The Shallow Decay Segment of GRB X-Ray Afterglow Revisited

Litao Zhao; Binbin Zhang; He Gao; Lin Lan; Houjun Lü; Bing Zhang

doi:10.3847/1538-4357/ab38c4

1. Introduction

Gamma-ray bursts (GRBs) are considered to be the most extreme explosive events in the universe, which contains two phenomenological emission phases: the prompt phase (with an initial prompt $\gamma$ -ray emission) and the afterglow phase (with a longer-lived broadband emission) (Zhang 2019). Although there are many uncertainties in the detailed physics of the prompt emission, mainly due to our poor understanding of the degree of magnetization of the GRB jet (Zhang 2014; Kumar & Zhang 2015), a generic synchrotron external-shock model has been constructed for interpreting the broadband afterglow data (Rees & Meszaros 1992, 1994; Meszaros & Rees 1993; Mészáros & Rees 1997; Gao et al. 2013; Wang et al. 2015).

In the pre-Swift era, the simple external-shock signal was found to successfully explain a bunch of late-time afterglow data (Waxman 1997; Wijers et al. 1997; Huang et al. 1999, 2000; Wijers & Galama 1999; Panaitescu & Kumar 2001, 2002; Yost et al. 2003). With the successful launch of the Neil Gehrels Swift Observatory, unprecedented new information about early-time afterglows was revealed (Burrows et al. 2005b; Tagliaferri et al. 2005; Nousek et al. 2006; O'Brien et al. 2006; Zhang et al. 2006; Evans et al. 2009), especially in the X-ray band, thanks mainly to the rapid slewing and precise localization capability of its onboard X-ray Telescope (XRT; Burrows et al. 2005a). After 6 months of data accumulating, a five-segment canonical X-ray afterglow light curve was proposed (Zhang et al. 2006), including a distinct rapidly decaying component, a shallow decay component, a normal decay component, a post jet break component and X-ray flares. With two years of data (with ∼100 GRBs), a series of systematic analysis of the Swift XRT data was performed, in order to explore the physical origin for each segment of the canonical light curve (Liang et al. 2007, 2008; Willingale et al. 2007; Zhang et al. 2007b; Evans et al. 2009).

After 14 years of operation, thousands of Swift GRBs were detected by XRT, it is of great interest to revisit the early analysis and see whether the previous results are still consistent with the larger sample. In this work, we focus on the analysis of the shallow decay component from Liang et al. (2007; hereafter L07). L07 made a comprehensive analysis of the properties of the shallow decay segment with a sample of 53 long Swift GRBs detected before 2007 February. Their statistic results are summarized as follows: (1) the distributions of shallow decay segment parameters are normal and lognormal, with $1{\mathrm{og}}_{10}{t}_{b}({\rm{s}})=4.09\pm 0.61$ , $1{\mathrm{og}}_{10}{S}_{{\rm{X}}}(\mathrm{erg}\ {\mathrm{cm}}^{-2})=-6.52\,\pm 0.69$ , ${{\rm{\Gamma }}}_{{\rm{X}},1}=2.09\pm 0.21$ , and ${\alpha }_{1}=0.35\pm 0.35;$ (2) The spectrum of the shallow decay phase is softer than that of the prompt gamma-ray phase, while the X-ray fluence and isotropic energy are almost linearly correlated with gamma-ray fluence and gamma-ray energy, respectively; (3) there is no significant spectral evolution between the shallow decay segment and its follow-up segment, and the follow-up segment in most bursts is consistent with the closure relations of external-shock models; (4) within the scenario of refreshed external shocks, the average energy injection index q ∼ 0, suggesting a roughly constant injection luminosity from the central engine; (5) there is an empirical multivariate relation between parameters ${E}_{\mathrm{iso},{\rm{X}}}-{E}_{p}^{{\prime} }-{t}_{b}^{{\prime} }$ (henceforth the prime marks properties in the burst rest frame). Based on these results, L07 proposed that the shallow decay segment in most bursts is consistent with an external forward shock origin with continuous energy injection from a long-lived central engine. For a small fraction of bursts, whose post-break phases significantly deviate from the external-shock models, the shallow decay phase might be of internal origin and demand a long-lived emission component directly from the central engine.

In this work, we revisit the results of L07 with Swift-observed GRBs between 2004 February and 2017 July. In addition, it has been discovered that there exists a rough anticorrelation between the rest frame X-ray plateau end time ( ${t}_{b}^{{\prime} }$ ) and X-ray luminosity L_X (Dainotti et al. 2008, 2010, 2011a). The slope is roughly −1 (Dainotti et al. 2013b). This suggests that the total plateau energy has relatively small scatter: a longer plateau tends to have a lower luminosity and vice versa. Dainotti et al. (2011b) analyzed and suggested correlations between L_X with several prompt emission parameters, including the isotropic energy ${E}_{\mathrm{iso},\gamma }$ . Xu & Huang (2012) claimed that a three-parameters collection, expressed as ${L}_{{\rm{X}}}\propto {t}_{b}^{0.87}{E}_{\mathrm{iso},\gamma }^{0.88}$ , becomes tighter. On the other hand, Dainotti et al. (2016, 2017) extend the ${L}_{{\rm{X}}}-{t}_{b}^{{\prime} }$ correlation by adding a third parameter (the peak luminosity in the prompt emission ${L}_{\mathrm{peak}}$ ), and find that the new ${L}_{{\rm{X}}}-{t}_{b}^{{\prime} }-{L}_{\mathrm{peak}}$ correlation becomes much tighter, which is 37% tighter than the Xu & Huang (2012) correlation. In this work, we also test these relations based on our new sample.

2. Data Reduction and Sample Selection

The XRT light curve data were downloaded from the Swift/XRT team website⁵ (Belokurov et al. 2009), and processed with HEASOFT v6.12. 1291 Swift GRBs were detected by Swift/XRT between 2004 February and 2017 July, with 625 GRBs having well-sampled XRT light curves, which contain at least six data points, excluding upper limits. We then fit the XRT light curves (in logarithmic scale) with the multisegment broken power-law (BPL) function. Here the multivariate adaptive regression spline (MARS) technique (e.g., Friedman 1991) is adopted, which can automatically determine both variable selection and functional form, resulting in an explanatory predictive model. It has been proven that MARS can automatically fit the XRT light curve with the multisegment BPL function (results are consistent with fitting results provided by the XRT GRB online catalog), detect and optimize all breaks, and record all break times and power-law indices for each segment (see Zhang et al. 2014 and Gao et al. 2017 for details). We find that the distribution of the index difference between adjacent segments for all GRBs could be fitted with a Gaussian function (with mean value 0.6 and standard deviation 0.3). To avoid the potential overfitting problem from the MARS technique, we treat the adjacent segments with index difference smaller than 0.3 as one component when calculating the segment time span.

We define segments with decay slopes shallower than 0.8 and time spans in log scale larger than 0.4 dex⁶ as the shallow decay component candidates. 284 GRBs are excluded due to the lack of any shallow decay candidates. Among the rest of the 341 GRBs, 67 bursts are further excluded because there are too few data points or there seem to be weak flare signatures in the shallow decay segments. Moreover, we excluded 54 GRBs whose shallow decay candidate segments lack a follow-up segment; namely, the shallow decay segment is the only segment, or the last segment, in the light curve. 18 GRBs are excluded because they consist of more than one nonadjacent shallow decay candidate, which will be discussed in a separate work. GRB 060218 is excluded because its early shallow decay behavior in the X-ray light curve should be mainly determined by the shock break-out emission instead of external-shock emission (Campana et al. 2006).

For the remaining 201 GRBs, we record the beginning time (t₁) of the shallow decay segment and the end time (t₂) of its follow-up segment. We then fit the light curve in the time interval [t₁, t₂] with a smoothed BPL function

$\begin{eqnarray}&&f(t)={f}_{0}{\left[{\left(\displaystyle \frac{t}{{t}_{b}}\right)}^{w{\alpha }_{1}}+{\left(\displaystyle \frac{t}{{t}_{b}}\right)}^{w{\alpha }_{2}}\right]}^{-1/w},\end{eqnarray} \tag{ 1 }$

with ${\alpha }_{1}$ and ${\alpha }_{2}$ representing the decay scopes of the pre-break segment and the post-break segment, and w determining the sharpness of the break. Here we adopt w = 3 as suggested in L07, in order to make a better comparison with their results. All fitting light curves are shown in Figure 1, and the best fitting parameters are collected in Table 1. The X-ray fluence (S_X) of the shallow decay segment is obtained by integrating the fitted light curve from t₁ to t_b. Note that here we further exclude three bursts in our sample because the difference between their ${\alpha }_{1}$ and ${\alpha }_{2}$ are smaller than 0.3.

**Figure 1.**
XRT light curves for the bursts in our sample. The solid red lines are the best fits with a smooth broken power law for the shallow decay phase and its follow-up decay phase. All the component figures are available in the figure set. (The complete figure set (198 images) is available.)
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Table 1. Part I: XRT Observations and Fitting Results for Our Sample

GRB Name	${\mathrm{log}}_{10}({t}_{1})$ ^a	${\mathrm{log}}_{10}({t}_{2})$ ^a	${\mathrm{log}}_{10}({t}_{{\rm{b}}})$ ^b	α_X,1^b	α_X,2^b	${\chi }^{2}/\mathrm{dof}$ ^b	S_X^c	Γ_X,1^d	Γ_X,2^d
	(s)	(s)	(s)				( ${10}^{-7}\,\mathrm{erg}\ {\mathrm{cm}}^{-2}$ )
050319	2.61	6.32	4.52	0.5 ± 0.03	1.54 ± 0.11	96.98/91	3.1 ± 0.46	${1.89}_{-0.07}^{+0.08}$	${2.12}_{-0.17}^{+0.18}$
050401	2.1	5.91	3.67	0.55 ± 0.02	1.43 ± 0.04	429.9/317	6.51 ± 0.8	1.91 ± 0.04	${1.82}_{-0.16}^{+0.17}$
050416A	2.26	6.77	3.22	0.39 ± 0.1	0.9 ± 0.02	111.4/91	0.31 ± 0.15	${1.98}_{-0.2}^{+0.21}$	${2.08}_{-0.14}^{+0.15}$
050505	3.5	6.2	4.31	0.68 ± 0.07	1.65 ± 0.06	233.1/188	2.32 ± 0.57	${1.97}_{-0.08}^{+0.09}$	${2.01}_{-0.08}^{+0.09}$
050607	2.84	6.21	3.99	0.57 ± 0.1	1.17 ± 0.08	19.9/20	0.15 ± 0.11	${1.71}_{-0.16}^{+0.21}$	${2.17}_{-0.43}^{+0.52}$
050712	3.68	6.19	4.82	0.65 ± 0.07	1.3 ± 0.1	75.22/41	0.74 ± 0.47	${2.04}_{-0.15}^{+0.19}$	${1.85}_{-0.18}^{+0.35}$
050713A	2.41	6.23	3.99	0.58 ± 0.03	1.29 ± 0.03	97.41/79	2.43 ± 0.64	${1.9}_{-0.18}^{+0.19}$	${2.18}_{-0.17}^{+0.18}$
050713B	3.06	6.14	4.04	−0.2 ± 0.1	1.02 ± 0.03	151/107	1.72 ± 0.22	${2.03}_{-0.14}^{+0.15}$	1.83 ± 0.13
050714B	3.75	5.89	4.56	0.22 ± 0.29	0.85	17.6/11	0.1 ± 0.05	${2.25}_{-0.53}^{+0.6}$	${2.7}_{-0.43}^{+0.49}$
050802	2.89	6.08	3.72	0.48 ± 0.07	1.56 ± 0.02	208.4/143	2.18 ± 0.31	${1.65}_{-0.09}^{+0.1}$	1.88 ± 0.08
050803	3.5	6.2	4.04	−0.46 ± 0.13	1.57 ± 0.02	307.2/104	1.51 ± 0.14	1.8 ± 0.14	2.07 ± 0.11
050814	3.82	5.96	4.95	0.67 ± 0.05	2.08 ± 0.16	44.62/40	0.76 ± 0.15	1.98 ± 0.16	${1.62}_{-0.14}^{+0.24}$
050822	3.02	6.66	4.27	0.29 ± 0.07	1.08 ± 0.03	131.8/91	$1\pm 0.23$	${1.9}_{-0.15}^{+0.16}$	${2.06}_{-0.12}^{+0.13}$
050824	3.75	6.41	4.82	0.24 ± 0.1	0.93 ± 0.08	67.98/37	0.67 ± 0.34	${2.12}_{-0.31}^{+0.34}$	${2.04}_{-0.17}^{+0.18}$
050826	4.03	5.1	4.61	0.27 ± 0.33	1.96 ± 0.71	1.045/5	0.12 ± 0.06	${1.73}_{-0.4}^{+0.49}$	${2.28}_{-0.66}^{+0.81}$
050915B	2.86	5.97	5.03	0.46 ± 0.04	1.6 ± 0.28	10.68/14	0.53 ± 0.19	${2.16}_{-0.32}^{+0.34}$	${1.99}_{-0.26}^{+0.6}$
050922B	3.22	6.37	5.31	0.35 ± 0.05	1.53 ± 0.12	36.68/16	1.65 ± 10.43	${2.34}_{-0.31}^{+0.33}$	${2.15}_{-0.31}^{+0.35}$
051109B	2.51	5.2	3.5	0.24 ± 0.13	1.28 ± 0.07	21.88/19	0.13 ± 0.04	${2.2}_{-0.32}^{+0.34}$	${1.81}_{-0.16}^{+0.35}$
051221A	2.86	6.03	4.63	0.56 ± 0.04	1.43 ± 0.07	92.32/55	0.66 ± 0.15	1.91 ± 0.15	${1.89}_{-0.22}^{+0.24}$
060108	2.66	5.57	4.11	0.26 ± 0.06	1.25 ± 0.08	46.49/25	0.36 ± 0.1	${1.99}_{-0.31}^{+0.34}$	${2.12}_{-0.27}^{+0.3}$
060109	2.8	5.57	3.79	−0.08 ± 0.07	1.49 ± 0.05	66.51/48	0.44 ± 0.05	${1.84}_{-0.24}^{+0.25}$	${2.23}_{-0.20}^{+0.21}$
060115	2.9	5.62	4.5	0.62 ± 0.05	1.2 ± 0.12	39.9/23	0.48 ± 0.28	${2.17}_{-0.13}^{+0.15}$	${2.16}_{-0.20}^{+0.31}$
060203	3.5	5. 59	4.46	0.77 ± 0.15	1.49 ± 0.22	55.94/34	0.08 ± 0.7	${2.06}_{-0.19}^{+0.2}$	${2.09}_{-0.28}^{+0.41}$
060204B	2.63	5. 8	3.9	0.63 ± 0.06	1.52 ± 0.05	79.3/59	1.23 ± 0.27	2.03 ± 0.11	${2.25}_{-0.35}^{+0.4}$
060219	2.35	5.75	4.35	0.51 ± 0.05	1.35 ± 0.11	13.94/14	0.19 ± 0.08	${2.84}_{-0.38}^{+0.41}$	${3.11}_{-0.72}^{+0.96}$
060306	2.57	5.58	3.51	0.31 ± 0.14	1.06 ± 0.03	94.32/76	0.58 ± 0.33	${1.76}_{-0.25}^{+0.27}$	2.1 ± 0.13
060312	2.39	5.46	4.37	0.72 ± 0.07	1.56 ± 0.46	54.06/21	0.5 ± 0.37	${1.95}_{-0.12}^{+0.15}$	${1.68}_{-0.21}^{+0.28}$
060413	3.07	5.3	4.41	0.17 ± 0.05	3.03 ± 0.1	393.1/145	7.16 ± 0.55	${1.39}_{-0.13}^{+0.14}$	${1.72}_{-0.36}^{+0.39}$
060428A	2.15	6.52	4.98	0.53 ± 0.02	1.37 ± 0.05	170/141	6.48 ± 1.15	${1.95}_{-0.13}^{+0.14}$	${1.92}_{-0.19}^{+0.2}$
060502A	2.35	6.2	4.57	0.59 ± 0.03	1.18 ± 0.05	64.41/62	2.22 ± 0.67	1.89 ± 0.13	${1.83}_{-0.16}^{+0.17}$
060510A	2.14	5.76	3.77	0.09 ± 0.05	1.5 ± 0.02	196.2/158	$11.4\pm 1$	${1.74}_{-0.08}^{+0.14}$	${1.84}_{-0.06}^{+0.07}$
060604	3.56	6.2	4.32	0.39 ± 0.09	1.29 ± 0.06	98.72/66	0.51 ± 0.14	${2.07}_{-0.16}^{+0.17}$	${2.08}_{-0.15}^{+0.16}$
060605	2.24	5.25	3.92	0.42 ± 0.03	2.06 ± 0.08	85.71/74	1.22 ± 0.1	${1.95}_{-0.11}^{+0.12}$	${2.10}_{-0.14}^{+0.15}$
060607A	2.68	5	4.11	0.44 ± 0.02	3.58 ± 0.08	253.9/173	12.01 ± 0.71	${1.52}_{-0.07}^{+0.08}$	1.56 ± 0.08
060614	3.66	6.4	4.69	0.09 ± 0.04	1.94 ± 0.04	181/150	$3\pm 0.2$	${1.74}_{-0.06}^{+0.1}$	${1.84}_{-0.12}^{+0.13}$
060707	2.58	6.53	5.09	0.62 ± 0.03	1.32 ± 0.12	59.43/33	1.91 ± 0.88	${2.12}_{-0.14}^{+0.15}$	${1.73}_{-0.32}^{+0.38}$
060712	2.55	6	3.97	0.34 ± 0.08	1.13 ± 0.05	23.93/22	0.17 ± 0.06	${2.57}_{-0.44}^{+0.51}$	${2.46}_{-0.29}^{+0.32}$
060714	2.47	6.11	3.67	0.38 ± 0.09	1.27 ± 0.04	81.86/47	0.88 ± 0.27	${1.88}_{-0.14}^{+0.15}$	${2.01}_{-0.18}^{+0.19}$
060719	2.71	5.88	3.87	0.48 ± 0.07	1.25 ± 0.04	60.03/50	0.55 ± 0.17	${2.24}_{-0.24}^{+0.26}$	${2.38}_{-0.2}^{+0.22}$
060729	2.83	7.1	4.8	0.12 ± 0.02	1.37 ± 0.01	1011/706	10.69 ± 0.38	1;96 ± 0.05	1.97 ± 0.05
060804	2.16	5.41	2.9	−0.31 ± 0.16	1.17 ± 0.06	38.76/28	0.5 ± 0.12	${1.82}_{-0.12}^{+0.29}$	${1.92}_{-0.14}^{+0.27}$
060805A	2.16	5.46	3.56	0.19 ± 0.17	1.5 ± 0.1	15.69/8	0.09 ± 0.03	${1.9}_{-0.32}^{+0.53}$	$2.11+{0.46}_{-0.38}$
060807	2.43	5.7	3.88	0.001 ± 0.04	1.64 ± 0.04	169.6/92	1.13 ± 0.08	1.9 ± 0.12	${2.36}_{-0.18}^{+0.19}$
060813	2	5.5	3.24	0.6 ± 0.03	1.3 ± 0.02	333/234	4.08 ± 0.75	1.83 ± 0.08	1.78 ± 0.09
060814	3.15	6.12	4.1	0.44 ± 0.08	1.36 ± 0.03	272.1/161	1.96 ± 0.34	1.97 ± 0.12	2.08 ± 0.09
060906	2.8	5.41	4.05	0.24 ± 0.06	1.74 ± 0.09	66.64/32	0.39 ± 0.05	${2.1}_{-0.22}^{+0.23}$	${1.76}_{-0.16}^{+0.22}$
060908	1.8	5.94	2.84	0.56 ± 0.07	1.44 ± 0.04	62.81/33	0.86 ± 0.23	${2.02}_{-0.37}^{+0.42}$	${2.09}_{-0.2}^{+0.21}$
060923C	2.88	6.08	5.32	0.54 ± 0.06	1.85 ± 0.29	14.88/16	1.11 ± 0.38	${2.74}_{-0.36}^{+0.4}$	${1.98}_{-0.36}^{+0.45}$

