MULTI-WAVELENGTH POLARIMETRY AND SPECTRAL STUDY OF THE M87 JET DURING 2002–2008^*

M87 was under intensive observations during its flare. During 2002–2008, M87 was observed by the VLA every five to six months at 8, 15, 22, and 43 GHz in A and B configurations. We extracted the VLA data from the NRAO⁶ data archive.

Data reduction was carried out using standard reduction techniques in the Astronomical Image Processing System (AIPS). Data were first calibrated in AIPS task CALIB, using 3C 286 as primary flux calibrator, 3C 138 as polarization position angle (PA) calibrator, and 1224 + 035 as instrument polarization calibrator, for phase only and amplitude & phase corrections. Subsequent runs of CALIB, using M87 as a self-calibrator, were performed to minimize the calibration errors. Antenna "D" terms were corrected using tasks PCAL and CLCOR with 1224 + 035 as polarization calibrator. Multi-source data sets were then separated into single source data sets using task SPLIT. After another run of self-calibration on single source data sets, Stokes I, Q, and U images were obtained using task IMAGR. The final images, thus obtained, have a beam size of ≈009 at 22 GHz and ≈014 at 15 GHz, in the B array configuration. The resolution of these images is comparable to that of optical images.

Next we used the procedure DOFARS to correct the polarization values using the RM, as described in Brentjens & de Bruyn (2005). This procedure reads the Q and U polarization cubes as inputs to run the task FARS, which evaluates the brightness distribution as a function of Faraday rotation using the measured brightness and given set of (wavelength)². FARS outputs the RM cubes, which were used as input for the task AFARS. This task produces a map of positions of maximum RM and flux densities. Next we used the output images of AFARS to correct and find the error maps of Stokes Q and U images, using task RFARS. The magnetic field position angle (MFPA) and FP maps were then obtained using RM-corrected Stokes Q and U images in task COMB.

Task PCNTR was used to plot the total flux contour maps of MFPA and FP. The radio maps at all epochs were convolved using task CONVL to the same resolution as the 15 GHz image. The errors in PA and FP were found by propagating errors in the Stokes Q and U images. Our K band (22 GHz) data have the highest signal-to-noise ratio (S/N) of all the VLA data and the U band (15 GHz) data have the most coverage over the time domain. In this paper, we use only these data for the comparison of the jet's radio and optical polarization structure. Optical images at all wavelengths were smoothed to the same resolution as 15 GHz radio images and were resampled at 0025/pixel to compare both bands at the same scale.

Before the onset of the HST-1 flare in 2001–2002, M87 was a regular target of HST since 1994, with observations occurring roughly every year. With more intense monitoring between 2002 December and 2007 November, the M87 jet was observed at four to five week intervals (Harris et al. 2009; Madrid 2009). All the observed epochs are listed in Table 1. The polarimetry was done in the F606W and F330W bands. The F606W polarimetry observations are used in this paper. The numbers in the bracket in front of the dates are the original sequence number of the observations used in Perlman et al. (2011).

The High-Resolution Channel (HRC) of the Advanced Camera for Surveys (ACS) was used for polarimetry observations of 17 of these epochs in the F606W band. The ACS HRC is a single-chip CCD camera, with a plate scale of 0028 × 0025 pixel⁻¹, corresponding to a field of view of about 28'' × 25'' and yielding a diffraction limited resolution of ≈006 for the F606W observations. These observations were reduced using methods of re-calibration following the ACS and WFPC2 Instrument Handbooks.

To prepare the observations for photometry and polarimetry, all the epochs of ACS/HRC F606W were combined to create a composite ("master") image. This improved the S/N of the background galaxy. The image was further modeled for galaxy emission using ellipse STSDAS, which was subtracted and split into three images, one corresponding to each polarizer on HST. For more details of individual HST observations and data reduction procedures, the reader may refer to Perlman et al. (2011).

To evaluate the relevance of the flux and polarization images and their measurements in terms of the different dynamic ranges of radio and optical data, we performed the error analysis on our radio images. The errors in flux, polarization, and PAs are the propagated statistical errors and systematic errors for the VLA instrument. The statistical error in flux (Stokes I, Q, and U) is calculated using the off-source rms noise, the number of pixels (N) in a box, and beam area. Note that the radio images are in units of flux density (Jy) per beam:

$$\begin{eqnarray}&&\mathrm{error}=\mathrm{rms}\ast \sqrt{\displaystyle \frac{N}{\mathrm{beam}\ \mathrm{area}}}.\end{eqnarray} \tag{ 1 }$$

As can be seen, the error is smaller for the larger regions as the error is reduced as 1/$\sqrt{N}$.

