Classification of Photospheric Emission in Short GRBs

We select short GRBs, i.e., GRBs with a duration T₉₀ < 2 s, detected by the GBM on board the Fermi Gamma-ray Space Telescope during the first 11 yr of its mission. We scan all the short bursts for which automatic spectral fits are performed on the time-resolved data around the peak flux within the time interval given in the GBM catalog (von Kienlin et al. 2014). We find a total of 147 short bursts for which spectral fits can be carried out and analyzed. All the data are taken from the Fermi/GBM burst catalog published at HEASARC.⁴ We further set a limit for at least one time bin to have a statistical significance (see Section 2.2 for the definition) S ≥ 15 in each pulse; we thus select 70 pulses from 68 short GRBs as a final sample. They are listed in Table 1.

Table 1.  A Sample of 70 Pulses from 68 Short GRBs Used in This Study

bn	T90	Detectors	Δ Tsrc	${\rm{\Delta }}{T}_{\mathrm{bkg},1}$	${\rm{\Delta }}{T}_{\mathrm{bkg},2}$	N	αmax	S	Epk	Flux
	(s)		(s)	(s)	(s)				(keV)	($\,\mathrm{erg}\,\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$)
081209981	0.19 ± 0.14	(n8)nbb1	−1.7 to 2.	−20. to −7.	7. to 13.	5	$-{0.42}_{-0.13}^{+0.09}$	19	${1080}_{-250}^{+190}$	${1.6}_{-0.9}^{+2.2}\times {10}^{-5}$
081216531	0.8 ± 0.4	n7(n8)nbb1	−1.7 to 2.	−20. to −7.	15. to 35.	8	$-{0.37}_{-0.07}^{+0.06}$	30	${1170}_{-110}^{+140}$	${1.6}_{-0.7}^{+1.1}\times {10}^{-5}$
081223419	0.58 ± 0.14	n6(n7)n9b1	−2. to 2.	−20. to −7.	15. t0 35.	4	$-{0.42}_{-0.17}^{+0.13}$	18	${200}_{-50}^{+30}$	${2.0}_{-1.2}^{+3.4}\times {10}^{-6}$
090108020	0.71 ± 0.14	(n1)n2n5b0	−2. to 2.	−20. to −7.	15. to 35.	6	$-{0.22}_{-0.15}^{+0.11}$	25	${140}_{-20}^{+10}$	${2.6}_{-1.3}^{+2.8}\times {10}^{-6}$
090227772	0.31 ± 0.02	n0(n1)n2b0	−2. to 2.	−20. to −7.	15. to 35.	9	$+{0.07}_{-0.07}^{+0.07}$	36	${1720}_{-140}^{+110}$	${8.2}_{-3.4}^{+5.7}\times {10}^{-5}$
090228204	0.45 ± 0.14	(n0)n1n3b0	−2. to 2.	−20. to −7.	15. to 35.	11	$-{0.01}_{-0.07}^{+0.06}$	45	${640}_{-50}^{+50}$	${13}_{-5.7}^{+10}\times {10}^{-5}$
090308734	1.67 ± 0.29	(n3)n6n7b0	−2. to 2.5	−20. to −7.	15. to 35.	6	$-{0.42}_{-0.10}^{+0.09}$	19	${750}_{-160}^{+100}$	${3.0}_{-1.6}^{+3.5}\times {10}^{-6}$
090328713	0.2 ± 1.0	(n9)nanbb1	−1. to 5.	−20. to −7.	15. to 35.	3	$-{0.87}_{-0.08}^{+0.06}$	17	${1700}_{-570}^{+310}$	${5.6}_{-2.4}^{+3.6}\times {10}^{-6}$
090510016	0.96 ± 0.14	(n6)n7n9b1	−1.7 to 2.	−20. to −7.	15. to 35.	9	$-{0.62}_{-0.05}^{+0.04}$	32	${2850}_{-340}^{+270}$	${2.1}_{-0.6}^{+1.0}\times {10}^{-5}$
090617208	0.19 ± 0.14	n0(n1)n3b0	−1. to 2.5	−20. to −7.	15. to 35.	4	$+{0.05}_{-0.19}^{+0.18}$	16	${920}_{-310}^{+120}$	${1.1}_{-0.9}^{+3.7}\times {10}^{-5}$
090802235	0.04 ± 0.02	n2(n5)b0	−3. to 8	−20. to −7.	20. to 40.	4	$-{0.48}_{-0.14}^{+0.14}$	21	${480}_{-150}^{+60}$	${1.4}_{-0.9}^{+2.7}\times {10}^{-5}$
090907808	0.8 ± 0.3	n6(n7)n9b1	−2. to 2.	−20. to −7.	15. to 35.	3	$-{0.09}_{-0.16}^{+0.15}$	17	${450}_{-100}^{+60}$	${1.8}_{-1.2}^{+4.4}\times {10}^{-6}$
