Some theorems on "arctic mirages": hillingar effect and superior mirages

In a stratified or a spherically symmetric atmosphere, we consider some cases where a curved light ray connects two points, $A$A and $S$S, on the sea or the ground, at the same height as in some cases of superior mirages or "arctic mirages." We study the range $D$D of this ray (i.e., the geodesic distance between $A$A and $S$S) as a function of ${\alpha _S}$α_S, the ray inclination with respect to the vertical at $A$A or $S$S. The fundamental expression of $D$D is an improper integral. We show that a change of variable, also used to calculate the astronomical refraction, usually gives a well-behaved integral. In a spherically symmetric atmosphere, this happens when the refraction coefficient exceeds unity on the entire ray, producing a hillingar effect (alias strong looming for an erect image) or superior mirages. We then find a nice expression of ${{{\rm d}D}/{{\rm d}{\alpha _S}}}$dD/dα_S, which is usually negative. Inverted images (i.e., mirages in the strict sense) show up when ${{{\rm d}D}/{{\rm d}{\alpha _S}}}$dD/dα_S is positive; for this, we find a necessary condition in each atmospheric model. The strength of these results comes from not requiring precise knowledge of the ray paths.