#  Using the Crab Nebula as Polarization Angle Calibrator for the Korean VLBI
Network

Minchul Kam Department of Physics and Astronomy, Seoul National University,
Gwanak-gu, Seoul 08826, Republic of Korea;  Sascha Trippe Department of
Physics and Astronomy, Seoul National University, Gwanak-gu, Seoul 08826,
Republic of Korea;  SNU Astronomy Research Center, Seoul National University,
Gwanak-gu, Seoul 08826, Korea  Do-Young Byun Korea Astronomy and Space Science
Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea
Jongho Park Korea Astronomy and Space Science Institute, Daedeok-daero 776,
Yuseong-gu, Daejeon 34055, Republic of Korea  Institute of Astronomy and
Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan  Sincheol
Kang Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-
gu, Daejeon 34055, Republic of Korea  Naeun Shin Department of Physics and
Astronomy, Seoul National University, Gwanak-gu, Seoul 08826, Republic of
Korea;  Korea Astronomy and Space Science Institute, Daedeok-daero 776,
Yuseong-gu, Daejeon 34055, Republic of Korea  Sang-Sung Lee Korea Astronomy
and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055,
Republic of Korea  Taehyun Jung Korea Astronomy and Space Science Institute,
Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Received October 13, 2022; accepted December 15, 2022)

##  1 Introduction

Radio observatories with circular polarization receivers, such as the Korean
VLBI Network (KVN), measure the electric vector position angle (EVPA) $\chi$
from the phase difference between the left circular polarization (LCP) and
right circular polarization (RCP) (Cotton, 1993). However, an arbitrary offset
in this phase difference is usually introduced by parts of the receiver such
as synthesizer or sampler, which results in an offset in the EVPA. Since this
offset tends to change when the system is turned on/off or when the frequency
setting is modified, it is necessary to observe a source with known EVPA, an
“absolute EVPA calibrator”, to correct the phase difference and measure the
intrinsic EVPA.

A good absolute EVPA calibrator should be bright, spatially compact, and
stable in polarization. Commonly, the AGN 3C 286 is considered as the source
which best satisfies these three conditions (Agudo et al., 2012; Perley &
Butler, 2013a, b). 3C 286 is a quasar with an angular size of a few arcseconds
at centimeter wavelengths (Akujor et al., 1995). The flux density $S_{\nu}$
from 1 to 50 GHz has been stable to within $\sim 1\%$ over 30 years (Perley &
Butler, 2013a) and exhibits a power-law spectrum with a spectral index
$\alpha\sim-0.61$ (defined by $S_{\nu}\propto\nu^{\alpha}$) (An et al., 2017).
Likewise, the polarized flux is very stable, albeit showing a slow secular
increase of less than 2% per century at any given frequency (Perley & Butler,
2013b).

The EVPA of 3C 286 has been studied across a wide frequency range. Bignell &
Seaquist (1973) measured $\chi=33.0^{\circ}\pm 0.9^{\circ}$ at 6.7 GHz and
$\chi=31.0^{\circ}\pm 1.3^{\circ}$ at 10.7 GHz with the 46-m Algonquin Radio
Telescope. Based on this, Perley & Butler (2013b) assumed that the EVPA of 3C
286 is 33∘ from 1 to 8 GHz and found that the EVPA rotates to $35.8^{\circ}\pm
0.1^{\circ}$ at 45 GHz. Polarization measurements with the IRAM Pico Veleta
30-meter Telescope (PV) (Agudo et al., 2012), Submillimeter Array (SMA)
(Marrone, 2006; Hull et al., 2016), the Combined Array for Research in
Millimeter-wave Astronomy (CARMA) (Hull et al., 2014, 2015), and the Atacama
Large Millimeter/submillimeter Array (ALMA) (Nagai et al., 2016) found that
this trend continues to higher frequencies, with the EVPA increasing up to
$41^{\circ}$. Likewise, the fractional polarization increases from 8.6% to 17%
with increasing frequency (Agudo et al., 2012; Hull et al., 2015, 2016; Nagai
et al., 2016).

Even though 3C 286 is compact and its EVPA is well-studied across a wide
frequency range, it is not bright enough to be used as the absolute EVPA
calibrator for the KVN which observes in the frequency range 22–129 GHz. The
total flux density is 1.5 Jy at 49 GHz (Perley & Butler, 2013a) and decreases
to below 1 Jy at higher frequencies (Agudo et al., 2012; Hull et al., 2015,
2016; Nagai et al., 2016). As a result, KVN cannot reliably detect the
polarized flux at $\geq$ 86 GHz, and when it does, the resulting EVPA
measurement comes with uncertainties larger than 5∘.

