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    Nonlinearity tables, was: Averaging
    From: Alexandre Eremenko
    Date: 2004 Oct 23, 16:26 -0500

    Here are the detailed non-linearity error tables.
    
    The left column is the maximal altitude of the body (at the time
    of the upper culmination) in degrees.
    The second column is the lower bound of the (LHA-) distance of the
    observed body from the meridian at the time of observation,
    in degrees.
    Other 2 columns give maximal non-linearity errors under these conditions.
    (That is by how much the graph of the altitude as a function
    of time differs from a straight line.)
    Column 3 is related to the interval of 5 minutes
    and column 4 to the interval of 4 minutes.
    (5 observations with 1 minute interval take 4 minutes!)
    The results are rounded to .1'
    
    As I said before, these results were obtained by computing the
    second derivative of the altitude with respect to time,
    for abouit 8 million equally spaced points in the parameter space,
    and then taking maximum over the specified parameter region.
    
    MAX. ALT        DIST    ERROR(5)          ERROR(4)
    
      60            0               0.3         0.2
      60            20              0.2         0.1
      70            0               0.4         0.3
      70            20              0.2         0.1
      80            0               0.4         0.3
      80            10              0.4         0.2
      80            20              0.2         0.1
      85            0               0.4         0.3
      85            10              0.4         0.2
      85            20              0.2         0.1
      85            45              0.1         0.0
    
    Examples:
    
    1. You made 5 observations with less than 1-minute intervals
    between them (so the total procedure took no more than 4 min)
    The body was roughly bearing NE. According to the last
    row of the table, the non-linearity error
    is zero. That is when you average the altitudes, you
    introduce an error which is less than the rounding error of
    the almanac.
    
    2. You made 5 observations with less than 1-minute intervals
    between them of the Sun at 10:15 a.m. local time, and
    your latitude was 30 degrees.
    Your non-linearity error does not exceed 0.1' according
    to the second row from below.
    
    3. You averaged 5 observations taken around noon,
    maybe 2 before noon and 3 after, but it took you no more
    than 4 minutes to do this.
    (I don't know why would one do this, but suppose you did)
    and the altitude was 65. According to row 3 your non-linearity error
    can be as large as 0.3'.
    
    One more simple rule of thumb:
    If you use a plastic (or other non-metal) sextant
    averaging the altitudes will ALWAYS increase precision.
    If you use the "Complete on board Cel Navigator" to reduce your sights
    averaging will ALWAYS increase your precision.
    
    Alex.
    
    
    

       
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