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Astron. Astrophys. 341, 902-911 (1999)

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5. Analytical representation

In 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 function

Stenflo 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 µ is given by


where [FORMULA] is a proportionality constant. However, since the continuum polarization is wavelength dependent, [FORMULA], 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 [FORMULA]. The emergent Stokes Q is proportional to the source function for Q, which scales as [FORMULA] due to the Rayleigh phase matrix, as can be seen from Eq. (9). These considerations imply that


Expressed in terms of fractional polarization we then get Eq. (15). The normalization of the intensity in terms of the disk center intensity makes [FORMULA] dimensionless.

In Fig. 6 we have noted that the contribution functions of Stokes I and Q do in fact partly overlap. Test runs have shown that the overlap even increases towards smaller µ values. This indicates that the Schuster-Schwarzschild model is not an ideal characterization. Still, since the peak of the Stokes Q contribution function is located above the point where the maximum of the contribution for Stokes I occurs, we can expect the Schuster-Schwarzschild model to be useful as a first approximation. To correct for higher order deviations we introduce a second parameter [FORMULA] into the function (15) as follows:


As in the case of [FORMULA], we assume the parameter [FORMULA] to be independent of µ, although it may vary with wavelength.

While the choice of the form (15) has been based on simple physical arguments, the introduction of [FORMULA] is motivated more by mathematical simplicity than by physical reasoning. The physics however enters as follows: We will see in the next section that [FORMULA] is positive and much smaller than unity. Therefore it is only relevant for small µ values and thereby accounts for the fact that the overlap of the Stokes I and Q contribution functions increases for smaller µ, which leads to larger deviations from the simple Schuster-Schwarzschild model. Furthermore the original function (15) diverges for [FORMULA]. This is clearly unphysical (Chandrasekhar 1960), since for sufficiently small µ the scattering layer becomes optically thick along the line of sight. The introduction of [FORMULA] improves upon this situation by keeping the polarization at [FORMULA] finite as long as [FORMULA] is positive.

We have performed least squares fits of the function (17) to our computations of the polarization CLV. This gives us values for [FORMULA] and [FORMULA] for different wavelengths and model atmospheres. The wavelength and model dependences of the two parameters will be given explicitly below.

Calculations have shown that [FORMULA], 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 [FORMULA] fixed. Thus only [FORMULA] is a free, model dependent parameter. In Fig. 7 the parameters [FORMULA] and [FORMULA] are shown as functions of wavelength and model atmosphere.

[FIGURE] Fig. 7. Wavelength dependence of the parameters [FORMULA] and [FORMULA], which are used in the analytical function approximating the CLV curves of the continuum polarization. (Note that the logarithm of [FORMULA] is shown.) To avoid overloading the plot we do not show all models. AYP2 yields a similar curve as AYFLUXT1 while the other model atmospheres lie approximately between the FALA7 and FALP3 curves.

5.2. Wavelength dependence of [FORMULA]

The wavelength dependence of [FORMULA] is well represented by a linear function (see Fig. 7),


where the coefficients [FORMULA] are the same for all models. Their values are listed in Table 1, the wavelength being expressed in Å.


Table 1. Coefficients describing the wavelength dependence of the fit parameters [FORMULA] and [FORMULA] (Eqs. (19) and (18)). The wavelength has to be given in units of Å. Note that [FORMULA], [FORMULA], and [FORMULA] are identical for all model atmospheres.

5.3. Wavelength dependence of [FORMULA]

The wavelength dependence of [FORMULA] is best described by the expression


Note that on the left hand side the base-10 logarithm of [FORMULA] is given and not [FORMULA] itself. The coefficients [FORMULA] are listed in Table 1, the wavelength being expressed in Å. As for [FORMULA] we have found that it is not necessary to let [FORMULA] 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 [FORMULA] 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 [FORMULA]. Table 1 might give the impression that [FORMULA] could be neglected because it is a factor [FORMULA] smaller than [FORMULA]. However, this huge factor is due to the circumstance that the wavelength is measured in Å. The terms containing [FORMULA], [FORMULA] and [FORMULA] in Eq. (19) are indeed of the same order of magnitude.

5.4. CLV curves

We 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.

[FIGURE] Fig. 8. Comparison between theoretical, computed CLV curves of the continuum polarization and the fitted analytical function of Eq. (17) using the wavelength dependence of the fit parameters given by the coefficients [FORMULA] and [FORMULA] in Table 1. fThe left panel is representative of the best fit, while the right panel shows one of the worst fits.

The introduction of [FORMULA] 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 [FORMULA]. 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.

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© European Southern Observatory (ESO) 1999

Online publication: December 16, 1998