Evidence of an Internal Dissipation Origin for the High-energy Prompt Emission of GRB 170214A

The Fermi Gamma-Ray Burst Monitor (GBM; energy coverage of 8 keV–40 MeV) was triggered by GRB 170214A at ${T}_{0}={\rm{15:34:26.92}}$ UT on 2017 February 14 (T₀, the GBM trigger time). The GBM light curve shows multiple overlapping peaks with a T₉₀ duration of about 123 s (Mailyan & Meegan 2017). Simultaneously, Fermi-LAT detected high-energy emission from GRB 170214A, at a location of R.A.= 256.33, decl. = −1.88 (J2000) (Racusin et al. 2017), which is consistent with that detected by Fermi-GBM. More than 160 photons above 100 MeV, with 13 of them above 1 GeV, are observed within 1000 s (Racusin et al. 2017), which makes it a good case to perform time-resolved analysis of high-energy emission.

Konus–Wind detected the multipeak light curve with a T₉₀ duration of about 150 s (Frederiks et al. 2017). Swift-XRT detected an afterglow emission close to the LAT position (Beardmore et al. 2017a, 2017b). Follow-up observations in the optical and/or NIR band are performed by RATIR (the Reionization and Transients Infrared Camera), NOT (the Nordic Optical Telescope), GROND, and Mondy (the AZT-33IK telescope in Sayan observatory) (Malesani et al. 2017; Mazaeva et al. 2017; Schady et al. 2017; Schady & Kruehler 2017; Troja et al. 2017a, 2017b). The ESO Very Large Telescope detected a faint optical afterglow and claimed a redshift of z = 2.53 (Kruehler et al. 2017).

Within 12° of the reported LAT position, R.A. = 256.33, decl. = −1.88 (J2000) (Racusin et al. 2017), the Pass 8 transient events are used in the energy range of 100 MeV–10 GeV. These data are analyzed using the Fermi ScienceTools package (v10r0p5) available from the Fermi Science Support Center (FSSC).⁷ Events with zenith angles >100° are excluded to reduce the contribution of Earth-limb gamma-rays. The instrument response function "P8R2_TRANSIENT020_V6" is used. GRB 170214A is modeled as a point source with the corresponding position, and the photon spectrum is assumed to be a power law, i.e., ${dN}/{dE}={N}_{0}{(E/100\mathrm{MeV})}^{-{{\rm{\Gamma }}}_{\mathrm{LAT}}}$, with the normalization factor (N₀) and photon index (${{\rm{\Gamma }}}_{\mathrm{LAT}}$) as free parameters. A background model comprises the galactic interstellar emission model ("gll_iem_v06.fits") and the extragalactic isotropic spectral template ("iso_P8R2_TRANSIENT020_V6_v06.txt"). For these diffuse components in the model, we calculate the response files using the gtdiffrsp tool. The livetime cube and exposure maps are generated by the gtltcube and gtexpmap tool. We run the gtlike tool to derive the best fit.

We first perform a blind search in three good time intervals, i.e., 0–900, 2500–7000, and 8500–13,000 s after the GBM trigger. The strong emission exists in the first 900 s, after which no significant emission is found.

Second, 10 and 100 s are employed as the resolved time bins before and after ${T}_{0}+200$ s, respectively, when performing the time-resolved analysis of the first 900 s of data. The nearby time bin is combined if the error of the energy flux in one time bin is larger than the central value. For the intensive emission period, i.e., 52–70 s, we divide this time interval into seven bins. Before 52 s, the LAT show marginally significant emission, which is treated as a single time bin. The likelihood results, i.e., the photon flux (${F}_{{\rm{L}}}$) and energy flux (${f}_{{\rm{L}}}$), are present in Table 1, where the total photon number within 12° (${N}_{\mathrm{ROI}}$), the predicted photon number (${N}_{{\rm{P}}}$), and the the test-statistic value (TS; the square root of TS approximately equals the detection significance (Mattox et al. 1996)), are also given.

Table 1.  Fermi-LAT Likelihood Results for GRB 170214A

T1–T2a	TSb	${N}_{\mathrm{ROI}}$ c	${N}_{{\rm{P}}}$ d	${{\rm{\Gamma }}}_{\mathrm{LAT}}$ e	${F}_{{\rm{L}}}(0.1\mbox{--}10\,\mathrm{GeV})$ e	${f}_{{\rm{L}}}(0.1\mbox{--}10\,\mathrm{GeV})$ e
(s)					$({10}^{-5}\,\mathrm{ph}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1})$	$({10}^{-8}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1})$
0–52	15	16	7.2	4.98 ± 2.62	3.01 ± 1.34	0.64 ± 0.34
52–62	11	4	4.0	6.96 ± 2.76	8.92 ± 4.44	1.71 ± 0.86
62–63	104	15	14.4	3.04 ± 0.54	285.06 ± 89.54	88.94 ± 29.37
63–64	308	16	16.0	4.05 ± 0.70	333.88 ± 182.45	79.51 ± 51.58
64–66	38	10	8.2	3.73 ± 0.91	84.86 ± 35.81	21.44 ± 9.35
66–67	20	5	5.0	3.93 ± 1.21	103.80 ± 46.62	25.21 ± 12.01
67–69	57	6	5.9	2.78 ± 0.73	57.75 ± 30.29	20.53 ± 11.44
69–70	19	5	3.7	3.19 ± 1.06	74.45 ± 40.72	21.82 ± 13.74
70–80	58	11	10.0	3.50 ± 0.79	20.37 ± 7.11	5.43 ± 2.26
80–90	196	11	11.0	2.50 ± 0.42	20.65 ± 6.29	8.95 ± 3.82
90–100	74	10	9.3	3.31 ± 0.70	18.65 ± 6.65	5.26 ± 2.07
100–110	58	6	5.2	1.54 ± 0.29	7.89 ± 3.79	11.89 ± 6.55
110–120	76	11	9.5	2.37 ± 0.43	16.90 ± 6.02	8.25 ± 3.96
120–130	92	12	11.1	2.01 ± 0.30	18.66 ± 6.11	13.81 ± 6.26
130–140	236	13	12.9	1.92 ± 0.25	21.39 ± 6.04	17.85 ± 7.34
140–150	97	16	13.0	2.20 ± 0.32	21.91 ± 6.38	12.75 ± 5.36
150–160	117	9	8.9	2.38 ± 0.44	15.35 ± 5.59	7.38 ± 3.63
160–170	108	7	7.0	2.35 ± 0.47	11.82 ± 4.51	5.88 ± 3.28
170–180	73	6	6.0	2.10 ± 0.43	9.37 ± 3.88	6.14 ± 3.79
180–190	147	8	8.0	2.27 ± 0.42	12.86 ± 4.60	6.91 ± 3.65
190-200	54	6	6.0	2.60 ± 0.60	9.89 ± 4.07	3.95 ± 2.22
200–300	142	43	39.7	2.67 ± 0.27	6.43 ± 1.17	2.45 ± 0.56
300–500	140	55	33.5	2.24 ± 0.21	2.58 ± 0.49	1.43 ± 0.38
500–700	65	38	20.8	2.50 ± 0.32	1.64 ± 0.41	0.71 ± 0.23
700–900	22	19	7.7	2.21 ± 0.43	0.59 ± 0.25	0.34 ± 0.20
190–900	380	161	106.2	2.44 ± 0.14	2.37 ± 0.54	1.07 ± 0.36

Notes.

^aThe start analysis time (T₁) and the end analysis time (T₂) in units of seconds. ^bTS is the test-statistic value, which is roughly equal to ${\sigma }^{2}$, where σ is the significance of the GRB detection. ^cThe observed LAT number counts within the region of interest (ROI), i.e., 12° of GRB center position. ^dThe predicted LAT number counts from GRB 170214A. ^ePhoton index (${{\rm{\Gamma }}}_{\mathrm{LAT}}$), photon flux (${F}_{{\rm{L}}}$), and energy flux (${f}_{{\rm{L}}}$) of GRB 170214A.

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Given the recommendation for selecting the detectors with high count rates above the background, the Time Tagged Event data from two Na i detectors (n0, n1) and one BGO detector (b0) are taken from FSSC and analyzed with the software package RMFIT version 4.3pr2.⁸ We select the energy range of 8–1000 keV for two Na i detectors and of 200 keV–10 MeV for a BGO detector. A first-order polynomial is applied to each detector to fit the background with flat count rate regions pre- and post-burst.

We select the same time bins as that used in the LAT analysis in the time interval of 52–160 s, after which there is no significant emission in the GBM band. Before 52 s, we perform a small time bins, i.e., 2 s per time bin, since it is a fast variable (hereafter FV) component in the GBM energy bands.⁹ The Band function is employed as the photon spectrum model in each time bin, which is described by Band et al. (1993):

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

where A is the normalization, ${E}_{b}=(\alpha -\beta ){E}_{p}/(2+\alpha )$, 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 ${E}^{2}N(E)$ representation. The energy fluxes (${f}_{{\rm{G}}}$) are obtained between 10 keV and 10 MeV, as shown in Table 2.

