The Fermi-GBM Gamma-Ray Burst Spectral Catalog: 10 yr of Data

Fermi GBM consists of 14 detector modules: 12 Sodium Iodide (NaI) detectors, covering the energies 8–1000 keV, and two Bismuth Germanate (BGO) detectors, covering 200 keV–40 MeV (Meegan et al. 2009). The NaI detectors are distributed in four clusters of three detectors on each corner of the spacecraft and oriented in such a way that enables the prime GRB scientific objectives of all-sky coverage and burst localization. The two BGO detectors are positioned on opposite sides of the spacecraft, also for full-sky coverage. The spectroscopy data products each have 128 channels of energy resolution, with CSPEC data being accumulated at fixed time intervals, and time-tagged event (TTE) data recording the time and energy of each count. The TTE data type is the most flexible, since it can be binned arbitrarily in time, and comprises the majority of data used for the analyses in this work. Since 2012 November 27, GBM has been operating in a mode in which TTE data are collected all the time. Previously, TTE data collection was initiated by a trigger, which occurs when the GBM flight software detects a significant rise above a preset threshold in the counting rates in two or more detectors in one of several energy and time ranges. When operating in this mode, TTE preburst data, which is being constantly accumulated in a ring buffer, is frozen and scheduled for transmission to accompany the triggered TTE data. Before the transition to continuous TTE data collection, if the burst was so bright as to fill up the finite-sized TTE ring buffer or the preburst TTE were somehow corrupted, it would be difficult to reconstruct the entire time history and, in such cases, CSPEC or CTIME data could be used.

Data selection is identical to that as described in Gruber et al. (2014). In brief, up to three NaI detectors with observing angles to the source less than 60^◦ are selected, along with the BGO detector that has the smallest observing angle of the burst. For each of these, standard energy ranges that avoid unmodeled effects, such as an electronic roll-off at low energies and high-energy overflow bins are selected. Each data set is binned according to whether the burst is long (1.024 s binning) or short (0.064 s binning), where the boundary between the long or short classes is defined by T₉₀ = 2.0 s. Next, a background model (polynomial in time) is chosen to fit regions of the light curve that bracket the emission interval. Although the individual energy channels are fitted separately, the background model shares the polynomial order (up to 4) chosen by the analyst to best represent the general trend of the nonburst portions of the data. The resulting model is then interpolated over the entire light curve, including the region(s) where the burst is active. The background uncertainty in each energy channel of each time bin is typically dominated by the uncertainties of the fitted temporal model parameters, except at the highest energies, where Poisson errors dominate.

In order to autonomously determine the time bins that comprise the burst source selections, we combine the individual NaI detector rate histories by summing over the selected detectors. This produces a single rate history (count s⁻¹) for each burst, where the source rates are added coherently and the background incoherently, thus improving the source statistics. We do the same with the interpolated background rate histories. We convert each of these into integrated light curves by multiplying each energy channel by the energy bin width and summing over the energy bins. The count rates are converted into counts by multiplying each time bin by the bin width. The signal-to-noise ratio (S/N) for each time bin is calculated by subtracting the background counts from the total counts, and dividing by the square root of the background counts. The source region is determined by those time bins that have a S/N in excess of 3.5σ, relative to the background model. The sum of the rate histories over these (possibly discrete) time bins defines the spectrum for the "fluence" (F) sample. The time bin with the highest S/N selects the spectrum for the "peak flux" (P) sample. The source selections are propagated to each detector's light curve, including that of the BGOs.

The sum of the duration of the selected time bins is defined as the "accumulation time," which serves as a proxy for the duration of the burst, as seen in panel (a) of Figure 1. Notably, this can be modeled as the sum of two Gaussians, much like the more familiar T₉₀ distribution (Koshut et al. 1996). Panel (b) shows the accumulation times for the short GRBs (T₉₀ < 2.0 s, Kouveliotou et al. 1993) as a separate histogram. Interestingly, there is very little overlap of the short GRB distribution (in gray) into the accumulation time distribution of long GRBs. Finally, panel (c) shows that there are substantial differences between T₉₀ and the accumulation time. One difference is that 10% of the burst fluence is omitted in the T₉₀ by design, resulting in a considerable number of bursts that fall below the line of equality (dashed). The other main difference is that the accumulation time omits quiescent portions of the light curve, so many bursts fall above the equality line (accumulation time shorter than T₉₀).

