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Frank Reed wrote, in NavList 1675,

 ...By the way, as I mentioned before in this  topic, I'm a
| big believer in averaging four sights whenever you're trying to do
lunars. And
| this was considered normal practice historically. That way, assuming
no
| underlying bias, if I have a typical error of 0.2 minutes of arc in
each
| observation, the final average angle will be accurate to 0.1 minutes
of arc.

That assumption, of "no underlying bias", is a big one, and an
unjustified one. You can halve the error by taking four observations,
but only when those observations are all statistically independent.

Let us consider in detail some components of error in taking a lunar
by four such observations.

1. Perception-errors in estimating the point of contact between the
two bodies. Each time, of the four, the observer will see exactly the
same picture, affected as it may be by irradiation due to brightness
differences. Whatever error he makes in that estimation is likely to
be the same each time, rather than varying randomly. It may differ
between one observer and another, but for the same observer any such
error will be systematic, not random.

2. Calibration-errors of the scale of the sextant. Presumably, all
observations in such a series, being closely spaced in angle, will be
subject to the same systematic error, corresponding to the region of
the arc that's in use. Part of that error may be offset by applying
the "box error", shown on the calibration certificate at (usually)
15-degree intervals, and pasted inside the sextant box. This is the
error, and the only error, that a sextant manufacturer will
acknowledge as being his responsibility.

3. Reading-errors, in getting the best estimate of the angle reading
of the sextant; interpolating by eye, to a small fraction of an arc-
minute, the divisions of its drum, perhaps with the aid of a Vernier.
Because a lunar distance is changing slowly, successive observations
will usually involve a different fraction, and so there will be some
degree of "random" scatter in this error, between one such observation
and the next.

4. Index-zero determination. Any error in measuring this will be
common to all four observations, and systematically subtracted from
each. It will involve similar error-components to those in types 1 and
3 above, but not 2. If an index-zero check is repeated several times,
there will not even be a "random" scatter as with type 3 above,
because exactly the same observation repeats each time.

It's pretty clear, then, that only a small fraction of the resulting
overall error will scatter statistically from one observation to the
next, and that most of the error will be repetitive. In which case,
the assumption that overall errors can be reduced by taking multiple
observations, by a factor equal to the square root of the number of
observations, becomes invalid.

It seems to me hopelessly optimistic to expect that the end-result of
all such errors combined could be kept within 0.1 arc-minutes, no
matter how many repetitions are made. That is what would be necessary,
as well as the elimination of all other sources of error (some, but
not all, of which have been considered in other postings) to achieve
the level of precision that Frank Reed has claimed, in his 6-mile
overall error.

George.

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



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