Reconciling observed GRB prompt spectra with synchrotron radiation ?
aa r X i v : . [ a s t r o - ph . H E ] O c t Astronomy&Astrophysicsmanuscript no. paper c (cid:13)
ESO 2018June 4, 2018
Reconciling observed GRB prompt spectrawith synchrotron radiation ?
Fr´ed´eric Daigne ⋆ , ˇZeljka Boˇsnjak , , and Guillaume Dubus , Institut d’Astrophysique de Paris, UMR 7095 Universit´e Pierre et Marie Curie – CNRS, 98bis boulevard Arago, 75014 Paris, France AIM (UMR 7158 CEA / DSM-CNRS-Universit´e Paris Diderot) Irfu / Service d’Astrophysique, Saclay, 91191 Gif-sur-Yvette Cedex,France Laboratoire d’Astrophysique de Grenoble, UMR 5571 Universit´e Joseph Fourier – CNRS, BP 53, 38041 Grenoble, France
ABSTRACT
Context.
Gamma-ray burst emission in the prompt phase is often interpreted as synchrotron radiation from high-energy electronsaccelerated in internal shocks. Fast synchrotron cooling of a power-law distribution of electrons leads to the prediction that the slopebelow the spectral peak has a photon index α = − / N ( E ) ∝ E α ). However, this di ff ers significantly from the observed medianvalue α ≈ −
1. This discrepancy has been used to argue against this scenario.
Aims.
We quantify the influence of inverse Compton (IC) and adiabatic cooling on the low energy slope to understand whether theseprocesses can reconcile the observed slopes with a synchrotron origin.
Methods.
We use a time-dependent code developed to calculate the GRB prompt emission within the internal shock model. The codefollows both the shock dynamics and electron energy losses and can be used to generate lightcurves and spectra. We investigate thedependence of the low-energy slope on the parameters of the model and identify parameter regions that lead to values α > − / Results.
Values of α between − / − ff er IC losses in the Klein-Nishina regime. This does notnecessarily imply a strong IC component in the Fermi / LAT range (GeV) because scatterings are only moderately e ffi cient. Steepslopes require that a large fraction (10-30%) of the dissipated energy is given to a small fraction ( < ∼ < ∼ . α up to − / ffi ciencies ( > / or with low magnetic fields. Conclusions.
Amending the standard fast cooling scenario to account for IC cooling naturally leads to values α up to −
1. Marginallyfast cooling may also account for values of α up to − /
3, although the conditions required are more di ffi cult to reach. About 20 %of GRBs show spectra with slopes α > − /
3. Other e ff ects, not investigated here, such as a thermal component in the electrondistribution or pair production by high-energy gamma-ray photons may further a ff ect α . Still, the majority of observed GRB promptspectra can be reconciled with a synchrotron origin, constraining the microphysics of mildly relativistic internal shocks. Key words. gamma-rays: bursts; shock-waves; radiation mechanisms: non-thermal
1. Introduction
The physical origin of the prompt emission in Gamma-RayBursts (hereafter GRBs) is still uncertain. The identificationof the dominant energy reservoir in the relativistic out-flow, of the mechanism responsible for its extraction andof the processes by which the dissipated energy is eventu-ally radiated remains a major unresolved issue. There arethree potential energy reservoirs : (i) thermal energy thatcan be radiated at the photosphere (M´esz´aros & Rees 2000;Daigne & Mochkovitch 2002; Giannios & Spruit 2007; Pe’er2008; Beloborodov 2010), (ii) kinetic energy that can beextracted by shock waves propagating within the outflowand then radiated by shock-accelerated electrons (internalshocks, Rees & M´esz´aros (1994); Kobayashi et al. (1997);Daigne & Mochkovitch (1998)), or (iii) magnetic energythat can be dissipated via the reconnection of field lines(Thompson 1994; Meszaros & Rees 1997; Spruit et al. 2001;Drenkhahn & Spruit 2002; Lyutikov & Blandford 2003;Giannios & Spruit 2005) and then radiated by acceleratedparticles. In the two last cases, the expected dominant radiative
Send o ff print requests to : F. Daigne ( [email protected] ) ⋆ Institut Universitaire de France processes are synchrotron radiation and inverse Comptonscattering.Observed GRB spectra can provide reliable constraintson the extraction mechanism and dominant radiative process.A typical GRB prompt emission spectrum is usually welldescribed by a phenomenological model (Band et al. 1993)where the photon flux follows N ( E ) ∝ E α at low energies and N ( E ) ∝ E β at high energies, with a smooth transition around E p , which is the peak energy of the ν F ν spectrum. Typicalvalues in GRBs observed by BATSE are α ∼ − β ∼ − . E p ∼
250 keVfor the peak energy (Preece et al. 2000). It is well known thatthe observed value α ∼ − α = − / α = − / ffi culty to reproduce the shortesttimescale variability (Rees & M´esz´aros 1994; Sari et al. 1996;Kobayashi et al. 1997). Daigne, Boˇsnjak & Dubus: Reconciling observed GRB prompt spectra with synchrotron radiation ?
On the other hand, synchrotron radiation is a very naturalexpectation for the emission from shock-accelerated electrons.In GRBs especially, it is most probably at work in afterglows.Observations of prompt GRBs by the LAT instrument onboard
Fermi indicate that most GRBs do not show an addi-tional component at high energy ( >
100 MeV) brighter oras bright as in the soft gamma-ray range (Abdo et al. 2009;Omodei et al. 2009). Prompt observations in the optical domainremain di ffi cult but do not show strong evidence in favorof a bright additional component at low energy, with somenotable exceptions like GRB 080319B (Racusin et al. 2008).This strongly favors synchrotron radiation compared to thesynchrotron self-Compton (SSC) process for the emissionobserved in the soft gamma-ray domain (keV–MeV). Indeed,the latter requires some fine tuning to avoid a strong componenteither in the optical-UV-soft X-rays domain, or in the GeVrange (Boˇsnjak et al. 2009; Zou et al. 2009; Piran et al. 2009).Compared to SSC, synchrotron radiation has also a betterability – at least in the internal shock framework – to reproducethe observed spectral evolution : e.g. hardness-intensity andhardness-fluence correlations, evolution of the pulse widthwith energy channel, time lags (Daigne & Mochkovitch 1998;Ramirez-Ruiz & Fenimore 2000; Daigne & Mochkovitch 2003,Boˇsnjak 2010 in preparation).In this paper, we investigate a solution to steepen the low-energy slope of the synchrotron component and possibly recon-cile the synchrotron process with observed GRB prompt spec-tra. This solution is related to the steepening of the low-energysynchrotron slope by moderately e ffi cient inverse Compton scat-terings in Klein-Nishina regime, as suggested by Derishev et al.(2001); Boˇsnjak et al. (2009); Nakar et al. (2009); see alsoRees (1967) where this is mentionned in the general con-text of SSC radiation. It is an alternative to the SSC scenario(see e.g. Panaitescu & M´esz´aros 2000; Baring & Braby 2004;Kumar & McMahon 2008), to the comptonization scenario(Liang et al. 1997; Ghisellini & Celotti 1999), or to other propo-sitions to modify the standard synchrotron radiation, relatedto the timescale of the acceleration process (Stern & Poutanen2004; Asano & Terasawa 2009), the pitch-angle distribution ofelectrons (Lloyd-Ronning & Petrosian 2002) or the small scalestructure of the magnetic field (Medvedev 2000; Pe’er & Zhang2006). A summary of the measurements of the low-energy pho-ton index α and its distribution in GRBs is given in Sect. 2. ThenSect. 3 describes how the standard synchrotron spectrum is af-fected by additional processes such as inverse Compton scatter-ings or adiabatic cooling. It allows to identify physical condi-tions – in terms of intensity of the magnetic field, distribution ofrelativistic electrons, etc. – that lead to low-energy slopes steeperthan the standard prediction α = − /
2. We discuss in Sect. 4 howsuch conditions could be found in GRB outflows. We computeexpected pulse lightcurves and spectral evolution in the frame-work of the internal shock model and show that steep slopes areindeed expected in a large region of the parameter space. Wesummarize our conclusions and discuss future possible develop-ments in Sect. 5.
