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    Re: Luni-Solar Distance
    From: George Huxtable
    Date: 2010 Oct 19, 16:23 +0100

    I had written, about the iteration process called for, when calculating
    rather than measuring, the altitudes of the bodies involved in a lunar
    distance observation,
    "You can see that the iteration is
    converging, but painfully slowly, improving by approximately a factor of
    two at each step".
    And Douglas asked whether it always converged, and if so, how quickly?
    As far as I understand what's going on, yes, it always converges. My
    example was deliberately chosen to show a situation that's worst-case, or
    nearly so, and in that worst-case, the resulting longititude error
    more-or-less halves at each iteration. That can happen over a rather wide
    range of possible geometries; it's not a particularly unusual state of
    affairs. It can be worst with a high Moon, seen from the Tropics. In many
    other plausible geometries, the end-result is little affected by the
    altitudes, in which case a single iteration might well suffice.
    We've discussed this matter on the list before, but I can't put my finger
    on the postings. As I remember, Frank expressed the view that reiteration
    would seldom if ever be necessary. If I have it right, that view seems to
    be in some doubt..
    Dougles asked further-
    "I might add: can the iteration convergence be improved with a correction
    factor as with the second method in Meeus's Kepler iteration method?"
    After the initial calculation, and one reiteration, one can deduce the
    sensitivity of the lunar distance to changes in assumed longitude, which
    then allows the end-result to be predicted without further iteration.
    The culprit in this seems to be our old friend, which we dubbed
    "parallactic retardation", which has been thrashed out on the list in the
    past. It can result in the apparent lunar distance changing at only
    about-half of the rate of the true lunar distance, 6' per hour instead of
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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