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Parallactic retardation of the Moon. A summary of the present position.

I find myself in a VERY odd position, just now.

1.

A year ago, I discussed the effect of rapidly-changing parallax (caused by
an observer's motion with the Earth's surface) on the measurement of lunar
distance. I pointed out that it (nearly) always acted to slow the apparent
motion of the Moon in the sky (from its true value), and in an extreme case
(Moon passing overhead) to halve it, roughly speaking. The list agreed on
naming the effect "parallactic retardation" of the Moon. A description of
the effect was discovered in a German text of the 1870s.

That posting "About Lunars, Part 4a", of 11 Jan 2003, is archived at-
www.irbs/com/lists/navigation/0301/0066.html

There was general agreement about this matter on the list, and Arthur
Pearson devised a program to show it up. And that remains the situation at
present. We all seem to agree, It's non-controversial.

==================

2.

I took things a step further in that posting, and argued that when you were
trying to measure the time from the position of the Moon, if the quantity
you were measuring (the apparent distance) was changing more slowly with
time (by half, say, in that extreme case), then the accuracy of the
resulting value for GMT would, similarly, be halved. The bugbear of the
lunar-distance method is always its requirement for extreme accuracy in the
measurement of angle, between the Moon and another body. Anything that made
the resulting GMT even less  sensitive to that measured angle should be
avoided. So I recommended as a result, that for maximum accuracy lunars
should be restricted, if possible, to situations when the altitude of the
Moon didn't exceed 30 deg or so.

This conclusion seemed to be accepted; at least, nobody seriously argued
with it.

==================

3.

There, matters stood for about a year, until Jan Kalivoda, in a penetrating
comment on 6 Jan 2004, in the thread "Real accuracy of the method of lunar
distances", caused me to think again.

My conclusion from that rethink is as follows-

Take the extreme case of the Moon passing overhead. Although (because of
changing parallax) the apparent motion of the Moon in the sky is then
roughly halved from its true value, an important correction has to be added
to it, which also varies with time. That correction is due to the rapidly
varying displacement, due to parallax, of the position of the Moon ALONG
ITS PATH. It can be accurately calculated, knowing the Moon's altitude.

That correction isn't explicitly obvious, because it's just a part (but the
major part) of the complex "clearing" process, which corrects for many
other matters at the same time; refraction for both bodies, parallax for
the other-body, and also the misalignment of the other-body from the path
of the Moon.

No doubt an expression could be derived for that displacement on its own,
but I haven't applied myself to that task. It would take into effect only
Moon parallax, not parallax of the other body, not refraction of either. It
would consider only displacement of the Moon along its path, not any
component at right-angles. It would presume that the other body is exactly
on the Moon's path somewhere, so its actual position doesn't need to be
considered. Seems a simple matter, really. Someone may wish to try it.

Anyway, that displacement, applied to the apparent position of the Moon
along its path, gives the true position of the Moon along its path, and I
am presuming that the displacement can be accurately known.

So, to the apparent lunar distance, slowly changing in our extreme example,
and with all the imperfections of sextant measurement, we must add (or
subtract) a precisely-known displacement which also changes with time. In
fact, it changes with time exactly enough to make up for any slowing of the
Moon's apparent position, so that after correction for that displacement
the resulting true lunar distance changes exactly at its normal rate of
about 0.5 deg per hour. That's self-obvious, of course, isn't it, seeing
that the slowing was due to changing parallax, and the displacement was to
allow for the effect of parallax.

The end result will be a quantity varying by 0.5 degrees per hour, and with
the angular uncertainty of the sextant observation. Variations in the
apparent motion of the Moon have been corrected out, and do not affect the
overall accuracy of determining time.

My conclusion now: that the accuracy of a lunar distance does NOT depend on
the apparent motion of the Moon, but its true motion; and there is no
advantage in accuracy in restricting Moon altitudes to less than 30 deg. If
this is right, then significant parts of my posting "About Lunars, part 4a"
need to be withdrawn and amended.

===============

4.

Now we get to the odd position that I am in now. Since suggesting that my
view of this matter has changed so radically, I've had messages from Fred
Hebard, Bill Noyce, Bruce Stark, Frank Reed, all urging me that I was right
in the first place. I am not convinced, yet, by these arguments, though my
mind remains open. As there's so much disagreement, on what ought to be a
straightforward (though complex) matter, it certainly seems worthwhile to
continue trying to resolve it and come to an agreement, if possible.

===============

5.

I suggest that those who oppose my new view of the matter are missing an
important point. The true lunar distance, when measured, is made up from
two components. One is the apparent lunar distance, as measured. The other
is the correction for displacement by parallax, which is disguised within
the clearing process, and which can change at a rate which is comparable
with the rate of change of the apparent lunar distance. When one of these
components changes more slowly, the other changes faster, to compensate.
You MUST think of these two components together, not dismiss the second as
being "only a correction". The end result is a corrected value which
changes at a steady rate.

Although the apparent lunar distance may at times change more slowly, you
can't use the apparent lunar distance until you have corrected it. The
slope of a plot of apparent lunar distance with time is, I maintain, quite
immaterial.

No doubt some contra-arguments will ensue, which will be welcome. How I
wish we could gather round a blackboard and draw some diagrams! It's the
one thing Nav-l lacks.

George.

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contact George Huxtable by email at george@huxtable.u-net.com, by phone at
01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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