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Ïîèñêîâûå ñëîâà: arp 220
Astronomy & Astrophysics manuscript no. reference2 June 29, 2009

c ESO 2009

The chemical abundance analysis of normal early A- and late B-type stars
L. Fossati1 , T. Ryabchikova1,2 , S. Bagnulo3 , E. Alecian4,5 , J. Grunhut5 , O. Kochukhov6 , and G. Wade
1 2 3 4 5 6

5

¨ ¨ ¨ Institut fur Astronomie, Universitat Wien, Turkenschanstrasse 17, 1180 Wien, Austria. e-mail: fossati@astro.univie.ac.at,ryabchik@astro.univie.ac.at Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya 48, 119017 Moscow, Russia. e-mail: ryabchik@inasan.ru Armagh Observatory, College Hill, Armagh BT61 9DG, Northern Ireland, UK. e-mail: sba@arm.ac.uk Observatoire de Paris-Meudon, LESIA, UMR 8111 du CNRS, 92195 Meudon Cedex, France. e-mail: evelyne.alecian@obspm.fr Physics Dept., Royal Military College of Canada, PO Box 17000, Station Forces, K7K 4B4, Kingston, Canada. e-mail: Jason.Grunhut@rmc.ca,Gregg.Wade@rmc.ca Department of Physics and Astronomy, Uppsala University, SE-751 20, Uppsala, Sweden. e-mail: Oleg.Kochukhov@fysast.uu.se
ABSTRACT

Context. Modern spectroscopy of early-type stars often aims at studying complex physical phenomena such as stellar pulsation, the

peculiarity of the composition of the photosphere, chemical stratification, the presence of a magnetic field, and its interplay with the stellar atmosphere and the circumstellar environment. Comparatively less attention is paid to identifying and studying the "normal" A- and B-type stars and testing how the basic atomic parameters and standard spectral analysis allow one to fit the observations. By contrast, this kind of study is paramount eventually for allowing one to correctly quantify the impact of the various physical processes that occur inside the atmospheres of A- and B-type stars. Aims. We wish to establish whether the chemical composition of the solar photosphere can be regarded as a reference for early Aand late B-type stars. Methods. We have obtained optical high-resolution, high signal-to-noise ratio spectra of three slowly rotating early-type stars (HD 145788, 21 Peg and Cet) that show no obvious sign of chemical peculiarity, and performed a very accurate LTE abundance analysis of up to 38 ions of 26 elements (for 21 Peg), using a vast amount of spectral lines visible in the spectral region covered by our spectra. Results. We provide an exhaustive description of the abundance characteristics of the three analysed stars with a critical review of the line parameters used to derive the abundances. We compiled a table of atomic data for more than 1100 measured lines that may be used in the future as a reference. The abundances we obtained for He, C, Al, S, V, Cr, Mn, Fe, Ni, Sr, Y, and Zr are compatible with the solar ones derived with recent 3D radiative-hydrodynamical simulations of the solar photosphere. The abundances of the remaining studied elements show some degree of discrepancy compared to the solar photosphere. Those of N, Na, Mg, Si, Ca, Ti, and Nd may well be ascribed to non-LTE effects; for P, Cl, Sc and Co, non-LTE effects are totally unknown; O, Ne, Ar, and Ba show discrepancies that cannot be ascribed to non-LTE effects. The discrepancies obtained for O (in two stars) and Ne agree with very recent non-LTE abundance analysis of early B-type stars in the solar neighbourhood.

