GRB 210121A: Observation of Photospheric Emissions from Different Regimes and the Evolution of the Outflow

Despite observations for about a half century, prompt gamma-ray burst (GRB) emission mechanisms are still a matter of interest. Two leading scenarios have been suggested to interpret the observed spectra of GRBs. One is synchrotron radiation, which invokes nonthermal emission of relativistic charged particles either from internal shocks or from internal magnetic dissipation processes (Lloyd & Petrosian 2000; Tavani et al. 2000; Baring & Braby 2004; Zhang 2018). The fast-cooling problem in relevance to the low-energy photon index α could be resolved by synchrotron radiation of electrons in a moderately fast-cooling regime (Uhm & Zhang 2014) or by a detailed treatment of the cooling electrons (Derishev et al. 2001; Geng et al. 2018). As a natural consequence of the fireball model, photospheric emission produced from highly relativistic outflows was previously considered as an explanation for prompt GRBs (Goodman 1986; Paczynski 1986), where the optical depth at the base of the outflow is much higher than unity (Piran 1999). The Planck spectrum related to the photospheric emission could be broadened in two ways. First, dissipation below the photosphere can heat electrons above the equilibrium temperature. These electrons emit synchrotron emission and Comptonize the thermal photons, thereby modifying the shape of the Planck spectrum (Pe'er et al. 2005, 2006; Rees & Mészáros 2005). Observational evidence for subphotospheric heating has been provided by Ryde et al. (2011). In addition, internal shocks below the photosphere (Rees & Mészáros 2005), magnetic reconnection (Thompson 1994; Giannios & Spruit 2005), and hadronic collision shocks (Beloborodov 2010; Vurm et al. 2011) can also cause dissipation. Second, the modification of the Planck spectrum could be caused by geometrical broadening. The photospheric radius is found to be a function of the angle to the line of sight of the photons of thermal emission observed (Abramowicz et al. 1991; Pe'er 2008). This means that the observed spectrum is a superposition of a series of blackbodies of different temperature, arising from different angles to the line of sight. Moreover, Pe'er (2008) showed that photons make their last scatterings at a distribution of radii and angles. The observer simultaneously sees photons emitted from a large range of radii and angles. Therefore, the observed spectrum is a superposition of comoving spectra (Ryde et al. 2010; Hou et al. 2018). Lundman et al. (2012) studied the nondissipative photospheric (NDP) emissions from a structured jet and reproduced the average low-energy photon index (α = −1) independent of viewing angle. The observed evolution patterns of the ν F_ν peak energy (E_p), including hard-to-soft and intensity tracking, could be reproduced by this model as well (e.g., Deng & Zhang 2014; Meng et al. 2019).

Three regimes are to be discussed in photospheric emission from a structured jet: (I) unsaturated emissions, which are dominant in the regime of unsaturated acceleration (the saturation radius is greater than the photonsphere radius, R_s > R_ph), (II) saturated emissions from the regime of R_s ≤ R_ph works throughout the wind profile, and (III) the intermediate photosphere (Song & Meng 2022), which represents the case where the regimes of R_s > R_ph and R_s ≤ R_ph work in lower and higher latitudes, respectively, and the contribution from the latter cannot be ignored. This has been never mentioned in previous studies in the spectral fit to photospheric emissions. In addition to the off-axis NDP model in the unsaturated regime (Wang et al. 2021), in this paper, we find that the on-axis NDP model that considers the intermediate photosphere is an alternative description for the emissions of GRB 210121A.

