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    Refraction formula comparison
    From: Paul Hirose
    Date: 2000 Oct 01, 2:37 PM

    The Explanatory Supplement to the Astronomical Almanac gives this
    low-precision formula for refraction below 15 degrees:
    R = (P / (T + 273)) * (.1594 + .0196 * H + .00002 * H^2) / (1 + .505
    * H + .0845 * H^2)
    where H = apparent altitude, P = air pressure in millibars, and T =
    degrees Celsius. Result is in degrees.
    The Explanatory Supplement also gives first-order approximations to
    correct celestial coordinates for refraction. I will not give them
    here, but they are simple formulas. You could use them to determine
    the refraction-corrected separation angle between stars.
    That was some sharp thinking by George Huxtable, to spot the problem
    lurking in the Bennett refraction formula. I'd expect the exact
    location of the singularity to depend on your machine. With my HP-15C
    calculator, it's at 89.92249993. Any change in the last decimal place
    avoids it.
    George is quite correct that a programmer should not shrug off a
    potential bomb like this. I once crashed an astronomy program by
    innocently entering exactly 90 degrees for a parameter.
    Here is a comparison of the refraction formulas:
     H  Bennett    Supp.     Keys   Smart
     0     34.5     34.1     35.0      -
     1     24.3     24.1     25.4   151.1
     2     18.2     18.1     18.8     2.0
     5      9.9      9.8      9.9     9.5
    10      5.4      5.3      5.3     5.3
    12      4.5      4.4      4.5     4.5
    15      3.6      3.6      3.6     3.6
    30      1.7      1.8      1.7     1.7
    Except for formula no. 2, all are equal at higher altitudes.
    The Explanatory Supplement formula is only supposed to be used below
    15 degrees, and the Smart formula from 15 up, but both are good for
    navigation outside those ranges. I included 12 degrees because that's
    where the maximum error of the Bennett formula is supposed to occur.

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