CiteSeerX — ASTRONOMY AND ASTROPHYSICS Structure and evolution of low-mass Population II stars

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ASTROPHYSICS AND

Structure and evolution of low-mass Population II stars

J. Montalb´an^1,2, F. D’Antona², and I. Mazzitelli³

1 Instituto de Astrof´ısica de Canarias, 38200 La Laguna, Tenerife, Spain (montalbn@coma.mporzio.astro.it)

2 Osservatorio di Roma, 00040 Monteporzio, Roma, Italy (dantona@coma.mporzio.astro.it)

3 Istituto di Astrofisica Spaziale, 00133 Roma, Italy (aton@hyperion.ias.rm.cnr.it) Received 18 October 1999 / Accepted 21 June 2000

Abstract. The focus of the present paper is on the detailed de- scription of the internal structures of low mass, population II stars, to clarify some issues about these stellar models and, mainly, their present reliability for observational comparisons.

We then explore 1) the role of the local convective model; 2) the differences between “grey” and “non grey” models, and between models in which the photospheric boundary conditions are set at different optical depths (τ_ph = 3 or 100); 3) the role of the equation of state (EoS), both in the atmospheric models and in the interior. One of the major conclusions of the paper is a cautionary note about the usage of the additive volume law in EoS calculations.

The dependence of the HR diagram locations and mass luminosity relations on metal and helium content are also discussed.

A few comparisons with globular cluster stars show that: 1) general consistency of distance scales and morphologies in the HR diagram is found, when comparing ground based measure- ments in the JohnsonB and V bands and observations in the HST bands; 2) a discrepancy between models and observations may exist for more metal rich clusters; 3) the plausible hypothesis that the mass function in the globular cluster NGC 6397 behaves smoothly until the lower limit of the main sequence poses constraints on the mass-luminosity relation at the lowest end of the main sequence.

The evolutionary tracks are available at the WEB location http://www.mporzio.astro.it.

Key words: stars: evolution – stars: low-mass, brown dwarfs – stars: luminosity function, mass function – stars: Population II – Galaxy: globular clusters: individual: NGC 6397, M92, M4

1. Introduction

The main aim of this paper is to clarify the uncertainties still weighing on the computation of low mass population II models, in spite of the recent improvements. Recent papers on the population II low-mass mean sequence containing a comparison with Globular Cluster HR diagrams include Baraffe et al.

(1997, BCAH97) who first show the results obtained by use Send offprint requests to: F. D’Antona

of the NextGen boundary conditions, and Cassisi et al. (2000) whose input physics closely resemble those of BCAH97. Our group (Silvestri et al. 1998) has recently published tracks, computed with the ATON2.0 code and grey atmospheres, from the pre-main sequence (PMS) down to the bottom of the MS to examine the problem of the theoretical luminosity function of Globular Cluster (GC) stars.

Here we present evolutionary tracks computed with grey and non-grey atmospheres for low-mass objects (M ≤ 0.7 M), for several metal abundances: [M/H] = –2.0, –1.5, –1.3 and –1.0. The computations were performed with an updated version of the ATON2.0 code (Ventura et al. 1998) including the “NextGen” non-grey (and non-dusty) atmosphere models by Allard & Hauschildt (priv. comm. thereafter AH97; NextGen models for 3000 K≤ Teff≤ 10000 K have been published by Hauschildt et al. 1999).

Reliability of the thermodynamics in the model atmospheres puts a lower mass limitM ≥ 0.085 M (Teff ≥ 2000 K) to our computations. Another limit is due to the surface gravities (logg ≥ 3.5) available in the grids; we can start modelling the Hayashi phase only after the deuterium burning stage. Contrac- tion is followed until either nuclear reactions or electron degeneracy take over. In the former case, the star reaches nearly com- plete thermal equilibrium (hydrogen-burning main sequence:

MS, when H-burning accounts for at least 99% of the stellar luminosity within a Hubble time). In the latter one, the mass being lower than the hydrogen-burning minimum mass (HBMM), surface luminosity losses force cooling and residual contraction. The object is “substellar” and is also known as a “Brown Dwarf”.

The exact determination of the the HBMM is tricky, depending on the physical, chemical and even numerical inputs in the code used to model stellar evolution (D’Antona 1987; Dor- man et al. 1989; Chabrier & Baraffe 1997 – hereinafter CB97).

Furthermore, in these objects convection spreads through the optically thin, dense and cool atmospheric layers, often dom- inated by molecules, where non-ideal gas effects take over in determining both thermodynamics and opacities. This affects not only the color-T_effrelations, but also the total luminosity which, in fully convective structures, is largely determined the surface heath transport. Below ∼0.5 M(T_eff≈ 5000 K), the photospheric boundary conditions begin to be critical for the

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whole structure. At lower masses (below∼ 0.13 M), when the pressure dissociation and ionization zone for a real gas is met, no matter how thin in mass the region involved is, problems with the evaluation of the adiabatic gradient give rise to uncertainties with the central temperature, the HBMM and the mass–luminosity relation.

The present results should be closer to those by CB97; the main differences are in the EoS used for the low-temperature and high-density plasma, in the local convective model and in the time-dependent chemical mixing scheme. Other minor differences in the nuclear network, in the electron screening etc., are expected to be immaterial to the main results for these structures.

In Sect. 2 we recall the main physical and chemical inputs in ATON2.0. Sect. 3 presents numerical results: internal structure, evolutionary tracks, theoretical HR diagrams and mass–

luminosity relationships. Here, also the effects of the atmosphere models, convection treatment and EoS are discussed.

Sect. 4 presents the transformations to the observational plane, and comparisons with magnitude–color diagrams for three globular clusters (M92, NGC 6397 and M4) observed by HST. Con- clusions are summarized in Sect. 5.

2. Theoretical models 2.1. Chemical inputs

Even if recent estimates by Izotov & Thuan (1998) of the primor- dial helium abundance seem to suggest thatYBig Bang=0.245, we will use the former value of YBig Bang=0.23 (Audouze 1986) for the helium mass fractionY in our models of lowest metallicity. For the various metal contents: [M/H]= –2.0, –1.5, – 1.3 and –1.0, we scaleY according to: ∆ Y/∆ Z ≈ 2 ± 0.5 (based on Y = 0.27–0.28). We also compute some tracks withY = 0.25 for comparison with CB97 and BCAH97. Note that a higher helium abundance would produce models with higher luminosity and temperature for stars with a radiative core (masses higher than∼ 0.4 msol) and for stars with masses lower than∼ 0.13 M, where degeneracy appears. For ∆Y =0.02,

∆ log L = 0.06 and ∆ log T_eff = 0.01 for 0.6 M, and for 0.1M,∆ log L = 0.08 and ∆ log T_eff = 0.02.

We always assume solar metal ratios (Grevesse & Noels 1993), neglecting enhancement of oxygen and other α- elements. Hence, in the comparisons with observations of low-Z globular clusters, tracks for metal contents slightly larger than those from the observed [Fe/H] ratios are chosen, consistently with Salaris et al. (1993).

