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

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3. Theory

We have derived recently evolutionary models aimed at describing the mechanical and thermal properties of LMS. These models are based on the most recent physics characteristic of low-mass star interiors, equation of state (Saumon, Chabrier & VanHorn 1995), enhancement factors of the nuclear rates (Chabrier 1997) and updated opacities (Iglesias & Rogers 1996; OPAL), the last generation of non-grey atmophere models (Allard & Hauschildt 1997, AH97) and accurate boundary conditions between the interior and the atmosphere profiles. This latter condition is crucial for an accurate description of LMS evolution, for which any diffusion approximation or grey treatment yield inaccurate results (Baraffe, Chabrier, Allard & Hauschildt 1995; Chabrier, Baraffe and Plez 1996; Chabrier & Baraffe, 1997). A complete description of the physics involved in these stellar models is presented in Chabrier and Baraffe (1997) and we refer the reader to this paper for detailed information.

Below [FORMULA], the stellar interior becomes fully convective: the evolution of LMS below this limit is rather insensitive to the mixing length parameter [FORMULA], and thus the models are not subject to any adjustable parameter. From this point of view, VLMS represent a formidable challenge for stellar evolution theory. Comparison with the observations is straightforward and reflects directly the accuracy of the physics and the treatment of the (self-consistent) boundary conditions involved in the calculations. Any VLMS model including an adjustable parameter or a grey atmosphere approximation in order to match the observations would reflect shortcomings in the theory and thus would lead to unreliable results. From this point of view, it is important to stress that, although agreement with the observation is a necessary condition to assess the validity of a stellar model, it is not a sufficient condition. This latter requires the assessment of the accuracy of the input physics and requires parameter-free, self-consistent calculations.

The first generation of the present models (Baraffe et al. 1995) were based on the "Base" grid of model atmospheres of Allard and Hauschildt (1995; AH95). These models improved significantly the comparison with the observed Pop I and Pop II M-dwarfs sequences (Monet et al. 1992), down to the bottom of the main-sequence, w.r.t. previous models (Baraffe et al. 1995). The AH95 models have been improved recently by including (i) a pressure broadening treatment of the lines, (ii) more complete molecular line lists, and (iii) by extending the Opacity-Sampling technique to the treatment of the molecular line absorption coefficients. This yields the so-called "NextGen" models ( Allard and Hauschildt 1997). The "NextGen" synthetic spectra and colors have been compared with Population-I M-dwarf observations by Jones et al. (1995, 1996), Schweizer et al. (1996), Leggett et al. (1996), Viti et al. (1996), and have been used for the analysis of the brown dwarf Gl229B by Allard et al. (1996). The present work is the first application of the "NextGen" model atmospheres to observations and evolutionary calculations of metal-poor populations.

A first (preliminary) set of the present improved stellar models has been shown to reproduce accurately the observed mass-magnitude (Henry & Mc Carthy 1993) and mass-spectral type (Kirkpatrick & Mc Carthy 1994) relationships for solar-metallicity LMS down to the bottom of the main sequence (Chabrier, Baraffe and Plez 1996; Baraffe & Chabrier 1996). The aim of the present work is to extend the comparison between theory and observation to metal-depleted abundances, characteristic of old disk and halo populations.

For this purpose, we have calculated evolutionary models for masses [FORMULA] down to the hydrogen-burning limit, based on the "NextGen" model atmospheres, for different low metal abundances characteristic of old globular clusters, namely [FORMULA] and -1.0. This low-metal grid represents a first step towards a complete generation of models covering solar metallicities. A major source of absorption in VLMS arises from the presence of TiO and [FORMULA] molecular bands in the optical and the infrared, respectively. Although tremendous progress has been accomplished over the past few years in this type of calculations, some uncertainty remains in the absorption coefficients of these molecules, which affect not only the spectrum, i.e. the colors, but also the profile of the atmosphere, and thus the evolution of late-type M dwarfs (see Allard et al. 1997a for a review of VLM stellar atmospheres modeling). Furthermore, the onset of grain formation, not included in the present models, may also affect the spectroscopic and structural properties of metal-rich VLMS below [FORMULA]  K (Tsuji et al. 1996a, b; Allard et al. 1996). We expect these shortcomings in the models to be of decreasing importance with decreasing metallicity. Strong double-metal bands (TiO, VO ), for example, are noticeably weaker in the spectra of metal-poor subdwarf (see e.g. Leggett 1992; Dahn et al. 1995), whereas they would dominate the spectrum of a solar metallicity object at this temperature. For this reason it is important to first examine the accuracy of the models for low-metal abundances and its evolution with increasing metallicity. The observed MS of the three afore-mentioned HST GCs represent a unique possibility to conduct such a project.