100223110	0.26 ± 0.09	n7(n8)b1	−2. to 2.1	−20. to −7.	15. to 35.	5	$-{0.14}_{-0.12}^{+0.11}$	20	${1250}_{-260}^{+150}$	${0.9}_{-0.5}^{+1.4}\times {10}^{-5}$
100629801	0.8 ± 0.4	(na)nbb1	−2. to 2.	−20. to −7.	15. to 35.	5	$-{0.76}_{-0.18}^{+0.15}$	16	${230}_{-80}^{+30}$	${2.8}_{-1.8}^{+5.6}\times {10}^{-6}$
100811108	0.39 ± 0.09	(n7)n9nbb1	−2. to 2.	−20. to −7.	15. to 35.	3	$-{0.14}_{-0.11}^{+0.09}$	19	${1140}_{-220}^{+140}$	${6.9}_{-3.8}^{+8.2}\times {10}^{-6}$
100827455	0.6 ± 0.4	n6(n7)n8b1	−2. to 2.	−20. to −7.	15. to 35.	5	$-{0.32}_{-0.11}^{+0.12}$	16	${780}_{-170}^{+110}$	${0.8}_{-0.5}^{+1.1}\times {10}^{-5}$
101216721	1.9 ± 0.6	n1n2(n5)b0	−2. to 3.	−20. to −7.	15. to 35.	8	$-{0.54}_{-0.10}^{+0.08}$	37	${180}_{-20}^{+20}$	${3.2}_{-1.4}^{+2.5}\times {10}^{-6}$
110212550	0.07 ± 0.04	n6(n7)n8b1	−1.8 to 1.7	−20. to −7.	15. to 35.	5	$-{0.38}_{-0.14}^{+0.11}$	18	${620}_{-150}^{+90}$	${1.7}_{-1.0}^{+2.9}\times {10}^{-5}$
110526715	0.45 ± 0.05	(n3)n4b0	−1.7 to 2.	−20. to −7.	15. to 35.	3	$-{0.88}_{-0.10}^{+0.09}$	17	${670}_{-240}^{+110}$	${2.2}_{-1.0}^{+1.8}\times {10}^{-6}$
110529034	0.51 ± 0.09	n6n7(n9)b1	−2. to 2.	−20. to −7.	15. to 35.	10	$-{0.52}_{-0.19}^{+0.15}$	15	${780}_{-350}^{+120}$	${1.0}_{-0.7}^{+2.4}\times {10}^{-5}$
110705151	0.19 ± 0.04	(n3)n4n5b0	−1.7 to 2.	−20. to −7.	15. to 35.	8	$-{0.09}_{-0.12}^{+0.11}$	23	${890}_{-180}^{+90}$	${1.6}_{-1.0}^{+2.2}\times {10}^{-5}$
111222619	0.29 ± 0.04	(n8)nbb1	−2. to 2.	−20. to −7.	15. to 35.	7	$-{0.21}_{-0.12}^{+0.11}$	27	${700}_{-110}^{+70}$	${1.5}_{-0.8}^{+1.9}\times {10}^{-5}$
120222021	1.09 ± 0.14	n3n4(n5)b0	−1.7 to 2.	−20. to −7.	15. to 35.	8	$-{0.31}_{-0.26}^{+0.18}$	34	${120}_{-30}^{+20}$	${1.7}_{-1.2}^{+4.9}\times {10}^{-6}$
120323507	0.39 ± 0.04	n0(n3)b0	−2. to 2.	−20. to −7.	10. to 30.	16	$-{0.73}_{-0.14}^{+0.15}$	25	${420}_{-160}^{+40}$	${3.1}_{-1.9}^{+5.6}\times {10}^{-5}$
120519721	1.1 ± 0.5	n7(n8)b1	−1.7 to 2.	−20. to −7.	15. to 35.	6	$-{0.45}_{-0.12}^{+0.09}$	23	${860}_{-190}^{+130}$	${3.9}_{-2.1}^{+5.5}\times {10}^{-6}$
120624309	0.64 ± 0.16	n1n2(na)b0	−2. to 2.	−20. to −7.	15. to 35.	11	$-{0.68}_{-0.05}^{+0.05}$	28	${3590}_{-540}^{+370}$	${2.4}_{-0.7}^{+1.0}\times {10}^{-5}$
120811014	0.45 ± 0.09	n7(n8)b1	−2. to 3.	−20. to −7.	15. to 35.	5	$+{0.01}_{-0.16}^{+0.12}$	21	${1170}_{-230}^{+180}$	${1.0}_{-0.6}^{+2.0}\times {10}^{-5}$
120817168	0.16 ± 0.11	n6(n7)n8b1	−1.5 to 1.7	−20. to −7.	15. to 35.	6	$-{0.50}_{-0.07}^{+0.06}$	30	${1530}_{-290}^{+190}$	${3.9}_{-1.5}^{+2.4}\times {10}^{-5}$
120830297	0.90 ± 0.23	(n0)n1n3b0	−1.7 to 2.	−20. to −7.	15. to 35.	5	$-{0.27}_{-0.10}^{+0.07}$	24	${1100}_{-180}^{+150}$	${3.6}_{-1.7}^{+3.2}\times {10}^{-6}$
121127914	0.6 ± 0.4	(n4)n8b1	−2. to 2.	−20. to −7.	15. to 35.	4	$-{0.53}_{-0.11}^{+0.10}$	19	${1080}_{-270}^{+170}$	${6.6}_{-3.6}^{+8.8}\times {10}^{-6}$
130416770	0.2 ± 0.4	n3(n4)n5b0	−1.7 to 8.	−20. to −7.	25. to 45.	3	$-{0.52}_{-0.09}^{+0.08}$	16	${1100}_{-230}^{+170}$	${1.0}_{-0.5}^{+1.0}\times {10}^{-5}$
