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

**About Lunars, part 3**

**From:**George Huxtable

**Date:**2002 Feb 13, 20:46 +0000

3.1 CHANGE OF PLAN. In part 2, there was a prediction of what part 3 would contain. However, I have recently had some useful feedback from listmembers, (especially Steven Wepster) which shows that rather more remains to be said about measuring and correcting the lunar distance. So part 3 will fill in some of those gaps. Part 4, which should appear in the next few weeks, will discuss the possibility of calculating the altitudes required with a lunar distance, instead of measuring them: also the question of observing Moon altitudes in place of lunar distances. It will show how to obtain local apparent time and use it with GMT to obtain longitude, as early navigators had to do. 3.2 UNDUE EMPHASIS ON ACCURACY IN MOON ALTITUDES. In part 1, under "How is a lunar distance measured?", I stated "The corrections are such that the Sun altitude is not needed to great accuracy, but the Moon altitude should be measured with precision." This overstated things. An error of a degree in Moon altitude will, at the most, give rise to an error of 1 minute in the parallax correction of the lunar distance, therefore the accuracy required in Moon altitude is no better than to 20 minutes or so. This is useful in night measurements of Moon-star or Moon-planet lunars, when the horizon has to picked out from the ripply reflection of moonlight below the Moon. For the refraction of Sun, stars, or planets, when altitudes are as low as 10 deg., altitude measurements to 20 min. of arc are needed, but from 20 deg up, 1 degree will suffice. Again, for stars or planets, this is helpful when the night-horizon is hard to make out. The conclusion is that the requirements for accuracy of the altitude measurements are usually rather easy to meet. This leaves the observer to concentrate all his precision on measuring the lunar distance. 3.3 UPSIDE-DOWN SEXTANT? In part 1, under "How is a lunar distance measured?", I said- "The horizon plays no part in the lunar distance observation itself, and the view, straight through the sextant, is instead used to observe the fainter of the bodies observed (usually the Moon). The brighter one (usually the Sun) is viewed in the index mirror. To see them both, the sextant may have to be tilted at what seems like a most unnatural angle, pointing up in the air, and over on its side, even upside-down." I should have added here that if you find the "sextant upside-down" posture particularly awkward (as many do), it will do no great harm to swap the two images over, in the two light-paths, and use the sextant in a more natural attitude. This presumes that your sextant offers a suitable choice of shades. 3.4 REFRACTION CORRECTIONS. (see parts 1 and 2) In clearing a lunar distance, it's necessary to separate the refraction corrections from those for dip and for semidiameter. In many tables these will have been combined, sometimes with Moon parallax also, to minimise the arithmetic for a normal sextant altitude sight. Such a combined refraction table is not useful for lunars. You can find the refraction correction, on its own, for any body, from the "mean refraction" table in Norie's, or in the star correction table in the pull-out card in the Nautical Almanac. 3.5 INVERSE LINEAR INTERPOLATION FOR TIME (see part 2). This unwieldy phrase describes the process of finding the exact GMT corresponding to a lunar distance, given predicted values of LD at two times. Although I stated that those times, T1 and T2, could be chosen at intervals 1, 2, or 3 hours apart, the example quoted presumed an interval of 3 hours. This is the maximum interval that's compatible with the assumption of a linear change, and is the interval on which the old lunar-distance tables were based. That 3-hour interval was chosen to minimise the size of the tables and to minimize the amount of calculation in their making (in the days when that mattered). If you are calculating lunar distances for yourself, it will do no harm to choose a shorter interval than 3 hours, if you can be sure that the GMT will end up within that interval. 3.6 ERROR IN EXAMPLE Please note that in part 2, an error occurred in the following passage, as uncovered by Greg Gilbert- "NOW TO DEAL WITH THE LUNAR DISTANCE, AT LAST! After correction for sextant errors, the Observed Lunar Distance between the near limbs of the Sun and Moon was 105 deg 50.5 min. There is no dip to consider because the horizon is not involved in the measurement." Where this reads 105deg 50.5 min, it should have been 106deg 50.5 min. Sorry about that. It was a transcription error, which doesn't affect the following parts of the calculation. 