In some cases, comparison is made with the first generation of models (Baraffe et al. 1995), based on the "Base" model atmospheres. This will illustrate the effect of the most recent progress accomplished in the treatment of LMS atmospheres. All the basic models assume a mixing length parameter, both in the interior and in the atmosphere, [FORMULA]. Since the value of the mixing length parameter is likely to depend to some extent on the metallicity, there is no reason for the value of [FORMULA] in globular clusters to be the same as for the Sun. Although the choice of [FORMULA] is inconsequential for fully convective stars, it will start bearing consequences on the models as soon as a radiative core starts to develop in the interior. For metallicities [FORMULA], this occurs for masses [FORMULA] [FORMULA] (see Chabrier & Baraffe 1997 for details). In order to examine the effect of the mixing length parameter for masses above this limit, we have carried out some calculations with [FORMULA], both in the interior and in the atmosphere (see Sect. 3.2 below). All the models assume a solar-mix abundance (Grevesse & Noels 1993). As discussed in the previous section, oxygen-enhancement in metal-poor stars is taken into account by using the eqn.(1) as the correspondence law between solar-mix and [FORMULA] -enhanced abundances. In order to test the accuracy of this procedure, a limited set of calculations with the exact [FORMULA] -enriched mixture have also been performed (Sect. 3.4).

The adopted helium fraction is [FORMULA] for all metallicities. The effect of helium abundance will also be examined below.

All models have been followed from the initial deuterium burning phase to a maximum age of 15 109 yr.

Before comparing the results with observations, we first examine some intrinsic properties of the models. Comparison is also made with other recent LMS models for metal-poor stars, namely D'Antona & Mazzitelli (1996; DM96) and the Teramo group of Dr. Castellani (Alexander et al. 1997). Although the EOS used by the Teramo group is the same as in the present calculations (SCVH), both groups use grey model atmospheres based on a [FORMULA] relationship, and match the interior and the atmosphere structures either at an optical depth [FORMULA] (DM96) or at the onset of convection (Teramo group).

3.1. Mass-effective temperature relationship

Fig. 1 shows the mass-effective temperature relationship for two metallicities, [FORMULA] ([FORMULA] ) (solid line) and [FORMULA] ([FORMULA] )(dash-dot line). We note the two well established changes in the slope at [FORMULA] 4500 K and 3800 K, respectively, for these metallicities. The first one corresponds to the onset of molecular formation in the atmosphere, and the related changes in the opacity (see e.g. Copeland, Jensen & Jorgensen 1970; Kroupa, Tout & Gilmore 1990). The second reflects the overwhelming importance of electron degeneracy in the stellar interior near the hydrogen burning limit (DM96; Chabrier & Baraffe 1997). As expected, the enhanced pressures of lower metallicity mixtures yield larger effective temperatures for a given mass (see Chabrier & Baraffe, 1997).

[FIGURE] Fig. 1. Mass - effective temperature relation for [FORMULA] =-1.5 (solid line) and [FORMULA] =-1.0 (dash-dot line). The full circles denote our calculations done with the Krisha-Swamy grey-like treatment (see text) for [FORMULA] =-1.5 while the dashed line show the results of the Teramo group (Alexander et al., 1997) for the same metallicity. The dotted line shows the results of D'Antona & Mazzitelli (1996; DM96) for [FORMULA] =-1.0.

Also shown in the figure are comparisons with DM96 and the Teramo group, for comparable metallicities. The DM96 models (dotted line) overestimate substantially the effective temperature, by up to [FORMULA] K below 0.5 [FORMULA]. This will yield substantially blue-shifted sequences in the CMDs. This is a direct consequence of using a [FORMULA] prescription (and a different EOS, and adiabatic gradient), which implies a grey atmosphere and radiative equilibrium, two conditions which become invalid under low-mass star characteristic conditions, as demonstrated in previous work (Chabrier, Baraffe & Plez 1996; Chabrier & Baraffe 1997). The Krishna-Swamy (1966,KS) formula used by the Teramo group is known to slightly improve this shortcoming. As shown in the figure, we reproduce the Teramo results (dashed line) when using the KS [FORMULA] relationship (filled circles). Though smaller, the departure of the mass- [FORMULA] relationship in that case is still significant, in particular near the brown dwarf domain, where it becomes too steep and predicts too large hydrogen burning minimum masses (see Chabrier & Baraffe, 1997). A detailed examination of these different prescriptions is given in Chabrier & Baraffe (1997). These comparisons show that, even though some improvement can be reached w.r.t. the basic Eddington approximation, any atmosphere-interior boundary treatment based on a prescribed [FORMULA] relation yields incorrect, substantially overestimated, effective temperatures for [FORMULA] K, i.e. [FORMULA]. Conversely, they underestimate the mass for a given temperature (luminosity) below this limit, i.e. at the bottom of the MS. This illustrates the unreliability of any grey-like treatment to describe accurately the non-grey effects and the presence of convection in optically thin layers of VLM stars. This bears important consequences for the cooling history and thus the evolution in general, and for the mass-luminosity relationship and thus the mass calibration in particular.

