4.1. Models constructed
The MoSEC code is used to construct a series of calibrated, evolutionary solar models. For all the models, the matching optical depth, , is chosen to be 2, while a global truncation error of better than is sought. From the considerations of Sect. 2, 1000 radial grid points is sufficient to meet this requirement in the space dimension, while Richardson extrapolation with provides sufficient accuracy in time. The models constructed are as follows:
In all cases, the heavy elements are initially chosen to be in the Grevesse & Noels (1993) proportions and iteration is performed to ensure that the final surface heavy-element mass fraction satisfies (see Sect. 3.2). It would be possible in the case of the models with heavy-element settling and diffusion to iterate the individual heavy-element abundances to ensure that the final surface abundances are in the Grevesse & Noels (1993) proportions, a procedure carried out by Richard et al. (1996). However, bearing in mind that neither the equation of state nor opacity calculation takes into account the variation of individual heavy-element abundances, we do not believe that this procedure would have a significant effect, and therefore do not include it at present.
Various properties of the computed models are listed in Table 3 and compared with the properties of a selection of other recent solar model calculations which include settling and diffusion of helium and heavy elements. , and denote the central hydrogen mass fraction, central temperature and central density respectively. denotes the initial (uniform) hydrogen mass fraction, denotes the final surface helium mass fraction, denotes the final surface heavy-element mass fraction, while denotes the fractional radius of the base of the convection zone. and represent the predicted neutrino capture rates for and detectors, given in solar neutrino units (1 SNU = capture atom-1 s-1).
Table 3. Comparison of the properties of solar models.
4.2. Opacity and equation of state interpolation
A series of models is constructed using the two different opacity interpolation methods (minimum norm, and birational splines) and various different numbers of fitting points (2,3 and 4); all other aspects of the physics (OPAL equation of state, no settling or diffusion) are kept constant. The calibrated hydrogen abundances and mixing-length parameters for these models are listed in Table 4. The third model listed corresponds to the "opal" model in table 3.
Table 4. The effect on the calibrated hydrogen abundance and mixing-length parameter of the choice of opacity interpolation method and number of interpolation points.
As can be seen from this table, there is a reasonably good agreement between models constructed using the two different interpolation algorithms with four interpolation points. The agreement is better for the initial hydrogen abundance than for the mixing-length parameter, which is as expected given that the latter is more sensitive to the opacity. The three models constructed with the birational-spline algorithm show, as expected, that as the number of interpolation points is reduced, the calibrated hydrogen abundance and mixing-length parameter exhibit increasingly large errors.
The effect of changing the fineness of the grid used in the equation of state tables is shown in Table 5. Both the models listed in this table are constructed without helium settling or diffusion, using the MHD equation of state; the first is constructed using equation of state tables on the same grid as the OPAL equation of state, while the second is a similar model constructed using MHD equation of state tables with a grid twice as fine. The differences between the two models in this case are smaller (much smaller in the case of the hydrogen abundance) than the differences described above between models calculated using two different opacity interpolation schemes. The errors introduced by interpolation (especially in the calibrated hydrogen abundance) are therefore dominated by the those arising from the opacity interpolation. The original choice of grid for the MHD equation of state tables (corresponding to that used in the OPAL tables) is therefore sufficiently fine.
Table 5. The effect on the calibrated hydrogen abundance and mixing-length parameter of changing the fineness of the equation of state grid.
4.3. Microscopic diffusion
It is well known that models without settling and diffusion of helium do not very well reproduce the seismically determined surface helium abundance and convection-zone depth of the sun (e.g. Christensen-Dalsgaard et al. 1993, Proffitt 1994), having surface helium abundances too high, and convection-zone depths too low. In this study, including settling and diffusion of helium decreases from 0.2634, in the case of the "opal" model, to 0.2365, in the case of the "ydiff" model, and decreases from 0.726 to 0.711. These are similar findings to those of Christensen-Dalsgaard et al. (1993). Comparison may be made with the seismically determined values of (Däppen et al. 1988, Vorontsov et al. 1991, Basu & Antia 1995, Dziembowski et al. 1994) for , and (Christensen-Dalsgaard et al. 1991b) for .
In addition to improving the agreement in convection-zone depth and surface helium abundance, helium settling and diffusion dramatically improve the agreement in sound-speed with the sun. This is shown in Fig. 3, where the dashed line represents the relative difference in squared sound speed, , between the "opal" model and the sun, and the dotted line represents the same quantity for the "ydiff" model. This improvement is principally due to the increased convection-zone depth, but also partly due to the reduced mean molecular weight just below the base of the convection zone. There are still significant features just below the base of the convection zone, where the sound speed is too low in the model, and near the centre, where the sound speed is too high in the model.
