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

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3. Evolutionary tracks

As mentioned in the introduction, the present study focusses on a mass-range limited to [FORMULA] and will consider the brown dwarf domain only scarcely. For larger masses, the physics of VLMS remains unchanged but variations of the mixing length parameter start to be consequential and require comparison with observations (such comparisons are considered in detail in Baraffe et al. 1997). On the other hand, a substantial improvement over existing brown dwarf models (e.g. Burrows et al. 1989; 1993), requires the derivation of non-grey atmosphere models with grains. As shown by the recent analysis of Tsuji et al. (1996), silicate and iron grains can contribute significantly to the opacity in the photosphere below [FORMULA] K (see also Lunine et al. 1986; Alexander and Fergusson 1994), a typical effective temperature for massive brown dwarfs with solar abundances (see Sect.  3.1).

In the following sub-sections, we present the evolution of the mechanical and thermal properties of objects ranging from 0.055 to 0.6 [FORMULA], over a metallicity-range [M/H]= -2 to 0. The general properties of VLMS and BDs have already been described by numerous authors (see Sect.  1). We will not redo such an analysis in detail but rather focus on selected masses to illustrate the differences arising from the new physics (EOS, nuclear rates, non-grey model atmospheres) described in the previous section. The different mechanical and thermal properties of the present models, for various metallicities, are presented in Tables 2-7.

3.1. Mechanical and thermal properties

3.1.1. Internal structure

Fig. 6 displays the behaviour of the central temperature, the radius and the degeneracy parameter along the main sequence (MS) for two metallicities, [M/H]=0 (solid line) and [M/H]=-1.5 (dashed line). For stars on the MS, the internal temperature is large enough in the stellar interior for the pressure to be dominated by classical contributions ([FORMULA]), so that hydrostatic equilibrium yields [FORMULA]. Below [FORMULA], the object is dense and cool enough for the electrons to become substantially degenerate ([FORMULA]), so that the electronic quantum contribution ([FORMULA]) overwhelms the ionic classical pressure. Eventually this will yield the well-known mass-radius relation for fully degenerate (zero-temperature) objects [FORMULA]. The brown dwarf domain lies between these two limits (classical and fully degenerate) and is characterized by [FORMULA] [FORMULA], about Jupiter radius, where [FORMULA] is the zero-temperature (fully degenerate) radius (see e.g. Stevenson, 1991). The transition between the stellar and sub-stellar domains is characterized by this ongoing electron degeneracy in the interior, as illustrated in Fig. 6. Note also that once degeneracy sets in, the temperature scales as [FORMULA], where the first and second term represent the classical and quantum gas contributions, respectively. This yields the rapid drop of the interior temperature (and effective temperature for a given [FORMULA] relation) near the sub-stellar transition, and thus the characteristic severe drop in the luminosity ([FORMULA]).

[FIGURE] Fig. 6. Degeneracy parameter [FORMULA], central temperature [FORMULA] (in units of [FORMULA] K) and radius ([FORMULA]), as function of mass for [M/H]=0 (solid line) and [M/H]=-1.5 (dash).

Fig. 7 displays the evolution of the radius for different masses. Although the radius is fixed mainly by the EOS, it does depend, to some extent, on the atmosphere treatment. The effect is negligible for solar metallicity, as shown in the figure for [FORMULA], but can yield [FORMULA] difference on the final radius for [FORMULA]. After a similar pre-MS contraction phase for all masses, the hydrogen-burning stars reach hydrostatic equilibrium whereas objects below the hydrogen-burning minimum mass keep contracting until reaching eventually the afore-mentioned asymptotic radius, characteristic of a strongly degenerate interior.

[FIGURE] Fig. 7. Evolution of the radius (in solar unit) for different masses, for solar metallicity. For 0.2 [FORMULA], comparison is shown with results obtained with the Krishna-Swamy T- [FORMULA] relationship (dash-dotted curve) and with the Eddington approximation (dotted line).

