Gamma-ray polarization of synchrotron-self-Compton process from a highly relativistic jet
aa r X i v : . [ a s t r o - ph . H E ] O c t Gamma-ray polarization of synchrotron-self-Compton processfrom a highly relativistic jet
Zhe Chang , , Hai-Nan Lin , ∗ Institute of High Energy PhysicsChinese Academy of Sciences, 100049 Beijing, China Theoretical Physics Center for Science FacilitiesChinese Academy of Sciences, 100049 Beijing, China
Abstract
The high polarization observed in the prompt phase of some gamma-ray bursts(GRBs) arouses extensive studies on the emission mechanism. In this paper, we inves-tigate the polarization properties of the synchrotron-self-Compton (SSC) process froma highly relativistic jet. A magnetic-dominated, baryon-loaded jet ejected from thecentral engine travels with a large Lorentz factor. Shells with slightly different veloc-ities collide with each other and produce shocks. The shocks accelerate electrons topower-law distribution, and at the same time, magnify the magnetic field. Electronsmove in the magnetic field and produce synchrotron photons. The synchrotron photonssuffer from the Compton scattering (CS) process and then are detected by an observerlocating slightly off-axis. We derive analytically the formulae of photon polarization inthe SSC process in two magnetic configurations: magnetic field in the shock plane andperpendicular to the shock plane. We show that photons induced by the SSC processcan be highly polarized, with the maximum polarization Π ∼
24% in the energy band[0 . ,
5] MeV. The polarization depends on the viewing angles, peaking in the plane per-pendicular to the magnetic field. In the energy band [0 . , .
5] MeV, in which most γ -ray polarimeters are active, the polarization is about twice of that in the Thomsonlimit, reaching to Π ∼ Subject headings: gamma-ray burst: general – polarization – radiation mechanism:non-thermal – scattering
1. Introduction
Recent polarimetric observations in the prompt phase (Coburn & Boggs 2003; McGlynn et al.2007; Kalemci et al. 2007; G¨otz et al. 2009; Yonetoku et al. 2011, 2012), as well as in the optical * [email protected] cir < . cir = 0 . ± . ±
20% (Coburn & Boggs 2003). Kalemci et al. (2007) reported thepolarization as high as 98% ±
33% in GRB 041219A, making it to be the most highly polarizedGRB that has ever been observed, despite the low statistical significance. McGlynn et al. (2007)re-checked the data of GRB 041219A, and found that the polarization is anti-correlated with pho-ton energy. The analysis on the data of GRB 110301A and 110721A also shows certainly polarized,with polarization 70% ± . σ ) and 84 +16 − %(3 . σ ), respectively (Yonetoku et al. 2012). Thetemporal variabilities of polarization have also been detected in some GRBs, such as GRB 041219A(G¨otz et al. 2009) and 100826A (Yonetoku et al. 2011). Especially, the polarization angle of GRB100828A shows a change of ∼ ◦ between two adjacent time intervals (Yonetoku et al. 2011). Inspite of large uncertainties exist, high polarization in the prompt phase of GRBs is still possible.The next generation gamma-ray polarimeter POLAR on board the Chinese Space LaboratoryTian-Gong II is expected to provide more polarimetric data with unprecedented precision.There are many theoretical interpretation for the origin of polarization. One of the mostpromising mechanisms to produce highly polarized photons is the synchrotron radiation. Forisotropic electrons whose energies follow the power-law distribution (with index p ), the maximumpolarization of synchrotron photons is well-known to be Π syn = ( p + 1) / ( p + 7 / comp = (1 − cos θ sc ) / (1 + cos θ sc ), where θ sc is the scatteringangle of the photon (Rybicki & Lightman 1979). Although photons can be completely polarized atthe specific angle θ sc = 90 ◦ , the probability is small since the cross section is minimum at this angle.Lazzati et al. (2004) investigated the polarization properties of an isotropic photon field scatteredby an electron jet (the so-called Compton drag model), and showed that the polarization, in somespecial cases, can be as large as that in the point-source limit. However, they only discussed inthe Thomson limit, and the seed photons were assumed to be unpolarized. In the energy bandsin which most γ -ray polarimeters are active (e.g., [50 , http://polar.ihep.ac.cn/cms/. ∼ γ -ray polarization in the SSCprocess from a highly relativistic jet. The SSC process is a natural prediction of the magnetic-dominated jet model (M´esz´aros & Rees 2011; Veres & M´esz´aros 2012). According to this model, ahighly relativistic and magnetized jet, which contains shells with slightly different velocities, ejectsfrom the central engine. Different Shells collide with each other and produce shocks. The shocksaccelerate electrons to be power-law distribution, and at the same time, magnify the magnetic field.Electrons moving in the uniform magnetic field radiate synchrotron photons, which followed byphoton-electron collision before escaping from the jet. Starting from the differential cross section ofphoton-electron scattering, we analytically derive the polarization of a photon after being scatteredby any electron. Then we integrate over the spectra of electrons and photons, thus the polarizationof the SSC process is obtained. We will show that, for isotropic and power-law electrons, photonsinduced by the SSC process can have high net polarization. The Klein-Nishina effect contributessignificantly to the polarization of the prompt emission of GRBs.The rest of the paper is arranged as follows. Section 2 is devoted to a short review on thephoton polarization in the CS process. In section 3, we calculate the polarization properties of the 4 –SSC process from a highly relativistic jet in two magnetic configurations. Finally, discussions andconclusions are given in section 4.
