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We have been discussing  "Refraction near the horizon", Publ. astron.
society of the Pacific, vol 102, pages 796 - 805 (July 1990), by Schaefer
and Liller. I have now made contact with an author of that paper, Brad
Schaefer. He acknowledges that the quantity being measured, by the sunset
timings, is indeed total refraction, in the path from the Sun, then tangent
to the ocean surface, then up to the observer. So that refraction is greater
than the horizontal refraction observed at the sea-surface, and if the
observer is on a mountain, significantly so.

It then seems wrong, to me, to put together all that data, some from high
altitudes, and some from near sea level, into a common data set, and analyse
it for scatter, without attempting some sort of correction for observer's
altitude. Schaefer defends it in these terms- "The effect is there but small
and largely swamped by the variations." I do not go along with that, and
have had a go at extracting better information from his data.

Schaefer and Liller quote an average horizontal refraction for 144
observations as 33.1' +/- 9.6', where the quoted error corresponds to 1
standard deviation, so about two-thirds of observations should lie within
that range. I have translated their values, given in degrees, to minutes.
This accords pretty well with Meeus' quoted result from that paper, that
"the refraction at the horizon fluctuates fluctuates by 0.3 degrees around a
mean value normally, and in some cases apparently much more".

What I have done next is to include only those sunset timings that have been
taken from altitudes of 120m. and below, thus excluding all the
high-mountain observations. And then, those remaining 83 observations give a
mean value of horizontal refraction of 31.0' +/- 4.9'. This is a
considerable reduction in scatter, which I surmise comes about for two
reasons. First, because there's now much less variation in the additional
length of light path, after skimming the horizon. And second, because nearly
all the light path is now over ocean, with little of its length over land,
with its more turbulent atmosphere.

There is an additional point to be made, for what it's worth, though I don't
wish to overstate it. A large part of that scatter is the result of a single
discordant observation, at 40m. altitude on Hawaii, which produced a
refraction of 67'. It's a dangerous business excluding data for no other
reason than that it seems out of line, especially when investigating
scatter. If we chose to exclude it, then the remaining 82 observations would
provide a mean refraction of 30.5' +/- 2.9', a big reduction in the scatter.
Not that I am suggesting we should do that. But it shows what an uncertain
business it is, trying to estimate scatter. And interestingly enough, it
corresponds rather well with what Greg Rudzinski wrote, back in [4628], as
follows- "Perform normal sight reduction and expect 6 minutes of arc
accuracy under normal weather conditions." (and to which I objected).

So, in the end, and with Bill Noyce's help, we seem to have got some useful
information from our scrutiny of that paper, in exposing some serious flaws,
and in correcting our (well, my) overestimate of refraction errors at the
horizon, at sea. Where did it all start?  "40South" asked- "Supposing one
was in a small boat with an accurate timepiece and the necessary tables, how
accurate could you determine your longitude by observing the rising or
setting of the sun or any other celestial body? " We seem to have progressed
well past that point, but thanks to 40South for triggering us off.

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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