2. A critical view on the observed distribution ofGRB prompt spectral properties
We examine here in detail the observed distribution of the lowenergy photon spectrum index α , as it provides a relevant crite-ria for the goodness of the emission model for GRB emission.To date the largest database of gamma-ray burst high time and energy resolution data was provided by the Burst and TransientSource Experiment (BATSE) ( ∼
20 keV - 2 MeV) on board the
Compton Gamma Ray Observatory . Kaneko et al. (2006) pre-sented a systematic spectral analysis of 8459 time-resolved spec-tra from 350 GRBs (including 17 short events) observed byBATSE; this sample includes also gamma–ray bursts that wereexamined in previously published catalogs of BATSE GRBs(e.g. Preece et al. 1998, 2000). The reported distribution of α isapparently not consistent with the predictions of the simple syn-chrotron model: Kaneko et al. (2006) showed that the medianvalue for the time resolved spectra α = –1.02 + . − . (long GRBs)and –0.87 + . − . (short GRBs). The values obtained for the timeintegrated spectra indicate somewhat softer spectra, with the re-spective median indices –1.15 + . − . and –0.99 + . − . for long andshort events respectively. The slope α is distributed roughly sym-metrically around the median value.The results obtained by other instruments are consistent withBATSE observations: Krimm et al. (2009) combined the Swift
Burst Alert Telescope (BAT) and
Suzaku
Wide band All-SkyMonitor (WAM) data covering the broad energy band 15 to5000 keV and report the distribution of α skewed toward slightlylower values, –1.23 ± High Energy Transient Explorer 2 (HETE-2) in the en-ergy band 2-400 keV find α = –1.08 ± Fermi Gamma–ray Space Telescope
GBM and LAT in the broad energy range ∼ >
100 GeV,e.g. the sample studied by Ghirlanda et al. (2010) of 12 GRBsobserved by
Fermi displays various values for α , ranging from–1.26 ± ± E peak may be oversampled. The results of thetime-resolved spectral analysis may be biased in the similarway: as the data were sampled more frequently during the in-tense episodes, the brighter portions of each burst may havemore impact in the final distribution of spectral parameters(Kaneko et al. 2006);2. the low-energy photon spectra indices tend to correlate withthe peak energy of the ν F ν spectrum, the slope α becom-ing softer when the peak energy is decreasing (Kaneko et al.2006; Crider et al. 1997; Lloyd-Ronning & Petrosian 2002;Ford et al. 1995; Preece et al. 1998). This e ff ect might be dueto a combination of the curvature of the spectrum around thepeak energy and the limited spectral energy range sampledby the instruments;3. as discussed by Preece et al. (1998) and Lloyd & Petrosian(2000) for the BATSE spectra, the data don’t always ap-proach the GRB spectral low energy power law within theinstrument energy range. If peak energy is close to the edgeof the instrumental energy window, the low energy spectralpower law may not have reached yet its asymptotic value. Inthat case lower values of α are determined (i.e. softer spec-tra). Kaneko et al. (2006) attempted to account for this e ff ectand applied as a better measure of the actual low energy be-havior the e ff ective index α e f f for BATSE data, defined as aigne, Boˇsnjak & Dubus: Reconciling observed GRB prompt spectra with synchrotron radiation ? 3 Fig. 1. Consequences on energetics of a steepening of the low-energy slope in the ”Band” function.
Left: the normalized ”Band”function is plotted for β = − .
25 and α = − . α = − f e of the energy that should be removed by any processleading to such a change of the low-energy photon index. Here f e ≃
19 %.
Center: the fraction f e is now plotted as a function of β ,assuming a low-energy photon index changing from α = − . Right: the fraction f e is now plotted as a function of α , whenthe photon index is changing from α = − . α . We assume β = − .
25. These three figures are plotted assuming that the processresponsible for the change of the slope α does not a ff ect the value of the break energy E b nor the high-energy tail of the spectrum.the tangential slope of the spectrum at 25 keV (the lower en-ergy limit of the BATSE window);4. the low-energy spectral index distributions of time-integrated and of time-resolved spectra are slightly di ff erent;it is expected due to the evolution of the spectral parame-ters during the integration time. The sharp spectral breaksare smeared over and the indices of time integrated spectraappear softer than in the case of the time-resolved spectra(Kaneko et al. 2006). These two last points could imply thatthe true low-energy spectral slopes in prompt GRBs are evensteeper than the observed median value.Another important aspect to examine in the observed distri-butions of spectral indices concerns the contribution of an in-dividual GRB to the overall distribution. Since the brighter andlonger events in general contribute with larger number of spectrain the BATSE spectroscopic catalog, we have computed distribu-tions of spectral parameters where each time bin of a given GRBis weighted by the corresponding fraction of the total fluence ofthe burst. In this way GRBs with di ff erent number of time re-solved spectra in their time histories have the same impact onthe overall distribution. Using the data by Kaneko et al. (2006),we find that: – Only 5% of GRBs have more than 50% of their spectra withvery soft low energy slope, α < –1.5. Such slopes are prob-ably related to the spectral curvature around the peak energyand do not raise a problem for the standard synchrotron sce-nario; –
70% of GRBs in the sample have more than 50% of theirspectra with a low energy photon index within the limitsof the synchrotron model, − / ≤ α ≤ − / α in therange − / ≤ α ≤ − – Only 20% of GRBs in the sample have more than 50% oftheir spectra with α > − /
3. However most of these spectrahave α close to − /
3. For instance, only 5% of GRBs in thesample have more than 50% of spectra with α > − . α is steeper than the expected slope for the fast cool-ing regime, α = − /
3. Synchrotron radiation in radiatively efficientregime
In this section, all quantities are given in the comoving frame ofthe emitting region.
