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<H2><A NAME="SECTION001163000000000000000">
Estimating the angle of rotation</A>
</H2>
<P>
As a consequence of the cross-disperser dispersion relation, a given line
repeated in two overlapped regions of the spectrum must be found at the same
y position if spectrum and detector are perfectly aligned.
<P>
In the following, we will call a pair two occurences of the same
line in the overlapped parts of two adjacent orders.
<P>
The above condition provides a geometrical way to estimate the angle, as:
<!-- MATH: $\alpha = {\Delta y}/{\Delta x}$ -->
<IMG
WIDTH="74" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img304.gif"
ALT="$\alpha = {\Delta y}/{\Delta x}$">,
with <IMG
WIDTH="60" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img305.gif"
ALT="$\Delta y$">
the difference in <I>y</I>
position of the two occurences of the pairs and <IMG
WIDTH="60" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img306.gif"
ALT="$\Delta x$">
the difference
of their <I>x</I> positions. This method requires overlaps between orders
and is used in the calibration method <TT>PAIR</TT>.
<P>
Another approach consists of
estimating the rotation angle as the parameter for which regression
analysis provides the smallest residual for a given set of observations.
<P>
<UL>
<LI>
<!-- MATH: $x_r = x \cos \alpha + y \sin \alpha$ -->
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SRC="img307.gif"
ALT="\( x_r = x \cos \alpha + y \sin \alpha \)">
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<!-- MATH: $m.\lambda = \sum_{i=0}^{N} a_{i}.x_{r}^{i}$ -->
<IMG
WIDTH="126" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
SRC="img308.gif"
ALT="\( m.\lambda = \sum_{i=0}^{N} a_{i}.x_{r}^{i} \)">
<LI><IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img309.gif"
ALT="\( \alpha \)">
such as the standard-error is minimized
</UL>
<P>
The determination of the optimum is performed analytically and is
involved in the calibration method <TT>ANGLE</TT>. This one-dimensional
representation of the dispersion relation is used as a starting point
in the loop and is replaced by a bivariate solution
<!-- MATH: $m.\lambda = f(x,m)$ -->
<IMG
WIDTH="100" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
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ALT="$ m.\lambda = f(x,m)$">as soon as a sufficient number of identifications has been performed.
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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