Statistical Study of Gamma-Ray Bursts with a Plateau Phase in the X-ray Afterglow

TL;DR

This study analyzes a large sample of Swift GRBs with plateau phases, confirming a key correlation among their X-ray luminosity, plateau end time, and prompt energy, supporting the magnetar central engine model and potential cosmological applications.

Contribution

It provides an updated, comprehensive correlation among GRB parameters using 174 bursts, including short GRBs and those with internal plateaus, enhancing understanding of their central engines.

Findings

Confirmed the $L_{X}$-$T_{a}$-$E_{γ,iso}$ correlation with a new fit.

Short GRBs follow the same correlation as long GRBs.

GRBs with internal plateaus also obey the correlation.

Abstract

A plateau phase in the X-ray afterglow is observed in a significant fraction of gamma-ray bursts (GRBs). Previously, it has been found that there exists a correlation among three key parameters concerning the plateau phase, i.e., the end time of the plateau phase in the GRB rest frame ($T_{a}$), the corresponding X-ray luminosity at the end time ($L_{X}$) and the isotropic energy of the prompt GRB ($E_{\gamma,\rm{iso}}$). In this study, we systematically search through all the \emph{Swift} GRBs with a plateau phase that occurred between 2005 May and 2018 August. We collect 174 GRBs, with redshifts available for all of them. For the whole sample, the correlation between $L_{X}$, $T_{a}$ and $E_{\gamma,\rm{iso}}$ is confirmed, with the best fit relation being $L_{X}\propto T_{a}^{-1.01}E_{\gamma,\rm{iso}}^{0.84}$. Such an updated three-parameter correlation still supports that the central leftover after GRBs is probably a millisecond magnetar. It is interesting to note that short GRBs with duration less than 2 s in our sample also follow the same correlation, which hints that the merger production of two neutron stars could be a high mass magnetar, but not necessarily a black hole. Moreover, GRBs having an "internal" plateau (i.e., with a following decay index being generally smaller than -3) also obey this correlation. It further strengthens the idea that the internal plateau is due to the delayed collapse of a high mass neutron star into a black hole. The updated three-parameter correlation indicates that GRBs with a plateau phase may act as a standard candle for cosmology study.

Tables

Table 1. Table 1: Seven Short GRBs Included in Our Sample

GRB name	$T_{90}$ (s)	$z$
GRB 051221A	1.4	0.55
GRB 061201	0.76	0.11
GRB 070809	1.3	0.22
GRB 090510	0.3	0.90
GRB 130603B	0.18	0.36
GRB 140903A	0.3	0.35
GRB 150423A	0.22	1.40

Table 2. Table 2: GRBs with an“Internal Plateau” in Our Sample

GRB name	${\alpha }_2$	$z$
GRB 050730	2.77	3.97
GRB 060607A	3.47	3.08
GRB 070110	8.95	2.35
GRB 100219A	4.64	4.7
GRB 100902A	4.69	4.5
GRB 111209A	15.11	0.68
GRB 111229A	3.15	1.38
GRB 120521C	3.03	6.0
GRB 120712A	3.26	4.17
GRB 130408A	3.79	3.76
GRB 170714A	4.97	0.79

Table 3. Table 3: Fitting Results and Some Key Parameters of the 174 GRBs in Our Sample

