Late Afterglow Bump/Plateau around the Jet Break: Signature of a free-to-shocked wind Environment in Gamma-ray Burst

Gamma-ray bursts (GRBs) are the most luminous sources of electromagnetic radiation known in the Universe. Observationally, GRBs generally appear as a brief and intense $\gamma$-rays followed by a long-lived afterglow emission. With the observations of Swift satellite, the diversity of light-curves in X-ray afterglows have been revealed (Gehrels et al., 2004; Burrows et al., 2005b). However, a canonical light-curve consisting of four power-law segments and a flaring component has been suggested, i.e., an initial steep decay with a typical slope $\lesssim-3$, a shallow decay with a typical slope $\sim-0.5$, a normal decay with a typical slope $\sim-1.2$, a late steep decay with a typical slope $\sim-2$, and one or several X-ray flares (Zhang et al., 2006; Nousek et al., 2006; O’Brien et al., 2006; Zhang et al., 2007). The initial steep decay component is believed to be the tail of the prompt emission due to the “curvature effect” of the high-latitude emission (Barthelmy et al., 2005; Liang et al., 2006; O’Brien et al., 2006; Lin et al., 2017a, b). The shallow, the normal, and the late steep decays are always attributed to the external shock (Zhang et al., 2006; Nousek et al., 2006; Panaitescu et al., 2006), which is developed when a relativistic jet propagates into the circum-burst medium. If there is no energy injection into the external shock, a normal decay phase would appear in the afterglows. However, the decay may become shallow during the phase with continuous energy injection into the external shock. The late steep decay corresponds to the emission of the external shock after the jet break. The X-ray flares generally show a sharp rise with a steep decay and thus could not be produced in the external shock (e.g., Romano et al., 2006; Falcone et al., 2006). It is suggested that most of X-ray flares have the same physical origin as the prompt $\gamma$-rays (Burrows et al., 2005a; Falcone et al., 2006, 2007; Zhang et al., 2006; Nousek et al., 2006; Liang et al., 2006; Chincarini et al., 2007, 2010; Hou et al., 2014; Wu et al., 2013; Yi et al., 2015, 2016; Mu et al., 2016a, b).

Multi-wavelength observations of afterglows have also revealed some puzzling features, which are beyond the expectation from the simple standard external shock scenario (Panaitescu et al., 2006; Panaitescu & Vestrand, 2011; Li et al., 2012; Liang et al., 2013; Wang et al., 2015). For example, some GRBs exhibit late simultaneous bumps in their optical and X-ray afterglows around the jet break, e.g., GRBs 100418A (Marshall et al., 2011; Laskar et al., 2015), 100901A (Gorbovskoy et al., 2012; Laskar et al., 2015), 120326A (Melandri et al., 2014; Hou et al., 2014; Laskar et al., 2015) , 120404A (Guidorzi et al., 2014; Laskar et al., 2015). Their optical afterglows seem to feature a long shallow decay before the late bumps. An exemplar of these afterglows is GRB 120326A, which exhibits simultaneous bumps in its optical and X-ray afterglows at around 35 ks after the burst trigger (Urata et al., 2014; Melandri et al., 2014). Hou et al. (2014) propose that a newborn millisecond pulsar with a strong wind is responsible for its late bumps. Melandri et al. (2014) explore various physical scenarios and conclude that the late bumps may be caused by a refreshed external-forward shock or a geometrical effect. By testing various physical processes, Laskar et al. (2015) argue that the energy injection into the external shock is the most plausible mechanism for its late bump. They also argue that GRB 120404A, with optical and X-ray bumps at $t_{\rm obs}\backsimeq 1000$ s, can be explained in the same scenario. Guidorzi et al. (2014) attribtuted the optical/X-ray bumps of GRB 120404A to a decelerating jet viewed close to the jet edge, combined with some early re-energization of the shock. For GRB 100814A, Geng et al. (2016) proposed an ultrarelativistic electron-positron pair wind model to explain the late optical bumps at $t_{\rm obs}\backsimeq 10^{4}$ s. Yu et al. (2015) invoked a magnetar with spin evolution to explain the afterglow emission. Nardini et al. (2014) attributed the feature to the late-time activity of the central engine in the observed afterglow emission. Apart from the late bumps, we also note an another puzzling feature that some bursts exhibit late simultaneous plateaux in their optical and X-ray afterglows around the jet break, e.g., GRBs 120729A, 051111, and 070318 (Huang et al., 2018).

The above mentioned late simultaneous bumps/plateaux are all directly followed by a steep decay, which is very different from the decay behavior of afterglows preceding the late simultaneous bumps/plateaux and thus is an intriguing phenomenon. The models involved to explain the late simultaneous bumps/plateaux should also explain the steep decay following the bumps/plateaux. Actually, this feature (i.e., the simultaneous bumps/plateaux directly followed by a steep decay) is strongly reminiscent of the afterglows in GRB 170817A, which is formed in an off-axis observed external-forward shock propagating in an interstellar medium (ISM, Troja et al., 2018; Lamb et al., 2019; Troja et al., 2019; Huang et al., 2019; Ren et al., 2020), of which the mass density is constant. The observations of GRB 170817A reveal an almost achromatic bump followed by a steep decay, which is very similar to the GRBs with late bumps mentioned above. Therefore, we would like to suggest that the late simultaneous bumps in GRBs 100814A, 120326A, and 120404A may be formed in an off-axis observed external-forward shock in a homogeneous medium. We would like to point out that the mechanism proposed to explain GRB 170817A has been recently extended to other bursts, e.g., GRBs 080503, 140903A, 150101B, and 160821B (Fraija et al., 2020). Since there is a long shallow decay before the late optical bump in these GRBs, which is different from that of GRB 170817A, the shallow decay may indicate another kind of the circum-burst medium, e.g., the free stellar wind medium.