2.2. Microphysics 2.2.1. Thermodynamics

One of the main differences between the present computations and CB97 ones is in the thermodynamics. Let us then sum- marize some features of the EoS employed in ATON2.0 code.

EoS tables are given as a function of temperature, pressure and H-abundance for any required value of Z. They are built up in

three steps. First, the EoS by Magni & Mazzitelli (1979, MM) is written for the five H-abundances:X = 1.0 − Z, X = 0.8 − Z, X = 0.5 − Z, X = 0.2 − Z; X = 0.0. A metal abundance different from zero is obtained by interpolating upon the H/He mixtures an “average” metal, according to the pure–carbon table by Graboske et al. (1973). This procedure would be far incor- rect for opacities, but is acceptable for thermodynamics (and better than allowing for the missing metals by increasing Y ) at least untilZ does not exceed a few %. The tables are then partially overwritten by the OPAL ones in the whole range for which these are available (Rogers et al. 1996), always for the same five H abundances as in MM and for the given metal abundance. Last, in the region above 3000 K, where OPAL tables are not available, the MM tables are overwritten by the Mihalas et al. (1988, MHD) ones for the proper H/He ratio andZ, but only atlog ρ ≤ −3, since they have been shown to be biased at larger densities (Saumon 1994). The various tables edges (see Fig. 13) are then placed either where the gas is still nearly ideal (transition between OPAL and MM), or where the free–energy minimization scheme is still nearly identical in the two cases (transition between MHD and MM) sinceT is too low to affect the internal partition function. In fact, no clear discontinuities have been detected on the final tables. Remember however that they are spaced inlog P by 0.5. Bicubic logarithmic interpolation on bothP and T are performed, and linear interpolation on X follows, to get the various thermodynamical quantities. The procedure then avoids interpolation between pure H and pure He tables. Let us comment on this latter statement.

For pure H, the MM EoS is far less updated than the one by Saumon et al. (1995, SCVH). This latter has been in fact constructed according to more recent data, mainly for configurational effects, and successfully compared to the results of lab- oratory experiments and Monte-Carlo simulations. The SCVH EoS is instead not necessarily superior to the MM one for pure He, since the former ignores both pressure ionization and ex- cited states of He andHe⁺, which are instead allowed for by MM. However, let us assume for ease the SCVH as ideally “perfect” both for pure H and pure He, and use it as a benchmark to check whether and how much, in the low-T , high-ρ regions of interest for objects close to the HBMM and not covered by OPAL or MHD tables (that is: in the logT ≤ 4, log P ≤ 12 region), the MM EoS is still affordable or not. For more accurate comparisons among the various EoSs, the reader is referred to Saumon (1994), who shows that in the regimes of interest here the two EoS are largely consistent to each other.

Fig. 1 shows the run of logρ vs. the H fractional abundance in mass at logT = 4, log P = 9, for both the MM (solid dots + solid line) and SCVH (stars + dotted line) EoSs. In these conditions, met in stars ofM ≥ 0.2 M, the performances are almost identical for pure He, where real–gas effects are immaterial. A bias about 6% inρ is instead present in the MM EoS for pure H, whereH₂already weakly starts feeling excluded volume interactions. As for intermediate H/He compositions, the MM points have been directly computed by minimizing the Free Energy for mixtures; SCVH line is instead computed from pure fluids, according to the Additive Volumes Law (AVL). Both the

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Fig. 1. Density vs. hydrogen abundance in mass fractionX at log T = 4, log P = 9, where non-ideal gas effects are still negligible. The EoS by Magni & Mazzitelli (squares and solid line) has been explicitly computed for H-He mixtures; the EoS by Saumon et al. is instead for pure H and pure He only. Interpolation is then performed according to the additive volumes law (stars and dotted line). When configurational effects are negligible, as in the present case, the two EoSs are almost coincident also for mixtures.

explicit computation and the interpolation law provide similar behaviors. Hence, there is no reason to suspect that, in the binary mixtures in the MM EoS, other “errors” are present than the ones relative to pure H or pure He.

In Fig. 2 we instead show what happens at log T = 4, log P = 12, that is: for structures close to the HBMM and in Brown Dwarfs. The MM EoS overestimatesρ for pure He by

∼ 8%, and underestimates it by ∼ 3% for pure H. For mixtures, the AVL interpolation badly fails, since it holds only when the two fluids do not interact with each other.

Then, in the lack of experimental data or Monte Carlo simulations for binary mixtures in these conditions, either the MM EoS a priori treats incorrectly mixtures, or we must accept the well known physical result that the AVL fails in extreme cases.

If it is so, the very fact that stars have an He–abundance about 25% leads to the conclusion that, even with the almost “perfect”

SCVH EoS, uncertainties in the thermodynamics of very-low- mass stars and Brown Dwarfs are likely to be larger than 10%

(actually,∼ 20% in the case above exemplified).

Our conclusions are therefore the following:

– The SCVH thermodynamics is by far the best choice for pure fluids, if non-ideal gas effects are to be accounted for (e.g. subatmospheres of H– and He–rich White Dwarfs).

– For very low mass MS structures and Brown Dwarfs (H-He mixtures), the AVL interpolation introduces a severe bias in the use of the SCVH EoS close to H- pressure ionization.

Until new computations in which mixtures of elements are

Fig. 2. The same as in the previous figure, but at logP = 12, where non-ideal gas effects (mainly covolume for molecular hydrogen) are relevant. These conditions are met in the subatmospheres of very low mass objects, at the boundary between stars and brown dwarfs. The Additive Volume Law fails in interpolating between pure chemical composition, and an EoS in which H–He mixtures are explicitly accounted for (such as in the MM one) is then required.

explicitly accounted for, the MM EoS (plus OPAL where available) is still usable.

So, at present, thermodynamics represents a non negligible source of uncertainty in the modelling of low-mass stellar structures, both in the interior and, as we will see later, in optical atmospheres.

2.2.2. Opacities

At temperaturesT ≥ 6000 K we adopt OPAL radiative opacities (Rogers & Iglesias 1993; solarZ-distribution from Grevesse &

Noels 1993). They are linearly extrapolated (log κ vs. log ρ) in high-ρ regions, and harmonically added to conductive opacities by Itoh & Kohyama (1993). Since the differences between relativistic, and Hubbard & Lampe (1969) non–relativistic conduc- tivities, are generally lower than 10% and never exceed 25%, the main problem with opacities is the high density extrapolation of the radiative tables. For the structures of interest here, the corresponding regions are however convective and nearly adiabatic, and uncertainties in˜κ do not affect the main results (apart from the case of the coolest atmospheres). At lower temperatures we use Alexander & Ferguson’s (1994) molecular opacities (plus electron conduction in full ionization) for the same H/He ratios andZ as in the OPAL case.

Opacities (and OPAL EoS) tables for Z = 2.10⁻⁴ and Z = 6.10⁻⁴ ([M/H]=–2.0 and –1.5 are not available in the original OPAL tables, so they have been obtained by interpola- tionlog ˜κ vs. log Z).

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2.2.3. Nuclear reactions

ATON2.0 includes 22 nuclear reactions for 14 elements. The cross-sections are from Caughlan & Fowler (1988); the low, intermediate and strong screening coefficients from Graboske et al.(1973).

