On the origin of GeV emission in gamma-ray bursts
aa r X i v : . [ a s t r o - ph . H E ] J a n Draft version October 29, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
ON THE ORIGIN OF GEV EMISSION IN GAMMA-RAY BURSTS
Andrei M. Beloborodov , Romain Hasco¨et and Indrek Vurm Physics Department and Columbia Astrophysics Laboratory, Columbia University, 538 West 120th Street, New York, NY 10027, USA;[email protected] Tartu Observatory, T˜oravere 61602, Tartumaa, Estonia
Draft version October 29, 2018
ABSTRACTThe most common progenitors of gamma-ray bursts (GRBs) are massive stars with strong stellarwinds. We show that the GRB blast wave in the wind should emit a bright GeV flash. It is producedby inverse Compton cooling of the thermal plasma behind the forward shock wave. The main part ofthe flash is shaped by scattering of the prompt MeV radiation (emitted at smaller radii) which streamsthrough the external blast wave. The inverse-Compton flash is bright due to the huge e ± enrichmentof the external medium. At late times, the blast wave switches to normal synchrotron-self-Comptoncooling. The mechanism is demonstrated by a detailed transfer simulation. The observed promptMeV radiation is taken as an input of the simulation; we use GRB 080916C as an example. Theresult reproduces the GeV flash observed by the Fermi telescope. It explains the delayed onset, thesteep rise, the peak flux, the time of the peak, the long smooth decline, and the spectral slope of GeVemission. The wind density required to reproduce all these features is typical of Wolf-Rayet stars. Oursimulation predicts strong TeV emission 1 min after the burst trigger; then a cutoff in the observedhigh-energy spectrum is expected from absorption by extragalactic background light. In addition, abright optical counterpart of the GeV flash is predicted for plausible values of the magnetic field; sucha double (optical+GeV) flash has been observed in GRB 130427A.
Subject headings: plasmas – radiation mechanisms: non-thermal – radiative transfer – scattering –gamma-rays: bursts, theory – relativity INTRODUCTIONThe luminosities of gamma-ray bursts (GRBs) peakin the soft gamma-ray band around 1 MeV (e.g.Goldstein et al. 2012). Observations by the LargeArea Telescope (LAT) onboard the
Fermi satellite(Atwood et al. 2009) show that some GRBs also giverise to a longer GeV flash, with a distinct light curve(Fermi-LAT Collaboration 2013). The energy emittedin the GeV band is smaller than that of the main(“prompt”) MeV radiation, typically by a factor ∼ Fermi
LAT provide supportto this picture:(1) In practically all GRBs detected by
Fermi
LAT(except a few cases with poor photon statistics) the peakof the GeV flash overlaps with the prompt MeV radiation(Fermi-LAT Collaboration 2013). The overlap impliesthat the GeV source experiences Compton cooling by theprompt MeV radiation (keV radiation in the rest frameof the source).(2) The GeV flash has a distinct light curve, differentfrom the prompt MeV burst. It quickly rises and thenshows a long monotonic decay, which lasts significantlylonger than the prompt MeV emission. This is expectedif the GeV flash is produced by the external blast wave.The blast wave has a larger radius and moves with asmaller Lorentz factor compared with the source of theprompt burst, and hence its emission can be spread overlonger observational times.(3) The onset of GeV emission is slightly delayed withrespect to the beginning of the prompt MeV burst. Thearrival time of photons emitted by the blast wave at ra-dius R is roughly given by t obs ∼ (1 + z ) (cid:18) Rv bw − Rc (cid:19) ≈ (1 + z ) R c . (1) A. M. Beloborodov, R. Hasco¨et, I. VurmHere t obs is measured by the clock of a distant observersince the first light signal from the beginning of the ex-plosion, Γ = (1 − v /c ) − / ≫ z is the cosmological redshift.The delay in the onset of GeV emission is expected ifthe blast-wave luminosity is suppressed at small radii. Itequals the time it takes the explosion to reach the radiuswhere it becomes a bright GeV source, which is typicallya few seconds. However, any model associating the GeV flash withthe external blast wave faces the following puzzle. Manyobserved GeV flashes reach the peak and start to de-cay at time T p much shorter than the duration of theprompt MeV burst, T GRB . For example, GRB 080916Chas T p ∼ . T GRB (Abdo et al. 2009). Why would thepeak of blast-wave radiation be much shorter than theprompt burst itself? Consider the standard model where T GRB corresponds to the duration of the ultra-relativisticejecta that emits the prompt burst. Then cT GRB (1+ z ) − is a measure of the ejecta thickness. The ejecta en-ergy is transferred to the blast wave through the reverseshock, which may be relativistic and can cross the ejectaas quickly as T cross ∼ T GRB (in observer time). Theejecta cannot transfer its energy at t obs ≪ T GRB , asthis would require a superluminal motion of the reverseshock, and hence the self-similar deceleration of the blastwave should not begin until t obs ∼ T GRB . Then the GeVflash is not expected to decay until t obs ∼ T GRB (e.g.Gao et al. 2009; He et al. 2011; Maxham et al. 2011).The problem becomes even more severe in explosionmodels with a non-relativistic reverse shock; then the de-celeration/decay stage is not expected until t obs ≫ T GRB .This puzzle is resolved by the fact that the blast wavepropagates in a medium with a quickly changing composi-tion.
As discussed in detail below, at radii
R < ∼ cmthe medium is extremely rich in e ± pairs, with Z ± > ∼ pairs per ion. Pairs are inevitably produced by theprompt MeV radiation propagating ahead of the blastwave (Thompson & Madau 2000; M´esz´aros et al. 2001;Beloborodov 2002, hereafter B02; Kumar & Panaitescu2004). The huge number of the prompt MeV photons( N MeV ∼ in isotropic equivalent for the brightestGRBs) implies exponential pair creation in a static opti-cally thin medium. In addition, radiation exerts a strongforce and significantly accelerates the external medium,which affects the strength of the forward shock and theevolution of its temperature.We show in this paper that the forward shock prop-agating in the pre-accelerated pair-enriched medium isan extremely efficient producer of GeV emission, regard-less of the details of the shock microphysics and its effi-ciency in nonthermal particle acceleration. This providesa robust mechanism for a GeV flash. As the blast waveexpands to larger radii where Z ± is reduced, its GeVluminosity decreases.A possible role of e ± loading for GeV emission waspreviously conjectured by Ghisellini et al. (2010), al-though their scenario differs from the model presentedhere. Ghisellini et al. (2010) assumed that the blast The prompt burst is emitted at a much smaller radius R MeV ,with a Lorentz factor Γ ej > ∼ Γ. Therefore, its delay ∼ (1 + z )( R MeV / c ) is much smaller. wave enters the self-similar deceleration stage in the pair-dominated zone and continues to radiate with the pair-assisted efficiency close to 100%. They explained theobserved decline of GeV emission by the steep reduc-tion of the dissipation power in the decelerating blastwave. As discussed above, the problem of this picture isthat the self-similar deceleration should not begin until t obs ∼ T GRB while the observed decline in many LATbursts starts at T p ≪ T GRB . Another difference con-cerns the emission mechanism: Ghisellini et al. (2010)associated GeV photons with synchrotron emission fromnonthermal particles. We find that the GeV flash is pro-duced by inverse Compton scattering of the prompt ra-diation by the thermal plasma behind the forward shock.In this paper, we study explosions in the wind mediumexpected from a massive progenitor (e.g. Chevalier & Li1999). We consider a Wolf-Rayet star with a typicalmass-loss rate ˙ M ≈ − M ⊙ yr − , which produces awind with density profile ρ ∝ R − . We calculate thedynamics, e ± density, and temperature of the blast waveand show that it must generate an inverse-Compton pair-dominated flash in the GeV band. Its light curve andspectrum can be calculated from first principles, using adirect simulation of radiative transfer.Preliminary estimates explaining the proposed mecha-nism are presented in Section 2. Then in Sections 3 and4 we describe the setup of our detailed calculations. Theresults are described in Section 5 using GRB 080916C asan example. In Section 6, we present analytical estimatesfor photon-photon ( γ - γ ) opacity. Then, in Section 7, wediscuss the expected synchrotron emission from the pair-loaded blast wave, and find that, in a broad range of themagnetization parameter ε B , the GeV flash is accompa-nied by a bright and brief optical flash. In Section 8we estimate the effect of the GeV flash on the externalmedium. Our results are summarized and discussed inSection 9. PRELIMINARY ESTIMATES2.1.
