Time-integrated polarizations in GRB prompt phase via the Multi-window interpretation

Although there had been decades after gamma-ray burst (GRB), which is the most catastrophic event in the universe, was first discovered, their origin is still indebate. The light curves of the GRBs show rich diversity, which adds the difficulty in finding their physical origin. However, the spectrum of the GRBs is often described by the Band function (Band et al., 1993), which is a broken power law, smoothly jointed at $(\alpha_{B}-\beta_{B})E_{p}/(2+\alpha_{B})$, where $E_{p}$ is the peak energy of the $\nu F_{\nu}$ spectrum and $\alpha_{B}$ ($\beta_{B}$) is the low-energy (high-energy) spectral index of the photon number flux. Observationally, there are mainly two kinds of the peak energy evolution patterns, i.e., the hard-to-soft and intensity-tracking patterns (Liang & Kargatis, 1996; Ford et al., 1995; Kaneko et al., 2006; Lu et al., 2010, 2012, 2018). And the origin of the Band function is also mysterious.

Three popular models are involved to interpret the observations of GRB prompt phase (Narayan et al., 1992; Paczynski & Xu, 1994; Rees & Meszaros, 1994; Thompson, 1994; Eichler & Levinson, 2000; Mészáros & Rees, 2000; Giannios, 2008; Zhang & Yan, 2011). In the scenario of the internal shock and the magnetic reconnection models, the radiation mechanisms are both the synchrotron emission (Daigne et al., 2009; Lan et al., 2021a; Giannios, 2008; Zhang & Yan, 2011). However, it is the multiple inverse-Compton scatterings in the high-energy gamma-rays for the photosphere model (Thompson, 1994; Lundman et al., 2018; Parsotan et al., 2020). Since all three models could explain the observed light curves and energy spectra in GRB prompt phase, these models and hence their corresponding radiation mechanism can not be distinguished by the traditional observations. Polarization, as a new window, is very sensitive to the radiation mechanism, magnetic field configuration (MFC) in the radiation region and the geometry (Waxman, 2003; Granot, 2003; Rossi et al., 2004; Lazzati, 2006; Toma et al., 2009; Lan et al., 2021b; Gill et al., 2021; Guan & Lan, 2023; Sui & Lan, 2024). Therefore, it can be used as a powerful tool to infer these properties of the radiating sources.

Because of the depolarization due to the multiple scatterings, the predicted polarization degree (PD) is roughly 0 in the gamma-ray band for the photosphere model (Lundman et al., 2018; Parsotan et al., 2020; Parsotan & Lazzati, 2022). However, the predicted time-integrated PD can be as high as $(40-50)\%$ for the synchrotron model in an ordered magnetic field (Toma et al., 2009; Gill et al., 2020, 2021; Guan & Lan, 2023; Sui & Lan, 2024). If the polarization angle (PA) rotates with time (or energy) and/or the magnetic field is mixed ¹¹1There are both the ordered component and the random component of the magnetic field (Lan et al., 2019). in the radiation region, there would be a reduction of the time- and energy-integrated PD. So the predicted time-integrated PD can range from $0\%$ to $50\%$ for the synchrotron models (Lan et al., 2019; Sui & Lan, 2024). Therefore, the precise PD observations are important in distinguishing the models and hence the corresponding radiation mechanisms during GRB prompt phase. If the gamma-ray emission in GRB prompt phase are unpolarized, the photosphere model survives and the synchrotron model with a totally random magnetic field could also interpret such observations. However, if the PD of the gamma-ray radiation are non-zero, the photosphere model is rejected.

The Gamma-Ray Burst Polarimeter (GAP) is a polarization detector to measure the linear polarizations in GRB prompt phase in the energy range of (70 keV, 300 keV) (Yonetoku et al., 2006; Murakami et al., 2010; Yonetoku et al., 2011). POLAR is also a dedicated GRB polarimeter and the linear polarizations were measured in the energy range of (50 keV, 500 keV) (Produit et al., 2018). The Cadmium Zinc Telluride Imager on board AstroSat is an in operation GRB polarization detector and its polarization measurement is done in the energy range of (100 keV, 600 keV) (Chattopadhyay et al., 2022). There should be sufficient photons to do polarization analysis, most of the current polarization observations are the time- and energy-integrated results (Yonetoku et al., 2011, 2012; Zhang et al., 2019; Kole et al., 2020; Chattopadhyay et al., 2022). The time-resolved observations are relatively rare and the time-resolved polarization analysis had been done in only 4 GRBs (Yonetoku et al., 2011; Zhang et al., 2019; Burgess et al., 2019; Kole et al., 2020; Sharma et al., 2019; Chattopadhyay et al., 2022).

