A Yoshida

A Pélangeon

J-L Atteia

Y E Nakagawa

K Hurley

R Vanderspek

M Suzuki

N Kawai

G Pizzichini

M Boër

J Braga

G Crew

T Q Donaghy

J P Dezalay

J Doty

E E Fenimore

M Galassi

C Graziani

J G Jernigan

D Q Lamb

A Levine

J Manchanda

F Martel

M Matsuoka

J-F Olive

G Prigozhin

G R Ricker

T Sakamoto

Y Shirasaki

S Sugita

K Takagishi

T Tamagawa

J Villasenor

S E Woosley

M Yamauchi

Intrinsic properties of a complete sample of HETE-2 Gamma-ray bursts A measure of the GRB rate in the Local Universe

Keywords: Gamma rays, bursts -X rays, bursts

published or not. The documents may come

Intrinsic properties of a complete sample of HETE-2 gamma-ray bursts. A measure of the GRB rate in the

Local Universe

Introduction

In recent years we have learned that long Gamma-ray bursts (T 90 > 2 sec.) are associated with the death of massive stars (M 20-30 M ⊙ ). This origin has been clearly established by the association of a few nearby GRBs with Type Ib/c supernovae (SNe Ib/c): GRB 980425 detected by Beppo-SAX (Pian et al. 1999) and SN 1998bw (Galama et al. 1998;Kulkarni et al. 1998), GRB 021211 detected by HETE-2 (Crew et al. 2002, 2003) and SN 2002lt (Della Valle et al. 2003), GRB 030329 also detected by HETE-2 (Vanderspek et al. 2003(Vanderspek et al. , 2004) ) and SN 2003dh (e.g. Matheson et al. 2003;Stanek et al. 2003;Hjorth et al. 2003a;Mazzali et al. 2003), GRB 031203 detected by INTEGRAL (Götz et al. 2003) and SN 2003lw (Malesani et al. 2004), GRB 050525A detected by Swift-BAT (Band et al. 2005) and SN 2005nc (Della Valle et al. 2006b). The association of the nearby GRB 060218 detected by Swift-BAT (Cusumano et al. 2006) with SN 2006aj (Masetti et al. 2006;Modjaz et al. 2006;Pian et al. 2006;Campana et al. 2006;[START_REF] Sollerman | [END_REF]Mirabal et al. 2006b;Cobb et al. 2006) provided a remarkable example of this paradigm. Even though a massive star's explosion can explain the production of both the GRB and the supernova (see for instance the collapsar model proposed by Woosley 1993; see also Woosley & Bloom 2006;Della Valle 2006a), these two phenomena are profoundly different in nature: the GRB is due to an ultra-relativistic outflow generated by a newborn black hole or magnetar, while the supernova is powered by the radioactive decay of 56 Ni within a massive shell of matter ejected at sub-relativistic velocities (e.g. Soderberg et al. 2006d). The question of the link between GRBs and SNe has recently become more complicated by the discovery of two low-redshift, long gamma-ray bursts -GRB 060505 at z =0.089 and GRB 060614 at z =0.125 -which were not associated with a supernova (Gehrels et al. 2006;Fynbo et al. 2006b;Della Valle et al 2006d;Gal-Yam et al. 2006). It is thus fair to say that despite very significant advances in our understanding of GRBs, the global picture continues to escape us and key issues remain to be elucidated. Concerning the connection between GRBs and supernovae, the observational clues are based on two facts: the clear association of some GRBs with Type Ib/c supernovae, and the evidence, based on statistical studies, that the majority of the 9000 Type Ib/c supernovae exploding each year in the universe do not produce GRBs. This last point was addressed by [START_REF] Berger | [END_REF], Soderberg et al. (2006aSoderberg et al. ( , 2006bSoderberg et al. ( , 2006cSoderberg et al. ( , 2006dSoderberg et al. ( , 2004b) ) and Della [START_REF] Valle | Interacting Binaries: Accretion[END_REF]Valle ( , 2006aValle ( , 2006c)).

While the SNe Ib/c population is rather well sampled in our local environment, the situation is quite different for GRBs, which need a complex series of detections to be fully characterized: detection of the high-energy signal in space, quick transmission to the ground, identification of the afterglow at X-ray, optical or radio wavelengths, and measurement of the redshift. This situation makes our understanding of the GRB population much less secure than for supernovae. Indeed, the observations by Beppo-SAX, HETE-2, and now Swift have demonstrated the diversity of the GRB population, which is composed of classical GRBs, short-duration GRBs, X-ray flashes (XRFs), and low-luminosity GRBs. In this classification, the 'classical GRBs' appear to be the 'high-luminosity' part (HL-GRBs) of a population which could consist mostly of lowluminosity or subluminous bursts (LL-GRBs), like GRB 980425 (Tinney at al. 1998) and GRB 060218 (Mirabal et al 2006a), or of X-ray flashes.

In this paper we take advantage of the broad energy coverage of HETE-2 instruments (2-400 keV) to discuss the global properties of the GRB population, and to infer a rate of GRB+XRF events which can be compared with theoretical predictions and with the known rate of Type Ib/c supernovae. This work is partly based on the forthcoming catalog of HETE-2 GRBs (Vanderspek et al. 2008). For a description of the HETE-2 mission and its instrumentation, see Ricker et al. (2001), Atteia et al. (2003a) and [START_REF] Villasenor | Gamma-Ray Burst and Afterglow Astronomy[END_REF]. Our work relies on the construction of a complete GRB sample containing all the long-duration bursts localized by the Wide Field X-Ray Monitor (WXM) and having a signal-to-noise ratio (SNR) larger than a given threshold in FREGATE (FREnch GAmma-ray TElescope) or in the WXM. For each burst in this sample, we compute the intrinsic properties at the source (T 90 , E peak , and E iso ) by correcting the observed properties for the effect of the redshift. When a spectroscopic redshift is not available (62 GRBs out of 82), we use the pseudo-redshift following the method of Pélangeon et al., i.e. the estimate of the redshift which is partly based on both the E peak -E iso (e.g. Amati et al. 2002) and E peak -L iso (Yonetoku et al. 2004) correlations, and derived from the spectral properties of the prompt emission (for more details see Pélangeon et al. 2006a andPélangeon &Atteia 2006b). We emphasize that this procedure has little impact on the final results since it introduces an additional uncertainty which is much smaller than the intrinsic dispersion of the parameters under study. In a second step we derive the 'visibility distance' for each burst (z max ), the distance at which the SNR of the burst reaches the threshold of our analysis, and the number of GRB within its visibility volume (N Vmax ). We then attribute to each GRB a weight W = 1/N Vmax , accounting both for the detection significance and the star formation history of the universe. Consequently, this method enables us to renormalize the global distribution, taking into account the true rate of occurrence of each type of GRB. We finally derive the GRB rate detected by HETE-2, and we discuss the relative importance of XRFs and classical GRBs in the overall GRB population.

This paper is organized as follows: Section 2 is devoted to the description of the HETE-2 GRB sample. Section 3 references and describes the tools necessary for our study. The three parameters studied here, E peak , T 90 , and E iso , are discussed in Sections 4, 5 and 6 respectively. The remaining three sections are dedicated to the interpretation of the results. In Section 7, we check the existence of correlations between the intrinsic parameters. In Section 8, we show that our study gives some clues to the nature of X-ray flashes in terms of intrinsic energetics and distance-scales. Finally, in Section 9 we discuss the rate of GRBs in the local universe and compare it with previous estimates obtained by other authors. A summary of the main results obtained in this paper is presented in Section 10.

Obtaining of a complete sample of HETE-2 bursts

Instrumentation

Since its launch in October 2000, the HETE-2 satellite (Ricker et al. 2001), dedicated to the detection and observation of Gamma-Ray Bursts, has detected 250 events classified as GRBs (Vanderspek et al. 2008). Thanks to the combination of its 3 instruments, the FREnch GAmma-ray TElescope (FREGATE, Atteia et al. 2003a), the Wide-field X-ray Camera (WXM, Shirasaki et al. 2003) and the Soft X-ray Camera (SXC, [START_REF] Villasenor | Gamma-Ray Burst and Afterglow Astronomy[END_REF], HETE-2 has a broad energy range covering the hard X-rays and γ-rays (2-400 keV), allowing the detection of both classical and soft GRBs (XRFs). Moreover, HETE-2 was designed to provide GRB positions to the community through the GRB Coordinates Network (GCN) [START_REF] Barthelmy | Proc. of Gamma-ray bursts: 5th Huntsville Symposium. AIP Conference Proceedings[END_REF] with an accuracy of one to several arcminutes, within seconds of the trigger. Consequently, multi-wavelength ground follow-up was done for most HETE-2 GRBs, leading to the determination of 25 spectroscopic redshifts.

Burst selection

It is of prime importance for our study to construct a GRB sample for which the detection criteria are fully understood and can be reproduced. This is essential for the determination of the visibility volume of each GRB in the sample. In addition it is necessary to be able to compute for each burst the three parameters of our study: E peak , T 90 , and the fluence. This section explains the construction of the sample1 .

Availability of the spectral data

We considered all the 250 events that constitute the total sample detected by HETE-2, i.e. both triggered and untriggered bursts, thus reducing the bias due to trigger algorithms. Then, we selected the bursts for which the angle of incidence was measured by any experiment. Its knowledge is necessary to perform the spectral analysis of the bursts and to derive their E peak and E iso . Of the 250 GRBs, 132 have no accurate localization and were rejected, leaving 118 GRBs. Among these bursts, we have also rejected GRB 040810 because this long and intense burst occurred as the instruments were being shut down. Consequently only the precursor was detected by HETE-2.

Technical problems

Of the remaining 117 GRBs, 2 bursts were rejected due to technical problems: GRB 010110 was detected and localized by the WXM when FREGATE was not working. It is thus impossible to constrain the spectral parameters with only the WXM data. The second burst rejected is GRB 021113. At the moment of its detection there were onboard software problems that did not permit the trigger to reach the ground. The spectral data are thus not available.

