GRB 170817A Afterglow from a Relativistic Electron-Positron Pair

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E × X aito ore(19;Nurnstars(1108) Neutron sources(1119); radiation 0 am-a uss69;Gaiainlwvs68;Hydrodyn waves(678); Gravitational bursts(629); Gamma-ray INTRODUCTION 10 . twocolumn 1rd n h iwn angle viewing the and rad, 11 1 .Tedtcino R 170817A GRB of detection The ). Fermi colo srnm n pc cec,NnigUniversity, Nanjing Science, Space and Astronomy of School 52 ole ta.2017 al. et Coulter r.Acnitnebtentecretdo-xsvle o GRB for values on-axis corrected the between consistence A erg. e + e .Aot17scnsaf- seconds 1.7 About ). − am-a us Moni- Burst Gamma-ray ; arwn oe ihasml o-a e iwdo-xscnreproduc can off-axis viewed jet top-hat simple a with model wind pair tl nAASTeX63 in style odti ta.2017 al. et Goldstein .Teeobser- These ). e + .Counter- ). ogLi Long e − arwn rmarpdyrttn,srnl antzdneu- magnetized strongly rotating, rapidly a from wind pair ) ,2 1, θ ABSTRACT v and ≈ ; 0 . 3rd h etfitn au of value best-fitting The rad. 23 iGoDai Zi-Gao g fCia ee 306 hn;[email protected] China; 230026, Hefei China, of ogy o rmrdot -asi ieetfo yia GRB typical a from different is X-rays to radio from ior ( and decreasing merger, rapidly the to after started days then 160 around in- peaked gradually was creasing, afterglow multi-wavelength a of nosity multi-messenger for possible astronomy. breakthrough and a merger BNS showing a products, of process the understand helps to counterparts electromagnetic com- following signal with gravitational-wave from bined The afterglow GRB shock. the relativistic with a spec- consistent power-law is non-thermal which a trum, by characterized be can 2020 2018a al. et Mooley detected was counterpart band ( radio a later, ( time AT2017gfo longer of position the emis- at X-ray sion detected ejecta Observatory X-ray the Chandra later, within synthesized nuclei ( neutron- process rapid of capture decay radioactive the by powered nova alnne l 2017 al. et Hallinan 2017 al. et Kasen B otne olwu bevtosrva httelumi- the that reveal observations follow-up Continued p η gUiest) iityo dcto,China Education, of Ministry University), ng ; ≈ . rj ta.2020 al. et Troja 0 1 10 ,1 3, . ≈ 6 − × 3 7adteiorpckntcenergy kinetic isotropic the and 47 ouealt-iesalwdcyin decay shallow late-time a roduce nPi idOsre Off-axis Observed Wind Pair on ajn 103 China 210023, Nanjing r,wihgv h e half-opening jet the give which ers, 10 snee ostsyobservational satisfy to needed is pouetepop msinof emission prompt the eproduce a hc n ehpsafterglow reshapes and shock nal 87.TeMro hi Monte chain Markov The 0817A. niu odciei h coming the in decline to ontinue 7i oglvdN,w show we NS, long-lived a is 17 13 n lcrmgei observa- electromagnetic and ; sotie norfitting, our in obtained is G ; une l 2018 al. et Ruan ine l 2017 al. et Pian mc(93;Non-thermal amics(1963); .TeXryadrdoemission radio and X-ray The ). .Temliwvlnt behav- multi-wavelength The ). 787 n typical and 170817A θ v scoet the to close is rj ta.2017 al. et Troja agtie l 2018 al. et Margutti ; .Aot9days 9 About ). ahtiie al. et Makhathini multi- e .A ). ; 2 Li & Dai afterglow. Considering the low luminosity of the prompt parent velocity of the radio source along the plane of the emission, the explanations for the afterglow luminos- sky v = (4.1 0.5)c was measured. Recently, Chan- app ± ity evolution usually fall into two types. One explana- dra X-ray Observatory continuous detected X-ray emis- tion is invoking a structured jet which have an angular sion from the location of GRB 170817A (Hajela et al. distribution in energy and Lorentz factor. The GRB 2020a,b, 2021a,b), which extended the afterglow data 170817A afterglow is well consistent with the emission to about 3.3 years after the merger. The late-time de- from structured jet viewed off-axis (Abbott et al. 2017c; rived unabsorbed X-ray flux is higher than what is ex- Troja et al. 2017; Lazzati et al. 2018; Lyman et al. 2018; pected from the structured jet model (Makhathini et al. Margutti et al. 2018; Mooley et al. 2018b; Troja et al. 2020; Troja et al. 2020; Hajela et al. 2020b, 2021a,b). 