Hi someled

Here is my last attempt to convey to you that indeed, in the references I provided the occurrence of the radius of the earth, R, does not indicate that anybody suggests that the mechanism of the refraction of light is influenced by R. It also should become clear to you why it pops up in the equations in the references. If that seems to you a contradiction in terms than please study the following with care to see the contradiction vanish in thin air (pun intended).

For the following see my attached image, I have it available only on my home computer and didn't know how to imbed into this post.

This image is a my counterpart to Fig. 1 in Ref.[3]. As in that figure we have the cord-length S stretching from point 1 to 2 with point 3 in the middle. The circular arc going from point 1 to 2 represents a beam of light under influence of refraction with point 2 being the target and point 1 the observer. Line 1-4 is tangential to the circular arc at point 1 and points in the direction the observer at point 1 perceives the target to be. Again, Delta_beta is the angle of refraction. Perpendicular to this tagent is the radius line 1-0. By way of similarily of right triangles you can prove that the angle Delta-beta occurs again between the lines 0-1 an 0-3.

Some trigonometry :Or with extremely small errors (less than 1%) according to the info in my previous post :

Delta_beta = S/(2*r)

Now use this relationship to eliminate Delta_beta from Eq.(7) in Ref.[3]. The resulting equation is :

S/(2*r) = -S/2 * (dN/dh) * 1.e-6

I assumed that the cord-length, line 1-3, is running horizontally. Hence cos(beta) = 1

The factor S/2 appears on both sides of the equation and therefore cancels out.

1/r = - (dN/dh) * 1.e-6

This equation still does not contain R=6370km which is what you rightfully demand to be. So, how does R=6370km come into being in eq.(10) in Ref.[3] you might ask.

Well, here is what scientist often do. In order to converse easily among each other it is better to present r with respect to some reference length, let's call that L. One can argue endlessly as to how big the value of L might be. If all the involved scientists were to live in the UK one (usually it is the one who publishes first on the subject at hand) might suggest the distance between London and Glasgow. If subsequent publishers in the field like that choice because they think that distance is relevant to the refraction of light in the context of surveying the landscape or the building of large structures the original suggestion for L will pervail, otherwise somebody else comes up with a different idea.

I suggest that L = distance between London and Glasgow is really not a good idea but maybe using the radius of the earth might not be a bad choice for two reasons. a) Everybody knows at least a good approximate value of that ( 6370km or the equivalent in miles or whatever units one prefers). b) r in above equation often comes out to be in the neighborhood of thousands of kilometers under common atmospheric conditions.

Last step to get to Eq.(10) in Ref.[3].

Multiply my last equation by R on both sides :

R/r = - R*(dN/dh) * 1.e-6

and abbreviate : k = R/r to obtain eq.(10) in Ref.[3]. Bingo.

And this is my last post trying to explain to you that in all the references I provided the radius of earth is simply used as a convenient way to display and communicate the effect of refraction of light due to temperature gradient in the atmosphere. Thank you for your time. Zack

sin(Delta_beta) = S/(2*r)