Energy Injection Driven by Precessing Jets in Gamma-Ray Burst Afterglows

To calculate the afterglow LCs of GRBs powered by a long-lived precessing jet, an approximate treatment for the precession process is adopted here. The spiral path of a precessing jet, represented by the blue curve in Figure 1, can be roughly considered to be ceaselessly filled by discrete sub-jets. The number of these discrete sub-jets is related to the jet half-opening angle θ_jet and the precession angle θ_pre, and one should ensure that the overlaps among the discrete sub-jets are as small as possible and that the sub-jets in each period are sufficient to cover the path.

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In the first precession period, due to the interaction between the sub-jets and the circumburst medium, the relativistic external forward shocks (EFSs) are generated and propagate through the medium. Because the age of the jet is much larger than the precession period, the subsequent sub-jets in each period carry out the energy injection, potentially catching up with the EFSs produced in the first period and colliding with them. The dynamics of the EFSs can be described by solving differential equations listed as follows ¹ (Zhang 2018),

$$\begin{eqnarray}&&\displaystyle \frac{d{\rm{\Gamma }}}{{dt}}=-\displaystyle \frac{{\rm{\Gamma }}({{\rm{\Gamma }}}^{2}-1)(\hat{\gamma }{\rm{\Gamma }}-\hat{\gamma }+1){c}^{2}\tfrac{{dm}}{{dt}}-{\rm{\Gamma }}(\hat{\gamma }-1)(\hat{\gamma }{{\rm{\Gamma }}}^{2}-\hat{\gamma }+1)\tfrac{3U}{R}\tfrac{{dR}}{{dt}}-{{\rm{\Gamma }}}^{2}\tfrac{{{dE}}_{\mathrm{inj}}}{{dt}}}{{{\rm{\Gamma }}}^{2}({M}_{0}+m){c}^{2}+({\hat{\gamma }}^{2}{{\rm{\Gamma }}}^{2}-{\hat{\gamma }}^{2}+3\hat{\gamma }-2)U},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{dU}}{{dt}}=(1-\epsilon )({\rm{\Gamma }}-1){c}^{2}\displaystyle \frac{{dm}}{{dt}}-(\hat{\gamma }-1)\left(\displaystyle \frac{3}{R}\displaystyle \frac{{dR}}{{dt}}-\displaystyle \frac{1}{{\rm{\Gamma }}}\displaystyle \frac{d{\rm{\Gamma }}}{{dt}}\right)U,\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{dm}}{{dt}}=4\pi {R}^{2}{{nm}}_{p}\displaystyle \frac{c\beta }{1-\beta },\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{dR}}{{dt}}=\displaystyle \frac{c\beta }{1-\beta },\end{eqnarray} \tag{ 4 }$$

where Γ , U, m, and R represent the bulk Lorentz factor, the internal energy, the sweep-up mass from the circumburst medium, and the distance from the central source, respectively. ${E}_{\mathrm{inj}}$, M₀, , n, and t are the injection energy, the initial mass, the radiation efficiency of electrons in the EFSs, the number density of the circumburst medium, and the time measured in the observer frame, respectively. Likewise, $\hat{\gamma }$ is the adiabatic index, which is obtained by the formula, $\hat{\gamma }\approx (5-1.21937\xi +0.18203{\xi }^{2}\,-0.96583{\xi }^{3}\,+\,2.32513{\xi }^{4}\,-\,2.39332{\xi }^{5}\,+\,1.07136{\xi }^{6})/3,$with $\xi \equiv {\rm{\Theta }}/(0.24+{\rm{\Theta }})$, ${\rm{\Theta }}\approx {\rm{\Gamma }}\beta ({\rm{\Gamma }}\beta +1.07{({\rm{\Gamma }}\beta )}^{2})/3(1+{\rm{\Gamma }}\beta \,+{1.07({\rm{\Gamma }}\beta )}^{2})$, and $\beta =\sqrt{1-1/{{\rm{\Gamma }}}^{2}}$ (e.g., Pe'er 2012). In addition, the function form of ${{dE}}_{\mathrm{inj}}/{dt}$ can be simply described as

$$\begin{eqnarray}\displaystyle \frac{{{dE}}_{\mathrm{inj}}}{{dt}}=\left\{\begin{array}{ll}2{P}_{\mathrm{jet}}/(1-\cos {\theta }_{\mathrm{jet}}), & {T}_{\mathrm{start}}\lt t\lt {T}_{\mathrm{end}},\\ 0, & \mathrm{others},\end{array}\right.\end{eqnarray} \tag{ 5 }$$

