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The math of clearing lunars
From: Frank Reed
Date: 2015 Jun 12, 20:47 -0700

There are three fundamentally distinct ways to clear lunars:

• a direct triangle solution.
• a series solution
• vertical circle forcing

In a direct triangle solution, which is really the "gold standard" and always the method of choice in software solutions (unless some historical method is being illustrated), we remove the effects of refraction and parallax by noting that these phenomena occur entirely in the vertical direction. That leaves the difference in azimuth between the Moon and the other body constant when transitioning from observed altitudes to cleared altitudes. So we use the observed altitudes, converted to zenith distances, and the observed lunar arc (the measured lunar distance) to calculate the difference in azimuth using the standard spherical triangle equation (the cosine formula). Call that difference dZ. Then we use the cleared altitudes (applying the usual altitude corrections to get Ho's) and this calculated difference in azimuth, dZ, to invert the process and calculate the cleared lunar distance.

The equations for this are simple enough. Start with the usual cosine formula applied to the triangle formed by the zenith, Moon, and other body, with LD, h1, and h2 for the observed lunar distance arc and the observed altitudes of the bodies:
cos LD = sin h1 · sin h2 + cos h1 · cos h2 · cos dZ.

Then turn that around and solve for cos dZ (we don't need the angle, dZ, itself, just its cosine):
cos dZ = (cos LD - sin h1 · sin h2) / (cos h1 · cos h2)

Set aside that value of cos(dZ). Then correct the altitudes for refraction and parallax, just as you would any altitudes of the Moon and other body. Call those corrected altitudes h1' and h2'. And remember our goal in this is to get the cleared or corrected lunar distance which we can call LD'. We can write down the same equation we started with except now it is based on the cleared quantities:
cos LD' = sin h1' · sin h2' + cos h1' · cos h2' · cos dZ

We plug in the known value of cos(dZ) from our previous calculation and thus we can solve for LD'. That's the cleared lunar distance, the thing we're looking for. Easy as could be if you have a computer of some sort but a very long calculation without one. Note in particular that the entire process has to be worked out to high precision since we want a result accurate to a small fraction of a minute of arc. The calculation should not admit any calculational error greater than about 5 seconds of arc.

In a series solution, the same math as above is replaced by a series expansion of terms which is truncated at second or third order since the additional terms are very small. The calculation consists primarily in working out that portion of the Moon's parallax which is directed along the lunar arc. This is the biggest correction to the lunar distance. This series expansion works because the altitude corrections are always small angles, so the correction to the lunar distance is on the order of one degree or less.

A quick analogy: I fire a cannonball into the air (well, into the sky --no air) from the surface of the Moon. It follows a ballistic trajectory and then lands some distance away across the lunar surface. The direct (accurate) solution for the motion of the cannonball is a Keplerian ellipse with one focus at the center of the Moon. The arcing path that we see is just that small portion of the elliptical orbit that happens to be above the Moon's surface. The same is true, in fact, for a ball thrown here on the Earth. When you throw a ball in your backyard, the path you see is the upper edge of a highly eccentric elliptical orbit with a focus at the center of the Earth. On Earth, air resistance changes the details quickly, so let's go back to the Moon. That Keplerian elliptical motion is quite complicated to work out, but if you want the exact result, and if computation is cheap, that's what you do. Yet we all know that a ballistic trajectory can be represented by a simple "series" of terms in the usual kinematic formulas:
y = y0 + vy0 · t + (g · t2) / 2 and
x = x0 + vx0 · t.
This is a series expansion, truncated after the first couple of terms, that represents that lunar cannonball's motion to high accuracy without worrying about the details of the Keplerian ellipse. It has terms directly proportional to time, t, and also a term quadratic in time.

Series solutions are normal in physics and engineering and other mathematical computations. The series expansions for clearing lunars have sometimes been misunderstood and mis-represented as (mere) approximations. They are, in fact, every bit as accurate as the direct triangle solution (from which they are derived) when applied in the correct circumstances. Just as the simple series for a ballistic cannonball on the Moon would become inaccurate when the trajectory covers hundreds of miles (a substantial fraction of the Moon's diameter), so a series expansion for clearing lunars will fail if it's applied when the lunar arc is very short (10° or less). And like the series expansion for a ballistic cannonball, the series expansion for lunars has terms which are directly proportional to the altitude corrections for the Moon and the other body, and also terms which are quadratic in the altitude corrections for the Moon and other body. I won't derive these here. I'll just state the general results. Where LD and LD' are the observed and cleared lunar distances and dh1 and dh2 are the altitude corrections for the moon and the other body, and cos(α) and cos(β) as well as x, y, and z are geometric factors dependent on the altitudes h1 and h2 and the observed LD, the cleared distance LD' is given by:
LD' = LD + (dh1 · cos α) +... (dh2 · cos β) + [x·dh12 + y·dh1·dh2 + z·dh22].
Note that this bears a resemblance to that kinematic equation for ballistic motion. There are terms directly proportional to the altitude corrections and also terms that are quadratic in the altitude corrections. This series expansion at first looks rather intimidating, but everything after the "..." which I have marked in bold text consists of very small angles which can all be lumped together and thrown into a compact lookup table. All that we have left to do is calculate the Moon's altitude correction multiplied by that geometric factor of cos(α) (in fact just the Moon's parallax in altitude, with the refraction part of the Moon's altitude correction also included in the lookup table). That leaves a very short equation:
LD' = LD + (HP · cos h1 · cos α) + T,
where the Moon's parallax in altitude is now explicit and T, for "Table", represents all those little details thrown into a table.

