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    Re: Historical Lunars : take in account 'delta-T' or ignore it ?
    From: Frank Reed
    Date: 2009 Dec 13, 18:50 -0800

    Antoine, you wrote:
    "Frank indicated that these topics might better be covered on NavList. Therefore and after extensive and in-depth reprocessing I am submitting everything to all of us in NavList."

    Thank you, Antoine! I see that there have already been replies from other people. George has already covered much of this, so I'm just going to throw in my own thoughts for the sake of another viewpoint.

    You also wrote:
    "When re-processing "historical" Lunars (i.e. "real world" Lunars observed during the period ---D.) with to-day computational power, should we take in account "delta-T" or ignore it?"

    Short answer: always include it.

    You wrote of lunar theories which, even as recently as the early 20th century,
    "included in particular an 'empirical term' exceeding 10 arcseconds in Longitude to best reconcile predictions with observations."

    This is actually quite important to the issue you're concerned with. If the mean celestial longitude (ecliptic longitude) of the Moon requires an empirical adjustment in the models and tables now and then, this is effectively indistinguishable from an adjustment in delta-T for values of delta-T less than about a minute. So long as the almanac publishers and ephemerides calculators included an empirical adjustment to the Moon's position, they were taking care of delta-T without knowing it.

    You also wrote:
    "First, we can state that, when they cleared their Lunars, the 18th, 19th and (if any) the early 20th centuries Navigators were not aware of the "necessity" of using 2 time scales."

    Navigators, per se, had no more reason to worry about, or know about, delta-T back then than navigators do today. It's the almanac publishers who would have needed to make any corrections.

    You proposed two calculational options for analyzing historical lunars:
    1) "If we want with our current "computation tools" check the quality
    of their Lunars, we should put ourselves in their exact same environment,
    i.e. reprocess their Lunar computations with delta-T = 0"
    and
    2) "If on the other hand we want to best derive their observed
    longitudes - and/or positions - we should certainly then take in account
    values of delta-T."

    I would put it a little differently. First let's deal with your second case. This is the one that we can all agree on. If we have some observations from 200 years ago, and we want to know how good they were, then we should use the best available positions for the celestial objects and of course that means that we should use the best estimates of delta-T available. There's no point fussing over whether you're using this or that JPL ephemeris of some French tables, they're all far more accurate than necessary for this task (anything published within the last ten years... before that you can expect approximations). You can take the positions of the celestial objects on the celestial sphere as givens, known quantities which can be tabulated once and for all, to an accuracy of one second of arc. Beyond that accuracy, you need to be more careful, but that doesn't matter for navigational calculations. The actual orientation of the Earth with respect to the celestial sphere is not known to quite the same accuracy. We don't know the exact value of delta-T, except in more recent decades. Even here, as you noted in your post, a difference of a couple of seconds in delta-T is not significant.

    Now back to your first case. If we want to work the calculations using the tools they had available, then what you want are the tables from the Nautical Almanac(s) for that year. The lunar models were progressively more accurate during the lunar period, so if you want to determine the inaccuracy arising from imperfect astronomical knowledge (which includes delta-T issues but only as a lesser factor) then you should get the almanac for that year. MANY of these are now available on google books. You could also satisfy yourself with knowing the range of possible errors in the tables for a given period if you don't need the exact values.

    Consider how this would have worked back in the 1840s. If you were a whaling captain from New Bedford (Mass., USA) going on a Pacific whaling voyage, you might buy two or even three years of Nautical Almanacs before departure. How do we know that the astronomers who compiled those almanacs had the Moon's position correct? How would they deal with the changing values of delta-T that we know about today? They would, of course, compare their predicted lunar positions with what they observed with their telescopes at Greenwich and other observatories a year or two before publication. At least, that's what they should have done... The lunar tables that they work from when compiling the published almanacs are adjustable. The constants which underly the lunar theory can be changed to bring the predictions inline with observations. But interestingly, for a whaling captain heading for the Pacific in the 1840s, the almanac publishers had NOT updated the tables sufficiently. Small errors, on the order of 10-15 seconds of arc (in the lunar distances) had crept in by this date (this is in addition to the general random errors due to the still-incomplete lunar theory). Presumably, this was because lunars had fallen into disuse much earlier on British vessels so there was less importance attached to accurate lunar distance tables. That means that there was an inescapable bias in lunar longitudes in the 1840s. It wasn't much, and its cause was partly due to delta-T, partly due to uncertainties in the Moon's mean longitude. And when the first edition of the American Ephemeris and Nautical Almanac was published in the 1850s, the Americans scored a few points in the world of astronomy since they had, in fact, gone to the extra trouble of updating the lunar tables based on recent observations. As a result, for a couple of years, the American almanac was better than the British standard. This small bias (partly delta-T) was smaller than the "random" error that still existed in the lunar theory. In other words, the tables might consistently bias your calculated longitudes by 4' to the west due to growing errors in the Moon's mean position in the tables, but those positions would also have a statistical error due to the lunar theory with a standard deviation of +/-7' east or west (just some sample numbers; the actual numbers varied over the decades).

    You gave an example of using a good modern estimate of delta-T for a modern lunar distance calculation and then running the calculation again with delta-T set to zero. This just wouldn't be done. You shouldn't be treating delta-T as a variable except for hypothetical cases some years into the future (e.g. tables for the year 2025) or in the early lunars period when modern analysis of old observations cannot give values for delta-T more accurate than +/- a few seconds. But in that early period, e.g. c.1770, this uncertainty is smaller than the uncertainty in the available lunar tables.

    -FER

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