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<H2><A NAME="SECTION0004153000000000000000">SLA_REFCOQ - Refraction Constants (fast)</A>
<A NAME="xref_SLA_REFCOQ"> </A><A NAME="SLA_REFCOQ"> </A>
</H2>
<DL>
<DT><STRONG>ACTION:</STRONG>
<DD>Determine the constants <I>a</I> and <I>b</I> in the
atmospheric refraction model
<IMG WIDTH="164" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img148.gif"
ALT="$\Delta \zeta = a \tan \zeta + b \tan^{3} \zeta$">, where <IMG WIDTH="11" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img149.gif"
ALT="$\zeta$"> is the <I>observed</I> zenith distance
(<I>i.e.</I> affected by refraction) and <IMG WIDTH="24" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img150.gif"
ALT="$\Delta \zeta$"> is
what to add to <IMG WIDTH="11" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img149.gif"
ALT="$\zeta$"> to give the <I>topocentric</I>
(<I>i.e. in vacuo</I>) zenith distance. (This is a fast
alternative to the sla_REFCO routine - see notes.)
<P> <DT><STRONG>CALL:</STRONG>
<DD><TT>CALL sla_REFCOQ (TDK, PMB, RH, WL, REFA, REFB)</TT>
<P> </DL>
<P> <DL>
<DT><STRONG>GIVEN:</STRONG>
<DD>
<BR>
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>TDK</EM></TD>
<TH ALIGN="LEFT"><B>D</B></TH>
<TD ALIGN="LEFT" NOWRAP>ambient temperature at the observer (degrees K)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>PMB</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP>pressure at the observer (mB)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>RH</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP>relative humidity at the observer (range 0-1)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>WL</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP>effective wavelength of the source (<IMG WIDTH="26" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img21.gif"
ALT="$\mu{\rm m}$">)</TD>
</TR>
</TABLE></DL>
<P> <DL>
<DT><STRONG>RETURNED:</STRONG>
<DD>
<BR>
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>REFA</EM></TD>
<TH ALIGN="LEFT"><B>D</B></TH>
<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>NOTES:</STRONG>
<DD><DL COMPACT>
<DT>1.
<DD>The radio refraction is chosen by specifying WL >100 <IMG WIDTH="26" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img21.gif"
ALT="$\mu{\rm m}$">.<DT>2.
<DD>The model is an approximation, for moderate zenith distances,
to the predictions of the sla_REFRO routine. The approximation
is maintained across a range of conditions, and applies to
both optical/IR and radio.
<DT>3.
<DD>The algorithm is a fast alternative to the sla_REFCO routine.
The latter calls the sla_REFRO routine itself: this involves
integrations through a model atmosphere, and is costly in
processor time. However, the model which is produced is precisely
correct for two zenith distances (<IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img154.gif"
ALT="$45^\circ$"> and <IMG WIDTH="40" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img161.gif"
ALT="$\sim\!76^\circ$">) and at other zenith distances is limited in accuracy only by the
<IMG WIDTH="164" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img148.gif"
ALT="$\Delta \zeta = a \tan \zeta + b \tan^{3} \zeta$"> formulation
itself. The present routine is not as accurate, though it
satisfies most practical requirements.
<DT>4.
<DD>The model omits the effects of (i) height above sea level (apart
from the reduced pressure itself), (ii) latitude (<I>i.e.</I> the
flattening of the Earth) and (iii) variations in tropospheric
lapse rate.
<DT>5.
