Constraining the Bulk Lorentz Factor of GRB Outflow in the Magnetic-dominated Jet Model Zhe Chang1,2,∗, Hai-Nan Lin1,†, Yunguo Jiang1,2,‡ 1Institute of High Energy Physics 3 1 Chinese Academy of Sciences, 100049 Beijing, China 0 2 2Theoretical Physics Center for Science Facilities n Chinese Academy of Sciences, 100049 Beijing, China a J 9 2 Abstract ] E Recent observations by the Fermi-LAT showed that there are delayed arrivals of H GeV photons relative to the onset of MeV photons in some GRBs. In order to avoid . h a large optical depth, the minimal value of the Lorentz factor has been estimated to p - be higher than 1000 in some brightest bursts. In this paper, we present a detailed o r calculation of the time delay between the MeV and GeV photons in the framework of t s the magnetic-dominated jet model. We find that the time delay strongly depends on a [ the saturated bulk Lorentz factor of the jet. Inspired by this fact, we use this model 3 to calculate the Lorentz factors of the four brightest Fermi bursts. The results indicate v that the Lorentz factors are much smaller than that obtained from the “single-zone” 2 7 scenario. The short burst GRB 090510 has a minimal Lorentz factor 385, while the 5 three long bursts GRB 080916c, GRB090902b and GRB 090926 have almost the same 3 . Lorentz factors, with an average value near 260. Another interesting result is that, for 5 0 long bursts, GeV photons are emitted after the bulk Lorentz factor saturates. For the 2 short GRB, however, MeV and GeV photons are emitted at the same phase, i.e., either 1 : in the expansion phase or in the coasting phase. v i X Subject headings: gamma ray bursts: general - ISM: jets and outflows r a 1. Introduction It is well known that photons with energy higher than m c2 ≈ 0.511 MeV in a local jet frame e may annihilate into electron-positron pairs. The large optical depth of γγ annihilation, as well as *[email protected] †[email protected] ‡[email protected] – 2 – the Compton scattering, will restrain these photons from escaping the jet. However, the observed GRB spectra often peak in the MeV range and sometimes extend to the GeV range. This is the so-called “Compactness problem”. Sincethe optical depth is proportionalto the inverse of the bulk Lorentz factor (Γ) of the jet, the “compactness problem” can be solved if we assume that the GRB outflow is moving with a large Γ (Rees 1966; Piran 1999). The requirement of the thin optical depth for the observed high energy photons sets a lower limit on Γ (Lithwick & Sari 2001). The Burst and Transient Source Experiment (BATSE) data indicate that the Band-like spectra of most bursts have a thermal component at the prompt phase (Ryde 2004), and the measurement of the temperature allows us to constrain both Γ and the initial size r of the flow (Pe’er et al. 2007). 0 The investigation of GRBs has entered into a new epoch since the launch of the Fermi satel- lite in June 2008. The Fermi LAT instrument has observed several GRBs with photon energy as high as tens GeV. Within the framework of the simplified “single zone” model, the GeV pho- tons set a very large lower limit on Γ (Lithwick & Sari 2001; Soderberg et al. 2003; Razzaque et al. 2004;Granot et al.2008;Abdo et al.2009a,b,c;Ackermann et al.2010,2011,2012;Ghisellini et al. 2010). For example, Abdo et al. (2009a) showed that the minimal Lorentz factor of GRB 080916c outflow was Γ ≈ 900. Ackermann et al. (2010) analyzed the spectra of GRB 090510 and showed min thatΓ ≈ 1200. Ghisellini et al.