained EAC factors. When significant EAC factors (i.e., significant deviation from unity) between the GBM and LAT data sets are required with the simplest models, those data sets are in much better agreement with the more complex ones. The strong corrections required with the simplest models seem to be unreasonable based on our current knowledge of the instruments; therefore, this reinforces the scenarios involving the more complex spectral shapes. The results of cross-calibration analysis are reported in greater detail in Appendix A.

Due to the very limited number of counts at high energies, we only used the LAT-LLE data in combination with GBM in the coarse-time analysis, and we did not use any LAT data in the fine-time analysis. We then estimated the probability that the Cstat improvements obtained with the most complex models compared to the simplest ones are not merely due to signal and/or background statistical fluctuations, by performing multiple sets of Monte Carlo (MC) simulations following the same procedure as discussed in Guiriec et al. (2011b, 2013; see Appendix B). In Sections 4 and 5 we compare the results of joint GBM and LAT data fits with those of the GBM-only ones for the time-integrated and the coarse-time analysis, respectively; the consistency of the joint GBM and LAT analysis with the GBM-only one is important to support the results of the fine-time analysis presented in Section 6, for which we only use GBM data.

We performed a time-integrated spectral analysis over a period corresponding to the most intense part of the prompt emission of the two GRBs in the keV–MeV energy range. (i.e., from T₀–0.10 s to T₀+71.00 s and from T₀ to T₀+20.00 s for GRBs 080916C and 090926A, respectively.) The fit results using the various models discussed above are reported in Tables 2 and 3 of Appendix C. We discuss below the fits and present the most relevant ones in Figures 4 and 5.

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4.1. GBM-only

4.2. GBM+LAT

We concluded above that both GRB 080916C and GRB 090926A were better fitted with a three-component model. However, it is impossible to conclude from the time-integrated spectra whether these separate components are artifacts due to a strong spectral evolution during the event duration, and whether they can be associated directly to physical processes. To address these questions, we performed a coarse-time spectral analysis selecting time intervals including the main structures observed in each burst LC (see Tables 4 and 5 of Appendix C for GRB 080916C and GRB 090926A, respectively.) This selection resulted in three and five time intervals for GRB 080916C and GRB 090926A, respectively.

We fitted either the GBM data alone or in combination with the LLE data. We did not include the LAT transient data in this analysis to avoid complications and possible biases due to the low number of photons and possible calibration inconsistencies. We also tested the need for EAC either among the GBM detectors or between the GBM and the LLE data; the results are reported in Appendix A.

5.1. GRB 080916C

5.2. GRB 090926A

5.3. Partial Conclusion

To better understand the behavior of the various spectral components as well as their evolution with time, we performed a very fine-time spectral analysis. At very fine timescales, GBM and LAT data are in very different count regimes; therefore, to avoid any complication related to this difference as well as to possible cross-calibration issue we only consider GBM data throughout this section.

With fine time intervals it is often difficult to unambiguously conclude that the most complex models are significantly better than the simpler ones based solely on the pure statistical considerations of the likelihood ratio test. Instead, our goal here is to verify that the complex shapes identified as significantly better than the Band function in the time-integrated and the coarse-time spectral analyses are also valid options at very fine timescales. It is important to keep in mind that even if the most complex models are not statistically better than the Band function based on the likelihood ratio test results, it does not necessarily mean that the contributions of the additional components (i.e., BB and PL) to the total flux are negligible and compatible with a null flux. As presented in the previous sections, the addition of a BB and a PL to the Band function significantly modifies the parameters of the Band function. Therefore, it is possible to have complex models that are statistically not favored compared to a Band function alone—based solely on the likelihood ratio test—but whose additional component fluxes are clearly positive. Moreover, we show in Appendix B that the results of the likelihood ratio test are not always sufficient to decide whether the simpler model is truly a better description of the data than a more complex one and that additional statistical tests are sometime necessary; following the procedure described in Appendix B, we conclude that C+BB+PL is globally a better description of the data than all the other simpler models. In this section, we will focus mostly on the consistency and continuity of the spectral results from time interval to time interval and on how the complex models modify the values of the spectral parameters of the Band function.

We have shown in panels (a1–6) of Figures 1 and 2 the two burst LCs in various energy bands with a 0.1 s time resolution; we show in panels (b1–6) the same LCs binned at the intervals we used for the fine-time spectral analysis (for the bin size determination see Appendix D). For this analysis we started by fitting to every time interval both Band-only and the model that we identified in the previous sections as the best choice, namely C+BB+PL. When all three components could not be fitted simultaneously, we kept as many of them as possible (i.e., if C+BB+PL could be fitted but only C+BB, we report the results from C+BB). An example of the three-component fit using C+BB+C for one time interval of GRB 090926A is presented in Figure 6. The complete fit results are reported in Appendix C in Table 6 for GRB 080916C, and in Table 7 for GRB 090926A (see also Appendix E).

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Figures 7 and 8 show the reconstructed photon LCs in multiple energy bands using the Band-only and the C+BB+PL analyses of GRBs 080916C and 090926A, respectively. These figures show the time evolution of the individual spectral components as well as the time evolution of the total emission, to be compared to the observed count LCs in the same energy bands. The evolution of the spectral parameters of the Band-only and C+BB+PL models are presented in Figures 9 and 10 for GRBs 080916C and 090926A, respectively, and the parameter distributions of the two fits are presented in Figures 11–14.

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Note that in the following we sometimes use the terminology Band when we talk about the C component of the C+BB+PL scenario; indeed, the C component of C+BB+PL corresponds globally to the traditional Band function and it is sometimes easier to refer to it as Band. The C and Band component are usually statistically equivalent as long as Band β is <−3.

6.1. Evolution of the Various Spectral Components in the C+BB+PL Scenario

6.2. Band Versus C+BB+PL

Figures 15(a) and (b) show the energy fluxes of the Band function (or C in the case of C+BB+PL) as a function of E_peak when fitting either Band-alone (blue) or C+BB+PL (red) to the time-resolved spectra of GRB 080916C (Figure 15(a)) and GRB 090926A (Figure 15(b)). We note that the correlation between the two quantities is much stronger with C+BB+PL. As reported for the first time in Guiriec et al. (2013), if this new ${F}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{NT}}$ relation is a specific property of the non-thermal component only (i.e., Band or C in the new multi-component model), it is expected that the fits of a Band function alone to GRBs that have additional spectral components will lead to large scatter in the data and in a weaker and biased relation or even in no correlation at all if the additional components are very intense.

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In Figure 15(c), the ${F}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{NT}}$ relations of several GRBs are overplotted. The results on short GRB 120323 A and long GRB 110721 A were already reported in Guiriec et al. (2013) and preliminary results of GRBs 080916C and 090926A using C+BB+PL were also shown in the same article. Here, the detailed analysis of GRBs 080916C and 090926A confirms the results reported in Guiriec et al. (2013): in the observer frame, the ${F}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{NT}}$ relations have similar slope for all GRBs whatever the duration of the burst. In addition, when corrected for redshift, all GRBs lie along the same ${L}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{rest},\mathrm{NT}}$ relation—with a very limited scatter—suggesting universality of the phenomenon (see Figure 15(d)). This new detailed analysis tightened the relation presented in Guiriec et al. (2013) even more.

The PL fit of the ${L}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{rest},\mathrm{NT}}$ relation for each individual GRB result is shown in colored dashed lines:

$$\begin{eqnarray*}&&{L}_{080916{\rm{C}},{\rm{i}}}^{\mathrm{NT}}=(8.7\pm 3.4){10}^{52}{\left(\displaystyle \frac{{E}_{\mathrm{peak},i}^{\mathrm{rest},\mathrm{NT}}}{100\;\mathrm{keV}}\right)}^{1.36\pm 0.12}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\end{eqnarray*}$$

$$\begin{eqnarray*}&&{L}_{090926{\rm{A}},i}^{\mathrm{NT}}=(10.0\pm 1.8){10}^{52}{\left(\displaystyle \frac{{E}_{\mathrm{peak},i}^{\mathrm{rest},\mathrm{NT}}}{100\;\mathrm{keV}}\right)}^{1.40\pm 0.07}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\end{eqnarray*}$$

$$\begin{eqnarray*}&&{L}_{120323{\rm{A}},i}^{\mathrm{NT}}=(9.3\pm 3.0){10}^{52}{\left(\displaystyle \frac{{E}_{\mathrm{peak},i}^{\mathrm{rest},\mathrm{NT}}}{100\;\mathrm{keV}}\right)}^{1.36\pm 0.15}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\end{eqnarray*}$$

$$\begin{eqnarray*}&&{L}_{110721{\rm{A}},i}^{\mathrm{NT}}=(8.4\pm 2.0){10}^{52}{\left(\displaystyle \frac{{E}_{\mathrm{peak},i}^{\mathrm{rest},\mathrm{NT}}}{100\;\mathrm{keV}}\right)}^{1.36\pm 0.12}\;\mathrm{erg}\;{{\rm{s}}}^{-1}.\end{eqnarray*}$$

The simultaneous fit of all the GRBs with a PL results in the solid black line:

$$\begin{eqnarray}&&{L}_{i}^{\mathrm{NT}}=(9.6\pm 1.1){10}^{52}{\left(\displaystyle \frac{{E}_{\mathrm{peak},i}^{\mathrm{rest},\mathrm{NT}}}{100\;\mathrm{keV}}\right)}^{1.38\pm 0.04}\;\mathrm{erg}\;{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 1 }$$

The strong compatibility of the results for all the tested GRBs reinforces the possibility of eventually using this relation as a redshift estimator.

We must note that the initial rising part of the first pulse of a burst usually does not follow the typical ${F}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{NT}}$ relation as already reported in Guiriec et al. (2013), but an anti-correlation is observed between ${F}_{i}^{\mathrm{NT}}$ and ${E}_{\mathrm{peak},i}^{\mathrm{NT}}$ instead.

In Section 6, we reported the constancy in the values of the α and the indices of the additional PL when fitting C+BB+PL to the time-resolved data of both GRBs. Not only did these values remain mostly constant during each burst (see panels (b) and (e) of Figures 9 and 10), but they are also nearly the same for both events (see panels (b) and (d) of Figures 12 and 14). Figures 16 and 17 show the Band-only and C+BB+PL combined parameter distributions, respectively, for GRBs 080916C and 090926A. This reinforces the conclusions reported in Section 6.2: the Band-only fits result in a narrow distribution of the E_peak values around 300 keV and in a broader distribution for α, with a large fraction of the latter values >−0.7. In contrast, the E_peak values for C+BB+PL are much more spread with a maximum of the distribution around 500 keV, the values of α are narrowly peaked around −0.7 and the α 1σ lower limits are all bellow −0.7 besides a few cases, and the values of the additional PL index are clustered between −1 and −2 with a distribution peak around −1.5.

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Since the slopes of the low-energy PL of C and of the additional PL in the C+BB+PL model do not change much with time and remain similar in the two bursts, those two parameters can be frozen to their typical values in the fits, reducing the complexity of the C+BB+PL model by two dof. This new model, therefore, which we will call C+BB+PL_5params hereafter, has only five parameters. C+BB+PL_5params has only one parameter more than Band-alone, which makes the two models very competitive from a statistical point of view.

We fitted C+BB+PL_5params to the time-resolved spectra and the results are reported in Table 6 of Appendix C and Figure 21 of Appendix E and Table 7 of Appendix C and Figure 22 of Appendix E for GRBs 080916C and 090926A, respectively. As expected, C+BB+PL_5params results in similar Cstat values as C+BB+PL except for a few cases where the additional PL slope appears to be a bit steeper than −1.5. However, this can be easily explained by the presence of a cutoff in the additional PL. Indeed, the C+BB+C_6params fits (i.e., C+BB+PL_5params with a exponential cutoff in the additional PL) seem to indicate the presence of a cutoff whose energy evolves smoothly with time. The cutoff could be hidden by the C component at times, and it could be at much higher energies at others. This result may question the use of the additional PL cutoff to estimate the Lorentz factor as proposed in Ackermann et al. (2011). Instead, such an evolution of the cutoff energy may be a signature of the emission mechanism(s) producing this component and not an effect of the γ–γ opacity.

It is very important to note that the number of time intervals in which the three components cannot be simultaneously fitted is very limited for C+BB+PL_5params compared to C+BB+PL. Even more interestingly, C+BB+PL_5params results indicate that the additional PL is present from the very beginning of the prompt emission in both GRBs. Moreover, the additional PL is the most intense component at very early times. This is supported by the very low values of α (i.e., <−1) obtained in the first time intervals when fitting C+BB+PL with all the parameters free (see Figures 9(b) and 10(b)). Indeed, in Section 6.1 we reported that we could not fit the C+BB+PL model at early times, but only C+BB. Intriguingly, in these time intervals, the values of α were significantly lower than average and closer to the typical index values of the additional PL. This can be explained if we consider the fit resulting from C+BB+PL_5params as the "true" model. With C+BB+PL_5params, the additional PL overpowers the other two components, and when fitting C+BB to the data, the value of α is a hybrid (i.e., ∼−1.1) between the real value of the C component (i.e., ∼−0.7) and the index of the additional PL that is not included in the fit (i.e., ∼−1.5). The same conclusions can be drawn for GRB 090926A to explain the low values of α obtained when fitting C+BB at late times (see Figure 10(b)). Indeed, in the C+BB+PL_5params scenario, the contribution coming from the additional PL is the most intense at late times (see panels (d) of Figure 8).

The reconstructed photon LCs resulting from C+BB+PL_5params are presented in panels (d) of Figures 7 and 8 for GRBs 080916C and 090926A, respectively. For both bursts, these LCs are in very good agreement with the observed count LCs, in particular the LLE LC between 20 and 100 MeV. Indeed, although we do not use the LAT-LLE data in our fits, the extrapolations of our model to higher energies reproduce very well the observations with a smooth evolution of the additional PL flux with time. The resemblance of the observed 20–100 MeV LCs with the reconstructed one is particularly striking for GRB 080916C: it is not only the intensity peak that is well recovered, but the long tails before and after it are also very adequately reproduced. Figure 18 shows the evolution of the energy content of each component with time (panels (a1) and (b1)) as well as their relative contributions to the total energy flux (panels (a2) and (b2)) between 8 keV and 100 MeV in the observer frame. The Band function contribution to the total energy flux dominates at early time while the additional PL contribution is dominant at later time; however, the contribution of the additional PL is also important at very early times during the first instant of the bursts. Because of the narrow shape of the thermal-like component, its contribution to the total energy flux is limited although it may reach a few tens of percent in some time intervals in the best case scenario.

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A larger sample of bursts will help to better estimate the typical values of α and of the additional PL index. While for the two GRBs studied in this article, as well as for many others, a value of α around −0.7 seems to be a good estimate, for others the value of α seems to peak around −1.2 (e.g., Guiriec et al. 2013); both values seem to be typical ones for α.

The ${F}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{NT}}$ relation can be used to better constrain the C+BB+PL_5params model and to even further decrease its complexity. Indeed, in the C+BB+PL_5params model the values of α, "β" (i.e., an exponential cutoff for C), and the index of the additional PL are frozen, therefore, the ${F}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{NT}}$ relation can be reduced to a correlation between the amplitude of the Band function (or C), A, and its peak energy E_peak. This relation between A and ${E}_{\mathrm{peak},i}$ can then be determined with an initial time-resolved analysis of the data for each burst with the C+BB+PL_5params model and subsequently be used as input to the C+BB+PL_5params model to tighten the fit results even more.

Moreover, if the ${L}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{rest},\mathrm{NT}}$ relation is confirmed and shown to hold for a large sample of bursts, then the relation between A and E_peak may be firmly determined beforehand for GRBs with known redshift and would allow us to constrain the C+BB+PL_5params model from the beginning; therefore, the C+BB+PL_5params model would then have only four independent parameters (i.e., C+BB+PL_4params) like the Band function alone. For GRBs with unknown redshift, z would be an additional parameter (i.e., five independent parameters and a constraint between A and E_peak) and it may be possible to evaluate the redshift of the burst from the fits of C+BB+${\mathrm{PL}}_{4\mathrm{params},{\rm{z}}}$ to each time-interval. Finally, it could also be possible to fit directly the cosmological parameters to the time-resolved spectra in the case of GRBs with known redshifts (i.e., C+BB+PL ${}_{4\mathrm{params},{{\rm{\Omega }}}_{{\rm{M}}},{{\rm{\Omega }}}_{{\rm{\Lambda }}}}$). Although preliminary results look promising, a comprehensive study is required to validate or not those relations prior to any possible use; those results will be presented in a follow-up article.

There are several theoretical possibilities for the physical origin of each of the three components identified in the spectra of GRB 080916C and GRB 090926A. However, the general framework of the discussion is well established: (i) GRBs are produced by ultra-relativistic outflows and (ii) the prompt emission has an internal origin. More precisely, (i) is required to avoid a strong γ–γ annihilation (e.g., Baring & Harding 1997; Lithwick & Sari 2001). A detailed calculation leads to constraints for the Lorentz factor and the radius of the emission site to allow the escape of photons at a given energy E (Granot et al. 2008; Hascoët et al. 2012). For the Band (or C) and BB components, this calculation does not provide a strong constraint on the emission site, which can be anywhere from the photosphere to the deceleration radius. On the other hand, as the PL component is extending at least up to 100 MeV, it must be produced above the photosphere. Condition (ii) is obtained from the study of the variability of the prompt emission, which cannot be reproduced by the external shock during the phase when the outflow is decelerated by the ambient medium (e.g., Sari & Piran 1997). This implies that the Band (or C) and BB components must be of internal origin. For the PL, our analysis clearly shows that variability below the one-second timescale is also present (see panels (c) and (d) of Figures 7 and 8, red curves), indicating that the origin is at least partially internal. On the other hand, the evolution of this component, which becomes brighter at the end of the burst, suggests that a link with the deceleration of the outflow should also be considered.

9.1. Origin of the Band (or C) and BB Components

9.2. Origin of the PL component

We summarize our results as follows:

• 1.  
  GRB prompt emission spectra are more complex than the shape resulting from the fit to a Band function alone. An adequate option to capture the complex shape of GRB prompt emission is a combination of at least three separate spectral components whose intensity and relative contribution to the total energy flux evolve with time. (i) The three components are not always present in all GRBs; or (ii) in some GRBs certain components strongly overpower the others making their identification more difficult. Figure 6 shows an example of the three-component model (i.e., (1) a non-thermal Band function with a high-energy PL index <−3 and which is statistically equivalent to a cutoff PL, (2) a thermal-like component adequately approximated with a BB component, and (3) a non-thermal additional PL with or without cutoff) in GRB 090926A. The uniqueness of the spectral decomposition in terms of three separate components cannot be proven but as want to stress that the proposed decomposition is not arbitrary and the smooth evolution of these components with time as well we the resulting ${F}_{i}^{\mathrm{NT}}$-${E}_{\mathrm{peak},i}^{\mathrm{NT}}$ and ${L}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{rest},\mathrm{NT}}$ relations reinforce its validity and makes it a very interesting scenario to probe the underlying GRB physics.
• 2.  
  The new three-component model is composed of (i) a smoothly broken PL with a constant low-energy index of ∼−0.7 (or ∼−1.2 as for GRB 120323A—see Guiriec et al. 2013) and a high-energy slope compatible with an exponential cutoff based on the quality of our current data (i.e., C), (ii) a thermal-like component adequately approximated with a BB component, and (iii) another PL component with a fixed index of ∼−1.5, whose cutoff energy is often challenging to identify. However, the cutoff would be below 100 MeV most of the time. When a cutoff is used, the index of the additional PL gets a bit closer to −1. It is important to note that this paper aims to establish the existence of at least three distinct spectral components, but the spectral shapes of the various components will be explored in greater detail in a following article.
• 3.  
  In the C+BB+PL model, (i) the C component can be interpreted as synchrotron radiation in an optically thin region above the photosphere, either from internal shocks or magnetic field dissipation regions, (ii) the BB component can be interpreted as the photosphere emission of a magnetized relativistic outflow, and (iii) the extra PL extending to high energies likely has an IC origin of some sort even though its extension to a much lower energy remains a mystery. This interpretation mostly relies on qualitative considerations that remain to be accurately quantified. The proposed interpretation should not prevent other theoretical scenarios to be tested to the new observational model.
• 4.  
  We succeed in reducing the number of parameters in the three-component model (i.e., C+BB+PL) from 7 down to 5 (i.e., C+BB+PL_5params) still keeping the same quality for the fits; with 5 free parameters C+BB+PL_5params is statistically competitive with the 4 parameters of the Band function. Additional constraints to the model will be added if the ${F}_{\mathrm{Band},i}$–${E}_{\mathrm{peak},i}$ and ${L}_{\mathrm{Band},i}$–${E}_{\mathrm{peak},i}^{\mathrm{rest}}$ relations are confirmed on a large sample of GRBs; this will result in an even simpler model.
• 5.  
  The C and BB components are intense at early times in a burst. The additional PL may kick off at very early times and even before the two other components and last much longer at low energies. The intensity peak of the additional PL seems to be often related to sharp and bright structures present in the LCs at all energies from 8 keV to tens of MeV and even GeV. The additional PL is the most intense component at late times at low energies (tens of keV). This additional PL may connect smoothly with the GeV LAT emission observed for thousands of seconds after the prompt phase and contemporaneously with the X-ray and optical afterglow emissions. Because the BB component is narrow, its contribution to the total energy flux between 8 keV and 100 MeV is very limited compared to the two other components (see Figure 18); however, the BB has an important relative contribution to the total emission at its spectral peak around 100–200 keV (see Figures 21 and 22 of Appendix E).
• 6.  
  From the data set we used, it is difficult to conclude if the additional PL is a single component. However, it is clear that there is a simultaneous evolution of the low- and high-energy fluxes which seems to propagate across the entire spectrum.
• 7.  
  There is a strong correlation between the energy flux of the non-thermal component—adequately fitted with either Band or C—and its E_peak within each burst. Taking into account the various components in the fit process reduces the scatter of this relation. When fitted with a PL, the ${F}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{NT}}$ relations have similar slopes for all GRBs. This relation may be useful to understand the mechanism responsible for the non-thermal prompt emission.
• 8.  
  In the central engine rest frame, the ${F}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{NT}}$ relation points toward a universal ${L}_{i}^{\mathrm{NT}}$–${E}_{\mathrm{peak},i}^{\mathrm{rest},\mathrm{NT}}$ relation. If confirmed, this relation could be used to estimate GRB redshifts as well as to constrain the cosmological parameters.

