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Hello Arthur, Fred, and All

I still owe Fred and Arthur a response following a correspondence from
February, when I had trouble with my computer. I apologize for the
delay.

Arthur asked:

"Am I correct in thinking of the change in distance [between two
celestial bodies] as a vector resulting from difference between the
vectors of their individual motion against the celestial sphere? I hope
I am not opening a can of worms here.  I had been focused on the single
dimension of the moon pulling
away from or closing with the moon, which for practical purposes is
fine, but I am interested in understanding the components."

In order to solve the problem of finding GMT from a lunar distance we
need no "insider information" about how this distance at a certain
moment comes to pass, i.e. the actual motions of the moon and the other
body in the sky. We satisfy ourselves with snapshots of the distances at
suitable intervals, taken from an almanac or self-computed, and hope
that linear interpolation will be adequate. Nevertheless, it may be fun
to investigate the individual motions of the two bodies in a few
examples, i.e. their individual speed and direction with respect to the
starry sphere.

For this purpose, let us look at two observations that Fred Hebard
posted. He asked:

"I have been computing the comparing distance between the moon and an
object (the one _not_ corrected for parallax and refraction), and
that has been varying. For instance, the moon and Jupiter were
separating at 35.3' per hour between 2:00 and 3:00 GMT on 12/27/02,
while the sun and moon were separating at 27.0' per hour between
20:00 and 21:00 GMT on 1/11/03. Not all this can be attributed to
the hourly movement of the sun. Is all the rest from their not moving
in the same direction or are there additional factors?"

There are several factors, but the main culprit is the "inconstant moon"
itself.

The moon's orbit is a pronounced ellipse (mean excentricity = 0.055), so
the actual angular speed at any given moment in time can vary
considerably from the sidereal mean motion of 32.94'/h. The moon speeds
up while approaching perigee and slows down while approaching apogee
once a month. In addition, the sun influences the motion of the moon in
several ways. Main factors are: the elongation between sun and moon, the
alignment of the sun with respect to the apses of the moon's orbit, and
the distance of the earth-moon system from the sun.

The complexity of the lunar motion is the very reason why it was so
difficult to come up with ephemerides that were precise enough to put
the long recognized theoretical possibility of using lunar distances
into actual practice. All of the above was known to Tycho Brahe, most of
it even to Ptolemy. But a sufficiently exact quantitative description
eluded us at least until Tobias Mayer.

Just to give you a rough idea of the possible extremes, I looked at the
moon's angular motion during the past 100 years with a computer program
and found values as low as 29.55'/h and as high as 38.35'/h.

Fred has caught the moon near the possible extremes. On 2002-12-27 02UT
the angular speed of the moon w.r.t. the starry sphere as it would have
been observed from the center of the earth was 35.18'/h. On 2003-01-11
20UT it amounted to 29.64'/h. Most of the discrepancy in the two
relative motions that Fred observed on different dates can be attributed
to the varying speed of the moon alone.

The sun's mean motion w.r.t the stars is 2.46' per hour. Again, the
actual motion varies due to the excentricity (e = 0.017) of the earth's
orbit, but much less so than the that of the moon. The fluctuation is
only +/- 3.3%. It causes the asymmetry of the Equation of Time. You can
get a visual idea about the magnitude of this effect by judging how much
a graphical representation of the EoT deviates from a perfectly
symmetric sinus-like shape.

On 01-11 the earth is near perihelion and the actual motion is at a
maximum of 2.55'/h. Note that this is the speed in the direction of the
ecliptic, not along the celestial equator. When you check this against
the tabulated values for GHA in the nautical almanac, you will see that
the SHA (or Right Ascension, if you prefer) of the sun at the same time
changes 2.71'/h. This makes sense: Think of a boat sailing at 2.55 knots
along the ecliptic line drawn on a globe. As the boat approaches 23deg
latitude, its change in longitude per hour will be 2.71', because of the
converging meridians. (With this we have just demonstrated the second,
much larger component of the EoT that is responsible for the sinusoidal
shape of the graph.)

