Light Curves of a Shock-breakout Material and a Relativistic Off-axis Jet from a Binary Neutron Star System

Binary neutron star (NS) mergers are thought to be natural candidates for gravitational waves (GWs), short gamma-ray bursts (sGRBs), mass ejections producing delayed radio emissions, and isotropic quasithermal optical/infrared counterparts, so-called kilonova/macronova (for reviews, see Nakar 2007; Berger 2014; Metzger 2017). Kilonova/macronova is related to a neutron-rich mass ejection ($\sim {10}^{-4}\mbox{--}{10}^{-2}\,{M}_{\odot }$), which presents a rapid neutron capture process (r-process) nucleosynthesis (Lattimer & Schramm 1974, 1976). This process synthesizes heavy and unstable nuclei—such as gold and platinum—and consequently heats rapidly the merger ejecta by the radioactive decay energy (Li & Paczyński 1998; Rosswog 2005; Metzger et al. 2010; Kasen et al. 2013; Metzger 2017). SGRBs, with duration less than 2 s, originate from internal collisions or magnetic dissipation within the beamed and relativistic outflow. Delayed radio emission is expected from the interactions of the ejected materials with the circumburst medium (Nakar & Piran 2011; Piran et al. 2013; Hotokezaka & Piran 2015). Kyutoku et al. (2014) proposed the possibility of detecting X-ray, optical, and radio fluxes from an ejected ultrarelativistic material decelerated early (from seconds to days) by the interstellar medium (ISM).

The GW transient GW170817, associated with a binary NS system with a merger time of 12:41:04 UTC, 2017 August 17, was detected by LIGO and Virgo experiments (Abbott et al. 2017a, 2017b). Immediately, GRB 170817A triggered the Gamma-ray Burst Monitor (GBM) on board the Fermi Gamma-ray Space Telescope at 12:41:06 UTC (Goldstein et al. 2017). The INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) detected an attenuated γ-ray flux with ∼3σ (Savchenko et al. 2017). GRB 170817A was followed-up by multiple ground-based telescopes in different bands. A bright optical i-band flux with magnitude 17.057 ± 0.0018 mag was detected by the 1 m Swope telescope at Las Campanas Observatory in Chile after 10.87 hr, followed by multiple optical and infrared ground-based telescopes. The X-ray and radio experiments followed-up this burst during the subsequent days after the merger without detecting any signal. Finally, GRB 170817A began to be detected in X-rays on the ninth day by Chandra (Margutti et al. 2017b; Troja et al. 2017b; Haggard et al. 2018), in the radio (3 and 6 GHz) bands on day nineteen by the Very Large Array and Atacama Large Millimeter/submillimeter Array (VLA and ALMA, respectively; Alexander et al. 2017), and in the optical band after ∼110 days (see; Margutti et al. 2018). It is worth highlighting that although optical afterglow was detected from the month of December, quasithermal optical emission associated with the kilonova was detected early, around 11 hr after the GBM trigger. The host galaxy associated with this event, NGC 4993, was located at a distance of ($z\simeq 0.01$) 40 Mpc (Coulter et al. 2018; Margutti et al. 2017a).

Whereas several authors have associated the early γ-ray photons to different emission mechanisms (Bromberg et al. 2018; Gottlieb et al. 2018b; Kisaka et al. 2018), the X-ray, optical, and radio afterglow have been related to the synchrotron forward-shock radiation, when the relativistic off-axis jet and/or cocoon are decelerated in a homogeneous low density medium in the range ${10}^{-4}$–${10}^{-2}\,{\mathrm{cm}}^{-3}$ (Alexander et al. 2017; Fraija et al. 2017a; Granot et al. 2018; Ioka & Nakamura 2018; Kasliwal et al. 2017; Lazzati et al. 2018; Margutti et al. 2017a; Mooley et al. 2018b; Murguia-Berthier et al. 2017; Piro & Kollmeier 2018; Wang & Huang 2018; Troja et al. 2018).

In this paper, we derive the forward-shock dynamics and the synchrotron light curves from the outermost shock-breakout material and the relativistic off-axis jet from a binary NS system. As a particular case, we explain the electromagnetic counterpart detected in the GW170817 transient. Our proposed model uses general arguments of the synchrotron afterglow theory introduced in Sari et al. (1998) and the results obtained numerically in Tan et al. (2001) about the ejected masses from GRB progenitors. This paper is arranged as follows: In Sections 2 and 3 we derive the synchrotron forward-shock model generated by the deceleration of the shock-breakout material and the relativistic off-axis jet, which were launched from a binary NS merger. In Section 4, we present the data used in this work and describe the electromagnetic counterpart of GW170817 with our synchrotron forward-shock model. In Section 5, we discuss the results and present our conclusions.

The convention ${Q}_{{\rm{x}}}=Q/{10}^{{\rm{x}}}$ in c.g.s. units and ${\rm{\hslash }}$ = c = 1 in natural units will be used. The values of cosmological parameters H₀ = 71 km s⁻¹ Mpc⁻¹, ${{\rm{\Omega }}}_{{\rm{m}}}=0.27$, and ${{\rm{\Omega }}}_{\lambda }=0.73$ are adopted (Spergel et al. 2003).

2.1. Properties and Considerations

The breakout burst signal properties depend on the mass (${M}_{{\rm{m}}}$), radius (${R}_{{\rm{m}}}$), and velocity (${\beta }_{{\rm{b}}}$) of the merger remnant. Right after the binary NS merger takes place, a shock formed at the interface between the NSs is initially launched from the core toward the crust at subrelativistic velocities (${\beta }_{{\rm{b}},\mathrm{in}}\sim 0.25;$ Kyutoku et al. 2014; Metzger et al. 2015). At this early phase, the shocked material cannot escape from the merged remnant because the initial velocity is less than the escape velocity ${\beta }_{{\rm{b}},\mathrm{es}}\simeq 0.83\,{\left(\tfrac{{M}_{{\rm{m}}}}{3{M}_{\odot }}\right)}^{\tfrac{1}{2}}{\left(\tfrac{15\mathrm{km}}{{R}_{{\rm{m}}}}\right)}^{\tfrac{1}{2}}$.

Although the shock velocity increases as the crust density decreases ($\rho \propto {r}^{-s}$ with $s\simeq 0.187$ for a polytropic index ${n}_{p}=3$), it will eventually reach the escape velocity. Once the shock has reached half of the escape velocity, it can leave the binary NS merger by converting thermal energy into kinetic energy. At this moment, the shock velocity has increased by a factor of $0.5{\beta }_{{\rm{b}},\mathrm{esc}}{/\beta }_{{\rm{b}},\mathrm{ini}}\simeq \tfrac{3}{2}$ and the crust density has decreased by $\simeq {\left(\tfrac{3}{2}\right)}^{-1/s}$. The ejected mass can be estimated as

$$\begin{eqnarray}\begin{array}{rcl}{M}_{b} & \simeq & 5\times {10}^{-5}\,{M}_{\odot }\left(\displaystyle \frac{{M}_{{\rm{c}}}}{{10}^{-2}{M}_{\odot }}\right)\left(\displaystyle \frac{{R}_{{\rm{b}}}}{1\,\mathrm{km}}\right)\left(\displaystyle \frac{15\,\mathrm{km}}{{R}_{{\rm{b}}}}\right)\,{\left(\displaystyle \frac{{\beta }_{{\rm{b}},\mathrm{ini}}}{0.25}\right)}^{\tfrac{5}{7}}\\ & & \times \ {\left(\displaystyle \frac{{\beta }_{{\rm{b}},\mathrm{esc}}}{0.83}\right)}^{-\tfrac{5}{7}},\end{array}\end{eqnarray} \tag{ 1 }$$

where M_c is the crust mass. Once the merger material has moved for long enough to achieve the near-Newtonian phase, the decelerated material propagates adiabatically with the same effective polytropic index (Meliani & Keppens 2010; van Eerten et al. 2010). Tan et al. (2001) investigated the acceleration of the shock waves to relativistic and subrelativistic velocities in the outer matter of an explosion. They found that when the energy of the explosion was concentrated in the outermost ejecta, the blast wave could generate a strong electromagnetic emission. Furthermore, the equivalent kinetic energy of the outermost matter can be expressed as a power-law velocity distribution given by

$$\begin{eqnarray}{E}_{{\rm{k}},{\rm{c}}}(\gtrsim {\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}})=\tilde{E}\left\{\begin{array}{ll}{\left({\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}\right)}^{-(1.58{\gamma }_{p}-1)} & \mathrm{for}\,({\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}})\gg 1,\\ {\left({\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}\right)}^{-(5.35{\gamma }_{p}-2)} & \mathrm{for}\,({\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}})\ll 1,\end{array}\right.\end{eqnarray} \tag{ 2 }$$

where $\tilde{E}$ is the fiducial energy and ${\gamma }_{p}=1+1/{n}_{p}$ with n_p is the polytropic index. For ${n}_{p}=3$, the kinetic equivalent energy is given by

$$\begin{eqnarray}&&{E}_{{\rm{k}},{\rm{c}}}(\gtrsim {\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}})\simeq \tilde{E}\,{\left({\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}\right)}^{-{\alpha }_{s}},\end{eqnarray} \tag{ 3 }$$

with ${\alpha }_{s}=1.1$ for ${\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}\gg 1$ and ${\alpha }_{s}=5.2$ for ${\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}\ll 1$.

Once the relativistic shock-breakout material sweeps up enough ISM, the electron population is cooled down emitting synchrotron radiation in the observer's line of sight. Assuming that all the energy is confined within an opening angle ${\theta }_{{\rm{c}}}\sim 1/{{\rm{\Gamma }}}_{{\rm{c}}}$ (Tan et al. 2001), the equivalent kinetic energy associated to the observed region becomes $\simeq \tfrac{{E}_{{\rm{k}},{\rm{c}}}(\gt {{\rm{\Gamma }}}_{{\rm{c}}})}{2{{\rm{\Gamma }}}_{{\rm{c}}}^{2}}$ (Nakar & Piran 2018), and therefore from Equation (3) the observed kinetic energy, ${E}_{{\rm{k}},{\rm{c}}}$, can be written as

$$\begin{eqnarray}&&{E}_{{\rm{k}},{\rm{c}}}(\gtrsim {\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}})\simeq \tilde{E}\,{\left({\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}\right)}^{-\delta },\end{eqnarray} \tag{ 4 }$$

with $\delta ={\alpha }_{s}+2$. The factor of 2 has been absorbed in the fiducial energy $\tilde{E}$. On the other hand, the electromagnetic energy released in the observer's direction can be estimated through the efficiency η and the equivalent kinetic energy ${E}_{\mathrm{obs},\gamma }\,\approx \eta \,{E}_{{\rm{k}},{\rm{c}}}(\gtrsim {\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}})=\eta \tilde{E}\,{\left({\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}\right)}^{-\delta }$. Therefore, the efficiency of kinetic to γ-ray energy conversion can be estimated as

$$\begin{eqnarray}&&\eta \simeq \displaystyle \frac{{E}_{\mathrm{iso},\gamma }\,{\left({\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}\right)}^{\delta }}{\,\tilde{E}},\end{eqnarray} \tag{ 5 }$$

where $\tilde{E}$ is a function of total kinetic energy.