1. Introduction
In the last decade there has been dramatic improvement in the tools for the analysis of optical stellar spectra, both from the observational and theoretical perspective. New high-resolution echelle instruments have come online, which cover much broader spectral ranges than older single-order spectrographs. Data quality has also substantially improved in terms of signalto-noise ratio (SNR), because of substantially greater instrument efficiency, and the use of large-size telescopes. Thanks to the vibrant observational activities of the past few years, and thanks to efficient and user-friendly data archive facilities, a huge highquality spectroscopic database is now available to the public. With the development of powerful and cheap computers, it has become practical to exploit these new data by performing spectral analysis using large spectral windows rather than selected spectral lines, at a level of realism heretofore impos-

sible. The high accuracy of observations and modelling techniques now allows, for example, stretching the realm of abundance analysis to faster rotators than was possible in the past, but also provides the possibility of learning more about the structure of the stellar atmospheres and the ongoing physical processes, especially when spectral synthesis fails to reproduce the observations. For instance, observed discrepancies between observed and synthetic spectra have allowed us to discover that the signature of chemical stratification is ubiquitous in the spectra of some chemically peculiar stars (Bagnulo et al. 2001; Wade et al. 2001; Ryabchikova et al. 2003) and to perform accurate modelling of this stratification in the atmospheres of Ap stars (for instance, Ryabchikova et al. 2005; Kochukhov et al. 2006). However, our inability to reproduce observations frequently stems for a very simple cause: that atomic data for individual spectral lines are incorrect. For solar type stars it is possible to construct a "reference" list of reliable spectral lines with reliable


2

Fossati et al.: The abundance analysis for normal A- and B-type stars.

atomic data through the comparison of synthetic spectra with the solar observed spectrum, since the solar abundances are accurately known. In many cases, however, the solar spectrum cannot provide the required information, because the temperature of the target stars is significantly different from that of the sun. This problem can be overcome by adopting an analogous reference at a temperature reasonably close to the target temperature. The method involves a selection of a set of suitable reference stars for which very high quality spectra are available. Then an accurate determination of the stellar photospheric parameters and an accurate abundance analysis are performed with the largest possible number of spectral lines and the best possible atomic data. Finally, those spectral lines exhibiting the largest discrepancies from the model fit are identified, and their atomic data revised by assuming that the average abundance (inferred from the complete sample of spectral lines of that element) is the correct one. In this process it is important to take effects into account that can potentially play a significant role in all stellar atmospheres, such as variations in the model structure from non-solar abundances and non-LTE effects. In this paper we address the problem of establishing references for effective temperatures around 10000­13000 K. This is in some respect the easiest temperature range to study, as well as one of the most interesting. This temperature is close to ideal because the spectra of stars in this interval are generally unaffected by severe blending. It is also relevant because stars in this temperature range display spectroscopic peculiarities (chemical abundance peculiarities, stratification, Zeeman effect, etc.) that reflect physical conditions and processes of interest for detailed investigation. A crucial prerequisite for studying and characterising these phenomena is the capacity to model the underlying stellar spectrum in detail, and this requires high quality atomic data. The highest degree of accuracy in abundance analysis is reached for sharp-lined stars. Unfortunately, these objects are quite rare among A- and B-type stars, which are generally characterised by high rotational velocities. Furthermore, most of the slowly rotating stars in the chosen temperature region belong to various groups of magnetic and non-magnetic, chemically peculiar objects. As a matter of fact, many previous studies aimed at determining the chemical composition of "normal" early A- and late B-type stars were based on samples "polluted" by moderately chemically peculiar stars. For instance, the work by Hempel & Holweger (2003) includes the sharp-lined HgMn star 53 Aur. Hill & Landstreet (1993) searched for compositional differences among A-type stars, but four out of six programme stars are in fact classified as hot Am stars on the basis of the abundances of the heavy elements Sr-Y-Zr-Ba, which are considered as diffusion indicators (i.e., Sirius, o Peg, and Leo, see Hempel & Holweger 2003). The complexity of the problem of distinguishing between normal A and marginal Am stars is further stressed by Adelman & Unsuree (2007). The aim of the present paper is to search for sharp-lined early A- or late Btype stars with a chemical composition as close as possible to the solar one. As a final outcome, one could assess whether the chemical composition of the solar photosphere may be considered at least in principle as a reference for the composition of the early A- and late B-type stars. If such a star is found, this will not imply that the solar composition is the most characteristic for the slowly rotating A- and B-type stars, but will be used as further evidence that any departure from the composition of the solar photosphere has to be explained in terms of diffusion or other physical mechanisms that are not active at the same efficiency level in the solar photosphere.