Beyond the purpose of the current article, this analysis using multiple spectral components may have a broader impact on other analyses:

• 1.  
  Since the cutoff in the additional PL seems to evolve with time—and most of the time bellow 100 MeV—and since it is sometimes hidden by other components, it may indicate that this cutoff is a property of the emission mechanism(s) producing the additional PL and not the result of γ–γ opacity. If it is the case, it is not correct to estimate the bulk Lorentz factor of the jet based on the additional PL cutoff energy.
• 2.  
  The presence of multiple components evolving independently with time and exhibiting highly variable relative fluxes may strongly bias the conclusion resulting from spectral lag analysis. Indeed, lags may be identified between high intensity structures belonging to different components.
• 3.  
  Such multi-component spectral analysis may help in the context of analysis conducted to constrain Lorentz invariance. Indeed, such analysis considers that photons are emitted from the same region and at the same time. By identifying the various components we may improve the association between low- and high-energy photons, as it would be the case e.g., for GRB 080916C. Based on the count LCs of Figure 7(a), we may be tempted to consider the association of the peak above 20 MeV around 6 s with peaks at lower energy happening at earlier (or later) times, especially if we do not have the knowledge of the low-energy LC below 20 keV. With the multiple component analysis, we see that the peak at high energy is directly correlated to the one at low energy. Therefore we can increase the accuracy of the Lorentz invariance measurement by reducing the uncertainties on the time dispersion.

We thank the referees for a thoughtful report that helped improve the article. The Fermi-LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States; the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique/Institut National de Physique Nucléaire et de Physique des Particules in France; the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy; the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK), and Japan Aerospace Exploration Agency (JAXA) in Japan; and the K. A. Wallenberg Foundation, the Swedish Research Council, and the Swedish National Space Board in Sweden.

Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'Études Spatiales in France.

To complete this project, S.G. was supported by the NASA Postdoctoral Program (NPP) at the NASA/Goddard Space Flight Center, administered by Oak Ridge Associated Universities through a contract with NASA as well as by the NASA grants NNH11ZDA001N and NNH13ZDA001N awarded to S.G. during the cycles 5 and 7 of the NASA Fermi Guest Investigator Program.

A.1. Time-integrated Spectral Analysis

A.2. Coarse-time Spectral Analysis

In every time intervals of the time-integrated, coarse-time and fine-time spectral analysis, we performed multiple sets of MC simulations (i) to compare all the tested models one to one, (ii) to estimate the goodness of fit of every tested model, and (iii) to estimate the trustworthiness of the measured spectral parameters for all the tested models. To generate the synthetic spectra of our simulation sets, we followed exactly the same procedure as already described in Guiriec et al. (2011b) and in the Appendix of Guiriec et al. (2013).

In the article, we will use the statistical terminology and call "best model" and "best fit" the model and the fit, respectively, that have the lowest Cstat value.

To assess which model is a "better description of the real data" set when comparing models M₁ and M₂, we performed ten statistical tests addressing the ten following questions (shortened for simplicity). In all tests we assume that both signal and background fluctuations are following Poisson statistics.

•   
  Test 1: estimate the odds of obtaining a ΔCstat value between M₁ and M₂ higher than the observed one if the true underlying model is M₁—with real fit parameters" $=\gt$ This is the typical "likelihood-ratio test" or "log-likelihood statistic test."
•   
  Test 2: estimate the odds of obtaining the observed parameters of M₂ when fitting M₂ to the data if the true underlying model is M₁—with real fit parameters.
•   
  Test 3: estimate the odds of obtaining the observed parameters of M₁ when fitting M₁ to the data if the true underlying model is M₁—with real fit parameters $=\gt$ This also serves to estimate the trustworthiness of the measured parameters.
•   
  Test 4: estimate the odds of obtaining the specific Cstat value when fitting M₁ to the data if the true underlying model is M₁—with real fit parameters $=\gt$ This is the typical goodness of fit test.
•   
  Test 5: estimate the odds of obtaining the specific Cstat value when fitting M₂ to the data if the true underlying model is M₁—with real fit parameters.
•   
  Test 6: estimate the odds of obtaining a ΔCstat value between M₁ and M₂ higher than the observed one if the true underlying model is M₂—with real fit $=\gt$ This is the typical "likelihood-ratio test" or "log-likelihood statistic test."
•   
  Test 7: estimate the odds of obtaining the observed parameters of M₂ when fitting M₂ to the data if the true underlying model is M₂—with real fit parameters $=\gt$ This also estimates the trustworthiness of the measured parameters.
•   
  Test 8: estimate the odds of obtaining the observed parameters of M₁ when fitting M₁ to the data if the true underlying model is M₂—with real fit parameters.
•   
  Test 9: estimate the odds of obtaining the specific Cstat value when fitting M₁ to the data if the true underlying model is M₂—with real fit parameters.
•   
  Test 10: estimate the odds of obtaining the specific Cstat value when fitting M₂ to the data if the true underlying model is M₂—with real fit parameters $=\gt$ This is the typical goodness of fit test.

We consider that M₂ is a better description of the data than M₁ if:

• 1.  
  Test 1 is conclusive (i.e., p-value < ∼10⁻⁵—see footnote ²² for the definition of p-value) and if Test 6 is inconclusive. (i.e., p-value $\gt \gt$ 10⁻⁵).
• 2.  
  Tests 3, 7, 8, 9 and 10 are inconclusive (i.e., p-value $\gt \gt$ 10⁻⁵) and Tests 4 and 5 are conclusive. (i.e., p-value $\lt \lt \lt$). In this case, Test 2 is also usually conclusive.

In addition, if M₂ is a "better description of the data" than M₁ because of criterion 1, we will also say that M₂ is "statistically significantly better" than M₁.

We note that as of today, the GBM instrument team does not anticipate any evidence for global strong instrumental systematic effects that have not been accounted for in the instrument response function yet; however, we cannot discard the fact that possible calibration problems discovered at later time may impact the results of the current analysis as well as all the analysis performed with the traditional Band function alone.

Hereafter we introduce two typical cases:

1. Comparison of the B+BB and C+BB+PL fits to the GBM data alone in the time interval lasting from T₀+4.3 s to T₀+7.5 s of the coarse-time analysis of GRB 080916C:

In the following, we will consider that B+BB and C+BB+PL correspond, respectively, to the M₁ and M₂ models from the description of the various tests presented earlier in this appendix. B+BB and C+BB+PL have six and seven free parameters, respectively, but they are not nested models.

The B+BB and C+BB+PL fits to the real data result in Cstat values of 508 and 495, respectively. In this time interval, C+BB is clearly a very bad fit to the data (i.e., extremely high Cstat value) and it is, therefore, not considered here. Indeed, when fitting B+BB to this time interval β is >−2.28 and clearly incompatible with the exponential cutoff of a C component.

We first present the results of the typical log-likelihood statistic test using B+BB with the parameter values resulting from the B+BB fit to the real data as the null-hypothesis (i.e., Test 1.) Despite the fact that C+BB+PL has one more free parameter than B+BB, the ${\rm{\Delta }}{\mathrm{Cstat}}_{{{\rm{M}}}_{1}-{{\rm{M}}}_{2}}$ distribution (i.e., ${\mathrm{Cstat}}_{{{\rm{M}}}_{1}}$-Cstat ${}_{{{\rm{M}}}_{2}}$) resulting from the B+BB and C+BB+PL fits to the ${10}^{-5}$ synthetic spectra has only negative values; this is possible because those two models are not nested. Since the two compared models are not nested, the ${\rm{\Delta }}{\mathrm{Cstat}}_{{{\rm{M}}}_{1}-{{\rm{M}}}_{2}}$ distribution does not respect Wilk's theorem (i.e., the ${\rm{\Delta }}{\mathrm{Cstat}}_{{{\rm{M}}}_{1}-{{\rm{M}}}_{2}}$ distribution does not converge to a ${\chi }^{2}$ distribution with 7 − 6 = 1 dof). Among the 10⁵ synthetic spectra, we did not encounter a single case for which ${\rm{\Delta }}{\mathrm{Cstat}}_{{{\rm{M}}}_{1}-{{\rm{M}}}_{2}}$ is larger than or equal to the ${\rm{\Delta }}{\mathrm{Cstat}}_{{{\rm{M}}}_{1}-{{\rm{M}}}_{2}}$ value resulting from the B+BB and C+BB+PL fits to the real data (i.e., the maximum value of the ${\rm{\Delta }}{\mathrm{Cstat}}_{{{\rm{M}}}_{1}-{{\rm{M}}}_{2}}$ distribution resulting from the simulation is $\lt \lt$ 508 − 495 = 13). Moreover, the ${\rm{\Delta }}{\mathrm{Cstat}}_{{{\rm{M}}}_{1}-{{\rm{M}}}_{2}}$ distribution has a very steep slope so it is extremely unlikely to get a ${\rm{\Delta }}{\mathrm{Cstat}}_{{{\rm{M}}}_{1}-{{\rm{M}}}_{2}}$ value even close to the observed one (i.e., 13). We can, therefore, confidently state that Test 1 is conclusive.

We now present the results of the log-likelihood statistic test but using C+BB+PL with the parameter values resulting from the C+BB+PL fit to the real data as the null hypothesis. (i.e., Test 6). The resulting ${\rm{\Delta }}{\mathrm{Cstat}}_{{{\rm{M}}}_{1}-{{\rm{M}}}_{2}}$ distribution only has positive values and the observed ${\rm{\Delta }}{\mathrm{Cstat}}_{{{\rm{M}}}_{1}-{{\rm{M}}}_{2}}$ (i.e., 13) is well within the 1σ of the distribution and very close to its most probable value. We can, therefore, confidently state that Test 6 is inconclusive.

Based on our previous descriptions of the test results, we can confidently conclude in this case that C+BB+PL is a "better description of the data" than B+BB and that in addition C+BB+PL is "statistically significantly better" than B+BB.

2. Comparison of the C+BB and C+BB+PL fits to the GBM data alone in the fine-time spectra of GRB 080916C.

Here we describe the overall results, but the significance depends on the time interval. However, we always observed the same trend, which increases the significance of the overall results compared to the significance obtained for each time-interval individually.

C+BB and C+BB+PL are nested-like models and C+BB+PL has two more free parameters than C+BB; in the following, we will consider that C+BB and C+BB+PL correspond, respectively, to the M₁ and M₂ models from the description of the various tests presented earlier in this appendix.

In the fine time intervals, the Cstat values resulting from the C+BB and C+BB+PL fits to the real data are usually very similar and Test 1 is typically inconclusive. Similarly, if the true underlying model is C+BB+PL with the parameters resulting from the C+BB+PL fit to the real data then we would also expect the value of ${\rm{\Delta }}{\mathrm{Cstat}}_{{{\rm{M}}}_{1}-{{\rm{M}}}_{2}}$ computed from the C+BB and C+BB+PL fits to the real data; therefore, Test 6 is also typically inconclusive.

We can often conclude that (i) if the true underlying model was C+BB with the parameter values resulting from the C+BB fit to the real data, then the measured Cstat values obtained when fitting C+BB (i.e., Test 4—goodness of fit) and C+BB+PL (i.e., Test 5) to the real data are much higher than expected, and (ii) if the true underlying model was C+BB+PL with the parameter values resulting from the C+BB+PL fit to the real data, then the measured Cstat values obtained when fitting C+BB (i.e., Test 9) and C+BB+PL (i.e., Test 10—goodness of fit) to the real data are typical of what we would expect. (i.e., within the 1σ confidence region of the Cstat distributions). We can then conclude that Tests 4 and 5 are conclusive while Tests 9 and 10 are inconclusive.