(2010)estimated thedecelerating timeof GeVemissions andob- min tainedtheLorentzfactor ofGRB090510 aslargeas2000. However, anefficientphysicalmechanism to boost the outflow to such a large Γ is still unclear. An interesting feature of the Fermi observations is that GeV photons often arrived seconds later than MeV photons (Abdo et al. 2009a,b,c; Ackermann et al. 2010, 2011). The explanation of this phenomenon should include both the intrinsic emission mechanism and the traveling process (Chang et al. 2012). In some quantum gravity theories, photons can interact with the quantum fluctuation of the space-time, so high energy photons travel slower than low energy ones. Although this effect is very small, it can cause adetectable time differenceafter photons travel a cosmological distance (Gambini et al. 1999; Ellis et al. 2008, 2011). Such Lorentz invariance violation (LIV) ef- fectsleadtoanaturaltimedelaybetweenGeVandMeVphotons(Schaefer1999;Abdo et al.2009c; Boggs et al. 2003; Nemiroff et al. 2011). As it was shown by Abdo et al. (2009c) and Chang et al. (2012), the LIV effect is very small, we will neglect this effect in the following. Without considering the LIV effects, the delayed arrival of GeV photons can also be explained by several GRB models (Duran & Kumar 2011; Boˇsnjak & Kumar 2011). Duran & Kumar (2011) assumedthatphotonsareemittedbyelectronsviasynchrotronradiation,ittakesmoretimeforelec- tronstobeaccelerated toalargeLorentzfactorinordertoradiateGeVphotons. Boˇsnjak & Kumar (2011) used the magnetic jet model, which was initially introduced by Drenkhahn (2002) and Drenkhahn & Spruit (2002), to account for this phenomenon. According to this model, the optical depth is larger for high energy photons than that for low energy photons. GeV photons can only escape at a larger radius where the optical depth is below unity. In the magnetic reconnection model, the Band-type spectra can be produced from the photo- – 3 – sphere through the magnetic dissipation (Giannios 2006; Giannios & Spruit 2007; Giannios 2008). Koers & Giannios (2007) first considered the neutron effects in this model, which was later used by M´esza´ros & Rees (2011) to interpret the GeV time delay. In the magnetic-dominated but baryon- loaded model (Koers & Giannios 2007; Beloborodov 2011; M´esza´ros & Rees 2011), MeV photons can escape the plasma at the photosphere radius, which correspond to the prompt emission. How- ever, GeV photons are produced by the nuclear inelastic collisions between protons and neutrons at a larger radius. In such two-zone scenario, the strong constraint on the bulk Lorentz factor can be loosened (Hascoet et al. 2011; Zhao et al. 2011; Zou et al. 2011). As pointed out by Zhao et al. (2011), the optical depth depends not only on the energy but also on the emission angle. An average Lorentz factor Γ ≈ 600 can be estimated for GRB 080916c, GRB 090510 and GRB min 090902b in the two-zone model. In a similar way, Zou et al. (2011) assumed that the GeV photons were emitted at a larger radius than the MeV photons, and gave an analytical formula for Γ by min calculating the optical depth of a GeV photon going through the MeV photons shell. Inthis paper,we usethe magnetic-dominated jet modeldiscussedby Koers & Giannios (2007) and M´esza´ros & Rees (2011) to constrain the bulk Lorentz factor of GRB outflows. We show that the Lorentz factor of the short burst GRB 090510 can be as small as 385, while that of the three long bursts converge to about 260. The rest of the paper is organized as follows: In section 2, we illustrate the magnetic-dominated jet model and the producing mechanism of MeV and GeV photons briefly. In section 3, we use the delayed arrival of GeV photons in four Fermi bursts to calculate the bulk Lorentz factor of GRB outflow. In section 4, we discuss the validity of this model. Finally, conclusions are given in section 5. 