Radio linear polarization of GRB afterglows: Instrumental Systematics in ALMA observations of GRB 171205A
AAccepted for publication in The Astrophysical Journal
Preprint typeset using L A TEX style emulateapj v. 12/16/11
RADIO LINEAR POLARIZATION OF GRB AFTERGLOWS: INSTRUMENTAL SYSTEMATICS IN ALMAOBSERVATIONS OF GRB 171205A
Tanmoy Laskar , Charles L. H. Hull , and Paulo Cortes Department of Physics, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom National Astronomical Observatory of Japan, NAOJ Chile, Alonso de C´ordova 3788, Office 61B, 7630422, Vitacura, Santiago, Chile Joint ALMA Observatory, Alonso de C´ordova 3107, Vitacura, Santiago, Chile National Radio Astronomy Observatory, Charlottesville, VA 22903, USA and * NAOJ Fellow
Accepted for publication in The Astrophysical Journal
ABSTRACTPolarization measurements of gamma-ray burst (GRB) afterglows are a promising means of probingthe structure, geometry, and magnetic composition of relativistic GRB jets. However, a precisetreatment of instrumental calibration is vital for a robust physical interpretation of polarization data,requiring tests of and validations against potential instrumental systematics. We illustrate this withALMA Band 3 (97.5 GHz) observations of GRB 171205A taken ≈ .
19 days after the burst, wherea detection of linear polarization was recently claimed. We describe a series of tests for evaluatingthe stability of polarization measurements with ALMA. Using these tests to re-analyze and evaluatethe archival ALMA data, we uncover systematics in the polarization calibration at the ≈ .
09% level.We derive a 3 σ upper limit on the linearly polarized intensity of P < . µ Jy, corresponding to anupper limit on the linear fractional polarization of Π L < . ≈ .
1% level from typical radio afterglows.
Keywords: gamma-ray burst: general – gamma-ray burst: individual (GRB 171205A) – polarization INTRODUCTIONPolarization studies of long-duration GRB afterglowsare expected to probe the presence of ordered magneticfields in their jetted outflows as well as the viewing ge-ometry (Granot 2003; Granot & K¨onigl 2003; Rossi et al.2004; Granot & Taylor 2005; Kobayashi 2017), yield-ing crucial constraints on the jet launching mechanismand the central engine (Lyubarsky 2009; Bromberg &Tchekhovskoy 2016). Whereas polarization studies in theoptical have revealed evidence for structured magneticfields in the outflow (Steele et al. 2009; Cucchiara et al.2011; Mundell et al. 2013; Wiersema et al. 2014), similarstudies at radio/millimeter (mm) frequencies have beenmore limited due to instrumental sensitivity constraints(Taylor et al. 1998; Frail et al. 2003; Taylor et al. 2004;Granot & Taylor 2005; van der Horst et al. 2014; Covino& Gotz 2016).The advent of the Atacama Large Millimeter/Sub-millimeter Array (ALMA) is changing the landscape, andhas resulted in the first detection of polarized emissionfrom GRBs in the radio/mm band, which has providedpreliminary constraints on the magnetic field structure inGRB jets (Laskar et al. 2019). Additionally, Urata et al.(2019) claimed a detection of (0 . ± . ≈ .
19 days after the burst with ALMA at 97.5 GHz. Byassuming an intrinsic polarization of ≈ f acc ≈ .
1. As polarization capabilities with ALMA continue toevolve since the initial commissioning effort (Nagai et al.2016), consistent analysis frameworks need to be de-ployed to interpret polarization observations, especiallyin the case of detections near the threshold of the cur-rent instrumental systematics. Here, we discuss strate-gies for testing data for these systematics in polarizationmeasurements of faint sources. We re-analyze the obser-vations reported in Urata et al. (2019), and demonstratethat the data suffer from unremovable, systematic cali-bration uncertainties.We report our derived upper limit on the polarizationof GRB 171205A in Section 2. We discuss the impli-cations of the upper limit on the magnetic field struc-ture, and compare with previous observations of polar-ized emission for GRB radio afterglows in Section 3. ALMA POLARIZATION OBSERVATIONS2.1.
QA2 calibration
We downloaded the raw data for full-Stokes ALMABand 3 (3mm) observations of GRB 171205A taken on2017 December 10 under project 2017.1.00801.T (PI:Urata) from the ALMA archive. The observations em-ployed J1127-1857 as bandpass and flux density calibra-tor, J1256-0547 as polarization calibrator, and J1130-1449 as complex gain calibrator. As a first step, we usedthe CASA (McMullin et al. 2007) calibration scripts, as-sociated with the data set and also available from theALMA archive, to regenerate the calibrated Quality As-surance 2 (QA2) measurement set. We made imagesin Stokes
IQU V from the full calibrated measurement a r X i v : . [ a s t r o - ph . H E ] A p r Laskar et al. −0.15 −0.10 −0.05 0.00 0.05 0.10
Stokes Q (mJy) −0.10−0.050.000.050.100.15 S t o k e s U ( m J y ) SB1 SB2 SB3
LSBUSBAll −0.15 −0.10 −0.05 0.00 0.05 0.10
Stokes Q (mJy) −0.10−0.050.000.050.100.15 S t o k e s U ( m J y ) SB1 SB2 SB3
LSBUSBAll
Figure 1.
Stokes Q versus Stokes U before (left) and after (right) self-calibration for GRB 171205A (circles) divided into lower sideband(diamonds) and upper sideband (squares), and further by time into the three executions of the scheduling block at 5.136–5.168 d (blue),5.174–5.207 d (black), and 5.217–5.245 d (orange). The polarization properties are neither consistent in frequency across ALMA Band3 (points with the same color), nor stable with time (points with same marker shape). The QU axis scales are equal and are identicalbetween the two panels. Upon self-calibration (see Section 2.2), the uncertainty on the individual measurements is reduced and the pointsshift closer to the origin (zero polarization). The measurements, which span 2.6 hours, exhibit an unexpectedly strong trend in time,corresponding to a rotation in the plane of polarization from ≈ − ◦ to ≈ ◦ for the self-calibrated data. set with CASA version 5.6.1 using a robust parameterof 0.0, and also independently from the lower sideband(LSB; 89 . . . . Q , U , and V images is ≈ . µ Jy, consistent with theexpected thermal noise given the observation duration.The GRB afterglow is well detected in Stokes I , with aflux density of 30 . ± .
