Gamma-ray burst prompt correlationsdownloads.hindawi.com/journals/aa/aip/4969503.pdf · Gamma-ray...

Review ArticleGamma-Ray Burst Prompt Correlations

M G Dainotti 123 R Del Vecchio3 andM Tarnopolski3

1Physics Department Stanford University Via Pueblo Mall 382 Stanford CA USA2INAF Istituto di Astrofisica Spaziale e Fisica Cosmica Via Gobetti 101 40129 Bologna Italy3Astronomical Observatory Jagiellonian University Orla 171 30-244 Krakow Poland

Correspondence should be addressed to M G Dainotti mdainottstanfordedu

Received 25 May 2016 Accepted 27 November 2016 Published 24 January 2018

Academic Editor Alberto J Castro-Tirado

Copyright copy 2018 M G Dainotti et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Themechanism responsible for the prompt emission of gamma-ray bursts (GRBs) is still a debated issueThe prompt phase-relatedGRB correlations can allow discriminating among the most plausible theoretical models explaining this emission We present anoverview of the observational two-parameter correlations their physical interpretations and their use as redshift estimators andpossibly as cosmological toolsThe nowadays challenge is to make GRBs the farthest stellar-scaled objects observed (up to redshift119911 = 94) standard candles through well established and robust correlations However GRBs spanning several orders of magnitudein their energetics are far from being standard candles We describe the advances in the prompt correlation research in the pastdecades with particular focus paid to the discoveries in the last 20 years

1 Introduction

Gamma-ray bursts (GRBs) are highly energetic eventswith the total isotropic energy released of the order of1048ndash1055 erg (for recent reviews see [1ndash6]) GRBs werediscovered by military satellites Vela in late 1960s and wererecognized early to be of extrasolar origin [7] A bimodalstructure (reported first by Mazets et al [8]) in the durationdistribution of GRBs detected by the Burst and TransientSource Experiment (BATSE) onboard the Compton Gamma-Ray Observatory (CGRO) [9] based on which GRBs arenowadays commonly classified into short (with durations11987990 lt 2 s SGRBs) and long (with 11987990 gt 2 s LGRBs)was found [10] BATSE observations allowed also confirmingthe hypothesis of Klebesadel et al [7] that GRBs are ofextragalactic origin due to isotropic angular distribution inthe sky combinedwith the fact that they exhibited an intensitydistribution that deviated strongly from the minus32 power law[9 11ndash14] This was later corroborated by establishing thefirst redshiftmeasurement taken forGRB970508 whichwith0835 lt 119911 ≲ 23 was placed at a cosmological distance of atleast 29Gpc [15] Despite initial isotropy SGRBs were shownto be distributed anisotropically on the sky while LGRBsare distributed isotropically [16ndash23] Cosmological conse-

quences of the anisotropic celestial distribution of SGRBswere discussed lately by Meszaros et al [24] and Meszarosand Rees [6] Finally the progenitors of LGRBs are associatedwith supernovae (SNe) [25ndash29] related to collapse of massivestars Progenitors of SGRBs are thought to be neutronstarndashblack hole (NSndashBH) or NSndashNSmergers [12 30ndash32] andno connection between SGRBs and SNe has been proven [33]

While the recent first direct detection of gravitationalwaves (GW) termed GW150914 by the Laser InterferometerGravitational Wave Observatory (LIGO) [34] interpretedas a merger of two stellar-mass BHs with masses 36+5minus4 119872⊙

and 29+4minus4 119872⊙ is by itself a discovery of prime importanceit becomes especially interesting in light of the finding ofConnaughton et al [35] who reported a weak transientsource lasting 1 s and detected by FermiGBM [36] only04 s after the GW150914 termed GW150914-GBM Its falsealarm probability is estimated to be 00022 The fluence inthe energy band 1 keVndash10MeV is computed to be 18+15minus10 times1049 erg sminus1 While these GW and GRB events are consistentin direction its connection is tentative due to relativelylarge uncertainties in their localization This association isunexpected as SGRBs have been thought to originate fromNSndashNS or NSndashBH mergers Moreover neither INTEGRAL[37] nor Swift [38] detected any signals that could be ascribed

HindawiAdvances in AstronomyVolume 2018 Article ID 4969503 31 pageshttpsdoiorg10115520184969503

2 Advances in Astronomy

to a GRB Even if it turns out that it is only a chancecoincidence [39] it has already triggered scenarios explaininghow a BHndashBH merger can become a GRB for example thenascent BH could generate a GRB via accretion of a mass≃ 10minus5 M⊙ [40] indicating its location in a dense medium(see also [41]) or two high-mass low-metallicity stars couldundergo an SN explosion and the matter ejected from thelast exploding star can formmdashafter some timemdashan accretiondisk producing an SGRB [42] Also the possible detection ofan afterglow that can be visible many months after the event[43] could shed light on the nature of the GW and SGRBassociation

From a phenomenological point of view a GRB iscomposed of the prompt emission which consists of high-energy photons such as 120574-rays and hard X-rays and theafterglow emission that is a long lasting multiwavelengthemission (X-ray optical and sometimes also radio) whichfollows the prompt phase The first afterglow observation(for GRB970228) was due to the BeppoSAX satellite [4445] Another class besides SGRBs and LGRBs that isintermediate in duration was proposed to be present inunivariate duration distributions [46ndash51] as well as in higherdimensional parameter spaces [51ndash56] On the other handthis elusive intermediate class might be a statistical featurethat can be explained by modeling the duration distributionwith skewed distributions instead of the commonly appliedstandard Gaussians [57ndash61] Additionally GRB classificationwas shown to be detector-dependent [1 62 63] Moreovera subclass classification of LGRBs was proposed [64] andNorris and Bonnell [65] discovered the existence of anintermediate class or SGRBs with extended emission thatshow mixed properties between SGRBs and LGRBs GRBswith very long durations (ultralong GRBs with 11987990 gt 1000 s)are statistically different than regular (ie with 11987990 lt 500 s)LGRBs [66] and hence might form a different class (see also[67ndash70]) Another relevant classification appears related tothe spectral features distinguishing normal GRBs from X-rayflashes (XRFs) The XRFs [71 72] are extragalactic transientX-ray sources with spatial distribution and spectral andtemporal characteristics similar to LGRBs The remarkableproperty that distinguishes XRFs from GRBs is that their]119865] prompt emission spectrum peaks at energies which areobserved to be typically an order of magnitude lower thanthe observed peak energies of GRBs XRFs are empiricallydefined by a greater fluence (time-integrated flux) in the X-ray band (2ndash30 keV) than in the 120574-ray band (30ndash400 keV)This classification is also relevant for the investigation of GRBcorrelations since some of them become stronger or weakerby introducing different GRB categories Grupe et al [73]using 754 Swift GRBs performed an exhaustive analysis ofseveral correlations as well as the GRB redshift distributiondiscovering that the bright bursts are more common in thehigh-119911 (ie 119911 ≳ 3) than in the local universe

This classification has further enhanced the knowledgeof the progenitor system from which GRBs originate Itwas soon after their discovery that LGRBs were thoughtto originate from distant star-forming galaxies Since thenLGRBs have been firmly associated with powerful core-collapse SNe and the association seems solid Nevertheless

there have been puzzling cases of LGRBs that were notassociated with bright SNe [74 75] This implies that it ispossible to observe GRBs without an associated bright SNeor there are other progenitors for LGRBs than core-collapseofmassive stars Another relevant uncertainty concerning theprogenitor systems for LGRBs is the role of metallicity 119885 Inthe collapsar model [27] LGRBs are only formed by massivestars with 119885119885⊙ below ≃ 01ndash03 However several GRBshave been located in very metal-rich systems [76] and it is animportant goal to understandwhether there are other ways toform LGRBs than through the collapsar scenario [77] Oneof the models used to explain the GRB phenomenon is theldquofireballrdquo model [78ndash80] in which a compact central engine(either the collapsed core of a massive star or the mergerproduct of an NSndashNS binary) launches a highly relativisticjetted electron-positron-baryon plasma Interactions of blobswithin the jet are believed to produce the prompt emissionInstead the interaction of the jet with the ambient materialcauses the afterglow phase However problems in explainingthe light curves within this model have been shown byWillingale et al [81] Specifically for ≃50 of GRBs theobserved afterglows are in agreement with the model but forthe rest the temporal and spectral indices do not conformand suggest a continued late energy injection Melandri et al[82] performed amultiwavelength analysis and found that theforward shock (FS) model does not explain almost 50 ofthe examined GRBs even after taking into account energyinjection Rykoff et al [83] showed that the fireball modeldoes not model correctly early afterglows Reference [84]analysed the prompt and afterglow light curves and pointedout that some GRBs required energy injection to explain theoutflows The crisis of the standard fireball models appearedwhen Swift [85] observations revealed a more complexbehaviour of the light curves than observed in the past [86ndash88] and pointed out that GRBs often follow ldquocanonicalrdquo lightcurves [89] Therefore the discovery of correlations amongrelevant physical parameters in the prompt phase is veryimportant in this context in order to use them as possiblemodel discriminators In fact many theoretical models havebeen presented in the literature in order to explain the widevariety of observations but each model has some advantagesas well as drawbacks and the use of the phenomenologicalcorrelations can boost the understanding of the mechanismresponsible for the prompt emission Moreover given themuch larger (compared to SNe) redshift range over whichGRBs can be observed it is tempting to include them ascosmological probes extending the redshift range by almostan order ofmagnitude further than the available SNe Ia GRBsare observed up to redshift 119911 = 94 [90] much more distantthan SNe Ia observed up to 119911 = 226 [91] and thereforethey can help to understand the nature of dark energy anddetermine the evolution of its equation of state at veryhigh 119911 However contrary to SNe Ia which originate fromwhite dwarves reaching the Chandrasekhar limit and alwaysreleasing the same amount of energy GRBs cannot yet beconsidered standard candles with their (isotropic-equivalent)energies spanning 8 orders of magnitude (see also [92] andreferences therein) Therefore finding universal relationsamong observable properties can help to standardize their

Advances in Astronomy 3

TfTＬＣＭ

T0

TＤ

T

TＪＥ

Figure 1 A sketch of the pulse displaying 119879ej and 119879peak (denoted by119879pk here) and the quantities 119879119891 and 1198790 = 119879119891 minus 119879rise (Figure afterWillingale et al [108] see Figure 1 therein)

energetics andor luminosities They can serve as a tracerof the history of the cosmic star formation rate [93ndash97]and provide invaluable information on the physics in theintergalactic medium [98ndash100] This is the reason why thestudy of GRB correlations is so relevant for understandingthe GRB emission mechanism for finding a good distanceindicator and for exploring the high-redshift universe [101]

This paper is organized in the following manner InSection 2 we explain the nomenclature and definitionsadopted in this work and in Section 3 we analyse the correla-tions between various prompt parameters We summarize inSection 4

2 Notations and Nomenclature

For clarity we report a summary of the nomenclature adoptedin the review 119871 119865 119864 119878 and 119879 indicate the luminositythe energy flux the energy the fluence and the timescalerespectively which can be measured in several wavelengthsMore specifically

(i) 11987990 is the time interval in which 90 of the GRBrsquosfluence is accumulated starting from the time atwhich 5 of the total fluence was detected [10]

(ii) 11987950 is defined similar to 11987990 as the time interval from25 to 75 of the total detected fluence

(iii) 11987945 is the time spanned by the brightest 45 of thetotal counts detected above background [102]

(iv) 119879peak is the time at which the pulse (ie a sharp riseand a slower smooth decay [103ndash106]) in the promptlight curve peaks (see Figure 1)

(v) 119879break is the time of a power law break in the afterglowlight curve [107 108] that is the time when theafterglow brightness has a power law decline thatsuddenly steepens due to the slowing down of the jetuntil the relativistic beaming angle roughly equals thejet-opening angle 120579jet [109]

(vi) 120591lag and 120591RT are the difference of arrival times tothe observer of the high-energy photons and lowenergy photons defined between 25ndash50 keV and

100ndash300 keV energy band and the shortest time overwhich the light curve increases by 50of the peak fluxof the pulse

(vii) 119879119901 is the end time prompt phase at which theexponential decay switches to a power law which isusually followed by a shallow decay called the plateauphase and 119879119886 is the time at the end of this plateauphase [81]

(viii) 119879119891 is the pulse width since the burst trigger at the time119879ej of the ejecta(ix) 119864peak 119864iso 119864120574 and 119864prompt are the peak energy that

is the energy at the peak of the ]119865] spectrum [110]the total isotropic energy emitted during the wholeburst (eg [111]) the total energy corrected for thebeaming factor [the latter two are connected via 119864120574 =(1 minus cos 120579jet)119864iso] and the isotropic energy emitted inthe prompt phase respectively

(x) 119865peak119865tot are the peak and the total fluxes respectively[112]

(xi) 119871119886 119871119883119901 and 119871119891 are the luminosities respective to 119879119886119879119901 (specified in the X-ray band) and 119879119891(xii) 119871 is the observed luminosity and specifically 119871peak

and 119871 iso are the peak luminosity (ie the luminosityat the pulse peak [113]) and the total isotropic lumi-nosity both in a given energy band More precisely119871peak is defined as follows

119871peak = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119865peak (1)

with119863119871(119911 Ω119872 ΩΛ) the luminosity distance given by

119863119871 (119911 Ω119872 ΩΛ) = 119888 (1 + 119911)1198670 int1199110

1198891199111015840radicΩ119872 (1 + 1199111015840)3 + ΩΛ

(2)

where Ω119872 and ΩΛ are the matter and dark energydensity parameters 1198670 is the present-day Hubbleconstant and 119911 is the redshift Similarly 119871 iso is givenby

119871 iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119865tot (3)

(xiii) 119878120574 119878obs 119878tot indicate the prompt fluence in the wholegamma band (ie from a few hundred keV to a fewMeV) the observed fluence in the range 50ndash300 keVand the total fluence in the 20 keVndash15MeV energyband

(xiv) 119881 is the variability of the GRBrsquos light curve It is com-puted by taking the difference between the observedlight curve and its smoothed version squaring thisdifference summing these squared differences overtime intervals and appropriately normalizing theresulting sum [102] Different smoothing filters maybe applied (see also [114] for a different approach)119881119891 denotes the variability for a certain fraction of thesmoothing timescale in the light curve


Most of the quantities described above are given in theobserver frame except for119864iso119864prompt 119871peak and 119871 iso whichare already defined in the rest frame With the upper indexldquolowastrdquo we explicitly denote the observables in the GRB restframeThe rest frame times are the observed times divided by

the cosmic time expansion for example the rest frame timein the prompt phase is denoted by 119879lowast119901 = 119879119901(1 + 119911) Theenergetics are transformed differently for example 119864lowastpeak =119864peak(1 + 119911)

The Band function [115] is a commonly applied phe-nomenological spectral profile such that

119873119864 (119864) = 119860norm times

( 119864100 keV)120572

exp(minus 1198641198640) 119864 le (120572 minus 120573) 1198640[ (120572 minus 120573) 1198640100 keV ]120572minus120573 ( 119864100 keV)

120573

exp (120572 minus 120573) 119864 ge (120572 minus 120573) 1198640(4)

where 119860norm is the normalization Here 120572 and 120573 are the low-and high-energy indices of the Band function respectively119873119864(119864) is in units of photons cmminus2 sminus1 keVminus1 For the cases120573 ltminus2 and 120572 gt minus2 119864peak can be derived as 119864peak = (2 + 120572)1198640which corresponds to the energy at the maximum flux in the]119865] spectra [115 116]

The Pearson correlation coefficient [117 118] is denoted by119903 the Spearman correlation coefficient [119] is denoted by 120588and the 119901 value (a probability that a correlation is drawn bychance) is denoted by 119875

Finally we mostly deal with correlations of the form 119910 =119886119909+119887 However when the intercept 119887 is neglected in the textbut its value is nonnegligible (or not known due to lacking inthe original paper) we use the notation 119910 sim 119886119909 to emphasizethe slope

3 The Prompt Correlations

Several physical relations between relevant quantities inGRBs were found since the 1990s In each paragraph belowwe follow the discovery of the correlation with the definitionof the quantities the discussions presented in literature andtheir physical interpretation

31 The 119871119901119890119886119896-120591119897119886119892 Correlation311 Literature Overview Liang and Kargatis [120] using 34bright GRBs detected by BATSE found that 119864peak dependslinearly on the previous flux emitted by the pulse that is thatthe rate of change of119864peak is proportional to the instantaneousluminosity Quantitatively

119871peak119873 = minusd119864peakd119905 (5)

where119873 is a normalization constant expressing the luminos-ity for each pulse within a burst and 119871peak was calculatedfrom the observed flux via (1)

The119871peak-120591lag correlationwas introduced for the first timeby Norris et al [113] who examined a sample of 174GRBsdetected by BATSE among which 6GRBs had an establishedredshift and those were used to find an anticorrelation

between 119871peak and 120591lag in the form of the following (seeFigure 2(a))

log119871peak = 5511 minus 114 log 120591lowastlag (6)

with 119871peak in units of 1053 erg sminus1 computed in the50ndash300 keV range and 120591lowastlag ismeasured in seconds A remark-ably consistent relationwas found by Schaefer et al [121] whoused a sample of 112 BATSE GRBs and reported that

log 119871peak = 5246 minus (114 plusmn 020) log 120591lag (7)

being in perfect agreement with the result of Norris et al[113] Here 119871peak is in units of 1051 erg sminus1 and 120591lag in secondsThis relation has been confirmedby several studies (eg [122ndash124])

Schaefer [125] showed that the 119871peak-120591lag relation isa consequence of the Liang and Kargatis [120] empiricalrelation from (5) and he derived this dependence to be ofthe form log 119871peak sim minus log 120591lag This correlation was usefulin the investigation of Kocevski and Liang [126] who used asample of 19 BATSE GRBs and the 119871peak-120591lag relation from[121] to infer their pseudoredshifts Their approach was tovary the guessed 119911 until it allowed matching the luminositydistance 119863119871 measured with the GRBrsquos energy flux and 119863119871that can be calculated from the guessed redshift within a flatΛCDMmodel until the agreement among the two convergedto within 10minus3 Next the rate of 119864peak decay as in [120]was measured Finally Kocevski and Liang [126] showed thatthe 119871peak is directly related to the GRBrsquos spectral evolutionHowever Hakkila et al [127] found a different slope minus062 plusmn004 and argued that the 119871peak-120591lag relation is a pulse ratherthan a burst property that is each pulse is characterized byits own 120591lag distinct for various pulses within a GRB

Tsutsui et al [128] using pseudoredshifts estimated viathe Yonetoku relation (see Section 362) for 565 BATSEGRBs found that the 119871peak-120591lag relation has a 120588 of only 038(see Figure 2(b)) However assuming that the luminosity isa function of both the redshift and the lag a new redshift-dependent 119871peak-120591lag relation was found as

log119871peak = 5088 + 253 log (1 + 119911) minus 0282 log 120591lag (8)


990123

971214

990510

970828

980703

970508

100

101

102

103

Isot

ropi

c Lum

inos

ity (1051

ergsＭminus1)

Lag Ch 1minush 3 (s)10010minus3 10minus2 10minus1

(a)

0001

001

01

1

10

100

1000

10000

Peak

Lum

inos

ity (1052

ergＭminus1)

1 10 10001

00758 (1 + z)253 ＦＡ0282

1000

(b)

Figure 2 (a) 119871peak versus 120591lowastlag distribution for six GRBs with measured redshifts The dashed line represents the power law fit to the lagtimes for ranges consisting of count rates larger than 01 times peak intensity (squares) yielding log119871peak sim minus114 log (120591lowastlag001 s) The lag time iscomputed using channel 1 (25ndash50 keV) and channel 3 (100ndash300 keV) of the BATSE instrument (Figure after Norris et al [113] see Figure 6therein AAS Reproduced with permission) (b) The 119871peak-120591lag distribution in the log 119871peak versus sim 253 log (1 + 119911) minus 0282 log 120591lag planeThe correlation coefficient is 120588 = 077 119875 = 79 times 10minus75 The solid line represents the best-fit line and two dashed lines delineate 1120590 deviation(Figure after Tsutsui et al [128] see Figure 4 therein Copyright 2008 AIP Publishing)

with119871peak in units of 1050 erg sminus1 120591lag in seconds120588 = 077 and119875 = 79 times 10minus75 Although the spectral lag is computed fromtwo channels of BATSE this new 119871peak-120591lag relation suggeststhat a future lag-luminosity relation defined within the Swiftdata should also depend on the redshift

Afterwards Sultana et al [129] presented a relationbetween the 119911- and 119896-corrected 120591lag for the Swift energy bands50ndash100 keV and 100ndash200 keV and 119871peak for a subset of 12Swift longGRBsThe 119911-correction takes into account the timedilatation effect by multiplying the observed lag by (1 + 119911)minus1to translate it into the rest frame The 119896-correction takesinto account a similar effect caused by energy bands beingdifferent in the observer and rest frames via multiplicationby (1 + 119911)033 [130] The net corrected 120591lowastlag is thence (1 +119911)minus067120591lag In addition Sultana et al [129] demonstrated thatthis correlation in the prompt phase can be extrapolated intothe 119871119886-119879lowast119886 relation [131ndash134] Sultana et al [129] found thefollowing (Note that Sultana et al [129] used 119871 iso to denotethe peak isotropic luminosity)

log 119871peak = (5487 plusmn 029)minus (119 plusmn 017) log [(1 + 119911)minus067 120591lag]

log119871119886 = (5157 plusmn 010) minus (110 plusmn 003) log119879lowast119886 (9)

with 120591lag inms119879lowast119886 in seconds and119871 in erg sminus1The correlationcoefficient is significant for these two relations (120588 = minus065 forthe 119871peak-120591lag and 120588 = minus088 for the 119871119886-119879lowast119886 relations) and hassurprisingly similar best-fit power law indices (minus119 plusmn 017and minus110 plusmn 003 resp) Although 120591lag and 119879lowast119886 representdifferent GRB time variables it appears distinctly that the119871peak-120591lag relation extrapolates into 119871119886-119879lowast119886 for timescales

120591lag ≃ 119879lowast119886 A discussion and comparison of this extrapolationwith the 119871119891-119879119891 relation are extensively presented in [135]

Ukwatta et al [136] confirmed that there is a correlationbetween 119871lowastpeak and the 119911- and 119896-corrected 120591lag among 31GRBsobserved by Swift with 119903 = minus068 119875 = 7 times 10minus2 and the slopeequal to minus14 plusmn 03 hence confirming the 119871peak-120591lag relationalthough with a large scatter This was followed by anotherconfirmation of this correlation [137] with the use of 43 SwiftGRBs with known redshift which yielded 119903 = minus082 119875 =55 times 10minus5 and a slope of minus12 plusmn 02 being consistent with theprevious results

Finally Margutti et al [138] established that the X-rayflares obey the same 119871peak-120591lowastlag relation (in the rest frameenergy band 03ndash10 keV) as GRBs and proposed that theirunderlying mechanism is similar

