Scattered Short Gamma-Ray Bursts as Electromagnetic Counterparts to Gravitational Waves and Implications of GW170817 and GRB 170817A

On 2017 August 17, gravitational waves (GWs) from a merger of two neutron stars (NSs) were finally discovered by the Laser Interferometer Gravitational-Wave Observatory (LIGO), along with Virgo, and dubbed as GW170817 (Abbott et al. 2017c). The Fermi Gamma-ray Burst Monitor (Fermi/GBM) and INTEGRAL also detected a short (duration ∼2 s) gamma-ray burst sGRB 170817A ∼1.7 s after the GWs ceased (Abbott et al. 2017b; Goldstein et al. 2017; Savchenko et al. 2017). Follow-up observations involving more than 3000 people identified the counterparts across the electromagnetic (EM) wavelengths (Abbott et al. 2017d), which also spotted the E/S0 host galaxy NGC 4993 about 40 Mpc away (Hjorth et al. 2017; Im et al. 2017). This event is really the historical breakthrough of the multimessenger astronomy.

An NS binary merger is long thought to be the most likely origin of sGRBs (e.g., Paczynski 1986; Eichler et al. 1989; Meszaros & Rees 1992; Narayan et al. 1992). The observational results, such as a wide variety of their host galaxy type and an absence of associated supernovae, are consistent with the merger scenario, but not conclusive evidence (e.g., Berger 2014). Simultaneous detections of GWs and an sGRB should be smoking-gun evidence for the merger scenario. This time, the apparent isotropic-equivalent energy E_iso ∼ 5 × 10⁴⁶ erg is significantly lower than that of ordinary sGRBs. This is also expected to some extent because an sGRB is caused by a relativistic jet with the emission beamed into a narrow solid angle. The GW observations allow the viewing angle θ_v ≲ 32° relative to the total angular momentum axis (Abbott et al. 2017a, 2017c). Thus, sGRB 170817A is likely the first off-axis sGRB ever observed.

Actually, an off-axis jet of a canonical sGRB seems to be consistent with all the EM signals currently observed in GW170817 (Abbott et al. 2017b; Granot et al. 2017; Ioka & Nakamura 2018). The off-axis de-beaming of the emission can easily decrease the apparent isotropic energy down to the observed value of ∼5 × 10⁴⁶ erg, where we should be careful that the de-beaming is less significant than the point-source case because the opening angle of the jet is comparable to the viewing angle (Ioka & Nakamura 2018). During the propagation in the material ejected at the merger, the jet also injects energy into the cocoon, and the jet-powered cocoon could be observed as the detected blue macronova (or kilonova) at ∼1 day (Abbott et al. 2017d; Andreoni et al. 2017; Arcavi et al. 2017; Coulter et al. 2017; Covino et al. 2017; Cowperthwaite et al. 2017; Díaz et al. 2017; Drout et al. 2017; Hu et al. 2017; Kasen et al. 2017; Kasliwal et al. 2017; Kilpatrick et al. 2017; Lipunov et al. 2017; McCully et al. 2017; Nicholl et al. 2017; Pian et al. 2017; Shappee et al. 2017; Siebert et al. 2017; Smartt et al. 2017; Soares-Santos et al. 2017; Tanaka et al. 2017; Utsumi et al. 2017; Valenti et al. 2017; Buckley et al. 2018; Matsumoto et al. 2018; Tominaga et al. 2018). The jet is finally decelerated by the interstellar medium, leading to the off-axis afterglow emission that can fit both the observed X-ray and radio signals (Alexander et al. 2017; Evans et al. 2017; Fraija et al. 2017; Haggard et al. 2017; Hallinan et al. 2017; Kim et al. 2017; Margutti et al. 2017; Troja et al. 2017). Such an off-axis jet model is the simplest and may be preferred by Occam's razor.

Nevertheless, the low luminosity of the prompt emission might be explained by other scenarios, such as emission from a structured jet with a wide-angle distribution (Lamb & Kobayashi 2016; Burgess et al. 2017a; Granot et al. 2018; Xiao et al. 2017; Jin et al. 2018; Zhang et al. 2018), breakout emission from a mildly relativistic cocoon (Gottlieb et al. 2018b; Piro & Kollmeier 2018; Pozanenko et al. 2018), and on-axis emission from a low-luminosity sGRB population (Murguia-Berthier et al. 2017a; He et al. 2018; Lamb & Kobayashi 2018; Yue et al. 2018; Zou et al. 2018). Some of these scenarios will be tested by the future radio and X-ray observations.

One possible tension against the off-axis jet scenario is the peak energy of the sGRB spectrum. Most of the currently proposed models also suggest a relatively soft peak energy, ∼1–10 keV (e.g., Bégué et al. 2017; Ioka & Nakamura 2018; Pozanenko et al. 2018). From the detailed analysis of the Fermi/GBM data, the observed peak energy of the main pulse is E_p ∼ 185 ± 62 keV for a Comptonized spectrum (a power law with an exponential cutoff; Burgess et al. 2017b; Goldstein et al. 2017), while the weak tail has a blackbody spectrum with kT = 10.3 ± 1.5 keV. Of course, we should be careful about these values because the observed flux of ∼1.9 photons cm⁻² s⁻¹ is just above the detection threshold of Fermi/GBM of ∼0.5 photons cm⁻² s⁻¹ (von Kienlin et al. 2017). Indeed, the peak energy is shifted softward to E_p ∼ 70 keV if the spectrum is fit by the Band function (Abbott et al. 2017b). In addition, the peak energy E_p ∼ 185 ± 62 keV is still consistent with any low peak energies within 3σ. Some analyses seem to adopt a prior that does not allow a low peak energy (≲10 keV). Nevertheless, the obtained peak energy could imply a different emission mechanism, which may be also working together with the off-axis de-beamed emission. This is worth exploring for future observations.

In this paper, we propose the Thomson scattering of the prompt emission as an EM counterpart to binary NS mergers. Scattering in GRBs has been discussed as a mechanism to make wide-angle emission (e.g., Nakamura 1998; Eichler & Levinson 1999; Kisaka et al. 2015). Figure 1 shows the schematic picture for scattering of prompt emission in a binary NS merger. At the merger of the NSs, strong GWs are emitted (t = 0). After the merger, a part of the NS mass is ejected (e.g., Hotokezaka et al. 2013), and a relativistic jet is launched from the central compact object. During the propagation in the merger ejecta, a part of the jet energy is injected into the cocoon. After the jet penetrates the merger ejecta, the photons in the jet can escape. If the viewing angle of θ_v relative to the jet axis is smaller than the jet opening angle Δθ, the escaping photons would be observed as the prompt emission of an sGRB. On the other hand, photons scattered by the cocoon could dominate the γ-ray flux for off-axis observers.

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The scattering model can naturally explain a peak energy in the main pulse as high as those of canonical sGRBs since the scattered photon energy is similar to the unscattered one unless the photon energy is ≳1 MeV. If the jet opening angle Δθ is wider than the beaming angle 1/Γ_j, where Γ_j is the bulk Lorentz factor of the jet, a fraction 2/Γ_jΔθ (e.g., the ratio of a ring ∼2πΔθ/Γ_j to a circle ∼πΔθ² of the solid angle) of the on-axis emission can be scattered at the jet–cocoon boundary to ∼4π directions. Then the isotropic luminosity of the scattered emission is

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{sc}} & \approx & {\epsilon }_{\mathrm{sc}}\displaystyle \frac{2}{{{\rm{\Gamma }}}_{{\rm{j}}}{\rm{\Delta }}\theta }\displaystyle \frac{{\rm{\Delta }}{\theta }^{2}}{2}{L}_{\mathrm{iso}}\\ & \approx & 3\times {10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\left(\displaystyle \frac{{\epsilon }_{\mathrm{sc}}}{0.1}\right)\left(\displaystyle \frac{{\rm{\Delta }}\theta }{0.3}\right)\\ & \times & {\left(\displaystyle \frac{{{\rm{\Gamma }}}_{{\rm{j}}}}{{10}^{3}}\right)}^{-1}\left(\displaystyle \frac{{L}_{\mathrm{iso}}}{{10}^{51}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right),\end{array}\end{eqnarray} \tag{ 1 }$$

where (Δθ²/2)L_iso is the geometrically corrected luminosity of the on-axis prompt emission and _sc is a correction factor depending on details (see next section). Interestingly, the above rough estimate is comparable to the observed luminosity of sGRB 170817A. Note that both the emissions of scattering and off-axis de-beaming can occur in a single sGRB, and they complement each other.