2018; Hajela et al. 2019; Lamb et al. 2019; Troja et al. However, the late-time radio observations did not show 2019). The second explanation is based on a classical this excess (Balasubramanian et al. 2021; Hajela et al. top-hat jet model with considering additional continu- 2021b). Therefore, it is necessary to revisit an off-axis ous energy injection into a jet (e.g. Geng et al. 2018; afterglow from an e+e− pair wind scenario supposing Li et al. 2018; Lamb et al. 2020). that the remnant of GW170817 is a long-lived massive On the other hand, the central remnant of GW170817 NS. which depends on the neutron star (NS) equation of This paper is organized as follows. In subsection 2.1 state (EoS) remains up in the air. Depending on we give the details of our numerical afterglow model whether the NS EoS is stiff enough, the remnant could from an e+e− pair wind scenario. In subsection 2.2 be a black hole, a temporal hypermassive NS, or a we describe the methods of fitting the data from GRB long-lived massive NS. Although there are theories 170817A with our model. Our fitting data includes against a long-lived massive NS as the merger rem- the thousand-day multi-wavelength afterglow and the nant (e.g. Granot et al. 2017; Margalit & Metzger 2017; VLBI proper motion. In Section 3 we describe our re- Ciolfi 2020), there is no direct evidence that rules out the sults. Our discussion and conclusions can be found in possibility that the post-merger remnant is a stable NS. Section 4. A concordance cosmology with parameters −1 −1 A relativistic jet could also be launched through νν¯ anni- H = 69.6km s Mpc , ΩM =0.286, and ΩΛ =0.714 hilation in a neutrino-dominated accretion flow (NDAF) is adopted in this paper. mechanism from such a stable NS with a hyper-accreting accretion disk (Zhang & Dai 2008, 2009, 2010). Besides, 2. METHOD the existence of an NS remnant can reduces the require- 2.1. The relativistic e+e− pair wind model ment on the ejecta mass in the kilonova model due to We assume that the central remnant of GW170817 is an additional energy injection from the NS (Li et al. a long-lived NS. A Poynting-flux-dominated outflow is 2018; Metzger et al. 2018; Yu et al. 2018). If the rem- launched from the NS, whose wind luminosity is domi- nant is a long-lived massive NS, it is suggested that a nated by the magnetic dipole spin-down luminosity, i.e. Poynting flux-dominated outflow would flow out from (Shapiro & Teukolsky 1983) the NS. This outflow can be accelerated due to magnetic dissipation (e.g. reconnection), and eventually domi- 2 6 4 BpRsΩ 44 2 −4 6 nated by the energy flux of ultra-relativistic wind con- Lw = 3 9.64 10 Bp,13P0,−3Rs,6 + − 6c ≃ × sists of electron-positron (e e ) pairs (Coroniti 1990; −2 t − Michel 1994; Kirk & Skjæraasen 2003). Dai (2004) re- 1+ obs erg s 1, (1) + − × τ alized that the continuous ultra-relativistic e e pair sd wind would interact with the GRB external shock and − where B = 1013B G, P = 10 3P − s, and R = reshape the afterglow emission signatures. Yu & Dai p p,13 0 0, 3 s 106R cm are the polar surface magnetic field strength, (2007) used this model to explain the shallow decay s,6 radius, and initial spin period of NS, respectively. t phase of the early GRB X-ray afterglows. Geng et al. obs is the time measured in the observer frame. τsd = (2018) modeled the first 150 days of GRB 170817A af- 7 −2 2 −6 2.05 10 I B P − R s is the characteristic spin- terglow lightcurves in this scenario. However, some new × 45 p,13 0, 3 s,6 down time scale, where I = 1045I g cm2 is the moment observational facts have been updated since then. Soon 45 of inertia of the NS. For simplicity, P − , R , and I afterwards, Mooley et al. (2018c) reported the radio ob- 0, 3 s,6 45 are taken to be unity throughout this work. servations of GRB 170817A afterglow using a Very Long As it propagates outwards, the Poynting-flux domi- Baseline Interferometry (VLBI), and found that the ra- nated outflow can be dissipated and accelerated by mag- dio source shows superluminal apparent motion between netic reconnection, which is eventually dominated by an 75 days and 230 days after the merger event. A mean ap- ultra-relativistic wind consisting of e+e− pairs with bulk Relativistic Electron-Positron Wind 3

