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    Re: karl appendix
    From: George Huxtable
    Date: 2010 Jun 26, 11:31 +0100

    Dave Walden wrote about John Karl's book "Celestial Navigation in the GPS
    Age", about curiosities he had discoved in the predicted sextant  angle
    between Antares and Nunki-
    
    "i see the major error has been corrrected without comment in the second
    printing"
    
    ===================
    
    This is a matter that cropped up some time ago, after the first edition
    appeared, in Navlist [3984] of 25 Nov 2007, in which I wrote-
    
    "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."
    
    ===================================
    
    Subsequently, John and I discussed the matter off-list. We realised that,
    though the predictions were correct for when the first star was rising, in
    the eastern sky, refraction of the second star was miscalculated when the
    first was falling, in the West, after culmination. This applied to all
    star-pair tables in that appendix. It's the discrepancy that Dave Walden
    noticed in his first posting..
    
    The result has been that for the second edition, that appendix has been
    completely rejigged, to avoid the problem, as Dave has noted..
    
    But Dave, and other observers wishing to work to high precision, should
    note that stellar aberration, which varies between star-pairs and cycles
    over the year, has not been taken into account, which can cause a maximum
    error of just over half an arc-minute, either way, in the case of the
    worst-affected pair, Betelgeuse - Spica. John explains this in some detail
    on page 240 of the second edition, and describes how a more precise
    calculation can be made in another way, if necessary
    
    In the first edition, however, aberration was treated in such a way that
    the maximum errors for certain star-pairs at certain times of year could
    have been doubled, so a significant improvement has been made for the
    second edition.
    
    George.
    
    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.
    
    
    
    
    

       
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