The physical conditions of the afterglow implied by MAGIC's sub-TeV observations of GRB 190114C
aa r X i v : . [ a s t r o - ph . H E ] J u l Draft version July 23, 2019
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The physical conditions of the afterglow implied by MAGIC’s sub-TeV observations of GRB 190114C
Evgeny Derishev and Tsvi Piran Institute of Applied Physics RAS, 46 Ulyanov st,603950Nizhny Novgorod, Russia Racah Institute of PhysicsThe Hebrew University of Jerusalem, Jerusalem 91904, Israel
ABSTRACTMAGIC’s observations of late sub-TeV photons from GRB190114C enable us, for the first time,to determine the details of the emission process in a GRB afterglow and to pin down the physicalparameters, such as the bulk Lorentz factor and the Lorentz factor of the emitting electrons as wellas some of the microphysical parameters. We find that the sub-TeV emission is Synchrotron-SelfCompton (SSC) radiation produced at the early afterglow stage. Combining the sub-TeV and X-rayobservations we narrow uncertainties in the conditions inside the emitting zone, almost eliminatingthem for some parameters. Seventy seconds after the trigger the external shock had Lorentz factor ≃ and the electrons producing the observed sub-TeV radiation had a Lorentz factor ≃ , sothat the sub-TeV radiation originates from Comptonization of X-ray photons at the border betweenThomson and Klein-Nishina regimes. The inferred conditions within the emitting zone are at odds withtheoretical expectations unless one assumes moderate (with τ ≃ ) absorption of sub-TeV photonsinside the source. With this correction the conditions are in good agreement with predictions of thepair-balance model, but are also acceptable for generic afterglow model as one of many possibilities.The different temporal evolution of the IC peak energy of these two models opens a way to discriminatebetween them once late-time detection in the TeV range become available. INTRODUCTIONThe bright gamma-ray burst, GRB 190114C, was de-tected by
Swift /BAT (Gropp et al. 2019),
Fermi /GBM(Hamburg et al. 2019) and Konus-Wind (Frederiks et al.2019). 50 seconds after the trigger, the MAGICCherenkov telescope detected a photon at energyabove 300 GeV with more than 20 σ significance(Mirzoyan 2019). While GRB γ -rays of several dozenGeV have been detected in the past by EGRET(Hurley et al. 1994) and Fermi /LAT (Abdo et al. 2009;Ackermann et al. 2014), this was the first ever detec-tion of a GRB by a Cherenkov telescope at sub-TeV.With a redshift of z = . (Selsing et al. 2019) thecorresponding energy in the source frame of the highestenergy photon was ≃ . TeV. Even though the informa-tion at this stage is limited to GCN circulars, we explorehere the origin of the sub-TeV emission and show that itcan lead to a deep insight on the conditions within theemitting regions. As we show here these observationsenables us to determine the bulk Lorentz factor and theLorentz factor of the emitting electrons as well as thedetails of the emission process. We can also put limitson the microphysical equipartiton parameters. This isthe first time that the conditions within the emitting region of a GRB have been determined with such aconfidence.In previous bursts, the EGRET and
Fermi /LAT GeVemission continued long after the prompt emission hadfaded away and have shown a gradual temporal decay.This has led to the suggestion that the GeV photonsarise from the afterglow (Kumar & Barniol Duran 2009;Ghisellini et al. 2010; Kumar & Barniol Duran 2010).As the MAGIC sub-TeV radiation shows a similar be-haviour, i.e. it was observed after the bulk of promptemission had faded away, we explore here the possibilitythat this is a part of the early afterglow.Within the standard afterglow model (Sari et al. 1998)the emission is produced via the synchrotron mecha-nism in an external shock. However, emission of sub-TeV photons via synchrotron mechanism is problem-atic within this model (Piran & Nakar 2010). Withany reasonable bulk Lorentz factor of the emitting re-gion, the observed photon energy violates the burn-off limit (e.g. Guilbert et al. 1983; de Jager et al. 1996;Aharonian 2000). The most natural emission mecha-nism is, therefore, Inverse Compton (IC). We do not con-sider here other emission mechanisms since Synchrotron-Self-Compton (SSC) is the simplest option.
Derishev & Piran
A synchrotron-emitting source must produce high-energy radiation through upscattering of synchrotronphotons by the same electrons. Thus, GeV and TeVradiation is expected from GRBs’ afterglows, bothat early and late stages (e.g. Meszaros et al. 1994;Waxman 1997; Wei & Lu 1998; Chiang & Dermer1999; Panaitescu & Kumar 2000; Sari & Esin 2001;Zhang & M´esz´aros 2001; Guetta & Granot 2003; Ando et al.2008; Fan & Piran 2008). For typical early afterglowparameters at least a few per cent of the total powermust be transferred to TeV photons and this com-ponent may even be dominant (Derishev et al. 2001).Yet multiple attempts to detect GRBs’ TeV emissionwith Cherenkov telescopes resulted only in upper limits(Aharonian et al. 2009; Albert et al. 2007; Horan et al.2007; Acciari et al. 2011; Hoischen et al. 2017; Berti2016). The striking contrast between the theoretical pre-dictions and the actual observations until now remaineda puzzle, implying that either the physics of the emit-ting zone in GRB afterglows is poorly understood (andno TeV radiation produced for one or another reason)or that TeV radiation was strongly absorbed in all casesof non-detection (see also Vurm & Beloborodov 2017,for a discussion). We demonstrate that the MAGICdetection solves the puzzle: GRB afterglows are indeedroutinely producing TeV emission, but it is attenuatedby internal absorption, that is usually strong enough toprevent detection.Consider an IC photon with energy E IC . The emit-ting electron must be energetic enough to upscatter it,namely: Γ γ e m e c ≥ E IC , where Γ is the bulk Lorentz fac-tor, γ e is the Lorentz factor of the accelerated electrons, m e the electron’s rest mass and c the speed of light.Thus, the sub-TeV observations imply that Γ γ e ≥ .The shock dynamics with reasonable circum-burst den-sities (see §
