GRB Spectra in the complex of synchrotron and Compton processes
Yunguo Jiang, Shao-Ming Hu, Xu Chen, Kai Li, Di-Fu Guo, Yu-Tong Li, Huai-Zhen Li, Hai-Nan Lin, Zhe Chang
aa r X i v : . [ a s t r o - ph . H E ] J a n GRB Spectra in the complex radiation of synchrotron andCompton processes
Yunguo Jiang , ‡ , Shao-Ming Hu , † , Xu Chen , Kai Li , Di-Fu Guo ,Yu-Tong Li , Huai-ZhenLi , Hai-Nan Lin , Zhe Chang , Shandong Provincial Key Laboratory of Optical Astronomy andSolar-Terrestrial Environment, Institute of Space Sciences,Shandong University at Weihai, 264209 Weihai, China Physics Department, Yuxi Normal University, 653100, Yuxi, China Institute of High Energy PhysicsChinese Academy of Sciences, 100049 Beijing, China Theoretical Physicsl Center for Science FacilitiesChinese Academy of Sciences, 100049 Beijing, China
Abstract
Under the steady state condition, the spectrum of electrons is investigatedby solving the continuity equation under the complex radiation of both the syn-chrotron and Compton processes. The resulted GRB spectrum is a broken powerlaw in both the fast and slow cooling phases. On the basis of this electronspectrum, the spectral indices of the Band function in four different phases arepresented. In the complex radiation frame, the detail investigation on physicalparameters reveals that both the reverse shock photosphere model and the for-ward shock with strong coupling model can answer the α ∼ − ‡ [email protected] † [email protected] Subject headings: gamma-ray burst: Band-like spectra: Klein-Nishina regime:Synchrotron radiation and Compton scattering
1. Introduction
Although a remarkable advance of investigations on GRBs was made in the past decades,the nature of the prompt emission of GRB is still unclear. Most spectra of GRBs can be de-scribed by the empirical Band function, whose format is a smoothly connected broken powerlaw (Band et al. 1993; Goldstein et al. 2012; Gruber et al. 2014). Many physical modelshave been built to explain the radiation mechanism in the prompt phase. Beloborodov(2010) discussed the thermal photosphere emission from the passively cooling jets. Inelasticnuclear collisions play an important role in producing the multiplicity of high energy photons(Beloborodov 2003). The resulted spectrum roughly is a broken power law except a highenergy cut off, and the range of the low energy spectral index is limited. Rees & M´esz´aros(2005) considered the leptonic dissipation model to account for the high luminosity of GRBs.One advantage of the photosphere model is the prediction of the observed peak photon en-ergies. However, the detailed spectral indices can not be read directly from the thermalradiation to fit observations. By including the reasonable ingredients like the bulk motionof the jet, the location of the dissipation and the viewing angles, the photosphere emissioncan also lead to a variety of non-thermal spectral shapes (Ryde et al. 2011). The more pop-ular idea about prompt emission of GRB is the relativistic fireball model (Rees & M´esz´aros1992; M´esz´aros 2002, 2006). An extreme fast outflow collides with the interstellar mediumand produces a violent shock. Electrons in the plasma will be accelerated by the shockto form a power-law energy distribution via Fermi acceleration (Peacock 1981). The syn-chrotron radiation (SR) of these electrons will lead to a broken power law spectrum in theoptically thin regime. The magnetic reconnection and turbulence model has been proposedby Zhang & Yan (2011) to explain the GRB emission. This model can overcome the lowefficiency and electron excess problems of the internal shock model. Thanks to the greatprogress on the detection of high energy astrophysical photons, especially the
Fermi and
Swift missions, these models can be well testified by the spectral analyses of GRBs.The
Fermi
Gamma-Ray Burst Monitor (GBM) spectral catalog of the first two yearsand four years indicated that the broken power law is still good in fitting most spectra ofGRBs (Goldstein et al. 2012; Gruber et al. 2014). Here, we denote the low and high energyspectral indices as α and β , respectively. For the high quality “BEST” samples, the lowenergy spectral indices α peak at −
1; Up to 17% samples violates the synchrotron − / − / α ∼ −
1, where photosphereemission of non-dissipative jets are considered. This suggests us that the thermal emissionplus the geometry effect can create the observed GRB spectra. Another remarkable modelwhich can solve the α ∼ − α is found to be relatedto the decaying slope of the magnetic filed by an analytic method. Further on, Zhao et al.(2014) presented the GRB spectrum with the decaying magnetic filed by considered boththe synchrotron and inverse Compton (IC) radiation. The IC component is also importantin generating the low energy spectral index. Duran et al. (2012) argued that the inclusion ofthe IC component will harden the spectrum. But, it is still difficult for α to approaches theobserved value −
1, and the maximal value of α is obtained in an extreme condition. In thispaper, we will show that the α ∼ − − α ∼ −
2. Continuity equation of electrons