Polarized flux and PA images were made in AIPS using the following equations:

$$\begin{eqnarray}&&P=\displaystyle \frac{\sqrt{{Q}^{2}+{U}^{2}}}{I},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\mathrm{PA}=\displaystyle \frac{1}{2}{\tan }^{-1}\left(\displaystyle \frac{U}{Q}\right).\end{eqnarray} \tag{ 3 }$$

To calculate the errors, we used the standard error propagation formulae. To find the error in polarized flux, we assumed that the errors in Stokes Q and U add up in quadrature as follows:

$$\begin{eqnarray}&&{\sigma }_{P}=\sqrt{{\sigma }_{Q}^{2}+{\sigma }_{U}^{2}}.\end{eqnarray} \tag{ 4 }$$

To find the error in PA, we used the following formula:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{PA}}=\displaystyle \frac{1}{2}\displaystyle \frac{\sqrt{({Q}^{2}\ast {\sigma }_{U}^{2})+({U}^{2}\ast {\sigma }_{Q}^{2})}}{{Q}^{2}+{U}^{2}}.\end{eqnarray} \tag{ 5 }$$

Tables 2 and 3 show the flux and polarimetry information for all the identified knots in M87's jet in radio and optical bands, respectively. The table lists the average flux, average polarization, and average PA of the magnetic field vectors in all the knots. The regions in Stokes Q and U used to obtain these values were obtained by putting boxes around each knot. We list the (X, Y) coordinates of these boxes in the respective tables.

Table 2.  Radio Flux and Polarization Data

Region	Xa	Ya	Flux Density (mJy)	Polarization (%)	Position Angle (deg)
Nucleus	769–781	429–441	4358.4 ± 0.9	1.3 ± 0.2	3 ± 1
HST-1	803–813	445–455	89.8 ± 0.8	8.1 ± 0.2	−7 ± 4
D-E	869–897	469–479	62.4 ± 1.2	23.6 ± 0.2	20 ± 3
D-M	905–923	473–489	28.4 ± 1.2	32.1 ± 0.2	18 ± 5
D-W	919–931	485–497	19.9 ± 0.9	19.5 ± 0.2	13 ± 9
E	973–1033	499–535	78.0 ± 3.3	11.4 ± 0.6	28 ± 13
F	1075–1133	537–585	137.8 ± 3.7	17.9 ± 0.7	17 ± 6
I	1175–1213	571–607	99.3 ± 2.6	8.6 ± 0.5	35 ± 10
A-shock	1223–1251	585–629	600.1 ± 2.5	22.7 ± 0.5	41 ± 1
A	1221–1295	581–643	1291.4 ± 4.7	10.3 ± 0.9	−36 ± 1
B1	1313–1341	617–657	337.8 ± 2.4	26.4 ± 0.4	32 ± 1
B2	1367–1397	625–679	149.5 ± 2.8	31.5 ± 0.5	−31 ± 2
C1	1431–1471	687–729	320.1 ± 2.9	24.4 ± 0.5	30 ± 1
C2	1479–1505	681–741	50.4 ± 2.8	56.7 ± 0.5	33 ± 3
G1	1475–1531	753–771	64.0 ± 2.3	28.5 ± 0.4	−11 ± 5
G2	1527–1559	725–759	105.4 ± 2.3	37.1 ± 0.4	−39 ± 2

Note.

^aBox coordinates (X, Y) are in pixels. The jet is ∼205 north of the x-axis, with a scale of 0025 pixel⁻¹.

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Table 3.  Optical Flux and Polarization Data

