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    Re: lat/long from meridian passage
    From: George Huxtable
    Date: 2011 Jan 24, 00:51 -0000

    John Karl noted that Patrick has neglected to correct for refraction. But
    he has taken the overall Sun correction, for lower-limb, relevant to the
    time-of-year, to be 14.8', and that correction includes both semidiameter
    and refraction. So I doubt if there's any omission on those grounds. The
    wording was somewhat misleading, in saying- "Because I am using an
    artificial horizon I made no dip or altitude corrections", because the
    altitude correction for refraction had been included.
    
    Jim Wilson has pointed out that "you definitely need to consider
    north-south vessel movement and declination change". In Patrick's case he
    is a stationary observer, so the first point doesn't arise. The second will
    be considered below.
    
    Patrick has told us that we need to make no allowance for index error or
    watch error. Can we take it, then, that any such errors were actually zero,
    or otherwise, that they have been allowed for first, before noting the
    corrected values?
    
    Patrick has taken Sun declination from the Almanac, quoting S 19� 38.5,
    though he doesn't state which Greenwich hour this corresponded to. Not
    having a 2011 Almanac in front of me, I'll assume (perhaps wrongly) that he
    took that figure from the nearest, and previous, whole hour GMT, 17:00 h.
    If that's wrong, I hope he will correct me, as it will affect what's stated
    in the next paragraph..
    
    Then he states "d=.06", which I presume should really be d=0.6. That
    indicates that the Sun is travelling North, at just over half-an-arc-minute
    in each hour, which has to be allowed for as a change in declination, from
    that at 17:00 GMT. He has worked out that local meridian passage will be
    around 17:17 GMT, so that extra 17 minutes has to be corrected for. That's
    easy to estimate, as a correction of 0.2', and it's clear that as the
    Southerly declination is reducing, at 17:17 the declination will have
    changed from S 19� 38.5 to S 19� 38.3 over that 17 minute interval. This
    differs from Patrick's value.
    
    If we consider the three middle observations to correspond all to the same
    altitude, we see a difference between their average, and that of the two
    outer observations, of 9.9'. But my calculation of the true difference
    between those altitudes indicates that it should be more like 8.7'. Which
    indicates to me that the observations are not all that precise, but have a
    scatter in them in the neighbourhood of +/- 0.5', with a scatter of the
    reflected double-altitudes being twice that. It looks as if there's room
    for a bit of improvement there, in observing technique.
    
    I'll leave it to Patrick to recalculate his latitude appropriately, but it
    looks as if it will move a bit further still from his GPS value. This
    leaves some doubt about his three central altitude observations.
    
    Timing and longitude.
    
    Patrick places a question mark against the 12:17 observation, perhaps
    because it shows a slightly smaller altitude then others taken a couple of
    minutes earlier and later. But over that period, of a couple of minutes of
    time either side of noon, I would not expect to see any change in altitude
    greater than about 0.1'. Those three observations closest to noon could
    just as well be regarded as three attempts to measure effectively the same
    angle, to be averaged accordingly. The differences between them could well
    be attributed to the inevitable scatter in making such observations. For
    the same reason, no real value should be placed in the timing of that
    fourth observation, noted as 12:19:23. Could Patrick really put his hand on
    his heart and declare that that was the moment when the Sun descended
    through the same altitude that it had at 12:15? That would be wishful
    thinking.
    
    So I would put no reliance on the averaged timing of the 12:15 and the
    12:19:23 observations. Combining their average, with the outer pair taken
    at 12:00 and at 12:34:20, just dilutes the precision of those outer
    observations, the only ones to carry real timing information. So forget the
    timing of the inner pair, just take the outer pair, giving a mid-value of
    12:17:10.
    
    However, that is the moment of maximum altitude, which is not quite the
    same as the moment of meridian passage, because the changing declination
    shifts the peak of the curve. Over the time-period of the observations, 34
    m 20s, the Sun's Southerly declination will have reduced by 0.3 arc-min.
    That in itself would have increased the Sun's altitude by 0.3'. We have to
    allow for that when correcting the symmetrical point of the altitude curve
    to find the moment of meridian passage. One way would be to correct each
    individual point for the changing declination at the moment of observation,
    then note the time of the resulting peak.
    
     Another approach is to determine the time of the maximum amplitude, then
    apply a correction, as explained in the following extract from a website I
    put together years ago about Lewis and Clark's celestial navigation-
    
    ================
    Max-to-meridian time-shift.
    As explained above, the mid-time between the am and pm observations would
    give the moment of maximum altitude, which was near, but not quite at, the
    moment of apparent noon, when the Sun would be on the meridian. The reason
    for the time-difference is because the Sun's declination is changing slowly
    during the day. At summer and winter solstices, the declination is
    unchanging, and the correction is zero. Near the equinoxes the Sun's
    declination is changing fastest, at about +24 arc-minutes per day in the
    spring, and at -24 in the autumn. This rate-of-change can be obtained from
    the difference in dec. values, in the Almanac, on successive days.
    
    The daily rate-of-change (with its sign) in arc-minutes per day, has to be
    multiplied by
    0.637(tan dec - tan lat)       :the term in brackets always being negative
    for Lewis and Clark.
    
    Appropriate signs for lat and dec are that North is positive, South
    negative. The result is the amount (in seconds of time) by which the moment
    of maximum altitude precedes the moment when the Sun is on the meridian (or
    apparent noon). This correction can take values up to about �17 seconds,
    for L&C's circumstances. For them, maximum altitude always preceded
    apparent noon, from Summer solstice to Winter solstice, and followed it the
    rest of the year.
    
    =====================
    
    In Patrick's case, the daily rate of change of declination, taken from the
    Almanac, is +13.9', and this results in a time correction of -9.8 seconds.
    So taking the maximum altitude to be at 12:17:10, that would make the
    meridian passage to be at 12:17:00. Which would reduce the Westerly
    longitude by 2.8' to put it at 76�21.8', in fortuitously good agreement
    with the GPS value of 76� 21.7'
    
    I should point out that the name of this thread, "lat/long from meridian
    passage", is misleading. The lat certainly comes from the moment of
    meridian passage. The longitude measurement is only possible becaused it's
    based on two observations taken WELL AWAY from meridian passage, and
    indeed, the further they are spaced, in time, from meridian passage, the
    better the result will be. This is a point that I have made many times
    before, and no doubt will need to do so into the future.
    
    George.
    
    contact George Huxtable, at  george{at}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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