Part II: XRT Observations and Fitting Results for Our Sample

060927	1.82	5	3.79	0.76 ± 0.05	2.49 ± 0.81	5.479/13	0.45 ± 0.16	${1.83}_{-0.11}^{+0.18}$	${2.53}_{-0.59}^{+1.7}$
061004	2.72	5.28	4.19	0.68 ± 0.07	1.64 ± 0.29	24.67/17	0.4 ± 0.24	${1.95}_{-0.19}^{+0.23}$	${2.32}_{-0.32}^{+0.68}$
061019	3.4	5.86	5.07	0.81 ± 0.05	1.96 ± 0.39	80.89/42	1.46 ± 0.73	${2.05}_{-0.23}^{+0.24}$	${2.03}_{-0.69}^{+0.87}$
061121	2.37	6.32	3.71	0.36 ± 0.03	1.4 ± 0.02	369.5/269	7.04 ± 0.84	${1.94}_{-0.12}^{+0.13}$	1.79 ± 0.06
061201	1.9	5.08	3.44	0.54 ± 0.07	2.05 ± 0.11	33.23/23	1.39 ± 0.27	${1.3}_{-0.17}^{+0.18}$	${2.02}_{-0.36}^{+0.4}$
061202	2.94	5.78	4.22	−0.1 ± 0.05	1.73 ± 0.03	202.2/116	2.55 ± 0.15	${1.97}_{-0.13}^{+0.14}$	${2.2}_{-0.14}^{+0.15}$
061222A	2.33	6.19	4.68	0.73 ± 0.01	1.72 ± 0.03	516.3/327	12.37 ± 0.89	1.84 ± 0.06	2.05 ± 0.1
070103	2.2	5.13	3.16	−0.23 ± 0.14	1.42 ± 0.06	31.95/24	0.21 ± 0.04	${1.91}_{-0.28}^{+0.29}$	${2.15}_{-0.28}^{+0.30}$
070110	3.5	4.5	4.31	0.066 ± 0.05	9.05 ± 0.52	111.1/107	2.45 ± 0.13	2.04 ± 0.08	${2.07}_{-0.17}^{+0.19}$
070129	3.15	6.21	4.25	0.15 ± 0.08	1.14 ± 0.03	102.4/80	0.62 ± 0.12	2.21 ± 0.16	${2.04}_{-0.14}^{+0.15}$
070306	2.73	6.05	4.48	0.11 ± 0.03	1.81 ± 0.04	193.4/140	5.24 ± 0.34	1.8 ± 0.09	${2.02}_{-0.16}^{+0.17}$
070328	2.18	5.92	2.84	0.25 ± 0.03	1.46 ± 0.01	683.4/531	5.57 ± 0.35	2.09 ± 0.04	1.9 ± 0.07
070420	2.51	5.72	3.51	0.23 ± 0.07	1.37 ± 0.02	230/139	2.53 ± 0.32	${2.03}_{-0.14}^{+0.15}$	${1.78}_{-0.22}^{+0.24}$
070429A	3.33	6.05	5.69	0.36 ± 0.08	3.03 ± 0.5	34.6/12	0.92 ± 0.14	${2.17}_{-0.27}^{+0.29}$	${2.19}_{-0.75}^{+1.11}$
070508	1.83	5.9	2.46	0.18 ± 0.32	0.95 ± 0.18	611/484	0.15 ± 0.55	1.8 ± 0.06	1.76 ± 0.04
070529	2.31	5.78	3.02	0.48 ± 0.16	1.25 ± 0.04	38.09/30	0.42 ± 0.17	${1.57}_{-0.18}^{+0.27}$	${2.11}_{-0.22}^{+0.24}$
070621	2.60	5.90	3.47	0.68 ± 0.19	1.14 ± 0.06	31.29/25	0.18 ± 0.83	${2.6}_{-0.23}^{+0.24}$	${2.25}_{-0.34}^{+0.38}$
070628	2	4.91	3.93	0.44 ± 0.02	1.29 ± 0.03	301.4/192	4.12 ± 0.26	1.9 ± 0.13	2.01 ± 0.11
070809	2.74	4.82	3.78	−0.06 ± 0.19	1.30 ± 0.15	16.71/11	0.16 ± 0.07	${1.4}_{-0.2}^{+0.39}$	${1.25}_{-0.17}^{+0.2}$
070810A	2.1	4.54	3.14	0.36 ± 0.25	1.32 ± 0.06	46.44/31	0.37 ± 0.24	${2.09}_{-0.24}^{+0.26}$	${2.06}_{-0.19}^{+0.2}$
071118	2.99	5.06	4.11	0.59 ± 0.06	2.18 ± 0.13	83.57/60	2.52 ± 0.4	1.36 ± 0.11	${1.53}_{-0.37}^{+0.4}$
080320	3	6.32	4.85	0.60 ± 0.04	1.25 ± 0.06	131/96	2.03 ± 0.6	1.97 ± 0.11	${2.18}_{-0.23}^{+0.25}$
080328	2.4	4.7	3.08	0.43 ± 0.09	1.19 ± 0.03	235.6/155	2.33 ± 0.64	1.72 ± 0.16	${1.94}_{-0.36}^{+0.39}$
080409	2.16	5.03	3.27	−0.01 ± 0.14	1.3 ± 0.14	15.24/8	0.14 ± 0.04	${2.01}_{-0.48}^{+0.54}$	${2}_{-0.51}^{+0.58}$
080413B	2.1	5.24	2.6	0.38 ± 0.12	1 ± 0.01	328.2/225	0.73 ± 0.27	${2.04}_{-0.12}^{+0.13}$	1.84 ± 0.07
080430	2.5	6.48	4.53	0.43 ± 0.02	1.16 ± 0.03	128.2/141	1.78 ± 0.28	1.97 ± 0.1	1.95 ± 0.13
080516	2.1	4.67	3.67	0.33 ± 0.09	1.07 ± 0.11	37.66/27	0.3 ± 0.1	${2.69}_{-0.47}^{+0.52}$	${2}_{-0.39}^{+0.43}$
080703	2	5.21	4.40	0.6 ± 0.03	2.36 ± 0.17	108.2/68	1.73 ± 0.32	${1.4}_{-0.06}^{+0.11}$	${1.29}_{-0.18}^{+0.21}$
080707	2.4	5.45	4.03	0.28 ± 0.07	1.17 ± 0.1	42.8/22	0.33 ± 0.14	${2.19}_{-0.26}^{+0.27}$	${1.7}_{-0.24}^{+0.26}$
080723A	2.53	5.67	3.4	−0.04 ± 0.13	1.06 ± 0.03	62.74/35	0.44 ± 0.1	${2.03}_{-0.23}^{+0.25}$	${1.71}_{-0.49}^{+0.55}$
080903	1.8	4.28	2.26	0.07 ± 0.26	1.57 ± 0.12	382.1/90	1.4 ± 0.53	${0.68}_{-0.06}^{+0.08}$	${1.21}_{-0.12}^{+0.21}$
080905B	2.27	4.75	4.16	0.47 ± 0.02	2.18 ± 0.25	103.5/69	6.19 ± 1.05	${1.68}_{-0.11}^{+0.12}$	${1.91}_{-0.18}^{+0.19}$
081007	2.43	6.24	4.62	0.67 ± 0.03	1.27 ± 0.06	81.85/58	1.83 ± 0.61	1.92 ± 0.13	${1.97}_{-0.24}^{+0.26}$
081029	3.4	5.84	4.21	0.4 ± 0.05	2.46 ± 0.09	125.3/80	1.03 ± 0.01	${1.82}_{-0.09}^{+0.1}$	${2.03}_{-0.2}^{+0.22}$
081210	3.71	5.83	5.3	0.64 ± 0.04	2.67 ± 0.57	16.29/21	1.4 ± 0.3	${1.86}_{-0.14}^{+0.19}$	${2.5}_{-0.54}^{+0.65}$
081230	2.53	5.94	4.19	0.48 ± 0.07	1.25 ± 0.1	57.91/39	0.48 ± 0.2	${2.07}_{-0.25}^{0.27}$	${1.95}_{-0.19}^{+0.21}$
090102	2.5	5.84	3.16	0.89 ± 0.05	1.45 ± 0.01	166.6/138	4.96 ± 0.67	${1.62}_{-0.23}^{+0.24}$	1.72 ± 0.08
090113	2	5.52	2.75	0.13 ± 0.16	1.29 ± 0.04	37.88/22	0.45 ± 0.11	${2.13}_{-0.35}^{+0.38}$	${2.2}_{-0.28}^{+0.31}$
090205	2.63	5.53	4.18	0.54 ± 0.06	1.98 ± 0.23	61.18/30	0.42 ± 0.12	${2.07}_{-0.15}^{+0.19}$	${2.02}_{-0.31}^{+0.34}$
090313	4.45	5.86	4.89	0.6 ± 0.18	2.3 ± 0.12	28.47/34	1.27 ± 0.33	${2.02}_{-0.19}^{+0.2}$	${2.27}_{-0.29}^{+0.32}$
090404	2.49	6.16	4.23	0.20 ± 0.07	1.25 ± 0.06	200.1/109	1.24 ± 0.33	${2.21}_{-0.14}^{+0.15}$	${2.65}_{-0.15}^{+0.16}$
090407	3.07	6.05	4.86	0.38 ± 0.03	1.67 ± 0.09	177.4/107	1.65 ± 0.22	${2.22}_{-0.12}^{+0.13}$	${2.11}_{-0.27}^{+0.29}$
090418A	2.09	5.64	3.49	0.47 ± 0.05	1.58 ± 0.04	109.7/101	3.3 ± 0.6	${2}_{-0.21}^{+0.22}$	1.98 ± 0.1
090424	2.43	6.70	2.97	0.57 ± 0.17	1.17 ± 0.02	517.5 /510	7.94 ± 0.45	1.9 ± 0.05	1.99 ± 0.08
090429B	2.1	5.64	2.9	−0.7 ± 0.23	1.31 ± 0.08	22.16/16	0.1 ± 0.02	${1.79}_{-0.26}^{+0.29}$	${2.02}_{-0.35}^{+0.4}$
090510	2	4.81	3.19	0.65 ± 0.04	2.18 ± 0.08	103.2/69	1.42 ± 0.22	${1.6}_{-0.1}^{+0.12}$	${2.05}_{-0.3}^{+0.33}$
090518	2.29	5.06	3.72	0.51 ± 0.09	1.14 ± 0.12	48.83/34	0.33 ± 0.22	${2.5}_{-0.36}^{+0.39}$	${2.38}_{-0.27}^{+0.29}$
090529	3.41	5.89	5.36	0.51 ± 0.1	1.69 ± 0.43	3.054/5	0.43 ± 0.24	${1.71}_{-0.23}^{+0.56}$	${1.78}_{-0.26}^{+0.32}$

Part III: XRT Observations and Fitting Results for Our Sample

090530	2.2	5.90	4.77	0.61 ± 0.02	1.35 ± 0.1	84.28/49	1.27 ± 0.35	${1.97}_{-0.13}^{+0.14}$	${1.87}_{-0.28}^{+0.3}$
090727	3.65	6.33	5.57	0.5 ± 0.05	1.56 ± 0.25	26.34/17	1.22 ± 0.45	${1.9}_{-0.12}^{+0.27}$	${2.1}_{-0.34}^{+0.63}$
090728	2.56	4.90	3.25	−0.03 ± 0.22	1.8 ± 0.09	22.97/22	0.37 ± 0.08	${1.74}_{-0.12}^{+0.15}$	${1.83}_{-0.25}^{+0.27}$
090813	2.03	5.82	2.73	0.23 ± 0.06	1.25 ± 0.01	403.4/286	1.78 ± 0.21	2.01 ± 0.07	1.79 ± 0.09
090904A	2.91	5.49	4.1	0.13 ± 0.12	1.31 ± 0.09	13.19/15	0.29 ± 0.07	${2.38}_{-0.4}^{+0.44}$	${2.04}_{-0.24}^{+0.35}$
091018	1.7	5.9	2.78	0.43 ± 0.05	1.2 ± 0.02	232/137	1.53 ± 0.31	${1.87}_{-0.17}^{+0.18}$	1.88 ± 0.09
091029	2.86	6.28	4.07	0.19 ± 0.04	1.15 ± 0.02	183.4/125	0.91 ± 0.13	2.1 ± 0.12	2.01 ± 0.11
091130B	3.58	6.13	4.70	0.29 ± 0.06	1.17 ± 0.05	147.7/96	0.78 ± 0.17	${2.3}_{-0.16}^{+0.17}$	${2.27}_{-0.18}^{+0.19}$
100111A	1.75	5.38	3.26	0.5 ± 0.09	1.05 ± 0.06	38.72/31	0.3 ± 0.17	${1.83}_{-0.21}^{+0.23}$	${1.84}_{-0.23}^{+0.24}$
100219A	2.95	5.07	4.53	0.56 ± 0.03	3.75 ± 0.3	28.34/22	1.82 ± 0.31	${1.44}_{-0.14}^{+0.16}$	${1.47}_{-0.37}^{+0.45}$
100302A	3.10	6.04	4.35	0.35 ± 0.12	0.95 ± 0.08	18.49/20	0.2 ± 0.1	${1.8}_{-0.25}^{+0.43}$	${1.87}_{-0.19}^{+0.21}$
100418A	3	6.47	4.91	−0.16 ± 0.06	1.44 ± 0.08	31.65/22	0.63 ± 0.1	${1.89}_{-0.34}^{+0.37}$	${1.93}_{-0.26}^{+0.29}$
100425A	2.53	5.7	4.51	0.54 ± 0.05	1.2 ± 0.13	20.87/19	0.35 ± 0.2	${2.24}_{-0.24}^{+0.27}$	${1.92}_{-0.29}^{+0.33}$
100508A	2.82	5.3	4.37	0.36 ± 0.05	2.83 ± 0.2	71.43/48	2.1 ± 0.36	1.3 ± 0.13	${1.62}_{-0.28}^{+0.3}$
100522A	2.28	5.59	4.31	0.5 ± 0.03	1.28 ± 0.06	128.3/98	1.8 ± 0.37	2.11 ± 0.15	${2.21}_{-0.18}^{+0.19}$
100614A	3.76	5.75	5.22	0.41 ± 0.05	2.28 ± 0.27	28.87/26	1.21 ± 0.18	${2.15}_{-0.22}^{+0.23}$	${2.14}_{-0.38}^{+0.51}$
100704A	2.67	6.29	4.53	0.67 ± 0.03	1.32 ± 0.05	202/155	3.52 ± 0.85	2.09 ± 0.12	${1.95}_{-0.12}^{+0.13}$
100725B	2.81	5.76	4.19	0.39 ± 0.06	1.4 ± 0.07	53.09/50	1.23 ± 0.24	${2.49}_{-0.2}^{+0.21}$	${2.59}_{-0.32}^{+0.34}$
100901A	3.59	5.98	4.58	−0.02 ± 0.03	1.5 ± 0.03	543.9/264	3.56 ± 0.16	2.02 ± 0.07	${2.15}_{-0.08}^{+0.09}$
100902A	3.18	6.17	5.93	0.71 ± 0.03	4.58 ± 0.96	84.64/53	3.67 ± 0.62	${2.31}_{-0.16}^{+0.17}$	${3.2}_{-1.8}^{+2.8}$
100905A	2.86	4.87	3.28	0.05 ± 0.4	1.12 ± 0.11	21/8	0.05 ± 0.03	${1.47}_{-0.3}^{+0.38}$	${1.85}_{-0.4}^{+0.49}$
100906A	2.57	5.62	4.1	0.7 ± 0.03	2.01 ± 0.05	344.3/127	4 ± 0.4	${1.9}_{-0.09}^{+0.1}$	${1.95}_{-0.14}^{+0.15}$
101117B	1.75	4.98	3.22	0.77 ± 0.06	1.29 ± 0.06	53.09/36	0.64 ± 0.39	${2.09}_{-0.22}^{+0.23}$	${1.99}_{-0.23}^{+0.24}$
110102A	2.72	6.4	4.09	0.49 ± 0.03	1.41 ± 0.02	397.7/294	5.12 ± 0.51	$2.06\pm 0,1$	1.98 ± 0.07
110106B	2.5	5.73	4.12	0.58 ± 0.05	1.41 ± 0.07	76.87/50	1.48 ± 0.41	$1.98\pm 0,16$	${1.66}_{-0.25}^{+0.26}$
110312A	2.75	5.96	4.91	0.53 ± 0.04	1.38 ± 0.2	31.74/25	2 ± 1.06	${2.31}_{-0.21}^{+0.22}$	${2.7}_{-0.59}^{+0.7}$
110319A	2.16	5.48	3.89	0.58 ± 0.04	1.34 ± 0.08	72.05/52	0.82 ± 0.24	${2}_{-0.18}^{+0.19}$	${2.45}_{-0.21}^{+0.22}$
110411A	2.38	5.1	3.53	0.43 ± 0.12	1.23 ± 0.1	57.95/35	0.31 ± 0.22	${2.9}_{-0.3}^{+0.32}$	${3.41}_{-0.46}^{+0.52}$
110530A	2.6	5.84	3.85	0.47 ± 0.07	1.22 ± 0.05	93.57/39	0.6 ± 0.18	${1.87}_{-0.19}^{+0.2}$	${2.09}_{-0.24}^{+0.26}$
110808A	2.69	5.79	5.22	0.57 ± 0.05	1.41 ± 0.41	8.851/10	0.57 ± 0.38	${2.32}_{-0.34}^{+0.37}$	${1.37}_{-0.26}^{+0.4}$
111008A	2.47	6.01	3.9	0.32 ± 0.04	1.31 ± 0.03	207.2/135	2.69 ± 0.42	1.83 ± 0.1	1.84 ± 0.09
111228A	2.62	6.47	4.13	0.4 ± 0.03	1.27 ± 0.02	188.3/149	3.56 ± 0.44	${1.96}_{-0.1}^{+0.11}$	2.01 ± 0.11
120106A	2.23	5.49	3.07	0.3 ± 0.15	1.1 ± 0.04	50.3/29	0.24 ± 0.1	${2.01}_{-0.29}^{+0.31}$	${2.02}_{-0.32}^{+0.34}$
120224A	2.43	5.74	3.29	−0.54 ± 0.11	0.98 ± 0.02	110.1/77	0.35 ± 0.04	${2.4}_{-0.28}^{+0.29}$	${2.02}_{-0.13}^{+0.14}$
120308A	2.64	4.28	3.75	0.52 ± 0.07	1.32 ± 0.13	107.6/86	1.68 ± 0.59	1.47 ± 0.1	${1.53}_{-0.13}^{+0.14}$
120324A	2.38	5.16	3.73	0.31 ± 0.06	1.12 ± 0.04	224.4/149	3.2 ± 0.85	${2.08}_{-0.25}^{+0.26}$	${2.08}_{-0.12}^{+0.13}$
120326A	2.37	6.21	4.71	−0.24 ± 0.02	2.04 ± 0.06	364.1/191	4.87 ± 0.18	1.77 ± 0.06	${2}_{-0.14}^{+0.15}$
120521C	2.75	5.15	4.06	0.35 ± 0.14	1.1 ± 0.2	5.505/7	0.17 ± 0.15	${1.78}_{-0.24}^{+0.31}$	${2.18}_{-0.59}^{+0.72}$
120703A	2	5.73	3.91	0.64 ± 0.04	1.22 ± 0.05	137/81	2.09 ± 0.75	${1.89}_{-0.13}^{+0.14}$	${1.89}_{-0.15}^{+0.16}$
120811C	2.4	5	3.34	0.43 ± 0.11	1.12 ± 0.08	61.36/45	0.96 ± 0.58	$1.68\pm 0,14$	${2.12}_{-0.2}^{+0.21}$
120907A	1.75	5.45	3.2	0.43 ± 0.07	1.08 ± 0.03	140.4/86	0.79 ± 0.24	${1.77}_{-0.16}^{+0.17}$	1.76 ± 0.11
121024A	2.69	5.63	4.64	0.88 ± 0.04	1.72 ± 0.27	66.73/48	1.76 ± 1.03	${1.85}_{-0.11}^{+0.12}$	${2.36}_{-0.77}^{+1.39}$
121031A	2.91	4.76	4.07	0.39 ± 0.06	1.05 ± 0.09	190.6/177	3.45 ± 1.35	${2.27}_{-0.16}^{+0.17}$	2.24 ± 0.1
121123A	3.11	5.26	4.56	0.73 ± 0.07	1.99 ± 0.17	54.85/39	0.89 ± 0.22	1.88 ± 0.11	${1.71}_{-0.25}^{+0.27}$
121125A	2.27	5.01	3.78	0.71 ± 0.05	1.35 ± 0.07	72.98/90	2.03 ± 1.03	${1.88}_{-0.16}^{+0.17}$	2.03 ± 0.12
121128A	2.13	5.19	3.2	0.55 ± 0.04	1.63 ± 0.04	126.6/95	2.46 ± 0.39	${1.79}_{-0.08}^{+0.12}$	1.89 ± 0.12
121211A	3.6	5.56	4.63	0.73 ± 0.06	1.55 ± 0.13	90.64/65	1.39 ± 0.53	${1.97}_{-0.11}^{+0.12}$	${1.84}_{-0.42}^{+0.47}$
130131A	2.83	5.61	3.88	0.63 ± 0.21	1.23 ± 0.16	11.51/17	0.16 ± 0.5	${2.12}_{-0.31}^{+0.33}$	${2.57}_{-0.44}^{+0.5}$

