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Alex Eremenko wrote-

>I agree with you that my "second lunar observation series"
>posted here was probably on the boundary of what should
>be permitted to average. It could be even "beyond this boundary",
>and I was just lucky to obtain improved results by the averaging.

I first suspected that might be the case, from the (small) inconsistency
between the longitude derived from  the averaged observations, and the
average of the individually-calculated longitudes. However, that could be
partly the result of quantising, to 0.1', of the inputs to Frank's program,
which he has since shown how to circumvent.

Even so, I wondered if there was some residual non-linearity, suspecting
first the curvature in the "clearing" corrections for refraction and for
parallax. However, I no longer think that those are significant, based on
some worst-case estimates I have made.

The worst case, in which the cleared lunar distance is as sensitive to
non-linearity as it can possibly be, occurs, as I see it, when the observer
is near the equator and the Sun and Moon both have zero declination, so
they follow each other along his prime vertical, the East-West line through
his zenith. This is also the simplest case to model, fortunately. In that
case, the non-linearity error depends only on the time-span of the
measurement, and on the mid-altitude: altitude of the Moon only, for
parallax, and for refraction, altitude of both bodies, but mainly the lower
of the two.

For example, in the case of Alex's lunar of 26 Oct., the time-span was
about 18 min and the Moon altitude was about 28deg, Altair being about
55deg. In that case my estimate is that averaging errors in the clearing
process can't exceed .017', corresponding to an error of 0.5' in longitude,
quite negligible for a lunar.

I'm now able, somewhat tentatively, to suggest an upper limit for the
time-span of a lunar observation, to ensure that the CLEARING error of the
lunar distance is kept within about 0.1 arc-minutes, so that it contributes
no more than 3' or so to the resulting longitude. It depends, more than
anything else, on the altitude of the LOWER of the two bodies concerned, at
the mid-time of the observation.

Altitude  Time-span
(degrees) (minutes)
   10        18
   15        28
   20        35
   25        40

I haven't taken it further, because I doubt that anyone would wish to spend
more than 40 minutes on a lunar observation.

My assumption is that over the time-span allowed, several lunar distances
are measured, at reasonably-equal spacings in time, and averaged.

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

Note that this recommendation is based solely on averaging errors in the
CLEARING process, so the altitudes are the important matter. I have not
(yet) made any assessment of averaging errors in the obtaining of an
averaged lunar distance, before clearing it. This may well restrict further
the allowable time-span of a lunar; we will have to see how it turns out.

I can identify two sources of non-linearity here, as follows.

1. Simplifying the Moon's path to be a great-circle around the sky, within
5 deg of the ecliptic, which passes a chosen star (or other body), at a
certain minimum passing distance, at a time of closest approach t. We know
that when it's a long way off, the Moon will be approaching at a rather
steady rate, which on close approach falls to zero at time t, then reverses
as the two bodies part. Clearly there must be a non-linearity in the plot
of lunar distance against time, which may limit the allowed time-span of an
observation. I would describe this as the Geometrical contribution to
averaging error.

2. Identifying the differences in real-life between the Moon's predicted
motion and the simplified path considered in (1) above. The main
discrepancy will be in the changing speed of the Moon due to the elliptical
nature of its orbit. There are indeed many other perturbations in the
Moon's motion but most will be negligible for this purpose.

I would describe this as the orbital contribution to averaging error.

===============.

I don't think it will be necessary to predict every detail of how the lunar
distance differs from linearity, but just consider worst-case conditions in
order to provide some sort of guidelines for observers.

Alex wrote-

>So, if this question was carefully investigated, and
>the results were published, why don't wee
>just look at these papers instead of inventing a bicycle again?

I don't recall these questions having been examined before, in any detail,
on the Nav-l list, or elsewhere, but I fully agree with Alex that if they
have, let's save ourselves a lot of trouble.

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

It may be interesting to read what Tobias Mayer had to say on this topic,
in his description (in Latin) of his repeating circle, which he sent, with
the instrument, to the Board of Longitude in winter 1754/5. Maskelyne
arranged for the text to be printed as part of "Tabulae Motuum Solis et
Lunae", in 1770, for which Mayer's widow was allowed part of the Longitude
Prize.

"Note 2.
Without putting too fine a point on it, so that the relative motion of the
moon with respect to the star remains constant, the time between the first
and the last operation ought not to be excessive. For otherwise if a larger
interval occurs, an average derived from the sum of all of the times and of
the distances, given the circumstances, would not be a permissible
assumption. The discrepancy in the approach and retreat of the moon from a
star within a space of half to one hour, is imperceptible, especially if
the moon is some distance from the star.  This interval of time, therefore,
while it lasts, is sufficient for the observations to be repeated six
times, or many more, should the need arise."

I thank Steven Wepster, who has studied Mayer in detail, for passing on a
copy of that interesting paper. Mayer's comments were very perceptive for
their time (though we ought to test them now), and his "repeating circle"
shows great ingenuity.

George.

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