content_image/398403.jpg content_image/398401.jpg content_image/398405.jpg content_image/398404.jpg content_image/398400.jpg content_image/398402.jpg content_image/398409.jpg content_image/398410.jpg content_image/398406.jpg content_image/398407.jpg content_image/398408.jpg content_image/398399.jpg
## Appendix: synchrotron source model used in pixel-based approach 

Following [10], we consider a uniform cloud of rel- ativistic electrons in the external uniform magnetic field with component B$_{⊥}$ perpendicular to the line of sight. We assume that the electron energy distribu- tion 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 

<img src='content_image/398403.jpg'>

<img src='content_image/398401.jpg'>

<img src='content_image/398405.jpg'>

<img src='content_image/398404.jpg'>

<img src='content_image/398400.jpg'>

<img src='content_image/398402.jpg'>

<img src='content_image/398409.jpg'>

<img src='content_image/398410.jpg'>

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 dis- tribution may be found from the spectral index α ( S$_{ν}$ ∼ ν $^{α}$) in the optically thin part of the spectrum: p = 1 − 2 α . 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$_{ν}$$_{1}$ = 1 may be found from ν$_{m}$ , p and τ$_{m}$ : ν$_{1}$ = τ 2 / ( p +4) m ν$_{m}$ . 

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. 

<img src='content_image/398406.jpg'>

<img src='content_image/398407.jpg'>

<img src='content_image/398408.jpg'>

<img src='content_image/398399.jpg'>