2.2. Analytical Model: Synchrotron Forward-shock Emission

2.2.1. Relativistic Phase (${{\rm{\Gamma }}}_{{\rm{c}}}\gt 1/{\theta }_{{\rm{c}}}$ and ${\beta }_{{\rm{c}}}=1$)

The forward-shock dynamics when the outflow with a constant equivalent kinetic energy is decelerated by the ISM has been exhaustively explored (see, e.g., Sari et al. 1998). Here, we derive the dynamics of the forward shock when kinetic energy can be described as a power-law velocity distribution parameterized in accordance with Equation (3). Taking into consideration the adiabatic evolution of the shock, the fiducial energy is given by $\tilde{E}=4/3\pi {{\rm{\Gamma }}}_{{\rm{c}}}^{2+\delta }{R}^{3}n\,{m}_{p}$ (Blandford & McKee 1976; Sari 1997) where n is the homogeneous ISM density, m_p is the proton mass, and $R=\tfrac{{{\rm{\Gamma }}}_{{\rm{c}}}^{2}\,t}{1+z}$ is the deceleration radius. The deceleration timescale and bulk Lorentz factor are given by

$$\begin{eqnarray}&&{t}_{\mathrm{dec}}\simeq 4.2\,\mathrm{days}\,\left(\displaystyle \frac{1+z}{1.02}\right)\,{n}_{-4}^{-\displaystyle \frac{1}{3}}\,{\tilde{E}}_{49}^{\displaystyle \frac{1}{3}}\,{{\rm{\Gamma }}}_{{\rm{c}},0.5}^{-\displaystyle \frac{\delta +8}{3}},\end{eqnarray} \tag{ 6 }$$

and

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{\rm{c}}}=3.1\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\displaystyle \frac{3}{\delta +8}}\,{n}_{-4}^{-\displaystyle \frac{1}{\delta +8}}\,{\tilde{E}}_{49}^{\displaystyle \frac{1}{\delta +8}}\,{t}_{15\,{\rm{d}}}^{-\displaystyle \frac{3}{\delta +8}},\end{eqnarray} \tag{ 7 }$$

respectively. Hereafter through this section the value of ${\alpha }_{s}=3.0$ (Hotokezaka & Piran 2015) is considered.

In the synchrotron forward-shock framework, the accelerated electron population is described by ${\gamma }_{{\rm{e}}}\geqslant {\gamma }_{{\rm{m}}}:N({\gamma }_{{\rm{e}}})\,d{\gamma }_{{\rm{e}}}\,\propto {\gamma }_{{\rm{e}}}^{-p}\,d{\gamma }_{{\rm{e}}}$ (Fraija 2014), where $p=2.2$ is the electron power index and ${\gamma }_{{\rm{m}}}$ is the minimum Lorentz factor given by

$$\begin{eqnarray}&&{\gamma }_{{\rm{m}}}=68.4\,{\epsilon }_{{\rm{e}},-1}{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\displaystyle \frac{3}{\delta +8}}\,{n}_{-4}^{-\displaystyle \frac{1}{\delta +8}}\,{\tilde{E}}_{49}^{\displaystyle \frac{1}{\delta +8}}\,{t}_{15\,{\rm{d}}}^{-\displaystyle \frac{3}{\delta +8}},\end{eqnarray} \tag{ 8 }$$

(e.g., see Fraija & Veres 2018) with ${\epsilon }_{{\rm{e}}}$ being the microphysical parameter associated to the energy fraction given to accelerate electrons (see Fraija et al. 2017c). Similarly, it is possible to define the microphysical parameter ${\epsilon }_{B}={U}_{B}/U$ (with ${U}_{B}=B{{\prime} }^{2}/8\pi$) associated with the fraction of the energy given to generate and/or amplify the comoving magnetic field, which is given by

$$\begin{eqnarray}&&B^{\prime} =0.4\,\mathrm{mG}\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\displaystyle \frac{3}{\delta +8}}\,{\epsilon }_{B,-3}^{\displaystyle \frac{1}{2}}\,{n}_{-4}^{\displaystyle \frac{\delta +6}{2(\delta +8)}}\,{\tilde{E}}_{49}^{\displaystyle \frac{1}{\delta +8}}\,{t}_{15\,{\rm{d}}}^{-\displaystyle \frac{3}{\delta +8}}.\end{eqnarray} \tag{ 9 }$$

Requiring the equality of the deceleration (Equation (6)) and synchrotron timescales, the cooling electron Lorentz factor becomes

$$\begin{eqnarray}&&{\gamma }_{{\rm{c}}}=2.7\times {10}^{8}\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\displaystyle \frac{\delta -1}{\delta +8}}\,{\epsilon }_{B,-3}^{-1}\,{n}_{-4}^{-\displaystyle \frac{\delta +5}{\delta +8}}\,{\tilde{E}}_{49}^{-\displaystyle \frac{3}{\delta +8}}\,{t}_{15\,{\rm{d}}}^{\displaystyle \frac{1-\delta }{\delta +8}}.\end{eqnarray} \tag{ 10 }$$

Comparing the acceleration and synchrotron timescales, the maximum Lorentz factor is

$$\begin{eqnarray}&&{\gamma }_{\max }=8.4\times {10}^{8}\,{\phi }_{-1}^{1/2}{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\displaystyle \frac{3}{2(\delta +8)}}\,{n}_{-4}^{\displaystyle \frac{1}{4}}\,{\tilde{E}}_{49}^{\displaystyle \frac{1}{2(\delta +8)}}\,{t}_{15\,{\rm{d}}}^{-\displaystyle \frac{3}{2(\delta +8)}},\end{eqnarray} \tag{ 11 }$$

with ϕ being the efficiency parameter (e.g., see Fraija 2015, Fraija et al. 2016b). Given the synchrotron process with Equations (6)–(10), the synchrotron spectral breaks can be written as

$$\begin{eqnarray}\begin{array}{rcl}{\epsilon }_{{\rm{m}}} & \simeq & 1.5\times {10}^{-3}\,\mathrm{GHz}\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\tfrac{4-\delta }{\delta +8}}\\ & & \times \ {\epsilon }_{{\rm{e}},-1}^{2}\,{\epsilon }_{B,-3}^{\tfrac{1}{2}}\,{n}_{-4}^{\tfrac{\delta }{2(\delta +8)}}\,{\tilde{E}}_{49}^{\tfrac{4}{\delta +8}}\times \ {t}_{15\,{\rm{d}}}^{-\tfrac{12}{\delta +8}}\\ {\epsilon }_{{\rm{c}}} & \simeq & 28.9\,\mathrm{keV}{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\tfrac{\delta -4}{\delta +8}}{\left(1+Y\right)}^{-2}\\ & & \times \ {\epsilon }_{B,-3}^{-\tfrac{3}{2}}\,{n}_{-4}^{-\tfrac{16+3\delta }{2(\delta +8)}}\,{\tilde{E}}_{49}^{-\tfrac{4}{\delta +8}}\times {t}_{15\,{\rm{d}}}^{-\tfrac{2\delta +4}{\delta +8}},\end{array}\end{eqnarray} \tag{ 12 }$$

where Y is the Compton parameter (i.e., see, Fraija et al. 2016a). For some purposes, the spectral breaks can be described through the luminosity, which is defined as $\tilde{L}=\tilde{E}/t$. The maximum flux estimated through the peak spectral power is given by

$$\begin{eqnarray}\begin{array}{rcl}{F}_{\max } & \simeq & 3\times {10}^{-3}\,\mathrm{mJy}\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\tfrac{8-2\delta }{\delta +8}}\,{\epsilon }_{B,-3}^{\tfrac{1}{2}}\,{n}_{-4}^{\tfrac{3\delta +8}{2(\delta +8)}}\,{D}_{26.5}^{-2}\,{\tilde{E}}_{49}^{\tfrac{8}{\delta +8}}\\ & & \times \,{t}_{15\,{\rm{d}}}^{\tfrac{3\delta }{\delta +8}},\end{array}\end{eqnarray} \tag{ 13 }$$

where D = 100 Mpc is the luminosity distance to the source. Using the observed synchrotron spectrum in the fast- and slow-cooling regimes with Equations (6–11), the synchrotron light curves in the fast-cooling regime can be written as

$$\begin{eqnarray}{F}_{\nu ,f}=\left\{\begin{array}{ll}{A}_{\mathrm{fl}}\,{t}_{15\,{\rm{d}}}^{\tfrac{11\delta +4}{3(\delta +8)}}\,{\left(\tfrac{{\epsilon }_{\gamma }}{6\mathrm{GHz}}\right)}^{\tfrac{1}{3}}, & {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{c}}},\\ {A}_{\mathrm{fm}}\,{t}_{15\,{\rm{d}}}^{\tfrac{2\delta -2}{\delta +8}}\,{\left(\tfrac{{\epsilon }_{\gamma }}{1\mathrm{eV}}\right)}^{-\tfrac{1}{2}}, & {\epsilon }_{{\rm{c}}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}}},\\ {A}_{\mathrm{fh}}\,{t}_{15\,{\rm{d}}}^{\tfrac{2\delta +4-6p}{\delta +8}}\,{\left(\tfrac{{\epsilon }_{\gamma }}{1\mathrm{keV}}\right)}^{-\tfrac{p}{2}}, & {\epsilon }_{{\rm{m}}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{\max },\end{array}\right.\end{eqnarray} \tag{ 14 }$$

with coefficients given by

$$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{fl}} & = & 2.4\times {10}^{-6}\,\mathrm{mJy}\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\tfrac{28-7\delta }{3(\delta +8)}}\\ & & \times \ {\left(1+Y\right)}^{\tfrac{2}{3}}\,{\epsilon }_{B,-3}\,{n}_{-4}^{\tfrac{20+6\delta }{3(\delta +8)}}\times \,{D}_{26.5}^{-2}\,{\tilde{E}}_{49}^{\tfrac{28}{3(\delta +8)}}\\ {A}_{\mathrm{fm}} & = & 0.6\,\mathrm{mJy}\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\tfrac{12-3\delta }{2(\delta +8)}}\\ & & \times \ {\left(1+Y\right)}^{-1}\,{\epsilon }_{B,-3}^{-\tfrac{1}{4}}\,{n}_{-4}^{\tfrac{3\delta }{4(\delta +8)}}\times \ {D}_{26.5}^{-2}\,{\tilde{E}}_{49}^{\tfrac{6}{\delta +8}}\\ {A}_{\mathrm{fh}} & = & 3.3\times {10}^{-9}\,\mathrm{mJy}\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\tfrac{(4-\delta )(p+2)}{2(\delta +8)}}\\ & & \times \ {\left(1+Y\right)}^{-1}\,{\epsilon }_{{\rm{e}},-1}^{p-1}\,{\epsilon }_{B,-3}^{\tfrac{p-2}{4}}\\ & & \times \ {n}_{-4}^{\tfrac{\delta (p+2)}{4(\delta +8)}}\,{D}_{26.5}^{-2}\,{\tilde{E}}_{49}^{\tfrac{4+2p}{\delta +8}}.\end{array}\end{eqnarray} \tag{ 15 }$$

