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

4. Influence of MHD EOS

In order to compare the results and the influence of MHD EOS, the required density, temperature and chemical composition are taken from a model of the present Sun (Model A, see Table 1). Henyey method is used for ionization of heavy elements, unless stated otherwise.

Table 1. Comparison of the present solar models with different EOS, opacities and data of the Sun.

Contribution of the Coulomb interaction, the degeneracy of the electrons and derivative of the partition function to the total pressure can be seen in Fig.1. For the Model A is plotted with respect to of temperature from surface to the center throughout the solar model, where P is the total pressure, and is any of fractional contribution to the pressure. Thin solid line is for the Coulomb pressure. The ratio is less than or equal to one, since the Coulomb term causes a decrease in pressure. At the surface the Coulomb pressure is zero. As the temperature and the density, and therefore the degree of ionization increase, fractional change becomes a few percent. Contribution of the Coulomb potential to the total pressure is maximum () between and , 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. As the temperature and the density increase the contribution becomes smaller, until partial degeneracy of the electrons sets in near the center of the Sun.

Fig. 1. The fractional contribution to the pressure due to Coulomb interaction, degeneracy of electrons and derivative of the partition functions of H, He, and He with respect to density, is plotted as a function of the logarithm of the temperature for a selected solar model (case A below).

The pressure of the degenerate-electrons calculated by Henyey fitting formula (also calculated by method of Yldz & Eryurt-Ezer (1992) using Guess functions, the difference is less than ), represented by dots, deviates from one at the central part. Here is subtracted from the total electron pressure. At the center of the Sun degeneracy parameter () is equal to which implies a slight degeneracy, and the contribution of degenerate-electrons is about .

Thick solid line in Fig.1. is a superposition of three Gaussian curves. From left to right, the first one is dominantly the density derivative of hydrogen partition function which is maximum () at . Second one is due to the density derivative of He partition function, which is maximum at about and its contribution to the total pressure is less than . Last one is due to the density derivative of He partition function, which is non-zero in an important part of the solar interior but with a lower contribution to the total pressure than the others. The base temperature of the convective zone in the Model A is about . This corresponds to a point ) at which He starts to appear.

The Saha equation, Eq (15), is solved exactly for H and He. Results obtained from Henyey method, the effective charges divided by the atomic numbers of each element, throughout the Model A, are given in Fig.2. Due to its low ionization potential hydrogen (solid line with asterisks) is rapidly completely ionized (at ). For the effective charge of He (solid line with circle) two phases are seen. Even at the center, none of the heavier elements, C (solid line), N (dashed line), O (solid line with triangle), Ne (dashed line with asterisks) are completely ionized. Owing to rough calculation, each line of the heavy elements cross each other near the center. On the other hand, at the center the radiation and the gas pressure (and energy) are sufficiently large, therefore the computed values are within an acceptable range. This is not the case at the surface where binding energy has an important role in the total energy. While H and He are neutral, each of the heavier elements is in the state of first ionization.

Fig. 2. The run of the effective charge of several elements as a function of the logarithm of the temperature in the model of Fig. 1. The Saha equation is solved for H and He, and Henyey's method is used for heavy elements.

A physically more reliable method is that of Gabriel & Yldz (1995). As seen in Fig.3, at the middle of the Sun, ionization degrees of the heavier elements are a little bit smaller than, but comparable to those given in Fig.2. At the center and at the surface, the values of ionization degree are not in contradiction with the basic truths of atomic physics. All atoms are neutral at the surface. Ionization of C starts before H, and ionization of N, O,and Ne starts after H but before He, as expected from the atomic physics.

Fig. 3. The run of the effective charge of several elements as a function of the logarithm of the temperature in the model of Fig. 1. The Saha equation is solved for H and He, and the method of Gabriel & Yldz (1995) is used for heavy elements.

The effect of the MHD EOS on the adiabatic gradient is shown in Fig.4. There we plot the ratio of the adiabatic gradient resulting from the Gabriel & Yldz (solid line) and from the Henyey method (dotted line) to the value obtained with EC EOS are given. Both are larger than the old gradient, and three peaks correspond ionization zones of H, He and He .

Fig. 4. The ratio of adiabatic gradients resulting from MHD EOS with Gabriel & Yldz (1995) (solid line) and with Henyey (dots) methods for ionization of heavier elements, to that obtained adopting the EOS of old routine in Ezer's code (EC) for the selected model of Fig. 1.

© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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