The evolution of the outflow is also extracted, to offer an interpretation of the mechanism that launches the jet. If we take the hyper-accreting black hole (BH) as the central engine, the GRB jet may be launched through two mechanisms. One is $\nu \overline{\nu }$ annihilation in a neutrino-dominated accretion flow (NDAF; e.g., Popham et al. 1999; Narayan et al. 2001; Kohri & Mineshige 2002; Matteo et al. 2002; Gu et al. 2006; Chen & Beloborodov 2007; Janiuk et al. 2007; Lei et al. 2009; Liu et al. 2010). This generates a fireball that is dominated by the thermal component. The jet is launched by neutrino annihilation ($\nu \overline{\nu }\to {e}^{+}{e}^{-}$), and ${\dot{E}}_{\nu \overline{\nu }}$ is the neutrino annihilation power. The other mechanism is the Blandford-Znajek mechanism (BZ; Blandford & Znajek 1977). The spin energy of the BH is tapped by a magnetic field and produces a Poynting flux. The correlations between the baryon loading parameter and the power for these two mechanisms are both positive, $\eta \propto {\dot{E}}_{\nu \overline{\nu }}^{0.26}$ (Lü et al. 2012) for the former, and ${\mu }_{0}\propto {\dot{E}}_{\mathrm{BZ}}^{0.17}$ (Yi et al. 2017) (μ₀ = η(1 + σ₀), where σ₀ is the ratio of the Poynting flux luminosity to the matter flux) for the latter. Thus, the index of the correlation offers a criterion.

This paper is organized as follows. In Section 2, the observation of GRB 210121A by different missions is introduced. In Section 3, the NDP model and R_ph in different regimes are introduced, especially in the intermediate photosphere. In Section 4, the methods for binning, background estimation, and spectral fitting are clarified. In Section 5, time-resolved analyses are performed, and the properties of the jet are extracted, while the regimes are determined. Then the discussion and conclusion are given in Section 6.

GRB 210121A was observed by Insight-HXMT (GCN: Xue et al. 2021), by the Gravitational wave high-energy Electromagnetic Counterpart All-sky Monitor (GECAM; GCN: Peng et al. 2021), and by the Fermi Gamma-ray Burst Monitor (Fermi/GBM) on 2021 January 21. The GRB triggered Insight-HXMT at 2021-01-21T18:41:48.750 UTC and GECAM at 2021-01-21T18:41:48.800 UTC. The first trigger is taken to be T₀ in the following analysis. The photon flux of GRB 210121A is shown in Figure 1, which is extracted from the joint analysis used in Song et al. (2022).

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The cutoff power-law (CPL) model is preferred for almost all the time-resolved slices, as shown in Wang et al. (2021). The values of α in the first epoch obtained from time-resolved analysis are greater than −2/3, as are those of the most part of the second epoch from T₀+2.8 s to T₀+14.9 s. According to Meng et al. (2021), the NDP spectra from a hybrid outflow with a moderate magnetization have a larger high-energy photon index (β), and are more compatible with the BAND function than the CPL model. Therefore, we prefer a pure hot fireball (or the Poynting flux completely thermalized below the photosphere) in the prompt phase.

We cannot justify the radiation mechanism from α alone (Meng et al. 2018; Burgess et al. 2020), thus a physical model fitting is needed.

A structured jet is supported by observation (e.g., studies of GRB 170817A; Lazzati et al. 2017; Bromberg et al. 2018; Geng et al. 2019), and the relativistic magnetohydrodynamic simulations for the GRB jet are also performed (e.g., Kathirgamaraju et al. 2019). In this analysis, we assume a structured jet with an opening angle θ_c and luminosity L₀. Motivated by the results of MacFadyen & Woosley (1999) and Zhang et al. (2003), e.g., the jet is structured with a constant inner and decreasing outer angular baryon loading parameter profile with the form for the GRB prompt emission phase (Dai & Gou 2001; Rossi et al. 2002; Zhang & Mészáros 2002; Kumar & Granot 2003),

$$\begin{eqnarray}&&{\left(\eta (\theta )-{\eta }_{\min }\right)}^{2}=\displaystyle \frac{{\left({\eta }_{0}-{\eta }_{\min }\right)}^{2}}{{\left(\theta /{\theta }_{c}\right)}^{2p}+1},\end{eqnarray} \tag{ 1 }$$

