HYDRODYNAMIC PROPERTIES OF GAMMA-RAY BURST OUTFLOWS DEDUCED FROM THE THERMAL COMPONENT

One of the major developments in the study of gamma-ray bursts (GRBs) in recent years has been the realization that a thermal component may be a key spectral ingredient. While the shape of most GRB spectra do not resemble a "Planck" function, in a non-negligible minority of GRBs a careful spectral analysis reveals a spectral component that is consistent with having a blackbody ("Planck") shape, accompanied by an additional, non-thermal part. Following pioneering work by Ryde (2004, 2005), such a component was clearly identified in 56 GRBs detected by BATSE (Ryde & Pe'er 2009) and is now identified in several Fermi GRBs as well. A few notable examples are GRB 090902B (Ryde et al. 2010), GRB 100507 (Ghirlanda et al. 2013), GRB 100724B (Guiriec et al. 2011), GRB 110721A (Axelsson et al. 2012; Iyyani et al. 2013), GRB 120323A (Guiriec et al. 2013), and GRB 101219B (Larsson et al. 2015).⁷

The existence of a thermal emission component may hold the key to understanding the prompt emission spectra. First, this component provides a physical explanation of at least part of the observed spectra. Furthermore, thermal photons serve as seed photons for inverse Compton (IC) scattering by energetic electrons; therefore these photons may play an important role in explaining the non-thermal part of the spectra as well. In fact, as it was recently shown by Axelsson & Borgonovo (2015), Yu et al. (2015b), while clear "Planck" spectra are only rarely observed, the narrowness of the spectral width of GRBs rules out a pure synchrotron origin in nearly 100% of the GRB spectra observed to date. Thus, it is possible that modified "Planck" spectra contribute to the observed emission in a very large fraction of GRBs.

Thermal photons decouple from the plasma at the photosphere, which is by definition the innermost region from which an electromagnetic signal can reach the observer. Therefore the properties of a Planck spectral component directly reveal the physical conditions at the photosphere, in those GRBs in which it can be directly identified. This feature is in contrast to the non-thermal spectral component, the origin of which is still uncertain. This is due to the fact that the radiative origin of the non-thermal component is still debatable, and its exact emission radius is very poorly constrained, both theoretically and observationally.

In the framework of the classical "fireball" model (Paczynski 1986; Rees & Meszaros 1992, 1994), the photospheric radius, r_ph, depends only on two free model parameters: the luminosity, L, and the Lorentz factor, Γ, at the photospheric radius⁸ (e.g., Paczynski 1990; Abramowicz et al. 1991; Mészáros & Rees 2000; Mészáros 2006, and references therein). The observed temperature weakly depends, in addition, on a third parameter, r₀, as ${T}^{\mathrm{ob}}\propto {r}_{0}^{1/6}.$ Here, r₀ is the acceleration radius, which is the radius where the acceleration of plasma to relativistic (kinetic) motion begins. Thus, by definition, at this radius the bulk Lorentz factor Γ(r₀) = 1, while at larger radii Γ(r) increases at the expense of the internal energy. In the classical "fireball" model, where the outflow expands freely and magnetic fields are sub-dominant, the growth is linear, ${\rm{\Gamma }}(r)\propto r$ below the saturation radius r_sat, and ${\rm{\Gamma }}(r)\propto {r}^{0}$ above this radius, as all the available energy is already in kinetic form.

The simple dependences of the temperature and photospheric radius on the luminosity, Γ, and r₀, have a strong implication. As was shown by Pe'er et al. (2007), one can use the measured values of the temperature and observed flux to directly measure r_ph and Γ. Using the fireball model scaling laws, one can then infer the value of r₀. Therefore, using only observational quantities one can deduce the entire (basic) dynamics of the earliest stages of the jet evolution within the framework of the "fireball" model. These dynamics, in turn, provide a strong tool in constraining models of GRB progenitors.

In order to perform these calculations, knowledge of the distance to the GRB is required, as the observed flux needs to be converted to luminosity. Moreover, reliable estimates of the dynamical parameters (Γ, r_ph, and r₀) rely on the assumption that the observed thermal component is not strongly distorted (e.g., by sub-photospheric dissipation; see Pe'er et al. 2006). Validating this last assumption is, nonetheless, relatively easy, as a significant thermal component F_th ≲ F_γ that can be directly observed necessitates that a strong distortion does not occur (see, however, Ahlgren et al. 2015).

Unfortunately, as of now, there are only very few GRBs which fulfill both requirements, namely, (1) a significant thermal component is clearly observed in their spectra, and (2) their luminosity distance is known. These observational constraints have limited, so far, the ability to carry out a statistical study of the outflow properties derived from photospheric emission.

