Testing the High-latitude Curvature Effect of Gamma-Ray Bursts with Fermi Data: Evidence of Bulk Acceleration in Prompt Emission

To delineate the characteristics of the pulses, several functional forms have been proposed (e.g., Kocevski et al. 2003; Norris et al. 2005). In order to adequately characterize a pulse shape, our next step is to employ an asymmetric fast-rising and exponential-decay function, the so-called FRED model (Kocevski et al. 2003), to fit the entire lightcurve of that pulse (Figure A1). The peak time of the pulse can be then determined. The function reads as

$$\begin{eqnarray}&&I(t)={I}_{p}{\left(\displaystyle \frac{t+{t}_{0}}{{t}_{p}+{t}_{0}}\right)}^{r}{\left[\displaystyle \frac{d}{r+d}+\displaystyle \frac{r}{r+d}{\left(\displaystyle \frac{t+{t}_{0}}{{t}_{p}+{t}_{0}}\right)}^{r+1}\right]}^{-\tfrac{r+d}{r+1}},\end{eqnarray} \tag{ 3 }$$

where I_p is the amplitude, t₀ and t_p are the zero time and the peak time of the pulse, and r and d are the rise and decay timescale parameters, respectively. The model invokes five parameters (I_p, t₀, t_p, r, and d). We also considered a broken power-law (BKPL) fit to the pulse (Appendix). In Figure A2 we present a comparison of the fitting results between the FRED model and the BKPL model.

In Table 2, we list the best-fit parameters by adopting the FRED model for our sample. We list the time resolution of the count rate (counts/sec) lightcurve used for each burst (Column 2), the start and stop times of the selected pulses (Column 3), and the corresponding significance S (Column 4), as well as the best-fit parameters for the FRED model (Columns 5–9) including the normalization I_p; the zero time t₀, which we fixed to zero for each case; the peak time t_p of the pulse; and the rise r and decay d timescale parameters. The reduced chi-squared χ²/dof (Column 10), the Akaike information criterion (AIC) statistic (Column 11), and the Bayesian information criterion (BIC) statistic (Column 12) are also presented. Note that the goodness of fit (GOF) can be evaluated by calculating the reduced chi-squared statistic when the uncertainties in the data have been obtained. For a set of N data points {x_i , y_i } with the estimated uncertainties {σ_i } in the y_i values, one has ${\chi }^{2}={{\rm{\Sigma }}}_{i=1}^{N}\tfrac{{({y}_{i}-\hat{{y}_{i}})}^{2}}{{\sigma }_{i}^{2}}$ and reduced ${\chi }_{\nu }^{2}={\chi }^{2}/\mathrm{dof}$, where dof =(N − N_varys) is the degrees of freedom, N is the number of data points, and N_varys is the number of variables in the fit. The bad fits (large ${\chi }_{\nu }^{2}$ values) indicate that these pulses cannot be well delineated by the FRED model. In Table 2, AIC is calculated by $N\mathrm{ln}({\chi }^{2}/N)+2{N}_{\mathrm{varys}}$, and BIC by $N\mathrm{ln}({\chi }^{2}/N)\,+\mathrm{ln}(N){N}_{\mathrm{varys}}$.

We use the energy flux lightcurves to measure the temporal indices. This is because the indices thus defined can be better compared with model predictions.

Our procedure to obtain the temporal indices includes the following steps:

• 1.  
  Calculate the energy flux in each selected time bin. In order to obtain the energy flux, one needs to perform the spectral fits. For a given burst in our final sample, we therefore use the typical spectral model, called the Band function model (Band et al. 1993), to fit the spectral data of each time bin (S > 15) selected by the BBlocks method, and the best-fit parameters are evaluated by adopting the maximum-likelihood estimation (MLE) technique. The energy flux in such narrow time bins thus can be also calculated from the best fits, with a k-correction (1–10⁴ keV) applied. ⁷
• 2.  
  Determine the entire time interval of the decay wing of the pulses. In order to determine the entire time interval of the decay wing of the pulses, one needs to determine the peak times of the pulses. The peak times of the pulses can be roughly obtained by using the FRED model to fit their pulse lightcurves as we discussed in Section 3.1. We find that the peak time determined by the FRED model for a good fraction of our sample can exactly match the true peaks of pulses (e.g., GRB 110920546). However, there are still some bursts whose peak times determined by the FRED model do not exactly describe the true peaks of the pulses. ⁸ Therefore, we use two selection criteria. First, for the cases where the peak times determined by the FRED model can exactly match the true peaks of pulses, we use these values (see the vertical yellow dashed lines in Figure 1). That is, as long as the peak time (t_p) of a certain pulse is obtained from the FRED model fits, the time window of the decaying wing of the pulse can be determined as t_p − t_stop, where t_stop is the end time of a pulse. The stop time of the decay wing of a certain pulse can be precisely determined by the stop time of the last time bin that satisfies S > 15. Second, for the cases whose peak times determined by the FRED model do not exactly describe the true peaks of the pulses, we inspect the peak times from their lightcurves by eye (see the vertical black dashed lines in Figure 1). We define this phase as "Phase I" throughout the paper.
• 3.  
  Determine the late-time interval of the decay wing of the pulses. Physically, the decay for prompt emission may not be fully controlled by the curvature effect. As shown in the theoretical modeling in Uhm & Zhang (2016b) and Uhm et al. (2018), the spectral lags are not caused by the curvature effect, and the temporal peaks of the pulses are often related to the time when the characteristic energy crosses the gamma-ray band as it decays with time. One possible test for this is to see whether the temporal peaks of the lightcurves for different GBM detectors that have different energy ranges occur at different times. We therefore compare the Na I (8 keV–1 MeV) and BGO (200 keV–40 MeV) lightcurves for each individual burst, as shown in Figure A3. We find that in many cases in our sample the peak times are clearly shifted between two different detectors (GRB 081224887, GRB 110721200, GRB 120426090, GRB 160216801, GRB 170921168, and GRB 171210493), indicating that the peaks of the pulses are indeed related to crossing of a spectral break. ⁹ For these bursts, the curvature effect does not kick in right after the peak. It may show up later in some bursts or would not show up at all in some others. When they show up, they may be related to the later part of the decay, usually not related to the decay right after the peak time. This brings an additional difficulty (other than the fact that the decay phase is usually short for prompt-emission pulses) in studying the curvature effect with the prompt-emission data. Besides testing the entire decay phase, we also adopt a more conservative approach by only testing the late-part time interval of the decay phase. Quantitatively, we only consider the last three time bins with S > 15. In practice, when a certain model is used to fit the data, the number of data points N should be greater than the number of variables N_varys of the model in order to get a good fitting result. The power-law model we use has two variables: amplitude and power-law index. This is why we include at least three data points in the fits. We define this phase as "Phase II" throughout the paper.
• 4.  
  After the time intervals are clearly defined in the aforementioned two cases, we then perform two fits ¹⁰ (see Figure 1): one uses a power-law model to fit the entire decay phase and obtain a temporal decay index defined as ${\hat{\alpha }}_{\mathrm{PL}}^{{\rm{I}}};$ the other uses a power-law model to fit the later part of the decay to obtain a temporal decay index defined as ${\hat{\alpha }}_{\mathrm{PL}}^{\mathrm{II}}$. The power-law function we use to fit the lightcurves in order to obtain the $\hat{\alpha }$ indices is given by

  $$\begin{eqnarray}&&{F}_{t}={F}_{t,0}\,{(t+{t}_{0})}^{-\hat{\alpha }},\end{eqnarray} \tag{ 4 }$$

  where F_t,0 is the amplitude and $\hat{\alpha }$ is the temporal slope. The t₀ parameter is fixed in the beginning of the pulse (t₀ = 0) for all cases in this task because this is physically more relevant (Zhang et al. 2006; Uhm & Zhang 2015). Note that the peak time t_p does not enter the problem of defining $\hat{\alpha }$, so the inaccurate determination of t_p in the pulse lightcurve fitting does not noticeably affect our results. All these lightcurve fits are performed using a pure Python package called lmfitt (Newville et al. 2016) by applying a nonlinear least-squares method using the Levenberg–Marquardt algorithm to fit a function to the data. Within lmfitt fits, we can set parameters with a varied or fixed value in the fit, or place an upper or lower bound on the value. The weight of parameter error is also easily taken into account in the fits. In Figure A2, we also use GRB 131231198 as an example case to compare the fitting results obtained from different Python packages (lmfit and scipy. optimize. curve_fit).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The start and stop times of each selected time interval (Column 2), the corresponding S value (Column 3), the adopted zero time t₀ (Column 4), the best-fit parameters, include the normalization (Column 5), the power-law index (Column 6), and the AIC and BIC statistics (Column 7) are listed in Table 3. For each burst, the entire decay phase is marked with (1) and the late-part decay phase is marked with (2).

Table 3. Results of Lightcurve and Spectral Fitting of the Decaying Wing of the Pulses

GRB	tstart ∼ tstop	S	t0	Lightcurve Power-law Fitting			Spectral Power-law Fitting			Spectral Cutoff Power-law Fitting
				Ft,0	$\hat{\alpha }$	AIC/BIC	Fν,0	$\hat{\beta }$	DIC/pDIC	N0,p	tp	$\hat{{\rm{\Gamma }}}$	Ec	DIC/pDIC
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)
081224887(1)	1.896 ∼ 12.502	88.10	0	(2.62 ± 0.06) × 10−6	1.81 ± 0.06	−159/−160	(1.62${}_{-0.04}^{+0.04}$) × 101	0.43${}_{-0.00}^{+0.00}$	8255/1.98	⋯	⋯	⋯	⋯	⋯
081224887(2)	5.424 ∼ 12.502	47.56	0	(8.16 ± 0.01) × 10−7	2.25 ± 0.00	−121/−123	(1.43${}_{-0.06}^{+0.06}$) × 101	0.52${}_{-0.01}^{+0.01}$	5533/2.00	⋯	⋯	⋯	⋯	⋯
090620400(1)	4.076 ∼ 12.289	47.80	0	(2.46 ± 0.05) × 10−6	3.02 ± 0.15	−161/−162	(1.27${}_{-0.05}^{+0.05}$) × 101	0.48${}_{-0.01}^{+0.01}$	6135/2.02	(5.43 ± 0.34) × 10−2	4.076	0.51 ± 0.04	117.3 ± 6.8	5116/2.83
090620400(2)	5.319 ∼ 12.289	35.56	0	(4.91 ± 0.34) × 10−7	2.78 ± 0.24	−101/−103	(1.34${}_{-0.07}^{+0.07}$) × 101	0.55${}_{-0.01}^{+0.01}$	5301/1.99	(5.25 ± 0.53) × 10−2	5.319	0.55 ± 0.06	87.9 ± 6.8	4827/2.57
090719063(1)	4.443 ∼ 14.562	128.19	0	(3.68 ± 0.17) × 10−6	3.21 ± 0.23	−207/−207	(3.73${}_{-0.08}^{+0.08}$) × 101	0.53${}_{-0.00}^{+0.00}$	6860/1.98	(8.73 ± 0.25) × 10−2	4.443	0.79 ± 0.02	182.8 ± 6.9	4292/2.95
090719063(2)	7.810 ∼ 14.562	66.34	0	(8.21 ± 0.34) × 10−7	2.39 ± 0.19	−132/−133	(5.05${}_{-0.24}^{+0.24}$) × 101	0.77${}_{-0.01}^{+0.01}$	4085/1.98	⋯	⋯	⋯	⋯	⋯
090804940(1)	1.279 ∼ 8.705	98.14	0	(1.09 ± 0.14) × 10−6	1.10 ± 0.16	−179/−179	(7.29${}_{-0.20}^{+0.19}$) × 101	0.73${}_{-0.01}^{+0.01}$	7458/1.99	⋯	⋯	⋯	⋯	⋯
090804940(2)	4.678 ∼ 8.705	40.50	0	(5.12 ± 0.58) × 10−7	3.09 ± 0.53	−97/−99	(7.21${}_{-0.48}^{+0.48}$) × 101	0.92${}_{-0.02}^{+0.02}$	4486/1.98	(54.40 ± 2.66) × 10−2	4.678	0.72 ± 0.09	49.4 ± 4.0	4167/2.73
100707032(1)	1.631 ∼ 28.780	131.02	0	(4.27 ± 0.05) × 10−6	1.57 ± 0.01	−320/−319	(2.42${}_{-0.04}^{+0.04}$) × 101	0.49${}_{-0.00}^{+0.00}$	9384/1.98	⋯	⋯	⋯	⋯	⋯
100707032(2)	14.210 ∼ 28.78	47.30	0	(4.01 ± 0.17) × 10−7	1.90 ± 0.15	−105/−107	(3.23${}_{-0.23}^{+0.23}$) × 101	0.82${}_{-0.02}^{+0.02}$	4646/1.97	⋯	⋯	⋯	⋯	⋯
110721200(1)	0.470 ∼ 25.000	76.49	0	(2.72 ± 0.09) × 10−6	1.46 ± 0.03	−243/−242	(1.06${}_{-0.02}^{+0.02}$) × 101	0.44${}_{-0.00}^{+0.00}$	7613/1.99	⋯	⋯	⋯	⋯	⋯
110721200(2)	6.252 ∼ 25.000	28.610	0	(4.07 ± 0.05) × 10−7	1.70 ± 0.02	−112/−114	5.51${}_{-0.34}^{+0.34}$	0.51${}_{-0.01}^{+0.01}$	6119/1.99	⋯	⋯	⋯	⋯	⋯
110920546(1)	9.966 ∼ 122.091	59.68	0	(2.75 ± 0.12) × 10−6	1.03 ± 0.08	−241/−241	8.71${}_{-0.20}^{+0.20}$	0.42${}_{-0.00}^{+0.00}$	12386/2.00	⋯	⋯	⋯	⋯	⋯
110920546(2)	55.534 ∼ 122.091	34.09	0	(3.39 ± 0.04) × 10−7	1.83 ± 0.04	−113/−115	8.00${}_{-0.37}^{+0.37}$	0.53${}_{-0.01}^{+0.01}$	9532/1.99	⋯	⋯	⋯	⋯	⋯
120323507(1)	0.094 ∼ 0.581	130.70	0	(15.32 ± 1.63) × 10−6	2.72 ± 0.19	−176/−177	(8.57${}_{-0.33}^{+0.33}$) × 102	0.90${}_{-0.01}^{+0.01}$	1643/2.00	⋯	⋯	⋯	⋯	⋯
120323507(2)	0.252 ∼ 0.581	66.33	0	(5.31 ± 0.57) × 10−6	1.46 ± 0.50	−83/−85	(6.13${}_{-0.13}^{+0.13}$) × 102	1.00${}_{-0.00}^{+0.00}$	1161/1.00	⋯	⋯	⋯	⋯	⋯
120426090(1)	1.044 ∼ 4.882	125.87	0	(4.94 ± 0.53) × 10−6	1.84 ± 0.28	−190/−190	(1.41${}_{-0.04}^{+0.04}$) × 102	0.70${}_{-0.01}^{+0.01}$	5338/2.04	⋯	⋯	⋯	⋯	⋯
120426090(2)	2.600 ∼ 4.882	35.11	0	(6.14 ± 1.14) × 10−7	3.67 ± 0.66	−94/−96	(1.25${}_{-0.05}^{+0.05}$) × 102	0.99${}_{-0.01}^{+0.01}$	2529/1.14	(9.19 ± 2.13) × 10−2	2.600	1.06 ± 0.12	52.9 ± 7.4	2408/−0.66
130305486(1)	4.632 ∼ 32.212	36.88	0	(3.12 ± 0.26) × 10−6	2.33 ± 0.18	−173/−173	4.02${}_{-0.12}^{+0.12}$	0.30${}_{-0.01}^{+0.01}$	8462/1.99	(1.53 ± 0.04) × 10−2	4.632	0.67 ± 0.03	545.7 ± 44.0	7110/2.92
130305486(2)	8.849 ∼ 32.212	16.99	0	(9.97 ± 0.87) × 10−7	1.24 ± 0.15	−96/−98	2.20${}_{-0.15}^{+0.15}$	0.33${}_{-0.01}^{+0.01}$	6953/2.00	⋯	⋯	⋯	⋯	⋯
130614997(1)	0.457 ∼ 6.210	64.91	0	(0.90 ± 0.10) × 10−6	0.76 ± 0.16	−123/−124	(7.23${}_{-0.32}^{+0.32}$) × 101	0.85${}_{-0.01}^{+0.01}$	4828/2.00	⋯	⋯	⋯	⋯	⋯
130614997(2)	2.030 ∼ 6.210	44.82	0	(6.37 ± 0.51) × 10−7	1.36 ± 0.21	−98/−100	(6.41${}_{-0.40}^{+0.40}$) × 101	0.89${}_{-0.02}^{+0.02}$	4217/1.97	⋯	⋯	⋯	⋯	⋯
131231198(1)	22.406 ∼ 59.114	298.10	0	(1.43 ± 0.20) × 10−6	4.01 ± 0.29	−555/−553	(1.22${}_{-0.01}^{+0.01}$) × 102	0.75${}_{-0.00}^{+0.00}$	12654/2.01	(7.56 ± 0.11) × 10−2	22.406	1.34 ± 0.01	239.8 ± 6.1	8716/2.97
131231198(2)	47.97 ∼ 59.114	31.39	0	(2.87 ± 0.60) × 10−7	9.00 ± 2.64	−98/−100	(4.10${}_{-0.13}^{+0.13}$) × 101	0.99${}_{-0.01}^{+0.01}$	5371/1.08	(0.98 ± 0.18) × 10−2	47.970	1.62 ± 0.09	132.7 ± 30.7	5314/0.75
