Detailed polarization measurements of the prompt emission of five Gamma-Ray Bursts

TL;DR

This study presents precise polarization measurements of five gamma-ray bursts using the POLAR instrument, providing insights into jet structures and magnetic fields, with results indicating low polarization levels and evolving polarization angles.

Contribution

First statistically meaningful polarization measurements of gamma-ray bursts with the POLAR instrument, clarifying polarization levels and evolution during bursts.

Findings

Gamma-ray emission is at most weakly polarized.

Polarization angle evolves within bursts.

Results challenge some existing models predicting higher polarization.

Abstract

Gamma-ray bursts are the strongest explosions in the Universe since the Big Bang, believed to be produced either in forming black holes at the end of massive star evolution or merging of compact objects. Spectral and timing properties of gamma-ray bursts suggest that the observed bright gamma-rays are produced in the most relativistic jets in the Universe; however, the physical properties, especially the structure and magnetic topologies in the jets are still not well known, despite several decades of studies. It is widely believed that precise measurements of the polarization properties of gamma-ray bursts should provide crucial information on the highly relativistic jets. As a result there have been many reports of gamma-ray burst polarization measurements with diverse results, see, however many such measurements suffered from substantial uncertainties, mostly systematic. After the first successful measurements by the GAP and COSI instruments, here we report a statistically meaningful sample of precise polarization measurements, obtained with the dedicated gamma-ray burst polarimeter, POLAR onboard China's Tiangong-2 spacelab. Our results suggest that the gamma-ray emission is at most polarized at a level lower than some popular models have predicted; although our results also show intrapulse evolution of the polarization angle. This indicates that the low polarization degrees could be due to an evolving polarization angle during a gamma-ray burst.

Figures

Tables

Table 1. Table 1: Summary of the five GRBs. The different properties of the 5 GRBs and of 2 time bins of GRB 170114A (*in units of erg/cm 2 in 10-1000 keV).

GRB	T90 (s)	Fluence*	PD	Prob(PD2%)	PDup(99%)	PA(deg.)	PA Change
161218A	6.76	$1.25\times {10}^{-5}$	9%	9%	45%	40	No
170101A	2.82	$1.27\times {10}^{-5}$	8%	13%	31%	164	No
170127C	0.21	$7.4\times {10}^{-6}$	11%	5.8%	67%	38	Unknown
170206A	1.2	$1.34\times {10}^{-5}$	10%	12%	31%	106	No
170114A	8.0	$1.93\times {10}^{-5}$	4%	14%	28%	164	Yes
170114Ap1	N$/$A	N$/$A	15%	8%	43%	122	N$/$A
170114Ap2	N$/$A	N$/$A	41%	0.49%	74%	17	N$/$A

Table 2. Table 2: Summary of all published GRB prompt emission polarization measurements. The quoted errors or limits are at the 1 ​ σ 1 𝜎 1\sigma level unless otherwise specified. (*The result of GRB 170114A is from the time binned analysis with 3 time bins where PD was assumed to be constant during the burst.)

GRB	Instr./Sat.	Pol. ($\%$)	Ref.
170206A	POLAR	($99\%$ confidence)	This Paper
170127C	POLAR	($99\%$ confidence)	This Paper
170114A	POLAR	($99\%$ confidence)	This Paper
170114A	POLAR	$28\pm 9\%$	This Paper, time binned*
170101A	POLAR	($99\%$ confidence)	This Paper
161218A	POLAR	($99\%$ confidence)	This Paper
160530A	COSI	($90\%$ confidence)	ref.  [2]
110721A	GAP/IKAROS	${84}_{-28}^{+16}$	ref.  [3]
110301A	GAP/IKAROS	$70\pm 22$	ref.  [3]
100826A	GAP/IKAROS	$27\pm 11$	ref.  [4]
021206	RHESSI	$80\pm 20$	ref.  [5]
021206	RHESSI	${41}_{-44}^{+57}$	ref.  [6]
140206A	IBIS/INTEGRAL	$\ge 28$ ($90\%$ confidence)	ref.  [7]
061122	IBIS/INTEGRAL	$\ge 33$ ($90\%$ confidence)	ref.  [8]
041219A	IBIS/INTEGRAL	$\le 4/43\pm 25$	ref.  [9]
041219A	SPI/INTEGRAL	$98\pm 33$	ref.  [10]
960924	BATSE/CGRO	$\ge 50$	ref.  [11]
930131	BATSE/CGRO	$\ge 35$	ref.  [11]

Table 3. Table 3: The resulting PD and PA values together with the corresponding χ 2 superscript 𝜒 2 \chi^{2} /NDF acquired when selecting different intervals between the signal and background time intervals.

time interval	PD. (%)	PA (deg.)	minimum ${\chi }^2$/NDF
5	4%	${164}^{\circ }$	1.19
45	3%	${164}^{\circ }$	1.08
90	3%	${164}^{\circ }$	1.18