Nevertheless, in recent years the number of GRBs in which a thermal component could be clearly identified is rapidly increasing, and is now at a few dozen. Furthermore, as we point out in the present work, uncertainties in the redshift have, in fact, only a weak effect on the deduced value of r₀. Consequentially, a good estimate of the value of r₀ can be achieved even for those GRBs for which the redshift is unknown. This enables the study of the hydrodynamical parameters in a reasonable sample of GRBs for which a thermal component was identified.

In this paper we analyze the existing data for those GRBs in which a distinct thermal component was clearly identified. Using the existing data, we deduce the values of the bulk Lorentz factor, Γ, and the acceleration radius, r₀. We find an average value of ${\rm{\Gamma }}\approx {10}^{2.5},$ which is similar to previous estimates, based on various methods. On the other hand, we find that $\langle {r}_{0}\rangle \approx {10}^{8}$ cm, higher than the common assumption of $\lesssim {10}^{7}\;{\rm{cm}}.$ These larger than expected r₀ values may be due to propagation effects of the jet, such as recollimation shocks, mass entrainment, or a non-conical structure, and may provide indirect evidence for the presence of a massive progenitor star.

This paper is organized as follows. In Section 2 we describe our sample selection and method of analysis. Our results are presented in Section 3. We discuss our findings in Section 4, before summarizing and concluding in Section 5.

Several authors have identified a thermal emission component in various GRBs (detected by both the BATSE instrument as well as by Fermi satellite), and studied their properties. As was shown by Pe'er et al. (2007), clear identification of both the temperature and flux in bursts with a known redshift z (luminosity distance d_L), can be used to infer the properties of the outflow within the framework of the "fireball" model⁹ via

$$\begin{eqnarray}&&{\rm{\Gamma }}={\left[{(1.06)(1+z)}^{2}{d}_{{\rm{L}}}\displaystyle \frac{{{YF}}^{\mathrm{ob}}{\sigma }_{T}}{2{m}_{p}{c}^{3}{\mathcal{R}}}\right]}^{1/4},\end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray}&&{r}_{0}=0.6\displaystyle \frac{{d}_{{\rm{L}}}}{{(1+z)}^{2}}{\left(\displaystyle \frac{{F}_{\mathrm{BB}}^{\mathrm{ob}.}}{{{YF}}^{\mathrm{ob}.}}\right)}^{3/2}{\mathcal{R}}{\rm{cm}}.\end{eqnarray} \tag{ 2 }$$

Here, σ_T is Thomson's cross section, ${\mathcal{R}}\equiv {({F}_{\mathrm{BB}}^{\mathrm{ob}}/\sigma {{T}^{\mathrm{ob}}}^{4})}^{1/2},$ ${F}_{\mathrm{BB}}^{\mathrm{ob}}$ is the observed blackbody flux, T^ob is the temperature of the thermal component, and σ is Stefan–Boltzmann constant. The total (thermal + non-thermal) flux is denoted by F^ob, and $Y\geqslant 1$ is the ratio between the total energy released in the explosion producing the GRB and the energy observed in γ-rays (both thermal and non-thermal). Measurements of the exact value of Y are difficult to conduct, though estimates can be done using afterglow observations (Cenko et al. 2011; Pe'er et al. 2012; Wygoda et al. 2015).

A key difficulty in identifying a thermal component and conducting the calculations is the fact that the signal varies with time. Thus, a time dependent analysis is required. As was shown by Ryde (2004, 2005) and Ryde & Pe'er (2009), for bursts that show relatively smooth, long pulses, and in which a thermal component could be identified, both the temperature and the flux show a very typical behavior: a broken power law in time. Typically, before ${t}_{\mathrm{brk}}^{\mathrm{ob}}\sim$ a few s, the temperature is roughly constant, while the thermal flux increases roughly as ${F}_{\mathrm{BB}}^{\mathrm{ob}}\propto {t}^{1/3}.$ At later times, both the temperature and the flux decay, T^ob ∝ t^−2/3 and F^ob_BB ∝ t⁻². Although the non-thermal and thermal fluxes are often correlated, the break time does not always coincide with the peak of the (non-thermal) flux. In many GRBs it is associated with the beginning of the rapid decay of the pulse (Ryde & Pe'er 2009).¹⁰

The exact interpretation of the early and late temporal behavior is still not fully clear. An interesting finding is that during the rise phase of the pulses (the first few seconds), the temperature is roughly constant. This implies via Equation (1) that the Lorentz factor is roughly constant (assuming that the ratio F_BB/F_tot does not vary much), though the total flux changes substantially. This no longer holds during the decay phase. A leading idea is that the late time behavior may be associated with a geometrical effect of (relativistic) "limb darkening" (Pe'er 2008; Pe'er & Ryde 2011; Lundman et al. 2013), though a non-spherical jet structure may be required (Deng & Zhang 2014). If this is the correct interpretation, it implies that reliable estimation of the hydrodynamic parameters can be done only if the data are taken at the break time, or earlier. Fortunately, in recent years enough data are available to carry out these time-dependent calculations for a substantial number of GRBs.

