A Spectral Analysis of Fermi-LLE Gamma-Ray Bursts

As we all know, gamma-ray bursts (GRBs) are the brightest explosions in the universe. It is generally believed that they are from magnetars or black holes resulting from the mergers of compact binaries (NS–NS or BH–NS) or the death of massive stars (Colgate 1974; Paczynski 1986; Eichler et al. 1989; Narayan et al. 1992; Woosley 1993; MacFadyen & Woosley 1999; Woosley & Bloom 2006; Kumar & Zhang 2015). The Band function (Band et al. 1993) can fit the gamma-ray burst spectra such as the time-integrated spectra and the time-resolved spectra, in which four parameters are contained: the low-energy power-law index α, the high-energy power-law index β, the peak energy E_p, and the normalized constant. It is proved that these parameters evolve with time instead of remaining constant. Many references, such as Golenetskii et al. (1983), Norris et al. (1986), Kargatis et al. (1994), Bhat et al. (1994), Ford et al. (1995), Crider et al. (1997), Kaneko et al. (2006), Peng et al. (2009) in the pre-Fermi era and Lu et al. (2012), Yu et al. (2016, 2019), Acuner & Ryde (2018), Li (2019) in the Fermi era have shown the evolutional characteristics of α and E_p in the Band function (Band et al. 1993). There are three types for the evolution patterns of peak energy E_p: (i) the "hard-to-soft" trend, when the value of E_p decreases monotonically (Norris et al. 1986; Bhat et al. 1994; Band 1997); (ii) those varying with flux, i.e., E_p will increase/decrease since the flux is increasing/decreasing, called the "flux-tracking" trend (Golenetskii et al. 1983; Ryde & Svensson 1999); (iii) the "soft-to-hard" trend or chaotic evolutions (Laros et al. 1985; Kargatis et al. 1994). Recently, Lu et al. (2012) and Yu et al. (2019) pointed out that the first two patterns are dominant. For the evolution of the low-energy photon index α, it does not show a strong general trend compared with E_p, although it evolves with time instead of remaining constant. However, the physical origin of the evolution patterns in E_p and α is not very clear. On the other hand, the analysis of a large sample of LLE GRBs for the parameters evolution and the parameter correlations is lacking, except for the single burst analysis, such as GRB 131231A in Li et al. (2019), which is a single-pulse burst, and GRB 180720B in Duan & Wang (2019), which is a multi-peaked burst in the prompt light curve.

Furthermore, the launch of the Fermi Space Gamma-ray Telescope (Fermi) in 2008 (Atwood et al. 2009) makes it possible to detect GRBs in a broad energy band both in the prompt emission and the afterglow phase. The Fermi satellite consists of the Gamma-ray Burst Monitor (GBM) and the Large Area Telescope (LAT) with the LAT Low Energy (LLE) detector. The GBM consists of 12 Na i detectors (8–900 keV) and two BGO (200 keV–40 MeV) detectors. Obviously, the energy range in GBM detection is from 8 keV to 40 MeV. The LAT can detect photons with an energy range from 100 MeV to 300 GeV. Moreover, the LLE can collect lower energy gamma-ray photons down to 10 MeV. About 2000 GRBs have been detected by Fermi in the last ten years while fewer of them were detected by Fermi-LAT, with the number just more than 100. In addition, GRBs detected by LLE are fewer than 100 according to the available data at the Fermi Science Support Center (FSSC).³ Ajello et al. (2019) note that only 74 GRBs were co-detected by the GBM and LAT (including LAT-LLE). We call them LLE GRBs. The photons cover eight orders of magnitude in the energy range for LLE bursts.

In this work, after performing a detailed time-resolved spectral analysis of the bright gamma-ray bursts with the detection of Fermi-LLE in the prompt phase, we present the time-resolved spectra around their peak flux, for which they all can be fitted well by the Band function. Then we give the evolution patterns of the peak energy E_p and low-energy spectral index α. The parameter correlations will also be presented in our analysis such as E_p–F, α–F, and E_p–α. Besides, we will make a statistical analysis for whether the low-energy power-law indices α exceed the synchrotron limit ($\alpha =-\tfrac{2}{3}$) given by Preece et al. (1998) in these slices. We will perform a simulation to identify whether the two strong positive correlations α–F and E_p–α for some bursts are intrinsic or artificial.