2. The magnetic-dominated jet model The hydrodynamics of the GRB outflow depends strongly on its geometry structure. In the magnetic-dominated jet model, the Lorentz factor of the outflow increases with radius roughly as (Drenkhahn 2002; Drenkhahn & Spruit 2002; Metzger et al. 2010; Granot et al. 2011)1 (r/r )1/3 r ≤ r , Γ(r)≃ 0 sat (1) η r > r , ( sat 1The bulk Lorentz factor of the flow in the magnetic reconnection model originally took the form Γ(r) ≈ η(r/rsat)1/3 for r < rsat and Γ(r) ≈ η for r > rsat (Drenkhahn 2002; Drenkhahn& Spruit 2002), where they considered that the flow starts with the Alfv´en speed at the initial radius r0. A compact form Γ(r) ∼= (r/r0)1/3 was takenbyKoers & Giannios (2007),wherer0 isalength scale definedbyspecificcombination oftheparameters. However, a Poyntingjet can also be accelerated efficiently without reconnection process (Granot et al. 2011), where Γ also takes the form Γ ∼ σ01/3(r/r0)1/3, but r0 denotes the width of the magnetic shell. Boˇsnjak & Kumar (2011) assumed that theformat in Eq.(1) is valid at least in theintervalof theThomson- andpair-production-photosphere radii, and r0 is roughly thesame orderof theradius wherethejet islaunched. Sinceit was unphysicalfor thejet to be accelerated to a large speed instantaneously, the Lorentz factor at the base r0 was taken to be of order unity. In thepresent work, we takethe idea of Boˇsnjak and Kumar, and write Γ as in Eq.(1). – 4 – where r is assumed to be the base of the outflow, and r is the saturation radius. η denotes the 0 sat ratio of the magnetic energy density to the baryon rest mass energy density at r initially. 0 The injected baryons include both protons and neutrons. Initially, the neutron-proton jet accelerates as a single fluid where neutrons and protons have elastic collisions. When the n−p collision time scale is longer than the expansion time scale, the neutron component will coast with a terminal bulk Lorentz factor Γ at a characteristic radius, while the proton component is still n accelerated. Thus,theneutroncomponentis embeddedinafaster proton flow, andthejetbecomes a compound flow (Beloborodov 2011). Thecrosssectionofn−pcollision isσ ≈ σ (c/v ),whereσ ≈ 3×10−26 cm2,andv isthe nuc π rel π rel relative speed of p to n. When v → c, the collision is inelastic. This occurs when the comoving rel expansion time t′ ≈ r/2cΓ becomes shorter than the comoving collision time t′ ≈ 1/n′σ c. exp nuc p π Here n′ = Lx/4πr2m c3ηΓ is the comoving proton number density, L is the isotropic equivalent p p luminosity, and x = n /(n + n ) is the proton fraction of the baryon density. This gives the p p n characteristic radius r /r = η6x/2ηΓ2, where η ≡ Lσ / 4πm c3r 1/6 ≃ 1.32 ×102L1/6r−1/6. π 0 π π π p 0 54 0,7 Here we have adopted the Q =Q ×10n convention. Making use of Eq. (1), one obtains n (cid:0) (cid:1) r η3(xη /2η)3/5 r < r , π π π sat = (2) r0 (ηπ6x/2η3 r > rsat. The pion production by the inelastic collisions is inevitable. A certain fraction of energy is carried away by neutrinos, which is an important prediction of the baryon loaded jet model. The π0 decay gives primary injected GeV photons. However, these photons undergo e± cascades and can not escape the opaque jet. Interactions in the plasma are complex, more details can be found in Beloborodov (2011). Suppose the final components in the jet contain photons with a Band-like spectrum, and the peak energy is around MeV. These photons start to be emitted when τ = n′σ r/2Γ ∼ 1, which T p T gives the Thomson photosphere radius, i.e., r /r = η6/2ηΓ2, where σ ≈6.65×10−25 cm2 is the ph 0 T T Thomson cross-section, and η ≡ Lσ /4πm c3r 1/6 ≃2.22×102L1/6r−1/6. Using Eq. (1) for Γ, T T p 0 54 0,7 one obtains (cid:0) (cid:1) r η3(η /2η)3/5 r < r , ph = T T sat (3) r η6/2η3 r > r . 