2. Compton scattering process
In this section, we will give a short review on the polarization properties of the CS process. Thissection includes two subsections. In subsection 2.1, we consider the process of a photon scatteredby an electron moving with any velocity, while subsection 2.2 deals with the isotropic electron case.The formulae presented in this section are valid not only in the Thomson region, but also in theKlein-Nishina region.
Suppose a photon with energy ε collides with an electron traveling with any velocity. Inthe laboratory frame, the electron initially goes along the ˆ l direction. After being scattered, thephoton is scattered to the ˆ n direction. We set a Cartesian coordinate system in the laboratoryframe, such that its origin is fixed to the collision point, the z -axis is along the direction of incidentphoton, the y -axis is in the scattering plane (the plane in which the incident and scatted photonsare contained), and ˆ x = ˆ y × ˆ z . In this coordinate system, the moving direction of the scatteredphoton can be written as ˆ n = sin θ sc ˆ y + cos θ sc ˆ z , (1)where θ sc is the scattering angle, i.e., the angle between the directions of incident and scatteredphotons. The direction of the incident electron can be written asˆ l = sin θ cos ϕ ˆ x + sin θ sin ϕ ˆ y + cos θ ˆ z , (2)where θ and ϕ are the polar and azimuthal angles of the incident electron, respectively. Theconservation of the four-momentum gives a constraint on the energy of the scattered photon, thatis (Akhiezer & Berestetskii 1965), ε = ε (1 − β e cos θ ) ε γ e m e c (1 − cos θ sc ) + (1 − β e cos θ ) , (3)where γ e is the Lorentz factor of the incident electron, β e = p − /γ e is the velocity of theelectron, and θ is the angle between ˆ n and ˆ l .The polarization of the scattered photon can be deduced starting from the differential crosssection of photon-electron scattering. The polarization-dependent cross section of photon-electronscattering, is given in the laboratory frame as (Berestetskii et al. 1982; Chang et al. 2014a,b) dσ = 14 r e d Ω (cid:18) ε ε (cid:19) (cid:20) F + F ( ξ + ξ ′ ) + F ξ ξ ′ + F ξ ξ ′ + F ξ ξ ′ (cid:21) , (4) 5 –where r e = e /m e c is the classical electron radius, d Ω = sin θdθdϕ , and the quantities F a ( a =0 , , , ,
33) are functions of the initial and final states of the photon and electron. The expressionof F a can be found at Chang et al. (2014a,b).The Stokes parameters, ξ i and ξ ′ i ( i = 1 , ,
3) in Eq.(4), stand for the polarization propertiesof the incident and scattered photons, respectively. The linear polarization of a photon can becompletely described by ξ and ξ . The remaining parameter, ξ , is a representation for thecircular polarization, positive for right-handed and negative for left-handed. As is mentioned in theintroduction, the circular polarization of GRB photons is small, so we will ignore it in this paper.The polarization of the incident photon, Π , can be conveniently written in terms of the Stokesparameters as Π = p ( ξ ) + ( ξ ) + ( ξ ) . (5)On the contrary, given a photon with linear polarization Π , we can write its Stokes parameters as ξ = Π sin 2 χ , ξ = 0 , ξ = Π cos 2 χ , (6)where the polarization angle χ ∈ [ − π/ , π/
