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Constraining the magnetic field in the parsec-scale jets of the brightest Fermi blazars with multifrequency VLBI observations

The spatially resolved broad-band spectroscopy with Very Long Baseline Interferometry (VLBI) is one of the few methods that can probe the physical conditions inside blazar jets. We report on measurements of the magnetic field strength in parsec-scale radio structures of selected bright Fermi blazars, based on fitting the synchrotron spectrum to VLBA images made at seven frequencies in a 4.6 -- 43.2 GHz range. Upper limits of B<= 10^-2 -- 10^2 G (observer's frame) could be placed on the magnetic field strength in 13 sources. Hard radio spectra (-0.5<= a<= +0.1, S_nu ~ nu^a) observed above the synchrotron peak may either be an indication of a hard energy spectrum of the emitting electron population or result from a significant inhomogeneity of the emitting region.


Introduction
Blazars are active galactic nuclei characterized by highly variable non-thermal continuum emission across the electromagnetic spectrum. The Large Area Telescope (LAT, [1]) on board the Fermi γ-ray observatory, launched on 2008 June 11, provides a wealth of new information about high-energy radiation of blazars. It is believed that this high-energy emission is intimately connected with the emission at lower energies (radio -optical) through the process of inverse Compton scattering. Very Long Baseline Interferometry (VLBI), with its spectacular angular resolution, provides the most detailed view of inner regions in blazars [2]. In most cases, VLBI observations reveal one-sided relativistically beamed parsec-scale jets originating in a bright, compact region called the "core".
The spatially resolved broad-band spectroscopy is one of the few methods that can probe the physical conditions inside blazar jets. Despite the great potential of this method, its applications are still relatively rare (see [3] and references therein), owing to difficulties in implementing it and interpreting results obtained [4]. In addition, multifrequency VLBI observations require large amounts of observing time and most VLBI arrays lack technical capability to perform simultaneous multifrequency observations. We utilize the Very Long Baseline Array (VLBA, [5]) which has a unique capability to conduct observations quasi-simultaneously at many frequencies, needed to constrain physical conditions in the regions where blazar radio emission originates. We reconstruct radio spectra of parsec-scale features in jets of selected γ-ray-bright blazars and compare them with the simple homogeneous synchrotron source model, which allows us to derive information about the magnetic field strength and particle energy distribution. During the first year of the Fermi scientific operations, we used the VLBA to observe 20 blazars, that were identified prior to the launch of Fermi as expected bright γ-ray emitters. In this paper, we present results for 17 of them. The data analysis for three remaining sources is still ongoing.

Observations and data reduction
The observations were conducted simultaneously at seven frequencies in a 4.6-43.2 GHz range. After the initial calibration in the AIPS package [6], the sources were self-calibrated and imaged independently at each frequency using the Difmap software [7]. Whenever feasible, the frequency-dependent core position (the "core shift" effect, see for a detailed discussion [8,9]) was taken into account. Image alignment is a major problem for spectral VLBI analysis because the self-calibration technique used to obtain high-quality images causes loss of information about the absolute position of a source. Positions of optically thin jet features or the core itself (if the "core shift" effect is negligible for a given source) are then used to align VLBI images obtained at different frequencies.
In order to reconstruct the spectrum in a given pixel of an image ("pixel-based approach"), the sum of CLEAN components around this pixel was used. The CLEAN components were summed in-eConf C091122 side a radius R = Θ(lowest frequency)/3, where Θ(lowest frequency) is the half-power beam width at the lowest observing frequency (4.6 GHz). This approach has a major advantage over the direct use of flux densities from reconstructed images because it avoids artifacts introduced by convolution with a Gaussian beam while preserving the angular resolution achieved.

Results
The radio spectra in the parsec-scale core regions (marked as shaded areas on Fig. 1) of the observed blazars have diverse properties. Most of them are nearly flat, with signs of curvature at lower frequencies, which we interpret as synchrotron selfabsorption. In one source (NRAO 530, Fig. 1r), the spectral turnover is particularly prominent and well sampled by our observations. Cores in three sources, 0716+714 ( In the following, we focus on sources that show a well defined spectral turnover, and thus can be used to determine the physical conditions in the emitting plasma. We fit a homogeneous synchrotron source model [10] (see the Appendix for more details) to the observed spectra in order to obtain the magnetic field strength and particle energy distribution. The sources for which the data reduction is complete are presented in Table I. For 13 sources, the model provided an adequate fit to the observed spectra ( Fig. 1) while four sources exhibit flat or inverted spectra ( Fig. 2) that cannot be described by the homogeneous synchrotron model.

Model reliability
The physical parameters presented in Table I correspond to typical values for emitting electrons inside a large area of a few milliarcseconds in size (shaded areas on Fig. 1). In most cases, only an upper limit on the magnetic field strength could be placed, because most regions where the spectral turnover is detected remain unresolved. The results are consistent with estimates obtained by a method based on model-fitting bright VLBI components with 2D Gaussian components (see Fig. 3 for results on BL Lacertae). The model-fitting based method is described in detail in [3] where it was applied to 3C 273 and the average magnetic field strength B ⊥ ∼ 10 −2 G was obtained for the 8 GHz core region which is comparable in size to the region probed in our research. This value is in a good agreement with our upper limit at B ⊥ ≤ 0.016 G given in Table I.

