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Appendix A: Polarization transfer in inhomogeneous medium

The radiation transfer equations in the quasi-longitudinal approximation are decoupled for extraordinary-ordinary (+/-) circular modes with radiation temperature $T_{\rm b}^{\pm }$ (Zheleznyakov [1970]).

$\displaystyle \frac{{\rm d}T_{\rm b}^{\pm }}{{\rm d}\tau ^{\pm }}$ = $\displaystyle T_{\rm e}-T_{\rm b}^{\pm },$  
$\displaystyle {\rm d}\tau ^{\pm }$ = $\displaystyle {\rm d}\tau ^{\rm o}\left(1\pm \frac{2\nu _{B}\cos \alpha }{\nu }\right),~~~%
\frac{\nu _{B}}{\nu }\ll 1$ (A1)

where ${\rm d}\tau ^{\rm o}=\mu ^{\rm o}{\rm d}l,\ \mu ^{\rm o}\simeq 0.2\cdot \nu
^{-2}N_{\rm e}^{2}T_{\rm e}^{-3/2}$ is the bremsstrahlung opacity in an isotropic plasma, $\nu _{B}=2.8\ 10^{6}B$ the electron gyrofrequency, $\nu =c/\lambda $ and $\alpha $ the angle between the magnetic field $\vec{B}$vector and the line of sight. For a plane-layer atmosphere $0<l<L_{\rm o}$Eq. (A1) can be integrated to obtain the observed brightness temperature

\begin{displaymath}T_{\rm b}^{\pm }=\int\limits_{0}^{\tau _{\rm o}^{\pm }}T_{\rm...
...{\rm o}^{\pm }}T_{\rm ext}^{\pm }(\tau =\tau
_{\rm o}^{\pm }).
\end{displaymath} (A2)

where $T_{\rm ext}^{\pm }$ is an incident radiation temperature at the bottom $l=L_{\rm o}$of the atmosphere.

For a single homogeneous layer $%
B(l)=B_{\rm o},~~~T_{\rm e}(l)=T_{\rm o},~~~N_{\rm e}(l)=N_{\rm o},$ $\ 0\leq l\leq L_{\rm o},$ with an optical depth $\tau _{\rm o}^{\pm }$ we obtain

\begin{displaymath}T_{\rm b}^{\pm }=T_{\rm o}-(T_{\rm o}-T_{\rm ext}^{\pm }){\rm e}^{-\tau _{\rm o}^{\pm }}.
\end{displaymath} (A3)

We consider the radiation transfer for the Stokes parameters $%
I=(T_{\rm b}^{+}+T_{\rm b}^{-})/2$ and circular polarization $%
V=(T_{\rm b}^{+}-T_{\rm b}^{-})/2$. After introducing definitions for $\beta =2\nu
_{B}\cos \alpha /\nu <<1,~~\ \tau _{\rm o}=(\tau _{\rm o}^{+}+\tau _{\rm o}^{-})/2$ and $%
T_{\rm ext}^{\pm }\equiv I_{\rm rad}\pm V_{\rm rad}$, we obtain in a linear approximation with respect to a small parameter $\beta _{\rm o}\tau _{\rm o}\exp
(-\tau _{\rm o})<<1$ the solutions for I and V as
I = $\displaystyle T_{\rm o}-(T_{\rm o}-I_{\rm rad}){\rm e}^{-\tau _{\rm o}},$  
V = $\displaystyle \beta _{\rm o}\tau _{\rm o}(T_{\rm o}-I_{\rm rad}){\rm e}^{-\tau _{\rm o}}+V_{\rm rad}{\rm e}^{-\tau _{\rm o}}.$ (A4)