In this work, we study the emission of the external-forward shock propagating in a free-to-shocked wind circum-burst environment, where the mass density in the shocked wind is almost constant (Weaver et al., 1977; Dyson & Williams, 1997; Chevalier & Li, 2000). The paper is originated as follows. The dynamics and radiation of the external-forward shock are presented in Section 2. The theoretical afterglows are presented and discussed in Section 3. As examples, we perform the fitting on the X-ray and optical afterglows of GRBs 120326A, 120404A, and 100814A, and the results and discussions are shown in Section 4. The conclusions are summarized in Section 5.

When a relativistic jet propagates in the circum-burst medium of a GRB, an external shock would be developed and thus produces a long-term broadband afterglow emission (Sari et al., 1998; Mészáros & Rees, 1999; Sari & Piran, 1999a, b). To describe the jet structure and the dynamics of the external-forward shock, we introduce a spherical coordinate ($R,\theta,\varphi$) with $R=0$ locating at the burst’s central engine and $\theta=0$ being along the jet axis. The jet flow moving at the direction of ($\theta$, $\varphi$) is represented with ($\theta$, $\varphi$)-jet. We assume the observer location at the direction of ($\theta_{\rm v}$, $\varphi_{\rm v}$) with $\varphi_{\rm v}=0$ and $\theta_{\rm v}<\pi/2$.

In our calculations, the jet moving toward us ¹¹1The central engine of a GRB usually launches a pair of outflows, i.e., a near-jet moving toward us and a counter-jet moving away from us. In general, the emission from the counter-jet is negligible compared with the from the near-jet. Then, we only consider the emission from the near-jet. is divided into $I\times L$ small patches along the $\theta$ and $\varphi$ directions in their linear space, i.e., $[0,\;\delta\theta],\;[\delta\theta,\;2\delta\theta],\;[2\delta\theta,\;3\delta\theta],\;\cdots,\;[(i-1)\delta\theta,\;i\delta\theta],\;\cdots,\;[(I-1)\delta\theta,\;I\delta\theta]$ with $\delta\theta={\theta_{\rm jet}}/I$ and $[0,\;\delta\varphi],\;[\delta\varphi,\;2\delta\varphi],\;[2\delta\varphi,\;3\delta\varphi],\;\cdots,$ $[(l-1)\delta\varphi,\;l\delta\varphi],\;\cdots,\;[(L-1)\delta\varphi,\;L\delta\varphi]$ with $\delta\varphi=\pi/L$. Here, $\theta_{\rm jet}$ is the opening angle of the jet. The dynamics and the emission of the external-forward shock is estimated independently in each patch. In addition, the following assumptions are adopted in estimating the dynamics and radiation of the external-forward shock: (1) The sideways expansion are ignored and the dynamics at each patch are assumed to be independent of other patches. The amount of sideways expansion is a debated topic, with numerical simulations suggesting a limited amount of spreading (e.g., Granot et al., 2001; Cannizzo et al., 2004; Zhang & MacFadyen, 2009; van Eerten et al., 2010; van Eerten & MacFadyen, 2012). In addition, it is hard to consider the sideways expansion in the context of a structured jet and the related afterglows fittings. (2) The micro-physical parameters for synchrotron emission, e.g., $\epsilon_{e}$ and $\epsilon_{B}$, are set as constants, where $\epsilon_{e}$ and $\epsilon_{B}$ are the fractions of the shock energy used to accelerate electrons and contributing to the magnetic energy, respectively. (3) The energy injection in the external shock is not considered in our work.

Circum-burst environment: For massive stars, associated with long GRBs, the stellar wind should be formed and the structure of the wind bubble has been widely studied (e.g., Scalo & Wheeler, 2001; Garcia-Segura et al., 1996; Castor et al., 1975). Usually, it is accepted that four regions, i.e., the free stellar wind, the shocked wind, the shocked ISM, and the unshocked ISM, will be formed as the stellar wind interacting with the ISM (Chevalier & Li, 2000). Since the latter two regions are too far to reach by the blast wave (please see figure 1 of Chevalier & Li, 2000), they are not considered here. The mass density of the stellar wind medium is given by ${\rho_{{}_{\rm{free-wind}}}}=5\times{10^{11}}{A_{*}}{R^{-2}}\rm g\cdot c{m^{-1}}$ (Chevalier & Li, 2000), where $A_{*}$ is a constant. However, the mass density in the shocked wind depends on the uncertain physics of the heat conduction. Heat conduction could be prevented by a magnetic field, which is expected to be toroidal in this region (Chevalier et al., 2004). Under this assumption, and using the fact that the internal speed of sound in this region is much higher than the expansion velocity, the density in shocked wind is approximately $R$-independent and equals to $\xi\rho_{\rm wind}(R_{\rm tr})$ with $\xi=4$ (Weaver et al., 1977; Dyson & Williams, 1997; Chevalier & Li, 2000). Please see Pe’er & Wijers (2006) for details. Here, $R_{\rm tr}$, i.e., the free-wind boundary, is the transition radius from the free-wind to the shocked-wind (Kong et al., 2010; Feng & Dai, 2011a; Ramirez-Ruiz et al., 2001) and depends on the age of the massive star wind and proper tracking of the shocked medium cooling. In summary, the mass density of the free-to-shocked wind circum-burst environment is described as

