MOJAVE XX. Persistent Linear Polarization Structure in Parsec-scale AGN Jets

There are 23 (5.3 per cent of the sample) essentially unpolarized sources (Table 3) at the sensitivity level of our stacked images. Eleven of them are radio galaxies, making up 46 per cent fraction of this source spectral type in the sample. One of these radio galaxies, 0055+300, is shown in Figure 2 (bottom right). Four of the radio galaxies, 0128+554, 1404+286, 1509+054 and 2021+614, are classified as compact symmetric objects (CSO). Our sample has one more CSO, i.e. the GPS radio galaxy 1345+125. Polarization emission in this source is detected in the southern jet feature, at about 45 mas from the core (Lister et al., 2003). However, the integrated fractional polarization over the entire source is relatively low, $m<0.9$ per cent. Unpolarized quasars and BL Lacs are much more rare, both represented by six sources, which makes 2 and 4 per cent of their fractions, respectively. For one of these quasars, 0710+196, polarization is detected in the core at most of the observing epochs, two of which, 2009 August 19 and 2014 January 25, show the strongest polarization with comparable $P$ flux but roughly orthogonal EVPA directions, thereby canceling each other in vector averaging and resulting in close to zero stacked $P$. Another quasar, 1722+401, demonstrates similar behaviour, with nearly perpendicular EVPAs at the first two and last two epochs. Two BL Lacs, 0329$+$654 and 2013$-$092, are the weakest sources in our sample, making detection of their polarization unlikely due to a sensitivity limit. Taking this into account, only 1 per cent of the quasars and 3 per cent of the BL Lacs from our sample are unpolarized. Some sources, e.g. 0615$-$172 and 2021$+$614, show faint and most likely spurious polarization in their stacked $P$-maps due to the $P$-signal being detected at one epoch only, which can be a result of poor leakage term solutions. We excluded these 23 sources from our further analysis.

The reason why some sources of different optical classes, redshifts, apparent speeds and viewing angles are so weakly polarized is unclear. Their polarization sensitivity reached both in single-epoch and stacked images is comparable to that of other AGNs in our sample. It is possible that the weakly polarized sources are subject to strong Faraday depolarization by an external screen associated with an obscuring torus due to large viewing angles of their jets, which is typically the case for radio galaxies (e.g. Aller et al., 2003; Middelberg et al., 2005). Internal plasma processes might also play a role, e.g. if a turbulent component of magnetic field in a jet dominates the ordered component on small scales (Marscher, 2014). In this scenario, the chaotic directions of magnetic field effectively cancel out polarization. It is further supported by a significantly higher fraction (almost every second object) of weakly polarized radio galaxies compared to that of blazars, as we probe the outflows on smaller (de-projected) linear separations from the true jet origin, where turbulence is expected to be stronger, and Faraday rotation could be higher because of denser thermal plasma.

We derived the integrated fractional polarization over the entire source as $m_{\mathrm{int}}=P_{\mathrm{tot}}/S_{\mathrm{tot}}$, where $S_{\mathrm{tot}}$ and $P_{\mathrm{tot}}$ are the total flux density and polarized flux, respectively, calculated over the image above the corresponding $4I_{\mathrm{rms}}$ and $4P_{\mathrm{rms}}$ levels. The values of $m_{\mathrm{int}}$ are distributed in a wide range, from 0.06 per cent for the galaxy NGC 1052 to 17.6 per cent for the quasar 3C 119. The distributions of $m_{\mathrm{int}}$ for the sources of different optical classes are shown in Figure 3. The least polarized sources are radio galaxies, with a median $m_{\mathrm{int}}=0.52\pm 0.24$ per cent. According to a Kolmogorov-Smirnov test, quasars and BL Lacs have statistically different distributions, with the medians of $m_{\mathrm{int}}$ equal to $1.93\pm 0.09$ and $3.05\pm 0.24$ per cent, respectively. BL Lacs being devided into different SED classes show that HSP sources are on average less polarized than ISP and LSP BL Lacs (Figure 4).