Our work is based on a very detailed and accurate study of a vast sample of spectral lines. As a by-product, we provide a list of more than 1100 spectral lines from which we have assessed the accuracy of the corresponding atomic data. Such a list may serve as a future reference for further abundance analysis studies of stars with a similar spectral type. This paper is organised as follows. Sect. 2 describes the target selection, observations and data reduction, Sect. 3 presents methods and results for the choice of the best fundamental parameters that describe the atmospheres of the programme stars, Sect. 4 presents the methods and results of the abundance analysis of the programme stars. Our results are finally discussed in Sect. 5.

2. Target selection, observations, and data reduction
For our analysis we need a late A-type or early B-type star with a sharp-line spectrum (hence the star must have a small sin i) and one exhibiting the least possible complications due to phenomena such as non-homogeneous surface distribution of chemical elements, pulsation, or a magnetic field. Finding such a target is not a simple task, because most of the early-type stars are fast rotators. Slowly rotating A- and B-type stars generally show some type of chemical peculiarity often associated to the presence of abundance patches, a magnetic field (which broadens, or even splits spectral lines), and chemical stratification. Even Vega, which has been considered for a long time as the prototype of a "normal", slow rotating A-type star, is in fact currently classified as a Boo star and discovered to actually be a fast rotating star seen pole-on, exhibiting, as such, distorted line profiles (Adelman & Gulliver 1990; Yoon et al. 2008). Based on our knowledge, we reached the conclusion that the most suitable target for our project is the B9 star 21 Peg (HD 209459), which is known from previous studies as a "normal" single star with sin i 4 km s-1 (see, e.g. Sadakane 1981). We felt it was necessary to consider additional targets of our spectral analysis, for two main reasons. Since we intend to provide an accurate reference for the typical abundances of the chemical elements in A- and B-type stars (and compare these values with the solar ones), we need to cross-check with further examples whether the results obtained for 21 Peg are similar to those of other "normal", slow rotating A-type stars. Second, to check the accuracy of the astrophysical measurements of the logg f values, which is a natural complement of the present work. Both these goals are best achieved with the use of abundance values that have been obtained with a homogeneous method, rather than from a mixed collection of data from the literature. Therefore, we have also analysed another two stars of similar temperature as 21 Peg, i.e., HD 145788 (HR 6041) and Cet (HR 811). Both stars fulfill our requirements, although are slightly less ideal than 21 Peg. HD 145788, suggested to us by Prof. Fekel, is a slowly rotating single star with sin i 8 km s-1 (Fekel 2003). Cet, a SB1 with sin i 20 km s-1 , shows an infrared excess, and is a suspected Herbig Ae/Be star (Malfait et al. 1998). Since its spectrum is not visibly contaminated by the companion, it still serves our purpose. Cet was also already used as a normal comparison star in the abundance study of chemically peculiar stars by Smith & Dworetsky (1993). The star 21 Peg was observed five times during two observing nights in August 2007, with the FIES instrument of the North Optical Telescope (NOT). FIES is a cross-dispersed high-


Fossati et al.: The abundance analysis for normal A- and B-type stars.