Number of GeV photons in the flash
Consider a blast wave that sweeps up the externalmedium. Let γ inj be the mean (thermal) Lorentz factorof hot electrons immediately behind the forward shock,and Γ be the bulk Lorentz factor of the shocked fluid.Subscript “inj” in γ inj stands for “injection” — the hotplasma is injected at the shock front and cools down be-hind it.The plasma is Compton cooled by the prompt GRBradiation that gradually leaks out of the explosion ejectaand streams radially through the external blast wave. Let E t ∼ E ′ t = E t . (2) A related technical remark: Ghisellini et al. (2010) used energyof the entire burst in the estimate of the pair-loading effect. Infact, when the GeV flash peaks, only a fraction ∼ T p /T GRB of theprompt burst is ahead of the blast wave, and E GRB contributingto its pair loading is reduced by a factor of ∼ T p /T GRB . The prompt photons are assumed to be emitted at a smallradius R MeV ≪ R , and their angles with respect to the radialdirection are θ ∼ ( R MeV /R ) Γ − ≪ Γ − . eV emission in gamma-ray bursts 3The hot electrons injected at the shock front lose energyby upscattering the target photons. The typical energyof upscattered photons in the fluid frame is E ′ IC ∼ γ e E ′ t (assuming Thomson scattering). The corresponding en-ergy of IC photons in the lab frame is E IC ≈ (2 / E ′ IC , E IC ∼ γ e E t . (3)One can see that GeV photons are generated when γ e ∼ (cid:18) E IC (cid:19) / (cid:18) E t (cid:19) − / . (4)Then one can verify that the scattering is in the Thom-son regime, γ e E ′ t /m e c <
1, although moderate Klein-Nishina corrections are beginning to appear at these en-ergies.As the electron injected with γ e = γ inj cools down, itproduces IC photons with decreasing E IC ∝ γ e . Theirnumber near a given energy E IC may be estimated as M ∼ γ e m e c E ′ IC ∼ m e c ( E ′ t E ′ IC ) / ∼ Γ m e c ( E t E IC ) / . (5)The multiplicity of photons with E IC ∼ γ inj ≫
50 is
M ∼ Γ / N GeV ∼ M N ± , (6)where N ± is the number of electrons/positrons swept-upby the shock, proportional to the total swept-up mass m , N ± = Z ± N p , N p = mµ e m p . (7)Here Z ± is the pair loading factor of the externalmedium, N p is the number of swept-up protons, and µ e depends on the chemical composition of the medium; µ e = 1 for hydrogen and µ e = 2 for heavier elements.The medium is expected to be a wind from a massiveprogenitor, which is losing mass before the explosion witha rate ˙ M . The mass of the wind medium contained in asphere of radius R is given by m ( R ) = ˙ M Rw , (8)where w is the wind velocity. The likely GRB progenitorsare Wolf-Rayet stars, whose observed winds have typical˙ M ∼ − M ⊙ yr − , w ∼ × cm s − , and µ e ≈ N p ∼ R ˙ M − , (9)where ˙ M − = ˙ M / − M ⊙ yr − and R = R/ cm.The value of Z ± can be exactly calculated using theobserved luminosity and spectrum of the prompt GRB(Section 3.1); it has enormous values Z ± ∼ at theearly stages of blast-wave expansion and then steeplydecreases with radius. In particular, for GRB 080916Cwe will show below that the GeV flash peaks at a well-defined radius R p ≈ cm where Z ± ∼ . Equa-tions (6) and (7) with Z ± ∼ give a rough estimate for the number of GeV photons, N GeV ∼ , (10)which is close to the isotropic equivalent of the brightGeV flashes observed by LAT. The high density of theprogenitor wind and the huge pair enrichment is whatmakes the inverse Compton mechanism capable of emit-ting a bright flash; models neglecting pair creation wouldfall far short in N GeV .Note that the prompt GRB radiation plays a key rolefor the GeV flash in two ways: (1) it provides targetphotons for IC scattering and (2) its interaction withthe external medium ahead of the shock ensures the e ± enrichment of the medium. The e ± pairs radiating GeVphotons behind the shock are created by the prompt MeVphotons propagating ahead of the shock. The total num-ber of the prompt photons in a burst like GRB 080916Cis huge, N MeV ∼ (isotropic equivalent). Almost allthese photons pass through the external medium unaf-fected, as the medium is optically thin. A small fractionof photons get scattered and converted to e ± pairs, sothe number of created pairs N ± ≪ N MeV . However, N ± greatly exceeds N p , by the factor Z ± ≫ Radiative efficiency in the GeV band
As will be demonstrated with detailed calculations be-low, the external blast wave inevitably passes througha stage with the pair-loading factor Z ± ∼ and pre-acceleration Lorentz factor γ ∼
10. It is an extremelyefficient producer of GeV emission at this stage. Threefactors contribute to the high efficiency:(1) The high pair-loading factor Z ± ∼ > m p /m e guarantees that most of the shock-dissipated energy isgiven to leptons.(2) At this stage, the shock-heated pairs have the ther-mal Lorentz factor γ inj ∼ Γ /γ ∼
50, so their IC cool-ing produces emission in the GeV band according toEquation (3). The relatively low value of γ inj is a re-sult of pair loading and pre-acceleration of the externalmedium. Note that pre-acceleration reduces the strengthof the forward shock: the fluid Lorentz factor jumps atthe shock front from γ ∼
10 to Γ, which corresponds toelectron heating to γ inj ∼ Γ /γ . (3) Inverse Compton cooling of the shocked pairs isfast, so they efficiently radiate their energy. The coolingtimescale of isotropic electrons with Lorentz factor γ e inthe fluid frame is given by t ′ IC = 3 m e c σ T U ′ γ e , (11)where U ′ = (2Γ) − U is the energy density of the col-limated prompt radiation in the fluid frame, and U = L GRB / πR c . The cooling timescale should be com-pared with the expansion timescale of the blast wave, t ′ exp = R/c Γ, t ′ IC t ′ exp = 12 π m e c R Γ σ T L GRB γ e ≈ γ e R L − (cid:18) Γ500 (cid:19) , (12) Energy transfer from the shocked ions to electrons is unable tosignificantly increase γ inj in the medium with Z ± ∼ , since theion abundance is smaller than m e /m p . This effect can, however,become significant soon after the peak of the flash, as Z ± decreases. A. M. Beloborodov, R. Hasco¨et, I. Vurmwhich gives t ′ IC < t ′ exp for γ e ≫
1. Compton cooling isfast for electrons emitting in the GeV band, γ e > ∼ γ e ≫ scatter photons with a smallerrate due to the Klein-Nishina correction, but their cool-ing is still fast.2.3. Lorentz factor of the blast wave and arrival timeof GeV photons
The arrival time of IC photons emitted at radius R p ∼ cm (peak of the GeV flash) depends on the Lorentzfactor of the blast wave, Γ, according to Equation (1).Note that R p can be significantly smaller than the radiuswhere the blast wave enters the self-similar deceleration.At this early stage, the blast-wave material is sandwichedbetween the forward and reverse shocks, and its Lorentzfactor Γ is regulated by the ram pressures in the twoshocks, P f and P r .An estimate for Γ may be obtained assuming pressurebalance P f ∼ P r . A convenient approximation for theshock pressure is given by (Beloborodov & Uhm 2006), P = 43 (cid:0) Γ − (cid:1) U up , (13)where Γ rel is the relative Lorentz factor of the upstreamand downstream, and U up = γ (1 + β ) γ heat ρc is theproper energy density of the upstream fluid; γ heat − γ (1 + β ) asrequired by the continuity equation (B02). This gives, P f ≈
43 Γ γ heat ρc γ (1 + β ) (cid:18) Z ± m e µ e m p (cid:19) , (14)where we used Γ rel ≈ Γ /γ (1 + β ) ≫
1. In the absenceof pre-heating and pre-acceleration ( γ heat = γ = 1) andmoderate pair loading ( Z ± ≪ m p /m e ), Equation (14)reduces to the standard relation P f = (4 / ρc .For the reverse shock one can use Equation (13) withΓ rel ≈ (1 / ej / Γ + Γ / Γ ej ) and U up = ρ ej c , P r ≈ (cid:18) Γ ej Γ − ΓΓ ej (cid:19) ρ ej c , (15)where ρ ej and Γ ej are the fluid mass density and Lorentzfactor of the ejecta. Then the pressure balance P f ∼ P r givesΓ ≈ Γ ej ( (cid:20) πAc γ heat (1 + Z ± m e /µ e m p ) L ej γ (1 + β ) (cid:21) / ) − / . (16)Here L ej = 4 πR Γ ρ ej c is the kinetic power of theejecta (isotropic equivalent) and we used the externaldensity profile ρ = AR − where A ≡ ˙ M / πw ∼ − g cm − . In the case of a relativistic reverse shock,Γ ≫ Γ , the expression for Γ simplifies and becomesindependent of Γ ej ,Γ ≈ (cid:20) L ej γ (1 + β )16 πAc γ heat (1 + Z ± m e /µ e m p ) (cid:21) / . (17)This equation gives Γ ∼
500 for the parameters ofGRB 080916C discussed in this paper. Note that γ is determined by the force exerted by the prompt radiationfront ahead of the blast wave. Our numerical calcula-tions give γ ∼ Z ± ∼ , and γ heat ≈ R p ≈ cm (see Section 5).Using Equations (1) and (17), one finds the arrival timeof the peak of the flash t obs ∼ −
10 s, which is consistentwith observations. The detailed calculations presentedbelow will give a more accurate estimate for the arrivaltime of the peak. We will also calculate the light curveof the GeV flash and show that its decay after the peakextends over much longer times.2.4.
Energy dissipated in the forward shock
As a final check, let us estimate the energy dissipated inthe forward shock near the radius R p ∼ cm. Sincemost of the dissipated energy E diss is radiated in GeVphotons, one expects a GeV flash of energy E flash ∼ E diss .The dissipation rate in the forward shock is approxi-mately given by, L diss ≈ πR (3 P f )Γ c ∼ πR (3 P r )Γ c ∼ L ej , (18)where we used Equation (15) and assumed Γ ej ≫ Γ. Theejecta power L ej is comparable to or larger than the ob-served luminosity of the prompt GRB, L GRB , dependingon the prompt emission efficiency ε rad , L ej L GRB = 1 − ε rad ε rad . (19)The peak luminosity of the flash L flash < ∼ L ej is compara-ble to L GRB that is observed before the peak of the flash.We will confirm this result with more detailed calcula-tions below. 2.5.
Summary
As the blast wave passes through the radius R p ∼ cm where γ ∼
10, the shock wave radiates mostof the dissipated energy in the GeV band, and the emit-ted radiation arrives at t obs ∼ −
10 s. This defines thepeak of the GeV flash. Below we present detailed calcu-lations that will give the light curve and spectrum of theflash, before and after the peak. SHOCK WAVE IN PAIR-LOADED MEDIUM3.1.
Pair loading
The prompt MeV radiation is nearly perfectly beamedin the radial direction in the blast-wave region, as it isemitted at much smaller radii. Those prompt photonsthat have already overtaken the forward shock propa-gate in the external medium, which has not yet learnedabout the explosion. Some of these photons scatter offthe ambient medium. Only a small fraction of photonsare scattered (the medium is optically thin), however thisfraction translates into a huge number of scattered pho-tons per ambient electron . Many of these photons quicklyconvert to e ± pairs. The conversion occurs because thescattered photons have large angles with respect to theprimary (collimated) GRB radiation, and the large anglelowers the energy threshold for the γ - γ reaction with thebeam, γ + γ → e + + e − .The created pairs also scatter the prompt photons,which leads to exponential e ± creation and a huge en-hancement of the electron density ahead of the forwardeV emission in gamma-ray bursts 5shock, by a factor Z ± exceeding 10 (B02). The e ± load-ing factor Z ± = n ± /n ≫ R < R load , where R load ≈ E / , cm , (20)and E GRB is the isotropic equivalent of the prompt GRBenergy ahead of the forward shock.
The main dimensionless parameter that controls Z ± atthe forward shock is proportional to the column densityof the GRB radiation ahead of the shock, ξ = σ T m e c E GRB πR = 650 E GRB , R − . (21)At observer times t obs ≪ T GRB , E GRB ahead of the shockis a fraction of the total prompt GRB energy (most ofwhich is still behind the shock). The pair loading factor Z ± ( ξ ) and the pre-acceleration Lorentz factor γ ( ξ ) de-pend only on the prompt radiation field and not on thedensity of the ambient medium (B02).We have extended the calculations of B02 in two ways:(1) B02 assumed a typical prompt GRB spectrum thatpeaks at E pk = m e c while the bright bursts detected byLAT have higher than average E pk . We have extendedthe model to bursts with high E pk ∼ −
10 MeV. (2) B02used the “cold approximation” assuming that the loaded e ± pairs are quickly cooled to a non-relativistic temper-ature, so that the scattering plasma may be assumed tobe cold. This approximation is accurate only for burstswith E pk ≪ L E = L pk E × (cid:26) ( E/E pk ) − α , E < E pk ( E/E pk ) − α , E > E pk (22)As a first test, we ran our code using the cold approxi-mation and found excellent agreement with Figures 1-3in B02. Note that Equation (4) in B02 misses the factor dǫ/dǫ sc which should have canceled the factor of (1+ β ) − in his Equations (42) and (43). However, the numericalresults in B02 are based on the correct equations, themissing factor dǫ/dǫ sc being a misprint that propagatedto Equations (42) and (43).Then we relaxed the cold approximation and obtained Z ± ( ξ ) and γ ( ξ ) for bursts with high E pk . Figure 1 showssample models with E pk = 1 , ,
10 MeV, α = 0 (pho-ton index − α = 1 . − . Z ± ( ξ ) and γ ( ξ ) do not depend on L pk E .For comparison, Figure 1 (left panel) also shows theresults obtained with the cold approximation, which aresignificantly different. MeV radiation scattered by thecold plasma is preferentially directed along radius (aKlein-Nishina effect), which reduces the efficiency of paircreation. One can see that relaxing the cold approxi-mation leads to significantly higher Z ± , mainly becausethe hot plasma scatters photons through larger angleswith respect to the primary collimated beam. The ther-mal Lorentz factor of the e ± plasma in the radiationfront reaches γ th ≈ γ ≈ γ th is reduced at larger ξ where γ ≫
1. 3.2.
