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Appendix A: Radiative transfer equation

Here, we provide a brief description of the equations we used to treat the radiative transfer and dichroic polarization in dusty environments. Under the assumption that no light is scattered into the line of sight, the radiative transfer equation can be written in the Stokes vector formalism as follows (Martin 1974):

(A.1)The mechanisms of dichroic polarization are straightforward to calculate by analytical functions since we consider the number density \begin{lxirformule}$n_{\rm{d}}$\end{lxirformule} of the dust to be constant in each cell. This condition is given since each cell in the model space of MC3D operates with a set of constant physical parameter. The matrix elements C_ext and ΔC_ext are the cross section for extinction and linear polarization, ΔC_circ for circular polarization due to birefringence. With the constant number density of the dust (n_d) the system of equations decays into two uncoupled systems (Whitney & Wolff 2002). The first system of equations solely describes the change of the I and Q parameter due to dichroic extinction. It can be solved by simple substitution and integration:

(A.2)The second system of equations can be handled as a complex eigenvalue problem. This leads to additional cosine and sine terms and a transfer between linear and circular polarization (A.3)Linear polarization arises from linear dichroism alone, while circular polarization depends on both a non-zero value in the U parameter and birefringence. Subsequently, circular polarization can occur in the case of non-parallel magnetic field lines along the line of sight.

Appendix B: Orientation of polarization

It is possible to determine the exact conditions for the 90° flip in a single cell of the model space. In general, a threshold for

this effect does not exist along the entire line of sight. In each cell of our model space we have the two opposing effects of dichroic extinction and thermal re-emission adding to the linear polarization perpendicular to each other. In the reference frame of the magnetic field the dichroic extinction provides a negative contribution to the Q parameter while thermal re-emission contributes positively to Q. In this orientation the U and V parameter remain zero. If we solve Eq. (A.2) for the Q_{i + 1} parameter, we can calculate the conditions when the two effects cancel each other out: (B.1)The contribution of thermal re-emission is determined by the temperature of the dust T_d, the number density n_d, the cross sections for absorption ΔC_abs,C_abs and the path length l. Inside each cell all the parameters and functions n_d, l, ΔC_abs, C_abs, C_ext, B_λ(T) are positive and constant, so one can solve Eq. (B.1). As we can derive from Eqs. (2) and (3), the contributions of I_i and Q_i in a single cell are as follows: For a wavelength of λ > 7 μm one can approximate C_abs,i ≈ C_ext,i. By introducing the optical depths for extinction (B.4)and polarization (B.5)we can derive an inequality for the Q parameter to change its sign as a function of the inverse hyperbolic tangent: (B.6)If the right-hand side is larger than 1, the polarization process is dominated by thermal re-emission and, in the reverse case, by dichroic extinction. However, in the calculated synthetic polarization maps the observed flip of 90° for the orientation of linear polarization depends on all the physical quantities along the entire line of sight and cannot be determined with this inequality.

© ESO, 2014