Revisiting the time-integrated polarizations of gamma-ray burst prompt phase

In the former studies, the time evolution information is missed in deducing the time-integrated polarizations of gamma-ray burst (GRB) prompt emission. Here, it is considered and the time-integrated polarizations is investigated through the summation of the time-resolved ones. The statistical properties of the distribution of the time-integrated polarization degree ($\Pi$) can be read from the $q-\Pi$ curve, where $q\equiv\theta_V/\theta_j$. $\theta_V$ and $\theta_j$ are the observational and jet half-opening angles, respectively. Hence, only the $q-\Pi$ curves are studied. In addition to a toroidal magnetic field in the radiation region, an aligned field is also discussed. We found the predicted time-integrated PD is around $(40-50)\%$ for High-energy Polarimetry Detector (HPD) on board POLAR-2 and is roughly $(30-40)\%$ for its Low-energy Polarimetry Detector (LPD). Therefore, $\Pi$ value detected by the HPD will be larger than that of the LPD in statistics and the result of the former estimations will underestimate the value of $\Pi$ in an ordered field. There are mainly two types of the $q-\Pi$ curve profiles, corresponding to two ordered magnetic field configurations.


INTRODUCTION
Gamma-ray bursts (GRBs) are the extensive γ-ray radiation at the cosmological distance, with extremely high isotropicequivalent luminosity of Liso ∼ 10 50 − 10 53 erg/s (Kumar & Zhang 2015).After more than two decades intensive studies, their emission mechanism (Rees & Meszaros 1994;Paczynski & Xu 1994;Giannios 2008; Zhang & Yan 2011;Granot 2016;Thompson 1994;Eichler & Levinson 1999, 2003;Lundman et al. 2018), jet structure (Rossi et al. 2004;Gill et al. 2020), and magnetic field configuration (Sari 1999;Granot & Königl 2003;Granot 2003;Toma et al. 2009;Lan et al. 2019) (MFC) in the emitting regions remain mysterious.The light curves of GRB prompt emission usually show abundant diversity.Different from the varieties of GRB prompt light curves, their spectra can be described by the Band function (Band et al. 1993).However, the origin of the Band function is also a mystery.The Band function is an empirical function, consisting of low-and high-energy power laws, smoothly connected around the peak energy Ep of the νFν spectrum (Band et al. 1993), which can not be interpreted by the fast-cooling synchrotron radiation in a constant magnetic field.Then the synchrotron origin of the Band spectrum in a decaying magnetic field was proposed (Pe'er & Zhang 2006;Derishev 2007;Zhao et al. 2014;Uhm & Zhang 2014).
To explain the observations of the GRB prompt emission, three popular models were proposed, i.e., the internal shock (IS) model (Rees & Meszaros 1994;Paczynski & Xu 1994), the magnetic reconnection (MR) model (Giannios 2008; Zhang & Yan 2011; Granot 2016), and the pho-tosphere model (Thompson 1994;Eichler & Levinson 1999, 2003;Lundman et al. 2013).The radiation mechanism of the first two models are the synchrotron emission, while it might be synchrotron emission in the low-energy X-ray band and be the inverse Compton scattering in the high-energy γ−ray band for the photosphere model (Lundman et al. 2018).The emission model of GRB prompt phase proposed by Uhm & Zhang (2015, 2016) and Uhm et al. (2018) was used here.It can mimic the emission from both the MR process and the IS process.
Up till now, most of the polarization observations in GRB prompt phase were given as the time-and energy-integrated ones, and the time-resolved polarization results are relatively rare (Yonetoku et al. 2011(Yonetoku et al. , 2012;;Zhang et al. 2019;Kole et al. 2020;Sharma et al. 2019;Chattopadhyay et al. 2022).Theoretically, time-resolved and time-integrated polarizations of GRB prompt emission were considered widely in the literature (Granot 2003;Lyutikov et al. 2003;Nakar et al. 2003;Lazzati et al. 2004;Granot & Taylor 2005;Toma et al. 2009;Lundman et al. 2014;Gill et al. 2020;Lan & Dai 2020;Cheng et al. 2020;Gill & Granot 2021;Lan et al. 2021a;Lan et al. 2021b;Guan & Lan 2023).In these above mentioned time-integrated models, the time-integrated Stokes parameters were directly constructed from the observed energy spectrum.So the time-evolving information of the polarization was not included.Recently, Gill & Granot (2024) considered the time-integrated polarizations in GRB prompt phase via the summation of the time-resolved ones with evolving Polarization angle (PA).
