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Issue
A&A
Volume 531, July 2011
Article Number A20
Number of page(s) 10
Section Extragalactic astronomy
DOI https://doi.org/10.1051/0004-6361/201116953
Published online 01 June 2011

© ESO, 2011

1. Introduction

Gamma-ray bursts (GRB) are the most luminous flashes of γ-rays known to humankind. It is generally believed that they originate from a compact source with highly relativistic collimated outflows (Γ > 100).

A large fraction of our knowledge of the prompt emission comes from the Burst And Transient Source Experiment (BATSE, Meegan et al. 1992) onboard the Compton Gamma-ray Observatory (CGRO, 1991−2000). Unfortunately, only a handful of BATSE bursts had a measured redshift because the BATSE error boxes were too large and thus, follow-up observations with X-ray and optical instrumentation was very limited. Moreover, the first afterglow was only detected in 1997, already near the end of the BATSE mission. The lack of distance measurements led to a focus of GRB studies in the observer frame without redshift corrections. Due to the cosmological origin of GRBs, such a correction is likely to be necessary to understand the intrinsic nature of these events.

With the two dedicated satellites, Beppo-SAX (Boella et al. 1997) and Swift (Gehrels et al. 2004), the situation has changed and afterglow and host galaxy spectroscopy has provided redshifts for more than 200 events by now. Unfortunately, the relatively narrow energy band of Beppo-SAX (0.1−300 keV) and Swift/BAT (15−150 keV) limits the constraints on the prompt emission spectrum because the peak energy, Ep, in the νFν spectrum of GRBs, can only be determined for low Ep values and is often unconstrained (Butler et al. 2007; Sakamoto et al. 2009).

In this work we will take advantage of the broad energy coverage of the Fermi/GBM (8 keV − 40 MeV) to study the primary spectral and temporal properties, such as Ep, T90, the time interval in which 90% of the burst fluence has been observed, and Eiso, the isotropic equivalent bolometric energy, in the rest-frame of the progenitors of 32 GRBs with measured redshift.

2. GRB sample and analysis

2.1. Instrumentation

The Gamma-ray Burst Monitor (GBM) is one of the instruments onboard the Fermi Gamma-ray Space Telescope (Atwood et al. 2009) launched on June 11, 2008. Specifically designed for GRB studies, GBM observes the whole unocculted sky with a total of 12 thallium-activated sodium iodide (NaI(Tl)) scintillation detectors covering the energy range from 8 keV to 1 MeV and two bismuth germanate scintillation detectors (BGO) sensitive to energies between 150 keV and 40 MeV (Meegan et al. 2009). Thus, GBM offers a unprecedented view of GRBs, covering more then 3 decades in energy.

2.2. Burst selection

The selection criterion for our sample is solely based on the redshift determination. We form a sample of 32 bursts detected by GBM up to October 16th, 2010, with known redshift (determined either spectroscopically or photometrically).

Our sample contains 4 short and 28 long GRBs. The redshift distribution of the GBM GRBs is shown in Fig. 1 together with a histogram of all 239 GRBs with redshift determinations to date1. A two-tailed Kolmogorov-Smirnov (KS) test between the full sample and the GBM-only sample shows that the two distributions are very similar (P = 84%). In conclusion, the GBM-only sample is representative of the full GRB sample with redshift.

thumbnail Fig. 1

Redshift distribution in % of GBM GRBs (blue solid line) compared to all 239 GRBs with measured redshift to date (red dashed line). Both samples contain long and short bursts.

2.3. Analysis of the GBM data

For the Ep determination, CSPEC data (Meegan et al. 2009) with a time resolution of 1.024 s (4.096 s pre-trigger) were used. For the short GRBs, i.e. those with T90 ≤ 2 s, time-tagged event (TTE, Meegan et al. 2009) data were used with a fine time resolution of 64 ms and the same channel boundaries as CSPEC data.

Detectors with source angles greater than 60° and those occulted by the spacecraft or solar panels were discarded. A maximum of three NaI detectors were used for each Ep determination. In four cases (GRB 090929A, GRB 090904B, GRB 090618, and GRB 081007) only one NaI detector could be used due to the reasons mentioned above. Where possible, both BGO detectors were included in the analysis if they were not occulted by the satellite during the prompt emission. Even if there was no apparent signal in the BGOs, they can help to determine an upper limit of the GRB signal in the spectra.

The spectral analysis was performed with the software package RMFIT2 (version 3.3rc8) and the GBM response matrices v1.8. To account for the changing orientation of the source with respect to the detectors caused by the slew of the spacecraft, the detector response matrices (DRM) were generated for every 2 degrees on the sky.

For each burst we fitted for every energy channel a low-order polynomial to a user defined background interval before and after the prompt emission and interpolated this fit across the source interval.

