3 Results for the solar age

We have computed solar models following the evolution from the zero-age main sequence with ages ranging from 4.40 to 5.00 Gyr in steps of 0.1 Gyr. Some basic quantities of a selection of models are summarized in Table 1.

Figure 1: The differences of $\Gamma _1$ (solid line) and $\rho$(dashed) between two models which either neglect (model 1) or contain the relativistic corrections (model 2) in the sense ${\rm (model~2 - model~1)/(model~1)}$.

For the higher ages the initial helium content has to be reduced to obtain the correct solar luminosity (compare models 1 and 5). Nevertheless, a larger lifetime leads to a steeper He profile toward the centre causing a larger central He abundance. The consequent increase of the opacity near the core demands an higher central temperature to produce the same amount of energy. This effect is further enhanced by diffusion which is operating longer for greater ages and is further increasing the central He content. Since the relativistic correction to $\Gamma _1$ increases with temperature (Eq. (1)), the inclusion of relativistic effects has a larger influence on older models. The relative differences in the profiles of $\Gamma _1$ and the density are shown in Fig. 1.

Models with greater $S_{\rm pp}(0)$, but the same age, have a smaller $T_{\rm c}$(see models 1 and 6 in Table 1), as the hydrogen burning in the core is more efficient.

In order to determine the seismic age, we calculated for all the solar models the small frequency separations $\delta\nu_{{\ell,n}}$ for $\ell = 0,1$ and $n \gg \ell$. These values have been compared with latest GOLF/SOHO data for $\ell = 0,1,2,3$, which have been obtained from long time series, and where the asymmetric line profile has been taken into account during the data reduction (Thiery et al. 2000). Only the frequencies of the mean multiplet (m=0) are used, as for them the influence of rotation is smallest.

For the analysis, the $\chi ^2$ method has been used, as in Dziembowski et al. (1999) or Schlattl et al. (1999);

$$% \chi^2_\ell=\frac{1}{ M-m}\sum_{ {n=m}}^{ M}\frac{(\delta\n... ...model}}}})^2}{\sigma^2_{{\ell,{n}}}+\sigma^2_{{\ell+2,{n-1}}}}$$ (2)

with M=31 for $\ell =0$and M=27 for $\ell=1$, and m being 10 in both cases. It is interesting to notice that including the relativistic corrections leads to a reduction of $\delta\nu_{{\ell,n}}$ of about $0.1~{\rm\mu Hz}$ for low frequencies (Fig. 2).

Figure 2: The differences of the quantity $\delta \nu _{n,\ell }$ between two models which either neglect (model 1) or contain the relativistic correction (model 2) in the sense ${\rm (model~2 - model~1)}$for an age of $4.20 \; {\rm Gyr}$ (solid line) and $4.70 \; {\rm Gyr}$ (dashed line).

 

 

Table 2: The best-fit age and the corresponding minimum of $\chi ^2$ for the grid with different equations of state and different values of $S_{\rm pp}(0)$in units of $10^{-25}~{\rm MeV~b}$.

		$\ell =0$		$\ell=1$
EOS	$S_{\rm pp}(0)$	$t_{\rm seis}$	$\chi_0^2$	$t_{\rm seis}$	$\chi_1^2$
OPAL96	3.89	$4.664\pm 0.088$	1.05	$4.672\pm 0.088$	1.66
OPAL01	3.89	$4.584\pm 0.088$	1.45	$4.624\pm 0.072$	1.66
MHD	3.89	$4.664\pm 0.080$	1.00	$4.680\pm 0.095$	1.65
MHD-R	3.89	$4.608\pm 0.040$	1.07	$4.640\pm 0.088$	1.25
OPAL01	4.00	$4.552\pm 0.080$	1.34	$4.584\pm 0.080$	1.47

The results for the $\chi ^2$-values in models with different ages are shown in Figs. 3 and 4. The best-fit age given by the minimal $\chi ^2$-value ( $\chi^2_{\rm min})$ and the error determined by the condition $\chi^2 - \chi^2_{\rm min} \le 1$are summarized in Table 2.

Figure 3: $\chi ^2_0$ for models with different age, neglecting (dashed line) or including the relativistic correction (solid line). The MHD-EOS has been used for the models in the left panel, the OPAL-EOS in the ones of the right panel.

Figure 4: Same as Fig. 3, but for $\chi ^2_1$.

Regardless of whether MHD or OPAL EOS is used, the best-fit age is reduced by about 0.05-0.08 $\; {\rm Gyr}$ when the relativistic corrections are included. The minimal value of $\chi_0^2$ is not significantly different for all the cases, although the models with OPAL96 EOS have a slightly smaller $\chi^2_{0,{\rm min}}$ than those obtained with OPAL01 EOS.

It is worth noticing that with $S_{\rm pp}(0)=4.00 \times 10^{-25}~{\rm MeV ~b}$the minimum $\chi ^2$-value slightly improves for both $\ell =0$ and $\ell=1$ (Table 2). Using OPAL01 EOS, which includes the relativistic corrections in a consistent way, we obtain in this case as the best-fit age $t_{\rm seis}= (4.57 \pm 0.11)$ Gyr, where we have taken the mean of the best-fit value for $\ell =0$ and $\ell=1$. This provides our most reliable value for the seismic solar age.