The synchrotron power of an electron with Lorentz factor γ isgiven by P syn ( γ ) = σ T c π B γ (1)where B is the magnetic field. If the source is relativisticallyexpanding, as expected in gamma-ray bursts, adiabatic coolingcompetes with radiation. This cooling process occurs on a typ-ical dynamical timescale t ex . It is then convenient to define Γ c as the Lorentz factor of electrons whose synchrotron radiativetimescale equals t ex (Sari et al. 1998) : Γ c = π m e c σ T B t ex . (2)If the GRB prompt emission comes from the radiation of rel-ativistic electrons, it is necessary, both to allow for the short-est timescales observed in GRB lightcurves and to minimize theconstraint on the energy budget, that these electrons are radia-tively e ffi cient, i.e. that their radiative timescale is shorter than t ex (Rees & M´esz´aros 1994; Sari et al. 1996; Kobayashi et al.1997). When only synchrotron radiation is considered, this isequivalent to the condition that most injected electrons by the ac-celeration process must have γ > Γ c . The resulting synchrotronspectrum has been described by Sari et al. (1998) when the ini-tial distribution of relativistic electrons is a power-law of slope Daigne, Boˇsnjak & Dubus: Reconciling observed GRB prompt spectra with synchrotron radiation ?
Fig. 2. The e ff ect of inverse Compton scatterings in Klein-Nishina regime on the fast cooling synchrotron spectrum. Thenormalized synchrotron spectrum defined by ν u ν | syn / n acce Γ m m e c is plotted as a function of the normalized frequency ν/ν m , as wellas the corresponding photon index d ln u ν / d ln ν −
1. All spectra in thick solid line are computed numerically using a detailed radiativecode including synchrotron radiation and inverse Compton scatterings (see text). Spectra in thin dotted line are computed withsynchrotron radiation only. For clarity purposes, all other processes (adiabatic cooling, synchrotron self-absorption, γγ annihilation)are neglected. A ratio Γ c / Γ m = − is assumed to ensure that all electrons are in fast cooling regime. The maximum Lorentz factorof electrons is fixed to Γ max = Γ min so the high energy cuto ff in the synchrotron spectrum appears at the same frequency inall cases plotted here. The four panels correspond to increasing values of the w m parameter, w m = .
01, 1, 100 and 10 fromthe top-left to the bottom-right panel, i.e. to a growing importance of Klein-Nishina corrections for inverse Compton scatterings.In each panel, the six curves in solid line correspond to increasing values of the Y parameter, Y = .
1, 1, 10, 10 , 10 and 10 ,i.e. to a growing e ffi ciency of inverse Compton scatterings. The table inserted in each panel lists the values of the ratio E ic / E syn of the inverse Compton component (not plotted here) over the synchrotron component. In the bottom-left panel ( w m = t ex (see text) is also plotted in dashed line for comparison. − p above a minimum Lorentz factor Γ m . If synchrotron self-absorption is neglected, three asymptotic branches are predicted ν u ν | syn n acce Γ m m e c ≃ (cid:16) ν c ν m (cid:17) / (cid:16) νν c (cid:17) / if ν < ν c (cid:16) νν m (cid:17) / if ν c ≤ ν ≤ ν m (cid:16) νν m (cid:17) − p − if ν > ν m , (3) where u ν is the final photon energy density at frequency ν and n acce is the initial density of relativistic electrons. The break fre- aigne, Boˇsnjak & Dubus: Reconciling observed GRB prompt spectra with synchrotron radiation ? 5 quency ν m (resp. ν c ) is defined as the synchrotron frequency foran electron with Lorentz factor Γ m (resp. Γ c ). This asymptoticspectrum shows clearly that the predicted photon spectral slope α below the peak of ν F ν is − /
2, in apparent contradition withobservations.
The mean value of α observed in the BATSE spectroscopic cat-alog (see Sect. 2) is close to −
1. As seen in Fig. 1, for typicalvalues of the high-energy photon index β between − − α = − . − −
40 % of the energy from thesynchrotron component into another component. A natural can-didate is inverse Compton scattering of synchrotron photons byrelativistic electrons. Indeed, this process necessarily takes placein the emitting region. Two parameters can be introduced to eval-uate the importance of inverse Compton scatterings. The first pa-rameter, w m , measures if scatterings occur mostly in Thomsonregime ( w m ≪
1) or if Klein-Nishina corrections are important.It is defined by w m = Γ m ǫ m . (4)The second parameter, Y Th , measures the intensity of the inverseCompton process in the Thomson regime and is defined by Y Th = σ T n acce ct syn ( Γ m ) × Γ , (5)where t syn ( γ ) = γ m e c P syn ( γ ) = t ex Γ c γ (6)is the synchrotron timescale of an electron with Lorentz factor γ .In Eq. 5 the first term, σ T n acce ct syn , is the Thomson optical depthassociated with relativistic electrons in fast cooling regime,and the second term, Γ , corresponds to the typical boost ofa photon scattered by an electron at Lorentz factor Γ m , if thescattering occurs in Thomson regime ( w m ≪ w m ≪ Y Th ≪
1, the ratio of the total energy E ic radiatedin the inverse Compton component over the total energy E syn radiated in the synchrotron component is simply given by E ic / E syn ≃ Y Th . Still in the Thomson regime ( w m ≪ Y Th is large, the e ff ective radiative timescale becomes ∼ t syn / Y Th and E ic / E syn ≃ √ Y Th . Finally, when Klein-Nishina correctionsare important ( w m > ∼ E ic / E syn ≪ Y Th (seeBoˇsnjak et al. 2009, for details).If the magnetic energy density represents a fraction ǫ B of thelocal energy density in the emitting region, and if the energy in-jected into accelerated relativistic electrons represents a fraction ǫ e of the same energy reservoir, Eq. 5 can be simplified to give Y Th = p − p − ǫ e ǫ B . (7)When considering only synchrotron radiation and inverseCompton scatterings, the spectral shape of the synchrotron com-ponent, i.e. ν u ν | syn / n acce Γ m m e c as a function of ν/ν m , dependsonly on these two parameters w m and Y Th , in the limit of ex-treme fast cooling ( Γ c ≪ Γ m ). It is well known that in Thomson regime ( w m ≪ γ like for the syn-chrotron power, the spectral shape of the synchrotron compo-nent is un-a ff ected by the scatterings (see e.g. Sari & Esin 2001;Boˇsnjak et al. 2009). To change the value of α , the physical con-ditions in the emitting regions must necessarily be such that w m > ∼
1. Then, Klein-Nishina corrections become importantfor most of the scatterings and the dependence of the electroncooling rate on γ di ff ers from γ . Derishev et al. (2001) haveshown that it results in a steeper slope α , that can potentiallyreach the value α = −
1. We have shown in a previous paper thatthis change of the synchrotron slope due to inverse Comptonscatterings in Klein-Nishina regime was indeed observed in de-tailed radiative calculations (Boˇsnjak et al. 2009). More recently,Nakar et al. (2009) have presented a complete analytical esti-mate of the asymptotic synchrotron spectrum in the presence ofinverse Compton scatterings, and have shown that the asymp-totic slope α = − ν F ν when Γ c ≪ Γ m (fast cooling), w m ≫ w / ≤ Y ≤ w . (8)Therefore the analytical work of Derishev et al. (2001) andNakar et al. (2009) and the numerical study of Boˇsnjak et al.(2009) indicate that the e ff ect of inverse Compton scatterings inKlein-Nishina regime on the synchrotron component seems tobe a promising possibility to explain observed values of α in therange [ − / − Y Th is increas-ing, for di ff erent values of w m (and for Γ c ≪ Γ m ). The spectraare computed using the radiative code described in Boˇsnjak et al.(2009) that solves simultaneously the equations of the time-evolution of the electron and photon distributions and that in-cludes most relevant processes (adiabatic cooling, synchrotronradiation and self-absorption, inverse Compton scatterings andphoton–photon annihilation). To focus on the e ff ect describedin this subsection, the spectra in Fig. 2 are computed includingonly synchrotron radiation and inverse Compton scatterings anddo not take into account the other processes, whose impact is dis-cussed later in the paper. The slope α is clearly steepening con-tinuously from − / −
