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

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2. Input physics

The Lyon evolutionary code has been originally developped at the Göttingen Observatory (Langer et al. 1989; Baraffe and El Eid, 1991 and references therein) and is based on one-dimensional implicit equations of stellar structure, solved with the Henyey method (Kippenhahn and Weigert, 1990). Convection is described by the mixing-length theory. Throughout the present paper, we use a mixing length equal to the pressure scale height, lmix = Hp as a reference calculation. As discussed in Sect.  3.2, the choice of this parameter is inconsequential for the evolution of objects below [FORMULA]. For larger masses, the dependence of the results on the mixing length is examined in BCAH97, along comparison with observations. The updated Livermore opacities (OPAL, Iglesias & Rogers 1996) are used for the inner structure ( T [FORMULA] 10 000K). The effect of the improved OPAL opacities compared to the previous generation (Rogers & Iglesias, 1992) on the evolution of VLM stars (BCAH95; Chabrier, Baraffe & Plez 1996) is found to be negligible, affecting the effective temperature by less than 1% and the luminosity by less than 3% for a given mass. For lower temperatures, we use the Alexander and Fergusson (1994) opacities. The helium fraction in the calculations is Y=0.275 for solar-like metallicities and Y=0.25 for metal-depleted abundances.

An adequate theory for stellar evolution requires i) an accurate EOS, ii) a correct treatment of the nuclear reaction rates, iii) accurate atmosphere models and iv) a correct treatment of the boundary conditions between the interior and the atmophere profiles along evolution. Each of these inputs is discussed in the following sub-sections.

2.1. The equation of state

Interior profiles of VLMS range from [FORMULA] K and [FORMULA] g cm-3 at the base of the photosphere (defined as [FORMULA]) to [FORMULA] K and [FORMULA] g cm-3 at the center for a 0.6 [FORMULA] star, and from [FORMULA] K and [FORMULA] g cm-3 to [FORMULA] K and [FORMULA] g cm-3 for a 0.1 [FORMULA], for solar metallicity. Within this temperature/density range, molecular hydrogen and atomic helium are stable in the outermost part of the stellar envelope, while most of the bulk of the star (more than 90% in mass) is under the form of a fully ionized H [FORMULA] /He [FORMULA] plasma. Therefore a correct EOS for VLMS must include a proper treatment not-only of temperature-ionization and dissociation, well described by the Saha-equations in an ideal gas, but most importantly of pressure -ionization and dissociation, as experienced along the internal density/temperature profile, a tremendously more complicated task. Moreover, under the central conditions of these stars, the fully ionized hydrogen-helium plasma is characterized by a plasma coupling parameter [FORMULA] for the classical ions (a is the mean inter-ionic distance, A is the atomic mass and [FORMULA] the mass -density) and by a quantum coupling parameter [FORMULA] ([FORMULA] is the electronic Bohr radius and [FORMULA] the electron mean molecular weigth) for the degenerate electrons. These parameters show that both the ions and the electrons are strongly correlated. A third characteristic parameter is the so-called degeneracy parameter [FORMULA], where [FORMULA] is the electron Fermi energy. The classical (Maxwell-Boltzman) limit corresponds to [FORMULA], whereas [FORMULA] corresponds to complete degeneracy. The afore-mentioned thermodynamic conditions yield [FORMULA] in the interior of VLMS along the characteristic mass range, implying that finite-temperature effects must be included to describe accurately the thermodynamic properties of the correlated electron gas. At last, the Thomas-Fermi wavelength [FORMULA] (where [FORMULA] denotes the electron particle density) is of the order of the mean inter-ionic distance a, so that the electron gas is polarized by the ionic field, and electron-ion coupling must be taken into account in the plasma hamiltonian. Such a detailed treatment of strongly correlated, polarisable classical and quantum plasmas, plus an accurate description of pressure partial ionization represent a severe challenge for theorists. Several steps towards the derivation of such an accurate EOS for VLMS have been done since the pioneering work of Salpeter (1961) and we refer the reader to Saumon (1994) and Saumon, Chabrier and VanHorn (1995) for a review and a comparison of the different existing EOS for VLMS. In the present calculations, we use the Saumon-Chabrier (SC) EOS (Saumon 1990; Chabrier 1990; Saumon & Chabrier 1991, 1992; Saumon, Chabrier & VanHorn 1995; SCVH), specially devoted to the description of low-mass stars, brown dwarfs and giant planets. This EOS presents a consistent treatment of pressure ionization and includes significant improvements with regard to previous calculations in the treatment of the correlations in dense plasmas. This EOS is tied to available Monte Carlo simulations and high-pressure shock-wave experimental data (see SCVH and references therein for details). As shown in Saumon (1994) and SCVH, significant differences exist between the SC EOS and other VLMS EOS, for the pressure-density relations and for the adiabatic gradients, so that we expect substantial differences in the derived stellar radii and entropy profiles. The SCVH EOS has been used previously to derive interior models for solar (Chabrier et al. 1992; Guillot et al. 1995) and extrasolar (Saumon et al. 1996) giant planets and for brown dwarfs (Burrows et al. 1993).

