3 Short discussion of the main results

3.1 HR diagram and lifetimes

The HR diagrams for pre-main sequence evolution and for the following phases are given in Figs. 1 and 2 respectively for both metallicities. For each stellar mass, Table 1 displays the lifetimes in the contraction phase and in the deuterium- and hydrogen-burning phases. Note that we did not complete the main sequence evolution computations for the less massive stars which have a H-burning phase longer than the age of the universe; for these stars, our last computed model corresponds to an age of 20 Gyr.

• In the stellar mass range we consider, the ignition of deuterium burning (indicated in Fig. 1) takes place in a fully convective interior. During pre-main sequence evolution, a radiative core develops. However a proper radiative branch is absent, since these stars maintain a convective envelope all along contraction until the ZAMS has been reached, and further on.
• From Table 1 one can see that the contraction time lasts less than 4 thousandths of the hydrogen-burning stage.
• Due to opacity effects, the entire evolution (pre-main sequence, main sequence and red giant branch) occurs at higher luminosity and effective temperature for a given stellar mass when the initial metallicity is smaller, or when the hydrogen content is lower for the same initial metallicity.
• As a consequence, the contraction phase and the deuterium- and hydrogen-burning phases for a given stellar mass are shorter at lower metallicity (see Table 1), and at lower hydrogen content for the same value of Z.

  

Figure 1: Theoretical HR diagrams for pre-main sequence evolution for all our models (from 0.4 to 1.0 ${M_{\odot}}$) for both metallicities. Triangles mark the ignition of deuterium burning. The radiative core appears at the square location, and the development of a small convective core is indicated by the circles (for Z=0.020, the tracks are given only for the models with Y=0.30 to avoid confusion)

  

Figure 2: Theoretical HR diagrams for stellar masses between 0.6 and 1 ${M_{\odot}}$, and position of the zero age main sequence (dashed line) for the entire mass interval

3.2 Influence of the equation of state

When we first compare the results obtained with the MHD and with the simple Geneva equations of state (see Figs. 3 and 4, and Table 2), we obtain essentially the same results than Lebreton & Däppen (1988). Firstly, the fact that MHD contains ${\rm H}_2$ molecules, and the simple Geneva code does not, is reflected in a shift essentially along the ZAMS. On the other hand, the Coulomb pressure correction, also contained in MHD, causes a slight shift of the ZAMS, clearly visible for higher masses, where there are no hydrogen molecules in the photosphere. This Coulomb effect has been well discussed in the case of helioseismology (e.g. Christensen-Dalsgaard et al. 1996). Conformal to the effect of the MHD equation of state to push the apparent position on the ZAMS upward, it also decreases the lifetime on the ZAMS (see Table 2).

  

Figure 3: Influence of the MHD equation of state on the main sequence evolutionary track of the 0.8 ${M_{\odot}}$ models. The solid, dotted and dashed lines correspond to models computed with the MHD, OPAL and the simple Geneva (Sect. 1) equations of state, respectively

  

Figure 4: Influence of the MHD equation of state on the main sequence evolutionary track of the 1.0 and 0.8 ${M_{\odot}}$ models for Z=0.020. The full and dashed lines correspond to models computed with the MHD and with the simple Geneva equations of state, respectively

 

Table 1: Lifetimes in contraction and nuclear phases (in units of 10⁶ yr), and ratio of the contraction time $t_{\rm c}$ to the hydrogen burning time $t_{\rm H}$. For the less massive stars which have a main sequence phase longer than the Hubble time, we stopped the computations at an age of 20 Gyr

 

Table 2: Lifetimes on the main sequence (in units of 10⁶ yr) for models computed with the MHD, the OPAL and the simple Geneva equation of state

For comparison, we have computed with the OPAL equation of state two 0.8 ${M_{\odot}}$ models (the lowest mass that can be computed with the current OPAL tables), for both metallicities. As can be seen in Fig. 3, the corresponding tracks are very close to those obtained with the MHD equation of state, the use of the OPAL equation of state leading to slightly higher effective temperature on the ZAMS. As far as their internal structure is concerned, the models computed with MHD equation of state have slightly deeper convection zones. The main sequence lifetime obtained with the MHD equation of state is slightly higher than the one obtained with the OPAL equation of state (Table 2). The comparison shows that down to 0.8 ${M_{\odot}}$ all is fine with the MHD pressure ionization. As mentioned in the introduction, Trampedach & Däppen (1998) predict a correct functioning of pressure ionization in MHD even for much smaller masses. With the present comparison, we have validated their prediction at least to 0.8 ${M_{\odot}}$.