Some natural phenomena due to Sunlight

Two everyday optical illusions caused by sunlight interacting with air and water are explained here using tools already developed in this chapter.

MIRAGE: on a hot, clear, sunny day along a level road, a shimmering 'pool of water' seems to appear some distance ahead, yet it recedes and vanishes as one approaches (only to seem to reappear further on). The physical reason: air in direct contact with the hot road is the hottest layer, and air gets progressively cooler with increasing height above it, so the refractive index of air INCREASES with height. A ray leaving the TOP of a distant object bends more and more toward the horizontal as it descends through this increasingly hot (lower refractive index) air near the road surface, and, for some further reason, then curves back UPWARD once it can bend no further -- so, to an observer, the ray appears to arrive from BELOW the road surface, exactly mimicking the geometry of reflection off an imaginary water surface at that spot. (Possible physical mechanisms for the actual upward bending very near the road include glancing-incidence reflection, unsteady convective motion of the hot air, or a rigorous treatment via Maxwell's equations for electromagnetic waves; true total internal reflection is NEVER involved in a mirage, since the relative refractive index between adjacent air layers is only just below 1, making the corresponding 'critical angle' approach $90^\circ$ rather than staying at some smaller, achievable value.)

RAINBOW: seven observed facts together define what any explanation must account for -- it is seen during rain, on the side of the sky OPPOSITE the Sun; it is visible only during mornings and evenings, never throughout the day; the common, PRIMARY rainbow shows red on its OUTER arch and violet on its inner arch; the rarer, fainter, concentric SECONDARY rainbow reverses this order (violet outer, red inner); it always appears as an arc of a circle; a COMPLETE circular rainbow can be seen only from a sufficiently high vantage point (such as an aeroplane); and true total internal reflection is never involved, only ordinary (partial) internal reflection.

A PRIMARY rainbow forms when sunlight enters the UPPER portion of a spherical raindrop at some angle of incidence, refracts and disperses into its constituent colours, undergoes exactly ONE internal reflection (ordinary, NOT total, since the relevant angle never reaches the water-air critical angle) off the drop's far inner surface, and then refracts a second time on finally exiting -- with the two extreme colours, violet and red, emerging inclined to the horizontal at about $41^\circ$ and $43^\circ$ respectively, so red appears on the OUTER edge of the visible bow and violet on the inner edge.

A SECONDARY rainbow instead forms from sunlight entering the LOWER portion of a raindrop and undergoing TWO successive internal reflections before finally emerging, at larger angles of about $51^\circ$ (red) and $53^\circ$ (violet) from the horizontal -- reversing the colour order (violet now on the outer edge, red on the inner) and appearing at a wider angle than, hence outside, the primary bow; it is also fainter, since two internal reflections lose more light at each bounce than one.

Why TIR is never involved: because the angle of incidence in air can never exceed $90^\circ$, the corresponding angle of refraction inside the drop (by Snell's law) always stays comfortably below the water-air critical angle ($\approx48^\circ35'$), so the internal reflection at the back of the drop is always PARTIAL, never total. …