## 3. TheoryWe 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
, the stellar interior becomes fully convective:
the evolution of LMS below this limit is rather insensitive to the
mixing length parameter
, and thus the models are not subject to
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 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
down to the hydrogen-burning limit, based on the
"NextGen" model atmospheres, for different low metal abundances
characteristic of old globular clusters, namely
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 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, . 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 in globular clusters to be the same as for the Sun. Although the choice of 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 , this occurs for masses (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 , 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 -enhanced abundances. In order to test the accuracy of this procedure, a limited set of calculations with the exact -enriched mixture have also been performed (Sect. 3.4). The adopted helium fraction is 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 10 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 relationship, and match the interior and the atmosphere structures either at an optical depth (DM96) or at the onset of convection (Teramo group). ## 3.1. Mass-effective temperature relationshipFig. 1 shows the mass-effective temperature relationship for two metallicities, ( ) (solid line) and ( )(dash-dot line). We note the two well established changes in the slope at 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).
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
K below 0.5
. This will yield substantially blue-shifted
sequences in the CMDs. This is a direct consequence of using a
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
relationship (filled circles). Though smaller,
the departure of the mass-
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, ## 3.2. Effect of the mixing lengthIn this section, we examine the effect of the mixing length
parameter
on the stellar models, for a given metallicity.
We must distinguish the effect of this parameter in the stellar
We first examine the effect of the mixing length parameter in the
We now examine the effect of convection in the stellar
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 to 2 yields variations on the effective temperature in the upper MS, for masses above , 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 ageThe effect of the age in a color-magnitude diagram is shown on Fig.
3a, for
for ages
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 ## 3.4. Effect of helium and -elementsAs mentioned above, the fiducial calculations have been conducted with a helium abundance . Fig. 3b compares these results (solid line; circles), for , with calculations done with ( ). 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
-element abundance enrichment of
for a
() mixture. Since the spectra of VLMS near the
bottom of the MS are governed by
-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 We have also examined the effect of
-enriched abundances in the The theoretical characteristics of the present models, effective
temperature, luminosity, gravity, bolometric magnitude and magnitudes
in
© European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |