SIAM News Blog

The Moon Sine

By Mark Levi

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.

[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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