It has been pointed out by CB97 that, in central conditions of low MS stars, the screening factors must also include the electronic contribution, being thus larger than the ionic ones.

However, in the regime of interest, nuclear cross-sections still grow up by orders of magnitude in a narrow range ofT . Any minimum structural feedback resets the luminosity of nuclearly- powered structures also if the screening changes. CB97 noticed in fact that their tracks are not affected by the improved screening factors. The effect could be instead significant close to the lithium burning minimum mass, where no structural feedback is expected but, as we shall see, in these conditions much larger uncertainties in stellar modelling exist (atmospheres, thermodynamics etc).

2.3. Macrophysics

2.3.1. Atmospheric structure and boundary conditions

The photospheric boundary conditions adopted by Silvestri et al. (1998) arose from grey atmospheres with theT(τ) relationship by Henyey et al. (1965), stopping integration atτ = 2/3.

Because of the low temperatures and high densities in the outer- most layers of very low mass stars, radiative absorption is domi- nated by molecules (H₂, H₂O, TiO, VO, etc.) and the outcom- ing flux is far different from the frequency-averaged distribution provided by grey models (Saumon et al. 1994). Moreover,H₂ recombination reduces the value of∇_ad, enhancing penetration of convection in the atmospheric layers (Auman 1969; Dorman et al. 1989; Saumon et al. 1994; Baraffe et al. 1995). Even if the T(τ) relationship is “corrected” to account for departure from greyness and for the efficiency of convection in the optical layers, Brett (1995), Allard & Hauschildt (1995, AH95), Saumon et al. (1994) and CB97 have shown that the convective flux in optically thin regions is overestimated, yielding flatter temperature gradients in the atmosphere and, hence, larger values of Teff. More generally, according to Saumon et al. (1994, AH95), Chabrier et al. (1996), any grey treatment seems to yield cooler and denser atmospheric profiles, overestimatingT_eff.

Since the thermal structure of the photosphere largely affects the internal structure (D’Antona & Mazzitelli 1985; D’Antona 1987), to put in evidence the effect of the atmospheric treatment we compute both non-grey and grey models, these latter with the T(τ) relationship by Henyey et al. (1965).

The NextGen models available to us for [M/H]= −1.0,

−1.3 and −1.5 cover the range 3.5 ≤ log g ≤ 6 and, the range 2000≥ Teff ≤10000 K. For [M/H]=–2.0, models down to Teff 900 K are also given.

The lower limit in gravity does not allow the initial deuterium burning in PMS to be followed. Another limit arises from the non-ideal gas effects in dense and cool atmospheres. We plot in Fig. 3ρ and T at the base of the [M/H]=–2.0 atmospheres,

Fig. 3. NextGenρ and T at the bottom of photosphere as a function ofTeffand for forτph = 3 (dotted lines) and τph= 100 (solid lines) two different logg values (5 and 6). Full squares correspond to Teff=1, 2, 2.5, 3, 3.5 and 4.10³K. The dashed line marks the separation between real and ideal gas. Also,ρ and T at the base of the photosphere for the 0.083 and 0.085MCB97 models are shown: [M/H]=–2.0, empty circles; [M/H]=–1.5, empty triangles and [M/H]=–1.0, empty squares.

for two different values oflog g (5 and 6) and for two choices of τph (3 and 100). Very low-Teffmodels enter the non-ideal gas region. Since the model atmospheres adopt ideal gas and a Saha-like thermodynamics (AH95), they should not be used in real-gas domain where pressure effects are relevant. For low- MS structures, models below 2000–2400 K are approaching the real-gas domain, the more so the larger becomesτ_ph. Thus, in what follows we always discuss models withτ_ph= 3 unless differently specified. We also show models withτ_ph= 100 mainly to make comparisons with CB97. Consistently, the lower lim- its toTeffdue to the thermodynamics in the model atmospheres are:

1. τph= 3, [M/H]=–2.0, Teff > 2200 K 2. τph= 3, [M/H]> −2.0, Teff ≥ 2000 K 3. τph= 100, [M/H]=–2.0, Teff > 2600 K

4. τph= 100, [M/H]> −2.0, Teff > 2200 − 2400 K So, for [M/H]=–2.0, our non-grey models of 0.085 Mcomputed with τph = 3 are still physically reliable, while only models with M ≥ 0.095 Mshould be accepted withτ_ph= 100. The choice of τ = 3 is also justified on physical grounds, as at this optical depth the radiative flux already closely approaches isotropy. In the following, structures for which the base of the atmosphere lies in the real-gas region are then meant for internal comparisons only. Their “absolute”

features are not physically reliable, and should not be taken into account in the comparisons with observations. In Fig. 3 we also mark the locations for the two CB97 models of 0.083 and

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Fig. 4. Evolution of luminosity vs. time for [M/H]=–2.0. Non- grey models (solid lines) are plotted for 0.4 M(developing a radiative core), 0.3, 0.2, 0.18, 0.15, 0.13, 0.11, 0.1, 0.095, 0.09 and 0.085M(fully convective). The last four grey models have been also plotted (dotted line).

0.085 M, for [M/H]=–1.0 (empty squares), [M/H]=–1.5 (empty triangles) and [M/H]=–2.0 (empty circles). At low Z, also these latter models lie close to or inside the real gas domain in atmosphere.

2.3.2. Convection

The ATON2.0 code allows us to treat convection either according to the MLT, or to the Full Spectrum Turbulence (FST) model (Canuto & Mazzitelli 1992; Canuto et al. 1996). Since the NextGen atmospheres adopt the MLT withα_MLT= l/H_p= 1, we do the same in the present computations. In the following sections we shall discuss the possible relevance of this choice for the interior of low-mass stellar models.

Lower MS fully convective stars are so dense that the convective gradient is nearly adiabatic. Hence, the structure and the location on the HR diagram of these objects are not affected by the choice ofαMLT, or even by the convective model. Increasing the star’s mass, the upper layers become less dense and hotter, convection less efficient, and the gradient becomes overadiabatic. For the highest masses considered here (M ≥ 0.5 M), the structure then depends on the treatment of overadiabatic convection and, by now, it is widely acknowledged thatα_MLT= 1 does not provide an acceptable solar fit. For comparison, we have also computed models with the FST treatment of convection, which provides a fair description of the turnoff colors of globular clusters (Mazzitelli et al. 1995; D’Antona et al. 1997).

When using the Allard-Hauschildt atmosphere models for say [M/H]=–2, one can see that for τ = 3 the convection flux represents only about 50% of the total flux. The profile is thus

Fig. 5. The same as in Fig. 4, but forTeff. starting from 0.7M.

not completely adiabatic. In that case, the connection with an interior adiabat is not correct and the results with such a boundary condition are meaningless. A model for superadiabaticity in inner layers, as in ATON2.0, is then required.