Up to now, more than one hundred bursts have been co-detected by the Fermi/GBM and LAT, but only 74 GRBs (Ajello et al. 2019) were detected by LLE, which can collect those lower energy gamma-ray photons down to 10 MeV in all of these bursts if there is no omission in our collection. This work makes use of all available LLE bursts observed until 2018 July 20. We remove   a pure blackbody burst GRB 090902B, three extremely bright bursts (GRBs 080916C, 130427A and 160625B) and two long bursts that have been studied in Li et al. (2019) (GRB 131231A) and Duan & Wang (2019) (GRB 180720B) in detail. These two long bursts originate from synchrotron emission in the prompt phase.

We downloaded data from the FSSC as described above. To complete this study, we take RMFIT as the tool for making the time-resolved spectral analysis. We perform a detailed time-resolved spectral analysis by using the TTE event data files of two Na i detectors and the corresponding BGO detector(s) on Fermi/GBM, but the use of LAT and LLE data was abandoned because of their lower impact for peak energy E_p and low-energy spectral index α. The background photon counts were estimated by fitting the light curve before and after the operated burst with a one-order background polynomial model. We selected all of the prompt phase as the source. We take the signal-to-noise ratio (S/N) as 40 in all of the slices for each burst and they all can be well fitted by the Band function (Band et al. 1993). To show the spectral evolution, the sample in our analysis includes only those bursts from which at least five time-resolved spectra can be produced from the data. Based on this, 32 GRBs have been excluded due to the insufficiency of the number of time-resolved spectra. Finally, we get a sample of 36 GRBs by filtering described as above. The reduced χ² has been taken into measuring the goodness of fit (GoF). The χ²/GoF is typically in the range of 0.75–1.5 in each slice.

In our work, we present the Band-fitting spectra for all of the bursts around their peak flux first. For the evolution patterns of α and E_p we will then identify them as "hard-to-soft" (h.t.s.), "soft-to-hard" (s.t.h.), "intensity-tracking" (i.t.), "rough-tracking" (r.t.), "anti-tracking" (a.t.), and "no", which means that it evolves without rule. It is notable that all "-tracking" patterns are based on the evolution of the energy flux. Finally, the statistical analysis of the linear dependence in the parameter correlations such as E_p–F, α–F, and E_p–α will be made by using   Pearson's correlation coefficient r. We also address whether the two observed correlations α–F and E_p–α are intrinsic or artificial by simulation.

The data analysis results have been presented in Tables 1–3, Figures 1–11. Table 1 shows the results of the time-resolved spectral fits at peak flux for all samples. Table 2 shows the results of the time-integrated spectral fits for all samples. The fitting results of the parameter correlations and the spectral evolution patterns of α and E_p have been shown in Table 3; simultaneously we also present the linear-fitting results from simulations for those bursts (23 GRBs) that exhibit a strong positive correlation in α–F and E_p–α correlations in this table. Figure 1 is a histogram of the maximal value of α in the detailed time-resolved spectra for each burst. Figure 2 presents those spectra with the best Band-fitting results around the peak flux for all of our bursts. Figure 3 shows a comparison between our fitting results and the results of the GBM catalog (Gruber et al. 2014; Narayana Bhat et al. 2016) at peak flux. Figure 4 presents a comparison between the histogram of α in the time-integrated spectra in our energy range and the BATSE energy range. Figure 5 shows a comparison between our time-integrated spectral analysis results and the corresponding results of the GBM catalog (Gruber et al. 2014). Figure 6 represents the temporal characteristics of energy flux for all bursts in our sample (the left-hand y-axis), along with the time evolution of E_p and α; both are marked with red stars in the right-hand y-axis. That is to say, Figure 6 shows the spectral evolutions for all of the bursts in our sample. The histograms of E_p  and α obtained by performing a detailed time-resolved spectral analysis have been shown in Figure 7. The correlations such as E_p–F, α–F, and E_p–α obtained from the time-resolved spectra are shown in Figure 8. Figure 9 shows the histograms of Pearson's correlation coefficient from the fitting results of parameter correlations such as E_p–F, α–F, and E_p–α. The last two figures, Figures 10, 11, are the linear-fitting results in α–F and E_p–α correlations from simulation for 23 GRBs.

