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<H3><A NAME="SECTION000513100000000000000">
Refraction</A>
</H3>
The final correction is for atmospheric refraction.
This effect, which depends on local meteorological conditions and
the effective colour of the source/detector combination,
increases the observed elevation of the source by a
significant effect even at moderate zenith distances, and near the
horizon by over <IMG WIDTH="25" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img301.gif"
ALT="$0^{\circ}
\hspace{-0.37em}.\hspace{0.02em}5$">. The amount of refraction can by
computed by calling the SLALIB routine
sla_REFRO;
however,
this requires as input the observed zenith distance, which is what
we are trying to predict. For high precision it is
therefore necessary to iterate, using the topocentric
zenith distance as the initial estimate of the
observed zenith distance.
<P>
The full
sla_REFRO refraction calculation is onerous, and for
zenith distances of less than, say, <IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img164.gif"
ALT="$75^\circ$"> the following
model can be used instead:
<P>
<P ALIGN="CENTER"><IMG WIDTH="255" HEIGHT="27"
SRC="img302.gif"
ALT="\begin{displaymath}
\zeta _{vac} \approx \zeta _{obs}
+ A \tan \zeta _{obs}
+ B \tan ^{3}\zeta _{obs} \end{displaymath}"></P>
where <IMG WIDTH="29" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img303.gif"
ALT="$\zeta _{vac}$"> is the topocentric
zenith distance (i.e. <I>in vacuo</I>),
<IMG WIDTH="28" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img184.gif"
ALT="$\zeta_{obs}$"> is the observed
zenith distance (i.e. affected by refraction), and <I>A</I> and <I>B</I> are
constants, about <IMG WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img304.gif"
ALT="$60\hspace{-0.05em}^{'\hspace{-0.1em}'}$">and
<IMG WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
SRC="img305.gif"
ALT="$-0\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.06$"> respectively for a sea-level site.
The two constants can be calculated for a given set of conditions
by calling either
sla_REFCO or
sla_REFCOQ.
<P>
sla_REFCO works by calling
sla_REFRO for two zenith distances and fitting <I>A</I> and <I>B</I>
to match. The calculation is onerous, but delivers accurate
results whatever the conditions.
sla_REFCOQ uses a direct formulation of <I>A</I> and <I>B</I> and
is much faster; it is slightly less accurate than
sla_REFCO but more than adequate for most practical purposes.
<P>
Like the full refraction model, the two-term formulation works in the wrong
direction for our purposes, predicting
the <I>in vacuo</I> (topocentric) zenith distance
given the refracted (observed) zenith distance,
rather than <I>vice versa</I>. The obvious approach of
interchanging <IMG WIDTH="29" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img303.gif"
ALT="$\zeta _{vac}$"> and <IMG WIDTH="28" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img184.gif"
ALT="$\zeta_{obs}$"> and
reversing the signs, though approximately
correct, gives avoidable errors which are just significant in
some applications; for
example about
<IMG WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img76.gif"
ALT="$0\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.2$"> at <IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img174.gif"
ALT="$70^\circ$"> zenith distance. A
much better result can easily be obtained, by using one Newton-Raphson
iteration as follows:
<P>
<P ALIGN="CENTER"><IMG WIDTH="313" HEIGHT="45"
SRC="img306.gif"
ALT="\begin{displaymath}
\zeta _{obs} \approx \zeta _{vac}
- \frac{A \tan \zeta _{va...
...
{1 + ( A + 3 B \tan ^{2}\zeta _{vac} ) \sec ^{2}\zeta _{vac}}\end{displaymath}"></P>
<P>
The effect of refraction can be applied to an unrefracted
zenith distance by calling
sla_REFZ or to an unrefracted
<IMG WIDTH="58" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img50.gif"
ALT="$[\,x,y,z\,]$"> by calling
sla_REFV.
Over most of the sky these two routines deliver almost identical
results, but beyond <IMG WIDTH="56" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img209.gif"
ALT="$\zeta=83^\circ$">sla_REFV
becomes unacceptably inaccurate while
sla_REFZ
remains usable. (However
sla_REFV
is significantly faster, which may be important in some applications.)
SLALIB also provides a routine for computing the airmass, the function
sla_AIRMAS.
<P>
The refraction ``constants'' returned by
sla_REFCO and
sla_REFCOQ
are slightly affected by colour, especially at the blue end
of the spectrum. Where values for more than one
wavelength are needed, rather than calling
sla_REFCO
several times it is more efficient to call
sla_REFCO
just once, for a selected ``base'' wavelength, and then to call
sla_ATMDSP
once for each wavelength of interest.
<P>
All the SLALIB refraction routines work for radio wavelengths as well
as the optical/IR band. The radio refraction is very dependent on
humidity, and an accurate value must be supplied. There is no
wavelength dependence, however. The choice of optical/IR or
radio is made by specifying a wavelength greater than <IMG WIDTH="51" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img307.gif"
ALT="$100\mu m$">for the radio case.
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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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