1. In Fig. 3, we have color-coded in thediagram w m – Y Th the value of the low-energy photon index of thesynchrotron spectrum. This slope is only a representative valueas the synchrotron spectrum below the peak shows some curva-ture with an evolving slope (see Fig. 2). In practice, we adoptfor α the clear maximum of the slope below ν m . If α = − / Y Th ≪
1) or Klein-Nishina corrections unimportant( w m ≪ ff erent in the quarter of the di-agram where w m ≥ Y Th ≥
1. In particular, a region where α evolves from − / − E ic / E syn are also plotted. In agree-ment with the analysis made at the beginning of this subsection,this ratio is typically in the range 0.1-1 in the region of inter-est (it is of course much smaller than Y Th due to Klein-Nishina In Nakar et al. (2009), the situation where the slope α = − (cid:16) Γ m / ˆ Γ m (cid:17) / < ǫ e /ǫ B < (cid:16) Γ m / ˆ Γ m (cid:17) , where the authors de-fine ˆ γ = m e c / h ν syn ( γ ) so that w m = Γ m / ˆ Γ m , leading to the conditionsgiven in Eq. 8. Daigne, Boˇsnjak & Dubus: Reconciling observed GRB prompt spectra with synchrotron radiation ? Fig. 3. The low-energy slope α of the fast cooling synchrotronspectrum in the presence of inverse Compton scatterings inKlein-Nishina regime. In the w m – Y Th plane, the value of thephoton index α of the synchrotron spectrum below the peak of ν F ν is color-coded. All spectra have been computed numericallyusing a detailed radiative code including synchrotron radiationand inverse Compton scatterings (see text). Black thick lines ofconstant ratio L ic / L syn are plotted on top of this diagram. Caseswhere L ic / L syn > α → − α ≃ − w m = − and Y > ∼ w m .The maximum value of α that can be reached dependson the assumptions about the electron acceleration process.In principle, one would wish to follow the full ’magneto-hydrodynamical’ evolution of the shocked region at the plasmascale. The framework used in the present study to follow thedynamics of shocks in a relativistic outflow limits us to makesimple assumptions on the electron injection timescale. Two ex-treme cases are possible. If electrons are injected regularly overa timescale t injec comparable to t ex (as assumed in Nakar et al.2009), the slope will never be steeper than α = − w m and Y parameters,see Nakar et al. 2009 and bottom-left panel in Fig. 2). If the in-jection occurs faster, the limit α = − Y th = and w m =
100 or 10000 in Fig. 2). Calculations presented in this pa-per corresponds to the regime where t injec ≪ t ex . The process described in the previous subsection o ff ers a physi-cal interpretation of the observed values of the low-energy pho-ton index in the range [ − / − α = − α andan additional explanation has to be found for the steeper slopes.When electrons are in slow cooling regime ( Γ c ≫ Γ m ), thepredicted value of the photon index below the peak of ν F ν is α = − / − ≤ α ≤ − / ffi ciency f rad = u γ / u acce ,where u acce is the initial energy density injected in relativisticelectrons and u γ the final energy density of the radiated photons.In the slow cooling regime f rad is low, which increases the re-quired energy budget to an uncomfortable level. Here, we ratherconsider the situation where electrons are in fast cooling regimebut not deeply in this regime, i.e. Γ c < ∼ Γ m rather than Γ c ≪ Γ m (”marginally fast cooling regime”).To illustrate this situation we plot in Fig. 4 the evolution ofthe synchrotron spectrum for an increasing ratio Γ c / Γ m and fordi ff erent values of ( w m ; Y Th ) representative of the di ff erent re-gions in the diagram of Fig. 3. As expected a break in the spec-trum appears at frequency ν c , e ff , i.e. at the synchrotron frequencyof electrons with Lorentz factor Γ c , e ff , whose radiative timescaleis equal to the dynamical timescale t ex . We have Γ c , e ff ≤ Γ c dueto inverse Compton scatterings.The photon index below ν c , e ff is − /
3. Therefore, when ν c , e ff is close to ν m the observed photonindex can be very close to this asymptotic value, even in fastcooling regime. This is well seen in Fig. 4 for w m =
100 and Y Th =
100 (top-right panel) or w m = and Y Th = (bottom-right panel) and for Γ c / Γ m = f rad ≃ . − . ffi ciency of inverse Compton scatterings is reduced, ν c , e ff is closer to ν c and the same e ff ect can be seen for lower val-ues of the ratio Γ c / Γ m . This is the case for instance for w m = . Y Th = . w m = and Y Th = Γ c / Γ m = . − f rad ≃ . − . → ff ect of adiabatic cooling for di ff erent values ofthe ratio Γ c / Γ m . The representative value of α is selected inthe same way as in Fig. 3, but the maximum is not always asclearly defined as in the Γ c / Γ m = Γ c / Γ m = . ν m disappears and α is given the asymptotic value − / ffi cult : depending on thelocation of the peak energy E p , and of the low-energy thresholdof the instrument, any value of α between − − / Γ c / Γ m = . α = − / w m − Y Th plane, together with a large radiative e ffi ciency.Even for Γ c / Γ m = α = − / ffi ciency is still larger than 66%. It is onlyfor Γ c / Γ m =
10 that the slow cooling regime dominates thediagram, high radiative e ffi ciency being found together with α = − / aigne, Boˇsnjak & Dubus: Reconciling observed GRB prompt spectra with synchrotron radiation ? 7 Fig. 4. The e ff ect of adiabatic cooling on the fast cooling synchrotron spectrum in presence of inverse Compton scatterings. The normalized synchrotron spectrum defined by ν u ν | syn / n acce Γ m m e c is plotted as a function of the normalized frequency ν/ν m ,as well as the corresponding photon index d ln u ν / d ln ν −
1. All spectra in thick solid line are computed numerically using adetailed radiative code including synchrotron radiation, inverse Compton scatterings and adiabatic cooling (see text). Spectra in thindotted line are computed without inverse Compton scatterings. All other processes (synchrotron self-absorption, γγ annihilation) areneglected. Each panel corresponds to a di ff erent set of parameters ( w m , Y Th ) indicated in the top-left corner. In each panel, spectraare plotted for increasing ratios Γ c / Γ m = .
01, 0 . Γ max / Γ min = . Spectra in dashed lines are radiatively ine ffi cient (slow cooling regime). The table inserted ineach panel lists the values of the radiative e ffi ciency f rad of the electrons. Other processes may influence the final spectral shape.Photon-photon annihilation produces a cuto ff at high energy(Granot et al. 2008; Boˇsnjak et al. 2009). For all cases presentedin this paper, we have checked that the opacity for this processwas extremely low below ∼
100 MeV. Photon photon annihi- lation can a ff ect the tail of the inverse Compton componentat high-energies. The fraction of the radiated energy which isreinjected in pairs via γγ → e + e − is usually, but not always,small in the examples shown in Sect. 4. For example, in thethree cases defined in Sect. 4, it is typically less than 10 − in case C, less than 0 .