where $\rho_{{\rm shocked-wind}}\equiv n_{0}m_{\rm p}{\rm cm^{-3}}=\xi\rho_{{}_{\rm free-wind}}(R_{\rm tr})$ and $m_{\rm p}$ is the proton mass. One should note that the value of $\xi$ can be different from 4. By fitting the afterglows, Jin et al. (2009a) have obtained $\xi\approx 4.07$ for GRB 081109A and Feng & Dai (2011b) have found $\xi=2.1$, 2.0, and 2.5 for GRBs 080916C, 080916C, and 090926A, respectively. For simplicity, $\xi=4$ is adopted in this paper.

Structured jet Description: In this paper, we mainly focus on an axisymmetric structured jet propagating in a free-to-shocked wind circum-burst environment. Our structured jet at the radius $R_{0}$ is read as

where $E_{\rm k,iso}/(4\pi)$ is the kinetic energy per solid angle at $\theta$, $E_{\rm k,iso,on}$ is corresponding to the value of $E_{\rm k,iso}$ at $\theta=0$, and $\theta_{\rm c}$ is the characteristic angle of $E_{\rm k,iso}$.

Dynamics of the external-forward shock: For the ($\theta$, $\varphi$)-jet, the evolution of the bulk Lorentz factor $\Gamma(\theta,R)$ is described as (Huang et al., 1999; Huang et al., 2000)

where $M_{0}(\theta)=E_{\rm k,iso}(\theta,R_{0})/(4\pi\Gamma_{0}c^{2})$ with $c$ being the light speed, and $m$ is the sweep-up mass of the external shock from $R_{0}=10^{13}$ cm to $R$ per solid angle and can be estimated with

In Equation (7), $\epsilon(\theta,R)=\epsilon_{e}\cdot\min[1,(\gamma^{\prime}_{\rm e,m}/\gamma^{\prime}_{\rm e,c})^{p-2}]$ is the radiation efficiency of the external-forward shock, where $\gamma^{\prime}_{\rm e,m}=\epsilon_{e}(p-2){m_{\rm{p}}}\Gamma/[(p-1){m_{\rm{e}}}]$ is the minimum Lorentz factor of the electrons accelerated in the shock, $\gamma^{\prime}_{\rm e,c}=6\pi m_{\rm e}c/(\sigma_{\rm T}\Gamma{B^{\prime}}^{2}t_{\rm obs}^{\rm on})$ is the efficient cooling Lorentz factor of electrons (Sari et al., 1998). Besides, $B^{\prime}(\theta,R)=(32\pi\rho\epsilon_{B})^{1/2}\Gamma c$ is the magnetic field behind the shock, $t_{{\rm obs}}^{\rm on}(\theta,R)=\int_{R_{0}}^{R}{(c-\upsilon)}{{dr}}/{c\upsilon}$ with $\upsilon(\theta,R)=c\beta$ and $\beta(\theta,R)=\sqrt{1-1/\Gamma^{2}}$, and $m_{\rm e}$, $p$, and $\sigma_{\rm T}$ are the electron mass, the power-law index of accelerated electrons in the shock, and the Thomson cross-section, respectively.

With Equations (3)-(8), one can estimate the value of $\Gamma$ at different $R$. Then, the observed time for a photon from ($\theta$, $\varphi$)-jet can be estimated with

where $\cos\Theta=(\sin{\theta}\cos{\varphi}{\rm{,}}\;\sin{\theta}\sin{\varphi},\;{\rm{cos}}{\theta})\cdot(\sin{\theta_{\rm{v}}},\;0,\;\cos{\theta_{\rm{v}}})=\sin{\theta}\cos{\varphi}\sin{\theta_{\rm{v}}}+\cos{\theta}\cos{\theta_{\rm{v}}}$, and ${\Theta}$ is the angle between the direction of ($\theta$, $\varphi$) and the line of sight, i.e., ($\theta_{\rm v}$, $\varphi_{\rm v}$) with $\varphi_{\rm v}=0$. In Equation (9), the fisrt two terms in the right-hand side in the square bracket is the arrival time of photons for an observer being in the direction of ($\theta_{i}$, $\varphi_{l}$). Then, if the observer being in the direction of ($\theta_{\rm v}$, $\varphi_{\rm v}$) with $\varphi_{\rm v}=0$, the last term should be added. For a given observer time $t_{\rm obs}$, one can obtain the corresponding value of $R=R_{\rm obs}(\theta,\varphi,\theta_{\rm v})$ based on Equation (9). The value of $R_{\rm obs}$ is the location of the external-forward shock for the ($\theta$, $\varphi$)-jet observed at $t_{\rm obs}$ and is used to calculate the observed flux from the ($\theta$, $\varphi$)-jet.