The SC EOS is a pure hydrogen and helium EOS, based on the so-called additive-volume-law (AVL) between the pure components (H and He). The accuracy of the AVL has been examined in detail by Fontaine, Graboske & VanHorn (1977). The invalidity of the AVL to describe accurately the thermodynamic properties of the mixture is significant only in the partial ionization region (see e.g. SCVH). As mentioned above, this concerns only a few percents of the stellar mass under LMS conditions. Given the negligible number -abundance of metals in stars ([FORMULA] by mass, i.e. [FORMULA] by number) we expect the presence of metals to be inconsequential on the EOS. Their contribution to the perfect gas term is just proportional to the number density ([FORMULA]), i.e. [FORMULA], whereas their non-ideal (correlation) contribution can be estimated either by the Debye-Huckel correction ([FORMULA]) for [FORMULA] or by the electrostatic (ion-sphere) term ([FORMULA]) for [FORMULA]. For solar metal-abundance (see e.g. Grevesse & Noels, 1993) this yields an estimated contribution to the EOS of [FORMULA] compared with the hydrogen+helium contribution.

Although this simple estimation shows that metals do not contribute appreciably to the EOS of VLMS, as long as the structure and the evolution are concerned, we have decided to conduct complete calculations by comparing models derived with the SC EOS and models based on the so-called MHD EOS (Hummer & Mihalas, 1988; Mihalas, Hummer & Däppen, 1988), which includes the contribution of heavy elements under solar abundances. Although the MHD EOS is devoted to stellar envelopes and weakly correlated plasmas, like the solar interior ([FORMULA]), and thus can not be applied to VLMS, it provides a useful tool for the present test. The test is even strengthened by comparing the complete MHD EOS with the pure hydrogen-helium ([FORMULA]) MHD EOS kindly provided by W. Däppen. Note that the MHD EOS for mixtures does not assume the additive-volume law between the various components, so that comparison with this EOS provides also a test for the validity of this approximation.

Fig. 1 shows in a HR diagram the results obtained with the MHD EOS for solar metallicity (open circles) and for [FORMULA] (triangles) up to 1 [FORMULA] 1. The difference is less than 1% in [FORMULA] and 4% in L. This demonstrates convincingly the negligible contribution of the metals to the EOS over the entire LMS mass range. As shown above, the contribution of metals to the EOS is proportional to a power of the charge Z and the atomic mass A. Therefore, when applying a metal free EOS to solar metallicity objects, the presence of metals can be mimicked by an equivalent helium fraction [FORMULA] in the EOS, at fixed hydrogen abundance. Varying X instead of Y would yield larger differences in [FORMULA] and L. Fig. 1 also displays results based on the SC EOS, with the afore-mentioned equivalent helium fraction (filled circles). Both the SC and MHD EOS yield very similar results. The differences between the SC and the complete ([FORMULA]) MHD EOS amount to less than 1.3% in [FORMULA] and 1% in L. Below 0.3 [FORMULA], however, the track based on the MHD EOS starts to deviate substantially from the one based on the SC EOS, a fairly reasonable limit for an EOS primarily devoted to solar conditions.

[FIGURE] Fig. 1. Theoretical HR-diagram for solar metallicity at t=5 Gyrs. Filled circles (solid line) Saumon-Chabrier EOS (SCVH, 1995) with [FORMULA]; triangles (dotted line): metal free (Z=0) MHD EOS with [FORMULA]; open circles: solar metallicity ( [FORMULA]) MHD EOS (Mihalas et al. 1988).

