Off-axis jet scenario for early afterglow emission of low-luminosity gamma-ray burst GRB 190829A
Yuri Sato, Kaori Obayashi, Ryo Yamazaki, Kohta Murase, Yutaka Ohira
aa r X i v : . [ a s t r o - ph . H E ] J a n Mon. Not. R. Astron. Soc. , 1–9 (2021) Printed 27 January 2021 (MN L A TEX style file v2.2)
Off-axis jet scenario for early afterglow emission oflow-luminosity gamma-ray burst GRB 190829A
Yuri Sato ⋆ , Kaori Obayashi † , Ryo Yamazaki , ‡ , Kohta Murase , , , § and Yutaka Ohira ¶ Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara 252-5258, Japan Institute of Laser Engineering, Osaka University, 2-6 Yamadaoka, Suita, Osaka 565-0871, Japan Department of Physics, Pennsylvania State University, University Park, Pennsylvania 16802, USA Department of Astronomy & Astrophysics, Pennsylvania State University, University Park, Pennsylvania 16802, USA Center for Multimessenger Astrophysics, Pennsylvania State University, University Park, Pennsylvania 16802, USA Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Kyoto 606-8502, Japan Department of Earth and Planetary Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan today
ABSTRACT
Recently, ground-based Imaging Atmospheric Cherenkov Telescopes have reportedthe detection of very-high-energy (VHE) gamma rays from some gamma-ray bursts(GRBs). One of them, GRB 190829A, was triggered by the
Swift satellite, and about2 × s after the burst onset the VHE gamma-ray emission was detected by H.E.S.S.with ∼ σ significance. This event had unusual features of having much smallerisotropic equivalent gamma-ray energy than typical long GRBs and achromatic peaksin X-ray and optical afterglow at about 1 . × s. Here we propose an off-axis jet sce-nario that explains these observational results. In this model, the relativistic beamingeffect is responsible for the apparently small isotropic gamma-ray energy and spec-tral peak energy. Using a jetted afterglow model, we find that the narrow jet, whichhas the initial Lorentz factor of 350 and the initial jet opening half-angle of 0.015 rad,viewed off-axis can describe the observed achromatic behavior in the X-ray and opticalafterglow. Another wide, baryon-loaded jet is necessary for the later-epoch X-ray andradio emissions. According to our model, the VHE gamma rays observed by H.E.S.S.at 2 × s may come from the narrow jet through the synchrotron self-Comptonprocess. Key words: gamma-ray bursts: individual: GRB 190829A — radiation mechanisms:non-thermal
Recently, very-high-energy (VHE) gamma-rays from somegamma-ray bursts (GRBs) were detected by ground-basedImaging Atmospheric Cherenkov Telescopes, such as theMajor Atmospheric Gamma Imaging Cherenkov (MAGIC)telescopes, and the High Energy Stereoscopic System(H.E.S.S.). A prototypical example so far is GRB 190114C,which was simultaneously detected with MAGIC and
Fermi
Large Area Telescope (MAGIC Collaboration, et al.2019a,b; Ajello et al. 2020). The observed spectrum in the ⋆ E-mail: [email protected] (YS) † E-mail: [email protected] (KO) ‡ E-mail: [email protected] (RY) § E-mail: [email protected] (KM) ¶ E-mail: [email protected] (YO)
VHE gamma-ray band is well explained by the synchrotronself-Compton (SSC) model (MAGIC Collaboration, et al.2019b; Derishev & Piran 2019; Fraija et al. 2019a,b,c;Wang, et al. 2019; Asano et al. 2020; Huang et al. 2020b).H.E.S.S. detected VHE gamma-rays from GRB 180720Babout 10 hours after the burst onset at 5.3 σ significancelevel, and the energy flux was νF ν ≈ × − erg s − cm − in the VHE band (Abdalla, et al. 2019). GRB 190829A wasalso detected by H.E.S.S. about 2 × s after the bursttrigger (de Naurois et al. 2019). Its significance is ∼ σ .Moreover, a possible detection of VHE gamma-ray emissionfrom a short GRB 160821B has been claimed by MAGIC(MAGIC Collaboration, et al. 2020; Zhang et al. 2020b). Itis expected that in the near future, the Cherenkov Tele-scope Array (CTA; Actis et al. 2011) will increase the num-ber of GRBs with VHE gamma-rays (Kakuwa, et al. 2012;Gilmore, et al. 2013; Inoue, et al. 2013). © Sato et al.