Finally, we also usually observe that (i) if the true underlying model is C+BB, it is likely that we obtain the measured parameters resulting from the C+BB fit to the real data (i.e., Test 3 inconclusive) but it is unlikely that we obtain the measured parameters resulting from the C+BB+PL fit to the real data (i.e., Test 2 conclusive). Conversely, if the true underlying model is C+BB+PL, it is likely that we obtain the measured parameters resulting from the C+BB+PL and C+BB fits to the real data (i.e., Tests 7 and 8 inconclusive, respectively).

Therefore, Tests 1 and 6 are inconclusive, but Tests 3, 7, 8, 9, and 10 are inconclusive and Tests 4 and 5 are conclusive so we can conclude that C+BB+PL is overall a "better description of the data."

Let us consider the specific case of the time interval lasting from T₀+8.5 s to T₀+9.5 s in GRB 080916C. (see Table 6 of Appendix C.) The C+BB and C+BB+PL fits to the real data result in the same Cstat values. (i.e., 526 for 472 and 470 dof for C+BB and C+BB+PL, respectively.) The result of Tests 1 and 6 are clearly inconclusive because C+BB and C+BB+PL are nested models. It is, therefore, not possible to conclude from Tests 1 and 6 solely that C+BB+PL is a better description of the data than C+BB. However, Test 4 indicates that if the true underlying model was C+BB, it would be extremely unlikely (i.e., <10⁻⁵) to get a Cstat value beyond 526 when fitting C+BB to the data; indeed, the Cstat distribution peaks narrowly around 485 when fitting C+BB to the synthetic spectra. The same conclusion is drawn for Test 5. In addition, if C+BB were the true underlying model, it would be very likely that we measure the spectral parameter values resulting from the C+BB fit to the real data (i.e., Test 3 inconclusive), but it would be extremely unlikely that we measure the spectral parameter values resulting from the C+BB+PL fit to the real data (i.e., Test 2 conclusive). This is particularly true for the α value of C and the value of the amplitude of the additional PL of the C+BB+PL model; indeed, the values obtained when fitting C+BB+PL to the real data are ∼−0.60 and $\gg 0$ for α and the amplitude, respectively, the values of α and of the amplitude resulting from the C+BB+PL fits to the C+BB synthetic spectra peak at ∼−1.4 and ∼0, respectively. None of the 10⁵ synthetic spectra based on the C+BB null-hypothesis result in parameter values compatible with the observed ones when fitting to C+BB+PL. Conversely, if the true underlying model was C+BB+PL, it would be very likely to obtain the observed Cstat values resulting from the C+BB and C+BB+PL fits to the real data. (i.e., 526—Tests 6, 9 and 10 are inconclusive). It would also be very likely to measure the spectral parameters resulting from the C+BB and C+BB+PL fits to the real data and this is especially true for α and the amplitude of the additional PL (i.e., Tests 7 and 8 are conclusive). We can then conclude that C+BB+PL is a "better description of the data than C+BB."

The values of the parameters resulting from the time-integrated spectral analysis of GRBs 080916C and 090926A as presented in Section 4 are reported in Tables 2 and 3, respectively.

The values of the parameters resulting from the coarse-time analysis of GRBs 080916C and 090926A as presented in Section 5 are reported in Tables 4 and 5, respectively.

The values of the parameters resulting from the fine-time analysis of GRBs 080916C and 090926A as presented in Section 6 are reported in Tables 6 and 7, respectively.

Table 2.  Parameters of the Various Tested Models Resulting from the Time-integrated Spectral Analysis of GRB 080916C from T₀–0.1 to T₀+71 s with their Asymmetrical 1σ Uncertainties

Models	Standard Model			Additional Model				EAC						Cstat/dof
	C or Band			BB	PL, C or Band
Parameters	E ${}_{\mathrm{peak}}$	α	β	kT	α	β	${E}_{0}$	n0/b0	n3/b0	n4/b0	L/b0	F/b0	B/b0
	(keV)			(keV)			(MeV)
b0+n0+n3+n4
Band	451	−1.03	-2.18	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	666/473
	±25	±0.02	±0.08	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+BB	1321	−1.30	−2.41	45.0	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	592/471
	${}_{-243}^{+366}$	±0.03	${}_{-0.30}^{+0.17}$	±1.6	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+PL	269	−0.35	−2.03	⋯	−1.84	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	612/471
	${}_{-20}^{+24}$	±0.16	±0.06	⋯	${}_{-0.13}^{+0.08}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+BB+PL	866	−1.06	−2.47	41.3	−1.77	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	587/469
	${}_{-178}^{+261}$	±0.15	${}_{-0.98}^{+0.25}$	±2.3	${}_{-0.51}^{+0.10}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
C+BB	1858	−1.33	⋯	44.9	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	597/472
	${}_{-295}^{+330}$	±0.02	⋯	±1.4	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
C+PL	362	−0.63	⋯	⋯	−1.63	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	650/472
	${}_{-21}^{+23}$	±0.09	⋯	⋯	±0.02	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
C+BB+PL	912	−1.04	⋯	41.7	−1.67	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	589/470
	${}_{-179}^{+307}$	±0.15	⋯	±2.4	±0.05	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
b0+n0+n3+n4 + EAC
Band	458	−1.03	−2.18	⋯	⋯	⋯	⋯	0.99	0.98	0.98	⋯	⋯	⋯	665/470
	±27	±0.02	±0.08	⋯	⋯	⋯	⋯	±0.03	±0.03	±0.03	⋯	⋯	⋯
B+BB	1311	−1.31	−2.43	44.8	⋯	⋯	⋯	1.07	1.05	1.06	⋯	⋯	⋯	588/468
	${}_{-238}^{+357}$	±0.03	${}_{-0.32}^{+0.17}$	±1.5	⋯	⋯	⋯	±0.04	±0.03	±0.03	⋯	⋯	⋯
B+PL	259	−0.30	−2.03	⋯	−1.84	⋯	⋯	1.05	1.04	1.04	⋯	⋯	⋯	609/468
	${}_{-20}^{+23}$	±0.17	±0.06	⋯	${}_{-0.11}^{+0.08}$	⋯	⋯	±0.03	±0.03	±0.03	⋯	⋯	⋯
B+BB+PL	898	−1.09	−2.47	41.7	−1.78	⋯	⋯	1.06	1.05	1.05	⋯	⋯	⋯	584/466
	${}_{-190}^{+290}$	±0.15	${}_{-1.13}^{+0.23}$	±2.3	${}_{-0.65}^{+0.11}$	⋯	⋯	±0.04	±0.04	±0.04	⋯	⋯	⋯
C+BB	1817	−1.34	⋯	44.7	⋯	⋯	⋯	1.07	1.06	1.06	⋯	⋯	⋯	594/469
	${}_{-289}^{+319}$	±0.02	⋯	±1.3	⋯	⋯	⋯	±0.03	±0.03	±0.03	⋯	⋯	⋯
C+PL	366	−0.64	⋯	⋯	−1.63	⋯	⋯	1.00	0.98	0.99	⋯	⋯	⋯	649/472
	±24	±0.09	⋯	⋯	±0.02	⋯	⋯	±0.03	±0.03	±0.03	⋯	⋯	⋯
C+BB+PL	945	−1.08	⋯	42.0	−1.67	⋯	⋯	1.06	1.05	1.05	⋯	⋯	⋯	586/467
	${}_{-194}^{+323}$	±0.15	⋯	±2.3	±0.05	⋯	⋯	±0.04	±0.03	±0.03	⋯	⋯	⋯
b0+n0+n3+n4+LLE+Front+Back
Band	459	−1.04	−2.19	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	728/541
	${}_{-21}^{+23}$	±0.01	±0.01	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+BB	1289	−1.30	−2.27	45.0	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	650/539
	${}_{-194}^{+228}$	±0.02	±0.03	±1.6	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+PL	333	−0.68	−2.17	⋯	−2.12	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	685/539
	±20	±0.08	±0.01	⋯	${}_{-0.24}^{+0.11}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+BB+PL	1026	−1.18	−2.25	42.2	−2.04	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	646/537
	${}_{-187}^{+280}$	±0.08	${}_{-0.06}^{+0.03}$	±2.2	${}_{-0.85}^{+0.14}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+BB+C	1042	−1.18	−2.27	42.2	−1.97	⋯	26640	⋯	⋯	⋯	⋯	⋯	⋯	646/536
	${}_{-200}^{+308}$	±0.09	±0.13	±2.2	${}_{-0.82}^{+0.11}$	⋯	${}_{-6450}^{+28200}$	⋯	⋯	⋯	⋯	⋯	⋯
b0+n0+n3+n4+LLE+Front+Back + EAC
Band	452	−1.03	−2.19	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.92	1.16	1.16	722/538
	±23	±0.01	±0.05	⋯	⋯	⋯	⋯	⋯	⋯	⋯	${}_{-0.20}^{+0.23}$	${}_{-0.36}^{+0.49}$	${}_{-0.35}^{+0.46}$
B+BB	1188	−1.29	−2.24	45.2	⋯	⋯	⋯	⋯	⋯	⋯	1.02	1.18	1.18	649/536
	${}_{-184}^{+241}$	±0.02	±0.06	±1.6	⋯	⋯	⋯	⋯	⋯	⋯	${}_{-0.19}^{+0.23}$	${}_{-0.35}^{+0.47}$	${}_{-0.34}^{+0.45}$
B+PL	302	−0.55	−2.08	⋯	−1.99	⋯	⋯	⋯	⋯	⋯	1.34	2.17	2.13	673/536
	±21	±0.11	±0.05	⋯	${}_{-0.16}^{+0.09}$	⋯	⋯	⋯	⋯	⋯	${}_{-0.26}^{+0.30}$	${}_{-0.62}^{+0.46}$	${}_{-0.46}^{+0.78}$
B+BB+PL	914	−1.12	−2.34	41.6	−1.91	⋯	⋯	⋯	⋯	⋯	0.77	0.92	0.91	643/534
	${}_{-184}^{+245}$	±0.10	${}_{-0.17}^{+0.14}$	±2.3	${}_{-0.13}^{+0.05}$	⋯	⋯	⋯	⋯	⋯	${}_{-0.26}^{+0.30}$	${}_{-0.37}^{+0.52}$	${}_{-0.30}^{+0.50}$
B+BB+C	975	−1.13	−2.45	41.8	−1.87	⋯	12100.0	⋯	⋯	⋯	0.67	0.85	0.84	641/533
	${}_{-200}^{+286}$	±0.10	${}_{-0.22}^{+0.16}$	±2.3	±0.06	⋯	${}_{-6450}^{+28200}$	⋯	⋯	⋯	${}_{-0.24}^{+0.29}$	${}_{-0.36}^{+0.53}$	${}_{-0.35}^{+0.50}$

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Table 3.  Parameters of the Various Tested Models Resulting from the Time-integrated Spectral Analysis of GRB 090926A from T₀ s to T₀+20 s with their Asymmetrical 1σ Uncertainties

