BAA / RMetS Joint Meeting, 2004 November 27

Rainbows

Mr Nayler remarked that of all meteorological phenomena, rainbows attracted perhaps the most interest. As well as scientists, they had also caught the imagination of many poets and philosophers over the centuries. From an astronomical prospective, Mr Nayler made the curious assertion, to be justified later, that whilst clouds were observed on many other planets and Moons in the solar system, with the possible exception of Titan, the same was not true of rainbows. Over the next half hour, he explained, he would be taking a historical approach, examining how humanity had come to understand their formation.

The first truly scientific study had been that of Aristotle, nearly 2500 years ago, but there were many basic facts which had been known since prehistoric times. A rainbow is an arc of many different colours, red on the outside, blue on the inside. The outside edge of the bow tends to be more brilliant than the inner edge. Sometimes a secondary bow can be seen just outside the first, with colours reversed. In the gap between the two bows, the sky is often noticeably darker than elsewhere, giving rise to the name Alexander's Dark Band, after Alexander of Aphrodisias who first noted the phenomenon in 200 AD. It had also been known since prehistoric times that certain conditions were necessary for a rainbow to be seen: the Sun had to be behind the observer, and there had to be clouds and rain in front. Finally, the speaker noted that rainbows were personal phenomena: the position of the bow was determined by the line between the observer's eye and the Sun behind him, and so everyone saw their own bow. And, contrary to popular myth, it was impossible to see a bow directly above oneself.

Aristotle's attempt to explain the formation of rainbows, around 350 BC, had been somewhat hampered by the untenable theory of light and colour within which he was trying to cast his theory: he essentially thought the eye sent out a ray, which interacted with the Sun's radiation. He had, however, been able to explain the circular shape of the bow by asserting that for all points on it, the distance to the observers' eye was in some constant ratio to the distance to the Sun. Despite the flawed theories upon which it was based, this work was to stand essentially unchallenged for nearly two millennia until the 17th century.

Aristotle had thought it was the entire cloud which gave rise to the bow. It was not until 1304 that it had first been proposed, by German monk Theodoric of Freiberg, that it was refraction within water droplets that caused the bow, though sadly this work was to be forgotten, and only uncovered in the 1830s. He backed up his argument by demonstrating the refraction of light through a circular flask of water. In 1266, Roger Bacon had also contributed to the field, by observing that the angular diameter of rainbows was always the same: 42° for the primary bow, and 51° for the secondary. Nowadays, it seemed curious that no one appeared to have made this measurement earlier.

However, the father of the modern understanding of rainbows was René Descartes, who in 1637 used his law of refraction, now known as Snell's Law in honour of its independent discoverer, to trace the paths of rays through spherical water droplets. Whilst most of the light would pass straight through the droplet, he realised a small proportion would undergo total internal reflection from the back surface of the droplet. He considered different rays of sunlight, entering the droplet at different angles to the surface, and calculated the angle at which they left the drop. It was found that all of these rays left the droplet going back in the direction they had come, at a slight angle to the original direction of travel. He further found that there was a maximum deflection from the original direction of 42°, and that at this angle there was a bunching together of the light rays – that is to say, many rays, having initially struck the droplet at different angles of incidence, all left it with nearly the same deflection. Rays at this angle, now known as Cartesian or caustic rays, added together to give the bright bow, circling 42° around the line connecting the Sun to the observer's eye. The formation of secondary bows was entirely analogous, but with two reflections within the drop, giving rise to a maximum deflection of 51°. The darkness of Alexander's Dark Band could also be explained: because no rays were deflected further than the caustic ray, none of the water droplets in the part of the sky outside the bow deflected light into the observer's eye.

However, the speaker remarked that Descartes' work still did not explain the colouration of the rainbow: it was monochromatic. This problem would have to wait until Newton's treatment in 1667. Around this time, Newton spent 18-months at Woolsthorphe, fleeing the plague, during which time he had realised that white light was not, as previously thought, pure. It was coloured light which was pure: white light was an impure mixture of colours. Whilst he had no particular interest in the phenomenon of rainbows, it provided a remarkably good test of his new theory. In particular, he realised that if the refractive index of the water droplets varied for different colours, as indeed he had experimentally found it to, the 42° radius of the rainbow would also be different for each colour, causing the observed splitting of the component colours in the Sun's light.

To close, the speaker remarked that it was theoretically possible to observe not just arcs of rainbows, but complete circular bows. To demonstrate this, he showed a rare instance of such an image, taken in Hawaii, with a complete circular primary bow, and near complete secondary. He remarked that this could not normally be observed, as the horizon normally truncated the bow. He also remarked that when the Sun was at an altitude above 42° in the sky, the entirety of the bow would be below the horizon, and so at most latitudes, no rainbows could be seen in the middle of the day in the summer months. Finally, in explanation of his initial assertion that no other planet had rainbows, he remarked that Descartes' calculation of the angular size of the rainbow depended finely upon the refractive index of water. For different refractive indexes, the resultant bows were of different sizes, as the caustic bunching of rays appeared at different deflections. But for refractive indices greater than around two, it was found that there was no caustic ray, and hence no rainbows. Therefore, on planets where the only droplets in the atmosphere would be of organic compounds, there would be no possibility of rainbows.

Following the applause for Mr Nayler's talk, Prof Collier introduced Mr Martin Mobberley to present his regular round-up of amateur astronomical news.