where ${T}_{\mathrm{start}}=((i-1){\rm{\Delta }}t\,+\,j\tau )$, ${T}_{\mathrm{end}}=(i{\rm{\Delta }}t+j\tau )$, i $(=1,2,\ldots ,\,k)$ represents the serial number of sub-jets within each period, j $(=1,2,\ldots ,\,{t}_{\mathrm{end}}/\tau -1)$ denotes the serial number of the precession period, τ is the precession period, and ${\rm{\Delta }}t=\tau /k$. In this work, the value of the mass density of the medium is considered to be a constant (i.e., interstellar medium), and the evolution of θ_jet is neglected because the sideways expansion is not important, as shown by numerical simulations (e.g., Zhang & MacFadyen 2009; Chen & Zhang 2021).

Two cases of the jet power P_jet are considered. A steady power is discussed in Case I, i.e., ${P}_{\mathrm{jet}}={P}_{\mathrm{jet}}^{0}$. In Case II, a power-law form

$$\begin{eqnarray}&&{P}_{\mathrm{jet}}(t)={P}_{\mathrm{jet}}^{0}{\left(t/{t}_{0}\right)}^{-\alpha }\end{eqnarray} \tag{ 6 }$$

and a smooth broken power-law form

$$\begin{eqnarray}&&{P}_{\mathrm{jet}}(t)={P}_{\mathrm{jet}}^{0}{\left[{\left(t/{t}_{0}\right)}^{{\alpha }_{{\rm{r}}}s}+{\left(t/{t}_{{\rm{b}}}\right)}^{{\alpha }_{{\rm{d}}}s}\right]}^{-1/s}\end{eqnarray} \tag{ 7 }$$

are taken into account, where α, ${\alpha }_{{\rm{r}}}$, and ${\alpha }_{{\rm{d}}}$ are all temporal slopes, ${t}_{{\rm{b}}}$ is the break time, s measures the sharpness of the break, ${t}_{0}\simeq {R}_{0}(1+z)/(2{{\rm{\Gamma }}}_{0}^{2}c)$ is the starting time of the afterglows, with an initial radius ${R}_{0}={10}^{14}\,\mathrm{cm}$, an initial Lorentz factor ${{\rm{\Gamma }}}_{0}=200$, and the redshift of the burst z, and ${P}_{\mathrm{jet}}^{0}$ is the jet power at t₀. Furthermore, in Case I, the value of the isotropic kinetic energy ${E}_{{\rm{k}},\mathrm{iso}}$ is equal to $2{P}_{\mathrm{jet}}^{0}{\rm{\Delta }}t/(1-\cos {\theta }_{\mathrm{jet}})$, while in Case II, it is equal to $2{\int }_{\left(i-1\right){\rm{\Delta }}t}^{i{\rm{\Delta }}t}{P}_{\mathrm{jet}}\,{dt}/(1-\cos {\theta }_{\mathrm{jet}})$. It should be noted that the afterglow starts from t₀; thus, the energy released before t₀ needs to be deducted in the first EFS process.

Following the evolution of the dynamics mentioned above, the evolution of the shock-accelerated electrons can be expressed as a function of the radius R, i.e.,

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial R}\left(\displaystyle \frac{{{dN}}_{{\rm{e}}}^{{\prime} }}{d{\gamma }_{{\rm{e}}}^{{\prime} }}\right)+\displaystyle \frac{\partial }{\partial {\gamma }_{{\rm{e}}}^{{\prime} }}\left({\dot{\gamma }}_{{\rm{e}}}^{{\prime} }\displaystyle \frac{{dt}^{\prime} }{{dR}}\displaystyle \frac{{{dN}}_{{\rm{e}}}^{{\prime} }}{d{\gamma }_{{\rm{e}}}^{{\prime} }}\right)={\hat{Q}}_{\mathrm{ISM}}^{{\prime} },\end{eqnarray} \tag{ 8 }$$

