Off-axis Synchrotron Light Curves from Full-time-domain Moving-mesh Simulations of Jets from Massive Stars

We perform standard top-hat Blandford–McKee (BM; Blandford & McKee 1976) simulations to check the robustness of the new AMR-enhanced moving-mesh scheme. For the BM solution, the Lorentz factor of the shock front Γ, the Lorentz factor of the fluid γ, the elapsed time t, and the jet radius R follow the relations (the speed of light c is set to one)

$$\begin{eqnarray}&&{\rm{\Gamma }}=\sqrt{2}\gamma ,l={\left({E}_{\mathrm{iso}}/{\rho }_{0}\right)}^{1/3},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&t=l{\left({\rm{\Gamma }}/\sqrt{17/8\pi }\right)}^{-2/3},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&R=(1-1/(8{{\rm{\Gamma }}}^{2}))t.\end{eqnarray} \tag{ 9 }$$

The half-opening angle of the top-hat BM jet (θ_jet) is set to θ_jet = 0.2. The isotropic equivalent energy is E_iso = 10⁵³ erg. The uniform ambient density ρ₀ is set to 1 proton per cubic centimeter, and the pressure is set to P₀ = 10⁻¹⁰ρ₀ following previous Eulerian AMR simulations (Zhang & MacFadyen 2009; van Eerten et al. 2010). The hydrodynamic simulation starts at the moment when the initial Lorentz factor of the fluid just behind the shock is γ = 20. In Figure 1, we demonstrate that the on-axis synchrotron light curves calculated from our BM simulation match well with results (as presented in Zhang & MacFadyen 2009) from simulations performed with the Eulerian AMR code RAM (Zhang & MacFadyen 2006). We calculate the broadband synchrotron radiation light curves utilizing a well-tested synchrotron radiation code (Zhang & MacFadyen 2009; van Eerten et al. 2010, see also Sari et al. 1998; De Colle et al. 2012a). The same radiation parameter values are used in both calculations: the electron equipartition factor _e = 0.1, the magnetic equipartition factor _B = 0.1, and the energy power-law index of relativistic electron p = 2.5. The flux distance scaling is set to $1/4\pi {d}_{28}^{2},{d}_{28}={10}^{28}\,\mathrm{cm}$ as in Zhang & MacFadyen (2009). We also adopt the same EOS (TM EOS) in this simulation for a fair comparison.

Zoom In Zoom Out Reset image size

For off-axis light curves, we compare with results from van Eerten et al. (2010). As shown in Figure 2, off-axis light curves from our BM simulation (solid lines) match well with results from van Eerten et al. (2010). One advantage of our new scheme is that we are able to perform accurate top-hat BM simulations at an earlier time with a Lorentz factor of hundreds instead of tens.

Zoom In Zoom Out Reset image size

Observational results indicate that GRB afterglow comes from ultra-relativistic jets with inferred Lorentz factors ∼100. Numerical simulations with initial Lorentz factors ∼100 are thus necessary to fully capture early afterglow emission. Here, we perform two top-hat BM simulations with different initial Lorentz factors: (1) t₀ = 4.37 × 10⁶ s, γ₀ = 100, R₀ = 1.31 × 10¹⁷ cm and θ_jet = 0.2 (denoted as BM-J0.2-G100 simulation). (2) t₀ = 1.28 × 10⁷ s, γ₀ = 20, R₀ = 3.83 × 10¹⁷ cm and θ_jet = 0.2 (denoted as BM-J0.2-G20 simulation), respectively. The RC EOS is adopted for both BM simulations (hereafter, all the simulations are performed with RC EOS). The synchrotron radiation parameter values are listed in Table 1. The on- and off-axis synchrotron light curves at frequency 10¹⁷ Hz from the BM-J0.2-G20 simulation and from the BM-J0.2-G100 simulation are overplotted in Figure 3. The early on-axis light curve calculated from the BM-J0.2-G100 simulation (dashed line) is an order of magnitude larger than that from the BM-J0.2-G20 simulation. The off-axis light curves from the BM-J0.2-G100 simulation rise up earlier. At a later time, the afterglow light curves from both simulations overlap with each other.