The number of the bursts detected by the polarization detector GAP is 3 and the averaged best-fit value of the time-integrated PD is $60\%$ for GAP bursts (Yonetoku et al., 2011, 2012). The observed time-integrated PDs of 15 bursts among the total 20 bursts detected by AstroSat are the upper limits and only 5 bursts are reported with concrete PD values (Chattopadhyay et al., 2022). The averaged best-fit value of the time-integrated PD of these 5 bursts is $57.83\%$. There are 14 GRBs with reported polarization detections by the POLAR (Zhang et al., 2019; Kole et al., 2020) and the averaged best-fit value of the time-integrated PD is $17.79\%$. The smallest lower limit of the detected time-integrated PD within $1\sigma$ confidence level is $16\%$ for GAP bursts and the highest upper limit of the detected PD within $1\sigma$ is $84\%$ for POLAR bursts. Since the observational errors are rather large and confidence levels (usually within $1\sigma$) are relatively low, the current data are in general not contradict.

Recently, time-integrated PDs in GRB prompt phase had been studied by Guan & Lan (2023). They used the observed parameters of the energy spectra to estimate the time-integrated PD, in which the time-resolved information was missed. However, up till now, there are 4 GRBs with time-resolved polarization observations. Among these 4 GRBs, there are 3 bursts with PA rotating abruptly at least once during the burst, which would lead to a reduction of the time-integrated PD. And also the variations of the parameters on the equal arrival time surface (EATS) would also impact the final time-integrated PDs (Sui & Lan, 2024). Therefore, the time-evolution effect should not be neglected in deducing the time-integrated PDs. The parameters are taken as their typical or fiducial values and vary in reasonable ranges to predicted the time-integrated PDs in Sui & Lan (2024). Actually, in addition to the polarization observations, both the observed light curves and energy spectra would put limits on the model parameters. Therefore, in order to predict more precise time-integrated PDs in GRB prompt phase, multi-window interpretations (i.e., the simultaneous interpretation/fitting of the light curve, $E_{p}$ curve, PD curve and the PA curve) are needed.

In this paper, the time-integrated PDs in GRB prompt phase are studied via the time-resolved multi-window fitting. The EATS efffect is considered and the time-resolved information is included. The observed PA rotations are recovered during the bursts. More important, most of the key model parameters are inferred from the fitting of the light curves, the energy spectra and the polarization curves. The predicted time-integrated PD for each burst is customized. In Section 2, the data processing is presented. In Section 3, we describe our model. In Section 4, our numerical results are presented. Finally, we give our conclusions and discussion in Section 5.

The light curves and $E_{p}$ curves are obtained from the Fermi/GBM data through the following procedure. $RMFit$ v4.3.2 is employed to perform data analysis (Gamma-ray astronomy Group, 2014). Observational database are downloaded in the official web site of $Fermi$ Gamma-ray Burst Monitor (GBM) ²²2https://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html. For each GRB, we select two sodium iodide (NaI) detectors that are most close to the GRB center as well as one bismuth germanate (BGO) detector. GBM time-tagged events (TTE) with 2 $\mu$s precision are used in the data reduction. Photon energy is selected from 8 keV to 1000 keV for NaI detector, while from 200 keV to 40 MeV for BGO detector. We select two intervals as the background intervals and employ a polynomial model with 0-3 order to fit the background. After that, we perform the time-averaged and time-resolved spectral fitting using a chi-square statistics. For the time-averaged spectrum, the GBM $T_{90}$ interval in the 50-300 keV energy band is selected as the analyzed duration (von Kienlin et al., 2020). For the time-resolved spectral analyses, we derive the time bins by setting a signal to noise ratio (SNR) between 12 and 100 among GBM $T_{90}$. A typical Band function is employed to fit all spectra (Band et al., 1993). Spectral parameters (e.g., $E_{p}$) and photon flux (between 8 keV and 40 MeV) of individual spectrum are calculated in the best-fit result.

The data of the polarizations are taken from the published papers (Yonetoku et al., 2011, 2012; Zhang et al., 2019; Kole et al., 2020; Chattopadhyay et al., 2022). The energy ranges for the polarization data are (70 keV, 300 keV), (50 keV, 500 keV) and (100 keV, 600 keV) for the bursts detected by the GAP, POLAR and AstroSat, respectively (Yonetoku et al., 2011, 2012; Zhang et al., 2019; Kole et al., 2020; Chattopadhyay et al., 2022). In our sample, there are total 23 bursts, which are co-detected by Fermi/GBM and polarization detectors. The main information of these bursts are shown in Table 1. The confidence level for the errors of the observed time-integrated PD ($PD_{obs}$) is $1\sigma$. It is $2\sigma$ for the upper limits of the $PD_{obs}$. The confidence level for the time-integrated photon spectral indices ($\alpha_{B}$ and $\beta_{B}$) are also $1\sigma$. In our sample, the average value of the observed time-integrated PD is around $60\%$ for GAP bursts, around $50\%$ for the AstroSat bursts and around $22\%$ for the POLAR bursts. The minimum lower limit of the observed time-integrated PD within $1\sigma$ is $16\%$ for GAP bursts, it is $28.32\%$ for AstroSat bursts. The maximum upper limit of the observed time-integrated PD within $1\sigma$ is $84\%$ for POLAR bursts. So because of large observational errors, the detection results of the three detectors are in general not contradictory. In addition, the errors of the current data are roughly within the confidence level of $1\sigma$. These results may be coincide with each other in higher confidence level.

