MEASURING THE BULK LORENTZ FACTORS OF GAMMA-RAY BURSTS WITH FERMI

Since the launch of the Fermi satellite, ∼250 GRBs per year have been detected with the Gamma-ray Burst Monitor (GBM). When sources are bright enough, the spacecraft will slew to the location of the burst and perform a pointed observation autonomously. So the prompt emissions of some GRBs were simultaneously observed with LAT. We search for such GRBs to make joint spectral analysis. A total of 49 GRBs are detected by the Fermi/LAT between 2011 August 1 and 2014 October 31, as listed in the LAT Burst Online Catalog.⁴ Focusing on the prompt phase, 28 of them were reported by the Fermi/LAT collaboration (through GCN circulars) to have LAT detection (>100 MeV) during the main gamma-ray emission phase, i.e., the time interval of GBM data analysis. Table 1 shows the information of these 28 GRBs, including the position derived from the LAT photons, the burst time interval used by the GBM team for spectral analysis (which is also the time interval used in our joint GBM/LAT analysis), and the LAT boresight angle at the GBM trigger time.

Table 1.  Properties of Fermi/LAT GRBs from 2011 August to 2014 October

GRB Name	Classa	T90b	T${}_{90}^{S}$ b	θc	R.A.d	Decl.d	LLE and GBMe	References
		(s)	(s)	(degree)	(degree)	(degree)
120107A	L	23.04	0.064	56	246.4	−69.93	b0, n6, n7	1
120316A	L	26.624	1.536	9	57.97	−56.46	LLE, b0, n0, n1	2
120709A	L	27.328	−0.128	22	318.41	−50.03	LLE, b1, n6, n7, n9	3
120830A	S	1.28	−0.384	38	88.42	28.81	b0, n0, n1, n3	4
130327B	L	31.233	2.048	47	218.09	−69.51	b0, n0, n1	5
130427A	L	138.242	4.096	48	173.15	27.71	LLE, b1, n9, n10	6
130502B	L	24.32	7.168	47	66.65	71.08	b1, n6, n7	7
130504C	L	73.217	8.704	47	91.72	3.85	LLE, b0, n2, n9	8
130518A	L	48.577	9.92	43	355.81	47.64	LLE, b1, n3, n7	9
130821A	L	87.041	3.584	37	314.1	−12	LLE, b1, n6, n9	10
130828A	L	136.45	13.312	40	259.83	28	b0, n0, n3	11
131014A	L	3.2	0.96	71.9	100.5	−19.1	LLE, b1, n9, na, nb	12
131018B	L	39.936	−1.024	12	304.41	23.11	b1, n6, n7	13
131029A	L	104.449	1.024	6	200.79	48.3	b0, n3, n5	14
131108A	L	18.496	0.448	27	156.47	9.9	LLE, b1, n3, n6	15
131209A	L	13.568	2.816	20	136.5	−33.2	b1, n6, n7	16
131231A	L	31.232	13.312	40	10.59	−1.85	LLE, b0, n0, n3	17
140102A	L	3.648	0.448	47	211.88	1.36	LLE, b1, n6, n7, n9, nb	18
140104B	L	188.417	9.216	25	218.81	−8.9	b1, n6, n7	19
140110A	L	9.472	−0.256	30	28.9	−36.26	LLE, b1, n6, n7, n9	20
140206B	L	116.738	8.256	45	315.26	−8.51	LLE, b0, n0, n1, n3	21
140323A	L	111.426	5.056	31	356.46	−79.87	b0, n0, n1	22
140402A	S	0.32	−0.128	13	207.47	5.87	b0, n1, n3	23
140523A	L	19.2	0.576	60	133.3	24.95	b0, n3, n4	24
140619B	S	2.816	−0.256	32	132.68	−9.66	LLE, b1, n6, n9	25
140723A	L	56.32	0	55	210.63	−3.73	b1, n9, na	26
140729A	L	55.553	0.512	26.2	193.95	15.35	LLE, b1, n6, n8, n9	27
141028A	L	31.489	6.656	25	322.7	−0.28	LLE, b1, n6, n7, n9	28
090926A	L	13.76	2.176	48.1	353.4	−66.32	LLE, b1, n3, n6, n7	29
100724B	L	114.69	8.192	48.9	119.89	76.55	LLE, b0, n0, n1	29

