Bulk Lorentz factors of GammaRay Bursts
Key Words.:
stars: gammaray bursts: general, Radiation mechanisms: non–thermal, Relativistic processesKnowledge of the bulk Lorentz factor of gammaray bursts (GRBs) allows us to compute their comoving frame properties shedding light on their physics. Upon collisions with the circumburst matter, the fireball of a GRB starts to decelerate, producing a peak or a break (depending on the circumburst density profile) in the light curve of the afterglow. Considering all bursts with known redshift and with an early coverage of their emission, we find 67 GRBs (including one short event) with a peak in their optical or GeV light curves at a time . For another 106 GRBs we set an upper limit . The measure of provides the bulk Lorentz factor of the fireball before deceleration. We show that is due to the dynamics of the fireball deceleration and not to the passage of a characteristic frequency of the synchrotron spectrum across the optical band. Considering the of 66 long GRBs and the 85 most constraining upper limits, we estimate or a lower limit . Using censored data analysis methods, we reconstruct the most likely distribution of . All are larger than the time when the prompt –ray emission peaks, and are much larger than the time when the fireball becomes transparent, that is, . The reconstructed distribution of has median value 300 (150) for a uniform (wind) circumburst density profile. In the comoving frame, long GRBs have typical isotropic energy, luminosity, and peak energy erg, erg s , and keV in the homogeneous (wind) case. We confirm that the significant correlations between and the rest frame isotropic energy (), luminosity () and peak energy () are not due to selection effects. When combined, they lead to the observed and correlations. Finally, assuming a typical opening angle of 5 degrees, we derive the distribution of the jet baryon loading which is centered around a few .
1 Introduction
The relativistic nature of GammaRay Bursts (GRBs) was originally posed on theoretical grounds (Goodman 1986; Paczynski 1986; Krolik & Pier 1991; Fenimore et al. 1993; Baring & Harding 1997; Lithwick & Sari 2001): the small size of the emitting region, as implied by the observed millisecond variability, would make the source opaque due to – pair production unless it expands with bulk Lorentz factor 100–1000 (e.g. Piran 1999). Refinements of this argument applied to specific GRBs (Abdo et al. 2009a, b; Ackermann et al. 2010a; Hascoët et al. 2012; Zhao et al. 2011; Zou & Piran 2010; Zou et al. 2011; Tang et al. 2015) led to some estimates of . A direct confirmation that GRB outflows are relativistic was found in 970508 (Frail et al. 1997): the suppression of the observed radio variability ascribed to scintillation induced by Galactic dust provided an estimate of the source relativistic expansion. Similarly, the longterm monitoring in the radio band of GRB 030329 (Pihlström et al. 2007; Taylor et al. 2004, 2005) allowed to set limits on the expansion rate (Mesler et al. 2012).
In the standard fireball scenario, after an optically thick acceleration phase the ejecta coast with constant bulk Lorentz factor before decelerating due to the interaction with the external medium. represents the maximum value attained by the outflow during this dynamical evolution. Direct estimates of became possible in the last decade thanks to the early follow up of the afterglow emission. The detection of an early afterglow peak, 150–200 s, in the NIR light curve of GRB 060418 and GRB 060607A provided one of the first estimates of (Molinari et al. 2007).
The recent development of networks of robotic telescopes (ROTSE–III: Akerlof et al. 2003; GROCSE: Park et al. 1997; TAROT: Klotz et al. 2009; SkyNet: Graff et al. 2014; WIDGET: Urata et al. 2011; MASTER: Lipunov et al. 2004; Pi of the Sky: Burd et al. 2005; RAPTOR: Vestrand et al. 2002; REM: Zerbi et al. 2001; Watcher: Ferrero et al. 2010) has allowed us to follow up the early optical emission of GRBs. Systematic studies (Liang et al. 2010; Lü et al. 2012; Ghirlanda et al. 2012, G12 hereafter) derived the distribution of and its possible correlation with other observables. G12 found:

that different distributions of are obtained according to the density profile of the circumburst medium;

the existence of a correlation (tighter with respect to that with );

the presence of a linear correlation .
As proposed by G12, the combination of these correlations provides a possible interpretation of the spectral energy correlations (Amati et al. 2002) and (Yonetoku et al. 2004) as the result of larger in bursts with larger luminosity/energy and peak energy. Possible interpretations of the correlations between and the GRB luminosity have been proposed in the context of neutrino or magneto–rotation powered jets (Lü et al. 2012; Lei et al. 2013). G12 (see also Ghirlanda et al. 2013a) showed that a possible relation between and the jet opening angle could also justify the correlation (here is the collimation corrected energy).
In order to estimate , we need to measure the onset time of the afterglow. If the circumburst medium is homogeneous, this is revealed by an early peak in the light curve, corresponding to the passage from the coasting to the deceleration phase of the fireball. On the other hand, if a density gradient due to the progenitor wind is present, the bolometric light curve is constant until the onset time. However, also in the wind case, a peak could be observed if pair production ahead of the fireball is a relevant effect as discussed in G12, for example.