05 in case A and between 0 . . Daigne, Boˇsnjak & Dubus: Reconciling observed GRB prompt spectra with synchrotron radiation ?
Fig. 5. The low-energy slope α of the synchrotron spectrum in the presence of inverse Compton scatterings in Klein-Nishinaregime, including the e ff ect of adiabatic cooling. Same as Fig. 3, now including adiabatic cooling for Γ c / Γ m = − , 10 − , 1and 10. Black solid lines of constant radiative e ffi ciency f rad are plotted on top of the three last diagrams. In the first panel, theradiative e ffi ciency is always close to 100 %. For larger ratio Γ c / Γ m , the radiative e ffi ciency remains larger than respectively 96 %( Γ c / Γ m = − ), 65 % ( Γ c / Γ m =
1) and 28 % ( Γ c / Γ m = γγ annihilation could be even more important.When the fraction of the energy in annihilated photons is nonnegligible, the resulting radiation of the created pairs coulda ff ect the spectral shape even at low energy, and modify theresults presented here. Despite its potentially interesting impact,we defer to future work the investigation of such cases, due tothe additional complexity it involves for the computation of theradiated spectrum. Numerical approaches to solve such a highlynon linear problem including thermalization e ff ects have been proposed by Pe’er & Waxman (2005); Asano & Inoue (2007);Belmont et al. (2008); Vurm & Poutanen (2009)At low energy, the synchrotron self-absorption can alsosteepen the spectrum. This e ff ect is included in simulations pre-sented in Sect. 4 and is always negligible in the soft gamma-ray domain, in agreement with the standard predictions forthe synchrotron fast cooling regime. Indeed the timescale for aigne, Boˇsnjak & Dubus: Reconciling observed GRB prompt spectra with synchrotron radiation ? 9 self-absorption at ν m is given by (see e.g. equation (28) inBoˇsnjak et al. 2009) : t a ( ν m ) t ex ≃ πν n acce c ≃ . × (cid:18) ν m (cid:19) (cid:18) t ex (cid:19) τ acce − ! − , (9)where 1 keV is taken for a typical value of the peak energyin the comoving frame and other parameters are given rep-resentative values for internal shocks. The timescale for self-absorption around ν m is therefore always much larger than allother timescales (dynamical or radiative) and the self-absorptionprocess is negligible in the soft gamma-ray range.In addition to the details of the radiative processes, the pre-cise shape of the electron distribution can also have an impacton the final spectrum. Here, we assume a power-law distribu-tion. More complex distributions showing several components(e.g. Maxwellian distribution + non-thermal tail) are observedin some simulations of particle acceleration in ultra-relativisticshocks (Spitkovsky 2008b,a; Martins et al. 2009). Such resultswould need to be confirmed for the mildly relativistic regimeof interest for the prompt GRB emission. In the ultra-relativisticregime relevant for the afterglow, Giannios & Spitkovsky (2009)have shown that the Maxwellian component could have an ob-servable signature. However Baring & Braby (2004) find that thenon-thermal electron population should dominate in the promptphase. We leave to a future work the study of the consequencesof more complex electron distributions on the observed GRBprompt spectra.
4. Constraints on the internal shock model
We have shown in Sect. 3 that the spectrum resulting fromsynchrotron radiation in the presence of inverse Comptonscatterings in Klein-Nishina regime can account for observedlow-energy photon index α = − / −
1, and that the additionale ff ect of adiabatic cooling with Γ c < ∼ Γ m can lead to steeperslopes up to α = − / ffi ciencyis still reasonably high ( f rad > ∼
50 %). In principle, this allowsto reconcile the synchrotron process with the observed spectralparameters in most GRB spectra (see Sect. 2). However it is stillnecessary to demonstrate that the physical conditions identifiedin Sect. 3 can be reached in the emitting regions within GRBoutflows. These conditions are approximatively w m > ∼ w / ≤ Y Th < ∼ w for changing the slope from α = − / − ff ect of IC scatterings in Klein-Nishina regime, and Γ m / Γ c ≃ . − α = − / ff ect of adiabatic cooling.We assume that the prompt gamma-ray emission is producedin a relativistic outflow ejected by a source at redshift z . We con-sider an emitting region at radius R within the outflow, with aLorentz factor Γ ∗ . We do not specify at this stage the physicalmechanism responsible for the energy dissipation in this region,leading to the presence of a magnetic field B and a population ofrelativistic electrons with a minimum Lorentz factor Γ m . We as-sume that the medium is optically thin, i.e. τ T = σ T n acce ct ex ≤ + z ) h ν m , obs ≃
380 keV Γ ∗ ! Γ m ! (cid:18) B (cid:19) . (10) Fig. 6. Exploration of the internal shock parameter space (1).
All cases fulfilling the conditions of (i) transparency; (ii) radia-tive e ffi ciency; and (iii) synchrotron dominance at low-energy(see text) are plotted with black dots in three planes : E p , obs vs α (top-left panel); E ic / E syn vs α (bottom-left panel); E p , obs vs E ic / E syn (top-right panel). The ratio E ic / E syn does not takeinto account the fraction of high energy photons that are sup-pressed by γγ annihilation. In addition, the cases where the syn-chrotron spectrum peaks in the gamma-ray range (10 keV ≤ E p , obs ≤
10 MeV) and the low-energy photon index is in therange − / ≤ α ≤ − − ≤ α ≤ − /
3) are plotted inred (resp. blue).The dynamical timescale relevant for adiabatic cooling can beestimated by t ex = . (cid:18) R cm (cid:19) Γ ∗ ! − . (11)From Eq. 2, this leads to Γ c Γ m ≃ . (cid:18) R cm (cid:19) − Γ ∗ ! (cid:18) B (cid:19) − Γ m ! − . (12)The two parameters governing inverse Compton scatterings aregiven by Eq. 4 and Eq. 5 : w m ≃ Γ m ! (cid:18) B (cid:19) (13)and Y Th ≃ (cid:18) τ T − (cid:19) (cid:18) R cm (cid:19) − Γ ∗ ! (cid:18) B (cid:19) − Γ m ! . (14)Lorentz factors Γ ∗ above 100 are necessary to avoid the pres-ence of a high-energy cuto ff in the spectrum due to γγ an-nihimation (see e.g. Lithwick & Sari 2001). Higher values are Fig. 7. Exploration of the internal shock parameter space (2).
The distribution of the three parameters defining the relativeimportance of inverse Compton scatterings, Klein-Nishina corrections and adiabatic cooling, Y th , w m and Γ c / Γ m are plotted forall cases presented in Fig. 6. In addition the distributions of the same parameters for cases where the synchrotron spectrum peaksin the gamma-ray range (10 keV ≤ E p , obs ≤
10 MeV) and the low-energy photon index is in the correct range are also plotted( − / ≤ α ≤ −
1: red; − ≤ α ≤ − /
3: blue).even required in some bursts detected by
Fermi / LAT. For in-stance Γ > ∼ −
900 has been derived by the
Fermi / LAT col-laboration for GRB 080916C (Abdo et al. 2009). Radii in therange 10 –10 cm are expected as the prompt GRB emissionshould mainly occur above the photospheric radius and belowthe deceleration radius. Then, the main constraint comes fromthe fact that in the proposed scenario gamma-ray photons mustbe produced directly by synchrotron emission. From Eq. 10, thisis always possible if electrons can be accelerated to very highLorentz factors. Then Eq. 13 and (14) show that the physicalconditions listed above and leading to − / ≤ α ≤ − α → − / The internal shock model allows a self-consistent evaluation ofthe physical conditions (i.e. Γ ∗ , R , B , Γ m , etc.) for each emittingregion. It is then possible to identify the pertinent range of theparameters leading to steep low-energy slopes : the propertiesof the relativistic outflow (Lorentz factor, kinetic energy) andthe parameters describing the microphysics at work in shockedregions (particle acceleration, magnetic field amplification).We consider first collisions between two equal-mass shells(Barraud et al. 2005; Boˇsnjak et al. 2009). More realistic out-flows are considered in the next subsection. The parametersdefining the dynamical properties of a collision are the meanLorentz factor in the outflow ¯ Γ , the ratio κ of the Lorentz factorof the faster shell over the Lorentz factor of the slower shell,the time separation τ between the ejection of the two shellsand the kinetic energy flux ˙ E injected in the outflow. These 4parameters allow to estimate the radius of the collision, as wellas the physical conditions in the shocked region (Lorentz factor Γ ∗ , comoving density ρ ∗ and comoving specific energy density ǫ ∗ ). In addition, four microphysics parameters are necessaryto estimate the distribution of relativistic electrons and themagnetic field : ǫ B and ǫ e are the fraction of the energy density ρ ∗ ǫ ∗ that is injected in the magnetic field and the relativisticelectrons, respectively. We assume that only a fraction ζ of theavailable electrons are accelerated in a non-thermal distributionand that this distribution is a power-law with index − p . The fulldescription of this model can be found in Boˇsnjak et al. (2009).We explore the parameter space of this model, assuminga constant value ǫ e = / p = .
5. A high value of ǫ e seems unavoidable in internalshocks to maintain a reasonable e ffi ciency of the process. Wehave checked that our results are not a ff ected much by taking ǫ e = .
1. As we are mainly interested in the low-energy photonindex α , the choice of p is not very important. The present value p = .
5, leads to a high-energy photon index β ≃ − .
25 closeto the mean value observed in BATSE spectroscopic catalog(Preece et al. 2000). We then compute the observed spectrumof 50400 collisions for log ¯
Γ = κ = τ = -2, -1, 0, 1 and 2; log ˙ E =
50, 51, 52, 53,54 and 55; log ζ = -4, -3, -2, -1 and 0; log ǫ B = − . → − . γγ annihilation.We keep only cases which fulfill the following conditions :(i) the shocked region is transparent ( σ T n ± ct ex < .
1, where n ± is the final lepton density in the shocked region, taking intoaccount pairs that were created by γγ annihilation but neglectingpair annihilation, see Boˇsnjak et al. 2009) ; (ii) electrons areradiatively e ffi cient ( f rad > .
5) ; (iii) synchrotron radiationis dominant at low-energy ( u ν | syn > u ν | ic at the frequencyof the synchrotron peak). This last condition eliminates a fewcases where inverse Compton scatterings are so e ffi cient that thesynchrotron component is hardly observed. After this selection, ∼
75% of cases are suppressed, because of a too large opticaldepth (condition (i): ∼
50% of cases), a too low radiativee ffi ciency (condition (ii): ∼
20% of cases) or a negligiblesynchrotron emission (condition (iii): ∼
10% of cases). Foreach spectrum, assuming a source redshift z =
1, we computethe observed peak energy of the synchrotron component E p , obs , aigne, Boˇsnjak & Dubus: Reconciling observed GRB prompt spectra with synchrotron radiation ? 11 the low-energy photon index α below the peak, and the ratio E ic / E syn of the inverse Compton component over the syn-chrotron component. All models fulfilling the three conditionslisted above are plotted in Fig. 6. This figure illustrates that theinternal shock model with a dominant synchrotron process inthe soft gamma-ray range (BATSE, Fermi / GBM) allows a largerange of low-energy photon index between − / − / α ≃ − ff ect ofIC scatterings in KN regime on the synchrotron spectrum; andvery low peak energies as expected in X-ray rich gamma-raybursts or X-ray flashes, Heise et al. 2001; Sakamoto et al. 2005,2008). All these cases have f rad > . f rad > . ≤ E p , obs ≤
10 MeV, the ratio E ic / E syn is typically in the range 10 − –1. This is in agreementwith the indication from the Fermi / LAT GRB detection ratethat most GRBs do not have a strong additional componentbetween 100 MeV and 10 GeV (Granot et al. 2010a). Notethat the density of points in Fig. 6 has no physical meaning asthe distribution of the physical parameters in GRB outflowsis unknown. This figure only illustrates the range of observedvalues that can be expected in the internal shock model.We plot in Fig. 7 the distributions of w m , Y Th and Γ c / Γ m forthe same models. The values of the low-energy photon index α are in full agreement with the analysis made in section Sect. 3.How are such values obtained ? The distributions of dynamical( ¯ Γ , κ , τ and ˙ E ) and microphysics ( ζ and ǫ B ) parameters for allmodels that fulfill the three conditions listed above and have inaddition a synchrotron spectrum that peaks in gamma-rays witha low-energy index in the expected range ( − / ≤ α ≤ − / ζ are needed to produce gamma-rays(typically ζ = − − − ), and that low values of ǫ B favorsteeper low-energy photon indexes. In a more realistic description of the internal shock model, eachpulse in GRBs with complex multi-pulses lightcurves is associ-ated with the propagation of a shock wave within the relativisticoutflow (Daigne & Mochkovitch 2000; Mimica et al. 2007;Mimica & Aloy 2010). This propagation implies an evolutionof the physical quantities in the shocked region, especially thedensity and therefore the magnetic field. This leads to a spectralevolution within each pulse that has already been partially de-scribed in Daigne & Mochkovitch (1998, 2003); Boˇsnjak et al.(2009). The model developed by Boˇsnjak et al. (2009) couplesa dynamical simulation of internal shock propagation within arelativistic outflow, and a detailed radiative code. This allowsto predict the lightcurves and spectral evolution in pulses fordi ff erent assumptions regarding the physical conditions inthe outflow. In order to compare the results with observeddistributions of spectral parameters, we face di ffi culties dueto several possible biases, as described in Sect. 2. To make afull comparison, one should generate noise in our syntheticbursts and take into account the response function of a giveninstrument before fitting the resulting spectrum by a Bandfunction. We did not follow this procedure as our primary goalis to identify the theoretical limits for the prediction of thelow-energy slope. We computed theoretical spectra over timebins of duration 0 .
25 s and measure the slope below the peak
Fig. 8. Case A : an example of a pulse generated by an in-ternal shock with synchrotron radiation in pure fast cool-ing.
The top panel plots the evolution of the three main pa-rameters shaping the radiated spectrum (see Sect. 3) : w m , Y Th and Γ c / Γ m . The four other panels show the lightcurves in dif-ferent energy channels corresponding to the GBM and the LATon board Fermi . The respective contributions of synchrotron andinverse Compton are also indicated. In the two GBM panels, theevolution of the low-energy slope and peak energy of the softgamma-ray component is also plotted. In this case, the standardfast cooling synchrotron slope α = − / t w = E and a Lorentz factor increasing from 100 to 400 (see Figure 1in Boˇsnjak et al. 2009). Constant microphysics parameters areassumed during the whole evolution. This is a simplifying as-sumption due to our poor knowledge of the physical processesat work in mildly relativistic shocks. However, as the diversity ofGRBs and their afterglows seem to indicate that these parametersare not universal, they are most probably evolving with shockconditions, which could impact the spectral evolution within apulse (Daigne & Mochkovitch 2003). We adopt here ǫ e = / p = . ζ and ǫ B to have the peak energy of thepulse well within the GBM range. We consider the followingexamples to illustrate the possible range of α : – Case A : ˙ E = erg / s, ǫ B = / ζ = × − . This caseis plotted in Fig. 8 and shows the standard slope α = − / E p , obs ≃
800 keV at the peak of the pulse. – Case B : ˙ E = erg / s, ǫ B = − and ζ = − . This caseis plotted in Fig. 9 and shows a steeper α ≃ −
1. The peakenergy is E p , obs ≃
700 keV at the peak of the pulse.Intermediate values of ǫ B between case A and B would leadto intermediate values of α between − / −
1. In bothexamples, it appears clearly that all spectral parameters are evolving during a given pulse. The evolution for the peak energyis stronger in case A than in case B, whereas the low-energyphoton index α evolves more strongly in the second case. Notethat in these examples, the direct emission from pulse ends at t obs ∼ . α can be expected in the range − / ≤ α ≤ − ffi cult to find microphysics parameters leading to even steeperslopes − ≤ α ≤ − /
3. This can be understood from the twoshell model presented in the previous subsection. To reach thenecessary condition Γ c / Γ m ∼ . −
1, it is necessary to havecollisions at lower radii, and / or with larger bulk Lorentz fac-tor, and / or to reduce the magnetic field (see Eq. 12). This can beachieved in di ff erent ways : decreasing the contrast κ , increas-ing the variability timescale τ , increasing the Lorentz factor ¯ Γ ,or reducing the kinetic energy flux ˙ E . In the following example,both the Lorentz factor and the kinetic energy flux have beenchanged : – Case C : the dynamics is the same as in case A and B ex-cept that the Lorentz factor has been multiplied by 3 and thekinetic energy flux reduced to ˙ E = × erg / s. The mi-crophysics parameters are ǫ B = . ζ = − . This caseis plotted in Fig. 10 and shows low-energy slopes α steeperthan − E p , obs ∼
170 keV at the peak of the pulse. The ra-diative e ffi ciency is still reasonably high ( ∼
60 %) howeverthe end of the evolution occurs in slow cooling regime whichresults in a more complex evolution of the peak energy in thetail of the pulse than in the two previous examples.This example illustrates that the ”marginally fast coolingregime” does provide low-energy slopes α > −
1. However, fol-lowing Fig. 6, the conditions require a smaller radius and / or alow magnetic field. This tends to favor less energetic internalshocks. Interestingly, as already pointed out in Boˇsnjak et al. (2009),the scenario presented in this section – internal shocks withdominant synchrotron radiation in the soft gamma-ray range –require high Lorentz factors for electrons, which, because ofKlein-Nishina corrections, always limits the e ffi ciency of inverseCompton scatterings. So the high-energy spectrum ( Fermi / LATrange) does not show any bright additional component simul-taneously with the peak of the pulse in the GBM range, whichseems in agreement with the GRB detection rate of
Fermi / LAT.However, as described in details in Boˇsnjak et al. (2009), thephysical conditions in the shocked region evolve during the in-ternal shock propagation in such a way that the w m parameterdecreases in the tail of the pulse (see top panel in Figs. 8–10).Inverse Compton scattering progressively enters the Thomsonregime and becomes more e ffi cient, especially in the low ǫ B (high Y Th ) case favored for steep low-energy slopes (see case Fig. 9. Case B : an example of a pulse generated by an inter-nal shock with synchrotron radiation in fast cooling regimea ff ected by non negligible inverse Compton scatterings inKlein-Nishina regime. Same as in Fig. 8. The slope α is steeper( − / < α < − w m but higher values of Y Th (see thetheoretical interpretation in Sect. 3).B in Fig. 9). This leads to the delayed emergence of an addi-tional component in the high-energy spectrum. We will investi-gate in the future if this e ff ect could explain the behaviour ob-served in Fermi / LAT GRB lightcurves where delays between theGeV and the keV-MeV emission are indeed observed (see for in-stance GRB 080916C, Abdo et al. 2009).
5. Discussion and conclusions
We present here a detailed discussion of the expected valuefor the low-energy slope α of the soft gamma-ray component(BATSE – Fermi / GBM range) in prompt GRBs, in the theo-retical framework where the kinetic energy of the outflow isextracted by internal shocks, and eventually injected in shock-accelerated electrons that radiate dominantly by synchrotronradiation. Our approach is based on the model developed inBoˇsnjak et al. (2009), which takes into account both a full treate-ment of the dynamics of internal shocks and a detailed radiativecalculation.1. We show that in a large region of the parameters space of theinternal shock model, the physical conditions in the emit-ting regions allow a combination of synchrotron radiationpeaking in the soft gamma-ray range, and moderately e ffi -cient inverse Compton scatterings in Klein-Nishina regime.Interestingly, these scatterings a ff ect the spectral shape of thesynchrotron component, due to a better e ffi ciency for lowfrequency photons. This results in a steepening of the low-energy photon index α , with α → − aigne, Boˇsnjak & Dubus: Reconciling observed GRB prompt spectra with synchrotron radiation ? 13 Fig. 10. Case C : an example of a pulse generated by aninternal shock with synchrotron radiation in ”marginallyfast cooling”.
Same as in Fig. 8 and 9. The slope α is steeper( − < α < − / Γ c / Γ m . This ratio is of the order of 0 . − − / ≤ α ≤ −
1, at the peak of the lightcurve. The examplespresented in the paper not only show high peak energies andsteep slopes at the peak of the lightcurve, but also a generalhard-to-soft spectral evolution, in agreement with observa-tions. This scenario constrains the microphysics in mildlyrelativistic shocks (shock acceleration and magnetic fieldamplification) : a large ( ǫ e ∼ . /
3) fraction of the dis-sipated energy should be preferentially injected into a small( ζ < ∼ .