The light curve when synchrotron emission lies in the slow-cooling regime is

$$\begin{eqnarray}{F}_{\nu ,s}=\left\{\begin{array}{ll}{A}_{\mathrm{sl}}\,{t}_{15\,{\rm{d}}}^{\tfrac{3\delta +4}{\delta +8}}\,{\left(\tfrac{{\epsilon }_{\gamma }}{6\mathrm{GHz}}\right)}^{\tfrac{1}{3}}, & {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}}},\\ {A}_{\mathrm{sm}}\,{t}_{15\,{\rm{d}}}^{\tfrac{3\delta -6(p-1)}{\delta +8}}\,{\left(\tfrac{{\epsilon }_{\gamma }}{1\mathrm{eV}}\right)}^{-\tfrac{p-1}{2}}, & {\epsilon }_{{\rm{m}}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{c}}},\\ {A}_{\mathrm{sh}}\,{t}_{15\,{\rm{d}}}^{\tfrac{2\delta +4-6p}{\delta +8}}\,{\left(\tfrac{{\epsilon }_{\gamma }}{1\mathrm{keV}}\right)}^{-\tfrac{p}{2}}, & {\epsilon }_{{\rm{c}}}\lt {\epsilon }_{\gamma },\end{array}\right.\end{eqnarray} \tag{ 16 }$$

with the coefficients given by

$$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{sl}} & = & 3.9\times {10}^{-2}\,\mathrm{mJy}\,{\left(\tfrac{1+z}{1.02}\right)}^{\tfrac{20-5\delta }{3(\delta +8)}}\\ & & \times {\epsilon }_{{\rm{e}},-1}^{-\tfrac{2}{3}}\,{\epsilon }_{B,-3}^{\tfrac{1}{3}}\,{n}_{-4}^{\tfrac{4\delta +12}{3(\delta +8)}}\,{D}_{26.5}^{-2}\times {\tilde{E}}_{49}^{\tfrac{20}{3(\delta +8)}}\\ {A}_{\mathrm{sm}} & = & 3.9\times \,{10}^{-8}\,\mathrm{mJy}\,{\left(\tfrac{1+z}{1.02}\right)}^{\tfrac{(4-\delta )(p+3)}{2(\delta +8)}}\\ & & \times {\epsilon }_{{\rm{e}},-1}^{p-1}\,{\epsilon }_{B,-3}^{\tfrac{p+1}{4}}\,{n}_{-4}^{\tfrac{16+\delta (p+5)}{4(\delta +8)}}\times \,{D}_{26.5}^{-2}\,{\tilde{E}}_{49}^{\tfrac{6+2p}{\delta +8}}\\ {A}_{\mathrm{sh}} & = & 3.3\times {10}^{-9}\,\mathrm{mJy}\,{\left(\tfrac{1+z}{1.02}\right)}^{\tfrac{(4-\delta )(p+2)}{2(\delta +8)}}{\left(1+Y\right)}^{-1}\\ & & \times {\epsilon }_{{\rm{e}},-1}^{p-1}\,{\epsilon }_{B,-3}^{\tfrac{p-2}{4}}\times {n}_{-4}^{\tfrac{\delta (p+2)}{4(\delta +8)}}\,{D}_{26.5}^{-2}\,{\tilde{E}}_{49}^{\tfrac{4+2p}{\delta +8}}.\end{array}\end{eqnarray} \tag{ 17 }$$

It is worth noting that when $\delta =0$, the observable quantities derived in Sari et al. (1998) and the light curves of the synchrotron forward-shock emission are recovered (e.g., see Fraija et al. 2016a).

2.2.2. Lateral Expansion (${{\rm{\Gamma }}}_{{\rm{c}}}\sim 1/{\theta }_{{\rm{c}}}$ and ${\beta }_{{\rm{c}}}\lesssim 1$)

In a realistic case, when ${{\rm{\Gamma }}}_{{\rm{c}}}$ drops below ${\theta }_{{\rm{c}}}$ material begins to slow down and expands laterally. The beaming cone of the radiation emitted broadens increasingly up until it reaches our field of view (${{\rm{\Gamma }}}_{{\rm{c}}}\sim {\theta }_{{\rm{c}}}^{-1};$ Rees 1999; Sari et al. 1999; Dermer et al. 2000; Granot et al. 2002, 2018). In the lateral expansion phase, the break should occur at

$$\begin{eqnarray}&&{t}_{\mathrm{br}}=22.8\,\mathrm{days}\,k\left(\displaystyle \frac{1+z}{1.02}\right)\,{n}_{-4}^{-\displaystyle \frac{1}{3}}\,{\tilde{E}}_{49}^{\displaystyle \frac{1}{3}}\,{\beta }_{{\rm{c}}}^{-\displaystyle \frac{{\alpha }_{s}}{3}}\,{\theta }_{c,{15}^{^\circ }}^{\displaystyle \frac{{\alpha }_{s}+6}{3}},\end{eqnarray} \tag{ 18 }$$

with ${\beta }_{{\rm{c}}}\simeq \sqrt{1-{\theta }_{{\rm{c}}}^{2}}$ and $k$ a parameter that is added to link the maximum value of the observed flux with the jet break by means of the opening and viewing angles (Granot et al. 2002; Nakar et al. 2002). Hereafter through this paper the value of $k=1$ will be assumed (Nakar et al. 2002; Granot et al. 2018).

Once the synchrotron flux reaches our field of view, the synchrotron spectral breaks and the maximum flux becomes

$$\begin{eqnarray}\begin{array}{rcl}{\epsilon }_{{\rm{m}}} & \simeq & 9.7\times {10}^{-3}\,\mathrm{GHz}\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\tfrac{6-{\alpha }_{s}}{{\alpha }_{s}+6}}\\ & & \times \ {\epsilon }_{{\rm{e}},-1}^{2}\,{\epsilon }_{B,-3}^{\tfrac{1}{2}}\,{n}_{-4}^{\tfrac{{\alpha }_{s}-2}{2(\alpha +6)}}\,{\beta }_{{\rm{c}}}^{-\tfrac{4{\alpha }_{s}}{{\alpha }_{s}+6}}\\ & & \times \ {\tilde{E}}_{49}^{\tfrac{4}{{\alpha }_{s}+6}}\,{t}_{15\,{\rm{d}}}^{-\tfrac{12}{{\alpha }_{s}+6}}\\ {\epsilon }_{{\rm{c}}} & \simeq & 16.6\,\mathrm{keV}{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\tfrac{{\alpha }_{s}-6}{{\alpha }_{s}+6}}\\ & & \times \ {\left(1+Y\right)}^{-2}\,{\epsilon }_{B,-3}^{-\tfrac{3}{2}}\,{n}_{-4}^{-\tfrac{(3{\alpha }_{s}+10)}{2({\alpha }_{s}+6)}}\,{\beta }_{{\rm{c}}}^{\tfrac{4{\alpha }_{s}}{{\alpha }_{s}+6}}\\ & & \times \ {\tilde{E}}_{49}^{-\tfrac{4}{{\alpha }_{s}+6}}\,{t}_{15\,{\rm{d}}}^{-\tfrac{2{\alpha }_{s}}{{\alpha }_{s}+6}},\end{array}\end{eqnarray} \tag{ 19 }$$

and

$$\begin{eqnarray}\begin{array}{rcl}{F}_{\max } & \simeq & 0.1\,\mathrm{mJy}\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\tfrac{12-2{\alpha }_{s}}{{\alpha }_{s}+6}}\,{\epsilon }_{B,-3}^{\tfrac{1}{2}}\,{n}_{-4}^{\tfrac{2-{\alpha }_{s}}{2({\alpha }_{s}+6)}}\,{D}_{26.5}^{-2}\,{\beta }_{{\rm{c}}}^{-\tfrac{8{\alpha }_{s}}{{\alpha }_{s}+6}}\\ & & \times \ {\tilde{E}}_{49}^{\tfrac{8}{{\alpha }_{s}+6}}\,{t}_{15\,{\rm{d}}}^{-\tfrac{3(2-{\alpha }_{s})}{{\alpha }_{s}+6}},\,\,\,\,\end{array}\end{eqnarray} \tag{ 20 }$$

respectively. Regarding the synchrotron spectrum, synchrotron spectral breaks and maximum flux, the synchrotron light curve in the slow-cooling regime becomes

$$\begin{eqnarray}{F}_{\nu }=\left\{\begin{array}{ll}{A}_{\mathrm{sl}}\,{t}_{15\,{\rm{d}}}^{\tfrac{3{\alpha }_{s}-2}{{\alpha }_{s}+6}}\,{\left(\tfrac{{\epsilon }_{\gamma }}{6\mathrm{GHz}}\right)}^{\tfrac{1}{3}}, & {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}}},\\ {A}_{\mathrm{sm}}\,{t}_{15\,{\rm{d}}}^{\tfrac{3({\alpha }_{s}-2p)}{{\alpha }_{s}+6}}\,{\left(\tfrac{{\epsilon }_{\gamma }}{1\mathrm{eV}}\right)}^{-\tfrac{p-1}{2}}, & {\epsilon }_{{\rm{m}}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{c}}},\\ {A}_{\mathrm{sh}}\,{t}_{15\,{\rm{d}}}^{\tfrac{2({\alpha }_{s}-3p)}{{\alpha }_{s}+6}}\,{\left(\tfrac{{\epsilon }_{\gamma }}{1\mathrm{keV}}\right)}^{-\tfrac{p}{2}}, & {\epsilon }_{{\rm{c}}}\lt {\epsilon }_{\gamma },\end{array}\right.\end{eqnarray} \tag{ 21 }$$

with the coefficients given by

$$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{sl}} & = & 0.8\,\mathrm{mJy}\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\tfrac{30-5{\alpha }_{s}}{3({\alpha }_{s}+6)}}\\ & & \times \ {\epsilon }_{{\rm{e}},-1}^{-\tfrac{2}{3}}\,{\epsilon }_{B,-3}^{\tfrac{1}{3}}\,{n}_{-4}^{\tfrac{4-2{\alpha }_{s}}{3({\alpha }_{s}+6)}}\,{D}_{26.5}^{-2}\\ & & \times \ {\beta }_{{\rm{c}}}^{-\tfrac{20{\alpha }_{s}}{3({\alpha }_{s}+6)}}\,{\tilde{E}}_{49}^{\tfrac{20}{3({\alpha }_{s}+6)}}\\ {A}_{\mathrm{sm}} & = & 4.5\times {10}^{-6}\mathrm{mJy}\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\tfrac{(6-{\alpha }_{s})(p+3)}{2({\alpha }_{s}+6)}}\\ & & \times \ {\epsilon }_{{\rm{e}},-1}^{p-1}\,{\epsilon }_{B,-3}^{\tfrac{p+1}{4}}\times {n}_{-4}^{\tfrac{({\alpha }_{s}-2)(p-3)}{4({\alpha }_{s}+6)}}\\ & & \times {D}_{26.5}^{-2}\,{\beta }_{{\rm{c}}}^{-\tfrac{{\alpha }_{\alpha }(6+2p)}{{\alpha }_{s}+6}}\,{\tilde{E}}_{49}^{\tfrac{6+2p}{{\alpha }_{s}+6}}\,\\ {A}_{\mathrm{sh}} & = & 2.9\times {10}^{-7}\,\mathrm{mJy}\,{\left(\displaystyle \frac{1+z}{1.02}\right)}^{\tfrac{(6-{\alpha }_{s})(p+2)}{2(\alpha +6)}}{\left(1+Y\right)}^{-1}\,{\epsilon }_{{\rm{e}},-1}^{p-1}\\ & & \times \ {\epsilon }_{B,-3}^{\tfrac{p-2}{4}}\,{n}_{-4}^{\tfrac{p({\alpha }_{s}-2)-2(3{\alpha }_{s}+2)}{4({\alpha }_{s}+6)}}\\ & & \times \ {D}_{26.5}^{-2}\,{\beta }_{{\rm{c}}}^{-\tfrac{{\alpha }_{s}(4+2p)}{{\alpha }_{s}+6}}\,{\tilde{E}}_{49}^{\tfrac{4+2p}{{\alpha }_{s}+6}}.\end{array}\end{eqnarray} \tag{ 22 }$$

For ${\alpha }_{s}=0$, the observable quantities are derived (Equations (19) and (20)) and the synchrotron light curve in the lateral expansion regime are recovered (i.e., see Rees 1999; Sari et al. 1999; Dermer et al. 2000; Granot et al. 2002, 2018).

