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<H2><A NAME="SECTION0004156000000000000000">SLA_REFZ - Apply Refraction to ZD</A>
<A NAME="xref_SLA_REFZ"> </A><A NAME="SLA_REFZ"> </A>
</H2>
<DL>
<DT><STRONG>ACTION:</STRONG>
<DD>Adjust an unrefracted zenith distance to include the effect of
atmospheric refraction, using the simple
<IMG WIDTH="164" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img148.gif"
ALT="$\Delta \zeta = a \tan \zeta + b \tan^{3} \zeta$"> model.
<P> <DT><STRONG>CALL:</STRONG>
<DD><TT>CALL sla_REFZ (ZU, REFA, REFB, ZR)</TT>
<P> </DL>
<P> <DL>
<DT><STRONG>GIVEN:</STRONG>
<DD>
<BR>
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>ZU</EM></TD>
<TH ALIGN="LEFT"><B>D</B></TH>
<TD ALIGN="LEFT" NOWRAP>unrefracted zenith distance of the source (radians)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>REFA</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP><IMG WIDTH="37" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img151.gif"
ALT="$\tan \zeta$"> coefficient (radians)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>REFB</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP><IMG WIDTH="44" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img152.gif"
ALT="$\tan^{3} \zeta$"> coefficient (radians)</TD>
</TR>
</TABLE></DL>
<P> <DL>
<DT><STRONG>RETURNED:</STRONG>
<DD>
<BR>
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>ZR</EM></TD>
<TH ALIGN="LEFT"><B>D</B></TH>
<TD ALIGN="LEFT" NOWRAP>refracted zenith distance (radians)</TD>
</TR>
</TABLE></DL>
<P> <DL>
<DT><STRONG>NOTES:</STRONG>
<DD><DL COMPACT>
<DT>1.
<DD>This routine applies the adjustment for refraction in the
opposite sense to the usual one - it takes an unrefracted
(<I>in vacuo</I>) position and produces an observed (refracted)
position, whereas the
<IMG WIDTH="164" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img148.gif"
ALT="$\Delta \zeta = a \tan \zeta + b \tan^{3} \zeta$"> model strictly
applies to the case where an observed position is to have the
refraction removed. The unrefracted to refracted case is
harder, and requires an inverted form of the text-book
refraction models; the formula used here is based on the
Newton-Raphson method. For the utmost numerical consistency
with the refracted to unrefracted model, two iterations are
carried out, achieving agreement at the 10<SUP>-11</SUP> arcsecond level
for <IMG WIDTH="56" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img208.gif"
ALT="$\zeta=80^\circ$">. The inherent accuracy of the model
is, of course, far worse than this - see the documentation for
sla_REFCO for more information.
<DT>2.
<DD>At <IMG WIDTH="56" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img209.gif"
ALT="$\zeta=83^\circ$">, the rapidly-worsening
<IMG WIDTH="164" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img148.gif"
ALT="$\Delta \zeta = a \tan \zeta + b \tan^{3} \zeta$"> model is abandoned and an empirical formula takes over:
<P>
<P ALIGN="CENTER"><IMG WIDTH="321" HEIGHT="49"
SRC="img210.gif"
ALT="\begin{displaymath}
\Delta \zeta = F \left(
\frac{0^\circ\hspace{-0.37em}.\hspa...
...hspace{0.02em}00202 E^2}
{1 + 0.28385 E +0.02390 E^2} \right) \end{displaymath}"></P>
where <IMG WIDTH="113" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img211.gif"
ALT="$E=90^\circ-\zeta_{true}$"> and <I>F</I> is a factor chosen to meet the
<IMG WIDTH="164" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img148.gif"
ALT="$\Delta \zeta = a \tan \zeta + b \tan^{3} \zeta$"> formula at <IMG WIDTH="56" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img209.gif"
ALT="$\zeta=83^\circ$">. Over a
wide range of observer heights and corresponding temperatures and
pressures, the following levels of accuracy are achieved,
relative to numerical integration through a model atmosphere:
<P> <DIV ALIGN="CENTER">
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="28" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img184.gif"
ALT="$\zeta_{obs}$"></TD>
<TH ALIGN="CENTER" NOWRAP><I>error</I></TH>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP> </TD>
<TD ALIGN="CENTER" NOWRAP> </TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img179.gif"
ALT="$80^\circ$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img185.gif"
ALT="$0\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.4$"></TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img186.gif"
ALT="$81^\circ$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img187.gif"
ALT="$0\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.8$"></TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img188.gif"
ALT="$82^\circ$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img212.gif"
ALT="$1\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.5$"></TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img190.gif"
ALT="$83^\circ$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img213.gif"
ALT="$3\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.2$"></TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img191.gif"
ALT="$84^\circ$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img214.gif"
ALT="$4\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.9$"></TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img193.gif"
ALT="$85^\circ$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img215.gif"
ALT="$5\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.8$"></TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img194.gif"
ALT="$86^\circ$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img216.gif"
ALT="$6\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.1$"></TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img196.gif"
ALT="$87^\circ$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img217.gif"
ALT="$7\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.1$"></TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img198.gif"
ALT="$88^\circ$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img43.gif"
ALT="$11\hspace{-0.05em}^{'\hspace{-0.1em}'}$"></TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img200.gif"
ALT="$89^\circ$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img218.gif"
ALT="$21\hspace{-0.05em}^{'\hspace{-0.1em}'}$"></TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img22.gif"
ALT="$90^{\circ}$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img219.gif"
ALT="$43\hspace{-0.05em}^{'\hspace{-0.1em}'}$"></TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img203.gif"
ALT="$91^\circ$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img220.gif"
ALT="$92\hspace{-0.05em}^{'\hspace{-0.1em}'}$"></TD>
<TD ALIGN="LEFT" NOWRAP>< high-altitude</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img205.gif"
ALT="$92^\circ$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG WIDTH="33" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img221.gif"
ALT="$220\hspace{-0.05em}^{'\hspace{-0.1em}'}$"></TD>
<TD ALIGN="LEFT" NOWRAP>< sites only</TD>
</TR>
</TABLE></DIV>
<DT>3.
<DD>See also the routine sla_REFV, which performs the adjustment in
<IMG WIDTH="58" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img50.gif"
ALT="$[\,x,y,z\,]$">, and with the emphasis on speed rather than numerical accuracy.
</DL></DL>
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<ADDRESS>
<I>SLALIB --- Positional Astronomy Library<BR>Starlink User Note 67<BR>P. T. Wallace<BR>12 October 1999<BR>E-mail:ptw@star.rl.ac.uk</I>
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