We traced out the total intensity ridgeline for each stacked map, following the approach described in Pushkarev et al. (2017). It is based on making an azimuthal slice around the core component at a given radius and finding a position of the weighted average in total intensity. The starting ridgeline point coincides with the core position and then the algorithm advances downstream for successively increasing core separations with a step of 0.5 mas, skipping the regions where emission is too faint, $<4I_{\mathrm{rms}}$. After that we applied a cubic spline interpolation to have a ridgeline point approximately every 0.1 mas along the ridgeline. In the cases when the counter-jet is detected, i.e. in the radio galaxies TXS 0128+554, NGC 1052, 3C 84, PMN J1511+0518, Cygnus A, quasar S4 0646+60 and BL Lac PKS B1413+135, we show the ridgeline for the approaching, brighter jet.

The first evidence that the degree of linear polarization of distinct jet features increases with their separation from the core was reported by Cawthorne et al. (1993), who analysed polarimetric VLBI data at 5 GHz for a small sample of AGN jets. Later, this trend was also detected at 22 and 43 GHz (Lister & Smith, 2000; Lister, 2001) for samples of 18 and 32 sources, respectively. At 15 GHz, the trend was found by Lister & Homan (2005), who studied the first-epoch MOJAVE data for the flux-density-limited complete sample of 133 sources. This observational result implies that the magnetic field, decreasing with distance from the true jet vertex as $B\propto r^{-k}$ ($k=2$ for the case of the poloidal field and $k=1$ for the toroidal field and a conical jet shape), becomes more ordered, if it is not entirely caused by, e.g. decreasing depolarization along the jet.

In Figure 5, we show the dependence of fractional polarization as a function of the projected, i.e. observed, distance down the jet along its ridgeline for the largest ever sample of over 400 sources. We confirm the general increase in $m$ downstream and discuss its features. First, the degree of linear polarization in the core region (within $\sim$0.5 beam) ranges from about 0.1 per cent to 10 per cent, with a median of about 1 per cent and 2 per cent for quasars and BL Lacs, respectively, whereas the radio galaxy cores are found to be low or completely unpolarized. The synchrotron emission of the core is partially opaque, leading to a further decrease in the fractional polarization both through opacity and possibly increased Faraday effects if they are present. The low $m_{\mathrm{core}}$ values are also caused by in-beam depolarization in the case of superposition from polarized emission of unresolved features with different EVPAs.

Beyond the core in the inner jet, $m$ increases quite rapidly, mostly due to a switch to an optically thin regime of synchrotron radiation and weakening of the beam depolarization effect. In the outer jet regions, the degree of polarization, on average, continues to grow with distance from the core. This can be explained by weakening shocks and accompanied turbulence of emitting plasma (Marscher, 2014), progressively smaller Faraday rotation (Hovatta et al., 2012; Kravchenko et al., 2017) and the spectral ageing effect (Kardashev, 1962). In the latter phenomenon, the high-energy electrons lose energy faster owing to synchrotron emission. This leads to steepening of both the energy and flux density spectra in the more remote jet parts. VLBI observations showed that the magnitude of the spectral ageing effect is about $\Delta\alpha\simeq-0.6$ for parsec-scale AGN jets (Pushkarev & Kovalev, 2012; Hovatta et al., 2014). In turn, this results in the growth of $\Delta m_{\mathrm{jet}}\sim 0.1$ because for the optically thin regime of synchrotron radiation from a region with an ordered magnetic field and randomly oriented pitch angles of relativistic electrons the corresponding maximum of $m$ is limited by $(p+1)/(p+7/3)$ (Pacholczyk, 1970), where $p=1-2\alpha$ is the power-law index in the energy spectrum of emitting particles $N(E)\propto E^{-p}$.