141028455(1)	11.565 ∼ 40.000	69.79	0	(2.96 ± 0.24) × 10−6	3.03 ± 0.52	−258/−257	8.35${}_{-0.23}^{+0.23}$	0.46${}_{-0.01}^{+0.01}$	7260/2.02	(1.73 ± 0.05) × 10−2	11.565	1.00 ± 0.02	364.8 ± 25.8	6401/2.92
141028455(2)	22.335 ∼ 40.000	21.28	0	(2.27 ± 0.04) × 10−7	3.38 ± 0.07	−114/−116	4.17${}_{-0.38}^{+0.37}$	0.54${}_{-0.02}^{+0.02}$	5685/1.96	(0.68 ± 0.08) × 10−2	22.335	1.08 ± 0.08	269.6 ± 60.7	5620/1.80
150213001(1)	2.227 ∼ 6.661	198.47	0	(2.56 ± 0.05) × 10−6	3.86 ± 0.05	−342/−341	(4.17${}_{-0.08}^{+0.08}$) × 102	0.93${}_{-0.01}^{+0.01}$	6143/2.04	(17.50 ± 0.59) × 10−2	2.227	1.33 ± 0.02	114.0 ± 4.0	4426/2.94
150213001(2)	4.085 ∼ 6.661	49.81	0	(8.86 ± 0.50) × 10−7	2.95 ± 0.30	−130/−131	(1.29${}_{-0.03}^{+0.03}$) × 102	1.00${}_{-0.00}^{+0.00}$	3350/1.03	⋯	⋯	⋯	⋯	⋯
150314205(1)	1.846 ∼ 14.999	176.97	0	(8.70 ± 0.73) × 10−6	1.09 ± 0.15	−287/−287	(3.93${}_{-0.06}^{+0.06}$) × 101	0.46${}_{-0.00}^{+0.00}$	11811/2.01	⋯	⋯	⋯	⋯	⋯
150314205(2)	7.847 ∼ 14.999	71.42	0	(1.86 ± 0.51) × 10−6	4.86 ± 1.58	−111/−112	(1.95${}_{-0.07}^{+0.07}$) × 101	0.48${}_{-0.01}^{+0.01}$	4482/1.98	(3.44 ± 0.11) × 10−2	7.847	1.03 ± 0.02	499.7 ± 44.1	3734/2.94
150510139(1)	0.889 ∼ 49.997	90.65	0	(8.14 ± 1.96) × 10−6	0.77 ± 0.18	−390/−389	9.19${}_{-0.17}^{+0.17}$	0.40${}_{-0.00}^{+0.00}$	10870/1.99	⋯	⋯	⋯	⋯	⋯
150510139(2)	28.736 ∼ 49.997	34.51	0	(5.90 ± 0.94) × 10−7	3.52 ± 0.75	−95/−97	7.31${}_{-0.43}^{+0.43}$	0.54${}_{-0.01}^{+0.01}$	6605/1.99	(0.90 ± 0.06) × 10−2	28.736	1.21 ± 0.04	575.0 ± 122.1	6447/2.29
150902733(1)	8.934 ∼ 25.000	112.07	0	(2.51 ± 0.71) × 10−6	4.28 ± 0.64	−260/−260	(1.65${}_{-0.03}^{+0.03}$) × 101	0.40${}_{-0.00}^{+0.00}$	11345/2.01	(4.68 ± 0.07) × 10−2	8.934	0.77 ± 0.01	375.8 ± 13.5	6656/2.97
150902733(2)	14.609 ∼ 25.000	32.82	0	(3.49 ± 0.64) × 10−7	5.62 ± 1.03	−97/−99	9.79${}_{-0.61}^{+0.61}$	0.56${}_{-0.01}^{+0.01}$	5601/2.01	(2.05 ± 0.20) × 10−2	14.609	0.87 ± 0.07	161.8 ± 19.8	5366/2.51
151021791(1)	0.797 ∼ 7.923	62.48	0	(8.56 ± 1.15) × 10−7	1.52 ± 0.12	−151/−152	(1.62${}_{-0.06}^{+0.06}$) × 101	0.50${}_{-0.01}^{+0.01}$	4361/1.99	⋯	⋯	⋯	⋯	⋯
151021791(2)	2.286 ∼ 7.923	36.55	0	(4.81 ± 0.39) × 10−7	1.65 ± 0.14	−100/−102	(1.47${}_{-0.10}^{+0.10}$) × 101	0.60${}_{-0.02}^{+0.02}$	3488/2.00	⋯	⋯	⋯	⋯	⋯
160216801(1)	5.031 ∼ 14.999	53.76	0	(1.18 ± 0.21) × 10−6	2.35 ± 0.33	−175/−175	(8.11${}_{-0.12}^{+0.12}$) × 101	1.00${}_{-0.00}^{+0.00}$	6954/1.00	⋯	⋯	⋯	⋯	⋯
160216801(2)	6.876 ∼ 14.999	21.46	0	(7.69 ± 0.27) × 10−7	2.65 ± 0.10	−103/−104	(3.40${}_{-0.12}^{+0.12}$) × 101	1.00${}_{-0.00}^{+0.00}$	5288/1.02	⋯	⋯	⋯	⋯	⋯
160530667(1)	6.661 ∼ 20.442	168.89	0	(4.36 ± 0.31) × 10−6	3.70 ± 0.28	−336/−335	(6.19${}_{-0.09}^{+0.09}$) × 101	0.58${}_{-0.00}^{+0.00}$	13223/2.00	(14.30 ± 0.33) × 10−2	6.661	0.77 ± 0.01	130.3 ± 3.1	6883/2.99
160530667(2)	12.961 ∼ 20.442	37.55	0	(3.15 ± 0.24) × 10−7	5.48 ± 0.43	−136/−137	(3.03${}_{-0.21}^{+0.22}$) × 101	0.81${}_{-0.02}^{+0.02}$	5215/2.00	(4.44 ± 0.68) × 10−2	12.961	0.86 ± 0.08	66.6 ± 6.9	5001/1.85
170114917(1)	2.047 ∼ 14.999	57.52	0	(1.01 ± 0.08) × 10−6	1.75 ± 0.09	−217/−217	(1.28${}_{-0.05}^{+0.05}$) × 101	0.52${}_{-0.01}^{+0.01}$	6211/2.02	⋯	⋯	⋯	⋯	⋯
170114917(2)	4.702 ∼ 14.999	30.73	0	(2.94 ± 0.12) × 10−7	2.09 ± 0.09	−107/−109	(1.01${}_{-0.09}^{+0.09}$) × 101	0.63${}_{-0.02}^{+0.02}$	5404/2.01	⋯	⋯	⋯	⋯	⋯
170921168(1)	4.353 ∼ 25.654	92.75	0	(2.03 ± 0.13) × 10−6	0.92 ± 0.15	−150/−150	(2.57${}_{-0.06}^{+0.06}$) × 10−2	0.97${}_{-0.01}^{+0.01}$	8470/1.97	⋯	⋯	⋯	⋯	⋯
170921168(2)	15.707 ∼ 25.654	43.76	0	(1.38 ± 0.03) × 10−6	2.40 ± 0.14	−101/−103	(1.75${}_{-0.02}^{+0.02}$) × 10−2	1.00${}_{-0.00}^{+0.00}$	6527/0.99	(3.81 ± 0.43) × 10−2	15.707	1.73 ± 0.05	105.0 ± 13.1	6147/2.05
171210493(1)	5.237 ∼ 137.109	74.10	0	(2.28 ± 0.02) × 10−6	1.20 ± 0.02	−310/−309	(1.11${}_{-0.03}^{+0.03}$) × 101	0.59${}_{-0.01}^{+0.01}$	10380/2.03	⋯	⋯	⋯	⋯	⋯
171210493(2)	64.334 ∼ 137.109	27.51	0	(1.32 ± 0.04) × 10−7	1.65 ± 0.11	−113/−114	9.59${}_{-0.72}^{+0.73}$	0.78${}_{-0.02}^{+0.02}$	8145/2.02	⋯	⋯	⋯	⋯	⋯
180305393(1)	3.449 ∼ 16.537	101.51	0	(2.72 ± 0.86) × 10−6	1.69 ± 0.40	−181/−181	(1.60${}_{-0.03}^{+0.03}$) × 101	0.36${}_{-0.00}^{+0.00}$	11378/1.98	⋯	⋯	⋯	⋯	⋯
180305393(2)	8.933 ∼ 16.537	35.63	0	(4.78 ± 0.33) × 10−7	3.72 ± 0.24	−101/−103	(1.27${}_{-0.07}^{+0.07}$) × 101	0.53${}_{-0.01}^{+0.01}$	5710/2.00	(6.05 ± 0.73) × 10−2	8.933	0.47 ± 0.07	80.5 ± 7.0	5312/2.31

