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

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3. Inference from seismically determined solar parameters

Solar age cannot be directly determined by means of helioseismology. In all the approaches, including this one, families of solar models with various assumed ages are calculated and [FORMULA] is determined by means of a comparison of more direct seismic observables. The most direct are the frequencies, but with no additional assumptions one may use the density, [FORMULA], or the squared isothermal sound speed, [FORMULA], determined by means of the frequency inversion. These two functions are linked by the hydrostatic equilibrium condition. From u, [FORMULA] and their derivatives one may determine a number of other useful structural functions. If, in addition, we assume equation of state (EOS) data, we may infer the values of [FORMULA] and [FORMULA]. The last two seismic observables were used by Weiss & Schlattl (1998) in their first attempt at the solar age determination. They subsequently considered also other quantities. There are various possibilities. We regard a comparison of the sound speed as most revealing. The value of [FORMULA] does not contain independent information and, since it is determined from the derivative of u, it is less accurate.

3.1. The sound speed

The result of the inversion for [FORMULA], the relative difference in [FORMULA] between the sun and model 0, is shown in Fig. 1, where [FORMULA] corresponds to the temperature minimum. In the same plot we show the difference in u between some other models (see Table 1) and model 0.

[FIGURE] Fig. 1. Relative differences in u between the sun and Model 0 determined by means of helioseismic inversion. Also shown are the differences between different models and Model 0. The vertical error bars (visible only for the inner most points) reflect only measurement errors. True uncertainty of the inversion is much greater (Degl'Innocenti et al., 1997).

The solar data are from the inversion of the frequency data obtained with the MDI instrument (Rhodes et al. 1997) and the GOLF instrument (Gabriel et al. 1997) on board of the SOHO spacecraft. The first data set contains modes with the [FORMULA] values from 0 to 250. We ignored the f-modes, and we were left with the frequencies of 1890 p-modes with [FORMULA] up to 184. The second set contains 153 frequency data for modes with [FORMULA] degrees up to 5. The data were combined into a set of 1945 p-mode frequencies. The inversion was done by means of the SOLA method (Pijpers & Thompson, 1992; Dziembowski et al., 1994).

One sees in Fig. 1 that the difference in u through most of the sun interior seems to favor higher age. However, the quantitative answer depends on the choice of the location in the sun's interior. In the region [FORMULA], u is almost independent of age. In the inner core the dependence on age is the strongest. Older models have higher helium abundance, hence higher mean molecular weight. This effect dominates the sound speed behavior. Unfortunately, results of seismic sounding of the inner core are unreliable.

An assessment of the solar age based on [FORMULA] is sensitive to the assumed metal abundance in the model. An increase of the [FORMULA] parameter by 10% has a similar effect on the sound speed in the outer part of the radiative interior as a 6% increase of age.

The implication about the age based on [FORMULA] depends also on other ingredients of the solar model construction such as opacity, nuclear reaction rates and diffusion coefficients. We will not consider all these effects in detail. In Fig. 2 we show few examples of the difference in u between models calculated assuming [FORMULA]. Model JCD (Christensen-Dalsgaard et al., 1996) is the closest to the sun. The improvement in the opacity data spoils this good agreement. However, as the comparison with Model 3 shows, the difference in opacity does not explain the whole difference between JCD and model 0. We suspect that the remaining difference in u may be caused by the difference in the treatment of the element settling. The difference between the model denoted FR97 (Ciacio et al., 1997) and model 0 in the outer part of the radiative interior is very small. A comparison of the plots in Figs. 1 and 2 shows that the revision the OPAL has resulted in changes of u similar to lowering [FORMULA] by 6%. Thus, with earlier OPAL opacities we will get solar age lower by 3.6% (0.16 Gy).

[FIGURE] Fig. 2. Relative differences in u between the sun and Model 0 determined by means of helioseismic inversion are compared with the differences between different models and Model 0. Model JCD (Christensen-Dalsgaard et al., 1996) and Model 3 were calculated with OPAL92 and that denoted FR97 with OPAL96 opacities.

In all the cases, values of [FORMULA] in the outer part of the radiative interior point to [FORMULA]. The difference is model dependent. We will quantify it in Sect. 3.1. Finally, let us point out that the result of inversion shown in Figs. 1 and 2 looks very similar to that of Brun et al. (1998) except for [FORMULA]. The implication concerning the solar age based on [FORMULA] from their inversion would therefore be similar to ours.

3.2. Helium abundance

The value of [FORMULA] as determined from the same data and with the same reference model is 0.249. It is by 0.006 larger than in our standard model and by 0.010 larger than in the model with age 5 Gy. The age inferred from [FORMULA] would be about 4 Gy. The number is in a reasonable agreement with Weiss & Schlattl (1998). Clearly, there are conflicting conclusions about [FORMULA] from [FORMULA] and [FORMULA]. Not surprisingly Weiss & Schlattl (1998) find rather large minimum values of [FORMULA] in their multi-parameter fits.

Adopting higher [FORMULA] values allows us to reduce the contradiction. We see in Table 1 that in model with [FORMULA], [FORMULA] is close to [FORMULA], and in Fig. 1 we see that [FORMULA] is closer to one inferred by the inversion. A similar, though smaller, effect is obtained by adopting the previous version of the OPAL opacities. Still, the most significant difference in u in the outermost part of the radiative interior cannot be removed by higher [FORMULA]. Modification in opacity is an option but it must be quite different from the return to earlier version of OPAL. Gough et al. (1996) suggested that the spike of [FORMULA] at [FORMULA] may be a consequence of neglecting a macroscopic mixing below the base of convective zone in the standard solar models. Models including this effect have been constructed by Richard et al. (1996). Such models explain the deficit of Li abundance in the sun's photosphere and yield better agreement with seismic determination of u near the base of convective zone. The effects leads also to an increase of Y in the envelope. Macroscopic mixing is a hypothetical effect and its description involves free parameters, so it is not included in the standard models. The effect most likely takes place. For present application this means that [FORMULA] and u in the outer part of the envelope are not safe probes of the solar age. In addition, there are difficulties with estimating uncertainties in seismic determination of Y following from inadequacies in the thermodynamical parameters.

3.3. Estimates of Estimates of tseis based on selected values of u and yph

For the sake of illustration of the discrepancies we will give estimates of [FORMULA] based on different observables. Unlike Weiss & Schlattl (1998), we will not try to fit simultaneously more than one parameter because our aim is only to quantify the problems with the assessment of the solar age with the method reviewed in this section. Furthermore, the meaning of the formal [FORMULA]-minimization procedure is problematic in the present case, as in fact Weiss & Schlattl (1998) emphasized.

In Table 2 we provide a list of the selected observables, Q with errors, its estimated [FORMULA] uncertainty [FORMULA], and the quantity

[EQUATION]

which measures sensitivity of each observable to the solar age. The values of [FORMULA] and [FORMULA] are from the inversion described in Sect. 3.1. The estimates of uncertainties, [FORMULA], are from Degl'Innocenti et al. (1997).

In Table 2 we list the values of the selected observables calculated in the three standard solar models.

In Table 4 we provide the values of t inferred from the differences between the sun and the models by using the various observables Q. The numbers mostly quantify only the effects discussed earlier in this section.


[TABLE]

Table 2. Selected seismic observables and their [FORMULA] uncertainties, [FORMULA].



[TABLE]

Table 3. Values of [FORMULA] and [FORMULA]



[TABLE]

Table 4. Helioseismic estimate of solar age (Gy), as inferred from the differences [FORMULA], calculated for different SSMs.


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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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