3.2. Effect of the mixing length

In this section, we examine the effect of the mixing length parameter [FORMULA] on the stellar models, for a given metallicity. We must distinguish the effect of this parameter in the stellar interior and in the atmosphere. Both effects reflect the importance of convective transport in two completely different regions of the star, namely the envelope and the photosphere, and thus do not necessarily bear the same consequences on the models.

We first examine the effect of the mixing length parameter in the atmosphere, by conducting evolutionary calculations with non-grey model atmospheres for [M/H]=-1.5, calculated with [FORMULA] and 2, while the value is kept unchanged in the interior. We have selected a range of effective temperatures and gravities ([FORMULA] = 4000 - 5800 K, log g = 4.5 - 5) corresponding to masses [FORMULA] [FORMULA]. The atmosphere profiles corresponding to both situations are shown in Fig. 2, where the onset of convection and the location of the optical depth [FORMULA] = 1 are indicated. As seen in the figure, the atmosphere profile is rather insensitive to a variation of [FORMULA] in the optically-thin region, as expected from the rather shallow convection zone in the atmosphere at this metallicity and effective temperature. The main consequence is that the spectrum, and thus the colors and magnitudes at a given [FORMULA], remain almost unaffected (less than 0.04 mag in the afore-mentioned range). Below [FORMULA] K, the effect of the mixing length in the atmosphere is inconsequential (see also Brett 1995). The evolutionary models calculated with both sets of model atmospheres in the selected range of effective temperatures differ by less than 50 K in [FORMULA]. The effect of the mixing length in the atmosphere thus bears no consequence on the evolution.

[FIGURE] Fig. 2. Atmosphere profiles for [FORMULA] =-1.5 and a mixing length [FORMULA] (solid line) and [FORMULA] (dash line), respectively, for selected effective temperatures. The open circles indicate the location of [FORMULA] and the crosses show the onset of convection.

We now examine the effect of convection in the stellar interior, by conducting calculations with [FORMULA] and 2 in the interior, keeping the fiducial value [FORMULA] in the atmosphere. This is illustrated in Fig. 3a by comparing the solid line (and empty circles), which corresponds to [FORMULA] in the interior, with the dashed line (triangles), which corresponds to [FORMULA], for [FORMULA] Gyr. As expected the effect is null or negligible for convection-dominated interiors, i.e. [FORMULA]. Above this limit, a larger mixing parameter, i.e. a more efficient convection, yields slightly hotter (bluer) models (D'Antona & Mazzitelli 1994; Chabrier & Baraffe, 1997), about 200-300 K for [FORMULA] K, i.e. [FORMULA], for the metallicity of interest, [FORMULA].

[FIGURE] Fig. 3a and b. a Effect of the age and of the mixing length in the stellar interior for [M/H]=-1.5. The models display the [FORMULA] Gyr isochrone when using a mixing length [FORMULA] (solid line; empty circles) and [FORMULA] (dashed line; triangles). The dotted line ([FORMULA] ) shows the 15 Gyr isochrone calculated with [FORMULA], whereas the dash-dotted curve (X ) corresponds to t=15 Gyrs and [FORMULA]. The masses indicated in the figure correspond to the 10 Gyrs isochrones (open circles and triangles). The upper mass corresponding to the 15 Gyrs isochrones is 0.75 [FORMULA]. b Effect of the helium fraction Y, for [FORMULA], with Y = 0.25 (solid curve, open circles) and Y = 0.23 (+).

In summary, the effect of the variation of the mixing length parameter on the stellar models remains weak for the lower MS and is essentially affected by the value in the stellar interior. A variation of [FORMULA] to 2 yields variations [FORMULA] on the effective temperature in the upper MS, for masses above [FORMULA], while the luminosity remains essentially unaffected. We will examine in section 4.1 whether comparison with the observed MS of GCs can help calibrating this value.