The growth of fractional polarization down the jet can also occur either through shear in a boundary layer (e.g. Laing & Bridle, 2014) or as the pitch of a helical magnetic field decreases with distance from the origin (Porth et al., 2011). As shown by Butuzova & Pushkarev (2022), the $m$-values on the jet axis gradually increase with decreasing the pitch angle, starting from about $55^{\circ}$. This was derived from modelling transverse slices of fractional polarization for a wide space of geometric and kinematic jet parameters, e.g. the viewing angle ranging from $2^{\circ}$ to $10^{\circ}$ in the observer’s frame and Doppler factors varying from 4 to about 15. This mechanism, likely contributing insignificantly in the innermost jet parts, can progressively add to the slow logarithmic increase in $m$ and ultimately dominate on large scales. The continuous growth in fractional polarization with the core separation, together with close alignment of EVPA with the local jet direction (typically seen in BL Lacs, more in subsection 4.5), allows us to argue for a large-scale ordered component, i.e. helical magnetic field rather than the presence of an isolated jet feature with high fractional polarization (e.g. Pushkarev & Gabuzda, 2000).

We also analysed individual dependencies of $m$ along the ridgelines. To get rid of a neighbouring-pixel dependence in a map arising due to convolution with the restoring beam, we interpolated the cubic spline of the ridgeline with a step of half a beam size. For the sources which had five or more such ridgeline points with $P>4\sigma_{P}$ beyond the core region, we calculated the Kendall’s $\tau$ correlation coefficient (Figure 6). In total, 215 sources satisfied these criteria, 87 of which show significant (with random chance probability $p<0.05$, hatched area) positive correlation, while only two objects, 0509+406 and 0603+476, have negative correlation.

As seen in Figure 5, the ridgeline $m$-values of BL Lacs at a given projected distance from the core are typically higher than those of quasars. This confirms earlier results obtained by Lister (2001) on a smaller source sample. To investigate it further, we first converted the angular distances to a linear projected scale for the sources with known redshifts and then corrected for projection by applying the viewing angles derived from the estimates of Doppler factors (Homan et al., 2021) and apparent jet speeds (Lister et al., 2021). The ridgeline $m$-values beyond the one-beam core region of 52 BL Lacs and 211 quasars vs de-projected distance in parsecs are given in Figure 7. It shows that the jet-axis fractional polarization of BL Lacs probed on smaller scales (up to decaparsecs) are considerably higher than those in quasars, but the corresponding slope is flatter, thus intersecting the steeper quasar trend on hectoparsec scales, showing comparable $m$-values. This can be explained by higher rotation measures (Hovatta et al., 2012) or initially larger helical B-field pitch angle or higher rates of energy losses in quasars compared to those of BL Lacertae objects. The HSP (17), ISP (9) and LSP (25) BL Lacs are characterised by on average decreasing viewing angles as found by Homan et al. (2021) and also increasing redshifts. They form different partly overlaid segments of the whole $m$-trend, showing generally higher fractional polarization of LSP BL Lacs. If we compare quasars and LSP BL Lacs which have matched redshifts $z\lesssim 0.5$, there is no systematic difference in fractional polarization along their jet ridgelines.

The brightness distribution of AGN jets on parsec scales in total intensity is typically dominated by a bright core for the sources of all optical classes. It is not the case for linearly polarized emission. Thus, the cores and innermost jet regions of radio galaxies are often unpolarized, and the median $P$-peak position is about 3 mas away from the apparent jet origin (Figure 8). BL Lacs and quasars have their polarization intensity peaks much closer to the core, at a median distance of about 0.2 mas downstream the jet. The non-zero offset is caused by opacity effects in the core region, so that polarization is dominated by the innermost jet components, the emission of which becomes optically thin. The $P$-peak position coincides with the core within the pixel size of 0.1 mas in only $\sim$6 per cent of cases for quasars and BL Lacs and none for radio galaxies.

Quasars are found to be strongest in polarization intensity, while BL Lacs have an almost two times weaker median $P$ peak, and radio galaxies are about an order of magnitude weaker. However, we remind that in terms of fractional polarization, the BL Lacs are stronger in the core. There are a few exceptional cases when quasars, like radio galaxies, do not reveal any polarization in the core but show a high SNR $P$ emission in another jet part. These are 1602+576 with the only polarized feature at about 5 mas from the core, 0821+621 and 1435+638 with detected polarization beyond the core, at separations of about 1 mas. Interestingly, no BL Lacs show such a phenomenon.

