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

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5. Solar age from small separations

The inner core is the region where the sound speed is most sensitive to the age. Inversion for u in this region is unreliable but this does not mean that oscillation frequencies are not affected by the sound speed modifications near the center. The quantities which are most sensitive to changes in the inner core are small separations


for [FORMULA] and 1. In fact it has been recognized long time ago that data on [FORMULA] may be used for measuring stellar ages (Ulrich, 1986; Christensen-Dalsgaard, 1988; Gough & Novotny, 1990).

In our set we have data on [FORMULA] for n from 10 to 32 and on [FORMULA] for n from 10 to 27. We now form three age indicators,


[FORMULA], which is defined is the same way as [FORMULA] but for [FORMULA], and [FORMULA], which includes small separations both for [FORMULA] and [FORMULA].

The behavior of the three indicators is shown in Fig. 4. The [FORMULA] minima occur now at the ages which are only somewhat larger than [FORMULA] and have values only somewhat higher than 1. Table 6 summarizes information about the minima for models with the standard and the enhanced value of [FORMULA]. In the latter case the minima occur still closer to [FORMULA], but the difference is small and cannot be regarded as significant. The ages [FORMULA] Gy are clearly disfavored. There is a rough agreement of our result with that of Guenther & Demarque (1997), who relied on comparison of frequencies for [FORMULA] up to 100 and small separations for [FORMULA] up to 10. Also in their comparisons the strong case for [FORMULA] comes from small separations at [FORMULA] and 1.

[FIGURE] Fig. 4. Determination of the solar age by fitting small frequency separations (see Eq. 4). The quantity [FORMULA] is defined in Eq. 5, [FORMULA] and [FORMULA] are defined immediately after.


Table 6. Seismic age from small separations

We believe that only in the case of inference based on the small separations it is justified to speak about "age determination" because only with these observables we attain [FORMULA]. Furthermore, only in this case the inference is truly robust to other uncertainties still present in the standard model construction. The overall uncertainty of the seismic measurement of the solar age with the data on small separations is not significantly larger than the formal errors quoted in Table 6. The effect of the [FORMULA] uncertainty, as we see in this table, is [FORMULA] Gy. Now we will review other uncertainties that may affect small separations.

Effect of uncertainties on the age indicators [FORMULA],[FORMULA], and [FORMULA] may be assesed from data in Table 7. The effect of the opacity is revealed by comparison of models 0 with models 3 and JCD and we see that it is small. As we discussed in Sect. 3, the difference in opacity does not explain the whole difference in the sound speed between the models 0 and JCD. We alluded that the treatment of the element settling may contribute. In any case the implication for [FORMULA] is certainly within the uncertainties quoted in Table 6. We should note that JCD model which is characterized by the lowest value of [FORMULA] yields also the values of [FORMULA] which are the closest to [FORMULA] on the basis of the seismic observables listed in Table 4.


Table 7. Seismic age indicators from small separations in various models at [FORMULA] Gy and [FORMULA]=0.0245.
Model 4 is the same as model 0, but with a 3.2% increase of the 3He+4He reaction cross-section and a 6% decrease of the 3He+3He reaction cross-section. Model 5 is the same as Model 0 but ignoring the effect of gravitational settling.

Ignoring gravitational settling altogether (see Model 5 in Table 7) has a significant effect on small separations. However, the effect is now part of the physics included in the standard modeling of the sun.

Calculated values of the small separations are affected by the nuclear reactions cross-sections. The most important effect is expected from changes in the branching ratio of the 3He+4He to the 3He+3He reaction. Its increase implies more neutrino energy losses, less economic hydrogen burning, and consequently less hydrogen in the center of the sun. Such models mimic ones with [FORMULA]. However with currently adopted uncertainties in the cross-sections (see Model 4 in Table 7) the consequences for the age indicators are not significant.

Mixing of hydrogen and helium reduces the µ-gradient in the core and thus has a similar effect as a lower age. This is not a standard effect and we feel that there is not enough justification to consider it as a source of uncertainty. Certainly macroscopic mixing at the base of the convective zone is of more concern because we have some evidence for it. The mixing affects gravitational settling and therefore may have an appreciable effect on small separations.

Small separations are influenced by the centrifugal and magnetic distortion (Dziembowski & Goupil, 1998). The effect of centrifugal distortion in the sun is small because it is a very slowly rotating star. However, at a rotation rate five times higher, the values of [FORMULA] for [FORMULA] are reduced by [FORMULA] µHz, which corresponds to about 0.5 Gy. Thus the effect has to be kept in mind when we will have small separations data for other stars. For the sun, the magnetic effects in high activity years are significant but they may easily be purged (Dziembowski & Goode, 1997). The problem does not concern the frequencies used in the present paper because we used data from 1996/97 season when the solar activity was at its minimum.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999