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    Determination of longitude and the prime vertical
    From: Geoffrey Kolbe
    Date: 2007 Apr 11, 06:51 +0100

    Take a particular position P with latitude L, where the azimuth of
    the sun is exactly 270 at a particular time. If the latitude is
    varied from that position by an amount delta L, then it seems the
    altitude varies as (Sin(delta L))^2, whereas the azimuth varies
    linearly with delta L.
    For example: When the sun has an altitude of 10 degrees, say, at the
    particular position P, then varying the latitude by delta L = +/- 5
    degrees will only change the altitude by about one minute,. The
    corresponding change in azimuth will be one degree.
    I would question Bill's statement that "the skew of the LOP helps to
    null out error". If the estimated position is 5 degrees North of the
    position P in the above example, the azimuth will be 269 degrees. The
    resulting position line drawn for this estimated position will drop
    to the East of the position P by about 4 miles.
    In such circumstances, it will be necessary to use the first fix as a
    new estimated position and reduce the sights again for a better
    estimated position.
    Where the sun is near the Prime Vertical (within a degree or so), It
    would seem that the best scenario is to assume the azimuth is exactly
    270 and draw a position line that is exactly North-South. Such a
    method will be more tolerant of large errors in latitude for the
    estimated position than the classic position line method.
    Recalling Joel's quote of Thomas Sumner:
    "And when the Latitude is uncertain, there are only two proper
    instants per day (when the sun bears exactly East or exactly West)
    when the solar altitude can be used to find the Longitude by
    Chronometer with accuracy.  If one makes an observation other than at
    those times....... the unavoidable errors in measurement may
    cause frequent errors which are very great."
    I have to ask whether Sumner was right...?
    Geoffrey Kolbe
    > > 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.
    Bill wrote (posting #2550)
    >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.
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