3

´ resolution echelle spectrograph that offers a maximum spectral resolution of R = 65 000, covering the entire spectral range 3700­7400 å. Data were reduced using a pipeline developed by D. Lyashko, which is based on the one described by Tsymbal et al. (2003). All bias and flat-field images were median-averaged before calibration, and the scattered light was subtracted by using a 2D background approximation. For cleaning cosmic ray hits, an algorithm that compares the direct and reversed observed spectral profiles was adopted. To determine the boundaries of echelle orders, the code uses a special template for each order position in each row across the dispersion axis. The shift of the row spectra relative to the template was derived by a crosscorrelation technique. Wavelength calibration of each image was based on a single ThAr exposure, recorded immediately after the respective stellar time series, and calibrated by a 2D approximation of the dispersion surface. An internal accuracy of 100 ms-1 was achieved by using several hundred ThAr lines in every echelle order. Each reduced spectrum has a SNR per pixel of about 300 at 5000 å. All five spectra are fully consistent among themselves, which confirms that the star is not variable. This allowed us to combine all data in a unique spectrum with a final SNR of about 700. Because of the very low sin i of 21 Peg, we made use of a very high-resolution spectrum (R = 120 000) obtained with the Gecko instrument (now decommissioned) of the CanadaFrance-Hawaii Telescope (CFHT, Landstreet 1998) to measure this parameter. The spectrum covers the ranges 4612­4640 å and 5160­5192 å, which are too short to perform a full spectral analysis, but sufficient to measure sin i with high accuracy. According to Hubrig et al. (2006) the mean longitudinal magnetic field of 21 Peg is -144±60 G, which excludes the possibility that the star has a structured magnetic field. The spectrum of HD 145788 was obtained with the cross´ dispersed echelle spectrograph HARPS instrument attached at the 3.6-m ESO La Silla telescope, with an exposure time of 120 s, and reduced with the online pipeline 1 . The reduced spectrum has a resolution of 115 000, and a SNR per pixel of about 200 at 5000 å. The spectral range is 3780­6910 å with a gap between 5300 å and 5330 å, because one echelle order is lost in the gap between the two chips of the CCD mosaic detector. The star Cet was observed with the ESPaDOnS instrument of the CHFT February, 20 and 21 2005. ESPaDOnS consists of a table-top, cross-dispersed echelle spectrograph fed via a double optical fiber directly from a Cassegrain-mounted polarisation analysis module. Both Stokes I and V spectra were obtained throughout the spectral range 3700 to 10400 å at a resolution of about 65 000. The spectra were reduced using the Libre-ESpRIT reduction package Donati et al. (1997, and in prep.). The two spectra (each obtained from the combination of four 120 s subexposures) were combined into a final spectrum that has an SNR per pixel of about 1200 at 5000 å. The observation of Cet enters in the context of a large spectropolarimetric survey of Herbig Ae/Be stars. Least-squares deconvolution (LSD, Donati et al. 1997) was applied to the spectra of Cet assuming a solar abundance line mask corresponding to an effective temperature of 13000 K. The resulting LSD profiles show a clean, relatively sharp mean Stokes I profile, corresponding to sin i = 20 ± 1 km s-1 , and no detection of
1 http://www.ls.eso.org/lasilla/sciops/3p6/harps/ software.html#pipe

any Stokes V signature indicative of a photospheric magnetic field. Integration of Stokes V across the line using Eq. (1) of Wade et al. (2000) yields longitudinal magnetic fields consistent with zero field and with formal 1 uncertainties of about 10 G. The high-resolution spectropolarimetric measurements therefore provide no evidence of magnetic fields in the photospheric layers of the star. All the spectra of the three stars were normalised by fitting a spline to carefully selected continuum points. For each object, radial velocities r were determined by computing the median of the results obtained by fitting synthetic line profiles of several individual carefully selected lines into the observed spectrum. The r values are listed in Table 1, and their uncertainty is of the order of 0.5 km s-1 . The spectra were then shifted to the wavelength rest frame. Selected spectral windows containing the observed blue He I lines, together with the synthetic profiles, are displayed in Fig. 1.

HD 145788 1.4 1.2

normalised flux

1 0.8 0.6

21 Peg

Cet

0.4 0.2

4385

4390

4470

4475
o

4920

4924

wavelength (A)
Fig. 1. Samples of the three He I lines: 4387 final synthetic profiles tra of HD 145788 and respectively. spectra of HD 145788, 21 Peg and Cet around å, 4471 å and 4921 å in comparison with our (dashed lines) calculated for each line. The spec Cet are shifted upwards and downwards of 0.5,

3. Fundamental parameters
Fundamental parameters for the atmospheric models were obtained using photometric indicators as a first guess. For their refined estimate, we performed a spectroscopic analysis of hydrogen lines and metal lines, and as a final step, compared the observed and computed energy distributions. The spectroscopic tuning of the fundamental parameters is needed since different photometries and calibrations would give different parameters and uncertainties. The spectroscopic analysis will provide a set of parameters that fit all the parameter indicators consistently, with less uncertainties. Model atmospheres were calculated with LLMODELS, an LTE code that uses direct sampling of the line opacities (Shulyak et al. 2004) and allows computing models with an individualised abundance pattern. Atomic parameters of spectral lines used for model atmosphere calculations were extracted from the VALD database (Piskunov et al. 1995; Kupka et al. 1999; Ryabchikova et al. 1999).


