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    From: Frank Reed
    Date: 2021 Nov 15, 04:02 -0800

    Going back ten days in the discussion, Dave Walden quoted a post by Paul Hirose regarding the delta-T error in the old "ICE" program, indicating that it amounted to only 0.1', a tenth of a minute in the position. That post was from 2014, and that was correct back then. Now, in 2021, the error has grown hugely ... to 0.15'. :)

    The short rule: take the delta-T error in seconds of time, divide by two and tag the result as "seconds of arc". This yields the error in the Moon's position on the celestial sphere, relative to the other stars and planets.

    This is actually quite easy to work out. Every second of delta-T error corresponds to half a second of arc error in the Moon's position on the celestial sphere. Delta-T, as most NavList readers know, is a clock error. There is a "true" time variable in the laws of physics. It's variously been known as ephemeris time and dynamical time in celestial mechanics computations, and a physical realization of it is atomic time. It's the same thing in terms of fundamental physics, and these various names obscure a simple truth: it has a really awesome name with no adjective, no qualifications. It's just plain "t". In the laws of physics, when you see "t", that's almost always what's implied. While we have to worry about relativistic differences, both from motion (SR) and from gravitation (GR), there is always a "t" in the cosmological rest frame to which any variations refer, and just like in classical physics, this is the pure "time" of the laws of physics.

    But we don't keep time by the pure "t" of the laws of physics. Historically we keep time by the orientation of the Earth relative to the Sun (mean solar time) and stars (sidereal time), and although we use atomic time on a daily basis, we still keep time based on the orientation of the Earth. Our common clocks eventually and slowly become desynchronized from the "t" of the universe because the rotation of the Earth is not uniform and is influenced by the tidal deceleration of the Moon and even the weather. So we need to tell our calculated astronomical positions that our input time, usually some variant of UT, needs a little offset, a "delta" to bring it in line with the universe's pure time variable. If we give it the wrong "delta-T" then our celestial coordinates will be slightly incorrect. One second error in time hardly matters at all. But ten seconds? Or thirty seconds? How much does that impact the positions of celestial bodies?

    Among the celestial bodies visible from the Earth, the Moon is by far the fastest in its motions across the sky, excluding occasional, rare close passes by small asteroids. The other celestial bodies are also impacted by errors in delta-T, but they're minor compared to the Moon. If you have ever thought about lunar distances navigation, or if you have attended one of my workshops on lunars, you know that 12 seconds of time corresponds to a tenth of a minute of arc error in the Moon's position. If you're trying to determine Greenwich time by lunars, every tenth of a minute error in your observed distance yields approximately 12 seconds error in the time. You can rederive this easily by remembering that the Moon's position changed by just about two degrees in one hour of time (that's a rough approximation but good enough for this purpose). Don't think you could remember that? How about remembering that the Moon covers 360° in 28 days? You divide that out and you find 1° in just about two hours. So you can remember that of figure it out on the back of an envelope. And the nice thing about this two-to-one relationship of hours to degrees is that it also applies at each step down the scale of hours, minutes, and seconds. The Moon moves about one degree in two hours of time. Equivalently it moves about one minute of arc in two minutes of time. And, still equivalently, it moves about one second of arc in two seconds of time. From either of the last ratios, you can see that the Moon moves just about a tenth of a minute of arc in 12 seconds of time, and since a tenth of a minute of arc is about the best that we can do, even with averaging, in manual celestial navigation, that's a good benchmark to know.

    The Moon's speed across the sky, 0.1' in 12 seconds, answers many questions in basic astronomy as well as celestial navigation. How careful do you have to be recording the GMT/UT when you're practicing backyard lunars? If you're casual with your timings, and you're wrong by 6 seconds, the difference is only 0.05', completely insignificant. So we can, in fact, be casual with timings like that. And we can also address the impact of an error in delta-T. It's really just a watch error. If an app (or a "program" as "ICE" would have been known when it was created in the late 1980s!) uses a delta-T value that's wrong by 12 seconds of time, the Moon's position on the celestial sphere will be wrong by 6 arcseconds (a tenth of a minute of arc).

    Back in 2014, around the date of the message that Dave Walden quoted, the delta-T error in the old "ICE" app was slightly more than 12 seconds. What's the corresponding error in the position? Just divide by two and tag it "of arc": 6 seconds of arc. Today in 2021, the delta-T error has grown to 18 seconds. So the error in the Moon's position is 9 seconds of arc or 0.15' (in tables that round to the nearest tenth, that error would display as 0.1' in half of entries and 0.2' in the other half, approximately). Should we care? No. This is minor, except for lunars. But it's growing, and more importantly there are numerous newer tools and apps that provide the required data.

    Given that 12 seconds error in delta-T yields only 0.1' error in the Moon's position, we can also see that excessive precision in quoted values of delta-T is pointless. Sometimes developers of tools and apps for celestial navigation will try to impress by listing delta-T to the nearest hundredth of a second or even more. This is nonsense. Changing delta-T by one whole second shifts the Moon's position by a mere half a second of arc. Even if you're contemplating high-accuracy lunars, the mountains and depressions along the Moon's limb are much bigger than that!

    Frank Reed
    Clockwork Mapping / ReedNavigation.com
    Conanicut Island USA

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