Evolution of the electron energy spectrum: In the X-ray and optical bands, the main radiation mechanism of the external-forward shock in GRBs is the synchrotron radiation of the sweep-up electrons (Sari et al., 1998; Sari & Piran, 1999a). We denote the instantaneous electron spectrum for the sweep-up electrons per solid angle in the ($\theta$, $\varphi$)-jet as $n^{\prime}_{e}({\gamma^{\prime}_{\rm e}},\theta,R)$, where $n^{\prime}_{e}d\gamma^{\prime}_{\rm e}$ is the number of electrons in $[\gamma^{\prime}_{\rm e},\gamma^{\prime}_{\rm e}+d\gamma^{\prime}_{\rm e}]$ with $\gamma^{\prime}_{\rm e}$ being the Lorentz factor of electrons. The evolution of $n^{\prime}_{\rm e}$ can be described as

where $\dot{\gamma}^{\prime}_{\rm e}$ is the cooling rate of electrons and $Q^{\prime}$ is the injection rate of electrons from the shock, i.e.,

Here, $p\,(>2)$ is the power-law index, ${\gamma^{\prime}_{\rm e,{\rm{max}}}}=\sqrt{9m_{\rm e}^{2}{c^{4}}/(8B^{\prime}{q_{e}^{3}})}$ with $q_{e}$ being the electron charge (e.g., Kumar et al., 2012), $Q^{\prime}_{0}$ is obtained by solving $\int_{{\gamma^{\prime}_{\rm e,m}}}^{{\gamma^{\prime}_{\rm e,\max}}}Q^{\prime}d{\gamma^{\prime}_{\rm e}}=({R^{2}}\rho/{m_{\rm{p}}})dR/dt^{\prime}$ with $dR/dt^{\prime}=c\beta\Gamma$, and $t^{\prime}(\theta,R)$ is the time elapsed in the comoving frame of the blast wave. The first and second terms in the right-hand side of Equation (11) are respectively the synchrotron radiative cooling and adiabatic cooling of electrons, and the inverse-Compton cooling is not considered here. We solve Equation (10) and $dR/dt^{\prime}=c\beta\Gamma$ for $n^{\prime}_{\rm e}({\gamma^{\prime}_{\rm e}},\theta,R)$ at different $R$. In our calculations, the fourth-order Runge-Kutta method is used. In addition, an appropriate time step $\Delta{t^{\prime}}<\min\left\{{\left|{\Delta{\gamma^{\prime}_{\rm e}}/{{\dot{\gamma}}_{\rm e}}}\right|}\right\}$ is adopted in our calculations, where ${\Delta\gamma^{\prime}_{\rm e}}$ is the width of our adopted energy grids for electrons (e.g., see appendix A of Geng et al., 2018).

Observed flux calculation: With the obtained $n^{\prime}_{\rm e}(\gamma^{\prime}_{\rm e},\theta,R)$, the spectral power of synchrotron radiation at a given frequency $\nu^{\prime}$ can be described as

where $F(x)=x\int_{x}^{+\infty}K_{5/3}(k)dk$ with $K_{5/3}(k)$ being the modified Bessel function of 5/3 order and $\nu^{\prime}_{\rm syn}=3q_{\rm e}B^{\prime}(\theta,R){\gamma^{\prime}_{\rm e}}^{2}/(4\pi m_{\rm e}c)$. Then, the observed flux density $\hat{f}_{\nu}(\theta,R)$ from the per solid angle of the jet flow in the direction of ($\theta$, $\varphi$) is

where $\nu$ is the observed photon frequency, $d_{\rm L}$ is the luminosity distance at the cosmological redshift $z$, and $D(\theta,\varphi,\theta_{\rm v},R)=1/[\Gamma(1-\beta\cos\Theta)]$ is the Doppler factor of the ($\theta$, $\varphi$)-jet relative to the observer. Then, the observed total flux density $f_{\nu}$ at $t_{\rm obs}$ is

where $\theta_{i}=(i-0.5)\delta\theta$ and $\varphi_{l}=(l-0.5)\delta\varphi$. The observed flux $F_{\rm XRT}$ in the X-ray band ($0.3-10$ keV) of the X-ray telescope (XRT) aboard Swift satellite is calculated with ${F_{{\rm{XRT}}}}=\int_{{\nu_{{}_{0.3}}}}^{{\nu_{{}_{10}}}}{{f_{\nu}}({t_{{\rm{obs}}}})d\nu}$, where $h\nu_{{}_{10}}=10$ keV and $h\nu_{{}_{0.3}}=0.3$ keV.

