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Astron. Astrophys. 361, 734-742 (2000)

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3. Results from the inversion code

In this section, we study if our modified uncombed model can explain some of the results obtained by WESa,b. In particular, we expect the horizontal flux tubes included in the model to be misinterpreted by the SIR inversion code as long as they remain unresolved. The SIR code has been thoroughly tested in the past (Ruiz Cobo & del Toro Iniesta, 1992; Westendorp Plaza et al., 1998). These last authors also retrieved successfully atmospheric models that included sharp discontinuities along the LOS (canopy-like atmospheres). But all these tests refer to homogeneous atmospheres. That is, they simulate observations where all LOS contributing to the observed profiles pass through the same atmosphere. With 1 ", or worse, resolution, it is clear that the structure of the penumbral magnetic field remains unresolved and that the observed profiles are coming from a combination of profiles formed in very different atmospheric conditions. We believe this is at the origin of the increase of the penumbral field strength and the sharp decrease of the inclination deduced by Westendorp Plaza and co-workers.

To prove this point, the profiles built in Sect. 2 (at disk center and with a filling factor of 0.5 for the background and tube models) are introduced as simulated observations into the SIR code. The synthetic profiles have noise added to simulate a S/N of 103 (similar to the ASP). The initial guess model is the background model and the number of nodes 2 used is similar to those in WESa,b. In Fig 3, we show the synthetic profiles (dots) and the fit produced by SIR (solid lines). The fit is as good as it can be for this S/N level (see the residual in the Stokes U panel). The blueshift (rest wavelength is given by the vertical dashed lines) produced by the Evershed outflow is evident in all Stokes profiles. The Stokes V profile coming from the model with the tube has an NCP of 1.6 mÅ and 2.9 mÅ for the FeI 6301 and 6302 lines respectively. The profiles in Fig. 3 have around half these values because the background profiles are almost perfectly antisymmetric.

[FIGURE] Fig. 3. Stokes parameters for the FeI 6301 and 6302 lines in the cold model with tubes at 150 km height. A filling factor of 50% is assumed for this model, the rest being the background model. Dots: synthetic data, solid lines: fit produced by SIR, dotted lines: residuals. Vertical dashed lines: rest wavelengths.

The interesting result here is the atmospheric model retrieved by SIR. It is shown in the four panels of Fig. 4 with dotted lines and error bars. The output SIR model is compared with a "mean" atmospheric model made after averaging the tube model and the background model (solid line). This average is first made in a geometrical height scale and afterwards its own optical depth scale is computed. Note that this model does not correspond to the model of any particular point in the atmosphere, it only represents average conditions within the resolution element that produced the synthetic profiles input to SIR. As can be seen in the right panels, the field strength (top) increases with height (around 600 G in the range [FORMULA]) and the field inclination (bottom) decreases abruptly from 74o to less than 40o at the top of the atmosphere. Thus, the synthetic profiles produced by the (unresolved) uncombed model are interpreted by SIR in a way that resembles very much the results of WESa,b. The SIR code only feels the effect of the tubes at the deeper layers and it fixes the atmospheric gradients accordingly. But it never retrieves the localized perturbation. This result also applied to the profiles coming from hot tubes and from tubes at a height of 250 km above the [FORMULA] level. We want to emphasize that this deceptive looking fit is not a fault of the SIR code. But rather, it represents the failure of one of the assumptions: that the atmosphere that generates the profiles is homogeneous. If we feed SIR with profiles generated from the tubes models but mixing no background contribution (i.e., a filling factor of one or, equivalently, a resolved uncombed penumbral model), SIR does a much better job in reproducing the perturbation represented by the tubes. Indeed, in the top part of the Field Strength panel of Fig. 4, we show the fit (thin dashed line, shifted upwards by 1500 G) produced when the filling factor is one. The thick dashed line is the same as in Fig. 1, also shifted by 1500 G.

[FIGURE] Fig. 4. Atmospheric models resulting from the inversion of the profiles in Fig. 3. Dotted lines: model derived by SIR and error bars. Solid line: model produced from averaging 50% of the model with a cold tube and 50% of the background model. See text for details.

