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2. The grids and the tidal friction in close binary systems

2.1. The models

The mass range covered is from 1 to tex2html_wrap_inline1112 for the three chemical compositions (X, Z) =(0.55, 0.03), (0.65, 0.03) and (0.75, 0.03). I denote these grids hereafter as X55, X65 and X75 respectively. Note that for X75, the abundance of helium is 0.22. This value is a little bit smaller than that used as primordial abundance of helium (about 0.24). However, this grid is useful to permit interpolations among stellar properties and isochrones as explained in the Introduction of the paper Claret & Giménez 1995. The value of Y for the models X65 is obtained by the relation tex2html_wrap_inline1144, withtex2html_wrap_inline1146 (the primordial He abundance) and tex2html_wrap_inline1148 taken equal to 0.24 and 2.7 respectively; two other values of Y were considered to take into account the uncertainties in tex2html_wrap_inline1152 and in the derivative tex2html_wrap_inline1148. In Figs. 1 (click here)-6 (click here) I show the HR and log tex2html_wrap_inline1156 diagrams. Figures 4 (click here), 5 (click here) and 6 (click here) are especially useful to be compared with binary star data given that the masses and radii in this kind of stars are generally the best determined parameters.

From the comparison among these figures one can note the influence of the mean molecular weight on the position of the tracks. Let us use simple homology relations, for example ideal gas and constant opacity, to interpretate such diagrams. Under such conditions, the radius of a model is proportional to tex2html_wrap_inline1158 where tex2html_wrap_inline1160 is the mean molecular weight, tex2html_wrap_inline1162 is the exponent of the density and tex2html_wrap_inline1164 is the exponent of the temperature in the nuclear energy generation rate. For the pp chain tex2html_wrap_inline1164 tex2html_wrap_inline1168, while tex2html_wrap_inline1170. For a fixed mass, it is easy to see that the models X55 - large mean molecular weight - have larger radii and smaller tex2html_wrap_inline1172 than the corresponding X75 grid. The aspect of Fig. 7 (click here) can be interpretated using the following argument: let be fXM the fraction of the stellar mass for which hydrogen is converted into helium during the lifetime of the star, being f smaller than 1, X is the initial hydrogen content per mass and M the total mass. The corresponding lifetime in the main-sequence will be given by the ratio of the total energy radiated to the average luminosity. In this way, tex2html_wrap_inline1178 fXM/L. A more direct comparison concerning the influence of the molecular weight on the tracks can be seen in Fig. 7 (click here) where the lifetimes of the hydrogen burning is shown for the three grids as a function of the logarithm of the stellar masses. An increase of 0.10 in the initial hydrogen content implies that the corresponding lifetimes are increased by about 1.7 times. On the other hand, Fig. 8 (click here) shows the effects of changing the chemical composition on the effective temperatures for the models at TAMS, also as a function of the logarithm of the stellar masses. Note that a variation of 0.10 in the initial helium abundance yields a difference about 0.08 in tex2html_wrap_inline1180.

Figure 1: HR diagram for the grid X55. Numbers attached denote tex2html_wrap_inline1182

Figure 2: HR diagram for the grid X65. Same remarks a Fig. 1 (click here)

Figure 3: HR diagram for the grid X75. Same remarks as Fig. 1 (click here)

Figure 4: tex2html_wrap_inline1184 diagram for the grid X55. Same remarks as Fig. 1 (click here)

Figure 5: tex2html_wrap_inline1184 diagram for the grid X65. Same remarks as Fig. 1 (click here)

Figure 6: tex2html_wrap_inline1184 diagram for the grid X75. Same remarks as Fig. 1 (click here)

Figure 7: The lifetimes for the hydrogen-burning phase for the three grids. The grid X55 is represented by a thick continuous line, X65 by a dashed one, and the continuous thin line denotes the grid X75

Figure 8: The effective temperatures at the TAMS for the three grids. Same line types as in Fig. 7 (click here)

2.2. Tidal evolution

Concerning tidal evolution, there are three mechanisms which are being used to study the circularization and synchronization of orbits in CBS. The first two mechanisms, which are based on the classical tidal theory, depend on the characteristics of the stellar core and envelope. On the other hand, Tassoul claimed that the distortions present in CBS cause hydrodynamical currents which tend to circularize their orbits. However, there is a strong discussion on the validity of the hydrodynamical mechanism (Rieutord 1992; Tassoul & Tassoul 1996). I do not discuss here this controversy. The interested readers can consult the original papers quoted above. For a comparison of predictions of the hydrodynamical and tidal-torque mechanisms with observations see Claret et al. (1995) and Claret & Cunha (1996).

