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Astron. Astrophys. 317, 919-924 (1997)

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2. Accuracy needed in p-mode frequencies to precisely sound the core

The Sun's p-mode frequencies are the most accurately known solar parameters. For over a 1000 modes, the precision is better than 3 [FORMULA]. For comparison, the precision in the determination of the Sun's mass is [FORMULA] and that for its radius [FORMULA]. For a typical 3 mHz p-mode, this precision translates to a 0.1 µHz uncertainty in mode frequency. We shall see that this is the kind of accuracy required in low-l frequencies to test stellar evolution theory. Many low-l modes have quoted errors which are smaller (Elsworth et al. 1994).

Admissible modifications in the input to a standard solar model lead to rather subtle changes in the core's sound speed at the level of a fraction of a percent. The kind of modifications we have in mind, like changing the solar age or certain nuclear reaction cross sections, leave an unmistakable signature only in the inner core ([FORMULA] ; Dziembowski et al. 1994). To detect such effects, one would need to reach better than a 1 [FORMULA] precision in the helioseismic determination of the square of the speed of sound. This latter quantity is the one that is typically chosen to be the primary seismic probe. We use [FORMULA], the square of the isothermal speed of sound, where p is the gas pressure and [FORMULA] is the local density.

We calculated the frequency change caused by a 1 [FORMULA] increase in u in the inner core([FORMULA]). The results are shown in Fig. 1 for the lowest degree oscillations, l =0-3, which are the ones detected in whole disk observations. In the figure, we focus on the whole disk modes found by Elsworth et al. (1994). We will fix out attention on these modes throughout this paper.

[FIGURE] Fig. 1. Frequency shift caused by a 1% increase in [FORMULA] throughout the inner core([FORMULA]) is shown as a function of mode frequency for modes with l between 0 and 3.

It may seem surprising in Fig. 1 that at low frequencies the l =1 modes are more sensitive to the inner core structure than the l =0 modes. This happens because the kernel for sound speed in the inner core goes through zero for low frequency l =0 modes, whilst remaining positive for the l =1 modes. One should recall that in the asymptotic approximation, the kernel is proportional to the energy density and, therefore it is positive definite. However, that approximation fails badly in the inner core.

The probing of the core is sensitive to the differences in [FORMULA] between modes of different l at similar frequencies. Except for the highest frequency modes, the differences are less than 0.5 µHz. Still, this picture sends a rather optimistic message about the prospects for probing the inner core because these differences are large compared to the observers' errors. However, we have to convince ourselves that what we measure as a frequency difference actually reflects a core structure at variance with the solar model rather than an effect arising from near the solar surface. We shall see that the near surface effects, which vary with the solar cycle, cause frequency shifts of up to 0.3 µHz.

Probing the rotation of the solar core is more problematic. Indeed, conflicting announcements have been made concerning core rotation. The results from the IRIS network, Fossat et al. (1995) indicate that the core ([FORMULA]) rotates some 40 [FORMULA] faster than the envelope. Whereas, results from the BISON network, Elsworth et al. (1995) point to the core rotating more slowly than the envelope.

Precision requirements for measuring just the mean value of rotation in the core are indeed quite high. In Fig. 2, we show the incremental increase in rotational splitting, as a function of frequency for the lowest l modes, that would result from the aforementioned 40 [FORMULA] spin-up in core rotation. For spherical rotation, the splittings themselves are the Zeeman-like uniform spacing between adjacent m -peaks in the fine structure of individual [FORMULA] -mulitplets. In whole disk observations, peaks having [FORMULA] -odd are not seen, and, therefore, the observed separation between neighboring peaks is twice the rotational splitting. The rotational splitting is close to 0.45 µHz. Fossat et al. (1995) quote an aggregated splitting for l =1 of 0.459 [FORMULA] 0.010µHz and 0.450 [FORMULA] 0.013 for l =2. Elsworth, et al. (1995) quote errors which are somewhat larger. From Fig. 2, it is clear that the errors in the aggregated splittings are comparable to the signal that would be caused by the 40 [FORMULA] change in rotation in the core. We will see that the effect of the near surface magnetic field on the frequency splittings can be as much as an order of magnitude larger.

[FIGURE] Fig. 2. Increase in the fine structure spacing caused by a 40% increase in rotation throughout the core([FORMULA]) is shown as a function of mode frequency for modes ranging from l =1 to 3.
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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998