Zoom In Zoom Out Reset image size

Table 1.  Synchrotron Radiation Parameters

Variable	BM models	Structured Jet (SJ) models
e	0.1	0.05
B	0.1	0.005
p	2.5	2.2
dL	2.05 × 1028 cm	2.05 × 1028 cm
z	1	1

Note. Two sets of microphysical parameter values are utilized for simulations presented in this study. The Blandford–McKee (BM) models include the BM-J0.2-G20, BM-J0.2-G100, and BM-J0.1-G100 simulations. The SJ models include the SJ0.1-EH, SJ0.1-EL, and SJ0.2-EH simulations. When we compare the SJ and BM models, the BM models use the same set of parameter values as the SJ models.

Download table as:  ASCIITypeset image

The top-hat jet model can explain the "jet break" phenomena observed in GRB afterglows. On-axis observers will start to see the edge of the jet when its Lorentz factor drops to γ ∼ 1/θ_jet. The missing flux will lead to a break in the slope of the observed light curves. Before the jet break time, the temporal slope of high-frequency light curves (ν > ν_c) scales as ${F}_{\nu }\propto {t}^{-3p/4+1/2}$. This should be the same for both spherical explosion and top-hat jets. After that, the light curve of the finite top-hat jet scales as F_ν ∝ t^−p (see, e.g., Rhoads 1999; Sari et al. 1999).

We perform a spherical BM simulation with parameters identical to the BM-J0.2-G100 model. The light curves from the spherical BM simulation (see the dashed lines in Figure 4) do not show any break. When we cut a conical segment with a half-opening angle θ = 0.2, and only add emission from this region, the calculated light curves (dotted–dashed line) display the expected temporal break due to pure relativistic beaming. For the light curve of the top-hat BM-J0.2-G100 simulation (solid line), the jet break happens at around the same time but with a steeper temporal slope. The extra decay then originates from a well-known hydrodynamic effect: lateral spreading of the jet (Rhoads 1999; Sari et al. 1999; Granot et al. 2001; Zhang & MacFadyen 2009; Wygoda et al. 2011; Granot & Piran 2012; van Eerten & MacFadyen 2012; Duffell & Laskar 2018). The comparison among these three sets of light curves indicates that the jet break phenomenon due to hydrodynamical effect is not negligible even for the relatively simplified model considered here.

Zoom In Zoom Out Reset image size

For off-axis light curves, a natural prediction from top-hat jet models is the existence of "orphan" afterglows. An observer located outside of the opening angle of relativistic jets will not be able to detect the early high-energy emission due to relativistic beaming. At a later time when the Lorentz factor of the jets reduces to γ = 1/(θ_obs − θ_jet), the off-axis observer starts to receive emission from the central jet at lower energies. It is then possible to detect the afterglow radiation without having detected prompt emission for off-axis observers. As shown in Figure 2, the off-axis afterglow emission show features a late rise-up on a timescale of days to months depending on the viewing angle. However, as we show in Figure 3, changing the initial Lorentz factor from 20 to 100 leads to an early rise-up for off-axis light curves. The explanation is that the BM-J0.2-G100 profile is set at an earlier time t₀ = 4.37 × 10⁶s, and at a smaller radius R₀ = 1.31 × 10¹⁷(cm).

For the remainder of the paper we utilize FTD simulations to study the on-axis and off-axis synchrotron light curves of LGRB jets. We discuss the features revealed from the light curves and their implications.

