© ESO, 2012

1. Introduction

Active galactic nuclei (AGN) have been studied extensively in the wavelength range from cm-radio to γ radiation in the last decades (see, e.g., Kembhavi & Narlikar 1999; or Krolik 1999, and references therein for a review). There is overwhelming observational evidence that the main source of their emission is accretion onto supermassive black holes (SMBH) with masses M_• ≃ 10^6...9   M_⊙ (e.g., Ferrarese & Ford 2005, and references therein). Important information on the physics of active galacic nuclei is encoded in their linear polarization m_L. Degrees and angles of polarization provide details on synchrotron emission, the geometry of emission regions, strength and orientation of magnetic fields, and (via Faraday rotation and/or depolarization) on particle densities and matter distributions of the surrounding or outflowing matter (see, e.g., Saikia & Salter 1988, and references therein).

In Trippe et al. (2010, hereafter Paper I), we presented the results of our polarimetric AGN survey with the IRAM Plateau de Bure Interferometer (PdBI; e.g., Winters & Neri 2011)¹. This survey observed 86 sources in the (observatory frame) frequency range 80–267 GHz. Sources were not resolved spatially, all properties observed were source-integrated ones. Paper I provided a statistical analysis of the entire survey sample. The main results of this analysis were

• For 73 out of the 86 AGN observed linear polarization was detected. The median degree of polarization was  ≈ 4%, the highest value ever measured was  ≈ 19%. These numbers are far below the value  ≳ 60% expected from “ideal” synchrotron emission from optically thin sources, pointing toward a substantial averaging of polarization due to limited spatial resolution and non-uniform B-field geometries, and/or a substantial contribution by optically thick emission from galactic cores.

• On average, BL Lacertae objects show a higher degree of polarization (m_L ≃ 7%) than Seyfert galaxies or QSOs (m_L ≃ 3% and m_L ≃ 5%, respectively). This “polarization level sequence” BL Lac  →  QSOs  →  Seyferts can be understood approximately in the context of the viewing angle unification scheme of AGN: a more direct view into a more active nucleus permits observation of more concentrated emission. This reduces the impact of averaging of the polarization through source geometry.

• We did not find evidence for correlations of polarization with rest-frame frequency or redshift z. The absence of a m_L − z correlation indicates that there has been no noticeable change of polarization properties since z ≃ 2.4.

• There is a trend toward higher degrees of polarization in sources with flat spectral mm/radio indices that indicate partially optically thick, core-dominated emission. This is somewhat unexpected because in general, optically thin AGN outflows are more polarized than optically thick cores. A possible explanation is – again – provided by source geometry: because the jet emission is distributed over much larger spatial areas than the nuclear emission, averaging of polarization through non-uniform magnetic field geometries could be more efficient in the outflows than in the cores, resulting in observation of the highest polarization levels in the most core-dominated sources.

In this article, Paper II, we focus on the linear polarization properties of AGN that have been observed multiple times and resolved in time and frequency. We selected sources from the sample analyzed in Paper I (1) for which significant polarization was detected at least 20 times, or (2) that had been observed at least 25 times and polarization had been detected at least 15 times. When selecting targets, these limits proved to be the minimum numbers necessary for reliable statistical statements. Observations were obtained between February 2007 and January 2011 at frequencies 82–253 GHz. Eventually, this left us with a sample of six targets: 0415+379, 0507+179, 0528+134, 0954+658, 1418+546, and 1637+574. We present the basic physical properties of our target sources in Table 1.

2. Observations and data processing

2.1. Polarization

In January 2007 (January 2008 for the 2 mm-band), the antennas of the PdBI were equipped with dual linear polarization Cassegrain focus receivers. Since then, it is possible to observe both orthogonal polarizations – “horizontal” (H) and “vertical” (V) with respect to the antenna frame – simultaneously. Observations can be carried out (non-simultaneously) in three atmospheric windows located around wavelengths of 1.3 mm, 2 mm, and 3 mm. Each of these bands covers a continuous range of frequencies; their ranges are 201–267 GHz for the 1.3-mm band, 129–174 GHz for the 2-mm band, and 80–116 GHz for the 3-mm band. Within a given band, any frequency is available for observations. Receivers covering a fourth wavelength band centered at 0.8 mm began operations at the end of 2010 but did not provide polarization data as of January 2011.

At the time of the observations presented here, the PdBI was not yet equipped for observations of all Stokes parameters. We collected linear polarization data on point sources via Earth rotation polarimetry, i.e. by monitoring the fluxes in the H and V channels as functions of parallactic angle ψ. For assessing the polarization of a source, we calculated the parameter $\mathit{q}\mathrm{\left(}\mathit{\psi }\mathrm{\right)}\mathrm{=}\frac{\mathit{V}\mathrm{-}\mathit{H}}{\mathit{V}\mathrm{+}\mathit{H}}\mathrm{\left(}\mathit{\psi }\mathrm{\right)}\mathrm{=}\frac{\mathit{Q}}{\mathit{I}}\cos \mathrm{\left(}\mathrm{2}\mathit{\psi }\mathrm{\right)}\mathrm{+}\frac{\mathit{U}}{\mathit{I}}\sin \mathrm{\left(}\mathrm{2}\mathit{\psi }\mathrm{\right)}$(1)from the fluxes H(ψ) and V(ψ). The second equality means that q(ψ) provides full information on linear polarization (see, e.g., Sault et al. 1996; Thompson et al. 2001; but also Beltrán et al. 2004) if a sufficient range of ψ is observed.

Owing to the nature of polarized light and because the PdBI antenna receivers are located in the Cassegrain foci, observing a polarized target results in q(ψ) being a cosinusoidal signal with a period of 180°. The functional form of q(ψ) is thus $\mathit{q}\mathrm{\left(}\mathit{\psi }\mathrm{\right)}\mathrm{\equiv }{\mathit{m}}_{\mathrm{L}}\cos \left[\mathrm{2}\mathrm{(}\mathit{\psi }\mathrm{-}\mathit{\chi }\mathrm{)}\right]\mathit{.}$(2)Here m_L is the fraction of linear polarization (ranging from 0 to 1; in the following, we will express m_L in units of %) and χ is the polarization angle (ranging from 0° to 180°). A more detailed discussion of the methodology is provided in Paper I. Our observational results are summarized in Table 4.

2.2. Flux densities

Source fluxes are initially recorded as antenna temperatures. Antenna temperatures are converted into physical flux densities (quoted in units of Jansky throughout this paper) by using empirical antenna efficiencies as conversion factors. Those factors are functions of frequencies and are located in the range from  ≈ 22 Jy K^-1 (for the 3-mm band) to  ≈37 Jy K^-1 (for the 1.3-mm band). To estimate the systematic uncertainties of our dataset, we analyzed observations of the radio continuum star MWC 349A (e.g. Tafoya et al. 2004) that were obtained and calibrated like the AGN measurements. Owing to its well-known flux properties MWC 349 serves as a “standard candle” for the PdBI. From the systematic scatter of the MWC 349 lightcurves we conclude that the systematic relative uncertainties of our AGN observations are  ≈ 10% for the 2-mm and 3-mm bands and  ≈ 16% for the 1.3-mm band. For more details regarding AGN flux monitoring with the PdBI, please see Trippe et al. (2011).

We note that the systematic accuracy of our flux data is limited by the automatic monitoring and calibration. When calibrating observations individually and interactively while using MWC 349 as reference, systematic errors can be reduced to  ≲ 5%; Krips et al. (in prep.) are going to discuss the recent developments regarding the PdBI flux calibration and the resulting improved performance.

