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

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4. Direct and almost direct use of measured frequencies

It 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 [FORMULA] and its errors.

Choosing, instead, a direct use of frequency differences we face another problem. The formal approach to determination of [FORMULA] is the minimization of

[EQUATION]

where the sum includes all [FORMULA] p-modes in the set, and [FORMULA] are measurement errors. The problem is revealed in Fig. 2, in which we may see that [FORMULA] 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 [FORMULA].

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 [FORMULA], i.e., above the lower turning point of all the p-modes in the set. The lower turning point is determined by the parameter [FORMULA]. Its maximum value for modes in our set is 0.1 and corresponds to the turning point [FORMULA]. Sufficiently far above the turning point, the relevant eigenfunctions, except for normalization, are [FORMULA]-independent. Therefore, we may expect that the part of the frequency differences due to the effects in the layers above [FORMULA] scale as [FORMULA], where [FORMULA] 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 [FORMULA] in a polynomial form to the frequency differences [FORMULA] and considered only the residual part of the differences

[EQUATION]

The quantity [FORMULA] 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 [FORMULA] functions, which is a modified [FORMULA] with [FORMULA] replaced [FORMULA]. The parameter s is the maximum value of the quantity [FORMULA] ([FORMULA] in µHz), which determines the lower turning point allowed in the set of modes. The case [FORMULA] corresponds to including all 1945 p-modes. The case [FORMULA] corresponds to a truncated set which includes only 956 modes with [FORMULA] In the latter case, we additionally remove effects of inadequate treatment of convection which are responsible for large values of [FORMULA] above 0.9. The minima of the modified [FORMULA] are pronounced and therefore we may, at least formally, determine the solar age and its uncertainty. Not surprisingly, the minimum is deeper for [FORMULA]. Still, the minimum value is [FORMULA]. One may see in Fig. 1 that [FORMULA] in the radiative interior cannot be compensated by an adjustment of the age.

[FIGURE] Fig. 3. Determination of the solar age by fitting p-mode frequencies. Values of [FORMULA] (left vertical axis, solid line) are calculated with Eq. 2. Values of [FORMULA] for [FORMULA] and [FORMULA] (right vertical axis, dashed lines) are calculated also with Eq. 2, but with [FORMULA] replaced by [FORMULA] (see Eq. 3). The choice [FORMULA] implies use of all p-mode frequencies and elimination of the near surface differences between the sun and the model. With [FORMULA] we use only modes with the lower turning point above [FORMULA] and we additionally eliminate effects of inadequacies in the treatment of convection.

In Table 5, we list the values of [FORMULA] (in Gy) determined as the minima of [FORMULA] and [FORMULA]. The errors are determined as the distances from [FORMULA], where [FORMULA].


[TABLE]

Table 5. Seismic age from p-mode frequencies


The results shown in Table 5 are consistent with implications from [FORMULA] discussed in the previous section. There are only few modes sensitive to u in the inner core, where [FORMULA] is not consistent with high [FORMULA]. Also, even with [FORMULA] there are not many modes sensitive to [FORMULA]. The results agree with those of Weiss & Schlattl (1998). All this does not mean that we should treat [FORMULA] given in Table 5 as realistic estimates of solar age. Rather, we think, the high values obtained for models with the standard metal abundance reflect an attempt to compensate such deficiencies of the model as too low opacity and/or neglect of macroscopic mixing beneath the base of convective envelope. With [FORMULA] we obtained [FORMULA] which is still higher but, within the error, consistent with [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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