Forward shock
The forward shock propagates in the pair-rich, pre-accelerated medium which is moving with γ <
Γ. Theshock thermalizes the relative Lorentz factor,Γ rel = Γ γ (1 − β bw β ) ≈ Γ γ (1 + β ) , (23)where β bw = (1 − Γ − ) / and β = (1 − γ − ) / . If thereis no energy exchange between e ± and ions, all shockedparticles acquire the thermal Lorentz factor γ inj ∼ Γ rel (assuming “cold” plasma ahead of the shock, γ th ∼ ε e ≤ e ± due to collective processes in the shock. Thenthe thermal Lorentz factor of shocked e ± is given by γ inj = Γ rel (cid:18) γ th + ε e µ e m p Z ± m e (cid:19) , (24)where µ e = 1 for hydrogen and µ e = 2 for heavier ions.The preheating by the prompt radiation gives γ th compa-rable to unity (Section 3.1); in Section 8 we will discussan extension of the model that can give γ th ≫ Z ± ≫ , the second term on the right-hand side of Equa-tion (24) is small compared with the first term, i.e. ionsare energetically unimportant. In this zone, the shockemission is produced by pairs with γ e ∼ Γ rel regardlessof the value of ε e ; the e ± pairs dominate the postshockenergy density and quickly radiate this energy away, lead-ing to nearly 100% radiative efficiency.The parameter ε e can become important where Z ± ≪ . Numerical simulations of electron-ion shocks with-out pairs show ε e ∼ . − . ε e forpair-loaded electron-ion shocks; it is possible that ε e de-pends on Z ± .The shock may also accelerate a small fraction of elec-trons/positrons to Lorentz factors much larger than γ inj ,forming a nonthermal electron population. We assumethat most of the shock energy is given to the quasi-thermal e ± -ion plasma, and neglect nonthermal particles.As will be seen below, they are not needed to produce theGeV flash, and are not expected to dominate the flashenergy output. 3.3. Blast-wave dynamics
The Lorentz factor Γ of the blast wave propagating inthe pre-accelerated medium with a given Lorentz factor γ ( R ) is calculated similarly to the standard model wherethe external medium is at rest. We are particularly inter-ested in the early stage, before the reverse shock crossesthe main part of the ejecta that carries most of the ex-plosion energy. An estimate for Γ at this stage was givenin Section 2.3.In our simulations we use a rather crude model forthe blast-wave dynamics. Our approach is similar tothe “mechanical” model of Beloborodov & Uhm (2006),where the blast-wave material is described by a singleLorentz factor Γ, and its evolution with time is derivedfrom energy and momentum conservation. The pre-acceleration of the external medium by radiation reducesthe pressure in the blast wave. The blast wave develops A. M. Beloborodov, R. Hasco¨et, I. Vurm Fig. 1.—
Pair loading factor Z ± ( ξ ) and pre-acceleration Lorentz factor γ ( ξ ) in the prompt radiation front propagating in the externalmedium (with µ e = 2) ahead of the blast wave. The radiation spectrum is assumed to be a broken power-law with the low-energy photonindex − − . Left: results for the radiation spectrum with E pk = 3 MeV; the exact calculation (solid curves) is compared with the coldapproximation (dotted curves). Right: results for the radiation spectra with E pk = 100 keV (dotted), 300 keV (short dashed), 1 MeV (longdashed), 3 MeV (solid), and 10 MeV (dash-dotted). Note that pair loading is very high ( Z ± ∼ ) at γ ∼
10 where the peak of the GeVflash is emitted (Section 2). where γ < Γ ej , closing the gap between the radiativelypre-accelerated external medium and the ejecta (B02).When the reverse shock becomes relativistic (Γ ≪ Γ ej )the value of Γ ej becomes unimportant — it has no influ-ence on Γ; this fact is also seen in the estimate (17).The relativistic reverse shock crosses the ejecta on anobserved timescale comparable to T GRB . At later timesthe energy supply to the blast wave from the ejecta drops,and the explosion dynamics switches to the self-similarregime; we follow this transition in our simulation. Theself-similar blast wave in a wind medium with a low ra-diative efficiency has Γ ∝ R − / , and with a high radia-tive efficiency Γ ∝ R − (Blandford & McKee 1976).As discussed above, radiative efficiency is close to 100%during the peak of the GeV flash; it can also be high atlater phases of the flash (see Section 8 below). The dy-namics of radiative blast waves involves subtle effects.The large energy losses of the post-shock plasma implyits quick and significant compression. In this regime, theforward shock has the Lorentz factor Γ FS ≈ Γ. There isa thin shell of fluid immediately behind the shock withLorentz factor 2 − / Γ FS (as required by the jump con-ditions), so the true profile of the fluid Lorentz factorbehind the shock is not flat — there must be a steepchange from 2 − / Γ to Γ. The corresponding velocityprofile is consistent with quick compression of the post-shock plasma — the expected result of strong radiativelosses. The characteristic thickness of the compressionlayer behind the shock is set by the cooling length.In the radiatively inefficient regime, the blast wave be- comes nearly adiabatic and Γ FS ≈ / Γ, i.e. the shockruns significantly faster, leaving more space for the post-shock material. Then the profile of the fluid Lorentzfactor behind the shock is smooth and flat.We model the transition between the radiative and adi-abatic regimes in a crude way, switching from Γ FS = Γ toΓ FS = 2 / Γ when radiative efficiency drops below 1 / RADIATIVE TRANSFERAs long as the GeV flash is dominated by IC scat-tering of the prompt radiation streaming through theblast wave, its light curve can be obtained by solvingradiative transfer for the prompt photons. The resultswill describe the main phase of the flash — its peakand early decay. Observations of GeV flashes by
Fermi
LAT are typically limited to this early phase; e.g. inGRB 080916C it lasts until t obs ∼
400 s (see below).Pair loading described in Section 3.1 can also bethought of as a result of radiative transfer of the promptphotons, but scattered in the external medium ahead ofthe blast wave. One can think of both pair loading andflash emission as two parts of one global transfer problemfor the prompt photons (Figure 2). To find an approx-imate solution to this problem, we divided it into twozones: ahead of the forward shock (zone I) and behindthe shock (zone II). Scattering in zone I controls the pairloading of the blast wave (as it generates MeV photonswith large angles). The GeV flash is mainly produced byeV emission in gamma-ray bursts 7
Fig. 2.—
Schematic illustration of the transfer problem. Redarrows show the prompt MeV radiation streaming from the ejectaand gradually overtaking the forward shock (FS). The prompt pho-tons can be scattered in the external medium ahead of the shock(zone I) or in the shock-heated plasma (zone II). The coordinate ̟ measures the distance from the leading edge of the radiation front;the unscattered prompt radiation arrives to the observer at time t obs = (1 + z ) ̟/c . The scattered photons arrive with a delay. scattering in the shock-heated zone II.The result of transfer in zone I was described in Sec-tion 3.1. The solution depends on the prompt radiationspectrum and should be obtained individually for a givenGRB. For a given spectral shape (i.e. given α , α , E pk )the obtained Z ± and γ at the forward shock are functionsof the GRB energy ahead of the shock, E GRB = Z t FS L GRB ( t ) dt, (25)where t = (1 + z ) − t obs and t FS is defined in Equa-tion (27) below. E GRB determines the value of param-eter ξ (see Equation (21)) and thus determines Z ± and γ . Note also that γ and Z ± enter our calculation of theblast-wave dynamics Γ( R ) (Section 2.3), thus the twocalculations are coupled and we perform them together,integrating over the history of the blast-wave expansion.Once we obtain solutions for Γ( R ), Z ± ( R ), and γ ( R ),we turn to the calculation of photon scattering behindthe shock (zone II). The blast wave is optically thin, soonly a small fraction of the prompt GRB photons is in-volved in the radiative transfer. In addition, multiple ICscattering is strongly suppressed by the Klein-Nishinaeffect at high energies, so one can safely use the sin-gle scattering approximation. One must, however, followthe transfer of scattered photons through the radiationfield, as many of them have high energies and can easilyconvert to e ± pairs, even though they have small angles θ ∼ Γ − . The secondary high-energy pairs are Comptoncooled by the prompt radiation, increasing the multiplic-ity of IC photons.Monte-Carlo technique is most suitable for this trans-fer problem. As the shock passes distance dR it sweepsup dN ± = Z ± ( R ) n p πR dR electrons/positrons, where n p ( R ) is the proton number density of the externalmedium. The shocked particles are heated to γ inj given Our simulation for GRB 080916C also takes into account that α , α , and E pk vary during the prompt emission, which affectsthe relation between E GRB and ξ and the dependence of ξ on R . by Equation (24). Effectively, dN ± hot particles are in-jected at the shock radius R FS , and we follow their cool-ing behind the shock, track the produced IC photons,any secondary products that may result from photon ab-sorption, and cooling of the secondary pairs.Particles and photons can be followed on the space-time diagram using lab-frame time t lab and radial posi-tion R as coordinates. Note that R is very close to ct lab everywhere in the relativistic blast wave (whose charac-teristic thickness R/ Γ ≪ R ). Therefore, instead of t lab ,it is convenient to use the coordinate ̟ defined by ̟ = ct lab − R. (26)Then ̟ = 0 corresponds to the first GRB photons thatwill be received at t obs = 0, and ̟ GRB = (1 + z ) − cT GRB corresponds to the end of the prompt GRB, t obs = T GRB (see Figure 2). As long as a particle has coordinate ̟ <̟
GRB , it is exposed to the prompt GRB photons and canscatter them. When coordinates (
R, ̟ ) are used insteadof (
R, t lab ), one can assume that all particles in the blastwave have the same radial position R , as the informationabout the small differences ∆ R ∼ R/ Γ is carried by thecoordinate ̟ . The blast-wave evolution is fully describedby functions of R , e.g. Z ± ( R ), Γ( R ), etc. The growingradius of the expanding blast wave, R ≈ ct lab , now playsthe role of a lab-frame time instead of coordinate t lab .The coordinate ̟ of the forward shock is given by ̟ FS ( R ) = ct FS = Z R dR ′ ( R ′ ) . (27)All shocked particles are advected by the expanding blastwave with Lorentz factor Γ, and their positions in theprompt radiation front, ̟ , evolve according to d̟ = dR . (28)Next, consider an IC photon scattered at R sc , ̟ sc through an angle θ sc (measured in the lab frame). Thescattered photon propagates along a straight line and itsangle relative to the radial direction decreases,sin θ ( R ) = R sc R sin θ sc . (29)The photon coordinate ̟ ( R ) grows according to d̟ = (1 − cos θ ) dR. (30)As the IC photon propagates, we evaluate γ - γ opacityalong the ray (see below) and check for absorption. Ifthe photon escapes, its arrival time is t obs ( R sc , ̟ sc , θ sc ) = (1 + z ) (cid:20) ̟ sc c + R sc c (1 − cos θ sc ) (cid:21) . (31)Every scattered photon is drawn from the prompt GRBradiation, which is assumed to be perfectly collimated atradii of interest, even when viewed from the rest frame ofthe blast wave. The luminosity L GRB ( t obs ) and spectrumof the prompt radiation are known from observations; inthe simulations we approximate the prompt spectrumby a broken power law. One can directly calculate theprompt radiation flux at any R and ̟ , F ( R, ̟ ) = L GRB ( t obs )4 πR , t obs = (1 + z ) ̟c . (32) A. M. Beloborodov, R. Hasco¨et, I. VurmThe photon scattering by an electron with a givenLorentz factor γ e is simulated using the exact Klein-Nishina cross section and drawing the target photonsfrom the prompt GRB spectrum.We assume that collective plasma effects maintain theisotropy of the electron distribution. This does not im-ply that the scattered radiation is isotropic in the fluidframe. The scattering rate for an electron moving withvelocity v is proportional to 1 − v · n where n is the unitvector in the radial direction (the photon direction beforescattering). Thus, the electron has a higher probabilityto scatter a photon when v · n <
0. As a result, IC radi-ation from isotropic relativistic electrons is significantlyanisotropic. The scattered photons have a higher proba-bility to carry a negative momentum in the fluid frame,which creates a “rocket effect” that tends to acceleratethe blast wave. This effect is neglected in our dynamicalmodel of the explosion (and should be included in fu-ture, more detailed models). However, the anisotropy ofIC radiation is accurately calculated in our Monte-Carlosimulation as we follow all scattering events individually.The anisotropy impacts the distribution of photon ar-rival times measured by a distant observer, leading to anadditional delay (see also Toma et al. 2009).The IC photons can escape or get absorbed by anotherphoton. The absorption opacity is discussed in detail inSection 6 below. Our Monte-Carlo simulation includesthe opacity provided by the main (unscattered) beam ofthe prompt radiation, κ γγ ( ǫ, θ ) ≈ α ) / σ T m e c F ǫ ( ǫ thr ) , (33)where θ is the angle of the IC photon, ǫ = E/m e c is itsdimensionless energy, and α = − d ln F ǫ /d ln ǫ is the spec-tral slope of target radiation evaluated near the threshold ǫ thr = 2 ǫ − (1 − cos θ ) − . As we follow each IC photon,we calculate the absorption opacity along its trajectoryand check for absorption. If the photon gets absorbedat some ̟ abs , we inject two new particles (an e ± pair)sharing the energy of the absorbed photon. The absorbedphotons indirectly contribute to the observed emission asthey create secondary e ± pairs whose IC emission mayescape. GEV FLASHWe have applied our transfer simulation toGRB 080916C, one of the first GRBs detected byLAT. It is an extremely bright burst, with isotropicenergy equivalent ∼ × erg (Abdo et al. 2009).The burst duration is T GRB ≈
100 s, which correspondsto ≈
20 s when corrected for cosmological redshift z ≈ .