WXM localization

Considering the 115 remaining GRBs with a known angle, we removed 24 GRBs for which the angle was obtained through the localization by the Inter Planetary Network (IPN) or the Swift satellite (Gehrels et al. 2004). Indeed, as the IPN and Swift have different sensitivities, we would have added a bias to our sample by considering them. Moreover, in order to have bursts well within the field of view of the WXM, we have considered an 'incident angle limit' of 45 • . This angle corresponds to a limit of detection for the WXM (Shirasaki et al. 2003), and to a decrease of the effective area for FREGATE (Atteia et al. 2003a). With this cut, only one burst is rejected: GRB 020201, whose angle is 55 • . This step leaves 90 GRBs in the sample (see Fig. 1).

Short-duration bursts

We also rejected the two bursts classified without ambiguity as short-duration bursts: GRB 020531 [START_REF] Lamb | proc. 3rd Rome Workshop on Gamma-Ray Bursts in the Afterglow Era[END_REF] and GRB 050709 [START_REF] Villasenor | [END_REF]. This is based on the fact that this class of burst is probably not associated with the same progenitors as the long GRBs. As one goal of this paper is to discuss the rate of long GRBs in the local universe and to compare it with the rate of SNe, we have excluded the short-duration bursts, leaving 88 bursts in the sample.

Threshold cut

Finally, in order to have homogeneous detection criteria for all the events of the sample, we removed the GRBs with a signalto-noise ratio (SNR) lower than a given threshold in FREGATE and in the WXM. These SNR were computed as follows:

-we measured the peak count rate of each burst. For FREGATE we used two different time resolutions (1.3 and 5.2 s) and two different energy ranges (6-80 and 30-400 keV) corresponding to the bands used by the instrument to trigger.

For the WXM we used three different time resolutions (1.2, 4.9 and 9.8 s) and the total energy range of the WXM (2-25 keV). -we computed a SNR for each burst in all the combinations of time resolution and energy range. We kept only the GRBs for which the SNR exceeded 7.9 in FREGATE or 5.8 in the WXM (in at least one combination of the time and energy ranges, see Fig. 2).

This last criterion removed 6 bursts with a low SNR: GRB 011103, XRF 020903, GRB 030323, GRB 030706, GRB 040131 and GRB 040228.

Our final sample contains 82 GRBs whose properties are given in Tables 2 and3.

The cases of XRF 020903 and GRB 030323

These two bursts have a spectroscopic redshift but they are rejected from our sample due to their SNR just below the thresholds of FREGATE and the WXM (Fig. 2). As they have spectroscopic redshifts -z = 0.25 for XRF 020903 (Soderberg et al. 2004a) and z = 3.372 for GRB 030323 (Vreeswijk et al. 2003) -we can argue that even if they had been present in our sample, the results presented in this paper would actually have been reinforced. Indeed, situated at a moderately high redshift, GRB 030323 has a large visibility volume and thus a low weight.

On the other hand, the nearby XRF 020903 has a small visibility volume and hence a large weight. Since this is an XRF with a low E peak , adding it to our GRB sample would have increased the predominance of X-ray flashes which is shown in Section 4, and strengthened the conclusions of this paper.

Deriving the intrinsic properties of HETE-2 GRBs

Burst parameters

We have used the spectral parameters and durations available in the HETE-2 Catalog (Vanderspek et al. 2008) which summa- rizes the information on GRBs detected by HETE-2 during the entire mission and thus complements the previous spectral analyses based on smaller samples of HETE-2 GRBs performed by Barraud et al. (2003Barraud et al. ( , 2004) ) and Sakamoto et al. (2005).

Burst distance

At the end of the 90s, the growing sample of GRBs with measured redshift allowed new types of studies: searches for correlations between physical quantities -characterizing the gamma-ray bursts' light-curves and/or spectra -in their rest frame. Following the discovery of correlations between E iso , the isotropic equivalent energy emitted by the burst, and various intrinsic GRB properties, it has been suggested that some GRB observables could be used as luminosity indicators, and hence as redshift indicators. The most used are the lag-luminosity (τ lag -L p,iso ) correlation (Norris, Marani & Bonnell 2000), the variability-luminosity (V-L p,iso ) correlation (e.g. Fenimore & Ramirez-Ruiz 2000;Reichart & al. 2001;Lloyd-Ronning, Fryer & Ramirez-Ruiz 2002a) and the E peak -E iso correlation -suggested by several authors (see e.g. Lloyd, Petrosian & Mallozzi 2000;Atteia 2000;Lloyd-Ronning & Ramirez-Ruiz 2002b)before Amati et al. (2002) firmly established it with a study of 12 GRBs with known redshift. Other correlations were found later, such as the E peak -L p,iso correlation (Yonetoku et al. 2004), the E peak -E γ correlation (Ghirlanda, Ghisellini & Lazzati 2004) and the E peak -L p,iso -T 0.45 correlation (Firmani et al. 2006). Recently, in his construction of the Hubble Diagram from a sample of 69 GRBs with redshifts, Schaefer (2007) showed that the use of five of these correlations simultaneously -lag-luminosity (τ lag -L p,iso , Norris et al. 2000); V-L p,iso (Fenimore & Ramirez-Ruiz 2000); E peak -L p,iso [START_REF] Schaefer | [END_REF]); E peak -E γ (Ghirlanda et al. 2004) and the minimum light-curves rise time versus luminosity (τ RT -L p,iso , Schaefer 2002)2 -led to more reliable luminosity distances than the use of only one distance indicator. The design of HETE-2 permitted the study of the energetics at work in GRBs, both in the observer and in the source frames. For example, the E peak -E iso correlation (Amati et al. 2002) was confirmed and extended at lower energy, thanks to the sample of HETE-2 X-ray rich GRBs (XRRs) and X-ray flashes (Amati 2003;Lamb, Donaghy & Graziani 2004a;Lamb et al. 2004b;Amati 2006), particularly XRF 020903 (Sakamoto et al. 2004). The HETE-2 sample is thus appropriate for the determination of pseudo-redshifts based on the E peak -E iso correlation. A first attempt in this direction was proposed by Atteia (2003b), which was later revised by Pélangeon et al. (2006a) to take into account the complex lightcurves of some long GRBs3 . The accuracy of this redshift indicator and its use in this study is discussed in the following paragraph.

Justification of the pseudo-redshifts used in this study

As shown in Table 3, 18 GRBs contained in our sample have a spectroscopic redshift and 2 have a photometric redshift (GRB 020127 and GRB 030115). We computed a pseudoredshift for the remaining 62 (among the 20 bursts having a known redshift, 14 were used for calibrating the pseudo-z). In order to test the impact of the use of pseudo-redshifts in this work:

-we computed the pseudo-redshifts of the 20 GRBs having a spectroscopic or photometric redshift, -for these same 20 GRBs we computed the ratio between the pseudo-redshift and the redshift, and the ratio of the pseudointrinsic properties derived with pseudo-redshifts to the intrinsic properties derived using the redshift (Fig. 3).

We note that the dispersion of the ratio between the pseudoredshifts and the redshifts is smaller than a factor of 2, except for two outliers: GRB 020819 and GRB 051022 for which the spectroscopic redshift differs from the pseudoredshift by factors of 2.95 and 2.15 respectively. For these 20 GRBs, the dispersion between the luminosity distance estimated with the pseudo-redshifts (D L,estim ) and the luminosity distance measured with the redshifts (D L,meas ) is:

σ D L = log(D L,estim /D L,meas ) = 0.
125 dex, i.e a factor of 1.34 4 . The dispersion is also smaller than a factor of 2 for the E peak (Fig. 3, top-right panel) and the T 90 parameters (Fig. 3, bottomleft panel). This is not surprising since these two quantities vary as (1 + z), reducing the impact of the redshift errors. E iso , which is more redshift-dependent than E peak and the duration, is more scattered and its dispersion is larger than a factor of 2. This corresponds to the bursts having the highest difference between their spectroscopic redshift and their pseudo-redshift: GRB 020819, GRB 040912B and GRB 051022 have a pseudo-z estimate two to three times higher than their redshift; and GRB 020124, GRB 030115 and GRB 050408 for which the pseudo-z is lower than the spectroscopic redshift by a factor of 2 (Fig. 3, bottom-right panel).

We consider that the good agreement of pseudo-z with spectroscopic redshifts for the 20 GRBs having a measured z is encouraging. One may question however if the pseudo-z values computed without knowing the spectroscopic redshift are as accurate as those computed when the redshift is known -i.e. do pseudo-z have predictive power? -Unfortunately, most of the bursts for which we have computed a pseudo-z do not have redshift measurements, so comparison with the redshift-estimates cannot be done. However, in some cases the pseudo-z values we issued in GCN Circulars were followed by a spectroscopic measurement. For instance, we determined the pseudo-redshifts of GRB 050525 5 and GRB 070125 6 before their redshifts 4 If we do not consider the two outliers GRB 020819 and GRB 051022 the dispersion is σ D L = 0.089 dex, i.e a factor of 1.23. 5 The first GCN Circular containing the computation of a pseudoredshift in nearly real time was done for this burst, using the spectral parameters obtained by Konus-WIND (Golenetskii et al 2005a). We found ẑ = 0.36 ± 0.10 (Atteia & Pélangeon 2005b) whereas the spectroscopic redshift obtained by spectroscopy of the host galaxy and published the day after was z = 0.606 (Foley et al. 2005). The difference between the pseudo-redshift and the redshift was due to an incorrect published fluence (Golenetskii et al. 2005b). As soon as the correct value was available, we recomputed the pseudo-redshift of this burst and found ẑ = 0.64 ± 0.10 (Atteia & Pélangeon 2005c). 6 We have computed a redshift-estimate of this bright burst using the spectral parameters derived from two different instruments, Rhessi (ẑ = In order to quantify the effect of the uncertainty from the pseudo-redshifts on the intrinsic distributions, we performed the following test: we artificially increased the errors on the pseudo-z values which are taken into account in the construction of the intrinsic parameters distributions (see Section 4.3). This extra dispersion takes on three values corresponding to factors of 1.4, 2.0 and 3.0. We tested the impact of this extra dispersion on the intrinsic E peak distributions ('simple' and 'unbiased'), and thus we show that even if the pseudo-z were only accurate to a factor of 2 or 3, this would not significantly change the intrinsic parameter distributions and all the subsequent results obtained in this paper (see Sect. 4). Butler et al. (2007) that the correlations between the intrinsic GRB parameters found in the pre-Swift era were due to instrumental biases (e.g. detector thresholds effects) rather than to real physical properties of the sources. This is the result of a study they performed on a sample of 77 Swift GRBs with measured redshifts, where they found that those GRBs are on average harder than the pre-Swift measurements, i.e. they have higher E peak and lower E iso . As a consequence, they conclude that these correlations cannot be used to estimate the GRB redshifts. Since no clear conclusion can be drawn from this debate, we have also considered this possibility and we calculated all the distributions twice: one case where we attribute a pseudoredshift to the 62 bursts which have no redshift, and the other case where we randomly attribute one of the 24 secure redshifts measured for the HETE-2 long GRBs. Therefore, for all the distributions we obtain we will comment on how they change between the two cases.