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Table 3.  Fitting Results of the Parameter Correlations and the Spectral Evolutions of E_p and α

GRB	Detectors	N	Ep–F	α–F	Ep–α	Spectral Evolutions	$\alpha \gt -\tfrac{2}{3}$	α–F	Ep–α
			r	r	r	Ep/α		r(S)	r(S)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
080825C	n9,na,b1	8	0.94	0.70	0.54	h.t.s./r.t.	yes	−0.38	−0.96
090328A	n7,n8,b1	8	0.70	0.93	0.83	h.t.s./i.t.	no	−0.20	−0.86
090626A	n0,n3,b0	20	0.61	0.69	0.01	r.t./r.t.	not all	−0.56	−0.88
090926A	n6,n7,b1	37	0.61	0.67	0.35	r.t./r.t.	not all	−0.36	−0.86
100724B	n0,n1,b0	30	0.59	0.35	−0.08	r.t./r.t.	not all	⋯	⋯
100826A	n7,n8,b1	24	0.93	0.08	−0.01	r.t./r.t.	not all	⋯	⋯
101014A	n6,n7,b1	21	0.86	0.83	0.62	r.t./r.t.	not all	0.28	−0.54
110721A	n6,n9,b1	7	0.62	0.76	0.07	h.t.s./s.t.h. to h.t.s.	no	−0.71	−0.88
120226A	n0,n1,b0	12	0.47	0.73	−0.11	r.t./r.t.	no	−0.57	−0.87
120624B	n1,n2,b0	5	0.52	0.61	0.94	h.t.s./h.t.s.	no	−0.40	−0.80
130502B	n6,n7,b1	25	0.64	0.75	0.24	r.t./r.t.	not all	−0.13	−0.67
130504C	n9,na,b1	29	0.54	0.45	−0.18	r.t./r.t.	no	⋯	⋯
130518A	n3,n7,b0,b1	19	0.61	0.69	0.32	r.t./r.t.	no	−0.71	−0.81
130821A	n6,n9,b1	11	0.67	0.71	−0.06	r.t./r.t.	not all	−0.002	−0.95
131108A	n3,n6,b0,b1	6	0.84	0.77	0.44	s.t.h. to h.t.s./r.t.	not all	−0.14	−0.32
140102A	n7,n9,b1	6	0.89	0.84	0.71	i.t./i.t.	not all	−0.002	−0.93
140206B	n0,n1,b0	23	0.67	0.58	0.38	r.t./r.t.	not all	⋯	⋯
141028A	n6,n9,b1	5	0.91	−0.07	0.18	i.t./h.t.s.	yes	⋯	⋯
150118B	n1,n2,b0	20	0.86	0.50	0.26	r.t./r.t.	not all	⋯	⋯
150202B	n0,n1,b0	7	0.72	−0.48	−0.69	r.t./a.t.	not all	⋯	⋯
150314A	n1,n9,b0,b1	17	0.05	0.95	0.05	no/r.t.	not all	−0.64	−0.89
150403A	n3,n4,b0	9	0.83	0.39	0.01	r.t./r.t.	not all	⋯	⋯
150510A	n0,n1,b0	11	0.56	0.95	0.55	s.t.h. to h.t.s./r.t.+h.t.s.	not all	0.27	−0.86
150627A	n3,n4,b0	39	0.66	0.75	0.59	r.t./r.t.	not all	−0.45	−0.79
150902A	n0,n3,b0	17	0.58	0.85	0.29	r.t./r.t.	not all	−0.68	−0.91
160509A	n0,n3,b0	39	0.46	0.83	0.39	r.t./r.t.	not all	−0.18	−0.96
160816A	n6,n7,b1	10	0.76	0.70	0.27	i.t./r.t.	not all	−0.08	−0.64
160821A	n6,n7,b1	130	0.43	0.81	0.08	r.t./r.t.	no	0.07	−0.72
160905A	n6,n9,b1	12	0.71	0.97	0.73	r.t./r.t.	no	0.65	−0.76
160910A	n1,n5,b0	13	0.83	0.17	−0.06	h.t.s./no	not all	⋯	⋯
170115B	n0,n1,b0	5	0.99	−0.95	−0.97	i.t./a.t.	yes	⋯	⋯
170214A	n0,n1,b0	24	0.30	0.73	−0.18	r.t./r.t.	not all	−0.64	−0.90
170510A	n9,na,b1	7	0.16	0.82	−0.02	no/r.t.	no	−0.43	−0.84
170808B	n1,n5,b0	31	0.81	0.33	0.27	r.t./r.t.	not all	⋯	⋯
171210A	n0,n1,b0	17	0.90	−0.50	−0.56	r.t.+h.t.s./no	not all	⋯	⋯
180305A	n1,n2,b0	8	0.83	−0.31	0.02	i.t./no	yes	⋯	⋯

