Spectral evolution in gamma-ray bursts: predictions of the internal shock model and comparison to observations
aa r X i v : . [ a s t r o - ph . H E ] A p r Astronomy&Astrophysicsmanuscript no. grbspectralevolution c (cid:13)
ESO 2018October 13, 2018
Spectral evolution in gamma-ray bursts: predictions of the internalshock model and comparison to observations ˇZeljka Boˇsnjak , , and Fr´ed´eric Daigne AIM (UMR 7158 CEA / DSM-CNRS-Universit´e Paris Diderot) Irfu / Service d’Astrophysique, Saclay, F-91191 Gif-sur-YvetteCedex, France UPMC-CNRS, UMR7095, Institut d’Astrophysique de Paris, F-75014, Paris, France Department of Physics, University of Rijeka, 51000 Rijeka, Croatiae-mail: [email protected] ; [email protected]
Received: 22.07.2013 – Accepted: 05.04.2014
ABSTRACT
Context.
Several trends have been identified in the prompt gamma-ray burst (GRB) emission: e.g. hard-to-soft evolution, pulse widthevolution with energy, time lags, hardness-intensity / -fluence correlations. Recently Fermi has significantly extended the spectralcoverage of GRB observations and improved the characterization of this spectral evolution.
Aims.
We want to study how internal shocks can reproduce these observations. In this model the emission comes from the synchrotronradiation of shock accelerated electrons, and the spectral evolution is governed by the evolution of the physical conditions in theshocked regions.
Methods.
We present a comprehensive set of simulations of a single pulse and investigate the impact of the model parameters, relatedto the shock microphysics and to the initial conditions in the ejecta.
Results.
We find a general qualitative agreement between the model and the various observations used for the comparison. All theseproperties or relations are governed by the evolution of the peak energy and photon indices of the spectrum. In addition, we identifythe conditions for a quantitative agreement. We find that the best agreement is obtained for (i) steep electron slopes ( p > ∼ . ff erent scenarios and find distinct properties – delayed onset,longer emission, and flat spectrum in some cases – suggesting that internal shocks could have a significant contribution to the promptLAT emission. Conclusions.
Spectral evolution is an important property of GRBs that is not easily reproduced in most models for the promptemission. We find that the main observed features can be accounted for in a quantitative way within the internal shock model.However the current uncertainties on shock acceleration in the mildly relativistic regime and relativistic ejection by compact sourcesprevent us from deciding if one or several of the proposed scenario are viable. It may be possible by combining observations over thewhole spectral range of
Fermi to identify in the future specific signatures imprinted by this uncertain underlying physics.
Key words.
Gamma-ray burst: general ; Shock waves ; Radiation mechanisms: non-thermal ; Methods: numerical
1. Introduction
Since the launch of the
Swift (2004) (Gehrels et al. 2004) and
Fermi (2008) satellites, there is a significantly growing sam-ple of gamma-ray bursts (GRBs) with a known redshift anda well characterized gamma-ray prompt emission (see e.g. therecent review by Gehrels & Razzaque 2013). The high-energydomain ( >
100 MeV) is currently explored by
Fermi -LAT(Atwood et al. 2009). The sample of detected bursts is still smallbut has allowed the identification of several important spec-tral and temporal properties (Omodei et al. 2009; Zhang et al.2011; Ackermann et al. 2013), that are summarized in Sect. 6.In the soft gamma-ray range, the GRB sample is much largerand not limited to the brightest bursts. Thanks to its large spec-tral range (8 keV-40 MeV),
Fermi -GBM (Meegan et al. 2009)has significantly improved the description of the GRB proper-ties in the keV-MeV range. This e ff ort follows the results al-ready obtained by several past or current missions, especiallyBATSE (Burst And Transient Source Experiment) on boardthe Compton Gamma-Ray Observatory (Kaneko et al. 2006),
Beppo-SAX (Guidorzi et al. 2011), and
HETE-2 (Lamb et al.2004; Sakamoto et al. 2005). Based on this large set of observa-tions, our current knowledge of the spectral and temporal proper-ties of the GRB prompt soft gamma-ray emission is summarizedin Sect. 2.The standard GRB model associates the prompt gamma-ray emission to internal dissipation within an ultra-relativisticoutflow ( Γ > ∼ ∼ ms), the internal shock model (Rees & Meszaros1994), where variations of the bulk Lorentz factor lead tothe formation of shock waves within the ejecta, was pro-posed for the extraction of the jet kinetic energy. The dis-sipated energy is distributed between protons, electrons, andmagnetic field; the prompt GRB emission model is due tothe synchrotron radiation of shock accelerated electrons, withan additional component due to inverse Compton scatterings. Boˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions
Detailed calculations of the expected light curves and spec-tra are available (Kobayashi et al. 1997; Daigne & Mochkovitch1998; Boˇsnjak et al. 2009; Asano & M´esz´aros 2011) and showa good agreement with observations except for a notable excep-tion, the low-energy photon index, which is usually observed tobe larger than the standard fast cooling synchrotron slope − / ffi ciency of theacceleration process and of the non-thermal emission above thephotosphere, this emission could be bright (M´esz´aros & Rees2000; Daigne & Mochkovitch 2002; Hasco¨et et al. 2013). Itproduces in principle a narrow quasi-Planckian component(Goodman 1986; Pe’er 2008; Beloborodov 2011); however dif-ferent possible sub-photospheric dissipation processes may af-fect the spectrum, especially due to the comptonization, so thatit appears as non-thermal (Thompson 1994; Rees & M´esz´aros2005; Pe’er et al. 2006; Giannios & Spruit 2007; Beloborodov2010; Vurm et al. 2011; Toma et al. 2011; Veres & M´esz´aros2012; Veres et al. 2013). The peak energy is governed by a de-tailed balance between the emission / absorption and scatteringprocesses (Vurm et al. 2013) and can reproduce the observedvalues (Beloborodov 2013, see however Zhang et al. 2012). Thelateral structure of the jet may also a ff ect the photospheric spec-trum (Lundman et al. 2013; Lazzati et al. 2013).Magnetized ejecta o ff er a third possibility. A large initialmagnetization may play a major role for the accelerationof the jet to relativistic speed (see e.g. Begelman & Li1994; Daigne & Drenkhahn 2002; Vlahakis & K¨onigl2003; Komissarov et al. 2009; Tchekhovskoy et al. 2010;Komissarov et al. 2010; Granot et al. 2011) and is al-ready invoked for this reason in some scenarios wherethe emission is due to the photosphere and / or internalshocks. However, if the ejecta is still magnetized at largedistance, magnetic reconnection can provide a new dis-sipation process (Spruit et al. 2001; Drenkhahn & Spruit2002; Lyutikov & Blandford 2003; Giannios & Spruit 2005;Zhang & Yan 2011; McKinney & Uzdensky 2012). Comparedto the previous possibilities, this model cannot provide yetdetailed predictions for the GRB light curves and spectra(see the preliminary calculation of the temporal properties byZhang & Zhang 2014).The photospheric emission should be present in all scenarios,even if very weak. On the other hand, magnetic reconnection re-quires a large magnetization at large distance which may pre-vent internal shock formation and propagation (Giannios et al.2008; Mimica & Aloy 2010; Narayan et al. 2011). Therefore,depending on the magnetization in the emission site, only oneof the two mechanisms should be at work. Recent observationsof two components in the soft gamma-ray spectrum of a fewbright Fermi / GBM bursts, one being quasi-Planckian and theother being non-thermal (Ryde et al. 2010; Guiriec et al. 2011;Axelsson et al. 2012; Guiriec et al. 2013), indicate that both thephotosphere and either internal shocks or reconnection may beindeed at work in GRBs (Hasco¨et et al. 2013).Aside from interpreting the light curves and spectra, a suc-cessful theoretical model should also reproduce the observedspectral evolution with time, which is is mainly governed bythe evolution of the peak energy of the spectrum. It can be re- lated either to the physics of the dissipative mechanism in theoutflow, or to the curvature of the emitting surface. In the firstcase, the spectral evolution is due to an intrinsic evolution ofthe physical conditions in the flow, whereas it is a geometri-cal e ff ect (delay, Doppler shift) in the second case. The spec-tral evolution in a pulse associated to the curvature e ff ect hasbeen studied by several authors and does not agree with obser-vations (Fenimore et al. 1996; Dermer 2004; Shen et al. 2005;Shenoy et al. 2013). Then, the spectral evolution has to be un-derstood from the physics of the dissipative mechanism and maytherefore represent an important test to discriminate between thedi ff erent possible prompt emission models listed above.Regarding the photospheric emission, the spectral evolutionhas been computed only in the case of non-dissipative photo-spheres (Daigne & Mochkovitch 2002; Pe’er 2008). As men-tioned above, this model cannot reproduce the observed spec-trum. In the case of a dissipative photosphere, the peak energyof the spectrum is fixed by a complex physics (Beloborodov2013), which makes di ffi cult a prediction of the spectral evo-lution. It is usually assumed that modulations in the propertiesat the base of the flow will lead to the observed evolution (seefor instance Giannios & Spruit (2007) in the case where the dis-sipation is associated to magnetic reconnection). However, dis-sipative photospheric models require that the dissipation occursjust below the photosphere for the spectrum to be a ff ected. Itis not obvious that a change in the central engine leading to adisplacement of the photosphere will a ff ect the dissipation pro-cess in the same way so that it remains well located. Therefore,it remains to be demonstrated that these models can reproducethe observed spectral evolution. In the context of an emissionproduced above the photosphere, several authors have inves-tigated the time development of the photon spectrum withoutspecifying the dissipation mechanism and relate the observedspectral evolution to the evolution of the electron / photon injec-tion rate and / or the decaying magnetic field (e.g. Liang 1997;Stern & Poutanen 2004; Asano et al. 2009; Asano & M´esz´aros2011). It is encouraging that a reasonable agreement with ob-servations is found in some cases. To reach a final conclusion,it is however necessary to carry such a study in the context ofa physical model for the dissipation, which gives a prescriptionfor the accelerated electrons and the magnetic field. It is still outof reach for reconnection models due to the lack of any spectralcalculation. It has been done for the internal shock model usinga very simple spectral calculation including only synchrotron ra-diation (Daigne & Mochkovitch 1998, 2002). Since these earlycalculations, the observational description of the spectral evolu-tion has improved a lot, as well as the modeling of the emissionfrom internal shocks. Therefore, we examine here the quantita-tive prediction of this model for the spectral evolution in GRBs.For the first time the detailed dynamical evolution is com-bined with the calculation of the radiative processes, and theoutcome is confronted to the large set of observed propertiessummarized in Sect. 2 (e.g. hard-to-soft evolution, pulse widthevolution with energy, time lags, hardness-intensity / fluence cor-relation). In Sect. 3, we present our approach, which is basedon the model developed in Boˇsnjak et al. (2009). FollowingDaigne et al. (2011) we define three reference cases , which arerepresentative of the di ff erent possible spectral shapes in thekeV-MeV range, and present a detailed comparison of their tem-poral and spectral properties with observations. Then we inves-tigate in Sect. 4 and Sect. 5 the e ff ect on our results of di ff erentassumptions for the microphysics and dynamics of the relativis-tic ejecta. The specific signatures in the Fermi -LAT range are oˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions 3 presented in Sect. 6. We discuss our results in Sect. 7 and con-clude in Sect. 8.