Region	Xa	Ya	Flux Density (μJy)	Polarization (%)	Position Angle (deg)
Nucleus	769–781	429–441	630.9 ± 0.1	3.1 ± 1.1	−17 ± 3
HST-1	803–813	445–455	562.7 ± 0.0	27.1 ± 1.0	−15 ± 3
D-E	869–897	469–479	42.0 ± 0.1	5.1 ± 6.7	−33 ± 3
D-M	905–923	473–489	14.3 ± 0.1	20.5 ± 4.0	−17 ± 3
D-W	919–931	485–497	11.3 ± 0.1	26.5 ± 3.3	−31 ± 3
E	973–1033	499–535	43.2 ± 0.2	10.8 ± 6.4	−27 ± 3
F	1075–1133	537–585	94.3 ± 0.2	12.8 ± 5.6	−38 ± 3
I	1175–1213	571–607	36.9 ± 0.2	22.0 ± 3.0	−26 ± 3
A-shock	1223–1251	585–629	33.4 ± 0.2	182.0 ± 2.1	−3 ± 3
A	1221–1295	581–643	839.6 ± 0.3	20.0 ± 1.0	19 ± 3
B1	1313–1341	617–657	203.2 ± 0.2	15.9 ± 1.2	−14 ± 3
B2	1367–1397	625–679	149.0 ± 0.2	22.6 ± 1.5	10 ± 3
C1	1431–1471	687–729	215.6 ± 0.2	8.0 ± 2.7	−9 ± 3
C2	1479–1505	681–741	52.0 ± 0.2	15.6 ± 4.1	−33 ± 3
G1	1475–1531	753–771	29.8 ± 0.1	21.9 ± 3.2	18 ± 3
G2	1527–1559	725–759	15.8 ± 0.2	26.9 ± 9.5	7 ± 3

Note.

^aBox coordinates (X, Y) are in pixels. The jet is ∼205 north of the +X-axis, with a scale of 0025 pixel⁻¹.

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Optical data were treated differently using the methods of debiasing. After the Stokes I, Q, and U images were obtained, we accounted for the well-known Rician bias in P (Serkowski 1962) using a Python code adapted from the STECF IRAF package (Hook et al. 2000). This code debiases the P image, following Wardle & Kronberg (1974), and calculates the error in the polarization PA accounting for the non-Gaussian nature of the distribution (see also Naghizadeh-Khouei & Clarke 1993). In the calculation, pixels with S/N <0.1 are excluded outright. Also, since the debiasing is done with the "most probable value" estimator, pixels with values of P that were negative or were above the Stokes I value (i.e., P > 100%) were blanked. Interested readers may refer to Cara et al. (2013) and Perlman et al. (2006, 2011) for further details on the application of this method.

The use of ACS/HRC polarizers with different orientation angles and the PA_V3 angle, which is the angle between the north and V3 axis of the telescope, rendered it necessary to correct the PA of the final image to obtain the real magnetic field PAs. The PA equation (Equation (3)) then gets modified to

$$\begin{eqnarray}&&\mathrm{PA}=\displaystyle \frac{1}{2}{\tan }^{-1}\left(\displaystyle \frac{U}{Q}\right)\ +\ \mathrm{PA}\_V3\ +\ \chi ,\end{eqnarray} \tag{ 6 }$$

where χ is the angle of the instrument in the focal plane. This converts the Stokes Q and U from the instrumental frame to the sky frame. A script used to do these corrections will also produce the polarization (P), FP, and PA images for each epoch and combine them all in one "master" image. We use this combined image in our analysis. To find the significant levels of polarized flux images, we make the vectors plots using the AIPS task PCNTR and setting the parameter PCUT = 3σ, where σ is the background noise in the Stokes Q and U images.

We show the polarization vector plots using the FP images in the left-hand side panels and those using polarized flux images on the right-hand side panels of Figures 3 through 7, except in Figure 5, where we do not show the FP plot because of the loss of detail due to poor S/N. The polarized flux images were obtained using the total flux and polarized flux. To display the magnetic field PA vectors, we used the PA image obtained from the Stokes Q and U images. The orientation of the vectors represent the direction of the local magnetic field, whereas their lengths represent the degree of polarization. The images show the MFPA vectors that are above the 3σ level in the polarized flux. The regions where we do not see any vectors within these figures, we believe, are the regions of lower polarization (P < 3σ) or depolarization (P ≈ 0).

We label the sub-components within the jet by visually inspecting the FP images on the left. The radio as well as optical images show a highly resolved flux and polarization structure of the jet, which was not seen in the old VLA observations presented in P99. By comparing our FP and polarized flux images, we can identify new sub-structures based on the detected total flux and the regions of significant polarized flux and hence claim that these new sub-structures are real, especially near the nucleus and HST-1, although we cannot comment if these are newly emerged or are just a result of better resolution.

The VLA and HST images show a wealth of information about the polarization and magnetic field structure of the M87 jet. The total flux and polarization images have many common general characteristics that are observed in both bands. Figure 1 shows false-color flux and FP images in the radio (top two panels) and optical (bottom two panels). The jet shows some striking similarities in terms of the total flux structure in the radio and the optical. In general, the radio jet shows a broader jet with more diffuse emission from near the surface as well as from the inter-knot regions, as compared to the optical trend previously discussed in Sparks et al. (1996).