3.3. Effect of the age

The effect of the age in a color-magnitude diagram is shown on Fig. 3a, for [FORMULA] for ages [FORMULA] and 15 Gyr. The time required to reach the zero-age main sequence is much smaller than the age of GCs over the entire stellar mass range so that all hydrogen burning objects do lie on the MS and the bottom of the GC MS is unaffected by age variation. On the other hand, metal-poor stars are significantly hotter and more luminous than their more metal-rich counterparts. Therefore, for a given mass, they burn more rapidly hydrogen in their core, and thus evolve more quickly off the MS. The more massive (i.e. hotter) the star, the larger the effect. This defines the turn-off point, i.e. the top of the MS. For fixed metallicity, the turn-off point will thus be reached for lower masses, i.e. fainter absolute magnitudes, as the age increases. This is illustrated in Fig. 3a where we compare the 10 Gyr and 15 Gyr isochrones calculated for different mixing length parameters, for the same metallicity. While the position in the HR diagram of masses below [FORMULA] remains unchanged, larger masses become substantially bluer and more luminous with age as they transform more central hydrogen into helium. The resulting increase of molecular weight yields further contraction and heating of the central layers, and thus an increase of the radiative flux ([FORMULA] ) and the luminosity. Eventually they evolve off the MS for a (turn-off)-mass [FORMULA] for [FORMULA] and -2 at t=10 Gyr and [FORMULA] at 15 Gyr. As shown in the figure, the upper MS is sensitive to age and mixing length variations. The calibration of the mixing length parameter on the turn-off point is then altered by age uncertainties. The precise determination of the turn-off point thus requires more detailed calculations, which are out the scope of the present study.

3.4. Effect of helium and [FORMULA] -elements

As mentioned above, the fiducial calculations have been conducted with a helium abundance [FORMULA]. Fig. 3b compares these results (solid line; circles), for [FORMULA], with calculations done with [FORMULA] ([FORMULA] ). The corresponding variations of the temperature (color) and the luminosity (magnitude) are negligible over the entire mass range, except near the turn-off point.

In order to examine the accuracy of the Ryan & Norris prescription (eqn.(1)), we have conducted calculations using model atmospheres computed with an [FORMULA] -element abundance enrichment of [FORMULA] for a [FORMULA] ([FORMULA]) mixture. Since the spectra of VLMS near the bottom of the MS are governed by [FORMULA] -elements, we expect this mixture to yield equivalent results to those obtained with the scaled-solar abundance mixture of [M/H]= -1.0. This test will also determine whether neglecting a relative under-abundance of Fe and other non- [FORMULA] elements in the scaled-solar abundances does affect or not the shape of the MS. We have computed model atmospheres with the afore-mentioned modified abundances for several temperatures and gravities in the range [FORMULA] K and [FORMULA]. We find that the atmosphere density and temperature profiles and the colors are essentially undistinguishable from the ones calculated with [FORMULA], [FORMULA], i.e. a scaled solar-mix, throughout the entire atmosphere.

We have also examined the effect of [FORMULA] -enriched abundances in the interior opacities (OPAL), thanks to appropriate opacities kindly provided by F. Rogers for [FORMULA] =-1.0 and [FORMULA]. Here too we find that the effect is negligible and does not modify noticeably the isochrones. This assesses the validity of the Ryan & Norris procedure to take into account the [FORMULA] -element enrichment for metal-poor stars, and demonstrates the consistency of our prescription, i.e. comparing solar-mix models with [FORMULA] to observations with [FORMULA]. Conversely, this demonstrates the inconsistency of comparing observations and solar-mix models for these stars with [FORMULA], as done sometimes in the literature.

The theoretical characteristics of the present models, effective temperature, luminosity, gravity, bolometric magnitude and magnitudes in VRIJHK for several metallicities and an age t = 10 Gyrs are given in Tables 2-5.

[TABLE]

Table 2. Physical properties and absolute magnitudes of low-mass stars for [FORMULA] and [FORMULA] = 10 Gyrs. The lowest mass corresponds to the hydrogen-burning limit. The mass m is in [FORMULA], [FORMULA] in K and the luminosity L in [FORMULA]. The VRI magnitudes are in the Johnson-Cousins system (Bessell 1990) and the JH K magnitudes in the CIT system (Leggett 1992). Note that the bolometric magnitude corresponds to [FORMULA] = 4.64.

[TABLE]

Table 3. Same as in Table 2 for [FORMULA]

[TABLE]

Table 4. Same as in Table 2 for [FORMULA]

[TABLE]

Table 5. Same as in Table 2 for [FORMULA]

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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