On the Implications of Late Internal Dissipation for Shallow-Decay Afterglow Emission and Associated High-Energy Gamma-Ray Signals
aa r X i v : . [ a s t r o - ph . H E ] M a y Preprint typeset using L A TEX style emulateapj v. 2/19/04
ON THE IMPLICATIONS OF LATE INTERNAL DISSIPATION FOR SHALLOW-DECAY AFTERGLOW EMISSION ANDASSOCIATED HIGH-ENERGY GAMMA-RAY SIGNALS K OHTA M URASE , K
ENJI T OMA , R YO Y AMAZAKI AND P ETER M ÉSZÁROS
ABSTRACTThe origin of the shallow-decay emission during early X-ray afterglows has been an open issue since thelaunch of the
Swift satellite. One of the appealing models is the late internal dissipation model, where X-rayemission during the shallow-decay phase is attributed to internal dissipation, analogous to the prompt gamma-ray emission. We discuss possible scenarios of the late prompt emission, such as late internal shocks, magneticreconnection, and photospheric dissipation. We also consider the consequences of late dissipation and a two-component (early and late) jet model for the high-energy (GeV-TeV) emission. We study not only synchrotronself-Compton (SSC) emission from the early and late jets but also external inverse-Compton (EIC) emission,which is naturally predicted in the late dissipation model. For the latter, we perform numerical calculationstaking into account the equal-arrival-time surface of EIC photons and show that the EIC component typicallyhas a peak at ∼ -
100 GeV which may dominate over the SSC components. We demonstrate that very highenergy gamma rays from both these components are detectable for nearby and/or energetic gamma-ray bursts,with current and future Cherenkov detectors such as MAGIC, VERITAS, CTA and HAWC, and possibly
Fermi .Although the expected event rate would not be large, detections should be useful as a test of the model. Multi-wavelength observations using both the ground-based telescopes and the
Swift and/or
Fermi satellites are alsoimportant to constrain the models.
Subject headings: gamma-rays burst: general — radiation mechanisms: non-thermal INTRODUCTION
An understanding of the mechanism controlling the earlyX-ray afterglow emission from gamma-ray bursts (GRBs)has been one of the most debated issues since the launchof the
Swift satellite. The canonical X-ray afterglow canbe classified into three phases: the steep-decay phase (see,e.g., Kumar & Panaitescu 2000; Yamazaki et al. 2006;Zhang et al. 2006), the shallow-decay (or plateau) phase andthe normal-decay phase (see, e.g., Nousek et al. 2006;O’Brien et al. 2006; Panaitescu et al. 2006;Zhang et al. 2006; Willingale et al. 2007, and referencestherein). In particular, the shallow-decay phase is difficultto explain by the standard external forward shock (FS)model (see reviews, e.g., Mészáros 2006; Zhang 2007).Numerous models have been proposed so far to explain it.Most frequently discussed models are modified externalshock models. One of the most popular interpretationsinvolves either a continuous energy injection into the ex-ternal FS, where the long-lasting central engine energyoutput has a smooth decline ∝ T - q , or short-lived centralengine ejects shells with a steep power-law distribution ofbulk Lorentz factors which can explain the shallow-decayemission (e.g., Rees & Mészáros 1998; Dai & Lu1998;Zhang & Mészáros 2001a; Zhang et al. 2006). Anotherversion invokes a time-dependent microphysical scenario inthe FS model, where changing parameters such as ǫ e lead tothe observed shallow-decay emission (e.g., Ioka et al. 2006). Center for Cosmology and AstroParticle Physics, Ohio State University,191 West Woodruff Avenue, Columbus, OH 43210, USA Department of Physics, Tokyo Institute of Technology, 2-12-1Ookayama, Meguro-ku, Tokyo 152-8550, Japan Department of Astronomy & Astrophysics, Pennsylvania State Univer-sity, University Park, PA 16802, USA Center for Particle Astrophysics, Pennsylvania State University, Univer-sity Park, PA 16802, USA Department of Physics and Mathematics, Aoyama Gakuin University,5-10-1 Fuchinobe, Sagamihara 252-5258, Japan
Dermer (2007) showed that the shallow decline may beexplained if ultrahigh-energy cosmic rays are efficientlyproduced, by the recovery of an adiabatic relativistic blastwave after its radiative phase due to efficient photomesonlosses and particle escape. The external reverse shock (RS)may also account for the shallow-decay emission, with anappropriate Γ distribution of the ejecta, if the RS emissiondominates over the FS emission in the X-ray band (e.g.,Genet et al. 2007; Uhm & Beloborodov 2007). On the otherhand, some authors have suggested explanations based ontwo- or multi-component jet scenarios. Panaitescu (2008a)showed that upscattering of FS photons by a relativistic shellcan outshine the standard FS emission. Recently, Yamazaki(2009) proposed an alternative interpretation, where X-raylight curves are explained by the difference between theX-ray onset time and the burst trigger time. Among geo-metrical models, two co-aligned jets with different openingangles, i.e., wide and narrow jets, can also lead to theshallow-decay emission (e.g., Eichler & Granot 2006), whilea multiple-sub-jets model was also proposed as one of theexplanations (Toma et al. 2006).Another attractive interpretation is that X-ray emis-sion is attributed to long-lasting internal dissipa-tion (Ghisellini et al. 2007; Kumar et al. 2008b). Thislong-lasting dissipation or late prompt emission modelcan explain the chromatic behavior, which is not so easyto explain in modified external shock models where theoptical flux would presumably track trends of the X-rayflux. This model is viable in the sense that the shallow-decay phase is not ubiquitous and some GRB afterglowsare explained simply by the standard FS model when thelate prompt emission is weak. Especially, some of theGRBs such as GRB 070110 have a plateau and a follow-ing steep decline, which strongly suggests that X-raysoriginate from the late internal dissipation rather than theFS emission (e.g., Troja et al. 2007; Liang et al. 2007b).This late internal dissipation model may also be consistent Murase et al.with the existence of X-ray flares in the early afterglowphase (e.g., Falcone et al. 2007; Chincarini et al. 2010).However, the situation is still inconclusive and unclear.For example, the lack of spectral evolution across thetransition from the plateau to the normal-decay phase andthe compliance of the closure relations in the normal-decay phase after the transition may rather suggest FSmodels (Liang et al. 2007b; Liang et al. 2009).On the other hand, recent novel results from the Fermi satellite have provided us with interesting clues for themechanisms of GRBs. Especially, the onboard LargeArea Telescope (LAT) has detected high-energy ( > GeV)gamma rays from a fraction of GRBs (Abdo et al. 2009a;Abdo et al. 2009b; Ackermann et al. 2010). Those detectionshave provided not only clues to the prompt emission mech-anism but have also provided the first detailed data aboutthe high-energy afterglow emission, which had been expectedfor many years (e.g., Dermer et al. 2000; Sari & Esin 2001;Zhang & Mészáros 2001a). In fact, late-time high-energygamma-ray emission from GRBs such as 080916C, 090510,and 090902B has been attributed to afterglow emission ratherthan the prompt emission (Kumar & Barniol Duran 2010;Ghisellini et al. 2010; He et al. 2010). Various theoreticalpossibilities for the high-energy emission mechanism havealso been discussed by numerous authors. The mostwidely discussed mechanisms are synchrotron and syn-chrotron self-Compton (SSC) emission (see review, e.g.,Fan & Piran 2008, and references therein). The exter-nal inverse-Compton (EIC) emission has been consideredin some cases where seed photons come from flares orprompt emission (e.g., Beloborodov 2005; Wang et al. 2006;Panaitescu 2008b; Murase et al. 2010). If protons and nucleiare accelerated up to very high energies, hadronic gamma-rayafterglows are also expected via the photomeson productionand ion synchrotron radiation (e.g., Böttcher & Dermer 1998;Pe’er & Waxman 2005; Murase 2007).Despite recent progress in observations of high-energygamma rays, the link between GRBs with shallow-decayemission and GRBs whose high-energy emission is detectedby
Fermi is uncertain, since due to the scarcity of simulta-neous detections with
Swift it is unclear whether GRBs de-tected by
Fermi /LAT do have the shallow-decay phase or not.Also, the detectability with
Fermi is limited at late times sothat it is not easy to distinguish among the various models forshallow-decay emission. In this sense, Cherenkov detectorssuch as MAGIC and VERITAS may be more important. Al-though detections of >
10 GeV photons from distant GRBsbecome difficult because of the attenuation by the extragalac-tic background light (EBL), Cherenkov telescopes could pro-vide many more photons than
Fermi when a nearby and/or en-ergetic burst occurs. Although very high-energy photons fromGRBs have not been firmly detected so far, the future CTAand AGIS arrays would significantly increase the chances toobserve high-energy GRB emission.Given a high enough detection rate of high-energy pho-tons by such observatories, high-energy gamma rays wouldprovide very useful probes of the origin of shallow-decayemission. For example, Murase et al. (2010) demonstratedthat EIC emission would be important to diagnose the prioremission model (Yamazaki 2009), which is one of the two-component (early and late) jet models. In this work, mo-tivated by the above prospects, we discuss theoretical pos-sibilities of late internal dissipation scenarios and investi-gate the associated high-energy emission. First, we review various late prompt emission scenarios, such as the late in-ternal shocks, dissipative photosphere, and magnetic dissi-pation scenarios. Second, we analytically study the high-energy gamma-ray emission expected in the internal dissi-pation model. In particular, we numerically calculate theEIC emission in detail, which plays an important role intwo-component jet models such as the late prompt emissionmodel. Throughout this work, cosmological parameters areset to H = 71 km s - Mpc - , Ω M = 0 .
3, and Ω Λ = 0 .
7, and weadopt the conventional notation Q = Q x × x . THEORETICAL POSSIBILITIES OF LATE PROMPT EMISSION
Observationally, a good fraction of GRB afterglowsshow shallow-decay emission from T ∼ . s to T ∼ . s (e.g., O’Brien et al. 2006; Willingale et al. 2007;Liang et al. 2009), which can be expressed as L LP ( T ) ∝ (cid:26) T - α fl ( T < T a ) T - α st ( T a ≤ T ) (1)Here T a ∼ s is the break time when the shallow-decayphase ceases, α fl ∼ . - . α st ∼ . - .