130504314	0.39 ± 0.18	(n3)n4b0	−2. to 1.7	−35. to −7.	15. to 30.	7	$-{0.09}_{-0.11}^{+0.08}$	21	${1370}_{-190}^{+150}$	${1.4}_{-0.7}^{+1.7}\times {10}^{-5}$
130628860	0.51 ± 0.14	n7n9(nb)b1	−2. to 2.1	−20. to −7.	15. to 35.	7	$-{0.18}_{-0.13}^{+0.14}$	18	${1120}_{-270}^{+150}$	${1.8}_{-1.2}^{+3.3}\times {10}^{-5}$
130701761	1.60 ± 0.14	(n9)nanbb1	−2. to 3.	−20. to −7.	15. to 35.	8	$-{0.24}_{-0.05}^{+0.18}$	18	${1200}_{-330}^{+30}$	${4.4}_{-3.0}^{+8.6}\times {10}^{-6}$
130804023	0.96 ± 0.09	n6(n7)n9b1	−2. to 2.	−8. to −5.	12. to 29.	10	$-{0.26}_{-0.10}^{+0.10}$	20	${850}_{-150}^{+109}$	${3.1}_{-1.7}^{+3.6}\times {10}^{-5}$
130912358	0.51 ± 0.14	n7(n8)nbb1	−2. to 2.	−20. to −7.	15. to 35.	8	$-{1.02}_{-0.09}^{+0.08}$	18	${1700}_{-870}^{+230}$	${7.2}_{-3.4}^{+6.6}\times {10}^{-6}$
131126163	0.2 ± 0.4	n2(n5)b0	−3 to 1.7	−25. to −10.	15. to 35.	4	$-{0.07}_{-0.16}^{+0.15}$	19	${750}_{-190}^{+100}$	${1.9}_{-1.3}^{+4.3}\times {10}^{-5}$
140209313	1.41 ± 0.27	n9(na)b1	0. to 4.	−20. to −7.	20. to 40.	13	$-{0.19}_{-0.10}^{+0.10}$	44	${220}_{-30}^{+20}$	${1.4}_{-0.7}^{+1.4}\times {10}^{-5}$
140807500	0.5 ± 0.2	n3(n4)n5b0	−2. to 2.	−20. to −7.	15. to 35.	4	$-{0.75}_{-0.09}^{+0.07}$	26	${750}_{-190}^{+110}$	${5.2}_{-2.2}^{+4.0}\times {10}^{-6}$
140901821	0.18 ± 0.04	n9(na)nbb1	−1.5 to 2.	−20. to −7.	15. to 35.	5	$-{0.22}_{-0.06}^{+0.05}$	36	${1200}_{-100}^{+90}$	${2.5}_{-0.9}^{+1.3}\times {10}^{-5}$
141011282	0.08 ± 0.04	n0(n1)n9b0	−1.8 to 1.7	−20. to −7.	15. to 35.	5	$-{0.46}_{-0.11}^{+0.10}$	20	${790}_{-180}^{+100}$	${2.7}_{-1.4}^{+3.4}\times {10}^{-5}$
141105406	1.28 ± 1.03	n6n7(n9)b1	−1.7 to 2.	−20. to −7.	15. to 35.	4	$-{0.42}_{-0.13}^{+0.11}$	18	${500}_{-120}^{+70}$	${2.2}_{-1.4}^{+3.4}\times {10}^{-6}$
141202470	1.41 ± 0.27	(n7)n8nbb1	−1.7 to 3.	−20. to −7.	15. to 35.	4	$-{0.32}_{-0.10}^{+0.08}$	20	${840}_{-150}^{+110}$	${3.5}_{-1.7}^{+3.6}\times {10}^{-6}$
141213300	0.8 ± 0.5	n1(n2)n5b0	−1.7 to 2.	−20. to −7.	15. to 35.	7	$-{0.87}_{-0.16}^{+0.14}$	15	${210}_{-90}^{+30}$	${1.7}_{-1.0}^{+2.9}\times {10}^{-6}$
150118927	0.3 ± 0.1	n7n8(nb)b1	−1.7 to 1.7	−20. to −7.	15. to 35.	6	$-{0.65}_{-0.10}^{+0.08}$	26	${620}_{-160}^{+90}$	${1.4}_{-0.7}^{+1.4}\times {10}^{-5}$
150810485	1.3 ± 1.0	n6n7(nb)b1	−1.7 to 2.	−20. to −7	15. to 35.	6	$-{0.61}_{-0.08}^{+0.07}$	15	${1530}_{-360}^{+250}$	${5.4}_{-2.5}^{+4.4}\times {10}^{-6}$
150811849	0.64 ± 0.14	(n4)n5b0	−2. to 2.	−20. to −7.	15. to 35.	6	$-{0.24}_{-0.11}^{+0.08}$	18	${1600}_{-270}^{+220}$	${8.3}_{-4.2}^{+9.3}\times {10}^{-6}$
150819440	0.96 ± 0.09	n2(na)b1	−0.2 to 0.3	−20. to −7.	15. to 35.	9	$+{0.25}_{-0.17}^{+0.14}$	33	${440}_{-70}^{+50}$	${6.2}_{-4.1}^{+12}\times {10}^{-5}$
150819440	0.96 ± 0.09	n2(na)b1	0.35 to 1.37	−20. to −7.	15. to 35.	11	$-{1.02}_{-0.05}^{+0.04}$	71	${330}_{-40}^{+30}$	${2.0}_{-0.5}^{+0.7}\times {10}^{-5}$
150922234	0.15 ± 0.04	n6(n9)nab1	−1.5 to 2.	−20. to −7.	15. to 35.	7	$-{0.24}_{-0.24}^{+0.16}$	15	${800}_{-270}^{+150}$	${1.1}_{-0.9}^{+4.3}\times {10}^{-5}$
150923864	1.79 ± 0.09	n6(n7)n9b1	−2. to 3.	−20. to −7.	15. to 35.	9	$+{0.06}_{-0.21}^{+0.25}$	15	${140}_{-30}^{+20}$	${1.5}_{-1.1}^{+4.0}\times {10}^{-6}$
