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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
12.

George.

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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