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Astron. Astrophys. 353, 666-690 (2000)

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2. Review of previous investigations

2.1. Neutral calcium, Ca I

The first fully detailed study of all useful calcium lines in the solar spectrum was published by Holweger (1972) who gave a complete evaluation of the relevant atomic data available at that time. The solar abundance of calcium resulting from his investigation is [FORMULA].

Later, at Oxford, G. Smith and his collaborators determined accurate laboratory oscillator strengths and damping parameters for collisions with helium around [FORMULA] for many useful lines of neutral calcium (Smith & O'Neill, 1975; O'Neill & Smith, 1980; Smith & Raggett, 1981; Smith, 1988). The helium damping parameters were semi-empirically extended to collisions with neutral hydrogen at solar temperatures (see O'Neill & Smith 1980 for details). These data were then used to perform absolute analyses of the Ca I lines in a small number of stars. In an iterative procedure these investigators evaluated the excitation and ionisation equilibrium from a set of weak Fe I and Fe II lines, derived a value of the microturbulence from medium strong Ca I lines and found the value of the surface gravity from the damping wings of the strong Ca I line at 6162 Å. For the solar case, Smith (1981) obtained an abundance of calcium [FORMULA] with the empirical solar model atmosphere of Holweger & Müller (1974), in complete agreement with the value already derived by Holweger (1972).

2.2. Statistical equilibrium in Ca I

In the case of the atmospheres of the Sun and Procyon, Watanabe & Steenbock (1985) investigated the statistical equilibrium of the level populations of a Ca model atom consisting in 16 levels of Ca I, 3 levels of Ca II (to allow for the ionization of the doubly excited levels of Ca I), plus the ground state of Ca III. They found no deviations from Local Thermodynamic Equilibrium (LTE) populations in the Ca II levels (the dominant ion in later type stellar atmospheres), and they mapped the abundance errors resulting from departures from LTE in the different lines of Ca I. They concluded that the global non-LTE effects on the determination of the calcium abundance are quite small in the case of the Sun, but are noticeable in the atmosphere of Procyon, although remaining small for weak lines ([FORMULA] mÅ).

Drake (1991), using new computational techniques, extended the work of Watanabe & Steenbock by looking at the behaviour of the formation of lines in the same model Ca atom under a much wider set of model atmospheric conditions. He considered sets of model atmospheres typical of: (1) solar type dwarfs with varying temperatures, (2) cool giants with varying gravities and (3) solar temperature metal-deficient dwarfs with varying metallicities. His results confirm the conclusions of Watanabe & Steenbock for the solar case. No significant non-LTE effect in the level populations of Ca II is found for any of the model atmospheres considered. Departures from LTE in Ca I are dominated by a general overionization showing an unexpectedly complicated behaviour with the basic stellar atmospheric parameters, often explained by the behaviour of the continuous absorption coefficient in the ultraviolet. The paper provides non-LTE abundance correction factors for the lines of each individual multiplet of Ca I for the different model atmospheres and one of its general conclusions is that "non-LTE effects in Ca I in solar type dwarf stars are likely to be insignificant, but these effects become increasingly significant toward lower gravity stars".

2.3. The ionized calcium infrared triplet lines

The H and K resonance lines of ionized calcium, Ca II, are the most characteristic features of the spectra of late type stars and they are widely used for the study of their chromospheric activity. The infrared triplet lines of Ca II (Ca II IRT) are also quite conspicuous features of the spectra of these stars. Since the advent of solid state detectors they are readily observed with high resolution, high signal-to-noise ratio and good photometric accuracy. The extended wings of these dark lines probe a wide range of photospheric layers, and are thus sensitive to the run of the temperature distribution with depth. Their cores are so opaque that they are formed in the uppermost atmospheric layers (chromospheres) and their central depths have also been shown to provide good indicators of chromospheric activity (see e.g., Linsky et al. 1979, Cayrel et al. 1983, Foing et al. 1989).

Further interest in the Ca II IRT arose when Jones et al. (1984) claimed on empirical grounds that the strength of these lines provided a very good luminosity indicator in the context of the evaluation of giant-to-dwarf ratios in stellar systems population synthesis; they advocated the existence of a single-valued relation between their strength and stellar surface gravity. This observation was further discussed and clarified by Diaz et al. (1989) who found instead a biparametric behaviour of the Ca II triplet with gravity and metallicity. Their findings were that, at high metallicities, gravity is the dominant parameter, whereas, at low metallicities, metal abundance is the leading parameter. For stars of spectral type later than K3, Zhou (1991) found some dependance of the IRT absorption on temperature; yet, as long as the metallicity differences are not too large, the IRT intensities allow a clear separation of giant from dwarf stars. Ginestet et al. (1994) set up a spectral classification scheme in the near infrared ([FORMULA] 8380-8780 Å) where the Ca II 8542 Å line plays a key role for the classification of stars of types G to M. Carquillat et al. (1997) published a useful atlas of spectra of stars of types F6 to M in the 8400-8800 Å region, at 2 Å resolution, illustrating the spectral changes with spectral type and luminosity class. The empirical approach is reconsidered in detail and summarized by Mallik (1997) who also provides an interesting atlas of spectra of the three IRT lines observed at a rather high resolution (0.4 Å) for stars of spectral types from F7 to M4 of all luminosity classes and a large range of metallicities.