where η is the angle-dependent baryon loading parameter, which is also the bulk Lorentz factor Γ in the saturated acceleration regime; η₀ is the maximum η and is also denoted as Γ₀; θ is the angle measured from the jet axis; θ_c is the half-opening angle for the jet core; p is the power-law index of the profile; and ${\eta }_{\min }=1.2$ is the minimum value of η, which differs from unity for numerical reasons. The exact value of ${\eta }_{\min }$ only affects the very low energy spectrum, many orders of magnitude below the observed peak energy (Lundman et al. 2012). This baryon loading parameter profile was in Lundman et al. (2012), Meng et al. (2018), and Meng et al. (2019).

The flux of the observed energy E_obs at the observer time t in the case of the continuous wind is deduced and explained in Section A.1. The angle-dependent photosphere radius R_ph, as the radius from which the optical depth for a photon that propagates in the radial direction is equal to unity, is defined as

$$\begin{eqnarray}&&{R}_{\mathrm{ph}}=\left\{\begin{array}{l}{\left(\displaystyle \frac{{\sigma }_{{\rm{T}}}}{6{m}_{{\rm{p}}}c}\displaystyle \frac{d\dot{M}}{d{\rm{\Omega }}}{r}_{0}^{2}\right)}^{1/3},\ {R}_{\mathrm{ph}}\ll {R}_{{\rm{s}}},\\ {\left(\displaystyle \frac{{\sigma }_{{\rm{T}}}}{2{m}_{{\rm{p}}}c}\displaystyle \frac{d\dot{M}}{d{\rm{\Omega }}}{r}_{0}^{2}\right)}^{1/3},\ {R}_{\mathrm{ph}}\lesssim {R}_{{\rm{s}}},\\ \displaystyle \frac{1}{(1+\beta )\beta {\eta }^{2}}\displaystyle \frac{{\sigma }_{{\rm{T}}}}{{m}_{{\rm{p}}}c}\displaystyle \frac{d\dot{M}}{d{\rm{\Omega }}},\ {R}_{\mathrm{ph}}\geqslant {R}_{{\rm{s}}},\end{array}\right.\end{eqnarray} \tag{ 2 }$$

where β is the velocity, r₀ is the radius of the central engine, and $d\dot{M}(\theta )/d{\rm{\Omega }}={L}_{0}/4\pi {c}^{2}\eta (\theta )$ is the angle-dependent mass outflow rate per solid angle; m_p is the mass of the proton, c is the light speed, and σ_T is electron Thomson cross section. The unsaturated regime used in Wang et al. (2021) is R_ph ≪ R_s. In this paper, an unsaturated emission of R_ph ≲ R_s is considered and added between the case of R_ph ≪ R_s and R_ph ≥ R_s (Song & Meng 2022).

R_s = η(θ)r₀ monotonically decreases with θ, while R_ph monotonically increases with θ. Thus, there exists a critical value θ_cri, and for θ ≥ θ_cri, it satisfies R_s(θ) ≤ R_ph(θ), namely,

$$\begin{eqnarray}&&\eta (\theta )\leqslant {\left(\displaystyle \frac{{\sigma }_{{\rm{T}}}}{2{m}_{{\rm{p}}}c}\displaystyle \frac{{L}_{0}}{4\pi {c}^{2}{r}_{0}}\right)}^{1/4}.\end{eqnarray} \tag{ 3 }$$

Note that for the intermediate photosphere, θ_cri satisfies 0 < θ_cri < 5/Γ₀, and the contribution from the saturated regime contributes to the prompt emission (Lundman et al. 2012), or the unsaturated emission is dominant. For an intermediate photosphere, R_ph is described by two forms: R_ph in the lower latitude of the unsaturated part is described by the second item of Equation (2), while that in the higher latitude takes the saturated form. If for a set of parameters, θ_cri is found to be 0 by Equation (3), the saturated regime works for throughout the jet profile. In our analysis, the regime could therefore be determined by the parameters obtained from the fitting to spectra.