3. Analysis

3.1. Variability of polarization and flux

Our analysis focuses on the properties of AGN for which polarization has been observed multiple ( ≥ 15) times. We take into account that the polarization is, a priori, a function of two independent parameters: time and frequency. Accordingly, we may present the polarization of our six target AGN in two-dimensional diagrams with the two dimensions being observing time and observing frequency, respectively. The result is shown in Fig. 1. The diagrams provide degrees of polarization and polarization angles for each measurement that detected significant polarization and upper limits on polarization levels otherwise.

In Fig. 2, we present the 3-mm lightcurves of our target sources. To prevent systematic errors caused by spectral slope effects, we limit our variability analysis to flux measurements obtained in the 3-mm band.

To assess variabilities quantitatively, we define the redshift-corrected fluctuation rate of a time-dependent parameter x like ${\mathit{\xi }}_{\mathit{x}}\mathrm{=}\frac{\mathrm{1}}{\mathit{N}}\mathrm{(}\mathrm{1}\mathrm{+}\mathit{z}\mathrm{)}\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}\mathrm{-}\mathrm{1}}\left|\frac{{\mathit{x}}_{\mathit{i}\mathrm{+}\mathrm{1}}\mathrm{-}{\mathit{x}}_{\mathit{i}}}{{\mathit{t}}_{\mathit{i}\mathrm{+}\mathrm{1}}\mathrm{-}{\mathit{t}}_{\mathit{i}}}\right|\mathit{,}$(3)where z is the redshift, x_i is the ith (i = 1,2,3,...) value of the dataset, t_i the time when x_i was obtained, and N the total number of measurements; |...| denotes the absolute value of the enclosed function. By construction, ξ_x is a measure of the rapidity of variations of the parameter x, taking explicitly into account the time axis of the dataset. Thus our parametrization provides additional information compared to commonly used, simpler variability estimators such as the σ_V parameter proposed by Hook et al. (1994). We note that our ξ is similar to a parameter introduced by Villforth et al. (2009), which they named “violence”.

In Table 2, we present fluctuation rates for the degree of linear polarization m_L, ξ_m, as well as for the 3-mm flux densities S_ν, ξ_S. In addition, we show normalized fluctuation rates ξ_m/ ⟨ m_L ⟩  and ξ_S/ ⟨ S_ν ⟩ , with  ⟨ ... ⟩  denoting the time average of the corresponding parameter. Finally, we also present the values of the fluctuation rate ratio $\mathit{\rho }\mathrm{=}\frac{{\mathit{\xi }}_{\mathit{m}}\mathit{/}\mathrm{⟨}{\mathit{m}}_{\mathrm{L}}\mathrm{⟩}}{{\mathit{\xi }}_{\mathit{S}}\mathit{/}\mathrm{⟨}{\mathit{S}}_{\mathit{\nu }}\mathrm{⟩}}\mathrm{·}$(4)For the two fluctuation rates, ξ_m and ξ_S, we calculated statistical errors numerically. Because all data points are affected by measurement errors, the calculated fluctuation rates deviate from zero even for signals with zero “true” variability. Our error analysis is based on a Monte Carlo scheme: we repeated each calculation 1000 times with different realizations of a given data set. The measurement error is defined as the central 68% (1σ) interval of the resulting distribution of ξ_x values. A realization of a data set is drawn by adding random, Gaussian errors – which are in accord with the individual statistical uncertainties – to all elements of a data set. For the ξ_m, errors are small compared to the parameter values and are quoted explicitly in Table 2. For the ξ_S, uncertainty ranges are asymmetric and reach the magnitudes of the parameter values; we therefore do not quote errors individually but restrict ourselves to the statement that the ξ_S values are accurate to factors  ≈ 2.

3.2. Correlations between flux and polarization

The interplay between source polarizations and fluxes promises new insights into the emission mechanisms. We therefore study the relation between fluxes S_ν on the one hand and polarization levels m_L, polarized fluxes P, and polarization angles χ on the other hand. Our observational procedures measure m_L according to Eq. (2) but not P. Thus we calculate linearly polarized fluxes like P = m_LS_ν by combining those – initially separate – polarization and flux measurements obtained simultaneously at the same frequency. Because flux densities are a function of observation frequency ν like S_ν ∝ ν ^− α, we apply spectral rescaling: ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}\mathrm{=}{\mathit{S}}_{\mathit{\nu }}{\left(\frac{\mathit{\nu }}{\mathrm{90} \mathrm{GHz}}\right)}^{\mathit{\alpha }}\mathrm{·}$(5)This procedure means that we scale all observed flux densities to an observing frequency ν = 90 GHz. We used the spectral index information from Paper I (see also Table 1). Accordingly, we computed rescaled polarized fluxes ${\mathit{P}}^{\mathrm{\ast }}\mathrm{=}{\mathit{m}}_{\mathrm{L}}{\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$. For the special case of 0415+379, we included only 3-mm flux densities into our calculations. For this AGN our flux data are affected by a combination of rapid flux variability and irregular sampling that introduce additional systematic errors when combining several wavelength bands; see also Trippe et al. (2011). For the remaining five sources we included all data available regardless of frequency. We note that rescaling is based on spectral slopes derived from an ensemble of flux measurements. Accordingly, rescaled parameters can only be used when analyzing the same ensemble of flux data collectively, though not for the analysis of individual or differences between individual measurements as in Sect. 3.1. Our subsequent correlation analysis is based on the rescaled parameters ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$ and P ^∗ .

In general, we quantify the correlation between two parameters x, y by means of the Pearson correlation coefficient $\mathit{r}\mathrm{\left(}\mathit{x,y}\mathrm{\right)}\mathrm{=}\frac{{\sum }_{\mathit{i}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{-}\mathrm{⟨}\mathit{x}\mathrm{⟩}\mathrm{)}\mathrm{(}{\mathit{y}}_{\mathit{i}}\mathrm{-}\mathrm{⟨}\mathit{y}\mathrm{⟩}\mathrm{)}}{\sqrt{{\sum }_{\mathit{i}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{-}\mathrm{⟨}\mathit{x}\mathrm{⟩}{\mathrm{)}}^{\mathrm{2}}{\sum }_{\mathit{i}}\mathrm{(}{\mathit{y}}_{\mathit{i}}\mathrm{-}\mathrm{⟨}\mathit{y}\mathrm{⟩}{\mathrm{)}}^{\mathrm{2}}}}$(6)(e.g., Snedecor & Cochran 1980). By construction, r values of ( − 1) +1 correspond to perfect (anti)correlation, a value of 0 means absence of any correlation. For each AGN, we computed correlation coefficients with x being ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$ and y being either m_L or P ^∗  or χ, thus providing²$\mathit{r}\mathrm{\left(}{\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}\mathit{,}{\mathit{m}}_{\mathrm{L}}\mathrm{\right)}$, $\mathit{r}\mathrm{\left(}{\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}\mathit{,}{\mathit{P}}^{\mathrm{\ast }}\mathrm{\right)}$, and $\mathit{r}\mathrm{\left(}{\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}\mathit{,\chi }\mathrm{\right)}$. We present the distributions and correlations of our parameters in Figs. 3–5.