G12 considered 28 GRBs with a clear peak in their optical light curve, and included three GRBs with a peak in their GeV light curves (as observed by the Large Area Telescope – LAT – on board Fermi). Early measurements are limited by the time needed to start the follow–up observations. The LAT (0.1–100 GeV), with its large field of view, performs observations simultaneously to the GRB prompt emission for GRBs happening within its field of view. The detection of an early peak in the GeV light curve, if interpreted as afterglow from the forward shock (e.g. Ghisellini et al. 2010; Kumar & Barniol Duran 2010), provides the estimate of the earliest (i.e. corresponding to the largest ). In the short GRB 090510, the LAT light curve peaks at 0.2 s corresponding to 2000 (Ghirlanda et al. 2010; Ackermann et al. 2010b). Recently it has been shown that upper limits on can be derived from the nondetection of GRBs by the LAT (Nava et al. 2017) and that such limits are consistent with lower limits and detections reported in the literature.
Precise and fast localisations of GRB counterparts, routinely performed by Swift, coupled to efficient follow up by robotic telescope networks, allowed us to follow the optical emission starting relatively soon after the GRB trigger. However, a delay of a few hundred seconds can also induce a bias against the measure of early–intermediate values (Hascoët et al. 2012). As argued by Hascoët et al. (2012), the distribution of , derived through measured , could lack intermediate to large values of (corresponding to intermediate to early values of ) and the – correlation could be a boundary, missing several bursts with large (i.e. because of the lack of early measurements).
For this reason, upper limits are essential to derive the distribution of in GRBs and its possible correlation with other prompt emission properties (, , ) and, in general, to study the comoving frame properties of the population. To this aim, in this paper (i) we collect the available bursts with an optical , expanding and revising the previously published samples, and (ii) we collect a sample of bursts with upper limits on . Through this censored data sample we reconstruct the distribution of accounting (for the first time) for upper limits. We then employ Monte Carlo methods to study the correlations between and the rest frame isotropic energy/luminosity and peak energy.
The sample selection and its properties are presented in §2, §3, and §4, respectively. The different formulae for the estimate of the bulk Lorentz factor appearing in the literature are presented and compared in §5. In §6 the distribution of and its correlation (§7) with , and are studied. Discussion and conclusions follow in §8. We assume a flat cosmology with .
2 The sample
We consider GRBs with measured redshift and well constrained spectral parameters of the prompt emission. For these events it is possible to estimate the isotropic energy and luminosity and the rest frame peak spectral energy (i.e. the peak of the spectrum).
can be estimated from the measure of the peak of the afterglow light curve interpreted as due to the deceleration of the fireball. We found in the literature 67 GRBs (66 long and 1 short) with an estimate of (see Tab. 6). Of these, 59 are obtained from the optical and 8 from the GeV light curves. Through a systematic search of the literature we collected 106 long GRBs whose optical light curve, within one day of the trigger, decays with no apparent . These GRBs provide upper limits . Details of the sample selection are reported in the following sections.
2.1 Afterglow onset
G12 studied a sample of 30 long GRBs, with measured from the optical (27 events) or from the GeV (3 events) light curves. We revise the sample of G12 with new data recently appearing in the literature, and we extend it, beyond GRB 110213A, including all new GRBs up to July 2016 with an optical or GeV afterglow light curve showing a peak .
Bursts with a peak in their early X–ray emission are not included in our final sample because the X–ray can be dominated (a) by an emission component of “internal” origin, for example, due to the long lasting central engine activity (e.g. Ghisellini et al. 2007; Genet et al. 2007; Ioka et al. 2006; Panaitescu 2008; Toma et al. 2006; Nardini et al. 2010) and/or (b) by bright flares (Margutti et al. 2010)^{1}^{1}1Liang et al. (2010), Lü et al. (2012) and Wu et al. (2011) include in their samples also bursts with a peak in the X–ray light curve thus resulting in a larger but less homogeneous and secure sample of ..
We excluded from our final sample (i) bursts with a multi–peaked optical light curve at early times^{2}^{2}2In these events might still be estimated, but only under some assumptions on the dynamical evolution of the burst outflow (e.g. GRB 090124 – Nappo et al. 2014).; and (ii) events with an optical peak preceded by a decaying light curve (e.g. GRB 100621A, GRB 080319C present in G12) since the early decay suggests the possible presence of a multi–peaked structure. The latter events, however, were included in the sample of (§2.2), considering the earliest epoch of their optical decay.
2.1.1 The gold sample
Table 6 lists all the GRBs we collected. The “Gold” sample is composed of sources with a complete set of information, namely measured (col. 6) and spectral parameters (col. 3–5). It contains 49 events: 48 long GRBs plus the short event 090510. GRBs of the Gold sample have the label “(g)” at the end of their names reported in col. 1 of Tab. 6. The redshift , rest frame peak energy , isotropic energy and luminosity ( and , respectively) are given in Tab. 6. Eight out of 49 GRBs have their measured from the GeV light curve as observed by the Fermi/LAT (labelled “L” or “SL” for the short GRB 090510).