where ${{dN}}_{{\rm{e}}}^{{\prime} }/d{\gamma }_{{\rm{e}}}^{}\,^{\prime}$ is the instantaneous electron energy spectrum, ${\gamma }_{{\rm{e}}}^{}\,^{\prime}$ is the Lorentz factor of the shock-accelerated electrons, ${\dot{\gamma }}_{{\rm{e}}}^{\prime}$ is the cooling rate of electrons with the Lorentz factor ${\gamma }_{{\rm{e}}}^{}\,^{\prime}$, ${dt}^{\prime} /{dR}$ = $1/{\rm{\Gamma }}c$, and ${\hat{Q}}_{\mathrm{ISM}}^{{\prime} }=\bar{K}{\gamma }_{{\rm{e}}}^{{\prime} -p}$, with $\bar{K}\approx 4\pi (p-1){R}^{2}{n}_{}{\gamma ^{\prime} }_{{\rm{e}},\min }^{p-1}$, is adopted to describe the injection behavior of newly shocked circumburst medium electrons (e.g., Fan et al. 2008; Huang et al. 2020), where p (>2) is the power-law index and ${\gamma }_{{\rm{e}},\min }\,^{\prime} \,\leqslant {\gamma }_{{\rm{e}}}^{{\prime} }\leqslant {\gamma }_{{\rm{e}},\max }^{}\,^{\prime}$ is adopted for ${\gamma }_{{\rm{e}}}^{}\,^{\prime}$. Note that quantities with superscript accent signs are defined in the comoving frame of the EFSs. Since the shocked electrons and the magnetic fields share the fractions _e and _B of the thermal energy density in the EFS downstream, the minimum Lorentz factor of the shock-accelerated electrons can be expressed as ${\gamma }_{{\rm{e}},\min }\,^{\prime} \,={\epsilon }_{{\rm{e}}}({\rm{\Gamma }}-1)(p-2){m}_{{\rm{p}}}/(p-1){m}_{{\rm{e}}}$, with ${m}_{{\rm{p}}}$ and ${m}_{{\rm{e}}}$ being the proton and electron masses, respectively, and the maximum Lorentz factor of electrons is ${\gamma }_{{\rm{e}},\max }^{{\prime} }=\sqrt{9{m}_{{\rm{e}}}^{2}{c}^{4}/8B^{\prime} {e}^{3}}$, with ${B}^{{\prime} }=\sqrt{32\pi {\rm{\Gamma }}({\rm{\Gamma }}-1){{nm}}_{{\rm{p}}}{\epsilon }_{B}{c}^{2}}$ being the magnetic field behind the EFSs and c being the speed of light (e.g., Kumar et al. 2012; Huang et al. 2020).

In order to calculate the observed flux of the synchrotron radiation from the sub-jets, each sub-jet is divided linearly into 300 × 1000 small emitters along ${\theta }^{{\prime} }$- and ${\phi }^{{\prime} }$-directions in the comoving spherical coordinate frame correlated to the comoving Cartesian coordinate system $({x}^{{\prime} },{y}^{{\prime} },{z}^{{\prime} })$, and wherein the sub-jet axis is aligned with the ${z}^{{\prime} }$-axis as shown in Figure 1. Due to the line-of-sight effect, the arrival time of photons from emitters with the radius R deviated from the line of sight is delayed, compared to that in the line of sight. Since the delays, ${t}_{\mathrm{de}}=R(1-\cos {\theta }_{\mathrm{em}})/c$, only depend on the cosine value of the angle between the off-sight emitters and the line of sight, which is depicted as

$$\begin{eqnarray}\,\begin{array}{rcl}\cos {\theta }_{\mathrm{em}} & = & \sin {\theta }_{\mathrm{obs}}\cos {\phi }_{\mathrm{obs}}\sin \theta \cos \phi \\ & & +\,\sin {\theta }_{\mathrm{obs}}\sin {\phi }_{\mathrm{obs}}\sin \theta \sin \phi +\cos {\theta }_{\mathrm{obs}}\cos \theta ,\end{array}\,\end{eqnarray} \tag{ 9 }$$

where $({\theta }_{\mathrm{obs}},{\phi }_{\mathrm{obs}})$, and $(\theta ,\phi )$ represent the coordinates of the observer and emitter in the observer frame, respectively. In this work, we set ${\phi }_{\mathrm{obs}}=0$ and the transition of the emitter coordinate from the comoving frame to the observer frame can be accomplished with the following relations:

$$\begin{eqnarray}\,\begin{array}{rcl}\sin \theta \cos \phi & = & \sin {\theta }_{\mathrm{pre}}\cos {\phi }_{\mathrm{pre}}\cos {\theta }^{{\prime} }\\ & & +\,\cos {\theta }_{\mathrm{pre}}\cos {\phi }_{\mathrm{pre}}\sin {\theta }^{{\prime} }\cos {\phi }^{{\prime} }\\ & & -\,\sin {\phi }_{\mathrm{pre}}\sin {\theta }^{{\prime} }\sin {\phi }^{{\prime} },\end{array}\,\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}\,\begin{array}{rcl}\sin \theta \sin \phi & = & \sin {\theta }_{\mathrm{pre}}\sin {\phi }_{\mathrm{pre}}\cos {\theta }^{{\prime} }\\ & & +\,\cos {\theta }_{\mathrm{pre}}\sin {\phi }_{\mathrm{pre}}\sin {\theta }^{{\prime} }\cos {\phi }^{{\prime} }\\ & & +\,\cos {\phi }_{\mathrm{pre}}\sin {\theta }^{{\prime} }\sin {\phi }^{{\prime} },\end{array}\,\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\cos \theta =\cos {\theta }_{\mathrm{pre}}\cos {\theta }^{{\prime} }-\sin {\theta }_{\mathrm{pre}}\sin {\theta }^{{\prime} }\cos {\phi }^{{\prime} },\end{eqnarray} \tag{ 12 }$$

where $({\theta }_{\mathrm{pre}},{\phi }_{\mathrm{pre}})$ is the coordinate of sub-jet axis in the observer frame. Thereby the observed time of an emitter at radius R is $(t+{t}_{\mathrm{de}})(1+z)$. In addition, the synchrotron radiation power at the frequency $\nu ^{\prime}$ can be calculated by the following formula (Rybicki & Lightman 1979):

$$\begin{eqnarray}&&{P}_{\mathrm{syn}}^{{\prime} }(\nu ^{\prime} )=\displaystyle \frac{\sqrt{3}{e}^{3}B^{\prime} }{{m}_{{\rm{e}}}{c}^{2}}{\int }_{{\gamma }_{{\rm{e}},\min }^{{\prime} }}^{{\gamma }_{{\rm{e}},\max }^{{\prime} }}\left(\displaystyle \frac{{{dN}}_{{\rm{e}}}^{{\prime} }}{d{\gamma }_{{\rm{e}}}^{{\prime} }}\right)F\left(\displaystyle \frac{\nu ^{\prime} }{{\nu }_{{\rm{c}}}^{{\prime} }}\right)d{\gamma }_{{\rm{e}}}^{{\prime} },\end{eqnarray} \tag{ 13 }$$

where ${\nu }_{{\rm{c}}}^{{\prime} }=3{eB}^{\prime} {\gamma }_{{\rm{e}}}^{{\prime} }{}^{2}/4\pi {m}_{{\rm{e}}}c$ and $F(\nu ^{\prime} /{\nu }_{{\rm{c}}}^{{\prime} })=(\nu ^{\prime} /{\nu }_{{\rm{c}}}^{{\prime} }){\int }_{{\nu }^{{\prime} }/{\nu }_{{\rm{c}}}^{{\prime} }}^{+\infty }{K}_{5/3}(x){dx}$, with ${K}_{5/3}(x)$ being a modified Bessel function of order 5/3. Hence, one can calculate the observed flux by summing the flux received from the emitters at the same observer time in a sub-jet, and the observed flux density can be derived as (e.g., Granot et al. 1999)

$$\begin{eqnarray}&&{S}_{{\nu }_{\mathrm{obs}}}=\displaystyle \frac{1+z}{4\pi {D}_{{\rm{L}}}^{2}}{\iint }_{\left(\mathrm{EATS}\right)}{P}_{\mathrm{syn}}^{{\prime} }(\nu ^{\prime} ){D}^{3}d{\rm{\Omega }},\end{eqnarray} \tag{ 14 }$$

where "EATS" is the equal-arrival time surface corresponding to the same observer time in a sub-jet, ${\nu }^{{\prime} }=(1+z){\nu }_{\mathrm{obs}}/D$ with $D=1/{\rm{\Gamma }}(1-\beta \cos {\theta }_{\mathrm{em}})$ being the Doppler factor of the emitters, and ${D}_{{\rm{L}}}$ is the luminosity distance in the standard ΛCDM cosmology model (${{\rm{\Omega }}}_{M}=0.27$, ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.73$, and H₀ = 71 km s⁻¹ Mpc⁻¹). Further, the total observed flux becomes the accumulation of the observed flux produced on each sub-jet at the same observed time.

