Magnetic Field Studies in BL Lacertae through Faraday Rotation and a Novel Astrometric Technique

It is thought that dynamically important helical magnetic fields twisted by the differential rotation of the black hole’s accretion disk or ergosphere play an important role in the launching, acceleration, and collimation of active galactic nuclei (AGN) jets. We present multi-frequency astrometric and polarimetric Very Long Baseline Array (VLBA) images at 15, 22, and 43 GHz, as well as Faraday rotation analyses of the jet in BL Lacertae as part of a sample of AGN jets aimed to probe the magnetic field structure at the innermost scales to test jet formation models. The novel astrometric technique applied allows us to obtain the absolute position at mm wavelengths without any external calibrator.

The location at which the jet becomes optically thin. Therefore its position shifts with observing frequency.
Motivation: Study the nature of core jets.
The radio core is a recollimation shock in the jet at a fixed location.
The mm-VLBI radio core and its connection with gamma-ray flares in AGN jets

Scientific context
The debate on the origin and location of the !-ray emission in jets of active galactic nuclei (AGN) has gained added interest after the launch of the Fermi satellite. Much of the current discussion lies in whether the !-ray flares are produced within the broad-line emission region (e.g., Tavecchio et al. 2010), or parsecs away from the central engine (e.g., Marscher et al. 2010;Agudo et al. 2011a;Ackermann et al. 2012).
Results of over three years of monthly monitoring of 34 blazars (the most luminous and variable BL Lac objects and flat-spectrum radio quasars) with the VLBA at 7mm by the Boston University blazar group (e.g., Marscher et al. 2012) show that most !-ray flares are simultaneous within errors with the appearance of a new superluminal component or a major outburst in the core of the jet, defined as the bright, compact feature in the upstream end of the jet (Marscher et al. 2008(Marscher et al. , 2010Jorstad et al. 2010;Agudo et al. 2011a). A burst in particle and magnetic rays, optical, infrared, and radio, together with mm-VLBI imaging, have shown that in several radio galaxies and blazars the core indeed is inferred to lie parsecs away from the central black hole (Marscher et al. 2002(Marscher et al. , 2010Chatterjee et al. 2011;Agudo et al. 2011aAgudo et al. , 2011b). This  with the weighted least-square method, then we derive the best-fit value as r RA (n) / n 20.94 6 0.09 . The fitted curve approaches the dashed line asymptotically at 41 6 12 mas eastwards of the 43-GHz core position (errors are 1s), which is equivalent to the projected separation 6 6 2R s for the black-hole mass M 5 6.0 3 10 9 solar masses (ref. 14) at a distance of 16.7 Mpc (ref. 15). The measured frequency dependence of the core shift roughly n 21 is consistent with a 'conical' jet with the radial profiles of the magnetic field strength and the electron number density varying as r 21 and r 22 , respectively 16 , with the assumption of a constant jet velocity. With regard to jet shape, a recent theoretical model shows that a jet seems to have a 'paraboloidal' shape near a central black hole 1 . If this is true of the M87 jet, then the location of the central engine is likely to be even closer to the 43-GHz radio core than the dashed line.

Central black hole and accretion flow
Radio core at different frequencies ! 5"#" 4 #" 3"#" 2"#" 1$ rc ( 5 ) rc ( 4 ) rc ( 1 ) rc ( 2 ) rc ( 3 ) 1 2 3 4 5 Figure 1 | Schematic diagram explaining the radio core shift of a jet. The diagram illustrates the core shift of a jet generated from the central black hole (a black dot) surrounded by the accretion disk (represented as a red ellipse), with the horizontal axis showing a distance from the black hole (r). The cores of a jet, the bright surfaces of optical depths being unity, are indicated as grey ellipses at the actual radio frequencies of VLBI measurements; darker colours indicate higher frequencies. The cores are located at the apparent origin of the jet in each frequency image. The optical depth t ssa for the synchrotron self-absorption is a function of the radio-emitting electron number density N e , the magnetic field strength B and the observing frequency n. Because N e and B have a radial profile in the jet, the radial position on the surface at which t ssa becomes unity shifts as a function of frequency. If we assume that N e and B have power-law profiles of r described as N e / r 2n and B / r 2m (n and m positive), the frequency dependence of the core position results in r(n) / n 2a . Here a is the positive power index described as a function of n and m (ref. 10). According to the relation, the cores shift towards the upper stream with increasing frequencies equipartition with the particle energy density is assumed. Figure 3 shows the sequence of total intensity images at the different frequencies, as well as the evolution of the core position with frequency. At cm-wavelengths (5 to 22 GHz) our simulations reproduce the opacity core-shift of a Blandford & Königl conical jet model, while at mmwavebands (43 and 86 GHz) the core position clearly departs from this behavior, revealing the core as an optically thin recollimation shock at a fixed jet location.
Polarization mapping has proven to be a powerful tool in establishing a clear relationship between stationary jet features and recollimation shocks (e.g., Agudo et al. 2012). Hence, to simulate the expected linearly polarized emission for the mm-VLBI core we have used the Turbulent Extreme Multi-Zone (TEMZ) model developed by Marscher (2012b). The TEMZ code divides the emission region beyond the conical recollimation shock into many cylindrical cells, as illustrated in Fig.4. Each 10th cell is assigned a uniform magnetic field whose direction is determined randomly, and the field vector is rotated smoothly between those cells. Relativistic electrons are injected into the cells at the shock front and subsequently lose energy from synchrotron and inverse Compton radiation as the plasma flows downstream. The resulting total and linearly polarized emission maps are shown in Fig. 5 for two different convolution beams. The "full" resolution images of Fig. 5a show the complex total and polarized emission structures arising from the conical shock. The convolved images of Fig. 5b, with a resolution in terms of the jet width similar to what we expect to obtain from our proposed observations at 86 GHz, reveal that, although the total intensity image only shows an unresolved structure (as commonly observed), the polarized information is capable of revealing the conical shock structure of the mm-VLBI core.

Scientific goals
We propose to test the hypothesis that the majority of !-ray flares in AGN jets are produced by the passing of new superluminal features through a recollimation shock at the end of the acceleration and collimation zone in a blazar jet by determining whether the properties of the mm-VLBI core are consistent with this scenario.
For this we propose polarimetric phase-reference VLBA observations of several !-ray emitting AGN jets at 5, 8.4, 15, 22, 43, and 86 GHz. To test whether there is any dependence on the mm-VLBI core nature with AGN type we propose to observe two quasars (3C273, CTA102), two BL Lac objects (BL Lac, 0716+714) and two radio galaxies (3C111, 3C120), whose jets are straight on scales ≲ 2 mas.
If our model is correct, we expect to see the opacity core frequency shift at cm-wavebands (5,8.4,15,and 22 GHz), with deviation from this behavior at mm-wavebands (43 and 86 GHz), as shown in Fig. 3. The highest resolution polarization images at 86 GHz should also help to reveal the recollimation shock nature of the mm-VLBI core, as shown in Fig. 5b. Measured core-shifts at cmwavebands are found to vary significantly from source to source  It is necessary to have astrometric measurements at mm wavelenght One of the main questions to answer in jets is the nature of the core.

Observation Strategy
Astrometric Technique applied to BL Lac.
Is a new approach to the Source Frequency Phase Transfer (SFPR) in which the ionospheric contribution is determine from the L (1.3 GHz), WC and K (22GHz) band. We have developed a program to fit the data.

Tropospheric contributions
We have to calculate the tropospheric contribution with the usual cm astrometry with the calibrator.
From 5 to 22 GHz The tropospheric phase contributions are proportional to the observing frequency Frequencyphasetransfer At 43 and 86 GHz