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

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4. The solar case

The absorption coefficient in the extended wings of very strong lines, such as the Ca II IRT, is proportional to the product [FORMULA] where N is the number density of the absorbing atom or ion, gf is the oscillator strength of the line and [FORMULA] is the damping parameter. Reliable gf values are known for the IRT lines. In the solar case, a reliable value of the calcium abundance and good state-of-the-art average descriptions of the physical conditions in the atmosphere are available. This allows us to determine with reasonable accuracy the damping parameter [FORMULA] by varying its value until we obtain a good fit of the relevant line in the observed solar spectrum. In this way Smith & Drake (1988) empirically derived accurate values of the damping constant for collisions with neutral hydrogen from a very careful analysis of center-limb Kitt Peak solar intensity spectra of the IRT lines, using the Holweger-Müller (1974) model atmosphere with the calcium abundance [FORMULA].

Studies of the Ca II IRT lines in stellar spectra are thus necessarily differential with respect to the Sun. This is why we need to analyse the solar case from good reference solar observations recorded under instrumental conditions as close to the stellar observations as possible. The analysis must be made with a solar model atmosphere consistent with the model atmosphere used for the analysis of the stellar spectra. Scaled solar empirical model atmospheres can be used only in a limited range of stellar atmospheric parameters around those of the Sun. Outside this range, we have to make use of grids of purely theoretical flux-constant model photospheres. Experience shows that the calcium abundance derived from the solar spectrum is somewhat dependent on the choice of the model solar atmosphere (see e.g. Smith 1981). Since [FORMULA] is the relevant information extracted from the wings of the Ca II IRT, we have to make certain that the damping constants derived from solar observations with different model photospheres are consistent with the calcium abundance corresponding to the particular model photosphere.

4.1. Computations

The computation of the synthetic spectra is carried out by means of the computer code ADRS which is an evolved FORTRAN implementation of the original ALGOL code described by Baschek et al. (1966). This is a fully detailed classical LTE treatment of line formation.

The model atmospheres are also computed in the framework of the LTE theory. A distribution [FORMULA] of the temperature with a reference optical depth scale (here, the monochromatic optical depth at [FORMULA] Å) is selected: in general it will be either a scaled solar empirical relation or a relation interpolated in a grid of flux-constant theoretical model photospheres. For a specified chemical composition and surface gravity the continuous opacities are computed and the pressures are obtained by integration of the hydrostatic equilibrium equation in plane-parallel geometry using a separate computer program. For the solar case I have taken into consideration five different temperature laws (two empirical and three theoretical). The first empirical temperature distribution was that of Holweger & Müller, 1974 (hereafter called Holweger model) which is almost unanimously considered to give the best account of the solar line spectrum in the framework of the LTE theory. Next, the temperature distribution of Maltby et al., 1986 (hereafter MACKKL model) was adopted: it is derived from the very well known empirical laws of Vernazza et al. (1981, VAL models) and includes a more realistic average chromospheric temperature rise. In this special case, as this model incorporates some NLTE features, the original tabulated pressures are used and are not recomputed as stated above. In the category of flux-constant theoretical model photospheres I have chosen the solar model in the grid presented by Eriksson et al. (1979): the models in this grid, which will be subsequently called GBEN models, are computed as described in Gustafsson et al. (1975) and were kindly provided in 1983 by Dr. Eriksson. I have also used the solar theoretical model given in Edvardsson et al. (1993): this model (EAGLNT) is computed basically in the same way as the GBEN models, but using a much more complete line list to account for the line blanketing effects. Finally, I have considered the widely used theoretical solar model atmosphere of Kurucz (1992). These different temperature distributions are compared in Fig. 3. Recent studies that have examined the microturbulence parameter for the Sun-as-a-star full-disk spectrum have found values in the range [FORMULA] to be appropriate (values of [FORMULA] for intensity spectra at disk center tend to be smaller and values for intensity at the limb are distinctly larger). A value of [FORMULA] has been adopted in this work.

[FIGURE] Fig. 3. Comparison of the solar temperature distributions used in this study. The left panel compares the Holweger-Müller model (solid line) with the MACKKL (dashed) and the Kurucz model (dotted). In the right panel the Holweger-Müller model is compared to the GBEN model (dashed) and to the EAGLNT (dotted).

