GRB 170817A Afterglow from a Relativistic Electron-Positron Pair Wind Observed Off-axis

On 2017 August 17 at 12:41:04 UTC, the Advanced Laser Interferometer Gravitational-Wave Observatory and Advanced Virgo Interferometer gravitational-wave detectors detected the first gravitational-wave (GW) event GW170817 from a binary neutron star (BNS) merger (Abbott et al., 2017a). About 1.7 seconds after the coalescence, Fermi Gamma-ray Burst Monitor and International Gamma-Ray Astrophysics Laboratory were triggered by an low-luminosity short gamma-ray burst (sGRB) GRB 170817A, independently (Abbott et al., 2017b; Goldstein et al., 2017; Savchenko et al., 2017). The detection of GRB 170817A confirmed that at least some sGRBs are associated with BNS merger events. Roughly 11 hours later, an optical counterpart AT2017gfo was discovered to be around the galaxy NGC 4993 (Coulter et al., 2017). Counterparts at ultraviolet and infrared band were also detected within one day (Soares-Santos et al., 2017). These observations support the hypothesis that AT 2017gfo is a kilonova powered by the radioactive decay of rapid neutron-capture process nuclei synthesized within the ejecta (Kasen et al., 2017; Pian et al., 2017). About 9 days later, Chandra X-ray Observatory detected X-ray emission at the position of AT2017gfo (Troja et al., 2017). A longer time later, a radio band counterpart was detected (Hallinan et al., 2017). The X-ray and radio emission can be characterized by a non-thermal power-law spectrum, which is consistent with the GRB afterglow from a relativistic shock. The gravitational-wave signal combined with following electromagnetic counterparts helps to understand the process of a BNS merger and possible products, showing a breakthrough for multi-messenger astronomy.

Continued follow-up observations reveal that the luminosity of a multi-wavelength afterglow was gradually increasing, peaked around 160 days after the merger, and then started to rapidly decreasing (Margutti et al., 2018; Mooley et al., 2018a; Ruan et al., 2018; Makhathini et al., 2020; Troja et al., 2020). The multi-wavelength behavior from radio to X-rays is different from a typical GRB afterglow. Considering the low luminosity of the prompt emission, the explanations for the afterglow luminosity evolution usually fall into two types. One explanation is invoking a structured jet which have an angular distribution in energy and Lorentz factor. The GRB 170817A afterglow is well consistent with the emission from structured jet viewed off-axis (Abbott et al., 2017c; Troja et al., 2017; Lazzati et al., 2018; Lyman et al., 2018; Margutti et al., 2018; Mooley et al., 2018b; Troja et al., 2018; Hajela et al., 2019; Lamb et al., 2019; Troja et al., 2019). The second explanation is based on a classical top-hat jet model with considering additional continuous energy injection into a jet (e.g. Geng et al. 2018; Li et al. 2018; Lamb et al. 2020).

