Dissecting the Energy Budget of a Gamma-Ray Burst Fireball
DDraft version February 10, 2021
Typeset using L A TEX twocolumn style in AASTeX62
Dissecting Energy Budget of a Gamma-Ray Burst Fireball
Bing Zhang, Yu Wang,
2, 3, 4 and Liang Li
2, 3, 4 Department of Physics and Astronomy, University of Nevada Las Vegas, Las Vegas, NV 89154, USA; [email protected] ICRANet, Piazza della Repubblica 10, 65122 Pescara, Italy INAF – Osservatorio Astronomico d’Abruzzo, Via M. Maggini snc, I-64100, Teramo, Italy Dip. di Fisica and ICRA, Sapienza Universita di Roma, Piazzale Aldo Moro 5, I-00185 Rome, Italy
ABSTRACTThe jet composition and radiative efficiency of GRBs are poorly constrained from the data. Ifthe jet composition is matter-dominated (i.e. a fireball), the GRB prompt emission spectra wouldinclude a dominant thermal component originating from the fireball photosphere, and a non-thermalcomponent presumably originating from internal shocks whose radii are greater than the photosphereradius. We propose a method to directly dissect the GRB fireball energy budget into three componentsand measure their values by combining the prompt emission and early afterglow data. The measuredparameters include the initial dimensionless specific enthalpy density ( η ), bulk Lorentz factors at thephotosphere radius (Γ ph ) and before fireball deceleration (Γ ), the amount of mass loading ( M ), as wellas the GRB radiative efficiency ( η γ ). All the parameters can be derived from the data for a GRB witha dominant thermal spectral component, a deceleration bump feature in the early afterglow lightcurve,and a measured redshift. The results only weakly depend on the density n of the interstellar mediumwhen the composition Y parameter (typically unity) is specified. Keywords:
Gamma-ray bursts – Relativistic fluid dynamics INTRODUCTIONThe jet composition of the gamma-ray bursts (GRBs)has been subject to debate (Kumar & Zhang 2015; Pe’er2015; Zhang 2018). The GRB prompt emission spectracan in principle help to diagnose the jet composition: theexistence of a bright thermal component would supporta matter-dominated fireball (M´esz´aros & Rees 2000),while the non-detection of such a component may sug-gest the dominance of a Poynting flux in the jet com-position (Zhang & Pe’er 2009) . Broad band observa-tions with GRB detectors, especially with the Gamma-ray Burst Monitor (GBM) and Large Area Telescope(LAT) on board the Fermi
Gamma-Ray Space Tele-scope, have collected rich data, which suggest that theGRB jet composition is likely diverse. Whereas someGRBs (e.g. GRB 090902B, Abdo et al. 2009a; Rydeet al. 2010; Pe’er et al. 2012, see Ryde 2005; Ryde &Pe’er 2009; Li 2019b for systematic searches) are consis-tent with being fireballs, a good fraction of bursts are A thermal component may still show up if the central enginemagnetization parameter σ is not extremely large so that σ atthe photosphere already drops to close to unity (e.g. Gao & Zhang2015; Beniamini & Giannios 2017). consistent with not having a thermal component (e.g.GRBs 080916C, 130606B, and many others, Abdo et al.2009b; Zhang et al. 2011, 2016; Oganesyan et al. 2017;Ravasio et al. 2019; Burgess et al. 2020). “Intermediate”GRBs with a dominant non-thermal component and asub-dominant thermal component have been discovered(e.g. GRB 100724B, GRB 110721A and several others,Guiriec et al. 2011, 2015; Axelsson et al. 2012), whichmay be understood within the framework of “hybrid”jets, i.e. the composition is a mixture of a matter com-ponent and a Poynting-flux component (Gao & Zhang2015; Li 2020). Some bursts (e.g. GRB 160625B) dis-played a significant change of jet composition among dif-ferent emission episodes within the same GRB (Zhanget al. 2018; Li 2019a), which may be consistent withsome central engine models (e.g. Metzger et al. 2011).Different jet compositions may imply different energydissipation (shocks vs. magnetic reconnection) and ra-diation (quasi-thermal vs. synchrotron) mechanisms.Another