The atomic constants first assumed for the computation of the synthetic spectra are essentially the same as those selected by Smith & Drake (1988) and are summarised in Table 1. The oscillator strengths come from measurements of the lifetime of the upper level and of branching ratios by Gallagher (1967) and are considered to be quite accurate.


[TABLE]

Table 1. The atomic constants for the Ca II IRT, after Smith & Drake (1988)


As stated before, for such dark lines as the Ca II IRT, the profile depends critically on the damping parameter. In the ADRS code, following Baschek et al. (1966), the damping by collisions with neutral atoms is treated in the classical impact van der Waals approximation. In a van der Waals potential, the frequency shift is given by

[EQUATION]

where [FORMULA] is the interaction constant and r is the interatomic separation. In this approximation, the damping parameter for collisions with neutral hydrogen atoms is

[EQUATION]

where A is a numerical constant, [FORMULA] is the number density of the perturbing particles (here, neutral hydrogen) and [FORMULA] is the relative velocity of the perturbing atom, [FORMULA], in such a way that [FORMULA]. This is expressed numerically, in c.g.s. units ([FORMULA]) as

[EQUATION]

Data on the interaction constant are not always available in the literature: in that case, it is often estimated from an approximate hydrogen-like formula ("Unsöld's formula"), multiplied by an arbitrary (adjustable) enhancement factor which will generally take a value between 1 and 5 (see in Fig. 8 a synthetic profile computed with the damping parameter predicted by Unsöld's formula, with unit enhancement factor).

The contribution of the collisions with neutral helium is accounted for by multiplying [FORMULA] by a factor

[EQUATION]

where p is the polarisability and µ the atomic weight.

Attention must be paid to some confusion in the literature concerning the symbols used. The interaction constant is usually refered to as [FORMULA] and it is often not immediately clear, even from the context, whether [FORMULA] or [FORMULA] is meant. In many papers or textbooks the damping constant [FORMULA] is the full width at half maximum (FWHM) of the Lorentzien profile of the absorption coefficient; but other authors, among which are G. Smith and collaborators, refer to [FORMULA] as being the half width at half maximum. Here, [FORMULA] will refer to the FWHM. These choices affect the value which must be used for the numerical constant A.

There is, however, evidence from both experimental and theoretical work, that the treatment based on pure van der Waals interaction is inadequate and leads to systematic underestimates of the damping constant. At high stellar photospheric temperatures it turns out that short-range repulsive forces largely dominate the broadening interactions (see e.g. O'Neill & Smith, 1980). These are often successfully described in the classical impact approximation by adding an [FORMULA] repulsive term to the interaction potential. All the recent developments point toward a [FORMULA] temperature dependence of the damping constant instead of the [FORMULA] expected for pure van der Waals interaction. For these reasons, following G. Smith and collaborators, I modified the ADRS code to allow optional treatment of the damping by collisions with neutral hydrogen atoms according to:

[EQUATION]

where the damping coefficient per unit perturber at [FORMULA], [FORMULA], and the exponent s are the input parameters. The pure van der Waals case is recovered with [FORMULA] and

[EQUATION]

From numerical experiments involving computations of the Ca II IRT lines with several model atmospheres for different effective temperatures, it turned out that lines synthetised with an identical value of [FORMULA] at [FORMULA], but either a [FORMULA] or a [FORMULA] dependence, gave indistinguishable profiles. Nevertheless, the synthetic profiles described hereafter were all computed assuming the more physical [FORMULA].

The radiative damping and electron Stark broadening are also treated in the same way as in Smith & Drake (1988). The radiative damping parameter is safely derived from the lifetime of the upper levels, the lower levels being metastable. The Stark broadening by collisions with electrons is treated in the classical impact approximation with a [FORMULA] dependence and a damping constant based on measurements by Chapelle & Sahal-Bréchot (1970). Similar relations hold as in the van der Waals case, with an interaction constant [FORMULA], in such a way that

[EQUATION]

[EQUATION]

with

[EQUATION]

