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    Re: sight reduction tables
    From: George Huxtable
    Date: 2007 Oct 1, 20:59 +0100

    Peter Fogg wrote-
    
    ...get a copy of *The Complete On-Board Celestial Navigator*,
    | subtitled Everything
    | But the Sextant, and have in the one slim volume all the data needed for 5
    | years; for sun, moon, planets and stars - including almanac and sight
    | reduction tables, and much else ...
    
    ==========
    
    Somehow, I guessed that Peter was going to push this once again. And
    Bennett's tables have least one other adherent in Alex Eremenko. Users
    should be aware of certain limitations, that we have discussed here ad
    nauseam in the past. Now, as there are several interested fresh members,
    perhaps it's time to cover some of that ground once again. I hope it can be
    done without the personal acrimony that was so evident last time round.
    
    Bennett provides a perfectly good source of 5-year predictions for Solar
    System and star bodies, rather cleverly presented, but to a limited
    precision. For example, to get the GHA of a star, four quantities have to be
    added together, each being tabulated to the nearest 1' of arc. Presuming
    that these have been tabulated with a maximum error of half an arcminute
    either way, the maximum error in the result will be no more than 2', and
    usually it will be less. Each sextant correction, dip and refraction, adds a
    similar error. And so on. A prudent navigator should assume that things are
    combining against him in the worst possible way, even if, usually, they
    won't be.
    
    In my view, such a limited precision is perfectly appropriate for use on a
    small craft at sea, when the horizon is so hard to pin down, as long as
    those limitations are kept in mind.
    
    Bennett's sight reduction tables, for obtaining calculated altitudes, seem
    to work perfectly well within those limitations, requiring only simple
    arithmetic, though they call for rather a lot of shuffling between pages.
    
    It's when we come to calculating azimuths that Bennett's tables fall down.
    It's true that as long as intercepts are kept short, by a good guess at the
    DR position, no great precision is called for in azimuths. Bennett has
    claimed for his azimuth tables, in the 2003-2007 edition- "No interpolation
    is required, and it is one of the simplest techniques for finding azimuth
    with an accuracy of one or two degrees". If that claim were met in all
    cases, the method would be perfectly acceptable. But I will show below that
    in certain circumstances, it's very far from being met. To be fair, Bennett
    agreed, some years ago, that those claims would be modified in some way, and
    I hope, in the new edition, that this has been done; or better still, his
    azimuth methods have somehow been improved.
    
    The difficulty arises because azimuths are calculated, within his table,
    from their sines, and for azimuths anywhere near East and West, small
    changes in those sines give rise to large changes in azimuth. It's
    compounded by the way that input to the table can be made only in integral
    degrees, so you have to round quantities up or down appropriately, to the
    nearest whole degree.
    
    I will give an example of the way things can go wrong. To be fair, it isn't
    a typical case; I have taken a bit of trouble to hit on an example where the
    table goes particularly haywire. It may be a worst-case, or there may be
    other combinations which are equally bad, or even worse. But you will see
    that there's nothing strange about the numbers I have put in, which
    represent perfectly ordinary observations. If you have a copy of Bennett,
    you can try it for yourself.
    
    Take a DR latitude to be N 60d 10 ', sighting a (mythical) star at dec N 55d
    31', with LHA 54d 29'. Bennett's tables give an altitude of 61d 29',
    compared with a precisely computed value, by a different methos, of 61d
    29.0. Nothing to complain about there, then.
    
    Next, round those quantities to the nearest minute, and enter the azimuth
    calculation, with dec 56, LHA 54, producing a lookup value of 452. Look for
    this value at alt= 61, and rejecting the ambiguities, the result is Az =
    291d. This compares with the true value, from a precise calculation, of Az =
    285.1d. Such a six-degree error in azimuth is a long way out, isn't it?
    
    But worse is to come.
    
    Next, take marginally different values, only a few minutes different, as
    follows-
    
    DR Lat, N 60d 25', star dec 55d 29', LHA 54d 31'. The altitude tables now
    give 61d 31', compared with 61d 30.8' from precise calculation. That, too is
    in excellent agreement.
    
    But what about the azimuths, when we plug in the new rounded values, dec =
    55, LHA = 55, now giving a lookup value of  469? With an altitude now
    rounded to 62d, the resulting azimuth is now in the band 270 to 272 degrees!
    But calculated precisely, by another method, the correct azimuth would be
    284.7d. An immense error of 13 to 15 degrees!
    
    So the end result is that by making small shifts in the starting
    assumptions, sufficiently small that the true azimuth of the star changes by
    less than half a degree, its azimuth deduced from the Bennett table shifts
    by all of 20 degrees! So much for the claim of "within one or two degrees".
    It's well understood that azimuths near due East and West are inaccurate to
    calculate, whenever an arc-sine method is used, but note that the example
    above was for a situation where the true azimuth was all of 15 degrees away
    from due East.
    
    I should add that there's an additional problem in Bennett's method, of
    resolving the resulting ambiguities in azimuth, not between East and West
    alternative values (that's simple), but between two possible azimuths
    equally spaced either side of due East (or due West), such as 249 / 291.. By
    a more suitable bit of maths, calculating azimuth from its tan, no such
    ambiguity occurs, but because Bennett gets az from its sine, he has to
    provide detailed instructions for choosing the right result.
    
    So I advise users to be on their guard when using the Bennet method for
    azimuths, even if in other respects they find the tables perfectly
    satisfactory.
    
    Bennett does offer an alternative graphical method for azimuths, the Weir
    azimuth diagram, but does not (in the 2003-2007 edition) give any guidance
    about when to use that instead of his table. I hope he has done so since.
    
    George.
    
    contact George Huxtable at george---.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
    
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