The Second AGILE MCAL Gamma-Ray Burst Catalog: 13 yr of Observations

The AGILE MiniCALorimeter (MCAL, Labanti et al. 2009) is a non-imaging gamma-ray scintillation detector, sensitive in the 400 keV–100 MeV energy range. It is composed of 30 CsI(Tl) scintillator bars (15 × 23 × 375 mm³ each), arranged in two orthogonal layers, providing a total on-axis geometrical area of ∼1400 cm². At both ends of each bar, the readout of the scintillation light is performed by two custom PIN photodiodes (PDs). Although being a segmented detector, MCAL is not capable of localizing GRBs, but it can only allow a rough (>30°) reconstruction of the direction of the incoming photons for some incoming angles. Moreover, the energy released by the incoming radiation produces a track in the detector bars which can be used to discriminate photons from high-energy particles.

MCAL is a self-triggered instrument, whose detection logic is based on the principle that a transient event exceeding a given threshold above the background rate issues an onboard data acquisition. The background count rate depends on the timescale and the energy range, and it is evaluated by using different ratemeters working on different search integration time (SIT) windows and energy ranges. In particular, hardware logics work on short-duration timescales (i.e., 0.293, 1, and 16 ms), whereas software logics work on longer-duration timescales (i.e., 64, 256, 1024, and 8192 ms), and both are evaluated in three energy ranges (low energy, 0.3–1.4 MeV; medium energy, 1.4–3 MeV; high energy, 3–100 MeV). The 0.293 ms, or "sub-ms," search window is a unique feature of MCAL, playing a crucial role in the detection of very short-duration events in the high-energy regime, such as submillisecond terrestrial gamma-ray flashes (TGFs, Marisaldi et al. 2010, 2014). Triggers are issued whenever a given threshold count rate is reached above the background. This threshold depends on the involved SIT duration. Hardware logics, which work on short-duration timescales, adopt a static trigger logic, where the threshold count rate is fixed and independent on the background rate. On the other hand, software logics, which work on longer-duration timescales, adopt a flexible trigger logic, where the threshold count rate varies depending on the background rate. Both static and flexible thresholds can be fully configured onground. Whenever one or more of these logic thresholds are exceeded, MCAL issues a Burst-START condition, starting a data acquisition. When all ratemeters return to a normal background level, MCAL issues a Burst-STOP condition (which can also be forced after a given amount of time, in order to prevent mass memory saturation in the case that a normal background level is not encountered). Burst-START and Burst-STOP conditions determine the duration of the trigger acquisition. MCAL stores data in a cyclic buffer, including pre- and post-burst data acquisitions of time intervals occurring immediately before and after the Burst-START and Burst-STOP: the duration of these acquisitions depend on the triggered logic timescale, as well. It is important to notice that the trigger logic is implemented in the AGILE data-handling unit and its parameters are flexible and fully configurable onground. Energy and time information of each triggered event is sent to telemetry as photon-by-photon, with 2 μs time resolution, in order to limit energy and time binning by only counting statistics.

MCAL can work in two operative modes: GRID mode, in which MCAL acts as a slave to the silicon tracker, issuing triggered data acquisition whenever the silicon tracker detects a signal, and BURST mode, in which MCAL works as an independent self-triggering detector. Both operative modes can work at the same time.

Although not having imaging capabilities, MCAL constitutes a suitable detector for GRB science. First of all, it provides a continuous monitoring of an ideal ∼4π all-sky, only constrained by Earth occultations, not requiring GRBs to lay within a given FoV (as for GRID and SuperAGILE detectors). Moreover, it offers the opportunity to cover the energy range between tens of mega electron volts and 100 MeV, poorly investigated by other space missions. Finally, it can operate as an independent self-triggering detector down to very short-duration timescales (∼300 μs), allowing the detection of very fast transients.

A first MCAL GRB catalog was released by Galli et al. (2013), covering the first two years of the AGILE mission in the so-called "pointing mode." This sample consisted of 84 bursts detected between 2007 and 2009 November, which are included in the present catalog. The AGILE MCAL is part of the third Inter-Planetary Network (IPN), contributing to GRB localization by means of triangulation with other space missions. Moreover, MCAL actively contributed to the LIGO-Virgo O2 (2017) and O3 (2018–2019) observational runs, promptly reacting to GW detections and delivering upper limit fluences in the 0.4–100 MeV energy range on different timescales (Verrecchia et al. 2019).

Data acquired by all AGILE detectors (i.e., Gamma-Ray Imaging Detector, GRID; MiniCALorimeter, MCAL; SuperAGILE, SA; and Anti-Coincidence, AC) are continuously recorded in telemetry, with 0.512 s (for SA) and 1.024 s (for GRID, MCAL, and AC) time resolution, independently on any trigger, to provide a continuous monitoring of the X- and gamma-ray background through orbital phases. Data of all detectors are used to build broad-band energy spectra, stored onboard to be directly telemetered down, one for each detection layer. Although designed to investigate the background modulation, the AGILE RMs clearly detect a large number of high-energy transients, such as GRBs, soft gamma repeaters, and solar flares, and it can work as an independent detectors as well. The coarse time resolution mostly makes RMs serve as a back-up or cross-check to other onboard detectors. From this perspective, MCAL RMs, operating in the same energy range of the MCAL detector, provide a useful tool to validate MCAL detections: long-lasting GRBs may be only partially acquired in MCAL-triggered data acquisitions, and their real duration can be only established by analyzing the associated RM continuous data stream. The AGILE RMs are routinely calibrated, comparing the detected GRBs with count rates and spectra reported for the same events detected by the IPN or other space missions.