Compared with GRB 190114C and 180720B,GRB 190829A has some peculiar observational prop-erties. The prompt gamma-ray emission (from ∼
10 keVto MeV band) consists of two temporally separatedcomponents (Chand et al. 2020). The burst started withless energetic emission (hereafter Episode 1 followingChand et al. 2020) with an isotropic equivalent gamma-rayenergy of E iso ,γ = 3 . × erg and a peak energy (thatis, the photon energy at which the νF ν -spectrum takes amaximum) E p = 120 keV. After quiescent time intervallasting about 40 s, the second brighter emission (Episode 2)with E iso ,γ = 1 . × erg and E p = 11 keV, appeared. Theobserved values of E iso ,γ and E p of Episode 2 are consistentwith Amati relation (Amati et al. 2002; Sakamoto et al.2008), while those of Episode 1 are in the region oflow-luminosity GRBs. Both Episode 1 and 2 have smaller E iso ,γ and E p than typical long GRBs, including the otherVHE gamma-ray events, GRB 190114C and 180720B (e.g.,Huang et al. 2020a). Indeed, GRB 190829A occurred sonearby with a redshift of z = 0 . . × s.Such an “achromatic” behavior is difficult to be explainedin standard afterglow model, in which the synchrotron emis-sion has the maximum when the typical frequency ν m crossesthe observation bands (Sari, Piran & Narayan 1998). In con-trast, the other VHE events, GRB 190114C and 180720B,showed monotonically decaying X-ray afterglow emission(Yamazaki et al. 2020; Fraija et al. 2019b). Possible inter-pretations of the achromatic bump are the X-ray flare withoptical counterpart (Chand et al. 2020; Zhang et al. 2020a;Zhao et al. 2020b) and the afterglow onset of baryon loadedoutflow with bulk Lorentz factor of about 30 (Fraija et al.2020).In this paper, we propose an off-axis jet scenario to ex-plain the observed properties of GRB 190829A. If the jet isviewed off-axis, the relativistic beaming effect causes appar-ently dim and soft prompt emission (Ioka & Nakamura 2001,2018; Yamazaki, Ioka, & Nakamura 2002, 2003, 2004a,b;Yamazaki, Yonetoku, & Nakamura 2003; Salafia et al. 2015,2016). Some low-luminosity GRBs may be explained bythis context (e.g., Yamazaki, Yonetoku, & Nakamura 2003;Ramirez-Ruiz et al. 2005). This model may also explain ob-served achromatic behavior of early X-ray and optical/IRafterglow with a maximum at 1 . × s. For the off-axisafterglow (e.g., Granot et al. 2002), the bulk Lorentz factorof the jet is initially so high that the afterglow emission isvery dim because of the relativistic beaming effect. As the jetdecelerates, the beaming effect becomes weak, resulting inthe emergence of a rising part in afterglow light curves. Afterthe peak of the emission, the jet has smaller Lorentz factorsso that the light curve only weakly depends on the viewingangle. This paper is organized as follows. In §
2, we constructour afterglow model following Huang et al. (2000). For sim-plicity, the jet is assumed to be uniform, and structuredjets (e.g., Rossi, Lazzati, & Rees 2002; Zhang & M´esz´aros2002; Zhang et al. 2004) are not considered. In §
3, we showthat our model explains the observed afterglow in the X- ray, optical, and radio bands. In order to explain the ob-served data, we need two jets with narrow and wide open-ing angles (see Fig. 1). The former is viewed off-axis, whilethe other is not. Such a two-compornet jet model (e.g.,Peng, K¨onigl, & Granot 2005; Racusin et al. 2008) might besupported by the fact that the prompt emission has twoindependent components. In §
4, VHE gamma-ray flux at2 × s is estimated. In §
5, using a simple model, we dis-cuss the on-axis prompt emission properties of the narrowjet. Section 6 is devoted to a discussion. In this paper, cos-mological parameters H = 71 km s − Mpc − , Ω M = 0 . Λ = 0 .