GRB Name	T${}_{}{}^a{}_{90}{}^{}$	$z^a$	$S$(15-150 keV)a	${\alpha }_1^b$	${\alpha }_2^b$	$\log {(T_a/{10}^3)}^b$	$\log {(L_X/{10}^{47})}^c$	$\log {(E_{\gamma ,\mathrm{iso}}^{\prime }/{10}^{53})}^d$	$\log {(E_{\gamma ,\mathrm{iso}}/{10}^{53})}^e$
	(s)		(${10}^{-7}$ erg/cm2)			(s)	(erg/s)	(erg)	(erg)
GRB 050315	95.6	1.949	32.2$\pm$1.46	-0.03$\pm$0.05	1.94$\pm$0.25	1.53$\pm$0.15	-0.29$\pm$0.32	-0.52$\pm$0.02	-0.46$\pm$0.02
GRB 050319	152.5	3.24	13.1$\pm$1.48	0.15$\pm$0.11	1.41$\pm$0.17	0.53$\pm$0.22	0.86$\pm$0.27	-0.53$\pm$0.05	-0.52$\pm$0.05
GRB 050401	33.3	2.9	82.2$\pm$3.06	0.55$\pm$0.03	1.63$\pm$0.28	0.21$\pm$0.16	1.38$\pm$0.17	0.19$\pm$0.02	-0.17$\pm$0.02
GRB 050416A	2.5	0.6535	3.67$\pm$0.37	0.32$\pm$0.09	1.09$\pm$0.09	0.2$\pm$0.2	-0.81$\pm$0.22	-2.38$\pm$0.04	-2.14$\pm$0.04
GRB 050505	58.9	4.27	24.9$\pm$1.79	0.06$\pm$0.14	1.78$\pm$0.11	0.37$\pm$0.08	1.29$\pm$0.2	-0.06$\pm$0.03	-0.49$\pm$0.03
GRB 050730	156.5	3.97	23.8$\pm$1.52	0.23$\pm$0.1	2.77$\pm$0.19	0.22$\pm$0.07	1.99$\pm$0.27	-0.13$\pm$0.03	-0.46$\pm$0.03
GRB 050801	19.4	1.38	3.1$\pm$0.48	0.44$\pm$0.45	1.25$\pm$0.15	-0.61$\pm$0.26	0.38$\pm$0.25	-1.81$\pm$0.06	-1.81$\pm$0.06
GRB 050802	19	1.71	20$\pm$1.57	0.5$\pm$0.19	1.6$\pm$0.08	0.29$\pm$0.12	0.67$\pm$0.34	-0.82$\pm$0.03	-1.02$\pm$0.03
GRB 050803	87.9	0.422	21.5$\pm$1.35	-0.09$\pm$0.11	1.63$\pm$0.05	0.95$\pm$0.03	-1$\pm$0.04	-2.01$\pm$0.03	-2.1$\pm$0.03
GRB 050814	150.9	5.3	20.1$\pm$2.2	0.06$\pm$0.14	1.81$\pm$0.34	0.66$\pm$0.27	0.52$\pm$0.43	-0.01$\pm$0.05	-0.17$\pm$0.05
GRB 050824	22.6	0.83	2.66$\pm$0.52	0.05$\pm$0.13	1.09$\pm$0.23	1.58$\pm$0.27	-1.67$\pm$0.24	-2.31$\pm$0.08	-2.11$\pm$0.08
GRB 050922C	4.5	2.199	16.2$\pm$0.54	0.61$\pm$0.09	1.4$\pm$0.07	-0.92$\pm$0.13	1.96$\pm$0.24	-0.72$\pm$0.01	-1.04$\pm$0.01
GRB 051016B	4	0.9364	1.7$\pm$0.22	0$\pm$0.09	1.6$\pm$0.18	0.95$\pm$0.18	-1.12$\pm$0.38	-2.4$\pm$0.05	-2.28$\pm$0.05
GRB 051109A	37.2	2.346	22$\pm$2.72	0.32$\pm$0.08	1.31$\pm$0.05	-0.19$\pm$0.11	1.49$\pm$0.19	-0.54$\pm$0.05	-0.8$\pm$0.05
GRB 051109B	14.3	0.08	2.56$\pm$0.41	-0.02$\pm$0.15	1.35$\pm$0.16	0.36$\pm$0.15	-3.29$\pm$0.32	-4.43$\pm$0.06	-4.43$\pm$0.06
GRB 051221A	1.4	0.547	11.5$\pm$0.35	0$\pm$0.22	1.43$\pm$0.11	1.17$\pm$0.12	-1.84$\pm$0.23	-2.04$\pm$0.01	-2.16$\pm$0.01
GRB 060108	14.3	2.03	3.69$\pm$0.37	0.14$\pm$0.13	1.46$\pm$0.22	0.77$\pm$0.2	-0.43$\pm$0.31	-1.42$\pm$0.04	-1.41$\pm$0.04
GRB 060115	139.6	3.53	17.1$\pm$1.5	0.22$\pm$0.18	1.28$\pm$0.17	0.38$\pm$0.16	0.33$\pm$0.28	-0.35$\pm$0.04	-1.01$\pm$0.04
GRB 060116	105.9	4	24.1$\pm$2.61	0.63$\pm$0.24	1.2$\pm$0.17	-0.66$\pm$0.27	1.2$\pm$0.25	-0.12$\pm$0.04	-0.52$\pm$0.04
GRB 060202	198.9	0.78	21.3$\pm$1.65	-0.08$\pm$0.29	0.88$\pm$0.03	0.27$\pm$0.14	-0.45$\pm$0.14	-1.46$\pm$0.03	-1.53$\pm$0.03
GRB 060206	7.6	4.05	8.31$\pm$0.42	0.22$\pm$0.17	1.03$\pm$0.07	-0.49$\pm$0.19	1.61$\pm$0.27	-0.57$\pm$0.02	-1.14$\pm$0.02
GRB 060210	255	3.91	76.6$\pm$4.09	0.07$\pm$0.16	1.47$\pm$0.08	0.24$\pm$0.1	1.58$\pm$0.22	0.37$\pm$0.02	0.04$\pm$0.02
GRB 060502A	28.4	1.51	23.1$\pm$1.02	0.2$\pm$0.11	1.18$\pm$0.08	0.49$\pm$0.15	0.03$\pm$0.26	-0.86$\pm$0.02	-1.08$\pm$0.02
GRB 060522	71.1	5.11	11.4$\pm$1.11	0.48$\pm$0.35	1.32$\pm$0.17	-0.89$\pm$0.24	1.74$\pm$0.3	-0.28$\pm$0.04	-0.63$\pm$0.04
GRB 060526	298.2	3.21	12.6$\pm$1.65	0.09$\pm$0.17	1.85$\pm$0.28	0.63$\pm$0.18	0.27$\pm$0.35	-0.55$\pm$0.05	-0.55$\pm$0.05
GRB 060604	95	2.1357	4.02$\pm$1.06	0.04$\pm$0.17	1.32$\pm$0.11	0.64$\pm$0.14	-0.01$\pm$0.24	-1.35$\pm$0.1	-1.34$\pm$0.1
GRB 060605	79.1	3.8	6.97$\pm$0.9	0.17$\pm$0.12	2.56$\pm$0.27	0.3$\pm$0.11	0.9$\pm$0.27	-0.69$\pm$0.05	-1$\pm$0.05
GRB 060607A	102.2	3.082	25.5$\pm$1.12	0.47$\pm$0.02	3.47$\pm$0.1	0.5$\pm$0.01	1.41$\pm$0.02	-0.28$\pm$0.02	-0.6$\pm$0.02
GRB 060614	108.7	0.13	204$\pm$3.63	-0.02$\pm$0.05	2.19$\pm$0.13	1.69$\pm$0.04	-2.79$\pm$0.15	-2.09$\pm$0.01	-2.09$\pm$0.01
GRB 060707	66.2	3.43	16$\pm$1.51	0.13$\pm$0.14	1.28$\pm$0.16	0.47$\pm$0.27	0.59$\pm$0.44	-0.4$\pm$0.04	-1.31$\pm$0.04
GRB 060708	10.2	2.3	4.94$\pm$0.37	0.1$\pm$0.16	1.33$\pm$0.1	0.07$\pm$0.14	0.6$\pm$0.25	-1.2$\pm$0.03	-1.37$\pm$0.03
GRB 060714	115	2.71	28.3$\pm$1.67	0.13$\pm$0.15	1.31$\pm$0.06	-0.01$\pm$0.1	0.97$\pm$0.16	-0.32$\pm$0.02	-0.36$\pm$0.02
GRB 060729	115.3	0.54	26.1$\pm$2.11	0.05$\pm$0.04	1.44$\pm$0.02	1.62$\pm$0.02	-0.9$\pm$0.06	-1.7$\pm$0.03	-1.75$\pm$0.03
GRB 060814	145.3	0.84	146$\pm$2.39	0.1$\pm$0.17	1.51$\pm$0.08	0.74$\pm$0.1	-0.38$\pm$0.22	-0.56$\pm$0.01	-0.68$\pm$0.01
GRB 060906	43.5	3.685	22.1$\pm$1.36	0.18$\pm$0.14	1.93$\pm$0.4	0.43$\pm$0.11	0.46$\pm$0.19	-0.21$\pm$0.03	-0.19$\pm$0.03
GRB 060908	19.3	1.8836	28$\pm$1.11	0.25$\pm$0.19	1.56$\pm$0.18	-0.7$\pm$0.21	1.37$\pm$0.25	-0.6$\pm$0.02	-1.06$\pm$0.02
GRB 060912A	5	0.937	13.5$\pm$0.62	0.5$\pm$0.2	1.14$\pm$0.06	-0.54$\pm$0.18	0.03$\pm$0.19	-1.5$\pm$0.02	-1.57$\pm$0.02
GRB 061021	46.2	0.3463	29.6$\pm$1.01	0.25$\pm$0.09	1.17$\pm$0.04	0.6$\pm$0.08	-1.11$\pm$0.17	-2.05$\pm$0.01	-2.14$\pm$0.01
GRB 061121	81.3	1.314	137$\pm$1.99	0.01$\pm$0.05	1.58$\pm$0.06	0.22$\pm$0.07	0.91$\pm$0.2	-0.2$\pm$0.01	-0.42$\pm$0.01
GRB 061201	0.76	0.111	3.34$\pm$0.27	0.32$\pm$0.16	2.14$\pm$0.28	0.36$\pm$0.13	-2.16$\pm$0.3	-4.02$\pm$0.03	-4.07$\pm$0.03
GRB 061222A	71.4	2.088	79.9$\pm$1.58	0.04$\pm$0.07	1.7$\pm$0.14	0.42$\pm$0.17	1.09$\pm$0.39	-0.07$\pm$0.01	-0.39$\pm$0.01