On the other hand, the central remnant of GW170817 which depends on the neutron star (NS) equation of state (EoS) remains up in the air. Depending on whether the NS EoS is stiff enough, the remnant could be a black hole, a temporal hypermassive NS, or a long-lived massive NS. Although there are theories against a long-lived massive NS as the merger remnant (e.g. Granot et al. 2017; Margalit & Metzger 2017; Ciolfi 2020), there is no direct evidence that rules out the possibility that the post-merger remnant is a stable NS. A relativistic jet could also be launched through $\nu\bar{\nu}$ annihilation in a neutrino-dominated accretion flow (NDAF) mechanism from such a stable NS with a hyper-accreting accretion disk (Zhang & Dai, 2008, 2009, 2010). Besides, the existence of an NS remnant can reduces the requirement on the ejecta mass in the kilonova model due to an additional energy injection from the NS (Li et al., 2018; Metzger et al., 2018; Yu et al., 2018). If the remnant is a long-lived massive NS, it is suggested that a Poynting flux-dominated outflow would flow out from the NS. This outflow can be accelerated due to magnetic dissipation (e.g. reconnection), and eventually dominated by the energy flux of ultra-relativistic wind consists of electron-positron ($e^{+}e^{-}$) pairs (Coroniti, 1990; Michel, 1994; Kirk & Skjæraasen, 2003). Dai (2004) realized that the continuous ultra-relativistic $e^{+}e^{-}$ pair wind would interact with the GRB external shock and reshape the afterglow emission signatures. Yu & Dai (2007) used this model to explain the shallow decay phase of the early GRB X-ray afterglows. Geng et al. (2018) modeled the first 150 days of GRB 170817A afterglow lightcurves in this scenario. However, some new observational facts have been updated since then. Soon afterwards, Mooley et al. (2018c) reported the radio observations of GRB 170817A afterglow using a Very Long Baseline Interferometry (VLBI), and found that the radio source shows superluminal apparent motion between 75 days and 230 days after the merger event. A mean apparent velocity of the radio source along the plane of the sky $v_{\rm app}=(4.1\pm 0.5)c$ was measured. Recently, Chandra X-ray Observatory continuous detected X-ray emission from the location of GRB 170817A (Hajela et al., 2020a, b, 2021a, 2021b), which extended the afterglow data to about 3.3 years after the merger. The late-time derived unabsorbed X-ray flux is higher than what is expected from the structured jet model (Makhathini et al., 2020; Troja et al., 2020; Hajela et al., 2020b, 2021a, 2021b). However, the late-time radio observations did not show this excess (Balasubramanian et al., 2021; Hajela et al., 2021b). Therefore, it is necessary to revisit an off-axis afterglow from an $e^{+}e^{-}$ pair wind scenario supposing that the remnant of GW170817 is a long-lived massive NS.

This paper is organized as follows. In subsection 2.1 we give the details of our numerical afterglow model from an $e^{+}e^{-}$ pair wind scenario. In subsection 2.2 we describe the methods of fitting the data from GRB 170817A with our model. Our fitting data includes the thousand-day multi-wavelength afterglow and the VLBI proper motion. In Section 3 we describe our results. Our discussion and conclusions can be found in Section 4. A concordance cosmology with parameters $H=69.6\rm\ km\ s^{-1}\ Mpc^{-1}$, $\Omega_{M}=0.286$, and $\Omega_{\Lambda}=0.714$ is adopted in this paper.

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 the $\chi_{\rm{dof}}^{2}\approx 3$ and $\approx 2$ for Prior A and B, indicating that a large parameter space for $\theta_{v}$ would improves the fitting quality. The corner plots shown in Figure 5 indicate that the posterior distribution of $\theta_{v}$ close to the lower limit of its prior, which means a smaller $\theta_{v}$ would lead to a better fit. After performing a full prior of $\theta_{v}$ ranging from 0 to 0.7, the posterior distribution of $\theta_{v}$ exhibits a normal distribution peaking at about 0.14 rad ($\approx 8$ deg), as shown in Figure 10. The derived $\theta_{v}\approx 8$ deg from Prior B is consistent with the viewing angle inferred from the detection of GW170817 ($\theta_{v}\leq 55$ deg at 90% confidence with a low-spin priors; Abbott et al. 2017a), but lower than the viewing angle inferred by combining GW and EM constraints (usually $\gtrsim 10$ deg; Finstad et al. 2018; Mandel 2018; Abbott et al. 2019). The smaller $\theta_{v}$ may pose a greater challenge to the estimation of the Hubble flow velocity for NGC 4993 or the constraint on Hubble constant $H_{0}$. Considering the reasonably good fit from Prior A, we conclude the results from Prior A are more reasonable than those from B. As shown in Appendix C, the results from Prior B are similar to Prior A, except for the derived smaller $\theta_{j}$ and $\theta_{v}$, and the difference in the apparent source size which we describe below. In the following we only discuss the results from Prior A.

