MNRAS,accepted GeV emission from Gamma-Ray Burst afterglows A. Panaitescu Space Science and Applications, MS D466, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 8 0 0 ABSTRACT 2 We calculate the GeV afterglow emission expected from a few mechanisms related to GRBs and theirafterglows.GiventhebrightnessoftheearlyX-rayafterglowemissionmeasuredbySwift/XRT, n a GLAST/LAT should detect the self-Compton emission from the forward-shock driven by the GRB J ejectaintothecircumburstmedium.NovelfeaturesdiscoveredbySwiftinX-rayafterglows(plateaus 0 andchromaticlight-curvebreaks)indicatetheexistenceofapair-enriched,relativisticoutflowlocated 1 behindthe forwardshock. Bulkandinverse-Compton upscattering of theprompt GRB emissionby suchoutflowsprovideanothersourceofGeVafterglowemissiondetectable byLAT.Thelarge-angle ] burstemissionandsynchrotronforward-shockemissionare,mostlikely,toodimathighphotonenergy h tobeobservedbyLAT.Thespectralslopeofthehigh-energyafterglowemissionanditsdecayrate(if p itcanbemeasured)allowtheidentificationofthemechanismproducingtheGeVtransientemission - o followingGRBs. r t Key words: radiationmechanisms:non-thermal-shockwaves-gamma-rays:bursts s a [ 2 v 1 SOURCES OF GEV AFTERGLOW not considered here. Other processes which rely on mech- 6 EMISSION anisms that are not directly related to the production of 3 theGRBandafterglowemissions(e.g.synchrotronemission 5 InGamma-Ray Bursts(GRBs), thereare twomain sources from VHE protons, synchrotron radiation from pairs and 1 ofGeVafterglowphotons.Oneisthepromptemissionitself, muons produced in photo-hadronic processes) and whose 2. which lasts for tγ = 3−300 s (e.g. Sakamoto et al 2007, high-energy emission is highly uncertain, are also ignored. Willingale etal2007), hasabrokenpower-lawspectrumF 1 ν peakingatǫ ∼100keV,abovewhichF ∝ν−βγ withβ ∈ 7 p ν γ 0 (0.6,2.3)anda25keV–1MeVfluenceφ=10−5±1 ergcm−2 v: (for long bursts) (e.g. Preece et al 2000). The other one is 1.1 Large-angle GRB emission the afterglow emission which, at X-rays, exhibits a plateau i X at 0.3–30 ks characterized by a 1 keV flux (νFν)1keV(t = Owing to relativistic beaming and geometrical curvatureof 1ks) = 10−11±1ergcm−2s−1, a power-law spectrum F ∝ spherical GRB source, during the burst, the observer re- r ν a ν−βx withβ ∈(0.5,2)(e.g.O’Brienetal2006,Willingaleet ceivesemission mainly from apatchmovingtoward theob- x al 2007) and a power-law decay Fν ∝t−αx with αx ∈(0,1) server and extending an angle opening θgrb = Γ−gr1b as seen (e.g. Nousek et al 2006). from the center of fireball, where Γ is the Lorentz fac- grb Inadditiontotheabove”direct”sourcesofhigh-energy tor of the source. Although the burst emission mechanism (GeV) afterglow photons, afraction of the hard X-raypho- ceases to operate after 10–100 s, the observer continues to tons from burst and soft X-ray photons from afterglow can receive emission from regions of the GRB source moving be upscattered in the GeV range either through inverse- at angles θ>θgrb relative to the center–observer axis (Ku- Compton off relativistic electrons or bulk-scattering off a mar & Panaitescu 2000). This ”large-angle” emission ar- faster part of the outflow. In this work, we evaluate the rives at observer at time t(θ) = tγ(θ/θgrb)2 and is rela- GeV afterglow fluxes expected from these direct and scat- tivistically boosted by a factor D(θ) = 2Γgrb/(1+Γ2grbθ2), tering mechanisms and compare them to the sensitivity of hence D(t) ≃ 2Γgrb(tγ/t) is progressively decreasing with theLarge AreaTelescope (LAT) at 1 GeV for a 1ksobser- observer time. From here, it can be shown that the large- vation, (ǫF ) = 10−9ergcm−2s−1, corresponding to 5 angle burst emission at photon energy ǫ above the burst ǫ LAT photonsbeing collected. spectralpeak energyǫp isFǫ(t)=Fǫ(grb)(t/tγ)−2−βγ,where Another mechanism directly related to those at work F(grb) = F (ǫ/ǫ )−βγ is the flux at energy ǫ during the ǫ p p in GRBs and their afterglows, which can produce afterglow burst and F is the burst average flux at the photon en- p emission above 100 MeV is the inverse-Compton scatter- ergy ǫ , which can be approximated by F ≃Φ/(ǫ t ). As- p p p γ ing of flaring internal-shock emission passing through the sumingthatthepower-lawspectrumofthepromptemission forward-shock (Fan & Piran 2006; Wang, Li & Mészáros extends up to above GeV photon energies, the large-angle 2006; Fan et al 2007, Galli & Piro 2007). That scenario is burst emission at ǫ>ǫ is given by p 2 A. Panaitescu φ ǫ 1−βγ t −2−βγ have found shock microphysical parameters, shock kinetic (ǫF )(lae) = . (1) ǫ tγ (cid:18)ǫp(cid:19) (cid:18)tγ(cid:19) energies,andambientmediumdensitiesforwhichνc