Larger than life
A proverbial absurdity is to believe that the Moon is made of green cheese.
The proverb's origins are lost, but in 1638 John Wilkins wrote in his science-fiction book The Discovery of a World in the Moone that common people thought it also absurd that the Moon could be large: "You may as soon perswade some Country peasants, that the Moon is made of greene Cheese (as wee say) as that 'tis bigger than his Cart-wheele, since both seem equally to contradict his sight, and he has not reason enough to lead him farther from his senses".
The class may enjoy 17th-century speculations on the size and habitability of the Moon, not to mention the spelling. However, the quotation provides teaching opportunities about the evidence of our senses, and about the use of reason to calculate the true size of the Moon. It also raises the question about the Moon illusion and the popular misunderstanding of the effects of atmospheric refraction.
The Moon illusion is a fascinating phenomenon that most students have noticed. The low Moon (or Sun) can appear almost twice as large as the high Moon and the effect is so compelling that it's often assumed to be a real optical phenomenon. However, refraction can't have the observed effect and the illusion is all in the brain. We don't see objects strictly in proportion to their image size. Instead we take into account the scale given by the surrounding scenery, the relative colour and brightness of the surrounds, the orientation of our gaze, and the location of the object in perceptual space. All of these factors help us to scale up the perceived size of distant objects close to the horizon, in comparison with those higher in the sky or closer to us. This scaling partly compensates for the reduced image size of distant objects and explains why photographs of distant hills look disappointingly small compared to real life. We notice the scaling much more for the Moon than for hills, because its image size is always about half a degree and doesn't shrink near the horizon. The Moon illusion, therefore, presents a good chance to educate students about refraction.
It's not easy to draw accurate ray diagrams for atmospheric effects, so start with the easier problem of explaining why objects are magnified when the eye is in air and the object in water. This situation occurs when wearing a face mask while snorkelling or when looking down into a pool with the eye close to the water surface. The refractive index of water is about 1.33, while that of air is about 1.0003. Light rays are bent when passing from water to air, in accordance with Snell's law: sin isin r = 1.33. This has the effect of enlarging an image at the eye by about four-thirds, and producing a virtual image of the object at three-quarters of its distance in water. Ask the students to draw appropriate ray diagrams on large sheets of graph paper and measure the size of the angular magnification with a protractor. For simplicity, ignore refraction within the eye and assume that the rays cross at the front of the eye. They could try this exercise twice - once with the eye fairly close to the airwater interface, and once with the eye further back. The magnification is reduced if the eye is further away in the air.
You can then point out that the magnification effect is exactly the wrong way round to account for the Moon illusion: the diver or fisherman looks from air into water (thin to thick medium), whereas the Moon watcher looks from air into space (thin to empty atmosphere). The latter case is more like a fish looking up through the surface of the water at a fisherman: the fisherman will be optically reduced to three-quarters of his size.
Ask pupils what size they think the Moon illusion is. Get them to compare their memory of the diameter of the low Moon with that of the high Moon, and ask them to vote for ratios of 1.3, 1.5, 1.8, 2.0, 2.3 or larger. Pick the median vote. It's likely to be 1.5 or 1.8. If optics is to provide the sole explanation of the illusion, refraction and perception should be similar in size. Yet not only is atmospheric minification the wrong effect, but it's minuscule in comparison with the effect of looking into water (1.3). The effect is so small that it's impossible to draw an accurate ray diagram.
Atmospheric refraction is, however, large enough to cause a noticeable change in the shape and colour of the setting Sun. The low Sun (or full Moon) becomes oval, with its vertical diameter shorter than its horizontal diameter. The Sun may actually be below the horizon at this time, but it's visible because the atmosphere refracts its rays upwards. The rays from the lower limb are refracted upwards more than those of the upper limb, because they pass through more atmosphere. This shrinks the vertical diameter by about 20 per cent in comparison with the horizontal diameter. The low Sun also looks red because the short wavelengths (blue) are scattered more by the atmosphere, leaving the long wavelengths (red) as dominant. Refraction of the last rays of the setting Sun sometimes produces a "green flash", because rays from the blue-green end of the spectrum are refracted more than those from the red end. Students may be able to draw schematic diagrams of these effects.
Refraction is probably enough for one lesson, but there is more to chew on in the "greene Cheese" quotation. How and when did people first make accurate estimates of the sizes of the Earth, Sun and Moon? How do we know what the Moon is made of, and what does this tell us about the origins of our solar system? And are there likely to be habitable planets in the universe?
Helen Ross is honorary reader in psychology at the University of Stirling, Scotland Email: firstname.lastname@example.org
The Mystery of the Moon Illusion: Exploring Size Perception. By Helen E Ross and Cornelis Plug. Oxford University Press. pound;29.95
Color and Light in Nature. By David K Lynch and William Livingston. Cambridge University Press. pound;13.97
Refractive index of water