In this paper, we consider the Thomson scattering of the sGRB prompt emission. In Section 2, we generally formulate the scattering mechanism to calculate the luminosity, duration, and time lag of the scattered emission for an off-axis observer with θ_v > Δθ. Then, in Section 3, we apply the formulae to an sGRB prompt emission from a binary NS merger. We consider a nonrelativistic scatterer or cocoon in Section 3.1 and a relativistic scatterer, which may be the head part of the merger ejecta produced at the onset of the merger (Kyutoku et al. 2014) or a mildly relativistic cocoon due to the imperfect mixing of the jet and the merger ejecta (e.g., Lazzati et al. 2017b; Nakar & Piran 2017; Gottlieb et al. 2018a), in Section 3.2. In Section 3.3, we again emphasize that the scattering model generally predicts the spectral peak energy E_p to be similar to the on-axis value if E_p ≲ MeV. Finally, we apply the scattering model to GRB 170817A/GW170817 and discuss the implications in Section 4. In Section 5, we discuss the constraints from the observations such as afterglow emission. We present the summary and discussions in Section 6. Hereafter, we use Q_x ≡ Q/10^x in cgs units.

We consider prompt emission at a radius r_emi from a relativistic jet with a Lorentz factor ${{\rm{\Gamma }}}_{{\rm{j}}}=1/\sqrt{1-{\beta }_{{\rm{j}}}^{2}}$ and an opening angle ${\rm{\Delta }}\theta \gg {{\rm{\Gamma }}}_{{\rm{j}}}^{-1}$. The observed isotropic luminosity for an on-axis observer is ${L}_{\mathrm{iso}}=4\pi {d}_{{\rm{L}}}^{2}{F}_{\mathrm{obs}}$, where ${d}_{{\rm{L}}}$ is the luminosity distance and F_obs is the observed flux. The observed duration of the prompt emission for an on-axis observer is usually determined by the intrinsic engine activity timescale t_dur. The jet is surrounded by a scatterer with an expansion velocity of β_cc, either relativistic or nonrelativistic. In order to scatter the prompt emission, the optical depth of the scatterer has to be of order unity,

$$\begin{eqnarray}&&\tau \sim 1,\end{eqnarray} \tag{ 2 }$$

and the emission radius has to be comparable to or smaller than the scattering radius,

$$\begin{eqnarray}&&{r}_{\mathrm{emi}}\lesssim {r}_{\mathrm{sc}}.\end{eqnarray} \tag{ 3 }$$

The isotropic luminosity of the scattered emission L_sc is obtained from the geometrically corrected luminosity (Δθ²/2)L_iso as

$$\begin{eqnarray}&&{L}_{\mathrm{sc}}\approx \displaystyle \frac{2}{{{\rm{\Gamma }}}_{{\rm{j}}}{\rm{\Delta }}\theta }\times \displaystyle \frac{{t}_{\mathrm{dur}}}{{T}_{\mathrm{dur},\mathrm{sc}}}\times {{\rm{\Gamma }}}_{{\rm{c}}}^{2}\times {\epsilon }_{\mathrm{sc}}\times \displaystyle \frac{{\rm{\Delta }}{\theta }^{2}}{2}{L}_{\mathrm{iso}}.\end{eqnarray} \tag{ 4 }$$

Here (a) the first factor 2/Γ_jΔθ is the fraction of the geometrically corrected luminosity (Δθ²/2)L_iso that can collide with the scatterer at the scattering radius, and is given by the ratio of the solid angles between the jet πΔθ² and an outer ring ∼2πΔθ/Γ_j with a width of the relativistic beaming angle ∼1/Γ_j. (b) The second factor is the ratio of the on-axis duration t_dur and the off-axis duration of the scattered emission T_dur,sc. The observed duration of the scattered emission is given by

$$\begin{eqnarray}&&{T}_{\mathrm{dur},\mathrm{sc}}\approx \max [{t}_{\mathrm{dur}},{\rm{\Delta }}T],\end{eqnarray} \tag{ 5 }$$

which may be longer than the jet activity timescale t_dur because the duration of a single scattered pulse ΔT is determined by the smaller Lorentz factor of the scatterer Γ_c than Γ_j as

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}T & \sim & \displaystyle \frac{{r}_{\mathrm{sc}}}{c{\beta }_{c}}[1-{\beta }_{{\rm{c}}}\cos ({\theta }_{{\rm{v}}}-{\rm{\Delta }}\theta )]\\ & \approx & \displaystyle \frac{{r}_{\mathrm{sc}}}{2c{\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}^{2}},\end{array}\end{eqnarray} \tag{ 6 }$$

in the case of ${{\rm{\Gamma }}}_{{\rm{c}}}^{-1}\gt {\theta }_{{\rm{v}}}-{\rm{\Delta }}\theta$. Note that the scattering occurs over a range of radii around r_sc, not a thin shell at r_sc. The total energy of $(2/{{\rm{\Gamma }}}_{{\rm{j}}}{\rm{\Delta }}\theta )({\rm{\Delta }}{\theta }^{2}/2){L}_{\mathrm{iso}}{t}_{\mathrm{dur}}$ is emitted with a duration of T_dur,sc, so that the luminosity is reduced by the factor of t_dur/T_dur,sc. (c) The third factor ${{\rm{\Gamma }}}_{{\rm{c}}}^{2}$ is the relativistic beaming factor for a relativistic scatterer because the scattered emission is beamed into a cone with an opening angle of ∼1/Γ_c. (d) The last factor _sc is a correction factor coming from, e.g., the ratio between the emission and scattering radii, r_emi/r_sc, the opacity of the scatterer, and so on. For example, Eichler & Levinson (1999) give _sc ∼ 10⁻³–10⁻¹ for r_emi/r_sc ∼ 0.1. Most of the energy is scattered within the angle

$$\begin{eqnarray}&&{\rm{\Delta }}{\theta }_{\mathrm{sc}}\approx \displaystyle \frac{1}{{{\rm{\Gamma }}}_{{\rm{c}}}}.\end{eqnarray} \tag{ 7 }$$

The jet may be collimated by the cocoon pressure. In the collimated case, the Lorentz factor of the jet during the propagation in the merger ejecta is suppressed (e.g., Bromberg et al. 2011; Mizuta & Ioka 2013). Since the scattered fraction is proportional to the beaming angle ∼1/Γ_j as in Equation (4), the scattered luminosity L_sc could be enhanced.

The time lag of the scattered emission behind the GW signal is given by

$$\begin{eqnarray}&&{\rm{\Delta }}{T}_{\mathrm{lag}}\approx {t}_{{\rm{j}}}+{t}_{\mathrm{br}}+{\rm{\Delta }}{T}_{\mathrm{scbr}},\end{eqnarray} \tag{ 8 }$$

where t_j is the time of the jet launch after the merger, and t_br is the timescale for the jet breakout from the merger ejecta.⁷ The last term in Equation (8) is the delay time of a single pulse caused by the curvature effect of the emission surface at the scatter-starting radius r_scbr,

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{T}_{\mathrm{scbr}} & \sim & \displaystyle \frac{{r}_{\mathrm{scbr}}}{c{\beta }_{c}}[1-{\beta }_{{\rm{c}}}\cos ({\theta }_{{\rm{v}}}-{\rm{\Delta }}\theta )]\\ & \approx & \displaystyle \frac{{r}_{\mathrm{scbr}}}{2c{\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}^{2}},\end{array}\end{eqnarray} \tag{ 9 }$$

in the case of ${{\rm{\Gamma }}}_{{\rm{c}}}^{-1}\gt {\theta }_{{\rm{v}}}-{\rm{\Delta }}\theta$. The jet can be launched either within a dynamical timescale after the merger (t_j ∼ 1–10 ms) or after a hypermassive NS, if any, collapses into a black hole (t_j ≳ 0.1 s). The jet breakout timescale should be t_br ≲ t_dur for a successful breakout (Nagakura et al. 2014).

In summary, for a given set of parameters of the prompt emission (r_emi, r_sc, r_scbr, L_iso, Γ_j, Δθ, t_dur, t_j, t_br, β_c) and a viewing angle θ_v, the properties of the scattered emission (L_sc, ${T}_{\mathrm{dur},\mathrm{sc}}$, ΔT_lag) can be calculated with Equations (3)–(8). In the next section, we set up the parameters in the context of an sGRB from a binary NS merger.

3.1. Nonrelativistic Scatterer

First, we consider scattering by nonrelativistic matter, namely, a nonrelativistic cocoon. During the propagation of the jet in the merger ejecta, a part of the energy is injected to the cocoon surrounding the jet, which can scatter the prompt emission. If the shocked jet and merger ejecta are efficiently mixed before the jet–cocoon breakout, the expansion velocity of the cocoon becomes subrelativistic, ∼0.2–0.4c (Ioka & Nakamura 2018), which is automatically adjusted to be slightly faster than the typical velocity of the dynamical ejecta (∼0.1c–0.2c; Hotokezaka et al. 2013; Sekiguchi et al. 2015; Foucart et al. 2016; Lehner et al. 2016; Radice et al. 2016; Bovard et al. 2017; Dietrich et al. 2017a, 2017b; Shibata et al. 2017).

The scattering radius is around the outer radius of the cocoon, where the optical depth for the Thomson scattering is of order unity in Equation (2). Since the irradiated cocoon is fully ionized near the jet–cocoon boundary, the electron number density of the cocoon is roughly $n\sim 3{M}_{{\rm{c}}}/(8\pi {m}_{{\rm{p}}}{\beta }_{{\rm{c}}}^{3}{c}^{3}{t}^{3}{f}_{\mathrm{cv}})$, where ${M}_{{\rm{c}}}\,={f}_{\mathrm{cm}}{M}_{{\rm{e}}}$ is the cocoon mass, ${M}_{{\rm{e}}}={10}^{-2}{M}_{{\rm{e}},-2}\,{M}_{\odot }$ is the ejecta mass, f_cm is the mass fraction of the cocoon, and f_cv is the fractional volume of the cocoon. Typical values of the fraction are f_cm ∼ f_cv ∼ 0.5 (Ioka & Nakamura 2018). The optical depth for the Thomson scattering is high enough,

$$\begin{eqnarray}&&\tau \sim {10}^{10}{t}_{0}^{-2}({f}_{\mathrm{cm}}/{f}_{\mathrm{cv}}){M}_{{\rm{e}},-2}{\beta }_{{\rm{c}},-0.5}^{-2},\end{eqnarray} \tag{ 10 }$$

which quickly drops below unity outside the cocoon radius. Since the scattering can start at the breakout time t = t_j + t_br and end at t = t_j + t_br + t_dur, the scattering radius ranges from

$$\begin{eqnarray}&&{r}_{\mathrm{scbr}}\sim {\beta }_{{\rm{c}}}c({t}_{{\rm{j}}}+{t}_{\mathrm{br}})\end{eqnarray} \tag{ 11 }$$

to

$$\begin{eqnarray}\begin{array}{rcl}{r}_{\mathrm{sc}} & \sim & {\beta }_{{\rm{c}}}c({t}_{{\rm{j}}}+{t}_{\mathrm{br}}+{t}_{\mathrm{dur}})\\ & \sim & 1\times {10}^{10}\,\mathrm{cm}\,{\beta }_{{\rm{c}},-0.5}{({t}_{{\rm{j}}}+{t}_{\mathrm{br}}+{t}_{\mathrm{dur}})}_{0}.\end{array}\end{eqnarray} \tag{ 12 }$$

In order to scatter the prompt emission, the emission radius has to be smaller than the above scattering radii r_sc as in Equation (3), and this can be realized by the photospheric prompt emission. The photospheric radius is determined by the optical depth, ${n}_{{\rm{j}}}^{{\prime} }{\sigma }_{{\rm{T}}}{r}_{\mathrm{ph}}^{{\prime} }\sim 1$, where ${n}_{{\rm{j}}}^{{\prime} }$ and ${r}_{\mathrm{ph}}^{{\prime} }$ are the baryon number density and the photospheric radius in the jet comoving frame, respectively. The mass conservation equation gives the number density, ${n}_{{\rm{j}}}^{{\prime} }\sim {L}_{\mathrm{iso}}/(4\pi {m}_{{\rm{p}}}{c}^{3}{r}_{\mathrm{ph}}^{2}\eta {{\rm{\Gamma }}}_{{\rm{j}}})$, where η is the photon-to-baryon ratio, σ_T is the Thomson cross section, and m_p is the proton mass. Then, the photospheric emission radius of the jet can be estimated as (Mészáros & Rees 2000; Mészáros 2006)

$$\begin{eqnarray}\begin{array}{rcl}{r}_{\mathrm{emi}} & \sim & {r}_{\mathrm{ph}}\sim \displaystyle \frac{{L}_{\mathrm{iso}}{\sigma }_{{\rm{T}}}}{4\pi {m}_{{\rm{p}}}{c}^{3}\eta {{\rm{\Gamma }}}_{{\rm{j}}}^{2}}\\ & \sim & 1\times {10}^{9}\,\mathrm{cm}\,{\eta }_{3}^{-1}{L}_{\mathrm{iso},51}{{\rm{\Gamma }}}_{{\rm{j}},3}^{-2}.\end{array}\end{eqnarray} \tag{ 13 }$$

Note that $\eta (\geqslant {{\rm{\Gamma }}}_{{\rm{j}}})$ is fairly unknown and the key parameter to understand the GRB physics (e.g., Mészáros & Rees 2000; Ioka et al. 2011). From Equations (12) and (13), two radii are comparable within reasonable ranges of parameters, in particular for the case of a high Lorentz factor $\eta \geqslant {{\rm{\Gamma }}}_{{\rm{j}}}\gtrsim 500$. Therefore, we consider ${r}_{\mathrm{emi}}\sim {r}_{\mathrm{sc}}$ in the nonrelativistic case. We note that the e^± pair production opacity for γ-rays with the peak energy ${E}_{{\rm{p}}}\sim 100\,{E}_{{\rm{p}},2}$ keV is sufficiently small at the photosphere r_emi ∼ 10¹⁰ cm (e.g., Lithwick & Sari 2001; Zhang & Mészáros 2001; Murase & Ioka 2008),

$$\begin{eqnarray}\begin{array}{rcl}{\tau }_{\gamma \gamma } & \sim & \displaystyle \frac{{L}_{\mathrm{iso}}{\sigma }_{{\rm{T}}}}{4\pi {E}_{{\rm{p}}}{{cr}}_{\mathrm{emi}}{{\rm{\Gamma }}}_{{\rm{j}}}^{2}}{\left(\displaystyle \frac{{{\rm{\Gamma }}}_{{\rm{j}}}{m}_{{\rm{e}}}{c}^{2}}{{E}_{{\rm{p}}}}\right)}^{2(1-{\beta }_{{\rm{B}}})}\\ & \sim & {10}^{-5}{L}_{\mathrm{iso},51}{{\rm{\Gamma }}}_{{\rm{j}},3}^{-5}{r}_{\mathrm{emi},10}^{-1}{E}_{{\rm{p}},2}^{2},\end{array}\end{eqnarray} \tag{ 14 }$$

where β_B ∼ 2.5 is the high-energy power-law index of the Band function (Band et al. 1993) and m_e is the electron mass.