Notes. References. (1) Zheng & Akerlof (2012), McBreen (2012); (2) Vianello et al. (2012); (3) Kocevski et al. (2012), Guiriec et al. (2012); (4) Vianello et al. (2012), Tierney (2012); (5) Ohno et al. (2013), Chaplin & Fitzpatrick (2013); (6) Zhu et al. (2013), von Kienlin (2013); (7) Hurley (2013), von Kienlin & Younes (2013); (8) Kocevski et al. (2013), Burgess et al. (2013); (9) Omodei & McEnery (2013), Xiong (2013); (10) Kocevski et al. (2013), Jenke (2013); (11) Vianello & Sonbas et al. (2013), Collazzi (2013); (12) Desiante et al. (2013), Fitzpatrick & Xiong (2013); (13) Vianello et al. (2013), Zhang (2013); (14) Racusin et al. (2013a), von Kienlin & Jenke (2013); (15) Racusin et al. (2013b), Younes (2013); (16) Vianello & Omodei (2013), von Kienlin & Meegan (2013); (17) Sonbas et al. (2013), Jenke & Xiong (2014); (18) Sonbas et al. (2014), Zhang & Bhat (2014); (19) Vianello et al. (2014), Xiong (2014); (20) Bissaldi et al. (2014), von Kienlin & Connaughton (2014a); (21) Bissaldi et al. (2014), von Kienlin (2014); (22) Vianello et al. (2014), Yu & von Kienlin (2014); (23) Bissaldi et al. (2014), Jenke & Yu (2014); (24) Vianello et al. (2014), von Kienlin & Connaughton (2014a); (25) Kocevski et al. (2014), Connaughton et al. (2014); (26) Bissaldi et al. (2014), Burns (2014); (27) Arimoto & Bissaldi (2014), Stanbro (2014); (28) Bissaldi et al. (2014), Roberts (2014); (29) Ackermann et al. (2013). ^aL means long burst and S means short burst. ^bGBM T90 duration and the start time from GBM trigger time cited from the GBM catalog, i.e., Goldstein et al. (2012), Paciesas et al. (2012), Gruber et al. (2014), von Kienlin et al. (2014). ^cOff-axis angle at the trigger time derived from reference in the ninth column. ^dLAT position from reference in the ninth column. ^eLLE represents the publicly LAT Low-energy data; others are the GBM detectors we used for spectral analysis.

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2.2.1. Data Preparation and Event Selection

2.2.2. Background Estimation and Spectrum Extraction

2.2.3. Spectral Models

GRB time-integrated spectra are usually fitted with a smoothed broken power-law (PL) function, the so-called Band function (Band et al. 1993). Possible superposition of a thermal component on the non-thermal spectrum was claimed in BATSE and Fermi GRBs (Ryde 2005; Ryde & Pe'er 2009; Ryde et al. 2010, 2011; Guiriec et al. 2011, 2013, 2015; Pe'er et al. 2012; Preece et al. 2014; Yu et al. 2015). Other non-Band models, such as the synchrotron model (Burgess et al. 2011, 2014; Preece et al. 2014), are also proposed. We do not include the thermal emission in the analysis, because the thermal emission with a single temperature is usually found in the careful time-resolved analysis, while our search for LAT spectral cutoff needs to be done in the time-integrated spectrum in order to have enough LAT photons. Furthermore, we also find that, when a thermal component can be adequately added in a time-integrated GRB spectrum, the LAT cutoff energy remains unchanged (although it changes the peak energy of the Band component). For some non-Band models, such as the PL model, smoothly broken power-law (SBPL) model, and Comptonized model (Goldstein et al. 2012; Gruber et al. 2014), we found that they do not improve the fit over the Band model significantly for the bursts in our sample. For these reasons, and considering that the Band model is a widely used phenomenological model, we use the Band model as the primary model in our analysis.⁶ Meanwhile, an extra PL component (sometimes with high-energy cutoff) was also found in some LAT-detected GRBs, such as GRB 090902B, GRB 090510, and GRB 090926A (Abdo et al. 2009a; Ackermann et al. 2010, 2011).

