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Astron. Astrophys. 327, 1054-1069 (1997)

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4. Results and discussion

4.1. Comparison with globular cluster main sequences

Figs. 4-6 show the main sequence CMDs of the three GCs mentioned in Sect. 2. As mentioned previously, the CMDs are shown in the [FORMULA] ([FORMULA]), [FORMULA] ([FORMULA]), and [FORMULA] filters in the WFPC2 Flight system to avoid possible errors due to uncertain photometric conversion into the standard Johnson-Cousins system. In all cases, the main sequence is well defined down to [FORMULA], [FORMULA]. Below this limit, the observed MS dissolves into the field and it becomes quite difficult to distinguish the cluster-MS from field stars.

The theoretical MS for the appropriate metallicity, as described in Sect. 2, are superimposed to the observations in each figure. For comparison in the Flight system, observed magnitudes have been dereddened with the corrections derived from the synthetic spectra (cf. Table 1). The first striking result is the excellent agreement between theory and observation for the three clusters, spanning a fairly large metallicity range from strongly metal-depleted abundances ([FORMULA]) to a tenth of solar metallicity ([FORMULA]). In particular, the changes of the slope in the observed MS are perfectly reproduced by the models, for the correct magnitude, color and metallicity. Since these changes stem from the very physical properties of the stellar interior and atmosphere, as discussed in the previous section, the present qualitative and quantitative agreement assesses the accuracy of the physical inputs in the present theory. The masses corresponding to these changes are indicated on the curves and their effective temperatures are given in Tables 2-5, for the various metallicities.

Let us now consider each cluster in turn in order of increasing metallicity.

[FORMULA] [FORMULA] (Fig. 4a,b): This is the most metal-depleted HST cluster presently observed, with [FORMULA] (Djorgovski 1993), i.e. [FORMULA] within the prescription adopted in the present paper. The MS of [FORMULA], as observed by DeMarchi & Paresce (1995), is shown in Fig. 4a, with the distance modulus adopted by these authors. Comparison is made with our models with [FORMULA] (solid line) and -2.0 (dashed line). Although both models merge in the upper MS, metallicity effect is clearly reflected in the lower MS where the peak of the spectral distribution falls in the V bandpass, i.e. below [FORMULA], [FORMULA], and [FORMULA] K. As seen in the figure, the agreement with the observations over the whole MS is excellent for [FORMULA] (the observationally-determined metallicity) although the models appear to be slightly too red by [FORMULA] mag or overluminous by [FORMULA] mag, in the upper MS. Even though this discrepancy is within the observational error bars in [FORMULA] and [FORMULA], which range from [FORMULA] mag for the upper MS to [FORMULA] mag for the lower MS (DeMarchi, private communication), we will examine the possibilities for this disagreement to be real. A slightly lower metallicity would leave the upper MS almost unchanged, as seen from the two theoretical sequences displayed in the figure. Although the offset in the upper MS might be compensated by a slightly larger mixing length (see Sect. 3.2), the fact that it is quite constant along the entire sequence whereas the mixing length only affects the upper MS (see Sect. 3.2) rather suggests either an overestimated reddening [FORMULA] (by [FORMULA] 0.05 mag) or an underestimated distance modulus. Indeed, assuming [FORMULA], i.e. [FORMULA] mag w.r.t. the value quoted by DeMarchi & Paresce, or a reddening fainter by [FORMULA] mag would bring theory and observation into perfect agreement. A 0.2 mag error on the distance modulus corresponds to [FORMULA] kpc, i.e. [FORMULA] 10% of the canonical value used presently (Djorgovski 1993). Such an error cannot be excluded for this remote cluster ([FORMULA] kpc).

[FIGURE] Fig. 4. a CMD for [FORMULA]. The data in the 606 and 814 filters are from DeMarchi & Paresce (1995). The data are dereddened with the corrections calculated in the present study, as given in Table 1. Solid line (o): [FORMULA] =-1.5; dashed line (+): [FORMULA] =-2.0. b Same as in a in the Johnson-Cousins system with the fit of Santiago et al. (1996) for the cluster (thick dash-dot).