0. Many GRBsshow the chromatic behavior, where optical and X-ray after-glows evolve in different ways, which tempts one to con-sider a two-component interpretation, i.e., X-ray and opti-cal emissions come from different emission regions. Someauthors argued that the shallow-decay X-ray emission maybe attributed to emission caused by internal dissipation sim-ilar to that of the prompt emission, while the normal-decayoptical emission is interpreted as an external FS compo-nent (Ghisellini et al. 2007; Kumar et al. 2008b). For exam-ple, Ghisellini et al. (2009) successfully fitted X-ray and op-tical afterglows of various bursts in this picture. The isotropicradiation energy of late prompt emission, E isoLP , is typically E isoLP ∼ (0 . - . × E isoGRB ∼ - erg, where E isoGRB is theisotropic radiation energy of prompt emission.Such an interpretation seems strongly supported for afraction of bursts. Some GRB afterglows show even aplateau rather than a shallow decay. For example, GRB070110 has the plateau, α fl ∼ .
09, and the following sud-den decline, α st ∼ ∼ -
50 % of GRBs (Falcone et al. 2007; Chincarini et al. 2010),and flares may be attributed to accidental events of strongerinternal dissipation, in the late internal dissipation sce-nario (Ghisellini et al. 2007; Kumar et al. 2008b). The dura-tion of flares ∆ T flare is shorter than the observation time t andits flux enhancement is striking (the energy fluence is about10 % of prompt emission), which suggests that they origi-nate from temporarily strong late internal dissipation by thelong-lasting central engine (Ioka et al. 2005). One puzzlingpoint is that the pulse width of flares increases linearly withtime (Chincarini et al. 2010), while those of prompt emissionhave various durations, showing no increasing pulse widthduring the burst. Possibly, this may reflect the behavior of thecentral engine. In the late internal dissipation model, one mayexpect some time variability in the apparently smooth X-rayigh-Energy GRB Afterglows in the Early and Late Jet Model 3light curve of the shallow-decay emission, where it may alsoinclude some information on the central engine. However,at present, it is difficult to measure well even if the emissionconsists of numerous events of small internal dissipation (e.g.,Ghisellini et al. 2007).If the shallow-decay emission originates from long-lasting internal dissipation, what activity of the cen-tral engine could be responsible for it? For thecentral engine of GRBs, two possibilities have beenmost frequently discussed, accretion of matter ontoa black hole (Kumar et al. 2008a) or a fast rotatingmagnetar (e.g., Dai & Lu1998; Zhang & Mészáros 2001a;Thompson et al. 2004; Yu et al. 2010). In the latter scenario,the break time T a can be attributed to the spin down time. L LP ( T ) ∝ L P ( T ) ∝ (cid:26) const . ( T < T a ) T - ( T a ≤ T ) (2)where L P is the spin down luminosity. In this scenario, theoutflow would initially be Poynting-dominated.In the former scenario, the shallow-decay behavior is at-tributed to the activity of the system of a black hole with anaccretion disk, e.g., mass fall back accretion onto the cen-tral black hole. The break time t a can be interpreted as theend time of mass fall back. Kumar et al. (2008b) proposedthat prompt emission is associated with the accretion of theinnermost region of the progenitor star, whose angular veloc-ity is small, while the outer envelope with the larger angularvelocity is responsible for the shallow-decay emission. Theoutflow luminosity is expected to be proportional to the massaccretion rate, and then the temporal index α fl is related to itsbehavior. After T a , we expect (Kumar et al. 2008a) L LP ( T ) ∝ ˙ M BH ( T ) ∝ (cid:20) + s - T - T a T ac (cid:21) - s + (3)where T ac is the accretion timescale and s ∼ -
1. The aboveexpression can explain both the rapid decline (when T ac < T a )and smooth transition (when T a < T ac ) at T ∼ T a .There is also another interpretation of the behavior after T a and the origin of T a . If the late jet continuously decelerates,one expects a jet break in observed light curves when Γ LP becomes θ - (Ghisellini et al. 2007).Note that the late internal dissipation could also ex-plain the steep-decay emission just after the prompt emis-sion phase, although it is usually attributed to the high-latitude prompt emission (e.g., Kumar & Panaitescu 2000;Yamazaki et al. 2006). For example, in the collapsar sce-nario, Kumar et al. (2008b) suggested that the accretion ofgas from the “transition” region between the core and theenvelope, where the density has a steep decline, may leadto the steep-decay emission. However, such possibility thatX-ray tails reflect the dying history of the central engine to-tally depends on models of the central engine, and the presentsituation is unclear since the apparent spectral evolution isalso affected by the intrinsic spectrum of the prompt emis-sion (Zhang et al. 2009).At present, it is difficult to discriminate among these pos-sibilities from observations. Therefore, for the discussionsbelow, we treat the temporal indices α fl and α st as just pa-rameters determined from observations. Also, we assume thatthe long-lasting internal dissipation occurs according to Equa-tion (1) without specifying the central engine. Late Internal Shock Scenario
In the classical scenario, the prompt emission is ex-plained by electromagnetic radiation from electrons accel-erated at internal shocks that occur in the optically thinrelativistic outflow (Rees & Mészáros 1994). Flares mayalso be explained similarly, where X-ray and/or ultravio-let photons are produced by relativistic electrons acceler-ated at late internal shocks (Fan & Wei 2005). The bulkLorentz factor responsible for flares is often thought to besmaller than that of prompt emission (e.g., Jin et al. 2010),and then, applying this scenario to late prompt emis-sion, the typical collision radius is estimated as r i ≈ Γ δ T var ≃ . × cm ( Γ LP / δ T var , (1 + z ) - . (How-ever, in some models such as the fast rotating magnetarmodel (Thompson et al. 2004), the bulk Lorentz factor of thelate jet may be much larger.)Let us consider the two-shell collision between fast andslow shells with Γ f and Γ s , respectively. The relative Lorentzfactor between the shells is Γ LP , sh ≈ ( Γ f / Γ s + Γ s / Γ f ) / ∼ Γ LP , is ≈ ( p Γ f / Γ s + p Γ s / Γ f ) / ∼ . Γ f ∼
13 and Γ s ∼ Γ LP ≈ p Γ f Γ s ∼ γ e , m ≈ ǫ e f e ( Γ LP , is - m p m e ≃ . × (cid:18) Γ LP , is - . (cid:19) ǫ e , - f - e , - , (4)where ǫ e is the fraction of the internal energy transferredto non-thermal electrons and f e is a number fraction ofaccelerated electrons. Introducing ǫ B which is the frac-tion of the internal energy transferred to the magnetic field,the comoving magnetic field is estimated as B ≃ . × G h ( Γ LP , is + / Γ LP , is - / i / ǫ / B , - L / k , r - i , ( Γ LP / - , and thenthe observed synchrotron peak energy is E b ≃ .
17 keV " ( Γ LP , is - / ( Γ LP , is + / / (3 / √ × f - e , - ǫ e , - ǫ / B , - L / k , r - i , (1 + z ) - . (5)The observed X-ray emission shows a hard spectrum in the X-ray band, F LP ∝ E - , which is attributed to synchrotron emis-sion by relativistic electrons with the spectral index of p ∼ β l ∼ Dissipative Photosphere Scenario
In the previous subsection, we discussed the late internalshock model based on the analogy to the prompt emission. Murase et al.However, the prompt emission mechanism itself is stillunder debate, and many possibilities have been suggested.Another popular scenario of the prompt emission is thephotospheric emission model, where quasi-thermal emissioncomes from around the photosphere ( τ T = n e σ T ( r i / Γ ) ∼
1) (e.g., Thompson 1994; Mészáros & Rees 2000;Mészáros et al. 2002; Rees & Mészáros 2005;Pe’er et al. 2006). Although there are various ver-sions (e.g., Ioka et al. 2007; Beloborodov 2010; Ioka 2010),we here consider the dissipative photosphere sce-nario (Rees & Mészáros 2005; Pe’er et al. 2006), whereinternal dissipation (via e.g., internal shocks or magneticreconnection) occurs around the photosphere. For the latejet making late prompt emission, the photospheric radius iswritten as r ph = (cid:18) ζ e L LP σ T ǫ r π Γ m p c (cid:19) ≃ . × cm ( ζ e / ǫ r ) L LP , ( Γ LP / - , (6)where ǫ r is the ratio of the radiation energy to the kinetic en-ergy carried by cold baryons, and ζ e is the ratio of the numberof electrons to the number of baryons, taking into accountthe possibility of copious pair production via internal dissipa-tion. The photospheric radius thus obtained would generallybe above the typical radius of a Wolf-Rayet star. Also, es-pecially when the late jet is baryon-rich compared to that forprompt emission, the photospheric radius is likely to be lo-cated above the coasting radius. The comoving temperatureat the photospheric radius is kT ≃
30 eV L / , r - / , . ( Γ LP / - / , (7)and the observed typical energy is E b ≃ .
48 keV L / , r - / , . (cid:18) ζ e /ǫ e (cid:19) / (1 + z ) - . (8)The observed X-ray emission shows a hard spectrum in theX-ray band, F LP ∝ E - , which requires some process suchas Comptonization by nonthermal electrons produced via in-ternal dissipation around the photosphere. In this scenario,the variability timescale would be relatively short, δ T var ∼
21 s r i , . ( Γ LP / - (1 + z ), although dissipation itself may lastfor a longer time.As mentioned before, flares would also be caused by ac-tivities of the long-lasting central engine. However, it mightnot be easy to explain flares in this scenario. Flares seem to becaused by occasional larger dissipation of relativistic outflowsbut those with Lorentz factors whose values are larger than thevalues of prompt emission but may be smaller than the valuesof late prompt emission, Γ flare ∼ -
50 (Jin et al. 2010). Onthe other hand, light curves of flares often show the expo-nential decay after the peak, which suggests relatively largeemission radii of r i ∼ × cm Γ , ∆ T flare , (1 + z ) - , ifthe decay of pulses is attributed to high-latitude emission. Thetypical radii seem above the photospheric radius, unless the jetis largely pair dominated. Here, one should keep in mind thatthe photospheric scenario and other scenarios are not mutuallyexclusive. For example, internal shocks may occur well abovethe photospheric radius as well as around the photospheric ra-dius (Rees & Mészáros 2005). Magnetic reconnection is alsoone of the possibilities. Magnetic Dissipation Scenario
Relativistic jets launched by the central engine may be ini-tially Poynting-dominated. If the outflow is still Poynting-dominated at the emission radii, without significant conver-sion into the kinetic energy, magnetic dissipation rather thanshock dissipation of the bulk kinetic energy may lead to pro-duction of nonthermal particles. Although detailed scenar-ios for this are still unavailable due to lack of our knowledgeon mechanisms of magnetic dissipation and associated parti-cle acceleration, prompt and/or late prompt emission may beproduced by internal dissipation of a significant fraction ofthe magnetic energy in the outflow. For example, Lyutikov(2006) argued that magnetic dissipation may occur aroundthe radius where the MHD approximation breaks down, if theoutflow is extremely magnetized. On the other hand, mag-netic fields are distorted by internal shocks at r i ∼ - cm,which may eventually lead to efficient magnetic reconnec-tion (Zhang & Yan 2011; McKinney & Uzdensky 2010).In this work, just for demonstrative purposes, we apply thejets-in-a-jet model (Giannios et al. 2009) for the late promptemission. In this scenario, the magnetic reconnection in ajet leads to many mini-blobs with relative Lorentz factors of ∼ √ σ . Indeed, radiation from such mini-blobs can reproducehighly variable light curves of prompt emission, though thereremain potential problems (see, e.g., Lazar et al. 2009). As-suming that the late jet has Γ LP ∼ σ ∼
30, the dissipation radius is estimated as r i ≃ . × cm ( Γ LP / σ . δ T var , (1 + z ) - .The typical electron Lorentz factor is estimated as γ e ∼ ǫ e √ σ m p m e ≃ . × σ / . ǫ e , - . (9)The magnetic field in the downstream blob can be B ∼ . × G L / B , r - i , . ( Γ LP / - (note that it does not haveto be Poynting dominated since a significant fraction of themagnetic energy is dissipated there). Then, the typical syn-chrotron peak energy is estimated as E b ≃ .
13 keV ǫ e , - σ / . L / B , r - i , . (1 + z ) - , (10)which seems consistent with observations. A hard spectrum, F LP ∝ E - , may be attributed to synchrotron emission fromnonthermal electrons accelerated at shocks caused by mini-blobs. Note that electrons are typically in the fast coolingregime, since γ e , c ≃ . L - B , r i , . ( Γ LP / (of course, the ac-tual electron Lorentz factor should be larger than unity). Here,the lower bulk Lorentz factor with the lower magnetization isassumed for the late jet compared to the case of prompt emis-sion, but it may not be the case in some models such as thefast rotating magnetar model (Metzger et al. 2010). In orderto have the synchrotron peak of ∼ . ǫ e might be required in this scenario.For example, if σ ∼ and r i ∼ cm, ǫ e ∼ - and theshort variability time are expected, depending on the scenario. ASSOCIATED HIGH-ENERGY EMISSION
Next, we consider consequences of the late internal dis-sipation model for high-energy emission. One can considertwo possibilities of high-energy emission. One is EIC emis-sion produced by electrons accelerated at the external shock If one applies this model to GRB prompt emission, we have E b ≃ .
57 MeV ǫ e , - σ / . L / B , r - i , . (1 + z ) - . igh-Energy GRB Afterglows in the Early and Late Jet Model 5caused by the early jet, which is responsible for the promptemission and the observed standard afterglow component.Late prompt photons from inner radii are naturally upscat-tered in the late internal dissipation model, and predictionsare not sensitive to details of late internal dissipation models.In this paper, we especially discuss this possibility in detail(see the next section). The other is the high-energy emissionfrom the emission radius at which internal dissipation occurs,e.g., SSC emission from the late jet. Obviously, predictionsof high-energy emission depend on each scenario. This pos-sibility is also discussed in this section. High-Energy Afterglow Emission
First, we discuss EIC emission caused by interactions withlate prompt photons and electrons accelerated at the externalshock of the early jet producing prompt emission. As demon-strated below, this EIC emission is useful as a test of the lateinternal dissipation model. We here give analytical consider-ations, but more detailed results with numerical calculationsare provided in the next section.We can think that one of the two components is the stan-dard afterglow component from the early jet. For an adi-abatic relativistic blast wave expanding into the interstel-lar medium (ISM) (Blandford & Mckee 1976), we obtain thebulk Lorentz factor as Γ ( T ) ≃ E / k , n - / T - / (1 + z ) / , (11)and the external shock radius is estimated as R ( T ) ≃ . × cm E / k , n - / T / (1 + z ) - / , (12)where E k is the isotropic kinetic energy of the ejecta and n isthe ISM density.Electrons would be accelerated at the external FS. The in-jection Lorentz factor of electrons is estimated as γ e , m ≃ . × ǫ e f , - f - e f ( g p / g . ) E / k , n - / T - / (1 + z ) / , (13)where g p = ( p - / ( p -
2) and p is the spectral index ofFS electrons. Here ǫ e f is the fraction of the internal en-ergy of the shocked ISM transferred to non-thermal elec-trons at the external FS, and f e f is the number fraction ofelectrons injected to the acceleration process at the externalFS (Eichler & Waxman 2005). The cooling Lorentz factor ofelectrons is estimated by t dyn = t cool , and we have γ e , c ≃ . × ǫ - B f , - E - / k , n - / T / (1 + z ) - / (1 + Y ) - (14)where t dyn = ˜∆ / c ≈ (4 /κ ) Γ cT is the dynamical timescale, t cool is the electron cooling timescale, and Y is the to-tal Compton Y parameter. Here, κ is set to 4 in thiswork (Panaitescu & Kumar 2004), and ǫ B f is the fraction ofthe internal energy of the shocked ISM transferred to thedownstream magnetic field. In the slow cooling case ( γ e , m <γ e , c ) with a constant Y , the steady electron distribution is d N e / d γ e ∝ γ - pe for γ e , m ≤ γ e < γ e , c and d N e / d γ e ∝ γ - p - e for γ e , c ≤ γ e . In the fast cooling case ( γ e , c < γ e , m ) with a con-stant Y , the steady electron distribution is d N e / d γ e ∝ γ - e for γ e , c ≤ γ e < γ e , m and d N e / d γ e ∝ γ - p - e for γ e , m ≤ γ e .These electrons upscatter late prompt photons at the vicin-ity of the FS. The expected EIC luminosity is very roughlywritten as L EIC ∼ min( Y EIC L LP , L e ) (e.g., Fan et al. 2008),where Y EIC is introduced as the ratio of the EIC energyflux to the seed photon energy flux. In the slow cool-ing case, noting that F EIC ( E ) ∼ R d γ e d τ e d γ e F LP ( γ e , E ), where γ e d τ e d γ e ∼ τ T ( γ e /γ e , m ) - p + for γ e , m ≤ γ e < γ e , c and γ e d τ e d γ e ∼ τ T ( γ e , c /γ e , m ) - p + ( γ e /γ e , c ) - p for γ e ≥ γ e , c , the resulting EICspectrum in the Thomson limit is expressed as EF EIC ( E ) ∝ E - β l ( E < E m EIC ) E (3 - p ) / ( E m EIC ≤ E < E c EIC ) E (2 - p ) / ( E c EIC ≤ E ) (15)where τ T ∼ ( σ T N e / π R ) is the Thomson optical depth, N e is the number of electrons, β l is the low-energy photon indexof late prompt emission, and E m EIC ≈ γ e , m E b ≃ . E b . ǫ e f , - f - e f (cid:18) g p g . (cid:19) E / k , n - / (cid:18) T + z (cid:19) - / (16) E c EIC ≈ γ e , c E b ≃
