Atmospheric refraction

Remarks

Since the refractive index of air decreases with height, rays bend towards the ground and $R \ge 0$. Refraction changes the altitude and leaves the azimuth unchanged. For a plane-parallel atmosphere, Snell's law gives $n_0 \sin z = \sin z_0$, where $z = 90^\circ - a$ and $n_0$ is the index of air at the ground. This depends only on the air at the ground, so for small zenith distances $R \approx (n_0 - 1)\tan z$. At standard pressure and temperature this is about $58''\tan z$. Example: at the horizon $R$ is about $34'$, more than the apparent radius of the Sun. Non-example: the twinkling of stars is a fast random fluctuation of the refraction, called scintillation, not the mean refraction $R$.