151222340	0.8 ± 0.4	(n4)b0	−1.7 to 2.	−20. to −7.	15. to 35.	6	$-{0.43}_{-0.12}^{+0.09}$	16	${1500}_{-320}^{+200}$	${4.6}_{-2.5}^{+6.6}\times {10}^{-6}$
151231568	0.8 ± 0.4	n6(n7)n8b1	−1.7 to 2.	−20. to −7.	15. to 35.	6	$-{0.58}_{-0.11}^{+0.10}$	19	${610}_{-180}^{+70}$	${4.1}_{-2.3}^{+4.7}\times {10}^{-6}$
160408268	1.1 ± 0.6	(n0)n1n3b0	−1.7 to 2.	−20. to −7.	15. to 35.	3	$-{0.77}_{-0.11}^{+0.08}$	17	${1070}_{-380}^{+180}$	${3.0}_{-1.5}^{+3.5}\times {10}^{-6}$
160612842	0.29 ± 0.23	n0(n1)b0	−1.7 to 1.7	−20. to −7.	15. to 35.	4	$-{0.78}_{-0.12}^{+0.08}$	15	${2150}_{-880}^{+510}$	${5.4}_{-2.6}^{+6.4}\times {10}^{-6}$
160726065	0.8 ± 0.4	n0(n1)n2b0	−1.7 to 2.	−20. to −7.	15. to 35.	6	$-{0.79}_{-0.11}^{+0.09}$	21	${460}_{-140}^{+70}$	${4.4}_{-2.0}^{+4.3}\times {10}^{-6}$
160806584	1.7 ± 0.5	(n8)nbb1	−2. to 2.	−20. to −7.	15. to 35.	6	$-{0.41}_{-0.20}^{+0.14}$	19	${190}_{-50}^{+30}$	${2.1}_{-1.4}^{+4.8}\times {10}^{-6}$
160822672	0.1 ± 0.4	n9(na)b1	−2. to 1.7	−20. to −7.	15. to 35.	5	$-{1.26}_{-0.06}^{+0.03}$	23	${290}_{-50}^{+40}$	${12}_{-3.0}^{+4.5}\times {10}^{-5}$
170127067	0.13 ± 0.05	(n4)b0	−2. to 2.	−35. to −15.	35. to 55.	5	$+{0.43}_{-0.11}^{+0.16}$	32	${900}_{-90}^{+70}$	${5.1}_{-3.5}^{+12}\times {10}^{-5}$
170206453	1.17 ± 0.10	(n9)nanbb1	−1.7 to 3.	−20. to −7.	15. to 35.	9	$-{0.13}_{-0.09}^{+0.07}$	39	${370}_{-40}^{+30}$	${1.5}_{-0.6}^{+1.2}\times {10}^{-5}$
170222209	1.67 ± 0.14	n2(n5)b0	−1.7 to 3.	−20. to −7.	20. to 40.	10	$-{1.11}_{-0.13}^{+0.10}$	15	${1550}_{-1120}^{+130}$	${5.8}_{-3.3}^{+7.5}\times {10}^{-6}$
170305256	0.45 ± 0.07	n0(n1)n2b0	−1.7 to 2.	−20. to −7.	15. to 35.	6	$-{0.23}_{-0.10}^{+0.11}$	25	${340}_{-50}^{+40}$	${7.9}_{-4.3}^{+8.0}\times {10}^{-6}$
170708046	0.15 ± 0.05	n7(n8)nbb1	−2 to 1.7	−20. to −7.	15. to 35.	5	$-{0.75}_{-0.10}^{+0.09}$	23	${310}_{-70}^{+40}$	${1.2}_{-0.6}^{+0.9}\times {10}^{-5}$
170816599	1.60 ± 0.14	n7(n8)nbb1	−2. to 3.	−20. to −7.	15. to 35.	5	$-{0.26}_{-0.07}^{+0.06}$	26	${1170}_{-120}^{+100}$	${1.1}_{-0.4}^{+0.6}\times {10}^{-5}$
171108656	0.03 ± 0.02	(na)nbb1	−5. to 1.7	−30. to −10.	10. to 30.	8	$-{0.11}_{-0.18}^{+0.16}$	25	${120}_{-20}^{+10}$	${9.7}_{-6.3}^{+16}\times {10}^{-6}$
171126235	1.47 ± 0.14	n0(n1)n5b0	−1.7 to 3.	−20. to −7.	15. to 35.	9	$+{0.09}_{-0.17}^{+0.14}$	29	${130}_{-20}^{+10}$	${4.4}_{-2.5}^{+6.2}\times {10}^{-6}$
180204109	1.15 ± 0.09	n3(n4)n5b0	−2. to 2.	−20. to −7.	15. to 35.	12	$-{0.75}_{-0.12}^{+0.12}$	16	${1300}_{-660}^{+180}$	${0.8}_{-0.4}^{+1.1}\times {10}^{-5}$
180703949	1.54 ± 0.09	n0n1(n3)b0	−0.30 to 0.68	−20. to −7.	20. to 40.	8	$-{0.35}_{-0.12}^{+0.10}$	39	${100}_{-10}^{+10}$	${4.7}_{-2.2}^{+3.6}\times {10}^{-6}$
180703949	1.54 ± 0.09	n0n1(n3)b0	0.7 to 2.5	−20. to −7.	20. to 40.	12	$-{0.24}_{-0.05}^{+0.06}$	8	${180}_{-10}^{+10}$	${1.5}_{-0.4}^{+0.6}\times {10}^{-5}$
180715741	1.7 ± 1.4	n3(n4)b0	−2. to 3.	−30. to −7.	20. to 45.	5	$-{0.31}_{-0.16}^{+0.16}$	16	${630}_{-220}^{+90}$	${1.8}_{-1.2}^{+4.0}\times {10}^{-6}$

