## 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 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. 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 D 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 D In the panel to the lower left we have plotted the ratio between
the polarization amplitudes in the cores of the D 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 D 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 D The D 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 D 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 (H 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 D From the detailed profile shapes of the polarized D 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 |