# Rainbow Physics Simulator - Raindrop Rays, 42° Angle, Alexander's Band

> Trace light through a raindrop, get the 42° primary and 51° secondary bows from water's dispersion, render a double rainbow, see polarization and halos.

- URL: https://qilingtools.com/en/physics/rainbow
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Drag the incoming ray across a raindrop and watch it refract in, reflect once (primary) or twice (secondary) and refract out. The minimum of the deviation curve is the rainbow angle, found wavelength by wavelength from water's dispersion: about 42.4° for red and 40.6° for violet. The sky view renders the double bow, the dark band and the brighter sky inside the primary around the antisolar point; raise the Sun above 42° and the primary sinks below the horizon. Fresnel brightness and polarization, the 22° halo, glory and fogbow are included too.

## How to use

1. On Ray in a raindrop, choose primary or secondary, drag the canvas or slider to set the ray height b, and switch between a single ray and a parallel beam
2. Compare with the deviation curve to see how the yellow Descartes ray marks the minimum deviation (rainbow angle)
3. On Rainbow in the sky, change the Sun elevation and watch the primary, secondary, dark band and colour order
4. On the brightness and halo tabs, read the energy and polarization of each order and the 22° and 46° halos

## Features

- Sellmeier dispersion for water; primary and secondary angles per wavelength
- Analytic cos i = √((n²−1)/(k(k+2))) cross-checked by numerical minimization
- Single-ray and parallel-beam (caustic) views, with white light shown in many colours at once
- Sky rendering from the solar spectrum and CIE colour matching, with Alexander's dark band
- Fresnel s and p reflectances give the brightness and polarization of each order
- Ice-crystal 22° and 46° halos, glory and fogbow (Airy broadening)

## FAQ

### Why is a rainbow always about 42° across?

The deviation D of a ray that refracts, reflects once and refracts out depends on where it hits the drop, and has a minimum. Near that minimum many rays leave at almost the same angle, so light piles up there. For water (n ≈ 1.333) the minimum deviation is about 138°, which is about 42° from the antisolar point (where your shadow's head points). So the rainbow is a circle of radius about 42° around the antisolar point.

### Why are the colours of the secondary bow reversed?

Light in the secondary bow reflects one extra time inside the drop, which flips the geometry: its minimum deviation is about 231°, or about 51° from the antisolar point. In the primary, red has the largest angle and sits outside; in the secondary, violet has the largest angle, so the secondary is violet outside and red inside. The extra reflection costs light, so it is only about 40% as bright.

### What is Alexander's dark band?

Primary light can only appear inside 42° (which is why the sky inside the primary is brighter), and secondary light only outside 51°. Between 42° and 51° there is neither, so the sky is noticeably darker. The band is named after Alexander of Aphrodisias, who described it around 200 AD.

### Why do you rarely see rainbows at midday?

A rainbow is centred on the antisolar point, which is as far below the horizon as the Sun is above it. With the Sun higher than about 42° the whole primary bow is below the horizon and invisible from the ground; the secondary disappears above about 51°. Rainbows therefore favour mornings and late afternoons, while from an aircraft you can see a full circle.

### Is rainbow light polarized?

Yes, by around 90% or more. For the primary Descartes ray the internal angle is about 40°, close to Brewster's angle (about 37°), so p-polarized light mostly leaks out and the reflected light is s-polarized, vibrating along the arc. Look through polarized sunglasses and tilt your head and part of the bow disappears.

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