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The data is then joint fit with RMfit ¹⁶ (currently at version 4.3.2, available at the Fermi Science Support Center), using a set of standard model functions (Section 2.4). For a fit statistic, we have chosen a variant of the Cash-statistic likelihood (Cash 1979), called C-Stat in RMfit and pstat in Xspec (Arnaud et al. 2011). C-Stat assumes the background model uncertainty to be negligible, which is a good approximation for the propagated uncertainties of a background model that is interpolated to a time interval much shorter than the intervals the model is based upon. Since the background uncertainties would ordinarily be combined with signal uncertainties using the quadratic sum, the signal always dominates the total uncertainties. One can account for the Gaussian uncertainties in the background correctly by using the pgstat statistic in Xspec. This statistic was not provided by RMfit at the time of the publication of the most recent GBM Spectroscopy Catalog (Gruber et al. 2014).

Performing spectral analysis successfully is highly dependent upon the correct modeling of the detector response matrices. The response matrices in turn are dependent upon the source position, relative to each detector normal. Since the Fermi spacecraft is in constant motion while in sky survey mode (the default mode), the detector response is a function of time. GBM uses OGIP ¹⁷ standard response Flexible Image Transport System (FITS) files for the response matrices. This standard allows for multiple RESPONSE extensions in a single file, to represent a time sequence. For long spectral accumulations (>20 s), each matrix is weighted by the fraction of the total counts for the corresponding period of time covered by the matrix and then the weighted matrices are summed together. The GBM response generator creates a new response matrix for every 2 degrees of slew. These files are distinguished from single-matrix files by the ".rsp2" filename extension. The standard GBM burst data product, however, has only contained the single-matrix files (with extension ".rsp") by default since the launch. Only the bursts longer than about 20 s require rsp2 files, so these were generated on an as-needed basis. Several types of errors were found while this catalog was being generated. The first and most important of these was that the rsp2 files were not systematically updated whenever a burst localization was changed. The improvement in location accuracy could have come from other spacecraft, days after the trigger, for example. Or else, the GBM team may decide to refine the location analysis after discussion of the burst trigger. Given the inherent latency in obtaining a "final" location, it was possible to create an initial set of response files using a localization that was superseded by a refined analysis. We have gone through the entire set of bursts in this catalog to fix this and other errors that had manifested. Where the new matrix files had significant differences, we have redone the spectral analyses. The updated response files are available at the HEASARC website.

For consistency between the several previous editions of GBM (and BATSE) Spectroscopy Catalogs, we chose four spectral models to fit the spectra of GRBs in our sample. All models are formulated in units of photon flux with energy (E) in keV and multiplied by a normalization constant A (photon s⁻¹ cm⁻² keV⁻¹). Both COMP and BAND are parameterized such that the characteristic energy is given as the energy of the peak in the differential ν f_ν spectrum (E_peak). The pivot energy (E_piv) normalizes the model to the energy range under consideration and helps reduce cross-correlation of other parameters. In all cases, E_piv is held fixed at 100 keV.

• 1.  
  PLAW: A single power law, with two free parameters: normalization (A) and power-law index (λ)

  $$\begin{eqnarray}&&{f}_{\mathrm{PLAW}}(E)=A{\left(\displaystyle \frac{E}{{E}_{{piv}}}\right)}^{\lambda }.\end{eqnarray} \tag{ 1 }$$
• 2.  
  COMP: An exponentially attenuated power law ("comptonized"), with normalization (A), low-energy power-law index (α), and characteristic energy (E_peak)