As for chemical mixing, CB97 pointed out that the deuterium lifetime against proton capture in the central layers of low-mass stars was comparable to the convective turnover time scale. The instantaneous mixing approximation could then lead to an overestimate of the nuclear rates. In the ATON code we adopt a more physically sound mixing scheme consisting of a diffusion equation coupled with the nuclear network. In the presence of both nuclear reactions and turbulent mixing, the local temporal variation of thei^th element follows the diffusion equation:

d X_i dt

=

∂ X_i

∂t

nucl+ ∂

∂ mr

(4πr²ρ²)²D∂ X_i

∂mr

, (1) whereD is the turbulent diffusive coefficient, provided by the convective model. This allows us to check CB97 claim.

3. Results and discussion 3.1. Evolutionary tracks

Figs. 4 and 5 show the run of L and Teffvs. time for the [M/H]=–2 sequence, and Fig. 6 shows the evolution of central T andρ for the same models. Initially, T_c increases following contraction, according to a polytrope–like relationT ∝ ρ^2/3. ForM ≥ 0.11M,T_cgets constant when the star reaches thermal equilibrium in the MS. For smaller masses,T_c reaches a maximum and then declines due the onset of degeneracy. Above 0.085M,T_c reaches a plateau, since the fast rise of central density keeps up the reaction rates and the MS is still reached.

For even smaller masses, the threshold of efficient H-burning

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Fig. 6. The run ofTcvs.ρcfor non-grey (dashed lines) and grey (dotted lines) 0.7, 0.6, 0.4, 0.35, 0.3, 0.25, 0.2, 0.18, 0.15, 0.13, 0.11, 0.1, 0.09 and 0.085Mtracks.

is not met, and the object evolves to a cool and completely de- generate configuration. Since the transition to substellar domain occurs below 0.085M, the EoS used in the atmospheres does not allow us to exactly determine the HBMM.

In Figs. 4–6 we also plot the curves for the lowest mass models computed with grey atmospheres (dotted curves). As these overestimate theT_eff, they reach degeneracy at larger masses and cool quicker. Therefore, the mass–luminosity and mass–

T_effrelationships will show an abrupt change of slope, and the HBMM will be larger. The differences are striking since, below 0.1M, steady H–burning is not ignited with grey atmospheres.

Evolution also depends on the metal content. Out of degeneracy, the lowerZ is, the hotter and more luminous are the objects when approaching the MS. The behavior is reversed after the onset of degeneracy, below∼ 0.1 M, since lower internal opacities accelerate the cooling. The maximum ofTc

is achieved at younger ages the lowerZ is: a 0.085 Mobject reaches the MS at ∼2 Gyr for [M/H]=–2.0, at ∼3 Gyr for [M/H]=–1.0. We give these values as relative estimates, due to the already discussed uncertainties in these extreme physical conditions.

3.2. Internal structure

Figs. 7 and 8 show the profileslog T −log P and log T −log ρ of the models with [M/H]=–2.0, from τ_phto the center, corresponding to τ_ph = 3 (dashed) and τ_ph = 100 (dotted). In Fig. 8, also the domains of partial ionization and dissociation, and the non-ideal gas region are shown. The lowest-mass structures (0.085 and 0.075M) cross the threshold of sudden pressure ionization (PPT), and the most of their subatmospheres lie inside the real gas region.

Fig. 7. Gravothermal structure (log T, log P) of models from the bot- tom of the photosphere to the center for 0.7, 0.5, 0.35, 0.2, 0.13, 0.1, 0.085 and 0.075Mat 10 Gyr. Dashed lines correspond to models with τph= 3; dotted lines to τph= 100. They coincide above ∼ 0.1 M

Fig. 8. T-ρ profiles for the same models as in Fig. 7. Shaded regions correspond to partial ionization and dissociation (50%) of hydrogen and helium (from SCVH), and to the region (real gas) where excluded volumeH2pressure overcomes ideal gas pressure (from MM). Pressure ionization (PPT) is also shown.

Shapes and slopes of the curves can be qualitatively inter- preted in terms of the processes occurring in various physical regions. For very lowZ, stars with masses ≥ 0.4 Mdevelop a radiative core in the MS and, in thelog T −log P plane, the central T–profiles flatten below the adiabatic one. For lower masses, being low–Z models dense and hot, central fluxes are peaked

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Fig. 9. Gravitational structure (log P, log ρ) for fully convective metal poor stars. The real–gas region is defined as in Fig. 8.

Fig. 10.log T − log P profiles for the 0.7 Minternal structures at 10Gyr (solid lines) and the corresponding atmospheres (dashes lines) from NextGen models. Three different treatments of convection in the interior: MLT (α = 1 and α = 1.6) and FST, and two different τph

(3 and 100) are used. Matching points are marked with empty squares for the atmospheres, with crosses for the integration from inside.

favoring convection. LargerZ structures ([M/H]=–1.0) can instead preserve a small radiative core in MS down to 0.35M. Below∼ 0.35 M, our MS structures are always found fully convective. Their interiors are also nearly adiabatic, since subatmospheric densities are large enough for convective energy transport to be very efficient. Then, the change of slope in the log T − log P plane below log T ≈ 4.5 is due to the decrease of∇adwhen crossing the region of partial ionization

and dissociation. At still lower temperatures, if logρ ≤ −5, recombination stops at neutral–H, and the adiabatic gradient rises again (see the 0.7 and 0.5Mcases). At the lower end of the MS, non–ideal gas effects strengthen recombination ofH₂and the adiabatic gradient remains quite low until the merging with the optical atmosphere. This is mainly true for the lowest-mass structures since, above logρ ∼ −2, they cross a region where real–gas effects are important also in determining the chemical equilibria. Unfortunately, no EoS up today is sufficiently reliable here, and physical uncertainties can largely affect the run of∇adin these outer regions. Given the fully convective structure, the run of∇adin the subatmosphere affects the thermal configuration and influences both the HBMM, and the surface characteristics.

The importance of non–ideal gas thermodynamics in these objects can also be seen in thelog P − log ρ profiles for stars below 0.4Mshown in Fig. 9. In between 0.4 and 0.15M, the profiles are close to a polytropic withP ∝ ρ^5/3, as expected for adiabatic structures. The various “kinks” present in the strat- ification correspond to changes in the molecular weight due to recombination of H to H₂, and, at large density, also of HI to H₂.

3.3. Dependence on convection treatment

For MS stars of 0.7–0.5 M, overadiabaticity in subphotospheric layers is different with either τ_ph = 3 or τ_ph = 100 and affects the whole structure. From Figs. 7 and 8 we see in fact that models withτ_ph = 3 have larger radiative cores than those withτph = 100. The difference increases with the stellar mass, since overadiabaticity in the outer layers grows larger.

We have computed 0.6 and 0.7Mmodels with two values of αMLT in the subatmosphere: 1.0 and 1.6, this latter corresponding the solar tuning required by the ATON2.0 code (Ven- tura et al. 1998). We also applied the FST convective model. As for the optical layers, Saumon et al. (1994) have shown that in the atmosphere also the temperature gradient is far from adiabatic, and is then sensitive to the convective model. This is also seen in Fig. 2 of BCAH97, where they compare atmospheric structures for two different values ofα_MLT.