Three model fits were applied to all time bins with a signal-to-noise (S/N) of at least 3.5 above the background model: a single power-law (PL), a power law function with an exponential high energy cutoff (COMP) and the Band function (Band et al. 1993a). For three GRBs (GRB 090424, GRB 090618, and GRB 090926A) an effective area correction was applied to the BGO with respect to the NaI detectors to account for systematics which dominate the statistical errors due to the brightness of these events. The best model fit is the function which provides the best Castor C-stat value (Cash 1979). An improvement by ΔC-stat  =  10 for every degree of freedom is required. The profile of the Cash statistics was used to estimate the 1σ asymmetric error.

The values obtained with this analysis method may be superseded by the GBM spectral catalogue released by the GBM team (Goldstein et al., in prep.).

3. The intrinsic properties

3.1. The peak energy (Ep)

3.1.1. Instrumental bias

A first important issue which needs to be addressed is whether the bursts observed by GBM are drawn from the same distribution as the bursts which were observed by BATSE. Because of the broader sensitivity of GBM to higher energies, there could be a significant deviation towards higher Ep values. To answer this question, we use the BATSE catalogue3 from which we extract the Ep values which were obtained from the time averaged spectra (fluence spectral fits). Ignoring the power-law (PL), Gaussian log and smoothly broken power law (SBPL) fits, we elected to the COMP model or Band function for the purpose of this test. We require ΔEp/Ep ≤ 0.4 and the low-energy power-law index, α has an absolute error σα ≤ 0.4. GRBs which did not fulfill these criteria were rejected from the sample. The Band function was always preferred over the COMP model in cases in which the high-energy power-law index β was constrained (σβ < 0.4). Both long and short GRBs were included in this analysis. The so obtained Ep distribution is shown in Fig. 2.

thumbnail Fig. 2

Ep values of 1367 BATSE/CGRO GRBs with a log-normal fit (black dotted histogram and line), 375 GBM GRBs and log-normal fit (red dashed histogram and line) and 30 GBM GRBs with redshift measurement (blue solid histogram). A KS test suggests that all three samples were drawn from the same distribution.

The same selection cut was applied to the 2-year GBM spectral catalogue. The red histogram in Fig. 2 shows this distribution. Both distributions peak at  ≈ 170 keV and show the same standard deviation. A KS test reveals that the difference between the two samples is not statistically meaningful (P  =  18%). This means that the two histograms are drawn from the same distribution, in agreement with Nava et al. (2010). We note that, Bissaldi et al. (2011) showed that the Ep distribution of some GBM-GRBs extends to higher energies compared to BATSE (Kaneko et al. 2006). However, this is of no surprise as Amati et al. (2002) demonstrated that bursts with higher fluence, i.e. higher Eiso, have, on average, higher Ep (see also Sect.4.1). Since Bissaldi et al. (2011) only use bright GRBs, it is to be expected that their Ep distribution is shifted to higher energies. We conclude that GBM, although being sensitive up to 40 MeV, does not find a previously undiscovered population of high-Ep GRBs, consistent with Harris & Share (1998).

The KS test was then applied to both the Ep distributions between the whole sample of GBM bursts and the 30 GBM bursts with measured redshift (for GRB 090519A and GRB 080928 a PL model fits the data best) and also to the Ep distribution of BATSE bursts and the Ep distribution of GBM bursts with measured redshift. In neither case the differences were statistically meaningful (P = 24% and P = 20%, respectively). Thus, all 3 histograms are very likely drawn from the same distribution. In conclusion, the sample of GBM GRBs with measured redshift presented here is representative for the whole population of GRBs which were ever observed by BATSE and GBM.

However, it should be stressed that GBM cannot measure Ep values which are lower than a certain limiting threshold. It is well known that Ep values of GRBs can go as low as a few keV. Pélangeon et al. (2008) for example find Ep values as low as  ~2 keV in GRBs that were observed by the High Energy Transient Explorer 2 (HETE-2, see e.g. Barraud et al. 2003, and references therein). These low energetic events have been classified as X-ray flashes (XRF) or X-ray rich bursts (XRB) (see e.g. Heise & in ’t Zand 2001; Sakamoto et al. 2005). However, it is very likely that XRFs and XRBs are nothing else than weak and long GRBs (see Kippen et al. 2004, and references therein).

The borderline Ep value is obviously located somewhere near the low-energy sensitivity of the NaIs, which has yet to be determined. Thus, in order to determine a potential bias in the Ep,rest distribution shown in Fig. 3, it is important to understand and quantify the limits of the GBM to measure Ep (be it either from the COMP model or Band function).

For this purpose, we created a set of simulated bursts with different initial spectral and temporal starting values. We input the source lifetime (tS, 1 s, 5 s, 10 s, 100 s) and the photon flux (f) in the 10 keV to 1 MeV range (1, 3 and 10 ph cm-2 s-1). For the simulation the Band function was chosen as the photon model with varying Ep (15, 17, 25, 50, 100 keV) but fixed α =  −0.8 and β =  −2.4. We simulate these bursts overlaid on real background data by using detector NaI 7 of GRB 090926A4. This results in 60 different burst models. For each model, we created 1000 bursts to account for Poissonian noise. This results in 60 000 spectra, each of which was then fitted with the Band function using the detector response matrix (DRM) of detector NaI 7 created for the location of GRB 090926A.

After the fitting procedure, we reject those bursts which have ΔEp/Ep ≥ 0.3 and σα ≤ 0.4. These rejected spectra are then defined as unconstrained. We did not apply this criterion to the high-energy power law index β. A spectral fit that has a constrained Ep and α but an unconstrained β is simply considered a COMP model. In Table 1 we report the mean and standard deviation of the output spectral parameters, Ep and α, of the simulated bursts.

thumbnail Fig. 3

Rest-frame distribution of Ep for 4 short (hatched histogram) and 26 long (empty histogram) GBM-GRBs. For two GRBs (GRB 090519A, GRB 080928) a simple PL model fits the data best, thus not providing an Ep value. The complete distribution has mean and median values of 1.1 MeV and 750 keV, respectively. A log-normal fit to the sample of long GRBs (dashed line) peaks at  ~800 keV and has a FWHM = 0.93 in log-space.

Table 1

Mean and standard deviation of the output spectral parameters of the simulated bursts.

The conclusions of this exercise are:

  • 1.

    GBM can recover Ep values only as lowas  ≈ 15 keV. The fact that the observed Ep distribution is indeed biased is in clear contradiction to e.g. Brainerd et al. (2000). They argued that the observed Ep distribution (by BATSE in this case) is actually the intrinsic one;

  • 2.

    the input Ep parameter can be recovered from the simulated spectra within the 2σ errors for almost all simulated flux levels and source lifetimes;

  • 3.

    GRBs with low fluxes, low Ep and short ts are more likely to be rejected than GRBs with higher fluxes, higher Ep or longer ts;

  • 4.

    the low-energy power law index α tends to get softer, i.e. to have lower values, for low Ep and low fluxes. This last point is particularly noteworthy. Crider et al. (1997), Lloyd-Ronning & Petrosian (2002) and later Kaneko et al. (2006) found a significant correlation between Ep and α in the time-resolved spectra of several GRBs. Supported by our simulations, we point out the possibility that a parameter reconstruction effect is at work in addition to any intrinsic correlation between Ep and α. This effect has already been brought forward by Preece et al. (1998); Lloyd & Petrosian (2000) and Lloyd-Ronning & Petrosian (2002). In short, it depends on how quickly α can reach its asymptotic value and how close Ep is located to the low-energy limit of the instrument’s energy bandpass. The closer Ep is situated to the detector’s sensitivity limit, the fewer is the number of photons in the low-energy portion of the spectrum. This makes it increasingly difficult to determine the asymptotic value of α. Instead, a more negative value of the low-energy power law index will be measured which is what is observed here.

3.1.2. The intrinsic Ep distribution

In Fig. 3 we present a histogram of Ep,rest of our sample of GBM GRBs where a Ep measurement was possible. The distribution of all bursts has mean and median values of 1.1 MeV and 750 keV, respectively. A log-normal fit to the sample of long bursts peaks at  ~ 800 keV. In the canonical scenario of GRB jets, turbulent magnetic fields build up behind the internal shock, and electrons produce a synchrotron power law spectrum. The typical rest-frame frequency νrest in the internal shock dissipation is (Zhang & Mészáros 2002), where L52 is the luminosity in 1052 erg/s and r13 the dissipation radius in units of 1013 cm. Our measurement of the rest-frame peak energy therefore leads to a constraint of  (in the above units).

Recently, Collazzi et al. (2011) reported that the width of the Ep,rest distribution must be close to zero with the peak value located close to the rest-mass energy of electrons at 511 keV. This effectively implies that all GRBs must be thermostated by some unknown physical mechanism. We tested this claim by determining the width of the Ep,rest distribution (σEp(1 + z) = 0.48) together with all the individual errors that add to the uncertainty of the Ep measurement in log-space (for details refer to Eq. (1) in Collazzi et al. 2011): (1)σChoice describes the different choices made by different analysts. Since all the bursts in this paper were analyzed consistently (selection of the time interval, energy range, etc.), we can set σChoice = 0. σDet is the error which results in not knowing the detector response perfectly. However, according to Collazzi et al., σDet is negligibly small, meaning that the calibrations of the detectors are usually well understood. σDef describes the differences that are obtained in Ep when using different photon models. As it was shown already by e.g. Band et al. (1993b) and Kaneko et al. (2006) the COMP model results in higher Ep values than the Band function. Collazzi et al. use σModel = 0.12 to account for this difference. Here, this value is adopted as σDef. The Poissonian errors are σPoisson ≈ 0.10 in log-space. Thus .

Inserting the just found values in Eq. (8) in Collazzi et al. gives (2)However, a zero width in the Ep distribution is synonymous with σEpint = 0. The conclusion is that the Ep,rest distribution does not have a zero width. This finding is in conflict with the implications and conclusions discussed in Collazzi et al.

3.2. GRB duration (T90)

For determining the duration of a GRB, we applied the method first introduced by Kouveliotou et al. (1993). They defined the burst duration as the time in which 90% of the burst counts is collected (T90). Here, we adopted the same definition; however, the burst’s fluence was used instead of the counts. The T90 value depends highly on the detector and on the energy interval in which it is determined (see e.g. Bissaldi et al. 2011). Additionally, since we are interested in durations in the rest-frame of the GRB, it is not sufficient to simply account for the time dilation due to cosmic expansion by dividing the measured durations by (1 + z). The energy band in which the T90 is determined needs to be redshift corrected as well. We determine the burst duration in fluence space in the rest-frame energy interval from 50 keV to 300 keV, i.e. in the observer frame energy interval from 50/(1 + z) keV to 300/(1 + z) keV.

thumbnail Fig. 4

T90,rest distribution of 4 short (red hatched histogram) and 28 long (blue histogram) GBM GRBs in the redshift corrected energy interval from 50 keV to 300 keV. The log-normal fit to the distribution (dashed line) of the long bursts peaks at  ~ 12 s and has a FWHM = 1.1 in log-space.

3.2.1. The intrinsic T90 distribution

In Fig. 4 we present the rest-frame distribution of T90 of our sample of 32 GBM GRBs. The number of short bursts is still too small to unambiguously recover a bimodal distinction of short and long events in the rest-frame. Henceforth, a distinction is made between the short and long class of GRBs using an ad-hoc definition of T90,rest ≤ 2 s for short bursts and T90,rest > 2 s for long bursts, although we do have neither observational nor physical evidence to support such a distinction in the GRB sample presented here. All GRBs in our sample that are defined as short in the observer-frame, also remain short with the here adopted definition. We note that GRB 100816A is peculiar in that it is classified as a short burst by GBM data, but it has a T90 = 2.9  ±  0.6 in the 15−350 keV energy range in Swift data with a low-level emission out to about T0+100 (Markwardt et al. 2010).

The mean value of the whole T90,rest distribution is 32.4 s and the median value is 10.8 s.

3.3. Isotropic energy (Eiso)

Eiso is calculated by (3)where dL is the luminosity distance and Sγ the fluence in the 1/(1 + z) keV to 10/(1 + z) MeV frame. We determine Sγ using the energy flux provided by the best-fit spectral parameters and multiplying it with the total time interval over which the fit was performed. Since we performed the fit for time intervals where the count rate exceeded a S/N ratio of 3.5, it happened that some time intervals of some bursts were not included in the fit (e.g. phases of quiescence where the count rate dropped back to the background level). These time intervals were not used to calculate the fluence. The Sγ distribution is shown in Fig. 5. The median value of the fluence distribution is 1.6  ×  10-5 erg cm-2 and the mean value is 5.9  ×  10-5 erg cm-2. A log-normal fit to the data peaks at 2.2  ×  10-5 erg cm-2.

thumbnail Fig. 5

Energy fluence distribution of 32 GBM GRBs determined in the energy range between 1/(1 + z) keV to 10/(1 + z) MeV range. The complete distribution has mean and median values of 1.6  ×  10-5   erg   cm-2 and 5.9  ×  10-5   erg   cm-2. A log-normal fit (dashed line) to the long GRBs peaks at 2.2  ×  10-5   erg   cm-2 and has a FWHM = 2.2 in log-space.