interesting subject related to the GRBprompt emission mechanism is the radiative efficiencyof a burst, which may be defined as (Lloyd-Ronning &Zhang 2004) η γ ≡ E γ E tot = E γ E γ + E k = L γ L w, , (1) a r X i v : . [ a s t r o - ph . H E ] F e b Zhang et al. where E γ , E k and E tot are isotropic-equivalent gamma-ray energy, afterglow kinetic energy, and total energy, re-spectively, and L γ and L w, are the isotropic-equivalentaverage gamma-ray luminosity and total wind lumi-nosity at the central engine, respectively. Consideringbeaming correction would lead to the same results, sinceall the energy/luminosity terms are multiplied by thesame beaming factor f b , which is not considered in thediscussion below. The E γ value can be well measuredfrom the data as long as the fluence is well measuredand redshift is known. The E k term, on the other hand,is usually estimated from the afterglow data throughmodeling. Its value depends on many uncertain shockmicrophysics parameters, mostly (cid:15) e (Freedman & Wax-man 2001), but also (cid:15) B and electron spectral index p aswell (Zhang et al. 2007; Wang et al. 2015). As a result,the derived GRB radiative efficiency has been subjectto large uncertainties, ranging from below 10% to morethan 90% (Zhang et al. 2007; Wang et al. 2015; Beni-amini et al. 2015; Li et al. 2018).The bulk Lorentz factor Γ of a GRB, which is relatedto the kinetic energy of the outflow, has been estimatedusing various methods. The maximum photon energyof prompt emission may be used to set a lower limit onΓ (e.g. Baring & Harding 1997; Lithwick & Sari 2001).However, a precise measurement cannot be made sincethe maximum energy also depends on emission radius,which is not well constrained (Gupta & Zhang 2008) .Two other methods can give better estimates of Γ: Thefirst makes use of the early afterglow lightcurve data. Ifa well-defined bump is identified in the early afterglowlightcurve, it can be interpreted as the fireball decelera-tion time. The Lorentz factor before deceleration (whichwe call Γ in the rest of the paper) can be estimated(Rees & M´esz´aros 1992; M´esz´aros & Rees 1993; Sari &Piran 1999), which depends on E k and the medium den-sity parameter (i.e. n for the constant medium modeland A ∗ for the wind model). Again E k needs to be esti-mated from the afterglow data or from the prompt emis-sion data assuming an efficiency parameter. Alterna-tively, if a strong thermal component is measured fromthe GRB prompt emission spectrum, one can estimatethe Lorentz factor at the photosphere radius (which wecall Γ ph in the rest of the paper) based on the stan-dard fireball photosphere model (Pe’er et al. 2007). TheGRB efficiency again needs to be assumed in order to Most work made use of the variability timescale to estimatethe emission radius, but the estimate is only relevant for the in-ternal shock model but does not apply to photosphere (e.g. Rees& M´esz´aros 2005) or magnetic dissipation (e.g. M´esz´aros & Rees1997; Lyutikov & Blandford 2003; Zhang & Yan 2011) models. perform the estimate. This simple method relies on theassumption of a matter dominated jet composition. Formore general hybrid jet models, more complicated di-agnoses are needed (Gao & Zhang 2015). Observation-ally, the Γ values derived from the afterglow decelerationmethod (Liang et al. 2010; L¨u et al. 2012; Ghirlandaet al. 2018) is somewhat smaller than those derived usingother methods (Racusin et al. 2009; Pe’er et al. 2015).In this paper, we propose a new method to diagnosefireball parameters by combining the deceleration andphotosphere methods. We show that with adequate ob-servations, one can measure several fireball parametersrelated to the energy budgets. In particular, the effi-ciency parameter that has to be assumed in previousmethods can be directly measured. The method is in-troduced in Section 2. Some examples are presented inSection 3. The results are summarized in Section 4 withsome discussion. THE METHOD2.1.
