The Moon Illusion Induced by Atmospheric Converging Light Effect

The mystery of the Moon illusion keeps pending, even if many physical and physiological explanations have been extensively attempted over thousands of years. Here I propose that the illusion may be related with the Earth’s atmospheric unusual optical effect. The Earth sphere shell atmosphere may form a convex lens to refract ray of light, this leads to that the ray of light emitted (reflected) from the Moon is greatly converged before it reach observer’s eyes, and thereby the retinal (camera) image of the Moon is commonly compressed. The difference of the convergence in atmosphere between the horizontal Moon and zenithal Moon results in that the former’s image is compressed much more than that of the latter, but reversely, the geometry of optics requires the horizontal Moon to be enlarged much more than the zenithal Moon when they are looked back into the sky. Over thousands of years many people were severely perplexed by a common illusion that the Moon appears to greater at the horizon than in the zenith. The same illusion occurs for other celestial bodies like the Sun and constellations. For an earthly observer to find the Moon illusion, such a process is due: a) ray of light is firstly reflected from the Moon’s surface to reach the apex of Earth’s atmosphere; b) and then it runs through atmosphere to enter observer’s eyes; c) a following excitation to perceptional system gives birth to a comparison of retinal image. This sequence involves at least physics (optics), meteorology, astronomy, and physiology. A physical explanation therefore was firstly advocated by Aristotle, Posidonius, Ptolemy, and others that the illusion is from an enlargement effect caused by the atmospheric refraction (reflection). In doing so, they often used atmosphere to make analogy with water. The atmospheric refraction requires that the horizon enlargement can be measured by instruments, however, no such enlargement has been found by any astronomical measurements over many centuries (McCready, 1986). The application of photography further reveals that the horizon Moon and zenith Moon taken with the same camera settings yield the same sized image. These evidences strongly frustrate the refraction theory and lead to that more eyes turn to seek for possible psychological explanation. The best-known representative of this attempt is apparent distance theory (first described by Cleomedes around AD 200) that proposes the horizon moon looks larger than the zenith moon just because it looks farther away. The inadequacy of this statement had been described last century by Boring (1962), Hershenson (1982), McCready (1965, 1986), and Restle (1970). A competing alternative is the angular size-contrast theory that believes the illusion to be from different contrasts between the Moon's subtense and the smaller and larger angular subtenses in its surroundings (Restle 1970, Baird, Wagner and Fuld 1990). But as argued by McCready and others, this theory has also difficulty explaining the old observation that the large-looking horizon moon will look smaller if one bends over and views it upside down (Washburn, 1894). In addition to these mentioned above, other attempt like angle of regard hypothesis was also developed. Ross and Plug (2002) investigated the whole history of speculation and research on the Moon illusion to conclude that “no single theory has emerged victorious". Undoubtedly, there must have missed something very important. Today I find that the unresolved illusion may attribute to an unusual optical effect of the Earth’s atmosphere. In general, object under water (denser medium) can be enlarged due to refraction when observed from outside (thinner medium). In turn, if object is observed from water, it will look smaller rather than larger. This knowledge has become a strong point (often employed by people) to disprove the refraction theory. It, however, is not applicable for the condition we look at the Moon. Gravity makes atmosphere form a spherical shell to enwrap the Earth’s body. The spherical shell atmosphere can optically form a convex (converging) lens between an earthly observer and the Moon. Under the condition, the imaging process of the Moon may be sketched with Figure 1. At the moment the Moon is zenithal with respect to observer S but horizontal with respect to both observer S and S. For S the ray of light from the Moon passes through atmosphere with a short path of bS/eS, while for S and S the ray of light from the Moon passes through atmosphere with a long path of aS/dS and cS/fS, respectively. The difference of optical