We analyzed the AGILE MCAL onboard triggers issued in the period from 2007 to 2020 November. This sample consisted of more than 10⁶ trigger acquisitions, resulting in a total effective exposure time of ∼100 days (corresponding to the sum of all MCAL-triggered data acquisitions). We point out that the majority (>99%) of these triggers are spurious, due to instrumental background noise and charged particles crossing the detector, triggering especially the short-duration hardware timescales (sub-ms, 1, and 16 ms). This is ascribed to the lowering of the onboard trigger thresholds, adopted to enhance the detection sensitivity to short-duration transients and to increase the exposure time. We carried out a cross-check with the list of GRBs reported by the IPN (Hurley et al. 2013; IPN webpage: http://www.ssl.berkeley.edu/ipn3/), which, for the same period, included 4578 bursts detected by various space missions. Given the large number of events to consider, it is important to estimate the chance match probability. The MCAL onboard trigger configuration underwent a number of modifications through the years, making the detection rate not uniform in time, as shown in Figure 1; for this reason, we cannot provide a stable MCAL GRB detection rate, but only a total number of triggers, equal to N_trg = 1047679 events. On the other hand, the IPN burst rate is quite stable since 2007/2008, as the number of space missions devoted to GRB detection remained quite constant: 4578 GRBs have been reported in 13 yr, corresponding to an average detection rate of ∼350 GRBs yr⁻¹, or r_IPN ∼ 1.1 × 10⁻⁵ Hz. GRBs exhibit different time profiles depending on the considered energy range. As a consequence, since MCAL operates in the >400 keV energy range, we expect that the MCAL trigger T₀s do not always coincide with T₀s reported by other space missions, typically more sensitive in the low-energy regime. For that reason, we adopted an a priori δ t = 60 s time window, in order to confidently ensure a match when the same burst is detected by MCAL and reported by IPN. The number of expected chance matches is equal to ${N}_{\exp }^{\mathrm{chance}}={N}_{\mathrm{trg}}\cdot \delta t\cdot {r}_{\mathrm{IPN}}\sim 700$. We point out that this number is underestimated, because we assumed a constant MCAL detection rate, but it is useful to qualitatively evaluate the impact of chance matches on our search. In Section 4, we carry out a visual inspection of MCAL data light curves for all matches retrieved from this cross-search, ending up with 787 events for which no significant signal is detected above the background: their number is compatible with ${N}_{\exp }^{\mathrm{chance}}$ and it represents the number of events that are rejected from the analysis.

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This first search is carried out by using MCAL data. We point out that scientific RMs offer another suitable tool to detect GRBs, even in the hard X-ray energy range, covering about three orders of magnitude in energy: a cross-search of IPN GRBs with the AGILE RMs is very promising and will be treated in a future work.

At the same time, we also performed an independent offline search of GRBs in the MCAL data. To do that, we used the same algorithm used for the AGILE routine data analysis and identification of bursts in MCAL data, to provide prompt communication via GCN Notices and Circulars to the scientific community (https://gcn.gsfc.nasa.gov/agile_mcal.html). The aim of the algorithm is to detect GRBs independently from the MCAL trigger time, carrying out a blind search in the MCAL data stream. It operates on different timescales (16, 32, 64, and 128 ms) and searches for events that can be confidently neglected as statistical fluctuations of the background. A more detailed description of the MCAL GRB algorithm can be found in Ursi et al. (2019).

Long GRBs might not be fully acquired in a single MCAL trigger: GRBs exhibiting more bursting episodes separated in time could trigger a single data acquisition on the first burst, without re-triggering on the following episodes; moreover, long GRBs could be triggered by MCAL on the burst onset, but the available onboard memory may not be sufficient to store the entire GRB data stream. Both these cases would produce incomplete or fragmented detections. In order to check the real duration of the detected GRBs and to verify whether the MCAL acquisition included the whole event, we investigated the corresponding MCAL RM data, which provide a continuous data stream in the same energy range of the MCAL detector, with a 1.024 s time resolution. For each GRB, we therefore studied the MCAL RM light curve in a time interval of T₀ ± 150 s, which allowed us to classify the triggered events as fully or partially acquired in the related onboard data acquisition.

We estimated the T₅₀ and T₉₀ time duration of our GRBs by adopting the algorithm reported by Koshut et al. (1996). An example of the algorithm applied to the short GRB 190606A is shown in Figure 3. The gray region indicates the portion of light curve on which the calculation has been performed: the background rate is evaluated in the red regions, before and after the event, and the burst count rate is evaluated in the blue region. T₅₀ and T₉₀ distributions of the N_full = 394 MCAL GRBs are reported in Figure 4. As N_full includes a certain number of candidates, we discriminate between the sample of confirmed bursts only (i.e., those retrieved in the cross-search with GRB lists, shown as a dashed line) and the sample of confirmed bursts plus our MCAL candidates (shown as a solid line).

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The time distribution of the confirmed sample results in 150 short GRBs and 217 long GRBs, corresponding to about 41% and 59% of the sample, respectively. On the other hand, the time distribution of the total sample results in 173 short GRBs and 221 long GRBs, corresponding to about 44% and 56% of the sample, respectively.

These distributions show a high fraction of short GRBs compared to those reported by other space missions, such as INTEGRAL (Bošnjak et al. 2014), Fermi GBM (Bhat et al. 2016), and Konus-Wind (Svinkin et al. 2016). This difference can be ascribed to several reasons. A first explanation is related to the detection efficiency of MCAL, which is not uniform through the years. Figure 5 shows the distribution in time of the duration of the GRBs triggered by MCAL: blue solid dots represent events fully acquired onboard, for which it was possible to evaluate a T₉₀ duration, whereas hollow dots represent partial or fragmented acquisitions, for which we report an approximate duration estimate based on MCAL RMs data. A depletion of fully acquired long-duration events in recent years can be noticed, due to the different trigger configurations adopted onboard. These changes were carried out to save mass memory during severe telemetry restrictions and to enhance the capability of MCAL to detect short GRBs during the LIGO-Virgo O2 and O3 runs (as short GRBs were the most promising candidates to be electromagnetic counterparts of GW events detected by LV). It is important to remark that this change did not affect the MCAL sensitivity to long GRBs, but only the possibility of fully detecting these transients. The partial data acquisition of long GRBs is not a problem of detection efficiency, but an issue regarding the post-burst duration and mass memory requirements.

