## 5. Results and discussionRecently 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 Two sets of the solar data are used in the calculations. In one set (D1), the age of the Sun is taken as , and its luminosity and radius are and , 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 and . The radius of the Sun is (Sackmann et al. 1993). In both of the sets the solar mass is . 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 , 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 is given
in the second column. The depth of the convective zone in terms of
solar radius, , is shown in the third column,
and base temperature of the convective zone, ,
is in the fourth column. Last five columns are the effective
temperature (), the central temperature
(), density(), abundance
of hydrogen by mass at the center (), and the
central pressure (), respectively (in
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 , and by if EC EOS is used. Increase in base temperature of convective zone is about with MHD EOS and 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 , and the base temperature of the convective zone is larger than 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 , 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 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 in case of CS, and 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 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 . 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 . 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 (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
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 . When one incorporates diffusion processes, Bahcall et al. (1995) emphasize that the decrease in mass fraction of He is about 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. © European Southern Observatory (ESO) 1997 Online publication: April 20, 1998 |