312 Physical Interpretation of the 119871119901119890119886119896-120591119897119886119892 Relation Thephysical assumption on which the work by Norris et al[113] was based is that the initial mechanism for the energyformation affects the development of the pulse much morethan dissipation From the study of several pulses in brightlong BATSE GRBs it was claimed that for pulses withprecisely defined shape the rise-to-decay ratio is le1 Inaddition when the ratio diminishes pulses show a tendencyto be broader and weaker

Salmonson [122] proposed that the 119871peak-120591lag relationarises from an entirely kinematic effect In this scenario anemitting region with constant (among the bursts) luminosityis the source of the GRBrsquos radiation He also claimed that vari-ations in the line-of-sight velocity should affect the observedluminosity proportionally to the Lorentz factor of the jetrsquosexpansion Γ = [1minus(V119888)2]minus12 (where V is the relative velocity


between the inertial reference frames and 119888 is the speed oflight) while the apparent 120591lag is proportional to 1Γ Thevariations in the velocity among the lines of sight is a result ofthe jetrsquos expansion velocity combined with the cosmologicalexpansion The differences of luminosity and lag betweendifferent bursts are due to the different velocities of theindividual emitting regions In this case the luminosity isexpected to be proportional to 1120591lag which is consistent withthe observed relationThis explanation however requires thecomoving luminosity to be nearly constant among the burstswhich is a very strong condition to be fulfilled Moreoverthis scenario has several other problems (as pointed out bySchaefer [125])

(1) It requires the Lorentz factor and luminosity to havethe same range of variation However the observed119871peak span more than three orders of magnitude (eg[121]) while the Lorentz factors span less than oneorder of magnitude (ie a factor of 5) [139]

(2) It follows that the observed luminosity should belinearly dependent on the jetrsquos Lorentz factor yetthis claim is not justified In fact a number ofcorrections are to be taken into account leading toa significantly nonlinear dependence The forwardmotion of the jet introduces by itself an additionalquadratic dependence [140]

Ioka and Nakamura [141] proposed another interpre-tation for the 119871peak-120591lag correlation From their analysis amodel in which the peak luminosity depends on the viewingangle is elaborated the viewing angle is the off-axis angularposition from which the observer examines the emissionIndeed it is found that a high-luminosity peak in GRBs withbrief spectral lag is due to an emitted jet with a smallerviewing angle than a fainter peak with extended lag It isalso claimed that the viewing angle can have implications onother correlations such as the luminosity-variability relationpresented in Section 32 As an additional result from thisstudy it was pointed out that XRFs can be seen as GRBsdetected from large angles with high spectral lag and smallvariability

On the other hand regarding the jet angle distributionsLiang et al [142] found an anticorrelation between the jet-opening angle and the isotropic kinetic energy among 179X-ray GRB light curves and the afterglow data of 57GRBsAssuming that the GRB rate follows the star formation rateand after a careful consideration of selection effects Luet al [143] found in a sample of 77GRBs an anticorrelationbetween the jet-opening angle 120579jet and the redshift in thefollowing form

log 120579jet = (minus090 plusmn 009) minus (094 plusmn 019) log (1 + 119911) (10)

with 120588 = 055 and 119875 lt 10minus4 Using a mock sample andbootstrap technique they showed that the observed 120579jet-119911relation is most likely due to instrumental selection effectsMoreover they argued that while other types of relationfor example 120591lag-119911 [144] or the redshift dependence of theshallow decay in X-ray afterglows by Stratta et al [145] mighthave connections with the jet geometry they are also likely

to stem from observational biases or sample selection effectsAlso Ryan et al [146] investigated the jet-opening angleproperties using a sample of 226 SwiftXRTGRBswith knownredshift They found that most of the observed afterglowswere observed off-axis hence the expected behaviour of theafterglow light curves can be significantly affected by theviewing angle

Zhang et al [33] argued on the basis of the kinematicmodel that the origin of the 119871peak-120591lag relation is due to amore intrinsic 119871peak-119881 relation (see Section 32) They alsogave an interpretation of the latter relation within the internalshock model (see Section 322) Recently Uhm and Zhang[147] constructed amodel based on the synchrotron radiationmechanism that explains the physical origin of the spectrallags and is consistent with observations

Another explanation for the origin of the 119871peak-120591lagrelation given by Sultana et al [129] involves only kinematiceffects In this case 119871peak and 120591lag depend on the quantity

119863 = 1Γ (1 minus 1205730 cos 120579) (1 + 119911) (11)

depicting the Doppler factor of a jet at a viewing angle 120579and with velocity 1205730 equiv V119888 at redshift 119911 In this study thereis no reference to the masses and forces involved and as aconsequence of the Doppler effect the factor119863 associates theGRB rest frame timescale 120591 with the observed time 119905 in thefollowing way

119905 = 120591119863 (12)

Therefore considering a decay timescale Δ120591 in the GRBrest frame (12) in the observer frame will give Δ119905 =Δ120591119863 Furthermore taking into account a spectrum given byΦ(119864) prop 119864minus120572 the peak luminosity (as already pointed out bySalmonson [122]) can be computed as

119871peak prop 119863120572 (13)

with 120572 asymp 1 In such a way (12) and (13) allow retrievingthe observed 119871peak-120591lag relation Finally the analogous cor-relation coefficients and best-fit slopes of the 119871peak-120591lag and119871119886-119879lowast119886 correlations obtained by Sultana et al [129] seem tohint toward a similar origin for these two relations

32 The 119871119901119890119886119896-119881 Correlation The first correlation between119871peak and 119881 was discovered by Fenimore and Ramirez-Ruiz[148] and was given as

log119871peak = 5649 + 335 log119881 (14)

with 119871peak measured in erg sminus1 Here the luminosity isper steradian in a specified (rest frame) energy bandpass(50ndash300 keV) averaged over 256ms First seven BATSEGRBs with a measured redshift were used to calibrate the119871peak-119881 relation Next the obtained relationship was appliedto 220 bright BATSEGRBs in order to obtain the luminositiesand distances and to infer that the GRB formation rate scalesas (1 + 119911)33plusmn03 Finally the authors emphasized the need ofconfirmation of the proposed 119871peak-119881 relation


ＦＩＡL

(ergＭminus1)

50

51

52

53

54

minus15 minus1 minus05

ＦＩＡ Vf=045

(a)GFM05 PL

05

1

15

2

25

3

35

ＦＩＡ(L)

ＦＩＡ (V)

minus06minus08minus1minus12minus14minus16minus18minus2

(b)

G05 PL

ＦＩＡ (V)

minus06 minus04 minus02minus08minus1minus12minus14minus16minus18minus2

minus05

0

05

1

15

2

25

3

35

ＦＩＡ(L)

(c)

Figure 3 (a) The variabilities 119881119891=045 and peak luminosities 119871peak of the data set excluding GRB980425 In this case 119881119891=045 indicatesthe variabilities for the 45 smoothing timescale of the light curve The solid and dotted lines are the best-fit line and 1120590 deviationrespectively in log 119871peak- log119881119891=045 plane (Figure after Reichart et al [102] see Figure 9 therein AAS Reproduced with permission)(b) The log 119871peak- log119881 plane for the sample of 32GRBs with measured redshift The best-fit lines and 1120590 deviations are also displayed solidlines are computed with the Reichart et al [102] method dashed-dotted lines with the DrsquoAgostini [153] method and the dashed lines arerecovered by Guidorzi et al [149] (Figure after Guidorzi et al [151] see Figure 1 therein) (c) log 119871peak- log119881 relation for the set of 551 BATSEGRBs The best-fit lines and 1120590 regions are also shown the solid lines are fitted with the Reichart et al [102] method the dashed-dotted lineswith the DrsquoAgostini [153] method and the dashed lines are recovered by Guidorzi et al [149] (Figure after Guidorzi et al [151] see Figure 2therein)

321 Literature Overview Reichart et al [102] used a totalof 20GRBs observed by CGROBATSE (13 bursts) theKONUSWind (5 bursts) the UlyssesGRB (1 burst) and theNEARXGRS (1 burst) finding

log 119871peak sim (33+11minus09) log119881 (15)

with 120588 = 08 and 119875 = 14 times 10minus4 (see Figure 3(a)) 119871peak wascomputed in the 50ndash300 keV observer-frame energy bandwhich corresponds roughly to the range 100ndash1000 keV inthe rest frame for 119911 ≃ 1-2 typical for GRBs in the sampleexamined The distribution of the samplersquos bursts in the

log 119871peak- log119881119891 plane appears to be well modeled by thefollowing parameterization

log119881119891 (119871) = log119881119891 + 119887 + 119898(log 119871peak minus log 119871peak) (16)

where 119887 = 0013+0075minus0092 is the intercept of the line 119898 =0302+0112minus0075 is its slope and 119881119891 and 119871peak are the medianvalues of 119881119891 and 119871peak for the bursts in the sample for whichspectroscopic redshifts peak fluxes and 64ms or betterresolution light curves are available

Later Guidorzi et al [149] updated the sample to32GRBs detected by different satellites that is BeppoSAXCGROBATSE HETE-2 and KONUS (see Figure 3(b)) The


existence of a correlation was confirmed but they found adramatically different relationshipwith respect to the originalone

log 119871peak = 336+089minus043 + 130+084minus044 log119881 (17)

with120588 = 0625 and119875 = 10minus4 and119871peak in units of 1050 erg sminus1However Reichart and Nysewander [150] using the same

sample claimed that this result was the outcome of animproper statistical methodology and confirmed the previ-ous work of Reichart et al [102] Indeed they showed that thedifference among their results and the ones from Guidorzi etal [149] was due to the fact that the variance of the samplein the fit in [149] was not taken into account They used anupdated data set finding that the fit was well described by theslope119898 = 34+09minus06 with a sample variance 120590119881 = 02 plusmn 004

Subsequently Guidorzi et al [151] using a sample of551 BATSE GRBs with pseudoredshifts derived using the119871peak-120591lag relation [152] tested the 119871peak-119881 correlation (seeFigure 3(c)) They also calculated the slope of the correlationof the samples using the methods implemented by Reichartet al [102] andDrsquoAgostini [153]The formermethod provideda value of the slope in the 119871peak-119881 correlation consistent withrespect to the previous works

log119871peak sim 35+06minus04 log119881 (18)

Instead the slope for this sample using the latter method ismuch lower than the value in [102]

log 119871peak sim 088+012minus013 log119881 (19)

The latter slope 119898 is consistent with the results obtained byGuidorzi et al [149] but inconsistent with the results derivedby Reichart and Nysewander [150]

Afterwards Rizzuto et al [154] tested this correlationwitha sample of 36 LGRBs detected by Swift in the 15ndash350 keVenergy range and known redshifts The sample consistedof bright GRBs with 119871peak gt 5 times 1050 erg sminus1 within100ndash1000 keV energy range In their study they adoptedtwo definitions of variability presented by Reichart et al[102] called 119881R and by Li and Paczynski [114] hereafter119881LP 119881R and 119881LP differ from each other with a differentsmoothing filter which in the second case selects only high-frequency variability Finally Rizzuto et al [154] confirmedthe correlation and its intrinsic dispersion around the best-fitting power law given by

log119871peak sim (23 plusmn 017) log119881LP (20)

with 120588 = 0758 and 119875 = 0011 andlog 119871peak sim (17 plusmn 04) log119881R (21)

with 120590log119871 = 058+015minus012 120588 = 0115 and 119875 = 0506Six low-luminosity GRBs (ie GRB050223 GRB050416AGRB050803 GRB051016B GRB060614 and GRB060729)out of a total of 36 in the sample are outliers of thecorrelation showing values of119881R higher than expectedThusthe correlation is not valid for low-luminosity GRBs

As is visible from this discussion the scatter in thisrelation is not negligible thus making it less reliable than thepreviously discussed ones However investigating the physi-cal explanation of this correlation is worth being depicted forfurther developments

322 Physical Interpretation of the 119871119901119890119886119896-119881 Relation Wehere briefly describe the internal and external shock model[80 155] in which the GRB is caused by emission from arelativistic expanding baryonic shell with a Lorentz bulkfactor Γ Let there be a spherical section with an openingangle 120579jet In general 120579jet can be greater than Γminus1 but theobserver can detect radiation coming only from the angularregion with size ≃ Γminus1 An external shock is formed whenthe expanding shell collides with the external medium Ingeneral there might be more than one shell and the internalshock takes place when a faster shell reaches a slower oneIn both cases one distinguishes an FS when the shockpropagates into the external shell or the externalmedium anda reverse shock (RS) when it propagates into the inner shell

Fenimore and Ramirez-Ruiz [148] pointed out that theunderlying cause of the 119871peak-119881 relation is unclear In thecontext of the internal shock model larger initial Γ factorstend to produce more efficient collisions After changingsome quantities such as the Γ factors the ambient densityandor the initial mass of the shells the observed variabilityvalues are not recovered Therefore the central engine seemsto play a relevant role in the explanation for the observed119871peak-119881 correlation In fact this correlationwas also exploredwithin the context of a model in which the GRB variability isdue to a change in the jet-opening angles and narrower jetshave faster outflows [156] As a result this model predictsbright luminosities small pulse lags and large variability aswell as an early jet break time for on-axis observed bursts Onthe other hand dimmer luminosities longer pulse lags flatterbursts and later jet break times will cause larger viewingangles

Guidorzi et al [151] gave an interpretation for the smallervalue of the correlation in the context of the jet-emissionscenario where a stronger dependence of Γ of the expandingshells on the jet-opening angle is expectedHowever Schaefer[157] attributed the origin of the 119871peak-119881 relation to be basedon relativistically shocked jets Indeed 119881 and 119871peak are bothfunctions of Γ where 119871peak is proportional to a high power ofΓ as was already demonstrated in the context of the 119871peak-120591lagrelation (see Section 312) and hence fast rise times and shortpulse durations imply high variability

33 The 119871 119894119904119900-120591119877119879 Correlation and Its Physical InterpretationSchaefer [158] predicted that 120591RT should be connected with119871 iso in a following manner

119871 iso prop 120591minus1198732RT (22)

with the exponent 119873 ≃ 3 (see also Schaefer [157 158])Therefore fast rises indicate high luminosities and slow riseslow luminosities 120591RT can be directly connected to the physicsof the shocked jet Indeed for a sudden collision of a materialwithin a jet (with the shock creating an individual pulse


50

51

52

53

54ＦＩＡ(L)

minus2 minus1 0 1minus3

ＦＩＡ[２４(1 + z)]

(a)

minus15 minus10 minus05 00 05minus20

ＦＩＡ[２４(1 + z)]

50

51

52

53

54

ＦＩＡ(L)

(b)

Figure 4 (a) The log 119871 iso- log 120591lowastRT relation with the best-fit line displayed The errors are given by the 1120590 confidence interval (Figure afterSchaefer [157] see Figure 5 therein AAS Reproduced with permission) (b) The log 119871 iso- log 120591lowastRT correlation with the best fit line (Figureafter Xiao and Schaefer [159] see Figure 3 therein AAS Reproduced with permission)

in the GRB light curve) 120591RT will be determined as themaximumdelay between the arrival time of photons from thecenter of the visible region versus their arrival time from itsedge

The angular opening of the emitted jet usually associatedwith Γ could cause this delay leading to a relation 120591RT propΓminus2 The radius at which the material is shocked affects theproportionality constant and the minimum radius underwhich the material cannot radiate efficiently anymore shouldbe the same for each GRB [139] In addition a large scatter isexpected depending on the distance fromwhich the collisionsare observed

With both 120591RT and 119871 iso being functions of Γ Schaefer[157] confirmed that log 119871 iso should be sim minus1198732 log 120591RT From69GRBs detected by BATSE and Swift the following relationwas obtained

log 119871 iso = 5354 minus 121 log 120591lowastRT (23)

with 119871 iso in erg sminus1 and 120591lowastRT measured in seconds The 1120590uncertainties in the intercept and slope are 120590119886 = 006 and120590119887 = 006 (see Figure 4(a)) The uncertainty in the log of theburst luminosity is

1205902log119871 iso = 1205902119886 + [120590119887 log 120591lowastRT01 s]

2 + (043119887120590RT120591RT )2

+ 1205902RTsys(24)

where Schaefer [157] takes into account the extra scatter 120590sysWhen 120590RTsys = 047 1205942 of the best-fit line is unity

Xiao and Schaefer [159] explained in detail the procedureof how they calculated 120591RT using 107GRBs with knownspectroscopic redshift observed by BATSE HETE KONUSand Swift (see Figure 4(b)) taking into account also thePoissonian noise Their analysis yielded


with the same units as in (23) As a consequence theflattening of the light curve before computing the rise time isan important step The problem is that the flattening shouldbe done carefully in fact if the light curve is flattened toomuch a rise time comparable with the smoothing-time binis obtained while if it is flattened not enough the Poissoniannoise dominates the apparent fastest rise time giving a toosmall rise timeTherefore for some of the dimmest bursts thePoissonian-noise dominant region and the smoothing-effectdominant region can coincide thus not yielding 120591RT valuesfor the weakest bursts Finally the physical interpretationof this correlation is given by Schaefer [157] It is shownthat the fastest rise in a light curve is related to the Lorentzfactor Γ simply due to the geometrical rise time for a regionsubtending an angle of 1Γ assuming that the minimumradius for which the optical depth of the jet material is oforder of unity remains constant The luminosity of the burstis also a power law of Γ which scales as Γ119873 for 3 lt 119873 lt 5Therefore the 120591RT-Γ and the119871 iso-Γ relations together yield theobserved 119871 iso-120591RT relation

34 The Γ0-119864prompt and Γ0-119871 iso Correlations andTheir PhysicalInterpretation Freedman andWaxman [160] in their analysisof the GRB emission considering a relativistic velocity for thefireball showed that the radiation detected by an observeris within an opening angle ≃ 1Γ(119905) Hence the totalfireball energy 119864 should be interpreted as the energy thatthe fireball would have carried if this is assumed sphericallysymmetric In particular it was claimed that the afterglowflux measurements in X-rays gave a strong evaluation forthe fireball energy per unit solid angle represented by 120598119890 =1205851198901198644120587 within the observable opening angle 1Γ(119905) where 120585119890is the electron energy fraction It was found that

Γ (119905) = 106 (1 + 1199112 )38 (119864prompt1198990 )18 119905minus38 (26)


100

1000

Γ 0

090510 080319B090902B

990123

080916C

01 1 10 100 1000001

EＣＭＩ (1052 erg)

(a)

101

102

103

Γ 0

10minus3 10minus2 10minus1 100 101 10210minus4

LＣＭＩ52

(b)

Figure 5 (a) log Γ0- log119864prompt relation with the addition of GRBs with an onset trend in the X-ray band (GRBs 070208 and 080319C redstars) Γ0 computed by RS peaks or probable afterglow peaks (pink open circles) lower values of Γ0 obtained from single power law decay lightcurves (pink solid triangles) and strong lower values of Γ0 for FermiLATGRBs 080916C 090902B and 090510 (red open triangles) calculatedfrom opacity limits with FermiLAT observationsThe solid line indicates the best fit of the Γ0-119864prompt relation log Γ0 = 226+025 log119864promptThe two dashed lines represent the 2120590 deviation where the standard deviation of the ratio Γ0119864025prompt for the data sample is 120590 = 011 (Figureafter Liang et al [164] see Figure 8 therein AAS Reproduced with permission) (b) log Γ0 versus log 119871 iso distribution The best-fit line isgiven by log Γ0 ≃ 240 + 030 log 119871 iso with 119903 = 079 The triangles represent the bursts with only lower values and the star indicates the onlyshort burst in the sample GRB090510 (Figure after Lu et al [143] see Figure 2 therein AAS Reproduced with permission)

where119864prompt is in units of 1053 erg 1198990 is the uniform ambientdensity of the expanding fireball in units of cmminus3 and 119905 is thetime of the fireball expansion in days Finally it was pointedout that 120585119890 from the afterglow observations should be close toequipartition namely 120585119890 ≃ 13 For example forGRB970508it was found that 120585119890 ≃ 02 [161ndash163] A similar conclusionthat is that it is also close to equipartition could be drawnfor GRB971214 however Wijers and Galama [162] pro-posed another interpretation for this GRBrsquos data demanding120585119890 ≃ 1

Liang et al [164] selected from the Swift catalogue20 optical and 12 X-ray GRBs showing the onset of theafterglow shaped by the deceleration of the fireball due to thecircumburst mediumThe optically selected GRBs were usedto fit a linear relation in the log Γ0- log119864prompt plane where Γ0is the initial Lorentz factor of the fireball and119864prompt is in unitsof 1052 erg (see Figure 5(a)) The best-fit line of the Γ0-119864promptrelation is given by

log Γ0 = (226 plusmn 003) + (025 plusmn 003) log119864prompt (27)

with 120588 = 089119875 lt 10minus4 and 120590 = 011which can bemeasuredwith the deviation of the ratio Γ0119864025prompt It was found thatmost of the GRBs with a lower limit of Γ0 are enclosed withinthe 2120590 region represented by the dashed lines in Figure 5(a)and it was pointed out that GRBs with a tentative Γ0 derivedfrom RS peaks or the afterglow peaks as well as those whichlower limits of Γ0 were derived from light curves with a singlepower law are systematically above the best-fit lineThe lowervalues of Γ0 obtained from a set of optical afterglow lightcurves with a decaying trend since the start of the detectionwere compatible with this correlation

Later this correlation was verified by Ghirlanda et al[165] and Lu et al [143] Ghirlanda et al [165] studyingthe spectral evolution of 13 SGRBs detected by FermiGBMinvestigated spectra resolved in the 8 keVndash35MeV energyrange and confirmed the results of Liang et al [164]

Lu et al [143] enlarged this sample reaching a totalof 51 GRBs with spectroscopically confirmed redshifts andengaged threemethods to constrain Γ0 (1) the afterglowonsetmethod [166] which considers 119879peak of the early afterglowlight curve as the deceleration time of the external FS (2)the pair opacity constraint method [167] which requires thatthe observed high-energy 120574-rays (ie those in theGeV range)are optically thin to electron-positron pair production thusleading to a lower limit on Γ0 of the emitting region (3) theearly external forward emissionmethod [168]where an upperlimit of Γ0 can be derived from the quiescent periods betweenthe prompt emission pulses in which the signal of externalshock has to go down the instrument thresholds Consideringsome aspects of the external shock emission the Γ0-119864promptcorrelation was statistically reanalysed using 38GRBs with Γ0calculated using method (1) (as the other two provide only arange of the Lorentz factors not a definite value) finding


with 119903 = 067 and 119864prompt in units of 1052 erg In additionapplying the beaming correction a relation between Γ0 and119871 iso using the same sample (see Figure 5(b)) was found tobe

log Γ0 = (240 plusmn 0002) + (030 plusmn 0002) log119871 iso (29)

with 119903 = 079 and 119871 iso in units of 1052 erg sminus1

Advances in Astronomy 11⟨E

p⟩

(keV

)

00 05 10 15 20 25minus05

ＦＩＡＣＨＮＨＭＣＮＳ (photons cＧminus2 Ｍminus1)