0 ( T sat The simulation of magnetohydrodynamics (MHD) shows that the jet can form a conical struc- ture (Tchekhovskoy et al. 2008). After the jet exits the stellar envelope, the inner jet runs faster than the outer sheath. Thus, the Lorentz factor tapers off towards the edges. In such structure, the neutrons from the outer sheath can drift into the inner core. The relative radial Lorentz factor ratio between neutrons and baryons is larger than 1, which ensures that the collisions are inelastic (M´esza´ros & Rees 2011). Suppose the jet has an open angle θ, the transverse pion optical depth – 5 – can be expressed as τ ≈ n′σ rθΓ= η6(r /r)(xθ/η). The jet becomes transversely optically thin π,⊥ p π π 0 (τ = 1) at r , which is defined as π,⊥ π,⊥ r xθ π,⊥ = η6 . (4) r π η 0 The dynamical evolution of the jet depends on η. If η is large, for instance η = 600η , the 600 saturation radiusr may belarger thanr ,r andr . MakinguseofΓ = (r/r )1/3,oneobtains sat π ph π,⊥ 0 the following characteristic radii: r ≃ 4.04×1012L3/5r2/5x3/5η−3/5 cm, π 54 0,7 0.5 600 r ≃ 3.98×1013L3/5r2/5η−3/5 cm,  ph 54 0,7 600 (5)  rπ,⊥ ≃ 4.41×1014L54η6−010x0.5θ−2 cm, r ≃ 2.16×1015r η3 cm. sat 0,7 600      On the other hand, if η issmall, r may become smaller than r ,r and r . In this case, one sat π ph π,⊥ obtains the corresponding radii: r ≃ 1.32×1013L x η3 cm, π 54 0.5 100 r ≃ 5.98×1014L η3 cm,  ph 54 100 (6)  rπ,⊥ ≃ 2.64×1015L54η1−010x0.5θ−2 cm, r ≃ 1.00×1013r η3 cm. sat 0,7 100      Here η is in unit of 100, i.e., η = 100η . In both cases above, one has the order r < r < r . 100 π ph π,⊥ TheGeVphotonsareassumedtobeproducedatr bytransversenuclearcollisions. However, π,⊥ the large optical depth prevents them to escape immediately. The spectra of produced photons depend on many parameters. Suppose that the Band spectrum peaks at E ∼ 1 MeV, then the p optical depth seen by a photon with energy E at radius r is approximately given by (Beloborodov 2011) 2×105 E −β−1 τ (E,r) ≈ r−1L η2β , (7) γγ 40−β−1 12 54 10 GeV 600 (cid:18) (cid:19) where β ≈ −2.5 is the spectrum index above the peak energy E , and r = r/1012 cm. Setting p 12 τ = 1, we get the γ−γ transparency radius, γγ r (E) ≈ 2.50×1013E3/2η−5L cm. (8) γγ 600 54 Or equivalently, r (E) ≈ 1.94×1017E3/2η−5L cm for the η ∼ 100 case. Here, E is the observed γγ 100 54 photon energy in unit of GeV. Thus, at r , multi-GeV photons will be copiously produced by the π,⊥ transverseindriftneutronscollidingwithjetcorebaryons. Intheradiusranger < r < r ,these π,⊥ γγ photons will annihilate into electron-positron pairs due to the large optical depth. Only beyond the radius r , the produced GeV photons can escape without obstructions. γγ – 6 – The time delay for a photon with energy E relative to the onset time of MeV photons, equates to the time it takes for the jet to propagate from r to r (E), ph γγ rγγ(E) dr ∆t= (1+z) , (9) 2Γ2c Zrph where z is the redshift. Theexplicit formula for ∆t dependson the order of r , r and r . From ph sat γγ Eqs. (5), (6) and (8), one can obtain r . r for the generic E & 10 GeV and η & 100. Using ph γγ Eq. (1) for Γ, one obtains the formats of ∆t for three different cases: 3(1+2cz)r0 (rrγ0γ)13 −(rrp0h)13 rph < rγγ < rsat; ∆t ≈ rγγ(2Eη)2−cr(cid:2)ph(1+z) (cid:3) rsat < rph < rγγ; (10) 3(1+2cz)r0 (rrsa0t)13 −(rrp0h)13  + rγγ(cid:2)(2Eη)2−crsat(1+z) (cid:3) rph < rsat < rγγ.      