09 mJy measured using CASA imfit . The Stokes I image is dynamic range limited,with an rms ≈ µ Jy . We also detect a point source inmaps of Stokes Q and U . Fitting for the linearly polar-ized flux density with the position fixed to that derivedfrom the Stokes I image, we obtain Q = − . ± . µ Jyand U = − . ± . µ Jy, in agreement with the valuesreported by Urata et al. (2019). However, we find thatthe Stokes Q measurements differ between the two side-bands by 32 µ Jy, corresponding to a difference in linearpolarization fraction, Π L ≈ .
1% relative to Stokes I .We tested for stability of polarization calibration by di-viding the data in time by each execution of the schedul-ing block (SB), as described in Laskar et al. (2019). Thisapproach reveals systematic trends in the QU time evolu-tion. Stokes Q appears to increase from − . ± . µ Jyto − . ± . µ Jy (a change of ≈ .
11% of Stokes I )over the course of the observations, while Stokes U ap-pears to increase from − . ± . µ Jy to − ± . µ Jy( ≈ .
19% of Stokes I ), where the uncertainties refer tothose associated with the point source fits with imfit ,and which are compatible with the expected thermalnoise in each SB execution of ≈ µ Jy. This variability The uncertainties reported by imfit follow the prescription ofCondon (1997). The expected theoretical rms for the full 3-hour observation is ≈ µ Jy. is especially strong in the USB, with Q and U apparentlychanging by ≈ .
23% and ≈ .
30% of Stokes I , respec-tively, over the course of the observations (Fig. 1). Themagnitude of these temporal changes are much largerthan the absolute value of the polarization detection pre-viously claimed by Urata et al. (2019) with these data.We also note the presence of significant signal in cir-cular polarization, with Stokes V = − . ± . µ Jy( ≈ .
23% of I ), at the same level as the previouslyclaimed linear polarization detection. Circular polariza-tion has only been reported once in a GRB afterglow(Wiersema et al. 2014), and its detection here is morelikely indicative of instrumental systematics than of anintrinsic origin. We note that the observed Stokes V iswithin the current systematic uncertainty for on-axis cir-cular polarization with ALMA ( ≈ . Q increasing from8 . ± .
04 mJy (1.11% of Stokes I ) to 9 . ± .
04 mJy(1.23%; a ≈ σ change, corresponding to 0.12% of I )and Stokes U increasing from − . ± .
05 mJy (1.86%of I ) to − . ± .
06 mJy (1.64%; a ≈ σ change,corresponding to 0.22% of I ). The gain calibrator alsoappears to exhibit a statistically significant circular po-larization signal, with V = − . ± .
05 mJy (0.24% of I ). These calibrators are not expected to be significantlycircularly polarized in the mm band, and thus the Stokes V measurement most likely indicates residual polariza-tion calibration errors. We discuss this further in Section2.4. 2.2. Detailed data analysis adio Linear Polarization of GRB Afterglows I Frequency (GH ) −0.020−0.015−0.010−0.0050.0000.0050.010 D - t e r m G a i n ( r e a l ) USB XY Frequency (GH ) −0.010−0.0050.0000.0050.0100.015 D - t e r m G a i n ( i m a g i n a r y ) USB XY Frequency (GHz) D - t e r m G a i n ( m a g n i t u d e ) USB XY Frequency (GHz) −0.03−0.02−0.010.000.010.020.03 D - t e r m G a i n ( r e a l ) LSB XY Frequency (GHz) −0.04−0.020.000.020.040.06 D - t e r m G a i n ( i m a g i n a r y ) LSB XY Frequency (GHz) D - t e r m G a i n ( m a g n i t u d e ) LSB XY Figure 2.
Real (left column), imaginary (center column), and magnitude (right column) of the derived complex polarization leakage(“ D -terms”) for antenna DA50 in the upper sideband (USB; upper row) and lower sideband (LSB; lower row) for both X (blue) and Y (orange) polarizations for a reduction using DA64 as reference antenna. The LSB leakage exhibits a peak at ≈
90 GHz and a large spikeabove ≈
93 GHz, while the USB leakage has a quasi-periodic structure. These structures are robust to choice of reference antenna used inthe polarization calibration (Fig. 3). D -term solutions for the other antennas exhibit similar trends. Table 1
Derived polarization properties of the polarization calibrator,J1256-0547Reference Method
Q U Π La χ a Antenna (%) (%) (%) (deg)DV06 qufromgain 2 . ± .
06 5 . ± .
04 6 .
31 35 . . ± .
12 5 . ± .
03 6 .
29 34 . − . ± .
09 0 . ± .
08 0 .
07 47 . . ± .
15 5 . ± .
09 6 .
37 35 . . ± .
12 5 . ± .
06 6 .
33 34 . − . ± .
26 0 . ± .
23 0 .