radiation from shock-accelerated electrons. The details of calculating the multi-wavelength lightcurves can be seen in Appendix A.

2.2. Modeling the GRB 170817A afterglow A numerical model is established to fit the GRB 170817A afterglow data including the multi-wavelength afterglow and the VLBI proper motion. The multi- wavelength afterglow data are taken from the following literature: Makhathini et al. (2020) collected and reprocessed the available radio, optical and X-ray data spanning from 0.5 days to 1231 days after merger, Balasubramanian et al. (2021) updated the radio observations to about 3.5 years after merger, and Hajela et al. (2020b, 2021a) updated the 0.3-10 keV X-ray data to 3.3 years after merger. Our fitting data include following specific bands: frequencies at 3 GHz and 6 GHz + − Figure 1. A cartoon picture of the relativistic e e pair from Karl G. Jansky Very Large Array (VLA), wave- wind model with a top-hat jet viewed off-axis, if the post- length at 600nm from Hubble Space Telescope (HST) merger remnant is a rapidly spinning magnetized NS. F606W, energy at 1 keV from Chandra X-ray Obser- vatory and XMM-Newton Observatory, and energy at 4 7 Lorentz factor Γw 10 10 (Atoyan 1999). We here 0.3-10 keV from Chandra X-ray Observatory, which can 4 ∼ − adopt Γw = 10 as a fiducial value in our calculations. be regarded as representative of the multi-wavelength af- Based on the luminosity Lw and the bulk Lorentz fac- terglow, as shown in Figure 2. In order to fit the VLBI + − tor Γw of the e e pair wind, and the assumption that proper motion, we develop a method used to calculate + − the e e pair wind is isotropic, the comoving electron the proper motion of the flux centroid, which can be seen number density can be estimated as in Appendix B. As part of fitting data, we construct a data point based on the mean apparent velocity of the ′ Lw nw 2 2 3 , (2) source, as shown in Figure 3. ≃ 4πR Γ mec w We develop a Fortran-based numerical model and where me is the mass of electron, c is the speed of light, implement the Markov Chain Monte Carlo (MCMC) and R is the distance from the central engine. techniques by using the emcee Python package As shown in Figure 1, most of the wind energy is (Foreman-Mackey et al. 2013). F2PY is employed to injected into the kilonova ejecta, only a small fraction provide a connection between Python and Fortran lan- ((1 cos θj )/2 where θj is the half opening angle of jet) guages. We perform the MCMC with 22 walkers for − − of the e+e pair wind propagates outside the ejecta in running at least 100,000 steps, until the step is longer + − the jet direction. when the ultra-relativistic e e pair than 50 times the integrated autocorrelation time τac, 5 wind interacts with its surrounding medium, a pair of i.e. Nstep = max(10 , 50τac) to make sure that the shocks will develop: a forward shock (FS) that propa- fitting is sufficiently converged (Foreman-Mackey et al. gates into the medium, and a reverse shock (RS) that 2013). Once the MCMC is done, the best-fitting val- propagates into the wind. Before this interaction, a ues and the 1σ uncertainties are computed as the 50th, forward shock has formed when the GRB jet interacts 16th, and 84th percentiles of the posterior samples. with the medium. For simplicity, we here assume that The free parameters in the fitting include: the half the two forward shocks eventually merged into one for- opening angle of jet (θj ), the viewing angle (θv), the ward shock. Thus, there are four regions separated by NS surface magnetic field strength at the polar cap re- two shocks: (1) the unshocked medium, (2) the shocked gion (Bp), the isotropic kinetic energy of the jet after jet and medium, (3) the shocked wind, and (4) the un- prompt emission (EK,iso), the initial Lorentz factor of shocked wind (Dai 2004). Regions 2 and 3 are separated the jet after prompt emission (Γ0), the number density by the contact discontinuity. of medium (n1), the fraction of the shock internal en- We solve a set of differential equations which describe ergy that is partitioned to magnetic fields in regions 2 the dynamics of such an FS-RS system and consider (ǫB,2) and 3 (ǫB,3), the fraction of the shock internal both synchrotron radiation and inverse Compton (IC) energy that is partitioned to electrons in region 2 (ǫe,2), 4 Li & Dai the electron energy spectral index in regions 2 (p2) and 3 (p3). The ǫe,3 is not regarded as a free parameter because the implicit condition ǫe,3 + ǫB,3 = 1 in region 3 (Dai 2004; Yu & Dai 2007; Geng et al. 2016, 2018). Uniform priors on θj , θv, log Bp, log EK,iso, log n1, log ǫB,2, log ǫB,3, log ǫe,2, p2, and p3 are adopted in this paper. In order to explore a parameter space as large as possible, we set a wide enough range for the priors, except for θv. Abbott et al. (2017a) derived θ 56◦ at 90% credible intervals with a low spin prior v ≤ by using the distance measured independently by GW. Finstad et al. (2018) obtained θv with the distance mea- surement from Cantiello et al. (2018) and the GW data from Abbott et al. (2017a). They derived the 90% con- +10 fidence region on the viewing angle is θ = 32− 1.7 v 13 ± degrees, and derived a conservative lower limit on the ◦ viewing angle θv 13 . Abbott et al. (2019) derived ≥ +15 1 the binary inclination angle θJN = 151−11 degrees with a low spin prior by jointing GW and electromagnetic (EM) observations. Conservatively, we limit the prior range for θ to 0.23 θ 0.7 (labeled Prior A). As a v ≤ v ≤ comparison, we also show the fitting results from priors with θ [0, 0.7] (labeled Prior B). The free parameters v ∈ and priors in our model can be seen in Table 1.

3. RESULTS Figures 2 and 3 show the data of the afterglow, and the best-fitting results from Prior A. In Appendix C, we show the fitting results from Prior B. We find that for both priors the overall quality of the fitting is good, except for the early-time X-ray lightcurve which exhibit − some deviation from the first data point. Table 1 shows Figure 2. Relativistic e+e pair wind model fit to the 2 multi-wavelength afterglow lightcurves of GRB 170817A. the χdof 3 and 2 for Prior A and B, indicating that ≈ ≈ The dashed and dotted curves represent the emission from a large parameter space for θv would improves the fitting quality. The corner plots shown in Figure 5 indicate that the FS and RS, respectively. The solid curves are the total emission lightcurves. the posterior distribution of θv close to the lower limit of its prior, which means a smaller θv would lead to a better flow velocity for NGC 4993 or the constraint on Hubble fit. After performing a full prior of θv ranging from 0 constant H0. Considering the reasonably good fit from to 0.7, the posterior distribution of θv exhibits a normal distribution peaking at about 0.14 rad ( 8 deg), as Prior A, we conclude the results from Prior A are more ≈ shown in Figure 10. The derived θ 8 deg from Prior reasonable than those from B. As shown in Appendix C, v ≈ B is consistent with the viewing angle inferred from the the results from Prior B are similar to Prior A, except detection of GW170817 (θ 55 deg at 90% confidence for the derived smaller θj and θv, and the difference in v ≤ with a low-spin priors; Abbott et al. 2017a), but lower the apparent source size which we describe below. In than the viewing angle inferred by combining GW and the following we only discuss the results from Prior A. EM constraints (usually & 10 deg; Finstad et al. 2018; In particular, our model can well reproduce the slowly rising phase of the early-time lightcurves. Instead of Mandel 2018; Abbott et al. 2019). The smaller θv may pose a greater challenge to the estimation of the Hubble invoking a structured jet, our model suggests that the initial slowly rising lightcurves are a consequence of the peak time difference between the FS emission and RS 1 Supposing the jet is aligned with the the angular momentum, emission. The FS emission reaches to a peak when the one can easily connect θv to the binary inclination angle θJN ≡ ◦ − Lorentz factor of the jet Γ 1/(θv θj ), while the according to θv min(θJN , 180 θJN ). ∼ − Relativistic Electron-Positron Wind 5

