Rainbow

The World of Atmospheric Optics – Part One
by John Flannery SDAS


My heart leaps up when I behold
A Rainbow in the sky:

-- The Rainbow by William Wordsworth

Painters, poets, and photographers have long tried to convey to us through pictures and prose the beauty of that delicate brushstroke of colour we term a rainbow. But rainbows are just one of the many classes of phenomena that fall into a grey area between meteorology and astronomy – the world of atmospheric optics.

Rainbows, along with the “ring” we sometimes see around the Moon, are our most familiar and frequent types of optical phenomena in the atmosphere but a new and refreshing appreciation of the daytime sky is revealed by becoming being aware of the many other classes that exist. Many of the complex and beautiful patterns result, in most cases, from the interplay of light with two simple substances – air and water.

Mythology
Early peoples must have marvelled at the sight of the mysterious arch of a rainbow spanning the heavens. Just as many celestial phenomena became part of their folklore, so too we find wonderful tales relating to rainbows.

To a tribe in South America the rainbow was a serpent who grew out of control and threatened Heaven and Earth. An army of millions of birds came to the rescue and pecked it to death. The serpent’s multicoloured blood flowed over its body and also dyed the plumage of the birds. Those birds that did not help remain, to this day, dark-coloured.

In Scandinavian cultures, the rainbow was a bridge between Heaven and Earth. This was the Bifrost of Norse mythology that lead from the realm of mortals (Midgard) to the abode of the gods (Asgard). The red of the rainbow was an eternal fire that protected Asgard against the giants (Jotnar). Bifrost is destroyed in the great battle Ragnarok at the end of time.

How rainbows form
Let us explore though a little of the physics of rainbows and their many forms so we become more aware of how to look for them.

The French philosopher Rene Descartes performed a simple experiment in 1637 to show how a rainbow was produced. However, he did not understand the nature of light and it was left to Isaac Newton to explain its composition thirty years later. Contrary to popular opinion, Newton did not see seven colours but five. He assumed that there were seven colours because it fitted in with the scheme of the Solar System at that time – Sun, Moon, and the five then known planets (excluding Earth). This magical number could also be identified with the seven knows of the music scale!

Basically, a rainbow is a reflection and refraction phenomenon. Sunlight shines on the raindrops and the light is reflected within the drops back towards the observer. It is also refracted because the light is passing from one medium to another – in this case, from water to air. The various wavelengths of the constituent colours of white light are refracted differently thus giving us a spectrum which we call a rainbow.

A primary rainbow is always centred around an imaginary point directly opposite to the Sun in the sky – the so-called anti-solar point. When the Sun is on the horizon, this point is on the opposite horizon, but as the Sun gains in altitude, this point is displaced below the horizon. A ray of sunlight is reflected and refracted within a raindrop, leaving at an angle of 42 degrees – this is the angle it subtends from the anti-solar point. From this, we can infer that for a primary rainbow to be visible the Sun must be less than 42 degrees above the horizon. A handy rule of thumb is the higher the Sun, the lower in the sky the rainbow appears. A rainbow at sunset arches high across the sky but may actually just display a range of reddish-pink hues, taking on the colours of the setting Sun.

If the beam of light undergoes two internal reflections in a water droplet it exits at an angle of 51 degrees to give a secondary, or second-order rainbow. If you see this bow you will see that the colour sequence is the reverse of that of the primary. In a primary bow, red is on the outside edge of the arch while in a secondary the same colour is on the inside edge.

I referred to rainbows as “primary” or “second-order” because you can get even higher-order bows occurring. People looked for these for centuries but without any success. It was when Edmund Halley computed the position of the third-order bow that a surprising result appeared; the third-order bow is 140 degrees from the anti-solar point. In other words, it is in the direction of the Sun and so people had been looking in the wrong place for this bow all along! The fourth-order bow also appears in a similar direction but a fifth-order bow is partially superimposed on the secondary bow. Third- and higher-order bows have rarely been observed in nature but some reliable sightings exist.

Each time the beam of light is reflected the intensity of the resultant bow is one-tenth that of the lesser-order one. Thus, a second-order bow is only one-tenth as bright as a primary rainbow. It’s an important point to remember and one reinforced when you see both rainbows.

The many aspects of rainbows
The strength of colours in a rainbow are directly related to the size of the water droplets reflecting and refracting light. If are the droplets are fine enough – such as in mist or fog – you may witness a ghostly arch around the anti-solar point. This is the fog bow, often seen looming ahead of you in mist as a car races by with headlights blazing. A strong streetlight can also serve as the light source to see fog bows.

Droplets between 1mm and 2mm in diameter give rise to the most vivid colours in a bow. Smaller than this and some of the colours are less pure. Droplets less than 1mm may also give rise to supernumerary arcs. These arcs are interference patterns created within the raindrops and cause colour overlap. Depending on how the rays mesh together, the interference can be constructive, in which case the rays produce a brightening, or destructive, in which case there is a reduction in brightness. The arcs appear at the outer and inner edges of the primary bow.

When you see a primary rainbow you may notice the sky inside the bow is brighter than the outside. This is because light reflected from a raindrop is scattered more towards the inside of the bow. In the secondary, light is scattered more towards the outside because of the two internal reflections. When the two bows appear together this preferential scattering gives rise to a dark strip of sky between the two bows that is termed Alexander’s Dark Band. The phenomenon is so-called after Alexander of Aphrodisias who first described it nearly two millennia ago. The stronger the sunlight and rain, the more intense the rainbow and the more likely you will see a prominent display of the dark band.

As you can see, rainbows come in many different forms but there are some even more fantastic aspects to rainbows.

What about reflected bows? Remember what was said earlier about the rainbow being centred on the anti-solar point? Well, if the light source is reflected in water, it has the effect of creating a second anti-solar point higher in the sky. If conditions are right a bow may form around this point – a reflection of a true rainbow. What a sight that must be.

Formed in the same way as the daytime bow, the light from the Moon cannot match the brilliance of the solar disk so a moon rainbow is but a pale relative of its daytime counterpart. While sometimes we can see some colour, the moon rainbow normally appears whitish; the colours are just too faint to distinguish properly except with a time-exposure photograph. It is because of this faintness that we normally only see a moon rainbow near time of Full Moon.

Rainbows in my rear-view mirror
Because a rainbow forms from water droplets falling as rain it means that the bow you see will be as distant as the rain shower. This leads to the intriguing question as to how close they can appear. I once saw part of a rainbow just 30 feet from me, projected against the white backdrop of a billboard. That meant the rain droplets were falling between me and the billboard with the Sun directly behind my back. A beautiful and personal sight as other commuters rushed by oblivious to the display.

We have all seen rainbows caused by spray watching the bow waves of a boat cutting through the water, standing near a waterfall, or creating our own rainbows with the garden hose. I witnessed a more recent spray rainbow driving home from Dublin to Nenagh before Christmas 2006. The road was wet after a recent rain shower and the Sun was sinking towards the western sky line. Glancing in my rear view mirror as I left Roscrea I noticed the spray being thrown up by my car was causing two separate splashes of colour. What a fantastic sight; it was almost as if the bows were racing me home! Obviously I couldn’t admire these for long with careful attention to what was ahead of me on the road being more important!

Finally, if you are an occasional hill-walker you may have seen your shadow projected against low cloud or a bank of fog. Look more closely and you may see a ring of light encircling the shadow. Sometime this circle may be coloured. This is a glory, or Brockenspectre, caused by the diffraction of light in water droplets. It can also be seen from an aircraft as it breaks through a layer of cloud after taking off.

We’ve covered a lot of ground in our discussion of bows but still only just scratched the surface in our study of their different forms. We did not cover coronas (not to be confused with the Sun’s corona visible during a total solar eclipse), aureoles, 360 degree bows, or iridescent cloud, to name some other phenomena. Let us not forget though what the true crock of gold is at the rainbow’s end; it’s what we experience as we watch all these magical sights that nature weaves, a source of riches that material wealth cannot match.