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I'm really pleased to note that this Lewis & Clark lunar problem, which has
been puzzling my brain for weeks now, has engaged the attention of some of
Nav-L's deep-thinkers. And seems to have puzzled them just as much. Indeed,
their postings read as though they were submissions to that much-missed
scientific periodical, "Journal of  Null Results".

But seriously, those negative responses, explaining the possible hypotheses
they have tinkered with to try to make sense of the L&C observations (and
failed) are a great encouragement to me. Some, but only some, of those
ideas I had considered already. There's nothing to beat a few minds
attacking a problem from different directions, and discussing their
findings.

From data derived quite independently, I can confirm Frank's table of
Moon-to-Aldebaran centre-to-centre apparent lunar distances an observer
should see over that night of 2-3 December. I made a similar plot of
distances (but to Moon limb) and allowing for a Moon semidiameter they
correspond well. They confirm that over the interval of about 2 hours
between Moon-Aldebaran observations A and C, the lunar-distance between
those bodies should be changing at about 0.418deg per hour, and not the
0.239deg per hour that Ken Muldrew deduced from the L&C observations.
There's a big difference!

In response, Frank Reed replied-

>Note that the rate of change in distance is not emphatically inconsistent
>with the reported observations. Lunars do sometimes change that slowly...

and here I disagree with Frank. Yes, apparent lunar distances sometimes CAN
change that slowly, if the Moon gets very high in the sky: but not on that
night at those times, as Frank's figures (and mine) confirm. Perhaps when
the Moon approaches zenith, several hours later, its apparent speed may
slow to correspond with observations, but that wouldn't happen until
several hours later.

I thought it might be interesting to find out how the apparent Moon moves,
against the star background, between observations A and C.

At time A (GMT 5h 28m 33.4s) I make the declination of the apparent Moon to
be 18.617deg, and its Sidereal Hour Angle (SHA) to be 228.828deg; which
corresponds to a Right Ascension (RA) of 8h 44m 41.0s.

At time C (1hr 58m later, GMT=7h 22m 32s), I get declination of apparent
Moon to be 18.326deg, and SHA = 227.985deg (or RA of 8h 48m 3.6sec).

Between those positions, I make it that in 118 minutes of time, the Moon's
calculated position has shifted by 0.85 degrees, so the apparent Moon
should be moving through the sky at a rate of 0.432deg par hour. Its
direction of motion is predominantly Eastwards, but it is dipping
significantly from due East, by about 20deg, at an azimuth direction of
250deg. Or we can think of the direction it's coming from, looking back on
the Moon's track through the sky, as about 70deg azimuth (measured
clockwise from due North).

For what follows, we will presume that the effects of changes in star
refraction are negligible, and for simplicity, ignore them.

Note that the calculated lunar distance to Aldebaran, over this period of
118 minutes between A and C, changes by 0.828deg, or at 0.421 deg per hour,
which is only slightly less than the speed of the Moon itself. We can
conclude that Aldebaran must be close to the great-circle path of the Moon
(at that moment). In fact its divergence from that course should be about
13deg. (from arc-cos (.421/.432)), to one side or another; but not well
defined.

Ken Muldrew observed-

>If we consider the two series with Aldebaran (A and C in your
>original message) as one series, then a least-squares fit gives an
>apparent motion of the moon of 0.239?/hr.


That puzzles me a bit. I have plotted the points out and drawn in a
fit-line by eye, and the slope I get is not 0.239 deg/hr, but nearer 0.293.
I can find nothing to bend, on that plot, to even approach Ken's value. So
I ask him to recheck his least-squares fit, please, in case it resolves
that discrepancy.

Even if I somewhat-arrogantly presume that my figure is the correct one,
then there is certainly a discrepancy with the calculated change of
lunar-distance. It could apply only to a star which diverged from the
great-circle path of the Moon by an angle of about 48deg (arc cos (.292/
.432)), to one side or the other. I can't understand how Ken has arrived at
a value of 36deg for that same divergence angle.