At the scattering radius in Equation (12), the duration of a single scattered pulse in Equation (6) is

$$\begin{eqnarray}&&{\rm{\Delta }}T\sim 1\,{\rm{s}}{({t}_{{\rm{j}}}+{t}_{\mathrm{br}}+{t}_{\mathrm{dur}})}_{0},\end{eqnarray} \tag{ 15 }$$

which is larger than the engine duration t_dur, and hence the observed duration of the scattered emission ${T}_{\mathrm{dur},\mathrm{sc}}$ in Equation (5) is given by

$$\begin{eqnarray}&&{T}_{\mathrm{dur},\mathrm{sc}}\sim {\rm{\Delta }}T\sim 1\,{\rm{s}}{({t}_{{\rm{j}}}+{t}_{\mathrm{br}}+{t}_{\mathrm{dur}})}_{0}.\end{eqnarray} \tag{ 16 }$$

Note that ${T}_{\mathrm{dur},\mathrm{sc}}\sim {t}_{\mathrm{dur}}$ if ${t}_{{\rm{j}}}\ll {t}_{\mathrm{br}}$ since t_br ≲ t_dur. A nonrelativistic scatterer scatters photons to a wide angle. Then, the luminosity of the scattered emission in Equation (4) is

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{sc}} & \sim & 3\times {10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\\ & & \times \ {\rm{\Delta }}{\theta }_{-0.5}{{\rm{\Gamma }}}_{{\rm{j}},3}^{-1}{\epsilon }_{\mathrm{sc},-1}{L}_{\mathrm{iso},51}({t}_{\mathrm{dur}}/{T}_{\mathrm{dur},\mathrm{sc}}).\end{array}\end{eqnarray} \tag{ 17 }$$

The corresponding γ-ray flux is ∼4 × 10⁻⁸ erg cm⁻² s⁻¹ at the GW detection horizon of LIGO O2, ∼80 Mpc (Abbott et al. 2016a), which is detectable by Swift/BAT.

The time lag for the detection of the scattered emission after the GWs in Equation (8) is determined by the scatter-starting radius in Equation (11) with Equation (9) as

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{T}_{\mathrm{lag}} & \sim & {t}_{{\rm{j}}}+{t}_{\mathrm{br}}+\displaystyle \frac{{r}_{\mathrm{scbr}}}{c{\beta }_{{\rm{c}}}}\\ & \sim & 2\,{\rm{s}}{({t}_{{\rm{j}}}+{t}_{\mathrm{br}})}_{0}.\end{array}\end{eqnarray} \tag{ 18 }$$

If ${t}_{{\rm{j}}}\ll {t}_{\mathrm{br}}$, the time lag is comparable to or shorter than the observed duration ${\rm{\Delta }}{T}_{\mathrm{lag}}\lesssim {T}_{\mathrm{dur},\mathrm{sc}}$ in Equation (16) since t_br ≲ t_dur. If otherwise ${t}_{{\rm{j}}}\gg {t}_{\mathrm{br}}$, the time lag is approximately twice the observed duration ${\rm{\Delta }}{T}_{\mathrm{lag}}\sim 2{T}_{\mathrm{dur},\mathrm{sc}}$. In any case, the time lag is comparable to or shorter than twice the observed duration ${\rm{\Delta }}{T}_{\mathrm{lag}}\lesssim 2{T}_{\mathrm{dur},\mathrm{sc}}$.

3.2. Relativistic Scatterer

A part of the scatterer could become relativistic. First, the head of the merger ejecta could be relativistic, which could be produced by the shock breakout at the onset of the binary NS merger (Kyutoku et al. 2014). Second, a part of the cocoon could be relativistic owing to the incomplete mixing of the shocked jet component and the shocked ejecta component (e.g., Lazzati et al. 2017b; Nakar & Piran 2017; Gottlieb et al. 2018a). The mass of the relativistic scatterer is uncertain. The fraction of the relativistic mass, which is ejected at the onset of the merger, could be ≲10⁻⁵ M_⊙ (Kyutoku et al. 2014). The mass of the relativistic cocoon highly depends on the degree of the turbulence in the cocoon. Some hydrodynamic simulations suggest a relativistic cocoon mass of ∼10⁻⁷–10⁻⁵ M_⊙ (Lazzati et al. 2017b; Gottlieb et al. 2018a).

Because of the relativistic motion of the scatterer, even if the jet breakout occurs at ∼10¹⁰ cm, the scattering radius could be far from the breakout point. At the scattering radius, the optical depth for the Thomson scattering is approximately τ ∼ 1, as discussed in Equations (2) and (10). In the relativistic case, the number density in the comoving frame is $n^{\prime} \,={M}_{{\rm{c}}}$/$[4\pi {m}_{{\rm{p}}}{r}_{\mathrm{sc}}^{2}({r}_{\mathrm{sc}}$/${{\rm{\Gamma }}}_{{\rm{c}}}^{2}){{\rm{\Gamma }}}_{{\rm{c}}}]$, so that the scattering radius is

$$\begin{eqnarray}&&{r}_{\mathrm{sc}}\sim {10}^{12}{M}_{{\rm{c}},-7}^{1/2}\,\mathrm{cm},\end{eqnarray} \tag{ 19 }$$

where M_c includes only the relativistic component.

The prompt emission can occur below these scattering radii in Equation (3). The radii of internal shocks are around $\sim {{\rm{\Gamma }}}_{{\rm{j}}}^{2}c{\rm{\Delta }}t\sim 3\times {10}^{11}\,\mathrm{cm}({\rm{\Delta }}t$/$\mathrm{ms}){({{\rm{\Gamma }}}_{{\rm{j}}}/{10}^{2})}^{2}$, where Δt is the variability timescale (e.g., Paczynski & Xu 1994; Rees & Meszaros 1994; Kobayashi et al. 1997). The photospheric radius in Equation (13) is large if η and Γ_j are small. Poynting-dominated jets generally have large emission radii (e.g., Giannios & Spruit 2005; Kumar & Zhang 2015; but see also Beniamini & Giannios 2017).

For large scattering radii, the observed duration of the scattered emission ${T}_{\mathrm{dur},\mathrm{sc}}$ in Equation (5) is mainly determined by the duration of a single pulse ΔT in Equation (6). If ΔT is longer than the on-axis sGRB duration t_dur, which is typically t_dur ∼ 0.1 s < 2 s, the observed duration is

$$\begin{eqnarray}&&{T}_{\mathrm{dur},\mathrm{sc}}\sim {\rm{\Delta }}T\sim 2\,{\rm{s}}\,{r}_{\mathrm{sc},12}{{\rm{\Gamma }}}_{{\rm{c}},0.5}^{-2},\end{eqnarray} \tag{ 20 }$$

not the engine duration t_dur. Note that we assume ${{\rm{\Gamma }}}_{c}^{-1}\,\gt {\theta }_{v}-{\rm{\Delta }}\theta$ in the above equation.