Table 2.  Fermi-GBM Results for GRB 170214A

T1–T2a	${f}_{{\rm{G}}}(10\,\mathrm{keV}-10\,\mathrm{MeV})$ b	${f}_{{\rm{G}}}(200\,\mathrm{keV}-1\,\mathrm{MeV})$ b	${f}_{{\rm{G}}}(8\,\mathrm{keV}-200\,\mathrm{MeV})$ b
(s)	$({10}^{-7}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1})$	$({10}^{-7}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1})$	$({10}^{-7}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1})$
0–2	8.23 ± 1.60	3.78 ± 0.61	1.73 ± 0.13
2–4	14.70 ± 3.79	3.69 ± 0.84	2.27 ± 0.14
4–6	21.40 ± 4.43	3.89 ± 0.12	2.99 ± 0.15
6–8	18.80 ± 4.80	7.92 ± 0.77	3.31 ± 0.15
8–10	27.60 ± 5.01	10.20 ± 0.75	4.10 ± 0.15
10–12	23.80 ± 4.99	9.41 ± 0.73	4.18 ± 0.15
12–14	29.40 ± 4.82	12.60 ± 0.82	5.77 ± 0.17
14–16	27.40 ± 1.81	12.10 ± 0.72	5.67 ± 0.16
16–18	35.40 ± 5.14	16.00 ± 2.68	6.95 ± 0.17
18–20	37.00 ± 1.76	16.50 ± 0.82	6.81 ± 0.18
20–22	41.40 ± 5.00	18.40 ± 0.84	8.06 ± 0.18
22–24	27.10 ± 4.62	12.40 ± 0.77	7.02 ± 0.17
24–26	16.40 ± 0.92	7.25 ± 0.67	5.32 ± 0.15
26–28	16.30 ± 1.44	<10.3	4.36 ± 0.15
28–30	20.50 ± 4.00	10.10 ± 0.71	6.69 ± 0.16
30–32	24.30 ± 4.26	8.00 ± 0.70	5.48 ± 0.16
32–34	20.80 ± 4.21	9.37 ± 0.70	6.24 ± 0.16
34–36	38.00 ± 2.01	16.30 ± 0.85	6.75 ± 0.17
36–38	39.00 ± 4.92	20.00 ± 0.89	8.85 ± 0.18
38–40	39.60 ± 1.87	16.80 ± 1.37	8.35 ± 0.17
40–42	39.80 ± 1.88	17.80 ± 0.97	7.08 ± 0.18
42–44	55.00 ± 5.00	25.00 ± 0.93	9.89 ± 0.19
44–46	68.70 ± 5.41	26.50 ± 2.87	9.58 ± 0.20
46–48	54.00 ± 5.15	19.20 ± 1.02	7.73 ± 0.19
48–50	47.50 ± 2.04	20.30 ± 1.13	8.28 ± 0.18
50–52	68.40 ± 4.10	23.80 ± 0.90	9.72 ± 0.20
52–62	39.00 ± 1.89	15.60 ± 0.34	7.66 ± 0.08
62–63	133.00 ± 6.13	33.70 ± 1.51	12.10 ± 0.29
63–64	98.10 ± 5.99	27.90 ± 1.70	9.93 ± 0.27
64–66	59.80 ± 3.90	20.60 ± 0.83	9.55 ± 0.19
66–67	56.20 ± 5.78	19.10 ± 1.25	8.86 ± 0.25
67–69	47.00 ± 3.84	14.90 ± 0.81	7.91 ± 0.18
69–70	29.40 ± 4.42	<16.7	5.01 ± 0.21
70–80	28.90 ± 1.82	12.70 ± 0.60	7.19 ± 0.08
80–90	7.02 ± 1.58	<5.1	2.16 ± 0.06
90–100	2.74 ± 1.39	0.44 ± 0.25	1.24 ± 0.05
100–110	6.97 ± 1.25	1.83 ± 0.42	2.25 ± 0.06
110–120	4.44 ± 0.61	<2.2	0.90 ± 0.05
120–130	9.33 ± 1.36	1.71 ± 0.30	1.86 ± 0.06
130–140	22.80 ± 1.80	7.61 ± 0.32	5.65 ± 0.07
140–150	6.08 ± 1.11	<4.2	1.97 ± 0.06
150–160	2.16 ± 1.06	<1.2	0.24 ± 0.06

Notes.

^aThe start analysis time (T₁) and the end analysis time (T₂) in units of seconds. ^bGBM energy flux in the corresponding energy range.

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For GRB 170214A, we build the light curves in two narrow energy bands, i.e., 8–200 keV and 200 keV–1 MeV. We employ a single power-law function (PL), i.e., $N{(E)={N}_{0}(E/100\mathrm{keV})}^{-{\alpha }_{\mathrm{PL}}}$ (where ${\alpha }_{\mathrm{PL}}$ is the PL decay index), to model the photon spectrum in the former energy band, because the derived peak energy in the whole GBM energy range (8 keV–10 MeV) is always larger than ∼200 keV. A Band function is used to model the photon spectrum for the latter energy band. The GBM fluxes in these two energy bands of GRB 170214A are present in Table 2.