In this work we collected the available data, and divided them into three categories. In category (I) we include seven GRBs that fulfill the entire set of conditions: (1) their redshifts are known; (2) a thermal component was reported in the literature; and (3) a time dependent analysis could be performed, and therefore the values of Γ and r₀ were inferred. These GRBs were analyzed in recent years by various authors. Here, we collected the published data and validated it. In case of multiple-pulsed GRB, we normally picked the first pulse. Two of these GRBs, GRB 970828 and GRB 990510, were detected by BATSE (Pe'er et al. 2007), and the other five were detected by Fermi. We summarize in Table 1 the derived values of the parameters, as well as the references from which they were taken.

Table 1.  List of GRBs in Our Category (I): Known Redshift, and (II): Fermi GRBs without Known Redshift

Category	Burst	z	T	FBB	${F}_{{\rm{BB}}}/{F}_{{\rm{tot}}}$	Γ	r0	References	Comments
			(keV)	(erg cm−2 s−1)			(cm)
I	970828	0.9578	78.5 ± 4	⋯	0.64	305 ± 28	2.9 ± 1.8 × 108	(1)	a,b
I	990510	1.619	46.5 ± 2	7.0 × 10−7	0.25	384 ± 71	1.7 ± 1.7 × 108	(1)	b
I	080810	3.355	62	1.6 × 10−7	0.28	570 ± (170)	2.3 ± (1.2) × 108	(2)	c,d
I	090902B	1.822	168	1.96 × 10−5	0.26	995 ± 75	5.2 ± 2.3 × 108	(3)	⋯
I	090926B	1.24	17.2 ± 1	3 × 10−7	0.92	110 ± 10	4.3 ± 0.9 × 109	(4)	e
I	101219B	0.55	19.1 ± 0.7	8.45 ± 0.03 × 10−8	≲1	138 ± 8	2.7 ± 1.6 × 107	(5)	⋯
I	110731A	2.83	85 ± 5	⋯	0.47	765 ± 200	3.46 ± 1.1 × 108	(6)	d
II	100724B	(1)	38 ± 4	2.6 × 10−7	0.04	325 ± 100	1.2 ± 0.6 × 107	(7)	⋯
II	110721A	(2)	30	⋯	0.09	450 ± 200	1 ± 0.4 × 107	(8)	f
II	110920	(2)	61.3 ± 0.7	⋯	0.3	442 ± (133)	2 ± 1 × 108	(9)	⋯
II	120323	(0.5)	11.5 ± 1.5	⋯	0.05	145 ± (20)	2.6 ± (0.9) × 109	(10)	g

Notes. The derived values of Γ and r₀ in this table are done under the assumption Y = 1. See the text for details.

^aData is based on published references. Some references omit data on F_BB, dF_BB, or dT. ^bGRBs 970828 and GRB 990510 were detected by BATSE instruments. All other GRBs in this list were detected by Fermi-GBM. ^cErrors in dr₀, dΓ are estimated based on the data provided in the reference. ^dIn addition to Fermi-GBM, this GRB was detected by the Swift-BAT. ^eIn addition to Fermi-GBM, this GRB was detected by MAXI. ^fErrors represent uncertainty in redshift as well. ^gShort GRB; errors estimated from data provided in Table 4 of Guiriec et al. (2013).

References. (1) Pe'er et al. (2007), (2) Page et al. (2009), (3) Pe'er et al. (2012), (4) Serino et al. (2011), (5) Larsson et al. (2015), (6) Basha (2013), (7) Guiriec et al. (2011), (8) Iyyani et al. (2013), (9) McGlynn & Fermi GBM Collaboration (2012), Iyyani et al. (2015), (10) Guiriec et al. (2013).

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In category (II) there are four GRBs detected by Fermi that both (1) show clear evidence of a significant thermal component, and (2) have a well defined temporal evolution, which enables a clear identification of the break time.¹¹ However, as opposed to GRBs in our category (I) sample, the redshifts of these GRBs are unknown. The dynamical properties of these GRBs can therefore be deduced only up to the uncertainty in the redshift. The derived parameters of these GRBs, as given by the various authors, are also shown in Table 1, with the assumed values taken for the redshifts by the different authors. In our analysis, we did not modify the assumed redshift, as it was taken as the mean redshift of GRBs detected by the relevant instrument (note that GRB 120323 is a short GRB). As we show below, this uncertainty does not affect our conclusions.