Thus, we assume the Band function or the Band plus a PL function for the fundamental GRB spectral models. To search for cutoff-like features at the high-energy end, three spectral models are considered, i.e.,

(a) Band model, that is,

$$\begin{eqnarray*}&&\begin{array}{c}{N}_{\mathrm{Band}}=B(E)\\ \quad =A\left\{\begin{array}{cc}{(E/100\;\mathrm{keV})}^{\alpha }{e}^{(-E(2+\alpha )/{E}_{p})} & \mathrm{if}\;E\lt {E}_{b}\\ {[(\alpha -\beta ){E}_{p}/(100\;\mathrm{keV}(2+\alpha ))]}^{(\alpha -\beta )} & \\ {e}^{\beta -\alpha }{(E/100\;\mathrm{keV})}^{\beta } & \mathrm{if}\;E\geqslant {E}_{b}\end{array}\right.\end{array}\end{eqnarray*}$$

where ${E}_{b}=(\alpha -\beta ){E}_{p}/(2+\alpha )$, α is the photon index at low energy, β is the photon index at high energy, and E_p is the peak energy in the ${E}^{2}B(E)$ representation.

(b) BandCut model, the Band function with an exponential cutoff:

$$\begin{eqnarray*}&&{N}_{\mathrm{BandCut}}=B(E){e}^{-E/{E}_{c}},\end{eqnarray*}$$

where E_c is the cutoff energy.

(c) Band+PLcut model, that is, the Band function plus a PL model with an exponential cutoff

$$\begin{eqnarray*}&&{N}_{\mathrm{Band}+\mathrm{PLCut}}=B(E)+k{(E/{E}_{\mathrm{piv}})}^{\lambda }{e}^{-E/{E}_{c}},\end{eqnarray*}$$

where E_piv is the pivot energy and E_c is the cutoff energy.

To take into account the uncertainties caused by intercalibration between the GBM and the LAT, we allow for an effective area correction during the combined fits. The calibration constant for the LAT is fixed to one, and data from other detectors are allowed to vary during the fits (Ackermann et al. 2013). The correction factors typically have values between 0.9 and 1.1 for the NaI detectors and between 0.7 and 1.3 for the BGO detectors.

We employ the following three criteria to determine the preferred spectral model: (1) the goodness of the fitting, which is measured by the reduced CSTAT (a smaller CSTAT value shows a better fit, and the difference, ΔCSTAT, is roughly equal to the square of significance of improvement; see Ackermann et al. 2011); here we claim a significant change with ΔCSTAT larger than 28; (2) the robustness of the model parameters, which is measured with the errors of the parameters; (3) whether a structure exists in the residual distribution.

We finally obtain the following results about the joint spectra of Fermi/LAT GRBs: 22 GRBs are adequately fitted with the Band function, and the remaining six GRBs show high-energy cutoff features: an exponential cutoff from the high-energy part of the Band component or from the extra PL component. The observed spectra with our fitting curve are shown in Figure 1, and the results are reported in Tables 2 and 3.

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2.3.1. Sample Fitted by the Band Function

2.3.2. Sample with High-energy Spectral Cutoff

If the spectral break/cutoff E_c is due to γγ absorption within the source, one can compute the bulk Lorentz factor (Γ) of the emitting region by taking ${\tau }_{\gamma \gamma }({E}_{c})=1$. We consider a simple one-zone model where the photon field in the emitting region is uniform, isotropic, and time independent in the comoving frame⁸ (see the supporting material for Abdo et al. 2009b). The target photons that annihilate with photons of energy E_c should have energy above ${E}_{t}={{\rm \Gamma }}^{2}{({m}_{e}{c}^{2})}^{2}/[{E}_{c}{(1+z)}^{2}]$, where z is the redshift. These photons come from the high-energy part of the Band function or the extra PL component, and their flux can be, respectively, parameterized as $f(E)=f({E}_{0}){(E/{E}_{0})}^{\beta }$ or $f(E)=f({E}_{0}){(E/{E}_{0})}^{\lambda }$ (in unit $\mathrm{photons}/({\mathrm{cm}}^{2}\;\mathrm{keV})$), where E₀ is some reference energy. Considering that photons with energy E_c collide with target photons with energy above E_t, we get the $\gamma \gamma$ absorption optical depth in the comoving frame (Gould & Schréder 1967),

$$\begin{eqnarray}{\tau }_{\gamma \gamma }({E}_{c}^{\prime }) & = & {{\displaystyle \int }}_{-1}^{1}d{\mu }^{\prime }\\ & & \times {{\displaystyle \int }}_{{E}_{t}^{\prime }}^{{E}_{\mathrm{max}}^{\prime }}{dE}\prime {n}_{\gamma }^{\prime }(E\prime ){\sigma }_{\gamma \gamma }({E}_{c}^{\prime },E\prime ,{\mu }^{\prime })(1-{\mu }^{\prime })W\prime \end{eqnarray} \tag{ 1 }$$

where ${E}_{c}^{\prime }=(1+z){E}_{c}/{\rm \Gamma }$ (hereafter the prime represents quantities in the comoving frame), ${\sigma }_{\gamma \gamma }$ is the absorption cross section, ${\mu }^{\prime }=\mathrm{cos}\theta \prime$, $\theta \prime$ is the angle between the colliding photon pair, W' is the shell width, and E_max is the maximum energy of the target photons. Here ${n}_{\gamma }^{\prime }(E\prime )$ is the number density of target photons in the comoving frame, which is given by (Abdo et al. 2009b)

$$\begin{eqnarray}&&{n}_{\gamma }^{\prime }(E\prime )={(\frac{{d}_{L}}{R})}^{2}\frac{{\rm \Gamma }f({E}_{0})}{{(1+z)}^{3}W\prime }{\left(\frac{E\prime }{{E}_{0}^{\prime }}\right)}^{\beta },\end{eqnarray} \tag{ 2 }$$

where R is the radiation radius and d_L is the luminosity distance. Introducing a dimensionless function $F(\beta )$, Abdo et al. (2009b) obtain a simplified expression of ${\tau }_{\gamma \gamma }$,

$$\begin{eqnarray}&&{\tau }_{\gamma \gamma }({E}_{c}^{\prime })={\sigma }_{T}{(\frac{{d}_{L}}{R})}^{2}\frac{{\rm \Gamma }{E}_{0}^{\prime }f({E}_{0})}{{(1+z)}^{3}}{(\frac{{E}_{c}^{\prime }{E}_{0}^{\prime }}{{m}_{e}^{2}{c}^{4}})}^{-\beta -1}F(\beta ),\end{eqnarray} \tag{ 3 }$$

where $F(\beta )\approx 0.597{(-\beta )}^{-2.30}$ for $-2.90\leqslant \beta \leqslant -1.0$ (Ackermann et al. 2011). The relation $R\simeq {{\rm \Gamma }}^{2}\;c\delta t/(1+z)$ is valid for the internal shock model, where $\delta t$ is the variability time. Setting ${\tau }_{\gamma \gamma }({E}_{c})=1$, the Lorentz factor Γ is given by

$$\begin{eqnarray}{\rm \Gamma } & = & \left[{\sigma }_{T}{({\displaystyle \frac{{d}_{L}}{c\delta t}})}^{2}{E}_{0}f({E}_{0})F(\beta ){(1+z)}^{-2(\beta +1)}\right.\\ & & \times {\left.{({\displaystyle \frac{{E}_{c}{E}_{0}}{{m}_{e}^{2}{c}^{4}}})}^{-\beta -1}\right]}^{1/(2(1-\beta ))}.\end{eqnarray} \tag{ 4 }$$

