CHANG-ES – XI. Circular polarization in the cores of nearby galaxies

Abstract

We detect five galaxies in the Continuum Halos in Nearby Galaxies – an EVLA Survey (CHANG-ES) sample that show circular polarization (CP) at L band in our high-resolution data sets. Two of the galaxies (NGC 4388 and NGC 4845) show strong Stokes V/I ≡ m_C ∼ 2 per cent, two (NGC 660 and NGC 3628) have values of m_C ∼ 0.3 per cent, and NGC 3079 is a marginal detection at m_C ∼ 0.2 per cent. The two strongest m_C galaxies also have the most luminous X-ray cores and the strongest internal absorption in X-rays. We have expanded on our previous Faraday conversion interpretation and analysis and provide analytical expressions for the expected V signal for a general case in which the cosmic ray (CR) electron energy spectral index can take on any value. We provide examples as to how such expressions could be used to estimate magnetic field strengths and the lower energy cut-off for CR electrons. Four of our detections are resolved, showing unique structures, including a jet in NGC 4388 and a CP ‘conversion disc’ in NGC 4845. The conversion disc is inclined to the galactic disc but is perpendicular to a possible outflow direction. Such CP structures have never before been seen in any galaxy to our knowledge. None of the galaxy cores show linear polarization at L band. Thus radio CP may provide a unique probe of the physical conditions in the cores of active galactic nuclei.

galaxies: active, galaxies: individual: NGC 4388, NGC 660, NGC 3628, NGC 3079, NGC 4845, galaxies: jets, galaxies: nuclei

1 INTRODUCTION

Circular polarization (CP) has been an elusive quantity to measure in extragalactic sources. When present, it is a weak signal with a typical value of m_C ∼ 0.2 per cent or less in Stokes |V/I|, where m_C ≡ V/I (Rayner, Norris & Sault 2000).¹ A strong signal could be considered |${>}0.3\hbox{ per cent}$| (Homan & Lister 2006). In addition, observing CP is technically challenging (Section 2.4) and CP, when measured, can be highly variable (see below).

When observed, the most successful mechanism for explaining CP and most closely matching the observations thus far is Faraday conversion (e.g. Jones 1988; Beckert & Falcke 2002; Macquart 2002; Beckert 2003; Enßlin 2003; O'Sullivan et al. 2013; Irwin et al. 2015). Initially linearly polarized emission, when travelling through a birefringent medium, can be converted to CP, provided the angle between the linearly polarized signal and the magnetic field varies along the line of sight. This condition can be achieved by normal Faraday rotation in a regular B field, or by a changing B field such as might be seen in a jet with a helical B component (Enßlin 2003; Irwin et al. 2015).

CP is often observed in or near the optically thick regime in total intensity (Rayner et al. 2000). This may be because Stokes V is predicted to have a steep negative spectral index in most cases (see Section 3.4). Thus low-frequency observations are more likely to result in a CP detection and this is where sources are more likely to be optically thick in total intensity. Although complex, observations of CP have the potential to probe the physical structure and properties of active galactic nuclei (AGNs) deep into their cores.

Almost all measurements of extragalactic CP have so far been made in AGNs, for example, as revealed in BL Lac objects (Gabuzda 2003), and blazars (Myserlis 2015), i.e. in sources for which strong AGN activity is clearly occurring and well known. For AGNs with jets, the circularly polarized signal is associated with the compact core (e.g. Homan & Wardle 2003). Rarely, CP can additionally be seen in inner bright jets; for example the Monitoring Of Jets in Active galactic nuclei with VLBA Experiments (MOJAVE) very long baseline interferometry (VLBI) survey at 15 GHz has found CP in 15 per cent of its 133 AGNs of which only are few examples of CP were detected in jets (Homan & Lister 2006). The signal is often time variable (Aller, Aller & Plotkin 2003), although stable systems have also been observed both in m_C (e.g. Myserlis et al. 2018, over time periods of up to 5.7 yr) and in the sign of m_C (e.g. Homan, Attridge & Wardle 2001, over a time period of 20 yr).

Prior to the Continuum Halos in Nearby Galaxies – an EVLA Survey (CHANG-ES) program (Irwin et al. 2012a),² the only nearby systems showing CP were M81 (Brunthaler et al. 2001; Brunthaler, Bower & Falcke 2006) and Sgr A* in the Milky Way itself (Bower et al. 2002; Bower 2003; Muñoz et al. 2012).³ The discovery of strong CP, at a level of |${\approx }2\hbox{ per cent}$| in a Virgo Cluster CHANG-ES galaxy, NGC 4845, however (Irwin et al. 2015) has opened up the possibility of exploring new nearby AGNs in detail, especially AGNs that are embedded in other emission.

NGC 4845 was observed serendipitously about 1 yr after a tidal disruption event (TDE) was detected in the hard X-ray regime; this event set the outburst time. The fact that CHANG-ES observations were broad-band yielding in-band spectral indices, and because multiple observations in different arrays were obtained, we were able to fit a simple AGN jet model with helical B fields to this source. Key to these measurements was the observation of steep negative spectral indices for Stokes V, α_V, which are predicted for Faraday conversion (Irwin et al. 2015, their equation E11) in the absence of complicated structural inhomogeneities (Jones & O'Dell 1977a). CP was detected at 1.6 GHz (hereafter L band) near the turn-over to optical thickness, but not at 5 GHz (hereafter C band) consistent with a steep α_V. The CP has now been confirmed through follow-up VLBI observations (Perlman et al. 2017) at about the same percentage level.

We have now done a thorough search through all 35 CHANG-ES galaxies at L band in our highest resolution (≈3 arcsec) B-configuration data, and here report on the detection of four more sources showing believable CP. Together with NGC 4845, there are now five CHANG-ES galaxies that show CP in their cores. Since optical spectroscopy surveys miss approximately half of the AGN population simply due to extinction through the host galaxy (Goulding & Alexander 2009), radio detection of CP is an additional tool for identifying AGNs, especially low-luminosity AGN (LLAGN; Ptak 2001), provided sensitive enough data are obtained. Since linear polarization (LP) may be completely undetectable in compact cores due to depolarization, indeed CP may be the only way of detecting low-luminosity radio AGNs with certainty when they are embedded in nuclear star-forming regions.

The identification of AGN is also important for the far-infrared (FIR)–radio continuum relation. For example, Wong et al. (2016) report that AGNs contribute significantly to this relation at 1.4 GHz; we also discuss the AGN effect on this relation in Li et al. (2016). Note that the detection of CP in a galaxy core is strong evidence that an AGN is present, but an AGN may still be present without necessarily showing CP. We will report, more generally, on AGNs in the CHANG-ES sample in a future paper.

In Section 2 we describe our observations and data reductions. Section 3 outlines our results and presents the X-ray data. Section 4 gives the discussion and Section 5 presents our conclusions. In addition, Appendix A shows the Faraday conversion development, progressing from the foundational work by Beckert & Falcke (2002).

2 OBSERVATIONS AND DATA REDUCTIONS

Observations were made with the Karl G. Jansky Very Large Array (JVLA) using standard CHANG-ES data reduction and imaging procedures. A complete description of the observing, data reduction, and imaging strategies can be found in Irwin et al. (2012b, 2013). A brief description follows.

2.1 Observations

B-configuration L-band observations were centred at 1.575 GHz with a bandwidth extending from 1.247 to 1.503 GHz and from 1.647 to 1.903 GHz for a total of 512 MHz. The central gap was set to avoid known interference. The L-band bandwidth contained 32 spectral windows, each with 64 channels for a total of 2048 channels; however, Hanning smoothing was applied so that the effective channel width was doubled.

The observations were carried out within a ‘scheduling block’ (SB), in the standard fashion that included a single scan on the primary gain and phase calibrator (hereafter the primary calibrator) of known flux density and source structure, and a single scan on a polarization leakage calibrator that is known to have negligible LP (OQ 208 was used for all galaxies except NGC 660 and NGC 891 that used 3C 84). The polarization calibrator (which was the primary gain and phase calibrator) is used only for determination of the LP and is not relevant for the CP (but see comments in Section 4) that we focus on here (see below). Observations on the source were flanked by observations of a secondary gain and phase calibrator (hereafter the secondary calibrator) that was close to the source on the sky. Other galaxies were included in the SB and observations of any given galaxy were interspersed throughout the SB so that a good uv distribution would result. The sources, their calibrators, and observing dates are listed in Table 1. Information about these galaxies can be found in Irwin et al. (2012a) and their distances are in Wiegert et al. (2015). We provide details, distances, etc. for the CP detected galaxies in Sections 4.1–4.5.

Table 1.

B-configuration L-band^a observing data.

