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I'm grateful to Arthur Pearson for making a useful simulation of the effect
of changing parallax of the Moon on the rate at Which the Moon appears to
move against its starry background. It has given me confidence in the
rather tentative notion that I put forward in "About Lunars, part 4" to
this list earlier this year, and dubbed "parallactic retardation". I am now
fairly sure that this is a serious effect that the observer of lunars ought
to be aware of (and which would have been useful knowledge in the heyday of
lunars).

The main problem with lunar distances is the slowness of the motion of the
Moon against its sky background, of the order of 30 arc-minutes (one Moon
diameter) in an hour. The Moon's (geocentric) angular speed varies
significantly between apogee (when it's furthest from the Earth) and
perigee (nearest), but 33 arc-minutes per hour is a rough mid-value. The
faster motion of 19 minutes per half-hour, shown up by Arthur's
simulations, is because they coincide, nearly, with perigee of the Moon.
Another simulation, made a fortnight later near apogee, would (I predict)
show a much slower geocentric motion for the Moon, and though the
retardation would then be less in real terms, it would become a greater
fraction of the Moon's motion.

Though the Moon moves through the stars much faster than any other object
in the sky, this rate-of-motion limits the accuracy of a measurement of
Greenwich time to something like 2 minutes of time for every arc-minute of
error in the lunar distance, and therefore that error of 1 arc-minute in
distance results in an error in longitude of about 30 minutes. It is this
magnification of errors by a factor of about 30 that is the bugbear of the
lunar distance method, and why such extreme precision of measurement is
called for.

What parallactic retardation shows is that the rapidly-changing parallax in
the view of the Moon by an observer on Earth can further slow the apparent
speed of the Moon through the stars, and by a very significant amount.
There is no such daily-parallax for stars, and very little for the Sun and
planets: it affects only the Moon, because it is so close to us. If the
apparent motion of the Moon is halved (and in extreme cases that can occur)
then the magnification factor of longitude error resulting from
lunar-distance error can increase from about 30 to about 60.

As Arthur's data shows, the effect of parallactic retardation is at its
worst when the Moon is near overhead, and is halved for Moon altitudes of
30�. So, as Arthur points out, a useful piece of advice for navigators by
lunars would be to avoid (if possible) those high lunar altitudes, but (I
would add) also avoiding low altitudes of 10� or less where refraction
errors become significant.

It's worth pointing out that these parallax effects doe not introduce any
actual ERROR, in themselves. The almanac gives the geocentric Moon
position, that is, as seen from an imaginary observer at the centre of the
Earth. That is the best the almanac can do, as it has no idea where the
observer will be. The corrections made in reducing the lunar distance allow
for the effects of refraction and parallax, for an observer somewhere on
the Earth's surface, and are capable of doing so very accurately. What the
parallactic retardation effect does is reduces the SENSITIVITY of the
resulting longitude to the measured lunar distance.

HERE'S A MIND-PICTURE

I keep a picture in my mind of the concept of lunar parallax, which might
help others. I imagine an observer, sitting at the Moon's centre (and you
can reduce the Moon to a point for this purpose), looking at the Earth,
which is enormous in his sky (subtending about 2�). Around the Earth, the
star background will always be visible agains the blackness of space. The
Earth's centre will appear to be moving against this star background,
making a circuit in a month. The line from the Moon to the Earth is exactly
the same as from the Earth to the Moon, but pointing in the opposite
direction. 180� away, so these two lines, though facing opposite
directions, change in exactly the same way.

Now imagine our Moon-man looking, not at the Earth's centre, but at a tiny
man on the Earth's surface, who is riding round with the Earth as it
rotates. The Earth is a ball which is rolling on its axis as it appears to
move around against the stars. But it is rolling "backwards", rather like
the wagon-wheels seemed to do on old western films. This will give the
Earthman, when on the side of the earth which can be seen by the Moonman, a
component of velocity, in the direction of apparent motion of the Earth,
which is opposed to that motion, and subtracts from it. The biggest such
effect would occur at the point directly below the Moon, from which the
Moon would be overhead. And someone on the other side of the Earth,
invisible from the Moon except if the Earth was transparent, would have a
velocity which added to that of the Earth.

So you can see that the man on Earth, as seen by the Moonman, would appear
to have a daily "rocking" motion, of up to 1� each way, which is
superimposed on the motion of the Earth's centre, with respect to the
starry background, of about 33 arc-minutes per hour.. And the motion of the
Moonman seen by the Earthman would vary in exactly the same way, but would
be in the opposite direction, differing by 180�. That is the motion we are
trying to deduce, the parallax-affected motion of the Moon's centre against
the stars..

I have over-simplified matters greatly by ignoring the tilt of the Earth's
axis with respect to the plane of the Moon's orbit, which can reach 28� or
so at times.

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

You will see from Arthur's graphs that although the equations in his
posting include the correction for parallax, in his simulation he has
decreed that the Earth has no atmosphere, and therefore no refraction, to
isolate the parallax contribution, for simplicity.

What Arthur has simulated is the effect of the expression for correcting a
geocentric altitude to give an apparent altitude as seen by an observer on
the Earth's surface. He quotes that formula as-

P = (1-0.0032*(Sin(Lat))^2) * (ATan(Cos(Mc)/((3438/HP)-Sin(Mc))))

which gives a parallax correction (in degrees) to subtract from the
geocentric altitude of the Moon (Mc) in degrees, calculated from the
Almanac. HP, the Horizontal Parallax of the Moon, is expected to be given
in minutes. Only if the  above expression is correct will the simulation be
valid. We have seen how easy it is to make errors in these matters, as in
first posting that expression to the list several months ago, I happened to
make two separate trancription errors, which Arthur Pearson managed to
uncover, and which I corrected in a posting last month.

What gives me great confidence, however, is that Arthur tells me he has
compared predictions from the above formula with Bruce Stark's volume of
Lunar Tables, in which Bruce has provided what he calls a "wrong-way"
parallax correction. Arthur confirms that he sees good agreement wherever
he has checked. As Bruce's table, and my formula, were derived quite
independently, then it seems more than likely that both of us are correct.

Thanks to Arthur Pearson for much painstaking work to confirm the reality
of parallactic retardation.

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

I have accessed Arthur's web page at
http://members.verizon.net/~vze3nfrm/files.html
but have not been able to download the files therein to my old Mac.
Instead, Arthur has kindly sent them to me directly as an attachment to an
off-list email, and should you find the same problem, no doubt he will do
the same for you, if you ask him nicely on  

The text on Arthur's recent mailing to the list, as read by my emailer
program, shows a few punctuation oddities, perhaps as a result of being put
together in a word-processor. If it's the same for you, you will find that
the meaning comes through perfectly well, except perhaps in the equations,
where (I take it) the character "�" (which is distinct from"^", "to the
power of"} is substituted in some places where there should be a minus
sign. Also it appears that "Mc," is printed where "Mc'" was intended. No
harm done, though.

George Huxtable.

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george@huxtable.u-net.com
George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
Tel. 01865 820222 or (int.) +44 1865 820222.
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