So, if you take a starmap, and mark on it the Moon at dec 18.617, RA 8h 44m
41s. Then draw two lines through that point: one at an angle of (70 - 48)
deg, or 22deg (measured clockwise from North, which is straight-up on the
map); the other at (70 + 48) deg, or 118 deg clockwise. Extend these lines
as great circles to an arc-length of 60deg 56.2' from the Moon. Then the
mystery star ought to lie somewhere near the end of one of those arcs.

Well, all I have here is Norton's star almanac, and I can't even hope to
approximate such big great-circle arcs on its map projection. What it needs
is someone with a decent-sized star-globe to see if there's some bright
star that fills the bill; though I doubt if there is.

Remember that many (but perhaps not all) star-globes show a
laterally-reversed view of the sky, and the pictures of the constellations.
This is because you and I see the bowl of the heavens from the inside, and
so does the star-map, but the star-globe views it from the outside.

Usefully, Frank has listed some of the ideas that have come into his mind
as possible sources of error, and which he has, after thought, rejected.

Here's another possibility. We suspect that observations A and C can't
possibly be of Aldebaran, because the lunar distance isn't changing fast
enough. Any star that did fit would have to be implausibly far from the
Moon's path. But perhaps the misidentification was a double one. Perhaps
observation A was between the Moon and one star, and observation C was to
another star, neither being Aldebaran. In that case, what seems to be a
slowly changing lunar distance over the intervening 2-hour interval would
be quite immaterial. Trouble is, we haven't found even one bright star
which fills the bill, yet, never mind two.

Frank concludes concludes-

>I have no real hypothesis, but maybe it's just plain junk data. I've seen in
>some 19th century whaleship log books occasional cases where it's very likely
>that data has been transcribed into the "official" log from someone's notes
>with blatant transcription errors. Maybe this case is nothing more than sloppy
>copying, after-the-fact.

In some ways I agree. We need to be suspicious.

The original manuscript that contains this section, which is described as
the "Eastern Journal", was discovered as recently as 1913, and is now held
in Philadelphia, in the library of the American Philosophical Society. It
was transcribed and edited by M.M.Quaife as "the Journals of captain
Meriwether Lewis...", pub in 1916 by the State Historical Society of
Wisconsin, and I have access to a copy in Oxford. It was transcribed afresh
by Gary Moulton for his 13-volume "Journals", and I have copied data from
Moulton into my postings to the list. And the "original manuscript" may
well have been transcribed from rough fiel notes, made on the night. So
there many steps in that process where errors could occur.

And yet- where could the transcription error possibly be? Not in the
determination of chronometer error from the following around-noon sight, I
am rather certain, because it fits in so well the gradual losing-of-time,
extrapolated from other equal-altitude observations made in the previous
few weeks on the Mississippi. And a time error would have to be quite
immense, a couple of hours or so, to fit in with the "Aldebaran"
observations. It's just not on.

As for the lunar distance observations, each of the three sets is quite
self-consistent, allowing for a scatter of a couple of minutes about the
mean value. Any common error, or transcription error, would have to apply
to the whole set of six readings. The errors of more than a degree in the
"Aldebaran" lunars are far too big to be accounted for by choosing the
wrong limb of the Moon, which at the most could give rise to an error of
30' or so. A systematic error in reading the sextant by exactly 1 degree,
or the chronometer by one or even two whole hours, wouldn't work, as I see
it.

In response to Bruce's suggestion, about possible misreading of the
vernier, I can only say that I have followed up a number of celestial
observations in the earlier part of their Mississippi travel, and though
L&C made many blunders in recording the details and analysing the results,
those sextant observations themselves (always by reflection) always seemed
to give the expected result, to within a minute or two.

The only answer that I can suggest, is that the wrong star was identified
in each case, A, B, and C, though if we are unable to suggest possible
"wrong" stars, it's unconvincing.

There's a real mystery here, which we haven't yet got to the bottom of, and
perforce I will just have to accept that and put this lunar stuff to one
side while I follow L&C up the Mississippi to their winter camp. Perhaps
their next lunars, taken several months later, may provide further clues.

Thanks for everyone's help; and don't stop pondering.

George.

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