The luminosity of the scattered emission L_sc is obtained from Equation (4) as

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{sc}} & \sim & 2\times {10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\\ & & \times \ {{\rm{\Gamma }}}_{{\rm{j}},2}^{-1}{\rm{\Delta }}{\theta }_{-0.5}{t}_{\mathrm{dur},-1}{{\rm{\Gamma }}}_{{\rm{c}},0.5}^{4}{r}_{\mathrm{sc},12}^{-1}{\epsilon }_{\mathrm{sc},-2}{L}_{\mathrm{iso},51}.\end{array}\end{eqnarray} \tag{ 21 }$$

If ΔT > t_dur, t_j, the observed time lag in Equation (8) is also

$$\begin{eqnarray}&&{\rm{\Delta }}{T}_{\mathrm{lag}}\sim {\rm{\Delta }}{T}_{\mathrm{scbr}}\sim 2\,{\rm{s}}\,{r}_{\mathrm{scbr},12}{{\rm{\Gamma }}}_{{\rm{c}},0.5}^{-2},\end{eqnarray} \tag{ 22 }$$

since t_br ≲ t_dur, and is comparable to the duration ${\rm{\Delta }}{T}_{\mathrm{lag}}\,\sim {T}_{\mathrm{dur},\mathrm{sc}}$ in Equation (20).

3.3. Spectrum

Figure 2 shows the energy spectrum of the scattered prompt emission (red solid curve). The basic features are the same for both the nonrelativistic and relativistic cases. For the nonrelativistic scatterer, the electrons see the same spectrum that the on-axis observers see. The Thomson scattering copies the original spectral shape below the $\sim {m}_{e}{c}^{2}\sim \mathrm{MeV}$ energy range. Above this energy, the scattered spectrum has a cutoff because the Klein–Nishina effect reduces the cross section and makes the scattering angles anisotropic. For the relativistic scatterer, the electrons see the redshifted spectrum in the comoving frame, while the scattered photons are blueshifted by the bulk motion of the scatterer in the observer frame. Since ${{\rm{\Gamma }}}_{{\rm{j}}}\gg {{\rm{\Gamma }}}_{{\rm{c}}}$, the energy of the scattered photon is

$$\begin{eqnarray}&&{E}_{\mathrm{sc}}=\displaystyle \frac{E}{2}\left[1+\displaystyle \frac{{\beta }_{{\rm{c}}}({\mu }_{\mathrm{sc}}-{\beta }_{{\rm{c}}})}{1-{\beta }_{{\rm{c}}}{\mu }_{\mathrm{sc}}}\right],\end{eqnarray} \tag{ 23 }$$

where E is the energy of the incident photon and ${\theta }_{\mathrm{sc}}={\cos }^{-1}{\mu }_{\mathrm{sc}}$ is the angle of the scattered photon direction relative to the motion of the scatterer. For ${\theta }_{\mathrm{sc}}\leqslant 1/{{\rm{\Gamma }}}_{{\rm{c}}}$, the energy of the photon changes to ${E}_{\mathrm{sc}}/E\sim 0.5-1$ by a scattering, resulting in a similar spectral shape to the nonrelativistic case.

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The peak energy E_p is similar to those of on-axis sGRBs. This is the characteristic of the scattering model, which is the main difference from other models such as the off-axis de-beamed emission and the emission from low-Γ_j jet/cocoon components as discussed in Section 6.

For on-axis sGRBs, there is an E_p–L_p correlation between the peak energy E_p and the peak luminosity L_p (Yonetoku et al. 2004; Tsutsui et al. 2013), as well as an E_p–E_iso correlation (Amati et al. 2002; Tsutsui et al. 2013). The off-axis scattered sGRBs appear below these correlations in the E_p–L_p and E_p–E_iso diagrams. Since ${L}_{\mathrm{sc}}\propto {L}_{\mathrm{iso}}$ in Equations (17) and (21), we may expect similar correlations like E_p–L_sc, probably with a larger dispersion caused by the dependence on many other parameters.

3.4. Monte Carlo Simulation

In order to estimate the isotropic energy of the scattered component in more realistic conditions, we perform Monte Carlo simulation for the photon propagation. Especially, there is no clear boundary between the jet and ejecta in reality. Some numerical simulations show that the energy and Lorentz factor are seen to smoothly vary from the jet to the cocoon (e.g., Zhang et al. 2003; Aloy et al. 2005; Murguia-Berthier et al. 2014, 2017b; Nagakura et al. 2014; Lazzati et al. 2018). Then, we assume an ultrarelativistic core with a uniform emissivity surrounded by the ejecta with mildly relativistic velocity. As a simple toy model, we linearly interpolate the difference of the Lorentz factor between the jet and the cocoon as

$$\begin{eqnarray}&&{\rm{\Gamma }}(\theta )={{\rm{\Gamma }}}_{{\rm{j}}}-({{\rm{\Gamma }}}_{{\rm{j}}}-{{\rm{\Gamma }}}_{{\rm{c}}})\displaystyle \frac{\theta -{\rm{\Delta }}\theta }{{\theta }_{\mathrm{int}}}\,(0\leqslant \theta -{\rm{\Delta }}\theta \leqslant {\theta }_{\mathrm{int}}),\end{eqnarray} \tag{ 24 }$$

where θ is the polar angle in a spherical polar coordinate. Here we introduce the angle of the interpolating region θ_int as a model parameter. Similar interpolation is also assumed for the electron number density. Using this model, we consider some smooth distributions for the velocity and the optical depth and investigate the effects on the isotropic energy of the scattered component.

The parameters of our model are summarized in Table 1. We assume that the jet and ejecta are steady radial flows with the Lorentz factor of Γ(θ) given by Equation (24), and the thermal motion of electrons in the flows is negligible. Photons are emitted to an isotropic direction in the jet comoving frame at the emission site. The emission site is at a given radius of r_emi and polar angle of θ ≤ Δθ. The total number of the generated photons in each simulation is about 10⁹. We calculate the scattering probability P for each photon,

$$\begin{eqnarray}&&P=1-\exp (-\tau ).\end{eqnarray} \tag{ 25 }$$

The optical depth τ for each photon is derived by the integration along the photon path from the emission point. We generate a random number in the range of [0:1] for each photon. If the scattering probability reaches the given random number, the photon is scattered. For the scattering process, we consider Thomson scattering. The probability of the photon scattered into each solid angle follows the differential cross section of Thomson scattering (e.g., Rybicki & Lightman 1979) in the comoving frame. We generate two additional random numbers to determine the photon direction after the scattering (Pozdnyakov et al. 1983). Then, we generate a new random number for a scattered photon and calculate the scattering probability from the scattering point. Because of the computational cost, we do not take into account the contribution of more than six-time-scattered photons to the total radiation energy. It is noted that the once- and twice-scattered photons dominantly contribute to the total number of escaping photons with the off-axis direction. Escaping photons are counted at the surface of a sphere with a radius of 100r_emi as a function of the propagation direction.