Note that we exclude an artificial offset in the calibration of the model magnitudes as source of discrepancy, since we use the magnitude zero-points kindly provided by G. DeMarchi. However, a calibration problem of the data in the [FORMULA] filter cannot be excluded, since a disagreement appears as well in the comparison of NGC6397 observed by the same group in the same filters. The disagreement however vanishes for the same cluster in [FORMULA] and [FORMULA] and for wCen observed by Elson et al. (1995) in [FORMULA] and [FORMULA] (see below). The latter agreement seems to reject an intrinsic problem of the atmosphere models in the spectral region covered by the [FORMULA] passband.

The data of DeMarchi & Paresce have been transformed into the standard [FORMULA] Johnson-Cousins system by Santiago et al. (1996) who used the dereddening correction quoted by the former authors (see Table 1) and the synthetic transformations of Holtzman et al. (1995). Fig. 4b compares the [FORMULA] MS fit derived by Santiago et al. to the present models in the same Johnson-Cousins color system, where the filter transmissions of Bessell (1990) have been applied to the models. The agreement is similar to that obtained in the previous Flight system, well within the error bars of the photometric [FORMULA] -to- [FORMULA] transformation, with the same slight offset in color. This comparison assesses the validity of the photometric synthetic transformations of Holtzman et al. (1995) in the present filters.

As shown in the figure and in Table 2, the faintest observed stars on the MS, [FORMULA] or [FORMULA], correspond to a mass [FORMULA], still well above the hydrogen burning limit. This latter, [FORMULA] for [FORMULA], is expected to correspond to [FORMULA], [FORMULA] or [FORMULA] and [FORMULA] (Table 2). At the bright end, the limit of the observations corresponds to the turn-off point [FORMULA] (DeMarchi & Paresce 1995) which corresponds to [FORMULA] depending on the age. Note that the Johnson-Cousins fit given by Santiago et al. (1996) starts at [FORMULA], which corresponds to a mass [FORMULA] and does not include the upper MS.

[FORMULA] [FORMULA] (Fig. 5a,b,c): This cluster has been observed separately by Paresce et al. (1995) and Cool et al. (1996) in different photometric filters, thus allowing comparison in the three WFPC2 filters mentioned previously. Fig. 5a shows the comparison of the observations and the models in [FORMULA] vs [FORMULA] for [FORMULA] and -2. The best agreement is obtained for the observed metallicity [FORMULA], although admitedly the theoretical sequence lies near the blue edge of the low MS ([FORMULA]). We note however that the error bars for this cluster range from [FORMULA] for the brightest stars to [FORMULA] 0.20 mag for the lower MS (cf. Paresce et al. 1995). The agreement displayed in Fig. 5a is therefore well within the error bars. Much better agreement is found in [FORMULA] vs [FORMULA], as shown on Fig. 5b, although an offset now appears for the intermediate part of the sequence ([FORMULA]), [FORMULA] mag in color and [FORMULA] mag in magnitude. As shown on Fig. 3a, this part of the MS is insensitive to the mixing length, and the disagreement would thus not be solved by a larger mixing length. In the same vein, models with [FORMULA] -enriched abundances would yield similar results, as discussed in Sect. 3.4. Models with a substantially lower metallicity would fail reproducing the shape of the MS both in [FORMULA] (cf. Fig. 5b) and [FORMULA] (see Fig. 5a).

[FIGURE] Fig. 5a-c. a CMD for [FORMULA] in the 606 and 814 filters. The data are from DeMarchi & Paresce (1995). The data are dereddened with the corrections calculated in Table 1. Solid line (o): [FORMULA] =-1.5; dashed line (+): [FORMULA] =-2.0. The distance modulus is the one quoted by Cool et al. (1995), i.e. [FORMULA] =11.7. b CMD for [FORMULA] in the 555 and 814 filters. The data are from Cool et al. (1995). The data are dereddened with the extinctions calculated in Table 1. Dot-dashed line ([FORMULA]): stellar models based on the "Base" model atmospheres (Baraffe et al. 1995) for [M/H]=-1.5; solid line (o): present stellar models based on the "NextGen" model atmospheres for [M/H]=-1.5; dashed line (+): [FORMULA] =-2.0 with the NextGen model atmospheres. The distance modulus is the same as in Fig. 5a, i.e. [FORMULA] =11.7. c Same as b, with the distance modulus quoted by DeMarchi & Paresce (1995), i.e. [FORMULA] =11.9 and the same extinction as in b. Solid line (o): [FORMULA] =-1.5; dashed line (+): [FORMULA] =-2.0. A comparison with the models of the Teramo group for [M/H]=-1.5 is shown (dash-dot)