4

Fossati et al.: The abundance analysis for normal A- and B-type stars.

Before applying the spectroscopic method, we estimated the star's sin i. For 21 Peg, a sin i value of 3.76 ± 0.35 km s-1 was derived from the fit with a synthetic spectrum to 21 carefully selected lines observed with the Gecko instrument. This value agrees very well with the 3.9 km s-1 value derived by Landstreet (1998). Achieving such high precision was possible thanks to the high quality of the spectrum and the low sin i. The sin i values for HD 145788 and Cet, 10.0 ± 0.5 km s-1 and 20.2 ± 0.9 km s-1 , respectively, were derived from fitting about 20 wellselected lines along the whole available spectral region. In the next sections, we describe the determination of the atmospheric parameters: T eff , effective gravity (log g), and microturbulent velocity (mic ), and their uncertainties. The fundamental parameters finally adopted for 21 Peg, HD 145788, and Cet are given in Table 1.
3.1. Photometric indicators

Since none of the three programme stars, HD 145788, 21 Peg or Cet are known to be photometrically variable or peculiar, we can use temperature and gravity calibrations of different photometric indices for normal stars to get a preliminary estimate of the atmospheric parameters. The effective temper¨ ature (T eff ) and gravity (log g) were derived from Stromgren photometry (Hauck & Mermilliod 1998) with calibrations by Moon & Dworetsky (1985) and by Napiwotzki et al. (1993), and from Geneva photometry (Rufener 1988) with the calibration by North & Nicolet (1990). The mean parameters from the three calibrations that were used as starting models are the following ones: T eff = 9675±75 K, log g = 3.72±0.03 for HD 145788; T eff = 10255±115 K, log g = 3.51 for 21 Peg; T eff = 13200±65 K log g = 3.77±0.15 for Cet. No error bar is given for the log g of 21 Peg since all three calibrations give the same value.
3.2. Spectroscopic indicators 3.2.1. Hydrogen lines

For a fully consistent abundance analysis, the photometric parameters have to be checked and eventually tuned according to spectroscopic indicators, such as hydrogen line profiles. In the temperature range where HD 145788, 21 Peg, and Cet lie, the hydrogen line wings are less sensitive to T eff than to log g variations, but temperature effects can still be visible in the part of the wings close to the line core. For this reason hydrogen lines are very important not only for our analysis, but in general for every consistent parameter determination. To spectroscopically derive the fundamental parameters from hydrogen lines, we fitted synthetic line profiles, calculated with SYNTH3 (Kochukhov 2007), to the observed profiles. SYNTH3 incorporates the code by Barklem et al. (2000)2 that takes into account not only selfbroadening but also Stark broadening (see their Sect. 3). For the latter, the default mode of SYNTH3, adopted in this work, uses an improved and extended HLINOP routine (Kurucz 1993). Figure 2 shows the comparison between the observed H line profile for 21 Peg and the synthetic profiles calculated with the adopted stellar parameters. In Fig. 2 we also added the synthetic line profiles calculated with 1 error bars on T eff (± 200 K; upper profile) and log g (± 0.1 dex; lower profile). The same profiles with the same uncertainties, but for H, H, and H (from left to right) for the three programme stars, are shown in Figs. 9, 8, and 10 (online material).
2