A rotating PA during the burst, which was already ob-served in 4 GRBs (Yonetoku et al. 2011;Zhang et al. 2019;Sharma et al. 2019;Kole et al. 2020;Chattopadhyay et al. 2022), can reduce the final time-integrated polarization degree (PD), as in the case of GRB 170114A (Zhang et al. 2019;Burgess et al. 2019;Kole et al. 2020).And the order degree of the magnetic field also decays during the main radiation episode, which would be the case for the MR process.
In addition, the energy spectrum would evolve during GRB prompt phase, which could affect the time-integrated PD of the jet emission (Guan & Lan 2023).Therefore, time-resolved polarization information will be important for the final timeintegrated ones and should be considered.
Here, we consider the time-and energy-resolved polarization using the polarization model proposed in Lan & Dai (2020) with a top-hat jet structure to investigate the timeand energy-integrated PD of GRB prompt phase.POLAR-2 is a forth coming mission dedicated to detect the GRB polarizations in both X-ray and γ−ray band (de Angelis & Polar-2 Collaboration 2022).The detection energy band of the two polarization detectors on board POLAR-2, the High-energy Polarimetry Detector (HPD) and Low-energy Polarimetry Detector (LPD), are 30-800 keV and 2-30 keV, respectively.We apply our model to predict the detection results of the HPD and LPD on board POLAR-2.The dependence of the results on the parameters are also investigated.This paper is arranged as follows.In Section 2, the polarization model is reviewed.The time-integrated PDs at energy band of LPD (2-30 keV) and HPD (30-800 keV) are presented in Section 3. Finally, we give our conclusion and discussion in Section 4.

THE MODEL
The emission model used here was proposed by Uhm & Zhang (2015, 2016) and Uhm et al. (2018).A thin, radially expanding relativistic jet shell continually emits photons from all positions in the shell.It begins to radiate at radius ron and stops at radius r of f .In the comoving frame of the shell, it is assumed that the radiation power has an isotropic angular distribution.The jet is assumed to have a top-hat structure with a half-opening angle of θj.Under the framework of the MR model, as the highly magnetized jet shell moves radially outward, because of the instabilities, the field with modulations would reconnect and release free magnetic energy to accelerate the jet and the electrons in it.There are two kinds of ordered magnetic field discussed in the literature of MR: aligned and toroidal (Spruit et al. 2001).The field lines of the aligned field are the latitude circles on the jet surface (Spruit et al. 2001;Lan et al. 2019), while they are concentric circles around the jet axis for the toroidal field (Spruit et al. 2001).The schematic diagrams of the two field configurations, the EATS, and the relevant angles are shown in Figure 1.
The Blandford-Znajek (BZ) mechanism would work for a black hole central engine and the field configuration in the jet at large radii would be toroidal (Blandford & Znajek 1977), while for a magnetar a striped wind is ejected via magnetic dipole radiation and MFC is aligned in the ejecta (Spruit et al. 2001).An aligned field is unstable to the tearing instability, while a toroidal field is highly unstable to the Kink instability.The turbulence would develop and the fields with different orientations will reconnect.Then the jet will be accelerated by the released free magnetic energy of the MR  process.The bulk Lorentz factor would vary in the form of power laws with respect to the radius (Drenkhahn 2002).
It was predicted that bulk acceleration patterns will vary with s = 0 for the toroidal field and with s = 1/3 for the aligned field under the MR model (Drenkhahn 2002).And the magnetic field strength in the co-moving frame of the radiating region will decrease with radius due to the adiabatic expansion of the jet shell and the MR process.
