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    Refraction -- A general formula for all heights and altitudes
    From: Frank Reed CT
    Date: 2005 Aug 25, 14:59 EDT

    For the calculating members of the list, here's  a general fast formula for
    refraction covering all observer heights and angles  below the horizon, too:
    'Alt is the  angular height (corrected for dip). Ht is the observer's height
    above sea  level.
    IF Alt < 0 THEN
    'use new below zero degrees altitude  formula:
    refx = EXP(3.537 - .369 * Alt + .051 * Alt * Alt)
    ELSEIF Alt <  15 THEN
    'use standard low altitude refraction formula (Bennett):
    refx =  .998 / TAN((Alt + 7.31 / (Alt + 4.4)) / kk)
    'use standard high  altitude refraction:
    refx = .972 / TAN(Alt / kk)
    END IF
    'refx as  calculated so far is the sea level refraction at standard
    temperature  and
    'pressure of 10 deg C and 1010 millibars. The result is in minutes of  arc.
    'Now adjust for non-standard tempreature and pressure.
    'Temp and  Press are sea level temperature in deg C and pressure in mbar:
    refx = refx *  (Press / 1010) * (283.15 / (273.15 + Temp))
    'Now scale for  height above sea level:
    'Primary scaling with height above sea level:
    refx  = refx * EXP(-ht / (11278 - ht / 13))
    'A correction in scaling for very low  angular altitudes, proportional to
    observer altitude:
    refx = refx - (ht /  10000) * EXP(-Alt / 14)
    'refx is in minutes of  arc.
    This procedure calculates refraction at all  altitudes from sea level to at
    least 8km at excellent accuracy levels. It is  based on (time-consuming)
    integrations using the method outlined in  Auer-Standish with an atmospheric model
    compatible with the refraction tables in  the Nautical Almanac (pre-2004),
    specifically a lapse rate of 7.25 deg C up to  11km altitude and constant
    temperature above that altitude. The fit to the  integrations is accurate to 0.15
    arcminutes or less except in cases where the  refraction is very large in which
    case the angular error may be a few tenths of  a minute of arc but is less than
    1% of the total refraction. Since red light and  blue light from a star have
    refractions that differ by more than 1%, this error  limit should be acceptable
    in almost every case.
    This calculation will  typically be thousands of times faster than running a
    complete integration. If  computing time is not an issue, the integration
    approach is  better.
    42.0N 87.7W, or 41.4N  72.1W.

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