It was recently proposed by

Throughout the paper we use the following notations:

-the spectroscopic or photometric redshifts are denoted by z.

-the pseudo-redshifts are denoted by ẑ.

-z sample refers to the value of the redshift used in our study:

this is the spectroscopic/photometric redshift (z) when it is available and the pseudo-redshift (ẑ) otherwise (see Table 2).

Parameter 1: E peak

Measuring the E peak of X-ray flashes

The peak energy of a burst is generally measured using the XSPEC software [START_REF] Arnaud | XSPEC, An X-Ray Spectral Fitting Package[END_REF] and fitting a phenomenological model -GRB (Band) Model (Band et al. 1993) or Cutoff Power Law Model -on the spectral data. However, for some X-ray flashes, the peak energy is lower than 10 keV, so the data do not constrain the four parameters of the Band function. For those bursts, the value obtained is too close to the lower boundary of FREGATE (6 keV) and of the WXM (2 keV). In this case, we observe a power-law spectrum instead of a Band function. To solve this problem, Sakamoto et al. (2004) showed that a modified function, called the Constrained Band Model (CBM), based on a three-dimensional subspace of the full fourdimensional Band function parameter space, could be used. We adopted this model to fit the data when it was necessary (see Table 2). However, in order to check whether such a model could add a bias to the estimate of E peak , we used XSPEC to simulate an XRF spectrum as follows:

-we use the data of a real XRF, in order to have real background statistics and instrumental response matrices. -we construct a fake burst that follows a Band model with typical values of -1.0 and -2.3 for the power-law indices at low and high energy respectively. These parameters correspond to the mean values for GRB spectra according to BATSE results (see e.g. Preece et al. 2000;Kaneko et al. 2006). The two other parameters, E peak and the normalization, are the variable parameters that let us produce fake XRFs of greater or lesser intensity. -we generate fake data files for XRFs of various brightnesses and various E peak . -we then perform a spectral analysis using a Band model and a Constrained Band model, and compare the parameters obtained.

We find that the Band model is valid for E peak down to 8 keV. For this limiting value, the errors on the spectral parameters are slightly larger than those obtained using the Constrained Band function, but they are still valid. For E peak below 8 keV, the Constrained Band model is preferable: it gives correct E peak values or provides at least reliable upper limits. In all cases we obtain consistent values between the CBM and the Band function when both functions are applicable, and these values are consistent with the ones we introduced to produce the fake XRFs. We thus conclude that the values determined with the Constrained Band model are not biased.

The observed E peak distribution

Several points can be noticed in Figure 4: first, the E obs peak distribution peaks at about 100 keV and extends over the entire energy range of HETE-2 (2-400 keV). Second, the two classes of bursts (classical GRBs and X-ray flashes) are clearly distinguished. As a comparison, the study of Kaneko et al. (2006) on the complete spectral catalog of bright BATSE Gamma-Ray Bursts, leads to an E obs peak distribution peaking at about 300 keV, extended at high energies up to several MeV and without significant events at very low energies. The paucity of soft (∼a few keV) and hard (∼MeV) bursts in our sample is probably not real. For the soft bursts, this is clearly visible by a simple comparison of the BATSE and HETE-2 E peak distributions. As BATSE triggers were generally done on the count rate between 50 and 300 keV, the ratio between the XRFs and the GRBs is low. For HETE-2, the energy range extends down to 2 keV and enables the detection of more XRFs but still prevents the detection of XRFs with E peak below a few keV. As far as the hard tail of the E peak distribution is concerned, the low sensitivity of the detectors at high energy is also one of the causes of the inefficient detection of hard bursts. Moreover, two bursts emitting the same energy with different hardness do not have the same number of photons: the softer GRB has more photons than the harder one. Consequently, BATSE-like or HETE2-like missions are probably not able to derive a true observed E peak distribution. The distributions obtained clearly show that the E obs peak distribution depends strongly on the energy range of the instruments involved.

The intrinsic E peak distribution

So far, the intrinsic E peak distribution has been very little discussed, mainly due to the difficulty of measuring both the E peak and the redshift of the bursts. On the one hand, BATSE, Beppo-SAX and HETE-2 did constrain the E obs peak thanks to their wide energy ranges, but the bursts they detected suffered from a lack of redshift determination. On the other hand, thanks to its performance and fast localization capability, Swift currently enables more redshift measurements, but the narrow energy range of the BAT instrument (15-150 keV) does not allow a good constraint of the E obs peak . As mentioned by Amati (2006), about 70 intrinsic E peak are available, but this sample contains bursts detected by different satellites. Our work is based on a sample containing all the bursts localized with a single satellite, hence avoiding selection effects inherent to the use of data from different instruments. We see in Figure 5 that the E intr peak distribution is broader than the E obs peak distribution. At high energies, some bursts have an E intr peak reaching a few MeV. Nevertheless, as we previously mentioned, the energy range of HETE-2 (2-400 keV) prevents us from drawing any conclusion about the high energy part of the E peak distribution.

At low energies, bursts having an E obs peak lower than 20 keV, and defined as X-ray flashes in the observer frame, still have a low E peak in their source frame. For most of those GRBs, the low intrinsic peak energy is due to their intrinsic faintness as they are at low or intermediate redshift. (This point is discussed in more detail in Section 8). Joining these two extreme energy ranges, the intrinsic E peak distribution of HETE-2 GRBs extends over 3 decades in energy. The true width of the distribution is probably even larger because bursts with E peak lower than 1 keV are not detected, and bursts with E peak greater than 1 MeV cannot be constrained by an HETE2-like mission. The bursts on the left part of the histogram correspond to the events that have a measured higher limit in E obs peak of about 4 keV (see Table 2). The energy range of HETE-2 (2-400 keV) is symbolized by the magenta horizontal solid line.

The unbiased E peak distribution

Construction

In order to derive the true E peak distribution, the distance-scale is not the only correction that has to be applied to the observed E peak distribution. All the bursts do not have the same brightness and are not detected at the same significance in the observer frame. We took this into account to correct the intrinsic E peak distribution obtained in the previous section.

To do this we used the following method:

(1) in order to take into account the errors on all the parameters used (spectral properties and redshift) we have computed 'smoothed distributions' by: -producing for each burst a set of 100 E obs peak,sim randomly selected within the 90% confidence level (c.l.) error range of the measured E obs peak given in Table 2 7 and 100 redshifts randomly selected within the 90% c.l. error range of their corresponding spectroscopic/photometric redshift or pseudoredshift z sim . Combined with the 100 simulated E obs peak,sim , we thus obtained a sample of 100 simulated E intr peak,sim . -representing each of the 8200 simulated bursts (b) with a normal distribution of the logarithmic E intr peak values (LE intr p,b ):

f b (le p,i ) = 1 σ LE intr p,b × exp - 1 2 le p,i -µ LE intr p,b σ LE intr p,b 2 (1)
where le p,i is the variable of the functions f b , µ LE intr p,b is the base 10 logarithm of E intr peak,sim related to the burst b and σ LE intr p,b is set to 0.05.

(2) then, we computed for the 8200 simulated GRBs a weight related to their detection significance. For that, we:

-determine the maximum redshift z max at which the source could have been detected by the instruments, by first comparing the SNR computed in Section 2 (see Table 2) to the SNR threshold of the HETE-2 instruments, and then by redshifting the sources until their peak photon flux reaches the SNR threshold of the trigger instrument, giving us both the visibility distance between the source and the satellite and the maximum redshift z max . -assume that the GRB rate follows the star formation rate. For this we have adopted the model SFR 2 of Porciani & Madau 7 For some XRFs, the lower error limit could not be constrained. In this case the value was set to E obs peak,min = 1 keV. For some other GRBs, it was the upper error limit which was not constrained. In this case we set the value to E obs peak,max = 1 MeV.

(2001) that reproduces a fast evolution between z = 0 and z = 1 and remains constant beyond z ≥ 2

R SFR 2 (z) ∝ 0.15h 65 exp(3.4z) exp(3.4z) + 22
(2)

-derive for each burst the number of GRB per year within its visibility volume

N Vmax ∝ z max 0 dz dV(z) dz R S FR 2 (z) 1 + z (3)
In this equation dV(z)/dz is the comoving element volume, described by

dV(z) dz = c H 0 4π dl 2 (z) (1 + z) 2 [Ω M (1 + z) 3 + Ω K (1 + z) 2 + Ω Λ ] 1/2 (4)
where H 0 is the Hubble constant, Ω K is the curvature contribution to the present density parameter (Ω K = 1-Ω M -Ω Λ ), Ω M is the matter density and Ω Λ is the vacuum density. Throughout this paper we have assumed a flat ΛCDM universe where (H 0 , Ω M , Ω Λ )=(65h 65 km s -1 Mpc -1 , 0.3, 0.7). This procedure allows us to give each burst a weight (W b ) inversely proportional to N Vmax . The rationale of weighting each burst by 1/N Vmax is the following: the visibility volume is different for each GRB of our sample. Moreover, each burst observed is randomly taken from all the bursters present in its visibility volume. In this way, rare bright bursts, having a large visibility volume, will have low weights, while faint local GRBs will have heigher weights. This procedure also takes into account the fact that the GRB rate evolves with redshift, leading to the fact that GRBs are about ten times more frequent at z ∼ 1 than at present. Note that an evolution of the GRB distributions with the redshift is not included in this procedure, although such evolution probably exists (e.g. Daigne, Rossi & Mochkovitch 2006). This case is not addressed here because we do not have a sufficient number of GRBs.