  $$\begin{eqnarray}&&{f}_{\mathrm{COMP}}(E)=A{\left(\displaystyle \frac{E}{{E}_{\mathrm{piv}}}\right)}^{\alpha }\exp \left[-\displaystyle \frac{(\alpha +2)E}{{E}_{\mathrm{peak}}}\right].\end{eqnarray} \tag{ 2 }$$
• 3.  
  BAND: The Band GRB function, with normalization (A), low-energy power-law index (α), high-energy power-law index (β), and characteristic energy (E_peak)

  $$\begin{eqnarray}{f}_{\mathrm{BAND}}(E)=A\left\{\begin{array}{ll}{\left(\displaystyle \frac{E}{100\mathrm{keV}}\right)}^{\alpha }\exp \left[-\displaystyle \frac{(\alpha +2)E}{{E}_{\mathrm{peak}}}\right], & E\geqslant \displaystyle \frac{(\alpha -\beta ){E}_{\mathrm{peak}}}{\alpha +2}\\ {\left(\displaystyle \frac{E}{100\mathrm{keV}}\right)}^{\beta }\exp \left(\beta -\alpha \right){\left[\displaystyle \frac{(\alpha -\beta ){E}_{\mathrm{peak}}}{100\mathrm{keV}(\alpha +2)}\right]}^{\alpha -\beta }, & E\lt \displaystyle \frac{(\alpha -\beta ){E}_{\mathrm{peak}}}{\alpha +2}.\end{array}\right.\end{eqnarray} \tag{ 3 }$$
• 4.  
  SBPL: A smoothly broken power law, with normalization (A), low-energy power-law index (λ₁), high-energy power-law index (λ₂), a characteristic break energy (E_break), and the break scale (Δ), in decades of energy. As in Gruber et al. (2014), we keep the value of Δ fixed at 0.3.

  $$\begin{eqnarray}&&{f}_{\mathrm{SBPL}}(E)=A{\left(\displaystyle \frac{E}{{E}_{\mathrm{piv}}}\right)}^{b}\ {10}^{(a-{a}_{\mathrm{piv}})},\end{eqnarray} \tag{ 4 }$$

  where:

  $$\begin{eqnarray*}\begin{array}{rcl}a & = & m{\rm{\Delta }}\mathrm{ln}\left(\displaystyle \frac{{e}^{q}+{e}^{-q}}{2}\right),\\ {a}_{\mathrm{piv}} & = & m{\rm{\Delta }}\mathrm{ln}\left(\displaystyle \frac{{e}^{{q}_{\mathrm{piv}}}+{e}^{-{q}_{\mathrm{piv}}}}{2}\right),\\ q & = & \displaystyle \frac{\mathrm{log}(E/{E}_{b})}{{\rm{\Delta }}},\\ {q}_{\mathrm{piv}} & = & \displaystyle \frac{\mathrm{log}({E}_{\mathrm{piv}}/{E}_{b})}{{\rm{\Delta }}},\\ m & = & \displaystyle \frac{{\lambda }_{2}-{\lambda }_{1}}{2},\\ \mathrm{and}\,b & = & \displaystyle \frac{{\lambda }_{2}+{\lambda }_{1}}{2}.\end{array}\end{eqnarray*}$$

The spectral parameter distributions presented in the following section are histogrammed probability density plots, which were created via Monte Carlo sampling from the probability density function (PDF) of each quantity from each GRB (Goldstein et al. 2016). In brief, for a quantity of interest from a total of N GRBs in our sample, we first determine the edges of our bins, then we take a sample from each of the N PDFs and place them in corresponding bins. This is done for a number of iterations (typically >1000), randomly sampling from the PDFs and recording the counts in each bin for each iteration. This process creates a PDF for each bin of the histogram, from which we choose the median as the centroid of the bin and the error bars represent the 68% credible interval centered at the median. This Monte Carlo sampling method allows us to represent the underlying distribution more accurately, especially at the extremes of the distribution where a combination of several low-probability densities can produce a finite probability density in the histogram.