Part IV: XRT Observations and Fitting Results for Our Sample

130603B	1.6	5.1	3.36	0.31 ± 0.05	1.53 ± 0.05	154.8/74	1.23 ± 0.17	${1.75}_{-0.19}^{+0.2}$	$2\pm 0.15$
130609B	2.74	5.49	3.65	0.64 ± 0.04	1.82 ± 0.03	373.4/278	9.69 ± 1.07	1.81 ± 0.05	${1.96}_{-0.09}^{+0.1}$
131018A	3.07	4.91	4.59	0.26 ± 0.05	1.35 ± 0.28	76.53/53	1.2 ± 0.31	${2.1}_{-0.16}^{+0.17}$	${1.76}_{-0.49}^{+0.57}$
131105A	2.61	5.87	3 .66	0.16 ± 0.11	1.19 ± 0.04	63.77/49	0.88 ± 0.22	1.92 ± 0.17	${1.99}_{-0.17}^{+0.18}$
140108A	2.88	5.63	3.8	0.33 ± 0.06	1.33 ± 0.03	240.5/150	4.5 ± 0.85	${1.76}_{-0.3}^{+0.32}$	${2.08}_{-0.16}^{+0.17}$
140114A	2.91	5.56	4.34	0.24 ± 0.06	1.25 ± 0.12	43.81/31	0.45 ± 0.13	${2.17}_{-0.19}^{+0.2}$	${1.7}_{-0.48}^{+0.56}$
140323A	2.34	5.07	3.95	0.56 ± 0.03	1.62 ± 0.08	162.3/105	6.07 ± 1.18	${2.1}_{-0.16}^{+0.17}$	${2.2}_{-0.61}^{+0.68}$
140331A	3.16	5.32	4.71	0.3 ± 0.06	2.07 ± 0.47	25.97/17	0.71 ± 0.24	${1.73}_{-0.2}^{+0.22}$	${2.19}_{-0.75}^{+1.22}$
140430A	2.66	5.66	4.68	0.6 ± 0.09	1.53 ± 0.4	45.08/37	1.17 ± 0.59	${2.19}_{-0.19}^{+0.2}$	${2.07}_{-0.24}^{+0.26}$
140512A	2.46	5.65	4.12	0.73 ± 0.02	1.57 ± 0.06	488.5/379	19.4 ± 3.03	1.8 ± 0.07	1.86 ± 0.1
140518A	2.8	4.27	3.28	−0.16 ± 0.31	1.38 ± 0.2	35.32/27	0.28 ± 0.11	${2.09}_{-0.19}^{+0.2}$	${1.89}_{-0.2}^{+0.21}$
140703A	2.34	5.48	4.16	0.62 ± 0.03	2.16 ± 0.09	122.8/76	6.82 ± 0.65	1.76 ± 0.11	1.94 ± 0.15
140706A	2.36	5.05	3.83	0.36 ± 0.07	1.25 ± 0.1	28.23/20	0.49 ± 0.16	${1.84}_{-0.23}^{+0.24}$	${1.77}_{-0.31}^{+0.34}$
140709A	2.43	5.5	4.22	0.56 ± 0.03	1.39 ± 0.07	72.03/74	3.63 ± 0.75	${1.84}_{-0.18}^{+0.19}$	${2.03}_{-0.19}^{+0.2}$
140916A	2.92	5.55	4.61	0.14 ± 0.02	2.05 ± 0.13	183.8/137	2.92 ± 0.23	2.09 ± 0.09	${2}_{-0.29}^{+0.32}$
141017A	2.41	5.6	3.32	0.07 ± 0.13	1.12 ± 0.03	144.1/79	0.8 ± 0.17	${2.13}_{-0.17}^{+0.18}$	2.01 ± 0.13
141031A	3.74	5.88	4.47	0.06 ± 0.18	1.07 ± 0.11	36.86/23	0.68 ± 0.3	${1.88}_{-0.25}^{+0.32}$	${1.9}_{-0.28}^{+0.31}$
141121A	4.45	6.17	5.52	0.46 ± 0.1	2.24 ± 0.18	25.1/20	2.86 ± 0.5	${1.87}_{-0.21}^{+0.26}$	${1.65}_{-0.13}^{+0.26}$
150120B	2.6	5.3	3.03	−0.23 ± 0.28	1.16 ± 0.04	61.48/40	0.15 ± 0.04	${1.72}_{-0.28}^{+0.3}$	${2.19}_{-0.19}^{+0.2}$
150201A	2.19	5.38	2.88	0.46 ± 0.04	1.27 ± 0.01	456.9/355	4.62 ± 0.73	1.84 ± 0.05	${2.28}_{-0.13}^{+0.14}$
150203A	2.19	5.38	4.59	0.687 ± 0.07	1.56 ± 0.9	54.18/11	0.36 ± 2.03	${2.01}_{-0.24}^{+0.36}$	${2.45}_{-0.81}^{+0.95}$
150323A	2.65	5.58	4.16	0.43 ± 0.05	1.26 ± 0.09	30.87/18	0.52 ± 0.17	${2.11}_{-0.23}^{+0.25}$	${1.77}_{-0.49}^{+0.57}$
150323C	2.9	5.8	3.85	0.38 ± 0.22	1.13 ± 0.14	9.361/12	0.07 ± 0.17	${2}_{-0.41}^{+0.46}$	${2.33}_{-0.39}^{+0.44}$
150424A	2.6	5.8	5.06	0.79 ± 0.04	1.43 ± 0.17	62.65/39	$52\pm 1.47\pm 0.88$	${1.96}_{-0.21}^{+0.22}$	${1.92}_{-0.18}^{+0.2}$
150607A	2.1	4.61	3.76	0.73 ± 0.05	1.21 ± 0.14	114.9/76	2.7 ± 2.59	${1.81}_{-0.14}^{+0.15}$	${1.79}_{-0.17}^{+0.18}$
150615A	2.87	5.5	4.12	−0.01 ± 0.13	0.79 ± 0.13	27.83/23	0.3 ± 0.18	${2.6}_{-0.33}^{+0.36}$	${2.4}_{-0.31}^{+0.41}$
150626A	2.87	4.81	4.15	0.21 ± 0.09	1.03 ± 0.19	25.99/28	0.51 ± 0.26	${2.42}_{-0.31}^{+0.34}$	${2.11}_{-0.29}^{+0.31}$
150711A	2.82	4.79	4.18	0.5 ± 0.07	1.43 ± 0.19	38.73/20	0.79 ± 0.32	${2.21}_{-0.27}^{+0.28}$	${1.82}_{-0.33}^{+0.36}$
150910A	3.03	5.5	3.94	0.61 ± 0.14	2.36 ± 0.16	201.9/101	11.9 ± 4.84	${1.79}_{-0.18}^{+0.19}$	1.78 ± 0.08
151027A	2.71	6.1	3.6	0.06 ± 0.03	1.65 ± 0.02	635.3/447	13.8 ± 0.61	2.03 ± 0.04	1.77 ± 0.12
151112A	3.63	5.83	4.76	0.64 ± 0.08	1.6 ± 0.25	34.09/38	1.09 ± 0.63	${2.28}_{-0.23}^{+0.24}$	${2.14}_{-0.23}^{+0.25}$
151228B	2.7	5.37	4.45	0.59 ± 0.06	1.7 ± 0.33	16.02/14	0.41 ± 0.31	${1.97}_{-0.23}^{+0.26}$	${1.94}_{-0.34}^{+0.41}$
151229A	1.9	4.66	2.44	0.27 ± 0.15	0.94 ± 0.03	108.3/75	0.42 ± 0.2	${1.53}_{-0.3}^{+0.33}$	2.05 ± 0.14
160119A	3.2	6.05	4.29	0.55 ± 0.08	1.43 ± 0.06	138/106	1.58 ± 0.39	${1.97}_{-0.15}^{+0.16}$	2.14 ± 0.13
160127A	2.35	5	3.64	0.83 ± 0.06	1.57 ± 0.1	38.51/41	0.79 ± 0.35	${1.8}_{-0.2}^{+0.21}$	${1.82}_{-0.18}^{+0.21}$
160227A	3.2	6.4	4.37	0.28 ± 0.09	1.2 ± 0.04	196/102	3.41 ± 0.92	1.67 ± 0.1	1.72 ± 0.1
160327A	2.53	5	3.62	0.3 ± 0.14	1.56 ± 0.09	56.85/28	0.42 ± 0.14	${1.64}_{-0.23}^{+0.26}$	${2.13}_{-0.21}^{+0.23}$
160412A	2.72	5.95	4.32	0.11 ± 0.08	1.01 ± 0.1	33.55/18	0.37 ± 0.15	${2.66}_{-0.45}^{+0.5}$	${1.76}_{-0.3}^{+0.32}$
160630A	2.22	5.41	3.2	0.06 ± 0.11	1.16 ± 0.03	92.93/73	0.97 ± 0.17	${1.81}_{-0.22}^{+0.23}$	${1.76}_{-0.13}^{+0.14}$
160804A	3.02	6.13	3.96	−0.28 ± 0.27	0.92 ± 0.05	75.62/43	0.37 ± 0.1	${1.97}_{-0.24}^{+0.25}$	${1.94}_{-0.16}^{+0.17}$
160824A	2.59	4.54	3.93	0.4 ± 0.05	1.45 ± 0.14	111.5/85	2.74 ± 0.68	${1.88}_{-0.21}^{+0.23}$	2.04 ± 0.17
161001A	2.45	5.36	3.53	0.69 ± 0.09	1.38 ± 0.05	65.41/41	1.46 ± 0.7	${1.83}_{-0.23}^{+0.24}$	${2.12}_{-0.23}^{+0.24}$
161011A	2.86	5.5	4.64	−0.1 ± 0.04	3.08 ± 0.18	131.9/48	1.71 ± 0.87	1.9 ± 0.15	${2.48}_{-0.43}^{+0.49}$
161108A	2.81	5.96	4.36	−0.02 ± 0.16	0.86 ± 0.07	33.44/24	0.43 ± 0.18	${1.95}_{-0.27}^{+0.29}$	${1.64}_{-0.25}^{+0.27}$
161129A	2.44	4.24	3.56	0.38 ± 0.11	2.18 ± 0.11	80.76/51	1.75 ± 0.37	${1.72}_{-0.18}^{+0.19}$	${1.82}_{-0.17}^{+0.18}$
161202A	2.53	5.16	4.17	0.75 ± 0.13	1.82 ± 0.36	34.26/31	2.29 ± 3.73	${1.72}_{-0.18}^{+0.19}$	${1.82}_{-0.17}^{+0.18}$
161214B	2.9	5.71	4.48	0.45 ± 0.06	1.32 ± 0.1	41.46/40	0.98 ± 0.31	${1.97}_{-0.17}^{+0.18}$	${2.26}_{-0.34}^{+0.38}$
170113A	2.41	5.94	3.64	0.48 ± 0.04	1.25 ± 0.02	211/148	2.87 ± 0.52	1.85 ± 0.12	1.79 ± 0.08

Part V: XRT Observations and Fitting Results for Our Sample

170202A	2.47	5.73	3.32	−0.12 ± 0.1	1.15 ± 0.04	50.64/39	0.66 ± 0.11	${2.12}_{-0.17}^{+0.18}$	${1.92}_{-0.17}^{+0.18}$
170208B	2.91	5.78	3.83	0.53 ± 0.14	1.07 ± 0.06	47/40	0.56 ± 0.75	${2.19}_{-0.22}^{+0.23}$	${2.54}_{-0.26}^{+0.28}$
170317A	2.18	5.16	3.32	0.44 ± 0.08	1.33 ± 0.08	56.84/45	0.39 ± 0.14	${1.97}_{-0.06}^{+0.17}$	${2.15}_{-0.23}^{+0.31}$
170330A	2	5.39	2.71	0.56 ± 0.08	1.33 ± 0.08	210.1/164	1.59 ± 0.14	${2.01}_{-0.26}^{+0.39}$	${1.92}_{-0.12}^{+0.13}$
170607A	3.13	6.29	4.25	0.29 ± 0.05	1.03 ± 0.03	204.7/166	2.86 ± 0.63	${2.08}_{-0.09}^{+0.1}$	1.93 ± 0.1
170711A	2.37	5.45	4.66	0.46 ± 0.03	1.48 ± 0.3	85.36/42	1.17 ± 0.51	${2.07}_{-0.35}^{+0.38}$	${2.11}_{-0.21}^{+0.22}$

Notes.

^at₁ and t₂ represent the beginning time (t₁) of the shallow decay segment and the end time (t₂) of its follow-up segment. ^bBreak time and decay slopes before and after the break, and the fit ${\chi }^{2}$ ${\chi }^{2}$ /dof. ^cThe X-ray fluence of the shallow decay segment. ^dPhoton index of shallow decay segment and its follow-up segment.

Download table as: ASCIITypeset images: 1 2 3 4

There is no spectral evolution during the shallow decay and the following normal decay phases (Liang et al. 2007), so we extract the time-integrated spectra in the time interval of [t₁, t_b] and [t_b, t₂]. We fit the X-ray spectrum with the Xspec package, and an absorbed power-law model is invoked to fit the spectrum, i.e., abs × zabs × zdust × power law, where zdust is the dust extinction of the GRB host galaxy, abs and zabs are the absorption models for the Milky Way and the GRB host galaxy, respectively. Then, we derived the photon indices Γ_X,1 and Γ_X,2 for the shallow decay and the following normal decay phases.