400

350

300

250

200

150

100

Figure 6 The average ]119865] peak energies as a function of intensityfor five groups of GRB spectra The vertical bars represent 1120590estimated error in the mean where the peak energy distributionswere assumed to be approximately Gaussian in logarithm of energyThe horizontal bars mark the bin widths (Figure afterMallozzi et al[110] see Figure 2 therein AAS Reproduced with permission)

Regarding the physical interpretation Liang et al [164]claimed that this correlation clearly shows the associationof 119864prompt with Γ0 angular structure and this result yieldedanother evidence for the fireball deceleration model InsteadLu et al [143] found that this relation is well explained bya neutrino-annihilation-powered jet during the emissionindicating a high accretion rate and not very fast BH spinBesides evidence for a jet dominated by a magnetic fieldhave already been presented [169ndash171] From the studies ofthe BH central engine models it was also indicated thatmagnetic fields are a fundamental feature [172] Neverthelessthe baryon loading mechanism in a strongly magnetized jetis more complex and it has still to be fully investigated

35 Correlations between the Energetics and the Peak Energy

351 The ⟨119864119901119890119886119896⟩-119865119901119890119886119896 and the 119864119901119890119886119896-119878119905119900119905 Correlations Mal-lozzi et al [110] analysed 399GRBs observed by BATSE anddiscovered a correlation between the logarithmic averagepeak energies ⟨119864peak⟩ and 119865peak Choosing as a selection cri-terion for the bursts 119865peak ge 1 ph cmminus2 sminus1 they derived 119865peakfrom the count rate data in 256ms time bins in the energyband 50ndash300 keV and used the 119864peak distribution derivedfrom theComptonized photonmodel (the differential photonnumber flux per unit energy)

d119873d119864 = 119860eminus119864(2+120573119878)119864peak ( 119864119864piv)

120573119878 (30)

with 119860 the normalization 120573119878 the spectral index and 119864piv =100 keVThen they grouped the sample into 5 different width119865peak bins of about 80 events each (see Figure 6) The bursts

were ranked such that group 1 had the lowest peak flux valuesand group 5 had the highest values They found a correlationwith 120588 = 090 and 119875 = 004 Lower intensity GRBs exhibiteda lower ⟨119864peak⟩

Later Lloyd et al [173] examined the119864peak-119878tot correlationwith 1000 simulated bursts in the same energy range asMallozzi et al [110] and found a strong correlation between119864peak and 119878tot (see Figure 7(a)) The relation between the twovariables was as follows

log119864peak sim 029 log 119878tot (31)

with the Kendall correlation coefficient [174] 120591 = 080 and119875 = 10minus13 In addition they compared it to the 119864peak-119865peakrelation (see Figure 7(b)) This relation was for the wholespectral sample and consistent with earlier results [110 175]However they selected a subsample composed of only themost luminous GRBs because spectral parameters obtainedfrom bursts near the detector threshold are not robustTherefore to better understand the selection effects relevantto 119864peak and burst strength they considered the followingselection criteria 119865peak ge 3 ph cmminus2 sminus1 119878obs ge 10minus6 erg cmminus2and 119878tot ge 5 times 10minus6 erg cmminus2 Due to the sensitivity over acertain energy band of all the detectors especially BATSEand to some restrictions to the trigger the selection effectsare inevitable However the subsample of the most luminousGRBs presents a weak 119864peak-119865peak correlation Instead a tight119864peak-119878tot correlation was found for the whole sample as wellas the subsample of the brightest GRBs Lloyd et al [173] paidmore attention to the 119864peak-119878tot correlation for the brightestGRBs because it is easier to deal with the truncation effects inthis case and the cosmological interpretation is simpler

This correlation has been the basis for the investigationof the Amati relation (see Section 352) and the Ghirlandarelation (see Section 353) Lloyd et al [173] concluded thatldquothe observed correlation can be explained by cosmologicalexpansion alone if the total radiated energy (in the 120574-rayrange) is constantrdquo In fact their finding does not depend onthe GRB rate density or on the distribution of other param-eters However the data from GRBs with known redshift areincompatible with a narrow distribution of radiated energy orluminosity

Following a different approach Goldstein et al [176]pointed out that the ratio 119864peak119878tot can serve as an indicatorof the ratio of the energy at which most of the 120574-rays areradiated to the total energy and claimed that the 119864peak-119878totrelation is a significant tool for classifying LGRBs and SGRBsThe fluence indicates the duration of the burst withoutproviding a biased value of 11987990 and 119864peak119878tot displays as aspectral hardness ratio an increased hardness for SGRBs inrespect to LGRBs in agreement with [10] This correlationis quite interesting since the energy ratio being dependentonly on the square of the luminosity distance gets rid ofthe cosmological dependence for the considered quantitiesTherefore it was evaluated that the energy ratio could be usedas a GRB classifier

Later Lu et al [177] with the results of time-resolvedspectral analysis computed the 119864peak-119878tot relation for 51LGRBs and 11 bright SGRBs observed with FermiGBM For


FＮＩＮ

0000110minus510minus610minus6 10minus5

EP

(100

keV

)

1

(a)

Ep

fp

11 10

(b)

Figure 7 119864peak versus (a) 119878tot and (b) 119865peak distributions for the complete (dashed elements) and sub (solid elements) spectral sample Theflux suggests a tight correlation at low values but not for the most luminous GRBsThe solid line represents a least squares fit compatible withthe correlation computed by statistical methods (Figures after Lloyd et al [173] see Figure 3 therein AAS Reproduced with permission)

each GRB they fitted a simple power law function Theymeasured its scatter with the distance of the data points fromthe best-fit lineThemeasured scatter of the119864peak-119878tot relationis 017 plusmn 008 This result was reported for the first time byGolenetskii et al [178] and later confirmed by Borgonovoand Ryde [179] Ghirlanda et al [180] Guiriec et al [181]Ghirlanda et al [165]

352 The 119864119901119890119886119896-119864119894119904119900 Correlation Evidence for a correlationbetween 119864peak and 119878tot was first found by Lloyd and Petrosian[182] and Lloyd et al [183] based on 46 BATSE events butthis relation was in the observer frame due to the paucity ofthe data with precise redshift measurement as was shownin previous paragraphs Evidence for a stronger correlationbetween 119864peak and 119864iso also called the Amati relation wasreported by Amati et al [111] based on a limited sample of12GRBswith known redshifts (9 with firm redshift and 3withplausible values) detected by BeppoSAX They found that

log119864peak sim (052 plusmn 006) log119864iso (32)

with 119903 = 0949 119875 = 0005 and 119864iso calculated as

119864iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119878tot (1 + 119911)minus2 (33)

Regarding the methodology considered instead of fittingthe observed spectra as done for example by Bloom et al[184] the GRB spectra were blue-shifted to the rest frames toobtain their intrinsic form Then the total emitted energy iscalculated by integrating the Band et al [115] spectral modelin 1ndash104 keV energy band and scaling for the luminositydistance This was computed employing a flat Friedmann-Lemaitre-Robertson-Walker cosmological model with 1198670 =65 km sminus1Mpcminus1 Ω119872 = 03 ΩΛ = 07 and taking intoaccount both the cosmological time dilation and spectralredshift

Amati et al [185] enlarged the set of Amati et al [111]by including 20GRBs from BeppoSAX with known redshiftfor which new spectral data (BeppoSAX events) or publishedbest-fitting spectral parameters (BATSE and HETE-2 events)were accessible The relation was found to be

log119864peak = (207 plusmn 003) + (035 plusmn 006) log119864iso (34)

with 119903 = 092 119875 = 11 times 10minus8 119864peak in keV and 119864isoin units of 1052 erg Therefore its statistical significanceincreased providing a correlation coefficient comparable tothat obtained by Amati et al [111] but based on a larger set

Based on HETE-2 measurements Lamb et al [186]and Sakamoto et al [187] verified the previous results andconsidered also XRFs finding out that the Amati relationremains valid over three orders of magnitude in 119864peak andfive orders of magnitude in 119864iso The increasing amount ofGRBs with measured redshift allowed verifying this relationand strengthen its validity as found by Ghirlanda et al [188]with 29 events (119903 = 0803 and 119875 = 76times10minus7 see Figure 9(a))

Ghirlanda et al [189] verified the 119864peak-119864iso correlationamong LGRBs considering a set of 442 BATSE GRBs withmeasured 119864peak and with pseudoredshifts computed viathe 119871peak-120591lag correlation It was shown that the scatter ofthe sample around the best-fitting line is comparable withthat of another set composed of 27GRBs with measuredspectroscopic redshifts This is because the weights of theoutliers were marginal It was noted that the relation for the442 BATSE GRBs has a slope slightly smaller (047) than theone obtained for the 27GRBs with measured spectroscopicredshifts (056)

Afterwards Amati [190] (see Figures 8(a) and 8(c))updated the study of the 119864peak-119864iso correlation consideringa sample of 41 LGRBsXRFs with firm values of 119911 and119864peak 12 GRBs with uncertain 119911 andor 119864peak 2 SGRBs withcertain values of 119911 and 119864peak and the subenergetic events


1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)

GRB 980425Long GRBsX-Ray FlashesShort GRBs

(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

BeppoSAXHETE-2KonusWIND

SwiftBATFermiGBM

(d)

Figure 8 (a) 119864peak-119864iso distribution for 41 GRBsXRFs with measured redshifts and 119864peak values Filled circles indicate Swift GRBs The solidline represents the best-fit line log119864peak = 198 + 049 log119864iso the dotted lines show the region within a vertical logarithmic deviation of 04The dashed line represents the best-fit line log119864peak = 189 + 057 log119864iso computed without taking into account the sample variance (Figureafter Amati [190] see Figure 2 therein) (b) The distribution of the sample in the 119864peak-119864iso plane The lines indicate the best-fit line and theplusmn2120590 confidence region for LGRBs and XRFs (Figure after Amati [193] see Figure 4 therein Copyright 2012 World Scientific PublishingCompany) (c) 119864peak-119864iso distribution of 12GRBs with uncertain values of 119911 andor 119864peak for the subenergetic event GRB980425 and for thetwo SGRBs 050709 and 051221 Filled circles represent Swift GRBs The solid line is the best-fit line log119864peak = 198 + 049 log119864iso the dottedlines mark the region within a vertical deviation in logarithmic scale of 04 The dashed line is the best-fit line log119864peak = 189 + 057 log119864isocomputed without taking into account the sample variance (Figure after Amati [190] see Figure 3 therein) (d) The 119864peak-119864iso distributionfor the LGRBsThe black line represents the best-fit line and for each point the color indicates the instrument which performed the spectralmeasurement (Figure after Amati and Valle [197] see Figure 4 therein Copyright 2013 World Scientific Publishing Company)

GRB980425SN1998bw and GRB031203SN2003lw The dif-ferent sets are displayed in Figure 8(b) Taking into accountalso the sample variance it was found that

log119864peak = 198+005minus004 + (049+006minus005) log119864iso (35)

with 120588 = 089 119875 = 7 times 10minus15 and units being the same asin (34) Moreover subenergetic GRBs (980425 and possibly

031203) and SGRBs were incompatible with the 119864peak-119864isorelation suggesting that it can be an important tool for dis-tinguishing different classes of GRBs Indeed the increasingnumber of GRBs with measured 119911 and 119864peak can providethe most reliable evidence for the existence of two or moresubclasses of outliers for the 119864peak-119864iso relation Moreoverthe relation is valid also for the particular subenergetic event


980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)

Figure 9 (a) 119864lowastpeak-119864120574 relation for GRBs with known redshift The filled circles represent 119864120574 for the events where a jet break was detectedGrey symbols indicate lowerupper limits The solid line represents the best fit that is log119864peak sim 5368 + 07 log119864120574 Open circles denote 119864isofor the GRBsThe dashed line represents the best fit to these points and the dash-dotted line is the relation shown by Amati et al [111] (Figureafter Ghirlanda et al [188] see Figure 1 therein AAS Reproduced with permission) (b) Rest frame plane of GRB energy The large blackdot indicates that all simulated GRBs were assigned 119864lowastpeak = 15 keV and 119864lowast120574 = 15 times 1048 erg Since Γ gt 1 but less than 8000 regions (I) areforbidden Since for all the simulated GRBs 120579jet le 90∘ they cannot stay in region (II)When Γ is small the beaming cone turns out to be largerthan the jet Therefore the isotropic-equivalent energy is given by log119864iso = log119864120574 + log(1 +1205730) + 2 log Γ lower than the energy computed bylog119864iso = log119864120574 minus log(1 minus cos 120579jet) This brings in a constraint log119864peak sim 13 times log119864iso and GRBs cannot lie to the right of this constraintHence region (III) is not allowed The black dots indicate the actual GRBs of the Swift sample The fit to the Swift sample is displayed as thedot-dashed line (Figure after Ghirlanda et al [209] see Figure 1 therein)

GRB060218 Finally the normalization considered by Amati[190] is consistent with those obtained by other instruments

Ghirlanda et al [191] confirmed the 119864peak-119864iso corre-lation for softer events (XRFs) The sample consisted of76GRBs observed by several satellites mainly HETE-2KONUSWind Swift and FermiGBM The most importantoutcome is a tight correlation with no new outliers (withrespect to the classical GRB980425 and GRB031203) in the119864peak-119864iso plane The obtained relation was


Amati et al [192] studied 95 Fermi GRBs with measured119911 and obtained an updated 119864peak-119864iso relation which read

log119864peak sim 057 log119864iso (37)

with 120588 = 088 and 119875 lt 10minus3 In particular they investigatedtwo GRBs (080916C and 090323) with very energetic promptemission and found that they follow the Epeak-119864iso relationwell On the other hand an SGRB 090510 also a very lumi-nous and energetic event was found not to obey the relationHence Amati et al [192] proposed that the correlation mightserve as a discriminating factor among high-energetic GRBsIn addition they claimed that the physics of the radiationprocess for really luminous and energetic GRBs is identical tothat for average-luminous and soft-dim long events (XRFs)because all these groups follow the Amati relation

Later Amati [193] provided an update of the analysisby Amati et al [194] with a larger sample of 120GRBs (see

Figure 8(b)) finding it to be consistent with the followingrelation

log119864peak = 2 + 05 log119864iso (38)

with units the same as in (34) and (35) Afterwards Qin andChen [195] analysed a sample of 153GRBs with measured 119911119864peak 119864iso and 11987990 observed by various instruments up to2012 May The distribution of the logarithmic deviation of119864peak from the Amati relation displayed a clear bimodalitywhich was well represented by amixture of two Gaussian dis-tributions Moreover it was suggested to use the logarithmicdeviation of the 119864peak value for distinguishing GRBs in the119864peak versus 119864iso plane This procedure separated GRBs intotwo classes the Amati type bursts which follow the Amatirelation and the non-Amati type bursts which do not followit For the Amati type bursts it was found that


with 119903 = 083 and 119875 lt 10minus36 while for non-Amati bursts


with 119903 = 091 and 119875 lt 10minus7 In both relations 119864peak is in keVand 119864iso is in units of 1052 erg

In addition it was pointed out that almost all Amati typebursts are LGRBs at higher energies as opposed to non-Amatitype bursts which are mostly SGRBs An improvement tothis classification procedure is that the two types of GRBsare clearly separated hence different GRBs can be easilyclassified


Heussaff et al [196] applying particular selection criteriafor the duration and the spectral indices obtained a set ofFermi GRBs and analysed their locations in the 119864peak-119864isoplane The sample composed of 43GRBs with known red-shifts yielded the following relation

log119864peak = 207 + 049 log119864iso (41)

with 120588 = 070 119875 = 17 times 10minus7 and the same units as inprevious relations of this type

Amati and Della Valle [197] pointed out that an enlargedsample of 156 LGRBs with known 119911 and 119864peak also followsthe Amati relation with a slope ≃ 05 (see Figure 8(d))Additionally Basak and Rao [198] showed that a time-resolved Amati relation also holds within each single GRBwith normalization and slope consistent with those obtainedwith time-averaged spectra and energeticsluminosity and iseven better than the time-integrated relation [199] Time-resolved 119864peak and 119864iso are obtained at different times duringthe prompt phase (see also [177 180 200] and Section 36)

353 The 119864119901119890119886119896-119864120574 Correlation The 119864peak-119864120574 relation (alsocalled the Ghirlanda relation) was first discovered byGhirlanda et al [188] who used 40GRBs with 119911 and 119864peakknown at their time of writing Considering the time 119879breakits value can be used to deduce 119864120574 from 119864iso Indeed even ifonly a little less thanhalf of the bursts have observed jet breaks(47) from [107] we know that

120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

where 119879break is measured in days 119899 is the density of thecircumburst medium in particles per cm3 120578120574 is the radiativeefficiency and 119864iso is in units of 1052 erg Here 120579jet is indegrees and it is the angular radius (the half opening angle)subtended by the jet For GRBs with no measured 119899 themedian value 119899 = 3 cmminus3 of the distribution of the computeddensities extending between 1 and 10 cmminus3 was considered[139 201ndash203]

Later Liang and Zhang [204] using a sample of 15GRBswith measured 119911 119864peak and 119879break considered a purelyphenomenological119879lowastbreak of the optical afterglow light curvesthus avoiding the assumption of any theoretical modelcontrary to what was done by Ghirlanda et al [188] Thefunctional form of this correlation is given by

log119864120574 = (085 plusmn 021) + (194 plusmn 017) log119864lowastpeakminus (124 plusmn 023) log119879lowastbreak (43)

where 119864120574 is in units of 1052 erg 119864lowastpeak in units of 100 keV119879lowastbreak is measured in days and 120588 = 096 and 119875 lt 10minus4

Nava et al [205] found that the Ghirlanda relationassuming a wind-like circumburst medium is as strong asthe one considering a homogeneous medium They analysedthe discrepancy between the correlations in the observedand in the comoving frame (with Lorentz factor identical tothe fireballrsquos one) Since both 119864peak and 119864120574 transform in the

same way the wind-like Ghirlanda relation remains linearalso in the comoving frame no matter what the Lorentzfactorrsquos distribution is The wind-like relation corresponds tobursts with the same number of photons emitted Instead forthe homogeneous density medium scenario it is common toconsider a tight relation between the Lorentz factor and thetotal energy thus limiting the emission models of the promptradiation Using 18GRBs with firm 119911 119864peak and 119879break Navaet al [205] found for the homogeneous density case

log119864lowastpeak100 keV = 045+002minus003

+ (069 plusmn 004) log 119864120574272 times 1052 erg (44)

with 120588 = 093 and 119875 = 23 times 10minus8 Instead for the wind case



with 120588 = 092 and 119875 = 69 times 10minus8Ghirlanda et al [206] tested the 119864peak-119864120574 correlation

using 33GRBs (16 new bursts detected by Swift with firm 119911and119864peak up toDecember 2006 and 17 pre-SwiftGRBs)Theyclaimed that for computing 119879break the following is required

(1) The detection of the jet break should be in the optical(2) The optical light curve should continue up to a time

longer than the 119879break(3) The host galaxy flux and the flux from a probable SN

should be removed(4) The break should not depend on the frequency in the

optical and a coincident break in theX-ray light curveis not necessary because the flux in X-rays could becontrolled by another feature

(5) The considered119879break should be different from the oneat the end of the plateau emission (the time119879119886 in [81])otherwise the feature affecting the X-ray flux is alsoinfluencing the optical one

Therefore considering all these restrictions the samplewas reduced to 16GRBs all compatible with the following119864peak-119864120574 relation

log119864peak100 keV = (048 plusmn 002)


No outliers were detected Therefore the reduced scatterof the 119864peak-119864120574 relation corroborates the use of GRBs asstandardizable candles


354 Physical Interpretation of the Energetics versus PeakEnergy Relations Lloyd et al [173] investigated the physicalexplanation of the 119864peak-119878tot correlation assuming the emis-sion process to be a synchrotron radiation from internal andexternal shocks Indeed they claimed that this correlation iseasily obtained considering a thin synchrotron radiation by apower law distribution of electrons with Γ larger than someminimum threshold value Γ119898 Moreover the internal shockmodel illustrates the tight 119864peak-119878tot relation and the emittedenergy better than the external shock model

Lloyd-Ronning and Petrosian [207] pointed out thatthe GRB particle acceleration is not a well analysed issueGenerally the main hypothesis is that the emitted particlesare accelerated via recurrent scatterings through the(internal) shocks They found that the recurrent crossings ofthe shock come from a power law distribution of the particleswith a precise index providing a large energy synchrotronphoton index Moreover the connection between 119864peakand the photon flux can be justified by the variation of themagnetic field or electron energy in the emission eventsFinally they claimed that in the majority of GRBs theacceleration of particles is not an isotropic mechanism butoccurs along the magnetic field lines

Amati et al [111] confirmed the findings of Lloydet al [183] that the log119864peaksim05 log119864iso relation is obtainedassuming an optically thin synchrotron shock model Thismodel considers electrons following the 119873(Γ) = 1198730Γminus119901 dis-tribution for Γ gt Γ119898 with Γ119898 GRB duration and1198730 constantin each GRB However the above assumptions are not fullyjustifiable In fact the duration is different in each GRB and119864iso might be smaller in the case of beamed emission

Amati [190] pointed out the impact that the correlationhas on the modeling of the prompt emission and on thepossible unification of the two classes of GRBs and XRFs Inaddition this correlation is often applied for checking GRBsynthesis models (eg [208 209])

In every model 119864peak and 119864iso depend on Γ and the119864peak-119864iso relation can help to relate the parameters of thesynchrotron shock model and inverse-Compton model [203208] Specifically Zhang and Meszaros [208] and Rees andMeszaros [210] found that for an electron distribution givenby a power law and produced by an internal shock in a fireballwith velocity Γ the peak energy is given as

log119864lowastpeak sim minus2 log Γ + 05 log 119871 minus log 119905] (47)

where 119871 is the total fireball luminosity and 119905] is the variabilitytimescale However to recover the 119864peak-119864iso relation fromthis relation Γ and 119905] should be similar for each GRB acondition that cannot be easily supported A further issuearises when one considers that 119871 prop Γ119873 with 119873 between 2and 3 in differentmodels [203 208 211] An explanation couldbe that direct or Comptonized thermal radiation from thefireball photosphere [208 210ndash219] can affect significantly theGRB prompt emission This can be a good interpretation ofthe really energetic spectra presented for many events [220ndash222] and the flat shape in GRB average spectra In such cases119864peak depends on the peak temperature 119879119887119887peak of photonsdistributed as by a blackbody and therefore it is associated

with the luminosity or emitted energy For Comptonizedradiation from the photosphere the relations are

log119864peak sim log Γ + log119879119887119887peak sim 2 log Γ minus 025 log119871 (48)

or

log119864peak sim log Γ + log119879119887119887peaksim minus05 log 1199030 + 025 log119871 (49)

where 1199030 is a particular distance between the central engineand the energy radiating area such that the Lorentz factorevolves as Γ ≃ 1199031199030 up to some saturation radius 119903119904 [210] Assuggested by Rees and Meszaros [210] in this scenario the119864peak-119864iso relation could be recovered for particular physicalcases just underneath the photosphere though it would relyon an undefined number of unknown parameters

