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Astron. Astrophys. 364, 157-164 (2000)

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4. Results and discussion

The Coulomb coupling leads to non-ideal effects in the EOS and modifies the thermodynamic functions. In the present work, we examine the contribution of the Coulomb coupling to the thermodynamic quantities of a fully ionized and weakly coupled H-He mixture, with abundances by mass of hydrogen and helium of [FORMULA] and [FORMULA], respectively.

Fig. 1 shows the total excess free energy [FORMULA] divided by [FORMULA] at [FORMULA] calculated according to Eqs (37), (42) and (60). In Fig. 1 we compare the value of [FORMULA] with values obtained from other theoretical expressions. As one would expect, our result is closed available representation of Stolzmann & Blöcker (1996) from a Padé approximation. For [FORMULA], the electron-electron exchange contribution is dominant; the Debye-Hückel approximation fails to account appropriately for the electron-electron exchange effects in the weak coupling regime, and hence it predicts a value of [FORMULA] even lower than the RPA values over a significant domain of [FORMULA]. It can also be seen from Fig. 1 that the simple Debye-Hückel approximation overestimates the Coulomb effects when the coupling becomes significant at moderately small [FORMULA]. However, it is easy to add the fitted formula to the Debye-Hückel approximation to obtain improved results.

[FIGURE] Fig. 1. Total excess free energy due to Coulomb coupling for a H-He mixture divided by the coupling parameter [FORMULA] at weak degeneracy [FORMULA]. The solid and dashed lines refer to the calculations of the present formula and Debye-Hückel approximation, respectively; the stars illustrate the Padé formula of Stolzmann & Blöcker (1996).

Fig. 2 shows the contributions of the electron-electron interaction [FORMULA] divided by [FORMULA] at [FORMULA] and 10. In Fig. 2 we note that the magnitude of the electron-electron exchange contribution decreases as [FORMULA] increases owing to reduction in the exchange energy. Since no electron exchange contribution (at fixed [FORMULA] regardless of [FORMULA] to the free energy is included in the Debye-Hückel approximation, the value of the Debye-Hückel term deviates widely from the present calculations. The result indicates that the electron-electron exchange effect and electron finite-temperature effect substantially modify the plasma properties.

[FIGURE] Fig. 2. Excess free energy arising from the electron-electron interaction [FORMULA] divided by [FORMULA] at [FORMULA] and 10, respectively.

The total excess pressure [FORMULA] due to Coulomb coupling is plotted in Fig. 3 at weak degeneracy, [FORMULA]. The Debye-Hückel pressure is calculated on the basis of a two-component plasma (TCP) for electron-ion interaction, and a one-component (OCP) for the electron and ion. The computed data are compared with the Debye-Hückel values. It can be seen that non-ideal contributions to the pressure increase systematically with increasing [FORMULA], and the Debye-Hückel approximation overestimates the Coulomb effects. Fig. 3 also reveals that the Coulomb coupling makes a negative contribution to the pressure term, and hence reduces the total pressure P.

[FIGURE] Fig. 3. Calculated Coulomb pressure [FORMULA] in units of [FORMULA], where [FORMULA], at weak degeneracy ([FORMULA]). Solid line: the present formula; dashed line: the Debye-Hückel approximation.

Fig. 4 shows the relative pressure, i.e., the ratio of the Coulomb pressure [FORMULA] to the pressure [FORMULA] corresponding to an ideal gas for the plasma parameters of the solar interior. The reference solar model was constructed with OPAL and Kurucz opacities (Iglesias et al. 1992; Kurucz 1991) and employed the EFF equation of state without any Coulomb term. In Fig. 4 we see, at the centre ([FORMULA]), the values of Debye-Hückel term without electron exchange contribution are smaller than our calculated values. On the other hand, in the intermediate regions, the results indicate a smaller Coulomb correction than that obtained with the Debye-Hückel theory. It seems that the terms of higher order in the Coulomb correction play an important role. Moreover, we find that the contribution to the pressure from the Coulomb correction is up to 1%, and thus it indeed reflects the fact that the Coulomb term has a significant effect on the thermodynamic properties of the plasma, at the centre and intermediate regions (Däppen 1998).

[FIGURE] Fig. 4. Ratio of Coulomb pressure to the ideal-gas pressure of the reference solar model with the EFF equation of state. The present formula (solid line:); the Debye-Hückel approximation (dashed line).

However, at the solar surface, either the present formula or the Debye-Hückel approximation overestimates Coulomb effects. It indicates that the influence of Coulomb term is not adequate by helioseismic constraint. At the solar surface, especially in the second helium ionization zone, complex physical effects, such as pressure ionization and the formation of bound states, should be taken into account.

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© European Southern Observatory (ESO) 2000

Online publication: December 15, 2000