### SLA_REFZ

Apply Refraction to ZD

ACTION:
Adjust an unrefracted zenith distance to include the effect of atmospheric refraction, using the simple $\Delta \zeta =atan\zeta +b{tan}^{3}\zeta$ model.
CALL:
CALL sla_REFZ (ZU, REFA, REFB, ZR)
##### GIVEN:
 ZU D unrefracted zenith distance of the source (radians) REFA D $tan\zeta$ coefficient (radians) REFB D ${tan}^{3}\zeta$ coefficient (radians)

##### RETURNED:
 ZR D refracted zenith distance (radians)

NOTES:
(1)
This routine applies the adjustment for refraction in the opposite sense to the usual one – it takes an unrefracted (in vacuo) position and produces an observed (refracted) position, whereas the $\Delta \zeta =atan\zeta +b{tan}^{3}\zeta$ 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 $1{0}^{-11}$ arcsecond level for $\zeta =8{0}^{\circ }$. The inherent accuracy of the model is, of course, far worse than this – see the documentation for sla_REFCO for more information.
(2)
At $\zeta =8{3}^{\circ }$, the rapidly-worsening $\Delta \zeta =atan\zeta +b{tan}^{3}\zeta$ model is abandoned and an empirical formula takes over:
$\Delta \zeta =F\left(\frac{{0}^{\circ }\phantom{\rule{-4.05144pt}{0ex}}.\phantom{\rule{0.21904pt}{0ex}}55445-{0}^{\circ }\phantom{\rule{-4.05144pt}{0ex}}.\phantom{\rule{0.21904pt}{0ex}}01133E+{0}^{\circ }\phantom{\rule{-4.05144pt}{0ex}}.\phantom{\rule{0.21904pt}{0ex}}00202{E}^{2}}{1+0.28385E+0.02390{E}^{2}}\right)$

where $E=9{0}^{\circ }-{\zeta }_{true}$ and $F$ is a factor chosen to meet the $\Delta \zeta =atan\zeta +b{tan}^{3}\zeta$ formula at $\zeta =8{3}^{\circ }$.

For optical/IR wavelengths, over a wide range of observer heights and corresponding temperatures and pressures, the following levels of accuracy (worst case) are achieved, relative to numerical integration through a model atmosphere:

 ${\zeta }_{obs}$ error $8{0}^{\circ }$ ′′07 $8{1}^{\circ }$ ′′13 $8{2}^{\circ }$ ′′24 $8{3}^{\circ }$ ′′47 $8{4}^{\circ }$ ′′62 $8{5}^{\circ }$ ′′64 $8{6}^{\circ }$ $8\phantom{\rule{-0.54753pt}{0ex}}$${}^{\prime }{\phantom{\rule{-1.09506pt}{0ex}}}^{\prime }$ $8{7}^{\circ }$ $10\phantom{\rule{-0.54753pt}{0ex}}$${}^{\prime }{\phantom{\rule{-1.09506pt}{0ex}}}^{\prime }$ $8{8}^{\circ }$ $15\phantom{\rule{-0.54753pt}{0ex}}$${}^{\prime }{\phantom{\rule{-1.09506pt}{0ex}}}^{\prime }$ $8{9}^{\circ }$ $30\phantom{\rule{-0.54753pt}{0ex}}$${}^{\prime }{\phantom{\rule{-1.09506pt}{0ex}}}^{\prime }$ $9{0}^{\circ }$ $60\phantom{\rule{-0.54753pt}{0ex}}$${}^{\prime }{\phantom{\rule{-1.09506pt}{0ex}}}^{\prime }$ $9{1}^{\circ }$ $150\phantom{\rule{-0.54753pt}{0ex}}$${}^{\prime }{\phantom{\rule{-1.09506pt}{0ex}}}^{\prime }$ $<$ high-altitude $9{2}^{\circ }$ $400\phantom{\rule{-0.54753pt}{0ex}}$${}^{\prime }{\phantom{\rule{-1.09506pt}{0ex}}}^{\prime }$ $<$ sites only
For radio wavelengths the errors are typically 50% larger than the optical figures and by $\zeta =8{5}^{\circ }$ are twice as bad, worsening rapidly below that. To maintain $1\phantom{\rule{-0.54753pt}{0ex}}$${}^{\prime }{\phantom{\rule{-1.09506pt}{0ex}}}^{\prime }$ accuracy down to $\zeta =8{5}^{\circ }$ at the Green Bank site, Condon (2004) has suggested amplifying the amount of refraction predicted by sla_REFZ below $10\circ \phantom{\rule{-4.05144pt}{0ex}}.\phantom{\rule{0.21904pt}{0ex}}8$ elevation by the factor $\left(1+0.00195\ast \left(10.8-{E}_{topo}\right)\right)$, where ${E}_{topo}$ is the unrefracted elevation in degrees.

The high-ZD model is scaled to match the normal model at the transition point; there is no glitch.

(3)
See also the routine sla_REFV, which performs the adjustment in $\left[\phantom{\rule{0.3em}{0ex}}x,y,z\phantom{\rule{0.3em}{0ex}}\right]$, and with the emphasis on speed rather than numerical accuracy.
REFERENCE:
Condon, J.J., Refraction Corrections for the GBT, PTCS/PN/35.2, NRAO Green Bank, 2004.