In Figure 2 we have plotted the flux and polarization profiles of the radio and optical (top and middle two panels), which quantifies the above differences in the locations of flux and polarization maxima. The flux profiles show slight differences in the locations of flux maxima in both bands. These differences are more prominent in the outer jet, i.e., 10''–20'' from the nucleus. The optical flux maxima of the knots are in several cases observed to be slightly downstream as compared to the radio by ∼05–1''. We also see similar differences in the locations of the polarization maxima (or minima) in both bands. There are several places within the jet where the maxima of the radio polarization fall in the same place as the optical polarization minima, e.g., in D-East, E, and F in the inner jet and in I, upstream and downstream ends of A, B2, and C2 in the outer jet. Close inspection of the bottom two panels shows that the flux and polarization do not necessarily follow each other, however; the locations of polarization maxima are shifted downstream by about ∼025–05, especially in the radio band, although the difference is not that significant in the case of the optical.

Both bands show a much more resolved FP structure as compared to similar previous studies. The FP of the radio is much more uniform as compared to the optical. As we will discuss later in this paper, we can clearly see this trend in the polarized flux images of individual knots as well. The optical images show regions of very high polarization and very low polarizations or depolarization close to each other, especially in the outer jet knots, A, B and C, whereas this difference in degree of polarization is much less prominent in the radio.

The FP seems to be significantly higher in regions of possible shocks in the jet, for example, knot HST-1, A-shock, and the downstream end of knot C. The diffuse emission in all the inter-knot regions seems to have higher polarization as well. In the radio band, the flux and polarization maximum often do not appear to be at the same location. In fact, the polarization maxima are located ≈025–05 of the flux maxima in some individual knots, e.g., in knot D-East and A-shock, the polarization maxima is 025 downstream of the total flux maxima. We have pointed out the flux and polarization maxima with red arrows in the radio maps (top two panels) in Figure 1 and these  can be very clearly seen in the bottom two panels of Figure 2, where we see the polarization maxima clearly shifted downstream of the flux maxima in both bands. This trend is much clearer in case of optical knots F, A, and C, shown by yellow arrows in the bottom two panels and is also discussed in their respective subsections.

One other important but not so obvious trend in Figure 1 is the apparent helical structure of the jet. This is seen in general in both radio and optical images. We see this much more clearly in the diagrams of MFPA vectors of individual knots (Figures 3 through 7). We discuss individual knots in detail in terms of the common trends discussed above as well as their flux and polarization structure next.

The inner jet consists of the nucleus and knots HST-1 and D along with the faint inter-knot emission. The inner jet extends out to about $4^{\prime\prime}$ from the nucleus. Figure 3, top and bottom, shows the stacked image of the nucleus and knot HST-1 in the radio and optical bands. On the left panel, we show the FP images and on the right panel are the polarized flux images.

Similarly, the total flux and polarization images of knot D are shown in Figure 4, top and bottom. We discuss the radio and optical total flux and polarization structure of the inner jet in the following section, starting with the nucleus and HST-1.

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3.2.1. Nucleus

3.2.2. Knot HST-1

3.2.3. Knot D

Knots E, F, and I lie between ∼50 and 110 from the nucleus of M87. Figure 5 show the polarized flux contour plots of this region. The knots show complex flux and polarization features. The S/N is not sufficient to resolve the individual sub-components as we see in the case of the inner jet, however. Most of the regions of these knots show very low polarization (well below 3σ) or regions of depolarization as can be seen from the figure, hence we do not show the FP plots here. As a result we cannot comment on their polarization structure; however, we describe their flux structures below.

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3.3.1. Knot E

3.3.2. Knot F

3.3.3. Knot I

The outer jet comprises knots A, B, C, and G, beyond which the jet disrupts and feeds matter into the eastern inner radio lobe. In Figures 6 and 7, the total flux is shown on the left and the polarized flux is shown on the right (radio on the top and optical at the bottom), both overlaid by the MFPA vectors.

