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Dan Allen, you wrote:
"Are you saying that  this approximation is sufficient to reproduce the
tables, or are you saying  that this is exactly equivalent to the
tables?

The latter. And  it enables us to go beyond the tables and assess, somewhat,
their  accuracy.

"What about all of the layers of the atmosphere and  Snell's
law?  Can Euclidian geometry and a simple linear factor really  be
sufficient?"

Yep. For the stuff involving so-called "terrestrial  refraction" (namely the
items in the subject line above, plus anything else you  can measure with a
sextant for coastal navigation and excluding standard  refraction of stars),
then this simple approach covers it all quite nicely. The  structure of the
layers of the atmosphere lying close to the ground plus the  standard laws of
refraction (not so much Snell's law per se) conspire to curve  light rays downward
at a rate that is directly proportional to the angular  distance traveled as
measured from the center of the Earth. That is, if I fire a  beam of light
horizontally (or even at some significant angle away from  horizontal) from my
apartment in Chicago, when it reaches an observer in Gary,  Indiana 25 miles way,
its direction will have rotated downward, away from a  straight line
trajectory, by an angle that is directly proportional to the  distance traveled. On
average, the constant of proportionality is about 0.15  minutes of arc per
nautical mile. Note that this is really a dimensionless  result: it's 0.15
arcminutes bending per 1.0 arcminute traveled as measured from  the center of the
Earth. Now, if something causes all light rays to curve  downward in this direct
proportionality fashion, then it is necessarily  equivalent to changing the
radius of the Earth and pretending that refraction  does not exist. The easiest
way to see this is to imagine the case where the  gradient of atmospheric
density is 7x higher than normal. In that case, light  rays are bent downward at a
rate of 1.0 --they bend towards the Earth's surface  1.0 arcminutes for every
1 nautical mile traveled. In other words, a horizontal  ray maintains constant
height above ground, just as if the Earth were flat as a  board. This
condition is rare, but it does happen. But this approach applies in  all cases, not
just this special case. Naturally, it has limitations, but it's  surprising how
well it works.

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

   

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