23 (Spergel et al. 2003) are adopted followingChand et al. (2020), whose values of E iso ,γ and E p are di-rectly used in this paper. Then, the luminosity distance toGRB 198029A is 0.35 Gpc. In this section, following Huang et al. (2000), we describe amodel of jet dynamics and associated synchrotron emission.Let t b and r be the time and radial coordinates, respectively,in the rest frame of the central engine located at the origin, r = 0. In this frame, the polar angle θ is set such that thecentral axis of the jet corresponds to θ = 0. We assume auniform jet with a thin shell emitting region at radius R .The jet velocity is βc = dRdt b , (1)where c is the speed of light, and the bulk Lorentz factoris Γ = 1 / p − β . In the central engine frame, the jet isejected from the central engine at t b = 0. Initially, the jethas the opening half-angle θ , isotropic-equivalent kineticenergy E iso , K and the initial bulk Lorentz factor Γ . Ambientinterstellar matter (ISM) is assumed to be uniform with thenumber density n . The jet decelerates via interactions withISM and forms a thin shell. The decrease of Γ is given by d Γ dm = − Γ − M ej + ǫm + 2(1 − ǫ )Γ m , (2)where M ej = E iso , K / Γ c and ǫ γ are ejecta mass and theradiative efficiency, respectively, and m is the swept-up mass(see, e.g., Huang et al. 2000). For our parameters adoptedin §
3, the value of ǫ γ is too small to affect our results. Itis geometrically related to the shell radius R and the jetopening half-angle θ j as dmdR = 2 πR (1 − cos θ j ) n m p , (3)where m p is the mass of the proton. We assume that the jetspreads laterally at the sound speed c s measured in the shellcomoving frame (see Pe’er 2012; Nava et al. 2013, for moredetailed treatments on the dynamical evolution of calcula-tions of the gas temperature), and set the increase of the jetopening half-angle as dθ j dt b = c s Γ R . (4)Solving Eqs. (1)–(4), we get the jet dynamics, that is, Γ and θ j as a function of time. Following the standard convention, © , 1–9 ff-axis jet scenario for GRB 190829A Figure 1.
Schematic view of our two-component jet model for GRB 190829A. The red and blue cones represent narrow and wide jets,respectively. Initial shapes of the jets are depicted with their initial opening half-angles. The black arrow shows the observer’s line ofsight. As the jets expand, they spread sideways, and at ∼ × s when H.E.S.S. detected VHE gamma-rays, the observer’s line of sightis inside the cone of the narrow jet. their time evolution is shown with the on-axis observer ( θ =0) time t which is related to t b by dtdt b = 1 − β . (5)In calculating synchrotron radiation, we assume thatmicrophysics parameters ǫ e and ǫ B , the energy fractions ofinternal energy going into radiating electrons and magneticfield, are constant. The electron energy distribution in theemitting thin shell has a power-law form with index p . Inthe slow cooling regime, the electron spectrum has a breakat the electron cooling Lorentz factor γ c , where we take intoaccount the SSC cooling in the Thomson limit as well as syn-chrotron energy losses (Dermer, Chiang, & Mitman 2000;Sari & Esin 2001; Zhang & M´esz´aros 2001). Then, it has aform N ( γ e ) ∝ γ e − p when γ m < γ e < γ c and N ( γ e ) ∝ γ e − p − when γ c < γ e .We assume that the observer’s line of sight is θ = θ v .The flux density F ν of the afterglow emission that arrives atthe observer time T is obtained by integrating the emissivityover the equal arrival time surface determined by Z − β cos Θ βc dR = T z , (6)where Θ is the angle between the radial directionat each emitter position and the line of sight (e.g.,Granot, Piran, & Sari 1999). In summary, parameters of thepresent model are isotropic-equivalent kinetic energy E iso , K ,initial Lorentz factor Γ , initial jet opening half-angle θ , ISM density n , microphysical parameters ǫ e and ǫ B , elec-tron power-law index p , and the viewing angle θ v . In this section, we show our numerical results of synchrotronafterglow emission in the X-ray (10 Hz), optical (V-band),and radio (1.3 and 15.5 GHz) bands, and compare themwith observation data of GRB 190829A. The X-ray data areextracted from the
Swift team website (Evans et al. 2007,2009) which provides us with the integrated energy flux inthe 0.3–10 keV band and the photon indices at some epoch.The index was around 2.2 at any time. On the other hand,we numerically calculate the energy flux density F ν =10 Hz .In order to compare theoretical and observational results,we convert the observed integrated energy flux to the fluxdensity at 10 Hz assuming that the photon index is 2.2 atany time. The optical V-band data (before the absorptioncorrection) are obtained from Chand et al. (2020). In ournumerical calculation, we take the V-band extinction A V =1 . ©000
Swift team website (Evans et al. 2007,2009) which provides us with the integrated energy flux inthe 0.3–10 keV band and the photon indices at some epoch.The index was around 2.2 at any time. On the other hand,we numerically calculate the energy flux density F ν =10 Hz .In order to compare theoretical and observational results,we convert the observed integrated energy flux to the fluxdensity at 10 Hz assuming that the photon index is 2.2 atany time. The optical V-band data (before the absorptioncorrection) are obtained from Chand et al. (2020). In ournumerical calculation, we take the V-band extinction A V =1 . ©000 , 1–9 Sato et al.