2.2.3. Nonrelativistic Phase (${\beta }_{{\rm{c}}}\ll 1$)

Once the decelerated material has swept enough ambient medium, it will go into a nonrelativistic phase. This transition affects the evolution of the material and in turn the synchrotron light curve. During this phase, the kinetic equivalent energy is ${E}_{{\rm{k}}}\propto {\beta }_{{\rm{c}}}^{2}{R}^{3}$ with the radius $R={\beta }_{{\rm{c}}}t$, the magnetic field is $B^{\prime} \propto {\beta }_{{\rm{c}}}$, and the minimum Lorentz factor is ${\gamma }_{{\rm{m}}}\propto {\beta }_{{\rm{c}}}^{2}$. Taking into account that the kinetic energy is given as a power-law distribution ${E}_{{\rm{k}}}\propto {\beta }_{{\rm{c}}}^{-{\alpha }_{s}}$ (Equation (2)), then the velocity evolves as ${\beta }_{{\rm{c}}}\propto {t}^{-\tfrac{3}{5+{\alpha }_{s}}}$. The synchrotron spectral breaks and maximum flux evolve as

$$\begin{eqnarray}\begin{array}{rcl}{\epsilon }_{{\rm{m}}} & \propto & {\gamma }_{{\rm{m}}}^{2}B^{\prime} \propto {\beta }_{{\rm{c}}}^{5}\propto {t}^{-\tfrac{15}{{\alpha }_{s}+5}},\\ {\epsilon }_{{\rm{c}}} & \propto & B{{\prime} }^{-3}{t}^{-2}\propto {\beta }_{{\rm{c}}}^{-3}{t}^{-2}\propto {t}^{-\tfrac{2{\alpha }_{s}+1}{{\alpha }_{s}+5}},\\ {F}_{\max } & \propto & {N}_{{\rm{e}}}P{{\prime} }_{\max }\propto {R}^{3}B^{\prime} \propto {\beta }_{{\rm{c}}}^{4}{t}^{3}\propto {t}^{\tfrac{3({\alpha }_{s}+1)}{{\alpha }_{s}+5}}.\end{array}\end{eqnarray} \tag{ 23 }$$

The synchrotron light curve in the slow-cooling regime when the decelerated material lies in the nonrelativistic phase becomes

$$\begin{eqnarray}{F}_{\nu }\propto \left\{\begin{array}{ll}{t}^{\tfrac{3{\alpha }_{s}+8}{{\alpha }_{s}+5}}\,{\epsilon }_{\gamma }^{\tfrac{1}{3}}, & {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}}},\\ {t}^{\tfrac{6{\alpha }_{s}-15p+21}{2({\alpha }_{s}+5)}}\,{\epsilon }_{\gamma }^{-\tfrac{p-1}{2}}, & {\epsilon }_{{\rm{m}}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{c}}},\\ {t}^{\tfrac{4{\alpha }_{s}-15p+20}{2({\alpha }_{s}+5)}}\,{\epsilon }_{\gamma }^{-\tfrac{p}{2}}, & {\epsilon }_{{\rm{c}}}\lt {\epsilon }_{\gamma }.\end{array}\right.\end{eqnarray} \tag{ 24 }$$

It is worth noting that when ${\alpha }_{s}=0$, the observable quantities derived and the synchrotron light curve in the nonrelativistic regime are recovered (i.e., see Wijers et al. 1997; Dai & Lu 1999; Huang et al. 1999; Livio & Waxman 2000; Huang & Cheng 2003).

2.3. Description of the Observable Quantities

2.3.1. X-Ray, Optical, and Radio Light Curves

Figures 1 and 2 display the resulting X-ray, optical, and radio light curves of the synchrotron forward-shock emission generated by a decelerated shock-breakout material for several parameter values. The light curves are presented for three electromagnetic bands: X-ray at 1 keV (upper panel), optical at 1 eV (medium panel), and radio at 6 GHz (lower panel), with the parameter values in the ranges of ${10}^{-6}\leqslant n\leqslant 1\,{\mathrm{cm}}^{-3}$, ${10}^{-4}\leqslant {\epsilon }_{B}\leqslant {10}^{-1}$, $2.2\leqslant p\leqslant 3.6$ and $2.0\leqslant {\alpha }_{s}\leqslant 3.5$, for the opening angle ${\theta }_{{\rm{c}}}=15^\circ$, the fiducial kinetic energy of $\tilde{E}={10}^{50}$ erg, the microphysical parameter given to accelerate electrons ${\epsilon }_{{\rm{e}}}=0.1$ and a source located at $D=100\,\mathrm{Mpc}$.

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The left- and right-hand panels in Figure 1 display the light curves considering the magnetic microphysical parameter ${\epsilon }_{B}={10}^{-2}$ and density $n={10}^{-2}\,{\mathrm{cm}}^{-3}$, respectively, for $p=2.2$, ${\alpha }_{s}=3.0$, $D=100\,\mathrm{Mpc}$, ${\theta }_{{\rm{c}}}=15^\circ$, $\tilde{E}={10}^{50}$ erg, and ${\epsilon }_{{\rm{e}}}=0.1$. Figure 2 shows the light curves considering the power indexes ${\alpha }_{s}=3.0$ (left-hand panels) and $p=2.2$ (right-hand panels), respectively, for ${\epsilon }_{B}={10}^{-2}$, $n={10}^{-2}\,{\mathrm{cm}}^{-3}$, $D=100\,\mathrm{Mpc}$, ${\theta }_{{\rm{c}}}=15^\circ$, $\tilde{E}={10}^{50}$ erg, and ${\epsilon }_{{\rm{e}}}=0.1$.

Figure 1 shows that X-ray, optical, and radio light curves have similar behaviors depending on which power-law segment of the spectrum is evolving. The left-hand panels show that for $n=1\,{\mathrm{cm}}^{-3}$, light curves do not present peaks, and as density decreases, the peak in the light curves are more notable. The right-hand panels display that radio flux is evolving in the same power law for ${10}^{-4}\leqslant {\epsilon }_{B}\leqslant {10}^{-1}\,{\mathrm{cm}}^{-3}$ whereas X-ray and optical fluxes evolve in different power-law segments of the synchrotron spectrum. For instance, the optical flux for ${\epsilon }_{B}=0.1$ changes the power-law segment during the jet break. After the jet break, it evolves in the second power law, while for other values of ${\epsilon }_{B}$ it keeps evolving in the first one.

Figure 2 shows that the fluxes are strongly dependent on the values of p and α_s. The right-hand panels show that at early times fluxes, in general, are dominated by those generated with small values of α_s whereas at later times are dominated by those with larger values. The left-hand panels show that as the electron power-law index increases, fluxes, in general, decrease, with the exception of the radio flux at ${\epsilon }_{\gamma }=6$ GHz for p = 2.2. This is due to the fact that when ${\epsilon }_{{\rm{m}}}$ scales as ${\left(\tfrac{p-1}{p-2}\right)}^{2}$, then radio flux for p = 2.6, 3.2, and 3.5 is in the regime ${\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}}}$ which in turn does not depend on p (as shown in these panels for early times) while for p = 2.2 is in the regime ${\epsilon }_{{\rm{m}}}\lt {\epsilon }_{\gamma }$. With the passage of time, ${\epsilon }_{{\rm{m}}}$ becomes less than 6 GHz and then the radio flux changes to the regime ${\epsilon }_{{\rm{m}}}\gt {\epsilon }_{\gamma }$, first for p = 2.6, then for p = 3.2 and finally for p = 3.6, as shown in these panels for later times. While radio and optical fluxes peak during the first hundred days, X-ray flux peaks depending the value of α_s. Furthermore, the right-hand panels show that as α_s increases, X-ray fluxes peak earlier.

2.3.2. Evolution of the Spectral Breaks

In the previous subsection the X-ray, optical, and radio light curves were illustrated for deceleration timescales ranging from hours to 1000 days. In this subsection we illustrate the evolution of the synchrotron spectral breaks during the first seconds. In this case, the deceleration time is around a few seconds and the bulk Lorentz factor is close to 100. The ranges of values for the microphysical parameters and ISM used in this analysis are those that allow the cooling spectral breaks to be observed in the energy range covered by Fermi-GBM, Swift-BAT, and INTEGRAL.

Figure 3 shows the evolution of synchrotron cooling and characteristic energies during the first 10 s for typical values of the magnetic microphysical parameter (${10}^{-2}\lt {\epsilon }_{B}\lt {10}^{-1}$; upper panel), the ISM (${10}^{-2}\lt n\lt 1\,{\mathrm{cm}}^{-3}$; medium panel), and the fiducial energy (${10}^{49}\lt \tilde{E}\lt {10}^{51}\,\mathrm{erg}$; lower panel). The left-hand panels display that ${\epsilon }_{{\rm{c}}}$ evolves in the γ-ray band as ${t}^{-1.09}$ (${\alpha }_{s}=3.0$) or ${t}^{-1.14}$ (${\alpha }_{s}=4.0$) and the right-hand panel shows that ${\epsilon }_{{\rm{m}}}$ evolves in the IR—optical band as ${t}^{-0.92}$ (${\alpha }_{s}=3.0$) or ${t}^{-0.85}$ (${\alpha }_{s}=4.0$). While the evolution of the cooling spectral break can be potentially detected by instruments such as Fermi-GBM and Swift-BAT, the characteristic spectral break could be detected by Swift-UVOT, 1-meter Swope telescope, and other optical telescopes. It is worth noting that our results are different from the evolution of synchrotron cooling ${t}^{-0.5}$ and characteristic ${t}^{-1.5}$ energies derived when the relativistic jet is decelerated by the ISM (e.g., see, Sari et al. 1998).

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We consider that a relativistic jet producing the long-lived afterglow emission is launched from a binary NS system. Additionally, we assume the relativistic jet is not aligned with the observer's line of sight (Fraija et al. 2017a; Granot et al. 2018; Ioka & Nakamura 2018; Lazzati et al. 2018; Murguia-Berthier et al. 2017). In order to obtain the quantities for the off-axis emission, the quantities derived in Sari et al. (1998) have to be modified using different boosts. For the off-axis case, given the evolution of the minimum electron Lorentz factor ${\gamma }_{{\rm{m}},\mathrm{off}}\propto {{\rm{\Gamma }}}_{{\rm{j}}}$, the magnetic field $B^{\prime} \propto {{\rm{\Gamma }}}_{{\rm{j}}}$, the cooling electron Lorentz factor ${\gamma }_{{\rm{c}},\mathrm{off}}\propto {\delta }_{D}^{-1}\,{{\rm{\Gamma }}}_{{\rm{j}}}^{-2}\,{t}^{-1}$, the number of radiating electrons ${N}_{{\rm{e}},\mathrm{off}}\propto {R}_{\mathrm{off}}^{3}\propto {({\delta }_{D}{{\rm{\Gamma }}}_{{\rm{j}}}t)}^{3}$ (Salmonson 2003), the solid angle ${{\rm{\Omega }}}_{\mathrm{off}}\propto {\delta }_{D}^{-2}$ (Rybicki & Lightman 1986), the maximum power ${P}_{{\nu }_{{\rm{m}}},\mathrm{off}}\propto {\delta }_{D}B^{\prime}$, the synchrotron energy breaks and the maximum synchrotron flux evolve as

$$\begin{eqnarray}\begin{array}{rcl}{\epsilon }_{{\rm{m}},\mathrm{off}} & \propto & {\delta }_{D}B^{\prime} {\gamma }_{{\rm{m}},\mathrm{off}}^{2}\propto {\delta }_{D}{{\rm{\Gamma }}}_{{\rm{j}}}^{3},\\ & & {\epsilon }_{{\rm{c}},\mathrm{off}}\propto {\delta }_{D}B^{\prime} {\gamma }_{{\rm{c}},\mathrm{off}}^{2}\propto {{\rm{\Gamma }}}_{{\rm{j}}}^{-3}{\delta }_{D}^{-1}{t}^{-2},\\ {F}_{\nu ,\max ,\mathrm{off}} & \propto & \displaystyle \frac{{N}_{{\rm{e}},\mathrm{off}}\,{P}_{{\nu }_{{\rm{m}}},\mathrm{off}}}{{{\rm{\Omega }}}_{\mathrm{off}}}\propto {\delta }_{D}^{3}\\ & & \times \ {N}_{{\rm{e}},\mathrm{off}}B^{\prime} \propto {\delta }_{D}^{6}{{\rm{\Gamma }}}_{{\rm{j}}}^{4}{t}^{3},\end{array}\end{eqnarray} \tag{ 25 }$$

respectively. Therefore, the flux density at a given energy evolves as ${F}_{\nu ,\mathrm{off}}\propto {\delta }_{D}^{\tfrac{17}{3}}{{\rm{\Gamma }}}_{{\rm{j}}}^{3}{t}^{3}$ for ${\epsilon }_{\gamma }\leqslant {\epsilon }_{{\rm{m}}}$, $\propto {\delta }_{D}^{\tfrac{p+11}{2}}{{\rm{\Gamma }}}_{{\rm{j}}}^{\tfrac{3p+5}{2}}{t}^{3}$ for ${\epsilon }_{{\rm{m}}}\leqslant {\epsilon }_{\gamma }\leqslant {\epsilon }_{{\rm{c}}}$ and $\propto {\delta }_{D}^{\tfrac{p+10}{2}}{{\rm{\Gamma }}}_{{\rm{j}}}^{\tfrac{3p+2}{2}}{t}^{2}$ for ${\epsilon }_{{\rm{c}}}\leqslant {\epsilon }_{\gamma }$, where the Doppler factor is ${\delta }_{D}={\delta }_{D}({\rm{\Delta }}\theta )\equiv \tfrac{1}{{{\rm{\Gamma }}}_{{\rm{j}}}(1-{\beta }_{{\rm{j}}}{\rm{\Delta }}\theta )}\approx \tfrac{2{{\rm{\Gamma }}}_{{\rm{j}}}}{1+{{\rm{\Gamma }}}_{{\rm{j}}}^{2}{\rm{\Delta }}{\theta }^{2}}$ for ${{\rm{\Gamma }}}_{{\rm{j}}}\gg 1$ and ${\rm{\Delta }}\theta \ll 1$ with ${\rm{\Delta }}\theta ={\theta }_{\mathrm{obs}}-{\theta }_{{\rm{j}}}$. The value of the flux density for ${\epsilon }_{{\rm{m}}}\leqslant {\epsilon }_{\gamma }\leqslant {\epsilon }_{{\rm{c}}}$ agrees with the flux reported in Rossi et al. (2002) and Nakar & Piran (2018). It is worth noting that when ${\rm{\Delta }}\theta =0^\circ$ and ${F}_{\nu ,\max ,\mathrm{off}}$ is divided by ${\rm{\Omega }}=4\pi {\delta }_{D}^{2}$ (Lamb et al. 2018), the relations derived in Sari et al. (1998) are recovered.

In order to find the evolution of the bulk Lorentz factor (${{\rm{\Gamma }}}_{{\rm{j}}}$) for a decelerated off-axis jet, the kinetic equivalent energy calculated through the Blandford–McKee condition ${E}_{{\rm{k}},{\rm{j}}}=\tfrac{4}{3}\pi {m}_{p}\,n\,{R}_{\mathrm{off}}^{3}\,{{\rm{\Gamma }}}_{{\rm{j}}}^{2}$ is obtained as follows. Taking into account the deceleration radius viewed off-axis ${R}_{\mathrm{off}}\,=\tfrac{{\delta }_{D}{{\rm{\Gamma }}}_{{\rm{j}}}}{(1+z)}t$ and⁴ the Doppler boost ${\delta }_{D}\simeq \tfrac{2}{{{\rm{\Gamma }}}_{{\rm{j}}}{\rm{\Delta }}{\theta }^{2}}$ for ${{\rm{\Gamma }}}_{{\rm{j}}}^{2}{\rm{\Delta }}{\theta }^{2}\gg 1$ with the energy limited to the opening angle ${E}_{{\rm{k}},{\rm{j}}}=\tfrac{\tilde{E}}{1-\cos {\theta }_{{\rm{j}}}}\,\simeq \tfrac{2\tilde{E}}{{\theta }_{{\rm{j}}}^{2}}$ (Granot et al. 2018), the kinetic energy becomes $\tilde{E}\,=\tfrac{16}{3}\pi {m}_{p}\,{(1+z)}^{-3}n\,{\theta }_{{\rm{j}}}^{2}\,{t}^{3}\,{{\rm{\Gamma }}}_{{\rm{j}}}^{2}{\rm{\Delta }}{\theta }^{-6}$.⁵ In this case, the bulk Lorentz factor evolves as

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{\rm{j}}}=8.3\,{\left(1+z\right)}^{\displaystyle \frac{3}{2}}\,{n}_{-4}^{-\displaystyle \frac{1}{2}}\,{\tilde{E}}_{50}^{\displaystyle \frac{1}{2}}\,{\theta }_{j,{5}^{^\circ }}^{-1}\,{\rm{\Delta }}{\theta }_{{15}^{^\circ }}^{3}\,{t}_{100{\rm{d}}}^{-\displaystyle \frac{3}{2}}.\end{eqnarray} \tag{ 26 }$$

Using the bulk Lorentz factor (Equation (26)) and Equations (25), we derive the relevant quantities of forward-shock synchrotron emission radiated in an off-axis jet. The minimum and cooling electron Lorentz factors are given by

$$\begin{eqnarray}\begin{array}{rcl}{\gamma }_{{\rm{m}},\mathrm{off}} & = & 212.9\,{\epsilon }_{{\rm{e}},-1}{\left(1+z\right)}^{\tfrac{3}{2}}\,{n}_{-4}^{-\tfrac{1}{2}}\\ & & \times \ {\tilde{E}}_{50}^{\tfrac{1}{2}}\,{\rm{\Delta }}{\theta }_{{15}^{^\circ }}^{3}\,{\theta }_{j,{5}^{^\circ }}^{-1}\,{t}_{100\,{\rm{d}}}^{-\tfrac{3}{2}},\\ {\gamma }_{{\rm{c}},\mathrm{off}} & = & 9.1\times {10}^{6}\,{\left(1+z\right)}^{-\tfrac{1}{2}}\,{\left(1+Y\right)}^{-1}{\epsilon }_{B,-4}^{-1}\,{n}_{-4}^{-\tfrac{1}{2}}\\ & & \times \ {\tilde{E}}_{50}^{-\tfrac{1}{2}}\,{\rm{\Delta }}{\theta }_{{15}^{^\circ }}^{-1}\,{\theta }_{j,{5}^{^\circ }}\times {t}_{100\,{\rm{d}}}^{\tfrac{1}{2}},\end{array}\end{eqnarray} \tag{ 27 }$$

which correspond to a comoving magnetic field given by

$$\begin{eqnarray}&&B^{\prime} =1.9\times {10}^{-2}\,\mathrm{mG}\,{\left(1+z\right)}^{\displaystyle \frac{3}{2}}\,{\epsilon }_{B,-4}^{\displaystyle \frac{1}{2}}\,{\tilde{E}}_{50}^{\displaystyle \frac{1}{2}}\,{\rm{\Delta }}{\theta }_{{15}^{^\circ }}^{3}\,{\theta }_{j,{5}^{^\circ }}^{-1}\,{t}_{100\,{\rm{d}}}^{-\displaystyle \frac{3}{2}}.\end{eqnarray} \tag{ 28 }$$

The synchrotron spectral breaks and the maximum flux can be written as

$$\begin{eqnarray}\begin{array}{rcl}{\epsilon }_{{\rm{m}},\mathrm{off}} & \simeq & 1.5\times {10}^{-2}\,\mathrm{GHz}\,{\left(1+z\right)}^{2}{\epsilon }_{{\rm{e}},-1}^{2}\,{\epsilon }_{B,-4}^{\tfrac{1}{2}}\,{n}_{-4}^{-\tfrac{1}{2}}\\ & & \times \ {\tilde{E}}_{50}\,{\rm{\Delta }}{\theta }_{{15}^{^\circ }}^{-4}\,{\theta }_{j,{5}^{^\circ }}^{-2}\times {t}_{100\,{\rm{d}}}^{-3},\\ {\epsilon }_{{\rm{c}},\mathrm{off}} & \simeq & 4.1\,\mathrm{keV}{\left(1+z\right)}^{-2}{\left(1+Y\right)}^{-2}\,{\epsilon }_{B,-4}^{-\tfrac{3}{2}}\,{n}_{-4}^{-\tfrac{1}{2}}\\ & & \times \ {\tilde{E}}_{50}^{-1}{\rm{\Delta }}{\theta }_{{15}^{^\circ }}^{-4}{\theta }_{j,{5}^{^\circ }}^{2}\times {t}_{100{\rm{d}}},\\ {F}_{\max ,\mathrm{off}} & \simeq & 5.6\times {10}^{3}\,\mathrm{mJy}\,{\left(1+z\right)}^{-4}\,{\epsilon }_{B,-4}^{\tfrac{1}{2}}\,{n}_{-4}^{\tfrac{5}{2}}\\ & & \times \ {D}_{26.5}^{-2}\,{\tilde{E}}_{50}^{-1}\,{\rm{\Delta }}{\theta }_{{15}^{^\circ }}^{-18}\times {\theta }_{j,{5}^{^\circ }}^{2}\,{t}_{100\,{\rm{d}}}^{6}.\end{array}\end{eqnarray} \tag{ 29 }$$