Galaxy	Obs. dateb	RAc	Dec.c	Primary cal.d	Second. cal.e
NGC 660	2012-06-24	|${\rm 01}^{\rm h}43^{\rm m}02{^{\rm s}_{.}}40$|	+13°38΄42|${^{\prime\prime}_{.}}$|2	3C 48	J0204+1514
NGC 891	2012-06-24	|${\rm 02}^{\rm h}22^{\rm m}33{^{\rm s}_{.}}41$|	+42°20΄56|${^{\prime\prime}_{.}}$|9	3C 48	J0230+4032
NGC 2613	2011-03-21	|${\rm 08}^{\rm h}33^{\rm m}22{^{\rm s}_{.}}84$|	−22°58΄25|${^{\prime\prime}_{.}}$|2	3C 286	J0856−2610
NGC 2683	2012-06-16	|${\rm 08}^{\rm h}52^{\rm m}41{^{\rm s}_{.}}35$|	+33°25΄18|${^{\prime\prime}_{.}}$|5	3C 286	J0837+2454
NGC 2820	2012-06-24	|${\rm 09}^{\rm h}21^{\rm m}45{^{\rm s}_{.}}58$|	+64°15΄28|${^{\prime\prime}_{.}}$|6	3C 286	J0921+6215
NGC 2992	2011-03-21	|${\rm 09}^{\rm h}45^{\rm m}42{^{\rm s}_{.}}00$|	−14°19΄35|${^{\prime\prime}_{.}}$|0	3C 286	J0943−0819
NGC 3003	2012-06-16	|${\rm 09}^{\rm h}48^{\rm m}36{^{\rm s}_{.}}05$|	+33°25΄17|${^{\prime\prime}_{.}}$|4	3C 286	J0956+2515
NGC 3044	2011-03-21	|${\rm 09}^{\rm h}53^{\rm m}40{^{\rm s}_{.}}88$|	+01°34΄46|${^{\prime\prime}_{.}}$|7	3C 286	J1007−0207
NGC 3079	2012-06-23	|${\rm 10}^{\rm h}01^{\rm m}57{^{\rm s}_{.}}80$|	+55°40΄47|${^{\prime\prime}_{.}}$|3	3C 286	J1035+5628
NGC 3432	2012-06-24	|${\rm 10}^{\rm h}52^{\rm m}31{^{\rm s}_{.}}13$|	+36°37΄07|${^{\prime\prime}_{.}}$|6	3C 286	J1130+3815
NGC 3448	2012-06-23	|${\rm 10}^{\rm h}54^{\rm m}39{^{\rm s}_{.}}24$|	+54°18΄18|${^{\prime\prime}_{.}}$|8	3C 286	J1035+5628
NGC 3556	2012-06-23	|${\rm 11}^{\rm h}11^{\rm m}30{^{\rm s}_{.}}97$|	+55°40΄26|${^{\prime\prime}_{.}}$|8	3C 286	J1035+5628
NGC 3628	2012-07-30	|${\rm 11}^{\rm h}20^{\rm m}17{^{\rm s}_{.}}01$|	+13°35΄22|${^{\prime\prime}_{.}}$|9	3C 286	J1120+1420
NGC 3735	2012-06-24	|${\rm 11}^{\rm h}35^{\rm m}57{^{\rm s}_{.}}30$|	+70°32΄08|${^{\prime\prime}_{.}}$|1	3C 286	J1313+6735
NGC 3877	2012-06-17	|${\rm 11}^{\rm h}46^{\rm m}07{^{\rm s}_{.}}80$|	+47°29΄41|${^{\prime\prime}_{.}}$|2	3C 286	J1219+4829
NGC 4013	2012-08-11	|${\rm 11}^{\rm h}58^{\rm m}31{^{\rm s}_{.}}38$|	+43°56΄47|${^{\prime\prime}_{.}}$|7	3C 286	J1146+3958
NGC 4096	2012-08-11	|${\rm 12}^{\rm h}06^{\rm m}01{^{\rm s}_{.}}13$|	+47°28΄42|${^{\prime\prime}_{.}}$|4	3C 286	J1146+3958
NGC 4157	2012-07-04	|${\rm 12}^{\rm h}11^{\rm m}04{^{\rm s}_{.}}37$|	+50°29΄04|${^{\prime\prime}_{.}}$|8	3C 286	J1219+4829
NGC 4192	2012-07-30	|${\rm 12}^{\rm h}13^{\rm m}48{^{\rm s}_{.}}29$|	+14°54΄01|${^{\prime\prime}_{.}}$|2	3C 286	J1254+1141
NGC 4217	2012-08-11	|${\rm 12}^{\rm h}15^{\rm m}50{^{\rm s}_{.}}90$|	+47°05΄30|${^{\prime\prime}_{.}}$|4	3C 286	J1219+4829
NGC 4244	2012-06-09	|${\rm 12}^{\rm h}17^{\rm m}29{^{\rm s}_{.}}66$|	+37°48΄25|${^{\prime\prime}_{.}}$|6	3C 286	J1227+3635
NGC 4302	2012-07-29	|${\rm 12}^{\rm h}21^{\rm m}42{^{\rm s}_{.}}48$|	+14°35΄53|${^{\prime\prime}_{.}}$|9	3C 286	J1254+1141
NGC 4388	2012-07-29	|${\rm 12}^{\rm h}25^{\rm m}46{^{\rm s}_{.}}75$|	+12°39΄43|${^{\prime\prime}_{.}}$|5	3C 286	J1254+1141
NGC 4438f	2012-07-29	|${\rm 12}^{\rm h}27^{\rm m}45{^{\rm s}_{.}}59$|	+13°00΄31|${^{\prime\prime}_{.}}$|8	3C 286	J1254+1141
NGC 4565	2012-06-03	|${\rm 12}^{\rm h}36^{\rm m}20{^{\rm s}_{.}}78$|	+25°59΄15|${^{\prime\prime}_{.}}$|6	3C 286	J1221+2813
NGC 4594	2011-03-17	|${\rm 12}^{\rm h}39^{\rm m}59{^{\rm s}_{.}}43$|	−11°37΄23|${^{\prime\prime}_{.}}$|0	3C 286	J1248−1959
NGC 4631	2012-06-03	|${\rm 12}^{\rm h}42^{\rm m}08{^{\rm s}_{.}}01$|	+32°32΄29|${^{\prime\prime}_{.}}$|4	3C 286	J1221+2813
NGC 4666	2012-06-10	|${\rm 12}^{\rm h}45^{\rm m}08{^{\rm s}_{.}}59$|	−00°27΄42|${^{\prime\prime}_{.}}$|8	3C 286	J1246−0730
NGC 4845	2012-06-11	|${\rm 12}^{\rm h}58^{\rm m}01{^{\rm s}_{.}}19$|	+01°34΄33|${^{\prime\prime}_{.}}$|0	3C 286	J1407+2827
NGC 5084	2011-03-17	|${\rm 13}^{\rm h}20^{\rm m}16{^{\rm s}_{.}}92$|	−21°49΄39|${^{\prime\prime}_{.}}$|3	3C 286	J0204+1514
NGC 5297	2012-06-10	|${\rm 13}^{\rm h}46^{\rm m}23{^{\rm s}_{.}}68$|	+43°52΄20|${^{\prime\prime}_{.}}$|5	3C 286	J1327+4326
NGC 5775	2011-04-05	|${\rm 14}^{\rm h}53^{\rm m}58{^{\rm s}_{.}}00$|	+03°32΄40|${^{\prime\prime}_{.}}$|1	3C 286	J1445+0958
NGC 5792	2011-04-05	|${\rm 14}^{\rm h}58^{\rm m}22{^{\rm s}_{.}}71$|	−01°05΄27|${^{\prime\prime}_{.}}$|9	3C 286	J1505+0326
NGC 5907	2011-03-08	|${\rm 15}^{\rm h}15^{\rm m}53{^{\rm s}_{.}}77$|	+56°19΄43|${^{\prime\prime}_{.}}$|6	3C 286	J1438+6211
UGC 10288	2011-04-05	|${\rm 16}^{\rm h}14^{\rm m}24{^{\rm s}_{.}}80$|	−00°12΄27|${^{\prime\prime}_{.}}$|1	3C 286	J1557−0001

Notes.^aThe central frequency for all observations and resulting maps was 1.575 GHz, unless otherwise indicated.

^bObserving dates, designated year-month-day (ut).

^cCentre of galaxy, from the NASA Extragalactic Database (NED); the pointing centre and field centre were set to these values.

^dPrimary gain and phase calibrator.

^eSecondary gain and phase calibrator.

^fOnly the upper half of the band produced reliable results for NGC 4438, hence the central frequency for maps of this galaxy was 1.775 GHz.

2.2 Signal detection at the JVLA

The JVLA uses circularly polarized right (R) and left (L) handed feeds and a correlated output from any two antennas RR, LL (the ‘parallel-hands’) and RL, LR (the ‘cross-hands’) is measured for each baseline. These are related to the Stokes parameters via

\begin{eqnarray} I & = & (RR + LL)/2, \end{eqnarray}

(1)

\begin{eqnarray} Q & = & (RL + LR)/2, \end{eqnarray}

(2)

\begin{eqnarray} U & = & (RL - LR)/2i, \end{eqnarray}

(3)

\begin{eqnarray} V & = & (RR - LL)/2, \end{eqnarray}

(4)

where it is understood that these signals are averages over some integration time (10 s for CHANG-ES). The absolute scale is set by the primary calibrator. LP and angle of a linearly polarized signal are then, respectively,

\begin{eqnarray} P_{{\rm lin}}&=&\sqrt{Q^2+U^2 - \sigma _{Q,U}^2}, \end{eqnarray}

(5)

\begin{eqnarray} \chi &=& \left(1/2\right)\arctan \left(U/Q\right), \end{eqnarray}

(6)

where σ_{Q, U} is the rms noise of the Q and U maps and does an approximate correction for Ricean bias (Simmons & Stewart 1985; Everett & Weisberg 2001; Vaillancourt 2006).⁴

The JVLA follows the IEEE standard such that a positive V signal corresponds to a counter-clockwise rotation of the |$\boldsymbol {E}$| vector when viewing the signal coming towards us and negative V corresponds to clockwise rotation (Perley 2016).

2.3 CHANG-ES standard calibrations and imaging

Data were reduced using the Common Astronomy Software Applications (casa) package⁵ (McMullin et al. 2007). Calibration steps are standard for wide band data are more thoroughly described in Irwin et al. (2013). Calibration of RR and LL is based on a known model for the primary gain and phase calibrator (Perley & Butler 2013), usually 3C 286 (Table 1). RR and LL are calibrated separately. Corrections were applied for antenna positions, delays, bandpasses, and gain and phase as a function of time, the latter via the secondary gain and phase calibrators. The data were Hanning smoothed and flagged manually and several iterations of the calibration steps were carried out when additional flagging occurred (and so throughout). In each determination of a new correction table, any previous tables were applied on the fly. Corrections were applied to all calibrators as well.

In addition, the cross-hands were also calibrated and imaged to obtain polarization and polarization angle images, but since LP is not the focus of this paper, we omit those details in our description. See Irwin et al. (2012b, 2013) for further calibration details.

Imaging in Stokes I was carried out using casa’s clean task, utilizing the multiscale multifrequency (ms–mfs) algorithm (Rau & Cornwell 2011), including a wide field option (Cornwell, Golap & Bhatnagar 2008) with 128 w-projection planes in most cases and Briggs robust = 0 uv weighting (Briggs 1995). During cleaning, an in-band spectral index, α, is fitted, but no spectral index curvature term is included. Self-calibration (e.g. Pearson & Readhead 1984) was also attempted for the total intensity images but results were only kept if the dynamic range improved. Total intensity images were then corrected for the primary beam (PB).

2.4 Special considerations for circular polarization

A Stokes V signal can be either positive or negative and the measurements are technically challenging. This is because the signal is intrinsically weak and possibly variable (Section 1) and also because one must measure a difference (rather than the average) of signals (equation 4); those correlated signals (RR or LL) are sensitive to weather disturbances and radio frequency interference (RFI). In addition, circularly polarized feeds are not the best choice for detection of circularly polarized signals because of possible instrumental effects that can carry through to create an apparent signal; linearly polarized feeds, such as used at the Australia Telescope Compact Array (ATCA) have advantages for observing CP (e.g. Rayner et al. 2000; O'Sullivan et al. 2013). Nevertheless, CP can still be measured with sufficient care; CP in M81, Sgr A*, and CHANG-ES NGC 4845 (Section 1) were measured at the Very Large Array (VLA). JVLA polarization characteristics are also known to be stable over time-scales of months with polarization dynamic ranges of 10³ possible (Sault & Perley 2013).

Recently Myserlis et al. (2018) have looked carefully at how to correct for instrumental effects using circular feeds, finding typical uncertainties of |${\approx }0.1\hbox{ per cent}$| in m_C using the Effelsberg 100-m telescope. However, CHANG-ES does not have sufficient calibrator coverage to replicate this approach.

A ‘gain transfer’ procedure has also been described in Homan & Wardle (1999), Homan et al. (2001), and Homan & Lister (2006). In this approach, a large number of sources are averaged and smoothed in time (RR and LL separately) to produce a smoothed set of gains that can be applied to all sources. The calibration is carried out iteratively so that outliers can be omitted prior to the final gain calibration. This approach can work well with a large number of sources that are observed together, for example 40 sources in 48 h as described in Homan et al. (2001). By contrast, each CHANG-ES B-array L-band observation was 2 consecutive hours in duration with the entire sample observed over the course of 17 months. Thus, individual calibration for each galaxy was required.

Therefore, of necessity, we have calibrated our sample in the standard fashion (Section 2.3), but have attempted to incorporate possible uncertainties into our error estimates (e.g. possible CP in the secondary calibrators) as described below. Our approach is similar to that adopted by Rayner et al. (2000, their section 3.3).

Errors can be separated into direction-independent errors and direction-dependent errors. The former relate to calibration errors prior to imaging and the latter relate to errors that enter into the imaging process (Bhatnagar et al. 2008).

Direction-independent errors in V include a possible calibration error due to the fact that the parallel hand leakage terms remain uncalibrated in our sample. Such errors have a maximum value of (Homan & Wardle 1999)

\begin{eqnarray} \frac{V}{I} = \frac{1}{\sqrt{N_{\rm s} N_{\rm a} \left(N_{\rm a}-1\right)}}\left(\sqrt{2\left(N_{\rm a} - 1\right)}\,m_{\rm L}\,D + D^2 \right), \end{eqnarray}

(7)

where N_s is the number of scans that are separated in parallactic angle, N_a is the number of antennas, m_L is the fractional linear polarization, and D is the fractional leakage term. Here N_a = 27. For our detections (Section 3.2), the parallactic angle coverage spans typically 109° (minimum of 90° and maximum of 160°) in six different scans; thus we take N_s = 6, or at least 15° separating scans. In no case do we detect any LP at the location of the V signal. Thus we assume an upper limit to m_L as the rms noise level of Q and U (which are the same) divided by the total intensity at the location of the V signal. Finally, we do not have a direct measurement of D, but have referred to a variety of sources that specify its value at the JVLA to be a few per cent. To overestimate this error, we take D = 0.05. For our detections, then, the largest error in V/I as a result of neglecting possible leakage in the parallel hands is 0.004 per cent. As this is significantly smaller than the other errors discussed below, it will not be considered further.

A remaining possible source of calibration error is if the calibrators themselves might be circularly polarized. We therefore imaged the (calibrated) primary and secondary calibrators (Table 1) for each source and included a measurement of the residual calibrator signal into the uncertainty analysis. We then take the uncertainty in V/I to be

\begin{eqnarray} \sigma _{V/I} = \sqrt{{\sigma _{{\rm gal}}}^2 + {\sigma _{{\rm prim}}}^2 + {\sigma _{{\rm sec}}}^2 }, \end{eqnarray}

(8)

where

\begin{eqnarray} \sigma _{{\rm gal}} &=& {\Delta V_{{\rm gal}}}/I_{{\rm gal}}, \end{eqnarray}

(9)

\begin{eqnarray} \sigma _{{\rm prim}} &=& {\Delta V_{{\rm prim}}}/I_{{\rm prim}}, \end{eqnarray}

(10)

\begin{eqnarray} \sigma _{{\rm sec}} &= {\Delta V_{{\rm sec}}}/I_{{\rm sec}}. \end{eqnarray}

(11)

In the above, the subscripts, gal, prim, and sec refer to the galaxy, the primary calibrator, and the secondary calibrator, respectively.