Table 4. Table 4: The resulting PD and PA values together with the corresponding χ 2 superscript 𝜒 2 \chi^{2} /NDF acquired when selecting different background time intervals in the analysis.

time interval	PD. (%)	PA (deg.)	minimum ${\chi }^2$/NDF
prior	5%	${164}^{\circ }$	1.15
post	7%	${164}^{\circ }$	1.27
post and prior	4%	${164}^{\circ }$	1.21

Table 5. Table 5: The resulting PD and PA values together with the corresponding χ 2 superscript 𝜒 2 \chi^{2} /NDF acquired when using spectral parameters from different instruments.

spectrum source	PD (%)	PA (deg.)	${\chi }^2$/NDF
Fermi-GBM	10%	${106}^{\circ }$	26.3/27
Konus-Wind	9%	${106}^{\circ }$	26.5/27

Table 6. Table 6: The resulting PD and PA values together with the corresponding χ 2 superscript 𝜒 2 \chi^{2} /NDF acquired under different incident angle conditions.

angle	PD (%)	PA (deg.)	${\chi }^2$/NDF
normal	4%	${164}^{\circ }$	32.3/27
$\theta$ + 3.7∘	3%	${154}^{\circ }$	33.5/27
$\varphi$ + 3.7∘	4%	${162}^{\circ }$	34.4/27

Table 7. Table 7: List of GRB parameters used for the polarization analysis. For all except one a Band function was used to model the spectrum, for GRB 170127C a single cut-off power law (*CPL) model was used.

GRB	161218A ref. [18]	170101A ref. [19]	170114A ref. [15]	170127C ref. [20]	170206A ref. [14]
Tstart	2016-12-18 T03:47:34.0	2017-01-01 T02:26:00.66	2017-01-14 T22:01:09	2017-01-27 T01:35:47.75	2017-02-06 T10:51:57.400
Tstop	2016-12-18 T03:47:40.5	2017-01-01 T02:26:02.16	2017-01-14 T22:01:18	2017-01-27 T01:35:47.95	2017-02-06 T10:51:59.400
Tinterval	5 seconds	5 seconds	5 seconds	90 seconds	5 seconds
RBOS	14	20	10	50	20
Instrument (Spectrum)	Konus-Wind	Konus-Wind	Fermi-GBM	Fermi-GBM	Fermi-GBM
Model	Band	Band	Band	CPL*	Band
$\alpha$	-0.28${}_{}{}^{+0.25}{}_{-0.21}{}^{}$	-1.44${}_{}{}^{+0.17}{}_{-0.13}{}^{}$	-0.84$\pm$0.04	-0.27$\pm$0.09	-0.28$\pm$0.04
$\beta$	-3.40${}_{}{}^{+0.43}{}_{-1.17}{}^{}$	-2.49${}_{}{}^{+0.23}{}_{-0.65}{}^{}$	-1.99$\pm$0.06	N/A	-2.55$\pm$0.12
$E_{\mathrm{peak}}$ (keV)	128${}_{}{}^{+8}{}_{-8}{}^{}$	123${}_{}{}^{+23}{}_{-21}{}^{}$	237$\pm$19	859$\pm$34	341$\pm$13
Instrument (Localization)	Swift-BAT	Swift-BAT	Fermi-GBM	Fermi-LAT	IPN
$\theta$	${24.3}^{\circ }$	${5.6}^{\circ }$	${26.5}^{\circ }$	${41.8}^{\circ }$	${18.3}^{\circ }$
$\varphi$	${356.1}^{\circ }$	${68.9}^{\circ }$	${5.4}^{\circ }$	${157.6}^{\circ }$	${147.2}^{\circ }$
Error       (Localization)	3′ (90%)	1.1′ (90%)	1∘ (1$\sigma$)	0.4∘ (90%)	1156 ${{(}^{\prime })}^2$ (3$\sigma$)

Table 8. Table 8: List of results from the polarization analysis.

GRB	161218A	170101A	170114A	170127C	170206A
EventsS	2079	1958	3709	685	2566
EventsB	3876	709	3971	98	796
M100	37%	38.5%	37.0%	30.0%	37.5%
MDP	43.0%	29.4%	27.4%	58.4%	25.9%
PD	9%	8%	4%	11%	10%
PA	${40}^{\circ }$	${164}^{\circ }$	${164}^{\circ }$	${38}^{\circ }$	${106}^{\circ }$
${\chi }^2$/NDF	20.6/27	33.7/27	32.3/27	5.3/12	26.3/27
Prob(PD2%)	9%	13%	5.8%	12%	14%
PDup (99%)	45%	31%	28%	67%	31%