GRB 070110	88.4	2.352	16.2$\pm$1.08	0.06$\pm$0.06	8.95$\pm$0.62	0.79$\pm$0.01	0.76$\pm$0.04	-0.67$\pm$0.03	-0.89$\pm$0.03
GRB 070129	460.6	2.3384	29.8$\pm$2.67	0.06$\pm$0.11	1.31$\pm$0.11	0.84$\pm$0.13	0.08$\pm$0.18	-0.41$\pm$0.04	-0.4$\pm$0.04
GRB 070208	47.7	1.165	4.45$\pm$1.01	0.13$\pm$0.14	1.67$\pm$0.42	0.59$\pm$0.21	-0.58$\pm$0.25	-1.79$\pm$0.09	-1.81$\pm$0.09
GRB 070306	209.5	1.497	53.8$\pm$2.86	0.07$\pm$0.05	1.92$\pm$0.1	1.09$\pm$0.04	0.19$\pm$0.1	-0.5$\pm$0.02	-0.64$\pm$0.02
GRB 070506	4.3	2.31	2.1$\pm$0.23	0.11$\pm$0.33	0.84$\pm$0.36	-0.23$\pm$0.33	0.51$\pm$0.23	-1.57$\pm$0.05	-1.71$\pm$0.05
GRB 070508	20.9	0.82	196$\pm$2.73	0.38$\pm$0.07	1.63$\pm$0.27	-0.06$\pm$0.25	0.93$\pm$0.16	-0.45$\pm$0.01	-0.69$\pm$0.01
GRB 070529	109.2	2.4996	25.7$\pm$2.45	0.18$\pm$0.21	1.35$\pm$0.1	-0.66$\pm$0.14	1.26$\pm$0.28	-0.42$\pm$0.04	-0.78$\pm$0.04
GRB 070714B	64	0.92	7.2$\pm$0.9	0.08$\pm$0.33	2.23$\pm$0.62	-0.13$\pm$0.22	-0.13$\pm$0.31	-1.79$\pm$0.05	-1.97$\pm$0.05
GRB 070721B	340	3.626	36$\pm$2	0.31$\pm$0.12	2.14$\pm$0.25	-0.2$\pm$0.15	1.44$\pm$0.36	-0.01$\pm$0.02	-0.45$\pm$0.02
GRB 070809	1.3	0.22	1$\pm$0.1	0$\pm$0.15	1.85$\pm$0.85	1.01$\pm$0.26	-2.82$\pm$0.36	-3.93$\pm$0.04	-3.96$\pm$0.04
GRB 070810A	11	2.17	6.9$\pm$0.6	0.13$\pm$0.16	1.45$\pm$0.15	-0.38$\pm$0.16	0.86$\pm$0.22	-1.1$\pm$0.04	-1.08$\pm$0.04
GRB 071020	4.2	2.145	23$\pm$1	0.79$\pm$0.07	1.43$\pm$0.41	0.5$\pm$0.2	0.59$\pm$0.34	-0.59$\pm$0.02	-1.03$\pm$0.02
GRB 080310	365	2.43	23$\pm$2	-0.01$\pm$0.08	1.73$\pm$0.13	0.47$\pm$0.07	0.51$\pm$0.21	-0.49$\pm$0.04	-0.32$\pm$0.04
GRB 080430	16.2	0.767	12$\pm$1	0.07$\pm$0.07	1.29$\pm$0.08	0.95$\pm$0.13	-0.93$\pm$0.2	-1.72$\pm$0.03	-1.79$\pm$0.03
GRB 080516	5.8	3.2	2.6$\pm$0.4	0.08$\pm$0.13	1.27$\pm$0.24	-0.02$\pm$0.2	0.79$\pm$0.27	-1.24$\pm$0.06	-1.35$\pm$0.06
GRB 080603B	60	2.69	24$\pm$1	0.13$\pm$0.23	1.21$\pm$0.27	-0.64$\pm$0.22	1.8$\pm$0.21	-0.4$\pm$0.02	-0.85$\pm$0.02
GRB 080605	20	1.6398	133$\pm$2	0.44$\pm$0.08	1.77$\pm$0.26	-0.22$\pm$0.12	1.52$\pm$0.24	-0.04$\pm$0.01	-0.41$\pm$0.01
GRB 080707	27.1	1.23	5.2$\pm$0.6	0.03$\pm$0.13	1.42$\pm$0.21	0.62$\pm$0.19	-0.7$\pm$0.38	-1.68$\pm$0.05	-1.76$\pm$0.05
GRB 080721	16.2	2.602	120$\pm$10	0.37$\pm$0.05	1.66$\pm$0.05	-0.51$\pm$0.09	2.74$\pm$0.23	0.27$\pm$0.03	-0.22$\pm$0.03
GRB 080810	106	3.35	46$\pm$2	0.08$\pm$0.15	1.6$\pm$0.11	-0.43$\pm$0.12	2$\pm$0.24	0.04$\pm$0.02	-0.38$\pm$0.02
GRB 080905B	128	2.374	18$\pm$2	0.1$\pm$0.15	1.46$\pm$0.07	0$\pm$0.11	1.42$\pm$0.18	-0.62$\pm$0.05	-0.73$\pm$0.05
GRB 081007	10	0.5295	7.1$\pm$0.8	0.21$\pm$0.11	1.18$\pm$0.08	0.41$\pm$0.16	-0.93$\pm$0.27	-2.28$\pm$0.05	-2.19$\pm$0.05
GRB 081008	185.5	1.9685	43$\pm$2	0.21$\pm$0.14	2.08$\pm$0.21	0.48$\pm$0.14	0.28$\pm$0.33	-0.38$\pm$0.02	-0.53$\pm$0.02
GRB 081029	270	3.8479	21$\pm$2	0.43$\pm$0.07	2.77$\pm$0.3	0.57$\pm$0.04	0.73$\pm$0.11	-0.21$\pm$0.04	-0.6$\pm$0.04
GRB 081221	34	2.26	181$\pm$3	0.39$\pm$0.11	1.3$\pm$0.02	-0.72$\pm$0.06	2.39$\pm$0.14	0.35$\pm$0.01	-0.06$\pm$0.01
GRB 090113	9.1	1.7493	7.6$\pm$0.4	-0.04$\pm$0.16	1.31$\pm$0.06	-0.71$\pm$0.11	1.3$\pm$0.13	-1.23$\pm$0.02	-1.4$\pm$0.02
GRB 090205	8.8	4.7	1.9$\pm$0.3	-0.05$\pm$0.15	1.79$\pm$0.32	-0.03$\pm$0.22	0.87$\pm$0.38	-1.12$\pm$0.06	-1$\pm$0.06
GRB 090313	79	3.375	14$\pm$2	0.04$\pm$0.42	2.11$\pm$0.27	0.91$\pm$0.16	0.61$\pm$0.38	-0.47$\pm$0.06	-0.53$\pm$0.06
GRB 090407	310	1.4485	11$\pm$2	0.03$\pm$0.08	2.21$\pm$0.27	1.56$\pm$0.14	-0.83$\pm$0.28	-1.22$\pm$0.07	-1.32$\pm$0.07
GRB 090418A	56	1.608	46$\pm$2	0.11$\pm$0.12	1.71$\pm$0.1	-0.07$\pm$0.09	1.1$\pm$0.23	-0.51$\pm$0.02	-0.73$\pm$0.02
GRB 090423	10.3	8	5.9$\pm$0.4	-0.14$\pm$0.14	1.49$\pm$0.11	-0.3$\pm$0.08	1.62$\pm$0.16	-0.29$\pm$0.03	-1.43$\pm$0.03
GRB 090510	0.3	0.903	3.4$\pm$0.4	0.62$\pm$0.07	2.31$\pm$0.25	-0.04$\pm$0.09	0.09$\pm$0.19	-2.13$\pm$0.05	-2.41$\pm$0.05
GRB 090516	210	4.109	90$\pm$6	0.07$\pm$0.23	1.87$\pm$0.14	0.32$\pm$0.07	1.25$\pm$0.19	0.47$\pm$0.03	0.36$\pm$0.03
GRB 090519	64	3.9	12$\pm$1	0.2$\pm$0.42	1.77$\pm$0.84	-0.24$\pm$0.34	0.12$\pm$0.31	-0.44$\pm$0.03	-1.12$\pm$0.03
GRB 090529	100	2.625	6.8$\pm$1.7	0.06$\pm$0.18	1.55$\pm$0.5	1.45$\pm$0.3	-0.89$\pm$0.48	-0.97$\pm$0.1	-0.97$\pm$0.1
GRB 090530	48	1.266	11$\pm$1	0.06$\pm$0.12	1.08$\pm$0.09	0.23$\pm$0.2	-0.29$\pm$0.26	-1.33$\pm$0.04	-1.47$\pm$0.04
GRB 090618	113.2	0.54	1050$\pm$10	0.48$\pm$0.06	1.59$\pm$0.06	0.5$\pm$0.16	0.2$\pm$0.33	-0.1$\pm$0	-0.2$\pm$0
GRB 090927	2.2	1.37	2$\pm$0.3	0$\pm$0.19	1.26$\pm$0.11	0.6$\pm$0.13	-0.49$\pm$0.2	-2$\pm$0.06	-2.08$\pm$0.06
GRB 091018	4.4	0.971	14$\pm$1	0.29$\pm$0.1	1.27$\pm$0.06	-0.53$\pm$0.16	0.84$\pm$0.2	-1.45$\pm$0.03	-1.52$\pm$0.03
GRB 091029	39.2	2.752	24$\pm$1	0.01$\pm$0.09	1.2$\pm$0.05	0.38$\pm$0.09	0.66$\pm$0.18	-0.38$\pm$0.02	-0.69$\pm$0.02
GRB 091109A	48	3.5	16$\pm$2	0.05$\pm$0.35	1.04$\pm$0.07	-0.76$\pm$0.19	1.06$\pm$0.21	-0.39$\pm$0.05	-0.84$\pm$0.05
GRB 091127	7.1	0.49	90$\pm$3	0.6$\pm$0.19	1.63$\pm$0.41	1.03$\pm$0.48	-0.22$\pm$0.23	-1.25$\pm$0.01	-1.24$\pm$0.01
GRB 091208B	14.9	1.0633	33$\pm$2	0.11$\pm$0.14	1.23$\pm$0.07	-0.4$\pm$0.12	0.58$\pm$0.29	-1$\pm$0.03	-1.08$\pm$0.03
GRB 100219A	18.8	4.7	3.7$\pm$0.6	0.21$\pm$0.26	4.64$\pm$1.44	0.83$\pm$0.12	0.28$\pm$0.35	-0.83$\pm$0.07	-1.33$\pm$0.07