Using the observed synchrotron spectrum in the slow-cooling regime with Equation (29), the synchrotron light curves can be written as

$$\begin{eqnarray}&&{F}_{\nu ,\mathrm{off}}=\left\{\begin{array}{l}{A}_{\mathrm{sl}}\,{t}_{100\,{\rm{d}}}^{7}\,{\left(\tfrac{{\epsilon }_{\gamma }}{6\mathrm{GHz}}\right)}^{\tfrac{1}{3}},{\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}}},\\ {A}_{\mathrm{sm}}\,{t}_{100\,{\rm{d}}}^{\tfrac{3(5-p)}{2}}\,{\left(\tfrac{{\epsilon }_{\gamma }}{1\mathrm{eV}}\right)}^{-\tfrac{p-1}{2}},{\epsilon }_{{\rm{m}}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{c}}},\,\,\\ {A}_{\mathrm{sh}}\,{t}_{100\,{\rm{d}}}^{\tfrac{16-3p}{2}}\,{\left(\tfrac{{\epsilon }_{\gamma }}{1\mathrm{keV}}\right)}^{-\tfrac{p}{2}},{\epsilon }_{{\rm{c}}}\lt {\epsilon }_{\gamma },\end{array}\right.\end{eqnarray} \tag{ 30 }$$

with the coefficients given by

$$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{sl}} & = & 1.8\times {10}^{4}\,\mathrm{mJy}\,{\left(1+z\right)}^{-\tfrac{14}{3}}\,{\epsilon }_{{\rm{e}},-1}^{-\tfrac{2}{3}}\\ & & \times \ {\epsilon }_{B,-4}^{\tfrac{1}{3}}\,{n}_{-4}^{\tfrac{8}{3}}\,{D}_{26.5}^{-2}\,{\tilde{E}}_{50}^{-\tfrac{4}{3}}\times {\rm{\Delta }}{\theta }_{{15}^{^\circ }}^{-\tfrac{58}{3}}\,{\theta }_{j,{5}^{^\circ }}^{\tfrac{8}{3}},\\ {A}_{\mathrm{sm}} & = & 1.5\times {10}^{-2}\,\mathrm{mJy}\,{\left(1+z\right)}^{p-5}\,{\epsilon }_{{\rm{e}},-1}^{p-1}\,{n}_{-4}^{-\tfrac{(p-11)}{4}}\\ & & \times \ {\epsilon }_{B,-4}^{\tfrac{p+1}{4}}\,{D}_{26.5}^{-2}\,{\tilde{E}}_{50}^{\tfrac{p-3}{2}}\times {\rm{\Delta }}{\theta }_{{15}^{^\circ }}^{2(p-10)}\,{\theta }_{j,{5}^{^\circ }}^{(1-p)},\\ {A}_{\mathrm{fh}} & = & 3.5\times {10}^{-4}\,\mathrm{mJy}\,{\left(1+z\right)}^{p-6}{\left(1+Y\right)}^{-1}\,{\epsilon }_{{\rm{e}},-1}^{p-1}\\ & & \times \ {\epsilon }_{B,-4}^{\tfrac{p-2}{4}}\,{D}_{26.5}^{-2}\times \ {n}_{-4}^{-\tfrac{(p-10)}{4}}\,{\tilde{E}}_{50}^{\tfrac{p-4}{2}}\\ & & \times {\rm{\Delta }}{\theta }_{{15}^{^\circ }}^{2(p-11)}\,{\theta }_{j,{5}^{^\circ }}^{(2-p)}.\end{array}\end{eqnarray} \tag{ 31 }$$

3.1. Lateral Expansion

In this case, the beaming cone of the radiation emitted off-axis, ${\rm{\Delta }}\theta$, broaden increasingly until this cone reaches our field of view (${{\rm{\Gamma }}}_{{\rm{j}}}\sim {\rm{\Delta }}{\theta }^{-1};$ Granot et al. 2002, 2018). In the lateral expansion phase, the break in the density flux should occur around

$$\begin{eqnarray}&&{t}_{\mathrm{br},\mathrm{off}}=205.6\,\mathrm{days}\,k\,\left(\displaystyle \frac{1+z}{1.02}\right)\,{n}_{-4}^{-\displaystyle \frac{1}{3}}\,{\tilde{E}}_{50}^{\displaystyle \frac{1}{3}}\,{\rm{\Delta }}{\theta }_{{15}^{^\circ }}^{2},\end{eqnarray} \tag{ 32 }$$

In this phase, the synchrotron spectral breaks and the maximum synchrotron fluxes are

$$\begin{eqnarray}\begin{array}{rcl}{\epsilon }_{{\rm{m}},\mathrm{off}} & \simeq & 7.6\times {10}^{-3}\,\mathrm{GHz}\,(1+z)\,{\epsilon }_{{\rm{e}},-1}^{2}\,{\epsilon }_{B,-4}^{\tfrac{1}{2}}\,{n}_{-4}^{-\tfrac{1}{6}}\\ & & \times \ {\tilde{E}}_{50}^{\tfrac{2}{3}}\,{t}_{100\,{\rm{d}}}^{-2},\\ {\epsilon }_{{\rm{c}},\mathrm{off}} & \simeq & 2.6\,\mathrm{keV}\,{\left(1+z\right)}^{-1}{\left(1+Y\right)}^{-2}\,{\epsilon }_{B,-4}^{-\tfrac{3}{2}}\,{n}_{-4}^{-\tfrac{5}{6}}\times {\tilde{E}}_{50}^{-\tfrac{2}{3}},\\ {F}_{\max ,\mathrm{off}} & \simeq & 2.5\times {10}^{3}\,\mathrm{mJy}\,{\left(1+z\right)}^{2}\,{\epsilon }_{B,-4}^{\tfrac{1}{2}}\,{n}_{-4}^{\tfrac{1}{6}}\,{D}_{26.5}^{-2}\,{\tilde{E}}_{50}^{\tfrac{4}{3}}\,{t}_{100\,{\rm{d}}}^{-1}.\end{array}\end{eqnarray} \tag{ 33 }$$

Given the synchrotron spectrum, the synchrotron spectral breaks and the maximum flux (Equation (33)) during this phase, the light curve when in the slow-cooling regime is

$$\begin{eqnarray}&&{F}_{\nu ,\mathrm{off}}=\left\{\begin{array}{l}{A}_{\mathrm{sl}}\,{t}_{100\,{\rm{d}}}^{-\tfrac{1}{3}}\,{\left(\tfrac{{\epsilon }_{\gamma }}{6\mathrm{GHz}}\right)}^{\tfrac{1}{3}},{\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}}},\\ {A}_{\mathrm{sm}}\,{t}_{100\,{\rm{d}}}^{-p}\,{\left(\tfrac{{\epsilon }_{\gamma }}{1\mathrm{eV}}\right)}^{-\tfrac{p-1}{2}},{\epsilon }_{{\rm{m}}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{c}}},\,\,\\ {A}_{\mathrm{sh}}\,{t}_{100\,{\rm{d}}}^{-p}\,{\left(\tfrac{{\epsilon }_{\gamma }}{1\mathrm{keV}}\right)}^{-\tfrac{p}{2}},{\epsilon }_{{\rm{c}}}\lt {\epsilon }_{\gamma },\end{array}\right.\end{eqnarray} \tag{ 34 }$$

with the coefficients given by

$$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{sl}} & = & 9.9\times {10}^{2}\,\mathrm{mJy}\,{\left(1+z\right)}^{\tfrac{5}{3}}\,{\epsilon }_{{\rm{e}},-1}^{-\tfrac{2}{3}}\\ & & \times \ {\epsilon }_{B,-4}^{\tfrac{1}{3}}\,{n}_{-4}^{\tfrac{2}{9}}\,{D}_{26.5}^{-2}\,{\tilde{E}}_{50}^{\tfrac{10}{9}},\\ {A}_{\mathrm{sm}} & = & 4.4\times \,{10}^{-4}\,\mathrm{mJy}\,{\left(1+z\right)}^{\tfrac{(p+3)}{2}}\,{\epsilon }_{{\rm{e}},-1}^{p-1}\\ & & \times \ {\epsilon }_{B,-4}^{\tfrac{p+1}{4}}\,{n}_{-4}^{-\tfrac{(p-3)}{12}}\times \,{D}_{26.5}^{-2}\,{\tilde{E}}_{50}^{\tfrac{(p+3)}{3}},\\ {A}_{\mathrm{sh}} & = & 1.7\times {10}^{-5}\,\mathrm{mJy}\,{\left(1+z\right)}^{\tfrac{(p+2)}{2}}{\left(1+Y\right)}^{-1}\\ & & \times \ {\epsilon }_{{\rm{e}},-1}^{p-1}\,{\epsilon }_{B,-4}^{\tfrac{p-2}{4}}\times \,{n}_{-4}^{-\tfrac{(p+2)}{12}}\,{D}_{26.5}^{-2}\,{\tilde{E}}_{50}^{\tfrac{(p+2)}{3}}.\end{array}\end{eqnarray} \tag{ 35 }$$

The observed fluxes agree with those reported in Granot et al. (2018).

4.1. Multiwavelength Afterglow Observations

4.2. Modeling the Afterglow Emission

Our afterglow model presented through the shock-breakout material and the off-axis jet is dependent on a set of eight parameters, ${{\rm{\Xi }}}_{\mathrm{fow},{\rm{b}}}$ = {$\tilde{E},n,p,{\theta }_{{\rm{j}}}$, ${\rm{\Delta }}\theta$, ${\varepsilon }_{B}$, ${\varepsilon }_{{\rm{e}}}$, α_s}.⁶ To find an adequate set of values inside the parameter space we utilize the Markov Chain Monte Carlo (MCMC) method, a Bayesian statistical technique that allows us to find best-fit values through a sampling process. By using these parameters, we determine a suitable prior distribution, to be utilized alongside a normal likelihood upon which an eight parameter σ is introduced, that allows us to generate samples for the posterior distributions of our on-axis model. We utilize the No-U-Turn Sampler (NUTS) from the PyMC3 python distribution (Salvatier et al. 2016) to generate a total of 21,000 samples, of which 7000 are utilized for tuning and subsequently discarded. In all simulations we have employed a set of normal, continuous distributions for our priors. With this specific choice we can simulate a more unbiased and uninformative set of priors.

Output is presented in the Figures 4–6, alongside Table 1. These figures are Corner Plots (Foreman-Mackey 2016), a specific plot where the diagonal is a one-dimensional kernel density estimation (KDE) describing the posterior distribution and the lower triangle is the bidimensional KDE. Figure 7 shows the light curves (above) and the spectral energy distributions (below) of the X-ray, optical, and radio bands of GRB 170817A with the model curve for the synchrotron radiation emitted in the deceleration of the shock-breakout material and relativistic off-axis jet. The best-fit value of each parameter is shown in green. Table 1 reports our chosen quantiles (0.15, 0.5, 0.85) retrieved from inference.