Inspecting the distributions in Figs. 3–5 shows that the presence or absence of correlations is far from obvious due to the substantial scatter within our data. To permit quantitative conclusions, we calculated significances by means of a permutation test. For each correlation coefficient r(x,y) we computed a sample of modified coefficients r(x,y′) with y′ being a copy of the initial parameter vector y with its elements rearranged in random order. Because the pairs (x,y′) are drawn randomly, the expected value for r(x,y′) is zero. From a total of 10 000 random realizations we determined the fraction p for which $\begin{array}{ccc}\mathit{r}\mathrm{\left(}\mathit{x,}{\mathit{y}}^{\mathrm{\prime }}\mathrm{\right)}& \mathrm{\ge }& \mathit{r}\mathrm{\left(}\mathit{x,y}\mathrm{\right)} \mathrm{if} \mathit{r}\mathrm{\left(}\mathit{x,y}\mathrm{\right)}\mathit{>}\mathrm{0}\mathit{,}\\ \mathit{r}\mathrm{\left(}\mathit{x,}{\mathit{y}}^{\mathrm{\prime }}\mathrm{\right)}& \mathrm{\le }& \mathit{r}\mathrm{\left(}\mathit{x,y}\mathrm{\right)} \mathrm{if} \mathit{r}\mathrm{\left(}\mathit{x,y}\mathrm{\right)}\mathrm{\le }\mathrm{0.}\end{array}$The fraction p is thus the false alarm probability, i.e. the probability that a value r(x,y) ≠ 0 is produced by a statistical fluctuation. Accordingly, our randomization test probes the null hypothesis “the parameters x and y are intrinsically uncorrelated”. In the following discussion as well as in Figs. 3–5, we quote significance levels s = 1 − p in units of %.

3.3. Relations between polarization and frequency

Because our observations are resolved in frequency, our data permit probing correlations between polarization properties and frequency. In Fig. 6, we present the degree of linear polarization m_L as function of observing frequency ν. We quote Pearson correlation coefficients and their significances computed analogously to the procedure outlined in Sect. 3.2. As already noted in Paper I (see especially the top panel of Fig. 6 therein), a positive correlation – if any – may be expected already because of observational bias: measurements obtained at higher frequencies tend to suffer from reduced signal-to-noise (S/N) ratios. This effect reduces the probability of detecting low values of m_L at high frequencies, resulting in an apparent increase of the average m_L with frequency.

A more promising approach is the analysis of the polarization angle χ. Other than for m_L, we are not aware of any mechanism that could introduce a bias affecting the observed values of χ. This is a convenient situation because it permits an analysis of Faraday rotation. For linearly polarized light propagating through a plasma permeated by a magnetic field, the polarization angle is related to the rest-frame wavelength³λ₀ like $\mathit{\chi }\mathrm{=}\mathrm{RM}\hspace{0.17em}\mathrm{\times }\hspace{0.17em}{\mathit{\lambda }}_{\mathrm{0}}^{\mathrm{2}}\mathit{,}$(7)where RM is the rotation measure $\frac{\mathrm{RM}}{\mathrm{rad}\hspace{0.17em}{\mathrm{m}}^{-2}}\mathrm{=}\mathrm{8.1}\hspace{0.17em}\mathrm{\times }\hspace{0.17em}{\mathrm{10}}^{\mathrm{5}}{\mathrm{\int }}_{\mathrm{l}\mathit{.}\mathrm{o}\mathit{.}\mathrm{s}\mathit{.}}\left(\frac{{\mathit{B}}_{\mathrm{\left|}\mathrm{\right|}}}{\mathrm{G}}\right)\left(\frac{{\mathit{n}}_{\mathrm{e}}}{\mathrm{c}{\mathrm{m}}^{-3}}\right)\mathrm{d}\left(\frac{\mathit{l}}{\mathrm{pc}}\right)\mathit{,}$(8)with B_|| being the strength of the magnetic field parallel to the line of sight (l.o.s.), n_e being the electron number density, and l being the coordinate directed along the l.o.s. (e.g., Wilson et al. 2010). Thus, deriving RM permits constraining the l.o.s.-integrated magnetic field strength and particle density of an active galaxy.

In Fig. 7 we present for each source observed polarization angles χ as function of squared rest-frame wavelength λ₀. Our data are affected by strong temporal variability. Where possible, we therefore limit our analysis to time windows for which the polarization appears coherent (compare Fig. 1, Table 4). We selected those time windows such that a minimum number of data points is excluded and the significance of a potential rotation measure signal is enhanced. We note that this “windowing” does not bias our results because it applies a selection in time but not in polarization angle or frequency. We quote Pearson correlation coefficients and their significance levels computed analogously to the procedure outlined in Sect. 3.2. For all sources except 0528+134 the observed polarization angles (after windowing) are distributed over ranges notably smaller than the permitted  [0,π]  intervals. For these AGN we attempted to derive rotation measures via linear fits to the data in accord with Eq. (7). Subsequently, we assumed that the observed distributions of χ vs. ${\mathit{\lambda }}_{\mathrm{0}}^{\mathrm{2}}$ are superpositions of two principal effects: (1) a linear trend due to Faraday rotation, and (2) scatter due to temporal variability of the AGN polarization that is independent of wavelength. In this case the RM is given by the slope of the straight line fitting the data best. We claim a successful rotation measure measurement if, and only if, the data show a linear correlation r between χ vs. ${\mathit{\lambda }}_{\mathrm{0}}^{\mathrm{2}}$ that is significantly different from zero.

We note that the statistical errors of the RM appear to be very small when regarding the scatter of the data. This is a consequence of our “two-effect assumption” outlined above. Our RM measurements used a χ² minimization algorithm⁴ that fits linear models to the data. For models with few parameters, the 1σ confidence intervals of the parameters are given by variations of χ² around its minimum of order unity. In our case however, the scatter of the data is much larger than the statistical error of any individual data point; therefore even the best-fitting solutions have χ² in the order of several thousands. From this it follows that the statistical errors we derive are meaningful if, and only if, our initial “two-effect assumption” is applicable. Obviously, the accuracy of our results is limited by systematic errors due to intrinsic temporal variability; statistical errors due to random measurement errors are hardly relevant. We therefore eventually quote for each RM value two errors: the statistical fit error and a systematic error. The systematic error essentially quantifies the uncertainty due to the temporal variability of χ. We derived systematic errors from the confidence levels s of the Pearson correlation coefficients, expressed in units of Gaussian σ. For a significance level s that corresponds to n × σ in Gaussian terms (with n being a positive real number), we quote RM/n as an estimate of the systematic error. Our results are summarized in Table 3.

4. Results

4.1. 0415+379 (3C 111)

For this Seyfert 1 galaxy located at z ≈ 0.05, 17 out of 34 polarimetric observations, obtained between March 2007 and January 2010, detected linearly polarized emission. The average degree of polarization is  ⟨ m_L ⟩  ≈ 2.3%. Initially, low polarization levels below 2% and polarization angles around 0° (modulo 0°/180° ambiguities) are predominant. In August 2007, the polarization changed to m_L ≈ 4 − 6% and χ ≈ 100°. After April 2008, polarization could only be detected at one occasion (April 2009, m_L ≈ 1%,χ ≈ 120°). The degree of polarization fluctuates at (normalized) rates of ξ_m ≈ 0.09% d^-1 and ξ_m/ ⟨ m_L ⟩  ≈ 0.04 d^-1, respectively.

The 3-mm flux density was highly variable, with the average value being  ⟨ S_ν ⟩  ≈ 7.5 Jy. In mid-2007, flux measurements commenced at values S_ν ≈ 10 Jy, rising to S_ν ≈ 15 Jy shortly thereafter. Since then, the flux declined to S_ν ≈ 2 Jy, with a short-lived intermediate high of S_ν ≈ 7 Jy observed at the end of 2008. The flux density fluctuates with ξ_S ≈ 0.7 Jy d^-1, the normalized fluctuation rate being ξ_S/ ⟨ S_ν ⟩  ≈ 0.1 d^-1. From a comparison of the fluctuation rates in flux and polarization, one obtains a fluctuation rate ratio of ρ ≈ 0.4.

Degree of polarization and rescaled flux show a strong (r =  −0.9) anticorrelation. With s = 99.91% (corresponding to  ≈ 3.3σ in terms of a Gaussian distribution), this anticorrelation is significant. An anticorrelation (r =  −0.51) is apparent also when comparing rescaled polarized flux P^∗ and ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$; however, as s ≈ 94% (corresponding to  ≈ 1.9σ), this signal is statistically insignificant. Additionally, polarization angle χ and ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$ show a strong (r = 0.91) correlation, which is significant at a level of s = 99.91% (≈3.3σ).

A search for a correlation between m_L and observing frequency ν provides a null result (r = 0.37, s = 92%). When comparing polarization angle χ versus squared rest-frame wavelength ${\mathit{\lambda }}_{\mathrm{0}}^{\mathrm{2}}$, we formally derive a rotation measure RM = (−4.0 ± 2.3) × 10³ rad m^-2 (statistical error only). As s = 71%, this signal is insignificant. Accordingly, we quote a 3σ upper limit on |RM| given by the absolute value of the RM formally derived plus three times its statistical error. Eventually, we find |RM| < 1.1 × 10⁴ rad m^-2.

4.2. 0507+179

This quasar, located at z ≈ 0.42, shows significant polarization in 21 out of 22 measurements obtained between September 2007 and January 2011. The average polarization level is  ⟨ m_L ⟩  ≈ 7.4%. In 2007, the degree of polarization is m_L ≈ 6%, declines to m_L ≈ 3% in the first half of 2008, rises again to levels up to m_L ≈ 11%, and decreases again to m_L ≈ 8%. In 2007 and 2008, the polarization angle varies in the range χ ≈ 50...110°, remaining around χ ≈ 100° thereafter. The degree of polarization fluctuates at rates of ξ_m ≈ 0.4 % d^-1 and ξ_m/ ⟨ m_L ⟩  ≈ 0.05 d^-1, respectively.

The 3-mm flux evolved smoothly, with values S_ν ≈ 0.7 Jy in 2007, peaking at S_ν ≈ 1.5 Jy at the beginning of 2009, and decreasing again to S_ν ≈ 0.5 Jy. The average flux density is found to be  ⟨ S_ν ⟩  ≈ 1 Jy. It fluctuates with ξ_S ≈ 0.1 Jy d^-1, corresponding to a normalized fluctuation rate of ξ_S/ ⟨ S_ν ⟩  ≈ 0.1 d^-1. The polarization – flux fluctuation rate ratio is ρ ≈ 0.7.

Polarization level and rescaled flux density appear to be uncorrelated (r = 0.22, s ≈ 79%). When comparing rescaled polarized flux and rescaled flux density, we find a positive correlation (r = 0.73), which is statistically significant at a level of s = 99.96%, corresponding to  ≈ 3.5σ. Polarization angle and rescaled flux are uncorrelated (r =  −0.08, s ≈ 61%).

From a comparison of m_L and observing frequency ν we do not find any significant correlation (r = 0.3, s ≈ 91%). We find a slightly positive correlation (r = 0.3) between polarization angle χ and the square of the rest-frame wavelength ${\mathit{\lambda }}_{\mathrm{0}}^{\mathrm{2}}$. We can formally derive a rotation measure of RM = (9.3 ± 1.6) × 10³ rad m^-2 (statistical error only). However, as the significance of the correlation is only s = 60% we have to reject this result as insignificant. Accordingly, we quote a 3σ upper limit of |RM| < 1.4 × 10⁴ rad m^-2.

4.3. 0528+134 (OG+134)

For this high-polarization quasar located at z ≈ 2.1, 18 out of 32 polarimetric observations, obtained between February 2007 and December 2008, detected linearly polarized emission. The average degree of polarization is  ⟨ m_L ⟩  ≈ 3.2%. During the observation timeline, polarization levels varied in the range m_L ≈ 2...6%, whereas the polarization angles varied in the range χ ≈ 0...140°. The degree of polarization fluctuates at rates of ξ_m ≈ 0.3% d^-1 and ξ_m/ ⟨ m_L ⟩  ≈ 0.1 d^-1, respectively.

The 3-mm flux density was at its peak of S_ν ≈ 5 Jy when observations commenced in 2007 and decreased steadily to levels of S_ν ≈ 1 Jy in 2010. The average of the flux density is  ⟨ S_ν ⟩  ≈ 3.5 Jy. The flux density fluctuates with ξ_S ≈ 1.1 Jy d^-1, the normalized fluctuation rate being ξ_S/ ⟨ S_ν ⟩  ≈ 0.3 d^-1. By comparing the fluctuation rates in flux and polarization, we obtain a fluctuation rate ratio of ρ ≈ 0.3.

The degree of polarization and the rescaled flux ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$ do not show a correlation (r =  −0.21, s ≈ 72%). When analyzing rescaled polarized flux P^∗ versus ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$, we see a positive correlation (r = 0.73). However, as s = 99.45%, being equivalent to  ≈ 2.8σ, this signal is only marginally significant. For the correlation between polarization angle and rescaled flux we find r =  −0.37 and s ≈ 87%.

Polarization level and observing frequency ν are uncorrelated (r =  −0.01). A comparison of polarization angle χ and squared rest-frame wavelength ${\mathit{\lambda }}_{\mathrm{0}}^{\mathrm{2}}$ finds that the values observed for χ occupy the entire range  [0,π] ; evidently, strong temporal variability renders our rotation measure analysis inconclusive even when using “windowing”.

4.4. 0954+658

This BL Lacertae object, located at z ≈ 0.37, shows significant polarization in 35 out of 37 measurements obtained between March 2007 and January 2011. The average polarization level is  ⟨ m_L ⟩  ≈ 7.4%. 0954+658 shows by far the highest degrees of polarization among our sources, ranging up to m_L ≈ 18% in 2007. In 2009, the polarization drops to m_L ≈ 5% and increases again up to m_L ≈ 10% in 2010/2011. The polarization angles remain around χ ≈ 0° for most of the observing timeline; the only notable exception occurs in 2009 when values of χ ≈ 10...30° occur. The degree of polarization fluctuates at a rate of ξ_m ≈ 0.5% d^-1 and a normalized rate ξ_m/ ⟨ m_L ⟩  ≈ 0.07 d^-1, respectively.

The average of the 3-mm flux density is  ⟨ S_ν ⟩  ≈ 1.2 Jy. The flux varies remains within the range S_ν ≈ 0.5...1.5 Jy throughout the observing timeline, the only exception being a short peak with S_ν ≈ 2.5 Jy occurring mid-2008. The flux fluctuation rate is ξ_S ≈ 0.1 Jy d^-1, corresponding to a normalized fluctuation rate of ξ_S/ ⟨ S_ν ⟩  ≈ 0.1 d^-1. The fluctuation rate ratio of polarization and flux is ρ ≈ 0.8.

Polarization level m_L and rescaled flux ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$ are uncorrelated (r =  −0.31, s ≈ 96%). Likewise, rescaled polarized flux P^∗ and ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$ are uncorrelated (r = 0.2, s ≈ 85%). Polarization angle χ and rescaled flux ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$ are uncorrelated (r = 0.36, s ≈ 97%).

A correlation analysis of m_L versus observing frequency ν unveils a positive correlation with r = 0.52. As s ≈ 99.9% ( ≈ 3.2σ), the correlation is statistically significant. We find a slightly negative correlation (r =  −0.21) between polarization angle χ and the square of the rest-frame wavelength ${\mathit{\lambda }}_{\mathrm{0}}^{\mathrm{2}}$. Regarding the polarization angle, we can formally derive a rotation measure of RM = (−14 ± 1) × 10³ rad m^-2 (statistical error only). However, as the significance of the correlation is s ≈ 99.2% ( ≈ 2.6σ), the correlation falls short of being statistically significant. Accordingly, we quote a 3σ upper limit of |RM| < 1.7 × 10⁴ rad m^-2.

4.5. 1418+546 (OQ+530)

For this BL Lacertae object located at z ≈ 0.15, 52 out of 76 polarimetric observations, obtained between January 2008 and January 2011, detected linearly polarized emission. The average degree of polarization is  ⟨ m_L ⟩  ≈ 5.0%. Polarization levels vary in the range m_L ≈ 3...10% with a tendency for the highest values to occur at the beginning and the lowest values to be found at the end of the observed period. The degree of polarization fluctuates at rates of ξ_m ≈ 0.3% d^-1 and ξ_m/ ⟨ m_L ⟩  ≈ 0.06 d^-1, respectively. The angles of polarization are located in the range χ ≈ 110...150°. From inspecting the corresponding time–frequency diagram (Fig. 1) we note a “swing pattern” for the polarization angles, with the highest values of χ to be found around the beginning of 2010.

The 3-mm lightcurve shows a double-peak morphology, with values ranging from S_ν ≈ 0.5 Jy at the beginnings of 2009 and 2011 to S_ν ≈ 1.3 Jy at the beginnings of 2008 and 2010. The average value of the flux is  ⟨ S_ν ⟩  ≈ 0.8 Jy. The flux density fluctuates with ξ_S ≈ 0.04 Jy d^-1, the normalized fluctuation rate being ξ_S/ ⟨ S_ν ⟩  ≈ 0.05 d^-1. From comparison of the fluctuation rates in flux and polarization, we obtain a fluctuation rate ratio of ρ ≈ 1.1.

When comparing m_L and ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$, we find a positive correlation (r = 0.57), which is significant as s ≈ 99.98% (i.e.  ≈ 3.7σ). An analysis of rescaled polarized flux P^∗ versus ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$ indicates a strong (r = 0.75) and significant (s ≈ 99.99%, corresponding to  ≈ 3.9σ) positive correlation. Polarization angle and flux are uncorrelated (r = 0.31, s ≈ 97%).

Polarization level and observing frequency ν show a positive correlation (r = 0.48), which appears to be significant (s ≈ 99.6%, i.e.  ≈ 2.9σ). We find a significant – with s ≈ 99.6%, meaning  ≈2.9σ – negative correlation (r =  −0.35) between polarization angle χ and the square of the rest-frame wavelength ${\mathit{\lambda }}_{\mathrm{0}}^{\mathrm{2}}$. We are able to derive a rotation measure of RM = (−39  ±  1_stat  ±  13_sys)  ×  10³ rad m^-2.

4.6. 1637+574 (OS+562)

This Seyfert 1 galaxy, located at z ≈ 0.75, shows significant polarization in 17 out of 26 measurements obtained between February 2007 and August 2009. The average polarization level is  ⟨ m_L ⟩  ≈ 3.0%. Degrees of polarization are found in the range m_L ≈ 1...5%, with the lowest values occurring at the beginning of the observed period. The angles of polarization swing from χ ≈ 150° to χ ≈ 90°. The degree of polarization fluctuates at a rate of ξ_m ≈ 0.2% d^-1 and a normalized rate ξ_m/ ⟨ m_L ⟩  ≈ 0.05 d^-1, respectively.

The 3-mm flux density is initially observed at values S_ν ≈ 1.2 Jy, increases up to S_ν ≈ 2 Jy in 2008 and decreases again to S_ν ≈ 1 Jy in 2009. The average flux is  ⟨ S_ν ⟩  ≈ 1.2 Jy. The flux fluctuation rate is ξ_S ≈ 0.17 Jy d^-1, corresponding to a normalized fluctuation rate of ξ_S/ ⟨ S_ν ⟩  ≈ 0.12 d^-1. The fluctuation rate ratio of polarization and flux is ρ ≈ 0.4.

Polarization level m_L and rescaled flux ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$ are uncorrelated (r = 0.1). When comparing rescaled polarized flux P^∗ and ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$, we find a positive correlation with r = 0.55; however, as s ≈ 97%, the correlation is insignificant. Likewise, polarization angle and flux are uncorrelated (r =  −0.22, s ≈ 76%).

An analysis of m_L versus observing frequency finds a positive correlation (r = 0.46), which is insignificant as s ≈ 97%. We find a strong, positive correlation (r = 0.78) between polarization angle χ and the square of the rest-frame wavelength ${\mathit{\lambda }}_{\mathrm{0}}^{\mathrm{2}}$. As s ≈ 99.98%, corresponding to  ≈ 3.7σ, the correlation is significant. The rotation measure we derive is RM = (42 ± 1_stat ± 11_sys) × 10⁴ rad m^-2.

5. Discussion

5.1. Polarization levels

We present the polarization properties of six mm/radio-luminous – with S_ν ≳ 0.8 Jy in the 3-mm band – AGN. Our sample is a mix of Seyfert galaxies, BL Lacertae objects, and quasars. An inspection of their mm/radio spectral indices (as listed in Table 1) indicates that the radiation of two sources – 0415+379 and 0528+134 – is optically thin and outflow-dominated, whereas the remaining four AGN show optically thick, core-dominated emission when applying the conventional division between the two cases placed at α = 0.5 (e.g., Krolik 1999).

The average degrees of polarization m_L we observe are located between  ≈ 2% and  ≈ 7%. This is substantially lower than the values one can derive from the theory of synchrotron emission from homogeneous and isotropic ensembles of relativistic electrons moving in uniform magnetic fields (e.g., Ginzburg & Syrovatskii 1965; Pacholczyk 1970; see also the corresponding discussion in Paper I). Given the range of α observed, we may expect theoretical polarization levels from m_L ≈ 13% (for α = 0.3, optically thick emission) to m_L ≈ 74% (for α = 0.9, optically thin emission). The most probable explanation is provided by geometric averaging: our observations do not resolve our targets spatially, whereas extended outflows (jets or halos) have been observed in all of them. More specifically, Lee et al. (2008) have been able to map four sources – 0415+379, 0528+134, 0954+658, and 1637+574 – using global VLBI observations at 86 GHz. In addition, Sokolovsky et al. (2011) observed an optically thick core as well as an optically thin jet in VLBA radio maps obtained in the range 1.4–15 GHz. These maps show jets for all sources observed, indicating that probably all our targets show optically thin outflows if observed with sufficient angular resolution. Accordingly we have to assume that any of our observations takes the average of multiple emission regions with different optical depths and different magnetic field orientations. Assuming we observe N emission zones with randomly distributed magnetic field orientations in any given source, we may expect source-integrated degrees of polarization ${\mathit{m}}_{\mathrm{L}}\mathrm{\to }{\mathit{m}}_{\mathrm{L}}^{\mathrm{\prime }}\mathrm{\approx }{\mathit{m}}_{\mathrm{L}}\mathit{/}\sqrt{\mathit{N}}$. From our observations (see Tables 1, 2) we can then estimate values between N ≈ 3 (for 0507+179 and 0954+658, assuming optically thick emission) and N ≈ 1000 (for 0415+379, assuming optically thin emission). The variation in N mirrors the variation in relevant emission region sizes: AGN radio cores are placed at spatial scales of parsecs whereas jets occur on scales of many kiloparsecs ( ≈ 80 kpc for the radio jet of 0415+379; Liu & Xie 1992). However, this simple “static” picture is complicated by any dynamical evolution of the emission regions; we discuss the potential signatures of shocks in the following subsections.