The Eiso distribution is shown in Fig. 6. The distribution for the long bursts has a median and mean value of 1.2  ×  1053 erg and 1.4  ×  1053 erg, respectively. Short bursts, on the other hand, have significantly lower values of 2.9  ×  1051 erg and 4.0  ×  1051 erg, respectively. A log-normal fit to the long bursts reveals a central value of 1053.1 erg. Because our sample is dominated by long GRBs a log-normal fit to the whole distribution results in an essentially unchanged peak value (1053 erg).

thumbnail Fig. 6

Eiso distribution for long (blue hatched histogram) and short (red hatched histogram). For long bursts the distribution has mean and median values of 1.4  ×  1053 erg and 1.2  ×  1053 erg, respectively. A log-normal fit to the long bursts (dashed line) peaks at 1.2  ×  1053 erg with a FWHM = 1.9 in log-space. Short bursts have a mean and median value of 4  ×  1051 erg and 2.9  ×  1051 erg, respectively. A log-normal fit to the combined sample (dash-dotted line) peaks at 9.8  ×  1052 erg with a FWHM = 2.2 in log-space.

4. Correlations

4.1. Amati relation

Amati et al. (2002) first showed that there is a tight correlation between Ep,rest and Eiso (the isotropic equivalent bolometric energy determined in the energy range between 1 keV to 10 MeV). This relation is now known as the “Amati relation”. In Fig. 7 we show the Amati relation for the 30 GBM GRBs with measured Ep,rest and Eiso. While there is an evident correlation between these two quantities (Spearman’s rank correlation of ρ = 0.74 with a chance probability of 1.7  ×  10-5) the extrinsic scatter of the long GRBs is larger by a factor of  ~2 in log -space compared to Amati (2010). Also, the best fit to our data is shifted to slightly larger Ep,rest values. The best fit power-law index to the long GRBs of our sample is 0.52  ±  0.06 which is in agreement with the indices obtained by e.g. Amati (2010); Ghirlanda et al. (2009, 2010). As has been shown by other authors in the past (see e.g. Amati et al. 2008; Ghirlanda et al. 2009; Amati 2010) short bursts do not follow the relation, being situated well outside the 2σ scatter around the best-fit. This is true also for the power-law fit derived here (see Fig. 7) except for GRB 100816A. However, as already stated above this burst may actually fall in an intermediate or hybrid class of short GRBs with extended emission (see e.g. Norris & Bonnell 2006; Zhang et al. 2009).

thumbnail Fig. 7

Amati relation for 4 short (red open triangles) and 26 long (blue open diamonds) GBM GRBs. Also shown is the best power-law fit to the data (black solid line) and the extrinsic 2σ scatter (black dashed lines) with the best power-law fit published by Amati (2010) (red dash-dotted line) with the 2σ scatter (red long dashed line).

4.2. Yonetoku relation

Yonetoku et al. (2004) found a tight correlation between the rest frame peak energy in the νFν spectrum Ep,rest and the 1-s peak luminosity (Lp) in GRBs (so called Yonetoku relation). The peak luminosity is calculated with (4)where dl is the luminosity distance and Fp the 1 s peak energy flux. We determine Fp in the energy range between 30 keV and 10 MeV in the rest frame of the GRB (Yonetoku et al. 2004). We use the GBM peak spectral catalogue (Goldstein et al. 2011) to determine the best fit spectral parameters of the brightest 1 s bin. We then determined the energy flux in the rest-frame of the GRB by integrating (5)where N0 is the normalization of the spectrum (in photons cm2 s-1), E the energy and Φ(E) the form of the spectrum (either Band or COMP). In order to determine the error on the energy flux, we calculate 1000 flux values for each GRB by varying the input parameters, N0, Ep, α and β according to the 1σ error on each parameter. We then use the median and the mean absolute deviation (MAD) around the median of the 1000 simulated bursts to determine the peak energy flux and the error on the latter, respectively.

We present this relation for 30 GBM GRBs in Fig. 8 and find a best-fit power law index of 0.58  ±  0.08. The Spearman’s rank correlation gives ρ = 0.7 with a chance probability of 2.3  ×  10-5. Our findings are in good agreement with Yonetoku et al. (2004); Ghirlanda et al. (2009) and Ghirlanda et al. (2010).

thumbnail Fig. 8

Yonetoku relation for 4 short (red open triangles) and 26 long (blue open diamonds) GBM GRBs. Also shown is the best power-law fit to the data (black dashed line).

4.3. T90,rest vs. redshift

The rest-frame T90 is plotted as a function of redshift in Fig. 9. Contrary to Pélangeon et al. (2008), who find a negative correlation between T90,rest and z, we do not find any evidence in the GBM data for such a correlation. Also, there is no correlation between T90,rest and z when accounting for Ep,rest. A Spearman rank test for all bursts gives a correlation coefficient of ρ = 0.04 with a chance probability of P = 0.81. However, the p-value in Pélangeon et al. is  ≈ 10-3 which makes it a weak case for such a correlation in the first place. Our results confirm the analyses with Swift detected GRBs (Greiner 2011). We do note a lack of short GRBs for z ≥ 2. This, however, is very likely a selection bias, because firstly, short GRBs at such high redshifts must be very luminous to be observed by GBM and secondly, short GRBs are subluminous in the optical band (Kann et al. 2008) and therefore it is difficult to obtain a redshift measurement.

thumbnail Fig. 9

Testing the cosmic evolution of T90,rest. No correlation is evident. Bursts with Ep,rest ≤ 500 keV are shown as crosses, bursts with Ep,rest > 500 keV shown as filled circles. The two bursts for which are best fit by a simple power-law are shown as open diamonds.