Note. Column 1: GRB names (bn). Column 2: Burst duration. Column 3: Detectors; the brightest detector is in brackets and is used to determine background and Bayesian blocks. Column 4: Source interval. Columns 5 and 6: Background intervals. Column 7: Number of Bayesian blocks during the source interval. Column 8: Maximum low-energy spectral index. Column 9: Significance of the time bin with α_max. Column 10: Corresponding peak energy (E_pk). Column 11: Corresponding flux.

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For the analysis, we follow the procedure of the Fermi/GBM GRB time-integrated (Goldstein et al. 2012; Gruber et al. 2014) and time-resolved catalogs (Yu et al. 2016, 2019). We select at most three sodium iodide (NaI) detectors and one bismuth germanium oxide (BGO) detector for the spectral analysis of each short GRB, see Table 1, Column 3. We use the response files RSP (except for two cases, GRB 090510 and GRB 170127, in which the RSP2 files are used) for each short GRB. We further use the standard Fermi/GBM energy ranges: 8–30 keV and 40 to ∼850 keV for the NaI detectors (avoiding the K edge at 33.17 keV),⁵ and ∼250 keV to 40 MeV for the BGO detectors.

We select the source interval from the first few seconds of the burst light curve where the first pulse is most prominent; see Table 1, Column 4. Indeed, most of the bursts in the sample are single-pulsed bursts. We use the NaI detector in which the largest photon counts per second were recorded from the burst to define the background intervals before and after the pulse; see Table 1, Columns 3, 5, and 6, respectively. These intervals are then applied to all detectors. As a standard procedure in GRB background fitting of GBM data, we fit a polynomial with the order of between 0 and 4 to the total count rate of each energy channel (128 channels for time-tagged Events, TTE) of each detectors. From this fit, the optimal order of the polynomial is determined by a likelihood ratio test. Then, this order of polynomial is interpolated through the source time interval to estimate the background photon count flux and its corresponding errors in each energy channel during the time of source activity.

We then rebin the light curves by applying the Bayesian block method (Scargle et al. 2013) to the unbinned TTE data. This method identifies intervals that are consistent with a constant Poisson rate. The light curves are thus rebinned into intervals over which the intensity change is small. The method uses the probability of a false positive of an intensity change p₀, which we set to p₀ = 0.01.⁶ This is the typical value employed for a GBM data analysis (e.g., Vianello et al. 2018; Burgess et al. 2019b; Yu et al. 2019). A consequence of the Bayesian block method is that the time bins will have variable widths and variable statistical significance. However, it ensures that the emission evolution is small within a time bin, which is essential in order to capture the instantaneous emission spectrum. We use the TTE data of the brightest NaI detector, and its binning is then transferred and applied to all other detectors. The total number of bins for each pulse is listed in Table 1, Column 7.

We further estimate the significance of the signal in each time bin. We employ the significance S given in Equation (15) in Vianello (2018), which is suitable for Poisson sources with Gaussian backgrounds. In particular, it is applicable for our analysis of GBM data because the background is not measured in an off-source interval, but is estimated through a polynomial fit, as described above. In this case, the typically employed S/N, given by $(n-b)/\sqrt{b}$, where n is the measurement and b is the background estimate, is not strictly valid and typically overestimates the significance measure (Vianello 2018). We find that the parameters from spectral modeling are typically well constrained when the statistical significance S ≥ 15 (see the Appendix and Yu et al. 2019 for further details). Therefore, we limit our analysis to time bins with at least this level of significance.

For the spectral model, we use the CPL model, which is a power law with an exponential cutoff, which has been extensively used because it is the best model for most GRBs in the Fermi GBM catalogs (Goldstein et al. 2012; Gruber et al. 2014; Yu et al. 2016, 2019). The CPL fit parameters are the normalization $K(\mathrm{ph}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}\,{\mathrm{keV}}^{-1}$), the low-energy power-law index α, and the cutoff energy ${E}_{{\rm{c}}}(\mathrm{keV})$. We further derive the CPL peak energy E_pk (keV) and the CPL energy flux $F\,(\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2})$.

To perform the time-resolved spectroscopy, we use the multi-mission maximum likelihood, 3ML, package (Vianello et al. 2015) and follow the method outlined in Yu et al. (2019). The spectral analysis is performed with a Bayesian approach. For the likelihood function we use a Poisson distribution for the source signal and a Gaussian distribution for the background signal. We employ prior distributions for the parameters that are based on the parameter ranges found in the GBM catalogs (e.g., Yu et al. 2016). While the normalization (K ∼ 10⁻¹¹–${10}^{3}\,\mathrm{ph}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}\,{\mathrm{keV}}^{-1}$) is assumed to have a log uniform prior, the low-energy index (α ∼ −3 to 2) and the cutoff energy (E_c ∼ 10–10,000 keV) are assumed to have uniform priors. We also investigate the sensitivity to the prior choices by trying different distributions of the priors. We find that the fit results are insensitive to the choice of prior distributions, mainly due to the high-significance level of the data that we are analyzing. A similar conclusion was drawn by Acuner et al. (2020).

The spectral analysis yields posterior probability distributions of the parameters by using their prior probability distributions and the likelihood function obtained from the data. For this, we used the Markov chain Monte Carlo (MCMC) technique. All parameter uncertainties quoted in this paper are characterized by the highest posterior density credible intervals.

In the 68 short GRBs listed in Table 1, we identified 70 distinct pulses. When divided into separate time bins, a total of 475 spectra within the GBM energy range (8 keV–40 MeV) can be analyzed. Out of these, 153 spectra have a statistical significance S ≥ 15. The results of the CPL model fits to these 153 spectra are presented in Figure 1. We show several of the relations: α − E_c (upper left panel), α − E_pk (upper right panel), F − E_pk (bottom left panel), and F − α (bottom right panel). We only observe a trend of a positive correlation between F − E_pk (Figure 1, bottom left panel).⁷

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We point out that most of the data with a very low spectral index −2.1 < α < −1.1 belong to a single burst, the brightest short GRB 120323 (see Figure 1). As a result, the values of α for the vast majority of bursts in our sample lie between −1.6 and +0.6 within 1σ uncertainty. This range is narrower than that observed in long GRBs, −2 < α < 1 (Yu et al. 2019). However, the range of E_pk ($40\,\mathrm{keV}\lt {E}_{\mathrm{pk}}\lt 6\,\mathrm{MeV}$) and flux ($8\times {10}^{-6}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}\lt F\lt 9\times {10}^{-3}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$) obtained from short GRBs is similar to the range obtained in long GRBs, $10\,\mathrm{keV}\lt {E}_{\mathrm{pk}}\lt 7\times {10}^{3}\,\mathrm{keV}$ and ${10}^{-7}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}\lt F\,\lt 5\times {10}^{-5}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$ (Yu et al. 2019).

