The Applicability of the Fourier Convoluton Theorem to the Analysis of Late-Type Stellar Spectra.

Solar flux and intensity measurements were obtained at Sacramento Peak Observatory to test the validity of the Fourier convolution method as a means of analyzing the spectral line shapes of late-type stars. Analysis of six iron lines near 6200 (ANGSTROM) shows that, in general, the convolution method is not a suitable approximation for the calculation of the flux profile. The convolution method does reasonably reproduce the line shape for some lines which appear not to vary across the disk of the sun, but does not properly calculate the central line depth of these lines. Even if a central depth correction could be found, it is difficult to predict, especially for stars other than the sun, which lines have nearly constant shapes and could be used with the convolution method. Therefore, explicit disk integrations are promoted as the only reliable method of spectral line analysis for late-type stars. Several methods of performing the disk integration are investigated. Although the Abt (1957) prescription appears suitable for the limited case studied, methods using annuli of equal area, equal flux or equal width (Soberblom, 1980) are considered better models. The model that is the easiest to use and most efficient computationally is the equal area model. Model atmosphere calculations yield values for the microturbulence and macroturbulence similar to those derived by observers. Since the depth dependence of the microturbulence is ignored in the calculations, the intensity profiles at disk center and the limb do not match the observed intensity profiles with only one set of velocity parameters. Use of these incorrectly calculated intensity profiles in the integration procedure to obtain the flux profile leads to incorrect estimates of the solar macroturbulence. It is suggested then that the previous discrepancies between resolved disk and integrated disk measurements of solar velocity fields are a result of the neglect of depth dependence in the microturbulence and therefore the neglect of center -to-limb variation of the line profiles. Center-to-limb variations of line depth and shape are considered for the modification of the rotational broadening function used in the convolution method. It has been suggested (Mihalas, 1979) that the parameterization of line depth and shape might be used successfully in conjunction with the convolution method for stars where the convolution method otherwise is expected to be an inappropriate approximation. It is shown that central depth variations alone mimic limb -darkening and do not effectively change the shape of the line broadening function, thus leading to the failure of this simple modification to reproduce observed line shapes. Stylized line shape variations are used to demonstrate why the convolution method works for fast rotators like A-type stars but not for the slow rotators like the sun.