WESa,b detected in their inversions what they called a "differential opacity effect" between the limb- and center-side penumbra. The reason to invoke a differential opacity was the fact that at [FORMULA] in the limb-side penumbra, SIR retrieved field vectors at 90o (or slightly larger) and redshifted LOS velocities while, at the center-side penumbra, smaller inclinations and blueshifted LOS velocities were found. Here, we investigate the origin of the differential opacity effect in the framework of the uncombed model. That, in general, one would probe different geometrical heights (for the same optical depths) in the limb- and center-side penumbra is easily seen from Fig. 5. The center- and limb-side LOS "see" a very similar atmosphere until they reach the top part of the tube. From there on, the path they follow inside it is different for each LOS. The limb-side LOS transverses a total length of [FORMULA] (d being the width of the tube). On the other hand, the center-side LOS has a larger path within the structure [FORMULA]. The tubes have a higher density than the background model and these different paths introduce a net difference in the optical depths seen at the two penumbral sides. In our case [FORMULA] and for the sunspot analyzed by WESa,b [FORMULA], the difference in the paths between the limb- and center-side penumbrae is only of around 5%, which is too small to explain the observed differences. A more significant path difference of 20% or larger is achieved for [FORMULA]. So, our realization of the uncombed penumbral model does not seem to generate a big enough differential opacity effect (we have quantified this to be of only a few kilometers). Note, however, that this result is very sensitive to the actual value of [FORMULA]. If this value is smaller than the one we use (as the footpoints of the elevated Evershed channels of Rimmele 1995would indicate) the effect can be more pronounced.

[FIGURE] Fig. 5. Representation of the path of a limb-side LOS and a center-side LOS. Both lines cross the penumbral background field and the penumbral flux tube but the cross-section of the tube seen by the center-side LOS is larger than for the limb-side LOS. The dashed line represents the solar vertical.

In order to understand, then, the differences observed by WESa,b, we have used our profiles synthesized at [FORMULA] in the center- and limb-side penumbra and analyzed them with SIR. In what follows, we refer only to the model with cold tubes at a height of 150 km. The actual profiles that were inverted had a 0.5 filling factor of background model added and noise to simulate a S/N of 103. The results are seen in Fig. 6. The top panels are the stratifications retrieved by SIR for the velocity (left) and magnetic field inclination (right). These figures display atmospheric models corrected for projection effects. That is, they provide the atmospheric quantities as they would be seen by an observer looking straight down into the penumbra. The limb-side stratifications (dotted lines) show the same qualitative effects as obtained by Westendorp Plaza and coworkers. In the layers around [FORMULA] one finds redshifted velocities and inclinations that are close to 90o. This is not observed in the center-side results (dashed lines). Both models are much more similar at layers around [FORMULA] in agreement with WESa,b. The average stratification (50% tube model, 50% background) is given in both cases by the thick solid line. As stated before, this difference cannot be understood in terms of observing different geometrical heights in each side of the penumbra. We believe the reason for these differences to be in the actual difference between the profiles synthesized in each penumbral side and the analysis carried out by SIR. In the bottom panels of Fig. 6, we plot the stratifications along the LOS that generated the profiles at the limb- (dotted) and center-sides (dashed). The LOS velocity at the limb-side shows a very strong gradient at the tube location which generates profiles that do not preserve the usual symmetry properties of the Stokes parameters (Q and U symmetric and V antisymmetric). Indeed, because of the large velocity gradient and the polarity change in the magnetic field inclination, the Stokes V profile coming from the tube atmosphere shows the well-known "cross-over effect" (Grigorjev & Katz 1972, 1975; but note that the background model does not). However, the LOS velocity in the center-side penumbra shows a very small velocity gradient (dashed line) and the tube profiles are almost perfectly symmetric (antisymmetric for Stokes V). This agrees with the fact that the center-side penumbral model at this µ value shows negligible NCP (see Fig. 2). The difference between the symmetries of the profiles at these two sides and the gradients needed by SIR to reproduce them explains the contrast between the results obtained at each penumbral side.

[FIGURE] Fig. 6. Top left: LOS Velocity as retrieved from SIR for the limb- (dotted) and center-side (dashed) penumbra. The models are projected so that the abcisae are measured along the local vertical. Thick solid line gives the mean stratification. Top right: The same but for Inclination. Bottom left: projected velocities along the LOS for the limb- (dotted) and center-side (dashed) penumbrae. Bottom right: Inclination projected along the LOS for the two cases.

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© European Southern Observatory (ESO) 2000

Online publication: October 2, 2000