## 3. Input physics## 3.1. OpacityIn 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
, and implemented by the Alexander &
Ferguson opacity table at low temperatures (OA). For the Sun, the
transition point ( 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, 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 energyThe 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, where summation is over all quantum states ## 3.2.1. Degeneracy of electronsThe 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 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 () and relativistic parameter () have extreme values. Therefore numerical methods should be employed for a semi-relativistic and semi-degenerate gas. As functions introduced by Guess (1966) are better suited for numerical calculations than Fermi-Dirac functions, we express as Like , the number density of electrons is an implicit function of and . In terms of the functions it is given as where is the number of electrons in volume
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: ## 3.2.2. Boltzmann gas and partition functionFree energy for the Boltzmann gas can be written as follows where and are number
of Now, energy and pressure can be found for a Boltzman gas by differentiating 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: where and . The second
quantity in the parenthesis in Eq.(10) is where
is the 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 where is the survival probability of the
state k for a 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 is continuous. ## 3.2.3. Coulomb interactionAt 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 where is proton charge and is Debye radius. Expression for pressure due to interaction is As seen in Eqs. (13) and (14) both and are negative, since the Coulomb interaction causes a decrease in the energy and the pressure. ## 3.2.4. Saha equationTo 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 of 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 EOSThe 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, (Gabriel 1994b). Then the problem reduces to solve the Saha equation for any given value of . We implemented the whole procedure in our stellar evolution code in the following way. Starting from the center, an initial guess of is obtained by assuming the complete ionization, that is each species has charge , which is sufficiently accurate for the center of a star. And then, in outer shells, the first guess of is taken as its last computed value in the previous shell, until the surface is reached. From the value of , the degeneracy parameter is found from Eq. (6). In the transition region , gas is partially degenerate and 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) Then, there is no need to solve the Saha equation. Two methods are
used to compute the effective charges of Since heavy elements are rare in the atmospheres of the stars only
the molecules of hydrogen, H © European Southern Observatory (ESO) 1997 Online publication: April 20, 1998 |