An External Shock Origin of GRB 141028A
J. Michael Burgess, Damien Bégué, Felix Ryde, Nicola Omodei, Asaf Pe'er, J. L. Racusin, A. Cucchiara
DDraft version September 24, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
AN EXTERNAL SHOCK ORIGIN OF GRB
J. Michael Burgess † , Damien B´egu´e ‡ , Felix Ryde , Nicola Omodei , Asaf Pe’er , J. L. Racusin ,A. Cucchiara Draft version September 24, 2018
ABSTRACTThe prompt emission of the long, smooth, and single-pulsed gamma-ray burst, GRB , isanalyzed under the guise of an external shock model. First, we fit the γ -ray spectrum with a two-component photon model, namely synchrotron+blackbody, and then fit the recovered evolution of thesynchrotron νF ν peak to an analytic model derived considering the emission of a relativistic blast-wave expanding into an external medium. The prediction of the model for the νF ν peak evolutionmatches well with the observations. We observe the blast-wave transitioning into the decelerationphase. Further we assume the expansion of the blast-wave to be nearly adiabatic, motivated bythe low magnetic field deduced from the observations. This allows us to recover within an order ofmagnitude the flux density at the νF ν peak, which is remarkable considering the simplicity of theanalytic model. Across all wavelengths, synchrotron emission from a single forward shock providesa sufficient solution for the observations. Under this scenario we argue that the distinction between prompt and afterglow emission is superfluous as both early and late time emission emanate from thesame source. While the external shock model is clearly not a universal solution, this analysis opensthe possibility that at least some fraction of GRBs can be explained with an external shock origin oftheir prompt phase. Subject headings: gamma-ray burst: individual (141028A) – radiation mechanisms: non-thermal –radiation mechanisms: thermal INTRODUCTION
Identifying the origin of the dynamical evolution ofgamma-ray burst (GRB) outflows is an unsolved issue,critical to the understanding of both the energetics andspectra of these events. One idea is that the emissionis the result of synchrotron radiation from an externalforward shock propagating into the external circumburstmedium (CBM) (Cavallo & Rees 1978; Rees & M´esz´aros1992; M´esz´aros & Rees 1993; Chiang & Dermer 1999;Dermer et al. 1999; Dermer & Mitman 1999). This mech-anism should produce smooth γ -ray pulses with dura-tions on the order of a few seconds for typical GRB pa-rameters. However, the short-time variability (on theorder of a few milliseconds) of many GRB light curvesruled out this postulate as a universal mechanism (e.g.Sari & Piran 1997; Kobayashi et al. 1997; Walker et al.2000) and gave favor to several alternative hypothesesincluding models that consist of rapid internal shocks inan unsteady outflow (Rees & M´esz´aros 1994) or mag-netic reconnection (e.g., Spruit et al. 2001; Drenkhahn& Spruit 2002; Zhang & Yan 2011) to account for theobserved emission. The Oskar Klein Centre for Cosmoparticle Physics, Al-baNova, SE-106 91 Stockholm, Sweden Department of Physics, KTH Royal Institute of Technology,AlbaNova University Center, SE-106 91 Stockholm, Sweden † [email protected] ‡ [email protected] W. W. Hansen Experimental Physics Laboratory, Kavli In-stitute for Particle Astrophysics and Cosmology, Department ofPhysics and SLAC National Accelerator Laboratory, StanfordUniversity, Stanford, CA 94305, USA. Physics Department, University College Cork, Cork, Ireland NASA Goddard Space Flight Center, Greenbelt, MD 20771,USA
Still, there do exist long and temporally smooth GRBswith typical variability time scales larger than a few sec-onds (see Golkhou & Butler 2014; Golkhou et al. 2015)that do not violate the variability constraints of the ex-ternal shock model and can be tested via their spec-tral evolution as to whether they conform to the well-established predictions made by this model. The sim-plicity of the model affords it the ability to be testedboth spectrally and temporally, a feature unique to theexternal shock model. The dynamics and spectra ofthe internal shock model have been simulated (Daigne& Mochkovitch 1998), but the dynamics rely on as-sumed and degenerate configurations from variations inthe wind (e.g. the radial distribution of the Lorentz fac-tors), forbidding the formulation of unique predictionsthat can be identified in the data. Therefore, it is cur-rently impossible to test the internal shock model in themanner presented here without severe degeneracies. Ad-ditionally, the internal shock model has trouble efficientlyconverting the internal kinetic energy of a GRB into ra-diation, which is challenging when trying to explain theextreme luminosities observed (Kobayashi et al. 1997).Herein, we analyze the bright, long, single-pulsedGRB and find several clues for an ex-ternal shock origin of its emission. We fit theGRB’s time-resolved spectra with a slow-cooled syn-chrotron+blackbody model (Burgess et al. 2014b) andexamine the evolution of the spectra. The evolution ofthe synchrotron νF ν peak ( E p ) is fit with an analyticphysical model predicted by Dermer et al. (1999). Fromthis fit, we obtain physical parameters such as the coast-ing Lorentz factor and CBM radial profile which can thenbe used to predict how the flux of the prompt emissionshould evolve. Comparing these predictions to the data a r X i v : . [ a s t r o - ph . H E ] J un enables us to test the validity of the model in severalways.The article is organized in the following manner. InSection 2, we introduce the formalism of the externalshock model to derive a function for E p ( t ) to fit to data.In Section 3, the observations and spectral analysis areintroduced. Section 4 details our application of the exter-nal shock model to the data. The parameters resultingfrom the analysis are used to make further predictionsregarding the flux evolution of the prompt phase of theGRB. From there, we analyze the photospheric emission.In Sections 5.1 and 5.2, we discuss the compatibility ofour results with the high and low-energy late-time obser-vations. THE EXTERNAL SHOCK MODEL
The external shock model is built upon the blast-waveevolution derived in Blandford & McKee (1976) whichtracks the evolution of a relativistically expanding fire-ball into an external medium. The equations can be ap-plied to GRBs by assuming some fraction of the electronsin the shocked external medium is accelerated to high en-ergies by the shock wave and radiates a fraction of thekinetic energy away via synchrotron radiation (Cavallo& Rees 1978; Rees & M´esz´aros 1992). In this work, weuse the analytic formalism developed in Dermer et al.(1999) to fit the spectral evolution of the emission. Webriefly review the main equations required and refer thereader to Dermer et al. (1999); Chiang & Dermer (1999)for more details on the model.The blast-wave is assumed to expand into an externalcircumburst medium (CBM) with a radial density evolu-tion modeled as a power law, n ( x ) = n x − η cm − (1)where n is the initial density and η describes the radialmorphology of the CBM such that η = 0 is a constantdensity and η = 2 describes a stellar wind. The dimen-sionless radial coordinate is x = r / r d where, following theconvention that a quantity w = w n n , r d = 5 . · (cid:34) (1 − η / ) E , n , Γ , (cid:35) / cm (2)is the radius at which the blast has swept up a significantamount of mass ( ∝ Γ − ) to begin decelerating (Rees &M´esz´aros 1992). Following the solution of Blandford &McKee (1976) the evolution of the bulk Lorentz factor(Γ) of the blast-wave is modeled as a broken power lawconsisting of a coasting phase followed by a decelerationphase: Γ( x ) = (cid:26) Γ x < x − g ≤ x (3)where Γ is the coasting Lorentz factor and g is the radia-tive regime index. For a constant density ( η = 0) CBM, g = 3 , / indicate the fully radiative and non-radiative(adiabatic) expansion regimes respectively (Blandford &McKee 1976). A fraction of the blast energy is dissipatedin the shock and accelerates electrons to high energies,which subsequently radiate this away via synchrotron ra-diation. Following the parameterization of Dermer et al. (1999), the temporal evolution of the νF ν peak energy, E p , of this synchrotron radiation can be modeled as E p ( t ) = E (cid:20) Γ( x )Γ (cid:21) x − η / keV (4)where E = 15 n / , q − Γ , z keV (5)is the observed E p at the observed deceleration time, t d = r d Γ c (1 + z ). (6)Here, c is the speed of light and q parameterizes themagnetic field and shock acceleration microphysics suchthat q ≡ [ (cid:15) B ( r s / )] / (cid:15) . Here (cid:15) B and (cid:15) e are the magneticand electron equipartition factors and r s (cid:39) (cid:15) B (cid:39) − and (cid:15) e (cid:46) . q (cid:39) − . While these values of (cid:15) B are lower than the typically assumed values of ∼ − ,recent studies (Lemoine et al. 2013; Santana et al. 2014)find that the values can be much lower and we thereforefollow these works. For the analysis, we will allow values (cid:15) B (cid:46) − to cover the range of typically assumed values.Additionally, we can write the measured νF ν peak lu-minosity as P p ( t ) = Π (cid:26) x − η ≤ x < x − η − g ≤ x < Γ / g (7)where Π ∝ (2 g − η ) m p c Γ n A g (1 + z ) erg s − . (8)Here, m p is the proton mass and A is the blast-wave areafactor. The proportionality in Equation 8 comes fromthe fact that in Dermer et al. (1999) an empirical photonmodel is used to create light curves and here we willbe using synchrotron emission directly to fit the time-resolved spectra of GRB (see Section 3).With these equations, the external shock model canpredict several observable quantities, namely, the evolu-tion of E p and P p which we test below in Section 4.2.The effects on the E p evolution of different parametersis shown in Figure 1. The parameters have the followingdominant effects: • Γ : sets the break time • η : sets the slope of early E p decay • g : sets the slope of late E p decay.These parameters, along with q which acts as a normal-ization, will be the free parameters that will be deter-mined by fitting the observed evolution of E p to Equa-tion 4. In reality, the range of values for g are determinedby the value of η (Dermer et al. 1999). Since we cannotknow a priori the value of η , we treat g as an independentparameter. It is important to note that this analytic ap-proximation to the blast-wave evolution fails to capturesome aspects that the full numerical solution exhibits − − − Time [s] − − − E p [ k e V ] changing Γ ( η = 0 , g = 2) changing η ( Γ = 600 , g = 2 )changing g ( Γ = 100 , η = 0 ) η =0 η =1 η =2 g =3/2 g =2 g =3 Γ =100 Γ =300 Γ =500 Figure 1.