3.7 ABOUT SOME PROBLEMS IN CLEARING THE LUNAR DISTANCE In "About Lunars. part 1." it was shown how to obtain a corrected Lunar Distance, D, using Young's method. It included the following comment, about Young's method- "CLEARING THE LUNAR DISTANCE, LONGHAND. The formula above is fine for electronic computation, but quite unsuitable for longhand calculation using logs. The trouble is that some of the quantities may go negative, and the log of a negative number is meaningless. In the era of lunar distances, the navigator relied on 5-figure logs and trig tables. The many different ways devised for clearing the Lunar distance were mostly devoted to expressing it in such a way that logs could be used for the solution, sometimes using auxiliary tables designed for the purpose. If readers find the need to do this clearance longhand using logs, my suggestion is to use Borda's method, to be found in Cotter. On request, I will spell it out for the list." ================ (end of quote) Bill Noyce responded by saying- >And I would like to take him up on his offer to spell out Borda's >method for clearing the distance longhand. Young's formula seems >to require converting out of logs to do additions. ================== For modern navigators, the availability of calculators and on-board computers has made lunars more accessible to the rest of us, and that is the approach I would recommend. But readers are entitled to choose a technique using tables and logs, the traditional method that was employed right through the era of lunars. There are several problems that crop up when trying to use Young's method using tables, rather than a calculator. Borda offers an alternative, but still involves popping in and out of logs, and more than once! However, I promised Borda's method, on request, and Bill Noyce has indeed requested it, so here goes- 3.8 BORDA'S METHOD This is more-or-less as quoted by Cotter. The quantities m, M, s, S, d, are given, as follows. d (observed lunar distance between centres) m (observed moon-centre altitude above true horizontal) M (true Moon-centre altitude above true horizontal: it has been corrected for parallax and refraction) s (obs Sun/body-centre altitude above true horizontal) S (true Sun/body-centre altitude above true horizontal: it has been corrected for parallax and refraction) The result D, will be the true lunar distance, which has been corrected for parallax and refraction. Firstly, an angle A has to be calculated. A is just an angle that's used as an intermediate step in the calculation. Whether it has any physical reality, I rather doubt. Obtain A as follows- First work out and write down (m+s+d)/2 and (m+s-d)/2 log cos A = {log cos((m+s+d)/2) + log cos((m+s-d)/2) +log cos M + log cos S + log sec m + log sec s}/2 This involves working out all those 6 log cos and log sec terms, adding them up, and dividing by two. Having calculated a value for the right-hand side of this equation, search for that value in the log cos table and find out what angle it corresponds to. This is angle A. You are now "out of logs" and back in the world of ordinary numbers, for a time. Now work out and write down A + (M+S)/2, and A ~ (M+S)/2. where "~" means "subtract the smaller from the larger" then go back into logs again, obtaining D from log sin (D/2) = {log sin (A + (M+S)/2) + log sin (A ~ (M+S)/2)}/2., So having calculated a value for the right-hand side of the equation, search for that value in the log sin table and find out what angle it corresponds to. Double the result. That gives you D, the corrected lunar distance, all corrections made. That second term of that last equation differs from what is given in Cotter, who puts a minus sign, which I have changed to "~". Following his notation would lead you into trying to obtain the log of a negative quantity. I think he has got that wrong. The way I have rewritten that expression now conforms with the following explanation in words of how to tackle the problem, as follows (from Cotter, but somewhat modified)- 1. Find M, S, m, s, and d, as above. D is the true lunar distance (in which the effects of parallax and refraction have been allowed for) that we wish to find. 2. Place under one another the apparent distance d and the apparent altitudes m and s: and take half their sum, L. From the half-sum L, subtract the apparent distance d. Under this place the true altudes M and S. 3. Take from tables log cosines of L, L-d, M, and S, log secant m, and log secant s. Add these six quantities and divide by 2. The result is the log cosine of A. So look up this quantity in the log cosine table, and find what angle corresponds to it. This is A. 4. Take half the sum of the true altitudes M and S. Call this B. Find the sum of, and the difference between, A and B. Add the log sines of the sum and the difference. Divide by 2. The result is the log sine of half the true lunar difference, that is D/2. So look up that result in the log sine table, find the angle that corresponds to it, and double it to obtain the corrected lunar distance D. (Cotter also missed out an important word in his paragraph 4, which I have reinstated.) 