Off-axis Emission from the Beame Afterglow of Gamma Ray Bursts an a Possible Interpretation of Slowly Declining X-Ray Afterglow of GRB980425 Takashi Nakamura Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606 ABSTRACT A formula for off-axis emissions from the beame afterglow of GRBs in a synchrotron moel is erive. The formula is applie to an inhomogeneous circumstellar matter obeying n r 2 suggeste for SN1998bw/GRB980425. The slow ecline rate as well as the X-ray flux similar to the observe values are obtaine for appropriate choice of the parameters so that the association of GRB980425 with SN1998bw is further strengthene. ApJ Letter in press) Subject heaings: gamma rays: bursts stars: supernovae 1. Introuction Recently the X-ray light curve of GRB980425 an follow up ata obtaine by the BeppoSAX have been reporte Pian et al. 1999). The NFI observation ientifie an X-Ray source S1) position-ally coincient with the unusual Type Ic supernova SN1998bw Galama et al. 1998, Kulkarni et al. 1998). Note here that the position of S1 is revise an the other source S2 is 4 away from SN1998bw. The X-ray flux of S1 was 3 10 13 erg s 1 cm 2 in April 26 an ecline a factor of two in six months. If this is an X-ray afterglow of GRB980425, the ecline rate t 0.2 ) is extremely slower than usual, which nees an interpretation. If GRB980425 is associate with SN1998bw, the istance is 40Mpc Galama et al. 1998) an the isotropic gamma ray total energy is unusually low 10 48 ergcomparewith>10 51 erg in other GRBs for which a reshift measurement is available. Such a low gamma ray luminosity might be interprete if the beam irection of GRB980425 is ifferent from the line of sightnakamura 1998, Eichler & Levinson 1998). In such an geometry we see only scattere gamma rays so that the luminosity is low an the energy of gamma rays is not high < 500keV Nakamura 1998, Eichler & Levinson 1998). This also interprets the fact that GRB980425 belongs to so calle Non High Energy class of GRBs Penelton et al. 1997). It is shown that if the angle between the beam an the line of sight is 30, the luminosity can be 10 3 smaller than usual Eichler & Levinson 1998). Such a beam moel is relevant now since an evience for the beaming has recently been suggeste from the rapi ecline of optical afterglow
2 of GRB990123 after ay 2 Kulkarni et al. 1999) an the rapi ecline of GRB980519 Halpern, Kemp, Piran & Bershay 1999). An angle of 30 has also been suggeste to interpret the linear optical polarization 0.5% in June 20 23; Kay, Halpern & Leighly, 1998) of SN1998bw. If the supernova ejecta is an oblate or a prolate shape with an aspect ratio 1 : 2 3 an the line of sight is 30 away from the symmetry axis, the observe linear polarization egree might be explaine Hoeflich, Wheeler & Wang 1998). From the raio observation of SN1998bw it is suggeste that the ensity istribution of the circumstellar matter follows n = n 0 r 2 0 /r2 with n 0 r 2 0 1034 /cm, which is consistent with the constant mass loss rate in the progenitor Li & Chevalier 1999). It is also claime that the energy of the orer of 10 50 erg shoul be injecte twice with the injection velocity of the orer of 0.5c, which strengthens the link between SN1998bw an GRB980425. In this Letter we try to interpret the slowly eclining X-ray afterglow of GRB980425 in the beam moel of GRBs assuming the line of sight is 30 away from the axis of the beam. We aopt the ensity istribution of the circumstellar matter suggeste by Li & Chevalier 1999. We will obtain the ecline rate an the X-ray flux similar to the observe values for appropriate choice of the parameters so that the association of GRB980425 with SN1998bw will be further strengthene. 2. Off-axis emission from the beame afterglow Granot, Piran & Sari 1998 as well as Woos & Loeb 1999 erive a general formula to compute the off-axis emission from beame GRBs. Here we aopt their formulations an notations. Let us use a spherical coorinate system r =r, θ, φ) where the coorinate are measure in the lab frame; let the θ =0axisz-axis) points to the etector an r = 0 be the central engine. Let also D be the istance to the source an α = r sin θ/d be the angle that a given ray makes with the normal to the etector. Then the observe flux is given by F ν T )= νd γβ 2π 0 αm νγ1+β) φ α 2 ν α 0 νγ1 β) ν 2 j ν [Ω, r,t + rµ c ] {1 µ 2 } 3/2, 1) where µ =1 ν /γν)/β. α m, T an j ν are the maximum value of α, the arrival time of a photon at the etector an the rest frame emissivity measure in erg s 1 cm 3 Hz 1 sr 1, respectively. Note here that means the physical quantity in the rest frame. We aopt the stanar fire ball moelpiran 1998). As an emissivity we exten a simple synchrotron moel Sari, Piran & Narayan 1998) use in a homogeneous ambient gas to the inhomogeneous circumstellar matter following n = n 0 r/ ). =2 correspons to the constant mass loss from the progenitor of GRBs. This kin of a moel has ever been stuie in a ifferent context Meszaros, Rees & Wijers 1998).
