# NavList:

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

**Rocky Mountain Lunar Distance**

**From:**Arthur Pearson

**Date:**2002 Dec 14, 15:39 -0500

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.i-DEADLINK-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.i-DEADLINK-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.