6 Results

Computing synthetic spectra is one step, distilling useful information from it is a second - and far more difficult - one. Fundamental stellar parameters for this sample of bright stars are a first direct result which can be deduced from this comparison between ISO-SWS data and synthetic spectra. In Papers III and IV of this series, these parameters will be discussed and confronted with other published stellar parameters.

Figure 3: Comparison between the ISO-SWS data of $\alpha$ Cen A (black) and the synthetic spectrum (grey) with stellar parameters $\mbox{$T_{{\rm eff}}$ }= 5830$ K, $\log g = 4.35$, M = 1.3 $M_{\odot }$, [Fe/H] = 0.25, $\mbox{$\xi_{\rm t}$ }= 1.0$ km s-1, $\mbox{${\rm ^{12}C/^{13}C}$ }~\!\! =~\!\! 89$, $\varepsilon$(C) = 8.74, $\varepsilon$(N) = 8.26, $\varepsilon$(O) = 9.13 and $\mbox{$\theta_{\rm d}$ }= 8.80$ mas. Some of the most prominent discrepancies between these two spectra are indicated by an arrow. A coloured version of this plot is available in the appendix as Fig. .1.

Figure 4: Comparison between band 1 and band 2 of the ISO-SWS data of $\gamma$ Dra (black) and the synthetic spectrum (grey) with stellar parameters $\mbox{$T_{{\rm eff}}$ }= 3960$ K, $\log g = 1.30$, M = 1.7 $M_{\odot }$, [Fe/H] = 0.00, $\mbox{$\xi_{\rm t}$ }= 2.0$ km s-1, $\mbox{${\rm ^{12}C/^{13}C}$ }= 10$, $\varepsilon$(C) = 8.15, $\varepsilon$(N) = 8.26, $\varepsilon$(O) = 8.93 and $\mbox{$\theta_{\rm d}$ }= 9.98$ mas. A coloured version of this plot is available in the appendix as Fig. .2.

A typical example of both a warm and cool star is given in Figs. 3 and 4 respectively. Different types of discrepancies do emerge. The size of the discrepancies between ISO-SWS observations and theoretical predictions varies a lot. Both for the warm and for the cool stars we see a general good agreement in shape between the ISO-SWS and theoretical data in band 1, with however (local) error peaks up to $\sim$8%. An exception is $\beta$ Peg for which the shape is wrong by $\sim$6% and local error peaks may go up to $\sim$15%. The agreement between observational and synthetic data is worse in band 2: a general mismatch by up to $\sim$15% may occur.

By scrutinising carefully the various discrepancies between the ISO-SWS data and the synthetic spectra of the standard stars in our sample, the origin of the different discrepancies was elucidated. First of all, a description on the general trends in discrepancies for the warm stars will be made, after which the cool stars will be discussed.

  
6.1 Warm stars: A0-G2

1.

Especially when concentrating on $\alpha$ Cen A (G2 V), one notifies quite a few spectral features which appear in the ISO-SWS spectrum, but are absent in the synthetic spectrum. By comparing the spectra of the other warm stars ($\beta$ Leo, $\alpha$ Car, $\alpha$ Lyr and $\alpha$ CMa) with each other, corresponding spectral features can be recognised in their ISO-SWS spectra, although this is somewhat more difficult for $\beta$ Leo due to the lower resolving power and lower signal-to-noise ratio. Also for the cooler stars in the sample, these spectral features are (weakly) present. Some of the most prominent ones are indicated by arrows in the spectrum of $\alpha$ Cen A in Fig. 3. Molecular absorption can not be a possible cause of/ contribution to e.g. the 3.9 $\mu$m feature, since this feature is seen for the warm stars in the sample, where no molecular absorption occurs at these wavelength ranges. The solar FTS-ATMOS spectrum proved to be extremely useful for the determination of the origin of these features. All spectral features, indicated by an arrow in Fig. 3, turned out to be caused by - strong - atomic lines (Mg, Si, Fe, Al, C, ...). This is illustrated for the wavelength range from 3.85 to 3.92 $\mu$m in Fig. 5. In panel a of Fig. 5, the high-resolution FTS-ATMOS spectrum of the Sun is compared with its synthetic spectrum in the wavelength range from 3.853 to 3.917 $\mu$m. The strongest lines are identified by using the line list of Geller (1992). At the ISO resolving power of 1000 in band 1E, these atomic lines are reduced to the features indicated in panel b. The same features can be recognised in the ISO-SWS spectrum of $\alpha$ Cen A in panel c. It is clear that these atomic features are not well calculated for the synthetic spectra of the Sun and $\alpha$ Cen A.