For the galaxy, ΔV_gal is the rms noise of the V map. For the point source calibrators, ΔV_prim and ΔV_sec are the rms values⁶ of a small residual signal at the map centre that, in all cases, was larger than the rms noise of the V map as a whole. The residual calibrator signal occurred and was measured over the region in which I had contiguously positive values (essentially within the region in which the synthesized beam falls to zero). In the event that a residual V is present in the calibrator, this error measurement should take such a signal into account. The galaxy signal was considered to be real only if |m_C| >  σ_V/I. The absolute value is used simply because CP can be positive or negative as indicated above.

This approach is meant to be conservative so that we do not mistake calibration errors for real signals. It is possible that real CP signals could still be present at lower levels but the observations would have to have been designed differently to detect them with confidence.

As for direction-dependent, or post-calibration errors, we first note that all of our sources were placed at the pointing centre and at the imaging field centre and each source with a detected signal is very small (≲20 arcsec, Section 3). The full width at half-maximum (FWHM) of the PB at L band is 25.8 arcmin (Perley 2016) so corrections for the PB were not necessary for V.

Direction-dependent errors that have not been taken into account in our standard imaging are those due to the azimuthally asymmetric primary beam and its rotation on the sky, and the well-known beam squint due to the fact that R and L feeds point to slightly different positions on the sky (Bhatnagar et al. 2008). These affect off-centre sources rather than those at the field centre. Beam squint, in particular, can introduce a false V signal, though not at the centre (Brisken 2003). Nevertheless, we made a variety of tests for two of our real signals using the awproject algorithm in casa⁷ that corrects for these effects. In each case, the central signal remains, although several minor off-centre signals do not.

A remaining potential source of error relates to the application of self-calibration when amplitude is allowed to vary. For the five sources showing CP, three of them included amplitude self-calibration. For these, we either imaged the non-self-calibrated data or did a self-calibration that derives gains from the average of R and L bands (gaintype=’t’ in casa gaincal). Again, the V signal remained.

Finally, we make no attempt to correct for a possible thermal contribution to Stokes I. The thermal contribution at L band is typically ≈10 per cent for galaxies as a whole (e.g. Condon 1992) and this is considerably less than our uncertainties in V/I (Table 3). We also argue in this paper for the Faraday conversion interpretation that requires that non-thermal particles dominate in our detected AGNs.

3 RESULTS

3.1 L-band non-detections of Stokes |$\boldsymbol {V}$|

Table 2 lists m_C = V/I upper limits for the galaxies that were not detected at B-configuration L band along with the beam sizes and measurement positions in the event that future observations may wish to compare with these data. In addition, we include CHANG-ES A that is a bright source almost directly behind UGC 10288 (Irwin et al. 2013).

Table 2.

L band upper limits^a.

Galaxy	Beam sizeb (arcsec, arcsec, °)	RAc ( per cent)	Dec.c	(V/I)L d
NGC 891	3.09, 2.88, 53.46	|${\rm 02}^{\rm h}22^{\rm m}33{^{\rm s}_{.}}23$|	+42°20΄58|${^{\prime\prime}_{.}}$|3	<0.36
NGC 2613	5.28, 2.97, 0.20	|${\rm 08}^{\rm h}33^{\rm m}22{^{\rm s}_{.}}79$|	−22°58΄25|${^{\prime\prime}_{.}}$|2	<8.77
NGC 2683	3.02, 2.95, 55.86	|${\rm 08}^{\rm h}52^{\rm m}41{^{\rm s}_{.}}31$|	+33°25΄19|${^{\prime\prime}_{.}}$|0	<1.40
NGC 2820	3.23, 3.17, 52.76	|${\rm 09}^{\rm h}21^{\rm m}45{^{\rm s}_{.}}87$|	+64°15΄27|${^{\prime\prime}_{.}}$|5	<1.87
NGC 2992	4.87, 3.57, 16.39	|${\rm 09}^{\rm h}45^{\rm m}41{^{\rm s}_{.}}93$|	−14°19΄36|${^{\prime\prime}_{.}}$|0	<0.09
NGC 3003	3.04, 2.96, 70.84	|${\rm 09}^{\rm h}48^{\rm m}36{^{\rm s}_{.}}69$|	+33°25΄17|${^{\prime\prime}_{.}}$|9	<19.0
NGC 3044	3.53, 3.34, 36.97	|${\rm 09}^{\rm h}53^{\rm m}40{^{\rm s}_{.}}88$|	+01°34΄46|${^{\prime\prime}_{.}}$|7	<0.45
NGC 3432	3.20, 3.12, 82.82	|${\rm 10}^{\rm h}52^{\rm m}30{^{\rm s}_{.}}96$|	+36°37΄08|${^{\prime\prime}_{.}}$|6	<11.1
NGC 3448	3.11, 2.95, 62.42	|${\rm 10}^{\rm h}54^{\rm m}39{^{\rm s}_{.}}19$|	+54°18΄20|${^{\prime\prime}_{.}}$|8	<0.58
NGC 3556	3.08, 2.96, 56.36	|${\rm 11}^{\rm h}11^{\rm m}30{^{\rm s}_{.}}43$|	+55°40΄26|${^{\prime\prime}_{.}}$|7	<0.88
NGC 3735	3.24, 3.11, 33.77	|${\rm 11}^{\rm h}35^{\rm m}57{^{\rm s}_{.}}20$|	+70°32΄07|${^{\prime\prime}_{.}}$|6	<0.77
NGC 3877	3.01, 2.83, 19.18	|${\rm 11}^{\rm h}46^{\rm m}07{^{\rm s}_{.}}70$|	+47°29΄39|${^{\prime\prime}_{.}}$|7	<0.87
NGC 4013	3.01, 2.90, −84.20	|${\rm 11}^{\rm h}58^{\rm m}31{^{\rm s}_{.}}38$|	+43°56΄51|${^{\prime\prime}_{.}}$|1	<0.34
NGC 4096	3.06, 2.94, −84.88	|${\rm 12}^{\rm h}06^{\rm m}01{^{\rm s}_{.}}25$|	+47°28΄40|${^{\prime\prime}_{.}}$|6	<13.0
NGC 4157e	3.01, 2.83, 26.37	|${\rm 12}^{\rm h}11^{\rm m}04{^{\rm s}_{.}}37$|	+50°29΄04|${^{\prime\prime}_{.}}$|8	<4.97
NGC 4192	3.21, 3.07, −7.49	|${\rm 12}^{\rm h}13^{\rm m}48{^{\rm s}_{.}}29$|	+14°54΄02|${^{\prime\prime}_{.}}$|1	<0.37
NGC 4217	3.05, 2.92, −84.62	|${\rm 12}^{\rm h}15^{\rm m}50{^{\rm s}_{.}}90$|	+47°05΄29|${^{\prime\prime}_{.}}$|4	<0.64
NGC 4244e	3.09, 3.00, 44.97	|${\rm 12}^{\rm h}17^{\rm m}29{^{\rm s}_{.}}66$|	+37°48΄25|${^{\prime\prime}_{.}}$|6	<41.4
NGC 4302	3.48, 3.10, −8.53	|${\rm 12}^{\rm h}21^{\rm m}42{^{\rm s}_{.}}31$|	+14°35΄52|${^{\prime\prime}_{.}}$|4	<1.18
NGC 4438	3.32, 2.91, −6.28	|${\rm 12}^{\rm h}27^{\rm m}45{^{\rm s}_{.}}52$|	+13°00΄33|${^{\prime\prime}_{.}}$|2	<0.06
NGC 4565	3.22, 2.96, 41.96	|${\rm 12}^{\rm h}36^{\rm m}20{^{\rm s}_{.}}78$|	+25°59΄15|${^{\prime\prime}_{.}}$|6	<0.95
NGC 4594	4,36, 3.25, −14,03	|${\rm 12}^{\rm h}39^{\rm m}59{^{\rm s}_{.}}43$|	−11°37΄23|${^{\prime\prime}_{.}}$|0	<0.18
NGC 4631	3.36, 3.03, 62.61	|${\rm 12}^{\rm h}42^{\rm m}07{^{\rm s}_{.}}87$|	+32°32΄34|${^{\prime\prime}_{.}}$|9	<0.54
NGC 4666	3.75, 3.43, 34.26	|${\rm 12}^{\rm h}45^{\rm m}08{^{\rm s}_{.}}62$|	−00°27΄42|${^{\prime\prime}_{.}}$|8	<0.31
NGC 5084	5.63, 2.95, −11,79	|${\rm 13}^{\rm h}20^{\rm m}16{^{\rm s}_{.}}85$|	−21°49΄38|${^{\prime\prime}_{.}}$|3	<0.36
NGC 5297	3.13, 2.99, 52.75	|${\rm 13}^{\rm h}46^{\rm m}23{^{\rm s}_{.}}67$|	+43°52΄20|${^{\prime\prime}_{.}}$|4	<10.8
NGC 5775	3.57, 3.40, 54.80	|${\rm 14}^{\rm h}53^{\rm m}57{^{\rm s}_{.}}34$|	+03°32΄43|${^{\prime\prime}_{.}}$|8	<0.69
NGC 5792e	3.89, 3.42, 48.63	|${\rm 14}^{\rm h}58^{\rm m}22{^{\rm s}_{.}}71$|	−01°05΄27|${^{\prime\prime}_{.}}$|9	<0.40
NGC 5907	3.35, 2.79, −4.64	|${\rm 15}^{\rm h}15^{\rm m}53{^{\rm s}_{.}}47$|	+56°19΄43|${^{\prime\prime}_{.}}$|6	<4.14
UGC 10288	3.67, 3.50, 57.18	|${\rm 16}^{\rm h}14^{\rm m}24{^{\rm s}_{.}}79$|	−00°12΄27|${^{\prime\prime}_{.}}$|2	<11.0
CHANG-ES Af	3.67, 3.50, 57.18	|${\rm 16}^{\rm h}14^{\rm m}23{^{\rm s}_{.}}28$|	−00°12΄11|${^{\prime\prime}_{.}}$|6	<0.15

Notes.^aUpper limits were calculated using equation (8) (see Section 2.4).

^bSynthesized beam parameters: major axis, minor axis, and position angle.

^cPosition of the peak in total intensity, I, which was closest to the NED position of Table 1 unless otherwise indicated. This is also where V was measured.

^dCP upper limits expressed as a fraction of the total intensity at the same position.

^eMeasurements were made at the NED centre. In these cases, total intensity emission is seen from the disc but little emission was seen at the centre.

^fCHANG-ES A is the bright double-lobed radio source behind the disc of UGC 10288 (Irwin et al. 2013). This measurement was made at the centre of the background source.

Measurements were made at the total intensity, I, peak that was closest to the NASA Extragalactic Database (NED) centre of the galaxy as given in Table 1 except for three sources that had no clear I peak near the core (see table footnotes) in which case we measured at the NED centre itself. An example is NGC 4244 that has a very high upper limit to V/I because of its low I value. With the exception of only two sources (next paragraph), no V signal was measured at all. Consequently, V/I was simply taken to be the rms noise value in V divided by I.

The two exceptions were NGC 4594 and NGC 5084 that showed weak V signals at 4.1 times and 3.0 times their respective rms V noise values. Once uncertainties in the calibration were factored in, however, the resulting V signals were not considered to be significant.

Of this list, at the corresponding location, only a single galaxy (NGC 4438) shows any LP, so at B-configuration L band, there is essentially no detectable LP for the CHANG-ES galaxies. This appears to be because the signal is weak in these small (≈3 arcsec) beams and also, when observing edge-on galaxies, there is significant depolarization along the line of sight. That CP can be detected for some galaxies (next section) whereas LP usually cannot, is an argument for including CP measurements in future observations of highly compact sources.

3.2 L-band detections of Stokes |$\boldsymbol {V}$|

Relevant data for the five galaxies that exceed the limits specified above are given in Table 3. In each case the V signal was measured at its peak value that is at or near the galaxy centre (see notes to the table). The level at which V has been detected ranges from 12ΔV to 154ΔV (ΔV being the rms noise). Notice that the signals are both positive and negative, as one would expect from natural signals rather than systematic offsets.

Table 3.

L-band detections.

