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

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3. Input physics

3.1. Opacity

In order to test the influence of the uncertainities in the radiative opacities we have used different sets of the opacity. The OPAL opacity is available for the temperatures larger than [FORMULA], and implemented by the Alexander & Ferguson opacity table at low temperatures (OA). For the Sun, the transition point ([FORMULA] K) is chosen such that it always belongs to convective region in all evolutionary phases of the Sun. So, small discrepancies may not cause any problem in the solution of finite-difference equations. We calculate the opacity by quadratic interpolation in chemical composition, T and [FORMULA]).

Model 1 and 2 are obtained using the radiative opacity of CS. The opacity tables include three H-He mixtures in which H/He =4; 1; 0. Required opacity for the solar material is found by logarithmic interpolation between the three sets of the opacity tables.

3.2. Equation of state: minimization of free energy

The free energy is minimized in order to obtain the Saha equation which is required to find the number of particles in each degree of ionization. This is the chemical picture in which atoms and molecules are described as separate entities.

Since Fermi and Bose gases obey different statistics, their free energies, F, are different. But, disregarding signs, they have the same form:

[EQUATION]

[EQUATION]

where summation is over all quantum states i, µ is the chemical potential, [FORMULA] is the energy of this state, and k is the Boltzman constant. For Fermi (Bose) gas upper (lower) sign in Eq. (3) applies. Total number of particles, N, is given by the summation of the average occupation number over all quantum states, [FORMULA].

3.2.1. Degeneracy of electrons

The free energy of a gas of particles with half integral spin is not as simple as that of the photon gas in those regions of the [FORMULA] plane where the gas is partially degenerate and partially relativistic. In the region of interest, there is no analytic solutions of the EOS except at some points in which both degeneracy parameter [FORMULA] ([FORMULA]) and relativistic parameter [FORMULA] ([FORMULA]) have extreme values. Therefore numerical methods should be employed for a semi-relativistic and semi-degenerate gas. As [FORMULA] functions introduced by Guess (1966) are better suited for numerical calculations than Fermi-Dirac functions, we express [FORMULA] as

[EQUATION]

Like [FORMULA], the number density of electrons [FORMULA] is an implicit function of [FORMULA] and [FORMULA]. In terms of the functions [FORMULA] it is given as

[EQUATION]

where [FORMULA] is the number of electrons in volume V. If [FORMULA] then perfect gas relation is recovered and [FORMULA] for the non-relativistic region can be given in terms of [FORMULA] and T:

[EQUATION]

which corresponds to slightly degenerate or non-degenerate case.

Then the energy and the pressure are given by the derivatives of the free energy with respect to the temperature and to the volume, respectively:

[EQUATION]

[EQUATION]

3.2.2. Boltzmann gas and partition function

Free energy for the Boltzmann gas can be written as follows

[EQUATION]

where [FORMULA] and [FORMULA] are number of i -type j -times ionized ions in volume V and partition function of these ions, respectively.

Now, energy and pressure can be found for a Boltzman gas by differentiating [FORMULA] given above. Since partition function is also a function of temperature and density (due to interaction between particles), its derivatives must also be taken into account:

[EQUATION]

and

[EQUATION]

where [FORMULA] and [FORMULA]. The second quantity in the parenthesis in Eq.(10) is [FORMULA] where [FORMULA] is the m th ionization potential of i -type ion

Influence of interaction between atoms (ions) may obliterate some outer states of an atom. Electrons occupying these states are not bounded to the atom any more. This is known as pressure ionization. MHD have shown how to include the effect of pressure ionization into the partition function based on Ünsold's theory (Clayton 1968). According to Ünsold's theory partition function can be written as

[EQUATION]

where [FORMULA] is the survival probability of the state k for a j -times ionized i -type atom with j less electron, [FORMULA] and [FORMULA] are the statistical weight and excitation energy of this state, respectively.

If the survival probability for any state is zero, neither this state nor higher states contribute to the partition function. Apart from thermodynamical consistency the survival probability given by MHD has important advantages. The partition function doesn't explode, and transition from bound to free states is continuous since [FORMULA] is continuous.

3.2.3. Coulomb interaction

At relatively high densities atoms are very close to each other. A bound electron may not preserve its original state under the influence of the Coulomb potential due to other charged particles in the plasma. The Coulomb interaction implies that slight changes occur in the energy and the pressure. An approximate expression for the Coulomb energy is given by Landau & Lifshitz (1969) as

[EQUATION]

where [FORMULA] is proton charge and [FORMULA] is Debye radius. Expression for pressure due to interaction is

[EQUATION]

As seen in Eqs. (13) and (14) both [FORMULA] and [FORMULA] are negative, since the Coulomb interaction causes a decrease in the energy and the pressure.

3.2.4. Saha equation

To find equilibrium number of particles it is necessary to minimize the free energy using stoichiometric relations describing ionization (or dissociation) processes. Then the Saha equation is obtained with ionization potential [FORMULA] of l -times ionized k -type atom:

[EQUATION]

where

[EQUATION]

with N being the total number of particles in the system including electrons.

Two remarks are called for Eq. (15) which includes degeneracy of electrons and the Coulomb interaction. First, any additional term to the free energy appears with its partial derivative with respect to the number density of particles and as an argument of the exponential in the Saha equation. Second, both of the terms coming from the free energies of the degenerate electrons and the Coulomb interaction behave like processes lowering ionization potential. Therefore, some of the early works on EOS (Harris 1964; Graboske et al. 1969) treated ionization potential somehow as a function of the thermodynamical variables.

3.3. Computational method of EOS

The total energy and pressure of the plasma are the summation of the partial contributions of the effects discussed in the previous sections and the radiation term. Most of the physical quantities appearing in the equations for energy and pressure of different kinds of gases can be identified, explicitly or implicitly, in terms of number density of electrons. Therefore the EOS is calculated by iterating over the number of moles of electrons per unit mass, [FORMULA] (Gabriel 1994b). Then the problem reduces to solve the Saha equation for any given value of [FORMULA].

We implemented the whole procedure in our stellar evolution code in the following way. Starting from the center, an initial guess of [FORMULA] is obtained by assuming the complete ionization, that is each species has charge [FORMULA], which is sufficiently accurate for the center of a star. And then, in outer shells, the first guess of [FORMULA] is taken as its last computed value in the previous shell, until the surface is reached.

From the value of [FORMULA], the degeneracy parameter [FORMULA] is found from Eq. (6). In the transition region [FORMULA], gas is partially degenerate and [FORMULA] can be recalculated from the fitting formula of Henyey (Gabriel 1994b; Yldz 1996).

Because of the low abundance of heavy elements in the solar mixture, probably it is unnecessary for those elements, to calculate the quantities such as partition functions and the number of elements at any degree of ionization, with high accuracy. Only for H and He very precise calculations need to be done.

In order to save computer time, all that is needed is to calculate the effective degree of ionization (i.e., effective charge of each element)

[EQUATION]

Then, there is no need to solve the Saha equation. Two methods are used to compute the effective charges of 12 C, 14 N, 16 O, 20 Ne and the binding energies of these atoms (ions). The first one is Henyey's fitting method (Gabriel 1994b; Yldz 1996) which uses [FORMULA] and the chemical potential of electron [FORMULA] as input parameters. The second method is due to Gabriel & Yldz (1995), which is similar to Henyey's method but more precise. It takes into account Debye radius as an additional parameter.

Since heavy elements are rare in the atmospheres of the stars only the molecules of hydrogen, H2, H [FORMULA], and H [FORMULA] are considered. The abundance and the energies of these molecules are solved by the method of Vardya (1961).

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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