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In response to my statement-

>> To get to the real lunar distance you MUST measure the apparent lunar
>> distance (whether it's changing with time or not), and then apply a
>> correction which, being precisely known, does not degrade the accuracy
>> of
>> that measurement. Because the resulting true Moon is always moving
>> about
>> the same speed across the sky, it can always be used to measure time
>> with
>> about the same accuracy.

Fred Hebard responded-

>This is enough to make my head spin!
>
>Now that it's phrased this way, parallactic retardation _seems_ to be
>an effect that could affect accuracy: if the apparent lunar distance
>were not changing at all for some period, then clearly the times at the
>beginning and end of that period could not be differentiated based upon
>the apparent distance, which would be the same.  The cleared distances
>would be different, but not the apparent ones.  We know the moon
>doesn't appear to stand still, but this degradation of accuracy would
>persist to a lesser extent if it were appearing to move more slowly,
>would it not?

The case that Fred refers to is completely hypothetical, but very
illustrative.  What would happen if the Earth happened to spin about twice
as fast as it does, so that the rapidly-changing parallax caused the Moon's
apparent motion against the stars to appear to come to a stop when the Moon
was overhead?

Then, because the apparent Moon isn't moving with respect to the stars,
what on earth would be the point of measuring its apparent position in
order to determine the time?

Answer: You don't find the time from the Moon's apparent position. You find
it from the Moon's true position. The almanac tabulated the distance of the
true Moon from various stars, which you can regard as landmarks or pointers
around its path, every three hours GMT. Knowing the Moon's true position,
you can interpolate between those values to arrive at GMT. So what we need
to know is how that true position varies with time. If we forget about the
effects of eccentricity of the Moon's orbit, then the true Moon will move
against the stars at a steady rate of about 0.5 degrees per hour.

But we can't observe the true position of the Moon. What we see in the sky
is the apparent position of the Moon, altered from the true position by the
changing parallax. When it rises, the apparent Moon is ahead of the true
Moon by about 1 degree. (this figure is the "Horizontal Parallax", or HP of
the Moon, which you can find tabulated in an almanac). When it sets, it's
about 1 degree behind. In between, through the day, it's falling back, on
the true Moon, and that falling-back is fastest around noon.

In the special circumstances we have chosen, that falling-back around noon
happens to be about equal, and opposite, to the motion of the true Moon
across the star background, so that with respect to the stars, the apparent
Moon has come to a stop. So what? We are not trying to discover the time
when the apparent Moon passes a certain value. We are trying to discover
when the true Moon passes a certain value. And we know, rather exactly, the
difference between the positions of the apparent Moon and the true Moon, by
observing the Moon's altitude, and then doing some calculation.  That gives
the parallax correction, the changing value of which caused the apparent
Moon to go slow in the first place. So, we add that known parallax
correction back in to the apparent position of the Moon. That step is
heavily disguised as part of the "clearing" process, but it is very real,
and very important. And that correction brings us back to the true Moon's
position, which will be changing steadily at 0.5 degrees per hour,
unaffected by parallax.

Interpolating for time, using that steady true motion, gives a constant
error in the resultant GMT, which will be quite unaffected by parallactic
retardation. Presuming that the error in the parallax correction is
negligible, then the accuracy in the true Moon's position around its path,
in angle, will be the same as the accuracy in measuring the apparent Moon's
position. Let's put that at 0.5 arc-minutes, just for argument.

With a rate-of-change of 0.5 degrees per hour in the motion of the true
Moon, this results in an error in GMT of just 1 minute.

And taken with a an accurate time-sight of some body moving at about 15 deg
per hour, that gives rise to an accuracy in the resulting longitude of
about 15 arc-minutes.

But the important point I am trying to make is that this accuracy is
unaffected by the parallactic retardation, contrary to the views I have
expressed over the last year.

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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