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    Lunars rate of change: geocentric and topocentric
    From: Frank Reed
    Date: 2022 Aug 16, 12:28 -0700

    In the Lichtenstein Lunar example that I outlined last week, several of you noticed that Procyon wasn't very well aligned with the Moon's motion. That is, Procyon is well off the ecliptic, and at such a short range, that means that Procyon is about 45° off-axis. While this does reduce the expected accuracy of a lunar observation for determining UT/GMT, it's not as much you might think. For most everything involving the Moon, you can assume that a tenth of a minute of arc in the Moon's position on the celestial sphere (relative to the background stars and planets) corresponds to about 12 seconds of time. The measured lunar angle from a star relatively close to the ecliptic changes at that rate. Being off-axis this Lichtenstein Lunar changed at a slower rate, a tenth of a minute of arc in about 12/cos(45°) or 17 seconds of time. So that's worse than a nice ecliptic-aligned star, but not severe.

    Antoine and a few others also noticed that there seems to be another source of decreased accuracy. The lunar angle is changing quite a bit more slowly! If you look at the locally observed or, so-called, "topocentric" angle between the Moon and Procyon, it changes even more slowly because the Moon's parallax in altitude is decreasing as it climbs in the sky. It's almost as if the lunar arc is being afflicted by some strange disease, unknown to the historical experts, unknown to the practical lunarians from two centuries ago. It's a sort of "parallactic retardation"!

    Is the fog clearing yet? Are vague memories emerging from the shadows? A long time ago, in a galaxy far, far away, the late George Huxtable wrote up a four-part essay on lunar distances or "lunars". That long time ago was actually twenty years ago, nearly two years before I joined the community. Parts 1-3 of his "About Lunars" series are good. Naturally there are some flaws, and George had no experience at all shooting lunars, so some of his statements are just wrong, but the series is still worth reviewing. Here's an index to those posts from twenty years ago. In the index you'll also find links to "About Lunars, part 4". This addition to his series was weak and contained a number of speculations and extrapolations that were simply incorrect. In particular there was one which he late found embarassing and wished he could expunge entirely. It was a section on "parallactic retardation". That was his --somewhat over-blown-- term for this phenomenon where the topocentric lunar distance changes much more slowly than the geocentric lunar distance which, at first glance, seems to imply that there are times lunars would produce significantly worse results for absolute time, even when the other body is perfectly aligned with the Moon's motion.

    The problem with this concept is that lunars or "lunar observations" are not just the observed lunar arc -- the angle measured by sextant from the Moon to the other body. Each complete lunar observation actually consists of three angles (possibly boiled down by averaging from multiple observations of each quantity). A complete lunar consists of the measured lunar arc and simultaneous observations of the altitudes of the two bodies. Those separate altitude observations cancel out the so-called "parallactic retardation". And hence, all you need to examine to understand the time resolution of a lunar observation are the geocentric lunar distances. Parallactic retardation isn't real! Well, mostly not (see below).

    George Huxtable in 2004 had been persuaded of the error of his ways by Jan Kalivoda (I checked in with Jan last year just to say hello and learned that he had long ago moved on from his nautical astronomy hobby and was now deeply involved in medieval literature). Once he realized that he was wrong about this parallactis retardation, George wanted the concept to die and be wiped from the record. But that, too, was a mistake. Parallactic retardation, or the slower rate of change of some topocentric lunars, does indeed matter when lunars are taken and reduced by an alternative method, which was, unfortunately, recommended by some mathematicians in the nineteenth century. And that's when the altitudes of the Moon and other body and calculated from a position derived from the longitude implied by the lunar distance. This throws out a key benefit of the observed altitudes which is their observational independence. If the altitudes of the bodies are calculated (and especially if they are iteratively re-calculated), then the longitude from the observed lunar changes the altitudes, and that leads to reduced accuracy when the topocentric lunar distance rate of change has been reduced by "parallactic retardation". 

    This latter exception does not imply that parallactic retardation is actually important. Rather, it is another argument against the ideas of the over-zealous mathematically-inclined lunar "experts" who advocated calculating lunar altitudes because that would be "more accurate". They were wrong, and as it turns out doubly wrong. Accuracy, sources of errors, error bars, and statistics were only barely understood in that era, and the idea really had no merit.

    You can play around with this and see how it affects the UT implied by the lunar using modern calculating apps. Specifically, in my clearing web app: Lichtenstein Lunar without altitudes. How much does the time have to change to get an error of a tenth of a minute of arc in the lunar? You should see that it's about 34 seconds of time per tenth of a minute of arc, which would be rather poor. Then try it with the altitudes as observed: Lichtenstein Lunar with altitudes. Notice that the sensitivity is now, in fact, about 17 seconds per tenth of a minute of arc, not so bad, and just as expected from the geocentric lunars.

    Frank Reed
    Clockwork Mapping / ReedNavigation.com
    Conanicut Island USA

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