For GRB 990123, GRB 080319B, and GRB 090102 reported in Tab. 6, it has been proposed that the early optical emission (and the observed peak) is produced by either the reverse shock (RS) (Bloom et al. 2009; Japelj et al. 2014; Sari & Piran 1999) or by a combination of forward and reverse shock (Gendre et al. 2010; Steele et al. 2009) or even by a twocomponent jet structure (e.g. Racusin et al. 2008, for GRB 080319B). We assume for these three GRBs that the peak is due to the outflow deceleration and include them in our sample.
2.1.2 The silver sample
In our search we found 18 events with but with poorly constrained prompt emission properties (, , ). In most of these cases, the Swift Burst Alert Telescope (BAT) limited (15–150 keV) energy band coupled with a relatively low flux of the source prevent us from constraining the peak of the spectrum () even when it lies within the BAT energy range. Most of these BAT spectra were fitted by a simple power law model. Sakamoto et al. 2011 showed that, also in these cases, the could be derived adopting an empirical correlation between the spectral index of the power law, fitted to the BAT spectrum, and . This empirical correlation was derived and calibrated with those bursts where BAT can measure . Alternatively, Butler et al. (2007, 2010) proposed a Bayesian method to recover the value of for BAT spectra fitted by the simple power law model.
We adopted the values of and calculated by Butler et al. (2007, 2010) for 15/18 GRBs in common with their list and the Sakamoto et al. (2011) relation for the remaining 3/18 events in order to exploit the measure of also for these 18 bursts. Sakamoto et al. 2011 and Butler et al. 2007, 2010 study the timeintegrated spectrum of GRBs. We estimated the luminosity =, where and are the peak flux and fluence, respectively, in the 15–150 keV energy range. GRBs of the “Silver” sample are labelled “(s)” in Tab. 6.
While we made this distinction explicit for clarity, in what follows we use the total sample of without any further distinction between the Gold and Silver samples.
2.2 Upper limits on
The afterglow onset is expected within one day for typical GRB parameters (see §4). Several different observational factors, however, can prevent the measure of . It is hard to construct a sample of upper limits . Hascoët et al. 2014 included some in their analysis but without a systematic selection criterion.
In this paper we collect from the literature all the GRBs with known and with an optical counterpart observed at least three times within one day of the trigger. If the light curve is decaying in time we set the upper limit corresponding to the earliest optical observation. Similar criteria apply if the longlived afterglow emission is detected in the GeV energy range by the LAT. For several recent bursts, highly sampled early light curves are available. We excluded events with complex optical emission at early times and selected only those with an indication of a decaying optical flux.
The 106 GRBs with are reported in Tab. 6. For the purposes of our analysis in the following we use a subsample of the 85 most constraining , that is, those with s which corresponds to five times the largest value of of the Gold+Silver sample.
3 Sample properties
In this Section we present the distribution of (in the observer and rest frame) and study the possible correlation of the rest frame with the observables of the prompt emission.
3.1 Distribution of the observer frame
Figure 1 shows the cumulative distribution (red line) of the observer frame afterglow peak time of long GRBs^{3}^{3}3The short GRB 090510 is not included in the distributions. Its onset time =0.2 s Ghirlanda et al. (2010) would place it in the lowest bin of the distribution.. The distribution of upper limits is shown by the dashed black line (with leftward arrows). The distribution of measured is consistent with that of the upper limits at the extremes, that is, below 30 s and above 1000 s. In particular, the low–end of the distribution of is mainly composed of bursts whose onset time is provided by the LAT data. Considering only GRBs with measured (red line in Fig. 1), the (log) average 230 s while upper limits (dashed black line in Fig. 1) have a (log) average 160 s. The relative position of the two distributions (red and black dashed) suggests that if we considered only measurements (as in G12; Liang et al. 2010; Lü et al. 2012) we would miss several intermediate–early onsets. This is confirmed also if we consider only the GRBs present in our sample which are part of the so called “BAT6” sample (Salvaterra et al. 2012). Indeed, this high flux cut sample of 58 Swift GRBs is 90% complete in redshift. There are 16 GRBs in our sample with measured and 34 with in common with the BAT6 sample (i.e. 86% of the sample). Their distribution (and the distribution of their ) is shown in the insert of Fig. 1. Similarly to the larger sample, the distribution of for the BAT6 sample is close to that of upper limits .
In our sample nearly half of the bursts have measured and half are upper limits. The distributions of and overlap considerably ensuring that random censoring is present. Survival analysis (Feigelson & Nelson 1985) can be used to reconstruct the true distribution of . We use the non–parametric Kaplan–Meier estimator (KM), as adapted by Feigelson & Nelson (1985) to deal with upper limits. The KM reconstructed CDF is shown by the solid black line in Fig. 1. The 95% confidence interval on this distribution (Miller 1981; Kalbfleish & Prentice 1980) is shown by the yellow shaded region in Fig. 1. The median value of the CDF is s (1 uncertainty). We verified that, considering a more stringent subsample of upper limits, that is, (), similar CDF and average values are obtained.