A phenomenological illustration of the magnetic-dominated jet model is depicted in Fig. 1. A magnetic-dominated but baryon-loaded jet is launched from a progenitor at the initial radius r . The bulk Lorentz factor of the jet evolves as Eq.(1). MeV photons are produced at the 0 radius r by nuclear collisions, π -decay, electron-positron annihilation, magnetic dissipation, and π 0 synchrotron radiation, etc.. But these photons can only escape at the photosphere radius r , ph where the Thomson optical depth decreases to below unity. Thus, multi-MeV photons are emitted at r and lead to the observed Band-type spectra (Veres & M´esza´ros 2012). The GeV photons are ph assumed to produce at r by transverse drift nuclear collisions, inverse Compton radiation, etc.. π,⊥ But these GeV photons are capable to escape only at a larger radius r due to the large optical γγ depth at r . The time it takes for the jet to propagate from r to r naturally leads to the π,⊥ ph γγ GeV time delay relative to the onset of MeV photons. 3. Constraints on the Lorentz factors From Eqs. (5), (6), (8) and (10), the time delay ∆t strongly depends on the terminal bulk Lorentz factor η. Inspired by this fact, we make useof the magnetic-dominated jet model discussed above to calculate η for four Fermi bursts, GRB 080916c, GRB 090510, GRB 090902b and GRB 090926, respectively. The observed parameters which are necessary in the calculation are listed in Table 1. Note that GRB 090510 is a short burst, while the other three are long bursts. In Table 1, E was high taken to bethe energy of the most energetic photon in each GRB. Oneexception is that the second energetic photon with E = 11.16 GeV in GRB 090902b was chosen, while the most energetic high 33.4 GeV photon arriving at 82 s was excluded. This is because the isolated photon is far apart from other GeV photons and it is quite possible that this individual event happened when the jet encountered the interstellar medium. – 7 – Fig. 1.— A phenomenological description of the magnetic-dominated jet model. r can locate sat before r , or beyond r , or at any place between them. ph γγ Table 1. GRB E T z E ∆t ios,54 90 high obs 080916c 8.8 66 4.35 13.22 12.94 090510 0.11 0.6 0.90 31.0 0.20 090902b 3.7 22 1.82 11.16 9.5 090926 2.2 13 2.11 19.6 21.5 Note. — The observed parameters of four Fermi- detected GRBs. E is the isotropic equivalent en- ios,54 ergy in unit of 1054 ergs. T is 90% the GRB du- 90 ration time in unit of second. z is the GRB redshift. E is the highest energy of photons for each burst in high unit of GeV. ∆t is the observed time delay between obs the highest energy photon relative to the onset of 100 MeV photons. The data were taken from Chang et al. (2012). – 8 – There are several parameters which are uncertain, such as the initial radius r , the jet open 0 angle θ and the ratio of proton number density to that of baryons x. Long bursts usually have time variability δt ≈ 10 ms, thus the initial radius is taken to be r ≈cδt ≈ 108 cm for long bursts. 0 The value of r for short bursts is usually assumed to be smaller than that of the long bursts, and 0 we set r ≈ 107 cm for short burst GRB 090510. A nominal value of jet open angle θ is taken to 0 be 0.01, and the proton fraction of the baryon density x is approximately 0.5, i.e., θ ≈ 1 and −2 x ≈1 (M´esza´ros & Rees 2011). When x approaches zero, this model reduces to the magnetic jet 0.5 model without the loaded baryons (Boˇsnjak & Kumar 2011). Aswas mentioned insection 2, theexplicit formulafor ∆tdependson theorder of r , r and ph sat r , which is not known previously. Thus, a self-consistent calculation should be taken carefully. γγ Case I: First, we consider the r > r > r case (see the first formula in Eq.