The Crab nebula, however, is much brighter than 3C 286. It is a supernova
remnant powered by a relativistic wind from the pulsar located at its center
(Hester, 2008). The integrated total flux of the Crab nebula is larger than
300 Jy throughout the KVN frequency range (Macías-Pérez et al., 2010; Weiland
et al., 2011) and exhibits a power-law spectrum with a spectral index
$\alpha\sim-0.3$ between 1 and 353 GHz (Baars et al., 1977; Macías-Pérez et
al., 2010; Arendt et al., 2011; Weiland et al., 2011; Ritacco et al., 2018)
and $\alpha\sim-0.7$ between $10^{4}$ and $10^{6}$ GHz (Macías-Pérez et al.,
2010). Excess emission observed above the synchrotron spectrum between
$10^{3}$ and $10^{4}$ GHz is explained by thermal emission of dust at a
temperature $T\sim 46$ K (Strom & Greidanus, 1992).

Aumont et al. (2010) mapped the Crab nebula with the instruments XPOL
installed at PV at 86 GHz. They measured an EVPA of $148.8^{\circ}\pm
0.2^{\circ}$ and a fractional polarization of $7.7\%\pm 0.2\%$ integrated over
the entire nebula. Weiland et al. (2011) observed the Crab nebula with WMAP at
22.70, 32.96, 40.64, 60.53, and 92.95 GHz for seven years and found an
integrated EVPA of $149^{\circ}$–$150^{\circ}$ and a fractional polarization
of $\sim 7\%$ across the observing frequency range.

Even though the Crab nebula is one of the brightest radio sources and its
integrated EVPA is well-studied, it cannot be directly used as an absolute
EVPA calibrator for KVN because the angular size of the nebula is larger than
the angular resolution of KVN. The average beam sizes of the KVN antennas are
129”, 63”, 31”, 29”, and 23” at 22, 43, 86, 94, and 129 GHz,
respectively,111https://radio.kasi.re.kr/status_report/files/KVN_Status_Report_2021.pdf
while the angular size of the Crab nebula is 7$\times$5 arcminutes. High
resolution polarization observations of the Crab nebula at millimeter
wavelengths show that the EVPA varies considerably with position (Aumont et
al., 2010; Ritacco et al., 2018). This implies that the EVPA integrated over
the KVN beam will be different from the EVPA integrated over the entire nebula
obtained from previous studies. In principle, one can infer the EVPA
integrated over the KVN beam if there is a spatially resolved linear
polarization image of the Crab nebula at each frequency. This is possible at
86 GHz because there is an XPOL map (Aumont et al., 2010), but not at other
frequencies. Therefore, it is essential to find the EVPA of the Crab nebula as
measured with KVN to use it as an absolute EVPA calibrator for KVN.

One way to infer the EVPA of the Crab nebula measured with KVN is using 3C 286
that is compact and has stable, well-known EVPA across the KVN frequency
range. The uncertainty in the EVPA of 3C 286 obtained from any single session
is too large, but this uncertainty can be reduced by combining data from
multiple observing sessions; this is permitted because the EVPA of 3C 286 is
expected to be stable in time (Perley & Butler, 2013b).

We present the results of four years of polarization observations of the Crab
nebula and 3C 286 obtained with KVN at 22, 43, 86, and 129 GHz. We also
present ALMA data of 3C 286 from the ALMA Calibrator Source
Catalogue222https://almascience.eso.org/sc to check the stability of 3C 286
and use it to find the EVPA of the Crab nebula seen by KVN. KVN observations
and ALMA archive data are summarized in Section 2. The polarization of the
Crab nebula and 3C 286 are presented in Section 3. In addition to the angular
resolution, we discuss other factors that contribute to the variation of the
EVPA with frequency in Section 4. A summary and conclusions of this study
follow in Section 5.

Table 1: Log of Crab nebula observations, by station and frequency.

Date | TN | US | YS | Crab
---|---|---|---|---
2017-02-24 | 86.2/— | 22.4/43.0 | 86.2/— | P
2017-03-25 | — | 22.4/43.0 | 86.2/129.4 | P
2017-04-23 | 86.2/129.4 | 22.4/43.0 | 86.2/129.4 | P
2017-05-31 | — | 22.4/43.0 | 86.2/129.4 | P
2017-09-24 | 86.2/129.4 | 22.4/43.0 | 86.2/129.4 | P
2017-10-27 | 22.4/43.0 | 86.2/129.4 | 86.2/129.4 | P
2017-11-19 | 22.4/43.0 | 22.4/43.0 | 86.2/129.4 | P
2017-12-21 | 86.2/129.4 | 22.4/43.0 | 86.2/— | P
2018-02-14 | 22.4/43.0 | 22.4/43.0 | 86.2/129.4 | P
2018-03-21 | 22.4/43.0 | — | 86.2/— | P
2018-05-05 | — | 22.4/43.0 | 86.2/— | P
2018-06-10 | 22.4/43.0 | — | 86.2/— | P
2018-10-01 | 22.4/43.0 | 94.0/— | 86.2/— | P
2018-11-02 | 22.4/43.0 | 94.0/141.0 | 86.2/129.4 | P
2018-12-02 | — | 94.0/— | — | P
2019-01-04 | — | — | 86.2/129.4 | P
2019-02-09 | 22.4/43.0 | 94.0/141.0 | 86.2/129.4 | P
2019-03-03 | 22.4/43.0 | 22.4/43.0 | 86.2/129.4 | P
2019-04-03 | 22.4/43.0 | 94.0/141.0 | 86.2/129.4 | P
2019-04-08 | 22.4/43.0 | 94.0/141.0 | 22.4/43.0 | P
2019-05-01 | 22.4/43.0 | 86.2/129.4 | 86.2/129.4 | P
2019-06-01 | 22.4/43.0 | 94.0/141.0 | 86.2/129.4 | P
2019-10-14 | 22.4/43.0 | — | 86.2/129.4 | P
2019-11-26 | 22.4/43.0 | — | 86.2/129.4 | P, I
2019-12-15 | 22.4/43.0 | — | 86.2/129.4 | P, I
2019-12-19 | 22.4/43.0 | 86.2/129.4 | 86.2/129.4 | P, I
2020-01-19 | 22.4/43.0 | — | 86.2/129.4 | P, I
2020-02-22 | 22.4/43.0 | 86.2/129.4 | 86.2/129.4 | P, I
2020-03-20 | 22.4/43.0 | 86.2/129.4 | 86.2/129.4 | P, I
2020-04-18 | 22.4/43.0 | 86.2/129.4 | 86.2/129.4 | P, I
2020-05-24 | — | 22.4/43.0 | — | P, I
2020-12-08 | 22.4/43.0 | 86.2/129.4 | — | P, I
2021-01-03 | 22.4/43.0 | 86.2/— | 22.4/43.0 | I
2021-01-22 | 22.4/43.0 | 22.4/43.0 | 86.2/129.4 | I
2021-02-10 | 22.4/43.0 | — | 86.2/129.4 | I
2021-02-13 | 22.4/43.0 | 94.0/141.0 | 86.2/129.4 | I