Table 1.  Model Parameters

Symbol		Value
remi	Photon emission radius	1012 cm
Δθ	Jet opening angle	0.2 rad
Γj	Jet Lorentz factor	200
Liso	Jet isotropic luminosity	1051 erg s−1
Mc	Cocoon mass	5 × 10−8 M⊙
Γc	Ejecta Lorentz factor	3
θint	Angle between jet and ejecta	$0.5{{\rm{\Gamma }}}_{{\rm{j}}}^{-1},1.0{{\rm{\Gamma }}}_{{\rm{j}}}^{-1},3.0{{\rm{\Gamma }}}_{{\rm{j}}}^{-1}$

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Figure 3 shows the photon isotropic energy as a function of the viewing angle. The vertical values are normalized by the isotropic energy of the unscattered component with θ_v = 0°. The normalized isotropic energy of the scattered component (thick solid curves) is ∼10⁻³–10⁻⁴, for the case of θ_int ≲ ${{\rm{\Gamma }}}_{{\rm{j}}}^{-1}$ and the viewing angle of θ_v ∼ 25°–30°, which gives the duration and time delay of ∼2 s (Equations (20) and (22)). The photons scattered by the ejecta with low Lorentz factor contribute to the radiation energy with a large viewing angle. Since the number of photons emitted from the jet decreases with an angle θ, the number of photons scattered by the ejecta increases for the narrower interface of θ_int. The isotropic energy is comparable to the analytical result of ${L}_{\mathrm{iso}}{\rm{\Delta }}{T}_{\mathrm{dur},\mathrm{sc}}$ in Equations (20) and (21). For the larger angle of θ_v ≳ 30°, isotropic energy is reduced because of the beaming effect (Equation (7)) and the enhanced optical depth for large-angle photons relative to the flow direction. We also show the isotropic energy of the unscattered component in Figure 3 (thin dashed curves). For the angle of θ_v ≳ 15°, the scattered component dominantly contributes to the isotropic photon energy. Although the isotropic energy of the scattered component with a range of direction of θ_v ≳ 15° is E_iso(θ_v)/E_iso(0) ≲ 10⁻⁴ in the case of ${\theta }_{\mathrm{int}}\,\gtrsim \,{{\rm{\Gamma }}}_{{\rm{j}}}^{-1}$, a detailed analysis including the dependence of other model parameters will be presented in a forthcoming paper.

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Now let us consider the implications of sGRB 170817A to the scattering model. In sGRB 170817A, the duration, time-averaged luminosity, and time lag of the prompt emission are T_dur,sc ∼ 2 s, L_sc ∼ 2.5 × 10⁴⁶ erg s⁻¹, and ΔT_lag ∼ 1.7 s, respectively (Goldstein et al. 2017). The estimated observed angle with respect to the orbital axis of the pre-merging binary system, which is supposed to be equal to the jet axis, is θ_v ≲ 32 deg (Abbott et al. 2017a, 2017c).

In the nonrelativistic case, from Equations (16) and (18), the observed duration ${T}_{\mathrm{dur},\mathrm{sc}}$ and time lag ${\rm{\Delta }}{T}_{\mathrm{lag}}$ are explained if the engine duration and the breakout timescale are t_dur ∼ t_br ∼ 2 s (>t_j) or the jet-launch time is t_j ∼ 2 s (>t_dur, t_br). From Equations (12) and (13) and the condition r_emi ≲ r_sc in Equation (3), the Lorentz factor of the jet should satisfy the condition $\eta \geqslant {{\rm{\Gamma }}}_{{\rm{j}}}\gtrsim 500$. Then, the observed luminosity is consistent with the scattering model within the reasonable ranges of the parameters: Γ_j ∼ 10³, Δθ ∼ 0.1–0.3 rad, _sc ∼ 0.1, and L_iso ∼ 10⁵¹ erg s⁻¹ in Equation (17). For this parameter set, the emission and scattering sites are r_sc ∼ r_emi ∼ 10¹⁰ cm and the photon-to-baryon ratio is η ∼ 10³, which would give constraints on the models of the prompt emission (e.g., Mészáros & Rees 2000; Ioka et al. 2011).

In the relativistic case, the observed duration and time lag can be explained if the scattering radius is r_sc ∼ 10¹² (Γ_c/3)² cm from Equations (20) and (22). If the Lorentz factor of the relativistic scatterer is Γ_c ∼ 1–10, the observed luminosity is also explained by the reasonable ranges of the parameters: Γ_j ∼ 10², Δθ ∼ 0.1–0.3 rad, t_dur ∼ 0.1 s, _sc ∼ 0.01, and L_iso ∼ 10⁵¹ erg s⁻¹ in Equation (21). From Equation (19) with Γ_c ∼ 1–10, the mass of the relativistic material should be M_c ∼ 10⁻⁸–10⁻⁶ M_⊙. Note that the opening angle of the scattered emission in Equation (7) is Δθ_sc ∼ 0.3 (Γ_c/3)⁻¹ rad, which is small for large Γ_c, and hence too large Γ_c is not preferred.

The observed peak energy of the main pulse E_p ∼ 185 ± 62 keV is consistent with the scattering model because previous sGRBs that are likely on-axis have comparable peak energies. In particular, the E_p–L_p correlation suggests L_p ∼ 2 × 10⁵¹ erg s⁻¹ for the main peak (Tsutsui et al. 2013), which is consistent with our choice of L_iso in the above discussion. The temperature of the weak tail kT = 10.3 ± 1.5 keV is not consistent with the scattering model, while the off-axis de-beamed emission can explain such low peak energies (Ioka & Nakamura 2018). Both the scattering model and the off-axis model coexist in a single sGRB, for example, if the Lorentz factor of the jet is high in the main pulse and low in the weak tail. The emission region could also range from the photosphere (for scattering) to the internal shocks (for off-axis de-beaming).

The energy and the Lorentz factor profiles of the jet and cocoon at the prompt emission phase are highly uncertain. In this circumstance, we should be open-minded about possible jet structures that are consistent with observations and theories. In Section 3.4, we consider the idealized structure between the jet and cocoon by using the parameter θ_int. This prescription is very useful for future studies. We show that if the jet has relatively steep angular distributions of the energy and Lorentz factor (θ_int ∼ ${{\rm{\Gamma }}}_{{\rm{j}}}^{-1}$), the properties of sGRB 170817A could be explained by the scattering emission of the canonical sGRB. On the other hand, if we adopt a smoothly varying jet (${\theta }_{\mathrm{int}}\gg {{\rm{\Gamma }}}_{{\rm{j}}}^{-1}$), most photons cannot reach to the low-Γ component (Γ < 10). Then, the radiation energy of the scattering emission for an off-axis observer would become much lower as shown in Figure 3 with ${\theta }_{\mathrm{int}}=3{{\rm{\Gamma }}}_{{\rm{j}}}^{-1}$. In this case, it is difficult to explain the observed properties of sGRB 170817A based on the scattering model. In the following, we discuss the jet–cocoon structure suggested by the numerical simulation results and the afterglow observations.

Some numerical simulations show that after the jet penetrates the ejecta, the width of θ_int is much wider than the inverse of the Lorentz factor of the ultrarelativistic component, ${{\rm{\Gamma }}}_{{\rm{j}}}^{-1}$ (e.g., Lazzati et al. 2018). However, it is difficult to calculate the steep angular distributions of the energy and Lorentz factor by current numerical simulations owing to the following numerical effects. First, in order to resolve the steep structure, an angular resolution of $d\theta \lesssim {{\rm{\Gamma }}}_{{\rm{j}}}^{-1}\sim 0.005\,\mathrm{rad}$ is required. The 3D numerical simulation is performed to avoid the plug effect, which is considered as a numerical artifact of the symmetry (Gottlieb et al. 2018a). Typical angular resolution at the base of the jet in 3D simulation is dθ ≳ 0.1 rad, which is not enough to resolve the structure in our model. Second, artificial baryon loading could happen when the contact discontinuity between the jet and cocoon crosses the grids via numerical diffusion (Mizuta & Ioka 2013). A certain level of baryon loading is unavoidable and could also smooth the angular distributions.

In addition, there are some uncertainties for the setup of numerical simulations such as the 3D structures of the ejecta and the magnetic field and the energy injection manner from the jet to the ejecta. For example, if the ejecta density is low enough, the jet can keep the initial structure, leading to a sharp boundary. Future, more sophisticated numerical simulations are required to clarify whether the steep angular distributions considered in our model are realized or not.