First, we consider a single jet viewed off-axis in order to dis-cuss the observed X-ray and optical bumps around T ∼ . × s. We adopt θ v = 0 .
031 rad, θ = 0 .
015 rad, E iso , K = 4 . × erg, Γ = 350, n = 0 .
01 cm − , ǫ e = 0 . ǫ B = 5 . × − and p = 2 .
44. The initial opening half-angleis small, so that we refer to “narrow jet” in the following.Solid lines in the left panel of Fig. 2 show our results. Ouroff-axis afterglow model well explains the observational re-sults of early X-ray and optical afterglow from about 8 × to 2 × s. An achromatic behavior in the X-ray and opticalbands is evident. The off-axis afterglow starts with a risingpart because of the relativistic beaming effect (Granot et al.2002). As the jet decelerates, the observed flux increases.When the jet Lorentz factor becomes Γ ∼ ( θ v − θ ) − = 65,the afterglow light curve takes a maximum. After that, theobserved flux is almost the same as that in the case of on-axisviewing ( θ v = 0: dashed lines in the left panel of Fig. 2). Ifwe assume the adiabatic evolution (Γ ∝ t − / ), the observertime of the flux maximum is analytically given by T pk ∼ (1 + z ) E iso , K πn m p c ! ( θ v − θ ) . (7)For our model parameters, we get T pk ∼ × s, which isconsistent with our numerical results within a factor of two.As shown in Fig. 3 (thick-red-solid and dot-dashed lines),the scaling Γ ∝ t − / is roughly a good approximation until t ∼ s, since the jet spreading effect is not significant (seethe thick red line in Fig. 4). This fact validates the estimateof T pk by Eq. (7). For comparison, the dashed lines in the leftpanel of Fig. 2 show the light curves in the on-axis viewingcase ( θ v = 0), in which the X-ray flux peaks much earlierthan the optical one (Sari, Piran & Narayan 1998).After several tens of thousand seconds after the burstonset, our numerically calculated X-ray light curve deviatesfrom the observed data (see the left panel of Fig. 2). Beforethat epoch, the sideway expansion of the jet is not significant(thick-red-solid line in Fig. 4). As the jet decelerates, the jetbecomes trans-relativistic (Γ .
10) around t ∼ s, andthen θ j begins rapid increase (Zhang & MacFadyen 2009) .Then, our numerical result shows that the jet dynamicsasymptotically reaches the scaling Γ ∝ t − / (black-dottedline in Fig. 3), at which the observed X-ray flux follows thescaling F ν ∝ t − p = t − . (Sari, Piran, & Halpern 1999).This slope is much steeper than observed. Hence, it is dif-ficult for the narrow jet to explain the observed late X-rayafterglow as well as radio emission at late times. In the nextsection § In the past literature, it used to be assumed that a relativis-tic jet rapidly decelerates and its opening angle increases expo-nentially just after the jet break time which is given by t jet ∼ (3 E iso , K / πn m p c ) / θ / (Sari, Piran, & Halpern 1999), andfor our model parameters, we get t jet ∼ × s. How-ever, as shown by relativistic hydrodynamics simulation byZhang & MacFadyen (2009), the lateral expansion is not signifi-cant until the trans-relativistic phase. Second, we consider a two-component jet model, in whichanother “wide jet” is introduced in addition to the narrowjet considered in § § θ = 0 . E iso , K = 2 . × erg,Γ = 20, ǫ e = 0 . ǫ B = 1 . × − , and p = 2 .
2. The val-ues of θ v and n are common for both jets. It is assumedthat the central axes of the two jets are identical ( θ = 0: seeFig. 1).One can find in the right panel of Figure 2 that earlyachromatic peaks in the X-ray and optical bands are ex-plained by the off-axis narrow jet emission (dashed lines inthe right panel), and that the late X-ray and radio afterglowis interpreted with the wide jet emission (dotted lines). Asshown in the thin-blue-solid line in Fig. 3, the wide jet doesnot decelerate until t ∼ × s, since it is heavy with a lowbulk Lorentz factor, Γ = 20. Our numerical result (dottedlines in the right panel of Fig. 2) shows that X-ray and opti-cal flux becomes maximum around this epoch. The wide jetbecomes trans-relativistic (Γ .