I shall concentrate our attention on the first two mechanisms. A star which belongs to a binary system is under the action of the disturbing potential due to its mate. This star reacts to the external potential through equilibrium and dynamical tides. The characteristic time scales for synchronization and circularization in the case of turburlent dissipation and radiative damping are given respectively by (see for example Zahn 1975 and 1989)





where the times scales are in years, the orbital period P in days and the mass, radius and luminosity are in solar units. The subscripts circ and sync refer to circularization and synchronization while turb and rad refer to turburlent dissipation and radiative damping respectively. The rotational angular frequency is represented by tex2html_wrap_inline1212 while e is the eccentricity of the orbit, q the mass ratio, tex2html_wrap_inline1116 the radius of gyration and tex2html_wrap_inline1220 is the orbital angular frequency. The meaning of E2 and tex2html_wrap_inline1120 will be explained in the next paragraphs.

The few papers that treat the oscillations driven by the tides do not consider the dissipation in the mentioned process (e.g. Cowling 1941). In more recent years Zahn (1975) studied the effects of the radiative damping in the dynamical tide. This mechanism, radiative damping, is the responsible for the synchronization and circularization of the orbits when the stars in a given system present convective cores and radiative envelopes. The gravity waves, due to the external potential, are damped in the external layers. This damping occurs because the radiative cooling time is of the order of the tidal period and in this way part of the angular momentum of rotation is transfered to the orbital motion. In order to characterize the time scales I use tex2html_wrap_inline1226. Such a parameter can be evaluated using the following equations:
where tex2html_wrap_inline1228 is the gamma function, x is the normalised radius of the configuration, the symbol f denotes the border of the convective core, s indicates surface values, the prime denotes the derivative with respect to x, R is the radius, M the mass, g the gravity and B is given by
where -gB is the square of the Brunt-Väisälä frequency.

The coefficient tex2html_wrap_inline1240 is given by
X is the solution of the differential equation
and Y(1) is the solution of the Clairault equation. The differential equations were solved for each configuration using a fourth order Runge-Kutta method. In this Paper I shall restrict the calculations to n = 2 since for larger n the contribution to the dynamic tide is very small. Such calculations deserve special attention. Let us rewrite -gB as
where tex2html_wrap_inline1252 is the usual tex2html_wrap_inline1254 ratio, G is the constant of gravitation, r the radial distance, Mr the mass at the distance r, P the total pressure, tex2html_wrap_inline1266, tex2html_wrap_inline1268 is the adiabatic gradient and tex2html_wrap_inline1270. As a massive star evolves the convective core recedes. The zone between the initial and the present convective core present a large chemical compositon gradient. In such zone the spatial gradient of chemical composition increases more and more and becomes wider. This feacture is well known when one analyses a diagram of propagation. As shown before, the tidal constant depends on the derivative of the Brunt-Väisäla frequency in the border of the convective core and the consequent numerical oscillations are clear in Fig. 9 (click here).

Figure 9: Schematic evolution of E2 (thick line) and tex2html_wrap_inline1120 (thin line) for a tex2html_wrap_inline1276 model with X=0.75 and Z=0.03. The apsidal motion constant is denoted by a dashed line. The attached labels are useful to identify the corresponding evolutionary phases since that in the lower left corner I represent the corresponding HR diagram

For stars with envelopes in convective equilibrium another mechanism was identified: the turbulent dissipation. This process is characterized by the parameter tex2html_wrap_inline1120 though in the last years the apsidal motion constant k2 was used as an approximation. In fact, Fig. 3 (click here) by Claret & Cunha (1997) shows that it is an acceptable approximation. A new approach for tex2html_wrap_inline1120 was obtained by Zahn 1989 using the mixing-length theory. He obtained
where tex2html_wrap_inline1288 denotes the bottom of the surface convective zone, tex2html_wrap_inline1290 is the mixing-length parameter and E describes a polytropic envelope. I have evaluated it through the equation
where tex2html_wrap_inline1294 is the mass of the convective envelope. The approximation described above is valid when the convective turnover time is smaller than the tidal period.

In Fig. 9 (click here) we can see the behaviour of the tidal torque constant and tidal coefficient for a tex2html_wrap_inline1276. Thick continuous line represents the evolution of E2, thin continuous line the evolution of tex2html_wrap_inline1120 and dashed line denotes the evolution of k2. The tidal torque constant decreases quickly as the model evolves. From homogeneous model to the hydrogen exhaustion it decays about 4 magnitudes. The contribution of the equilibrium tide is very small in the main-sequence, as expected. In this specific case we have computed this contribution only for models with effective temperatures below 8000 K. This contribution increases with the depth of the convective outer layers. A rapid analysis of Fig. 9 (click here) during the phases d-f shows that indeed the apsidal motion constant k2 is a good approximation to tex2html_wrap_inline1120.

In the next papers of this series we will analyse the influence of the chemical compositions and core overshooting on the tex2html_wrap_inline1120 and E2. The present models are available in electronic form at the CDS or directly from the author. Also, I can provide to interested readers specific calculations or details which were not given here.


Drs. M.A. López Valverde and G. Meynet are gratefull acknowledged by their useful suggestions. The Spanish DGYCIT (PB93-0134) is gratefully acknowledged for support during the development of this work.

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