Although the thermal component observed in the spectra of GRBs in our category (II) sample is statistically significant, it is relatively weak: in 3/4 GRBs, the flux in the thermal component is less than 10% of the total flux observed in γ-rays. This poses a challenge to the analysis method: as was shown by Zhang & Pe'er (2009), Hascoët et al. (2013), Gao & Zhang (2015), the photospheric component is suppressed if the flow is highly magnetized. Thus, weak thermal fluxes may be an indication of highly magnetized outflow, in which case the dynamical calculations presented in Equations (1) and (2) are modified (see Gao & Zhang 2015, for a full treatment in this case). Nonetheless, the magnetization parameter is unknown, and a weak thermal component does not necessitate a high magnetization; it could result, e.g., from ${r}_{\mathrm{ph}}\gg {r}_{\mathrm{sat}}.$ We therefore decided to include these bursts in our sample. As will be shown below, the values obtained for both r₀ and Γ for these bursts are similar to those obtained for the GRBs in the rest of our sample, which may indicate similar dynamics.

In category (III) we used 36 GRBs detected by BATSE, taken from the sample of Ryde & Pe'er (2009). This is the largest single sample of GRBs in which a temporal analysis was carried out and a thermal component was clearly identified. Out of 56 GRBs in the Ryde & Pe'er (2009) sample, we selected those GRBs in which a clear break time in the temporal behavior of both the temperature and flux was identified. Furthermore, the break times of the temperature and flux evolution were consistent with each other. Thus, although a break time was identified in all GRBs in our sample, the sub-sample of GRBs in this category is homogeneous, as all GRBs are detected by the same instrument and identical selection criteria were used.

The sample of our category (III) GRBs is given in Table 2. The values of the temperature, thermal flux, and ratio of thermal to total flux are given at the break time; the derived values of Γ, r₀, and the photospheric radius, r_ph are under the assumption of z = 1.

Table 2.  Sample of Our Category III GRBs: from Ryde & Pe'er (2009)