10) at t ∼ s and finallyenters to the Newtonian phase at t & s. We find thatfor 10 s . t . s, the absorption frequency ν a , typicalfrequency ν m and cooling frequency ν c obeys the relation ν a < ν m < ν c . The value of ν c is located between the opti-cal and the X-ray bands, and ν m is lower than the opticalband. Then, the X-ray and optical fluxes follow the scalings F ν ∝ t − (3 p − / = t − . and F ν ∝ t − p − / = t − . , re-spectively (e.g., Gao et al. 2013), which is consistent withthe observation. The typical frequency ν m decreases withtime, and at t ∼ × s, it crosses 15.5 GHz, at which the15.5 GHz light curve has a peak. After that, the flux followsthe scaling F ν ∝ t − p − / = t − . (e.g., Gao et al. 2013).Subsequently, ν m intersects 1.3 GHz at t ∼ × s, andthe 1.3 GHz flux takes maximum, after which the flux de-cays in the same manner. These radio behavior is roughlyconsistent with the observation. Note that if the initial bulkLorentz factor of the wide jet Γ is larger than 20, our nu-merical X-ray light curve is brighter than the observed datain earlier epoch. × SECONDS
In this section, we estimate the VHE gamma-ray flux at2 × s along with our two-component jet model consideredin §
3. For simplicity, the SSC flux at hν = 0 . F ν at2 × s as seed photons for SSC emission. First, we considerthe narrow jet, which has the bulk Lorentz factor Γ ≃ B ≃ . × − G, the min-imum electron Lorentz factor γ m ≃ . × , the elec-tron cooling Lorentz factor γ c ≃ . × [14 / (1 + Y )],the typical frequency ν m ≃ . × Hz, the cooling fre-quency ν c ≃ . × [14 / (1 + Y )] Hz and the peak flux © , 1–9 ff-axis jet scenario for GRB 190829A -6 -5 -4 -3 -2 -1 X-rayV-band1.3 GHz ( ×
10) 15.5 GHz F l ux [ m J y ] Time from burst onset : T [s] 10 -6 -5 -4 -3 -2 -1 X-rayV-band1.3 GHz ( ×
10) 15.5 GHz F l ux [ m J y ] Time from burst onset : T [s]10 -6 -5 -4 -3 -2 -1 X-rayV-band1.3 GHz ( ×
10) 15.5 GHz F l ux [ m J y ] Time from burst onset : T [s]
Figure 2.
Afterglow light curves in the X-ray (10 Hz: red), optical (V-band: blue) and radio bands (1.3 GHz: orange, 15.5 GHz:green), which is compared with the observed data of GRB 190829A (X-ray: red points, V-band: blue triangles, 1.3 GHz: orange filled-circles, 15.5 GHz: green squares). In the left panel, solid and dashed lines show the emission from the narrow jet ( θ = 0 .
015 rad, E iso , K = 4 . × erg, Γ = 350, n = 0 .
01 cm − , ǫ e = 0 . ǫ B = 5 . × − and p = 2 . θ v = 0 .
031 rad)and on-axis ( θ v = 0), respectively. In the right panel, we show the results of our two-component jet model — solid lines are the sum ofthe narrow (dashed lines) and wide (dotted lines) jets. The latter has parameters, θ v = 0 .
031 rad, θ = 0 . E iso , K = 2 . × erg,Γ = 20, n = 0 .
01 cm − , ǫ e = 0 . ǫ B = 1 . × − and p = 2 . -1 Γ β , R / c m On-axis observer time : t [s]
Figure 3.
The four-velocity Γ β (solid lines) and radius R (dashedlines) of narrow (thick-red lines) and wide (thin-blue lines) jets asa function of the on-axis observer time t . The black dot-dashedand dotted lines represent analytical scalings Γ β ∝ t − / (adi-abatic evolution without sideway expansion) and Γ β ∝ t − / (adiabatic evolution with sideway expansion), respectively. F max = F ν = ν m ≃ . ×
10 mJy. Then, the break frequen-cies for the SSC emission (Sari & Esin 2001) are given by ν IC m ≈ γ m ν m ≃ . × Hz and ν IC c ≈ γ c ν c ≃ . × [14 / (1 + Y )] Hz. One can find that the observationphoton energy hν = 0 . p ν IC m ν IC c < ν < ν IC c ,so that the SSC flux is calculated by F SSC ν ≈ . Rσ T n F max ( p − p + 1) νν IC m ! (1 − p ) / × " p + 3)( p + 2) − p + 1)( p + 2) + ln ν IC c ν ! , (8)where σ T is the Thomson cross section. Hence, the SSC θ j [r a d ] On-axis observer time : t [s]
Figure 4.