should be between optical and soft X-ray at 1 ks (Panaitescu Fortherepresentativeburstquantitiesgivenin§1,thelarge- 2005). This suggests that, if other afterglows have similar angle flux at photon energy ǫ=1ǫ9 GeV is forward-shock parameters, their cooling frequency is never (ǫFǫ)(lae)=10−6(βγ+1)φ−5tγβ,γ1+1(cid:18)ǫǫp9,5(cid:19)βγ−1(cid:16)1kts(cid:17)−2−βγcemrg2s(2) ttsiohooantmsthuhocehuslpdabecbotvereuFmtheo∝fsotνfht−eXβfxo-−rraw1y/a2rrdaa-nbsgohevo.eckTνask,yinnitcghfironolttloorwoansccetomhuainstt-, ǫ c where the notation xn = x/(10ncgs) was used, the photon for νc around 100 keV, the GeV flux is a factor ∼ 100 energy being measured in eV. lower than that given in equation (4). Then, the GeV flux Equation (2) showsthat,at 1ksafter trigger, theGeV from the forward-shock synchrotron emission would be de- large-angle emission may be detected by LAT at 1 GeV for tectedbyLATonlyforthehardest(βx =0.5)andbrightest (i) the hardest bursts (βγ = 0.5), for which (ǫFǫ)1GeV = ((νFν)1keV = 10−10ergcm−2s−1) afterglows observed by 10−9φ−5tγ1.,51ergcm−2s−1, or for (ii) the harder (βγ = 1), XRT. longest, and brightest bursts (t >100 s, φ>10−4ergcm−2), IftheX-rayafterglow istheforward-shock synchrotron γ for which (ǫFǫ)1GeV = 10−9φ−4t2γ,2ergcm−2s−1. Bursts in emission upscattered in a different part of the outflow (see eithercategory areextremelyrare(perhapslessthan1per- §1.4),thentheupscatteredνccouldbeabove1MeVandthe cent of BATSE GRBs have those properties), thusis seems likelihood of detecting GeV photons from the same mech- veryunlikely that thelarge-angle burstemission could pro- anism as that yielding the X-ray afterglow will be larger, duce1GeVphotonsthatLATcandetectat 1ksfrom trig- althoughstillrestrictedtoaminorityofafterglows withthe ger. Owing to its very fast decay, there is a much better hardest X-ray emission. chance to detect the large-angle emission at earlier times (e.g. at 100 s). 1.3 Self-Compton scattering of afterglow emission Weconsider now theupscattering of theX-raysynchrotron 1.2 Primary afterglow emission forward-shockemissionbythesameelectrons,themodelfor GeVafterglow emission thathasreceived mostattention so If the soft X-ray (1–10 keV) power-law spectrum measured far(e.g.Mészáros&Rees1994,Panaitescu&Mészáros1997, by theX-RayTelescope (XRT) on Swift extendsover more Dermer, Chiang & Mitman 2000, Zhang & Mészáros 2001, than5ordersofmagnitudeinphotonenergyabove10keV, Pe’er&Waxman2005,Fanetal2007).Galli &Piro(2007) then theobserved X-rayflux offer a detailed treatment of this emission and comparison (νF ) =10−11±1(t/1ks)−αx ergcm−2s−1 (3) withLAT’sdetectioncapability.Incontrasttotheseworks, ν 1keV wewanttoestimatetheGeVinverse-Comptonfluxexpected impliesaGeVafterglow fluxǫFǫ=(νFν)1keV(ǫ/1keV)1−βx for the observed X-ray synchrotron forward-shock flux by which is making minimal use of the usual afterglow parameters. (ǫFǫ)(aglow) =10−(5+6βx)(νFν)1keV,−11ǫ91−βx ergcm−2s−1 .(4) scattAerteda epmhoistsoionnen(ǫe(rigc)y),atbhoeveflutxheofpethake ifnrevqeurseen-cCyomofpttohne p Therefore, LAT-detectable GeV emission arising from the forward-shock emission is samemechanismasthatfortheX-rayafterglow emission is (ǫF )(ic) =(ǫF )(ic)(ǫ/ǫ(ic))1−βx (5) obtained for β <(4±1)/6. Suchhardspectra aremeasured ǫ ǫ p p x for about 10 percent of Swift afterglows. where F(ic) is the peak flux of the inverse-Compton after- ǫ,p IftheX-rayafterglow isthesynchrotronemission from glow spectrum. The spectral quantities of the upscattered theforward-shockdrivenbytheGRBejectaintothecircum- emission can berelated to those of thesynchrotron’s: burstmedium(Mészáros&Rees1997),thentheassumption thatthesoft X-rayspectrumF ∝ν−βx extendsuptoGeV ǫ(pic) =γe2ǫ(psy) , Fǫ(,ipc) =τeFǫ(,spy) (6) ǫ energies is quite likely wrong, particularly for the harder whereγ istheLorentzfactoroftheelectronswhichradiate e afterglow spectra (βx<2/3): atthepeakfrequencyǫ(psy) andτe istheopticalthicknessof (1)IfthesoftX-raybandwereabovethe”cooling” fre- theforward-shocktoelectronscattering.TheobservedX-ray quency(νc)ofthesynchrotronspectrum,thentheelectrons emissionat1keVis(νFν)1keV =(ǫFǫ)(psy)(1keV/ǫ(psy))1−βx, accelerated by the forward-shock would have a power-law thus distribution with energy (γ m c2) flatter than dn /dγ ∝ e e e e (ǫF )(ic) =(νF ) (ǫ/1keV)1−βxγ2βxτ . (7) γ−4/3, which implies that dn /dγ must have a cut-off at ǫ ν 1keV e e e e e some electron energy for which the synchrotron character- If the GRB progenitor is a Wolf-Rayet star, the mass istic frequency is above X-ray. However, that the electrons swept-up by the forward-shock