We classify fitted burst models as GOOD if the parameter error of all model parameters are within certain limits. We have chosen our threshold such that ≈70% of the parameter uncertainties across the board satisfy the cutoff. Figures 2–4, depicting the cumulative distribution function (CDF) for errors of E_peak, β, and λ₂, respectively, were used as motivation for this approach. Note that for many GRBs there can be several models that qualify as GOOD.

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We simultaneously require the following criteria to be satisfied in order for a model to be considered GOOD:

• 1.  
  Amplitude: Positive relative error <0.44 and Negative relative error <0.83.
• 2.  
  Low-energy Index: Positive error <0.37 and Negative error <0.51.
• 3.  
  High-energy Index: Positive error <1.0 and Negative error <0.53.
• 4.  
  E_peak: Positive relative error <0.35 and Negative relative error <0.43.
• 5.  
  E_break: Positive relative error <1.0 and Negative relative error <1.5.

We also identify a BEST model in order to determine which of the GOOD models is the best representation of the burst emission. In addition to the aforesaid constraints for the GOOD sample, we compare the differences in C-Stat (ΔC-Stat) per degree of freedom between the various models. The idea of the BEST parameter sample is to obtain the best estimate of the observed properties of a GRB. Besides the necessity of having constrained parameters, already required for the GOOD sample, we compare the difference in C-Stat (ΔC-Stat) per degree of freedom between the various models in order to assess if a statistically more complex model, i.e., a model with more free-fit parameters, is preferred over a simpler model. If the ΔC-Stat observed in the data exceeds a critical value (ΔC-Stat_crit), then the statistically more complex model is preferred. Following the analysis done in Gruber et al. (2014), we use ΔC-Stat_crit = 8.58 for PLAW versus COMP and ΔC-Stat_crit = 11.83 for COMP versus BAND. Since BAND and SBPL have the same number of degrees of freedom, the model with the lower C-Stat was preferred among them. Applying these criteria, the number of bursts that classify as GOOD and BEST for each model can be seen in Table 1, alongside a comparison with the previous spectral catalog (Gruber et al. 2014).

The time-integrated spectral distributions depict the overall emission properties and do not take into account any spectral evolution. Figure 5 shows the E_peak/Fluence ratio (in units of area; Goldstein et al. 2010) distribution for all COMP GOOD F spectral fits, with short GRBs highlighted in gray. This "energy ratio" plot further highlights the robustness of bimodality observed in GRB duration. The low-energy indices α or λ₁ (from COMP, BAND, or SBPL, as appropriate), as shown in Figure 6(a), are distributed about a −1.1 power law typical of most GRBs. About 17% of the BEST low-energy indices (Figure 6(b)) have α > − 2/3, violating the synchrotron "line of death" (Preece et al. 1998), while 82% of the indices have α > − 3/2, violating the synchrotron slow-cooling limit (Cohen et al. 1997). The distribution of high-energy indices β or λ₂ (from BAND or SBPL) in Figure 6(c), peak at a slope of about −2.1 and have a long tail toward steeper indices. The very steep high-energy indices indicate that the spectrum of these GRBs closely mimic a COMP model, which is equivalent to a BAND function with a high-energy index of − ∞ . Figure 7 shows the difference between the time-integrated low- and high-energy spectral indices, ΔS = (α − β). This quantity is useful as the synchrotron shock model (SSM; Baring 2006) makes predictions of this value in a number of cases and the power-law index, p, of the electron distribution can be inferred from ΔS.

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The GOOD and BEST E_peak distributions (shown in Figures 6(e), (f)) generally peak around 150–200 keV and cover just over two orders of magnitude, which is consistent with previous findings (Goldstein et al. 2012; Gruber et al. 2014). As discussed in Kaneko et al. (2006), although the SBPL is parameterized with E_b , the E_peak can be derived from the functional form. We have calculated the E_peak for all bursts with a low-energy index shallower than −2 and a high-energy index steeper than −2. The value of E_peak can strongly affect the measurement of the low-energy index of the spectrum, as shown in Figure 8. A general trend appears to show that spectra with smaller E_peak values also have smaller values of the low-energy power-law index. Asymmetric uncertainties for the SBPL E_peak have not been calculated for this catalog.