3.1.2. Luminosity

Figs. 8a-b exhibit [FORMULA] for different masses, for [FORMULA] and [FORMULA], respectively. Initial deuterium burning proceeds very quickly, at the very early stages of evolution, and lasts about [FORMULA] years. Our calculations were done with an initial deuterium abundance [FORMULA] in mass fraction, which corresponds to the average abundance in the interstellar matter (cf. Linsky et al. 1993). A value [FORMULA] increases the deuterium burning timescale by a factor [FORMULA] 2 and the luminosity by 10% to 50% during this phase. We verified that this effect is inconsequential for the rest of the evolution. As clearly shown in the figures, for masses above [FORMULA] for [M/H]=0 and [FORMULA] for [M/H]=-1.5, the internal energy provided by nuclear burning quickly balances the contraction gravitational energy, and the lowest-mass star reaches complete thermal equilibrium ([FORMULA], where [FORMULA] is the nuclear energy rate), after [FORMULA] Gyr, for both metallicities. The lowest mass for which thermal equilibrium is reached defines the so-called hydrogen-burning minimum mass (HBMM), and the related hydrogen-burning minimum luminosity (HBML). These values are given in Tables 2-7, for various metallicities. Note the quick decrease of luminosity with time for objects below the HBMM, with [FORMULA] (cf. Burrows et al. 1989; Stevenson 1991). As mentioned above, we have not explored the BD domain and we stopped the calculations at 10 Gyrs for MS stars. Cooler models require non-grey atmospheres with grains.

[FIGURE] Fig. 8a and b. a Evolution of the luminosity for different masses for [FORMULA]. Solid lines: total luminosity. Nuclear luminosity [FORMULA]: dotted line: 0.5 [FORMULA] and 0.075 [FORMULA]; dashed line: [FORMULA]; dash-dot line: [FORMULA]. b Same as a for [FORMULA] Nuclear luminosity [FORMULA]: dotted line: 0.5 [FORMULA]; dashed line: [FORMULA]; dash-dot line: [FORMULA].

As already shown by Chabrier et al. (1996), stellar models based on non-grey model atmospheres yield smaller HBMM than grey models, a direct consequence of the lower effective temperature and luminosity, as discussed in the previous section. The larger the luminosity, for a given mass, the larger the required central temperature to reach thermal equilibrium, which in turns implies a larger contraction (density) and degeneracy.

As illustrated on Fig. 8, slightly below [FORMULA] (resp. 0.083 [FORMULA]) for [M/H]=0 (resp. [M/H] [FORMULA] -1), nuclear ignition still takes place in the central part of the star, but cannot balance steadily the ongoing gravitational contraction. This defines the massive brown dwarfs. The evolution equation thus reads [FORMULA], where the second term on the right hand side of the equation stems from the contraction energy plus the internal energy released along evolution. As shown in Fig. 7, contraction is fairly small after [FORMULA] yr, so that most of the luminosity arises from the thermal content. Below about [FORMULA] (resp. [FORMULA]) for [M/H]=0 (resp. [M/H] [FORMULA] -0.5), the energetic contribution arising from hydrogen-burning, though still present for the most massive objects, is order of magnitudes smaller than the internal energy, which provides essentially all the energy of the star ([FORMULA]).

As seen in the figures, objects with lower metallicity evolve at larger luminosities and effective temperatures, a well-known result. The effects of metallicity on the atmosphere structure have been discussed extensively by Brett (1995) and AH95 but can be apprehended with intuitive arguments. The lower the metallicity, the lower the opacity and the more transparent the atmosphere. The same optical depth thus lies at deeper levels, i.e. at higher pressure ([FORMULA]). Therefore, for a given mass ([FORMULA]), the [FORMULA] interior profile matches, for a given optical depth [FORMULA], an atmosphere profile with larger [FORMULA] (see Fig. 5). This yields a larger luminosity, since the radius barely depends on the atmosphere, as shown previously.

Objects above 0.15 [FORMULA] reach the MS in [FORMULA] yrs, while it takes [FORMULA] yrs (resp. [FORMULA] yrs) for [FORMULA] (resp. [FORMULA]) for objects in the range [FORMULA]. Objects above [FORMULA] (resp. [FORMULA]) for [FORMULA] (resp. [FORMULA]) start evolving off the MS at [FORMULA] Gyr.

As shown on Fig. 8, an object on the pre-MS contraction phase which will eventually become a H-burning star ([FORMULA]) can have the same luminosity and effective temperature as a bona-fide brown dwarf.