4.4. Ep,rest vs. redshift

In order to explain the detection rate of GRBs at high-z, Salvaterra & Chincarini (2007) conclude that high-z GRBs must be more common (e.g. Daigne et al. 2006; Wang & Dai 2011) and/or intrinsically more luminous (Salvaterra et al. 2009) than bursts at low-z (but see Butler et al. 2010). As already mentioned above, Yonetoku et al. (2004) found a tight correlation between the 1-s peak-luminosity (Lp) and Ep,rest in GRBs. Assuming that the luminosity function of GRBs indeed evolves with redshift and that the Yonetoku relation is valid, we would also expect a positive correlation of Ep,rest with z.

In Fig. 10 we present Ep,rest vs. z. As was shown in Sect. 3.1.1 and in Table 1, GBM can reliably measure Ep down to  ~15 keV. The solid line indicates this redshift-corrected lower limit. The Spearman’s rank correlation, using only the long GRBs, is ρ = 0.58 with a chance probability of P = 2  ×  10-3. When including the short GRBs, the correlation coefficient effectively remains unchanged, whereas the chance probability increases to P = 4  ×  10-3, making a correlation slightly less likely.

thumbnail Fig. 10

Testing the cosmic evolution of Ep,rest for long (diamonds) and short (triangles) GRBs. The solid line indicates the redshift corrected lower limit for GBM to measure Ep which is currently estimated to be  ~15 keV in the observer frame (see Table 1). Bursts with high (green), intermediate (blue) and low (red) peak photon flux (in ph cm-2 s-1) are labeled.

However, this correlation can be explained entirely by selection effects: GRBs do not populate the empty area in Fig. 10 (low Ep,rest and z > 1) because they simply can not be detected by GBM. Even though GBM could recover a low Ep value of such GRBs, as was shown above, the detection of such events is very challenging because of the low photon fluxes of these events. As one can see in Fig. 10 the lower boundary of the apparent correlation is composed of the bursts that have relatively low peak photon fluxes. This is already an indication that these events reside at the lower fluence limit for GBM to both trigger on these events.

Since we actually know the intrinsic parameters of the 32 GRBs, one can test up to which maximum redshift, zmax these bursts could have been detected, i.e. for which GBM would have triggered. GBM has many trigger algorithms (various trigger time scales for various energy ranges). For the purpose of this test, we focus on the 50 keV to 300 range which is the classical trigger energy range for a GRB and a timescale of a maximum of 4.096 s for long GRBs and 1.024 s for short GRBs. In order to shift a GRB to a higher redshift, three observables change:

  • 1.

    the duration ;

  • 2.

    the peak energy ;

  • 3.

    the flux of the GRB.

While it is straightforward to account for the changes of Ep and T90, the proper treatment of the flux is more complex. RMFIT outputs the spectral parameters, including the normalization (N0 in ph cm-2 s-1 keV-1) of the spectrum, which can be recognized as a proxy for the flux of a GRB. Therefore, in order to decrease the flux when shifting the GRB to ever higher redshifts, the normalization has to be decreased accordingly. This was done as follows:

The photon luminosity of a GRB, in the energy band from E1 to E2 is defined as follows (6)where dL is the luminosity distance and z0 the redshift of the burst. The integral describes the photon flux of the burst in the E1 to E2 energy range at the rest-frame of the burst. N0 is the normalization of the spectrum and Φ1(E) is the shape of the spectrum (Band or COMP) with Ep = Ep(z0), i.e. Φ2 = Φ2(E,Ep(z0)). The luminosity of a burst is independent of redshift. This, in turn, means that the above equation is valid also when exchanging z0 with zmax. One gets (7)where Φ2 = Φ2(E,Ep(zmax)).

Setting the two equations equal and solving for Nzmax one is left with (8)It can be shown, for both the COMP model and the Band function, that the fraction of the two integrals is nothing else than (9)where α is the low-energy power law index.