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The peak flux spectrum depicts the brightest portion of each burst, on a fixed timescale of 1.024 s for long GRBs and 64 ms for short GRBs. The time bin with the highest significance is chosen for the peak flux (P) sample. The low-energy indices, shown in Figure 9(a), have a median value of about −1.3 and show a bimodal distribution. This is due to the fact that most GRBs of the P sample are best fit by the PLAW function, as less photon fluence accumulation leads to a decrease in S/N. About 22% of the BEST low-energy indices (Figure 9(b)) have α > − 2/3, violating the synchrotron "line of death," while 70% of the indices have α > − 3/2, violating the synchrotron cooling limit; both of these are significantly larger percentages than those from the F spectra. The high-energy indices in Figure 9(c) peak at about −2.3 and again have a long tail toward steeper indices. The lower number of GOOD spectral fits compared to the F spectra is likely due to the poorer statistics resulting from shorter integration times. Shown in Figure 10 is the ΔS distribution for the P spectra, which suffers a deficit in values compared to the F spectral fits largely due to the inability of the data to sufficiently constrain the high-energy power-law index.

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In Figures 9(e), (f), we show the E_peak distribution for the P spectra. The E_peak distribution for the BEST sample peaks at around 250 keV and covers just over two orders of magnitude, which is consistent with previous findings (Goldstein et al. 2012; Gruber et al. 2014). It should be noted that the data over the short timescales in the P spectra do not often favor either the BAND or the SBPL model, resulting in large errors on parameters. Figure 11 shows a correlation between the ${E}_{peak}$ and low-energy power-law index and its uncertainty, which is similar to the one seen in the ${F}$ spectral fits. Table 2 is instructive to identify the similarities and differences in parameter values between the two types of spectra in this catalog. Table 3 compares the parameter values we obtained to a few similar studies. The divergent methodologies and samples used in each study must be taken into consideration while viewing these results.

Table 2. The Median Parameter Values and the 68% CL of the Distribution of the GOOD Sample

Model	Low-energy	High-energy	Epeak	Ebreak	Photon Flux	Energy Flux
	Index	Index	(keV)	(keV)	(photons s−1 cm−2)	(10−7 erg s−1 cm−2)
				Fluence Spectra
PLAW	$-{1.55}_{-0.20}^{+0.18}$	⋯	⋯	⋯	${2.54}_{-1.14}^{+3.98}$	${3.42}_{-1.51}^{+7.38}$
COMP	$-{0.93}_{-0.31}^{+0.39}$	⋯	${191}_{-97}^{+309}$	⋯	${2.62}_{-1.19}^{+4.23}$	${3.06}_{-1.62}^{+8.61}$
BAND	$-{0.84}_{-0.26}^{+0.31}$	$-{2.29}_{-0.42}^{+0.29}$	${159}_{-72}^{+178}$	${111}_{-41}^{+120}$	${3.46}_{-1.68}^{+4.98}$	${4.76}_{-2.56}^{+8.27}$
SBPL	$-{1.00}_{-0.26}^{+0.28}$	$-{2.32}_{-0.48}^{+0.32}$	${161}_{-84}^{+237}$	${112}_{-58}^{+115}$	${3.19}_{-1.53}^{+4.41}$	${4.15}_{-2.12}^{+8.39}$
BEST	$-{1.08}_{-0.44}^{+0.45}$	$-{2.20}_{-0.29}^{+0.26}$	${180}_{-88}^{+307}$	${107}_{-49}^{+88}$	${2.37}_{-1.05}^{+3.83}$	${2.94}_{-1.39}^{+7.90}$
				Peak Flux Spectra
PLAW	$-{1.50}_{-0.20}^{+0.17}$	⋯	⋯	⋯	${4.81}_{-2.73}^{+9.52}$	${7.36}_{-4.34}^{+16.29}$
COMP	$-{0.69}_{-0.30}^{+0.37}$	⋯	${242}_{-127}^{+338}$	⋯	${8.68}_{-4.76}^{+13.79}$	${13.25}_{-8.17}^{+36.61}$
BAND	$-{0.57}_{-0.27}^{+0.33}$	$-{2.39}_{-0.37}^{+0.29}$	${222}_{-100}^{+248}$	${149}_{-55}^{+176}$	${16.31}_{-8.43}^{+27.66}$	${27.37}_{-17.09}^{+60.25}$
SBPL	$-{0.78}_{-0.24}^{+0.27}$	$-{2.40}_{-0.49}^{+0.31}$	${214}_{-105}^{+255}$	${140}_{-58}^{+140}$	${13.49}_{-7.69}^{+21.35}$	${22.56}_{-12.75}^{+50.34}$
BEST	$-{1.30}_{-0.33}^{+0.77}$	$-{2.34}_{-0.36}^{+0.28}$	${233}_{-117}^{+316}$	${163}_{-65}^{+156}$	${4.62}_{-2.55}^{+8.90}$	${6.46}_{-3.52}^{+17.82}$

