Astron. Astrophys. 356, 200-208 (2000)

3. Model atmosphere

To determine atmospheric parameters and abundances for a star with peculiarities similar to HD 166473 is quite difficult. In Sect. 1 it was already mentioned that this star has the most peculiar photometric colours next to Przybylski's star (HD 101065) which probably are a result of a large overabundance of rare earth elements (see Sect. 4.3). Only HD 101065 appears to be even more abnormal in this respect.

To determine the starting values for the model atmosphere of HD 166473 we used the software package TEMPLOGG (Rogers 1995) which takes advantage of calibrations of the Strömgren system (Crawford 1979, and subsequent calibrations) and we derived a = 7900 K and = 4.4. For the initial model atmosphere we used [M/H] = +1.00, which is close enough to the photometric estimate of [M/H] = +1.27 and which is a standard abundance value for Kurucz's scaled solar model atmospheres (Kurucz 1993). For the microturbulent velocity an artificially high value of = 4.0 km s^{- 1} was chosen, simply to compensate for missing magnetic intensification caused by a field of 8.6 kG. The naming convention used for the models includes , , [M/H], and , for example: 7900g44p10k4.

The final synthetic spectrum was computed with the SYNTHMAG code (Piskunov 1999), based on a radiation transfer in the presence of a magnetic field. The large microturbulence was used only for model calculations, while for spectrum synthesis the microturbulence was set to zero, because a strong magnetic field supresses any turbulent motions, which is corroborated by the investigation of other roAp stars (see Paper I - Paper IV).

After several iterations of the whole synthesis procedure (for technical details see Gelbmann et al. 1997, Gelbmann 1998), we arrived at a compromise model with the following parameters: = 7700 K, = 4.2, and [M/H] = +0.50. Unfortunately, this model does not satisfy all criteria for a consistent spectrum analysis. Although it reproduces reasonably well the H line, the Strömgren colours, and provides an ionization equilibrium, it still cannot remove a slight dependency of the abundance on the lower levels exitation energy of spectral lines for both Fe I and Fe II . A higher temperature would be needed to remove this trend, but simultaneously it would degrade the fit to photometric indices and the H line profile. On the other hand, a Kurucz model atmosphere with =2 km s^-1, = 7250 K, = 4.5, and [M/H] = +1.00 formally reproduces well the Strömgren indices, but does not fit any more the H line profile. The value for = 4.2, determined spectroscopically with the procedure previously described and using a higher microturbulence, is more appropriate for a main sequence star and hence was chosen for the "final" model atmosphere.

Another serious shortcoming of the traditional approach comes from the peculiar chemical composition of the atmosphere. The deviation of the abundance pattern of HD 166473 from the solar values affects the [M/H] determination, which is calibrated for main sequence stars with scaled solar abundances. Particularly, the rare earth elements (REE) are important, because they have spectra with more spectral lines than the iron peak elements and hence contribute much to the line opacities. Unfortunately, our knowledge of the atomic parameters of REE is limited to a few thousand lines of the neutral and first ionized species, while we include millions of iron peak element lines. Moreover, the total number of REEs exceeds the total number of iron peak elements.

The following experiment illustrates the importance of the specific chemical abundance in model atmosphere calculations. On average, REE are overabundant in HD 166473 by +2.8 dex. We calculated opacity distribution functions (ODFs) for Przybylski's star which is extremely overabundant in REE by about +4.0 dex and deficiencient in CNO. In order to compensate for the missing line opacities due to insufficiently known REE spectral lines we increased the overabundance of the iron peak elements to +1.5 dex. All calculations were performed with the procedure developed by Kupka & Piskunov (Kupka & Piskunov 1998; Piskunov & Kupka 2000).

Table 1 compares the observed Strömgren indices of HD 166473 with three sets of synthetic indices. The first set is produced from a scaled solar abundance Kurucz' model with = 7700 K, = 4.2, [M/H] = +0.50, and = 0 km s^-1, the second differs only by a = 4 km s^-1. The third set is obtained with a model of = 7550 K, = 4.2, and = 1 km s^-1 but with an ODF developed for Przybylski's star which resembles the abundance pattern of HD 166473.

Table 1. Comparison of observed and calculated Strömgren indices of HD 166473 for Kurucz's model atmospheres using scaled solar abundances (scaled) and using a more realistic, but not yet optimized opacity distribution function (ODF) table.

The results presented in Table 1 illustrate the possibility to reproduce the observed colour indices (except for in this case) even for most peculiar objects provided that the peculiar chemical composition of CP stars correctly is taken into account. An analysis based on the 7550g42_ODF model still cannot remove the previously mentioned trend of line abundances versus lower excitation energy levels. The ODF model, however, provides the same average Fe peak element abundance as our adopted "final" model.

