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    Re: Star-star distances for arc error
    From: George Huxtable
    Date: 2009 Jun 26, 21:27 +0100

    Frank Reed wrote-
    "you may want to take a peek at "appendix G: sextant arc error tables" in
    John Karl's book "Celestial Navigation in the GPS Age". The preface to these
    tables begins, "The following tables are handy for checking sextant arc
    error. They give star-star distances adjusted for the refraction..." And for
    those NavList members who have a copy of this book, if you look at those
    tables, you can see large blocks in the center of each table where the
    refracted distance barely changes. Those are the cases where both objects
    are above 45 degrees and fit the case which I've described in the first
    message in the thread. I should add that Ken Gebhart, who is co-publisher of
    John Karl's book and the world's largest distributor of sextants today,
    considers this the best text in celestial navigation currently on the
    market. I think it's pretty good, too! :-)"
    I think it's pretty good, too, as a whole, but one of its serious weaknesses
    is in that star-star distance table, appendix G. Anyone using it should be
    aware of the following problem, which I explained on [4062], as follows-
    "With the discussion about inter-star differences, I remembered that John
    Karl's new book, "Celestial Navigation in the GPS age", devoted several
    pages to helping users to calibrate or check their own sextants that way. He
    selected 12 pairs of bright stars (with rather a Northern-hemisphere bias),
    to provide a suitable spread of angles to calibrate, ranging from Bellatrix
    to Betelgeuse, at about 7 deg 30', as far as Betelgeuse to Spica, at over
    113 deg. For each such pair, he provides a table, showing how the refraction
    alters the odd minutes and fractions of that separation, based on the
    observer's latitude, and on the altitude of the first-named star. In the
    explanation he claims- "Since the observer's latitude and the star's
    altitude determine the altitudes of any other star ... the altitude of the
    second star is not needed". On the face of it, it seems a good simple
    scheme, dead easy for a user to implement.
    But on reflection, I'm not convinced. I have been worrying about that
    statement. I don't think it is true. It's all a bit more complicated than
    that, I fear.
    Given a latitude, and a star with known declination, and an observed
    altitude, it's true that one can deduce a local hour angle. That local hour
    angle will be the same in amount, corresponding to that altitude, whether
    the star is rising or falling in the sky, before or after culmination, but
    will be opposite in sign. And there will therefore be two completely
    different Greenwich hour angles. And therefore two completely different
    possible values for the local hour angle, and thus the altitude, of the
    second star. Therefore, as I see it, there should be two different tables
    for the refraction correction, depending on whether the first star is to the
    East or the West of the observer. The table as given, for, say, Bellatrix to
    Betelgeuse, tells only half the story. I haven't investigated the matter
    deeply enough to discover which half.
    Am I missing something, somewhere? Have I misunderstood? Can anyone help?
    John Karl himself, perhaps, if he still tunes in to Navlist, though we
    haven't heard from him recently."
    That was in November 2007. Since then, John Karl and I have been in touch.
    He has accepted that the star-star distances would only be correct during
    the rising part of path of star 1, to the Eastwards, and has completely
    revised that table for the second edition of his book. That edition is
    presumably out, by now, but I haven't seen it.
    In addition, there's another source of erroir in those predictions. No
    account has been taken of annual aberration, which varies cyclically over
    the year, and in the worst case can amount to errors in star-star angle
    amounting to 40 arc-seconds (though not quite that high for any star pairs
    chosen by John).
    So, anyone wishing to make high-precision checks on a sextant using
    star-star distances would be well advised to calculate, and correct for, the
    refractions from first principles, rather than follow Frank's recommendation
    of Table G in Karl's book. I wonder if Frank has ever tried it?
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Navigation List archive: www.fer3.com/arc
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