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Astron. Astrophys. 343, 990-996 (1999)

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6. Conclusions

There is no evidence from helioseismology that the solar age is different from [FORMULA]. Though it is true that by adopting higher values for the solar age one may achieve a better agreement for most of seismic data, this cannot be regarded as an argument that the sun is older than the meteoritic data indicate. What seismic testing of current standard solar models reveals is the need for increase of the sound speed in the outer part of the radiative zone by about half percent ([FORMULA] in u). The required change may be partially achieved by an age increase above 5 Gy but it may also be caused by an increase of opacity in the relevant region. The required opacity increase may result from some still ignored effects in the OPAL calculations but also may indicate that the metal content in the outer part of the radiative zone is higher. We showed that adopting [FORMULA], which is by 10% higher than the standard one but still within the error bars of determination, we derive [FORMULA] below 5 Gy and only marginally inconsistent with [FORMULA].

In fact, helioseismology provides a strong support for the assumption [FORMULA]. We showed that small frequency separations [FORMULA] and [FORMULA] determined from the data are in a drastic disagreement with the models older than 4.9 Gy and they are in a good agreement with the models calculated assuming [FORMULA]. The age of the sun determined from the best seismic data and with use of our standard models, which were calculated with the latest OPAL opacity data and the standard metallicity parameter [FORMULA], is

[EQUATION]

Outside the error range, we have [FORMULA].

The small separations are only weakly affected by uncertainties in the opacity. Still, models with enhanced opacity in the outer part of the radiative zone yield values of [FORMULA] even closer to [FORMULA]. Thus we conclude that the inadequacies of the current solar models cannot be reconciled by departing from the standard assumption about solar age but the resolution must be searched in opacity enhancement.

Our answer to the question how accurately we can determine age of the sun using stellar evolution theory and helioseismic data, posed by Paczynski (1997), is more optimistic than the answer of Weiss & Schlattl (1998).

The error bars given above may be somewhat underestimated. Taking into account the uncertainties beyond those included in the formal errors, the accuracy of the astrophysical estimate the solar age is, in our opinion, [FORMULA] Gy or 4%, which is significantly better than 0.5 Gy, as suggested by Weiss & Schlattl (1998).

The cause for the discrepant estimates is in the use of different observables. We believe that only the small separations are good probes of the solar age based on p-mode frequency data. Other ones, like frequencies themselves, seismically inferred sound speed, and photospheric helium abundance are too sensitive to the opacity to be regarded as reliable tools for measuring solar and stellar ages.

We examined various effects that may contribute to the uncertainty of the age determination from the small separations. None of the uncertainties in the physics included in modern standard solar models was found very significant. However, we identified few effects beyond standard model that may influence the small separations. Perhaps the most important is a macroscopic mixing in the outermost part of the radiative interior. We considered also the effects of the centrifugal and magnetic forces and we pointed out that while they are not important for our seismic estimate of the solar age they must be kept in mind in interpretation of data on the small separations from years of high magnetic activity as well as the data for stars rotating more rapidly than the sun.

All the seismic observables we discussed here are still available only for the sun. The observables that we are likely to have in not too distant future for other stars are the small separations. Measuring these parameters is one of the main goals of the three currently prepared or planned space asteroseismic missions. It is very fortunate that, as we have shown, the small separations are the best seismic age indicators derived from p-mode frequencies. There is a potential for measuring stellar ages based on g-modes, which are excited in a number of stars. However, also in this case it is essential to check robustness of the seismic dating.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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