In Figs. 10 and 11 we show the run oflog T vs. log P and oflog T vs. log ρ for the 0.7 Mstructure, for different choices of the convective model and two different values of τ_ph. To avoid discrepances between the atmosphere and the interior, CB97 suggested use of τph = 100. However, with this value and withαMLT = 1, we already find a discontinuity in ∇ at the match with the subatmosphere, the behavior being instead more continuous withτph= 3. Both Teffand the internal structure of the star are then affected by the choice ofτph. ForαMLT= 1.6, despite the difference with the MLT tuning in the atmosphere models, the match is better, and neither the internal structure nor Teffis sensitive toτph. We have also tested the FST convective model. In this case, the run oflog T vs. log P is more continuous withτ_ph= 100, but the whole structure is more sensitive to the choice ofτ_ph.

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Fig. 11. The same as Fig. 10, but forlog T − log ρ profiles. All the models show some density inversion due to overadiabaticity.

For these relatively large masses, then, the influence of the convective model on the results is still comparable to the effect of the atmospheric model. Switching from grey to non–grey structures, the difference isTeffis in fact about 180 K. This is also the order of magnitude of the differences found between non–grey models withα_MLT=1 and 1.6, the larger beingτ_ph, the larger also the difference. The effect on the luminosity is instead always lower.

Let us now switch to the discussion of the time–dependent mixing coupled with nuclear evolution. In masses ≤0.3 M(fully convective), the deuterium lifetime against proton capture in the central layers is definitely shorter than the convective turnover time scale (CB97). This conclusion is independent of the convective model adopted. However, our models do not show any luminosity differences between the case in which we solve (Eq. 1) and in the instantaneous mixing approximation (see Ventura & Zeppieri 1998 for a similar result for population I structures). The same homeostatic mechanism responsible for the reset ofTc’s in the presence of different screening also com- pensates for different D–distributions, and the surface luminosity remains almost unchanged. Moreover, above∼ 3 × 10⁶K, the run of the p+p and D+p nuclear cross sections vs. T is almost parallel, and the equilibrium concentration of D is nearly constant for a large fraction of the structure. The mixing scheme is then not so important for the nuclear energy output as suggested by CB97.

3.4. Influence of photospheric boundary conditions on the stellar model

We compute, for each mass and [M/H], non-grey models with different values of τ_ph. In Figs. 12 and 13 we show the runs log T −log P and log T −log ρ for the 0.2 and 0.08 Mmodels at 10 Gyr, both forτph = 3 and =100. We observe small dis-

Fig. 12. Matching oflog T − log P profiles from the interior (solid lines) and from the atmosphere (dashed line) for models of 0.2 and 0.08Mat 10 Gyr using bothτph = 3 and 100. For 0.2 Mmodels coincide. For 0.08Mwe get different structures depending onτph. Crosses are (T, P ) at τphfrom the interior, empty squares are from the non-grey atmospheres.

Fig. 13. The same as Fig. 12, but for the T–ρ profiles. Inconsistency between the EoS used in atmospheres and in the interior is made evident by a discontinuity inρ. We also show the domains corresponding to different EoS used in the computation: vertical shade OPAL, horizontal shade MHD; and the rest MM.

continuties in log ρ (≈ 20%) when we use τ_ph = 100. At mass 0.2Mthis is a consequence of different mean molecular weights between MHD EoS and AH97’s EoS in the ideal gas domain. A smaller discontinuity (5%) appears also if the SCVH EoS is used (Cassisi 2000, private communication). Thislog ρ

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Fig. 14. HR diagrams for non-grey models at 10 Gyr. Masses marked:

0.7, 0.6, 0.5, 0.4, 0.35, 0.3, 0.2, 0.15, 0.13, 0.11, 0.1, 0.09 and 0.085M.

Fig. 15. HR diagrams from our grey (dashed lines) and non-grey (solid lines) models. For non-grey models withτph = 3, the squares correspond to 0.7, 0.6, 0.5, 0.4, 0.35, 0.3, 0.2, 0.15, 0.13, 0.11, 0.1, 0.09 and 0.085M. For grey models, the last masses marked are 0.09M([M/H]=–1.0), and 0.1 Mfor the other metallicities. The vertical positions of the curves for [M/H]> −2.0 are displaced in luminosity. For [M/H]=–1.5, ∆(log L/L) = 0.5; for [M/H]=–1.3,

∆(log L/L) = 1 and for [M/H]=–1.0, ∆(log L/L) = 1.5.

discontinuity has no relevant effect either on the internal structure or on the surface properties. More severe is the problem for the lowest-mass structures. The larger isτ_ph, the deeper in non–

ideal gas region is the match between atmosphere and interior.

There is a mismatch in density. In addition, the slope in both the log T − log ρ and log T − log P planes suddenly changes when

Fig. 16. HR diagrams from BCAH97 models (dashed lines) and our non-grey models (solid lines) computed with the same inputs:τph = 100, Y = 0.25 and age=10Gyr. The points on the curves (squares: our models; asterisks: BCAH97 models) are for masses: 0.7, 0.6, 0.5, 0.4, 0.35, 0.3, 0.2, 0.15, 0.13, 0.11, 0.1 and 0.09M. For BCAH97, 0.085 and 0.083Mare also shown. The same vertical displacements of the curves as in Fig. 15 have been used.

switching from the ideal-gas atmosphere to the real-gas interior.

In fact, not only theP = P (T, ρ) behavior changes, but even the∇_ad= ∇_ad(T, ρ) run is affected. Also notice that, if the fit to the interior is performed in thelog T − log ρ plane instead of thelog T − log P one (as in CB97), a small discontinuity in pressure is bound to arise, violating hydrostatic equilibrium.

From these comparisons we conclude that:

– the larger isτ_phchosen as boundary condition, the lower are ρcandTc, and the lower is the value of HBMM found;

– for very low mass stars and Brown Dwarfs, the match be- tween model atmospheres (ideal gas EoS) and internal structures needs to be performed at lowest values ofτphas possible. Otherwise, problems with the thermodynamics in the photosphere overcomes other sources of uncertainty.

3.5. HR diagram

The HR location of the present models is shown in Figs. 14, 15 and 16. The curves follow the well known double–kink behavior (D’Antona 1987; D’Antona & Mazzitelli 1996). The more luminous kink (FK, atT_eff∼ 4500 K for [M/H]=–2.0, T_eff∼ 4300 K for [M/H]=–1.0) shows up when starting crossing the H₂ molecular dissociation region (Copeland et al. 1970) and the value of the adiabatic gradient drops. The second kink (SK) appears when approaching degeneracy, and the object tends to a nearly constant radius track in the HR diagram.

The FK and SK divide the mass domain into three regions, sensitive to different physical inputs. Above the FK, for masses larger than 0.5M, the luminosity of the star is not largely af-

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Fig. 17. Differences between HR diagrams from BCAH97 models and our non-grey models computed with the same inputs (τph = 100, Y = 0.25).