35. Abdo et al. (2009) fitted the prompt emissionof GRB 080916C by the Band function in five consecu-tive time bins. We use the prompt emission describedby these fits at
E <
100 MeV as an input of our transfersimulation.The main parameter of the problem is the external den-sity. We consider the progenitor wind with mass density ρ ( R ) = AR , A = ˙ M πw . (34)We find that A ≈ × g cm − gives a GeV flashconsistent with LAT observations, and therefore in all Fig. 3.—
Lorentz factor of the blast wave (Γ) and the pre-accelerated medium ahead of the blast wave ( γ ) in GRB 080916C. R p is the radius where the GeV flash peaks. The wind densityparameter is A = 3 × g cm − . Fig. 4.—
Pair loading factor of the forward shock. figures we show the explosion model with this A . Theejecta is assumed to have a high Lorentz factor Γ ej =1200 and carry energy five times that of the prompt GRBradiation, L ej = 5 L GRB . The blast wave is not sensitiveto the value of Γ ej when Γ ≪ Γ ej (Section 2.3).Note that the blast wave is optically thin in the regionof main interest, R > ∼ cm. Its Thomson opticaldepth at radius R is given by τ ± ≈ Z ± σ T Aµ e m p R ≈ × − (cid:18) Z ± (cid:19) A R − . (35)Hereafter we assume µ e = 2 (a progenitor wind that iseV emission in gamma-ray bursts 9 Fig. 5.—
Cooling tracks of the shocked particles in the expand-ing blast wave (with ε e = 1) exposed to the prompt radiation.Each track starts at the shock front with the thermal Lorentz fac-tor γ e = γ inj (shown by the solid curve). Compton cooling by theprompt radiation operates (and dominates) at radii R < R ; thecorresponding tracks are shown by filled squares. The figure showsone realization of the tracks randomly drawn from our Monte-Carlosimulation. Occasional big jumps (the result of large energy lossin Klein-Nishina scattering) introduce a significant random com-ponent, allowing the tracks to cross. At radii R > R the promptradiation decouples from the blast wave and no longer can cool it.If SSC radiation is neglected, the blast wave becomes adiabatic;dotted lines show the result of adiabatic cooling. made of elements heavier than hydrogen).5.1. Blast wave dynamics, shock heating and cooling
Figures 3 and 4 show the blast-wave dynamics Γ( R ),pair loading Z ± ( R ), and pre-acceleration Lorentz factor γ ( R ). The displayed model assumes ε e = 1; similar re-sults are obtained for ε e = 0 . e ± pairs at radii R < cm; Z ± ≈ at10 cm. The prompt radiation accelerates the externalmedium to a relativistic speed at radii R < × cm.The Lorentz factor of the blast wave slowly decreasesfrom 700 at R = 10 cm to 300 at R ∼ cm. One cannotice jumps in the derivative d Γ /dR . These jumps arecaused by the rough description of the observed promptradiation taken from Abdo et al. (2009) — the burst wasdivided into five time bins of constant luminosities L GRB .Our simulation assumes L GRB = 0 . L ej (which corre-sponds to a constant radiative efficiency, ε rad = 1 / L ej = 5 L GRB . The pressure in the reverse shockjumps as it crosses the boundary of each shell, which af-fects the blast-wave dynamics. The reverse shock reachesthe end of the ejecta at R ∼ cm and then the blastwave switches to the self-similar deceleration. At a com-parable radius, Compton cooling of the forward shockbecomes inefficient (as nearly all prompt radiation hasovertaken the forward shock and decouples from it), andthe blast wave becomes adiabatic. In this model, we ne-glected synchrotron self-Compton (SSC) cooling of the Fig. 6.—
Theoretical light curve and data above 100 MeV forGRB 080916C. The wind density parameter is A = 3 × g cm − .To illustrate the effect of ε e , we ran the simulation for three cases: ε e = 0 (dotted curve), 0 . blast wave, because for GRB 080916C it becomes impor-tant only at late times t obs >
300 s, where the LAT dataends.Figure 5 shows the cooling tracks of the shock-heatedparticles on the R - γ e plane. The particles are cooling fastas long as the forward shock overlaps with the prompt ra-diation front, in agreement with Equation (11). Our sim-ulation assumes that the prompt GRB ends at ̟ GRB /c =(1 + z ) − T GRB ≈
19 s. The last prompt photons over-take the forward shock at radius R ≈ . × cm, andCompton cooling by the prompt radiation ends.5.2. Light curve
Figure 6 shows the light curve of high-energy emission( E obs >
100 MeV) predicted by the transfer simulation,and compares it with the LAT data. The peak of theGeV flash at t obs ∼ R p indicated in Figures 3 and 4.The shock wave is a weak producer of GeV emission atradii R < R p because the shock is weak — it propagatesin the medium pre-accelerated by the prompt radiationpressure to a large Lorentz factor γ , which reduces theram pressure in the shock and the thermal Lorentz factorof shocked particles γ inj (Equation 24). The IC emissionof the forward shock appears in the GeV band when γ decreases to ∼
10 and γ inj reaches ∼
50. This conditiondetermines the radius R p where the GeV flash peaks. Asthe shock expands to larger radii R > R p , γ inj becomesmuch greater than 50 and the multiplicity of GeV pho-tons saturates at M < ∼
10 (see Section 2.1). Then thedecrease of the pair loading factor Z ± (Figure 4) leads The smaller Z ± ahead of the shock is partially compensatedby the production of secondary particles in the e ± cascade behindthe shock, which results from γ - γ absorption of high-energy ICphotons. R > R p , at t obs ≪ T GRB , well before the reverse shockcrosses the ejecta, i.e. well before the blast wave entersthe stage of self-similar deceleration. This resolves thepuzzle discussed in Section 1.The production of GeV photons continues as long asthe shock-heated plasma finds targets for inverse Comp-ton scattering. Prompt photons serve as targets until ̟ FS = ̟ GRB , i.e. until the blast wave reaches the ra-dius R where the prompt emission completely overtakesthe blast wave, R ≈ c T GRB z . (36)Photons scattered at radius R arrive with a significantdelay after the last prompt photons, depending on thescattering angle θ , t obs ( θ ) = T GRB + (1 + z )(1 − cos θ ) R c ≈ T GRB (cid:2) (1 − cos θ ) (cid:3) . (37)Here Γ FS ≈ Γ for a radiative forward shock and Γ =2Γ for a shock with a reduced radiative efficiency. Thearrival time given by Equation (37) can be much longerthan T GRB . For isotropic scattering, the average scat-tering angle in the fluid frame ˜ θ = π/ θ ≈ β bw and 1 − cos θ ≈ (2Γ ) − . This would give t obs ≈ T GRB if the shock is radiatively inefficient at R ,and t obs ≈ T GRB if it is efficient. In fact, even when thehot electrons are isotropic in the fluid frame, the scat-tering is anisotropic — the probability of “backward”scattering (˜ θ > π/
2) is larger than the probability of“forward” scattering (˜ θ > π/ P (cos ˜ θ ) = (1 − cos ˜ θ ) / θ , delaying the averagearrival time of scattered photons. As a result, a changein the GeV light curve associated with the end of the tar-get prompt radiation at R may be expected at observertime t ∼ (3 − T GRB . (38)The scattering regime significantly changes over thecourse of the flash. The peak at t obs ∼ T p is emittedin approximately Thomson regime. Indeed, at R p theshock wave heats the e ± pairs to γ inj ∼
50 while the tar-get radiation density in the fluid frame peaks at E ′ pk ∼ (2Γ) − E pk ∼ γ inj E ′ pk /m e c < γ inj grows by a few orders of magnitude,and the scattering of photons with E t ∼ E pk is sup-pressed by the Klein-Nishina effects. Then the shockwave is mainly cooled by softer photons of energy E t < ∼ E KN ∼ Γ γ inj m e c , (39)and cooling occurs in a regime that is intermediate be-tween the Thomson and Klein-Nishina limits. In this regime, significant luminosity is given to IC photonswith energies E IC comparable to the electron energy, andhence the typical E IC weakly depends on the target ra-diation spectrum. As a result, the light curve shown inFigure 6 at t obs > T p is not very sensitive to the spec-trum of radiation that provides targets for IC scattering(we verified this by varying the target radiation in ourtransfer simulation). The remaining important conditionis that the electrons have enough time to radiate theirenergy, i.e. cooling is faster than the expansion of theblast wave. This condition is satisfied (see Section 2.2and Figure 5).The hot electrons see a significant scattering opticaldepth in the target photons of energies E t ∼ E KN . Notethat the same photons are near the threshold for γ - γ re-action with the IC photons of energy E IC ∼ Γ γ e m e c .This implies that the IC photons see an interesting op-tical depth to γ - γ absorption (the γ - γ cross section σ γγ > ∼ . σ T is comparable to Compton cross section). Inour simulation, we observed significant absorption of ICphotons and emission from secondary pairs at t obs > T p ,which has a modest impact on the light curve in Fig-ure 6. It more significantly affects the emission at ener-gies E ≫ Spectrum
Figure 7 shows the spectrum of high-energy emissionpredicted by the transfer simulation at t obs ∼
2, 8, and70 s. The spectrum is shaped by fast Compton coolingof the shock-heated e ± , partial absorption of IC photonsby photon-photon collisions, γ + γ → e + + e − , and cool-ing of the secondary pairs. The spectrum received nearthe peak of the flash ( t obs ∼ EL E ∼ const, mainly because of the fast evolu-tion of γ inj with radius, which implies a quick growthof the maximum IC photon energy from < ∼ > ∼
100 GeV. As the blast wave expands by a factor of 2around R p ≈ cm, γ inj changes by a factor of ∼ t obs (which vary with the pho-ton angles).After the peak, t obs > T p , a large fraction of the blast-wave power is emitted at energies E > ∼
100 GeV. Absorp-tion is significant for photons with energies
E >
10 GeV;however, it never completely suppresses the high-energyemission. This is an interesting feature of radiative trans-fer through the pair-loaded blast wave. It is related tothe fact that the flash peaks when the radiation front hasa well defined value of ξ ∼
300 (see Section 5.4) and ξ gradually decreases after the peak. The parameter ξ is ameasure of the column density of prompt photons, andits preferred value ξ ∼
300 corresponds to a preferredvalue of the optical depth to γ - γ absorption, τ γγ , whichturns out to be comparable to unity. The opacity seenby the high-energy IC photons is dominated by the un-scattered, beamed prompt radiation with photon indexclose to − α ≈ E ≫
10 GeV, and itsdependence on the emission angle θ is given by τ γγ ( x ) ≈ . x ǫ − ξ, (40)where x = θ Γ ∼ ǫ pk = E pk /m e c ∼
10 ineV emission in gamma-ray bursts 11
Fig. 7.—
Spectrum of GRB 080916C in three time windowsaround t obs ∼
2, 8, and 70 s from the transfer simulation with ε e = 1. Photon energy has been corrected for cosmological red-shift z = 4 . E = (1 + z ) E obs . Vertical dashed line shows thelower boundary of LAT data (1 + z ) ×
100 MeV. The high energypart (above 100 MeV) shows the IC emission from the blast wave,the result of our transfer calculations. The gray strips show theGBM data in three time bins A, C, E that roughly correspond to t obs ∼
2, 8, and 70 s. The width of the strips indicates the 1- σ un-certainty of the spectral fits by the Band function (Sylvain Guiriec,private communication). GRB 080916C. We used Equation (60) derived in Sec-tion 6 below and substituted α = α = 0 and α = 1 .