The model used here for fitting the time-resolved multi-window observations is same to that in Wang et al. (2024). We assume that GRB at redshift $z$ originates from the synchrotron radiation of the multiple thin emitting shells. These shells, with different physical properties, are injected discontinuously because of the activities of the GRB central engine. They would expand radially at relativistic velocities. The process of the magnetic reconnection in each individual shell would deplete the magnetic energy and provides the energy for the acceleration and radiation of the shell. The relativistic electrons would be injected isotropically into each shell with an rate of $R_{inj}$ and radiate the synchrotron photons in the magnetic fields. Here, the single-energy electrons are assumed (Uhm et al., 2018). For “i” model (corresponding to the observed hard-to-soft $E_{p}$ evolution pattern), the variation of the electron Lorentz factor with radius is assumed to be $\gamma_{ch}(r)=\gamma_{ch}^{0}(r/r_{0})^{g}$, where $\gamma_{ch}^{0}$ is the value of electron Lorentz factor at ${r_{0}}$. While for “m” model (the corresponding $E_{p}$ evolution pattern is intensity-tracking), we take $\gamma_{ch}(r)=\gamma_{ch}^{m}(r/r_{m})^{g}$ for $r\leq r_{m}$ and $\gamma_{ch}(r)=\gamma_{ch}^{m}(r/r_{m})^{-g}$ for $r>r_{m}$, where $\gamma_{ch}^{m}$ is value of the electron Lorentz factor at the normalization radius $r_{m}$. Unless specified, we take $g=-0.2$ for the “i” model and $g=1.0$ for the “m” model, respectively (Uhm et al., 2018).

All the shells will begin to radiate at radius $r_{on}$ at burst source time $t_{on}$ and stop at radius $r_{off}$. The dynamics of the shells would depend on the central engine (Drenkhahn, 2002). For a perpendicular rotator ³³3The magnetic axis of the magnetar is perpendicular to its rotational axis., the magnetic field in its ejecta is aligned and the reconnection of such field would lead to the bulk acceleration of the jet shell. For a black hole central engine, the magnetic field in jet is toroidal and the reconnection of the toroidal field would lead to a roughly constant-velocity jet shell at large radii. The bulk Lorentz factor of each shell then reads $\Gamma(r)=\Gamma_{0}(r/r_{0})^{s}$, where $\Gamma_{0}$ is the bulk Lorentz factor of the shell at the normalization radius $r_{0}$ and $s=1/3$ (0) for the aligned-fields (the toroidal-fields) case (Drenkhahn, 2002). Meanwhile, the magnetic field in the shell decays with radius $B^{\prime}(r)=B^{\prime}_{0}(r/r_{0})^{-b}$, where $B^{\prime}_{0}=30$ G is the strength of the magnetic field at $r_{0}$ and the decay index $b$ is taken as 1.0. For simplicity, the MFC in each jet shell is assumed to be large-scale ordered. Except the MFC, the other physical properties are assumed to be independent between the emitting shells of one burst.

The time-resolved and energy-averaged flux density of the emission from a burst is

where $h{\nu_{1}}=8$ keV, $h{\nu_{2}}=40$ MeV. The $\nu$ is the frequency and $F_{\nu}$ is the time- and energy-resolved flux density of the single burst and the corresponding expression can be found in Wang et al. (2024).

The predicted time-resolved and energy-integrated PD and PA (preliminary) of a burst with aligned fields in all the radiation shells (aligned-fields case) read,

where $F=\int^{\nu_{2}}_{\nu_{1}}F_{\nu}d\nu$, $Q=\int^{\nu_{2}}_{\nu_{1}}Q_{\nu}d\nu$ and $U=\int^{\nu_{2}}_{\nu_{1}}U_{\nu}d\nu$ are the time-resolved and energy-integrated Stokes parameters. And $h{\nu_{1}}$ and $h{\nu_{2}}$ here are the lower- and upper-limit of the corresponding polarization detectors. And $Q_{\nu}$ and $U_{\nu}$ are the time- and energy-resolved Stokes parameters Q and U of a single burst, respectively. Their expressions can be found in Wang et al. (2024). If $Q>0$, the final PA ($PA$) of a burst is $PA=PA_{pre}$. If $Q<0$, the final PA is $PA=PA_{pre}+\pi/2$ when $U>0$ and is $PA=PA_{pre}-\pi/2$ when $U<0$ (Lan et al., 2018).