Note. Column (1) lists the GRB name; Column (2) lists the start and stop times of the decay phases (in units of s); Column (3) lists the statistical significance S; Column (4) lists the model parameter t₀ as described in Equations (4), (7), and (8), which we fixed to zero; Columns (5)–(7) list the best-fit parameters for the power-law model in Equation (4): the normalization F_t,0 (in units of erg cm⁻² s⁻¹), the power-law index $\hat{\alpha }$, and the AIC and BIC statistics; Columns (8)–(10) list the best-fit parameters for the power-law model as shown in Equation (5): normalization F_ν,0 (in units of phs cm⁻² s⁻¹ keV⁻¹), the power-law index $\hat{\beta }$, and the DIC and p_DIC statistics; Columns (11)–(15) list the best-fit parameters for the cutoff power-law model as presented in Equations (6)–(8): normalization N_0,p (in units of phs cm⁻² s⁻¹ keV⁻¹), parameter t_p, which we fixed at the beginning of the decay phases, the cutoff power-law index Γ, the $\hat{\beta }$ index derived from Γ, and the AIC and BIC statistics. Note that (1) marks the entire decay phase of the pulses and (2) marks the late-part decay phase of the pulses. Note that we did not apply the chi-squared test for our sample, because the sample size in our selected bursts is not large enough; the chi-squared test usually requires a relatively large sample size.