For the following reasons we did not use our ODF model (adapted for Przybylski's star) for the abundance analysis of HD 166473:
- Missing line opacities from REE were substituted by enhanced lines of the iron peak elements, but both groups of elements have a different distribution of lines over the entire spectral range.
- A direct comparison of abundances for HD 166473 with the abundances determined for other roAp stars, but which are based on scaled solar composition models, would be difficult.

The Balmer hydrogen lines are sensitive to . The profiles of H and H were calculated for a set of models, compared with the observations, and a possible range of = 7500 K to 8000 K was determined. Hence, the formal error in our effective temperature determination is 250 K. The error of the surface gravity is estimated to 0.2 dex. Fig. 1 shows a comparison of the observed H line with three different synthetic profiles.

Fig. 1. Comparison between the observed H line profile and three different synthetic profiles. The solid line represents the model 7700g42p05k4, the dashed line - 8000g42p05k4, and the dash-dotted line - 7500g42p05k4.

3.1. Magnetic field modelling

The strong magnetic field in HD 166473 has to be taken into account in an abundance analysis. Radiation transfer calculations in the presence of a magnetic field were implemented in the spectrum synthesis code (Piskunov 1999) and applied to Eu lines in magnetic CP stars (Ryabchikova et al. 1999b) as a test.

First we had to determine a magnetic field configuration which reproduces the Zeeman pattern of the lines observed in the high resolution spectrum which ranges from 6123 to 6178 Å. As observations of only a single phase are available we cannot determine a complete magnetic field model, and we have to accept that our "solution" will be far from unique. As a matter of fact, a large variety of more-or-less complicated field geometries with various multipole components allows the reproduction of the Stokes I component as is observed in our spectrum within the observational errors. We considered it reasonable for our abundance analysis to choose the magnetic model which is the simplest and has a minimum of free parameters.

To determine such a model we used the program INVERS10 (Piskunov 1999) which originally was written to solve the inversion problem and to perform magnetic Doppler imaging. We applied this code in the "direct" mode, i.e. we assumed a magnetic field geometry and computed the disk integrated spectrum. It was not possible to reproduce the high resolution spectrum with a centered dipole and even off-sets of 0.5 radii were not sufficient, hence we investigated more complex models. A decentered dipole (polar field strength of 10 kG) with the magnetic center close to the stellar surface, the magnetic axes perpendicular to the rotation axes, and with a radial field component of +1 kG superposed (Fig. 2) produced a satisfactory fit. Such a field geometry definitely does not represent the actual geometry for HD 166473, but on the level of a phase snap-shot disk average it reproduces well the Stokes I components of our observations. A refinement of this model was considered not to be critical for the interpretation of the low resolution spectra which, however, were the basis of our abundance analysis.

Fig. 2. Magnetic surface field vectors derived from our model which is used to compute the Stokes I component for our synthetic spectra

The chosen 3-D model with 200 surface elements fits both observed spectra: the high resolution (instrumental broadening of 0.08 Å) and small spectral range spectrum with clearly visible Zeeman pattern and also the spectrum with lower resolution (instrumental broadening of 0.3 Å), but wide spectral range (Fig. 3.1). For the latter spectrum we found negligible differences between a synthesis based on a detailed 3D magnetic field model, and on a simpler model which assumes a plan-parallel atmosphere with a radial magnetic field component of 7 kG and a field component parallel to the surface of 5 kG at every surface point. As the final abundance analysis had to use a spectral range as wide as possible, the relevant issue was to model efficiently the low resolution spectrum. Hence, to save significant computing time this analysis was carried out with the program SYNTHMAG (Piskunov 1999) which uses the simplified magnetic field geometry.

Fig. 3. Thin line with +: high resolution observations, thin line with x: low resolution spectrum, full lines: synthetic spectra convolved with the instrumental profiles for, respectively, the high and low resolution spectra. Note the line close to 6148.8 which is missing in our atomic line data base.

3.2. Abundance analysis

Line lists for all chemical elements were generated with the current version of VALD-2 (Kupka et al. 1999, Ryabchikova et al. 1999a), and lines in the Balmer wings as well as in spectral regions crowded with telluric lines were ignored. The most recent references for the atomic data of individual species will be given in the corresponding sections. We did not analyze the spectral region below 5000 Å because of severe blending. The optimum fit of synthetic lines to the observations resulted in a total of 366 individual "line" abundances which were appropriately averaged to determine the final abundances of 33 elements.

An example for the spectral region around 6148 Å is given in Fig. 3.1. The large diversity of Zeeman patterns which have to be take into account is obvious. Differences of the fit to the observations for some of the lines may also be caused by an inhomogeneous surface distribution of the elements. For our synthesis we used the same magnetic field geometry for the entire spectrum and for all the elements.

© European Southern Observatory (ESO) 2000

Online publication: March 28, 2000
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