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A second explanation is ascribed to a physical reason: MCAL operates above 400 keV and it is therefore more sensitive to GRBs with a hard spectrum, typically short GRBs. MCAL is sensitive in a limited energy range, which does not include the X-ray regime, in which most GRBs emit the largest fraction of their energy. The burst emission above 400 keV may therefore last less with respect to the overall duration of the event in a wider energy range. It is interesting to compare the T₉₀s obtained by MCAL to those provided by Fermi GBM (operative in the 10 keV–40 MeV energy range), for a common data sample. We considered 272 GRBs detected by both MCAL and GBM and compared the corresponding durations obtained by the two detectors, in their energy ranges. As shown in Figure 6, MCAL T₉₀s are slightly shorter than those reported by GBM: this is true especially for long GRBs, which exhibit shorter durations in the MCAL detections, contributing to the populating of the short GRB region. This behavior is even more evident when comparing MCAL data with SA data. GRBs in the SA data usually exhibit longer durations with respect to those reported by MCAL and MCAL RMs. In Figure 7, we show an example of this for GRB 200829A: in the SA RMs data, the event lasts 15.3 ± 0.5 s, whereas in the MCAL RMs data it lasts 8.1 ± 0.5 s. In the plot, the SA count rate is rescaled to better highlight the burst duration difference in the two energy ranges. The energy range in which a burst is detected may affect not only the total time duration observed, but also the T₀: a GRB onset may occur at slightly different times depending on the energy range. We already pointed out such issue in the previous section, when we discussed the necessity of a δ t match window for the cross-check between MCAL triggers and IPN GRBs.

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The shortest-duration GRB detected by MCAL is GRB 090522, with T₅₀ = 0.008 s and T₉₀ = 0.012 s, whereas the longest-duration one is GRB 110820B, with T₅₀ = 128 s and T₉₀ = 156 s. The short GRBs subsample peaks in T₅₀ =0.25 s and T₉₀ = 0.36 s, whereas the long GRBs subsample peaks in T₅₀ = 7.69 s and T₉₀ = 14.77 s.

Table 2 reports the 503 MCAL GRBs, with trigger time in UTC and related information. For each event, we define four flags corresponding to the detections of the scientific RMs of SA, AC, and MCAL (columns RM SA, RM AC, and RM MCAL, respectively) and of the MCAL detector with high time resolution (column MCAL). Each flag characterizes the detection type and is assigned as follows:

• 1.  
  Y, complete detection: events that can be considered for timing and, if localization is available, spectral analysis;
• 2.  
  I, partial detection: events that are not fully acquired in the trigger, providing only partial data collections and incomplete time duration estimates (which are starred in Table 2);
• 3.  
  F, fragmented detection: events whose light curve presents gaps due to saturations of the MCAL memory cyclic buffer, providing only partial detections and incomplete time duration estimates (which are starred in Table 2);
• 4.  
  N, no detection: events that are not detectable above the background rate;
• 5.  
  n, no data available: events that are not covered by data for the time interval under analysis (due to Earth occultations, passages in the South Atlantic Anomaly, or data losses).

Events with MCAL = Y are the N_full = 394 GRBs previously discussed, for which the related onboard acquisitions are complete and for which it was possible to estimate T₅₀ and T₉₀ time durations, as well as the total number of counts released in the MCAL detector. On the other hand, for events with MCAL = I or MCAL = F, we can only provide partial estimates of T_50,90 durations and number of counts, based on the available data. The duration of such events is marked with a star in Table 2. An example of a GRB with all flags equal to Y (i.e., GRB 130606B), fully acquired by MCAL and clearly detected in all RMs data is shown in Figure 8. As for the example in Figure 7, in the MCAL high-energy range, the event exhibits a shorter duration with respect to the softer SA and AC energy range.

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Figure 1 shows how the MCAL detection rate changed in 13 yr. Such variation does not depend only on the trigger configuration changes, but also on other issues undergone by the satellite in its lifetime. Such changes make the MCAL burst sample not uniform. In order to provide more homogenoeus subsets of the MCAL GRBs, we introduce a flag indicating different configurations, running in different periods of the AGILE lifetime, during which the onboard trigger condition and sensitivity to GRBs can be considered stable. In particular, the configuration flag may assume the following values:

• 1.  
  C1 (from 2007 April to 2009 November): the satellite was in its 2 yr nominal phase and fully operational, all passages were served at the ground station, and all triggered GRBs were completely acquired onboard;
• 2.  
  C2 (from 2009 December to 2015 February): after the reaction wheel failure, the satellite started spinning around its Sun-pointing axis, making the detection of long-duration transients more difficult, and requiring onboard configuration changes to test the detector;
• 3.  
  C3 (from 2015 March to 2016 August): the AC veto shield was inhibited to increase the detection of TGFs (Marisaldi et al. 2015), with the drawback effect of an enhancement of the background rate, which required a change of the onboard configuration to make the trigger logics more conservative;
• 4.  
  C4 (from 2016 August, alternating with C3, on demand): a new onboard configuration was introduced, aimed at increasing the sensitivity to short-duration, weak signals (e.g., subthreshold events or GRB precursors), in view of the LIGO-Virgo follow-up campaign. Such configuration is available on demand and it is adopted whenever possible, depending on telemetry requirements; otherwise, the baseline C3 configuration is adopted.

