Astron. Astrophys. 363, 289-294 (2000)

4. Comparison with line parameter correlations

Theoretical correlations , , and can be compared with results of spectral observations made with high spatial resolution, namely with correlations , , and , if we suppose that a) we know exactly the region of the atmosphere where the spectral line is formed and b) if the region which contributes mainly to the line formation process is narrow. Both assumptions are supported by results of Kucera et al. (1998). Here denote spatial variations of residual intensity in the line core and correspond to spatial variations of Doppler shifts.

To test whether spectral lines reproduce the main real correlative relationships of the model photosphere we simulated a set of 13 Fe I and 6 Fe II lines using our time-dependent 2-D models. The line parameters are given in Table 1. Together with line wavelength () and low excitation potential (EPL), Table 1 represents an estimation of line formation heights. denotes the effective height of line formation for the line equivalent width (weighted over the whole profile); is the effective height of line formation for central line depth; is the geometrical height at line center optical depth = 1.0. Here denote the integral optical depth in the line and in the continuum at the wavelength of the line center. and were calculated with depression contribution and Unsöld-Pecker weighting functions. All these weighting quantities we computed for a 1-D model which was obtained by spatially and temporally averaging our sequence of 2-D models.

Table 1. Spectral lines used for simulation.

Fig. 5 displays correlations, derived from these simulations. They are shown in dependence on (Fig. 5a-c) and (Fig. 5d-f). If we compare between and (Fig. 2 - Fig. 5a and d), and (Fig. 4 - Fig. 5b and e), and and (Fig. 2 - Fig. 5c and f) it is easy to conclude that a) spectral line correlations correctly reproduce real correlative relationships of the model atmosphere if b) we use the scale of geometrical heights for our sample of spectral lines. Therefore, this scale is more suitable for diagnostic purpose in the context of our 2-D models.

Fig. 5a-i. Correlations found from simulated a -f and observed g -i variations. Error bars are estimates of the standard deviations - they reflect a scatter of the correlation coefficients measured from the set of spectrograms. The open circles and dashed lines represent Fe II lines, filled circles with solid lines are correlations obtained with Fe I lines. The squares stem from Fe I observations of Hanslmeier et al. (1990); triangles from Fe I observations of Kucera et al. (1995) and those presented here; and diamonds from Fe I lines investigated by Balthasar et al. (1990).

To test these simulated correlations we present in Figs. 5g-i correlation coefficients calculated for five Fe I lines from spectral observations described in Sect. 2. The numerical values of correlations are given in Table 2 where means and differences have been derived from two subsequent exposures taken in the same slit position. The height , determined in the same way as in Table 1, is given in Table 2 too. Moreover in Figs. 5g-i some previously published data are plotted from papers of Hanslmeier et al. (1990), Balthasar et al. (1990), and Kucera et al. (1995).

Table 2. Correlations of the spectral line characteristics derived from the observations.

Although these observations do not describe possible relationships in detail, they are not in disagreement with the theoretical prediction. It is very important that show a tendency to decrease their negative correlative relationship in the upper photosphere in agreement with model predictions.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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