Frequencies are in units of GHz. ‘P’ and ‘I’ represent the pulsar position and
total intensity peak position, respectively.

##  2 Observations and Data Reduction

###  2.1 KVN

Both the Crab nebula and 3C 286 have been used as absolute EVPA calibrators
for the Plasma-physics of Active Galactic Nuclei (PAGaN) blazar monitoring
project, a KVN Key Science Project, started in 2017 (Park et al., 2018).
Single-dish observations of these sources are conducted close to (mostly
within 1$\sim$2 days) each VLBI observing run for the EVPA calibration. Each
station operates at two frequencies simultaneously in dual-polarization mode
(Lee et al., 2011). KVN Yonsei (YS) and KVN Tamna (TN) have mostly been used
at 86/129 GHz and 22/43 GHz, respectively. KVN Ulsan (US) has observed at
either 22/43 GHz or 86/129 GHz depending on the weather condition. From
November 2018 to June 2019, US was mostly used at 94/141 GHz. In this study,
we included PAGaN runs from February 2017 to February 2021; the observing
dates and frequencies are summarized in Table 1.

Observations are executed in scans of 40 minutes. Each scan consists of cross-
scan pointing, sky dipping curve measurement, and 8 sets of position-switching
cycles described in Kang et al. (2015). Each position-switching cycle
alternately observes the target source and a reference sky patch. We adopt the
standard deviation of the EVPA values obtained from the 8 sets of position
switching measurements as the random error of the EVPA. Jupiter and the radio
galaxy 3C 84 are assumed to be unpolarized on arcsecond scales, and we
observed at least one of them in each session for the purpose of instrumental
polarization calibration. We reduced the data with the KVN single-dish data
processing pipeline described in Kang et al. (2015). We allocated at least two
scans to each of the Crab nebula and 3C 286 in each session to increase the
signal-to-noise ratio.

For the Crab nebula, we observed either the total intensity peak at 150 GHz
located at equatorial coordinates (J2000) R.A. $=$ 05h34m32.3804s and Dec $=$
22∘00’44.0982” (Ritacco et al., 2018) or the pulsar position at RA $=$
05h34m31.971s and Dec $=$ 22∘00’52.06”. From November 2019 to December 2020,
we observed each time both positions to investigate the EVPA difference
between them. Either position was observed two to four times at elevations
between $30^{\circ}$ and $70^{\circ}$ where the normalized gain becomes larger
than 0.8.
333https://radio.kasi.re.kr/status_report/files/KVN_status_report_2021.pdf

###  2.2 ALMA

The ALMA Calibrator Source Catalogue provides linear polarization measurements
of 3C 286 obtained with the 7-m Array (Misato Fukagawa, priv. commun.). To
confirm the stability of 3C 286 and to use them as reference values for
finding the EVPA of the Crab nebula seen by KVN, we collected the 3C 286 data
obtained from January 18, 2019 (MJD 58501) to December 8, 2021 (MJD 59556);
see Figure 1. The data were obtained in Band 3 (91.5 and 103.5 GHz), Band 6
(233 GHz), and Band 7 (343.5 and 349.5 GHz). In case of Band 7, we combined
the data at 343.5 and 349.5 GHz and used the geometrical mean of the two
frequencies, 346.5 GHz, as proxy for the observing frequency.

##  3 Results

###  3.1 The Stability of 3C 286

Before using the EVPA of 3C 286 to find $\chi_{\rm obs}$ for the total
intensity peak of the Crab nebula (hereafter I-peak), we first need to check
its stability. The EVPA of 3C 286 has been assumed to be stable based on the
fact that the total and polarized flux from 1 to 50 GHz have been stable for
over 30 years (Perley & Butler, 2013a, b). This allows us to compare the EVPA
of 3C 286 measured with KVN with that measured with VLA 10–20 years ago
(Perley & Butler, 2013b) at $\leq 50$ GHz. At higher frequencies, Agudo et al.
(2012) measured the EVPA of 3C 286 with PV from September 2006 to January 2012
and showed that the EVPA of 3C 286 is stable for $\sim$5 years at 86 GHz and
$\sim$2 years at 230 GHz. Since this is shorter than the time gap of 10–15
years between the KVN and PV observing periods, we used ALMA data to test the
stability of 3C 286 at $\geq 86$ GHz over a longer time scale.

The EVPA and fractional polarization of 3C 286 measured with ALMA do not show
any systematic trend with time (see Appendix A). In addition, we found that
the standard deviations of the EVPA are close to the mean of the measurement
uncertainties at all four ALMA frequencies. This suggests that the EVPA of 3C
286 measured with ALMA is stable over the 3-year observing period at
frequencies up to 346.5 GHz. In addition, the weighted mean and its standard
error, $37.0^{\circ}\pm 0.1^{\circ}$ at 91.5 GHz, agrees with the value
$37.3^{\circ}\pm 0.8^{\circ}$ measured with PV at 86 GHz (Agudo et al., 2012),
despite the time gap of $\sim 10$ years between the two datasets (Figure 1).
This suggests that the EVPA of 3C 286 has been stable for more than a decade
at $\sim$86 GHz. This enables us to estimate the EVPA of the Crab nebula seen
by KVN by comparing the EVPA of 3C 286 measured with KVN (see Appendix A) at
86 GHz with that measured with PV at 86 GHz 10–15 years ago.

Figure 1: The EVPA of 3C 286 measured with ALMA (this work), VLA (Perley &
Butler, 2013b), IRAM (Agudo et al., 2012), SMA (Marrone, 2006; Hull et al.,
2016), CARMA (Hull et al., 2014, 2015), and ALMA (Nagai et al., 2016) are
shown for reference. Table 2: Our EVPA values for the Crab nebula (both I-peak
and pulsar position) for KVN, along the reference values for 3C 286.

$\nu$ [GHz] |  $\chi_{\rm ref,3C286}$ [∘] |  $\chi_{\rm peak}$ [∘] |  $\chi_{\rm pulsar}$ [∘]
---|---|---|---
22.4 | 35.0 $\pm$ 0.2a | $154.2\pm 0.3$ | 154.4 $\pm$ 0.4
43.0 | 35.8 $\pm$ 0.1b | $151.0\pm 0.2$ | 150.7 $\pm$ 0.4
86.2 | 37.3 $\pm$ 0.8c | $150.0\pm 1.0$ | 149.0 $\pm$ 1.0
94.0 | 37.0 $\pm$ 0.1d | $151.3\pm 1.1$ | —
129.4 | — |  $149.0\pm 1.6$e | 150.2 $\pm$ 2.0e

a VLA 23 GHz (Perley & Butler, 2013b)
b VLA 45 GHz (Perley & Butler, 2013b)
c XPOL 86 GHz (Agudo et al., 2012)
d ALMA 91.5 GHz (this work)
e XPOL 86 GHz (Ritacco et al., 2018)

###  3.2 The EVPA of the Crab Nebula

Given that the EVPA measured with circular polarization receivers has an
unknown systematic offset $\chi_{\rm off}$ introduced by parts of the receiver
system, what we measure is the EVPA difference $\Delta\chi$ between 3C 286 and
the Crab nebula:

$\displaystyle\Delta\chi$ $\displaystyle=(\chi_{\rm off}+\chi_{\rm
3c286})-(\chi_{\rm off}+\chi_{\rm crab})$ (1) $\displaystyle=\chi_{\rm
3c286}-\chi_{\rm crab}$

where $\chi_{\rm 3c286}$ and $\chi_{\rm crab}$ are the EVPA values of 3C 286
and the Crab nebula respectively. The measurement uncertainties in $\chi_{\rm
crab}$ are negligible compared to those of 3C 286 because the Crab nebula is
much brighter than 3C 286. Therefore, we assume those to be negligible so that
we can convert $\Delta\chi$ to $\chi_{\rm 3c286}$. Then, we can determine
$\chi_{\rm crab}$ that makes $\langle\Delta\chi\rangle-\chi_{\rm crab}$ equal
to the known EVPA values of 3C 286 at each frequency, where
$\langle\Delta\chi\rangle$ is the weighted mean of $\Delta\chi$ observed in
multiple epochs at each frequency. We adopt the standard errors of mean of the
EVPA of 3C 286 as the uncertainties of $\chi_{\rm crab}$ because we neglected
the EVPA uncertainties of the Crab nebula as they are much smaller than those
of 3C 286.