GRB 100302A	17.9	4.813	3.1$\pm$0.4	0.03$\pm$0.15	0.96$\pm$0.1	0.22$\pm$0.2	0.31$\pm$0.25	-0.89$\pm$0.05	-1.1$\pm$0.05
GRB 100418A	7	0.6235	3.4$\pm$0.5	-0.16$\pm$0.11	1.43$\pm$0.15	1.69$\pm$0.12	-1.84$\pm$0.11	-2.46$\pm$0.06	-2.42$\pm$0.06
GRB 100424A	104	2.465	15$\pm$1	-0.23$\pm$0.25	2.36$\pm$0.13	-1.08$\pm$0.05	2.51$\pm$0.21	-0.67$\pm$0.03	-0.76$\pm$0.03
GRB 100425A	37	1.755	4.7$\pm$0.9	0.22$\pm$0.13	1.18$\pm$0.16	0.55$\pm$0.22	-0.54$\pm$0.3	-1.43$\pm$0.08	-1.25$\pm$0.08
GRB 100513A	84	4.772	14$\pm$1	0.24$\pm$0.24	1.14$\pm$0.2	0.02$\pm$0.29	0.77$\pm$0.28	-0.24$\pm$0.03	-0.53$\pm$0.03
GRB 100615A	39	1.398	50$\pm$1	0.16$\pm$0.07	1.45$\pm$0.23	0.86$\pm$0.17	0.38$\pm$0.22	-0.59$\pm$0.01	-0.64$\pm$0.01
GRB 100621A	63.6	0.542	210$\pm$0	0.47$\pm$0.08	1.52$\pm$0.11	1.2$\pm$0.18	-0.86$\pm$0.3	-0.79$\pm$0	-0.81$\pm$0
GRB 100704A	197.5	3.6	60$\pm$2	0.25$\pm$0.09	1.41$\pm$0.09	0.36$\pm$0.14	1.13$\pm$0.25	0.2$\pm$0.01	0.03$\pm$0.01
GRB 100814A	174.5	1.44	90$\pm$2	0.37$\pm$0.06	2.4$\pm$0.27	1.84$\pm$0.07	-0.47$\pm$0.39	-0.31$\pm$0.01	-0.52$\pm$0.01
GRB 100901A	439	1.408	21$\pm$3	0.01$\pm$0.04	1.47$\pm$0.04	1.19$\pm$0.02	0.1$\pm$0.02	-0.96$\pm$0.06	-1.14$\pm$0.06
GRB 100902A	428.8	4.5	32$\pm$2	0.65$\pm$0.03	4.69$\pm$1.59	2.18$\pm$0.07	-0.34$\pm$0.16	0.08$\pm$0.03	0.07$\pm$0.03
GRB 100906A	114.4	1.727	120$\pm$0	0.38$\pm$0.13	2.5$\pm$0.32	0.7$\pm$0.15	0.3$\pm$0.39	-0.04$\pm$0	-0.13$\pm$0
GRB 101219B	34	0.5519	21$\pm$4	0.11$\pm$0.24	0.8$\pm$0.11	1.22$\pm$0.23	-2.08$\pm$0.22	-1.77$\pm$0.08	-1.86$\pm$0.08
GRB 110213A	48	1.46	59$\pm$4	-0.41$\pm$0.26	1.95$\pm$0.06	0.01$\pm$0.08	1.42$\pm$0.3	-0.48$\pm$0.03	-0.55$\pm$0.03
GRB 110715A	13	0.82	118$\pm$2	0.35$\pm$0.1	0.98$\pm$0.01	-0.84$\pm$0.09	1.32$\pm$0.1	-0.67$\pm$0.01	-0.87$\pm$0.01
GRB 110808A	48	1.348	3.3$\pm$0.8	0.03$\pm$0.19	1.18$\pm$0.23	1.11$\pm$0.27	-1.12$\pm$0.3	-1.8$\pm$0.09	-1.68$\pm$0.09
GRB 111008A	63.46	5	53$\pm$3	0.01$\pm$0.07	1.39$\pm$0.07	-0.06$\pm$0.08	1.79$\pm$0.19	0.37$\pm$0.02	0.26$\pm$0.02
GRB 111123A	290	3.1516	73$\pm$3	0.24$\pm$0.22	1.86$\pm$0.3	0.7$\pm$0.14	0.51$\pm$0.24	0.2$\pm$0.02	0$\pm$0.02
GRB 111209A	-	0.677	360$\pm$10	0.33$\pm$0.03	15.11$\pm$2.02	1.5$\pm$0.06	-1.57$\pm$0.48	-0.36$\pm$0.01	-0.47$\pm$0.01
GRB 111228A	101.2	0.71627	85$\pm$2	0.14$\pm$0.11	1.32$\pm$0.06	0.77$\pm$0.09	-0.4$\pm$0.16	-0.93$\pm$0.01	-0.87$\pm$0.01
GRB 111229A	25.4	1.3805	3.4$\pm$0.7	-0.14$\pm$0.09	3.15$\pm$1.13	0.57$\pm$0.09	-0.14$\pm$0.25	-1.77$\pm$0.08	-1.82$\pm$0.08
GRB 120118B	23.26	2.943	18$\pm$1	-0.17$\pm$0.18	1.25$\pm$0.22	0.04$\pm$0.21	0.79$\pm$0.21	-0.46$\pm$0.02	-0.41$\pm$0.02
GRB 120326A	69.6	1.798	26$\pm$3	-0.23$\pm$0.13	1.99$\pm$0.1	1.26$\pm$0.03	0.26$\pm$0.05	-0.67$\pm$0.05	-0.93$\pm$0.05
GRB 120327A	62.9	2.813	36$\pm$1	0.08$\pm$0.17	1.55$\pm$0.12	-0.4$\pm$0.12	1.46$\pm$0.25	-0.19$\pm$0.01	-0.47$\pm$0.01
GRB 120404A	38.7	2.876	16$\pm$1	0.05$\pm$0.17	2.05$\pm$0.28	-0.04$\pm$0.12	0.91$\pm$0.2	-0.53$\pm$0.03	-0.62$\pm$0.03
GRB 120422A	5.35	0.28	2.3$\pm$0.4	0.16$\pm$0.12	1.31$\pm$0.39	2.07$\pm$0.25	-3.46$\pm$0.33	-3.35$\pm$0.07	-3.44$\pm$0.07
GRB 120521C	26.7	6	11$\pm$1	0.07$\pm$0.19	3.03$\pm$1.57	0.51$\pm$0.13	0.31$\pm$0.27	-0.2$\pm$0.04	-0.42$\pm$0.04
GRB 120712A	14.7	4.1745	18$\pm$1	-0.04$\pm$0.41	3.26$\pm$1.67	1.65$\pm$0.15	-0.55$\pm$0.32	-0.22$\pm$0.02	-0.67$\pm$0.02
GRB 120802A	50	3.796	19$\pm$3	0.1$\pm$0.11	0.87$\pm$0.37	0.18$\pm$0.26	0.72$\pm$0.27	-0.26$\pm$0.06	-0.8$\pm$0.06
GRB 120811C	26.8	2.671	30$\pm$3	0.3$\pm$0.13	1.26$\pm$0.17	-0.14$\pm$0.17	1.16$\pm$0.18	-0.31$\pm$0.04	-0.65$\pm$0.04
GRB 120922A	173	3.1	62$\pm$7	0.09$\pm$0.24	1.09$\pm$0.06	-0.27$\pm$0.15	1.3$\pm$0.15	0.11$\pm$0.05	-0.14$\pm$0.05
GRB 121024A	69	2.298	11$\pm$1	0.24$\pm$0.29	1.52$\pm$0.28	0.59$\pm$0.18	0.18$\pm$0.2	-0.86$\pm$0.04	-1.16$\pm$0.04
GRB 121128A	23.3	2.2	69$\pm$4	0.2$\pm$0.19	1.67$\pm$0.12	-0.42$\pm$0.1	1.58$\pm$0.25	-0.09$\pm$0.02	-0.43$\pm$0.02
GRB 121211A	182	1.023	13$\pm$2.67	0.09$\pm$0.22	1.32$\pm$0.15	0.64$\pm$0.16	-0.46$\pm$0.22	-1.44$\pm$0.08	-1.33$\pm$0.08
GRB 130131B	4.3	2.539	3.4$\pm$0.4	0.23$\pm$0.33	1.47$\pm$0.41	-0.56$\pm$0.26	1.03$\pm$0.3	-1.29$\pm$0.05	-1.76$\pm$0.05
GRB 130408A	28	3.758	23$\pm$4	0.28$\pm$0.32	3.79$\pm$0.74	0.76$\pm$0.07	0.6$\pm$0.24	-0.18$\pm$0.07	-0.67$\pm$0.07
GRB 130420A	123.5	1.297	71$\pm$3	0.32$\pm$0.13	1.08$\pm$0.07	0.2$\pm$0.16	0.08$\pm$0.22	-0.5$\pm$0.02	-0.67$\pm$0.02
GRB 130511A	5.43	1.3033	2.2$\pm$0.4	0.38$\pm$0.13	1.55$\pm$0.19	-0.37$\pm$0.22	0.08$\pm$0.41	-2$\pm$0.07	-2.24$\pm$0.07
GRB 130514A	204	3.6	91$\pm$2	0.01$\pm$0.36	1.23$\pm$0.07	-0.18$\pm$0.2	1.26$\pm$0.24	0.39$\pm$0.01	0.25$\pm$0.01
GRB 130603B	0.18	0.3564	6.3$\pm$0.3	0.14$\pm$0.11	1.92$\pm$0.23	0.37$\pm$0.16	-1.06$\pm$0.33	-2.69$\pm$0.02	-2.85$\pm$0.02
GRB 130606A	276.58	5.913	29$\pm$2	-0.01$\pm$0.19	1.91$\pm$0.25	0.14$\pm$0.12	1.3$\pm$0.23	0.22$\pm$0.03	-0.19$\pm$0.03
GRB 130612A	4	2.006	2.3$\pm$0.5	0.11$\pm$0.27	1.44$\pm$0.34	0.06$\pm$0.24	-0.17$\pm$0.26	-1.64$\pm$0.09	-2.66$\pm$0.09
GRB 131030A	41.1	1.293	290$\pm$0	0.54$\pm$0.12	1.21$\pm$0.03	-0.75$\pm$0.09	1.83$\pm$0.13	0.11$\pm$0	-0.14$\pm$0
GRB 131103A	17.3	0.599	8.2$\pm$1	0.03$\pm$0.19	1.07$\pm$0.04	-0.23$\pm$0.14	-0.16$\pm$0.14	-2.11$\pm$0.05	-2.12$\pm$0.05
GRB 131105A	112.3	1.686	71$\pm$5	0.15$\pm$0.13	1.22$\pm$0.06	0.23$\pm$0.1	0.64$\pm$0.1	-0.29$\pm$0.03	-0.52$\pm$0.03