4) suggests that the bulk Lorenz factor at thetime of the observations can hardly be above ∼ . Thisimmediately implies that the typical electron’s Lorentzfactor satisfies γ e ≥ .The X-ray flux observed in GRB 190114C is larger butcomparable to the sub-TeV flux observed at the sametime (see § §
7. This suggests that the seed photons are X-rays,that we take to be E X ∼ keV. In this case γ e ≃ satisfies the Thomson IC relation E IC , Thomson ≃ γ e E X and with Γ ≃ also the KN one E IC , KN ≃ Γ γ e m e c (see § §
6) shows that it is in theformer and just slightly below the transition region.When discussing the implications of our results weconsider two afterglow models. Both share the sameshock deceleration dynamics. The first, widely used,“generic” model (see e.g. Sari et al. 1998; Piran 1999)assumes that the downstream electrons carry a constantfraction, ǫ e of the total downstream energy. This leadsto an average Lorentz factor of the electrons, γ e that isproportional to the bulk Lorentz factor Γ .The second is the “pair balance”model (Derishev & Piran2016), which makes a specific prediction about the av-erage Lorentz factor of radiating electrons along withfew other predictions. This model includes an “acceler-ator”, which supplies energy to radiating particles, andan “emitter”, which drains energy from the particles andtransfers it to synchrotron and IC radiation. The inter-action between the two is in the form of two-photon pairproduction due to internal absorption of high-energy ICphotons by low-energy target photons (of synchrotronorigin). The pair balance model also specifies the energycoming from the magnetic field decay as the source ofpower for accelerating particles, that leads to the predic-tion that the Compton y parameter in the emitting zoneis a few. The need to balance the accelerator’s power bypair loading results in the requirement that the emittingzone is not entirely transparent to its own IC radiation.It also drives the Lorentz factor of radiating electronsto a value, which corresponds to the border betweenThomson and KN Comptonization regimes of their ownsynchrotron radiation.We begin with a brief summary of the relevant obser-vations in §
2. These are the preliminary results as de-scribed in GCNs. Clearly, exact numerical values maychange in the final release of observational data, but thiswill not undermine the validity of our analysis. Someof the conclusions may change, though, if the ratio be-tween the sub-TeV and X-ray luminosities will be revisedsignificantly. We outline in § § § § § § § §
10 and we conclude in § SUMMARY OF OBSERVATIONS mplication of Sub-TeV Emission
Redshift:
The associated optical transient revealed aredshift of . (Selsing et al. 2019; Castro-Tirado et al.2019). Prompt emission:
The prompt emission of GRB190114C consists of initial hard-spectrum multi-peakedpulse with duration ≃ s and a weaker and softer pulsestarting ≃ s after the trigger (Frederiks et al. 2019).The peak luminosity was L peak ≃ . × erg/s(Frederiks et al. 2019). Energy release:
According to Fermi/GBM data(Hamburg et al. 2019), the prompt energy radiated byGRB 190114C is E isorad = × erg (isotropic equiva-lent). Konus-Wind team reports (Frederiks et al. 2019)somewhat smaller value E isorad = . × erg. Duration:
The reported duration of GRB 190114Cis T ≃ s in − keV energy range, accord-ing to Fermi/GBM team (Hamburg et al. 2019), and T ≃ s in − keV energy range, according toSwift-BAT team (Krimm et al. 2019). The larger num-ber was also claimed to be a possible underestimate. Thelarge disagreement between the two estimates of T mayindicate that the burst’s emission at t & s is domi-nated by slowly decaying afterglow component and thatafterglow’s contribution to the overall radiated energy iswell above 10 per cent. The extended emission:
As a reference point, wetake the moment 70 seconds after the trigger. By thistime the prompt emission of GRB 190114C has fadedaway and Swift-XRT and MAGIC observations startedat around this time. Judging from the Konus-Windlightcurve , the burst’s luminosity at this moment wasapproximately × − of the peak value, that cor-responds to L sec ≃ × erg/s. In the avail-able plots, the Swift-BAT lightcurve appears satu-rated at 70 seconds after the trigger, but extrapola-tion from ÷ s time interval gives a compara-ble number, L sec / L peak ∼ × − . The energy ra- http://gcn.gsfc.nasa.gov/notices s/883832/BA/ diated at later time, t > s, can be estimated as E af terglow ∼ L sec ×
70 sec ≃ . × erg, that is ≃ . E isorad . This is likely an underestimate, especially ifthe lightcurve’s decay law is not much steeper than ∝ t − .The average flux of GRB 190114C in Swift-XRT energyrange in the time interval ÷ s after the triggerwas F X ≃ . × − erg/s/cm (corresponds to fluence ≃ . × − erg/cm ) . Comparing this flux to the flu-ence reported by Konus-Wind team ( . × − erg/cm ,(Frederiks et al. 2019) and their estimate for E isorad , we es-timate the average X-ray luminosity in the time interval ÷ s after the trigger (in the observer’s frame) as L iso X ≃ × erg/s (isotropic equivalent; in the progen-itor’s frame). GeV observations:
GRB 190114C was in the Fermi-LAT field of view for 150 seconds since the trigger(Kocevski et al. 2019). The estimated energy flux above100 MeV during this period is ≃ × − erg/s/cm , thatconstitutes a fair fraction of the flux at smaller pho-ton energies (see Ravasio et al. 2019, for a discussionof GeV emission). The highest observed photon energyis 22.9 GeV. This event was observed 15 seconds afterthe trigger and most likely should be attributed to theprompt emission. In this paper we do not discuss theorigin of the observed GeV emission and we focus on thetwo dominant afterglow components, the X-rays and thesub-TeV emission. TeV observations:
The MAGIC Cherenkov tele-scope detected sub-TeV gamma-ray emission from GRB190114C (Mirzoyan 2019). The observation started 50seconds after the trigger and resulted in detection ofpoint-like source with the significance > σ in the first20 minutes. The source was reported to fade quickly.Due to poor observational conditions (large zenith an-gle ≃ ◦ and the presence of partial moon) the en-ergy threshold was ≃ GeV. For MAGIC sensitiv-ity, we estimate that σ detection in 20 minutes cor-responds to the fluence ≃ × − erg/cm . Giventhe redshift of GRB 190114C, it is beyond the gamma-ray horizon even at the threshold energy (about 1/4 of300 GeV gamma-rays reach the Earth from the GRB’sdistance) and therefore only fairly narrow spectral range − GeV contributed to the MAGIC fluence. TheTeV component is probably − times wider (in loga-rithmic units). After correction for absorption and spec-trum’s width we estimate that the intrinsic TeV fluenceis ∼ − erg/cm .We focus the calculations on a single epoch, the ob-servations at s after the trigger, corresponding to Derishev & Piran s in the local frame. In our numerical estimates belowwe use the following values, based on this observationalsummary.Quantity Value in progenitor’s frame(for z = . )Time since explosion t = sEnergy of sub-TeV photons E IC = GeVPrompt radiated energy(isotropic equivalent) E isorad = × ergAverage isotropic equivalentX-ray luminosity at t = s L iso X = × erg/sRatio of sub-TeV to X-ray luminosities η IC = . In the following, unless stated otherwise all quantitiesare measured in the source frame. All energies and lu-minosities are isotropic equivalent. We express quan-tities denoted by ˆ in terms of the observed values inGRB 190114C, e.g. ˆ E IC ≡ E IC /
500 GeV . With ÷ per cent radiative efficiency at the prompt phase, thesenumbers correspond to kinetic energy of ejecta at theafterglow phase E isotot ≃ erg (isotropic equivalent).The bolometric luminosity of GRB 190114C is largerthan the X-ray luminosity, which we infer from theSwift-XRT X-ray data. It includes contribution fromsoft gamma-ray, MeV, GeV and TeV spectral domains,which is comparable to that of X-ray domain. Some-what arbitrarily we estimate the bolometric luminosityas L isobol ≡ η bol L iso X ≃ × erg/s, and we will use thecorrection factor η bol ≃ as a parameter. THE MODELThe late observations of the sub-TeV component sug-gests that it arose from the afterglow. We consider,therefore, an external shock model. Given the scarcityof currently available data a simple one zone model is ac-ceptable. The shock is then characterized by its Lorentzfactor, Γ , radius, R , which can be expressed in terms of Γ and the time since the explosion, t , and the surroundingmatter density, ρ . For simplicity we consider a single en-ergy electron population, characterized by the electron’sLorentz factor γ e , but we stress that the results are valideven for more general electron distributions. As justifiedlater in this section we focus here on IC emission as thesource of the sub-TeV emission.It is common (see e.g. Piran 1999) to characterize thecondition within the emitting region, the downstream,using the local equipartition parameters ǫ e and ǫ B thatrelate the electron’s energy density and the magneticenergy density to the total downstream energy density, e . However, one can use other parameters to charac-terize the conditions. In particular those parameterscan be interchanged with the Compton y parameter,with t cool / t dyn the ratio between the electrons’ radiativecooling time t cool , and the shock’s dynamical timescale t dyn = R /( Γ c ) , and the overall radiation efficiency ǫ r .Given that y is easy to derive from observations for190114C, it will be illuminating to use y at times insteadof one of the microphysical parameters to characterizethe system.There are only two efficient emission mechanismsfor external shock: synchrotron and IC from electronsand/or positrons. Synchrotron is strongly disfavouredas the source of the sub-TeV photons. In the simplestmodel, the radiating electrons/positrons are acceleratedby Fermi mechanism (diffusive shock acceleration, shearflow acceleration or acceleration in turbulent electro-magnetic fields) and the rate of energy gain is limited to ∼ eBc . Equating this rate with the rate of synchrotronlosses gives the largest energy a particle can achieve and,therefore, the largest energy of synchrotron photons,the so called burn-off limit (e.g. Guilbert et al. 1983;de Jager et al. 1996; Aharonian 2000): E ∼ m e c / α ,where α is the fine structure constant. If the sub-TeV photons were Lorentz-boosted synchrotron pho-tons, then the bulk Lorentz factor must be larger than5000. For the time of observations this implies unrealis-tically low density of material around the GRB source.In principle, there are several ways to surpass theburn-off limit for synchrotron photons. All use the ideaof accelerating electrons in one place, with weaker mag-netic field, and then making them radiate in other re-gions with a stronger magnetic field (e.g. Kumar et al.2012). One such mechanism is ultra-fast reconnectionwith formation of pinch-like structures, where local mag-netic field can be much stronger than the average value(see e.g. Kirk 2004; Uzdensky et al. 2011; Cerutti et al.2012; Kagan et al. 2016). However, it can occur onlyin magnetically-dominated environments, which are un-likely for external shocks. The converter accelerationmechanism (Derishev et al. 2003; Stern 2003) also pro-vides the necessary non-local acceleration. However, inthis case the highest-energy photons are less beamedthan the low-energy photons (Derishev et al. 2007), andthe sub-TeV synchrotron radiation generated due to con-verter acceleration should be time-dilated with respectto softer spectral ranges, unlike the observations ofGRB 190114C.We consider therefore, in the following, Inverse Comp-ton in the context of SSC, as there is an observed sig-nificant flux of X-ray photons, produced presumably bysynchrotron mechanism and those are the natural seeds mplication of Sub-TeV Emission § THE BLAST WAVEWe use the theory of an adiabatic blast wave to expressthe physical conditions within the emitting regions interms of three quantities, the isotropic equivalent bolo-metric luminosity L isobol ≡ η bol L iso X , the time in the sourceframe t , and the shock’s Lorentz factor Γ . These expres-sions are well known (e.g. Piran 1999) (usually in termsof other