Equations

$${\chi }^2=\sum _{i=1}^n\frac{(X_i-Y_i{)}^2}{{\epsilon }_i^2+{\sigma }_i^2},$$

$$P({\mathbf{B}}_i|\mathbf{A})=\frac{P(\mathbf{A}|{\mathbf{B}}_i)P({\mathbf{B}}_i)}{{\sum }_{j=1}^{n}P(\mathbf{A}|{\mathbf{B}}_j)P({\mathbf{B}}_j)}.$$

$$h(\mathbf{B}|\mathbf{A})=\frac{f(\mathbf{A}|\mathbf{B})\pi (\mathbf{B})}{{\int }_{\mathbf{B}}f(\mathbf{A}|\mathbf{B})\pi (\mathbf{B})d\mathbf{B}}.$$

$$f(\mathbf{A}|\mathbf{B})d\mathbf{A}\Rightarrow P(\mathbf{A}|{\mathbf{B}}_i)\pi (\mathbf{B})d\mathbf{B}\Rightarrow P({\mathbf{B}}_i)h(\mathbf{B}|\mathbf{A})d\mathbf{A}\Rightarrow P({\mathbf{B}}_i|\mathbf{A}),$$

Full text

Detailed polarization measurements of the prompt emission of five Gamma-Ray Bursts

Shuang-Nan Zhang Corresponding Author. These authors contributed equally: Shuang-Nan Zhang, Merlin Kole.

Merlin Kole∗

Tian-Wei Bao

Tadeusz Batsch

Tancredi Bernasconi

Franck Cadoux

Jun-Ying Chai

Zi-Gao Dai

Yong-Wei Dong

Neal Gauvin

Wojtek Hajdas

Mi-Xiang Lan

Han-Cheng Li

Lu Li

Zheng-Heng Li

Jiang-Tao Liu

Xin Liu

Radoslaw Marcinkowski

Nicolas Produit

Silvio Orsi

Martin Pohl

Dominik Rybka

Hao-Li Shi

Li-Ming Song

Jian-Chao Sun

Jacek Szabelski

Teresa Tymieniecka

Rui-Jie Wang

Yuan-Hao Wang

Xing Wen

Bo-Bing Wu

Xin Wu

Xue-Feng Wu

Hua-Lin Xiao

Shao-Lin Xiong

Lai-Yu Zhang

Li Zhang

Xiao-Feng Zhang

Yong-Jie Zhang

Anna Zwolinska

Gamma-ray bursts are the strongest explosions in the Universe since the Big Bang, believed to be produced either in forming black holes at the end of massive star evolution  [1, 2, 3] or merging of compact objects [4]. Spectral and timing properties of gamma-ray bursts suggest that the observed bright gamma-rays are produced in the most relativistic jets in the Universe [4]; however, the physical properties, especially the structure and magnetic topologies in the jets are still not well known, despite several decades of studies. It is widely believed that precise measurements of the polarization properties of gamma-ray bursts should provide crucial information on the highly relativistic jets [5]. As a result there have been many reports of gamma-ray burst polarization measurements with diverse results, see [1], however many such measurements suffered from substantial uncertainties, mostly systematic  [7, and references therein]. After the first successful measurements by the GAP and COSI instruments  [2, 3, 4], here we report a statistically meaningful sample of precise polarization measurements, obtained with the dedicated gamma-ray burst polarimeter, POLAR onboard China’s Tiangong-2 spacelab. Our results suggest that the gamma-ray emission is at most polarized at a level lower than some popular models have predicted; although our results also show intrapulse evolution of the polarization angle. This indicates that the low polarization degrees could be due to an evolving polarization angle during a gamma-ray burst.

POLAR is a dedicated Gamma-ray Burst (GRB) polarization detection experiment onboard China’s Tiangong-2 spacelab [11], launched on Sept. 15th, 2016 and stopped operation on March 31, 2017. POLAR detected 55 GRBs with high significance. In order to make statistically significant GRB polarization measurements and yet with negligible systematic errors, we select a subsample of five GRBs for detailed analysis of their polarization properties; please refer to the supplementary information (SI) for the sample selection criteria and the properties of the five selected GRBs. We employ a straight forward $\chi^{2}$ based analysis, similar to that successfully employed in [4], to study the polarization properties of the five GRBs, while a Bayesian method is employed to accurately determine the credible regions of the measurements. The studies rely on extensive ground and in-orbit calibration data and Monte-Carlo simulations matching the calibration data [12, 13]. Please refer to the methods section for details of the methodology and analysis.

In Figure 1, we show the measured modulation curves of the five GRBs integrated over the whole GRB duration, together with the best fitting simulated modulation curves from linear polarization and fitting residuals. All fittings are statistically acceptable with no significant systematic deviations. In Figure 2, we show the 2-D posterior distributions of the five GRBs, i.e., the posterior probability as functions of both polarization angle (PA) and degree (PD). Clearly the measured PD is correlated with PA for all five GRBs; we thus calculate the cumulative probabilities of PD marginalized over the PA for all five GRBs, as shown in Figure 3. In Table 1 we summarize the main results for the five GRBs, including for each GRB the name, T90 value [14] (the time interval over which $90\%$ of the total background-subtracted counts are observed), the fluence, the most likely value of PD, the compatibility of the time integrated polarization with an unpolarized flux, the upper limit of the PD, and the most likely PA and evidence for a change in the PA.