Earlier, studying the core polarization properties, we found that the EVPAs in the VLBI cores of BL Lacs are better aligned with the innermost jet direction and less variable than in quasars (Hodge et al., 2018). Here, we analyse the EVPA orientation with respect to the local jet direction along the ridgeline. In Figure 9, we plot two-dimensional histograms of $|\mathrm{EVPA}-\mathrm{jetPA}|$ vs core distance separately for different optical classes of the sources. BL Lacs do show a tendency for EVPA to follow the jet. It is clearly seen in about 60 per cent of BL Lacs. This confirms earlier results taken by Gabuzda et al. (2000) on the polarized core and jet components from a sample of 24 BL Lacertae objects. The offset of the density peak of the unsigned EVPA deviations from zero is likely caused by a limited accuracy of the local jet direction from ridgelines and noise error on the EVPAs, rather being a real physical effect. Indeed, if the sign of EVPA deviation is taken into account the density peak spreads around zero nearly symmetrically. Only 14 objects (10 per cent) currently classified as BL Lacs, namely, 0141$+$268, 0214$+$083, 0313$+$411, 0518$+$211, 0723$-$008, 0735$+$178, 0845$-$068, 0851$+$202, 1011$+$496, 1133$+$704, 1215$+$303, 1219$+$285, 1514$-$241, 2010$+$463, manifest EVPA roughly orthogonal to the jet direction. If we divide BL Lacs into SED classes, the sources with the aligned EVPA are fractionally distributed very much close to the whole BL Lac sub-sample (55 per cent of LSP, 26 per cent of ISP, and 26 per cent of HSP). However, those characterised by the EVPA transverse to the local jet position angle have a roughly double relative excess of HSP and the corresponding lack of ISP sources. About 26 per cent of BL Lacs show either non-monotonic orientation of EVPA along the jet or reveal polarization in the core region only, with EVPA not aligned with the jet. Some of these sources might represent EVPA aligned with the innermost jet direction if the outflow is bent there but not resolved. The remaining 4 per cent (six sources) are unpolarized objects, listed in Table 3.

The $|\mathrm{EVPA}-\mathrm{jetPA}|$ in quasars show all possible values, with a clear predominance of a peak near $90^{\circ}$ and a weaker secondary peak at values close to $0^{\circ}$, in contrast to BL Lacs. There is also an indication that EVPAs in quasars and radio galaxies become preferentially transverse to the jet in the outer ridgeline regions. This analysis suggests that $|\mathrm{EVPA}-\mathrm{jetPA}|$ might be a good discriminator between flat-spectrum radio quasars and BL Lacertae objects.

We also produced a series of all intermediate stacked maps for every source by successively adding all available later epochs to the first one to analyse the changes in the image parameters. As expected, the noise level both in $I$ and $P$ reduces the following $N_{\mathrm{epoch}}^{-0.5}$ dependency (Figure 10, top). The evolution of $\sigma_{P}$ against the time interval of epochs in a stacked map is shown in Figure 10 (bottom). The noise reaches values down to about 30 $\mu$Jy beam^-1. The three stripes from upper to mid and lower reflect observations carried out with initially different bit-rates of 128 (until mid-2007, brighter sources), 512 and 2048 Mb s^-1 (since 2014, weaker sources). Stacking reduced the rms noise by a factor of a few for most cases and up to ten times for the most frequently observed sources. The corresponding dependencies for $\sigma_{I}$ look qualitatively similar.

