## 3. Physical assumptions## 3.1. Solar age and luminosityEstimates 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 abundancesThe 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 generationThermonuclear 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 stateThere 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
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, 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.
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 ## 3.5. OpacityAs 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 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 |