Thermal Mirage

Thermal Mirage is never where you look. They are where the light came from.

On a hot day the road heats the thin layer of air lying on it, and hot air is less dense than the cooler air above. Light is slightly slower in air than in vacuum: the refractive index of air is about 1.0003, and that extra 0.0003 shrinks as the air thins. The second formula on this page, the Gladstone–Dale relation, says exactly this: n − 1 is proportional to the density, so at fixed pressure it falls as 1/T. At standard temperature and pressure n − 1 ≈ 2.9×10⁻⁴.

Now follow a ray of skylight heading down toward the road at a shallow angle. Each layer it enters is hotter and faster than the last, and Snell's law, which for flat layers becomes the first formula, n cos α = constant (α is the ray's angle above the horizontal), tips the ray a little flatter at every step. If it starts shallow enough, it never reaches the road. It curves through a lowest point and climbs back up, into your eye. Your brain traces the ray back in a straight line to a spot on the road, and finds a piece of sky there. Bright, bluish, shimmering, flat: it reads as water.

The numbers are small. Between air at 15 °C and air at 50 °C, the refractive index changes by about three parts in a hundred thousand. The third formula turns that into an angle: only rays flatter than α_max ≈ √(2Δn) are turned back. For Δn ≈ 2.5×10⁻⁵, a temperature difference of about 30 K, √(5×10⁻⁵) ≈ 0.007 rad ≈ 0.4°: less than half a degree. This is why the puddle always lies far down the road and retreats as you drive toward it. The shimmer is turbulence: rising hot air keeps changing the bending.

Feynman explains this mirage in Chapter 26 of the first volume of his Lectures, using Fermat's principle: light takes the path of least time, and over a hot road the quickest route runs low.

Thermal Mirage never lies. They report the sky, from the wrong address.