In our earlier study of jet shapes and their opening angles (Pushkarev et al., 2017), we found that the apparent jet opening angle increases if more epochs are added to the stacked map. This growth is not monotonic, it plateaus on scales of about five years, implying that stacking revealed a full cross-section of the inner jet. In the current paper, we analyse the effect of stacking on linearly polarized emission. Combining data of different epochs results in higher image sensitivity, enlarging the areas of the revealed jet structure in total and linearly polarized intensity, $S_{I}$ and $S_{P}$, where $S_{I}$ is the area within the bottom ($4\sigma_{I}$) total intensity contour and $S_{P}$ is the area within an overlap of the bottom linearly polarized ($4\sigma_{P}$) and total intensity ($4\sigma_{I}$) contours. Thus, the image $P$-filling factor of a jet defined as $S_{P}/S_{I}$ is limited by 1. In addition to the number of epochs in a stack, their temporal spacing also matters. If the time interval $\tau$ between the first and last epoch is wide enough it allows reaching the maximum $S_{P}/S_{I}$ value. Figure 11 presents the corresponding jet $P$-filling factor. It increases with $\tau$ up to a certain level, which is distributed in a wide range, with a typical value of about 1/3 and then becomes quasi-constant or slightly decreases due to a source evolution. We selected 53 sources with $\tau>10$ yr and $N_{\mathrm{epoch}}>20$ and found that the time interval, at which the maximum $S_{P}/S_{I}$ is reached, ranges from 4 to 17 yr, with a median of $10.5\pm 2.7$ yr.

To visualise how a stacking map evolves by progressively adding more epochs, we made gif-animated cumulative images. In Figure 12, we show the cumulative stacked map of the bright LSP quasar 1641+399 having 51 epochs distributed over 21 yr, as an example. The cumulative stacked maps for other sources are available from the MOJAVE website²²2https://www.cv.nrao.edu/MOJAVE/allsources.html; select a source of interest and click on ’Cumulative Polarization Stacking’ link.

There are three different EVPA patterns revealed in the stacked maps: (i) with electric vectors transverse to the jet, (ii) aligned with the outflow local direction and (iii) showing a fountain-like distribution, with EVPA following the jet near its axis and gradually rotating to the orthogonal direction at the jet edges. There are three main types of $m$-cuts across the jet, represented by horizontal (quasi-constant $m$), U- or W-shaped profiles. Possible combinations of EVPA patterns and $m$-cut types beyond the core are not equally likely. Thus, the most common case seen in every third source is the U-shaped $m$-slices (Figure 13), with a dip on the axis and higher values at jet edges. This is accompanied by EVPAs either all transverse to the jet (e.g. 0214$+$083, 0430$+$052, 0836$+$710) or distributed in a fountain-like manner (see, e.g. 1641$+$399, 1920$-$211, 2155$-$152). This combination of EVPA and $m$-slices is typically seen in quasars with polarized emission detected across the entire width of the jet. Changing of the field order from the centre to the edge of a jet and manifesting U-shaped $m$-cuts can occur naturally in at least two scenarios. First, a turbulent or tangled magnetic field that has been shock-ordered transverse to the flow with shear near the edges of the jet creating a competing longitudinal magnetic field. Second, a helical magnetic field can also produce this pattern as a result of superposition of linearly polarized emission from regions with different EVPA with the most efficient cancellation near the jet axis, depending on the pitch angle (except for special combinations of pitch angle and viewing angle, that in projection can yield a parallel field on one side of the jet and a transverse field on the other (e.g. Lyutikov et al., 2005)). Another evidence for the helical B-field scenario is asymmetry of the $m$ and $P$ transverse cuts (Aloy et al., 2000; Lyutikov et al., 2005; Clausen-Brown et al., 2011; Zamaninasab et al., 2013; Fuentes et al., 2018; Kramer & MacDonald, 2021). The latter often show two unequal peaks. We also note that in the case of the fountain-like EVPA pattern, the centre dip in the $m$-profile is deeper than for the cases of purely orthogonal EVPA distribution, sometimes reaching a close to zero level. This kind of EVPA distribution across the jet was obtained by Murphy et al. (2013) for a jet model with a helical magnetic field with a relatively large pitch angle $\gtrsim 60^{\circ}$.

The second often-seen type of $m$-cut is flat, with no prominent trends (Figure 14, left). It is accompanied by EVPA well aligned with the jet direction and observed 2.5 times more commonly in BL Lacs (e.g. 0303$+$490, 1717$+$178, 2157$+$213) than in quasars (e.g. 1253$-$055, 1441$+$252, 1754$+$155). Polarization intensity is quite symmetric and has one peak on the jet axis. Sources with this type of $m$-profile reveal a narrow $P$-distribution in a transverse direction, typically within two beams. The only source showing a flat $m$-profile but EVPA transverse to the jet is the quasar 1700+685.