On the theoretical side, the findings of Jones et al. (1984) led G. Smith, J.J. Drake and co-workers in particular to include the Ca II IRT lines as key features in the line list for their studies of the spectrum of calcium in late F to early K type stars. On the basis of LTE spectrum synthesis calculations Smith & Drake (1987) investigated the sensivity of the wings of the strongest line of the Ca II IRT at 8542 Å to the basic stellar atmospheric parameters for dwarf and subgiant solar-type stars. They found in fact little sensitivity to the surface gravity, a much larger effect from metallicity increasing with temperature, and a sensitivity to effective temperature which starts becoming significant only at the higher end of the temperature interval considered [FORMULA] [FORMULA] [FORMULA]. They were also able to obtain good fits to the observed wings of this line in spectra of the stars [FORMULA] and [FORMULA]. Smith & Drake (1988) obtained excellent, very detailed fits to the solar intensity profiles of the wings of the three Ca II IRT lines at two µ-positions on the solar disk from LTE synthetic profiles calculated using the solar empirical model photosphere of Holweger & Müller (1974). Adopting the reliable oscillator strengths given by Gallagher (1967) and the calcium abundance from Smith (1981), they derive accurate hydrogen broadening parameters for the three lines.

Smith & Drake (1990) again investigated the sensitivity of the Ca II 8542 Å line, but this time for a range of atmospheric parameters representative of cool giant stars. Again, the sensitivity of the computed profile to metallicity is the largest. Although larger for the lower values of [FORMULA] than in the case of dwarfs, the sensitivity to surface gravity is still weak. As for the effects of changes in effective temperature they remain quite small. Erdelyi-Mendes & Barbuy (1991) calculated synthetic spectra of the Ca II IRT region for stars of cooler effective temperature [FORMULA] [FORMULA] [FORMULA], but with a "low resolution" approach devised for direct comparison with the empirical data of Diaz et al. (1989), where they take into account the effect of numerous atomic and molecular lines in the spectral region. They show in particular the importance of the contribution to total absorption of CN and TiO bands at the lower temperatures. Their Fig. 1 illustrates the impossibility of locating the real continuum in M stars. They investigate the sensitivity of global absorption in the region of the Ca II IRT lines to the basic stellar atmospheric parameters, confirming, with restrictions, the empirical conclusions of Diaz et al.

These studies have described the response of the Ca II IRT lines to changes in the basic photospheric parameters within a given grid of model stellar photospheres. However the theory tells us that the detailed shape of the extended wings of very strong lines, such as the Ca II IRT, reflects the variation of the source function with the depth in the atmosphere. When the conditions of LTE apply, this is a direct function of the distribution of temperature with depth. The observed shape of very dark lines should thus allow us to check the validity of the temperature distribution adopted for the analysis of the spectrum of a given star. In the special case of the Ca II line at 8542 Å, this property was applied by Drake & Smith (1991) to build an empirical model atmosphere for their very detailed study of the spectra of calcium and iron in the K0 giant Pollux. It has also been used to assess that scaled solar empirical Holweger-Müller temperature distributions were adequate for the study of the chromospherically active dwarfs [FORMULA] (Drake & Smith 1993) and [FORMULA] (Ruck & Smith 1995), whereas a theoretical flux-constant MARCS model atmosphere (i.e. computed according to Gustafsson et al. 1975) was more appropriate for the analysis of the spectrum of the subdwarf Groombridge 1830 (Smith et al. 1992). From high resolution spectra of Ca II [FORMULA] in four dwarfs and two giants in the Hyades cluster, Smith & Ruck (1997) derived the metallicity of the dwarfs and the gravity of the giants (the metallicity of the latter being determined, independently of gravity, by the Mg I [FORMULA] line).

2.4. Departures from LTE in the Ca II IRT

Within the entire range of spectral types under consideration (late F, G and early K dwarfs and giants), Ca II turns out to be the dominant ionization stage. The investigations of the statiscal equilibrium in Ca I by Watanabe & Steenbock (1985) and Drake (1991) have already suggested that the global population of Ca II is never significantly altered by departures from LTE. Jorgensen et al. (1992) looked more specifically at the effects of departures from LTE on the Ca II IRT lines for a complete set of MARCS model photospheres. For that purpose they used a model calcium atom comprising 8 levels in Ca I, 5 levels in Ca II, plus the ground state of Ca III. Unfortunately, there is no discussion of the effects on the detailed profile of these lines: they discuss only the effects on the combined integrated equivalent widths. Nor do they examine the effect of an eventual chromospheric temperature rise, known to affect the central core intensities, considering that such an effect will not alter significantly the global equivalent width. However, they conclude that, if the line cores are indeed out of LTE, the Ca II IRT line wings are formed in conditions close to LTE, and the effect of departures from LTE on the equivalent widths is always negligible. The largest non-LTE effects are found at high temperatures and gravities. They also point out that the dependence of the equivalent width on gravity is higher at the higher metallicities and that, at low metallicities, the dependence on [FORMULA] is greater than that on [FORMULA]. The interesting point they make is that the complicated behaviour of the equivalent widths with the variations of the stellar atmospheric parameters is largely dependent on whether the natural or the collisional broadening mechanism is dominant in the atmosphere considered. It is worth mentioning that these authors adopted a value for the collision broadening parameters that differs from the value derived by Smith & Drake (1988) by 25% or so; this may affect the behaviour of the calculated line strengths.

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Online publication: December 17, 1999