We conclude that for the PD measurement, the most probable PD values for all five GRBs are between 4% to 11%. The 99% upper limit ranges between 28% to 45% for four GRBs and 67% for GRB 170127C, which has the lowest fluence and thus the poorest statistics for polarization measurement. Although the analysis provides non-zero polarization degrees, it should be noted that linear polarization measurements always provide positive results, see for example [15, and references therein] for a discussion. The low polarization degrees found in the analysis presented here do not allow us to fully reject the hypothesis that every GRB is non-polarized. However, from Figure 3 we can find for each GRB the probability that PD is less than a certain positive value, which turns out to be between 5.8% and 14% for 2% PD. Since all the five GRBs show statistically consistent PD, we can combine the measurements to give joint lower and upper limits. As can be found in the bottom right panel of Figure 3, the average PD is 10% and the probability is 0.1% that the polarization degrees of all the five GRBs are lower than 5% or higher than 16%. Our results indicate for the first time with high precision that the prompt gamma-ray emission of GRBs is not highly polarized, as some models have predicted. Furthermore the results favor a low polarization level and can reject the hypothesis that all GRBs are unpolarized while several individual GRB polarization levels are still found to be consistent with zero polarization. Further evidence for a low but non-zero PD values was found when performing time dependent analysis.

Previously it has been reported that the PA of GRBs can change during the GRB [4]. We thus divide each of all five GRBs into two equal time bins and make the same analysis, to examine if the PA changed during each burst. As shown in detail in the methods section, a significant PA change was not observed for three GRBs, can not be constrained for GRB 170127C due to low statistics but was clearly detected for GRB 170114A. Interestingly, GRB 170114A is a single pulse GRB and also has the lowest PD (4%) among all five GRBs integrated over the whole GRB periods. Performing a time resolved analysis, similar to that applied previously by the GAP collaboration on the long multi-peak GRB 100826A, we find that the polarization properties of this GRB are best described by a constant PD of $28\%$ with a PA which changes significantly during three 2 second time bins. While the time integrated analysis of this GRB is consistent with an unpolarized flux, the time resolved analysis results in a PD which is inconsistent with an unpolarized flux with a $99.7\%$ confidence level. While the measurement results of GAP for GRB 100826A already showed strong evidence of changes in the PA for different peaks inside of a long GRB, the measurement of GRB 170114A shows an intrapulse evolution of the PA. This implies that the measured low PD for all GRBs in the sample presented here integrated over the whole burst duration might be due to rapid changes of PA during the bursts. This however cannot be well constrained with the current instruments.

In summary, with the dedicated GRB polarization instrument POLAR onboard China’s Tiangong-2 spacelab, we have measured the polarization properties of five GRBs with high precision, so far the largest sample of GRB polarization measurements with high statistical significance. For the first time we find that averaged over the whole bursts, GRBs are at most modestly polarized, i.e, PD$\approx$10%. For the brightest GRB in the sample an unpolarized flux can be rejected using time resolved analysis and a PD of $28\pm 9\%$ is found. Furthermore for the first time an intrapulse evolution of the PA is detected which has important implications for the time integrated PD measurements, as these low measurement values of the PD of all GRBs might be due to rapid PA changes during GRBs. Statistically most of previous prompt polarization emission measurements of GRBs, though some of quite high polarization degrees at face values, are consistent with our results, given their larger uncertainties in most cases. A summary of all previous reported GRB prompt emission polarization measurements is given in Supplementary Table 1 of the Supplementary Material.

Regarding the theoretical interpretation there are three main factors that affect the polarization properties significantly, these are the emission mechanism [16, 17, 18], the magnetic field configuration (MFC) [19, 5, 20, and references therein], and the jet geometry [21, 22]. Several popular models have been proposed to interpret GRBs, e.g., internal shock [23, 24], dissipative photosphere [25, 26], and internal-collision-induced magnetic reconnection and turbulence (ICMART) [27]. Depending on the MFC in the collision shell, the predicted polarization degree (PD) of the internal shock model can vary from a few% to 70%, about 10% PD for the mixed MFC and up to about 70% for the large-scale ordered MFC. For the dissipative photosphere model, the predicted PD is relatively low in the $\gamma$-ray band [28]. A decaying PD is predicted during each pulse in the ICMART model; the maximum value at the beginning of the pulse can reach up to 60% and the minimum value at the end decreases to a few %-10% as presented in  [27]. However, the detailed numerical simulations show that $\sim 10\%$ PD is also possible for the ICMART model [29]. As our results overall show a relatively low PD at first sight they appear to agree with the dissipative photosphere model. However, this model has difficulty to explain the PD of $28\%$ detected for GRB 170114A using the time binned analysis as well as the change in PA which appears to be of a continuous nature. The average PD level of 10% is possible according to the ICMART model, depending on the MFC in the emission region. For the internal shock model, the predicted PD can range from few% to 70%, depending on the MFC in the collision shells. The observed $\sim 10\%$ PD with POLAR suggests that the MFC is most likely mixed during the GRB prompt phase. Alternatively, a violent change of the PA can also result in a decrease of the time-integrated PD, which should have significant implications to all GRB models. To further understand the physics and astrophysics of the most relativistic jets produced by the strongest explosions in the Universe, future GRB polarization instruments are required to provide time-resolved polarization properties of much larger samples of GRBs.