3. L07 Revisited

3.1. Distributions of the Characteristic Properties of the Shallow Decay Phase

For our sample, we find that the distributions of the characteristic properties of the shallow decay phase (e.g., t_b, S_X, Γ_X,1, and α_X,1) accords with normal or lognormal distribution,⁷ with ${\mathrm{log}}_{10}({t}_{b})=4\pm 0.72$ , ${\mathrm{log}}_{10}({S}_{{\rm{X}}}(\mathrm{erg}\,{\mathrm{cm}}^{-2}))=-6.97\,\pm 0.56$ , ${{\rm{\Gamma }}}_{{\rm{X}},1}=1.96\pm 0.26$ , and ${\alpha }_{{\rm{X}},1}=0.43\pm 0.22$ . For comparison, we plot our result together with L07's result in Figure 2. It is found that when a larger sample is involved, the distributions for all the parameters become slightly broader, but the peaks of each distribution barely change, for instance, the mean value for distributions of ${\mathrm{log}}_{10}({t}_{b})$ , ${\mathrm{log}}_{10}({S}_{{\rm{X}}})$ , Γ_X,1, and α_X,1 differs by 3%, 7%, 6%, and 23% respectively, between L07 results and our results.

**Figure 2.** Distributions of the characteristic properties of the shallow decay phase for GRBs in our sample, comparing with L07's results. The solid red and pink lines are the results of Gaussian fits.
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3.2. Relationship of the Shallow Decay Phase to the Prompt Gamma-Ray Phase

In order to investigate the relationship of the shallow decay phase to the prompt gamma-ray phase, we first collect relevant parameters of prompt gamma-ray emission for our sample, including the duration (T₉₀), fluence ( ${S}_{\gamma }$ ), photon indices (Γ_γ), and the peak energy of the $\nu {F}_{\nu }$ spectrum (E_p) through published articles and GCN Circulars (the results are presented in Table 2). For a BAT spectrum that can be fitted by a single power law, we estimate E_p through the correlation between the BAT photon index Γ and E_p (Zhang et al. 2007a; Sakamoto et al. 2009c; Virgili et al. 2012), namely,

$\begin{eqnarray}&&{\mathrm{log}}_{10}({E}_{p})=(4.34\pm 0.475)-(1.32\pm 0.219){\mathrm{log}}_{10}({\rm{\Gamma }}).\end{eqnarray} \tag{ 2 }$

Table 2. Part I: BAT Observations and Redshift for Our Sample

GRB Name	${T}_{90}({\rm{s}})$	${S}_{\gamma }$ ^a	${{\rm{\Gamma }}}_{\gamma ,1}$ ^b	E_p^b	BAT Ref^c	z	z Ref^c	Group of GRBs^d
	(s)	( ${10}^{-7}\,\mathrm{erg}\ {\mathrm{cm}}^{-2}$ )		(keV)
050319	10 ± 2	8 ± 0.8	2.2 ± 0.2	33.41 ± 12.22	3119	3.24	3136	case¹(regime 2)
050401	33 ± 2	140 ± 14	1.5 ± 0.06	133.14 ± 29.35	3173	2.9	3176	case²
050416A	2.4 ± 0.2	69	2.9 ± 0.2	12.32 ± 4.45	3273	0.6528	3542	case¹(regime 1)
050505	60 ± 2	41 ± 4	1.5 ± 0.1	133.14 ± 38.15	3364	4.27	3368	case¹(regime 2)
050607	26.5	8.9 ± 1.2	1.93 ± 0.14	53.6 ± 16.4	3525	...	...	case¹(regime 1)
050712	48 ± 2	18	1.56 ± 0.18	115.56 ± 42.64	3576	...	...	case¹(regime 2)
050713A	70 ± 10	91±6	1.58 ± 0.07	110.37 ± 25.84	3597	...	...	case¹(regime 1)
050713B	54.2	82±10	1.56 ± 0.13	115.56 ± 34.64	3600	...	...	case¹(regime 2)
050714B	46.7	6.5 ± 1.4	2.6 ± 0.3	18.28 ± 8.08	3615	2.438	...	case¹(regime 1)
050802	13 ± 2	28 ± 3	1.6 ± 0.1	105.47 ± 28.37	3737	1.7102	3749	case¹(regime 2)
050803	85 ± 10	39 ± 3	1.5 ± 0.1	133.14 ± 35.98	3757	...	...	case¹(regime 2)
050814	65 ± 20	21.7 ± 3.6	1.8 ± 0.17	68.94 ± 24.5	3803	5.3	3809	case¹(regime 3)
050822	102 ± 2	34 ± 3	2.5 ± 0.1	21.06 ± 6.04	3856	1.434	H12	case¹(regime 1)
050824	25 ± 5	2.8 ± 0.5	2.7 ± 0.4	15.95 ± 12.45	3871	0.8278	3874	case¹(regime 1)
050826	35 ± 8	4.3 ± 0.7	1.2 ± 0.3	297.96 ± 61.25	3888	0.296	5982	case¹(regime 2)
050915B	40 ± 1	34 ± 1	1.89 ± 0.06	57.81 ± 13.27	3993	...	...	case¹(regime 2)
050922B	80 ± 10	18 ± 3	1.89 ± 0.23	57.81 ± 24.92	4019	...	...	case¹(regime 2)
051109B	15 ± 1	2.7 ± 0.4	1.96 ± 0.25	50.69 ± 24.63	4237	...	...	case¹(regime 2)
051221A	1.4	32 ± 1	1.08 ± 0.14	${403}_{-72}^{+93}$	4363 4394	0.5465	4384	case¹(regime 2)
060108	14.4 ± 1	3.7 ± 0.4	2.01 ± 0.17	46.29 ± 15.58	4445	2.03	4539	case¹(regime 1)
060109	116 ± 3	6.4 ± 1	1.96 ± 0.25	50.69 ± 24.47	4476	...	...	case¹(regime 1)
060115	142 ± 5	19 ± 1	1.76 ± 0.12	74.77 ± 20.42	4518	3.5328	4520	case¹(regime 1)
060203	60 ± 10	8.5 ± 1.2	1.62 ± 0.23	100.84 ± 45.17	4656	...	...	case¹(regime 2)
060204B	134 ± 5	30 ± 2	0.82 ± 0.4	96.8 ± 41	4671	...	...	case¹(regime 1)
060219	62 ± 2	4.2 ± 0.8	2.6 ± 0.4	18.28 ± 11.64	4811	...	...	case¹(regime 1)
060306	61 ± 2	22 ± 1	1.85 ± 0.1	62.45 ± 16.47	4851	1.55	P13	case¹(regime 1)
060312	43 ± 3	18 ± 1	1.35 ± 0.34	62.8 ± 21.5	4864	...	...	case¹(regime 3)
060413	150 ± 10	36 ± 1	1.67 ± 0.08	90.36 ± 22.8	4961	...	...	case²
060428A	$39.4\pm 2$	14 ± 1	2.04 ± 0.11	43.88 ± 13.27	5022	...	...	case¹(regime 2)
060502A	33 ± 5	22 ± 1	1.43 ± 0.08	158.21 ± 39.79	5053	1.5026	5052	case¹(regime 2)
060510A	21	255 ± 7	1.661 ± 0.072	${186}_{-24}^{+36}$	5113	...	...	case¹(regime 2)
060604	10 ± 3	1.3 ± 0.3	1.9 ± 0.41	56.72 ± 31.64	5214	2.1357	5218	case¹(regime 1)
060605	15 ± 2	4.6 ± 0.4	1.34 ± 0.15	200.06 ± 75.23	5231	3.773	5226	case¹(regime 3)
060607A	100 ± 5	26 ± 1	1.45 ± 0.07	150.47 ± 35.27	5242	3.0749	5237	case⁴
060614	102	409 ± 18	1.57 ± 0.12	${302}_{-85}^{+214}$	5264	0.1257	5276	case¹(regime 3)
060707	68 ± 5	17 ± 2	0.66 ± 0.63	${66}_{-10}^{+25}$	5289	3.4240	5298	case¹(regime 2)
060712	26 ± 5	13 ± 2	1.64 ± 0.32	96.47 ± 25.27	5304	...	...	case¹(regime 1)
060714	115 ± 5	30 ± 2	1.99 ± 0.1	47.99 ± 12.52	5334	2.7108	5320	case¹(regime 2)
060719	55 ± 5	16 ± 1	2 ± 0.11	47.13 ± 12.88	5349	1.5320	K12	case¹(regime 1)
060729	116 ± 10	27 ± 2	1.86 ± 0.14	61.24 ± 17.75	5370	0.5428	F09	case¹(regime 2)
060804	16 ± 2	5.1 ± 0.9	1.78 ± 0.28	71.78 ± 56.05	5395	...	...	case¹(regime 2)
060805A	5.4 ± 0.5	0.74 ± 0.2	2.23 ± 0.42	31.81 ± 12.26	5403	2.44	...	case¹(regime 2)
060807	34 ± 4	7.3 ± 0.9	1.57 ± 0.2	112.93 ± 12.14	5421	...	...	case¹(regime 1)
060813	14.9 ± 0.5	55 ± 1	1 ± 0.09	176 ± 83	5443	...	...	case¹(regime 2)
060814	134 ± 4	269 ± 12	1.43 ± 0.16	${257}_{-58}^{+122}$	5460	1.9229	K12	case¹(regime 1)
060906	$43.6\pm 1$	22.1 ± 1.4	2.02 ± 0.11	45.47 ± 12.14	5534 5538	3.6856	5535	case¹(regime 3)
060908	19.3 ± 0.3	29 ± 1	1.33 ± 0.07	205.54 ± 48.55	5551	1.8836	F09	case¹(regime 2)
060923C	76 ± 2	16 ± 2	2.72 ± 0.24	15.53 ± 5.36	5611	...	...	case¹(regime 3)

Part II: BAT Observations and Redshift for Our Sample

060927	22.6 ± 0.3	11 ± 1	0.93 ± 0.38	71.7 ± 17.6	5639	5.467	5651	case¹(regime 2)
061004	6.2 ± 0.3	5.7 ± 0.3	1.81 ± 0.10	67.57 ± 18.13	5964	...	...	case¹(regime 1)
061019	191 ± 3	17 ± 2	1.85 ± 0.26	62.45 ± 31.63	5732	...	...	case¹(regime 3)
061121	81 ± 5	137 ± 2	1.32 ± 0.05	${606}_{-72}^{+90}$	5831 5837	1.3145	5826	case¹(regime 2)
061201	0.8 ± 0.1	3.3 ± 0.3	0.36 ± 0.65	${873}_{-284}^{+458}$	5882 5890	0.111	5952	case¹(regime 3)
061202	91 ± 5	35 ± 1	1.63 ± 0.27	98.63 ± 23.83	5887	...	...	case¹(regime 2)
061222A	72 ± 3	83 ± 2	1.02 ± 0.11	${324}_{-41}^{+54}$	5984	2.088	P09	case¹(regime 2)
070103	19 ± 1	3.4 ± 0.5	2 ± 0.2	47.13 ± 17.93	5991	2.6208	K12	case¹(regime 1)
070110	85 ± 5	16 ± 1	1.57 ± 0.12	112.93 ± 35.26	6007	2.3521	6010	case⁴
070129	460 ± 20	31 ± 3	2.05 ± 0.16	43.11 ± 14.57	6058	2.3384	K12	case¹(regime 1)
070306	$209.5\pm 10$	53.80 ± 2.86	1.72 ± 0.1	84.74 ± 22.84	6173	1.49594	6202	case¹(regime 3)
070328	69 ± 5	90.6 ± 1.79	1.09 ± 0.08	421.59 ± 111.87	6225 6230	...	...	case¹(regime 2)
070420	77 ± 4	140 ± 4.48	1.08 ± 0.13	${167}_{-16}^{+20}$	6342 6344	...	...	case¹(regime 2)
070429A	163 ± 5	9.1 ± 1.43	2.11 ± 0.27	38.85 ± 15.97	6362	...	...	case¹(regime 3)
070508	21 ± 1	196 ± 2.73	0.81 ± 0.07	188 ± 8	6390 6403	0.82	6398	case¹(regime 1)
070529	$109.2\pm 3$	25.70 ± 2.45	1.38 ± 0.16	179.9 ± 57.53	6468	2.4996	6470	case¹(regime 1)
070621	33.3	33 ± 1	1.57 ± 0.06	112.93 ± 24.8	6560 6571	...	...	case¹(regime 1)
070628	$39.1\pm 2$	35 ± 2	1.91 ± 0.09	55.65 ± 14.86	6589	...	...	case¹(regime 2)
070809	1.3 ± 0.1	1 ± 0.1	1.69 ± 0.22	86.56 ± 38.97	6732	0.2187	7889	case¹(regime 3)
070810A	11 ± 1	6.9 ± 0.6	2.04 ± 0.14	43.88 ± 13.64	6748	2.17	6741	case¹(regime 1)
071118	71 ± 20	5 ± 1	1.63 ± 0.29	98.63 ± 15.24	7109	...	...	case²
080320	14 ± 2	2.7 ± 0.4	1.7 ± 0.22	84.74 ± 36.47	7505	...	...	case¹(regime 1)
080328	90.6 ± 1.5	94 ± 2	1.52 ± 0.04	126.92 ± 25.43	7533	...	...	case¹(regime 2)
080409	$20.2\pm 8$	6.1 ± 0.7	2.1 ± 0.2	39.52 ± 13.18	7582	...	...	case¹(regime 2)
080413B	8 ± 1	32 ± 1	1.26 ± 0.27	73.3 ± 15.8	7606	1.1014	7601	case¹(regime 1)
080430	16.2 ± 2.4	12 ± 1	1.73 ± 0.09	79.55 ± 20.16	7656	0.767	7654	case¹(regime 2)
080516	5.8 ± 0.3	2.6 ± 0.4	1.82 ± 0.27	66.24 ± 41.3	7736	3.2	7747	case¹(regime 1)
080703	3.4 ± 0.8	2 ± 0.3	1.53 ± 0.22	123.95 ± 63.34	7938	...	...	case³(regime 4)
080707	27.1 ± 1.1	5.2 ± 0.6	1.77 ± 0.19	73.25 ± 29.4	7953	1.2322	7949	case¹(regime 2)
080723A	$17.3\pm 3$	3.3 ± 0.4	1.77 ± 0.21	73.25 ± 28.43	8007	...	...	case¹(regime 2)
080903	66 ± 11	14 ± 1	0.84 ± 0.52	60 ± 15	8176	...	...	case¹(regime 3)
080905B	128 ± 16	18 ± 2	1.78 ± 0.15	71.78 ± 23.48	8188	2.3739	8191	case¹(regime 3)
081007	10 ± 4.5	7.1 ± 0.8	2.51 ± 0.2	20.76 ± 7.15	8338	0.5295	8335	case¹(regime 2)
081029	270 ± 45	21 ± 2	1.43 ± 0.18	158.21 ± 63.37	8447	3.8479	8438	case¹(regime 3)
081210	146 ± 8	18 ± 2	1.42 ± 0.14	162.27 ± 55.22	8649	...	...	case¹(regime 3)
081230	60.7 ± 13.8	8.2 ± 0.8	1.97 ± 0.16	49.77 ± 16.49	8759	...	...	case¹(regime 2)
090102	27 ± 2.2	0.68 ± 0.02	1.36 ± 0.08	189.64 ± 47.5	8769	1.547	8766	case¹(regime 3)
090113	9.1 ± 0.9	7.6 ± 0.4	1.6 ± 0.1	105.47 ± 28.92	8808	1.7493	K12	case¹(regime 1)
090205	8.8 ± 1.8	1.9 ± 0.3	2.15 ± 0.23	36.3 ± 12.57	8886	4.6497	8892	case¹(regime 3)
090313	78 ± 19	14 ± 2	1.91 ± 0.29	55.65 ± 28.37	8986	3.375	8994	case¹(regime 3)
090404	84 ± 14	30 ± 1	2.32 ± 0.08	27.58 ± 7.22	9089	3	P13	case¹(regime 1)
090407	310 ± 70	11 ± 2	1.73 ± 0.29	79.55 ± 36.98	9104	1.4485	K12	case¹(regime 2)
090418A	56 ± 5	46 ± 2	1.48 ± 0.07	139.75 ± 32.45	9157	1.608	9151	case¹(regime 2)
090424	52	201	0.9 ± 0.02	177 ± 3	9230	0.544	9243	case¹(regime 2)
090429B	$5.5\pm 1$	3.1 ± 0.3	0.47 ± 0.77	42.1 ± 5.6	9290	9.4	C11	case¹(regime 2)
090510	0.3 ± 0.1	3.4 ± 0.4	0.98 ± 0.2	618.98 ± 30.97	9337	0.903	9353	case¹(regime 3)
090518	6.9 ± 1.7	4.7 ± 0.4	1.54 ± 0.14	121.07 ± 40.95	9393	...	..	case¹(regime 1)
090529	100	6.8 ± 1.7	2 ± 0.3	47.13 ± 26.19	9434	2.625	9457	case¹(regime 3)