We use this solution to construct a solution of the transfer equations in an inhomogeneous medium by summing up multi-layer solutions. For known spatial distributions of $T_{\rm e}(l),\ N_{\rm e}(l),$ and B(l) we introduce the frequency-independent radio depth t(l) as a function of the current depth l, based on an isotropic optical depth $\tau (l)$

\begin{displaymath}t(l)=\frac{1}{\lambda ^{2}}\tau (l)=\frac{1}{\lambda ^{2}}%
\int\limits_{0}^{l}\mu ^{\rm o}(l^{\prime }){\rm d}l^{\prime }
\end{displaymath} (A5)

and express all spatial distributions as functions of the radio depth t. We decompose an atmosphere with a total radio depth $t_{\rm o}\equiv t(l=L_{\rm o})$into a sum of n homogeneous layers with solutions given by (A4):

\begin{displaymath}I_{k}=T_{k}-(T_{k}-I_{k+1}){\rm e}^{-\tau _{k}},V_{k}=\beta _{k}\tau
_{k}(T_{k}-I_{k+1}){\rm e}^{-\tau _{k}}
\end{displaymath} (A6)

where $\tau _{k}=\lambda ^{2}t_{k}$ the radio depth, Tk the kinetic temperature for the k-th layer, and $I_{k},\ I_{k+1}$ the radiation temperature at the top and bottom of the k-th layer. For the total intensity at the top of the atmosphere we have to solve a pair of coupled equations for the Stokes parameters I and V
I = $\displaystyle T_{\rm o}-\sum\limits_{k=0}^{n}(T_{k}-T_{k+1})\exp
\left(-\sum\limits_{i=0}^{k}\tau _{i}\right),$  
V = $\displaystyle \sum\limits_{k=0}^{n}\beta _{k}\tau _{k}(T_{k}-I_{k+1})\exp
\left(-\sum\limits_{i=0}^{k}\tau _{i}\right).$ (A7)

To decouple the equations, we should express the (Tk-Ik+1) in terms of Tk by means of recurrent use of Eqs. (A6) as
(Tk-Ik+1) $\textstyle \equiv$ $\displaystyle \triangle T_{k}+\sum\limits_{i=1}^{\infty
}\triangle T_{k+i}\exp (-\sum\limits_{j=1}^{i}\tau _{k+j}),$ (A8)
$\displaystyle \triangle T_{k}$ $\textstyle \equiv$ Tk-Tk+1.  

After decoupling the Eqs. (A7) by means of substituting the term (Tk-Ik+1), in the limit of infinitesimal thin layers we can replace summation by integration, and in the limit of $t_{\rm tot}>>\lambda ^{-2}$, find the solution of the radiation transfer equation:

\begin{displaymath}I(\lambda )=\lambda ^{2}\int\limits_{0}^{\infty }T_{\rm e}(t){\rm e}^{-\lambda ^{2}t}{\rm d}t
\end{displaymath} (A9)

$\displaystyle V(\lambda )$ = $\displaystyle \lambda ^{2}\int\limits_{0}^{\infty }\beta (t)\triangle
T(t){\rm e}^{-\lambda ^{2}t}{\rm d}t,$  
$\displaystyle \triangle T(t)$ = $\displaystyle \int\limits_{0}^{\infty }\frac{{\rm d}T_{\rm e}(t+t^{\prime })}{{\rm d}t}%
{\rm e}^{-\lambda ^{2}t^{\prime }}{\rm d}t^{\prime }.$ (A10)

In the case of a homogeneous magnetic field distribution $B(l)=B_{\rm o}$ the radiation transfer Eq. (A1) depends on frequency and magnetic field only through the combination $\lambda ^{2}(1\pm \frac{1}{2}\beta _{\rm o})$. Therefore, its solution is ruled by a scaling law

\begin{displaymath}T_{\rm b}^{\pm }(\lambda ^{2})=I(\lambda ^{2}(1\pm \frac{1}{2...
...beta _{\rm o}\lambda ^{2}\frac{{\rm d}I}{{\rm d}\lambda ^{2}},
\end{displaymath} (A11)

and the polarization spectrum is determined by the differential equation

\begin{displaymath}V(\lambda )=\beta _{\rm o}\lambda ^{2}\frac{{\rm d}I}{{\rm d}...
... e}(t)}{{\rm d}t}{\rm e}^{-\lambda ^{2}t}\lambda
^{2}{\rm d}t.
\end{displaymath} (A12)

These equations have been derived before in a similar form (Bogod & Gelfreikh [1980]; Grebinskij [1985]) without discussions of its applicability.

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