The stellar progenitor before collapse utilized in the simulations follows the analytical model in Duffell & MacFadyen (2015). This model approximates the output of a MESA (Paxton et al. 2011, 2013) simulation in which a low-metallicity rapidly rotating star evolves to a Wolf-Rayet star. The density as a function of radius is

$$\begin{eqnarray}&&\rho (r,0)=\displaystyle \frac{{\rho }_{c}{\left(\max (1-r/{R}_{3},0)\right)}^{{\rm{n}}}}{1+{\left(r/{R}_{1}\right)}^{{k}_{1}}/(1+{\left(r/{R}_{2}\right)}^{{k}_{2}})}+{\rho }_{\mathrm{wind}}{\left(r/{R}_{3}\right)}^{-2}.\end{eqnarray} \tag{ 10 }$$

The parameters in the above equation are listed in Table 1 of Duffell & MacFadyen (2015) and are included here for clarity (see Table 2). The velocity and pressure are initially set to negligible values. Self-gravity and stellar rotation are not included in the simulations.

Table 2.  Stellar Parameters

Variable	Definition	Value
M0	Characteristic Mass Scale	2 × 1033 g
R0	Characteristic Length Scale	7 × 1010 cm
ρc	Central Density	$3\times {10}^{7}{M}_{0}/{R}_{0}^{3}$
R1	First Break Radius	0.0017 R0
R2	Second Break Radius	0.0125 R0
R3	Outer Radius	0.65 R0
k1	First Break Slope	3.24
k2	Second Break Slope	2.57
n	Atmosphere Cutoff Slope	16.7
ρwind	Wind Density	${10}^{-9}{M}_{0}/{R}_{0}^{3}$

Note. Courtesy of Table 1 in Duffell & MacFadyen (2015).

Download table as:  ASCIITypeset image

The jet engine is initiated at around the radius r₀ = 0.01R₀ (7 × 10⁸ cm) using a source term. From this distance, the density field of the progenitor in which the jet propagates is comparable to that of the E25 model in Heger et al. (2000) and the 16TI model in Woosley & Heger (2006) (see Figure 5). The jet engine model utilizes the nozzle function g(r, θ) in Duffell & MacFadyen (2015). For clarity, the expressions are included in the following:

$$\begin{eqnarray}&&g(r,\theta )\equiv (r/{r}_{0}){e}^{-{\left(r/{r}_{0}\right)}^{2}/2}{e}^{(\cos \theta -1)/{\theta }_{0}^{2}}/{N}_{0},\end{eqnarray} \tag{ 11 }$$

where N₀ is the normalization of g via the integration over $r\in [0,\infty ],\theta \in [0,\pi /2]$:

$$\begin{eqnarray}&&{N}_{0}\equiv 4\pi {r}_{0}^{3}(1.\,-\,{e}^{-1/{\theta }_{0}^{2}}){\theta }_{0}^{2}.\end{eqnarray} \tag{ 12 }$$

Zoom In Zoom Out Reset image size

The source terms in Equations (1) and (2) are

$$\begin{eqnarray}&&{S}^{0}={L}_{0}{e}^{-t/{\tau }_{0}}g(r,\theta ),\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{S}^{r}={S}^{0}\sqrt{1-1/{\gamma }_{0}^{2}},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{S}_{D}={S}^{0}/{\eta }_{0}.\end{eqnarray} \tag{ 15 }$$

This jet engine features a smoothly decaying tail with an average engine duration τ₀ = 10 s. The engine completely shuts down at around 20 s (see discussions of engine duration in e.g., Lazzati et al. 2013). The simulation (denoted as SJ0.1-EH (Energy High) hereafter) performed in this study differs from Duffell & MacFadyen (2015) in two ways. First, we utilize the incorporated AMR scheme, which enforces the criteria Δr/r < 1/(16Γ²), to better resolve the relativistic shell. Second, we adopt the newly implemented RC EOS instead of the original ideal gas EOS. We also perform additional simulations. One simulation has relatively low jet engine energy (denoted as SJ0.1-EL) and another simulation has a different jet engine half-opening angle θ₀ = 0.2 (denoted as SJ0.2-EH). The jet engine parameter values for these three models are listed in Table 3.