Fig. 18. Central conditions ofT (crosses) and ρ (squares) as a function of mass for models at 10Gyr. Dashed lines correspond to CB97 models;

solid lines is for present models with the same Z, Y andτph.

fected either by the atmospheric boundary conditions (Fig. 15), or by the EoS (Fig. 16). For these masses convection can be still overadiabatic in the outer layers (Sect. 3.3), andTeff(not luminosity) is affected by the convective model adopted. We recall the attention upon the fact that in the present models, the region where the FK appears is still almost completely inside the OPAL EoS. Since this latter is at least as good as the SCVH one, but explicitly treats H/He mixtures and metals, we have reasons to think that our models down to∼ 0.4 Mare not less updated than BCAH97 ones.

For models in between the FK and the SK (0.4 ≥ M/M ≥ 0.11), the photospheric boundary conditions play a role, while

Fig. 19. The MLR from our non-grey models and its dependence on metallicity.

the convective model becomes less and less significant. Stars grow up fully convective, and the luminosity output is mainly constrained by the “thermostat” acting in the photosphere. The steepness of the MS in this region is also a function of the∇_ad, thus depending on the EoS. In this range of mass, the differences of luminosity between sets of models is almost negligible.

However the differences inT_eff(Fig. 17) can reach up to∼ 4%

(≈ 150K) at 0.11M. In the HR diagram, this leads to a steeper slope of our sequences with respect to those of CB97 (Fig. 16).

These different shapes of HR diagram also reflect different central conditions. Fig. 18 shows theρ_c−T_cplane for the lowest- mass structures. Models by CB97 (dashed lines) are compared to ours (solid lines), computed with the same NextGen atmospheres (τ = 100) and Y = 0.25. The runs of ρc vs. mass behave differently in the two sets. Below ∼ 0.13 M, when the non–ideal gas effects in the subatmosphere make the MM EoS far different from the AVL–interpolated one by SCVH,ρc

grows faster (andTcdrops faster) than in CB97, when decreas- ing the mass. Opposite is the run ofT_c, at least until electron degeneracy takes over.

3.6. Mass–luminosity and mass–Teffrelationships

The mass–luminosity relationship (MLR) is crucial for a correct translation of the luminosity functions of given stellar samples into the mass functions (MF), mainly for low-mass stars. From Figs. 19–20 we see that the general shape of the MLR is very similar for all sets of models. However, when trying to under- stand the MF at the lower end of the main sequence, even subtle differences in the shape can be relevant. Since what matters is not the MLR itself, but its derivative. The luminosity function will show a maximum at each change of slope of the MLR.

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Fig. 20. The MLR from the BCAH97 models (dashed lines) and from our non-grey models (dotted lines) using the same inputs. A vertical displacement of the curves have been used for ¿–2.0. For [M/H]=–

1.5,∆(log L/L) = 1; for [M/H]=–1.3, ∆(log L/L) = 2 and for [M/H]=–1.0, ∆(log L/L) = 3.

Fig. 21. The MLRs from non-grey models computed with SCVH EoS (Cassisi et al. and BCAH97), and with MM EoS (this paper).

In Fig. 19 we present the dependence of the MLR on the metallicity. Down to 0.11M, the luminosities reflect the con- sequences of a lower opacity in the subphotospheric layers for the lower-metallicity models. Once degeneracy appears, a lower opacity will produce a quicker cooling of the objects, and the relation between the luminosity andZ is reversed.

Figs. 19–21 show mass–luminosity relations at 10 Gyr. Be- fore degeneracy sets in, the luminosity only slightly depends on the atmosphere model or EoS. Both inputs are instead critical

Fig. 22. Mass-Teffrelationship from BCAH97 models (dashed lines) and from our non-grey models (solid lines) using the same inputs.

A vertical displacement of the curves have been used for [M/H]>

−2.0. For [M/H]=–1.5, ∆(log Teff)=0.15; for [M/H]=–1.3, ∆(log Teff)=0.3 and for [M/H]=–1.0, ∆(log Teff)=0.45.

on the location of the change in slope due to the onset of degeneracy, and on the amplitude of this change in slope. Also, a dependence on the helium content is found: below 0.13M, the lower is the mass, the larger is the difference in luminosity andTeffbetweenY = 0.25 and Y = 0.23 models. Of course, this will produce variations in the derivative of the MLR. The change in slope corresponding to the first “kink”, between 0.4 and 0.5M, will be larger forY = 0.25 than for Y = 0.23.

On the contrary, for the second kink, the change of slope produced when degeneracy appears will be larger forY = 0.23 than for 0.25 ((∆L/∆m)₂₅< (∆L/∆m)₂₃). In fact, the MLR at low masses seems to be very sensitive to different physical inputs. And it is not clear now to attribute differences in MLR (and therefore in HBMM) to a particular input, since models computed with the same EoS and atmosphere models provide quite different MLRs. Fig. 21 show MLRs from our models with Y = 0.23 and 0.25, from CB97 (Y = 0.25) and from Cassisi et al. (2000) (Y = 0.23) who also use SCVH EoS and NextGen atmosphere models.

Figs. 22 and 23 display the effective temperature mass relationship, and its dependence on the EoS and metallicity.

4. Observational plane

In the next sections we will test our grids of models on some observational data for Globular Clusters with metallicities in the range [M/H]= −1.0 to −2.0. More detailed comparisons be- tween theoretical and observational color-magnitude diagrams will be presented in a forthcoming work (D’Antona et al. 2000, in preparation). Here we wish to address two main problems: i) are the parameters derived from optical data for GCs distance

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Fig. 23. Mass-Teffrelationship from our non-grey models and its dependence on metallicity.

modulus and reddening compatible with the description of the lower-mass main sequence given by the present models? ii) Can the luminosity function of low-mass stars in GCs be used as a test for the lowest mass stellar models?

For our comparisons at low metallicity we will use full isochrones from the bottom of the MS to the RGB, obtained from two sets of models. ForM > 0.6 Mwe use Silvestri et al.

(1998) models computed with grey atmosphere boundary conditions and FST description of convection, transformed into the observational plane using Castelli (1998) color-T_effrelations.

For lower masses, we use the present non-grey models transformed according to AH97 colors. We had to resort to this procedure because AH97 models are not available for low enough logg values necessary for the turnoff (TO) region. Furthermore, the TO is sensitive to the convection description, and AH97 only consider MLT withα = 1. Fortunately, at 0.6 M(MV∼ 6) the models do not depend on convection treatment nor on the boundary conditions in the atmosphere (at least for the low metallicity models), so that this is a suitable point to make the junction between non-grey models and grey-models with FST convection.

4.1. Observational color-magnitude diagram:

Globular clusters M92, NGC6397, and M4:

HR diagram locations

Observational data for GC low mass stars have been generally obtained in recent years with the HST Wide Field Cameras in the bands m₅₅₅,m₆₀₆andm₈₁₄. The colorm₅₅₅− m₈₁₄ corresponds fairly well to the Johnson–V minus Cousin–I (see also BCAH97). The filter bands are in fact similar, the check on the model atmospheres of very low mass stars showing that convolution of the spectra with theV or m555passbands, and withI or m814provide very close magnitudes. The Holtzmann

et al. (1995) transformations generally adopted to pass from HST bands to the standard bands are also consistent with these marginal differences in the colors. Differences betweenV − I andm₅₅₅− m₈₁₄are not larger than 0.02 mag tillV − I=1.5 and are not larger than 0.04 mag tillV − I=2.7. It is certainly more correct to compare colors and models in the same passbands, and we do this when the data are available. However, if we use directlyV and V −I isochrones and compare them with m₅₅₅vs.m₅₅₅− m₈₁₄data, or withV versus V − I data obtained by Holtzmann et al. (1995) transformations fromm₅₅₅ andm814 (in the absence of the original data set) we do not expect significant differences.

