4. Theoretical results
We computed evolutionary sequences of stars with masses 0.5, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3 and from the top of the Hayashi track, corresponding to central temperatures of the order of K, until the models reached the zero age main-sequence. The one solar mass models were further evolved up to the present age of years, a value chosen according to the "Global Oscillations Network Group solar model comparison project" (Christensen-Dalsgaard 1991; Gabriel 1991). All the models were computed hydrodynamically (at constant mass) until central temperatures reached values of the order of K, and hydrostatically afterwards.
For comparison we consider five sets of models each with different input physics concerning the high- and low-temperature opacities, the EOS and the treatment of superadiabatic convection. Summarising:
These choices are meant to represent the main uncertainties in the physics used to compute the stellar models. Models M2 and M3 explore the effect of uncertainties in the opacity coefficients: OPAL opacities are in general higher than the LAOL ones, specially around 30 000 K where the differences amounts to a factor 2 to 3 for the solar metallicity (Rogers & Iglesias 1992). On the other hand, Neuforge low-temperature opacities can differ from the Kurucz set by some 15 (Neuforge 1993). Model M5 explores the uncertainties in the treatment of superadiabatic convection by considering the energy flux transported by turbulent convection according to the CM model. In our study we choose to adopt the CM model with rather than , where z refers to the distance to the top of the convective zone, because our model computed using fails to reproduce the effective temperature of the Sun by 6 (cf. Civelek & Kiziloglu 1995), which is incompatible with the methodology used (see further). Anyway, working with , apart from being easier from the numerical point of view, is likely to give all the same an idea of what we could find with a convection model different from MLT.
All theoretical uncertainties referred to so far affect directly the temperature gradient throughout the star and are expected to influence the position of models in the Hertzsprung-Russell diagram (HRD), specially their effective temperatures along the Hayashi track. The model M4 is intended to account for the uncertainties in the EOS. The inclusion of Coulomb corrections can reduce the pressure and energy at the center of the Sun by as much as 5 . On the other hand, the correct evaluation of the continuum depression due to Coulomb interactions is crucial for an accurate computation of the partition functions, which in turn determine the ionisation state of the gas, specially for low-temperature configurations like PMS low-mass stars. Uncertainties in the EOS, mimicked by neglecting the effect of Coulomb interactions, are thus expected to lead to relatively important uncertainties in the theoretical results.
Therefore, by comparing the results obtained with each set of models we expect to be able to evaluate the variations due to different input physics and/or convection algorithms. This allows to discuss their importance with respect to the observational uncertainties.
For each of these models we started out fitting the solar radius and luminosity at 4.75 109 years with in order to determine the values of the hydrogen and helium initial mass fractions, and the mixing parameter . We adopted (Ulrich & Rhodes 1983) and (Bahcall & Ulrich 1988). The precision achieved in the fitting of and is better than 0.2 and 0.5 percent, respectively. Table 2 presents the results obtained for these parameters as well as for other physical quantities concerning the center of the Sun and the lower boundary of the convective envelope. The table also includes the corresponding results obtained by Charbonnel & Lebreton (1993, hereafter CL) and Morel et al. (1995) for comparison.
Table 2. Parameters for the Sun using different models: hydrogen and helium initial mass fractions, convection parameter , metallicity Z, radius and temperature at the lower boundary of the convective envelope, central temperature, density and hydrogen mass fraction. CL93 stands for the Charbonnel & Lebreton (1993) solar model, while M95 stands for the Morel et al. (1995) solar model
Our models computed with the MLT lead to values around 1.5, while the model computed with the CM model, M5, has below unity. This agrees with the predictions of Canuto & Mazzitelli (1991). All these models have an initial helium abundance around 0.26, except the model computed without Coulomb corrections (M4) which has a higher value, 0.27. This result is due to the fact that Coulomb corrections decrease the pressure, specially in the central regions, and this has to be compensated by a decrease of the mean molecular weight so that the solar luminosity is still maintained at solar age (e.g. CL). For this reason models including Coulomb corrections present lower central pressures, densities and temperatures, as well as a lower initial helium content with respect to the models computed without Coulomb corrections.
The results obtained with our reference model show some important differences with respect to the CL model, specially concerning the initial hydrogen and helium mass fractions. Since the CL model was computed using a set of opacity tables very similar to the one we used for M3, and taking into account that the results obtained with M1 and M3 are basically the same, we conclude that the differences found cannot be due to the different opacity sets used. As to the EOS, the CL model uses the so-called MHD EOS (Hummer & Mihalas 1988; Mihalas et al. 1988; Däppen et al. 1988) which is based on the "chemical picture" and includes Coulomb corrections using the Debye-Hückel model. Thus, the results obtained with this EOS for the Sun should not differ in a significant way from the ones of our EOS. Consequently the differences found should be mainly due to the higher Z value used by CL.
On the contrary, the results obtained with the Morel et al. (1995) model are very similar to ours. The opacities used are the same as in the CL model, while the EOS uses the Eggleton et al. (1973) formalism with the inclusion of Coulomb corrections. The atmosphere is computed using laws derived from the atmosphere models calculated with the Kurucz (1991) ATLAS9 code. The differences found, specially concerning the value of the convection parameter , should probably be due to the differences in the computation of the atmosphere.
In Fig. 1 we compare the HRD location of models M2 to M5 relatively to the reference model, including the corresponding isochrones. We see that models M1, M2 and M3 are not very different from each other (Figs. 1a and 1b). The M1 and M3 tracks and isochrones are almost coincident. This results from the fact that the Neuforge and Kurucz opacities differ at most by some 10-15 for the range of temperatures of interest. Model M3 leads to tracks slightly redder than M1. On the contrary, M4 and M5 show considerable differences when compared with the reference model (Figs. 1c and 1d). M4 leads to hotter tracks, while M5 produces tracks clearly shifted towards lower effective temperatures for ages in excess of approximately one million years.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998