The emission model discussed above is not only suitable to the MR process, but also can mimic the emission of the internal shock (IS) model.In the scenario of the IS, the electrons are accelerated by the ISs and injected into the shell continuously until the shock crossed the shell.The magnetization would be lower than 10 −3 in order for the shocks to accelerate particles and the magnetic field would be dominated by the shock-generated small-scale random field (Sironi & Spitkovsky 2011).The decay of the ordered magnetic field for the MR process should be faster than b = 1 because besides the adiabatic expansion of the field with the jet shell it will also reconnect to deplete the free magnetic energy.So the field would be mixed with both ordered component and time-integrated polarizations 3 random component in the MR region.Another difference between the MR model and the IS model is that the bulk velocity of the shell will be constant for the IS (i.e., s = 0), while it can increase with radius for the MR model.
The polarization of the above emission model was predicted by Lan & Dai (2020), which was used here for the timeresolved polarization calculation.The expressions of the timeand energy-resolved Stokes parameters are slightly different from that in Lan & Dai (2020), we type these formula in the Appendix for reference.Then the energy-integrated Stokes parameters are derived with the following formula.
where Fν , Qν , and Uν are the time-and energy-resolved Stokes parameters.ν represent the observational frequency.hν1 and hν2 are the lower-and upper-limits of the corresponding detector, respectively.h is the Planck constant.
Since the flux and polarized flux are very tiny without T90, we neglect these contributions to the time-integrated polarization.Therefore, the time-integrated Stokes parameters are obtained by integrating the time-resolved ones within the T90.
where T5 and T95 are the times when the time-accumulated flux reaches 5% and 95% of the total flux during the burst duration, respectively.And T90 = T95 − T5. t obs is the observational time.
For an aligned field in the emission region, the timeand energy-integrated Stokes parameters Q and Ū are both nonzero, the final time-and energy-integrated PD (Π) and PA (χ) of the GRB emission can be calculated as: For a toroidal field, one of the Stokes parameters ( Ū ) is zero, then the Π is and the χ for Q > 0 will have a 90 • difference with that for Q < 0.
It should be noted that the magnetic field in the emission region of the MR model would be very likely to be mixed because of the instability and/or the turbulent developed in the plasma.Since the profile of the PD curve with a mixed field will be very similar to that with the corresponding ordered field (Lan et al. 2019;Lan & Dai 2020), here we only assume that magnetic field is large-scale ordered in the radiation region.Hence, the time-and energy-integrated PD predictions (Π) in this paper would be the theoretical upper limits.

NUMERICAL RESULTS
Because the current polarization analysis can only be done for bright bursts, in general the observation angle θV , which is the angle between the line of sight and the jet axis, for the bursts with polarization detection should be within the jet cone, i.e., with q ≡ θV /θj ≤ 1.In statistics, the value of Π at the plateau stage of the q − Π curve with typical parameters when q < 1 would be equal to the mean value of the timeintegrated PD of the simulated bursts (Toma et al. 2009;Lan et al. 2021a).So the predicted q − Π curve with indicated typical parameters, especially for q ≤ 1, will be important.Although the flux density will drop dramatically when q > 1 and it is impractically to measure the polarization at large off-axis angles, the q − Π curves with q > 1 are also studied for reference here.
Single-energy electrons are assumed in the model here, as in the former studies (Uhm & Zhang 2015, 2016;Uhm et al. 2018;Lan & Dai 2020), their Lorentz factor is denoted as γ ch .Two γ ch patterns are considered, corresponding to two general Ep evolution mode (i.e., hard-to-soft and intensitytracking modes).
First, we discuss the influences of the parameter s and the γ ch patterns on the time-and energy-integrated PD (Π).The results are shown in Figure 2. Total 8 models are compared, SOAi with s = 0, SOAi with s = 0.35, SOAm with s = 0, SOAm with s = 0.35, SOTi with s = 0, SOTi with s = 0.35, SOTm with s = 0, and SOTm with s = 0.35.Here, the "SO" represents the synchrotron emission in an ordered magnetic field, "A" or "T" are for the aligned or the toroidal field, and "i" or "m" correspond to the "i" (with a hard-to-soft Ep evolution pattern) or "m" (with an intensity-tracking Ep mode) mode.