11 59 . a Π L is the linear polarization fraction and χ = arctan ( U/Q ) is the po-larization (electric field vector) position angle. qufromgain and xyamb do not provide uncertainties on these quantities
Given the apparent instability of polarization proper-ties of the target and phase calibrator with both timeand frequency in the QA2 results, we perform a full in-dependent reduction of the data. We import the rawASDM datasets into CASA, followed by flagging of non-interferometric (e.g. pointing, atmospheric calibration,and sideband ratio) data. We apply the system temper-ature (Tsys) and water vapor radiometer (wvr) calibra-tions to the data, and concatenate the three executionsof the scheduling block (SB) into a single CASA mea-surement set.We perform interferometric and polarization calibra-tion using standard techniques, beginning with derivingthe bandpass phase and amplitude calibration, in that or-der. We use DV06 as reference antenna, and validate ourcalibration by repeating the entire analysis separately us-ing a nearby antenna with a different architecture, DA64.For the polarization calibration, we first derive the com-plex gain solutions on the polarization calibrator, andthen derive an a priori estimate of its Stokes Q and U from the ratio of complex gains using the python utility qufromgain from the ALMA polarization helpers mod-ule ( almapolhelpers.py ; see CASA documentation fordetails). The parallactic angle of the polarization calibra-tor decreases from ≈ ◦ to ≈ ◦ over the course ofthe observations, providing adequate coverage for disen-tangling the source and instrumental polarization. Thederived fractional Q and U values for the polarizationcalibrator are consistent across all four spectral windowsand across the use of the two different reference anten-nas, although we note that using DV06 yields a lowerestimated uncertainty on Q and U (Table 1). We notethat these are fractional polarization values, since theywere derived assuming unity Stokes I .To derive the cross-hand delays, we use scan 61 onthe polarization calibrator as the scan with the strongestpolarization signal, selected based on a plot of the com-plex polarization ratio for this calibrator as a function oftime . We next solve for the XY phase of the referenceantenna, the channel-averaged polarization of the polar-ization calibrator, and the instrumental polarization us-ing the XYf+QU mode in CASA’s gaincal task. Thenet instrumental polarization averaged across all base-lines (as reported by gaincal ) varies from ≈ .
06% to ≈ .
2% over the four spectral windows. We resolve the QU phase ambiguity with the python utility xyamb us-ing the fractional Q and U derived earlier. We list thefinal derived values for the fractional polarization of thepolarization calibrator in Table 1.We use these derived polarization properties to refinethe complex gain solution on the polarization calibrator.We run qufromgain again on the resulting calibrationtable, which yields a residual polarization statistically See https://casaguides.nrao.edu/index.php/3C286_Polarization for a description of this process.
Laskar et al.
Frequency (GHz) D - t e r m G a i n ( m a g n i t u d e ) Ref=DV06Ref=DA64
Frequency (GHz) D - t e r m G a i n ( m a g n i t u d e ) Ref=DV06Ref=DA64
Figure 3.
Magnitude of the polarization leakage for antennaDA50 in the lower sideband (left) and upper sideband (right) de-rived using independent reductions with two different reference an-tennas, DV06 (blue) and DA64 (orange). The derived D -terms arerobust to choice of reference antenna used. D -term solutions forthe other antennas exhibit similar trends. indistinguishable from zero, and demonstrates that thesource polarization has been successfully removed fromthe gain solutions. However, we note that the final resid-ual polarization, determined by running qufromgain onthe calibrated calibrator data, is ≈ .
1% (Table 1), sug-gesting that the minimum systematic uncertainty in po-larization measurements from this dataset is at least ofthis order. Antenna DV22 exhibits large ( ≈ D terms”) using polcal . The derived leakage terms ex-hibit a strong increase up to ≈
7% for several antennasat the upper edge ( (cid:38)
93 GHz) of the LSB, in additionto a weaker peak at ≈
90 GHz (Fig. 2). This behavior isseen in both reductions , i.e., independent of the refer-ence antenna used for calibration (Fig. 3). The leakageappears lower and more consistent across channels in theUSB, but does exhibit a quasi-periodic structure, as pre-viously also noted in the 3C286 Science Verification dataof ALMA Band 6 polarization observations (Nagai et al.2016).We set the flux density of J1127-1857 using measure-ments near the time of the GRB observations listed theALMA calibrator catalog, from which we derive a spec-tral index of β = − . ± .
01 and a flux density of ≈ .
05 Jy at a reference frequency of 91.5 GHz. The de-rived flux density of the polarization calibrator (J1256-0547) is 12 . ± . β = − . ± . . ± . β = − . ± . Imaging
We combine and image the calibrated measurement setusing tclean in CASA with a robust parameter of 0.0and one Taylor term (i.e. nterms=1). The clean beam is0 . (cid:48)(cid:48) × . (cid:48)(cid:48)
20 at a position angle of 85 ◦ The afterglow iswell-detected with a flux density of 30 . ± . imfit in CASA.No significant polarization signal is detected at the po-sition of the afterglow in Stokes Q , U , or in the P im- We also tested our analysis by flagging these channels, but thisdid not significantly change the results of the subsequent imaging. age. Our initial estimate for the point source flux den-sity is ≈
4% lower than the self-calibrated and sideband-combined flux density reported by Urata et al. (2019).However, we caution against a direct flux comparison,since Urata et al. (2019) do not report the flux densityor spectral properties of the flux calibrator that they as-sumed for the analysis.We note the presence of significant ( ≈ I image, both for the GRB andthe phase calibrator, indicating residual calibration er-rors, potentially due to atmospheric phase decoherence .We correct for these by performing two rounds of phase-only self-calibration with solution intervals of 10 min and2 min on both the GRB afterglow and phase calibratordata. We split the data into upper and lower sidebandsfor this step in order to reduce the fractional bandwidthfrom ≈
16% for the full dataset to ≈
4% per sideband,and thus minimize the effect of the frequency structure ofthe source on the calibration solutions. This is especiallyimportant for the calibrator, which exhibits a fitted spec-tral index (from the gain solutions) of ≈ − .