http://www.astro.uu.se/barklem/hlinop.html

The three hydrogen lines (H, H, and H) used to spectroscopically improve the fundamental parameters for HD 145788 gave slightly different results both for T eff and for log g. As final values, we took their mean (T eff = 9750 K; log g = 3.7). This discrepancy is visible in Fig. 9 (online material), but it lies within the errors given for T eff and log g. The spectrum of HD 145788 also allowed a good normalisation of the region bluer than H. We synthesised this region to check the quality of our final fundamental parameters finding a very good fit for the three hydrogen lines H, H , and H8 . For 21 Peg we obtained the same temperature estimates from H and H (T eff = 10400 K) and by 200 K less from the fitting of H. We adopted a final T eff of 10400 K. To fit all three hydrogen lines, we need slightly different values of log g: 3.47 for H, 3.54 for H and 3.57 for H. We finally adopted log g = 3.55 taking possible continuum normalisation problems into account, in particular, for H. The temperature determination for Cet was more difficult thanks to the weak effect that this parameter has on the hydrogen lines at about 13000 K and to the slightly peculiar shape of the H line. As explained in Sect. 5, Cet probably shows a small emission signature in the region around the core of H possibly because of a circumstellar disk. This region is the one where T eff effects are visible, making it almost impossible to obtain a good temperature determination from this line. H and H gave best temperatures of 12700 K and 12900 K, respectively, leading to a final adopted value of 12800 K. Confirmation of this value was then given by the spectrophotometry (see Sect. 3.3). The results of LTE abundance analysis (Sect. 4) show that Cet has little He overabundance that leads to an overestimation of log g if the He abundance is not taken into account in the model atmosphere calculation (Auer et al. 1966). For this reason we derived the first set of fundamental parameters (T eff = 12800 K; log g = 3.80) and then the He abundance (log(NHe /Ntot ) = -0.97 dex). As a next step we recalculated a set of model atmospheres with the derived He abundance and re-fit the hydrogen line profiles. The best-fit gave us the same temperature, but weak effective gravity (log g = 3.75). As expected, the He overabundance is acting as pressure, requiring an adjustment of log g to be balanced. We obtained the He abundance from the fitting of the blue He line wings. The blue He lines are, in general, considered as showing very little non-LTE effect (Leone & Lanzafame 1998), and as we only used the line wings, this leads us to conclude that our results should not be affected by non-LTE effects and that the He is overabundant in Cet. The best fit to the blue He I lines of Cet is shown in Fig. 1. The example of Cet is important because it clearly shows the effect of the single element abundance on the parameter determination, not only for chemically peculiar stars (for which this effect is well known and often, but not always, taken into account), but also for chemically "normal" stars. The set of parameters that best fit the hydrogen line profiles could not be unique. For 21 Peg, we checked that using a combination of a lower temperature and lower gravity or else higher temperature and higher gravity increases the standard deviation of the fit of the H line wings by 25%. The result is that a different combination of T eff and log g could in principle provide a similar fit. For this reason the derived fundamental parameters should be checked with other indicators, such as the analysis of metallic lines (ionisation and excitation equilibrium) and the fitting of the spectral energy distribution. The latter is more important because ionisation and excitation equilibrium should be strictly used only with a full non-LTE treatment of the line formation.


Fossati et al.: The abundance analysis for normal A- and B-type stars. Table 1. Adopted atmospheric parameters for the analysed stars. Star Name 21 Peg HD 145788 Cet T eff [K] 10400 9750 12800 log g [cgs] 3.55 3.70 3.75 mic [km s-1 ] 0.5 1.3 1.0 mic [km s-1 ] 0.4 0.2 0.5 sin i [km s-1 ] 3.76 10.0 20.2 sin i [km s-1 ] 0.35 0.5 0.9 r [km s-1 ] 0.5 -13.9 12.5

5

The uncertainties on T eff , log g, and r are 200 K, 0.1 dex, and 0.5 km s-1 , respectively.