This is better illustrated in Fig. 2, which displays the adiabatic gradient along the density-profile in a 0.2 [FORMULA] and a 0.6 [FORMULA] star for the SC and MHD EOS. For the 0.6 [FORMULA] star, slight discrepancies between the SC and MHD adiabatic gradient appear in the regime of partial ionisation of hydrogen and helium (log [FORMULA] [FORMULA] -4 to -0.5 and log T [FORMULA] 4.2 to 5.3). The differences become substantial for the 0.2 [FORMULA] star because of the strong departure from ideality in a large part of the interior. The main discrepancies appear for log [FORMULA], with log T [FORMULA] and [FORMULA], which marks the onset of hydrogen pressure and temperature partial ionisation. The unphysical negative value of the adiabatic gradient in the MHD EOS for log [FORMULA] clearly illustrates the invalidity of the MHD EOS to describe the interior of dense systems, as clearly stated by its authors (see Hummer & Mihalas, 1988). Similar discrepancies occur also for other LMS EOS (see Saumon 1994; SCVH), which will affect substantially the structure and the evolution of these convective objects. We emphasize the excellent agreement between both EOS in the domain of validity of the MHD EOS, even though the SC EOS does not include heavy elements. This clearly demonstrates the negligible effect of metals on the adiabatic gradient. This latter is essentially determined by hydrogen and helium pressure- and temperature- ionisation and/or molecular dissociation.

[FIGURE] Fig. 2. Adiabatic gradient as a function of density for the structure of a 0.6 [FORMULA] (solid and dashed curves) and 0.1 [FORMULA] (dash-dot and dotted curves). The solid and dash-dotted curves correspond to the Saumon-Chabrier EOS (SCVH, 1995). The dashed and dotted curves to the MHD (1988) EOS.

These calculations clearly demonstrate that metal-free EOS can be safely used to describe the structure and the evolution of LMS with a solar abundance of heavy elements, providing the use of an effective helium fraction to mimic the effect of metals. It also assesses the validity of the Saumon-Chabrier EOS, devoted primarily to dense and cool objects, for solar-like masses 2.

2.2. The nuclear reaction rates

The thermonuclear processes relevant from the energetic viewpoint under the central temperatures and densities characteristic of VLMS are given by the PPI chain:




The destruction of [FORMULA] by reaction (3) is important only for T [FORMULA] K i.e masses M [FORMULA] for ages [FORMULA] 10 Gyrs, since the lifetime of this isotope against destruction becomes eventually smaller than a few Gyrs. In the present calculations, we have also examined the burning of light elements, namely Li, Be and B, whose abundances provide a powerfull diagnostic to identify the mass of VLMS and brown dwarfs (see Sect. 3.3). Our nuclear network includes the main nuclear-burning reactions of [FORMULA] and [FORMULA] (cf. Nelson et al. 1993). We will focus on the depletion of the most abundant isotopes [FORMULA] and [FORMULA], described by the following reactions:




The rates for these reactions are taken from Caughlan and Fowler (1988). These rates correspond to the reactions in the vacuum, or in an almost perfect gas where kinetic energy largely dominates the interaction energy. As already mentioned, such conditions are inappropriate for VLMS. In dense plasmas, the strongly correlated surrounding particles act collectively to screen the bare Coulomb repulsion between two fusing particles. This will favor the reaction and then enhance substantially the reaction rate with respect to its value in the vaccuum. As mentioned in Sect. 2.1, under the central conditions characteristic of low-mass stars, the electrons are only partially degenerate and are polarized by the ionic field. This responsive electronic background will also screen the nuclear reactions and must be included in the calculations. 3 Under the conditions of interest, both enhancement factors, ionic and electronic, are of the same order, i.e. a few units (Chabrier 1997). Different treatments of these enhancement factors have been derived, again following the pioneering work of Salpeter (1954). A complete treatment of the ionic screening contribution over the whole stellar interior density-range, from the low-density Debye-Huckel limit to the high-density ion-sphere limit was first derived by DeWitt et al. (1973) and Graboske et al. (1973). The inclusion of electron polarisability, in the limit of strongly degenerate electrons, i.e. [FORMULA] and [FORMULA], was performed by Yakovlev and Shalybkov (1989). An improved treatment of the ionic factor, and the extension of the electron response to finite degeneracy, as found in the interior of VLMS, was performed recently by Chabrier (1997). The difference between the Graboske et al. and the Chabrier results, which illustrates both the improvement in the calculation of the ionic factor and the effect of electron polarisability, is shown on Fig. 3 along temperature profiles characteristic of VLMS, for two central densities, for Li-burning ([FORMULA]). Substantial differences appear, in particular in the intermediate-screening regime ([FORMULA]) characteristic of LMS and BD interiors. The larger the charge, the larger the effect ([FORMULA]). Such differences translate into differences in the abundances as a function of time and mass, as will be examined in Sect. 3.3. Note that the inclusion of electron polarisability was found to decrease substantially the deuterium-burning minimum mass (Saumon et al. 1996).