Figure 2 shows the GRB afterglow LCs of Case I in the 10 keV band since triggering of the GRB for the precession periods τ = 10 and 100 s, the precession angles θ_pre = 5° and 10°, and the initial jet powers P_jet = 10⁴⁹ and ${10}^{50}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. The black, red, green, blue, and magenta lines in every panel denote the results for θ_obs = 0°, 5°, 10°, 15°, and 20°, respectively. In this case, one can find that the shallow decay or plateau segments (hereafter called plateaus) appear in some LCs. Some of the plateaus have tiny positive slopes.

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The plateaus under the condition of a short precession period in the panels in the left column appear earlier than those in the right column, and the values of the fluxes with ${P}_{\mathrm{jet}}\,={10}^{49}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ are definitely an order of magnitude lower than those with ${P}_{\mathrm{jet}}={10}^{50}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. More importantly, when the values of the precession angles are near those of the observer angles (i.e., the jets sweep across the line of sight), the flux is higher. By contrast, the flux is lower and no plateau structure (only a bump shape) emerges when the difference between these two angles is larger, exceeding the range of the jet opening angle. Moreover, the oscillations appearing in the early stage of the plateaus might be caused to great extent by the discrete approximations of the jet precession.

Figure 3 shows the LCs of GRB afterglows under the power-law evolution of the jet power shown in Equation (6), with τ = 10 s, t_end = 2 ks, and ${P}_{\mathrm{jet}}^{0}={10}^{50}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. Panels (a)–(c) are the results with θ_pre = 5° for temporal slopes α of 1, 0.5, and 0.1, respectively, and panels (e)–(g) are the same as panels (a)–(c), respectively, except θ_pre = 10°. The black, red, and green lines in panel (d) are the same as those of the LCs with θ_obs = 5° in panels (a)–(c), and the lines in panel (h) are the same as those of the LCs with θ_obs = 5° in panels (e)–(g), respectively. In this case, one can find that regardless of whether θ_pre = 5° or 10°, once the values of α are small enough, e.g., panels (c) and (g), the plateaus very clearly emerge. The slope of the plateau decreases quickly with increasing α until the giant bump replaces the plateau when α = 1, as shown in panels (a) and (e). In addition, comparing panel (d) with (h), one can see that although the precessing angle is changed from 5° to 10°, a similar plateau still exists in the LCs under the same viewing angle θ_obs = 5°, which implies that the evolution of the precessing angle does not significantly affect the plateau structure of the LCs unless the angle between the line of sight and the axis of the first sub-jet is greater than the opening angle of the sub-jets. Figure 4 is the same as Figure 3, except τ = 100 s. Despite the difference in the precessing periods, one can see that the results implicated in Figure 4 are similar to those in Figure 3.

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Figure 5 shows the LCs of GRB afterglows with τ = 10 s, t_end = 2 ks, and θ_pre = 10° under a smooth broken power-law evolution of the jet power, as shown in Equation (7), with ${P}_{\mathrm{jet}}^{0}={10}^{50}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, where the parameter values of the smooth broken power-law function are α_r = −1/2, α_d = 5/3, s = 1, and t_b = 1 ks. The black, red, green, blue, and magenta lines denote the results for θ_obs = 0°, 5°, 10°, 15°, and 20°, respectively. One can see that a plateau appears in the LC when the viewing angle $5^\circ \leqslant {\theta }_{\mathrm{obs}}\leqslant 15^\circ$, but due to the quick decrease in the broken power-law jet power after 1 ks, the duration of this plateau is shorter than those in the cases with the steady or slow decay (α < 0.5) jet powers.

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From the above figures, we note that the precession period does not impact the LC shape; instead, the appearance of the plateaus nearly depends on the time evolution of the jet powers and the relations among the viewing angles, the jet half-opening angles, and the precession angles.

No matter which cases of the jet precession substantially increase the size of the jet head caused by the continuous sub-jets along the spiral route to power the observable energy injections in GRB afterglows even the structured jets considered. Furthermore, the jet precession model also can apply perfectly to understand the spectral evolution of the afterglow LCs.