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3.1. Band-fitting Results at Peak Flux for All of the Bursts

3.2. Evolution Patterns of E_p and α

3.3. Parameter Correlations

3.4. Whether the Two Observed Strong Positive Correlations are Intrinsic or Artificial

In this work, after performing a detailed time-resolved spectral analysis of the bright gamma-ray bursts with the detection of Fermi-LLE in the prompt phase, we presented all the spectra with the best Band-fitting results at peak flux for our bursts. To confirm whether our results are consistent with the Fermi team's results, we compared our results with the Fermi GRB spectral catalog. Then we gave the evolution patterns of the peak energy E_p and low-energy spectral index α. Also, the statistical analysis for whether the low-energy power-law indices α exceed the synchrotron limit were given. Finally, the parameter correlations such as E_p–F, α–F, and E_p–α were also presented in the analysis. To address whether the two observed correlations α–F and E_p–α are intrinsic or artificial, we performed a simulation.

Meanwhile, some interesting phenomena were found in our Fermi-LLE bursts, such as:

• 1.  
  A single Band function is enough to perform the spectral fitting for every burst around their peak flux.
• 2.  
  77.8% of the bursts have an α_max, which is larger than the synchrotron limit ($-\tfrac{2}{3}$) in our bursts.
• 3.  
  As we all know, the typical value of the low-energy photon index α is ∼−1.0 for the time-integrated spectrum, while the typical value of α in our sample is ∼−0.9.
• 4.  
  A good fraction of GRBs follow the "hard-to-soft" trend (about two-thirds), and the rest should be the "flux-tracking" pattern (about one-third) in the literature for E_p evolution. However, it is obvious that the "flux-tracking" pattern is very popular for most of the bursts in our study including "intensity-tracking" (five GRBs) and "rough-tracking" (22 GRBs). The total number is 27, which means that 75% of the bursts exhibit the "flux-tracking" pattern. Additionally, the low-energy photon index α does not show a strong general trend compared with E_p although it also evolves with time instead of remaining constant as seen in the literature. We find that 28 GRBs exhibit a "flux-tracking" pattern, which includes "intensity-tracking" (two GRBs) and "rough-tracking" (26 GRBs) in our study. In brief, 77.8% of our bursts exhibit the "flux-tracking" pattern.
• 5.  
  For the parameter correlations, from Section 3.3, a majority of bursts exhibit a strong (very strong) positive correlation (69.4%) between E_p and F (energy flux). We find that 63.9% of our bursts have a strong (very strong) positive correlation between α and F. But there is no clear behavior in the E_p–α correlation in our sample. Finally, it is noteworthy that a very strong negative correlation has been exhibited both for α–F and E_p–α correlations for GRB 170115B.
• 6.  
  The two observed strong positive correlations (α–F and E_p–α) are artificial in our sample except for GRB 160905A in its α–F correlation.

Over the last fifty years, research in the field of gamma-ray bursts has made a lot of progress, but there are still some open questions (e.g., Zhang 2011, 2018; Dai et al. 2017; Pe'er 2019). One of the questions is about the radiation mechanism in the prompt emission, that debates whether the GRB prompt emission is produced by the synchrotron radiation or the emission from the photosphere (Vereshchagin 2014; Pe'Er & Ryde 2017; Oganesyan et al. 2018, 2019; Ravasio 2019). However, a unified model has not been provided even though the physical models such as the synchrotron model (Zhang et al. 2016) and subphotospheric dissipation model (Ahlgren et al. 2019) have been used to make the spectral fitting.

As we all know, the Band component in most observed gamma-ray burst spectra seems to be thought of as synchrotron in origin. Two possible cases should be considered; the first one is for the internal shock model (Paczynski & Xu 1994; Rees & Meszaros 1994), which invokes a small radius. The second case invokes a large internal magnetic dissipation radius, the so-called Internal-Collision-induced MAgnetic Reconnection and Turbulence (ICMART) model (Zhang & Yan 2011). For the internal shock model, the peak energy ${E}_{p}\propto {L}^{1/2}{\gamma }_{e,{ch}}^{2}{R}^{-1}{\left(1+z\right)}^{-1}$ can be derived from the synchrotron model in Zhang & Mészáros (2002), where L is the "wind" luminosity of the ejecta, ${\gamma }_{e,{ch}}$ is the typical electron Lorentz factor of the emission region, R is the emission radius, and z is the redshift of the burst. Then, the tracking behavior will emerge because of the natural relation of E_p ∝ L^1/2. A hard-to-soft evolution pattern of peak energy E_p is predicted for the ICMART model (Zhang & Yan 2011; Uhm & Zhang 2014). On the other hand, Uhm et al. (2018) also pointed out that the "flux-tracking" behavior could be reproduced within the ICMART model if other factors such as bulk acceleration are taken into account. Furthermore, Zhang et al. (2016) demonstrated that the synchrotron model can reproduce the E_p-tracking pattern through the data analysis for GRB 130606B. Therefore, the "flux-tracking" behavior of E_p can be made with both of these two synchrotron models. In short, a hard-to-soft pattern and tracking behavior of E_p can be reproduced successfully in the synchrotron model.

