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<a name="S15"></a>Online material</h2>
<h3 class="sec2">
<a name="S16"></a>Appendix A: Radiative transfer equation</h3>
<a name="SD1"></a><p>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 (<a name="InR40"></a><a href="/articles/aa/full_html/2014/06/aa23116-13.html#R40">Martin 1974</a>): </p>
<p><a name="FD15"></a><span class="img-equation"><img src="aa23116-13-eq138.png"><span class="label-eq">(A.1)</span></span>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
              <span class="simple-math"><i>C</i><sub>ext</sub></span> and <span class="simple-math">&Delta;<i>C</i><sub>ext</sub></span> are the
            cross section for extinction and linear polarization, <span class="simple-math">&Delta;<i>C</i><sub>circ</sub></span> for
            circular polarization due to birefringence. With the constant number density of the dust
              (<span class="simple-math"><i>n</i><sub>d</sub></span>) the system of equations decays into
            two uncoupled systems (<a name="InR52"></a><a href="/articles/aa/full_html/2014/06/aa23116-13.html#R52">Whitney &amp; Wolff
              2002</a>). The first system of equations solely describes the change of the
              <span class="simple-math"><i>I</i></span> and
              <span class="simple-math"><i>Q</i></span>
            parameter due to dichroic extinction. It can be solved by simple substitution and
            integration: </p>
<p><a name="FD16"></a><span class="img-equation"><img src="aa23116-13-eq142.png"><span class="label-eq">(A.2)</span></span>The second system of
            equations can be handled as a complex eigenvalue problem. This leads to additional
              <span class="simple-math">cosine</span> and <span class="simple-math">sine</span> terms and a transfer between linear
            and circular polarization <a name="FD17"></a><span class="img-equation"><img src="aa23116-13-eq145.png"><span class="label-eq">(A.3)</span></span>Linear
            polarization arises from linear dichroism alone, while circular polarization depends on
            both a non-zero value in the <span class="simple-math"><i>U</i></span> parameter and birefringence. Subsequently,
            circular polarization can occur in the case of non-parallel magnetic field lines along
            the line of sight. </p>
<h3 class="sec2">
<a name="S17"></a>Appendix B: Orientation of polarization</h3>
<a name="SD2"></a><p>It is possible to determine the exact conditions for the <span class="simple-math">90&deg;</span> flip in a single cell of
            the model space. In general, a threshold for </p>
<p>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
              <span class="simple-math"><i>Q</i></span>
            parameter while thermal re-emission contributes positively to <span class="simple-math"><i>Q</i></span>. In this orientation
            the <span class="simple-math"><i>U</i></span> and
              <span class="simple-math"><i>V</i></span>
            parameter remain zero. If we solve Eq. (<a href="#FD16">A.2</a>) for the <span class="simple-math"><i>Q</i><sub><i>i</i> + 1</sub></span> parameter,
            we can calculate the conditions when the two effects cancel each other out:
              <a name="FD18"></a><span class="img-equation"><img src="aa23116-13-eq147.png"><span class="label-eq">(B.1)</span></span>The contribution of
            thermal re-emission is determined by the temperature of the dust <span class="simple-math"><i>T</i><sub>d</sub></span>, the number
            density <span class="simple-math"><i>n</i><sub>d</sub></span>, the cross sections for absorption
              <span class="simple-math">&Delta;<i>C</i><sub>abs</sub>,<i>C</i><sub>abs</sub></span>
            and the path length <span class="simple-math"><i>l</i></span>. Inside each cell all the parameters and
            functions <span class="simple-math"><i>n</i><sub>d</sub></span>, <span class="simple-math"><i>l</i></span>, <span class="simple-math">&Delta;<i>C</i><sub>abs</sub></span>,
              <span class="simple-math"><i>C</i><sub>abs</sub></span>, <span class="simple-math"><i>C</i><sub>ext</sub></span>,
              <span class="simple-math"><i>B</i><sub><i>&lambda;</i></sub>(<i>T</i>)</span>
            are positive and constant, so one can solve Eq. (<a href="#FD18">B.1</a>). As we can derive from Eqs. (<a href="#FD2">2</a>) and (<a href="#FD2">3</a>), the
            contributions of <span class="simple-math"><i>I</i><sub><i>i</i></sub></span> and
              <span class="simple-math"><i>Q</i><sub><i>i</i></sub></span> in a single cell
            are as follows: <a name=""></a><span class="img-equation"><img src="aa23116-13-eq153.png"></span>For
            a wavelength of <span class="simple-math"><i>&lambda;</i> &gt; 7 <i>&mu;</i>m</span>
            one can approximate <span class="simple-math"><i>C</i><sub>abs,i</sub> &asymp;
                  <i>C</i><sub>ext,i</sub></span>. By
            introducing the optical depths for extinction
                  <a name=""></a><span class="img-equation"><img src="aa23116-13-eq156.png"><span class="label-eq">(B.4)</span></span>and polarization
                <a name=""></a><span class="img-equation"><img src="aa23116-13-eq157.png"><span class="label-eq">(B.5)</span></span>we can derive an
            inequality for the <span class="simple-math"><i>Q</i></span> parameter to change its sign as a function of the
            inverse hyperbolic tangent: <a name="FD22"></a><span class="img-equation"><img src="aa23116-13-eq158.png"><span class="label-eq">(B.6)</span></span>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 <span class="simple-math">90&deg;</span> 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. </p>
<br><p><i>&copy; ESO, 2014</i></p>
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