is Mfs = (dM/dt)R/v = radiating at 1 GeV must have an energy larger by a fac- 6×1028R16 g, assuming a typical mass-loss rate dM/dt = tor 103 than those radiating at 1 keV, suggests that the 10−5M⊙yr−1 and wind velocity v=103kms−1). Conserva- synchrotronfrequencycut-offassociatedwiththemaximum tion of the forward-shock energy Efs during its interaction electron energy is below 1 GeV. with the stellar wind, Efs = Γ2fsMfsc2, and the relation (2) If the soft X-ray range is below ν , then the GeV between the shocks radius and the arrival time of photons c flux depends on the exact location of ν . For 10 GRB af- emitted by the region moving at an angle Γ−1 relative to c fs terglows with a good coverage at radio, optical, and X-ray, the direction toward the observer, R ≃ Γ2 ct, lead to a fs fs throughmodellingtheirmultiwavelengthmeasurements,we Lorentzfactor of theshocked circumburst medium GeV afterglows 3 Γfs=42(Efs,53/A∗)1/4(t/1ks)−1/4 (8) flaresrequirealong-livedcentralengine,expellingrelativis- tic outflows at lab-frame times comparable to the observer and radius time when the flares are seen. Rfs=2.3×1016(Efs,53/A∗)1/2(t/1ks)1/2 cm (9) During the X-ray plateau, the 0.3–10 keV afterglow fluxdecaysslowerthanexpectedfortheforward-shocksyn- for a burst at redshift z = 2, where A∗ ≡ chrotron emission, which prompted speculations that (at (dM/dt)/(10−5M⊙yr−1)×(103kms−1/v)(forGalacticWR least one) of the basic assumptions is invalid. One plausi- stars, the wind parameter A∗ varies from 0.3 to 2 – Nugis blescenarioisthatenergyisinjectedintotheforward-shock &Lamers2000).Therefore,theforward-shockopticalthick- by means of some late ejecta which catch-up with the for- ness to electron scattering is wardshockasthelatterundergoesdecelerationbysweeping- τ = σeMfs ≃10−5E−1/2A3/2(t/1ks)−1/2 (10) up the ambient medium (Nousek et al 2006, Panaitescu et e 4πmpRf2s fs,53 ∗ al 2006a, Zhang et al 2006). In that case, the plateau end marks a change in (e.g. cessation of) the power injected in where σ is the Thomson cross-section for electron scatter- e the forward-shock. However, this interpretation encounters ing. the following problem: a change in the forward-shock dy- The hardest X-ray spectra (β ≃ 1/2) measured by x namics should bemanifested in theafterglow light-curveat XRT indicate that, at least for some afterglow, the cool- all frequencies, yet about half of the plateau ends which ing frequency of the synchrotron emission is above X-ray. havebeenmonitoredintheopticaldonotoccurintheopti- Forsimplicity,weassumethatthisgenerallyvalid,henceγ e calaswell,i.e.arechromatic(Watsonetal2006,Panaitescu ofequation(7))isthetypicalLorentzfactoroftheelectrons et al 2006b). Analternative explanation for suchchromatic acceleratedbytheforward-shock,parameterizedbythefrac- X-ray light-curve breaks – the passage of a spectral break tion of the post-shock energy that the electrons acquire (if through the X-ray – is ruled out by that the hardness of they all had same Lorentz factor γ ) : e the X-ray continuum does not evolve across the light-curve γemec2 =εeΓfsmpc2 (11) break (Willingale et al 2007, Liang et al 2007). whereΓ m c2isitsinternalenergyperproton.Then,equa- Thus,theexistenceofX-rayflaresandplateaussuggest fs p a sustained ejection of relativistic material, but the chro- tion (8) leads to matic X-ray breaks cannot be explained by the injection of γe=3800(εe/0.05)(Efs,53/A∗)1/4(t/1ks)−1/4 (12) energy intotheforward-shock produced bythelate outflow catching-upwith theshock. Instead,thosechromatic X-ray where the fractional electron energy was normalized to the breaks suggest that there is a contribution to the afterglow maximumvaluefoundbyPanaitescu(2005)frommodelling fluxthat does not arise from theforward shock. the afterglow emission. XRT observations show that the As shown by Panaitescu (2007), bulk scattering of the peak frequency of the synchrotron forward-shock spectrum (ǫ(sy))isbelow0.3keVbefore1ksaftertriggerthus,forthe forward-shockemissionofftheextendedoutflowrequiredby p X-ray flares can account for the observed decoupling of the electronLorentzfactorofequation(12),thepeakfrequency optical and X-ray emissions because the scattered emission of the upscattered emission (ǫ(ic)) at 1 ks should be below p islikelytoovershinethatcomingdirectlyfromtheforward- theGeV range. shockonlyathigher(X-ray)butnotatlower(optical)pho- Combiningequations(7),(10),and(12),weobtainthat, ton energies. For thisto happen, thescattering outflow, in- for theX-rayfluxtypically measured byXRT(equation 3), nertotheforward-shock,must(1)moveatahigherLorentz theflux of theforward-shock self-Compton fluxshould be factor (whichisanaturalconsequenceoftheforward-shock (ǫFǫ)(ic) ≤10−10+1.2βx 0ε.0e5 2βxEfβsx,2−531A∗3−2βx dmeecdeiluermat)ioanndc,aquuseitdebliykeiltys,i(n2t)ersahcotuioldn bweitehntrhicehecdircwuimthbulerpst- (cid:16) (cid:17) ×(νFν)1keV,−11ǫ91−βx 1kts −βx2+1 cemrg2s (13) t(ownhsichabsouvgegewshtsatthiastetxhpeecstceadtteforirngnoorumtflalo,wbiasrynoonticacecjeelcetra- (cid:16) (cid:17) ated from within the collapsing star but, instead, it results with a weak dependence on the forward-shock’s kinetic en- from the dissipation of magnetic fields at large distances ergy. Therefore, for the typical βx = 1 measured by XRT, fromtheprogenitor–theelectromagneticmodelofLyutikov X-ray afterglows brighter than average should be accompa- 2006a).Thefirstconditionaboveimpliesthatthebrightness nied by inverse-Compton emission detectable toLAT. of the scattered emission received by observer at time t de- pendsontheLorentzfactorandmass-flux(i.e.opticalthick- ness) in the scattering outflow at a distance ct behind the 1.4 Bulk-scattering of GRB emission forward-shock, consequentlytheobserver-frame duration of Most of the X-ray afterglows monitored by Swift exhibit theX-raylight-curveplateau isequaltothelab-frame light flares and light-curve plateaus at 0.1–10 ks after trigger. travel-timebetweentheinneredgeofthescatteringoutflow Mostflaresevolveonatimescaleshorterthanthetimewhen and theGRB source. theyoccur(Burrowsetal2007,Chincarinietal2007),which InadditiontoX-rayplateausandchromaticbreaks,the indicatesthattheyarenotfromtheforward-shock.Instead, scattering model can also explain the short duration of X- flares have been attributed (e.g. Zhang et al 2006) to the ray flares (Shen et al 2007 have investigated the brightness samemechanism(perhapssynchrotronand/orself-Compton of X-ray flares resulting from scattering the burst emission emissionfrominternalshocksinarelativistic,unstableout- offgeometricallythin,relativisticoutflows)andtheobserved flow – Rees & Mészáros 1994) that produces the highly- diversityofpost-plateaudecays,someofwhichdisplayvery variable burst emission. Therefore, this scenario for X-ray fastdecays(t−3 tot−9).Theformerarisesfromthatthere- 4 A. Panaitescu flection of the photons emitted by theforward-shock (mov- is both the angular spread in the photon arrival time and ing at Γ ) off a surface moving at a higher Lorentz factor the time to sweep-up the photons released by the GRB fs (Γ )reducesobserver-frametimescalesbyafactorΓ2 /Γ2 . source).Foraradially-extendedscatteringoutflow,thescat- sc fs sc The latter feature is naturally accommodated because the teredGRBemissionisfurtherspreadoveranobserver-frame decay of the scattered emission depends on the radial dis- equal to the light travel-time from the inner to the outer tribution of mass-flux and Lorentz factor in the scattering edgesofthescatteringoutflow.AstheX-rayplateauresults outflow. from upscatteringby thesame outflow(butof theforward- A pair-rich outflow located behind the forward-shock shock emission), thegeometrical thickness of thescattering provides two ways to produce GeV afterglow photons. One outflow is approximately equal to the duration t of the X is the scattering of X-ray forward-shock photons to higher X-ray plateau. Therefore, for Γsc >∼ Γgrb, a burst duration energies.IftheX-ray-to-GeVphotonsarisefromthismech- tγ <∼300s,andanX-rayplateaulastingfortX ∼10ks,the anism then the fluxes at low and high photon energies are photonarrival-time spread arising from theradial extentof relatedthroughequation(4).Asnotedin§1.2,inthismodel, the scattering outflow is dominant and the received flux is the upscattered cooling frequency could be above 1 MeV, lower than that given in equation (15) by a factor δt /t . sc X thus the scattered forward-shock emission of the brightest The scattering outflow lags by t <10 ks behind the X and hardest X-ray afterglows may be detected by LAT. GRBsource,whoseradiusisR /c=Γ2 t =100Γ2 t γ grb γ grb,2 γ,1 GeV photons are also produced through scattering of ks ⋆ . Thus, theradius of the scattering outflow can be ap- theprompt,GRBemission.Thedependenceofthescattered proximatedbythatoftheGRBsource,hencetheevolution GRB emission on the various burst and scattering outflow of thescatterer’s optical thickness (τ ∝R−2) is e γ parameters can be obtained as follows. For seed photons moving radially inward and scattered photons moving ra- t2 σ E 1/2 2.7s E 1/2 τ = 0, t ≡ e sc = sc,53 (16) dially outward along the observer–center of explosion, the e t2 0 (cid:18)4πΓscγeΓ4grbmec4(cid:19) Γ2grb,2(cid:18)(Γscγe)4(cid:19) Dopplerfactorforthescatterer–GRBsourcerelativemotion