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

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3. Atomic diffusion and MS-fitting distances

The MS-fitting technique is the `classical' method to derive distances to GC (see, e.g., Sandage 1970). The basic idea is very simple. Suppose that precise parallaxes of neighbouring subdwarfs are available; for a given GC metallicity [FORMULA], it is possible to construct an empirical template MS by considering subdwarfs with metallicity [FORMULA] close to [FORMULA] and applying to their colours small shifts (obtained using the derivative [FORMULA][Fe/H] as derived from theoretical isochrones) for reducing them to a mono-metallicity sequence with [FORMULA]. The fit of this empirical MS to the observed GC one (reddening-corrected) provides the cluster distance modulus.

The main underlying assumption is that the MS of subdwarfs with a certain value of [Fe/H] is coincident with the MS of GC with the same metallicity. If atomic diffusion is at work in Halo stars, the underlying assumption of this technique is no longer rigorously satisfied. The point is that (as previously discussed) the spectroscopical metallicity of a GC is determined from its RGB stars; this metallicity is very close to the primordial GC chemical composition, but is not (due to the effect of diffusion) the MS one. Therefore, when fitting the local subdwarfs MS to the MS of a GC with the same observed metallicity, one is introducing an error in the derived distance modulus.

The release of the Hipparcos catalogue has enlarged the number of metal poor subdwarfs with precise parallaxes which can be now used for applying this technique to the Galactic GC, and several authors (see, e.g, Reid 1997; Gratton et al. 1997; Chaboyer et al. 1998; Carretta et al. 1999) have recently derived distances (and ages) of GC using the MS-fitting and subdwarfs with Hipparcos parallaxes. In particular, Carretta et al. (1999) have carefully analyzed the total error budget associated with the MS-fitting, but the effect of atomic diffusion in subdwarfs is nowhere mentioned.

We display in Fig. 5 standard, C and D isochrones transformed to the [FORMULA] plane according to the transformations described in Weiss & Salaris (1999), for [Fe/H]=-2.3 and -1.0. In the case of D isochrones, when computing the bolometric corrections (already used in the previous section) and the [FORMULA]-colour conversion, we have taken into account the fact that the surface [Fe/H] value is not constant along the MS (it is actually decreasing). It must be noticed that the surface helium content for stars around the TO of the isochrones with diffusion is always much lower (generally Y is in the range [FORMULA]0.1-0.2 for ages between 8 and 12 Gyr, as discussed in the previous section) than the helium content used in the model atmospheres producing the adopted colour transformations. However, this does not introduce a substantial systematic error since, according to Carney (1981), the He abundance does not appreciably affect the flux distribution at temperatures appropriate to GC dwarfs and subgiants.

[FIGURE] Fig. 5. Colour Magnitude Diagram of standard (solid line) and D (dashed line) isochrones with [FORMULA]8 Gyr and the displayed metallicities; also plotted are C isochrones (dotted lines) for the same metallicities but [FORMULA]8 and 12 Gyr.

The behaviour of the isochrones in the observational [FORMULA] plane closely follows the results in the theoretical HR diagram. Standard isochrones (solid line) are systematically bluer than the diffusive ones. The C isochrones are the reddest ones, and they are progressively redder than the standard or the D ones for increasing age (the effect is stronger for large metallicities). In Fig. 6 the lower MS for [Fe/H]=-1.0 and [FORMULA]8,12 Gyr is shown; when [FORMULA]6 standard isochrones are unaffected by age (a fact that is well known), while D isochrones are insensitive to the age only for [FORMULA]7 and C isochrones are always affected by the age, at least down to [FORMULA]=7.3 (because of the metal decrease with time due to diffusion).

[FIGURE] Fig. 6. Colour Magnitude Diagram of the low MS of standard (solid line), D (dashed line) and C isochrones (dotted lines) with [FORMULA]8 and 12 Gyr and [Fe/H]=-1.0.

The MS of a GC with an observed RGB metallicity [Fe/H]=-1.0 is given by the D isochrones in Fig. 6, while the MS of local subdwarfs with the same observed metallicity is given by the C isochrones. As it is evident the two MS are generally not coincident .

Another difference with respect to the standard case (and another potential source of systematic errors on the actual MS-fitting distances) is the value of the derivative [FORMULA]/[FORMULA] [Fe/H]. This is the only information needed from theoretical isochrones to be employed in the MS-fitting technique; it is used for shifting the subdwarfs to a mono-metallicity sequence corresponding to the observed cluster [Fe/H]. Since the difference in colour between the standard MS and the C MS depends on the metallicity (see Fig. 5) this will have an impact on [FORMULA][Fe/H] for the subdwarfs.

