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<H2><A NAME="SECTION001182000000000000000"> </A>
<A NAME="blaze-correction"> </A>
<BR>
Fitting the Blaze Function
</H2>
<P>
A different approach is also available at this stage, as an
alternative to the method described in the last Section. It consists
of a correction for the blaze function by using a suitable model of
the blaze effect introduced by the echelle grating. In this approach,
no correction for the chromatic response of the instrument is applied.
It is noteworthy however to say that the standard star correction is a
much more efficient way to perform the instrumental response
correction. The model assumes a plane grating, used in near-Littrow
mode. The blaze function <I>R</I> at wavelength <IMG
WIDTH="42" HEIGHT="56" ALIGN="BOTTOM" BORDER="0"
SRC="img323.gif"
ALT="$\lambda$">
is approximated
by
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
R(\lambda) = {\sin^2 \pi \alpha X \over (\pi \alpha X)^2}
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:echelle-10"> </A><IMG
WIDTH="165" HEIGHT="63"
SRC="img324.gif"
ALT="\begin{displaymath}R(\lambda) = {\sin^2 \pi \alpha X \over (\pi \alpha X)^2}
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(7.5)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
where <IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img325.gif"
ALT="$\alpha$">
is a grating `constant' with value between 0.5 and 1, and
<!-- MATH: $X=m(1-\lambda_c(m)/\lambda)$ -->
<IMG
WIDTH="164" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img326.gif"
ALT="$X=m(1-\lambda_c(m)/\lambda)$">,
in which <I>m</I> is the order number, and
<!-- MATH: $\lambda_c(m)$ -->
<IMG
WIDTH="19" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img327.gif"
ALT="$\lambda_c(m)$">
is the central wavelength of order <I>m</I>. Both parameters are
related through the grating `constant' <I>k</I> by
<!-- MATH: $k=m\lambda_c(m)$ -->
<IMG
WIDTH="78" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img328.gif"
ALT="$k=m\lambda_c(m)$">.
This correction is done with the command <TT>RIPPLE/ECHELLE</TT>; the command
includes three methods to compute the parameters <I>k</I> and <IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img329.gif"
ALT="$\alpha$">:
<P>
The first one, method <TT>SINC</TT>, is a modification of the method
suggested by Ahmad, 1981, <I>NASA IUE Newsletter</I>, <B>14</B>, 129.
This algorithm approximates the blaze function by a sinc square and
finds the function parameters by a <I>non-linear least squares fit
to the order profile</I>. The method is suitable for objects without
strong emission or absorption features and can be used to get a first
estimation of the blaze parameters.
<P>
The second method, named <TT>OVER</TT>, is
based on Barker, 1984, <I>Astron.J.</I>, <B>89</B>, 899. This method uses the
overlapping region of adjacent orders to estimate, in a few iterations,
the parameter <I>k</I> of the blaze function which is, as before, assumed to be
a sinc square. The method works well, provided that orders are overlapping
and that there is a very good estimation of the parameter <IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img330.gif"
ALT="$\alpha$">,
assumed to be a contant.
<P>
The third method, <TT>FIT</TT>,
is an extension of the previous one. It uses, as before,
the overlapping region of the adjacent orders but has the advantage of
assuming that both parameters <I>k</I> and <IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img331.gif"
ALT="$\alpha$">
can vary. The method
minimises the difference of the corrected orders in each of the
overlapping intervals.
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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