<DD>The model has been tested using the following range of conditions:
<DL COMPACT>
<DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>lapse rates 0.0055, 0.0065, 0.0075 degrees K per metre
<DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>latitudes <IMG WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img1.gif"
ALT="$0^{\circ}$">, <IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img162.gif"
ALT="$25^\circ$">, <IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img163.gif"
ALT="$50^\circ$">, <IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img164.gif"
ALT="$75^\circ$"> <DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>heights 0, 2500, 5000 metres above sea level
<DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>pressures mean for height -10% to +5% in steps of 5%
<DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>temperatures <IMG WIDTH="38" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img165.gif"
ALT="$-10^\circ$"> to <IMG WIDTH="38" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img166.gif"
ALT="$+20^\circ$"> with respect to
<IMG WIDTH="34" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img167.gif"
ALT="$280^\circ$">K at sea level
<DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>relative humidity 0, 0.5, 1
<DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>wavelength 0.4, 0.6, ... <IMG WIDTH="34" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img168.gif"
ALT="$2\mu{\rm m}$">, + radio
<DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>zenith distances <IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img39.gif"
ALT="$15^\circ$">, <IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img154.gif"
ALT="$45^\circ$">, <IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img164.gif"
ALT="$75^\circ$"></DL>
For the above conditions, the comparison with sla_REFRO
was as follows:
<P> <BR>
<BR>
<BR>
<P>
<TABLE CELLPADDING=3 BORDER="1">
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP> </TD>
<TH ALIGN="RIGHT" NOWRAP><I>worst</I></TH>
<TH ALIGN="RIGHT" NOWRAP><I>RMS</I></TH>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>optical/IR</TD>
<TD ALIGN="RIGHT" NOWRAP>62</TD>
<TD ALIGN="RIGHT" NOWRAP>8</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>radio</TD>
<TD ALIGN="RIGHT" NOWRAP>319</TD>
<TD ALIGN="RIGHT" NOWRAP>49</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP> </TD>
<TD ALIGN="RIGHT" NOWRAP>mas</TD>
<TD ALIGN="RIGHT" NOWRAP>mas</TD>
</TR>
</TABLE>
<P> <BR>
<BR>
<BR>
<BR>
<P>
For this particular set of conditions:
<DL COMPACT>
<DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>lapse rate <IMG WIDTH="86" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img169.gif"
ALT="$6.5^\circ K km^{-1}$"><DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>latitude <IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img163.gif"
ALT="$50^\circ$"> <DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>sea level
<DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>pressure 1005mB
<DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>temperature <IMG WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img170.gif"
ALT="$7^\circ$">C
<DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>humidity 80%
<DT><IMG WIDTH="7" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img72.gif"
ALT="$\cdot$"><DD>wavelength 5740A
</DL>
the results were as follows:
<P> <BR>
<BR>
<BR>
<P>
<TABLE CELLPADDING=3 BORDER="1">
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP COLSPAN=1><IMG WIDTH="11" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img149.gif"
ALT="$\zeta$"></TD>
<TD ALIGN="CENTER" NOWRAP COLSPAN=1>sla_REFRO</TD>
<TD ALIGN="CENTER" NOWRAP COLSPAN=1>sla_REFCOQ</TD>
<TD ALIGN="CENTER" NOWRAP COLSPAN=1>Saastamoinen</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>10</TD>
<TD ALIGN="RIGHT" NOWRAP>10.27</TD>
<TD ALIGN="RIGHT" NOWRAP>10.27</TD>
<TD ALIGN="RIGHT" NOWRAP>10.27</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>20</TD>
<TD ALIGN="RIGHT" NOWRAP>21.19</TD>
<TD ALIGN="RIGHT" NOWRAP>21.20</TD>
<TD ALIGN="RIGHT" NOWRAP>21.19</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>30</TD>
<TD ALIGN="RIGHT" NOWRAP>33.61</TD>
<TD ALIGN="RIGHT" NOWRAP>33.61</TD>
<TD ALIGN="RIGHT" NOWRAP>33.60</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>40</TD>
<TD ALIGN="RIGHT" NOWRAP>48.82</TD>
<TD ALIGN="RIGHT" NOWRAP>48.83</TD>
<TD ALIGN="RIGHT" NOWRAP>48.81</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>45</TD>
<TD ALIGN="RIGHT" NOWRAP>58.16</TD>
<TD ALIGN="RIGHT" NOWRAP>58.18</TD>
<TD ALIGN="RIGHT" NOWRAP>58.16</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>50</TD>
