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I've had a go at calculating the effect of Moon's parallax (only) it terms
of its component along the path of the Moon, with respect to the starry
background. That, I suggest, is the only component of importance in
determining the effect of parallax on lunar-distance to another body
exactly along the Moon's path. There's another component, at right-angles
to the Moon's path, that only slightly affects the lunar distances;, tha I
will disregard.

Ignore any parallax of the other-body, and ignore refraction, in order to
isolate the effect of Moon parallax.

Try following this word-picture.

Draw a standard PZX spherical triangle, with which many listmembers will be
very familiar. P points along the Earth's axis above the North pole, Z
points along the observer's zenith (right above his head), and in this case
X points along the true direction of the Moon's centre (from the Earth's
centre)

In this spherical triangle PZX, all the sides are of course parts of great
circles.

The angle at P is HA, the Hour Angle of the Moon; the opposite side p is
the co-alt of the Moon as seen by the observer (i.e. 90 - alt).
The angle at Z is the Azimuth of the Moon, which we won't be using; the
opposite side is the co-dec of the Moon, always between 60 and 120 degrees.
The angle at X is what will interest us; the opposite side is the co-lat of
the observer (i.e. 90 - lat).

As the Earth turns, the Moon will be moving in a circle centred at P. If,
but only if, the Moon's declination is zero (co-dec = 90), that will be a
great circle. Instantaneously, when The Moon is at X, its velocity will be
at right-angles to PX

Parallax depresses the Moon's altitude (or increases its co-alt) by an
amount P = HP cos alt.  HP is the horizontal parallax of the Moon, varying
between apogee and perigee, but never departing far from 1 degree.

This parallax displaces the apparent Moon from X, in the direction ZX. What
we are looking for, however, is just the component of that displacement in
the direction of travel of the Moon; that is, the component at right angles
to PX. This component is obtained from
P sin X.

Sin X can be found from the sine rule for spherical triangles, which gives-

sin X / sin co-lat = sin HA / sin co-alt,

which can be rewritten as-

sin X = sin HA cos lat / cos alt

So the parallax component along the Moon's path , P Sin X, is-

HP cos alt sin X, or

Displacement by parallax of the Moon along its path = HP sin HA cos lat.
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I am surprised that this has fallen out so neatly, to be independent of
Moon dec, and hope that there isn't an error somewhere. That's one reason
for posting this result now, to see if anyone can pick a hole in it, before
taking the argument further.

In some respects, it seems reasonable, in that there's no displacement by
parallax of the Moon along its path when HA is zero (Moon on meridian), and
then the rate-of-change of that displacement is a maximum. Also there's no
displacement at the poles.

Comments are welcome.

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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