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    Re: GHA of the Moon
    From: Frank Reed
    Date: 2020 Oct 2, 22:12 -0700

    David C, you wrote:
    "Back in the days when "computer" was an occupation were these equations used?"

    No, not those polynomial equations. And in fact the "polynomials" from HMNAO are not in any real sense "computing" the Moon's position. They are just a more involved form of the simple interpolation scheme that I described. Those polynomials are created by "fitting" to proper almanac data that has been computed from the laws of physics (meaning, almost entirely, just Isaac Newton's good-ole inverse-square law of gravitation). All modern almanacs and better modern apps (like mine, of course!) draw their data directly from the JPL numerical integrations of the Solar System. The polynomials are smoothed curves that get you from one day to the next using the exact values of the coordinates (and their rates of change) at midnight each day from those JPL numerical integrations (or secondary sources derived from them). I have suggested that it's sufficient to get hold of the coordinates for every hour, and then you can just use easy, bug-proof linear interpolation. But really, they're very similar. 

    You asked:
    "Or were simpler, presumably less accurate methods used?"

    The older methods were somewhat closer to the underlying celestial mechanics, and calculating the Moon's position from the "mean longitude" and "longitude of the ascending node" and "orbital eccentricity" and all that is yet another way to proceed. The quality of the results natually improved, and they improved in the earliest period primarily to satisfy the needs of lunarians. If you want the longitude by lunar distances, a one minute of arc error in the Moon's position yields a two minute of time error in the Greenwich time. Or dividing by ten, a tenth of a minute of arc corresponds to twelve seconds of time. In the earliest periods of lunar tables, an error of half a minute of arc was not rare. Within thirty years, errors had been reduced to less than half that. 

    Speaking of accuracy, those polynomials with five coefficients for every day promise to yield the Moon's declination to +/- 0.003 seconds of arc (if I remember correctly). That's great! It's also un-necessary for celestial navigation calculations. In "typical" modern celestial navigation, one could accept errors on the order of 30 seconds of arc or at worst 3 seconds of arc. So the polynomials are on the order of 1000 to 10,000 times over-kill! It's like ordering a nuclear strike on a rowboat. :) Note that my suggested solution, simple interpolation between hourly values, is not quite as accurate, and you could expect errors as large as one second of arc from that method (but it's easy to extend to quadratic level ...or cubic, or quartic, etc., and then you're back to those polynomials).

    Frank Reed

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