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Astron. Astrophys. 364, 879-886 (2000)

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5. Results and comparison

Coulomb coupling and pressure ionization lead to influence of nonideality on the equation of state. In order to examine the importance of nonideal effects on the EOS and compare the results with those of other EOS, we have employed four solar models for this paper. All models have been constructed with OPAL opacities (Iglesias et al. 1992) at [FORMULA] and Kurucz tables (Kurucz 1991) at lower temperatures. Table 1 summarizes the main properties of the models and displays the values of the ratio [FORMULA] of metallicity to initial hydrogen abundance, the central temperature and density. The nonideal effects in the different equation of state are listed in Table 2.


Table 1. Reference solar models


Table 2. Non-ideal effects in the equation of state

The simplicity of our improved EOS formalism allows us to obtain the solution easily and investigate the properties of thermodynamic functions. Contributions of Coulomb coupling, pressure ionization and electron degeneracy to the total pressure can be seen in Fig. 1. The ratio of non-ideal part to ideal pressure, e.g., [FORMULA], is plotted in Fig. 1 with respect to [FORMULA] of temperature from surface to the center throughout the solar model. Here [FORMULA] is the total pressure of Model 1 employed EFF EOS without including any nonideal effects, and [FORMULA] is any of the fractional contributions to the pressure calculated according to Eqs. (38) and (48). In Fig. 1 the dashed line indicates the Coulomb pressure for fully ionized H-He mixture, the dotted line is the Coulomb pressure under Debye-Hückle approximation, and the solid line is the pressure correction due to Coulomb coupling and pressure ionization for the partially ionized plasma.

[FIGURE] Fig. 1. The fractional contribution arising from nonideal effects to the ideal pressure of the reference solar model 1 which employed the EFF EOS. The solid line refers to the calculation of pressure correction including Coulomb coupling and pressure ionization for the partially ionized plasma, in the sense [FORMULA]; the dashed and dotted lines denote pure Coulomb pressure correction, [FORMULA], for the fully ionized plasms by means of the present formula and Debye-Hückel approximation, respectively.

Comparison of the Coulomb pressure corrections in Fig. 1 shows the following points: (1) At the solar surface, either the present formula or the Debye-Hückle approximation overestimates Coulomb effects. Contribution of the Coulomb potential to the total pressure is maximum ([FORMULA] and 4.7), at which point the most abundant species (hydrogen and helium) are completely and singly ionized, respectively. This means that the Debye radius is sufficiently small. (2) In the intermediate regions, the Debye-Hückle approximation also overestimates the Coulomb effects. (3) As the temperature and the density increase, the Coulomb contribution becomes smaller, until partial degeneracy of the electrons sets in near the center of the Sun. At the centre, the values of Debye-Hückle term without electron exchange contribution (dotted line) are smaller than our calculated values. As a result, we may conclude that the terms of higher order in the Coulomb correction play an important role even under the weakly-coupled limit.

Pressure ionization is caused by interparticle interactions and contributes to the pressure [FORMULA]. The fully ionized ions, e.g., [FORMULA] and [FORMULA], have no contribution to the pressure, because they have no bound systems interacting with surroundings. The comparison of solid line and dash line in Fig. 1. shows the difference of the pressure corrections mainly at low temperature. The contribution from pressure ionization starts to play a role and increases the pressure before pressure ionization takes place. In contrast, Coulomb correction becomes important only after pressure ionization takes place. As the degree of ionization increases, a great number of charged particles are produced, therefore fractional change, i.e., [FORMULA], increases a few percent. The increase of the pressure due to pressure ionization compensates the decrease of the pressure due to Coulomb coupling in the region of pressure ionization. Consequently, the pressure correction which corresponds to a partially ionized plasma is smaller than that of a fully ionized gas. The Coulomb coupling makes a negative pressure correction with respect to the ideal-gas value, while pressure ionization makes a positive contribution to it.

Our improved EOS has simple and explicit expression for its practical use, however, its reliability requires comparison with the EOS data calculated by other EOS. Fig. 2 shows the profile of [FORMULA] for the two different formalism of the EOS with Model 1 as the reference model, where the relative pressure difference is [FORMULA] with the pressure labeled "s" corresponding to our EOS or the MHD EOS. In Fig. 2, we can see that the MHD and our improved EOS are similar throughout the Sun except near the surface. This discovery reveal that the Coulomb interaction term is the dominant nonideal correction in solar interior. Comparision of the pressure corrections in Fig. 2 shows that the central pressure is smaller in our EOS than in the MHD EOS. Although the MHD EOS takes into account the Coulomb correction to the pressure in the Debye-Hückel approximation, it fails to account for the contributions of the electron-electron interaction and screened potential in the weak coupling central regime. At the surface, the MHD EOS includes several factors ignored by us, e.g., the formation of [FORMULA] and [FORMULA], and excited states in bound systems. Therefore, it seems much more realistic than our simple EOS.

[FIGURE] Fig. 2. The relative pressure difference with respect to Model 1 with the EFF EOS. The solid and dashed lines correspond to the pressure change [FORMULA] and [FORMULA], respectively.

The adiabatic sound speed [FORMULA] is very sensitive to changes in the equation of state. The relative difference in the sound speed with respect to the depth inside both models are shown in Fig. 3. Comparision of the sound speeds corresponding to the EFF EOS and our EOS shows that the computation of solar model has been truly improved through improved equation of state in physics. From Fig. 3, we also can see that the overall agreement between our analytical formalism and the results obtained by MHD is rather good. The divergence is found mainly at the surface because of the occurrence of many complicated physics processes. Under conditions of weak coupling and weak degeneracy, however, our EOS yields quite reasonable results and time-saving calculations.

[FIGURE] Fig. 3. The relative difference in the sound speed against fractional redius between Model 4 and 2 (solid line), between Model 4 and Model 3 (dotted line) and between Model 4 and 1 (dashed line).

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001