In this paper, we mainly focused on the afterglows of GRBs in a free-to-shocked wind circum-burst environment. In Figure 1, we shows the light-curves (solid lines) of the radiation from the external-forward shock propagating in a free-to-shocked wind circum-burst environment, where $E_{\rm k,iso,on}=10^{52}$ erg, $\Gamma_{0}=250$, $\theta_{\rm c}=5^{\circ}$, $\theta_{\rm jet}=8\theta_{\rm c}$, $p=2.2$, $\epsilon_{e}=0.1$, $\epsilon_{B}=10^{-4}$, $A_{*}=0.01$, $R_{\rm{tr}}=10^{17}\rm{cm}$, and $z=1$ are adopted to calculate the dynamics and emission of the external-forward shock²²2 In general, the value of $A_{*}\sim 1$ is assumed. However, the value of $A_{*}<1$ is generally reported in fitting the afterglows of GRBs, e.g., $A_{*}\backsimeq 0.005-0.032$ for GRB 050319 (Kamble et al., 2007), $A_{*}\backsimeq 0.02$ for GRB 081109A (Jin et al., 2009b), $A_{*}\backsimeq 0.2$ for GRB 160625B(Fraija et al., 2017), and $A_{*}\backsimeq 0.06$ for GRB 190114C (Fraija et al., 2019). Since the value of $A_{*}\sim 0.01$ is obtained from these fittings, $A_{*}\sim 0.01$ is adopted to discuss the theoretical light-curves for an external-forward shock propagating in a free-to-shocked wind circum-burst environment.. One can find that the XRT/optical light-curves (solid lines) from a low viewing angle (e.g., $\theta_{\rm v}\lesssim\theta_{\rm c}$) can be decomposed into three phases: \footnotesize{i}⃝ the free-wind-phase, which appears in the early stage and is formed during the external-forward shock propagating in the free-wind environment; \footnotesize{ii}⃝ the shocked-wind-phase, which appears in the middle section bridging the free-wind-phase to the post-jet-break-phase and is formed during the external-forward shock propagating in the uniform shocked-wind environment; \footnotesize{iii}⃝ the post-jet-break-phase, which appears in the late stage as a steep decay by reflecting the situation that the jet core is fully visible for the observer, i.e., $\Gamma(\theta_{\rm c},R)\cdot\sin(\theta_{\rm c}+\theta_{\rm v})\lesssim 1$ ³³3For a low value of $\theta_{\rm c}+\theta_{\rm v}$, $\Gamma_{\rm c}\cdot\sin(\theta_{\rm c}+\theta_{\rm v})$ is reduced to $\Gamma_{\rm c}\cdot(\theta_{\rm c}+\theta_{\rm v})$, which is generally used in discussing the jet-break behavior. For a high value of $\theta_{\rm c}+\theta_{\rm v}$, however, the jet-break appears in the situation with $\Gamma_{\rm c}\cdot\sin(\theta_{\rm c}+\theta_{\rm v})\sim 1$ rather than $\Gamma_{\rm c}\cdot(\theta_{\rm c}+\theta_{\rm v})\sim 1$. . Here, $\Gamma(\theta_{\rm c},R)$ is the bulk Lorentz factor of the jet flow in the direction with $\theta=\theta_{\rm c}$, $\theta_{\rm c}$ is the characteristic angle of $E_{\rm k,iso}$, and the direction of ($\theta_{\rm v}$, $\varphi_{\rm v}$) with $\varphi_{\rm v}=0$ is the line of sight.

Figure 1 reveals that the transition from a free-wind to a homogeneous shocked-wind generally witnesses the flatting/brightening of the light-curves. By increasing the viewing angle, the changes in light-curves of Figure 1 (solid lines) can be summarized as follows. \footnotesize{1}⃝ Firstly, the free-wind-phase and the shocked-wind-phase become shallow while the post-jet-break phase remains its morphology. \footnotesize{2}⃝ At a certain viewing angle, e.g., $\theta_{\rm v}\sim 4\theta_{\rm c}$, the late simultaneous plateaux in the optical and X-ray afterglows appear. It should be pointed out that these plateaux are all directly followed by a post-jet-break-phase, which is similar to the late plateau found in the afterglows of GRBs 120729A, 051111, and 070318 (Huang et al., 2018). \footnotesize{3}⃝ Further increasing the viewing angle, the free-wind-phase turns into a rising pattern. Meanwhile, the free-wind-phase and post-jet-break-phase remain their decaying morphology. Then, the late simultaneous bumps in the optical and X-ray afterglows appear. It should be pointed out that these late bumps are directly followed by a post-jet-break-phase and preceded by a shallow decay, which is very similar to the late simultaneous bumps found in some bursts, e.g., GRBs 100418A, 100901A, 120326A, and 120404A. The behaviors of \footnotesize{2}⃝ and \footnotesize{3}⃝, i.e., the late simultaneous bumps/plateaux followed by a post-jet-break-phase in the optical and X-ray afterglows, are the key findings of this work. In Figure 2, we also shows X-ray afterglows (solid lines) of the external-forward shock in a free-to-shocked wind environment for a jet with $\theta_{\rm jet}=\theta_{\rm c}$, where the values of the other parameters are the same as those in Figure 1. One can find that the free-wind-phase in Figure 2 is shallower than that in Figure 1 for the light-curves from large viewing angles. Then, the opening angle of the jet may affect the light-curves before the late bump/plateau. It is worthy to point out that the only difference in producing the light-curves of Figure 1 or 2 is the viewing angle $\theta_{\rm v}$. Thus, the appearance of the late simultaneous bumps/plateaux in Figure 1 or 2 is related to the differences in the viewing angle rather than in the dynamics of the external shock.