GRB 140114A	139.7	3	32$\pm$1	0.03$\pm$0.13	1.48$\pm$0.35	0.77$\pm$0.18	0.21$\pm$0.22	-0.2$\pm$0.01	-0.16$\pm$0.01
GRB 140206A	93.6	2.73	160$\pm$3	0.19$\pm$0.08	1.37$\pm$0.05	-0.3$\pm$0.1	2.06$\pm$0.17	0.43$\pm$0.01	-0.11$\pm$0.01
GRB 140213A	60	1.2076	120$\pm$0	0.7$\pm$0.08	1.58$\pm$0.13	1.28$\pm$0.17	-0.18$\pm$0.33	-0.33$\pm$0	-0.4$\pm$0
GRB 140304A	15.6	5.283	12$\pm$1	-0.79$\pm$0.67	2.38$\pm$0.2	-0.99$\pm$0.12	2.85$\pm$0.38	-0.24$\pm$0.03	-0.81$\pm$0.03
GRB 140430A	173.6	1.6	11$\pm$2	0.03$\pm$0.22	1.13$\pm$0.25	0.59$\pm$0.22	-0.22$\pm$0.28	-1.14$\pm$0.07	-1.14$\pm$0.07
GRB 140512A	154.8	0.725	140$\pm$3	0.66$\pm$0.04	1.77$\pm$0.14	0.99$\pm$0.1	-0.12$\pm$0.19	-0.71$\pm$0.01	-0.84$\pm$0.01
GRB 140518A	60.5	4.707	10$\pm$1	0.04$\pm$0.15	1.7$\pm$0.33	-0.31$\pm$0.12	1.32$\pm$0.24	-0.39$\pm$0.04	-1.21$\pm$0.04
GRB 140614A	720	4.233	13$\pm$4	0.4$\pm$0.18	1.64$\pm$0.26	0.07$\pm$0.21	0.76$\pm$0.3	-0.35$\pm$0.12	-0.71$\pm$0.12
GRB 140629A	42	2.275	24$\pm$2	0.37$\pm$0.12	1.87$\pm$0.17	-0.01$\pm$0.21	0.95$\pm$0.39	-0.52$\pm$0.03	-0.6$\pm$0.03
GRB 140703A	67.1	3.14	39$\pm$3	0.41$\pm$0.14	2.38$\pm$0.44	0.53$\pm$0.13	1.04$\pm$0.36	-0.08$\pm$0.03	-0.24$\pm$0.03
GRB 140903A	0.3	0.351	1.4$\pm$0.1	0.03$\pm$0.1	1.56$\pm$0.34	0.94$\pm$0.19	-1.67$\pm$0.21	-3.36$\pm$0.03	-3.36$\pm$0.03
GRB 141004A	3.92	0.57	6.7$\pm$0.3	0.24$\pm$0.16	1.87$\pm$0.26	0.03$\pm$0.17	-0.77$\pm$0.3	-2.24$\pm$0.02	-2.27$\pm$0.02
GRB 141026A	146	3.35	13$\pm$1	0.03$\pm$0.15	1.65$\pm$0.51	1$\pm$0.19	0.01$\pm$0.24	-0.51$\pm$0.03	-0.29$\pm$0.03
GRB 141121A	549.9	1.47	53$\pm$4	0.06$\pm$0.16	2.48$\pm$0.5	2.1$\pm$0.11	-1.17$\pm$0.26	-0.52$\pm$0.03	-0.63$\pm$0.03
GRB 150323A	149.6	0.593	61$\pm$2	0.14$\pm$0.17	1.31$\pm$0.22	0.84$\pm$0.2	-1.45$\pm$0.29	-1.25$\pm$0.01	-1.28$\pm$0.01
GRB 150403A	40.9	2.06	170$\pm$3	0.29$\pm$0.03	1.44$\pm$0.02	-0.31$\pm$0.03	2.48$\pm$0.09	0.25$\pm$0.01	-0.12$\pm$0.01
GRB 150423A	0.22	1.394	0.63$\pm$0.1	0.08$\pm$0.23	1.73$\pm$0.75	-0.14$\pm$0.49	-0.58$\pm$0.51	-2.49$\pm$0.06	-2.93$\pm$0.06
GRB 150424A	91	3	15$\pm$1	0.35$\pm$0.09	1.51$\pm$0.19	0.91$\pm$0.19	-0.03$\pm$0.35	-0.53$\pm$0.03	-0.99$\pm$0.03
GRB 150910A	112.2	1.359	48$\pm$4	0.51$\pm$0.07	2.47$\pm$0.11	0.52$\pm$0.04	0.9$\pm$0.13	-0.63$\pm$0.03	-0.85$\pm$0.03
GRB 151027A	129.69	0.81	78$\pm$2	0.02$\pm$0.05	1.77$\pm$0.05	0.37$\pm$0.03	0.85$\pm$0.08	-0.86$\pm$0.01	-0.93$\pm$0.01
GRB 151027B	80	4.063	15$\pm$3	0.02$\pm$0.18	1.33$\pm$0.19	0.36$\pm$0.21	0.7$\pm$0.26	-0.32$\pm$0.08	-0.44$\pm$0.08
GRB 151112A	19.32	4.1	9.4$\pm$1.2	-0.01$\pm$0.13	1.67$\pm$0.3	0.68$\pm$0.21	0.63$\pm$0.36	-0.51$\pm$0.05	-0.67$\pm$0.05
GRB 151215A	17.8	2.59	3.1$\pm$0.7	0.21$\pm$0.22	1.6$\pm$0.67	0.11$\pm$0.4	0.32$\pm$0.29	-1.32$\pm$0.09	-1.32$\pm$0.09
GRB 160121A	12	1.96	6.1$\pm$0.5	0.02$\pm$0.11	1.58$\pm$0.61	0.78$\pm$0.25	-0.12$\pm$0.45	-1.23$\pm$0.03	-1.34$\pm$0.03
GRB 160227A	316.5	2.38	31$\pm$2	0.04$\pm$0.1	1.43$\pm$0.14	0.89$\pm$0.14	0.41$\pm$0.24	-0.38$\pm$0.03	-1.04$\pm$0.03
GRB 160303A	5	2.3	1.5$\pm$0.3	0$\pm$0.21	1.68$\pm$0.76	0.65$\pm$0.34	-0.78$\pm$0.53	-1.72$\pm$0.08	-2.23$\pm$0.08
GRB 160314A	8.73	0.726	2.8$\pm$0.4	0.14$\pm$0.28	2.27$\pm$1.13	2.09$\pm$0.22	-2.56$\pm$0.39	-2.4$\pm$0.06	-2.52$\pm$0.06
GRB 160327A	28	4.99	14$\pm$1	-0.22$\pm$0.31	1.61$\pm$0.15	-0.38$\pm$0.13	1.53$\pm$0.18	-0.21$\pm$0.03	-0.33$\pm$0.03
GRB 160804A	144.2	0.736	114$\pm$3	0.26$\pm$0.17	0.94$\pm$0.12	0.78$\pm$0.2	-1.07$\pm$0.34	-0.78$\pm$0.01	-0.93$\pm$0.01
GRB 161108A	105.1	1.159	11$\pm$1	0.24$\pm$0.12	0.95$\pm$0.19	1.29$\pm$0.21	-1.08$\pm$0.39	-1.4$\pm$0.04	-1.45$\pm$0.04
GRB 161117A	125.7	1.549	200$\pm$0	0.1$\pm$0.43	1.2$\pm$0.06	0.42$\pm$0.12	0.44$\pm$0.15	0.1$\pm$0	-0.23$\pm$0
GRB 170113A	20.66	1.968	6.7$\pm$0.7	0.08$\pm$0.12	1.29$\pm$0.06	-0.08$\pm$0.1	1.08$\pm$0.17	-1.19$\pm$0.04	-1.78$\pm$0.04
GRB 170202A	46.2	3.65	33$\pm$1	-0.42$\pm$0.28	1.15$\pm$0.05	-0.43$\pm$0.1	1.61$\pm$0.19	-0.05$\pm$0.01	-0.26$\pm$0.01
GRB 170519A	216.4	0.818	11$\pm$2	0.03$\pm$0.17	1.44$\pm$0.19	0.47$\pm$0.16	-0.32$\pm$0.26	-1.7$\pm$0.07	-1.72$\pm$0.07
GRB 170607A	-	0.557	75$\pm$3	0.03$\pm$0.1	1.09$\pm$0.07	0.89$\pm$0.1	-0.8$\pm$0.16	-1.21$\pm$0.02	-1.29$\pm$0.02
GRB 170705A	217.3	2.01	95$\pm$3	0.29$\pm$0.07	1.29$\pm$0.09	0.78$\pm$0.12	0.68$\pm$0.2	-0.02$\pm$0.01	-0.19$\pm$0.01
GRB 170714A	-	0.793	28$\pm$3	0.54$\pm$0.01	4.97$\pm$0.16	0.89$\pm$0.01	0.97$\pm$0.03	-1.33$\pm$0.04	-1.39$\pm$0.04
GRB 171205A	189.4	0.0368	36$\pm$3	-0.05$\pm$0.09	1.14$\pm$0.15	2.02$\pm$0.13	-4.63$\pm$0.1	-3.96$\pm$0.03	-3.97$\pm$0.03
GRB 171222A	174.8	2.409	19$\pm$2	0.02$\pm$0.13	1.21$\pm$0.42	1.47$\pm$0.33	-0.72$\pm$0.33	-0.58$\pm$0.04	-0.55$\pm$0.04
GRB 180115A	40.9	2.487	7.6$\pm$1.1	0.15$\pm$0.14	1.41$\pm$0.13	0$\pm$0.14	0.68$\pm$0.24	-0.96$\pm$0.06	-1.14$\pm$0.06
GRB 180325A	94.1	2.25	65$\pm$2	0.45$\pm$0.11	2.38$\pm$0.69	0.16$\pm$0.27	1.46$\pm$0.32	-0.1$\pm$0.01	-0.52$\pm$0.01
GRB 180329B	210	1.998	33$\pm$3	0.03$\pm$0.13	1.56$\pm$0.16	0.26$\pm$0.12	0.42$\pm$0.17	-0.49$\pm$0.04	-0.98$\pm$0.04
GRB 180404A	35.2	1	13$\pm$1	0.02$\pm$0.13	1.48$\pm$0.38	0.89$\pm$0.33	-1.09$\pm$0.57	-1.46$\pm$0.03	-1.47$\pm$0.03
GRB 180720B	-	0.654	860$\pm$10	0.67$\pm$0.09	1.99$\pm$0.45	1.33$\pm$0.17	0.32$\pm$0.49	-0.01$\pm$0.01	-0.15$\pm$0.01