Thus, (10)For GBM to trigger, two detectors need to be above the trigger threshold. Therefore, real background information of the second brightest detector of every burst was used. With this background data, we shift each bursts in steps of Δz = 0.25 to higher redshifts and simulate 1000 bursts for each redshift step (changing the source lifetime, input Ep and normalization as described above, additionally adding Poissonian noise to the best-fit parameters) by forward folding the photon model through the detector response matrix, created at the time and location of the real GRBs. To determine if GBM would have triggered, we determine the signal-to-noise ratio (SNR) with (11)where, Δt is the trigger time scale (Δt = 4.096 s for long GRBs and Δt = 1.024 s for short GRBs), cs and cb are the counts/s of the source and background, respectively. If more than 90% of the simulated bursts have SNR ≥ 4.5 it can safely be assumed that the burst would have been detected at this redshift. We plot Ep,rest, which obviously remains constant at all redshifts, vs. the range of the actually measured to maximum redshift in Fig. 11. We already know that the here presented sample of GRBs is representative of all bursts detected by GBM (see Sect. 3.1.1). Therefore, we note that the determination of the lowest measurable Ep value is not as crucial as the determination of the detector sensitivity. None of the bursts in our sample populates the empty region at 2 ≤ z ≤ 8 and 50 ≤ Ep,rest    [keV]  ≤ 500 even when shifting them at their maximum detectable redshift. We conclude that the correlation between Ep,rest and z can be explained entirely by the sensitivity limitations of the instrument.

thumbnail Fig. 11

Same as Fig. 10 but with a line indicating the maximum possible redshift, zmax that these 30 GRBs can have to still be detectable by GBM.

4.4.1. Low number statistics?

Another way to test if the Ep − z correlation is genuine is by bearing in mind that it arises simply due to the lack of GRBs with measured redshift and Ep < 100 keV. In the present redshift sample there are only 4 GRBs with such a low Ep value. Considering the Ep distribution for 488 GBM bursts of the first two years of operations (Fig. 2) it is possible to estimate as to how many bursts with Ep < 100 keV are expected if 32 GRBs are drawn randomly from the full GBM sample. After a run of 104 drawings, a distribution is obtained which peaks at  ~7. This means that, on average, 7 out of 32 GRBs are expected to have Ep < 100 keV. According to our simulation the probability of observing only 4 such GRBs, as we do in our actual sample, is P  =  8%, which is within the 2σ limit and thus not significant. Therefore, more GRBs with redshift measurement are required to understand if the “blank area” at low Ep values between 2 < z < 5 in Fig. 10 is simply underrepresented or not populated at all.

Table 2

Rest-frame parameters of 32 GBM GRBs.

5. Summary and conclusions

The Fermi/GBM is a key instrument to study the temporal and spectral properties of GRBs. For bursts which have redshift measurements it becomes even more important since it allows GRBs to be studied in their rest-frame. Here, we presented such a study for 32 GRBs observed by GBM, focusing on both the temporal and spectral properties, as well as on intra-parameter relations within these quantities.

The Ep,rest distribution of the GBM GRB sample covers an energy range from tens of keV up to several MeV, peaking at  ≈ 800 keV. Despite the broader energy coverage of GBM compared to BATSE, no high-Ep population is found. However, the GBM Ep distribution is strongly biased against XRFs or XRBs which have Ep values that fall below 15 keV. Additionally, we confirm a previously reported parameter reconstruction effect, namely that the low-energy power law index α tends to get softer when Ep is close to the lower end of the detector energy range.

Using the canonical internal shock model for GRBs, the mean Ep,rest (≈ 800 keV) implies a dissipation radius of  cm.

Another finding of this work is that the width of the Ep distribution is, in fact, not close to zero. Such a claim has recently been brought forward by Collazzi et al. (2011) who argue that all GRBs are thermostated and thus, have the same Ep,rest. If true, this would require an unknown physical mechanism that holds GRBs at a constant Ep,rest value. However, it could be shown here that such a mechanism is not required because our sample has an Ep,rest distribution that ranges over several decades in energy.

We find that the T90,rest stretches from tenths of a second to several hundreds of seconds with a median value of  ≈ 10 s.

The Eiso distribution ranges from 1051 erg to 1055 erg, peaking at  ≈ 1053 erg. We confirm the Ep,rest – Eiso correlation and find a power law index of 0.52, consistent with the values reported in the literature but with a significantly larger scatter around the best-fit. We also confirm a strong correlation between the 1 s peak luminosity of the burst and its Ep,rest with a best-fit power law index of 0.58.

We looked for additional correlations between the parameters. We did not find any evidence for a cosmic evolution of T90,rest. Although a correlation between Ep,rest and z is not entirely unexpected from theoretical considerations and looks intriguing on a Ep  −  z plot, we conclude that the apparent relationship is simply arising due to the detector sensitivity.

2

RMFIT for GBM and LAT analysis was developed by the GBM Team and is publicly available at fermi.gsfc.nasa.gov/ssc/data/analysis/

4

The choice of NaI 7 and the choice for this specific GRB is completely arbitrary. We could have chosen any other detector that observed any other real GRB.

