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Astron. Astrophys. 327, 1039-1053 (1997)

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4. Mass-dependence of the photospheric quantities

4.1. Mass-luminosity relationship

Fig. 12 shows the mass-luminosity (ML) relationship for VLMS for three metallicities, [FORMULA], [FORMULA] and [FORMULA], for [FORMULA] 10 Gyrs. The zero-metal 4case sets the upper limit for the luminosity for a given mass.

[FIGURE] Fig. 12. Mass-luminosity relationship for [M/H]=0, [FORMULA] -1.5 and the zero-metal case Z=0. The previous results of Baraffe et al. (1995) based on the Base atmosphere models are also shown ( dashed curve for [M/H]=0 and dash-dotted curve for [M/H]=-1.5).

We first note the well-known wavy behaviour of the ML relation (see e.g. DM94). The change of slope below [FORMULA] is due to the formation of H2 molecules in the atmosphere (Auman 1969; Kroupa, Tout & Gilmore, 1990), which occurs at higher [FORMULA] for decreasing metallicity, because of the denser atmosphere (see SCVH). The steepening of the ML relation near the lower-mass end reflects the onset of ongoing degeneracy in the stellar interior, as demonstrated in Sect.  3.1.1. The previous stellar models of BCAH95, based on the Base AH95 atmosphere models, are shown for comparison. We note that for solar-metallicity, the substantial improvement in the most recent atmosphere models translate into significant differences ([FORMULA] in L and [FORMULA] K in [FORMULA]), the Straight-Mean approximation used in the Base models leading to an overall overestimated opacity. This difference between the models vanishes for lower metallicities (see Allard et al., 1997b). Note that the present models have been shown to reproduce accurately the mass- [FORMULA] and mass- [FORMULA] relationship determined observationaly by Henry & Mc Carthy (1993) down to the bottom of the MS (Chabrier et al., 1996; Allard et al., 1997a). The ML relations for different metallicities are given in Tables 2-7. The comparison with other models/approximations is devoted to the next subsection.

4.2. Mass-effective temperature relationship

Figs. 13 display the mass-effective temperature relationships for sub-solar (Fig. 13a) and solar (Fig. 13b) metallicities. A shown in Fig. 13a, our zero-metal models reproduce correctly the models of Saumon et al. (1994)(filled circles). Figs. 13 also display the [FORMULA] relation obtained with the Krishna-Swamy [FORMULA] relationship, with (dotted line) and without (dot-dashed line) convection in the optically thin region (cf. Sect.  2.5). The figures clearly show the overestimated effective temperature obtained by grey models using a diffusion approximation (Burrows et al. 1989, ([FORMULA]); DM94 (squares); DM96 (triangles)), as demonstrated in Sect.  2.5.

[FIGURE] Fig. 13 a Mass-effective temperature relationship for low-metallicity. Solid lines: present models for [FORMULA] and [FORMULA], from top to bottom. Dashed line: non-grey models based on the Base atmosphere models for [FORMULA] (BCAH95). Dotted line and dot-dashed line: models for [M/H]=-1.5 based on the Krishna-Swamy [FORMULA] relationship with (dot) and without (dash-dot) convection in the optically thin region. Crosses: the [M/H]=-1.5 models of Alexander et al. (1996). Triangles: the [M/H]=-1.0 models of D'Antona and Mazzitelli (1996). Full circles: the Z=0 models of Saumon et al. (1994). b Same as a for solar metallicity. Solid and dashed lines as in a. Dash-dotted line: Eddington approximation. Dotted line (and crosses): Krishna-Swamy relationship. Comparison with previous works: Burrows et al. (1989), models G ([FORMULA]); D'Antona & Mazzitelli (1994), models with the Alexander opacities and the MLT (full squares); Dorman et al. (1989), models with the FGV EOS (full circles).

As already mentioned, the Krishna-Swamy relationship with convection arbitrarily suppressed at [FORMULA] (Alexander et al. 1996, (X)) leads to less severe discrepancy in the region where convection does penetrate into the optically thin layers (2500 K [FORMULA] 5000 K). This paradoxal and inconsistent situation clearly illustrates the dubious reliability of such a treatment, and reflects the unreliable representation of the effects of atmospheric convection within a grey-approximation. This is clearly illustrated by the rather unphysical atmospheric profile obtained within this approximation, as shown in Fig. 5b. For solar metallicity, however, both KS grey approximations yield a similarly good match to the innermost part of the atmosphere profile (see Sect.  2.5). It is the reason why the KS treatment with convection in the optically thin region, as used in Dorman et al. (1989) (filled circles), yields a reasonable agreement at solar metallicity, whereas it yields severe discrepancies for lower metallicities. This reflects the significant overestimation of the convective flux as density and pressure increase with decreasing metallicity. Models based on the Eddington approximation predict even higher [FORMULA] at a given mass (see Fig. 5).

The unreliability of any [FORMULA] relationship for VLMS becomes even more severe near the bottom of the MS ([FORMULA]), as shown on the figures. They yield too steep [FORMULA] relationships in the stellar-to-substellar transition region and thus too large HBMMs, by [FORMULA], as discussed in Sect.  3.1.2. The difference between grey and non-grey calculations vanishes for [FORMULA] K, i.e. [FORMULA] for metal-depleted abundances. For a 0.8 [FORMULA] star with [FORMULA], the difference between models based on non-grey AH97 atmospheres and on grey models calculated with Alexander and Fergusson (1994) Rosseland opacities amounts to [FORMULA] 1-2% in [FORMULA] and less than 1% in L.

The different mass- [FORMULA] relations are given in Tables 2-7. Differences between effective temperatures as a function of the metallicity, for a given mass, [FORMULA], decrease with mass. The 0.5 [FORMULA] star with [M/H]=-1.5 is [FORMULA] 800 K hotter than its solar metallicity counterpart, whereas the difference reduces to [FORMULA] for the 0.09 [FORMULA]. This stems from the decreasing sensitivity of the atmosphere structure to metal abundance with decreasing [FORMULA] (see e.g. Allard, 1990; Brett, 1995; AH97).

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998