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    "THE ATTAINMENT OF PRECISION IN CELESTIAL NAVIGATION by Robert Gordon. Journal of the Institute of Navigation. 1964.
    From: Douglas Denny
    Date: 2009 Jun 27, 16:30 -0700

    Further to my last posting about general issues Re. solar eye damage and 
    telescope aperture/magnification  herewith is the promised report:-
    " The Attainment of Precision in Celestial Navigation"  by   Robert Gordon,  
    Department of Engineering and Applied Science,  Yale University.
    Published by the Institute of Navigation in their  'Journal of the Institute 
    of Navigation'  in 1964  when sextants were very much in common usage as it 
    was then only in the  very  earliest beginnings of the satellite era.   It is 
    the only definitive scientific report that I know exists on this subject of 
    sextant accuracy in practical use. 
    It is quite comprehensive and deals with real experimental evidence and 
    considerations of errors  from  500 observations made on land and sea.  
    Gordon makes clear that this is not a full scientific investigation  and is 
    "not really adequate to discharge this task to complete satisfaction"  but is 
    nevertheless happy that they will be  "...of considerable interest, even 
    though a more valuable survey would have been made if a far larger number of 
    observations... had been made".
    And it is indeed of considerable interest.
    I find the following particularly interesting with relevance to recent discussions:-
    Accuracy of Measuring Inter-Stellar Distances.
    page 129,  refers to methods of determining index error, and specifically 
    mentions inter-stellar angles as a means of checking sextant accuracy.  He 
    concludes it is impracticable as the superimposition of stars cannot be made 
    accurately enough,  and "the error in the readings is unacceptably large".  
    He places the error as large as 5 minutes of arc, which even I find 
    surprising and considerably larger than I would have thought.  He also makes 
    the point that I have mentioned of different eye acuities having an effect: 
    namely the ability of  judging continuity of a line as opposed to 
    superposition of points.
    Using direct and reflected horizon is the best method for this reason,  giving 
    a standard deviation (sigma) of 0.28 minutes of arc under best conditions 
    from 165 observations made.  For those who do not understand the maths of 
    standard deviation,  it is a measure of the spread or expected results 
    obtained by statistical methods.  
    2x Standard Deviation gives the expected range in 68% of observations.
    4 x sigma  gives the expected range in 95% of observations.
    So the expected accuracy of finding index error by horizon co-incidence is 
    within 0.56 minutes of arc for 68 % of observations; and 1.12 minutes of arc 
    in 95% of cases.
    Resolving Power of Telescopes.
    The practically determined resolving power of a telescope on a sextant as 
    opposed to the theoretical RP is discussed on page 133.
    What is clear is that a higher power magnification gives much better resolving 
    power practically (as might be intuitively  expected),  but the interesting 
    thing is the difference in theoretical to practical found. It is quite a 
    large amount - I think due to (and confirming)  resolving power issues within 
    the eye rather than the telescope.
    With a 3X telescope the theoretical RP is 3.4 seconds of arc;  but the 
    practical measured was 1.5 minutes of arc.
    With a 6X telescope the theoretical RP  is 4.7 seconds of arc;  but the 
    practical measured was 30 seconds of arc.
    Random Errors of Daylight Observations.
    Page  131. 
    This part of the report  for most people here will be the most interesting I 
    expect.  It gives a good idea what the expected accuracy of observations 
    using the Sun or Moon is likely to be in practical use.
    Observations were made on land; on a small boat yawing at anchor; reaching in 
    a force 3 wind at sea;  Beating into force 4 wind etc ...
    I note the error in most cases including the  'ideal' conditions was a sigma 
    of around 0.33 minutes of arc.  This means an accuracy range of 0.66 minutes 
    of arc is to be expected in 68% of observations;  and 1.32 minutes of arc in 
    95% of all observations.
    This is fully what I would have expected without any reference to this report, 
     as my contention is that the limit to sextant observational accuracy is not 
    in the sextant but in the eye itself.  
    The limiting resolution is to be found in eye acuity (or resolving power),  
    due to diffraction effects and aberrations. Various things can affect eye 
    resolving power including  pupil size; contrast of target and background;  
    and can be shown also to be dependant on  the type of 'target' observed due 
    to psycho/physiological effects.
    Hecht and Mintz found the resolving power of a line against a uniformly bright 
    background is in the order of 0.5 minute of arc subtended at the eye - a 
    curiously similar result to the practical sextant tests.   Could this be the 
    limiting value for 'normal' observations in daylight?
    I am sure you will find this report interesting, and it does at least give you 
    all some real experimental figures to debate.
    The sextant they used incidentally is the same that I have - a Henry Hughes 
    and Sons sextant. Theirs was checked carefully of course.
    My Hughes sextant has a certificate which states zero error at zero degrees 
    and one minute 30 seconds at all measurement points after that. I will have 
    to assess what is going on between zero and the first test point to find out 
    what is going on.  I shall use my Elliot engineering dividing head though for 
    that job when I can find the enthusiasm to make a mount for the sextant, but 
    even then the dividing head only has a certified accuracy of better than 45 
    seconds of arc! not really enough for me. One day I shall invest in an 
    optical dividing head for this kind of test.
    Douglas Denny.
    Chichester. England.

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