Most works investigate the photon spectra in the prompt phase with the assumptionthat the injected electrons have a power-law distribution and cool down instantaneously. Theobserved emissions probably originate from electrons in different emission states and regions 4 –(Duran et al. 2012). After interacting or radiating photons, the spectrum of electrons in theplasma changes. The electron evolution is governed by the continuity equation, which is anreduced version of the Fokker-Planck equation for electrons. The ignored terms are relatedto the dispersion and escaping effects. The Larmor radius of electrons is much smaller thanthe emission region, and the dispersion effect is not important in the GRB scenario. In thiswork, we follow Duran et al. (2012) to solve the continuity equation analytically. In the localjet frame , the continuity equation is written as ∂∂t N ( γ ′ ) + ∂∂γ ′ (cid:2) ˙ γ ′ N ( γ ′ ) (cid:3) = S ( γ ′ ) , (1)where γ ′ denotes the Lorentz factor of electrons. S ( γ ′ ) denotes the energy distribution ofthe source electrons. It is widely accepted that the shock-accelerated electrons have a powerlaw distribution, i.e., S ( γ ′ ) ∝ (cid:26) γ ′− p γ ′ m ≤ γ ′ ≤ γ ′ max , γ ′ < γ ′ m , or γ ′ > γ ′ max . (2)For shock accelerated electrons, γ ′ m = ε e p − p − M p M e Γ (Γ is the bulk Lorentz factor of the jet, ε e is the constant fraction of the shock energy) is the minimal Lorentz factor of electrons (Sari1998). The maximum energy for electrons gained in the shock acceleration via Fermi process γ ′ max is written as γ ′ max ≈ p πe/σ T B ′ ∼ × B ′− / (Kumar et al. 2012; Fan & Piran2008).For a steady case, one sets ˙ N ( γ ′ ) = 0. The electron will lose energy via synchrotronradiation and IC scattering processes. The cooling of the electron is described by − ˙ γ ′ = σ T B ′ γ ′ πM e c + σ KN Lγ ′ πR Γ M e c , (3)where σ KN is the cross section valid in the KN range. Defining η ≡ γ ′ hν seed /M e c ( ν seed isthe frequency of the seed photon), η describes how deep an electron is in the KN regime. Inthe Thomson regime, one has η ≪
1, while one has η ≥ σ KN = σ T f ( η ), f ( η ) is written as (Duran et al. 2012) f ( η ) = 34 (cid:20) ηη (cid:18) η (1 + η )1 + 2 η − ln(1 + 2 η ) (cid:19) + ln(1 + 2 η )2 η − η (1 + 2 η ) (cid:21) . (4)However, one electron loses energy ∼ γ ′ M e c in the KN regime, and the power ofCompton scattering is proportional to γ ′ , rather than γ ′ in the second term of Equation (3) Here we use the prime sign to mark parameters in the jet frame. f ( η ) function, i.e.,˜ f ( η ) ≡ f ( η )1 + η . (5)The general Compton-Y parameter is defined as Y C ≡ ˜ f ( η ) U ′ γ U ′ B , (6)where U ′ γ = L γ / πR Γ c and U ′ B = B ′ / π refer to the photon and the magnetic field energydensity, respectively. Y C describes the ratio between the power of the Compton process P ′ IC and that of the synchrotron radiation P ′ syn in the local frame. Now the cooling rate of theelectron is rewritten as − ˙ γ ′ = γ ′ T ′ syn (1 + Y C ) . (7)where T ′ syn ≡ πM e c/σ T B ′ . In the complex of the synchrotron and IC processes, electronswill lose energy significantly if the cooling time t ′ c is smaller than the dynamical time t ′ dyn ≈ R/ Γ c . When they are the same, the critical Lorentz factor of electrons is expressed as γ ′ c ≈ π Γ M e c σ T RB ′
11 + Y C . (8)Duran et al. (2012) mainly investigated the spectral slope in the low energy range. Thus,the electron spectrum was solved analytically by the continuity equation in the low energyrange. We will complete the analytical solutions of the continuity equation in the full energyrange. We also consider both the fast and slow cooling phases, analogous to the full spectrumof the synchrotron radition (Sari 1998; Fan & Piran 2008).Considering the stationary case, one sets ˙ N ( γ ′ ) = 0. The continuity equation is writtenas ∂∂γ ′ (cid:2) ˙ γ ′ N ( γ ′ ) (cid:3) = S ( γ ′ ) . (9)The analytical solution is given by N ( γ ′ ) ∝ ˙ γ ′− R S ( γ ′ ) dγ ′ . In the fast cooling phase, γ ′ c < γ ′ < γ ′ m , one has N ( γ ′ ) ∝ ˙ γ ′− . The corresponding spectral index of electrons in thisenergy range is given by (Duran et al. 2012) p ≡ (cid:12)(cid:12)(cid:12)(cid:12) d ln N ( γ ′ ) d ln γ ′ (cid:12)(cid:12)(cid:12)(cid:12) = 2 + d ln ˜ f ( η ) d ln η Y C Y C . (10) 6 –The formula of d ln ˜ f ( η ) /d ln η can be approximated in the two limits, i.e., d ln ˜ f ( η ) d ln η ≈ (cid:26) − η η ≪ , − η ≫ . (11)One observes that the value of d ln ˜ f ( η ) /d ln η is always negative. Since the Compton param-eter Y C is always positive, the range of p is 0 < p <
2. This result have a significant impacton the spectral indices of GRBs.In the slow cooling phase, i.e., γ ′ m < γ ′ < γ ′ c , the electrons lose no significant energyvia radiation. So, the spectral distribution of electrons is roughly invariant. This leadsto N ( γ ′ ) ∝ γ ′− p . The distribution of electrons in the high energy regime is also changedaccording to the continuity equation. For γ ′ ≥ max( γ ′ m , γ ′ c ), since the source has the form S ( γ ′ ) ∝ γ ′− p , the corresponding power-law index can be obtained analytically p ≡ (cid:12)(cid:12)(cid:12)(cid:12) d ln N ( γ ′ ) d ln γ ′ (cid:12)(cid:12)(cid:12)(cid:12) = p + 1 + d ln ˜ f ( η ) d ln η Y C Y C . (12)Following the argument previously, the range of p is ( p − < p < ( p + 1). Therefore, ourwork here completes the analysis given by Duran et al. (2012). Collecting these results, thedistribution of electrons is written as N ( γ ′ ) ∝ γ ′− p γ ′ c < γ ′ < γ ′ m ,γ ′− p γ ′ m < γ ′ < γ ′ c ,γ ′− p max { γ ′ m , γ ′ c } < γ ′ < γ ′ max . (13)The cutoff factor γ ′ max is not analyzed here, since it occupies a negligible fraction. Setting Y C = 0, which means that the Compton processes are ignored, Equation (13) reduces to thesynchrotron radiation case. From Equation (13), it is evident that the inclusion of Comptonprocesses changes the spectra of electrons, and further affect the observed photon spectra.
