3 The stellar spectral library

In the following, we will present the stellar spectral library put together for this work. For the sake of reference, Fig. 1 presents the distribution of all spectra in the $\log\mbox{$T_{\rm eff}$ }- \log g$ plane.

Figure 1: Distribution of the $\mbox{\rm [{\rm M}/{\rm H}]}=0$ spectra incorporated in our stellar library (large symbols) in the $\log\mbox{$T_{\rm eff}$ }- \log g$ plane, compared to the position of stellar models of solar metallicity (small dots; these models are isochrones to be discussed later in Sect. 5.1). The spectra are taken from Castelli et al. (1997; crosses), Fluks et al. (1994; circles), Allard et al. (2000a; squares), and pure blackbody (triangles). Fluks et al. spectra have been arbitrarily located at $\log g=0$. Similar distributions hold for all metallicities between $\mbox{\rm [{\rm M}/{\rm H}]}=-2.5$ and +0.5(see text).

   
3.1 Kurucz atmospheres

Earlier Padova isochrones were based on the Kurucz (1993) libraries of ATLAS9 synthetic atmospheres. As discussed in a series of papers by Castelli et al. (1997), Bessell et al. (1998), and Castelli (1999), these models are superseded by now. Firstly, small discontinuities associated to the scheme of "approximate overshooting'' initially adopted by Kurucz have been corrected (cf. Bessell et al. 1998). Secondly, no-overshooting models have been demonstrated to produce $T_{\rm eff}$-colour relations in better agreement with empirical ones, at least for stars hotter than the Sun (Castelli et al. 1997).

3.1.1 The more recent models

In the present work, we adopt the ATLAS9 no-overshoot models that have been calculated by Castelli et al. (1997). They correspond to the "NOVER'' files available at http://cfaku5.harvard.edu/grids.html. The metallicities cover the values $\mbox{\rm [{\rm M}/{\rm H}]}=-2.5$, -2.0, -1.5, -1.0, -0.5, 0.0, and +0.5, with solar-scaled abundance ratios. A microturbulent velocity $\xi=2~{\rm km~s}^{-1}$, and a mixing length parameter $\alpha=1.25$, are adopted. Notice that these models are now being extended so as to include also $\alpha$-enhanced chemical mixtures, which represents a potentially important improvement for our future works.

Kurucz models cover quite well the region of the $\log\mbox{$T_{\rm eff}$ }$ vs. $\log g$ plane actually occupied by stars, at least in the $3500~{\rm K} \le T \le 50~000~{\rm K}$, $0\le \log g \le5$ intervals (see Fig. 1). However, it has to be extended to both lower and higher $T_{\rm eff}$s, as will be detailed below.

It is important to recall that Kurucz (ATLAS9) spectra are widely used in the field of synthetic photometry, mainly because of their wide coverage of stellar parameters and easy availability. Moreover, there are also good indications in the literature that these spectra do a good job in synthetic photometry, provided that we are dealing with broad-band systems. Compelling examples of this can be found in Bessell et al. (1998), who compares the UBVRIJHKL results obtained from the recent ATLAS9 spectra to empirical relations derived with the infrared flux method, lunar occultations, interferometry, and eclipsing binaries. Their results indicate that the 1998 ATLAS9 models are well suited to synthetic photometry, but for small errors, generally lower than 0.1 mag in colours, that we do not consider as critical. In fact, we are more interested in the overall dependencies of colours and magnitudes with stellar parameters - probably well represented by present synthetic spectra - than on details of this order of magnitude.

Additionally, Worthey (1994) presented extensive comparisons between Kurucz (1993) spectra and stars in the low-resolution spectral library by Gunn & Stryker (1983), obtaining generally a good match for wavelengths redder than the B pass-band. Worthey's Fig. 9 also presents a comparison between Kurucz (1993) solar spectra and Neckel & Labs (1984) data, with excellent results (errors lower than 0.1 mag) all the way from the UV up to the near-IR. Since the ATLAS9 1998 spectra differ just little from the Kurucz (1993) version (a few percent in extreme cases), these results are to be considered still valid.

3.1.2 Some caveats on ATLAS9 spectra

The previously mentioned works point to a reasonably good agreement between ATLAS9 spectra and those of real stars of near-solar metallicity, especially in the visual and near-infrared pass-bands. However, there are many known inadequacies in these spectra, which should be kept in mind as well. Here, we give just a brief list of the potential problems, concentrating on those which may be more affecting our synthetic colours.