2. GRB temporal and spectral properties:observations
There are several global trends in GRB spectra and lightcurves that have been identified in the prompt emission ob-servations by various missions during the past three decades.Spectral variations were already observed by the KONUS ex-periment providing time resolved data between 40 and 700 keV(Golenetskii et al. 1983). BATSE provided the largest databaseof high temporal and spectral resolution prompt GRB data andallowed detailed studies of the correlations between spectraland temporal properties (e.g. Band et al. 1993; Ford et al. 1995;Norris et al. 1996; Kaneko et al. 2006). Since GRB peak en-ergies are usually above the higher energy limit of the
Swift -BAT( ∼
150 keV), it is di ffi cult to examine the analogous cor-relations for the sample of Swift
GRBs. Due to the large num-ber of events with determined redshifts, this sample is howeverof great interest to test spectral and temporal properties in thesource frame (Krimm et al. 2009; Ukwatta et al. 2010).
Fermi -GBM data are providing new insights in the temporal and spec-tral behavior of GRBs, extending the spectral coverage to high-energies and to energies below the BATSE low-energy threshold(e.g. Lu et al. 2012; Bhat et al. 2012). We list here the commonlyobserved trends in the spectral and temporal properties to whichwe are referring in the subsequent sections when comparing theinternal shock model with observations. We distinguish betweenshort and long GRBs when necessary. It should be noted thatgenerally, GBM data indicate that the temporal and spectral evo-lution of short GRBs are very similar to long ones, but with lightcurves contracted in time and with harder spectra due to higherpeak energies (Guiriec et al. 2010; Bhat et al. 2012). obs.
When the burst has apparent sepa-rated pulses in the time histories, a fast rise and an exponentialdecay of the pulse is often observed (Fishman & Meegan 1995).Nemiro ff et al. (1994) showed that the individual long pulses inGRBs are time-asymmetric. The most thorough study of pulsesin long GRB light curves was provided by Norris et al. (1996)(see also e.g. Quilligan et al. 2002; Hakkila & Preece 2011) us-ing 64 ms resolution BATSE data. They found a typical rise-to-decay time ratio ∼ obs. Norris et al.(1996) found that the dominant trend in the pulse shapes ob-served in di ff erent energy channels is a faster onset at higherenergies and a longer decay at lower energies. The dependencebetween the energy E obs and the pulse width is approximatelya power law, W ( E obs ) ∝ E − a obs with a ≃ .
40, in a sample oflong BATSE bursts (Fenimore et al. 1995; Norris et al. 1996).The same evolution with a ≃ .
40 is also found in sample of longpulses with large time lags (Norris et al. 2005). Bissaldi et al.(2011) found the same trend for
Fermi bursts with a ≃ .
40 (seealso Bhat et al. 2012). obs.
Time lags are commonly observed in GRBpulses: pulses tend to peak earlier at higher energy in the softgamma-ray range (Norris et al. 1996). Time lags were studiedfor a large sample of BATSE GRBs (Band 1997; Norris 2002;Hakkila et al. 2008). Short lags ( <
350 ms) dominate the BATSEsample, even if a long lag ( > Swift and
HETE-2 short GRBs (see also Yi et al. 2006). Guiriec et al. (2010) con-firmed negligible spectral lags below 1 MeV for three brightshort GRBs observed by
Fermi . obs. Norris et al. (1986) examineda handful of bursts observed by the
Solar Maximum Mission satellite between 50 and 300 keV, and found that the pulseemission evolved from hard to soft with the hardness maxi-mum preceeding the peak of the intensity. More detailed studiesfollowed using BATSE data (Bhat et al. 1994; Ford et al. 1995;Band 1997): it was found that the spectral peak energy E p , obs isrising or slightly preceding the pulse intensity increase, and issoftening during the pulse decay (hard-to-soft evolution withina pulse). The later pulses in burst time history were also foundto be softer than earlier ones (global hard-to-soft evolution). For Fermi
GBM bursts, Lu et al. (2012) reported the same hard-to-soft evolution in the variation with time of the spectral peak en-ergy in the majority of GRBs, but also found cases where thepeaking energy is simply tracking the intensity. These burstsshow usually more symmetric pulses. Short
Fermi
GRBs areusually found to follow the ’intensity tracking’ pattern. obs.
Golenetskii et al. (1983) reported the discovery of a corre-lation between the instantaneous luminosity and the temperature kT characterizing the photon spectrum, L ∝ ( kT ) γ , with γ ≈ ∼
50% of the events with a largerspread for the exponent, γ ≈ F ( t ) as F ( t ) ∝ E . , obs (hard-ness intensity correlation, or HIC). Borgonovo & Ryde (2001)studied this HIC using the value of EF E at the peak energy torepresent the intensity, and found it was proportional to E η p , obs .This makes the correlation less dependent on the observationalspectral window. The mean value of η was found to be 2 . γ = N ( t ), and found E p , obs ∝ N ( t ) δ , with δ ∼ F ( t ), E p , obs ∝ F ( t ) κ , with κ ≃ . − . obs. This correla-tion was discovered by Liang & Kargatis (1996). It describesthe exponential decay of the spectral hardness as a function ofthe cumulative photon fluence Φ ( t ), E p , obs ( t ) ∝ exp( − Φ ( t ) / Φ ).Crider et al. (1999) tested a reformulated correlation using en-ergy fluence instead of the photon fluence, and confirmed the de-cay pattern. Ryde & Svensson (2000) used this correlation com-bined with the HIC correlation to obtain a self-consistent de-scription for the temporal behavior of the instantaneous photonflux and got a good agreement with BATSE data. obs. Kouveliotou et al.(1993) reported that short GRBs are harder than long ones.Hardness was characterized by the ratio of the total counts in thetwo BATSE energy channels (usually 100-300 keV and 50-100keV energy range). Ghirlanda et al. (2004) argued that the hard-ness of the short events is owing to a harder low energy spectralslope (photon index α ) in short bursts, rather than to a higher Boˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions
Case Dynamics Microphysics Spec. @ max. Spectro-temporal properties Figs.
Ejection E kin , iso Γ ( t ) ¯ Γ ζ ǫ B p E p , obs α τ r /τ d a ( W ( E )) δ (HIC) κ (HIC)[erg] [keV]A ˙E = cst . × smooth 340 3 . × − / . − . .
38 0 .
29 2 .
28 2 . . × − . − . .
39 0 .
30 2 .
15 1 .
97 15,17varying 744 − . .
31 0 .
28 2 .
23 1 .
55 10,18,16,17varying 2 . − . .
30 0 .
29 2 .
12 1 .
48 15,174 . × − . − . .
41 0 . / / . × − . − . .
46 0 . / / ˙E = cst . × smooth 340 1 . × − − . − . .
43 0 . / / . × − . − . .
54 0 .
24 0 .
97 0 .
89 5,6,7,15,171 . × − . − . .
54 0 .
27 1 .
23 1 .
05 5,6,71 . × − . − . .
54 0 .
27 1 .
31 1 . . × − . − . .
54 0 .
28 1 .
32 1 . − . .
33 0 .
24 0 .
96 0 .
80 8,10,11,9,18,16,17varying 2 . − . .
32 0 .
27 1 .
27 0 .
97 15,171 . ×
360 varying 691 − . .
37 0 . / / . ×
360 varying 2 . − . .
36 0 .
26 0 .
92 0 .
78 15,175 . × sharp 2 . × − − . .
68 0 . / / . × sharp varying 772 − . .
04 0 . / / M = cst 1 . × . × − − . .
75 0 .
16 0 .
13 0 .
17 14,12,18,16˙ M = cst 1 . × varying 630 − . .
60 0 . / / ˙E = cst . × smooth 1020 1 . × − − . − . .
55 0 . / / Table 1. Parameters of all the GRB pulse models discussed in the paper.
The three reference cases defined in Sect. 3 are listedin bold face. For other models discussed in Sect. 4 and Sect. 5, we list only the input parameters that are modified compared to thereference case. The first columns list the parameters for the dynamics and the microphysics (see text). In all cases, ǫ e = /
3. Thelast columns list a few properties of the corresponding simulated GRB pulse : spectral properties at the maximum of the GBM lightcurve (peak energy and low-energy photon index), and four indicators of the spectral and temporal properties: ratio of the rise timeover the decay time of the pulse (BATSE 2 + a for the evolution of the pulse width W ( E ) with energy ( W ( E ) ∝ E − a ),slopes of the hardness-intensity correlation ( δ is the slope when using the photon flux and κ the energy flux), see text. Cases whereit is not possible to define the slopes of the HIC are indicated with ’ / ’. For reference, typical observed values are τ r /τ d ≃ . − . a ≃ . − . δ ≃ . − . κ ≃ . − . E p , obs . This is not confirmed by the detailed anal-ysis of three bright GBM short GRBs by Guiriec et al. (2010),which shows that the hardness of short bursts is due both to hardlow-energy photon indexes α and high peak energies E p , obs .
