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    Re: sight reduction tables
    From: Ronald van Riet
    Date: 2007 Oct 02, 04:29 -0700

    I am a relative newcomer to the celestial navigation arena so forgive
    me if what I say has been said a zillion times before or makes no
    sense at all, but George points out (rightly it seems) that there can
    be large errors in azimuth with an example where the azimuth is up to
    15 degrees off.
    Why is that so bad? By how much does that displace the resulting LOP
    and the fix using one or two more LOPs?
    And if you want to be more precise, why not use the new fix as the
    assumed position for a next round of calculations? Since this AP would
    be much nearer the actual position, an azimuth offset by 15 deg won't
    make any significant difference.
    Then again, if we know under what circumstances the error is large,
    why not shoot some celestial objects that do not exhibit these large
    errors? assuming of course, that these are available.....
    On Oct 1, 9:59 pm, "George Huxtable" 
    > 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!
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