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Table 1.  Median and Symmetrical 35% Quantiles (0.15, 0.5, 0.85), Truncated at the Third Decimal, after Describing the X-rays and Radio Wavelengths at 3 and 6 GHz with Our Afterglow Model as Described in Sections 2 and 3

Parameters		Median
	Radio (3 GHz)	Radio (6 GHz)	X-Ray (1 keV)
$\tilde{E}\,({10}^{49}\,\mathrm{erg})$	${3.999}_{-0.100}^{+0.100}$	${3.999}_{-0.099}^{+0.100}$	${4.000}_{-0.101}^{+0.098}$
$n\,({10}^{-4}\,{\mathrm{cm}}^{-3}$ )	${1.000}_{-0.010}^{+0.010}$	${1.000}_{-0.010}^{+0.010}$	${1.023}_{-0.098}^{+0.099}$
$p$	${2.217}_{-0.051}^{+0.050}$	${2.216}_{-0.096}^{+0.102}$	${2.193}_{-0.083}^{+0.100}$
${\theta }_{{\rm{j}}}$ (deg)	${3.002}_{-0.069}^{+0.066}$	${3.000}_{-0.069}^{+0.070}$	${3.001}_{-0.070}^{+0.068}$
${\rm{\Delta }}\theta$ (deg)	${15.996}_{-0.066}^{+0.071}$	${15.988}_{-0.068}^{+0.068}$	${15.995}_{-0.066}^{+0.069}$
${\varepsilon }_{B}\,({10}^{-4})$	${2.003}_{-0.099}^{+0.099}$	${2.003}_{-0.099}^{+0.100}$	${2.007}_{-0.097}^{+0.098}$
${\varepsilon }_{{\rm{e}}}\,({10}^{-1})$	${1.002}_{-0.049}^{+0.050}$	${1.000}_{-0.049}^{+0.049}$	${1.003}_{-0.049}^{+0.050}$
${\alpha }_{s}$	${2.300}_{-0.097}^{+0.099}$	${2.300}_{-0.097}^{+0.099}$	${2.299}_{-0.098}^{+0.099}$

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In accordance with the values obtained after describing the multiwavelength light curves and the SEDs at 15 ± 2, 110 ± 5, and 145 ± 20 days of GRB 170817A, we found that (i) for the shock-breakout material the bulk Lorentz factor becomes ${{\rm{\Gamma }}}_{{\rm{c}}}\simeq 3.1{\left(t/15\mathrm{days}\right)}^{-0.24}$, the equivalent kinetic energy and the observed electromagnetic energy ${E}_{{\rm{k}},{\rm{c}}}=3\times {10}^{49}\,\mathrm{erg}\,{\left({\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}\right)}^{-4.3}\simeq 3.8\times {10}^{47}\,\mathrm{erg}$ and ${E}_{\mathrm{obs},\gamma }\,=5.4\times {10}^{46}\,\mathrm{erg}$, respectively, which corresponds to an efficiency of $\sim 16 \%$ and (ii) for the relativistic off-axis jet the bulk Lorentz factor becomes ${{\rm{\Gamma }}}_{{\rm{j}}}\simeq 4.6{\left(t/140\mathrm{days}\right)}^{-\tfrac{3}{2}}$ the equivalent kinetic energy and the released electromagnetic energy are ${E}_{{\rm{k}},{\rm{j}}}\,\simeq \tfrac{2\tilde{E}}{{\theta }_{{\rm{j}}}^{2}}\simeq 7.01\times {10}^{51}\,\mathrm{erg}\,$ and ${E}_{\gamma }=5.4\,\times {10}^{46}\mathrm{erg}{(1+{\rm{\Delta }}{\theta }^{2}{{\rm{\Gamma }}}_{{\rm{j}}}^{2})}^{3}\simeq$ $4.8\times {10}^{51}\,\mathrm{erg}$, respectively, which corresponds to an efficiency of $\sim 11 \%$. For the shock-breakout material, the flux evolves as ${F}_{\nu }\propto {t}^{0.64}$ for ${\epsilon }_{{\rm{m}}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{c}}}$ and ${F}_{\nu }\propto {t}^{0.09}$ for ${\epsilon }_{{\rm{c}}}\lt {\epsilon }_{\gamma }$. The cooling spectral break ${\epsilon }_{{\rm{c}}}\sim 11.2\,\mathrm{keV}$ is above the X-ray band and the characteristic break ${\epsilon }_{{\rm{m}}}=0.02\,\mathrm{GHz}$ is below the radio band at 15 days. The X-ray, optical, and radio fluxes peak at ∼30 days, and later they evolve as ${F}_{\nu }\propto {t}^{-0.76}$. At ∼500 s, the X-ray flux will vary from $\propto {t}^{-0.76}$ to $\propto {t}^{-1.03}$ whereas the optical and radio fluxes will continue evolving as ${F}_{\nu }\propto {t}^{-0.76}$. During the nonrelativistic phase (${\beta }_{{\rm{c}}}\ll 1$), the flux will evolve in accordance with the synchrotron spectrum given in Equation (24). For the relativistic off-axis jet, the cooling spectral break, ${\epsilon }_{{\rm{c}}}\sim 3.5\,\mathrm{keV}$, is above the X-ray band and the characteristic break, ${\epsilon }_{{\rm{m}}}=0.03\,\mathrm{GHz}$, is below the radio band at 100 days. During this period, the observed flux increases as ${F}_{\nu }\propto {t}^{4.2}$ (Equation (34)), as predicted in Nakar & Piran (2018). The X-ray, optical, and radio fluxes peak at ∼140 days, and later they evolve as ${F}_{\nu }\propto {t}^{-2.2}$.

On the other hand, taking into consideration the GBM data during the first seconds, Veres et al. (2018) showed the evolution of the main peak (${E}_{\mathrm{peak}}$) as a function of time, and the luminosity as a function of ${E}_{\mathrm{peak}}$. Using simple power laws in both cases ${E}_{\mathrm{peak}}\propto {(t-{t}_{0})}^{k}$ and $L\propto {E}_{\mathrm{peak}}^{q}$, they obtained the best-fit values of $k=-0.97\pm 0.35$ for ${t}_{0}=-0.15\pm 0.04\,{\rm{s}}$ and $q=0.90\,\pm 0.10$. The values of k and q are consistent with the evolution of the cooling spectral break (Equation (29)) ${\epsilon }_{{\rm{c}}}\propto {t}^{-(1.01-1.13)}$ and $\tilde{L}\propto {\epsilon }_{{\rm{c}}}^{\tfrac{\delta +8}{2\delta +4}}={\epsilon }_{{\rm{c}}}^{0.88-0.93}$ for $3.0\lt {\alpha }_{s}\lt 4.0$. In order to find the values of parameters that reproduce this evolution, Figure 8 is presented. This figure exhibits the energy peak as a function of time since GW trigger. The red solid line is the fitted simple power law ${E}_{\mathrm{peak}}={A}_{0}{(t-{t}_{0})}^{\alpha }$ with ${A}_{0}=30.2\pm 7.97\,\mathrm{keV}$, ${t}_{0}=-0.15\pm 0.04\,{\rm{s}}$, $\alpha =-0.97\pm 0.35$, and ${\chi }^{2}/\mathrm{ndf}=0.44/4$. Dashed, dotted, and dashed–dotted lines represent the cooling spectral break of our theoretical model for ${\alpha }_{s}=2.3$ and $n=0.6\,{\mathrm{cm}}^{-3}$ (green line), ${\alpha }_{s}=3.2$ and $n=4\,{\mathrm{cm}}^{-3}$ (gold line), and ${\alpha }_{s}=4.2$ and $n=12\,{\mathrm{cm}}^{-3}$ (blue line), respectively. The values used are $\tilde{E}=3\times {10}^{49}\,\mathrm{erg}$, ${\epsilon }_{B}\,={10}^{-1}$, ${\epsilon }_{{\rm{e}}}={10}^{-1}$, and $p=2.2$. In all of the cases, the cooling spectral break can successfully describe the early evolution of ${E}_{\mathrm{peak}}$ for $1.2\lt {\alpha }_{s}\lt 4.2$ and $0.6\lt n\lt 10\,{\mathrm{cm}}^{-3}$. If this is the case, the bulk Lorentz factor begins to change from ∼12 at a few seconds to ∼3 around 100 days. If the cooling spectral break evolution enters the X-ray band, then the density of the circumburst medium must be much higher than the one found to explain the afterglow at tens of days. The high and low values of the ISM found at early and late times, respectively, could be explained through a stratified medium around the merger as discussed in other sGRBs (Parsons et al. 2009; Norris et al. 2011; Wang & Huang 2018).

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In the framework of the binary NS system, we have derived the dynamics of the forward shock and the synchrotron light curves from the outermost (shock-breakout) material and the relativistic off-axis jet. The resulting equivalent kinetic energy for the shock-breakout material is given by $\tilde{E}{\left({\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}\right)}^{-\delta }$ and for the relativistic jet is $2\tilde{E}/{\theta }_{{\rm{j}}}^{2}$. We have analyzed the case in which the shock-breakout material and the relativistic jet are decelerated by a homogeneous medium and evolve in the fully adiabatic regime. The main differences between both ejected materials lie in the fact that the jet is moving at ultrarelativistic velocities, is narrowly collimated and is observed from a viewing angle. Taking into account the velocity regime, the values obtained of α_s for the mildly relativistic shock-breakout material (${\alpha }_{s}=2.3$) agrees with the description presented by Tan et al. (2001) and the works previously described in Hotokezaka & Piran (2015) and Kyutoku et al. (2014). Considering the observed times of the lateral expansion for the shock-breakout material (Equation (18)) and the relativistic off-axis jet (Equation (32)), the observed flux of shock-breakout material peaks first due to the emission is released on-axis from a material described by a power-law velocity distribution. It is worth mentioning that for $\delta =0$, the observable quantities and light curves derived in Sari et al. (1998), Fraija et al. (2016a), Dai & Lu (1999), Huang & Cheng (2003), Dermer et al. (2000), Granot et al. (2002), and Rees (1999) are recovered. In addition, we have analyzed the shock-breakout material considering the values in the typical ranges: the medium density (${10}^{-6}\leqslant n\leqslant 1\,{\mathrm{cm}}^{-3}$), magnetic microphysical parameter (${10}^{-4}\leqslant {\epsilon }_{B}\leqslant {10}^{-1}$), power indices ($2.2\leqslant p\leqslant 3.6$) and ($1.5\leqslant {\alpha }_{s}\leqslant 3.5$) for a fiducial kinetic energy ($\tilde{E}={10}^{49}$–${10}^{50}\,\mathrm{erg}$), and microphysical parameter (${\epsilon }_{{\rm{e}}}={10}^{-1}$) for an event located at a luminosity distance of $D=100\,\mathrm{Mpc}$.

For the shock-breakout material we have found that the cooling spectral break evolves in the γ-ray bands and the characteristic break in the IR—optical bands. The cooling (characteristic) spectral break evolves as ${\epsilon }_{{\rm{c}}}\propto {t}^{-(1.0-1.1)}$ (${\epsilon }_{{\rm{m}}}\propto {t}^{-(0.9-1.0)}$), instead of the typical evolution ${\epsilon }_{{\rm{c}}}\propto {t}^{-0.5}$ (${\epsilon }_{{\rm{m}}}\propto {t}^{-1.5}$) suggested by the decelerated jet (Sari et al. 1998). The analysis of the early spectral evolution of the tails as suggested by some authors (e.g., see Giblin et al. 1999; Fraija et al. 2012, 2017b) could illustrate whether the evolution of ${E}_{\mathrm{peak}}$ as early observed in γ-ray and optical bands could be generated by external shocks during the prompt phase. In addition, this analysis could reveal the type of scenario (e.g., internal or external shocks), circumburst medium (e.g., homogeneous or stratified), the regime (e.g., adiabatic or radiative), and the geometry of material that has been decelerated.

Considering the multiwavelength campaign dedicated to follow-up the electromagnetic counterpart of GW170817 (Abbott et al. 2017c) and future campaigns, the light curves to be observed in X-rays at 1 keV, optical band at 1 eV, and radio wavelength at 6 GHz were derived for the shock-breakout material and the relativistic off-axis jet. We found that fluxes have similar behaviors depending on which power-law segment of the spectrum are evolving. We have shown that they are strongly dependent on the values of p and ${\alpha }_{s};$, and at early times fluxes, in general, are dominated by those generated with small values of α_s whereas at later times they are dominated by those with larger values.

