# NavList:

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

**Picturing lunar distances.**

**From:**George Huxtable

**Date:**2003 May 15, 16:18 +0100

Picturing Lunar distances. Kieran Kelly asks about rigorous and approximate solutions to the clearing of lunar distances. I am sure that Kieran understands these matters well, but some others seem to find difficulty in picturing what goes on when a lunar distance is being "cleared". Clearing, in this sense, means being corrected, very precisely, for the combined effects of refraction and parallax. Explaining such matters for the list is made somewhat harder by its discouragement of graphic attachments. In this case, however, flat pictures don't help greatly: it's impossible to transmit a solid model. You will have to make (or imagine) your own, as follows- Take a nice, big, round, orange. Bisect it into two halves so the navel is at the centre of one half. You can eat the other half. Indeed, you can eat both halves, as long as you leave a half-skin intact. Put the half-skin, navel up, on a flat plate. Imagine a tiny observer (you), just at the centre point of this orange hemisphere, in the plane of the plate, which represents your true horizontal. The orange half-skin represents your view of the sky. The navel is at your zenith point Z. Between the plate and the navel, measured around the skin, is 90 degrees, so that gives you a scale of angle. Somewhere in the sky you can see the Moon and the other body you observe: we will presume here that it's the Sun, but you should keep in mind that it could just as well be a planet or a star, as long as it's somewhere near the Moon's path. This orange half-globe is quite a good analogy to the sky seen by an observer, as long as one difference is kept in mind. The observer sees the bowl of the sky from the inside, looking out. You view the orange from the outside, looking in. This causes some differences: if, to an observer, the Sun is to the left of the Moon, viewed on the outer surface it will appear to be on its right. This doesn't detract from the orange model in our present application. What a lunar observer does is to measure, at a known time, and as precisely as possible, the apparent angular distance, diagonally in the sky, between the center of the Moon and the centre of the Sun. This will lie somewhere between 0 and 180 degrees. We will call this angle d. To arrive at that angle he has had to make corrections for semidiameter and index error, and we will presume that's been done. At the same time, he measures the apparent altitudes, above the true horizontal, of the centres of the Moon (m) and the Sun (s) (having corrected where necessary for index error and dip, and semidiameter, but not for refraction or parallax). We can now mark these in on the outer skin of our orange. Put a dot somewhere on the half-orange that's at an angle m degrees up from the plate. This represents the centre of the apparent Moon Mark it with an m. Now you need to find where to mark the apparent Sun. You know that it's an angle s up from the horizontal, so lightly mark in a horizontal line, of constant altitude s degrees up from the plate, and the Sun must lie on that. Now set a pair of dividers, or knot a piece of thread, to subtend an arc equal to the apparent lunar distance d. Swing it about centre m (to left or right, it doesn't matter) until it intersects the horizontal line. That point is where the apparent Sun lies. Mark it with an s. Join the three points to form a triangle Zms. All three sides of this triangle are parts of great-circles, so here we have a spherical triangle, which can be solved using well-known formulae. It's very similar to the standard PZX astronomical triangle, though it represents different quantities. What we need to know is the angle at Z, the difference in azimuth of Sun and Moon that the observer sees. It's not very complicated. We get Z from- cos Z = (cos d - sin s sin m) / (cos s cos m) (equation 1) in which Z is angle between azimuths of Moon and Sun, d is apparent lunar distance m is apparent moon altitude s is apparent Sun altitude. The "clearing" operation involves calculating what the corrected lunar distance D should be, if it was seen by a mythical observer at the centre of the Earth, because that's what the predicted lunar distances in the almanac are given for. We have to allow for the effects on the apparent altitudes of the Moon and Sun, caused by parallax and refraction. The correction to the apparent moon altitude always increases it, by an amount corr(m), the Moon parallax always greatly exceeding the refraction, which works in the opposite direction. So the corrected Moon altitude M = m + corr(m). On the other hand, the Sun's apparent alatitude id always reduced by the amount of the correction, because for the Sun refraction always exceeds parallax, so the corrected Sun altitude S = s - corr(s). Defined in this way, these corrections, which work in