3. Physical assumptions
3.1. Solar age and luminosity
Estimates of the age of the oldest meteorites set a lower bound on the age of the solar system as a whole, and on the age of the sun in particular. Without access to all the meteoritic data, Guenther (1989) recommended a value of 4.49 Gyr. More recent studies suggest the value should be nearer 4.6 Gyr (e.g. Bahcall et al. 1995), and it is this value which is adopted here. At any rate, the correct value for the solar age remains one of the more uncertain ingredients of solar modelling.
The luminosity is also a somewhat uncertain quantity due to the difficulty of calibrating satellite radiometers, and also due to its inherent long-term variability. The value adopted here is the same as that used by Bahcall et al. (1995), namely .
3.2. Element abundances
The relative abundances of elements in the solar interior are determined by a combination of meteorite analysis and photospheric line strength measurement. The former gives the initial, homogeneous, composition of the primeval solar nebula, while the latter reflects the present-day composition of the outer layers. Apart from the depleted elements and , the agreement between the two has improved greatly during recent years as improvements have been made in the atomic data which affect photospheric abundance measurements. This enables us to have greater confidence in our knowledge both of the relative abundances of the heavy elements, and also of , the ratio of heavy-element abundance to hydrogen abundance. Current best estimates are given for these quantities by Grevesse & Noels (1993), and are adopted in the present work; the recommended value for is 0.0245.
3.3. Energy generation
Thermonuclear energy generation is by the three branches of the proton-proton chain and by the CNO cycle. The energy production per unit mass is computed using a subroutine written by Bahcall (Bahcall & Pinsonneault 1992a, 1992b), with cross sections taken from Bahcall & Pinsonneault (1992a), and energy releases for each reaction taken from Bahcall & Ulrich (1988).
3.4. Equation of state
There are two broad approaches to the problem of finding the thermodynamic properties of a partially-ionized plasma. The chemical-picture approach, of which the Saha equation is an example, is based on the principle of free-energy minimization. The partition function, , of the plasma is assumed to be factorizable into a part corresponding to the internal excitation states of individual particles, and a part corresponding to their translational states. The free energy, , where k is Boltzmann's constant, is then minimized with respect to variations in the occupation numbers which satisfy appropriate stoichiometric constraints. The Saha equation is the simplest realization of this procedure, but suffers from the disadvantage that it predicts unphysical recombination of ions and electrons in the solar centre. There have been several attempts to alleviate this problem. In the CEFF equation of state, Eggleton et al. (1973) introduced an extra term into the free energy which forced complete ionization in the solar centre, although there was no physical justification for this addition. Mihalas et al. (1988) considered an occupation-probability formalism which effectively provided a density-dependent cut-off in the internal part of the partition function. This again predicted almost complete ionization in the solar centre. The Mihalas, Hummer and Däppen (MHD) equation of state was published in the form of tables for a single heavy-element abundance of , and for a heavy-element mixture of carbon, nitrogen, oxygen and iron.
In this study, a new MHD-like equation of state, MHD-E, is calculated using a code written by the author, with energy-level data kindly provided by W. Däppen. Relativistic effects are included in the calculation of the electron free energy (they were not in either OPAL or MHD), which will be seen to have a significant effect on the computed models. The relative heavy-element abundances are set equal to those used in the calculation of the OPAL tables, while the total heavy-element abundance, Z, may be varied, overcoming the limitation of the original MHD tables.
The other approach to the equation of state is known as the physical picture. It does away with the concept of atoms, considering only fundamental particles such as nuclei and electrons. Interactions between particles are taken into account using the techniques of many-body theory. The only realization of these ideas has been carried out by the OPAL group at Livermore, with the publication of preliminary tables in 1994 and subsequently of tables with a finer mesh (Rogers et al. 1996). These tables are computed using the Grevesse (1993) abundances for carbon, nitrogen and oxygen, with the abundances of all the other heavy elements being added to the abundance of neon (see Table 2); they cover , and . Various comparisons have been made between the OPAL and MHD equations of state (Däppen et al. 1990, Däppen 1992), with the conclusion that they are remarkably similar over the bulk of the solar interior, with OPAL also predicting almost complete ionization at the solar centre; a full explanation for this has yet to be found.
Table 2. Relative heavy-element abundances.
Evolutionary models are computed using both equation of state formalisms. The pressure and other thermodynamic quantities are evaluated as functions of the density, temperature, hydrogen and overall heavy-element mass fractions by means of interpolating tables. Some studies have ignored the variation of the overall heavy-element mass fraction in the calculation of the equation of state, using instead a constant, prescribed value (e.g. Morel et al. 1997). We perform a comparison of models calculated with and without this assumption in order to test its significance. In our best model, we choose only to ignore the variation of the individual heavy-element abundances.
As was the case for the equation of state, the opacity is obtained from interpolating tables. The tables are constructed using the OPAL opacities (Rogers & Iglesias 1992), calculated with the Grevesse (1993) heavy-element mixture, except at low temperatures (), where the Kurucz (1991) low-temperature opacities are substituted. The interpolation is carried out using a package written by G. Houdek (Houdek & Rogl 1996), which allows a choice between a minimum-norm and a birational-splines algorithm. There is also a freedom of choice in the number of fitting points used in the X and Z interpolations (2,3 or 4). In the first instance the minimum-norm algorithm is used with four fitting points; the effect of other choices on the calibrated hydrogen abundance and mixing-length parameter is investigated in Sect. 4.2.
The heavy-element abundance used to interpolate the opacity tables only includes the abundance changes due to element segregation, since nuclear burning by the CNO cycle has little effect on the opacity (Proffitt 1994). As with the equation of state, the variation of the individual heavy-element abundances is ignored; as suggested by Morel et al. (1997), this may well have a significant effect on the computed models relative to the high precision of p mode frequency data.
© European Southern Observatory (ESO) 1998
Online publication: May 15, 1998