(10)). The cal- sat γγ ph culated saturation bulk Lorentz factor and characteristic radii are listed in Table 2. The character- istic radii of the short burst GRB 090510 follow the order r >r > r , which is self-consistent. sat γγ ph However, for three long bursts, the results indicate r < r , which are in contradiction with the sat γγ assumption. The short burst GRB 090510 has a Lorentz factor about 720, which is much lower than the prediction of one-zone models. For instance, Abdo et al. (2009b) presented that the bulk Lorentz factor of GRB 090510 was as large as 1200. Our results indicate that GeV photons in the short burst are emitted before the Lorentz factor of the jet saturates. Case II: Then, we consider the r < r < r case (see the second formula in Eq.(10)). sat ph γγ The results are given in Table 3. For all of the four bursts, we have r < r . Thus, the π,⊥ ph transverse nuclear collisions happen inside the photosphere, and GeV photons are converted to the e± cascades. The overlap of the producing regimes of GeV and MeV photons is possible, but GeV photons are attenuated until r . The spectrum of GRB 090510 is fitted well by the Band function γγ plus a power-law component which dominates in the band above 30 MeV (Zhao et al. 2011), this can be explained well by the Magnetic-dominated jet model. If these arguments are true, the bulk Lorentz factor of GRB 090510 is further reduced to 385. In Table 3, one also notice that for GRB 080916c, r < r , which is not self-consistent. The bulk Lorentz factor of GRB 090902b and ph sat GRB 090926 in this case are calculated to be 245 and 252, respectively. Case III: Finally, we consider the r < r < r case (see the third formula in Eq.(10)). ph sat γγ The results are listed in Table 4. The data of GRB 090510 are absent, because any value of η can not fit ∆t = 0.2 s by the formula. The minimal value of ∆t is 0.34 s locating at η ≈ 1. The obs 600 data of three long bursts fit well in this case. The bulk Lorentz factors are 270, 252 and 258 for GRB 080916c, GRB 090902b and GRB 090926, respectively. Sofar,wehaveobtainedtheself-consistentLorentzfactorsandcharacteristic radiiforallofthe four bursts, and summarized the Lorentz factors in Table 5. From the calculation above, the two conditions r < r and r < r always hold. For short burst GRB 090510, the self-consistent π ph π,⊥ γγ cases are case I and II, and the corresponding Lorentz factors are 720 and 385, respectively. For the three long bursts, case III is valid for all of them, but they are all excluded in the case I. The – 9 – Table 2. GRB η r /cm r / cm r /cm r /cm r /cm 600 ph γγ sat π π,⊥ 080916c 0.39 5.24×1013 1.78×1016 1.28×1015 5.33×1012 1.51×1014 090510 1.20 1.29×1013 3.18×1014 3.73×1015 1.31×1012 6.74×1013 090902b 0.32 6.80×1013 4.67×1016 7.08×1014 6.90×1012 2.32×1014 090926 0.26 7.73×1013 3.09×1017 3.80×1014 7.84×1012 2.87×1014 Note. — The calculated saturation bulk Lorentz factors and characteristic radii under the assumptionthatr > r > r . Wechooser = 1forshortburstGRB090510 andr = 10 sat γγ ph 0,7 0,7 for other three long bursts. The characteristic radii of three long bursts are not self-consistent. Table 3. GRB η r /cm r / cm r /cm r /cm r /cm 100 ph γγ sat π π,⊥ 080916c 2.58 1.37×1015 1.09×1016 1.72×1015 5.02×1012 1.36×1014 090510 3.86 6.31×1015 7.16×1015 5.75×1014 1.91×1012 1.25×1014 090902b 2.45 1.48×1015 1.38×1016 1.47×1015 5.96×1012 1.81×1014 090926 2.52 1.62×1015 2.80×1016 1.60×1015 5.88×1012 1.77×1014 Note. — The calculated saturation bulk Lorentz factors