path in atmosphere between S and S (S) may yield different refraction effect for the ray of light. Theoretically speaking, a long path for the ray of light in medium corresponds to a big refraction, while a short path does a small refraction on the assumption that the medium is uniformly isotropic. In this present case, we formulate that several points (a, b, c, d, e, and f, for example) on the Moon emit many rays of light at the same time. We further assumed that the distance of point a and d represent a maximal subtense of the Moon, and that the ray of light emitted from them due to atmospheric refraction may just enter S’s eyes (camera). Therefore, the ray of light emitted from the zenithal Moon with the same amplitude subtense cannot be observed by S because it undergoes a smaller atmospheric refraction. In other words, the maximal subtense of the Moon that S can see at the time is represented by the ray of light emitted from point b and e, which corresponds to a small linear size relative to the distance of point a and d. For S the ray of light dd in atmosphere runs path dS whose length is larger than that of the path aS that is run by the ray of light aa, this results in that dd in atmosphere is refracted much more than aa. A natural result is that dS is converged much more than aS before they enter the eyes (camera). This unevenness of refraction (convergence) leads the retinal (camera) image of the Moon to be inward compressed and to become flatten. For instance, due to angle α < β, this leads to dk < ka. But note that, this retinal (camera) image of the Moon is different from what an observer is looking at the Moon. In the eyes of S (S), the ray of light emitted from point a (b) looks like from the direction of aa (bb) rather than from the direction of aa (bb). In this case, the visual (perceived) linear size indirectly represented by arc ad (cf) and angular size∠aSd (cSf) of the horizontal Moon in the eyes of S (S) is geometrically larger than that of the zenithal Moon in the eyes of S, this leads to that the horizontal Moon looks like greater than the zenithal Moon. Atmosphere density is gradually decreasing with the increase of the height from the Earth’s surface, this leads to that the atmosphere lens formed cannot keep uniformly isotropic. In addition, atmosphere thickness is also currently uncertain. The two give rise to a difficulty determining optical parameter for the lens. But as the distance of the Moon and the Earth is far larger than the radius of curvature of the atmosphere lens, according to the imaging physics of a lens, the image of the Moon may geometrically locate at the position of the focus of the atmosphere lens. And then, the focus of the atmosphere lens may be conceptually derived by means of the enlarging ratio of the Moon’s image between the horizontal and zenithal Moon. Refer to Figure 1, we assumed the focus of the atmosphere lens to locate at the position F, which is far away from the Moon and beyond the Earth’s center (point O). For the image of the Moon formed in the eyes (camera) of S and S respectively, both optics and mathematics (geometry) require that eb/eb ≈ SF/sF, and cf/cf ≈ HF/sF. Here we assumed that the length of eb is equal to that of cf, and then cf/ eb ≈ SF/ HF, cf/ eb therefore indicates the enlargement ratio of linear size between the horizontal Moon and the zenithal Moon, where eb and cf indirectly represent the linear size of the Moon that can be observed by S and S respectively, eb and cf represent the size of retinal (camera) image respectively, SF (HF) represents the vertical distance between S (S) and focus (F), while sF is the distance between focus (F) and atmosphere apex that is closest to the Moon. Atmosphere thickness sS, the Earth-Moon distance OM, and the Earth radius (equivalent to OS or OS) here are respectively assumed to be 100 km, 380,000 km, and 6,371 km. Due to a geometry of similar triangle, OH/OS=OS/OM, and then OH is worked out to be approximately 107 km. Enright (1975) used instrument to measure the enlargement ratio of the Moon to be 1.5 to 2.0. As a result, SF/ HF ≈ cf/ eb = 1.5~2.0, where SF= SO+OF, HF=HO+OF, and OF is the focal length of the atmosphere lens. The final expression will be (6371+OF)/(107+OF)= 1.5~2.0, OF is calculated to be 6,157~12,422 km. The unevenness of atmosphere distribution in the vertical direction forms a gradient-refraction for the atmosphere lens, and the atmosphere also cannot form a pure convex lens because its interior is filled with solid and liquid materials that cut off ray of light to pass through, these may greatly compromises the result calculated above.