3.2 Distribution of the rest frame
Figure 2 shows the cumulative distribution of in the rest frame. Colour and symbols are the same as in Fig.1. In particular we note that also in the rest frame the cumulative distribution of measured (solid red line) is close to the distribution of upper limits (leftward arrows). The KM estimator leads to a reconstructed rest frame distribution (solid blackyellow shaded curve) which is distributed between 1 and sec with an average value of sec. The insert of Fig. 2 shows the distribution of the onset time of the GRBs belonging to the complete Swift sample. Again the measured distribution (solid blue line) is close to the distribution of upper limits, suggesting the presence of a selection bias against the measurement of the earliest values which is, however, not due to the requirement of the measure of the redshift.
3.3 Comparison between , , and
One assumption for the estimate of from the measure of (see §4) is that most of the kinetic energy of the ejecta has been transferred to the blast wave (so called “thin shell approximation” – Hascoët et al. 2014) which is decelerated by the circumburst medium. Therefore, we should expect that be larger than the duration of the prompt emission, estimated by . To check this hypothesis, we collected for the bursts of our sample: its distribution is shown by the grey dotted line in Fig. 1. A scatter plot showing versus the observer frame is shown in the top panel of Fig. 3: bursts with measured are shown by the red filled circles (GRBs with derived from the GeV–LAT light curve are shown by the star symbols), upper limits are also shown by the black (green for LAT bursts) symbols. The majority (80%) of GRBs lie below the equality line (dashed line in Fig. 3) having . 20% of the bursts have . A generalised Spearman’s rank correlation test (accounting also for upper limits – Isobe et al. (1986, 1990)) indicates no significant correlation between and (at level of confidence).
The prompt emission of GRBs can be highly structured with multiple peaks separated by quiescent times. While is representative of the overall duration of the burst, another interesting timescale is the peak time of the prompt emission light curve . This time corresponds to the emission of a considerable fraction of energy during the prompt and it is worth comparing it with . The distribution of is shown by the dot–dashed line in Fig. 1. The bottom panel of Fig. 3 compares with . Noteworthily, no GRB has . There are only two upper limits from the early follow up of the GeV light curve (green arrows in the bottom panel of Fig. 3) which lie above the equality line. However, the large uncertainties in these two bursts (090328 and 091003) on their GeV light curve at early times (Panaitescu 2017) make them also compatible with having . Again, the Spearman’s generalised test results in no significant correlation between and .
3.4 Empirical correlations
We study the correlation between the onset time in the rest frame and the energetic of GRBs. Figure 4 shows / versus the prompt emission isotropic energy , isotropic luminosity and rest frame peak energy .
GRBs with measured (red circles and green stars in Fig. 4) show significant correlations (chance probabilities – Tab. 1) shown by the red solid lines (obtained by a least square fit with the bisector method) in the panels of Fig. 4. The correlation parameters (slope and normalisation) obtained only with are reported in the left part of Tab. 1.
Correlation  only  &  

/(1+z) vs  –0.54  3  –0.710.06  39.703.28  –0.20  1  –0.950.01  51.611.00  
/(1+z) vs  –0.62  1  –0.670.10  36.895.10  –0.44  2  –0.870.02  46.771.00  
/(1+z) vs  –0.54  3  –1.250.12  5.110.33  –0.24  3  –1.250.03  4.700.10 
Upper limits (black downward arrows in Fig. 4) are distributed in the same region of the planes occupied by . Figure 4 shows also the lower limits on (grey upward arrows) derived assuming that .
The 85 GRBs without a measured should have their onset corresponding to the vertical interval limited by the grey and black arrows in Fig. 4. Indeed, the reconstructed distribution of (shown by the solid black line in Fig. 1) is bracketed by the cumulative distribution of on the left–hand side (dot–dashed grey line in Fig. 1) and by the distribution of on the right–hand side (dashed black line in Fig. 1).
In order to evaluate the correlations of Fig. 4 combining measured and upper/lower limits we adopted a Monte Carlo approach. We assume that the KM estimator provides the distribution of of the population of GRBs shown by the solid black line in Fig. 1. For each of the 85 GRBs with upper limits we extract randomly from the reconstructed distribution a value of (requiring that the extracted value falls within the range – where runs from 1 to 85) and combine them with the 66 GRBs with measured to compute the correlation (using the bisector method). We repeat this random extraction obtaining random samples and compute the average values of the Spearman’s rank correlation coefficient, of its chance probability, and of the slope and normalisation of the correlations (fitted to the 10 randomly generated samples). In Fig. 4, the average correlation obtained through this Monte Carlo method is shown by the dotdashed black line. The average values of the rank correlation coefficient, its probability, and the correlation parameters (slope and normalisation) are reported in the right section of Tab. 1.
These results show that significant correlations exist between the observables (i.e. fully empirical at this stage). A larger energy/luminosity/peak energy corresponds to an earlier . The distribution of upper () and lower () limits in the planes of Fig. 4 show that these planes cannot be uniformly filled with points, further supporting the existence of these correlations.
4 On the origin of the afterglow peak time
In the following we assume that the afterglow peak is produced by the fireball deceleration. However, other effects can produce an early peak in the afterglow light curve: can be due to the passage across the observation band of the characteristic frequencies of the synchrotron spectrum. In this case, however, any of the characteristic synchrotron frequencies should lie very close to the observation band at the time of the peak.
