Astron. Astrophys. 345, 156-162 (1999)

4. Effects of microscopic diffusion on the structure of main-sequence models

Due to the gravitational settling, all species but 1 H sink; therefore as time goes on, the amount of hydrogen at the surface is enhanced while the amount of helium and heavy elements decreases. Then a first effect of microscopic diffusion is the decrease of the surface value of [Fe/H] with respect to time. The variation of Z modifies the opacity and then changes slightly the temperature gradient in the radiative zones, as well as the position of their boundaries.

In the mean time, the sinking of helium with respect to hydrogen produces a decrease of the mean molecular weight µ in the envelope as, assuming here a complete ionisation:

In the outer layers of a model computed with microscopic diffusion , and then as , see Fig. 1.

Fig. 1. Profiles of abundances, in mass, of hydrogen X (full), helium Y (dash), heavy elements Z (dot-dash) and mean molecular weight µ (dot) for two models evolved up to 10 Gy, respectively with (heavy) and without (thin) microscopic diffusion.

Fig. 2. The same, see Fig. 1, for normalized pressure (full), temperature (dash) and luminosity (dot-dash). The factors of normalization are respectively dyne cm-2 for the pressure, K for the temperature and erg cm-2 s-1 for the luminosity.

This variation of the mean molecular weight produces an increase of the radius and consequently a decrease of the effective temperature. But, the variation of the metal content introduces a concomitant variation of the opacity in the radiative regions, which tends to counterbalance this effect.

With the initial abundances given in Table 1, we have computed respectively sets of models, i) without diffusion, ii) with diffusion of helium only, iii) with diffusion of all species.

The main characteristics of four models of masses 0.85, 0.8 and 0.75, at the age of 10 Gyr computed without and with microscopic diffusion of all species are displayed in Table 2 and in Table 3 for the models of 0.7 and 0.6 ¹.

Table 2. Characteristic of models of (Q), (N), (R), (P) and (S) at 10Gy. The models evolved with microscopic diffusion are specified by the labels `dc' and `d' respectively for calibrated and non-calibrated (see text). is the bolometric magnitude, (); and are respectively the surface abundances, in mass unit, of hydrogen and helium; and are respectively the initial surface abundances, in mass unit, of hydrogen and heavy elements of calibrated models "dc"; and are respectively the surface values of [Fe/H] at zero age main-sequence and at 10Gy; in Kelvin, is the component on the axis of the global shift GS (see text) for 10 Gy.

Table 3. Same as Table 2 for models of (P) and (S).

During the main-sequence, models with diffusion are cooler than models without diffusion; this is essentially due to the changes of the mean molecular weight. As already stressed by Castellani et al. (1997) comparison with unphysical models, including diffusion of helium only, shows that a significant contribution to the displacement in the HR diagram is due to helium settling, in particular for low metal content as illustrated on Fig. 3.

Fig. 3. The main-sequence HR evolutionary tracks for 0.8, , without diffusion (thin full), with diffusion of helium only (dash-dot-dash), with diffusion of all elements (dashed), and finally with the calibration of surface [Fe/H] (heavy full) (the initial metallicity has been computed in order to achieve, at 10 Gyr, the initial surface metallicity of the other three models). On each evolutionary track the open circles correspond to ages of 3 Gy, 6 Gy and 9 Gy respectively. For 9 Gy, AB is the diffusion shift DS , BC the calibration shift CS and AC the global shift GS (see text).

In the following, we will call "diffusion shift" DS , the translation in the HR diagram from a "standard model" to a model including microscopic diffusion, at the same mass and the same age . It consists in:

• a reduction of the effective temperature, decreasing with mass from 100 K at 0.85, to almost zero at 0.6; the stronger increase when [Fe/H] decreases is explained by the larger influence of helium stratification at low metallicities;

• an augmentation of the luminosity of the order of 0.01 to 0.02 dex, almost independent of [Fe/H], very sightly decreasing with mass, almost certainly due to the higher temperature and higher helium content of the central core of models with diffusion.

The time dependence of the process shows up through the increasing separation between the tracks as evolution proceeds.

© European Southern Observatory (ESO) 1999

Online publication: April 12, 1999
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