### SLA_REFCOQ

Refraction Constants (fast)

ACTION:
Determine the constants $a$ and $b$ in the atmospheric refraction model $\Delta \zeta =atan\zeta +b{tan}^{3}\zeta$, where $\zeta$ is the observed zenith distance (i.e. affected by refraction) and $\Delta \zeta$ is what to add to $\zeta$ to give the topocentric (i.e. in vacuo) zenith distance. (This is a fast alternative to the sla_REFCO routine – see notes.)
CALL:
CALL sla_REFCOQ (TDK, PMB, RH, WL, REFA, REFB)
##### GIVEN:
 TDK D ambient temperature at the observer (K) PMB D pressure at the observer (mb) RH D relative humidity at the observer (range 0 – 1) WL D effective wavelength of the source ($\mu m$)

##### RETURNED:
 REFA D $tan\zeta$ coefficient (radians) REFB D ${tan}^{3}\zeta$ coefficient (radians)

NOTES:
(1)
The radio refraction is chosen by specifying WL $>100$ $\mu m$.
(2)
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.
(3)
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 ($4{5}^{\circ }$ and $\sim \phantom{\rule{0.3em}{0ex}}7{6}^{\circ }$) and at other zenith distances is limited in accuracy only by the $\Delta \zeta =atan\zeta +b{tan}^{3}\zeta$ formulation itself. The present routine is not as accurate, though it satisfies most practical requirements.
(4)
The model omits the effects of (i) height above sea level (apart from the reduced pressure itself), (ii) latitude (i.e. the flattening of the Earth) and (iii) variations in tropospheric lapse rate.
(5)
The model has been tested using the following range of conditions:
• lapse rates 0.0055, 0.0065, 0.0075 K per metre
• latitudes ${0}^{\circ }$, $2{5}^{\circ }$, $5{0}^{\circ }$, $7{5}^{\circ }$
• heights 0, 2500, 5000 metres above sea level
• pressures mean for height $-10$% to $+5$% in steps of $5$%
• temperatures $-1{0}^{\circ }$ to $+2{0}^{\circ }$ with respect to $280$K at sea level
• relative humidity 0, 0.5, 1
• wavelength 0.4, 0.6, … $2\mu m$, + radio
• zenith distances $1{5}^{\circ }$, $4{5}^{\circ }$, $7{5}^{\circ }$

For the above conditions, the comparison with sla_REFRO was as follows:

 worst RMS optical/IR 62 8 radio 319 49 mas mas

For this particular set of conditions:

• lapse rate 6.5 K km${}^{-1}$
• latitude $5{0}^{\circ }$
• sea level
• pressure 1005 mb
• temperature ${7}^{\circ }$C
• humidity 80%
• wavelength 5740 Ȧ

the results were as follows:

 $\zeta$ sla_REFRO sla_REFCOQ Saastamoinen 10 10.27 10.27 10.27 20 21.19 21.20 21.19 30 33.61 33.61 33.60 40 48.82 48.83 48.81 45 58.16 58.18 58.16 50 69.28 69.30 69.27 55 82.97 82.99 82.95 60 100.51 100.54 100.50 65 124.23 124.26 124.20 70 158.63 158.68 158.61 72 177.32 177.37 177.31 74 200.35 200.38 200.32 76 229.45 229.43 229.42 78 267.44 267.29 267.41 80 319.13 318.55 319.10 deg arcsec arcsec arcsec

The values for Saastamoinen’s formula (which includes terms up to ${tan}^{5}$) are taken from Hohenkerk & Sinclair (1985).

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 ′′001 resolution used.

(6)
Outlandish input parameters are silently limited to mathematically safe values. Zero pressure is permissible, and causes zeroes to be returned.
(7)
The algorithm draws on several sources, as follows:
• 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).
• The formula for the water vapour pressure, given the saturation pressure and the relative humidity is from Crane (1976), expression 2.5.5.
• 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 Hohenkerk & Sinclair (1985) and Rueger (2002).
• The formula for $\beta \phantom{\rule{1em}{0ex}}\left(={H}_{0}/{r}_{0}\right)$ 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.
• The formulae for the refraction constants as a function of $n-1$ and $\beta$ are from Green (1987), expression 4.31.

The first three items are as used in the sla_REFRO routine.

REFERENCES:
(1)
Crane, R.K., Meeks, M.L. (ed), “Refraction Effects in the Neutral Atmosphere”, Methods of Experimental Physics: Astrophysics 12B, Academic Press, 1976.
(2)