Also for high-energetic GRBs (ie 119864iso asymp 1055 erg) thenonthermal synchrotron emission model can explain the119864peak-119864iso correlation This can be possible by consideringeither the minimum Lorentz factor and the normalization ofthe power law distribution of the emitting electrons constantin each GRB or constraints on the slope of the relationbetween Γ and the luminosity [183 208]

Panaitescu [223] used 76GRBs with measured redshiftsto analyse the case in which the 119864peak-119864iso relation for LGRBsis due to the external shock generated by a relativistic outflowinteracting with the ambient medium He considered theeffect of each parameter defining the 119864peak-119864iso relation onthe radial distribution of the external medium density andpointed out that the log119864peak sim 05 log119864iso relation isrecovered if the external medium is radially stratified Forsome combinations of radiative (synchrotron or inverse-Compton) and dissipation (such as RS or FS) mechanismsit is concluded that the external medium requires a particledensity distributed distinctly from 119877minus2 with 119877 being the dis-tance at which the GRB radiation is generatedThis tendencyshould be commonly associated with uniform mass-loss rateand final velocity

Mochkovitch and Nava [224] checked whether the119864peak-119864iso relation can be recovered in a casewhen the promptemission is due to internal shocks or alternatively if the cor-relation can give some limits for the internal shock scenariodefined through the impact of only two shells SimulatedGRB samples were obtained considering different modelparameter distributions such as the emitted power in therelativistic emission and Γ Simulated 119864peak-119864iso distributionswere plotted for each case and analysed together with theobserved relation (based on 58GRBs) The sample containedonly luminous Swift GRBs with 119865peak gt 26 ph cmminus2 sminus1 in the15ndash150 keV energy band In conclusion a correspondencebetween the model and data was found but exclusively if thefollowing restrictions for the dynamics of the emission andfor the dispersion of the energy are assumed

(1) The majority of the dispersed energy should beradiated in few electrons

(2) The spread between the highest and the lowestLorentz factor should be small


(3) If themean Lorentz factor grows as Γ prop 12 (where is the rate of injected energy or mean emitted powerin the relativistic outflow) the 119864peak-119864iso relation isnot retrieved and119864peak is diminishingwith larger119864isoHowever the 119864peak-119864iso relation can be regained ifΓ prop 12 is a lower constraint for a particular

(4) When the timescale or the width of the variabilityof the Lorentz factor is associated with Γ 119864peak-119864isorelation is recovered

For the Ghirlanda relation [188] with the assumptionthat the line of sight is within the jet angle the 119864peak-119864120574relation indicates its invariance when moving from the restframe to the comoving frame As a result the number ofradiated photons in each GRBs is comparable and shouldbe about 1057 The last characteristic could be importantfor understanding the dynamics of GRBs and the radiativemechanisms (see also Figure 9(b))

Collazzi et al [225] found that the mean 119864lowastpeak is near to511 keV the electron rest-mass energy 1198981198901198882 Therefore it isclaimed that the tight shape of the 119864peak distribution does notstem only from selection effects No studied mechanism candrive this effect however with the 119864lowastpeak compatible with theeffective temperature of the 120574-ray radiating area the almostconstant temperature needs some mechanism similar to athermostat keeping the temperature at a steady value Itwas suggested that such a mechanism could be an electron-positron annihilation

Ghirlanda et al [209] using a simulated sample analysedif different intrinsic distributions of Γ and 120579jet can replicatea grid of observational constraints With the assumptionthat in the comoving frame each GRB has similar 119864peak and119864120574 it was found that the distributions of Γ and 120579jet cannotbe power laws Instead the highest concordance betweensimulation and data is given by log-normal distributionsand a connection between their maxima like 12057925jetmaxΓmax =const In this work 120579jet and Γ are important quantities forthe calculation of the GRB energetics Indeed from a sampleof asymp30GRBs with known 120579jet or Γ it was found that the119864120574 distribution is centered at 1050ndash1051 erg and it is tightlyrelated to 119864peak It was obtained that

log119864peak sim log1198641205745 minus 21205730 (50)

Present values of Γ and 120579jet rely on incomplete data sets andtheir distributions could be affected by biases NeverthelessGhirlanda et al [209] claimed that greater values of Γ arerelated to smaller 120579jet values that is the faster a GRB thenarrower its jet

Furthermore GRBs fulfilling the condition sin 120579jet lt 1Γmight not display any jet break in the afterglow light curveand Ghirlanda et al [209] predicted that this group shouldcomprise asymp6 of the on-axis GRBs Finally their work iscrucial as it allowed finding that the local rate of GRBs isasymp03 of the local SNe Ibc rate and asymp43 of the localhypernovae(ie SNe Ibc with wide-lines) rate

36 Correlations between the Luminosity and the Peak Energy

361 The 119871 119894119904119900-119864119901119890119886119896 Correlation The 119871 iso-119864peak relation wasdiscovered by Schaefer [203] who used 84GRBs with known119864peak from the BATSE catalogue [121] and 20GRBs withluminosities based on optically measured redshift [111 226]It was found that (see Figure 10) for the 20GRBs

log119864peak sim (038 plusmn 011) log119871 iso (51)

with 119903 = 090 and 119875 = 3 times 10minus8 and among the 84GRBs therelation was


The underlying idea is that 119871 iso varies as a power of Γ as wehave already discussed in Section 312 and119864peak also varies assome other power of Γ so that119864peak and119871 iso will be correlatedto each other through their dependence on Γ For the generalcase where the luminosity varies as Γ119873 and119864peak varies as Γ119872and therefore log119864peak will vary as (119872 + 1)119873 times log119871 iso

Frontera et al [200] using a sample of 9GRBs detectedsimultaneously with the Wide Field Camera (WFC) onboard the BeppoSAX satellite and by the BATSE instrumentreported the results of a systematic study of the broad-band (2ndash2000 keV) time-resolved prompt emission spectraHowever only 4 of those GRBs (970111 980329 990123990510) were bright enough to allow a fine time-resolvedspectral analysis resulting in a total of 40 spectra Finally thestudy of the time-resolved dependence (see also the end ofSection 352) of 119864peak on the corresponding 119871 iso was possiblefor two bursts with known redshift (ie 990123 and 990510)and found using the least squares method (see Figure 11)

log119864lowastpeak sim (066 plusmn 003) log119871 iso (53)

with 120588 = 094 and 119875 = 157 times 10minus13Afterwards Nava et al [227] using a sample of 46 Swift

GRBs with measured 119911 and 119864peak found a strong 119871 iso-119864peakcorrelation with a functional form of

log119864lowastpeak = minus (2533 plusmn 326) + (053 plusmn 006) log 119871 iso (54)

with 120588 = 065 and 119875 = 10minus6 119864peak is in keV and 119871 iso is inunits of 1051 erg sminus1 Furthermore using 12GRBs with onlyan upper limit on 119911 (3 events) or no redshift at all (3 events)or with a lower limit on 119864peak (3 events) or no estimate at all(3 events) they found that these bursts also obey the obtained119871 iso-119864peak relation362 The 119871119901119890119886119896-119864119901119890119886119896 Correlation It was also found thatthe Amati relation holds even if 119864iso is substituted with 119871 isoand 119871peak which is not surprising given that these ldquoenergyindicatorsrdquo are strongly correlated To this end the Yonetokucorrelation ([116] see Figure 12(a)) relates 119864peak with 119871peakThe relation was obtained employing 11 GRBs with knownredshifts detected by BATSE together with BeppoSAX GRBsfrom [111]This relation uses 119871peak of the burst instead of 119871 iso


100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)

Figure 10 Direct fit of log119871 iso- log119864peak dataThis is shown here fortwo independent data sets for which the luminosities are derived bytwo independent methods The first data set consists of 20 burstswith spectroscopically measured redshifts (the open circles) Thesecond one is for 84 bursts (whose binned values are shown asfilled diamonds and the horizontal bars are the bin widths) whoseluminosity (and then redshift) were determinedwith the spectral lagand variability light curve parameters Both data sets show a highlysignificant and similar power law relations (Figure after Schaefer[203] see Figure 3 therein AAS Reproduced with permission)

10 10001

L (1052 erg Ｍminus1)

10

100

1000

10000

Epi

(keV

)

Figure 11 119864lowastpeak versus 119871 iso obtained from data for GRBs 990123and 990510 The solid line is the best-fit power law (Figure afterFrontera et al [200] see Figure 6 therein AAS Reproduced withpermission)

and it is tighter than previous prompt correlations The best-fit line is given by

log119871peak sim (20 plusmn 02) log119864lowastpeak (55)

with 119903 = 0958 119875 = 531 times 10minus9 and the uncertainties are 1120590errorThis relation agrees well with the standard synchrotronmodel [183 208] Finally it has been used to estimate pseu-doredshifts of 689 BATSE LGRBs with unknown distancesand to derive their formation rate as a function of 119911

Ghirlanda et al [228] selected 36 bright SGRBs detectedby BATSE with an 119865peak on the 64m119904 timescale in the

50ndash300 keV energy range exceeding 10 ph cmminus2 sminus1 In 7cases the signal-to-noise-ratio was too low to reliably con-strain the spectral best-fit parameters One case yieldedmissing data Hence the sample consisted of 28 events Dueto unknown redshifts 119864lowastpeak 119864iso and 119871peak were expressedas functions of the redshift in the range 119911 isin [0001 10] Itwas found that SGRBs are unlikely to obey theAmati relation119864iso-119864lowastpeak but the results were consistent with the 119871peak-119864lowastpeakrelation of Yonetoku et al [116] Hence assuming that thisrelation indeed holds for SGRBs their pseudoredshifts wereestimated and found to have a similar distribution as LGRBswith a slightly smaller average redshift

Afterwards Yonetoku et al [229] investigated the promptemission of 101 GRBs with measured redshifts and a reported119865peak detected until the end of 2009 The sample comes fromevents detected in a number of independent missions thesatellites used for this purpose are KONUS Swift HXD-WAM and RHESSI Using this data set the 119864peak-119871peakcorrelation was revised and its functional form could bewritten as

log 119871peak = (5243 plusmn 0037)+ (160 plusmn 0082) log119864lowastpeak (56)

with 119903 = 0889 for 99 degrees of freedom and an associated119875 = 218 times 10minus35 119871peak is expressed in erg sminus1 and 119864lowastpeakin units of 355 keV To provide reference to previous worksthe 1ndash104 keV energy band in the GRB rest frame was usedto calculate the bolometric energy and 119871peak Finally it wasdemonstrated that this relation is intrinsic to GRBs andaffected by the truncation effects imposed by the detectorthreshold

Lu and Liang [230] using time-resolved spectral data fora sample of 30 pulses in 27 bright GRBs detected by BATSEinvestigated the 119871peak-119864peak relation in the decay phases ofthese pulses (see Figure 12(b)) Quite all of the pulses followeda narrow 119871peak-119864peak relation given by


with 119903 = 091 and 119875 lt 10minus4 but the power law index variedThe statistical or observational effects could not account forthe large scatter of the power law index and it was suggestedto be an intrinsic feature indicating that no relation commonfor all GRB pulses 119871peak-119864peak would be expected Howeverin the light of Fermi observations that revealed deviationsfrom the Band function ([181 231ndash234] see also [235]) it wasproposed recently that the GRB spectra should be modelednot with the Band function itself (constituting a nonthermalcomponent) but with additional blackbody (BB thermal)and power law (PL nonthermal) components [216 218 219236] The nonthermal component was well described withinthe context of synchrotron radiation from particles in the jetwhile the thermal componentwas interpreted by the emissionfrom the jet photosphere The PL component was claimed tooriginate most likely from the inverse-Compton processTheresults point toward a universal relation between 119871peak and119864lowastpeak related to the nonthermal components


Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01

Present WorkAmati et al (2002)

(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)

Figure 12 (a)The log 119871peak- log119864peak relationThe open squaresmark the BATSE data BeppoSAX events which are converted into the energyrange of 30ndash10000 keV are shown as filled squares and the cross points The solid line indicates the best-fit line (Figure after Yonetoku et al[116] see Figure 1 therein AAS Reproduced with permission) (b) log 119871peak versus log119864peak for 276 time-resolved spectra within the decaypulses for the sampleThe solid line stands for the best fit to the data (Figure after Lu and Liang [230] see Figure 4 therein Copyright 2010Springer)

Tsutsui et al [237] analysed 13 SGRB candidates (ie anSGRB with 119879lowast90 lt 2 s) from among which they selected 8events considering them as secure ones An SGRB candidateis regarded as a misguided SGRB if it is located within the3120590int dispersion region from the best-fit 119864lowastpeak-119864iso functionof the correlation for LGRBs while the others are regarded assecure SGRBs The relation obtained with secure GRBs is thefollowing

log119871peak = (5229 plusmn 0066) + (159 plusmn 011) log119864lowastpeak (58)

with 119903 = 098 and 119875 = 15 times 10minus5 where 119864lowastpeak (in unitsof 7745 keV) is from the time-integrated spectrum while119871peak (in erg sminus1) was taken as the luminosity integratedfor 64ms at the peak considering the shorter duration ofSGRBs Application of this relation to 71 bright BATSESGRBs resulted in pseudoredshifts distributed in the range119911 isin [0097 2258] with ⟨119911⟩ = 105 which is apparentlylower than ⟨119911⟩ = 22 for LGRBs Finally Yonetoku et al[238] using 72 SGRBs with well determined spectral featuresas observed by BATSE determined their pseudoredshiftsand luminosities by employing the 119871peak-119864peak correlationfor SGRBs found by Tsutsui et al [237] It was foundthat the obtained redshift distribution for 119911 le 1 wasin agreement with that of 22 Swift SGRBs indicating thereliability of the redshift determination via the 119864lowastpeak-119871peakrelation

363 Physical Interpretation of the Luminosity versus PeakEnergy Relations As pointed out by Schaefer et al [121]and Schaefer [203] 119864peak and 119871 iso are correlated because oftheir dependence on Γ The 119871 iso-119864peak relation could shedlight on the structure of the ultrarelativistic outflow the

shock acceleration and the magnetic field generation [239]However since only few SGRBs are included in the samplesused the correlations and interpretations are currently onlyapplicable to LGRBs

Schaefer et al [121] and Schaefer [203] claimed thatthe values of 119864peak are approximately constant for all thebursts with 119911 ge 5 However with the launch of the Swiftsatellite in the end of 2004 the hunt for ldquostandard candlesrdquovia a number of GRB correlations is still ongoing Thusthe great challenge is to find universal constancy in someGRB parameters despite the substantial diversity exhibitedby their light curves If this goal is achieved GRBs mightprove to be a useful cosmological tool [240]

Liang et al [241] defined a parameter 120596 =(119871 iso1052 erg sminus1)05(119864peak200 keV) anddiscussed possibleimplications of the 119864peak-119871 iso relation for the fireball modelsThey found that 120596 is limited to the range ≃ 01ndash1 Theyconstrained some parameters such as the combined internalshock parameter 120577119894 for the internal as well as externalshock models with an assumption of uncorrelated modelparameters Their distributions suggest that the productionof prompt 120574-rays within internal shocks dominated by kineticenergy is in agreement with the standard internal shockmodel Similarly in case when the 120574-rays come from externalshocks dominated by magnetic dissipation These resultsimply that both models can provide a physical interpretationof the 119871 iso prop 1198642peak relation as well as the parameter 120596

To explain the origin of this correlation Mendoza et al[242] considered simple laws of mass and linear momen-tum conservation on the emission surface to give a fulldescription of the working surface flow parameterized bythe initial velocity and mass injection rate They assumeda source-ejecting matter in a preferred direction 119909 with


a velocity V(119905) and a mass ejection rate (119905) both dependenton time 119905 as measured from the jetrsquos source that is they stud-ied the case of a uniform release of mass and the luminositywas measured considering simple periodic oscillations of theparticle velocity a common assumption in the internal shockmodel scenario

Due to the presence of a velocity shear with a considerablevariation in Γ at the boundary of the spine and sheathregion a fraction of the injected photons is accelerated via aFermi-like acceleration mechanism such that a high-energypower law tail is formed in the resultant spectrum Ito etal [243] showed in particular that if a velocity shear with aconsiderable variance in Γ is present the high-energy partof the observed GRB photon spectrum can be explained bythis photon accelerationmechanismTheaccelerated photonsmay also account for the origin of the extra hard powerlaw component above the bump of the thermal-like peakseen in some peculiar GRBs (090510 090902B 090926A)It was demonstrated that time-integrated spectra can alsoreproduce the low energy spectra of GRBs consistently due toa multitemperature effect when time evolution of the outflowis considered

Regarding the Yonetoku relation its implications arerelated to the GRB formation rate and the luminosity func-tion of GRBs In fact the analysis of Yonetoku et al [116]showed that the existence of the luminosity evolution ofGRBs assuming as a function a simple power lawdependenceon the redshift such as 119892(119911) = (1 + 119911)185 may indicate theevolution of GRB progenitor itself (mass) or the jet evolutionTo study the evolution of jet-opening angle they consideredtwo assumptions either the maximum jet-opening angledecreases or the total jet energy increases In the former casethe GRB formation rate obtained may be an underestimationsince the chance probability of observing the high-redshiftGRBs will decrease If so the evolution of the ratio of theGRB formation rate to the star formation rate becomes morerapid On the other hand in the latter case GRB formationrate provides a reasonable estimate

Recently Frontera et al [244] building on the spectralmodel of the prompt emission of Titarchuk et al [245] gavea physical interpretation of the origin of the time-resolved119871 iso-119864peak relation The model consists of an expandingplasma shell result of the star explosion and a thermal bathof soft photons Frontera et al [244] showed analytically thatin the asymptotic case of the optical depth 120591 ≫ 1 the relationlog 119871 iso- log119864peak indeed has a slope of 12 This in turn isevidence for the physical origin of the Amati relation (seeSection 352)

37 Comparisons between 119864119901119890119886119896-119864119894119904119900 and 119864119901119890119886119896-119871119901119890119886119896 Corre-lation For a more complete dissertation we compare the119864peak-119864iso correlation with the119864peak-119871peak correlation To thisend Ghirlanda et al [246] derived the 119864peak-119871peak relationwith a sample of 22GRBs with known 119911 and well determinedspectral properties This relation has a slope of 051 similarto the one proposed by Yonetoku et al [116] with 12GRBsalthough its scatter is much larger than the one originallyfound

Tsutsui et al [247] investigated these two relations usingonly data from the prompt phase of 33 low-redshift GRBswith 119911 le 16 In both cases the correlation coefficientwas high but a significant scatter was also present Nexta partial linear correlation degree which is the degree ofassociation between two random variables was found to be120588119871peak 119864iso 119864peak = 038 Here 120588123 means the correlationcoefficient between the first and the second parameter afterfixing the third parameter This fact indicates that twodistance indicatorsmay be independent from each other evenif they are characterized by the same physical quantity 119864peakand similar quantities 119871peak and 119864iso To correct the largedispersion of the Yonetoku correlation Tsutsui et al [247]introduced a luminosity time constant 119879119871 defined by 119879119871 =119864iso119871peak as a third parameter and a new correlation wasestablished in the following form

log119871peak = (minus387 plusmn 019) + (182 plusmn 008) log119864peakminus (034 plusmn 009) log119879119871 (59)

with 119903 = 094 and 119875 = 10minus10 Here 119871peak is in units of1052 erg sminus1 119864peak is in keV and 119879119871 in seconds In this waythe systematic errors were reduced by about 40 and theplane represented by this correlation might be regarded as aldquofundamental planerdquo of GRBs

Later Tsutsui et al [248] reconsidered the correlationsamong 119864peak 119871peak and 119864iso using the database constructedby Yonetoku et al [229] which consisted of 109GRBs withknown redshifts and 119864peak 119871peak and 119864iso well determinedThe events are divided into two groups by their data qualityOne (gold data set) consisted of GRBs with 119864peak determinedby the Band function with four free parameters GRBs inthe other group (bronze data set) had relatively poor energyspectra so that their 119864peak were determined by the Bandfunction with three free parameters (ie one spectral indexwas fixed) or by the cut-off power law (CPL)model with threefree parameters Using only the gold data set the intrinsicdispersion 120590int in log 119871peak is 013 for the119864peak-119879119871-119871peak cor-relation and 022 for the 119864peak-119871peak correlation In additionGRBs in the bronze data set had systematically larger 119864peakthan expected by the correlations constructed with the golddata set This indicates that the quality of the sample is animportant issue for the scatter of correlations among 119864peak119871peak and 119864iso

The difference between the 119864peak-119871peak correlation forLGRBs from [180] and the one from [229] is due to thepresence of GRB060218 In the former it was consideredan ordinary LGRB while in the latter it was consideredan outlier by a statistical argument Because GRB060218 islocated far from the 119871peak-119864peak correlation in [229] (morethan 8120590) it makes the best-fit line much steeper

Regarding the high-energetic GRBs Ghirlanda et al [180]considered 13GRBs detected by Fermi up to the end of July2009 and with known redshift They found a tight relation

log119864lowastpeak sim 04 log 119871 iso (60)


with a scatter of 120590 = 026 A similarly tight relation existsbetween 119864lowastpeak and 119864iso

log119864lowastpeak sim 05 log119864iso (61)

The time-integrated spectra of 8 Fermi GRBs with measuredredshift were consistent with both the 119864peak-119864iso and the119864peak-119871 iso correlations defined by 100 pre-Fermi bursts

Regarding the study of SGRBs within the context of thesetwo correlations Tsutsui et al [237] used 8 SGRBs out of13 SGRB candidates to check whether the 119864peak-119864iso and119864peak-119871peak correlations exist for SGRBs as well It was foundthat the119864peak-119864iso correlation seemed to hold in the followingform

log119864iso = (5142 plusmn 015) + (158 plusmn 028) log119864lowastpeak (62)

with 119903 = 091 119875 = 15 times 10minus3 119864iso in erg sminus1 and 119864lowastpeakin units of 7745 keV They also found that the 119864peak-119871peakcorrelationwith a functional form as in (58) is tighter than the119864peak-119864iso one Both correlations for SGRBs indicate that theyare less luminous than LGRBs for the same 119864peak by factors≃ 100 (for 119864peak-119864iso) and ≃5 (for 119864peak-119871peak) It was the firsttime that the existence of distinct 119864peak-119864iso and 119864peak-119871peakcorrelations for SGRBs was argued

38 The 119871119883119901-119879lowast119901 Correlation and Its Physical InterpretationUsing data gathered by Swift Willingale et al [81] proposed aunique phenomenological function to estimate some relevantparameters of both the prompt and afterglow emission Bothcomponents are well fitted by the same functional form

119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)

The index 119894 can take the values 119901 or 119886 to indicate the promptand afterglow respectivelyThe complete light curve119891tot(119905) =119891119901(119905) + 119891119886(119905) is described by two sets of four parameterseach 119879119894 119865119894 120572119894 119905119894 where 120572119894 is the temporal power law decayindex the time 119905119894 is the initial rise timescale 119865119894 is the fluxand 119879119894 is the break time Figure 13 schematically illustratesthis function

