3. Center-to-limb variation of profile parameters
3.1. Extracted parameter relations
Inspection of the 11 diagrams in Fig. 1 immediately reveals several qualitative center-to-limb properties of the profiles. The relative importance of the core polarization peaks increases while the surrounding minima decrease and turn negative as we move away from the limb. The ratio between the D1 and D2 core amplitudes remains approximately constant, and the width of the D2 core peak tends to get narrower further from the limb.
To bring out these properties in quantitative detail we have extracted a number of key profile parameters from the spectra in Fig. 1 and plotted them vs. µ in Fig. 4. The strongest polarization maximum that is not affected by the Hanle effect is the maximum in the blue wing of the D2 line. Its measured center-to-limb variation is represented by the filled circles in the upper left panel of Fig. 4. While the solid curve is a fit to these data points, the almost indistinguishable dashed curve represents 12 times the theoretical continuum polarization at the wavelength of the D2 line, computed from the theory of Fluri & Stenflo (1999) with a model for the average quiet Sun due to Fontenla et al. (1993). The solid curve has been obtained by forming the ratio between the data points and and then fitting a straight line to these ratio points, since they are found to have a linear dependence on µ. The fit then gives us the following expression for the blue wing polarization,
This is the formula used to plot the solid curve in the upper left panel.
In the panel to the upper right we have plotted the ratios between the D2 core polarization amplitude and (filled circles), as well as the ratios between the average of the two surrounding polarization minima, , and (open circles). The dashed and dotted curves are second-order polynomial fits to all the points, given by
For comparison we have in addition plotted with asterisks the similarly determined normalized average of the polarization minima that surround the core peak in the D1 line. We have here limited ourselves to the region , since only for these µ values the structure of the D1 polarization peak is sufficiently undisturbed by cross-talk effects. In this range we find that does not vary with limb distance and has an average value of about -0.11.
In the panel to the lower left we have plotted the ratio between the polarization amplitudes in the cores of the D1 and the D2 lines. This ratio is found to be approximately independent of µ, with a value of about 0.27. The dashed horizontal line represents the average of the points with . The ratio between the polarization maximum in the D1 red wing and the corresponding maximum in the D2 blue wing (not plotted here) is about the same ().
Finally the panel to the lower right provides line width information. The filled circles give the total width at half level of the core polarization peak in the D2 line. The dashed line is a linear fit to the points with . It is given by
The half level used for the width determination is the level that is halfways between and . Similarly the asterisks give the corresponding total width for the D1 line at the half level between and , but limited to the region to avoid the influence of cross-talk effects. The dotted line is a fit given by
The D2 and D1 polarization peaks are much narrower than the corresponding intensity profiles. For comparison we give as open triangles the total width at half the minimum level of the D1 intensity profile, which in contrast to the D2 intensity profile is not significantly affected by blends. The fit in the form of a dashed-dotted line is given by
The polarization peaks are thus narrower by a factor of two or more as compared with the intensity profile. This factor increases towards disk center.
3.2. Influence of the finite spectral resolution
Since the core polarization peaks in the D1 and D2 lines are so narrow, they can only be revealed in observations with sufficient spectral resolution. In our observations the dominating factor that determines the spectral resolution is the geometrical width of the entrance slit, which when optically mapped onto the spectral focus is close to 40 mÅ. The effective smearing width is expected to be somewhat larger than this, since there are other factors (including pixel width and data reduction procedures) that also contribute to the spectral resolution.
To get a better feeling for the amount of spectral smearing in our data we have determined the total width at half the minimum level of a telluric water (H2O) line at 5891.660 Å and plotted the results as the crosses and the horizontal solid line in the lower right panel of Fig. 4. The half width is found to be 85 mÅ, independent of µ. We have then similarly determined the half width from an FTS spectrum at disk center. The FTS spectrum, which may be regarded as fully spectrally resolved, gives a half width of 55 mÅ for the same H2O line (which in the FTS spectrum had about the same strength as in our ZIMPOL spectra and therefore had about the same saturation effects).
The amount of smearing due to the convolution of the instrumental profile with the line profile depends critically on the shapes of these profiles. For dispersion profiles the widths add linearly, for Gaussian profiles they add quadratically. The shape of the telluric line is in the FTS spectrum more dispersion-like than Gaussian. With the measured total half widths of 85 and 55 mÅ in the two cases, it would follow that the half width of the instrumental profile is 30 mÅ if the profiles were dispersion-like, 65 mÅ if they were Gaussian. In practice neither of these extreme cases apply but we have something in between.
For a better quantitative estimate of the smearing function we assume that its shape is rectangular, and then vary its width as the single free parameter. A rectangular profile is chosen both for simplicity, and because the main smearing contribution comes from the projected slit width. Comparing the telluric line profile of the smoothed FTS spectrum with the same profile of our present data set, we find a good agreement when the width of the rectangular smoothing window is about 50 mÅ. This value is consistent with our previous estimate based on the slit width and considering secondary effects that may include contributions from pixel size and data reduction procedures.
From these considerations and our comparison between the smeared and unsmeared telluric line widths we estimate that the true values for the widths of the D1 core peaks are smaller by about 25 mÅ than the values given by Eq. (5), while for the broader D2 core peak the reduction may be expected to be somewhat smaller. As we expect the peak area to be conserved, a reduced width implies a correspondingly increased peak amplitude. Since the narrower D1 peak is more affected than the D2 peak, the ratio between the D1 and D2 amplitudes that was displayed in the lower left panel of Fig. 4 is expected to increase somewhat, in favor of the D1 amplitude. This accentuates the enigma that accompanies the D1 line, which in standard quantum-mechanical scattering theory has been considered to be unpolarizable.
From the detailed profile shapes of the polarized D1 line displayed in Fig. 3 we notice that the dips forming the minima that surround the core peak are narrower than the core peak itself. We may therefore expect that the depths of these minima would increase more than the corresponding increase of the core maximum when the spectral resolution is increased. This applies to the D2 line as well.
Given the uncertainty in the shape and width of the instrumental smearing profile we will not here try to make a deconvolution of the spectra, since such deconvolutions tend to be numerically unstable (e.g. due to division by numbers that are nearly zero in the Fourier domain). Instead we recommend that when trying to model the present observational data, the theoretical profiles should first be smoothed with a rectangular window of width 50 mÅ. The so smoothed model profiles can then be used to extract the various profile parameters that may be directly compared with the empirically determined center-to-limb variations that we have presented here.
© European Southern Observatory (ESO) 2000
Online publication: March 9, 2000