One should note that Equation (3) is obtained when the upper limit of the second integral in Equation (1), E_max, is taken to be $\infty ,$ which is valid only when the energy of target photons that annihilate with E_c is well below the cutoff energy, i.e.,

$$\begin{eqnarray}&&{E}_{c}\gg {{\rm \Gamma }}^{2}{m}_{e}^{2}{c}^{4}/[{E}_{c}{(1+z)}^{2}]\end{eqnarray} \tag{ 5 }$$

(Li 2010; Zhao et al. 2011). This condition is usually satisfied when the cutoff energy E_c is larger than a few hundred MeV. However, for bursts with lower E_c, such as some bursts in our sample, this condition is not satisfied anymore. For these low E_c bursts, the energy of target photons should be comparable to E_c (i.e., ${E}_{c}\gtrsim {{\rm \Gamma }}^{2}{m}_{e}^{2}{c}^{4}/[{E}_{c}{(1+z)}^{2}]$), and then Γ is estimated to be (Li 2010)

$$\begin{eqnarray}&&{\rm \Gamma }\approx \frac{{E}_{c}}{{m}_{e}{c}^{2}}(1+z).\end{eqnarray} \tag{ 6 }$$

Using the above method, we can now calculate Γ for each burst with spectral cutoffs. For GRB 090926A, since the time-resolved spectra of the maximum spikes show spectral cutoffs, we use the cutoff energy for the time-resolved spectrum to calculate Γ. For four GRBs that do not have redshift measurements, we assume redshifts of z = 1 for them. The variability time $\delta t=0.1$ s is adopted for GRB 131108A, based on the 2 ms resolution light curve of the bright NaI detector. For GRB 090926A, $\delta t=0.15$ s is adopted according to Ackermann et al. (2011). We first use Equation (4) to calculate Γ and then check whether Equation (5) is satisfied. We find that only two bursts (GRB 131108A and GRB 090926A), which have relatively larger E_c, satisfy this condition. The other seven bursts all have ${E}_{c}\lesssim 100$ MeV, and Equation (5) is not satisfied for them, so their Lorentz factors Γ are calculated with Equation (6). We note that this estimate of Γ suffers from less assumptions, as they are independent of the internal shock model assumption and the estimate of the variability timescale. The results of Γ are presented in Table 4. The values of Γ in our sample range from 90 to 900, providing direct evidence that GRBs are powered by ultrarelativistic outflow.

It has been usually suggested that, even if no spectral cutoff is measured, the observed highest-energy photon can be used to place a lower limit on the bulk Lorentz factor, assuming that the absorption optical depth ${\tau }_{\gamma \gamma }({E}_{\mathrm{max}})\lesssim 1$ for the maximum energy photon (Krolik & Pier 1991; Fenimore et al. 1993; Woods & Loeb 1995; Baring & Harding 1997; Hascoët et al. 2012). However, from our sample that has measured cutoffs, one can see that the absorption optical depth equals unity for the cutoff energy (i.e., ${\tau }_{\gamma \gamma }({E}_{c})=1$) and the absorption optical depth for the maximum energy photon is larger than unity (i.e., ${\tau }_{\gamma \gamma }({E}_{\mathrm{max}})\gt 1$). For this reason, the usual approach that uses ${\tau }_{\gamma \gamma }({E}_{\mathrm{max}})\lesssim 1$ for the highest-energy photon to estimate the lower limits on the bulk Lorentz factors is inaccurate.

Another method for estimating the bulk Lorentz factors is from the peak in the early optical afterglow light curve, assuming that this peak is caused by the afterglow onset, at which the jet is decelerated (Sari & Piran 1999; Liang et al. 2010; Racusin et al. 2011; Hascoët et al. 2014). Although most of the LAT bursts do not have early optical afterglow data, such an estimate may be possible in some cases. The Lorentz factors can also be determined from the thermal component in the prompt emission, assuming that it comes from the fireball photosphere (Pe'er et al. 2007). But this method depends on the unknown composition of GRB outflow and the efficiency of the dissipation mechanism responsible for the non-thermal component (Gao & Zhang 2014; Peng et al. 2014).

