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Astron. Astrophys. 355, 299-307 (2000)

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2. The models

The stellar evolution computations have been performed using the same code and the same input physics as in Salaris & Weiss (1998 - hereinafter SW98). We just recall that we have used the opacities by Iglesias & Rogers (1996) and Alexander & Ferguson (1994) for an [FORMULA]-enhanced heavy elements mixture ([FORMULA]=0.4; the details of the distribution are given in SW98), and that the mixing length calibration allows to reproduce the RGB effective temperatures of a selected sample of GC in the ([FORMULA]) plane, as derived by Frogel et al. (1983), which is the observational constraint for Halo stars. In case of models with diffusion, since the RGB position is almost unchanged with respect to standard isochrones, the same mixing length value satisfies the metal-poor RGB observational constraint. Therefore, we did not change the mixing length calibration with respect to the case of standard isochrones. This permits also to clearly show the influence of diffusion on the stellar models without any contribution from the variation of other parameters. We stress also that large part of our results does not depend on the mixing length calibration.

For the present calculations we have included the diffusion of helium and heavy elements following Thoul et al. (1994); their formalism and the input physics used in our models have been already successfully tested on the Sun (see, e.g., Ciacio et al. 1997). The variations of the abundances of H, He, C, N, O and Fe are followed all along the structures; all other elements are assumed to diffuse in the same way as fully-ionized iron (see, e.g., Thoul et al. 1994 for a comparison among different diffusion formalisms). The local changes of metal abundance are taken into account also in the opacity computation, by calculating the actual global metallicity at each mesh point, and then interpolating among tables with different Z. In this way one does not take into account the small changes in the metal distribution due to differences in the diffusion velocities of C,N,O and Fe; however, for the mass range we are dealing with ([FORMULA]1.0 [FORMULA]) the differences are small and our procedure does not introduce any significant error in the opacity (see, e.g., the detailed discussion in Turcotte et al. 1998).

We have computed a set of MS models (and isochrones) with diffusion and initial metallicities [Fe/H]=-2.3, -2.0, -1.7, -1.3, -1.0, -0.7, -0.6 ([FORMULA]=0.23 and [FORMULA]=3 as in SW98). In addition we have also computed a set of isochrones for 8 and 12 Gyr which displays as actual surface metallicity the set of values previously given (we will call them `calibrated' diffusive isochrones, following the nomenclature by MB99). When computing the latter isochrones we had to ensure that, for the selected ages, all the different evolving masses showed the prescribed surface metallicity. This means that we had to employ an iterative procedure for finding the exact value of the initial metallicity (larger than the final one) to be used for each case (we kept fixed [FORMULA]=3 when deriving the initial helium abundance).

In Fig. 1 we show a comparison in the HR diagram among the sets of isochrones computed, and the standard ones by SW98, with [FORMULA]8 and 12 Gyr and [Fe/H]=-2.3 and -1.0. For the standard (solid lines) and `calibrated' diffusive ones (dotted lines - hereinafter C) the labelled value of [Fe/H] represents the actual surface metallicity, which is constant along the isochrone, while for the non calibrated isochrones with diffusion (dashed line - hereinafter D) it represents only the initial surface metallicity.

[FIGURE] Fig. 1. HR diagram of standard (solid line), D (see text - dashed line) and C isochrones (see text - dotted line) with [FORMULA]8,12 Gyr, [Fe/H]=-2.3 (upper panel) and [Fe/H]=-1.0 (bottom panel).

Our results closely resemble the results by MB99; specifically, for a fixed age and luminosity the standard isochrones are hotter than the D and C ones. The C isochrones are the reddest among the three sets. The [FORMULA] difference at fixed luminosity among the three sets of isochrones increases with increasing metallicity. As in MB99 we find that the shift toward higher [FORMULA] of the C isochrones with respect to the standard ones increases for increasing luminosities; this changes slightly the slope of the MS, making it more vertical for the C isochrones. At [Fe/H]=-2.3 the C and D isochrones are almost coincident all along the lower MS. Table 1 displays a quantitative evaluation of the [FORMULA] difference, at selected luminosities along the lower MS, between standard and C isochrones with t=8 and 12 Gyr and metallicities [Fe/H]=-1.0, -1.7, -2.3.