The synchrotron injection frequency is (Panaitescu & Kumar 2000):
(1) 
where and represent the fraction of energy shared between electrons and magnetic field at the shock and is the efficiency of conversion of kinetic energy to radiation (i.e. represents the kinetic energy in units of ergs in the blast wave). Here is measured in days in the source rest frame. The above expression for is valid either if the circumburst medium has constant density or if its density decreases with the distance from the source as (wind medium); in the latter case only the normalisation constant is larger by a factor 2.
Figure 5 shows (red and blue symbols for the homogeneous or wind medium case and stars for the LAT bursts) with respect to the optical frequency (solid horizontal line). We assumed typical values of the shock parameters: and and an efficiency 20% (Nava et al. 2014; Beniamini et al. 2015). The value is consistent with Beniamini & van der Horst (2017) and Nava et al. (2014) who recently found a narrow distribution of this parameter as inferred from the analysis of the radio and GeV afterglow, respectively. is less constrained and has a wider dispersion, between and (Granot & van der Horst 2014; Santana et al. 2014; Zhang et al. 2015; Beniamini et al. 2016), which translates into a factor of 10 for the value of (vertical lines in Fig. 5) obtained assuming (open circles in Fig. 5) . In order to account for this uncertainty we show, as vertical lines in Fig. 5 (for the open circles only for clarity), the possible range of frequencies that are obtained assuming . In all bursts, the injection frequency, when the afterglow peaks (i.e. at ), is above the optical band and it cannot produce the peak as we see it^{4}^{4}4We note also that for the LAT bursts (star symbols) the injection frequency is a factor 10 below the GeV band, and also in these cases the peak of the LAT light curve can be interpreted as the deceleration peak (Nava et al. 2017). .
However, one may argue that the above argument depends on the assumed typical values of and . Consider, for example, the homogeneous case . In order to “force” Hz at and reproduce the observed flux at the peak , one would require a density of the ISM:
(2) 
where is the luminosity distance in units of cm and is the rest frame peak time, now expressed in units of 100 seconds. For typical peak flux of 10 mJy we should have densities cm.
The other synchrotron characteristic frequency which could evolve and produce a peak when passing across the optical band is the cooling frequency . For the homogeneous medium Hz , where is the number density of the circum burst medium and the Compton parameter (Panaitescu & Kumar 2000). In this case, the portion of the synchrotron spectrum which could produce a peak is , under the assumption that , and the flux evolution should be (Panaitescu 2017; Sari et al. 1998). This is much shallower than the rising slopes of the afterglow emission of most of the bursts before (Liang et al. 2013). In the wind case, the cooling break increases with time and, if transitioning accross the observing band from below, it should produce a decaying afterglow flux and not a peak.
Therefore, we are confident that cannot be produced by any of the synchrotron frequencies crossing the optical band and it can be interpreted as due to the deceleration of the fireball and used to estimate .
5 Estimate of
In this Section we revise, in chronological order, the different methods and formulae proposed for estimating the bulk Lorentz factor (Ghirlanda et al. 2012; Ghisellini et al. 2010; Molinari et al. 2007; Nava et al. 2017; Sari & Piran 1999). The scope is to compare these methods and quantify their differences. We consider only the case of an adiabatic evolution of the fireball propagating in an external medium with a power law density profile . The general case of a fully radiative (and intermediate) emission regime is discussed in Nava et al. (2013). In §5 we present the results, that is, estimates of , for the two popular cases (homogeneous medium) and (wind medium). The estimate and comparison of in these two scenarios for a sample of 30 GRBs was presented, for the first time, in G12.
During the coasting phase the bulk Lorentz factor is constant () and the bolometric light curve of the afterglow scales as . After the deceleration time, starts to decrease and, in the adiabatic case, the light curve scales as independently from the value of .
Therefore, for a homogeneous medium () the light curve has a peak, while in the wind case () the light curve should be flat before and steeper afterwards^{5}^{5}5 On the other hand, as discussed in G12 and Nappo et al. (2014), there are ways to obtain a peak also in the case. This is why we consider both the homogeneous and the wind case for all bursts. Observationally, most early light curves indeed show a peak. . is the bulk Lorentz factor corresponding to the coasting phase. It is expected that the outflow is discontinuous with a distribution of bulk Lorentz factors (e.g. to develop internal shocks). represents the average bulk Lorentz factor of the outflow during the coasting phase.
5.1 Sari & Piran (1999)
In (Sari & Piran 1999, hereafter SP99) is derived assuming that corresponds to the fireball reaching the deceleration radius . This is defined as the distance from the central engine, where the mass of the interstellar medium , swept up by the fireball, equals :
(3) 
where is the isotropic equivalent kinetic energy of the fireball after the prompt phase.
SP99 assume as the link between the deceleration time and . This assumption corresponds implicitly to considering that the fireball travels up to with a constant bulk Lorentz factor equal to , or, in other words, that the deceleration starts instantaneously at this radius. Instead, the deceleration of the fireball starts before . This approximation underestimates the deceleration time and, consequently, underestimates :
(4) 
where . The original formula reported in SP99 is valid only for a homogeneous medium. Here, Eq. 2 has been generalised for a generic power law density profile medium.