Burst	Trigger	T	FBB	${F}_{{\rm{BB}}}/{F}_{{\rm{tot}}}$	Γ	r0	rph
		(keV)	(erg cm−2 s−1)			(cm)	(cm)
910807	647	57.3 ± 2.2	1.7 ± 0.3 × 10−6	0.64	257 ± 10	6.23 ± 1.54 × 108	4.88 ± 0.60 × 1011
910814	678	173.1 ± 12.6	5.7 ± 2.0 × 10−6	0.51	549 ± 39	8.96 ± 4.68 × 107	2.09 ± 0.44 × 1011
911016	907	62.5 ± 2.5	9.5 ± 2.0 × 10−7	0.67	247 ± 10	4.22 ± 1.03 × 108	2.96 ± 0.38 × 1011
911031	973	59 ± 6.9	6.9 ± 4.7 × 10−7	0.18	334 ± 55	7.00 ± 8.33 × 107	3.53 ± 1.23 × 1011
920525	1625	148.5 ± 10.4	1.2 ± 0.4 × 10−5	0.43	583 ± 42	1.38 ± 0.78 × 108	4.41 ± 0.89 × 1011
920718	1709	44 ± 2.3	1.7 ± 0.5 × 10−6	0.28	278 ± 18	3.02 ± 1.66 × 108	9.06 ± 1.41 × 1011
921003	1974	15.9 ± 1.3	7.7 ± 3.3 × 10−7	0.59	125 ± 12	5.08 ± 3.14 × 109	2.12 ± 0.58 × 1012
921123	2067	56.1 ± 1.8	2.8 ± 0.5 × 10−6	0.40	306 ± 11	4.13 ± 1.19 × 108	7.87 ± 0.76 × 1011
921207	2083	94.8 ± 3.2	2.0 ± 0.1 × 10−5	0.61	458 ± 11	7.37 ± 1.48 × 108	1.11 ± 0.09 × 1012
930112	2127	111.7 ± 7.5	2.9 ± 1.0 × 10−6	0.42	426 ± 30	1.13 ± 0.63 × 108	2.78 ± 0.54 × 1011
930214	2193	100.7 ± 7.3	6.2 ± 2.4 × 10−7	0.81	283 ± 23	1.73 ± 0.62 × 108	1.05 ± 0.26 × 1011
930612	2387	40.2 ± 4.3	4.7 ± 2.7 × 10−7	0.42	208 ± 26	4.15 ± 3.93 × 108	4.22 ± 1.47 × 1011
940410	2919	61.4 ± 14.1	2.3 ± 2.7 × 10−7	0.11	320 ± 102	1.46 ± 4.03 × 107	1.94 ± 1.26 × 1011
940708	3067	73.2 ± 9.8	1.3 ± 0.9 × 10−6	0.19	377 ± 66	5.51 ± 8.25 × 107	3.80 ± 1.47 × 1011
941023	3256	46.5 ± 7.5	3.3 ± 1.4 × 10−7	0.61	196 ± 29	4.31 ± 3.29 × 108	2.55 ± 0.81 × 1011
941026	3257	55.4 ± 6.4	6.1 ± 2.2 × 10−7	0.54	233 ± 24	3.29 ± 2.04 × 108	2.89 ± 0.78 × 1011
941121	3290	41.1 ± 2	1.3 ± 0.3 × 10−6	0.34	247 ± 14	4.05 ± 1.94 × 108	8.00 ± 1.18 × 1011
950403	3492	108.5 ± 4.1	3.3 ± 0.6 × 10−5	0.35	598 ± 25	3.02 ± 1.03 × 108	1.39 ± 0.15 × 1012
950624	3648	28.9 ± 2.6	1.3 ± 0.7 × 10−7	0.38	150 ± 16	3.06 ± 2.76 × 108	3.06 ± 0.89 × 1011
950701	3658	67.6 ± 3	3.0 ± 0.7 × 10−6	0.42	334 ± 16	3.13 ± 1.22 × 108	6.09 ± 0.82 × 1011
951016	3870	28.2 ± 3.8	5.9 ± 4.7 × 10−7	0.10	252 ± 53	0.94 ± 1.70 × 108	1.17 ± 0.48 × 1012
951102	3891	66.9 ± 4.9	2.8 ± 1.0 × 10−6	0.23	380 ± 35	1.28 ± 1.00 × 108	6.76 ± 1.42 × 1011
951213	3954	52.9 ± 4.6	9.4 ± 4.3 × 10−7	0.22	300 ± 34	1.09 ± 1.05 × 108	4.97 ± 1.27 × 1011
951228	4157	24.4 ± 0.6	4.5 ± 0.7 × 10−7	0.98	128 ± 4	3.40 ± 0.31 × 109	6.96 ± 0.68 × 1011
960124	4556	60.6 ± 5.5	2.3 ± 0.9 × 10−6	0.22	364 ± 39	1.40 ± 1.19 × 108	7.29 ± 1.71 × 1011
960530	5478	44.1 ± 3.6	4.7 ± 0.4 × 10−7	0.38	220 ± 9	2.55 ± 0.43 × 108	3.78 ± 0.37 × 1011
960605	5486	64.9 ± 3.2	1.4 ± 0.4 × 10−6	0.54	279 ± 15	3.48 ± 1.33 × 108	3.80 ± 0.60 × 1011
960804	5563	47.9 ± 3	2.7 ± 0.9 × 10−6	0.26	311 ± 25	2.93 ± 1.98 × 108	1.06 ± 0.20 × 1012
960912	5601	68.1 ± 6.6	7.6 ± 3.8 × 10−7	0.33	300 ± 34	1.08 ± 1.01 × 108	2.70 ± 0.77 × 1011
960924	5614	155.4 ± 8	7.3 ± 1.0 × 10−4	0.94	820 ± 24	3.17 ± 0.48 × 109	4.45 ± 0.65 × 1012
961102	5654	62.2 ± 3.3	8.1 ± 2.3 × 10−7	0.35	283 ± 17	1.51 ± 0.76 × 108	3.17 ± 0.50 × 1011
970223	6100	98.3 ± 7	3.7 ± 1.3 × 10−6	0.32	441 ± 35	1.13 ± 0.75 × 108	4.23 ± 0.86 × 1011
970925	6397	44.6 ± 2.3	7.6 ± 2.1 × 10−7	0.33	243 ± 15	2.51 ± 1.30 × 108	5.10 ± 0.80 × 1011
980306	6630	80.6 ± 4.6	5.5 ± 1.1 × 10−6	0.84	334 ± 13	7.95 ± 2.09 × 108	5.83 ± 0.82 × 1011
990102	7293	54.4 ± 2.2	7.0 ± 1.6 × 10−7	0.58	229 ± 10	3.85 ± 1.18 × 108	3.10 ± 0.42 × 1011
990102	7295	100.7 ± 16.1	1.4 ± 0.6 × 10−6	0.68	330 ± 45	2.20 ± 1.46 × 108	1.90 ± 0.64 × 1011

Note. See the text for details.

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While the redshifts of all the GRBs in our categories (II) and (III) are unknown, we point out that the additional uncertainty in the estimate of r₀ due to the lack of a precise redshift is not very large. This is due to the fact that ${r}_{0}\propto {d}_{{\rm{L}}}/{(1+z)}^{2},$ and, for the range of redshifts typical for pre-Swift GRBs, 0.5 ≲ z ≲ 2.5, one finds $0.76\leqslant {d}_{{\rm{L}}}/{{d}_{{\rm{L}}}}_{(z=1)}\times {(2/(1+z))}^{2}\leqslant 1.06.$ A similar calculation shows that the added uncertainty in the estimate of Γ is $0.7\leqslant {\rm{\Gamma }}(z)/{{\rm{\Gamma }}}_{(z=1)}\leqslant 1.75,$ for $0.5\leqslant z\leqslant 2.5.$

