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    Exorcizing the Evil Effects of Parallactic Retardation
    From: Fred Hebard
    Date: 2004 Mar 14, 10:16 -0500

    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 were calculated from 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 hours in the table below. The retardation can be seen by
    comparing the difference between the "observed sextant reading" of
    lines 1 and 4, 17'55" of arc, with that of lines 5 & 8, 29'52" of arc.
    The Moon is only moving through 17'55" of apparent arc from 5:00 hours
    to 6:00 hours (lines 1 & 4), when it is near zenith, compared to 29'52"
    of arc from 0:00 hours to 1:00 hours (lines 5 & 8), when it is near the
    The thinking has been that since the moon is moving through such a
    smaller swath of arc (17'55"), 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 35 (30"/17'55"), rather than 1 part in 60
    (30"/29'52").  This is the supposed "evil effect of parallactic
    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 parallax is fixed by observation
    of the moon's 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 53.3 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 5:00:00 between lines 1 and 2, then
    the observed sextant reading has to increase by 33'10" of arc, from
    37*0'57" to 37*34'7", to move the time by lunar up by one hour.   You
    can also see that the observed sextant reading changes 33'20" 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
    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


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