When the results presented in Figure 1 are compared to the results obtained by Yu et al. (2019), we conclude that most of the bins obtained from short GRBs generally have harder values of the spectral index α, higher energies, and higher fluxes than those in long GRBs.

We show the distribution of the low-energy spectral indexes, α, in the left panel of Figure 2. The histogram contains 153 spectra. The green curve is a smoothed version of the distribution, using the kernel density estimation (KDE), for which the errors are taken into account (see Silverman 1986 for the KDE definition). We use the average of the asymmetric errors as the standard deviation of the Gaussian kernels in the KDE. This is a reasonable choice because the highest posterior density credible intervals of the α parameter are roughly symmetric around its mean values. We compare the values of the low-energy spectral index α (considering 1σ lower limit) with the synchrotron line-of-death value (Preece et al. 1998; Kaneko et al. 2006) α = −2/3, which is shown by the red dashed line in Figure 2. We find that 56% of the analyzed spectra (within a 1σ error) violate the criteria set by the synchrotron line-of-death.

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Global parameter distributions, such as this α-distribution, contain a varying number of spectra from each individual burst, and the spectral index typically varies between time bins of the same burst. Therefore it is difficult to draw firm conclusions on the emission mechanism based on the entire sample because there is a bias toward strong bursts with many time bins (e.g., GRB 120323; see the purple line in Figure 1).

The best way to constrain the emission mechanism during a pulse/burst therefore is to select the time bin that contains the highest value of the spectral index α in each pulse/burst. The reason for this is that physical models typically have an upper limit on how hard the spectra can become. Therefore it is enough that one single bin violates such a limit for the corresponding emission model to be rejected by the data, under the assumption that a single emission mechanism is the source of the observed signal in the entire duration of the burst (e.g., Acuner et al. 2019; Ryde et al. 2019; Yu et al. 2019). Some time-resolved spectral catalogs present the evolution of the parameters over all time bins (e.g., Kaneko et al. 2006; Yu et al. 2016, 2019). Their parameter relations for individual GRBs are then interpreted by physical models (e.g., a qualitative photospheric emission scenario, Ryde et al. 2019).

For each of the 70 pulses in the 68 short bursts in our sample we therefore select the time bin that contains the maximum (hardest) value of the low-energy spectral index, which is denoted by α_max. The spectral index α_max and its corresponding peak energy (E_pk), flux, and statistical significance (S) are listed in the last four columns of Table 1. We find that for most pulses, α_max occurs close to or at the peak of the light curve. As such, it contains the most valuable information of the spectra. This is demonstrated in GRB 140209 shown in Figure 3: the temporal evolution of α and of the peak energy E_pk are shown, overlaid on the energy flux light curve in gray.

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We now show in Figure 2 (right panel) the distribution of α_max in each of the 70 pulses from 68 short GRBs in the sample together with the KDE of the distribution. The red dashed line again shows the synchrotron line-of-death, α = −2/3. We find that 70% (within a 1σ error) of the pulses violate the line-of-death criterion, and are therefore better interpreted with a model that is not synchrotron, such as the photospheric model. This fraction is larger than the fraction obtained in the case of long GRBs, 60% within a 1σ error (Yu et al. 2019). We also note that the softest value is α_max = −1.26 for GRB 160822.

We show in Figure 4 the relation of α_max and E_pk in all 70 pulses from 68 short GRBs in the sample. The light blue line corresponds to the values of α that are found when a CPL function is fit to synthetic data, generated by an NDP spectrum peaking at different energies, E_pk, as described in Acuner et al. (2019) and shown in their Figure 3. These α-values (light blue line) are significantly lower than the asymptotic slope of the theoretical NDP spectrum (α ∼ 0.4, Beloborodov 2010; Bégué et al. 2013; Ito et al. 2013; Lundman et al. 2013) due to (i) the limited energy band of the detector and (ii) the limitation of the CPL function to correctly model the shape of the true spectrum. The shape of the NDP is shown in Figure 1 in Ryde et al. (2017), and an analytical approximation is given in Equation (1) in Acuner et al. (2019).

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From Figure 4, we find that 36% of the observed points (25/70) are consistent with being above the NDP line within 1σ error, and are therefore consistent with having a dominant quasi-blackbody component. This fraction is significantly larger than the fraction found in the study of pulses from long GRBs, 26% (for single or multiple pulses bursts) and 28% (for only single-pulsed bursts) from the two catalogs of Yu et al. (2016, 2019), respectively; see Acuner et al. (2019) for further details.

Even though most of the short GRBs in the sample are single-pulse bursts, two bursts in our sample (GRB 150819 and GRB 180703) are found to have two separate pulses (these are marked by blue and green points, respectively, in Figure 4). While the first pulse of GRB 150819 (blue in Figure 4) shows a hard spectral index, ${\alpha }_{\max }={0.25}_{-0.17}^{+0.14}$, the spectral slope of the second pulse is much softer, ${\alpha }_{\max }=-{1.02}_{-0.05}^{+0.04}$. This might be an indication for a change in leading emission mechanism, e.g., from photospheric emission to synchrotron (Zhang et al. 2018). On the other hand, both pulses of GRB 180703 (green in Figure 4) are very hard, and they are both compatible with the NDP line. This might be an example of a burst in which a single emission mechanism is responsible for the full duration of a burst. In this case, the dominant emission mechanism throughout the full duration of the burst is likely photospheric emission (Acuner & Ryde 2018).