variables) and are given here for completeness.The necessary expressions are summarized below.We consider a uniform medium around the progenitor(ISM for short) and a stellar wind (wind for short). Thedensity of circum-burst medium, ρ ( R ) , is ρ = Û M π R v w ( wind ) ,ρ ( ISM ) , (1)where ρ is the local density of circum-burst medium, Û M and v w are the mass-loss rate and the wind velocity.The radius of the blast wave, R , and its Lorentz factor Γ are related to the observed time t as: R ≃ ( ) Γ ct . (2)Here and in many expressions below, the wind and ISMcases differ only by numerical factors. We present theresults as a single expression preceded by column of twocoefficients: the upper one for a wind and the lower onefor an ISM. The shock’s Lorentz factor Γ at a given timeis expressed in terms of E isotot : Γ ≃ (cid:18) E isotot v w Û Mc t (cid:19) / , ( wind ) (cid:18) E isotot π ρ c t (cid:19) / , ( ISM ) . (3)We define the radiative efficiency ǫ r as the ratio ofoutgoing radiation energy flux to the upstream energyflux: ǫ r = η bol L iso X / π R Γ ρ c . The magnetic field cariesa fraction ǫ B of the comoving-frame energy density e = Γ ρ c and the comoving-frame magnetic field strengthis: B = ( πǫ B e ) / ≃ ( / ) Γ ǫ B η bol L iso X ǫ r c t ! / . (4) Finally, we write the isotropic equivalent energy of theshock as: E isotot ≃ ( / ) η bol L iso X t ǫ r . (5) OPACITYThe fact that the sub-TeV photons have not been ab-sorbed at the source is not trivial. Thus, before turn-ing to the radiation mechanism we consider the implica-tions of this simple observation. The sub-TeV photonsare emitted along with lower-energy X-ray photons. Re-gardless of origin of the X-ray radiation – the afterglowemission, the trailing part of prompt emission, or both– it could make the source opaque for the sub-TeV pho-tons of energy E IC due to two-photon pair production.The main contribution to the opacity comes from pho-tons of energy ∼ E a = Γ ( m e c ) / E IC ≃ Γ × . (inthe observer’s frame this energy is E obs a = E a /( + z ) ≃ Γ × . ). Let η a be the fraction of the X-ray lumi-nosity emitted at energies around E a , then the opticaldepth for absorption of the sub-TeV photons is τ γγ ≃ Γ σ γγ n a R ≃ ( / ) σ γγ η a L iso X E IC π Γ tc ( m e c ) ≃ ( . . ) η a ˆ L iso X ˆ E IC Γ ˆ t (6)where σ γγ ≃ . σ T is the value of two-photon pairproduction cross-section near its peak, calculated as-suming isotropic distribution of target photons. Recallthat ˆ X denotes the value corresponding to the one ob-served in GRB 190114C (see §
2) and here and elsewhere Γ ≡ Γ / . Note that this result was obtained here us-ing the observed parameters. However it is more generalsee § τ γγ ≃ σ γγ η a E IC c π ( m e c ) (cid:16) v w c (cid:17) − / (cid:0) L iso X t (cid:1) Û M / (cid:0) E isotot (cid:1) / t / , ( wind ) (cid:18) π (cid:19) / σ γγ η a E IC c / ( m e c ) (cid:0) L iso X t (cid:1) ρ / (cid:0) E isotot (cid:1) / t / , ( ISM ) . (7)Clearly a source capable of emitting sub-TeV radiationmust have τ γγ . . This implies that there is biasagainst observing sub-TeV and more energetic emissionin dense circum-burst environments both for the windmodel, where τ γγ ∝ Û M / , and for the ISM model, where τ γγ ∝ ρ / . For the wind model there is an observationalbias against weak bursts, τ γγ ∝ (cid:0) E isotot (cid:1) − / , whereas for Derishev & Piran the ISM model there is a feeble bias in favor of weakbursts, τ γγ ∝ (cid:0) E isotot (cid:1) / . The two models differ also inthe time dependence of the two-photon absorption op-tical depth: it slowly decreases with time for the windmodel, τ γγ ∝ t − / , and – even slower – increases withtime for the ISM model, τ γγ ∝ t / . Note that whenestimating the time dependence we have approximatedthe shock luminosity decrease with time as / t and wehave ignored the dependence of η a on time, E isotot and Û M (or ρ ). We don’t expect those factors to be significantenough to change qualitatively our results.The requirement τ γγ . sets a limit on the Lorentzfactor: Γ & ( ) η a ˆ L iso X ˆ E IC ˆ t ! / , (8)Note that η a is a function of Γ but because of the weakdependence of Γ on all other parameters this can be ig-nored. For Γ ≃ the energy of the absorbing photonsis E obs a ≃ keV and η a is not much below unity, there-fore the transparency condition (8) is Γ & . Becauseof the strong dependence of the opacity on Γ , the latterconclusion will not change significantly if the source ismoderately opaque ( τ γγ ≃ ÷ ), as suggested by ouranalysis of radiative efficiency in §
9. Instead, this wouldmake the estimate more certain: Γ ≃ ÷ for amoderately opaque source.We derived Eq. (6) assuming that the low energy (X-ray) photons, that absorb the high energy (sub-TeV)photons, are emitted by the same source. If the low en-ergy photons are prompt radiation that is emitted froma smaller radii then they propagate in small angles rel-ative to the shock normal and this accordingly reducesthe interaction rate. However, given the very weak de-pendence in Eq. 8 on the X-ray luminosity (1/6 power)this limit will be more or less valid even if only a smallfraction of the X-ray photons is produced by the elec-trons emitting the high energy radiation.The lower limit on the shock’s Lorentz factor, set bythe transparency condition (Eq. 8), in combination withshock deceleration law (Eq. 3) yields an upper limit onthe external density. This corresponds to an upper limiton the mass loss rate for the wind case Û M < Û M upp = E isotot v w c t π tc ( m e c ) σ γγ η a L iso X E IC ! / (9) ≃ × − E isotot , v w , . ˆ t / (cid:16) η a ˆ L iso X ˆ E IC (cid:17) / M ⊙ / yr , where v w , . = v w / . cm/s, and to an upper limit onthe density of circum-burst medium for the ISM case ρ < ρ , upp = E isotot π c t π tc ( m e c ) σ γγ η a L iso X E IC ! / (10) ≃ E isotot , ˆ t / (cid:16) η a ˆ L iso X ˆ E IC (cid:17) / m p / cm . In both equations E isotot , = E isotot / erg. Given thesevalues, which are within the range that is typically ex-pected in both cases, and the weak dependence of Γ onthe external density (see Eq. 3) there is no much free-dom in the value of Γ . Namely Γ cannot be much