Model  

SP99  
M07  
G10  
G12  
N13  
N14 
5.2 Molinari et al. (2007)
(Molinari et al. 2007, hereafter M07) introduce a new formula for , obtained from SP99 by modifying the assumption on . They, realistically, consider that the deceleration begins before so that . Under this assumption, and . However, they assume obtaining:
(5) 
This estimate of is a factor 2 larger than that obtained by SP99: as discussed in Nava et al. (2013), M07 overestimate^{6}^{6}6Liang et al. (2010) adopt the same equation as M07. the deceleration radius by a factor and, consequently, the value of is overestimated by the same factor. This is consistent with the results of numerical onedimensional (1D) simulations of the blast wave deceleration (Fukushima et al. 2017). Their results suggest that should be a factor 2.8 smaller than that derived (e.g. by Liang et al. 2010) through Eq. 5.
5.3 Ghisellini et al. (2010)
A new method to calculate the afterglow peak time is presented in (Ghisellini et al. 2010, hereafter G10). This method does not rely on the definition of the deceleration radius as in SP99 and M07. G10 derive by equating the two different analytic expressions for the bolometric luminosity as a function of the time during the coasting phase and during the deceleration phase^{7}^{7}7G10 derive in the case of an adiabatic or a fully radiative evolution of the fireball that propagates in a homogeneous medium. For our purposes we consider only the adiabatic case..
The relation linking the radius and the time is assumed to be: , where during the coasting phase and during the deceleration one. In the latter case, its value depends on the relation between the bulk Lorentz factor and the radius : for an adiabatic fireball, for instance, integration of , assuming (according to the self–similar solution of (Blandford & McKee 1976, hereafter BM76), we obtain .
G10 assume a relation between and which is formally identical to the BM76 solution but with a different normalisation factor:
(6) 
where is the interstellar mass swept by the fireball up to the radius .
In G10, the authors are interested in the determination of the peak time of the light curve, but that expression can also be used to determine the initial Lorentz factor . The expression, generalised for a power law profile of the external medium density, is:
(7) 
This estimate of is even lower than that of SP99 and, therefore, we can reasonably presume that also this value of will be underestimated with respect to the real one^{8}^{8}8Although Eq. 7 is obtained with an incorrect normalisation of the relation between and , we report also this derivation since it was the first to propose a different method to derive the time of the peak of the afterglow. Cfr. with Eq. 8 showing the correct normalisation..
5.4 Ghirlanda et al. (2012)
Ghirlanda et al. (2012) derive another formula to estimate , based on the method proposed in G10, that is, intersecting the asymptotic behaviours of the bolometric light curve during the coasting phase with that during the deceleration phase. In order to describe in the deceleration regime, G12 use the BM76 solution with the correct (with respect to G10) normalisation factor:
(8) 
The relation between radius and time is that presented in G10:
(9) 
where () corresponds to the coasting (deceleration) phase. The authors use these relations to obtain analytically the bolometric light curves before and after the peak and extrapolate them to get the intersection time which is used to infer :
(10) 
The main difference with respect to the formula of G10 comes from the normalisation factor of the BM76 solution and corresponds to a factor .
5.5 Nava et al. (2013)
(Nava et al. 2013, hereafter N13) propose a new model to describe the dynamic evolution of the fireball during the afterglow emission. With this model, valid for an adiabatic or full and semi–radiative regime, N13 compute the bolometric afterglow light curves and derive a new analytic formula for . They rely on the same method (intersection of coasting/deceleration phase luminosity solution) already used by G10 and G12, but with a more realistic description of the dynamics of the fireball, provide an analytic formula for the estimate of in the case of a purely adiabatic evolution (Eq. 11), and a set of numerical coefficients to be used for the full or semi–radiative evolution.
In the rest of the present work we will adopt the formula of N13 to compute so we report here their equation:
(11) 
The difference with respect to the formula of G12 is due to the different relation between the shock radius and the observed time . All the preceding derivations assume that most of the observed emission comes from the single point of the expanding fireball that is moving exactly toward the observer, along the line of sight. Actually, the emitted radiation that arrives at time to the observer does not come from a single point, but rather from a complex surface (Equal Arrival Time Surface, EATS) that does not coincide with the surface of the shock front. N13 avoid the rather complex computation of the EATSs, adopting a relation proposed by Waxman (1997) to relate radii to times. For simplicity, in the ultra–relativistic approximation:
(12) 
where is the shock Lorentz factor. The time is the sum of a radial time , that is the delay of the shock front with respect to the light travel time at radius and an angular time , that is the delay of photons emitted at the same radius but at larger angles with respect to the line of sight. Using the correct dynamics, N13 obtain the relation between observed time and shock front radius:
(13) 
N13 use these relations to obtain analytically the bolometric light curve during the coasting and the deceleration phase and, from their intersection time, estimate through Eq. 11.