Table 4. Table 4: The Bromberg criteria test on the “internal plateau” GRBs. Except for two GRBs with T 90 subscript 𝑇 90 T_{90} unavailable (GRBs 111209A and 170714A), all other events are explicitly identified as long GRBs.

GRB name	$T_{90}$(s)	PL	$f_{NC}$	Type
GRB 050730	156.5	1.53	0.08 ()	Long
GRB 060607A	102.2	1.47	0.10 ()	Long
GRB 070110	88.4	1.58	0.10 ()	Long
GRB 100219A	18.8	1.34	0.20 ()	Long
GRB 100902A	428.8	1.98	0.06 ()	Long
GRB 111209A	–	1.48	–	–
GRB 111229A	25.4	1.85	0.17 ()	Long
GRB 120521C	26.7	1.73	0.17 ()	Long
GRB 120712A	14.7	1.36	0.23 ()	Long
GRB 130408A	28.0	1.28	0.17 ()	Long
GRB 170714A	–	1.76	–	–

Table 5. Table 5: Kendall’s Tau test for a few pairs of parameters. The coefficients indicate that no significant correlations existed between these parameters.

	$\tau$	$p$
log$T_a$ - $z$	-0.24	2.55e-6
log$E_{\gamma ,\mathrm{iso}}$ - log$T_a$	-0.11	0.038
${\alpha }_1$ - ${\alpha }_2$	0.03	0.57
log$L_X$ - ${\alpha }_1$	0.11	0.024
log$T_a$ - ${\alpha }_1$	-0.11	0.028
log$E_{\gamma ,\mathrm{iso}}$ - ${\alpha }_1$	0.12	0.019
log$L_X$ - ${\alpha }_2$	0.04	0.41
log$T_a$ - ${\alpha }_2$	0.16	0.002
log$E_{\gamma ,\mathrm{iso}}$ - ${\alpha }_2$	0.13	0.014
log$L_X$ - log$T_{90}$	0.06	0.22
log$T_a$ - log$T_{90}$	0.17	0.0008