Download table as:  ASCIITypeset images: 1 2

The GRB prompt-emission spectra are likely curved. However, since the simplest curvature-effect model (Equation (1)) applies to single power-law spectral models, we first apply a simple power-law fit to the time bins where the curvature effect is tested:

$$\begin{eqnarray}&&{F}_{\nu }={F}_{\nu ,0}\,{\nu }^{-\hat{\beta }},\end{eqnarray} \tag{ 5 }$$

where F_ν,0 is the amplitude and $\hat{\beta }$ is the spectral index. The spectral analysis is performed using a pure Python package called the Multi-Mission Maximum Likelihood Framework (3ML; Vianello et al. 2015). The best model parameters can be evaluated using a given model to fit the data by applying either the MLE technique or the full Bayesian approach. Usually the best-fit results obtained from both methods are the same. ¹¹

We attempt two fits using the simple power-law model. One is to select the entire decay phase as the time interval to perform the spectral fit. The spectral index obtained this way is defined as ${\hat{\beta }}_{\mathrm{PL}}^{{\rm{I}}}$. The other is to select the later part of the decay as the time interval. The spectral index thus obtained is defined as ${\hat{\beta }}_{\mathrm{PL}}^{\mathrm{II}}$.

For each spectral fit, we employ a fully Bayesian approach to explore the best parameter space and to obtain the best-fit parameters. The best-fit parameters, including the normalization (Column 8) and the power-law index (Column 9), as well as the deviance information criterion (DIC; Moreno et al. 2013; Column 10) and p_DIC (Gelman et al. 2014; Column 10), are tabulated in Table 3.

The aforementioned discussion invokes the simplest curvature-effect model, which assumes that the instantaneous spectrum of the prompt-emission tail is a simple power law. In this case, the predicted temporal decay and the spectral indices satisfy the simplest closure relation (Equation (1)). However, the instantaneous spectrum upon the cessation of prompt emission is likely not a simple power law, but it may follow a non-power-law model such as the Band function (e.g., Band et al. 1993). The characteristic frequency ν_c may not be far outside the GBM spectral window. In this case, testing the curvature effect would become more complicated.

We also test the curvature effect using the more complicated model as described in Zhang et al. (2009). We consider that for each time bin the photon flux can be described by a power-law spectrum with an exponential cutoff. This spectrum has one parameter less than the Band function and is found to be adequate to describe the GRB spectra during the decay phase ¹² :

$$\begin{eqnarray}&&N(E,t)={N}_{0}(t){\left(\displaystyle \frac{E}{{E}_{\mathrm{piv}}}\right)}^{-\hat{{\rm{\Gamma }}}}\exp \left\{-\left[\displaystyle \frac{E}{{E}_{c}(t)}\right]\right\},\end{eqnarray} \tag{ 6 }$$

where $\hat{{\rm{\Gamma }}}=\hat{\beta }+1$ is the power-law photon index, E_piv is the pivot energy fixed at 100 keV, and ${N}_{0}(t)\,={N}_{0,{\rm{p}}}{[(t-{t}_{0})/({t}_{{\rm{p}}}-{t}_{0})]}^{-(1+\hat{{\rm{\Gamma }}})}$ is the time-dependent photon flux (in units of photons keV⁻¹ cm⁻² s⁻¹) at 100 keV (see also Equation (7) in Zhang et al. 2009). For such a spectrum, the standard curvature effect predicts

$$\begin{eqnarray}&&{E}_{{\rm{c}}}(t)={E}_{{\rm{c}},{\rm{p}}}{\left(\displaystyle \frac{t-{t}_{0}}{{t}_{{\rm{p}}}-{t}_{0}}\right)}^{-1}\end{eqnarray} \tag{ 7 }$$

where E_c,p = E_c (t_p), t₀ is fixed to zero, and t_p is the beginning of the decay of the pulses; and

$$\begin{eqnarray}&&{F}_{\nu ,c}(t)={F}_{\nu ,c,p}{\left(\displaystyle \frac{t-{t}_{0}}{{t}_{{\rm{p}}}-{t}_{0}}\right)}^{-2}\end{eqnarray} \tag{ 8 }$$

where F_ν,c (t) = E_c(t)N_c(t) and F_ν,c,p = E_c,p N_c,p, where ${N}_{{\rm{c}}}{(t)=N({E}_{{\rm{c}}},t)={N}_{0}(t)({E}_{{\rm{c}}}/{E}_{\mathrm{piv}})}^{-\hat{{\rm{\Gamma }}}}\exp (-1)$, which is calculated using Equation (6) when E is at cutoff energy E_c, and ${N}_{{\rm{c}},{\rm{p}}}=N{({E}_{{\rm{c}}},{t}_{{\rm{p}}})={N}_{0}({t}_{{\rm{p}}})({E}_{{\rm{c}}}/{E}_{\mathrm{piv}})}^{-\hat{{\rm{\Gamma }}}}\exp (-1)$, which is calculated at time t_p and cutoff energy E_c.