Figure 3 shows the best-fitting results for apparent velocity. A consistency between the model and the data suggests that our model can also reproduce the superluminal apparent motion between 75 days and 230 days after the merger. We also plot the dimensionless apparent velocity βapp,j and Lorentz factor Γj of the location which is along the edge of the jet and closest to the line of sight (LOS) in Figure 3. One can see βapp,j reaches the maximum value β = Γ2 1 Γ when app,j,max j − ≈ j Γj = 1/ sin(θv θj ) 8.4 (Zhangq 2018). The dimen- − ≈ sionless apparent velocity of the flux centroid βapp trace βapp,j at early times, but βapp is clearly larger than βapp,j at late times. As shown in Figure 3, we also plot the apparent velocity of the FS flux centroid βapp,FS and the RS flux centroid βapp,RS. There is almost no difference − Figure 3. Relativistic e+e pair wind model fit to the VLBI between βapp,j, βapp,FS, and βapp,RS, which means that proper motion of GRB 170817A afterglow. The mean di- βapp is larger than βapp,j, which is not because of the − mensionless apparent velocity of the radio afterglow source RS emission from the e+e pair wind. In order to il- ± βapp = 4.1 0.5 between 75 days and 230 days after the lustrate this discrepancy, we show the predicted source merger from Mooley et al. (2018c) yields the data point used radio images at 75 days, 207.4 days, and 230 days seen to fit. The green, red, and blue solid curves show the evolu- by the observer in Figure 4. The black plus sign marks tion of dimensionless apparent velocity of the flux centroid the location which is closest to the LOS, and the white based on our best-fitting parameters when considering FS emission, RS emission and total emission, respectively. The plus sign marks the flux centroid. One can see the proper blue dashed and dotted curves denote βapp,j and Γj. motion of the flux centroid is caused by three variations: the movement of the jet relative to observer, the varia- peak time of the RS emission is roughly equal to the tion of the radio afterglow image size, and the changes spin-down time scale of the NS (Geng et al. 2016). In in the location of the flux centroid relative to the jet. our fitting, the peak times of FS and RS emission are All these variations would keep the flux centroid away about 50 days and 150 days respectively, and the super- from observer for an off-axis configuration. Therefore, a position of these two components flatten the radio X- relatively large apparent velocity of the flux centroid is − ray lightcurves during this period, although emission of expected in late times. each component rises sharply. The late-time X-ray af- Ghirlanda et al. (2019) reported GRB 170817A radio terglow of GRB 170817A shows a clear excessive emis- observations using VLBI with effective angular resolu- tion is 1.5 3.5 mas, found that the source radio im- sion as compared with the estimated emission from the × off-axis structured jet model (Hajela et al. 2021b). This age appears compact and apparently unresolved, and excess could be explained by invoking additional emis- estimated that the apparent size of the radio source is sion component (e.g. kilonova afterglow; Hajela et al. constrained to be smaller than 2.5 mas (90% confidence 2019; Troja et al. 2020; Hajela et al. 2021b, accretion- level) at 207.4 days after the burst. In order to decide powered emission; Hajela et al. 2021b; Ishizaki et al. the size of our predicted source radio images, we adopt 2021, or energy injection from the NS; Troja et al. 2020; an elliptical Gaussian function to fit the images. The Hajela et al. 2021b). Our model can reproduce a late- MCMC method is adopted to reach the best fit to the time shallower decay (but cannot reproduce a late-time images. We derive the size of our predicted images from Result A and B at 207.4 days are about 2.4 0.4 mas rise; as shown in Figure 2) in the X-ray lightcurve with- × and 3.4 1.2 mas, respectively. This is consistent with out invoking an additional energy source. Moreover, our × model can give a natural explanation of the late-time the expectation that a larger viewing angle leads to a harder spectrum (Hajela et al. 2021b) due to the change smaller size of image due to the projection effect. The of composition and a relatively small p from the FS (the predicted size from Result A is consistent with the ob- transition of the blast wave dynamics from the relativis- servational constraint, whereas the size from Result B tic phase to the sub-relativistic phase can also lead to is slightly larger. Although we cannot rule out Result B a shallower decay in the late-time lightcurves without due to the size of the real observed image of our model is spectral evolution). hard to decide, Result A is more promising than Result B. 6 Li & Dai