We estimated the additional uncertainty of GRBs in our category (III) due to the unknown redshifts as follows.¹² We used the pre-Swift distribution of GRB redshifts given by Jakobsson et al. (2006). We performed a Monte-Carlo simulation, simulating 10⁶ GRBs drawn from this distribution, and calculated the ratios ${d}_{{\rm{L}}}/{(1+z)}^{2}$ and ${({d}_{{\rm{L}}}\times {(1+z)}^{2})}^{1/4}$ of each GRB in our simulation, normalized to these values for z = 1. In our calculation, we assumed a standard cosmology (flat universe with Ω_m = 0.286 and ${H}_{0}=69.6\;\mathrm{km}\;{{\rm{s}}}^{-1}\;{\mathrm{Mpc}}^{-1}$). Based on the simulated results, we conclude that the average values of r₀ and Γ are $\langle {r}_{0}\rangle /{{r}_{0}}_{z=1}=0.863,$ and $\langle {\rm{\Gamma }}\rangle /{{\rm{\Gamma }}}_{z=1}=1.117.$ The standard deviations due to the uncertain redshifts are σ(r₀) = 0.2533 and σ(Γ) = 0.531. These values are not surprising given the above analysis and the fact that the mean redshift in the sample of Jakobsson et al. (2006) is $\langle z\rangle =1.345.$

The values of Γ and r₀ presented in Tables 1, 2 are derived under the assumption of Y = 1. This is in order to be consistent with the results presented in the various references from which the data are adopted. However, in presenting the results in the figures below, we adopt a somewhat higher value of Y = 2. As explained above, the exact value of Y is very difficult to measure. Measurements based on afterglow observations reveal mixed results. Some works found very high efficiency in γ-ray production, implying Y ≲ a few (Cenko et al. 2011; Pe'er et al. 2012). On the other hand, several works found inefficient radiation, implying larger values of Y (Santana et al. 2014; Wang et al. 2015). A large value of Y would increase the measured value of the Lorentz factor by tens of %, as Γ ∝ Y^1/4, while decreasing r₀ by a factor of a few, as ${r}_{0}\propto {Y}^{-3/2}.$ As the highest values of Γ we obtain are ≳10³, while the lowest value of r₀ is ∼10^6.5 cm (see below), we deduce that the value of Y cannot be much greater than unity. In the analysis below, we therefore take as a fiducial value Y = 2.

The inferred values of the Lorentz factor Γ and the acceleration radius r₀ from our sample are presented in Figure 1. The green points represent GRBs with known redshift (our category I GRBs), GRBs in our category (II) are presented by the blue points, and category (III) GRBs are shown by the magenta points. The error bars represent statistical errors; additional errors due to the uncertain redshifts are shown by the dashed (red) dots. The blue stars represent the values of the parameters of GRBs in our category (II), with assumed redshift z = 1. As GRBs in this category form a small fraction of our sample, the uncertainty in the redshift of these GRBs does not affect any of our conclusions. In the plot, we assume a fixed value of Y = 2. We further show in Figure 2 histograms of the values of Γ and r₀.

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The most interesting result from these plots is the range of parameter values. We find that the range of values of the Lorentz factor stretches between 130 and 1200 (with 1σ errors inclusive), with an average value of $\langle {\rm{\Gamma }}\rangle =370,$ or $\langle {\mathrm{log}}_{10}({\rm{\Gamma }})\rangle =2.57$ and standard deviation of $\sigma ({\mathrm{log}}_{10}({\rm{\Gamma }}))=0.21.$ These results are similar to the values of Γ inferred for bright GRBs by other methods which are in wide use, such as opacity arguments (Krolik & Pier 1991; Woods & Loeb 1995; Lithwick & Sari 2001), the deceleration time of the emission from the forward shock (Sari & Piran 1999); see also Racusin et al. (2011), or the onset of the afterglow (Liang et al. 2010). Furthermore, the results of Figure 2 indicate that the distribution of Γ is fairly well fitted by a Gaussian, suggesting that selection biases are not playing a major role (see further discussion below).

The initial acceleration radius, r₀, ranges between $4\times {10}^{6}\leqslant {r}_{0}\leqslant 2\times {10}^{9}\;{\rm{cm}}.$ The average value is $\langle {r}_{0}\rangle =9.1\times {10}^{7}\;{\rm{cm}},$ or $\langle {\mathrm{log}}_{10}({r}_{0})\rangle =7.96$ and standard deviation of $\sigma ({\mathrm{log}}_{10}({r}_{0}))=0.56.$ Similarly to the distribution of Γ, the distribution of ${\mathrm{log}}_{10}({r}_{0})$ is also close to a Gaussian (see Figure 2).¹³ The average value found corresponds to a minimum variability time of $\delta t={r}_{0}/c\sim 3\times {10}^{-3}$ s, and is about 30 times the gravitational radius of 10 M_⊙ black hole. While this value is fully consistent with our data, as all the GRBs in our sample are long, and their variability time is much longer than 10 ms, this value is different from the commonly assumed value of the acceleration radius, ≈ few gravitational radii (e.g., Mészáros 2006; Kumar & Zhang 2014 and references therein).