The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to Shuang-Nan Zhang (zhangsn@ihep.ac.cn) and/or Merlin Kole (Merlin.Kole@unige.ch).

Acknowledgments

We gratefully acknowledge the financial support from the National Basic Research Program (973 Program) of China (Grant No. 2014CB845800), the Joint Research Fund in Astronomy under the cooperative agreement between the National Natural Science Foundation of China and the Chinese Academy of Sciences (Grant No. U1631242), the National Natural Science Foundation of China (Grant No. 11503028, 11403028), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB23040400), the Swiss Space Office of the State Secretariat for Education, Research and Innovation (ESA PRODEX Programme), the National Science Center of Poland (Grant No. 2015/17/N/ST9/03556), and the Youth Innovation Promotion Association of Chinese Academy of Sciences (Grant No. 2014009). We furthermore thank Dr. J.M. Burgess of MPE, Garching, Germany, for providing the energy spectra for 170114A.

Authors Contributions

The author list is in alphabetical order except the two first names, who are the first authors. T.W. Bao, T.Batsch, T. Bernasconi, F. Cadoux, J.Y. Chai, Y.W. Dong, N. Gauvin, W. Hajdas, M. Kole, H.C. Li, L. Li, Z.H. Li, J.T. Liu, X. Liu, R. Marcinkowski, S. Orsi, M. Pohl, N. Produit, D. Rybka, H.L. Shi, L.M. Song, J.C. Sun, J. Szabelski, T. Tymieniecka, R.J. Wang, X. Wen, B.B. Wu, X. Wu, H.L. Xiao, S.L. Xiong, L.Y. Zhang, L. Zhang, S.N. Zhang, X.F. Zhang, Y.J. Zhang and A. Zwolinska contributed to the development of the mission concept and/or construction and testing of POLAR. M. Kole, Z.H. Li, N. Produit, J.C. Sun, Y.H. Wang, S.L. Xiong and S.N. Zhang were involved in the presented analysis. Z.G. Dai, M.X. Lan and X.F. Wu contributed to the theoretical discussions in this manuscript. The manuscript was produced by M. Kole, Z.H. Li, J.C. Sun, Y.H. Wang and S.N. Zhang. The PIs of the POLAR collaboration are S.N. Zhang, M. Pohl and X. Wu.

Author Information

*Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China

*Shuang-Nan Zhang, Tian-Wei Bao, Jun-Ying Chai, Yong-Wei Dong, Han-Cheng Li, Lu Li, Zheng-Heng Li, Jiang-Tao Liu, Xin Liu, Hao-Li Shi, Li-Ming Song, Jian-Chao Sun, Rui-Jie Wang, Yuan-Hao Wang, Xing Wen, Bo-Bing Wu, Hua-Lin Xiao, Shao-Lin Xiong, Lai-Yu Zhang, Li Zhang, Xiao-Feng Zhang, Yong-Jie Zhang

*University of Chinese Academy of Sciences, Beijing 100049, China

*Shuang-Nan Zhang, Jun-Ying Chai, Han-Cheng Li, Zheng-Heng Li, Xin Liu, Li-Ming Song, Yuan-Hao Wang, Xing Wen

*Department of Nuclear and Particle Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1205 Geneva, Switzerland

*Merlin Kole, Franck Cadoux, Silvio Orsi, Martin Pohl, Xin Wu

National Centre for Nuclear Research, ul. A. Soltana 7, 05-400 Otwock, Swierk, Poland

Tadeusz Batsch, Dominik Rybka, Jacek Szabelski, Teresa Tymieniecka, Anna Zwolinska

University of Geneva, Geneva Observatory, ISDC, 16, Chemin d’Ecogia, 1290 Versoix, Switzerland

Tancredi Bernasconi, Neal Gauvin, Nicolas Produit

School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China

Zi-Gao Dai, Mi-Xiang Lan

Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, China

Zi-Gao Dai

*Paul Scherrer Institut, 5232, Villigen, Switzerland

*Wojtek Hajdas, Radoslaw Marcinkowski, Hua-Lin Xiao

Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China

Mi-Xiang Lan, Xue-Feng Wu

Methods

Sample selection

The GRBs analyzed in this paper are selected from the POLAR GRB catalog, presented in ref.  [2], based on the following criteria:

• •

The GRB has been observed by detectors other than POLAR and good measurement of both the spectrum and location are provided by other instruments.

• •

The fluence of the GRB, as provided by other instruments in the $10-1000\,\mathrm{keV}$ energy range, exceeds $5\times 10^{-6}\mathrm{erg/cm^{2}}$.