The known EVPA values of 3C 286 at the frequencies closest to the KVN
frequencies are $35.0^{\circ}\pm 0.2^{\circ}$ and $35.8^{\circ}\pm
0.1^{\circ}$ at 23 and 45 GHz (Perley & Butler, 2013b), $37.3^{\circ}\pm
0.8^{\circ}$ at 86 GHz (Agudo et al., 2012), and $37.0^{\circ}\pm 0.1^{\circ}$
and $36.8^{\circ}\pm 0.3^{\circ}$ at 91.5 and 103.5 GHz (from the ALMA data),
respectively. Using these values, $\chi_{\rm obs}$ that we obtained for the
Crab nebula I-peak are 154.2∘ $\pm$ 0.3∘, 151.0∘ $\pm$ 0.2∘, 150.0∘ $\pm$
1.0∘, and 151.3∘ $\pm$ 1.1∘ at 22, 43, 86, and 94 GHz (Table 2 and Figure 2).
Additionally, we measured the EVPA at the pulsar position relative to the EVPA
at I-peak (Table 3), eventually resulting in EVPA values for the pulsar
position of 154.4∘ $\pm$ 0.4∘, 150.7∘ $\pm$ 0.4∘, and 149.0∘ $\pm$ 1.0∘ at 22,
43, and 86 GHz, respectively (Table 2 and Figure 2). The EVPA difference
between the two positions is less than $1^{\circ}$ across the KVN frequency
range (Table 3). This observation is in fair agreement with previous studies
(Ritacco et al., 2018).

At 129 GHz, the number of EVPA measurements of 3C 286 with KVN is too small to
determine $\chi_{\rm obs}$ for the I-peak reliably. Considering that the
angular resolution of XPOL at 86 GHz, 27”, is close to that of KVN at 129 GHz,
23”, we adopted the values $149.0^{\circ}\pm 1.6^{\circ}$ and
$150.2^{\circ}\pm 2.0^{\circ}$ obtained with XPOL at 86 GHz (Ritacco et al.,
2018) as the reference values for the Crab I-peak and pulsar position at 129
GHz, respectively.

Figure 2: The EVPA of I-peak and pulsar position of the Crab nebula as observed by KVN. Table 3: The mean and standard error of the EVPA at I-peak with respect to the EVPA at the pulsar position. $[^{\circ}]$ | 22 GHz | 43 GHz | 86 GHz | 129 GHz
---|---|---|---|---
$\chi_{\rm rel}$ | -0.27$\pm$0.23 | 0.28$\pm$0.34 | 0.98$\pm$0.31 | 0.52$\pm$0.59

###  3.3 The Fractional Polarization of the Crab Nebula

The fractional polarization of the Crab nebula tends to increase with
frequency at both positions (Table 4). This is consistent with previous
observations which found that the fractional polarization is larger at the
central region where the pulsar and the I-peak are located (Aumont et al.,
2010; Ritacco et al., 2018). The fractional polarization measured with KVN at
86 and 129 GHz is 1%–2% lower than that measured with the PV instruments XPOL
at 86 GHz and NIKA at 150 GHz even though their angular resolutions and
frequencies are similar (see Appendix B). This is currently under
investigation and will be discussed in a future study. Previous studies using
POLKA (Wiesemeyer et al., 2014), XPOL (Aumont et al., 2010), SCUPOL (Matthews
et al., 2009), and NIKA (Ritacco et al., 2018) found that the EVPA at I-peak
and at the pulsar position are almost identical within the measurement
uncertainties, even though the absolute values are slightly different since
the instruments operated at different frequencies (see Table 1 in Ritacco et
al., 2018).

Table 4: Fractional polarization at the pulsar position and I-peak of the Crab nebula.  $\nu$ [GHz] | Pulsar [%] | I-peak [%]
---|---|---
22.4 | 12.62 $\pm$ 0.01 | 12.86 $\pm$ 0.01
43.0 | 14.18 $\pm$ 0.01 | 13.39 $\pm$ 0.01
86.2 | 15.64 $\pm$ 0.04 | 17.89 $\pm$ 0.04
129.4 | 15.75 $\pm$ 0.20 | 16.36 $\pm$ 0.14

##  4 Discussion

Our analysis has reliably measured the EVPA for the Crab nebula, both at
I-peak and the pulsar position, across the frequency range accessible to KVN
observations. The values we provide here can serve as a calibration reference
for current and future polarization studies based on KVN data.

The EVPA values at all frequencies and positions are located near 150∘.
Notably, the EVPA values of the Crab nebula (at both positions) at 129 GHz,
which we adopted from XPOL results, are consistent with the KVN values at 43,
86, and 94 GHz within twice the measurement uncertainty. Possible exceptions
are the EVPA values at 22 GHz; $154.2^{\circ}\pm 0.3^{\circ}$ at the I-peak
position and $154.4^{\circ}\pm 0.4^{\circ}$ at the pulsar position deviate
from the EVPA values at the rest of the frequencies by ten times the
measurement uncertainty. Here we explore mechanisms that might introduce a
frequency dependency and the resulting limits on any feasible systematic
trend.