3.9 BORDA'S METHOD: AN EXAMPLE Let's try Borda's method for real using the numbers below, for Steven Wepster's Sun-lunar in the Atlantic on 2001 April 02, just as were used in part 2 with Young's method. I will stick to the degrees-and-minutes notation rather than decimal degrees, as that's what is needed for looking up the tables. d=107 deg 22.9 min (observed lunar distance between centres) m= 49 deg 52.5 min (observed moon-centre altitude above true horizontal) M= 50 deg 29.9 min (true Moon-centre altitude above true horizontal) s= 21 deg 10.4 min (obs Sun/body-centre altitude above true horizontal) S= 21 deg 08.1 min (true Sun/body-centre altitude above true horizontal) Following the written-out instructions in 3.8, we have now completed step 1. now for steps 2 and 3 d 107deg 22.9 m 049deg 52.5 log sec = 10.19080 s 021deg 10.4 log sec = 10.03036 sum (d+m+s) 178deg 25.8 half-sum L 089deg 12.9 log cos = 8.13673 L-d -018deg 10.0 log cos = 9.97779 (same as log cos +18deg 10.0) M 050deg 29.9 log cos = 9.80353 S 021deg 08.1 log cos = 9.96976 sum = 58.10897 Halve = 29.05448 cast off 20= 9.05448 from log cos table, Angle A = 083deg 29.4min ===================== next, step 4. M (copied) 050deg 29.9 S (copied) 021deg 08.1 sum 071deg 38.0 half-sum, =B 035deg 49.0 A (copied) 083deg 29.4 sum, A and B. 119deg 18.4 log sin = 9.94052 diff., A and B. 047deg 40.4 log sin = 9.86884 sum = 19.80936 halved = 9.90468 from log sin table, Angle D/2 = 053deg 24.6 so Angle D = 106deg 49.2 ======================= This result, of 106deg 49.2min, for the corrected lunar distance, can be compared with the value obtained (in part 2) for D using Young's method with a calculator, which was 106deg 49.3. So rather good agreement: nothing to complain about there! However, few navigators would find clearing the lunar distance by Borda's method to be a lot of fun, I admit. 3.10 AN EASIER APPROACH TO CLEARING THE LUNAR DISTANCE. The two methods considered so far, Young's and Borda's, are "mathematically exact". Both these methods end up by multiplying the observed lunar distance d by a calculated factor, which is always very near 1. The observed distance d can be of the order of 100 deg or so. To obtain the answer D to within 0.1 min, that multiplying factor of about 1 has to be known to 1 part in 100,000 or thereabouts. In other words the error in that multiplying factor should be within .00001. That is why 5-figure logs are required, and every step in the calculation made with scrupulous attention to accuracy. But what if we can correct d, to obtain D, by ADDING a correction to it, rather than using a multiplier? The correction is never more than about 60 minutes, so to make that correction to provide D to 0.1 minutes, it's only necessary to evaluate that added correction to 0.1 in 60 minutes. An error of 1 part in 1,000 of that added amount now becomes acceptable. That is an accuracy that might even be approached by the careful wielder of a slide-rule that shows trig functions. Throughout the 19th century, much effort went into devising a solution to clearing the lunar distance which involves such an added correction, rather than a multiplying factor. The trouble has been that such solutions have involved an approximation to the geometry, not an exact solution. They rely on the fact that both angles are small, the parallax and the refraction. In practice, those angles always are acceptably small. If observations are made only when altitudes exceed 10 degrees, refractions are always less than 5.5 min. Parallax of the Moon never much exceeds 60 minutes. If we make an additional rule that lunar distances are measured only when they exceed 10 degrees the requirements are met for good accuracy of the "approximate solution". 