95 GeV E b . ǫ - B f , - E - / k , n - / (cid:18) T + z (cid:19) / (cid:18) + Y (cid:19) (17)The contribution below E m SSC mainly comes from interactionsbetween electrons with ∼ γ e , m and photons with E < E b , whilethe contribution in the range E m SSC ≤ E < E c SSC comes frominteractions between electrons with γ e , m < γ e ≤ γ e , c and pho-tons with ∼ E b . The EIC flux at E c EIC is also estimated from E c EIC F c EIC ≈ xY EIC ( E c F c ), where the EIC Compton Y parame-ter, Y EIC , is introduced as the ratio of the EIC energy loss rateto the synchrotron energy loss rate (and it is different from Y EIC ). Here, x . E m KN ≈ Γ γ e , m m e c / (1 + z ) (18) E c KN ≈ Γ γ e , c m e c / (1 + z ) (19) E b KN ≈ Γ m e c / E b / (1 + z ) (20)When the KN effect becomes important, the EIC spectrum hasbreaks. Here, let us introduce E KN , as the first break energydue to the KN effect. When the KN break exists above theEIC peak, instead of Equation (15), we have EF EIC ( E ) ∝ E - β l ( E < E m EIC ) E (3 - p ) / ( E m EIC ≤ E < E c EIC ) E (2 - p ) / ( E c EIC ≤ E < E KN , ) E β l - p ( E KN , ≤ E ) (21)Here E KN , = E b KN ≃ . E b . ) - E / k , n - / T - / (1 + z ) - / . (22)This case is typical for our adopted parameters, and the EICemission at E > E KN , is dominated by radiation from elec-trons with γ e ∼ E / Γ m e c (1 + z ) interacting with seed photonswith the energy of ∼ Γ m e c / E (1 + z ) via the Thomson scat-tering.If the KN break appears below E c EIC , we obtain EF EIC ( E ) ∝ E - β l ( E < E m EIC ) E (3 - p ) / ( E m EIC ≤ E < E KN , ) E β l - p ( E KN , ≤ E ) (23) Murase et al.where E KN , = E c KN ≃
690 GeV ǫ - B f , - E - / k , n - / T - / (1 + z ) - / + Y . (24)If γ e , m and/or E b are too large, one expects the deep KNregime. In this case, we have EF EIC ( E ) ∝ E - β l ( E < E KN , ) E β l - p + ( E KN , ≤ E < E KN , ) E β l - p ( E KN , ≤ E ) (25)Here, E KN , = E m KN ≃ . ǫ e f , - f - e f ( g p / g . ) E / k , n - / T - / (1 + z ) - / (26)and E KN , ≡ Γ γ e , c m e c / (1 + z ) is the second KN break. ThisEIC spectrum is anticipated in the prior emission model forshallow-decay emission (Murase et al. 2010).We are interested especially in cases where the EIC fluxexceeds the afterglow SSC flux. For this purpose, we next es-timate the SSC flux. The SSC emission has been studied bymany authors (see reviews, e.g., Fan & Piran 2008), so thatwe here discuss it just briefly. The characteristic energiesof the SSC emission are obtained as (e.g., Sari & Esin 2001;Zhang & Mészáros 2001b) E m SSC ≃ . g p / g . ) f - e f ǫ e f , - ǫ / B f , - × E / k , n - / T - / (1 + z ) / (27) E c SSC ≃
490 GeV (cid:18) + Y (cid:19) - ǫ - / B f , - × E - / k , n - / T - / (1 + z ) - / . (28)For the slow cooling case that we are interested in, the SSCspectrum in the Thomson limit is expressed as EF SSC ( E ) ∝ E / ( E < E m SSC ) E (3 - p ) / ( E m SSC ≤ E < E c SSC ) E (2 - p ) / ( E c SSC ≤ E ) (29)Note that only the first SSC component is important, since thesecond SSC component is typically negligible due to the KNsuppression. The energy flux at the SSC peak is also evaluatedas E c SSC F c SSC ≃ . × - GeV cm - s - Y SSC (cid:18) + Y (cid:19) p - d - L , . × (cid:18) g p g . (cid:19) p - f - pe f ǫ p - e f , - ǫ p - B f , - E p k , n p - (cid:18) T + z (cid:19) - p , (30)by which we can normalize the SSC spectrum. As a re-sult, the EIC flux and SSC flux are roughly related as E c EIC F c EIC / E c SSC F c SSC ∼ xY EIC / Y SSC . See Section 4 for detaileddiscussions on the relative importance of each component.The KN effect may become important in the cases we con-sider here. For our typical parameters, the KN break is locatedabove E c SSC , which is E KN ≈ Γ (1 + z ) m e c E c ≃ . ǫ / B f , - E / k , n / T / (1 + z ) - / (cid:18) + Y (cid:19) . (31)In general, SSC spectra can be complicated and consist of sev-eral breaks (Nakar et al. 2009; Wang et al. 2010). Hence, we numerically calculate the SSC emission taking into accountthe KN effect.We can also calculate light curves, once the dynamical evo-lution of the blast wave is given. In the next section, we showthe resulting light curves of the EIC and SSC emission. Notethat the temporal behavior would change after the jet breaktime of T j ∼ s (Rhoads 1999; Sari et al. 1999) (where Γ θ j ∼ t j . High-Energy Late Prompt Emission
High-energy emission is expected from the late jet itself, asmentioned before. For example, one can expect SSC emis-sion as well as synchrotron emission if electrons are accel-erated in the magnetized region. If protons are also acceler-ated up to very high energies, hadronic gamma rays are pro-duced via photomeson and photopair production, and proton-synchrotron radiation. Predictions depend on models, whichare quite uncertain. Hence, in this subsection, we just provideanalytical considerations on several interesting cases.In the photospheric scenario, the injection Lorentz fac-tor of electrons should be γ e , m ∼ F LP ∝ E - by Comptonization (e.g.,Thompson 1994; Ioka et al. 2007). The IC spectrum may beextended up to high energies, but high-energy photons cannotavoid attenuation by pair-production. Murase & Ioka (2008)showed that the pair-production break (or cutoff) should bearound E cut ≈ Γ LP m e c / (1 + z ) in the pair-photospheric sce-nario (for β h = 2), which suggests that high-energy emissionabove GeV is not expected in the one-zone case. In the multi-zone case, relativistic electrons may be produced at outerradii. For example, internal dissipation may occur above thephotospheric radius, leading to EIC emission with the typi-cal energy of ∼ γ e , m E b ∼ GeV. However, we will not discusshere such more complicated possibilities.In the magnetic dissipation or late-internal shock scenarios,the typical emission radii are much larger, so that it is easier toexpect high-energy gamma rays that escape from the source.Here, as a demonstrative example, we consider the SSC emis-sion in the jets-in-a-jet model described in the previous sec-tion, which is sufficient for our purposes in this work. First,in the Thomson limit, the typical SSC energy is estimated as E b SSC = 2 γ e , m E b ≃
210 MeV ǫ e , - σ / . L / B , r - i , . (1 + z ) - . (32)Introducing the Compton Y parameter, Y LP , the SSC flux at E b SSC is written as E b SSC F b SSC ≈ Y LP E b F b LP ∼ . × - GeV cm - s - Y LP L e , (1 + Y LP ) 1 + zd L , . . (33)In the Thomson limit, Y LP can be approximated as Y LP ≈ - + √ + ǫ e / ǫ B . But, the KN effect may actually become impor-tant at sufficiently high-energies. When the KN break existsabove E b SSC , we have EF SSC ( E ) ∝ E - β l ( E < E b SSC ) E - β h ( E b SSC ≤ E < E KN , ) E β l + - β h ( E KN , ≤ E ) (34)The KN break is given by E KN , = ( Γ em m e c / (1 + z ) E b ) E b ≃ . Γ LP / ǫ - e , - σ - / . L - / B , r i , . (1 + z ) - . If E KN , < igh-Energy GRB Afterglows in the Early and Late Jet Model 7 E b SSC , the spectrum is in the deep KN regime, and we obtain EF SSC ( E ) ∝ (cid:26) E - β l ( E < E KN , ) E β l + - β h ( E KN , ≤ E ) (35)where E KN , = Γ em γ e , m m e c / (1 + z ) ≃
12 GeV ǫ e , - ( Γ LP / σ . (1 + z ) - . In the fast coolingcase, the resulting spectra can be more complicated espe-cially when Y LP in the Thomson limit is so large that thedistribution of electrons is affected by the KN effect (see,e.g., Nakar et al. 2009; Bosnjak et al. 2009). But the aboveexpressions are reasonable for moderately small values of Y LP .In the late internal dissipation scenario, the pair-creation process is crucial for high-energy gamma-ray emis-sion. The optical depth for pair production is esti-mated as (e.g., Lithwick & Sari 2001; Murase & Ioka 2008;Gupta & Zhang 2008) τ γγ ≈ . σ T l L b LP π r i Γ em c (1 + z ) E b (1 + z ) EE b Γ m e c ! β - , (36)where l is the comoving width. Assuming l ∼ r i / Γ em ,the pair-production break (or cutoff) is estimated as E cut ≃ . L b LP , ) - β - r β - i , . ( Γ LP / ββ - σ ββ - . ( E b . ) ββ - (1 + z ) - ββ - in the magnetic dissipation model. At energies higher thanthis energy, the spectrum is suppressed or may have a cutoff.In Figures 1 and 2, we show SSC spectra which are calcu-lated analytically using Equation (34). The parameters aredescribed in the caption of Figure 1. The pair-productionopacity is taken into account by 1 / (1 + τ γγ ) (Baring 2006).From Figures 1 and 2, we see that it is difficult to detect high-energy late prompt emission from distant bursts, but is possi-ble for nearby bursts. The SSC peak is expected around theGeV range, which may be reached by Fermi if GRBs occur at z . .