5. Asignificant fraction of the high-energy photons are emit-ted within the “escape cone” θ < ∼ x esc / Γ where τ γγ < ∼ t obs ∼ − s (Fig-ure 7) and leads to the overall two-hump appearance ofthe high-energy spectrum.The high-energy spectrum in Figure 7 cuts off at en-ergy E max which increases with time and reaches theTeV band at t obs ∼ γ inj ∼ atlate stages of the flash when Z ± is reduced. The ther-mal particles produce IC photons of maximum energy E max ∼ Γ γ inj m e c < ∼
10 TeV. Emission above E max ispossible if the post-shock plasma contains a nonther-mal component accelerated at the shock; it would not,however, make a large contribution to the flash energyand would not significantly change the GeV emission ob-served by LAT.Figure 7 also shows the prompt emission ob-served by Gamma-ray Burst Monitor (GBM) below100 MeV. Recent analysis of the GBM and LAT datashows clear evidence for two separate spectral com-ponents that dominate below and above 100 MeV(Fermi-LAT Collaboration 2013, Guiriec et al. in prepa-ration). This agrees with the theoretical expectation that Fig. 8.—
Flash spectrum at t obs ∼ t obs ∼
70 s (lower panel) for ε e = 1 and 0.1. Solid curves showthe model with no magnetic field ( ε B = 0) and dotted curves showthe model with ε B = 10 − . Photon energy has been corrected forcosmological redshift z = 4 . E = (1 + z ) E obs . Vertical dashedline shows the lower boundary of LAT data (1 + z ) ×
100 MeV. the prompt MeV emission comes from a separate (inter-nal) source at small radii. Note that its spectrum mayextend to high energies and contribute to the flux de-tected by LAT, mixing with the IC emission from theexternal shock wave. However, the external shock is thestronger source in the GeV band, especially at late timeswhen the prompt emission declines.As seen in Figure 7, the predicted GeV emissionfrom the pair-loaded external shock starts very soft andquickly hardens as the flash reaches its peak. The av-erage spectral slope between (1 + z ) ×
100 MeV and(1 + z ) × ∼ − ε e = 1 and ε e = 0 .
1. The value of ε e makes asignificant difference for the spectrum at high energies E ≫ ε e implies a higher γ inj .5.4. Analytical estimates for R p , Γ , and A The radius and Lorentz factor of the blast wave can bequickly reconstructed from the observed GeV flash usingthe following estimates. The estimates approximatelyagree with our numerical results for GRB 080916C, showhow the results depend on the GRB parameters, and maybe applied to other GRBs with a detected GeV flash.Let us neglect the variability of the MeV prompt ra-2 A. M. Beloborodov, R. Hasco¨et, I. Vurmdiation; then the shock wave is exposed to radiation ofconstant luminosity L GRB and constant spectrum. Theradiation front ahead of the forward shock has the energygiven by Equation (25), which may be written as E GRB = L GRB t FS . (41)Here t FS = ̟ FS c ≈ R c (42)is the time coordinate of the forward shock, which isrelated to the arrival time of the GeV photons by t obs ∼ (1 + z ) R Γ c ≈ z ) t FS . (43)The main parameter ξ that governs pair loading and pre-acceleration of the external medium (Equation (21)) is ξ ≈ L t FS R − ≈ L (cid:18) t obs z (cid:19) − (cid:18) Γ500 (cid:19) − , (44)where L = L GRB / erg s − .The value of ξ at the peak of the flash can be estimatedusing the approximate relation (see B02 and Figure 1), γ ≈ (cid:18) ξξ acc (cid:19) , ξ acc ≈ − , (45)valid in the region of main interest, 1 < ξ/ξ acc < E IC ∼ γ inj ∼ E IC /E pk ) / (Sec-tion 2), which corresponds toΓ γ ∼ , (46)yielding ξ ≈ ξ acc (cid:18) Γ500 (cid:19) / . (47)Combining Equations (44) and (47), we obtain the radiusand Lorentz factor of the blast wave when it emits thepeak of the GeV flash ( t obs = T p ), R p ≈ L / (cid:18) T p (1 + z ) s (cid:19) / cm , (48)Γ( R p ) ≈ L / (cid:18) T p (1 + z ) s (cid:19) − / , (49)where T p is the observed arrival time of the peak.Using the obtained Γ and Equation (17) one can esti-mate the parameter A = ˙ M / πw of the wind medium, A ≈ L ej γ πc Γ (cid:18) Z ± m e µ e m p (cid:19) − ≈ − ε rad ε rad L / (cid:18) T p (1 + z ) s (cid:19) / g cm − . (50)These estimates assumed that the reverse shock is ultra-relativistic (Γ ej ≫ Γ); it is straightforward to obtain amore general estimate of A using Equation (16) insteadof Equation (17). PHOTON-PHOTON ABSORPTIONThe target photons providing opacity for the GeV flashcan be divided into two categories: (1) the almost per-fectly collimated prompt radiation (Section 6.1), and (2)scattered prompt photons (Sections 6.2 and 6.3). Thedensity of scattered radiation is relatively small — theexternal medium and the blast wave are optically thineven after e ± loading, — however, it may provide aninteresting contribution to the γ - γ opacity, because thescattered photons have larger angles and higher energies.6.1. Unscattered prompt radiation
Let us first evaluate the γ - γ opacity provided by theunscattered prompt radiation, which we assume to beperfectly collimated at radii where the GeV flash is pro-duced. The absorption optical depth seen by a high-energy photon of dimensionless energy ǫ = E/m e c prop-agating at some angle θ along its path s is given by τ γγ ( ǫ ) = Z Z F ǫ ( ǫ t ) m e c σ γγ ( ǫ cm ) (1 − µ ) d ln ǫ t ds, (51)where σ γγ is the cross section for reaction γ + γ → e + + e − in the center-of-momentum frame of the two collidingphotons, ǫ cm is the photon energy in this frame, and µ = cos θ describes the angle between the two photons inthe lab frame. The spectral flux of the target photons is F ǫ ( ǫ t ) = L ǫ ( ǫ t )4 πR , (52)where L ǫ ( ǫ t ) = L pk ǫ (cid:18) ǫ t ǫ pk (cid:19) − α (53)is the spectral luminosity of the prompt radiation and ǫ pk is the peak/break energy of the prompt GRB spectrum.For a broken power-law spectrum with indices α and α , L pk ǫ is related to the bolometric luminosity L GRB by L GRB = ( α − α )(1 − α )( α − L pk ǫ ǫ pk . (54)Using the relation 2 ǫ = ǫǫ t (1 − µ ) to express ǫ t interms of ǫ cm and evaluating the integral over ǫ cm , onefinds τ γγ = ψ σ T Z L pk ǫ πm e c R (cid:18) ǫ thr ǫ pk (cid:19) − α (1 − µ ) ds, (55)where ǫ thr = 2 ǫ (1 − µ ) , (56)and the numerical factor ψ ( α ) can be approximated as(Svensson 1987), ψ ( α ) ≈ α ) / , (57)which is accurate to within 0.3% in the range 0 < α <
6. The quantity ψσ T has the meaning of effective crosssection for absorption. The spectral slope α = α if ǫ thr ≪ ǫ pk and α = α if ǫ thr > ǫ pk .Consider a high-energy photon generated by IC scat-tering at radius R IC with angle θ IC relative to the radialeV emission in gamma-ray bursts 13direction. As the photon propagates, its angle changesaccording to Equation (29). This change is related tothe path element ds by ds = − R dθ/ sin θ , and one canexpress the integral in Equation (55) as an integral over0 < θ < θ IC , which yields (in the small-angle approxima-tion θ IC ≪ τ γγ ( ǫ, θ IC ) = σ T L pk ǫ πm e c ψ ( α )2 α +1 (2 α + 3) ( ǫ pk ǫ ) α θ α +2IC R IC . (58)Note that τ γγ → θ IC →
0. The condition τ γγ < θ IC < θ esc ( ǫ ) for IC photons of agiven energy ǫ .It is useful to rewrite Equation (58) as τ γγ = σ T L pk ǫ θ πm e c R IC ψ ( α )(2 α + 3) (cid:18) ǫ thr ǫ pk (cid:19) − α , (59)where ǫ thr ≈ ǫθ ) − is the threshold energy evaluatedat the emission radius R IC . High-energy photons pro-duced by the plasma moving with a bulk Lorentz factorΓ have the characteristic beaming angle θ IC ∼ Γ − (orsomewhat larger, because of the anisotropy effect dis-cussed after Equation (32)). It is convenient to describethe photon angle using the variable x = Γ θ IC , which iscomparable to unity for a typical IC photon. Then theoptical depth may be written as τ γγ ≈ ξ x L pk ǫ L GRB ψ ( α )(2 α + 3) (cid:18) ǫ thr ǫ pk (cid:19) − α . (60)Here ξ is the main physical parameter of the promptradiation front given by Equation (21), and we estimated E GRB ahead of the forward shock as E GRB ≈ L GRB t FS with t FS ≈ R/ c . The peak of the GeV flash occurswhere ξ ∼
300 (Section 5.4).IC photons of energy ǫ < ǫ = 4Γ /ǫ pk x interact withprompt photons ǫ t > ǫ thr > ǫ pk and α = α ; this gives τ γγ <
1. Absorption is significant for IC photons with ǫ > ǫ . These photons can interact with the low-energypart of the prompt spectrum ǫ t < ǫ pk where α = α .Note that α ≈ −
1) is typical for GRBs,including GRB 080916C. Then τ γγ weakly varies with ǫ for ǫ > ǫ , and its value is close to unity for ξ ∼ α < τ γγ is maximum at ǫ = ǫ anddecreases at higher energies. For GRBs with α > τ γγ continues to grow with ǫ > ǫ and becomes well aboveunity. Then the size of the escape cone θ esc decreases asa power-law with ǫ , and so does the fraction of escapingphotons. This implies a steeper spectrum where τ γγ ≫ Prompt radiation scattered ahead of the forwardshock
High-energy photons from the forward shock have topass through the prompt radiation that has been scat-tered ahead of the shock by the pair-loaded and pre-accelerated ambient medium. The specific intensity of the scattered radiation can be expressed as (B02) I sc ( ǫ sc , µ sc , ̟ ) ǫ sc = Z ̟ d̟ ′ − µ sc F ǫ ( ǫ ) ǫ Z ± n π dσdµ sc dǫ dǫ sc . (61)Here F ǫ is the spectral flux of prompt radiation, ǫ is theprompt photon energy (before scattering), µ sc = cos θ sc describes the scattering angle, and ǫ sc is the photon en-ergy after scattering; Z ± ( ̟ ′ ) is the pair loading factor,and n is the external electron density before e ± load-ing. The integral is taken over the Lagrangian coordi-nate ̟ = ct − R that measures the distance inside theprompt radiation front; d̟/ (1 − µ sc ) is the elementarypath length along the scattered photon trajectory in thelab frame.We are interested in the optical depth τ γγ created bythe scattered radiation, as seen by a high-energy photonof energy ǫ emitted by the shock wave. The photon hasan angle θ ∼ Γ − , which is much smaller than the typicalangles of the target photons θ sc ∼ γ − (where γ = (1 − β ) − / is the Lorentz factor of the pair-loaded mediumaccelerated by the radiation front). Therefore, here thehigh-energy IC photon may be approximated as perfectlycollimated in the radial direction, θ = 0. Then, τ γγ ( ǫ ) = 2 πR Z Z I sc ( ǫ sc , µ sc ) ǫ sc m e c σ γγ ( ǫ cm )(1 − µ sc ) dµ sc dǫ sc . (62)Following B02, we will make the simplifying assumptionthat the prompt radiation is scattered at 90 ◦ in the localrest frame of the medium (which corresponds to µ sc = β in the lab frame), and approximate the Thomson cross-section as dσ/dµ sc ≈ σ T δ ( µ sc − β ). Then we obtain, τ γγ ( ǫ ) = ψ σ T R n L pk ǫ L GRB Z ξ dξ ′ Z ± ( ξ ′ ) (cid:18) ǫ thr ǫ pk (cid:19) − α , (63)where ǫ thr = 2(1 + β ) /ǫ (1 − β ) is the pair-productionthreshold energy for the prompt photon (before scatter-ing) for interaction with a high-energy photon ǫ , and ξ = ̟σ T L GRB / πm e c R .The optical depth given by Equation (63) can beunderstood as follows. The column density of elec-trons exposed to the prompt radiation is ∼ R Z ± n (accounting for pair loading). Each electron at coor-dinate ̟ in the radiation front has scattered approx-imately ξ/ǫ pk photons, and hence the column densityof scattered photons is ∼ n R Z ± ( ξ ) ξ/ǫ pk . A fraction( ǫ pk L pk ǫ /L GRB )( ǫ thr /ǫ pk ) − α of these photons are near thethreshold for pair production, where the average γ - γ cross section is large, σ γγ ∼ ψ σ T .Consider a simplified analytical model of the radiationfront in the region where 1 < γ .
30 (B02), γ = (cid:18) ξξ acc (cid:19) , Z ± = Z acc (cid:18) ξξ acc (cid:19) , (64)where ξ acc ≈ −
200 (the more accurate front structureis shown in Figure 1). Then one can evaluate the integral The factor dǫ /dǫ sc is missing in Equation (4) in B02. Z ± dξ = Z acc ξ acc dγ/ ǫ thr ≈ γ /ǫ , which yields τ γγ ( ǫ ) = ψ ( α ) Z acc ξ acc − α )2 α L pk ǫ L GRB ( ǫ pk ǫ ) α γ − α τ , (65)where τ = σ T R n ( R ) is the Thomson optical depththrough the progenitor wind and γ is the pre-accelerationLorentz factor at the location of the forward shock.The power-law segment of the Band spectrum that pro-vides the dominant contribution to τ γγ is determined bycomparing ǫ pk and ǫ thr ; the lower-energy segment domi-nates if ǫ > γ ǫ pk . (66)Most of the GeV flash is emitted at radii where the pre-acceleration Lorentz factor γ < ∼
10, and the condition(66) is satisfied for ǫ > ∼ . Then α = α in Equa-tion (65). For the typical α ≈
0, one finds that τ γγ athigh energies does not depend on ǫ and its value is small, τ γγ <
1, for τ ∼ × − expected for the progenitorwind at the flash radius R p ∼ cm. In particular,for α = 0 and α = 1 . L pk ǫ /L GRB = (3 ǫ pk ) − and τ γγ ∼ ǫ − γ τ < . (67)Our conclusion that the scattered radiation providesa small τ γγ < τ ∝ R − . Less luminous bursts alsohave smaller ǫ pk . Second, the estimates in Section 6.3 inB02 confused the photon index with the energy index ofthe prompt GRB spectrum, leading to an overestimationof τ γγ .6.3. Prompt radiation scattered behind the forwardshock
The plasma immediately behind the forward shock hasan ultra-relativistic temperature and here scattering pro-duces high-energy IC photons. The high-energy photonsmay interact between themselves. An exact calculationof this “self-absorption” of the GeV flash would require afull nonlinear simulation of radiative transfer. A simpleestimate suggests that the self-absorption effect is notstrong in GRB 080916C. The isotropic equivalent of thephoton number in the flash is N GeV ∼ , and thecolumn density of GeV photons is ∼ N GeV / πR . Thisgives an upper bound on the absorption optical depthprovided by the GeV photons, τ γγ < ∼ σ γγ N GeV πR ≈ . (cid:18) N GeV (cid:19) R − , (68)where we estimated the effective cross section σ γγ = ψ σ T and assumed the spectral index α ∼ ∼ ψ ( α ) ∼ . τ ± = Z ± τ (pair annihilation is negligi-ble). It scatters the prompt photons with a moderate change in photon energy and a typical scattering angle θ sc ∼ Γ − . Some of these photons may overtake theGeV photons emitted immediately behind the shock andcontribute to the absorption opacity seen by the GeVphotons. Their contribution is small compared to τ γγ of photons scattered ahead of the shock (Section 6.2).The numbers of photons scattered ahead and behind theshock are comparable, however the angles of photonsscattered ahead of the shock are much larger, makingthem more important targets for photon-photon absorp-tion. 6.4. Summary
The unscatterred collimated prompt radiation domi-nates the γ - γ opacity seen by the GeV photons. The cor-responding optical depth τ γγ is evaluated in Section 6.1;it is shown to be small at energies E ≪
30 GeV andcomparable to unity at higher energies. Prompt photonsscattered ahead or behind the shock provide an addi-tional small contribution to τ γγ , which may be neglected,at least for the GeV flash in GRB 080916C. SYNCHROTRON EMISSIONThe presence of a magnetic field in the blast wave canhave three observational effects. (1) If the field is strong,synchrotron losses of the shocked plasma can competewith its IC cooling by the prompt radiation; this wouldweaken the GeV flash. (2) Synchrotron losses give emis-sion in softer bands, e.g. optical or X-rays, providingan additional test for the pair-dominated flash mecha-nism. (3) Synchrotron photons may become the maintargets for IC scattering by the high-energy electrons inthe blast wave, which can affect the observed light curveand spectrum of high-energy emission.7.1.
Cooling rate and the characteristic photon energy
The competition between synchrotron cooling andCompton cooling by the prompt radiation was discussedby Beloborodov (2005b). The two contributions to thecooling rate of isotropic electrons with a thermal Lorentzfactor γ e ≫ E syn = − σ T U ′ B cγ e , ˙ E IC ≈ − σ T U ′ T cγ e , (69)where U ′ B is the magnetic energy density, and U ′ T is theenergy density in the prompt photons of energy E < E KN (Equation (39)) which can be scattered with approxi-mately Thomson cross section; U ′ B and U ′ T are mea-sured in the fluid frame. We include only the unscat-tered prompt radiation in U ′ T , assuming that it domi-nates Compton cooling of the blast wave; the density ofsynchrotron radiation from the blast wave itself is as-sumed to be relatively small. Then, U ′ T ≈ f T U ′ , f T ≈ E KN ≫ E pk (cid:18) E KN E pk (cid:19) − α +1 E KN < E pk (70)where U ′ = L GRB πc R Γ (71)is the energy density of the prompt radiation in the fluidframe, and α is the spectral index of radiation at photonenergies E < E pk .eV emission in gamma-ray bursts 15The magnetic energy density behind the shock may beexpressed in the standard form using the parameter ε B , U ′ B = 3 ε B P f = ε B ρc Γ γ (1 + β ) , (72)where ρ is the mass density of the external medium and γ = (1 − β ) − / is its pre-acceleration Lorentz factor;we neglected the increase in ρ due to e ± pairs loadedahead of the shock. We focus here on the main phaseof the GeV flash before the reverse shock has crossedthe ejecta. Then Equation (17) may be used to obtainanother expression for U ′ B , U ′ B ≈ ε B L ej π c R Γ . (73)The ratio of synchrotron and Compton cooling rates isthen given by ˙ E syn ˙ E IC ≈ U ′ B U ′ T ≈ ε B L ej f T L GRB . (74)The numerical factor f T is comparable to unity at thepeak of the GeV flash, when the forward shock heatsthe plasma to γ inj ∼ . After the peak, γ inj increases,however the flash light curve shown in Figure 6 is stilldominated by particles cooled to γ e ∼ , with f T ∼ E s ≈ . γ e ~ eB ′ m e c , (75)where B ′ = (8 πU ′ B ) / is the magnetic field measured inthe fluid frame. Using Equation (73) one obtains, E s ∼ ε / (cid:16) γ e (cid:17) (cid:18) L ej erg s − (cid:19) / R − eV . (76)Most of the synchrotron power is emitted by parti-cles with γ e ∼ γ inj . As the blast wave expands from R ∼ cm to 10 cm, γ inj ( R ) evolves from low val-ues ∼ ∼ (at the peak of the GeV flash) to ∼ − , see Equation (24) and Figure 5. As a result, E s ( γ inj ) evolves by a huge factor ∼ , and hence theblast wave must produce broad-band synchrotron radia-tion. The emitted synchrotron power may be estimatedusing Equation (74) with f T that corresponds to γ inj .Moderately high ε B > ∼ − would imply strong syn-chrotron emission in the hard X-ray band. It can eas-ily conflict with the observed radiation spectrum, whichcan be used to infer an upper limit ε maxB ∼ − forGRB 080916C. 7.2. Optical flash
If one is interested in radiation in a fixed spectral band,e.g. optical E ∼ z ) eV, the observed emission willbe dominated by particles that have cooled behind theshock to Lorentz factor γ e = γ opt such that E s ( γ opt ) ∼ z ) eV. From Equation (76) one finds γ opt ∼ ( ε B / − ) − / L − / , R / (1 + z ) / . (77) A more accurate expression for γ opt may be obtainedfrom Equation (75) using Equation (72), γ opt ≈ Γ (cid:20) γ (1 + β ) ε B ρc (cid:21) / (1 + z ) / . (78)In the blast wave with pure thermal heating, opticalemission remains negligible until γ inj ( R ) exceeds γ opt ;the optical light curve is expected to reach its peak atthis point. This happens soon after the peak of the GeVflash.The subsequent decay of the optical flash can be de-scribed using the following estimate for the optical lumi-nosity, L opt ∼ EL E ∼ dN ± dt Γ γ opt m e c f syn , (79)where t = (1 + z ) − t obs , N ± is the number of elec-trons/positrons cooling behind the shock, and f syn = ˙ E syn ( γ opt )˙ E IC ( γ opt ) + ˙ E syn ( γ opt ) ≈ U ′ B U ′ T ( γ opt ) . (80)Equation (79) states that each particle emits in the op-tical band a fraction ∼ f syn / ∼ Γ γ opt m e c , as γ e decreases from γ opt to γ opt / ∼ Γ γ opt m e c / f syn ≪ L opt ∼ R Z ± [ γ (1 + β ) ε B A ] / × L ej L GRB ǫ pk (1 + z ) erg s − . (81)Here we used dN ± /dt ∼ Z ± (4 πR ρ/µ e m p t ) and t ∼ R/c Γ . Equation (81) shows that the decay of the op-tical flash is controlled by the evolution of the factor Z ± R [ γ (1+ β )] / with time t . This evolution is