Adjust an unrefracted Cartesian vector to include the effect of atmospheric refraction, using the simple A tan Z $+$ B tan$\ast$$\ast$3 Z model.

void palRefv ( double vu[3], double refa, double refb, double vr[3] );

vu[3] = double (Given)

refa = double (Given)

refb = double (Given)

vr[3] = double (Returned)

• 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 A tan Z $+$ B tan$\ast$$\ast$3 Z 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 algorithm used here is equivalent to one iteration of the Newton-Raphson method applied to the above formula.

• Though optimized for speed rather than precision, the present routine achieves consistency with the refracted-to-unrefracted A tan Z $+$ B tan$\ast$$\ast$3 Z model at better than 1 microarcsecond within 30 degrees of the zenith and remains within 1 milliarcsecond to beyond ZD 70 degrees. The inherent accuracy of the model is, of course, far worse than this - see the documentation for palRefco for more information.

• At low elevations (below about 3 degrees) the refraction correction is held back to prevent arithmetic problems and wildly wrong results. For optical/IR wavelengths, over a wide range of observer heights and corresponding temperatures and pressures, the following levels of accuracy (arcsec, worst case) are achieved, relative to numerical integration through a model atmosphere:

ZD error

80 0.7 81 1.3 82 2.5 83 5 84 10 85 20 86 55 87 160 88 360 89 640 90 1100 91 1700 } relevant only to 92 2600 } high-elevation sites

The results for radio are slightly worse over most of the range, becoming significantly worse below ZD=88 and unusable beyond ZD=90.

• See also the routine palRefz, which performs the adjustment to the zenith distance rather than in Cartesian Az/El coordinates. The present routine is faster than palRefz and, except very low down, is equally accurate for all practical purposes. However, beyond about ZD 84 degrees palRefz should be used, and for the utmost accuracy iterative use of palRefro should be considered.