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

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7. Dependence on height in the atmosphere

The center-to-limb measurements mix two different types of information: (i) the variation of the vertical and horizontal velocity components, [FORMULA] and [FORMULA], due to the projection effect, and (ii) the increasing height for observations off disk center due to the inclined line of sight which moves the optical depth, where the line core is formed, outwards. This opacity effect depends on the spectral line used for the velocity measurements. The MDI velocity data are measured with the Ni I line at 676.8 nm. We have calculated the photospheric line-depression contribution function of this line as described by Grossmann-Doerth (1994). The three curves correspond to line center, line wing at half maximum and far wing at an intensity of 95% of the adjacent continuum. The contribution function is rather broad, which makes the line rather insensitive to an inclination of the line of sight. MDI uses a range of [FORMULA] pm centered around the line core for the Doppler measurements (Scherrer et al. 1995), so the velocity measurements are height-averaged over a significant range of optical depths.

[FIGURE] Fig. 8. Photospheric line-depression contribution function of the Ni I line at 676.8 nm, for the line core and for [FORMULA] and [FORMULA] pm from the line center

The projection effects and the effects of non-vertical motion can be described by the simple geometrical model outlined above, in combination with the theoretical model of adiabatic waves in an isothermal atmosphere. For a first discussion it may suffice to assume that the optical depth varies in proportion to [FORMULA]. The range [FORMULA] then corresponds to a height range of [FORMULA] km; to be specific, we take the 115-km range from [FORMULA] to [FORMULA] in model C of Vernazza et al. (1981). We interpolate this range for the values [FORMULA] in steps of [FORMULA]. With these values, and for a given mode and given horizontal wave number, we then determine the vertical wave number and the angle [FORMULA] according to (10) and (11), and finally the line-of-sight power according to (8). The factor [FORMULA] in that equation is


This factor comprises the amplitude increase that is due to the decrease of the density, and the amplitude decrease that is due to the evanescent character of the waves. In all cases considered here the net effect is [FORMULA]; this factor alone would therefore cause an increase of the mode power with decreasing [FORMULA], in particular for the high frequencies above the acoustic cut-off where [FORMULA].

The second factor of (8) depends explicitly on [FORMULA]. This factor causes an opposite trend in [FORMULA]. Indeed this trend of decreasing line-of-sight amplitude always wins, except for the f mode very close to the limb.

For the two horizontal wave numbers [FORMULA] Mm-1 and [FORMULA] Mm-1, the centers of the two [FORMULA] ranges considered in the data analysis above, we have calculated the line-of-sight power as a function of the mode and position on the disk. For [FORMULA] Mm-1 the results are shown in Fig. 9; this figure must be compared to the observational results shown in Fig. 4b. There is qualitative agreement in that the power decrease towards the limb is weakest for the f mode, and that there is a reversal in the sequence of p modes at some intermediate mode number: At [FORMULA], for example, the power (relative to disk center) has a minimum at p2 in the observational result, and at p4 in the theoretical model. The occurrence of such a minimum is a consequence of the combined effects of the inclination [FORMULA] of the oscillation velocity vector and the dependence on height in the atmosphere. We conclude that both effects must be taken into account in an analysis of the center-to-limb variation of the wave spectrum, although a more detailed analysis might be necessary for a better agreement.

[FIGURE] Fig. 9. Theoretical power distribution for [FORMULA] Mm-1

For the larger horizontal wave number [FORMULA] Mm-1 a similar comparison is more difficult. At this wave number there is almost no significant power in modes above p4, which is the reason why these modes are not represented in Fig. 5. In addition, the theoretical model of evanescent waves in an isothermal atmosphere obviously is not applicable for modes with a frequency above the acoustic cut-off.

Generally, the observed velocity appears to be more vertical in the data than predicted by our simple model. Possible reasons are plentiful: As far as the model is concerned, the real solar atmosphere is not isothermal, and the waves may behave in a non-adiabatic manner. For the determination of the dependence on height a detailed model for the center-to-limb variation of the Ni I 676.8-nm line profile may be required, possibly in combination with a velocity weighting function such as proposed by Beckers & Milkey (1975).

Schou & Bogart (1998), using MDI data obtained from May to July 1996, also investigated the contribution of the horizontal velocity component to the observed power at various positions on the disc. They found good agreement for the difference between the f and p1 anisotropies with values expected from the approximation [FORMULA] (their Fig. 7, a and b, where [FORMULA] corresponds to our [FORMULA]); for the difference between p1 and p2 they suggest a similar agreement, but in this case their `expected' result (the solid curves in their Fig. 7, c and d) appears to be too large by a factor of [FORMULA]. Schou & Bogart conjecture that intrumental imperfections may lead to an imperfect point-spread function and so prevent the extraction of good absolute anisotropy values. This may in fact be another reason for the sometimes bad agreement between model and measurement in the present study. The difficulty may occur in particular near the limb, where the resolution becomes inferior, and in particular for the f mode with its rather small amplitude. In any case, for the differences between modes we obtain reasonable results, even for modes higher than p2, - cf. Figs. 9 and 4b.

We finally remark that the present study confirms another result of Stix & Wöhl (1974): Horizontal sound waves, which appeared rather prominent in models that attributed the solar oscillations to local excitation by granules (Meyer & Schmidt 1967, Stix 1970), are absent. We find no power along the location of this mode, [FORMULA], cf. the lower right corner of Fig. 2. This result seems interesting in view of the revival of modified local excitation models (Goode et al. 1992, Rimmele et al. 1995).

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© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999