ATLAS9 spectra are based on 1D static and plan-parallel LTE model atmospheres, which use a huge database of atomic line data (Kurucz 1995). The line list is known not to be accurate: In fact, Bell et al. (1994) show that the solar spectra calculated using Kurucz list of atomic data present many unobserved lines; moreover, the number of lines which are too strong exceeds those which are too weak. The problem can be appreciated by looking at the high-resolution spectral plots presented by Bell et al. (1994), but could hardly be noticeable in low-resolution plots (such as in the comparisons presented in Worthey's 1994 Fig. 9, and in Castelli et al. 1997 Fig. 2).

Also, Bell et al. (2001) show that a motivated increase in the Fe  I bound-free opacity cause a significant improvement in the fitting of the solar spectrum in the 3000-4000 Å wavelength region, affecting the entire UV region as well. Such increased sources of continuous opacity are still missing in ATLAS9 atmospheres.

These results indicate that ATLAS9 spectra will produce worse results when applied to (i) narrow-band photometric systems, in which individual metallic lines can more significantly affect the colours, and (ii) in the UV region, especially shortward of 2720 Å (see Bell et al. 2001). In both cases, the errors caused by wrong atomica data are such that we can expect not only systematic and $T_{\rm eff}$-dependent offsets in synthetic colours, but also a somewhat wrong dependence on metallicity. Clearly, these points are worth being properly investigated by means of detailed spectral comparisons.

Regarding the present work, the above-mentioned problems (i) critically determine the inadequacy of synthetic colours computed for the Strömgren system (Girardi et al., in preparation), and (ii) may possibly cause significant errors in our synthetic HST/WFPC2 UV colours.

Other potential problems worth of mention are:

• The deficiencies in 1D atmosphere models and in the mixing length description of convection, as compared to exploratory 3D model atmospheres (e.g. Asplund et al. 2000). They should affect mainly the intermediate-strong lines with significant non-thermal broadening and in stars cooler than $\sim$6000 K. The consequences of spatial and thermal inhomogeneities in the atmosphere of the Sun and Procyon have been investigated by Allende Prieto et al. (2001, 2002);
• Inadequacies in the hydrogen line computations. Especially affected are high members of the Balmer series in GK stars - due to inhomogeneities in real atmospheres and inadequate treatment of hydrogen line broadening in cool stars (see Barklem et al. 2000a,b) - and the core of Balmer lines and the region around the Paschen discontinuity and longward in A stars (Colina et al. 1996). In synthetic photometry, these inadequacies are expected to slightly affect the U and B magnitudes for A-F stars, but to have a greater effect in Strömgren indices;
• The inadequacy of using scaled-solar metal ratios in giants where the abundance of CNO elements has been altered by dredge-up events. Similarly, the inadequancy of using scaled-solar abundances for halo and bulge stars, instead of $\alpha$-enhanced ones.

Obviously, some improvement upon these points is expected in future releases of ATLAS spectra (see Castelli & Kurucz 2001), and of other extended spectral grids as well. Fortunately, the work by Bessell et al. (1998) gives us some confidence that present broad-band magnitudes and colours (from U to K) are modelled with an accuracy that is already acceptable for many applications.

Finally, we remark that some authors (Lejeune et al. 1997, 1998) propose the application of a posteriori transformations to Kurucz (1993) spectra, as a function of wavelength and $T_{\rm eff}$, such as to reduce the errors of the derived synthetic UBVRIJHKL photometry. In our opinion, such transformations are questionable because they do not correct the cause of the discrepancies - majorly identifiable in the imperfect modelling of absorption lines - and the case for applying them to stars of all surface metallicities and gravities is far from compelling.

   
3.2 Extension to higher temperatures

For $\mbox{$T_{\rm eff}$ }>50~000$ K, we simply assume black-body spectra. This is probably a good approximation for wavelengths $\lambda>912$ Å. In fact, we find always a reasonably smooth transition in the computed $BC_{S_\lambda}$s as we cross the $\mbox{$T_{\rm eff}$ }=50~000$ K temperature boundary.

   
3.3 Extension to M giants

Synthetic spectra for M giants have still many problems - mainly in their ultraviolet-blue region - that partially derive from incomplete opacity lists of molecules such as TiO, VO and H₂O (see e.g. Plez 1999; Alvarez & Plez 1998; Alvarez et al. 2000; and Houdashelt et al. 2000a,b to appreciate the state of the art in the field).

Therefore, we prefer to use the empirical M giant spectra from Fluks et al. (1994; or "intrinsic'' spectra as referred in their paper). They cover the wavelength interval from 3800 Å to 9000 Å. Outside this interval, the empirical spectra have been extendend with the "best fit'' synthetic spectra computed by the same authors.