Both Figs. 5a and 5b are displayed with the distance modulus quoted by Cool et al. (1996), i.e. [FORMULA]. The choice of the value quoted by Paresce et al. (1995), [FORMULA], brings the tracks for [FORMULA] exactly on the observed sequence for the [FORMULA] vs [FORMULA] CMD, as illustrated on Fig. 5c. Such an undetermination in the distance modulus of [FORMULA] corresponds to a difference of 200 pc, i.e. about [FORMULA]. In the [FORMULA] and [FORMULA] filters, the models are slightly too blue, with an offset in color [FORMULA] mag, which can well stem from the same calibration problem in [FORMULA] mentioned previously for the cluster M15. The predicted sequence with the metallicity [M/H]=-1.5 remains however well within the error bars of the observed MS.

It is interesting to analyse the agreement obtained by the Teramo group models for this cluster. We first note that their models, while based on a solar-mix, correspond to [FORMULA] rather than [FORMULA]. As discussed in the previous section, this leads to inconsistent comparisons, and is reflected by the fact that: i) they fit the MS with a rather low value of the metallicity for this cluster ([FORMULA]), and ii) they adopt a reddening correction [FORMULA] which differs significantly from the value prescribed by Cool et al. (1996) and by the present calculations (see Table 1). Using the correct extinction would redshift significantly their models for [M/H]=-1.5 w.r.t. the observations, as shown on Fig. 5c (dash-dotted line).

Note that for this cluster, the observations are still above the hydrogen-burning limit. The HBMM [FORMULA] corresponds to [FORMULA], [FORMULA] for [FORMULA], whereas the bottom of the observed MS, [FORMULA], [FORMULA], corresponds to [FORMULA]. The observations of Cool et al. (1996) extend to brighter magnitudes and reach the turn-off point, [FORMULA], [FORMULA].

A limited set of the first generation of the present models at [FORMULA], based on the "Base" model atmospheres (Baraffe et al. 1995), is also shown on Fig. 5b for [FORMULA] (full circles, dash-dot line). These models have a slightly different trend w.r.t. the present ones. This stems from a general overestimation of the molecular blanketing, the main source of absorption in the coolest VLMS, because of the straight-mean approximation in the "Base" models, and thus an underestimation of the flux in the V-band ( Chabrier et al. 1996; Allard & Hauschildt 1997). The better agreement with the present models clearly illustrates the recent improvements in the treatment of the molecular opacities, especially for TiO (AH97), which strongly affect the atmosphere profile, and thus the evolution. However, the agreement between the previous models and observations was already quite satisfactory ([FORMULA] mag) and for the first time reproduced accurately the bottom of the observed MS for metal-poor LMS.

[FORMULA] [FORMULA] (Fig. 6): this cluster has been observed by Elson et al. (1995) in the F606W and F814W filters. The observed metallicity usually quoted for this cluster is [FORMULA], i.e. [FORMULA] (cf Elson et al. 1995 and references therein), although a value [FORMULA], i.e. [FORMULA] has been suggested recently (Norris et al., 1996). The data presented in Fig. 6 have been recalibrated (Elson, private communication), since the calibrations of Holtzmann et al. (1995) used in Elson et al. (1995) were preliminary at this time. The zero points for both HST filters are updated (cf. Holtzmann et al. 1995), as well as the transformation into the Johnson-Cousins system. Here again, the match between theory and observation is almost perfect, as shown in Fig. 6, in both the HST instrumental and the Jonhson-Cousins systems. Note that both systems predict the same masses for the upper MS ([FORMULA]) and the lower MS ([FORMULA]), well within the error bars due to the photometric conversion from [FORMULA] into [FORMULA] (Holtzman et al. 1995). Once again, this assesses the accuracy of the present photometric conversions of HST magnitudes into Johnson-Cousins magnitudes.

[FIGURE] Fig. 6. a CMD for [FORMULA]. The data in 606 and 814 filters are from Elson et al. (1995). The models correspond to [M/H]=-1.3 (solid line, open circle) and [M/H]=-1.0 (dashed line, [FORMULA]) with [FORMULA]. Full circles correspond to [M/H]=-1.0 and [FORMULA] b CMD for [FORMULA] in the Johnson-Cousins system. The data from Elson et al. (1995) have been recalibrated with updated transformations (Elson, private communication). Same models as in a.