4.1. Binning Method of Light Curves for Time-resolved Spectra

4.2. Background Estimation and Spectrum Fitting Method

In this section, time-averaged spectral fitting for two epochs as well as time-resolved analyses for fine bins are performed with the NDP model. The properties of the jet and the evolution of the outflow are extracted.

5.1. Time-averaged Results of Two Epochs

Table 1 and Figure 2 show the fit results, MCMC samples, and spectra for the two epochs. It is found that θ_c is at 0.01 with small uncertainties for both epochs. The values of r₀ are well consistent with 10^7.61 cm. p becomes larger in epoch 2 than in epoch 1. The luminosity of the outflow changes with time, and becomes lower in epoch 2 than in epoch 1, which has an almost similar trend as the flux. For the first epoch, χ² = 248.2 and BIC = 262.14 with a degree of freedom (dof) of 203, and χ² = 360.0 and BIC = 375.15 with a dof of 323 in the second epoch. We also performed the fit with an on-axis NDP model dominant in the regime of R_ph ≪ R_s. However, the fit is not good, with a BIC equal to 904.85 and 2014.26 for the two epochs with the same dofs. Thus, for an on-axis observation, the NDP model in the regime of R_ph ≪ R_s is not preferred and is inconsistent with the spectra.

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Table 1. Fit Results with the On-axis NDP Model of Epochs 1 and 2

Time Bins(s)	log (r0( cm))	η0	p	θc	log (L0(erg s−1))	z	BIC	$\tfrac{{\chi }^{2}}{{dof}}$	θcri(10−3)
epoch1:[-0.01,2.19]	7.43${}_{-0.44}^{+0.43}$	268.1${}_{-34.6}^{+66.8}$	1.07${}_{-0.43}^{+1.44}$	0.010${}_{-0.003}^{+0.003}$	50.62${}_{-0.26}^{+0.42}$	0.37${}_{-0.09}^{+0.06}$	262.14	$\tfrac{248.2}{203}$	0
bin 0:[-0.01,0.43]	⋯	262.1${}_{-10.2}^{+10.2}$	⋯	⋯	50.42${}_{-0.02}^{+0.02}$	⋯	240.60	$\tfrac{236.0}{196}$	2.4
bin 1:[0.43,0.87]	⋯	341.8${}_{-23.2}^{+34.8}$	⋯	⋯	50.65${}_{-0.02}^{+0.02}$	⋯	258.76	$\tfrac{254.2}{196}$	6.0
bin 2:[0.87,1.31]	⋯	280.2${}_{-4.7}^{+13.4}$	⋯	⋯	50.57${}_{-0.01}^{+0.03}$	⋯	314.46	$\tfrac{309.9}{196}$	1.7
bin 3:[1.31,1.75]	⋯	222.3${}_{-3.4}^{+3.4}$	⋯	⋯	50.53${}_{-0.02}^{+0.01}$	⋯	372.02	$\tfrac{367.4}{196}$	0
bin 4:[1.75,2.19]	⋯	198.4${}_{-3.0}^{+6.0}$	⋯	⋯	50.46${}_{-0.02}^{+0.02}$	⋯	251.48	$\tfrac{246.9}{196}$	0
epoch 2:[2.8-14.9]	7.61${}_{-0.31}^{+0.21}$	184.7${}_{-33.75}^{+44.17}$	1.27${}_{-0.25}^{+0.17}$	0.010${}_{-0.003}^{+0.003}$	50.15${}_{-0.50}^{+0.35}$	0.34${}_{-0.10}^{+0.11}$	375.15	$\tfrac{360.0}{323}$	0
bin 5:[2.80,3.70]	⋯	211.2${}_{-6.2}^{+3.1}$	⋯	⋯	50.35${}_{-0.01}^{+0.01}$	⋯	336.59	$\tfrac{331.6}{316}$	0
bin 6:[3.70,4.60]	⋯	201.6${}_{-6.1}^{+3.0}$	⋯	⋯	50.37${}_{-0.02}^{+0.02}$	⋯	331.57	$\tfrac{326.6}{316}$	0
bin 7:[4.60,5.50]	⋯	194.5${}_{-4.1}^{+8.1}$	⋯	⋯	50.06${}_{-0.02}^{+0.02}$	⋯	385.30	$\tfrac{380.3}{316}$	0
bin 8:[5.50,6.40]	⋯	176.7${}_{-8.1}^{+9.6}$	⋯	⋯	50.01${}_{-0.02}^{+0.03}$	⋯	431.45	$\tfrac{426.4}{316}$	0
bin 9:[6.40,7.30]	⋯	167.4${}_{-5.7}^{+2.9}$	⋯	⋯	50.04${}_{-0.03}^{+0.03}$	⋯	469.49	$\tfrac{464.5}{316}$	0
bin 10:[7.30,8.20]	⋯	191.2${}_{-14.2}^{+9.5}$	⋯	⋯	49.97${}_{-0.03}^{+0.02}$	⋯	441.54	$\tfrac{436.5}{316}$	0
bin 11:[10.90,11.80]	⋯	188.7${}_{-14.0}^{+14.2}$	⋯	⋯	49.80${}_{-0.02}^{+0.02}$	⋯	475.14	$\tfrac{470.1}{316}$	3.0
bin 12:[11.80,12.70]	⋯	141.1${}_{-4.9}^{+2.5}$	⋯	⋯	49.81${}_{-0.05}^{+0.03}$	⋯	374.59	$\tfrac{369.6}{316}$	0
bin 13:[12.70,13.60]	⋯	153.3${}_{-10.2}^{+5.9}$	⋯	⋯	49.78${}_{-0.03}^{+0.03}$	⋯	366.11	$\tfrac{361.1}{316}$	0
bin 14:[14.00,14.90]	⋯	147.7${}_{-10.7}^{+7.2}$	⋯	⋯	49.69${}_{-0.03}^{+0.05}$	⋯	379.15	$\tfrac{374.1}{316}$	0