For a burst with toroidal fields in all its emitting shells (toroidal-fields case), because of the axial symmetry of the emission region and the reference axis selected here, the Stokes parameter $U_{\nu}$ equals to zero. The time-resolved and energy-integrated PD is defined as follows.

When $PD$ changes its sign, the corresponding PA would change abruptly by 90°.

Finally, the time-integrated and energy-integrated PD of a burst for the aligned-fields case read as follows.

where $\bar{F}=\int^{T_{95}}_{T_{5}}Fdt/T_{90}$, $\bar{Q}=\int^{T_{95}}_{T_{5}}Qdt/T_{90}$, and $\bar{U}=\int^{T_{95}}_{T_{5}}Udt/T_{90}$ are the time-averaged and energy-integrated Stokes parameters. The accumulated flux reaches $5\%$ ($95\%$) of the total flux of the burst at the time of $T_{5}$ ($T_{95}$), and $T_{90}=T_{95}-T_{5}$. For the toroidal-fields case, the time-integrated and energy-integrated PD is

In this work, the time-integrated polarizations of 23 GRBs are predicted via the multi-window fitting. For the aligned-fields case, the directions of the large-scale ordered aligned magnetic fields in the shells of the individual burst are set to be the same for the bursts without observed PA rotation(s). So the calculated PDs for these bursts are the upper limits. For the large-scale ordered toroidal-fields case, both the predicted time-resolved and time-integrated PDs here are also the upper limits. Actually, for the bursts without PA rotation observations the predicted PD upper limits are similar for the two MFC cases. Although there is degeneracy between the parameters, the predicted time-integrated PDs with different sets of the parameters are roughly unchanged. Our results for the time-integrated PD are in general robust.

The 4 out of total 23 bursts have the time-resolved PA observations. For GRB 170206A, because of a comparable time-integrated PD to its time-resolved ones, the time-resolved PA is inferred to be a constant. And the observed best fit values of the time-resolved and time-integrated PDs are both smaller than the corresponding predicted upper limits. Therefore, it is inferred that mixed magnetic fields would be in the radiation regions of this burst. Among these 4 bursts, time-resolved PAs of 3 bursts show abrupt rotation(s), which would lead to a reduction of the time-integrated PDs. The observed PA rotations and the best fit values of the time-integrated PD of these 3 bursts could be recovered by the aligned-fields case with different orientations of the fields in the shells. The toroidal-fields case could not interpret such polarization observations of these 3 bursts. If the current polarization measurements for these bursts with abrupt PA rotations are roughly credible, to match with the observed time-integrated PD and the time-resolved abrupt PA rotations, the MFC in the radiation regions of these bursts would be dominated by large-scale ordered aligned-field component.

The predicted average values of the time-integrated PD here are around $44\%$ for the aligned-fields case and around $49\%$ for the toroidal-fields case, both of which are higher than the results of $37.4\%$ in Guan & Lan (2023). Compared to the constant time-resolved PAs predicted in the toroidal-fields case, the predicted PA rotations of the aligned-fields case lead to a smaller average value of the time-integrated PD. The results of the time-integrated estimation method used in Guan & Lan (2023); Toma et al. (2009) are relatively smaller than the results here. Because the predicted time-resolved PA is a constant during the main radiation epoch of each burst for the toroidal-fields case, the result here is comparable to that of $50\%$ in Sui & Lan (2024). Compared with the typical or fiducial parameters used in Sui & Lan (2024), the model parameters are inferred from the multi-window fitting here. Although the time-integrated PDs are both obtained via the time-resolved ones here and in Sui & Lan (2024), the results here should be more accurate, especially for the bursts with PA rotations.

For most of the bursts (22 out of total 23 bursts) in our sample, the predicted time- and energy-integrated PDs could match with the corresponding observations. There is no time-resolved PA observations or the time-resolved PA is indicated to be a constant for 19 bursts among these 22 bursts. Both the aligned-fields case and the toroidal-fields case could interpret the observational data of these 19 bursts equally well. However, the calculated upper limits for the time-integrated PD of one burst in our sample are lower than the corresponding observed lower limit of $1\sigma$ confidence level. Since the observational errors of the polarizations are large and the confidence level is relatively low, whether or not the models of the synchrotron emission in ordered magnetic fields are rejected for the this burst is not sure. Finally, the synchrotron radiation in ordered magnetic field(s) could interpret the multi-window observations of most bursts in our sample. Future accurate polarization measurements will put strict constrains on the MFC in the radiation region and the origin of the GRBs.