We used the same distance modulus as Elson et al. (1995), i.e. [FORMULA]. Adopting the slightly higher value 13.92 quoted by Santiago et al. (1996) will shift the blue edge of the observed MS on the theoretical isochrone with [M/H]=-1.3, yielding a less good agreement. Our reddening corrections (cf. Table 1) are in excellent agreement with the values quoted by Elson et al. (1995). Calculations with a mixing length [FORMULA] for the [M/H]=-1 isochrone are also shown in Fig. 6. Note that the observations remain compatible with a value [FORMULA]. Since the same remark applies to the lower-metallicity clusters analysed previously, it would be hazardous to try to derive robust conclusions about a possible dependence of the mixing length on metallicity. The observed width of the lower MS does not allow a precise determination of the metallicity of the cluster from the theoretical isochrones. The shape of the MS is well reproduced with both [M/H]=-1.3 and -1. This result is consistent with the spread in metallicity determined by Norris et al. (1996).

The properties of the models for the afore-mentioned different metallicities, in particular the mass-color-magnitude relationships, are displayed in Tables 2-5. The lowest mass in the table corresponds to the hydrogen-burning minimum mass (HBMM) for each metallicity. For an age [FORMULA] 10 Gyr and metallicities [M/H] [FORMULA], the magnitude of the most massive brown dwarf is [FORMULA] 2 mag in I and K, [FORMULA] 3 mag in V and [FORMULA] mag in J and H fainter than the one corresponding to the HBMM. This sets the scale of the detection limit for the search for brown dwarfs in globular clusters with future space-based observations.

As already mentioned and clearly seen from the figures and from Tables 2-5, the more metal-depleted clusters have bluer main sequence colors, a consequence of the increasing effective temperature at a given mass with decreasing metallicity, since the same optical depth corresponds to a denser layer in an increasingly transparent atmosphere (see e.g. AH95; Chabrier & Baraffe 1997). Metallicity effects are most apparent in the intermediate MS, i.e. [FORMULA] and [FORMULA] K, where the peak of the spectral distribution falls near the V bandpass. This stems essentially from the increasing TiO-opacity, which absorbs mainly in the optical and reddens the [FORMULA] color. Thus, for a given [FORMULA] , a metal poor object will appear bluer than the more metal-rich counterpart. This effect is strengthened at fixed mass by the fact that the lower the metallicity, the hotter (bluer) the star.

4.2. Color-magnitude diagram in the near infrared

The observations of GCs in near IR colors will soon be possible with the next generation of HST observations, i.e. the NICMOS camera, and in a more remote future, with the european Very-Large-Telescope (VLT). The NICMOS filters include Wide (W), Medium (M) and Narrow (N) bandpasses from 1.1 to 2.4 [FORMULA] m. In Figs. 7a-b we show our models at t=10 Gyrs for [M/H]=-2, -1.5 and -1.0 in the NICMOS wide filters F110W, F160W and F187W. For comparison, the same isochrones are displayed in the J ([FORMULA]) and H ([FORMULA]) magnitudes (cf. Fig. 7a, dotted curves), defined in the CIT system (Leggett 1992). The masses listed in Table 2-5 are indicated by the signs ([FORMULA] and circles) on the curves (except 0.13 [FORMULA] excluded for sake of clarity). We note the ongoing competition, in these infrared colors, between the reddening due to the decreasing temperature and increasing metallic molecular absorption in the optical and the increasing collision-induced absorption of H2 in the infrared (cf. Saumon et al., 1994; Allard & Hauschildt, 1995) which shifts back the flux to shorter wavelengths. This leads to quasi-constant color sequences from [FORMULA] to [FORMULA] [FORMULA], corresponding to [FORMULA] K, for which H2 becomes stable in the atmosphere, to [FORMULA] K, as predicted also for zero-metallicity (Saumon et al., 1994). Below this limit, molecular hydrogen becomes dominant, the density keeps increasing ( cf. Chabrier & Baraffe, 1997) and H2 CIA-absorption becomes the dominant effect ([FORMULA] in first order, see Guillot et al., 1994). This causes the blue loop at the very bottom of the MS in IR colors, as seen in Figs. 7, whereas the optical colors redden monotonically with decreasing mass, as shown in the tables. The blue loop becomes more dramatic with decreasing metallicities, since the lower the metallicity the denser the atmosphere. This general trend is very similar in all NICMOS filters covering the above-mentioned wavelength range and is clearly a photometric signature of the stellar to sub-stellar transition, whose physical source is the large increase of the density in this region and the ongoing H2 molecular recombination and collision-induced absorption. The limit magnitudes required to reach the very bottom of the MS are [FORMULA] for [M/H]=-2, [FORMULA] for [M/H]=-1.5 and [FORMULA] for [M/H]=-1, and are essentially the same for all NICMOS filters from F110 to F240. The more massive brown dwarfs will be about 2 magnitudes fainter.