For the structured jet adopted in Figures 1 and 2, the deceleration radius of the on-axis jet flow is in the free-wind medium, where the deceleration radius of the ($0$, $0$)-jet is $1.17\times 10^{14}\rm cm$ based on Equation (A10). Then, there is no broad late bumps in the afterglows for an on-axis observer. In fact, the late simultaneous bumps/plateaux appear only in the cases with high viewing angle for the free-to-shocked-wind-afterglows of Figure 1 or 2. We would like to point out that if the deceleration radius of the on-axis jet flow lies beyond or at around the free-wind boundary, a broad bump preceded by a long plateau or shallow decay can appear in the afterglows even in the cases with low viewing angle. In Figure 3, we shows the light-curves of the radiation from the external-forward shock propagating in a free-to-shocked wind environment, where a structured jet with $E_{\rm k,iso,on}=10^{55}$erg, $\Gamma_{0}=100$, $\theta_{\rm jet}=8\theta_{\rm c}$, $p=2.2$, $\epsilon_{e}=0.1$, $\epsilon_{B}=10^{-4}$, $A_{*}=0.01$, $R_{\rm{tr}}=10^{17}\rm{cm}$, and $\theta_{\rm c}=5^{\circ}$ (top panel) or $\theta_{\rm c}=0.5^{\circ}$ (bottom panel) is adopted. For the given structured jet in Figure 3, the deceleration radius of the ($0$, $0$)-jet in the free-wind medium is $7.33\times 10^{17}\rm cm$, which is significantly larger than $R_{\rm tr}$. Then, the deceleration radius of the on-axis jet flow would lie beyond the free-wind boundary. In Figure 3, late broad bumps preceded by a long plateau or shallow decay indeed appear in the afterglows for the cases with low viewing angle. In addition, the late broad bump is followed by a normal/steep decay if a structured jet with high/low characteristic angle $\theta_{\rm c}$ (e.g, $\theta_{\rm c}=5^{\circ}$/$\theta_{\rm c}=0.5^{\circ}$) is adopted.

Now, we can conclude that the late simultaneous broad bumps directly followed by a steep decay and preceded by a long plateau or shallow decay may sound the external-forward shock propagating in a free-to-shocked wind environment. In addition, one of the following conditions should be satisfied: \footnotesize{1}⃝ the observer should have a high viewing angle with respect to the jet axis; \footnotesize{2}⃝ the structured jet should have a low characteristic angle and the deceleration radius of the on-axis jet flow is at around or beyond the free-wind boundary. The reason for the appearance of the broad bumps in condition \footnotesize{2}⃝ or Figure 3 is clear. However, the reason for the appearance of the bumps/plateaux in condition \footnotesize{1}⃝ or Figures 1 and 2 is unclear. Then, the following subsections, i.e., Subsection 3.1-3.3, are dedicated to discuss the late simultaneous bumps/plateau in Figures 1 and 2.

As examples, we perform the fitting on the X-ray and optical afterglows of GRBs 120326A, 120404A, and 100814A. The light-curves from prompt emission to the late afterglow of these bursts are shown in Figure 5, where the late bumps directly followed by a post-jet-break-phase and preceded by a plateau or shallow decay can be easy found in these light-curves.

• •

  GRBs 120326A is a long GRB at redshift $z=1.798$ with unusual X-ray and optical afterglows. The late simultaneous bumps followed by a post-jet-break-phase in the optical and X-ray afterglows can be easily found at around $t_{\rm obs}\sim 4\times 10^{4}$ s. Some authors proposed that the energy injection model may be responsible to this late bump (e.g., Melandri et al., 2014; Hou et al., 2014; Laskar et al., 2015). Since the observed bumps is very similar to those found in Figures 1-3, we would like to model the late simultaneous bumps based on an off-axis observed external-forward shock in a free-to-shocked wind environment. The XRT data at $t_{\rm obs}\gtrsim 3\times 10^{3}$ s and the optical data at $t_{\rm obs}\gtrsim 10^{2}$ s are used in our fitting.

• •

  GRB 120404A, with redshift $z=2.876$, is a long GRB with a significant rebrightening in the optical and near-infrared bands at $t_{\rm obs}\sim 2\times 10^{3}$ s. The X-ray observations around this time also show a bump although the data are sparse owing to the orbital gap of Swift (Guidorzi et al., 2014). Interestingly, the bumps are all directly followed by a post-jet-break-phase. Laskar et al. (2015) proposed that the energy injection model may be responsible to this late bump. However, we would like to model these two late simultaneous bumps based on an off-axis observed external-forward shock in a free-to-shocked wind environment since these bumps are similar to those in Figures 1-3. The XRT data at $t_{\rm obs}>550$ s and the optical data at $t_{\rm obs}\gtrsim 200$ s are used for our fitting.

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  GRB 100814A is a long GRB with the redshift $z=1.44$. The optical rebrightening appears at $10^{5}$ s. De Pasquale et al. (2015) presented a broadband observations of GRB 100814A and they attribute the late optical rebrightening to a long-lived external-reverse shock and external-forward shock. Besides, Geng et al. (2016) used an ultrarelativistic $e^{+}e^{-}$ wind injection model to explain the late rebrightening. It should be noted that the optical/X-ray afterglows all turn into the steep decay after the peak time of the optical bump. Then, we would like to believe that there may be a simultaneous late bumps in the optical and X-ray afterglows and the rise of the X-ray bump is covered by the late central engine activities. The XRT data at $t_{\rm obs}\gtrsim 9\times 10^{4}$ s and the optical data at $t_{\rm obs}\gtrsim 10^{3}$ s are also modelled as the emission from an off-axis observed external-forward shock in a free-to-shocked wind environment.

