## 4. Direct and almost direct use of measured frequenciesIt is unfortunate that the parameters of seismic models which exhibit greatest sensitivity to solar age are, for various reasons, unreliable. The sound speed in the inner core cannot be precisely measured because the inversion is not accurate enough. Other parameters are formally very accurate but we cannot trust model predictions. Since the nature of the uncertainties is so diversified, we are reluctant to quote any quantity as a best value of and its errors. Choosing, instead, a direct use of frequency differences we face another problem. The formal approach to determination of is the minimization of where the sum includes all p-modes in the set, and are measurement errors. The problem is revealed in Fig. 2, in which we may see that depends only very weakly on age. There is a minimum near 5.2 Gy, but it is very shallow and does not allow a trustworthy estimate of . This problem is a consequence of the fact that the main part of the frequency differences between the sun and the model has nothing to do with the differences in the internal structure but rather it is caused by inadequacies in the treatment of oscillations in the outer layers, where the neglect of nonadiabatic effects and dynamical effects of convection is not justified. These inadequacies are significant in the outermost layers above , i.e., above the lower turning point of all the p-modes in the set. The lower turning point is determined by the parameter . Its maximum value for modes in our set is 0.1 and corresponds to the turning point . Sufficiently far above the turning point, the relevant eigenfunctions, except for normalization, are -independent. Therefore, we may expect that the part of the frequency differences due to the effects in the layers above scale as , where is the mode inertia calculated upon assuming the same normalization of the eigenfunctions in the photosphere. In order to eliminate these near-surface contaminations, we fitted in a polynomial form to the frequency differences and considered only the residual part of the differences The quantity is the part of the
frequency difference that may be attributed only to the difference in
the internal structure. In Fig. 3 we plot two
functions, which is a modified
with
replaced
. The parameter
In Table 5, we list the values of (in Gy) determined as the minima of and . The errors are determined as the distances from , where .
The results shown in Table 5 are consistent with implications
from discussed in the previous
section. There are only few modes sensitive to © European Southern Observatory (ESO) 1999 Online publication: March 1, 1999 |