The radiation transfer equations in the quasi-longitudinal approximation are
decoupled for extraordinary-ordinary (+/-) circular modes with radiation
temperature
(Zheleznyakov [1970]).

= | |||

= | (A1) |

where is the bremsstrahlung opacity in an isotropic plasma, the electron gyrofrequency, and the angle between the magnetic field vector and the line of sight. For a plane-layer atmosphere Eq. (A1) can be integrated to obtain the observed brightness temperature

(A2) |

where is an incident radiation temperature at the bottom of the atmosphere.

For a single homogeneous layer with an optical depth we obtain

(A3) |

We consider the radiation transfer for the Stokes parameters and circular polarization . After introducing definitions for and , we obtain in a linear approximation with respect to a small parameter the solutions for

I |
= | ||

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 and

(A5) |

and express all spatial distributions as functions of the radio depth

(A6) |

where the radio depth,

I |
= | ||

V |
= | (A7) |

To decouple the equations, we should express the (

(T_{k}-I_{k+1}) |
(A8) | ||

T_{k}-T_{k+1}. |

After decoupling the Eqs. (A7) by means of substituting the term (

(A9) |

= | |||

= | (A10) |

In the case of a homogeneous magnetic field distribution the radiation transfer Eq. (A1) depends on frequency and magnetic field only through the combination . Therefore, its solution is ruled by a scaling law

(A11) |

and the polarization spectrum is determined by the differential equation

(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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