5, and thusa potential variation in Stokes I intensity of ≈
8% acrossthe ALMA band. We solve for a single gain solutionfor both polarizations (gain mode ‘T’) using gaincal inCASA, in order to avoid introducing a phase offset be-tween the X and Y polarizations. Additionally, we setthe reference antenna mode to strict to enforce the useof a single reference antenna during the self-calibration.We continue the use of the same reference antenna forself-calibration as that employed during the earlier cali-bration steps.We fit the Stokes I image with a point source modelusing CASA imfit , followed by fits to the QU V P imageswith the position and beam parameters fixed to that de-rived from the Stokes I image. We perform point sourcefits at each step during the phase-only self-calibration,and present these, together with the Stokes I map rms,in Table 2 for reference. The phase self-calibration re-veals low-level ( ≈ ≈ I (Table 2). Furthermore, these symmet-ric residuals are not completely removable even with 30 s For reference, the phase calibrator J1130-1449 is 5 . ◦ We also average the data to a 6s integration time and decimateby 2 channels in order to reduce the data volume. The resultingtotal beam smearing across the 25 (cid:48)(cid:48) × (cid:48)(cid:48) image is ≈ . (cid:48)(cid:48) , which isa fraction of the 0 . (cid:48)(cid:48) cell size, much smaller than the synthesizedbeam, and negligible for a point source near the field center. adio Linear Polarization of GRB Afterglows I Table 2
Impact of self-calibration on ALMA Band 3 (97.5 GHz) Polarization Observations of GRB 171205ASideband Selfcal Selfcal
I I rms
Q U V P
Type Interval a (mJy) ( µ Jy) ( µ Jy) ( µ Jy) ( µ Jy) ( µ Jy)LSB None . . . 31 . ± .
11 38.0 − . ± . . ± . − . ± . . ± . . ± .
03 15.8 − . ± . . ± . − . ± . . ± . . ± .
02 11.4 − . ± . . ± . − . ± . . ± . . ± .
03 11.2 14 . ± . . ± . − . ± . . ± . b . ± .
03 10.5 4 . ± . . ± . − . ± . . ± . . ± .
12 35.3 − . ± . − . ± . − . ± . . ± . . ± .
04 14.7 − . ± . − . ± . − . ± . . ± . . ± .
03 12.2 − . ± . − . ± . − . ± . . ± . . ± .
03 11.3 − . ± . − . ± . − . ± . . ± . b . ± .
03 10.6 − . ± . − . ± . − . ± . . ± . . ± .
09 29.3 − . ± . . ± . − . ± . . ± . . ± .
03 7.7 − . ± . . ± . − . ± . . ± . a Cross-hand phase fixed for 20-minute solutions, and left free for 30-second solutions. b For comparison with the analysisof Urata et al. (2019). amplitude and phase self-calibration, suggesting that theerrors may be baseline-based, rather than antenna-based.Finally, we note that the minimum theoretical solutioninterval for self-calibration ( t solint ), which is given by I peak σ I > √ N − (cid:114) t int t solint , (1)where is t int ≈ . × s is the total integration time onsource, N = 43 is the number of antennas in the array, I peak ≈
30 mJy is the peak intensity of the source used forself-calibration, and σ I ≈ . t solint (cid:38)
35 s. Thus,the 30 s solution interval used by Urata et al. (2019) isshorter than the minimum possible t solint where stablesolutions may be expected.We perform point source fits on our final images (am-plitude and phase self-calibrated to 20 min, with thecross-hand phase fixed), as well as on images made using30 s amplitude and phase self calibration, where the X and Y gains were allowed to vary independently. We findthat reducing the solution interval and fitting the cross-hand phase yields only a marginal increase in Stokes I flux density, from 33 . ± .
03 mJy to 33 . ± .
03 mJy inthe lower sideband, and from 32 . ± .
03 to 32 . ± . uv -data into a sin-gle measurement set, and image the entire 4 GHz datasetsimultaneously. Except for Stokes U in the LSB, no sig-nificant ( (cid:38) σ ) emission is visible in the Stokes QU P images. On the other hand, significant ( ≈ σ ) circularpolarization again appears at the ≈ .
23% level.2.4.
Polarization measurements
We are unable to reproduce the polarization measure-ments of Urata et al. (2019) in our analysis. In the lowersideband, our measurements of Stokes Q are statisticallyindistinguishable from zero, whereas Stokes U appearspositive, rather than negative as found by the previousauthors. In the upper sideband, the 30 s amplitude self-calibration yields an extremely large change in Stokes Q relative to the 20 min calibration; the Stokes Q flux den-sity changes from − . ± . µ Jy to − . ± . µ Jy,highlighting the danger in leaving the cross-hand phasefree while self-calibrating weakly polarized sources. We note that self-calibration moves the QU data pointscloser to the origin in the Q - U plane, corresponding tozero polarization (Fig. 1).In all cases, our images reveal an unexpected detec-tion in Stokes V = − . ± . µ Jy in the LSB and V = − . ± . µ Jy in the USB, corresponding to acircular polarization at the level of ≈ . ≈ . V is almost certainly spuriousand most likely indicates residual (unremovable) calibra-tion errors. These may arise, for instance, from time-variable XY phase or standing waves in the orthomodetransducers. Another possibility is that the polarizationcalibrator has non-zero circular polarization. Standardpolarization calibration assumes negligible V in the po-larization calibrator. Thus, non-zero V in the calibratormay corrupt the calibration solution, and the calibrator’s V may subsequently appear in calibrated science targetdata. We note that I to V conversion due to beam squintis expected to be negligible close to the primary beamaxis. The level of spurious circular polarization is sim-ilar to that of the claimed linear polarization detectionin Urata et al. (2019); however, as those authors do notpresent Stokes V images or photometry, we cannot per-form a direct comparison in Stokes V . We stress that thesystematic calibration errors causing a spurious Stokes V signal may or may not be the same errors causing thespurious Q and U detections we see in the data.We further test for calibration stability by dividing thedata into time bins by each of the three executions ofthe scheduling block. (Table 3). We find that the polar-ization measurements exhibit significant time variability.Stokes Q increases by ≈ .
3% of Stokes I , changing signduring the observation from negative to positive (Fig. 4).The change is ≈ σ relative to the typical (statistical)measurement uncertainty in Q . The variation is espe-cially pronounced in the LSB ( ≈ .
4% of I ), and is evenlarger in the LSB for Stokes Q ( ≈ .
5% of I ) prior toself-calibration. At the same time, the polarization prop-erties exhibit very different structures in the lower andupper sideband. For instance, Stokes U is positive inthe first scheduling block in the LSB (89 ± µ Jy), but
Laskar et al.