(3) the total smoothed distribution (SD) is the sum of the individual functions f b (le p,i ), normalized by the corresponding burst's weight W b

SD (LE intr peak,sample ) = 1 n simu × n burst b=1 W b × f b (le p,i ) (5)
with n simu the number of simulations (100) for each burst and n burst the number of bursts contained in our sample (82).

Results

The resulting unbiased E peak distribution for our sample of bursts with this method is shown in Figure 6. In this Figure we see that the E peak distribution has dramatically changed, with a clear domination of the bursts having low intrinsic E peak , i.e. the Xray flashes.

As explained in Section 3, we tested whether a high accuracy for the pseudo-redshifts was necessary, i.e. whether the results obtained could depend on these estimated distance scales. To do this, we performed the same analysis as described throughout this Section, but this time for simulated redshifts randomly selected within the error ranges based on the initial ones and artificially increased by factors of 1.4, 2.0 and 3.0. We find that both the distribution of intrinsic E peak and the unbiased distribution of intrinsic E peak do not significantly depend on the accuracy of the pseudo-z within a factor 2 or 3 (Fig. 6). This result can be explained by the intrinsic dispersion of the parameters studied which is so large that the uncertainties in the distance scales do not strongly affect the results. Moreover, we also performed this analysis by assuming for the distance of each of the 62 GRBs without secure redshift a set of 100 redshifts randomly taken from the total redshift distribution of HETE-2 long bursts (24 values). The results obtained in this case differ significantly because the predominence of low E intr peak bursts has disappeared. With this method, the distribution is flat and broad. Here again, the distribution's width may not reflect the true one, but it contributes interesting information to the unbiased E peak distribution, which is probably situated -'in reality' -between the two curves presented in Fig. 6 (black solid line and blue dashed line).

Parameter 2: T 90

In this section we study the duration distribution of the complete sample of HETE-2 GRBs. We adopted T 90 as the common definition for all the bursts. We derive in the following sub-sections the observed, intrinsic and unbiased log T 90 distributions using the same method as described for the study of the intrinsic log E peak distribution (Sect. 4).

The observed T 90 distribution

T 90 is defined as the duration for which 90% of the counts in a given energy range are detected. Recall that in this study we only focus on long-duration GRBs. Over the 4 FREGATE bands (A: 6-40 keV; B: 6-80 keV; C: 30-400 keV; D: >400 keV) we chose the one that contained both the classical GRBs and the XRFs, i.e. the band B (6-80 keV). The log T 90 distribution in this energy range (see Fig. 7) extends from 0.33 to 2.73. The mean value of this distribution is 1.40 and the median value 1.33.

The intrinsic T 90 distribution

The histograms are shown in Figure 8. We caution that the observed T 90 are measured in the same energy range in the observer frame (see Table 2). Since this energy range depends on the redshift of the source, it is different for each burst and this has an impact on the intrinsic duration measurement. The histogram of the intrinsic log T 90 in the (6-80 keV) energy range extends from -0.26 to 2.40. The mean value of this distribution is 1.01 and the median value is 1.00. Contrary to the log E peak distribution, the intrinsic log T 90 distribution is not broader than the observed log T 90 distribution and is extended over about the same range (∼2.5 decades).

The unbiased T 90 distribution

Correcting the intrinsic T 90 for the N Vmax (see Part 4.4) we obtained the unbiased T 90 distribution (Fig. 9). We note that the GRB population is dominated by events with an intrinsic duration of about 10 s (for an energy range of 6-80 keV in the observer frame). Nevertheless, a non-negligable number of GRBs have a long intrinsic duration lasting about 300 s.

If we consider the case 2 study (Fig. 9), we obtain a significantly different unbiased T 90 distribution, more flat than the one obtained in case 1, i.e. no typical intrinsic duration for the GRB appears when considering distances randomly taken within the total redshift distribution of HETE-2 GRBs. The smoothed distribution based on the 20 bursts with spectroscopic/photometric redshifts (z) is represented by the lower solid red curve; the one for the complete sample is given by the upper solid black line. The blue dashed SD corresponds to case 2 (see Part 3.3), i.e. the sample of 6200 simulated GRBs (corresponding to the 62 GRBs of our sample without secure redshift) which were randomly given a redshift from the HETE-2 redshift distribution.

Parameter 3: isotropic energy

The observed fluence

Taking the fluences (i.e. the fluxes integrated over the duration of the bursts' spectra) in the HETE-2 Catalog (Vanderspek et al. 2008) and applying the same method described in Section 4, we show in Figure 10 the histograms of the logarithmic fluence in the energy range 2-30 keV. The distribution of the log S X extends between -7.12 and -4.24. The mean and median values of this distribution are -5.90 and -6.00, respectively.

The isotropic energy

The equivalent of the fluence measured in the observer frame is the energy radiated in the source frame. One easy way to estimate this energy is to assume an isotropic emission (E iso ) integrated over a fixed energy-range (1-10 4 keV in the observer frame, see e.g. Amati et al. 2002, Amati 2006). We computed this quantity for the 82 bursts and obtained the distribution shown in Fig. 11. This distribution of log E iso extends from 49.50 to 54.24, and has a mean value of 52.44 and a median value of 52.62. Two classes of bursts are evident, with a separation at about 51.5. Most of the bursts have a log E iso lying between 52 and 54 (the classical GRBs) and the second sample has a log E iso lower than 51.

The unbiased isotropic energy

The correction of these E iso for the N Vmax of each burst gives us the unbiased E iso distribution (see Fig. 12). This distribution decreases with E iso , and the best-fit power-law for the complete sample distribution, estimated between E iso =10 50 erg and E iso =10 54 erg is 10 -7.94 × E -0.84 iso,52 , where E iso,52 is in units of 10 52 erg. As for the two previous cases studied, this distribution is dramatically different from the observed and intrinsic ones. It shows a clear predominence of bursts with low E iso . This distribution hence strengthens the result we obtained in Section 4, i.e. the predominence of X-ray flashes in the overall GRB population. Here again we can discuss the impact of our second case calculation. In this case, the shape of the distribution is similar to the one obtained in case 1.

Correlations

Having determined the intrinsic properties of the HETE-2 GRBs, we test whether any correlations are found between these values. We performed a pairwise comparison of the four intrinsic parameters (E peak , T 90 , E iso , z sample ) to search for possible correlations. Six plots are obtained and shown in Figure 13. The corresponding correlation coefficients and associated p-values testing the hypothesis of no correlation are presented in Table 1. We note Table 1. Correlation coefficients (R) obtained for the pairwise comparison of the four intrinsic parameters discussed in this paper (E peak , T 90 , E iso , z sample ). The lower and upper bounds are at a 95% confidence interval. The corresponding p-values test the non-correlation by computing the probability of having a correlation as large as the observed value by chance. that the E intr peak -E iso (Amati) correlation is found (1 st panel) and that the sub-sample of XRFs fills the gap between XRF 020903 and the cluster of intrinsic classical GRBs (see next Section). This is not really surprising since our pseudo-redshift estimates assume that the E intr peak -E iso relation is valid. It is nevertheless interesting to note that nearly all of the 82 GRBs in our sample are consistent with this correlation with no strong outlier. While it is not the purpose of this paper to discuss the reality of E intr peakz sample and E iso -z sample correlations, we note that these correla- In each panel the GRBs that have a measured redshift (z) are represented with the green filled circles, and the ones having a pseudo-redshift (ẑ) are symbolized with the blue open circles. In the top left panel, by way of comparison, we show (solid line) the best-fit power-law E intr peak =77× 0.57 iso,52 (with E intr peak in keV and E iso,52 in units of 10 52 erg) found by Amati (2006) and also the vertical logarithmic deviation of 0.4 (dotted lines) displayed in Amati (2006). In the top right panel the dashed curve marks the redshift evolution of the E peak , assuming a burst with E (z=0) peak = 40 keV (see Stratta et al. 2007).

tions could be partially or totally explained by selection effects which practically prevent the detection of low-luminosity GRBs at high redshift. In the three last plots, no other 'simple' tight correlation is found, but these plots interestingly show the instrumental limits affecting the detection of the bursts (duration, energetics and redshift). For instance, we can clearly see that HETE-2 was lucky to detect the bright GRB 030329 (see panels 2 and 6). Hence Figure 13 can be used to show the characteristics of the bursts missed by HETE-2, i.e. mainly the GRBs at low-redshift with high E intr peak or high E iso ; and conversely the GRBs with low E peak occuring at high redshift.

The nature of X-ray flashes

The demonstration of the existence of X-ray flashes [START_REF] Heise | Proc. 2nd Rome Workshop: Gamma-Ray Bursts in the Afterglow Era[END_REF][START_REF] Kippen | Proc. 2nd Rome Workshop: Gamma-Ray Bursts in the Afterglow Era[END_REF] was immediately followed by several theories attempting to explain their nature and to determine whether they belong to the same class as the classical GRBs, or whether they are different cosmological events. Thanks to various studies of samples of X-ray flashes, several clues now allow us to unambiguously associate XRFs with the classical GRBs. XRFs are GRBs with lower E peak and higher fluxes in X-rays than in γ-rays (e.g. [START_REF] Kippen | [END_REF]Kippen et al. , 2004;;Barraud et al. 2003;Sakamoto et al. 2005). The other spectral parameters of the prompt emission and the temporal behaviour of XRF afterglows at various wavelengths are identical to those of classical GRBs (e.g. D 'Alessio, Piro & Rossi 2006). In our complete sample, we have 22 bursts classified as XRFs. 12. Unbiased E iso distributions for the complete sample of 82 GRBs localized by HETE-2. The smoothed distribution based on the 20 bursts with a spectroscopic/photometric redshift (z) is represented by the lower solid red curve; the one for the complete sample is shown by the upper solid black line. The blue dashed SD corresponds to case 2 (see Part 3.3), i.e. the sample of 6200 simulated GRBs (corresponding to the 62 GRBs of our sample without secure redshifts) which were randomly given a redshift from the HETE-2 redshift distribution. The bestfit power-law (green dashed line) for the complete sample, estimated between E iso =10 50 and 10 54 erg is 10 -7.94 × E iso 10 52 -0.84 .

In the following section we discuss some results obtained with this sample.

The redshift distribution of HETE-2-XRFs

To explain the origin of the XRFs, [START_REF] Heise | Proc. 2nd Rome Workshop: Gamma-Ray Bursts in the Afterglow Era[END_REF] proposed that they are classical GRBs occuring at high redshift. If we consider the redshift distribution of the HETE-2 complete sample, we note that the sub-sample of XRFs has a mean distance scale < z sample > HETE-2 XRFs = 0.82, lower than the mean redshift of the 'classical' GRBs: < z sample > HETE-2 GRBs = 2.09 (see Fig. 14). The mean redshift of the XRFs contained in our sample is also lower than the mean redshift of the Swift XRFs derived by Gendre, Galli & Piro (2007): < z > Swift XRFs = 1.40 for a sample of 9 XRFs with known redshifts. Consequently, we reject this hypothesis, as was previously done in other studies, e.g. by Barraud et al. (2003) who highlighted this fact with the similar duration distributions they obtained for the XRFs and the long classical GRBs.

Do X-ray flashes form a distinct population ?

We consider in this paragraph the question posed, e.g., in Stratta et al. (2007): do the XRFs form a continuum extending the 'classical GRBs' to soft energies, or do they form a distinct population, the intrinsic XRFs (i-XRFs)? Fig. 13 (panel 1) shows the Amati relation for the complete sample of 82 HETE-2 bursts and Fig. 13 (panel 2) plots the E intr peak versus the distance-scales of the sources. We do not find any clear evidence for a distinct population and the results indicate a continuum between the classical GRBs and the XRFs. Indeed, (cdf). The cdf of the HETE-2 complete sample is shown with the green solid line. The two cumulative distributions for the 'classical' GRBs and the XRFs are shown with the blue dashed line and the red dotted line, respectively. We note that most of the 22 XRFs are definitely situated at intermediate or low redshift. For comparison, we also considered all the long-duration bursts with redshifts detected by Swift-BAT from the beginning of the mission to GRB 080210 (the last Swift burst with a spectroscopic redshift measured to date). The corresponding cdf is the black solid line.

according to the HETE-2 sample, several XRFs are situated in the gap between the group of intrinsic classical GRBs and the two intrinsic XRFs 020903 (Sakamoto et al. 2004) and 060218 (Amati et al. 2007), as is XRF 050416, a soft burst detected by Swift (Sakamoto et al. 2006a(Sakamoto et al. , 2006b)). However, half of the subsample of the observed XRFs are part of the cluster of intrinsic classical GRBs when considered in their source frame. This fact was also noted by Stratta et al. (2007). The various corrections (distance scale, number of GRB within the visibility volume) applied to the distributions obtained in the observer frame allowed us to obtain the unbiased distributions of the intrinsic parameters, i.e. distributions that reflect more closely the true distributions of these parameters and the dominant characteristics of the gamma-ray burst population. This study also allows us to derive the rate of GRBs detected and localized by HETE-2 (R H2 0 ). For that purpose, we consider that each GRB in our sample contributes to the local rate in proportion to:

The rate of GRBs in the Local Universe

h b = N Vloc (z = 0.1) N Vmax (z = z b,max ) (6)
with the N V (z) computed according to Eq. 3, and we obtain the rate of HETE-2 GRBs during the mission,

τ = 1 V loc n burst b=1 h b (7)
which is 3.30 Gpc -3 .

In order to normalize this rate per year, we took into account the effective monitoring time of the WXM, obtained from:

T m = T mission × T ǫ 4π × S cov (8)
where T m is the effective monitoring time of the WXM, T mission = 69 months is the duration of the mission, T ǫ = 37% is the mean observation efficiency during T mission , and S cov = 2π(1 -cos 45 • ) = 1.84 sr the sky-coverage of the WXM8 . The effective monitoring time of the WXM is hence T m = 0.31 yr. Using this, the rate of GRBs in the Local Universe per Gpc 3 and per year can be found to be ∼10.6 Gpc -3 yr -1 . This result is a lower limit because as shown in Section 4, the bursts that dominate the overall population are the ones with low E intr peak , and as we showed in Section 7, HETE-2 missed the bursts with intrinsic E peak lower than 1 or 2 keV as well as bursts occuring at very high redshifts. If we consider the unbiased distributions obtained in case 2 (shown as dotted blue lines throughout the different Figures), we obtain 1.02 Gpc -3 , corresponding to a rate of ∼3.3 Gpc -3 yr -1 . As shown in Section 4, the population of gamma-ray bursts is dominated by the X-ray flashes. This is understandable as XRFs are soft but also faint in the observer frame, according to the hardness-intensity relation derived by Barraud et al. (2003). Therefore, if the rate of detected bursts is in fact higher for classical GRBs than for XRFs, we can guess that this is only due to instrumental limitations, such as the size and the sensitivity of the detectors.

Comparison with the rates of GRBs obtained in previous studies

Several studies have derived local rates of GRBs using various methods mainly based on theoretical considerations. According to the results obtained so far, the predicted rate of GRBs in the Universe seems to be between unity (or less) GRB and several hundreds per Gpc 3 and per year. These results seem to be related to the nature of the bursts considered in the studies, i.e (1) 'classical GRBs' also called 'high-luminosity GRBs' (HL-GRBs) and/or (2) sub-luminous (low-luminosity) GRBs (LL-GRBs).

(1) Several authors obtained a GRB rate close to 1 Gpc -3 yr -1 . The method they used is generally based on the determination of the luminosity function (LF) by fitting a phenomenological model to the log(N) -log(P) relation of well-known GRB catalogs. Assuming that GRBs trace the star formation rate (SFR), a model of the SFR is adopted for it and the rate of GRBs is determined. For instance, Guetta et al. (2004Guetta et al. ( , 2005) ) use the 2204 GRBs of the GUSBAD Catalog (Schmidt 2004) and a sample consisting of 595 long GRBs detected by BATSE. Following Schmidt (2001), they use these samples to find the luminosity function by fitting a single power-law model to the BATSE observed peak flux distribution. In addition, they assumed that the GRB rate follows the SFR and hence adopted either the SFR model SFR 2 of Porciani & Madau (2001) or the SFR model of Rowan-Robinson (1999). The estimated rate lies between 0.1 and 1.1 Gpc -3 yr -1 . This rate is in agreement with the one found by Schmidt (2001): 0.5 Gpc -3 yr -1 with 1391 GRBs contained in the BATSE DISCLA 2 Catalog (Schmidt 1999) and the one recently found by Liang et al. (2007) (1.12 +0.43 -0.20 Gpc -3 yr -1 ) with a sample of 45 HL Swift-GRBs. The rate of HL HETE-2 GRBs is also consistent with all these results, since we obtain a lower limit R H2 0 > ∼ 0.45 Gpc -3 yr -1 (see Part 9.3).

(2) GRB 060218, detected by Swift-BAT (Cusumano et al. 2006), was found to be at very low-redshift z =0.0331 (Mirabal et al. 2006a). Moreover, its association with the supernova SN 2006aj (Pian et al. 2006) recalled the characteristics of GRB 980425, the closest GRB ever identified, at a redshift z =0.0085 (Tinney et al. 1998). These GRBs associated with SNe are peculiar in the sense that they constitute the two nearest long duration bursts detected in eight years. Liang et al. (2007) show that the detection of these two events implies a local rate of low-luminosity GRBs (LL-GRBs) of R LL ∼ 800 Gpc -3 yr -1 , in agreement with Guetta & Della Valle (2007) who find a rate of GRB 980425-like events of ∼ 430 Gpc -3 yr -1 and a rate of GRB 060218-like events of ∼ 350 Gpc -3 yr -1 . The rate of LL-GRBs, which is considerably higher than the rate of HL-GRBs, could be explained in two ways: first, the log(N) -log(P) distribution of BATSE bursts often used to fit the GRB luminosity function cannot constrain it down to such low luminosities, because a GRB 060218-like event could not trigger BATSE. Second, the use of a phenomenological model for the GRB LF requires a choice for the extrema values L min and L max . However, due to instrumental limitations, the true L min of GRBs is unknown. Thus, the number of low-luminosity GRB is quite uncertain.

Discussion

We showed that the study of the complete sample of GRBs localized by the WXM on-board HETE-2 led to a rate R H2 0 > ∼ 10.6 Gpc -3 yr -1 . Our result is independent of any assumption on the luminosity function, as it is based on measured data. We note that this measured rate is at least 10 times higher than the HL-GRB rate and 10 times lower than the LL-GRB one. How can we explain this 'intermediate' result?

-(R H2 0 < R LL 0 ):
We first note that we have no event like GRB 980425 or GRB 060218 in our sample. Previous studies showed that it was impossible for an HETE-like mission with a WXM-like camera to detect a burst like XRF 060218. This burst occured at a distance d =145 Mpc, and Soderberg et al. (2006d) show that the WXM could have detected it at a maximum distance of 110 Mpc. For a burst similar to GRB 980425 occurring at d = 36.1 Mpc, the WXM could have detected it to a distance d = 60 Mpc. However, Guetta et al. (2004) show that with their model of the LF, the local rate of nearby bright bursts observable by HETE-2 within this distance is only 0.057 yr -1 .

-(R H2

0 > R HL 0 ): If we now imagine that HETE-2 was a mission that could not detect X-ray flashes, we can try to re-do all the steps to derive the rate of GRBs detected and localized by it (see Part. 9.1) in the sample of 60 'classical' GRBs. We obtain R H2 0 > ∼ 0.45 Gpc -3 yr -1 . This result is in good agreement with the rates usually obtained when only the HL-GRBS are considered:

∼[0.1-1] Gpc -3 yr -1 .
Consequently, we infer that the difference between the rate of HL-GRBs obtained with BATSE and the rate measured with HETE-2 is due to the X-ray flashes. These soft events cannot be detected in the BATSE energy range (50-300 keV). With our method based on observed properties of GRBs, XRFs and classical GRBs can be distinguished. The average lower redshift of XRFs leads to higher weights for the XRFs than for classical GRBs 9 . In fact, it appears that XRFs constitute an intermediate class in terms of frequency between the HL-GRBs and the LL-GRBs, and the rate of XRFs could be higher than the rate of classical GRBs by a factor of ∼10-100.

Despite their higher softness, faintness and frequency, our study brings the XRFs closer to the classical GRBs. This is not the case for GRB 980425 or XRF 060218-like events that are undetectable and out of reach of a HETE-2-like mission. -5 × 10 3 Gpc -3 yr -1 measured by Cappellaro, Evans & Turatto (1999) and Dahlen et al. (2004), the ratio is 0.1%. This low ratio highlights the fact that not all the type Ib/c SNe produce GRBs. The main explanation invokes a mildly relativistic jet, which is the source of the GRB, but that does not appear in all SNe Ib/c. Indeed, if the optical luminosities of GRBs and SNe can be considered to be similar, optical spectroscopy reveals broad absorption lines in SNe spectra associated with GRBs, indicative of fast ejecta. These peculiar highly energetic SNe, often designed Hypernovae (HNe) would not represent more than 3% of the SNe Ib/c, as obtained by the studies based on the comparison of radio luminosities of GRB afterglows and SNe Ib/c (see e.g. [START_REF] Berger | [END_REF]Soderberg et al. 2006b;[START_REF] Soderberg | Proc. for 'Supernova 1987A: 20 Years After: Supernovae and Gamma-Ray Bursters' AIPC[END_REF]. From previous studies of the predicted local rate of GRBs, and with our measured local rate, we now have a ratio between the SNe Ib/c and the GRBs lying in the range ∼[0.1%-3%]. This small ratio could imply that the escape of the ultra-relativistic jet from the progenitor star is not the only ingredient needed to produce a GRB. In addition, several studies of stellar rotation, binarity, asymmetry and metallicity have shown that the progenitors must have some special characteristics in order to produce a gamma-ray burst.

Summary

In this paper we have presented the study of a sample containing all the gamma-ray bursts detected by HETE-2 above a certain threshold and localized by the WXM. Our main goal was (1) to provide intrinsic distributions of the main global properties associated with the GRBs, i.e. taking into account the cosmological effects. The parameters studied are the peak energy (E peak ), the duration (T 90 ) and the total energy (E iso ). The second purpose of this paper is (2) to derive unbiased distributions of these properties, taking into account the spatial density of GRBs, and to measure the rate of HETE-2 GRBs in the local universe.

(1) The main conclusions are the following: 9 Recall that the weight taken for each burst is inversely proportional to the number of GRBs within its visibility volume and is obviously redshift-dependent.

-the intrinsic E peak distribution is quite broad, containing at the extreme sides of the energy range a large sample of Xray flashes and few 'classical' hard bursts. Nevertheless, we caution that the true width of this distribution is probably even broader than derived here, since the detector characteristics prevented HETE-2 from detecting bursts with an intrinsic peak energy of 1 or 2 keV or, at the opposite end, bursts with an intrinsic peak energy of a few MeV. -the intrinsic T 90 distribution shows that HETE-2 did not localize very long duration bursts (T 90 >1000 s). This may be a limitation of the orbit because due to its antisolar position, the detectors are turned off every half-orbit when they have the Earth in their field of view, i.e. every ∼ 45 min. On the other hand, some events have an intrinsic duration below 2 seconds and might be considered as 'intrinsically short'. -the intrinsic E iso distribution extends over more than four decades in energy. Most of the observed bursts have E iso higher than 10 52 erg: the 'classical' GRBs. The distribution extends down to lower E iso (10 50 -10 51 erg), corresponding to the sample of HETE-2 XRFs.

(2) The main conclusions are the following:

-the 'unbiased' distributions are dramatically different from their uncorrected shapes. Indeed, the unbiased E peak distribution shows a predominence of bursts with intrinsic E peak lower than 10 keV and the unbiased E iso distribution shows a predominence of bursts with E iso lower than 10 50 -10 51 erg.

The E iso distribution can be fitted between 10 50 and 10 54 erg by a simple power-law with a slope of -0.84. However, we caution that the 'simple' distributions of the intrinsic parameters probably suffer from biases. The same point may be made for the shapes of the 'unbiased' distributions: the true ones are probably also different, as shown by the two studies performed using different distances for the bursts without secure redshifts -case 1: pseudo-redshifts and case 2: known spectroscopic redshifts randomly attributed -which lead to different shapes for the E peak and the T 90 distributions. This emphasizes the predominence of X-ray flashes and the existence of a typical intrinsic duration for the GRBs in the first case, whereas a nearly constant number of GRBs through both the E peak energy range and the durations is highlighted in the second case. We may guess that the true distributions are situated between the results of these two cases. -a possible application of our study is the determination of the rate of GRBs measured by HETE-2 in the Local Universe. We obtained a lower limit of 10.6 Gpc -3 yr -1 , which is intermediate in magnitude between the rates usually found when a population of High-Luminosity GRBs is considered (∼1 Gpc -3 yr -1 ) or when the Low-Luminosity GRBs are also taken into account ( > ∼ 100 Gpc -3 yr -1 ). We explain this result with two arguments. First, HETE-2 did not detect LL-GRBs like GRB 980425 or XRF 060218 -mainly due to the small size of its detectors -so the rate cannot be as high as the one found for LL-GRBs. Second, it is the sample of XRFs detected by HETE-2 that makes the difference between the rate we obtained and the one found for HL-GRBs. As a consequence, thanks to the construction of unbiased distributions of the intrinsic properties of HETE-2 GRBS, we experimentally showed in this paper the dominance in terms of frequency of the X-ray flashes over the 'classical GRBs', a frequency which is nevertheless lower than the sub-luminous bursts, and much smaller than the Type Ib/c supernovae. The GRB names in the format YYMMDD are given in column 1. Columns 2 to 5 contain the spectral parameters measured for each burst: the slope of the power-laws at low-energy (α) and at high-energy when possible (β); the peak energy (E obs peak ) in keV and the fluence in the energy range 2-30 keV (S X ), in units of 10 -7 ergs cm -2 . The T 90 durations in the FREGATE band B (6-80 keV) are reported in column 6. The signalto-noise ratio (SNR) corresponding to the best combination of energy range and time resolution (see Section 2) are in column 7. The z sample (i.e. the spectroscopic/photometric redshifts when available or the pseudo-redshifts otherwise) are reported in column 8. The last column contains the references for the redshifts. 2.5 +0.6 -0.6

GRB

18.9 ± 5.13 6.28 1 +0.5 -0.5

[0] 021104 -1.1 -2.9

[0] 031109A -1.17

Fig. 1 .

 1 Fig. 1. Boresight angle cumulative distribution of the 115 GRBs localized by HETE-2/WXM, the Interplanetary Network or Swift. The dark dashed vertical line represents the 'angle limit' of 45 • , corresponding to the limit of detection by the WXM. The GRBs localized by the WXM are represented with black open circles, the ones localized by the IPN are in blue filled triangles, and the ones localized by Swift are in dark-red filled stars.

Fig. 2 .

 2 Fig. 2. Signal-to-noise ratio (SNR) cumulative distribution of the 88 GRBs localized by the WXM and within 45 • of the boresight. The threshold of FREGATE (SNR=7.9), is represented with the right-most vertical magenta dashed line, whereas the threshold of the WXM (SNR=5.8) is represented with the leftmost vertical magenta dashed line. The different symbols show the configuration of time resolution and energy range leading to the highest SNR for each burst: a given shape is associated with a given energy range and a given color corresponds to a given time resolution: [WXM/2-25 keV/1.2 s]= light-green filled circles; [WXM/2-25 keV/4.9 s]= light-green open circles; [WXM/2-25 keV/9.8 s]= dark filled circles; [FREG/6-80 keV/1.3 s]= red filled triangles; [FREG/6-80 keV/5.2 s]= red open triangles; [FREG/30-400 keV/1.3 s]= blue filled squares; [FREG/30-400 keV/5.2 s]= blue open squares. Note that there is only one light-green filled circle and one light-green open circle (left part of the figure) due to the fact that only one burst has its highest SNR in the configuration [WXM/2-25 keV/1.2 s] and one burst in the configuration [WXM/2-25 keV/4.9 s].

Fig. 3 .

 3 Fig.3. Ratio between the pseudo-redshift and the redshift for the 20 GRBs contained in our sample for which both quantities are available (upper-left panel). The ratio of the 3 pseudointrinsic and intrinsic parameters studied are also shown, respectively E peak (upper-right panel), T 90 (lower-left panel) and E iso (lower-right panel). In all plots, the filled region corresponds to a deviation from the equality between the two parameters by a factor of 2, and the error bars are at the 1σ level (stat.+syst.)

Fig. 4 .