3 We assume that electrons are accelerate in the shock to a power-law istribution of Lorenz factor γ e with a minimum Lorenz factor γ m : Nγ e ) γe p γ e,γ e >γ m. γ m =Gγ where γ is the Lorenz factor of the shocke flui an G = ɛ e p 2)/p 1)m p /m e with ɛ e being the efficiency of the acceleration. Since p = 2 2.5 is suggeste from observations, G = 60ɛ e /0.1)3p 2)/p 1)). The magnetic fiel strength is given as B = B 0 γr/ ) /2 where B 0 = 32πm p n 0 ɛ B. ɛ B measures the ratio of the magnetic fiel energy to the total thermal energy. The rest frame raiation power P ) an the characteristic synchrotron frequency ν γ e )) from a ranomly oriente electron with Lorenz factor γ e 1inamagneticfielB are given by P = P 0 γ 2 γe 2 r/ ) an ν γ e )=ν 0 γγer/r 2 0 ) /2 where P 0 = σ T B0 2 c/6π an ν 0 = eb 0 /2πm e c =3.5 10 5 Hzn 0 /1cm 3 ) 1/2 ɛ B /0.1) 1/2. The peak spectral power flux oes not epen on γ e an occurs at ν γ e )anis given by P ν,max = P /ν = P 0 γ/ν 0 r/ ) /2. The critical gamma factor γ c is efine by γ c m e c 2 = P γ c )τ = P γ c )r/γc) anisgivenbyγ c =F/γr/ ) 1 where F = m e c 3 /P 0.Then the energy spectra epen on two frequencies efine by ν m ν γ m )=ν 0 G 2 γ 3 r/ ) /2 an ν c ν γ c )=ν 0 F 2 γ 1 r/ ) 3/2 2 Sari, Piran & Narayan 1998). There are two ifferent cases in the energy spectra epening on whether ν m >ν c fast cooling) orν m<ν c slow cooling);sari, Piran & Narayan 1998). In both cases the emissivity j ν isexpresseintheformas j ν γn 0 =P ν,max 4π r ) fν ), 2) where fν )isgivenforfast cooling case as ν /ν c) 1/3 ν fν c >ν )= ν /ν c) 1/2 ν m >ν >ν c ν m /ν c ) 1/2 ν /ν m ) p/2 ν >ν m, while for slow cooling case we have a smilar formulasari, Piran & Narayan 1998). Let θ an θ v be the beaming half-angle an the angle between the irection to the etector an the axis of the emission cone. In case of θ v = 0, Eq. 1) is evaluate as F ν T )= r3 n 0 3D 2 β r ) I; I= For γ 1an θ<1, the integral is expresse as γ 2 θ 2 νγ1 β cos θ) νγ1 β) νν 2ν 2 P ν,max fν ). 3) s I = 1 s 2 γp ν,maxfνγs/2). 4) If γ θ 1,Ioes not epen on θ. This is expecte since in this case the etector etects the raiation only from the half-angle of γ 1 an can not observe the ege of the beam. For =0, Eq. 4) gives the same results as in Sari, Piran & Narayan 1998. 1 1 In reality there is 10% or so ifference epening on the parameter p. However such a ifference is consistent in a simple moel of emissivity in Sari, Piran & Narayan 1998.