There have been suggestions that the Lorentz factors correlate with other quantities of the GRB jets, such as the isotropic gamma-ray luminosity or energy (Liang et al. 2010; Ghirlanda et al. 2012; Lü et al. 2012). The Lorentz factors in all the references are determined from the afterglow onset time, at which the jet is decelerated, so the value depends on the details of the dynamics and the circumburst environment. The Lorentz factors determined through the absorption cutoff in high-energy photons are more straightforward and reliable. We test the relation ${\rm \Gamma }-{L}_{\gamma ,\mathrm{iso}}$ and ${\rm \Gamma }-{E}_{\gamma ,\mathrm{iso}}$ using our sample, where ${L}_{\gamma ,\mathrm{iso}}$ is the averaged, isotropic gamma-ray luminosity in 10–1000 keV and ${E}_{\gamma ,\mathrm{iso}}$ is the isotropic gamma-ray energy in 10–1000 keV. The results are shown in Figure 2. We find the relation

$$\begin{eqnarray}&&{\rm \Gamma }={10}^{1.65\pm 0.20}{L}_{\gamma ,\mathrm{iso},51}^{0.52\pm 0.13},\end{eqnarray} \tag{ 7 }$$

with a Pearson correlation coefficient of r = 0.844 and null hypothesis probability of 0.008, which indicates a tight positive correlation. Removing the four GRBs that do not have redshift measurements, we examine whether the correlation remains. Although the sample gets smaller, we find that a correlation between Γ and ${L}_{\gamma ,\mathrm{iso}}$ still remains. Similarly, for the eight GRBs, we find that

$$\begin{eqnarray}&&{\rm \Gamma }={10}^{1.01\pm 0.56}{E}_{\gamma ,\mathrm{iso},52}^{0.88\pm 0.35},\end{eqnarray} \tag{ 8 }$$

with a Pearson correlation coefficient of r = 0.707 and null hypothesis probability of 0.050. Although our results generally agree with earlier suggestions that more powerful GRBs move faster, the correlation slopes are different. We note that the number of GRBs in our sample is limited, and the correlation remains to be tested with a large sample in the future.

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GRB jets are accelerated at the early stage while the internal energy of the fireball is gradually converted to the kinetic energy. After the acceleration, the jet is expected to have a Lorentz factor equal to the initial dimensionless entropy $\eta ={L}_{0}/(\stackrel{\dot{}}{M}{c}^{2})$, where L₀ and $\stackrel{\dot{}}{M}$ are, respectively, the total energy and mass outflow rates. Considering the relation in Equation (7), the mass outflow rates should follow that $\stackrel{\dot{}}{M}\propto {L}_{\gamma ,\mathrm{iso}}^{0.48\pm 0.13}$ (assuming that ${L}_{\gamma ,\mathrm{iso}}\propto {L}_{0}$). This puts useful constraints on any central engine models for GRBs.

We perform a complete analysis of the LAT-detected GRBs since 2011 August, i.e., the bursts that are not included in the first Fermi/LAT GRB catalog (Ackermann et al. 2013). Our aim is to search for cutoff-like spectral features in the high-energy gamma-ray emission, as has been seen in GRB 090926A. We find six GRBs showing such spectral features, with the cutoff energies ranging from ∼10 to ∼500 MeV. Assuming a simple one-zone model for the MeV–GeV emission, we compute the bulk Lorentz factors of the emitting region of these bursts. Motivated by earlier suggestions that the Lorentz factors may correlate with other burst quantities, such as the isotropic gamma-ray luminosity or energy (Liang et al. 2010; Ghirlanda et al. 2012; Lü et al. 2012), we test these relations with our sample. It is found that the Lorentz factors are well correlated with the isotropic gamma-ray luminosity of the bursts, suggesting that more powerful GRB outflows move faster.