Electron energy spectrum
The spectra above the synchrotron turnover are nearly flat or slightly inverted in core regions of all sources investigated in this work. This may result from blending of a few emitting regions with different peak frequencies (an example of such blending may be found in the core region of BL Lacertae, see the top right panel in Fig. 3, see also [3] for 3C 273). Alternatively, a nearly flat spectrum may imply a hard energy spectrum of the emitting electrons (as may be the case for the component B1 in the jet of BL Lacertae whose spectrum is presented in the lower right panel of Fig. 3; see also Table II). The hard electron spectrum is difficult to explain by the conventional firstorder Fermi acceleration mechanism. However, the hard spectrum can be produced by the second-order Fermi mechanism ("stochastic acceleration") [11].

Summary and prospects
We have placed upper limits of B ⊥ ≤ 10 −2 -10 2 G (observer's frame) on the magnetic field strength in 13 out of 17 Fermi blazars (Table I). Spectra of four blazars could not be described by the simple homogeneous synchrotron source model, and no constraints on the magnetic field and particle energy spectrum could be obtained for these sources. Hard spectra (−0.5 ≤ α ≤ +0.1, S ν ∼ ν α ) observed in the blazar cores above the synchrotron peak may either indicate a hard energy spectrum of the relativistic electron population in the jet or result from significant inhomogeneity (of the magnetic field and plasma parameters or just optical depth) across the emitting region.
Since γ-ray emission in blazars is suggested to originate from regions spatially close to the VLBI core [12], the estimates of the magnetic field strength and electron energy distribution presented in Table I could be used to constrain broad-band Spectral Energy Distribution (SED) models. Simultaneously with the VLBA observations described here, Swift Xray/ultraviolet/optical target of opportunity observations were performed. When combined with the Fermi γ-ray data this will provide a unique data set for the SED modeling. Analysis of this data set is presently underway. Note, that for the optically thin part of the synchrotron spectrum p = 1 − 2α, where α is defined as Sν ∼ ν α . b The estimates correspond to the region of parsec-scale radio core. c The values are in the observer's frame. d The homogeneous synchrotron source model is not applicable for this spectrum.

Appendix: synchrotron source model used in pixel-based approach
Following [10], we consider a uniform cloud of relativistic electrons in the external uniform magnetic field with component B ⊥ perpendicular to the line of sight. We assume that the electron energy distribution is described by a power law: where N 0 and p are constants. For this case, we can write the radiation transfer equation where I ν is the specific intensity at the frequency ν, ds is the differential element of path length, j ν is the emission coefficient (emissivity), κ ν is the absorption (extinction) coefficient. If j ν and κ ν are constant along the line of sight, the solution of the radiation transfer equation is where L is the size of the electron cloud along the line of sight, and κ ν L = τ ν is the optical depth. Following [10], this solution can be written in the form: where where e is the electron charge, m e is the electron mass, and c is the speed of light in vacuum. The power law index p of the electron energy distribution may be found from the spectral index α (S ν ∼ ν α ) in the optically thin part of the spectrum: p = 1 − 2α. The optical depth τ m corresponding to the observed spectral peak frequency ν m may be found from equation exp[τ m ] = 1 + p+4 5 τ m . The frequency ν 1 corresponding to τ ν1 = 1 may be found from ν m , p and τ m : From equations (1), (2) and (3), we can derive the perpendicular component of the magnetic field: Note, that if the size of an emitting region is smaller then the angular resolution, only a lower limit on the intensity and, correspondingly, an upper limit on the magnetic field B ⊥ can be derived.  Figure 1: VLBI images at 8.4 GHz and radio continuum spectra of sources for which the synchrotron selfabsorption turnover frequency was successfully measured. The shaded area on the images marks the region where the synchrotron turnover was detected. The color of the shaded area represents the upper limit of the magnetic field strength (B ⊥ ). The values of B ⊥ and p presented in Table I Figure 2: VLBA spectra of sources whose spectra cannot be adequately described by the simple homogeneous model. The spectra correspond to parsec-scale core regions. Green lines represent a power law fit. It is possible to fit a homogeneous synchrotron source model (blue curve) to the observed spectrum of 0716+714, however, a simple power law provides better (lower χ 2 ) fit.

eConf C091122
1 mas BL Lac 43 GHz Figure 3: Results of the model-fitting based analysis of BL Lacertae (2200+420). The contours represent the CLEAN image at 43 GHz, the beam size and the linear scale are indicated at the lower left panel. The source structure was approximated with Gaussian components (see Table II). The fitting was conducted in the the uv-plane. The two inner components (C1 its position is marked by the red arrow, C2 marked by the blue arrow) are unresolved while the third component (B1 marked by two black arrows) is resolved at 43 GHz. Fig. 1x presents the spectrum of this source obtained in the course of pixel-based analysis. This spectrum corresponds to the combined spectrum (black) of components C1 (red) and C2 (blue) presented in the upper right panel of the current figure. The peak frequency for B1 is not well constrained, however, following [13] we can put an upper limit on the magnetic field strength in this component B < 0.06 G (observer's frame). Sm -synchrotron self-absorption peak flux density. α -spectral index in the optically thin part of the spectrum.