Rotation of Polarization Angle in Gamma-Ray Burst Prompt Phase$-$II. The Influence of The Parameters

A simple physical picture of the radiation region is assumed here, as that in Uhm & Zhang (2015, 2016) and Uhm et al. (2018). A thin relativistic jet shell expands radially outward in space. The magnetic reconnection process in jet would simultaneously accelerate the electrons and the jet itself. The accelerated electrons in the magnetic field would emit synchrotron photons uniformly from all positions within the jet shell. In the comoving frame of the shell, the emission generated at each position is assumed to be isotropic. This jet shell with a top-hat structure starts to emit at the radius $r_{on}$ and stops at $r_{off}$.

The polarization model we use here is same as Lan & Dai (2020); Sui & Lan (2024), and the detailed calculation formula can be found there. Since PA of the top-hat jet with a toroidal MF or a axisymmetric random MF could only change abruptly by $90^{\circ}$ or stay as a constant, while it could evolve gradually for an aligned field, the MF configuration considered here is a large-scale ordered aligned MF, which would be in the wind from a perpendicular rotator (i.e., the pulsar with its rotational axis perpendicular to its magnetic axis (Spruit et al., 2001)). The time-resolved PD ($PD$) and preliminary PA ($PA_{pre}$) of the radiation from a jet with an aligned field in its emission region can be expressed as follows.

where $F_{\nu}$ is the flux density, $Q_{\nu}$ and $U_{\nu}$ are the Stokes parameters Q and U, respectively. And the formula for these three Stokes parameters can be found in Lan & Dai (2020); Sui & Lan (2024). If $Q_{\nu}>0$, then the PA of the jet radiation is $PA=PA_{pre}$. If $Q_{\nu}<0$, then $PA=PA_{pre}+\pi/2$ for $U_{\nu}>0$ and $PA=PA_{pre}-\pi/2$ for $U_{\nu}<0$ (Lan et al., 2018).

One difference from Lan & Dai (2020) is that the local PD used here is a broken power law.

Here, the off-axis observations are also considered, the equal arrival time surface should include the off-axis observational geometry, which had already been done in Wang & Lan (2023).

where $\theta_{0}$ equals to 0 for on-axis observations, while it is $\theta_{V}-\theta_{j}$ for off-axis observations.

In the comoving frame, mainly due to the expansion of the jet shell, the MF strength in the radiation region decays with radius (Uhm & Zhang, 2014; Drenkhahn, 2002; Lan et al., 2021)

Since the decay index of the MF ($b$) is unimportant for PA rotation (Wang & Lan, 2023), here we only consider two models with $b=1$ (i.e., $[2b_{i}]$ and $[2b_{m}]$) in Uhm et al. (2018). The two models are parameterized, and the only difference between the “i” and “m” models is that their electron Lorentz factor $\gamma_{ch}$ varies differently with radius. The variation of $\gamma_{ch}$ in the $[2b_{i}]$ model is a single power law, and the corresponding peak-energy evolution mode is hard-to-soft.

where $\gamma_{ch}^{0}=5\times 10^{4}$, and $g=-0.2$ (Uhm et al., 2018). While it is a broken power law for the “m” model, and the peak-energy evolution mode is intensity-tracking.

where we take $\gamma_{ch}^{m}=2\times 10^{5}$, and $g=1.0$ (Uhm et al., 2018).

The variation of the bulk Lorentz factor $\Gamma$ is also roughly a power law with radius, as predicted in Drenkhahn (2002).

For an aligned field in the radiation region, $s$ equals to 0.35 (Drenkhahn, 2002).

With the model described in Section 2.1, we numerically calculate the detailed PA evolutions for different parameters. We define $\triangle$PA to be $\triangle$PA$\equiv$P$\textrm{A}_{max}-$P$\textrm{A}_{min}$, where P$\textrm{A}_{max}$ and P$\textrm{A}_{min}$ are the maximum and minimum values of the time-resolved PA within $T_{90}$, respectively. $T_{90}$ is the duration of time with the accumulated flux accounting from 5$\%$ to 95$\%$ of the total flux. And we only record the PA rotation with $\triangle$PA greater than $10^{\circ}$. The parameters we use are as follows: $\alpha_{B}=-0.8$, $\beta_{B}=-2.3$, $s=0.35$, $r_{on}=10^{14}$ cm, $r_{off}=3\times 10^{16}$ cm, $r_{0}=10^{15}$ cm, $r_{m}=2\times 10^{15}$ cm, $B_{0}^{\prime}=30$ G, $R_{inj}=10^{47}$ ${s}^{-1}$, and $\delta=\pi/6$ (Uhm et al., 2018). $R_{inj}$ is the injection rate of electrons in the shell, $\delta$ is the direction of the aligned MF, and $\theta_{j}$ is half-opening angle of the jet.

