Next message: Frank Reed: "Re: dip, dip short, distance off with buildings, etc."
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