• •

The incoming angle with respect to the POLAR instrument zenith, $\theta$, is below $45^{\circ}$.

The first selection criterion minimizes the systematic errors in the polarization results induced by uncertainties in the spectrum and location of the GRB. The second and third criteria ensure a large number of events and a large modulation factor ($M_{100}$, the amplitude of the modulation measured for a $100\%$ polarized flux) and therefore a high statistical significance of the measurement. The third criterion furthermore reduces the influence of photons scattered off the objects in the vicinity of POLAR, thereby removing additional systematic errors.

This selection resulted in the following six GRBs eligible for study: 161129A, 161218A, 170101A, 170114A, 170127C and 170206A. The light curves of these GRBs can be found in ref.  [3]. GRB 161129A was however preceded by a large solar flare. It is therefore not selected as systematics induced by potential low level irradiation from the last part of the flare require further studies. Therefore, only the latter five GRBs are included for this work. The light curves of the five GRBs are shown in Supplementary Figure 1. In these light curves only events are selected where at least two bars are above trigger level as only these events can be used for polarization studies. Details of the event selection including the trigger algorithm can be found in ref.  [12]. Furthermore the different time bins studied for GRB 170114A, discussed later in this document, are indicated in the bottom right panel of the figure.

Methodology of GRB polarization analysis

A full detailed instrument calibration of the POLAR detector in-orbit is a prerequisite for the polarization study. The results of such a study are presented in ref. [12]. The polarization analysis is performed by producing one modulation curve from the measured data. This modulation curve is fitted to a simulated instrument response consisting of 6060 simulated modulation curves to find the best fitting polarization degree (PD) and polarization angle (PA), described in more detail in the next section.

The simulated response is produced using the Monte Carlo (MC) software as presented in ref. [13]. It should be noted that some updates to the MC software have been implemented since the publication of ref. [13] based on a deeper analysis of the in-orbit data. These updates mainly concern the inclusion of a detailed mass model of the Tiangong-2 (TG-2) space lab, on which POLAR is mounted, and a better parameterization of the non-linearity of the energy gain based on the findings presented in ref. [12]. The measured and simulated data are processed using the methods described in ref. [12].

The event selection as well as the data processing of both the measured and simulated data is equivalent. Events selected for analysis consist of all the clean events as described in ref. [12]. Additionally no channels with an energy deposition above overflow are selected. Finally the two channels in POLAR with the highest energy depositions, as measured in keV, are selected. The angle between the two bars is taken to be the scattering angle. The first bar of the two is that with the highest energy deposition within the whole event. The second bar is selected as that with the highest energy deposition among those bars not adjacent to the first bar. If the second channel is not found with the non-adjacent criterion the event is not selected. The non-adjacent criterion is used to improve $M_{100}$ and to reduce the influence of the fluctuations of crosstalk signals. In order to reduce the effect of the energy threshold non-uniformity, an additional energy threshold of 15 keV is applied in the event selection on the energy deposition of the second channel.

For the measured modulation curve the selected signal time intervals are based on the T90 values measured by POLAR. The background time interval is selected as two time intervals which are respectively before and after the GRB as described in more detail in the supplementary material and indicated in Supplementary Figure 1. The modulation curve of the background is subtracted from the modulation curve produced using data in the selected signal time interval. Here the relative difference between the length of the signal and background intervals is taken into account, as well as the error propagation for each bin. As the background is relatively stable, as illustrated in Supplementary Figure 6, the influence of the background selection on the final results was found to be negligible in a dedicated study described in detail in a dedicated section in the supplementary materials. The results are summarized in Supplementary Tables 2 and 3.

For each GRB a range of simulations is performed. In each the GRB is simulated using photons emitted from a circular plane with a radius of 250 mm, with the incident direction calculated based on the best available measured location for the GRB from other instruments. The influence of systematic uncertainties in the location of the GRB on our final results was found to be small through a dedicated study described in a dedicated section of the Supplementary Materials. Its results are summarized in Supplementary Table 5. The size of the emitting plane is sufficient to illuminate the full instrument as well as the materials surrounding POLAR. It should furthermore be noted that it was found that, by once removing them in the simulations, the materials surrounding POLAR have no significant influence on the final polarization measurement for the studied GRBs.