Before being compared with the observed spectrum, the synthetic spectra have to be convolved with different macroscopic broadening profiles. On our CFH solar observations, the spectrograph's instrumental profile appears to dominate the macro-broadening contributions. Other contributions come from macroturbulence (width [FORMULA]) or rotation ([FORMULA]). At the resolution of typical CFH stellar spectra, numerous synthetic spectrum calculations in the solar case have shown that the global macro-broadening processes can adequately be represented by a unique convolution with a gaussian profile, the width of which, [FORMULA], may be adjusted so as to reproduce as well as possible the observed breadth of well defined unblended weak atomic lines lying in the observed spectral region. In the present case, the only really adequate lines for that purpose are the Fe I lines at [FORMULA] and [FORMULA] Å. Best-fitting synthetic profiles are obtained with the convolution by a gaussian profile having a characterestic width [FORMULA]. This result does not depend on the adopted model atmosphere and a change in the adopted microturbulence from [FORMULA] to [FORMULA] does not change [FORMULA] by more than [FORMULA]. Should we have to deal with a star with appreciable rotation, this would have to be taken explicitely into account because the rotational broadening is rather elliptical, far from gaussian. In the same way, when the synthetic spectra are to be compared with the Solar Flux Atlas, we find that the combined effects of macroturbulence and instrumental broadening are well represented by convoluting the computed flux profiles by a gaussian of width [FORMULA].

It turned out that the synthetic profiles of the very broad Ca II IRT lines are rather insensitive to the precise values adopted for the instrumental, Stark damping, micro- or macro-turbulence broadening parameters. They are however, as expected, quite sensitive to the calcium abundance and to the value of the hydrogen damping parameter [FORMULA].

4.2. Solar calcium abundance

When interpreting spectra, we generally use temperature distributions interpolated in the GBEN grid of flux-constant model atmospheres. Preliminary computations of the Ca II IRT profiles were carried out using the "canonical" calcium abundance [FORMULA] and the values of the damping parameter provided by Smith & Drake (1988). The synthetic profiles calculated with the Holweger-Müller temperature distribution correctly matched the observed profile, but those calculated using the GBEN solar model atmosphere produced quite unsatisfactory results: the computed wings of the IRT lines were much too deep compared with the observed profiles. This evidently resulted from assuming a calcium abundance which was not consistent with the choice of the model temperature distribution. Therefore, the calcium abundance must first be redetermined for each of the selected model atmospheres. To that purpose the Ca I lines equivalent widths (for intensities at the center of the solar disk) listed in Table 1 of Smith (1981) were used. A number of lines considered as discrepant by the author were discarded, as well as the lines with equivalent widths [FORMULA] mÅ, reducing the number of "useful" Ca I lines to 14. Most of these lines are of medium strength so that their equivalent width is sensitive to the adopted damping and microturbulence parameters. Good estimates of the oscillator strengths and damping parameters are given by Smith (1981). For each solar model atmosphere the value of the microturbulence (assumed to be constant with depth) was adjusted to cancel any systematic dependence on the equivalent width of the individual abundances derived from each line. The resulting average values of the calcium abundance and of the microturbulence [FORMULA] are given in Table 2. Let us remark that if, with the Holweger-Müller model, we use "default" damping parameters calculated from Unsöld's formula with a unit enhancement factor (as often done in abundance studies), we obtain [FORMULA] and [FORMULA], i.e. values which are appreciably different from the values listed in Table 2. As already observed by Smith (1981) we see that the calcium abundance is somewhat dependant on the choice of the model atmosphere. The value obtained for the abundance with the Holweger model is slightly higher (0.02 dex) than that given by Smith; this probably comes from using somewhat higher continuous opacities and a reduced line list.


[TABLE]

Table 2. Calcium abundance, microturbulence and hydrogen damping parameters derived with the five solar model atmospheres considered.


4.3. Perturbation by hydrogen Paschen lines

The Ca II IRT lines are generally considered to be a useful diagnostic tool in the range of late F to M stars. Hydrogen Paschen lines happen to fall in the red wings of the CaII IRT lines and start to contribute significantly at the hotter end of this range: P16 [FORMULA] lies in the wing of Ca II [FORMULA], P15 [FORMULA] falls in that of Ca II [FORMULA] and P13 [FORMULA] blends with Ca II [FORMULA]. In the hotter stars, the Paschen lines dominate the Ca II lines: this is quite well illustrated by Fig. 1 of Ginestet et al. (1994) which shows this spectral region in A5 stars of different luminosities. The little perturbed P14 [FORMULA] line falls in between the two strongest Ca II lines: it thus allows to estimate the importance of the perturbation of the Ca II IRT by the hydrogen Paschen lines. Inspection of the Atlas of Carquillat et al. (1997) shows that P14 is readily identified in stars as cool as G0, with a strong positive luminosity effect. Therefore, the contribution of the Paschen lines must be taken into account for the quantitative interpretation of the Ca II IRT line wings in stars earlier than the Sun.

In the solar case, Smith & Drake (1988) argue, on the basis of the non-detection of P14, that the hydrogen contribution to the observed profiles of the Ca II IRT lines is negligible. And, indeed, synthetic profiles fitting the observed blue wing of [FORMULA] or [FORMULA] do not leave room for a significant additional absorption in the red wing. Only does the [FORMULA] line show a slight, but significant, extra absorption in the red wing which can be attributed to P13.

R. and G. Cayrel, and collaborators, use the outer profile of the Balmer [FORMULA] line as a temperature criterion. For that purpose they compute synthetic Balmer line profiles with the help of an LTE computer code initially written by Praderie (1967), and later updated and implemented by R. Cayrel, A. Talavera and M. Spite. This computer program, called HYDRO, also computes profiles of the Paschen lines. I have used it to try and assess more quantitatively the contribution of the Paschen line absorption to the Ca II IRT line profiles in the Sun and hotter stars. The optical depths in the Paschen lines at selected frequencies, as computed by program HYDRO, are written to a file. Monochromatic line optical depths from the several Paschen lines that can contribute are separately summed up; components with central wavelengths distant by more than [FORMULA] 35 Å from the current frequency point are extrapolated according to a [FORMULA] law. The spectrum synthesis program ADRS has been modified to read the file of the total Paschen line optical depths and add them to the total computed (continuous plus other lines) optical depths.

Synthetic Ca II [FORMULA] and [FORMULA] profiles have been computed taking into account the total contribution of P14 through P17. In the solar case, as illustrated by Fig. 2, the profiles computed with the Holweger model show only a very slight red wing asymmetry, and the main effect of the Paschen lines absorption is to lower significantly the apparent continuum location. Thus, as explained in Sect. 3.2, solar spectra observed under "stellar" conditions (over rather short wavelength ranges) must be locally renormalized to ensure consistency in the comparison between computed and observed profiles. We can indeed check that the renormalized continuum of our CFH spectrum coincides with the Solar Flux Atlas definition (which is based on consideration of very long wavelength scans) if and only if we take explicitly the Paschen lines into account in the calculations. This demonstrates the validity of our renormalization procedure and shows that the computed flux in the far wings of the Paschen lines is correct. According to the Eddington-Barbier approximation the intensity at disk center reflects the value of the source function at total optical depth [FORMULA] and the measured flux is typical of the value of the source function at [FORMULA]. One finds that the Paschen lines, at the central wavelength of P15, contribute to the total optical depth [FORMULA] by only 1.2% at [FORMULA] and 0.9% at [FORMULA]. The maximum contribution of the Paschen absorption at [FORMULA] in the [FORMULA] line wing is 1.6% at [FORMULA]; at [FORMULA] the corresponding maximum is 1.0% at [FORMULA]. We thus see that, in the solar case, the contribution of the Paschen lines to the Ca II IRT features is almost negligible. Their effect is essentially the same as that of a slight increase of the continuous absorption coefficient.

We shall see, however, in Sect. 5.1, that the hydrogen absorption is no longer negligible for stars hotter and/or more luminous than the Sun (see Table 3). This must be taken into account when calculating synthetic profiles as well as for the continuum normalization.


[TABLE]

Table 3. Percent contribution of the Paschen lines to the total optical depth at [FORMULA], near the center of the P15 line [FORMULA].