Figure 5: In panel a) the high-resolution FTS-ATMOS spectrum of the Sun is compared with its synthetic spectrum based on the Holweger-Müller model (1974) and computed by using the atomic line list of Hirata & Horaguchi (1995). Panel  b) shows the same comparison as panel  a), but now at a resolving power of 1000 for the wavelength range going from 3.52-4.08 $\mu$m (band 1E). The data of band 1E of the ISO-SWS spectrum of $\alpha$ Cen A (at a resolving power of 1000) are plotted in panel  c).

For Fig. 5, the atomic line list of Hirata & Horaguchi (1995) was used to generate the synthetic spectrum. This line list has as starting files the compilation by Kurucz & Peytremann (1975) and Kurucz (1989) who lists the semi-empirical gf-values for many ions. Energy levels were adopted from recent compilations (Sugar & Corliss 1985) or individual works. The line list of Kelly (1983) was merged into the file. The gf-values are taken from several compilations (e.g. Fuhr et al. 1988a, 1988b; Reader et al. 1980; Wiese et al. 1966, 1969; Morton 1991, 1992). Published results of the Opacity Project (Seaton 1995) were also included. Comparing the atomic line list of Hirata & Horaguchi (1995) with the identifications as given by Geller (1992) made clear that quite some lines are misidentified or not included in the IR atomic line list of Hirata & Horaguchi (1995).

The usage of VALD (Vienna Atomic Line Database: Piskunov et al. 1995; Ryabchikova et al. 1997; Kupka et al. 1999) and the line list of van Hoof (1998) did not solve the problem. Obviously, the oscillator strengths of the atomic lines in the infrared are not known sufficiently well.

In order to test this hypothesis, Sauval (2002, priv. comm.) has constructed a new atomic line list by deducing new oscillator strengths from the high-resolution ATMOS spectra of the Sun (625-4800 cm^-1). A preliminary comparison using this new linelist is given in Fig. 6 (which should be compared with Fig. 5). The contents, nature, limitations, uncertainties, ... of this new line list will be discussed in a forthcoming paper. But it is already obvious from a confrontation between Figs. 5 and 6 that these new oscillator strengths from Sauval are more accurate.

Figure 6: In panel a) the high-resolution ATMOS spectrum of the Sun is compared with its synthetic spectrum based on the Holweger-Müller model (1974) and computed by using the atomic line list of Sauval (2002, priv. comm.). Panel  b) shows the same comparison as panel  a), but now at a resolving power of 1000 for the wavelength range going from 3.52-4.08 $\mu$m (band 1E). The data of band 1E of the ISO-SWS spectrum of $\alpha$ Cen A (at a resolving power of 1000) are plotted in panel  c).

It is plausible that the same reason, i.e. wrong and missing oscillator strengths of atomic lines in the infrared, together with noise, is the origin of the observed discrepancies between the ISO-SWS and synthetic spectra for the other warm stars in the sample, because

• incomplete atomic data already caused a problem for the synthetic spectrum of $\alpha$ Cen A, and
• the stars concerned are hotter than $\alpha$ Cen A, so atoms (maybe of higher excitation or ionisation) will determine even more the spectral signature of these stars.

The lack of reliable atomic data rendered the determination of the fundamental stellar parameters for the warm stars from the ISO-SWS data impossible. Another consequence concerns the continuum, which was very difficult to determine. Therefore, the uncertainty on the angular diameter of these warm stars is more pronounced.

2.