Galaxy	|$V_{{{\rm max}}_L}{}^{a}$|	|$\Delta V_{{{\rm gal}}_L}{}^{b}$|	|$I_{{{\rm max}}_L}{}^{a}$|	Beam sizec	(Vmax/Imax)Ld	(SV/SI)Le
	(⁠|$\mu$|Jy beam−1)	(⁠|$\mu$|Jy beam−1)	(mJy beam−1)	(arcsec, arcsec, °)	(per cent)	(per cent)
NGC 660	+715.0	50.0	245.0	3.39, 3.27, 44.43	+0.29 ±  0.07	+0.31 ±  0.07
NGC 3079	−221.9	19.0	122.7	3.14, 3.00, 58.44	−0.18 ±  0.15	−0.17 ±  0.2
NGC 3628	−198.7	13.5	80.0	3.21, 3.13, 3.73	−0.25 ±  0.08	−0.27 ±  0.08
NGC 4388	−583.0	25.0	25.2	3.57, 3.22, −2.09	−2.31 ±  0.10	−2.6 ±  0.3
Jet	−145.8	25.0	4.48	3.57, 3.22, −2.09	−3.26 ±  0.10	−3.4 ±  0.3
NGC 4845	+3700	24.0	209.3	3.55, 3.34, 27.73	+1.77 ±  0.11	+2.3 ±  0.2

Notes.^aV_max and I_max are measured at the same position. They are also at the same NED centre as given in Table 1 to within 0.7 arcsec, except for (a) NGC 660 whose peaks are located 3 arcsec to the north-west of the NED centre, (b) NGC 3628 that are 3 arcsec south of the NED centre, and (c) the NGC 4388 Jet that is measured at a local peak at RA = 12|$^{\rm h}25^{\rm m}46{^{\rm s}_{.}}$|85, Dec. = 12°39΄50|${^{\prime\prime}_{.}}$|5 and is marked with a blue ‘+’ in Fig. 4.

^bV map noise of the image (see Section 2.4).

^cSynthesized beam parameters: major axis, minor axis, and position angle.

^dCP percentage with uncertainty calculated from equation (8).

^eCP percentage using the ratio of flux densities (rather than specific intensities) measured over a half-power beamwidth centred at |V_max|. Uncertainties are propagated from the previous column with increases, if required, for variations that result from adjusting the central position by approximately a cell size.

The weakest signal, and most marginal detection is NGC 3079; its V signal is measured at the 12ΔV level. Two galaxies, however, show very strong V/I in their cores, of order 2 per cent, namely NGC 4388 and NGC 4845. For NGC 4845, there is a small difference between the ratio of the peak V/I and the ratio of flux densities, S_V/S_I, because the total intensity distribution is slightly more peaked than the CP distribution at the centre. Recall that the signal is only considered to be real if V/I exceeds the errors as defined in equation (8); hence signal-to-noise ratio (S/N) alone for V is not taken to definitively determine whether a CP signal is significant.

V and I images are shown in Fig. 4. Although CP has been reported previously in NGC 4845 (Irwin et al. 2015), we present the V image for the first time here.

An interesting result is that all sources except for NGC 3079 show V emission that is resolved and shows structure, sometimes rather complex. The most striking case is NGC 4388 that shows CP in the direction of its northern jet. As indicated in Section 1, few examples of CP associated with jets exist in the literature. We will discuss each galaxy further in Sections 4.1–4.5.

For each detection, we also checked for CP at C band reduced in the same way as described for L band. The results, with other relevant data, are given in Table 4. In no case do we detect any CP at the higher frequency. This is an important result because it implies that the V spectral index is very steep, consistent with the interpretation of Faraday conversion as discussed further in Section 3.4.

Table 4.

C-band upper limits for detected sources^a.

Galaxy	Obs. dateb	Beam sizec	|$\Delta \,V_{{{\rm gal}}_C}{}^{d}$|	(V/I)Ce
		(arcsec, arcsec, °)	(⁠|$\mu$|Jy beam−1)	(per cent)
NGC 660	2012-01-27	3.15, 2.64, −55.72	14.2	<0.015
	2012-01-28
NGC 3079	2012-02-15	2.68, 2.59, −88.97	3.35	<0.081
NGC 3628	2012-02-29	3.68, 2.74, −78.98	3.10	<0.022
NGC 4388f	2012-04-08	2.79, 2.72, −13.15	3.20	<0.039
NGC 4845	2012-02-23	3.15, 2.76, 35.56	13.6	<0.052
	2012-02-25

Notes.^aPositions are the same as the L-band detection positions. The central frequency is 6.00 GHz.

^bObserving dates, designated year-month-day (ut). If two dates are given, the observing was split into two sessions.

^cSynthesized beam parameters: major axis, minor axis, and position angle.

^dV map noise level (see Section 2.4).

^eCP percentage upper limits calculated from equation (8).

^fValues for the jet are the same.

We also do not detect any LP at the location of the CP signal, again consistent with the Faraday conversion mechanism that we discuss at length in this paper though Faraday depolarization is also likely at play (Irwin et al. 2015).

3.3 Spectral index measurements

We distinguish between band-to-band spectral indices and in-band spectral indices. Our spectral index results are listed in Table 5.

Table 5.

Spectral indices^a.

Galaxy	|$\alpha _{I_L}{}^{b}$|	|$\alpha _{V_L}{}^{c}$|	|$\alpha _{I_C}{}^{d}$|	pe	ΔTf (d)	αV(L − C)g
NGC 660	+1.23	−1.4	−0.314	1.63	149, 148	<− 2.9
NGC 3079	−0.41	–	+0.295	–	129	<− 3.1
NGC 3628	−0.31	+2.7	−0.675	2.35	152	<− 3.1
NGC 4388	−0.94	−2.7	−0.818	2.64	112	<− 3.9
Jet	−1.05	–	−0.805	2.61	112	<− 2.9
NGC 4845h	+0.81	−3.4	−0.493	1.99	109, 107	<− 4.2

Notes.^aPositions are the same as in Table 3. Values are in-band spectral indices except for the last column. A horizontal line means that there was insufficient signal-to-noise to provide a reliable value. Estimated errors in α_IL, α_VL, and α_IC are ≈5, 20, and 1 per cent, respectively.

^bTotal intensity spectral index at L band measured from the spectral index maps that were formed as described in Wiegert et al. (2015). Values are the average over the L-band half-power beam width centred at |V_max|.

^cStokes V spectral index at L band. Signals that are positive (see last column of Table 3) were measured in the same way as for |$\alpha _{I_L}$|⁠. Signals that are negative were fitted as described in Section 3.3 and illustrated in Fig. 1. Dashes indicate that there is insufficient signal in the subbands for a reliable result.

^dTotal intensity spectral index at C band measured in the same way as |$\alpha _{I_L}$|⁠.

^eEnergy spectral index of relativistic particles (N(E)  ∝  E^−p) assuming that the C-band spectral index (except for NGC 3079) is optically thin, i.e. |$p=1\,-\,2\alpha _{I_C}$|⁠.

^fDuration between the earlier C-band observations and later L-band observations. Galaxies that had split observations (Table 4) have two entries.

^gUpper limits to the L- to C-band spectral indices using equation (12).

^hFrom Irwin et al. (2015).

Band-to-band spectral indices, α_V(L − C), are measured between the centre of L band and the centre of C band. We do not detect CP at C band, so we can only provide upper limits to α_V(L − C) using

\begin{eqnarray} \frac{|V_L|}{\Delta V_{{{\rm gal}}_C}} =\left(\frac{\nu _L}{\nu _C}\right)^{\alpha _V (L-C)}, \end{eqnarray}

(12)

where V_L is the detected L-band signal (Table 3) and |$\Delta V_{{{\rm gal}}_C}$| is the rms noise at C band (Table 5) since there is no emission or any sign of residual signals above the rms noise at C band. These upper limits are also estimates because of the possibility that the CP is variable, hence we note the dates and number of days that have elapsed between L- and C-band observations. For example, for NGC 4845, variability of order 20 per cent is observed in L-band quantities over a time period of 164 days, and the flux density in Stokes I at C band varies by 18 per cent over only 67 days (Irwin et al. 2015).

In-band spectral indices are generated when spectral fitting is carried out during the mapping and cleaning process. The wide bands used in CHANG-ES permit such measurements, providing a slope for the centre of any given band. The default cut-off is 5σ for the formation of spectral index maps.

However, due to a casa limitation,⁸ we must determine in-band spectral indices manually for any V signal that is negative. Consequently, for such cases, we have split the band into four equal frequency sections and imaged each section separately. The resulting maps are then smoothed to the resolution of the map at the lowest frequency end of the band. We then examine whether the signal at each of the four frequencies has a S/N >5σ. If not, we conclude that |$\alpha _{V_L}$| cannot be reliably obtained. If so, we then determine α_V via a curve fit using the Levenberg–Marquardt algorithm (Levenberg 1944; Marcquart 1963).

An example is shown in Fig. 1 for NGC 4388, where negative values (which simply describe the direction of the CP) have been inverted to positive in order to show the plot in the standard fashion. The red curve (indistinguishable from the green) corresponds to |$V=a\nu ^{\alpha _{V_L}}$| and the green curve will be discussed in Section 3.4.

Figure 1.

L-band V signal (inverted to be positive) as a function of frequency for NGC 4388, where the band has been broken into four frequency sections to see the spectral dependence. Red and green curves (virtually indistinguishable) represent best fits of the form, |$V=a\nu ^{\alpha _V}$|⁠, where a = 784.8 and α_V = −2.7 (red, reduced χ² = 1.3), and V = C₁ν⁻¹ + C₂ν⁻³, where C₁ = 54.7 and C₂ = 773.8 (green, reduced χ² = 1.4), for ν in GHz and S_V in |$\mu$|Jy. For the last point, only 5/8 of the frequency band was useable.

In Table 5, we also list the spectral indices for Stokes I in each band. This allows us to check whether CP is occurring in an optically thick or optically thin regime and also provides an estimate of p in the optically thin regime, should either be the case.

3.4 The Faraday conversion interpretation

Faraday conversion has not yet been established for all sources and source complexities may also be present. For example, optical depth effects can lead to a variety of spectral indices, as pointed out by Jones & O'Dell (1977a), and variable (including inverted) Stokes V spectral indices have been observed in M81 (Brunthaler et al. 2006) and Sgr A* (Bower 2003). Nevertheless, given that this interpretation has achieved the most success at explaining the observations to date (Section 1), we have developed and expanded the analysis of Beckert & Falcke (2002) in order to provide analytical solutions for a variety of relatively straightforward cases.

3.4.1 Faraday conversion predictions for p = 2

In Irwin et al. (2015), appendix E, we determined the frequency dependence of Stokes V for the case in which the power-law spectral index of relativistic electrons p = 2 (where N(E) = N₀ E^−p) for the energy distribution of electrons. This choice of p was adopted because the observed radio continuum total intensity in-band spectral index |$\alpha _{I_{\rm C}}\approx -0.5$| in the C-band observations of NGC 4845. This implies that, at C band, the core of NGC 4845 is optically thin in which case α_I = (1 − p)/2. At L band, however, the core is transitioning to being optically thick and that is where CP is observed.

In our homogeneous source with the Faraday rotation coefficient small compared to the conversion and absorption coefficients (Irwin et al. 2015), but nevertheless optically thin, the flux density of Stokes V, S_V, has a frequency dependence of

\begin{eqnarray} S_V&=& \frac{C_1}{\nu }+\frac{C_2}{\nu ^3}\quad (p=2). \end{eqnarray}

(13)

Here the C_n values are functions of various physical parameters. The first term is an emission term and the second is the conversion term. The conversion term requires that relativistic particles dominate.

In fact, as found in Jones & O'Dell (1977a), Jones (1988), and Irwin et al. (2015), CP of the magnitude detected here requires a lower electron energy cut-off, γ₀ ≈  γ_e, where γ_e is the Lorentz factor of an electron radiating at the peak radio frequency. However, this does not explain the lack of observed LP (Jones & O'Dell 1977a), which must be due to a surrounding depolarizing sheath, in our view.

In Jones & O'Dell (1977b), the percentage CP emerging from an optically thin boundary is calculated under the conditions mentioned above. Generally the amplitude and frequency dependence is similar to what we observe and predict (equation 14), but there can be sudden cut-offs at high frequency probably due to a lack of Faraday rotation. In a standard jet model (Jones 1988; Irwin et al. 2015) the rotation of a helical magnetic field also serves to rotate the plane of polarization.

The observed L band in-band spectral indices for NGC 4845, α_V, range from −2.2 to −3.4, depending on JVLA configuration (i.e. spatial resolution) clearly showing that, in this interpretation, the medium is dominated by relativistic particles. O'Sullivan et al. (2013) also found an average ν⁻³ dependence for the quasar PKS B2126−158. That is, the steep spectral dependence provides the evidence that Faraday conversion is the dominant mechanism and, for NGC 4845, also explains why CP is observed at L band but not at C band.