Our fitting is performed based on the Markov Chain Monte Carlo (MCMC) method to produce posterior predictions for the model parameters. MCMC method is widely used in finding a best set of parameters for a specified model, e.g., GRB 080413B (Geng et al., 2016); GRBs 100418A, 100901A, 120326A, and 120404A (Laskar et al., 2015). In our work, fitting the afterglows of GRBs 120326A, 120404A, and 100814A with the MCMC method is to test whether or not the late simultaneous bumps followed by a post-jet-break-phase can be explained with an off-axis observed external-forward shock in a free-to-shocked wind environment. The posterior probability density functions for the physical parameters, i.e., $E_{\rm k,iso,on}$, $\Gamma_{0}$, $\theta_{\rm c}$, $\theta_{\rm jet}/\theta_{\rm c}$, $\theta_{\rm v}/\theta_{\rm c}$, $p$, $\epsilon_{e}$, $\epsilon_{B}$, $A_{*}$, and $R_{\rm tr}$, are presented in Figure 8, where only the fitting result of GRB 120326A is shown as an example. The optimal result from MCMC fitting is shown in Figure 7 with blue line (XRT) and red line (optical), and the obtained parameters at the $1\sigma$ confidence level are reported in Table 1, where the values of the transition radius $R_{\rm{tr}}$ (i.e., $1.05\times 10^{17}$cm, $6.31\times 10^{16}$cm, and $3.98\times 10^{17}$cm for GRBs 120326A, 120404A and 100814A, respectively) are consistent with those found in other bursts (e.g., Kong et al., 2010; Feng & Dai, 2011a; Ramirez-Ruiz et al., 2001; Li et al., 2020). It can be found that both the X-ray afterglow and the optical afterglow of these bumps can be well modelled with an off-axis observed external-forward shock in a free-to-shocked wind environment.

1. 1.

   The fitting results reported in Table 1 reveals that the on-axis observed isotropic kinetic energy $E_{\rm k,iso,on}$ of these bursts (i.e., $\sim 5\times 10^{54}\rm erg$, $6\times 10^{53}\rm erg$, and $8\times 10^{54}\rm erg$ for GRBs 120326A, 120404A, and 100814A, respectively) are significantly high but consistent with those found in other bursts (see the figure 10 in Racusin et al., 2011 and the table 1 in Yi et al., 2017). The total kinetic energy of the structured jet is also estimated and shown in the last row of Table 1. Compared with the beaming corrected kinetic energy reported in the table 2 of Yi et al. (2017), the total kinetic energy of our obtained structured jet is also consistent with those found in other bursts. However, it should be noted that the on-axis observed luminosity of GRBs 120326A and 100814A may be significantly high since the value of $\theta_{\rm v}\sim 4\theta_{\rm c}$ is obtained from our MCMC fittings. According to the structure of the jet, the on-axis observed luminosity of these two bursts may be around $10^{54}\rm erg\cdot s^{-1}$, which lies in the high-luminosity range of GRBs. Here, the time-averaged luminosity of GRBs 120326A and 100814A are estimated to be around $\sim 2\times 10^{51}\rm erg\cdot s^{-1}$ (Melandri et al., 2014) and $10^{51}\rm erg\cdot s^{-1}$ (Nardini et al., 2014), respectively.

2. 2.

   The theoretical light-curves does not well fit the transition behavior from the wind-phase to the ISM-like phase in the afterglows of GRB 120326A. It may imply that the density jump factor $\xi$ from the wind medium to the ISM-like medium may be less than 4, i.e., $\xi<4$. Figure 1 reveals that the external-forward shock propagating in a shocked-wind environment can yield a plateau before the late simultaneous bumps in the case with $\theta_{\rm v}=4\theta_{\rm c}$. Then, we try to fit the afterglows of GRB 120326A with the emission of the external-forward shock propagating in a homogeneous medium, where the priors of $\log_{10}(E_{\rm k,iso,on}/\rm erg)$, $\log_{10}\Gamma_{0}$, $\theta_{\rm c}$, $\theta_{\rm jet}/\theta_{\rm c}$, $\theta_{\rm v}/\theta_{\rm c}$, $p$, $\log_{10}\epsilon_{e}$, $\log_{10}\epsilon_{B}$, and $\log_{10}n_{0}$ are set as uniform distribution in the range of (52, 55), (1.5, 3.0), (0.3, 8.0), (2.5, 4.5), (0.0, 8.0), (2.1, 2.9), (-3.2, -0.5), (-6.5, -2.0), and (0.0, 1.7) in our MCMC fittings, respectively. However, no well fitting result is found. The fitting result with minimum reduced $\chi^{2}$ is shown with dashed lines in the top panel of Figure 7. It can be found that the early plateau of GRB 120326A could not be well fitted based on the emission of the external-forward shock in a homogeneous medium.

3. 3.