###  4.1 Angular Resolution

The EVPA of the Crab nebula is distributed around $140^{\circ}$–$150^{\circ}$
in the vicinity of I-peak, and changes by $30^{\circ}$ to $100^{\circ}$ toward
the edge of the nebula (see Figure 3 in Ritacco et al., 2018). Therefore,
different EVPA values will be observed with different spatial resolutions even
when observing the same position. Indeed, the EVPA at I-peak obtained with KVN
at 86 GHz with 32” angular resolution, $150.0^{\circ}\pm 1.0^{\circ}$, is
consistent with the value $149.0^{\circ}\pm 1.6^{\circ}$ obtained with XPOL at
86 GHz with 27” angular resolution. However, the difference between values
measured with KVN and XPOL increases with increasing KVN beam size. This
suggests that the spatial resolution may noticeably affect the observed EVPA
value as function of frequency.

One more thing we should take into account is the expansion of the Crab
nebula. It is expanding at 1500 km s-1 which corresponds to 0.15” per year at
a distance of 2 kpc (Bietenholz et al., 1991). This corresponds to an angular
expansion of about 1.5” since the XPOL map was obtained in January 2009. This
would be negligible for the measurements for the entire nebula which extended
over 7 by 5 arcminutes. The beam of KVN is much smaller than the angular size
of the Crab nebula. The smallest beam size of KVN is 23” at 129 GHz. The
expansion of 1.5” corresponds to $\sim$0.4% of the KVN 129 GHz beam in area.
Therefore, we conclude that the effect of the expansion is negligible when we
compare the EVPA values obtained with KVN and IRAM.

###  4.2 Spectral Index

Arendt et al. (2011) mapped the Crab nebula with the GISMO bolometer camera on
PV and constructed spectral index maps at a resolution of 20” using the
wavelength pairs (i) 2 mm and 6 cm, and (ii) 2 mm and 21 cm. For both
frequency pairs, the spectral index becomes closer to zero toward the center
of the nebula, with significant substructure on angular scales smaller than
the KVN beam sizes. This implies that the integrated EVPA within the beam may
change, depending on the observing frequency. Ritacco et al. (2018) collected
maps of the Crab nebula obtained from different instruments at different
frequencies, and degraded them to the 27” resolution of XPOL to remove the
resolution effect from the observed EVPA. The EVPA at the I-peak and the
pulsar position differs depending on the frequency even though the resolutions
are the same (see Table 1 in Ritacco et al., 2018). This suggests that
spectral index variations may contribute to variations of the EVPA with
frequency, if any.

###  4.3 Instrumental Polarization

An ideal circular polarization receiver responds to only one desired
polarization. However, a real circular polarization receiver also responds to
the undesired orthogonal polarization, resulting in instrumental polarization
(Roberts et al., 1994). The instrumental polarization consists of on-axis and
off-axis components. The on-axis instrumental polarization is a component at
the center of the antenna beam that is primarily caused by the polarization
splitter which splits the received signal into RCP and LCP (Napier, 1999).
This can be treated as constant across the beam and can be removed by standard
instrumental polarization calibration processes. On the other hand, the off-
axis instrumental polarization is caused by (i) the reflection at the surface
of the paraboloid main reflector (Rudge et al., 1982; Napier, 1999) and (ii)
complicated quasi-optics for simultaneous multi-frequency observation, and
their contribution varies across the beam.

The response function to the desired polarization, the co-polarization beam,
has a circular symmetry with a maximum at the center. On the other hand, the
response function to the off-axis instrumental polarization, the cross-
polarization beam, has a vertical symmetry with two primary beams, mainly due
to the contribution of ellipsoidal mirrors in the quasi-optics. The amplitude
of the cross-polarization beam is smaller than that of the co-polarization at
the center by more than 50 dB, indicating that the off-axis instrumental
polarization does not affect observations of compact objects such as blazars.
However, it might need to be considered for observations of sources which are
larger than the primary cross-polarization beams at all frequencies of KVN,
such as the Crab nebula, because the two primary beams of cross-polarization
reach their peak values at approximately the location of the half power of the
primary co-polarization beam.

One way to test the effect of cross-polarization on $\chi_{\rm obs}$ for the
Crab nebula is convolving the XPOL map444https://wiki.cosmos.esa.int/planck-
legacy-archive/index.php/External_maps (Aumont et al., 2010), the only
available map of the Crab nebula at KVN frequencies, with the co-polarization
and cross-polarization beams of KVN, and comparing the total flux values
obtained with the two beams at I-peak. We confirmed that the ratio between the
flux obtained with cross-polarization beam and that obtained with co-
polarization beam is less than 0.015% at 22–86 GHz. According to the error
propagation of polarization, this flux ratio corresponds to the EVPA of 0.03∘
at all KVN freuqencies. Therefore, we conclude that the effect of the
instrumental polarization is negligible.