Energy budget decomposition
Very generally, the effective energy per baryon at thecentral engine can be defined by the parameter µ ≡ η (1 + σ ) (cid:39) η, (2)where η ≡ ( n w, m p c + e + p ) / ( n w, m p c ) = 1 +ˆ γe / ( n w, m p c ), σ , n w, , e , p are the dimensionlessspecific enthalpy density (also called dimensionless en-tropy in the literature, e.g. M´esz´aros & Rees 2000),the magnetization parameter, number density, internalenergy density, and pressure of the fireball wind at thecentral engine, respectively, and ˆ γ = 4 / η (cid:29)
1. The last ap-proximation in Equation (2) applies to a pure fireballwith σ (cid:39)
0, which is the regime discussed in this pa-per. During the subsequent evolution of the fireball, theeffective energy per baryon can be defined by µ = Γ( R )Θ( R ) , (3)which is conserved unless radiation is leaked out fromthe fireball. Here Γ( R ) is the bulk Lorentz factor of thefireball as a function of the radius R from the centralengine, and Θ( R ) = 1 + ˆ γe ( R ) / [ n w ( R ) m p c ] is the di-mensionless specific enthalpy density as a function of R .Figure 1 shows a cartoon picture of the evolution of µ (only up to the deceleration radius R dec , beyond whichit is no longer of interest) and Γ (throughout the accel-eration, coasting, dissipation, and deceleration phases)as a function of R . One can see that before the deceler-ation radius, the µ parameter undergoes two significantdrops: The first drop occurs at the photosphere radius issecting a GRB fireball µ value drops from η to Γ ph .The second drop occurs at the internal shock radii wheresignificant dissipation of the fireball kinetic energy oc-curs and additional photon energy (in the form of syn-chrotron radiation) is released from the fireball. The µ value drops from Γ ph to Γ before entering the deceler-ation phase.For a fireball with an isotropic equivalent total mass M , the initial, total energy of the fireball is E tot = ηM c . (4)The energy emitted in thermal emission from the pho-tosphere is E th = ( η − Γ ph ) M c ; (5)that emitted in non-thermal emission from internalshocks is E nth = (Γ ph − Γ ) M c ; (6)and the total emitted energy is E γ = E th + E nth = ( η − Γ ) M c . (7)The kinetic energy left in the afterglow is E k = Γ M c , (8)so that the radiative efficiency (1) becomes η γ = η − Γ η . (9)2.2. Prompt emission constraint
The fireball initially undergoes a rapid accelerationwith Γ ∝ R due to the internal pressure of the fireball(M´esz´aros et al. 1993; Piran et al. 1993; Kobayashi et al.1999). It coasts at a radius R c = Γ c R at which accelera-tion essentially stops, where R is the initial radius of thefireball, and Γ c is the coasting Lorentz factor. In orderto constrain Lorentz factor using the thermal emissioninformation, the photosphere radius R ph needs to begreater than R c . In previous treatments (e.g. M´esz´aros& Rees 2000; Pe’er et al. 2007), Γ c is approximated as η (for the regime we are interested in, i.e. R ph > R c ). Wenote that the fireball Lorentz factor never fully achieves η , as the fireball contains a significant amount of inter-nal energy, especially below R ph . Numerical simulations(Kobayashi et al. 1999) showed that acceleration doesnot stop abruptly, but undergoes a smooth transitionaround R c . See also Figure 1. As a result, a more rea-sonable approximation would be that the Lorentz factorof the fireball only reaches Γ ph at R ph , when the fireballbecomes transparent. After discharging photons at R ph , the internal energy becomes negligibly small so that µ becomes close to the bulk Lorentz factor Γ = Γ ph , whichcoasts with this value afterwards. As a result, one mayapproximately treat the fireball dynamics as having aneffective coasting Lorentz factor Γ c ∼ Γ ph and an effec-tive coasting radius at R c ∼ Γ ph R .For R ph > R c (i.e. Γ ph < Γ ph , ∗ ), the observer-frame(without the (1 + z ) correction from cosmological ex-pansion) luminosity and temperature of the photosphereemission can be estimated as (M´esz´aros & Rees 2000,but with η replaced by Γ ph , and L w, replaced by L w , ph ) L ph L w , ph (cid:39) (cid:18) Γ ph Γ ph, ∗ (cid:19) / = (cid:18) R ph R c (cid:19) − / = (cid:18) r ph R (cid:19) − / , (10) T ph T (cid:39) (cid:18) Γ ph Γ ph , ∗ (cid:19) / = (cid:18) R ph R c (cid:19) − / = (cid:18) r ph R (cid:19) − / , (11)where L ph is the photosphere emission luminosity (i.e.the luminosity of the thermal spectral component), L w , ph is the kinetic luminosity of the wind at the pho-tosphere, which is related to the total wind luminositythrough L w , ph = L w, (Γ ph /η ), r ph = R ph Γ ph (12)is the radius of the projected photosphere area for arelativistically moving fireball,Γ ph , ∗ = (cid:18) L w , ph Y σ T πm p c R (cid:19) / (cid:39) (cid:18) L w , ph , Y R , (cid:19) / (13)is the critical Γ ph above which R ph becomes smaller than R c so that the method discussed here no longer applies,and T (cid:39) (cid:18) L w, πR σ B (cid:19) / (cid:39) . × K (cid:18) L w, , R , (cid:19) / (14)is the initial temperature at the central engine. Here m p is the proton mass, c is the speed of light, σ T isthe Thomson cross section, σ B is the Stefan-Boltzmannconstant, Y is the lepton-to-baryon number ratio, whichequals unity for a pure hydrogen fireball but could begreater (for a pair-loaded fireball) or slightly smaller (fora neutron-rich fireball without pair loading) than unity.Both L w , ph and L w, are normalized to 10 erg s − (hereafter the convention Q = 10 n Q n is adopted in cgsunits). Notice that in Eq. (14) we have neglected a coef-ficient of order unity, which depends on the compositionof the outflow at the jet base (Kumar & Zhang 2015).Other coefficients of the order unity are also neglectedin our derivations below. Zhang et al. R μ & ΓΓ ph Γ η T h e r m a l E m i ss i o n N o n -t h e r m a l E m i ss i o n A f t e r g l o w E m i ss o n R c E th/ M c R IS R ph R dec Γμ E nth/ M c Figure 1.
