Insights into the physics of gamma-ray bursts from the high-energy extension of their prompt emission spectra

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Table 3.

Bulk Lorentz factors Γ for each GRB that was co-detected by GBM and LAT and analyzed in this work, applying the compactness argument.

Name	Method	Γ	R [cm]	t_var [ms]
110721	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 412.7	> 3.41 × 10¹⁴	100
				(assumed)

160625	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	$283 . 3_{- 9.4}^{+ 8.5}$ $283.3_{-9.4}^{+8.5}$	$2 . 00_{- 0.13}^{+ 0.12} \times 10^{14}$ $2.00_{-0.13}^{+0.12}\times10^{14}$	100
	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 745.1	> 6.22 × 10¹⁴	100
080916	Aft. onset (ISM)	1410 ± 151	> 3.73 × 10¹³ [Eq. (8)]	> 1.67
	Aft. onset (Wind)	660 ± 49	> 1.10 × 10¹⁵ [Eq. (8)]	> 225

170214	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	$333 . 8_{- 17.2}^{+ 22.0}$ $333.8_{-17.2}^{+22.0}$	$1 . 89_{- 0.19}^{+ 0.26} \times 10^{14}$ $1.89_{-0.19}^{+0.26}\times10^{14}$	100

100724	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	$203 . 9_{- 10.7}^{+ 8.8}$ $203.9_{-10.7}^{+8.8}$	$0 . 83_{- 0.08}^{+ 0.07} \times 10^{14}$ $0.83_{-0.08}^{+0.07}\times10^{14}$	100

140206	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	$257 . 5_{- 7.6}^{+ 9.5}$ $257.5_{-7.6}^{+9.5}$	$1 . 33_{- 0.08}^{+ 0.10} \times 10^{14}$ $1.33_{-0.08}^{+0.10}\times10^{14}$	100

131108	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 398.1	> 2.80 × 10¹⁴	100

141028	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 231.8	> 0.97 × 10¹⁴	100

100116	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 278.0	> 1.54 × 10¹⁴	100

160821	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	$307 . 5_{- 7.9}^{+ 7.6}$ $307.5_{-7.9}^{+7.6}$	$1 . 89_{- 0.10}^{+ 0.10} \times 10^{14}$ $1.89_{-0.10}^{+0.10}\times10^{14}$	100

130504	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 136.2	> 0.37 × 10¹⁴	100

110328	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 201.8	> 0.81 × 10¹⁴	100

160509	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	$228 . 3_{- 6.3}^{+ 6.1}$ $228.3_{-6.3}^{+6.1}$	$1 . 44_{- 0.08}^{+ 0.08} \times 10^{14}$ $1.44_{-0.08}^{+0.08}\times10^{14}$	100

180720	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	$129 . 8_{- 2.8}^{+ 2.4}$ $129.8_{-2.8}^{+2.4}$	$0 . 61_{- 0.03}^{+ 0.02} \times 10^{14}$ $0.61_{-0.03}^{+0.02}\times10^{14}$	100

151006	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 229.5	> 1.05 × 10¹⁴	100

160905	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	$176 . 6_{- 7.3}^{+ 10.5}$ $176.6_{-7.3}^{+10.5}$	$0 . 62_{- 0.05}^{+ 0.08} \times 10^{14}$ $0.62_{-0.05}^{+0.08}\times10^{14}$	100

150902	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 254.6	> 1.30 × 10¹⁴	100

090328	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	$143 . 5_{- 5.2}^{+ 5.2}$ $143.5_{-5.2}^{+5.2}$	$0 . 71_{- 0.05}^{+ 0.05} \times 10^{14}$ $0.71_{-0.05}^{+0.05}\times10^{14}$	100

100826	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 282.0	> 1.6 × 10¹⁴	100
	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 495.3	> 3.84 × 10¹⁴	100
110731	Aft. onset (ISM)	971.2 ± 12.1	> 1.60 × 10¹³ [Eq. (8)]	> 1.08
	Aft. onset (Wind)	331.5 ± 8.3	> 2.57 × 10¹⁵ [Eq. (8)]	> 1494

160910	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 289.6	> 1.68 × 10¹⁴	100

150202	$γ γ {\overset{e}{\to}}^{+} e^{-}$ $\gamma\gamma \xrightarrow e^{+}e^{-}$	> 171.4	> 0.59 × 10¹⁴	100

Notes. Depending on the presence or absence of a cutoff in the high-energy spectrum, these values are measurements or lower–limits, respectively. For each GRB, the table list the name, the method applied to derive Γ, the corresponding value of bulk Lorentz factor Γ, the implied value of the distance R from the central engine and the variability timescale, which we assumed to be 100 ms for each GRB. For the only two GRB in our sample (GRB 080916C and GRB 110131) with also an independent estimate of Γ derived from the afterglow onset in Ghirlanda et al. 2018, we reported those Γ values for the ISM and wind–like cases and the implied lower–limits on the distance R when combined with the pair–production opacity argument (following Eq. (8)). In those two cases, we also report the different variability timescale implied by the corresponding values of R and Γ, following t_var = R(1 + z)/(2cΓ²).

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