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In a previous posting I wrote-

> the questions remain, related to our previous long discussion on
>"averaging". Is there a non-linearity effect which will upset the result of
>a longitude deduced by averaging lunar distances, especially of a series
>that's protracted in time? Is there in that case an advantage in clearing
>each one, then averaging the result? If such a difference could be glimpsed
>in Alex's series, though trivial in amount, are there situations where it
>can become important? If the answer to these questions is "yes", then I
>think the culprit will be found in the non-linearity of the Moon parallax
>correction.

====================

I was suggesting that non-linearity in the Moon's parallax correction
(which varies as cos (alt)) would impose a limit on the time-span of a set
of lunar distance measurements.

Having made some rough estimates, it now seems that I was wrong: or at
least, the allowable time-span of a set of lunar distances is quite a long
one, IF (and this is an important proviso) parallax is the only important
factor.

I have taken the worst-case situation, which also happens to be the
simplest to calculate, to be this- The observer is on the equator, and the
Sun and Moon both have a constant declination of zero. Both the Moon and
the Sun will then track across the observer's sky, from East to West,
through his zenith. The Sun is well to the West of the Moon, and as the
Moon's altitude increases, after rising in the East, the angle between Sun
and Moon changes due to parallax by an amount that's exactly equal (but
opposite) to the Moon's parallax correction. The Sun's parallax is taken as
negligible.

If the parallax correction varied linearly with time, then the correction
for a series of observations, averaged over time, would be exactly the same
as the correction for a single observation at the mid-point in time. But,
the correction follows HP cos alt (where HP, horizontal parallax, isn't
very far from 1 degree) which is curved, not straight. And it's most curved
when the altitude is small, and nearly straight when the Moon is overhead.
So the non-linearity in the parallax correction will have most effect at
low Moon altitudes. For other reasons (refraction) angles less than about
10 degrees should be avoided anyway, so I have taken the worst-case to be
when the Moon'a altitude is 10 degrees.

I have considered a set of closely-spaced lunar distances, spanning a time
before and after the mid-time when the Moon passes through 10deg. altitude.
I've asked how long that time-span can be, before the difference between
the average parallax correction, and the single value of parallax
correction at its mid-time, reaches 0.1 arc-minutes. That's the difference
which would give rise to an error in longitude of 3 arc-minutes. Smaller
errors than that are regarded as insignificant in the context of a lunar
distance, which is by no means an exact science.

My result is that if that set of measurements is kept within a time-period
of 45 minutes, between 22.5 minutes before and 22.5 minutes after the
moment of 10 degrees Moon altitude, and are scattered in time reasonably
uniformly over that period, the averaged distances and times can be used in
clearing the lunar distance, and the resulting error in longitude due to
that averaging will not exceed 3 arc-minutes. For higher Moon altitudes and
different geometries, the allowable time-span will be greater, so that
45-minute limit can be taken as reasonably safe in all cases.

The above estimate is rather tentative at present, and comments or
corrections would be most welcome.

Of course, in that worst-case example I have chosen, altitudes would be
significaltly less than 10 degrees, over the early part of that time-span.

And BEWARE! My model has taken account only of the parallax correction. Not
(yet) of non-linearities in the true motion of the Moon in the sky (which I
expect to be small on this time-scale, but that needs checking). More
important, I have taken no account (yet) of refraction, in the apparent
positions of Sun and Moon, and refraction corrections become large (and
VERY non-linear) at low altitudes. So it may be (and I think this is
likely) that when altitudes are low, refraction, rather than parallax, will
set the overall limit on the allowable length of a set of lunar distance
observations.

Of course, if each observation is treated and cleared individually (as is
now possible and easy), rather than being gathered together and averaged
first, then the resulting longitudes can always be averaged with confidence
(for a static observer).

George.

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contact George Huxtable by email at george@huxtable.u-net.com, by phone at
01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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