4 Net circular polarization, $\vec{N(\psi)}$, along an azimuthal section

Figure 3: Left panel: $N(\psi )$ for Fe I 1564.8 nm, with $\theta =15^\circ$. Solid line: With anomalous dispersion; dashed line: without anomalous dispersion. Right panel: Same as left panel, but for Fe I 630.2 nm.

With the model described in the preceding section, we calculate synthetic V-profiles along an azimuthal section for the specified radial position (12000 km). Figure 3 displays the results for Fe I 1564.8 nm (left panel) and for Fe I 630.2 nm (right panel). The calculations are performed with (solid line) and without (dashed line) the effects of anomalous dispersion for $\theta =15^\circ$. For both lines, $N(\psi )$ is symmetric w.r.t. the x-axis ( $\psi=0^\circ$ and $180^\circ$) if the effects of anomalous dispersion are not taken into account. Including anomalous dispersion, this symmetry is broken. For Fe I 630.2 the antisymmetric component is small relative to the symmetric component. However, for Fe I 1564.8 the antisymmetric component dominates $N(\psi )$.

4.1 Symmetry properties of N $\mathsf{(\psi)}$

To understand the antisymmetric component in $N(\psi )$, we consider the symmetry properties of the discontinuities $\triangle v(\psi )$, $\triangle \gamma (\psi )$, and $\triangle \phi (\psi )$ at the interface between the flux tube and the background, which cause the asymmetry in $V(\lambda)$. In our model, $\triangle B$ is negligible at the interface and B only slightly decreases with height. Its influence can therefore be neglected in the following discussion, but we note that $\triangle B(\psi)$ is symmetric and would not alter the following argument.

Figure 4: The azimuthal dependence of the difference in azimuth $\triangle \phi (\psi ) =\phi _{\rm t}-\phi _{\rm b}$ between the flow channel and the background magnetic field w.r.t. the LOS is displayed for a heliocentric angle of 15$^\circ$. The background inclination is 65$^\circ$ while the flow channel is horizontal.

   
Properties of $\triangle \phi (\psi )$, $\triangle \gamma (\psi )$, and $\triangle v(\psi )$:

Taking into account that $\psi = \phi'$ for an axially symmetric sunspot with no azimuthal components, it is seen from Eqs. (1)-(3) that $\gamma(\psi)$ and $v(\psi)$ are symmetric while $\phi(\psi)$ is antisymmetric w.r.t. the transformation $\psi \rightarrow -\psi$. As a consequence, $\triangle\phi(\psi)= \phi_{\rm t}(\psi) - \phi_{\rm b}(\psi)$ is antisymmetric, while $\triangle \gamma (\psi )$ and $\triangle v(\psi )$ are symmetric. In Fig. 4, $\triangle \phi (\psi )$ is plotted for a horizontal tube ( $\gamma'_{\rm t}=90^\circ$), a background magnetic field with an inclination of $\gamma'_{\rm b}=65^\circ$, and a heliocentric angle of $\theta =15^\circ$.

   
$\triangle \phi$ is capable of breaking the symmetry:

It has been demonstrated analytically by LL96 that along the LOS a discontinuity in the azimuth of the magnetic field vector is capable to produce a non-vanishing net circular polarization, N. Their formulas reflect that a discontinuity in $\phi$ along the LOS can produce a non-zero N, if and only if anomalous dispersion is included in the transfer equation of polarized light. Moreover, they demonstrate that the effect of $\triangle \phi$ on N is proportional to $\sin(2\triangle\phi)$, implying that $N=N(\triangle\phi)$ is an antisymmetric function.

Since $\triangle \gamma (\psi )$ and $\triangle v(\psi )$ are both symmetric, $N=N(\triangle\gamma(\psi),\triangle v(\psi))$ must also be symmetric w.r.t. $\psi$. Hence, only $\triangle \phi$ is capable to introduce an antisymmetric component in $N(\psi )$, i.e., $N(\psi )$ is composed of a symmetric contribution from $\triangle \gamma (\psi )$ and $\triangle v(\psi )$ (and from $\triangle B(\psi)$, if present) and of an antisymmetric contribution from $\triangle \phi (\psi )$. The latter contributes to N only if anomalous dispersion is included.

It can be seen in Fig. 3 that the values for N with and without anomalous dispersion are not identical where $\triangle\phi=0$, i.e., for $\psi=0^\circ,180^\circ$. This means that N, which is solely produced by $\triangle \gamma$ at these locations, depends on whether anomalous dispersion is included or not. In other words, switching on the anomalous dispersion introduces both, a symmetric contribution to $N(\psi )$ and the antisymmetric contribution that is caused by $\triangle \phi$.

   
4.2 Difference between Fe I 1564.8 nm and Fe I 630.2 nm

Having shown that $\triangle \phi (\psi )$ causes the antisymmetric component in $N(\psi )$, the question remains, why this effect is small for Fe I 630.2 nm and rather large for Fe I 1564.8 nm. Again, the work of LL96 is of help. They have found an analytical solution for a model with a single discontinuity along the LOS. In their Eqs. (18) and (19), they isolate the effects of $\triangle \gamma$ and $\triangle \phi$ on N (respectively, $\triangle\theta$, $\triangle\varphi$, v in their article). From these equations it is apparent that the weights of the symmetric contribution from $\triangle \gamma$ and the antisymmetric contribution from $\triangle \phi$ depend on the ratio between the wavelength shift due to the Doppler effect and the magnitude of the Zeeman splitting. Hence, the large difference in wavelength between Fe I 630.2 and Fe I 1564.8 is responsible for the significant difference between the two lines, since the Doppler effect depends linearly on wavelength while the Zeeman splitting is proportional to $\lambda^2$. Inserting numbers that correspond to our model into the solution of LL96, Müller (2001) estimates that the $\triangle \phi$-effect should dominate N for Fe I 1564.8 and that the $\triangle \gamma$-effect is more important for Fe I 630.2. Hence, although we cannot separate the effects of $\triangle \gamma$ and $\triangle \phi$ on N in our numerical model, the results presented in Fig. 3 can be understood on the basis of the analytical work of LL96.

4.3 Comparison with observation

Maps of the net circular polarization of sunspot penumbrae have been published by Westendorp Plaza et al. (2001b) in Fe I 630.2 nm and by Schlichenmaier & Collados (2001) in Fe I 1564.8 nm. These measurements reveal that in penumbrae, $N(\psi )$ is essentially symmetric for Fe  I 630.2 nm and antisymmetric for Fe I 1564.8 nm. Our theoretical results, based on synthetic lines that emanate from the moving tube model, are in full agreement with these measurements.