However, the whole procedure reveals a problem: if we simply merge empirical and synthetic spectra from Fluks et al. (1994), the resulting synthetic $B\!-\!V$ and $U\!-\!B$ colours just badly correlate with the measured colours for the same stars (which were also obtained by Fluks et al. 1994). This problem probably derives from a bad flux calibration at the blue extremity of the observed spectra and/or from the imperfect match between synthetic and observed spectra at 3800 Å. In order to circumvent (at least partially) the problem, we simply multiply each M-giant spectrum blueward of 4000 Å (with a smooth transition in the range from 4000 Å to 4800 Å) by a constant, typically between 0.8 and 1.2, so that the synthetic colours recover the observed behaviour of the $B\!-\!V$ vs. $V\!-\!K$ data. The first two panels of Fig. 2 show the results.

Figure 2: Colour vs. $V\!-\!K$ relations for giants. The connected open circles represent the relation obtained from $\mbox{\rm [{\rm M}/{\rm H}]}=0$ ATLAS9 spectra located along the $\mbox{$T_{\rm eff}$ }= 3250 + 500~\log g$ line (the typical location for RGB stars) in the diagram of Fig. 1. The connected open squares correspond to the relation obtained from the M-giant spectra from Fluks et al. (1994), completed and modified at $\lambda <4800$ Å as detailed in the text. The dashed lines represent the empirical relations for F0-K5 solar-metallicity giants from Alonso et al. (1999b), whereas the dotted lines are the synthetic relations for K0-M7 giants from Houdashelt et al. (2000a). The empirical data for M giants (small dots) are from Fluks et al. (1994). As far as possible, all observations have been converted to the same photometric system as used in our synthetic photometry (i.e. the "Bessell'' UBVRIJHK system; see text).

Actually, Fig. 2 presents six different colour vs. $V\!-\!K$ diagrams that are useful to understand the situation for giants. Care has been taken in expressing data and models in the same photometric system, the "Bessell'' UBVRIJHK one, that we will detail later in Sect. 4.1. For M giants, the empirical photometric data from Fluks et al. (1994; small dots) can be compared with the results of our synthetic photometry. Noteworthy, there is a reasonably good match between the synthetic and observed relations for most colours. This has been imposed for $U\!-\!B$ and $B\!-\!V$, whereas is a natural result for all colours involving wavelengths longer than $\sim$4800 Å. The only clear exception is the $V\!-\!R$ colour, for which differences of $\sim$0.4 mag are found for all giants of spectral type later than M4 ( $\mbox{$V\!-\!K$ }\ga5$). The reason for this discrepancy is not clear, but may lie in the use of R filters with different transmission curves. Also the predictions for $J\!-\!K$ do not fit well all the photometric data, somewhat failing for the spectral types later than M7 ( $\mbox{$V\!-\!K$ }\ga8$). However, since these latters are quite rare, such mismatch does not pose a serious problem.

For the sake of comparison, Fig. 2 also presents the relations obtained by means of the M-giant models from Houdashelt et al. (2000a), in the case of solar metallicity. Together with other recent examples (e.g. Plez 1999; Alvarez et al. 2000), they represent state-of-the-art computations of cool oxygen-rich stellar atmospheres. As can be appreciated in the figure, Houdashelt et al. models reproduce well the empirical data as far as $V\!-\!K\la6$ (spectral types earlier than M5), but start departing from these for cooler stars. A similar situation holds if we look at different $\mbox{$T_{\rm eff}$ }$-colour relations, as can be seen in Figs. 13 and 14 of Houdashelt et al. (2000a), where they compare their $\mbox{$T_{\rm eff}$ }$-colour relations with those obtained with Fluks et al. (1994) spectra and data for field giants. Also in this case, it seems that Fluks et al. (1994) spectra do better reproduce the empirical relations for the spectral types later than M4.

Once we have defined the library of M-giant spectra, we associate effective temperatures to them by using the scale favoured by Fluks et al. (1994). In this scale, M giants cover the temperature interval from 3 850 K (MK type M0) to 2500 K (MK type M10). We recall that Fluks et al. (1994) $T_{\rm eff}$ values are derived from a careful fitting of the observed spectra with synthetic model atmospheres of solar metallicity. Their scale is also in excellent agreement with the empirical one from Ridgway et al. (1980), which covers spectral types earlier than M6.

After the proper $T_{\rm eff}$ is attributed, each one of our modified spectra is completely re-scaled by a constant, so that the total flux vs. $T_{\rm eff}$ relation - i.e. $F_{\rm bol}=\sigma T_{\rm eff}^4$ - is recovered.