The afterglow emission has been detected in sGRB 170817A, which could give constraints on the jet structure (Alexander et al. 2017; Evans et al. 2017; Fraija et al. 2017; Haggard et al. 2017; Hallinan et al. 2017; Kim et al. 2017; Margutti et al. 2017; Troja et al. 2017). The observed light curves show a gradual rise in radio, optical, and X-ray bands up to ∼150 days (Alexander et al. 2018; D'Avanzo et al. 2018; Dobie et al. 2018; Ghirlanda et al. 2018; Lyman et al. 2018; Margutti et al. 2018; Mooley et al. 2018a, 2018b; Nynka et al. 2018; Pooley et al. 2018; Resmi et al. 2018; Ruan et al. 2018; Troja et al. 2018). The gradual rise implies the continuous energy injection into the observed region, which is consistent with the structured jet with smoothly varying kinetic energy distribution in angular direction.

However, the afterglow observations cannot in principle constrain the Lorentz factor distribution at the prompt emission phase, because the afterglow emission comes from the decelerated shocks and does not keep the information about the initial Lorentz factor of the jet and cocoon (e.g., Sari 1997). Even if the kinetic energy of the jet has a smooth angular distribution, the scattering could occur if there is a steep distribution in the Lorentz factor.

For the energy distribution, the angular structure may be different between the prompt emission phase and the afterglow emission phase. In about half of sGRBs, the isotropic photon energy of the prompt emission is lower than that of the later activities, so-called extended emission (Kisaka et al. 2017). The injected energy from the jet component, which causes the extended emission, may dominate the observed afterglow emission. In this case, the afterglow emission does not constrain the jet structure at the prompt emission phase.

Even if the jet component of the prompt emission is the dominant energy source of the observed afterglow emission, the structures may be different between the afterglow phase and prompt emission phase. During the propagation in the circum-merger medium, the energy distribution becomes smooth even if the initial jet has a sharp profile because the material at the boundary of the jet expands to a wider angle. This is actually observed in the numerical simulation of afterglows (Granot et al. 2001). As in Figure 2 of Granot et al. (2002), the numerical calculations show gradual rising light curves of afterglow even from a top-hat jet, which is caused by the side expansion of the material from the jet. Therefore, the afterglow observation cannot rule out the existence of the jet with steep angular distributions at the prompt emission phase.

Furthermore, as shown in Figure 4, the energy distribution in our model is consistent with and even implied by the observations. We compare the isotropic radiation energy of the prompt emission from the structured jet constrained by the afterglow observations with the observed value of sGRB 170817A. Here, we assume that the angular structure at the afterglow emission phase is the same as that at the prompt emission phase. We use the radiation efficiency of _γ and the isotropic kinetic energy distribution of the structured jet to estimate the isotropic radiation energy of the prompt emission. We adopt the structured jet model suggested by Abbott et al. (2018; see also Margutti et al. 2018; Xie et al. 2018), the radiation efficiency of _γ = 0.15, the circum-merger density n_ism ∼ 10⁻⁴ cm⁻³, and the viewing angle of θ_v ∼ 20°, which are consistent with the constraints by the observed afterglow emission, including the recent VLBI observations (Ghirlanda et al. 2018; Mooley et al. 2018a). However, as seen in Figure 4, a simple extrapolation of the jet structure (black dashed curve) to the observed viewing angle exceeds the observed one in sGRB 170817A (blue asterisk) unless the radiation efficiency of _γ is much smaller than that in typical prompt emission of GRBs (e.g., Freedman & Waxman 2001; Beniamini et al. 2016). This suggests a steeper structure than a simple extrapolation in the outer part of the jet at the prompt emission phase. In fact, the afterglow observations cannot give constraints on the structure around the viewing angle, in particular $| {\theta }_{{\rm{v}}}-\theta | \lesssim 1/{\rm{\Gamma }}\sim 10^\circ$, because this part contributes to the earlier emission and does not affect the emission after the first detection of the afterglow, i.e., >9 days after the merger (e.g., Troja et al. 2017). A steeper energy distribution such as our model (red curve) and narrow Gaussian jet models (Lyman et al. 2018; Ruan et al. 2018; Troja et al. 2018) is implied for the outer region of the jet.⁸ The detection of the earlier afterglow emission is important to clarify the detailed structure of the jet and cocoon.

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We propose that the Thomson-scattered emission of a prompt sGRB is a promising EM counterpart to a binary NS merger, which is observable from a large viewing angle. The GRB jet is surrounded by a cocoon, either subrelativistic or mildly relativistic, which could scatter the prompt emission into a wide angle of ∼1/Γ_c. The striking feature of the scattering model is that the spectral peak energy E_p is almost independent of the viewing angle and similar to the on-axis values if E_p ≲ MeV as in Figure 2. The isotropic-equivalent luminosity of the scattered emission is also estimated as L_sc ∼ 10⁴⁶–10⁴⁷ erg s⁻¹ in Equations (17) and (21) for fiducial parameters if the prompt emission occurs around the scattering radius. These features are consistent with sGRB 170817A, in particular the main pulse of the prompt emission, supporting the scattering model. We also clarify what determines the duration of the scattered emission and its time lag behind the GWs in Equations (16) and (18) for the nonrelativistic case and in Equations (20) and (22) for the relativistic case. These estimates also reproduce the observed duration of ∼2 s and time lag of ∼1.7 s in sGRB 170817A with typical sGRB parameters. We also perform the Monte Carlo simulation and find that the luminosity of the scattering component is consistent with the observations. The scattering model requires the steep angular distributions of the Lorentz factor structure between the jet and cocoon, which is consistent with the late-time afterglow observations (see discussion in Section 5). These successful results support that the binary NS merger in GW170817 is associated with a typical sGRB jet, which produces sGRB 170817A and other EM counterparts. They also suggest that the scattering emission is ubiquitous in off-axis sGRBs and provides a clear target for future simultaneous observations with GWs.

We should remind that the scattering emission coexists with the off-axis de-beamed emission, complementing each other. In particular, both models suggest a typical off-axis sGRB jet for sGRB 170817A. The possible evolution of the observed peak energies from the main pulse (E_p ∼ 185 ± 62 keV) to the weak tail (kT ∼ 10.3 ± 1.5 keV) may be explained by invoking the scattering and the off-axis de-beamed emissions, respectively.

The remarkable characteristic of the scattering model is that the spectral shape is similar to the on-axis one with a cutoff around ∼MeV energy due to the Klein–Nishina effect as in Figure 2. In particular, the peak energy E_p distribution is similar to the on-axis one with an accumulated bump around MeV energy. We also expect E_p–L_sc and E_p–E_iso correlations with similar slopes, fainter normalizations, but larger dispersions than the on-axis correlations. These spectral features are the main difference from the other models such as the off-axis de-beamed emission from an sGRB jet (e.g., Ioka & Nakamura 2018) and the emission from a low-Γ_j jet/cocoon with a wide opening angle (e.g., Lamb & Kobayashi 2017; Lazzati et al. 2017a, 2017b; Kathirgamaraju et al. 2018). The off-axis jet model generally predicts low peak energies of ∼10 keV (e.g., Ioka & Nakamura 2001, 2018). In the low-Γ_j jet and mildly relativistic cocoon models, if the emission is purely thermal radiation, the peak energy is also significantly soft (∼1–10 keV) compared with the on-axis prompt emission (Lazzati et al. 2017b). Therefore, we can test models by observing the peak energy distributions. The other observables such as the duration and time lag after GWs could also be helpful for discriminating models.