The energy spectrum of the simulated photons follows a Band function [4] using the best published parameters for the GRB. The influence of the choice of the used spectrum, when different spectra were available, was found to be negligible through a study described in a dedicated section of the Supplementary Materials. The results of this study are summarized in Supplementary Table 4. It should be noted here that the spectrum of GRB 170127C is best fitted using a single cutoff power-law and is therefore also modeled as such in the POLAR analysis. The lower limit for the energy range for simulated photons is 10 keV, below which the effective area of POLAR becomes negligible. The upper limit is set to 1000 keV for all GRBs except for 170127C, where this is extended up to 1500 keV where, according to the spectra provided by Fermi-GBM, the flux drops below $1\%$ of that at 10 keV. The photons are then simulated with an origin from a random position in the circular plane and a momentum vector based on the GRB location with respect to POLAR. For each GRB a total of 61 simulations are performed each with 5 million photons. A total of 60 simulations are performed for the 100% polarized photons, differing in the polarization angles (PA) with step sizes of 3 degrees. One simulation is performed for the unpolarized photons. The modulation curves of the polarization degrees (PD) between 0–100% can be generated by mixing those of the unpolarized flux and the 100% polarized flux. Therefore, using the modulation curves of the 61 simulations, a total of 6060 different simulated modulation curves was generated in the 2-D plane of PA and PD with a step of 1% in the PD direction and 3 degrees in the PA direction. The modulation curves produced this way, as well as those from the measured data, are normalized by the total number of events within the modulation curve. Subsequently the $60\times 101$ array of simulated modulation curves are used to find the best fitted PA and PD for the measured modulation curve using the least $\chi^{2}$ method as discussed in the next section. This array of simulated modulation curves is also used for the Bayesian analysis method, discussed later in this document, to generate the posterior probability distribution of the true PA and PD.

The results of this analysis when applied on full GRBs is detailed in Supplementary Table 7 while the input parameters and their origins are summarized in Supplementary Table 6.

Least $\chi^{2}$ analysis method

In order to determine the PA and PD of each GRB the measured modulation curve is fitted with the simulated instrument response using the least $\chi^{2}$ method. One $\chi^{2}$ value between the measured modulation curve and one of the 6060 simulated modulation curves as mentioned above can be calculated using Eq (1),

[TABLE]

where $X_{i}$ and $Y_{i}$ are the counts of the bins of the measured and simulated modulation curves respectively, $\varepsilon_{i}^{2}$ and $\sigma_{i}^{2}$ are the uncertainties of $X_{i}$ and $Y_{i}$ respectively, and $n$ is the number of bins in the modulation curves. Note that both the measured and simulated modulation curves are normalized before calculating the $\chi^{2}$ value. The uncertainties of the bins in the measured modulation curve are taken to be the statistical errors. On top of the statistical uncertainties in the simulated modulation curves these also contain systematic uncertainties resulting both from the spectral and location parameters of the GRB as well as from uncertainties in the calibration parameters used in the simulations. The details of the systematic error determination is described in the supplementary material.

Using the 6060 simulated modulation curves and the single measured modulation curve with their corresponding errors the $60\times 101$ array of $\chi^{2}$ values corresponding to different PAs and PDs can be calculated. As an example Supplementary Figure 2 shows the map of $\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm min}$ for GRB 170206A. The best fitted PA and PD of this GRB are those corresponding to the $\chi^{2}_{\rm min}$.

As discussed in ref. [5], the confidence area of the PA–PD measurement can be determined using the value of $\Delta\chi^{2}$. In Supplementary Figure 2, the three black contours from left to right correspond to $\Delta\chi^{2}=2.28,4.61$ and $9.21$ which are the upper quantiles with probabilities 32%, 10% and 1% for the $\chi^{2}$ distribution of 2 degree of freedom (d.o.f.). In order to get the upper limit of PD without considering the value of PA, the upper quantile of the 1 d.o.f. $\chi^{2}$ distribution is used. The red contour corresponds to $\Delta\chi^{2}=6.64$ which is the upper quantile with probability 1% for the 1 d.o.f. $\chi^{2}$ distribution. The maximum PD on the red contour is the upper limit of PD with confidence level 99%, which is approximately 34%.

Bayesian analysis method

Suppose $\bm{A}=(\hat{p},\hat{\phi})$ is the measurement of PD and PA using the least $\chi^{2}$ method as discussed in the previous section, and $\bm{B}_{i}=(p,\phi)$ is the true PD and PA of the source. Then $P(\bm{A}|\bm{B}_{i})$ is the probability that the measurement of PD and PA is $\bm{A}$ under the condition that the true PD and PA of the source is $\bm{B}_{i}$. If the prior probability of the true PD and PA of the source is $P(\bm{B}_{i})$, the posterior probability of $\bm{B}_{i}$, under the condition that the measurement of PD and PA is $\bm{A}$, can be acquired using the Bayesian equation as presented by Eq. (3).

[TABLE]

Here $P(\bm{A}|\bm{B}_{i})$ for different bins of $\bm{B}_{i}$ can be acquired by performing a series of measurement simulations, the procedure for which is described in the supplementary material. Some examples of the distribution of $P(\bm{A}|\bm{B}_{i})$ for GRB 170206A are shown in the Supplementary Figure 3. As there is no information of the true PD and PA of the source before the measurement, the prior probability of $\bm{B}_{i}$ is taken to be uniform in the range of 0% and 100% for PD and $0^{\circ}$ and $180^{\circ}$ for PA and 0 outside this range. After producing the distribution of $P(\bm{A}|\bm{B}_{i})$ for all different bins of $\bm{B}_{i}$, using simulations, the posterior probability distribution of the true PA and PD of the source under the condition of the single measurement can be directly calculated with Eq. (3). The Supplementary Figure 4 shows the result of $P(\bm{B}_{i}|\bm{A})$ for GRB 170206A, and Figure 2 show the results of all the 5 selected GRBs.