In a particular case, we have considered the multiwavelength afterglow observations detected from GW170817 and found the best-fit values of a set of eight parameters, ${{\rm{\Xi }}}_{\mathrm{fow},{\rm{b}}}$ = {$\tilde{E},n,p,{\theta }_{{\rm{j}}}$, ${\rm{\Delta }}\theta$, ${\varepsilon }_{B}$, ${\varepsilon }_{{\rm{e}}}$, α_s} for our afterglow model using the MCMC method. For the shock-breakout material, we found that the bulk Lorentz factor becomes ${{\rm{\Gamma }}}_{{\rm{c}}}\simeq 3.1{\left(t/15\mathrm{days}\right)}^{-0.24}$ and the equivalent kinetic energy $\simeq 3.31\,\times {10}^{47}\,\mathrm{erg}$, which corresponds to an efficiency of $\sim 16 \%$. The cooling spectral break ${\epsilon }_{{\rm{c}}}\,\sim 19.2\,\mathrm{keV}$ is above the X-ray band and the characteristic break ${\epsilon }_{{\rm{m}}}=1.2\times {10}^{-3}$ GHz is below the radio band at 15 days. The X-ray, optical, and radio fluxes peak at ∼30 days, and later they evolve as ${F}_{\nu }\propto {t}^{-0.76}$. At ∼500 s, the X-ray flux will vary from $\propto {t}^{-0.76}$ to $\propto {t}^{-1.03}$ whereas the optical and radio fluxes will continue evolving as ${F}_{\nu }\propto {t}^{-0.76}$. For the relativistic jet, we found that the bulk Lorentz factor becomes ${{\rm{\Gamma }}}_{{\rm{j}}}\simeq 4.6{\left(t/140\mathrm{days}\right)}^{-\tfrac{3}{2}}$. The cooling spectral break ${\epsilon }_{{\rm{c}}}\sim 3.5\,\mathrm{keV}$ is above the X-ray band and the characteristic break ${\epsilon }_{{\rm{m}}}=0.03\,\mathrm{GHz}$ is below the radio band at 100 days. During this period, the observed flux increases as ${F}_{\nu }\propto {t}^{4.2}$ (Equation (34)), as predicted in Nakar & Piran (2018). The X-ray, optical, and radio fluxes peak at ∼140 days, and later they evolve as ${F}_{\nu }\propto {t}^{-2.2}$.

Recently, Mooley et al. (2018a) reported new detections in radio wavelengths collected with Very Long Baseline Interferometry. These observations exhibited for almost 150 days (between 75 and 230 days post-merger) superluminal motion with apparent speed of ∼4. This provided direct evidence that the binary NS system in GW170817 launched a relativistic narrowly collimated jet with an opening angle $\lesssim {5}^{^\circ }$, a bulk Lorentz factor of ∼4 (at the time of measurement), observed from a viewing angle of ${20}^{^\circ }\pm 5^\circ$. Our model is consistent with the results shown by Mooley et al. (2018a) in that the earlier emission is dominated by the slower shock-breakout material, and the later emission (≳80 days post-merger) by a relativistic off-axis jet. Taking into account the values of ${\rm{\Delta }}\theta \simeq 16^\circ$ and ${\theta }_{{\rm{j}}}\simeq 5^\circ$ reported in Table 1, the value of the viewing angle ${\theta }_{\mathrm{obs}}\sim 21^\circ$ is found, which agrees with that reported in Mooley et al. (2018a). The observed flux generated by the deceleration of the mildly relativistic shock-breakout material dominates at early times (≲50 days) and the relativistic off-axis jet dominates at later times (≳80 days). This behavior is due to the fact that the mildly relativistic shock-breakout material is seen on-axis (${\theta }_{{\rm{c}}}\simeq 16^\circ$) whereas the relativistic jet is off-axis with ${\theta }_{\mathrm{obs}}\gt 2{\theta }_{{\rm{j}}}$. It is worth mentioning that the values of opening angle, the bulk Lorentz factor and the viewing angle reported by Mooley et al. (2018a) also agree with those found in our model.

The binary NS system ejects several materials during the merger. In addition to a relativistic jet and a shock-breakout material, a dynamical ejecta and/or neutrino-driven wind are also launched. Since the relativistic jet makes its way out inside the dynamical ejecta, the energy deposited laterally could create a cocoon. Depending on the duration and the energy deposited by the relativistic jet in the dynamical ejecta, the cocoon may or may not be formed (Murguia-Berthier et al. 2014; Nagakura et al. 2014; Nakar & Piran 2017; Lazzati et al. 2017a, 2017b). If the relativistic jet is launched before the ejecta begins to expand, then its propagation through the dynamical ejecta cannot inflate a cocoon and hence it will be neglected (Gottlieb et al. 2018a). We argue that in GW170817 there was no delay between the explosion (i.e., the ejection of the shock-breakout ejecta) and the ejection of the relativistic jet, so the cocoon emission is neglected. In the previous case the delay of 1.74 ± 0.05 s (Abbott et al. 2017b) found between the NS merger GW chirp signal and the γ-ray flux detected by GBM could be interpreted in terms of the extra path length that radiation travels from the edge of the off-axis jet to an observer in comparison with the GW, which is emitted in the observer's direction. This geometrical delay expressed in terms of ${\rm{\Delta }}\theta =16^\circ$ and the distance from the central engine to the emitting region R_γ is given by (see Figure 1 shown in Granot et al. 2017)

$$\begin{eqnarray}&&{t}_{{\rm{\Delta }}\theta }={R}_{\gamma }[1-\cos {\rm{\Delta }}\theta ]\simeq 1.71\,{\rm{s}}\,{R}_{12.1}.\end{eqnarray} \tag{ 36 }$$

It is worth noting that although the jet moves at speed slightly less than the speed of light from the acceleration phase to the internal shocks take place R_γ, this extra delay is neglected ${t}_{\gamma }\simeq {R}_{\gamma }{{\rm{\Gamma }}}_{{\rm{j}}}^{-2}\simeq 4.6\times {10}^{-3}\,{\rm{s}}\,{R}_{\gamma ,12.1}{{\rm{\Gamma }}}_{{\rm{j}},2}^{-2}$. Therefore, the observed delay between the GW signal and the γ-ray flux can be explained in the framework of a geometrical delay ${t}_{{\rm{\Delta }}\theta }$.

Tan et al. (2001) studied a transrelativistic acceleration model, in the context of a supernova explosion, and modeled the kinetic energy of the expelled outer material. These authors found that the equivalent kinetic energy of the outermost material could be described through a power-law velocity distribution and also showed that part of it would be given to the circumburst medium, generating a strong electromagnetic emission. Kyutoku et al. (2014) applied the transrelativistic acceleration model to describe the shock-breakout material ejected in the binary NS merger. They showed the kinetic energy distribution of the shock-breakout material for different polytropic indexes n = 3, 4, and 6 and masses ejected by the shock breakout ${M}_{\mathrm{sh}}={10}^{-4},{10}^{-5}$ and ${10}^{-6}\,{M}_{\mathrm{sun}}$. Given the values reported in Table 1, the equivalent kinetic energy is ${E}_{{\rm{k}},{\rm{c}}}(\gtrsim {\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}})\simeq 3\times {10}^{49}\,\mathrm{erg}\,{\left({\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}}\right)}^{-2.3}=3.1\times {10}^{48}\,\mathrm{erg}$, for ${{\rm{\Gamma }}}_{{\rm{c}}}\simeq 3$. This value agrees with the most optimistic scenario reported in Figure 2 (n = 3 and ${M}_{\mathrm{sh}}={10}^{-4}\,{M}_{\mathrm{sun}}$) by Kyutoku et al. (2014). It is worth noting that although values of higher kinetic energies are difficult to resolve, by recent 3D merger simulations, relevant implications for the NS equation of state must be analyzed with caution.

Margutti et al. (2017a) and Alexander et al. (2017) studied the X-ray and radio light curves of GRB 170817A in a context of standard synchrotron emission from the forward-shock model. Authors concluded that the on-axis afterglow emitted by a jet was ruled out arguing that although this model can describe the X-ray light curve, it fails to comply with the upper limits in the radio light curve, which vary as $\propto {t}^{\tfrac{1}{2}}$. We propose that the X-ray, optical, and radio fluxes are not emitted from a decelerated jet but from the fraction of the outermost layer moving toward us, which evolves with a steeper slope ${F}_{\nu }\propto {t}^{0.64}$ before the break. The evolution of this model over a few days is below the upper limits and consistent with the observations.

The dynamics of different masses ejected from the merger with significant kinetic energies have been investigated as possible electromagnetic emitters (Hotokezaka et al. 2013; Kyutoku et al. 2014; Hotokezaka & Piran 2015). Authors suggested that the electromagnetic signatures associated with the deceleration of these relativistic and subrelativistic masses by the circumburst medium could be detected from γ-rays to radio wavelengths and could be observed at nearly all the viewing angles. For instance, Kyutoku et al. (2014) considered a power-law velocity distribution with $E\propto {{\rm{\Gamma }}}^{-1.1}$ for ${n}_{p}=3$ and proposed that the shock-breakout material ejected at ultrarelativistic velocities (${\rm{\Gamma }}\simeq 40\mbox{--}400$) could be decelerated emitting early photons by synchrotron radiation, which would be detected in current X-ray and radio instruments. Hotokezaka & Piran (2015) studied the dynamics and the radio components emitted by different ejected masses including a dynamical ejected mass and a cocoon. Assuming a breakout material ejected at subrelativistic velocities ${\beta }_{{\rm{c}}}{\rm{\Gamma }}\simeq 1$ for a velocity distribution of ${E}_{{\rm{k}},{\rm{c}}}\propto {({\beta }_{{\rm{c}}}{{\rm{\Gamma }}}_{{\rm{c}}})}^{-3}$ for ${n}_{p}=3$, authors showed that in all cases an early electromagnetic component from the decelerated material is expected at the radio wavelengths. The light curves derived in this paper are different from those derived in the above papers. In this paper, we derive the synchrotron light curves generated from the mildly relativistic shock-breakout material and the relativistic off-axis jet when both are decelerated by a homogeneous density for the adiabatic index ${n}_{p}=3$. We have shown that at early times before 50 days, the emission that originated from the decelerated shock-breakout material dominated, and at later times, longer than ≳80 days, the emission was dominated by the off-axis jet. For instance, X-ray flux derived in Kyutoku et al. (2014) only decrease with time as $\propto {t}^{-0.42}$, and in our case X-ray fluxes increase as $\propto {t}^{0.64}$ at early times. It is worth noting that masses ejected from a collapsar scenario to describe low-luminosity GRBs have also been considered (e.g., see Barniol Duran et al. 2015).

While writing this paper we became aware of a recent preprint (Nakar & Piran 2018) that explains that the increase in X-ray and radio flux observed in GW170817 could be explained in terms of the synchrotron radiation that originated in a decelerated material moving at larger angles. From the observations they obtained values of the bulk Lorentz factor and the isotropic equivalent energy similar to those reported in this paper for the GW170817 event.

Electromagnetic counterpart observations from a binary NS system associated with gravitational waves cast the merger scenario in a new light. Similar analyses to the one developed in this paper with futures observations can shed light on the properties of the outer ejected materials.

We thank Enrico Ramirez-Ruiz, Jonathan Granot, Dafne Guetta, Fabio de Colle, Rodolfo Barniol-Duran, and Diego Lopez-Camara for useful discussions. N.F. acknowledges financial support from UNAM-DGAPA-PAPIIT through grants IA102917 and IA102019. A.C.C.D.E.S.P. acknowledges that this study was financed in part by the Coordenao de Aperfeioa-mento de Pessoal de Nıvel Superior—Brasil (CAPES)—Finance Code 001 and also thanks the Professor Dr. C. G. Bernal for tutoring and useful discussions. P.V. thanks Fermi grants NNM11AA01A and 80NSSC17K0750.