Are these differences large enough to affect substantially the MS-fitting GC distances? It depends on the subdwarf sample. To explain this point let's consider, as an example, subdwarfs with [Fe/H][FORMULA]1.3 and [FORMULA], which hypothetically have to be employed for deriving the distance to a GC with [Fe/H][FORMULA]0.7. The variation of [FORMULA][Fe/H] due to diffusion causes a shift of the empirical subdwarfs MS at the cluster metallicity by [FORMULA]+0.02 with respect to the standard case. This, by itself, would induce a GC distance modulus larger by [FORMULA]0.1 mag, since the MS slope [FORMULA] is equal to about 5.5. However, one must correct for the vertical [FORMULA] difference between subdwarfs and GC MS, which tends to reduce the derived distance modulus by [FORMULA]0.05-0.08 mag in the age range 8-12 Gyr. The final combined effect is to have distances unchanged or increased at most by 0.05 mag with respect to the standard case. However, in the hypothesis that for determining the MS-fitting distance to a GC with [Fe/H][FORMULA]1.3 one can use only subdwarfs with [Fe/H][FORMULA]0.7, the situation is quite different, since the use of the diffusive C isochrones would cause a decrease of the distance modulus by [FORMULA]0.10-0.13 mag.

In the following we will study the effect of diffusion on the MS-fitting distances obtained using subdwarfs with accurate Hipparcos parallaxes. We have considered, as a test (the results are summarised in Table 2), four clusters included in the analysis by Gratton et al. (1997), namely M92 ([Fe/H][FORMULA]), M5 and NGC288 ([Fe/H][FORMULA]), 47Tuc ([Fe/H][FORMULA]). The subdwarfs [FORMULA], [FORMULA] and [Fe/H] values come from Table 2 of Gratton et al. (1997); the clusters reddenings and metallicities are from the quoted paper, as well as the observational clusters MS lines. For each cluster we have considered only bona fide single stars fainter than V=6 (to avoid evolutionary effects for the standard isochrones, as well as the influence of the mixing-length calibration), with [FORMULA] and in the same metallicity range as in Gratton et al. (1997).


Table 2. MS-fitting distance moduli ([FORMULA]) of selected clusters.

In the case of the standard models by SW98 we recover basically the same distances by Gratton et al (1997), whose results were obtained by using a value for [FORMULA] [Fe/H] derived from different isochrones, and considering subdwarfs also in the range 5.5[FORMULA]6.0. When deriving the MS-fitting distances taking into account diffusion, we have (as outlined in the previous example) corrected the subdwarfs colours by using the [FORMULA] [Fe/H] values derived from the C models, and we have also accounted for the difference in brightness at fixed colour between the subdwarfs MS (C isochrones) and the clusters one (D isochrones). Since there are small evolutionary effects for the D isochrones (representing the GC) even when 6[FORMULA]7, we have taken into account 4 different possibilities. In the first two cases we have assumed for the clusters age [FORMULA]=8 Gyr with subdwarfs ages [FORMULA]=8 and 12 Gyr, and in the second two cases we considered [FORMULA]=12 Gyr and again [FORMULA]=8 and 12 Gyr.

As it is clear from Table 2, there are no appreciable modifications to the distance moduli derived from standard isochrones. The differences with respect to the standard case are small and generally within the small formal error bars associated to the fit (the error bar takes into account only the error on the fit due to the uncertainties on the subdwarfs [FORMULA] and [FORMULA]). This is a quite important point, since it confirms the robustness of the published Hipparcos MS-fitting distances which did not take into account the effect of atomic diffusion on GC and field subdwarfs evolution.

The reason for this occurrence is that - thanks to the Hipparcos results - the sample of lower MS metal poor subdwarfs with accurate parallaxes has substantially increased with respect to the recent past. In performing the MS fitting we have used objects whose metallicity is close to the actual GC metallicity; in this case, as it is evident, the colour correction to be applied to the subdwarfs is small, and even the occurrence of a sizeable change of [FORMULA] [Fe/H] does not modify appreciably the final distance. Moreover, the subdwarfs are all sufficiently faint so that the difference between the GC (D isochrones) and subdwarfs (C isochrones) MS is generally kept at the lowest possible value (this difference generally increases for increasing luminosity).

In conclusion, the effect of diffusion on the two main distance determination methods for GC stars, namely MS-fitting and HB fitting, is practically negligible, since also the HB luminosities are negligibly influenced by diffusion. The final effect on the GC age estimates is therefore just a reduction by about 1 Gyr due to the change of the TO brightness.

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Online publication: March 17, 2000