Jet opening half-angle θ j as a function of the on-axisobserver time t . Thick-red-solid and thin-blue-solid curves are fornarrow and wide jets, respectively. Two horizontal dotted anddashed lines are the initial values (0.015 rad and 0.1 rad for thenarrow and wide jets, respectively). energy flux from the narrow jet is estimated as νF SSC ν ∼ . × − erg s − cm − . Since the jet energy is large, wehave a lot of seed photons from its own synchrotron radiationto get detectable SSC emission. In reality, the Klein-Nishinaeffect becomes important below ν IC c . The Y parameter at γ c is significantly reduced due to the Klein-Nishina effect, sothe VHE gamma-ray flux is expected to have a peak aroundTeV energies below ν IC c in the Thomson limit. Correspond-ingly, the value of ν c would be underestimated.Similarly, we calculate the SSC flux from the wide jet.At 2 × s, we have Γ ≃ B ≃ . × − G, γ m ≃ . × , γ c ≃ . × [10 / (1 + Y )], ν m ≃ . × Hz, ν c ≃ . × [10 / (1 + Y )] Hz, and F max ≃ . ν IC m ≃ . × Hz and ν IC c ≃ . × [10 / (1 + Y )] Hz, resulting in νF SSC ν ∼ × − © , 1–9 Sato et al. erg s − cm − at hν = 0 . × s. Justaround this epoch, the wide jet is in the transition from thefree expansion to the adiabatic deceleration phase. Indeed,as shown by the dotted lines in the right panel of Fig. 2, wesee a rising part in X-ray and optical bands. After 2 × s,the SSC flux from the wide jet might increase just for awhile. However, it would soon start to decrease and keepsubdominant.We discuss the time dependence of the VHE gamma-rayflux between ∼ and ∼ s in the Thomson limit. Inthis epoch, the observed flux for θ v = 0 .
031 hardly changesfrom that for θ v = 0 (for narrow jets, see the left panelof Fig. 2), so that the standard analytical calculation foron-axis observer is a good approximation for the presentcase. We find that for both jets, the break frequencies of thenarrow and wide jets at hν = 0 . ν IC m < ν <ν IC c at any time, so that the SSC flux is given by F SSC ν ∝ Rσ T n F max ( ν/ν IC m ) (1 − p ) / . For the narrow jet, the Lorentzfactor is approximated to follow the scaling Γ ∝ t − / (seethe thick-red-dashed line in Fig. 3). Then for synchrotroncomponent, we have γ m ∝ t − / , ν m ∝ t − and F max ∝ t − (Sari, Piran, & Halpern 1999), so that ν IC m ∝ t − and F SSC ν ∝ t − (3 p − / = t − . . On the other hand, as discussedin the previous paragraph, the VHE flux from the wide jethas brighten, following the analytical scaling F SSC ν ∝ t until t ∼ × s at the transition from the free expansion tothe adiabatic deceleration phase. After this time, the fluxdecays. For the jet dynamics Γ ∝ t − / ( ∝ t − / ), we getthe VHE flux F SSC ν ∝ t − (9 p − / = t − . ( ∝ t − (3 p − / = t − . ). Such a time evolution could have been observed withgood statistics by more sensitive detectors like CTA. The prompt emission of GRB 190829A had smaller values ofthe peak energy E p and the isotropic-equivalent gamma-rayenergy E iso ,γ than typical long GRBs. In this section, we dis-cuss whether E p and E iso ,γ from our narrow jet were typicalor not if it would have been viewed on-axis ( θ v ≈ = 1 / p − β . The jetis uniform, whose intrinsic emission properties do not varywith angle, and has a sharp edge at the opening half-angle θ . Then, the viewing-angle dependence of the peak en-ergy, E p ( θ v ), and isotropic-equivalent gamma-ray energy, E iso ,γ ( θ v ), can be analytically calculated (Donaghy 2006;Graziani, Lamb, & Donaghy 2006), and we obtain R = E p ( θ v ) E p (0)= 2(1 − β )(1 − β cos θ )2 − β (1 + cos θ ) × f ( β − cos θ v ) − f ( β cos θ − cos θ v ) g ( β − cos θ v ) − g ( β cos θ − cos θ v ) , (9) R = E iso ,γ ( θ v ) E iso ,γ (0)= (1 − β ) (1 − β cos θ ) β (1 − cos θ )[2 − β (1 + cos θ )] × [ f ( β − cos θ v ) − f ( β cos θ − cos θ v )] , (10)where, functions f and g are given by f ( z ) = Γ (2Γ − z + (3Γ sin θ v − z + 2 cos θ v sin θ v | z + Γ − sin θ v | , (11)and g ( z ) = 2Γ z + 2 cos θ v | z + Γ − sin θ v | , (12)respectively (see also Urata et al. 2015). For parameters ofour narrow jet given in § = 350, θ = 0 .
