Apply Refraction to ZD

Adjust an unrefracted zenith distance to include the effect of atmospheric refraction, using the simple Δζ = a tan ζ + b tan 3ζ model.



unrefracted zenith distance of the source (radians)



tan ζ coefficient (radians)



tan 3ζ coefficient (radians)




refracted zenith distance (radians)

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 Δζ = a tan ζ + b tan 3ζ 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 1011 arcsecond level for ζ = 80. The inherent accuracy of the model is, of course, far worse than this – see the documentation for sla_REFCO for more information.
At ζ = 83, the rapidly-worsening Δζ = a tan ζ + b tan 3ζ model is abandoned and an empirical formula takes over:
Δζ = F 0.55445 0.01133E + 0.00202E2 1 + 0.28385E + 0.02390E2

where E = 90ζtrue and F is a factor chosen to meet the Δζ = a tan ζ + b tan 3ζ formula at ζ = 83.

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:

ζobs error
80 ′′07
81 ′′13
82 ′′24
83 ′′47
84 ′′62
85 ′′64
86 8
87 10
88 15
89 30
90 60
91 150 < high-altitude
92 400 < sites only
For radio wavelengths the errors are typically 50% larger than the optical figures and by ζ = 85 are twice as bad, worsening rapidly below that. To maintain 1 accuracy down to ζ = 85 at the Green Bank site, Condon (2004) has suggested amplifying the amount of refraction predicted by sla_REFZ below 10.8 elevation by the factor (1 + 0.00195 (10.8 Etopo)), where Etopo 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.

See also the routine sla_REFV, which performs the adjustment in [x, y, z], and with the emphasis on speed rather than numerical accuracy.
Condon, J.J., Refraction Corrections for the GBT, PTCS/PN/35.2, NRAO Green Bank, 2004.