4 Now let us consier another extreme case of 1 >θ v > θ>γ 1. In this case the line of sight from r = 0 oes not pass through the beam. Then Eq. 1) is evaluate as F ν T )= r3 0 n 0P 0 12πD 2 r νγ1 β cosθv+ θ)) ) 32 )/2 γνν I; I = 2φ v ν 0 νγ1 β cosθ v θ)) ν 2 fν ), 5) where cos φ v =cos θ cos θ cos θ v )/sin θ v sin θ. Eq.5) is a general formula for off-axis emission from beame GRBs in a synchrotron moel. In the next section we shall apply the formula to GRB980425. 3. Application to GRB980425 Now let us consier =2 case which is suggeste from SN1998bw Li & Chevalier 1999). Then Eq. 5) is rewritten as F ν T )= r3 0 n 0P 1 β cosθv+ θ)) 0 12πD 2 I; I = ν 0 1 β cosθ v θ)) s 2φ v fνγs). 6) s2 The position of the shock is relate to the arrival time T as r = ct/1 β cos θ v ) 1.9 10 16 cmθ v /30 ) 2 T where T is the arrival time in ays. γ is given for the aiabatic case by γ = m p c 2 n 0 r0 2 = 63 r2π θ2)1/2 10 51 erg )1/2 1 β cos θ v θ 2 ) 1/2 1cm 3 ) 1/2 10 17 cm ) 1 T 1/2, 7) where is the total energy in the beam. The two characteristic energy hν m an hν c at the etector are given by hν m =2 10 2 ev G 60 )2 1 β cos θ v ) 1 γ 0.1 20 )2 r ) 1 ɛ B 0.1 )1/2 1cm 3 )1/2 8) hν c =8.7 10 3 ev 1 β cos θ v ) 1 γ 0.1 20 ) 2 r ) ɛ B n 0 0.1 ) 3/2 1cm 3 ) 3/2 10 17 cm ) 2 9) Now consier the observe raiation in optical an X-ray bans. I is evaluate for θ v 30 an fast cooling case as I =2 4+p/2 1 θv 2+p Then the energy flux in a ban of ν <ν<ν u is given by νu n 0 θ θ v ) 2 G p 1 r ) 1 p/2 γ p 2 F ν 0 ν )p/2. 10) F b F ν ν =1.74 10 12 erg s 1 cm 2 2 C p ν p 2 ν 0 ) p/2 1 ν 0 ) p/2 1 ) ν ν u n 0 D 10 17 cm )4 p 1cm 3 )2 p/2 40Mpc ) 2 θ v 30 ) 4 2 θ ) 4 p θ v 10 51 erg )p/2 1 T 2 p, 11) n 0
5 where C p =2 3p 2)/2 12140 p 2) G/60) p 1 Let us compute F b in X-ray ban ν =1.6keV, ν u = 10keV) an V ban ν =5 10 14 Hz, ν u =6 10 14 Hz) for p =2.2anp=2.5. For p =2.2 F b =6.5 10 13 erg s 1 cm 2 1.4 10 13 erg s 1 cm 2 ; V ban) 10 17 cm )1.8 1cm 3 )0.95 40Mpc ) 2 θ v 30 ) 4 ɛ e 0.1 )1.2 ɛ B 0.1 )0.05 2 θ ) 1.8 θ v 10 51 erg )0.1 T 0.2. 12) In this case the eclining rate is almost the same as the observe value while the absolute value of the X-ray flux is similar to the observe one an can be ajuste by choosing the appropriate values of various parameters such as θ v, θ, ɛ e,ɛ B,,n 0 an. For p=2.5 F b =8.0 10 13 erg s 1 cm 2 3.0 10 13 erg s 1 cm 2 ; V ban) 10 17 cm )1.5 1cm 3 )0.875 40Mpc ) 2 θ v 30 ) 4 ɛ e 0.1 )1.5 ɛ B 0.1 )0.125 2 θ ) 1.5 θ v 10 51 erg )0.25 T 0.5. 13) In this case, the amplitue of the X-ray flux can be similar to the observe value but the ecline rate is somewhat too rapi. For comparison, we shall compute the energy flux for an aligne case θ v = 0). Since in this case the position of the shock is relate to the arrival time of the raiation by r =2γ 2 ct, γ is given by γ = 13 10 51 erg )1/4 θ n 0 15 ) 1/2 1cm 3 ) 1/4 10 17 cm ) 1/2 T 1/4 14) F b is given for p =2.2as F b =7.8 10 11 erg s 1 cm 2 1.7 10 11 erg s 1 cm 2 ; V ban) 10 17 cm )0.5 1cm 3 )0.3 40Mpc ) 2 θ 15 ) 1.6 ɛ e 0.1 )1.2 ɛ B 0.1 )0.1 10 51 erg )0.8 T 0.9. 