[TABLE]

Table 1. [FORMULA] difference (in K) between standard and C isochrones ([FORMULA](standard)-[FORMULA](C)) for three selected metallicities ([Fe/H]=-2.3, -1.7, -1.0) and three luminosities along the lower MS.


Fig. 2 shows the difference [FORMULA] at fixed age between standard and C isochrones for three selected values of the evolving mass along the MS, metallicities [Fe/H]=-1.0 and -1.7, [FORMULA]8 and 12 Gyr. For the [Fe/H]=-1.7 isochrones only data with [FORMULA]8 Gyr are shown, since stars with M=0.8[FORMULA] are evolved off the MS at [FORMULA]12 Gyr. The displayed data are comparable with analogous quantities in Fig. 6 of MB99; our results appear consistent with MB99, when taking into account that their [FORMULA] values correspond to an age of 10 Gyr.

[FIGURE] Fig. 2. Difference [FORMULA] (K) between standard and calibrated diffusive C isochrones for three selected values of the evolving mass, with metallicities [Fe/H]=-1.0 ([FORMULA]8 and 12 Gyr) and -1.7 ([FORMULA]8 Gyr). The crossing between the [Fe/H]=-1.7 models and the more metal rich ones is due to evolutionary effects.

In Fig. 3 the differences [FORMULA][Fe/H] ([FORMULA]) between the initial metallicity (helium content) and the TO surface values of D isochrones are displayed for two selected initial metallicities and various ages. As a general trend, [FORMULA][Fe/H] and [FORMULA] increase for decreasing age (in the age range we are dealing with) and for decreasing initial metallicity (at fixed age). In the age range 6-14 Gyr [FORMULA][Fe/H] is generally between 0.1 and 1.0 - apart for the most metal poor isochrones which show a larger metal depletion ([FORMULA][Fe/H][FORMULA]1.8 at t=6 Gyr for an initial [Fe/H]=-2.3) -, while [FORMULA] ranges between [FORMULA]0.05 and [FORMULA]0.2; the surface Y is practically zero for the [Fe/H]=-2.3 isochrones when t is less than about 8 Gyr.

[FIGURE] Fig. 3. He and metal depletion at the TO (difference between the initial and the TO values) as a function of the age for selected D isochrones with the displayed initial metallicities. Filled circles show selected TO masses indicated in units of [FORMULA] (upper panel).

This increase of the TO surface metallicity (and helium) depletion with decreasing age could appear at first surprising, since diffusion has less time to work when age is lower; however, one must also take into account that convective envelopes are progressively thinner for stars populating the TO at decreasing age, thus increasing the rate of depletion of helium and metals (notice that more metal poor models have thinner surface convective regions). It is the competition between these two factors which determines the final trend of the TO surface abundances with time.

In Fig. 4 we compare our C isochrones (bolometric luminosities were transformed to [FORMULA] using the bolometric corrections described in Weiss & Salaris 1999; we just recall that, after the necessary calibration of the zero point to reproduce the solar [FORMULA] value, our adopted bolometric corrections from Buser & Kurucz 1978, 1992 agree quite well with the empirical determinations by Alonso et al. 1996a) with a sample of metal-poor subdwarfs with accurate Hipparcos parallaxes ([FORMULA]) listed by Carretta et al. (1999), and [FORMULA] derived from the Infrared Flux Method (IRFM - Alonso et al. 1996b). The goal is to check if isochrones with full efficiency of diffusion are compatible with the HR diagram of field subdwarfs with accurate parallaxes and empirical [FORMULA] determinations (therefore eliminating the influence of the adopted colour-transformations).