opposite directions, are always treated as positive quantities. This isn't entirely logical, but it's been done that way for so many years it's hard to change it now. The effect of these corrections is to jack up the position of the Moon, toward Z, by an amount corr(m), and drop the position of the Sun, further from Z, by an amount corr(s). Note that parallax and refraction act only to raise and lower these positions along vertical lines, and don't give rise to any sideways shift, so the azimuth angle between Moon and Sun stays exactly as it was before. You can now put in a new dot for the corrected Moon, up a bit towards Z, and for the corrected Sun, down a (much smaller) bit away from Z. These shifts are both small angles, a degree or less for corr(m), and no more than a few minutes for corr(s), far too small to show on the skin of an orange, so to show the principle you will have to exaggerate them greatly. Now draw in a new skewed great-circle line joining these corrected Moon and Sun positions, and the exact length of that line is the corrected lunar distance D, which we need to find. If you have drawn it right, it should cross over the original lunar distance line at just one point between the Moon and Sun positions. So now there's a different spherical triangle, embracing exactly the same zenith angle Z, but with the lengths slightly different from before. A similar formula applies, as follows- cos Z = (cos D - sin S sin M) / (cos S cos M) (equation 2) where, as stated before, M = m + corr(m), and S = s - corr(s) Because the right-hand sides of equations 1 and 2 are both equal to cos Z, they are equal to each other, and we can write (cos D - sin S sin M) / (cos S cos M) = (cos d - sin s sin m) / (cos s cos m), or (cos D - sin S sin M) = (cos d - sin s sin m) (cos S cos M) / (cos s cos m), or cos D = ((cos d - sin s sin m) (cos S cos M) / (cos s cos m)) + (sin S sin M), or D = arccos (((cos d - sin s sin m) (cos S cos M) / (cos s cos m)) + (sin S sin M)) (equation 3). The whole complex business of clearing the lunar distance boils down to obtaining a value for D, to high accuracy, from equation 3. No approximations have been made in deriving it: it's completely rigorous. Nowadays, it can be computed as it stands to 10 digit accuracy on a pocket calculator in a couple of minutes. Not so in previous centuries, however. The only aid available was log tables and log trig tables, and these were needed to 6-figures or better to preserve enough precision. However, logs only help in multiplications and divisions, not in additions and subtractions. As Bruce Stark has explained, whenever an add or subtract crops up, it's necessary to take antilogs, do the addition, then take logs again (though he has used clever dodges to bypass that problem). Also, cos d comes into the expression, and because it goes negative when d exceeds 90 degrees, log cos d becomes meaningless above 90 degrees. For those reasons, equation 3 was ill-suited to solution using logs. Many methods were invented which minimised such complications, and equation 3 was bent and twisted to an extraordinary extent to achieve this. In many cases, these methods remained rigorous, without approximations: the precision was limited only by the precision of the tables available. These included Borda, Dunthorne, Delambre, Raper, Young, Kraaft. Other methods, however, arrived at D by adding or subtracting a small correction to the measured value d. This couldn't be done by a rigorous geometrical process, but by devising small triangles to fit onto the ends of the line d which provide amounts to add and subtract to equate it to D. Because those correction triangles are always small, no more than 1 degree in extent, they can be closely approximated by plane triangles, using ordinary plane geometry. If it was necessary, a further level of correction could be applied to arrive at even higher precision. The great advantage of such non-rigorous methods was this. Because the correction was a small quantity to be added to or subtracted from d, it didn't need to be calculated to a high accuracy (as a fraction of itself), so even four-figure tables would suffice. If the navigator didn't carry those bulky and expensive six-figure or seven-figure tables, he would in fact end up with a more accurate answer using a non-rigorous method that a rigorous one. Lyons provided one of the earliest non-rigorous methods, followed by Merrifield, Airy, Hall. C.H.Cotter, "A History of Nautical Astronomy", Hollis and Carter 1968, provides the best assessment of all these methods of clearing the distance, though as the Nav-L list is by now well aware, he is prone to make careless errors. George Huxtable. ================================================================ contact George Huxtable by email at george@huxtable.u-net.com, by phone at 01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. ================================================================