<TD ALIGN="RIGHT" NOWRAP>69.28</TD>
<TD ALIGN="RIGHT" NOWRAP>69.30</TD>
<TD ALIGN="RIGHT" NOWRAP>69.27</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>55</TD>
<TD ALIGN="RIGHT" NOWRAP>82.97</TD>
<TD ALIGN="RIGHT" NOWRAP>82.99</TD>
<TD ALIGN="RIGHT" NOWRAP>82.95</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>60</TD>
<TD ALIGN="RIGHT" NOWRAP>100.51</TD>
<TD ALIGN="RIGHT" NOWRAP>100.54</TD>
<TD ALIGN="RIGHT" NOWRAP>100.50</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>65</TD>
<TD ALIGN="RIGHT" NOWRAP>124.23</TD>
<TD ALIGN="RIGHT" NOWRAP>124.26</TD>
<TD ALIGN="RIGHT" NOWRAP>124.20</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>70</TD>
<TD ALIGN="RIGHT" NOWRAP>158.63</TD>
<TD ALIGN="RIGHT" NOWRAP>158.68</TD>
<TD ALIGN="RIGHT" NOWRAP>158.61</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>72</TD>
<TD ALIGN="RIGHT" NOWRAP>177.32</TD>
<TD ALIGN="RIGHT" NOWRAP>177.37</TD>
<TD ALIGN="RIGHT" NOWRAP>177.31</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>74</TD>
<TD ALIGN="RIGHT" NOWRAP>200.35</TD>
<TD ALIGN="RIGHT" NOWRAP>200.38</TD>
<TD ALIGN="RIGHT" NOWRAP>200.32</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>76</TD>
<TD ALIGN="RIGHT" NOWRAP>229.45</TD>
<TD ALIGN="RIGHT" NOWRAP>229.43</TD>
<TD ALIGN="RIGHT" NOWRAP>229.42</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>78</TD>
<TD ALIGN="RIGHT" NOWRAP>267.44</TD>
<TD ALIGN="RIGHT" NOWRAP>267.29</TD>
<TD ALIGN="RIGHT" NOWRAP>267.41</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>80</TD>
<TD ALIGN="RIGHT" NOWRAP>319.13</TD>
<TD ALIGN="RIGHT" NOWRAP>318.55</TD>
<TD ALIGN="RIGHT" NOWRAP>319.10</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="RIGHT" NOWRAP>deg</TD>
<TD ALIGN="RIGHT" NOWRAP>arcsec</TD>
<TD ALIGN="RIGHT" NOWRAP>arcsec</TD>
<TD ALIGN="RIGHT" NOWRAP>arcsec</TD>
</TR>
</TABLE>
<P> <BR>
<BR>
<BR>
<BR>
<P>
The values for Saastamoinen's formula (which includes terms
up to <IMG WIDTH="33" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img171.gif"
ALT="$\tan^5$">) are taken from Hohenkerk and Sinclair (1985).
<P>
The results from the much slower but more accurate sla_REFCO
routine have not been included in the tabulation as they are
identical to those in the sla_REFRO column to the
<IMG WIDTH="31" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img158.gif"
ALT="$0\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.01$"><P>
resolution used.
<DT>6.
<DD>Outlandish input parameters are silently limited
to mathematically safe values. Zero pressure is permissible,
and causes zeroes to be returned.
<DT>7.
<DD>The algorithm draws on several sources, as follows:
<UL>
<LI> The formula for the saturation vapour pressure of water as
a function of temperature and temperature is taken from
expressions A4.5-A4.7 of Gill (1982).
<LI> The formula for the water vapour pressure, given the
saturation pressure and the relative humidity is from
Crane (1976), expression 2.5.5.
<LI> The refractivity of air is a function of temperature,
total pressure, water-vapour pressure and, in the case
of optical/IR but not radio, wavelength. The formulae
for the two cases are developed from the Essen and Froome
expressions adopted in Resolution 1 of the 12th International
Geodesy Association General Assembly (1963).
</UL>
The above three items are as used in the sla_REFRO routine.
<UL>
<LI> The formula for <IMG WIDTH="91" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img172.gif"
ALT="$\beta~(=H_0/r_0)$"> is
an adaption of expression 9 from Stone (1996). The
adaptations, arrived at empirically, consist of (i) a
small adjustment to the coefficient and (ii) a humidity
term for the radio case only.
<LI> The formulae for the refraction constants as a function of
<I>n</I>-1 and <IMG WIDTH="12" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img173.gif"
ALT="$\beta$"> are from Green (1987), expression 4.31.
</UL></DL></DL>
<P> <DL>
<DT><STRONG>REFERENCES:</STRONG>
<DD><DL COMPACT>
<DT>1.
<DD>Crane, R.K., Meeks, M.L. (ed), ``Refraction Effects in
the Neutral Atmosphere'',
<I>Methods of Experimental Physics: Astrophysics 12B,</I>
Academic Press, 1976.
<DT>2.
<DD>Gill, Adrian E., <I>Atmosphere-Ocean Dynamics,</I>
Academic Press, 1982.
<DT>3.
<DD>Hohenkerk, C.Y., & Sinclair, A.T., NAO Technical Note
No. 63, 1985.
<DT>4.
<DD>International Geodesy Association General Assembly, Bulletin
Géodésique <B>70</B> p390, 1963.
<DT>5.
<DD>Stone, Ronald C., P.A.S.P. <B>108</B> 1051-1058, 1996.
<DT>6.
<DD>Green, R.M., <I>Spherical Astronomy,</I> Cambridge
University Press, 1987.
</DL></DL>
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<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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