Note. In the time-resolved results from bin 0 to 14, a dash represents the fixed values in the fitting: p = 1.0 for bins 0–4 in the first epoch and p = 1.27 for bins 5–14 for the second epoch. For both epochs, logr₀ = 7.61, θ_c = 0.01, z = 0.37.

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These two results corresponds to θ_cri = 0, which means that the emission from the saturated regime is dominant. When changes in L₀ and η₀ during the epochs are considered, the emissions may be from the intermediate photosphere or from the unsaturated regime, which is determined in the fine time-resolved results.

5.2. Time-resolved Results for Fine Bins

Since the uncertainties of the fit results from the time-averaged spectra do not seem to be large, we could assume that r₀, p, and θ_c have small changes during each epoch and z stays constant for the whole GRB duration. Therefore, they could be fixed in the time-resolved fitting to suppress the uncertainty in extracting the correlation of Γ₀ − L₀. The time-resolved analysis is performed with free η₀ and L₀, while the other parameters are fixed to p = 1.0 for bins 0–4 in the first epoch and to p = 1.27 for bins 5–14 for the second epoch. For both epochs, logr₀ = 7.61, θ_c = 0.01, and z = 0.37. The fit results are listed in Table 1, where four bins have nonzero θ_cri. In all the bins, bin 1 has the largest η₀ and flux, and also the largest θ_cri. We can conclude that it receives a larger contribution from the unsaturated regime than the other bins. The relation of θ_cri and L₀ for each set of (r₀, p, θ_c) of bin 1 and bin 5 are shown in Figure 3 as dotted red lines. The values of the luminosity measured in the time-resolved analysis are represented by vertical dot–dashed blue lines. As shown in Figure 3(a), the corresponding θ_cri of bin 1, denoted as the red star, is lower than θ_c (denoted by the solid green line) as well as 5/Γ₀ (denoted by the dashed orange line), which means that the emission is from the intermediate photosphere. For comparison, the emission in other bins, e.g., in bin 5, is shown in Figure 3(b). The vertical blue line does not intersect the dotted red line that represents θ_cri, which means that the corresponding θ_cri is 0, and the emission is from the saturated regime. The time-resolved spectra are shown in Figure 4(a), where the luminosity and hardness generally decrease with time.