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The overall morphology of the outer knots, A, B, C, and G, is quite different from the rest of the jet. The difference in the thickness of the radio and optical jet is clearly evident from Figures 6 and 7. The radio jet is much thicker and disorganized, while the optical jet shows much more defined structure, near the bend as well as in the individual knots. This structure may suggest the presence of a layered surface emitting different energy electrons from different physical regions as modeled in P99. This may be caused by the internal magnetic field structure changing in the outer jet. P99 report a similar difference in the radio and optical morphology of the inner and outer jet knots. Owen et al. (1989) and Bicknell & Begelman (1996) suggested this could be due to the Kelvin–Helmholtz instabilities causing the shocks within the jet and disrupting the magnetic field structure. We discuss the features in each of these knots in detail below.

3.4.1. Knot A

3.4.2. Knot B

3.4.3. Knot C

3.4.4. Knot G

The total radio flux variations of the nucleus and HST-1 along with the large flare in HST-1 around 2005 are similar to the optical variability of Perlman et al. (2011; their Figures 1 and 2) and the X-ray variability of Harris et al. (2009; their Figure 9). The optical–UV data also show two small flares in the nucleus just before and after the flare in HST-1 (Perlman et al. 2011). The majority of optical observations were taken during 2004–2006 when knot HST-1 was already bright and hence the full dynamic range of the variability of HST-1 was not observed in the optical. On the other hand, radio data were taken during 2002–2008 and span over the time domain of HST-1's flare.

Both HST-1 and the nucleus show a much similar behavior in polarization and MFPA variability in radio to what can be seen in the middle and lower panels of Figure 8, respectively. Polarization behavior of the nucleus is very consistent with values between 1% and 4%, as compared to the larger variations observed in the optical which were 1%–13%. During this time, the optical MFPA changed by as much as 90° whereas in the radio it remained more stable between 80° and 90°. HST-1 is much more highly polarized than the nucleus in both optical and radio. In the optical, its polarization ranged between 20% and 45%, with little variability, compared to close to 20% in the radio with small variability, and was strongly correlated with flux with nearly constant EVPA $\approx \,-62^\circ$.

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Similar to the nucleus, the MFPA of HST-1 stayed close to 80° on average. This value differs from the optical MFPA (≈30°) (Perlman et al. 2011), suggesting these particles may represent very different populations in space and may indicate optical emission during the flare being much more dominated by the flaring component. Polarization behavior of HST-1 at the epoch 2007 June is significantly higher than at the other epochs. This epoch is very close to the second flare in the nucleus as well as to a smaller second flare in HST-1. The immediate next epoch observed after 2007 June was seven months later, in 2008 January, which has FP values consistent with the rest of the epochs.

In Figure 9 we have plotted FP versus flux at 22 GHz on the left for the nucleus and on the right for HST-1. Neither the nucleus nor HST-1 shows any evidence of correlation between polarization and flux. During the earlier epochs (points 1 through 5), the polarization of HST-1 increases linearly with the increase in its flux; however, in the last two epochs (points 6 and 7), the polarization does not seem to have any correlation with the flux value. On the other hand, the nucleus shows a very different behavior during the same period. The nuclear polarization does not have any correlation with its flux whatsoever and changed randomly. Perlman et al. (2011) explain the correlation between flux and polarization in terms of a looping. They indicate this looping as hard lags (clockwise looping) and soft lags (counterclockwise looping) in their Figures 3 (for the nucleus) and 4 (for HST-1). They explain the correlation with the help of the connection between acceleration and cooling timescales controlling the spectral evolutions of radiating particles.

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We plot the radio–optical spectral indices using flux values at 22 GHz and F606W (≈4.95 × 10⁵ GHz). We compare them to the optical SED of Perlman et al. (2011), Figures 3 and 4. Figure 10 (top) shows the evolution of spectral index ${\alpha }_{\mathrm{ro}}$ versus total radio flux for the nucleus (on the left) and HST-1 (on the right). We assumed the relation between the spectral index α and flux to be ${S}_{\nu }$ ∝ ${\nu }^{\alpha }$. As can be seen, there is no direct evidence of correlation between spectral index and flux. For the nucleus, the spectral index evolves independent of the total flux. At the beginning of the observations, up to epoch 5, it does not show any correlation with the flux whatsoever, and oscillates between −0.85 and −0.75; however, from epoch 5 onward, it shows a monotonic increase from a low of ∼−0.9 to a high of ∼−0.75.

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In the case of knot HST-1, ${\alpha }_{\mathrm{ro}}$ shows a similar behavior. The spectral index stayed close to −0.45 before a sharp increase in it around the 2005 flare when the spectral index increased to ∼−0.35. At epoch 5, we see a sudden drop in the spectral index; however, beyond this, we see a monotonic increase in its value until the last epoch 8. During all this time, the spectral index of HST-1 varied between ∼−0.5 and −0.35.