largerthan the opacity limit given in Eq. 8. COMPTONIZATION REGIMESThe IC mechanism comes in two varieties: either itoperates in Thomson regime, where the energy of theelectrons/positrons greatly exceeds the energy of the up-scattered photons, or in KN regime, where the energyof the upscattered photons approximately equals the en-ergy of the electrons/positrons. The observation of GRB190114C at sub-TeV energy allows us to discriminatebetween these two options. Let E sy be the photonenergy at the synchrotron peak of the SED and γ e the(comoving-frame) Lorentz factor of electrons, that emitssynchrotron photons with this energy. We define γ cr asthe critical electron Lorentz factor that satisfies the re-lation m e c = γ cr E sy = γ ~ ω B ⇒ γ cr = (cid:18) B cr B (cid:19) / , (11)where B cr ≃ . × G is the Schwinger field strength.Electrons with Lorentz factor γ e < γ cr Comptonize theirown synchrotron radiation in the Thomson regime, andfor γ e & γ cr Comptonization proceeds in the KN regime.The largest energy of the IC photons, which can be pro-duced in Thomson regime, is E cr IC = Γ γ cr m e c = Γ (cid:18) B cr B (cid:19) / m e c . (12)If this energy is larger than the energy of the observed ICphotons then Comptonization is in the Thomson regime,otherwise it is in the KN regime.Substituting the magnetic field strength expected inthe external shock (Eq. 4) into Eq. 12 we find that E cr IC ≃ ( / / ) Γ B ǫ r t c ǫ B η bol L iso X ! / m e c (13) ≃ ( . . ) Γ (cid:18) ǫ r ǫ B η bol (cid:19) / ˆ t / ( ˆ L iso X ) / TeV mplication of Sub-TeV Emission ǫ r and ǫ B (this ratio is probably not far fromunity, as suggested by the comparable luminosities insynchrotron and IC radiation), equation 13 serves as alimit on the Lorentz factor of the external shock thatseparates the two Comptonization regimes, E IC < E cr IC and E IC > E cr IC . If the shock’s Lorentz factor is largerthan Γ KN ≃ ( − / − / ) (cid:18) E IC m e c (cid:19) / ǫ B η bol L iso X ǫ r B t c ! / (14) ≃ ( ) (cid:18) ǫ B η bol ǫ r (cid:19) / ˆ E / IC ( ˆ L iso X ) / ˆ t / then the IC radiation is produced in the Thomsonregime.For GRB 190114C the transparency condition (8) im-plies that Γ > Γ KN and hence the observed sub-TeV radi-ation is produced in the Thomson regime, but not veryfar from the KN limit. The fact that Γ KN is close to thelimit set by the transparency condition (8) is a mere co-incidence for the generic afterglow model that employsthe classical equipartion parameters (Sari et al. 1998),i.e., the peak of synchrotron spectrum is by chance closeto the energy of photons, which contribute most to theopacity. On the contrary, it is a basic prediction forpair-balance model (Derishev & Piran 2016). THE LORENTZ FACTOR OF THE RADIATINGELECTRONSOne may calculate the comoving-frame Lorentz factor, γ e , of the electrons that produce the IC photons in twoways: assuming that Comptonization proceeds in KNregime, that gives E IC ≃ Γ γ e , KN m e c ⇒ γ e , KN ≃ E IC Γ m e c , (15)or assuming that electrons are producing IC photons byupscattering of their own synchrotron photons in theThomson regime , that gives E IC ≃ Γ γ e , Th BB cr m e c ⇒ γ e , Th ≃ (cid:18) E IC Γ m e c B cr B (cid:19) / . (16)From the last equation, substituting the magnetic fieldstrength B from Eq. (4), we obtain γ e , Th ≃ ( / / ) (cid:18) Γ B cr E IC m e c (cid:19) / ǫ r c t ǫ B η bol L iso X ! / . (17) Strictly speaking, the statement that the largest contributionto the seed photons in Thomson regime is due to self-producedsynchrotron photons is true if the synchrotron SED around thefrequency γ e ω B has a convex shape in log-log plot. Comparison of the above equation with Eq. (14) revealsa simple relation γ e , Th ≃ (cid:18) ΓΓ KN (cid:19) / γ e , KN . (18)The actual value of the electron’s Lorentz factor is γ e = max (cid:2) γ e , Th , γ e , KN (cid:3) . (19)By making this choice, one immediately knows theregime of Comptonization.Substituting the lower limit (Eq. 8) on the shock’sLorentz factor Γ into Eq. (18) and recalling that Γ can-not be much larger, we find that γ e , Th ≃ ( . ÷ ) γ e , KN .This means, that the observed sub-TeV radiation wasproduced in the Thomson regime (though rather closeto KN regime) and hence γ e ≃ γ e , Th ≃ ( . . ) × (cid:18) ǫ r ǫ B η bol (cid:19) / Γ / (cid:16) ˆ E IC ˆ t (cid:17) / (cid:16) ˆ L iso X (cid:17) / . (20)Given the weak ( γ e , Th ∝ Γ / ) dependence on the shock’sLorentz factor and the fairly narrow allowed range of Γ ,this expression provides a rather good estimate for γ e . THE COOLING RATEThe cooling parameter is the ratio of the radiativecooling time, t cool , to the shock dynamical timescale, t dyn ≃ R /( Γ c ) . Using the magnetic field strength (Eq. 4)and the shock radius (Eq. 2) we have: t cool t dyn = π m e c γ e σ T B ( + η IC ) Γ cR (21) ≃ ( ) π Γ m e c γ e σ T ( + η IC ) ǫ r c t ǫ B η bol L iso X ! . The slowest cooling corresponds to the smaller electron’sLorentz factor γ e = γ e , KN (see Eq. 15): (cid:18) t cool t dyn (cid:19) max ≃ ( ) π Γ (cid:0) m e c (cid:1) σ T ( + η IC ) E IC ǫ r c t ǫ B η bol L iso X ! . (22)Using Eqs. (3) and (5) to substitute the shock’sLorentz factor Γ and kinetic energy E isotot , we find thefast cooling condition (cid:18) t cool / t dyn (cid:19) max < for a wind: Û M > v w c π (cid:0) m e c (cid:1) t σ T c ( + η IC ) E IC ! / η bol L iso X ǫ B ǫ r ! / (23) ≃ × − v w , . ˆ t ( + η IC ) ˆ E IC ! / η bol ˆ L iso X ǫ B ǫ r ! / M ⊙ / yr Derishev & Piran and for an ISM: ρ > c t / (cid:18) m e σ T ( + η IC ) E IC (cid:19) / π ǫ r ǫ B η bol L iso X ! / (24) ≃ . × − t / ( + η IC ) ˆ E IC ! / ǫ r ǫ B η bol ˆ L iso X ! / m p / cm . Given these small values we conclude that fast coolingregime for GRB 190114C at time moment t = s isassured.The fastest cooling corresponds to the larger electron’sLorentz factor γ e = γ e , Th (see Eq. 17). We then multiplyEq. (22) by γ e , KN / γ e , Th to obtain (cid:18) t cool / t dyn (cid:19) min and sub-stitute Γ from the equation for the two-photon opticaldepth (6) to obtain: (cid:18) t cool t dyn (cid:19) min ≃ τ γγ (cid:18) γ e , KN γ e , Th (cid:19) σ γγ σ T η a ( + η IC ) L iso X L isobol ǫ r ǫ B ≃ .
02 1 τ γγ (cid:18) γ e , KN γ e , Th (cid:19) η a ǫ r ( + η IC ) η bol ǫ B (25)for both wind and ISM cases. The similarity of theexpression for (cid:18) t cool / t dyn (cid:19) max (Eq. 22) to expression (6)for the photon absorption optical depth, which eventu-ally leads to the above simple relation, is not coinci-dental. Both describe electromagnetic interaction ofenergetic particles (electrons in one case and photons inthe other) having the same energy with the same back-ground low-energy photons. The only difference is in thecross-sections for electron-photon and photon-photon in-teractions. Thus, if Comptonization operates in the KNregime or close to it, then the IC photons arising fromfast cooling electrons should have an optical depth topair creation with the low energy seed photons thatis larger than y (see e.g Moderski et al. 2005; Derishev2009).In the above discussion the cooling rate was consideredwithin the SSC model. However, it is important to notethat the large flux of low energy X-rays ensures that theIC emitting electrons would be in fast cooling regardlessof the origin of the X-ray photons and a slow cooling ICregime is ruled out. THE RADIATION EFFICIENCY AND ITSIMPLICATIONSAssuming turbulent, i.e., isotropic on large scales,magnetic field in the downstream the intensity of thesynchrotron radiation at the front of a plane-parallelemitting region at an angle θ to its normal is: I sy ( θ ) ≃ ( θ ) σ T γ e n e e B cl em π = I sy ( ) cos ( θ ) = y e B c π cos ( θ ) , Here l em is the thickness of the emitting region measuredin the shock comoving frame: l em ≃ ct cool < R / Γ for t cool < t dyn (fast cooling) , R / Γ for t cool / t dyn > (slow cooling) , (26)and y ≡ σ T γ e n e l em (27)is the Compton y parameter. The energy density of thesynchrotron radiation inside the emitting region is e sy ≃ π c ∫ π / I sy ( θ ) sin θ d θ ≃ π Λ I sy ( ) c ≃ Λ y e B . (28)The integral in this equation has a logarithmic diver-gence at θ → π / , that is an artifact of the plane ge-ometry approximation. However, the shock has a finitecurvature and hence the integral is finite. We take thisinto account introducing the geometrical factor Λ ≃ + ln (cid:18) R Γ l em (cid:19) , (29)that reproduces both asymptotic limits, Λ ≃ for l em ≃ R / Γ and Λ ≃ ln ( R / Γ l em ) for l em ≪ R / Γ .The synchrotron radiation flux at the shock front is F sy ≃ π ∫ π / cos θ I sy ( θ ) sin θ d θ ≃ π I sy ( ) ≃ y e B c . (30)Comparing it to the energy flux, associated with thedownstream plasma, F = ( c / ) e = ( c / ) e B / ǫ B for adownstream velocity equal to c / , we introduce syn-chrotron radiative efficiency (Sari et al. 1996): ǫ sy ≡ F sy F ≃ y ǫ B . (31)Note that the radiative efficiency can also be expressedin terms of ǫ e : ǫ sy < ǫ r = ǫ e for t cool < t dyn (fast cooling) , t dyn / t cool for t cool > t dyn (slow cooling) . (32)While the latter expression for the efficiency, ǫ r = ǫ e min ( , t dyn / t cool ) ,is more familiar, the expression ǫ sy = y ǫ B is also useful This flux is calculated for a static emission zone. In the case ofGRB afterglows, the emitting zone is associated with the down-stream plasma, which recedes from the shock front, and hencefewer photons move in forward direction in the shock plane. Onthe other hand, most of the photons, which appear to move back-wards in the shock frame, actually move in forward direction inthe progenitor’s frame and overtake the shock at later time whenit decelerates. mplication of Sub-TeV Emission y can be directly estimated from the observable η IC .Calculating the IC radiation flux at the shock frontin the same way as for synchrotron radiation, where themagnetic field energy density is replaced by the energydensity of synchrotron radiation with an additional fac-tor κ KN ≤ that accounts for the KN effect we obtain F IC ≃ κ KN y e sy c ≃ κ KN Λ y e B c . (33)Therefore, the IC radiative efficiency is (Sari et al.1996) : ǫ IC ≡ F IC F ≃ κ KN Λ y ǫ B ; η IC ≡ F IC F sy ≃ κ KN Λ y . (34)Note that within the Thomson regime y and η IC are thesame up to the logarithmic geometrical factor Λ / . Theoverall radiative efficiency is ǫ r = ǫ sy + ǫ IC ≤ ǫ e . The equal-ity ǫ r = ǫ e holds for the fast cooling regime.If L isoIC , inferred from the available observational data,is treated as the intrinsic IC luminosity of the externalshock, then the Compton y parameter can be estimatedfrom the ratio of sub-TeV (IC) to X-ray (synchrotron)luminosities (see Eq. 34). Assuming a geometrical factor Λ ∼ and keeping in mind that Comptonization pro-ceeds in nearly Thomson regime, we arrive at y ∼ . .To first order in y , the external shock efficiency can beestimated as ǫ r ≃ y ǫ B ∼ . ǫ B .Using the above estimates for the radiative efficiencyand combining it with the expression for the shock’sisotropic equivalent kinetic energy (see Eq. 5) we findthat ǫ B ≃ ( / ) η bol L iso X t (cid:0) + η IC (cid:1) y E isotot ≃ ( . . ) η bol ˆ L iso X ˆ t ˆ η IC E isotot , , (35)where the last approximate equality is valid for smallvalues of η IC as in the case of GRB 190114C. The aboveequation suggests either a large magnetization or a verylarge value of E isotot and consequently a low efficiency ofthe prompt phase.Large magnetization is unexpected as the shock prop-agates into an unmagnetized medium. In particular,PIC simulations consistently show ǫ B ∼ − for ashock propagating into unmagnetized medium (e.g.Sironi & Spitkovsky 2011; Sironi et al. 2015). Thisvalue implies E isotot ∼ erg, the implied real energyis uncomfortably large, even after adding typical beam-ing corrections. This also implies a radiative efficiency Note the geometrical factor Λ that was included here relativeto(Sari et al. 1996) of only a few percent at the prompt phase. Numeri-cal simulations also suggest that the magnetic field’senergy share is several times less than that of the ac-celerated electrons (e.g. Spitkovsky 2008; Kumar et al.2015). Again this is at odds with our estimate for GRB190114C unless the electrons radiate in the slow coolingregime, which is ruled out (see Eqs. (23) and (24) anddiscussion thereafter).A possible resolution of this apparent problem is thatthe sub-TeV radiation is stronger than what we use as acanonical value and y is larger. This can happen if thesub-TeV radiation is strongly absorbed within the emit-ting zone or if our estimate of the sub-TeV luminosityfrom the available GCN data was too low or both. Alarger intrinsic sub-TeV luminosity (and hence a largerCompton y ) would resolve both problems of too largeshock’s kinetic energy and too low ǫ e (relative to ǫ B ).While a careful examination of the observational datacould reveal a better estimate for the ratio of sub-TeVto X-ray fluxes it may be much more difficult to assessdirectly whether there was some level of internal self-absorption of the sub-TeV photons or not.Yet another possibility is that at an observer time of t = s the X-ray radiation is still dominated by theprompt emission and the external shock’s contributionto the observed X-ray and sub-TeV fluxes is small. Thiswould relax the requirements for the shock’s kinetic en-ergy. But the estimate of Compton y parameter wouldremain essentially unchanged, hence ǫ B > ǫ e will stillhold.An independent way of estimating the parameters ofthe emitting zone is based on the relation between radia-tive efficiency and the total energy of the radiating par-ticles (electrons and/or positrons). Given their Lorentzfactor γ e and their number per baryon, ξ e (note thathere ξ e can be larger than unity if there is a signifi-cant pair loading as a result of internal absorption ofIC photons) one can set an upper limit to the radiativeefficiency: ǫ r ≤ ǫ e ≡ ξ e γ e m e /( Γ m p ) (36)that becomes an equality in the fast cooling case relevantto GRB 190114C.Using this relation with Eqs. 31 and 34 we find thatin the fast cooling regime ǫ B ≃ ξ e γ e m e y (cid:0) + η IC (cid:1) Γ m p ≃ . ξ e γ e , ˆ η IC Γ . (37)For the parameters of GRB 190114C we get ǫ B ≃ . ξ e .If all the electrons from the circum-burst medium areaccelerated, then ξ e = . for a Wolf-Rayet stellar windcase and ξ e = . for the ISM case. This would im-ply a rather large ǫ B ≃ . ÷ . . If on the other hand0 Derishev & Piran ǫ B is small, as implied by the PIC simulations, thenthe fraction of accelerated of electrons, ξ e , should besmall as well. This is also observed in PIC simulations(Sironi & Spitkovsky 2011).Combined with Eq. 5, Eq. 37 gives a lower limit onthe kinetic energy of the shock. Comparing this valuewith the estimated isotropic equivalent prompt γ -rayenergy ( × erg from section 2) the ISM scenariois consistent with a prompt radiation efficiency up to ≃ per cent. A smaller prompt efficiency would im-ply that only a fraction of the available electrons is ac-celerated by the external shock. In the wind scenariothe prompt efficiency is limited to . per cent, un-less there are additional electron-positron pairs that areproduced within the external shock and are acceleratedalong with electrons from the wind. If one assumes thatonly a fraction of the observed X-ray luminosity is dueto external shock (as discussed earlier in this section),then the shock’s contribution to IC luminosity and hencethe requirements on the shock’s kinetic energy would beproportionally smaller.The parameters, which we determined for the earlyafterglow phase of GRB 190114C, are consistent withthe generic afterglow model, albeit with a larger thanexpected ǫ B value and with and ǫ B > ǫ e (if one assumesthat the observed ratio of IC to synchrotron luminosityis intrinsic to the source). At the same time they fit wellinto more specific predictions of the pair-balance model.Two key predictions of this model are: (i) The IC peakis produced at the border between KN and Thomsonregimes; (ii) η IC = a few. Both are satisfied here (seeEqs. 14 and 18 and discussions there for the first condi-tion). As noted earlier the unexpectedly large inferredmagnetization together with the inequality ǫ B > ǫ e sug-gest that there is moderate internal absorption of sub-TeV photons. If so this will increase the estimate ofthe intrinsic Compton y parameter to ≃ a few, getting itcloser to the range predicted by the pair balance model. TEMPORAL EVOLUTION OF IC PEAKPOSITION: GENERIC VS. PAIR-BALANCEMODELSPredictions of the temporal evolution of the IC peakare drastically different in these two models, making itpossible to distinguish between the two if observationsat later time become available. In the simplest scenariofor both models, the microphysical parameters ǫ B and ǫ e remain constant. In the case of fast cooling and Comp-tonization in Thomson regime, this implies that η IC doesnot change with time. But the two models differ in thepredicted evolution of the IC peak energy, E IC , p ∝ Γ γ e , p B (in Thomson regime). In the generic model γ e , p is proportional to Γ (see Eq.36). Therefore, E IC , p ∝ Γ B ∝ Γ t − / ∝ ( t − , ( wind ) t − / , ( ISM ) , (38)where we substituted the magnetic filed strength B fromEq. (4) and then the shock Lorentz factor Γ from Eq. (3).Both for the wind and for the ISM cases the genericmodel predicts a fast decrease of the peak IC energy.For example, if a GRB starts with E IC , p ≃ TeV at 100seconds after the explosion, then one hour later the ICpeak would be located at ≃ GeV.In the pair-balance model the peak Lorentz factorof the radiating electrons is determined by the pair-production condition and the number of energetic elec-trons (positrons) is regulated in such a way that γ e , p ≃ γ cr ∝ B − / (see Eq. 11). Therefore, E IC , p ∝ Γ B − / ∝ Γ t / ∝ ( const , ( wind ) t − / , ( ISM ) . (39)The pair-balance model predicts that the peak IC energydoes not change with time in the wind case. In theISM case, the model predicts weak evolution IC peaktowards lower energies. For the same hypothetical GRB,which starts with E IC , p ≃ TeV at 100 seconds after theexplosion, the IC peak would be above ≃ GeV evenat 10 hours after the explosion and will still be accessiblefor Cherenkov telescopes, provided that the flux, thatdecreases like t − , doesn’t fall below the sensitivity limit. CONCLUSIONSMAGIC’s observations of the sub-TeV emission fromGRB 190114C opened a new window on the emissionprocess in GRBs’ afterglows. Within the SSC frame-work this emission has to be assigned to the IC com-ponent. It is the first time when this component wasunequivocally observed. With this information at ourdisposal we are able to constrain the conditions withinthe emitting region of a GRB to a better precision thanhas been ever possible.Our analysis is based on the preliminary data de-scribed in GCNs. We expect that our results will holdunless these values will be significantly revised in therefined analysis. Given the available data we use asingle zone model and we do not attempt to reproducethe whole spectrum. Instead we focus on the two domi-nant components, the sub-TeV radiation and the lowerenergy X-rays, that turn out to be the seed photons forthe IC process producing