Heavy-element settling and diffusion have been included in many of the recent solar model calculations, including all the models listed in the lower half of Table 3. These processes are included in model "zdiff", increasing by 0.0057 and reducing by 0.001, in very close agreement with the results of Proffitt (1994). The corresponding form of , shown by the thick solid line in Fig. 3, is very similar to that found for the "ydiff" model, with a maximum difference of about in sound speed between the two models at around (which is, nevertheless, significant compared to the errors in the inversion, shown by the thin solid lines). Consequently, unlike helium, heavy-element settling and diffusion do not significantly improve the agreement in sound speed between solar models and the sun.
The flux of neutrinos arising from the decay of is strongly affected by the inclusion in the evolution calculation of the diffusion and settling of helium and heavy-elements. Gravitational settling leads to an increase in mean molecular weight at the centre, and a consequent increase in the central temperature. This leads to an increase in the flux of neutrinos (since the latter goes roughly like at the solar centre). Bahcall & Pinsonneault (1992b) found that helium settling alone brought about an increase of about in the flux measured by the chlorine neutrino capture experiments; Proffitt (1994) found an increase in this flux of about , the discrepancy being due to the different formulations of the diffusion equations for helium. In this study, we find that the flux goes from to , an increase of about ; this is very close to the value obtained by Proffitt (1994).
We find that helium and heavy-element settling combined produce an increase in the flux of about . There is considerable variation in the corresponding increase found by other authors, varying from (Richard et al. 1996) to (Bahcall & Pinsonneault 1995); Turck-Chièze & Brun (1997) find an increase of , closest to our result. The reasons for these wide discrepancies are not yet clear. In terms of the actual flux prediction, all the models listed in Table 3 predict a somewhat higher flux than the model "zdiff", although the calculations of Turck-Chièze & Brun (1997) predict a lower flux of 7.2 SNU.
4.4. The equation of state
The model "zdiffop" is constructed only taking into account the variation of the overall heavy-element abundance, Z, in the interpolation of the opacity tables. This is a similar assumption to that made by Morel et al. (1997). The relative difference in squared sound speed between this model and the sun is shown by the dotted line in Fig. 4, and compared with the same quantity for four other models, "zdiff" (solid line), "mhd" (dot-dashed line), "mix" (dashed line) and "zdiffre" (dot-dot-dot-dashed line). The relative differences in squared sound speed between the models "zdiffop", "mix", "mhd" and "zdiffre" and the model "zdiff" are plotted in Fig. 5 using the same line styles. As can be seen, the difference in between "zdiffop" and "zdiff" is almost within the standard errors in the inversion (shown by the thin solid lines), and the neglect of the variation of the overall heavy-element abundance in the calculation of the equation of state is therefore valid. It therefore seems that, contrary to the claims of Morel et al. (1997), equation of state data taking into account detailed changes in the heavy-element mixture are unnecessary.
Comparison is made between the OPAL and the MHD-E equations of state using the "mhd" model, which differs from the "zdiff" model only in using the MHD-E instead of the OPAL equation of state. Early comparisons of the OPAL and MHD equations of state (Däppen et al. 1990, Däppen 1992) found very close agreement over most of the solar interior. More recently, Guenther et al. (1996) computed models using both equation of state formalisms, finding very close agreement both in global properties and in local physical quantities, while Rogers et al. (1996) found slight differences in the location of the hydrogen and helium ionization zones leading to a so-called "pressure spike" in the MHD equation of state at temperatures of .
In the present calculation, the two models "mhd" and "zdiff" do not agree as closely as those of Guenther et al. (1996), with larger differences in initial hydrogen abundance, mixing-length parameter, and also in the sound speed. However, better agreement is found when the "mhd" model is compared to the "zdiffre" model, which uses the OPAL equation of state but includes a correction for relativistic effects. This shows that the inclusion of the relativistic correction in the new MHD-E equation of state accounts for much of the difference found between the "mhd" and "zdiff" models. The relativistic correction has the effect of increasing the electron pressure at the centre, thus reducing the hydrogen abundance there in the calibrated model. This has a significant effect on the sound speed throughout the radiative interior, reducing it by a maximum of about near the base of the convection zone.
4.5. Sub convection-zone mixing
A simple model of the sub convection-zone mixing associated with the solar tachocline (Spiegel & Zahn 1992, Elliott 1997) is included in the model "mix". Instead of assuming complete mixing to the base of the convection zone, as in the standard solar model prescription, the region of mixing is assumed to extend a distance of below the base of the convection zone; this value is somewhat lower than the thickness of the tachocline as inferred from rotation-rate inversions (Kosovichev 1996). The effect of the extra mixing is to redistribute gravitationally settled helium back into the convection zone, thereby reducing the mean molecular weight just below the base of the convection zone. This, in turn, increases the sound speed in that region as is shown by the dashed line in Fig. 5. Deeper down, the recalibration of the model reverses the situation, causing the sound speed to be lower in "mix" than in "zdiff". The overall effect of the extra mixing is to improve slightly the agreement in sound speed with the sun: the sound speed increase in the "mix" model relative to the "zdiff" model would need to be displaced downward to account fully for the hump in just below the base of the convection zone.
© European Southern Observatory (ESO) 1998
Online publication: May 15, 1998