In order to perform spectral analysis of the MCAL GRBs, it is necessary to retrieve the instrument response matrices corresponding to the angle under which the bursts are observed. In order to do this, sky localizations of the events are needed. As MCAL is a non-imaging detector, localizations only can be retrieved from available SuperAGILE detections, or externally, from other space missions, or IPN triangulation. We carried out a search in different databases from various satellites with imaging or localization capabilities: in particular, we adopted the Fermi GBM, Fermi LAT, and Fermi GBM Trigger lists, as well as the Swift BAT and Swift XRT lists (all reported in https://heasarc.gsfc.nasa.gov/W3Browse/), and the INTEGRAL ISGRI list (https://www.isdc.unige.ch/integral/science/grb). We retrieved sky coordinates for 276 of our GRBs, as shown in Figure 9, for which we could simulate the corresponding response matrices. Taking into consideration the available count statistics, spectral analysis was possible for 258 of these bursts.

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4.2.1. Spectral Fit with a Power-law Model

GRB spectra can often be described by means of a Band model, a smooth-joint broken power law (PL) with a peak energy E_p (Band et al. 1993). This value corresponds to the maximum of the ν F ν spectrum: it divides the PL at low energies, described by a low-energy photon index α, from the PL at high energies, described by a high-energy photon index β. Typically, α>−2 and β<−2, in order to ensure the presence of an E_p, which usually ranges in the tens to few hundreds of kilo electron volts energy range, but can extend up to mega electron volt energies for the hardest GRBs. As MCAL operates above 400 keV, we expect that most E_ps will lay outside the MCAL energy range, and that the time-integrated spectrum of most events would only appear as a simple PL with photon index ∼β. As a consequence, we carried out a systematic spectral analysis fitting each spectrum using a single PL model $A\tfrac{{E}^{\beta }}{{E}_{0}}$, where A is a normalization constant and E₀ = 100 keV. The spectral analysis was carried out using the XSpec spectral fitting package (version 12.9.0, Arnaud 1996); as the MCAL average background rate ranges between 200 and 700 Hz, depending on the year and on the orbital position, and since the detected events may exhibit different durations and count rates, we adopted different XSpec statistics to perform the spectral analysis. In particular, all events exhibiting less than 1000 background-subtracted counts have been treated with the XSpec Cstat statistic, a modified version of the Cash statistic used for Poisson data on a Poisson background; in the case of Poisson data over a Gaussian background, we adopted the PGstat statistic. In all other cases, events have been treated using the standard chi statistics. We carried out the spectral analysis in the 0.4–50 MeV energy range.

For ${N}_{\mathrm{full}}^{\mathrm{sp},\mathrm{PL}}=258$ events out of 276 localized bursts, it was possible to perform a spectral analysis with a reliable ${\chi }_{\mathrm{red}}^{2}$ and a β ranging between −4 and 0. These events are reported, with related information, in Table 3. The localizations reported in the table are obtained from different space missions or IPN triangulations and are flagged as SA (AGILE SuperAgile), GBM (Fermi GBM), LAT (Fermi LAT), BAT (Swift BAT), XRT (Swift XRT), INT (INTEGRAL ISGRI), or IPN. The 0.4–50 MeV interval corresponds to a number of MCAL spectral channels N_channels = 79, so that the number of degrees of freedom (dof) is accordingly set for a two-parameter model as N_dof = 75. The average photon index obtained from this systematic analysis is 〈β〉 = −2.3, and the related distribution is shown in Figure 10. The fluxes (90% confidence level) obtained by these spectral fittings range between 3.9 × 10⁻⁷ and 1.0 × 10⁻² erg cm⁻² s⁻¹, whereas the corresponding fluences (90% confidence level) range from 5.5 × 10⁻⁸ and 1.3 × 10⁻² erg cm⁻². GRB fluxes are shown in Figure 11, plotted with respect to the T₉₀, together with the corresponding flux upper limits (ULs, i.e., the minimum detectable fluxes for the MCAL timescales). MCAL ULs depend on many factors: energy range, involved trigger logic timescale (which have different thresholds), spectral shape (hardness) of the event under analysis, onboard configuration (described in Ursi et al. 2019), background rate, and angles θ, ϕ under which the event is observed (and associated response matrix). As the GRB fluxes reported in the figure refer to events that occurred in 13 yr, we cannot report a single UL value for each timescale, as ULs vary from period to period, depending on technical issues undergone by the satellite in the years and related onboard configuration changes. For each trigger timescale, we therefore report the maximum flux ULs obtained for the more constraining and conservative configurations (red arrows and bars), as well as the minimum flux ULs obtained for the best configurations (blue arrows and bars), achieved in the AGILE lifetime. ULs are evaluated in the 0.4–100 MeV energy range, for each logic timescale, simulating GRBs with a PL with a mean 〈β〉 = −2.3. These values are in good agreement with the detected bursts, constraining the region in the flux-T₉₀ parameter space in which GRBs can be detected by MCAL. We point out that ULs refer to the GRB fluxes capable of triggering the MCAL detector, which do not always correspond to events that can be fully acquired onboard and for which a spectral analysis can be performed. As a consequence, especially for the longest-duration timescales, a gap is present between the UL value and the actually detected GRBs. In the fluence distribution represented in Figure 12, a bimodal shape is evident, with a local minimum at ∼(0.5–1) × 10⁻⁵ erg cm⁻², between the short and long GRB populations. The low-fluence regime is mostly constituted by short GRBs, whereas a sharp decrease in the high-fluence (≳5 × 10⁻⁴ erg cm⁻²) events is present, due to the limited number of long GRBs detected by MCAL.

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Table 3. List of ${N}_{\mathrm{full}}^{\mathrm{sp},\mathrm{PL}}=258$ Localized GRBs whose Time-integrated Spectrum is Fitted with a Single Power-law Model in the 0.4–50 MeV Energy Range

NAME	local.	l, b	θ, ϕ	stat.	β	${\chi }_{\mathrm{red}}^{2}$	Flux (0.4–50 MeV)	Fluence (0.4–50 MeV)
	by	(deg)	(deg)			(75 dof)	(erg cm−2 s−1)	(erg cm−2)
GRB071227A	XRT	267.48, −46.75	110.20, 209.59	cstat	$-{2.11}_{-0.35}^{+0.27}$	1.40	(5.36 ± 0.54) E-06	(6.34 ± 0.63) E-06
GRB080212B	IPN	200.76, 27.06	78.51, 209.05	pgstat	$-{3.20}_{-0.22}^{+0.20}$	0.97	(2.78 ± 0.28) E-06	(6.68 ± 0.67) E-06
GRB080303B	IPN	3.86, −1.11	90.27, 270.27	chi	$-{2.71}_{-0.28}^{+0.24}$	0.93	(5.48 ± 0.55) E-06	(4.21 ± 0.42) E-05
GRB080319C	XRT	83.33, 35.44	84.38, 342.10	chi	$-{2.10}_{-0.55}^{+0.39}$	1.19	(6.40 ± 0.64) E-06	(7.16 ± 0.72) E-06
GRB080328A	XRT	161.85, 6.17	155.52, 20.63	chi	$-{1.67}_{-0.33}^{+0.27}$	1.14	(1.90 ± 0.19) E-06	(2.29 ± 0.23) E-05
GRB080407A	IPN	249.13, 78.50	62.56, 267.28	chi	$-{2.87}_{-0.29}^{+0.26}$	1.58	(2.48 ± 0.25) E-06	(2.61 ± 0.26) E-05
GRB080507B	IPN	90.47, 31.57	10.34, 0.26	cstat	$-{3.29}_{-2.02}^{+0.83}$	0.88	(2.92 ± 0.29) E-06	(2.80 ± 0.28) E-07
GRB080514B	SA	54.58, −34.45	37.61, 86.44	chi	$-{1.65}_{-0.11}^{+0.10}$	1.40	(4.92 ± 0.49) E-06	(2.36 ± 0.24) E-05
GRB080528A	IPN	176.18, −7.39	93.58, 4.72	cstat	$-{2.16}_{-0.34}^{+0.25}$	0.98	(2.92 ± 0.29) E-05	(3.50 ± 0.35) E-06
GRB080530A	IPN	288.15, 62.08	126.21, 269.99	cstat	$-{2.09}_{-0.20}^{+0.15}$	1.18	(1.45 ± 0.14) E-05	(5.82 ± 0.58) E-06

Note. For each event, name, galactic coordinates, statistics adopted for the spectral analysis, β photon index (with related 1σ errors), ${\chi }_{\mathrm{red}}^{2}$, and corresponding fluxes and fluences (90% confidence interval) are reported.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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It is interesting to notice that 95 of these bursts exhibit PLs with β > −2, which correspond to events whose spectral energy densities ν F_ν have a positive slope extending up to the highest energies. It should be noticed that the results reported in Table 3 are obtained by an automatic spectral fit performed on the whole sample of localized GRBs, integrated on the whole duration of the bursts: these results provide an overall spectral picture of each GRB, without treating in detail the individual evolution of each event, and without investigating the possible existence of more spectral episodes or extra components. As a consequence, the PLs with β > −2 obtained from the fit of the 95 bursts reported above may describe the event as a whole, as well as represent episodes arising only at the beginning of the prompt, or subsequently throughout the spectral evolution.

We analyzed the log N–log F brightness distribution of our GRB sample, where fluxes F are obtained from the spectral fits with the PL model. Such distribution provides information on the intrinsic and spatial properties of GRBs. The N ∼ F^−3/2 trend usually retrieved at large fluxes, with events following a −3/2 slope in the log N–log F plane, shows a clear deviation at fainter fluxes (i.e., F < 4 × 10⁻⁶ erg cm⁻² s⁻¹), as reported in Figure 13: since fewer faint bursts than expected are observed, the spatial distribution of GRBs cannot be consistent with a homogeneous Euclidean universe.

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In general, the MCAL GRB population decline curve is compatible with those reported by BATSE (Hurley 1991; Kouveliotou et al. 1993) and Fermi GBM (Nava et al. 2011; Bhat et al. 2016), although the departure from the −3/2 slope occurs at different flux values, depending on number statistics and on the sensitivity of the instrument. It is important to note that the different MCAL trigger configurations adopted during the AGILE lifetime make the integral distribution of the flux not uniform. In particular, the C3 configuration discussed in Section 4.1, which is used on demand to overcome telemetry requirements, is less sensitive to low-flux bursts: as GRBs detected in the C3 configuration are about 17% of the total, this issue could introduce a bias in the overall log N–log F distribution. All the C2, C3, and C4 configurations make the full acquisition of long GRBs more difficult, biasing the analyzed sample toward short-duration events, which constitute a large fraction (40%) of the total, with respect to that reported by other detectors (Bošnjak et al. 2014; Bhat et al. 2016; Svinkin et al. 2016). Another consequence of the different configuration settings is the slight break in the log N–log F slope, present at about (0.5–1.0) × 10⁻⁵ erg cm⁻², already pointed out in Figure 12. The deficit at high fluxes can be ascribed both to the fixed limited energy range adopted for the spectral analysis and to small number statistics: a similar discrepancy is reported also in the plots reported by BATSE (Kommers et al. 2000) and Fermi GBM (Bhat et al. 2016). On the other hand, as MCAL is an all-sky monitor, the plot is not affected by on-axis area variations. The only effects that could affect a continuous and homogeneous MCAL monitoring of the sky are represented by Earth occultations, which constantly hide about 35% of the sky, and by passages into the South Atlantic Anomaly, where detectors are switched off. However, as Figure 13 includes GRBs detected over a 13 yr time span, we can assume that these effects are smoothed out over such a large time interval and do not significantly affect the plot.