With Equations (7) and (8), one can also get a direct relation between F_ν,c (t) and E_c (t):

$$\begin{eqnarray}&&{F}_{\nu ,c}(t)=\displaystyle \frac{{N}_{{\rm{c}},{\rm{p}}}}{{E}_{{\rm{c}},{\rm{p}}}}{E}_{{\rm{c}}}^{2}(t).\end{eqnarray} \tag{ 9 }$$

From the data, the time-dependent parameters E_c (t) and F_ν,c (t) can be directly measured. One can then directly compare the data against the model predictions in Equations (7)–(9).

For the case of power-law spectra, as discussed above, we measure the temporal indices for two phases (Phase I and II) and their corresponding spectral indices (using a time-integrated spectrum throughout the decay phase). The results are as follows:

• 1.  
  Entire decay phase (Phase I): The parameter set (${\hat{\alpha }}_{\mathrm{PL}}^{{\rm{I}}}-{\hat{\beta }}_{\mathrm{PL}}^{{\rm{I}}}$) is presented as orange dots in Figure 2. Eight out of 24 cases satisfy the inequality $\hat{\alpha }\geqslant 2+\hat{\beta }$. These bursts are GRB 090620400, GRB 090719063, GRB 130305486, GRB 131231198, GRB 141028455, GRB 150213001, GRB 150902733, and GRB 160530667. Other bursts are below the line, suggesting that not the entire decay segment can be attributed to the curvature effect for these bursts, which is quite reasonable in view of the modeling presented in Uhm & Zhang (2016b) and Uhm et al. (2018).
• 2.  
  Late-part decay phase (Phase II): The parameter set (${\hat{\alpha }}_{\mathrm{PL}}^{\mathrm{II}}-{\hat{\beta }}_{\mathrm{PL}}^{\mathrm{II}}$) is presented as blue dots in Figure 2. Upward of 11 out of 24 cases now satisfy the inequality $\hat{\alpha }\geqslant 2+\hat{\beta }$. These bursts include GRB 090620400, GRB 090804940, GRB 120426090, GRB 131231198, GRB 141028455, GRB 150314205, GRB 150510139, GRB 150902733, GRB 160530667, GRB 170921168, and GRB 180305393. This suggests that three additional bursts have the curvature effect showing up during the last three data points, while the remaining 13 bursts still do not have the HLE turned on by the end of the observed pulse.

Zoom In Zoom Out Reset image size

One immediate observation is that a good fraction of our sample has entered the $\hat{\alpha }\gt 2+\hat{\beta }$ regime. Since the HLE curvature effect defines the steepest decay index allowed in a GRB pulse, the results strongly suggest that the emission region is undergoing bulk acceleration in the region where prompt emission is released. We calculated the distance of this region from the central engine, R_GRB, using Equation (2) and found that they are typically ∼10¹⁵–10¹⁶ cm for a typical Lorentz factor Γ ∼ 100 (Table 4). In this region, it is impossible to have thermally driven bulk acceleration. The only possibility is that the jet is Poynting-flux dominated in the region, and the GRB emission is powered by the dissipation of a Poynting flux (Zhang & Yan 2011). About one-half of the dissipated energy is released as GRB emission, while the other one-half is used to accelerate the ejecta. This conclusion is consistent with previous results from prompt-emission spectral-lag analysis (Uhm & Zhang 2016b) and the curvature-effect test of X-ray flares (Jia et al. 2016; Uhm & Zhang 2016a).

Table 4. Estimation of GRB Emission Radius Using High-latitude Emission

GRB	Γ2	z	${t}_{\mathrm{HLE}}^{{\rm{I}}}$	${R}_{\mathrm{GRB}}^{{\rm{I}}}$	${t}_{\mathrm{HLE}}^{\mathrm{II}}$	${R}_{\mathrm{GRB}}^{\mathrm{II}}$
	(used)	(used)	(s)	(cm)	(s)	(cm)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
090620400	1.0	1.0	4.11	1.2 × 1015	3.48	1.0 × 1015
090719063	1.0	1.0	5.06	1.5 × 1015	⋯	⋯
090804940	1.0	1.0	⋯	⋯	2.01	0.6 × 1015
120426090	1.0	1.0	⋯	⋯	1.14	0.3 × 1015
130305486	1.0	1.0	13.79	4.1 × 1015	⋯	⋯
131231198	1.0	0.642	22.36	6.7 × 1015	6.79	2.0 × 1015
141028455	1.0	2.33	8.54	2.6 × 1015	5.30	1.6 × 1015
150213001	1.0	1.0	2.22	0.7 × 1015	⋯	⋯
150314205	1.0	1.758	⋯	⋯	2.59	0.8 × 1015
150510139	1.0	1.0	⋯	⋯	10.63	3.2 × 1015
150902733	1.0	1.0	8.03	2.4 × 1015	5.20	1.6 × 1015
160530667	1.0	1.0	6.89	2.1 × 1015	3.74	1.1 × 1015
170921168	1.0	1.0	⋯	⋯	4.97	1.5 × 1015
180305393	1.0	1.0	⋯	⋯	3.80	1.1 × 1015

Note. Column (1) lists the GRB name. Column (2) lists the Γ values used, where we adopted a typical value (Γ₂ = 1) for all cases. Column (3) lists the redshift used; a majority of bursts in our sample have no redshift observations, so we adopt a typical value (z = 1) instead. Column (4) lists the duration of the HLE in the source frame for "Phase I," which is calculated using the observed HLE duration divided by (1+z). Column (5) lists the GRB emission radius R_GRB for Phase I, derived using Equation (2). Again, Column (6) lists the duration of the HLE in the source frame for Phase II, and Column (5) lists the GRB emission radius R_GRB for Phase II.

