Refraction

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 $0^{\circ} \hspace{-0.37em}.\hspace{0.02em}5$. 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.

The full sla_REFRO refraction calculation is onerous, and for zenith distances of less than, say, $75^\circ$ the following model can be used instead:

$$\zeta _{vac} \approx \zeta _{obs} + A \tan \zeta _{obs} + B \tan ^{3}\zeta _{obs}$$

where $\zeta _{vac}$ is the topocentric zenith distance (i.e. in vacuo), $\zeta_{obs}$ is the observed zenith distance (i.e. affected by refraction), and A and B are constants, about $60\hspace{-0.05em}^{'\hspace{-0.1em}'}$and $-0\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.06$ 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.

sla_REFCO works by calling sla_REFRO for two zenith distances and fitting A and B to match. The calculation is onerous, but delivers accurate results whatever the conditions. sla_REFCOQ uses a direct formulation of A and B and is much faster; it is slightly less accurate than sla_REFCO but more than adequate for most practical purposes.

Like the full refraction model, the two-term formulation works in the wrong direction for our purposes, predicting the in vacuo (topocentric) zenith distance given the refracted (observed) zenith distance, rather than vice versa. The obvious approach of interchanging $\zeta _{vac}$ and $\zeta_{obs}$ and reversing the signs, though approximately correct, gives avoidable errors which are just significant in some applications; for example about $0\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.2$ at $70^\circ$ zenith distance. A much better result can easily be obtained, by using one Newton-Raphson iteration as follows:

$$\zeta _{obs} \approx \zeta _{vac} - \frac{A \tan \zeta _{va... ... {1 + ( A + 3 B \tan ^{2}\zeta _{vac} ) \sec ^{2}\zeta _{vac}}$$

The effect of refraction can be applied to an unrefracted zenith distance by calling sla_REFZ or to an unrefracted $[\,x,y,z\,]$ by calling sla_REFV. Over most of the sky these two routines deliver almost identical results, but beyond $\zeta=83^\circ$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.

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.

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 $100\mu m$for the radio case.