Part III: BAT Observations and Redshift for Our Sample

090530	48 ± 36	11 ± 1	1.61 ± 0.17	103.12 ± 34.85	9443	1.266	15571	case¹(regime 2)
090727	302 ± 23	14 ± 2	1.24 ± 0.24	264.7 ± 140.61	9724	...	...	case¹(regime 2)
090728	59 ± 18	10 ± 2	2.05 ± 0.26	43.11 ± 19.87	9736	...	...	case¹(regime 3)
090813	9	13 ± 1	1.43 ± 0.08	158.21 ± 39.97	9792	...	...	case¹(regime 2)
090904A	122 ± 10	30 ± 2	2.01 ± 0.1	46.29 ± 12.36	9888	...	...	case¹(regime 2)
091018	4.4 ± 0.6	14 ± 1	1.77 ± 0.24	${19.3}_{-11}^{+18}$	10040	0.971	10038	case¹(regime 2)
091029	$39.2\pm 5$	24 ± 1	1.46 ± 0.27	61.4 ± 17.5	10103	2.752	10100	case¹(regime 1)
091130B	112.5 ± 17.1	13 ± 1	2.15 ± 0.15	36.3 ± 12.26	10221	...	...	case¹(regime 1)
100111A	12.9 ± 2.1	6.7 ± 0.5	1.69 ± 0.13	86.56 ± 25.02	10322	...	...	case¹(regime 2)
100219A	$18.8\pm 5$	3.7 ± 0.6	1.34 ± 0.25	$200\pm 93.9$	10434	4.6667	10445	case³(regime 4)
100302A	17.9 ± 1.7	3.1 ± 0.4	1.72 ± 0.19	81.23 ± 29.1	10462	4.813	10466	case¹(regime 1)
100418A	7 ± 1	3.4 ± 0.5	2.16 ± 0.25	35.7 ± 13.96	10615	0.6235	10620	case¹(regime 2)
100425A	37 ± 2.4	4.7 ± 0.9	2.42 ± 0.32	23.68 ± 12.53	10685	1.755	10684	case¹(regime 2)
100508A	52 ± 10	7 ± 1.1	1.23 ± 0.25	272.55 ± 146.57	10732	...	...	case²
100522A	35.3 ± 1.8	21 ± 1	1.89 ± 0.08	57.81 ± 14.68	10788	...	...	case¹(regime 1)
100614A	225 ± 55	27 ± 2	1.88 ± 0.15	58.92 ± 18.98	10852	...	...	case¹(regime 3)
100704A	197.5 ± 23.3	60 ± 2	1.73 ± 0.06	79.55 ± 17.95	10932	...	...	case¹(regime 2)
100725B	200±30	68 ± 2	1.89 ± 0.06	57.81 ± 13.41	10993	...	...	case¹(regime 1)
100901A	439±33	21 ± 3	1.59 ± 0.21	107.88 ± 53.74	11169	1.408	11164	case¹(regime 2)
100902A	428.8±43	32 ± 2	1.98 ± 0.13	48.87 ± 14.16	11202	...	...	case²
100905A	3.4 ± 0.5	1.7 ± 0.2	1.09 ± 0.19	421.59 ± 40.24		...	...	case¹(regime 2)
100906A	114.4 ± 1.6	120	1.78 ± 0.03	71.78 ± 14.63	11233	1.727	11230	case¹(regime 3)
101117B	5.2 ± 1.8	11 ± 1	1.5 ± 0.09	133.14 ± 34.9	11414	...	...	case¹(regime 2)
110102A	264 ± 8	165±3	1.6 ± 0.04	105.47 ± 21.62	11511	...	...	case¹(regime 2)
110106B	24.8 ± 4.8	20±1	1.76 ± 0.11	74.77 ± 20.26	11533	0.618	11538	case¹(regime 3)
110312A	28.7 ± 7.6	8.2±1.3	2.32 ± 0.26	27.58 ± 10.97	11783	...	...	case¹(regime 1)
110319A	19.3 ± 1.6	14±1	1.31 ± 0.43	$21.9\pm 7$	11811	...	...	case¹(regime 1)
110411A	80.3 ± 5.2	33±2	1.51 ± 0.31	$41\pm 8.1$	11921	...	...	case²
110530A	19.6 ± 3.1	3.3±0.5	2.06 ± 0.24	42.36 ± 16.22	12049	...	...	case¹(regime 1)
110808A	48 ± 23	3.3±0.8	2.32 ± 0.43	27.58 ± 49.72	12262	1.348	12258	case¹(regime 3)
111008A	63.46 ± 2.19	53±3	1.86 ± 0.09	61.24 ± 16.32	12424	4.9898	12431	case¹(regime 2)
111228A	101.2 ± 5.42	85±2	2.27 ± 0.06	29.84 ± 7.59	12749	0.716	12761	case¹(regime 2)
120106A	61.6 ± 4.1	9.7±1.1	1.53 ± 0.17	123.95 ± 45.81	12815	...	...	case¹(regime 1)
120224A	8.13 ± 1.99	2.4±0.5	2.25 ± 0.36	$30.8\pm 32$	12983	...	...	case¹(regime 1)
120308A	60.6 ± 17.1	12±1	1.71 ± 0.13	82.96 ± 23.8	13022	...	...	case¹(regime 3)
120324A	118 ± 10	101±3	1.34 ± 0.04	200.06 ± 38.03	13096	...	...	case¹(regime 1)
120326A	69.6 ± 8.3	26±3	1.41 ± 0.34	41.1 ± 6.9	13120	1.798	13118	case¹(regime 3)
120521C	26.7 ± 4.4	11±1	1.73 ± 0.11	79.55 ± 22.68	13333	6	13348	case¹(regime 1)
120703A	25.2 ± 4.3	35±1	1.51 ± 0.06	129.98 ± 27.54	13414	...	...	case¹(regime 2)
120811C	26.8 ± 3.7	30±3	1.4 ± 0.3	42.9 ± 5.7	13634	2.671	13628	case¹(regime 1)
120907A	16.9 ± 8.9	6.7±1.1	1.73 ± 0.25	79.55 ± 42.57	13720	0.97	13723	case¹(regime 2)
121024A	69 ± 32	11±1	1.41 ± 0.22	166.46 ± 100.22	13899	0.97	13890	case¹(regime 1)
121031A	226 ± 19	78±2	1.5 ± 0.05	133.14 ± 28.49	13949	...	...	case¹(regime 1)
121123A	317 ± 14	150±10	0.96 ± 0.2	$65\pm 5.1$	13990	...	...	case¹(regime 3)
121125A	53.2 ± 4.6	47±1	1.52 ± 0.06	126.92 ± 27.68	13996	...	...	case¹(regime 2)
121128A	23.3 ± 1.6	69±4	1.32 ± 0.18	64.5 ± 6.8	14011	2.2	14009	case¹(regime 3)
121211A	182 ± 39	13±2.67	2.36 ± 0.26	25.93 ± 10.67	14067	1.023	14059	case¹(regime 3)
130131A	51.6 ± 2.4	3.1±0.6	2.12 ± 0.32	38.19 ± 25.72	14163	...	...	case¹(regime 1)

Part IV: BAT Observations and Redshift for Our Sample

130603B	0.18 ± 0.02	6.3±0.3	0.82 ± 0.07	1177.96 ± 322.01	14741	0.356	14744	case¹(regime 2)
130609B	210.6 ± 15.1	160	1.32 ± 0.04	211.22 ± 40.09	14867	...	...	case¹(regime 3)
131018A	73.22 ± 18.97	11±1	2.24 ± 0.14	31.3 ± 9.47	15354	...	...	case¹(regime 2)
131105A	112.3 ± 4.1	71±5	1.45 ± 0.11	150.47 ± 44.64	15459	1.686	15450	case¹(regime 2)
140108A	94 ± 4	70±2	1.62 ± 0.05	100.84 ± 22.11	15717	...	...	case¹(regime 1)
140114A	139.7 ± 16.4	32±1	2.06 ± 0.09	42.36 ± 11.23	15738	...	...	case¹(regime 2)
140323A	104.9 ± 2.6	160	1.35 ± 0.17	127.1 ± 59.2	16029	...	...	case¹(regime 2)
140331A	209 ± 33	6.7 ± 1.3	2.01 ± 0.29	46.29 ± 26.12	16063	...	...	case¹(regime 3)
140430A	173.6 ± 3.7	11 ± 2	2 ± 0.22	47.13 ± 20.29	16200	1.6	16194	case¹(regime 2)
140512A	154.8 ± 4.88	140 ± 3	1.45 ± 0.04	150.47 ± 29.96	16258	0.725	16310	case¹(regime 3)
140518A	60.5 ± 2.4	10 ± 1	0.92 ± 0.61	43.9 ± 7.6	16306	4.707	16301	case¹(regime 2)
140703A	67.1 ± 67.9	39 ± 3	1.74 ± 0.13	77.91 ± 23.87	16509	3.14	16505	case¹(regime 3)
140706A	48.3 ± 4.3	20 ± 1	1.8 ± 0.1	68.94 ± 18.46	16539	...	...	case¹(regime 2)
140709A	98.6 ± 7.7	53 ± 2	1.23 ± 0.27	$78\pm 20.4$	16553	...	...	case¹(regime 2)
140916A	80.1 ± 15.6	17 ± 3	2.15 ± 0.27	36.3 ± 16.26	16827	...	...	case¹(regime 3)
141017A	55.7 ± 2.8	31 ± 1	1.05 ± 0.28	80.4 ± 17.2	16927	...	...	case¹(regime 1)
141031A	920 ± 10	29 ± 5	1.21 ± 0.13	289.16 ± 99.79	17010	...	...	case¹(regime 1)
141121A	549.9 ± 37.4	53 ± 4	1.73 ± 0.13	79.55 ± 22.45	17083	1.47	17081	case¹(regime 3)
150120B	24.30 ± 3.39	5.4 ± 0.8	2.20 ± 0.25	33.41 ± 13.05	17330	...	...	case¹(regime 1)
150201A	26.1±5.2	7.8 ± 1.2	2.32 ± 0.24	27.58 ± 11.34	17374	...	...	case¹(regime 1)
150203A	25.8±5	9.1 ± 0.6	1.9 ± 0.12	56.72 ± 15.74	17410	...	...	case¹(regime 1)
150323A	149.6±8.9	61 ± 2	1.85 ± 0.07	62.45 ± 14.78	17628	0.593	17616	case¹(regime 2)
150323C	159.4±54	13 ± 3	2.09 ± 0.32	$40.2\pm 24$	17637	...	...	case¹(regime 1)
150424A	91±22	15 ± 1	1.23 ± 0.15	272.55 ± 104.2	17761	0.3	17758	case¹(regime 2)
150607A	26.3 ± 0.7	23 ± 2	0.99 ± 0.41	84.9 ± 40.1	17907	...	...	case¹(regime 2)
150615A	27.6 ± 5.1	5.5 ± 0.7	1.84 ± 0.21	63.68 ± 27.83	17930	...	...	case¹(regime 1)
150626A	599.5 ± 99.3	18 ± 2	1.37 ± 0.25	92.5 ± 45.9	17941	...	...	case¹(regime 1)
150711A	64.2 ± 14.2	43 ± 2	1.79 ± 0.08	70.34 ± 17.49	18020	...	...	case¹(regime 2)
150910A	112.2 ± 38.3	48 ± 4	1.42 ± 0.12	162.27 ± 45.97	18268	1.359	18273	case³(regime 4)
151027A	129.69 ± 5.55	78 ± 2	1.72 ± 0.05	81.23 ± 17.55	18496	0.81	18487	case¹(regime 3)
151112A	19.32 ± 31.24	9.4 ± 1.2	1.77 ± 0.21	73.25 ± 28.15	18593	4.1	18603	case¹(regime 2)
151228B	48 ± 16	24 ± 1	1.09 ± 0.34	$78.1\pm 27$	18752	...	...	case¹(regime 3)
151229A	1.78 ± 0.44	5.9 ± 0.4	1.84 ± 0.1	63.68 ± 18.03	18751	...	...	case¹(regime 1)
160119A	116 ± 14	71 ± 2	1.68 ± 0.05	83.44 ± 19.62	18899	...	...	case¹(regime 1)
160127A	6.16 ± 0.94	3.3 ± 0.5	0.99 ± 1.11	25.6	18944	...	...	case¹(regime 3)
160227A	316.5 ± 75.4	31 ± 2	0.75 ± 0.51	65.8 ± 16.4	19106	2.38	19109	case¹(regime 2)
160327A	28 ± 9	14 ± 1	1.84 ± 0.11	63.68 ± 17.11	19240	4.99	19245	case¹(regime 2)
160412A	$31.6\pm 4$	19 ± 1	1.88 ± 0.08	58.92 ± 14.57	19301	...	...	case¹(regime 2)
160630A	29.5 ± 5.51	12 ± 1	1.32 ± 0.14	211.22 ± 76.23	19639	...	...	case¹(regime 2)
160804A	144.2 ± 19.2	114 ± 3	1.4 ± 0.23	52.7 ± 5.4	19765	0.736	19773	case¹(regime 1)
160824A	99.3 ± 7.8	26 ± 2	1.64 ± 0.11	96.47 ± 22.41	19861	...	...	case¹(regime 2)
161001A	2.6 ± 0.4	6.7 ± 0.4	1.14 ± 0.09	358.57 ± 99.67	19974	...	...	case¹(regime 1)
161011A	4.3 ± 1.3	2.3 ± 0.4	1.15 ± 0.35	347.44 ± 43.24	20032	...	...	case¹(regime 3)
161108A	105.1 ± 11.9	11 ± 1	1.85 ± 0.17	62.45 ± 22.88	20151	1.159	20150	case¹(regime 1)
161129A	35.53 ± 2.09	36 ± 1	1.57 ± 0.06	112.93 ± 24.12	20220	0.645	20245	case¹(regime 3)
161202A	181	86 ± 3	1.43 ± 0.3	${222}_{-63}^{+201}$	20238	...	...	case²
161214B	24.8 ± 3.1	21 ± 1	1.76 ± 0.09	$74.77\pm 19$	20270	...	...	case¹(regime 1)
170113A	20.66 ± 4.4	6.7 ± 0.7	0.75 ± 0.59	73.3 ± 33.6	20456	1.968	20458	case¹(regime 2)

Part V: BAT Observations and Redshift for Our Sample

170202A	46.2 ± 11.5	33 ± 1	1.68 ± 0.07	88.44 ± 20.87	20596	3.645	20584	case¹(regime 2)
170208B	128.07 ± 5.75	61 ± 2	1.18 ± 0.21	97.8 ± 27.4	20654	...	...	case¹(regime 1)
170317A	11.94 ± 4.87	13 ± 1	0.8 ± 0.49	58.8 ± 10.5	20896	...	...	case¹(regime 1)
170330A	144.73 ± 3.51	56 ± 3	1.38 ± 0.09	179.9 ± 45.88	20968	...	...	case¹(regime 2)
170607A	23	75 ± 3	1.61 ± 0.07	103.12 ± 23.51	21218	...	...	case¹(regime 1)
170711A	31.3 ± 7.6	14 ± 1	1.75 ± 0.1	76.32 ± 20.03	21331	...	...	case¹(regime 2)

Notes.

^a ${S}_{\gamma }$ ${S}_{\gamma }$ represents the fluence of gamma-rays in the 15–150 keV band. ^b ${{\rm{\Gamma }}}_{\gamma ,1}$ ${{\rm{\Gamma }}}_{\gamma ,1}$ represents the power-law index of the time-averaged spectrum. ^cThe references of BAT observations and redshift for our sample. ^dExternal shock model regimes for individual bursts (adjusted by closure relations with temporal and spectrum properties of shallow decay's follow-up phase).

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Download table as: ASCIITypeset images: 1 2 3 4

Within our sample, 97 GRBs have redshift measurements and their isotropic-equivalent radiation energies ( ${E}_{\mathrm{iso},\gamma }$ , ${E}_{\mathrm{iso},{\rm{X}}}$ ) in the prompt phase and in the shallow decay phase could be calculated as

$\begin{eqnarray}&&{E}_{\mathrm{iso},(\gamma ,{\rm{X}})}=\displaystyle \frac{4\pi {D}_{L}^{2}{S}_{(\gamma ,{\rm{X}})}}{1+z},\end{eqnarray} \tag{ 3 }$

where D_L is the luminosity distance. Here H₀ = 71 km ${{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1},{{\rm{\Omega }}}_{M}=0.3,{{\rm{\Omega }}}_{A}=0.7$ are adopted for calculating D_L. We then calculate the bolometric energy ${E}_{\mathrm{iso},(\gamma ,{\rm{X}})}^{b}$ in the 1 ∼ 10⁴ keV band with the k-correction method proposed by Bloom et al. (2001),

$\begin{eqnarray}&&{E}_{\mathrm{iso},(\gamma ,{\rm{X}})}^{b}={{kE}}_{\mathrm{iso},(\gamma ,{\rm{X}})},\end{eqnarray} \tag{ 4 }$

where

$\begin{eqnarray}&&k=\displaystyle \frac{{\int }_{1/(1+z)}^{{10}^{4}/(1+z)}{EN}(E){dE}}{{\int }_{15}^{150}{EN}(E){dE}},\end{eqnarray} \tag{ 5 }$

assuming the gamma-ray spectrum for all GRBs following Band function with photon indices −1 and −2.3 before and after E_p.

Figure 3 shows parameter relationships of the shallow decay phase to the prompt gamma-ray phase, including t_b–T₉₀, S_X– ${S}_{\gamma }$ , ${{\rm{\Gamma }}}_{{\rm{X}},1}-{{\rm{\Gamma }}}_{\gamma }$ , and ${E}_{\mathrm{iso},{\rm{X}}}$ – ${E}_{\mathrm{iso},\gamma }$ relationships. We find that with a larger sample, Γ_X,1 and Γ_γ still show no correlation, with Γ_X,1 being systemically larger than Γ_γ. The result is consistent with the relevant findings in Dainotti et al. (2015). The tentative correlations of duration, energy fluences, and isotropic energies between the gamma-ray and X-ray phases still exist, with the best fit as ${\mathrm{log}}_{10}({t}_{b})=(0.43\pm 0.08){\mathrm{log}}_{10}({{\rm{T}}}_{90})+(3.33\pm 0.15)$ (Spearman correlation coefficient r = 0.39, significance level $p\lt {10}^{-4}$ for N = 198, fraction of the variance ${R}^{2}=14.1 \%$ , and significance of Anderson–Darling test ${p}_{\mathrm{AD}}=0.647$ ), ${\mathrm{log}}_{10}({S}_{{\rm{X}}})\,=\,(0.56\,\pm \,0.06){\mathrm{log}}_{10}({S}_{\gamma })\,+\,(-3.74\,\pm \,0.37)$ (r = 0.55, $p\lt {10}^{-4}$ for N = 198, ${R}^{2}=35.3 \%$ and ${p}_{\mathrm{AD}}=0.44$ ), and ${\mathrm{log}}_{10}({E}_{\mathrm{iso},{\rm{X}}})\,=\,(0.788\,\pm \,0.05){\mathrm{log}}_{10}({E}_{\mathrm{iso},\gamma })+(9.85\pm 2.58)$ (r = 0.77, $p\lt {10}^{-4}$ for N = 198, ${R}^{2}=72.4 \%$ and ${p}_{\mathrm{AD}}\,=0.566$ ). With the exception of ${E}_{\mathrm{iso},{\rm{X}}}$ – ${E}_{\mathrm{iso},\gamma }$ , the correlations of t_b–T₉₀ and ${E}_{\mathrm{iso},{\rm{X}}}$ – ${E}_{\mathrm{iso},\gamma }$ become weaker with a larger sample, and the correlation coefficients were reduced by approximately 18% (for duration) and 21% (for energy fluence). The fitting results and Cook distance for the fitting are shown in Figure 3. Here we also apply the bivariate linear regression procedure to fit the data (see Kelly 2007 for details). We find that the best-fit results for different regression methods are consistent. The intrinsic scatter to the population ${\sigma }_{\mathrm{int}}$ is shown in Figure 3. In order to compare with L07, we also check the possible linear correlations for the quantities in the two phases by defining 2σ linear correlation regions⁸ (see Figure 3 for details), and we find that most of our GRBs fall in this region, suggesting that the radiation during the shallow decay phase might indeed be correlated with that in the prompt gamma-ray phase.