An independent check of our results is provided by Agudo et al. (2010). This study provides single-epoch measurements of linear polarization obtained between 2005 and 2006 at 86 GHz for five of our targets. The authors found degrees of polarization of  < 1.5% (i.e., an upper limit only), 2.6 ± 0.5%, 5.2 ± 0.5%, 2.8 ± 0.6%, and  < 1.5% for 0415+379, 0528+134, 0954+658, 1418+546, and 1637+574, respectively. In addition, Agudo et al. (2010) derived polarization angles of 91 ± 6°, 8 ± 3°, and 108 ± 6° for 0528+134, 0954+658, and 1418+546, respectively. These values may be compared with our results given in Tables 2 and 4 as well as Fig. 1; taking into account the temporal variability, both datasets agree well. This indicates that neither Agudo et al. (2010) nor our study seem to have observed a source in a temporary atypical or extreme state. This reduces the probability that any of the two studies is affected by temporal selection effects.

5.2. Variability

Variability of AGN polarization is well-known in the cm/radio regime (e.g., Aller et al. 1985) and has been probed in the mm/radio regime for a few selected sources (e.g., Jorstad et al. 2005) as well as in the optical (e.g., Villforth et al. 2009). Our variability analysis (see Eqs. (3), (4) and Table 2) treats variability in source flux S_ν and degree of polarization m_L jointly (see also Figs. 1, 2). Both parameters fluctuate at relative rates of few per cent per day. For the flux densities, those fluctuation rates have been observed in cm/radio as well as mm/radio bands and have been discussed in the context of potential intra-day variability (e.g., Fuhrmann et al. 2008). Rapid flux variability is the basis of the famous historical statement that the size of the (relevant parts of the) individual emission regions cannot exceed a few light days (Smith & Hoffleit 1963). Our analysis makes it possible to extend this statement to the observed polarization: because the fluctuation rates of flux density and degree of polarization are compatible, the spatial scales of the corresponding emission regions should be compatible as well. We may quantify this by means of the fluctuation rate ratios ρ as given by Eq. (4). Our analysis finds values ρ ≈ 0.3 − 1.1, permitting two principal conclusions:

• 1.

  Within their uncertainties of factors  ≈ 2, the ρ values are on the order of unity for all our six target sources. This suggests that ratios ρ ≈ 1 are indeed characteristic for AGN emission (at least in mm/radio) – pointing toward a strong structural correlation between total and polarized flux.

• 2.

  Even though all ratios ρ are consistent with being on the order of unity, the individual values tend toward numbers somewhat smaller than one⁵. This indicates that the size scales relevant for the polarized emission – like turbulence or eddy scales of the magnetic fields involved – are compatible with, but not smaller than, the relevant size scales for the overall emission.

Rapid – referring here to timescales shorter than a few months – variability of polarization is supposedly caused by shocks propagating through magnetized plasma (e.g., Hughes et al. 1985). In this case, the degree of linear polarization of synchrotron radiation from a partially compressed plasma is reduced by a factor μ compared to the case without compression (Hughes et al. 1985; Cawthorne & Wardle 1988), where $\mathit{\mu }\mathrm{=}\frac{\mathit{\delta }}{\mathrm{2}\mathrm{-}\mathit{\delta }}$(9)with the shock parameter $\mathit{\delta }\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{k}}^{\mathrm{2}}\mathrm{)}{\cos }^{\mathrm{2}}\mathit{ϵ}\mathit{.}$(10)Here 0 ≤ k ≤ 1 is the compression factor, meaning the factor by which the length of the affected region is reduced by compression. The angle ϵ is the angle between the line of sight and the plane of compression in the frame of the emitter. From Eq. (10) follows 0 ≤ δ ≤ 1. Additional information from, e.g., time-resolved radio interferometric maps, is not available, making it impossible to disentangle k and ϵ in δ; accordingly, δ is our only observable. As discussed in Sect. 5.1, we probably observe polarization from multiple emission regions; in this case, any attempt to constrain the parameters of Eqs. (9) and (10) actually provides source-averaged, “typical” values. Assuming we observe different values for μ at two points of time “1” and “2”, we may write $\mathrm{\Delta }\mathit{\mu }\mathrm{=}{\mathit{\mu }}_{\mathrm{1}}\mathrm{-}{\mathit{\mu }}_{\mathrm{2}}\mathrm{=}\frac{{\mathit{\delta }}_{\mathrm{1}}}{\mathrm{2}\mathrm{-}{\mathit{\delta }}_{\mathrm{1}}}\mathrm{-}\frac{{\mathit{\delta }}_{\mathrm{2}}}{\mathrm{2}\mathrm{-}{\mathit{\delta }}_{\mathrm{2}}}\mathrm{=}\frac{\mathrm{2}\mathrm{(}{\mathit{\delta }}_{\mathrm{1}}\mathrm{-}{\mathit{\delta }}_{\mathrm{2}}\mathrm{)}}{\mathrm{(}\mathrm{2}\mathrm{-}{\mathit{\delta }}_{\mathrm{1}}\mathrm{)}\mathrm{(}\mathrm{2}\mathrm{-}{\mathit{\delta }}_{\mathrm{2}}\mathrm{)}}\mathrm{·}$(11)Our observations do not provide the absolute, intrinsic degrees of polarization – meaning the polarization levels corresponding to the case when shocks are absent – of individual emission regions. Instead, we observe source-averaged (except, possibly, 1418+546; see Sect. 5.3) polarization levels that fluctuate with time. From this we attempt to constrain at least average, “typical” shock parameters δ. For a given source, we use the time average of the degree of polarization  ⟨ m_L ⟩  as an approximate reference. We then use the normalized standard deviation of the degree of polarization σ_m/ ⟨ m_L ⟩  as an estimate for Δμ, meaning $\mathrm{\Delta }\mathit{\mu }\mathrm{\approx }\frac{{\mathit{\sigma }}_{\mathit{m}}}{\mathrm{⟨}{\mathit{m}}_{\mathrm{L}}\mathrm{⟩}}\mathrm{·}$(12)Evidently, Δμ depends on the individual values of δ₁ and δ₂ as well as on their difference δ₁ − δ₂ (see Eq. (11)). Accordingly, it is not possible to derive single values for δ_1,2 but ranges in a plane spanned by δ₁ and δ₂. In Fig. 8 we present such a plane with the ranges marked that are permitted for σ_m/ ⟨ m_L ⟩  ≈ 0.33 − 0.65 as observed for our sources (compare Table 2). Our result points toward the occurrence of fairly violent shocks and/or multiple shocks with varying geometries within our sources. According to Eq. (10) and regarding, e.g., the pair δ₁ ≈ 1 and δ₂ ≈ 0.8 in Fig. 8, our observations indicate the occurrence of k ≲ 0.9, or of variations in ϵ by  ≳ 10°, or combinations of both.

5.3. Correlations between flux and polarization

Immediate correlations between rescaled (Eq. (5)) polarization parameters and source flux (Figs. 3–5) can occur in several varieties. Because we might observe multiple emission components within the same source, we can write the observed linear polarization vector P as ${P}\mathrm{=}\sum _{\mathit{i}}{P}\mathit{i}$(13)with two-dimensional vectors P = (Q,U) with Stokes parameters Q and U. To separate out the variability in source flux S_ν, we also write a normalized polarization ${P}\mathrm{\equiv }\frac{{P}}{\mathit{I}}\mathrm{=}\frac{{\sum }_{\mathit{i}}{p}\mathit{i}{\mathit{I}}_{\mathit{i}}}{{\sum }_{\mathit{i}}{\mathit{I}}_{\mathit{i}}}$(14)with the Stokes parameter I being identical to S_ν; the index i indicates the contribution from the ith emission component. Here p = (Q/I,U/I) ≡ (q,u), which relates to our observed parameters like ${\mathit{m}}_{\mathrm{L}}^{\mathrm{2}}\mathrm{=}{\mathit{q}}^{\mathrm{2}}\mathrm{+}{\mathit{u}}^{\mathrm{2}}$ and χ = 0.5tan^-1(u/q). In general, one may assume that all parameters involved can vary independently, leaving the polarization parameters m_L, χ and rescaled fluxes ${\mathit{S}}_{\mathit{\nu }}^{\mathrm{\ast }}$ uncorrelated; indeed, this is what our data suggest for most of our targets. Nevertheless, we are able to probe at least two special cases:

• 1.