Equations

$$F_X(t)=F_{X0}{\left[{\left(\frac{t}{T_0}\right)}^{{\alpha }_1\omega }+{\left(\frac{t}{T_0}\right)}^{{\alpha }_2\omega }\right]}^{-1/\omega }.$$

$$L_X=\frac{4\pi D_{\mathrm{L}}^2(z)F_{X0}}{(1+z{)}^{1-{\beta }_X}},$$

$$E_{\gamma ,\mathrm{iso}}^{\prime }=\frac{4\pi D_{\mathrm{L}}^2(z)S}{1+z},$$

$$E_{\gamma ,\mathrm{iso}}=E_{\gamma ,\mathrm{iso}}^{\prime }\times \frac{{\int }_{15/1+z}^{150/1+z}E\Phi (E)dE}{{\int }_{15}^{150}E\Phi (E)dE},$$

$$\log (L_X/10^{47}\mathrm{erg}\cdot {\mathrm{s}}^{-1})=a+b\log (T_a/10^3\mathrm{s})+c\log (E_{\gamma ,\mathrm{iso}}/10^{53}\mathrm{erg}),$$

$$\mathcal{L}\left(a,b,c,{\sigma }_{\mathrm{int}}\right)\propto \prod _i\frac{\exp \left[-(y_i-a-b{x}_{1,i}-c{x}_{2,i}{)}^2/(2({\sigma }_{\mathrm{int}}^2+{\sigma }_{y_i}^2+b^2{\sigma }_{x_{1,i}}^2+c^2{\sigma }_{x_{2,i}}^2))\right]}{\sqrt{{\sigma }_{\mathrm{int}}^2+{\sigma }_{y_i}^2+b^2{\sigma }_{x_{1,i}}^2+c^2{\sigma }_{x_{2,i}}^2}},$$

$$\log (L_X/10^{47}\mathrm{erg}\cdot {\mathrm{s}}^{-1})=(1.11\pm 0.04)+(-1.01\pm 0.05)\log (T_a/10^3\mathrm{s})+(0.84\pm 0.04)\log (E_{\gamma ,\mathrm{iso}}/10^{53}\mathrm{erg}).$$

Full text

gamma-ray burst: general – methods: statistical

1 Introduction

Gamma-ray bursts (GRBs) are erratic $\gamma$-ray flashes in the universe, lasting from milliseconds to as long as thousands of seconds111Some ultra-long GRBs even last for tens of thousands of seconds (Levan et al., 2014)., usually with a non-thermal spectrum, isotropically distributed on the sky (Band et al., 1993; Kouveliotou et al., 1993; Meegan et al., 1992). The isotropic energy release of the prompt GRB emission ranges from $10^{48}$ to $10^{55}$ erg (Kumar & Zhang, 2015), making GRBs the most energetic stellar explosions in our universe. In the era of BeppoSAX (Fishman & Meegan, 1995), multi-wavelength (from X-ray to radio) afterglows following the prompt emission were detected, which help to localize GRBs precisely. The first redshift measurement was made for GRB 970508 (Metzger et al., 1997). Its spectrum revealed a redshift of 0.835, which formally confirms its cosmological origin. GRBs can even be observed up to $z\sim$ 10 with current detectors (Cucchiara et al., 2011). After decades of observations and researches, people have obtained a general picture of GRB physics. The widely accepted model to explain the origin of GRBs is the so called “fireball” model (Rees & Meszaros, 1992; Piran et al., 1993; Wijers et al., 1997; Mészáros, 2006). The prompt emission is considered to be produced by internal shocks due to the interaction of ejecta in the fireball, while the broadband afterglow is produced by the interaction between the GRB ejecta and the circumburst medium (Mészáros & Rees, 1997; Vietri, 1997; Tavani, 1997; Waxman, 1997; Sari, 1997; Huang et al., 1999, 2000).

It has long been found that GRBs can be grouped into two distinct classes by considering their durations and spectral hardness (Kouveliotou et al., 1993; Bromberg et al., 2013). Long GRBs typically last for 20 – 30 s, while short GRBs typically last for 0.2 – 0.3 s. Besides, short GRBs are on average harder than long GRBs in hardness ratio. Long GRBs, with a good number being identified to be associated with core-collapse supernovae (Galama et al., 1998; Hjorth et al., 2003; Stanek et al., 2003; Campana et al., 2006; Xu et al., 2013), are generally believed to originate from the deaths of massive stars. In contrast, the favored scenario for short GRBs is the coalescence of two compact stars, especially two neutron stars (Paczynski, 1986; Eichler et al., 1989, NS-NS), or a neutron star and a black hole (Paczynski, 1991, NS-BH). In both the collapsar models and the coalescence models, the violent explosion would generate a stellar-size, hyper-accreting black hole, or a rapidly-spinning, strongly-magnetized neutron star (Usov, 1992; Thompson, 1994; Dai & Lu, 1998; Popham et al., 1999; Rosswog et al., 2003; Lei et al., 2013). Recently, the detection of gravitational waves with the advanced Laser Interferometer Gravitational Wave Observatory (LIGO) opened a “multi-messenger” era in astronomy studies(Abbott et al., 2016a, b, 2017a, 2017b). The association of the gravitational wave event GW170817 and the short event of GRB 170817A confirms the binary neutron star coalescence scenario for short GRBs (Abbott et al., 2017a; Goldstein et al., 2017).