Models	Standard Model			Additional Model				EAC							Cstat/dof
	C or Band			BB	PL, C or Band
Parameters	${E}_{\mathrm{peak}}$	α	β	kT	α	β	${E}_{0}$	b1/b0	n3/b0	n6/b0	n7/b0	L/b0	F/b0	B/b0
	(keV)			(keV)			(MeV)
b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV
Band	294	−0.75	−2.44	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1138/566
	±5	±0.01	±0.03	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+BB	450	−1.01	−2.58	39.4	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	999/564
	±16	±0.02	±0.06	±0.8	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+PL	259	−0.51	−2.37	⋯	−2.02	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1057/564
	±5	±0.04	±0.03	⋯	${}_{-0.24}^{+0.13}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+BB+PL	436	−0.96	−3.37	37.3	−1.62	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	974/562
	±15	±0.05	${}_{-0.68}^{+0.29}$	±1.1	${}_{-0.04}^{+0.06}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
C+BB+PL	441	−0.90	⋯	37.2	−1.61	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	976/563
	±13	±0.05	⋯	±1.0	${}_{-0.03}^{+0.05}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
C+BB+C	425	−0.90	⋯	37.6	−1.51	⋯	21.050	⋯	⋯	⋯	⋯	⋯	⋯	⋯	967/562
	±14	±0.06	⋯	±1.2	${}_{-0.05}^{+0.09}$	⋯	${}_{-6.440}^{+12.800}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯
b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + EAC
Band	294	−0.75	−2.44	⋯	⋯	⋯	⋯	1.02	1.03	0.99	0.99	⋯	⋯	⋯	1078/562
	±5	±0.01	±0.03	⋯	⋯	⋯	⋯	±0.07	±0.07	±0.07	±0.07	⋯	⋯	⋯
B+BB	475	−1.02	−2.52	38.9	⋯	⋯	⋯	1.05	0.96	0.92	0.92	⋯	⋯	⋯	937/560
	±22	±0.02	±0.07	±0.7	⋯	⋯	⋯	±0.07	±0.07	±0.07	±0.07	⋯	⋯	⋯
B+PL	257	−0.51	−2.33	⋯	−2.00	⋯	⋯	0.99	0.96	0.92	0.92	⋯	⋯	⋯	1000/560
	±6	±0.04	±0.04	⋯	${}_{-0.24}^{+0.11}$	⋯	⋯	±0.07	±0.07	±0.07	±0.07	⋯	⋯	⋯
B+BB+PL	443	−0.92	−3.40	37.3	−1.60	⋯	⋯	1.10	1.08	1.04	1.03	⋯	⋯	⋯	916/558
	±20	±0.05	${}_{-0.75}^{+0.33}$	±1.1	${}_{-0.05}^{+0.07}$	⋯	⋯	±0.07	±0.08	±0.08	±0.08	⋯	⋯	⋯
C+BB+PL	446	−0.91	⋯	37.1	−1.60	⋯	⋯	1.11	1.09	1.05	1.04	⋯	⋯	⋯	918/559
	±18	±0.04	⋯	±1.0	${}_{-0.04}^{+0.05}$	⋯	⋯	±0.07	±0.08	±0.08	±0.08	⋯	⋯	⋯
C+BB+C	432	−0.91	⋯	37.6	−1.50	⋯	21.130	1.10	1.07	1.03	1.03	⋯	⋯	⋯	909/558
	±20	±0.06	⋯	±1.2	${}_{-0.06}^{+0.15}$	⋯	${}_{-6.560}^{+12.900}$	±0.07	±0.08	±0.08	±0.08	⋯	⋯	⋯
b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + LLE + Front + Back
Band	280	−0.73	−2.29	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1234/633
	±4	±0.01	±0.01	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+BB	405	−0.99	−2.28	40.9	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1100/631
	±18	±0.02	±0.01	±0.8	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+PL	251	−0.46	−2.31	⋯	−1.95	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1121/631
	±4	±0.03	±0.02	⋯	${}_{-0.05}^{+0.03}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+C	254	−0.46	−2.38	⋯	−1.85	⋯	1319	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1110/630
	±5	±0.03	±0.03	⋯	±0.02	⋯	${}_{-503}^{+948}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+BB+PL	370	−0.84	−2.34	38.0	−1.88	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1091/629
	±23	±0.06	±0.03	±1.5	±0.04	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+BB+C	415	−0.87	−2.68	36.9	−1.73	⋯	789.600	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1056/628
	±18	±0.04	${}_{-0.13}^{+0.10}$	±1.1	±0.03	⋯	${}_{-145.000}^{+207.000}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯
B+BB+B2	430	−0.90	<−4	37.6	−1.53	−2.29	33.140	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1016/627
	±14	±0.05	⋯	±1.1	${}_{-0.06}^{+0.09}$	±0.05	${}_{-9.600}^{+22.400}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯
C+BB+C	451	−0.86	⋯	36.0	−1.73	⋯	617.7	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1084/629
	±11	±0.03	⋯	±0.9	±0.02	⋯	${}_{-82.9}^{+104.0}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯
C+BB+B2	430	−0.90	⋯	37.6	−1.53	−2.29	33.770	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1016/628
	±14	±0.05	⋯	±1.1	${}_{-0.05}^{+0.09}$	±0.05	${}_{-10.200}^{+15.600}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯
b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + LLE + Front + Back + EAC
Band	293	−0.75	−2.42	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.47	0.39	0.41	1189/630
	±4	±0.01	±0.03	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	±0.06	±0.07	±0.07
B+BB	439	−1.00	−2.49	39.7	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.38	0.28	0.30	1056/628
	±17	±0.02	±0.04	±0.8	⋯	⋯	⋯	⋯	⋯	⋯	⋯	±0.07	±0.08	±0.08
B+PL	259	−0.52	−2.36	⋯	−2.05	⋯	⋯	⋯	⋯	⋯	⋯	0.64	0.62	0.64	1105/628
	±5	±0.04	±0.03	⋯	${}_{-0.16}^{+0.08}$	⋯	⋯	⋯	⋯	⋯	⋯	±0.08	${}_{-0.11}^{+0.15}$	${}_{-0.11}^{+0.15}$
B+C	258	−0.49	−2.39	⋯	−1.90	⋯	2564	⋯	⋯	⋯	⋯	0.74	0.84	0.87	1102/627
	±5	±0.03	±0.03	⋯	${}_{-0.08}^{+0.05}$	⋯	${}_{-1140}^{+4330}$	⋯	⋯	⋯	⋯	${}_{-0.11}^{+0.14}$	${}_{-0.21}^{+0.29}$	${}_{-0.21}^{+0.29}$
B+BB+PL	425	−0.93	−2.55	37.8	−1.94	⋯	⋯	⋯	⋯	⋯	⋯	0.35	0.31	0.33	1049/626
	±19	±0.04	±0.06	±1.1	±0.05	⋯	⋯	⋯	⋯	⋯	⋯	±0.07	±0.10	±0.10
B+BB+C	430	−0.88	−3.20	36.8	−1.66	⋯	1098	⋯	⋯	⋯	⋯	1.46	2.86	2.91	1029/625
	±13	±0.04	${}_{-0.47}^{+0.27}$	±1.0	±0.04	⋯	${}_{-212}^{+301}$	⋯	⋯	⋯	⋯	±0.38	${}_{-0.87}^{+0.96}$	${}_{-0.89}^{+0.94}$
B+BB+B2	425	−0.90	<−10	37.6	−1.51	−2.27	20.030	⋯	⋯	⋯	⋯	0.83	0.88	0.91	1014/624
	±15	±0.06	⋯	±1.2	${}^{\pm }0.07$	±0.08	${}_{-8.040}^{+19.600}$	⋯	⋯	⋯	⋯	${}_{-0.17}^{+0.21}$	${}_{-0.25}^{+0.32}$	${}_{-0.25}^{+0.32}$
C+BB+C	439	−0.87	⋯	36.7	−1.63	⋯	1041	⋯	⋯	⋯	⋯	1.92	3.88	3.93	1032/626
	±12	±0.04	⋯	±1.0	±0.03	⋯	${}_{-194}^{+273}$	⋯	⋯	⋯	⋯	±0.28	${}_{-0.73}^{+0.88}$	${}_{-0.71}^{+0.84}$
C+BB+B2	425	−0.90	⋯	37.6	−1.52	−2.27	20.580	⋯	⋯	⋯	⋯	0.82	0.86	0.90	1014/625
	±15	±0.05	⋯	±1.2	${}_{-0.05}^{+0.09}$	±0.08	${}_{-8.410}^{+17.100}$	⋯	⋯	⋯	⋯	${}_{-0.15}^{+0.21}$	${}_{-0.23}^{+0.32}$	${}_{-0.23}^{+0.32}$

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Table 4.  Parameters of the Various Tested Models Resulting from the Coarse-time Spectral Analysis of GRB 080916C with their Asymmetrical 1σ Uncertainties