The magnitudem606, on the contrary, covers a wide band, which is more difficult to relate to a standard magnitude without introducing uncertainties of several hundredths of magnitude.

Comparisons with this band must then always employ magnitudes obtained from the model atmospheres, as we will show for M92.

Observational comparisons also require independent infor- mations on the reddening and distance moduli of the clusters.

In some cases, the observed reddening is uncertain by some hundredths of magnitude. More important, distances listed in the literature are generally based on an assumed relation between the luminosity of the horizontal branch stars (or the RR Lyrae variables) and the cluster metallicity. As these relations can differ by∼0.3 mag or more at the lowest metallicities, also the match of the main sequence HR diagram is affected by this large uncertainty.

However, there is today an increasing agreement, based both on theoretical models (Mazzitelli et al. 1995; Weiss et al. 1996; Caloi et al. 1997) and observational grounds (Reid 1997; Gratton et al. 1997), that the distances of GCs are consistent with the “long” distance scale (Walker 1992; Sandage 1993). According to this choice, the distance moduli are constrained within∼0.1 mag. For the very low metallicity cluster M92, even the “shortest” distance modulus, found by Pont et al. (1998),(m − M)_V = 14.67 ± 0.08) is only ∼ 0.13 mag smaller than the value quoted by Gratton et al. (1997) (14.8± 0.06). Pont et al. (1998) consider that, if their correction to the binary subdwarfs luminosity is too large, the modulus increases to 14.74± 0.08, in good agreement with the other values. The

“short” modulus was in fact∼ 14.6 (e.g. Buonanno et al. 1989).

We consider the data for three clusters of increasing metallicity, in which we implicitly allow for anα enhancement by increasing the “equivalent”–Z value: M92 ([M/H]= −2) (An- dreuzzi et al. 2000; Piotto 1998, priv. comm.), NGC 6397 ([M/H]=–1.5) (King et al. 1998) and M4 ([M/H]= −1) (Richer et al. 1997). The data are shown in Figs. 24, 25, 26 and 27 together with the present models and those by BCAH97. In our comparisons we have chosen distance moduli and redden- ings which allow a good match of the luminous part of the HR diagram (horizontal branch, turnoff and upper main sequence) from ground based data (inV , B − V ), for the clusters M92 and NGC6397. We then check how these values also fit the low main sequence from HST data. For the cluster M4 we use the

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BCAH97 -2.0

Fig. 24. The data are taken from Andreuzzi et al. (2000), in the bands m555andm814, and are shown in the standardV versus V − I plane.

Our model isochrone of 12Gyr forY = 0.23, Z = 2×10⁻⁴is shown, together with the corresponding BCAH97 models (dashed curve) for the metallicity [M/H]= −2.

Fig. 25. The data set for M92 is taken from Piotto (1998, priv. comm.) in the HST bandsm606andm814. A reddening of E(B-V)=0.03 and a distance modulus of 14.7 have been assumed, consistent with optical data and horizontal branch models (Caloi et al. 1997). The curves are the isochrones for 12Gyr in the same magnitude bands as the data.

reddening and distance determined by Richer et al. (1997) by fitting the White Dwarf sequence.

In Fig. 24 the data from Andreuzzi et al. (2000) are compared to the models directly in theV , V − I plane. A full isochrone of 12Gyr is shown. The most luminous objects in the data set suffer from saturation, and thus we must not consider them as

BCAH97 -1.5

Fig. 26. NGC6397 data from King et al. (1998) compared with our models for [M/H]= −1.5 and BCAH97 models. The distance modulus and reddening assumed are also compatible with the optical data (although NGC6397 distance modulus is more uncertain, due to the peculiarity of the very blue horizontal branch). The BCAH97 data reproduce better the general shape of the low MS, especially if a slightly larger reddening is assumed (E(B − V ) = 0.18).

BCAH97 -1.0

Fig. 27.V versus V −I diagram for the low main sequence of M4. Data are from Richer et al. (1997), whose reddening and distance modulus (which allow a quite good fit of the white dwarf sequence –data and line at the bottom left) are assumed. The models for [M/H]= −1 are shown (continuous line) together with BCAH97 models (dashed).

The slope and location of the region between the two kinks is not well reproduced by either set of models. Both the photomety calibration and the model atmosphere colors can have a role in this mismatch.

representative of the turnoff color and luminosity location. Be- low these points, the main sequence of M92 is satisfactorily fit

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by the models, in particular the regionMV = 6 − 7, where the models do not depend on the adopted boundary conditions or on the convection. The slope between the FK and SK also shows a good agreement, although the observations do not reach low enough luminosities to help discriminating between present and BCAH97 models. The comparison with Piotto (1998, priv.

comm.) data in the passbands 606 and 814 is shown in Fig. 25, and is also very good in the MS regionM_V = 6 − 7, but the region between the FK and SK is somewhat more discrepant.

For NGC 6397, again the upper part of the MS is well reproduced. The region between the two kinks has a slope which looks more in line with the BCAH97 models. Also notice that a slightly larger reddening [E(B − V ) = 0.18] would provide a better consistency of the BCAH97 sequence with the data.

Remember that, while the SK occurs where the MM EoS is employed for the interior, the FK occurs at masses for which our models employ in the interior mainly the modern Rogers et al. (1996) EoS.

In the case of M4 the upper MS reasonably fits, although not so well as for the other clusters. However, the data have larger errors than the rest of the sequence, probably being closer to saturation. The low MS is not reproduced well, either by our models, or by BCAH97 ones. This discrepancy of the lower main sequence can be due to several reasons: in part it may reflect problems in the calibration of the HST photometry, or its conversion to the standard system. Problems in fitting precisely the lower main sequence in clusters of moderate metallicity have already been noticed by Cassisi et al. (2000) for the data of the cluster NGC 6752. One should also remember that the present generation of model atmospheres does not reproduce well the population I in theV versus V − I plane (Baraffe et al. 1998) so that the poor fit with NGC6752 (Cassisi et al. 2000) and our poor fit with M4 may be the first sign, at low metallicity, of this discrepancy, which might be imputed to some missing opacity in the optical range.

Therefore, it is not yet clear if we should regard the superior agreement of BCAH97 models in the case of NGC 6397 as a strong point in favour of their EoS — compared both to the Rogers et al. (1996) EoS for what concerns the FK morphology, and to the MM EoS for the SK, or if the missing opacities, which prove to be so important at larger metallicity, begin to have an effect already at [M/H]=–1.5. We discuss further the subject by looking at the mass function issue.