for a pair-dominated outflow of kinetic energy E . The is D =Γ /Γ . Taking into account that (i) the observed sc sc grb product Γ γ was scaled to 104, for which the scattered scattered and seed photon energies are a larger by a factor sc e spectrum peaks at 1 GeV (equation 14, with Γ = 100 Γ andΓ ,respectively,thaninthecorrespondingsource grb sc grb and ǫ ≃100 keV). By integrating 1−e−τe over the evolu- frame, (ii) bulk-scattering increases the energy of the seed p tiongiveninequation(16),itcanbeshownthattheaverage photonbyafactorD,and(iii)ifthescatteringelectronsare fraction of scattered photons is 2t /t , for t ∼ 100. A similarly large burst radius is obtained by Ku- mar et al (2007) for a set of 10 GRBs from the the GRB tail sourceenhancestheintensityoftheincidentfluxbyafactor epoch and from that the forward-shock is already decelerating D3;however, in thescatterer’s frame, thehighly collimated whenthe X-rayafterglow emissionemerges. Then, the 10−3−1 incident flux, arriving from within D−2 around the radial s variability timescale typically observed can be accommodated directionofmotion(owingtorelativisticbeaming)isnearly if the GRB emissionarises from hot-spots whose angular extent isotropically redistributed by electron scattering, hence the is (much) smaller than the visible area of opening Γ−1. Alter- grb net increase in intensity produced by scattering is just a natively, if the burstemissionarises fromthe entire Γ−1 visible factor D. area,theburstvariabilitytimescaleandtheassumedGgRrBb radius The GRB emission scattered by a surface is spread (in requireaΓgrb largerbyafactor3–100thanconsideredhereand the observer frame) over a time δtsc = (Γgrb/Γsc)2tγ (this acorrespondinglylargerΓsc. GeV afterglows 5 10−7 10−9 Prompt nnll//nnpp>=3100 LAE aatt a1n Gy eεV>εp(sy) −2−1m s) 10−8 Scnal/tntepr=e1d FS 10−10−2−1m s) V (erg c 10−9 GRB emission βx=1 FSSca ettmeriessdion 10−11V (erg c e e G k 1 LAE 1 F at ε10−10 βγ=1.5 LAT 5γ X−ray 10−12F at ε ε βFS=1 LAT 1γ plateau ε 10−11 x 10−13 101 102 103 104 102 103 104 time (s) time (s) Figure1.Right panel:X-rayplateauand1GeVemissionfromscatteringtheforward-shock(FS)synchrotronemissionbyadelayed, lepton-rich (nl(epton) ≫np(roton)), relativistic outflow. Also shown are the X-ray direct FS emissions (for spectral slope βx = 1) and the delayed large-angleemission (LAE) releasedduringthe burst but arrivinglater at observer. The forward-shockenergy is 1053 ergs (comparabletotheGRBoutput) andthecircumburstmediumhastheradialdensitydistributionexpected foraWolf-Rayetwind.The scatteringoutflowproperties(seebelow)werechosensothatthescatteredfluxovershinesthatoftheforwardshockandyieldsafluxof 10−11ergcm−2s−1at1keVwithaspectralslopeβx=1(averagevaluesmeasuredbyXRT).Inthiscase,neitherthedirectsynchrotron FSnorthescatteredemissionaredetectablebyLAT.Left panel:GeVemissionresultingfromscatteringthe∼100keVburstemission canbedetected byLATifthescatteringoutflowispair-enriched(nl>10np),asrequiredtoproduceanX-rayplateau.Reddottedlines labelled”LAT”indicatethe1GeVfluxcorrespondingtoLATdetecting1and5photonsforanintegrationtimeequaltothetimegiven onabscissa.Theburstspectralpeakfluxwassetto2mJywhich,fora10sburstwithaFǫ∝ǫ1/3belowits100keVpeakandFǫ∝ǫ−1.5 aboveit,correspondstoa25–1000keVfluenceof10−5ergcm−2 (averageparametersforBATSElongbursts).TheLorentzfactorofthe GRBsource(aheadofthescatteringoutflow)is100.Bothpanels:Thescatteringoutflowhasakineticenergyof1053 ergs,ascattering parameter Γγe =104 and is radiallyextended, beinginitially102−104 light-seconds behind the forwardshock. This gap introduces a delaybetweenthearrivaltimesofthedirectandscatteredburstemissions.Theradialdistributionofscatteringoutflow’sLorentzfactor andmasswereassumedtobeuniform. theobserver,therelativisticboostassociatedwiththeGRB 2 PAIR-FORMATION OPACITY FOR GEV source–scatterer relativemotion ismuchsmaller. Inthenu- PHOTONS merical integration of the scattered emission, we take into ForatestphotonofenergyǫemittedatradiusRbyasource account that dependence of relativistic beaming of the in- moving at Lorentz factor Γ, most of the optical thickness coming GRB emission on the angle at which the photon to pair-formation on photons from same source arises as moves(relativetotheradialdirection),aswellasotherfac- the test photon travels a distance of order R (because of tors(e.g. thedecrease of thescatterer’s optical thicknessas the dilution of photons – n ∝ R−2). The relativistic mo- it expands, the effect of the source’s geometrical curvature γ tion of the source collimates most photons within an angle on thephoton arrival time and energy). The numerical cal- θ = Γ−1 around the radial direction. For this reason, dur- culationofthescatteredGRBemissionconfirmsthedepen- ing a small radial displacement dr, the test photon crosses denceofthescatteredfluxonthemodelparametersgivenin a distance dr/(2Γ2) of the photon front (defined by the lo- equation(18),setsitsnormalizationandtemporalevolution cation of photons moving at angle Γ−1 relative to the test (Figure 1): photon), thus the corresponding infinitesimal optical thickness to pair-formation dτ =(dr/2Γ2)σ n(ǫ), where n(ǫ) γγ γγ (ǫFǫ)(sc)=10−9φtγ−,15Ets1Xc/,,2453(ΓscΓγg2eβr)bγ24,β+2γ2−0.5(cid:18)ǫǫp9,5(cid:19)1−βγ(cid:16)1kts(cid:17)−1cemrg2s.