The factor of cos(α) is what I refer to as a "corner cosine". If we draw a diagram at the Moon's position in the sky with a vertical arc towards the zenith and another arc (the lunar distance itself) towards the other body, then α is just the angle between those two. The cosine of that angle tells us how much of the Moon's altitude correction impacts the lunar arc. It tells us what fraction of the Moon's parallax in altitude lies in the direction pointed towards the other body. If that angle is near zero or 180 degrees, then the parallax correction lies in the same direction as the lunar distance arc and it is added or subtracted 100% to the observed distance. If, on the other hand, that angle is near 0 (implying, not that the bodies are at the same altitude, but that the lunar distance arc "leaves" the Moon horizontally) then the altitude correction has no impact on clearing the lunar distance: 0%.

If you draw the spherical triangle, it's not hard to figure out the value of that "corner cosine". And when we multiply it by the expression for the Moon's parallax in altitude, we're left with:
LD' = LD + HP · cos h1 · (sin h1 - cos LD · sin h2)/(cos h1 · sin LD) + T,
or, simplifying,
LD' = LD + [HP · sin h1 / sin LD] - [HP · sin h/ tan LD] + T .

At this point we're left with two very short terms to calculate by logarithms or otherwise (the two quantities set off in square brackets) and one quantity to look up in a table. This is a short, easy, and efficient calculation which can be performed in five minutes or less. This method, which was developed over several decades in the early 19th century, was brought to its most effective form by David Thompson who published a small set of tables realizing this method in the 1820s. It was popular because it was short and efficient. It was so popular that Nathaniel Bowditch borrowed it or procured it (in fact, "stole" it) for the 1837 edition of the New American Practical Navigator. This was the last edition published before he died. Bowditch claimed that his son had re-calculated all of the values for the correction table (the quantity T), but this is highly dubious.

Finally we come to vertical circle forcing, a modern method of clearing lunars which works in a remarkable number of cases thanks to the insensitivity of the lunars clearing process to the altitude of the Moon (which I have discussed many times before, but feel free to ask for details!). This vertical circle forcing, which is something that I developed, allows us to clear many lunars with no trigonometric calculations at all --just simple addition and subtraction of angles. And that's great. But it suffers from the same "problem" that afflicts any other modern paper method for solving celestial navigation problems -- no one needs it anymore. As I have joked on several occasions, what I must invent next is a time machine! If I had a time machine, I could take this "vertical circle forcing" method back to Bowditch or Maskelyne, and they could spread the word and then navigators would be spared some of the effort of lunars math in a great many useful cases. Without a time machine, these modern methods, no matter how clever, are curiosities or amusements and little more. Calculators and computers exist. And calculators are as reliable as any paper tables. While historical methods worked on paper sometimes make sense, since they give us insight into history, what point is there insisting that we do calculation on paper, using a modern technique like this creation of mine, in the year 2015? ... That's an open question! One answer: it's fun. And maybe that's enough.

How can it be that there are only three ways to work lunars, as I said at the top? Where is Borda's method? Where is Lyons' method?! What about Bruce Stark's tables?? What about versine methods? And so on and so on... The fact is that all of these other variants are merely computational variants. Nearly all are equivalent by trigonometric identities to one of the methods above. For example, when Nathaniel Bowditch first introduced a modified method of working lunars in the first edition of his Navigator back in 1802, he was simply replacing the cosine formula version of the calculation of the "corner cosines" with a different formula which is "identical" in the sense of a trigonometric identity. His modification changed nothing except the steps of the "recipe" in the process. Variations by trigonometric identities are endless, and also they are fundamentally trivial apart from the practical details like counting the number of tabular entries. Once upon a time, those calculational "recipe" details mattered. And they are interesting in the history of the mathematics of nautical astronomy. But they have no real significance. Imagine someone taking the equation sin x = y and telling that they have invented a new way to solve it as cos(90°-x) = y. Of course, that would be unimpressive. It's a simple trigonometric identity. Unfortunately in the history of lunars, endless trigonometric substitutions, not so different from this simple example, were peddled as "new" solutions to the problem of clearing lunars. And inevitably they were sold as methods that would "finally" make the math easy for anyone. Trouble is, the math was already easy. The Thompson variant of the series solution, outlined above, was dirt simple. Math wasn't the problem with lunars... Lunars had their day and passed into history, "never to be resurrection-ised" as Lecky put it.

Well, okay, they shot lunars --real lunar distance sights-- with handheld sextants from NASA's Skylab space station in 1973, but that's a story for another day...

Frank Reed
Conanicut Island USA

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