We now have a tool for probing the physical properties deep within AGN that are otherwise inaccessible in total intensity observations. In the case of NGC 4845 (see Irwin et al. 2015), our AGN jet model provided additional constraints on the physical parameters. To explain the magnitude of the observed CP and its spectrum at 1.5 GHz, our estimated values were B = 0.04 G, the relativistic electron density, n_er = 100 cm⁻³, L = 10¹⁷ cm, and initial linearly polarized Stokes U₀ = 60 mJy. The electron spectrum is a power law with lower cut-off near γ_e for electrons radiating in the 1.5 GHz band. As stated above, an additional depolarizing Faraday screen was also required to reduce the observed linear polarization to observable values; in an edge-on galaxy, such reduction is to be expected.

3.4.2 Faraday conversion predictions for a general p

We have now developed the CP predictions for the general case in which the power-law spectral index for relativistic electrons, p, can have a range of values and provide straightforward analytical results in Appendix A.

The emission term (equation A16) gives a ν⁻¹ dependence as in equation (13) for p = 2. This is, however, flatter than observed and so we concentrate mainly on the steeper conversion term.

The conversion term (equations A17, A18, and A19 for p < 2, p = 2, and p > 2, respectively) reveals the steeper frequency dependence of Faraday conversion and shows the physical dependences explicitly. For the p = 2 case, we also now include a weaker logarithmic term that was omitted from Irwin et al. (2015) appendix E.

A general consequence is that CP should have a frequency dependence of ν^{−(2 + p/2)} that goes as ν⁻³ when p = 2 as found earlier. If p < 2 and the second term in equation (A17) is negligible, then the frequency dependence would be ν^{−(2 + p/2)} as above. If p > 2 and the second term in equation (A19) is negligible, then the frequency dependence would be ν⁻³, also as found for p = 2.

The terms in square brackets, however, can modulate this behaviour, depending on the magnitude of the term, x ≡ 0.0028γ₀²B_⊥/ν₉, where B_⊥ is the perpendicular magnetic field, ν₉ (GHz) is the observing frequency, and γ₀ is the Lorentz factor of the lower energy cut-off of the relativistic electrons. If |$x^{1-p/2}\,{\rm is}\,\mathcal {O}\left(1\right)$|⁹ for example, then the magnitude of CP can decline and eventually reverse sign. We provide possible examples for which this modulating term may not be negligible in Sections 3.5.2 and 3.5.3. If the observed frequency dependence follows the description given in the previous paragraph, though, then this modulating term is likely small.

3.5 Reconciliation of theory with observations

The important and most clear-cut parameter, in the context of the CHANG-ES data, involves the frequency dependences given in Table 5. A clear result is that the band-to-band spectral indices in every case are very steep, with an average of α_V(L − C) < −3.4 for the galaxy cores. To explain this spectral dependence with an emission term (equation A16) would require an electron energy index of p = 6.8 (equation A16), an unphysically steep value and unsupported by the data in the table. We conclude that Faraday conversion can account for these steep spectral indices, although there are some peculiarities as discussed below.

Four sources provide Stokes V in-band spectral index measurements, α_VL. Of these, the two strongest sources, NGC 4388 (CP of −2.6 per cent) and NGC 4845 (CP of +2.3 per cent) also show very steep in-band spectral indices: α_VL of −2.7 and −3.4, respectively (typical uncertainties of |${\approx } 20\hbox{ per cent}$|⁠, Table 3). These two are the most clear-cut cases. NGC 4845 has been extensively examined in the context of a jet model in Irwin et al. (2015) and need not be examined further here.

In the next three subsections, we consider each of the remaining three galaxies, namely the strong CP source, NGC 4388, and the weaker sources, NGC 660 and NGC 3628 (CP of 0.31 and 0.27 per cent, respectively).

3.5.1 Faraday conversion in NGC 4388

For NGC 4388, the conversion falls into the category p > 2 (equation A19). Although we do not have information on all of the physical parameters that enter into a CP calculation, we can ask what reasonable values might fit the observations. For example, at L band, no linear polarization is measured at the core, precluding an estimate of a lower limit for U₀. In principle, we could make an estimate of the amount of L-band LP that is to be Faraday converted by observing the LP at C band along with its spectral index, and then extrapolating to L band. Unfortunately, the C-band LP at the nucleus is also very weak, likely due to significant Faraday depolarization; with a degree of polarization at the nucleus of only |${\approx }0.2\hbox{ per cent}$| in C band (Damas-Segovia et al. 2016) and a peak C-band value of 9.0 mJy beam⁻¹ (for the same spatial resolution), this corresponds to only 18 |$\mu$|Jy beam⁻¹ at C band – much less than the observed CP at L band. In other words, the observed C-band LP extrapolated to L band is far too weak to put meaningful constraints on U₀. Since Stokes V at L band is |$|{S_V}_L|=355\,\mu$|Jy, there must originally have been at least this amount of linearly polarized signal for conversion.

We do, however, have a limit on the line-of-sight distance through the source, L, since VLBI observations by Giroletti & Panessa (2009) provide a source size of 6 mas = 0.48 pc. We also know p = 2.64, ν₉ = 1.5, and can adopt a reasonable value of |$\theta =\pi /4$|⁠. We do not know the other quantities in equation (A19) but, as an example, we can adopt values similar to those from the model of NGC 4845 (Irwin et al. 2015). Thus for n_e = 200 cm⁻³, B = 0.04 G, and γ₀ = 100, we find a (not unique) solution of |$|U_0/{S_V}_L|\,= 14\hbox{ per cent}$| conversion. Note that no reasonable combination of values produces the |$B=67\,\mu$|G that is measured for the nucleus by Damas-Segovia et al. (2016), so it is clear that the CP is probing deep into the core of the AGN where the magnetic field is much higher.

CP is observed in the jet of NGC 4388 also and is remarkably strong, even stronger than at the core (⁠|$m_{\rm C}=3.4\hbox{ per cent}$|⁠, Table 3). We have measured jet values at the position of the peak V intensity in the jet, marked with a blue ‘plus’ in Fig. 4. We note that, again, the band-to-band spectral index is steep, as expected for Faraday conversion, and spectral indices where measurable (Table 5) are similar to values in the core. Further discussion is in Section 4.4.

3.5.2 Faraday conversion in NGC 660

NGC 660 is an interesting case because its in-band spectral index (α_VL = − 1.4) is much flatter than its band-to-band spectral index (−2.9), indicating curvature in the Stokes V spectrum (Table 5). For this galaxy, p = 1.63 so the relevant equation is equation (A17). We have no other information to help determine its physical parameters but note that a flatter distribution of relativistic particles generally results in stronger CP, all else being equal. On the other hand, the second term in the square brackets must not be negligible for this source since, if it were, then S_Vconv ∝ ν^−2.8 that is contrary to observations. Thus we should see S_Vconv ∝ ν₉^−2.8[1 −  x^0.19], where x was defined in Section 3.4.2.

Strictly speaking, x cannot be equal to 1 or CP will disappear. However, if this term is to modify the slope then, to order of magnitude, we can let x ≈ 1, or equivalently, 0.0028γ₀²Bsin θ = ν₉. For |$\theta =\pi /4$|⁠, then γ₀²B ≈ 758. If γ₀ = 100, we require B ≈ 0.08 G, for example. Compared to the previous example, then, one can put tighter restrictions on the physical parameters when there are departures from the predicted frequency behaviour for conversion. Note that x increases with frequency within the band and that will flatten the spectral index, as observed.

3.5.3 Faraday conversion in NGC 3628

NGC 3628, like the other galaxies, has a steep band-to-band spectral index (Table 5), arguing for the Faraday conversion interpretation. However, its Stokes V in-band spectral index is positive (α_VL = +2.7) which is highly discordant in comparison to the declining spectral indices seen for the other sources. It may be that absorption effects can turn over the spectral index, making it positive. Indeed, the measurement of α_VL is based on splitting a weak signal into four parts, as described in Section 3.3 and is prone to errors. Nevertheless, this is a good example to show how the spectrum cannot only be flattened (as for NGC 660) but can actually become inverted. In Fig. 2 we plot the term, F(ν₉), as given in equation (A21) that folds together the frequency-dependent terms from equation (A19). In this way, we can examine cases for which x is |$\mathcal {O}\left(1\right)$|⁠.

Figure 2.

The function F(ν₉) (equation A21) as a function of frequency across L band. The values of γ₀ and B are marked on each of the solid curves (black, blue, green, and red). For any of these curves, one could obtain an equivalent result if γ₀ is halved and B is quadrupled, as illustrated for the green curve. Note that F(ν₉) is actually increasing when x is |$\mathcal {O}\left(1\right)$|⁠, in contrast to the normal steeply declining behaviour for Faraday conversion. The red dashed curve plots the observed behaviour for NGC 3628 (with arbitrary scaling) as described in Section 3.5.3.

Here we see a series of curves for different values of γ₀ and B. The result is striking in that increasing values with frequency result, similar to what is observed for NGC 3628 (red dashed curve). They are inverted with respect to the steep negative behaviour that is typically seen in Faraday conversion, a result that is solely due to the fact that x is |$\mathcal {O}\left(1\right)$|⁠. It is clear that the curvature of F(ν₉) does not match the fitted curve for the galaxy; however, given the uncertainties in this exercise, we take this as a good demonstration of the kind of spectral behaviour that could result. If we now repeat the exercise carried out for NGC 660 by letting x ≈ 1, we would find similar values for B and γ₀.

3.6 X-ray emission from the galaxy cores

As has been shown in Irwin et al. (2015) there is a close link between the radio and X-ray emission in NGC 4845. The radio observations were obtained approximately 1 yr after a tidal disruption event that was observed in the hard X-ray regime. A declining light curve was observed in the radio and a model developed to quantitatively explain the observations as well as offer a possible link between the X-ray and radio regime. This source showed strong CP, suggesting that it is important to measure the X-ray characteristics of our CP detections for current and subsequent analyses. All galaxies showing CP have been observed and detected in X-rays. However, we have now reanalysed the X-ray data in a consistent fashion.

For the X-ray analysis of the core emission, XMM–Newton archive data were used. The data were processed using the sas¹⁰ 15.0.0 package (Gabriel et al. 2004) with standard reduction procedures. The tasks epchain and emchain helped to obtain event lists for two EPIC¹¹-MOS cameras (Turner et al. 2001) and the EPIC-pn camera (Strüder et al. 2001). The event lists were then carefully filtered for periods of intense background radiation by creating light curves of high-energy emission. With these light curves good time interval (GTI) tables were produced and used to remove the data when high count rates were observed.

Next, the spectral analysis was performed. The spectra of the central regions of all five galaxies were created using all three EPIC cameras. The position and sizes of the regions were chosen to both match the radio centre of a galaxy and to include the brightest X-ray emission. The background spectra were obtained using blank sky event lists (see Carter & Read 2007). These were filtered using the same procedures as for the source event lists. For each spectrum, response matrices and effective area files were produced. Next, including these ancillary files, spectra from all three EPIC cameras and the corresponding background blank sky spectra were merged using the sas task epicspeccombine into a final background subtracted source spectrum. In the case of NGC 660 and NGC 3079, where two observations were available for each galaxy, the resulting spectra were created in the same way, as epicspeccombine allows merging of a larger number of spectra. The spectra were then fitted using xspec 12 (Arnaud 1996).

Because of the limited resolution of the XMM–Newton data and consequently the sizes of the core regions used (between 25 and 61 arcsec in diameter), one expects a contribution from the gaseous (thermal) component in the extracted spectra. Therefore, all spectra except one, were fitted with models that included both the gaseous and the central source components, represented by a mekal model and a power-law model, respectively. A mekal model is a model of an emission spectrum from hot diffuse gas based on the calculations of Mewe and Kaastra (Mewe, Gronenschild & van den Oord 1985; Kaastra 1992). The only spectrum that did not require a gaseous model component was that of NGC 4845, most likely due to the very high brightness of the central source, which made the emission from the hot gas emission negligible. For all sources except NGC 3079, an additional component of the model needed to be used to account for a high absorption in the core. For this galaxy, the contribution from the thermal component reached 24 per cent of the total emission, while for the other objects it was not higher than 6 per cent of the total X-ray core emission. It is important to note that the fluxes are measured in the range, 0.3–12 keV, uniformly for all galaxies.

The fluxes of the power-law component derived directly from the model fits were then used to calculate the luminosities, which together with basic information on the used XMM–Newton data sets and absorption values (both foreground and fitted for the central regions) are presented in Table 6.

Table 6.

