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<H2><A NAME="SECTION001162000000000000000">
The Echelle Relation</A>
</H2>
<P>
The wavelength calibration involves a physical equation, the echelle relation
and regression analysis to achieve estimates of the dispersion relation. Provided
that the echelle dispersion is performed with a grating, any echelle spectrum can
be calibrated usually with four lines used as pre-identifications and
a catalog of laboratory wavelengths associated to the calibration lamp. The
achieved accuracy is usually in the range 0.2 - 0.02 pixel. Accuracy can be
improved by selecting lines of a sufficient signal-to-noise ratio and using
a line catalog sorted for blends for the specific spectral resolution of the
instrument.
<P>
The echelle relation derives from the grating dispersion relation : <BR>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\sin i + \sin \theta = k . m . \lambda
\end{displaymath} -->
<IMG
WIDTH="190" HEIGHT="19"
SRC="img299.gif"
ALT="\begin{displaymath}\sin i + \sin \theta = k . m . \lambda \end{displaymath}">
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<BR CLEAR="ALL">
<P></P>
<P>
with <I>k</I> the grating constant, <I>m</I> the order number, and <IMG
WIDTH="42" HEIGHT="56" ALIGN="BOTTOM" BORDER="0"
SRC="img300.gif"
ALT="$\lambda$">
the wavelength.
The cross-disperser displaces successive orders vertically with respect
to one another. For a given position <I>x</I> on the frame, we have : <BR>
<P>
<DIV ALIGN="CENTER">
<!-- MATH: $m . \lambda = cste(x)$ -->
<IMG
WIDTH="95" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img301.gif"
ALT="\( m . \lambda = cste(x) \)">
(Echelle Relation)
</DIV>
<P>
The acurracy of this relation is limited by optical aberrations and
optical misalignments, which make it only useful to initialise the
calibration process by reducing the number of identifications necessary
to determine this one-dimensional relation, expressed as a polynomial
of low degree <I>N</I> like:
<P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\lambda(m,x) = \frac{1}{m}.\sum_{i=0}^{N} a_{mi}.x^{i}
\end{displaymath} -->
<IMG
WIDTH="222" HEIGHT="74"
SRC="img302.gif"
ALT="\begin{displaymath}\lambda(m,x) = \frac{1}{m}.\sum_{i=0}^{N} a_{mi}.x^{i} \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P>
The two major limits of accuracy of the echelle relation are:
<UL>
<LI><B>Optical aberrations:</B> the echelle relation does not
include the effect of optical
aberrations which will displace the lines in the frame and then
become a source of unaccuracy when attempting to estimate the echelle
relation parameters from a calibration frame. This contributor however will be
partially removed by using an appropriate model to fit the echelle
relation, like a polynomial of sufficient degree.
<P>
<LI><B>Optical misalignments:</B> Optical misalignments occur between
echelle grating and cross-disperser between this latter and detector.
The effective misalignment angle can be up to a few degrees (usually less than 3).
Over many hundreds pixels, the misalignment error amounts to systematic
errors of many pixels, far beyond the seeked accuracy. it is therefore
necessary to correct for any rotation of the detector.
</UL>
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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