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Bill B, you wrote:
"My questions. If I recall the GP would be approximately 5400 nautical miles 
(nm) away when I see Sun set (using appropriate eye protection of course). In 
my mental picture, I am not wrapping my vision along the surface of the 
Earth. I have a line of sight from my eye position to the Sun, which will run 
out of atmosphere *long* before 5400 nm "

Aha. A good little technical point there, Bill! As Christian noted in a later 
message, I wasn't taking his wording too literally. I was merely agreeing 
that if we had detailed observational information on the state of the 
atmosphere between us and the object we're viewing, then the problem is 
straight-forward physics: a numerical integration of the differential 
equations for the law of refraction in a continuous medium. It's solvable 
with enough data.

But you're right in pointing out that the points we would require for this 
integration do not extend to the GP of the Sun but rather they extend along a 
line inclining up through the atmosphere. The details in the high atmosphere 
don't much affect the refraction so let's suppose we stop when we reach an 
altitude of 40,000 feet. So if I am standing on the roof of a building 100 
feet above sea level, and I look out and see the lower limb of the Sun just 
on the horizon (sea horizon), how many "weather balloons" (observation 
points) would I need? Just for fun, let's suppose I place a detector every 
500 feet between me and the top of the troposphere (around 40,000 feet) all 
the way out towards the lower limb of the Sun. From my location to the 
horizon is a distance of about 10n.m. (sqrt(ht in feet)). From that spot, 
continuing on a straight ray to an altitude of 40,000 feet is about 200 n.m. 
(sqrt again). So that's 210 miles or, doing the math, around 2500 detectors 
or "weather balloons". We're very lucky that all of that detail is irrelevant 
to most refraction. For altitudes above 3 or at worst 5 degrees, nothing 
matters except the temperature and pressure right at the observer's location 
at the bottom of the atmosphere. It's worth remembering that this fortunate 
circumstance is what made celestial navigation possible. If we were dependent 
on the observation of altitudes very close to the horizon, the accuracy of 
celestial navigation would be ten times worse and sometimes considerably 
worse than that.

-FER




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