the sub-TeV photons. An ex-ternal shock with a bulk Lorentz factor Γ ≃ andelectrons accelerated to γ e ≃ can explain the obser-vations with a SSC model in which the IC process is in mplication of Sub-TeV Emission (cid:0) Γ ǫ r (cid:1) − , thatmeans the upstream acquires enough momentum fromsecondary pairs to start moving at relativistic speed evenbefore the shock comes. This forces one to use a mod-ified shock solution, as discussed in (Derishev & Piran2016).The detection of sub-TeV photons implies that thesource’s optical depth with respect to two-photon pairproduction is at most a few. Note that we cannot ex-clude absorption of sub-TeV radiation at a moderatelevel within the source itself. The target photons for ab-sorption are in the X-rays. The pair annihilation opac-ity is alleviated by the Lorentz boost, just like in thecommon compactness argument (e.g. Baring & Harding1997; Piran 1999; Lithwick & Sari 2001). It turns outthat for both wind and ISM a minimal bulk Lorentz fac-tor of order Γ ≃ at the time of observation is neededto allow escape of sub-TeV photons. As usual in com-pactness arguments the dependence of this limit on thedifferent parameters is rather low and the limit is veryrobust. For the same reason the limit doesn’t vary muchif we require an optical depth of a few instead. On theother hand the shock deceleration dynamics implies, forreasonable circum-burst densities, that the bulk Lorentzfactor must be Γ . at the time of the observations.Combined with the opacity limit we find that the bulkLorentz factor of the afterglow is Γ ≃ .The electrons must be energetic enough to produce thesub-TeV photons. With Γ ≃ this implies γ e & .An upper limit γ e . . ÷ . × derives from conditionthat the emission process is SSC in the Thomson regime.Once more, the two limits bracket γ e nicely from aboveand from below. Thus, the IC operates in the Thomsonregime but very close to the Thomson/KN boundary.The seed photons are X-rays and they are, indeed, thesynchrotron emission produced by γ e ≃ electrons.Remarkably the observed flux of the X-ray photons iscompatible with this interpretation. Furthermore, ouranalysis indicates that the observed sub-TeV emission isnear the peak of the IC component.We obtained the values for Γ and γ e in a way, thatdoes not use spectral information (i.e., we were not mak-ing spectral fits). Yet we arrived at pretty certain esti- mates. This was possible because we could constrainthe IC mechanism to Thomson regime of operation. Inthis regime the electrons which are responsible for thepeak of synchrotron SED comptonize mostly their ownsynchrotron radiation. This alleviates the major uncer-tainty of SSC modelling – a possibility that the seedphotons for the main (in the sense of energetics) part ofelectron distribution are produced by some lower-energyelectrons.If the ratio between IC and synchrotron luminosities η IC ≃ . , as we used in our estimates, reflects the in-trinsic conditions in the emitting zone, then either theGRB’s kinetic energy was in excess of erg and its ra-diative efficiency was below several per cent or the shockmagnetization is large, with ǫ B ∼ . . In either casethe energy share of the radiating electrons is ≃ timessmaller than that of the magnetic field. These find-ings concerning the microphysical equipartition param-eters depart from both theoretical expectations and re-sults of PIC simulations (e.g. Sironi & Spitkovsky 2011;Sironi et al. 2015). However, they can be made consis-tent with those expectations assuming a moderate (with τ γγ ≃ ) intrinsic absorption of the sub-TeV radiation.If the sub-TeV radiation from GRB 190114C was in-deed partially self-absorbed, as suggested by our anal-ysis, then we can speculate that other bursts regularlyescape detection by Cherenkov telescopes just becausethe sources are typically self-absorbed in the TeV range.Indeed, the opacity argument sets an upper limit onthe surrounding matter density. While this limit is notvery stringent for an ISM, it is rather low for a wind, Û M < . × − E isotot , v w , . M ⊙ / yr , and the majority ofprogenitors may fail to pass the self-absorption filter (seealso Vurm & Beloborodov 2017).Under the assumption of moderate intrinsic absorp-tion of the sub-TeV radiation, the conditions in the emit-ting zone fit nicely into the predictions of pair-balancemodel: Comptonization proceeds at the border betweenThomson and KN regimes; internal absorption of ICphotons provides secondary pairs for further accelera-tion and emission; the Compton y parameter is of orderunity. The same conditions are possible for a genericmodel as well, though there is no special preference forthis region in the parameter space. A clear distinctionbetween the generic and the pair-balance models canbe made if late-time observations of TeV emission be-come available: the generic model predicts a rapid de-cline of peak IC energy with time, whereas the pair-balance model predicts that the peak IC energy staysapproximately constant in time.The main uncertainty in the interpretation of ourresults arises from the uncertainty in the IC-to-2 Derishev & Piran synchrotron luminosity ratio η IC that we inferred fromthe preliminary data to be ≈ . . This has lead tothe conclusion that ǫ B > ǫ e and to the conclusion that ǫ B is rather large compared to expectations. Howeverthe analysis outlined here doesn’t depend on this value.Clearly the qualitative conclusions will have to be re-vised if it turns out that η IC > . However the rest of theanalysis concerning the conditions within the emittingregions still holds.Future observations of GRBs in the sub-TeV rangewill provide further insight into the conditions withinGRBs’ emitting zones. In particular we will be able toexplore the range of microphsyical parameters that arisein GRBs afterglow. Once the sub-TeV spectra become available, they may shed more light on whether there issignificant internal absorption or not, that is critical tosome parts of the analysis. The observations will enableus to distinguish between different acceleration mech-anisms and explore the microphsyics of shock accelera-tions. Beyond GRBs, these results will have impact on awhole suite of other astrophysical phenomena involvingrelativistic shocks. ACKNOWLEDGEMENTSThis research is supported by the Russian ScienceFoundation grant No 16-12-10528 (ED), by an advancedERC grant (TREX) and by the I-Core center of excel-lence of the CHE-ISF (TP).REFERENCES
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