  We receive radiation from a single predominant emission component with an arbitrary but approximately fixed degree and angle of polarization. Any change in source flux should be accompanied by a proportional change in polarized flux. In this case we would observe a strong correlation between flux density and polarized flux. The flux would not correlate with degree or angle of polarization.

• 2.

  We receive radiation from two (or more) emission components with each component having an arbitrary but approximately constant q_i and u_i. The observed polarization signal is a superposition of the individual components. Any increase (or decrease) in the luminosity of one component increases (decreases) its contribution to the average signal observed. In this scenario, we would observe a tight correlation between flux and degree of polarization as well as between flux and polarization angle⁶.

Remarkably, our data indicate the occurrence of those special cases in three objects. In 0415+379, we observe a signature as predicted by scenario 2. While the source flux increases by a factor  ≈3, the degree of polarization drops by a factor  ≈ 4, leading to a strong anticorrelation between the two parameters. Simple arithmetics shows that, when assuming that the degrees of polarization are approximately fixed, we observe increased emission from (at least) one weakly or non-polarized component added to the emission from (at least) one component with a higher level of linear polarization. Our interpretation is supported by recent time-resolved radio interferometric observations. Grossberger et al. (2011) were able to associate the flux peak observed in 2007 with the ejection of a new jet component that adds its emission to various other, separate jet components recognized in radio interferometric maps obtained with the Very Long Baseline Array (VLBA).

For 0507+179, we observe a signature in agreement with scenario 1: while the source flux increases by a factor  ≈3, the polarized flux increases by a factor  ≈ 3, too. There is no significant correlation between flux on the one hand and degree or angle of polarization on the other hand. Radio interferometric VLBA maps obtained by Fey & Charlot (1997) at 8.6 GHz show a dominating core and a fainter jet component about 3 mas west of the core. However, as this map was obtained in 1995, any comparison between the data of Fey & Charlot (1997) and our results has to be taken with care. We note that our “one-component scenario” does not contradict the discussion in Sect. 5.1. The scenarios are reconciled when assuming large-scale regions with coordinated luminosity variations – like shock fronts – that have substructure defined by, e.g., characteristic ordering scales of magnetic fields.

In 1418+546, we observe a more complex signature. While the source flux increases by a factor  ≈ 3, the polarized flux inceases by a factor  ≈ 8, resulting in a significant correlation between the two parameters in agreement with scenario 1. Additionally, there is also a significant, though more scattered, positive correlation between flux and degree of polarization, with m_L rising by a factor  ≈ 4. There is no significant correlation between flux and polarization angle. In essence, we seem to observe a single, dominating emission component that tends toward higher polarization levels at higher flux densities. We note the agreement of the behavior of 1418+546 with the signatures expected from recent model calculations based on “shock in jet” scenarios (Hughes et al. 2011). Regarding the lightcurve, our observations indeed appear to cover two outbursts occurring within about two years. Our observations find (a) a correlated variation of flux and polarization, (b) changes of the polarization angle by tens of degrees, and (c) no correlation between flux and polarization angle. We see the best agreement of our data with the scenario of an oblique shock front observed at an angle  > 60° to the jet axis (see Fig. 7f of Hughes et al. 2011). We also note that the calculations of Hughes et al. (2011) assume a spectral index α = 0.25; this may be compared to the observational result α = 0.2 ± 0.1 (see Table 1 and Paper I). Overall, our observations appear to provide new support for the “oblique shock” scenario.

5.4. Rotation measures

We analyzed the rotation measures of our target AGN according to the procedure outlined in Sect. 3.3. We are able to derive statistically significant rotation measures for 1418+546 and 1637+574 (see Fig. 7 and Sects. 4.5, 4.6). We note that for these two sources we included all data available into our calculation because “time windowing” did not increase the significance of the result or reduce the scatter caused by temporal variability. Our calculations assume that any Faraday rotation detected is caused by matter in or within the vicinity of the emitters. A priori, there are two additional sources of Faraday rotation that might distort our analysis: the terrestrial ionosphere and Galactic interstellar matter. For the frequency range of our observations, ν > 80 GHz, the ionosphere is able to rotate the polarization angle of electromagnetic radiation by less than 0.01° (Thompson et al. 2001), meaning its influence may be neglected. Galactic interstellar matter introduces rotation measures |RM| ≲ 200 at radio frequencies (Taylor et al. 2009). This number is about one order of magnitude smaller than the statistical errors of our rotation measure values, which renders the influence of interstellar matter irrelevant.

We find RM = (−39 ± 1_stat ± 13_sys) × 10³ rad m^-2 and RM = (42 ± 1_stat ± 11_sys) × 10⁴ rad m^-2 for 1418+546 and 1637+574, respectively. For 0415+379, 0507+179, and 0954+658 we derive 3σ upper limits on |RM| of 1.1 × 10⁴ rad m^-2, 1.4 × 10⁴ rad m^-2, and 1.7 × 10⁴ rad m^-2, respectively. Evidently, our observations are subject to strong selection effects. As already noted by Agudo et al. (2010), Eq. (7) implies that, at mm-wavelengths, changes in polarization angle as small as  ≈5° require |RM| ≈ 7 × 10³ rad m^-2. In addition, our polarization angle data show substantial scatter caused by rapid temporal variability. Therefore rotation measures smaller than a few 10⁴ rad m^-2 probably remain unnoticed simply because they do not imprint a statistically significant signal into a $\mathit{\chi }\mathrm{-}{\mathit{\lambda }}_{\mathrm{0}}^{\mathrm{2}}$ diagram. This is illustrated by the case of 1418+546, where an RM ≈  − 3.9 × 10⁴ rad m^-2 signal still comes with a statistical false-alarm probability of  ≈ 0.4%, barely corresponding to a 3σ significance threshold. Rotation measure values derived from cm/radio observations are usually located in ranges from a few 10 rad m^-2 to a few 1000 rad m^-2 at most (e.g., O’Sullivan & Gabuzda 2009; Taylor et al. 2009). At mm/radio frequencies, AGN polarization and Faraday rotations are poorly probed, with the most comprehensive study to date probably being the work by Jorstad et al. (2007). Those studies observe values up to |RM| ≈ 3.4 × 10⁴ rad m^-2 (for CTA 102; Jorstad et al. 2007). Accordingly, we report in this study the highest rotation measures observed to date in AGN.

From Eq. (8) we may estimate the product of magnetic field strength and electron density averaged along the line of sight  ⟨ B_||n_e ⟩ . For emission region sizes on the order of one parsec (compare, e.g., O’Sullivan & Gabuzda 2009, for the case of 1418+546) and RM of order 10⁵ rad m^-2, we find  ⟨ B_||n_e ⟩  ~ 0.1 G cm^-3. For realistic electron densities n_e ~ 10³ cm^-3 (e.g., Koski 1978; Bennert et al. 2009) this corresponds to average magnetic field strengths B_|| ~ 10^-4 G.

Observing increasing rotation measures when going from cm/radio to mm/radio frequencies may be expected from Eq. (8) for optically thick sources. Higher frequencies probe regions closer to the central engine with higher electron densities and stronger magnetic fields. For realistic emission region geometries, one may assume n_e ∝ l ^− a and B_|| ∝ l^-1, with n_e being the electron density, B_|| being the magnetic field parallel to the line of sight, l being the distance from the core where the observed emission predominantly originates, and a being a positive real number  (Jorstad et al. 2007). For outflows with spherical or conical geometries, a = 2. From Eq. (8) one derives immediately |RM| ∝ l ^− a. For optically thick sources, distance l and frequency ν are related like l ∝ ν^-1, leading to a “core shift” effect. For the specific case of 0507+179, Sokolovsky et al. (2011) explicitly observed a core shift for the frequency range 1.4–15 GHz. Combining the various relations leads to $\mathrm{\left|}\mathrm{RM}\mathrm{\right|}\mathrm{\propto }{\mathit{\nu }}^{\mathit{a}}\mathit{,}$(15)when inserting values for a in agreement with observations, i.e. a ≈ 1−3 (O’Sullivan & Gabuzda 2009), one may indeed expect |RM| of order 10⁵ rad m^-2. This is additionally supported by observations of Sagittarius A*, the radio source commonly equated with the supermassive black hole at the center of the Milky Way (e.g., Genzel et al. 2010, and references therein). Here rotation measures up to |RM| ≈ 5  ×  10⁵ rad m^-2 at frequencies around 340 GHz have been reported (Marrone et al. 2006; Macquart et al. 2006).

For 1418+546 it is possible to compare our result with RM observations obtained at cm/radio frequencies and from these to derive a. O’Sullivan & Gabuzda (2009) report RM =  −501  ±  48 rad m^-2 at observed frequencies around 15 GHz. This can be related to our result, RM = (−39 ± 1_stat ± 13_sys) × 10³ rad m^-2 at ν ≈ 150 GHz. From Eq. (15) we obtain a = 1.9 with a formal statistical error of only  ±  0.04. From the systematic error in |RM| and the uncertainties in defining the two reference frequencies used in our calculation we expect an additional systematic error of approximately 0.3. Eventually, we obtain a = 1.9 ± 0.3 with the error being predominantly of systematic nature. This may be compared to the theoretical value a = 2 for spherical or conical outflows. Our rotation measure analysis therefore indicates that for the specific case of 1418+546 the assumption of a simple spherical or conical outflow geometry is sufficient.

6. Summary and conclusions

We have analyzed the linear polarization of six AGN – 0415+379 (3C 111), 0507+179, 0528+134 (OG+134), 0954+658, 1418+546 (OQ+530), and 1637+574 (OS+562) – in the observatory frame frequency range 80–253 GHz, corresponding to a range of rest frame frequencies 88–705 GHz. Our data were obtained in the years 2007–2011 and provide information resolved in time and frequency for all targets. Our sample includes two optically thin, outflow-dominated and four optically thick, core-dominated emitters. This study arrives at the following principal conclusions:

• 1.

  In agreement with the results of Paper I, the observed degrees of polarization m_L are lower than the levels expected from the theory of synchrotron emission by up to one order of magnitude. This indicates that the polarization signal is effectively averaged out. Assuming radiation from N emission regions with random orientations and geometries, we expect ${\mathit{m}}_{\mathrm{L}}\mathrm{\to }{\mathit{m}}_{\mathrm{L}}^{\mathrm{\prime }}\mathrm{=}{\mathit{m}}_{\mathrm{L}}\mathit{/}\sqrt{\mathit{N}}$. Our results point to values from N ≈ 3 (for two optically thick emitters) to N ≈ 1000 (0415+379, an optically thin emitter).

• 2.

  Degrees of polarization as well as fluxes vary with relative rates of a few per cent per day. We find fluctuation rate ratios ρ (Eq. (4)) on the order of unity though typically slightly lower than one. This indicates that the size scales relevant for the polarized emission are compatible with, but probably not smaller than, the size scales relevant for the overall emission.

• 3.

  The degrees of polarization in our sample fluctuate with normalized standard deviations σ_m/ ⟨ m_L ⟩  ≈ 0.33−0.65. We identify this variability approximately with source-averaged fluctuations expected from shock compression of magnetized plasmas. Our calculations point toward fairly strong shocks and/or complex, variable shock geometries, with compression factors k ≲ 0.9 and/or changes in viewing angles  ≳ 10°.

• 4.

  A search for correlations between source fluxes and polarization parameters identifies two special cases. For 0415+379, we observe a strong anticorrelation between flux and degree of polarization. This indicates the presence of (at least) two distinct emission regions with different levels of polarization. For 0507+179 and 1418+546, we find a positive correlation between flux and polarized flux. Here the data point toward emission from a single, predominant component.

• 5.

  For 1418+546 we find (a) a correlated variation of flux and polarization, (b) changes of the polarization angle by tens of degrees, and (c) no correlation between flux and polarization angle. Comparison with the “oblique shock” models by Hughes et al. (2011) indicates good agreement with the presence of an oblique shock front observed at an angle  > 60° to the jet axis.

• 6.

  We derive rotation measures of RM = (−39 ± 1_stat ± 13_sys) × 10³ rad m^-2 and RM = (42 ± 1_stat ± 11_sys) × 10⁴ rad m^-2 for 1418+546 and 1637+574, respectively. These are the highest values reported to date for AGN except for Sagittarius A*. Our results point to magnetic field strengths on the order of  ~10^-4 G. For 0415+379, 0507+179, and 0954+658 we are able to derive upper limits |RM| < 1.7 × 10⁴ rad m^-2.

• 7.

  Comparing our rotation measure values with those obtained by other studies at lower frequencies, we are able to constrain the emission region geometry of 1418+546. Exploiting the relation |RM| ∝ ν^a we find a = 1.9 ± 0.3, in good agreement with a = 2 as expected for a spherical or conical outflow.

• 8.

  We do not see a relation between the observed polarization properties of our sources and their physical parameters such as black hole masses or kpc-morphology, as summarized in Table 1.

This paper is the second of a set of two articles. Earth rotation polarimetry of AGN is a by-product of regular PdBI operations; accordingly, the database is going to grow steadily in the future. We therefore expect that a new study – analogous to the one outlined in this article as well as in Paper I – carried out in a few years from now promises new insights already because of substantially improved statistics. In addition, the PdBI crew currently evaluates the implementation of an observing mode that permits observations of all four Stokes parameters. Evidently, such an observing mode would permit additional studies if implemented eventually.

Our work demonstrates the power of mm/radio polarization studies of AGN based on data resolved in time and frequency. Given the strong temporal variability of AGN polarization, it becomes clear that more accurate measurements of differential parameters such as rotation measures require simultaneous multi-frequency observations. New facilities such as the Korean VLBI Network (KVN; e.g., Kim et al. 2010), which permits single-polarization observations simultaneously at four frequencies – namely 22, 43, 86, and 129 GHz – or dual-polarization

observations simultaneously at two of the aforementioned frequencies, will be important tools for new discoveries.

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Acknowledgments

We are grateful to the entire IRAM and PdBI staff, whose hard work over many years made this study possible. Our work has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We have also made use of data from the MOJAVE database that is maintained by the MOJAVE team (Lister et al. 2009). For data processing and analysis we applied the software package GILDAS, developed and maintained by the GILDAS team (http://www.iram.fr/IRAMFR/GILDAS/), as well as the software package DPUSER (http://www.mpe.mpg.de/~ott/dpuser/index.html), developed and maintained by Thomas Ott at MPE Garching. Last but not least, we are grateful to the referee, E. Ros, whose review improved the quality of our paper.

References

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