Demonstrating the effect of the physical parameters onthe evolution of E p with time. It is easy to see that Γ has thestrongest effect on the timescale of the burst, while η affects theearly evolution of E p and g the late evolution. (Chiang & Dermer 1999). Most notable is the smoothtransition from coasting to decelerating at the deceler-ation radius ( r d ). The formalism we adopt will serveas a proof of concept that will be further applied to alarger sample and improved upon with a full numericaltreatment. OBSERVATIONS OF GRB
Data Acquisition
Rapid variability poses a problem for the externalshock model; therefore, bright, long, single-pulsed GRBsprovide the most viable candidates for being producedby external shocks. GRB is an example of thisclass of GRB (see Figure 2). GRB was discov-ered by the
Fermi
Gamma-ray Burst Monitor (GBM)(Roberts 2014; Meegan et al. 2009) and the bright GRBtriggered an autonomous repoint of the
Fermi spacecraftto optimize the Large Area Telescope (LAT) (Atwoodet al. 2009) for follow-up observations. In ground anal-ysis, LAT also localized and detected the GRB (Bissaldiet al. 2014) until ∼ s after the trigger. The lo-calization led to a target of opportunity observation bythe narrow-field Swift instruments (the X-ray Telescope,XRT; and the Ultraviolet Optical Telescope, UVOT)(Gehrels et al. 2004). The X-ray and optical afterglowwas detected by XRT (Kennea 2014) and UVOT (Siegel& Pagani 2014), and subsequently by many differentground-based facilities, including a measurement of aredshift of z=2.332 with the Very Large Telescope/X-Shooter instrument (Malesani 2014). We collected op-tical/NIR photometry from GCN circulars, and con-structed a multi-band SED using GROND data (Kann2014), and light curves using r’ and i’ filters (GROND,RATIR: Troja (2014), P60: Cenko (2014), LCOGTN:Kopac (2014)). The XRT light curve and spectra wereobtained from the XRT Team Repository (Evans et al.2007, 2009). We triggered a pre-approved late-timeChandra target of opportunity observation to constrainthe properties of the break in the X-ray afterglow hintedat by XRT. The 40 ks Chandra observation was ana-lyzed with CIAO v4.6, yielding a faint detection whichwas converted to flux using the fit to the XRT spectrum.The X-ray and optical afterglow observations began at ∼ s after the GBM trigger. It is therefore impossible to observe the continuous evolution of the flux from theprompt to afterglow phase at low-energy.
10 15 20 25
Time [s] . . . . . . . . L [ e r g s − ] Figure 2.
The luminosity lightcurve of GRB consistingof the synchrotron ( purple ) and blackbody ( red ) components.
Spectral and Temporal Analysis
For the prompt emission, we use GBM time-taggedevent (TTE) data and
F ermi
LAT low-energy (LLE)data to perform spectral analysis. The LLE techniqueis an analysis method designed to study bright transientphenomena in the 30 MeV - 1 GeV energy range, andwas successfully applied to
Fermi -LAT GRBs (Acker-mann et al. 2013) and solar flares (Ackermann et al. 2012;Ajello et al. 2014). The idea behind LLE is to maximizethe effective area below ∼ during theprompt emission. We find the dominant component tobe t var = 24 . / (1 + z ) s and an insignificant fast compo-nent with t var = 3 . / (1 + z ) s. These values place theburst safely within the range of what can be expectedby an external shock. We use time bins of 1 s to besure to bin below the fast variability component. To fitthe time-resolved spectra, we employ a two-componentmodel consisting of synchrotron emission from an incom-pletely cooled electron distribution (slow-cooled) and ablackbody (Burgess et al. 2011, 2014b) (see Figure 3). InBurgess et al. (2014b), it was shown that single pulsedGRBs have spectra that are compatible with slow-cooledsynchrotron sometimes with and sometimes without theaddition of a blackbody. This motivates our choice ofthe synchrotron+blackbody photon model for fitting thespectra of the GRB . We note that not all GRBspectra are compatible with this photon model. Thereexist GRBs such as GRB with a clear domi-nant photospheric component originating from subpho-tospheric dissipation (Pe’er et al. 2005; Ryde et al. 2010).The synchrotron model implemented is fully physicalin the sense that the spectral shape comes from the syn-chrotron emissivity and electron distribution alone. Thespectral fitting should not be confused with fitting theempirical Band function (Band et al. 1993) with its vari- Energy [keV] − − ν F ν [ e r g s − c m − k e V − ] (a) 10 Energy [keV] − − ν F ν [ e r g s − c m − k e V − ] (b) Figure 3.
The time-resolved vF v spectra in 1 s intervals of the synchrotron (5-27 s) (a) and blackbody (5-15 s) (b) components from theGBM+LLE fits. The synchrotron spectrum evolves in time from purple to white while the blackbody evolves from red to yellow . able low-energy index ( α ). The blackbody is only in-cluded in time bins that statistically require it via a likeli-hood ratio test. We find that this statistical requirementproduces a continuous presence of the blackbody for thefirst part of the observation. The data are well fit by thetwo-component model (see Table 1 for a summary of thespectral fits) and we observe two key features that moti-vate our investigation of the GRB under the framework ofthe external shock model: E p evolves as a broken powerlaw in time and the blackbody is only significant for aduration of ∼
10 seconds compared to the total promptduration of ∼
25 s (see Section 4.4).We calculate the total k-corrected energy (Hogg et al.2002) in the synchrotron component in the 30 keV-300MeV interval by summation over each time bin: E synch =4 πd (cid:80) F synch ∆ t i ∼ . · erg. The total isotropic en-ergy of the burst is estimated to be several times larger.In the following, we take the total isotropic energy (ki-netic + radiative) of the blast-wave to be E iso ∼ = 10 ergwhile noting that this is an extremely high value. Never-theless, the GRB is extremely bright and no mechanismis known that is efficient enough to convert the entirerest mass of the progenitor to radiation.For the late time GeV emission, we performed an un-binned likelihood analysis of the LAT data to recoverthe energy flux ( F E ) and photon index ( γ ph ) of theemission (see Table 2) with the gtlike program dis-tributed with the Fermi
ScienceTools . We selected P7REP SOURCE V15 photon events from a 15 ◦ circular re-gion centered at the Swift XRT position (R.A.=322 . ◦ − . ◦
23, J2000) and within 105 ◦ from the localzenith (to reduce contamination from the Earth limb).Events with measured energy from 100 MeV to 10 GeVare included in our analysis (the highest energy event as-sociated with this GRB has an energy of 3.8 GeV and ar-rives 157.5 seconds after the GBM trigger time). Furtherdetails on the LAT analysis are discussed in Appendix A. EXTERNAL SHOCK ANALYSIS E p Evolution
After performing spectral fits to the data we can testthe external shock model by fitting the evolution of therecovered synchrotron E p with Equation 4. The fit isperformed with a Bayesian analysis tool built upon the MULTINEST (Feroz et al. 2009) software which allows usto fully explore the correlated parameter space of themodel. The free parameters in the fit are Γ , η , g , and q − to which we assign flat priors that are consistentwith physical expectations (Γ ∈ { , } , η ∈ { , } g ∈ { , } , q − ∈ { , } ). We also note that it is notpossible to fit for the density n , which we set at n = { , , } cm − with different values mainly affectingthe value of Γ recovered. Calculations in the text andfigures assume n = 100 cm − but are complementedwith calculations at the other values in associated tables.Figure 4 shows the fit of the E p evolution and the best-fitparameters are detailed in Table 3.We find that g (cid:39) .
