Why Is That Rainbow White? The Droplets Are Too Small for Colour

Light entering a spherical raindrop is bent on the way in, reflected from the back surface and bent again on the way out, emerging concentrated at an angle of roughly forty two degrees from the direction it came. Because the amount of bending depends slightly on wavelength, each colour emerges at a marginally different angle, and those angles are separated enough to be seen as distinct bands. That separation depends on the light behaving as though it travels in straight rays within the drop, which holds when the drop is large compared with the wavelength of the light.

Fog droplets are far smaller and the physics changes:

• •Raindrops are around a millimetre across, fog droplets a hundredth of that
• •At that size the light spreads out rather than following a ray path
• •Each colour is smeared across a range of angles instead of one
• •The smeared bands overlap and add back up to white
• •The whole arc becomes broader and fainter than a rainbow
• •Faint colour survives at the edges, reddish outside and bluish inside

The conditions are specific and are the reason most people have never seen one. There must be fog or thin low cloud in front of the observer, the sun must be behind the observer and fairly low, and crucially the fog must be thin enough for sunlight to reach the droplets, which means standing at the edge of a fog bank rather than inside it. Hilltops above a valley filled with fog are ideal, as are coastal mornings and the tops of mountains. Pilots and mountaineers see them frequently. The arc appears in the same place a rainbow would, centred on the point directly opposite the sun.

Sailors have a name of their own for these arcs and their accounts are the oldest descriptions available. Sea fog off cold currents produces them regularly, and the pale arc standing over the water with the low sun behind was familiar enough to acquire the names sea-dog and white rainbow in English maritime usage, with equivalents in other languages. The arcs are frequently reported as brighter over water than over land, which fits the fog there being thinner and more uniform. Aircraft and ship reports remain one of the better sources of observations, since the conditions required are commonest over sea and at altitude.

Several other pale arcs and rings come from the same family of effects and are worth distinguishing. A glory is a set of coloured rings centred exactly on the shadow of the observer's own head, seen against cloud from an aircraft or a mountain, and arises from a different interaction with droplets of similar size. A brocken spectre is the enormously magnified shadow of the observer cast onto fog, frequently ringed by a glory, and the apparent size is an illusion of distance rather than any magnification. Ice crystals rather than droplets produce a quite separate family of halos, arcs and sun dogs, which sit at different angles and can be told apart by position alone.

Light bent and reflected inside a droplet emerges at about forty two degrees, with each colour at a slightly different angle, which is what makes a rainbow. In fog the droplets are a hundred times smaller, so light spreads rather than following a ray path, each colour smears across many angles and the overlapping bands add back to white. Seeing one requires standing at the edge of the fog with the sun low behind.