Finally, we face the problem of defining the transition between the M-giant spectra, and the ATLAS9 ones which are available for temperatures higher than 3500 K. To this aim, it is helpful to examine Fig. 2, where we also include:

• the synthetic photometry for a sequence of ATLAS9 spectra for M giants (empty circles). These spectra are located along the line $\mbox{$T_{\rm eff}$ }= 3250 + 500~\log g$, which represents fairly well the locus of low-mass red giants in a $\mbox{$T_{\rm eff}$ }$ vs. $\log g$ diagram (see Fig. 1);
• the mean empirical relations for F0-K5 giants of $\mbox{\rm [{\rm Fe}/{\rm H}]}=0$as derived from Alonso et al. (1999b) fitting formulas (dashed lines).

An important fact to be noticed is that our synthetic photometry reproduces Alonso et al. (1999b) relations in all colours remarkably well.

From inspecting this and other similar plots, we can conclude that the mismatch between Kurucz ATLAS9 and Fluks et al. (1994) spectra starts at about $\mbox{$T_{\rm eff}$ }=3850$ and increases slowly as the temperature decreases down to 3500 K (i.e. from $\mbox{$V\!-\!K$ }\simeq3.5$ to $\mbox{$V\!-\!K$ }\simeq4.7$). Hence, we adopt a smooth transition between these two spectral sources over this temperature interval. The same M giant spectra are assumed for all metallicities.

The complete procedure ensures reasonable colour vs. $V\!-\!K$ relations for all giants of near-solar metallicity (Fig. 2). Nevertheless, this kind of approach cannot be completely satisfactory, first because the original Fluks et al. (1994) spectra have been artificially corrected at wavelengths shorter than 4800 Å in order to produce reasonable $B\!-\!V$ and $U\!-\!B$, and second because we do not dispose of similar M-giant spectra for metallicities very different from solar. Better empirical and theoretical spectra for M giants seem to be urgently needed. Anyway, in the context of the present work the problem is not dramatic because M giants cooler than $\mbox{$T_{\rm eff}$ }\sim3500$ K are only found in the RGB-tip and TP-AGB phases of high metallicity stellar populations, and constitute just a tiny fraction of the number of red giants. The problem could be critical, instead, when we consider integrated properties of stellar populations, because M giants, despite their small numbers, have high luminosities and contribute a sizeable fraction of the integrated light.

   
3.4 Extension to M+L+T dwarfs

Although the modelling of cool dwarfs atmospheres presents challenges comparable to those found in late-M giants (e.g. the inadequacy of TiO and H₂O line lists, and dust formation; see Tsuji et al. 1996, 1999; Leggett et al. 2000), present results compare reasonably well with observational spectral data (see e.g. Fig. 9 in both Leggett et al. 2000 and 2001). A review on the subject can be found in Allard et al. (1997).

An extended library of synthetic spectra for cool dwarfs (of types M and later) is provided by Allard et al. (2000a; see ftp://ftp.ens-Lyon.fr/pub/users/CRAL/fallard). We use their set of "BDdusty1999'' atmospheres (see also Chabrier et al. 2000; Allard et al. 2000b, 2001), that should supersede the "NextGen'' models from the same group (Hauschildt et al. 1999) due to the consideration of better opacity lists and dust formation. Dust can significantly affect the coolest atmospheres, corresponding to dwarfs of spectral types L and T.

The selected spectra cover the $T_{\rm eff}$ intervals:

• from 4000 K to 2800 K ("AMES'' models) for metallicities $\mbox{\rm [{\rm M}/{\rm H}]}=0.0$, -0.5, -1.0, -1.5, and both $\log g=5.0$ and 5.5;
• from 2800 K down to 500 K ("AMES-dusty'' models) only for $\mbox{\rm [{\rm M}/{\rm H}]}=0.0$, and $\log g$ values between 3.5 and 6.0.

These spectra are presented with a extremely high resolution, that by far exceeds the one necessary in our work. Thus, we have convolved the flux per unit frequency $F_\nu$ with a Gaussian filter of $\sigma_\nu=2.4\times10^{-18}~ {\rm Hz}$, that corresponds to a FWHM of 20 Å at $\lambda=5550$ Å. The resulting spectra were then reported to the same grid of wavelengths of Kurucz' spectra.

We find that there is a good agreement between ATLAS9 and BDdusty1999 spectra in the $T_{\rm eff}$ range between $\sim$3800 K and 4000 K. Then, we set the transition between ATLAS9 and BDdusty1999 spectra at $\sim$3900 K. This choice guarantees smooth $T_{\rm eff}$ vs. colour relations for dwarfs.