4.2.2. Spectral Fit with a Band Model

We also performed an automatic spectral analysis by using a Band model. We expect that most of MCAL bursts will exhibit an E_p below the energy threshold of 400 keV. Nevertheless, we ended up with ${N}_{\mathrm{full}}^{\mathrm{sp},\mathrm{Band}}=43$ GRBs which can be described by a Band model with E_p ≥ 400 keV, α > −2, and β < −2, reported in Table 4. Distributions of peak energy E_p, low-energy photon index α, and high-energy photon index β are shown in Figure 14. We adopted the same criteria used for the spectral analysis with a PL model. The fits with a four-parameters model result in a number of dof equal to N_dof = 73, in the 0.4–50 MeV energy range. Our sample of Band-fittable GRBs exhibits a mean peak energy 〈E_p〉 = 640 keV, a mean 〈α〉 = −0.6, and a mean 〈β〉 = −2.5. Although the MCAL sample is biased toward higher-energy GRBs, the 〈α〉 and 〈β〉 mean values are compatible with those reported by Poolakkil et al. (2021) for a large sample of Fermi GBM GRBs, with an 〈α〉 ranging between −0.8 (for long GRBs) and −0.5 (for short GRBs), and 〈β〉 ranging between −2.4 (for long GRBs) and −2.6 (for short GRBs). On the other hand, the MCAL 〈E_p〉 is slightly higher than that reported for GBM bursts, ranging between 144 keV (for long GRBs) and 413 keV (for short GRBs). Among the GRBs fitted with a Band model, we recall the ultra-long GRB 080407A (Pal'shin et al. 2012), the hard-spectrum GRB 131028A whose E_p ∼ MeV is also reported by Fermi and Konus-Wind (Golenetskii et al. 2013; von Kienlin 2013), the high-energy GRB 130427A detected by Fermi LAT and extensively treated in Ackermann et al. (2013), and the GRB 190114C, first event detected at VHE by the MAGIC telescope MAGIC Collaboration et al. (2019), Ajello et al. (2020), Ursi et al. (2020). An example of a GRB spectrum fitted with a Band model (GRB 110721A) is reported in Figure 15.

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Table 4. List of ${N}_{\mathrm{full}}^{\mathrm{sp},\mathrm{Band}}=43$ Localized GRBs Whose Time-integrated Spectrum is Fitted with a Band Model with E_p ≥ 400 keV in the 0.4–50 MeV Energy Range

