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Astron. Astrophys. 321, 643-651 (1997)

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4. Line profile calculations

We calculated synthetic line profiles and fitted them to observed line profiles in order to derive the magnetic field strength. Therefore we used a Stokes diagnostic code described by Grossmann-Doerth et al. (1988) and Grossmann-Doerth (1993). This version employs the DELO-method for solving the Unno-Rachovsky equations (Rees et al. 1989) numerically and provides the line depression contribution functions necessary to infer the formation height of the absorption line under consideration. The computations were all done with plane parallel models of either the quiet or the active (e.g. umbra, penumbra) solar atmosphere. We used the photospheric model T93 of Schleicher (1976), the penumbra model of Ding & Fang (1989), the umbral model M4 of Kollatschny et al. (1980) and in addition two umbral models IAC-C and IAC-H corresponding to "cool" and "hot" spots (Collados et al. 1994).

Following the matrix technique described by Balthasar & Schmidt (1993), we computed synthetic line profiles for field values from 0 to 3100 G with 50 G increments and for angles between 0 and 90 degrees with 10 degree steps. In all calculations we applied a field gradient of 2 G/km (Pahlke & Wiehr 1990). This results in a matrix of line profiles, whose entries are compared with the observed profiles.

4.1. Oscillator strengths

Unlike to the [FORMULA]  630.25 nm line, where all relevant atomic parameters, in particular the value for the combined oscillator strength and abundance [FORMULA] and the Van-der-Waals fudge factor are well known, we have less information of those parameters for the [FORMULA]  846.85 nm line and the Ti I blend.

In a first approach the value for the weighted oscillator strength of the neutral titanium line given by Vakulenko & Savanov (1990) was adopted. We assumed the solar Ti abundance to be at log [FORMULA] =5.08 (Blackwell et al. 1987). For the Fe content in the solar atmosphere a value of log [FORMULA] =7.50 was adopted (Ross & Aller 1976). The Van-der-Waals fudge factors for both lines at [FORMULA]  846.85 nm were set to a value of 2 because the excitation energy is rather low. Since we have no clue for the corresponding log [FORMULA] value of the Fe I  [FORMULA]  846.85 nm line, we compared synthetic line profiles calculated with different loggf with those of the Kitt-Peak spectral atlas using the model T93. In order to compensate for unresolved velocity fields the synthetic profiles at [FORMULA]  846.85 nm have been convolved with a Gaussian corresponding to 2.0 km/s. A further convolution with a Gaussian of 0.7 pm was applied in order to match the effective spectral resolution of the spectrograph. This comparison yields a value loggf =-2.24 for the iron line at [FORMULA]  846.85 nm. Applying the obtained values to calculations with the umbral model M4 shows that the synthetic profiles fit the observed ones rather badly. A further variation of the loggf values for both lines was therefore performed until finally an acceptable agreement between observed and synthetic line profiles was found for both the quiet sun and for the umbra. The atomic parameters are listed in Table 2.


Table 2. Atomic parameters of the lines at [FORMULA]  846.85 nm .

4.2. Formation height of the lines

The determination of height levels in UDs is a difficult task, limited by the lack of information about the actual temperature and pressure stratification for the different umbral regions. In order to overcome this, the observed umbral and UD line profiles are compared with the computed synthetic profiles from the different model atmospheres. We regarded those model atmospheres for further use which reproduce best the observed profiles and give the right continuum intensity ratio of umbra to photosphere. These were used to derive the height levels of the continuum and the line core. M4 reflects best the situation inside the darkest part of the umbra while the profiles of the quiet sun are best reproduced by the model T93. Fig. 3 (middle) shows an umbral line profile taken from the darkest part of the umbra (dashed) and the corresponding synthetic profile (solid). The direct measurement yields 2700 G, whereas the comparison with synthetic line profiles gives an actual field strength of about 2900 G.

[FIGURE] Fig. 3. Left: the temperature stratifications of the used umbral models M4 (Kollatschny et al. 1980) and IAC-H (Collados et al. 1994) as function of optical depth (continuum, 500 nm). Middle: an umbral line profile of the [FORMULA]  846.85 nm line (dotted) and the synthetic line profile calculated with the M4 model (solid). Right: the line depression contribution function of the line core (dotted) and the contribution function of the emergent continuum intensity (solid) of the [FORMULA]  846.85 nm line as function of optical depth. Both CFs are calculated with the model M4.

Contribution functions (CF) were used to extract the information about the line-forming layers. The weighted center h of the CF is determined for the continuum ([FORMULA]) and the line core ([FORMULA]). All calculations concerning the continuum are done with the CFs for the emergent intensity, while for the line core the CFs for the line depression have been used. Two typical CFs are displayed in Fig. 3 (right). The relationship between the optical depth [FORMULA] and the geometrical height h is given by the model atmospheres.

Table 3 summarizes the results derived from line profiles representative for the quiet sun and the umbra. In the umbra as well in the unperturbed photosphere the continuum of both lines forms near the level where the continuum optical depth [FORMULA] (for a wavelength of 500 nm) equals unity, whereas the line core at least outside the spot is formed in high layers. Single line calculations for the [FORMULA]  846.85 nm line indicate that this is mainly due to the neutral iron line.


Table 3. Height levels of the observed lines inside the umbra and outside the spot. Calculations for the quiet sun and the umbra were done with the model T93 and M4 respectively; [FORMULA] - continuum, [FORMULA] - line core.

The comparison with the other model atmospheres reveals that the IAC-H model of Collados et al. (1994) matches best the UD line profiles. The temperature stratifications used for the calculations of the synthetic line profiles are displayed in Fig. 3 (left). The resulting formation heights are listed in Table 4. Both, the emergent contiuum intensity and the line depression of the CUD2 ([FORMULA]  630.25 nm) originate from deeper layers than those from the PUDs and the CUD1 ([FORMULA]  846.85 nm). A direct comparison of geometric height scales in and outside umbral dots is not possible. This could be only done knowing the Wilson depression of the different umbral regions.


Table 4. Formation heights of the UDs, derived from the IAC-H model; [FORMULA] - continuum, [FORMULA] - line core.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998