Table 1 shows the best-fitting parameters and their 1σ uncertainties by our fitting. The best-fitting parameters give θ 6◦, which is consistent with the typical jet j ≈ opening angle (e.g. Frail et al. 2001; Wang et al. 2015, 2018). The derived θ 13◦ reaches the lower limit v ≈ on the viewing angle by combining GW and EM constraints (e.g. Finstad et al. 2018; Abbott et al. 2019). An NS with B 1.6 1013 G is needed in our fit- p ≈ × ting. Here Bp is mainly determined by τsd, while τsd is roughly equal to the peak time of the flux from the RS (Geng et al. 2016), and also roughly equal to the peak time of the afterglow ( 150 days) in our fit- ∼ ting. Therefore, B 1013 G can be determined ac- p ∼ cording to τ 150 days. The isotropic kinetic energy sd ≈ E 2 1052 erg corresponding to the true kinetic K,iso ≈ × energy of jet E = E (1 cos θ )/2 5.6 1049 erg, K K,iso − j ≈ × which is comparable to the kinetic energy in the jet core inferred from structured jet models (e.g. Hajela et al. 2019; Lamb et al. 2019; Troja et al. 2019; Ryan et al. 2020). The initial Lorentz factor Γ 47 is lower than 0 ≈ that obtained by the structured jet models. Besides, the Figure 4. −5 The relative position of 4.5 GHz radio images at microphysics parameters we derived are ǫB,2 7 10 , 75 days (upper), 207.4 days (middle), and 230 days (lower) −6 ≈ × ǫB,3 3 10 , ǫe,2 0.07, ǫe,3 1, p2 2.0, p3 2.2. seen by the observer. The white plus sign is the location of ≈ × ≈ ≈ ≈ ≈ Figure 5 displays the corner plots showing the results the flux centroid. The black plus sign is the location closest to the LOS. Purple ellipses are the best-fitting elliptical of our MCMC parameter estimation. There is a strong Gaussian showing the sizes (full width at half maximum) of degeneracy between EK,iso and n1, EK,iso and ǫB,2, n1 the images. and ǫB,2, which leads to poor limitations on these parameters. 4. DISCUSSION & CONCLUSIONS Supposing the GRB 170817A prompt emission origi- Table 1. Free parameters, priors, and best-fitting results in our model. nates from internal dissipation of the jet energy (however, a cocoon shock breakout as the origin of the Parameter Prior A Result A Prior B Result B prompt emission was suggested by some other au- +0.01 +0.01 thors, e.g. Kasliwal et al. 2017; Bromberg et al. 2018; θj (rad) [0, 1] 0.11−0.01 [0, 1] 0.05−0.01 +0.00 +0.02 Gottlieb et al. 2018), the observed off-axis values of θv (rad) [0.23, 0.7] 0.23−0.00 [0, 0.7] 0.14−0.02 +0.03 +0.04 physical quantities of prompt emission can be corrected log[Bp (G)] [10, 14] 13.20−0.03 [10, 14] 13.25−0.04 +0.81 +0.75 to an on-axis values via the angle-dependent Doppler log[EK,iso (erg)] [49, 54] 52.27−0.89 [49, 54] 52.87−0.79 +0.12 +0.13 factor. For the GRB duration T90 and the peak of the log Γ0 [1, 3] 1.67−0.18 [1, 3] 1.79−0.18 −3 − − +0.81 − − +0.79 GRB energy spectrum Ep, one has log[n1 (cm )] [ 6, 0] 3.91−0.90 [ 6, 0] 4.35−0.79 +2.04 +1.84 log ǫB,2 [−7, −0.5] −4.17 [−7, −0.5] −3.93 T90,off Ep,on (0) 1 β cos(θv θj ) −1.94 −1.75 = = D = − − , − − − +0.07 − − − +0.33 log ǫB,3 [ 7, 0.5] 5.51−0.08 [ 7, 0.5] 4.36−0.25 T90,on Ep,off (θv θj ) 1 β − − − +0.41 − − − +0.56 D − − (3) log ǫe,2 [ 7, 0.5] 1.14−0.41 [ 7, 0.5] 2.47−0.55 +0.01 +0.03 while for the isotropic γ-ray energy of prompt emission p2 [2, 3] 2.02−0.01 [2, 3] 2.03−0.02 +0.05 +0.03 Eγ,iso, one has p3 [2, 3] 2.21−0.03 [2, 3] 2.16−0.02 2 χ /dof - 150.95/46 - 101.11/46 E (0) 2 1 β cos(θ θ ) 2 γ,iso,on D = − v − j . Note—The uncertainties of the best-fitting parameters are measured E ≈ (θ θ ) 1 β γ,iso,off v j as 1σ confidence ranges. D − − (4) Using the observed quantities for GRB 170817A, T 2 s, E 200 keV, E = 5.3 90,off ≈ p,off ≈ γ,iso,off × 1046 erg (Abbott et al. 2017c), and our derived parameters, θ 0.11, θ 0.23, Γ 47, one can derive j ≈ v ≈ 0 ≈ Relativistic Electron-Positron Wind 7

