NavList:

   

Reply

On Sat, 10 Jan 2004 00:45:09 +0000, George Huxtable wrote:

>This effect of parallax is superimposed on top of the normal motion of the
>Moon against the starry sky, and as that motion is always toward the East,
>you can see straightaway that the effect of the rocking due to parallax on
>the true motion of the Moon is always to make it appear less. Sometimes
>significantly so. In an extreme case, when the Moon passes overhead, it can
>roughly halve the true motion, so it's a very important effect.
>
George, forgive my presumption, but I wonder if you would please
indicate where my interpretation may have gone off the track.

I think of the problem as being in two parts also.  As an analogy in
the absence of a blackboard, think of a person on one of those now
banned bits of playground equipment that is a big, level, circular
platform that pivots in the center and goes around and around.

Part I
Put yourself on the platform in a swivel chair.  Look up at a lamp
standard about thirty platform diameters away; or closer, to
exaggerate the effect.  Have someone impart a gentle, constant motion
to the platform.  Notice that the lamp moves in an ellipsoidal pattern
against a distant background ... mostly side to side.  Both these
movements (up and down as well as side to side) are roughly harmonic
or sinusoidal if plotted against time.  For a platform constantly
turning counterclockwise as seen from above, also note: (1) the lamp
will turn clockwise against a distant background if it is above the
plane of the eye of the observer, (2) the lamp will trace a straight
side to side line if it is the plane of the observer's eye, (3) the
lamp will appear to trace a counterclockwise elliptical path against a
distant stationary background if it is below the plane of the eye of
the observer.

Incidentally, I have heard the term "parallax scroll" used in computer
games for the movement of a foreground against a background to give a
three dimensional effect to the older games.  To avoid confusion with
the correct term, I will use "parallax scroll" for this apparent
motion.

If I understand what you say, the actual (or true ) location of the
lamp (or moon) against the distant background (or stars) is stationary
in my example so far (or known and listed in the almanac) for any
given instant as calculated to the center of the platform (or the
center of the earth).

From surveying practice, the apparent position of the street lamp's
elliptical "parallax scroll" against the background (say a big brick
wall with a few spot lamps on it) can best be fixed against that
background with angular distance measurements from three equidistant
objects radiating out at 120 degrees from the street lamp.  (The stars
are not that accommodating for the moon ... at least not in the last
two months of overcast I have had over here ... would be difficult for
a moon position close to the horizon, and may not show well against an
illuminated limb of the moon.)

Locating the Part I or "parallax scroll" position with a fixed
precision and accuracy of observation would give the angular position
of the platform to a varying degree of accuracy, greatest when nearest
the street lamp, and least at the far left or the far right.  Greatest
when the vector arrow length (arc distance divided by time) of the
parallax is longest.  Translating this to Celestial navigation, it
would seem the accuracy of position finding would be greatest when the
observed moon crosses the meridian.

Part II
Superimposed on this "parallax scroll" of the lamp or moon motion is a
movement of the whole background, in this example, it would not be
moving very much, about one degree per revolution of the platform,
about five feet at one hundred yards.  Any observation of the
background around the position of the street lamp would not be
inherently much more difficult at any position of the platform ... but
refraction would make the far east or far west observations more
suspect.

Finally, there is the movement of the lamp.  It moves from right to
left at the rate of about twelve degrees per revolution of the
platform, and usually up or down a bit.  If we take away the parallax
scroll of Part I (by shifting to the center of the platform), there is
no difference in the accuracy of locating the lamp against the
background no matter where it is ... and the motion of the street lamp
during the time of observation would describe an almost straight line
against the background.  For the moon, there is the problem of
refraction near the horizons which spoils the observational accuracy
in spite of a constant observational precision.

Argument
If you want a good fix of the lamp against the background, or the moon
against the star field, take it when the apparent motion of the lamp
(moon) is greatest; in other words, when the vector length per second
of observation, as determined by the greatest vector sum of the *two*
"parallax scroll motions", the lamp (moon) orbital motion, and the
slow background (or celestial) motion.  Don't worry about how fast the
vector sum arrow is rotating or changing direction, only about how
long it is, because that will determine the smallest error in position
of the platform (longitude) for a given precision of an observation
set of the angular distance between the street lamp (moon) and three
distinct bricks (stars).  Again, I am defining the vector sum as
angular distance over time.  Working backward, if the angular distance
over time is zero, the position is effectively unknowable because the
potential error is so great.

Unfortunately, the vectors for the horizontal motion of the orbiting
moon and the rotating earth oppose each other most at the meridian
crossing, and depending on your latitude and the time of the year
there may not be much vertical motion.  Too bad atmospheric refraction
degrades the accuracy of an observation close to the rising or setting
moon by an unknown amount, otherwise that should be the best time to
take a lunar.  Also, the likelihood of having three stars to check
against the moon is not good, because at least half are below the
horizon.

Conclusion
All in all, it might be best to take the lunar whenever you can get
the greatest number of well positioned stars for observation, as the
apparent motion is still appreciable even at the meridian crossing.

If the earth were transparent, the apparent motion would be greatest
when the moon is on the meridian 180 degrees from the observer
(anti-meridian crossing?), and *that* would be the best time, as the
vector lengths of the individual major motions would combine to make
the longest arrow.  The moon could also be seen to be making the
complete epicyclic path against the background stars instead of the
"bob" you describe, which is the top half.

Question
Since I am so weak on the celestial math and the standard formulae, I
don't know how accurately the stars above or below the moon's path can
be used to help determine it's absolute location in the celestial
sphere from the observed sextant angles ... and from there determine
the location of the observer on the actual earth.  It seems to me the
stars most in line with the vector of the moon's *orbital* motion
would give the most accurate results from the tables, not the ones
most in line with the longest vector of the apparent motion, but I am
not at all sure about that.  (Not that there is ever the perfect
choice of stars in the first place.)  Have I made a reasonably good
stab at an analogy and its conceptual analysis?

I hope I have made my thoughts clear enough to critique.


--
Richard ...

   

Reply