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The relations between η₀ and L₀ are shown in Figure 4. ${{\rm{\Gamma }}}_{0}\propto {L}_{0}^{0.25\pm 0.5}$ is extracted from the time-resolved results denoted by black dots. The time-averaged results for the two epochs represented by red squares are also plotted, and they are well consistent with this trend.

In a previous study, Wang et al. (2021) performed a fit on the first epoch in GRB 210121A with the NDP model in the unsaturated regime (R_ph ≪ R_s), and the observation is determined to be greatly off-axis with the extracted θ_v ≃ 15/Γ₀. In this paper, we use the NDP model and consider the intermediate photospheric emissions to describe two epochs of GRB 210121A. We give an alternative description with an on-axis observation. The obtained luminosity, log (L₀(erg ${{\rm{s}}}^{-1}))={50.62}_{-0.26}^{+0.42}$, is naturally lower than that of the off-axis, log (L₀( erg ${{\rm{s}}}^{-1}))={51.94}_{-0.50}^{+0.84}$ (this value is from Wang et al. 2021), and the uncertainties of this result are smaller as well. The extracted redshift in this work, $z\sim {0.37}_{-0.09}^{+0.06}$, is also consistent with the prediction of the photosphere death line, z ∼ [0.3, 3.0] (Zhang et al. 2012; Wang et al. 2021).

The intermediate photosphere always has a moderate α between these two regimes (Song & Meng 2022), where α is extracted from spectra below the peak energy with an exponential cutoff power law. This could explain that bin 1 has the largest $\alpha =-{0.22}_{-0.07}^{+0.07}$ (this value is from Wang et al. 2021), while the others have a smaller θ_cri and receive a larger contribution from the saturated regimes. According to Wang et al. (2021), it seems that the off-axis unsaturated NDP model could also give a smaller α than the on-axis unsaturated NDP model, therefore, there may be two solutions for the observation, and the on-axis NDP model that considers the intermediate photonsphere gives an alternative solution.

Liang et al. (2010) presented a correlation of ${{\rm{\Gamma }}}_{0}\propto {E}_{\mathrm{iso},\gamma }^{0.25}$. Lü et al. (2012) discovered an even tighter correlation ${{\rm{\Gamma }}}_{0}\propto {L}_{\mathrm{iso},\gamma }^{0.30}$ from 50 GRBs and proposed an interpretation. Considering the beaming factor (${f}_{{\rm{b}}}\propto {L}_{\mathrm{iso},\gamma }^{-0.145}$) and the similar efficiency of the jet kinetic energy via an internal shock in dissipation to prompt emissions of GRBs, it has ${{\rm{\Gamma }}}_{0}\propto {L}_{\mathrm{iso},\gamma }^{0.22}$. However, Yi et al. (2017) found that the data are more consistent with the latter mechanism, the index∼0.14. In this analysis, Γ₀ − L₀ of GRB210121A is obtained from the time-resolved analysis, and there are some advantages. First, we do not need to consider the efficiency of the prompt emission, and ${L}_{0}\propto \dot{E}$ ($\dot{E}$ is the power). Second, we obtain the baryon loading parameter directly from the fit result, and consider the regime in this procedure. The Lorentz factor of the outflow may be lower than the baryon loading parameter if the emission is from the unsaturated regime. ${\eta }_{0}\propto {L}_{0}^{0.25\pm 0.5}$ is extracted from a time-resolved analysis, and it is more consistent with the mechanism of $\nu \overline{\nu }$ annihilation.

