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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Moon - Antares
From: Frank Reed
Date: 2009 Jan 26, 19:07 -0800

You wrote:
"Mr. Reed, I believe that civil discourse requires a bit of formality,
particularly due to some of the invective found in the archives.  Topics that
involve intellectual discourse can become testy, so for now, I will stick to
the nee-plus-ultra formality.  "

"I have been considering the statement by Mr. Reed that the changes in
elevation at the moon's limb may result in "seconds" of variation with
respect to the time of an event"

I do have one request: please do not refer to me in the third person. I
PERSONALLY consider that impolite. Ok? :-)

You calculated the Moon's distance as:
"356,043 km = 14959871 km * 0.00238"

That's a bit low. The Moon is about 240,000 miles or 385,000 km away. Since
the orbital eccentricity is around 0.055, the apogee and perigee distances
are about 5.5% greater or less than this (somewhat variable because the
Moon's motion is very far from a simple Keplerian ellipse).

You wrote:
"16.775 arc-minutes = DEGREES(arctangent (1737.4 km / 356043 km))* 60"

Just FYI, you don't need to dig out trig functions when you're calculating
small angles unless you need an extraordinarily exact value. For small angles
just take the distance across and divide by the distance out. Then multiply
by 3438. That gives you the result in minutes of arc. It's convenient and
quick, and you don't need a calculator. It's also very useful in marine

And you wrote:
"The angular rate of the moon is given in the Nautical Almanac,  in the
increments section, as 14 degrees 19 minutes per hour (examine the increment
for 59 minutes and 60 seconds)."

Now here you have made a mistake. That number you've taken from the almanac is
the base rate at which the Moon changes its GHA (the total rate, on average,
is about 10 minutes of arc per hour faster since you have to add "v"). But we
don't want the rate of change of GHA. We need to know the rate of change of
the Moon's position relative to the stars or in other words its rate of
change of SHA. That is a MUCH small number. Subtract the Moon's average rate
of 14.5 deg per hour from 15 degrees per hour... The rate of change of the
Moon's position relative to Antares or any other star is around 0.5 degrees
per hour, or dividing both by 3600, around 0.5 seconds of arc per second of
time (the actual rate varies somewhat both because of the Moon's varying
orbital speed and also because of changing parallax during an observation).

You wrote:
"Next we should find the highest mountains on the moon."

Actually what you need are the differences in altitude between the highland
features near the Moon's limb and the lowland plains. The lowland plains are
places like Mare Smythii and Mare Orientale. The highlands aren't usually
named but there are some very high areas both north and south of Mare
Orientale. The difference in altitude between the plains and highlands is
about 10km. So the mountain heights you found should be doubled to get a real
sense of the extremes of topography near the limb. Also note that the "limb"
is a bit of a "moving target" because of lunar librations.

And you wrote:
"Time, seconds = Angle Subtended * Moons Angular rate
Mons Huygens        0.222 seconds

Again, because of the mistake I noted above, you need to multiply these
numbers by about 30. You can easily get a difference in occultation timings
of three seconds due to limb features.

And:
"In short, the tallest mountains on the moon, even if on the limb, fall far
short of "seconds" of time."

As I hope you can see now, several seconds would not be an unusual timing
difference due to limb features. For occultations, there's also another
factor. Imagine a case where Antares is just barely occulted. That would
happen when the Moon's center passes, for example, 15 minutes of arc just
north or south of the star. The star will be seen to approach the limb at a
shallow angle. In that case, a small difference in angular height of limb
features can have an even larger impact on the timing of occultations. If the
occultation is "grazing", a star can disappear and then briefly re-appear as
it lines up with a deep valley near one of the Moon's poles.

And you concluded:
"From a different standpoint, lunar calculation for longitude could not have
been successful had the limb provided such variation in precision as seconds
of time.  Observers, of necessity, bring the object to the moon at different
limb locations as a function of the close proximity of the moon to the earth.
If seconds of variation existed, then the longitude could not have been
determined with the degree of accuracy necessary."

Not true. At best, observers in the late 18th/early 19th centuries expected to
determine GMT using lunar distances within 15 or 20 seconds, and even 30 or
60 seconds of time was considered acceptable error. The few seconds, plus or
minus, caused by the variations along the lunar limb did not matter to them.
Whether it matters to modern lunar distance enthusiasts is an open
question...

-FER

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