An indicative description of the evolution of µ and Γ in a GRB fireball. Both axes are in logarithmic scales. Inreality, internal shocks may spread in a wide range of radii. The observed flux of the photosphere blackbody com-ponent is F obbb = (4 πr σ B T ph ) / (4 πD ). Using Equation(11) and noticing L w, = 4 πD F ob γ η − γ ( F ob γ is the ob-served total gamma-ray flux), one can derive (Pe’er et al.2007) R (cid:39) D L (1 + z ) η / R , (15)where R ≡ (cid:18) F obbb σ B T (cid:19) / (cid:39) r ph D L (1 + z ) , (16) η th ≡ η γ F obbb F ob γ = E th E tot , (17)and T ob = T ph / (1+ z ) is the effective temperature of theobserved thermal spectrum.Making use of Equation (10) and noticing L w , ph =4 πD F ob γ f − γ , where f γ = L γ L w , ph = η − Γ Γ ph , (18) The photosphere spectrum is not exactly a blackbody, butdoes not significantly deviate from it (Pe’er 2012; Deng & Zhang2014). one can further deriveΓ ph (cid:39) (cid:34) (1 + z ) D L Y σ T F ob γ m p c R f / γ η / γ (cid:35) / = (cid:34) (1 + z ) D L Y σ T F ob γ m p c R η / ( η − Γ )Γ / (cid:35) / . (19)One can see that the parameters η and Y in Equation (4)of Pe’er et al. (2007) are replaced by Γ ph and f / γ /η / γ ,respectively. In the second equation, Equation (18) hasbeen used. Solving for Γ ph , one can further deriveΓ ph = (cid:34) (1 + z ) D L Y σ T F ob γ m p c R η / η − Γ (cid:35) / . (20)2.3. Afterglow constraint
For a constant density interstellar medium , onecan estimate Γ using the observed deceleration time t dec . The deceleration radius can be estimated with We do not discuss the case of a wind medium (Dai & Lu 1998;M´esz´aros et al. 1998; Chevalier & Li 1999) in this paper. Afterglowobservations suggest that the majority of GRBs, especially thosewith the clear deceleration signature, are consistent with havinga constant density medium (Zhang et al. 2007; Liang et al. 2010). issecting a GRB fireball π/ R nm p c = E k / (ˆ γ Γ Γ dec ), where Γ dec =Γ /
2. This gives the deceleration radius R dec =(3 E k / π ˆ γ Γ nm p c ) / (cid:39) (6 . × cm) E / k, Γ − / , n − / .The deceleration time in the observer frame can becalculated as t dec = (cid:82) R dec (1 + z ) / (2Γ( r ) c ) dr (cid:39) . z ) R dec / Γ c . Reversely solving it, one finallygets (Zhang 2018)Γ (cid:39) . / (cid:18) E k (1 + z ) π ˆ γnm p c t (cid:19) / (cid:39) t − / , (cid:18) z (cid:19) / (cid:18) E k, n (cid:19) / = 170 t − / , (cid:18) z (cid:19) / (cid:18) E γ, n (cid:19) / (cid:18) Γ η − Γ (cid:19) / . (21)2.4. Dissecting fireball energy budget
The five unknown parameters that characterize aGRB fireball, i.e. η , Γ ph , Γ , η γ , and M can be inprinciple solved with Equations (5), (6), (9), (20) and(21), using the observed quantities E th , E nth , E γ , F ob γ , F obbb , T ob , t dec and z . There are only two free param-eters. One is Y , which depends on the composition ofthe fireball (pairs, protons and neutrons), but a reason-able estimate is Y ∼