(19) titshestethdepihllauobtti-oofnrnamcoafentdaferognremstitypphaoiofrttsoh.neIsnttaaernggdreatutpisnihngogtfortnohmsewuRiptphteowrh2liRimchiottvheoerf ≃ (1/4)σ for the cross-section σ for pair-formation, we e γγ obtain The abovedecay of thescattered emission is obtained for a uanndifoLrmoresnctazttfearcitnogroΓutflaorwe,cinonwsthaincht.tVhaereiojeucstademcaaysssdcaMn/bdet τγγ(ǫ)<∼ σeN32[>πRǫt2(ǫ)] (20) sc obtained if these quantities vary with the depth ct in the whereN(>ǫ )isthenumberofphotonsabovethethreshold t scattering outflow, e.g. for dM/dt ∝ tm and Γsc ∝ tg, the energy ǫt for pair-formation on a test photon of energy ǫ, scatteredfluxaboveǫ(psc)evolvesasFǫ(sc) ∝tm+(2βγ−0.5)g−1. containedinaslabofthicknessR/(2Γ2)aheadoftheshellat Equation (19) indicates that LAT will detect the radiusR(i.e.thenumberofphotonsemittedfromobserver- scattered GRB emission for (1) brighter bursts (φ/t > frame time t(R) to 2t(R)). γ 10−6ergcm−2s−1) and/or (2) afterglows with shorter We estimate now the number N(> ǫ ) of target pho- t plateaus (tX<10 ks). tons emitted by a source of flux (ǫFǫ)1GeV = 10−9(ǫFǫ)−9 6 A. Panaitescu ergcm−2s−1 (normalization chosen for 5 photons collected emission spectrum is at only 200 keV. We note that, if the byLATina1ksobservation)duringt−2t,foranobserver- forward-shock microphysical parameters inferred by mod- frame test photon of energy ǫ=1ǫ GeV. elling the broadband emission of 10 afterglows (Panaitescu 9 For a high-energy flux with the above properties, the 2005) are representative, then we expect the peak of the number of photons with lab-frame energy above (z +1)ǫ inverse-Compton spectrum to be above 1 MeV at 1 ks. For emitted from observer time t to 2t is atenfoldincreaseoftheupscatteredpeakenergy,theoptical N[>(z+1)ǫ]=4πd2l(z)ǫFǫ t =1054.3 z+12 (ǫFǫ)−9 t (21) steheicmksneqsusitoefeliqkuelaytitohna(t2,5e)vednecfroeratshesebhyaradfeascttoobrs1e0rβvxe,dtXhu-rsaiyt (z+1)ǫ z+1 3 ǫ 1ks (cid:16) (cid:17) 9 afterglows (β ≃1.5), theself-Compton scattering emission x whered ≃5×1027(z+1)2cmistheluminositydistance;this is optically thin topair-formation at 1 GeV. l approximation is within 25 percent of the correct value for The test photon may also form pair on a forward- z∈(0.5,5).Theconditionforpair-formationǫ ǫ (1− shock synchrotron photon. If the X-ray plateau emission target test cosθ) ≥ 2(m c2)2 and that most target photons move at is synchrotron from forward-shock, then the number of e angle θ <∼ Γ−1 relative to the test photon, imply that the these photons can be calculated by using the typical X-ray lab-frameminimumenergyǫ ofatargetphotonwithwhich plateau flux (equation 3) in equation (21). Then, the num- t the test photon of observer-frame energy ǫ can form a pair ber of target photons above the threshold energy given in is equation (24) is N (>ǫ ) = 1058.3−2.3βx[(z +1)/3]1.5βx+2 sy t ǫt(ǫ)=350[(z+1)/3]−1Γ2ǫ−91 eV. (22) t(νicFaνl)1tkheiVck,−n1e1sEsf−stβ,o5x3/2p(ati/r1-fkosr)m21a(1t+ioβnx)ǫoβ9nx,ffororwmarwdh-sehreoctkhesyonp-- For a Fǫ ∝ ǫ−β spectrum, the number of photons with en- chrotron photons is ergy aboveǫ is N[>ǫt(ǫ)]=t[ǫt/(z+1)ǫ]−βN[>(z+1)ǫ] τic,sy <10−2.3βx−0.6(cid:16)z+3 1(cid:17)23βx+3(cid:18)EAfs∗,53(cid:19)21βx+1 =1054.3+7β[(z+1)/3]2β+2Γ−2β(ǫFǫ)−9(t/1ks)ǫ29β−1.(23) ×(νFν)1keV,−11(t/1ks)12(βx−1)ǫβ9x . (26) To complete the calculation of the optical thickness to pair-formation, the radius and Lorentz factor of the GeV Equations(25)and(26),withtheGeVfluxofequation(13), source must be specified. We restrict attention to the two showthatpair-formationonforward-shocksynchrotronpho- mechanismswhicharemostlikelytoyieldaLAT-detectable tons could be more important than on inverse-Compton emission at 1 GeV: inverse-Compton scatterings in the for- photons for the harder X-ray afterglows with βx<0.7, pro- ward shock (§1.3) andbulk-scatteringoftheburstemission videdthattheforward-shocksynchrotronspectrumextends in a delayed outflow (§1.4). abovethethreshold energy ǫt of equation (24) and theself- Compton spectrum peaks below thesame ǫ . t The Klein-Nishina effect reduces the scattered flux at 2.1 Self-Compton forward-shock emission photon energies approaching the observer frame energy of thescatteringelectron,i.e.belowΓ γ m c2/(z+1).Equa- For the forward-shock Lorentz factor given in equation (8), fs e e tions (8) and (12) yield a KN cut-off energy ǫ∗ = 30 the lab-frame threshold energy of the target photon (equation 22) with which a test photon can annihilate is [(z + 1)/3]−1/2(εe/0.05)(Efs,53/A∗t3)1/2GeV. Because the electronsacceleratedattheforward-shockhaveapower-law ǫt =0.6[(z+1)/3]−1/2(Efs,53/A∗)1/2(t/1ks)−1/2ǫ−91 MeV(24) distribution with energy above that given in equation (12), theactual cut-offof theinverse-Compton spectrum may be which is around 200 keV in the observer frame. The peak well above ǫ∗. energyoftheinverse-Comptonspectrum(ǫ(ic))isverylikely p below1GeVbutmaybeabovethethresholdenergyǫ .For t ǫ(pic)>ǫt, the number of photons above the threshold ǫt in- 2.2 Scattered burst emission creases with decreasing spectral peak energy ǫ(ic), thus an p For a scattering outflow that is more relativistic than the upperlimitontheopticalthicknesstopair-formation isob- tained by assuming that ǫ(ic)<ǫ . Then, the numberof tar- GRB source (Γsc >∼ 100Γgrb,2), equation (22) shows that p t thethresholdenergyforthetargetphotonisǫ =350Γ2 ǫ get photons is obtained by substituting the forward-shock t sc,3 9 MeV. SubstitutingtheGRB source radius Lorentz factor of equation (8) in equation (23): N(> ǫ )< t 1(t0/514.k3s+)321.7(ββxx+[(1z)+ǫ21βx)/−31].1.T5βoxg+et2h(ǫeFrǫw)−it9h(Ath∗e/Efofrsw,5a3r)d21-βsxhock ra- Rgrb=Γ2grbzc+tγ1 =1015 Γ2grb,2(z+tγ,11)/3 cm (27) 9 dius given by equation (9), the optical thickness to pair- inequation(20)andusingequation(23)withΓ=Γ ,the grb formation of equation (20) is optical thickness to pair-formation for the scattered GRB emission is τic,ic <103.7βx−4.6(cid:16)z+3 1(cid:17)23βx+3(ǫFǫ)−9(cid:18)EAfs∗,53(cid:19)12βx+1 τsc,sc <∼10βγ−1.9(cid:16)z+3 1(cid:17)2βγ+4Γ2β(γǫΓF4ǫ)−9t2 1ktsǫ29βγ−1.(28) sc,3 grb,2 γ,1 ×(t/1ks)21(βx−1)ǫ2βx−1. (25) 9 Inaddition,thescatteredphotonscancreatepairswith Thus, pair-opacity for the inverse-Compton forward-shock GRBphotons.Thelab-frameenergyofaGRBphotonemit- emission at 1 GeV could be important in soft X-ray af- tedatanangleθ>Γ−1 relativetotheradialdirectionofflow grb terglows (β >5/4) if the peak energy of the upscattered isafactor(Γ θ)2 smaller thanifthephotonwerereleased x grb GeV afterglows 7 at θ<Γ−1. Then the condition for pair-formation leads to properties),keepingtoaminimumtheuseofspecificmodel grb a minimum energy of the target photon (as measured by parameters that are not well determined. theobserver if that photon were emitted at θ<Γ−gr1b) that is Exceptforthehardestorlongest&brightestGRBs,the independentof θ: large-angle emission released during the burst and arriving ǫ(grb) =0.60[(z+1)/3]−2Γ−2 ǫ−1 MeV. (29) lateratobservershouldnotbedetectabletoLAT.Withthe t grb,2 9 exception of the hardest X-ray afterglows, detectable GeV Further, it can be shown that, owing to the scattering out- emissionisnotexpectedfromthesameemissionprocessthat flow beingbehind theGRB source, thescattered GeV pho- yieldstheX-rayemission,particularlyiftheX-rayafterglow tons can interact only with burst photons that have been is synchrotron emission from theforward shock. emitted at an angle larger than θmin(tlag)=(ctlag/Rγ)1/2, Theothertwomechanismsdiscussedrelyontheupscat- withctlag beingtheseparationbetweentheplaceofscatter- tering (either in the forward-shock or an other part of the ing and the GRB source and Rγ =Γ2grbct the radius of the relativistic outflow) of lower energy photons into the GeV latter. ForΓsc>Γgrb,photonsscattered bythefluid at ctlag range and offer better prospects for detection with LAT. behind the GRB source arrive at observer at time t=t , lag The self-Compton scattering of forward-shock syn- thusθ (t)≃(t/t )1/2Γ−1 =0.1(t /t )1/2Γ−1 .Taking min γ grb 3 γ,1 grb,2 chrotron photonsis expected to be above LAT’s 1 ks sensi- into account that the lab-frame distribution GRB photons tivity for X-ray afterglows with plateaus brighter than the with angle θ relative to the radial direction of motion is average.Thismechanismcanbeidentifiedbasedonthatthe dN /dΩ ∝ [Γ (1−Bcosθ)]−2 (with B the speed the grb grb spectral slope of the GeV emission should be the same as GRBsource),thephotonsemittedat θ>θ areafraction min for the X-ray afterglow (β = β ). In addition the de- GeV x f(t)=(Γ θ )−2 =t /t (30) cay rates of the fluxes at