Figure 5. A corner plot showing the results of our MCMC parameter estimation for the relativistic e+e− pair wind model. Our best-fitting parameters and corresponding 1σ uncertainties are shown with the black dashed lines in the histograms on the diagonal. the on-axis values, T 0.06 s, E 6.6 MeV, ble to those sGRBs at low redshifts. For example, the 90,on ≈ p,on ≈ E = 5.7 1049 erg. The derived T falls into short GRB 150101B jointly detected by Fermi/GBM γ,iso,on × 90,on the T distribution of sGRBs (e.g., Kouveliotou et al. and Swift/BAT, has a redshift z = 0.1343 0.0030, 90 ± 1993). The derived Ep,on is larger than typical value which is the second nearby sGRB. At this redshift, the (several hundred keV), but is still consistent with the isotropic energy released in the 10 1000 keV energy ∼ − wide E distribution of sGRBs (e.g. Gruber et al. 2014). band is E 1.3 1049 erg (Fong et al. 2016). GRB p γ,iso ≈ × After the correction, the energy released during the 160821B, the lowest redshift sGRB identified by Swift, prompt emission phase of GRB 170817A is compara- has a redshift z = 0.162 (Levan et al. 2016), and its 8 Li & Dai isotropic energy is E 2.1 1050 erg in the 8 1000 In this paper, we fit the multi-wavelength after- γ,iso ≈ × − keV range (Lüet al. 2017). The consistence between the glow lightcurves and the VLBI proper motion of GRB on-axis values for GRB 170817A and typical values ob- 170817A based on the relativistic e+e− pair wind model. served for sGRBs suggests that our model can reproduce The MCMC method is adopted to obtain the best-fitting the prompt emission of GRB 170817A. parameters. We find that the overall quality of the fit- The derived NS surface magnetic field strength is ting is good, indicating that our relativistic e+e− pair B 1013 G in our fitting, which is more than one wind model can explain the GRB 170817A afterglow. p ∼ order of magnitude higher than previous results. For We obtain a set of best-fitting parameters using the prior example, Ai et al. (2018) constrained Bp to be lower of viewing angle ranging from 0.23 rad to 0.7 rad pro- than 1012 G2 if a long-lived NS survives after the vided by the GW and EM obervations. The best-fitting ∼ merger. The constraint on B is mainly contributed by value θ 0.23 is close to the lower limit of the prior, in- p v ≈ the kilonova observations. One can directly derive the dicating that our model prefers a smaller viewing angle. constraint on B based on the “Arnett Law”: the peak If one allows the prior of 0 θ 0.7, the best-fitting p ≤ v ≤ luminosity of the kilonova equals the heating rate at the value of viewing angle can be as low as 0.14 rad. Be- ≈ peak time, i.e. L Q˙ (t ) (Arnett 1982). The sides, our model can also reproduce the prompt emission peak ≈ peak kilonova associated with GW170817 has a peak lumi- of GRB 170817A. A NS with B 1.6 1013 G is needed p ≈ × nosity L 1042 erg at 0.5 days after merger. For in our fitting. Combining the derived B and the kilo- peak ∼ p a long-lived NS as post-merger remnant, the heating of nova observations, we find that the fraction of the spin- the kilonova ejecta usually comes from two components: down luminosity which is thermalized and available to radioactive r-process heating and magnetic dipole spin- power the kilonova can be as low as 10−3. Our model ∼ down heating, i.e. Q˙ = Q˙ ra + Q˙ md. Therefore, one can reproduce a late-time shallow decay in the X-ray can write Q˙ md(tpeak)= ηLsd(tpeak) . Lpeak, where Lsd lightcurve and predicts that the X-ray and radio flux is the spin-down luminosity, and η is a fraction of Lsd will continue to decline in the next few years. In con- that is used to heat the ejecta. Since τsd t peak, trast, Hajela et al. (2021b) predicts that the X-ray and − ≫ one has L = 9.64 1044B2 . η 1L . If one radio flux will keep increasing for at least a few years sd,0 × p,13 peak adopts 0.1 <η < 1 as suggested by Yu et al. (2013), within the framework of the kilonova afterglow model, then B . 1011 1012 G, which is generally consistent and the X-ray flux will remain constant or decay with in- p − with the result from Ai et al. (2018). However, there dex 5/3 in the fall-back accretion model (Ishizaki et al. − are no simulations providing the exact value of η so far, 2021). Further observations of GRB 170817A afterglow and η could be much smaller than unity for the fol- may help verify or disprove our model. lowing reasons: (1) if a fraction of spin-down energy is reflected by the ejecta walls and the large pair optical depth through the nebula behind the ejecta, a low effi- ACKNOWLEDGMENTS ciency of the spin-down luminosity used to thermaliza- We thank the referee for helpful comments that have tion is expected, as suggested by Metzger & Piro (2014); allowed us to significantly improve our manuscript. (2) a fraction of spin-down energy could be converted to This work was supported by the National Key Re- the kinetic energy of the kilonva ejecta rather than heat search and Development Program of China (grant No. the ejecta (e.g. Wang et al. 2016); (3) The spin-down 2017YFA0402600), the National SKA Program of China powered outflow could be collimated along the GRB jet (grant No. 2020SKA0120300), and the National Natural direction due to the interaction between the outflow and Science Foundation of China (grant No. 11833003). the ejecta (e.g. Bucciantini et al. 2012). According to the derived Bp and the peak luminosity of kilonova associated with GW170817, η can be as low as 10−3 in ∼ Software: emcee (Foreman-Mackey et al. 2013) our estimation.

2 This constraint requires the ellipticity of the NS is smaller than ∼ 10−4, which is consistent with the implicit condition of our model. By default, the rotational energy loss is dominated by magnetic dipole radiation rather than GW radiation in our model, i.e. the luminosity of GW emission should be less than the luminosity of magnetic dipole emission, which allows us to derive −5 13 the upper limit of ellipticity . 5 × 10 for Bp ≈ 1.5 × 10 G. Relativistic Electron-Positron Wind 9

APPENDIX

A. DYNAMICS AND RADIATION There are four distinct regions when an ultra-relativistic e+e− pair wind interact with the jet and medium: (1) the unshocked medium, (2) the shocked jet and medium, (3) the shocked wind, (4) the unshocked wind. In the following, the quantities of region i are denoted by subscript i. The comoving- and observer-frame quantities are marked with and without a superscript prime (′), respectively. Neglecting the internal structure of the blastwave (Blandford & McKee 1976), and assuming that regions 2 and 3 move with the same Lorentz factor, i.e. Γ2 =Γ3 = Γ, the dynamics of such an FS-RS system can be solved by the energy conservation. Let us ﬁrst deﬁne a spherical coordinate system (r, θ, φ), where r is the distance from the coordinate origin, θ and φ are the latitudinal and azimuthal angles, respectively. The GRB central engine is located at coordinate origin, and the GRB jet axis is along the direction of θ = 0. The observer lies on the φ = π/2 plane, and θv is the angle between the LOS and the GRB jet axis. We divide the θ [0,θ ] and φ [0, 2π] into M and N parts, so that the GRB jet ∈ j ∈ can be discretized into M N grids and each grid has a solid angle dΩ = sin θdθdφ. For a top-hat jet model which is × adopted in this paper, the dynamical evolution for each grid is identical. For any grid, the dynamical evolution per unit solid angle can be described as follow. Considering radiative loss, the total kinetic energy of region 2 is

E = (Γ 1)(M + m )c2 + (1 ǫ )Γ(Γ 1)m c2, (A1) 2 − ej 2 − 2 − 2 where Mej and m2 are the rest masses of the initial GRB ejecta and FS swept-up medium per unit solid angle. ′−1 ′−1 ′−1 ǫ2 = ǫetsyn /(tsyn + tex ) is the radiation eﬃciency of region 2, where ǫe is the fraction of the shock internal energy ′ ′ that is partitioned to electrons, tsyn and tex are the synchrotron cooling timescale and the comoving-frame expansion timescale, respectively (Dai & Lu 1999). Energy conservation requires that the change of the total kinetic energy of region 2 should be equal to the work done by region 3 to region 2 subtract a fraction of thermal energy which is radiated from region 2: ′ dE = R2p dR ǫ Γ(Γ 1)dm c2, (A2) 2 3 − 2 − 2 where the comoving pressure of region 3 should be calculated by