3. The spectra of photons
The photon spectra in the prompt phase of GRB can be studied on the base of theelectron spectrum in Equation (13). First, we discuss the low energy synchrotron spectrum.In the fast cooling phase, the distribution of electrons is composed of two connected powerlaws with indices − p and − p , respectively. It is well known that the flux of synchrotronradiation for the low energy range is proportional to ν / . One way to estimate the flux inthe low energy regime due to the synchrotron is given by Duran et al. (2012), i.e., F syn ν = A ( B ′ ) Z ∞ γ ′ ν dγ ′ N ( γ ′ ) (cid:20) νν ( γ ′ ) (cid:21) / , (14) 7 –where γ ′ ν = p πM e cν/eB ′ , and ν ( γ ′ ) is the frequency corresponding to γ ′ . The integration R ∞ γ ′ ν is done for γ ′ > γ ′ ν . If γ ′ ν is smaller than γ ′ c , then the integration is composed of treeparts, i.e., R γ ′ c γ ′ ν + R γ ′ m γ ′ c + R γ ′ max γ ′ m . The result of the integration is independent of γ ′ ν , since thereis no electrons distributing for γ ′ < γ ′ c . Thus, one always has F ν ∝ ν / . Duran et al. (2012)considered that there is no lower cutoff for the electron spectrum, and obtain a formula ofthe spectral index. In the presence of bounded electron spectrum, their formula is valid onlyfor γ ′ c . We present the formula of the spectral index at this critical frequency, i.e., α c ≡ d ln F syn ν c d ln ν c = 13 − (5 / q ) / − ( p − / ( p + 2 / q )( γ ′ ν /γ ′ m ) / q . (15)where we define q ≡ d ln ˜ f ( η ) d ln η Y C Y C for concision. The formula here agrees with Duran et al.(2012) except that we take use of the formula of p in Equation (12). α c is meaningful todescribing the spectral curvature. But α c can not describe the full spectral slope at theturning point, since only the low energy tail of the synchrotron emission is considered toobtain α .When γ ′ c < γ ′ ν < γ ′ m , the integration contributes to the spectral index. However,Equation (14) counts only a small part of the synchrotron radiation power. The photonspectrum should be derived via the relation F ν dν ∝ N ( γ ′ ) P ′ syn dγ ′ and ν ∝ γ ′ (Fan & Piran2008). By considering these relations and combining the synchrotron self-absorption, thefull photon spectrum in the fast cooling phase is written as F syn ν ∝ ν ν < ν a ,ν ν a < ν < ν c ,ν − p − ν c < ν < ν m ,ν − p − ν m < ν < ν max . (16) ν a refers to the absorption cut off frequency. The presence of Compton process changesthe second index of the spectrum, and this breaks the “line of death” of the index of thesynchrotron radiation.Now we discuss the spectrum in the slow lcooling case. According to Equation (13), thespectral indices of electrons are − p and − p for the low and high energy range, respectively.Therefore, the synchrotron induced spectrum of photons is written as F syn ν ∝ ν ν < ν a ,ν ν a < ν < ν m ,ν − p − ν m < ν < ν c ,ν − p − ν c < ν < ν max . (17) 8 –Compared to Equation (16), we only replace p with p in the slow cooling phase. The typicalvalue of p is larger than 2 for the shock accelerated electrons. We also have 0 < p <
2, seelast section. Thus, the fast cooling case has a harder spectral index than the slow cooling casein the intermediate energy range. For the highest energy range, the spectrum becomes soft.The reason is that the most energetic electrons mainly lose energy via Compton processes,and their synchrotron emission is not as efficient as the pure synchrotron radiation case.The Compton process will produce a spectrum in the energy range above the synchrotronspectrum. The source photons of the Compton processes can be of the synchrotron origin.In this case, our scenario mainly discusses the synchrotron self-Compton (SSC) process. Weestimate the energy of scattered photons in order to understand whether these photons canbe observed in GRBs. The typical bulk Lorentz factor of GRB jets in the prompt phase isseveral hundreds. The observed peak energy of most GRBs is about hundred keV. In thelocal jet frame, the energy of synchrotron photons peaks at the order of keV. Consideringthe moderate shock acceleration efficiency ǫ ∼ .
1, and p = 2 .