Acknowledgments

We thank Jonathan Granot for useful discussions. A.J.v.d.H. was supported by NASA grant NNH07ZDA001-GLAST. SF acknowledges the support of the Irish Research Council for Science, Engineering and Technology, cofunded by Marie Curie Actions under FP7. The GBM project is supported by the German Bundesministerium für Wirtschaft und Technologie (BMWi) via the Deutsches Zentrum für Luft- und Raumfahrt (DLR) under the contract numbers 50 QV 0301 and 50 OG 0502.

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All Tables

Table 1

Mean and standard deviation of the output spectral parameters of the simulated bursts.

In the text
Table 2

Rest-frame parameters of 32 GBM GRBs.

In the text

All Figures

thumbnail Fig. 1

Redshift distribution in % of GBM GRBs (blue solid line) compared to all 239 GRBs with measured redshift to date (red dashed line). Both samples contain long and short bursts.
In the text
thumbnail Fig. 2

Ep values of 1367 BATSE/CGRO GRBs with a log-normal fit (black dotted histogram and line), 375 GBM GRBs and log-normal fit (red dashed histogram and line) and 30 GBM GRBs with redshift measurement (blue solid histogram). A KS test suggests that all three samples were drawn from the same distribution.
In the text
thumbnail Fig. 3

Rest-frame distribution of Ep for 4 short (hatched histogram) and 26 long (empty histogram) GBM-GRBs. For two GRBs (GRB 090519A, GRB 080928) a simple PL model fits the data best, thus not providing an Ep value. The complete distribution has mean and median values of 1.1 MeV and 750 keV, respectively. A log-normal fit to the sample of long GRBs (dashed line) peaks at  ~800 keV and has a FWHM = 0.93 in log-space.
In the text
thumbnail Fig. 4

T90,rest distribution of 4 short (red hatched histogram) and 28 long (blue histogram) GBM GRBs in the redshift corrected energy interval from 50 keV to 300 keV. The log-normal fit to the distribution (dashed line) of the long bursts peaks at  ~ 12 s and has a FWHM = 1.1 in log-space.
In the text
thumbnail Fig. 5

Energy fluence distribution of 32 GBM GRBs determined in the energy range between 1/(1 + z) keV to 10/(1 + z) MeV range. The complete distribution has mean and median values of 1.6  ×  10-5   erg   cm-2 and 5.9  ×  10-5   erg   cm-2. A log-normal fit (dashed line) to the long GRBs peaks at 2.2  ×  10-5   erg   cm-2 and has a FWHM = 2.2 in log-space.
In the text
thumbnail Fig. 6

Eiso distribution for long (blue hatched histogram) and short (red hatched histogram). For long bursts the distribution has mean and median values of 1.4  ×  1053 erg and 1.2  ×  1053 erg, respectively. A log-normal fit to the long bursts (dashed line) peaks at 1.2  ×  1053 erg with a FWHM = 1.9 in log-space. Short bursts have a mean and median value of 4  ×  1051 erg and 2.9  ×  1051 erg, respectively. A log-normal fit to the combined sample (dash-dotted line) peaks at 9.8  ×  1052 erg with a FWHM = 2.2 in log-space.
In the text
thumbnail Fig. 7

Amati relation for 4 short (red open triangles) and 26 long (blue open diamonds) GBM GRBs. Also shown is the best power-law fit to the data (black solid line) and the extrinsic 2σ scatter (black dashed lines) with the best power-law fit published by Amati (2010) (red dash-dotted line) with the 2σ scatter (red long dashed line).
In the text
thumbnail Fig. 8

Yonetoku relation for 4 short (red open triangles) and 26 long (blue open diamonds) GBM GRBs. Also shown is the best power-law fit to the data (black dashed line).
In the text
thumbnail Fig. 9

Testing the cosmic evolution of T90,rest. No correlation is evident. Bursts with Ep,rest ≤ 500 keV are shown as crosses, bursts with Ep,rest > 500 keV shown as filled circles. The two bursts for which are best fit by a simple power-law are shown as open diamonds.
In the text
thumbnail Fig. 10

Testing the cosmic evolution of Ep,rest for long (diamonds) and short (triangles) GRBs. The solid line indicates the redshift corrected lower limit for GBM to measure Ep which is currently estimated to be  ~15 keV in the observer frame (see Table 1). Bursts with high (green), intermediate (blue) and low (red) peak photon flux (in ph cm-2 s-1) are labeled.
In the text
thumbnail Fig. 11

Same as Fig. 10 but with a line indicating the maximum possible redshift, zmax that these 30 GRBs can have to still be detectable by GBM.
In the text

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