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Subject: Rocky Mountain Lunar Distance
From: Arthur Pearson (arthurpearson@XXX.XXX)
Date: Sat Dec 14 2002 - 15:39:31 EST
Inspired by recent threads on this list, and by additional reading about
Lewis and Clark and David Thompson, I decided to test the early 1800’s
methodology for determining position on a recent ski tour in the
Colorado Rockies. With the help of various postings to this list and
some off-list information from George Huxtable, I constructed a
spreadsheet to make the calculations, loaded it on my palm pilot, and
headed off toward the continental divide with a sextant, a cheap bubble
horizon and common wristwatch set roughly to zone time. Apart from
using modern almanac data and a spreadsheet to speed the calculations, I
followed exactly the procedures used by the early explorers. With some
good luck, I got some good results. What is more remarkable is how
forgiving the procedure is toward flawed assumptions and even sloppy sun
sights that lead up to the lunar distance observation. In short, it is
a wonderfully robust procedure for the rough conditions of its early
practitioners, and not a bad procedure for today if you can make the
calculations conveniently.
The procedure I used is essentially the “old timers” method we discussed
in “Use of Sun Sights for Local time, and Lunars for Longitude”
(http://www.irbs.com/lists/navigation/0210/0069.html) wherein longitude
is obtained by: 1) noon sight for latitude; 2) afternoon or morning sun
sight to determine local time (LT); 3) lunar distance to determine
Greenwich time (GT); 4) longitude taken as the difference between LT and
GT. The procedure also relies on the insights of “Calculated Altitudes
for Lunars” (http://www.irbs.com/lists/navigation/0210/0172.html) where
we agreed that with the proper procedure, calculated altitudes for the
sun and moon could be used to clear the lunar distance with sufficient
accuracy to get a good longitude on the first iteration.
The best documentation I have found on the use of these methods concerns
Lewis and Clark (1803-1806) and David Thompson (1790-1812). Richard
Preston’s excellent article on Lewis and Clark has been mentioned on
this list before and is available at
http://www.aps-pub.com/proceedings/jun00/Preston.pdf. He provides an
outline of their procedures and a full discussion of the instructions
provided by astronomer Robert Patterson in the “Astronomical Notebook”
which Lewis carried on his journey. George Huxtable kindly supplied an
email transcript of the “Astronomical Notebook” which provided detailed
examples of the procedures including the proper method for calculating
altitudes.
David Thompson explored western Canada and his navigational procedures
are documented by J. Gottfred at
http://www.northwestjournal.ca/dtnav.html. Thompson’s use of calculated
altitudes is elaborately reconstructed by Gottfried who provides a
comprehensive set of diagrams and trigonometric formulas in explanation
of the technique. It is interesting to note that Thompson worked his
own sights to a full solution in the field. L&C only recorded their
observations and then turned them over to Ferdinand Hassler, a
mathematician at West Point who spent 10 years working them before
giving up in frustration. Preston suggests that his inability to work
the data may have been in part due to the controversy and confusion
surrounding the calculated altitude method.
My spreadsheet for calculation follows Preston’s description of the
procedure. The details and any errors are my own:
1. Assume a longitude. This determines a corresponding estimate of the
GT of Local Apparent Noon (LAN) and an estimated correction from LT to
GT.
2. Take a noon sight at LAN. From estimated GT of LAN, estimate
declination sun and calculate latitude.
3. Take an afternoon sun sight and note watch time (WT).
4. Assume WT is roughly equal LT, determine estimated GT of afternoon
sun sight.
5. With latitude and altitude in hand, estimate declination sun,
calculate LHA sun and convert to time to determine actual LT of sight.
6. Determine correction to WT to arrive at LT.
7. Take a lunar distance and note WT.
8. Correct WT to LT using correction determined in step 6. Correct LT to
estimated GT of lunar distance.
9. Take latitude, estimated declination of sun, convert time of lunar
since LAN to LHA of the sun, and calculate altitude of sun.
10. Take latitude, estimated declination of moon, find LHA moon as (LHA
sun +/- difference between GHA sun and GHA moon), and calculate altitude
of moon.
11. Clear the lunar distance.
12. Determine actual GT.
13. Convert difference between GT and LT to longitude.
The description above glosses over some complications of modern almanac
data and the equation of time, but the framework is complete.
My test of the method took place Dec. 8, 2002 near Breckenridge,
Colorado. I skied with friends up a trail that threaded toward the
continental divide to a high point at about 10,500 feet (3,200 meters).