5, one obtains γ ′ m ∼ η parameter is calculated to be η ∼
12. So the SSC process happens in the KN regime.The scattering process will transit half of the electron energy to the photon. Thus, theobserved photon energy can be as high as 150 GeV, which can be recorded by the LATmonitor. However, such high energy photons can be attenuated by three significant effects.The first one is due to the cross section in the KN range, which strongly suppresses thecollision chance. Secondly, the optical depth for such high energy photons can be large inthe prompt phase (Boˇsnjak & Kumar 2012; Beloborodov 2010; Chang et al. 2012a). Thiswill lead to the time lag phenomenons for GeV photons in GRB Abdo et al. (2009a,b);Ackermann et al. (2010, 2011). The production of such photon collision is pairs of electronand position. This may enlarge the radiation populations in the jet slightly. The third effectis caused by the extragalactic background lights (EBL). The absorption can be described bythe model dependent gamma-ray opacity (Archambault et al. 2014). This effect suggests usthat GRBs can enlarge the positron population in the universe. All these effects reduce thechance of VHE photons to arrive the detector.The power of the Compton processes in the GRB is about the same as the synchrotron, asindicated in the next section. Although the cross section suppresses the collision probability,the Compton scattering can not be ignored only by this reason. The Compton inducedspectrum is most evident in blazars, which is also produced the radiation in jets of AGNs.The spectral energy distribution (SED) of Blazars shows a bimodal pattern (Bonnoli et al.).The first peak in SED, usually in the optical to the soft X-ray range, is due to the synchrotronradiation, while the second peak in the high energy range is of the SSC or external Comptonradiation. If electrons in the jet have a complex spectrum as in Equation (13), one canexpect that the spectral indices of photons are strongly related. Since we concern the GRB 9 –spectrum in this work, the study of the complex spectrum for blazars will be given in anotherwork (Jiang et al., in prep)
4. GRB spectra
Most GRB spectra are described successfully by the empirical Band function. Now, weaim to obtain the spectral indices of the Band function by using the F ν spectra in Equations(16) and (17). The empirical Band function is a smoothly jointed broken power law, whichreads N ( E ) = A × ( ( E ) α e − ( α +2) EEp , E < E c , ( E ) β e β − α ( E c ) α − β , E ≥ E c . (18) A is the normalization factor at 100 keV in units of photons s − cm − keV − (Yu et al. 2014).The relation of E p and E c is E p = E p ( α − β ) / (2 + α ). Some bursts show a double brokenpower law spectra. In order to analyze such anomalous spectra, we introduce the thirdspectral index γ . Following Yu et al. (2014), the empirical function for the double brokenpower law is written as N ( E ) = A × ( EE n ) α , E < E b , ( E b E n ) α − β ( EE n ) β , E b ≤ E < E b . ( E b E n ) α − β ( E b E n ) β ( EE n ) γ , E ≥ E b . (19)Here A is the normalization flux at certain energy E n .By the relation E N ( E ) ∼ νF synν , one obtains the expression of α , β and γ as functionsof p and p in both the fast and the slow cooling phase. There is still another phase that ν c and ν m is too close to distinguish, which is named as the “marginal” phase or the “both”phase (Yu et al. 2014). Since p and p are functions of p and q , one can express p and q interms of α and β for a given phase. In this way, p and q can be obtained from observations.For the Band function, one remaining important question is that E p corresponds towhich frequency, ν m or ν c . Since E p is the peak energy of the νF ν spectrum, one concretecriteria is that the spectral index of νF ν spectrum is positive for E < E p and negative for E > E p . Note that this criteria is not valid for the double broken power law. In the fastcooling phase, E p is equal to hν m , because 0 < p <
2. Meanwhile, p should be larger than3. This can be used as a consistent condition to constrain the physical parameters. In theslow cooling phase, E p is produced by electrons with γ ′ m for p >
3. If p < E p correspondsto hν c . So, there are two possible scenarios to explain the Band function in the slow coolingphase. A double broken power law is possible in both the fast and slow cooling phases. Both 10 – E b and E b are uniquely determined if the absorption part is not included in the spectra.We will not discuss the synchrotron absorption effect unless some extreme hard low energyspectrum needs explanation. Collecting all these results, formulae of spectral indices aregiven in Table 1 and 2.
5. Applications5.1. The α ∼ − problem With the known expression of spectral indices, we would like to discuss the α ∼ − α ∼ −
1. Since p > α in the slow cooling “a” case is less than − /
2. Thus, we numericallyinvestigate the p value for different parameters. We define a new parameter, i.e., ξ ≡ U ′ γ U ′ B = 2 L γ R Γ cB ′ , (20)which describes the ratio of the source photon energy density over the magnetic field energydensity. We then plot p as a function of η for different values of ξ in Figure 1. It is evidentthat the minimal value of p decreases when ξ increases. This means that the Comptonprocesses will strongly flatten the electron spectrum in the low energy range. Note that theminimal value of p can be 0 if ξ goes to infinity. Correspondingly, the value of α in the bandspectrum can reach − .
5. Duran et al. (2012) considered α in Equation (15), and pointedout that the extreme value of α is −
1. In our analysis, α ∼ −
1, or equally p ∼
1, is not alimit but a median value in the whole parameter spaces. The range of p can be used to fitmore GRB spectra.In order to show the parameter space more clearly, we plot the contour of p in Figure 2,where the x and y axes are log η and log ξ , respectively. The green region shows a “boomerang”pattern in Figure 2, denoting for the p ∼ ξ canbe several tens up to 10 n ( n ≥
4) when η is several. In the right wing, log η and log ξ showa linear relation. We also want to know the value of Compton parameter of Y C in the sameparameter space, since it denotes which radiation mechanism is dominant. The contour plotof Y C is given in Figure 3. One can observe that Y C is larger than 1 in the warm red andyellow color region, and is very small in the green and blue region. The separating zonebetween the warm and cold region is very close to the right wing of the boomerang, if Figure3 and 2 are overlapped. So, p can vary significantly when powers of the synchrotron andCompton scattering are comparable.One question arises naturally, if these parameter values reflect the true physical con- 11 –Table 1. Spectral indices in Band functionSpectral indices α β S E p p q Fast cooling − p +12 − p +12 p − hν m S + 1 − α − − p +12 − p +12 p − p hν c − α − − S − − − p +12 p − hν m S + . . . Marginal case − − p +12 p − hν m,c S − − q . . . Note. — The parameter S is defined as S ≡ α − β . In the “Slow cooling a” case,one has p <
3; while one has p > p and q .Table 2. Spectral indices in the double broken power lawSpectral indices α β γ p q E b E b Fast cooling − − p +12 − p +12 β − γ ) + 1 − β − hν c hν m Slow cooling − − p +12 − p +12 − β − β − γ ) − hν m hν c Note. — A double broken power law is possible in both the fast and slow cooling phases. p denotes the spectral index of injected electrons. q denotes the correction from the Comptonprocesses. E b and E b are the broken energies in the double broken power law spectrum. 12 –ditions in GRBs. We estimate the η and ξ values for typical GRBs. η has already beenestimated to be 12 in the last section. Here, we choose to calculate η = γ ′ p E ′ p /M e c forreference, i.e., η ≈ z ) γ ′ p, Γ − (cid:18) E p . (cid:19) . (21)The convention Q = 10 n Q n is taken. The most common observed E p is 300 keV (Goldstein et al.2012; Gruber et al. 2014). In the fast cooling phase, γ ′ p is the minimal Lorentz factor γ ′ m ofinjected electrons, which is a free parameter unless some specific acceleration mechanismsare specified. For the shocked electrons, one has γ ′ m ∼ = 610 ε e Γ (Sari 1998). Thus, η is in theorder of 10 for ε e ∼ .
1. In the synchrotron radiation, the magnetic field B ′ can be estimatedas B ′ ≈ . × (1 + z )Γ − γ ′− p, (cid:18) E p . (cid:19) Gauss . (22)Once the magnetic field B ′ is known, ξ is calculated from the definition, i.e., ξ ≈ . × − (1 + z ) − L R − γ ′ p, (cid:18) E p . (cid:19) − . (23)It seems that ξ is very small. This can be realized in the magnetic dominated jet model(Chang et al. 2012b). The way to increase ξ is quite limited. R is constrained by the variationtime of GRBs. E p and L γ are observed variables. When we increase γ ′ p , ξ is significantlyenlarged. But, this also increases the value of η . One can avoid to increase η by consideringthe low energy external photons, which is the case in the external Compton model. In Figure1, one observes that p is smaller than 2 unless log ξ is positive. Therefore, the left wing ofthe green zone is difficult to be located in the SSC model.In the external Compton (EC) model, by increasing γ ′ p and decreasing E ′ source in the sametime, one can realize that η is unchanged while ξ is greatly enhanced. When γ ′ p ∼ × and E source ∼
10 keV, one has η ∼
10 and ξ ∼ B ′ ∼
10 Gauss,which may be the shocked magnetic field in the jet. The resulted p is in the green zone.Correspondingly, the Y C contour tells us that the power of Compton is roughly the samewith that of synchrotron, which only doubles the energy budget. By this constraint, we candiscard the left arm of the boomerang, where the corresponding Y C is much larger than 1.In some bursts, a black-body (BB) component plus the Band function can better fit theobserved spectra Yu et al. (2014). The BB bump in the spectra is found to be 10 keV.Combining these information, the prompt emission of GRB can be explained by the reverseshock plus photosphere model. A reverse shock compresses the magnetic field and accelerateselectrons in the jet, then electrons collide with the thermal photons and give synchrotron 13 –radiation in the same time. This produces the observed E p and low energy spectral index.The high value of γ ′ p may also be due to particle acceleration in the magnetic reconnec-tion (Zhang & M´esz´aros 2002). So, photospherical emission plus the magnetic reconnectionmodel is a also a hopeful theory to explain the α ∼ − γ ′ p increases to 10 , larger ξ and η can be realized. In Equation (23), ξ dependson γ ′ p . One observes that the p = 1 contour is a straight line for log η >
2. One needs afine tuning of γ ′ p to land on the green zone. This strongly constrains all the relevant physicalparameters. The relation between log ξ and log η can also be described by a straight line.One finds that log ξ = 4log η + b , where b ∼ −
11 denotes all other parameters except γ ′ p inEquations (21) and (23). We have marked this line with the white color in Figure 2. Thecrossing point of these two lines will determine the precise value of γ ′ p . We figure out roughlythat p locates at 1 when γ ′ p ≈ . × . The corresponding coordinate of this point is(log η, log ξ ) = (5 . , .
69) in Figure 1. The same point in Figure 3 corresponds to Y C ≈ p ∼ Y C ∼ p = 1.Substituting γ ′ p ≈ . × back into Equations (21) and (22), one obtains that B ′ ≈ − Gauss and η ≈ . × . The Compton scattering processes are in the deep KN regime.In the circum-stellar medium (CSM), B ′ is at the order of µ -Gauss (Kumar 2009). Theshocked magnetic field can be amplified to hundred times larger. So, the derived magneticfield B ′ coincides with external shock model. With the price of γ ′ p ∼ . × , the complexradiation mechanism can be a reasonable solution to the α ∼ − γ ′ m challenges the shock acceleration mechanism of electrons. The highenergy electrons may be produced by the strong coupling between electrons and protons inthe explosive jet, where the temperature is extremely high (Zhang & M´esz´aros 2002). Thenthese electrons are accelerated by the forward shock. A magnetic dominated outflow collidingwith the CSM offers necessary gradients to explain all parameters obtained here, and needsfuture studies.We have used the typical parameters of GRB to calculate, but they are different fromburst to burst. α is also different from burst to burst. The inclusion of Compton processesoffers a new method to fit the observed spectra. Recently, it was found that α ∼ − . α ∼ −
1, since − − / − /
2. Our analysis here also indicates that α can be any value between − / − /
2, if Compton processes are included.
In Table 1, we have four possible phases available to explain the GRB spectra. Phasetransitions between them may occur when γ ′ m and γ ′ c vary with time. γ ′ m is determined byhow electrons are injected, i.e. the acceleration processes. Except the slow cooling phase a, γ ′ m is related to the peak energy E p . By the Equation (20), the peak energy E p of synchrotronorigin is expresses as E p = 11 + z √ ~ q e M e c / ξ − / L / γ R − γ ′ m . (24)Here we include the redshift to count for the cosmological effects. Obviously, one observesthat E p ∝ L / γ , which is the Yonetoku relation (Yonetoku et al. 2004). If we change L γ tothe isotropic energy E iso , this relation is the Amati relation (Amati et al. 2002, 2008). TheAmati relation is evident by statistics of many bursts (Amati et al. 2008; Yu et al. 2014),other parameters should be similar among different bursts. In the fast cooling and the slowcooling b phases, one can estimate γ ′ m by the following relation, γ ′ m = (1 + z ) / ( M e c / ~ q e ) / ξ / L − / γ E / p R / . (25)In the “slow cooling a” case, E p is related to γ ′ c . The critical Lorentz factor in Equation (8)is calculated to be γ ′ c = 3 πM e c σ T
11 + Y C ξL − γ Γ R, (26)which is obtained by requiring t ′ cool ∼ t ′ dyn . The electrons with γ ′ c radiate synchrotron photonswith energy E c , i.e., E c = 9 √ π ~ q e M e c / σ T
11 + z (cid:0)
11 + Y C (cid:1) ξ / L − / γ Γ R. (27)One observes that E c ∝ L − / γ . E c strongly depends on Γ, so it can vary fast during oneburst. This relation is used to explain the anti-relation between the low energy peak andthe bolometric luminosity in blazars, if the E c is the observed low peak energy (Lyu et al.2014).Both γ ′ m and γ ′ c can be changed during the prompt emission. γ ′ m signifies the injection ofthe photon energy, which marks the kinematic of the jet. It can be increased in the beginning 15 –pulse of the burst. At the afterglow phase, no more energy is injected, γ ′ m will be decreased.Assuming γ ′ c is invariant for a steady state, the variation of γ ′ m leads to the phase transition.Most probably, a slow to fast transition occurs in the flux arising phase, and a fast to slowphase transition happens in the flux decaying phase.The variation of γ ′ c depends on the radiation power. Suppose that the system is in aslow cooling phase in the beginning, which means γ ′ c > γ ′ m . And the synchrotron radiationis in a steady state. As the photon density increases (the SSC case), the Compton scatteringbecomes important. The Compton scattering will increase the radiation efficiency, and γ ′ c becomes smaller, see Equation (29). When γ ′ c is smaller than γ ′ m , a slow to fast phasetransition will occur. Electrons with γ ′ m will quickly radiate away their energy, and cooldown to the energy γ ′ c . Thus, the major population of electrons will accumulate at γ ′ c . Othercases of phase transition are also possible if γ ′ m and γ ′ c both vary. All these phase transitionsare indicated by the change of spectral indices and the peak energy. They can be used toexplain the time-resolved spectra in many bursts. With the observed α , β and E p , one cananalyze the time resolved spectra of GRBs.As a concrete example, we consider the time resolved spactrum of GRB 080916C. GRB080916C is a long burst well known for its high redshift z ∼ .
35 and extreme luminosity L γ ∼ erg (Abdo et al. 2009a). The minimal bulk Lorentz factor was estimated to bearound 600 to meet the optical depth constraint τ γγ < t , the emission radius R is estimated to be around ∼ cm. Thepeak energy evolves in time. In the first time interval a (0 ∼ .
58 s since the trigger time), E p is about 440 keV. It goes up to 1 . . ∼ .
68 s), andthen decreases. With these parameters, one can obtain that γ ′ p ≈ . × (1 + z ) / ξ / ( E p / L − / γ, R / , (28) γ ′ c ≈ . × − ξ
11 + Y C L − γ, Γ R . (29)The minimal Lorentz factor γ ′ m meets the expected value of Lorentz factor for shockedelectrons, i.e., 610 ε e Γ ∼ . × ε e, − Γ (Sari 1998). The critical Lorentz factor γ ′ c is muchless than one. This is misleading since γ ′ c ≥ γ ′ c depends onthe assumption t ′ cool ∼ t ′ dyn . If t ′ cool ≪ t ′ dyn , γ ′ c can be much larger than one, or even largerthan γ ′ p . The phase of the system is undetermined by just comparing the derived values.In the time interval a, α and β are observed to be − . ± .
04 and − . ± . α and β are − . ± .
02 and − . ± . p = 5 . ± .
32 and q = − . ± .
08. Thespectral index of electrons is very large. Considering the slowing cooling b phase, one obtainsthat the spectral index of electrons is p = 4 . ± .
24. In the marginal phase, one obtains p + q = 3 . ± .
24. If Compton process is not significant, or equally q ≈
0, one has p = 3 . ± .
24. In the second time interval, the emission zone are most probably in the fastcooling phase. With known α and β , one obtains that p = 3 . ± . q = − . ± . p , the variation of q can account for the spectralindices in both the first and second time intervals. In both these two phases, E p correspondsalways to γ ′ m . The revolution of E p agrees with that of the time resolved luminosity. So, wehave a high level confidence to claim that a phase transition occurs in the prompt emissionof GRB 080916C, i.e., transition from the marginal phase to the fast cooling phase. The time resolved spectra offers us more information on the understanding of the ra-diation mechanism of the prompt emission. Yu et al. (2014) presented the time resolvedspectra of eight most energetic bursts with high signal-to-noise level. The distributions ofspectral indices and E p peak at − . − .
