# NavList:

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

**Re: October Lunar**

**From:**Frank Reed

**Date:**2008 Oct 07, 00:00 -0400

I wrote previously, "The short distance is not an issue (two reasons: the altitudes are calculated so their accuracy is not a problem, and also we don't have to interpolate between geocentric distances three hours apart, which was an issue historically but not today)." And George, you replied: "I don't understand the relevance of "the altitudes are calculated so their accuracy is not a problem" to the acceptability of a short lunar distance. Anyway, even if the altitudes had been measured rather than calculated, that measurement can be made to quite sufficient accuracy." The accuracy of the altitudes is critical for short distance lunars. The error in clearing a lunar distance arising from errors in the altitudes that enter the clearing process is given approximately by: errLD = (1/60)*[errH2*cos(H2)/sin(LD) - errH1*cos(H1)/tan(LD)], where errLD represents the error in clearing the distance, LD is the observed lunar distance, errH2 is the error in the altitude of the other body (Sun, star, or planet), H2 is the altitude of the other body, errH1 is the error in the altitude of the Moon, and H1 is the altitude of the Moon. For "typical cases" of lunar observations, the error in the lunar clearing process is about a tenth of a minute of arc for a five arcminute error in either altitude, which gives plenty of leeway. As I've noted a number of times before, the most interesting thing here, a bit of a "miracle", is that the error in the Moon's altitude becomes insignificant when the LD is close to 90 degrees --you can be wrong by a degree or two and it will do no harm. At the other extreme, as in this specific case, when the lunar distances are small enough, sin(LD) and tan(LD) are approximately given by LD when LD is "in radians". If also, the altitudes are nearly the same (as they are to some degree whenever the LD is small enough, and as they were in practice in this specific case), then cos(H2)=cos(H1), and then we have: errLD = (1/60)*(errH2 - errH1)*cos(H1)/LD. Supposing that we measure LD in degrees, and assuming cos(H1) is nearly one, this reduces to approximately errLD = (errH2 - errH1)/LD. This means that if we have an error of 2 minutes of arc in either altitude, which is well within the realm of possibility for measured altitudes, and the LD is 12 degrees, then the resulting error in the clearing process is about 0.15 minutes of arc (an important difference in clearing lunars). Also notice that a COMMON error in both altitudes is harmless. If, for example, there is some odd refraction affecting both bodies by the same amount or we have miscalculated the dip, or the IC is wrong for the instrument used to measure the altitudes, it leads to no error in the clearing process. Nothing about the above formulas for the error in the clearing process resulting from an error in the altitudes depends on the altitudes being measured. They also apply when the altitudes are calculated. We calculate altitudes from latitude and local apparent time (two inputs yield two outputs). If you think through the various cases, an error in either of the inputs will generally affect both altitudes by about the same amount when the lunar distance is short. And as noted above, a common error does not affect the outcome of the clearing process for short distances. So if we calculate the altitudes, the whole problem of getting accurate altitudes for short distance lunars evaporates. To summarize, if you intend to shoot short distance lunars, you need to be aware that the altitudes have a critical effect on the clearing process. The altitudes either have to be measured simultaneously to an accuracy of a minute of arc or better (more so if the distance is shorter than in this case) or they should be calculated. Naturally for a modern observer, it is convenient to calculate them. And you wrote: "that left unaddressed the other problem about short-distance lunars; that a planet, even though never far from the Moon's orbital plane, can get way out-of-line with the Moon's direction of travel on near approach." I left that 'unaddressed' for two reasons. First and foremost, that is completely irrelevant to this observation and the accuracy question arising from it. Second, as it happens, in this particular case, the Moon and Venus were only moderately out of line. This degree of misalignment was acceptable historically (by the standards of the historical Nautical Almanacs, as published). And: "As we have discussed before, the estimated longitude error, stated in Frank's reduction program, doesn't allow for that angular offset, so paints an over-optimistic picture in such a situation. He is considering a revision, which would be useful." No, I'm not considering a revision 'per se'. I have considered adding a details page for people with interest in possibilities, but at present I'm happy with what it does, as is. Ya see, the users of this site consist of two very different groups. The first group is much larger and consists of people who are just learning about lunars and who are interested in shooting a lunar once or twice and getting a general idea of what it means. The "approx longitude error" number is for them. They've shot a lunar, and they say to themselves 'ok, that was fun, but what would this have meant historically?' They don't care if the particular lunar geometry of the sight they've taken is optimal for finding longitude. They just want a general indication of how well they would have done with that same level of accuracy in a traditional lunars case. Rather than over-stating the accuracy, I've frequently felt that the number under-states it since we normally quote navigational accuracy in miles rather than minutes of longitude. The second group of users consists of people with some expertise in lunars like the lunarian members of NavList, including myself. This group is perhaps ten or even fifty times smaller in number, but we're much more likely to be repeat customers. For us, there are many possibilities. Just a few ways we might use lunars: 1) We have a historical lunar distance observation from 1809. It includes the observed distance and the observed altitudes. We enter the data, and adjust the position until all of the observations have zero error. This is a simultaneous solution of the lunar problem and the time sight. There is no single error in longitude. 2) We have a lunar observation from 1850 serving as a check on a chronometer which is probably nearly correct. In this case, we really should be talking about the error in GMT though the equivalent error in longitude is fundamentally the same thing. Here we don't worry about the time sight, and this case is closest to the "pure" case where an error in the observed distance gives a direct error in the resulting longitude. 3) We're measuring lunars simply to test our sextants and our skills and the longitude is quite irrelevant. 4) We have measured a lunar distance at known GMT and we are trying to determine a "lunar distance line of position". In this case, an error in the measured distance leads to an error in the LOP (typically 60 times larger than the error in the LD) but dependent on the geometry. There IS a sensitivity to longitude, a corresponding "longitude error" in this case, but in general these LOPs are not aligned with lines of longitude so it would be misleading to give a pure longitude error in this case. 5) And all the other things I haven't thought of right now. :-) -FER --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@googlegroups.com To unsubscribe, email NavList-unsubscribe@googlegroups.com -~----------~----~----~----~------~----~------~--~---