There's not much to it. Here's the differential triangle:

The element of path length **ds** along the ray
corresponds to the increment **dR**
in distance from the center of the Earth, at **C**. The local zenith
distance at a distance **R** from the center is **z**; at
**R + dR**, it's
**z + dz**.
But because the local vertical at the latter point has been rotated by
**d**θ (the angle at the center of the Earth, between the two
local verticals), the differential of refraction **dr** is not just
**dz**, but

so that

From the little differential triangle, we see that

so

We will shortly need **dR/R** (see below), so let's rearrange this to
give

Now, the refractive invariant is

which is a constant. So we can set its logarithmic differential equal to zero:

or

Now plug in the expression found earlier for **dR/R**, and use the fact
that **dθ = dr − dz**:

or, cancelling the two **dz** terms,

so that

(Note that there is a minus sign because **n** decreases as **R**
increases, going outward from the observer toward space.)
This is the most useful form of the refraction differential. It shows how
extremely sensitive the refraction is to the local zenith distance: where
the ray is horizontal, **(tan z)** becomes infinite.

If you want to see how the integration is actually carried out, see the integration page, which contains plots of the integrand for several different zenith distances.

Copyright © 2003 – 2006, 2012 Andrew T. Young

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