There are two different variation profiles of the q − Π curve for these 8 models in Figure 2, corresponding to the two different ordered MFCs in the radiation region.The two profiles are independent of the bulk Lorentz factor pattern (s), the Ep evolution mode, and the observational energy band.Π values Figure 2. PD (Π) variations with q ≡ θ V /θ j .The left and right panels correspond to the results of the HPD (30-800 keV) and LPD (2-30 keV), respectively.The upper and lower panels are for the aligned and toroidal field in the emission region, respectively.In each upper panel, the red squares, green circles, blue diamonds, and pink stars represent the models of SOAi with s = 0, SOAi with s = 0.35, SOAm with s = 0, and SOAm with s = 0.35.In each lower panel, the red squares, green circles, blue diamonds, and pink stars represent the models of SOTi with s = 0, SOTi with s = 0.35, SOTm with s = 0, and SOTm with s = 0.35.
around q = 0 will increase rapidly with q for the toroidal field because the axial symmetry of the observed emission region was broken speedy, while for an aligned field there is no such symmetry and the asymmetry offered by the aligned field keeps roughly unchanged, hence Π maintains a roughly constant value for q < 1.And also because of different asymmetry, Π will increase with q to roughly 70% when q ≫ 1 after a sudden decay for the aligned field, while it will roughly decay shallowly with q to zero when q ≫ 1 after a sudden decay for the toroidal field.
To interpret the different variation trends of the aligned and toroidal fields at large q value (q ≫ 1), we plot the local polarization distribution on the plane of sky for the SOAi and SOTi models with a constant velocity jet shell (i.e., s = 0) at the peak time of the light curve of q = 3.The observational energy is taken as 400 keV for reference 1 .The other parameter are taken as their fixed values.For an aligned field, the local PD (P D(θ, ϕ)) and PA (P A(θ, ϕ)) of the emission from a point-like region (Sari 1999) are defined as follows.
1 The local polarization distribution is similar for the observational energy of 15 keV.
where fν (θ, ϕ), qν (θ, ϕ) and uν (θ, ϕ) are the Stokes parameters of the point-like region.While for a toroidal field, one of the Stokes parameters (uν (θ, ϕ)) is zero, then the local PD for a point-like region is Its local PA will rotate abruptly by 90 • when the local PD changes its sign.
For the aligned field, its local PD (P D(θ, ϕ)) is a constant and equals to (βs + 1)/(βs + 5/3) = 0.77 for the observational energy of 400 keV.Since the syntropy of the local PA (P A(θ, ϕ)) is good at the regions where the local flux (fν (θ, ϕ)) remains a relatively high value, the PD of the jet radiation (Π) with an aligned field would be as large as 60% at q = 3 for HPD.While for the toroidal field, the local flux also decrease with θ and the polarized flux are almost been cancelled.So the PD of the jet radiation with a toroidal field would be small.
There are only small differences in concrete Π value in each panel, and the influences of the bulk Lorentz factor pattern (s) and the Ep evolution mode on the time-and energyintegrated PD are very limited.Therefore, in the following we will focus on the "i" model with s = 0 to discuss the influences of other parameters on Π.It should be noted that Π will be larger for HPD than LPD at the plateau of the q − Π curve when q < 1.Since in statistics the value of Π at the plateau stage of the q − Π curve with typical parameters when q < 1 would be equal to the observed mean value of the time-integrated PD (Lan et al. 2021a), the detected mean value of HPD will be larger than that of LPD.This result is consistent with that in Guan & Lan (2023).