3. Spectral evolution in the internal shock model
Several steps are necessary to model the prompt GRB emissionfrom internal shocks. From the initial conditions in the ultra-relativistic outflow ejected by the central engine, the dynamicalevolution must be calculated. This allows to know how many in-ternal shocks will form and propagate within the outflow and tocompute the time-evolution of the physical conditions in eachof the shocked regions (Lorentz factor, mass and energy density,...). Then, the distribution of shock-accelerated electrons and theintensity of the shock-amplified magnetic field must be evalu-ated. This is the most uncertain step and is usually done using avery simple parametrization of the local microphysics. Knowingthe distribution of relativistic electrons accelerated at each shockand the magnetic field, it is then possible to compute the emis-sion produced in the comoving frame of each shocked region,taking into account the relevant radiative processes. Finally, thecontributions of each emitting region are summed up taking intoaccount relativistic e ff ects (Doppler boosting, relativistic beam-ing), the curvature of the emitting surface (integration on equal-arriving time surfaces) and cosmological e ff ects (redshift, timedilation). This full procedure allows to predict light curves and time-evolving spectra for synthetic GRBs and therefore to makea detailed comparison with observations.To follow this procedure, we use the model described inBoˇsnjak et al. (2009). We assume that the outflow at large dis-tance from the source has a negligible magnetization, which canbe achieved either in the standard fireball or in an e ffi cient mag-netic acceleration scenario, which is preferred by observations(Hasco¨et et al. 2013). A moderate magnetization ( σ > ∼ . − . ff ect the spectrum of internal shocks, especially at high en-ergy (Mimica & Aloy 2012). The model parameters are of twotypes: (i) the initial conditions for the dynamics of the relativisticoutflow, given by the duration of the relativistic ejection t w , theinitial distribution of the Lorentz factor Γ ( t ej ), where 0 ≤ t ej ≤ t w is the time of the ejection, and the initial kinetic power ˙ E ( t j );(ii) the microphysics parameters : it is assumed that a fraction ǫ B of the dissipated energy in a shocked region is injected into theamplified magnetic field, and that a fraction ǫ e of the energy is in-jected into a fraction ζ of the electrons to produce a non-thermalpopulation. The distribution of these accelerated relativistic elec-trons is a power-law with a slope − p .We refer to Boˇsnjak et al. (2009) for a detailed descriptionof the model, which is based on a multi-shell approximationfor the dynamics and a radiative code that solves simultane-ously the time evolution of the electron and photon distribu-tions in the comoving frame of the emitting material, taking intoaccount all the relevant processes : synchrotron radiation andself-absorption, inverse Compton scattering (including Klein-Nishina regime), γγ annihilation, and adiabatic cooling. oˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions 5 Fig. 1. Three reference cases: normalized light curves in the four BATSE channels. (blue line: 20-50 keV; black line: 50-100keV; green line: 100-300 keV; red line: 300-2000 keV) for the cases A, B & C (see text).Depending on the choice of microphysics parameters, thetypical Lorentz factor of the shock-accelerated electrons and theintensity of the magnetic field can be rather di ff erent, even forsimilar relativistic outflows and dynamical evolutions. Then, thedominant radiative process in the soft gamma-ray range (BATSEor GBM) could be either direct synchrotron radiation or inverseCompton scattering of low-energy synchrotron photon (SSC).Boˇsnjak et al. (2009) have shown that the second case wouldpredict a bright additional component in the GeV, due to the sec-ond inverse Compton scatterings. Such a peak does not seem tobe observed by Fermi and the SSC scenario is probably ruledout, as discussed for instance by Piran et al. (2009). Thereforewe focus here on the scenario where the prompt GRB emissionis dominated by synchrotron radiation from shock-acceleratedelectrons in internal shocks. In this case, there are in princi-ple two components in the spectrum, one, peaking in the softgamma-ray range, due to synchrotron radiation, and a secondone, peaking at high energy, associated to inverse Compton scat-terings, which are very likely in the Klein-Nishina regime.
Even in the synchrotron scenario, the spectral shape of theprompt emission still depends strongly on the assumptions forthe microphysics parameters. Assuming that electrons are radia-tively e ffi cient (synchrotron fast cooling, Sari et al. 1998), whichis required both by the variability timescale and the energetics ofthe prompt emission, there are three main possibilities, that havebeen described by Daigne et al. (2011) and illustrated with threereference cases. Each of these cases corresponds to an exampleof a single pulse burst that should be seen as a building block formore complex light curves. The initial Lorentz factor is risingduring the ejection, according to the simple law Γ ( t ej ) = Γ max +Γ min − Γ max − Γ min cos (cid:16) π t ej . t w (cid:17) for 0 ≤ t ej t w ≤ . Γ max for 0 . ≤ t ej t w ≤ , (1)with Γ min =
100 (resp. 300) and Γ max =
400 (resp. 1200) in casesA and B (resp. case C): see Fig. 1 in Boˇsnjak et al. (2009). Theduration of the ejection is t w = E = erg . s − (resp. 5 × erg . s − )in cases A and B (resp. in case C). Then, the collision of the‘slow’ and ‘rapid’ parts in the ejecta lead to the formation of twointernal shock waves, a short lived ‘forward’ shock and a ‘re- verse’ shock that crosses most of the ejecta and dominates theemission. The three reference cases di ff er mainly by the micro-physics: in all three cases, ǫ e = / p = .
5, but the twoother microphysics parameters ǫ B and ζ are di ff erent. Case A. Pure fast cooling synchrotron case, with ǫ B = ǫ e = / and ζ = . . For ǫ B > ∼ ǫ e , inverse Compton scatterings arevery ine ffi cient. Then the radiated spectrum does not show anadditional component at high energy and is very close to thestandard fast cooling synchrotron spectrum with a low-energyphoton index α = − / Case B. Fast cooling synchrotron case a ff ected by inverseCompton scattering in Klein Nishina regime, with ǫ B = − and ζ = . . For ǫ B ≪ ǫ e , inverse Compton scattering be-comes e ffi cient and leads to a second spectral component at highenergy, which remains however weak due to the Klein-Nishinaregime. In addition, as the scatterings are more e ffi cient for low-energy photons in this regime, the low-energy photon index ofthe synchrotron component is modified and becomes larger thanthe standard fast cooling value: − / < ∼ α < ∼ − Case C. Marginally fast cooling case, with ǫ B = . and ζ = . . In some conditions, it is possible that the cooling fre-quency ν c becomes very close to the synchrotron frequency ν m .In such a situation where ν c < ∼ ν m , radiation is still e ffi cient(fast cooling), but the intermediate region of the spectrum where α = − / α = − / E p , obs and the low-energy photon index α of the syn-chrotron component (GBM range), assuming a source redshift z =
1. We have plotted for each case the normalized light curvesin the four BATSE channels in Fig. 1 and the spectral evolutionin Fig. 2. In the three cases, spectral evolution is found, with aglobal trend for the peak-energy to follow the intensity, and witha hard-to-soft evolution during the pulse decay.
Boˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions
Fig. 2. Three reference cases: spectral evolution.
The timeevolution of the observed peak energy E p , obs (top) and low-energy photon index α (bottom) is plotted for the three referencecases: A (black), B (red) and C (blue). In the top pannel, theenergy range of the four BATSE channels is also indicated. Three possible time scales can in principle govern the observedevolution: the radiative timescale for the cooling of shock ac-celerated electrons, the hydrodynamical timescale associated tothe propagation of internal shocks, and the time scale associ-ated to the curvature e ff ect or high-latitude emission. The radia-tive time scale has to be the shortest to allow for the observedshort time scale variability (Rees & Meszaros 1994; Sari et al.1996; Kobayashi et al. 1997) and can not be the main driverof the observed spectral evolution. The curvature e ff ect leadsto a strong temporal and spectral evolution which is not ob-served during most of the prompt emission (Fenimore et al.1996; Dermer 2004; Shenoy et al. 2013), but is most probablyresponsible for the early steep decay found in the X-ray after-glow by Swift / XRT (see Hasco¨et et al. 2012, for a recent discus-sion in the context of di ff erent GRB prompt emission models),as demonstrated in several studies (see e.g. Kumar & Panaitescu2000; Liang et al. 2006; Butler & Kocevski 2007; Zhang et al.2007; Qin 2008; Genet & Granot 2009; Willingale et al. 2010).Therefore, the temporal and spectral evolution of GRB pulsesin the internal shock model has to be mainly governed by thehydrodynamical timescale (Daigne & Mochkovitch 2003).When an internal shock is propagating within the outflow,the corresponding bolometric luminosity is given by L bol = f rad ǫ e π R Γ ∗ ρ ∗ ǫ ∗ c , (2)where R is the shock radius, Γ ∗ is the Lorentz factor of theshocked material, ρ ∗ and ǫ ∗ are the mass and the specific internalenergy density in the shocked region (comoving frame), and f rad the radiative e ffi ciency, which is close to 1 for synchrotron radi-ation in fast cooling regime. For typical values representative of Fig. 3. Three reference cases: dependence of the pulse shapeon energy.
Top: pulse width as a function of energy.
Bottom: position of the pulse maximum in channel i with respect to thelowest considered energy band, 20-50 keV (BATSE channel 1).Dashed lines show the energy bands for which the light curveswere calculated. The first 4 energy bands correspond to the samefour BATSE channels used in Fig. 1. The shaded regions indicatethe spectral coverage of Fermi
GBM and LAT detectors. Blackdots correspond to reference case A, red crosses to case B, andblue square symbols to case C.cases A, B and C close to their maximum, we have L bol ≃ . × f rad ǫ e / ! (cid:18) R × cm (cid:19) × Γ ∗ ! ρ ∗ − g . cm − ! ǫ ∗ / c . ! erg . s − , (3)Fig. 12 shows the time evolution of Γ ∗ , ρ ∗ and ǫ ∗ in cases Aand B. Case C would show a similar behavior. In these threeexamples, the bolometric luminosity is initially rising when theshock forms (increase of Γ ∗ and ǫ ∗ ), reaches a maximum, andthen decreases again due to the radial expansion (decrease of ǫ ∗ and ρ ∗ ): see Boˇsnjak et al. (2009) for a more detailed discussion.This leads to the pulse shape of the light curve. In more realisticcases, a large number of shock waves will form and propagatein the flow and several of them can contribute at the same time,leading to a complex, multi-pulses, light curve.In addition to the evolution of the bolometric luminosity,the spectral evolution a ff ect the details of the pulse shape ina given energy band. When the synchrotron peak is not toomuch a ff ected by inverse Compton scattering in Klein-Nishinaregime (see below), the observed peak energy can be expressedas E p , obs ≃ Γ ∗ h ν ′ m , where ν ′ m is the peak of the synchrotron spec-trum in the comoving frame, leading to E p , obs ≃
380 keV1 + z Γ ∗ ! Γ m ! (cid:18) B (cid:19) , (4)where B is the intensity of the magnetic field (comoving frame)and Γ m the minimum Lorentz factor of the power-law distribu- oˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions 7 tion of accelerated electrons. These two last quantities can beestimated from the microphysics parameters, leading to E p , obs ≃ . + z Γ ∗ ! ρ ∗ − g . cm − ! / ǫ ∗ / c . ! / × ǫ B / ! / ( p − / ( p − / ! ǫ e / ! (cid:18) ζ . (cid:19) − , (5)using typical values for Γ ∗ , ρ ∗ and ǫ ∗ at the time correspond-ing to the maximum of the light curve in case A or B. Thespectrum around E p , obs is very close to the standard fast cool-ing synchrotron spectrum (i.e. α = − /
2) in case A and showsa larger low-energy photon index in case B ( α ≃ − .
