## 5. Analytical representationIn order to simplify comparisons with empirical data, we determined an analytical expression, similar to the expression introduced by Stenflo et al. (1997), that approximates the CLV curves of the continuum polarization for all visual wavelengths and all nine solar model atmospheres. Since this function is largely based on physical considerations, it provides us with a better understanding of the formation of the continuum polarization in the solar atmosphere. Here we investigate the applicability and accuracy of our analytical representation. Later our intention is to apply it to observed center-to-limb variations of the continuum polarization to determine the zero level of the polarization scale and to use it for diagnostic work. ## 5.1. Analytical functionStenflo et al. (1997) introduced a function that should describe
the CLV of the continuum polarization when the scattering layer is
optically thin and located above the layer where the major part of the
intensity is formed. This function of where is a proportionality constant. However, since the continuum polarization is wavelength dependent, , being a measure of the degree of polarization, must be a function of wavelength. Let us summarize the reasons why the particular form of function
(15) was chosen. A plane-parallel atmosphere is assumed, in which
scattering is confined to a layer above the slab where the intensity
is produced. This corresponds to the Schuster-Schwarzschild model (cf.
Collins 1989). The path length within the optically thin scattering
layer scales as . The emergent Stokes
Expressed in terms of fractional polarization we then get Eq. (15). The normalization of the intensity in terms of the disk center intensity makes dimensionless. In Fig. 6 we have noted that the contribution functions of Stokes
As in the case of , we assume the
parameter to be independent of
While the choice of the form (15) has been based on simple physical
arguments, the introduction of is
motivated more by mathematical simplicity than by physical reasoning.
The physics however enters as follows: We will see in the next section
that is positive and much smaller
than unity. Therefore it is only relevant for small We have performed least squares fits of the function (17) to our computations of the polarization CLV. This gives us values for and for different wavelengths and model atmospheres. The wavelength and model dependences of the two parameters will be given explicitly below. Calculations have shown that , although needed to improve the fit, is fairly insensitive to the choice of model atmosphere. Therefore the function (17) is fitted to the theoretical CLV curves with fixed. Thus only is a free, model dependent parameter. In Fig. 7 the parameters and are shown as functions of wavelength and model atmosphere.
## 5.2. Wavelength dependence ofThe wavelength dependence of is well represented by a linear function (see Fig. 7), where the coefficients are the same for all models. Their values are listed in Table 1, the wavelength being expressed in Å.
## 5.3. Wavelength dependence ofThe wavelength dependence of is best described by the expression Note that on the left hand side the base-10 logarithm of is given and not itself. The coefficients are listed in Table 1, the wavelength being expressed in Å. As for we have found that it is not necessary to let freely vary from model to model, but we can keep it at a fixed value for all the models. It is absolutely necessary to keep all four significant digits of as given in the table. The reason is that the center-to-limb curves obtained by the analytical function are very sensitive to tiny changes in . Table 1 might give the impression that could be neglected because it is a factor smaller than . However, this huge factor is due to the circumstance that the wavelength is measured in Å. The terms containing , and in Eq. (19) are indeed of the same order of magnitude. ## 5.4. CLV curvesWe tested Eq. (17) against our computations of the continuum polarization CLV for the different model atmospheres. Fig. 8 shows the results of this test for two different model atmospheres and two fixed wavelengths. The left panel is representative of the best fit whereas the right panel shows one of the worst fits.
The introduction of has improved the representation of the theoretical data drastically. For most model atmospheres and wavelengths major deviations of the analytical function from the theoretical curve occur only very close to the limb, for . Since observations that close to the limb (within one arcsec) are very difficult because of seeing, and because the analytical function (17) was introduced primarily to simplify comparisons with observations, we can conclude that the functional form adopted to describe the CLV of continuum polarization is appropriate. © European Southern Observatory (ESO) 1999 Online publication: December 16, 1998 |