**Figure 3.** Parameter relationship between the shallow decay phase and the prompt emission phase, including t_b– ${T}_{90}$ , S_X– ${S}_{\gamma }$ , ${{\rm{\Gamma }}}_{{\rm{X}},1}-{{\rm{\Gamma }}}_{\gamma }$ , and ${E}_{\mathrm{iso},{\rm{X}}}$ – ${E}_{\mathrm{iso},\gamma }$ . The black solid line represents the best fit with the least square regression method. The red solid line presents the best fit with the bivariate linear regression method and the pink shadowed region shows the intrinsic scatter to the population $3{\sigma }_{\mathrm{int}}$ . The gray lines mark the best fitting results (solid) and its 2σ linear correlation region (dotted), which is defined with $y=x+A\pm 2{\delta }_{A}$ , where y and x are the two quantities in question and A and ${\sigma }_{A}$ are the mean and its 1σ standard error for the y–x correlation. The orange solid lines show y = x.
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3.3. Test Physical Origin of the Shallow Decay Segment

We first check whether there is any spectral evolution between shallow decay segment and its follow-up segment. The left panel of Figure 4 plots Γ_X,2 as a function of Γ_X,1. We find that all GRBs in our sample fall in the 2σ linear correlation regions. The middle of Figure 4 shows the comparison between the distributions of Γ_X,1 and Γ_X,2. For L07's sample, the Kolmogorov–Smirnov test suggests that these two distributions are consistent with the significance level of 0.96. With the larger sample, we find that the two distributions are still consistent, but the significance level of this consistency decreases to 0.40. As suggested by L07, here we define a new parameter as

$\begin{eqnarray}&&\mu =\displaystyle \frac{{{\rm{\Gamma }}}_{{\rm{X}},2}-{{\rm{\Gamma }}}_{{\rm{X}},1}}{\sqrt{\delta {{\rm{\Gamma }}}_{{\rm{X}},1}^{2}+\delta {{\rm{\Gamma }}}_{{\rm{X}},2}^{2}}},\end{eqnarray} \tag{ 6 }$

to verify this consistency within the observational uncertainty for individual bursts. The right panel of Figure 4 shows μ distribution. We find that 84% of GRBs in our sample have $\mu \leqslant 1$ , and only 2 GRBs (GRB 080903 and GRB 150201A) show significant spectral evolution ( $\mu \gt 3$ ). Considering all of this evidence, we confirm L07's conclusion that there is no significant spectral evolution between the shallow decay segment and its follow-up segment, indicating that the shallow decay phase is a refreshed forward shock (Dai & Lu 1998; Rees & Mészáros 1998; Zhang & Mészáros 2001; Nousek et al. 2006; Zhang et al. 2006).

**Figure 4.** Left panel: comparison between Γ_X,1 and Γ_X,2. The orange lines represent ${{\rm{\Gamma }}}_{{\rm{X}},1}={{\rm{\Gamma }}}_{{\rm{X}},2}$ (solid) and its 2σ region (dotted). Middle panel: the distribution of Γ_X,1 and Γ_X,2, with solid lines showing our results and dotted lines showing L07's results. Right panel: the distribution of μ.
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In this scenario, the shallow decay's follow-up segment should be consistent with the predictions of the forward shock models, namely the observed spectral index ${\beta }_{{\rm{X}},2}={{\rm{\Gamma }}}_{{\rm{X}},2}-1$ and temporal decay index ${\alpha }_{2}$ should follow the so-called "closure relations" (Gao et al. 2013, for a review). Although the closure correlations vary for different spectral regimes, different cooling schemes or different properties of the ambient medium, as illustrated in L07, three regimes are relevant here:

1.
regime 1: ${\nu }_{x}\gt \max ({\nu }_{m},{\nu }_{c})$ (for either ISM or wind medium), where ${\alpha }_{2}=(3{\beta }_{2}-1)/2;$
2.
regime 2: ${\nu }_{m}\lt {\nu }_{x}\lt {\nu }_{c}$ (for ISM medium), where ${\alpha }_{2}=3\beta /2;$
3.
regime 3: ${\nu }_{m}\lt {\nu }_{x}\lt {\nu }_{c}$ (for wind medium), where ${\alpha }_{2}=(3{\beta }_{2}+1)/2$ .

We define a new parameter ${\phi }_{i}\,(i=1,2,3)$ to test how well individual bursts being consistent with the closure relations under regime $(i=1,2,3)$ :

$\begin{eqnarray}&&{\phi }_{i}=\displaystyle \frac{| {\alpha }^{\mathrm{obs}}-{\alpha }_{i}({\beta }^{\mathrm{obs}})| }{\sqrt{{\left(\delta {\alpha }^{\mathrm{obs}}\right)}^{2}+{\left[\delta {\alpha }_{i}\left({\beta }^{\mathrm{obs}}\right)\right]}^{2}}},\end{eqnarray} \tag{ 7 }$

where ${\alpha }^{\mathrm{obs}}$ and ${\alpha }_{i}({\beta }^{\mathrm{obs}})$ are the temporal decay slope from the observations and from the closure relations, and $\delta {\alpha }^{\mathrm{obs}}$ and $\delta ({\alpha }_{i}({\beta }^{\mathrm{obs}}))$ represent their uncertainties. For an individual burst, if there exists ${\phi }_{i}\lt 3$ , we conclude that this burst belongs to the regime i model within 3σ significance. For those bursts that have more than one regime satisfying ${\phi }_{i}\lt 3$ , we choose the regime with the smallest ϕ value.

Overall, we find that 66/198 bursts belong to regime 1, 82/198 bursts belong to regime 2, and 45/198 bursts belong to regime 3. For the other 5 bursts, we test their consistency with the curvature effect curve $\alpha =\beta +2$ (Kumar & Panaitescu 2000; Panaitescu et al. 2006) with the ϕ parameter. We find that three bursts cannot be excluded within 3σ significance by the curvature effect regime. These bursts could still be interpreted with the external-shock model once the jet break effect is invoked. We name regime 4 for these bursts. The other two bursts that could be excluded within 3σ significance by the curvature effect regime are difficult to interpret under an external-shock framework. As suggested by many previous works, the early X-ray plateau for these cases should be of internal origin and is directly connected to a long-lasting central engine (Troja et al. 2007; Rowlinson et al. 2010, 2013, 2014; Dall'Osso et al. 2011; Gompertz et al. 2013, 2014, 2015; Lü & Zhang 2014; Lü et al. 2015; Rea et al. 2015; Gibson et al. 2017, 2018; Gompertz & Fruchter 2017; Li et al. 2018; Stratta et al. 2018). We did not assign these bursts into either regime. Above all, we divided the GRBs of our sample into four groups, case¹, case², case³, and ${\mathrm{case}}^{4}$ . Which case of each GRB in our sample belongs to and which regime is consistent with are list in Table 2. The relationships of α_X,2 and ${\beta }_{{\rm{X}},2}$ in difference cases are shown in Figure 5.

**Figure 5.** Temporal decay index α_X,2 as a function of the spectral index ${\beta }_{{\rm{X}},2}$ for shallow decay's follow-up segment, compared with the closure correlations of various spectral regimes. The gray dots show GRBs belonging to regimes 1–3 (case¹). Pink dots show GRBs whose shallow decay segment and follow-up segment belongs to different spectrum regimes (case²). Black dots show GRBs that do not belong to regime 1–3 but that cannot be excluded within 3σ significance by the curvature effect regime (case³). Green dots show GRBs that could be excluded within 3σ significance by the curvature effect regime (case⁴).
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**Figure 5.** Temporal decay index α_X,2 as a function of the spectral index ${\beta }_{{\rm{X}},2}$ for shallow decay's follow-up segment, compared with the closure correlations of various spectral regimes. The gray dots show GRBs belonging to regimes 1–3 (case¹). Pink dots show GRBs whose shallow decay segment and follow-up segment belongs to different spectrum regimes (case²). Black dots show GRBs that do not belong to regime 1–3 but that cannot be excluded within 3σ significance by the curvature effect regime (case³). Green dots show GRBs that could be excluded within 3σ significance by the curvature effect regime (case⁴).
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The normal decay phases for 196/198 bursts in our sample are consistent with the external-shock models, inferring that their shallow decay phase might also be of external origin and may be related to continuous energy injection from the central engine. Here we assume that the central engine has a power-law luminosity release history as $L\left(t\right)={L}_{0}{\left(\tfrac{t}{{t}_{0}}\right)}^{-q}$ , so that the injected energy would be ${E}_{\mathrm{inj}}=\tfrac{{L}_{0}{t}_{0}^{q}}{1-q}{t}^{1-q}$ (Zhang & Mészáros 2001). When ${E}_{\mathrm{inj}}$ is larger than the impulsively injected energy during the prompt emission phase, the dynamics of the external shock wave would be related to q as ${\rm{\Gamma }}\propto {t}^{-\tfrac{q+2}{8}}$ ( ${\rm{\Gamma }}\propto {t}^{-\tfrac{q}{4}}$ ) and $R\propto {t}^{\tfrac{2-q}{4}}$ ( $R\propto {t}^{\tfrac{2-q}{2}}$ ) for the ISM (wind) case. Consequently, we have ${\nu }_{m}\propto {t}^{-1-q/2}$ ( ${t}^{-1-q/2}$ ), ${\nu }_{c}\propto {t}^{-1+q/2}$ ( ${t}^{1-q/2}$ ), and ${F}_{\nu ,\max }\propto {t}^{1-q}$ ( ${t}^{-q/2}$ ) for the ISM (wind) models for p > 2, and ${\nu }_{m}\propto {t}^{-\tfrac{(q+2)(p+2)}{8(p-1)}}$ $({t}^{\tfrac{4+{pq}}{4(1-q)}})$ , ${\nu }_{c}\propto {t}^{-1+q/2}$ ( ${t}^{1-q/2}$ ), and ${F}_{\nu ,\max }\propto {t}^{1-q}$ ( ${t}^{-q/2}$ ) for the ISM (wind) models for $1\lt p\lt 2$ , where p is the electronic power-law index, and ${F}_{\nu ,\max }$ is the observed peak flux. The relationship between decay slope α and energy injection index q for different spectral regimes could thus be derived (see the results in Tables 13–16 of Gao et al. 2013).

In this case, the decay indices difference between the shallow decay segment and its follow-up segment in different regimes would be (Gao et al. 2013).

$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}\alpha & = & \displaystyle \frac{1}{16}(p+14)(1-q),\mathrm{regime}\ 1,1\lt p\lt 2\\ {\rm{\Delta }}\alpha & = & \displaystyle \frac{1}{16}(p+18)(1-q),\mathrm{regime}\ 2,1\lt p\lt 2\\ {\rm{\Delta }}\alpha & = & \displaystyle \frac{1}{8}(p+4)(1-q),\mathrm{regime}\ 3,1\lt p\lt 2\\ {\rm{\Delta }}\alpha & = & \displaystyle \frac{1}{4}(p+2)(1-q),\mathrm{regime}\ 1,p\gt 2\\ {\rm{\Delta }}\alpha & = & \displaystyle \frac{1}{4}(p+3)(1-q),\mathrm{regime}\ 2,p\gt 2\\ {\rm{\Delta }}\alpha & = & \displaystyle \frac{1}{4}(p+1)(1-q),\mathrm{regime}\ 3,p\gt 2,\end{array}\end{eqnarray} \tag{ 8 }$

where p could be derived from the observed spectral index, depending on the observed spectral regime. Given the regime models for each burst (here we only test bursts in regimes 1–3), its energy injection parameter q could be thus derived based on α_X,1 and α_X,2. The upper panels of Figure 6 show the distributions of these GRBs in the two-dimensional q–Δα and q–p planes along with the contours of constant p and Δα. The distribution of derived p and q values are plotted in the lower panels of Figure 6. We find that q values are mainly distributed between −0.5 and 0.5, with an average value of 0.16 ± 0.12, which is consistent with the model prediction (q value around zero) where a spinning-down pulsar acts as the energy injection central engine (Dai & Lu 1998; Zhang & Mészáros 2001). With the derived q value, we can further test whether the α_X,1 and ${\beta }_{{\rm{X}},1}$ satisfy the corresponding closure relations with energy injection (see Table 14 and 16 in Gao et al. 2013) with the ϕ parameter. It turns out the shallow decay phases for all 194 GRBs satisfy the closure relations with energy injection, and their corresponding spectral regime for 189/196 (96.4%) bursts are consistent with their follow-up phases, which is expected because there is no systematic spectral evolution between these two segments.⁹ The results reinforce the conclusion that the shallow decay segment in most bursts is consistent with an external forward shock origin, probably due to a continuous energy injection from a long-lived central engine.

**Figure 6.** Upper panels: distributions of bursts in regimes 1–3 in the two-dimensional q–Δα and q–p planes, along with the model prediction for $\upsilon \gt \max ({\upsilon }_{m},{\upsilon }_{c})$ (thick solid lines) and ${\upsilon }_{m}\lt \upsilon \lt {\upsilon }_{c}$ (thin solid lines). Lower panels: distributions of p values and q values.
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**Figure 6.** Upper panels: distributions of bursts in regimes 1–3 in the two-dimensional q–Δα and q–p planes, along with the model prediction for $\upsilon \gt \max ({\upsilon }_{m},{\upsilon }_{c})$ (thick solid lines) and ${\upsilon }_{m}\lt \upsilon \lt {\upsilon }_{c}$ (thin solid lines). Lower panels: distributions of p values and q values.
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3.4. Empirical Relations Revisited

3.4.1. Empirical Relation among ${E}_{\mathrm{iso}},{E}_{p}^{{\prime} }$ , and $t{{\prime} }_{b}$

Based on their sample, L07 have investigated the relations among the energies ${E}_{\mathrm{iso},{\rm{X}}}$ and ${E}_{\mathrm{iso},\gamma }$ and other parameters ${E}_{p}^{{\prime} }$ and ${t}_{b}^{{\prime} }$ , with a regression model as

$\begin{eqnarray}&&{\mathrm{log}}_{10}({E}_{\mathrm{iso},({\rm{X}},\gamma )})={\kappa }_{0}+{\kappa }_{1}{\mathrm{log}}_{10}({E}_{p}^{{\prime} })+{\kappa }_{2}{\mathrm{log}}_{10}({{\rm{t}}}_{b}^{{\prime} }),\end{eqnarray} \tag{ 9 }$

where ${\kappa }_{0}$ , ${\kappa }_{1}$ , and ${\kappa }_{2}$ are multiple regression coefficients. The probability from a t-test (p_t) was invoked to measure the significance of the dependence of each variable on the model and the probability from an F-test (p_F) was invoked to justify the global regression of the three-parameter relation. For the ${E}_{\mathrm{iso},{\rm{X}}}-{E}_{p}^{{\prime} }-{t}_{b}^{{\prime} }$ relation, L07 yields ${\kappa }_{0}=44.0\pm 1.1$ ( ${p}_{t}\lt {10}^{-4}$ ) and ${\kappa }_{1}=1.82\pm 0.33\,({p}_{t}\lt {10}^{-4})$ , ${\kappa }_{2}=0.61\,\pm 0.18\,({p}_{t}\lt 3\times {10}^{-3})$ , and the p_F is less than 10⁻⁴. Their results suggest a strong correlation between ${E}_{\mathrm{iso},{\rm{X}}}$ and ${E}_{p}^{{\prime} }$ but a tentative correlation between ${E}_{\mathrm{iso},{\rm{X}}}$ and ${t}_{b}^{{\prime} }$ , and thus a tentative global three-parameter correlation. We apply the regression model to our sample, we yield ${\kappa }_{0}=49.2\pm 0.77$ ( ${p}_{t}\lt {10}^{-4}$ ), ${\kappa }_{1}=0.77\pm 0.22\,({p}_{t}=0.001)$ , and ${\kappa }_{2}=-0.051\pm 0.12({p}_{t}=0.683)$ , with ${p}_{F}=0.002$ , ${R}^{2}=12.5 \%$ , and ${p}_{\mathrm{AD}}\,=0.555$ . We find that with a larger sample, a strong correlation between ${E}_{\mathrm{iso},{\rm{X}}}$ and ${E}_{p}^{{\prime} }$ still exists but the tentative correlation between ${E}_{\mathrm{iso},{\rm{X}}}$ and ${t}_{b}^{{\prime} }$ has completely disappeared, so that the global three-parameter correlation becomes less significant. Note that in Dainotti et al. (2011b) it was already clear that ${E}_{\mathrm{iso},{\rm{X}}}-{t}_{b}^{{\prime} }$ was a weak correlation, while ${L}_{\mathrm{peak}}-{t}_{b}^{{\prime} }$ was a stronger correlation. Dainotti et al. (2016) thus proposed the ${L}_{{\rm{X}}}-{L}_{\mathrm{peak}}-{t}_{b}^{{\prime} }$ correlation given the already established two-parameter ${L}_{\mathrm{peak}}-{t}_{b}^{{\prime} }$ (e.g., Dainotti et al. 2008) and ${L}_{\mathrm{peak}}-{L}_{{\rm{X}}}$ correlations (Dainotti et al. 2011b, 2015). With an extended sample (updated to 2016 July) of long GRBs with X-ray plateau, Dainotti et al. (2017) yielded ${\mathrm{log}}_{10}{L}_{X}=(17.67\,\pm 5.7)-(0.83\pm 0.10){\mathrm{log}}_{10}{t}_{b}^{{\prime} }+(0.64\pm 0.11){\mathrm{log}}_{10}{L}_{\mathrm{peak}}$ , with the Pearson correlation coefficient r = 0.90 with a probability of the same sample occurring by chance, $P=1.75\times {10}^{-17}$ .