1) due toinverse Compton scattering in Klein Nishina regime. As shownin Daigne et al. (2011), this e ff ect is expected for large values ofboth the Compton parameter given by Y Th = p − p − ǫ e ǫ B (6)and the parameter w m defined by w m = Γ m h ν ′ m m e c . (7)Note that the Compton parameter Y Th defined in Eq. (6) is com-puted assuming the Thomson regime for the scatterings. The ef-fective Compton parameter in the simulations presented here isalways smaller due to the Klein-Nishina corrections. Close tothe maximum of the pulse light curve, Y Th ≃
110 (resp. 0 . w m ≃
60 (resp. 40) for case B (resp. case A).From Eqs. (5-7), the evolution of the physical conditions ( Γ ∗ , ρ ∗ and ǫ ∗ ) in the shocked region during the propagation of theshock wave leads to an evolution of the peak energy and thespectral shape (particularly the photon index α ) and is thereforeat the origin of the observed spectral evolution. The combina-tion of the evolution of bolometric power L bol and this spectralevolution allows to understand the details of the pulse shape in agiven energy channel. The luminosity at a given photon energycan be written L ( E obs ) = L bol E p , obs S E obs E p , obs ! , (8)where S is the spectral shape, normalized by R ∞ S ( x ) dx = (cid:2) E , obs ; E , obs (cid:3) is given by F ≃ L bol π D × Z E , obs / E p , obs E , obs / E p , obs S ( x ) dx . (9)The light curve is shaped by the two terms in Eq. (9). The evo-lution of the first term has been described above and is responsi-ble for the general shape of the light curve, with a clear peak.The spectral correction contained in the second term is morecomplicated : it depends on the energy channel and is respon-sible for the time lags, evolution of the pulse shape with energy,etc. (Daigne & Mochkovitch 1998, 2003; Hafizi & Mochkovitch2007; Boc¸i et al. 2010).The temporal and spectral evolution which has just been de-scribed is valid as long as the internal shock phase is active andthe observed emission is dominated by the contribution of on-axis radiation from shocked regions. After the last internal shockhas disappeared, the observed emission is due to the high latitudeemission and is therefore governed by the geometry of the shellsand the relativistic Doppler e ff ects, which lead to the asymptotic evolution E p ∝ / t obs . The corresponding bolometric flux de-creases as F bol ∝ / t leading to a spectral evolution whichis much too fast to reproduce the properties of observed bursts,as pointed out by Fenimore et al. (1996). For instance the HICwould have a slope δ < . δ ≃ . −
1. This clearly indicates that the spectral evolution inGRB pulses is not governed by such geometrical e ff ects but bythe physics of the internal dissipative and radiative process in theoutflow as described above. We now compare more quantitatively the spectral evolutionfound in reference cases A, B and C to the available observa-tions. Fig. 1 shows the normalized light curve in the four BATSEchannels for each case, and Fig. 2 shows the corresponding spec-tral evolution. The break at t HLE , obs ≃ . t obs < t HLE , obs .There is a good qualitative agreement with observations (seeSect. 2). In particular, it is found that the pulse light curve isasymetric with a fast rise and a slow decay ( obs. ); this as-symetry is stronger at lower energy ( obs. ) and the width ofthe light curve is broader at lower energy ( obs. ); the pulselight curve peaks earlier at higher energy ( obs. ); the peak en-ergy decreases in the decay phase of the pulse ( obs. ).The more quantitative comparison is less satisfactory:– The shape of the pulse in channel 1 and 2 for case A showsa double peak which does not seem to be usually observed inGRBs. This is due to a too strong spectral evolution in this case(see Fig. 2). The peak energy has a maximum E p , obs ≃
800 keVclose to the first maximum of the light curve at t obs ≃ . E p , obs ≃
10 keV at the end of theon-axis emission at t HLE , obs . The rapid decrease of E p , obs leads toan increase of the second term in Eq. (9) (spectral correction) forchannels 2 and 1 which are successively crossed by E p , obs . Thiscompensates the decrease of the bolometric luminosity to createa second peak in the light curve for these two channels.– In the three cases, the pulse width does not decrease enoughwith energy. This is illustrated in Fig. 3 (upper panel), where W ( E ) is plotted as a function of the mean energy of the channel,defined by E = √ E min E max where E min and E max are the lowerand higher energy bounds. In agreement with observations, it isfound that the width follows approximatively a power-law evo-lution W ( E ) ∝ E − a . However, the value of the index a , listedin Tab. 1, is usually a little too small compared to observations( obs. ). The best agreement is found for case A.– Time lags between the di ff erent channels are too large, as il-lustrated in Fig. 3 (lower panel). A quantitative comparison withobservations is more delicate as we do not measure the lag bythe maximum of correlation like in the method described by e.g.Band (1997), which is usually applied to GRB data. We ratherplot the di ff erence between the time of the maximum of the lightcurve in a given channel, and the time of the maximum in chan-nel 1. The observed trend is reproduced (channel 4 peaks first,channel 1 peaks the last), but, especially in case A, the typicallags are too long compared to observations ( obs. ). Boˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions
Fig. 4. Three reference case: hardness-intensity and hardness-fluence correlations.
The peak energy energy is plotted as afunction of the photon flux (left panel) and photon fluence (right panel) between 20 keV and 2 MeV for the three reference cases, A(black), B (red) and C (blue). The peak energy, photon flux and photon fluence are normalized by their respective maximum values.The dashed lines show the behavior during the rise of the pulse, thick lines correspond to the decay phase, and the thin lines arethe high latitude emission. This last stage is unlikely to be observed in complex lightcurves, except during the early steep decay inX-rays.
Fig. 5. Impact of the electron slope p : spectral evolution. Thetime evolution of the observed peak energy E p , obs (top) and low-energy photon index α (bottom) in case B is plotted for di ff erentvalues of the relativistic electron slope p . The evolution for p = . p = . p = .
9. In each case, the valueof ζ is adjusted to keep the same observed peak energy of thetime-integrated spectrum (see Tab. 1).– The HIC is qualitatively reproduced, as illustrated in Fig. 4(left panel), where the peak energy E p , obs is plotted as a functionof the photon flux N in the 20–2000 keV range in log–log scale.The peak energy increases during the rise of the light curve,reaching a maximum which precedes the maximum of the in-tensity; then it decreases during the pulse decay (hard to softevolution, obs. ). However, the quantitative behavior is not re- produced. During the pulse decay, the peak energy should follow E p , obs ∝ N δ with δ ≃ . − E p , obs ∝ F κ with κ ≃ . − . N and F being the photon and energy fluxes ( obs. ). This isnot found in our simulations. Cases B and C do not show a sim-ple power-law behavior during the decay phase. Case A is closerto the expected evolution, but the slopes δ and κ are too largecompared with BATSE and GBM observations.– The same disagreement in found for the HFC, as illustratedin Fig. 4 (right panel). Again, cases B and C do not really showthe expected behavior log E p , obs ∝ Φ , whereas the agreement isbetter for case A, with a quasi exponentially decay for the peakenergy as a function of the photon fluence ( obs. ).In the three cases, a careful analysis shows that the disagree-ments listed above are due to the fact that the spectral evolu-tion, even if it reproduces qualitatively the hard-to-soft evolution( obs. ), is usually to strong (the peak energy, and sometimesthe spectral slopes, vary too much). We note that cases B andC have the strongest disagreement with the observed HIC andHFC. This is due to a peculiar spectral evolution in this case (seeFig. 2): the peak energy is initially decreasing during the pulsedecay, as expected, but then does not evolve any more (case B:it is only slightly increasing at the end of the pulse) or startsto increase instead of decreasing (case C), which is usually notobserved. This unexpected behaviour is analysed in § oˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions 9 Fig. 6. Impact of the electron slope p : hardness-intensity cor-relation. The HIC diagram in case B is plotted for di ff erent val-ues of the relativistic electron slope p .assumptions for the initial conditions in the outflow that impactthe dynamics of the internal shock phase. We now investigate inSect. 4 and Sect. 5 how these factors a ff ect the predicted spectralevolution and if the observed evolution can be better reproduced.
4. Impact of the uncertainties on the microphysics
Due to a lack of better, physically motivated, prescriptions, themicrophysics in the shocked region (magnetic field amplifica-tion, electron acceleration) is simply parametrized by ǫ B , ǫ e , ζ and p in our model of the internal shock phase. Such an over-simplistic description may be at the origin of some disagree-ments between the simulated and observed spectral evolutionpointed out in the previous section. We investigate here this pos-sibility and focus on the two most relevant cases, i.e. A and B. Unfortunately, the current physical understanding of shock ac-celeration in the mildly relativistic regime does not allow to pre-dict the value of the slope p of the electron distribution. A ’stan-dard’ value p = . p a ff ectsthe predicted temporal and spectral properties of the simulatedpulse. The most dramatic change is for case B. For low values of p , case B shows a very peculiar spectral evolution in the decayphase of the pulse, the peak energy starting to increase after aninitial decrease. The more standard simple hard to soft evolutionis recovered above a threshold p ≃ .
7, as shown in Fig. 5 wherethe time evolution of E p , obs and α is plotted for case B for dif-ferent values of p . Moreover, we find that increasing p improvesthe quantitative comparison between the predicted and observedspectral evolution in cases A and B: see for instance the values ofthe index a and the slopes δ and κ in Tab. 1 and the correspondingFig. 7 (pulse width, time lags) and Fig. 6 (HIC).The peculiar evolution of the peak energy for p < ∼ . Fig. 7. Impact of the electron slope p : pulse width and timelags. Same as in Fig. 3 for Case B with four di ff erent values ofthe electron slope p . Color code is the same as in Fig. 5. In thelower panel, the time lags are the same for the four cases exceptfor the highest energy channel where the p = . p = . p = . p = . ff ected by inverse Compton scattering inKlein Nishina regime (see Fig. 2 in Daigne et al. (2011)). Forlarge values of the Compton parameter Y Th which favors thescatterings, not only the low-energy slope of the synchrotronspectrum is a ff ected, but also the peak, which moves towardshigher energy, with the spectrum around the peak becomingvery flat. The synchrotron peak energy in the comoving frameis not any more simply proportional to B Γ and the standardspectral evolution governed by Eq. (5) is lost. For higher p , wefind numerically that the peak is not shifted any more, even forvery large values of Y Th . This is also confirmed by the semi-analytical calculation of the Klein-Nishina e ff ects on opticallythin synchrotron and synchrotron self-Compton spectrum madeby Nakar et al. (2009): see the discussion of their case IIc, whichis relevant for our reference case B . They find a threshold at p =
3, slightly larger than p ≃ .
7. This di ff erence is probablydue to the additional approximations which are necessary to al-low for the analytical treatment, compared to the full numericalresolution we use here. A strong but common assumption is to assume constant micro-physics parameters in a GRB. The fact that di ff erent values of theparameters are needed to fit observations from di ff erent GRBs(see for instance the case of GRB afterglows as in the study byPanaitescu & Kumar 2001) indicates on the other hand that thereare no universal values. Then, it is highly probable that ǫ B , ǫ e , ζ and p depend on the shock conditions and evolve during a GRB.In Daigne & Mochkovitch (2003), where only synchrotron ra-diation was included in a simple way, it has been shown that See Daigne et al. (2011) for a correspondence between the nota-tions of Nakar et al. (2009) and Daigne et al. (2011) and a comparison.0 Boˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions
Fig. 9. Impact of a varying accelerated electron fraction ζ : hardness-intensity correlation. Same as in Fig. 4 for case B with avarying parameter ζ during the propagation of internal shocks (see text), either with the same ejection duration as in the referencecase B (black) or with an extended high-Lorentz factor tail in the ejecta (red) : see § Fig. 10. Impact of a varying accelerated electron fraction ζ : pulse shape. Same as in Fig. 1 for case A (left panel) and B (rightpanel) assuming a varying parameter ζ during the propagation of internal shocks (see text). Conventions are the same as in Fig. 1.The corresponding time-evolving spectrum for case B is shown in Fig. 8.assuming such variations of the microphysics parameters couldgreatly improve the comparison between the predicted and theobserved spectral evolution in a pulse. As there are no physicallymotivated prescriptions for such variations, we cannot fully ex-plore this possibility. We only illustrate the e ff ect in cases A andB, assuming a simple variation law for one parameter, i.e. ζ ∝ ǫ ∗ as suggested by Bykov & Meszaros (1996), so that a larger frac-tion of electrons is accelerated when the shock is more violent.We have normalized ζ to have a similar peak energy of the time-integrated spectrum as in the reference cases (see Tab. 1). FromEq. (5), it appears clearly that this prescription will reduce thevariations of E p , obs during the pulse, as it is now only propor- tional to ǫ / ∗ rather than ǫ / ∗ . This is confirmed by our detailedradiative calculation, as illustrated for case B in Fig. 8 (spectrum)and in Fig. 11 (left panel: spectral evolution). The peak energydecreases only by a factor ∼ α re-mains close to − . oˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions 11 Fig. 11. Impact of a varying accelerated electron fraction ζ : spectral evolution – pulse width and time lags. Curves are plottedfor case B with a varying parameter ζ during internal shocks (see text), either with the same ejection duration as in the referencecase B (black) or with an extended high-Lorentz factor tail in the ejecta (red) : see § Left panel: thetime evolution of the peak energy and the low-energy photon index. The dashed vertical line indicates the start of the high latitudeemission.
Right panel: evolution of the pulse width and time lags, as in Fig. 3.varying electron fraction ζ has an overall good agreement withBATSE and GBM observations. Especially, the predicted spec-tral evolution reproduces now qualitatively and quantitatively theobservational constraints obs. to obs. described in Sect. 2.The assumption ζ ∝ ǫ ∗ is suggested by Bykov & Meszaros(1996), but one may expect variations of other microphysics pa-rameters as well. We expect that any modification of the micro-physics leading to a reduced dependence of the peak energy tothe shock conditions will produce a similar improvement as de-scribed here.We conclude that the disagreement between the observedspectral evolution in GRBs and the predictions of the simplestversion of the internal shock model illustrated by our referencecases A, B and C can be largely due to over-simplifying as-sumptions regarding the microphysics in the emitting shockedregions. Our current knowledge of the physics of mildly rela-tivistic shocks does not allow yet to improve this description butwe have illustrated that both a qualitative and quantitative agree-ment can be achieved, for instance if the fraction of acceleratedelectron is varying with the shock strength.
5. Impact of the uncertainties on the dynamics
The physics of the central engine, and of the acceleration of therelativistic outflow, is not well understood. This does not allowto predict the initial conditions in the jet before the internal shockphase. Typically, a pulse is due to the collision between two re-gions with di ff erent Lorentz factors (in the reference cases, the’slow’ region correspond to Γ ( t ej ) = Γ min → Γ max , and the ’rapid’region to Γ = Γ max , see Eq. (1)). Two internal shocks form, a’forward’ and a ’reverse’ shock (see Fig. 12). In the referencecases, the emission is entirely dominated by the ’reverse’ internalshock. This assumption has the advantage of simplicity to sim-ulate a single pulse burst (as a building block for more complexlight curves) but is not physically motivated. We now investigate how these assumptions can a ff ect the dynamics of the internalshock phase and therefore the spectral evolution in pulses. In the reference cases, the Lorentz factor during the relativisticejection increases continuously from Γ min to Γ max . The shape ofthe transition can impact the radius where the internal shocksform and the initial strength of the shocks. It will therefore af-fect the pulse shape and the spectral evolution, especially in theearly phase. To investigate this e ff ect, we have simulated case Bassuming a much steeper transition: Γ ( t ej ) = ( Γ min for 0 ≤ t ej ≤ . t w Γ max for 0 . t w ≤ t ej ≤ t w , (10)where the value 0 . t w has be chosen to have the same meanLorentz factor than in the reference cases. For such initial con-ditions, the internal shock phase starts earlier and the shocks areimmediately stronger than in the reference cases as shown inFig. 12. The corresponding e ffi ciency is increased and therefore,we have adjusted the injected kinetic power to keep the same ra-diated energy as in the reference cases (see Tab. 1). As expected,the fact that the shock is initially stronger a ff ects the rising partof the pulse: see Fig. 13 (light curves, pulse width and time lags)and Tab. 1. The main change is the increase of the time lags,which become too large compared to observations. However, wechecked that, as for the reference case, a varying fraction of ac-celerated electrons ζ ∝ ǫ ∗ solves this problem (Fig. 13, middlepanel). We conclude that the main e ff ect of the shape of the ini-tial distribution of the Lorentz factor is on the rising part of thepulse, especially at high energy as discussed in Sect. 6.Another e ff ect is also related to the initial distribution of theLorentz factor. As can be seen in Fig. 1, the decaying part ofthe light curve is interrupted at t HLE , obs with a break towards asteeper decline. Such breaks are usually not observed, which Fig. 12. Evolution of the physical conditions in the shocked region. left panel:
The evolution of the Lorentz factor Γ ∗ (top),the specific energy density ǫ ∗ (middle) and the mass density ρ ∗ (bottom) is plotted for the reference case B (case A would showexactly the same evolution), case B with a sharp distribution of the initial Lorentz factor (see text) and case B with a constantejected mass flux ˙ M rather than a constant injected kinetic power ˙ E (see text). For each case, two curves are seen, correspondingto the propagation of a ’forward’ and a ’reverse’ internal shocks; Middle panel: corresponding spectral evolution (same color code)assuming a constant fraction ζ of accelerated electrons; Right panel: corresponding spectral evolution (same color code) assuminga varying fraction ζ of accelerated electrons (see text). Fig. 13. Impact of the initial distribution of the Lorentz factor in the outflow.
Left and middle panels: normalized light curvescorresponding to the 4 BATSE energy channels for case B computed with the sharp initial distribution of the Lorentz factor givenby Eq. (10) and a constant (left) or varying (middle) accelerated electron fraction (see text). Conventions are the same as in Fig. 1;
Right panel: pulse width and pulse maximum position as a function of energy for the same cases (filled circles: constant ζ , opencircles: varying ζ ). The shaded regions and lines have the same meaning as in Fig. 3.can be easily understood as complex GRB light curves showthe superimposition of many pulses, which makes the observa-tion of the very end of the decay of a pulse di ffi cult. We notehowever that t HLE , obs is directly related to the radius where thepropagation of the internal shocks responsible for the pulse ends(Hasco¨et et al. 2012). In the reference case, it can easily occurat later times by simply increasing the duration of the phase ofthe relativistic ejection where Γ =
Γ =
400 phase in the ejectionlasts for 2.2 s instead of 1.2 s. The pulse is exactly the same ex-cept for the break in the decay at t HLE , obs ≃ ff ects the HIC and the HFC (Fig. 9), but at late timeswhere the flux is too low to be considered in observed HIC dia-grams. We do not discuss here two other factors also related tothe initial distribution of the Lorentz factor: (i) the contrast Γ max / Γ min has a direct impact on the collision radius and on thestrength of the shock. Very high contrasts may appear unreal-istic for central engine models and very low contrasts lead totoo low internal shock e ffi ciencies. We have favored here a in-termediate value leading to a total e ffi ciency of a few percents(ratio of the radiated energy over the kinetic energy): see alsoBoˇsnjak et al. (2009) and figure 8.d therein; (ii) the mean value¯ Γ of the initial Lorentz factor in the ejecta. The main e ff ects ofthis parameter are on the peak energy ( E p , obs decreases if ¯ Γ in-creases, all other parameters being constant, Barraud et al. 2005;Boˇsnjak et al. 2009) and at high energy ( γγ annihilation, seeSect.6). oˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions 13 Fig. 14. Impact of the shape of the injected kinetic power in the outflow.
Left and middle panels: normalized light curvescorresponding to the 4 BATSE energy channels for case B computed by assuming a constant ejected mass flux ˙ M rather thana constant injected kinetic power ˙ E (see text) and a constant (left) or varying (middle) accelerated electron fraction (see text).Conventions are the same as in Fig. 1; Right panel: pulse width and pulse maximum position as a function of energy for the samecases (filled circles: constant ζ , open circles: varying ζ ). The shaded regions and lines have the same meaning as in Fig. 3. Fig. 8. Impact of a varying accelerated electron fraction ζ :spectral evolution. Time resolved spectra are plotted for case Bassuming a varying parameter ζ during the propagation of inter-nal shocks (see text). The corresponding light curves are shownin the right panel of Fig. 10. Each curve corresponds to a timeinterval of 0 .