3, which is less than what is ex-pected if the blast-wave decelerates adiabatically ( g =1 . We used version 09-34-02 available from the
Fermi
ScienceSupport Center http://fermi.gsfc.nasa.gov/ssc/ Time [s] E p [ k e V ] Figure 4.
Bayesian fit of the E p evolution of GRB . Themaximum-likelihood point is indicated by the blue curve and pos-terior samples are in orange . The fitted parameters are indicatedin Table 3. assumption, we numerically solved for the evolution of Γwith radius via the equations of energy and momentumconservation and examined the transition phase in Fig-ure 5. There is clearly a region that corresponds to ourrecovered value of g (cid:39) .
3. Late (hundreds of secondsafter the trigger) time observations smoothly connect-ing the prompt and afterglow emission would allow usto measure the asymptotic value of g . Yet, with the re-covered fit parameters, we can calculate several physicalproperties of the outflow including r d = 3 . · cm andthe resulting predicted P p ( t ) evolution though we willassume g (cid:39) . radius [cm] Γ g=1.3 g=1.5 Figure 5.
The blast-wave Lorentz factor evolution from our nu-merical simulation of the external shock model ( blue dashed line ).We illustrate that we are most likely finding the blast-wave stillevolving in the transition phase, which accounts for the shallowvalue of g found in the fit of E p . Peak Flux - E p plane Using the recovered parameters from the fit, we canuse Equations 4 and 7 to predict the evolution in the P p - E p plane. In Figure 6, the data from GRB is plotted with the predicted curve from the externalshock model derived from the E p fits. Noting the dis-cussion in Section 4.1, negative fluxes would be obtained Table 1
Results of the time-resolved spectral analysis of GBM+LLE data (10 keV - 300 MeV).T start [s] T stop [s] F synch E [erg s − ] b E p [keV] c p d F BB E [erg s − ] kT [keV]6.0 7.0 (1 . ± . · − ±
336 10 . a (1 . ± . · − . ± . . ± . · − ±
472 5 . ± .
21 (1 . ± . · − . ± . . ± . · − ±
310 5 . ± .
58 (2 . ± . · − . ± . . ± . · − ±
310 4 . ± .
30 (1 . ± . · − . ± . . ± . · − ±
108 4 . ± .
18 (4 . ± . · − . ± . . ± . · − ±
79 4 . ± .
21 (3 . ± . · − . ± . . ± . · − ±
71 4 . ± .
20 (6 . ± . · − . ± . . ± . · − ±
55 4 . ± .
18 (3 . ± . · − . ± . . ± . · − ±
46 3 . ± .
13 (1 . ± . · − . ± . . ± . · − ±
34 3 . ± .
12 (2 . ± . · − . ± . . ± . · − ±
16 3 . ± . · · · · · · . ± . · − ±
21 3 . ± . · · · · · · . ± . · − ±
20 3 . ± . · · · · · · . ± . · − ±
18 3 . ± . · · · · · · . ± . · − ±
18 3 . ± . · · · · · · . ± . · − ±
23 3 . ± . · · · · · · . ± . · − ±
23 3 . ± . · · · · · · . ± . · − ±
20 3 . ± . · · · · · · . ± . · − ±
28 3 . ± . · · · · · · . ± . · − ±
37 3 . ± . · · · · · · . ± . · − ±
44 3 . ± . · · · · · · . ± . · − ±
44 3 . ± . · · · · · · a fixed b synchrotron energy flux c synchrotron νF ν peak d e − spectral index Table 2
Results of the time-resolved spectral analysis of LAT data (100 MeV - 10 GeV) where TS is the value of the Test Statistic.T start [s] T stop [s] TS γ ph F E [erg s − cm − ]13.3 23.7 26.3 − ± ± × − − ± ± × − − ± ± × − · · · < × − · · · < × − · · · < × − Table 3
The best-fit parameters of the E p evolution fit for each assumed value of n as well as the inferred r d . n [cm − ] Γ η q − g r d [cm]1 1125 . +16 . − . . +0 . − . . +0 . − . . +0 . − . . ·
10 844 . +12 . − . . +0 . − . . +0 . − . . +0 . − . . ·
100 632 . +7 . − . . +0 . − . . +0 . − . . +0 . − . . · for g < . g via the fit to the E p evolution is not actually measuringthe asymptotic behavior of the blast-wave evolution asis intended, we set g = 1 . g does not affect the other parameters in thefit as g only modifies the late time evolution of E p (seeFigure 1). The analytic model given by Equation 7 and Equation8 is consistent with the data up to a scale factor of 9which can easily be due to insufficient knowledge of allintrinsic parameters, on the simplified evolution law forthe flux, and on the assumed value of g . It is pertinentto note that the parameters used in the P p ( t ) predictioncome from the fitted E p evolution alone. In addition,the predicted P p ( t ) is independent of the observed fluxvalues in the data. The close agreement between thepredicted and measured P p ( t ) values is evidence that the P p [ e r g s − ] Decelerationphase CoastingphaseModeldeviation E p [keV] L [ e r g s − ] Figure 6.