Time from T0		Models	Standard Model			Additional Model			EAC				Cstat/dof
			C or Band			BB	PL or C
Tstart	Tstop	Parameters	${E}_{\mathrm{peak}}$	α	β	kT	α	${E}_{0}$	n0/b0	n3/b0	n4/b0	L/b0
(s)	(s)		(keV)			(keV)		(MeV)
b0+n0+n3+n4
−0.1	+4.3	Band	454±38	−0.69±0.04	−2.25 ${}_{-0.16}^{+0.12}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	505/473
		B+BB	${1192}_{-184}^{+214}$	−0.97±0.04	<−5	39.8±2.6	⋯	⋯	⋯	⋯	⋯	⋯	477/471
		C+BB	${1189}_{-182}^{+215}$	−0.97±0.04	⋯	39.8±2.7	⋯	⋯	⋯	⋯	⋯	⋯	477/472
b0+n0+n3+n4 + EAC
		Band	459±42	−0.69±0.04	−2.26 ${}_{-0.16}^{+0.12}$	⋯	⋯	⋯	1.03±0.06	0.98±0.05	0.98±0.05	⋯	502/470
		B+BB	${1183}_{-182}^{+210}$	−0.97±0.04	<−5	40.0±2.6	⋯	⋯	1.08±0.06	1.03±0.05	1.03±0.05	⋯	573/468
		C+BB	${1180}_{-180}^{+212}$	−0.97±0.04	⋯	40.0±2.6	⋯	⋯	1.08±0.06	1.03±0.05	1.03±0.05	⋯	573/469
b0+n0+n3+n4+LLE
		Band	496±33	−0.72±0.03	−2.67±0.08	⋯	⋯	⋯	⋯	⋯	⋯	⋯	524/484
		B+BB	${1069}_{-176}^{+221}$	−0.95±0.05	−2.95 ${}_{-0.15}^{+0.12}$	39.4±2.8	⋯	⋯	⋯	⋯	⋯	⋯	493/482
		C+BB	${1193}_{-186}^{+217}$	−0.97±0.04	⋯	39.8±2.7	⋯	⋯	⋯	⋯	⋯	⋯	493/483
b0+n0+n3+n4+LLE + EAC
		Band	451±38	−0.69±0.04	−2.23 ${}_{-0.14}^{+0.11}$	⋯	⋯	⋯	⋯	⋯	⋯	${7.8}_{-3.9}^{+6.6}$	517/483
		B+BB	${1051}_{-193}^{+226}$	−0.94±0.05	−2.80 ${}_{-0.48}^{+0.32}$	39.4±2.8	⋯	⋯	⋯	⋯	⋯	${1.58}_{-1.17}^{+2.97}$	493/481
		C+BB	${1190}_{-181}^{+220}$	−0.97 ${}_{-0.12}^{+0.04}$	⋯	39.8±2.7	⋯	⋯	⋯	⋯	⋯	${1.58}_{-1.10}^{+3.00}$	493/483
b0+n0+n3+n4
+4.3	+7.5	Band	${1291}_{-250}^{+309}$	−1.15±0.03	−2.01 ${}_{-0.13}^{+0.10}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	519/473
		B+BB	${2464}_{-574}^{+670}$	−1.23±0.03	−2.10 ${}_{-0.18}^{+0.13}$	${44.4}_{-5.9}^{+6.7}$	⋯	⋯	⋯	⋯	⋯	⋯	508/471
		B+PL	${967}_{-178}^{+247}$	−0.97±0.13	<−5	⋯	−1.47 ${}_{-0.05}^{+0.10}$	⋯	⋯	⋯	⋯	⋯	517/471
		C+BB+PL	${1554}_{-268}^{+331}$	−0.70±0.10	⋯	39.4±3.8	−1.56±0.04	⋯	⋯	⋯	⋯	⋯	495/470
b0+n0+n3+n4 + EAC
		Band	${1299}_{-266}^{+337}$	−1.15±0.03	−2.01 ${}_{-0.13}^{+0.10}$	⋯	⋯	⋯	1.06±0.07	0.96±0.06	1.02±0.06	⋯	511/470
		B+BB	${2383}_{-549}^{+647}$	−1.24±0.03	−2.12 ${}_{-0.18}^{+0.13}$	${44.1}_{-5.3}^{+5.8}$	⋯	⋯	1.13±0.08	1.04±0.07	1.10±0.07	⋯	500/468
		B+PL	${990}_{-185}^{+259}$	−0.97±0.13	<−5	⋯	−1.47 ${}_{-0.05}^{+0.10}$	⋯	1.05±0.07	0.95±0.06	1.01±0.06	⋯	510/468
		C+BB+PL	${1520}_{-256}^{+315}$	−0.70±0.10	⋯	39.5±3.5	−1.58±0.04	⋯	1.14±0.08	1.04±0.07	1.10±0.07	⋯	487/467
b0+n0+n3+n4+LLE
		Band	${1400}_{-243}^{+312}$	−1.16±0.03	−2.08±0.04	⋯	⋯	⋯	⋯	⋯	⋯	⋯	546/484
		B+BB	${2474}_{-459}^{+790}$	−1.23±0.03	−2.12 ${}_{-0.06}^{+0.04}$	${44.1}_{-5.7}^{+6.7}$	⋯	⋯	⋯	⋯	⋯	⋯	534/482
		B+C	${1092}_{-281}^{+536}$	−1.03±0.15	<−2.5	⋯	−1.48±0.05	${104.3}_{-46.7}^{+36.8}$	⋯	⋯	⋯	⋯	538/481
		B+BB+C	${1655}_{-287}^{+318}$	−0.70±0.10	<−2.5	39.3±3.6	−1.58 ${}_{-0.06}^{+0.02}$	${126.8}_{-27.1}^{+53.5}$	⋯	⋯	⋯	⋯	515/479
		C+C2	${1044}_{-196}^{+286}$	−0.99±0.13	⋯	⋯	−1.48 ${}_{-0.05}^{+0.12}$	${98.54}_{-17.6}^{+24.3}$	⋯	⋯	⋯	⋯	539/482
		C+BB+PL	${1786}_{-305}^{+365}$	−0.70±0.10	⋯	39.1±3.3	−1.67±0.01	⋯	⋯	⋯	⋯	⋯	543/481
		C+BB+C	${1608}_{-275}^{+337}$	−0.70±0.10	⋯	39.3±3.7	−1.58±0.02	${117.3}_{-21.9}^{+33.2}$	⋯	⋯	⋯	⋯	516/480
b0+n0+n3+n4+LLE + EAC
		Band	${1418}_{-247}^{+315}$	−1.16±0.03	−2.10±0.10	⋯	⋯	⋯	⋯	⋯	⋯	${0.88}_{-0.26}^{+0.34}$	545/483
		B+BB	${2615}_{-602}^{+618}$	−1.24±0.03	−2.17±0.11	${44.4}_{-5.9}^{+6.4}$	⋯	⋯	⋯	⋯	⋯	${0.91}_{-0.25}^{+0.33}$	534/481
		B+C	${1011}_{-228}^{+250}$	−1.03 ${}_{-0.16}^{+0.18}$	<−5	⋯	−1.42±0.05	${101.0}_{-20.0}^{+26.9}$	⋯	⋯	⋯	1.68±0.38	535/480
		B+BB+C	${1569}_{-295}^{+334}$	−0.70±0.10	<−3	${39.5}_{-3.6}^{+4.0}$	−1.54 ${}_{-0.06}^{+0.04}$	${121.8}_{-22.9}^{+39.7}$	⋯	⋯	⋯	${1.34}_{-0.42}^{+0.44}$	515/478
		C+C	981±200	−1.00±0.17	⋯	⋯	−1.43±0.05	102.6±26.4	⋯	⋯	⋯	${1.69}_{-0.31}^{+0.37}$	535/481
		C+BB+PL	${1739}_{-311}^{+390}$	−0.70±0.10	⋯	39.3±3.5	−1.64 ${}_{-0.14}^{+0.06}$	⋯	⋯	⋯	⋯	${1.27}_{-0.89}^{+0.79}$	543/480
		C+BB+C	${1526}_{-268}^{+335}$	−0.70±0.10	⋯	39.5±3.8	−1.54±0.04	${120.0}_{-22.4}^{+33.4}$	⋯	⋯	⋯	${1.41}_{-0.36}^{+0.41}$	515/479
b0+n0+n3+n4
+7.5	+71.0	Band	413±26	−1.07±0.02	−2.29 ${}_{-0.17}^{+0.11}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	618/473
		B+BB	${967}_{-210}^{+326}$	−1.34±0.04	−2.32 ${}_{-0.29}^{+0.17}$	45.5±2.0	⋯	⋯	⋯	⋯	⋯	⋯	569/471
		B+PL	263±25	−0.40±0.19	−2.16 ${}_{-0.14}^{+0.09}$	⋯	−1.84 ${}_{-0.16}^{+0.09}$	⋯	⋯	⋯	⋯	⋯	579/471
		C+BB	${1633}_{-458}^{+523}$	−1.39±0.03	⋯	44.8±1.7	⋯	⋯	⋯	⋯	⋯	⋯	573/472
		C+C	${1959}_{-371}^{+443}$	−1.42±0.02	⋯	⋯	+0.95±0.22	64±7 keV	⋯	⋯	⋯	⋯	573/471
		C+BB+PL	${681}_{-155}^{+470}$	−1.10 ${}_{-0.27}^{+0.20}$	⋯	42.9±3.4	−1.69 ${}_{-0.07}^{+0.23}$	⋯	⋯	⋯	⋯	⋯	570/470
		C+BB+C	${455}_{-138}^{+215}$	−0.72±0.42	⋯	${41.5}_{-4.4}^{+3.9}$	−1.53±0.08	${7.247}_{-3.220}^{+11.600}$	⋯	⋯	⋯	⋯	567/469
b0+n0+n3+n4 + EAC
		Band	${419}_{-28}^{+31}$	−1.07±0.02	−2.28 ${}_{-0.17}^{+0.11}$	⋯	⋯	⋯	0.98±0.04	0.98±0.04	0.98±0.04	⋯	618/470
		B+BB	${961}_{-208}^{+330}$	−1.35±0.04	−2.34 ${}_{-0.32}^{+0.18}$	45.2±1.9	⋯	⋯	$1.06\pm 0,05$	1.06±0.05	1.06±0.05	⋯	568/468
		B+PL	255±25	−0.36 ${}_{-0.18}^{+0.12}$	−2.16 ${}_{-0.14}^{+0.09}$	⋯	−1.84 ${}_{-0.14}^{+0.09}$	⋯	1.04±0.04	1.04±0.05	1.04±0.04	⋯	578/468
		C+BB	${1595}_{-448}^{+499}$	−1.40±0.04	⋯	44.6±1.6	⋯	⋯	1.06±0.05	1.07±0.05	1.06±0.05	⋯	571/469
		C+C	${1773}_{-329}^{+389}$	−1.42±0.01	⋯	⋯	+1.17±0.23	57±6 keV	1.05±0.05	1.06±0.05	1.05±0.05	⋯	571/468
		C+BB+PL	${714}_{-174}^{+473}$	−1.15 ${}_{-0.16}^{+0.22}$	⋯	43.3±3.1	−1.69 ${}_{-0.07}^{+0.35}$	⋯	1.05±0.05	1.06±0.05	1.05±0.05	⋯	569/467
		C+BB+C	${472}_{-148}^{+242}$	−0.80 ${}_{-0.41}^{+0.81}$	⋯	${42.1}_{-4.1}^{+3.6}$	−1.54 ${}_{-0.09}^{+0.21}$	${7.343}_{-3.380}^{+12.700}$	1.05±0.05	1.06±0.05	1.05±0.05	⋯	566/466
b0+n0+n3+n4+LLE
		Band	402±23	−1.07±0.02	−2.17±0.01	⋯	⋯	⋯	⋯	⋯	⋯	⋯	632/484
		B+BB	${866}_{-171}^{+233}$	−1.33±0.04	−2.19±0.03	46.1±2.1	⋯	⋯	⋯	⋯	⋯	⋯	583/482
		B+PL	${283}_{-16}^{+18}$	−0.54±0.11	−2.20 ${}_{-0.08}^{+0.05}$	⋯	−1.93 ${}_{-0.15}^{+0.06}$	⋯	⋯	⋯	⋯	⋯	592/482
		B+BB+PL	${661}_{-147}^{+294}$	−1.14 ${}_{-0.13}^{+0.16}$	−2.28 ${}_{-0.13}^{+0.11}$	43.1±3.2	−1.84 ${}_{-0.38}^{+0.14}$	⋯	⋯	⋯	⋯	⋯	581/480
		B+BB+C	${691}_{-180}^{+308}$	−1.14±0.19	−2.56 ${}_{-1.11}^{+0.46}$	43.1±3.5	−1.74±0.02	${362.6}_{-121.0}^{+333.0}$	⋯	⋯	⋯	⋯	580/479
		C+C	336±20	−0.66±0.09	⋯	⋯	−1.75±0.02	${247.8}_{-62.5}^{+121.0}$	⋯	⋯	⋯	⋯	608/482
		C+BB+PL	${646}_{-112}^{+193}$	−1.02±0.13	⋯	41.0±2.8	−1.79±0.02	⋯	⋯	⋯	⋯	⋯	593/481
		C+BB+C	${738}_{-176}^{+535}$	−1.15±0.20	⋯	43.2±3.2	−1.71±0.05	${237.9}_{-97.1}^{+121.0}$	⋯	⋯	⋯	⋯	581/480
b0+n0+n3+n4+LLE + EAC
		Band	403±24	−1.07±0.02	−2.18±0.07	⋯	⋯	⋯	⋯	⋯	⋯	${0.96}_{-0.27}^{+0.35}$	632/483
		B+BB	${873}_{-176}^{+246}$	−1.33±0.04	−2.15±0.08	46.0±2.1	⋯	⋯	⋯	⋯	⋯	${1.17}_{-0.33}^{+0.42}$	583/481
		B+PL	271±22	−0.48 ${}_{-0.14}^{+0.09}$	−2.14 ${}_{-0.10}^{+0.08}$	⋯	−1.90 ${}_{-0.14}^{+0.07}$	⋯	⋯	⋯	⋯	${1.37}_{-0.38}^{+0.48}$	591/481
		B+BB+PL	${662}_{-152}^{+307}$	−1.14±0.15	−2.28 ${}_{-0.27}^{+0.17}$	43.1±3.3	−1.84 ${}_{-0.48}^{+0.15}$	⋯	⋯	⋯	⋯	${0.99}_{-0.42}^{+0.47}$	581/479
		B+BB+C	${677}_{-175}^{+300}$	−1.12±0.22	−2.58 ${}_{-1.83}^{+0.38}$	43.1±3.5	−1.73±0.02	${320.7}_{-94.7}^{+223.0}$	⋯	⋯	⋯	${1.15}_{-0.51}^{+0.74}$	580/478
		C+C	306±22	−0.52±0.12	⋯	⋯	−1.67±0.03	${214.6}_{-47.9}^{+81.5}$	⋯	⋯	⋯	${2.27}_{-0.66}^{+0.73}$	604/481
		C+BB+PL	${678}_{-129}^{+162}$	−1.07 ${}_{-0.27}^{+0.11}$	⋯	41.4±3.2	−1.85 ${}_{-0.25}^{+0.05}$	⋯	⋯	⋯	⋯	1.12±0.50	592/480
		C+BB+C	${668}_{-152}^{+420}$	−1.09±0.23	⋯	42.9±3.4	−1.68 ${}_{-0.06}^{+0.16}$	${226.1}_{-60.0}^{+102.0}$	⋯	⋯	⋯	${1.52}_{-0.65}^{+0.71}$	581/479

Download table as:  ASCIITypeset images: 1 2

Table 5.  Parameters of the Various Tested Models Resulting from the Coarse-time Spectral Analysis of GRB 090926A with their Asymmetrical 1σ Uncertainties

Time from T0

Models

Standard Model

Additional Model

EAC

Cstat/dof

C or Band

BB

PL or C

Tstart

Tstop

Parameters

${E}_{\mathrm{peak}}$

α

β

kT

α

${E}_{0}$

b1/b0

n3/b0

n6/b0

n7/b0

L/b0

(s)

(s)

(keV)

(keV)

(MeV)

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV

0.0

+5.0

Band

304±7

−0.45±0.02

−2.32±0.03

⋯

⋯

⋯

⋯

⋯

⋯

⋯

⋯

762/566

B+BB

529±23

−0.74±0.03

−2.68±0.09

36.4±1.2

⋯

⋯

⋯

⋯

⋯

⋯

⋯

695/564

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + EAC

Band

309±7

−0.45±0.02

−2.48±0.06

⋯

⋯

⋯

1.05±0.07

1.35±0.12

1.29±0.12

1.29±0.12

⋯

726/562

B+BB

496±29

−0.71±0.03

−2.74±0.10

36.1±1.4

⋯

⋯

1.10±0.08

1.22±0.11

1.16±0.10

1.16±0.10

⋯

670/560

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + LLE

Band

324±7

−0.48±0.02

−2.48±0.03

⋯

⋯

⋯

⋯

⋯

⋯

⋯

⋯

803/573

B+BB

527±23

−0.74±0.03

−2.66±0.04

36.4±1.3

⋯

⋯

⋯

⋯

⋯

⋯

⋯

706/571

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + LLE + EAC

Band

305±7

−0.45±0.02

−2.32±0.03

⋯

⋯

⋯

⋯

⋯

⋯

⋯

${3.22}_{-0.63}^{+0.81}$

775/572

B+BB

531±23

−0.74±0.03

−2.71±0.09

36.4±1.2

⋯

⋯

⋯

⋯

⋯

⋯

${0.79}_{-0.27}^{+0.36}$

705/570

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV

+5.0

+9.0

Band

347±9

−0.70±0.01

−2.58±0.07

⋯

⋯

⋯

⋯

⋯

⋯

⋯

⋯

757/566

B+BB

443±26

−0.91±0.03

−2.53±0.09

49.4±2.1

⋯

⋯

⋯

⋯

⋯

⋯

⋯

720/564

B+PL

314±12

−0.46±0.05

−2.71 ${}_{-0.14}^{+0.11}$

⋯

−1.70 ${}_{-0.07}^{+0.05}$

⋯

⋯

⋯

⋯

⋯

⋯

724/564

B+BB+PL

${413}_{-20}^{+29}$

−0.73 ${}_{-0.11}^{+0.08}$

−3.10 ${}_{-0.40}^{+0.20}$

44.1±3.6

−1.62 ${}_{-0.05}^{+0.10}$

⋯

⋯

⋯

⋯

⋯

⋯

707/562

C+BB+PL

${439}_{-20}^{+23}$

−0.78±0.08

⋯

44.6±3.0

−1.57 ${}_{-0.05}^{+0.08}$

⋯

⋯

⋯

⋯

⋯

⋯

711/563

C+BB+C

425±26

−0.77±0.11

⋯

45.0±3.4

−1.50 ${}_{-0.07}^{+0.15}$

${25.8}_{-11.9}^{+45.6}$

⋯

⋯

⋯

⋯

⋯

709/562

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + EAC

Band

347±9

−0.70±0.01

−2.43±0.08

⋯

⋯

⋯

0.95±0.10

0.82±0.10

0.78±0.09

0.79±0.09

⋯

732/562

B+BB

${504}_{-39}^{+44}$

−0.95±0.04

−2.41±0.09

46.5±1.8

⋯

⋯

0.94±0.10

0.77±0.09

0.74±0.08

0.74±0.09

⋯

691/560

B+PL

308±12

−0.45±0.05

−2.51 ${}_{-0.17}^{+0.14}$

⋯

−1.73 ${}_{-0.13}^{+0.06}$

⋯

0.94±0.10

0.86±0.11

0.82±0.11

0.83±0.11

⋯

705/560

B+BB+PL

${472}_{-40}^{+53}$

−0.86±0.11

−2.94 ${}_{-0.54}^{+0.29}$

${44.8}_{-3.0}^{+2.7}$

−1.53 ${}_{-0.09}^{+0.54}$

⋯

0.99±0.10

0.85±0.10

0.81±0.10

0.82±0.10

⋯

685/558

C+BB+PL

${498}_{-36}^{+42}$

−0.87±0.08

⋯

44.5±2.3

−1.51 ${}_{-0.07}^{+0.12}$

⋯

1.01±0.11

0.85±0.11

0.81±0.10

0.82±0.10

⋯

688/559

C+BB+C

${484}_{-40}^{+48}$

−0.88±0.10

⋯

45.3±2.5

−1.39 ${}_{-0.12}^{+0.57}$

${24.1}_{-13.8}^{+37.1}$

1.00±0.10

0.85±0.11

0.81±0.10

0.81±0.10

⋯

685/558

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + LLE

Band

322±7

−0.67±0.01

−2.31±0.02

⋯

⋯

⋯

⋯

⋯

⋯

⋯

⋯

804/573

B+BB

401±32

−0.91±0.04

−2.26±0.02

${53.0}_{-2.5}^{+3.4}$

⋯

⋯

⋯

⋯

⋯

⋯

⋯

745/571

B+PL

311±11

−0.46±0.05

−2.65±0.09

⋯

−1.73±0.02

⋯

⋯

⋯

⋯

⋯

⋯

728/571

B+BB+PL

410±25

−0.72±0.08

−2.88 ${}_{-0.20}^{+0.15}$

43.7±3.5

−1.68 ${}_{-0.03}^{+0.06}$

⋯

⋯

⋯

⋯

⋯

⋯

714/569

B+BB+C

414±24

−0.74±0.09

−3.16 ${}_{-0.38}^{+0.23}$

44.2±3.6

−1.59 ${}_{-0.05}^{+0.10}$

${101.6}_{-31.7}^{+76.7}$

⋯

⋯

⋯

⋯

⋯

708/568

C+BB+C

436±23

−0.77±0.08

⋯

44.6±3.0

−1.55 ${}_{-0.05}^{+0.08}$

${69.63}_{-15.2}^{+25.4}$

⋯

⋯

⋯

⋯

⋯

712/569

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + LLE + EAC

Band

345±9

−0.70±0.01

−2.56±0.06

⋯

⋯

⋯

⋯

⋯

⋯

⋯

0.25±0.07

762/572

B+BB

439±27

−0.91±0.03

−2.50±0.07

49.7±2.1

⋯

⋯

⋯

⋯

⋯

⋯

${0.31}_{-0.35}^{+0.10}$

724/570

B+PL

311±11

−0.46±0.05

−2.64±0.12

⋯

−1.74±0.07

⋯

⋯

⋯

⋯

⋯

0.94±0.12

728/570

B+BB+PL

412±23

−0.72±0.08

−3.08 ${}_{-0.32}^{+0.24}$

43.5±3.4

−1.64±0.05

⋯

⋯

⋯

⋯

⋯

${1.50}_{-0.48}^{+0.50}$

713/568

B+BB+C

414±24

−0.74±0.10

−3.21 ${}_{-0.46}^{+1.32}$

44.3±3.6

−1.58 ${}_{-0.06}^{+0.11}$

${111.8}_{-39.6}^{+116.0}$

⋯

⋯

⋯

⋯

${1.14}_{-0.30}^{+0.39}$

708/567

C+BB+C

435±23

−0.78±0.09

⋯

44.9±3.1

−1.54 ${}_{-0.06}^{+0.11}$

${91.33}_{-28.5}^{+66.2}$

⋯

⋯

⋯

⋯

${1.35}_{-0.32}^{+0.41}$

711/568

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV

+9.0

+11.0

Band

254±7

−0.81±0.02

−2.79±0.13

⋯

⋯

⋯

⋯

⋯

⋯

⋯

⋯

640/566

B+BB

331±36

−1.16±0.05

−2.46 ${}_{-0.16}^{+0.11}$

43.4±1.7

⋯

⋯

⋯

⋯

⋯

⋯

⋯

569/564

B+PL

211±7

−0.21±0.08

−2.96 ${}_{-0.20}^{+0.15}$

⋯

−1.82±0.05

⋯

⋯

⋯

⋯

⋯

⋯

560/564

B+BB+PL

313±24

−0.87±0.17

<−13

39.8±2.4

−1.73±0.06

⋯

⋯

⋯

⋯

⋯

⋯

555/562

C+BB+PL

313±23

−0.87±0.15

⋯

39.8±2.4

−1.73 ${}_{-0.04}^{+0.07}$

⋯

⋯

⋯

⋯

⋯

⋯

555/563

C+BB+C

307±26

−0.85±0.17

⋯

39.9±2.5

−1.71 ${}_{-0.06}^{+0.09}$

${77.45}_{-56.10}^{+6290}$

⋯

⋯

⋯

⋯

⋯

555/562

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + EAC

Band

256±7

−0.81±0.02

−3.02 ${}_{-0.06}^{+0.09}$

⋯

⋯

⋯

${1.07}_{-0.17}^{+0.25}$

${1.32}_{-0.30}^{+0.41}$

${1.31}_{-0.30}^{+0.41}$

${1.26}_{-0.29}^{+0.39}$

⋯

625/562

B+BB

${327}_{-35}^{+39}$

−1.15±0.05

−2.46 ${}_{-0.20}^{+0.15}$

43.5±1.8

⋯

⋯

0.91±0.19

0.96±0.23

0.95±0.23

0.92±0.23

⋯

557/560

B+PL

216±9

−0.25±0.10

−3.16 ${}_{-0.49}^{+0.25}$

⋯

−1.84±0.05

⋯

0.89±0.18

${1.12}_{-0.26}^{+0.38}$

${1.12}_{-0.26}^{+0.38}$

${1.07}_{-0.25}^{+0.37}$

⋯

547/560

C+BB+PL

283±21

−0.73±0.15

⋯

39.5±3.2

−1.80±0.06

⋯

1.00±0.22

${1.38}_{-0.32}^{+0.40}$

${1.38}_{-0.32}^{+0.39}$

${1.33}_{-0.30}^{+0.38}$

⋯

540/559

C+BB+C

281±22

−0.72±0.17

⋯

39.5±3.3

−1.78±0.07

${116.8}_{-80.60}^{+246000}$

1.00±0.22

1.38±0.37

1.37±0.37

1.32±0.37

⋯

539/558

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + LLE

Band

230±7

−0.78±0.02

−2.24±0.02

⋯

⋯

⋯

⋯

⋯

⋯

⋯

⋯

748/573

B+BB

162±21

−1.03±0.05

−2.03±0.02

52.4±1.6

⋯

⋯

⋯

⋯

⋯

⋯

⋯

591/571

B+PL

211±6

−0.20±0.08

−3.06±0.16

⋯

−1.81±0.03

⋯

⋯

⋯

⋯

⋯

⋯

564/571

C+BB+PL

313±22

−0.86±0.15

⋯

39.7±2.3

−1.74±0.04

⋯

⋯

⋯

⋯

⋯

⋯

559/569

C+BB+C

311±23

−0.87±0.16

⋯

40.0±2.5

−1.73 ${}_{-0.04}^{+0.08}$

>229.3

⋯

⋯

⋯

⋯

⋯

558/569

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + LLE + EAC

Band

250±8

−0.80±0.02

−2.68±0.10

⋯

⋯

⋯

⋯

⋯

⋯

⋯

0.07±0.03

653/572

B+BB

319±36

−1.16±0.05

−2.38±0.10

43.9±2.0

⋯

⋯

⋯

⋯

⋯

⋯

${0.23}_{-0.08}^{+0.11}$

576/570

B+PL

211±7

−0.20±0.09

−2.98±0.18

⋯

−1.82±0.05

⋯

⋯

⋯

⋯

⋯

${0.81}_{-0.34}^{+0.24}$

564/570

C+BB+PL

312±23

−0.86±0.15

⋯

39.7±2.4

−1.74±0.05

⋯

⋯

⋯

⋯

⋯

${1.05}_{-0.26}^{+0.30}$

559/569

C+BB+C

312±23

−0.87±0.15

⋯

39.9±2.4

−1.73 ${}_{-0.05}^{+0.08}$

>166.8

⋯

⋯

⋯

⋯

${0.91}_{-0.27}^{+0.33}$

558/568

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV

+11.0

+13.0

Band

198±7

−0.80±0.03

−2.46±0.07

⋯

⋯

⋯

⋯

⋯

⋯

⋯

⋯

608/566

B+BB

316±35

−1.14±0.06

−2.54 ${}_{-0.20}^{+0.13}$

31.4±1.3

⋯

⋯

⋯

⋯

⋯

⋯

⋯

584/564

B+PL

174±9

−0.41±0.13

−2.43 ${}_{-0.13}^{+0.09}$

⋯

−1.95 ${}_{-0.33}^{+0.12}$

⋯

⋯

⋯

⋯

⋯

⋯

594/564

C+BB+PL

324±28

−1.10 ${}_{-0.07}^{+0.12}$

⋯

30.8±1.4

−1.61 ${}_{-0.11}^{+0.21}$

⋯

⋯

⋯

⋯

⋯

⋯

582/563

C+BB+C

300±33

−1.08±0.08

⋯

31.0±1.3

−1.5 (fix)

${14.25}_{-6.44}^{+29.00}$

⋯

⋯

⋯

⋯

⋯

580/563

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + EAC

Band

198±8

−0.79±0.03

−2.45±0.11

⋯

⋯

⋯

${1.39}_{-0.30}^{+0.36}$

${1.26}_{-0.28}^{+0.33}$

${1.18}_{-0.25}^{+0.31}$

${1.18}_{-0.26}^{+0.31}$

⋯

585/562

B+BB

${322}_{-35}^{+44}$

−1.14±0.06

−2.55 ${}_{-0.32}^{+0.19}$

31.2±1.3

⋯

⋯

${1.45}_{-0.31}^{+0.41}$

${1.30}_{-0.30}^{+0.40}$

${1.21}_{-0.28}^{+0.37}$

${1.22}_{-0.28}^{+0.37}$

⋯

562/560

B+PL

169±9

−0.37±0.14

−2.35±0.11

⋯

−1.92 ${}_{-0.30}^{+0.12}$

⋯

1.29±0.33

${1.02}_{-0.21}^{+0.26}$

${0.95}_{-0.20}^{+0.24}$

${0.95}_{-0.20}^{+0.24}$

⋯

571/560

C+BB+PL

316±30

−1.08 ${}_{-0.07}^{+0.12}$

⋯

30.6±1.5

−1.64 ${}_{-0.13}^{+0.24}$

⋯

${1.67}_{-0.36}^{+0.47}$

${1.70}_{-0.38}^{+0.52}$

${1.58}_{-0.36}^{+0.49}$

${1.59}_{-0.36}^{+0.49}$

⋯

558/559

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + LLE

Band

183±6

−0.76±0.03

−2.23±0.02

⋯

⋯

⋯

⋯

⋯

⋯

⋯

⋯

640/573

B+BB

${267}_{-39}^{+49}$

−1.12±0.08

−2.17±0.03

${33.2}_{-1.5}^{+2.1}$

⋯

⋯

⋯

⋯

⋯

⋯

⋯

606/571

B+PL

173±9

−0.38±0.13

−2.53±0.11

⋯

−1.81±0.03

⋯

⋯

⋯

⋯

⋯

⋯

602/571

C+BB+PL

${324}_{-24}^{+33}$

−1.04±0.10

⋯

30.2±1.6

−1.72 ${}_{-0.04}^{+0.13}$

⋯

⋯

⋯

⋯

⋯

⋯

591/570

C+BB+C

${320}_{-26}^{+32}$

−1.12 ${}_{-0.06}^{+0.12}$

⋯

31.0±1.5

−1.51 ${}_{-0.16}^{+0.36}$

${84.23}_{-38.70}^{+118.00}$

⋯

⋯

⋯

⋯

⋯

587/569

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + LLE + EAC

Band

197±7

−0.79±0.03

−2.45±0.07

⋯

⋯

⋯

⋯

⋯

⋯

⋯

0.22±0.08

615/572

B+BB

310±36

−1.13±0.05

−2.49±0.13

31.6±1.3

⋯

⋯

⋯

⋯

⋯

⋯

${0.19}_{-0.09}^{+0.13}$

591/570

B+PL

174±9

−0.41±0.12

−2.42±0.10

⋯

−1.95 ${}_{-0.27}^{+0.11}$

⋯

⋯

⋯

⋯

⋯

${0.45}_{-0.18}^{+0.35}$

600/570

C+BB+PL

322±27

−1.07±0.10

⋯

30.5±1.5

−1.66±0.11

⋯

⋯

⋯

⋯

⋯

${1.27}_{-0.48}^{+0.61}$

590/569

C+BB+C

320±30

−1.11 ${}_{-0.07}^{+0.12}$

⋯

30.9±1.5

−1.54 ${}_{-0.14}^{+0.35}$

${77.26}_{-32.2}^{+113.0}$

⋯

⋯

⋯

⋯

${0.85}_{-0.34}^{+0.51}$

587/568

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV

+13.0

+20.0

Band

161±12

−1.10±0.04

−2.33±0.15

⋯

⋯

⋯

⋯

⋯

⋯

⋯

⋯

706/566

B+BB

${241}_{-60}^{+134}$

−1.39±0.10

−2.25 ${}_{-0.77}^{+0.15}$

27.2±2.5

⋯

⋯

⋯

⋯

⋯

⋯

⋯

696/564

B+PL

136±11

−0.53 ${}_{-0.29}^{+0.24}$

−2.34 ${}_{-0.25}^{+0.13}$

⋯

−1.97 ${}_{-0.58}^{+0.13}$

⋯

⋯

⋯

⋯

⋯

⋯

697/564

C+PL

145±11

−0.57 ${}_{-0.22}^{+0.25}$

⋯

⋯

−1.77±0.04

⋯

⋯

⋯

⋯

⋯

⋯

705/565

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + EAC

Band

154±11

−1.08±0.04

−2.17±0.10

⋯

⋯

⋯

${1.65}_{-1.12}^{+3.27}$

${0.63}_{-0.55}^{+0.37}$

${0.63}_{-0.54}^{+0.37}$

${0.61}_{-0.53}^{+0.36}$

⋯

698/562

B+BB

${145}_{-114}^{+124}$

−1.34 ${}_{-0.14}^{+0.32}$

−1.85 ${}_{-0.13}^{+0.18}$

${29.9}_{-3.6}^{+2.7}$

⋯

⋯

${2.12}_{-1.52}^{+5.75}$

${0.49}_{-0.57}^{+0.20}$

${0.49}_{-0.57}^{+0.20}$

${0.47}_{-0.55}^{+0.19}$

⋯

684/560

B+PL

${115}_{-5}^{+9}$

+0.10±0.40

<−7

⋯

−1.65±0.03

⋯

${2.56}_{-1.32}^{+14.5}$

${0.38}_{-0.18}^{+0.11}$

${0.37}_{-0.18}^{+0.11}$

${0.36}_{-0.29}^{+0.11}$

⋯

685/560

C+PL

114±8

+0.11 ${}_{-0.36}^{+0.41}$

⋯

⋯

−1.65±0.03

⋯

${1.95}_{-1.38}^{+10.70}$

${0.29}_{-0.32}^{+0.16}$

${0.29}_{-0.33}^{+0.15}$

${0.28}_{-0.36}^{+0.15}$

⋯

685/561

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + LLE

Band

149±10

−1.08±0.04

−2.10±0.02

⋯

⋯

⋯

⋯

⋯

⋯

⋯

⋯

720/573

B+BB

${159}_{-37}^{+121}$

−1.29 ${}_{-0.16}^{+0.09}$

−2.04±0.04

31.24.7

⋯

⋯

⋯

⋯

⋯

⋯

⋯

704/571

B+PL

133±9

−0.43±0.24

−2.47 ${}_{-0.19}^{+0.14}$

⋯

−1.86±0.03

⋯

⋯

⋯

⋯

⋯

⋯

704/571

C+BB+C

${179}_{-62}^{+40}$

−0.7 (fix)

⋯

${23.2}_{-2.8}^{+3.8}$

−1.80±0.05

${140.7}_{-64.4}^{+623.0}$

⋯

⋯

⋯

⋯

⋯

708/570

b0+b1+n3+n6+n7 with b0 > 756.48 keV and b1 > 609.51 keV + LLE + EAC

Band

${153}_{-9}^{+25}$

−1.08 ${}_{-0.07}^{+0.04}$

−2.22 ${}_{-0.29}^{+0.05}$

⋯

⋯

⋯

⋯

⋯

⋯

⋯

${0.51}_{-0.41}^{+0.13}$

715/572

B+BB

${234}_{-71}^{+84}$

−1.38±0.10

−2.20±0.16

${27.4}_{-2.0}^{+4.0}$

⋯

⋯

⋯

⋯

⋯

⋯

${0.46}_{-0.32}^{+0.39}$

703/570

B+PL

${134}_{-8}^{+12}$

−0.46±0.25

−2.40±0.20

⋯

−1.90 ${}_{-0.32}^{+0.08}$

⋯

⋯

⋯

⋯

⋯

${0.70}_{-0.39}^{+0.60}$

703/570

C+BB+C

${174}_{-19}^{+39}$

−0.7 (fix)

⋯

${23.6}_{-2.9}^{+4.3}$

−1.78±0.05

${207.9}_{-123.0}^{+73.3}$

⋯

⋯

⋯

⋯

${1.31}_{-0.52}^{+0.81}$

708/569

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