###  4.4 Faraday Rotation

When linearly polarized radiation passes through a magnetized plasma, the EVPA
rotates due to Faraday rotation. The amount of Faraday rotation is
proportional to the integral of the product of the electron density $n_{\rm
e}$ and magnetic field $\vec{B}$ along the line of sight $\vec{l}$, the
rotation measure (RM)

${\rm RM}\propto\int n_{\rm e}\vec{B}\cdot d\vec{l}$ (2) $\chi_{\rm
obs}=\chi_{0}+\lambda^{2}\,\rm RM$ (3)

where $\chi_{\rm obs}$ and $\chi_{0}$ are the observed and intrinsic EVPA,
respectively. The large-scale RM over the Crab nebula is between $-21$ rad m-2
and $-43$ rad m-2 from centimeter to millimeter wavelengths (Burn, 1966;
Manchester, 1971; Rankin et al., 1988; Bietenholz et al., 1991; Weiland et
al., 2011). This corresponds to Faraday rotation of less than $0.5^{\circ}$
between 22 and 129 GHz, indicating that the effect of Faraday rotation is
negligible within the KVN frequency range.

##  5 Summary and Conclusions

In this paper, we have reported the first cross-calibrated EVPA measurements
of the Crab nebula at its total intensity peak (I-peak) and pulsar positions
by KVN.

In order to obtain the intrinsic EVPA of the Crab nebula as observed by KVN,
we used the EVPA of the standard calibrator 3C 286 as reference.

We first tested its stability using archival ALMA polarization measurements.
The EVPA and fractional polarization of 3C 286 measured with ALMA between
January 2019 and December 2021 are stable up to frequencies of 346.5 GHz at
least within the 3-year observation period. The average EVPA at each ALMA
frequency is consistent with the values obtained with other instruments over a
wide frequency range. In particular, the value $37.0^{\circ}\pm 0.1^{\circ}$
measured with ALMA at 91.5 GHz is virtually identical to $37.3^{\circ}\pm
0.8^{\circ}$ obtained with the Pico Veleta observatory at 86 GHz between
September 2006 and January 2012. This suggests that the EVPA of 3C 286 is
stable for more than a decade at least for the KVN frequency range, and enable
us to use it to estimate the EVPA of the Crab nebula seen by KVN.

Using the EVPA values of 3C 286 as reference, we have shown that the EVPA at
the Crab nebula I-peak seen by KVN is $154.2^{\circ}\pm 0.3^{\circ}$,
$151.0^{\circ}\pm 0.2^{\circ}$, $150.0^{\circ}\pm 1.0^{\circ}$, and
$151.3^{\circ}\pm 1.1^{\circ}$ at 22, 43, 86, and 94 GHz, respectively. Using
our measurements of the EVPA difference between the I-peak and the pulsar
position, we have also found that the EVPA values at the pulsar position are
$154.4^{\circ}\pm 0.4^{\circ}$, $150.7^{\circ}\pm 0.4^{\circ}$, and
$149.0^{\circ}\pm 1.0^{\circ}$ at 22, 43, and 86 GHz respectively. Due to the
lack of EVPA measurements of 3C 286 at 129 GHz, we adopted the XPOL values
$149.0^{\circ}\pm 1.6^{\circ}$ and $150.2^{\circ}\pm 2.0^{\circ}$ for the Crab
I-peak and pulsar position at 129 GHz, respectively, considering the fact that
the resolution and frequency of XPOL at 86 GHz are close to that of KVN at 129
GHz. Indeed, the values obtained with XPOL are in agreement with that obtained
with KVN at 43, 86, and 94 GHz, except for the values at 22 GHz.

There are several factors that might contribute to a variation of the EVPA
with frequency. Since the angular size of the Crab nebula is larger than the
beams of the KVN, the integrated EVPA within the beam of KVN could be
different at different frequencies. In addition, the spectral index is steeper
in the outer region of the Crab nebula, which also may affect the observed
EVPA at different frequencies. When considering the effects of instrumental
polarization and Faraday rotation, we found them to be negligible.

Our analysis has demonstrated that the Crab nebula, which is a well-known EVPA
calibrator in radio astronomy, can be used for this purpose also by the KVN.
The specific (frequency-dependent) EVPA values we find can serve as references
for current and future polarimetric studies with the KVN.

Figure 3: The EVPA and fractional polarization of 3C 286 measured with ALMA.
Gray-dotted lines on the left represent the weighted mean of the EVPA at each
frequency. Gray-dotted lines on the right represent the weighted mean of the
fractional polarization at each frequency.

###### Acknowledgements.

We are grateful to the staff of the KVN who helped to operate the array and to
correlate the data. We thank Jonathan Aumont and Alessia Ritacco for providing
the XPOL data and useful discussions. PAGaN is a KVN Key Science Program. The
KVN is a facility operated by the Korea Astronomy and Space Science Institute
(KASI). The KVN observations and correlations are supported through the high-
speed network connections among the KVN sites provided by the Korea Research
Environment Open NETwork (KREONET), which is managed and operated by the Korea
Institute of Science and Technology Information (KISTI). We acknowledge
financial support from the National Research Foundation of Korea (NRF) grant
2022R1F1A1075115. J.P. acknowledges ﬁnancial support from the Korean National
Research Foundation (NRF) via Global PhD Fellowship grant 2014H1A2A1018695 and
support through the EACOA Fellowship awarded by the East Asia Core
Observatories Association, which consists of the Academia Sinica Institute of
Astronomy and Astrophysics, the National Astronomical Observatory of Japan,
Center for Astronomical Mega-Science, Chinese Academy of Sciences, and the
Korea Astronomy and Space Science Institute. This paper makes use of the
following ALMA data: ADS/JAO.ALMA#2011.0.00001.CAL. ALMA is a partnership of
ESO (representing its member states), NSF (USA) and NINS (Japan), together
with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, AUI/NRAO and NAOJ.

