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    Re: The Complete On-Board Celestial Navigator Second Edition/ Astronomical Refraction
    From: George Huxtable
    Date: 2003 May 3, 10:38 +0100

    May I join Peter Fogg in welcoming George Bennett on board as a new member.
    Many members will use his clever and simple semi-empirical formula, which
    has been discussed before on this list, in their own computed corrections
    for refraction. It can be found quoted in Meeus, "Astronomical algorithms",
    and no doubt many other places.
    If alt is given in degrees, then Bennett's equation for refraction, to
    subtract, in minutes, is given by-
    R = 1 / tan (alt + (7.31/(alt + 4.4))),
    which can be rewritten, if you prefer it, as-
    R = tan (90 - alt - (7.31 / (alt + 4.4))), which is exactly the same thing.
    These expressions have two, very minor, drawbacks. The refraction at 90 deg
    alt isn't quite exactly zero, which it simply has to be by symmetry. And at
    a certain angle in the working range, an infinity arises during the
    calculation, which may throw pocket calculators and some computers, though
    others may be able to overcome it.
    Some of us use a tinkered-with version, which is-
    R = tan (90 - alt - (7.31 / (alt + 4.4)) + .000861 * alt) or, if you prefer,
    R = tan (90 - (.999139 * alt) - (7.31 / (alt + 4.4)))
    An infinity can still arise, but now it's at a negative altitude, just
    below the horizon, where it's out of the working range. It also takes the
    refraction at vertically-up to (effectively) zero. Otherwise, the
    refraction predictions are effectively the same as in the simpler formula.
    If George Bennett has any comments on that tinkering, it would be good to
    read them.
    Intrigued by comments on this list, I recently invested in a copy of George
    Benntett's "The Complete On-Board Celestial Navigator Second Edition". This
    contains five years of tables for all the navigational bodies, condensed by
    working to a relaxed precision of the nearest 1 minute of arc, rather than
    the 0.1 minutes of the normal almanac. Everything, interpolation,
    correction, sight reduction, is done to that reduced precision, so these
    approximations might sometimes combine to put the resulting position-line 3
    miles out, maybe  a bit more.
    I think that such a level of precision is entirely appropriate to the level
    of accuracy with which we can observe altitudes from the unstable deck of a
    small boat, from so near to the wave-surface. It's quite good enough to
    "get-you-home" in an emergency, and avoid major hazards on the way, which
    is Bennett's stated aim: but more, it could be used on a regular basis for
    celestial navigation of a small craft.
    However, I have reservations about Bennett's table for obtaining azimuths,
    and warn users of the book to be aware that for a range of azimuths near to
    due East and due west, the procedure introduces azimuth errors that can be
    far greater than the "one or two degrees" that are claimed. Azimuths are
    calculated in terms of sin az, and this makes for simple tabulation, but is
    inaccurate and also ambiguous near 90 and 270 degrees. Bennett includes
    instructions for resolving the ambiguity, but the inaccuracies remain.
    For azimuths within 15 or 20 deg of East or West, navigators would be well
    advised to use corrected compass bearings of the object instead, or else
    the Weir azimuth diagram that Bennett provides, and avoid the azimuth
    table. It would be interesting to know of George Bennett's reactions to
    these comments.
    George Huxtable.
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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