## 2. Input physicsThe 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, l 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 stateInterior profiles of VLMS range from
K and
g cm 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 Although this simple estimation shows that metals do not contribute
appreciably to the EOS of VLMS, as long as the Fig. 1 shows in a HR diagram the results obtained with the MHD EOS
for solar metallicity (open circles) and for
(triangles) up to 1
This is better illustrated in Fig. 2, which displays the adiabatic
gradient along the density-profile in a 0.2
and a 0.6
star for the SC and MHD EOS. For the 0.6
star, slight discrepancies between the SC and
MHD adiabatic gradient appear in the regime of partial ionisation of
hydrogen and helium (log
-4 to -0.5 and log T
4.2 to 5.3). The differences become
substantial for the 0.2
star because of the strong departure from
ideality in a large part of the interior. The main discrepancies
appear for log
, with log T
and
, 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
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
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. The nuclear reaction ratesThe 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 by reaction (3) is important only for T K i.e masses M for ages 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 and (cf. Nelson et al. 1993). We will focus on the depletion of the most abundant isotopes and , 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.
## 2.3. Deuterium-burning on the main sequenceThe
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 is the convective mixing diffusion coefficient, and are the respective rates of reactions (1) and (2), and and denote respectively the hydrogen and the deuterium abundances by number (, following the notations of Clayton 1968). As long as , the deuterium abundance in each burning layer will be close to its nuclear quasi-equilibrium value, as given by 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 reaction is then given by: where
is the energy of the
reaction. Note that during the ## 2.4. Model atmospheresThe 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 ( K) molecules become stable (H The present evolutionary calculations are based on the latest
generation of LMS non-grey atmosphere models at finite metallicity
(Allard & Hauschildt 1997; AH97), labeled 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
## 2.5. Boundary conditionsThe 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 ) 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. ) relationships have the generic form: with different possible q() functions (Mihalas, 1978). The most simple form is based on the Eddington approximation, which assumes that the radiation field is isotropic, which yields q()=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() 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
) relationship requires not only a grey
approximation All these arguments show convincingly that a description of M-dwarf
atmospheres based on grey models and
relationships is physically incorrect. The
consequence on the Figs. 5a-b show the temperature-pressure profile of
In that case, both KS profiles depart substantially from the
correct one, even though the model based on the KS relation
Even when the temperature in the atmosphere is low enough so that convection does not penetrate anymore into the optically thin region ( K), strong departure from greyness still invalidates the use of a relationship. This is illustrated in Fig. 5c for solar metallicity models with = 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 . This is confirmed on Fig. 5c where the two grey profiles, with and without grain, yield essentially the same atmospheric structure at 2000, i.e. . 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 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- relationships, as will be discussed in Sect. 4. © European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |