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Up: Modeling echelle spectrographs

B. Evaluation of the rotated echelle relation

In order to obtain an analytical least-squares solution we use the simplified rotated echelle relation. The residual tex2html_wrap_inline1488 of a given line of index tex2html_wrap_inline1490 at position (xi,yi) and of wavelength tex2html_wrap_inline1494, and the relative order number pi, is given by

One has to consider the case where the order number m is also an unknown quantity since it has useful practical applications. We rewrite, using tex2html_wrap_inline1500

By determining the partial derivatives of Ri2 with respect to each parameter (m,a,b,c) we obtain a system of the form A x = B with A a tex2html_wrap_inline1510 matrix and B a vector such as



with the quantities Sx, etc... being defined as




and similar formulae for Sy, tex2html_wrap_inline1518, tex2html_wrap_inline1520, tex2html_wrap_inline1522, tex2html_wrap_inline1524, tex2html_wrap_inline1526, Sy2, tex2html_wrap_inline1530, tex2html_wrap_inline1532, tex2html_wrap_inline1534.

The full system was solved using mathematical packages, and solutions for subsets with fixed parameters were determined with a view towards robust techniques for practical calibration procedures.

Copyright by the European Southern Observatory (ESO)