fast; whenapproximated by a power law t − a its slope is a ∼ − ε B ∼ − .Its peak occurs where Z ± ∼ and can reach an opticalluminosity L opt ∼ erg s − .In summary, the peak luminosity of the optical flash isachieved when γ inj exceeds γ opt . This typically happensat t obs ∼ z ) s. The optical flash can be extremelybright, but it quickly decays. We find that its luminositydrops by a factor of 10 − as t obs grows by a factor of 10,mainly because of the decreasing pair loading factor Z ± .At later times the prompt radiation decouples from theblast wave and the Compton cooling ends, which impliesthe end of the fast decay; then the optical flash shouldevolve to normal optical afterglow.Note that the e ± pairs collected at R < ∼ cm areCompton cooled to a low temperature and do not con-tribute to the afterglow emission at late times. This isin contrast to explosion models where the prompt radi-ation quickly decouples from the blast wave and Comp-ton cooling is inefficient; in this case the blast wavewould carry slowly cooling pairs and the synchrotronafterglow would have a long “memory” of pair loading(Beloborodov 2005a).6 A. M. Beloborodov, R. Hasco¨et, I. Vurm IMPACT OF THE GEV FLASH ON THEEXTERNAL MEDIUMOur transfer simulations described in Section 5 showthat some of the produced high-energy photons do not es-cape — they are absorbed by the prompt radiation beamand convert to e ± pairs. Most of the conversion eventsoccur behind the forward shock and join the shockedplasma moving with Lorentz factor Γ, however a smallfraction convert ahead of the shock and join the exter-nal medium, which moves with a much smaller Lorentzfactor γ . These rare events create particles of very highenergies (GeV-TeV) in the external medium, depositingtheir energy and momentum. Thus, the GeV flash it-self creates additional pre-heating and pre-accelerationof the external medium, which was not taken into ac-count in our model of the radiation front in Section 3.1.We now estimate this effect and its implications.8.1. Fraction of the flash power deposited ahead of theshock
First, let us roughly estimate the fraction of the flashpower that converts to e ± pairs ahead of the shock wave.Only photons with sufficiently small angles can overtakethe forward shock, θ < θ max = Γ − . (82)For the simplest estimate, we picture the flash source asan infinitesimally thin shell behind the shock (the fast-cooling limit) and assume that only photons emitted with θ < θ max have a chance to convert ahead of the shock.The absorption optical depth τ γγ seen by these photonsis given by Equation (58); it increases with θ and is max-imum at θ max . The deposited power ahead of the shockmay be written as L ± = ζ Z τ γγ ( ǫ, θ max ) L ǫ dǫ, (83)where τ γγ ( ǫ, θ max ) ≈ ψ ( α )2 α +1 (2 α + 3) σ T L pk ǫ ( ǫ pk ǫ ) α πm e c R Γ α +2FS ,L ǫ is the flash spectrum, and ζ = 0 . − . α , is determined as follows. The maintarget photons contributing to τ γγ have energies ǫ t ∼ ǫ thr = 4 ǫ (1 − cos θ ) ≈ ǫ , (84)which should be compared with ǫ pk ∼
10. This gives, α = (cid:26) α , ǫ < ǫ − Γ α , ǫ > ǫ − Γ (85)where the characteristic ǫ = 8Γ /ǫ pk corresponds tophoton energy ǫ m e c ∼ GeV. The flash spectrumextends above ǫ after the peak time T p , when γ inj ex-ceeds ∼ Γ FS ; then photons with ǫ > ǫ make the maincontribution to the integral in Equation (83), and τ γγ should be evaluated with α = α . In particular, for α = 0 we obtain L ± L flash ∼ . ζ σ T L pk ǫ πm e c R Γ , (86)where we assumed that a large fraction of the flash lumi-nosity L flash is emitted above ∼ GeV; this assump-tion is satisfied in the self-consistent model, as we showbelow. 8.2.
Pre-heating and pre-acceleration
The injection of power L ± into the external mediumcan be described as inelastic collision which heats andaccelerates the medium. Consider an external mass shell dm = 4 πR ρ dR = 4 πA dR. (87)It first interacts with the prompt radiation and then it isexposed to the high-energy flash photons, which depositenergy, dE ± ∼ L ± dR c . (88)This energy is deposited in the form of ultra-relativistic e ± pairs, which are expected to immediately share theirmomentum dE ± /c with the medium through collectiveprocesses (B02). The GeV flash accelerates the mediumto a high Lorentz factor γ ′ ≫ G ≡ dE ± dmc = L ± πc Γ A ≫ . (89)The deposited energy dE ± is shared between the bulkkinetic energy of the accelerated medium and its inter-nal energy (i.e. heat). The ultra-relativistic pairs canscatter the prompt radiation ahead of the forward shock;however, since the pairs are isotropic in the fluid frame,the produced high-energy photons have large angles andquickly convert to e ± pairs, which join the medium. For simplicity, let us consider radii where the pre-acceleration by the prompt radiation is not significant(
R > × cm, see Figure 3), so that we can isolatethe effect of the GeV flash. We can evaluate the Lorentzfactor gained by the shell, γ ′ , and its new rest-mass dm ′ (which includes the deposited heat) from the energy andmomentum conservation laws, dm + dE ± c = γ ′ dm ′ , (90) dE ± c = γ ′ β ′ dm ′ . (91)This gives, γ heat ≡ dm ′ dm = (2 G + 1) / , (92) γ ′ β ′ = Gγ heat . (93) This cascade in the external medium has a moderate effecton pair multiplicity Z ± . The high-energy particles injected by theflash radiation are relatively close to the forward shock and havetime for a moderate number of scatterings before they are sweptby the shock. A dedicated numerical simulation will be needed toquantify this effect. eV emission in gamma-ray bursts 17Also note the relation, γ heat = γ ′ (1 + β ′ ) . (94)It is easy to see that G ≫ γ heat ≫ γ ′ ≫
1. Indeed, substituting Equa-tion (86) into Equation (89) and using the simple esti-mate for the blast-wave Lorentz factor Γ ∼ L ej / πc A (see Equation (17) and Equation (96) below), one obtains G ∼ . ζσ T L pk ǫ πm e c R Γ Γ L flash L ej ∼ . ζξ Γ Γ L pk ǫ L GRB L flash L ej (95)which gives a typical G ∼ − . The value of G is strongly reduced at smaller radii where the promptradiation pre-accelerates the external medium to γ ≫ G ≫ γ inj > and γ < Effect on the blast wave Lorentz factor
We now estimate the effect of pre-acceleration and pre-heating by the flash radiation on the blast-wave Lorentzfactor Γ. Similar to Section 2.3 we consider sufficientlyearly times ( t obs < T GRB ) and use the pressure balancebetween the forward and reverse shock, P f ∼ P r , for arough estimate. On the other hand, to isolate the ef-fect of the flash, we consider late enough times when theprompt radiation does not significantly pre-accelerate themedium, γ ≈
1. Then Equation (17), with γ replaced by γ ′ and Z ± ≪ µ e m p /m e , givesΓ ≈ L ej πc A . (96)The result is the same as if there were no effect of theflash on the external medium — the terms γ ′ (1 + β ′ )and γ heat cancel (see Equation (94)). The enhancementof the shock pressure due to the increased fluid mass bythe factor of γ heat is compensated by the reduction ofpressure due to the fluid pre-acceleration to γ ′ .We conclude that the blast-wave dynamics should notbe strongly changed by the flash impact on the exter-nal medium. More detailed calculations will, however,be needed at smaller radii where the effect of the flashradiation on the external medium interferes with thatof the prompt radiation, increasing the pre-accelerationLorentz factor from γ ≫ γ ′ .8.4. Effect on radiative efficiency
The deposited heat implies a huge energy per electronahead of the shock, γ th m e c . In the region of interest,where G ≫ µ e m p /m e ≫ Z ± ≫
1, one finds γ th ≈ γ heat µ e m p Z ± m e ≫ . (97)When the hot fluid passes through the shock, the ther-mal Lorentz factor of particles increases to γ inj given byEquation (24). Using Equation (94), one obtains γ inj ≈ Γ µ e m p Z ± m e . (98)This relation shows that all the energy available for dis-sipation in the blast wave ( Z ± γ inj m e c /µ e m p ≈ Γ c per unit external mass) has been converted into the heat ofpairs behind the shock. It implies the effective ε e = 1, re-gardless of the efficiency of energy transfer from the ionsto pairs at the shock front. The high-energy particles be-hind the shock radiate most of their energy and produceradiation beamed within angle θ ∼ Γ − . Our transfersimulations in Section 5 and analysis in Section 6 showthat a large fraction of this radiation avoids γ - γ absorp-tion and escapes, leading to a high radiative efficiency ofthe blast wave. DISCUSSION9.1.
Mechanism of the GeV flash
The external shock of the GRB explosion in a denseprogenitor wind generates a bright GeV flash due to in-verse Compton (IC) cooling of the shock-heated plasma.We showed that scattering of the prompt MeV radia-tion streaming through the external blast wave is the keymechanism during the main phase of the flash, shapingits peak and early decay.Most MeV photons stream without any interaction,however a small fraction get scattered, and many of thescattered photons (in particular those scattered in the ex-ternal medium ahead of the forward shock) collide withother MeV photons and convert to e ± pairs. This leads toa dramatic enhancement of electron density in the blastwave, by a factor of Z ± ∼ at radii R ∼ cm,and hence a dramatic increase in the number of promptphotons scattered in the blast wave. In addition, theGRB radiation pressure significantly pre-accelerates theexternal medium ahead of the forward shock. This ef-fect reduces the strength of the shock and regulates thespectrum of its inverse-Compton radiation.We have examined the inverse-Compton pair-dominated flash using a direct radiative transfersimulation. As an example, we calculated the flashexpected from GRB 080916C, one of the few brightestGRBs well observed by LAT. When the reverse shock isrelativistic, the dynamics and emission of the forwardshock is indifferent to the precise Lorentz factor of theejecta Γ ej ; only the ejecta power L ej is important. L ej can be estimated from the observed GRB luminosityassuming a plausible radiative efficiency of the promptemission ε rad <
1. The main remaining parameter ofthe blast wave is the density of the external mediumwhich depends on the progenitor mass-loss rate ˙ M .We find that ˙ M ≈ − M ⊙ yr − , which is typicalfor Wolf-Rayet stars, gives a GeV flash in strikingagreement with observations (Figure 6). Our resultsexplain the previously puzzling features of the GeVlight curve including the early peak and the long decay.The light curve is shaped by the pre-acceleration andpair-loading effects; the peak is reached where γ ∼ Z ± ∼ , when most of the shock energy is emittedin IC photons of energy E IC ∼ (Γ /γ ) MeV, in the GeVband.The predicted spectrum in the GeV band has the pho-ton index ∼ − E ≫
10 GeV, the spectrum is af-fected by γ - γ absorption. However, absorption does notstrongly suppress the emission even at very high ener-gies E >
100 GeV. Our analysis in Section 6 shows that8 A. M. Beloborodov, R. Hasco¨et, I. Vurmthe main source of γ - γ opacity seen by the GeV photonsis the unscattered prompt radiation; the correspondingoptical depth τ γγ is given by Equation (58), which isself-regulated to a moderate value comparable to unity.As a result, we predict escaping gamma-rays at energies E ≫
10 GeV, up to the TeV range, where the flash canbe detected by the atmospheric Cherenkov telescopes.When comparing the model with the LAT data we as-sumed that all observed GeV emission comes from theblast wave. In fact, at early times, the high-energy tailof the prompt emission may contribute to the observedGeV light curve near the peak the flash. Variability de-tected at early times provides evidence for such a con-tribution. After subtraction of the prompt emission, thetrue light curve of the GeV flash may have a somewhatlower peak, perhaps by a factor ∼
2. Then our best-fit model will need to be revised, resulting in moderatechanges in A , R p , and Γ.Given the similar light curves of the GeV flashesin many GRBs, it appears likely that all of themare produced by the same mechanism. This includesGRB 090510 that was attributed to the short GRB class,which is usually associated with a different type of pro-genitors. It could be that GRB 090510 is an “impostor”and its progenitor had a significant wind before the explo-sion. A wind medium was also suggested by Panaitescu(2011) based on the afterglow properties of GRB 090510.Our preliminary analysis of the GeV flash in GRB 090510confirms the requirement of a high external density at R ∼ cm, suggesting a wind medium. However, theformal constraints on the density profile in this case arenot tight and will be investigated in a future work. Incontrast, the IC flash in GRB 080916C requires the den-sity profile to be close to R − ; a uniform medium wouldgive a GeV light curve much flatter than observed.9.2. Approximations used and possible extensions
From a technical point of view, this paper examinedthe coupled problem of radiative transfer and blast-wavedynamics in a wind medium. The problem can be solvedexactly from first principles, although in this paper weused some approximations. Below we summarize our ap-proximations, discuss the accuracy of our results, andoutline directions for future work.(1) We conservatively assumed that the postshockplasma is dominated by the thermal e ± population. Thisassumption is broadly consistent with observations of col-lisionless shocks in the solar system and supernovae, aswell as numerical simulations of relativistic shocks (e.g.Sironi & Spitkovsky 2009). Our calculations made noadditional assumptions concerning particle accelerationin the shock wave. The likely presence of a small num-ber of nonthermal particles would weakly change the pre-dicted light curve shown in Figure 6 (as discussed inSection 5) except possibly at the earliest stages, beforethe peak of the flash. We used the simplest possibleapproximation where the shocked particles acquire themono-energetic distribution δ ( γ e − γ inj ) with γ inj givenby Equation (24). Detailed future models can use a morerealistic distribution, e.g. Maxwellian, and include non-thermal particles.(2) Our calculations had to invoke one phenomeno-logical parameter ε e . The shock wave heats ions andelectrons/positrons, and ε e is the fraction of the ion en- ergy that is immediately (due to collective plasma effects)passed to e ± . This parameter is not relevant at the peakof the flash, however its value can affect the decay afterthe peak (see Figure 6). Future particle-in-cell simula-tions of pair-loaded shocks may provide an estimate for ε e . In Section 8, we showed that the blast wave after thepeak of the GeV flash enters a peculiar radiative regimewhich can be described as emission with effective ε e = 1.For comparison, Figure 6 also presents the GeV flashesobtained with ε e = 0 and 0 .
1; it shows that variations in ε e would have a modest effect on the light curve. Com-parison with the LAT data in Figure 6 gives no preferenceto any ε e at times t obs <
40 s. At later times, the datafavors ε e > .
1. The value of ε e makes a significant differ-ence for the flash spectrum at high energies E ≫ R > R , where R isgiven by Equation (36). In reality, some target photonsare available for the blast wave even at R > R (theyare provided by a weaker/softer tail of the prompt ra-diation and by the synchrotron emission from the blastwave). The high-energy emission will continue as long asthe target radiation field is able to drain an interestingfraction of the shock energy via Compton cooling. Thus,the observed light curve of the GeV flash can extend tomuch longer observational times than shown in Figure 6.As the radiation density decreases behind the promptradiation front, the transition from fast to slow coolingregime will affect the GeV light curve.(4) We used a simplified “mechanical” model for theblast-wave dynamics, which treats the shocked gas as onehot body. It is equivalent to assuming a flat profile of thefluid Lorentz factor behind the forward shock. Futuredetailed models of GeV flashes will be based on full hy-drodynamical simulations. We found that the light curveof the GeV flash near its maximum is quite sensitive tosmall refinements in Γ( R ), even when these refinementsare at ∼
10% level. Thus, careful hydrodynamical sim-ulations will help improve the accuracy of the explosionreconstruction from the observed GeV emission.(5) We calculated in detail how the scattering of GRBradiation and pair creation in the external medium im-pacts the forward shock. However, we did not studythe dynamical effect of pair creation behind the shock.Many of the photons scattered in the external mediumpropagate into the blast wave and the unshocked ejecta,and create pairs there with a rate similar to that aheadof the blast wave. As these pairs are picked up bythe relativistic flow, they exert a significant drag andheat it. Our preliminary estimates suggest that this ef-fect is important for the blast-wave dynamics at earlytimes, and will reduce the Lorentz factor Γ at small radii R = 10 − cm. It can strongly affect the rise ofthe GeV light curve. We defer the full calculation to afuture work; it will also include the “rocket effect” due toanisotropy of IC emission, which will give a push to theblast wave. All these effects will likely change the rise tothe peak and possibly the peak itself. Therefore, we onlytrust our best-fit value of the wind density parameter A within a factor of ∼ e ± ahead of the blast wave and deposithuge energy and momentum. Thus, the full non-linearproblem must include the impact of the GeV flash on theexternal medium, not only the impact of the prompt ra-diation. Our analysis of this effect in Section 8 suggeststhat it does not significantly change the ram pressure inthe forward shock. However, it has another importantimplication: it leads to the effective ε e = 1 and enforcesthe high radiative efficiency of the blast wave. Detailednonlinear simulations of this effect are deferred to a fu-ture work.Such simulations will also allow one to explore the fol-lowing possibility. The high-energy pairs created in theexternal medium by the IC flash photons may not becompletely cooled before the shock reaches them andboosts their energy even more, producing extremely en-ergetic particles. These particles in turn produce moreenergetic photons, some of which can again convert aheadof the shock, injecting new very-high-energy pairs. Thus,the following cycle is possible for a small number of par-ticles/photons: shock-heating → emission of high-energyphotons → photon conversion to e ± ahead of the shock → shock heating. As a result, ultra-high-energy particlescould be generated. This bootstrap mechanism is simi-lar to “photon breeding” proposed by Stern & Poutanen(2006). 9.3. Future observational tests
The predicted peak time of the GeV flash, T p , de-pends on the density parameter A (Section 5.4). Al-though many bursts detected by LAT have T p ≪ T GRB ,some may have T p ∼ T GRB . It will be useful to studysuch bursts for the following reason. Our calculationspredict that the flash peaks in the GeV band, and itsemission below 100 MeV is weak and has a hard spectralslope (Figure 7). This weak emission can only be seenwhen the bright prompt emission turns off. A flash with T p ∼ T GRB would still be near its peak at t obs > T GRB ,and the measurement of its spectrum could be extendedbelow 100 MeV to test our prediction in this energy band.Future analysis of the entire sample of LAT bursts willallow one to estimate the wind density, the radius andLorentz factor of the blast-wave, and the efficiency ofthe prompt emission for a number of GRBs. Our prelimi-nary analysis of the published LAT catalogue of 35 bursts(Fermi-LAT Collaboration 2013) suggests that the den-sity parameter A ∼ − g cm − is typical forGRBs with detected GeV flashes.The total energy of the GeV flash is roughly propor-tional to the product of its peak luminosity L p and itspeak time T p , which scales with A . We conclude thatthe flash is likely to be detected in GRBs that are brightand exploding in dense stellar winds. This may explainwhy only ∼
10% of GRBs are found to produce strongemission in the GeV band. Note also that a relatively lowwind density is suggested by the analysis of optical after-glows in a sample of bursts, none of which was detectedby LAT (Hascoet et al. 2013).Observations of the GeV flash determine not only A but also R p and the blast-wave Lorentz factor at R p (Section 5.4). In particular, for GRB 080916C we found R p ≈ cm and Γ( R p ) ≈ This completely de-fines the blast wave, and one can extrapolate its dynam-ics at later times when the optical and X-ray afterglowemission is observed. This opens new prospects for un-derstanding afterglow emission of GRBs.The prediction of bright emission above 100 GeV (Fig-ure 7) can be tested with ground-based telescopes. Inparticular, the High Altitude Water Cherenkov telescope(Taboada & Gilmore 2013) and the Cherenkov TelescopeArray (Inoue et al. 2013) should be able to observe thisemission. We expect that the intrinsic cutoff of the high-energy spectrum at t obs > ∼ Optical flash
We argued in Section 7.2 that the magnetic field in theblast wave may be measured through observations of thelow-energy (synchrotron) counterpart of the GeV flash,in particular in the optical band. A small magnetizationparameter ε B would not affect the GeV flash and stillgive bright optical emission which scales as ε / .For instance ε B ∼ − gives an optical counter-part that reaches the peak luminosity comparable to10 erg s − in ∼ z ) s, followed by a steep decayphase, roughly as t − . This fast decay is mainly con-trolled by the quickly decreasing pair-loading of the ex-ternal medium as the blast wave expands past ∼ cm.Most of the shock energy is lost to the fast Comptoncooling, and only a small fraction is given to the opticalsynchrotron emission.The expected optical flash is very similar to the flashobserved in GRB 990123 (Akerlof et al. 1999). Note thatit reached its peak well before the end of the promptemission, which is consistent with efficient Compton cool-ing of the flash-producing electrons (Beloborodov 2005b).Unfortunately, GRB 990123 could not be observed athigh energies (it was too far off axis for EGRET, theonly available GeV telescope at the time). If our inter-pretation of the optical flash in GRB 990123 is correct,it should have been accompanied by a bright GeV flash.Such double (optical+GeV) flashes may be detectedby future simultaneous observations by Fermi and opticalrobotic telescopes at times t obs ∼ (10 − z ) s afterthe burst trigger. Our calculations predict that the peakof the optical flash is slightly delayed compared with theGeV peak and decays faster.When this work was completed, the first detection of adouble optical+GeV flash was reported in GRB 130427A(Vestrand et al. 2013). It confirms the predictions of ourmodel. A detailed study of the flash in GRB 130427Aand its implications will be published elsewhere (Vurmet al., in preparation).We are grateful to Nicola Omodei and Sylvain Guiriecfor providing LAT and GBM data for GRB 080916C.This work was supported by NSF grant AST-1008334 This value is in conflict with Abdo et al. (2009) who concludedthat the GeV source moves with Γ >0 A. M. Beloborodov, R. Hasco¨et, I. Vurmand NASA Fermi Cycle 6 grant NNX 13AP246.