and characteristic radii under the assumptionthatr < r < r . TheparametersarethesameasinTable2. Thecharacteristic sat ph γγ radii of the GRB 080916c are not self-consistent. Table 4. GRB η r /cm r / cm r /cm r /cm r /cm 600 ph γγ sat π π,⊥ 080916c 0.45 4.82×1013 8.68×1015 1.97×1015 4.89×1012 1.31×1014 090902b 0.42 5.77×1013 1.20×1016 1.60×1015 5.86×1012 1.77×1014 090926 0.43 5.71×1013 2.50×1016 1.72×1015 5.80×1012 1.74×1014 Note. — The calculated saturation bulk Lorentz factor and characteristic radii for three long bursts under the assumption that r < r < r . The parameters are the same as in ph sat γγ Table 2. – 10 – allowed Lorentz factors for the three long bursts seem to converge to an average value about 260. The saturation bulk Lorentz factor of the short burst GRB 090510 is reduced sharply, but still higher than that of long bursts. Another interesting feature is that GeV photons are emitted after the bulk Lorentz factor saturates for long bursts, i.e. in the coasting phase. For the short burst, however, both MeV and GeV photons are emitted either in the expansion phase or in the coasting phase. Case I can not happen for the long bursts, while case III can not happen for the short bursts. 4. Discussion Besides the time delay, another important feature of GeV emissions is that they last much longer than the sub-MeV photons (Gao et al. 2009; Kumar 2009; Ghirlanda 2010; Ghisellini et al. 2010). For instance, the duration time of the sub-MeV photons is 55 seconds in GRB 080916c, while photons with energy > 100 MeV last about 1400 seconds (Kumar 2009). The observed decline of flux can be explained by the synchrotron radiation in the external shock (ES), i.e., F ∝ t(3β+2)/4νβ/2 (β = −2.4 for GRB 080916c). The data of the initial 55 seconds are able to ν explain theobserved X-ray andoptical fluxof theafterglow oneday later. Thus,theGeV emissions have an afterglow origin. The spectrum and the light curve of the GRB 090510 were also explained by the synchrotron radiation in the ES model (Ghirlanda 2010). Ghisellini et al. (2010) studied the light curves of 11 GRBs detected by LAT, and concluded that LAT fluxes decay in a common way F ∝ t−1.5 for ν the four brightest GRBs studied in this paper. The LAT fluxes can be interpreted as the fireball emission in the radioactive regime. The spectra of the GeV emissions in some bursts showed a different power law from the Band function. Thus, the spectra and the light curves present strong evidences that GeV emissions have different origin with the sub-MeV emissions. As hinted by Ghisellini et al. (2010), one can divide the “total emission time” of sub-MeV and GeV emissions into two parts2: one is the overlap regime where both the sub-MeV and GeV photons are present; anotheristheregimewhereonlyLATphotonsexist. Thelattercanbenamedastheearlyafterglow. Notice that our calculation about GeV time delay is valid in the overlap regime, the GeV emissions in the early afterglow are not discussed. Once the outflow collides with the environment medium, the forward shock can also occur in the magnetic-dominated jet model. Both electrons and protons can be accelerated by the shock and form a power law spectral distribution. The characteristic frequency of the synchrotron radiation follows ν = Γγ2 qB/2πm c, thesynchrotron e,p e,p radiation of protons can be ignored compared to that of electrons. The produced photons have the same power law spectrum with electrons. Since the optical depth is small at so large radius, 2X-ray and optical radiation usually arrive one day later, i.e., in the afterglow phase. Wedo not includethem in the“total emission time”.