3.2. Full convection limit

Solar-like stars are essentially radiative, except for a small convective region in the outermost part of the envelope, due to hydrogen partial ionisation, and sometimes for a small convective core where nuclear burning takes place. As the mass decreases, the internal temperature decreases ([FORMULA]), the inner radiative region shrinks and vanishes eventually for a certain mass below which the star becomes entirely convective (see e.g. D'Antona & Mazzitelli 1985, Dorman et al. 1989). This transition mass [FORMULA] has been determined with grey atmosphere models, which become invalid below [FORMULA] K, and thus must be recalculated accurately. Fig. 9 shows the interior structure of stars with [FORMULA] as a function of time for [M/H]=0. The pre-MS contraction phase proceeds at constant [FORMULA], i.e. constant [FORMULA] ([FORMULA]). After [FORMULA] years, a radiative core develops and grows. The physical reason is the decreasing opacity after the last bump due to metal absorption (mainly Fe,), for [FORMULA] (this temperature decreases with metallicity) (cf. Rogers and Iglesias 1992, Fig. 2). The radiative core thus appears earlier for the more massive, hotter, stars. The minimum mass for the onset of radiation in the core is found to be [FORMULA], for all the studied metallicites ([FORMULA]. After [FORMULA] years, the star reaches thermal equilibrium, nuclear fusion proceeds at the center, and a small convective core develops for a certain time, depending on the mass and the metallicity, bracketting the central radiative region between two convective zones. We verified that the growth of the central convective core is governed by the [FORMULA] reaction. The nuclear energy released by the [FORMULA] and [FORMULA] reactions is insufficient to generate convective instability. As long as the reaction given by eqn.(2) dominates, the [FORMULA] abundance, and thus the convective core, increases. The situation reverses as soon as the central temperature is high enough for [FORMULA] to reach its equilibrium abundance (eqns. (2)+(3)), wich decreases with increasing temperature (see e.g. Clayton 1968, Fig. 5.4).

[FIGURE] Fig. 9. Evolution of the radiative zone [FORMULA] for [M/H]=0. The upper curves determine the bottom of the convective enveloppe and the lower curves the top of the convective core for 0.4 [FORMULA] (dash-dot), 0.5 [FORMULA] (solid line) and 0.6 [FORMULA] (dash).

The extension of the afore-mentioned radiative region decreases with temperature, and thus with mass, as shown in the figure. For the afore-mentioned limit-mass [FORMULA], this inner radiative zone remains only for [FORMULA]. For all other (greater and smaller) metallicities, it vanishes as soon as the convective core appears. In this case the 0.35 [FORMULA] star will become fully convective again after [FORMULA] yrs. This rather complicated dependence on metallicity stems from the subtle competiton between the decreasing opacity, which favors radiation, and the increasing pressure and luminosity which inhibate radiation and favor convection, with decreasing metallicity ([FORMULA]). This yields the minimum value for [FORMULA] for [FORMULA].

Table 1 gives the position of the bottom of the convective enveloppe as a function of mass and metallicity. We also give the results for the masses of the binary system YY-Gem. Note that grey models, which have higher luminosity (see Sect.  4.1), thus have a larger [FORMULA], which favors convection and thus yields larger convective envelopes and larger convective cores and [FORMULA]. As will be shown below, the onset of a radiative core, and the retraction of the bottom of the convective zone to outer, cooler regions bears important consequences on the abundance of light elements in the envelope.

[TABLE]

Table 1. Bottom of the convective enveloppe [FORMULA] normalised to the radius of the star R as a function of mass and metallicity, for an age of 10 Gyrs. Comparison is made with grey models based on the Krishna-Swamy prescription for the 0.4 [FORMULA] and [M/H]=-1.5. The values 0.57 and 0.62 [FORMULA] correspond to the eclipsing binary system YY-Gem (Leung and Schneider 1978)

3.3. Abundance of light elements

Observation of lithium in the atmosphere of VLMS is a powerful diagnostic for the identification of genuine brown dwarfs, as proposed initially by Rebolo and collaborators (Rebolo, Martin & Magazzù 1992). The first theoretical analysis of light-element burning in VLMS, and the expected abundances along evolution, were carried by Pozio (1991) and Nelson, Rappaport and Chiang (1993). As for the convective limit, this analysis must be redone with updated EOS, screening factors and atmosphere models. The initial abundances were taken to be [FORMULA], [FORMULA] and [FORMULA], as in Nelson et al. (1993). The modification of the abundances along evolution, i.e. the depletion factor, is given by [FORMULA], where [FORMULA] is the abundance of element i at a given time and [FORMULA] is the afore-mentioned initial abundance. The burning temperatures for these elements (in the vacuum) are [FORMULA] K, [FORMULA] K and [FORMULA] K, respectively.