15) while for p =2.5 F b =2.7 10 11 erg s 1 cm 2 1.7 10 11 erg s 1 cm 2 ; V ban) 10 17 cm )0.5 1cm 3 )0.375 40Mpc ) 2 θ 15 ) 1.75 ɛ e 0.1 )1.5 ɛ B 0.1 )0.125 10 51 erg )0.875 T 1.125. 16) Let us apply Eq. 16) to GRB990123. There was a break in the optical light curve at T =2. This may be ue to the wiening of the beaming half-angle Rhoas 1997, Kulkarni et al. 1999). From the conition of γ = θ 1 at T =2, θis expresse by the other parameters as θ =1.8 10 51 erg ) 1/2 10 17 cm ) n 0 1cm 3 )0.5 17)
6 Aopting D =10Gpc as the luminosity istance to GRB990123 z = 1.6; Bloom et al. 1999a) we have F b =1.8 10 14 erg s 1 cm 2 1.1 10 14 erg s 1 cm 2 ; V ban) 10 17 cm ) 1.125 1cm 3 ) 0.5 10Gpc ) 2 ɛ e 0.1 )1.5 ɛ B 0.1 )0.125 10 51 erg )1.75 T 1.125. 18) For example, if 10 52 erg with the other parameters unchange, the ecline rate an amplitue of the flux in X-ray an the optical bans are similar to the observe values. However even if 10 51 erg, by appropriate choice of the other parameters such as,n 0,ɛ e an ɛ B,wemayalso have the observe amplitue. 4. Discussion Since a possible association of GRBs with Type Ib/Ic supernovae is suggeste only for GRB980425, we ha better wait for more events to confirm the conjecture. However recently another possible association of GRB with the supernova is suggeste for GRB980326. The ecline of the afterglow was levele off at 20 ays after the GRB, which was consiere as the host galaxy. Keck observe the host galaxy nine months after the burst again but coul not fin the host galaxy. A possible explanation for this peculiar event is that the apparent level off at 20 ays is ue to the peak of supernova like SN1998bwBloom et al. 1999b). If this is the case, at least two examples of association of GRBs with supernovae exist. For GRB980425 in our moel the break in the light curve when γ = θ 1 shoul occur at n 0 T = 544 10 51 erg ) θ v 30 )2 1cm 3 ) 1 10 17 cm ) 2. 19) The break time oes not epen on θ but on,θ v,n 0 an. Before the break time, Eq.12) an 13) can be use so that the optical luminosity is much lower than the luminosity of SN1998bw for T < 180 Iwamoto et al. 1998). However the ecline rate is so low that the luminosity from the afterglow might overcome the luminosity of the supernova an the host galaxy before T 544. Therefore it is important to observe SN1998bw up to 2 years after the supernova event. In conclusion, if the line of sight is 30 away from the axis of the beam for GRB980425, the ecline rate an the X-ray flux similar to the observe values are obtaine for appropriate choice of the parameters. This strengthens the link between GRB980425 an SN1998bw. This work was supporte in part by Grant-in-Ai of Scientific Research of the Ministry of Eucation, Culture, an Sports, No.11640274 an 09NP0801.
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