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

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. $=$ 05^h34^m32.3804^s and Dec $=$ 22^∘00’44.0982” (Ritacco et al., 2018) or the pulsar position at RA $=$ 05^h34^m31.971^s 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

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.

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.

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:

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.

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).

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.

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.

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.

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)

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.