After integrating the 2-D posterior probability distribution of PD and PA, as shown in Figure 2 along the PA direction, a 1-D posterior probability distribution of PD is acquired as illustrated in Supplementary Figure 5. This 1-D distribution can be used to find the upper limit of PD with a certain credible level by integrating it from right to left and to find the lower limit by integrating it from left to right, as shown in Figure 3. From Figure 3(b), it can be seen that the upper limit of PD with credible level 99% is about 31%, which is very similar to that obtained with the least $\chi^{2}$ method.

Evolution of polarization angle

For all GRBs with exception of 170127C, which is too short and lacks statistics, a time binned analysis is also performed. For this purpose the data from the GRBs is split into equal time intervals. The GRBs 161218A, 170101A and 170206A are split into two intervals. GRB 170114A, which consists of a bright peak lasting approximately 6 seconds followed by 3 seconds with few events, is divided into three time bins. These time intervals for 170114A are indicated in the bottom right panel of Supplementary Figure 1. For this GRB only the two time bins in the peak are discussed here as the MDP for the third bin is significantly larger than 100%, indicating that no constraining measurement on the polarization properties of this interval are possible.

The analysis is performed using two different approaches for each GRB. First the data from both time bins are analyzed independently following the same procedure as that applied in the analysis of the time integrated GRBs. For the two time bins the incoming angles as used in the simulations is corrected for the minor changes induced by the rotation of POLAR with respect to the GRB during the burst. The second applied method is the same as that used by the GAP collaboration for GRB 100826  [4]. In this analysis the two time intervals are analyzed simultaneously while the PD of both intervals is forced to be equal, while the PA are allowed to vary. This study therefore takes the assumption that the PD is constant throughout the GRB while the PA can vary with time.

For GRBs 161218A, 170101A and 170206A the analyses using both methods is found to give consistent results for both time intervals to the analysis of the full time integrated GRB. No significant change in the angle is therefore found during these GRBs. It should however be noted that this analysis is only performed when dividing the full GRB in two equal time intervals; we can not exclude that the polarization angle varies on shorter time scales, since the data of POLAR lack statistics to do such studies.

For GRB 170114A a significant evolution of the polarization angle is found using both analysis methods. In the analysis the changes in the incoming angle of the GRB during the burst as well as spectral evolution of the GRB is taken into account. Such spectral evolution was not reported in the GCN by Fermi-GBM [15], the only other instrument apart from POLAR that reported a detection of this GRB. However, analysis of the Fermi-GBM data shows that a significant spectral evolution occurred during the GRB. Using independent analysis of both bins and using new spectra for the two time bins resulted in the modulation curves presented in Supplementary Figure 7 and 9 and the $\Delta\chi^{2}$ distributions shown in Supplementary Figures 8 and 10. The first time interval is consistent with an unpolarized or lowly polarized flux and gives an upper limit of a polarization degree of 42%, while the second interval only has a $0.49\%$ probability for PD lower than $2\%$. The 99% upper limit for the polarization degree of the second part is 74%.

It was furthermore found that the PD of both time intervals is compatible within $1\sigma$ with a PD around 25%, albeit with very different polarization angles. We can therefore perform the second analysis approach where PD for both bins is forced to be equal. The result of this study on GRB 170114A is a PD of 24% throughout the GRB while the PA of the first time bin is $116^{\circ}$ and that of the second time bin is $11^{\circ}$ with a $\chi^{2}/$NDF of $59.6/55$. The resulting $\Delta\chi^{2}$ map for the first time bin is shown in Supplementary Figure 11. The $\Delta\chi^{2}$ map of the second time bin is the same as this but shifted along the PA axis. The results exclude an unpolarized flux with a 99.2% confidence level.

The statistics for this GRB allow the peak to be divided into three time bins, each of 2 seconds. Using the 3 time bins the same analysis results in a PD of 28% with PA=$98^{\circ}$ for the first time bin, PA=$152^{\circ}$ for the second time bin and PA=$38^{\circ}$ for the final time bin. The $\chi^{2}/$NDF for this analysis is 27.6/29, note that the number of bins in the modulation curves is reduced here in order to allow for sufficient statistics in each bin. The final result in the form of a $\Delta\chi^{2}$ map for the first time bin is shown in Supplementary Figure 12. This analysis results in a higher exclusion level with PD=28% with a $1\sigma$ error of 9% and excludes an unpolarized flux with 99.7% confidence.

Data Availability Statement

All the data that support the plots within this paper and other findings of this study are available from the POLAR Collaboration (merlin.kole@unige.ch) upon reasonable request.