   There seems to be a plateau at $t_{\rm obs}\sim 600$ s in the optical afterglow of GRB 100814A, which may indicate an energy injection into the external shock. If so, our model could not well describe the afterglows of GRB 100814A in its early phase. This may be the reason for the deviation of our theoretical results relative to the observations in the early phase. In addition, the X-ray and optical afterglows of GRB 100814A show a chromatic behavior, which could not be explained only with our scenario (i.e., an off-axis observed structured jet propagating into a free-to-shocked wind environment) since the appearance of the late simultaneous bump/plateau in our scenario is quasi-achromatic. Other emission rather than that of the external-forward shock may contribute to the long X-ray shallow decay.

4. 4.

   For the afterglows of GRB 120404, the fitting result shows that the line of sight lies within the jet core, i.e., $\theta_{\rm v}=0.68\theta_{\rm c}$. However, the late simultaneous bumps do not appear in the free-to-shocked-wind-afterglows of Figures 1 and 2 for $\theta_{\rm v}<\theta_{\rm c}$. Then, the appearance of the bumps in GRB 120404 should be related to the condition \footnotesize{2}⃝, i.e., the structured jet has a low characteristic angle and the deceleration radius of the on-axis jet flow is at around or beyond the free-wind boundary. Based on the fitting result of this burst, the deceleration radius of the ($0$, $0$)-jet can be estimated and is $2.43\times 10^{16}$ cm, which is at around the transition radius $R_{\rm tr}=6.31\times 10^{16}$ cm of this burst. Then, the fitting result is consistent with the condition \footnotesize{2}⃝.

In this paper, the emission of the external-forward shock in a free-to-shocked wind circum-burst environment is studied. We mainly focus on the light-curves of the afterglows in the late phase. A late simultaneous broad bumps/plateaux followed by a post-jet-break-phase and preceded by a plateau or shallow decay is found in the obtained afterglows with one of the following conditions: \footnotesize{1}⃝ the observer should have a high viewing angle with respect to the jet axis; \footnotesize{2}⃝ the structured jet should have a low characteristic angle and the deceleration radius of the on-axis jet flow is at around or beyond the free-wind boundary. Our obtained bump/plateau in the afterglows is similar to those found in the afterglows of some bursts, e.g., the late bump found in GRBs 100418A, 100901A, 120326A, 120404A (see Laskar et al., 2015), 080413B, 091029, 100814A, and 090426 (see Geng et al., 2016), and the late plateau found in GRBs 120729A, 051111, and 070318 (see Huang et al., 2018). As examples, we perform the fitting on the optical and X-ray afterglows of GRB 100814A, GRB 120326A and GRB 120404A based on the Markov Chain Monte Carlo method. It is found that the external-forward shock propagating in a free-to-shocked wind circum-burst environment for an off-axis observer can well explain the late phase of the optical and X-ray afterglows in these bursts. Since our obtained bumps/plateaux appear simultaneous in the optical and X-ray afterglows, our scenario could not fully account for chromatic bumps/plateaux, e.g., GRB 100814A. Then, we suggest that a late broad bump/plateau followed by a post-jet-break-phase and preceded by a plateau or shallow decay may sound the transition of the circum-burst environment from a free-wind to a homogeneous shocked-wind medium.

In this work, a Gaussian structured jet is adopted to fit the late phase of afterglows in GRBs 100814A, 120326A and 120404A. However, we would like to point out that our conclusion about the circum-burst environment (i.e., the free-to-shocked wind environment) in these bursts may be robust. The reasons are shown as follows. Firstly, the late simultaneous bumps found in these bursts are very similar to those in GRB 170817A, which is formed in an off-axis observed external-forward shock propagating in an interstellar medium (Troja et al., 2018; Lamb et al., 2019; Troja et al., 2019; Huang et al., 2019; Ren et al., 2020). This implies that the late bumps in GRBs 100814A, 120326A, and 120404A may be formed in an off-axis observed external-forward shock in a homogeneous medium. Interestingly, there seems to be no plateau/shallow decay in the early phase of GRB 170817A’s afterglows (see the figure 1 of Ren et al., 2020, especially the X-ray afterglow in $t_{\rm obs}\sim 2-100$ day). This behavior may also appear in the afterglows of GRBs 120326A and 120404A if their late simultaneous bumps appear with the same mechanism as that in GRB 170817A. In this paradigm, the plateau/shallow decay preceding the bumps in these two bursts may require an another explanation and the external-forward shock in the free-wind environment may be a natural candidate, especially for long GRBs. Secondly, the external-forward shock with other kinds of structured jet are also studied. For example, Huang et al. (2019) has studied the jet structure owing to the jet precession and the corresponding afterglows in a homogeneous environment. Very complex structured jets have been found. Granot et al. (2018) studies the afterglows from 2D relativistic hydrodynamic simulations of a GRB jet propagating in a homogeneous medium. It seems to be the case that the late simultaneous bumps generally appear without preceding by a plateau/shallow decay in the afterglows for a structured jet propagating in a homogeneous medium (e.g., the figures 3 and 5 of Huang et al., 2019, the figures 6 and 7 of Granot et al., 2018). However, our work could not rule out the scenario that other kinds of structured jet propagating only in a homogeneous medium could explain the observed late simultaneous bumps directly followed by a steep decay and preceded by a plateau/shallow decay in these burst. Our scenario, i.e., an off-axis observed structured jet propagating into a free-to-shocked wind environment, only provides a possible way to decipher these bumps.