As for hydrogen, burning ignition temperatures translate into minimum burning masses. Using the ion+electron screening factors mentioned in Sect. 2.2, we get the following values, for solar metallicity: [FORMULA], [FORMULA] and [FORMULA]. For comparison, Nelson et al. (1993) find that [FORMULA] of 7 Li is burned in a 0.059 [FORMULA], whereas this value is already reached in our 0.055 [FORMULA] model. D'Antona & Mazzitelli (1994) obtain [FORMULA]. This less efficient nuclear burning in Nelson et al. and DM94 stems on one hand from the grey approximation, which yields larger L and [FORMULA] and thus central densities, which favors the onset of degeneracy (cf. Sect.  3.1.2 and Fig. 10 below), and also from the smaller Graboske et al. (1973) screening factors (see Fig. 3).

These effects, and the metallicity dependence, are illustrated in Figs. 10a-b which display the evolution of central temperature and lithium-abundance, respectively, for different metallicities. Comparison is made with a solar metallicity model of DM94 (cf. their Table 7). The denser metal poor stars (and grey models) reach the limit of degeneracy earlier ([FORMULA]), yielding a lower maximum central temperature for metal poor objects. The direct consequence is an increasing minimum burning mass with decreasing metallicity, [FORMULA] for [M/H] [FORMULA] instead of 0.055 for [M/H] [FORMULA].

[FIGURE] Fig. 10a and b. a Evolution of the central temperature in a 0.06 [FORMULA] brown dwarf. Solid line: [FORMULA]; dot-dash line: [FORMULA]; ([FORMULA]): [FORMULA]. Dotted line: 0.06 [FORMULA] of DM94 for [M/H]=0. Dashed line: calculations with the Graboske et al. (1973) screening factors, for [FORMULA]. Note that the solid and dashed curves are undistinguishable, and illustrate the negligible effect of the electronic screening factor on the evolution of LMS. b Same as a for the abundance of Lithium with regard to its initial abundance. Same legends as in a.

Fig. 11 shows the abundances of Li, Be and B as a function of mass and metallicity. The gaps correspond to fully convective interiors, as described in Sect. 3.2. In that case, convective mixing brings the elements present in the envelope down to the central burning region where they are destroyed. Above 0.4 [FORMULA], the central radiative core appears and the bottom of the convection zone retracts to cooler regions, as described previously. As mentioned above, this depends mainly on the central opacity: the larger the opacity, and thus the metal-abundance, the larger the central temperature required to allow radiative transport ([FORMULA]). This yields more efficient depletion with increasing metallicity, as illustrated in the figure. Lithium is totally destroyed in the mass range 0.075-0.6 [FORMULA] for [M/H]=0, whereas it reappears for M [FORMULA] for [M/H] [FORMULA] -1. However, below [M/H]=-1.5, the situation seems to reverse, as the [M/H]=-2 case shows a slightly higher level of depletion. At such low metallicities, the dependence of the opacity on the metallicity for T [FORMULA] decreases rapidly and the dominant effect is now the higher luminosity at [M/H]=-2, which implies a larger central temperature to favor radiation ([FORMULA]). The depletion factors in the stellar interiors are given in Tables 2-7. The lithium depletion factors in the brown dwarf regime are given in Chabrier et al. (1996). A complete description of light elements depletion in this regime, which implies evolutionary models based on dusty atmosphere models, is under progress.

[FIGURE] Fig. 11. Abundances of light elements as a function of mass and metallicity at [FORMULA] 10 Gyrs. Only results in stars are shown ([FORMULA] for [M/H]=0 and [FORMULA] for [M/H] [FORMULA] -1). Abundances normalized to their initial value are shown for Li (solid,O), Be (dot,X) and B(dash, [FORMULA]).

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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