7. Very high energy gamma rays above ∼
30 GeV arealso detectable with the future CTA, although its detectablitydepends on the pair-creation opacity both inside and outsidethe source. In Figures 1 and 2, the KN break is seen aroundTeV but the attenuation by pair-creation masks it. The lightcurves of the high-energy gamma rays basically follow theobserved X-ray light curve of the late prompt emission.One can also calculate the SSC emission in the late in-ternal shock scenario similarly to how it was done in theprevious paragraph, by changing parameters. In this para-graph, we briefly discuss the hadronic emission, althoughdetailed studies are beyond the scope of this work. In thelate internal shock scenario, not only electron but also pro-tons may be accelerated up to very high energies. Even inthe magnetic dissipation scenario, protons may be acceler-ated (Giannios 2010), although a large baryon loading maynot be expected. Hadronic emission in the late internal dissi-pation model was considered and discussed in Murase (2007).The Lorentz factor of the late jet might be relatively small,and the late jet might be more baryon-rich compared to theearly jet making prompt emission. Then, as shown in Murase(2007) and Murase & Nagataki (2006), copious soft photonfields in the late jet lead to a high meson production efficiencygiven by f p γ ∼ . L b LP , r i , . ( Γ em / (1 + z ) E b . ( E p / E bp ) β - , (37)where the multi-pion production effect (which is a factor ofthree) is taken into account. The expected neutrino flux is -10-9-8-7-6-5 -8 -6 -4 -2 0 2 4 l og ( E g F g [ G e V c m - s - ] ) log(E g [GeV])SSCSynFermi/LATCTA F IG . 1.— Spectra of synchrotron and SSC emission in the magnetic dis-sipation model for late prompt emission at T = 10 s. The source redshiftis taken to z = 1. Assumed parameters are: L B | T ′ a = L e | T ′ a = 10 . erg s - , σ = 10 . , Γ LP = 5, and r i = 10 . cm. The thick solid curve represents anSSC spectrum taking into account attenuation by pair-creation both insideand outside the source. An SSC spectrum shown as the thin sold curve in-cludes only the source attenuation, while the thin dotted curve spectrum doesnot include either of them. The Fermi /LAT and CTA sensitivities (with theduty factor of 30 %) are also overlayed (CTA Consortium 2010). The LATsensitivity curves in the sky survey mode are used for the long time obser-vations, although the possible continuous observations by LAT may improvethe detectability by a factor of 3-5 (e.g., Gou & Mészáros 2007). -10-9-8-7-6-5 -8 -6 -4 -2 0 2 4 l og ( E g F g [ G e V c m - s - ] ) log(E g [GeV])SSCSynFermi/LATCTA F IG . 2.— Same as Figure 1, but T = 10 . s and z = 0 . comparable to or maybe larger than that of prompt emis-sion, since the meson production efficiency is high whilethe total radiation energy of flares or late prompt emissionis ∼
10 % of that of prompt emission (Falcone et al. 2007;Chincarini et al. 2010) (and the kinetic or magnetic energymay be larger if the radiation efficiency is low). Hadronicgamma rays are also expected as well as neutrinos. The ef-ficient pair-production in the source induces electromagneticcascades. Assuming an E - spectrum for cascade emission,the gamma-ray flux is crudely estimated as EF p γ ∼ + z π d L L CR R∼ . × - GeV cm - s - L CR , R + zd L , . , (38)where R is the conversion factor from the total energy amount Murase et al.of protons into the energy amount of protons per energydecade, R ∼
20 for p = 2. Although the detailed calculationis beyond the scope of this work, this suggests the potentialimportance of hadronic gamma rays for nearby GRBs.Although predictions for the high-energy late prompt emis-sion are model-dependent, once high-energy gamma rays aredetected in sufficient amounts, they would be useful to dis-tinguish between the various uncertain mechanisms of lateprompt emission. NUMERICAL RESULTS OF HIGH-ENERGY AFTERGLOWEMISSION
In the previous section, we gave analytical estimates of EICand SSC emission produced by relativistic electrons acceler-ated at the external FS. As we discussed, the EIC emissiondoes not depend on details of late internal dissipation mech-anisms, and is useful as a good probe of the different scenar-ios. In this section we calculate numerically the EIC emis-sion, which provides significantly more accurate results thanthe analytical estimates. This is because: (1) the EIC emis-sion is anisotropic, which leads to suppression by a factor of x ; (2) the KN suppression becomes important at high energiesabove E KN , ; (3) the influence on the electron distribution iscomplicated if the EIC/SSC cooling is efficient and the KNeffect is relevant.In order to calculate the EIC emission, we need to considerthe equal-arrival-time surface of upscattered photons. The ex-pression for the EIC emission is written as (see Appendix A) F EIC ( T ) = 32 σ T Z drr (1 - cos ˜ θ ) Z d γ e dn e d γ e ˜∆ Z dy (1 - ξ ) × (cid:20) - y + y + ξ - ξ ) (cid:21) F LP ( r ) G ( ε )(1 + Γ θ ) (39)where y ≡ ξ m e c - cos ˜ θ ) γ e ε (1 - ξ ) and ξ ≡ (1 + z )(1 + Γ θ ) E Γ γ e m e c . The scatter-ing angles θ and ˜ θ of EIC photons are measured in the centralengine frame and the comoving frame, respectively. The func-tion G ( ε ) represents the spectral shape of seed photons withenergy ε in the comoving frame (e.g., ε b = (1 + z ) E b / Γ ). Inthe case of a broken power-law spectrum for late prompt emis-sion, it is G ( ε ) = ( ε/ε b ) - β l + for ε < ε b and G ( ε ) = ( ε/ε b ) - β h + for ε b ≤ ε , respectively.The input parameters required for the calculations are basi-cally determined by afterglow observations at X-ray and op-tical bands. We set typical parameters following Ghiselliniet al. (2009). As for the electron distribution, we exploitthe standard external FS model (e.g., Mészáros & Rees 1997;Sari et al. 1998) and adopt the following fiducial parameterset: E k = 10 . erg, n = 1 cm - , ǫ e f = 10 - , ǫ B f = 10 - and p = 2 . β l = 1 for E < E b and β h = 2 . E b ≤ E , with E ′ b = 10 - . eV. Thebreak time is set to T ′ a = 10 s and the late prompt luminosityat T ′ a is taken as L b LP | T ′ a = 10 - . erg s - (which means E LP , X /ǫ e f E k ∼ α fl = 0 . α st = 1 .
5. Also, assuming r i = 10 . cm, the high-energy cutoff due to pair creation is determined from E cut ≃ . L b LP , ) - β - r β - i , . ( Γ em / ββ - ( E b . ) ββ - (1 + z ) - ββ - (e.g.,Gupta & Zhang 2008; Murase & Ioka 2008) with the atten-uation factor of 1 / [1 + τ γγ ( E )] (Baring 2006), and the -10-9-8-7-6-5 -2 -1 0 1 2 3 4 l og ( E g F g [ G e V c m - s - ] ) log(E g [GeV]) EICSSCSynFermi/LATCTA F IG . 3.— Gamma-ray spectra of EIC emission in the late internal dis-sipation model for GRB afterglows, caused by Compton scatterings of X-ray photons by electrons accelerated at the external shock. Calculation arenumerically performed according to Equation (39), taking into account theequal-arrival-time surface. The observation time is set to T = 10 s and thesource redshift is taken as z = 0 .
3. Relevant parameters for the late jet are L b LP | T ′ a = 10 . erg s - , T ′ a = 10 s, E ′ b = 0 . α fl = 0 .
2, and α st = 1 . E k = 10 . erg, ǫ ef = 10 - , ǫ Bf = 10 - , n = 1 cm - , and p = 2 .
4. For comparison, we also showthe assumed synchrotron spectrum and the resulting SSC spectrum. Thickcurves represent cases where the EBL attenuation is taken into account, whilethin ones do not. Note that the attenuation by pair creation in the source isconsidered. The
Fermi /LAT and CTA sensitivities (with the duty factor of30 %) are also overlayed (CTA Consortium 2010). low-energy cutoff due to synchrotron self-absorption is givenfrom the blackbody limit (e.g., Shen & Zhang 2009). But,note that those cutoff energies are not relevant for our results.Jet opening angles of both the jets are set to θ j = 0 . /θ j .In this section, we discuss the results on high-energy af-terglow emission, that is, EIC and SSC components from theearly jet. One should keep in mind that SSC emission fromthe late jet, which was discussed in the previous section, mayalso exist. In the late internal shock and magnetic dissipationscenarios, one could expect ∼ -
10 GeV gamma rays via theSSC mechanism, which are potentially important for
Fermi if nearby and/or energetic GRBs occur. But its predictionsare highly model dependent, and very high energy emissionis not expected when r i and/or Γ em are small enough (e.g.,in the dissipative photosphere scenario), so that it will not beshown here. On the other hand, the EIC emission consideredhere is independent of various late internal dissipation scenar-ios. Even if the SSC emission from the late jet exists, thisEIC and/or SSC components from the early jet will typicallybe dominant at very high energies. Therefore, our results onthe EIC emission provide the most conservative high-energypredictions of the late internal dissipation model.The resulting spectra for our typical parameter sets areshown in Figures 3 and 4. As expected in the previous sec-tion, the EIC peak is located at E c EIC ∼ -
100 GeV. In ourcases, the EIC peak energy is comparable to the SSC peak en-ergy at T ∼ T a , which can be understood from E c EIC / E c SSC ∼ E b / E c . When the EIC emission is dominant, its spectrum isroughly expressed by Equations (21). (When the SSC emis-sion is dominant, its spectrum is roughly expressed by Equa-tion (29).) As expected before, the KN suppression becomesimportant above ∼ -
10 TeV but it is difficult to be observedigh-Energy GRB Afterglows in the Early and Late Jet Model 9 -10-9-8-7-6-5-10 -8 -6 -4 -2 0 2 4 l og ( E g F g [ G e V c m - s - ] ) log(E g [GeV]) EICSSCSynLP F IG . 4.— Spectra of early and late jets in the late dissipation model, con-sidered in this work. Syn and SSC come from synchrotron and SSC emis-sion by relativistic electrons accelerated at the external shock of the earlyjet. LP represents the assumed seed photon spectrum from the late jet, whichis responsible for shallow-decay X-ray emission, and EIC is the EIC emis-sion by Compton scatterings of X-ray photons by electrons accelerated at theexternal shock. The observation time is set to T = 10 . s and the sourceredshift is taken as z = 0 .
3. Here, relevant parameters for the late jet are L b LP | T ′ a = 10 erg s - , T ′ a = 10 s, E ′ b = 0 . α fl = 0 .
2, and α st = 1 . -10-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 l og ( t - [ s - ] ) log( g e )EICSSCSynDyn F IG . 5.— Electron cooling timescales by the EIC, SSC, and synchrotronprocesses at the external shock radius of r = 10 . cm are shown. For com-parison, the dynamical timescale is also shown. One can see that the corre-sponding γ e , c ∼ . . Source parameters are the same as those used in thecaption of Figure 4. due to the EBL attenuation.For these parameter sets of E isoLP , X /ǫ e f E k ∼ p ∼ . T ∼ T a . This can be understood by comparing electroncooling timescales. An example of three timescales is shownin Figure 5, where we can see that γ e , c ∼ - at r ∼ . cm(or T ∼ s). When the EIC and SSC cooling times can beestimated in the Thomson limit, we obtain t - t - ∼ L LP , | T ′ a (7 g p / - p f - pe f ǫ - pe f , - ǫ - pB f , - E - p / k , n (2 - p ) / × (1 + Y ) - p (1 + z ) - p / T - α LP + p / . (40) -10-9-8-7-6-5 1.5 2 2.5 3 3.5 4 4.5 5 l og ( E g F g [ G e V c m - s - ] ) log(T [s]) EIC at 1 GeVSSC at 1 GeVSyn at 1 GeVFermi/LAT at 1 GeV F IG . 6.— Gamma-ray light curves of EIC emission at 1 GeV in the lateinternal dissipation model for GRB afterglows, caused by Compton scatter-ings of X-ray photons by electrons accelerated at the external shock. Forcomparison, light curves of synchrotron and SSC emission are also shown.The parameter set is the same as that used in the caption of Figure 3. The Fermi /LAT sensitivity is overlaid. Note that the attenuation by pair creationboth inside and outside the source is taken into account. -10-9-8-7-6-5 1.5 2 2.5 3 3.5 4 4.5 5 l og ( E g F g [ G e V c m - s - ] ) log(T [s])EIC at 100 GeVSSC at 100 GeVCTA at 100 GeV F IG . 7.— Same as Figure 6, but at 100 GeV. The CTA sensitivity (with theduty factor of 30 %) is overlaid instead of the Fermi one (CTA Consortium2010).
Then, the ratio of the EIC flux to the SSC flux is roughlyestimated as ∼ xt SSC / t EIC . Note that, for sufficiently large γ e , c and/or E b / E c , the results are affected by the KN effect.In the case shown in Figure 5, the synchrotron cooling isdominant. If the EIC cooling is more important than the SSCcooling and the synchrotron cooling, the afterglow emissionfrom the early jet is affected by the EIC cooling. However,this occurs only when the late prompt emission from the latejet is bright enough. The associated afterglow emission fromthe early jet, produced by electrons with ∼ γ e , c , is typicallymasked by the emission from the late jet, so that it seems diffi-cult to observe the EIC influence at the optical or X-ray band.The resulting light curves are shown in Figures 6, 7 and8. The SSC flux evolves as EF SSC ∝ T - p / at E > E c SSC . Onthe other hand, the EIC flux has shallower light curves, butits time evolution is different from that in the X-ray band(see Figure 8). In this sense, the EIC emission in the lateinternal dissipation model can be distinguished from the pre-dictions of other models, such as SSC emission from the0 Murase et al. -10-9-8-7-6-5 1.5 2 2.5 3 3.5 4 4.5 5 l og ( E g F g [ G e V c m - s - ] ) log(T [s])EIC at 100 GeVSSC at 100 GeVLP at 1 keVLP at 1 eVSyn at 1 keVSyn at 1 eV F IG . 8.— Light curves of early and late jets in the late dissipation model atvarious energy bands. Syn and SSC come from synchrotron and SSC emis-sion by relativistic electrons accelerated at the external shock of the early jet.LP represents the assumed seed photon emission from the late jet, which isresponsible for shallow-decay X-ray emission, and EIC is the EIC emissionby Compton scatterings of late prompt photons by electrons accelerated atthe external shock. The parameter set is the same as that used in the captionof Figure 4. Note that the attenuation by pair creation both inside and outsidethe source is taken into account. late jet or SSC afterglow emission in modified FS mod-els. The time evolution of the EIC emission is understoodfrom Y EIC = t - ( γ e , c ) / t - ( γ e , c ). If electrons with γ e , c arein the Thomson regime (which is not always true), we ex-pect Y EIC ∝ L LP / R Γ B ∝ T - α LP + . On the other hand,the synchrotron luminosity in the slow cooling case obeys L c SSC ∝ T - p / from Equation (30). Then, we roughly expect E c EIC F c EIC ∼ xY EIC ( E c F c ) ∝ T - α LP + - p / , which declines morerapidly than the shallow decay emission. For example, for p ∼ . α LP ∼ .
2, we have EF EIC ∝ T - . . When the KNeffect plays a role, the temporal index is somewhat steeper,which seems consistent with the numerical results. The breaktime of the shallow-decay emission is T a ∼ s, but the EICflux does not decline for a while even after T a . This is be-cause seed photons interacting with FS electrons come frombackward ( θ ∼ θ = 0, which significantly contribute to the EIC flux,are delayed compared to non-scattered photons from θ ∼ R / Γ c ∼ T ,which is understood from the fact that EIC emission inducedby an impulsive seed photon emission lasts until we observephotons entering the FS region with θ ∼ / Γ .From Figures 3 and 7, for our typical parameter sets, theEIC emission is expected at energies larger than 10 GeV.As we can see, the EBL attenuation is moderate for nearbyGRBs, though it becomes crucial for distant bursts (see be-low). At such very high energies, observations by Cherenkovtelescopes such as MAGIC, VERITAS, HAWC, and CTAare more promising. Although no clear detections havebeen obtained so far (Abdo et al. 2007; Albert et al. 2007;Aharonian et al. 2009; Aleksi´c et al. 2010), future observa-tions with HAWC and CTA would improve the chances forthis, and either detections or non-detections are important totest the model. Detections by Fermi are limited at late times,but they are being made in the earlier afterglow phase. How-ever, note that the synchrotron or SSC emission is more im-portant than the EIC emission at the earlier phase (especially -14-12-10-8-6 -1 0 1 2 3 4 l og ( E g F g [ G e V c m - s - ] ) log(E g [GeV])EICEIC(intrinsic)SSCSSC(intrinsic)Fermi/LATCTA F IG . 9.— Gamma-ray spectra of EIC and SSC emission in the late internaldissipation emission model at T = 10 . s (thin) and T = 10 . s (thick), butthe source redshift is taken as z = 1. Source parameters are the same as thatused in the caption of Figure 3. The EBL attenuation is included in EIC andSSC, but not included in EIC (intrinsic) and SSC (intrinsic). One can see thatit is crucial for detections by Cherenkov telescopes. -10-9-8-7-6-5 1.5 2 2.5 3 3.5 4 4.5 5 l og ( E g F g [ G e V c m - s - ] ) log(T [s]) EIC at 1 GeVSSC at 1 GeVSyn at 1 GeVFermi/LAT at 1 GeV F IG . 10.— Gamma-ray light curves of EIC emission at 1 GeV in the lateinternal dissipation model for GRB afterglows. For comparison, light curvesof synchrotron and SSC emission are also shown. The parameter set is thesame as that used for Figure 9. Here, the attenuation by pair creation bothinside and outside the source is taken into account. just after the prompt emission), which can be expected fromEquation (40). Also, the SSC emission from the late jet,which has been discussed in the previous section, can be rel-evant in the GeV range (e.g., in the magnetic dissipation sce-nario), while the EIC emission and/or SSC emission from theearly jet will be dominant in the 100 GeV range.We have considered in our calculations the EBL attenu-ation, using the low-IR model developed by Kneiske et al.(2004). Detecting gamma rays at very high energies above100 GeV is prevented by this EBL attenuation. Even for aburst at z = 0 .
3, we have seen that the EBL attenuation largelydegrades the resulting fluxes at &
300 GeV. For a burst athigher redshifts, the situation becomes worse. In Figures 9,10 and 11, results for z = 1 are shown, where the EBL attenu-ation becomes crucial at &
100 GeV. The EIC and SSC peakenergies are higher than the cutoff by the EBL attenuation,so that both of the EIC and SSC fluxes are largely degraded.However, detections around ∼
10 GeV appear still promisingat earlier times, even though they are difficult at late times.igh-Energy GRB Afterglows in the Early and Late Jet Model 11 -10-9-8-7-6-5 1.5 2 2.5 3 3.5 4 4.5 5 l og ( E g F g [ G e V c m - s - ] ) log(T [s])EIC at 100 GeVSSC at 100 GeVCTA at 100 GeV F IG . 11.— Same as Figure 10 but at 100 GeV. Note that gamma rays absorbed by the EBL must produceenergetic pairs, which lead to IC emission by scatterings withthe EBL photons. In our calculations, this pair echo emis-sion is not included since it is beyond the scope of this work,although it could affect the observed afterglow emission ifthe intergalactic magnetic field in voids is weak enough (e.g.,Razzaque et al. 2004; Murase et al. 2009).
Discussion on Parameter-Dependence
We have demonstrated that the EIC emission dominatesover the SSC emission in the late internal dissipation modelfor shallow-decay emission. Importantly, predictions of theEIC emission are straightforward, once X-ray and optical af-terglows are well observed. The parameters necessary for cal-culations of high-energy emission are determined via fittingwith the two-component (early and late) jet model as done inGhisellini et al. (2009). The parameter dependence of therelative importance of the EIC emission to the SSC emissionis seen from Equation (40). The most important quantity is E isoLP /ǫ e f E k (which is expected by setting p ∼ T & T a ), but canbe less important for smaller values. In fact, there is large di-versity among observed X-ray and optical afterglows so thatit would be natural to expect that high-energy afterglows alsoexhibit a high diversity, depending on E isoLP /ǫ e f E k .In our calculations, we have assumed ǫ B f = 10 - , but theEIC and SSC peaks are rather sensitive to ǫ B f (see Equa-tions (17) and (28)). We see that E c EIC / E c SSC ∝ ǫ / B f , so thatthe EIC peak is more likely to be higher than the SSC peakfor larger ǫ B f . This implies that the EIC component is morefrequently dominant over the SSC one at high energies. Notethat the EIC peak energy can be around 1 -
10 GeV rather than0 . - ǫ B f ∼ - .Another potentially relevant parameter is E b . For typicalvalues used in this work, the results on the EIC emission arenot so sensitive to this quantity, up to a modest factor (see Fig-ure 12). But this may not be the case if the EIC cooling occursin the KN regime. If E b is so large that the EIC cooling occursin the KN regime while the SSC cooling does in the Thom-son regime, the EIC emission would be more suppressed. Sofar, we have assumed that E b does not depend on time. Al-though this may not be true, it is difficult to determine its time -10-9-8-7-6-5 -3 -2 -1 0 1 2 3 4 l og ( E g F g [ G e V c m - s - ] ) log(E g [GeV]) F IG . 12.— EIC spectra calculated with different values of E b . Intrinsic EICspectra (where the EBL attenuation is not taken into account) are shown at T = 10 . s (thin) and T = 10 . s (thick). The source redshift is set to z = 0 . E ′ b = 10 . eV and the dashed curvesare for E ′ b = 0 . L b LP / L b LP | T ′ a ) / , while the other parameters are thesame. -10-9-8-7-6-5 1.5 2 2.5 3 3.5 4 4.5 5 l og ( E g F g [ G e V c m - s - ] ) log(T [s]) F IG . 13.— EIC light curves (thick) at 100 GeV, calculated with differentassumptions on E b and Γ LP . For comparison, SSC light curves (thin) arealso shown. The source redshift is set to z = 0 .
3. For the solid curves, theused parameter set is the same as that in the caption of Figure 3. The dashedcurves are for E ′ b = 0 . L b LP / L b LP | T ′ a ) / rather than E ′ b =const. Thedotted curves are for α st = 2 . α st = 1 .
5. The dot-dashed curvesare for the case with the evolving E ′ b and α st = 2 .
8. The attenuation by paircreation both inside and outside the source is taken into account. evolution from observations. To see the influence of this un-certainty on results, we also calculate the EIC emission withthe break energy of E b ( T ) ∝ L / ( T ). However, as seen fromFigures 12 and 13, the results are hardly changed, because theEIC emission mainly occurs in the Thomson regime. At latertimes, the EIC flux with E b ( T ) ∝ L / ( T ) is a bit larger thanthat with E b =const. This is because lower E b at late timescan compensate the KN effect due to increase of γ e , c .The low-energy photon index of the late prompt emission, β l , is observationally uncertain. But this becomes crucial forthe EIC spectrum at relatively low energies of E . E m EIC , sothat we expect that our results are not affected by this. Inaddition, the EIC flux is larger if β l > α st = 2 . T a . For the jetopening angle and bulk Lorentz factor of the late jet, we haveassumed Γ LP > /θ LP . This may not be the case, as discussedin Ghisellini et al. (2007). If the late jet is decelerated withtime, we expect that the observed light curve has the breakwhen Γ LP becomes ∼ /θ LP . This break may be the originof t a , although it is not clear why the late jet is deceleratedcontinuously. In this case, only a fraction of seed photons caninteract with FS electrons after T a . But this just correspondsto a change of α , which is already taken into account in ob-servable parameters.The jet opening angles and axes of the two jets are also as-sumed to be the same. However, we can still expect the EICemission even if the jets are a bit misaligned. If either edge ofthe early jet is on the line of sight, photons from the late jetstill come to the observer through the early jet (independentlyof the prompt emission mechanism), but the resulting EIC fluxis reduced by a factor of two at most. Note that the importantassumption used in this work is ( r i / r ) Γ θ LP ≪
1, which is typ-ically valid in our model. When this condition does not hold,more detailed calculations are required.
Specific Cases
Bursts with a sudden decline in their X-ray afterglowsare of particular interest. For example, GRB 070110has a steep decline of α st = 9 after the plateau of α fl =0 .
09 (Troja et al. 2007). In Figures 14 and 15, we show thespecific case of GRB afterglows with such a plateau (with α fl = 0 and α st = 10), to see the EIC emission induced by theplateau X-ray emission. It is obvious that the EIC spectrum issimilar to that shown in Figure 3, since a similar seed photonspectrum is assumed. On the other hand, the EIC light curveis different from those shown in Figure 4, reflecting differ-ent X-ray light curves. Before T ∼ T a , the EIC light curve issteeper than the X-ray one, as discussed before. However, thisis not the case after ∼ T a . This is because the EIC emissionis similar to high-latitude emission, so that the EIC emissiondoes not show a sudden decline even though the seed photonemission ends abruptly. This was the behavior seen for animpulsive seed photon emission, as demonstrated in the prioremission model (Murase et al. 2010). Detecting such a sig-nature of high-latitude emission associated with the suddendecline after the plateau would be useful as evidence of thelate internal dissipation model.Another case of interest is that of the GRBs that were ob-served by Fermi which may be represented by the late dissi-pation models. The high-energy emission detected by
Fermi may originate from the external shock. For example, Kumar& Barniol Duran (2010) argued that long-lasting GeV emis-sion comes from electrons accelerated at the FS caused by anadiabatic relativistic blast wave expanding into a low densityISM ( n ∼ - cm - ), with a low magnetic field ( ǫ B f ∼ - ).On the other hand, Ghisellini et al. (2010) argued that long-lasting GeV emission may be explained by a radiative rel-ativistic blast wave, with ǫ e f ∼ p ∼
2. Although theorigin of GeV emission especially at the very early stageis still under debate (He et al. 2010; Liu & Wang 2011), thelate-time GeV emission is likely to be regarded as afterglows.Just for demonstrative purposes, we also calculate the EICemission for a burst like GRB 090902B. Unfortunately, therehave been no bursts that have canonical early afterglow lightcurves simultaneously observed by
Fermi and
Swift , so thatwe just show the result for parameters provided in Cenko et al. -10-9-8-7-6-5 -2 -1 0 1 2 3 4 l og ( E g F g [ G e V c m - s - ] ) log(E g [GeV]) EICSSCSynFermi/LATCTA F IG . 14.— Gamma-ray spectra of EIC emission from the GRB afterglowwith the plateau X-ray emission at T = 10 . s (thin) and T = 10 . s (thick).For comparison, synchrotron and SSC emission from the standard afterglowcomponent are also shown. The source redshift is set to z = 0 .
3. Relevant pa-rameters for the late jet are L b LP | T ′ a = 10 . erg s - , T ′ a = 10 s, E ′ b = 0 . α fl = 0, and α st = 10. Relevant parameters for the standard afterglow compo-nent are the same as those in the caption of Figure 3. The attenuation by paircreation both inside and outside the source is taken into account. -10-9-8-7-6-5 1.5 2 2.5 3 3.5 4 4.5 5 l og ( E g F g [ G e V c m - s - ] ) log(T [s])EIC at 100 GeVSSC at 100 GeVLP at 1 keVSyn at 1 keVCTA at 100 GeV F IG . 15.— Light curves of early and late jets in the late dissipation model atvarious energy bands. Syn and SSC come from synchrotron and SSC emis-sion by relativistic electrons accelerated at the external shock of the early jet.LP represents the assumed seed plateau emission from the late jet, and EIC isthe EIC emission by Compton scatterings of late prompt photons by electronsaccelerated at the external shock. The parameter set is the same as that usedfor Figure 14. We can see that the EIC light curve is much shallower thanthat of late prompt emission after T a . The attenuation by pair creation bothinside and outside the source is taken into account. (2010) in Figure 16 (see also Pandey et al. 2010) (but the red-shift is set to z = 1 here). Those parameters are indicated fromlate-time observations but not exact ones for explaining long-lasting GeV emission, since implied γ e , c is smaller than thoseused in Kumar & Barniol Duran (2010). In fact, Liu & Wang(2010) showed that the GeV emission may rather be explainedby an additional jet component (although the two-componentjet model used there is different from that considered here).Here, we are not pursuing possibilities to explain the long-lasting GeV emission with the late internal dissipation model,so these parameters are suitable enough for the present pur-pose. As can be seen, for our parameters on late promptemission, the EIC component is dominant over the SSC oneat late times. The EIC component is especially important atigh-Energy GRB Afterglows in the Early and Late Jet Model 13 -10-9-8-7-6-5 -2 -1 0 1 2 3 4 l og ( E g F g [ G e V c m - s - ] ) log(E g [GeV]) EICSSCSynFermi/LATCTA F IG . 16.— Gamma-ray spectra of EIC emission for a burst with GRB090902B-like afterglow parameters, at T = 10 . s (thin) and T = 10 . s(thick). For comparison, synchrotron and SSC emission from the standardafterglow component are also shown. The source redshift is set to z = 1. Rel-evant parameters for the late jet are L b LP | T ′ a = 1 . × erg s - , T ′ a = 10 s, E ′ b = 0 . α fl = 0, and α st = 2. Relevant parameters for the stan-dard afterglow component are E k = 6 . × erg, ǫ ef = 0 . ǫ Bf = 0 . n = 5 . × - cm - , and p = 2 .
22. The attenuation by pair creation bothinside and outside the source is taken into account. high energies of ∼ -
100 GeV. At lower energies, the syn-chrotron component dominates over the others, although thecurve shown is fairly optimistic (since γ e , M = q π e σ T B η is usedassuming that the upstream magnetic field is the downstreamone and η ∼ Fermi observations. On the other hand, high-energy late prompt emission could potentially be relevant inthe GeV range. One may expect that it shows the shallow-decay behavior when the shallow-decay emission comes fromthe late internal dissipation. However, the observational sit-uation is currently unclear since simultaneous detections by
Fermi and
Swift are required. Possibly, for bursts detectedby
Fermi /LAT, T a is large enough and it may become impor-tant only at late times, or T a is small enough but it may bemasked. Or, declining high-energy emission could happen,if the steep-decay emission comes from the late internal dis-sipation. Also, Fermi /LAT bursts tend to be most energeticones, and it has not been settled whether accelerated electronsare in the fast or slow cooling regime (Ghisellini et al. 2010;Kumar & Barniol Duran 2010). Future simultaneous detec-tions of high-energy gamma rays from GRBs with canonicalafterglow light curves are anticipated. SUMMARY AND DISCUSSION
In this paper, we have studied the possibility that theshallow-decay or plateau emission originates from late inter-nal dissipation in the late jet driven by the long-lasting centralengine (e.g., mass fall back onto a black hole or rotationalenergy loss of fast rotating magnetars). We have discussedvarious theoretical scenarios of the emission mechanism, lateinternal shock, magnetic dissipation, and photospheric sce-narios. There are few clues to the origin of the late promptemission, and all the three scenarios seem compatible withobservations at present. We have also investigated the associated high-energy emis-sion in the late internal dissipation model and discussed twopossibilities: high-energy late prompt emission and high-energy afterglow emission. The former comes from internaldissipation in the late jet, and the predictions depend on thespecific scenarios. For example, in the photospheric scenario,high-energy emission may be produced by the IC process but > GeV emission is not expected in the one-zone case due tothe large pair-creation opacity around the photosphere. Onthe other hand, the late internal shock and magnetic dissipa-tion scenarios may lead to ∼ -
10 GeV gamma rays by theSSC mechanism, which are important for
Fermi . As demon-strated in this work, detections by
Fermi and possibly CTA areexpected for nearby and/or energetic GRBs, which would beuseful for revealing the mechanism of late prompt emission.The latter possibility includes the SSC and EIC emis-sions produced by electrons accelerated at the external shock,which will be especially relevant in the very high energyrange. Especially, the EIC emission, which is high-energyafterglow emission induced by late prompt photons, is not sosensitive to details of models and should be useful as a test ofthe existence of late internal dissipation during the shallow-decay phase. In this work, we have investigated the EIC emis-sion both analytically and numerically and demonstrated thatthe EIC flux may become larger than the SSC flux aroundthe end time of the shallow-decay phase. The EIC peak istypically expected at ∼ -
100 GeV, and the EIC emissiontypically has a steeper light curve than in the X-ray one, buta shallower one when the X-ray light curve shows a suddendecline. Hence, it would be possible to distinguish it from theother possibilities such as SSC components from the early andlate jets. We also expect that it is easier for the synchrotronand SSC components to dominate at very earlier times.Although the detectability depends on the parametersand on the EBL, ground-based gamma-ray observatories,such as MAGIC, VERITAS, HESS, CTA and HAWC,would be important tools in the search for such signals.Very high energy gamma rays from GRBs have not beenfirmly observed so far (Abdo et al. 2007; Albert et al. 2007;Aharonian et al. 2009; Aleksi´c et al. 2010) and the event rateof nearby bursts is not large. (For example, the rate of GRBsoccurring at within z ∼ . ∼ a few eventsper year (e.g., Liang et al. 2007a).) Nevertheless, once suffi-ciently fast follow-up observations are successful for nearbyevents, Cherenkov telescopes with a low-energy threshold( ∼
30 GeV) may allow us to have good photon statisticsthanks to their high sensitivities. Theoretical predictions ofthe EIC emission are testable once parameters are specifiedfrom observations, and the strategy for testing the model is asfollows. First, one determines the relevant standard afterglowparameters for the early flow. When afterglows are well ob-served at optical (and/or X-ray) bands, the parameters such as E k , p , ǫ B f and ǫ e f are determined in the context of the stan-dard external FS theory. At the same time, parameters on thelate prompt emission, such as L b LP and E b and T a , can also bedetermined from observations at X-ray (and/or optical) bands.With those parameters, both the EIC and SSC emissions arecalculated and can be compared to high-energy observations.Even non-detections would provide useful constraints on themodels, especially for GRBs with a strong plateau or shallow-decay emission.It is also important to keep in mind that GRB after-glows seem to be fairly diverse (e.g., Ghisellini et al. 2009;Liang et al. 2009). Although GRBs with a shallow-decay4 Murase et al.emission may be explained by a late jet from the long-lastingcentral engine, some GRBs do not show the shallow decayand can be explained by the standard afterglow model, wherethe synchrotron or SSC emission is expected to be dominant.For this reason, multi-wavelength observations from radio togamma rays are important for comprehensive studies of GRBafterglows. ACKNOWLEDGMENTS
K.M. acknowledges financial support by a Grant-in-Aidfrom JSPS, from CCAPP and from PSU. K.T. and P.M. acknowledge partial support from NASA NNX08AL40G,NASA NNXAT72G, NSF PHY-0757155 and U.R.A. 10-S-017. This research was also supported by Grant-in-Aid fromthe Ministry of Education, Culture, Sports, Science and Tech-nology MEXT of Japan, no. 19047004 and 21740184 (R.Y.).The numerical calculations were carried out on Altix3700BX2 at YITP in Kyoto University. P.M. acknowledges thehospitality of the Institute for Advanced Study, Princeton,during part of this project.