If the jet is accelerated thermally as $\Gamma$ is proportional to $r^{1}$, the later energy dissipation in the jet is very likely due to the internal shocks. In this scenario, the acceleration radius could be smaller than the radiation radius of GRB prompt phase. The thermal energy translates to the bulk kinetic energy of the jet first and then the bulk kinetic energy is dissipated at a larger radius through the shocks. However, in the scenario of the magnetic reconnection model, the jet is dominated by Poynting flux. Initially, the magnetic Reynolds number of the jet might be larger than 1, so the magnetic field is frozen in the jet and magnetic reconnection process can not happen. With the increase of the jet radius, its volume will increase, leading to a decrease of the particle number density. The conductivity of the jet, which is proportional to the particle number density, would also decrease. Hence its magnetic Reynolds would also decrease. Further more, the instabilities of the plasma would also develop during the jet propagation. Finally, at large radius the magnetic reconnection happens and the free magnetic energy is depleted. Actually, the energy for radiation and bulk acceleration of the jet in the magnetic reconnection model are both from the magnetic reconnection process, i.e., the acceleration radius is roughly same as the radiation radius in the scenario of the magnetic reconnection model. Since both the interpretations of the spectral lags in GRB prompt phase (Uhm & Zhang, 2016) and of the high-latitude emission in GRB prompt phase (Li & Zhang, 2021) suggested that the radiation region of GRB prompt emission is at large radius from the central engine ($10^{15}$–$10^{16}$ cm) and undergos the bulk acceleration, here we assume the radiation or the acceleration radii could be around $10^{15}$ cm.

It is worth noting that PA defined in the Section 2.1 is within the range of ($-90^{\circ}$, $90^{\circ}$), which would lead to some abrupt $180^{\circ}$ jumps in the PA curves. Since the abrupt $180^{\circ}$ PA jump is meaningless and the polarization direction is unchanged, in this case we will first eliminate these abrupt $180^{\circ}$ jumps by adding or subtracting $180^{\circ}$ to keep the continuity of the PA curves at these jump points. Then $\triangle$PA$\equiv PA_{max}-PA_{min}$ is calculated with these handled PA curves. After this treatment, the maximum $\triangle$PA during the burst duration is $90^{\circ}$.

Figure 1 shows the calculation results for the [$2b_{i}$] model. Its $\gamma_{ch}$ varies as a single power-law with radius (see Equation 6) and the corresponding peak-energy evolution mode is hard-to-soft. Here, we take $\Gamma_{0}=250$ and $\theta_{j}=0.1$ rad as typical parameters. In the first column, the observational angle $\theta_{V}$ is less than or equal to $\theta_{j}$, while in the second column, $\theta_{V}$ is slightly greater than $\theta_{j}$. We find that for on-axis observations(i.e.,$q\equiv\frac{\theta_{V}}{\theta_{j}}\leq 1$), the PA remains unchanged within $T_{90}$. For off-axis observations (i.e.,$q>1$), $\triangle$PA$>10^{\circ}$ is within the range of $1.01\leq q\leq 1.2$, and when $q>1.3$, the PA remains unchanged within $T_{90}$.

Figure 2 shows the results for the [$2b_{m}$] model. The only difference between the [$2b_{i}$] and [$2b_{m}$] models is their variation of $\gamma_{ch}$ with radius. The $\gamma_{ch}$ of the $[2b_{m}]$ model varies as a broken power-law (see Equation 7) and the evolution mode of the peak energy is intensity-tracking. And also we take $\Gamma_{0}=250$ and $\theta_{j}=0.1$ rad. We find that the evolution of $\gamma_{ch}$ has almost no effect on the $q$ ranges for $\triangle$PA$>10^{\circ}$. For example, for the [$2b_{m}$] model, $\triangle$PA$>10^{\circ}$ is within the range of $1.01\leq q\leq 1.3$.

The influences of $B^{\prime}$, $r_{on}$ and $r_{off}$ on the PA rotations are also studied and we find that the changes of these parameters also have slight effect on the q ranges for $\triangle$PA$>10^{\circ}$. The values of both $\Gamma_{0}$ and $r_{0}$ will affect the values of the bulk Lorentz factor $\Gamma$ within range from $r_{on}$ to $r_{off}$. And one profile of $\Gamma$ would correspond to lots of the combinations of ($\Gamma_{0},r_{0}$). The changes of $\Gamma$ due to the changes of $r_{0}$ could be mimicked by the changes of $\Gamma_{0}$. So we take $r_{0}=10^{15}$ cm and only let $\Gamma_{0}$ vary. Therefore, in the following we will use [$2b_{i}$] model as an example and fix the values of $B^{\prime}$, $r_{on}$, $r_{off}$ and $r_{0}$ to explore the influence of the other parameters on the PA rotations.