[FIGURE] Fig. 3. Screening factors as a function of temperature for Lithium-burning (eqn. (4)), for (a) [FORMULA] g cm-3 and (b) [FORMULA] g cm-3. Solid line: present ion+electron screening factors (Chabrier, 1997); dash-dot: ionic factor only; dot: electron factor only; dashed line: Graboske et al. (1973).

2.3. Deuterium-burning on the main sequence

The initial D-burning phase ends after [FORMULA] yrs and is inconsequential for the rest of the evolution and the position on the Main Sequence (Burrows, Hubbard & Lunine, 1989 and Sect.  3 below). We focus in this section on the deuterium production/destruction rate along the PPI reactions, given by eqn. (1) and (2), which is essential for the nuclear energy generation required to reach thermal equilibrium. For stars below [FORMULA], which are entirely convective (see Sect.  3.2), the deuterium lifetime against proton capture [FORMULA] is found to be much smaller than the mixing timescale in the central regions, where energy production takes place. The mixing time in these regions is estimated from the mixing- length theory (MLT), [FORMULA], where [FORMULA] is the mean velocity of turbulent eddies. These fully convective stars are essentially adiabatic throughout most of their interior, with a degree of superadiabaticity [FORMULA] variing from [FORMULA] 10-8 to [FORMULA] from the center to 99% of the mass. Thus the mixing-length parameter is inconsequential, and a reasonable estimate for the mixing length is the pressure scale height [FORMULA]. This yields a mixing timescale [FORMULA] s, to be compared with [FORMULA] s in the central part of the star where nuclear energy production takes place. This is illustrated in Fig.4 where both timescales are compared in a 0.075 [FORMULA] star evolving on the main sequence.

[FIGURE] Fig. 4. Comparison of the deuterium burning lifetime against p -capture (solid line) and the mixing timescale (dashed line) in a 0.075 [FORMULA], with solar metallicity. Deuterium burning occurs in the region [FORMULA] K.

Since deuterium is burned much more quickly than it is mixed, a deuterium abundance gradient will develop in the central layers. This process can be described by the stationary solution of the following diffusion equation, since the diffusion and the nuclear timescales are orders of magnitudes smaller than the evolution time:


Here [FORMULA] is the convective mixing diffusion coefficient, [FORMULA] and [FORMULA] are the respective rates of reactions (1) and (2), and [FORMULA] and [FORMULA] denote respectively the hydrogen and the deuterium abundances by number ([FORMULA], following the notations of Clayton 1968). As long as [FORMULA], the deuterium abundance [FORMULA] in each burning layer will be close to its nuclear quasi-equilibrium value, as given by [FORMULA] in Eq.(7). The abundance of deuterium is relevant only in the central region, where we adopt the equilibrium value. The complicated task of solving Eq.(7) is thus not necessary in this case. The nuclear energy production-rate of the [FORMULA] reaction is then given by:


where [FORMULA] is the energy of the [FORMULA] reaction. Note that during the initial deuterium burning phase, this situation does not occur and the deuterium abundance is calculated like other species in the network, under the usual instantaneous mixing approximation, which corresponds to [FORMULA]. In that case abundances are homogeneous throughout the whole burning core, i.e. [FORMULA] and the concentration of deuterium is given by an average value over the convective zone [FORMULA]. When [FORMULA], instantaneous mixing will thus overestimate the deuterium-concentration in the central layers. This can be understood intuitively since instantaneous mixing will provide too much deuterium, i.e. more than produced by nuclear equilibrium, at the bottom, i.e. the hottest part, of the burning region. This yields an overestimation of the nuclear energy production at a given temperature and [FORMULA], and thus of the total luminosity of the star. This effect is not drastic for stars M [FORMULA], but increases the luminosity by [FORMULA] for 0.1 [FORMULA] and by [FORMULA] for 0.075 [FORMULA] and thus bears important consequences for a correct determination of the stellar to sub-stellar transition.