To better interpret the role of each fluid component, we use three passive scalars X_i to track the mass fraction of each individual component filling the cells. The subscript i labels each individual component: 1 for a stellar progenitor, 2 for ISM material, and 3 for jet engine material. Initially, X₁(X₂) is set to 1(0) inside the progenitor and 0(1) in the ISM. Three auxiliary equations are solved accordingly (Duffell & MacFadyen 2013):

$$\begin{eqnarray}&&{\partial }_{\mu }({X}_{i}\rho {u}^{\mu })={S}_{D}.\end{eqnarray} \tag{ 16 }$$

The conserved mass for each component (${M}_{i}=\int {X}_{i}\rho {u}^{0}{dV}$) is updated based on the density flux through the cell boundary and the addition of source term (for the injection of jet engine material). Dividing the individual conserved mass M_i by the total conserved mass ($M=\int \rho {u}^{0}{dV}$) gives us the new primitive passive scalar X_i.

4.2.1. Launch of the Jet Engine

Figure 6 shows the early evolution of the jet engine flow upon injection. The ram pressure generates a bow shock. Dense stellar material gets pushed to the side, forming a cocoon that confines the engine outflow. A high-density wedge of stellar material develops at the head of the jet. It shreds jet engine materials from the head. These engine materials curl back, forming vortexes. These vortexes then detach from the jet and get swept backward relative to jet propagation. This vortex shedding phenomena is, generally speaking, similar to that found in previous simulations (Scheck et al. 2002; Mizuta et al. 2004; Morsony et al. 2007). The speed of the jet head keeps increasing and soon exceeds the local sound speed (at the time t = 1.5s, the Lorentz factor of the head of the jet is ∼2, as shown in Figure 6). The backflow becomes quasi-straight to the main jet (Mizuta et al. 2010). At early times, the bow shock has a narrow head and a wide tail. When it approaches the surface, the jet head expands in the low-density envelope, as shown in the next subsection (see also, e.g., Zhang et al. 2003, 2004b). We use spherical coordinate to conduct simulations. As jets propagate outward, the width of the cell increases. The features of the inner part of the jet (close to pole) may not be fully resolved. We resolve the relativistic shell in the radial direction via the previously described AMR scheme. As the ultra-relativistic jet shell penetrates the progenitor, the stellar material that lies on top of the jet easily gets pushed aside. No strong "plug" instability has been seen (Lazzati et al. 2010; Mizuta & Ioka 2013; Gottlieb et al. 2018a; Xie et al. 2018).

Zoom In Zoom Out Reset image size

4.2.2. Propagation of the Jet

Figures 7 and 8 show snapshots of the simulation domain at different stages of jet evolution for the SJ0.1-EH model. Each snapshot displays the contour of log density, normalized Lorentz factor, and normalized mass fraction of jet engine/stellar material. At lab frame time t = 2.5 s, the jet outflow begins to expand in the low-density wind. The shock wraps around the star as shown in the snapshots at t = 4 s and t ≈ 6 s. At a later time, t = 20 s, the shock accelerates and approaches its terminal Lorentz factor ∼10². The engine completely turns off at around this time. Along the polar axis, a relativistic blob forms behind the shock front. Through internal collisions, the relativistic blob forms an ultra-relativistic thin shell. At first, the relativistic shell is hidden behind the photosphere (indicated by the magenta line in Figure 8). We define the photosphere as the place where the optical depth is unity. We estimate the optical depth according to:

$$\begin{eqnarray}&&\tau ={\int }_{{r}_{\mathrm{ph}}}^{\infty }{\sigma }_{T}{\rm{\Gamma }}(1-\beta \cos \theta ){ndl},\end{eqnarray} \tag{ 17 }$$

where σ_T is the Thomson scattering cross section, β is the absolute value of the velocity normalized by speed of light, Γ is the Lorentz factor of the gas, θ is the angle between the velocity vector and the line of sight, and n is the electron number density (Abramowicz et al. 1991; Mizuta et al. 2011). In detail, we first initiate a sufficient number of tracing rays at the given observing angle. Each tracing ray will enter the spherical domain from a position of the outer boundary. We then calculate the crossed length in each cell the ray intercepts and perform the integration of optical depth (Equation (17)), assuming the density is uniform within the cell. The contribution of optical depth from materials outside of the simulation domain is added using an analytical expression. Following this procedure, we are able to get the optical depth for all of the cells in the simulation domain at each snapshot. Note that this procedure is a simplified version in terms of the calculation of actual photosphere. Photons are propagating in a turbulent, evolving density background. Integration of optical depth over continuous snapshots is then preferred. In this work, we focus on the study of optically thin synchrotron radiation light curves. Once the emitting shell breaks out of the photosphere, the photosphere position has no impact on the shape of the light curve. We find that the characteristics of synchrotron light curves are not sensitive to the exact definition of photosphere. For a detailed treatment of photons breaking out of the photosphere, we refer readers to the study of photospheric emission (e.g., Lazzati et al. 2011; Mizuta et al. 2011; De Colle et al. 2018b; Gottlieb et al. 2018b).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

A wind profile is adopted here to describe the density field of the surrounding interstellar medium (ISM): ρ_ISM = Ar⁻², A = 24 × (5 × 10¹¹g cm⁻¹). The photosphere is initially located at a radius r_ph = 4.8 × 10¹² cm. At around ∼2 × 10² s, the shock front breaks through the original photosphere. The photosphere then advances outward with the jet. Eventually, at around $t\sim 5\times {10}^{3}\,{\rm{s}},r\sim {10}^{14}\,\mathrm{cm}$, the photosphere begins to fall behind the relativistic shell. The process of the jet breaking out of the photosphere covers the dynamical distance where prompt emission is estimated to occur (e.g., Piran 1999; Kumar & Zhang 2015). At t = 3 × 10⁵ s, the photosphere falls far behind the relativistic shell, and Kelvin–Helmholtz instability is seen behind the shock front. At t ∼ 2 × 10⁹ s, the jet reaches a distance of ∼3 pc. At this time, the jet and stellar material form a highly aspherical structure and have become fully Newtonian.

The angular structure of the jets features an ultra-relativistic core, primarily composed of jet engine material. The core is surrounded by a mildly relativistic sheath (see Figure 8). The angular distribution of the total energy (excluding rest mass energy) dE/dΩ, and energy-averaged Lorentz factor Γ for the emerged jet, are shown in Figure 9. During the coasting period ∼10²−10⁶ s, the jet angular structure does not change significantly. They can be well fit by a USJ model in which dE/dΩ and Γ varies as a power law of polar angle (Granot & Kumar 2003; Kumar & Granot 2003; Granot & van der Horst 2014):

$$\begin{eqnarray}&&{\rm{\Theta }}=\sqrt{1+{\left(\theta /{\theta }_{0}\right)}^{2}},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&\epsilon (\theta )={\epsilon }_{0}{{\rm{\Theta }}}^{-\alpha },\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{\rm{\Gamma }}(\theta )=1+({{\rm{\Gamma }}}_{0}-1){{\rm{\Theta }}}^{-\beta },\end{eqnarray} \tag{ 20 }$$

