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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
horizon.

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
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 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
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

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