[FIGURE] Fig. 7. a Isochrones at t=10 Gyrs in the NICMOS filters (solid curves) F110W and F160W for three different metallicity [M/H]=-2, -1.5 and -1 (from left to right). The dotted curves correspond to the same isochrones in the CIT system [FORMULA] -(J-H). The signs on the curves correspond to the masses (except 0.13 [FORMULA]) tabulated in Tables 2-5. b Same as in a in the NICMOS filters F110W and F187W.

4.3. Comparison with the halo subdwarf sequence

Fig. 8 displays different observations of halo field stars in the standard Johnson-Cousins system. The filled circles are the subdwarf sequence of Monet et al. (1992), the crosses the more complete subdwarf sequence of Dahn et al. (1995), and the triangles correspond to a sub-sample of Leggett's (1992) halo stars. The halo classification was determined photometrically (Leggett 1992) and kinematically: the stars in the three afore-mentioned samples have tangential velocities [FORMULA] km.s-1 (Monet et al.), [FORMULA] 160 km.s-1 (Dahn et al.) and [FORMULA] 180 km.s-1 (present sub-sample of Leggett). All these observations appear to be fairly consistent, the Monet et al. sample representing the most extreme halo fraction of the Dahn et al. sample, the Leggett's sample containing only a few genuine halo stars. We stress that the linear fit proposed by Leggett's (1992) is a rather poor representation of the distribution of the true halo objects and is strongly misleading. We also emphasize that linear fits are not correct to fit LMS sequences in the HR diagram, since they do not reproduce the wavy behaviour of the sequences, wich reflects intrinsic physical properties of these stars, as discussed in Sect. 3.1. This non linearity has already been stressed for the mass-luminosity (D'Antona & Mazzitelli, 1994; Chabrier et al., 1996) and the mass-spectral class relations (Baraffe & Chabrier, 1996). We first note the important spread in color, over 1 mag, which reflects the large spread in metallicity in the sample.

[FIGURE] Fig. 8. Subdwarf halo field stars from the data of Monet et al. (1992) (full circles), Dahn et al. (1995) ([FORMULA]) and Leggett (1992) (triangles), in the standard Jonhson-Cousins system. All the stars shown here from the Leggett's sample have tangential velocities [FORMULA] km.s-1. Solid lines: present models for [FORMULA] =-2,-1.5, -1.3 and -1 (from left to right) for t=10 Gyr. The empty circles correspond to the masses indicated in the figure.

The solid lines indicate the present LMS sequences for [FORMULA] and -1.0 from left to right. The Monet et al. sample is consistent with an average metallicity [FORMULA] to -1.5. It is more hazardous to try to infer an average value for Leggett's sample, given the limited number of objects and the large dispersion. The Dahn at al. sample clearly includes the two previous ones, with objects ranging from [FORMULA] to [FORMULA]. This reflects the difficulty to determine the precise origin of an object from its kinematic properties only.

As seen in the figure, these field stars are fairly consistent with the theoretical sequences determined for the clusters, for similar metallicities. This is particularly obvious for the sequences of [FORMULA] ([M/H]=-1.5) and [FORMULA] ([M/H]=-1.3), which match perfectly the Monet et al. average sequence. This strongly suggests, contrarily to what has been suggested by Santiago et al. (1996), that there is no significant difference in the structure and the evolution of globular star clusters and field halo stars. The discrepancy between the CMD of Leggett (1992) and those from the HST led Santiago et al. (1996) to invoke the possibility of a calibration problem of the HST data. However, the agreement that we find between our theoretical models and the globular cluster CMDs in both the HST data and the Johnson-Cousins system (e.g for [FORMULA] and wCen), excludes this hypothesis. It rather stems from the large metallicity dispersion in Leggett's sample and from the misleading fit of this sample.

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Online publication: April 6, 1998