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]).

$$\frac{{\rm d}T_{\rm b}^{\pm }}{{\rm d}\tau ^{\pm }} T_{\rm e}-T_{\rm b}^{\pm },$$ =

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

$$T_{\rm b}^{\pm }=\int\limits_{0}^{\tau _{\rm o}^{\pm }}T_{\rm... ...{\rm o}^{\pm }}T_{\rm ext}^{\pm }(\tau =\tau _{\rm o}^{\pm }).$$ (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

$$T_{\rm b}^{\pm }=T_{\rm o}-(T_{\rm o}-T_{\rm ext}^{\pm }){\rm e}^{-\tau _{\rm o}^{\pm }}.$$ (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

$$T_{\rm o}-(T_{\rm o}-I_{\rm rad}){\rm e}^{-\tau _{\rm o}},$$ I =

$$\beta _{\rm o}\tau _{\rm o}(T_{\rm o}-I_{\rm rad}){\rm e}^{-\tau _{\rm o}}+V_{\rm rad}{\rm e}^{-\tau _{\rm o}}.$$ V = (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 $\tau (l)$

$$t(l)=\frac{1}{\lambda ^{2}}\tau (l)=\frac{1}{\lambda ^{2}}% \int\limits_{0}^{l}\mu ^{\rm o}(l^{\prime }){\rm d}l^{\prime }$$ (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):

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

where $\tau _{k}=\lambda ^{2}t_{k}$ the radio depth, T_k 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

$$T_{\rm o}-\sum\limits_{k=0}^{n}(T_{k}-T_{k+1})\exp \left(-\sum\limits_{i=0}^{k}\tau _{i}\right),$$ I =

$$\sum\limits_{k=0}^{n}\beta _{k}\tau _{k}(T_{k}-I_{k+1})\exp \left(-\sum\limits_{i=0}^{k}\tau _{i}\right).$$ V = (A7)

To decouple the equations, we should express the (T_k-I_k+1) in terms of T_k by means of recurrent use of Eqs. (A6) as

$$\equiv \triangle T_{k}+\sum\limits_{i=1}^{\infty }\triangle T_{k+i}\exp (-\sum\limits_{j=1}^{i}\tau _{k+j}),$$ (T_k-I_k+1) (A8)

$$\triangle T_{k} \equiv$$ T_k-T_k+1.

After decoupling the Eqs. (A7) by means of substituting the term (T_k-I_k+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:

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

$$V(\lambda ) \lambda ^{2}\int\limits_{0}^{\infty }\beta (t)\triangle T(t){\rm e}^{-\lambda ^{2}t}{\rm d}t,$$ =

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

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

and the polarization spectrum is determined by the differential equation

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

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