 4 Fig. 4. Distribution of the E obspeak measured for the complete sample of 82 GRBs detected and localized by HETE-2. The subsample's histogram of bursts with secure redshifts (z) is shown in red, and the total sample's histogram is shown in grey. The bursts on the left part of the histogram correspond to the events that have a measured higher limit in E obs peak of about 4 keV (see Table2). The energy range of HETE-2 (2-400 keV) is symbolized by the magenta horizontal solid line.

Fig. 5 .

 5 Fig. 5. Distribution of E intrpeak for the complete sample of 82 GRBs detected and localized by HETE-2. The histogram related to the sub-sample of bursts with secure redshifts (z) is shown in red, and the total sample's histogram is shown in grey. The blue dashed histogram corresponds to the studied case 2 (see Part 3.3), i.e. the sample of 62 GRBs without secure redshift that are randomly given a redshift from the HETE-2 redshift distribution.

Fig. 6 .

 6 Fig.6. Unbiased distributions of the intrinsic E peak for the complete sample of 82 GRBs detected and localized by HETE-2. The smoothed distribution (SD) based on the 20 bursts having a spectroscopic/photometric redshift (z) is represented by the lower solid red curve and the one corresponding to the complete sample is shown with the upper solid black curve. The three black dotted smoothed distributions show the test performed to quantify the impact of the pseudo-redshift accuracy in this study (all the pseudo-redshifts are given an error range artificially increased by factors of 1.4, 2.0 and 3.0). The blue dashed SD corresponds to case 2 (see Part 3.3), i.e. the sample of 6200 simulated GRBs (corresponding to the 62 GRBs of our sample without secure redshift) which are randomly given a redshift from the HETE-2 redshift distribution.

Fig. 7 .

 7 Fig. 7. Distributions of the log T 90 measured in the observer frame and in band B of FREGATE (6-80 keV) for the complete sample of 82 GRBs localized by HETE-2. The histogram related to the sub-sample of bursts with secure redshifts (z) is shown in red, and that of the total sample is shown in grey.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Intrinsic log T 90 distributions for the complete sample of 82 GRBs detected and localized by HETE-2. The histogram of the sub-sample of bursts with secure redshifts (z) is shown in red, and the total sample histogram is shown in grey. The blue dashed histogram corresponds to the case 2 (see Part 3.3), i.e. to the 62 GRBs without secure redshift that were randomly given a redshift from the HETE-2 redshift distribution.

Fig. 10 .

 10 Fig. 10. Distributions of the logarithmic fluence in the X energy range (2-30 keV) for the complete sample of 82 GRBs localized by HETE-2. The histogram for the sub-sample of bursts with secure redshifts (z) is shown in red, and the total sample histogram is shown in grey.

Fig. 11 .

 11 Fig. 11. E iso distributions for the complete sample of 82 GRBs localized by HETE-2. The histogram of the sub-sample of bursts with secure redshifts (z) is shown in red, and the histogram of the total sample is shown in grey. The blue dashed histogram corresponds to case 2 (see Part 3.3), i.e. to the sample of 62 GRBs without secure redshifts that were randomly given a redshift from the HETE-2 redshift distribution.

Fig. 13 .

 13 Fig. 13. Correlations between the intrinsic parameters studied: E intr peak , T intr 90 , E iso and the distance-scale z sample . In each panel the GRBs that have a measured redshift (z) are represented with the green filled circles, and the ones having a pseudo-redshift (ẑ) are symbolized with the blue open circles. In the top left panel, by way of comparison, we show (solid line) the best-fit power-law E intr peak =77× 0.57 iso,52 (with E intr peak in keV and E iso,52 in units of 10 52 erg) found byAmati (2006) and also the vertical logarithmic deviation of 0.4 (dotted lines) displayed inAmati (2006). In the top right panel the dashed curve marks the redshift evolution of the E peak , assuming a burst with E (z=0) peak = 40 keV (seeStratta et al. 2007).

  Fig.12. Unbiased E iso distributions for the complete sample of 82 GRBs localized by HETE-2. The smoothed distribution based on the 20 bursts with a spectroscopic/photometric redshift (z) is represented by the lower solid red curve; the one for the complete sample is shown by the upper solid black line. The blue dashed SD corresponds to case 2 (see Part 3.3), i.e. the sample of 6200 simulated GRBs (corresponding to the 62 GRBs of our sample without secure redshifts) which were randomly given a redshift from the HETE-2 redshift distribution. The bestfit power-law (green dashed line) for the complete sample, esti-

Fig. 14 .

 14 Fig.14. Four distance-scale cumulative distribution functions (cdf). The cdf of the HETE-2 complete sample is shown with the green solid line. The two cumulative distributions for the 'classical' GRBs and the XRFs are shown with the blue dashed line and the red dotted line, respectively. We note that most of the 22 XRFs are definitely situated at intermediate or low redshift. For comparison, we also considered all the long-duration bursts with redshifts detected by Swift-BAT from the beginning of the mission to GRB 080210 (the last Swift burst with a spectroscopic redshift measured to date). The corresponding cdf is the black solid line.

9. 1 .

 1 The rate of GRBs in the Local Universe, as measured by HETE-2

9. 4 .

 4 Comparison with the rate of Type Ib/c SNe If we compare the measured rate of GRBs based on the complete HETE-2 sample and the rate of SNe Ib/c considered in Soderberg et al. (2006d), R SNeIb/c 0 = 9 +3

Table 2 .

 2 Observed properties and distances used in this study.

Table 3 .

 3 Intrinsic properties of our complete sample of HETE-2 localized GRBs. For each burst, the values reported in columns 2, 3 and 4 are base 10 logarithms which enable us to obtain the individual lognormal distributions of the E peak , T 90,B and E iso . The z max used to derive the corresponding V max are shown in column 5. The resulting V/V max are reported in the last column. ± 0.19 0.779 ± 0.012 010213...... 0.593 ± 0.03 1.239 ± 0.028 -2.172 ± 0.023 0.39 ± 0.08 0.091 ± 0.009 010225...... ± 0.74 0.455 ± 0.032 020625...... 1.141 ± 0.135 0.991 ± 0.096 -1.498 ± 0.126 0.53 ± 0.29 0.671 ± 0.035 020801...... 2.054 ± 0.129 2.339 ± 0.117 1.241 ± 0.222 1.68 ± 0.85 0.507 ± 0.049 020812...... 2.752 ± 0.182 0.606 ± 0.103 1.063 ± 0.121 4.78 ± 1.39 0.716 ± 0.035 020813...... 2.773 ± 0.042 1.588 ± 0.002 1.982 ± 0.016 3.8 ± 0.03 0.169 ± 0.002 020819...... 1.858 ± 0.079 1.304 ± 0.067 -0.251 ± 0.076 0.96 ± 0.02 0.124 ± 0.002 021004......