2] is the angle between the polarization vector and the x -axis in the xy plane.From the quantum electrodynamics, we can derive the Stokes parameters of the secondaryphoton. They are given as (Berestetskii et al. 1982) ξ f1 = ξ F F + ξ F , ξ f2 = ξ F F + ξ F , ξ f3 = F + ξ F F + ξ F . (7)The secondary photon is still circularly unpolarized if the incident photon is circularly unpolarized.In particular, if the incident photon is unpolarized (neither linearly nor circularly), we have ξ f1 = ξ f2 = 0, and ξ f3 = F /F . In the electron rest case, ξ f3 is positive definitely, which means that thepolarization vector of the secondary photon is always perpendicular to the scattering plane. Thisis a well-known property of the CS process. Similar to the incident photon, the polarization of thesecondary photon, Π, can be derived from Eq.(5) by replacing ξ i with ξ f i . In this subsection, we consider the polarization of a photon after being scattered by isotropicelectrons. We assume that the energy of the electrons follows the power-law distribution, i.e., N e ( γ e ) dγ e ∝ γ − pe dγ e . Once the single CS process is studied clearly, the polarization of a photonscattered by isotropic electrons can be easily obtained by averaging over the energy and orientationalangle of the incident electrons.Similar to the single scattering case, the Stokes parameters of a photon after being scattered byisotropic electrons, h ξ f i i , can be derived from Eq.(7) by replacing F a with the averaged components 6 – h F a i , which are defined to be (Chang et al. 2014a,b) h F a ( ε , θ sc ) i ≡ C Z γ γ N e ( γ e ) dγ e Z π sin θ dθ Z π dϕ (cid:18) ε ε (cid:19) F a . (8)Here, C ≡ R γ γ N e ( γ e ) dγ e R π sin θ dθ R π dϕ is the normalization factor. It is a nuisance factor,since the stokes parameters of the scattered photon are the ratios of the linear combination of h F a i (see Eq.(7)), thus the normalization factor is completely canceled out. In Eq.(8), we have assumedthat the Lorentz factors of the incident electrons are in the range γ e ∈ [ γ , γ ]. Therefore, thepolarization of the photon after being scattered by isotropic electrons, is given as h Π( ε , θ sc ) i = q h ξ f1 i + h ξ f2 i + h ξ f3 i . (9)In the Thomson limit, i.e., ε ≪ m e c , the above formulae are much simpler. In the leadingorder approximation, Eq.(3) is reduced to ε = ε (1 − β e cos θ )1 − β e cos θ . (10)The polarization of the photon after scattering is simplified to (Chang et al. 2014b) h Π( θ sc ) i = Π h F ih F i , (11)where F = ε ε + ε ε − sin θ sc , (12)and F = 2 γ e (1 − β e cos θ ) (cid:18) − β e sin θ sin ϕ − β e cos θ tan θ sc (cid:19) . (13)We can see that the polarization of the scattered photon is independent of photon energy and initialpolarization angle. Differing from the single scattering case, the scattered photon is polarized onlyif the incident photon is polarized. This is because the isotropic distribution of the electrons cancelsout the polarization. Nevertheless, if the incident photon is initially polarized (e.g., synchrotronphoton), the net polarization of the scattered photon cannot be completely canceled out.
3. SSC process from a highly relativistic jet
In this section, we calculate the polarization of photons in the SSC process from a highly rela-tivistic jet. Consider a highly relativistic, magnetic-dominated, and baryon-loaded jet ejected fromthe central engine travels with a large Lorentz factor. Shells with slightly different velocities collidewith each other and produce shocks. The shocks accelerate electrons to power-law distribution,and at the same time, magnify the magnetic field. Electrons move in the uniform magnetic field 7 –and produce synchrotron photons. The synchrotron photons collide with the seed electrons andthen escape from the jet, and are detected by the observer at a certain direction. For simplicity,we assume that both the synchrotron cooling and Compton cooling are slow enough such thatthe electrons always keep the power-law distribution. We consider two different cases which havebeen extensively studied in literatures: (1) magnetic field in the shock plane, and (2) magneticfield perpendicular to the shock plane. These two globally ordered magnetic configurations can beadvected by the jet from the central engine (Spruit et al. 2001; Fendt & Ouyed 2004).