Time (days) −0.3−0.2−0.10.00.10.20.3 F a c t i o n a l P o l a i z a t i o n ( % ) LSB
Stokes QStokes UStokes V
Time (days) −0.3−0.2−0.10.00.10.20.3 F a c t i o n a l P o l a i z a t i o n ( % ) ALL
Stokes QStokes UStokes V 5.14 5.16 5.18 5.20 5.22 5.24
Time (day ) −0.3−0.2−0.10.00.10.20.3 F r a c t i o n a l P o l a r i z a t i o n ( % ) USB
Stoke QStoke UStoke V
Figure 4.
Fractional Stokes
QUV measurements for GRB 171205A (using self-calibrated data) as a function of time relative to the
Swift trigger time (center), divided into LSB (left) and USB (right). Each time bin corresponds to one execution of the scheduling block. Thefractional uncertainty from the Stokes I measurement is two orders of magnitude smaller than that from the QUV measurements, and isthus ignored. Error bars in the time direction correspond to the span of the data imaged.
Time (day ) −2.0−1.5−1.0−0.50.00.51.01.52.0 F r a c t i o n a l P o l a r i z a t i o n ( % ) LSB
Stoke QStoke UStoke V 5.14 5.16 5.18 5.20 5.22 5.24
Time (day ) −2.0−1.5−1.0−0.50.00.51.01.52.0 F r a c t i o n a l P o l a r i z a t i o n ( % ) ALL
Stoke QStoke UStoke V 5.14 5.16 5.18 5.20 5.22 5.24
Time (days) −2.0−1.5−1.0−0.50.00.51.01.52.0 F r a c i o n a l P o l a r i z a i o n ( % ) USB
S okes QS okes US okes V5.14 5.16 5.18 5.20 5.22 5.24
Time (days) −0.4−0.3−0.2−0.10.00.10.20.30.4 D i ff e r e t i a l F r a c t i o a l P o l a r i z a t i o ( % ) LSB
Stokes QStokes UStokes V 5.14 5.16 5.18 5.20 5.22 5.24
Time (days) −0.4−0.3−0.2−0.10.00.10.20.30.4 D i ff e r e t i a l F r a c t i o a l P o l a r i z a t i o ( % ) ALL
Stokes QStokes UStokes V
Time (days) −0.4−0.3−0.2−0.10.00.10.20.30.4 D i ff e r e n t i a l F r a c t i n a l P l a r i z a t i n ( % ) USB
Stokes QStokes UStokes V
Figure 5.
Upper panels: same as Fig. 4, but for the gain calibrator J1130-1449. The (statistical) uncertainties on each point are typicallysmaller than the thickness of the line used to plot the horizontal error bar. We find a non-zero Stokes V , as well as significant evolutionof Stokes QUV with time. The latter effect is clearer with the respective mean values of
QUV removed (lower panels; mean subtractedindependently for each subplot and polarization). All plots in the same row are on the same scale. negative in the USB ( − ± µ Jy). The difference of ≈ µ Jy between LSB and USB, a factor of ≈ ≈
3% differencein their Stokes I . Our measurements of Stokes U for theGRB decrease toward zero with time in both sidebands.This trend is robust to self-calibration (Fig. 1). The timescale of this evolution is ≈ . ≈ . ≈
2% wouldimply unphysically rapid changes, α ≈ −
70 for an ex-pected power law temporal evolution, P ∝ t α , ruling outintrinsic changes and implying instabilities in the polar-ization calibration.We search for systematic calibration errors by repeat-ing the above analysis for the gain calibrator, J1130-1449.We self-calibrate the data separately in the two sidebandsin the same manner as for GRB 171205A. The imagesreveal statistically significant circular polarization at thelevel V ≈ .
25% (sideband-averaged), similar to that ob-tained in the GRB data. This source has been variouslycategorized as an optical quasi-stellar object (QSO) and blazar (Massaro et al. 2009; Mignard et al. 2016). QSOsand blazars have been observed to circular polarization atthe ≈ .
1% level at cm wavelengths (Rayner et al. 2000).However, the circular polarization fraction is expected tofall with frequency as
V /I ∝ ν α V with − α V ≈ V at the mm wavelengths employed here. Indeed, veryfew blazars have detected circular polarization at mmwavelengths (Agudo et al. 2010, 2018; however, see alsoThum et al. 2018). Thus, the consistent detected Stokes V for the gain calibrator may imply residual uncorrectedinstrumental polarization in the data.We search the gain calibrator data for systematics byinvestigating variability in the polarization properties intime and frequency. As in the case of the GRB, we findthat Stokes Q and U vary by up to 0 .
4% of Stokes I overtime, and the variation is as strong as ≈ .
7% of Stokes I in the LSB (Fig. 5). Intrinsic variations on the time scaleof ≈ . adio Linear Polarization of GRB Afterglows I Table 3
Stability of ALMA Band 3 Polarization Observations to Time and Frequency SlicingTarget Sideband Frequency SB a I I rms
Q U V P (GHz) Execution (mJy) ( µ Jy) ( µ Jy) ( µ Jy) ( µ Jy) ( µ Jy)GRB 171205A LSB 91.463 1 33 . ± .
03 16.6 − . ± . . ± . − . ± . . ± . . ± .
04 16.1 23 . ± . . ± . − . ± . . ± . . ± .
04 17.4 56 . ± . . ± . − . ± . . ± . . ± .
03 17.4 − . ± . − . ± . − . ± . . ± . . ± .
04 16.9 − . ± . − . ± . − . ± . . ± . . ± .
04 18.3 − . ± . − . ± . − . ± . . ± . . ± .
03 12.5 − . ± . . ± . − . ± . . ± . . ± .
03 12.2 − . ± . . ± . − . ± . . ± . . ± .
03 13.8 21 . ± . . ± . − . ± . . ± . Q (mJy) U (mJy) V (mJy) P (mJy)Gain Calibrator LSB 91.463 1 847 . ± .
08 69.8 7 . ± . − . ± . − . ± .
05 15 . ± . . ± .
08 69.4 12 . ± . − . ± . − . ± .
05 17 . ± . . ± .
08 68.0 13 . ± . − . ± . − . ± .