	GRB	log E intr peak	log T intr 90,B	log E iso,52	z max	V/V max
	001226......	1.648 ± 0.091 1.12 2.003 ± 0.122 0.403 ± 0.059 1.66 ± 0.037 0.516 ± 0.112 -0.173 ± 0.08 1.5 ± 0.3	1
	010326......	2.341 ± 0.111 -0.264 ± 0.106 0.518 ± 0.115	4.85 ± 1.47 0.552 ± 0.049
	010612......	3.306 ± 0.168 0.916 ± 0.095	1.457 ± 0.11	9.22 ± 2.2	0.61 ± 0.052
	010613......	1.891 ± 0.087 1.952 ± 0.034	0.711 ± 0.12	1.42 ± 0.28 0.173 ± 0.016
	010629......	1.938 ± 0.06	0.925 ± 0.053	0.357 ± 0.054	1.86 ± 0.49 0.228 ± 0.029
	010921......	2.128 ± 0.053 1.177 ± 0.014 -0.076 ± 0.028 1.18 ± 0.01 0.104 ± 0.001
	010928......	3.293 ± 0.073 0.841 ± 0.048	1.814 ± 0.12	9.23 ± 1.43 0.406 ± 0.027
	011019......	1.525 ± 0.178 1.168 ± 0.089 -1.324 ± 0.123 0.65 ± 0.34	0.63 ± 0.038
	011130......	0.43 ± 0.162	0.958 ± 0.068	-2.26 ± 0.06	0.19 ± 0.05 0.705 ± 0.038
	011212......	0.471 ± 0.177 1.476 ± 0.063 -1.703 ± 0.061 0.38 ± 0.16 0.679 ± 0.028
	020124......	2.531 ± 0.053 1.086 ± 0.008	1.999 ± 0.083	5.58 ± 0.06 0.532 ± 0.005
	020127......	2.481 ± 0.082 0.399 ± 0.032	0.553 ± 0.059	3.45 ± 0.38 0.376 ± 0.016
	020305......	2.995 ± 0.17	1.118 ± 0.058	1.22 ± 0.107	4.11 ± 0.91	0.367 ± 0.03
	020317......	1.98 ± 0.093	0.36 ± 0.06	0.23 ± 0.185	2.59 ± 0.41 0.727 ± 0.016
	020331......	2.48 ± 0.077 3.83 2.492 ± 0.104 1.755 ± 0.057 1.189 ± 0.073 1.17 ± 0.013 0.754 ± 0.087 3.1 ± 0.02	0.655 ± 0.006
	021016......	3.037 ± 0.17	1.358 ± 0.115	1.517 ± 0.096	5.01 ± 1.83 0.451 ± 0.064
	021021......	1.56 ± 0.16	0.961 ± 0.102 -0.751 ± 0.143	1.08 ± 0.3	0.927 ± 0.009
	021104......	1.84 ± 0.109	1.002 ± 0.06	0.097 ± 0.116	1.25 ± 0.21	0.77 ± 0.012
	021112......	2.517 ± 0.148 -0.216 ± 0.137 0.482 ± 0.168	4.74 ± 1.22 0.875 ± 0.015
	021211......	1.977 ± 0.03	0.321 ± 0.015	0.348 ± 0.043	2.93 ± 0.02 0.149 ± 0.002
	030115......	2.484 ± 0.094 0.802 ± 0.043	0.592 ± 0.03	4.04 ± 0.11 0.436 ± 0.006
	030226......	2.481 ± 0.053 1.409 ± 0.014	0.923 ± 0.065	2.9 ± 0.03	0.556 ± 0.004
	030324......	3.237 ± 0.255 0.374 ± 0.097	0.847 ± 0.035	6.05 ± 1.26 0.586 ± 0.032
	030328......	2.515 ± 0.029 1.739 ± 0.005	1.381 ± 0.013	4.08 ± 0.02 0.226 ± 0.001
	030329......	1.915 ± 0.008 1.345 ± 0.004	0.03 ± 0.008	1.16 ± 0.00 0.007 ± 0.000
	030416......	0.295 ± 0.173	1.1 ± 0.042	-2.501 ± 0.139 0.21 ± 0.03	0.228 ± 0.01
	030418......	2.309 ± 0.147 1.558 ± 0.113	1.175 ± 0.055	3.19 ± 1.07 0.906 ± 0.015
	030429......	2.122 ± 0.069 0.554 ± 0.054	0.54 ± 0.049	3.39 ± 0.02 0.718 ± 0.004
	030519......	2.409 ± 0.036 0.841 ± 0.018	1.21 ± 0.067	3.74 ± 0.29 0.075 ± 0.005
	030528......	1.753 ± 0.043	1.549 ± 0.02	0.483 ± 0.035	1.34 ± 0.01 0.307 ± 0.003
	030723......	0.793 ± 0.279 0.795 ± 0.045 -1.258 ± 0.105	0.6 ± 0.09	0.802 ± 0.013
	030725......	2.282 ± 0.045 1.965 ± 0.036	0.752 ± 0.016	2.6 ± 0.33	0.132 ± 0.012
	030821......	2.372 ± 0.063 0.845 ± 0.048	0.719 ± 0.058	3 ± 0.53	0.447 ± 0.024
	030823......	1.717 ± 0.071 1.439 ± 0.043	0.101 ± 0.054	1.14 ± 0.23	0.53 ± 0.022
	030824......	0.74 ± 0.17	0.915 ± 0.041 -1.659 ± 0.089	0.29 ± 0.1	0.551 ± 0.03
	030913......	2.996 ± 0.147 -0.012 ± 0.137 1.023 ± 0.189	8.19 ± 2.21 0.772 ± 0.032
	031026......	3.595 ± 0.234 0.947 ± 0.093	2.178 ± 0.136	6.89 ± 1.69 0.969 ± 0.004
	031109A...... 2.564 ± 0.044	1.47 ± 0.024	1.02 ± 0.025	2.93 ± 0.38 0.129 ± 0.011
	031109B...... 2.231 ± 0.174 2.145 ± 0.129	0.664 ± 0.099	3.09 ± 1.18	0.94 ± 0.01
	031111A...... 3.116 ± 0.045 0.402 ± 0.037	1.288 ± 0.053 10.51 ± 1.17 0.182 ± 0.018
	031111B...... 0.752 ± 0.272 1.359 ± 0.027	-1.808 ± 0.14	0.19 ± 0.05	1
	031203......	2.683 ± 0.074 0.517 ± 0.057	1.661 ± 0.057	6.63 ± 1.31	0.26 ± 0.034
	031220......	2.204 ± 0.19	0.585 ± 0.117	0.013 ± 0.062	2.08 ± 0.82	0.64 ± 0.048
	040319......	2.183 ± 0.129	0.37 ± 0.124	0.5 ± 0.13	2.9 ± 1.31	0.421 ± 0.062
	040423......	1.845 ± 0.085 1.314 ± 0.078	0.607 ± 0.102	1.8 ± 0.58	0.507 ± 0.041
	040425......	3.001 ± 0.064 1.634 ± 0.038	1.307 ± 0.07	2.73 ± 0.35 0.739 ± 0.009
	040511......	2.421 ± 0.062	1.22 ± 0.05	1.255 ± 0.076	3.46 ± 0.63	0.366 ± 0.03
	040701......	0.405 ± 0.146 0.955 ± 0.122 -2.096 ± 0.074	0.3 ± 0.01	0.384 ± 0.013
	040709......	2.15 ± 0.051	1.618 ± 0.041	0.898 ± 0.035	2.49 ± 0.5	0.182 ± 0.021
	040802......	2.43 ± 0.077	0.033 ± 0.062	0.955 ± 0.115	3.74 ± 0.79 0.358 ± 0.036

Table 3 .

 3 (continued) 040825A...... 2.056 ± 0.133 1.287 ± 0.068 -0.088 ± 0.069 1.17 ± 0.35 0.772 ± 0.02 040825B...... 1.942 ± 0.123 0.701 ± 0.098 0.698 ± 0.039 2.49 ± 0.52 0.838 ± 0.014 040912A...... 1.247 ± 0.095 0.851 ± 0.064 -0.782 ± 0.24 0.61 ± 0.36 0.162 ± 0.043 040912B...... 1.62 ± 0.194 1.68 ± 0.015 0.327 ± 0.131 1.82 ± 0.01 0.754 ± 0.007 040916...... 0.465 ± 0.163 2.401 ± 0.052 -1.329 ± 0.052 0.42 ± 0.18 0.856 ± 0.028 040924...... ± 0.46 0.203 ± 0.014 060121...... 2.526 ± 0.043 -0.045 ± 0.033 1.499 ± 0.069 5.37 ± 0.62 0.253 ± 0.018

	GRB	log E intr peak	log T intr 90,B	log E iso,52	z max	V/V max
		1.88 ± 0.014	0.259 ± 0.006	0.292 ± 0.029	2.5 ± 0.02	0.129 ± 0.001
	041004......	1.938 ± 0.059 1.504 ± 0.037	0.632 ± 0.235	1.13 ± 0.25 0.213 ± 0.019
	041006......	1.915 ± 0.016	1.11 ± 0.004	0.219 ± 0.013	2.37 ± 0.02 0.092 ± 0.001
	041016......	3.263 ± 0.288 0.689 ± 0.096	1.64 ± 0.243	4.95 ± 1.35	0.7 ± 0.031
	041127......	1.809 ± 0.091 1.417 ± 0.067	0.458 ± 0.08	1.53 ± 0.49 0.341 ± 0.038
	041211......	2.804 ± 0.188	1.42 ± 0.054	1.496 ± 0.263	8.66 ± 1.5	0.389 ± 0.029
	050123......	2.013 ± 0.092 0.765 ± 0.085	0.434 ± 0.142	2.73 ± 0.86 0.395 ± 0.045
	050209......	3.311 ± 0.274 0.669 ± 0.111	1.2 ± 0.212	4.24 ± 1.36 0.663 ± 0.043
	050408......	1.778 ± 0.083 1.103 ± 0.005	0.537 ± 0.007	2.68 ± 0.01 0.261 ± 0.002
	050509......	1.104 ± 0.345	1.12 ± 0.095	-0.836 ± 0.125 0.64 ± 0.28 0.807 ± 0.026
	050729......	2.455 ± 0.084 0.172 ± 0.059	1.515 ± 0.214	4.87 ± 0.86 0.447 ± 0.026
	050807......	2.25 ± 0.224	0.781 ± 0.117	0.15 ± 0.117	1.14 ± 0.67 0.411 ± 0.066
	050922......	2.656 ± 0.073 0.285 ± 0.062	0.742 ± 0.132	4.59 ± 0.08 0.378 ± 0.007
	051021......	2.386 ± 0.067 1.184 ± 0.053	0.685 ± 0.061	3.04 ± 0.65 0.289 ± 0.028
	051022......	2.585 ± 0.022 1.995 ± 0.012	1.356 ± 0.014	3.71 ± 0.05 0.065 ± 0.001
	051028......	3.103 ± 0.152	0.509 ± 0.1	1.273 ± 0.125	6.72 ± 2.05 0.522 ± 0.049
	051211......	2.862 ± 0.101 -0.076 ± 0.085	2.24 ± 0.377	7.71 ± 1.85 0.652 ± 0.035
	060115......	2.397 ± 0.056	0.91 ± 0.025	1.096 ± 0.065	4.63

Send offprint requests to: A. Pélangeon & J-L. Atteia e-mail: [alexandre.pelangeon/atteia]@ast.obs-mip.fr

The order of the steps described in this Section is not important; we could have performed them in any order and obtained the same sample.

See e.g.Ghirlanda, Ghisellini & Firmani (2006) orSchaefer (2007) for a complete summary of these correlations.

See also http://www.ast.obs-mip.fr/grb/pz

1.63 ± 0.80,Pélangeon & Atteia 2007a) and Konus-WIND (ẑ = 1.34 ± 0.30,Pélangeon & Atteia 2007b). Both values were consistent with a redshift of ∼1.5. The spectroscopic redshift z ∼ 1.54 was measured several days after our estimate(Fox et al. 2007; see alsoCenko et al. 2008).

Recall that throughout this study, we only consider the GRBs localized by the WXM.

Pélangeon et al.: Intrinsic properties of a complete sample of HETE-2 GRBs

Acknowledgements. The HETE-2 mission was supported in the US by NASA contract NASW-4690, in France by CNES contract 793-01-8479 and in Japan in part by the Ministry of Education, Culture, Sports, Science and Technology and by Grant-in-Aid for Scientific Research on Priority Areas 19047001. The authors acknowledge the valuable support of the HETE-2 Operation Team. AP is supported in France by the Ministry of National Education, Research and Technology. YEN is supported by the JSPS Research Fellowships for Young Scientists. This work is supported in part by a special postdoctoral researchers program in RIKEN. KH is grateful for support under MIT Contract SC-R-293291. G. Pizzichini aknowledges financial support as part of ASI contract I/088/06/0. Finally, the authors acknowledge the referee for his/her valuable and relevant remarks that helped improving the content of this paper.

4.9 ± 0.61 18.83 4.83 +1.9 -1.9

[0] 060115 -0.74 +0.12 -0.1 -1.81