While the sun was burning through the high cirrus clouds, the moon was
obscured and the prospects looked bleak. We reached a meadow just before
noon and I took the LAN site. I find my bubble horizon to be a real
challenge, and in the excitement of the moment and the sharing of the
"view" with my mountaineer pals, my sight was 7' off compared to the
latitude on the topographic map. Not a good start, but not an impediment
to a reasonable longitude as we shall see.
We continued on working east and about a 1/2 mile south toward the
divide before emerging into a clearing just over two hours later. The
clouds cleared and the quarter moon emerged into a bright, blue sky.
Pressed to begin the return trip, I took only four distances with a
standard scope (Ds=~58°), switched to the bubble horizon and got just
one altitude of the falling sun (Hs =~16°). I can't judge the accuracy
of the afternoon sun as there was no authoritative local time to compare
it to. As we shall see, any inaccuracy in this sight has a limited
effect on the determination of longitude.
I worked out my results for the location of the lunar distance
observations, a spot known as Bakers Tank which is shown at
N 39° 26.5'
W 105° 59.8'
on the USGS topo maps. As stated, my noon latitude sight was 7' off, and
thus my observed latitude, adjusted for DR between lunch and Bakers
Tank, was 7' too far north. This flawed latitude became an input into
both the sun sight for LT, and the clearing of the lunar. I retained the
error in my subsequent calculations to assess the accuracy of the entire
sequence.
My first two lunars were suspect as I got identical angles measured more
than a minute apart. Solved individually, they yielded longitudes 80'
and 90' too far east (very poor compared to Preston's calculation of
L&C's results). The latter two distances seemed more appropriate in
their distribution. Graphing the sights and plotting slope of calculated
Da for the approximate hour favored the last two. They produced
longitudes within 16' and 3' of truth respectively. Accepting the flawed
latitude and averaging the two latter distances, we get a fix of:
N 39° 33.5'
W 105° 53.0'
My luck, skill or inconsistency is less interesting than an analysis of
the tolerance of this procedure to flawed assumptions and early errors.
As my calculations are done in a spreadsheet, I have the luxury of
easily recalculating the results to test their sensitivity to changes in
the inputs.
As usual, the lunar itself must be of the highest precision. A 1' error
in the measured distance shifts the derived longitude of Bakers Tank
about 30' (greater distance shifts to the west, lesser to the east).
There is no escape from the rigors of lunar precision.
However, one need not start with an accurate estimate of longitude. My
initial calculation assumed W 110°, about 185 miles west of the truth.
After the first iteration, I shifted my assumed longitude to W 106°.
The second iteration longitude shifted a mere 0.1' west. Swinging my
assumed longitude from W 110° to W 100° (a 460 mile shift) shifts the
result a whopping 0.2' west. The procedure is remarkable tolerant of
error in the initial assumption of longitude.
Error in the LAN sight has a greater but still modest impact. A 1' error
in here shifts the derived longitude about 1.1'. Had I gotten my LAN
exactly correct (correcting my 7' error) my derived longitude would have
been about 8' further east. This is a greater error in longitude, but a
lesser error in distance (we trade a 7 mile error in latitude for a 6
mile error in longitude). Better to get the latitude right, but all is
not lost if you don't.
Error in the sun sight for time has a bit more leverage on longitude. A
1' error here shifts the derived longitude about 1.3'. I have no way of
knowing how well I did on this front.
In summary, the potential contribution to error in longitude was roughly
as follows for Bakers Tank on Dec. 8, 2002:
1' error in lunar distance = 30' error in longitude.
1' error in sun sight for time = 1.3' error in longitude.
1' error in LAN sight = 1.1' error in longitude.
5° error in assumed longitude = 0.1' error in longitude.
I think this test is a strong affirmation that the old time method for
determining longitude, including the use of an assumed longitude and
calculated altitudes based thereupon, is remarkably robust and utterly
suitable for the use it was given by early explorers. The longitude
assumption has remarkably little impact on the results even for the most
disoriented explorer. Greater skill and a decent reflecting horizon
would leave little excuse for the sun sights to be outside of a minute
or two of accuracy. This eliminates any significant error in the
calculations that clear the lunar distance. As is the case with other
methods, the precision of the lunar is the all important determinant of
accuracy.
I would be most interested in any articles or books that document the
use of these procedures by sea captains of the era as thoroughly as
Preston and Gottfred document their use in the heart of the continent.
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