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George H, you wrote:
"That's a perfectly  valid way to assess refraction, but not a particularly
new one."

Oh, of  course. Let's be clear -- all of this was worked out a very long time
ago.  Terrestrial refraction was a serious practical problem for terrestrial
survey  work and many people probably wrote about it (I've never looked into
it), and to  that extent, there is nothing new under the Sun! But it's
certainly new to this  list as far as I am aware.

Just about one year ago, there was a rather  long discussion where people
were trying to puzzle out the origin of the  equation underlying Bowditch's Table
XV. It didn't interest me at the time, but  it does now. No one seemed to
know how to derive it last year or, more  importantly, assess its limitations.
What I am saying is that this simple  technique of changing the curvature of the
Earth can be used to analyze not just  Table XV [see my reply to Bill] but
everything else where terrestrial refraction  impacts navigation. For example,
consider "anomalous dip". The primary source of  dip anomalies (though not the
only source) is the variation in the scale height  in the layer of the
atmophere close to the ocean which in turn changes the  effective curvature of the
Earth. If I calculate dip based on a lapse rate of  -6.5 deg per kilometer
(implying a scale height of about 10km), I find that the  dip at 5 meters height of
eye is 3.9 minutes of arc. If instead the lapse rate  is -34.1 deg/km, the
atmosphere has constant density and the dip is exactly  equal to the geometric
value which 4.3 minutes of arc. And if the lapse rate is  +25deg/km, then the
dip at 5 meters if 3.4 minutes of arc. Naturally a  calculation like this
assumes that the lapse rate is constant and the atmosphere  is more or less the
same on the whole path from horizon to observer (so it can't  handle really
exotic refraction, like mirages).

There's also a conceptual  aspect to this. Next time you're looking at a
distant ocean scene with a few  boats off in the distance, perhaps a lighthouse,
and beyond the horizon some low  hills of an island, ask yourself how the scene
would change if the Earth's  curvature were a little greater or a little
less. That's exactly what you would  see under conditions of variable refraction.
A temperature inversion would lift  the distant hills just as if the Earth was
nearly flat. Note that this sort of  analysis does not apply to distant
objects that are higher than a few hundred  meters (so no mountains) and it has the
same limitations that I mentioned above,  but it covers a very large portion
of the possible variation in  refraction.

Finally, the practical value of this is not that we can start  calculating
dip and all the rest based on the specific temperature profile  since, of
course, we don't usually have access to the temperature profile.  Instead it
provides a means of assessing possible errors arising from using the  tables
"naively", and it leaves open the option of calculating different  versions of the
tables when circumstances might require  them.

-FER
42.0N 87.7W, or 41.4N  72.1W.
www.HistoricalAtlas.com/lunars

   

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