01) fraction of electrons to produce a non-thermalpopulation with high Lorentz factors, and the fraction of theenergy which is injected in the magnetic field should remainlow ( ǫ B < ∼ − ) to favor inverse Compton scatterings. Thecurrent knowledge of the microphysics in mildly relativisticshocks is unfortunately still rather poor and does not allowa comparison of these constraints with some theoretical ex-pectations. The prediction that only a small fraction ζ of elec-trons are injected into a non-thermal power-law distributionleads naturally to a new question that we plan to investigatein the future : does the remaining quasi-thermal populationof electrons contribute to the emission ?2. We also identify a regime of marginally fast-cooling syn-chrotron radiation with Γ c < ∼ Γ m which leads to even steeperslopes α → − / ffi ciency of the process ( f rad > ∼ . α > − / or large bulkLorentz factors, and / or weak magnetic fields.The present study shows that for a large region of theparameter space, internal shocks lead to spectra dominated by a bright synchrotron component in the soft gamma-ray range,with a steep slope low-energy photon-index − / ≤ α ≤ − Fermi / LAT observations and GRB detectionrate than other scenario, such as standard SSC in Thomsonregime, where bright components are predicted at high energy.On the other hand, even if steeper slopes in the range − ≤ α ≤ − / ffi ciency f rad ≃ . − . f rad → α steeper than − /
3, whereas such slopesare measured in a non negligible fraction of GRB spectra(Ryde 2004; Ghirlanda et al. 2003). About 20 % of GRBshave more than 50 % of their spectra with such very steepslopes. One possibility in such cases would be the appear-ance of a quasi-thermal component of photospheric origin(M´esz´aros & Rees 2000; Daigne & Mochkovitch 2002; Pe’er2008; Beloborodov 2010; Pe’er et al. 2010). The emissionfrom both the photosphere and internal shocks have a similarduration, the latter having only a very short lag behind the first.The intensity of the photospheric emission depends strongly onthe unknown mechanism responsible for the acceleration of therelativistic outflow. A pure fireball would lead to a dominantthermal emission, whereas mechanisms such as the ”magneticrocket” recently suggested by Granot et al. (2010b) would onthe other hand correspond to much colder jets with only a faintphotospheric emission. In addition, the way it will superimposeon the non-thermal emission from internal shocks depends onthe details of the initial distribution of the Lorentz factor inthe flow (see Daigne & Mochkovitch 2002). Very steep slopescould be observed in the time bins where the photosphericemission is dominant.Other e ff ects could be a source of additional complexity inthe prompt GRB spectrum within the scenario proposed in thispaper. As it is required that only a fraction of available electronsare shock accelerated into a non-thermal power-law distribution,possible extra components in the spectrum could be associatedwith the remaining thermal population of electrons. Preliminaryinvestigations show that the synchrotron radiation from theseelectrons is easily self-absorbed, in agreement with Zou et al.(2009). On the other hand, due to lower electron Lorentz fac-tors, inverse Compton scatterings by these electrons usually oc-cur in Thomson regime, which is e ffi cient. This could lead toadditional components at low and / or high energy that o ff er in-teresting perspectives for the interpretation of the complex be-haviour observed in GRBs detected by Fermi / LAT. We also findthat in some cases, a non negligible fraction of the radiated en-ergy is re-injected in pairs due to γγ annihilation. These pairscan radiate and scatter photons as well. These second order ef- fects are not included in the present version of the model andcould impact the final spectra shape. Acknowledgements.
The authors thank R. Mochkovitch, A. Pe’er, P. Kumar andE. Nakar for valuable discussions on this work. The authors thank Y. Kaneko forher help regarding BATSE GRB spectral results. The authors acknowledge sup-port from the Agence Nationale de la Recherche via contract ANR–JC05–44822.Z.B. and F.D. acknowledge the French Space Agency (CNES) for financial sup-port. G.D. acknowledges support from the European Community via contractERC–StG–200911.
References
Abdo, A. A., Ackermann, M., Arimoto, M., et al. 2009, Science, 323, 1688Asano, K. & Inoue, S. 2007, ApJ, 671, 645Asano, K. & Terasawa, T. 2009, ApJ, 705, 1714Band, D., Matteson, J., Ford, L., et al. 1993, ApJ, 413, 281Baring, M. G. & Braby, M. L. 2004, ApJ, 613, 460Barraud, C., Daigne, F., Mochkovitch, R., & Atteia, J. L. 2005, A&A, 440, 809Belmont, R., Malzac, J., & Marcowith, A. 2008, A&A, 491, 617Beloborodov, A. M. 2010, MNRAS, 407, 1033Borgonovo, L. & Ryde, F. 2001, ApJ, 548, 770Boˇsnjak, ˇZ., Daigne, F., & Dubus, G. 2009, A&A, 498, 677Crider, A., Liang, E. P., Smith, I. A., et al. 1997, ApJ, 479, L39 + Daigne, F. & Mochkovitch, R. 1998, MNRAS, 296, 275Daigne, F. & Mochkovitch, R. 2000, A&A, 358, 1157Daigne, F. & Mochkovitch, R. 2002, MNRAS, 336, 1271Daigne, F. & Mochkovitch, R. 2003, MNRAS, 342, 587Derishev, E. V., Kocharovsky, V. V., & Kocharovsky, V. V. 2001, A&A, 372,1071Drenkhahn, G. & Spruit, H. C. 2002, A&A, 391, 1141Ford, L. A., Band, D. L., Matteson, J. L., et al. 1995, ApJ, 439, 307Ghirlanda, G., Celotti, A., & Ghisellini, G. 2003, A&A, 406, 879Ghirlanda, G., Nava, L., & Ghisellini, G. 2010, A&A, 511, A43 + Ghisellini, G. & Celotti, A. 1999, A&AS, 138, 527Ghisellini, G., Celotti, A., & Lazzati, D. 2000, MNRAS, 313, L1Giannios, D. & Spitkovsky, A. 2009, MNRAS, 400, 330Giannios, D. & Spruit, H. C. 2005, A&A, 430, 1Giannios, D. & Spruit, H. C. 2007, A&A, 469, 1Granot, J., Cohen-Tanugi, J., & do Couto e Silva, E. 2008, ApJ, 677, 92Granot, J., for the Fermi LAT Collaboration, & the GBM Collaboration. 2010a,in Italian Physical Society, Vol. 102, The Shocking Universe - Gamma-RayBursts and High Energy Shock phenomena, ed. G. Chincarini, P. d’Avanzo,R. Margutti & R. Salvaterra, 177–190, arXiv:1003.2452Granot, J., Komissarov, S., & Spitkovsky, A. 2010b, ArXiv e-printsHeise, J., in’t Zand, J., Kippen, R. M., & Woods, P. M. 2001, in Gamma-rayBursts in the Afterglow Era, ed. E. Costa, F. Frontera, & J. Hjorth, 16– + Kaneko, Y., Preece, R. D., Briggs, M. S., et al. 2006, ApJS, 166, 298Kobayashi, S., Piran, T., & Sari, R. 1997, ApJ, 490, 92Krimm, H. A., Yamaoka, K., Sugita, S., et al. 2009, ApJ, 704, 1405Kumar, P. & McMahon, E. 2008, MNRAS, 384, 33Liang, E., Kusunose, M., Smith, I. A., & Crider, A. 1997, ApJ, 479, L35 + Lithwick, Y. & Sari, R. 2001, ApJ, 555, 540Lloyd, N. M. & Petrosian, V. 2000, ApJ, 543, 722Lloyd-Ronning, N. M. & Petrosian, V. 2002, ApJ, 565, 182Lyutikov, M. & Blandford, R. 2003, arXiv:astro-ph / ++