Meanwhile, the photosphere model can also produce an E_p-tracking pattern and a hard-to-soft pattern of E_p successfully (Deng & Zhang 2014; Meng et al. 2019), but this model predicts a hard-to-soft pattern of α instead of α-tracking behavior. It is difficult to produce the observed α-tracking behavior in this model. On one hand, the predicted α value (α ∼ +0.4) is much harder than that observed (Deng & Zhang 2014). The introduction of a special jet structure is necessary to reproduce a typical α ∼ −1 (Lundman et al. 2013). On the other hand, this model invokes an even smaller emission radius than the internal shock model, so, the contrived conditions from the central engine are needed to reproduce the tracking pattern of α. However, few bursts exhibit a hard-to-soft pattern in our sample. Besides, Ahlgren et al. (2019) used the physical subphotospheric model to fit the Fermi data (including six LLE bursts in our sample; GRBs 090926A, 130518A, 141028A, 150314A, 150403A, 160509A), only 171 out of 634 spectra are accepted (17 out of 135 spectra for the six LLE bursts). As a result, we infer that the great majority of bursts in our sample are dominated by the synchrotron component even though the photosphere component is still not excluded in their prompt phases.

Additionally, the patterns of the peak energy E_p evolution have close connections to the spectral lags (Uhm et al. 2018). In general, the light curves at higher energies peak earlier than those at lower energies, named positive spectral lags. In contrast, for the negative spectral lags, the higher energy emission slightly lags behind the lower energy emission (Uhm & Zhang 2016). Earlier studies in the literature show that only small fraction bursts show negative lags or no spectral lags (Norris et al. 1996, 2000; Liang et al. 2006; Ukwatta et al. 2012). Uhm et al. (2018) studied and provided the connections between the patterns of the E_p evolution and the types of spectral lags (positive or negative lags). According to Uhm et al. (2018), the positive spectral lags can occur if the peak energy exhibits a hard-to-soft evolution pattern, but the negative type cannot occur. When the E_p presents a flux-tracking behavior, both the positive and the negative types of spectral lags can occur. The clue to differentiate between the positive lags and the negative lags for the E_p-tracking pattern comes from the peak location of the flux curve. The peak location of the flux curve slightly lags behind the peak of E_p curve for the former, whereas there is no longer a visible lag between them for the latter (Uhm et al. 2018). Assuming that those bursts that exhibit a hard-to-soft pattern or flux-tracking pattern of peak energy E_p occur as spectral lags, then the positive type of spectral lags will occur at the six bursts that exhibit a hard-to-soft behavior of E_p (GRBs 080825C, 090328A, 110721A, 120624B, 160910A, 171210A). The positive type of spectral lags will also occur at the 12 GRBs because of their peak location of flux curves slightly lags behind their peak of E_p curves (GRBs 090926A, 100826A, 130502B, 130504C, 130518A, 140206B, 150118B, 150627A, 160509A, 160821A, 170214A, 170808B). The negative lags will occur at the rest of the bursts because there is no visible lag between the two peaks (GRBs 090626A, 100724B, 101014A, 120226A, 130821A, 140102A, 141028A, 150202B, 150403A, 160816A, 160905A, 170115B, 180305A).

We thank the anonymous referee for helpful suggestions. We also thank Lei-Ming Du, Zhao-Yang Peng, and Dao-Zhou Wang for their help. We acknowledge the use of the public data from the Fermi data archives. This work is supported by the National Natural Science Foundation of China (grant Nos. 11673006, U1938201, and 11533003), the Guangxi Science Foundation (grant Nos. 2016GXNSFFA380006, 2017AD22006, 2017GXNSFBA198206, and 2018GXNSFGA281007), the One-Hundred-Talents Program of Guangxi colleges, and High level innovation team and outstanding scholar program in Guangxi colleges.