At the End of the Rainbow

A few years ago three generations of my clan were strolling along the sea wall between Whitstable and Herne Bay on the north Kent coast, which looks a lot like the view in this painting by my talented wife …

Seeing a rainbow ahead, with one end seemingly touching the coast, my eldest grandson took off. Deaf to all calls from the adults-of-limited-imagination, he just kept on going; the only option was to follow, which I duly took it upon myself to do. I finally caught up with him after a few hundred metres, disgracefully out of breath, and persuaded him to stop and to tell me where it was he wanted to go. The answer was beautifully simple: to get to the gold at the end of the rainbow just as he’d seen cartoon characters do on TV. It ended in tears, lots of them. Only time will tell whether this example of reality’s disappointments will leave a long-term legacy in his mind; what I do know is that it near-traumatised me and I am unlikely ever to forget the event. However, on a more upbeat note, I recalled this ‘trauma’ only after deciding I’d like to write something about rainbows – and that decision arose from taking part in a recent Teachers’ Tea Party at a local art gallery on the theme of colour.

This photograph of a double rainbow at the Helford river estuary in Cornwall (UK) comes from fellow scientist Jon Butterworth – used here with his kind permission. (Jon and I used to work together whilst we both served on the Science & Technology Facility Council’s Science Board – for my appraisal of this and other committees I have known and loved/loathed, see here. He writes excellent and highly readable blogs on physics for The Guardian .) In the post associated with his photo’ (here) Jon covers some of the fascinating physics behind this beautiful natural phenomenon. It’s all about sunlight reflecting from and refracting within raindrops and emerging towards the eye at a specific, fixed angle. He’s also very careful to point out that what looks like it might be the rainbow’s reflection in the river is actually nothing of the sort; it’s associated with light coming from a different set of raindrops in the sky which happens to arrive at the water’s surface at precisely the right angle to give the effect we see. In this post I want to re-cover only a little of the area covered so expertly by Jon, I’m more interested in addressing one or two other aspects of rainbows.

As I hinted above, it’s motivated and inspired by a modest recent collaboration with the Turner Contemporary gallery in which I shared with local primary school teachers a few ideas on colour, inspired by an exhibition of work by JMW Turner. I wrote about this project here . Rainbows, as a manifestation of the spectrum of colours within visible sunlight, figured within the event only in passing since I needed some ‘show-and-tell’ that wasn’t weather-dependant and chose to rely on the use of the spectrum available from a prism instead. However, Turner himself was as fascinated by the rainbow as anyone – even if his depictions are often more emotionally suggestive than ‘accurate’ renditions, e.g. here – so the thought occurs to me that one might indeed say a little more on the matter. Specifically, why was it that my grandson could never, ever have reached ‘the end of the rainbow’ and what does it all have to do with the speed of light?

There is never truly a ‘beginning’ to a topic in science like this, but we’ll start with a description of an experiment I did many years ago using a rather basic item of laboratory equipment called a spectrometer. In essence, after a lot of trial-and-error, I managed to create the rainbow’s spectrum of colours from a single droplet of water which was suspended from the tip of a glass pipette. My intention was to explore the possible use of this setup as an aid to teaching, but it was far too tricky to get the conditions ‘just right’ for it to be a practicable proposition; so, whilst being pleased with myself for getting the hoped-for result, there was never a follow-up. Equally unfortunately given the topic of this post, I didn’t have the foresight to photograph the results – this was pre-digital remember, so it would have taken a little more effort than nowadays – and I no longer have access to such kit, so a diagram will have to suffice. The spectrometer is a simple, yet useful device: a source of light is shone into a collimator one side, which allows one to control the size and shape of the light beam that falls onto whatever sample is at the centre of the turntable; the light that emerges from our sample can be viewed using the telescope. Key to such experiments is the ability accurately to measure the angle of the telescope as it views this emerging light.

Hopefully, my simplistic diagram below will help me to describe what is happening inside the water droplet – and by implication within each and every raindrop giving rise to a full-blown natural rainbow. The sunlight, depicted as yellow in this diagram (top left) but in practice a mixture of all the visible colours* (and a great many others beside – like infra-red, ultra violet and so on), strikes the droplet. Some of the light is reflected of course, we couldn’t see the raindrop at all otherwise, but some of the light is transmitted into the water and this is where it starts to get interesting. When light enters a transparent material it gets refracted – ‘bent’ – and it turns out that different colours of light get refracted by different amounts: the red end of the spectrum less than orange, which is refracted less than yellow, which is refracted less than … violet. There’s a lot of physics behind this, but the essence of the matter is that, whilst the speed of all types light is the same everywhere in a vacuum (and almost the same in a gas, like air) it changes when it’s travelling through a material, and the degree to which it changes depends on the colour (technically, the wavelength) of the light. Our sunlight, now with its constituent colours dispersing due to the effects of refraction in the water, then reaches the far side of the droplet. As before, some of this light may leave the raindrop, but a significant amount may be reflected from the inner surface layer of water as though from a mirror. This reflected light now travels back across the droplet and may leave the water,# being refracted one more time as the slightly different speeds of the various colours once more become near-identical back in the air. This whole glorious process leaves us with the original sunlight now dispersed – ‘split’ – into its various colours.

Convinced? Sadly, there’s a need to complicate things for you at this point since, accurate though my description is in the context of a single droplet, the real thing arises from the combined optical effects of countless thousands of raindrops. The problem we now face is that our raindrops may be at altitudes of 1000-2000 m and this means that if we’re seeing the red end of the rainbow emitted by a given raindrop at our eye then the violet light from that drop will be travelling past us several tens of metres above our head. (Feel free to check my trigonometry, but the 2º gap between the direction of the red light and the violet leads to quite a separation over such distances.) The violet light we see must therefore come from other raindrops at a lower altitude. The existence of a rainbow just became a lot more remarkable. Not only do we need the properties of an approximately spherical droplet of water outlined above, but we need our tens of thousands of raindrops to exist in a cloud to permit violet light to reach the eye from raindrops towards the bottom and for the other colours, through to red, to reach the eye from drops higher up.

But why a bow – why the arc in the sky? Well, that follows directly from the precise angles we’ve already introduced above: for instance, the 40º angle between the sun’s rays and the violet light reaching our eye is all that is needed, and that can be satisfied all the way from the horizon on our left to the horizon on our right as long as we have raindrops at the appropriate altitude and distance. The raindrops emit the refracted light as a cone shape, not as a single line as most diagrams would depict it; we see only a part of the circular end of the cone either because the cloud isn’t extensive enough or, quite simply, the ground gets in the way.

We’ve got to the point of being able to say why my grandson could never have reached the end of the rainbow he chased along the coast. As he ran towards the rainbow he had first spotted, the angle between the raindrops in the cloud and the Sun remained almost the same for the duration; however, he was continually changing the angle between his eyes and those raindrops. As a consequence, that first rainbow had in reality disappeared as he ran, to be replaced by a new one wherein the necessary precise angles were again formed. The ‘end of the rainbow’ had therefore moved away from him at precisely the speed he was running. In the diagram above I have tried to illustrate this in the case of the red part of the rainbow: the required angle of 42 degrees for him to see the red light can only be maintained if it originates in raindrops at precisely the right distance from his eyes. Hopefully, one day, the beauty of the rainbow and of the physics behind its origins will be enough to overcome the disappointment of his failure to find that pot of gold … perhaps my memory of the traumatic incident will dissipate at the same time ;-)


Footnotes
*. The sun emits light of a great many wavelengths (colours), but the peak in its emissions happens to fall in the yellow part of the visible spectrum: that is why it ‘looks’ yellow to us. Other stars peak at different wavelengths; astronomers are familiar with red dwarf stars and blue stars – some are brightest in the infra-red or in the ultra violet wavelengths. It’s all to do with the surface temperature of the star; the physics behind this is not at all dissimilar to the way one estimates the temperature of a glowing metal: dull red, through orange, yellow and to ‘white hot’. Yellow corresponds to a surface temperature of around 5,500ºC.
#. Some of the light may actually be reflected for a second time and then leave the droplet at another point after further refraction. This is what gives rise to secondary rainbows, and it explains why the order of the colours is reversed in the secondary bow (see here or here or here).