The P p − E p plane of GRB ( top ) where P p is the νF ν flux calculated at E p . The predicted evolution ( green )is produced by substituting the recovered parameters of the E p evolution fit into Equation 7. The flux given by Equation 7 isscaled by a factor of 9 to match the data. At late times (pinkregion), the observed flux decays faster than the model predicts.The decay phase synchrotron HIC ( blue ) ( bottom ). The fitted valueof the slope is ζ = 1 . ± .
22. The decay phase is selected as theportion of the lightcurve that monotonically decreases with timei.e. from 10-27 s. observed flux at the peak and the observed νF ν peakenergy are linked in a manner predicted by the exter-nal shock model. We do note the deviation from thepredicted curve at late times. This therefore providesstrong evidence that these two independent quantitiesare linked in a manner predicted by the external shockmodel. The Luminosity - E p plane The commonly observed correlation in the decay phaseof many GRB pulses between luminosity and E p in theform, L ∝ E ζ p , (9)sometimes referred to as the hardness-intensity correla-tion (HIC) (Golenetskii et al. 1983), is potentially an-other clue to the radiation mechanism responsible forthe observed emission. It is quite simple to relate theluminosity and νF ν peak energy of many common ra-diation mechanisms analytically. However, the spreadin HIC power law slopes observed across many GRBsis difficult to explain with one mechanism (Borgonovo& Ryde 2001). However, single pulse GRBs fit with asynchrotron photon model have been found to have HICpower law slopes of ∼ . L and E p during the deceleration phase of a blast-wave. Wehave x ∝ t / (2 g +1) , Γ ∝ t − g/ (2 g +1) , E p ∝ Γ Bγ , and L ∝ Γ B γ . Here γ min is the minimum electron en-ergy in an assumed shock accelerated power law distri-bution (though the result is insensitive to the fact thatwe assume a power law) and B is the strength of themagnetic field. There are two regimes of cooling for elec- trons: fast and slow (Sari et al. 1998). Each regime canbe characterized by how γ min evolves with time such that γ min ∝ (cid:26) Γ ∝ t − g/ (2 g +1) slow cooling( x Γ) − ∝ t − g/ (2 g +1) fast cooling . (10)Since L ∝ B Γ E p , we can write L ∝ (cid:26) E / p slow cooling E g p fast cooling (11)When fitting the L − E p data in the decay phase (10-27s) we find ζ = 1 . ± .
22 (see Figure 6). This is closerto the value expected for the slow-cooling regime whichfalls in line with the use of slow-cooling synchrotron tofit the time-resolved spectra. An a posteriori justificationfor the use of slow-cooling synchrotron to fit the spectrais discussed in Section 5.1.
The photosphere
We interpret the observed thermal component in theframework of photospheric emission, i.e., when the out-flow becomes transparent at a radius ∼ − cm,i.e., much below r d . Because of highly relativistic mo-tion, the observed time delay between photons from thephotosphere and those from the external shock is small.As the thermal component is not dominant, the errorsin the calculation of outflow parameters derived in Pe’eret al. (2007) are large and we therefore did not use thisformalism herein.We note that the blackbody is not statistically signifi-cant (below 3- σ confidence level) in the first five secondsafter trigger. However, photons from the photosphere areexpected to arrive slightly before synchrotron photons.The delay between the photospheric photons and thoseat the peak of the external shock light curve can be esti-mated as ( r d − r ph ) /c Γ (cid:39) . t d .Thus, we added five seconds to the duration of the photo-spheric emission. The identification of the photosphericemission can give important constraints. First, the du-ration of the photospheric component (∆ t ph ) sets thewidth of the expanding outflow ( W ) at the photosphere(see B´egu´e et al. 2013; B´egu´e & Iyyani 2014; Vereshcha-gin 2014), ∆ t ph z = R ph c + Wc + R ph c (12)where the first term accounts for the expansion time ofthe outflow up to the photosphere, the second term isthe light-crossing time of the outflow and the last termis the angular timescale at the photosphere. For the ob-served duration at hand and for the Lorentz factor val-ues expected in GRB physics, the second term in theright-hand side of the equation dominates. It implies W ∼ c ∆ t ph / (1 + z ) ∼ . · cm.As the width W , the Lorentz factor Γ and the totalenergy of the blast wave are constrained, the state of thereverse shock can be a posteriori studied. Following thediscussion in Dermer & Menon (2009), we define ξ ∼ (cid:0) Γ300 (cid:1) / W / (cid:18) E n (cid:19) / , (13)such that x NR = ξr d is the radius at which the reverseshock becomes relativistic. We find ξ = 0 . ξ is the same for each combination of Γ and n (see Table4). DISCUSSION
Compatibility with the Late-time X-ray/opticalEmission
Under the assumption of an external shock in theprompt phase, the afterglow is produced by the sameprocess (that is the deceleration of a blast wave by theCBM), and we do not expect any break in the light curve,as already seen in several GRBs observed by
Swif t (Zhang et al. 2006). Unfortunately, the XRT observa-tions only cover a short time range beginning a few ∼ s after the prompt emission (See Figure 7). At most, wecan say that XRT late time observations are compatiblewith the emission from a decelerating blast-wave in thesemi-radiative regime, i.e., a region between 1 . < g < s < t b < . · s.Identifying the break as a jet break, the jet opening anglecan be estimated to be (Ghirlanda et al. 2004) Time [s] − − − − − − − F l u x D e n s i t y [ m J y ] r’ bandi’ bandXRTChandra Figure 7.