Following the same approach as adopted in [131] Qiand Lu [249] investigated the prompt emission proper-ties of 107GRB light curves detected by the XRT instru-ment onboard the Swift satellite in the X-ray energy band(03ndash10 keV) They found that there is a correlation between119871119883119901 and119879lowast119901 Among the 107GRBs they used only 47 becausesome of the events did not have a firm redshift and some didnot present reliable spectral parameters in the prompt decayphase Among the 47GRBs only 37 had 119879lowast119901 gt 2 s and 3 ofthem had 119879lowast119901 gt 100 s

The functional form for this correlation could be writtenin the following way

log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2

ＦＩＡ10(Ｍ＝Ｍ)

Figure 13 Functional form of the decay and the fitted parametersThe prompt component (green curve) has no rise because time zerois set at the peakThe afterglow component (blue curve) rises at time119879119886 as shown (Figure after Willingale et al [81] see Figure 1 therein AAS Reproduced with permission)

where 119871119883119901 is in erg sminus1 and 119879lowast119901 is in seconds The fits wereperformed via the DrsquoAgostini [153] fitting method applied tothe following data sets

(1) The total sample of 47GRBs (see Figure 14(a))(2) 37GRBs with 119879lowast119901 gt 2 s (see Figure 14(b))(3) 34GRBs with 2 s lt 119879lowast119901 lt 100 s (see Figure 14(c))

The results of these fittings turned out to give different formsof (64) In case (1) 119886 = 5091 plusmn 023 and 119887 = minus089 plusmn 019were obtained The slope 119887 is different in cases (2) and (3)119887 = minus173 and 119887 = minus074 respectively The best fit withthe smallest 120590int comes from case (3) Remarkably in thiscase the slope 119887 is close to the slope (minus074+020minus019) of a similarlog 119871119883- log119879lowast119886 relation [131]

Qi and Lu [249] noticed a broken linear relation of the119871119883119901-119879lowast119901 correlation More specifically an evidence of curva-ture appears in Figure 14(b) One can see from Figure 14(a)that if the best-fit line is extended to the range of 119879lowast119901 lt 2 s allthe GRBs with 119879lowast119901 lt 2 s are located below this line Howeverthe small sample of GRBs used in their analysis is still notsufficient to conclude whether the change in the slope is realor just a selection bias caused by outliers If there is a changein the slope this may suggest that GRBs could be classifiedinto two groups long and short based on their values of 119879lowast119901instead of 11987990 since 119879lowast119901 is an estimate of the GRB durationbased on temporal features of the light curves and 11987990 is ameasure based on the energy This idea has actually beenproposed for the first time by OrsquoBrien and Willingale [250]It is worth noting that while 11987990 and 119879119901 are both estimatesof the GRB duration the correlation does not hold if 119879119901 isreplaced with 11987990 For an analysis of an extended sample andcomparison of 11987945 versus 119879119901 also see [251]


minus1 0 1 2 3 4minus2

ＦＩＡ[Tp(1 + z)]

44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)

Figure 14 (a) log 119871119883119901 (in erg sminus1) versus log119879lowast119901 (in s) for the whole sample of 47GRBsThe red dots represent SGRBs (ie 11987990 lt 2 s) (Figureafter Qi and Lu [249] see Figure 1 therein AAS Reproduced with permission) (b) Best fit of the log 119871119883119901 (in erg sminus1) versus log119879lowast119901 (in s)relation following equation (64) and the corresponding 2120590 confidence region Only GRBs with 119879lowast119901 gt 2 s are included in the fit (Figure afterQi and Lu [249] see Figure 2 therein AAS Reproduced with permission) (c) Best fit of the log 119871119883119901 (in erg sminus1) versus log119879lowast119901 (in s) relationfollowing equation (64) and the corresponding 2120590 confidence region In this case only the 34GRBs with 2 s lt 119879lowast119901 lt 100 s are included in thefit (Figure after Qi and Lu [249] see Figure 3 therein AAS Reproduced with permission)

Regarding the physical interpretation the change of theslope in the 119871119883119901-119879lowast119901 relation at different values of 119879lowast119901 in [249]can be due to the presence of few GRBs with a large 119879lowast119901 but it might also be due to different emission mechanismsUnfortunately the paucity of the sample prevents puttingforward any conclusion due to the presence of (potential)outliers in the data set A more detailed analysis is necessaryto further validate this correlation and better understand itsphysical interpretation

39 The 119871119891-119879119891 Correlation and Its Physical Interpretation Inmost GRBs a rapid decay phase (RDP) soon after the promptemission is observed [89] and this RDP appears to continue

smoothly after the prompt in terms of both temporal andspectral variations [86] This indicates that the RDP couldbe the prompt emissionrsquos tail and a number of models havebeen proposed to take it into account (see [88]) in particularthe high latitude emission (HLE) This model states thatonce the prompt emission from a spherical shell turns off atsome radius then the photons reach the observer from anglesapparently larger (relative to the line of sight) due to the addedpath length caused by the curvature of the emitting regionThe Doppler factor of these late-arriving photons is smaller

A successful attempt to individually fit all the distinctpulses in the prompt phase and in the late X-ray flaresobserved by the complete SwiftBAT + XRT light curves hasbeen performed by Willingale et al [108] using a physically


ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)

Figure 15 (a) 119871119891 (in erg sminus1) versus 119879lowast119891 (b) 119871119891 (in erg sminus1) versus 119864peak (Figures after Willingale et al [108] see Figure 16 therein)

motivated pulse profileThis fitting is an improved procedurecompared to the Willingale et al [81] one The pulse profilehas the following functional form

119875 = [min(119879 minus 119879ej119879119891 1)120572+2 minus (119879119891 minus 119879rise119879119891 )120572+2]

sdot [1 minus (119879119891 minus 119879rise119879119891 )120572+2]minus1

(119879 minus 119879ej119879119891 )minus1 (65)

where 1198790 = 119879119891 minus 119879rise (with 119879rise the rise time of thepulse) is the arrival time of the first photon emitted fromthe shell It is assumed here that the emission comes from anultrarelativistic thin shell spreading over a finite range of radiialong the line of sight in the observer frame measured withrespect to the ejection time 119879ej From these assumptions it ispossible tomodel the rise of the pulse through 120572119879rise and119879119891(see also Figure 1)

The combination of the pulse profile function119875(119905 119879119891 119879rise) and the blue shift of the spectral profile 119861(119909)produces the rise and fall of the pulse 119861(119909) is approximatedwith the Band function in the following form

119861 (119909) = 119861normtimes 119909(120572minus1)119890minus119909 119909 le 120572 minus 120573119909(120573minus1) (120572 minus 120573)(120572minus120573) 119890minus(120572minus120573) 119909 gt 120572 minus 120573

(66)

where 119909 = (119864119864119891)[(119879 minus 119879ej)119879119891]minus1 with 119864119891 being the energyat the spectral break and 119861norm is the normalization

Using this motivated pulse profile Willingale et al [108]found that within a sample of 12GRBs observed by Swift in

the BAT and XRT energy bands 119871119891 is anticorrelated with 119879lowast119891in the following way

log 119871119891 sim minus (20 plusmn 02) log119879lowast119891 (67)

Therefore high-luminosity pulses occur shortly after ejec-tion while low-luminosity pulses appear at later time (seeFigure 15(a)) Moreover Willingale et al [108] also found acorrelation between 119871119891 and 119864peak as shown in Figure 15(b)This is in agreement with the known correlation between119871peak for the whole burst and 119864peak of the spectrum duringthe time 11987990 [116 237] for comparison with the 119871peak-119864peakcorrelation see also Section 362

In the 12 light curves considered by Willingale et al[108] 49 pulses were analysed Although several pulses witha hard peak could not be correctly fitted the overall fittingto the RDP was satisfactory and the HLE model was shownto be able to take into account phase of the GRB emissionHowever it is worthmentioning the hard pulse in GRB061121which requires a spectral index 120573119878 = 24 larger than the valueexpected for synchrotron emission that is 120573119878 = 1

Lee et al [252] and Quilligan et al [253] discussedanalogous correlations although these authors consideredthe width of a pulse rather than 119879119891 which is in fact closelycorrelated with pulse width Many authors afterwards [254ndash264] have used the motivated pulse profile of Willingale et al[108] for various studies on the prompt emission propertiesof the pulses

Regarding the physical interpretation in [108] the fluxdensity of each prompt emission pulse is depicted by ananalytical expression derived under the assumption thatthe radiation comes from a thin shell as we have alreadydescribed The decay after the peak involves the HLE [265]along the considered shell which is delayed andmodifiedwitha different Doppler factor due to the curvature of the surface[266 267]


4 Summary

In this work we have reviewed the bivariate correlationsamong a number of GRB prompt phase parameters and theircharacteristics It is important tomention that several of thesecorrelations have the problem of double truncation whichaffects the parameters Some relations have also been testedto prove their intrinsic nature like the 119864peak-119878tot 119864peak-119864isoand 119871peak-119864peak relations For the others we are not aware oftheir intrinsic forms and consequently how far the use of theobserved relations can influence the evaluation of the theoret-ical models and the ldquobestrdquo cosmological settings Thereforethe evaluation of the intrinsic correlations is crucial for thedetermination of the most plausible model to explain theprompt emission In fact though there are several theoreticalinterpretations describing each correlation in many casesmore than one is viable thus showing that the emissionprocesses that rule GRBs still need to be further investigatedThese correlations might also serve as discriminating factorsamong different GRB classes as several of themhold differentforms for SGRBs and LGRBs hence providing insight into thegenerating mechanisms Hopefully those correlations couldlead to new standard candles allowing exploring the high-redshift universe

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

M Tarnopolski acknowledges support in a form of a specialscholarship of Marian Smoluchowski Scientific ConsortiumldquoMatter-Energy-Futurerdquo from KNOW funding Grant noKNOW48SSPC2015 The work of R Del Vecchio wassupported by the Polish National Science Centre throughGrant DEC-201204AST900083

References

[1] E Nakar ldquoShort-hard gamma-ray burstsrdquo Physics Reports vol442 no 1ndash6 pp 166ndash236 2007

[2] B Zhang ldquoOpen questions in GRB physicsrdquo Comptes RendusPhysique vol 12 no 3 pp 206ndash225 2011

[3] N Gehrels and S Razzaque ldquo120574-ray bursts in the swift-Fermierardquo Frontiers of Physics vol 8 no 6 pp 661ndash678 2013

[4] E Berger ldquoShort-duration gamma-ray burstsrdquo Annual Reviewof Astronomy and Astrophysics vol 52 no 1 pp 43ndash105 2014

[5] P Kumar and B Zhang ldquoThe physics of gamma-ray bursts andrelativistic jetsrdquo Physics Reports vol 561 pp 1ndash109 2015

[6] P Meszaros and M J Rees ldquoGamma-ray burstsrdquo GeneralRelativity and Gravitation

[7] RW Klebesadel I B Strong and R A Olson ldquoObservations ofgamma-ray bursts of cosmic originrdquoThe Astrophysical Journalvol 182 pp L85ndashL88 1973

[8] E P Mazets S V Golenetskii V N IlrsquoInskii et al ldquoCatalogof cosmic gamma-ray bursts from the KONUS experimentdatamdashparts I and IIrdquoAstrophysics and Space Science vol 80 no1 pp 3ndash83 1981

[9] C Meegan G Fishman R Wilson et al ldquoThe spatial distribu-tion of gamma-ray bursts observed by BATSErdquo in Proceedingsof the Compton Gamma-Ray Observatory pp 681ndash685 USA

[10] C Kouveliotou C A Meegan G J Fishman et al ldquoIdentifi-cation of two classes of gamma-ray burstsrdquo The AstrophysicalJournal vol 413 no 2 pp 101ndash104 1993

[11] B Paczynski ldquoOn the Galactic origin of gamma-ray burstsrdquoActa Astron vol 348 pp 485ndash494 1990

[12] B Paczynski ldquoCosmological gamma-ray burstsrdquo Acta Astro-nomica vol 410 pp 257ndash267 1991

[13] G J Fishman ldquo120574-ray burstsrdquo Annual Review of Astronomy andAstrophysics vol 33 no 1 pp 415ndash458

[14] M S Briggs W S Paciesas G N Pendleton et al ldquoBATSEobservations of the large-scale isotropy of gamma-ray burstsrdquoThe Astrophysical Journal vol 459 p 40 1996

[15] M R Metzger S G Djorgovski S R Kulkarni et al ldquoSpectralconstraints on the redshift of the optical counterpart to the 120574-ray burst of 8 May 1997rdquoNature vol 387 no 6636 pp 878ndash8801997

[16] L G Balazs AMeszaros and I Horvath ldquoAnisotropy of the skydistribution of gamma-ray burstsrdquo p 339 0 1ndash6 339 November1998

[17] A Meszaros Z Bagoly I Horvath L G Balazs and R VavrekldquoA remarkable angular distribution of the intermediate subclassof gamma-ray burstsrdquoThe Astrophysical Journal vol 539 no 1pp 98ndash101 2000

[18] A Meszaros Z Bagoly and R Vavrek ldquoOn the existence of theintrinsic anisotropies in the angular distributions of gamma-rayburstsrdquoAstronomy amp Astrophysics vol 354 no 1 pp 1ndash6 2000

[19] A Meszaros and J Stocek ldquoAnisotropy in the angular distribu-tion of the long gamma-ray burstsrdquo Astronomy amp Astrophysicsvol 403 no 2 pp 443ndash448 2003

[20] M Magliocchetti G Ghirlanda and A Celotti ldquoEvidence foranisotropy in the distribution of short-lived gamma-ray burstsrdquoMonthly Notices of the Royal Astronomical Society vol 343 no1 pp 255ndash258 2003

[21] A Bernui I S Ferreira and C A Wuensche ldquoOn the large-scale angular distribution of short gamma-ray burstsrdquo TheAstrophysical Journal vol 673 no 2 pp 968ndash971 2008

[22] R Vavrek L G Balazs A Meszaros I Horvath and Z BagolyldquoTesting the randomness in the sky-distribution of gamma-rayburstsrdquo Monthly Notices of the Royal Astronomical Society vol391 no 4 pp 1741ndash1748 2008

[23] M Tarnopolski ldquoTesting the anisotropy in the angular distribu-tion of FermiGBM gamma-ray burstsrdquo Monthly Notices of theRoyal Astronomical Society vol 472 no 4 pp 4819ndash4831 2017

[24] A Meszaros L G Balazs Z Bagoly and P Veres ldquoImpact oncosmology of the celestial anisotropy of the short gamma-rayburstsrdquo Baltic Astronomy vol 18 no 3-4 pp 293ndash296 2009

[25] J Hjorth J Sollerman P Moslashller et al ldquoA very energeticsupernova associated with the 120574-ray burst of 29 March 2003rdquoNature vol 423 no 6942 pp 847ndash850 2003

[26] DMalesani G Tagliaferri G Chincarini et al ldquoSN 2003lw andGRB 031203 a bright supernova for a faint gamma-ray burstrdquoThe Astrophysical Journal vol 609 no 1 pp L5ndashL8 2004

[27] S EWoosley and J S Bloom ldquoThe supernova-gamma-ray burstconnectionrdquoAnnual Review of Astronomy and Astrophysics vol44 pp 507ndash556 2006

[28] M Sparre J Sollerman J P U Fynbo et al ldquoSpectroscopicevidence for SN 2010ma associated with GRB 101219Brdquo TheAstrophysical Journal Letters vol 735 no 1 p L24 2011


[29] S Schulze D Malesani A Cucchiara et al ldquoGRB 120422ASN2012bz bridging the gap between low- and high-luminositygamma-ray burstsrdquo Astronomy amp Astrophysics vol 566 articleA102 2014

[30] D Eichler M Livio T Piran and D N Schramm ldquoNucle-osynthesis neutrino bursts and 120574-rays from coalescing neutronstarsrdquo Nature vol 340 no 6229 pp 126ndash128 1989

[31] R Narayan B Paczynski and T Piran ldquoGamma-ray burstsas the death throes of massive binary starsrdquo The AstrophysicalJournal vol 395 no 2 pp 83ndash86 1992

[32] E Nakar and T Piran ldquoOutliers to the peak energy-isotropicenergy relation in gamma-ray burstsrdquo Monthly Notices of theRoyal Astronomical Society vol 360 no 1 pp L73ndashL76 2005

[33] B Zhang B-B Zhang F J Virgili et al ldquoDiscerning the physicalorigins of cosmological gamma-ray bursts based on multipleobservational criteria the cases of z = 67GRB 080913 z =82GRB 090423 and some shorthard GRBsrdquoTheAstrophysicalJournal vol 703 no 2 pp 1696ndash1724 2009

[34] B P Abbott R Abbott T D Abbott et al ldquoObservation ofgravitational waves from a binary black hole mergerrdquo PhysicalReview Letters vol 116 no 6 061102 16 pages 2016

[35] V Connaughton E Burns A Goldstein et al ldquoFermi GBMobservations of ligo gravitational-wave event GW150914rdquo TheAstrophysical Journal Letters vol 826 no 1 article no L6 2016

[36] P N Bhat C A Meegan A Von Kienlin et al ldquoThe thirdfermi gbm gamma-ray burst catalog the first six yearsrdquo TheAstrophysical Journal Supplement Series vol 223 no 2 articleno 28 2016

[37] V Savchenko C Ferrigno S Mereghetti et al ldquoIntegral upperlimits on gamma-ray emission associated with the gravitationalwave event gw150914rdquo The Astrophysical Journal Letters vol820 no 2 article no L36 2016

[38] P A Evans J A Kennea S D Barthelmy et al ldquoSwift follow-up of the gravitational wave sourceGW150914rdquoMNRASLettersvol 460 no 1 pp L40ndashL44 2016

[39] M Lyutikov ldquoFermi GBM signal contemporaneous withGW150914mdashan unlikely associationrdquo

[40] X Li F-W Zhang Q Yuan et al ldquoImplications of the tentativeassociation between gw150914 and a fermi-gbm transientrdquoTheAstrophysical Journal Letters vol 827 no 1 article no L16 2016

[41] A Loeb ldquoElectromagnetic counterparts to black hole mergersdetected by ligordquoThe Astrophysical Journal Letters vol 819 no2 article no L21 2016

[42] R Perna D Lazzati and B Giacomazzo ldquoShort gamma-raybursts from the merger of two black holesrdquo The AstrophysicalJournal Letters vol 821 no 1 article no L18 2016

[43] B J Morsony J C Workman and D M Ryan ldquoModelingthe afterglow of the possible Fermi-GBM event associated withGW150914rdquo The Astrophysical Journal Letters vol 825 no 2article no L24 2016

[44] E Costa F Frontera J Heise et al ldquoDiscovery of an X-rayafterglow associated with the 120574-ray burst of 28 February 1997rdquoNature vol 387 no 6635 pp 783ndash785 1997

[45] J van Paradijs P J Groot T Galama et al ldquoTransient opticalemission from the error box of the 120574-ray burst of 28 february1997rdquo Nature vol 386 no 6626 pp 686ndash689 1997

[46] I Horvath ldquoA third class of gamma-ray burstsrdquoThe Astrophys-ical Journal vol 508 no 2 pp 757ndash759 1998

[47] I Horvath ldquoA further study of the BATSE Gamma-Ray Burstduration distributionrdquo Astronomy amp Astrophysics vol 392 no3 pp 791ndash793 2002

[48] I Horvath L G Balazs Z Bagoly and P Veres ldquoClassificationof Swiftrsquos gamma-ray burstsrdquo Astronomy amp Astrophysics vol489 no 1 pp L1ndashL4 2008

[49] I Horvath ldquoClassification of BeppoSAXrsquos gamma-ray burstsrdquoAstrophysics and Space Science vol 323 no 1 pp 83ndash86 2009

[50] D Huja A Meszaros and J Rıpa ldquoA comparison of thegamma-ray bursts detected by BATSE and Swiftrdquo Astronomy ampAstrophysics vol 504 no 1 pp 67ndash71 2009

[51] J Rıpa A Meszaros C Wigger D Huja R Hudec and WHajdas ldquoSearch for gamma-ray burst classes with the RHESSIsatelliterdquo Astronomy amp Astrophysics vol 498 no 2 pp 399ndash406 2009

[52] S Mukherjee E D Feigelson G J Babu F Murtagh CFralev and A Raftery ldquoThree types of gamma-ray burstsrdquo TheAstrophysical Journal vol 508 no 1 pp 314ndash327 1998

[53] I Horvath L G Balazs Z Bagoly F Ryde and A Meszaros ldquoAnew definition of the intermediate group of gamma-ray burstsrdquoAstronomy amp Astrophysics vol 447 no 1 pp 23ndash30 2006

[54] I Horavth Z Bagoly L G Balazs A De Ugarte PostigoP Veres and A Meszaros ldquoDetailed classification of swiftrsquosgamma-ray burstsrdquo The Astrophysical Journal vol 713 no 1pp 552ndash557 2010

[55] P Veres Z Bagoly I Horvath AMeszaros and L G Balazs ldquoAdistinct peak-flux distribution of the third class of gamma-raybursts a possible signature of X-ray flashesrdquoThe AstrophysicalJournal vol 725 no 2 pp 1955ndash1964 2010

[56] C Koen and A Bere ldquoOn multiple classes of gamma-raybursts as deduced from autocorrelation functions or bivariatedurationhardness ratio distributionsrdquo Monthly Notices of theRoyal Astronomical Society vol 420 no 1 pp 405ndash415 2012

[57] H Zitouni N Guessoum W J Azzam and R MochkovitchldquoStatistical study of observed and intrinsic durations amongBATSE and SwiftBAT GRBsrdquo Astrophysics and Space Sciencevol 357 no 1 2015

[58] M Tarnopolski ldquoAnalysis of Fermi gamma-ray burst durationdistributionrdquo Astronomy and Astrophysics vol 581 2015

[59] M Tarnopolski ldquoAnalysis of gamma-ray burst duration distri-bution usingmixtures of skewed distributionsrdquoMonthly Noticesof the Royal Astronomical Society vol 458 no 2 pp 2024ndash20312016

[60] M Tarnopolski ldquoAnalysis of the observed and intrinsic dura-tions of gamma-ray bursts with known redshiftrdquo Astrophysicsand Space Science vol 361 no 3 article no 125 2016

[61] M Tarnopolski ldquoAnalysis of the observed and intrinsic dura-tions of SwiftBAT gamma-ray burstsrdquoNew Astronomy vol 46pp 54ndash59 2016

[62] O Bromberg E Nakar T Piran and R Sari ldquoShort versus longand collapsars versus non-collapsars a quantitative classifica-tion of gamma-ray burstsrdquoThe Astrophysical Journal vol 764no 2 article 179 2013

[63] M Tarnopolski ldquoOn the limit between short and long GRBsrdquoAstrophysics and Space Science vol 359 no 1 2015

[64] H Gao Y Lu and S N Zhang ldquoA new class of 120574-ray burstsfrom stellar disruptions by intermediate-mass black holesrdquoTheAstrophysical Journal vol 717 no 1 pp 268ndash276 2010

[65] J P Norris and J T Bonnell ldquoShort gamma-ray bursts withextended emissionrdquo The Astrophysical Journal vol 643 no 1article 266 2006

[66] M Boer B Gendre and G Stratta ldquoAre ultra-long gamma-ray bursts differentrdquo The Astrophysical Journal vol 800 no1 article no 16 2015


[67] F J Virgili C G Mundell V PalrsquoShin et al ldquoGRB 091024A andthe nature of ultra-long gamma-ray burstsrdquo The AstrophysicalJournal vol 778 no 1 article 54 2013

[68] B-B Zhang B Zhang K Murase V Connaughton and MS Briggs ldquoHow long does a burst burstrdquo The AstrophysicalJournal vol 787 no 1 article 66 2014

[69] A J Levan N R Tanvir R L C Starling et al ldquoA newpopulation of ultra-long duration gamma-ray burstsrdquo TheAstrophysical Journal vol 781 no 1 article 13 2014

[70] A J Levan ldquoSwift discoveries of new populations of extremelylong duration high energy transientrdquo Journal of High EnergyAstrophysics vol 7 pp 44ndash55 2015

[71] J Heise J I Zand R M Kippen and P M Woods ldquoX-rayflashes and X-ray rich gamma ray burstsrdquo Gamma-Ray Burstsin the Afterglow Era pp 16ndash21 2001

[72] R M Kippen P M Woods J Heise J I Zand R D Preeceand M S Briggs ldquoBATSE observations of fast X-ray transientsdetected by BeppoSAX-WFCrdquo in Gamma-ray Bursts in theAfterglow Era E Costa F Frontera and J Hjorth Eds p 222001

[73] D Grupe J A Nousek P Veres B-B Zhang and N GehrelsldquoEvidence for new relations between gamma-ray burst promptand x-ray afterglow emission from 9 years of swiftrdquo TheAstrophysical Journal Supplement Series vol 209 no 2 articleno 20 2013

[74] J P U Fynbo D Watson C C Thone et al ldquoNo supernovaeassociated with two long-duration 120574-ray burstsrdquo Nature vol444 no 7122 pp 1047ndash1049 2006

[75] M Della Valle D Malesani J S Bloom et al ldquoHypernovasignatures in the late rebrightening ofGRB050525ArdquoTheAstro-physical Journal Letters vol 642 no 2 pp L103ndashL106 2006

[76] D A Perley N R Tanvir J Hjorth et al ldquoThe swift GRBhost galaxy legacy survey ii rest-frame near-ir luminositydistribution and evidence for a near-solarmetallicity thresholdrdquoThe Astrophysical Journal vol 817 no 1 article no 8 2016

[77] J Greiner P A Mazzali and D A Kann ldquoA very luminousmagnetar-powered supernova associated with an ultra-long120574-ray burstrdquo Nature vol 523 no 7559 pp 189ndash192 2015

[78] R A M J Wijers M J Rees and P Meszaros ldquoShocked byGRB 970228 the afterglow of a cosmological fireballrdquoMonthlyNotices of the Royal Astronomical Society vol 288 no 4 ppL51ndashL56 1997

[79] P Meszaros ldquoTheoretical models of gamma-ray burstsrdquoin Proceedings of the Gamma-Ray BURSTS pp 647ndash656Huntsville Alabama (USA)

[80] P Meszros ldquoGamma-ray burstsrdquo Reports on Progress in Physics[81] R Willingale P T OrsquoBrien J P Osborne et al ldquoTesting the

standard fireball model of gamma-ray bursts using late X-rayafterglows measured by Swiftrdquo The Astrophysical Journal vol662 no 2 I pp 1093ndash1110 2007

[82] A Melandri C G Mundell S Kobayashi et al ldquoThe early-time optical properties of gamma-ray burst afterglowsrdquo TheAstrophysical Journal vol 686 no 2 pp 1209ndash1230 2008

[83] E S Rykoff F Aharonian and CW Akerlof ldquoLooking into thefireball Rotse-III and Swift observations of early gamma-rayburst afterglowsrdquo The Astrophysical Journal vol 702 no 1 p489 2009

[84] S R Oates M J Page P Schady et al ldquoA statistical comparisonof the opticalUV and X-ray afterglows of gamma-ray burstsusing the swift ultraviolet optical and X-ray telescopesrdquoMonthly Notices of the Royal Astronomical Society vol 412 no1 pp 561ndash579 2011

[85] N Gehrels G Chincarini P Giommi et al ldquoThe swift gamma-ray burst missionrdquo The Astrophysical Jornal vol 611 pp1005ndash1020 August 2004

[86] P T OrsquoBrien R Willingale J Osborne et al ldquoThe early X-rayemission from GRBsrdquo The Astrophysical Journal Letters vol647 no 2 pp 1213ndash1237 2006

[87] T Sakamoto J E Hill R Yamazaki et al ldquoEvidence ofexponential decay emission in the gamma-ray burstsrdquo TheAstrophysical Journal vol 669 no 2 pp 1115ndash1129 2007

[88] B-B Zhang E N-W Liang and B Zhang ldquoA comprehensiveanalysis of Swift XRT data I Apparent spectral evolution ofgamma-ray burst X-ray tailsrdquo The Astrophysical Journal vol666 no 2 I pp 1002ndash1011 2007

[89] J A Nousek C Kouveliotou and D Grupe ldquoEvidence fora canonical gamma-ray burst afterglow light curve in theSwift XRT Datardquo The Astrophysical Journal vol 642 no 1 pp389ndash400 2006

[90] A Cucchiara A J Levan D B Fox et al ldquoA photometricredshift of 119911 sim 94 for GRB 090429Brdquo The AstrophysicalJournal vol 736 no 1 article 7 2011

[91] S A Rodney A G Riess D M Scolnic et al ldquoErratum TwoSNe Ia at redshift sim2 Improved classification and redshiftdetermination with medium-band infrared imagingrdquo TheAstronomical Journal vol 151 no 2 2016

[92] H Lin X Li S Wang and Z Chang ldquoAre long gamma-ray bursts standard candlesrdquo Monthly Notices of the RoyalAstronomical Society vol 453 no 1 pp 128ndash132 2015

[93] T Totani ldquoCosmological gamma-ray bursts and evolution ofgalaxiesrdquoTheAstrophysical Journal vol 486 no 2 pp L71ndashL74

[94] C Porciani and P Madau ldquoOn the association of gamma-raybursts with massive stars Implications for number counts andlensing statisticsrdquoThe Astrophysical Journal vol 548 no 2 pp522ndash531 2001

[95] V Bromm and A Loeb ldquoHigh-redshift 120574-ray bursts frompopulation III progenitorsrdquoTheAstrophysical Journal vol 642no 1 I pp 382ndash388 2006

[96] M D Kistler H Yuksel J F Beacom A M Hopkins and J SB Wyithe ldquoThe star formation rate in the reionization era asindicated by gamma-ray burstsrdquoThe Astrophysical Journal vol705 no 2 pp L104ndashL108 2009

[97] R S De Souza N Yoshida and K Ioka ldquoPopulations III1 andIII2 gamma-ray bursts constraints on the event rate for futureradio and X-ray surveysrdquo Astronomy amp Astrophysics vol 533article A32 2011

[98] R Barkana and A Loeb ldquoGamma-ray bursts versus quasarsLy120572 signatures of reionization versus cosmological infallrdquo TheAstrophysical Journal vol 601 no 1 I pp 64ndash77 2004

[99] K Ioka and P Meszaros ldquoRadio afterglows of gamma-raybursts and hypernovae at high redshift and their potential for21 centimeter absorption studiesrdquo The Astrophysical Journal vol 619 no 2 pp 684ndash696 2005

[100] S Inoue K Omukai and B Ciardi ldquoThe radio to infraredemission of very high redshift gamma-ray bursts probing earlystar formation throughmolecular and atomic absorption linesrdquoMonthly Notices of the Royal Astronomical Society vol 380 no4 pp 1715ndash1728 2007

[101] R Salvaterra ldquoHigh redshift gamma-ray burstsrdquo Journal ofHigh Energy Astrophysics vol 7 pp 35ndash43 2015

[102] D E Reichart D Q Lamb E E Fenimore E Ramirez-RuizT L Cline and K Hurley ldquoA possible cepheid-like luminosityestimator for the long gamma-ray burstsrdquo The AstrophysicalJournal vol 552 no 1 pp 57ndash71 2001


[103] G J Fishman C A Meegan R B Wilson et al ldquoThe firstbatse gamma-ray burst catalogrdquo The Astrophysical JournalSupplement Series vol 92 no 1 pp 229ndash283 1994

[104] J P Norris R J Nemiroff and J T Bonnell ldquoAttributes ofpulses in long bright gamma-ray burstsrdquo The AstrophysicalJournal vol 459 p 393 1996

[105] B E Stern and R Svensson ldquoEvidence for ldquochain reactionrdquoin the time profiles of gamma-ray burstsrdquo The AstrophysicalJournal vol 469 no 2 pp L109ndashL113

[106] F Ryde and R Svensson ldquoOn the variety of the spectral andtemporal behavior of long gamma-ray burst pulsesrdquo TheAstrophysical Journal vol 566 no 1 I pp 210ndash228 2002

[107] R Sari T Piran and J P Halpern ldquoJets in gamma-ray burstsrdquoThe Astrophysical Journal Letters vol 519 no 1 pp L17ndashL201999

[108] R Willingale F Genet J Granot and P T OrsquoBrien ldquoThespectral-temporal properties of the prompt pulses and rapiddecay phase of gamma-ray burstsrdquoMonthly Notices of the RoyalAstronomical Society vol 403 no 3 pp 1296ndash1316 2010

[109] J E Rhoads ldquoHow to tell a jet from a balloon a proposed testfor beaming in gamma-ray burstsrdquo The Astrophysical JournalLetters vol 487 no 1 p L1 1997

[110] R S Mallozzi W S Paciesas G N Pendleton et al ldquoThe nu Fnu Peak Energy Distributions of Gamma-Ray Bursts Observedby BATSErdquoThe Astrophysical Journal vol 454 p 597 1995

[111] L Amati F Frontera M Tavani et al ldquoIntrinsic spectra andenergetics of BeppoSAX gamma-ray bursts with known red-shiftsrdquoAstronomyampAstrophysics vol 390 no 1 pp 81ndash89 2002

[112] T T Lee and V Petrosian ldquoDistributions of peak flux andduration for gamma-ray burstsrdquoThe Astrophysical Journal vol470 p 479 1996

[113] J P Norris G F Marani and J T Bonnell ldquoConnectionbetween energy-dependent lags and peak luminosity ingamma-ray burstsrdquo The Astrophysical Journal vol 534 no 1pp 248ndash257 2000

[114] L Li and B Paczynski ldquoImproved correlation between thevariability and peak luminosity of gamma-ray burstsrdquoMonthlyNotices of the Royal Astronomical Society vol 366 no 1 pp219ndash226 2006

[115] D Band J Matteson L Ford et al ldquoBATSE observationsof gamma-ray burst spectra I Spectral diversityrdquo TheAstrophysical Journal Letters vol 413 no 1 pp 281ndash292 1993

[116] D Yonetoku T Murakami T Nakamura R Yamazaki A KInoue and K Ioka ldquoGamma-ray burst formation rate inferredfrom the spectral peak energy-peak luminosity relationrdquo TheAstrophysical Journal vol 609 no 2 pp 935ndash951 2004

[117] M G Kendall and A StuartThe Advanced Theory of Statisticsvol 2 of Inference and Relationship Macmillan New York NYUSA 1973

[118] P R Bevington and D K Robinson Data Reduction and ErrorAnalysis for the Physical Sciences McGraw-Hill Boston MassUSA 2nd edition 1992

[119] C Spearman ldquoThe proof and measurement of associationbetween two thingsrdquo The American Journal of Psychology vol15 no 1 p 72 1904

[120] E Liang and V Kargatis ldquoDependence of the spectral evolutionof 120574-ray bursts on their photon fluencerdquo Nature vol 381 no6577 pp 49ndash51 1996

[121] B E Schaefer M Deng and D L Band ldquoRedshifts andluminosities for 112 gamma-ray burstsrdquo The AstrophysicalJournal vol 563 no 2 pp L123ndashL127 2001

[122] J D Salmonson ldquoOn the kinematic origin of the luminosity-pulse lag relationship in gamma-ray burstsrdquo The AstrophysicalJournal vol 544 no 2 pp L115ndashL117 2000

[123] F Daigne and R Mochkovitch ldquoThe physics of pulses ingamma-ray bursts Emission processes temporal profiles andtime-lagsrdquo Monthly Notices of the Royal Astronomical Societyvol 342 no 2 pp 587ndash592 2003

[124] Z Zhang G Z Xie J G Deng and W Jin ldquoRevisiting thecharacteristics of the spectral lags in short gamma-ray burstsrdquoMonthly Notices of the Royal Astronomical Society vol 373 no2 pp 729ndash732 2006

[125] B E Schaefer ldquoExplaining the gamma-ray burst lagluminosityrelationrdquoThe Astrophysical Journal vol 602 no 1 pp 306ndash3112004

[126] D Kocevski and E Liang ldquoThe connection between spectralevolution and gamma-ray burst lagrdquoThe Astrophysical Journalvol 594 no 1 pp 385ndash389 2003

[127] J Hakkila T W Giblin J P Norris P C Fragile and J TBonnell ldquoCorrelations between lag luminosity and durationin gamma-ray burst pulsesrdquoThe Astrophysical Journal vol 677no 2 pp L81ndashL84 2008

[128] R Tsutsui T Nakamura D Yonetoku T Murakami S Tanabeand Y Kodama ldquoRedshift dependent lag-luminosity relationin 565 baste gamma ray burstsrdquo in Proceedings of the Santa FeConference on Gamma-Ray Bursts 2007 GRB 2007 pp 28ndash31USA November 2007

[129] J Sultana D Kazanas and K Fukumura ldquoLuminositycorrelations for gamma-ray bursts and implications for theirprompt and afterglow emissionmechanismsrdquoTheAstrophysicalJournal vol 758 no 1 p 32 2012

[130] N Gehrels J P Norris S D Barthelmy et al ldquoA new 120574-rayburst classification scheme from GRB 060614rdquo Nature vol444 no 7122 pp 1044ndash1046 2006

[131] M G Dainotti V F Cardone and S Capozziello ldquoA time-luminosity correlation for 120574 -ray bursts in the X-raysrdquoMonthlyNotices of the Royal Astronomical Society vol 391 no 1 ppL79ndashL83 2008

[132] M G Dainotti R Willingale S Capozziello V F Cardoneand M Ostrowski ldquoDiscovery of a tight correlation forgamma-ray burst afterglows with ldquocanonicalrdquo light curvesrdquoTheAstrophysical Journal vol 722 no 2 pp L215ndashL219 2010

[133] M G Dainotti V F Cardone S Capozziello M Ostrowskiand R Willingale ldquoStudy of possible systematics in the 119871119883-119879

119886

correlation of gamma-ray burstsrdquo The Astrophysical Journalvol 730 no 2 article no 135 2011

[134] M G Dainotti V F Cardone E Piedipalumbo and SCapozziello ldquoSlope evolution of GRB correlations andcosmologyrdquo Monthly Notices of the Royal Astronomical Societyvol 436 no 1 pp 82ndash88 2013

[135] M Dainotti V Petrosian R Willingale P OrsquoBrien MOstrowski and S Nagataki ldquoLuminosity-time and luminosity-luminosity correlations for GRB prompt and afterglow plateauemissionsrdquo Monthly Notices of the Royal Astronomical Societyvol 451 no 4 pp 3898ndash3908 2015

[136] T N Ukwatta M Stamatikos K S Dhuga et al ldquoSpectral lagsand the lag-luminosity relation an investigation with swift batgamma-ray burstsrdquo The Astrophysical Journal Letters vol 711no 2 pp 1073ndash1086 2010

[137] T N Ukwatta K S Dhuga M Stamatikos et al ldquoThe lag-luminosity relation in the GRB source frame An investigationwith Swift BAT burstsrdquo Monthly Notices of the RoyalAstronomical Society vol 419 no 1 pp 614ndash623 2012


[138] R Margutti C Guidorzi G Chincarini et al ldquoLag-luminosityrelation in 120574-ray burst X-ray flares a direct link to the promptemissionrdquo Monthly Notices of the Royal Astronomical Societyvol 406 no 4 pp 2149ndash2167 2010

[139] A Panaitescu and P Kumar ldquoProperties of relativistic jets ingamma-ray burst afterglowsrdquoThe Astrophysical Journal Lettersvol 571 no 2 pp 779ndash789 2002

[140] E E Fenimore C D Madras and S Nayakshin ldquoExpandingrelativistic shells and gamma-ray burst temporal structurerdquoTheAstrophysical Journal vol 473 no 2 pp 998ndash1012 1996

[141] K Ioka and T Nakamura ldquoPeak luminosity-spectral lagrelation caused by the viewing angle of the collimated gamma-ray burstsrdquoThe Astrophysical Journal Letters vol 554 no 2 ppL163ndashL167 2001

[142] E-W Liang J L Racusin B Zhang B-B Zhang and D NBurrows ldquoA comprehensive analysis of Swift XRT data III Jetbreak candidates in X-ray and optical afterglow light curvesrdquoThe Astrophysical Journal vol 675 no 1 pp 528ndash552 2008

[143] J Lu Y-C Zou W-H Lei et al ldquoLorentz-factorndashisotropic-luminosityenergy correlations of gamma-ray bursts and theirinterpretationrdquo The Astrophysical Journal vol 751 no 1 articleno 49 2012

[144] T-F Yi G-Z Xie and F-W Zhang ldquoA close correlationbetween the spectral lags and redshifts of gamma-ray burstsrdquoChinese Journal of Astronomy and Astrophysics vol 8 no 1 pp81ndash86 2008

[145] G Stratta D Guetta V DrsquoElia M Perri S Covino and LStella ldquoEvidence for an anticorrelation between the duration ofthe shallow decay phase of GRB X-ray afterglows and redshiftrdquoAstronomy amp Astrophysics vol 494 no 2 p -L12 2009

[146] G Ryan H van Eerten A MacFadyen and B-B ZhangldquoGamma-ray bursts are observed off-axisrdquo The AstrophysicalJournal vol 799 no 1 article 3 2015

[147] Z L Uhm and B Zhang ldquoToward an understanding of GRBprompt emission mechanism I the origin of spectral lagsrdquoTheAstrophysical Journal vol 825 no 2 article no 97 2016

[148] E E Fenimore and E Ramirez-Ruiz ldquoRedshifts For 220 BATSE120574-Ray Bursts Determined by Variability and the CosmologicalConsequencesrdquo

[149] C Guidorzi F Frontera E Montanari et al ldquoThe gamma-rayburst variability-peak luminosity correlation New resultsrdquoMonthly Notices of the Royal Astronomical Society vol 363 no1 pp 315ndash325 2005

[150] D E Reichart and M C Nysewander ldquoGRB Variability-Luminosity Correlation Confirmedrdquo ArXiv Astrophysicse-prints August 2005

[151] C Guidorzi F Frontera E Montanari et al ldquoThe slope ofthe gamma-ray burst variabilitypeak luminosity correlationrdquoMonthly Notices of the Royal Astronomical Society vol 371 no2 pp 843ndash851 2006

[152] C Guidorzi ldquoTesting the gamma-ray burst variabilitypeakluminosity correlation using the pseudo-redshifts of a largesample of BATSE gamma-ray burstsrdquo Monthly Notices of theRoyal Astronomical Society vol 364 no 1 pp 163ndash168 2005

[153] G DrsquoAgostini ldquoFits and especially linear fits with errorson both axesrdquo Extra Variance of The Data Points And OtherComplications Arxiv Physics E-Prints November 2005

[154] D Rizzuto C Guidorzi P Romano et al ldquoTesting thegamma-ray burst variabilitypeak luminosity correlation ona Swift homogeneous samplerdquo Monthly Notices of the RoyalAstronomical Society vol 379 no 2 pp 619ndash628 2007

[155] T Piran ldquoThe physics of gamma-ray burstsrdquo Reviews of ModernPhysics vol 76 no 4 pp 1143ndash1210 2004

[156] J D Salmonson and T J Galama ldquoDiscovery of a tight correla-tion between pulse LAGluminosity and jet-break times A con-nection between gamma-ray bursts and afterglow propertiesrdquoThe Astrophysical Journal vol 569 no 2 I pp 682ndash688 2002

[157] B E Schaefer ldquoThe Hubble Diagram to Redshift gt6 from 69rdquoGamma-Ray Bursts vol 6600 Article ID 511742 pp 16ndash46May 2007

[158] B Schaefer ldquoFour luminosity indicators for gamma-ray burstsrdquoin Proceedings of the COSPAR Scientifc Assembly volume 34 ofCOSPAR Meeting p 1141 Houston Texas USA Oct 2002

[159] L Xiao and B E Schaefer ldquoEstimating redshifts for longgamma-ray burstsrdquo The Astrophysical Journal vol 707 no 1pp 387ndash403 2009

[160] D L Freedman and E Waxman ldquoOn the energy of gamma-rayburstsrdquo The Astrophysical Journal vol 547 no 2 pp 922ndash9282001

[161] E Waxman ldquo120574-Ray burst afterglow confirming thecosmological fireball modelrdquo The Astrophysical Journalvol 489 no 1 pp L33ndashL36

[162] R AM JWijers and T J Galama ldquoPhysical parameters of GRB970508 andGRB 971214 from their afterglow synchrotron emis-sionrdquoTheAstrophysical Journal vol 523 no 1 pp 177ndash186 1999

[163] J Granot T Piran and R Sari ldquoImages and spectra from theinterior of a relativistic fireballrdquo The Astrophysical Journal vol513 no 2 pp 679ndash689 1999

[164] E-W Liang S-X Yi J Zhang H-J Lu and B-B Zhang ldquoCon-straining gamma-ray burst initial lorentz factor with the after-glow onset feature and discovery of a tight Γ0-E119892119886119898119898119886119894119904119900 Corre-lationrdquoTheAstrophysical Journal vol 725 pp 2209ndash2224 2010

[165] G Ghirlanda G Ghisellini L Nava and D Burlon ldquoSpectralevolution of FermiGBM short gamma-ray burstsrdquo MonthlyNotices of the Royal Astronomical Society vol 410 no 1 ppL47ndashL51 2011

[166] R Sari and T Piran ldquoGRB 990123 the optical flash and thefireball modelrdquo The Astrophysical Journal vol 517 no 2 ppL109ndashL112 1999

[167] Y Lithwick and R Sari ldquoLower limits on Lorentz factors ingamma-ray burstsrdquo The Astrophysical Journal Letters vol 555no 1 pp 540ndash545 2001

[168] Y-C Zou and T Piran ldquoLorentz factor constraint from the veryearly external shock of the gamma-ray burst ejectardquo MonthlyNotices of the Royal Astronomical Society vol 402 no 3 pp1854ndash1862 2010

[169] B Zhang and A Persquoer ldquoEvidence of an initially magneticallydominated outflow in GRB 080916CrdquoTheAstrophysical JournalLetters vol 700 no 2 pp L65ndashL68 2009

[170] Y Fan ldquoThe spectrum of 120574-ray burst a cluerdquoMonthly Notices ofthe Royal Astronomical Society vol 403 no 1 pp 483ndash490 2010