In particular, our model can well reproduce the slowly rising phase of the early-time lightcurves. Instead of 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 emission. The FS emission reaches to a peak when the Lorentz factor of the jet $\Gamma\sim 1/(\theta_{v}-\theta_{j})$, while the peak time of the RS emission is roughly equal to the spin-down time scale of the NS (Geng et al., 2016). In our fitting, the peak times of FS and RS emission are about 50 days and 150 days respectively, and the superposition of these two components flatten the radio$-$X-ray lightcurves during this period, although emission of each component rises sharply. The late-time X-ray afterglow of GRB 170817A shows a clear excessive emission as compared with the estimated emission from the off-axis structured jet model (Hajela et al., 2021b). This excess could be explained by invoking additional emission component (e.g. kilonova afterglow; Hajela et al. 2019; Troja et al. 2020; Hajela et al. 2021b, accretion-powered emission; Hajela et al. 2021b; Ishizaki et al. 2021, or energy injection from the NS; Troja et al. 2020; Hajela et al. 2021b). Our model can reproduce a late-time shallower decay (but cannot reproduce a late-time rise; as shown in Figure 2) in the X-ray lightcurve without invoking an additional energy source. Moreover, our model can give a natural explanation of the late-time harder spectrum (Hajela et al., 2021b) due to the change of composition and a relatively small $p$ from the FS (the transition of the blast wave dynamics from the relativistic phase to the sub-relativistic phase can also lead to a shallower decay in the late-time lightcurves without spectral evolution).

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 $\beta_{\rm app,j}$ and Lorentz factor $\Gamma_{\rm j}$ 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 $\beta_{\rm app,j}$ reaches the maximum value $\beta_{\rm app,j,max}=\sqrt{\Gamma_{\rm j}^{2}-1}\approx\Gamma_{\rm j}$ when $\Gamma_{\rm j}=1/\sin(\theta_{v}-\theta_{j})\approx 8.4$ (Zhang, 2018). The dimensionless apparent velocity of the flux centroid $\beta_{\rm app}$ trace $\beta_{\rm app,j}$ at early times, but $\beta_{\rm app}$ is clearly larger than $\beta_{\rm app,j}$ at late times. As shown in Figure 3, we also plot the apparent velocity of the FS flux centroid $\beta_{\rm app,FS}$ and the RS flux centroid $\beta_{\rm app,RS}$. There is almost no difference between $\beta_{\rm app,j}$, $\beta_{\rm app,FS}$, and $\beta_{\rm app,RS}$, which means that $\beta_{\rm app}$ is larger than $\beta_{\rm app,j}$, which is not because of the RS emission from the $e^{+}e^{-}$ pair wind. In order to illustrate this discrepancy, we show the predicted source radio images at 75 days, 207.4 days, and 230 days seen by the observer in Figure 4. The black plus sign marks the location which is closest to the LOS, and the white plus sign marks the flux centroid. One can see the proper motion of the flux centroid is caused by three variations: the movement of the jet relative to observer, the variation of the radio afterglow image size, and the changes in the location of the flux centroid relative to the jet. All these variations would keep the flux centroid away from observer for an off-axis configuration. Therefore, a relatively large apparent velocity of the flux centroid is expected in late times.

Ghirlanda et al. (2019) reported GRB 170817A radio observations using VLBI with effective angular resolution is $1.5\times 3.5$ mas, found that the source radio image appears compact and apparently unresolved, and estimated that the apparent size of the radio source is constrained to be smaller than 2.5 mas (90% confidence level) at 207.4 days after the burst. In order to decide the size of our predicted source radio images, we adopt an elliptical Gaussian function to fit the images. The MCMC method is adopted to reach the best fit to the images. We derive the size of our predicted images from Result A and B at 207.4 days are about $2.4\times 0.4$ mas and $3.4\times 1.2$ mas, respectively. This is consistent with the expectation that a larger viewing angle leads to a smaller size of image due to the projection effect. The predicted size from Result A is consistent with the observational constraint, whereas the size from Result B is slightly larger. Although we cannot rule out Result B due to the size of the real observed image of our model is hard to decide, Result A is more promising than Result B.