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Table 3. The Median Parameter Values and the 68% CL of the BEST Model Fits

Data Set	Low-energy	High-energy	Epeak	Ebreak	Photon Flux	Energy Flux
	Index	Index	(keV)	(keV)	(photons s−1 cm−2)	(10−7 erg s−1 cm−2)
			Fluence Spectra
This Catalog BEST	$-{1.08}_{-0.44}^{+0.45}$	$-{2.20}_{-0.29}^{+0.26}$	${180}_{-88}^{+307}$	${107}_{-49}^{+88}$	${2.37}_{-1.05}^{+3.83}$	${2.94}_{-1.39}^{+7.90}$
Gruber et al. (2014)	$-{1.08}_{-0.44}^{+0.43}$	$-{2.14}_{-0.37}^{+0.27}$	${196}_{-100}^{+336}$	${103}_{-63}^{+129}$	${2.38}_{-1.05}^{+3.68}$	${3.03}_{-1.40}^{+7.41}$
Goldstein et al. (2012)	$-{1.05}_{-0.45}^{+0.44}$	$-{2.25}_{-0.73}^{+0.34}$	${205}_{-121}^{+359}$	${123}_{-80}^{+240}$	${2.92}_{-1.31}^{+3.96}$	${4.03}_{-2.13}^{+9.38}$
Kaneko et al. (2006)	$-{1.14}_{-0.22}^{+0.20}$	$-{2.33}_{-0.26}^{+0.24}$	${251}_{-68}^{+122}$	${204}_{-56}^{+76}$	⋯	⋯
			Peak Flux Spectra
This Catalog BEST	$-{1.30}_{-0.33}^{+0.77}$	$-{2.34}_{-0.36}^{+0.28}$	${233}_{-117}^{+316}$	${163}_{-65}^{+156}$	${4.62}_{-2.55}^{+8.90}$	${6.46}_{-3.52}^{+17.82}$
Gruber et al. (2014)	$-{1.32}_{-0.33}^{+0.74}$	$-{2.24}_{-0.38}^{+0.26}$	${261}_{-130}^{+364}$	${133}_{-39}^{+349}$	${4.57}_{-2.49}^{+8.82}$	${6.49}_{-3.46}^{+17.52}$
Goldstein et al. (2012)	$-{1.12}_{-0.50}^{+0.61}$	$-{2.27}_{-0.50}^{+0.44}$	${223}_{-126}^{+352}$	${172}_{-100}^{+254}$	${5.39}_{-2.87}^{+10.18}$	${8.35}_{-4.98}^{+22.61}$
Nava et al. (2011)	$(-{0.56}_{-0.37}^{+0.40})$ a	$-{2.39}_{-0.62}^{+0.23}$	${225}_{-122}^{+391}$	⋯	⋯	${13.5}_{-10.1}^{+79.8}$
Kaneko et al. (2006)	$-{1.02}_{-0.28}^{+0.26}$	$-{2.33}_{-0.31}^{+0.26}$	${281}_{-99}^{+139}$	${205}_{-55}^{+72}$	⋯	⋯

Note.

