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    Re: [9955]time of meridian passage accuracy
    From: Antoine Cou�tte
    Date: 2009 Sep 27, 05:37 -0700

    Dear Jim and George,
    Thank you George for so kindly providing me with Jim's paper.
    Thank you Jim and Congratulations for your Navigation paper. It is one of the 
    Best and most Comprehensive I have seen dealing with a "graphs only" 
    approach. I will now on keep it on file as a quite precious addition to my 
    Naval Academy Navigation course and to the Book by M. Mortimer Rogoff you 
    made a reference to. Longitude error quite high sensitivity to observation 
    errors is also very clearly mentioned as it was in Mortimer Rogoff's Book.
    I still need to dig out in further depth the (other) graphical method 
    mentionned to me a few days ago with a referenced http link on this NavList 
    site and which was published recently (this summer I think).
    The interest of graphical methods is to avoid computations. They also show 
    that LAN's can be much more extended than the restrictions I was earlier 
    taught with limitations to zenit distances less than 20?.
    It could be noticed that in the method you published, which is based on the 
    differentiation of the well known/fundamental Height formula, such 
    differentiation is first order. IN LAN's first order is generally well 
    sufficient to deal with the majority of practical purposes, but it does have 
    definite limitations if and when you tackle observations closer and closer to 
    "Computation only" methods using first order terms have a limitation because 
    of the presence of significant UNEVEN ORDER TERMS in the higher order 
    development of hights vs time when we deal with cases with either changing 
    Body declination or N/S observer speed component.
    For this very reason, and as a cherry on the cake, if we consider LAN methods 
    based exclusively based on Computations alone (no graphs), Douglas Denny and 
    George are BOTH right on what seems an apparently Contradictary and 
    irreductible statement/argument between them, with Douglass stating "The 
    Height curve IS skewed" and George stating with equal force "the Height curve 
    is NOT skewed, only displaced left or right". If you consider cases 
    sufficiently close from LAN time and to an accuracy of 0.1 arc minute for 
    example, then the Height curve is NOT skewed as George points out, ... but if 
    you deal with examples of heights sufficiently far away from Culmination 
    time, then ODD terms will eventually come to play in some significant manner 
    and the height curve when referenced to elapsed time IS definitely skewed.
    Peace be with you all !   :-))
    I would still guess that ALL first order methods - including the graphical 
    ones such as yours Jim - might share the same practical use limitations for 
    near zenit culminations with high N/S Obervers speeds and or significant 
    changes in declination. It might be worth verifying this with examples. I 
    should have between 10 and 15 past examples of LAN with cumlminations above 
    85?. We could check results with your graphical method Jim.
    Now, for most practical purposes, and assuming that you want your LAN's 
    results as accurate as achievable (which we all aim at), I would offer the 
    following comments from experience :
    1) - Try getting an as wider "heights span" as possible in order to improve 
    Longitude accuracy. "Heights span" here means the difference between 
    culmination height and you lowest observed height in your set of height 
    shots. A One arcminute observation error on observed heights has less 
    relative effect on longitude accuracy if you have a full 8 to 10 degree 
    height span than if you have only a 3 degree height span.
    2) - Whenever achievable, prefer culminations closer to zenit. With low 
    culminations heights, the "heights vs time" curve becomes more "flat" and it 
    degrades accurate longitude determination. In addition, for low culmination 
    heights, the "heights span" criteria adressed in 1) herabove requires longer 
    observations time spans, therefore increasing both the constraint duration 
    for a constant bottom speed, and the effect of speed knowledge uncertainties.
    3 ) - Shoot as high LAN's as possible, then ? Not quite, because of at least 2 reasons :
    3.1) - Sextant swinging will become more difficult and possibly more inaccurate close to zenit, and
    3.2) in the cases with high N/S speeds and/or high declination speeds close to 
    zenit, even if primary consequence is to move "sideways" (i.e advance or 
    delay) the "heights vs time" curve, such curve may eventually become 
    significantly skewed. Ignoring such skewness (right term ?) may result into 
    appreciable errors mostly on Longitudes determination. Accuracies on 
    Longitudes remain a "weak point" of LAN's methods since most often the 
    observed span of azimuths necessarily remains quite narrow around the 
    Have a Great day, ALL of you !
    Antoine M. "Kermit" Cou?tte
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