Knowing that the sun always moves into the same general direction as the
moon, we subtract the sun's 2.55' from the moon's 29.64' and obtain
27.09' for the hourly increase in distance between these two bodies.
That's fairly close to reality, but an exact computation gives 27.04'
for the instant of Fred's observation. That's because the moon can
deviate from the ecliptic up to 5deg. In our particular case the moon
did not head straight away from the sun, but in a direction of 176deg.

Now let's move to Fred's second observation. Looking at the data from
2003-12-27 involving Jupiter, we notice an interesting phenomenon. The
increase in distance, which I compute to be 35.20'/h, is slightly
greater than the speed of the moon!

This can happen only with planets, but it is not at all unusual with
them. It is a consequence of their ability to enter into apparent
retrograde motion. (Other than by looking at the sky for several days in
a row, you can check this by inspecting the v-correction at the bottom
of Jupiter's column in the Nautical Almanac. On the given date, it is
greater than 2.46', meaning that the GHA of Jupiter is increasing faster
than that of Aries. Consequently, Jupiter moves west over the stellar
background). Obviously, this is only of theoretical interest, as the
effect is smaller than the rounding error that can occur in manual
computation using tables.

To sum up: When Fred observed the distance sun/moon, the speed of the
moon already happened to be low and the relative speed was further
decreased by an especially fast sun. When he observed moon/Jupiter, the
moon was already moving fast and Jupiter moved away in retrograde
motion.

Is there ever any practical need to consider the motion of the moon as a
vector? Maybe, for sight preparation. We said in the introduction that
once we have a distance observation, it is not necessary to consider
vectors, since all we need is to interpolate the snapshots. But how do
we know that the distance function behaves nicely so that linear
interpolation is adequate? And how can we get guidance of which bodies
to observe? The answer is to check the motion of the moon against the
positions of the other bodies.

As long as the almanacs contained distance tabulations, their authors
did the thinking for us. If the star (or planet) was listed on a certain
date, it probably was a good star to observe. The almanac maker's
selection criteria included:

1. The star should be near the ecliptic.
2. The stars should be evenly spaced.
3. The star should be bright.
4. The star should not be too far away from the moon (< ca. 120 deg) .
5. The star should not be too close to the sun.
6. The star must be between the moon and the sun, or else it must be on
the moon's side.
7. The star must not be too close to the moon.

Some of these are pretty obvious. I want to draw your attention to 1) 4)
and 7). These three criteria provide a rough method of eliminating stars
that are not situated in a favorable direction with respect to the
velocity vector of the moon. Ideally, the moon should move straight
towards the star or away from it. There are two disadvantages, if it
doesn't: First, the distance changes only with the cosine of the
relative bearing of the star (compare to "velocity made good" when
tacking upwind); second, the distance will not change in a linear
fashion, complicating interpolation. 1) und 4) take care of this.

As to 7), there is really nothing wrong with a star being too close to
the moon, per se; but the likelihood that it does not lie dead ahead (or
astern) is sharply increased. The only way of finding out for sure is to
inspect the particular vector diagram: Course and speed versus relative
bearing. When almanacs were computed manually, one would not bother with
such subtleties. A broader brush like criterion 7) was required. But a
modern algorithm for star selection, be it for an almanac or an ad hoc
sight planning program could be based entirely on the analysis of the
vectors involved.

So far we have only talked about true motion. This post is already
getting long, so I will mention just in passing that the apparent motion
of the bodies could also be decomposed into their true motion and a
second component that is due to refraction and parallax. The true stars
may be fixed, but the apparent stars aren't. They describe little
elliptical semi arcs, convex towards the horizon between rising and
setting. The circumpolar stars describe full ellipses during a siderial
day. The moon does a similar thing, but in the opposite direction,
because of parallax. Could we stop the true motion of the moon for an
evening, the effect would be quite noticeable, considering that parallax
is twice the diameter. Superimposing the dance for parallax with the
true motion, we find that parallax changes course and speed of the
apparent moon. The change in course is at best a degree or so. Not
really relevant for lunar distances, but too bad if it should turn out
as a collision avoidance maneuver spoiling an expected grazing
occultation.

Best regards

Herbert Prinz

   

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