When the sun is either rising or setting and appears to be just on the
horizon, it is in fact below the horizon. The explanation for this seeming
paradox is that light from the sun bends slightly when entering the earth’s
atmosphere, as shown in Fig. \(\textbf{P33.51}\). Since our perception is based
on the idea that light travels in straight lines, we perceive the light to be
coming from an apparent position that is an angle \(\delta\) above the sun's
true position. (a) Make the simplifying assumptions that the atmosphere has
uniform density, and hence uniform index of refraction \(n\), and extends to a
height \(h\) above the earth's surface, at which point it abruptly stops. Show
that the angle \(\delta\) is given by $${ \delta = \mathrm{arcsin} ({{nR}\over{R
+ h}}}) - \mathrm{arcsin}({{{R}\over R + h}}) $$ where \(R\) = 6378 km is the
radius of the earth. (b) Calculate \(\delta\) using \(n\) = 1.0003 and \(h\) = 20
km. How does this compare to the angular radius of the sun, which is about one
quarter of a degree? (In actuality a light ray from the sun bends gradually,
not abruptly, since the density and refractive index of the atmosphere change
gradually with altitude.)