In summary, the NDP model that considers the intermediate photosphere is first extracted in GRB 210121A, and it could well explain the changes in α during the pulses. The correlation of η₀ − Γ₀ is extracted from the time-resolved analysis and implies that the jet of GRB 210121A may stem mainly from the $\nu \overline{\nu }$ annihilation.

The authors thank for support from the National Program on Key Research and from the Development Project (2021YFA0718500). The authors are very grateful for the GRB data of Fermi/GBM, HXMT, and GECAM. We are very grateful for the comments and suggestions of the anonymous referees. X.-Y. S. thanks Yan-Zhi Meng for the suggestion on the algorithm of spectra fitting procedure and Wen-Xi Peng for the support during the work.

A.1. The Flux of Photospheric Emissions from Different Regimes

As discussed, for instance, in Lundman et al. (2012), Deng & Zhang (2014), and Meng et al. (2018), the flux of the observed energy E_obs at the observer time t in the case of a continuous wind could be shown as in Equation (A1),

$$\begin{eqnarray}\begin{array}{rcl}{F}_{E}^{\mathrm{obs}}({\theta }_{{\rm{v}}},{E}_{\mathrm{obs}},t) & = & \displaystyle \frac{1}{4\pi {d}_{{\rm{L}}}^{2}}\displaystyle \int \int (1+\beta ){D}^{2}\displaystyle \frac{d{\dot{N}}_{\gamma }}{d{\rm{\Omega }}}\times \displaystyle \frac{{R}_{\mathrm{ph}}}{{r}^{2}}\\ & & \times \,\exp \left(-\displaystyle \frac{{R}_{\mathrm{ph}}}{r}\right)\times \left\{E\displaystyle \frac{{dP}}{{dE}}\right\}\times \displaystyle \frac{\beta c}{u}d{\rm{\Omega }}{dr},\\ r & = & \displaystyle \frac{\beta {ct}}{u},E={E}_{\mathrm{obs}}(1+z),\end{array}\end{eqnarray} \tag{ A1 }$$

where the velocity $\beta =\tfrac{v}{c}$ and the Doppler factor $D\,={[{\rm{\Gamma }}(1-\beta \cos {\theta }_{\mathrm{LOS}})]}^{-1}$ both depend on the angle θ to the jet axis of symmetry, in which θ_LOS is the angle to the line of sight (LOS) of the observer. The viewing angle θ_v is the angle of the jet axis of symmetry to the LOS. If it is on-axis, we have θ_v = 0 and θ_LOS = θ. $d{\dot{N}}_{\gamma }/d{\rm{\Omega }}={\dot{N}}_{\gamma }/4\pi$ and ${\dot{N}}_{\gamma }=L/2.7{k}_{{\rm{B}}}{T}_{0}$, where L is the total outflow luminosity, ${T}_{0}={(L/4\pi {r}_{0}^{2}{ac})}^{1/4}$ is the base outflow temperature, and a is the radiation constant. dP/dE describes the probability for a photon to have an observer frame energy between E and E + dE within the volume element dV, and it is a comoving Planck distribution with the comoving temperature ${T}^{{\prime} }(r,{\rm{\Omega }})$, as shown in Equation (A2),

$$\begin{eqnarray}&&\displaystyle \frac{{dP}}{{dE}}=\displaystyle \frac{1}{2.40{\left({k}_{{\rm{B}}}{T}_{\mathrm{ob}}\right)}^{3}}\displaystyle \frac{{E}^{2}}{\exp (E/{k}_{{\rm{B}}}{T}_{\mathrm{ob}})-1},\end{eqnarray} \tag{ A2 }$$