1. The second parameter is thedensity parameter n , which may be further constrainedvia afterglow modeling (e.g. Panaitescu & Kumar 2001,2002). Even if it is not constrained, the solutions onlyweakly depend on it. One may take a standard value n = 1 cm − when solving the problem.There is no analytical solution to the problem. Onecan numerically solve the problem using a root-findingalgorithm. From Equations (5) or (6), one can solve M = E γ ( η − Γ ) c , (22)Γ ph = ηE nth + Γ E th E γ . (23)From Equation (21), one can derive η = 3 . (cid:18) Γ (cid:19) − t − , (cid:18) z (cid:19) (cid:18) E γ, n (cid:19) +Γ . (24)Inserting Equations (23) and (24) to Equation (20), Γ can be then solved by assigning typical values for Y and n . Once Γ is solved, η can be solved from Equation(24); Γ ph and M can be solved from Equations (23) and(22), respectively, and η γ can be solved from Equation(9). EXAMPLESIn order to perform the diagnosis proposed in this pa-per, a GRB needs to satisfy the following three require-ments: • The burst needs to have a matter-dominated com-position with a distinct thermal spectral compo-nent. One may use the contrast between the ther-mal and non-thermal components to estimate themagnetization parameter σ at the central enginebased on the hybrid-jet diagnostic method pro-posed by Gao & Zhang (2015) (see Li 2020 fora systematic analysis of the GRB data using themethod). If σ is close to 0, the burst would be afireball. • The burst needs to have early afterglow data thatshow a distinct bump that is consistent with decel-eration of a fireball in a constant density medium(e.g. Molinari et al. 2007; Liang et al. 2010). • The burst needs to have a measured redshift.Few GRBs satisfy these constraints in the currentdatabase. We have gone over the currently detectedGRBs from the archives, but could not find an idealcase with all three criteria satisfied. One GRB to whichthis method may be applied is GRB 190114C, which isstudied elsewhere (L. Li et al. 2021, in prep.).Instead of performing case studies, in the following weperform calculations for some example cases and explorethe dependence of the results on various parameters. Forexample, we consider a GRB at z = 1 with the followingobserved quantities: E th = 10 erg, E nth = 5 × erg, F ob γ = 10 − ergs − cm − , F obbb = 6 × − ergs − cm − , T = 100 keV, and t dec = 20 s. According to the for-malism discussed in Section 2, following fireball param-eters can be derived: η (cid:39) ph (cid:39) (cid:39) η γ (cid:39) . M (cid:39) . × − M (cid:12) .In general, the results are mainly defined by three en-ergy values (only two are independent), i.e. E th , E nth ,and E γ = E th + E nth . This is because given a GRBduration T and a redshift, the energy parameters ( E th and E γ ) can be approximately translated to the fluxparameters ( F obbb and F ob γ ) . The observed temperature T ob is also related to F obbb through r ph . Figure 2 showsthe contours of η , Γ ph , Γ , η γ , E k and E tot in the E γ − p plane, where p ≡ E th /E γ is the thermal emission frac-tion. The following parameters, i.e. z = 1, n = 1 cm − , Y = 1, T = 15 s, t dec = 30 s, and T ob = 60 keV, are E th , E nth and E γ include the energies during the entire T of GRB prompt emission, whereas F γ and F bb are measured dur-ing the time intervals when the thermal emission presents. Fortypical GRBs, the prompt emission lightcurves show a rough fast-rise-exponential-decay behavior and the thermal emission usuallyappears at the most luminous peak region. For a theoretical esti-mation, we may calculate the flux at the peak region as ∼ T , e.g., F ob γ ∼ z ) E γ / πD T . Zhang et al.
Figure 2.
Contour plots of η , Γ ph , Γ , η γ , E k and E tot in the E γ − p plane. o b o b o b o b o b o b Figure 3.