05 18 . ± . . ± .
07 61.3 8 . ± . − . ± . − . ± .
05 15 . ± . . ± .
06 66.2 8 . ± . − . ± . − . ± .
05 15 . ± . . ± .
06 62.5 9 . ± . − . ± . − . ± .
06 15 . ± . . ± . . ± . − . ± . − . ± .
05 15 . ± . . ± . . ± . − . ± . − . ± .
05 16 . ± . . ± . . ± . − . ± . − . ± .
05 17 . ± . a The times of the three SB executions (considering target and gain calibrator scans only) are: 5.135–5.169, 5.174–5.208, and 5.211–5.245 days,respectively. can cause variability on much shorter (hour) time scales,this effect is expected to be negligible at mm wavelengths(Quirrenbach 1992; Goodman & Narayan 2006). Thus,the observed strong variability of the polarization prop-erties of the gain calibrator are most likely instrumentaland not intrinsic to the source. One possible origin forthese systematics may be time-varying XY phase. How-ever, investigating this requires second-order calibrationcorrections, which are beyond the scope of this work.2.5. Systematic calibration uncertainty
In light of the observed variability of the GRB andgain calibrator data in time and frequency, we believethe systematic calibration uncertainty for this dataset islarger than the nominal 3 σ value of 0.1% quoted in theALMA Cycle 4 Technical Handbook , relevant “for thebrightest calibrators”. Whereas the Handbook does notclarify this term precisely, calibrators with polarizationfraction (cid:38)
10% are available to ALMA , and thus, witha fractional polarization of ≈ . L ≈ .
16% of Stokes I for the cali-brator when both sidebands are combined (this number is∆Π L ≈ .
21% prior to self-calibration). If the systematicerror is a random (Gaussian) process, then this numberwould be an overestimate of the intrinsic standard devi-ation of that random process. The expectation value ofthe difference between the maximum and minimum (i.e.,the range ) of three numbers drawn from a unit normaldistribution is 3 π − / ≈ .
69 (Schwarz 2006). Thus, we https://almascience.nrao.edu/documents-and-tools/cycle4/alma-technical-handbook This quantity follows a Gumbel distribution. estimate an additional 1 σ systematic calibration uncer-tainty of ≈ . / . ≈ .
09% for these observations.In conjunction with the statistical uncertainty of ≈ . σ ) uncertainty in the polarization measurement is ≈ . . ± . µ Jy (undebiased), and thus the detectionof polarization in this event is only significant at ≈ . σ .Since P/σ P (cid:46) √
2, the maximum likelihood estimate for P is ˆ P = 0 upon correcting for Rician bias (Vaillancourt2006). Even for the total linearly polarized density of ≈ µ Jy reported in Urata et al. (2019), the addition ofa 0 .
09% systematic uncertainty renders the measurementat best a ≈ . σ detection. Given the significant variabil-ity observed and our inability to reproduce the earlierauthors’ results using an independent analysis, we con-sider these data to provide a 3 σ upper limit of (cid:46) . P (cid:46) . µ Jy) on the linear polariza-tion of GRB 171205A for the remainder of this work. DISCUSSIONThe precise interpretation of the polarization upperlimit depends strongly upon whether the emission arisesfrom shocked jet material (i.e., the reverse shock; RS)or from the shocked ambient environment (the forwardshock; FS), and upon the magnetic field structure in theregion of emission (Ghisellini & Lazzati 1999; Granot &K¨onigl 2003; Rossi et al. 2004; Granot & Taylor 2005).A detailed study of the afterglow emission and its de-composition into forward and reverse shock componentsis beyond the scope of this work, but we briefly discussboth scenarios. In the case of radiation powered by FSemission and where polarization is the result of viewing aregion with shock-produced magnetic fields off-axis, thetemporal evolution of the polarization fraction typicallyexhibits two peaks; however, the polarization fractioncan be very low, especially when the viewing geometry
Laskar et al.
Optical Fractional Linear Polarization,
L, opt C u m u l a t i v e D i s t r i b u t i o n A A L M A mm u pp e r li m i t Kaplan-Meier estimate of
L,opt
Figure 6.
Kaplan-Meier cumulative distribution function of theoptical linear polarization of GRB afterglows observed between 2.6and 10.4 days (i.e., within a factor of 2 in time relative to the ALMAobservations of GRB 171205A), including polarization upper limits,from Covino & Gotz (2016). Between 4–27% of optical afterglowpolarization measurements are lower than the ALMA 3mm upperlimit of < .
30% for GRB 171205A (intersection of the dashed linewith the shaded region). Thus, we cannot rule out that the linearpolarization in this burst is intrinsically low. is close to being on-axis (Rossi et al. 2004). Thus, wecannot rule this scenario out.3.1.
No strong evidence for thermal electrons
A suppression of the polarization by Faraday depolar-ization due to a quasi-thermal population of electronsnot accelerated at the FS, as argued by Urata et al.(2019), is an interesting possibility (Toma et al. 2008).In their analysis of this burst, Urata et al. (2019) con-trast their reported ALMA Band 3 measurement of Π L =(0 . ± . L ≈ .
2% (with-out error bars; they also do not describe how they removeany potential RS contamination). They ascribe the dif-ference between the measured and the “typical” opticalpolarization to the presence of quasi-thermal electrons.Whereas such a population should indeed exist (Eichler& Waxman 2005; Sironi & Spitkovsky 2011), we cautionthat (i) there is no evidence that radio polarization mea-surements track the optical polarization (indeed, thereis exactly one radio polarization detection of a GRB af-terglow to date, with Π L , opt ≈ L , radio ; however, thedetected optical polarization for that event is likely dom-inated by extrinsic dust scattering; Laskar et al. 2019;Jordana-Mitjans N. and Mundell); and (ii) the Urataet al. (2019) analysis ignores the optical polarization up-per limits. Including these upper limits, we find that asmany as 27% of optical polarization observations madewithin a factor of 2 in time of the time of these ALMA ob-servations (5.19 days, corresponding to the range ≈ . Constraints on magnetic field geometry
We now discuss the observed upper limit on the polar-ization at ≈ .