015 rad and θ v = 0 .
031 rad), we get R = 3 . × − and R = 1 . × − .If the narrow jet emitted Episode 1 of observed promptemission (see § E p ( θ v = 0 . E iso ,γ ( θ v = 0 . . × erg (Chand et al. 2020), thenon-axis quantities are obtained as E p (0) = E p ( θ v ) /R =3 . E iso ,γ (0) = E iso ,γ ( θ v ) /R = 2 . × erg.These values are within the range for bursts detected sofar (e.g., Zhao et al. 2020a). The isotropic equivalent ki-netic energy of the narrow jet just after the prompt emis-sion is E iso , K = 4 . × erg (see § η γ = E iso ,γ (0) / ( E iso ,γ (0) + E iso , K ) ≈ .
4. On the other hand, ifthe narrow jet is responsible for Episode 2 (that is, E p ( θ v =0 . E iso ,γ ( θ v = 0 . . × erg),we obtain E p (0) = 340 keV and E iso ,γ (0) = 1 . × erg,which are again similar to typical long GRBs. In this case,the efficiency is η γ ≈ . § § η γ is almost typical, however on-axis E p (0) is lo-cated at the highest end of the distribution for long GRBs.On the other hand, if the narrow jet produced Episode 2,then on-axis E p (0) is smaller though η γ is somewhat higher(but it is still comparable, and one can say that the valueis reasonable considering very simple approximation of ourprompt emission model). Episode 1 and 2 may be emittedfrom narrow and wide jets, respectively. Note that if the widejet emits Episode 2, its efficiency is small, η γ ≈ × − ,so that it might be natural that the narrow jet causes bothEpisode 1 and 2.In this section, we simply assumed that the promptemission was caused by a top-hat shaped jet, and ob-tained the ratios, R = E p ( θ v ) /E p (0) ∼ − and R = E iso ,γ ( θ v ) /E iso ,γ (0) ∼ − , for our narrow jet. For off-axisjet emission, these values depend on the profile of angulardistribution of the bulk Lorentz factor and intrinsic emissiv-ity near the periphery of the jet. If the jet is structured likea Gaussian or power-law profile, then R and R may belarger in the off-axis case (e.g., Salafia et al. 2015), so thaton-axis E p (0) and E iso ,γ (0) may be smaller than the presentestimates. © , 1–9 ff-axis jet scenario for GRB 190829A We have investigated an off-axis jet scenario in which wehave invoked a two-component jet model to explain theobservational results of GRB 190829A. According to ourmodel, the early X-ray and optical afterglow was off-axisemission from the narrow jet, which may also be responsi-ble for VHE gamma-rays detected at ∼ × s, and thelate X-ray and radio afterglow came from the wide jet (Fig-ure 1). Since the narrow jet was viewed off-axis, the promptemission was dim and soft due to the relativistic beamingeffect. On the other hand, the wide jet had the isotropic-equivalent kinetic energy E iso , K ∼ erg which was muchlarger than the observed isotropic equivalent gamma-ray en-ergy E iso ,γ ∼ − erg. If the wide jet has a typical valueof the efficiency of the prompt emission, our result wouldbecome inconsistent with the observational result because itis seen on-axis. Since the initial bulk Lorentz factor of thewide jet is Γ = 20, the jet is likely to be dirty (i.e., highlyloaded by baryons) and it may have a large optical depth.It may be as small as η γ = E iso ,γ / ( E iso ,γ + E iso , K ) . − unlike typical bright GRBs with high Lorentz factors.There are still some observed components that arebrighter than the prediction of our jet model. They may beother components. For example, very early ( T . × s)X-ray emission should be the contribution from late promptemission like flares. The observed optical flux later than ∼ × s is a supernova component (Hu et al. 2020). Atthe late epoch ( T ∼ s), the 15.5 GHz radio flux also ex-ceeds our numerical result, which could be other componentssuch as counter-jet emission.As seen in the right panel of Fig. 2, our theoretical ra-dio fluxes in both 1.3 and 15.5 GHz sometimes overshot theobserved ones. However, the excess is only within a factorof two, and this difference may come from the uncertaintyof our simple model. More realistic modeling may solvethis problem. For example, structured jets such as Gaus-sian jets instead of uniform jets would decrease the radiofluxes keeping the X-ray and optical brightness unchanged(e.g., Cunningham et al. 2020).We have also estimated the VHE gamma-ray flux at2 × s and have found that the narrow jet dominates theobserved gamma-ray emission. Since the synchrotron radi-ation is bright enough due to the large jet energy, the ob-served VHE gamma-ray