^a Low-energy index of the peak flux spectra with curved function only.

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Over the ten years of operations covered in this catalog, GBM triggered on 395 short GRBs, 17% of the total number of bursts. The idea that short GRBs and long GRBs represent two distinct populations was bolstered by the comparison between their hardness ratios (Kouveliotou et al. 1993; Bhat et al. 2016). Short GRBs are significantly harder, as determined by the ratio of the counts in two broad energy bands (25–100, 100–300 keV) (Kouveliotou et al. 1993). Spectral fit parameters should reflect this dichotomy in hardness in two ways. First, the median values for E_peak should be significantly different, with the higher value being associated with short bursts. Second, a low-energy power law index that is higher than another (e.g., −1 versus −2) is said to be harder, as a positive uptick requires an increase in higher-energy photons, all other things being equal. Here, we can verify both of these by comparing the median fitted spectral parameters between short and long bursts in Table 4. This is in agreement with results from early on in the mission (Nava et al. 2011).

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Table 4. The Median Parameter Values and the 68% CL for all Long and Short GRBs

		Long GRBs			Short GRBs
Model	Low-energy Index	High-energy Index	Epeak (keV)	Low-energy Index	High-energy Index	Epeak (keV)
Fluence Spectra
PLAW	$-{1.58}_{-0.18}^{+0.15}$	⋯	⋯	$-{1.35}_{-0.16}^{+0.09}$	⋯	⋯
COMP	$-{1.01}_{-0.35}^{+0.39}$	⋯	${205}_{-109}^{+374}$	$-{0.59}_{-0.34}^{+0.49}$	⋯	${534}_{-312}^{+660}$
BAND	$-{0.84}_{-0.35}^{+0.48}$	$-{2.41}_{-3.96}^{+0.49}$	${144}_{-85}^{+229}$	$-{0.46}_{-0.37}^{+0.68}$	$-{2.98}_{-6.56}^{+1.04}$	${413}_{-263}^{+651}$
SBPL	$-{1.03}_{-0.34}^{+0.47}$	$-{2.40}_{-1.49}^{+0.48}$	${160}_{-86}^{+255}$	$-{0.66}_{-0.30}^{+0.53}$	$-{2.79}_{-6.04}^{+0.87}$	${476}_{-280}^{+480}$
Peak Flux Spectra
PLAW	$-{1.52}_{-0.20}^{+0.14}$	⋯	⋯	$-{1.33}_{-0.15}^{+0.11}$	⋯	⋯
COMP	$-{0.80}_{-0.43}^{+0.45}$	⋯	${224}_{-124}^{+435}$	$-{0.39}_{-0.46}^{+0.63}$	⋯	${532}_{-316}^{+732}$
BAND	$-{0.64}_{-0.42}^{+0.63}$	$-{2.69}_{-5.23}^{+0.75}$	${166}_{-98}^{+295}$	$-{0.22}_{-0.51}^{+0.92}$	$-{4.18}_{-8.54}^{+2.26}$	${426}_{-279}^{+635}$
SBPL	$-{0.85}_{-0.42}^{+0.55}$	$-{2.62}_{-5.84}^{+0.68}$	${181}_{-96}^{+272}$	$-{0.49}_{-0.40}^{+0.81}$	$-{3.26}_{-12.9}^{+1.34}$	${415}_{-236}^{+552}$

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The hard nature of short bursts is even more dramatic when considering the distributions of the fitted parameters. Figures 12 and 13 compare E_peak between long and short GRBs for the fluence and peak flux spectral fits, respectively. In order to improve the sample size of the short burst population, we present fits from the total ensemble of bursts; one for each of the three models that have an energy-related parameter (COMP, BAND, and SBPL). Similarly, Figures 14 and 15 compare the low-energy indices between long and short GRBs for all four models (including PLAW) from the fluence and peak flux spectral fits, respectively. Clearly, it is highly improbable that the long and short distributions are the same for the E_peak and low-energy index distributions. Figures 16 and 17 compare the high-energy indices between long and short GRBs for BAND and SBPL from the fluence and peak flux spectral fits, respectively. Although the presence of a high-energy power-law component is a clear signal of "hardness," the fitted index itself seems to be invariant between the two classes of GRBs.