The scattering layer could be accelerated by the illumination of the prompt emission. If the kinetic energy of the scatterer in the illuminated region is less than the energy injected by the radiation, the scattering layer is accelerated. The radiation energy that illuminates the ejecta is

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{rad}} & \sim & \displaystyle \frac{2}{{\rm{\Delta }}\theta {{\rm{\Gamma }}}_{{\rm{j}}}}\displaystyle \frac{{\rm{\Delta }}{\theta }^{2}}{2}{E}_{\mathrm{iso}}{\epsilon }_{\mathrm{acc}}\\ & \sim & 3\times {10}^{46}{\rm{\Delta }}{\theta }_{-0.5}{{\rm{\Gamma }}}_{{\rm{j}},2}^{-1}{\epsilon }_{\mathrm{acc},-1}{E}_{\mathrm{iso},50}\,\mathrm{erg},\end{array}\end{eqnarray} \tag{ 26 }$$

where _acc is the energy gain normalized by the electron rest mass energy for single Thomson scattering, which is (E_p/m_ec²)² ∼ 0.1 for photons with E_p ∼ 200 keV. In the nonrelativistic scatterer case, the kinetic energy of the illuminated ejecta is

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{kin}} & \sim & \displaystyle \frac{1}{2}{f}_{\mathrm{cm}}{M}_{{\rm{e}}}{\beta }_{{\rm{c}}}^{2}{c}^{2}\displaystyle \frac{3{\rm{\Delta }}\theta }{4\pi {{\rm{\Gamma }}}_{{\rm{j}}}}\\ & \sim & 4\times {10}^{47}{\rm{\Delta }}{\theta }_{-0.5}{{\rm{\Gamma }}}_{{\rm{j}},2}^{-1}{f}_{\mathrm{cm},-0.3}{M}_{{\rm{e}},-2}{\beta }_{{\rm{c}},-0.5}^{2}\,\mathrm{erg},\end{array}\end{eqnarray} \tag{ 27 }$$

so that the acceleration by the radiation is negligible. On the other hand, in the relativistic scatterer case, the kinetic energy of the illuminated ejecta is

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{kin}} & \sim & {{\rm{\Gamma }}}_{{\rm{c}}}{M}_{{\rm{c}}}{c}^{2}\displaystyle \frac{3{\rm{\Delta }}\theta }{4\pi {{\rm{\Gamma }}}_{{\rm{j}}}}\\ & \sim & 4\times {10}^{44}{\rm{\Delta }}{\theta }_{-0.5}{{\rm{\Gamma }}}_{{\rm{j}},2}^{-1}{{\rm{\Gamma }}}_{{\rm{c}},0.5}{M}_{{\rm{c}},-7}\,\mathrm{erg},\end{array}\end{eqnarray} \tag{ 28 }$$

which is less than the radiation energy, E_rad. Then, the scattered layer could be accelerated. During the acceleration, the illuminated ejecta interact with a part of the outer region of ejecta (θ > Δθ + 1/Γ_j). Then, a part of the injected energy would be distributed. We assume that the ejecta with an angle of ${{\rm{\Gamma }}}_{\mathrm{acc}}^{-1}$ are accelerated up to Γ_acc as a result of the interaction with the outer region of the ejecta,

$$\begin{eqnarray}&&{E}_{\mathrm{rad}}={{\rm{\Gamma }}}_{\mathrm{acc}}\displaystyle \frac{{M}_{{\rm{c}}}{c}^{2}}{2{{\rm{\Gamma }}}_{\mathrm{acc}}^{2}},\end{eqnarray} \tag{ 29 }$$

where $\sim 1/(2{{\rm{\Gamma }}}_{\mathrm{acc}}^{2})$ is the volume fraction of the accelerated ejecta. Then, the Lorentz factor of the accelerated ejecta is

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{\mathrm{acc}} & \sim & \displaystyle \frac{{M}_{{\rm{c}}}{c}^{2}}{2{E}_{\mathrm{rad}}}\\ & \sim & 3{\rm{\Delta }}{\theta }_{-0.5}^{-1}{{\rm{\Gamma }}}_{{\rm{j}},2}{\epsilon }_{\mathrm{acc},-1}^{-1}{M}_{{\rm{c}},-7}{E}_{\mathrm{iso},50}^{-1},\end{array}\end{eqnarray} \tag{ 30 }$$

which is similar to the pre-accelerated value as we consider in Section 3. In this case, the radiative acceleration does not significantly change the results in Section 3. The time-dependent radiation-hydrodynamics simulations are required to determine the detailed profile of the accelerated ejecta. This will be addressed in future works.

The multiscattering components could also contribute to the soft weak tail. If photons are trapped by the expanding cocoon via scattering, the adiabatic cooling decreases the photon temperature or peak energy while keeping the photon number. This is an interesting future problem.

The success of the scattering model also has interesting implications for the emission mechanism of sGRBs. For the nonrelativistic scatterer, the emission and scattering sites have to be ≲10¹⁰ cm to scatter the prompt emission in Equations (3) and (12). Based on the photosphere model, the Lorentz factor and the photon-to-baryon ratio should be ${{\rm{\Gamma }}}_{{\rm{j}}}\sim {10}^{3}$ and η ∼ 10³ from Equation (13) and discussions in Section 4. For the relativistic scatterer, the scattering site is preferred not to be far away, r_sc ≲ 10¹² cm, from Equation (19) and discussions in Section 4. In either case, the emission radius is not very large as in the case of some Poynting-dominated models, favoring the photosphere model and the internal shock model.

Although scattered emission may have been already detected in the previous observations, they would be most likely classified as redshift-unknown events, because they are faint and their early afterglows are also faint owing to the off-axis viewing angle. The afterglow emission of the scatterer is also faint before its deceleration time, which is late because of its low Lorentz factor.

The event rate of the scattered prompt emission detected by a γ-ray detector is estimated as

$$\begin{eqnarray}&&{ \mathcal R }\sim 0.3{{ \mathcal R }}_{\mathrm{merge},3.5}{F}_{\mathrm{lim},-7}^{-3/2}{L}_{\mathrm{sc},46}^{3/2}{\rm{\Delta }}{{\rm{\Omega }}}_{4\pi }\,{\mathrm{yr}}^{-1},\end{eqnarray} \tag{ 31 }$$

where we assume isotropic emission, ${{ \mathcal R }}_{\mathrm{merge},3.5}\equiv {{ \mathcal R }}_{\mathrm{merge}}/{10}^{3.5}$ Gpc⁻³ yr⁻¹ is the merger rate of the binary NS, F_lim is the detector sensitivity limit in 1 s integration time, and ${\rm{\Delta }}{{\rm{\Omega }}}_{4\pi }\equiv {\rm{\Delta }}{\rm{\Omega }}/(4\pi \,\mathrm{sr})$ is the detector field of view. Using the upper limit on the binary NS merger rate derived from the LIGO O1 (∼3 × 10³ yr⁻¹ Gpc⁻³; Abbott et al. 2016b), approximately three events would be detected during ∼10 yr of observations of Fermi/GBM with a full-sky field of view and the flux limit F_lim ∼ 10⁻⁷ erg cm⁻² (Narayana Bhat et al. 2016). For Swift/BAT with the field of view ΔΩ = 1.4 sr and the flux limit F_lim ∼ 10⁻⁸ erg cm⁻² (Krimm et al. 2013), ∼12 events would be detected during the ∼13 yr of observations. The simultaneous observation with GWs will help us not to miss the event to follow up, revealing the off-axis nature of the sGRBs and binary NS mergers.

We are grateful to the anonymous referee for constructive comments. We would like to thank Kuniaki Masai, Yutaka Ohira, Masaru Shibata, Masaomi Tanaka, Shuta Tanaka, Ryo Yamazaki, and Michitoshi Yoshida for discussions. This work is partly supported by "New Developments in Astrophysics through Multi-Messenger Observations of Gravitational Wave Sources," No. 24103006 (K.I., T.N.), KAKENHI Nos. 16J06773, 18H01246 (S.K.), Nos. 26287051, 26247042, 17H01126, 17H06131, 17H06362, 17H06357 (K.I.), No. 17K14248 (K.K.), and No. 15H02087 (T.N.) by the Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.