where θ₀, ₀, α, β, and Γ₀ are free fitting parameters. We fit the three performed simulations: SJ0.1-EH, SJ0.1-EL, and SJ0.2-EH, and list their fitting parameter values in Table 4. As shown in Figure 9, the half-opening angle of the central core is θ_core ∼ 0.1 for the SJ0.1-EH and SJ0.1-EL simulations, and θ_core ∼ 0.16 for the SJ0.2-EH simulation. The total energy of the jet core is E_jet ∼ 10⁵² erg for SJ0.1-EH and SJ0.2-EH, and E_jet ∼ 10⁵¹ erg for SJ0.1-EL falls within the inferred range of GRB kinetic energy (see, e.g., Frail et al. 2001; Panaitescu & Kumar 2002; Berger et al. 2003a; Lloyd-Ronning & Zhang 2004; Goldstein et al. 2016). The angular power-law decay index α in all of the simulations is significantly larger than 2, the typical value adopted in the USJ model (Rossi et al. 2002; Zhang & Mészáros 2002a; Granot & van der Horst 2014). An observational correlation between the isotropic emitted energy E_iso and the spectral peak energy E_p: E_iso ∝ E_p² has been discovered (Amati et al. 2002). This relationship extends from GRBs to X-ray flashes (XRFs; see, e.g., Lamb et al. 2005). In the unified GRB-XRF model, XRFs are the result of a highly collimated GRB jet viewed off-axis (Yamazaki et al. 2002; Lamb et al. 2005). The observed E_p(E_iso) range is more than 2(4) orders of magnitude (see, e.g., Zhang et al. 2004a). For the USJ model with α = 2, the relation ${E}_{p}\propto {E}_{\mathrm{iso}}^{1/2}\propto {\theta }^{-1}$ implies that the viewing angles of XRFs need to be at least two orders of magnitude larger than those of GRBs. This puts strong constraints on the USJ model with α = 2 (or vice-versa the unified GRB-XRF model). Zhang et al. (2004a) showed that a quasi-universal Gaussian-like structured jet model with a steeper angular profile can reconcile this. The structured jets that emerged in our simulations feature a steep angular profile with α ∼ 8. This profile can also explain the wide range of observed values for E_iso given the limited off-axis observer angles. We only need to have a viewing angle several times larger than θ_core to get an XRF whose E_iso is 10²−10⁴ times lower than the typical GRB E_iso (Zhang et al. 2004a). Note that, in the study of constraints on the structure of LGRB jets, Beniamini & Nakar 2019 found that there has to be either a very steep angular structure for the jet energy (as what we find here), or else the efficiency of gamma-ray production should decrease strongly at high latitudes.

Zoom In Zoom Out Reset image size

Table 4.  Fitting Parameter Values for the Angular Structure of Emerged Jets

Variable	SJ0.1-EH	SJ0.1-EL	SJ0.2-EH
θcore	0.1	0.09	0.16
0	1.3 × 1054 erg	7.8 × 1052 erg	2.6 × 1053 erg
α	8.9	7.0	6.7
Γ0	73	20	25
β	5.1	3.1	7.9

Note. θ_core is defined as the half-opening angle of the central core for structured jets. ₀, α, Γ₀, β are fitting parameters from Equations (18)–(20).

Download table as:  ASCIITypeset image

As seen in Figure 10, the on-axis light curves start with an early pulse and are followed by three major segments: an early time decay, a shallow decay, and a late steeper decay. These light-curve components share similarities with canonical X-ray afterglows observed by the Swift X-ray Telescope (XRT) (Gehrels et al. 2009; Kumar & Zhang 2015). The timescale of the early pulse is tens of seconds (see Figure 11 for an example), and falls within the duration of observed LGRB prompt emission. The light-curve decomposition shows that the on-axis multi-frequency light curves from shock-heated ISM (dashed lines) are flat for the entire duration of the prompt and shallow decay phases. We thus find that the pulse mainly originates from shock-heated stellar material. Long and temporally smooth GRBs with typical variability timescales larger than a few seconds are observed and sometimes considered to arise from an external shock (Burgess et al. 2016; Huang et al. 2018).

Zoom In Zoom Out Reset image size

On-axis light curves from FTD simulations provide hints that, while a unified external shock model can explain both prompt and afterglow emissions, the prompt γ-ray emission of observed single-pulsed GRBs may still come from shock-heated stellar materials instead of freshly shock-heated ISM materials (e.g., Burgess et al. 2018). Note that GRB prompt emission involves more complicated physics (e.g., photospheric emission, magnetic reconnection). Here, we simply note the implications of the hydrodynamic results.

