NavList:

Ed,

We usually split the process of clearing lunars into two parts. This was done historically, and it's also the way I describe the process.

The first part is what I call "pre-clearing". For the altitudes we do an approximate adjustment for dip and semi-diameter. No more than approximate is necessary since the main part of the clearing process (next step) is not very sensitive to the altitudes (which you can see by contemplating those graphical tables created by George Margetts over 200 years ago!). We also adjust the lunar distance itself for the semi-diameters of the Moon and Sun, if the other body is the Sun, paying very careful attention to the fine details and adding in the augmentation since anytime we "touch" the lunar distance angle itself, we have to work to the highest achievable accuracy. The key to the pre-clearing step, and the reason it works as a separate step is that we are correcting the distance for adjustments that are purely along the line between the centers of the bodies. Pre-clearing is simple addition and subtraction with no significant geometric or trigonometric calculations.

The second and famous part of clearing lunars, the big show, is the process of correcting for parallax and refraction. In the early history of lunars, this was thought to be an insurmountable problem. The logic was that the position of the Moon in the sky depended slightly on one's location, so how could it be used to determine longitude when you surely had to know your longitude at the outset to get that correction? But then, shown the way by a French astronomer in the 17th century (can't dig up his name right now), the nautical astronomers, such as they were, realized that it all depended on the Moon's altitude, which we can measure as part of the process of "taking" a lunar. We don't need to know where we are to correct the Moon's position relative to the Sun. We only need to know how high the Moon is in our sky, and simiarly we need to know the altitude of the Sun or other body. Given those altitudes, the correction process goes quite nicely because the corrections for altitudes are entirely in a direction toward the zenith. Any correction to the position of the Sun or Moon that acts purely aong the direction from horizon to zenith can be folded in and cleared during this process. Refraction moves the stars and Sun and Moon up, always up. The Moon's parallax shifts its position downward, always perfectly downward away from the zenith. And the parallaxes of other nearby bodies like Venus (and even the Sun, though we can often ignore that) are in the purely vertical direction. The math concept that you're very familiar with covers all of these separate adjustments to the lunar distance with ease. They act entirely in the vertical direction, along the arcs from the horizon to the zenith for both bodies. No need to get into mathematical details here. The key is to see this commonality.

We have thus split the various adjustments to the lunar distance into two broad categories. Anything that happens simply along the line between the centers is equivalent to an adjustment to the objects' semi-diameters so we can handle anything in that category, like the augmentation and the refraction of the semi-diameters, in the pre-clearing step. Anything that happens purely in the vertical direction, like refraction in position and parallax in alltitude is part of the normal clearing process.

So what about that small correction for the Earth's oblateness? Where does it fit in? Does it act primarily along the line between the centers of the Sun and Moon? Or does it act in the vertical direction? Well, neither! And that's why the math for that little adjustment is just a bit tricky to wrap your mind around. But do you really need to do so at this point? It's a really small adjustment. You could just ignore it if you like. Bowditch ignored it. There was no adjustment for the oblateness of the Earth in lunars, even as an option, in the vast majority of the methods that were published and used in the period when lunars saw significant navigational use, in the late 18th and the first half of the 19th century. It wasn't until Chauvenet's method was inserted in Bowditch after 1880 (long after its practical value had expired) that the topic was really explored, and you can read a straight-forward discussion of the math behind it in Chauvenet's "Spherical and Practical Astronomy" --which you want if you don't have a copy yet (that's the one that's just "Chauvenet" espcially to the generation 10-20 years older than me for whom it was a unique resource, thanks in large part to the Dover reprints).

Frank Reed
PS: My Lunars class, for anyone interested, April 21-22 this year: http://www.reednavigation.com/lunars-class/.