A priori, we do not expect any correlation between the values of Γ and of r₀, as these are two independent parameters of the fireball model. Nonetheless, it is interesting to search for such a correlation, since if it exists it could put interesting constraints on the nature of the progenitor. This is due to the fact that if indeed r₀ is related to the gravitational radius, then ${r}_{0}\propto M,$ where M is the mass of the central object, while ${\rm{\Gamma }}\propto L/\dot{M}{c}^{2},$ where $\dot{M}$ is the mass ejection rate.

Linear fitting (with points weighted by the errors in both directions) reveals best fitted values of ${\mathrm{log}}_{10}({\rm{\Gamma }})={a}_{1}+{a}_{2}{\mathrm{log}}_{10}({r}_{0}),$ with a₁ = 3.31 ± 0.66 and ${a}_{2}=-0.10\pm 0.08.$ The correlation is thus very weak, with intrinsic scatter 0.44 ± 0.05 (Spearman rank −0.31 with significance of 0.035, equivalent to 2.1σ). While this analysis is done on a non-homogeneous sample, a similar analysis carried out on GRBs in our category (III) only, which do form a homogeneous sample, is not significantly different. We can conclude that our data does not point to any correlation between the values of r₀ and Γ.

4.1. Possible Selection Bias ?

The derived values of Γ span about one order of magnitude, between 10² and 10³. This range is similar to the typical values of Γ inferred from other methods, as discussed above. The values of r₀ span a wider range of roughly three orders of magnitude, between 10^6.5 and 10^9.5 cm. These values are higher than the typically assumed value of ${r}_{0}\lesssim {10}^{7}\;{\rm{cm}}.$

The values of Γ and r₀ are derived from the identified values of the observed temperature, T^ob., as well as the fluxes ${F}_{\mathrm{BB}}^{\mathrm{ob}.}$ and ${F}_{\mathrm{tot}}^{\mathrm{ob}.}.$ Typical values of the observed GRB fluxes are in the range of ${F}_{\mathrm{tot}}^{\mathrm{ob}.}\sim {10}^{-7}{-10}^{-4}\;\mathrm{erg}\;{\mathrm{cm}}^{-2}\;{{\rm{s}}}^{-1}$ (Kaneko et al. 2006; Gruber et al. 2014; von Kienlin et al. 2014, see also Tables 1, 2). The ratio of thermal to total flux cannot be less than a few % (typically, it is of the order of 50%, see Table 2), otherwise the thermal component could not have been clearly identified. Furthermore, the BATSE detector is sensitive at the range 20 keV–2 MeV, while the GBM detector has even broader sensitivity range, 8 keV–40 MeV. Thus, a thermal peak could be identified for the BATSE bursts at temperatures ${10}^{3}\lesssim {T}^{\mathrm{ob}}\lesssim {10}^{6}\;{\rm{eV}}.$ Using these observational constraints, we plot in Figure 3 the range of r₀ and Γ that could have been derived from observations of temperature and fluxes in this range using Equations (1), (2), under the assumption of z = 1.

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Clearly, the possible range of values of both Γ and r₀ are much greater than the observed ones. Values of Γ are possible in the range ${10}^{1.5}\lesssim {\rm{\Gamma }}\lesssim {10}^{3.5},$ and those of r₀ could span a much broader range, ${10}^{5.5}\lesssim {r}_{0}\lesssim {10}^{13}$ cm (see Figure 3). The addition of uncertainty in the redshift would imply that in principle, an even broader range of parameter values could be obtained. However, this is not observed: the inferred values of Γ and r₀ span only a small part of the range allowed by the detector's capabilities. Combined with the fact that the histograms of both Γ and r₀ are close to Gaussian, we thus conclude that selection effects are likely to be ruled out.

Our sample is not homogeneous, as we use data obtained by two different instruments (Fermi and BATSE detector). Furthermore, for the GRBs in our category (I) and (II) we rely on analysis carried by various authors. Thus, there is a risk of bias in the results. This risk does not exist for GRBs in our category (III), as all the data is taken from the BATSE instruments, and all the analysis was carried out by us. Our category (III) GRBs constitute the largest fraction of GRBs in our sample ($36/47\sim 75\%$). When conducting a separate analysis to the data in our category (III) GRBs only, we obtain similar results to those obtained when analyzing the data in categories (I) and (II) (see Figures 1 and 2), indicating that selection bias, if any, does not significantly affect the results.

An additional source of uncertainty is the unknown value of $Y\geqslant 1,$ the ratio between energy released in the explosion and the energy observed in γ-rays. As discussed above, the value we chose here, Y = 2, is a realistic approximation, based on (1) estimates from the literature; (2) the fact that the values of Γ obtained using this value are consistent with measurements based on other methods; and (3) the fact that a higher value would imply, in some cases, values of r₀ lower than the gravitational radius, which is unphysical. Nonetheless, we stress that because the derived values of both Γ and r₀ depend on Y, they are obviously sensitive to the uncertainty in the value of this parameter.