13, and 374 . β ∼ E p , α ∼ E p , and F p ∼ E p relations for the three bursts, respectively. The peak flux F p is calculated by F p = E p N ( E p ), see Equation (18). We use the linear relation y = a + bx to fit the these relations, and results are presented in notes of figures. According to theserelations, the phases of spectra will be analyzed in the following.GRB 100724B is composed of two pulses. Note that the second pulse is composed of 17 –several relative small peaks in the light curve, but we still consider them to be in one pulsefor convenience of classification. The plots in Figure 4 indicate that β becomes smaller when E p increases, while α is almost independent of E p . The small slopes of two fitting lines for β ∼ E p relation tell us that the Compton component may exist but not signify. The value of α also approaches − / hν c and hν m can be reflected by the F p ∼ E p relations, which are not well linearly fitted.In the marginal phase, we can not determine p and q in principle, but we can obtain theirconstrains via the relation p + q = − β −
2. The relation of p + q versus E p is plotted inFigure 6. The linear fittings are not good for the second pulse, this probably is due to ourrough classification of pulses.GRB 100826A contains one big pulse and one small pulse. The plot of β ∼ E p relationindicates that β decreases when E p increases, and also α have the same trend. The F p ∼ E p relation reveals that E p is proportional to the E p N ( E p ) flux, which is a sign of the fastcooling phase. Thus, both pulses of GRB 100826A are in the fast cooling phase. In thecomplex radiation mechanism, the difference between α and β , i.e., S = α − β is invariantin principle. We plot the p ∼ E p relation in Figure 9, where p denotes the spectral index ofinjected electrons. It is evident that p remains almost invariant when E p varies. The q ∼ E p relation is also plotted in Figure 9. Except some exceptional points, there is a significantcorrelation between q and E p . The intercepts of the fitting lines are near −
2. The two lineshave different slopes, which indicates the intrinsic physical parameters are different for thetwo pulses. The correlation between q and E p indicates that the Compton scattering is inthe deep KN regime. The different trend of p and q dependence on E p can be considered asan evidence for the complex radiation mechanism.The light curves in GRB 130606B, see Fig.A.2. in (Yu et al. 2014), have four pulses intotal. In Table 3, we combined the latter two pulses as one. This may bring some large errorsin linear fitting. In Figure 10, β and α have different dependence on E p for three pulses. Inpulse 1, the α and β are independent of E p , and α approaches − /
3. The F ∼ E p relationillustrates that pulse 1 is in the slow cooling b or marginal case. In the slow cooling b case,the injected electrons should have a spectral index p >
3, this phase can not be excluded.The nearly zero slope of the linear fitting of β ∼ E p relation indicates that the synchrotronradiation is dominant. In pulse b, both β and α show a slightly increasing trend when E p increases. However, the linear fitting has a large error. The most possible phase in pulse 2is the fast cooling with the pure synchrotron radiation, since the F ∼ E p relation shows anice linear fitting, see Figure 11. Pulse 3 have a large error correction for the β ∼ E p fitting,while α has the similar behavior with pulse 1. The F ∼ E p relation indicates a good linearfitting with slope 1 . ± . F p ∼ E p relations indicate that E p mostly corresponds to hν m ,the slow cooling a phase is not evident in these three bursts. The spectra of GRB 100826Acontains the feature of complex emission in a high significant manner. Such kind of feature isless evident in GRB 100724B and GRB 130606B. Frontera et al. (2012) found the correlationbetween α and E p in GRB 980329, which is also a burst with single one pulse. One alsoobserves that the flux rising phase and flux decaying phase have no differences in theserelations during one pulse. This hints us that it is a better method to classify the timeresolved spectra according to pulses. The F p ∼ E p relations in their work also show the nicelinear fittings in GRB 970111, GRB 980329, GRB 990123, and GRB 990510 (Frontera et al.2012). Our results agree with theirs. The marginal phase seems to be popular in the bursts.In this case, γ ′ m and γ ′ c is close to each other. Yu et al. (2014) also revealed that there is auniversal break ratio between ν m and ν c , which is less than 10. Since γ ′ m corresponds to theacceleration processes, while γ ′ c is related to the emission power of electrons, the agreementof them indicates the balance between the the acceleration and radiation processes. This isreflected by the fact that the Band function is suitable to describe the spectra in differenttime slices during one burst.
6. Discussion and conclusion
By considering both the SR and the Compton processes, the continuity equation ofelectrons is investigated. We find that the spectrum of electrons is a broken power law inboth the fast and slow cooling phases. The inclusion of the Compton processes changes thespectral indices of photons. Analytical expression of spectral indices in four different phasesof GRBs is given, which enriches the theoretical implications of the Band function. The α ∼ − β ∼ E p and α ∼ E p shows the signature of the complex radiationin the fast cooling phase, especially in GRB 100826A. The pure synchrotron radiation can notexplain the revolution of spectral indices. Frontera et al. (2012) has reported the dependenceof α on E p in GRB 980329, which is also of one pulse burst. We guess that the spectral indicesdependence on the peak energy can be found in the pulse dependent spectra. Titarchuk et al.