The bottom two plots of Figure 10 show the evolution of ${\alpha }_{\mathrm{ro}}$ versus the optical flux. As can be seen, there is no clear correlation between the spectral index and the optical flux either. Perlman et al. (2011) observed a "looping" behavior during the maximum of HST-1's flaring in 2005 in their plot of ${\alpha }_{\mathrm{UV} \mbox{-} {\rm{O}}}$ versus the optical flux plot (their Figure 4); however, we do not see any such trend in the ${\alpha }_{\mathrm{ro}}$ plot of HST-1 or the nucleus. This may point toward the possibility that the radio and optical electrons may originate in different regions within the jet.

Figure 11 summarizes the spectral index variability of the nucleus and HST-1 in a single plot over the period of observations. The spectral index of HST-1 shows larger variations as compared to the nucleus, especially close to its flaring in 2005. The ${\alpha }_{\mathrm{ro}}$ of HST-1 varied by about 30% between −0.38 in mid-2005 to −0.51 in early 2006, whereas that of the nucleus shows variations close to 15% in the same time.

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Several regions of the M87 jet have been labeled as shocks by a wide variety of authors, including, for example, Owen et al. (1989), Bicknell & Begelman (1996), Perlman et al. (1999), Cawthorne (2006), and Nalewajko & Sikora (2012). While most previous works have considered perpendicular shock features, in which the magnetic field is strongly compressed perpendicular to the jet flow, such features can also be oblique or conical (e.g., Cawthorne & Cobb 1990; Bicknell & Begelman 1996). In addition, if the overall jet magnetic field structure has a helical element, the effect on the polarization morphology by any of these features can be considerably more complicated. The high resolution of these images allows us to have a nuanced discussion of these issues.

The two strongest shock features are knots HST-1 and A. Knot HST-1 is believed to be the location of a recollimation shock, as detailed in a number of previous papers, including Cawthorne (2006), Bromberg & Levinson (2009), Nakamura et al. (2010), and Asada & Nakamura (2012). Recollimation shocks are believed to occur in a wide variety of jet systems as a result of interactions between the jet and the surrounding interstellar medium (ISM) (Mizuno et al. 2015) near the Bondi radius, where the local ISM becomes dominated by the influence of the nuclear supermassive black hole. This is observed not only in AGNs, but also in X-ray binaries and protostellar objects. As a result, M87 is not the only AGN where recent work has noted the presence of a recollimation shock. Other examples include 3C 120 (Agudo et al. 2012), BL Lac (Cohen et al. 2014), CTA 102 (Fromm 2015), and 1803+784 (Cawthorne et al. 2013). However, of these objects, M87 lies by far the closest to us, thus giving us a unique opportunity to study these structures, believed to be universal, in detail.

Models for generating relativistic jets from magnetized accretion disks almost unanimously consider the magnetic field geometry of the nuclear jet to have a dynamic, helical morphology (e.g., Tchekhovskoy et al. 2012; Dexter et al. 2014; Sa̧dowski & Narayan 2015; Tchekhovskoy 2015). The compression of the flow in a recollimation shock should change the morphology of the magnetic field structure. In M87, we see a combination of features in the nuclear and HST-1 regions (Figure 3). In particular, we see helical undulations in the polarization PA in the region between the nucleus and HST-1 (features Nuc-$\alpha ,\beta ,\gamma$ in the radio image). Within the flux maximum of HST-1, we see a significant increase in polarization, particularly in the optical, as well as a nearly perpendicular magnetic field direction. The radio 22 GHz image, while showing increased polarization at the flux maximum, does not show a perpendicular magnetic field direction. However, this may be due to an increased RM at HST-1, as discussed by Chen et al. (2011), who used the same VLA data as us, but concentrated only on the nuclear region of M87. Their images (see their Figure 1) show a nearly perpendicular magnetic field orientation at the highest radio frequency (43 GHz), which would have the lowest Faraday rotation. Downstream of HST-1's flux maximum, the radio image once again shows helical undulations in the magnetic field vectors (features HST-1 $\alpha ,\beta ,\gamma ,\delta ,\epsilon$). These features are consistent with the compression of a helical magnetic field (seen both upstream and downstream of HST-1) in the recollimation shock, where the perpendicular component would become dominant. To explain the higher degrees of perpendicular polarization, Cawthorne (2006) assumed that there has to a combination of chaotic and poloidal magnetic field present in the jet at this location, a notion that is consistent with our data.