We then calculated the PA rotations with different values of $\theta_{j}$, $\Gamma_{0}$ and $q$. A part of the detailed time-resolved results are shown in the Appendix. We found all three parameters have significant influence on the PA rotations within $T_{90}$. As shown in Figure 3, when the product value $\theta_{j}\Gamma_{0}$ is fixed, regardless of the concrete values of both $\theta_{j}$ and $\Gamma_{0}$, $\triangle$PA remains roughly unchanged for a fixed $q$ value. For example, for $\theta_{j}\Gamma_{0}=1$ with $q=2$, the values of $\triangle$PA are $66.99^{\circ}$ for $(\theta_{j},\Gamma_{0})=(0.02rad,50)$, $65.88^{\circ}$ for $(\theta_{j},\Gamma_{0})=(0.01rad,100)$, and $64.92^{\circ}$ for $(\theta_{j},\Gamma_{0})=(0.002rad,500)$. Therefore, for fixed values of both $\theta_{j}\Gamma_{0}$ and $q$, $\triangle$ PA is roughly the same.

To demonstrate our above conclusion, we fix the product value of $\theta_{j}\Gamma_{0}$ and calculate the $\triangle$PA for various combinations of ($\theta_{j},\Gamma_{0}$). The results are shown in Table 1. When $\theta_{j}\Gamma_{0}=0.1$, three sets of ($\theta_{j},\Gamma_{0}$) are considered and the differences of the $\triangle$PAs between the various combinations are all within $10^{\circ}$ for one $q$ value. For $\theta_{j}\Gamma_{0}=1$, six sets of ($\theta_{j},\Gamma_{0}$) are considered. The differences of $\triangle$PAs for most combinations are within $10^{\circ}$ except for $(\theta_{j},\Gamma_{0})=(0.001\rm{\ rad},1000)$ with $q=2.5$ and 3. Actually, the differences of the $\triangle$PAs between $(\theta_{j},\Gamma_{0})=(0.001rad,1000)$ and other combinations for each $q$ value considered here are the maximum.

For $\theta_{j}\Gamma_{0}=10$, the differences of the $\triangle$PAs for the majority combinations are within $10^{\circ}$. However, they are larger than $10^{\circ}$ between $(\theta_{j},\Gamma_{0})=(0.01rad,1000)$ and other combinations for $q=1.2$ and 1.3. When $q=1.3$, the differences of the $\triangle$PAs are relatively large and the maximum difference reaches $\sim 11.8^{\circ}$. We only simply take $\triangle$PA$=19.92^{\circ}$ with parameter set of $(\theta_{j},\Gamma_{0})=(0.04rad,250)$ (close to the typical set of $(\theta_{j},\Gamma_{0})=(0.1rad,250)$) for $q=1.3$, as shown in Figure 3. For $\theta_{j}\Gamma_{0}=100$, the differences of the $\triangle$PAs between the six sets of ($\theta_{j},\Gamma_{0}$) are all within $10^{\circ}$ for each $q$ value. Although there are some exceptions, the differences of the $\triangle$PAs will be within $10^{\circ}$ for the majority combinations of ($\theta_{j},\Gamma_{0}$) with the same values of both $\theta_{j}\Gamma_{0}$ and $q$,. Therefore, the conclusion is approximately held.

In Figure 3, as the $\theta_{j}\Gamma_{0}$ value increases, the $q$ range for $\triangle$PA$>10^{\circ}$ becomes smaller. When $q$ range for $\triangle$PA$>10^{\circ}$ reduces to $1.0<q\leq 1.2$, as $\theta_{j}\Gamma_{0}$ value increases, its range remains roughly unchanged and meanwhile the $\triangle$PA becomes larger. It is found that $\triangle$PA will reach roughly $90^{\circ}$ for $\theta_{j}\Gamma_{0}>50$ and $1<q\leq 1.2$. For a fixed value of $\theta_{j}\Gamma_{0}$, the range of $q$ with $\triangle$PA$>10^{\circ}$ begins at $q=1+1/(10\theta_{j}\Gamma_{0})$ and ends at $q=1.2+4/(\theta_{j}\Gamma_{0})$. The $\triangle$PA increase and then decrease with $q$ for a fixed $\theta_{j}\Gamma_{0}$ value when $\theta_{j}\Gamma_{0}\leq 10$, while it will increase with $q$ when $\theta_{j}\Gamma_{0}>10$. For a fixed value of $q$ with $q>1.2$, $\triangle$PA also increase and then decrease with $\theta_{j}\Gamma_{0}$. For a fixed value of $q$ with $q\leq 1.2$, $\triangle$PA will roughly increase with $\theta_{j}\Gamma_{0}$.