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    Re: Quad regression for Lunar Distances?
    From: Frank Reed
    Date: 2018 Nov 8, 10:30 -0800

    Lars,

    Although it would have been a modest amount of work historically, it's trivial today. And since we're already talking about regression calculations, I assume we're not too worried about calculation expense.

    How can we get the rate of change of the apparent lunar distance? Let's ignore the refractions of both bodies since they change slowly except when the Moon or other body is lower than perhaps ten degrees. We'll also ignore the parallax of the other body for the same reason. That leaves only the Moon's parallax correction, which is HP·cos(h). We can treat HP as a constant over a typical observation interval, and therefore the rate of change of the altitude correction is just HP·sin(h)·(dh/dt). The altitude and the rate of change of altitude can both be determined by observation. Finally we need the fraction of that rate of change that acts along the lunar arc. This is the same process of calculating the "corner cosine" at the Moon which I have described many times with respect to series solutions of lunars. Regardless of what we call it, it's a simple bit of spherical trig: get the cosine of the angle at the Moon's corner in the ZMS triangle (Z: zenith, M: Moon, S: Sun or star). That gives us the rate of change of the altitude-varying part of the observed lunar distance. Work up a practical case, and you'll see typical numbers like 0.1'/min (a tenth of a minute of arc per minute of time) going as high as 0.2'/min near the zenith under the right circumstances.

    We can get the rate of change of the geocentric lunars by consulting our almanac tables. This rate will come to be roughly half a degree per hour. This is a case where the prop.log of the rate of change (found in the British Nautical Almanac and its followers only after 1834) would be merely a distraction. You would simply calculate the geocentric rate directly from change in published distance divided by time. Typically, this will yield about 0.56'/min, a bit higher when the Moon is at perigee and lower when the Moon is at apogee, and also when the other body is not directly aligned with the Moon's motion across the celestial sphere. There's no complexity to this part of the calculation: just take a pair of tabulated distances, subtract them, and divide by the time interval.

    The two rates, one derived from the apparent altitude correction, the other from the tabulated geocentric distances, combine to give the apparent rate of change of the observed lunar distances. That's the slope we expect to see when we graph out our sights. Noise in the data can easily create an impression of a different slope, and that can throw off the selection of a best point, and it can also make it much harder to spot a genuine outlier.

    Also notice that if computation is cheap, you could just calculate the apparent rate of change (the expected slope of the observations) based on an estimated position without having to bother with measuring the Moon's altitude or its rate of change.

    You mentioned that dreadfully cumbersome phrase "parallactic retardation". This was a mouthful of jargon that George Huxtable invented when he thought he was onto something important, something profound about lunars deserving a big, weighty name. This was back in 2002 when he was operating largely in a vacuum. Eventually he realized that this was not significant, and he tried hard to disavow the whole thing. He was so embarassed by the issue that he missed out on one aspect that actually was relevant to lunars (when altitudes are calculated instead of observed). But the beast won't die. Regrettably, George added a section about his mistaken discovery to his multi-part document about lunars. While the first couple of parts are good, the latter parts are weak. They depend on poor sources, like the awful chapter on lunars in Cotter's "History of Nautical Astronomy" and they also depend on various misconceptions which George had about lunars including this infamous "parallactic retardation". I'm sure he would write a radically different document if he were alive today and could approach it fresh --without the influence of some of those poor sources.

    Frank Reed

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