25 s, starting with the bluest curve (0.5–0.75 s) andfinishing with the reddest one (9.75–10 s). The time bin corre-sponding to the pulse maximum is indicated (1.5–1.75 s). Thetime-integrated spectrum is also plotted as a thick black line. Athin dashed line of photon index α = − The reference cases are computed using the simple assumptionthat the injected kinetic power ˙ E is constant during the rela- tivistic ejection, which corresponds to an ejected mass flux thatevolves as ˙ M ∝ / Γ . Other assumptions are of course possibleand can again a ff ect the dynamics and the predicted spectral evo-lution. To investigate this possibility, we have simulated case Bassuming a constant ejected mass flux ˙ M , i.e. ˙ E ∝ Γ . We fixedthe value of ˙ M so that the total radiated energy is the same inboth cases and we adjusted ζ to have similar peak energies. Aclear drawback of the ˙ M = cst assumption, as already explainedin Kobayashi et al. (1997); Daigne & Mochkovitch (1998) is asmaller internal shock e ffi ciency, which leads to increase ˙ E tohave the same GRB fluence (see Tab. 1). As seen in Fig. 12, theimpact on the dynamics is weaker than in the case studied in § obs. ) and the evolution of the pulsewidth with energy ( obs. ). The HIC ( obs. ) is also slightlyimproved. There is also a weak impact on the high-energy emis-sion in the LAT range that is discussed in Sect. 6. We have not commented yet the evolution of the pulse prop-erties with duration. Observations show that pulses of shortduration are more symmetric ( obs. ), have very short, orzero, time-lags ( obs. ), and are harder ( obs. ), for a largepart due to higher peak energies as shown by the analysis ofthree bright GBM short GRBs (Guiriec et al. 2010). The inter-nal shock model reproduces qualitatively well these observations(Daigne & Mochkovitch 1998). Short GRBs have similar lumi-nosities compared to long GRBs (Nakar 2007), then it is rea-sonable to assume similar kinetic power ˙ E . Short GRBs emitMeV photons like long GRBs (see e.g. Guiriec et al. 2010), andeven GeV photons in the case of GRB 090510 (Ackermann et al.2010). Then, one would also expect similar Lorentz factors.The main di ff erence seems to be simply limited to the shorterduration, and more generally a compression of all variabilitytime scales (Guiriec et al. 2010; Bhat et al. 2012). If all inputparameters are kept constant except for the timescales, a sim-ple – two-shell collision– model of the internal shock phase(see e.g. Barraud et al. 2005), shows that the radius and comov- Fig. 15. E ff ect of the duration of the ejection. The peak en-ergy at the maximum of the pulse (top panel), the di ff erence be-tween the time of maximum in BATSE channel 3 and 1 (secondpanel), the index of the power-law giving the evolution of thepulse width with energy a = − d ln W / d ln E (third panel), theratio of the rise and decay time of the pulse in BATSE channel2 + H (ratio of the photonfluence in BATSE channel 3 over the photon fluence in BATSEchannel 2) are plotted as a function of the duration T in BATSEchannel 2 + ζ , and case B with a constant (red) or varying (magenta) ζ , with p = . t w which is varied from 2 ms to 200 s.ing mass density evolve as R ∝ t var and ρ ∗ ∝ t − , where t var is the variability timescale, and that Γ ∗ and ǫ ∗ are unchanged.Then Eq. (3) shows that L bol is not a ff ected but Eq. (5) indicatethat E p , obs ∝ t − : shorter pulses are naturally expected to havehigher peak energies. An increase of E p , obs has a direct impact onthe second factor in Eq. (9) (spectral correction) and then a ff ectmany observed features, such as the time lags, the pulse width,etc., as it has been illustrated in all the examples shown in thispaper. A secondary e ff ect can also play a role. The parameter w m also evolves with the variability timescale, as w m ∝ t − , so theimportance of Klein-Nishina corrections should increase whenthe duration decreases, which can a ff ect the general spectralshape and especially the low-energy photon index: see Figure8.d in Boˇsnjak et al. (2009).To test in more details the predicted pulse evolution with du-ration, we have computed a series of synthetic pulses keeping thesame parameters as in the reference cases A et B ( Γ min , Γ max , ˙ E and microphysics parameters), except for the duration t w whichvaries from 2 ms to 200 s. We have also performed the same se-ries of simulations for cases A and B with a varying fraction ofaccelerated electrons ζ , as discussed in § p = . p = . p = . ff erent features of the pulse as a function of the du- ration T measured between 50 and 300 keV (BATSE channels2 + ffi cult to test with ob-servations as very long pulses are rare); (iii) for the same rea-son, short pulses have the same width in all BATSE channels, a →
0; (iv) for the same reason, short pulses have more symmet-ric shapes (i.e. the pulse decay time becomes comparable to therise time). We notice that for constant ζ , the e ff ect is too strong,with the shortest pulses having a rise time time longer than thedecay time, which is sometimes observed but remains rare. Thecase with a varying ζ shows a much better agreement with ob-servations; (v) short pulses have a larger hardness ratio H . Thisindicator was used to identify the hardness-duration relation inKouveliotou et al. (1993). It depends mainly on E p , obs and there-fore the observed trend is reproduced. At very short durations,as E p , obs is above the spectral range of BATSE channel 3, thehardness ratio becomes constant, and has a value that dependsonly on α , i.e. H → (cid:16) − α − − α (cid:17) / (cid:16) − α − − α (cid:17) . Itis therefore distinct for case A ( α ≃ − . H → .
02) andcase B ( α ≃ − . . H → .
33 to 1 . obs. , and that the otherproperties (symmetry of short pulses, obs. and obs. ; van-ishing lags for short GRBs, obs. ) are also explained as a con-sequence of higher peak energies at short duration.
6. High energy signatures ( >
100 MeV)
We have described in the previous section how the di ff erentassumptions in the internal shock model (dynamics or micro-physics) can a ff ect the temporal and spectral properties of pulsesin the soft gamma-ray range. Is it possible to distinguish amongthe di ff erent possibilities from the high energy emission above100 MeV? The few bursts detected by Fermi -LAT show severalinteresting features (Ackermann et al. 2013): (i) LAT GRBs areamong the brightest ones detected by the GBM, with the excep-tion of few cases (e.g. GRB 081024 or GRB 090531). The mea-sured E iso in the subsample of bursts with a measured redshiftshows that the LAT bursts are intrinsically brighter. The ratioof fluences in the high ( >
100 MeV) and low ( < < ∼ >
10 GeV)are coming from the highest fluence GRBs (080916C, 090510,090902B); (ii) several LAT GRBs require an extra power-lawcomponent in addition to the Band model in the high energyportion of the spectrum. It can have a significant contributionto the total energy budget (10 - 30%) and becomes prominentat energies E obs > ∼
100 MeV. The slope of the power-law lieswithin the range − − .
6; (iii) the emission above 100 MeVsystematically starts later with respect to the GBM light curve.The ratio of the time delay over the total duration in the GBM islarger for longer bursts. In long GRBs, e.g. GRB 080916C, thetypical delay is of the order of a few seconds, while it is less thanone second in short GRBs, e.g. respectively ∼ . .
05 s oˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions 15 in GRB 090510 or GRB 081024B; (iv) the emission in the LATis long lasting compared to the GBM. It decays smoothly withtime and can be fitted with a power-law, F ν ∝ t − α with α closeto 1 in most of the cases. A break in the decay of the extendedemission is detected in GRB 090510, GRB 090902B, and GRB090926A, with a transition from α ≃ . . ffi cult to reconcile with an external ori-gin. As emphasized by Beloborodov et al. (2013), there is anadditional theoretical argument against the scenario where thewhole LAT emission (prompt and long lasting) would be associ-ated to the external shock (rise and decay): when using large timeintegration bins, the LAT flux starts to decay well before the endof the prompt emission in GBM, whereas the self-similar stageof the blast wave cannot be reached at such early times (typi-cally, not earlier than the duration of the prompt emission).In the following we examine which of the properties ofthe early high-energy emission in GRBs can be accommodatedwithin the internal shock model and if the LAT observations mayo ff er a way to distinguish among the di ff erent scenarios stud-ied in the two previous sections. We do not consider the pos-sible contribution of shock-accelerated protons to the emission,as various studies have shown that it requires extreme parame-ters to be dominant in the LAT range, due to a weak e ffi ciency(Asano et al. 2009; Asano & M´esz´aros 2012). We do not includeadditional processes that may be important, such as the scatter-ing of photospheric photons by shock accelerated electrons ininternal shocks (Toma et al. 2011) or the scatterings of promptphotons in the pair-enriched shocked external medium at earlystages of the deceleration (Beloborodov et al. 2013). As illustrated in Fig. 16, cases A and B, with a constant or a vary-ing ζ , have very di ff erent high-energy spectra as the e ffi ciency ofinverse Compton scatterings strongly depend on ǫ B /ǫ e . The in-verse Compton component is negligible in case A, whereas itcreates a well defined additional component at high-energy incase B. This additional component is stronger when ζ is con-stant. As shown in § ζ ∝ ǫ ∗ maintains a highervalue of the peak energy during the decay, and then a less ef-ficient inverse Compton emission. It is interesting to note thatthe additional component in the GeV range is very flat in the ν F ν spectrum (see Fig. 16 bottom-left panel) and would proba-bly be fitted by a power-law with a photon index close to −
2, asobserved in several LAT bursts (Ackermann et al. 2013).These examples cannot be directly compared to
Fermi -LATbursts as they radiate ∼ erg, whereas LAT bursts are muchbrighter (Ackermann et al. 2013). In addition, we did not try to Fig. 16. High-energy emission: spectra.
The time-integratedspectrum (0 – 15 s) is plotted from the keV to the GeV rangefor the same cases as in Fig. 18.
Top left panel: reference casesA and B;
Bottom left panel: cases A and B with a varying accel-erated electron fraction ζ ; Top right panel: case B with a sharpinitial distribution of the Lorentz factor (see text) and a constantor a varying accelerated electron fraction ζ ; Bottom right panel: case B with a constant ejected mass flux (see text) and a constantor a varying accelerated electron fraction ζ .adjust the model parameters to improve the peak energy of theadditional component. In our examples, the additional compo-nent typically appears above 1-10 GeV, whereas it is alreadydetected at lower energy in LAT bursts. The shape of the addi-tional component and its peak energy are determined in a com-plex manner by the relative e ffi ciency of the synchrotron andinverse Compton emission, the slope p of the shock-acceleratedelectrons, and the γγ annihilation. This is illustrated in Fig. 17where the spectrum in cases A and B is plotted for two di ff er-ent values of p , which directly impacts the photon index β ofthe high-energy part of the dominant (synchrotron) component.Increasing p and β allows to observe the emergence of the addi-tional component at lower energy and a ff ects its measured slope.Due to the high peak energies of the inverse Compton com-ponent in our reference cases, the light curves above 1 GeV aremainly governed by the synchrotron radiation and peak approx-imatively at the same time as the soft gamma-ray component,with only a very small delay (see Fig. 18), contrary to the ob-served delayed onset of the GeV emission (Ackermann et al.2013). To increase this delay, one should either increase the γγ annihilation in the early phase by decreasing ¯ Γ as illus-trated in Hasco¨et et al. (2012), or adjust the parameters so thatthe inverse Compton emission peaks at lower energy (see e.g.Asano & M´esz´aros 2012), or both. Nevertheless, in case B witha constant ζ , where the inverse Compton emission is the most ef-ficient, the additional component starts to be visible in the lightcurve during the pulse decay (see Fig. 18, right, top panel). Onealso sees a small high-energy precursor that appears because theshock is initially weak, with a low peak energy and a high inverseCompton e ffi ciency (Boˇsnjak et al. 2009). This precursor, never Fig. 17. High-energy emission: e ff ect of the electron slope p . The time-integrated spectrum (0 – 15 s) is plotted from the keVto the GeV range for reference case A (left panels) and B (rightpanels), either assuming a constant (top panels) or a varying(bottom panels) fraction ζ of accelerated electrons, with p = . . The assumptions for the dynamics have a large impact on thehigh energy emission. In the case ˙ M = cst (rather than ˙ E = cst),the inverse Compton emission is more e ffi cient during the pulsedecay (but not during the rise, as in reference case B). This is dueto a more rapid decrease of the peak energy during the decay (seeFig. 12) and therefore a more rapid decrease of the Klein Nishinacorrections. This improves the light curves (Fig. 18 bottom pan-els), which show a more intense tail due to inverse Comptonemission, and a larger delay between the peaks of the LAT andGBM light curves, however still too small to explain the ob-served delayed onset. The additional component in the spectrumis already detected between 1 GeV and 10 GeV (Fig. 16).The case where the initial distribution of the Lorentz factorhas a sharp transition from Γ min to Γ max has the strongest impact.In this case, the shocks are immediately violent so that the weakprecursor observed in the LAT in other cases does not appear(see Fig. 18): the peak energy of the ’forward’ internal shockis indeed immediately very high and the corresponding inverseCompton emission is suppressed by Klein-Nishina corrections.This is only at late times that the inverse Compton emission be-comes bright when more scatterings occur in Thomson regime.However, an important di ff erence in this case is the fact that theemission of the ’forward’ internal shock lasts longer and is notnegligible (see Fig. 12). It is even dominant in the LAT for thefirst seconds. Due to a lower peak energy, the corresponding ad-ditional high energy component is well seen in the LAT, eitherwith a constant accelerated electron fraction ζ , or even more witha varying ζ , which is the only simulated case in all the examples presented in this paper where the choice of parameters imme-diately leads to a peak energy of the IC component at 10 GeV(see Fig. 16). For this reason, this case is the brightest in the LATrange and illustrates well that the high energy emission from in-ternal shocks is not only sensitive to the details of the assump-tions regarding the microphysics but also to the dynamics.Interestingly, we note that in most of the scenarios discussedin Sect. 6, the plots showing the time lag with respect to the low-energy channel as a function of the energy (Figs. 3, 7, 11, 13,14) shows a U-shape, the light curves initially peaking earlierwhen the energy is increasing, with a reversal of this trend above ∼ −
100 MeV. Such a behavior is found in GBM + LAT data,as studied by Foley et al. (2011); Foley (2012).We conclude that the high energy emission from internalshocks is highly sensitive to the details of the assumptions re-garding both the microphysics and the dynamics, and can there-fore provide valuable diagnostics to distinguish among the var-ious scenarios discussed in this paper. However a direct com-parison of our results with observations reveals to be delicateas LAT GRBs are among the brightest, with isotropic energiesmuch larger than the average ’typical’ value considered here. Asthis paper is mainly focussed on the temporal and spectral prop-erties in the soft gamma-ray range, we leave to a forthcomingstudy a more detailed comparison to
Fermi data, which will bebased on simulated bursts with more extreme parameters, espe-cially regarding the total injected energy and the Lorentz factor.