Download table as:  ASCIITypeset image

A few bursts (GRB 081224887, GRB 090719063, GRB 100707032, GRB 110721200, and GRB 110920546) have been reported in some previous studies (Iyyani et al. 2013, 2015, 2016; Li 2019b) to require an additional thermal component in order to produce acceptable spectral fits. The thermal component is also included in our analysis for these bursts. For a self-consistency test, it is worth noting that these GRBs do not qualify for our Phase II sample and only one burst (GRB 090719063) is included in our Phase I sample. The results imply that the emission in these bursts may be dominated by other mechanisms (e.g., photosphere emission). The existence of a thermal component is consistent with a lower magnetization in the jet (Gao & Zhang 2015).

We notice that six cases (GRB 090804940, GRB 120426090, GRB 150314205, GRB 150510139, GRB 170921168, and GRB 180305393) are not included in the Phase I sample but are included in the Phase II sample, indicating that the curvature effect may only dominate the later part of emission for these bursts. It is also interesting to note that three cases (GRB 090719063, GRB 130305486, and GRB 150213001) are included in the Phase I sample but not in the Phase II sample. These may be spurious cases, which may have contamination from another emission episode. Our analysis below confirms this speculation.

In total, 14 bursts (including eight cases in the Phase I sample and 11 cases in the Phase II sample, noticing that some cases appear in both samples) meet the HLE-dominated criterion based on the power-law spectral analysis. These bursts are our primary interest. Our next step is to study these bursts in detail by investigating their compliance with the curvature-effect predictions in the more complicated cutoff power-law model using a time-dependent analysis.

To test whether the CPL can account for the observed data as well, we adopt the following procedures:

• 1.  
  We first apply the CPL model to fit the spectral data for these cases using the same episodes as the PL model to check whether the CPL model can improve the spectral fit results compared with the PL model. We find that the CPL fits are much better than the PL fits for all these cases by comparing the DIC statistic. We report our results in Table 3. For each individual fit, we fix t₀ to zero and t_p to the starting time of Phase I or Phase II. The best-fit parameters, including t₀ (fixed, Column 4), N_0,p (Column 11), t_p (fixed, Column 12), Γ index (Column 13), and cutoff energy E_c (Column 14), as well as the DIC (Column 15) and p_DIC statistics (Column 15), are listed in Table 3.
• 2.  
  Theoretically, we consider the evolution of E_c and F_ν,c according to Equations (7) and (8) as predicted by the HLE curvature-effect theory (for a constant Γ). The predicted parameter evolution curves for both F_ν,c (t) and E_c(t) are plotted in the left panel of Figure 3 for each case to be directly compared with the data. In the right panel of Figure 3, we plot the theoretically predicted E_c − F_ν,c relation for each case to be directly compared with the observations.
• 3.  
  The observed parameters for each time slice, including N₀(t), Γ, and E_c(t), have been obtained by applying Step (1) in Section 3.2. Since we consider the case at the characteristic energy E_c, one needs to obtain F_ν,c (t) and E_c(t). The characteristic energy E_c is straightforwardly obtained, and F_ν,c (t) is derived using Equation (8). For this step, N_c,p is calculated at peak time t_p with characteristic energy E_c using Equation (6).
• 4.  
  Test the model with observed data. Through Step (3), the observed data points are available in the forms of [F_ν,c (t), t], [E_c(t), t], and [F_ν,c (t), E_c(t)]. The [F_ν,c (t), t], [E_c(t), t] data points are plotted in the left panel of Figure 3, and the [F_ν,c (t), E_c(t)] data points are plotted in the right panel of Figure 3 for each burst. They are directly compared with the model predictions.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

From the left panel in Figure 3, we can see that, except for several apparent cases that violate the predictions (090719063, 090804940, 130305486, 150213001, 150902733, 170921168), all other data points are generally consistent with the model predictions. The data of some bursts (090620400, 120426090, 150510139) match the constant Γ predictions well, suggesting that they are consistent with HLE emission with no significant acceleration. Some other cases (131231198, 141028455, 150314205, 160530667, 180305393) have either E_c(t) or F_ν,c (t) below the model prediction lines, consistent with the bulk acceleration in the emission region. For both cases, the [F_ν,c (t), E_c(t)] test generally satisfies the model prediction (Equation (9)) within error. This is consistent with Z. Uhm & B. Zhang (2018, unpublished) and D. Tak et al. (2020, in preparation), who first performed such a test and showed that Equation (9) is generally valid regardless of bulk Lorentz factor evolution in the emission region.

It is interesting to note that the three cases (GRB 090719063, GRB 130305486, and GRB 150213001) that are in the Phase I sample but not in the Phase II sample indeed do not satisfy the simple model predictions in the [F_ν,c (t), E_c(t)] test, supporting that the cases are spurious.