Although important progresses have been made toward understanding GRBs, some crucial issues still remain unsolved. With a cosmological origin, GRBs could be used as a powerful tool to probe the early universe. Nevertheless, compared to type Ia supernovae (SNe Ia), GRBs are not perfect standard candles. The luminosities of GRBs cover several orders of magnitude. Their diversity, together with the complex classifications and triggering mechanisms, prevents direct applications of GRBs in cosmology. Interestingly, a few correlations have been found between various GRB parameters, either on GRB prompt emission (Amati et al., 2002; Norris et al., 2000; Ghirlanda et al., 2004, 2009; Yonetoku et al., 2004; Tsutsui et al., 2009; Willingale et al., 2010; Geng & Huang, 2013; Zhang et al., 2018), or on the afterglow (Liang & Zhang, 2005; Oates et al., 2009, 2012; Dainotti et al., 2010, 2016, 2017; Xu & Huang, 2012; Liang et al., 2015). These correlations can provide important clues for studying the physical mechanisms of the GRB central engines. They can also hopefully make GRBs as some kinds of standard candles so that these energetic events can be used to limit cosmological parameters (Fenimore & Ramirez-Ruiz, 2000; Schaefer, 2003, 2007; Dai, 2004; Ghirlanda et al., 2004, 2006; Wang & Dai, 2006; Amati et al., 2008; Wang et al., 2011, 2015). In this aspect, a lot of work still needs to be done because GRBs originate from various mechanisms.

There is a special subclass of GRBs, i.e., GRBs with a plateau phase in the X-ray light curve. They are characterized by a shallow decay phase followed by a normal decay in X-ray afterglow. The X-ray light curve of the shallow decay phase is usually very flat so that it typically shows up as a plateau, while the power-lay index of the subsequent normal decay is usually $\sim-1$. It is usually believed that the plateau phase is due to some kinds of extra energy injection into the external shock, possibly from a long-lasting central engine (Dai & Lu, 1998; Zhang & Mészáros, 2001; Rowlinson et al., 2010, 2013; Bucciantini et al., 2012; Gompertz et al., 2013; Geng et al., 2016). Thus the X-ray plateau observations can help constrain various central engine models. For example, Li et al. (2018) analyzed the Swift/XRT light curves of 101 GRBs that have a plateau phase (and with the redshift being available). They compared the energetics with the maximum energy budget of magnetars ($\sim 2\times 10^{52}$ erg), trying to determine whether the central engine is a magnetar or a black hole. More interestingly, it is also found that there exists another special kind of plateau, known as the “internal plateau”, characterized by a plateau followed by an extremely rapid decay (decay index usually $<-3$). Such a rapid decay at the end of the “internal plateau” is much steeper than that due to the evolution of the synchrotron emission from a decelerating forward shock, but has to invoke some internal dissipation processes. One promising interpretation within the magnetar framework is that the “internal plateau” is due to the continuous energy supply from a “supra-massive” neutron star. The neutron star will finally collapse into a black hole, leading to a sudden switching off of the central engine, which manifests as the subsequent steep decay phase. The collapse of the supra-massive neutron star may either be triggered by spinning down, or by fall-back accretion, e.g., GRB 070110 as studied by Chen et al. (2017) and GRB 170714A as studied by Hou et al. (2018). Comparing with the “internal plateau”, a plateau followed by a normal decay is usually called an “external plateau” due to the fact that it can be well explained by the deceleration of the external shock.

Several interesting correlations are obtained for GRBs with a plateau phase. Dainotti et al. (2008) first found an anti-correlation between the end time of the plateau phase in the GRB rest frame ($T_{a}$) and the corresponding X-ray luminosity at that moment ($L_{X}$). By analyzing a sample of 77 GRBs, they derived the relation as $L_{X}\propto T_{a}^{-1.06\pm 0.27}$(Dainotti et al., 2010). Xu & Huang (2012, hereafter, XH2012) further found a much tighter three-parameter correlation among $T_{a}$, $L_{X}$ and the isotropic energy in the prompt emission ($E_{\gamma,\rm{iso}}$, here we denote it as the L-T-E correlation), e.g., $L_{X}\propto T_{a}^{-0.87\pm 0.09}E_{\gamma,\rm{iso}}^{0.88\pm 0.08}$. Dainotti et al. (2017) later also presented a $L_{X}-T_{a}-L_{\gamma,\rm{peak}}$ correlation between $T_{a}$, $L_{X}$, and the peak luminosity in the prompt emission ($L_{\gamma,\rm{peak}}$), $L_{X}\propto T_{a}^{-0.83\pm 0.10}L_{\rm peak}^{0.64\pm 0.11}$. Interestingly, Si et al. (2018) recently analyzed 50 GRBs which have a plateau phase in the optical afterglow light curve. They also found a similar three-parameter correlation for their sample. Their analysis indicates $L_{b,z}\propto T_{b,z}^{-0.9}E_{\gamma,{\rm iso}}^{0.4}$ and $L_{b,z}\propto T_{b,z}^{-0.9}E_{\rm p,i}^{0.5}$, where $T_{b,z}$ is the break time of the plateau phase in the optical band, $L_{b,z}$ is the corresponding optical luminosity during the plateau phase, and $E_{\rm p,i}$ is the peak energy of the prompt emission. For a comprehensive overview of various GRB correlations, one may refer to Wang et al. (2019), or Zhao et al. (2019).

In this study, we have collected a sample of 174 Swift GRBs that have a plateau phase in the X-ray afterglow light curve. These GRBs, all with the redshift being measured, occurred between March 2005 and August 2018. This sample provides a good opportunity for various statistical analysis. We fit the light curve of each GRB to get relevant parameters of the plateau phase, and then explore potential correlations between various pairs of parameters. Especially, the L-T-E correlation is extensively examined with this enlarged sample, and the physics behind the L-T-E correlation is investigated. The structure of our article is organized as follows. We describe the sample selection and data reduction processes in Section 2. The data are statistically analyzed and the results are presented in Section 3. In Section 4, we briefly summarize our conclusions and discuss the implications.

2 Sample Selection and Data Analysis

After the successful launch of Swift in 2004 (Gehrels et al., 2004), many interesting features were soon discovered in the X-ray afterglows of some GRBs, including X-ray flares and the plateau phase (Zhang et al., 2006; Nousek et al., 2006). In this paper, we mainly analyze Swift GRBs with a plateau phase in the X-ray light curve. Our sample are selected from the GRBs occurred between March 2005 and August 2018, all with the redshift being measured. The observed X-ray data are taken from the Swift GRB light curve repository (Evans et al., 2007, 2009), with the spectrum range of the light curve being 0.3 — 10 keV. We select our GRBs by the following four criteria: (1) There should be an obvious flat segment in the X-ray light curve that could be reliably identified as a plateau phase. To be more specific, we require that the power-law index of the plateau phase should be in a range of $-1.0$ — $+1.0$. We exclude any GRBs that do not meet this requirement. (2) There are abundant observational data points during the plateau phase. This is to ensure that the light curve can be well defined during the fitting process so that we could correctly extract the key parameters relevant to the plateau phase without any difficulties. (3) There are no flares observed during the plateau phase. Again it is to ensure a smooth fit of the plateau phase. (4) The redshift should be available for the event, so that we could calculate the isotropic $\gamma$-ray energy release ($E_{\rm\gamma,iso}$). Also, with the redshift, we will be able to derive the intrinsic $T_{a}$ parameter by considering the time dilation effect. With these four criteria, we finally obtained a sample of 174 GRBs. The redshift range of our sample is 0.04 — 8.0. We note that seven short GRBs ($T_{90}<2$ s) are interestingly included in this sample, as listed in Table 2.