4.4. Solar synthetic spectra

Synthetic profiles of the first two Ca II IRT lines, [FORMULA] and [FORMULA], were computed with each of the five selected solar model atmospheres, with due account for Paschen line absorption. For each model, the value of the hydrogen damping parameter [FORMULA] of each line was changed until a "best possible fit" of the observed solar profile was obtained. This raises the question of the choice of "best fitting" criteria. As already stated, the central core of the IRT lines is formed under conditions departing from LTE and, thus, it is not expected to be correctly represented by LTE synthetic profiles. These criteria must therefore be restricted to the outer wings which are formed in layers where LTE prevails, i.e., as experience shows, that part of the wings where the depression in the profile, [FORMULA], is smaller than [FORMULA] (see the following discussion of the fits obtained with the Holweger or MACKKL empirical model atmospheres). For accurate fits, another difficulty comes from the slight asymmetries of the observed profiles: it is clear that the cores are somewhat red-shifted relative to the wings and that the bisectors of each profile at different depressions do not fall at exactly the same wavelength. The synthetic profiles, computed for basically static model atmospheres, are unable to reproduce these asymmetries. This makes it difficult to compare observed with computed depressions at an exact given wavelength. Uncertainties in the wavelength calibration of the observations and in the radial velocity shift of the star add to this difficulty. Alternatively, if we assume that the radiation on either side of the line center at a given depression is formed in the same layers (with the same motions) the comparison between the observed and computed line widths at selected values of the depression is not affected by these uncertainties. This method has been chosen here. For the empirical determination of [FORMULA] the synthetic profiles were primarily compared with the quasi-noiseless Solar Flux Atlas. The selected values of the line depression are [FORMULA] 0.185, 0.21, 0.24, 0.28 and 0.31 for [FORMULA], and [FORMULA] 0.175, 0.19, 0.22, 0.30 for [FORMULA]. In the special case of the [FORMULA] feature a correct representation can only be obtained by taking also into account the blending Fe I line at [FORMULA] Å; the gf value of the Fe I line is adjusted so as to reproduce the observed value of its central depth.

The "best fit" [FORMULA] values found in this way are shown in Table 2. They are averages of the values fitting the profile width at each selected line depression; the rms uncertainties quoted in the table come from the dispersion of the values of the damping constant derived at each of the selected depressions (only an ideal fit would produce a unique value of [FORMULA] at each of them). Thus the values in Table 2 reflect best fits in a range of depressions restricted to [FORMULA], which means that the quoted uncertainties are consistently underevaluated. Let [FORMULA] be the value of [FORMULA] such that the width of the computed profile at the depression [FORMULA] is equal to the observed value. In general the actual fits tend to imply a clear monotonous trend of [FORMULA] with [FORMULA]. Accordingly, a measure of the quality of the fit may be given as, for example, the relative difference [FORMULA]. The relative difference [FORMULA] is given in Table 2 for each of the adopted model atmospheres.

4.4.1. Holweger-Müller model

The synthetic spectra (thick solid lines) obtained with the Holweger model are illustrated in Fig. 4 for the first two IRT lines and are compared with the observed profiles (Solar Flux Atlas - thin lines). The synthetic spectra quite accurately represent the observed profiles in the line wings, at distances from the line center [FORMULA] Å for the [FORMULA] Å line, and [FORMULA] Å for the [FORMULA] Å line, i.e. respectively [FORMULA] and [FORMULA]. As expected, the line cores are not adequately represented by these LTE computations. In both lines the observed line core is broader and slightly red-shifted relative to the computed core. Curiously the observed core is less deep than computed for the [FORMULA] line, whereas it is deeper in the case of the [FORMULA] line. The latter behaviour is indicative of a source function ruled by diffusion processes; the apparently different behaviour of the [FORMULA] line may be explained by a more effective filling in by chromospheric emission in conformity with what was observed by Shine & Linsky (1972) on spectra of solar plages of different degrees of activity. The greater reactivity of [FORMULA] to the degree of activity is not easy to explain: to account for it, Shine & Linsky (1974) must play with the temperature distribution in the chromosphere, the depth distribution of microvelocities and the effect of macrovelocities.

[FIGURE] Fig. 4. Synthetic spectra computed with the Holweger model. Thin solid line: observed spectrum. Thick solid line: profile computed with the full Holweger model. Dashed line: profile computed with a Holweger model truncated upwards at [FORMULA]. The location of the center of perturbing Paschen lines is indicated by arrows.

These LTE computations show that the center of the two lines is completely opaque already at the surface of the model atmosphere. The solar model atmosphere given by Holweger & Müller (1974) starts at the reference optical depth (monochromatic optical depth at 0.5 µ) [FORMULA]. Experimenting with a model extrapolated upwards, we find that, in LTE, the total optical depth at line center is unity at [FORMULA] for [FORMULA], and [FORMULA] for [FORMULA]. In LTE, the intensity at the line center is thus fixed by the Planck function at the temperature of the uppermost layer, and, therefore, depends on the "starting" (uppermost) reference optical depth of the model atmosphere. The Holweger model is constructed empirically and aims at reproducing as well as possible the continuum and line data of the observed solar spectrum in the visible and near infrared; it does not feature a chromospheric temperature rise and, although it extends up to [FORMULA] it is not considered to be physically meaningful above [FORMULA] where the actual chromospheric temperature rise occurs. Most theoretical flux-constant model photospheres usually start at [FORMULA] and do not include chromospheric layers. If we want to compare results obtained by means of the Holweger model with results produced by means of some theoretical photosphere, we should use a version of the Holweger model for which the layers above [FORMULA] have been cut away. In Fig. 4 the profiles computed using such a "truncated" Holweger photosphere are represented by dashed lines. They depart significantly from the profiles computed with the full Holweger model only in the very central core, within [FORMULA] Å from the center of [FORMULA] and [FORMULA] Å for [FORMULA]. This demonstrates that these central regions of the Ca II lines are indeed formed in chromospheric layers.