5.6 Nappo et al. (2014)
(Nappo et al. 2014, hereafter N14) do not introduce directly a new formula to estimate , but show a new way (valid only in the ultra–relativistic regime ) to convert the shock radius into the observer time , bypassing the problem of the calculation of the EATSs. They assume that most of the observed emission is produced in a ring with aperture angle around the line of sight. The differential form of the relation between observed time and radius can be written as , that differs from the analogous relations of G10 and G12 by a factor .
Using this simplified relation coupled to the dynamics of N13 we derive a new expression for :
(14) 
We show in the following paragraph that this expression provides results that are very similar to those obtained with the formula of N13, proving that the approximation on the observed times is compatible with that suggested by N13 and before by Waxman (1997).
5.7 Comparison between different methods
All the previous expressions have the same dependencies on the values of , and . They differ only by a numeric factor and can be summarised in one single expression:
(15) 
where is a numeric factor that depends on the chosen method and on the power law index of the external medium density profile. In Tab. 2 we list the different values of for the various models and we show the comparison between the different estimates of for both a homogeneous and a wind medium. All the possible estimates of are within a factor of the estimate of N13; in particular the value provided by the N14 formula (Eq. 14) is similar to N13 within a few percent.
6 Results
Through Eq. 11 we estimate:

the bulk Lorentz factors of the 67 GRBs with measured ;

lower limits for the 85 bursts with upper limits on the onset time ;

upper limit for the 85 bursts with lower limits on the onset time =.
We consider both a homogeneous density ISM () and a wind density profile (). For the first case, we assume cm. For the second case, where is the massloss rate and the wind velocity. For typical values (e.g. Chevalier & Li 1999) yr and km s , the normalisation of the wind case is cm.
In both cases we assume that the radiative efficiency of the prompt phase is 20% and estimate the kinetic energy of the blast wave in Eq. 11 as . The assumed typical value for is similar to that reported in Nava et al. (2014) and Beniamini et al. (2015) who also find a small scatter of this parameter. We notice that assuming different values of and within a factor of 10 and 3 with respect to those adopted in our analysis would introduce a systematic difference in the estimate of corresponding to a factor 1.5 (2.3) for s=0 (s=2).
6.1 Distribution of
Fig. 6 shows the cumulative distribution of . The solid red line is the distribution of for the 66 GRBs with a measure of . The cumulative distribution of is shown by the black dashed line (with rightward arrows) in Fig. 6. The cumulative distribution of is shown by the dotted grey line (with leftward arrows) in Fig. 6. We note that while is derived from the optical light curves decaying without any sign of the onset (i.e. providing ), the limit is derived assuming that the onset time happens after the peak of the prompt emission (i.e. ).
A theoretical upper limit on can be derived from the the transparency radius, that is, (Daigne & Mochkovitch 2002). The maximum bulk Lorentz factor attainable, if the acceleration is due to the internal pressure of the fireball (i.e. ) is:
(16) 
where is the Thomson cross section and is the radius where the fireball is launched. We assume cm. This is consistent with the value obtained from the modelling of the photospheric emission in a few GRBs Ghirlanda et al. (2013b). The cumulative distribution of upper limits on obtained through Eq. 16, substituting for each GRB its , is shown by the green solid line in Fig. 6 (with the leftward green arrows). This distribution represents the most conservative limit on .^{9}^{9}9Changing the assumed value of within a factor of 10 shifts the green curve by a factor 1.8.
Similarly to the cumulative distributions of , shown in Fig. 1, also the distributions of (red solid line) and the distribution of lower limits (black dashed line) are very close to each other. While at low and high values of the two curves are consistent with one another, for intermediate values of the distribution of lower limits is very close to that of measured . In the wind case (right panel of Fig. 6), the lower limits distribution violates the distribution of measured . This suggests that the distribution of obtained only with measured suffers from the observational bias related to the lack of GRBs with very early optical observations.
We used the KM estimator to reconstruct the distribution of , combining measurements and lower limits, similarly to what has been done in §3.1 for . The solid black line (with its 95% uncertainty) in Fig. 6 shows the most likely distribution of for the population of long GRBs under the assumption of a homogeneous ISM (left panel) and for a wind medium (right panel).
The median values of (reported in Tab.3) are 320 and 150 for the homogeneous and wind case, respectively, and they are consistent within their 1 confidence intervals. G12 found smaller average values of (i.e. 138 and 66 in the homogeneous and wind case, respectively) because of the smaller sample size (30 GRBs) and the noninclusion of limits on . Indeed, while the intermediate/small values of are reasonably well sampled by the measurements of , the bias against the measure of large is due to the lack of small measurements (the smallest are actually provided by the still few LAT detections).
The reconstructed distribution of (black line in Fig. 6) is consistent with the distribution of upper limits derived assuming (dotted grey distribution) in the homogeneous case. For the wind medium there could be a fraction of GRBs (20%) whose is smaller than the peak of the prompt emission. However, Fig. 6 shows that, both in the homogeneous and wind case, the reconstructed distribution is consistent with the limiting distribution (green line) derived assuming that the deceleration occurs after transparency is reached.
[68% c.i.]  [68% c.i.]  