Secondly, the hydrogen lines are conspicuous. For example, the synthetic hydrogen Pfund lines are almost always predicted as being too strong for main-sequence stars, while they are predicted as being too weak for the supergiant $\alpha$ Car. This indicates a problem with the generation of the synthetic hydrogen lines, which is corroborated when the high-resolution FTS-ATMOS spectrum of the Sun is compared with its synthetic spectrum (see Fig. 7). In the TURBOSPECTRUM program (Plez et al. 1992; Plez et al. 1993) the hydrogen line-profile calculation, adopted from the SYNTHE code of Kurucz, includes the Stark, the Doppler, the van der Waals and the resonance broadening. For the first four lines in every series, the fine structure is also included for the core calculations. The problematic computation of the self broadening of hydrogen lines (Barklem et al. 2000) and the unproper treatment of the convection by using the mixing-length theory (see, e.g., Asplund et al. 2000 where realistic ab-initio 3D radiative-hydrodynamical convection simulations have been used) may account for part of this discrepancy.

Figure 7: FTS-ATMOS intensity spectrum of the Sun (black) at a resolving power $\lambda / \Delta \lambda$ of 60 000. The synthetic spectrum (grey) has been computed using the Holweger-Müller model (Holweger & Müller 1974) with $\mbox{$\xi_{\rm t}$ }= 1$ km s-1. The Bracket $\beta$ line is indicated by an arrow. Most of the other features are CO $\Delta v = 2$ lines, some are atomic lines.

3.

Compared to the ISO-SWS data, the synthetic spectra of warm stars display a higher synthetic flux between the H5-9 and H5-8 hydrogen line (see Fig. 8). From other SWS observations available in the ISO data-archive, we could deduce that this "pseudo-continuum'' starts arising for stars hotter than K2 ($\sim$4500 K). Since this effect is systematically seen for all warm stars in the sample, the origin of this discrepancy can not be a wrong multiplication factor. Note that band 1D and band 1E overlap each other over quite a large wavelength range, so that the multiplication factor of band 1E can be accurately determined. Moreover such an effect was not seen for the cooler K and M giants, the spectrum of which is dominated by OH features in this wavelength range, so we could reduce the problem as having an atomic origin. A scrutiny on the hydrogen lines shows that the high-excitation Humphreys-lines (from H6-18 on) - and Pfund-lines - are always calculated as too weak. Moreover, the Humphreys ionisation edge occurs at 3.2823 $\mu$m. Since the discrepancy does not appear above the limit (i.e. at shorter wavelengths) and is disappearing beyond the Brackett-$\alpha$ line, the conclusion is reached that a problem with the computation of the complex line profile of the crowded Humphreys hydrogen lines towards the series limit causes this discrepancy. The higher the transition, i.e. going from H6-18 to H6-93, the more the $\log gf$-value should be increased in order to get a good match between observations and theoretical computations: increasing $\log gf$ with +0.3 dex gives us a good match for the H6-18 line in $\alpha$ Car, while we should increase $\log gf$ with +1.0 dex for the H6-16 line in $\alpha$ Car. We may conclude that a complex problem with the computation of the pressure broadening will be on the origin of the discussed discrepancy.

Figure 8: Comparison between the observed ISO-SWS spectrum of $\alpha$ Car and its synthetic spectrum in the wavelength region from 3.20 to 4.08 $\mu$m. The Brackett-$\alpha$, two Pfund-lines (H5-8 and H5-9) and a number of Humphreys-lines are identified. The dashed line indicates the Humphreys ionisation edge.

4.

From 3.48 $\mu$m on, fringes at the end of band 1D affect the ISO-SWS spectra of almost all stars in the sample.

5.

A clear discrepancy is visible at the beginning of band 1A. For the warm stars, the H5-22 and H5-23 lines emerge in that part of the spectrum. An analogous discrepancy is also seen for the cool stars, though it is somewhat more difficult to recognise due to the presence of many CO features (Fig. 9). Being present in the continuum of both warm and cool stars, this discrepancy is attributed to problems with the Relative Spectral Response Function (RSRF). A broad-band correction was already applied at the short-wavelength edge of band 1A (Vandenbussche et al. 2001), but the problem seems not to be fully removed. At the band edges, the system response is always small. Since the data are divided by the RSRF, a small problem with the RSRF at these places may introduce a pronounced error at the band edge.