Fig.1 shows the case that the magnetic field is contained in the shock plane. Consider a photon xy z δ θθ θ sc n (cid:11) θ (cid:15)(cid:3) ϕ (cid:12) n (cid:11) θ , (cid:3) ϕ (cid:12) (cid:63)(cid:63) Ο observer obs B Fig. 1.—
A schematic representation of the SSC process in the case that the magnetic field is contained inthe shock plane. We choose a Cartesian coordinate system such that the z -axis is in the jet direction, the x -axis is parallel to the magnetic field, and the xyz axes form the right-handed set. with energy ε travels along any direction ˆ n and collides with isotropic electrons. After scattering,the photon goes towards the ˆ n direction and then is detected by an observer. We set a Cartesiancoordinate system in the jet comoving frame, such that the z -axis is in the jet direction, the x -axisis parallel to the magnetic field, and ˆ y = ˆ z × ˆ x . The directions of the incident and scattered photonscan be written as ˆ n = sin θ cos ϕ ˆ x + sin θ sin ϕ ˆ y + cos θ ˆ z , (14)ˆ n = sin θ obs cos ϕ obs ˆ x + sin θ obs sin ϕ obs ˆ y + cos θ obs ˆ z , (15) 8 –where θ and ϕ ( θ obs and ϕ obs ) are the polar and azimuthal angles of the incident (scattered) photon,respectively. The scattering angle is given ascos θ sc = ˆ n · ˆ n = sin θ cos ϕ sin θ obs cos ϕ obs + sin θ sin ϕ sin θ obs sin ϕ obs + cos θ cos θ obs . (16)If the electrons are isotropic and their energies follow the power-law distribution ( N e ( γ e ) dγ e ∝ γ − pe dγ e ), the spectrum of synchrotron photons is also power-law, but not isotropic. Most photonsare emitted in the plane perpendicular to the magnetic field (Rybicki & Lightman 1979), N γ ( ε , θ, ϕ ) ∝ ε − p − (sin δ ) p +12 , (17)where δ is the pitch angle of the synchrotron photon with respect to the direction of the magneticfield (the x -axis in our case), that is,cos δ = ˆ x · ˆ n = sin θ cos ϕ. (18)The polarization of the photons induced by the SSC process can be easily obtained by averagingover the photon spectrum, i.e., hh Π( θ obs , ϕ obs ) ii = R h Π( ε , θ sc ) i N γ ( ε , θ, ϕ ) sin θdθdϕdε R N γ ( ε , θ, ϕ ) sin θdθdϕdε , (19)where h Π( ε , θ sc ) i is given by Eq.(9), or in the Thomson limit, by Eq.(11).Up to now, we are working in the jet comoving frame. The jet moves towards the observerhighly relativistically. In order to transform the above formulae to the observer frame, the followingrelation is useful, cos θ obs = cos ¯ θ obs − β jet − β jet cos ¯ θ obs , ϕ obs = ¯ ϕ obs , (20)where β jet = (1 − / Γ ) / is the velocity of the jet with respect to the observer in unit of lightspeed, and Γ is the Lorentz factor of the jet. Hereafter, the quantities in the observer frame aredenoted with a bar. Since the polarization is Lorentz invariant (Cocke & Holm 1972), we have hh ¯Π(¯ θ obs , ¯ ϕ obs ) ii = hh Π( θ obs , ϕ obs ) ii . (21)Fig.2 shows the evolution of the polarization with the viewing angles ¯ θ obs in four differentenergy bands: ε = [5 , . , . , .
5] MeV, and in the Thomson limit. Different curvesstand for different azimuthal angles ¯ ϕ obs . In the plot, we have taken the index of electron spectrumto be p = 3. Therefore, the polarization of the synchrotron photons is about Π ≈ ≈
200 (Chang et al. 2012). We have assumed that the Lorentz factorsof incident electrons are in the range γ e ∈ [1 , χ has been 9 – ¯ θ obs Γ ¯ Π (a) ε = [5, 50] MeV ¯ ϕ obs = 0 ¯ ϕ obs = π / ¯ ϕ obs = π / ¯ ϕ obs = π / ¯ ϕ obs = π / ¯ θ obs Γ ¯ Π (b) ε = [0.5, 5] MeV ¯ ϕ obs = 0 ¯ ϕ obs = π / ¯ ϕ obs = π / ¯ ϕ obs = π / ¯ ϕ obs = π / ¯ θ obs Γ ¯ Π (c) ε = [0.05, 0.5] MeV ¯ ϕ obs = 0 ¯ ϕ obs = π / ¯ ϕ obs = π / ¯ ϕ obs = π / ¯ ϕ obs = π / ¯ θ obs Γ ¯ Π (d) Thomson limit ¯ ϕ obs = 0 ¯ ϕ obs = π / ¯ ϕ obs = π / ¯ ϕ obs = π / ¯ ϕ obs = π / Fig. 2.—
The polarization of photons as a function of viewing angles in the case that the magnetic field iscontained in the shock plane. The polarization in four energy bands are showed in different panels. averaged. As is showed in Fig.2, the polarization of photons in the energy band ε = [0 . ,
5] MeV islarger than that in any other bands. This is because the polarization as a function of photon energypeaks at about ε ∼ ε = [0 . , .