4.2. NGC 6397: Luminosity and mass functions

As above shown (Fig. 21), although differences in the models due to the EoS adopted are not too large, different computations, even with similar physical inputs (EoS and atmosphere models), provide quite different MLRs close to the low-mass end. Therefore we expect differences to arise in the computation of the MF from the luminosity function (LF) of the low main sequence stars in GCs. Differences may be subtle, as they are due not to the precise MLR, but to its derivative. Notice that we will deal only with the present day MF, which can be modified with respect to the initial MF due to the cluster dy-

namical evolution. In fact, what we are interested in by now is the ‘continuity’ of the MF and not its precise behaviour.

We use the present, non-grey stellar models for masses from 0.7 Mdown to the lower end of the main sequence to derive the MF of the nearby GC NGC 6397. Second-epoch Hubble Space Telescope observations of this cluster allowed the separation of the faintest cluster stars from the field (King et al. 1998), thus extending the luminosity function of this GC far enough to approach the limit of hydrogen burning (HBMM) on the main sequence. The main sequence (MS) of this cluster seems to peter out at the faint end. This is certainly the best available observational LF to be compared with the theoretical models down to the lowest masses. King et al. (1998) discuss with care the possible errors in the determination of their LF, especially in the lowest luminosity bins. The LF shows a rapid decline at the low mass end. This is confirmed by the NICMOSJ and H band observations by De Marchi et al. (2000), but the detailed behaviour at the low mass end can only be looked at from the optical data, which, further, have the quoted advantage of having been separated from the field stars by proper motion selection.

Thus, while keeping in mind that further observational work will be necessary to completely assess this problem, we will assume, for the purpose of a full discussion, that the formal error bars given in the LF of King et al. (1998) are fully reliable.

As a general argument we state that, if the MLR is correct, the resulting MF should behave smoothly near the HBMM, because the process of star formation is unaware of the later restric- tions imposed by the physics of energy generation. Though fun- damentally qualitative, this smoothness criterion is a potentially sensitive test (see the “golden rule” given by D’Antona 1998).

Because of the way that the LF maps into the MF, i.e. MF = LF

× slope(MLR), a smooth MF can result from the NGC 6397 LF only if the stellar models close to the HBMM provide a steep enough MLR. King et al. (1998) and King & Piotto (2000) derive the MF at the low-mass end using the MLRs from BCAH97 and from Alexander et al. (1997). In both cases they find that the MF drops at the latest two points of the luminosity function, around 0.1 M. Even if it is not possible to exclude that the MF is really bending at 0.1M, King & Piotto (2000). also share our point of view and consider that such a feature in the MF is less likely than an insufficient slope of the model MLRs.

This line of reasoning does not imply that the global MF of this (or other) clusters has a particular shape, e.g. a unique power law for the whole range of masses involved. Actually, recent work by Paresce & De Marchi (2000) attempts to un- derstand the LFs of several GCs in terms of log-normal mass function. This approach to the “global” MF, from the turnoff down to the lowest masses we are interested in, is not relevant for our present purposes: a log-normal distribution can be in any case subdivided into pieces of power law distributions having different exponents for each mass interval (see e.g. the discussion by De Marchi et al. (2000) relative to the MF of NGC 6397).

We only make the hypothesis that, if the MF has a given expo- nent close to the HBMM, we do not expect it to change abruptly in a small interval of masses adjacent to it.

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The LF of NGC 6397 has been analyzed by Silvestri et al. (1998) who derived the MF both by adopting their own low mass models based on grey atmospheres and the non–

grey BCAH97 models. In both cases they show that the bulk of the MF can be described by a unique power law of the form d N/dM = kM^−(x+1)with indexx = −0.5, apart from the most luminous bins close to the turnoff, which would require a smallerx, and apart from the latest mass bins.¹

Here we repeat the derivation of the MF by use of the present models. Theoretical LFs are computed for agest₀ = 10, 12, 14, 16 and 18 Gyr for several power-law indices (x = 0, x = 0.5 and x = −0.5), and for two values of metallicity in the range appropriate for this cluster (Z = 3 × 10⁻⁴and6 × 10⁻⁴, corresponding to [M/H]=–1.8 and -1.5). We show results for observational LFs evaluated using

d N dm_I = dN

dM ×

dM dm_I

t0

. (2)

The upper panel in Fig. 28 shows the comparison with the LF for NGC 6397. To convert the apparent to absolute magnitudes, we have first assumed distance moduli in the range(m−M)_I = 12. − 12.5. The distance modulus can have a significant impact on the resulting MF because its value controls the alignment of the theoretical slopes with the observed points. The LFs with x = 0.5 and 0.0 are not able to fit the observational results for any value of the distance modulus, therefore they are ruled out.

Among the results forx = −0.5 we select the distance moduli which provide the best fits for both values of the metal content:

these values also agree with those determined by fitting the HR diagrams (D’Antona et al. 2000, in preparation). In the upper panel the results are shown for each of the four combinations of distance moduli and metal content which provide similar quality of fits. The theoretical LFs (TLFs) agree quite well with the observational luminosity function (OLF) except for two points that are clearly outside the theoretical predictions, at∼ 0.7 M (close to the turn-off) and at the point corresponding to the last non-empty bin (∼ 0.1 M). The discrepancy at the turn-off agrees with the interpretation by Paresce & De Marchi (2000) that a unique power law actually cannot describe the MF in the entire mass range from 0.8 M to the HBMM. At the lowest luminosity bin (mI ∼ 24) the problem looks different: the TLF does not drop rapidly enough.

These disagreements are translated into the corresponding features in the MF, as shown in the lower panel of Fig. 28: the broad features are similar in all the cases, rising gradually with a small slope and dropping abruptly for the last observational point. The discrepancy at logM/M ≈ −0.15 corresponds to the turn-off region where the MF could actually change slope.

1 The result by Silvestri et al. (1998) on the substantial consistency of the MF of NGC 6397 with a unique power law is attributed by DeMarchi et al. (2000) to the use of grey models, but Silvestri et al.

show in their Fig. 14 that the same result is obtained with the BCAH97 non-grey models. They further show that the deviation of the four latest points from a power law obtained by Chabrier & M´era (1997) is in part due to their use of a very short distance modulus.

X=-0.5

10Gyr 12Gyr 14Gyr 16Gyr 18Gyr

NGC6397 t=12Gyr

t=14Gyr

t=12Gyr

t=14Gyr

X=-0.5

Fig. 28. Upper panel: Theoretical luminosity function from our non- grey models for different values of age, distance modulus and metallicity. The points correspond to the observational luminosity function for NGC 6397 by King et al. (1998). Lower panel: Mass functions derived from the theoretical luminosity functions in the upper panel.

Adopting our models, only one point (the last one) of the OLF does not match into a slowly rising MF, while the last two were not matched in previous MF determinations.

The striking decline of the OLF at the faint end is almost completely compensated by the steepness of the theoretical MLR near the HBMM apart from the latest bin.

Whereas King et al. (1998) found that the MFs obtained using the MLRs from BCAH97 and Alexander et al. (1997) drop suddenly in the last two points, our new MLR maintains the same slightly rising tendency and drops abruptly only in the last point. This change of slope happens in a range of masses