7. Discussion
The spectro-temporal evolution in the internal shock model isgoverned by the hydrodynamics : the physical conditions in theshocked regions vary on the hydrodynamical timescale associ-ated to the propagation of the internal shocks. This evolutiona ff ects in a complex manner the respective e ffi ciency of the ra-diative processes (synchrotron radiation, inverse Compton scat-terings) as well as the peak energy and spectral shape of eachcomponent. The model parameters can be divided in two groups:assumptions for the microphysics and for the dynamics. Bothcan strongly a ff ect the spectro-temporal evolution in GRBs. The dissipation of the energy in the shocked region is pa-rameterized by ( ǫ e , ζ, p ) describing the energy injection in therelativistic electrons distribution, and ǫ B describing the am-plification of the magnetic field. The values of these micro-physics parameters are broadly constrained by the observa-tions. As GRBs are extremely bright, a high ǫ e is requiredto avoid an energy crisis. As Fermi -LAT observations are notcompatible with a SSC spectrum (Piran et al. 2009), the softgamma-rays must be directly produced by synchrotron radia-tion, which requires a low fraction of accelerated electrons ζ < ∼ − − − (Daigne & Mochkovitch 1998; Boˇsnjak et al. 2009;Daigne et al. 2011; Beniamini & Piran 2013). The fact that theobserved low energy index α is usually larger than the standardsynchrotron fast cooling value − / ǫ B , typ-ically ǫ B < ∼ − − − (Daigne et al. 2011; Barniol Duran et al.2012), if such photon indices are mainly due to the e ff ect of in-verse Compton scattering in the Klein Nishina regime. The factthat many bursts show also a steep high-energy photon index β or are even well fitted with a power-law + an exponential cut-o ff (Kaneko et al. 2006; Goldstein et al. 2012) implies that theelectron slope p can be larger than the usually considered value p ≃ . − . oˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions 17 Fig. 18. High-energy emission: light curves.
The light curves in the soft gamma-ray range (260 keV – 5 MeV, left figure) and inthe high-energy gamma-ray range ( > ff erent cases discussed in the paper. For the high-energylight curves, a thin solid line indicates that the synchrotron emission is dominant above 1 GeV whereas a thick solid line indicatesthat the inverse Compton emission is dominant. Top panel: reference cases A and B;
Second panel: cases A and B with a varyingaccelerated electron fraction ζ ; Third panel: case B with a sharp initial distribution of the Lorentz factor (see text) and a constantor a varying accelerated electron fraction ζ ; Bottom panel: case B with a constant ejected mass flux (see text) and a constant or avarying accelerated electron fraction ζ .We found that the spectro-temporal evolution predicted bythe internal shock model is qualitatively in agreement with theobservations. This is illustrated by the two reference cases A andB defined in Daigne et al. (2011) and corresponding to ǫ e = / ζ , p = . ǫ B , leading respec-tively to a standard α ≃ − . α ≃ − . ff ect of the electron distri-bution slope (Fig 6), and found that p ≃ p = α < ∼ − β . However, the evo-lution of the peak energy remains usually too rapid comparedto observations. This problem had already been identified byDaigne & Mochkovitch (1998, 2003) based on a much simplertreatment of the radiative processes. They also suggested thatthis may be related to the common assumption of constant mi-crophysics parameters during the evolution of the shocks, whichmay appear unrealistic. To investigate the impact of these as-sumptions, they considered a simple prescription for varying mi-crophysics parameters – in the absence of still missing physi-cally motivated prescriptions based on shock acceleration theory– where the fraction of accelerated electrons is evolving with theshock Lorentz factor, such as ζ ∝ ǫ ∗ . We simulated the spectro-temporal evolution predicted by the internal shock model under such an assumption. This indeed leads to a much better quantita-tive agreement: the evolution of the peak energy is slower, and,as it governs most of the other properties, the general agreementis much better for the hardness intensity correlation, the evolu-tion of the pulse shape and time of pulse maximum with energychannels, etc. (see Sect. 4 and Figs. 9 and 11). There are no theoretical arguments why microphysics parame-ters should be universal in mildly relativistic shocks (see e.g.Bykov et al. 2012). Even in the ultra-relativistic regime, GRBafterglows already show the opposite, as a broad distributionof parameters is necessary to fit the observations (see e.g.Panaitescu & Kumar 2001; Cenko et al. 2010). In absence of awell established shock theory, we have tested here variationsfollowing the prescription ζ ∝ ǫ ∗ , which is suggested by thework of Bykov & Meszaros (1996). Our result that varying mi-crophysics parameters improves the quantitative agreement be-tween the predictions of the internal shock model and the ob-served spectro-temporal evolution observed in GRBs is thereforeencouraging. On the other hand, some of the typical values of themicrophysics parameters in the simulations presented in this pa-per may appear unrealistic, compared to recent progress in shockacceleration modelling, especially results from large Particle-In-Cell (PIC) simulations. Steep p ≃ . ǫ e ≃ . − . ζ ≃ − − − and low magnetic field energy frac-tion ǫ B ≃ − may appear in contradiction with shock simula-tions, as mentioned for instance by Barniol Duran et al. (2012);Beloborodov (2013); Beniamini & Piran (2013). This calls forseveral comments : (i) current PIC simulations are limited to ultra-relativistic shocksand do not describe yet the parameter space of mildly relativisticshocks such as in internal shocks, i.e. with typical shock Lorentzfactors γ sh < ∼
2. A direct comparison is therefore di ffi cult. For γ sh =
15, PIC simulations show that acceleration does not oc-cur for magnetized ( σ > ∼ − ) perpendicular shocks, but is ob-served either for weakly magnetized or quasi-parallel ”sublumi-nal” shocks with typically ǫ e ∼ . ζ ∼ − and p ≃ . − . ζ ∼ − .Theoretical investigations of the energy transfer from protonsto electrons in mildly relativistic shocks also predict that onlya fraction of electrons are accelerated, with ζ as low as 10 − (Bykov & Meszaros 1996). Therefore the values of the acceler-ated electron fraction ζ in the simulations discussed here are notin strong contradictions with shock acceleration modelling, butare usually too small. The low values of ζ in our simulationsare necessary to reach high peak energies for the synchrotroncomponent. However, a detailed comparison between the ballis-tic (”solid shells” model) approach used here for the dynamicsof internal shocks with a more precise calculation based on a 1DLagragian special relativistic hydrocode shows that the agree-ment between the two calculations is usually very good, exceptfor the mass density ρ ∗ and specific internal energy density ǫ ∗ in the shocked region, which are underestimated by the simplemodel (Daigne & Mochkovitch 2000). For similar Lorentz fac-tor Γ ∗ microphysics parameters ǫ e , ζ , p , and ǫ B , larger values of ρ ∗ and ǫ ∗ lead to a larger peak energy. As ρ ∗ is typically under-estimated by at least a factor ∼ and ǫ ∗ by a factor ∼ ζ deduced from the simple dynamical model may beunderestimated by a factor > ∼ / × / ≃
30 as E p , obs ∝ ρ . ∗ ζ − (see Eq. (5)). Taking into account this e ff ect, the cases listed inTab. 1 would correspond to the range ζ ∼ (1 −
10) %, in bet-ter agreement with theoretical predictions. In addition, we havesimulated very smooth single pulse bursts for simplicity and tobetter identify the spectro-temporal evolution but more variableoutflows would lead to more e ffi cient collisions with higher val-ues of the dissipated specific internal energy ǫ ∗ , allowing to reachhigh peak energies for larger values of ζ ;(iii) shock acceleration is accompanied by the amplification ofthe magnetic field in the shocked region. Both processes cannotbe dissociated. Therefore, the low values of ǫ B considered in ref-erence case B and the derived cases may appear unrealistic, asdiscussed in Barniol Duran et al. (2012). However, one shouldremember that ǫ B should be understood here as fixing the typicalstrength of the magnetic field seen by radiative electrons, i.e. on alength scale fixed by the electron radiative timescale. This lengthscale is much larger than the plasma scale and ǫ B is therefore notonly determined by the amplification at the shock, but also bythe evolution of the magnetic field on larger scales. Recently,Lemoine (2013) demonstrated that if the large magnetic fieldgenerated in a thin microturbulent layer at the shock front de-cays over some hundreds of skin depths – as suggested by recentsimulations (Keshet et al. 2009) – the e ff ective ǫ B deduced fromobservations may be much lower than the value predicted by PICsimulations at the shock (see also Derishev 2007; Kumar et al.2012; Uhm & Zhang 2013). In addition, the low value of ǫ B usedin case B is required to favor inverse Compton scatterings inKlein-Nishina regime and to increase the low-energy slope ofthe synchrotron spectrum (see Daigne et al. (2011) for a discus-sion of the detailed conditions). As discussed below ( § α > ∼ − ǫ B would be relaxed and higher values could be considered. The dynamics of the relativistic outflow is determined by theinitial conditions described by the variation of the bulk Lorentzfactor of the flow Γ ( t ej ), the kinetic power ˙ E ( t ej ), and the dura-tion of the relativistic ejection t w . As the spectral evolution ismainly governed by the details of the propagation of the inter-nal shock waves, any change in the initial Lorentz factor or ki-netic power directly a ff ects the light curve shape (see Figs. 13and 14). Typically, we find that steeper variations of the Lorentzfactor lead to internal shocks which are immediately e ffi cient,with a peak energy which is already high at early times in thepulse. The biggest impact is an improvement of the light curveat high energy (GeV range) compared to Fermi observations, asillustrated in Fig. 18. Changing the assumptions on the injectedkinetic power also impacts the results, mostly at high energy, andcan a ff ect the overall e ffi ciency of the internal shock phase.Unfortunately, the current understanding of GRB central en-gines and of the relativistic ejection phase does not allow a de-tailed prediction of the input parameters Γ ( t ej ) and ˙ E ( t ej ). We caninvestigate which assumptions favor the best agreement with ob-servations, but we can not conclude if these assumptions are re-alistic and – when di ff erent assumptions lead to a similar agree-ment – which assumption should be preferred.We have also tested an interesting property of the internalshock model : the dependence of the temporal and spectral prop-erties on the duration of a pulse. For that purpose, we have sim-ulated a series of pulses keeping all parameters constant exceptfor the total duration of the relativistic ejection, t w . It is very en-couraging to observe that, despite its simplicity (in reality, varia-tions of t w are probably accompanied by variations of other inputparameters), the model reproduces well the observations : shortpulses become more symmetric, have smaller or zero time lags,have a higher hardness ratio (see Fig. 15). This is mainly dueto the fact that the peak energy is higher for a shorter variabilitytimescale, a clear prediction of the internal shock model. At veryshort duration, most of the pulse light curve in the soft gamma-ray range occurs in the same portion of the spectrum (belowthe peak energy), which explains why the lags vanish and thehardness ratio tends to be constant. All these properties of shortpulses have been observed in real GRBs since the BATSE era,and have been confirmed by Fermi (Guiriec et al. 2011, 2013),which in addition has shown the dominant e ff ect of higher peakenergies to explain this evolution in short pulses. There are several potential additional e ff ects that are not takeninto account in this work but should be examined in the future.– We have shown in Sect. 6 that the di ff erent sub-scenarios of theinternal shock model may di ff er by their predictions for the high-energy gamma-ray emission. However, a special modeling ef-fort is necessary to compare these predictions to observations, as Fermi -LAT bursts are among the brightest GRBs ever detected,whereas the pulses simulated here have average properties.– Recent