Parameters of the XMM–Newton observations and parameters derived from the model fits to the core regions.

Galaxy	ObsID	Obs. date	ΔTa	Foreground NHb	Internal NHc	Luminosityc
			(d)	(1020 cm−2)	(1022 cm−2)	(106 L⊙)
NGC 660	0093641001	2001-01-07	4186	4.64	0.60|$^{+0.64}_{-0.46}$|	1.22|$^{+2.83}_{-0.52}$|
	0671430101	2011-07-18	342
NGC 3079	0110930201	2001-04-13	4089	0.89	–	2.30|$^{+0.69}_{-0.51}$|
	0147760101	2003-10-14	3175
NGC 3628	0110980101	2000-11-27	4263	1.97	0.33 ± 0.04	1.89|$^{+0.34}_{-0.29}$|
NGC 4388	0110930701	2002-12-12	3518	2.58	26.45|$^{+1.53}_{-1.47}$|	599|$^{+298}_{-191}$|
NGC 4845	0658400601	2011-01-22	507	1.44	6.93 ± 0.10	1400|$^{+102}_{-98}$|

Notes.^aDuration between the earlier XMM observations and later B-array L-band observations.

^bWeighted average value after Leiden–Argentine–Bonn (LAB) Survey of Galactic H i (Kalberla et al. 2005).

^cLuminosity in the 0.3–12 keV band, derived from the model fits to the spectra. See text for details.

Fig. 3 presents the relation between X-ray luminosities of the central sources and luminosities of the CP component produced in the core regions of the galaxies. As can be seen by the error bars, the relation is quite approximate but provides a baseline for possible future studies. As was indicated at the beginning of Section 3.6, there appears to be a close link between the X-ray data and CP. Here we have ensured that all X-ray data were reduced in the same way so as to make a proper comparison. In Section 4.6 we discuss the important result of the internal absorption.

Figure 3.

Luminosity of the circularly polarized signal (obtained from the flux density, S_V) against X-ray luminosity, both in units of the solar luminosity.

4 DISCUSSION

In the next sections, we refer to Fig. 4 and discuss each of the galaxies in Tables 3 and 6, followed by a discussion of the sample as a whole.

Figure 4.

Galaxies with CP at L band. Black contours show the V signal and colour represents the total intensity (labelled ‘I’ in a small rectangle at upper right), with two dashed contours in blue showing the 95 and 20 per cent I levels. The centre of the galaxy is at the centre of the 95 per cent blue dashed contour. The galaxy name is at upper right with labelling as to whether the CP is positive (+ve) or negative (−ve). The beam is shown as a white circle at lower left. The first V contour is at 2σ, where σ is measured near the source. Weak residual sidelobes are still evident in a few cases. V contours are: 100, 150, 250, 400, and 600 |$\mu$|Jy beam⁻¹ for NGC 660; −38, −75, −130, and −200 |$\mu$|Jy beam⁻¹ for NGC 3079; −27, −45, −75, and −120 |$\mu$|Jy beam⁻¹ for NGC 3628; −50, −100, −200, −350, and −500 |$\mu$|Jy beam⁻¹ for NGC 4388; and 48, 100, 250, 500, 1000, 2000, and 3300 |$\mu$|Jy beam⁻¹ for NGC 4845. Other dashed lines and annotations are discussed in Section 4.

4.1 NGC 660

NGC 660 (D = 12.3 Mpc; Wiegert et al. 2015) is a polar ring galaxy that has undergone a merger (Braine, Combes & Casoli 1993; Alton et al. 1998). Previous radio continuum data have been obtained by van Driel et al. (1995), Filho, Barthel & Ho (2002), and Filho et al. (2004). Its nuclear type is H ii/LINER¹² (NED; Irwin et al. 2012a). However, an activity type of Sy 2 has also been recorded (Véron-Cetty & Veron 2006) and it is becoming clear that nuclear activity levels and classifications can change with time (e.g. LaMassa et al. 2015).

Argo et al. (2015) found that NGC 660 has undergone a spectacular radio outburst that occurred sometime between 2008 and 2010.7 (see also Minchin et al. 2013) and our L-band data were taken after this outburst (2012 June, Table 1). Their 2013 October high-resolution EVN¹³ image shows two new components, likely related to this outburst, one to the west and one to the north-east of the central bright source, suggesting jet-like outflow on scales of ≈1 pc.

The X-ray data of Argo et al. (2015), aside from emission along the main north-east (NE) to south-west (SW) disc (see our Fig. 4 a, NE–SW dashed line), appear to show an extension perpendicular to the disc on scales of ≈2 arcsec (120 pc). On a similar scale, we also see a SE ‘bulge’ in the CP image contours, which we have emphasized with a north-west (NW) to south-east (SE) dashed line. If these features are related, then it is likely that our SE bulge is also related to jet-like outflow (rather than a starburst) as suggested by Argo et al. (2015). However, such emission must be associated with a previous outflow event since highly superluminal motion would be required otherwise. Our SE bulge/jet feature seen in CP is now one of the few sources known for which CP is seen in an extended structure; CP is usually only observed in a galaxy's core (Section 1).

The two X-ray data points of the core (Table 6) were taken before (first line) and after (second line) the putative radio outburst described by Argo et al. (2015). The first measurement (in 2001) was long before the radio outburst. The second measurement (in 2011) was made between ≈1294 and 340 d after the outburst, given the uncertainty in the outburst window noted above. Since the X-ray luminosities of the two measurements are not very different (within the error bars quoted) then if there had been an X-ray outburst associated with the radio outburst, it is likely that the X-ray outburst had already declined significantly by the time of the second measurement.

As noted in Table 3, the total intensity and CP centres are offset to the NW of the optical centre as given in NED. A similar offset in the same direction has been noted by Stevens, Amure & Gear (2005) for the submm centre. The true centre of the galaxy is likely at the radio and submm peaks. We find the radio peak to be at RA = 01^h43^m02|${^{\rm s}_{.}}$|33, Dec. = +13°38΄44|${^{\prime\prime}_{.}}$|7 to an accuracy of 0.3 arcsec in both coordinates at the observing date given in Table 1.

4.2 NGC 3079

NGC 3079 (D = 20.6 Mpc; Wiegert et al. 2015) has a unique radio structure in the form of two kpc-scale radio lobes on either side of the NW–SE major axis (Fig. 4b, colour image), the first detected by de Bruyn (1977). The AGN in this galaxy has been detected in VLBI, showing linear features related to a jet, luminous H₂O maser emission, and a time variable component (see summary in Kondratko, Greenhill & Moran 2005). X-ray observations support the presence of an obscured AGN (Iyomoto et al. 2001; Cecil, Bland-Hawthorn & Veilleux 2002). Middelberg et al. (2007) have detected variably brightening components in VLBI due to jet–cloud interactions. Recently, Shafi et al. (2015) have found a rich interplay between NGC 3079 and its nearby companions.

This is our weakest CP signal (Fig. 4b) and only the central core is represented. We were unable to measure a reliable in-band spectral index in Stokes V. However, the total intensity spectral index is negative at L band and positive at C band. This suggests that at least two components must be present within our 3 arcsec (300 pc) beam; for example, one with a negative spectral index dominating at L band and one with a positive spectral index becoming dominant at C band. Middelberg et al. (2007) have mapped the multiple VLBI components in this source, showing that some have positive and some negative spectral indices that also vary with time.

4.3 NGC 3628

NGC 3628 (D = 8.5 Mpc; Wiegert et al. 2015) is a member of the Leo Triplet, and has an H ii/LINER nuclear classification as listed in NED. Because of its high inclination and prominent dust lane, however, its AGN is highly obscured. Goulding & Alexander (2009) list it as an ‘optically unidentified’ AGN that they identified via the high excitation emission line, [Ne v] |$\lambda 14.32 \,\mu$|m. The AGN was previously identified, however, via the presence of a hard nuclear X-ray source (Yaqoob et al. 1995; González-Martín et al. 2006, and Table 6). Flohic et al. (2006) also fit a diffuse X-ray spectrum in the central kpc of this galaxy. Dong & De Robertis (2006) find a black hole mass of 2 ×  10⁷ M_⊙.

Our observations show some evidence for a resolved source with two extensions emerging NE to SW from the nucleus (Fig. 4c) and extending ≈10 arcsec (≈400 pc). The soft X-ray Chandra data also show emission in these directions at about the same scale and the hard X-ray emission shows the SW extension (Tsai et al. 2012, their fig. 8). There is an enhancement in roughly this direction in the inner 4 arcsec of the galaxy as seen in the mid-IR (Asmus et al. 2014). Given the additional evidence for alignments in the NE–SW direction, our CP results suggest that a jet is in this direction.

4.4 NGC 4388

NGC 4388 (D = 16.6 Mpc; Wiegert et al. 2015) is in the Virgo Cluster and harbours an outflow lobe, i.e. the vertical jet that extends away from the plane to the north, that is clearly seen in the colour total intensity image of Fig. 4(d). Note that the plane of the galaxy is east–west so the jet is approximately perpendicular to the plane in projection. The total intensity and LP of NGC 4388 have been reported by Damas-Segovia et al. (2016) for C band only, revealing new linearly polarized features and their interaction with the intergalactic medium (IGM). NGC 4388 is a hard X-ray source (Iwasawa et al. 2003; Krivonos et al. 2015, and Table 6) and has been detected in VLBI at 1.6 GHz (Giroletti & Panessa 2009).

This galaxy shows the strongest m_C = V/I in the CHANG-ES sample and the core of this galaxy shows the ‘classic’ CP spectrum characteristic of Faraday conversion (Section 3.5.1). Since the LP of the AGN is entirely depolarized at L band and almost entirely depolarized at C band, CP in NGC 4388 appears to be the only way to probe the physical properties of the AGN at these frequencies. Known and adopted physical parameters produce results that are quite reasonable, given our development in Appendix A.

Remarkably, there is CP along the northern jet as well, as high as ≈3 per cent at the peak V position of the jet that is marked with a blue ‘+’ in Fig. 4(d). This is probably the clearest case yet of observations of CP in a jet. At this position, there is |$-146\,\mu$|Jy beam⁻¹ in Stokes V, yet there is no B-configuration L-band linearly polarized emission at all, to an rms level of |$12\,\mu$|Jy beam⁻¹, supporting the Faraday conversion interpretation. At C band, there is a strongly linearly polarized lobe centred at the position marked with a black ‘x’ in our figure (see also fig. 3 of Damas-Segovia et al. 2016). However, this emission falls off strongly towards the south such that there is only marginal LP at the position of our blue ‘+’. Thus, the C-band LP is anticorrelated spatially with the CP at L band. Again, this supports the Faraday conversion interpretation.

The emission is not strong enough in the jet to measure V in-band spectral indices from our data. However, given how extensively this galaxy has been observed by others, there may be additional constraints that can be applied to this region in the future. More sensitive observations with the application of the same kind of analysis as in Section 3.5.1 may yield further details of the physical conditions in this jet.

4.5 NGC 4845

NGC 4845 (D = 17 Mpc) has been extensively studied in Irwin et al. (2015) who discovered the CP and provided an initial development of the Faraday conversion model for the case p = 2 (now superseded by our Appendix A). In this paper, a jet model was fit to the core that explained the observed variability in the multiple CHANG-ES observations, and also assisted with the CP interpretation.

Our B-array L-band radio data were taken after a hard X-ray outburst detected by Nikolajuk & Walter (2013) who interpreted the outburst as being due to the tidal disruption of a Jupiter-mass object by a black hole. The peak of the X-ray curve has been well determined to be on 2011 January 22, 507 d prior to our B-array L-band observations. We have now reanalysed the data corresponding to this X-ray peak (Table 6), ensuring that the analysis has been done in the same fashion for all of our CP-detected galaxies.

Follow-up VLBI observations have now confirmed the CP at the ≈2 per cent level (Perlman et al. 2017), in agreement with the CHANG-ES observations. The VLBI observations have resolved the CP, showing that it is elongated NW–SE with a size ≈5 mas (0.4 pc). At the same angle in the NW direction, there is a feature at L band that is possibly related to nuclear outflow. Our observations are of much lower resolution (3.4 arcsec, or 280 pc), but there is a hint of this angle in our CP map as well, with contours that bulge out in the same direction. We have drawn a NW–SE dashed line through this slight bulge in our V image (Fig. 4e, see below). The fact that the level of CP is the same between VLBI and our CHANG-ES data suggests that most of the CHANG-ES-detected CP is concentrated into a small region associated with the AGN. Indeed, this is consistent with our Faraday conversion parameters in the context of the jet model.

Fig. 4(e) shows a map of the CP for NGC 4845 for the first time, since this map had not yet been shown in Irwin et al. (2015). Although most of the CP is concentrated in a small region at the core, we also see that it is resolved, similar to most of the other galaxies. An interesting new result is that the CP occurs mainly in an elongation that extends from the NE to the SW at an angle that is tilted with respect to the total intensity. We have marked this disc by a NE–SW dashed line in Fig. 4(e) and have labelled it: the ‘conversion disc’. The conversion disc has an extent of ≈15 arcsec (1.2 kpc). This is almost as large as the total intensity disc (1.8 kpc; Irwin et al. 2015) that delineates the galaxy's major axis.

Although the conversion disc is tilted with respect to the galactic disc, it is perpendicular to the likely outflow direction as described above. Consequently, we suggest that it is the conversion disc that is associated with the outflow alignment. The core size, within which most of the total intensity flux is measured, is less than 41 pc (Irwin et al. 2015). Thus any accretion disc is no doubt much smaller than the conversion disc; possibly the inner ‘edge’ of the conversion disc is actively feeding the AGN and aligning the outflow in this galaxy. Recently McKinney, Tchekhovskoy & Blandford (2013) have developed models showing various alignments of accretion discs with black hole spins or with galaxy discs, depending on disc thickness and distance from the black hole. An example of a misaligned accretion disc in a nearby LLAGN is NGC 4258 (Herrnstein et al. 2005).

The tilted conversion disc of NGC 4845 also requires that a kpc-scale magnetic field be present that is thus tilted with respect to the major axis. For such a disc to exist requires that relativistic electrons not only dominate (as they likely do in the inner total intensity disc as well) but the physical parameters in this disc are such that conversion can occur.

To our knowledge, this is the first time that such a feature has been seen in any galaxy and we have coined the term, ‘conversion disc’, to reflect our interpretation that the CP mechanism is conversion, as well as the fact that this disc-like feature has a large extent.

4.6 Summary and intercomparison

Let us adopt ≈0.2 per cent (Table 3 and Section 1) as the value at which one could detect CP, had the calibrator errors and rms not been so high. If that were the case, then only nine CHANG-ES galaxies had observations that were sensitive enough to have detected CP (Tables 2 and 3), of which five were actually detected. Although the statistics are low, this is a 55 per cent detection rate, irrespective of any knowledge of the presence or absence of an AGN. It is tempting to extrapolate and suggest that, with sufficiently sensitive and high fidelity observations, the majority of the galaxies that have AGN would also display CP. The final count of AGNs in the CHANG-ES sample is left to another paper, but it may very well be that all AGNs in nearby galaxies harbour a circularly polarized component at some time.

A Stokes V signal is, itself, compelling evidence for the presence of an AGN since star-forming regions are unable to explain such a CP signal. There are other galaxies in the CHANG-ES sample that are also known to have AGNs yet we do not detect CP in them. It is not always clear why this is the case, but a strong contributor is surely the limitation of high upper limits in the observations (Table 2). Another possibility is that CP is known to be highly variable, so applying an across-the-board 0.2 per cent limit on CP may not reflect the reality of the physical conditions in these sources.

All of our CP galaxies show a hard X-ray source in their cores and all also display further evidence for an AGN (LINER, VLBI detections, etc., Section 4). In Table 6 we list values for the cores of the CP-detected sources from XMM–Newton data in a consistent fashion and for the same energy range. In Fig. 3 we plot the CP signal against the X-ray signal from the galaxy cores, both in solar luminosity units. In all cases, the XMM data were taken significantly before the radio data, the shortest time separation being approximately 1 yr.

We only know of two outbursts: a radio outburst in NGC 660 that occurred between 2008 and 2010.7 and the X-ray outburst of NGC 4845. For the remaining sources, we do not have any information about outbursts, but suspect that future monitoring programs may shed more light on whether variability could be a clue that CP will be detected.

One important conclusion from Fig. 3, however, is that the two galaxies that (by far) have the strongest X-ray cores are also the galaxies that show the strongest m_C; both NGC 4388 and NGC 4845 have values of order 2 per cent in absolute value (Table 3). The internal absorption specified in this table is also interesting. Only NGC 3079 does not show an internal absorption component and this is also our most marginal signal. On the other hand, NGC 4388 and NGC 4845, our strongest sources, also show the strongest internal absorption. Internal absorption is relatively easy to determine in X-rays and, again, this result suggests a link between the X-ray emission and CP.

In four out of five cases, CP shows structure, sometimes clearly jet-like morphology (Fig. 4). The fifth case, NGC 3079, is a marginal detection with no structure observed. This is the first time that CP structures have been observed in nearby galaxies and opens up the possibility of probing physical conditions, not only in the core, but also in jets or accretion discs.

5 CONCLUSIONS

CP has the potential to probe deep into the cores of galaxies and provide information on the magnetic field strength and cosmic ray electron energy distribution. CP is also a clear indication of the presence of an AGN (or LLAGN) and may be helpful in sorting out whether larger scale activity is AGN or starburst related. Since LP may not be detectable due to Faraday depolarization, CP may in fact be the only way to probe the physical conditions deep into cores at these low frequencies. Ideally, higher frequency measurements of LP might be extrapolated to the lower frequencies (as we suggest in Section 3.5.1) in order to make an estimate of U₀ (equations A17, A18, or A19), as long as the connection is clear. If there are multiple components, however, such a connection may not be straightforward.

We have carefully searched through the CHANG-ES high-resolution (B-configuration) L-band data for CP signals. Many of the galaxies do not have sufficiently high S/N and sufficiently low calibration errors to detect CP. However, we find five nearby galaxies with detectable (and sometimes very strong) CP, namely NGC 660, NGC 3079, NGC 3628, NGC 4388, and NGC 4845 (NGC 3079, however, is only a marginal detection). To our knowledge, this more than doubles the number of known AGNs with CP-producing central regions in spiral galaxies, the previous ones being only the Milky Way and M81.

CP is observationally challenging to measure and also is difficult to interpret; however, the most successful mechanism to date is Faraday conversion. We show that all of our detections have steep L- to C-band spectral indices [α_V(L −  C) ≲ −3] that are typically expected from Faraday conversion.

We have provided some examples as to how the Faraday conversion interpretation is most closely aligned with the data and can lead to estimates of the physical parameters of AGNs (Section 3.5). Moreover, we have expanded the analysis of Irwin et al. (2015) to provide analytical formulae for a general p, which is the energy spectral index of cosmic ray electrons and is a measurable quantity because it can be determined from total intensity measurements in the optically thin limit (here taken as being C band). These equations are given in equations (A17)–(A19); equation (A18) also supersedes and is more accurate than the earlier equation given in Irwin et al. (2015).

An unexpected result is that the CP is resolved for four of our sources (Fig. 4). For the first time, CP is seen in structures in nearby galaxies. For example, CP is seen along the northern jet in NGC 4388 at a very high level (≈3 per cent). Such measurements present us with a new probe of the physical conditions in nearby jets.

Also, for the first time, we have seen what we have named a conversion disc in NGC 4845 (Fig. 4e). This disc is at an angle to the galaxy's major axis. It is roughly perpendicular to an apparent outflow direction (cf. Perlman et al. 2017) suggesting that the inner part of this disc may be related to an accretion disc that aligns an outflow. However, the conversion disc is seen only in CP. This suggests that an originally linearly polarized signal associated with an extended accretion disc has been converted to CP in a region that has properties adequate for this conversion to occur. It also suggests that many more LLAGNs in the nearby universe may reveal their accretion discs via CP measurements only.

Given the frequency of detections in the CHANG-ES survey and that it was not set up for CP measurements, these results raise the possibility that CP may actually be quite a normal, although possibly variable characteristic of AGNs that are embedded in spiral galaxies. Our X-ray measurements also indicate that galaxies with the strongest X-ray core emission are also the galaxies that show the strongest CP (⁠|$m_{\rm C}\approx 2\hbox{ per cent}$| in absolute value) and the strongest internal absorption. Thus, targeting strong X-ray cores in nearby spiral galaxies could result in a high frequency of CP detections.

ACKNOWLEDGEMENTS

This work has been supported by a Discovery Grant to the first author by the Natural Sciences and Engineering Research Council of Canada. YS acknowledges generous support by the Hans–Boeckler Foundation. The CHANG-ES project at Ruhr-University Bochum has been supported by DFG through FOR1254. This research has made use of the NASA/IPAC Extragalactic Database (NED) that is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

Facilities: VLA.

Footnotes

1

We distinguish between ‘CP’ that refers to any detection of Stokes V and the ratio, m_C ≡ V/I.

2

This survey targets 35 nearby edge-on galaxies at L band (1.5 GHz) and C band (6.0 GHz) in full polarization. See Irwin et al. (2012a) for details.

3

The radio jet in NGC 1275 also shows strong m_C (Homan & Wardle 2004), though at a distance of ∼70 Mpc, we do not classify it as ‘nearby’.

4

Some authors increase σ_{q, U} by a factor of 1.4 that we have not done in this zeroth-order approximation. For a more sophisticated analysis, see Müller, Beck & Krause (2017).

5

http://casa.nrao.edu; using version 4.7.2 (r39762) for V or earlier versions for I.

6

Note that we are using Δ for rms noise values of any image, whereas we are using σ to denote the uncertainty in the ratio, V/I.

7

casa version 4.7.2-REL (r39762) was used.

8

In-band spectral indices are not computed when the signal is negative.

9

Of order one.

10

Science Analysis System.

11

European Photon Imaging Camera.

12

Low-ionization nuclear emission-line region.

13

European VLBI Network.

APPENDIX A: CIRCULAR POLARIZATION BY RELATIVISTIC CONVERSION

In Irwin et al. (2015, hereafter CHANG-ES V), a useful approximation for the production of CP in a GHz-peaked radio source was given. This assumes the dominance of relativistic particles over thermal particles and correspondingly weak Faraday rotation in a magnetized plasma. The mechanism is the differential rotation in the plasma of the two modes in elliptically polarized synchrotron radiation. The formulae were given for the special case when the power-law energy distribution of the electrons was E⁻². See CHANG-ES V appendix E for details of that development in which we were primarily concerned with the frequency dependence of CP.

We now generalize this result to the energy distribution E^−p (i.e. N(E)  ∝  E^−p) of the relativistic electrons. We consider the Faraday thin case since, as we showed in CHANG-ES V, once a source becomes Faraday thick, the conversion term becomes very small (see CHANG-ES V appendix E for a more technical definition of ‘Faraday thick’ and ‘Faraday thin’ in this context). Thus, along a line of sight, the conversion is occurring mainly prior to Faraday thickness.

The approximate result for the flux density, S_V, includes an emission term (subscript ‘em’) and a conversion term (subscript ‘conv’) that is given in equation (E11) of CHANG-ES V and maintains the form

\begin{eqnarray} S_V= S_{V_{{\rm em}}}+S_{V_{{\rm conv}}}=\Omega _{\rm s} \eta _V L + U_0 \kappa _{\rm c} L = \Omega _{\rm s}\frac{\tilde{\eta }_V}{\nu _9}L+\frac{U_0\tilde{\kappa }_{\rm c}L}{\nu _9^3}. \end{eqnarray}

(A1)

In this expression Ω_s is the source solid angle, U₀ is the linearly polarized flux entering the plasma medium, L is the line-of-sight distance through the source, ν₉ is the observed frequency in GHz, η_V is the emission coefficient for S_V in the relativistic medium, and κ_c is the conversion coefficient from U₀ to |$S_{V_{{\rm conv}}}$|⁠. With this formalism, based on Beckert & Falcke (2002), the conversion coefficient in Stokes Q is zero so we need only U₀ to describe the initial linearly polarized flux. |$\tilde{X}$| indicates the value of a quantity, X, at 1 GHz (e.g. |$\tilde{\eta }_V$| is the value of η_V at 1 GHz). The coefficients, however, are function of both frequency and p. We now consider these coefficients, |$\tilde{\kappa }_{\rm c}$| and |$\tilde{\eta }_V$|⁠.