19 days in the context of the magnetic fieldgeometry in the jet powering GRB 171205A. In general,the observed polarization degree is a function of the ratioof the off-axis viewing angle ( θ ) to the opening angle ofthe jet ( θ jet ), and the time relative to the jet break time, t jet (Rhoads 1999; Sari et al. 1999). The X-ray light curvefor the afterglow of GRB 171205A exhibits a shallow, un-broken power law decay with α ≈ − .
06 to (cid:38)
35 days ,indicating that t jet (cid:38)
35 days. Thus, we consider theobservation time of t obs = 5 .
19 days to correspond to anupper limit on the ratio t obs /t jet (cid:46) .
15 days.Together with coeval Atacama Compact Array (ACA)345 GHz observations, the ALMA 97.5 GHz data in-dicate an optically thin spectrum in the mm-band at ≈ .
19 days, for which we calculate β mm = − . ± . β = − . ± . ≈ . β cm = 1 . ± .
03 (Urata et al. 2019).These observations indicate that both the synchrotronpeak frequency ( ν m ) and self-absorption break ( ν a ) are ata frequency lower than ALMA Band 3. Furthermore, theVLA spectrum is shallower than the fully self-absorbedexpectation of 2 ≤ β cm ≤ .
5, implying that ν a is inthe cm band at ≈ ν m . Thereforedepolarization due to synchrotron self-absorption in theALMA bands is unlikely, indicating that the polarizationof the observed radiation is intrinsically low.We note that RS emission has been seen in ALMAobservations of GRB afterglows as late as ≈ θ B , the observed polar-ization would be suppressed by a factor of the num-ber of patches visible, N ≈ (Γ θ B ) − , where Γ is thejet Lorentz factor at the time of observations (Nakar& Oren 2004; Granot & Taylor 2005). This implies θ B (cid:46) Π L , lim Γ − Π − ≈ × − Γ − rad, where we havetaken Π L , max = (1 − β ) / (5 / − β ) ≈ .
68 (Granot &Taylor 2005) for β ≈ − .
43 (Urata et al. 2019). Thislimit is consistent with the value of θ B ≈ − rad in-ferred from polarization observations of the reverse shockin GRB 190114C (Laskar et al. 2019). Thus, if the emis-sion arises from the reverse shock, this may indicate auniversal magnetic field coherence scale.The low degree of polarization disfavors models of po-larization produced by toroidal magnetic fields in GRBjets. To see this, we compare the models of Granot &Taylor (2005) together with the data in Fig. 7. We ex- adio Linear Polarization of GRB Afterglows I −2 −1 t/t jet F r a c t i o n a l P o l a r i z a t i o n ( % ) GRB 171205A BM ISMWind 10 −2 −1 t/t jet F r a c t i o n a l P o l a r i z a t i o n ( % ) GRB 171205A FS ISMWind
Figure 7.
The expected polarization signature from uniform jets with toroidal magnetic fields expanding into constant density (solidlines) and wind-like environments (dashed lines) for a Blandford-McKee (BM) evolution (left) and FS-like evolution (right), together withthe measured polarization fraction upper limit for GRB 171205A (red point). The lower limit on t jet corresponds to t/t jet (cid:46) .
15 for theGRB. The three lines are for θ obs /θ jet = 0 .
2, 0.1, and 0.05, from highest to lowest, respectively. A toroidal magnetic field geometry isdifficult to reconcile with the upper limit for most viewing angles if the emission arises from a RS. plore a range of off-axis angles and both constant den-sity and wind-like progenitor environments. For theLorentz factor evolution of the ejecta after deceleration(Γ ∝ R − g ), we consider two scenarios: a minimum valueof g = (3 − k ) /
2, corresponding to the evolution of thefluid just behind the forward shock, and g = 7 / − k ,a maximum value expected for a reverse shock, corre-sponding to the Blandford-McKee self-similar solution(Kobayashi 2000; Granot & Taylor 2005) . In the for-mer case, we find that a toroidal field would producetoo high a polarization degree regardless of viewing an-gle or the circumburst geometry. In the latter case, θ obs /θ jet ≈ .
05 is marginally allowed by the data; how-ever, this would require a very precise alignment of thejet axis with the line of sight, and is therefore unlikely .Finally, we note that a “universal structured jet” modelwith a toroidal magnetic field can be ruled out, sinceit would produce a much higher degree of polarization,Π L (cid:38)
30% at t (cid:46) t jet (Lazzati et al. 2004).3.3. Radio Polarization of GRB Afterglows
We now compare this derived upper limit to val-ues previously reported for radio observations ofGRB afterglows. Our compiled sample of radio lin-ear polarization observations includes one detection(GRB 190114C; Laskar et al. 2019), and several upperlimits (GRB 980329, Taylor et al. 1998; GRB 980703,Frail et al. 2003; GRBs 990123, 991216, and 020405,Granot & Taylor 2005; GRB 030329, Taylor et al. 2004;and GRB 130427A, van der Horst et al. 2014; andGRB 171205A, this work). We convert upper limitslisted at different confidence intervals to a uniform 3 σ limit for comparison across events. We multiply thequoted fractional polarization upper limits by the Stokes I flux density to estimate the upper limit in flux densityunits, and plot these separately at C-band ( ≈ . ≈ . ≈ . Here k is the power law index of the radial density profile. This would require a chance alignment probability of ≈ × − for a typical opening angle of ≈ ◦ . GRB 171205A exhibited the brightest Stokes I fluxdensity of our sample at the time of the polarization ob-servations. Thus, our upper limit on the polarized fluxof GRB 171205A, while not the strongest in absoluteflux terms, yields the deepest upper limit on the frac-tional polarization of Π L < .
