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How big's your bow ?

QUESTION: What determines the size of rainbow? They obviously vary as shown by double rainbows.

ANSWER 1 : The size of a rainbow is fixed by the way the Sun's rays go through the raindrops. When a light ray strikes a raindrop, part of it is reflected and lost and part is refracted into the drop. When this ray hits the back of the drop, part of it is refracted out and part is reflected back to the front surface. Part of this reflected ray is again reflected and part is refracted back out.

If the original ray hits near the centre, it will be deflected by 180o and return along the same path. This is how catseyes work, but you will never see sunlight reflected this way because of the shadow cast by your head.

But what happens if the original ray hits the raindrop off- centre? As the point of contact moves away from the centre it reaches a point where many rays return virtually in the same line, and reinforce each other to make a bright return at 41o from the sunline - the line from the Sun to the raindrop. These returns happen at all points around the sun- line, and combine to form a bright cone of angle 41o with its axis on the sun-line (see raindrop B in the diagram).

The light ray is split into its component wavelengths by the raindrop, and different colours are refracted by different amounts - red less, blue more. So the bright cone shows rainbow colours, with red on the outside.

If you look at a sunlit sky, full of raindrops, your eye will be on the surface of the bright cone of raindrops 41o from your antisunline - the line running from your eye to the top of your shadow on the ground. So you will see the rainbow as a circle that is 41o from the antisun- line, with the red on the outside. The original rays which hit the drop at the wrong place to form the rainbow will produce a very faint return, always less than 40o from the antisun-line, and so inside the rainbow. This makes the sky appear darker above the bow.

However, a secondary bow can form outside the primary. It is caused by a double reflection of rays striking raindrops. Some of the lost reflected light mentioned in the first paragraph can be reflected twice in the raindrop (see raindrop A in the diagram) and therefore still reaches an observer on the ground as it finally exits the drop at an angle of around 52o from the antisun-line.The fact it is reflected twice means the red will now be on the inside of the cone, and fainter.

ANSWER 2: The variation in apparent size of rainbows is due to several factors. If the Sun is higher in the sky then less of the rainbow's arc will be above the horizon (where it is more visible), and hence it will seem smaller - even though it is still 41o from the antisolar point. The antisolar point is the point where an imaginary ray connecting the Sun and the observer meets the ground, coinciding with the top of the observer's shadow. If the Sun is above the horizon, the antisolar point is below the horizon,if the sun is below the horizon the antisolar point will be in the sky.

Similarly, the extent and distance of the water droplets (from the observer) can give rise to partial arcs, which obviously appear smaller than a full bow. Finally a rainbow's relative size is subject to the same optical illusion that makes the Moon appear larger when it is lower down in the sky - we can more readily compare its size to the objects on the horizon. So a rainbow behind some houses may appear smaller than a rainbow spanning the open countryside.

ANSWER3: The relative position of the Sun, the observer and the bow, lying in a straight line in that order, was understood by the ancient Greeks. Aristotle reported that the bow is part of the circumference of the base of a cone with the Sun at the apex and the eye of the observer on the line from apex to the centre on the base.

This description was extended by medieval philosophers working between 1200 and 1300 which culminated in the writing of Theodric of Frieberg in the first decade of the next century. Theodric showed that the rainbow was formed by individual droplets of water when sunlight falls on rain or mist.

More than 300 years later Rene Descartes set out the basis for modern science in his Discourse on Method (1637). Among the appendices is an extended explanation of the rainbow. In this he identifies 41o 47' as the angle between sunlight entering the drops and the maximum intensity of the red rays leaving to form the primary bow. For the secondary bow, Descartes calculated the angle as 51o 37'.- New Scientist

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