Then the influences of the magnetic field strength on the time-and energy-integrated PD are investigated.The results are shown in Figure 4. Three parameter sets of (B ′ 0 , b) are considered (Uhm & Zhang 2014), i.e., (30 G, 1), (30 G, 1.5), (100 G, 1).Again, there are two q − Π profiles corresponding to two MFCs.And different values of both B ′ 0 and decay index b will not affect the q − Π profile and have limited influences on the Π value.We note that, for example for the HPD, there are ≤ 10% difference for the time-integrated PDs at the plateau phase when q < 1 for the different combinations of (B ′ 0 , b).Because the time-integrated flux and polarized flux are mainly from the peak time of the light curve and the PA is roughly a constant for on-axis observations, time-integrated PD of the single-pulse burst is roughly equal to the timeresolved PD at the time of the light-curve peak.So we use the PD at peak time to investigate.We found that the influence of different combinations of (B ′ 0 , b) on the distribution of the local flux density on the EATS will be larger than that for the local polarized flux2 , which will result in a different time-resolved PD at the peak time for different combinations of (B ′ 0 , b), so does the time-integrated one.And we will simply take B ′ 0 = 30 G and b = 1 in our following analysis.In Figure 5, the influences of different sets of (Γ0, θj) with a constant Γ0θj value are studied.Since the estimated values of θj and Γ0 are around 0.1 rad and 100 (Lloyd-Ronning et al. 2019;Rouco Escorial et al. 2023;Ghirlanda et al. 2018), here we take Γ0θj = 10.As our above studies, depending on the two MFCs, there are two q − Π profiles and different set of (Γ0, θj) with a Γ0θj value of 10 will not affect the q − Π profile and has a limited influence on Π value.Same as the case with different combinations of (B ′ 0 , b), there are also ≤ 10% difference for the time-integrated PDs at the plateau phase when q < 1 for the different sets of (Γ0, θj).With the study, same as that for the combinations of (B ′ 0 , b), the influence of different sets of (Γ0, θj) on the distribution of the local flux density on the EATS will be larger than that for the local polarized flux, which will result in a different timeresolved PD at the peak time for different sets of (Γ0, θj), so does the time-integrated one.The variations of the q − Π curve with both Γ0 and θj are also investigated, respectively.Independent to the concrete values of Γ0 or θj, the profiles of the q − Π curves are similar if the values of Γ0θj are the same.
In statistics, the predicted PDs with a large-scale ordered magnetic field in its radiation region will concentrate around (40 − 50)% for HPD and around (30 − 40)% for LPD, while it is around (20 − 40)% for HPD and around (15 − 30)% for LPD in the former estimations (Guan & Lan 2023).The predicted PD here is higher than that of the former estimation.That is because the variations of magnetic field strength, bulk Lorentz factor, and the Lorentz factor of electrons in the comoving frame with radius would also affect the final timeand energy-integrated PD, as shown in Figures.2, 4-5.

CONCLUSION AND DISCUSSION
The time evolution information was missed in the former studies of the time-integrated PD in GRB prompt phase (Granot 2003;Lyutikov et al. 2003;Nakar et al. 2003;Granot & Taylor 2005;Toma et al. 2009;Guan & Lan 2023).Here, it is considered with the time-resolved polarization model proposed in Lan & Dai (2020).The model can mimic both the emission from the MR model with an accelerating or a constant-velocity shell and IS model with a constant-velocity shell.Here, in addition to a toroidal field, a large-scale aligned field is also considered, which is also one of the popular MFC in the literature (Spruit et al. 2001;Drenkhahn 2002).The profile of the q − Π curve for an aligned field is very different SOTi ( 0 = 1000, j = 0.01 rad ) Figure 5. Same as Figure 2, but the product of Γ 0 θ j is a constant value of 10 and only the models with "i" mode and s = 0 are considered.
time-integrated polarizations 7 from that of a toroidal field.The results here are consistent with that in Toma et al. ( 2009) and Guan & Lan (2023).The statistical properties of the time-integrated PD can be read roughly from the q − Π curve (Toma et al. 2009;Lan et al. 2021a).The time-integrated PDs of the most simulated bursts will concentrate on the Π value at the plateau stage of the q−Π curve with typical values of the parameters when q < 1 (Lan et al. 2021a).Most of the GRBs used for polarization analyse are very bright, indicating on-axis observations.The observed time-integrated PDs should also concentrate on the PD values at the plateau of the q − Π curve with typical values of the parameters for q < 1.It is known that the local PD in an ordered field will be larger for high-energy spectral index β than low-energy spectral index α (Rybicki & Lightman 1979).Since the high-energy photons with large spectral index (i.e., β) become less when Ep increases, Π of the jet emission will decrease with Ep (Toma et al. 2009;Guan & Lan 2023).Hence, the statistical properties do not need to be studied specifically.
