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

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4. Frequency shift in low degree modes induced by the NSPA

Having determined [FORMULA], we can easily evaluate the corresponding frequency shift for low degree modes using Eq. (3). But now we cannot use Eq. (5), instead we use the exact expression for Q. For the moment, we ignore [FORMULA] which would lead to l -independent frequency shifts. Such shifts do not affect the results of inversions for either structure or rotation. Fig. 4 shows the frequency shifts for the l =1-3 modes for selected m -values. In addition to the centroid shifts, we show shifts for m -values of the modes visible in whole disk observations. The size of each shift, [FORMULA], should be compared to the observational errors and to the effect of the artificial 1 [FORMULA] sound speed perturbation shown in Fig. 1. The errors quoted by Elsworth et al. (1994) for modes in the vicinity of 3 mHz are typically 0.04, 0.05, 0.08, 0.15 µHz for l =0, 1, 2 and 3, respectively. From Fig. 4, we see that the shifts in the 1986 observations were within the errors, while in 1988 the shift is comparable to the errors. In 1989 and 1990, years of high activity, the plotted shifts due the NSPA exceed the observational errors by factors as large as three. A comparison with Fig. 1 forces the conclusion that structural inversions may be significantly corrupted if one does not account for the NSPA. That is, one might well mistakenly interpret the NSPA as a perturbation of the speed of sound in the inner core. The largest shifts are observed at the highest frequencies which reflects the decrease of inertia with increasing frequency. Note from Fig. 1, that the high frequency modes are the most important ones for probing the innermost part of the Sun.

[FIGURE] Fig. 4. The frequency shift, [FORMULA], is shown as a function of mode frequency for l from 1 to 3. The shifts are for the centroid(m =0) and the modes visible in whole disk observations-those with ([FORMULA]) being even.

Care must be taken in extracting centroid frequencies from whole disk measurements. The NSPA has a strong m -dependence. In particular, for l =1 the sign of the shift is opposite for m =0 and 1. For l =2 and 3, there is an ambiguity because we see contributions from two different m -values and the contributions have opposite signs in the whole disk observations. How the various m -components contribute to measured frequencies depends on their amplitudes. If energy equipartition is assumed, then the amplitude ratios may be calculated (Christensen-Dalsgaard and Gough, 1982; Christensen-Dalsgaard, 1989). In detail, the result depends on what type of helioseismic observations are made. However, modes of higher m always have larger predicted amplitudes.

In practice, NSPA only corrupts the probing of the core. Information about the envelope relies mostly on high-l modes in which the perturbation behaves as though it were spherically symmetric. For the core, we rely on low-l modes. In this case, the centroids are shifted in an l -dependent way. Furthermore, much of our low-l information comes from whole disk observations for which peaks are seen only if [FORMULA] is even, whereas higher l data come from resolved disk observations from which all peaks, in principle, can be identified in a multiplet. For the former kind of observations, extraction of the centroid frequency is not straightforward.

4.1. Comparsion with direct observations of centroid shifts in whole disk observations

Frequency shifts in whole disk observations have been measured by Anguera Gubau et al. (1992) and by Elsworth et al. (1994). We can compare the results of their direct observations with our inferences from the BBSO data using the 1986.5 data as a reference. In Table 1, we give the average frequency shift relative to 1986.5 from the BISON data for 1988.5, 1989.5 and 1990.5. In Table 2, we show the predicted average frequency shift for the same BISON modes, but with result being calculated from the BBSO data. Here we make use of [FORMULA], as shown in Fig. 3. The averages are from modes in the middle of the 5 min band of oscillations. For l =0-1, the averages are over modes ranging in n from 17 to 24 and for l =2 the averages are over n =16-23.


Table 1. Mean frequency shifts relative to 1986.5 from BISON data


Table 2. Mean frequency shifts relative to 1986.5 calculated from BBSO data

From Anguera Gubau et al. (1992) the average shifts of frequencies between periods of high activity and low activity are 0.17, 0.46 and 0.23 µHz for l =0, 1 and 2, respectively. The corresponding averages of Elsworth et al. (1994) are 0.19, 0.19 and 0.31 µHz.

There is a very large spread in the shifts among individual modes with respect to n for all low-l data both from Anguera Gubau et al.(1992) and Elsworth et al. (1994). With caveat in mind, we conclude that there is an overall agreement between our calculated values and those from whole disk observations. Certainly, there is no evidence that the frequency shifts measured are due to anything but the NSPA. A similar conclusion was reached by Elsworth et al. (1994).

4.2. Effect of the NSPA on fine structure

In Figs. 5 and 6, we show the effect on the fine structure arising from the NSPA for l =2 and 3. Except for the lowest order modes, we see that in years of higher activity the NSPA visibily distorts the fine structure pattern from the Zeeman-like uniform spacing predicted by the linear effect of rotation. We note that the symmetric departure from uniform spacing arising from the latitudinal dependence of the rotation rate in the convective envelope is small enough that it could not be noticed in the figures. For these figures, we assumed a constant splitting due to rotation of 0.45 µHz, and added the calculated effect of centrifugal distortion.

[FIGURE] Fig. 5. The ([FORMULA]) even fine structure peaks which are those detected in whole disk observations are shown for l =2 for n =12, 17 and 22 as a function of frequency. The non-uniform spacing arises from the NSPA. Modes with higher n are more affected by the NSPA.
[FIGURE] Fig. 6. The ([FORMULA]) even fine structure peaks which are those detected in whole disk observations are shown for l =3 for n =13, 17 and 22 as a function of frequency. The non-uniform spacing arises from the NSPA. Modes with higher n are more affected by the NSPA.

This latter effect is well-approximated by


One can easily calculate that the size of the effect never exceeds 0.01 µHz, and is below the current accuracy in frequency for individual modes, but it is of the size of the errors in the splittings measured by Fossat et al. (1993).

This visible departure from uniformity would not effect the determination of the internal rotation if we could rely only on measurements of separation between fine structure components with the same value of [FORMULA]. That is, the frequency differences defined by [FORMULA]. However, for low degree modes from whole disk observations one does not rely on individual peaks, but rather the fit is to the whole multiplet, and therefore the NSPA must be taken into account.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998