Synthetic profiles have also been computed with the Holweger model in the van der Waals damping approximation, with the same value of [FORMULA] at the reference temperature of 5000 K, but a [FORMULA] instead of a [FORMULA] dependence. They turn out to be indistinguishable from the profiles discussed above. Therefore the IRT lines can be correctly described with the more commonly used van der Waals approximation. The corresponding interaction constant [FORMULA] (see the correspondence formula in Sect. 4.1) is given in Table 2, as well as the enhancement factor [FORMULA] of [FORMULA] relative to the Unsöld approximation. We see that this factor is unusually high for the Ca II IRT.

4.4.2. MACKKL model

The MACKKL model (Maltby et al., 1986) is an empirical model of the average solar atmosphere which includes a chromospheric temperature rise starting at [FORMULA], upwards to [FORMULA]. The computed and observed profiles of the [FORMULA] line are compared in Fig. 5. The thin solid line shows the observed spectrum and the thick solid line shows the profile computed by means of the full MACKKL atmosphere; again, the dashed line shows the profile computed by means of a MACKKL model starting at [FORMULA] and consisting only of its photospheric part (temperature increasing monotonically with optical depth).

[FIGURE] Fig. 5. Synthetic profiles computed with the MACKKL model. Thick solid line: full model, with chromosphere; dashed line: photospheric part of the MACKKL model only, starting at [FORMULA]. The thin line shows the observed profile.

As expected in LTE, the chromospheric temperature rise results in a strong emission peak in the core of the computed profile. But, as already shown by, e.g., Shine & Linsky (1974) or Jorgensen et al. (1992), the LTE approximation is not expected to be valid in the core of the IRT lines. The value of the test-parameter [FORMULA] indicates, however, that the fit to the observed line wings is good and even better than with the Holweger model. The comparison between the observed and computed profiles basically confirms the conclusions drawn in the case of the Holweger model, i.e. that the core of the line, within [FORMULA] Å, is formed in the chromosphere under conditions strongly deviating from LTE. Outside this interval the good quality of the fit to the observed profile strongly suggests that the line wings are formed in the photosphere under conditions very close to LTE. The case of the [FORMULA] line, which will not be shown here, is exactly similar to that of [FORMULA] and confirms the discussion given above. Even though the calcium abundances derived with the Holweger and the MACKKL models are slightly different, the hydrogen damping constants derived with both models are quasi identical.

4.4.3. GBEN theoretical model

The fit to the observed line wings obtained with the theoretical flux-constant GBEN model photosphere (Eriksson et al. 1979, Gustafsson et al. 1975) is much less satisfactory, as illustrated by a much higher value of the test-parameter [FORMULA]. The "best fit" synthetic profile of the [FORMULA] line computed with the GBEN model and the corresponding damping parameter given in Table 2 is shown in Fig. 6. It can be seen that while the predicted profile has the same width as the observed one at [FORMULA], it is clearly too broad for [FORMULA] and too narrow for [FORMULA]. It is still possible to give an average value of the damping parameter, but with somewhat larger error bounds. In that respect, and within the accuracies typical of fine analyses (see Sect. 4.4.6), the empirical model temperature distributions give more adequate results than the theoretical flux conxtant GBEN temperature distribution. Synthetic profiles for the [FORMULA] line present exactly the same problems as for [FORMULA].

[FIGURE] Fig. 6. Synthetic profile computed with the GBEN model (thick solid line). The observed spectrum is drawn as a thin continuous line. For comparison the profile computed with the Holweger model starting at [FORMULA] is superimposed as a broken line.