If indeed the observed spectra above the NDP line have a photospheric origin, then one can use the data to calculate the coasting Lorentz factor, η. Here we estimate the Lorentz factor, η, for 25 pulses from 24 short GRBs above the NDP line by using Equations (1) and (4) in Pe'er et al. (2007). As the redshifts of most bursts in our sample are unknown, we have assumed redshift z = 1. We further assume that the flux, F, in the analyzed time bin is equal to the blackbody flux, ∼F_BB, and that the observed temperature is related to the peak energy via E_pk ∼ 1.48T^obs. The flux and peak energy (E_pk) for each short GRB obtained in our analysis for the corresponding α_max time bin are presented in Table 1. For comparison, we also compute η for all 12 long GRBs found above the NDP line in the samples of Yu et al. (2019) and Acuner et al. (2019).

The distributions of the Lorentz factor η for 25 pulses from 24 short GRBs and 12 long GRBs above the NDP line are presented in Figure 5. We find that the mean Lorentz factor (η_mean = 775) of the short GRBs is similar, although somewhat higher, than that of the long GRBs (η_mean = 416) (Pe'er et al. 2007; Racusin et al. 2011; Chen et al. 2018; Ghirlanda et al. 2018).

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Surprisingly, we find a bimodal distribution in the values of η for short GRBs (Figure 5, left panel). There are 11 objects in peak 1 and 14 objects in peak 2. However, no such bimodal distribution is found in the analysis of long GRBs. The low peak coincides with the values obtained for long GRBs, while the high peak is a factor of ∼3 higher. When we cut the sample at η_c = 700, we find that the pulses with high Lorentz factors, defined as peak 2, have a correspondingly higher E_pk (Figure 5, right panel). While this by itself may not be surprising, we point out that no such clear correlation is observed when the entire population of 70 pulses is analyzed (see Figure 4).

To validate the existence of this bimodality, we applied the dip test (Hartigan & Hartigan 1985) in the R package⁸ to the sample of 25 pulses. The dip test results in 0.092, implying that the Lorentz factor distribution is bimodal at a confidence level of 95%.

For the two groups from the bimodal η distribution (Figure 5), a strong positive correlation between η and E_pk is found (Figure 6, top left panel). This can be explained as due to the computational dependence, as $\eta \sim {E}_{\mathrm{pk}}^{1/2}{F}^{1/8}$, where F is the flux in each of the analyzed time bins. The formula is adopted from Pe'er et al. (2007) by identifying the peak energy with the blackbody temperature. The full dependence is $\eta \sim {E}_{\mathrm{pk}}^{1/2}{F}^{1/8}{(1+z)}^{1/2}{d}_{L}^{1/4}$. Therefore, it is important to note that the computation is not very sensitive to the uncertainty on the distance.

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However, unexpectedly, we also find anticorrelations between η and T₉₀ and between T₉₀ and E_pk (Figure 6). The latter correlation is between observed quantities and thereby independent of any model for deriving their values, unlike η. We therefore fit this correlation with a power-law function ${T}_{90}\propto {E}_{\mathrm{pk}}^{-s}$ using Bayesian inference and account for the measurement errors in both parameters. We make use of MCMC algorithms to explore the posterior distribution of the fit (see, e.g., Kelly 2007). This is shown by the gray lines (Figure 6, bottom left panel) which are 1000 randomly selected samples from the MCMC sampling and shows the degree of spread in the posterior distribution of the slope. The blue line shows the mean of the posterior distribution and has a slope of s = 0.50. The corresponding standard deviation 0.19.

We do not observe any correlation between T₉₀ and the spectral slope α_max for bursts above the NDP line (Figure 6, bottom right panel). Similarly, no such correlation is found when the entire sample of 70 pulses is considered. However, when we consider all the sources having E_pk > 800 keV (presented in Figure 4), we do observe an anticorrelation between T₉₀ and α_max, which is presented in Figure 7. This anticorrelation implies that sources with harder spectra have shorter T₉₀. This suggests that there might always exist a thermal emission at short times that is accompanied by other emission processes such as synchrotron at later times. If this interpretation is correct, the lack of thermal emission in a given GRB might be explained by the lack of observed or studied spectra at sufficiently short time.

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The anticorrelation we found between E_pk and T₉₀ of short GRBs with high E_pk motivated us to study a possible correlation between the HR and T₉₀. Following Kouveliotou et al. (1993), we calculate the HR using the two typical energy bands, 100–300 and 50–100 keV. To integrate the spectra, we use the CPL fit parameters, α_max and E_c, for the 70 spectra in our sample. For comparison, we also calculated the HR for the spectra with the maximum value of α in each of the 38 pulses from 37 long GRBs in the catalog by Yu et al. (2019). These pulses were selected from single-pulsed long bursts that have at least five time bins in which the statistical significance is S ≥ 20.

The HR–T₉₀ relation is shown in Figure 8 (left panel) for both short and long GRBs. Short bursts with α_max above the NDP model prediction are displayed in black while those below the prediction are plotted in red. For the long bursts, the colors are blue and purple, respectively. While the T₉₀ selection criteria enable us to clearly discriminate the long and short GRB population, we do not observe any additional correlation in this plot.

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The HR–T₉₀ relation for the 25 pulses from 24 short GRBs above the NDP line is shown in the right panel in Figure 8. The color-coding (peak 1: black, and peak 2: green) is the same as that used for the bimodal η distribution in Figure 5. Now, a clear separation is observed: GRBs with lower peak energy have a low Lorentz factor, lower HR, and longer T₉₀.

Indeed, all the parameters of these GRBs in the first peak of the bimodal η distribution (in Figure 5, black) seem to form a continuous distribution of the parameters of the population of long GRBs. This is in contrast to GRBs in the second peak of the bimodal η distribution (in Figure 5, green), which have a higher HR than the GRBs below the NDP line (45 pulses) and long GRBs (38 pulses). This result implies that the duration T₉₀ as a single criterion does not make a good separation between the two populations; as we show, the short GRBs may instead be composed of two separate populations, one that forms a continuation of the long GRB population, and another, separate population.