5] MeV, in which most γ -ray polarimeters are active,the maximum polarization is about twice of that in the Thomson limit. This implies that the Klein-Nishina effect should be considered in the theoretical calculations of the γ -ray polarization. Foreach energy band, the polarization reaches its maximum at ¯ ϕ obs = π/
2, corresponding to the planeperpendicular to the magnetic field. In this plane, the polarization is independent of ¯ θ obs . Anothernoticeable feature is that, for a fixed azimuthal angle ¯ ϕ obs , the polarization has a minimum at¯ θ obs Γ ≈
1. In the jet frame, this angle corresponds to θ obs ≈ π/
2, i.e., the shock plane ( xy plane).Generally speaking, the observer whose line-of-sight is perpendicular (parallel) to the magnetic fieldsees the highest (lowest) polarization. This subsection is similar to the last subsection, except that the magnetic field is perpendicularto the shock plane (see Fig.3). We set a Cartesian coordinate system in the jet comoving frame,such that the z -axis is in the jet direction, the y -axis is in the plane which contains the directionsof jet and line-of-sight, and ˆ x = ˆ y × ˆ z . In such a coordinate system, the directions of incident and 10 – xy z θθ θ sc n (cid:11) θ (cid:15)(cid:3) ϕ (cid:12) n (cid:11) θ , (cid:3) π/2 (cid:12) (cid:63)(cid:63) Ο observer obs B Fig. 3.—
A schematic representation of the SSC process in the case that the magnetic field is perpendicularto the shock plane. We choose a Cartesian coordinate system such that the z -axis is in the jet direction,the y -axis is in the plane which contains the directions of jet and line-of-sight, and the xyz axes form theright-handed set. scattered photons can be written asˆ n = sin θ cos ϕ ˆ x + sin θ sin ϕ ˆ y + cos θ ˆ z , (22)ˆ n = sin θ obs ˆ y + cos θ obs ˆ z , (23)respectively, and the scattering angle is given ascos θ sc = ˆ n · ˆ n = sin θ sin ϕ sin θ obs + cos θ cos θ obs . (24)For isotropic and power-law ( N e ( γ e ) dγ e ∝ γ − pe dγ e ) electrons, the spectrum of synchrotronphotons is (Rybicki & Lightman 1979) N γ ( ε , θ ) ∝ ε − p − (sin θ ) p +12 . (25)After integrating over the photon spectrum, we obtain the polarization of the SSC process as afunction of viewing angle θ obs , that is, hh Π( θ obs ) ii = R h Π( ε , θ sc ) i N γ ( ε , θ ) sin θdθdϕdε R N γ ( ε , θ ) sin θdθdϕdε . (26)Note that in this case, due to the axis-symmetry, the polarization is independent of the azimuthalangle ϕ obs . With the help of Eqs.(20) and (21), we can transform the polarization from the jetframe to the observer frame. 11 –In Fig.4, we plot the polarization of the SSC photons as a function of viewing angle ¯ θ obs infour energy bands: ε = [5 , . , . , .
5] MeV, and in the Thomson limit. In the plot, the ¯ θ obs Γ ¯ Π ε = [5, 50] MeV ε = [0.5, 5] MeV ε = [0.05, 0.5] MeVThomson limit Fig. 4.—
The polarization of photons as a function of viewing angles in the case that the magnetic field isperpendicular to the shock plane. Different lines stands for different energy bands. Note that in this case,the polarization is independent of ¯ ϕ obs . parameters are chosen to be the same to the last subsection, i.e., p = 3, Γ = 200 and γ e ∈ [1 , χ . Similarto the last case, photons in the energy band ε = [0 . ,
5] MeV have the highest polarization, whichcan be as high as 24%. Photons with energy much higher than 50 MeV are almost unpolarized(which is not plotted in this figure). The polarization in the energy band ε = [0 . , .
5] MeV isapproximately twice of that in the Thomson limit. In each energy band, the polarization has apeak at the viewing angle ¯ θ obs Γ ≈
1, corresponding to θ obs ≈ π/ xy plane).
4. Discussions and conclusions
In this paper, we have presented an analytical calculation of γ -ray polarization induced by theSSC process from a highly relativistic jet in the Klein-Nishina region. We investigated the scenariothat isotropic electrons radiate synchrotron photons in the globally uniform magnetic field. Thesynchrotron photons are scattered by the seed electrons and then escape from the jet. Afterintegrating over the electron distribution and photon spectrum, the polarization of the SSC processwas obtained. We calculated two cases which have been extensively discussed in literatures: (1) 12 –magnetic field in the shock plane, and (2) magnetic field perpendicular to the shock plane. Thesetwo globally uniform magnetic configurations, although cannot be produced by the shock, can beadvected by the jet from the central engine. Instead of firstly working in the electron-rest frame,then transforming to the jet frame, as was done by most authors, in this paper we directly work inthe jet frame. This has the advantage of avoiding the complicate Lorentz transformation betweenthese two frames. The general formulae presented in this paper are valid in the Klein-Nishinaregion, as well as in the Thomson region. The formulism is useful in calculating the polarization ofthe SSC process in astrophysical processes, such as GRBs and AGNs.We numerically calculated the polarization of the SSC process from isotropic electrons withpower-law distribution. We found that, photons induced by the SSC process can be highly polarized,despite that the seed electrons are isotropic. In both magnetic configurations, photons in the energyband ε = [0 . ,
5] MeV have the highest polarization, reaching to about 24%. This is due to thefact that the polarization peaks at ε ∼ ε = [0 . , .
5] MeV (mostpolarimetric observations in the prompt phase of GRBs are performed in this energy band) isabout 20%, twice of that in the Thomson limit. This implies that the Klein-Nishina effect, whichis often neglected in literatures, should be carefully considered. In both magnetic configurations,the observer whose line-of-sight is perpendicular to the magnetic field sees the highest polarization.On the contrary, if we see along the direction of the magnetic field, we see the lowest polarization.Magnetic field plays an important role in the polarization effect. If isotropic photons are scatteredby isotropic electrons, the polarization certainly vanishes due to the symmetry. The existence ofmagnetic field breaks the symmetry of photon distribution, thus the polarization arises.The temporal variabilities of polarization observed in GRB 041219A and GRB 100826A maybe partially due to the evolution of the bulk Lorentz factor Γ. Especially, if the jet acceleratessmoothly (e.g., the magnetic-dominated jet model predicts that Γ ∝ r / ), the polarization anglecan be changed 90 ◦ suddenly at a critical value of Γ (Chang & Lin 2014). Therefore, the observationof polarization may help us to learn the evolution process of the outflow. Moreover, recent studies on γ -ray polarization in the photospheric emission models and some comptonized emission models showthe anti-correlation between the polarization degree and the luminosity (Lundman et al. 2014).Interestingly, in the case that the magnetic field is contained in the shock plane, we find thatthe polarization degree increases as the viewing angle increases, if photons are emitted at a largeviewing angle (¯ θ obs Γ &
1) The larger viewing angle leads to the lower luminosity. Thus, brightGRBs show low polarization, while dimmer ones show high polarization. However, in the case thatthe magnetic field is perpendicular to the shock plane, the situation is completely opposite. Thefuture observation of polarization-luminosity relation provides a way to distinguish the magneticconfigurations in the emission region.We stress that the real astrophysical process is much more complicate than the scenariosinvestigated in this paper. Firstly, we regarded the emission region as a point source. For a typicalGRB of redshift z ∼
1, it locates at a distance about Gpc away from us. This distance is much larger 13 –than the size of emission region, thus the point-source limit is a good approximation. Secondly,we only considered the slow cooling case, i.e., we assumed that the radiation is slow enough suchthat the electrons and photons keep their original spectra. Otherwise, after radiating synchrotronphotons, the spectrum of electrons is changed, and this further affects the photon spectrum. Thiscorrelated process heavily complicates the calculation, or even makes it impossible. Thirdly, forsimplicity, we only considered the single-scattering process. In fact, multi-scattering processes oftentake place. After one scattering process, the polarization and spectrum of photons are all changed.Such a complicate situation can only be studied using the Monte Carlo simulations. Finally,the e + e − pair production and annihilation above the threshold may also affect the polarizationsignificantly. All of these issues remain to be the interesting future work.We are grateful to X. Li, P. Wang, S. Wang and D. Zhao for useful discussion. This work hasbeen funded by the National Natural Science Fund of China under Grant No. 11375203. REFERENCES
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