where k_B is the Boltzmann constant, T_ob(r, Ω) = D(Ω) · ${T}^{{\prime} }(r,{\rm{\Omega }})$ is the observer frame temperature, and in the saturated acceleration regime (R_s < R_ph), it is defined by (Mészáros & Rees 2000; Deng & Zhang 2014)

$$\begin{eqnarray}{T}^{{\prime} }(r,{\rm{\Omega }})=\left\{\begin{array}{ll}\displaystyle \frac{{T}_{0}}{{\rm{\Gamma }}({\rm{\Omega }})}, & r\lt {R}_{s}({\rm{\Omega }})\lt {R}_{\mathrm{ph}}({\rm{\Omega }}),\\ \displaystyle \frac{{T}_{0}{[r/{R}_{s}({\rm{\Omega }})]}^{-2/3}}{{\rm{\Gamma }}({\rm{\Omega }})}, & {R}_{s}({\rm{\Omega }})\lt r\lt {R}_{\mathrm{ph}}({\rm{\Omega }}),\\ \displaystyle \frac{{T}_{0}{[{R}_{\mathrm{ph}}({\rm{\Omega }})/{R}_{s}({\rm{\Omega }})]}^{-2/3}}{{\rm{\Gamma }}({\rm{\Omega }})}, & {R}_{s}({\rm{\Omega }})\lt {R}_{\mathrm{ph}}({\rm{\Omega }})\lt r.\end{array}\right.\end{eqnarray} \tag{ A3 }$$

In the unsaturated acceleration case, the comoving temperature ${T}^{{\prime} }(r)$ is given by

$$\begin{eqnarray}&&{T}^{{\prime} }(r)={T}_{0}/{\rm{\Gamma }},\end{eqnarray} \tag{ A4 }$$

where Γ is R_ph/r₀. The spectra are always obtained within a time interval in the time-averaged or time-resolved analysis. The luminosity is taken to be constant as an averaged value, and the spectrum is taken as a time-integrated spectrum in this time interval.

Continuous wind could be assumed to consist of many thin layers from an impulsive injection at its injection time $\hat{t}$, and the wind luminosity at $\hat{t}$ is denoted as ${L}_{{\rm{w}}}(\hat{t})$. The flux of the observed energy E_obs at the observer time t is given by

$$\begin{eqnarray}&&\begin{array}{l}{F}_{{E}_{\mathrm{obs}}}^{\mathrm{obs}}({\theta }_{{\rm{v}}},{E}_{\mathrm{obs}},t)\\ \ =\,\left\{\begin{array}{ll}{\displaystyle \int }_{0}^{t}{F}_{E}^{\mathrm{obs}}({\theta }_{{\rm{v}}},t,\hat{t},{L}_{{\rm{w}}}(\hat{t}))d\hat{t}, & t\lt {t}_{{\rm{D}}},\\ {\displaystyle \int }_{0}^{{t}_{{\rm{D}}}}{F}_{E}^{\mathrm{obs}}({\theta }_{{\rm{v}}},t,\hat{t},{L}_{{\rm{w}}}(\hat{t}))d\hat{t}, & t\geqslant {t}_{{\rm{D}}},\end{array}\right.\end{array}\end{eqnarray} \tag{ A5 }$$

where t_D is the duration of emission of the central engine, and $r=\tfrac{\beta c(t-\hat{t})}{u}$ in ${F}_{E}^{\mathrm{obs}}({\theta }_{{\rm{v}}},t,\hat{t},{L}_{{\rm{w}}}(\hat{t}))$ in Equation (A1). The observed GRB has a duration, thus, it is reasonable that the central engine produces a continuous wind. For the case of the constant wind luminosity, after the central engine is abruptly shut down (t > t_D), the flux drops sharply. Moreover, the observed light curves of the GRBs show a relatively slow change in luminosity, unlike the steep rise and fall (almost within 10⁻³ s), thus, a continuous wind with variable luminosity is used to simulate the GRB pulse.