The light curves of GRB 170214A in the LAT (0.1–10 GeV) and GBM (10 keV–10 MeV) bands are plotted in Figure 1, which can be described with four phases.

• 1.  
  0–52 s: The LAT emission is too weak to be subdivided in this period, while the GBM emission shows a few pulse structures.
• 2.  
  52–80 s (Period 1): One fast variable (FV) component is found in both the LAT and GBM bands with fast rising and fast decaying behaviors. We fit it with an empirical smoothly broken power-law function (SBPL):

  $$\begin{eqnarray}&&f(t)={f}_{0}{\left[{\left(\displaystyle \frac{t}{{t}_{p}}\right)}^{-{\alpha }_{r}s}+{\left(\displaystyle \frac{t}{{t}_{p}}\right)}^{-{\alpha }_{d}s}\right]}^{-1/s},\end{eqnarray} \tag{ 1 }$$

  where f₀ is the normalization in units of $\mathrm{erg}\ {\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$, t_p is the peak time of the FV component, and the temporal indices of the rising part and the decaying part are ${\alpha }_{r}$ and ${\alpha }_{d}$, respectively. Here, s determines the smoothness of the peak, which is fixed at 10 for the fast variable components, following the suggestions in Liang et al. (2008). For both the GBM and LAT bands, the value of ${\alpha }_{r}$ cannot be constrained given only two flux points in the rising part, and hence we fix their values based on the connection of the two data points in the two energy bands, i.e., ${\alpha }_{r}=40$ in the LAT band and ${\alpha }_{r}=20$ in the GBM band. As shown in Table 3, both light curves decay steeply, i.e., ${\alpha }_{d}=-24\pm 14.5$ in the LAT band and ${\alpha }_{d}=-8.5\pm 0.4$ in the GBM band. Although the error bar of ${\alpha }_{d}$ is quite large in the LAT band, the result clearly shows a quick flux drop after the peak. Such strong variabilities imply that they are mostly likely to be related to the central engine of GRB 170214A. The peak times t_p in both energy bands are consistent with each other with uncertainties, which are around 61.8 s after GBM trigger.
• 3.  
  90–160 s (Period 2): One FV component appears in the light curve either in the GBM or LAT band. By fitting them with an SBPL function, ${\alpha }_{r}$ is found to be 3.0 ± 1.1 for the LAT emission and 10.3 ± 1.0 for the GBM emission, which are a bit too rapid for the external reverse shock model. This is also proved by the large decay indices (${\alpha }_{d}$) in both energy bands, which are −10.7 ± 7.4 and −28.9 ± 17.8 for the LAT and GBM emission, respectively, although the resultant error bars are quite large. The peak times in both energy bands are consistently around 139 s after GBM trigger.
• 4.  
  160–900 s (Extended phase): No GBM emission is detected in this phase, which implies that GRB 170214A enters the so-called afterglow phase. The LAT light curve shows a power-law decay with a decay index (${\alpha }_{\mathrm{LAT}}$) of −1.6 ± 0.2. The LAT photon index (${{\rm{\Gamma }}}_{\mathrm{LAT}}$) in the time interval 190–900 s is found to be −2.4±0.1, translating to a spectral index ${\beta }_{\mathrm{LAT}}={{\rm{\Gamma }}}_{\mathrm{LAT}}+1=-1.4\pm 0.1$. In the external shock model, we have the synchrotron flux at high energy ${f}_{\mathrm{LAT}}\propto {\nu }^{\beta }{t}^{\alpha }$. Considering that the LAT energy band ($\gt 100\,\mathrm{MeV}$) is usually above the external synchrotron cooling energy ($h{\nu }_{c}$, where h is the plank constant), we can derive the injection electron spectrum power index p to be ∼2.8 from ${f}_{\mathrm{LAT}}\propto {\nu }^{-p/2}$. This predicts a power-law index of about −1.6 for the light curve in the external shock model, i.e., ${f}_{\mathrm{LAT}}\propto {t}^{(2\mbox{--}3p)/4}$ (Sari et al. 1998), which is consistent with the observed one, which is −1.6 ± 0.2. Thus, we conclude that the late LAT emission can be well explained by the external shock model.

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Table 3.  Temporal Behaviors of GRB 170214A in the LAT and GBM Band

			LAT				GBM
T1–T2	Period	${\alpha }_{r}$ a	${\alpha }_{d}$ a	tpa	sa	${\alpha }_{r}$ a	${\alpha }_{d}$ a	tpa	sa
(s)				(s)				(s)
52–80	1	40 (fixed)	−24 ± 14.5	62.3 ± 1.1	10	20 (fixed)	−8.5 ± 0.4	60.7 ± 0.4	10
90–160	2	3.0 ± 1.1	−10.7 ± 7.4	140.5 ± 14.9	10	10.3 ± 1.0	−28.9 ± 17.8	137.8 ± 3.4	10
0–900	3	2.1 ± 0.4	−1.6 ± 0.2	145 ± 21.9	3	⋯	⋯	⋯

Note.

^aThe parameters of the smoothly broken power-law function (SBPL); ${\alpha }_{r}$ is the rising index before the peak time of t_p, after which the temporal decay index is ${\alpha }_{d}$, and s is the smoothness of the break.

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We then perform a global fit to the LAT light curve in the whole detection interval, i.e., 0–900 s. Based on the above analysis, we decompose the three components from the LAT light curve, i.e., Period 1, Period 2 and an underlying component, and Period 3, with each of them modeled by an SPBL. The results are present in Table 3 and plotted in Figure 1. For Period 1 and Period 2, the results are similar to that discussed above. As for Period 3 (with sharpness s of 3), the peak time is around 145 s, which can be explained as the dynamic deceleration time of the ejecta. The rising temporal index is around 2.1, which is consistent with the expected index (∼2.0) in the external shock model (Kumar & Barniol Duran 2009; Ghisellini et al. 2010). Apparently, the flux of Period 3 is comparable with that of Period 2 at ∼160.8 s after GBM trigger, after which the afterglow emission takes over. This time is also consistent with the end time of GBM emission.

In Period 1 and Period 2, we perform a joint spectral fit employing the GBM and LAT data between 8 keV and 10 GeV. The Castor Statistic (CSTAT) is used in the spectral fit, as in other bright LAT GRBs (Abdo et al. 2009a; Ackermann et al. 2010, 2011). For the same degrees of freedom (DOF), the smaller the CSTAT value is, the better the photon model is for the data. The results are presented in Table 4, where the photon index ${{\rm{\Gamma }}}_{\mathrm{LAT}}$ derived from the LAT data only is also presented for comparison with the results of the GBM+LAT joint fit. The results are also shown in Figure 2.

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Table 4.  Spectral Analysis Results of GRB 170214A in Two Periods

		GBM+LAT					LAT
		8 keV–10 GeV					0.1–10 GeV
T1–T2	Modela	αa	βa	Epa	Ecb	CSTAT/DOFc	${{\rm{\Gamma }}}_{\mathrm{LAT}}$ d
(s)				(keV)	(MeV)
52–80	Band	−0.74 ± 0.02	−2.34 ± 0.06	361 ± 12	⋯	600/348	3.49 ± 0.28
⋯	BandCut	−0.73 ± 0.01	−2.25 ± 0.02	355 ± 6	224 ± 58	575/347	⋯
90–160	Band	−1.30 ± 0.03	−2.25 ± 0.11	292 ± 37	⋯	714/348	2.12 ± 0.13

Notes.

^aThe spectral parameters of the Band model; α is the photon index below the peak energy of E_p, above which the photon index is β. ^bThe high-energy exponential cutoff energy. ^cThe Castor Statistic (CSTAT) value and the degree of freedom (DOF). ^dPhoton index of the LAT data only.

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For the intensive period of 52–80 s, the Band parameters, α, β and E_p, are found to be −0.74 ± 0.02, −2.34 ± 0.06, and 361 ± 12 keV, respectively. The CSTAT value is 600, with a DOF of 348. The ${{\rm{\Gamma }}}_{\mathrm{LAT}}$ from the LAT data only is much softer than the β from the GBM+LAT joint fit in this period, i.e., $-{{\rm{\Gamma }}}_{\mathrm{LAT}}$ of −3.49 ± 0.28 compared with β of −2.34 ± 0.06. Thus, we test a BandCut model, which is described as the Band with a high-energy cutoff, i.e., ${e}^{-E/{E}_{c}}$, where E_c is the exponential cutoff energy. With one more parameter, the BandCut will be regarded as a more preferred model than a single Band if Δ(CSTAT) is larger than 28 (Ackermann et al. 2013; Tang et al. 2015). However, only Δ(CSTAT) ∼ 25 is found for the BandCut model with a cutoff energy of 224 ± 58 MeV. Therefore, we consider that they are the equally good models in this period. Alternatively, when a power-law function is added to the Band model, i.e., Band+PL, the fit becomes even worse than a single Band with a larger CSTAT value.

For the second period of 90–160 s, α, β, and E_p in the Band model are −1.30 ± 0.03, −2.25 ± 0.11, and 292 ± 37 keV respectively, with the CSTAT/DOF of 714/348. The photon index of the LAT data is 2.12 ± 0.13, which is consistent with the high-energy photon index β of the GBM+LAT joint fit. The spectrum fit in this period cannot be improved significantly by either employing another fitting function or adding an additional component to the Band function, i.e., with a larger CSTAT value.