2.4. Model atmospheres

The low temperature and high pressure in the photosphere of M-dwarfs raise severe problems for the computation of accurate atmosphere models. For these low effective temperatures ([FORMULA] K) molecules become stable (H2, H2 O, TiO, VO,...), and constitute the main source of absorption along characteristic wavelengths. The presence of these molecular bands complicates tremendously the treatment of radiative transfer, not only because of the numerous transitions to be included in the calculations, but also because the molecular absorption coefficients strongly depend on the frequency and a grey-approximation, as used for more massive stars, is no longer valid. Moreover, the high density in M-dwarf atmospheres yields the presence of collision-induced absorption, an extra degree of complication. These points have been recognized long ago and motivated various developments in the modelling of M-dwarf atmospheres since the pioneering work of Tsuji (1966). Substantial improvement in this field has blossomed in recent years with the work of Allard and collaborators (Allard 1990; Allard and Hauschildt 1995a; 1997), Brett (1995), Tsuji and collaborators (Tsuji et al. 1996) and Saumon (Saumon et al. 1994), due to recent interest in the extreme lower main sequence and the necessity to derive accurate atmophere models to identify the luminosity and the colors of M-dwarfs and brown dwarfs. We refer the reader to the recent review by Allard et al. (1997b) for details.

The present evolutionary calculations are based on the latest generation of LMS non-grey atmosphere models at finite metallicity (Allard & Hauschildt 1997; AH97), labeled NextGen. In order to illustrate the most recent improvements in LMS atmosphere theory, we will make comparisons with stellar models based on the previous so-called Base models (Allard and Hauschildt 1995a; AH95), as used in the calculations of Baraffe et al. (1995). A preliminary version of the NextGen models was used by Chabrier, Baraffe & Plez (1996) and Baraffe & Chabrier (1996) and compared with results based on the other source of non-grey atmosphere models presently available, computed by Brett and Plez (Brett 1995; Plez 1995, private communication; BP95). This latter set, however, is restricted to solar metallicity. As shown in Chabrier et al. (1996) the BP95 models lead to [FORMULA] intermediate between the Base and NextGen models, for a given mass. Note that the AH97 models used in the present work include improved molecular opacity treatment compared to the straight mean method used in the Base models, and the more recent water linelist of Miller et al. (1994) (see AH97 for details). A summary of the main differences in the input of these atmosphere models is outlined in Chabrier et al. (1996).

As shown e.g. by Jones et al. (1995), the models still predict too strong infrared water bands despite the inclusion of these new (even if still incomplete and preliminary) water linelists (see AH97). This shortcoming, and the remaining uncertainties in the calculation of the TiO absorption coefficients, represent the main limitation of present VLMS atmosphere models for solar-like metallicities. A second limitation comes from grain formation below [FORMULA] (Tsuji, Ohnaka & Aoki; 1996) which is likely to affect the spectra and the atmosphere structure of the coolest M-dwarfs, and brown dwarfs, for solar-metallicity. Work in this direction is under progress.

2.5. Boundary conditions

The last but not least problem arising in the modelization of VLMS is the determination of accurate outer boundary conditions (BC) to solve the set of internal structure equations. All previous VLMS models relied on grey atmosphere models. The BC were based either on a [FORMULA]) relationship (Burrows et al. 1989; Dorman et al. 1989; D'Antona & Mazzitelli 1994, 1996; Alexander et al. 1996) or were obtained by solving the radiative transfer equations (Burrows et al. 1993). In order to make consistent comparison with these models, and to demonstrate the limits of a grey approximation for VLMS, we first give a short overview of the various procedures used in the literature.

[FORMULA]) relationships have the generic form:


with different possible q([FORMULA]) functions (Mihalas, 1978). The most simple form is based on the Eddington approximation, which assumes that the radiation field is isotropic, which yields q([FORMULA])=2/3, as used in Burrows et al. (1989). An exact solution of the grey problem (Mihalas 1978) gives actually a function which departs slightly from 2/3, but this correction is inconsequential on the resulting evolutionary models (Baraffe & Chabrier 1995).

As mentioned in the previous sub-section, the strong frequency-dependence of the molecular absorption coefficients yields synthetic spectra which depart severely from a frequency-averaged energy distribution (see e.g. Allard 1990; Saumon et al. 1994). Modifications of the function q([FORMULA]) have been derived in the past in order to mimic departures from greyness (Henyey et al. 1965, as used by D'Antona & Mazzitelli 1994; Krishna-Swamy 1966, as used by Dorman et al. 1989). These corrections, however, are based at least partly on ad-hoc calibrations to the Sun and do not rely on reliable grounds.