Contour plots of η , Γ ph , Γ , η γ , E k and E tot in the T ob − p plane. adopted in the calculations. One can see that the effi-ciency η γ is reasonably high, between ∼ (25% − η , Γ , E k , and E tot are all insensitive to the thermalemission fraction p but positively scale with E γ . Onlythe Γ ph contour positively scales with both E γ and p .Fixing E γ , Γ ph decreases as p increases. This is fullyconsistent with intuition.Figure 3 shows the contours of η , Γ ph , Γ , η γ , E k and E tot in the T ob − p plane. The following parameters, i.e. z = 1, n = 1 cm − , Y = 1, T = 15 s, t dec = 30 s, and E γ = 10 erg, are adopted for the calculations. Thepatterns are more complicated, which is a result of thecomplicated relationship between r ph and various energybudget parameters. The bottom-left panel again showsthat usually the fireball radiative efficiency η γ is high,i.e. ∼ (20% − T ob . Given a measured T ob , η γ increases as the thermal issecting a GRB fireball p increases to high values. This is due to thesignificant increase of η in these cases. CONCLUSIONS AND DISCUSSIONWe have proposed a method to dissect the energy bud-get of a GRB fireball making use of the constraints de-rived from the thermal and non-thermal emission com-ponents in the prompt emission spectrum and the decel-eration bump feature in the early afterglow lightcurveof a GRB. The key point is that the blackbody spec-tral component observed in the prompt emission phaseand the early afterglow bump are measuring the bulkLorentz factor of the fireball at two different stages, i.e.Γ ph and Γ , respectively. Both are lower than the ini-tial dimensionless specific enthalpy density of the fireball η . With observational quantities such as E th , E nth , E γ , F ob γ , F obbb , T ob , t dec and z , one can directly measure sev-eral crutial fireball parameters, including η , Γ ph , Γ , η γ ,and M .In order to apply the method, the three criteria dis-cussed in Section 3 are needed. The lack of GRBs satis-fying all three criteria is the combination of the rarenessof fireballs and some observational selection effects. Forexample, the GRBs with well-studied prompt emissionspectra were usually detected by Fermi , whereas thosewith early afterglow and redshift measurements wereusually detected by
Swift . On the other hand, burststhat can satisfy all three constraints may be regularlydiscovered by the upcoming Chinese-French GRB detec-tor SVOM (Wei et al. 2016), which has the capabilityof obtaining both broad-band prompt emission spectra(using ECLAIRS and GRM) and early optical afterglowlightcurves (using VT). Many of these bursts will have redshift measurements with the detection of early af-terglows. The diagnosis proposed in this paper can beroutinely applied to those bursts.There are some caveats when applying the methodproposed here. First, we have applied the stan-dard fireball photosphere-internal-shock model (Rees& M´esz´aros 1994; M´esz´aros & Rees 2000; Daigne &Mochkovitch 2002) that invokes two distinct emissionsites. Some models interpret both thermal and non-thermal emissions as arising from the photosphere region(e.g. Vurm et al. 2011; Veres et al. 2012). Our methoddoes not apply to those models. Second, if the centralengine carries significant magnetization ( σ (cid:29) Aartsen, M., Ackermann, M., Adams, J., et al. 2017,Advances in Space Research,doi:https://doi.org/10.1016/j.asr.2017.05.030Abdo, A. A., Ackermann, M., Ajello, M., et al. 2009a,ApJL, 706, L138Abdo, A. A., Ackermann, M., Arimoto, M., et al. 2009b,Science, 323, 1688Asano, K., & M´esz´aros, P. 2011, ApJ, 739, 103Axelsson, M., Baldini, L., Barbiellini, G., et al. 2012, ApJL,757, L31Baring, M. G., & Harding, A. K. 1997, ApJ, 491, 663Beniamini, P., & Giannios, D. 2017, MNRAS, 468, 3202Beniamini, P., Nava, L., Duran, R. B., & Piran, T. 2015,MNRAS, 454, 1073 Burgess, J. M., B´egu´e, D., Greiner, J., et al. 2020, NatureAstronomy, 4, 174Chevalier, R. A., & Li, Z.-Y. 1999, ApJL, 520, L29Dai, Z. G., & Lu, T. 1998, MNRAS, 298, 87Daigne, F., & Mochkovitch, R. 2002, MNRAS, 336, 1271Deng, W., & Zhang, B. 2014, ApJ, 785, 112Freedman, D. L., & Waxman, E. 2001, ApJ, 547, 922Gao, H., & Zhang, B. 2015, ApJ, 801, 103Ghirlanda, G., Nappo, F., Ghisellini, G., et al. 2018, A&A,609, A112Guiriec, S., Connaughton, V., Briggs, M. S., et al. 2011,ApJL, 727, L33Guiriec, S., Kouveliotou, C., Daigne, F., et al. 2015, ApJ,807, 148Gupta, N., & Zhang, B. 2008, MNRAS, 384, L11