The third type of $m$-slice is W-shaped (Figure 14, right). This kind of profile is sort of intermediate between the U-shaped and horizontal $m$-cut types and is quite rare. We visually identified only a few BL Lacs (e.g. 1307+121, 2200+420) and quasars (e.g. 1611+343, 2308+341) showing this type of $m$-profiles. Their EVPAs follow the jet in the ridgeline area and show progressive rotation to the orthogonal direction at the jet edges if polarization is detected there. Sources with this $m$-cut type show wide jets, up to ten beams in a transverse direction both in total intensity and linear polarization. As shown by Zakamska et al. (2008), this combination of transverse $m$ and EVPA profiles can be produced by a toroidal or highly twisted large-scale B-field.

In Figure 15, we plot transverse $m$-cuts beyond the core region for 307 sources except those 58 with quasi-constant $m$-profiles. The cuts are drawn through every ridgeline point typically separated by 0.1 mas. A 10 mas long cut across the jet begins at a point at 5 mas from the ridgeline in $\textrm{PA}_{\textrm{cut}}=\textrm{PA}_{\textrm{jet}}-90^{\circ}$. The negative offsets are those before the slice crosses the ridgeline. As seen, the dominant fraction of sources from our sample shows a clear increase in fractional polarization towards the jet edges.

We summarise the described connections between the observed EVPA patterns and transverse $m$, $P$-cut types in Table 4. In Figure 16, we present a density distribution of the EVPA orientation with respect to the jet direction for all sources against transverse separation from the ridgeline. It shows that the EVPAs largely either follow the jet within a narrow central channel (typically seen in BL Lacs) or are orthogonal to the outflow across its whole extent, as mainly seen in quasars and radio galaxies.

The observed diversity of polarization patterns listed in Table 4 can be produced by a helical magnetic field with a different pitch angle and certain geometric and kinematic parameters of the outflow surrounded by a sheath of different thickness and speed relative to the jet spine (Butuzova & Pushkarev, 2022). In particular, one can speculate that in the sources with EVPA aligned with the local jet direction, we detect a narrow stripe of linearly polarized emission around the outflow axis and not at the edges, since it comes from a jet spine, while a sheath is either geometrically thin or its emission is too weak. In contrast, when EVPA is transverse to the jet and polarization is detected across the entire jet width, this can occur when the sheath emission dominates, for example due to its geometric thickness. Indeed, as the jet propagates in the dense environment, the shear layer with a longitudinal B-field (transverse EVPA) can be formed due to interaction with the ambient medium (Laing & Bridle, 2014).

Another piece of evidence for the scenario of jet interaction with a dense surrounding medium comes from jet bending. While straight jets typically show quite symmetric $I$ and $P$ cuts, jet bending changes the situation. Out of the visually selected 72 curved jets with detected polarization in the region(s) where the jet is bent, 53 show asymmetry in the transverse $P$-profiles, with an enhanced linearly polarized emission at the outer side of a curved outflow and transverse EVPA in nearly all these cases (see, e.g., Figure 13, right).

There are eight radio-bright blazars (three BL Lacs and five quasars, see Table 5) from our sample, which showed a significant positional association with the high-energy neutrino events detected by IceCube and Baikal-GVD (IceCube Collaboration et al., 2018; Plavin et al., 2020, 2022; Zhirkov et al., 2021; Baikal-GVD Collaboration et al., 2022). We constructed the most probable list of neutrino associations with VLBI-bright blazars according to the rationale in the references provided. These objects show different polarization properties and EVPA patterns, from BL Lac 1749+096 with polarization detected only in the central channel to those with a rich polarization distribution across the entire jet, e.g. in 0506+056 or 0735+178. Yet, we do not see specific features in linear polarization on parsec scales attributed to the highly probable blazar-neutrino associations.