Knot A, shown in Figure 6, and the surrounding region is also highly interesting. Bicknell & Begelman (1996) analyzed it as the site of a strong, highly oblique shock feature, which is how Nakamura and collaborators (Nakamura & Meier 2004, 2014; Nakamura et al. 2010) have described it, and knot C as a pair of fast-mode shocks. Our data throw interesting light on this. In particular, knot A, which, using previous optical imaging data Sparks et al. (1996) has revealed to have a highly complex internal structure, is now seen to have a complex polarization structure as well. The maximum optical polarization occurs well upstream (by 04) of the flux maximum. This is also where the magnetic field direction is first seen to rotate. This rotation is seen in both the optical and radio, but the radio does not show increased polarization in this region. Downstream of this, the flux maximum region is seen to have increased polarization as well (and in fact this is where the maximum radio polarization is located). Surveying the polarization maps in this region, one sees four features, each with slightly different magnetic field orientations, giving the impression of either magnetic "sheets" or filaments that are wound with a slightly different PA. We also see reduced polarization downstream, as well as a couple of features with almost zero polarization with nearly circularly symmetric magnetic field vectors that surround it. This is consistent with a compression of the overall helical magnetic field at knot A, but we find it implausible to think of A and C as a connected shock system as discussed by Nakamura & Meier (2014). Not only is there a large (400 pc) distance between these two knot complexes, but also the very different morphology of the magnetic field vectors argues strongly against it. Note also that in knot A we also see higher optical polarization along the jet edges (only statistically significant in knots A and B), further suggesting interactions with the surrounding ISM. A likely shock feature is also seen in knot C (and possibly G). This region, shown in Figure 7, has a conical shape, and the polarization map shows a nearly perpendicular magnetic field orientation and increased polarization at both upstream and downstream ends of the flux maximum region, with a polarization minimum between and vectors along the knot edges which follow the morphology of the flux contours. This suggests a double (or triple, if G is included) shock structure that might contribute to the breakup of the jet downstream of knot G, something that has been noted before in a variety of papers. Tchekhovskoy & Bromberg (2016) have pointed out that the kink instability that is important in FR I jets (see also below, Section 5.2) can lead to features like this on kiloparsec scales. Our polarization map is consistent with this idea.

In another study, Chen et al. (2011) claim high Faraday rotations in the inner jet of M87, i.e., in the nucleus and HST-1. They analyze the same data as ours, taken between 2003 and 2007 at 8, 15, and 22 GHz. Their studies show quite significant internal Faraday rotations in HST-1 at 8 GHz radio observations along with significant variations in the EVPA and FP during the period of observations. They claim that the variability in FP and observed helical undulations in the polarization structure of HST-1 was most possibly caused by internal Faraday rotation during the time of the flare in HST-1. We do not find significant Faraday rotations in our observations, hence we can neither confirm nor deny their claims, although the observed differences in the MFPA vector orientations in the radio and optical may be explained with their results.

A number of regions of the jet have a helical morphology and magnetic field vectors. Some of these regions are shock-like and/or connected with shock-like features, such as the features upstream and downstream of the flux maximum of knot HST-1, or the region downstream of knot A (both discussed above), but other features, such as knot D and knot B, do not appear to be strongly shock related. Here, we will discuss first the non-shock-related helical features, and then attempt to bring them and the helical features in the more shock-like knots into a coherent picture.

The apparent helical morphology of knot D is evident in both its flux morphology as well as its polarization vectors. The knot gives the appearance of being a braided, filamentary structure. P99 described it as being shock-like, and the differences observed by those authors between the optical and radio polarization vectors were one of the reasons behind their suggested model of a stratified energetic structure. As can be seen in Figure 3, our increased resolution throws a significant amount of new light on this issue, although it should also be mentioned that it is likely that the motion of some components (which are seen to be as fast as $\sim 5c$; Meyer et al. 2013), could play a significant role, producing differences of as large as 02 in the positions of the fastest components over 10 years. The polarization vectors of knot D also give a strongly helical appearance. In the brightest regions these appear to follow the flux contours. One exception to this is the flux maximum region of knot D-East, which is very low polarization in the optical and shows some signs of cancellation in that band, and the upstream half of its flux maximum region in the radio. This is significantly more detail than could be seen in P99, and in fact comparing our images to theirs we see that the low polarization optical regions are larger in the more recent data. This suggests that the polarized structures are moving down the jet flow, and while cancellation may play a non-negligible role in the differences between the optical and radio, it also seems clear that there are spectral differences as well, such as those described in the P99 model. Indeed, as discussed in Meyer et al. (2013), the fastest moving parts of the knot D complex are located in its eastern part, apparently being ejected from a feature at its upstream end. That feature is at the approximate position of D-Eastα in Figure 4. A second exception is seen in the apparent double structure that appears to develop in D-Middle, which is seen primarily in the radio. However, when seen in the broader context of the total flux morphology it reinforces its apparent braided nature.