The X-ray and optical afterglow of GRB .A power law fit best describes the optical data while a brokenpower law fit, indicative of a jet break, best describes the X-raydata. The yellow shaded region indicates the 90% uncertainty inthe break time due to the sparse sampling of data prior to theChandra follow-up observation. θ j = 0 . (cid:18) t b z (cid:19) / (cid:18) η γ n E γ, iso , (cid:19) / (14)where t b is the break time in days and η γ is the radiativeefficiency taken to be 0.1 corresponding to our assump-tion of E = 10 erg. Thus, the opening angle rangesfrom 2 . ◦ < θ j < . ◦ , when considering the fitted pa-rameters obtained in Section 4.1. Therefore, the radia-tive energy budget of the burst is considerably reducedto few ∼ erg (see Table 5).Finally, additional constraints can be obtained fromsimultaneous optical and X-ray observations at around Table 4
Values of (cid:15) B from the afterglow and E p assuming different valuesof n as well as the location of the relativistic reverse shock ( ξ ). n Γ (cid:15) B via ν c (cid:15) B via E p ξ < . · − ∼ . · − < . · − ∼ . · − < . · − ∼ . · − Table 5
Corrected Emitted Burst Energy ( E cor ) n [cm − ] t b (s) θ j (degrees) E cor (erg)1 1 · . · . · . ·
10 1 · . · . · . ·
100 1 · . · . · . ·
50 ks after the trigger. The spectrum is consistent witha single power-law of index β = 1 . ± .
07. Therefore, wecan deduce that the cooling frequency is above the XRTfrequency: ν c > ν XRT . From the expansion of a blast-wave in a constant density CBM, ν c can be estimated as(Panaitescu & Kumar 2000a) ν c = 3 . · E − / n − ( Y + 1) − (cid:15) − / , − T − / Hz , (15)where (cid:15) B parameterizes the magnetic field in the shockedISM, T d is the time in days and Y is the Compton pa-rameter, that we chose to be zero for simplicity. Thisleads to an upper limit (cid:15) B < . · − taking T d = 1 dayand assuming n = 100 and E = 100 which impliesΓ = 632 (see however, Table 4 for values correspondingto different parameter choices).Also, this value can be cross-checked by consideringthe peak energy in the first seconds of the prompt phase,which can be evaluated as: E p (cid:39) (cid:112) π(cid:15) B m p c n Γ B crit Γ γ (16)where B crit = 4 . · G is the critical magnetic field, κ parameterizes the minimum Lorentz factor γ min of theaccelerated electrons such that γ min = κ Γ( m p / m e ), m e isthe mass of an electron, and E p is given in units of themass of an electron. Using E p ∼ . n = 100as obtained from the data, and assuming κ = 1 gives (cid:15) B ∼ . · − (see also Table 4). This value is consistentwith the upper limit obtained from the late afterglow .Finally, the cooling time of an electron with Lorentzfactor γ min can be compared to the expansion time ofthe blast-wave at the observed luminosity peak which However we note that this value is very sensitive to κ whichis fairly unconstrained: as an example, with κ = 0 .
1, it becomes (cid:15) B ∼ . · − , incompatible with the upper limit obtained fromthe afterglow. is on the order of the t d . We find that they are com-parable, leading to efficient energy extraction from theelectrons, without drastically changing the electron dis-tribution function, i.e. , creating an additional power lawover several orders of magnitude at low-energy charac-teristic of fast-cooled (or completely cooled) electrons. LAT late time GeV emission
F ermi
LAT observed a GeV component over the du-ration of 10 − s. Unfortunately, there were no simul-taneous observations at other wavelengths from which abroadband spectrum could be obtained. Nevertheless,we can use the analytic flux evolution derived in Der-mer et al. (1999) to check if the observed LAT fluxes arecompatible with the model. We must first address a fewcaveats: the analytic model assumes no synchrotron self-Compton (SSC) emission and the fluxes are sensitive toour lack of knowledge about the total burst energy (whichwe assume to be 10 erg for the purpose of calculation)as well as the degeneracies in the recovered parametersfrom the E p evolution fit.In the past years, a debate concerning the origin ofthe late time (few seconds after the trigger) high-energyLAT emission has taken place, opposing a synchrotronmechanism (Kumar & Barniol Duran 2009, 2010), SSC(Panaitescu & Kumar 2000b), and pair-loading of theCBM (Beloborodov 2005) as the possible candidates forthis emission. First, we can assess the SSC component,and show that with the parameters at hand from theprompt emission the SSC flux in the LAT bandpass ismuch less than the synchrotron flux. We can constrainthe SSC flux in the LAT bandpass by following the dis-cussion in Beniamini & Piran (2013) (see Appendix Bfor further details). Via Equation B12, the ratio of SSCflux to synchrotron flux in the LAT bandpass ( F SSC ) canbe computed by substituting values derived in Section5.1 to yield F SSC (cid:39) − . The numerical result showsthat the SSC flux in the LAT band is much less thanthe synchrotron flux. However, the computation relies on the assumption that electrons are fast cooled eitherby synchrotron or by inverse Compton of the producedsynchrotron photons, that is to say that all the energy isemitted by one or the other mechanism. It comes withtwo consequences:1. the Compton parameter is an upper limit as shownin Sari et al. (1996). Therefore the result ofEq.(B12) is only an upper limit,2. as a result of the assumption of fast cooling,the electron distribution function extends to smallLorentz factors. These electrons are not in theKlein-Nishina regime and can efficiently upscat-tered synchrotron photon in the LAT band. How-ever, we found that electrons with Lorentz factor γ min are not strongly cooled over a dynamical time.Therefore the electron distribution function doesnot extend substantially below γ min , reducing theinverse Compton flux.We now consider that the GeV emission is the resultof synchrotron emission from the same forward-shock re-sponsible for the lower energy prompt phase. Using theanalytic estimate for the blast-wave evolution, we esti- mate the expected flux of the synchrotron emitting blast-wave and compare it to the integrated (100 MeV - 10GeV) LAT energy flux ( F E ) by integrating Equation 1from Dermer et al. (1999) across the LAT bandpass (seeFigure 8). Because of our lack of knowledge about theintrinsic total energy of the blast-wave and the degenera-cies in the fit parameters ( q , n ), we vary the parametersacross a broad range and find that the model gives con-sistent limits within an order of magnitude. We alsonote that this model is simple and will not be as accu-rate as a full numerical solution. Additionally, using thepower-law electron index ( p ) found from GBM+LLE syn-chrotron fits, we compute the synchrotron photon indicesvia the transformation γ ph = − ( p + 1) / Time [s] − − − − − − − F E [ e r g s − c m − ] E = 5 · erg q = 0 . E = 1 · erg q = 0 . Figure 8.
