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    Re: determination of longitude and the prime vertical
    From: Bill B
    Date: 2007 Apr 10, 16:56 -0400

    > From: "George Huxtable" 
    > Yes, that's the point of it. When the Sun is on or near the prime vertical,
    > its altitude depends only on its hour angle and declination, and is exactly
    > the same whatever the observer's latitude. So, knowing declination, and
    > measuring altitude, gives hour angle, and then from the chronometer,
    > longitude. The latitude didn't need to be known.
    Excellent and scholarly explanations by George.  Although George does state
    that if very close to Az 90 or 270, altitude is very insensitive to
    latitude, I am having trouble with the above paragraph.
    If hour angle and declination are held constant, the altitude will vary
    depending on latitude.  sin Hc = sin L  sin dec + cos L cos LHA cos dec.
    It follows that the hour angle when the sun is on the prime vertical varies
    with latitude.
    What is important is that latitude being off by a degree will have little
    affect in longitude in the bigger picture.  In the scenarios below, AP being
    a degree being off true position will only affect longitude by 0d 0!4.
    In this case the skew of the LOP helps to null out error. In other cases the
    error may be compounded but < 1 arc minute.  The variables are whether the
    body is east or west, LOP is slightly greater or less than 90/270, and
    actual latitude is north or south of AP.
    In my trials using a time sight to calculate longitude directly versus the
    intercept method (Hc vs. Ho) I find any error (intercept method) is
    nominally 50% higher when using a time sight.
    The following information was taken from:
    Date:     1 May, 2007
    UT:       20:41:45
    Position: N40 W60
    Hc:       24d 00!2
    Az:       270.00
    LHA:      71d 09!5
    dec:      15d 09!5
    Date:     1 May, 2007
    UT:       20:41:45
    Position: N41 W60
    Hc:       24d 00!0
    Az:       269.56
    LHA:      71d 09!5
    dec:      15d 09!5
    Date:     1 May, 2007
    UT:       20:44:27
    Position: N41 W60
    Hc:       23d 29!4
    Az:       270.00
    LHA:      71d 50!4
    dec:      15d 09!5
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