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Astron. Astrophys. 346, 633-640 (1999)

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5. Center-to-limb variation

The projection procedure described above removes the foreshortening effect of the observed area and creates arrays of equal size measured on the solar surface. The measured Doppler shifts have not been transformed, i.e. we analyze the power spectra of the line-of-sight velocity.

In order to calculate the mode power, we first determined a trial frequency of the eight lowest modes at an initial wave number, say [FORMULA] Mm-1.

In intervals containing 11 frequency points centered around each trial value we then calculated the precise location of the ridges from the Fourier transform of the ridge profile within the interval: for each ridge the phase angle of the first Fourier component defines the location of the ridge maximum. This procedure is fast and insensitive to noise. The computed ridge location was then used as trial value for the adjacent [FORMULA] positions both towards larger and smaller wave numbers. The precise locations of all ridges at disk center were then used as trial values for the power spectra of the data fields outside the disk center. Although we did not expect to find a significant variation of the ridge locations across the disk, we checked the ridge positions using the method described above. The velocity power was integrated along each ridge in two intervals of the horizontal wave number, namely [FORMULA] Mm-1 and [FORMULA] Mm-1, and in a frequency interval of 0.18 mHz, centered at the ridge maximum. Fig. 3 shows the frequency integration intervals for the wave number [FORMULA] Mm-1 (averaged over two [FORMULA] points, i.e., over [FORMULA] Mm-1). We performed the same procedure also with a slightly wider interval, with no noticeable effect on the results, the additional power just shifts the curves vertically in Figs. 4a and 5a. Of course, foreshortening occurs near the limb. This effect permits full resolution of the wave number 1 Mm-1 only down to [FORMULA], and of the wave number 0.5 Mm-1 down to [FORMULA]. Figs. 4a and 5a show a clear power increase below these [FORMULA] values; hence, from our point of view, this increase is mainly noise, divided by [FORMULA].

[FIGURE] Fig. 3. Power spectrum at a wave number of [FORMULA] Mm-1 at disk center. The integration intervals of the ridges are indicated by vertical lines

[FIGURE] Fig. 4a. Center-to-limb variation of the power for the f mode and the lowest seven p modes for the low-[FORMULA] range ([FORMULA] Mm-1). Dotted lines show results for North and South hemisphere of each mode.

[FIGURE] Fig. 4b. Same as Fig. 4a, but in a linear representation. Each curve is normalized to its value at disk center.

[FIGURE] Fig. 5a. Center-to-limb variation of the power of the f mode and the lowest seven p modes, for the high-[FORMULA] range ([FORMULA] Mm-1). Dotted lines show results for North and South hemisphere of each mode.

[FIGURE] Fig. 5b. Same as Fig. 5a, but in a linear representation. Each curve is normalized to its value at disk center.

Fig. 4a displays the CLV of the power of the f mode and the lowest seven p modes for the low-[FORMULA] interval. The modes p2 and p3 have the largest power values. The horizontal line corresponds to a vertical motion with constant v, cf. Eq. (8) below. The ordinate value for this curve is arbitrarily chosen. The variation of the f mode and of the modes p4 to p6 is rather close to such a horizontal line, whereas the power of the lower p modes decreases faster towards the limb. Fig. 4b shows the same results in a linear representation, with each curve normalized to its value at disk center. The increase of power very close to the limb should be treated with some caution: the ridges become very weak towards the limb (because of the decreasing line-of-sight component); in addition, the ridges are very close to each other at low wave numbers and some "crosstalk" may occur. Figs. 5a and 5b show the CLV of the mode power for the high wave number region. As compared to Fig. 4a, the mode p4 is weaker in Fig. 5a by about one order of magnitude. The higher modes are still weaker, and are not included in Fig. 5 for this reason.

The northern and southern hemispheres have been analyzed independently with exactly the same procedure. Figs. 4 and 5 show North-South averaged curves; the results of each hemisphere are used as an estimate for the accuracy of the measurements (indicated by dotted lines around each mode), since there is no physical reason to expect a North-South asymmetry. There is, however, a moderate North-South difference of the f mode in the high-[FORMULA] range. This asymmetry includes the increase of the f-mode power at low latitude, which occurs only for the northern hemisphere (see Fig. 5). We have no explanation for this asymmetry.

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© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999
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