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Hi George, you wrote:
"I can see, then, how Frank's proposal could give one position line,
placed at right angles to the azimuth of the Moon, on the chart,
because the Moon's altitude has been deduced from its parallax. But
it's only the Moon's altitude that can be deduced by this method,
because it has such a large parallax. No information is provided on
the altitudes of stars.

I haven't yet understood Frank's explanation of how a second position
line can be deduced by measuring the distance between the Moon and a
second star, and, I too, ask for a more detailed explanation with a
numerical example. "

First, with respect to a numerical example, refer to the Moon-Aldebaran
and Moon-Pollux sights that I posted on Thursday. All the information
that you need is there. To "work" the sights, use any software that
allows you to clear a lunar distance and also calculates the objects'
altitudes for you (my web site does this... you could also use Arthur
Pearson's old spreadsheet... maybe you have your own software that does
this). Go to integral longitudes on either side of the DR and vary the
latitude at each until the observed distance for the Moon-Aldebaran
sight matches the calculated apparent distance. Take those two points
and draw a line through them. That's your lunar line of position. Do
the same for the Moon-Pollux sight. Cross the two lines of position and
read off your fix. While you're at it, vary the position a few miles on
either side of the LOP (pick points that are on a line perpendicular to
the LOP) to find out how wide the band of confidence is around that
LOP.

Now for the theoretical background...

I first started thinking about this while puzzling over Ken Gebhart's
"Sun squash" idea. He proposed using the apparent flattening of the Sun
at sunset or sunrise as a surrogate for the altitude. Basically, this
idea says that we can measure the apparent flattening even when we
can't see the horizon and then we can look up the corresponding
altitude by using the altitude corrections tables in reverse. That's a
good idea for a low accuracy altitude, and if measured with a sextant
it could really be effective, but I started to wonder whether there
might be a way to do this that is more general and, with any luck, more
accurate. I considered the possibility of measuring the angles between
the stars of Orion's belt when they're rising (since they're visible
from almost everywhere on Earth). The variation in refraction could be
turned around to give their altitudes even when the horizon is not
clearly visible. But the trouble there is that stars are not generally
visible at low altitudes, especially when the horizon is in doubt. So
why not use the Moon? It has large changes in its altitude correction
with altitude (except from about 7 to 15 degrees). Could I somehow
measure the Moon's altitude correction and do a reverse-lookup in the
altitude correction to determine the Moon's altitude... And then I
realized: that's a problem that is nearly identical to the historical
problem of lunar distance observations.

When I work a lunar distance observation, I need the Moon's altitude
correction (parallax+refraction) to clear it and compare with the
expected true distance in order to determine GMT. If we already know
the GMT, lunar distance observations like this are not necessary, but
we can invert the problem and use the difference between the observed
and true distance to read out the Moon's altitude correction. At this
point in my thinking, it appeared that I could only get one altitude
correction, and therefore one altitude and one line of position. But
this entire process can easily be re-expressed in the modern language
of lines of position, circles of position, and more generally "cones of
position".

When I measure an ordinary altitude of a star here on Earth, I generate
a circle of position, which is approximated locally by a line of
position. But if I am not exactly on the Earth's surface, that circle
of position has some extension in space. After correcting for
refraction and dip, a measured altitude places me on a "cone of
position" where the apex of the cone is located at the center of the
Earth. The intersection of that cone of position with the surface of
the Earth is the usual circle of position.

When I measure the distance of a star from the center of the Moon
(assuming now that I have corrected for semi-diameter, which happens to
be equivalent to correcting for dip), I am now creating cones of
position centered in an entirely different location. Imagine measuring
the distance from Aldebaran to the Moon and finding that it's 12.5
degrees from your location. Where else on Earth and in space would the
angle be the same at that instant in time? Clearly, the angle would the
same at all points on a large conic surface with its center located at
the center of the Moon. This is a cone of position just like the
"ordinary" cones of position in standard celestial navigation. At the
Earth's distance from the Moon, the diameter of the cone is generally
much larger than the Earth, so the cone slices across the Earth in a
nearly straight arc. Even when the distance from the star to the Moon
is zero (star grazing the Moon's limb) the diameter of the cone of
position where it intersects the Earth's surface is about 2000 miles
(nearly equal to the diameter of the Moon). Note that for a lunar
distance of 90 degrees, the cone degenerates to a plane.  Also notice
that if the cone of position intersects the Earth's surface at a
shallow angle (which occurs when the Moon is low in the sky and the
star is more or less above it) then the intersection with the Earth's
surface will be a small circle arc, and also in this case a small error
in angle leads to a very large error in the line of position.

Given one cone of position based on a lunar distance observation, it is
fairly obvious that I can make two observations and get two lines of
position (or more if I feel like it). The two cones intersect on long
lines radiating from the center of the Moon. Where those lines reach
the Earth's surface is our positional fix. Just as in standard
celestial navigation, if the directions to the two stars are nearly
perpendicular, the lines of position will be nearly perpendicular. I
find that this general approach based on cones of postion makes it much
easier to see the limits and capabilities of the method. For example,
suppose the Moon is 30 degrees high. On the basis of a parallax
calculation, the accuracy of the method should be degraded by a factor
of two at this altitude. And it is if the cone of position in roughly
horizonal where it intersects our position. This happens when the
observed star is directly above or below the Moon. But if the star is
to the left or right of the Moon, then the cone of position is nearly
vertical as it passes through our observing location and the
corresponding line of position is very nearly as accurate as it would
be when the Moon is high overhead (so one line of position would be
accurate, the other inaccurate).

A picture (or two) is worth a few hundred words. I'm going to post some
3d graphics of these cones of position in a follow-up message. I'll
leave this message "text-only"...

-FER


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