178 [142, 240]  85 [65, 117]  
&  320 [200, 664]  155 [100, 256] 
–(points)  0.86  2  3.020.24  –0.590.10  0.92  1.42  2.780.17  –0.400.05  

(points+LL)  0.62  1.2  2.600.16  –0.670.06  0.80  1.9  2.870.05  –0.250.02  
–(points)  0.81  3.0  2.850.22  –0.200.10  0.93  1.0  2.660.13  –1.130.04  
(points+LL)  0.46  2.8  2.190.10  –0.020.16  0.73  1.2  2.660.05  –0.830.02  
–(points)  0.73  5  1.470.13  –0.240.06  0.80  3.3  1.410.11  0.250.04  
(points+LL)  0.42  8  1.280.03  –0.400.03  0.61  1  1.430.03  0.070.01 
7 Correlations
G12 found correlations between the bulk Lorentz factor and the prompt emission properties of GRBs: , and, with a larger scatter, . Interestingly, combining these correlations leads to and which are the (Amati et al. 2002) and Yonetoku (Yonetoku et al. 2004) correlations. G12 showed that, in order to reproduce also the correlation (Ghirlanda et al. 2007), the bulk Lorentz factor and the jet opening angle should be =const.
In this Section, with the 66 long GRBs with measured (a factor 3 larger sample than that used in G12) plus 85 lower/upper limits, we analyse the correlations of (both in the homogeneous and wind case) with , and .
Figure 7 shows the correlation between (for the homogeneous and wind density circumburst medium, left and right panels, respectively) and . Lower limits are shown by rightward black arrows and occupy the same region of the data points with estimated (red symbols). The green symbols show the LAT bursts which have the largest values of and . Upper limits obtained requiring that the onset of the afterglow happens after the main emission peak of the prompt light curve are shown by the grey (leftward) arrows.
The shadowed grey region shown in Fig. 7 shows the limit obtained requiring that the deceleration happens after the transparency radius, that is, . Two different limits are shown corresponding to different (by a factor of 10) assumptions for , that is, the radius where the jet is launched (see Eq. 16). The correlation between and in the wind profile (bottom panel of Fig. 7) is less scattered than in the homogeneous case (upper panel of Fig. 7). The correlations between and the isotropic energy and the peak energy are shown in Figs. 8 and 9, respectively. Symbols are the same as in Fig. 7. Also in these cases, the correlation in the wind case is less scattered than in the homogeneous case.
The correlations shown in Figs. 7, 8, and 9 were analysed computing the Spearman’s rank correlation coefficient and its probability and fitting the data, with the bisector method, with a linear model:
(17) 
where is either or . Similarly, for the correlation between and we adopted the model:
(18) 
The results are given in Tab. 4 both for the homogeneous and for the wind case. Correlation analysis considering only estimates of are shown by the red solid line in Figs. 7, 8, and 9.
In order to reconstruct the correlation considering measured and upper/lower limits, we adopted the same Monte Carlo procedure described in §3. We generate samples composed by the GRBs with measured and assigning to the 85 GRBs without measured a value of randomly extracted from the reconstructed distribution of shown in Fig. 6. We then analyse the correlations of and the energetic variables within these random samples and report the central values of the correlation parameters (coefficient, probability, slope and normalisation) in Tab. 4. In all cases we find significant correlations also when upper/lower limits are accounted for with this Monte Carlo method. The corresponding correlation lines are shown with the dot–dashed lines in Figs. 7, 8, and 9. We note that the correlations found with only or reconstructed accounting also for limits are very similar.
Figure 7 (and similarly Figs. 8 and 9) shows that the planes are not uniformly filled with data points, contrary to what is claimed by Hascoët et al. (2014). Indeed, the right part of the planes of Figs. 7 and 8 corresponding to large and any possible value of and , respectively, are limited by the excluded region, that is, deceleration should happen after the fireball transparency. Moreover, the upper limits obtained requiring that the afterglow onset time is after the main prompt emission peak leads to the upper limits shown by the grey downward arrows in Figs. 7 and 8, which are even more constraining than the shaded regions. This confirms that the bulk Lorentz factor is indeed strongly correlated with the prompt emission properties (, and ).
Rest Frame  Comoving Frame  

median [68% c.i.]  median [68% c.i.]  median [68% c.i.]  
Isotropic energy [erg]  52.88 [52.25, 53.74]  50.50 [49.85, 51.33]  50.89 [50.33, 51.57]  
Isotropic luminosity [erg s]  52.40 [51.60, 53.16]  47.43 [46.79, 48.03]  48.18 [47.70, 48.63]  
Peak energy [keV]  2.64 [2.18, 3.08]  –0.008 [–0.42, 0.43]  0.35 [0.02, 0.69] 
We compute the comoving frame peak energy and isotropic energy with the equations derived in G12: and . The luminosity . Primed quantities refer to the comoving frame. For GRBs with a a lower limit transforms into an upper limit on , and . Vice versa, lower limits provide upper limits which transform the rest frame observables into lower limits in the comoving frame. Through the Monte Carlo method adopted in the previous Sections we derive the distributions of the comoving frame ,