The off-axis light curves (seen in Figure 10) exhibit clear temporal breaks as well. The break time depends on the viewing angle. Wang et al. (2015) fit the broken power-law (BPL) model to optical and X-ray light curves of 85 GRBs, and found that the break times range from a few 10² s to 10³ days after the GRB prompt emission. The break times seen in the on-axis light curves in Figure 10 fall in this range. For off-axis light curves, an achromatic rebrightening component may appear around the break time and is then followed by a steeper decay. This is what happens for off-axis light curves from narrow jet simulations SJ0.1-EH and SJ0.1-EL, but not from the wider jet simulation SJ0.2-EH. Rebrightening features occur in observations of long GRBs (e.g., GRB070311 Guidorzi et al. 2007, GRB081028 Margutti et al. 2010, GRB100814A De Pasquale et al. 2015, GRB120326A Melandri et al. 2014; Hou et al. 2014), short GRBs (e.g., GRB050724 Campana et al. 2006, GRB080503 Gao et al. 2015), and XRFs (e.g., XRF030723 Huang et al. 2004). Various mechanisms have been proposed to explain these rebrightenings. Here we list three: the density jump model (Lazzati et al. 2002; Dai & Wu 2003; Tam et al. 2005; Geng et al. 2014; Uhm & Zhang 2014), the refreshed-shock or energy injection model (Dai & Lu 1998; Rees & Mészáros 1998; Kumar & Piran 2000; Sari & Mészáros 2000; Zhang & Mészáros 2001, 2002b; Granot & Kumar 2006; Zhang et al. 2006; Dall'Osso et al. 2011; Uhm et al. 2012; Laskar et al. 2015), and the two-component jet model (Berger et al. 2003b; Huang et al. 2004, 2006). Other models can be found in, e.g., Kong et al. (2010). The FTD jet simulations demonstrate that structured jets can naturally drive the rebrightening afterglow component for off-axis observers. The ratio of its temporal width to its peak time is ΔT/T ∼ 1. This feature is clearly different from X-ray flares that characterize the short rise time δT/T ≪ 1 (e.g., Burrows et al. 2005; Fan & Wei 2005; Liang et al. 2006; Zhang et al. 2006). Before the rebrightening, the flux decays as segments of power laws. This early decaying component for off-axis light curves originates from the shocked ISM, which is different from the rapidly fading "merger flash"—the non-thermal cooling emission of shock-heated merger ejecta and jet engine materials propagating in a low-density environment. The magnitude of the early X-ray "merger flash" for GRB170817A has been estimated to lie below the instrument-detection limit of Swift when its X-ray Telescope made the first observation of the merger site (Xie et al. 2018). Comparison of the shapes of the off-axis light curves presented here and those in Xie et al. (2018) shows that even though structured jets are produced in both scenarios, different setups of ISM density and progenitor profile can drive distinct off-axis light curves. In the work of Xie et al. (2018), the late off-axis afterglow light curves keep increasing before the external shock decelerated. In this new piece of work, we adopt a dense wind ISM profile. The off-axis external shock decelerates after breaking out of the photosphere (revealed in the decreased value of Lorentz factor in Figures 8 and 9. The off-axis light curves in this case decay with time at first. The late afterglow emission (including the rebrightening components) may be overshadowed by ongoing supernova emission (see Figure 10).

The long time monitoring of Type Ib/c SNe does not present evidence for a steeply rising light curve (Soderberg et al. 2006; Bietenholz et al. 2014; Ghirlanda et al. 2014; De Colle et al. 2018a). However, broad-lined Type Ibc supernovae (SNIbc-BL), including those lacking prompt GRB emission, may be producing off-axis afterglow components that, if disentangled from supernova emission, would indicate that they harbor off-axis GRBs (see, e.g., Modjaz et al. 2019).