3.11 LETCHER'S METHOD. In "Self-contained celestial navigation using H.O.208" (1977), John S Letcher describes such an approximate method for clearing the lunar distance. The way he proposes for its use would be suitable only for lunar distances up to 90 degrees, whereas any lunar observer would certainly require its application to angles up to 120 degrees. I will therefore give a modified calculation method that will cover the full range of useful sextant angles. I am not sure where this method comes from, as Letcher does not quote any reference, nor does he show how it is derived from first-principles. It might be his own invention, perhaps. To me, it seems very clever. I will christen it "Letcher's method" until we find out more about its provenance. If anyone recognises it from another publication, I would be interested to learn. Let's investigate Letcher's Method. We will assume that all sextant observations have been corrected for index error. As before, it starts with the observed lunar distance d, after correction for semidiameters. This is to be corrected by adding, separately,- a) The correction P, for the combined parallaxes of the Moon and the Sun, which may be a positive or negative amount, never greater than about 60 min. b) The correction R, for the refractions of Moon and Sun, always positive, never more than 11 min. Both P and R need to be calculated to within 0.1 min. These two corrections require knowledge of- d, the observed lunar distance between centres (i.e. corrected for semidiameters). m, the observed altitude of the Moon's centre above the true horizon (i.e. corrected for dip and semidiameter) s, the observed altitude of the centre of the Sun (or other body), above the true horizon (i.e. corrected for dip and, if necessary, semidiameter). HP, the Moon's horizontal parallax, in minutes of arc, at the hour of observation. The HP of the Sun (or other body) is ignored, thus losing a bit of precision. It is no longer necessary (as it was in the other methods) to make parallax and refraction corrections to m and s individually, as M and S are no longer needed. These corrections are taken into account automatically, as part of the clearance procedure. That is partly why the method is so much simpler. Let's go through it. First, obtain B = (cos d sin m - sin s) / sin d (B is just an intermediate angle, used in the computation). Then the parallax correction in minutes is- P = HP*B + (HP)^2*[(cos m)^2 - B^2)]/ (6900*tan d) P may be a positive or negative number, in minutes, to be added or subtracted from d. The second term of P is usually a very small quantity, but still needs to be evaluated. Now for the refraction term R. This, very cleverly, includes its own built-in calculation of the way refraction changes with altitude. R = .95* (sin s/sin m + sin m/sin s - 2*cos d) / sin d ,in minutes of arc. So the end result is D = d + P + R , giving the Corrected Lunar Distance. 3.7. LETCHER: WORKED EXAMPLE Let's compare this method with the others, using Steven Wepster's Atlantic observations of 2001 Apr 02 once again. The inputs we need are, as before- d = 107.382 deg (observed lunar distance between centres) m = 049.875 deg (observed moon-centre altitude above true horizontal) s = 021.173 deg (obs Sun or body-centre altitude above true horizontal) HP = 59.4 min. By pocket calculator we get- B= -0.6178 so P = -36.7 - 0.1 = -36.8 min, and R = 3.2 min As d (in minutes) is 107deg 22.9, then for the Corrected Lunar Distance, D= d + P + R, or 107deg 22.9 - 36.8 + 3.2 Therefore D = 106deg 49.3 by Letcher's method. =============== This should be compared with 106 deg 49.2, by Borda's method using tables, and 106deg 49.3 using Young's method with a calculator. Of these, I would choose the last, 106deg 49.3, as being the more precise. Really, what better agreement could anyone wish for, between three different methods? However, don't expect to always obtain that same precision using Letcher's method. He concedes that by ignoring some of the contributions such as the Sun's parallax and the ellipsoidal figure-of-the-Earth, errors may combine up to a total of 0.3 minutes. Even so, when compared with the likely errors inherent in measuring the lunar distance, due to the motion of a small craft, a possible error of 0.3 minutes in the clearing of it may be very acceptable. My vote would go to Letcher. But it's your choice... My thanks to Bill Murdoch for alerting me to this useful Letcher publication. If you can find a secondhand copy, I think it would be a worthwhile buy. It was written in the days when scientific calculators existed but were uncommon; hence the emphasis on using tables. ================= Part 4, which should appear in the next few weeks, will discuss the possibility of calculating the altitudes required with a lunar distance, instead of measuring them: also the question of observing Moon altitudes in place of lunar distances. It will show how to obtain local apparent time and use it with GMT to obtain longitude, as early navigators had to do. George Huxtable. ------------------------------ george---.u-net.com George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. Tel. 01865 820222 or (int.) +44 1865 820222. ------------------------------