3.2.2. Metallic lines

The metallic-line spectrum may also provide constraints on the atmospheric parameters. If no deviation from the local thermodynamic equilibrium (LTE) is expected, the minimisation of the correlation between individual line abundances and excitation potential, for a certain element/ion, allows one to check the determined T eff . Then the balance between different ionisation stages of the same element provides a check for log g. The microturbulent velocity, mic is then determined by minimising the correlation between individual abundances and equivalent widths for a certain element. Determining the fundamental parameters performed in this way has to be done iteratively since, for example, a variation in T eff leads to a change in the best log g and mic .
-4

-4.2 -4.4 -4.6 -4.8 -5 10 20 30 40 50 60 70 80
o

Fe II Fe I 90 100 110

equivalent width (mA) Fe abundance
-4 -4.2 -4.4 -4.6 -4.8 -5 1 2 3 4 5 6 7 8 9 10 11

The uncertainties listed in Table 1 are given considering 2 on the error bar of the derived slopes. Considering a 1 error bar, the uncertainties on mic are of 0.1 km s-1 for HD 145788 and 21 Peg and of 0.2 km s-1 for Cet. According to previous works, deviation from LTE of the Fe II lines is expected to be very small ( 0.02 dex Gigas 1986; Rentzsch-Holm 1996) for the analysed stars, such that the absence of any clear correlation between the line Fe II abundance and the excitation potential confirms the T eff derived from the hydrogen lines. For the Fe I lines, deviations from LTE of about +0.3 dex are given by Gigas (1986) and Rentzsch-Holm (1996). However, both Gigas (1986) and Rentzsch-Holm (1996), as well as Hempel & Holweger (2003), used the same model atom, which includes 79 Fe I and 20 Fe II energy levels. We note that the highest energy level in their model atom for Fe II has an excitation energy of about 6 eV, while the ionisation potential is 16.17 eV. Such a model atom does not provide collisional coupling of Fe II to Fe III, which operates for the majority of iron atoms in line formation layers below log 5000 = -1. Unfortunately, the existing NLTE calculations for Fe are not accurate enough to be applied now to our stars. Clearly, an extended energy-level model atom is needed for a reliable non-LTE analysis of Fe. The ionisation equilibrium for different elements/ions (or its violation) can be seen in Table 4 and is discussed in Sect. 4.
3.3. Spectrophotometry

Fe abundance

excitation potential (eV)
Fig. 3. Iron abundance vs. equivalent widths (upper panel) and excitation potential (lower panel) for 21 Peg. The open circles indicate Fe I, while the open triangles indicate Fe II.

Figure 3 shows the correlations of Fe I and Fe II abundances with the equivalent widths (upper panel) and with the excitation potential (lower panel) for 21 Peg. The correlation with the equivalent widths is shown for a mic of 0.5 km s-1 , which we infer to be the best value for 21 Peg, since the slope of the linear fit for Fe I is -4.897 â 10-3 ± 1.560 â 10-3 må-1 and for Fe II is 2.146 â 10-5 ± 4.058 â 10-4 må-1 . Here we gave a preference to the result obtained from Fe II because of the higher number of measured Fe II lines in a wider range of equivalent widths. The same analysis was made for HD 145788 and for Cet. The error bar on mic was calculated using the error bar of the slope (abundance vs. equivalent width) derived from a set of different mic .

For a complete self-consistent analysis of any star, one should reproduce the observed spectral energy distribution with the adopted parameters for a model atmosphere. In the optical spectral region, spectrophotometry was only available for 21 Peg and Cet, while in the ultraviolet, IUE spectra were available for all three stars. For Cet ultraviolet spectrophotometry from the TD1 satellite (Jamar et al. 1976) was also available, along with the flux calibrated spectra from STIS at HST (Gregg et al. 2004). The comparison between the observed flux distributions and the model fluxes calculated with the adopted atmospheric parameters for 21 Peg and Cet is shown in Fig. 4. For HD 145788 we estimated a reddening E(B-V)0.2 from the dust maps of Schlegel et al. (1998). The comparison of reddened model fluxes with the available IUE spectrum, Johnson UBV photometry (Nicolet 1978), Geneva photometry3 and 2MASS photometry (Cutri et al. 2003) is shown in Fig. 11 (online material). This plots supports the value of E(B-V)=0.2 and shows good agreement between all the observations and the model fluxes, confirming the obtained fundamental parameters, and also the importance of considering reddening in the analysis of relatively nearby stars such as HD 145788. The optical spectrophotometry was taken from Adelman et al. (1989) and Breger (1976). All flu