4.2. How Ubiquitous is Thermal Emission in GRBs ?

4.3. Possible Evidence of a Massive Progenitor

In this work, we analyzed a sample of 47 GRBs which show a significant thermal emission component. We split our sample into three categories: (I) 7 GRBs with known redshifts; (II) 4 GRBs detected by Fermi, in which the redshifts are unknown; and (III) 36 GRBs detected by BATSE, selected from the sample of Ryde & Pe'er (2009) based on clear identification of the break time. We analyzed the dynamical properties of these GRBs based on standard "fireball" model assumptions using the method derived in Pe'er et al. (2007). We found a mean Lorentz factor of $\langle {\rm{\Gamma }}\rangle =370,$ consistent with values obtained by different methods. However, we further found that the acceleration radius r₀ ranges between $4\times {10}^{6}\leqslant {r}_{0}\leqslant 2\times {10}^{9}\;{\rm{cm}},$ with mean value $\langle {r}_{0}\rangle \simeq {10}^{8}\;{\rm{cm}}.$ This is ≈1 order of magnitude above the commonly assumed value of ∼10⁷ cm. We found only a very weak correlation between the values of Γ and r₀, which cannot be used to put further constraints on the properties of the progenitor. We showed that the derived values of r₀ are only weakly sensitive to the uncertainty in the redshifts.

We showed in Section 4.1 that selection biases are unlikely to affect our obtained results. We further argued (Section 4.2) that despite the fact that GRBs with a pure thermal emission component are relatively rare, the existence of such a component is likely to be very ubiquitous, and possibly it exists in nearly all GRBs, albeit in most of them it is distorted. We claimed that within the context of the standard fireball model, it is more likely to detect a pure thermal component in long, smooth GRBs, that are less variable, and as a result internal shocks occur above the photosphere. Finally, we argued in Section 4.3, that the most likely interpretation of the values of r₀ is that they are propagation effects in massive progenitor stars, such as collimation shocks, mass entrainment, or non-conical expansion.

An alternative scenario that can explain the relatively large values of r₀ is that the acceleration in fact begins at radii much greater than the gravitational radius of the newly formed black hole. This could be due to an initially large jet opening angle, followed by a strong poloidal recollimation at $\sim {10}^{2}-{10}^{3}{r}_{g}$ that accelerates the jet. Some evidence for a similar scenario exists in a different object, the active galaxy M87 (Junor et al. 1999). This scenario requires very strong poloidal magnetic fields, whose existence in a GRB environment is uncertain.

The calculation of r₀ and Γ in Equations (1), (2) were done under the assumption of the "classical" (weakly magnetized) "fireball" model. The values of Γ that we found are consistent with the values found using other methods, that are independent of some of the underlying "fireball" model assumptions, such as afterglow measurements (e.g., Racusin et al. 2011). These values are more than an order of magnitude higher than the values of Γ inferred in active galactic nuclei (AGNs) and X-ray binaries. Thus, while magnetic acceleration (e.g., via the Blandford–Znajek process; Blandford & Znajek 1977) is likely to occur in XRBs and AGNs, currently there is no evidence that a similar mechanism is at work in GRBs as well.

In addition to the GRBs considered in this work, thermal emission was found recently in several GRBs in the X-ray band, extending well into the afterglow phase. These include both low luminosity GRBs such as GRB 060218 (Campana et al. 2006; Shcherbakov et al. 2013) or GRB 100316D (Starling et al. 2011) and also many GRBs with typical luminosities (e.g., Page et al. 2011; Sparre & Starling 2012; Starling et al. 2012; Friis & Watson 2013; Bellm et al. 2014; Piro et al. 2014; Schulze et al. 2014). Common to all GRBs in this category are (1) the fact that the thermal component is observed well into the afterglow phase; and (2) the inferred values of the Lorentz factors are at least an order of magnitude lower than that of GRBs in our sample: Γ ≲ few tens, and in some cases much lower, ${\rm{\Gamma }}\gtrsim 1.$ These results indicate that most likely the physical origin of the thermal component in bursts in this sample is different than bursts in our sample, the leading models being supernovae shock breakout and emission from the emerging cocoon. Given this different origin, we excluded these bursts from the analysis carried out here.

Finally, we stress that as the calculations here are carried under the standard assumptions of the classical "fireball" model, any inconsistency that may be found in the future between the values of r₀ derived by our method and those derived in alternative ways (e.g., via variability time argument) would question the validity of the entire "fireball" model.

A.P. wishes to thank Miguel Aloy for useful discussions. We would like to thank Mohammed Basha, Kim Page, and Motoko Serino for providing us with direct access to their data. We would further like to thank the referee, Dr. Bing Zhang, for useful comments, that helped us improve this manuscript. This research was partially supported by the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement ${{\rm{n}}}^{\circ }$ 618499. This work was supported in part by National Science Foundation grant No. PHYS-1066293 and the hospitality of the Aspen Center for Physics.