In the 17th Century, everyone assumed that the atmosphere terminated
abruptly in a smooth surface where it met “the æther,” just like
the smooth surface where still water meets the “subtler” medium of air.
(The æther was supposed to be subtler still than air; so, why not?
The gas laws were not known.)
This picture of a homogeneous layer of air surrounding the Earth
goes back at least to a work on optics attributed to
**Ptolemy**,
which says that “at the surface between air and ether, there is a
refraction of the visual ray according
to the difference in density between these two media.”

This view had been explicitly adopted by
**Kepler**
in his failed attempt to understand refraction, and it was adopted by
Giovanni Domenico Cassini as well.

Unlike Kepler, Cassini knew the sine law of refraction. And it was a simple exercise in elementary trigonometry for him to apply it to the uniform atmosphere, thus making the first quantitative model for astronomical refraction.

Cassini's model is so simple that anyone who had trigonometry in high school can follow it. Here's how it works:

In the diagram at the left, the heavy arc represents the surface of the
Earth, with radius R and center at **C**, and the thinner arc at a
height h above it is the surface of the (uniform) atmosphere.
The observer at **O** sees a star (off to the right somewhere) at
apparent zenith distance z_{0}, in the direction of the dashed
line.

The ray OP from the star entered the atmosphere at the point **P**.
Because the local vertical there is in the direction from **C** to
**P**, which is different from the local vertical through **O**,
the ray meets the vertical at **P** at an angle z_{1}, which is
less than z_{0}.

Because of the refraction that occurs at **P**, the ray direction
outside the atmosphere is different: the ray meets the local vertical
there at an angle z_{2},
which is bigger than z_{1}.
If the refractive index of air is n, the law of refraction tells us that

So much for the law of refraction. The nice thing is that we can
calculate the sine of z_{1} from the sine of the observed zenith
distance, z_{0} .
If you recall the Law of Sines for plane triangles, and apply it
to the triangle OPC, you'll see that

But the sine of the angle COP is the same as the sine of its
supplement, z_{0} . So we have:

or

Now that we have sin z_{1} , the law of refraction gives us
sin z_{2} :

and consequently

But (as you can see from the diagram above) the refraction, r, is just the difference of these two angles:

So, because we already had an expression for sin z_{1} a few lines
above:

That's it! We're done. This is Cassini's refraction model.

Copyright © 2003 – 2006, 2012 Andrew T. Young

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