The assumption of a temperature stratification following a [FORMULA]) relationship requires not only a grey approximation but also the assumption of radiative equilibrium, implying that all the energy in the optically thin layers is transported by radiation. However, below [FORMULA] K, molecular hydrogen recombination in the envelope (H+H [FORMULA] H2) reduces the entropy and thus the adiabatic gradient (see SCVH). This favors the onset of convective instability in the atmosphere so that convection penetrates deeply into the optically thin layers (Auman 1969; Dorman et al. 1989; Allard 1990; Saumon et al. 1994; Baraffe et al. 1995). Radiative equilibrium is no longer satisfied and flux conservation in the atmosphere now reads [FORMULA]. Though rigorously inconsistent with the use of a [FORMULA] relationship, by definition, modifications of eqn.(9) have been proposed to account for convective transport in the optically thin layers. Henyey et al. (1965) prescribed a correction to the calculation of the temperature gradient and to the convective efficiency in optically thin layers which is equivalent to a correction of the diffusion approximation. This procedure was used by Dorman et al. (1989). On the other hand, some authors have just neglected the presence of convection in the optically thin regions of the atmosphere (Alexander et al. 1996).

All these arguments show convincingly that a description of M-dwarf atmospheres based on grey models and [FORMULA] relationships is physically incorrect. The consequence on the evolution and the mass-calibration can be determined by comparing stellar models based on the various afore-mentioned grey treatments with the ones based on non-grey atmosphere models and proper BCs. These latter are calculated as follows. We first generate 2D-splines of the atmosphere temperature-density profiles in a ([FORMULA])-plane, for a given metallicity. The connection between the atmosphere and the interior profiles is made at [FORMULA], the corresponding (T- [FORMULA]) values being used as the BC for the Henyey integration. This choice is motivated by the fact that i) at this optical depth, all atmophere models are adiabatic and can be matched with the interior adiabat, and ii) [FORMULA] corresponds to a photospheric radius [FORMULA], where [FORMULA] is the stellar radius, so that the Stefan-Boltzman equation [FORMULA] holds accurately. We verified that varying this BC from [FORMULA] to 100 does not affect significantly the results. Convection starts to dominate, i.e carries more than 50% of the energy in the atmosphere for [FORMULA] (Allard, 1990; Brett, 1995). Therefore, [FORMULA] = 100 is a safe limit to avoid discrepancy between the treatment of convection in the atmosphere and in the interior. Note that [FORMULA] corresponds to a pressure range [FORMULA] to [FORMULA] bar, depending on the temperature, gravity and metallicity. Along this important pressure-range, the dominant source of absorption shifts, going inward in the atmosphere, from water absorption to TiO, and eventually CIA H2 absorption, whereas the molecular line width changes from thermal-broadening to pressure broadening (AH95; Brett, 1995). For a fixed mass and composition, there is only one atmosphere temperature-density (or pressure) profile with a given effective temperature and gravity wich matches the interior profile for the afore-mentioned BC. This determines the complete stellar model for this mass and composition. The effective temperature, the colors and the bolometric corrections are given by the atmosphere model.

Figs. 5a-b show the temperature-pressure profile of NextGen atmospheres for [FORMULA] = 3500 K and log g = 5 for [M/H]=0 (Fig. 5a) and [M/H]=-1 (Fig. 5b). The location of the onset of convection is shown in the figures. For both metallicities, convection reaches the optically thin region ([FORMULA] = 0.02 for [M/H]=0 and [FORMULA] = 0.06 for [M/H]=-1). Also shown are profiles derived from grey models and [FORMULA] relationships using the Eddington approximation (dotted line), the Krishna-Swamy (KS) relation with a correction for the presence of convection in optically thin layers (cf. Henyey et al. 1965) (dash-dot line) and with convection arbitrarily stopped at T [FORMULA] (dashed line). The Eddington approximation yields severely erroneous results. The atmosphere profile is substantially cooler and denser than the non-grey one above [FORMULA], so that a hotter atmosphere model is required to match the internal adiabat, yielding too hot stellar models, as demonstrated in Chabrier et al. (1996). For solar metallicity, the two profiles based on the KS relation are almost undistinguishable and yield atmosphere models very close to the non-grey one. This agreement, however, strongly depends on the thermodynamic conditions, pressure and temperature, along the atmosphere profile. This is shown convincingly in Fig. 5b, for a denser metal-poor atmosphere. The higher pressure of metal-poor models favors the convective flux in the present grey models, and thus yields a flatter temperature gradient.