The LAT GeV integrated energy fluxes (100 MeV - 10GeV) are compared with a range of expected fluxes from the ana-lytic model of Dermer et al. (1999) that includes only synchrotronemission. Suitable agreement can be found for a valid range ofassumed blast-wave parameters.10 Time [s] − . − . − . − . − . − . − . − . γ ph GBM+LLELAT
Figure 9.
The photon indices from the GBM+LLE fits and LATfits. A clear evolution is seen into the LAT band pass with theindices all in line with what could be expected from synchrotron-emitting high-energy electrons accelerated into a power lawmomentum-distribution.
Therefore, we conclude that the high energy emissionfrom the LAT is compatible with synchrotron emission0alone, and does not require an additional mechanism. Itwas proposed by Kumar & Barniol Duran (2009, 2010),in which the prompt MeV phase is assumed to be theresult of an unspecified mechanism, that the GeV LATemission arises from a separate exernal shock with a syn-chrotron peak on the order of a few 100 MeV. Conditionson the density and the magnetic field are then derivedsuch that this external shock is not more luminous thanthe prompt emission in the sub-MeV band. However, forGRB we show that the entire emission can beexplained by synchrotron emission with an MeV peak en-ergy from a single forward shock without fine-tuning ofthe parameters, i.e., both the MeV and the delayed GeVcan be explained self-consistently by an external shock.
Summary of Prompt Phase
In this section, we concisely summarize the novel ap-proach we have employed to test the external shockmodel in the prompt phase. Almost as important asthe observed E p evolution is the fact that we can fitthe spectrum with a physical slow-cooling synchrotronmodel. Even though it is possible to fit a physical syn-chrotron model to the spectra (as opposed to an empir-ical Band function from which the comparison to phys-ical models must be inferred and can lead to problemsas shown in Burgess et al. (2014a)), it must be shownthat the photon model is sound, i.e, that the dynamicalevolution can lead to slow-cooling synchrotron emission.In Section 5.1, we confirmed that both the afterglow andprompt phase are consistent with slow-cooling emission.The evolution of E p in time can be fit with the analyticprediction of Dermer et al. (1999). While the blast-waveappears to be transitioning to the asymptotic limits, ( g < .
5) we tentatively conclude that it is evolving towardsan adiabatic regime. Full temporal coverage of the fluxand spectra through the late time evolution will help toresolve this issue. Unfortunately, few single pulsed GRBshave been observed with such a temporal and wavelengthcoverage. This unique opportunity presents itself onlydue to the current multi-mission capabilities and begsfor continued multi-wavelength coverage.If we take the fitted parameters of the E p evolution fit,we can predict the evolution of the P p − E p plane which isconsistent with the data in both the rise and decay phaseof the pulse. The HIC of the prompt decay phase of thesynchrotron luminosity is consistent with what would beexpected from slow-cooling synchrotron in a deceleratingexternal shock.We use the observation of the photospheric componentto estimate both the width of the blast-wave as well aswhen the relativistic reverse shock crosses the forwardshock. From our calculations, the reverse shock will havevery little impact on the evolution of the forward shockand we therefore neglect it. CONCLUSION
In this article, we have shown that the prompt phaseof GRB is consistent with originating from anexternal shock that emits synchrotron radiation from anelectron distribution that did not have enough time tocool completely (slow-cooled). Combining the results ofthe E p evolution fits with the other clues from the data,an external shock origin of GRB is a very likelyscenario. Not only do we find that the prompt emission is explained by synchrotron emission from a forward shock,but we find clues for a late time high-energy emissionresulting from the extension of this synchrotron emis-sion rather than an SSC component. We want to stressthat in this scenario the delineation between “prompt”and “afterglow” is superfluous as the early and late timeemission both originate from the same mechanism. Weare currently applying this analysis to other long, singlepulse GRBs and find that they too have their νF ν peakand P p ( t ) which evolved consistently with the predictionsof an external shock. These results will be presented ina forthcoming publication.We propose that external shocks are still a viable can-didate to explain the prompt dynamics of GRBs. How-ever, the most likely scenario is that they are a subset ofmultiple dynamical frameworks including internal shocksthat further subdivide into categories based on opacityand/or magnetic content of the outflow. We note thatthis is not an entirely new idea. Panaitescu & M´esz´aros(1998) numerically investigated the parameter space ofthe external shock model and proposed that smooth,single-pulsed (and possibly multi-pulsed) GRBs can be-long to a subclass of GRBs that are the results of exter-nal shocks. We have now shown this quantitatively byapplying the model to the data.The analysis of individual GRBs is crucial to identi-fying what is likely a host of different emission mech-anisms. For example, GRB exhibits dynamicsand spectra that are consistent with subphotospheric dis-sipation (Ryde et al. 2010). The fact that both externalshocks and photospheric emission are observationally vi-able candidates for explaining GRB emission implies thatwe should use caution when trying to apply one mecha-nism as an explanation for all GRBs based of propertiesfrom catalogs and should instead focus on how to relatethe different emission mechanisms into a unified frame-work.The authors are very grateful for insightful discus-sions with Chuck Dermer, Peter M´esz´aros, GregoryVereshchagin and Peter Veres that helped to improvethe manuscript. This work made use of data suppliedby the UK Swift Science Data Centre at the Univer-sity of Leicester, data obtained from the Chandra DataArchive, and software provided by the Chandra X-rayCenter (CXC). JLR and AC acknowledge support forthis work from NASA through Chandra Award NumberGO4-15073Z issued by the Chandra X-ray ObservatoryCenter under contract NAS8-03060.The Fermi
LAT Collaboration acknowledges generousongoing support from a number of agencies and insti-tutes that have supported both the development and theoperation of the LAT as well as scientific data analysis.These include the National Aeronautics and Space Ad-ministration and the Department of Energy in the UnitedStates, the Commissariat `a l’Energie Atomique and theCentre National de la Recherche Scientifique / InstitutNational de Physique Nucl´eaire et de Physique des Par-ticules in France, the Agenzia Spaziale Italiana and theIstituto Nazionale di Fisica Nucleare in Italy, the Min-istry of Education, Culture, Sports, Science and Technol-ogy (MEXT), High Energy Accelerator Research Organi-zation (KEK) and Japan Aerospace Exploration Agency1(JAXA) in Japan, and the K. A. Wallenberg Foundation,the Swedish Research Council and the Swedish NationalSpace Board in Sweden.Additional support for science analysis during the op-erations phase is gratefully acknowledged from the Is-tituto Nazionale di Astrofisica in Italy and the CentreNational d’´Etudes Spatiales in France.This research made use of
Astropy , a community-developed core Python package for Astronomy (AstropyCollaboration et al. 2013) as well
Matplotlib , an opensource Python graphics environment (Hunter 2007). Wealso thank the developer of pymultinest for enablingthe use of
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LAT ANALYSIS
The model used in the likelihood fit is composed of the Galactic diffuse emission produced by cosmic-ray interactionwith gas and radiation fields and the isotropic diffuse emission. In addition, we add all the point sources in the ROI withspectral models and parameters from the 2FGL catalog (Nolan et al. 2012). While the normalization of the Galactictemplate and the spectral parameters of all the 2FGL sources are frozen to their nominal values, the normalizationof the isotropic template is left free to vary in order to absorb statistical fluctuations. The GRB location is fixed,and its spectrum is described by a power law dN/dE ∝ E γ ph with γ ph the photon index (note that typically γ ph < χ n dof / n dof is the number ofdegrees of freedom associated with the GRB. The factor of 1/2 in front of the TS PDF formula results from allowingonly positive source fluxes.If the resulting TS value is lower than an arbitrary threshold ( T S <
10) we merge the corresponding time bin withthe next one, and we repeat the likelihood analysis. This step is iterated until one of two conditions is satisfied: 1) wereach the end of a GTI before reaching
T S = 10, in which case we compute the value of the 95% CL upper limit (UL)for the flux evaluated using a photon index of −
2; 2) we reach
T S >
10, in which case we evaluate the best-fit valuesof the flux and the spectral index along with their 1 σ errors. HIGH-ENERGY SSC FRACTION
Following Beniamini & Piran (2013), the SSC and Klein-Nishina peak frequencies are defined respectively as ν min = E p h − (B1) ν SSC = γ ν min (B2) ν KN = Γ γ min m e c h − (B3)where h is Planck’s constant. The total SSC flux is linked to the total synchrotron flux at the peak ( F synch ν min ) by F SSC ν min F synch ν min = Y Λ KN (B4)where Y is the Compton parameter and, Λ KN = (cid:40) (cid:16) ν SSC ν KN (cid:17) − / ν SSC ν KN >
11 otherwise . (B5)In the derivation of the Y via Sari et al. (1996), the Klein-Nishina correction of the Compton cross-section ( σ KN ∝ ln(2 x ) /x where x is the photon energy in the rest frame of the electron in m e c units) was ignored. Here, in orderto estimate Y we include this correction by considering electrons with Lorentz factor γ min and their correspondingsynchrotron photons. Therefore, we write Y = (cid:18) (cid:15) e (cid:15) B (cid:19) / · ln (cid:16) γ min hν min m e c (cid:17) γ min hν min m e c / . (B6)The total upscattered flux in the LAT bandpass can then be written as F LATSSC ν min F synch ν min = F SSC ν min F synch ν min · Λ W (B7)where Λ W = (cid:40) ν SSC , ν KN ) < ν max (cid:16) min( ν SSC , ν KN ) ν max (cid:17) − α − otherwise . (B8)Here, α refers to the low-energy photon index of the Band function which is -2/3 in our case and ν max is the maximumsynchrotron frequency corresponding to the maximum Lorentz factor of the accelerated electrons (de Jager et al. 1996): γ max = 4 · (cid:18) B (cid:19) − / (B9)In order to estimate the synchrotron flux in the LAT band pass, we write F synch ν = (cid:18) νν min (cid:19) − s − F synch ν min (B10)3where s is the high-energy electron power law index which we take s ≡ . F SSC ≡ F LATSSC F LATsynch = F LATSSC (cid:82)
LAT dν (cid:16) νν min (cid:17) − s − F synch ν min (B11)This can be further simplified via Equations B4 and B7 to F SSC = ν min (cid:82) LAT dν (cid:16) νν min (cid:17) − s − · Λ W · Λ KN · Y . (B12)This is in fact an upper limit on the fraction of SSC due to the assumption that electrons radiate all their energy ina dynamical time either by synchrotron and/or inverse-Compton and that there is a significant fraction of electronsbelow γ minmin