[FIGURE] Fig. 4. [FORMULA]-[FORMULA] diagram of C isochrones and a sample of subdwarfs with accurate Hipparcos parallaxes and IRFM effective temperatures. The average subdwarfs metallicity is given in the labels at the top right corner of the three panels; isochrones metallicities and ages are shown at the left bottom corner of each panel. Subdwarfs are represented by circles; error bars on their values of [FORMULA] and log([FORMULA]) are also displayed. In the upper and middle panel solid lines correspond to the C isochrones with the lower metallicity between the given values. In the bottom panel C isochrones are represented with solid lines, while the standard isochrones with [Fe/H]=-1.0 and t=10,16 Gyr are also displayed (dotted lines) for comparisons.

Lebreton et al. (1999) recently performed this kind of comparison for subdwarfs with metallicities in the range -1.0[FORMULA][Fe/H][FORMULA]0.3, and found that for -1.0[FORMULA][Fe/H][FORMULA]0.5 the inclusion of the full efficiency of diffusion is necessary for reproducing observational data. We extend their analysis by studying the case of lower metallicities. Our adopted subdwarfs spectroscopic [Fe/H] values come from Carretta et al. (1999), and are homogeneous with the Carretta & Gratton (1997) metallicity scale for GC; even if the metallicities adopted by Alonso et al. (1996b) for applying the IRFM method are different (generally lower by [FORMULA] 0.2), the sensitivity of the derived [FORMULA] to the input metallicity is so low (Alonso et al. 1996b) that no appreciable inconsistency is introduced by our choice of the [Fe/H] scale. For two stars we found differences as high as [FORMULA]1.0 between the metallicity used by Alonso et al. (1996b) and the Carretta et al. (1999), and we did not consider them. The [Fe/H] values for the subdwarfs displayed in the pictures are in the ranges, respectively, between -1.55 and -1.69 (top panel), -1.24 and -1.28 (middle panel), -0.98 and-1.02 (bottom panel). The error bar on [Fe/H] is typically by 0.10-0.15, while the errors on [FORMULA] and [FORMULA] are shown in the figure.

The displayed MS isochrones have ages equal to 8 and 12 Gyr respectively, corresponding to approximately the upper and lower limit of the ages determined by SW98 for a large sample of Galactic GC. As it is evident from the figure, the agreement between observations and theoretical models is satisfactory. Our calibrated diffusive C isochrones (which are the correct ones to be compared with field subdwarfs of a given metallicity) reproduce satisfactorily the observations. The best agreement is for subdwarfs with average metallicity [Fe/H]=-1.0 and -1.6; at the intermediate metallicity -1.25 the models appear to be slightly too hot, but when taking into account the error bar on the temperature and metallicity of the subdwarfs the discrepancy does not appear to be significant. In the case of [Fe/H]=-1.0 standard isochrones are also shown; it is clear that in spite of the good agreement between C models and data, one cannot use these results as a definitive proof that diffusion is fully efficient in Halo subdwarfs. Standard isochrones can also reasonably reproduce the observational data (even if for different ages), at least given the present sample of objects and observational uncertainties on [FORMULA] and [FORMULA].

Regarding this last point one should notice that Cayrel et al. (1997) have shown how the MS location of standard isochrones appears to be too hot by [FORMULA]120-140 K in comparison with a sample of metal poor subdwarfs with accurate Hipparcos parallaxes, when using the empirical [FORMULA] determinations by Alonso et al. (1996b), and [Fe/H] values from the compilations by Cayrel de Strobel et al. (1997). This difference with respect to our conclusion that at [Fe/H]=-1.0 standard isochrones are still able to reproduce within the errors the position of local subdwarfs, is due mainly to the different [Fe/H] scale we have employed. We used the Carretta et al. (1999) metallicities which are on a scale homogeneous with the GC metallicities we will use in the next section. For the stars with [Fe/H][FORMULA]1.0 our adopted [Fe/H] values are about 0.2-0.3 larger than the corresponding values given by Cayrel de Strobel et al. (1997 - we averaged the high S/N data); these larger metallicities reduce the discrepancy between standard isochrones and observations.

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Online publication: March 17, 2000
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