30% (including systemat-ics). This imposes a factor of 3 stronger constraint onthe intrinsic polarization of radio afterglows than previ-ously performed (GRB 030329; Taylor et al. 2004). Asdiscussed in Urata et al. (2019), the emission appearsto be optically thin at 97.5 GHz at 5.19 days. There-fore, depolarization due to synchrotron self-absorption isunlikely to be the cause for the non-detection of polar-ized emission (Toma et al. 2008; Granot & van der Horst2014), suggesting that the absence of strongly polarizedemission is intrinsic to the source.We note that the polarization upper limits (and themeasurement in the case of GRB 190114C) all lie in therange of ≈ µ Jy; the difference in the polarizationfractions arises from the large spread (over two orders ofmagnitude) in Stokes I flux densities in the respectivebands at the time of observation. Of these, the ALMAobservations of GRB 190114C represent the earliest post-burst polarization-sensitive observations obtained for anymm-band GRB afterglow. The fractional polarizationlimits in the cm-band are all higher (i.e., worse) thanthose obtained in the mm-band, indicating the need toimprove instrument sensitivity and stability at these fre-quencies in order to probe polarized emission from GRBafterglows.According to the ngVLA reference design, the 1 σ pointsource sensitivity at 8 GHz in 1 hour of on-source in-tegration is ≈ . µ Jy. The polarization sensitivityin the current design is expected to be better thanΠ L ≈ . ≈ µ Jy GRB afterglow (Chandra & Frail 2012)polarized at 0 .
1% will be ≈ Laskar et al. −1 Time afte bu st (days) L i n e a p o l a i z a t i o n ( μ J y ) −1 Time after burst (days) F r a c t i o n a l o l a r i z a t i o n ( % ) −1 Fractional polarization (%) )2 )1 L i n e a % p o a % i ( a t i o n ( μ J y ) I = m J y I = m J y I = μ J y I = μ J y ngVLA (1 h%)SKA1-mid (1 hr)VLA (1 hr)ALMA (3 hr)4.8 GHz8.5 GHz97.5 GHz Figure 8.
Linear polarized intensity (left) and fractional linear polarization (center) as a function of observation time for GRB radioafterglows at ≈ . ≈ . ≈ . I values from 10 µ Jy to 10 mJy. For typicalGRB radio afterglows ( I ≈ µ Jy), polarization observations will be possible with SKA1-mid at the level of Π L ≈ (cid:46) .
1% level. cal GRB radio afterglow.A detection of linearly polarized radio emission unam-biguously associated with the afterglow forward shockwould provide the first constraints on the magnetic fieldstructure and viewing geometry for long-duration GRBs(Granot & K¨onigl 2003). In particular, the evolution ofthis quantity across the jet break is a sensitive measureof the degree of order in the magnetic fields, the jet struc-ture, and the off-axis viewing angle (Rossi et al. 2004).Thus, we suggest that a more robust interpretation ofafterglow polarization requires sensitive measurements(with detections) at multiple epochs. Such observations,while challenging for typical GRBs with ALMA in themm band, may be routinely tractable with the ngVLAand full SKA. CONCLUSIONSWe have presented a series of tests useful for estimat-ing the impact of systematic calibration errors in ALMApolarization data. In particular, we recommend basicsanity checks of (i) dividing the data in time and fre-quency to test for calibration stability, and (ii) checkingthe gain calibrator or using test calibrators, if available,to verify and quantify the success of polarization calibra-tion. While these tests have been performed at 3 mmhere, they are widely applicable to observations at anyfrequency.We have re-analyzed ALMA Band 3 (3 mm) full con-tinuum polarization observations of GRB 171205A taken ≈ .
19 days after the burst and performed detailed veri-fication steps to test the stability of polarization calibra-tion. In contrast to previous work (Urata et al. 2019),we do not detect significant linear polarization from theradio afterglow. We find a higher systematic uncertaintythan assumed by Urata et al. (2019), and infer a 3 σ upperlimit of P (cid:46) . µ Jy, corresponding to Π L (cid:46) .
30% ofStokes I , for which we derive a value of 32 . ± .
03 mJy(statistical error). The upper limit on Π L is consistentwith the range of optical linear polarization observed forGRB afterglows, and thus not immediately indicative ofthe presence of a population of thermal electrons. If theemission arises in the reverse-shocked region, the upperlimit rules out a toroidal magnetic field geometry for most viewing angles, and is consistent with random mag-netic field patches of coherence length, θ B (cid:46) × − rad.We have compiled observations of polarized emission inGRB radio afterglows from the literature, and demon-strate that the current observations and limits of lin-ear polarized intensity span a narrow range, likely dueto signal-to-noise limitations. We expect that improve-ments in cm-band polarization sensitivity and stability,such as with the ngVLA and full SKA, will open a newavenue for pursuit of GRB jet structure and magnetiza-tion in the future.We thank the anonymous referee for their sugges-tions, which improved this manuscript. TL thanksR. Margutti and P. Schady for helpful discussions.CLHH acknowledges the support of both the NAOJ Fel-lowship as well as JSPS KAKENHI grant 18K13586.This paper makes use of the following ALMA data:ADS/JAO.ALMA Agudo, I., Thum, C., Wiesemeyer, H., & Krichbaum, T. P. 2010,ApJS, 189, 1 2.4Agudo, I., Thum, C., Molina, S. N., et al. 2018, MNRAS, 474,1427 2.4Bromberg, O., & Tchekhovskoy, A. 2016, MNRAS, 456, 1739 1Chandra, P., & Frail, D. A. 2012, ApJ, 746, 156 3.3Condon, J. J. 1997, PASP, 109, 166 1Covino, S., & Gotz, D. 2016, Astronomical and AstrophysicalTransactions, 29, 205 1, 6, 3.1Cucchiara, A., Cenko, S. B., Bloom, J. S., et al. 2011, ApJ, 743,154 1Dent, W. A. 1965, Science, 148, 1458 2.4Eichler, D., & Waxman, E. 2005, ApJ, 627, 861 3.1Frail, D. A., Yost, S. A., Berger, E., et al. 2003, ApJ, 590, 992 1,3.3, 8Ghisellini, G., & Lazzati, D. 1999, MNRAS, 309, L7 3 adio Linear Polarization of GRB Afterglows I11