Knot B's appearance is also striking. The brightest regions (Figure 6) also appear to have a filamentary structure, but it is less clear that they are helical. The inclination of these brightest features as compared to knot A, its neighbor, is striking: the brightest region extends for about 2'' and extends for nearly the entire width of the flow, starting in its southern half at the upstream end and appearing to spread to the entire flow by knot B's downstream end. This region is fairly clearly demarcated by the polarization vectors and sketched out by the red arrows in Figure 6. The upstream end of that region shows a polarization minimum in the radio, and multiple regions of low polarization in the optical, combined with other regions of higher polarization but filamentary magnetic field features seen in the radio. The differences are significant, and suggest that there are some spine/sheath issues to the jet structure in this region as well. The downstream end of knot B, historically called B2, shows radio polarization vectors that follow the flux contours, but the optical polarization vectors show somewhat more structure, more clearly delineating the apparent change in direction seen at knot B2 and also not including the apparent radio polarization minimum seen near the centroid of B2's flux contours. The appearance of the vectors suggests that the latter is due to cancellation. Also, as seen in knot A, the edge regions of the knot have a somewhat higher FP than other parts. This suggests a continuing interaction with the ISM in knot B, but while it is clear that knots A and B are spatially contiguous, the changing inclination and complex polarization morphology suggests that the two features are complex dynamically as well, which is not really consistent with a single shock complex.

It is also worth pointing out that previous workers have mentioned that knot E has a filamentary, helical structure (in particular Hardee & Eilek 2011). While the polarization vectors (Figure 5) in this region (particularly in the optical for knot F) are suggestive in this regard, most parts of these knots fall below the 3σ significance level both in the radio and optical. The sole exceptions to this are the brightest parts of this region. We do see both perpendicular vectors near a flux maximum e.g., in knot A, but also regions where cancellation is likely playing a role due to circularly symmetric vector patterns near the knot's upstream end, e.g., in knot D-East and B2, where P99 noticed a polarization minimum. In P99 we discussed a significant role for the Kelvin–Helmholtz instability in this region as well as for knots A and B; however, due to the low significance we do not discuss it further. Data with higher S/N are needed to study these in detail.

In Section 5.1 above we discussed the prominent helical features seen upstream and downstream of the shock-like components HST-1 and A. These features are not dissimilar to the ones described here and point out that kinks may be very important dynamically in jets on large scales. Tchekhovskoy & Bromberg (2016) have pointed out that the kink instability is important in FR I jets (see also Section 5.2). Their model elucidates some of the issues brought out by the model of Hardee & Eilek (2011), which attempted to model the entirety of the inner jet as a combination of braided helical features. These changes can cause instabilities (like Kelvin–Helmholtz instability), which in turn can cause shearing of field lines near the boundary between the jet surface and ISM. Their global 3D MHD simulations of low and high power AGN jets show that such instabilities have been produced within the jet as a result. Magnetic fields, being the natural driving force for launching the jets, can suppress instabilities like Kelvin–Helmholtz and initiate the current-driven kink instabilities. The kink instabilities, especially of the kink mode m = 1, play an important role in causing the jet to move sideways and developing the helical motions, possibly as seen in our radio polarization images. The potential of the growth of these kink modes depends on the Alfvén wave travel time across the jet and also on the ambient density. The tightly collimated jets, like those in M87, are more susceptible to kink instabilities. In such jets, the instabilities can rapidly form and disrupt the jet and cause them to decelerate or stall. Knot D, for example, may be an example of a kink developing downstream of the knot HST-1 recollimation shock, while knot B may be an example of the continuing interaction of the jet with the galactic ISM downstream of the main knot A shock region. Interestingly, Meyer et al. (2013) showed that the superluminal velocity vectors in the outer jet, mainly in knots A and B, appear to line up in helical pattern. We are in the process of attempting to model the geometry of this region but this is work in progress. We will present the results of this work in a future paper.