[171] B Zhang andH Yan ldquoThe internal-collision-inducedmagneticreconnection and turbulence (ICMART) model of gamma-rayburstsrdquoThe Astrophysical Journal vol 726 no 2 p 90 2011

[172] W H Lei D X Wang L Zhang Z M Gan Y C Zou andY Xie ldquoMagnetically torqued neutrino-dominated accretionflows for gamma-ray burstsrdquo The Astrophysical Journal vol700 no 2 pp 1970ndash1976 2009

[173] N M Lloyd V Petrosian and R S Mallozzi ldquoCosmologicalversus Intrinsic The Correlation between Intensity and thePeak of the ]F Iiexclrdquo The Astrophysical Journal vol 534 no 1 pp227ndash238 2000


[174] M G Kendall ldquoA New Measure of Rank CorrelationrdquoBiometrika vol 30 no 1-2 p 81 1938

[175] R S Mallozzi G N Pendleton W S Paciesas R D Preeceand M S Briggs ldquoGamma-ray burst spectra and the hardness-intensity correlationrdquo in Proceedings of the GAMMA-RAYBURSTS vol 428 pp 273ndash277 Huntsville Alabama (USA)

[176] A Goldstein R D Preece and M S Briggs ldquoA newdiscriminator for gamma-ray burst classification The 119864peak-fluence energy ratiordquo The Astrophysical Journal vol 721 no 2pp 1329ndash1332 2010

[177] R-J Lu J-J Wei E-W Liang et al ldquoA comprehensive analysisof fermi gamma-ray burst data II 119864119901 evolution patterns andimplications for the observed spectrum-luminosity relationsrdquoThe Astrophysical Journal vol 756 no 2 article no 112 2012

[178] S V Golenetskii E P Mazets R L Aptekar and V N IlyinskiildquoCorrelation between luminosity and temperature in 120574-rayburst sourcesrdquo Nature vol 306 no 5942 pp 451ndash453 1983

[179] L Borgonovo and F Ryde ldquoOn the hardness-intensitycorrelation in gamma-ray burst pulsesrdquo The AstrophysicalJournal vol 548 no 2 pp 770ndash786 2001

[180] G Ghirlanda L Nava and G Ghisellini ldquoSpectral-luminosityrelation within individual Fermi gamma rays burstsrdquoAstronomy amp Astrophysics vol 511 no 1 article A43 2010

[181] S Guiriec M S Briggs V Connaugthon et al ldquoTime-resolvedspectroscopy of the three brightest and hardest short gamma-ray bursts observed with the fermi gamma-ray burst monitorrdquoThe Astrophysical Journal vol 725 no 1 pp 225ndash241 2010

[182] N M Lloyd and V Petrosian ldquoDistribution of spectral charac-teristics and the cosmological evolution of gamma-ray burstsrdquoThe Astrophysical Journal vol 511 no 2 pp 550ndash561 1999

[183] N M Lloyd V Petrosian and R D Preece ldquoSynchrotronemission as the source of GRB spectra Part II Observationsrdquoin Proceedings of the The fifth huntsville gamma-ray burstsymposium pp 155ndash159 Huntsville Alabama (USA)

[184] J S Bloom D A Frail and R Sari ldquoThe prompt energy releaseof gamma-ray bursts using a cosmological k-correctionrdquo TheAstronomical Journal vol 121 no 6 pp 2879ndash2888 2001

[185] L Amati F Frontera JM Castro Ceron et al ldquoThe Prompt andAfterglow Emission of GRB 001109 Measured by BeppoSAXrdquoin A Workshop Celebrating the First Year of the HETE MissionAmerican Institute of Physics Conference Series J J M and RK Ricker Eds vol 662 pp 387ndash389 April 2003

[186] D Q Lamb T Q Donaghy and C Graziani ldquoA unified jetmodel of X-ray flashes and 120574-ray burstsrdquo New AstronomyReviews vol 48 no 5-6 pp 459ndash464 2004

[187] T Sakamoto D Q Lamb C Graziani et al ldquoHigh energytransient explorer 2 observations of the extremely soft X-rayflash XRF 020903rdquo The Astrophysical Journal vol 602 no 2 Ipp 875ndash885 2004

[188] G Ghirlanda G Ghisellini and D Lazzati ldquoThe collimation-corrected gamma-ray burst energies correlate with the peakenergy of their V119865V spectrumrdquo The Astrophysical Journal vol616 no 1 pp 331ndash338 2004

[189] G Ghirlanda G Ghisellini and C Firmani ldquoProbing theexistence of the 119864peak-119864iso correlation in long gamma rayburstsrdquo Monthly Notices of the Royal Astronomical Society vol361 no 1 pp L10ndashL14 2005

[190] L Amati ldquoThe Epi-Eiso correlation in gamma-ray burstsupdated observational status re-analysis and mainimplicationsrdquo Monthly Notices of the Royal AstronomicalSociety vol 372 no 1 pp 233ndash245 2006

[191] G Ghirlanda L Nava G Ghisellini C Firmani and J ICabrera ldquoThe 119864peak-119864iso plane of long gamma-ray bursts andselection effectsrdquo Monthly Notices of the Royal AstronomicalSociety vol 387 no 1 pp 319ndash330 2008

[192] L Amati F Frontera and C Guidorzi ldquoExtremely energeticFermi gamma-ray bursts obey spectral energy correlationsrdquoAstronomy amp Astrophysics vol 508 no 1 pp 173ndash180 2009

[193] L Amati ldquoCosmology with the 119864p119894-119864iso correlation of gamma-ray burstsrdquo International Journal of Modern Physics ConferenceSeries vol 12 pp 19ndash27 2012

[194] L Amati C Guidorzi F Frontera et al ldquoMeasuring thecosmological parameters with the E119901119894-E119894119904119900 correlation ofgamma-ray burstsrdquo Monthly Notices of the Royal AstronomicalSociety vol 391 no 2 pp 577ndash584 2008

[195] Y-P Qin and Z-F Chen ldquoStatistical classification of gamma-ray bursts based on the amati relationrdquo Monthly Notices of theRoyal Astronomical Society vol 430 no 1 pp 163ndash173 2013

[196] V Heussaff J-L Atteia and Y Zolnierowski ldquoThe Epeak-Eisorelation revisited with fermi GRBs resolving a long-standingdebaterdquo Astronomy amp Astrophysics vol 557 article 100 2013

[197] L Amati and M Della Valle ldquoMeasuring cosmologicalparameters with gamma ray burstsrdquo International Journal ofModern Physics D vol 22 no 14 Article ID 1330028 2013

[198] R Basak and A R Rao ldquoErratum Correlation between theisotropic energy and the peak energy at zero fluence for theindividual pulses of gamma-ray bursts Toward a universalphysical correlation for the prompt emissionrdquoTheAstrophysicalJournal vol 754 no 1 article no 79 2012

[199] R Basak and A R Rao ldquoPulse-wise Amati correlation in Fermigamma-ray burstsrdquo Monthly Notices of the Royal AstronomicalSociety vol 436 no 4 pp 3082ndash3088 2013

[200] F Frontera L Amati C Guidorzi R Landi and J InrsquoT ZandldquoErratum Broadband time-resolved 119864p119894-119871 iso correlation in120574-ray bursts (Astrophysical Journal (2012) 754 (138))rdquo TheAstrophysical Journal vol 757 no 1 article no 107 2012

[201] D A Frail S R Kulkarni R Sari et al ldquoThe radio afterglowfrom grb 980519 a test of the jet and circumstellar modelsrdquoTheAstrophysical Journal vol 534 no 2 pp 559ndash564 2000

[202] S A Yost D A Frail F A Harrison et al ldquoThe broadbandafterglow of GRB 980329rdquo The Astrophysical Journal vol 577no 1 I pp 155ndash163 2002

[203] B E Schaefer ldquoExplaining the gamma-ray burst 119864peakdistributionrdquo The Astrophysical Journal vol 583 no 2 ppL71ndashL74 2003

[204] E Liang and B Zhang ldquoModel-independent multivariablegamma-ray burst luminosity indicator and its possiblecosmological implicationsrdquo The Astrophysical Journal vol 633no 2 pp 611ndash623 2005

[205] L Nava G Ghisellini G Ghirlanda F Tavecchio and CFirmani ldquoOn the interpretation of spectral-energy correlationsin long gamma-ray burstsrdquo Astronomy amp Astrophysics vol450 no 2 pp 471ndash481 2006

[206] G Ghirlanda L Nava G Ghisellini and C FirmanildquoConfirming the 120574-ray burst spectral-energy correlationsin the era of multiple time breaksrdquo Astronomy amp Astrophysics vol 466 no 1 pp 127ndash136 2007

[207] N M Lloyd-Ronning and V Petrosian ldquoInterpreting thebehavior of time-resolved gamma-ray burst spectrardquo TheAstrophysical Journal Letters vol 565 no 1 pp 182ndash194 2002

[208] B Zhang and P Meszaros ldquoAn analysis of gamma-ray burstspectral break modelsrdquo The Astrophysical Journal vol 581 no2 pp 1236ndash1247 2002


[209] G Ghirlanda G Ghisellini R Salvaterra et al ldquoThe fasterthe narrower Characteristic bulk velocities and jet openingangles of gamma-ray burstsrdquo Monthly Notices of the RoyalAstronomical Society vol 428 no 2 pp 1410ndash1423 2013

[210] M J Rees and P Meszaros ldquoDissipative photosphere models ofgamma-ray bursts and X-ray flashesrdquoTheAstrophysical Journalvol 628 no 2 p 847 2005

[211] E Ramirez-Ruiz ldquoPhotospheric signatures imprinted on the120574-ray burst spectrardquoMonthly Notices of the Royal AstronomicalSociety vol 363 no 1 pp L61ndashL65 2005

[212] F Ryde ldquoIs thermal emission in gamma-ray bursts ubiquitousrdquoThe Astrophysical Journal Letters vol 625 no 2 pp L95ndashL982005

[213] A M Beloborodov ldquoCollisional mechanism for gamma-rayburst emissionrdquo Monthly Notices of the Royal AstronomicalSociety vol 407 no 2 pp 1033ndash1047 2010

[214] S Guiriec V Connaughton M S Briggs et al ldquoDetectionof a thermal spectral component in the prompt emission ofGRB 100724BrdquoThe Astrophysical Journal Letters vol 727 no 2article L33 2011

[215] R Hascoet F Daigne and R Mochkovitch ldquoPrompt thermalemission in gamma-ray burstsrdquo Astronomy amp Astrophysics vol551 article A124 2013

[216] S Guiriec F Daigne R Hascoet et al ldquoEvidence for a photo-spheric component in the prompt emission of the short GRB120323a and its effects on the GRB hardness-luminosity rela-tionrdquoThe Astrophysical Journal vol 770 no 1 article 32 2013

[217] I Vurm and A M Beloborodov ldquoRadiative transfer models forgamma-ray burstsrdquo The Astrophysical Journal vol 831 no 2article no 175 2016

[218] S Guiriec C Kouveliotou F Daigne et al ldquoToward a betterunderstanding of the grb phenomenon a new model for grbprompt emission and its effects on the new L119894

119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905

relationrdquo The Astrophysical Journal vol 807 no 2 article no148 2015

[219] S Guiriec R Mochkovitch T Piran et al ldquoGRB 131014AA laboratory for studying the thermal-like and non-thermalemissions in gamma-ray bursts and the new L119899119879ℎ119894-119864119899119879ℎ119903119890119904119905119901119890119886119896119894relationrdquoThe Astrophysical Journal vol 814 no 1 article no 102015

[220] F Frontera L Amati E Costa et al ldquoPrompt and delayedemission properties of 120574-ray bursts observed with BeppoSAXrdquoThe Astrophysical Journal vol 127 no 1 pp 59ndash78 2000

[221] R D Preece M S Briggs R S Mallozzi G N Pendleton WS Paciesas and D L Band ldquoThe BATSE gamma-ray Burstspectral catalog I High time resolution spectroscopy of brightBursts using high energy resolution datardquo The AstrophysicalJournal vol 126 no 1 pp 19ndash36 2000

[222] G Ghirlanda A Celotti and G Ghisellini ldquoExtremely hardGRB spectra prune down the forest of emission modelsrdquoAstronomy amp Astrophysics vol 406 no 3 pp 879ndash892 2003

[223] A Panaitescu ldquoAn external-shock origin of the relation forgamma-ray burstsrdquo Monthly Notices of the Royal AstronomicalSociety vol 393 no 3 pp 1010ndash1015 March 2009

[224] R Mochkovitch and L Nava ldquoThe 119864p-119864iso relation and theinternal shock model Aprdquo Astronomy and Astrophysics vol557 pp 10ndash1051 May 2015

[225] A C Collazzi B E Schaefer and J A Moree ldquoThe total errorsin measuring 119864peak for 120574-ray burstsrdquo The Astrophysical Journalvol 729 no 2 article 89 2011

[226] B E Schaefer ldquoGamma-ray burst Hubble diagram to z = 45rdquoThe Astrophysical Journal Letters vol 583 no 2 pp L67ndashL702003

[227] L Nava R Salvaterra G Ghirlanda et al ldquoA complete sampleof bright Swift long gamma-ray bursts testing the spectral-energy correlationsrdquoMonthly Notices of the Royal AstronomicalSociety vol 421 no 2 pp 1256ndash1264 2012

[228] G Ghirlanda G Ghisellini and A Celotti ldquoThe spectra ofshort 120574-ray burstsrdquo Astronomy amp Astrophysics vol 422 no 3pp L55ndashL58 2004

[229] D Yonetoku T Murakami R Tsutsui T Nakamura YMorihara and K Takahashi ldquoPossible origins of dispersion ofthe peak energy-brightness correlations of gamma-ray burstsrdquoPublications of the Astronomical Society of Japan vol 62 no 6pp 1495ndash1507 2010

[230] R Lu and E Liang ldquoLuminosity-peak energy relation in thedecay phases of gamma-ray burst pulsesrdquo Science China PhysicsMechanics amp Astronomy vol 53 no 1 pp 163ndash170 2010

[231] A A Abdo M Ackermann M Ajello et al Fermi Observationsof GRB 090902B A Distinct Spectral Component in the Promptand Delayed Emission 7060 L138L144 0 L138ndashL144 706November 2009

[232] M Ackermann K Asano W B Atwood et al A Short-Hard Gamma-ray Burst with an Additional Hard Power-lawComponent from 10 keV TO GeV Energies vol 7160 01178ndash1190 716 June 2010

[233] M Ackermann M Ajello K Asano et al Detection of aSpectral Break in the Extra Hard Component of GRB 090926A0 114 729 March 2011

[234] M AckermannM Ajello A Allafort et alTheFirst Fermi-LATCatalog of Sources above 10 GeV 0 34 209 December

[235] H-N Lin X Li and Z Chang ldquoEffect of gamma-ray burst(GRB) spectra on the empirical luminosity correlations andthe GRB Hubble diagramrdquo Monthly Notices of the RoyalAstronomical Society vol 459 no 3 pp 2501ndash2512 2016

[236] S Guiriec M M Gonzalez J R Sacahui C KouveliotouN Gehrels and J McEnery ldquoCGROBATSE Data supportthe new paradigm for GRB prompt emission and the newL119894119899119905ℎ-E119901119890119886119896119894

119899119905ℎ119903119890119904119905 relationrdquo The Astrophysical Journal vol 819no 1 article no 79 2016

[237] R Tsutsui D Yonetoku T Nakamura K Takahashi and YMorihara ldquoPossible existence of the 119864119901-119871119901 and 119864119901-119864119894119904119900 corre-lations for short gamma-ray bursts with a factor 5-100 dimmerthan those for long gamma-ray burstsrdquo Monthly Notices of theRoyal Astronomical Society vol 431 no 2 pp 1398ndash1404 2013

[238] D Yonetoku T Nakamura T Sawano K Takahashi and AToyanago ldquoShort gamma-ray burst formation rate frombatse data using 119864119901-119871119901 correlation and the minimumgravitational-wave event rate of a coalescing compact binaryrdquoThe Astrophysical Journal vol 789 no 1 article no 65 2014

[239] N M Lloyd-Ronning and E Ramirez-Ruiz ldquoOn the spectralenergy dependence of gamma-ray burst variabilityrdquo TheAstrophysical Journal vol 576 no 1 I pp 101ndash106 2002

[240] F Wang Z Dai and E Liang ldquoGamma-ray burst cosmologyrdquoNew Astronomy Reviews vol 67 pp 1ndash17 2015

[241] EW Liang ZGDai andX FWu ldquoThe luminosity-119864119901 relationwithin gamma-ray bursts and the implications for fireball mod-elsrdquoTheAstrophysical Journal vol 606 no 1 pp L29ndashL32 2004

[242] S Mendoza J C Hidalgo D Olvera and J I Cabrera ldquoInternalshocks in relativistic jets with time-dependent sourcesrdquoMonthly Notices of the Royal Astronomical Society vol 395 no3 pp 1403ndash1408 2009


[243] H Ito S Nagataki M Ono et al ldquoPhotospheric emission fromstratified jetsrdquo The Astrophysical Journal vol 777 no 1 article62 2013

[244] F Frontera L Amati R Farinelli et al ldquoPossible physicalexplanation of the intrinsic 119864p119894-ldquointensityrdquo correlationcommonly used to ldquostandardizerdquo GRBsrdquo International Journalof Modern Physics D vol 25 no 5 Article ID 1630014 2016

[245] L Titarchuk R Farinelli F Frontera and L Amati ldquoAnupscattering spectral formation model for the prompt emissionof gamma-ray burstsrdquo The Astrophysical Journal vol 752 no2 article no 116 2012

[246] G Ghirlanda G Ghisellini C Firmani A Celotti and ZBosnjak ldquoThe peak luminosity-peak energy correlation ingamma-ray burstsrdquo Monthly Notices of the Royal AstronomicalSociety vol 360 no 1 pp L45ndashL49 2005

[247] R Tsutsui T Nakamura D Yonetoku T Murakami YKodama and K Takahashi ldquoCosmological constraints fromcalibrated Yonetoku and Amati relation suggest fundamentalplane of gamma-ray burstsrdquo Journal of Cosmology andAstroparticle Physics vol 2009 no 8 article no 015 2009

[248] R Tsutsui T Nakamura D Yonetoku T Murakami and KTakahashi ldquoIntrisic Dispersion of Correlations among Ep Lprdquoand Eiso of Gamma Ray Bursts depends on the quality of DataSet ArXiv e-prints December 2010

[249] S Qi and T Lu ldquoA new luminosity relation for gamma-raybursts and its implicationrdquo The Astrophysical Journal vol 717no 2 pp 1274ndash1278 2010

[250] P T OrsquoBrien and R Willingale ldquoUsing Swift observations ofprompt and afterglow emission to classify GRBsrdquo PhilosophicalTransactions of the Royal Society A Mathematical Physical ampEngineering Sciences vol 365 no 1854 pp 1179ndash1188 2007

[251] M G Dainotti M Ostrowski and R Willingale ldquoTowardsa standard gamma-ray burst Tight correlations between theprompt and the afterglow plateau phase emissionrdquo MonthlyNotices of the Royal Astronomical Society vol 418 no 4 pp2202ndash2206 2011

[252] A Lee E D Bloom and V Petrosian ldquoProperties of Gamma-Ray Burst Time Profiles Using Pulse Decomposition AnalysisrdquoThe Astrophysical Jornal vol 131 November 2000

[253] F Quilligan B McBreen L Hanlon S McBreen K J Hurleyand D Watson ldquoTemporal properties of gamma ray burstsas signatures of jets from the central enginerdquo Astronomy ampAstrophysics vol 385 no 2 pp 377ndash398 2002

[254] O M Littlejohns N R Tanvir R Willingale P A Evans P TOrsquoBrien and A J Levan ldquoAre gamma-ray bursts the same athigh redshift and low redshiftrdquo Monthly Notices of the RoyalAstronomical Society vol 436 no 4 Article ID stt1841 pp3640ndash3655 2013

[255] Z Bosnjak and F Daigne ldquoSpectral evolution in gamma-raybursts Predictions of the internal shockmodel and comparisonto observationsrdquo Astronomy amp Astrophysics vol 568 articleno A45 2014

[256] P A Evans R Willingale J P Osborne et al ldquoGRB 130925Aan ultralong gamma ray burst with a dust-echo afterglow andimplications for the origin of the ultralong GRBsrdquo MonthlyNotices of the Royal Astronomical Society vol 444 no 1 pp250ndash267 2014

[257] J Hakkila and R D Preece ldquoGamma-ray burst pulse shapesEvidence for embedded shock signaturesrdquo The AstrophysicalJournal vol 783 no 2 article no 88 2014

[258] T Laskar E Berger N Tanvir et al ldquoGRB 120521C at z sim 6 andthe properties of high-redshift 120574-ray burstsrdquo The AstrophysicalJournal vol 781 no 1 article no 1 2014

[259] O M Littlejohns and N R Butler ldquoInvestigating signaturesof cosmological time dilation in duration measures of promptgamma-ray burst light curvesrdquo Monthly Notices of the RoyalAstronomical Society vol 444 no 4 pp 3948ndash3960 2014

[260] A Roychoudhury S K Sarkar and A Bhadra ldquoSpectral lag fea-tures of GRB 060814 from swift bat and Suzaku observationsrdquoThe Astrophysical Journal vol 782 no 2 article no 105 2014

[261] C Ceccobello and P Kumar ldquoInverse-Compton drag on ahighly magnetized GRB jet in stellar enveloperdquoMonthly Noticesof the Royal Astronomical Society vol 449 no 3 pp 2566ndash25752015

[262] D Kazanas J L Racusin J Sultana and A Mastichiadis ldquoTheStatistics of the Prompt-to-Afterglow GRB Flux Ratios and theSupercritical Pile GRB Modelrdquo

[263] T Laskar E Berger R Margutti et al ldquoEnergy injection ingamma-ray burst afterglowsrdquo The Astrophysical Journal vol814 no 1 article no 1 2015

[264] Z Y Peng Y Yin T F Yi Y Y Bao and H Wu ldquoAcomprehensive comparative study of temporal propertiesbetween X-ray flares and GRB pulsesrdquo Astrophysics and SpaceScience vol 355 no 1 pp 95ndash103 2015

[265] F Genet and J Granot ldquoRealistic analytic model for the promptand high-latitude emission in GRBsrdquo Monthly Notices of theRoyal Astronomical Society vol 399 no 3 pp 1328ndash1346 2009

[266] F Ryde and V Petrosian ldquoGamma-ray burst spectra and lightcurves as signatures of a relativistically expanding plasmardquoTheAstrophysical Journal vol 578 no 1 I pp 290ndash303 2002

[267] C D Dermer ldquoRapid X-ray declines and plateaus in Swift GRBlight curves explained by a highly radiative blast waverdquo TheAstrophysical Journal vol 664 no 1 I pp 384ndash396 2007

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components


Shock and Vibration


High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in



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OpticsInternational Journal of