Figure 9: Band 1A of Vega (A0 V), $\alpha$ Car (F0 II) and $\alpha$ Tau (K5 III) are displayed. The ISO-SWS spectrum is plotted in black, the synthetic spectrum in grey. A clear discrepancy is visible at the beginning of this band ( $\lambda \protect \la 2.407$ $\mu$m).

6.

Memory effects of the detectors make the calibration of band 2 for all the stars very difficult. These memory effects are more severe for the cool stars, since the CO and SiO absorptions cause a steep increase (decrease) in flux for the up (down) scan (Fig. 10). Band-border ratios are determined from the composite SEDs by Cohen or from the synthetic spectrum. The RSRFs for the sub-bands will therefore only be modelled well once there is a fool-proof method to correct SWS data for detector memory effects. Two other examples of this problematic RSRF modelling are visible in band 2B and band 2C. For almost all warm and cool stars, there is a "dip'' around 6 $\mu$m and a jump in flux level around 9.3 $\mu$m (Fig. 10). This latter problem is a residue of the correction for an instrumental absorption feature, documented by Vandenbussche et al. (2001). The short-wavelength part of band 2A (where the CO $\Delta v = 1$absorption starts) and of band 2C (with SiO $\Delta v = 1$) are thus useless for passing a quantitative judgement upon the parameters influencing the observed spectrum in this wavelength range, like $T_{{\rm eff}}$, $\varepsilon$(C), $\varepsilon$(O), ... From the longer wavelength parts of these same bands, one will not be able to estimate directly fundamental stellar parameters, but these data enable us to see if there are no contradictions between this part of the ISO-SWS spectrum and the synthetic spectrum generated using the parameters adopted from literature or estimated from the ISO-SWS data in band 1.

Figure 10: Band 2 for Sirius (A1 V) and $\gamma$ Dra (K5 III) are displayed. The ISO-SWS spectrum is plotted in black, the synthetic spectrum in grey. Problems with the RSRF are indicated by an arrow. A coloured version of this plot is available in the appendix as Fig. .3.

  
6.2 Cool stars: G2-M2

1.

The situation changes completely when going to the cool stars of the sample. While the spectrum of the warm stars is dominated by atomic-line features, molecules determine the spectral signature of the cool stars. A few of the - problematic - atomic features (see Sect. 6.1) can still be identified in these cool stars, e.g. the Mg-Si-Al-Ti-Fe spectral feature around 3.97 $\mu$m (Fig. 5) remains visible for the whole sample, even in $\beta$ Peg.

2.

One of the most prominent molecular features in band 1 is the first-overtone band of carbon monoxide (CO, $\Delta v = 2$) around 2.4 $\mu$m. Already from $\alpha$ Cen A on, CO lines emerge in band 1A, although the atomic features are still more dominant for this star. It is striking that for all stars in the sample with spectral type later than G2, the strongest CO features (=band heads of $^{12}{\rm {CO}}$ 5-3, 6-4, 7-5, 8-6, and of $^{13}{\rm {CO}}$ 4-2, 5-3, 6-4 and 7-5) are always predicted too strong compared to the ISO-SWS observation (see, e.g., band 1A of $\gamma$ Dra in Fig. 11, where both SWS and theoretical data are rebinned to a resolving power of 1500, being the most conservative theoretical resolution for band 1A, Leech et al. 2002). The only exception is seen for the band head of $^{13}{\rm {CO}}$ 4-2 since the problem with the RSRF at the beginning of band 1A (see point 5. in previous section) dominates in this wavelength range.

Figure 11: Comparison between the ISO-SWS observation and the synthetic spectrum of $\gamma$ Dra at a resolving power of 1500 (being the most conservative theoretical resolution for band 1A Leech et al. 2002). A coloured version of this plot is available in the appendix as Fig. .4.

In these oxygen-rich stars, the CO lines are a direct measure of the C abundance. From the present spectra, this carbon abundance can be estimated in two ways:

• the strength of the CO spectral features is directly related to $\varepsilon$(C);
• the presence of many CO absorption lines causes a global depression in the spectrum up to $\sim$2.9 $\mu$m (see Fig. 4 in Paper I), which may be used to determine $\varepsilon$(C).

