The possibility of using helioseismology to constrain the solar age has been discussed by several authors in the past. Very recently Dziembowski et al. (1999) have shown that the most robust and accurate method is provided by the small frequency separation analysis (SFSA), $\delta\nu_{{\ell,n}}=\nu_{{\ell,n}}-\nu_{{\ell+2,n-1}}$ , for spherical harmonic degrees $\ell = 0,1$ and radial order $n \gg \ell$ (Tassoul 1980).

The important property of this quantity is its strong sensitivity to the sound-speed gradient near the solar centre and its weak dependence on the details of the treatment of the outer layers. Despite our ignorance of a reliable convection model for the solar envelope we are therefore able to verify how well our models are able to reproduce the deep radiative regions, in particular the solar core. Since the properties of the core are mainly determined by the present central hydrogen abundance, and the latter is influenced by the solar age, SFSA is a reliable tool to examine the seismic age of the Sun.

Adopting the OPAL equation of state (Rogers et al. 1996) a seismic age of $(4.66 \pm 0.11) \; {\rm Gyr}$ has been obtained by Dziembowski et al. (1999), which is consistent with the meteoritic age $(4.57\pm 0.02) \; {\rm Gyr}$ of Bahcall et al. (1995).

The aim of this paper is to show that an important ingredient in this type of analysis is the usage of an accurate equation of state (EOS). In particular, by the inclusion of the special relativistic corrections, like in the updated version of the OPAL EOS, the helioseismic age of the Sun is reduced to $(4.57\pm 0.11)\ {\rm Gyr}$ , which is in remarkable agreement with the meteoritic value.

Elliott & Kosovichev (1998) have demonstrated that the inclusion of relativistic corrections in the EOS leads to a better agreement between the solar models and the seismic Sun. By inverting SOI-MDI/SOHO p-mode frequencies they found that the solar adiabatic exponent $\Gamma _1$ is much better reproduced by solar models including the relativistic contribution to the Fermi-Dirac statistics. Since the improved EOS causes a decrease of $0.2\%$ in the adiabatic index $\Gamma _1$ in the solar centre, the sound speed ( $\propto \sqrt{\Gamma_1}$ ) is reduced by about 0.1%. Therefore, the influence of the relativistic corrections should also be visible in the small frequency separations $\delta\nu_{{\ell,n}}$ . Indeed, Bonanno et al. (2001) have found that including this effect in the value of $\Gamma _1$ improves the agreement in $\delta\nu_{{\ell,n}}$ between solar models and observations, thereby confirming the results of Elliott & Kosovichev (1998).

Table 1: Characteristic quantities of selected solar models. The indices 0, ${\rm ph}$ , ${\rm cz}$ , and c denote initial, photospheric, bottom of convective envelope, and centre, respectively. MHD-R is the abbreviation for the MHD EOS containing the relativistic corrections in $\Gamma _1$ .
Model ${\rm\frac{age}{Gyr}}$ EOS ${\it Y}_0$ ${\it Z}_0$ ${\it Y} _{\rm ph}$ ${\it Z} _{\rm ph}$ $\frac{r_{\rm cz}}{R_{\rm ph}}$ ${\it X}_{\rm c}$ ${\it Y}_{\rm c}$ $\frac{\rho_{\rm c}}{{\rm gcm^{-3}}}$ $\frac{T_{\rm c}}{10^6~ {\rm K}}$ $\frac{S_{\rm pp}(0)}{{\rm 10^{-25}\;MeV~ b}}$

1 4.58 OPAL 01 0.2755 0.01995 0.2453 0.01805 0.7132 0.3353 0.6432 152.87 15.73 3.89

2 4.58 OPAL 96 0.2749 0.01995 0.2449 0.01806 0.7132 0.3289 0.6428 152.70 15.72 3.89

3 4.60 OPAL 01 0.2752 0.01995 0.2451 0.01805 0.7125 0.3342 0.6443 153.16 15.73 3.89

4 4.60 MHD-R 0.2757 0.01997 0.2452 0.01805 0.7141 0.3341 0.6444 153.22 15.74 3.89

5 5.00 OPAL 01 0.2714 0.02013 0.2405 0.01816 0.7082 0.3133 0.6650 159.82 15.84 3.89

6 4.58 OPAL 01 0.2758 0.01989 0.2460 0.01803 0.7118 0.3362 0.6423 151.35 15.66 4.00

**Table 1:** Characteristic quantities of selected solar models. The indices 0, ${\rm ph}$ , ${\rm cz}$ , and c denote initial, photospheric, bottom of convective envelope, and centre, respectively. MHD-R is the abbreviation for the MHD EOS containing the relativistic corrections in $\Gamma _1$ .
Model	${\rm\frac{age}{Gyr}}$	EOS	${\it Y}_0$	${\it Z}_0$	${\it Y} _{\rm ph}$	${\it Z} _{\rm ph}$	$\frac{r_{\rm cz}}{R_{\rm ph}}$	${\it X}_{\rm c}$	${\it Y}_{\rm c}$	$\frac{\rho_{\rm c}}{{\rm gcm^{-3}}}$	$\frac{T_{\rm c}}{10^6~ {\rm K}}$	$\frac{S_{\rm pp}(0)}{{\rm 10^{-25}\;MeV~ b}}$
1	4.58	OPAL 01	0.2755	0.01995	0.2453	0.01805	0.7132	0.3353	0.6432	152.87	15.73	3.89
2	4.58	OPAL 96	0.2749	0.01995	0.2449	0.01806	0.7132	0.3289	0.6428	152.70	15.72	3.89
3	4.60	OPAL 01	0.2752	0.01995	0.2451	0.01805	0.7125	0.3342	0.6443	153.16	15.73	3.89
4	4.60	MHD-R	0.2757	0.01997	0.2452	0.01805	0.7141	0.3341	0.6444	153.22	15.74	3.89
5	5.00	OPAL 01	0.2714	0.02013	0.2405	0.01816	0.7082	0.3133	0.6650	159.82	15.84	3.89
6	4.58	OPAL 01	0.2758	0.01989	0.2460	0.01803	0.7118	0.3362	0.6423	151.35	15.66	4.00

In addition to the age, the central hydrogen abundance is also crucially dependent on the precise value of $S_{\rm pp}(0)$ , the zero-energy astrophysical S-factor for the proton-proton fusion cross section. Schlattl et al. (1999) and Antia & Chitre (1999) have shown, using the old version of the OPAL EOS, that an increase of $S_{\rm pp}(0)$ by about 4% with respect to Adelberger et al.'s (1998) value yields a better agreement with the observed frequencies for an age of 4.57 Gyr. For this reason we consider in our analysis also different values of $S_{\rm pp}(0)$ .

Including the updated OPAL EOS the best agreement between meteoritic and seismic age could be achieved with Adelberger et al.'s (1998) $S_{\rm pp}(0)=4.00 \times 10^{-25}~{\rm MeV ~b}$ . Hence, by taking into account the relativistic corrections in the EOS there is no need for an artificial increase of $S_{\rm pp}(0)$ , as suggested by previous works, in order to obtain a better agreement between seismic and meteoritic age.

The code and physics used to compute the various solar models are described briefly in the next section, followed by the consequences for the seismic age obtained by means of the SFSA (Sect. 3). In the final part the results are discussed.

1 Introduction