4.4.4. EAGLNT theoretical model

The extensive study by Edvardsson et al. (1993) of the abundances of the elements in 189 F and G dwarfs is based on a grid of theoretical model photospheres which is basically an improvement of the GBEN grid. They are computed with essentially the same computer code but using a more complete line list to account for the line blanketing effects. The solar model of this new grid (Table 6 of their paper) has been used to compute synthetic profiles of the Ca II IRT lines and the result for the [FORMULA] line is shown in Fig. 7. As shown by the value of the test-parameter [FORMULA], they provide a much better match of the observed profiles than that obtained with the GBEN model, although the quality of the fit is not yet as good as with the empirical models. The extended line list of Edvardsson et al. thus leads to significantly improved photospheric temperature distributions.

[FIGURE] Fig. 7. Synthetic profile computed with the EAGLNT model (thick line). The profile computed with the GBEN (dashed line) is shown for comparison. The observed profile is represented by the thin line.

4.4.5. Kurucz theoretical model

With the preceding theoretical model atmospheres, when [FORMULA] is chosen to match exactly the width of the observed profiles at a depression [FORMULA] of, say 0.22, the predicted profiles are too narrow at deeper depressions ([FORMULA]). As seen in Fig. 8, the opposite is true of profiles obtained with the Kurucz (1992) solar model photosphere: the predicted widths are somewhat too wide at the deeper depressions, resulting in a negative value of [FORMULA]. Thus, in the LTE approximation, the Kurucz model does not allow as good a representation of the Ca II IRT lines as that obtained with the empirical models. The absolute value of [FORMULA] shows that the [FORMULA] line is as correctly represented with the Kurucz model atmosphere as with the EAGLNT; for the [FORMULA] line, however, the Kurucz model fares substantially worse than the EAGLNT.

[FIGURE] Fig. 8. Synthetic profile computed with the Kurucz model. For illustration purposes the dashed line shows a synthetic profile calculated with the Kurucz model and the van der Waals interaction constant in the Unsöld approximation, with unit enhancement factor.

4.4.6. Comparison with the CFH solar profiles

The same fitting procedures have also been carried out, but comparing, this time, the synthetic profiles computed with the different model atmospheres to the properly renormalized "stellar quality" CFH solar spectrum instead of the Solar Flux Atlas. The fits obtained have the same quality as the previous ones and the values derived for [FORMULA] are consistent with those found in Table 2, although with a slightly higher uncertainty (by less than [FORMULA]) due to the non-negligible noise in the CFH data ([FORMULA]). In the presence of noise, at each depression [FORMULA], [FORMULA] is affected by a certain uncertainty. When interpreting stellar spectra, if the derived [FORMULA]'s are equal within their error bars on the selected range of [FORMULA], we will consider that a given model provides a satisfactory representation of the Ca II IRT. From this point of view, when the synthetic profiles described above are compared with the CFH spectrum, the Holweger, MACKKL and Edvardsson models qualify as sufficiently adequate models of the solar atmosphere, whereas the GBEN and Kurucz models don't.

Other computations have also been made, neglecting the Paschen lines contribution. If a self-consistent treatment of the continuum normalization is carried out, they lead to [FORMULA] values only slightly smaller than those in Table 2 (by [FORMULA], which is about the size of the uncertainties).

4.4.7. Conclusions of the solar computations

In summary, we confirm that the derived calcium abundance as well as the interaction constants for broadening by collisions with neutral hydrogen atoms depend somewhat on the model photosphere used for the computations. The consequence is that, if we want to analyse stellar spectra of the Ca II IRT lines differentially with respect to the Sun, we need to refer to the abundance and hydrogen damping constants determined in the solar case by means of a solar model atmosphere consistent with the model atmospheres used for the program stars. The very low uncertainties on the solar calcium abundance quoted in Table 2 refer only to uncertainties in the line atomic parameters and equivalent width measurements. In an effort to include the contribution of the choice of the model atmosphere to the uncertainties, we may restrict ourselves to averaging the results obtained with the Holweger, MACKKL and EAGLNT models (the only ones correctly describing the CFH observation), which leads us to recommend the following value for the solar calcium abundance [FORMULA], and, for the hydrogen damping constants: [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA]. Within the error limits, these values are consistent with the damping constants obtained by Smith & Drake (1988) from very high quality solar intensity spectra, using the Holweger-Müller model atmosphere.

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Online publication: December 17, 1999
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