Computing a synthetic spectrum with the carbon abundance determined from this last criterion, results however in the (strongest) computed CO spectral features being too strong compared to the ISO-SWS observation, most visible in the strong bandheads (2-4%). It has to be noted that this mismatch occurs in band 1A, where the standard deviation of the rebinned spectrum is larger than for the other sub-bands (see Fig. 6 in Paper I) and that the error is within the quoted accuracy of ISO-SWS in band 1A (Leech et al. 2002). However, this mismatch is not random, in the sense that the observed CO features are always weaker than the synthetic ones.

A first step towards solving this problem was the use of the high-resolution FTS-KP spectrum of $\alpha$ Boo (see Sect. 5). As was explained in Sect. 5 a high-resolution synthetic spectrum was generated for $\alpha$ Boo and rebinned to a resolution of 60 000. The agreement between the observed FTS-KP and synthetic spectrum is extremely good! One example was already shown in Fig. 16, another one is depicted in Fig. 17 on page . In this latter figure, the problematic subtraction of the atmospheric contribution causes the spurious features around 2.46700 $\mu$m for the summer FTS-KP spectrum and around 2.46735 $\mu$m for the winter FTS-KP spectrum. When scrutinising carefully the first-overtone CO lines in the FTS-KP spectrum, it is obvious that all the ¹²CO 2-0 lines, and almost all the ¹²CO 3-1 lines, are predicted as too weak (by 1-2%) and not as too strong as was suggested in previous paragraph from the comparison between ISO-SWS and synthetic spectra! Also the fundamental CO lines are predicted as being a few percent too weak. For a better judgement of the KP-SWS-synthetic correspondence, the FTS-KP spectrum was rebinned to the ISO resolving power. This was not so straightforward due to the presence of the spurious features originating from the problematic subtraction of the atmospheric contribution. In order to conserve the flux, these spurious features were replaced by the flux value of the synthetic spectrum. This adaptation is acceptable for the following reasons:

• most of the changes were made in regions that are not strongly contributing to the most prominent features;
• if for a particular feature the first argument does not apply, the change mentioned above does still not introduce additional errors. The FTS-KP-spectrum will be locally shifted towards the ISO spectrum, since the synthetic predictions are (in this particular case) systematically in between the ISO-SWS and FTS-KP spectrum. Thus, if statistically relevant differences are still detected, they will always correspond to an underestimate of the real differences between the ISO-SWS and FTS-KP spectra.

Figure 18 on page shows the comparison between the ISO-SWS, the FTS-KP and the synthetic spectrum for $\alpha$ Boo at a resolving power of 1500. It is clear that the differences between the strength of the CO features in the ISO-SWS and FTS-KP spectrum are significant. This is a strong indication for problems in the calibration process. The question now arises where this calibration problem originates from.

Firstly, it has to be noted that the flux values are unreliable in the wavelength region from 2.38 to 2.40 $\mu$m due to problems with the RSRF of band 1A; main features here include the CO 2-0 P18 and the CO 2-0 P21 lines.

Secondly, no correlation is found with the local minima and maxima in the RSRF of band 1A.

Figure 12: Comparison between the AOT06 observation (revolution 538, black) and the AOT01 speed-4 observation (grey) of $\gamma$ Dra. Both are rebinned to a resolving power of 1500. A coloured version of this plot is available in the appendix as Fig. .5.

Defining the spectral resolution and instrumental profile for the ISO-SWS grating spectrometers, is not straightforward (Lorente 1998; Lutz et al. 1999). Only for an AOT02 observation has the instrumental profile now been derived quite accurately (Lutz et al. 1999). The AOT01 mode introduces an additional smoothing which, due to the intricacies of SWS-data acquisition, is different from a simple boxcar smooth. The simulation of SWS-AOT01 scans gives non-gaussian profiles, as can be seen in Fig. 6 by Lorente (1998). Nevertheless, the theoretical profile of a speed-4 observation approximates a gaussian profile very closely. Therefore, since the instrumental profile of an AOT01 is still not exactly known, the synthetic data were convolved with a gaussian with ${FWHM}=\lambda$/resolution. This incorrect gaussian instrumental profile introduces an error which will be most visible on the strongest lines.

