A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Short-cut lunars. was: Clearing lunars
From: George Huxtable
Date: 2010 Aug 28, 00:57 +0100
From: George Huxtable
Date: 2010 Aug 28, 00:57 +0100
Perhaps it's worthwhile discussing, in a bit more detail, Frank's discovery that short-cut methods are possible, simplifying the clearance of lunars, over quite a range of possible geometries. I think he has it right. It's highly inventive. But it isn't any sort of replacement for the real thing. I had asked- "Now we come to quite different ground, a matter which Frank has raised before. It is quite true that special geometries arise in which longitude can be derived from a lunar distance observation, without involving trig. And it's also true that the processing of lunars in that way may be tolerant of quite wide deviations from those exact geometries. But how tolerant? How much deviation from that geometry can be accepted, and still keep within an error-band of an arc-minute or so, outside which a lunar loses much of its value?" Frank's answer was this- "A whole arc-minute?? Well, quite a large number of cases! I previously focused on cases where the resulting error was only a TENTH of a minute, and even then there are far more cases than you would expect. It's not just small, first-order deviations from the exact vertical circle case that work, but much larger deviations. You asked: "So, what criteria does our navigator adopt, in order to decide whether such a short-cut is acceptable, or whether a full-blown trig solution is needed?" There is a very simple test. You compare the difference between the observed altitude of the Moon and the adjusted altitude (adjusted to force vertical circle geometry). If that difference is smaller than 6*tan(LD)/cos(h_moon) then any error resulting from treating it as a vertical circle case is a tenth of a minute of arc or less. " Thanks for a direct answer to my question. So, let's sum up what our navigator has to do, when he decides that the geometry is such that Moon and other-body appear to be somewhere near 180º apart in azimuth, so a likely candidate for a short-cut, trig-free, lunar. Having measured sextant altitudes of moon and Sun-or-star, and lunar distance between the appropriate limbs- Correct measured distance for index error and combined semidiameters, to get apparent distance d between their centres Correct altitudes for index error, dip, and semidiameters, to get apparent altitudes of centres above the horixontal of Moon m and Sun-or-star s. From refraction and parallax of other-body, where relevant, applied to s, find true altitude S All this has to be done whether of not a short-cut procedure can be taken. ================= Next comes the procedure to see if a short-cut can apply- Find the difference between m and the quantity 180 - s - d, the value it must have if the azimuths differed by exactly 180º Compare that difference with the quantity 6' *tan d / cos m, by calculation or by table. If it's more, the short-cut has to be abandoned, and normal trig clearing procedure adopted. If it's less, the short-cut can apply. In which case- Discard the calculated value of m, and substitute m' = 180 - s - d. From refraction and parallax of Moon, applied to m', find true Moon altitude M Then find cleared lunar distance D simply, from D = 180º - S - M. ================== But if the short-cut didn't apply, then preserve the original value of m. From refraction and parallax of Moon, applied to m, find true Moon altitude M Then find cleared lunar distance D from cos D = (cos d - sin s sin m) (cos S cos M) / (cos s cos m) + sin S sin M That final step should present few difficulties to anyone with a modern calculator, but was the very devil to calculate using logs, in the days when it had to be done that way. ================= I've written it all out to check whether I've understood the procedure. No doubt Frank will put me right if I've got it wrong. It should be clear that the short-cut could save quite a bit of work if the appropriate conditions apply. But the test is non-trivial, and if the test failed often, aborting that extra work, and necessitating the full procedure anyway, then overall there might not be much saving. So it depends on the navigator being able to judge well in advance whether an observation is a suitable candidate or not. No doubt, that skill could come with experience. But now, consider the situation facing the navigator who has never learned the full trig procedure for clearing a lunar. What happens if the test fails? He has no alternative but to pack up, and wait for another day. But away from the tropics, the opposite-azimuth situation will arise only once a month. That doesn't matter, to a hobbyist, who just wishes to test his skill when the occasion arises. He could go home and try again another day. But Frank was imagining himself back in the 1780s, when lunars were starting to be used in earnest by professional navigators at sea. They knew how to work them. They may not have understood what they were doing, but it was a familiar, well-memorised routine. And when they neede a lunar, they needed it there and then. Lunars were unavailable for so much of the time anyway: when it was cloudy, when the Moon wasn't up, or when it was too near the Sun. So such a short-cut procedure wasn't an alternative, to anyone doing real navigation. The full trig procedure had to be available, at his fingertips, for when it was needed. Maybe Frank's appearance, with his new procedure, in 1780, would have been welcomed by navigators. But I have my doubts. George. contact George Huxtable, at email@example.com or at +44 1865 820222 (from UK, 01865 820222) or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.