

Estimating Contrail Height
Name: Joe
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
Early in the morning I see the contrails of airplanes just above
the horizon. How far away is the airplane? For this problem I assume that the
airplane is flying at 30,000 feet. From my viewpoint the angle from the horizon
is 30 degrees. To me the angle appears to be 30 degrees but with the curvature
of earth, the angle will be greater. Is there an approximation I can use such
as: If the angle I see is 30 degrees the plane is 13 miles, If the angle I see
is 45 degrees the plane is 9 miles, etc.?
Replies:
Yes there are analyses of apparent altitude vs. angle of observation above the
horizon, taking into account the curvature of the Earth. However, things are
not so simple at first sight because the atmosphere "bends" the light and
another correction for the index of refraction of the atmosphere, in addition
to the Earth's curvature, is needed. The derivations require some trigonometry
which is too long to reproduce here, but you can find the details at the
following web sites:
http://tchester.org/sgm/analysis/peaks/how_to_get_view_params.html
http://tchester.org/sgm/analysis/peaks/refraction.html
The topic of "real" vs. "apparent" altitude is problem treated in many
contexts  meteorology, ballistic calculations, astronomy, and navigation 
just to mention a few. Each approaches the analysis a bit differently
depending upon the needs of each field, so there is no one simple formula,
that I am aware of.
Vince Calder
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Update: June 2012

