# NavList:

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

**Re: Latitude by Lunar Distance**

**From:**Frank Reed CT

**Date:**2006 Oct 07, 17:54 -0700

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 --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To unsubscribe, send email to NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---