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    Re: Fw: Letcher page 103
    From: George Huxtable
    Date: 2010 Feb 11, 11:58 -0000

    Greg Rudzinski, referring to Letcher's book  "Self-Contained Celestial 
    Navigation with HO 208", pointed to by Andres at-
    http://www.devill.net/Infos/Astro/Celestial-navigation-with-HO208.pdf ,
    wrote-
    
    "Page 103 has a most clever way to check a chronometer at sea. When doing 
    the Dreisonstok method I only solve for Hc then use a standard 10" K&E slide 
    rule to solve for azimuth using the formula cos dec x sin meridian angle / 
    cos Hc."
    
    In response, I pointed out- "However, that procedure is VERY vulnerable to 
    an error in assumed dip; particularly so, because of the use of opposite 
    azimuths. Each arc-minute of dip error results in an error of about 4 
    minutes of time."
    
    And Greg has just asked me a fair question- "At what point would you use 
    Letcher's opposing Sun Moon method to reset a stopped chronometer? "
    
    My answer is that I wouldn't use it. I would instead, measure the lunar 
    distance between the Moon and the Sun, and obtain the time from it in the 
    traditional way. This has the enormous advantage that the horizon isn't 
    involved, so there's no dip, with its inevitable uncertainties, to bother 
    about.
    
    Perhaps we can think up a simplified example, in which the observer is on 
    the equator, and both Sun and Moon have zero declination (which we will 
    pretend to be unchanging). The Sun is rising in the Sky, exactly due East, 
    at 15� per hour. The Moon is falling, due West, at 15.5� per hour.
    
    If we measured the direct lunar-distance between them, that would be 
    increasing at 0.5� per hour, and by comparing that with Moon and Sun 
    predictions, for Greenwich, we could deduce Greenwich Time and set the 
    chronometer accordingly.  What is crucial for this purpose (because 0.5� per 
    hour is so slow) is to measure that angle with the greatest possible 
    precision. And that's where a direct measure of lunar distance scores, in 
    removing the horizon's uncertainties.
    
    We could also measure the Sun-Moon distance by measuring the Sun's altitude 
    above the Eastern horizon, and the Moon's above the Western horizon, and 
    subtracting the sum from 180�. That, effectively, is what Letcher is doing. 
    And because those altitudes are summed, so is any altitude error, including 
    the horizon dip. Presuming that dip is the same all round the horizon, any 
    dip error is doubled.
    
    Another, and somewhat better way of doing the same job, of deducing the 
    lunar distance from altitudes, would be to measure both Moon and Sun up from 
    the same horizon, either East or West. That can be done if such 
    back-of-the-head altitudes, greater than 90�, come within the 120� angular 
    range of a sextant. It's an unfamiliar, and somewhat tricky way to use a 
    sextant, but perfectly possible.
    
    Now the two altitudes, instead of one rising and one falling, will both be 
    increasing (if measured from the Eastern horizon) or both decreasing (if 
    from the West) and now the relevant quantity is their difference. Therefore, 
    any common error (principally the dip) is nulled out and has no effect.on 
    the result.
    
    The plot that Letcher provides, of Sun altitude increasing and Moon altitude 
    decreasing, showing that the difference between them is changing very 
    rapidly with time (at 30� per hour, or so); gives an illusory notion of 
    precision. What those rapid changes are reflecting is simply the Earth's 
    rotation, which provides the local time; useless for setting a chronmometer 
    if the longitude is unknown.
    
    Indeed, Letcher's text, on page 102, confirms my own assessment. He states 
    "If it is felt that each observation has been taken and worked to a 
    precision of 0.5', then the plot can't be wrong by more than 1.0', 
    corresponding to two minutes of time, or 30' either way in the logitude that 
    results from using the chronometer's indicated time for GMT".  To presume an 
    overall altitude error, up from the horizon, including that of dip, to be no 
    more that 0.5', is optimistic to say the least. However, it corresponds 
    exactly with my own assessment that "Each arc-minute of dip error results in 
    an error of about 4 minutes of time."
    
    That isn't the only problem with Letcher's procedure, but it will do to be 
    going on with.
    
    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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