##  Appendix A The Stability of 3C 286

To determine if there is any systematic trend with time in the polarization of
3C 286 over the 3-year observing period of ALMA, we compared the standard
deviations of the EVPA and the fractional polarization with the average values
of their measurement uncertainties at each frequency. We found that the
standard deviations of the EVPA and the fractional polarization are close to
the mean of the measurement uncertainties at all four ALMA frequencies,
suggesting that both the EVPA and the fractional polarization are stable at
least for 3 years up to 346.5 GHz (see Figure 3 and Table 5).

Table 5: Polarization measurements of 3C 286 with ALMA.

$\nu$ | $\chi$ | m | sχa |  $\langle\varepsilon_{\chi}\rangle$b | smc |  $\langle\varepsilon_{\rm m}\rangle$d
---|---|---|---|---|---|---
$[$GHz$]$ | [∘] | [%] | [∘] | [∘] | [%] | $[\%]$
91.5 | $37.0\pm 0.1$ | $14.3\pm 0.1$ | 2.8 | 2.1 | 1.5 | 1.2
103.5 | $36.8\pm 0.1$ | $14.7\pm 0.1$ | 2.4 | 2.2 | 1.2 | 1.3
233 | $37.5\pm 0.2$ | $16.3\pm 0.2$ | 2.1 | 2.2 | 1.3 | 1.3
346.5 | $40.5\pm 0.3$ | $16.5\pm 0.2$ | 3.6 | 2.6 | 2.7 | 2.0

a Standard deviation of the EVPA.
b Mean EVPA error.
c Standard deviation of the fractional polarization.
d Mean fractional polarization error.

Table 6: Fractional polarization measurements of 3C 286.

$\nu[$GHz$]$ | mkvn [%] | mref [%]
---|---|---
22.4 | 10.5 $\pm$ 0.1 | 12.6a
43.0 | 10.9 $\pm$ 0.1 | 13.2b
86.2 | 11.3 $\pm$ 0.2 | 13.5 $\pm$ 0.3c
94.0 | 12.6 $\pm$ 0.4 | 14.3 $\pm$ 0.1d
129.3 | 10.9 $\pm$ 0.8 | 14.7 $\pm$ 0.1e

a VLA 22.4 GHz (Perley & Butler, 2013b)
b VLA 43.5 GHz (Perley & Butler, 2013b)
c XPOL 86 GHz (Agudo et al., 2012)
d ALMA 91.5 GHz (this work)
e ALMA 103.5 GHz (this work)

Figure 4: Fractional polarization at pulsar and I-peak. XPOL, NIKA, POLKA,
SCUPOL results are taken from Table 1 in Ritacco et al. (2018).  Figure 5: The
fractional polarization of 3C 286 obtained with KVN (this work), ALMA (this
work), VLA (Perley & Butler, 2013b), IRAM (Agudo et al., 2012), SMA (Marrone,
2006; Hull et al., 2016), and ALMA (Nagai et al., 2016).

##  Appendix B Fractional Polarization

The fractional polarization at the I-peak and pulsar positions of the Crab
nebula are presented in Figure 4. The beam sizes of KVN at 86 and 129 GHz are
32” and 23”, respectively. These are very close to the 27” and 18” beam sizes
of XPOL at 86 GHz and NIKA at 150 GHz, respectively. The apertures of the KVN
and IRAM Pico Veleta antennas are 21 m and 30 m, respectively. The KVN
antennas are shaped Cassegrain reflectors whose aperture efficiencies,
compared to the usual combination of a parabolic main and a hyperbolic sub-
reflector, are increased by slightly modifying the reflector surfaces (Kim et
al., 2011); the IRAM telescope uses a Nasmyth-focus configuration (Baars et
al., 1987). This makes the angular resolution of KVN close to that of IRAM at
a given frequency, despite the different apertures.

Nevertheless, the fractional polarization of 16%–18% measured with KVN at 86.2
and 129.4 GHz (see Figure 4) is somewhat lower than the values 17%–21%
measured with XPOL at 86 GHz and NIKA at 150 GHz. This discrepancy in the
fractional polarization corresponds to a loss of polarized flux of 11%–14% at
86 GHz and 12%–20% at 129 GHz. Likewise, the fractional polarization of 3C 286
measured with KVN is lower than that measured with other instruments (see
Figure 5 and Table 6). In terms of polarized emission, the loss corresponds to
12%–17% at 22–94 GHz and $26\%$ at 129 GHz.

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