Figure 13: Top: Comparison between the AOT01 speed-4 observation of $\gamma$ Dra rebinned at 1) a resolving power of 1500 (black) and 2) a resolving power of 1300 (grey). Bottom: Comparison between the ISO-SWS AOT01 observation in band 1A (black) and its synthetic spectrum (grey) at a resolving power of 1300. A coloured version of this plot is available in the appendix as Fig. .6.

Only for $\gamma$ Dra an AOT06 observation, scanning this wavelength range, was available in the ISO data-archive. The comparison between the AOT01 speed-4 and AOT06 observation, both rebinned to a resolving power of 1500, is an indication that the resolving power of an AOT01 speed-4 observation in band 1A is lower than 1500 (Fig. 12). The theoretical resolving power for an AOT01 speed-4 observation in band 1A is $\ge$1500. Figure 5 in Lorente (1998) shows, however, a large deviation from this value, attributed to the fainter continuum of the source used for measuring the instrumental profile and the less accurate fitting in this band. The use of a resolving power of 1300 - instead of the theoretical resolving power of 1500 - for band 1A yields a) almost no difference from the AOT01 SWS data at a resolving power of 1500 and b) a better match between the SWS and synthetic data for the strongest CO features (Fig. 13). This observational resolving power of 1300 is also in good agreement with the observational value given by Lorente (1998) in her Fig. 5.

The incorrect use of a Gaussian instrumental profile, together with too high a - theoretical - resolving power of 1500 form the origin of the discrepancy seen for the strongest CO features. Therefore, the resolving power of band 1A was taken to be 1300 instead of 1500. Part of the other discrepancies seen in band 1A may be explained by problems with 1. the accuracy of the oscillator strengths of atomic transitions in the near-IR (see point 1. in this section) or 2. the RSRF in the beginning of band 1A (see point 5. in previous section).

Figure 14: Summer and winter FTS-KP spectra of $\alpha$ Boo at a resolving power $\lambda / \Delta \lambda$ of 60 000. They are compared with the synthetic spectrum for $\alpha$ Boo with stellar parameters $\mbox{$T_{{\rm eff}}$ }= 4350$ K, $\log g = 1.50$, M = 0.75 $M_{\odot }$, [Fe/H] = -0.50, $\mbox{$\xi_{\rm t}$ }= 1.7$ km s-1, $\mbox{${\rm ^{12}C/^{13}C}$ }= 7$, $\varepsilon$(C) = 7.96, $\varepsilon$(N) = 7.61, $\varepsilon$(O) = 8.68, $\varepsilon$(Mg) = 7.33 and $\Gamma _{\rm t} = 3$ km s-1. The OH 1-0 lines are predicted as too weak. A coloured version of this plot is available in the appendix as Fig. .7.

3.

Concentrating in the OH-lines, we see that for both the high-resolution FTS-KP (Figs. 14 and 16) and the medium-resolution SWS spectra (Fig. 19), the strongest lines (OH 1-0 and OH 2-1 lines) are predicted as too weak, while the other OH lines match very well. Since the same effect occurs for these two different observations, it is plausible to assume that the origin of the problem is situated in the theoretical model or in the synthetic-spectrum computation. Wrong oscillator strengths for the OH lines could, e.g., cause that kind of problems. Being based on different electric dipole moment functions (EDMFs), the OH-line lists of Sauval (Mélen et al. 1995), Partridge & Schwenke (1997) and Goldman et al. (1998), do however all show the same trend. The difference in $\log gf$-value for the main branches is small (<0.01) between Goldman and Sauval, and is somewhat larger (<0.05) between Schwenke and Sauval. Thus, no systematic error seems to occur in the oscillator strenghts. Since a similar discrepancy was also noted for the low-excitation CO lines, some assumptions on which the models are based, e.g. homogeneity, hydrostatic equilibrium, are cleary questionable for these stars.

4.

From 3.48 $\mu$m on, fringes at the end of band 1D affect the ISO-SWS spectra of almost all stars in the sample.

5.

Also for the cool stars, the same remark as given in point 6 for the warm stars concerning the memory effects applies. Clearly, the fundamental bands of CO and SiO are present for almost all cool stars, though no accurate quantitative scientific interpretation can be made due to these memory effects.