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We use the BAND model as discussed in Section 2.4, which provides the peak energy (E_peak), amplitude, and the indices of α and β as well as the measured fluence in the Fermi-GBM bandpass (10–1000 keV). For the GRBs with a known redshift (z), we calculated the k-corrected, isotropic-equivalent gamma-ray energy of the GRB in the comoving bolometric bandpass of 1 keV–10 MeV and the estimated uncertainty on the prompt energy release.

In order to calculate the k-correction, we use the fluence in the Fermi-GBM bandpass and expand the spectral model given in (Band et al. 1993) to the comoving bolometric bandpass of 1 keV–10 MeV (Bloom et al. 2001). The k-value is defined as

$$\begin{eqnarray}&&k=\displaystyle \frac{{S}_{\left[\tfrac{{E}_{1}}{1+z},\tfrac{{E}_{2}}{1+z}\right]}}{{S}_{[{e}_{1},{e}_{2}]}}\end{eqnarray} \tag{ 5 }$$

where S is the fluence for a range of given energies, E₁ = 1 keV, E₂ = 10 MeV, e₁ = 10 keV, e₂ = 1000 keV. The isotropic energy can then be calculated by

$$\begin{eqnarray}&&{E}_{\mathrm{iso}}=\displaystyle \frac{4\pi {D}_{{\ell }}^{2}}{1+z}{S}_{\mathrm{obs}}k\end{eqnarray} \tag{ 6 }$$

with D_ℓ being the luminosity distance and S_obs being the observed fluence. The distribution of the E_iso is presented in Figure 19 and shows that the long GRBs have E_iso centered around 10⁵³ erg. The limited number of short GRBs with the redshift does not provide much insight into their distribution but the outlier in this plot is GRB 170817A, which was found to have a lower redshift than the other GRBs.

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The peak rest-frame energy (E^rest _peak) is determined by ${E}_{\mathrm{peak}}^{\mathrm{rest}}={E}_{\mathrm{peak}}^{\mathrm{obs}}(1+z)$, where E^obs _peak is the observed peak energy from the F spectra. The distribution of E^rest _peak is shown in Figure 19 to span from 10 keV–3000 keV. Neither short nor long GRBs appear to exhibit a preference for a particular rest frame E_peak but there does appear to be a separate group of GRBs with E_peak > 1 MeV. Using the spectra of the brightest time bin, we can determine the isotropic luminosity for the GRBs with redshift using

$$\begin{eqnarray}&&{L}_{{\rm{i}}{\rm{s}}{\rm{o}}}=4\pi {D}_{\ell }^{2}{S}_{[E1,E2]}.\end{eqnarray} \tag{ 7 }$$

The distribution of L_iso, shown in Figure 19, shows that neither the short nor the long GRBs have preference for L_iso. However, GRB 170817A has a L_iso a few order of magnitudes lower than the rest. The E_iso, E_rest ^peak, and L_iso with their uncertainties are shown in Table 5.

The COMP spectral model (Section 2.4) is also used to determine the E_iso, E^rest _peak, and L_iso in a similar way to Section 4.1 and using the spectral shape from Equation (2). The values and uncertainties for E_iso, E^rest _peak, and L_isoare presented in Table 5.

The distribution of the E_iso in Figure 20, shows that the short and long GRBs do not have a preference for the E_iso. However, GRB 170817A is again at the low end of the E_iso distribution. The distribution of E^rest _peak using the COMP model spans from 10–3000 keV with a group of GRBs with E_peak > 1 MeV, similar to the BAND model. The L_iso distribution for the COMP model also shows no preference for a value and GRB 170817A has a lower value than the rest of the GRBs.

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