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# The Moon Sine

Figure 1. v>0 if illuminated and v<0 if not. In other words, the positive direction is away from the sun.

### The Sun-Moon-Eye Angle

On a long car ride from State College to Boston in late August, my wife and I were accompanied by a waning gibbous moon — a disk low on the horizon with a bit nibbled off, as in Figure 1a. During the forced idleness of the long ride I realized how easy it is to tell the sun-moon-eye angle $$\theta$$ of Figure 2 from the face of the moon: namely,

$\cos\theta=\frac{v}{r}, \tag1$

where $$v$$ and $$r$$ are marked in Figure 1. Here, $$v$$ may either be positive or negative, as stated in the caption. This sign convention gives an acute $$\theta$$ for the gibbous moon and an obtuse $$\theta$$ for the crescent moon, in agreement with common sense. Figure 3 explains the proof of $$(1)$$. For the harvest moon, $$\theta$$ vanishes, and therefore so does sine on harvest moon.

Figure 2. The sun-moon-eye angle.

### The Terminator

The great circle on the moon that separates light from dark is called the lunar terminator. To our eye the terminator is an ellipse, since it is a parallel projection of a circle. Where are the foci of this ellipse? The answer is given by the same sun-moon-eye angle $$\theta$$, as Figure 4 shows. And how do these foci move in time? It turns out that they execute harmonic motion if we neglect the eccentricities of the orbits of Earth and the moon. I leave out the proof of these claims.

Figure 3. The proof of (1).

Figure 4. The foci of the lunar terminator are given by the same θ as in Figure 2.

### The Lunar Tilt Illusion

To conclude, I would like to mention a somewhat related Moon Tilt Illusion pointed out to me by Nick Trefethen: the tilt of the crescent seems wrong, and the moon should look fuller. Very nice discussions of this are available in [1] and [2].

The figures in this article were provided by the author.

References
[1] Berry, M.V. (2015). The squint Moon and the witch ball. New J. Phys., 17, 060201.
[2] Trefethen, L.N. (2011). The other moon illusion. In Trefethen’s Index Cards (p. 270). Hackensack, NJ: World Scientific.

Mark Levi ([email protected]) is a professor of mathematics at the Pennsylvania State University.

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