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| Paper: |
Determination of Broadening Functions Using the Singular-Value Decomposition (SVD) Technique |
| Volume: |
185, Precise Stellar Radial Velocities, IAU Colloquium 170 |
| Page: |
82 |
| Authors: |
Rucinski, S. |
| Abstract: |
BF's determined with the SVD technique are much better defined than the cross-correlation functions (CCF): While the geometric information in CCF is doubly broadened by the natural and instrumental broadenings from the template and the program spectrum (they combine quadratically), a broadening function - as a result of a true deconvolution - contains only the part which is different in both spectra. Representation of the convolution linking the sharp and broadened spectra by a linear, over-determined set of equations is particularly conducive to the SVD technique which is able to remove the linear dependencies (mostly due to the mapping of a featureless continuum) and to have control over the noise and numerical inversion errors. The SVD solution gives a clear insight into the vector bases in both spaces of the observed spectra and of the broadening functions offering full understanding of the amount of broadening information derivable from the spectra. Normally, the supposedly over-determined system turns out to be poorly conditioned, but this deficiency can be fully controlled by the SVD process. The approach also avoids the well-known traps of the Fourier-quotient deconvolution which almost always requires application of frequency filtering. The BF determinations though the SVD approach require more computer work, but offer a far superior results to the CCF's. They do not presume anything about the contributing physical processes (such as binarity in TODCOR, especially if applied without knowledge of the two contributing spectra) and are able to reproduce any BFs for binary, spotted or stellar-group (clusters, galaxies) spectra. They cannot replace the spectrum modeling and synthesis, but can be a useful tool in their preparation, as they integrate the geometrical information from a spectrum, while modeling must be usually done on a per-line basis. |
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