NAME	stat.	α	β	Ep	${\chi }_{\mathrm{red}}^{2}$	Flux (0.4–50 MeV)	Fluence (0.4–50 MeV)
				(keV)	(73 dof)	(erg cm−2 s−1)	(erg cm−2)
GRB080407A	chi	$-{0.23}_{-0.21}^{+0.15}$	$-{3.12}_{-0.04}^{+0.01}$	${686}_{-577}^{+280}$	1.59	(2.28 ± 0.23) E-06	(2.39 ± 0.24) E-05
GRB080723D	chi	${0.23}_{-0.14}^{+0.23}$	$-{2.87}_{-0.32}^{+0.04}$	${601}_{-515}^{+248}$	0.99	(8.88 ± 0.89) E-07	(3.36 ± 0.34) E-05
GRB080916C	chi	$-{0.58}_{-0.23}^{+0.06}$	$-{2.30}_{-0.43}^{+0.04}$	${721}_{-533}^{+278}$	0.81	(4.30 ± 0.43) E-06	(8.16 ± 0.82) E-05
GRB081004A	cstat	$-{1.01}_{-0.22}^{+0.16}$	$-{2.31}_{-0.97}^{+0.89}$	${925}_{-635}^{+346}$	0.88	(8.90 ± 0.89) E-06	(5.70 ± 0.57) E-07
GRB081207A	chi	$-{1.06}_{-0.17}^{+0.22}$	$-{2.11}_{-0.16}^{+0.14}$	${516}_{-416}^{+207}$	1.13	(1.42 ± 0.14) E-06	(5.47 ± 0.55) E-05
GRB081209A	cstat	$-{0.54}_{-0.25}^{+0.06}$	$-{2.47}_{-0.12}^{+0.34}$	${770}_{-570}^{+297}$	0.95	(5.72 ± 0.57) E-06	(1.10 ± 0.11) E-06
GRB081222A	chi	$-{0.59}_{-1.50}^{+0.59}$	$-{3.08}_{-0.42}^{+0.41}$	${778}_{-620}^{+310}$	1.17	(1.75 ± 0.18) E-06	(9.32 ± 0.93) E-06
GRB081224A	chi	${0.02}_{-0.09}^{+0.22}$	$-{3.28}_{-0.11}^{+0.28}$	${732}_{-612}^{+298}$	1.09	(1.65 ± 0.17) E-06	(4.33 ± 0.43) E-06
GRB090328B	cstat	$-{0.86}_{-0.22}^{+0.03}$	$-{2.83}_{-0.18}^{+0.23}$	${1963}_{-1193}^{+701}$	0.78	(8.54 ± 0.85) E-06	(5.46 ± 0.55) E-07
GRB090401B	chi	$-{0.91}_{-0.18}^{+0.16}$	$-{2.23}_{-0.07}^{+0.14}$	${890}_{-612}^{+333}$	0.94	(4.98 ± 0.50) E-06	(2.23 ± 0.22) E-05
GRB090427A	chi	$-{0.90}_{-0.10}^{+0.09}$	$-{2.25}_{-0.44}^{+0.27}$	${608}_{-472}^{+240}$	1.54	(8.76 ± 0.88) E-06	(1.01 ± 0.10) E-05
GRB090618A	chi	$-{0.91}_{-0.15}^{+0.08}$	$-{3.17}_{-0.28}^{+0.24}$	${701}_{-588}^{+286}$	1.31	(8.46 ± 0.85) E-07	(3.77 ± 0.38) E-05
GRB090720B	chi	$-{0.59}_{-0.29}^{+0.28}$	$-{2.29}_{-0.78}^{+2.28}$	${638}_{-490}^{+250}$	1.32	(3.96 ± 0.40) E-06	(1.24 ± 0.12) E-05
GRB090809B	chi	${0.74}_{-0.15}^{+0.16}$	$-{2.83}_{-1.55}^{+2.35}$	${669}_{-546}^{+270}$	1.17	(3.34 ± 0.33) E-06	(2.09 ± 0.21) E-05
GRB091109B	cstat	${0.14}_{-0.34}^{+0.02}$	$-{2.42}_{-0.42}^{+0.20}$	${994}_{-678}^{+371}$	1.11	(1.67 ± 0.17) E-05	(8.02 ± 0.80) E-07
GRB100612A	cstat	$-{0.84}_{-0.16}^{+0.50}$	$-{2.05}_{-1.55}^{+2.04}$	${500}_{-403}^{+200}$	1.31	(5.20 ± 0.52) E-06	(1.17 ± 0.12) E-06
GRB100724B	chi	$-{0.82}_{-0.02}^{+0.05}$	$-{2.10}_{-0.30}^{+0.18}$	${444}_{-379}^{+183}$	1.72	(5.00 ± 0.50) E-06	(1.81 ± 0.18) E-04
GRB100811A	cstat	$-{0.07}_{-0.15}^{+0.06}$	$-{2.63}_{-0.00}^{+0.48}$	${885}_{-639}^{+338}$	0.79	(8.02 ± 0.80) E-06	(1.67 ± 0.17) E-06
GRB101219A	cstat	$-{0.84}_{-0.16}^{+0.83}$	$-{2.07}_{-0.55}^{+1.98}$	${428}_{-369}^{+177}$	1.10	(6.54 ± 0.65) E-06	(1.88 ± 0.19) E-06
GRB110529A	cstat	$-{0.80}_{-0.26}^{+0.16}$	$-{2.10}_{-0.43}^{+0.22}$	${472}_{-393}^{+192}$	0.94	(9.20 ± 0.92) E-06	(2.65 ± 0.27) E-06
GRB110721A	chi	$-{0.83}_{-0.75}^{+0.80}$	$-{2.10}_{-0.55}^{+0.55}$	${1412}_{-863}^{+505}$	1.35	(8.00 ± 0.80) E-06	(3.33 ± 0.33) E-05
GRB120226B	chi	$-{0.82}_{-0.81}^{+0.82}$	$-{2.07}_{-1.05}^{+2.05}$	${481}_{-395}^{+194}$	1.21	(3.46 ± 0.35) E-06	(1.86 ± 0.19) E-05
GRB120512A	chi	$-{0.59}_{-0.07}^{+0.23}$	$-{3.01}_{-0.12}^{+0.33}$	${709}_{-580}^{+286}$	1.15	(4.14 ± 0.41) E-07	(3.79 ± 0.38) E-06
GRB130228B	cstat	$-{0.86}_{-0.89}^{+0.42}$	$-{2.20}_{-0.22}^{+0.22}$	${480}_{-405}^{+196}$	0.72	(7.40 ± 0.74) E-06	(4.97 ± 0.50) E-06
GRB130306A	chi	$-{1.03}_{-0.16}^{+0.15}$	$-{2.72}_{-0.18}^{+0.15}$	${918}_{-663}^{+351}$	0.89	(6.78 ± 0.68) E-07	(1.81 ± 0.18) E-05
GRB130427A	chi	$-{0.95}_{-0.12}^{+0.09}$	$-{3.17}_{-0.33}^{+0.28}$	${1282}_{-878}^{+480}$	2.62	(6.32 ± 0.63) E-05	(2.99 ± 0.30) E-04
GRB130828A	chi	$-{0.28}_{-0.12}^{+0.19}$	$-{2.54}_{-0.15}^{+0.44}$	${797}_{-589}^{+308}$	1.10	(5.34 ± 0.53) E-07	(1.01 ± 0.10) E-05
GRB131028A	chi	$-{0.36}_{-0.16}^{+0.02}$	$-{2.35}_{-0.03}^{+0.15}$	${884}_{-618}^{+333}$	2.15	(1.67 ± 0.17) E-05	(8.43 ± 0.84) E-05
GRB140508A	chi	${1.35}_{-0.32}^{+0.11}$	$-{2.84}_{-0.39}^{+0.54}$	${1259}_{-842}^{+467}$	1.08	(1.58 ± 0.16) E-06	(1.99 ± 0.20) E-05
GRB140930B	cstat	$-{0.96}_{-0.31}^{+0.06}$	$-{2.42}_{-0.31}^{+0.27}$	${1455}_{-909}^{+525}$	0.95	(1.09 ± 0.11) E-05	(4.54 ± 0.45) E-06
GRB141012A	chi	$-{0.51}_{-0.03}^{+0.06}$	$-{2.63}_{-0.02}^{+0.34}$	${1045}_{-719}^{+392}$	1.01	(4.06 ± 0.41) E-07	(4.88 ± 0.49) E-06
GRB150403A	chi	$-{1.11}_{-0.31}^{+0.05}$	$-{2.76}_{-0.38}^{+0.03}$	${1585}_{-999}^{+574}$	1.19	(6.74 ± 0.67) E-06	(3.67 ± 0.37) E-05
GRB171011B	chi	$-{1.27}_{-0.02}^{+0.24}$	$-{2.65}_{-0.33}^{+0.33}$	${1379}_{-888}^{+503}$	0.98	(9.84 ± 0.98) E-06	(5.03 ± 0.50) E-06
GRB171011A	chi	$-{0.52}_{-0.19}^{+0.19}$	$-{2.58}_{-0.25}^{+0.02}$	${725}_{-556}^{+284}$	0.81	(1.50 ± 0.15) E-06	(1.92 ± 0.19) E-06
GRB180204A	cstat	$-{0.60}_{-0.25}^{+0.59}$	$-{2.37}_{-1.55}^{+2.32}$	${803}_{-579}^{+307}$	1.36	(1.29 ± 0.13) E-05	(1.03 ± 0.10) E-06
GRB180720A	chi	$-{0.48}_{-0.90}^{+0.14}$	$-{2.90}_{-0.32}^{+0.32}$	${1945}_{-1190}^{+696}$	0.79	(5.30 ± 0.53) E-04	(5.09 ± 0.51) E-04
GRB180728A	chi	$-{0.57}_{-0.16}^{+0.15}$	$-{2.57}_{-0.25}^{+1.98}$	${503}_{-444}^{+210}$	1.16	(1.88 ± 0.19) E-06	(4.87 ± 0.49) E-06
GRB190114C	chi	$-{0.44}_{-0.30}^{+0.17}$	$-{2.79}_{-0.23}^{+0.00}$	${941}_{-679}^{+360}$	1.21	(5.52 ± 0.55) E-05	(1.24 ± 0.12) E-04
GRB190305A	chi	$-{0.39}_{-0.21}^{+0.12}$	$-{3.00}_{-0.34}^{+0.42}$	${1313}_{-881}^{+487}$	1.22	(8.00 ± 0.80) E-03	(5.12 ± 0.51) E-03
GRB190606A	cstat	${0.21}_{-0.20}^{+0.22}$	$-{2.11}_{-0.44}^{+0.39}$	${959}_{-637}^{+354}$	1.19	(4.12 ± 0.41) E-05	(3.63 ± 0.36) E-06
GRB190810A	cstat	$-{0.89}_{-0.16}^{+0.16}$	$-{2.04}_{-1.50}^{+2.30}$	${564}_{-435}^{+222}$	0.76	(5.38 ± 0.54) E-06	(5.80 ± 0.58) E-07
GRB191221B	chi	$-{1.27}_{-0.28}^{+0.09}$	$-{2.52}_{-0.15}^{+0.53}$	${1317}_{-847}^{+481}$	0.93	(4.72 ± 0.47) E-06	(2.36 ± 0.24) E-05
GRB201103B	chi	$-{0.34}_{-0.25}^{+0.25}$	$-{2.44}_{-1.11}^{+2.44}$	${902}_{-634}^{+341}$	0.73	(5.18 ± 0.52) E-06	(1.01 ± 0.10) E-05

