Why the rainbow sits at 42°
Drag a ray into a raindrop and watch sunlight bunch up at one turning angle, which is where the rainbow's edge appears.
Made by Claude, an AI model. It follows rays through a perfect sphere of water; the checks further down compare it with the closed-form answer from Descartes.
Controls
Inside the drop
How far each ray turns
What you see from the ground
Why an edge, and why there
A ray that enters at height b turns by a total angle D before it leaves the drop. Near the entry height where D is smallest, the curve is flat, so a whole bundle of neighbouring rays leaves at almost the same angle and adds up. Rays with a smaller D do not exist, so light stops abruptly there: that is the edge of the bow. Water bends violet more than red, so each colour has its own minimum and the edge is spread into a spectrum.
Setting dD/di = 0 in Snell's law gives Descartes' closed form, cos² i = (n² − 1) / (k (k + 2)), where k counts the reflections inside ( for the primary bow, for the secondary). The page does not use this formula to draw anything in the drop picture: it traces each ray with vectors and Snell's law, and the closed form is only the answer the scan is held against.
The same formula says when there is nothing to find. If the drop does not bend light at all, n² − 1 is not positive, D only falls as the ray moves outward, and there is no minimum and no bow. Use the button to see that.
The page checks itself
Fault drills
Refractive index of water against a second source
What this does not show
- Brightness. Rays are counted, not weighted, and the bunching count is a count of the drawn rays only.
- Wave effects. Light is treated as rays, so the faint extra bands inside the primary bow, and the blurring of the edges and of the dark band, are missing. Light reflected straight off the outside of the drop still falls in the dark band, so it is darker, not black.
- Real rain: drops are modelled as perfect spheres of equal size, and the Sun is a point.
- Polarisation, and the sky's own light.
- The history in the next paragraph is from memory and was not checked here.
Descartes published the closed form in 1637; the dark band is named for Alexander of Aphrodisias, who described it in antiquity. Refractive indices: Daimon and Masumura (2007) Sellmeier fit for water at 20.0 °C, checked against the Hale and Querry (1973) table for water at 25 °C. Both were read from the refractiveindex.info database (public domain, CC0).