The conversion coefficient (Sazonov 1969; Beckert & Falcke 2002) for ultrarelativistic particles is given by (p ≠ 2)

\begin{eqnarray} \tilde{\kappa }_{\rm c}=-\pi \frac{\nu _{\rm p}^2\nu _B^2}{\lambda _9} \sin ^2{\theta } f(p,\nu ), \end{eqnarray}

(A2)

where θ is the angle between the line of sight and the magnetic field, B (the perpendicular field is Bsin (θ)), and λ₉ = c/(10⁹ Hz). Note that θ ranges from 0° to <90° since we only see emission from particles that have a component of velocity moving towards us. The squared plasma frequency

\begin{eqnarray} \nu _{\rm p}^2=\frac{n_{\rm e} e^2}{\pi m_{\rm e}}, \end{eqnarray}

(A3)

and the electron gyrofrequency

\begin{eqnarray} \nu _B=\frac{eB}{2\pi m_{\rm e} c} \end{eqnarray}

(A4)

are to be measured in GHz. The constants, e, m_e, and c are the electric charge, the electric mass, and the speed of light, respectively. Following Beckert & Falcke (2002), the electron density in the plasma frequency, n_e, comprises both thermal and non-thermal electrons with the non-thermal component dominating. The power-law function of equation (A2), f(p, ν), is (p ≠ 2)

\begin{eqnarray} f(p,\nu )\equiv \frac{2}{p-2}\left(\frac{\nu _9}{\nu _B\sin \theta }\right)^{(1-p/2)}\left(\left(\frac{\gamma _0^2\nu _B\sin \theta }{\nu _9}\right)^{(1-p/2)}-1\right), \end{eqnarray}

(A5)

where γ₀ is the lower energy cut-off of the relativistic particles. Here and throughout this development, we have set the pitch angle equal to the angle between the line of sight and the magnetic field since, for ultrarelativistic particles, they are about the same.

The emission coefficient may be written for a general energy power law as (e.g. Beckert & Falcke 2002, their equation D11)

\begin{eqnarray} \tilde{\eta }_V=-\eta _0 \cos {\theta }(\nu _B\sin {\theta })^{p/2}\nu _9^{(1-p/2)}g(p), \end{eqnarray}

(A6)

where

\begin{eqnarray} \eta _0\equiv \pi \nu _{\rm p}^2\nu _B\frac{m_{\rm e} c^2}{\lambda _9^3}. \end{eqnarray}

(A7)

Again, we will measure all frequencies in GHz. The function g(p) is

\begin{eqnarray} g(p)\equiv \frac{3^{(p-1)/2}(p+2)}{2p}\Gamma \left(\frac{p}{4}+\frac{2}{3}\right)\Gamma \left(\frac{p}{4}+\frac{1}{3}\right), \end{eqnarray}

(A8)

where Γ represents the gamma function. These results assume the energy distribution function to be in the form (Beckert & Falcke 2002)

\begin{eqnarray} \frac{{\rm d}N(\gamma )}{{\rm d}\gamma }=n_{{\rm er}}\frac{|p-1|}{\gamma _0^{(1-p)}}\gamma ^{-p}, \end{eqnarray}

(A9)

where n_er is the relativistic electron density.

For the conversion term, it is possible to simplify the function f of equation (A5) under certain conditions. If the peak frequency γ²ν_B is used in this equation for ν and if γ₀ ≪ γ, then for p < 2 the first term in the bracket may be ignored. In that case we see from equations (A2) and (A5) that

\begin{eqnarray} \tilde{\kappa }_{\rm c} =-\frac{2\pi \nu _{\rm p}^2\nu _B^2}{(2-p)\lambda _9} \sin ^2{\theta }\left(\frac{\nu _9}{\nu _B\sin \theta }\right)^{(1-p/2)}, \end{eqnarray}

(A10)

and so provided that p < 2, |$\kappa _{\rm c} = \tilde{\kappa }_{\rm c}/{\nu _9}^3 \propto \nu _9^{-(2+p/2)}$| rather than |$\nu _9^{-3}$|⁠. Should p > 2 be the case and the frequency ratio remain the same, then the first term in the brackets dominates and f = const. Hence the |$\nu _9^{-3}$| behaviour is maintained.

We may note finally that if |$\nu _9\approx \gamma _0^2\nu _B$|⁠, then over a band width Δν/ν ≪ 1 f may increase strongly away from zero as ν increases. This would be reflected in the sudden onset of CP across the bandwidth and possible sign changes. (An example is given in Fig. 2 and discussed in Section 3.5.3.)

The emission term generalizes slightly given equation (A6) to give the frequency dependence:

\begin{eqnarray} \eta _V=\frac{\tilde{\eta }_V}{\nu _9}\propto \nu _9^{-p/2}, \end{eqnarray}

(A11)

which holds for all p.

In summary the formal expression for |$S_V=S_{V_{{\rm em}}}+S_{V_{{\rm conv}}}$| in this model takes the following forms for |$S_{V_{{\rm em}}}$| and |$S_{V_{{\rm conv}}}$|⁠:

\begin{eqnarray} S_{V_{{\rm em}}}&=&-\Omega _{\rm s}\eta _0 L\cos {\theta }(\nu _B\sin {\theta })^{p/2}\nu _9^{(-p/2)}g(p)\quad (p<2, p=2, p>2), \end{eqnarray}

(A12)

so the emissive term maintains the same form independent of p. However, for the conversion term we have

\begin{eqnarray} S_{V_{{\rm conv}}} - 2\pi U_0L\frac{\nu _{\rm p}^2 (\nu _B\sin \theta )^{(1+p/2)}}{(2-p)\lambda _9}\left(\frac{1}{\nu _9}\right)^{(2+p/2)}\left[1-\left(\frac{{\gamma _0}^2\nu _B\sin \theta }{\nu _9}\right)^{1-p/2} \right]\quad (p<2), \end{eqnarray}

(A13)

with the second term in the square brackets only becoming important if γ₀ν_Bsin θ ≈ ν₉. As noted above, if γ₀ν_Bsin θ < ν₉, then that term is negligible and |$S_{V_{{\rm conv}}}\,\propto \,\nu _9^{-(2+p/2)}$|⁠. The equation is the same if p > 2 but we express it somewhat differently since it is convenient to have p/2 − 1 > 0,

\begin{eqnarray} S_{V_{{\rm conv}}}=- 2\pi U_0L\frac{\nu _{\rm p}^2(\nu _B\sin \theta )^{(1+p/2)}}{(p - 2)\lambda _9}\left(\frac{1}{\nu _9}\right)^{(2+p/2)} \left[ \left( \frac{\nu _9}{{\gamma _0}^2\nu _B\sin \theta } \right)^{(p/2-1)} - 1 \right]\quad (p>2). \end{eqnarray}

(A14)

Here the second term in the square brackets (−1) becomes important only if the first term ≈1. As noted above, if γ₀ν_Bsin θ < ν₉, then |$S_{V_{{\rm conv}}}\,\propto \,\nu _9^{-3}$|⁠. For the limiting case, p = 2,

\begin{eqnarray} S_{V_{{\rm conv}}}=2\pi U_0 L\frac{ {\nu _{\rm p}}^2\left(\nu _B\sin \theta \right)^2}{\lambda _9}\,\frac{1}{{\nu _9}^3}\, {\rm ln}\left(\gamma _0\sqrt{\left( \frac{\nu _B\sin \theta }{\nu _9} \right)} \right)\quad (p=2) \end{eqnarray}

(A15)

in units consistent with equations (A13) and (A14). This equation is equivalent to the result given in appendix E of CHANG-ES V except that, in the earlier development, we were primarily concerned with the frequency dependence and so did not include the weak logarithmic term that we now include, as well as a factor |$\pi$|⁠. B in that equation was taken to be the perpendicular field that is Bsin (θ) here.

We now provide practical numerical expressions.

For our estimates, we take the Stokes parameter, U₀, as an input value to the relativistic conversion region. It may be that there is a purely relativistic core producing this value interior to the conversion region. However, it would be more compelling to produce it internally. Such LP can be generated by Faraday rotation or a rotating magnetic field as revealed in Jones (1988) but does not occur in our homogeneous magnetic field with a small Faraday rotation.

In the remainder of this appendix, n_e can effectively be taken as n_er since relativistic particles dominate.

For |$S_{V_{{\rm em}}}$| (mJy), with L (pc), ν₉ (GHz), B (G), n_e (cm⁻³), ν₉ (GHz), and the source size ϕ (arcsec),

\begin{eqnarray} S_{V_{{\rm em}}}&=& -122\,\phi ^2 \,L\,n_{\rm e} B \cos \theta \left( \frac{2.8\times 10^6 B\sin \theta }{\nu _9} \right)^{p/2} \,g(p)\quad ({\rm any}\ p). \end{eqnarray}

(A16)

This equation corrects the magnitude of the emission term in appendix E of CHANG-ES V. If p = 2, then the frequency dependence goes as ν⁻¹, as found previously. Adopting values similar to those of NGC 4845, if ϕ = 0.001, L = 0.03, n_e = 100, B = 0.04, ν₉ = 1.5, |$\theta =\pi /4$|⁠, and p = 2, we find |$S_{V_{{\rm em}}}\approx 1$| mJy (in absolute value).

For |$S_{V_{{\rm conv}}}$| (mJy), with L (pc), ν₉ (GHz), B (G), and n_e (cm⁻³), where U₀ (mJy) is the linearly polarized flux density to be converted:

\begin{eqnarray} S_{V_{{\rm conv}}}&=& -\frac{5.21\,\times \,10^{7} U_0 \,L \,n_{\rm e}\, (0.0028 B\sin \theta )^{1+p/2}\,{\nu _9}^{-(2+p/2)}}{2-p}\left[ 1\,-\, \left(\frac{{0.0028\gamma _0}^2B\sin \theta }{\nu _9}\right)^{1-p/2} \right]\quad (p<2), \end{eqnarray}

(A17)

\begin{eqnarray} S_{V_{{\rm conv}}}&=& 4.078\,\times \,10^{2} U_0 \,L \,n_{\rm e}\, (B\sin \theta )^{2}\,{\nu _9}^{-3}{\rm ln}\left[ 0.053 \sqrt{ \frac{{\gamma _0}^2B\sin \theta }{\nu _9} } \right]\quad (p=2), \end{eqnarray}

(A18)

\begin{eqnarray} S_{V_{{\rm conv}}}&=& -\frac{5.21\,\times \,10^{7} U_0 \,L \,n_{\rm e}\, (0.0028 B\sin \theta )^{1+p/2}\,{\nu _9}^{-(2+p/2)}}{p-2} \left[ \left(\frac{\nu _9}{0.0028{\gamma _0}^2 B\sin \theta }\right)^{p/2-1} - 1 \right]\quad (p>2). \end{eqnarray}

(A19)

Note that in the text (Section 3.4.2 and throughout) we let

\begin{eqnarray} x\,\equiv \, \frac{0.0028{\gamma _0}^2 B\,\sin \theta }{\nu _9}. \end{eqnarray}

(A20)

The combination of all terms that multiply U₀ must be <1 because |$S_{V_{{\rm conv}}}$| must be <U₀ (in absolute value). The above equations are not valid for a combination of terms >U₀ since this development assumes that the Stokes V opacity is small (see CHANG-ES V appendix E for a more precise definition).

For future reference, we isolate the frequency-dependent terms in equation (A19) and define the function

\begin{eqnarray} F_{\nu _9}={\nu _9}^{-(2+p/2)} \left[ \left(\frac{\nu _9}{0.0028{\gamma _0}^2 B\sin \theta }\right)^{p/2-1} - 1 \right]. \end{eqnarray}

(A21)

This is the function that is plotted in Fig. 2.

Again, adopting values similar to those found for NGC 4845, for L = 0.01 pc, n_e = 100 cm⁻³, B = 0.04 G, γ₀ = 100, |$\theta =\pi /4$|⁠, and ν₉ = 1.5, then S_V/U₀ = −0.34 for p = 1.5, S_V/U₀ = −0.031 for p = 2.0, and S_V/U₀ = −0.0029 for p = 2.5. Strong and significant changes, including sign changes, can occur as the two terms in square brackets for p < 2, p > 2 become comparable. The logarithmic term of equation (A18) can also be positive or negative. The result is, of course, entirely dependent on the magnitude of the linearly polarized flux to be converted so, for example, a polarized flux of 50 mJy could produce a circularly polarized flux of 1.6 mJy using the above values if p = 2. The strongest conversion, however, occurs for p < 2 (corresponding to a flatter electron energy distribution) in the above approximations. A linearly polarized flux of 50 mJy could produce 17 mJy of CP if p = 1.5 in this example. As has been found by others, the conversion term likely outweighs the emission term in most real situations.

Finally, a comment on the sign of the CP. It is important to note that Faraday conversion depends on the perpendicular component of the magnetic field, as opposed to the line-of-sight component that is important for normal Faraday rotation. Conversion does not depend on the sign of the charge nor on the orientation of the B-field as long as it has a transverse component, as noted in Beckert (2003). It is the sign of Stokes U to begin with and the conversion depth, κ_cL (equation A1) that determines the sign of the CP.