flux, νF ν ∼ − erg s − cm − , isable to be expected by SSC mechanism. In this paper, weindependently calculate two emission components from twojets. External inverse Compton with seed photons comingfrom the companion jet might be effective (e.g., Zhang et al.2020a). Such an interaction between two jets remains to befuture work.Late-time ( T ∼ − s) X-ray synchrotron emissionfrom the wide jet is about a factor of two smaller than ob-served data (see the red solid line in the right panel of Fig. 2).In calculating the synchrotron radiation, we have assumedthe Thomson limit to derive ν c for simplicity. If we con-sider the Klein-Nishina effect (Nakar, Ando, & Sari 2009;Murase et al. 2010; Wang et al. 2010; Murase et al. 2011;Jacovich, Beniamini, & van der Horst 2020; Zhang et al.2020a), the Compton Y parameter becomes smaller, so that ν c becomes larger. Then, the X-ray flux increases if ν c isaround the X-ray band. As a limiting test case, we have calculated the X-ray synchrotron emission setting Y = 0all the time. In this case, the X-ray flux actually becomeslarger but by less than ten times. It is expected that in-clusion of the Klein-Nishina effect causes the increase ofthe hard X-ray flux. Other possibilities to have a larger X-ray flux in the late epoch include delayed energy injection(e.g., Zhang et al. 2006) and/or a low-energy part of SSCor external inverse-Compton emission (e.g., Fan et al. 2008;Zhang & M´esz´aros 2001; Zhang et al. 2020a).The initial Lorentz factor of our narrow jet is Γ =350, which may be similar to or slightly smaller thanthose of long GRBs with VHE gamma-ray detection. ForGRBs 190114C and 180720B, the afterglow onset peaktime may imply the initial Lorentz factor of ≈
500 and ≈ ǫ B is on the order of 10 − , which is also similar to theother two long VHE events (Ajello et al. 2020; Fraija et al.2019a,b,c; Wang, et al. 2019; Jordana-Mitjans et al. 2020).At present, although the number of VHE events is small,these values are common for events with detectable VHEgamma-rays. If there is no magnetic field amplification, ǫ B is about 10 − ( n / . − ) − ( B ISM / µ G) , where B ISM is the magnetic field strength for the ambient medium.Therefore, the magnetic field in the emission region ofthose GRB afterglows is expected to be amplified. Al-though the mechanism has not been understood yet (e.g.,Tomita, Ohira, & Yamazaki 2019), more detailed observa-tions of VHE gamma-rays would provide us a new hint ofthe magnetic field amplification mechanism (e.g., Lemoine2015).The initial opening half-angle of the narrow jet is θ =0 .
015 rad. This is near the lower limit of previously measuredvalues for long GRBs, however, it is still larger than thesmallest one (Zhao et al. 2020a). In our model, the narrowjet is seen off-axis, resulting in dim prompt emission. Nev-ertheless, this event was observed since it occurred nearby.Hence, similar but distant ( z ≫ .
1) events must be viewedon-axis to be detected. However, a small solid angle of thenarrow jet decreases the detection rate, which may explainthe small number of VHE gamma-ray events that have beendetected so far.Compared with other long GRBs with radio detec-tion, GRB 190829A showed a lower radio afterglow lumi-nosity (Rhodes et al. 2020), which allows us to adopt a lowambient density of n = 0 .
01 cm − . However, there maybe two classes in long GRBs, radio-loud and radio-quietevents (Zhang et al. 2020c). Although radio-loud GRBs haveslightly larger isotropic-equivalent energies E iso ,γ of theprompt gamma-ray emission, the E iso ,γ distributions for thetwo classes look similar (see Fig. 11 of Zhang et al. 2020c).It might be possible that long GRBs arise in the rarefiedmedium. Such an environment appears when the wind of aprogenitor star is strong, or the bursts occur in the super-bubble made by OB association. ACKNOWLEDGMENTS
We thank Katsuaki Asano, Kunihito Ioka,Kazumi Kashiyama, Takanori Sakamoto, Motoko Serino,Shuta J. Tanaka, and Kenji Toma for valuable comments. © , 1–9 Sato et al.
This research was partially supported by JSPS KAKENHIGrant Nos. 18H01232 (RY), 20H01901 (KM), 20H05852(KM) and JP19H01893 (YO). R.Y. deeply appreciatesAoyama Gakuin University Research Institute for helpingour research by the fund. The work of K.M. is supportedby NSF Grant No. AST-1908689. Y.O. is supported byLeading Initiative for Excellent Young Researchers, MEXT,Japan.
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