To better interpret the features of light curves from FTD simulations, we decompose the on-axis and off-axis light curves based on the lateral angle and Lorentz factor distribution of emitting materials in the shell (shown in Figure 12 and Figure 13). For on-axis light curves, the materials confined within a lateral angle of 0.2 determine the light-curve shape for the first ∼10¹ days, covering the prompt to early normal decay phases. The flattening part of the late normal decay (T > 10 days) mainly comes from high-latitude emission (θ_obs > 0.2) (see Figure 12). The first ∼0.1 day light curves, especially the early pulse, are emitted by materials with high Lorentz factors (Γ > 10), while the flattening part of the late normal decay comes from sub-relativistic materials with Γ < 1.2 (see Figure 13). For off-axis light curves, the early decay part originates from the angular region that is close to the observers' light of sight. For example, at the off-axis viewing angle θ_obs = 0.4, the materials within the angular region 0.2 < θ < 0.4 drive the early decay. As time goes on, the emission from the central region 0.0 < θ < 0.2 gradually becomes important. It flattens the light curve and eventually drives the rebrightening components (see off-axis light curves from SJ0.1-EH and SJ0.1-EL simulations). After rebrightening, the off-axis light curves go through a steeper decay, followed by a later flattening part. The sub-relativistic materials with Γ < 1.2 again drive the late flattening part (see Figure 13). Note that, generally speaking, the material in the shell expands laterally and slows down during its propagation. It is almost certain that the initially ultra-relativistic materials (Γ > 10) within the central region will first contribute to the early on-axis pulsed emission and then contribute to the off-axis rebrightening components as it slows down to the intermediate Lorentz factor region 2 < Γ < 10 (see the solid orange and dotted magenta lines in Figure 13).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The off-axis light curves in Figure 10 rise very early on. This differs significantly from the prediction of late rise-ups in top-hat BM jet models. In Figure 14, we compare the on- and off-axis light curves among three scenarios for two cases: (1) light curves (red solid line) calculated from FTD simulations—case 1: SJ0.1-EL (on the top) and case 2: SJ0.2-EH (on the bottom). (2) light curves calculated from part of the FTD simulations covering lab frame time period t > 4.37 × 10⁶ s (blue dotted–dashed line); and (3) light curves calculated from top-hat BM simulations (green dashed line) with an initial Lorentz factor of 100, jet half-opening angle 0.1 (on the top; BM-J0.1-G100 model) and 0.2 (on the bottom; BM-J0.2-G100 model). All of the light curves are calculated with the same radiation microphysical parameters listed in Table 1. Scenarios (2) and (3) use the same initial lab frame time t = 4.37 × 10⁶ s to start the calculation of synchrotron emission. The off-axis light curves from these two scenarios display qualitatively similar patterns. They all rise on a timescale of days. The rise time depends on the viewing angle. More profound differences occur between Scenarios (1) and Scenario (3). First, the off-axis light curves from Scenario (1) rise up almost instantaneously—on a timescale of seconds up to a few minutes. There are two major reasons for this early emission feature. First, the afterglow emission from FTD simulations begins at a time much earlier than that for typical BM timescales. In FTD jet simulations, the shock front begins to surpass the photosphere at the radius r_ph ∼ 10¹³ cm, and starts to emit observable synchrotron radiation. It is much smaller than the initial position of typical top-hat BM models ∼10¹⁷cm. Second, the emerged jet is structured and has a mildly relativistic sheath extending to a large lateral angle. The emission driven by the relativistic sheath is not accounted for in the top-hat jet models. Based on the above analysis, we point out that the afterglow emission calculated from top-hat BM jet models has missing components in time and space.

Zoom In Zoom Out Reset image size

Here we revisit the concept of "orphan afterglow": the observation of a late rising afterglow light curve without an accompanying prompt γ-ray burst for off-axis observers. Specific surveys have been designed and performed to search for OAs in the X-ray (e.g., Grindlay 1999; Greiner et al. 2000), optical (e.g., Rau et al. 2006; Malacrino et al. 2007), and radio bands (e.g., Levinson et al. 2002; Rykoff et al. 2005; Gal-Yam et al. 2006; Bannister et al. 2011; Bell et al. 2011; Frail et al. 2012). No OAs have been conclusively detected so far (Ghirlanda et al. 2014, 2015). There are, however, OA candidates that attract attention (e.g., Law et al. 2018).

The off-axis afterglow light curves presented in this work advocate new templates for use in the search for off-axis afterglows or OAs. The overall shapes of the off-axis light curves from FTD simulations are joint results of relativistic beaming and hydrodynamical effects.