Here, the predicted PAs are roughly constant within T90 for on-axis observations with q ≤ 1 and the magnetic field is assumed to be large-scale ordered.Therefore, the predicted time-integrated PD of GRB prompt phase is the upper limit, which is around (40 − 50)% with typical parameters for HPD and is larger than that of (20 − 40)% the former studies (Toma et al. 2009;Guan & Lan 2023).Hence, the upper limit of Π calculated here could also interpret most of the current time-integrated PDs observed by GAP, POLAR, and AstroSat (Yonetoku et al. 2011(Yonetoku et al. , 2012;;Zhang et al. 2019;Kole et al. 2020;Sharma et al. 2019;Chattopadhyay et al. 2022).The predicted upper limit of Π at 250 keV is around 25% in Lan et al. (2021a) (see Fig. 1 in their paper), that is because their derivation of the comoving peak frequency (hν ′ 0 = Ep(1 + z)/D) is wrong.However, the profiles of their q − Π curve for the aligned and toroidal fields are similar to our results here.The Π value here for the LPD at the plateau stage of the q −Π curve will be relatively smaller ((30−40)%) than that for the HPD, and the result for LPD here is also larger than that of (15 − 30)% in Guan & Lan (2023).
The profile of the q − Π curve depends mainly on both the MFC in the emitting region.Because the asymmetry offered by the large-scale ordered aligned magnetic field is roughly unchanged when q ≤ 1, Π values around q = 0 are also constants as in the Π plateau phase when 0 < q < 1 for an aligned field, while it is very different for a toroidal field with a sharp rise of Π starting from 0 when q ≥ 0 followed by a plateau phase when q < 1.Also due to different asymmetries, roughly when q > 1+1/(Γ0θj), Π will increase with q to about 70% for the aligned field, while it decays shallowly to zero for the toroidal field.Although the comoving peak frequency is incorrect in Lan et al. (2021a), the profiles of their q − Π curve for the aligned and toroidal field are similar to the results here.
Although the calculation methods are different, our profile of the q − Π curve for the toroidal field is consistent with that in Toma et al. (2009).Although both the time-integrated flux density and the time-integrated Ep are mainly determined by emission episode around the light-curve peak, and PA for on-axis observations is a constant within T90 (Wang & Lan 2023), the variations of the parameters on the EATS would also affect the value of Π.So the predicted Π here is larger than that estimated via the former time-integrated method.
Therefore, the calculation method of the time-integrated PD proposed in Granot (2003); Lyutikov et al. (2003); Nakar et al. (2003); Granot & Taylor (2005);Toma et al. (2009) and Guan & Lan (2023) can only be a preliminary estimation.Although the predicted PAs are almost constants during the burst duration for the on-axis observations, it will rotate for slightly off-axis observations (Wang & Lan 2023), for which the flux is roughly comparable to the on-axis one.Hence, the time-resolved method has the potential to interpret the observations with PA rotation (Wang & Lan 2023) and will be investigated in detail in our future work.

Figure 1 .
Figure 1.(a) Schematic diagrams for the two kinds of the ordered magnetic field in the jet surface.The blue arrows show the field directions.And the cross represents the jet axis.(b) Schematic cross-sections of the EATSes at different times for on-axis (top) and off-axis (bottom) observations.The radiation starts from ton and ceases at t of f .

Figure 3 .
Figure 3.The distributions of the local flux (fν (θ, ϕ)) and local polarizations on the plane of sky for SOAi model (a) and SOTi model (b).The red and black pluses show the projections of the LOS and jet axis.The red-dashed and thick-solid-black circles represent the projections of the 1/Γ 0 cone and the jet cone.(a).The local PD of the emission from a point-like region is a constant of 0.77.The directions of the equivalent-length black short-lines show the directions of the local PA (P A(θ, ϕ)).The green dashed line represents the direction of the aligned field.(b).Upper panel shows the distribution of the local flux (fν (θ, ϕ)).The lower panel shows the distribution of the local PD (P D(θ, ϕ)).The local PA (P A(θ, ϕ)) with a red color is rotated by 90 • compared with that of blue color.The green dashed circle shows the field lines of the toroidal field.