Note. For each event, name, statistics adopted for the spectral analysis, α and β photon indices, E_p, with related 1σ errors, ${\chi }_{\mathrm{red}}^{2}$, and corresponding fluxes and fluences (90% confidence interval) are reported.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table 5. List of the Eight GRBs Fitted with a Band Model, with Related Redshifts, Flux in the 1–10,000 keV Energy Range (90% Confidence Level), E_p,i (with 1σ error), and E_iso

NAME	z	Flux (1 keV–10 MeV) (erg cm−2 s−1)	Ep,i (keV)	Eiso (erg)
GRB090618A	0.54	(1.04 ± 0.10)E-06	${701}_{-588}^{+286}$	5.69E+52
GRB110721A	0.382	(5.84 ± 0.58)E-06	${1412}_{-863}^{+505}$	1.53E+52
GRB130427A	0.34	(6.10 ± 0.61)E-05	${1282}_{-878}^{+480}$	1.41E+53
GRB140508A	1.028	(9.82 ± 0.98)E-07	${1259}_{-842}^{+467}$	5.23E+52
GRB150403A	2.06	(3.80 ± 0.38)E-06	${1585}_{-999}^{+574}$	3.79E+53
GRB180728A	0.117	(2.62 ± 0.26)E-06	${503}_{-444}^{+210}$	3.84E+50
GRB190114C	0.42	(4.48 ± 0.45)E-05	${941}_{-679}^{+360}$	7.63E+52
GRB191221B	1.148	(2.98 ± 0.30)E-06	${1317}_{-847}^{+481}$	8.41E+52

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The 95 GRBs fitted with a PL with β > −2 and the 43 GRBs fitted with a Band model with E_p ≥ 400 keV constitute the subsample of MCAL GRBs with the highest energies: it is interesting to notice that 20 of these bursts have associated Fermi LAT detections. These events are particularly relevant, as they allow to better investigate the high-energy component of GRBs, and will be object of a forthcoming study.

Among the ${N}_{\mathrm{full}}^{\mathrm{sp},\mathrm{Band}}=43$ GRBs fitted with a Band model, eight events have been reported with known redshifts, provided by X-ray or optical observations of their afterglow, as reported in Table 5. For these events, we estimated the related rest-frame parameters, by adopting a standard cosmological model with H₀ = 67.3 km s⁻¹ Mpc⁻¹, Ω_M = 0.315, and Ω_Λ = 0.685. For each burst, we evaluate the intrinsic peak energy of the time-integrated spectrum E_p,i = (1 + z)E_p and the isotropic equivalent energy E_iso released between 1 keV and 10 MeV on the whole duration of the event. We checked our sample by using the Amati relation ${E}_{{\rm{p}},i}=K{({E}_{\mathrm{iso}}/{10}^{52})}^{m}$: a best-fit of our GRB sample results in a slope of m = 0.130 ± 0.067, shown in Figure 16 with respect to redshift z, which is slightly more gentle than typical values m ∼ 0.45 reported in (Amati et al. 2002; Azzam & Alothman 2013), due to the nonuniform hardness of this sample.

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