Fermi / GBM observations have shown a disagree-ment in the soft gamma-ray range between the observed spec-trum and the phenomenological Band function (Band et al. oˇsnjak & Daigne: Spectral evolution in GRBs: internal shock predictions 19 α , as it is usually found inthese bright GBM bursts that adding a new spectral component atlow energy to better reproduce the spectral shape leads to smallervalues of α . As discussed above, this relaxes the constraint onone microphysics parameter, ǫ B . A promising interpretation forthese new observations is that an extra component associated tothe photospheric emission is detected, in agreement with theoret-ical predictions (Guiriec et al. 2011, 2013). A possible diagnos-tic to distinguish among the di ff erent scenarios discussed herewould be to simultaneously simulate the photospheric and inter-nal shock emission, as the predicted spectral evolution for thesetwo components has not the same dependence on the propertiesof the relativistic outflow (Hasco¨et et al. 2013);– To be able to explore a large range of the parameter space,some simplifications have been made in the present calculations.There are several possible improvements which may be investi-gated in the future : (i) what is the contribution to the emissionof the thermal electrons which are not shock-accelerated (seee.g Giannios & Spitkovsky 2009). This depends of course on thefraction ǫ the of the dissipated energy which remains in the fraction1 − ζ of electrons that are not accelerated. We have checked thatfor ǫ the /ǫ e < ∼ .
1, the additional component due to the emission ofthe thermal electrons does not a ff ect the gamma-ray spectrum,and therefore does not change the results of the present paper.On the other hand, it may contribute in certain conditions to theprompt optical emission. We note that the absence of a clear sig-nature of thermal electrons in afterglow observations may indi-cate that the ratio ǫ the /ǫ e is not very large in relativistic shocks; (ii)what is the contribution to the emission of the secondary leptonsproduced by γγ annihilation. As shown by Asano & M´esz´aros(2011), this could have an important contribution to the observedextra power-law component identified by Fermi / LAT. In addi-tion, a more precise calculation of the γγ annihilation may helpin better reproducing the delayed onset of the GeV light curve,also identified by Fermi (Hasco¨et et al. 2012); (iii) what is thee ff ect of the injection timescale of the accelerated particles ? Aslow injection may improve the spectral shape at low energy, asinvestigated recently by Asano & Terasawa (2009); (iv) what isthe e ff ect of a decaying magnetic field behind the shock front ?Such an evolution is expected from shock acceleration modellingand may improve the shape of the synchrotron spectrum (in-creasing low-energy photon index) without implying as low val-ues of ǫ B as what is considered in this paper (see e.g. Derishev2007; Wang et al. 2013).
8. Conclusions
Motivated by the results from the
Fermi satellite which signif-icantly extends the spectral coverage of the GRB phenomenonand improves particularly the spectral analysis of the promptemission, we investigated in this paper the origin of the ob-served spectral evolution in GRBs. We presented the results ofa set of numerical simulations of the GRB prompt emission inthe framework of the internal shock model. We made a detailedcomparison of the model predictions with the observed tempo-ral and spectral GRB properties in the soft gamma-ray range.We focussed on the simplest case of a single pulse burst asso-ciated to the synchrotron radiation from shock-accelerated elec-trons in the internal shocks formed after the collision betweena ’fast’ and a ’slow’ region in an ultra-relativistic ejecta. Weconsidered three reference case with a duration of 2 − . × , 1 . × and 1 . × erg, a peak energy of 730, 640 and 160 keV, anda low-energy photon index of -1.5, -1.1 and -0.7.We show that many observed properties or common trends –namely (i) the pulse asymmetry, (ii) the energy dependent pulseasymmetry (evolution of the pulse width with energy channel),(iii) the time lags between the light curves in di ff erent energychannels, (iv) the hard-to-soft evolution within pulses, (v) thehardness-intensity correlation, (vi) the hardness-fluence correla-tion – can be accounted for and are governed by the details ofthe spectral evolution, i.e. the evolution of the peak-energy andthe spectral slopes.We showed that there is a qualitative agreement between themodel results for our three reference cases and the large set ofobservations listed above. With a comprehensive set of simu-lations, we demonstrated that a quantitative agreement can beachieved under some constraints on the model parameters. Wedistinguished between the e ff ects of the microphysics (detailsof the energy distribution in shocked regions) and the dynami-cal parameters (initial conditions in the outflow). We found thatthe agreement with the observed spectral evolution can be sig-nificantly improved if (i) the distribution of shock-acceleratedelectrons is steeper than what is usually assumed, with a slope p > ∼ .
7; (ii) the microphysics parameters vary with the shockconditions in a manner that reduces the dependency of the peakenergy on the shock conditions. It is illustrated here by the casewhere the fraction of accelerated electrons increases for strongershocks; (iii) the initial variations of the Lorentz factor in the out-flow are steeper. An additional advantage of this assumption isthe increase of the e ffi ciency of internal shocks; (iv) the rela-tivistic ejection proceeds with a constant mass flux rather than aconstant kinetic energy flux. A drawback of this last possibilityis a reduced e ffi ciency of the shocks. As the microphysics pa-rameters are not well constrained by the current stage of shockacceleration modelling in the mildly relativistic regime relevantfor internal shocks, and as the initial conditions in the outflow arealso poorly constrained due to many uncertainties regarding themechanism responsible for the relativistic ejection by the centralengine, we cannot conclude if one of these four possibilities maybe expected or should be preferred.We also specifically investigated the impact of the dura-tion of the relativistic ejection, as many of the properties listedabove are known to evolve with pulse duration. The internalshock model naturally predicts a larger peak energy for shortpulses, and possibly a harder photon index due to a deeper Klein-Nishina regime for inverse Compton scatterings. We showed that– in agreement with observations – this leads to a hardness-duration correlation and to the following consequences: pulsesbecome more symmetric, with almost no evolution of the pulsewidth with energy, and with very short or zero lags. The promptemission from short GRBs could then be due to the same mech-anism as in long GRBs, but for di ff erent model parameters dueto the fact that all timescales are contracted, probably because ofa di ff erent central engine.Finally, we investigated the signature at high-energy ( Fermi -LAT range). In this domain, the observed flux is made of thehigh-energy tail of the synchrotron component and a new com-ponent produced by inverse Compton scattering. A direct com-parison with
Fermi -LAT results is not possible as LAT burstsare among the brightest whereas we have simulated here aver-age pulses. However, we note a qualitative agreement with data:due to the evolving e ffi ciency of the scatterings – they usuallyoccur in the Klein-Nishina regime at early times and enter theThomson regime during the pulse decay – the resulting emission at high-energy can di ff er significantly from the keV-MeV range;specifically, the rise of the light curve is delayed and the emis-sion lasts longer. This leads to a U-shape curve when plottingtime lags with respect to the low-energy channel as a function ofenergy, in agreement with GBM + LAT observations. However,we do not have a quantitative agreement : the onset of the high-energy light curve is not delayed enough. Interestingly, some ofthe e ff ects listed above – a steeper electron slope, a varying elec-tron acceleration fraction, and especially steeper variations of theinitial Lorentz factor – have also a positive impact on the proper-ties of the high-energy emission. The time-integrated spectrumat high-energy depends strongly on the e ffi ciency of the inverseCompton scatterings. In some cases, it is found to be very closeof the extrapolation of the MeV component, possibly with a cut-o ff at high-energy; in other cases, it clearly shows an additionalcomponent, which can either be rising (photon index greater than −
2) or flat (photon index close to − ff erences between the various scenarios discussed in the pa-per, this motivates a specific comparison to Fermi -LAT burstswhich will hopefully provide diagnostics to distinguish amongthe various theoretical possibilities.This study illustrates the capacity of the internal shock modelto reproduce most of the observed properties of the GRB promptemission related to the spectral evolution, both for long and shortbursts. Our conclusions are limited by many uncertainties inthe ingredients of the model, namely the details of the micro-physics in mildly relativistic shocks and the initial conditionsin the GRB relativistic outflows. However, in a more optimisticview, we showed that this poorly understood physics may havea detectable imprint in GRB data, which should allow for someprogress in the future.
Acknowledgements.
The authors thank R. Mochkovitch for many valuable dis-cussions on this work, and a careful reading of the manuscript. F.D. and Z.B.acknowledge the French Space Agency (CNES) for financial support.
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