[FIGURE] Fig. 5a-c. [FORMULA] atmosphere profile according to the non-grey models of Allard & Hauschildt (1997) (solid line). Dash-dotted line: grey models obtained with the Krishna-Swamy (1966) T- [FORMULA] relationship with convection included in the optically thin region. Dashed line: idem with convection arbitrarily stopped at T [FORMULA], i.e. [FORMULA]. Dotted curve: Eddington approximation. a [FORMULA] = 3500 K, [FORMULA], [M/H]=0. The points indicate the onset of convection, corresponding to [FORMULA] = 0.02 for the non-grey model (circle), [FORMULA] = 0.06 for the Krishna-Swamy case (square) and [FORMULA] = 0.25 for Eddington (cross). The dashed and dot-dashed curves are undistinguishable in that case. For the dashed curve, convection sets in at T = [FORMULA], by definition. b [FORMULA] = 3500 K, log g =5, [M/H]=-1. Convection occurs at [FORMULA] = 0.06 for the non-grey model, [FORMULA] = 0.08 for the Krishna-Swamy case and [FORMULA] = 0.24 for Eddington. Note the departure between the dashed and dash-dotted curve when convection sets in in the optically thin region. c [FORMULA] = 2000 K, log g =5.5, [M/H]=0. In this case, the dashed curve corresponds to a model based on the Krishna-Swamy relation with grainless Rosseland opacities (D. Alexander, private communication). Convection starts well below the photosphere ([FORMULA]) and the departure between grey and non-grey models stems essentially from strong non-grey effects.

In that case, both KS profiles depart substantially from the correct one, even though the model based on the KS relation without convection in optically thin region, although physically inconsistent, yields less severe disagreement ([FORMULA] K in [FORMULA]). It must be kept in mind, however, that the convection correction in a grey atmosphere based on a [FORMULA]) relationship does not reflect adequately the influence of convection on a correct non-grey model (see e.g. Brett 1995, AH95; AH97). It also demonstrates that the Henyey et al. (1965) correction to the radiative diffusion approximation, when using a [FORMULA] relationship, overestimates the convective flux in optically thin regions. This yieds flatter temperature gradient in the atmosphere and thus larger effective temperature for a given mass. The models of Burrows et al. (1993), although based on grey atmosphere models, do not rely on a [FORMULA] relation but use BC based on the resolution of the transfer equations. Although cooler than the Burrows et al. (1989) models, they still yield too large effective temperatures compared with the non-grey models. As shown above this stems very likely from an overestimation of convection efficiency in the atmosphere.

Even when the temperature in the atmosphere is low enough so that convection does not penetrate anymore into the optically thin region ([FORMULA] K), strong departure from greyness still invalidates the use of a [FORMULA] relationship. This is illustrated in Fig. 5c for solar metallicity models with [FORMULA] = 2000 K and log g = 5.5.

The effect of grain formation in the atmosphere on the evolution was considered in Chabrier et al. (1996), by comparing, within a grey approximation, stellar models based on the Alexander and Fergusson (1994) Rosseland opacities and on similar dust-free opacities kindly provided by Dave Alexander. Grains were found to affect the evolution only below [FORMULA]. This is confirmed on Fig. 5c where the two grey profiles, with and without grain, yield essentially the same atmospheric structure at [FORMULA] 2000, i.e. [FORMULA]. These calculations will be reconsidered once non-grey atmosphere models with grains will be available.

These calculations show convincingly that procedures based on a grey approximation and a [FORMULA] relation for the derivation of VLMS evolutionary models are extremely unreliable, even though they may yield, under specific thermodynamic conditions, to (fortuitous) agreement with consistent non-grey calculations. As a general result, a grey treatment yields cooler and denser atmosphere profiles below the photosphere (cf. Saumon et al. 1994; Allard and Hauschildt 1995b), and thus overestimates the effective temperature for a given mass (Chabrier et al., 1996). Therefore, they lead to erroneous mass-luminosity and mass- [FORMULA] relationships, as will be discussed in Sect.  4.

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Online publication: April 6, 1998