The result implies that the prompt GBM emission and the prompt LAT emission of GRB 170214A may come from the same region, and disfavors the existence of other spectral components in these two periods.

3.3.1. Method

First, the keV/MeV–GeV correlation between the GBM energy flux (${f}_{{\rm{L}}}$) and the LAT energy flux (${f}_{{\rm{G}}}$) is tested in a certain time period from T₁ to T₂ with several time bins, i.e., ${N}_{\mathrm{bin}}$. Assuming ${f}_{{\rm{L}}}({T}_{i})$ and ${f}_{{\rm{G}}}({T}_{i})$ are the LAT flux and GBM flux at the time of T_i, the linear equation can be represented as

$$\begin{eqnarray}&&{f}_{{\rm{L}}}({T}_{i})=A+B\times {f}_{{\rm{G}}}({T}_{i}).\end{eqnarray} \tag{ 2 }$$

According to Pearson's correlation, the coefficient R can be represented

$$\begin{eqnarray}&&R=\displaystyle \frac{{\displaystyle \sum }_{1}^{{N}_{\mathrm{bin}}}({f}_{{\rm{G}}}({T}_{i})-{\bar{f}}_{{\rm{G}}})({f}_{{\rm{L}}}({T}_{i})-{\bar{f}}_{{\rm{L}}})}{\sqrt{{\displaystyle \sum }_{1}^{{N}_{\mathrm{bin}}}{({f}_{{\rm{G}}}({T}_{i})-{\bar{f}}_{{\rm{G}}})}^{2}}\sqrt{{\displaystyle \sum }_{1}^{{N}_{\mathrm{bin}}}{({f}_{{\rm{L}}}({T}_{i})-{\bar{f}}_{{\rm{L}}})}^{2}}},\end{eqnarray} \tag{ 3 }$$

where the ${\bar{f}}_{{\rm{L}}}$ and ${\bar{f}}_{{\rm{G}}}$ represent, respectively, the average fluxes of the LAT band and the GBM band in the chosen time interval. We also calculate the p value of the null hypothesis, which can be described as the confidence level of $1-p$ for the keV/MeV–GeV correlation, using the software Origin. A strong correlation can be claimed when $R\gt 0.8$ while a moderate correlation can be claimed when $0.5\lt R\lt 0.8$ (Newton & Rudestam 1999). In the above two cases, the p value should be smaller than 0.05, which represents the 95% confidence level of the correlation. Since the R value in each fit is larger than 0 (R = 0, no correlation) in our analysis, other cases are defined as a weak correlation. We first test the correlation in the whole prompt emission duration, which covers the time period of both GBM and LAT detection (labeled as "Trace" hereafter). Second, we try to determine whether the correlation exists in some subperiods, such as in the FV components.

The ratio ${\mathscr{L}}$ is calculated between the duration of the FWHM of the SBPL-fit result and the whole duration of the FV component. The FWHM duration ω is derived from the time spanning half of the SBPL peak flux in the LAT light curve. The period T (=${T}_{2}-{T}_{1}$) is the lower-limit value of the duration of the FV component, which often lasts for a longer duration than T as shown in Figures 1 and 4. We regard it as a rapid variability of the LAT FV component if (1) the post-peak decay index ${\alpha }_{d}$ is sharper than −3.0, since such a sharp decay index cannot be explained by external forward shocks (Kobayashi & Zhang 2003; Burrows et al. 2005), and (2) the ratio ${\mathscr{L}}$ is significantly smaller than 1.0, which implies a rapid variability timescale (Ackermann et al. 2011).

3.3.2. Result

First, we test the correlation between the LAT (0.1–10 GeV) and GBM (10 keV–10 MeV) light curves during the prompt phase. In the Trace period, R is 0.90 with $p\lt {10}^{-4}$, which implies a strong correlation between the GBM and LAT emission. In both Period 1 and Period 2, strong correlations are also found with R of 0.95 and 0.82, respectively, as shown in Table 5.

Table 5.  Correlation Analysis Results of GRB 170214A and Five Other Bright LAT GRBs