For the ${E}_{\mathrm{iso},\gamma }-{E}_{p}^{{\prime} }-{t}_{b}^{{\prime} }$ relation, L07 yields ${\kappa }_{0}\,=48.3\pm 0.8$ ( ${p}_{t}\lt {10}^{-4}$ ) and ${\kappa }_{1}=1.7\pm 0.25\,({p}_{t}\lt {10}^{-4})$ , ${\kappa }_{2}=0.07\pm 0.13\,({p}_{t}=0.486)$ , and the p_F is less than 10⁻⁴. Their results suggest a strong correlation between ${E}_{\mathrm{iso},\gamma }$ and ${E}_{p}^{{\prime} }$ but no correlation between ${E}_{\mathrm{iso},\gamma }$ and ${t}_{b}^{{\prime} }$ . A tentative global three-parameter correlation might exist. With a larger sample, we yield ${\kappa }_{0}=51\pm 0.76$ ( ${p}_{t}\lt {10}^{-4}$ ), ${\kappa }_{1}=0.883\,\pm 0.22\,({p}_{t}\lt {10}^{-4})$ , and ${\kappa }_{2}=-0.19\pm 0.12\,({p}_{t}=0.11)$ , with ${p}_{F}\lt {10}^{-4}$ , ${R}^{2}=19.5 \%$ , and ${p}_{\mathrm{AD}}=0.403$ . We confirm that a strong correlation between ${E}_{\mathrm{iso},\gamma }$ and ${E}_{p}^{{\prime} }$ and a tentative global three-parameter correlation exist and there is no correlation between ${E}_{\mathrm{iso},\gamma }$ and ${t}_{b}^{{\prime} }$ . The data and regression modeling results are shown in Figure 7. It is worth noticing that Dainotti et al. (2011b) also investigated the ${E}_{\mathrm{iso},\gamma }^{b}$ – ${t}_{b}^{{\prime} }$ relation with 62 long GRBs with known redshift, and the correlation coefficient and the random occurrence probability p for their sample are $r=-0.19$ and p = 0.1, also inferring that the correlation between ${E}_{\mathrm{iso},\gamma }$ and ${t}_{b}^{{\prime} }$ was very weak.

**Figure 7.** Left panel: the correlation between ${{\rm{\Sigma }}}_{{\rm{X}}}={\mathrm{log}}_{10}({E}_{\mathrm{iso},{\rm{X}}})-{\kappa }_{1}{\mathrm{log}}_{10}({t}_{b}^{{\prime} })$ and ${E}_{p}^{{\prime} }$ . Right panel: the correlation between ${{\rm{\Sigma }}}_{\gamma }={\mathrm{log}}_{10}({E}_{\mathrm{iso},\gamma }^{b})-{\kappa }_{2}{\mathrm{log}}_{10}({t}_{b}^{{\prime} })$ and ${E}_{p}^{{\prime} }$ . The black lines represent the best fitting results (solid) and its 3σ uncertainty region (dotted) with least square regression method. The red line presents the best fitting results with the bivariate linear regression method and the pink shadowed region shows the intrinsic scatter to the population $3{\sigma }_{\mathrm{int}}$ .
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**Figure 7.** Left panel: the correlation between ${{\rm{\Sigma }}}_{{\rm{X}}}={\mathrm{log}}_{10}({E}_{\mathrm{iso},{\rm{X}}})-{\kappa }_{1}{\mathrm{log}}_{10}({t}_{b}^{{\prime} })$ and ${E}_{p}^{{\prime} }$ . Right panel: the correlation between ${{\rm{\Sigma }}}_{\gamma }={\mathrm{log}}_{10}({E}_{\mathrm{iso},\gamma }^{b})-{\kappa }_{2}{\mathrm{log}}_{10}({t}_{b}^{{\prime} })$ and ${E}_{p}^{{\prime} }$ . The black lines represent the best fitting results (solid) and its 3σ uncertainty region (dotted) with least square regression method. The red line presents the best fitting results with the bivariate linear regression method and the pink shadowed region shows the intrinsic scatter to the population $3{\sigma }_{\mathrm{int}}$ .
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3.4.2. Revisit Correlations between ${t}_{b}^{{\prime} }$ (t_b), L_X, and ${E}_{\mathrm{iso}\gamma }$

Dainotti et al. (2008) discovered a formal anticorrelation between X-ray luminosity at the end of plateau L_X and rest frame plateau end time ${t}_{b}^{{\prime} }$ , described as ${\mathrm{log}}_{10}({L}_{{\rm{X}}})\,={\kappa }_{0}+{\kappa }_{1}{\mathrm{log}}_{10}({{\rm{t}}}_{{\rm{b}}}^{{\prime} })$ . Dainotti et al. (2010) fitted the correlation between L_X and ${t}_{b}^{{\prime} }$ and achieved the results ${\kappa }_{0}=51.06\pm 1.02$ , ${\kappa }_{1}=-{1.06}_{-0.28}^{+0.27}$ . It is of interest to justify this empirical relationship with our updated sample. With the multivariate linear regression method, we find that the anticorrelation indeed exists between L_X and ${t}_{b}^{{\prime} }$ , with ${\kappa }_{0}=50.7\pm 0.39$ ( ${p}_{t}\lt {10}^{-4}$ ) and ${\kappa }_{1}=-1.15\pm 0.1({p}_{t}\lt {10}^{-4})$ , where ${p}_{F}\lt {10}^{-4}$ , ${R}^{2}\,=54.9 \%$ , and ${p}_{\mathrm{AD}}=0.262$ . In addition, the L_X for individual GRBs in our sample are listed in Table 3. Basically speaking, our results in general agree well with Dainotti et al. (2013a), indicating that an intrinsic relationship between L_X and $({t}_{b}^{{\prime} })$ indeed exist. This correlation has been tested against selection bias robustly (Dainotti et al. 2013a). It is worth noticing that L_X is roughly inversely proportional to the timescale of the energy injection, inferring that the energy reservoir should be roughly a constant. This is consistent with the energy injection model, where the central engine is a newborn magnetar (Dai 2004).

Table 3. Part I: Rest-frame Properties for Bursts with Known Redshifts

GRB Name	${\mathrm{log}}_{10}({T}_{90}^{{\prime} })$	${\mathrm{log}}_{10}({E}_{p}^{{\prime} })$	${\mathrm{log}}_{10}({E}_{\mathrm{iso},\gamma })$	${\mathrm{log}}_{10}({E}_{\mathrm{iso},\gamma }^{b})$	${\mathrm{log}}_{10}({E}_{\mathrm{iso},{\rm{X}}})$	${\mathrm{log}}_{10}({t}_{b}^{{\prime} })$	${\mathrm{log}}_{10}(({L}_{{\rm{X}}})$
	(s)	(keV)	(erg)	(erg)	(erg)	(s)	(erg s⁻¹)
050319	0.37 ± 0.08	2.15 ± 0.18	52.23 ± 0.04	52.58 ± 0.08	51.82 ± 0.06	3.89 ± 0.06	47.05 ± 0.05
050401	0.93 ± 0.03	2.72 ± 0.1	53.39 ± 0.04	53.77 ± 0.04	52.06 ± 0.05	3.08 ± 0.04	48.13 ± 0.03
050416A	0.16 ± 0.03	1.31 ± 0.17	50.47 ± 0.03	50.91 ± 0.06	49.52 ± 0.17	3 ± 0.14	46.21 ± 0.11
050505	1.06 ± 0.01	2.85 ± 0.13	53.13 ± 0.04	53.55 ± 0.05	51.88 ± 0.1	3.59 ± 0.07	47.43 ± 0.08
050714B	1.13	1.8 ± 0.22	51.93 ± 0.09	52.3 ± 0.09	50.13 ± 0.18	4.03 ± 0.19	45.53 ± 0.13
050802	0.68 ± 0.06	2.46 ± 0.13	52.29 ± 0.04	52.65 ± 0.06	51.18 ± 0.18	3.29 ± 0.04	47.38 ± 0.06
050814	1.01 ± 0.12	2.64 ± 0.17	52.99 ± 0.07	53.33 ± 0.11	51.54 ± 0.08	4.15 ± 0.06	46.35 ± 0.08
050822	1.62 ± 0.01	1.71 ± 0.14	52.24 ± 0.04	52.59 ± 0.04	50.7 ± 0.09	3.89 ± 0.08	46.34 ± 0.07
050824	1.14 ± 0.08	1.46 ± 0.27	50.6 ± 0.09	51 ± 0.11	50.07 ± 0.17	4.56 ± 0.16	45.2 ± 0.1
050826	1.43 ± 0.09	2.59 ± 0.38	49.95 ± 0.07	50.55 ± 0.5	48.39 ± 0.18	4.51 ± 0.15	43.83 ± 0.15
051109B	1.14 ± 0.03	1.74 ± 0.23	48.58 ± 0.06	48.92 ± 0.09	47.25 ± 0.12	3.46 ± 0.1	43.72 ± 0.09
051221A	−0.04	2.79 ± 0.14	51.38 ± 0.01	52.09 ± 0.07	49.69 ± 0.09	4.45 ± 0.08	44.78 ± 0.08
060108	0.68 ± 0.03	2.15 ± 0.17	51.55 ± 0.05	51.89 ± 0.05	50.53 ± 0.1	3.63 ± 0.1	46.34 ± 0.08
060115	1.5 ± 0.02	2.53 ± 0.14	52.67 ± 0.02	53 ± 0.11	51.07 ± 0.19	3.84 ± 0.18	46.27 ± 0.16
060306	1.38 ± 0.01	2.2 ± 0.13	52.11 ± 0.02	52.46 ± 0.08	50.54 ± 0.17	3.11 ± 0.16	46.97 ± 0.14
060502A	1.12 ± 0.06	2.6 ± 0.12	52.09 ± 0.02	52.53 ± 0.05	51.08 ± 0.11	4.17 ± 0.11	46.2 ± 0.1
060604	0.5 ± 0.11	2.25 ± 0.34	51.13 ± 0.09	51.46 ± 0.18	50.72 ± 0.11	3.83 ± 0.08	46.37 ± 0.08
060605	0.5 ± 0.05	2.98 ± 0.19	52.1 ± 0.04	52.57 ± 0.1	51.52 ± 0.03	3.25 ± 0.03	47.41 ± 0.03
060607A	1.39 ± 0.02	2.79 ± 0.11	52.71 ± 0.02	53.16 ± 0.05	52.37 ± 0.03	3.5 ± 0.01	48.08 ± 0.02
060614	1.95	2.53 ± 0.26	51.16 ± 0.02	51.71 ± 0.13	49.03 ± 0.03	4.64 ± 0.02	44.34 ± 0.03
060707	1.19 ± 0.03	2.47 ± 0.62	52.6 ± 0.05	52.95 ± 0.09	51.64 ± 0.16	4.45 ± 0.16	46.18 ± 0.15
060714	1.49 ± 0.02	2.25 ± 0.13	52.68 ± 0.03	52.99 ± 0.01	51.13 ± 0.11	3.1 ± 0.1	47.35 ± 0.11
060719	1.34 ± 0.04	2.08 ± 0.13	51.97 ± 0.03	52.28 ± 0.01	50.49 ± 0.12	3.47 ± 0.1	46.47 ± 0.1
060729	1.88 ± 0.04	1.98 ± 0.15	51.3 ± 0.03	51.63 ± 0.07	50.9 ± 0.02	4.61 ± 0.01	46.04 ± 0.01
060805A	0.2 ± 0.04	2.04 ± 0.31	50.98 ± 0.11	51.33 ± 0.22	50.06 ± 0.16	3.03 ± 0.14	46.45 ± 0.15
060814	1.66 ± 0.01	2.88 ± 0.49	53.37 ± 0.02	53.89 ± 0.1	51.23 ± 0.07	3.64 ± 0.05	47.02 ± 0.06
060906	0.97 ± 0.01	2.33 ± 0.14	52.76 ± 0.03	53.09 ± 0.01	51.01 ± 0.05	3.38 ± 0.05	46.89 ± 0.05
060908	0.83 ± 0.01	2.77 ± 0.11	52.39 ± 0.01	52.84 ± 0.05	50.85 ± 0.11	2.38 ± 0.08	47.85 ± 0.09
060927	0.54 ± 0.01	2.67 ± 0.62	52.72 ± 0.04	53.06 ± 0.03	51.32 ± 0.14	2.98 ± 0.12	47.1 ± 0.15
061121	1.54 ± 0.03	3.15 ± 0.14	52.77 ± 0.01	53.58 ± 0.04	51.48 ± 0.05	3.34 ± 0.04	47.65 ± 0.05
061201	−0.14 ± 0.05	2.99 ± 0.2	48.96 ± 0.04	49.86 ± 0.12	48.58 ± 0.08	3.4 ± 0.06	44.9 ± 0.09
061222A	1.37 ± 0.02	3 ± 0.21	52.92 ± 0.01	53.57 ± 0.05	52.1 ± 0.03	4.19 ± 0.03	46.96 ± 0.03
070103	0.72 ± 0.02	2.23 ± 0.19	51.70 ± 0.06	52.03 ± 0.06	50.49 ± 0.08	2.6 ± 0.06	47.44 ± 0.05
070110	1.4 ± 0.02	2.58 ± 0.15	52.3 ± 0.03	52.66 ± 0.07	51.48 ± 0.02	3.79 ± 0.01	47.22 ± 0.02
070129	2.14 ± 0.02	2.16 ± 0.16	52.58 ± 0.04	52.90 ± 0.15	50.88 ± 0.08	3.73 ± 0.06	46.61 ± 0.05
070306	0.48 ± 0.09	2.33 ± 0.13	50.92 ± 0.25	51.26 ± 0.09	51.46 ± 0.03	4.08 ± 0.02	46.94 ± 0.02
070508	1.06 ± 0.02	2.67 ± 0.33	52.52 ± 0.01	53.07 ± 0.11	50.79 ± 0.33	2.2 ± 0.24	48.35 ± 0.16
070529	1.49 ± 0.01	2.8 ± 0.2	52.55 ± 0.04	53 ± 0.11	50.75 ± 0.15	2.48 ± 0.11	47.69 ± 0.12
070809	0.03 ± 0.03	2.02 ± 0.22	49.05 ± 0.04	49.39 ± 0.25	48.24 ± 0.15	3.69 ± 0.13	44.53 ± 0.11
070810A	0.54 ± 0.04	2.14 ± 0.15	51.87 ± 0.04	52.21 ± 0.04	50.61 ± 0.21	2.64 ± 0.17	47.38 ± 0.19
080413B	0.58 ± 0.05	2.19 ± 0.18	52 ± 0.01	52.34 ± 0.02	50.34 ± 0.14	2.28 ± 0.1	47.83 ± 0.08
080430	0.96 ± 0.06	2.15 ± 0.12	51.25 ± 0.04	51.6 ± 0.07	50.42 ± 0.07	4.28 ± 0.06	45.68 ± 0.05
080516	0.14 ± 0.02	2.44 ± 0.25	51.73 ± 0.06	52.07 ± 0.14	50.79 ± 0.17	3.12 ± 0.16	47 ± 0.14
080707	1.08 ± 0.02	2.21 ± 0.2	51.3 ± 0.05	52.61 ± 0.12	50.1 ± 0.16	3.68 ± 0.14	45.97 ± 0.12
080905B	1.58 ± 0.05	2.38 ± 0.16	52.35 ± 0.05	52.7 ± 0.12	51.88 ± 0.07	3.64 ± 0.07	47.5 ± 0.05
081007	0.82 ± 0.16	1.5 ± 0.17	50.7 ± 0.05	51.06 ± 0.07	50.1 ± 0.3	4.44 ± 0.12	45.09 ± 0.12
081029	1.75 ± 0.07	2.88 ± 0.21	52.77 ± 0.04	53.22 ± 0.15	51.46 ± 0.03	3.53 ± 0.02	47.2 ± 0.03
090102	1.03 ± 0.03	2.68 ± 0.12	50.6 ± 0.02	51.06 ± 0.05	51.46 ± 0.06	2.75 ± 0.03	48.16 ± 0.04