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APPENDIX
FORMULAS OF EIC EMISSIONHere, we derive formulas of EIC emission in GRB afterglows. The observed flux from the shell expanding toward us relativis-tically is (e.g., Granot et al. 1999; Woods & Loeb 1999) F ( T ) = 1 + zd L Z d φ Z d cos θ Z dr r ˜ j ε Γ (1 - β cos θ ) , (A1)where ˜ j ε is the comoving emissivity and ε is the seed photon energy in the comoving frame. Hereafter, we also use ˜ E =(1 + z ) E Γ (1 - β cos θ ) and T = (1 + z )( ˆ T - r cos θ/ c ). Especially, the comoving EIC emissivity is written as (e.g., Toma et al. 2009) ˜ j ε = 32 σ T (1 - cos ˜ θ ) Z d γ e dn e d γ e Z dy ˜ J seed ε (1 - ξ ) (cid:20) - y + y + ξ - ξ ) (cid:21) , (A2)where ˜ J seed ε = 12 Γ (cid:18) π r d L + z F seed (cid:19) . (A3)Here, y ≡ ξ m e c - cos ˜ θ ) γ e ε (1 - ξ ) , ξ ≡ ˜ E γ e m e c , and scattering angles θ and ˜ θ are measured in the central engine frame and the comovingframe, respectively. The range of y is - cos ˜ θ ) γ e (1 - ξ ) ≤ y ≤
1. Note that Equation (A2) is easily obtained from dN (1)EIC d ˜ Ed ˜ T d ˜Ω ≈ πγ e σ T c Z d ε ε dn seed d ε " - ξ b ˜ θ (1 - ξ ) + ξ b ˜ θ (1 - ξ ) + ξ - ξ ) (A4)where b ˜ θ = 2(1 - cos ˜ θ ) γ e ε/ m e c (Aharonian & Atoyan 1981; Fan & Piran 2008).First, we shall derive the formula for an impulsive seed photon spectrum. We also assume that seed photons come from r i ≪ r . In the case of instantaneous emission (at t ) from an infinitely thin shell (at R ), by using the replacement of ˜ j ε → ˜ j ε δ ( ˆ T - ˆ T ) t dyn δ ( r - R ) ˜∆ , we obtain F EIC ( T ) = 32 σ T (1 - cos ˜ θ ) Z d γ e dn e d γ e ˜∆ κ Z dy ¯ F seed | T (1 + Γ θ ) (1 - ξ ) (cid:20) - y + y + ξ - ξ ) (cid:21) (A5)where t dyn = ˜∆ / c is the comoving dynamical timescale and ˜∆ = R /κ Γ is the comoving shell thickness. Here, θ ( T ) = 2 (cid:20) - cR (cid:18) ˆ T - T + z (cid:19)(cid:21) (A6)In the case of a broken power-law seed spectrum, we can write ¯ F seed | T ≡ ¯ F b seed | t G ( ε ), where ¯ F b seed | T = L b seed Γ∆ T π d L E b t dyn (1 + z ) which issmeared over the dynamical timescale of the shell. Note that ∆ T is the duration of impulsive seed photon emission in the oberverframe. Equation (A5) is essentially the same as Equation (5) used in Murase et al. (2010) .Next, we shall derive the formula for continuous seed photon emission. This is obtained by the similar procedure. Performingthe replacement of ˜ j ε → ˜ j ε δ ( ˜ r - ˜ R ( ˆ T )) ˜∆ ( ˆ T ) leads to F EIC ( T ) = 32 σ T Z dr (1 - cos ˜ θ ) Z d γ e dn e d γ e ˜∆ Z dy (1 - ξ ) (cid:20) - y + y + ξ - ξ ) (cid:21) r β F b seed ( T ) G ( ε ) Γ (1 - β cos θ ) , (A7)where ˜∆ = r /κ Γ and θ = θ ( r ) is given by cos θ = cr (cid:18)Z r dr c β - T + z (cid:19) , (A8)and ˜ θ = ˜ θ ( r ) is obtained via the Lorentz transformation. When Γ θ ≫
1, we obtain Equation (39).DISTRIBUTION OF NONTHERMAL ELECTRONSIn order to calculate both the EIC and SSC emission, we use the following electron distribution for γ e ≥ γ e , m , which wouldapproximately mimic the distribution of relativistic electrons in the dynamical timescale, dn e d γ e ∝ min[1 , f - ] γ - pe , (B1)where p is the spectral index of accelerated electrons and f cool ≡ t dyn / t cool is the effective optical depth for energy losses. In theslow cooling case with t cool = t syn (where t syn is the synchrotron cooling timescale), we have dn e / d γ e ∝ γ - pe for γ e , m ≤ γ e < γ e , c There was an unimportant typo in that paper, but calculations were performed using the correct expression, dropping off κ . dn e / d γ e ∝ γ - p - e for γ e ≥ γ e , c . In the fast cooling case, we set p = 1 for γ e , c ≤ γ e < γ e , m , which reproduces ∝ γ - e if t cool = t syn .The value of γ e , c is determined by finding solutions of (e.g., Nakar et al. 2009; Wang et al. 2010) t - = t - ( γ e ) + t - ( γ e ) + t - ( γ e ) , (B2)where the IC loss timescales are evaluated from t - = c γ e ( γ e - Z d µ (1 - µ ) Z d ε dn seed d ε d µ ( K IC σ IC ) (B3)where K IC is the electron inelasticity for the IC process (which is calculated from Equation (C2)) and σ IC is the IC cross sectionwhich is given by the KN formula.The normalization is determined by Z d γ e dn e d γ e (4 π r ˜∆ ) = N e = 4 π - k nr , (B4)where k = 0 for the ISM and k = 2 for the wind medium. (A somewhat different normalization, N e ≈ π r (4 Γ n )( r / Γ ) was usedin Murase et al. 2010.) SSC EMISSION AND PAIR PRODUCTIONIn this work, we also calculate the SSC emission for comparison. For simplicity, we simply calculate the observedSSC flux from the comoving SSC power per comoving energy. The comoving SSC power per comoving energy is givenby (Blumenthal & Gould 1970) ˜ E dN
SSC d ˜ Ed ˜ T = Z d γ e d N e d γ e Z d ε dn syn d ε ˜ E (cid:28) d σ IC d ˜ E c ′ (cid:29) (C1)where c ′ = c (1 - µ ) and (cid:28) d σ IC d ˜ E c ′ (cid:29) = 34 σ T c γ e ε (cid:20) + v - v + v w (1 - v )2(1 + vw ) + v ln v (cid:21) , (C2)and v ≡ ˜ E εγ e (1 - ξ ) and w ≡ εγ e m e c . Note that numerically calculated SSC fluxes have convex curves, which lead to larger fluxescompared to analytically calculated SSC segments (Sari & Esin 2001), and the Klein-Nishina effect becomes often importantabove the TeV range (e.g., Wang et al. 2010). As for the seed photon spectrum, the analytical synchrotron spectrum is used inthis work, which is expressed as three segments both in the fast and slow cooling cases (Sari et al. 1998).High-energy gamma rays may suffer from pair-production process with target photons in the source. We also take into accountthe resulting gamma-ray attenuation in the source. The optical depth for the pair production is expressed as τ γγ ( ˜ E ) = ˜∆ Z d µ (1 - µ ) Z d ε dn syn d ε σ T (1 - β ) (cid:20) β CM ( β - + (3 - β ) ln (cid:18) + β CM - β CM (cid:19)(cid:21) , (C3)where β CM = p (1 - m e c / S ) and S is the Mandelstam variable. In this work, pair attenuation in the source is taken into ac-count by introducing the suppression factor 1 / (1 + τ γγγγ