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      SUBROUTINE slRFCQ (TDK, PMB, RH, WL, REFA, REFB)

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      R F C Q
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  Determine the constants A and B in the atmospheric refraction
  model dZ = A tan Z + B tan**3 Z.  This is a fast alternative
  to the slRFCO routine - see notes.

  Z is the "observed" zenith distance (i.e. affected by refraction)
  and dZ is what to add to Z to give the "topocentric" (i.e. in vacuo)
  zenith distance.

  Given:
    TDK      d      ambient temperature at the observer (deg K)
    PMB      d      pressure at the observer (millibar)
    RH       d      relative humidity at the observer (range 0-1)
    WL       d      effective wavelength of the source (micrometre)

  Returned:
    REFA     d      tan Z coefficient (radian)
    REFB     d      tan**3 Z coefficient (radian)

  The radio refraction is chosen by specifying WL > 100 micrometres.

  Notes:

  1  The model is an approximation, for moderate zenith distances,
     to the predictions of the slRFRO routine.  The approximation
     is maintained across a range of conditions, and applies to
     both optical/IR and radio.

  2  The algorithm is a fast alternative to the slRFCO routine.
     The latter calls the slRFRO 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 distance (45 degrees and about 76 degrees)
     and at other zenith distances is limited in accuracy only by the
     A tan Z + B tan**3 Z formulation itself.  The present routine
     is not as accurate, though it satisfies most practical
     requirements.

  3  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.

     The model was tested using the following range of conditions:

       lapse rates 0.0055, 0.0065, 0.0075 deg/metre
       latitudes 0, 25, 50, 75 degrees
       heights 0, 2500, 5000 metres ASL
       pressures mean for height -10% to +5% in steps of 5%
       temperatures -10 deg to +20 deg with respect to 280 deg at SL
       relative humidity 0, 0.5, 1
       wavelengths 0.4, 0.6, ... 2 micron, + radio
       zenith distances 15, 45, 75 degrees

     The accuracy with respect to direct use of the slRFRO routine
     was as follows:

                            worst         RMS

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

     For this particular set of conditions:

       lapse rate 0.0065 degK/metre
       latitude 50 degrees
       sea level
       pressure 1005 mB
       temperature 280.15 degK
       humidity 80%
       wavelength 5740 Angstroms

     the results were as follows:

       ZD        slRFRO   slRFCQ  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 and Sinclair (1985).

     The results from the much slower but more accurate slRFCO
     routine have not been included in the tabulation as they are
     identical to those in the slRFRO column to the 0.01 arcsec
     resolution used.

  4  Outlandish input parameters are silently limited to mathematically
     safe values.  Zero pressure is permissible, and causes zeroes to
     be returned.

  5  The algorithm draws on several sources, as follows:

     a) 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).

     b) The formula for the water vapour pressure, given the
        saturation pressure and the relative humidity, is from
        Crane (1976), expression 2.5.5.

     c) 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 the Essen and Froome
        expressions adopted in Resolution 1 of the 12th International
        Geodesy Association General Assembly (1963).

     The above three items are as used in the slRFRO routine.

     d) The formula for beta, the ratio of the scale height of the
        atmosphere to the geocentric distance of the observer, 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.

     e) The formulae for the refraction constants as a function of
        n-1 and beta are from Green (1987), expression 4.31.

  References:

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

     Gill, Adrian E., "Atmosphere-Ocean Dynamics", Academic Press, 1982.

     Hohenkerk, C.Y., & Sinclair, A.T., NAO Technical Note No. 63, 1985.

     International Geodesy Association General Assembly, Bulletin
     Geodesique 70 p390, 1963.

     Stone, Ronald C., P.A.S.P. 108 1051-1058, 1996.

     Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987.

  P.T.Wallace   Starlink   4 June 1997

  Copyright (C) 1997 Rutherford Appleton Laboratory
  Copyright (C) 1995 Association of Universities for Research in Astronomy Inc.

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