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Astron. Astrophys. 326, 187-194 (1997)

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5. Results and discussion

Recently we have gathered very important information about the internal structure of the Sun by helioseismological investigations. Distance of the convective zone from the center of the Sun is determined from p -mode oscillations as [FORMULA] by Christensen-Dalsgaard et al. (1991). Photospheric abundance of He by mass is found as [FORMULA] (Hernandez & Christensen-Dalsgaard 1994). There are several attempts to determine the central density of the Sun from helioseismological data. Results obtained by Dziembowski et al. (1994) are in between [FORMULA], while Gough & Kosovichev (1988) quote [FORMULA] and Vorontsov & Shibahashi (1991) quote a range of [FORMULA].

Two sets of the solar data are used in the calculations. In one set (D1), the age of the Sun is taken as [FORMULA], and its luminosity and radius are [FORMULA] and [FORMULA], respectively. In the other set (D2) (case C below), the luminosity and the age of the Sun are taken from Bahcall et al. (1995), namely [FORMULA] and [FORMULA]. The radius of the Sun is [FORMULA] (Sackmann et al. 1993). In both of the sets the solar mass is [FORMULA].

For different combinations of EOS's and opacities and for different solar data, starting from threshold of stability point at which gravitational and internal energies are nearly the same, we constructed a series of evolutionary models for the Sun. Each model given in Table 1 is obtained by changing initial mass fraction of hydrogen, helium and convective parameter [FORMULA], in order to fit the Sun's luminosity and radius to the present data, with different accuracies. The first column in Table 1 represents the initial abundance of He by mass, and [FORMULA] is given in the second column. The depth of the convective zone in terms of solar radius, [FORMULA], is shown in the third column, and base temperature of the convective zone, [FORMULA], is in the fourth column. Last five columns are the effective temperature ([FORMULA]), the central temperature ([FORMULA]), density([FORMULA]), abundance of hydrogen by mass at the center ([FORMULA]), and the central pressure ([FORMULA]), respectively (in cgs). All the models, except Model C, is with D1. The first three models are constructed with an accuracy of a few percent, and the fourth model with an accuracy of [FORMULA]. Models A, B and C are more precise models (accuracy is [FORMULA]). The difference between the Model A and the Model B is the method of heavy element ionization. The latter (also Model 4) uses Henyey's method (MHD+H), the former is obtained by the method of Gabriel & Yldz (MHD+GY) as the Model C.

When we compare each of the first four models with the model of the same EOS but different opacity (Model 1 with 3, and Model 2 with 4), it is seen that OA opacities strongly influence the structure of the Sun. The enhancement of the opacity (OPAL) below the convective zone enlarges the zone's size toward the center of the Sun. When MHD is used, OA opacities enlarge the convective zone by about [FORMULA], and by [FORMULA] if EC EOS is used. Increase in base temperature of convective zone is about [FORMULA] with MHD EOS and [FORMULA] with EC EOS whenever OA opacities are employed in place of CS opacity. While He abundance by mass is exceeding the helioseismological result, the central density is about [FORMULA], and the base temperature of the convective zone is larger than [FORMULA] in the models with OA. Because of increase in opacity, less energy reaches the surface. This is compensated by an increase in He abundance. The convective parameter alpha increases by about [FORMULA], when OA is employed in place of CS.

If we compare Model 1 with 2, and Model 3 with 4, it becomes obvious that the depth and the base temperature of the convective zone is not very sensitive to the EOS. The values of He abundance and [FORMULA] are changed by EOS. Incorporation of the Coulomb interaction reduces the energy and pressure by some fraction. In order to compensate this reduction in pressure, mass fraction of He (or molecular weight per free particle) decreases by about [FORMULA] in case of CS, and [FORMULA] in case of OA opacities. The only difference between the ways that Model A (Gabriel & Yldz) and B (Henyey) are obtained, is the calculation method of the ionization of heavy elements and their internal energies. There are small differences between these models. The convective parameter [FORMULA] of Model A is a little bit larger than that of Model B, since the pressure scale height (the number density of electrons) of Model A in outer layers is less than that of Model B.

The small differences between Models B and 4 are due to low accuracy of Model 4. The Models A (D1) and C (D2) have different solar data. The Model C has lower luminosity and higher age than the Model A.

In Table 1, we also give models constructed by Charbonel & Lebreton (CL) (1993), Bahcall & Ulrich (BU) (1988), Turck-Chieze et al. (TCCD) (1988), Cox et al. (CGK) (1989) , Sackmann et al. (SBF) (1990), Lebreton & Däppen (LD) (1988), and Ciacio et al. (CDR)(1996). Similar to our models, all the models obtained by different authors are standard models, that is there is no rotation, no diffusion process and magnetic field is negligible. Except model of CGK, the initial mass fraction of He in Models A, B and C is a little bit larger than those found by other researchers. The extreme model of CGK, which has the largest value of He mass fraction, base temperature of the convective zone, and the central temperature, has an age of [FORMULA]. The EOS of CGK is very similar to EC EOS, and Iben's fitting formula (1975) is used for opacity. The model of CDR which has the same solar data as Model C except the solar mass and heavy element abundance, is obtained by using both the EOS and the opacity of OPAL. The significant difference between these two models is in He abundance due to their data of high solar mass and low metallicity. The other models take the age of the Sun as [FORMULA]. Therefore, their He mass fractions are closer to, but less than, that of our models with OA and MHD EOS. Almost all the models use the same nuclear reaction rates of Caughlan & Fowler. Model of CK, which uses OPAL opacity and tables of MHD EOS, is close to our best models. There is a small difference in the value of [FORMULA] (the difference in He mass fraction is due to different age of the Sun). This difference stems possibly from their different heavy element abundance and low opacity table. They use different initial mixture for heavy elements. Model of BU includes the Coulomb interaction in EOS and the Los Alamos Opacity (Cox et al. 1991).

The physical variables which determine the structure of the Sun is summarized in Table 2. These values (in cgs) are from the Model A which has the accuracy of [FORMULA] and is obtained by using MHD+GY and recent opacities. For accuracy of the model, number of shells is increased to 739, during the evolution. The energy that the Sun radiates is produced within the inner [FORMULA] core. Convective zone is between shells 256 and 727, where radiative temperature gradient exceeds adiabatic temperature gradient. Its mass is only [FORMULA] of the total mass of the Sun, and its distance to the center is [FORMULA]. The total internal energy in outer shells is negative, since binding energy of an atom having electronic configuration is considered negative.


Table 2. The present Sun model (Model A) with OPAL opacity and MHD EOS by using method of Gabriel and Yldz for ionization of heavy elements

Our conclusion is that solar models obtained by MHD EOS and OPAL opacities are in closer agreement with the results the helioseismology. The remaining small differences can be removed by taking into consideration the diffusion process of He and heavy elements. With MHD EOS and OPAL opacity, and with diffusion process, Basu & Thompson (BT) (1996) gives the distance from bottom of the convective zone to the center as [FORMULA]. When one incorporates diffusion processes, Bahcall et al. (1995) emphasize that the decrease in mass fraction of He is about [FORMULA] which gives better agreement with the observation. From Table 1, one sees that the main effect upon the initial He abundance is provided by the adoption of the recent opacities which tend to increase the abundance, while the MHD EOS has a smaller effect in the opposite direction

Structure of the Sun is not very sensitive to the methods of Henyey (Model B) and Gabriel & Yldz (Model A) for the ionization of heavy elements. But the method of Gabriel & Yldz is better than Henyey method for rapid fitting processes, and, near the surface of the Sun, it gives neutral heavy elements as expected.

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998