these two energies should satisfy grb min γ α =α +(β +1)/2 (equation 13). Pair-formation may GeV x x of the numberof GRB photons reducetheself-Compton scatteringfluxat1GeV(equation 4πd2(z) z+1 2 25) if the peak energy of the self-Compton emission spec- Ngrb = (z+l1)2tγFp =1059.3 3 φ−5ǫ−p,15 . (31) trum is below 100 keV or of the X-ray plateau emission is (cid:16) (cid:17) hard and extendswell above 100 keV (equation 26). Thus, thenumberof GRB photonsabove thethreshold en- Bulk-scattering of the forward-shock emission by a de- ergy ǫ(grb) that can reach the test photon is N(> ǫ(grb)) = t t layed outflow can explain the X-ray plateaus observed in f(ǫt(grb)/ǫp)−βγNgrb. Then,for thescatterer radiusgiven in many Swift afterglows if the scattering outflow is pair-rich. equation(27),equations(20)and(29)–(31),lead tothefol- ThesamemechanismmayproduceGeVemission thatLAT lowing optical thicknesstopair-formation onburst photons can detect if the afterglow X-ray emission is brighter and τsc,grb<∼101.5−0.8βγ(cid:16)z+3 1(cid:17)2βγ+4φtγ−,15Γǫ2βpβ,γ5γ−+14(cid:16)1kts(cid:17)−1ǫβ−γ9 .(32) hslaiaorrnlydemfroartyhbaaunlrssaotvseprbraorgidgeuh.ctSeecradttehtteaerncitnaagbvloeerfaGtgheeeVabnuedrmsatisftspeirorongml,opwptasretwmiciitush-- grb,2 short-lived X-ray plateaus (equation 19). The upscattered Equations(28) and(32) show thatGeVafterglow pho- emission should have the same spectral slope as the burst tons may be lost to pair-formation on upscattered photons (β =β )butcan exhibitavariety ofdecays, depending oronburstemission.Theenergyabovewhichthisprocesses GeV γ on theradial distribution of mass and Lorentzfactor in the reduces the high-energy flux is strongly dependent on the scattering outflow. For this mechanism, pair-formation on Lorentz factor of the GRB source and, for pair-creation on either the burst photons or the upscattered emission itself the scattered emission, also on the Lorentz factor of the reducestheGeVfluxaboveaphotonenergythatisstrongly scattering outflow. dependent on the Lorentz factors of the GRB source and The KN effect reduces the GeV flux at photon ener- scatteringoutflow(equations28and32).Consequently,the gies approaching ǫ∗ = Γscγemec2/(z+1) = 2[(z+1)/3]−1 detectionofphotonsabove1GeVduringtheearlyafterglow (Γscγe)4GeV. The KN cutoff energy ǫ∗ is above the peak canbeusedtosetlowerlimitsontheseLorentzfactors,sim- energy ǫ(sc) of the scattered GRB emission (equation 14) p ilar to how it was done for the GRB source Lorentz factor if the scattering parameter Γ γ is less than Γ2 (m c2) sc e grb e usingthedetectionof0.1–10GeVphotonsduringtheburst [(z + 1)ǫ ]−1 = 2 × 104[(z + 1)/3]−1Γ2 ǫ−1, for which p grb,2 p,5 byCGRO/EGRET(e.g.Fenimore,Epstein&Ho1993,Bar- ǫ(psc)<3[(z+1)/3]−2Γ2grb,2ǫ−p,15GeV.Thecut-offatǫ∗issharp ing & Harding 1997, Lithwick & Sari 2001). onlyifthescatteringelectronsarecold(γe=1);inthiscase, Foraneffectiveareaof8000cm2,LAT’ssensitivitycor- owing to that Γ is most likely below 104, the KN effect sc responds to several GeV photons detected in 1 ks. Thus, truncates thescattered spectrum aboveits peak energy. formostGeVafterglows,thehigh-energylight-curvewillbe toocrudeforadeterminationofitspower-lawdecayrate.As thespectralslopecanbedeterminedwithamodestnumber of collected photons, comparing the slope of the GeV spec- 3 CONCLUSIONS trum with those of the burst and X-ray afterglow emission We have assessed the ability of four mechanisms related provides a better way to identify the mechanism that pro- to GRBs and their afterglows to produce LAT-detectable ducedthehigh-energyafterglow. ForGRB940217,theonly GeV emission during the early afterglow phase (t ∼ 1 ks). burstforwhichGeVafterglowemissionwasdetectedsofar, WenotethattheexpectedGeVfluxeswerecalculated from EGRETobservations(Hurleyet al1994) showahardcom- afterglow and GRB observables (X-ray plateau flux, dura- ponent at 0.1–4 GeV, with F ∝ǫ1/3 during the burst and, ǫ tion,andspectralslope;burstfluence,duration,andspectral possibly, during the afterglow, at 5 ks. The hardness of the 8 A. Panaitescu GeV emission suggests that it is the burst emission below its spectral peak upscattered bya lagged outflow. We conclude by noting that, within the framework of the GeV afterglow emissions analyzed here, LAT will de- tecthighenergyphotonsunderfavourableconditions.GRBs dimmer or harder than the average and X-ray afterglows with dim and long plateaus could be accompanied by GeV emission that is too faint to be detected by LAT. 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GeV emission from Gamma-Ray Burst afterglows

Similar GeV emission from Gamma-Ray Burst afterglows