Part II: Rest-frame Properties for Bursts with Known Redshifts

090113	0.52 ± 0.04	2.46 ± 0.13	51.75 ± 0.02	52.12 ± 0.05	50.51 ± 0.1	2.31 ± 0.08	47.81 ± 0.07
090205	0.19 ± 0.08	2.31 ± 0.2	51.85 ± 0.06	52.18 ± 0.1	51.19 ± 0.11	3.43 ± 0.1	46.77 ± 0.11
090313	1.25 ± 0.09	2.39 ± 0.26	52.5 ± 0.06	52.80 ± 0.1	51.46 ± 0.1	4.25 ± 0.06	46.64 ± 0.09
090404	1.32 ± 0.07	2.04 ± 0.13	52.75 ± 0.01	53.08 ± 0.05	51.36 ± 0.1	3.63 ± 0.08	47.06 ± 0.07
090407	2.1 ± 0.09	2.29 ± 0.27	51.75 ± 0.07	52.12 ± 0.27	50.93 ± 0.05	4.47 ± 0.05	45.89 ± 0.04
090418A	1.33 ± 0.04	2.56 ± 0.11	52.46 ± 0.02	52.87 ± 0.04	51.31 ± 0.08	3.07 ± 0.06	47.64 ± 0.08
090424	1.53	2.44 ± 0.01	52.35 ± 0.25	52.8 ± 0.01	50.78 ± 0.18	2.79 ± 0.17	47.73 ± 0.16
090429B	−0.28 ± 0.07	2.64 ± 0.56	52.51 ± 0.04	52.83 ± 0.01	50.98 ± 0.1	1.88 ± 0.08	48.34 ± 0.04
090510	−0.8 ± 0.12	3.07 ± 0.3	50.84 ± 0.05	51.46 ± 0.13	50.47 ± 0.07	2.91 ± 0.05	47.03 ± 0.06
090529	1.44	2.23 ± 0.25	52 ± 0.1	52.34 ± 0.11	50.81 ± 0.19	2.91 ± 0.05	47.03 ± 0.18
090530	1.33 ± 0.24	2.37 ± 0.19	51.64 ± 0.04	52 ± 0.26	50.7 ± 0.11	4.42 ± 0.1	45.47 ± 0.09
091018	0.35 ± 0.06	1.58 ± 0.57	51.52 ± 0.03	51.91 ± 0.38	50.56 ± 0.08	2.49 ± 0.07	47.65 ± 0.06
091029	1.02 ± 0.05	2.36 ± 0.41	52.59 ± 0.02	52.93 ± 0.05	51.17 ± 0.06	3.5 ± 0.05	47.06 ± 0.04
100219A	0.52 ± 0.1	3.05 ± 0.29	52.14 ± 0.07	52.61 ± 0.31	51.83 ± 0.07	3.77 ± 0.03	47.05 ± 0.06
100302A	0.49 ± 0.04	2.67 ± 0.2	52.09 ± 0.05	52.4 ± 0.15	50.94 ± 0.22	3.58 ± 0.22	46.47 ± 0.16
100418A	0.63 ± 0.06	1.76 ± 0.21	50.52 ± 0.6	50.83 ± 0.13	49.79 ± 0.22	4.7 ± 0.06	44.95 ± 0.05
100425A	1.13 ± 0.03	1.81 ± 0.24	51.54 ± 0.08	51.91 ± 0.21	50.41 ± 0.18	4.07 ± 0.18	45.63 ± 0.14
100901A	2.26 ± 0.03	2.41 ± 0.22	52 ± 0.06	52.35 ± 0.32	51.24 ± 0.02	4.2 ± 0.02	46.72 ± 0.01
100906A	1.62 ± 0.01	2.29 ± 0.1	53.12 ± 0.25	53.46 ± 0.04	51.46 ± 0.04	3.66 ± 0.03	47.02 ± 0.05
110106B	1.19 ± 0.08	2.08 ± 0.14	51.28 ± 0.02	51.62 ± 0.1	50.14 ± 0.11	3.91 ± 0.1	45.76 ± 0.1
110808A	1.31 ± 0.17	1.81 ± 0.3	51.17 ± 0.09	51.52 ± 0.24	50.43 ± 0.21	4.85 ± 0.21	44.86 ± 0.16
111008A	1.03 ± 0.01	2.56 ± 0.12	53.35 ± 0.02	53.68 ± 0.01	52.05 ± 0.06	3.12 ± 0.06	48.04 ± 0.05
111228A	1.77 ± 0.02	1.71 ± 0.12	52.04 ± 0.01	52.4 ± 0.07	50.66 ± 0.05	3.89 ± 0.05	46.37 ± 0.04
120326A	1.4 ± 0.05	2.06 ± 0.19	52.3 ± 0.05	52.64 ± 0.01	51.57 ± 0.05	4.27 ± 0.01	46.96 ± 0.01
120521C	0.58 ± 0.07	2.75 ± 0.13	52.78 ± 0.04	53.12 ± 0.12	51 ± 0.24	3.22 ± 0.23	46.81 ± 0.17
120811C	0.86 ± 0.06	2.2 ± 0.2	52.67 ± 0.04	52.96 ± 0.01	51.17 ± 0.2	2.77 ± 0.17	47.74 ± 0.13
120907A	0.93 ± 0.18	2.2 ± 0.2	51.2 ± 0.07	51.53 ± 0.2	50.27 ± 0.2	4.35 ± 0.17	45.45 ± 0.1
121024A	1.54 ± 0.17	2.52 ± 0.25	51.42 ± 0.04	51.87 ± 0.25	50.63 ± 0.2	4.35 ± 0.17	45.45 ± 0.18
121128A	0.86 ± 0.03	2.31 ± 0.14	52.88 ± 0.03	53.22 ± 0.01	51.43 ± 0.07	2.7 ± 0.05	48.06 ± 0.06
121211A	1.95 ± 0.08	1.72 ± 0.21	51.53 ± 0.08	51.88 ± 0.19	50.56 ± 0.14	4.32 ± 0.11	45.69 ± 0.11
130603B	−0.88 ± 0.05	3.2 ± 0.15	50.29 ± 0.02	51.29 ± 0.11	49.58 ± 0.06	3.23 ± 0.05	46.08 ± 0.06
131105A	1.62 ± 0.02	2.61 ± 0.14	52.69 ± 0.03	53.14 ± 0.07	50.77 ± 0.1	3.23 ± 0.08	47.1 ± 0.08
140430A	1.82 ± 0.01	2.09 ± 0.2	51.83 ± 0.08	52.16 ± 0.07	50.86 ± 0.17	4.26 ± 0.16	45.86 ± 0.16
140512A	1.95 ± 0.01	2.41 ± 0.09	52.27 ± 0.01	52.71 ± 0.04	51.4 ± 0.06	3.88 ± 0.06	46.92 ± 0.06
140518A	1.03 ± 0.02	2.4 ± 0.4	52.58 ± 0.04	52.92 ± 0.01	50.94 ± 0.16	2.53 ± 0.12	47.86 ± 0.08
140703A	1.21 ± 0.3	2.51 ± 0.15	52.9 ± 0.03	53.21 ± 0.1	52.13 ± 0.04	3.54 ± 0.03	47.66 ± 0.04
141121A	2.35 ± 0.03	2.3 ± 0.15	52.45 ± 0.03	52.78 ± 0.12	51.18 ± 0.07	5.13 ± 0.05	45.52 ± 0.07
150323A	1.97 ± 0.03	2 ± 0.11	51.73 ± 0.01	52.06 ± 0.01	49.66 ± 0.13	3.96 ± 0.11	45.32 ± 0.09
150424A	1.85 ± 0.09	2.55 ± 0.2	50.51 ± 0.03	51.09 ± 0.12	49.51 ± 0.2	4.94 ± 0.18	43.96 ± 0.17
150910A	1.68 ± 0.13	2.58 ± 0.15	52.34 ± 0.04	52.81 ± 0.08	51.73 ± 0.15	3.56 ± 0.07	47.65 ± 0.12
151027A	1.86 ± 0.02	2.17 ± 0.1	52.11 ± 0.04	52.46 ± 0.03	51.34 ± 0.02	3.34 ± 0.02	47.78 ± 0.02
151112A	0.58 ± 0.42	2.57 ± 0.21	52.46 ± 0.05	52.81 ± 0.17	51.54 ± 0.18	4.06 ± 0.16	46.55 ± 0.15
160227A	1.97 ± 0.09	2.35 ± 0.33	52.59 ± 0.03	52.90 ± 0.04	51.63 ± 0.1	3.85 ± 0.09	47.18 ± 0.08
160327A	0.67 ± 0.12	2.58 ± 0.14	52.78 ± 0.03	53.09 ± 0.01	51.23 ± 0.13	2.84 ± 0.11	47.55 ± 0.13
160804A	1.92 ± 0.05	1.96 ± 0.07	52.19 ± 0.01	52.49 ± 0.01	49.7 ± 0.1	3.72 ± 0.07	45.88 ± 0.04
161108A	1.69 ± 0.05	2.13 ± 0.17	51.57 ± 0.04	51.91 ± 0.09	50.16 ± 0.15	4.03 ± 0.14	45.82 ± 0.08
161129A	1.33 ± 0.02	2.27 ± 0.1	51.57 ± 0.01	51.96 ± 0.03	50.26 ± 0.09	3.34 ± 0.07	46.59 ± 0.11
170113A	0.84 ± 0.08	2.34 ± 0.49	51.78 ± 0.04	52.13 ± 0.13	51.41 ± 0.07	3.16 ± 0.06	47.61 ± 0.06
170202A	1 ± 0.1	2.61 ± 0.11	52.93 ± 0.01	53.28 ± 0.04	51.22 ± 0.07	2.65 ± 0.06	48.01 ± 0.04

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Xu & Huang (2012) introduced a third parameter ${E}_{\mathrm{iso},\gamma }$ to the above relation (replacing ${t}_{b}^{{\prime} }$ by t_b) and claimed that the three-parameter correlation would be tighter. The relation could be expressed as ${L}_{{\rm{X}}}={\kappa }_{0}+{\kappa }_{1}{\mathrm{log}}_{10}({t}_{b})+{\kappa }_{2}{\mathrm{log}}_{10}({E}_{\mathrm{iso},\gamma })$ , with ${\kappa }_{0}=4.14$ , ${\kappa }_{1}=-0.87$ , and ${\kappa }_{2}=0.88$ . We have also justified this empirical relationship with our updated sample. With multiple regression analysis methods, we find that the three-parameter correlation indeed exists, with ${\kappa }_{0}=9.26\,\pm 2.59\,({p}_{t}\lt {10}^{-4})$ , ${\kappa }_{1}=-0.97\pm 0.07\,({p}_{t}\lt {10}^{-4})$ , and ${\kappa }_{2}\,=0.79\pm 0.05$ ( ${p}_{t}\lt {10}^{-4}$ ), where ${p}_{F}\lt {10}^{-4}$ , ${R}^{2}=73.3 \%$ , and ${p}_{\mathrm{AD}}=0.372$ . The data and fitting results regarding this relationship are shown in Figure 8. It is interesting to note that besides the Xu & Huang (2012) correlation, more recently Dainotti et al. (2016, 2017) showed that the ${L}_{{\rm{X}}}-{L}_{\mathrm{peak}}-{t}_{b}^{{\prime} }$ correlation is 37% tighter than the Xu & Huang (2012) correlation, making the ${L}_{{\rm{X}}}-{L}_{\mathrm{peak}}-{t}_{b}^{{\prime} }$ correlation the tighest correlation in the literature so far involving the plateau emission.

**Figure 8.** Left panel: the correlation between L_X and ${t}_{b}^{{\prime} }$ . Middle panel: the correlation between ${{\rm{\Sigma }}}_{{E}_{\mathrm{iso},\gamma }}={\mathrm{log}}_{10}({L}_{{\rm{X}}})-{\kappa }_{1}{\mathrm{log}}_{10}({t}_{b})$ and ${E}_{\mathrm{iso},\gamma }$ . Right panel: the correlation between ${{\rm{\Sigma }}}_{{t}_{b}}={\mathrm{log}}_{10}({L}_{{\rm{X}}})-{\kappa }_{2}{\mathrm{log}}_{10}({E}_{\mathrm{iso},\gamma })$ and t_b. The black lines represent the best fitting results (solid) and its 3σ uncertainty region (dotted) with the least square regression method. The red line presents the best fitting results with the bivariate linear regression method and the pink shadowed region shows the intrinsic scatter to the population $3{\sigma }_{\mathrm{int}}$ .
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4. Conclusions and Discussion

With Swift/XRT light curves of GRBs between 2004 February and 2017 July, we revisit the analysis of the shallow decay component from Liang et al. (2007). Our results and the comparison with L07's results are summarized as follows:

1.
We find that with a larger sample, the distributions of the characteristic properties of the shallow decay phase (e.g., t_b, S_X, Γ_X,1, and α_X,1) still accord with normal or lognormal distribution with ${\mathrm{log}}_{10}({t}_{b})=4\pm 0.72$ , ${\mathrm{log}}_{10}({S}_{{\rm{X}}}(\mathrm{erg}\,{\mathrm{cm}}^{-2}))=-6.97\pm 0.56$ , ${{\rm{\Gamma }}}_{{\rm{X}},1}=1.96\pm 0.26$ , and ${\alpha }_{{\rm{X}},1}=0.43\pm 0.22$ . Comparing with L07's results, the distributions for all the parameters become slightly broader but the peaks of each distribution barely change.
2.
We find that with a larger sample, Γ_X,1 and Γ_γ still show no correlation, with Γ_X,1 systemically larger than Γ_γ. The tentative correlations of durations, energy fluences, and isotropic energies between the gamma-ray and X-ray phases still exist, but all correlations become significantly weaker than L07's results.
3.
We find that for most GRBs, there is no significant spectral evolution between the shallow decay segment and its follow-up segment, and the follow-up segment for most bursts in our sample are consistent with the external-shock models. These two findings are consistent with L07's results, which infer that the shallow decay phase for most GRBs should be of external origin and may be related to continuous energy injection from the central engine.
4.
Assuming that the central engine has a power-law luminosity release history as $L\left(t\right)={L}_{0}{\left(\tfrac{t}{{t}_{0}}\right)}^{-q}$ , we find that q values are mainly distributed between −0.5 and 0.5, with an average value of 0.16 ± 0.12, which is consistent with L07's results and is well consistent with the model prediction (q value around zero) where a spinning-down pulsar acts as the energy injection central engine (Dai & Lu 1998; Zhang & Mészáros 2001). With the derived q value, we find that the shallow decay phases for 196/198 GRBs satisfy the closure relations with energy injection, and their corresponding spectral regime for most bursts (96.4%) is consistent with their follow-up phases.
5.
We find that with a larger sample, L07 suggested that a correlation between ${E}_{\mathrm{iso},{\rm{X}}}$ and ${E}_{p}^{{\prime} }$ still exists but the tentative correlation between ${E}_{\mathrm{iso},{\rm{X}}}$ and ${t}_{b}^{{\prime} }$ is completely disappeared, so that the global three-parameter correlation becomes less significant. On the other hand, we confirm that a strong correlation between ${E}_{\mathrm{iso},\gamma }$ and ${E}_{p}^{{\prime} }$ and a tentative global three-parameter correlation ( ${E}_{\mathrm{iso},\gamma }-{E}_{p}^{{\prime} }-{t}_{b}^{{\prime} }$ ) exist but there is no correlation between ${E}_{\mathrm{iso},\gamma }$ and ${t}_{b}^{{\prime} }$ .
6.
We find that the anticorrelation indeed exists between L_X and $({t}_{b}^{{\prime} })$ (as suggested by Dainotti et al. 2008), with ${\kappa }_{0}=50.7\pm 0.39$ ( ${p}_{t}\lt {10}^{-4}$ ) and ${\kappa }_{1}=-1.15\,\pm 0.1\,({p}_{t}\lt {10}^{-4})$ , where the p_F is less than 10⁻⁴. Xu & Huang (2012) introduced a three-parameter correlation by inserting ${E}_{\mathrm{iso},\gamma }$ into the above relation (and replacing ${t}_{b}^{{\prime} }$ by t_b) and we find that with our updated sample, the three-parameter correlation indeed exists, with ${\kappa }_{0}=9.26\,\pm 2.59\,({p}_{t}\lt {10}^{-4})$ , ${\kappa }_{1}=-0.97\pm 0.07\,({p}_{t}\lt {10}^{-4})$ , and ${\kappa }_{2}=0.79\pm 0.05$ ( ${p}_{t}\lt {10}^{-4}$ ), where the p_F is ${p}_{F}\,\lt {10}^{-4}$ . More recently Dainotti et al. (2016, 2017) showed that the ${L}_{{\rm{X}}}-{L}_{\mathrm{peak}}-{t}_{b}^{{\prime} }$ correlation is 37% tighter than the Xu & Huang (2012) correlation, making the ${L}_{{\rm{X}}}-{L}_{\mathrm{peak}}-{t}_{b}^{{\prime} }$ correlation the tightest correlation in the literature involving plateau emission. In addition, we note that this correlation is the result of two correlations that have been tested for selection bias.

In conclusion, with an updated sample, our results are consistent with most of L07's results and confirm their suggestion that the shallow decay segment in most bursts is consistent with an external forward shock origin, probably due to a continuous energy injection from a long-lived central engine.

This work was supported by the National Natural Science Foundation of China (under grant Nos. 11722324, 11603003, 11633001, 11690024, and 11603006), and the Strategic Priority Research Program of the Chinese Academy of Sciences (grant No. XDB23040100). L.H.J. acknowledges support by the GuangXi Science Foundation (grant Nos. 2017GXNSFFA198008 and 2016GXNSFCB380005), the One-Hundred-Talents Program of GuangXi colleges, and Bagui Young Scholars Program of GuangXi. B.B.Z. acknowledges support from National Thousand Young Talents program of China and National Key Research and Development Program of China (2018YFA0404204) and The National Natural Science Foundation of China (grant No. 11833003).

Software: XSPEC (Arnaud 1996), HEAsoft (v6.12; NASA High Energy Astrophysics Science Archive Research Center (HEASARC), 2014), root (v5.34; Brun & Rademakers 1997), Linmix_err (Kelly 2007).

The Shallow Decay Segment of GRB X-Ray Afterglow Revisited

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ORCID iDs

Dates

Abstract

1. Introduction

2. Data Reduction and Sample Selection

3. L07 Revisited

3.1. Distributions of the Characteristic Properties of the Shallow Decay Phase

3.2. Relationship of the Shallow Decay Phase to the Prompt Gamma-Ray Phase

3.3. Test Physical Origin of the Shallow Decay Segment

3.4. Empirical Relations Revisited

3.4.1. Empirical Relation among ${E}_{\mathrm{iso}},{E}_{p}^{{\prime} }$ , and $t{{\prime} }_{b}$

3.4.2. Revisit Correlations between ${t}_{b}^{{\prime} }$ (t_b), L_X, and ${E}_{\mathrm{iso}\gamma }$

4. Conclusions and Discussion

Footnotes

The Shallow Decay Segment of GRB X-Ray Afterglow Revisited

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Permissions

Share this article

Author e-mails

Author affiliations

ORCID iDs

Dates

Abstract

1. Introduction

2. Data Reduction and Sample Selection

3. L07 Revisited

3.1. Distributions of the Characteristic Properties of the Shallow Decay Phase

3.2. Relationship of the Shallow Decay Phase to the Prompt Gamma-Ray Phase

3.3. Test Physical Origin of the Shallow Decay Segment

3.4. Empirical Relations Revisited

3.4.1. Empirical Relation among {E}_{\mathrm{iso}},{E}_{p}^{{\prime} }, and t{{\prime} }_{b}

3.4.2. Revisit Correlations between {t}_{b}^{{\prime} } (tb), LX, and {E}_{\mathrm{iso}\gamma }

4. Conclusions and Discussion

Footnotes

3.4.1. Empirical Relation among ${E}_{\mathrm{iso}},{E}_{p}^{{\prime} }$ , and $t{{\prime} }_{b}$

3.4.2. Revisit Correlations between ${t}_{b}^{{\prime} }$ (t_b), L_X, and ${E}_{\mathrm{iso}\gamma }$