<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <!--Converted with LaTeX2HTML 98.1p1 release (March 2nd, 1998) originally by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds * revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan * with significant contributions from: Jens Lippmann, Marek Rouchal, Martin Wilck and others --> <HTML> <HEAD> <TITLE>Estimating the angle of rotation</TITLE> <META NAME="description" CONTENT="Estimating the angle of rotation"> <META NAME="keywords" CONTENT="vol2"> <META NAME="resource-type" CONTENT="document"> <META NAME="distribution" CONTENT="global"> <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1"> <LINK REL="STYLESHEET" HREF="vol2.css"> <LINK REL="next" HREF="node162.html"> <LINK REL="previous" HREF="node160.html"> <LINK REL="up" HREF="node158.html"> <LINK REL="next" HREF="node162.html"> </HEAD> <BODY > <!--Navigation Panel--> <A NAME="tex2html3447" HREF="node162.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="icons.gif/next_motif.gif"></A> <A NAME="tex2html3444" HREF="node158.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="icons.gif/up_motif.gif"></A> <A NAME="tex2html3438" HREF="node160.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="icons.gif/previous_motif.gif"></A> <A NAME="tex2html3446" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="icons.gif/contents_motif.gif"></A> <BR> <B> Next:</B> <A NAME="tex2html3448" HREF="node162.html">Identification loop</A> <B> Up:</B> <A NAME="tex2html3445" HREF="node158.html">Wavelength Calibration</A> <B> Previous:</B> <A NAME="tex2html3439" HREF="node160.html">The Echelle Relation</A> <BR> <BR> <!--End of Navigation Panel--> <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: $x_r = x \cos \alpha + y \sin \alpha$ --> <IMG WIDTH="161" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="img307.gif" ALT="\( x_r = x \cos \alpha + y \sin \alpha \)"> <LI> <!-- 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" SRC="img311.gif" ALT="$ m.\lambda = f(x,m)$">as soon as a sufficient number of identifications has been performed. <P> <HR> <!--Navigation Panel--> <A NAME="tex2html3447" HREF="node162.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="icons.gif/next_motif.gif"></A> <A NAME="tex2html3444" HREF="node158.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="icons.gif/up_motif.gif"></A> <A NAME="tex2html3438" HREF="node160.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="icons.gif/previous_motif.gif"></A> <A NAME="tex2html3446" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="icons.gif/contents_motif.gif"></A> <BR> <B> Next:</B> <A NAME="tex2html3448" HREF="node162.html">Identification loop</A> <B> Up:</B> <A NAME="tex2html3445" HREF="node158.html">Wavelength Calibration</A> <B> Previous:</B> <A NAME="tex2html3439" HREF="node160.html">The Echelle Relation</A> <!--End of Navigation Panel--> <ADDRESS> <I>Petra Nass</I> <BR><I>1999-06-15</I> </ADDRESS> </BODY> </HTML>