We find a weak positive correlations between F and E_pk (in Figure 9, left panel) and between F and α_max (in Figure 9, right panel) for bursts above the NDP line (in the two groups seen in the bimodal η distribution). However, no clear correlation is seen between these parameters for the bursts below the NDP line. This by itself is an interesting result. In the literature, several publications claim that there is a strong correlation between the luminosity, L_peak, or the isotropic energy, E_iso, and peak energy, E_pk (e.g., Yonetoku et al. 2004; Amati 2006; Ghirlanda et al. 2009). These claims are based on a large sample of long GRBs. In contrast, here we do not find any such correlation when the entire sample of short GRB pulses is considered, but we do find a correlation when we consider only those short GRBs whose spectral slope are above the NDP line.

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In this work, we fit the Fermi/GBM data using the phenomenological CPL model (this is also known as the Comptonized model). Several empirical models are commonly used in the literature for the spectral analysis of GRBs. In addition to the CPL model (Kaneko et al. 2006), these include the Band model (Band et al. 1993) as well as the smoothly broken power-law model (Ryde 1999). Yu et al. (2016) showed that the CPL is the preferred model for the majority (70%) of bursts, according to the Castor C-Statistic (CSTAT).⁹ In addition, a consistent result was found by Yu et al. (2019) based on the deviance information criterion (DIC) in Bayesian statistics (Spiegelhalter et al. 2002). This means that the results of the CPL model for these bursts have lower DIC and higher effective number of parameters, p_DIC > 0 (Gelman et al. 2014), than those of the Band model. Additionally, the resulting parameters for the CPL fits are constrained within the prior ranges more often than those obtained from the BAND function fits; see also Burgess et al. (2019a, 2019b).

It was recently argued by Burgess et al. (2019a) that a direct fit of the data to a synchrotron model enables overcoming the line-of-death criteria in many GRBs. However, this idea suffers several severe drawbacks. First, the bursts selected in that work are limited to bursts in the Yu et al. (2016) catalog, with the additional constraints of being single-pulse GRBs and having known redshifts. This selection is different from the short pulses considered here. Second, the values of the parameters found in their fits require an unacceptably high ratio of explosion energy to ambient mass density, of more than 7 orders of magnitude higher $(E/{10}^{53}\,\mathrm{erg})/({n}_{\mathrm{ism}}/1\,{\mathrm{cm}}^{-3})\gtrsim 4\times {10}^{7}$ than the highest observed so far. In order to overcome this problem, Burgess et al. (2019a) suggested an additional acceleration of particles within the relativistically expanding jet ("jet within a jet"); however, no such mechanism that can lead to relativistic expansion within an already relativistically expanding jet is known. We thus find that this model is still incomplete, and an interpretation of an empirical fit still provides better insight. Another suggestion was given by Ghisellini et al. (2020) based on the low-energy break in the prompt spectrum of GRBs. They argued that the emission process is still synchrotron radiation, but produced by protons, and that it cannot be completely cool.

When we consider the entire sample of 70 pulses analyzed in the short GRB population, we do not observe a correlation between the burst duration T₉₀ and the peak energy E_pk. Similarly, for the long GRBs that we considered (both below and above the NDP line), no clear T₉₀ − E_pk correlation was detected. However, when we consider only those short bursts that have a hard value of the spectral index, α, such that they are above the NDP line (Figures 4, 5), we do find an inverse correlation between T₉₀ and peak energy, ${T}_{90}\propto {E}_{\mathrm{pk}}^{-s}$, where the slope of the power-law function is s = 0.50 and the corresponding standard deviation is 0.19.

A quantitative relationship between the temporal and spectral structure in GRBs has been considered by several authors in the past. However, these works treated only the long GRBs. Richardson et al. (1996) and Bissaldi et al. (2011) found a negative correlation between T₉₀ and peak energy, ${T}_{90}\propto {E}_{\mathrm{pk}}^{-s}$ where s ≃ 0.4. Their samples contained only the bright, long GRB population. When considering the entire set of the long GRB population, Qin et al. (2013) report a similar correlation, but with weaker dependence, s = 0.2.

Here we report for the first time such a correlation in the sample of short GRBs. This could not have been done in the past because the sample was small, and because a suitable method for studying the spectra was lacking. The similarity between the correlation found here for short GRBs above the NDP line and for the bright long GRBs (which tend to have a harder spectral index, α; see Bissaldi et al. 2011), as well as the fact that we do not detect any correlation for bursts below the NDP line, suggests a possible correlation between the emission mechanism and the burst duration. Bursts above the NDP line are consistent with originating from the photosphere, hence the photons directly probe the inner engine. In contrast, bursts whose spectra are below the NDP line may have additional radiative mechanisms, such as synchrotron emission, which originate from the outer regions of the outflow (outside the photosphere) and as such do not necessarily directly follow the duration of the inner engine. If this interpretation is correct, it points to a possible correlation between the duration of the inner engine and the temperature, or total energy, of the released photons. This further points to the importance of a spectral analysis in analyzing possible correlations in the GRB population.

A second correlation we find is between T₉₀ and α_max when we consider a cut at higher peak energy (${E}_{\mathrm{pk}}\gt 800\,\mathrm{keV}$) (see Figure 7). This (anti-) correlation further suggests a dual emission mechanism: short-duration GRBs might be dominated by a thermal component, while an additional emission process may lead to shallower spectra and be characterized by a longer duration.

The results we find therefore strongly support the idea that the spectra of both short and long GRBs contain (at least) two separate components: a photospheric emission component that correlates directly with the inner engine activity, and a second component, possibly having a synchrotron origin, that is longer in nature and less steep.