Table 1 shows the best-fitting parameters and their $1\sigma$ uncertainties by our fitting. The best-fitting parameters give $\theta_{j}\approx 6^{\circ}$, which is consistent with the typical jet opening angle (e.g. Frail et al. 2001; Wang et al. 2015, 2018). The derived $\theta_{v}\approx 13^{\circ}$ reaches the lower limit on the viewing angle by combining GW and EM constraints (e.g. Finstad et al. 2018; Abbott et al. 2019). An NS with $B_{p}\approx 1.6\times 10^{13}\rm\ G$ is needed in our fitting. Here $B_{p}$ is mainly determined by $\tau_{\rm sd}$, while $\tau_{\rm sd}$ 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 ($\sim 150$ days) in our fitting. Therefore, $B_{p}\sim 10^{13}\rm\ G$ can be determined according to $\tau_{\rm sd}\approx 150\rm\ days$. The isotropic kinetic energy $E_{\rm K,iso}\approx 2\times 10^{52}\rm\ erg$ corresponding to the true kinetic energy of jet $E_{\rm K}=E_{\rm K,iso}(1-\cos\theta_{j})/2\approx 5.6\times 10^{49}\rm\ erg$, 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 $\Gamma_{0}\approx 47$ is lower than that obtained by the structured jet models. Besides, the microphysics parameters we derived are $\epsilon_{B,2}\approx 7\times 10^{-5}$, $\epsilon_{B,3}\approx 3\times 10^{-6}$, $\epsilon_{e,2}\approx 0.07$, $\epsilon_{e,3}\approx 1$, $p_{2}\approx 2.0$, $p_{3}\approx 2.2$. Figure 5 displays the corner plots showing the results of our MCMC parameter estimation. There is a strong degeneracy between $E_{\rm K,iso}$ and $n_{1}$, $E_{\rm K,iso}$ and $\epsilon_{B,2}$, $n_{1}$ and $\epsilon_{B,2}$, which leads to poor limitations on these parameters.

Supposing the GRB 170817A prompt emission originates from internal dissipation of the jet energy (however, a cocoon shock breakout as the origin of the prompt emission was suggested by some other authors, e.g. Kasliwal et al. 2017; Bromberg et al. 2018; Gottlieb et al. 2018), the observed off-axis values of physical quantities of prompt emission can be corrected to an on-axis values via the angle-dependent Doppler factor. For the GRB duration $T_{90}$ and the peak of the GRB energy spectrum $E_{p}$, one has

while for the isotropic $\gamma$-ray energy of prompt emission $E_{\gamma,\rm iso}$, one has

Using the observed quantities for GRB 170817A, $T_{90,\rm off}\approx 2\rm\ s$, $E_{p,\rm off}\approx 200\rm\ keV$, $E_{\gamma,\rm iso,off}=5.3\times 10^{46}\rm\ erg$ (Abbott et al., 2017c), and our derived parameters, $\theta_{j}\approx 0.11$, $\theta_{v}\approx 0.23$, $\Gamma_{0}\approx 47$, one can derive the on-axis values, $T_{90,\rm on}\approx 0.06\rm\ s$, $E_{p,\rm on}\approx 6.6\rm\ MeV$, $E_{\gamma,\rm iso,on}=5.7\times 10^{49}\rm\ erg$. The derived $T_{90,\rm on}$ falls into the $T_{90}$ distribution of sGRBs (e.g., Kouveliotou et al. 1993). The derived $E_{p,\rm on}$ is larger than typical value (several hundred keV), but is still consistent with the wide $E_{p}$ distribution of sGRBs (e.g. Gruber et al. 2014). After the correction, the energy released during the prompt emission phase of GRB 170817A is comparable to those sGRBs at low redshifts. For example, the short GRB 150101B jointly detected by Fermi/GBM and Swift/BAT, has a redshift $z=0.1343\pm 0.0030$, which is the second nearby sGRB. At this redshift, the isotropic energy released in the $\sim 10-1000$ keV energy band is $E_{\gamma,\rm iso}\approx 1.3\times 10^{49}\rm\ erg$ (Fong et al., 2016). GRB 160821B, the lowest redshift sGRB identified by Swift, has a redshift $z=0.162$ (Levan et al., 2016), and its isotropic energy is $E_{\gamma,\rm iso}\approx 2.1\times 10^{50}\rm\ erg$ in the $8-1000$ keV range (Lü et al., 2017). The consistence between the on-axis values for GRB 170817A and typical values observed for sGRBs suggests that our model can reproduce the prompt emission of GRB 170817A.

