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Volume 518, July-August 2010
Herschel: the first science highlights
Article Number A2
Number of page(s) 11
Section The Sun
Published online 18 August 2010

Online Material

\end{figure} Figure 10:

Scatter plots of the longitudinal magnetic field, Bz (first two rows), of the LOS velocity (third row), and of the magnetic field inclination (fourth row), as functions of their corresponding values in the MHD simulations at $\log\tau=-1$. The first column refers to results obtained with the 630.15 nm line; the second column displays results obtained with the 630.25 nm line; results with the two lines at the same time are shown in the third column. Color and line codes are the same as in Fig. 8. The mean value of the rms difference between the inferences and the simulations at $\log\tau=-1$ is given in the upper left corner of each panel.

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Appendix A: Milne-Eddington vs. classical proxies

The study of the solar atmosphere relies on the availability of precise magnetic fields and LOS velocities. Thus, one needs robust diagnostics in order to extract this information from the Stokes spectra. Classical methods such as tachogram techniques, the weak field approximation, and the center-of-gravity technique represent an alternative to Stokes inversions.

For some of these methods, the random errors induced by photon noise have been estimated not to exceed $\sim$20 m s-1 in the case of the LOS velocity or $\sim$10 G for the magnetic flux (see e.g., Scherrer et al. 1995; Scherrer & SDO/HMI Team 2002; Martínez Pillet 2007). However, like in the case of ME inversions, systematic uncertainties coming from the hypotheses underlying the technique are expected to be larger than the random errors themselves. A thorough study, similar to the one we have performed for the ME technique, is thus in order. We carry out such an analysis in this Appendix for the Fourier tachometer technique (Brown 1981; Beckers & Brown 1978)[*], the center-of-gravity method (Rees & Semel 1979; Semel 1967), and the weak field approximation (Landi Degl'Innocenti & Landolfi 2004; Landi degl'Innocenti 1992).

Both the center-of-gravity method and the weak-field approximation are applied to the whole profiles while the Fourier tachometer uses only four wavelength samples across the Stokes I profile (-9, -3, 3, and 9 pm). The center-of-gravity technique extracts the longitudinal component of the magnetic field, Bz, from the separation between the barycenters of the Stokes I+V and I-Vprofiles. Bz can also be obtained with the weak-field approximation through a proportionality between the Stokes V profile and the wavelength derivative of Stokes I. The transverse component of the field, in turn, is derived through a proportionality between Stokes L and the second wavelength derivative of Stokes I[*]. Regression fits are used between the circular (linear) polarization profiles and the first (second) derivatives of the intensity profiles for increased accuracy. Then, the magnetic inclination is obtained from the ratio between the transverse and longitudinal components of the field.

Figure 10 summarizes the results. Each column refers to a different set of lines: Fe I 630.15 nm (left), Fe I 630.25 nm (middle), and the two lines simultaneously considered (i.e., ME inversion; right). The labels on the ordinates are self-explanatory, while the abscissae give the values of the corresponding quantities at $\log\tau=-1$.

The less accurate method turns our to be the weak-field approximation: the inferred magnetic inclinations show larger rms fluctuations than the ME ones, and the longitudinal component of the field displays a clear saturation for fields stronger than 1000-1100 G when calculated with the line at 630.25 nm. For Bz values above 1 kG, the weak field inferences resulting from the 630.15 nm line seem to be closer to the MHD values at $\log\tau=-1$than the ME ones. Nevertheless, they present a larger scatter. The center-of-gravity method looks very robust and, indeed, it presents less scatter than ME inversions for the longitudinal field component and the LOS velocity (only the results from the 630.15 nm line are shown; the results for 630.15 nm are very similar). The good performance of the center-of-gravity method was noticed earlier by Cauzzi et al. (1993) and Uitenbroek (2003). Unfortunately, this technique does not provide information about the field inclination. The LOS velocities resulting from the tachometer are fairly comparable to those of the ME inversion. The scatter is similar in both cases.

In summary, the ME inversion seems to be the more complete and accurate technique, although none of the others can be discarded. In particular, a combination of the center-of-gravity technique for calculating Bz and  $v_{\rm LOS}$ along with the weak field approximation for the magnetic inclination may represent a suitable alternative which is much less expensive in terms of computing resources. It is important to note, however, that the results of our study are only valid when the magnetic field is spatially resolved. Further investigation is needed to check the applicability of these techniques when the field is unresolved. This additional investigation is important in view of the theoretical deviations predicted by Landi Degl'Innocenti & Landolfi (2004).

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