GRB Name	GBM(E1–E2)	LAT(E1–E2)	T1–T2	Period	${N}_{\mathrm{bin}}$	R	p	Correlationa
	(keV)	(GeV)	(s)
170214A	10–10,000	0.1–10	52–160	Trace	16	0.90	$\lt {10}^{-4}$	Strong
⋯	⋯	⋯	52–80	1	8	0.95	$3.7\times {10}^{-3}$	Strong
⋯	⋯	⋯	90–160	2	7	0.81	$2.9\times {10}^{-2}$	Strong
⋯	8–200	⋯	52–160	Trace	16	0.68	$4.0\times {10}^{-3}$	Moderate
⋯	⋯	⋯	52–80	1	8	0.75	$3.3\times {10}^{-2}$	Moderate
⋯	⋯	⋯	90–160	2	7	0.85	$1.5\times {10}^{-2}$	Strong
⋯	200–1000	⋯	52–160	Trace	11	0.81	$2.7\times {10}^{-3}$	Strong
⋯	⋯	⋯	52–80	1	7	0.96	$7.4\times {10}^{-4}$	Strong
⋯	⋯	⋯	90–160	2	4	0.83	0.17	Weak
080916C	10–10,000	0.1–10	3.7–53.3	Trace	24	0.59	$2.6\times {10}^{-3}$	Moderate
090510	10–10,000	⋯	0.3–0.9	Trace	4	0.09	0.91	Weak
090902B	10–10,000	⋯	0–23	Trace	27	0.73	$\lt {10}^{-4}$	Moderate
⋯	⋯	⋯	0–12.5	1	13	0.76	$2.6\times {10}^{-3}$	Moderate
⋯	⋯	⋯	12.5–23	2	14	0.53	$4.6\times {10}^{-2}$	Moderate
090926A	10–10,000	⋯	2–16.5	Trace	26	0.12	0.56	Weak
⋯	⋯	⋯	5.5–8.5	1	6	0.89	$1.7\times {10}^{-2}$	Strong
130427A	10–10,000	⋯	0–200	Trace	20	0.16	0.51	Weak

Note.

^a $R\gt 0.8$ for strong positive correlation ($p\lt 0.05$); (2) $0.5\lt R\lt 0.8$ for moderate positive correlation ($p\lt 0.05$); (3) $0\lt R\lt 0.5$ for weak positive correlation (Newton & Rudestam 1999).

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Second, the correlations of the LAT emission (0.1–10 GeV) and the two sub-energy bands of the GBM emission, i.e., 8–200 keV and 200 keV–1 MeV, are tested. For the case of 8–200 keV, the LAT-detected emission moderately correlates with that detected in the GBM band in the Trace period, while in Period 1 and Period 2, they show a moderate and a strong correlation, respectively, which can be found in Table 5. As for 200 keV–1 MeV, strong correlations between the emission in the LAT and GBM energy band are found in both the Trace period and Period 1. However, a weak correlation, with a p value of 0.17 is found in Period 2 due to the non-detection of the 200 keV–1 MeV emission in the four time bins of this period.

Third, we study the GeV variability of the two LAT FV components during Period 1 and Period 2, the results of which are given in Table 6. The first FV component with an ${\alpha }_{d}$ of −23.88 ± 14.5 and ${\mathscr{L}}\lt 0.09$exhibits rapid variability. For the second FV component, we find that it has rapid variability with an ${\alpha }_{d}$ of −10.66 ± 7.36 and ${\mathscr{L}}\lt 0.60$, although with a large uncertainty on the decay power-law index.

Table 6.  GeV Variability of the Possible FV Components in Five Other LAT GRBs

GRBa	${T}_{1}-{T}_{2}$	${\alpha }_{r}$ b	${\alpha }_{d}$ b	tpb	sb	ωc	${\mathscr{L}}$ d	Variabilitye
	(s)	(s)
170214A_1	52–80	40.00 (fixed)	−23.88 ± 14.5	62.3 ± 1.1	10	2.4 ± 0.5	<0.09	Y
170214A_2	90–160	2.96 ± 1.06	−10.66 ± 7.36	140.5 ± 14.9	10	41.8 ± 7.0	<0.60	Y
080916C	3.7–9.7	10.00 ± 3.85	−5.38 ± 1.83	5.9 ± 0.2	10	1.3 ± 0.2	<0.22	Y
090510	0.3–1.5	14.55 ± 4.91	−4.49 ± 0.39	0.8 ± 0.02	10	0.2 ± 0.02	<0.17	Y
090902B_1	0–12.5	2.47 ± 0.84	−7.23 ± 3.60	10.3 ± 0.4	10	3.8 ± 0.5	<0.30	Y
090902B_2	12.5–23	5.62 ± 2.56	−3.24 ± 1.73	15.9 ± 0.6	10	5.6 ± 0.9	<0.53	Y
090926A_1	2–8.5	4.04 ± 0.92	−10.63 ± 4.25	7.0 ± 0.2	10	1.7 ± 0.3	<0.26	Y
090926A_2a	8.5–11.5	19.47 ± 7.61	−13.32 ± 5.07	9.9 ± 0.2	10	0.9 ± 0.07	<0.31	Y
090926A_2b	11.5–13.5	72 (fixed)	−29.08 ± 11.53	11.9 ± 0.2	10	0.4 ± 0.05	<0.22	Y
090926A_2c	13.5–16.5	25.17 ± 4.89	−14.99 ± 4.78	14.5 ± 0.1	10	1.2 ± 0.1	<0.39	Y
130427A	0–70	1.10 ± 0.18	−1.71 ± 0.24	20.5 ± 1.5	3	23.0 ± 1.8	<0.33	N

Notes.

^aThe subscript represents the index of the FV component. ^bThe parameters of the smoothly broken power-law function (SBPL); ${\alpha }_{r}$ is the rising index below the peak time of t_p, above which the decay index is ${\alpha }_{d}$, and s is the smoothness of the break. ^cDuration of the FWHM in the light curve of the FV component. ^dRatio between ω and $T(={T}_{2}-{T}_{1})$. ^eRapid variability when the central value of ${\alpha }_{d}$ is smaller than −3 and ${\mathscr{L}}\lt 1.0$.

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The above results on the temporal correlation of keV/MeV–GeV and the temporal variability of the LAT FV components suggest that the prompt high-energy emission in GRB 170214A cannot be produced in the external shock region, but may share the same internal origin as the GBM emission.

In the Trace period, the GBM and LAT light curves show a moderate correlation with an R of 0.59 and a p of 0.0026.

We search the GeV variability in the first 10 s, i.e., between 3.7 and 9.7 s. The GeV emission with ${\alpha }_{d}=-5.38\pm 1.83$ and ${\mathscr{L}}\lt 0.22$ infers rapid variability in this period. The resultant peak time of 5.9 ± 0.2 is consistent with the prominent peak in the time interval of 3.6–7.7 s (Abdo et al. 2009b).

GRB 090510 is a short GRB. We find that there is a weak correlation in the Trace period, i.e., 0.3–0.9 s.

The emission in the LAT band shows a rapid rise in the Trace period, after which it exhibits a fast decay. Thus, we extended the time interval to a longer period as a possible FV component, i.e., 0.3–1.5 s. In this time interval, the LAT light curve shows rapid variability with ${\alpha }_{d}=-4.49\pm 0.39$ and ${\mathscr{L}}\lt 0.17$. The discrepancy between the peak times in the GBM and the LAT light curves is about 0.15 s, which is comparable with the time lag between the GBM and LAT light curves derived in Ackermann et al. (2010), i.e., 0.25 ± 0.05 s.

Moderate correlations are found between the GBM and LAT light curves in the Trace period, Period 1 (0–12.5 s), and Period 2 (12.5–23 s).

The LAT light curve is subdivided into two possible FV components for variability analysis, i.e., 0–12.5 s and 12.5–23 s, which is the same as Period 1 and Period 2. For the first FV component, the variability is rapid with ${\alpha }_{d}=-7.23\pm 3.60$ and ${\mathscr{L}}\lt 0.30$. The peak time of the first FV components (10.3 ± 0.4 s) is consistent with that discovered in Abdo et al. (2009a), which is around 9 s. For the second FV component, the resultant ${\alpha }_{d}$ with a large uncertainty, i.e., ${\alpha }_{d}=-3.24\pm 1.73$, and ${\mathscr{L}}\lt 0.53$ can indicate rapid variability.

A weak correlation is found between the GBM and LAT light curves in the Trace period, although both light curves show rapid variabilities. However, we find a strong correlation in the first 8.5 s, with an R of 0.89.

The LAT light curve indeed shows fast variability. Thus, two possible FV components are employed to search for the GeV variability, i.e., 2–8.5 s and 8.5–16.5 s. For the first FV component, the variability is rapid with an ${\alpha }_{d}$ of −10.63 ± 4.25 and ${\mathscr{L}}\lt 0.26$. The peak time (7.0 ± 0.2 s) of the first FV component is located at the time interval "b" (3.3–9.8 s; Ackermann et al. 2011). For the second FV component, we find that the LAT light curve could be decomposed into substructures. Three SBPL components are employed to fit the GeV light curve; the results are shown in Table 6. All of them exhibit rapid variabilities with ${\alpha }_{d}$ much sharper than −3 and ${\mathscr{L}}\lt 0.39$. One of the peak times, i.e., 9.9 ± 0.2 s, is consistent with the peak time in the time interval "c" (9.8–10.5 s) in Ackermann et al. (2011).

In the Trace period, the correlation between the LAT-detected and GBM-detected emissions is weak. This is consistent with the conclusion drawn by Ackermann et al. (2014), i.e., the LAT-detected emission does not appear to be temporally correlated with the GBM emission beyond the initial spike at the GBM trigger.

We then perform the analysis in the first 70 s of the LAT-detected emission, a possible FV component, to study the GeV variability. ${\alpha }_{d}=1.71\pm 0.24$ implies that the variability is not rapid enough to support an internal origin in this period.