The derived NS surface magnetic field strength is $B_{p}\sim 10^{13}\rm\ G$ in our fitting, which is more than one order of magnitude higher than previous results. For example, Ai et al. (2018) constrained $B_{p}$ to be lower than $\sim 10^{12}\rm\ G$²²2This constraint requires the ellipticity of the NS is smaller than $\sim 10^{-4}$, 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 the upper limit of ellipticity $\lesssim 5\times 10^{-5}$ for $B_{p}\approx 1.5\times 10^{13}\rm\ G$. if a long-lived NS survives after the merger. The constraint on $B_{p}$ is mainly contributed by the kilonova observations. One can directly derive the constraint on $B_{p}$ based on the “Arnett Law”: the peak luminosity of the kilonova equals the heating rate at the peak time, i.e. $L_{\rm peak}\approx\dot{Q}(t_{\rm peak})$ (Arnett, 1982). The kilonova associated with GW170817 has a peak luminosity $L_{\rm peak}\sim 10^{42}\rm\ erg$ at 0.5 days after merger. For a long-lived NS as post-merger remnant, the heating of the kilonova ejecta usually comes from two components: radioactive $r$-process heating and magnetic dipole spin-down heating, i.e. $\dot{Q}=\dot{Q}_{\rm ra}+\dot{Q}_{\rm md}$. Therefore, one can write $\dot{Q}_{\rm md}(t_{\rm peak})=\eta L_{\rm sd}(t_{\rm peak})\lesssim L_{\rm peak}$, where $L_{\rm sd}$ is the spin-down luminosity, and $\eta$ is a fraction of $L_{\rm sd}$ that is used to heat the ejecta. Since $\tau_{\rm sd}\gg t_{\rm\ peak}$, one has $L_{\rm sd,0}=9.64\times 10^{44}B_{p,13}^{2}\lesssim\eta^{-1}L_{\rm peak}$. If one adopts $0.1<\eta<1$ as suggested by Yu et al. (2013), then $B_{p}\lesssim 10^{11}-10^{12}\rm\ G$, which is generally consistent with the result from Ai et al. (2018). However, there are no simulations providing the exact value of $\eta$ so far, and $\eta$ could be much smaller than unity for the following 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 efficiency of the spin-down luminosity used to thermalization is expected, as suggested by Metzger & Piro (2014); (2) a fraction of spin-down energy could be converted to the kinetic energy of the kilonva ejecta rather than heat the ejecta (e.g. Wang et al. 2016); (3) The spin-down powered outflow could be collimated along the GRB jet direction due to the interaction between the outflow and the ejecta (e.g. Bucciantini et al. 2012). According to the derived $B_{p}$ and the peak luminosity of kilonova associated with GW170817, $\eta$ can be as low as $\sim 10^{-3}$ in our estimation.

In this paper, we fit the multi-wavelength afterglow lightcurves and the VLBI proper motion of GRB 170817A based on the relativistic $e^{+}e^{-}$ pair wind model. The MCMC method is adopted to obtain the best-fitting parameters. We find that the overall quality of the fitting is good, indicating that our relativistic $e^{+}e^{-}$ pair wind model can explain the GRB 170817A afterglow. We obtain a set of best-fitting parameters using the prior of viewing angle ranging from 0.23 rad to 0.7 rad provided by the GW and EM obervations. The best-fitting value $\theta_{v}\approx 0.23$ is close to the lower limit of the prior, indicating that our model prefers a smaller viewing angle. If one allows the prior of $0\leq\theta_{v}\leq 0.7$, the best-fitting value of viewing angle can be as low as $\approx 0.14$ rad. Besides, our model can also reproduce the prompt emission of GRB 170817A. A NS with $B_{p}\approx 1.6\times 10^{13}\rm\ G$ is needed in our fitting. Combining the derived $B_{p}$ and the kilonova observations, we find that the fraction of the spindown luminosity which is thermalized and available to power the kilonova can be as low as $\sim 10^{-3}$. Our model can reproduce a late-time shallow decay in the X-ray lightcurve and predicts that the X-ray and radio flux will continue to decline in the next few years. In contrast, Hajela et al. (2021b) predicts that the X-ray and radio flux will keep increasing for at least a few years within the framework of the kilonova afterglow model, and the X-ray flux will remain constant or decay with index $-5/3$ in the fall-back accretion model (Ishizaki et al., 2021). Further observations of GRB 170817A afterglow may help verify or disprove our model.