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    Re: Exorcizing the Evil Effects of Parallactic Retardation (Modified by Fred Hebard)
    From: Fred Hebard
    Date: 2004 Mar 20, 16:10 -0500

    This is a repost of my previous message, with the table included as
    text in the body of the file, as well as a pdf, as requested by George
    Huxtable.  (The pdf was made from a jpg, so the text is not editable).
    My previous version of the table contained a blunder, which, however,
    did not affect the net result.  The blunder was that my ephemerides for
    Jupiter were advanced by one hour.  I guess I should have noticed
    something was up when the distance between the Moon and Jupiter was
    reputed to have changed 17 degrees in 5 hours!  This time, I double
    checked the calculations with the U.S. Naval Observatory site's
    computer (http://aa.usno.navy.mil/).
    
    In January, George Huxtable surprised us all by saying that he no
    longer felt that parallactic retardation affected the accuracy of
    observation of time by lunar distance.  I've finally had time to look
    at this numerically, and, by golly, George is right, which should come
    as little surprise to most of us.
    
    The table below gives some simulated observed distances between the
    Moon and Jupiter on 1/14/04 at 1*31.3'S latitude and 0*0.0'W longitude.
      The altitudes of the Moon and Jupiter, and the Moon's refraction,
    parallax and semi-diameter plus Jupiter's refraction were calculated
    using the time in the first column.  The horizontal parallax of the
    Moon was taken from the Nautical almanac and used to calculate the real
    parallax for the Moon at that altitude and latitude, using Young's
    Method of Clearing that George Huxtable presented to the list in
    2002-2003.
    
    Observations near the equator give the greatest "parallactic
    retardation," combined with the time when the moon is near zenith,
    which is 5:00:01 hours in the table below. The retardation can be seen
    by comparing the difference between the "observed sextant reading" of
    lines 1 and 4, 19'34" of arc, with that of lines 5 & 8, 29'52" of arc.
    The Moon is only moving through 19'34" of apparent arc between 5:00:01
    hours and 5:59:59 hours (lines 1 & 4), when it is near zenith, compared
    to 29'52" of arc between 0:00:01 hours and 1:59:59 hours (lines 5 & 8),
    when it is near the horizon.
    
    The thinking has been that if the moon is moving through such a smaller
    swath of apparent arc (19'34"), then any errors in measurement of
    distance will be magnified.  If the error is 30" of arc, one will be
    measuring to only 1 part in 39 (30"/19'34"), rather than 1 part in 60
    (30"/29'52").  This is the supposed "evil effect of parallactic
    retardation."
    
    However, George has been trying to point out to us that there are two
    components to measuring a lunar distance, one the distance itself, and
    two the altitude of the bodies.  The Moon's parallax is fixed by
    observation of its altitude, not the distance, and it is the large
    shift in the _COMPUTED_ parallax between 5:00:01 and 5:59:59 that gives
    rise to the retardation.  Between 5:00:01 and 5:59:59, the computed
    parallax of the Moon increases from 3.2' to 11.6' of arc, almost a
    4-fold difference.  In contrast, between 0:00:01 and 0:59:59, the
    parallax decreases from 56.3 to 51.2, only a 10% difference.
    
    But if one holds the parallax constant, in this table by holding the
    assumed time constant, such as at 5:00:01 between lines 1 and 2, then
    the observed sextant reading has to increase by 34'7" of arc, from
    22*28'12" to 23*2'19", to move the time by lunar up by one hour.   You
    can also see that the observed sextant reading decreases 34'27" between
    lines 3 and 4 of the table to move the lunar time back one hour from
    5:59:59 to 5:00:01.  The error has dropped back to 1 part in 60.  AND
    IN A REAL LUNAR, PARALLAX IS HELD CONSTANT BECAUSE IT IS FIXED BY THE
    ALTITUDE MEASUREMENT.
    
    So this is what George has been trying to tell us.  Perhaps he can
    explain these numbers more clearly than I have; but it was the numbers
    themselves that convinced me.
    
    Fred Hebard
    
    Moon-Jupiter at 1d31.3'S,0d0.0'W on 1/14/04
    
    time assumed  observed sextant  time by   Hc Moon  Refraction  Parallax
      SemiDiameter  Hc Jupiter  Refraction
    to calculate       reading        lunar   decimal        Moon      Moon
              Moon     decimal     Jupiter
    HPM, SD, etc  degree  min sec             degrees         min       min
               min     degrees         min
    5:00:01          22   28   12   5:00:01   86.85463        0.1       3.2
              16.1    70.77158         0.3
    5:00:01          23    2   19   5:59:58   86.85463        0.1       3.2
              16.1    70.77158         0.3
    5:59:59          22   13   19   5:00:01   78.61850        0.2      11.6
              16.1    56.36603         0.7
    5:59:59          22   47   46   5:59:58   78.61850        0.2      11.6
              16.1    56.36603         0.7
    0:00:01          20   28   53   0:00:01   14.15560        4.1      56.3
              15.8    32.29832         1.6
    0:00:01          20   58   45   0:55:53   14.15560        4.1      56.3
              15.8    32.29832         1.6
    0:59:59          20   28   53   0:04:27   28.69067        1.9      51.2
              15.8    47.15395         0.9
    0:59:59          20   58   45   0:59:59   28.69067        1.9      51.2
              15.8    47.15395         0.9
    
    
    
    

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