AstronomyAdvances in

Antennas andPropagation

International Journal of




Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics


Volume 2018


Mathematical PhysicsAdvances in


ChemistryAdvances in


Journal of

Chemistry


Advances inPhysical Chemistry


RotatingMachinery



Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom


to a GRB Even if it turns out that it is only a chancecoincidence [39] it has already triggered scenarios explaininghow a BHndashBH merger can become a GRB for example thenascent BH could generate a GRB via accretion of a mass≃ 10minus5 M⊙ [40] indicating its location in a dense medium(see also [41]) or two high-mass low-metallicity stars couldundergo an SN explosion and the matter ejected from thelast exploding star can formmdashafter some timemdashan accretiondisk producing an SGRB [42] Also the possible detection ofan afterglow that can be visible many months after the event[43] could shed light on the nature of the GW and SGRBassociation

From a phenomenological point of view a GRB iscomposed of the prompt emission which consists of high-energy photons such as 120574-rays and hard X-rays and theafterglow emission that is a long lasting multiwavelengthemission (X-ray optical and sometimes also radio) whichfollows the prompt phase The first afterglow observation(for GRB970228) was due to the BeppoSAX satellite [4445] Another class besides SGRBs and LGRBs that isintermediate in duration was proposed to be present inunivariate duration distributions [46ndash51] as well as in higherdimensional parameter spaces [51ndash56] On the other handthis elusive intermediate class might be a statistical featurethat can be explained by modeling the duration distributionwith skewed distributions instead of the commonly appliedstandard Gaussians [57ndash61] Additionally GRB classificationwas shown to be detector-dependent [1 62 63] Moreovera subclass classification of LGRBs was proposed [64] andNorris and Bonnell [65] discovered the existence of anintermediate class or SGRBs with extended emission thatshow mixed properties between SGRBs and LGRBs GRBswith very long durations (ultralong GRBs with 11987990 gt 1000 s)are statistically different than regular (ie with 11987990 lt 500 s)LGRBs [66] and hence might form a different class (see also[67ndash70]) Another relevant classification appears related tothe spectral features distinguishing normal GRBs from X-rayflashes (XRFs) The XRFs [71 72] are extragalactic transientX-ray sources with spatial distribution and spectral andtemporal characteristics similar to LGRBs The remarkableproperty that distinguishes XRFs from GRBs is that their]119865] prompt emission spectrum peaks at energies which areobserved to be typically an order of magnitude lower thanthe observed peak energies of GRBs XRFs are empiricallydefined by a greater fluence (time-integrated flux) in the X-ray band (2ndash30 keV) than in the 120574-ray band (30ndash400 keV)This classification is also relevant for the investigation of GRBcorrelations since some of them become stronger or weakerby introducing different GRB categories Grupe et al [73]using 754 Swift GRBs performed an exhaustive analysis ofseveral correlations as well as the GRB redshift distributiondiscovering that the bright bursts are more common in thehigh-119911 (ie 119911 ≳ 3) than in the local universe

This classification has further enhanced the knowledgeof the progenitor system from which GRBs originate Itwas soon after their discovery that LGRBs were thoughtto originate from distant star-forming galaxies Since thenLGRBs have been firmly associated with powerful core-collapse SNe and the association seems solid Nevertheless

there have been puzzling cases of LGRBs that were notassociated with bright SNe [74 75] This implies that it ispossible to observe GRBs without an associated bright SNeor there are other progenitors for LGRBs than core-collapseofmassive stars Another relevant uncertainty concerning theprogenitor systems for LGRBs is the role of metallicity 119885 Inthe collapsar model [27] LGRBs are only formed by massivestars with 119885119885⊙ below ≃ 01ndash03 However several GRBshave been located in very metal-rich systems [76] and it is animportant goal to understandwhether there are other ways toform LGRBs than through the collapsar scenario [77] Oneof the models used to explain the GRB phenomenon is theldquofireballrdquo model [78ndash80] in which a compact central engine(either the collapsed core of a massive star or the mergerproduct of an NSndashNS binary) launches a highly relativisticjetted electron-positron-baryon plasma Interactions of blobswithin the jet are believed to produce the prompt emissionInstead the interaction of the jet with the ambient materialcauses the afterglow phase However problems in explainingthe light curves within this model have been shown byWillingale et al [81] Specifically for ≃50 of GRBs theobserved afterglows are in agreement with the model but forthe rest the temporal and spectral indices do not conformand suggest a continued late energy injection Melandri et al[82] performed amultiwavelength analysis and found that theforward shock (FS) model does not explain almost 50 ofthe examined GRBs even after taking into account energyinjection Rykoff et al [83] showed that the fireball modeldoes not model correctly early afterglows Reference [84]analysed the prompt and afterglow light curves and pointedout that some GRBs required energy injection to explain theoutflows The crisis of the standard fireball models appearedwhen Swift [85] observations revealed a more complexbehaviour of the light curves than observed in the past [86ndash88] and pointed out that GRBs often follow ldquocanonicalrdquo lightcurves [89] Therefore the discovery of correlations amongrelevant physical parameters in the prompt phase is veryimportant in this context in order to use them as possiblemodel discriminators In fact many theoretical models havebeen presented in the literature in order to explain the widevariety of observations but each model has some advantagesas well as drawbacks and the use of the phenomenologicalcorrelations can boost the understanding of the mechanismresponsible for the prompt emission Moreover given themuch larger (compared to SNe) redshift range over whichGRBs can be observed it is tempting to include them ascosmological probes extending the redshift range by almostan order ofmagnitude further than the available SNe Ia GRBsare observed up to redshift 119911 = 94 [90] much more distantthan SNe Ia observed up to 119911 = 226 [91] and thereforethey can help to understand the nature of dark energy anddetermine the evolution of its equation of state at veryhigh 119911 However contrary to SNe Ia which originate fromwhite dwarves reaching the Chandrasekhar limit and alwaysreleasing the same amount of energy GRBs cannot yet beconsidered standard candles with their (isotropic-equivalent)energies spanning 8 orders of magnitude (see also [92] andreferences therein) Therefore finding universal relationsamong observable properties can help to standardize their


TfTＬＣＭ

T0

TＤ

T

TＪＥ



















119871peak = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119865peak (1)


119863119871 (119911 Ω119872 ΩΛ) = 119888 (1 + 119911)1198670 int1199110

1198891199111015840radicΩ119872 (1 + 1199111015840)3 + ΩΛ

(2)


119871 iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119865tot (3)







119873119864 (119864) = 119860norm times

( 119864100 keV)120572


120573




















990123

971214

990510

970828

980703

970508

100

101

102

103

Isot

ropi

c Lum

inos

ity (1051

ergsＭminus1)


(a)

0001

001

01

1

10

100

1000

10000

Peak

Lum

inos

ity (1052

ergＭminus1)

1 10 10001

00758 (1 + z)253 ＦＡ0282

1000

(b)























119863 = 1Γ (1 minus 1205730 cos 120579) (1 + 119911) (11)


119905 = 120591119863 (12)


119871peak prop 119863120572 (13)



log119871peak = 5649 + 335 log119881 (14)



ＦＩＡL

(ergＭminus1)

50

51

52

53

54


ＦＩＡ Vf=045

(a)GFM05 PL

05

1

15

2

25

3

35

ＦＩＡ(L)

ＦＩＡ (V)


(b)

G05 PL

ＦＩＡ (V)


minus05

0

05

1

15

2

25

3

35

ＦＩＡ(L)

(c)




























119871 iso prop 120591minus1198732RT (22)



50

51

52

53

54ＦＩＡ(L)


ＦＩＡ[２４(1 + z)]

(a)


ＦＩＡ[２４(1 + z)]

50

51

52

53

54

ＦＩＡ(L)

(b)








2 + (043119887120590RT120591RT )2

+ 1205902RTsys(24)






Γ (119905) = 106 (1 + 1199112 )38 (119864prompt1198990 )18 119905minus38 (26)


100

1000

Γ 0

090510 080319B090902B

990123

080916C

01 1 10 100 1000001


(a)

101

102

103

Γ 0


LＣＭＩ52

(b)













p⟩

(keV

)

00 05 10 15 20 25minus05


400

350

300

250

200

150

100






120573119878 (30)










FＮＩＮ


EP

(100

keV

)

1

(a)

Ep

fp

11 10

(b)






119864iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119878tot (1 + 119911)minus2 (33)









1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)


(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)


SwiftBATFermiGBM

(d)







980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






TfTＬＣＭ

T0

TＤ

T

TＪＥ



















119871peak = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119865peak (1)


119863119871 (119911 Ω119872 ΩΛ) = 119888 (1 + 119911)1198670 int1199110

1198891199111015840radicΩ119872 (1 + 1199111015840)3 + ΩΛ

(2)


119871 iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119865tot (3)







119873119864 (119864) = 119860norm times

( 119864100 keV)120572


120573




















990123

971214

990510

970828

980703

970508

100

101

102

103

Isot

ropi

c Lum

inos

ity (1051

ergsＭminus1)


(a)

0001

001

01

1

10

100

1000

10000

Peak

Lum

inos

ity (1052

ergＭminus1)

1 10 10001

00758 (1 + z)253 ＦＡ0282

1000

(b)























119863 = 1Γ (1 minus 1205730 cos 120579) (1 + 119911) (11)


119905 = 120591119863 (12)


119871peak prop 119863120572 (13)



log119871peak = 5649 + 335 log119881 (14)



ＦＩＡL

(ergＭminus1)

50

51

52

53

54


ＦＩＡ Vf=045

(a)GFM05 PL

05

1

15

2

25

3

35

ＦＩＡ(L)

ＦＩＡ (V)


(b)

G05 PL

ＦＩＡ (V)


minus05

0

05

1

15

2

25

3

35

ＦＩＡ(L)

(c)




























119871 iso prop 120591minus1198732RT (22)



50

51

52

53

54ＦＩＡ(L)


ＦＩＡ[２４(1 + z)]

(a)


ＦＩＡ[２４(1 + z)]

50

51

52

53

54

ＦＩＡ(L)

(b)








2 + (043119887120590RT120591RT )2

+ 1205902RTsys(24)






Γ (119905) = 106 (1 + 1199112 )38 (119864prompt1198990 )18 119905minus38 (26)


100

1000

Γ 0

090510 080319B090902B

990123

080916C

01 1 10 100 1000001


(a)

101

102

103

Γ 0


LＣＭＩ52

(b)













p⟩

(keV

)

00 05 10 15 20 25minus05


400

350

300

250

200

150

100






120573119878 (30)










FＮＩＮ


EP

(100

keV

)

1

(a)

Ep

fp

11 10

(b)






119864iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119878tot (1 + 119911)minus2 (33)









1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)


(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)


SwiftBATFermiGBM

(d)







980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery









119873119864 (119864) = 119860norm times

( 119864100 keV)120572


120573




















990123

971214

990510

970828

980703

970508

100

101

102

103

Isot

ropi

c Lum

inos

ity (1051

ergsＭminus1)


(a)

0001

001

01

1

10

100

1000

10000

Peak

Lum

inos

ity (1052

ergＭminus1)

1 10 10001

00758 (1 + z)253 ＦＡ0282

1000

(b)























119863 = 1Γ (1 minus 1205730 cos 120579) (1 + 119911) (11)


119905 = 120591119863 (12)


119871peak prop 119863120572 (13)



log119871peak = 5649 + 335 log119881 (14)



ＦＩＡL

(ergＭminus1)

50

51

52

53

54


ＦＩＡ Vf=045

(a)GFM05 PL

05

1

15

2

25

3

35

ＦＩＡ(L)

ＦＩＡ (V)


(b)

G05 PL

ＦＩＡ (V)


minus05

0

05

1

15

2

25

3

35

ＦＩＡ(L)

(c)




























119871 iso prop 120591minus1198732RT (22)



50

51

52

53

54ＦＩＡ(L)


ＦＩＡ[２４(1 + z)]

(a)


ＦＩＡ[２４(1 + z)]

50

51

52

53

54

ＦＩＡ(L)

(b)








2 + (043119887120590RT120591RT )2

+ 1205902RTsys(24)






Γ (119905) = 106 (1 + 1199112 )38 (119864prompt1198990 )18 119905minus38 (26)


100

1000

Γ 0

090510 080319B090902B

990123

080916C

01 1 10 100 1000001


(a)

101

102

103

Γ 0


LＣＭＩ52

(b)













p⟩

(keV

)

00 05 10 15 20 25minus05


400

350

300

250

200

150

100






120573119878 (30)










FＮＩＮ


EP

(100

keV

)

1

(a)

Ep

fp

11 10

(b)






119864iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119878tot (1 + 119911)minus2 (33)









1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)


(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)


SwiftBATFermiGBM

(d)







980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






990123

971214

990510

970828

980703

970508

100

101

102

103

Isot

ropi

c Lum

inos

ity (1051

ergsＭminus1)


(a)

0001

001

01

1

10

100

1000

10000

Peak

Lum

inos

ity (1052

ergＭminus1)

1 10 10001

00758 (1 + z)253 ＦＡ0282

1000

(b)























119863 = 1Γ (1 minus 1205730 cos 120579) (1 + 119911) (11)


119905 = 120591119863 (12)


119871peak prop 119863120572 (13)



log119871peak = 5649 + 335 log119881 (14)



ＦＩＡL

(ergＭminus1)

50

51

52

53

54


ＦＩＡ Vf=045

(a)GFM05 PL

05

1

15

2

25

3

35

ＦＩＡ(L)

ＦＩＡ (V)


(b)

G05 PL

ＦＩＡ (V)


minus05

0

05

1

15

2

25

3

35

ＦＩＡ(L)

(c)




























119871 iso prop 120591minus1198732RT (22)



50

51

52

53

54ＦＩＡ(L)


ＦＩＡ[２４(1 + z)]

(a)


ＦＩＡ[２４(1 + z)]

50

51

52

53

54

ＦＩＡ(L)

(b)








2 + (043119887120590RT120591RT )2

+ 1205902RTsys(24)






Γ (119905) = 106 (1 + 1199112 )38 (119864prompt1198990 )18 119905minus38 (26)


100

1000

Γ 0

090510 080319B090902B

990123

080916C

01 1 10 100 1000001


(a)

101

102

103

Γ 0


LＣＭＩ52

(b)













p⟩

(keV

)

00 05 10 15 20 25minus05


400

350

300

250

200

150

100






120573119878 (30)










FＮＩＮ


EP

(100

keV

)

1

(a)

Ep

fp

11 10

(b)






119864iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119878tot (1 + 119911)minus2 (33)









1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)


(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)


SwiftBATFermiGBM

(d)







980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery
















119863 = 1Γ (1 minus 1205730 cos 120579) (1 + 119911) (11)


119905 = 120591119863 (12)


119871peak prop 119863120572 (13)



log119871peak = 5649 + 335 log119881 (14)



ＦＩＡL

(ergＭminus1)

50

51

52

53

54


ＦＩＡ Vf=045

(a)GFM05 PL

05

1

15

2

25

3

35

ＦＩＡ(L)

ＦＩＡ (V)


(b)

G05 PL

ＦＩＡ (V)


minus05

0

05

1

15

2

25

3

35

ＦＩＡ(L)

(c)




























119871 iso prop 120591minus1198732RT (22)



50

51

52

53

54ＦＩＡ(L)


ＦＩＡ[２４(1 + z)]

(a)


ＦＩＡ[２４(1 + z)]

50

51

52

53

54

ＦＩＡ(L)

(b)








2 + (043119887120590RT120591RT )2

+ 1205902RTsys(24)






Γ (119905) = 106 (1 + 1199112 )38 (119864prompt1198990 )18 119905minus38 (26)


100

1000

Γ 0

090510 080319B090902B

990123

080916C

01 1 10 100 1000001


(a)

101

102

103

Γ 0


LＣＭＩ52

(b)













p⟩

(keV

)

00 05 10 15 20 25minus05


400

350

300

250

200

150

100






120573119878 (30)










FＮＩＮ


EP

(100

keV

)

1

(a)

Ep

fp

11 10

(b)






119864iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119878tot (1 + 119911)minus2 (33)









1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)


(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)


SwiftBATFermiGBM

(d)







980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






ＦＩＡL

(ergＭminus1)

50

51

52

53

54


ＦＩＡ Vf=045

(a)GFM05 PL

05

1

15

2

25

3

35

ＦＩＡ(L)

ＦＩＡ (V)


(b)

G05 PL

ＦＩＡ (V)


minus05

0

05

1

15

2

25

3

35

ＦＩＡ(L)

(c)




























119871 iso prop 120591minus1198732RT (22)



50

51

52

53

54ＦＩＡ(L)


ＦＩＡ[２４(1 + z)]

(a)


ＦＩＡ[２４(1 + z)]

50

51

52

53

54

ＦＩＡ(L)

(b)








2 + (043119887120590RT120591RT )2

+ 1205902RTsys(24)






Γ (119905) = 106 (1 + 1199112 )38 (119864prompt1198990 )18 119905minus38 (26)


100

1000

Γ 0

090510 080319B090902B

990123

080916C

01 1 10 100 1000001


(a)

101

102

103

Γ 0


LＣＭＩ52

(b)













p⟩

(keV

)

00 05 10 15 20 25minus05


400

350

300

250

200

150

100






120573119878 (30)










FＮＩＮ


EP

(100

keV

)

1

(a)

Ep

fp

11 10

(b)






119864iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119878tot (1 + 119911)minus2 (33)









1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)


(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)


SwiftBATFermiGBM

(d)







980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery
























119871 iso prop 120591minus1198732RT (22)



50

51

52

53

54ＦＩＡ(L)


ＦＩＡ[２４(1 + z)]

(a)


ＦＩＡ[２４(1 + z)]

50

51

52

53

54

ＦＩＡ(L)

(b)








2 + (043119887120590RT120591RT )2

+ 1205902RTsys(24)






Γ (119905) = 106 (1 + 1199112 )38 (119864prompt1198990 )18 119905minus38 (26)


100

1000

Γ 0

090510 080319B090902B

990123

080916C

01 1 10 100 1000001


(a)

101

102

103

Γ 0


LＣＭＩ52

(b)













p⟩

(keV

)

00 05 10 15 20 25minus05


400

350

300

250

200

150

100






120573119878 (30)










FＮＩＮ


EP

(100

keV

)

1

(a)

Ep

fp

11 10

(b)






119864iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119878tot (1 + 119911)minus2 (33)









1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)


(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)


SwiftBATFermiGBM

(d)







980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






50

51

52

53

54ＦＩＡ(L)


ＦＩＡ[２４(1 + z)]

(a)


ＦＩＡ[２４(1 + z)]

50

51

52

53

54

ＦＩＡ(L)

(b)








2 + (043119887120590RT120591RT )2

+ 1205902RTsys(24)






Γ (119905) = 106 (1 + 1199112 )38 (119864prompt1198990 )18 119905minus38 (26)


100

1000

Γ 0

090510 080319B090902B

990123

080916C

01 1 10 100 1000001


(a)

101

102

103

Γ 0


LＣＭＩ52

(b)













p⟩

(keV

)

00 05 10 15 20 25minus05


400

350

300

250

200

150

100






120573119878 (30)










FＮＩＮ


EP

(100

keV

)

1

(a)

Ep

fp

11 10

(b)






119864iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119878tot (1 + 119911)minus2 (33)









1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)


(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)


SwiftBATFermiGBM

(d)







980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






100

1000

Γ 0

090510 080319B090902B

990123

080916C

01 1 10 100 1000001


(a)

101

102

103

Γ 0


LＣＭＩ52

(b)













p⟩

(keV

)

00 05 10 15 20 25minus05


400

350

300

250

200

150

100






120573119878 (30)










FＮＩＮ


EP

(100

keV

)

1

(a)

Ep

fp

11 10

(b)






119864iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119878tot (1 + 119911)minus2 (33)









1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)


(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)


SwiftBATFermiGBM

(d)







980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






p⟩

(keV

)

00 05 10 15 20 25minus05


400

350

300

250

200

150

100






120573119878 (30)










FＮＩＮ


EP

(100

keV

)

1

(a)

Ep

fp

11 10

(b)






119864iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119878tot (1 + 119911)minus2 (33)









1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)


(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)


SwiftBATFermiGBM

(d)







980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






FＮＩＮ


EP

(100

keV

)

1

(a)

Ep

fp

11 10

(b)






119864iso = 4120587119863119871 (119911 Ω119872 ΩΛ)2 119878tot (1 + 119911)minus2 (33)









1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)


(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)


SwiftBATFermiGBM

(d)







980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(a)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Ep

(keV

)


(b)

980425

031203

050709

051221

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)

(c)

1049 1050 1051 1052 1053 10541048

EＣＭＩ (erg)

1

10

100

1000

10000

Epi

(keV

)


SwiftBATFermiGBM

(d)







980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






980425030329

XRF020903

XRF03072310

100

1000EＪＥ

(keV

)

1049 1050 1051 1052 1053 1054 10551048

Energy (erg)

tＤＮ~3 days

(a)

(I)

(I)

(II)

(III)

ΓＧＣＨ = 1

E ＪpropE

ΓＧＲ

EＪprop E

13

ＣＭＩ

50 52 54 5648

ＦＩＡ E (erg)

0

1

2

3

4

5

ＦＩＡEp

(keV

)

1Γge ＭＣ

ＨＤＮ

(b)










log119864peak = 2 + 05 log119864iso (38)









log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







log119864peak = 207 + 049 log119864iso (41)




120579jet = 0161 (119879break1 + 119911 )38 (119899120578120574119864iso)18 (42)

































or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery















or






























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery



























100

1000

10000

51 52 53 5450

ＦＩＡ(L)

Elowast ＪＥ(1+z)


10 10001


10

100

1000

10000

Epi

(keV

)














Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






Isot

ropi

c Lum

inos

ity (1052

ergＭ

minus1)

100 1000

Ep(1 + z) (keV)

100

10

1

01


(a)

ＦＩＡ(L

pＬＡＭminus1)

ＦＩＡ(EpＥ６)

2 3 4

54

52

50

(b)































119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery


























119891119894 (119905) = 119865119894119890120572119894(1minus119905119879119894)119890minus119905119894119905 119905 lt 119879119894119865119894 ( 119905119879119894)

minus120572119894 119890minus119905119894119905 119905 ge 119879119894 (63)




log 119871119883119901 = 119886 + 119887 log119879lowast119901 (64)

ata

Ta Fa

Ｊ

TＪ FＪ

minus8

minus6

minus4

minus2

0

ＦＩＡ10

(flux

)

0 2 4 6minus2










44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








44

46

48

50

52

54

ＦＩＡL８p

(a)



44

46

48

50

52

54

ＦＩＡL８p

(b)

44

46

48

50

52

54

ＦＩＡL８p



(c)







ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






ＦＩＡ10(L

fer

gsＭminus1)

46

48

50

Tf(1 + z) (s)1041000100101

54

52

(a)

EＪＥ(1 + z) (keV)10 100 1000

ＦＩＡ10(L

fer

gsＭminus1)

50

52

(b)





(119879 minus 119879ej119879119891 )minus1 (65)




(66)










4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






4 Summary




Acknowledgments


References









































































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery

















































































































119886


























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery

























































































119873119879-E119901119890119886119896119894119899119905ℎ119903119890119904119905
























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery


































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





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