3 The grids and the structural parameters

The basic information on the input physics of the models can be found in Claret 1995 (Paper I). The masses of the models shown here range from 1 to 40 $M_{\odot}$. Convective core overshooting was taken into account for models with . For the adopted initial metal content of Z = 0.004, we have computed models with an initial content of hydrogen of 0.65, 0.744 and 0.80. The middle value of 0.744 was selected on the basis of the enrichment law $Y = Y_{\rm p} + (\Delta Y/\Delta Z)$ Z for a primordial helium abundance of 0.24 and an enrichment ratio of 3. The grids with X=0.65 and 0.80 provide the necessary range for parametric studies and interpolation of different chemical composition the latter being obviously artificial from the astrophysical point of view since the corresponding helium content is not compatible with the generally accepted values of the primordial helium abundance. However, it is useful to interpolate the properties of models with intermediary chemical composition.

Figures 1-3 show our usual HR diagrams for the three grids (X65, X744, X80) where the influence of the mean molecular weight on the morphology of the tracks is clear. In fact, by using simple homology relationships it is possible to understand qualitatively the behaviour of radius, effective temperature and luminosity of the models (see previous papers of this series).

The less massive models were computed until just before the helium flash, while the more massive ones were followed up to the core helium burning (in some cases, even until the first stages of carbon burning). However, for the sake of clarity, the blue loops for the more massive models were not plotted in Figs. 1-3. As it is well known, lower metallicity models present more extended blue loops, and we wanted to check their behaviour for fixed values of the mass and the metal content. Figure 4 shows the effect of changing the initial abundance of helium. The extension of the blue loops is presented for different values of initial helium for models of a 6 solar masses star. The blue loops are more extended as the value of Y increases. The existence and extension of blue loops depend on factors such as the central helium burning time scale, the corresponding one for shell burning and, of course, on the available amount of helium. Results shown in Fig. 4 were obtained by integration of the stellar structure equations using a grid of almost 4000 points in order to avoid problems with the chemical composition profiles.

  

Figure 1: HR diagram for the grid X65. Numbers attached denote $\log M$

  

Figure 2: HR diagram for the grid X744. Same remarks a Fig. 1

  

Figure 3: HR diagram for the grid X80. Same remarks as Fig. 1

  

Figure 4: Influence of helium content in the blue loops. Models with 6 $M_{\odot}$. Solid line denotes model with (X, Y) = (0.55, 0.446); dotted indicates (0.60, 0.396) and dashed line represents models with (0.65, 346)

  

Figure 5: The lifetimes for the hydrogen-burning phase for the three grids. The grid X65 is represented by a thick continuous line, X744 by a dashed one, and the continuous thin line denotes the grid X80

  

Figure 6: The parameter $\lambda_2$ as a function of the age for models of 1 $M_{\odot}$ during the main-sequence. The continuous line represents a model with X=0.65, dotted one represents a model with X=0.744 and the dashed line denotes a model with X=0.80

  

Figure 7: Effect of the overshooting on the tidal constant E2 during the main-sequence. Model with core overhooting ($\alpha_{\rm ov}=0.20$) is represented by continuous line and dashed line denote model without overshooting. Both models are for X=0.744 and the masses are 10 $M_{\odot}$

We can see in Fig. 5 the dependency of the main-sequence lifetime on the initial hydrogen content for the three grids presented here. As expected, for a fixed metallicity, the largest lifetime corresponds to the largest initial hydrogen content.

Concerning the structural stellar parameters, we have computed the apsidal motion constants (integrating the Radau equation), the moment of inertia and the potential energy. As commented in Paper II we have also introduced in our stellar evolutionary code the corresponding equations to compute internal parameters which are used to calculate the synchronization and circularizations times in binary systems. These parameters are related with the equilibrium and the dynamical tides. In the case of the equilibrium tides the turbulent dissipation was identified as the agent of synchronization and circularization of orbits in binaries presenting convective envelopes. In order to compute the time scales for synchronization and circularization, we have to calculate the constant $\lambda_2$ which depends on the physical properties of the stellar envelope. Indeed, following the mixing-length approximation, the constant $\lambda_2$ can be written as (Zahn 1989)

$${\lambda_{2}} \propto {\int_{x\rm _b}^{1}} {x^{22/3}}{(1-x)^2 {\rm d}x}$$ (1)

where $x\rm _b$ indicates the bottom of the convective envelope in normalized units. Some years ago, the apsidal motion constant k₂ was used instead of $\lambda_2$. As we have shown in previous papers (Claret & Cunha 1997, Fig. 3 and in the Paper II, Fig. 9) this is a good approximation, at least during the Main-Sequence. Figure 6 shows how $\lambda_2$ depends on the chemical composition. Since this parameter depends on the depth of the convective envelope, the hydrogen content drives its behaviour. It is interesting to note that within the mixing-length theory the differences due to changes in the initial chemical composition can reach almost an order of magnitude.

The mechanism used to investigate the tidal evolution of stars with convective core and radiative envelopes is the radiative damping which is characterized by the tidal torque constant E₂. The calculation of such a parameter is more problematic since it depends, among other intermediary calculations, on the derivative of the Brunt-Väisäla frequency just in the boundary of the convective core. For little evolved models, near the ZAMS, the computations are relatively simple. When a star in these conditions evolves, the convective core recedes and a large chemical composition gradient appears. Such gradients are responsible for the numerical difficulties to compute the tidal torque constant.

As E₂ depends strongly on the physical conditions just at the boundary of the convective core, one should expect differences between models with and without core overshooting. Figure 7 illustrates the situation for models with 10 $M_{\odot}$. The model computed adopting core overshooting present values of E₂ slightly larger than the standard one (about 0.1 dex as a average). This difference is not distinguishable at the present data quality of synchronization and circularization levels.

In order to maintain a coherent format with respect to the previous models of this series published in the WWW page of CDS at Strasbourg the values of $\lambda_2$ and E₂ are not given. Interested readers can send their request sending a message to claret@iaa.es.

Acknowledgements

The Spanish DGYCIT (PB94-0007 and PB96-0840) is gratefully acknowledged for support during the development of this work.