(2012) argued that the GRB spectra are formed two upscattering processes. The resultedspectral index of Comptonization has relations with the bulk motion of the outflow and theotical depth. Such model also predicts a correlation between E p and L iso for time resolvedspectra(Frontera et al. 2012). However, β and α are not related in the Comptonizationtheory. This can be considered as a special feature of the complex radiation in the fastcooling phase.In the complex radiation scenario, the derived p for the injected electrons is larger than3, e.g., p ≈ . p is from 2 . . p canbe as hard as 1. The analytical way to obtain this hard spectra is also solved by thecontinuity equation in the unsteady state, where the injection, acceleration and escapingterms are considered. It was shown that the first Fermi acceleration can not produce apower law spectra without the injection (Guo et al. 2014). These numerical results are notverified in observation, especially from GRB. The conflict here is a two sides thing. Oneside is that the soft injected spectrum questions the shock acceleration paradigm of GRB,specific physical configurations are needed to explain the steeper spectral index. Even inthe pure synchrotron radiation, the derived p from Band fitting does not agree with theshock acceleration completely. The other side is that our analysis is questioned. In ourderivation of the continuity equation, we consider only the single power law injection. Ifthe injected electrons have a broken power law spectrum, the resulted photon spectrum isextended to double broken power law. Then, the Band function may be fitted in the shockacceleration model. The more complicate spectrum can be obtained by complicating theinjected spectrum. We did not consider the acceleration term and the time evolution term.If these terms are included, the power law solution still exists from analytical argument, andother corrections to the index can be expected.In our work, the Compton induced spectra are not discussed. Roughly, the spectralindex in the high energy range induced by Compton processes are p , since the electrons share 20 –their energy with photons. The observation of the Compton induced spectrum is difficult inGRBs, see Section 3. One can expect that this high energy spectral index depends on thesynchrotron peak energy and the IC peak energy. The Compton spectra are most evident inblazars. Abdo et al. (2010b) found a harder-brighter tendency in the gamma ray band forcertain specific objects of the subclasses of blazars, i.e., FSRQs and LBLs. Lyu et al. (2014)indicated that FSRQs and LBLs are in the slow cooling phase, because there is an anti-correlation between luminosity and peak energy. Combining them together, one can inferthat the spectral index in the gamma ray band becomes softer when the peak energy increasesfor FSRQs and LBLs. For a sample of bright blazars, a correlation between the spectral indexand the peak energy was found (Abdo et al. 2010a). The linear fitting of the correlation islog ν ICp = −
4Γ + 31 . p values for different ξ ξ =0.1 ξ =1 ξ =10 ξ =100 ξ =1000 ξ =0.1 ξ =1 ξ =10 ξ =100 ξ =1000 ξ =0.1 ξ =1 ξ =10 ξ =100 ξ =1000 ξ =0.1 ξ =1 ξ =10 ξ =100 ξ =1000 ξ =0.1 ξ =1 ξ =10 ξ =100 ξ =1000 −2 −1 0 1 2 3ln η p Fig. 1.— The index p as a function of η for ξ = 0 . , , ,
100 and 1000, respectively. 22 –Fig. 2.— The contour plot of p in the log η and log ξ plane. 23 –Fig. 3.— The contour plot of Compton Y C parameter. The numbers in the contour linesdenote values of log Y C . 24 –Table 3. The pulses of GRBsGRB Pulse Time interval (s) Number of spectra Phases100724B 1 − . ∼ .
089 10 Marginal2 41 . ∼ .
458 23 Marginal100826A 1 − . ∼ .
792 36 Fast cooling2 40 . ∼ .
549 14 Fast cooling130606B 1 − . ∼ .
459 7 Slow cooling b/Marginal2 11 . ∼ .
218 11 Fast cooling3 26 . ∼ .
824 13 Slow cooling b/MarginalNote. — GRB pulses and their phasesFig. 4.— The plots of β and α versus E p in GRB 100724B. In the left panel, the fitting linesare described by β = − . ± . − (0 . ± . E p (black line) and β = − . ± . − (0 . ± . E p (red line) in pulse 1 and 2, respectively. In the right panel, thelinear fittings of α are given by α = − . ± . − (0 . ± . E p (black line) and α = − . ± . − (0 . ± . E p (red line), respectively. The value of E p is given inunit of MeV in convention. 25 –Fig. 5.— The plot of F p ∼ E p relation in GRB 100724B. The fitting lines are given bylog F p = − . ± .
197 + (1 . ± . E p (black) and log F p = − . ± .
181 +(2 . ± . E p (red), respectively. 26 –Fig. 6.— The plot of p + q versus E p relation in GRB 100724B. In pulse one, the linearfitting is given by p + q = 0 . ± .
130 + (0 . ± . E p , In pulse two, the fitting line isgiven by p + q = 1 . ± .
237 + (0 . ± . E p . 27 –Fig. 7.— The plot of β and α versus E p relation in GRB 100826A. In the left panel, thefitting lines are described by β = − . ± . − (0 . ± . E p (black) and β = − . ± . − (1 . ± . E p (red), respectively. In the left panel, the linear fittings of α are givenby α = − . ± . − (0 . ± . E p (black) and α = − . ± . − (2 . ± . E p (red), respectively. 28 –Fig. 8.— The plot of F p ∼ E p relation in GRB 100826A. The fitting lines are given by log F p = − . ± .
077 + (1 . ± . E p and log F p = − . ± .
151 + (1 . ± . E p ,respectively. 29 –Fig. 9.— The plots of p and q versus E p relation in GRB 100826A. In the left panel, theline fittings of p versus E p are described by p = 3 . ± .
196 + (0 . ± . E p and3 . ± . − (1 . ± . E p in pulse 1 and 2, respectively. In the right panel, the linearfitting of q are given by q = − . ± .
135 + (1 . ± . E p and q = − . ± .
233 +(4 . ± . E p in pulse 1 and 2, respectively.Fig. 10.— The dependence of β and α on E p in GRB 130606B. In the left panel, thethree fitting lines in pulse 1, 2, and 3 are β = − . ± . − (0 . ± . E p , β = − . ± .
360 + (0 . ± . E p , and β = − . ± . − (0 . ± . E p , respectively.In the right panel, the fitting lines of α are given by − . ± . − (0 . ± . E p , − . ± .
136 + (0 . ± . E p , and − . ± . − (0 . ± . E p in pulse 1, 2, and3, respectively. 30 –Fig. 11.— The plot of flux on ∼ E p in GRB 130606B. The three fitting lines are described bylog F p = − . ± . . ± . E p , log F p = − . ± . . ± . E p ,and log F p = − . ± .
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