′ ′ p = (1 ǫ )(Γ 1)(ˆγ Γ + 1)n m c2. (A3) 3 − 3 34 − 3 34 4 e

Hereγ ˆ3 and ǫ3 are the adiabatic index and the radiation eﬃciency of region 3, Γ34 = Γ43 (1/2)(Γ3/Γ4 +Γ4/Γ3) ′ ′ + − ≃ stands for the relative Lorentz factor between region 3 and 4, n4 = nw is the comoving e e number density of region 4. Combining equations (A1), (A2), and (A3), the diﬀerential equation dΓ/dR can be written in the form

2 ′ 2 dΓ R (1 ǫ3)(Γ34 1)(ˆγ3Γ34 + 1)n me (Γ 1)n1mp = − − 4 − − . (A4) dR M + ǫ m + 2(1 ǫ )Γm ej 2 2 − 2 2

To solve the diﬀerential equation dΓ/dR, one needs to know the evolution of m2 with distance R, which can be written as dm 2 = R2n m . (A5) dR 1 p Furthermore, the rest mass of region 3 is obtained by (Dai & Lu 2002)

dm Γ β ′ 3 = R2 34 34 n m . (A6) dR Γ β 4 e 3 3

Here m3 is the rest mass of RS swept-up medium per unit solid angle, β3 is the dimensionless speed of region 3, and β (Γ2 Γ2)/(Γ2 +Γ2) is the relative dimensionless speed between region 3 and 4. 34 ≃ 3 − 4 3 4 In order to calculate the synchrotron and IC emission, we assume that a fraction of shock internal energy that is partitioned to magnetic ﬁelds ǫB,i and electrons ǫe,i, and assume that electrons in regions 2 and 3 can be accelerated ′− by FS and RS with a power-law distribution in energy dN ′ /dγ′ γ pi (γ′ γ′ γ′ ). Here p is the electron e,i e,i ∝ e,i m,i ≤ e,i ≤ M,i i 10 Li & Dai

′ ′ energy spectral index in region i, γm,2 = (mp/me)ǫe,2(Γ2 1)(p2 2)/(p2 1) and γm,3 = ǫe,3(Γ43 1)(p3 2)/(p3 − − − ′ − ′ − − 1) are the comoving-frame minimum electron Lorentz factors in regions 2 and 3, γM,i 6πqe/[σT Bi(1 + Yi)] 8 ′ ≃ ≃ 10 Bi(1 + Yi) is the maximum Lorentz factors in region i. The comoving magnetic fieldp strength in regions 2 and 3 are B′ = 8πǫ n m c2(Γ 1)(γ ˆ Γ + 1)/(ˆγ 1) and B′ = 8πǫ n′ m c2(Γ 1)(γ ˆ Γ + 1)/(ˆγ 1), p 2 B,2 1 p 2 − 2 2 2 − 3 B,3 4 e 34 − 3 34 3 − respectively. The Compton Y parameter, defined by the ratio of the IC power to the synchrotron power, can be p p written as Y = ( 1+ 1+4η η ǫ /ǫ )/2 (Fan & Piran 2006; He et al. 2009), where η is the fraction i − rad,i KN,i e,i B,i rad,i of electron energy that is radiated, and η is the fraction of synchrotron photons below the KN limit frequency p KN,i (Nakar & Granot 2007). The final electron energy has a broken power-law distribution, since the electrons are cooled by synchrotron and IC radiation. By comparing the electron minimum Lorentz factor γm and the electron cooling Lorentz factor γc that can be written as ′ 6πmec(1 + z) γc,i = ′2 , (A7) σT Bi Γtobs(1 + Yi) two different regimes can be derived. If γ′ γ′ , all the electrons have cooled, this is the fast cooling regime, one c,i ≤ m,i has ′ ′−2 ′ ′ ′ dNe,i γe,i γc,i γe,i γm,i ′ ′−(p+1) ′ ≤ ′ ≤ ′ . (A8) dγe,i ∝ ( γ γ <γ γ e,i m,i e,i ≤ M,i ′ ′ If γm,i <γc,i, only a fraction of electrons have cooled, this is the slow cooling regime, one has

′ ′−p ′ ′ ′ dNe,i γe,i γm,i γe,i γc,i ′ ′−(p+1) ′ ≤ ′ ≤ ′ . (A9) dγe,i ∝ ( γ γ <γ γ e,i c,i e,i ≤ M,i Once the electron distribution is determined, the synchrotron radiation power per unit solid angle of each gird in the region i at the frequency ν′ can be calculated as (Rybicki & Lightman 1979)

′ ′ 3 ′ γM,i ′ ′ ′ √3qe Bi dNe,i ν ′ Pi,syn(ν )= 2 ′ F ′ dγe,i. (A10) mec min(γ′ ,γ′ ) dγ ν Z m,i c,i e,i ch Here ν′ = (1+ z)ν / is the synchrotron frequency in the comoving frame, where ν is the observed frequency, z is obs D obs the redshift, and 1/Γ(1 β cos α) is the Doppler factor with cos α = cos θ cos θ + sin θ sin φ sin θ in our assumed D≡ − v v geometrical setting (α is the angle between the velocity direction of each gird and the LOS). qe is the electron charge, ∞ ′ ′2 ′ ′ ′ ′ ′ + ν =3γ qeB /(4πmec) is the critical frequency of synchrotron radiation, F (ν /ν ) = (ν /ν ) ′ ′ K5/3(x)dx, and ch e,i i c ch ν /νch K5/3(x) is the modiﬁed Bessel function of 5/3 order. R Besides the synchrotron radiation, we also consider the IC radiation from shock-accelerated electrons. The IC radiation include two parts: the self-Compton (SSC) radiation, and combined IC (CIC) radiation, i.e., photons in region i are scattered by electrons in regions j (i = j). The IC radiation power (per unit solid angle of each gird at ′ 6 the frequency ν ) of electrons in the region i scatter photons from region j can be calculated as (i = j for SSC and i = j for CIC; Blumenthal & Gould 1970; Yu & Dai 2007) 6 ′ ′ ∞ ′ ′ γM,i ′ ′ dNe,i ′ ′ ν fνs,j Pi,IC(ν )=3σT ′ dγe,i dνs,j ′2 ′2 g(x, y), (A11) γ′ dγ ν′ 4γ ν Z min,i e,i Z s,j,min e,i s,j ′ ′ ′ ′ 2 ′ ′ 2 ′ ′ 2 ′ where γmin,i = max[min[γc,i,γm,i],hν /(mec )], νs,j,min = ν mec /[4γe,i(γe,imec hν )]. g(x, y)=2y ln y + (1+ 2 2 − 2y)(1 y)+ x y (1 y) with x =4γ′ hν′ /m c2 and y = hν′/[x(γ′ m c2 hν′)]. − 2(1+xy) − e,i s,j e e,i e − Finally, we integrate equations (A10) and (A11) over the equal arrival time surface (EATS) to get the total ﬂux density at the observed frequency νobs:

1+ z 3 ′ ′ ′ ′ Fνobs = 2 [Pi,syn(ν )+ Pi,IC(ν )]dΩ (A12) 4πDL D (EATS)Z

θj 2π 1+ z 3 ′ ′ ′ ′ = 2 [Pi,syn(ν )+ Pi,IC(ν )] sin θdθdφ. (A13) 4πDL 0 0 D Z(EATS)Z Relativistic Electron-Positron Wind 11

Figure 6. Two coordinate systems before (blue) and after (orange) transformation. The jet direction along the z-axis, while ′ the LOS is aligned with z -axis.

For a given observed time tobs, the emission radius Rα at each grid (i.e. the EATS) is obtained by

Rα 1 β cos α t = (1+ z) − dr const. (A14) obs βc ≡ Z0 B. PROPER MOTION OF THE FLUX CENTROID In order to calculate the proper motion of the flux centroid, one needs to project the position of each grid at each moment onto the plane perpendicular to the LOS. We first convert the spherical coordinate system (R,θ,φ) to Cartesian coordinate system (x,y,z) = (R sin θ cos φ, R sin θ sin φ, R cos θ). In the Cartesian coordinate system, the GRB jet axis is aligned with z direction, and the observer lies on the y z plane. We perform a coordinate rotation ′ ′ ′ − ′ about the x axis from (x,y,z) to (x ,y ,z ), after which LOS is aligned with z axis, and the angle between the z axis ′ ′ ′ ′ and the z axis is θv, as shown in Figure 6. The coordinate transformation from (x,y,z) to (x ,y ,z ) is given by ′ x = x, (B15) ′ y = y cos θv z sin θv, (B16) ′ − z = z cos θv + y sin θv. (B17) r′ ′ ′ For a given observer time tobs, the projected position of each grid on the EATS is given by g = (xg ,yg) = (R sin θ cos φ, R cos θ sin θ R sin θ sin φ cos θ ). Since the flux centroid is defined as the mean location of a distri- α α v − α v bution of flux density in space, the location of flux centroid on the plane perpendicular to the LOS can be expressed as r′ dF (x′ ,y′ ) r′ dF (x′ ,y′ ) r′ ′ ′ g νobs g g g νobs g g fc = (xfc,yfc)= ′ ′ = , (B18) R dFνobs (xg,yg) R Fνobs ′ ′ where dFνobs (xg ,yg) is the flux density of each gridR on the EATS (one can easily calculate the surface brightness of ′ ′ each location via dFνobs (xg,yg)/dS⊥, where dS⊥ is the area of each grid projected into the plane of the sky). Because ′ ′ of the jet is symmetric about the y z plane, the flux centroid can also described by (0,yfc)= (0, ygdFνobs /Fνobs ). − ′ For the observer, the angular position of each grid is (0,y /D ), where D = D /(1 + z)2 is the angular diameter fc A A L R distance. Finally, the average apparent velocity of the flux centroid over a time interval t t can be written as obs,j − obs,i y′ y′ v = fc,j − fc,i , (B19) app t t obs,j − obs,i and the dimensionless velocity βapp = vapp/c.

C. FITTING RESULTS FROM PRIOR B We also show the ﬁtting results from priors with θ [0, 0.7] (Prior B) in Figure 7, 8, 9, 10. v ∈ 12 Li & Dai

− Figure 7. Relativistic e+e pair wind model ﬁt to the multi-wavelength afterglow lightcurves of GRB 170817A. The dashed and dotted curves represent the emission from the FS and RS, respectively. The solid curves are the total emission lightcurves. Relativistic Electron-Positron Wind 13

− Figure 8. Relativistic e+e pair wind model fit to the VLBI proper motion of GRB 170817A afterglow. The mean dimensionless apparent velocity of the radio afterglow source βapp = 4.1±0.5 between 75 days and 230 days after the merger from Mooley et al. (2018c) yields the data point used to fit. The green, red, and blue solid curve shows the evolution of dimensionless apparent velocity of the flux centroid based on our best-fitting parameters when considering FS emission, RS emission and total emission, respectively. The blue dashed and dotted curves denote βapp,j and Γj.

Figure 9. The relative position of 4.5 GHz radio images at 75 days (upper), 207.4 days (middle), and 230 days (lower) seen by the observer. The white plus sign is the location of the ﬂux centroid. The black plus sign is the location closest to the LOS. Purple ellipses are the best-ﬁtting elliptical Gaussian showing the sizes (full width at half maximum) of the images. 14 Li & Dai

Figure 10. A corner plot showing the results of our MCMC parameter estimation for the relativistic e+e− pair wind model. Our best-ﬁtting parameters and corresponding 1σ uncertainties are shown with the black dashed lines in the histograms on the diagonal. Relativistic Electron-Positron Wind 15

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